BOILING HEAT TRANSFER Modern Developments and Advances

Edited by

R. T. Lahey, J f.

Center for Multiphase Research Rensselaer Polytechnic Institute

Troy, NY, USA

1992

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L1brary of Congress Cataloglng-1n-Publlcatlon Data

Boiling heat transfer modern developments and advances edited by

R. To Lahey.

p. CII.

Includes blbl10graphlcal references.

ISBN 0-444-89499-3 (alk. paper)

1. Heat--TransIl1ss10n. 2. Ebullition. Lahey. R1chard T.

TJ260.B575 1992

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ISBN 0 444 89499 3

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PREFACE

The chapters in this book have evolved from lectures which were given in a Center for Multiphase Research (CMR) sponsored short course at Rensselaer. This course, "Modern Developments in Boiling Heat Transfer and Two-Phase Flow", is intended to provide industrial, government and academic researchers with state-of-the-art research findings in the area of multiphase now and heat transfer technology. Moreover, this course has been focused on technology transfer, that is, indicating how recent significant results may be used for practical applications.

This book is intended to serve the same basic purpose as the course. The chapters give detailed technical material that will hopefully be useful to engineers and scientists who work in the field of multiphase flow and heat transfer.

The authors of all chapters in this book are members of the CMR at Rensselaer. This research center currently involves 20 faculty members from science and engineering, and more than 100 graduate students and staff, working together synergistically to advance the state-of-the-art in multiphase science. Some of the fruits of their labor, as well as the work of others in the field, are contained in this book.

�- --. -- -- -- -- ----- --_ .. -�-�-- -----"Iviii CMR

R.T. Lahey, Jr. The Edward E. Hood, Jr. Professor of Engineering Director - Center for Multiphase Research Rensselaer Polytechnic Institute Troy, New York - USA

vii

TABLE OF CONTENTS

OF 1W()'PBASE FLOW (0. C. Jones) 1

Absttad 1

1. INTRODUCTION 1

2. NOTATION 2

2.1 Independent Variables 2

2.2 Dependent Variables 2

2.3 Volume Fluxes 6

2.4 Comparison Between Quality and Void Fraction 8

2.5 Mixture Relations 8 3. FLOW PATTERNS AND REGIMES 13

3.1 Dispersed Flows 13 3.2 Separated Flows 14

3.3 Vertical Flow 14

3.4 Horizontal Flows 16 3.5 Flow Pattern Maps 18 3.6 Objective Flow Pattern Identification ID

4. MIXTURE MODELS ID 4.1 Homogeneous Flow Model 22 4.2 Drift Flux Model Z3 4.3 Two-Fluid Model � «�� �

ANALYTICAL MODELING OF MULTIPHASE FLOWS (DA Drew) 31 1. INTRODUCTION 31 2. MULTIPHASE CONTINU BALANCE EQUATIONS 31 3. AVERAGING :ti

3.1 Local Balance Equations 40 3.1.1 Jump Conditions 41

4. ENSEMBLE AVERAGING 42 4.1 Other Averages 43 4.2 Averaging Procedures 44 4.3 Averaged Equations 45 4.4 Definition of Average Variables 46 4.5 Averaged Equations m

4.5.1 Jump Conditions m 5. CLOSURE CONDITIONS 52

5.1 Completeness of the Formulation 52 5.2 Constitutive Equations 52

5.2.1 Guiding Principles 53 5.2.2 Objectivity 53

5.3 Inviscid Flow Around a Sphere 66 5.3.1 Averaged Velocities 66 5.3.2 Averaged Pressures 59 5.3.3 Interfacial Force 59 5.3.4 Dispersed Phase Stress ED 5.3.5 Momentum Jump Condition ED

viii

5.3.6 Reynolds Stresses 5.4 Constitutive Assumptions

5.4.1 Stress 5.4.2 Interfacial Force 5.4.3 Momentum Source from Surface Tension

6. SOME CONSEQUENCES OF THE FORMULATIONS 6.1 Discussion of the Force on a Sphere

61 62 63 ED EB EB EB 72 82 8i

6.2 Nature of the Equations 7. CONCLUSION 8. REFERENCES

THE PREDICTION OF PHASE DISrRIBUI'ION AND SEPARATION PHENOMENA USING TWO·FLUID MODElS (R.T. Lahey, Jr.) 85 Abstract 85 1. INTRODUCTION 85 2. DISCUSSION - PHASE DISTRffiUTION DATA 86 3. DISCUSSION - THE ANALYSIS OF PHASE DISTRffiUTION m

3.1 Turbulence Modeling 102 3.2 Boundary Conditions 104

4. DISCUSSION - THE ANALYSIS OF PHASE SEPARATION PHENOMENA 1m

5. PHASE SEPARATION ANALYSIS 113 6. COMPARISONS WITH MEASUREMENTS 115 7 . ANALYSIS OF PRESSURE DROP 118 8. SUMMARY AND CONCLUSIONS 118 REFERENCES lID

WAVE PROPAGATION PHENOMENA IN TWO-PHASE FLOW

(R.T. Lahey, Jr. ) 123 Abstract 123 1. INTRODUCTION 123 2. DISCUSSION 123 3. ANALYSIS lID

3.1 The Dispersion Relation 138 3.2 Prediction of Propagating Pressure Pulses 139 3.3 The Relationship to Critical Flow 145

4. THE LINEAR ANALYSIS OF VOID WAVE PROPAGATION 146 5. NONLINEAR ANALYSIS OF vom WAVE PROPAGATION 161

5.1 Characteristics, Shocks and Kinematic Wave Speeds 163 5.2 Nonlinear Void Wave Profiles 165 5.3 Nonlinear Void Waves and Their Stability 165

6. NOMENCLATURE 171 REFERENCES 173

CRITICAL FLOW: Basic Considerations and Limitations in the Homogeneous Equilibrium Model (O.C. Jones) 175 Abstract 175 1. INTRODUCTION 175 2. HOMOGENEOUS EQUILIBRIUM 176

ix

2.1 Basic Considerations 176 2.2 Isentropic Homogeneous Equilibrium (Two-Phase) 178 2.3 Generalized Pressure Gradient 179

3. LIMITATIONS IN THE HOMOGENEOUS EQUILmRIUM MODEL 00 3.1 Flashing Fano Flow 00 3.2 Flashing Rayleigh Flow 182 3.3 Flow with Simple Area Change 183

4. GENERAL SUMMARY lH1 5. NOMENCLATURE 187 6. REFERENCES 187

NONEQUILIBRIUM PHASE CHANGE - L Flashing Inception, Critical Flow, and Void Development in Ducts (D.C. Jones) 189 Abstract 189 1. INTRODUCTION 10 2. THERMOFLUID DYNAMICS OF REAL FLUIDS IN THE

NUCLEATION ZONE 191 2.1 Flashing Inception 13 2.2 Mechanics and Thermodynamics of Surface Nucleation 195 2.3 Bulk Nucleation Dynamics 20 2.4 Bubble Number Density 2fJ1 2.5 Superheat 2fJ1 2.6 Void Development in the Nucleation Zone � 2. 7 Critical Mass Flow Rates 210 2.8 Overall Summary for Nucleation in Flowing. Liquids 211

3. VOID DEVELOPMENT DOWNSTREAM OF THE NUCLEATION ZONE 211 3.1 Flow Regimes 211 3.2 Two-Phase Flow Modeling 213 3.3 Nucleation Kinetics 218 3.4 Numerical Methods 219 3.5 Numerical Model 2m 3.6 Comparison with Void Development Data 2m 3.7 Overall Sumary--Downstream Void Development 2Z3

4. GENERAL SUMMARY 224 5. NOMENCLATURE 2$ 6. REFERENCES 2'28

TWO-PHASE FLOW DYNAMICS (M.Z. Podowski) 235 1. INTRODUCTION 235 2. TRANSIENT IN BOILING LOOPS 235

2.1 Typical Boiling Loop Configurations 235 2.2 Modeling of Boiling Loop Dynamics '131

3. EXAMPLES OF TWO-PHASE FLOW TRANSIENTS 248 4. EFFECT OF LATERAL DISTRmUTION OF FLOW PARAMETERS

ON BOILING CHANL DYNAMICS 258 5. REFERENCES 2m

x

INSTABILr IN TWO-PHASE SYSI'EMS(M.Z. Podowski) 271 1. INSTABILITY MODES 271 2. LINEAR ANALYSIS OF TWO-PHASE FLOW INSTABILITIES 274 3. NONLINEAR PHENOMENA 20 4. STABILITY MARGINS rol REFERENCES :m APPENDIX A :m

APCATIONS OF FRACTAL AND CHAOS THEORY IN THE FIElD OF MULTIPHASE FLOW &; HEAT TRANSFER (R.T. Lahey, Jr.) 317 Abstract 317 1. INTRODUCTION 317 2. FRACTALS 318 3. BIFURCATION THEORY 328

3.1 Static Bifurcations 328 3.2 Dynamic Bifurcations 33 3.3 Self-Similarity and Mixed Bifurcations 331

4. CHAOS THEORY 339 5. THE ANALYSIS OF CHAOS IN SINGLE-PHASE NATURAL

CONVECTION LOOPS 359 6. APPLICATIONS OF CHAOS THEORY - THE ANALYSIS OF

NONLINEAR DENSITY-WAVE INSTABILITIES IN BOILING CHANNELS 371

7. CLOSURE 382 NOMENCLATURE 384 REFERENCES �

ELEMENTS OF BOILING HEAT TRANSFER (AE. Bergles) 389 Abstract 389 1. INTRODUCTION 389 2. POOL BOILING 30

2.1 The Boiling Curve 30 2.2 Natural Convection 392 2.3 Nucleation 392 2.4 Saturated Nucleate Pool Boiling 40 2.5 Peak Nucleate Boiling Heat Flux 413 2.6 Transition and Film Boiling 418 2.7 Influence of Subcooling on the Boiling Curve 4a) 2.8 Construction of the Complete Boiling Curve 42 2.9 Crossflow Effects on Boiling from Cylinders 4Z3

3. FLOW INSIDE TUBES 425 3.1 Flow Patterns 425 3.2 Subcooled Boiling 4Zl 3.3 Forced Convection Vaporization 4.'J) 3.4 Critical Heat Flux or Dryout 432 3.5 Transition and Film Boiling 43

4. TWO-PHASE FLOW AND HEAT TRANSFER UNDER MICRO-GRAVITY CONDITIONS 43 4.1 Introduction 43

4.2 Interface Configuration and Dynamics 4.3 Pool Boiling 4.4 Forced Convection Phase Change

5. CONCLUDING REMARKS 6. NOMENCLATURE REFERENCES

NONEQU1LIBRIUM PHASE CHANGE - 2. Relaxation Models,

xi

435 4.'J) 438 440 441 443

General Applications, and Post;.Dryout Heat Transfer (D.C. Jones) 447 Abstract 447 1. INTRODUCTION 447 2. GENERAL NONEQUILIBRIUM RELAXATION THEORY 44B

2.1 General Balance Equation and Kochine's Relation 44B 2.2 Phase-Change Mass Flux 449 2.3 The Fundamental Paradox 451 2.4 The Quasi-One-Dimensional Mass Conservation and the

Volumetric Source Term 451 2.5 Nonequilibrium Relaxation 453 2.6 Relationship Between the Relaxation Potential and Temperatures 45

3. APPLICATION TO POST-DRYOUT HEAT TRANSFER 456 3.1 Historical Review 456 3.2 Nonequilibrium Relaxation Applied to Dispersed Droplet Flows 40 3.3 Superheat Relaxation 461 3.4 Implementation 462 3.5 Correlation oC the Superheat Relaxation Number 46

4. GENERAL SUMMARY 476 5. NOMENCLATURE 477 6. REFERENCES 479

SHELIDE BOILING AND TWO-PHASE FLOW (M.K Jensen) Abstract 1. INTRODUCTION 2. FLOW PATTERNS 3. PRESSURE DROP

3.1 Void Fraction 3.2 Two-Phase Friction Multiplier

4. HEAT TRANSFER COEFFICIENTS 5. CRITICAL HEAT FLUX CONDITION 6. SIMULATION OF CROSSFLOW BOILING 7. TUBE BUNDLES WITH ENHANCED TUBES 8. CONCLUSIONS 9. NOMENCLATURE 10. REFERENCES

THE EFCI' OF FOULING ON BOILING HEAT TRANSFER (E.F.e. Somerscales) Abstract 1. INTRODUCTION

1.1 Definition oC Fouling

483 483 483 485 487 48 491 495 501 506 508 510 510 511

515 515 515 515

xu

1.2 Objectives of the Lecture 516 1.3 Cost of Fouling 516 1.4 Observed Effects of Fouling 517 1.5 Importance of Fouling 518 1.6 Categories of Fouling 521

2. FUNDAMENTAL PROCESSES OF FOULING 5Z3 2.1 Introduction 5Z3 2.2 Kern-Seaton Model 5Z3 2-3 The Phases of Fouling 524 2-4 Growth Processes 525 2-5 Processes in the Deposit 532 2.6 Removal Processes 534 2.7 Limitations of Fouling Models s:r

3. EMPIRICAL FOULING MODELS s:r 3.1 Introduction s:r 3.2 Falling Rate Fouling 538

4. DESIGN OF HEAT TRANSFER EQUIPMENT SUBJECT TO FOULING 542 4.1 Introduction 542 4.2 Fouling Thermal Resistances and Fouling Factors 543 4.3 The Cleanliness Factor and Percent Oversurface 54 4.4 The Effect of Fouling on Pressure Drop 545 4.5 Design Features that Minimize Fouling 546

5. FOULING AND BOILING 547 5.1 Introduction 547 5.2 Precipitation Fouling 547 5.3 Corrosion Fouling 551 5.4 Particulate Fouling 552 5.5 Chemical Reaction Fouling 557

6. SUMY AND CONCLUSIONS 58) NOMENCLATURE 58) REFERENCES �

INTERMOLECULAR AND SURFACE FORCES WITH APUCATIONS IN CHANGE-OF·PHASE HEAT TRANSFER (P.C. Wayner, Jr.) sm 1. INTRODUCTION sm 2. THEORETICAL BACKGROUND 573

2.1 Equilibrium Vapor Pressure of a Liquid Film 573 2.2 Interfacial Mass Flux 578 2-3 Fluid Mechanics 582

3. APPLICATIONS 583 3.1 An Evaporating Ultra-Thin Film 683 3.2 Nucleation 58 3.3 Effect of Disjoining Pressure on Diffusion in an Arnold Cell 589 3.4 Marangoni Flows 592 3.5 Effect of Conduction Resistance 594 3.6 Cavitation 8X)

4. THE VAN DER WAALS DISPERSION FORCE 001

4.1 Derivation of the Nonretarded van der Waals Interaction Free Energy (per unit area) Between Two Flat Surfaces Across a Vacuum. W = -Al12 xa2

4.2 Surface-Surface Interaction with Number Densities. Pi = Pj = P

4.3 The Force Law for Two Flat Surfaces Separated by a Vacuum Using the Hamaker Constant Concept

4.4 Calculation of van der Waals Forces from the DLP Theory 4.5 Approximate Model 4.6 Numerical Example: Hamaker Co�stant 4.7 Combining Rules: Hamaker Constant 4.8 Surface Energy

5. SUMMARY NOMENCLATURE LITERATURE CITED

XIII

001

812

Elements of Two-Phase Flow

Owen C. Jones

Professor of Nuclear Engineering and Engineering Physics

Rensselaer Polytechnic Institute, Troy, NY 12180-3590

Abstract This chapter introduces the subject of two-phase flow. Starting with the notation, independent

variables and dependent variables are defined including velocities, temperatures, pressures, vol­ume concentrations, mass concentrations, and volume fluxes. Mixture relations are defined and dynamic and thermal quantities introduced. Flow patterns and regimes are discussed for both dis­persed and separated flows. and for vertical and horizontal ducts. Typical flow patten maps are mentioned and an objective method of flow pattern definition introduced.Types of modeling are discussed including the homogeneous flow model, the drift-flux model, and the two-fluid mod­el. The concept of averaging a heterogeneous mixture of phases is introduced in preparation for later chapters.

1. INTRODUCTION

Two-phase flow is the simultaneous flow of two separate states of matter. These states can be any combination of gas, liquid, or solid, and can occur with or without simultaneous change from one state to another. Such change can occur with condensation, melting, sublimation, boiling, or the like. Since more than one phase can occur simultaneously, and since phase change can occur in the flow field. nomenclature is necessarily more complex than when only one phase flows by itself. This chapter will introduce the concepts and notation which are used by a wide variety of

persons dealing with two-phase flows.

Two-phase flow suffers from all the complications of single-phase flows. In addition, numer­ous additional difficulties are encountered due to the interaction of the phases and the deforma­

tion which can occur at phase boundaries. In single-phase flows, there are only two predominant

flow regimes considered: laminar and turbulent. In two-phase flows, however, not onl y can these regimes occur separately in each phase, but also the sn ucture of the phase distribution can change giving rise to many other flow regime considerations. This chapter will also introduce the ways in which the phases can be structured, and describe ways in which these flo� regimes can be classi­fied.

Because phases can interact and structure themselves in different regimes, it is only natural

that this structuring has led to the use of different models for describing these flows. Further, it is

2

also natural that the flow regimes have achieved usefulness in discriminating between the use of one model and another. Several of the most common flow models will be introduced and their basic areas of usefulness described in this chapter.

2. NOTATION

2.1. Independent Variables

The common independent parameters found in two-phase flows are identical to those in sing­le-phase flow: space coordinates and time. Space coordinates can be denoted as a vector, x or -; , or by the individual coordinates x, y, and z. Time is usually denoted by "t." Dimensionless coordi-nates can be � , 1] , and', whereas dimensionless time can be or

2.2. Dependent Variables

Common dependent parameters are also those identical to single-phase flow: velocity, pres­sure, temperature. Since these can be different in each phase, it is common to use subscripts to denote the phase: s-solid, I-liquid, v-vapor. Sometimes a vapor is distinguished from a gas by using the subscript "g" for the latter. In the particular case where a phase is at saturation conditions and distinctions are desired between the saturated condition and another condition, the common thermodynamic notation is generally used: g-saturated vapor; I-saturated liquid. Since each phase will occupy only a fraction of the total area or volume, a volume concentration must also be considered.

the velocity vector is generally written as v or v The coordinate-directed veloci­ties are generally denotedu, v. orw for thex ,y-, orz directed components of the velocity. These are modified by the appropriate subscript to denote the phase to which the component applies. Thus, UI represents the x-directed liquid velocity whereas Wv represents the z-directed vapor velocity. When a velocity is denoted without a subscript, it generally means that the phases are as­sumed to flow with identical velocity at a point.

When considering more than one phase existing simultaneously in a conduit or stream tube, the possibility immediately arises that the velocities are not identical. For instance, air bubbling up through a static column of liquid has obvious differences between the air bubble velocity and the liquid velocity. In a near-horizontal duct such as a storm sewer line, the phases could be com­pletely separated. In this case, the liquid would flow by gravity down the slope of the line, where­as the air might be relatively stagnant. In a vertical round tube having a thin film fo water draining down the sides, air could be flowing up the tube in the opposite direction.

In each case, the velocity of the two-phases may be completely different. The only thing that is required in a continuum viewpoint is that there be continuity of mass, momentum, and energy at the point of contact between the phases, at the interface, and that the no slip condition hold as well. This implies that tangential velocities of each phase be the same and tangential velocity gradients be identical at an interface.

3

Normal phase velocities at an interface are not necessarily identical. An interface can store no mass since it has no volume. With mass transfer at the interface such as might occur with evapora­tion or condensation. the mass flux across the interface would thus be conserved. The velocities would, then. differ by the density ratio between the two phases. Without mass exchange. the nor­mal phasic velocities at an interface are the same and identical to the normal interface velocity.

This jump in normal velocity at an interface when mass transfer occurs. which is due solely to differing phasic densities, then leads directly to jumps in normal momentum flux and normal ki­

netic energy flux as well. The jump in normal momentum flux, it will be seen later. leads directly to differences in pressure between the phases. Mass transfer furtherrequires that both momentum and energy gradients exist normal to the interface.

If the flow field is considered as being averaged locally in space and time, one phase can ap­pear to move relative to another. When the relative velocity between phases is considered, the subscript "r" denotes this quantity. since in many conditions the less dense phase will precede the more dense phase, the relative velocity is usually chosen positive in this instance. Thus, for a gas­liquid mixture, Ur = Uv - u" or Ur = ug - ufo

When a velocity is written without a"subscript. it is usually assumed that the phases flow with identical local velocity such that Ur = O. The flow is then said to be in a state of mechanical equilib­rium at that part of the flow field. Note that spatial and temporal variations can still be presumed to exist.

If. on the other hand, the phase velocities are not identical at a section in the flow, the flow is in mechanical nonequilibrium and the phase velocities must be considered separately. Mechanical nonequilibrium is usually only considered in cases where the difference in velocity between the phases presents a difficulty from a design or analysis viewpoint. Such situations would exist if the relative velocity is less than 10% of the mean mixture velocity.

A somewhat archaic term, but one which still finds use in some circles, is the slip ratio. The slip ratio. $, is generally considered to be the ratio between the vapor and liquid velocities: $ = vv1v" and can encompass the range s C (_00, +00).

The temperature is generally written as "T' with appropriate subsCJ1pt to de­note the phase. When the temperatures are assumed identical at a point. the flow is said to be in thermodynamic equilibrium at that point. If the average temperatures are assumed identical in a

plane or cross section. the flow is said to be in thermal equilibrium at that section.

If a system is in a state of thermal nonequilibrium at a given plane. the temperatures of each phase are not equal and must be considered separately. Thermal nonequilibrium is usually only considered in cases where the difference in temperature between the phases presents a difficulty from a design or analysis viewpoint. Such cases would occur if the thennal nonequilibrium leads to temperature differences of more than just a few degrees. and where ignoring these differences

could lead to difficulties in predicting limiting conditions in engineering equipment.

The pressure in a two-phase system is generally denoted by up." Again. subscripts

are used to denote the pressures in each phase if they are considered separately. However. except

4

where detailed analytical considerations are required, the phase pressures are usually assumed identical.

One of the important differences between single- and two-phase flows is the need to quantify the relative amounts of each phase. 'This is done through the concept of concentrations. There are two-different types of volume concentrations in general usage: stat­ic; dynamic or kinematic.

The static concentration, a, commonly termed the void fraction, is simply defined as the vol­ume occupied by the vapor relative to that of the mixture. Thus,

Vv Vy 1 V Vy+VI (1)

where Vk is the volume occupied by phase-k within the total volume, V. It is usual, with appropriate short-time averaging of field quantities, for the void fraction to be considered as a space-time variable and so the volumes in Eq. (1) become differential. If the flow field is quasi­one-dimensional, the differential volume ratio becomes an area ratio so that

Av 1 a=-= --

A l+� Av

and

Al I (l-a)=-= -­

A I +Av AI

(2)

(3)

where both the liquid fraction and void fraction are specified. From this, it is obvious that the phase area ratios are

Ay a -=--Al I-a

(4)

The kinematic void concentration, �, is the ratio of the volumetric flow rate of the vapor to the total volumetric flow and is given by

(5)

From a comparison of (2) and (5) the relationship between static and kinematic void fractions is found to be

1 {3 = 1 a u"

and (6)

5

It is thus seen that in vertical upflow, when the average vapor velocity is usually greater than

the liquid velocity due to buoyancy, that the kinematic, or flowing, void fraction, �, is generally

greater than the static void fraction, a. This is simply because the faster-moving vapor requires less area than it would if flowing at slower than average velocity.

From these two comparative relationships, it is easy to see why the slip ratio became an early convenience. More recent usage has tended more to the relative velocity as a measure of the dif­ference between vapor and liquid velocities since the relative velocity is always finite, and in

many cases small relative to the mixture velocity which will be defined shortly.

There are two mass concentrations which have found usefulness: flow­ing concentration, x; static concentration, C. Since the majority of situations encountered in two­

phase analysis involve the flow of material, the flowing concentration, or quality, is of more gen­eral utility. The quality is simply defined as the flowing mass fraction of vapor relative to the mix­ture. Thus,

mv xorx =--m (7)

where Tilk is the mass flow rate for phase-k. The latter, X , is used when there is a potential for

conflict with spatial coordinates. When the flow field is in thennal equilibrium, the quality is

identical with the thermodynamic quantity and is, thus, a state variable definable in terms of other

state variables such as specific internal energy, specific enthalpy, or specific volume.

The static mass concentration, C, is the mass ratio of vapor to the total mass at a point Thus

(8)

where Mk is the mass of phase-k in the volume holding total mass M. In the case of thermal equilibrium in a nonflowing system, the two mass concentrations are identical.

The mass concentrations are similar to the volume concentrations with the added exception that the phase densities are involved in the mass concentration. Thus, again considering a quasi­one-dimensional viewpoint,

C = = 1 + R!.� 1 + l!v Av a p.

and the flowing mass concentration is given by

(9)

(10)

6

Thus, the connections between static and flowing quality are similar to those for static and flow void fraction. Thus,

1

1 + � X Uj and X =

1 + (l-C) � . C u,.

( 1 1)

From these two relationships, it is easy to see that the flowing quality is generally smaller than the static mass concentration. The later has found little usage in modem two-phase flow analysis which generally attends to consideration of flowing systems.

By using Eq. (3) with the defini­tion of flowing quality given in Eq. (7), one obtains

and ( 1 2)

2.3. Volume Fluxes

With single-phase flows in conduits, one is normally able to determine the average velocity in

a conduit from the measurable quantities: mass flow rate, Til ; density, p; flow area, A. Mass flow rate and thermodynamic state are two things which may be controlled parameters. Thus,

m U=-.

eA ( 13)

In two-phase flows, each phase is, in many circumstances, separately controlled and/or measured. One would be tempted. therefore, to determine a velocity for each phase based on this flow rate.

The difficulty in the preceding thought for two-phase flows is that neither phase occupies the entire cross-sectional area of the conduit. Thus, to determine the velocity of one phase or the oth­er, the area occupied by that particular phase mu.st be used instead of the total area. thus.

my and Uy= -- .

evAv ( 14)

The individual flow areas are not fixed but vary with flow conditions. Multiplying and divid­ing each of these equations by the ratio of the phase area to the total area, and taking into account Eqs. (2) and (3), the following result is obtained relating phase velocity to individual phase flow rates, densities, void fraction, and total flow area:

Till ( 1 - a)e,A

mv and Uv=--. aevA

(15)

7

This shows that the void fraction must be taken into account for actual determination of the kinematic velocity of each phase.

It is interesting that both the liquid and vapor velocities have a term which looks like the veloc­

ity each would have if flowing by itself in the conduit: m /pA. these terms are called superficial velocities or volume fluxes. The latter term has come into more modem usage and is the one used herein. The literature uses various symbols for these terms. Herein, the symbol "j" shall be uti­lized with appropriate subscripts. Thus,

. ml . mv )t = - and h =-·

Q1A QvA (16)

A comparison of (15) and (16) yields the relationships between the phasic volume fluxes and the velocities given by:

(17)

Of course, since Eq. (16) shows that thej's are calculated directly from the mass flow rates as if the fluids were flowing alone in the conduit, they can also be calculated from the volume flow

rates, Q, since Q = m /p for each phase. Thus,

(18)

But certainly, the total volume flow rate is the sum of the individual volume flow rates so that

. Q/+ Qv . . ] (19)

It is thus seen that the total volume flux is just the sum of the individual phase volume fluxes. Note that the same can not be said for the kinematic velocities.

The fluxes are easy quantities with which to work since they can usually be calculated from known parameters: i.e., the things which are controlled with knobs, wheels, levers, etc., and di­rectl y measured with gages, meters, etc. On the other hand, they do not necessarily provide a good measure of what is actually happening in the conduit unless the phase volume fraction for the phase is near unity. For instance, a liquid volume flux flowing at 1 mls in a duct having a void

fraction of 99% would be traveling at an average velocity of 100 m/s. However, the velocities themselves provide a good indication of the physical situation from both a kinematic and dynam­

ic viewpoint. But these are more difficult to determine. It is clear, then, why the void fraction is one of the key parameters in two-phase flows.

It was mentioned earlier that the slip ratio is a quantity which has found application in the past but is seeing less frequent usage as the relative velocity itself becomes a more commonly seen measure of the vapor-liquid velocity differences. Nevertheless, the slip ratio is still considered on

8

occasion. It is easily seen that this ratio can be written completely in tenns of phasic densities, and mass and volume concentrations. By taking the ratio of vapor velocity to liquid velocity in terms of the phase flow rates, this relationship is obtained as

or u. ( X ) ( (II 1-a ) -;= T:x

2.4. Comparison Between Quality and Void Fraction

(20)

For one who is used to thinking in terms of thermodynamic parameters, the quality is the natu­ral indicator used to specify the relative amount of vapor in the mixture. Celtainl y this is still true. Nevertheless, in many instances, the quality does not provide a physical indication of the situa­tion inside a conduit caring two-phase, gas- or vapor liquid flow. This is because of the differ­ence in densities of the phases.

Consider a mixture of air and water, with the air at a density of 1.0 kg/m3, and water at a density of 10 kg/m3. Let's say that both the water and the air flow with a superficial velocity or volume flux of 1.5 mls with a void fraction of 0.38, or 38% of the cross sectional area taken up by the air flow. The quantities calculated are:

• liquid velocity:

• gas velocity:

• liquid mass flux:

• gas mass flux:

• quality:

Ui = 1.510.62 = 2.42 mls

Uv = 1.510.38 = 3.95 mls;

ffl, fA= 10 x 1.5 = 1500 kg/s-m2

fflv fA = 1 x 1 5 = 1.5 kg/s-m2

X = 1.5 I ( 1.5 + 1500 ) = 0.00999

Thus, it would seem that the mass fraction of vapor flowing in the duct is negligible when, in fact, the vapor fills almost 40% of the pipe and has a velocity almost twice that of the liquid. For this reason, when considering the kinematic and dynamic aspects of two-phase, gas-liquid flows, it is the practice to consider void fraction rather than quality. On the other hand, when mass flows and thermodynamics are the main concern, quality is the preferred indicator.

2.5. Mixture Relations

In many instances, the mixture as an entity needs to be addressed rather than separate phases. There have been numerous cases in the past where certain average quanti­ties have been defined for density, enthalpy, etc. Generally, however, all these are artifices except

those which make sense form the viewpoint of the conserved quantities of mass, momentum, and

energy.

The mass of a mixture in any differential volume Adz is simply

9

dm = e",Adz = dm, + dmv = [(1 -a)e, + aev]Adz (21)

from which the definition of mixture density is seen to be

(22)

The mass flow of the mixture is certainly the sum of the mass flow rates of each phase. Thus.

m=m,+mv (23)

where the subscript "m" refers to the mixture. From the previous definitions of the phase flow rates it is seen that

so that the mixture velocity is readily obtained as

(l-a)e,u,+aevuv

(l-a)e,+aev

which is obviously a mass-weighted average velocity.

(24)

(25)

The mass flow rate is also obtained directly from the volume fluxes since in general m = pQ.

and also m = pAj. Thus. from Eqs. (17) and (24) it is easily seen that

(26)

where G is called the mass flux or mass velocity. By comparing (26) with (24) it is also seen that

(27)

which is the flow of mass per unit area of the duct or stream tube. Note that while calculation of the mass flux in terms of the mixture density and velocity is difficult without having the appropriate quantities at hand. calculation in tenns of the volume fluxes is a simple process once the "hand wheel" values are known. Of course the mass flux is determined from the total mass

flow rate as G = m fA so that the quality expressions on the right of (27) follow directly from the definition.

Consider the example discussed above. The mixture density and velocity are:

• em = (0.62 ,10) + (0.38 '1) = 620.38 kg/m3

• Um = [(0.62 ·100 '2.42) + (0.38 ·1 ·3.95)over620.38j = 2.4203 mls

On the other hand, the volume flux of the mixture is

10

• j = jl + jv = 1.5 + 1.5 = 3.0 mls

The mass flux of the mixture is given in terms of the volume fluxes and phase densities by

• G = (100 -l.5) + (1 -1.5) = 1501.5 kg/s-m2

and in terms of the mixture density and mixture velocity as

• G = (620.38 - 2.4203) = 1501.5 kg/s-m2•

To this point, only kinematic quantities have been dis­

cussed. However, dynamic and thermal quantities must also be considered.

From a dynamic viewpoint, the momentum flux is the quantity most usually encountered. Mo­mentum is, of course, mass times velocity, velocity in this case being momentum per unit mass. Momentum flux, or momentum flow per unit area, is simply the sum of the mass flux of each component times the component velocity. Thus,

(28)

The terms in the numerators of the right side of (28) are sometimes termed the superficial momentum fluxes and are commonly-encountered coordinates used for delineation of flow

pattern boundSlies, a topic which will be discussed shortly. From Eq. (27), it is easily seen that

the momentum flux written in terms of quality is

(29)

From a thermal viewpoint, the energy flux is the quantity most generally encountered. This energy flux consists of internal energy, flow work, kinetic energy, and potential energy. Similar to the calculation of momentum flux, the total energy flux is simply the sum of the individual com­ponent fluxes.

If the energy content is considered to be

1 E = h+-u2+gz

2

where h is the enthalpy, then the energy flux is simply

E = (l a)eluIE1+a(2vUvEv = [(l X)EI+XEv]G.

(30)

(31)

Of course the enthalpy, h, is given in the usual manner in terms of the internal energy and flow

work, h = u + pv where in this case u is specific internal energy and v is specific volume. Note

from Eqs. (30) and (31), that the kinetic energy flux is given by

. 1 3 1 3 K=-(I a)(21ul +-aQvUv. 2 2 (32)

From this discussion it is seen that the mass flux, momentum flux, and kinetic energy flux look almost identical except for the power of the phasic velocity in each case being 1, 2, and 3, respec­tively.

The case where the velocities of the different phases are not equal has been previously consid­ered. This difference may be the source of differences in pressure between phases which, in some cases, can be substantial.

The case where temperatures of the different phases are not equal has not been considered. If the equilibrium condition is such that the liquid and vapor would coexist simultaneously, then the mixture temperature would be the local saturation temperature: i.e., Tm=Ts.

In some instances, however, the phasic temperatures may not be equal at any cross section in the flow. In this case, the average temperature for each phase in a cross-sectional area normal to the flow direction shall be considered, and this average temperature is that known as the phase mixing-cup or bulk temperature. The mean temperature for the mixture must still be that asso­ciated with the thermodynamic state: saturated, subcooled, or superheated according to the local equivalent equilibrium conditions.

The bulk phase temperature represents the average temperature the phase would have if sepa­rated at that location from the other phase and brought to a thoroughly-mixed equilibrium state without heat loss or gain relative to the surroundings. TIris temperature, then, represents the ther­modynamic temperature for the specific phase, which, together with the pressure, would be nec­essary and sufficient to determine all other thermodynamic properties of the phase at that location in the duct.

While thermodynamic equilibrium will in all things be assumed, thermal nonequilibrium may exist. The difference between the two is as follows.

Thermodynamic equilibrium exists when all thermodynamic properties follow directly from a specification of two independent properties. Thermal equilibrium exists when both phases have identical temperatures in a given region. A change in the energy content of the mixture due to heat addition orrejection may change both the local quality of the flowing mixture and the phasic tem­perature. Any difference in temperature between the two would be considered thermal nonequili­brium.

Note that in some circumstances, the concept of thermal nonequilibrium together with ther­modynamic equilibrium can present a conundrum. Such is the case when liquid is superheated, or vapor subcooled. In these conditions, the phase is in a metastable state where thermodynamic properties are not really defined on the basis of equilibrium thermodynamics. For such cases, it is usual to consider that the temperature governs the departure from equilibrium and calculate prop­erties as if they were saturated values at the given temperature. Where considerable differences in saturation pressure according to the local temperature and the local pressure exist, the effects of phase compressibility are also considered in calculation of the phasic properties.

IT two phases having initially different temperatures are brought into intimate thermal contact with each other and allowed to coexist for an infinite time, heat exchange would occur between

1 2

the two at a rate governed by the laws of heat transfer which would occur at interfaces. The two phases would eventually come into thermal equilibrium with each other.

Changes in the energy content of a mixture are governed by the first law of thermodynamics. The mixture enthalpy is given by

(33)

where the subscript e on the quality indicates the equilibrium value under which circumstances both phases have the same temperature, the saturation temperature, and thef- and g-subscripts indicate saturation values for the liquid and vapor enthalpies.

H the actual bulk liquid and vapor temperatures differ from saturation, then there will be a dif­

ference between the actual quality, X, and the equilibrium valueu, given by rewriting Eq. (212) as

(34)

Thus, only if there is a difference between the actual temperature of the liquid and/or vapor, and saturation temperature, can there be a diference between actual and equilibrium qualities. In fact, even if there are diferences, the actual and equilibrium qualities may be identical if the effects of vapor superheat and liquid subcooling cancel each other.

From (213), if the vapor is superheated and the liquid is at saturation, the vapor temperature is

given by

(35)

On the other hand, if the vapor is at saturation and the liquid is subcooled, the liquid temperature is

(36)

In the former case, vapor superheat would result in the equilibrium quality exceeding the actual quality while in the latter case, liquid subcooling would result in the actual qUality exceeding the equilibrium value.

In all cases, it is generally assumed that the phases have identical temperatures at an interface. Furthermore, energy continuity is generaly assumed at an interface since, without the ability to

store mass, an interface can not store energy. The only exception to this is the consideration of surface tension effects where surface energy may change. The assumption of identical interfacial

phasic temperatures, then, simultaneously with nonequilibrium, means that temperature gradi­ents must occur in one or both phases. TIris is, of course, a dynamic situation which would result in

relaxation of both phases to a mutual equilibrium condition without the addition or rejection of

heat from the mixture.

3. FLOW PATTERNS AND REGIMES

The different visible ways in which the phases can become distributed from a geometric view­point are termedflow patterns.The ways in which phase distributions lead to differences in physi­cal behavior requiring differing modeling are termed flow regimes. Thus, differing flow regimes

imply differences in modeling approaches. Differing flow patterns imply a visible difference in

the structure of the flow. The two are not necessarily the same.

Whereas in single-phase flows, only two dominant flow regimes exist, laminar and turbulent, many differing flow patterns and regimes exist in multiphase flows. On the surface, one could

consider all combinations of laminar and turbulent flow for each phase. For instance, a tube of

honey and air, initially settled with the air on the top, would have a transient two-phase flow when inverted. The air would slowly rise through the honey which would drain down around the outside

of the bubble. It is difficult to imagine �ither phase to have anything but streamline behavior, lam

inar in each phase.

On the other hand, a high speed mixture of gas and liquid such as air and water flowing concur­

rently in a pipe might lead to the water flowing on the walls of the pipe with waves, and the air

flowing in the core of the geometry, perhaps with droplets mixed in with the air. In this case, one would expect both phases to be simultaneously turbulent.

One could imagine, however, other situations where one phase would be laminar-like while

the other would behave like a turbulent fluid. On the other hand, the movement of one phase

through another, even if there is streamline flow, can produce locally instantaneous fluctuations

due to the passage effects--say the wakes around bubbles as they rise through a stagnant liquid.

Some researchers treat this as turbulence and attempt to average the effects in the time domain in a

manner similar to the averaging in single-phase flows which leads to Reynolds stresses.

Regardless of whether the flow is vertical, horizontal, or something in between, there are two

major classifications of flow regimes/patterns which may be used universally. These are sepa­rated and dispersed. Traditionally, these have been modeled by different techniques.

3.1. Dispersed Flows

Dispersed flows exist when one phase is unifOlmly mixed in another to the extent that when

examined in the large they may appear as a quasi-homogeneous mixture or emulsion. Numerous small bubbles mixed in a liquid is one example of a dispersed flow regime. C� bubbles uniform­

ly percolated upward through a glass of beer form a dispersed flow regime. A spray of droplets

which form a mist such as in a combustion nozzle or a spray fire nozzle form a dispersed flow

regime where droplets are the dispersed elements. Sufficiently high gas flows with a liquid in a

pipe will cause the liquid to disperse in the gas forming a mist flow which is also dispersed.

1 4

3.2. Separated Flows

Separated flows exist when both phases exist in continuous regions where all elements of each phase are connected. Water running in a river or stream when considered in conjunction with the surrounding air above and the interlace surlace separating the two form a separated, two-phase flow. Condensate forming on a vertical wall as a film and draining down the wall also is an exam­ple of a separated flow.

Gas percolating up through a slow-moving liquid in a pipe would produce a dispersed flow of gas bubbles in liq uid (bubbly flow). Larger amounts of gas flow would cause large bubbles to flow intermittently with bubbly-liquid slugs (slug flow) and become chaotic (churn bubbly). Suffi­ciently high gas flows in a pipe with water will cause the large bubbles to merge and the water to

adhere to the wall and form an annulus (annular) which is also an example of a separated two­phase flow. Increasing the gas flow and velocity still further would tend to cause waves on the surlace of the film (wavy annular) which would then be sheared off into the gas stream (annular mist). If the gas flow was increased sufficiently, all the liquid would be completely sheared off the wall and entrained in the gas (mist flow) causing a transition back from separated to dispersed flow.

The cases described represent different flow regimes and patterns. There are many others. These patterns differ somewhat when the flow is vertical or horizontal, and these two cases shall be considered separately.

3.3. Vertical Flow

One can consider that there are three dominant flow patterns in vertical gas-liquid flows:

bubble, annular, mist. There are transitions between each, and several subclassifications within these transitions. In this case as the flow transitions from bubble to mist, it turns from dispersed (bubbly mixture). to separated (annular pattern). and back into dispersed flow (mist).

1. In bubbly flows (Fig. l a), vapor or gas is distributed as discrete entities or

bubbles in a continuum of liquid. Various sizes of bubbles may exist depending on numer­ous factors. Parameters which affect bubble sizes include the manner in which gas is in­

jected into the flow, the turbulence in the liquid phase, the size and distribution of nucle­

ation sites, and numerous other factors. the sizes are difficult to determine a priori.

Bubbles may be spherical, but more often than not, they are irgularly shaped due to vari­

ous forces acting through the liquid. The less viscous the liquid, the more freedom the

bubbles have to take on differing shapes. Bubbly flows have sizes which are generally

much smaller than the smallest dimension of the cross section.

While there are numerous methods proposed to determine the limits of bubbly flows. none

are generally satisfactory. A good rule of thumb, however, is that the upper limit of bubbly

flow in ducts is between 20% and 30% void fraction.

2. With sufficiently high flow rates of vapor or gas relative to the liquid, the

liquid is pushed aside and flows along the walls of the conduit forming an annulus

(a) bubbly (b) annular

(d) bubbly annular transition -- slug and churn

. ' '.'

(c) mist

(e) annular-mist transition

Figure 1. Flow patterns in vertical flows.

1 5

(Fig. Ib) while the gas flows in the

core of the flow. In annular flow,

waves will be present due to the shear­

ing action of the gas and the surface

tension of the liquid. The thickness of

the film, and the velocity of the liquid

are determined by the forces involved.

With sufficiently high velocities with

large turbulence, there may be bubbles

entrained in the liquid film. Alternate­ly, there may be droplets entrained in

the gas core. This is a combination of

separated and dispersed flow regimes.

Anular flow is a separated flow re­

gime which exists between two dis­

persed flow regimes: bubbly flow at

low void fractions; mist flow at high

void fractions. In ducts, annular flow

generally forms between 70% and

80% void fraction, the lower values

being observed with high Jiquid

phase flow rates.

3. When the vapor/gas velocities are sufficiently high relative to the liquid, the

liquid can be entirely sheared from, or prevented from wetting. the wall and will be broken

into droplets forming a mist (Fig. Ic). As in bubbly flow, the size and distribution of drop­

lets can depend on a number of factors. Like bubbly flow, mist flow is a dispersed flow but

with the phases reversed--continuous gas/vapor and dispersed liquid. The lower limit for

mist flow in ducts is generally taken to be approximately 95% void fraction.

4. The change from bubbly flows to annu­

lar flows involves a transition which extends from approximately 20% void fraction to

80% void fraction (Fig. Id). Starting with bubbly flow. as the void fraction is increased, the bubbles tend to interact with each other more and more.

When bubble volumes begin to elongate and take on lateral dimensions approaching the

smallest dimension of the duct. it is said that slug flow begins. Continued agglomeration

results in growth of the length of the void rather than lateral growth. These elongated gas/

vapor regions are separated by liquid slugs which will generally contain small bubbles,

some of which existed prior to the formation of slug flow, and some of which formed by

extrusion and breakup of the trailing edges of the elongated bubbles. Under some circum­

stances, there is sufficient turbulence to significantly distort the elongated bubbles and

form a very chaotic mixture of large and small bubbles--chum turbulent flow.

16

As the void fraction i s increased by increasing gas flow, the dominant, elongated bubbles grow lengthwise and merge together forming the upper limit to slug flow. This generally occurs between 70% and 80% void fraction.

It is easy to see that slug flow cares characteristics of the two extremes of the transition zone. On the one hand, a lateral cross section within the large bubbles would look like an­nular flow. On the other hand, the slugs between bubbles look like bubbly flow.

There is evidence to suggest that physical phenomena characteristic of each of the end re­gimes blend in the slug and churn flow regime transitions from one to the other. As the void fraction increases through the approximate range of 20% to 80%, the characteristics of bubbly flow are lost and those representing annular flow are gained.

An increase in the turbulence levels in the fluid with increasing flow rates may result in the breakup of the larger bubbles in slug flow, or simply prevent their formation altogether. This condition is termed chum flow or churn turbulent flow. Nevertheless, bubbles having a size at times consistent with the small dimension of the duct will exist. This chaotic mix­ture exists with the vapor being churned about due to the turbulent actions of the liquid. The upper limit to churn flow generally occurs between 70% and 80% voids, the lower values occurring at higher values of liquid velocity.

If the mass flux is increased to the range of 2500-300 kg/s-m2, the churn flow becomes extremely chaotic and difficult to photograph with any degree of resolution. Some re­searchers have chosen to represent the flow regime above this range as homogeneous and, for many purposes, this is a reasonable representation. However, the flow still maintains many of the alternating characteristics of chum flows in this region.

5. It is uncommon for mist flows in a duct to exist at void f actions less than 95%. On the other hand, a mixture of annular flow and mist flow, mist-annular flow, is commonly found to exist above 95% void fractionsunless a disturbance causes the liquid to dewet the wall at lower void fractions . One defini tion of this combination which is encountered on occasion is wispy-annUlar.

A commonly anticipated disturbance is the onset of the critical heat flux condition which, in the case of annular flow or annular-mist flow, causes film dry-out leading to mist flow. On the other hand, if the walls of the conduit are hydrophobic in nature, surface tension would act to dewet the wall at lower void fractions and droplet flow (implying larger drop­lets than in mist flow) might form at lower void fractions. Thus, depending on the sUiface

characteristics of the duct, it is seen that shear effects can overcome surface tension to cause transition to mist flows at unusually low void fractions or without significant di�tur­bance.

3.4. Horizontal Flows Flow patterns in horizontal flows are similar to those in vertical flows except gravity plays a

role in stratification of the vapor. If the geometry of the duct and velocity of the mixture are such

1 7

that the transport time for the mixture through the duct, Uum, is larger than the transport time for lateral separation, dlvd, stratification will occur and be important and will affect the flow struc­ture. (In this instance. L and d are the length and lateral dimension of the duct and where Um and Vd are longitudinal mixture and lateral dis�ive velocities, respectively.) For stratification to be an important consideration, L Vdl dVm > > 1. Where this parameter is on the order of unity or smaller, flow segregation due to lateral drift is of decreasing importance and the flows tend to have less and less lateral skew in the profiles. Aside from lateral distortions in the profiles of phase velocity and concentration, the flow patterns are similar to those encountered in vertical flow (Fig. 2).

(a) bubbly flow

(b) semi-annular and annular flow

(c) mist flow

(d) bubbly-annular transition

(d) annular-mist transition

Figure 2. Flow patterns in horizontal flows.

1. This regime is identical to vertical flow except the bubbles rise laterally in

the duct. If the stratification condition is met, LVdldvm> > I, a high concentration of bubbles

will form in the upper regions of the duct and may agglomerate leading to slug flow.

1 8

2. This regime is also identical to vertical flow except grav­ity tends to keep the liquid from the upperregions ofthe duct or make the liquid drain down around the sides if deposited in the upper regions. Thus, part of the duct lateral periphery may be completely dry most or all of the time.

Stratified flow is herein considered to be a limi ting subset of this flow pattern which occurs when all the liquid resides at the bottom of the duct. The gas flow may be insufficient to cause any curvature of the interface. Unlike vertical flow, horizontal, stratified flow can occur when there is negligible gas flow as evidenced by water draining in a storm sewer.

As the gas flow increases, large waves may occur and the surface of the liquid will climb the walls of the duct. A limit of annular flow may be encountered due to the intennittent impact of these waves with the upper surface beginning the transition to bubbly flow. This subset of semi-anular flow is sometimes termed wavy flow.

3. This regime is identical to that which occw"s in vertical flow. Shear and inertial forces completely dominate the gravitational forces and little lateral asymmetry in the moisture concentration profile is expected.

4. Agglomeration of small bubbles in bubbly flows may lead to elongated bubbles as in vertical flows. If the stratification parameter is much greater than unity, these elongated bubbles will concentrate in the upper regions of the duct. With low flow rates, the gas bubbles would tend to expand laterally to nearly equal the duct hori­zontal dimension and have the appearance of a somewhat laterally-skewed slug flow.

With larger flow rates, turbulence is larger and the flow may appear more as chum turbu­lent flow, again with some lateral skew if the stratification conditions are met. Note, how­ever, that the larger the Reynolds numbers in the duct, the more the turbulence which as­sists lateral mixing, and the less lateral asymmetry will occur.

5 . Because of the large velocities which lead to this transition in the first place, gravitational separation is generally not encountered and so this transition is virtually identical to that encountered in vertical flow.

3.S. Flow Pattern Maps

There are many flow pattern maps which have been developed over the years. Most have the same thing in common. They are based on subjective observations by individual workers. or at most, a consensus of the observations of several researchers. They provide useful indications of what the flow field may look like in the particular conditions and in the given geometry. Typical of such maps are the two shown in Fig. 3 for vertical and hOlizontal flows as given by Collier (Con­vective Boiling and Condensation, McGraw Hill, New York, 1972).

The scale parameters include the liquid and gas densities, PI and PS' and volume fluxes for liquid and gas,iandjg, and also the parameters Gf, Gg , '£I, and A. the G's are the mass fluxes of the two phases whereas the other two parameters are property combinations. Thus, for a given ther-

106 106

10' 1 0 '

1 0 4 104

10] 10'

� e.o 101

Annular

I I ,

, , , ,

" Wispy : Annular \

... 101 _ _ _ _ J _ _ _ _ _ _ _ _ _ _ . Churn� � _ J � a.

10

.\

� � I , .. I 10 " J I , Bubbly

I \ I , I , I ' .. / Bubbly-Slug

10·\ Ne

Slug

1 10 Lb/s.fl2

10 2

101

10] 104

10] 104

pd f 2

10' 10 '

106

10 6

(a) vertical flows (Hewitt and Roberts)

2.0 \00 \0 SO

� A. 0..5

0.2 0.\

� O.OJ .1 Ne :0- t ..J 002 .\ ..I0Il

kg/$.m2 Lb/s-fl2

100 20 � 100 20 SO 100 lO

Gf ¥ (b�. horizontal flows (Baker)

Figure 3. Typical flow pattern maps.

1 9

20

modynarnic state, the scale parameters are similar. However, for vertical flows, the parameters represent momentum fluxes whereas for horizontal flows. they represent mass fluxes.

3.6. Objective Flow Pattern Identification

There have been few attempts to develop useful objective flow pattem identifiers. All have utilized the fluctuating nature of the flow to one degree or another. The obvious quantities which fluctuate are void fraction and pressure, and these are the quantities which have been utilized for this purpose.

The most definitive is the fluctuating nature of the void fraction in a cross section of the duct. In bubbly flow, the cross section averaged void fraction will be a low value and will undergo minor fluctuations about the mean void fraction representative of the temporal passage of the bubbles across the plane. Thus, the probability density of the void fraction would be expected to be a sharp peak centered at the mean void fraction with a standard deviation representative of the rms void fluctuation magnitude. This is what is seen in practice (Fig. 4a).

In annular flows, the cross section average void fraction will be a high value and will undergo minor fluctuations due to the passage of waves th ough the cross section (Fig. 4b). The probabili­ty density function of void fraction would have a sharp peak at a high value of void fraction, Ct, with standard deviation representative of the rms void fluctuation magnitude due to wave pas­sage.

In slug and churn flows, a combination of bubbly and annular flows would be expected. The probability density function would exhibit two peaks conesponding to the alternate appearance of the two flow regimes. For low flow rates and velocities, the peaks would be widely separated corresponding to the clear intermittency representative of the periodic switch fom slugs to major bubbles and back again.

As the velocities increase and the flow becomes more chum-like, and tends toward the homo­geneous, the distinction between the bubbly and annular characteristics begins to blur. Neverthe­less, even for the case noted which had a mass flux of well over 300 kg/s-m2, there is a clear indication of remaining intermittency which must be taken into account in considerations of such flows.

It is clear from the probability density functions and accompanying photographs that the bubbly-armular transition is a combination of the two-separate patterns. The relative heights of each peak in the multiple-peaked PDF's represent the time fraction each pattern is present at the cross section under observation. Thus, the residence time fraction for bubbly flow would repre­sent the time during which a given property of the flow would exhibit bubbly flow-like behavior. Similar remarks can be said for the annular flow residence time fraction.

4. MIXTURE MODELS

It was mentioned earlier that there were two dominant flow regimes: dispersed and separated. Each has a model which has been developed to describe the behavior of the flow in its respective

2 1

1 2 1 2

o -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.4 0.2 0 0.2 0.4 0.6 0.8

Void Fraction Void Fraction (a) bubbly flow (b) annular flow

6 6 t: t:

5 .9 5 0 u °E t: t: ::3 4 &! 4 �

� �e .c; 3 t: 3 � 0 0 0 0 .� 2 .q 2 ::c :c � t';I .r;, ,J:J £ 0 .. p. 0 0

-0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Void Fraction Void Fraction (c) slug flow (d) churn flow

Figure 4. Probability densities for flow regime identification.

regime. For the dispersed flow regime, the drift flux model has elements bon-owed from the

theory of molecular drift in matter. This model treats the two-phase flow as a mixtw e but ac­counts for the relative motion of one phase within the other and couples the two by correlating the

interaction effects. For the sepa ated flow regime, the two-fluid model treats each phase as a sepa­

rate continuum and couples the two with intelfacial boundary conditions.

For the case where there is no slip or relative velocity between the phases. a particularly simple model is obtained termed the homogeneous flow mode.

22

4.1. Homogeneous Flow Model

This model assumes the mixture is completely mixed or homogeneous and may be treated ex­actly as a single-phase fluid. thus. since Ui = Uv• a = 13. Eq. (6). C = x. Eq. (1 1). and X and a are related to each other only through the equation of state. Eqs. ( 12) and (20). Of course. the mixture velocity is identical to the phase velocity. Eq. (25). Also. the mixture density remains the same but now is a state vaJiable.

The difficulty with the homogeneous model is associated with detelmination of transport properties for the mixture since there are no physical laws upon which to base these relationships. To calculate the frictional pressure gradient some method is needed to obtain the friction coeffi­cient. Consider the D' Arcy equation for pressure gradient in tenns of the friction factor as

dp L u2 -=/-(2 -dz D 2

(37)

where/is the friction factor and D the hydraulic diameter of the duct. For two-phase flows. the friction factor would be obtained from the Reynolds number and the Moody diagram or equivalent. For two-phase flows. one must specify how to calculate the mixture viscosity utilized by the Reynolds number.

It has been common practice since the mid 1940's to relate the two-phase frictional pressure gradient to that which would be encountered with liquid on) y flowing in the duct at the same mass flux. This is because the early development of liquid-vapor flow technology arose due to the par­allel development of nuclear power where vapor is fonned by heating a liquid causing increasing quality and void fraction and decreasing mixture density. Rewriting (37) in telms of the mass flux gives,

(38)

The quantity in parentheses is tenned the two-phase fliction multiplier and given the term $2/0 or $]0 . Various methods have been proposed to detelmine this quantity.

The simplest is to assume thatfi� is identical tono so that the homogeneous two-phase multi­plier is simply the density ratio of liquid to two-phase values. This result is usually low and fric­tion is underestimated. A better method is to assume the multiphase character of the flow makes the duct appear wholly rough and use a constant friction factor. A value of 0.02 has been deter­mined to provide a more reasonable estimate but itself tends to be on the low side.

A third method is to try to empirically detennine a mixture viscosity from which a Reynolds number is obtained from the mass flux, Re = GD/�.?�. The friction factor would then be deter­mined from the Moody chart. In addition. various correlations have been given for the two-phase multiplier, the earliest seeming to be that due to Martinelli and Nelson. The reader is directed to one of the excellent text books on multiphase flow for fUlther information on this topic.

23

One of the most useful places where homogeneous flow theory can be used is in the calculation of sonic velocities and critical flows. This will be covered later in the bok so will not be discussed herein.

4.2. Drift Flux Model

In the dispersed flow regime. the dispersed phase tends to travel relative to the continuous phase, drifting through the continuum much in the same manner as molecules drift through a continuum during a diffusion process. The model which has been developed to describe this situation is termed the drift flux model after its molecular counterpart. Much of the concepts, notation, and terminology were bOlTowed from that literature.

The difficulty in the drift flux model is that the details of the flow are not determined, but rather are averaged. Correlations must be developed and utilized for the average effects of one phase moving relative to another in the flow field, thus coupling the phases in the field.

Recall that both the superficial and actual velocities have been previously defined, the superficial velocity being termed the volumetric flux of a particular phase. The total volumetric flux of the mixture,j, was simply the sum of the individual phasic volume fluxes. Consider the difference between the gas velocity and the total volume flux, Vgj . This quantity is

Vgj = ug -j = ug - Ug + j,) (39)

so that

Vgj = (1 - a)(ug - UI) = ( 1 - a)u, (40)

which is a positive quantity as long as the vapor velocity leads the liquid velocity. Thus, the vapor velocity leads the volume flux of the mixture by an amount proportional to the relative velocity reduced by the liquid volume fraction. The vapor or gas is said to drift relative to the center of

volume of the mixture so that the quantity Vgj is termed the drift velocity of the gas.

In a similar way, the drift velocity of the liquid is given by

Vlj = - au, (4 1 )

which i s a negative quantity for the case where the gas leads the liquid; i.e., the liquid lags the mixture which lags the gas. Thus, the gas drifts forward and the liquid drifts backward relative to the motion of the center of volume of the mixture.

Volume fluxes can also be defined relative to the drift velocities. These drift fluxes are thus defmed as

jgl = aVgj = a(l - a)u, (42)

and

24

(43)

The drift fluxes are simply the volume fluxes of the individual components relative to a surface moving at the average volume flux of the mixture. Equation (43) shows that the volume flux of the liquid is equal in magnitude and opposite in direction to that for the gas.

Equation (42) for the vapor drift flux may be rewritten wholly in terms of volume fluxes and the void fraction as

(44)

so that in terms of the volume flux of the gas (44) is

(45)

This shows that the volume flux of the gas is due to both a homogeneous component, the volume flux of the mixture, and a drift flux of the gas relative to the mixture. The drift flux is a direct analog of the molecular diffusion flux in gases.

A similar expression is found for the liquid as

h = ( l - a)j -jgj.

The expressions for the volume fractions in terms of the fluxes are

jl ( jgl) and ( l - a) = -:- 1 + --:-) )1

(46)

(47)

showing that the void fraction is diminished from the value obtained from the flow rates, the kinematic void concentration, 13, due to the drift flux of the vapor relative to the center of volume of the mixture. From another viewpoint, since the vapor moves faster than the mixture average,it requires less flow area than it would moving at the same speed.

In both cases, it is seen that if there is no relative velocity between the gas and the liquid, the drift flux vanishes and the void fraction is determined directly from the measurable volume fluxes of each phase.

To see how the drift affects the density of the mixture, Eq. (22) may be written in terms of the volume fluxes and drift flux using (47) so that

(48)

which shows that the mixture density can be considered the volume flux-weighted density with a correction due to the drift flux effect which causes the vapor to occupy less than homogeneous volume and the liquid to occupy more.

2S

Finally, it is common that the relative velocity, U" is assumed to be a function of the terminal velocity of a single discrete entity and the phasic volume fraction, the latter accounting for the crowding and buoyancy effects. The latter can be viewed as the trend toward the limiting case where the fluid the entity is passing through becomes gradually made up of the same material, thus increasingly offsetting the gravitational settling of the entity. In the limit as the less dense phase is completely displaced, the entity is buoyed by a force due to the weight of fluid displaced which is exactly the weight of the entity itself.

For the gas phase, then, the relative velocity can be written as

(49)

which shows that v, � 00 as ex � 1. The dlift flux can then be written in terms of the telminal velocity of the single bubble as

(50)

and behaves as shown in Fig Sa.

The terminal velocity for single gas bubbles in an infinite extent of liquid is well known (cJ. Wallis, G.B., One Dimensional Two-Phase Flow, McGraw hill, New York, 1969). thus, Eq. (50) provides a method of correlating the drift flux for given situations by simply determining the value of n which best fits the data. Similar results could be found for liquid droplets in a gas or less dense liquid medium, or for solid particles in gas or liquid.

Finally, consider the expression for the drift flux in terms of the individual volume fluxes of the phases:

(5 1 )

which shows thatjgl i s a linear function of the void fraction having an intercept at ex = a ofjg and an intercept at ex = 1 having a value of -k

Equation (50) can be considered the equation which govems the physical behavior of the drift flux or phenomena line. Equation (5 1), on the other hand, is the operating line for a given system. Together, they represent a set of simultaneous equations which specify the actual operating con­ditions which will be achieved.

Examine Fig. 5a. The curved line represents Eq. (50). In Fig. 5b, however, the straight lines represent several potential operating lines, all having the same value for the gas volume flux,jg . In the first case for.iJ.i . the liquid flows upward cocurrently with the gas. Note that as the liquid up­flow increases, the unity void fraction intercept becomes more negative and the line becomes steeper with negative slope. In the second case, since -.iJ,2is positive,.iJ,2 is negative and the liquid flows down the duct in countercurrent flow relative to the gas upflow. For the cases ofjf,3 andh",4 . the rate of liquid downflow increases for each case relative to case for h",2 .

26

Figure 5c shows the combination of (a) physical behavior the previous two figures where the op-

erating lines are shown in conjunction with the phenomena line. Recall that ;.: the straight lines represent several po- :I tential operating conditions where the �

4: flows of liquid and gas are indepen- � dently set.The point(s) of intersection of a given operating line with the phe- 0

nomena line detelmine the possible op-erational states for the system. Line #l , being cocunent upflow of liquid and 0 Void Fraction gas, has only one possible operating state. An increase in the liquid flow rate

(b) operating lines would steepen the negative slope of the jl,4 operating line and move the intersec- jg tion point to the left, decreasing the void fraction. A decrease in the flow ;.: :I rate would lessen the slope of the oper- � -jl,3 .: ating line. increasing the void fraction. � -j1,2

Line #2 represents countercurrent 0

flow, liquid downflow, gas upflow. In this case, there are two possible opera-tional states, one at low void fraction 0 Void Fraction and one at a higher value. The actual void fraction which would occur in (c) intersection of operating lines with practice would depend on how the con- phenomena line -jl.4 ditions were approached; i .e., the meth jg ad of obtaining the operational state. If ;.: the duct is initially dry. and liquid

:I fi: jl,3

downflow begins very slowly from the .: top, the higher void condition would be � -jl,2 expected to be reached. On the other

0 hand, if the duct began with very low downflow of liquid at the specified val-ue but with a duct full of liquid, and then the gas flow was slowly increased, 0 Void Fraction the lower void state would be that ob- Figure 5. Physical behavior and operational tained. characteristics of the drift flux model.

Line #3 is similar to that for operational state #2 except that the two possible operating condi­tions have coalesced to a single solution. 1bis solution represents the extreme limit of liquid

27

downflow with the given upflow of gas. Conversely, this state represents the maximum upflow of gas with the given liquid drainage down the duct.

Further increase in the gas upflow or liquid downflow would result in an impossible operation­al state. In practice, what happens is that with an increase in gas upflow from state #2, downflow would no longer be possible and all the liquid would be swept up and out of the duct, thus transitioning immediately to a zero liquid flux condition and unity void fraction. For obvious reasons,

this condition is termed flooding. Alternatively, if the liquid downflow were to be increased, the gas would be swept down and out the bottom of the duct with transition to 100% liquid downflow

and zero gas flux.

The challenge in the drift flux model is the appropriate determination of the exponent n for use with Eq. (50), and to determine the limits on the use of this exponent. In practice, as the void frac­

tion changes, flow regimes change and thus the value of the exponent can change markedly, thus altering the operational state of the system.

4.3. Two-Fluid Model

In the separated flow regime, the two-fluid model, which considers each phase separately from the other, has found acceptance. Indeed, due to the relative ease of numerically p rogram

ming single-phase equations for each phase, and providing results, this model has been accepted

even in the case of dispersed flows.

The basis for the two-fluid model is the application of the Navier-Stokes equations sepai-ately

in each phase. If this were to be undertaken instantaneously, the equations for the liquid and gas

must be separately coupled to existing solid boundaries as appropriate. In addition, these equa­tions must be matched at liquid-gas interfaces with appropriate values of the dependent variables

and their fluxes specified as matching conditions on these boundaries. In practice, this is not gen­erally undertaken instantaneously, but rather some attempt is made to average the field equations and the matching conditions .

The obvious difficulty, then, in the use of the two-fluid model is that the coupling between the

phases must be done locally and instantaneously at the interfaces_ Knowledge for closure of a field description of separated two-phase flow must be obtained by inteyfacial balance equations which account for the mass, momentum, and energy transfer at interfaces which in most cases are poorly defined in space and time. Thus, while early and easy computational results have been possible using this model, these results have been based on inappropriately simplified assump­

tions of interfacial physics and assumptions of multiple arbitrary coeffieients which have little or no basis in fact.

While the drift flux model will not be discussed to any great extent within the balance of this book, the two-fluid model will be discussed extensively. For this reason, no additional descrip­

tion of the latter model shall be given here.

4.4. Averaging

Due to the variation between phases at a point, multiphase flows tend to have significant fluc­

tuations in both space and time. For instance, at a point in space-time, (x,t), the void fraction can

28

be considered as a binaIyfunction equivalent to the Kronecker delta, Og(x,t). That is, when the gas-phase exists at a the point, Og(x,t) = 1 , whereas when the liquid phase exists at a point, 08 (x.t) = 0.

The void fraction at a point in space and time can be considered as the short -time average of 08 given by

(52)

with "t'p being the period of the high frequency fluctuations of interest in the process itself.

The average in Eq. (52) is taken over a time span which is long compared with the void fluctua­tions and short with respect to the transient system behavior. In this way, the averaging is accom­plished in much the same manner as that used for determining turbulence in single-phase fluids. In fact, however, the void fluctuations can have substantial energy in the low frequency domain so that it may be impossible to determine a meaningful average from (52). In this case, the two phase flow problem should be treated as in initial value problem and the transient behavior detelmined in the absense of averaging.

When the average in (52) has no meaning, one can average in space to obtain the time varying cross sectional average void fraction given by

< aCt) >= 2-f a(x, t)dA A A (53)

which defines the area-average of any quantity as well as that required for determining the average void fraction in a cross sectional area. Thus, the area-averaged velocity would be given by

< u >=�f udA. A A (54)

Consider now the implications in averaging relative to the drift flux mode. Equation (45) re­lates the gas flux to the total volume flux and the drift flux. Averaging term-by-term yields

<jg >=< aj > +<jgl > (55)

If a mean drift velocity Vgj is defined as

(56)

and a distribution parameter Co is defined as

TABLE 1 . Values for drift-flux calculation of void fraction.

Flow Regime Void Fraction Distribution (l Parameter, Co

Bubbly flow 0.0 < (l < 0.25 1 .25

Slug/chum flow 0.25 < (l < 0.75 1 .15

Annular flow 0.75 < (l < 0.95 1 .05

Mist flow 0.95 < (l < 1 .0 1 .00

• ( � ) 1/4 Vgj = 1 .53

• ( d6() ) 1/2 Vgj = 0.35 �

• Vgj = 0.5 - 5 m/s in steam/air and water

• ( � ) 1/4 = 1 .53

< aj > Co = .

< a >< j >

then

< j g >= Co < a >< j > +< a > v gj

so that

jg --= Co < J > + vgj. < a >

Drift Velocity vgf

Eq. (57)

Eq. (58)

Eq. (59)

Eq. (60)

29

(57)

(58)

(59)

(60)

(61)

(62)

(63)

Equation (63) has become relatively important since the quantity on the left, when plotted as a function of the mixture volume flux, tends to be a straight line for a given flow regime, having a slope of Co and an intercept of v gj. Sharp breaks in the slope of the line have been observed when the flow regimes transition from one to another. Con-elations of both slope and intercept have been determined for various flow regimes as reported in the literature and various text books on two-phase flow. The values in Table 1 are reasonable distillations of these results.

30

Except for the distribution parameter and the averaged drift velocity, all tenns are directly

measurable. The drift velocity has been correlated and the distribution parameter detennined for several flow regimes.

Once the values of the distribution parameter and dlift velocity have been determined, it is a

simple matter to determine the void fraction for vwious combinations of gas and liquid flow rates

through simple rewTangement of Eq. (63) to yield

< j > < a >= g Co < j > + vgj

(64)

The values of distribution parameter and drift velocity which are reasonable and can be used to detennine the void fraction are given in Table 1 . Note that the values for annular flow are only

approximate as this is a separated flow pattem where the drift velocity is not generally constant. However, the values provided will generally suffice for cocurrent annular flow in cases where the mixture velocity is significantly larger than the dlift velocity. Note further that Eqs. (57) and

(60) are identical except for the difference in the phase density in the denominator.

Note that iteration may be required since the drift and distribution parameters chosen are de­

pendent on the void fraction. This is generally simple to do and with some practice becomes an

infrequent necessity.

Note, too, that if the drift velocity is small relative to the modified total volume flux, the void

fraction is given approximately by

Qg f3 < a >= -- = -. CoQ Co

(65)

It is thus seen that even without local slip, there are differences between the kinematic static

void fractions. These differences are due to concentration and velocity profiles, and their interac

tionas shown by Eq. (6 1 ) . From Table 1 , it is seen that Co varies upward to approximately 1 .25 for

simple adiabatic flows. Therefore, the void fraction will always be less than the kinematic void fraction by as much as 10-20%. This is because the voids tend to concentrate in the regions of

high velocity so w·e caITied preferentially. (Caution should be taken in diabatic situations as the flow regimes can be considerably distorted relative to those encountered in gas-liquid flows and

the distribution parameter significantly affected. Such is the case, for instance, in subcooled boil­

ing flows where all the voids are immediately adjacent to the heated surface.)

ANALYTICAL MODELING OF MULTIPHASE FLOWS

Donald A. Drew Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy. New York 12180-3590

1. INTRODUCTION

3 1

The occurrence of multiphase flows is very widespread. Indeed, vapor/liquid flows are common in power production and utilization technologies and in chemi­cal processing. Likewise, solid/fluid slur flows, fluidized beds and various com­bustion processes are other common examples of important multiphase flows.

It is vital for industrial applications that predicitive solutions to equations describing the motions of multiphase flows be readily obtained.

This chapter presents a theoretical framework for modeling unsteady multi­dimensional multiphase flows. The models that are described here are based on the idea that a continuum description of the gas and liquid is useful. Thus, we shall derive a model for the evolution of the volume fraction of gas, the densities of the gas and liquid, the velocities and temperatures of the gas and liquid.

The derivation has several parts. First, we shall derive equations of balance of mass, momentum, and energy for both phases. We shall perform this deriva­tion two different ways, using the CONTROL VOLUME approach, and using the AVERAGING approach. Then we shall derive closure laws. To do this, shall ap­ply the average operation to the solution for the flow arc;mnd a single sphere. We then discuss a more general approach to closure conditions. Finally, we present some considerations of the nature of the equations.

2. MULTIPHASE CONTINU BALANCE EQUATIONS

In this section we shall derive the equations of balance of mass, momentum, and energy for each phase in a two-phase material. We shall use the CONTROL VOLUME approach, where we consider separately the amount of mass, momen­twn, and energy of each phase in a fixed control volume, and account for how it changes.

In order to describe the properties of multi phase mixtures, we introduce the idea of volume fraction, a,,(x, t) for each constituent, or phase. If V is an arbitrary

32

control volume having the point x inside, and Vi.(t) is the volume of phase Ie inside V, then

Ar.(V) = V.(x, t) V

(1)

is the ratio of volume of phase Ie to total volume in V. Then the volume fraction is defined to be

OJc(x, t) = lim Ar.(r, x) v-a (2)

The density of phase Ie is defined as follows. The mass of phase Ie inside V is given by Mr.(V). The mass density or phase Ie is defined by

1. MJc(V) pr. = Im --­v-o Vj.(x, t)

(3)

Note that this mass density is the mass PER UNIT VOLUME OF PHASE Ie. The mass of phase Ie per unit total volume is given by or.pr..

The mass of phase Ie inside an arbitrary control volume is given by

fv OJcPr. dV (4)

The mass of phase Ie inside of V can change due to two different effects. First, mass can move into or out of V across the boundary avo The flux of mass of phase Ie, the mass per unit area per unit time, across an element of surface with unit normal n, is given by Or.pJc Vr. . n. This mass flux defines the defines the center of mass velocity or barycentric velocity VJc. Also, mass of phase Ie is increased at a rate rr., the rate of production of mass of phase Ie per unit volume, due to phase change or chemical reaction. ,This is the rate at which material of the other phases change to material of phase Ie. Then bal ance of mass of phase Ie is given by

-dd Or.Pr. dV + OJcpJcVJc · n dS = rJc dV, (5)

U sing the divergence theorem on J8V Or.pJc VI: • n dS and taking the time derivative inside the integration over the (fixed) volume V gives

(6)

33

If we assume that the integrand in eq. (6) is continuous in V, then by the Dubois­Reymond Lemma, the integrand vanishes. Thus, we have the equation of balance of mass:

(7)

The momentum of phase k inside of V is defined to be

(8)

The momentum of phase k inside of V can change due to several different effects. First, momentum can move into or out of V with the mass that moves in or out of V This is expressed as

(9)

In addition, the material outside of V exerts a force on the material inside of V through the boundary av of V. This momentum flux is given as

( 10)

where tk is the traction. The traction exerted depends on the orientation of the surface element dS. This dependence is denoted by

(11)

Note that tAt is the force per unit area of phase k acting through av. It seems rea­sonable to assume that the force per unit (mixture) area is equal to ctk times the force per unit area of phase k. Also, the momentum of phase k can be increased inside of V due to body sources. The body source has three parts. The first is the external body source, given by

(12)

where gil is the external acceleration (due to gravity and other body forces). The second body source term is the interfacial force, representing the interaction of the materials; more specifically, it is the force on phase k due to the intimate contacts with the other phases. It will be taken to be of the form

34

(13)

where M", is the interfacial force per unit volume. The third body source term is the rate of momentum gain due to phase change or chemical reactions, and is given by

(14)

where Vki is the interfacial velocity of phase k. Balance of momentum of phase k is given by

( 15)

We apply the divergence theorem to the first surface integral to obtain

(16)

n n

Figure 1. A "Flake"

If we then apply the momentum equation (15) to a volume with largest linear

35

dimension I and let I be small, the volume integrals in eq. ( 15) are of order 13, while the surface integrals are order z2. Consequently, the surface integrals must be zero for sufciently small volumes. If we apply this argument to a "flake" with one dimension much smaller than the other two, we note that all volume integrals in eq. (15) go to zero, while the integrals of the flux term collapse to integrals over the front and back sides of the flake. See Figure 1. We then have

(17)

Since the face S of the flake is arbitrary, we see that

( 18)

We then apply the momentum balance equation to a tetrahedron with one face having normal n and the other three faces having normals to the coordinate directions e l o e2 , and ea. If the tetrahedron is sufficiently small, the volume integrals will be negligible compared to the surface integral. See Figure 2. We have

n

Figure 2. A Tetrahedron

(19)

36

Note that eq. (18) gives

t,,( ell = -t.( el l , t,,( e2) = -t,,(�) ,

t,,( -ea) = -t..,( ea)

Also, the areas of Si t S2, and Sa are related to the area of Sn by

SI = Sn(el · n) , S2 = Sn(e2 . n) , Sa = Sn(ea ' n)

If we use the integral mean value theorem on eq. (19), we have

(20)

(21)

(22)

Thus, the traction vector for phase k is linear in the normal n, and defines the stress tensor T" so that

(23)

Note that the stress tensor represents the force per unit area of phase k across a surface element having unit normal in the appropriate coordinate direction. The combination Q" T", represents the force per unit total area.

If we apply the divergence theorem to the remaining surface integral, take the time derivative inside the volume integral , and assume that the integrand is continuous, we can obtain the equation of balance of linear momentum. We have

(24)

The energy of phase k inside of V is defined to be

(25)

where u. is the internal energy. Note that the term jv� represents the kinetic energy of the mean motion; if there are velocity fluctuations due to the relative motions of the materials, that kinetic energy will be included in the internal en-

37

ergy Uk. In analogy with the momentum, energy of phase k inside of V can change due to several different effects. First, energy can move into or out of V with the mass that moves in or out of V. The rate of change of energy due to this is

(26)

The material outside of V does work on the material inside of V through 8V. This causes a rate of change of energy equal to

(27)

In addition, energy can flow through 8V in the form of a heat flux. We assume that the heat flux per unit (mixture) area is equal to O!k times the heat flux per uni t area of phase k. This rate of change of energy is given by

(28)

Also, energy of phase k can be increased inside of V due to the work due body forces, an energy source, and change of phase. The energy source due to the working of the body force is

(29)

The energy source is taken to be

(30)

where Tic is the energy source per unit mass. The energy changes inside of V due to the working of the interfacial forces and due to the heat flux through the interface. This source is taken to be

(31)

The energy source due to phase change is taken to be

(32)

Balance of energy of phase k is given by

38

(33)

An argument similar to that given for the stress tensor gives

QIe(X, t; n) = n . q,. (34)

where qle is the energy flux vector for phase k. Then, applying the divergence the­orem, taking the time derivative inside the integral, and assuming the resulting integrand is continuous yields the equation for the balance of energy

(35)

In the momentum equation (24), the exchange of momentum has the same form as a body force, and in the energy equation (35), the exchange of energy has the same form as the body heating ("radiation"). In our treatment of the dynamics we assume that the body force and body heating are specified externally, while the exchanges of momentum and energy are specified by constitutive equations. In fact, motivating the forms of such equations has a central role in the theory of multiphase mixtures. In order to give appropriate forms for the necessary constitutive equations, it is helpful to have a different version of the derivation of the equations of motion. This is done in the next section.

3. AVERAGING

The CONTROL VOLUME approach to deriving the equations of balance of mass, momentum, and energy sufers from a problem of interpretation. Note that with the definition of a., eq. (2), is such that when V -+ 0, if the point x happens to lie in phase Ie, then the value of ale so defined is equal to 1, while if x is outside of phase Ie, then ale = O. Thus, taking the mathematical operations seriously leads to quantities that are not appropriate to describe macroscopic flow properties. Indeed, the quantity ala defined this way is discontinuous. There are arguments that attempt to fix this difficulty by not letting the volume V get too

39

small. The approach that we shall take here is that we shall assume that the appropriate fields to use to describe the flow should be average variables.

As an example of how averaging gives appropriate information in multiphase flows, consider the following situation for the motion of particles in gas turbines. Gas turbines blades may be eroded by particulate matter suspended in the gas stream which impacts on turbine blades. The trajectories of individual particles moving through the gas turbine are very complicated, and depend on where and when the particles enter the device. Fortunately, predictions of the exact time and place of the impact of a single particle is not normally required. A prediction of interest to the designer is the expected value of the particle flux near compo­nents susceptible to erosion in the gas turbine. Thus, average, or expected values, of the concentration and velocities of the particles are of interest. The local con­centration of particles is proportional to the probability that particles will be at the various points in the device at various times. The particle velocity field will be the mean velocity that the particle will have if they are at that position in the device. With this information, the design engineer will be able to assess the places where erosion due to particle impact may occur. Note that there may be no times that there will be many particles in some representative control volume. Thus, averaging schemes that depend on the concept of many representative par­ticles will fail. Ensemble averaging, on the other hand, is appropriate. In this case, the ensemble is the set of motions of a single particle through the device, given that it started at a random point at the inlet (with some distribution of position associated with the dynamics of the particle moving through the inlet flow) at a random time during the transient flow through the device. It is clear that the solution for the average concentration and average velocity gives little information about the behavior of a single particle in the device; however, the information is quite appropriate for assessing the probability of damage to the device.

Often in the literature, averages such as time averages [ 1] , space averages [2] , and combinations of such averages [3] are used. Such averages sometimes have utility in certain specific situations but have shortcomings.

We shall re-derive the equations of balance of mass, momentum and energy using AVERAGING techniques.

A prime characteristic of multiphase flows is that there is uncertainty in the exact locations of particular constituents at any given time. This means that, for equivalent gross flows, there will be uncertainty in the locations of particular constituents for all times. For instance, consider a suspension of small particles in a liquid. The exact distribution of the locations of the particles is immaterial as long as they are reasonably "spread out." We would not allow the particles to be lumped in some way that is not consistent with the initial conditions appropriate for the flow.

Consider the set of all experiments with the same boundary conditions, and initial conditions with some (undefined) properties that we would like to associate

40

with the motion and distribution of the particles and the fluid. We call this set an ensemble. Such ensembles are reasonable sets over which to perform averages because variations in the details of the Hows are assured in all situations, while at the same time variations in the gross flows cannot oc.

We take the approach here that we wish to predict averages over ensembles of flows. If, indeed, an average is justified, it does not matter that we allow the particles, bubbles or droplets to be rearranged in space within reason.

We shal derive the balance equations by applying an averaging operation to the equations of motion for two continua separated by an interface across which the densities, velocities, etc. may jump. We then define appropriate averaged variables, and write the averaged equations in terms of them. As a special ex­ample, we shall consider these equations for two materials, specifically, we shall consider a particle/fluid mixture. We will then consider constitutive equations for the flow of a dilute mixture of spherical bubbles.

3.1 Local Balance Equations The exact equations of motion, valid inside each material are

Mass Balance

op - + V ' pv = O at '

Momentum Balance

apv at + V . pvv = V . T + pg ,

Energy Balance

where p is the density, v is the velocity, T is the stress tensor

T = -pI + T

(36)

(37)

(38)

(39)

Here p is the pressure and T is the shear stress; u is the internal energy; q is the heat flux; and r is the distributed (per unit mass) heating source.

These equations are assumed to hold in the interiors of each material involved in the flow.

4 1

The canonical form o f the equations o f motion for the flow of a pure substance, or for the flow within a region consisting entirely of one phase in a multiphase mixture, can be written conveniently as

8p'l - + V · p'lv = V · J + p!

8t (40)

where 'l is the quantity conserved, J is its molecular flux, and ! is its source density. The so-called molecular or difive fluxes J involve transport properties. For example, the heat-flux vector involves thermal conductivity; the stress, or momentum flux vector, involves the viscosity.

3.1 .1 Jump Conditions At an interface between phases, properties are discontinuous, although mass,

momentum and energy must be conserved. Neglecting storage terms on the in­terface, the jump conditions valid across the interface are Mass Jump

(41)

Momentum Jump

[(pv(V - Vi) + T) . n] = mi (42)

Energy Jump

[(p ( u + �v2) (v - Vi) + (T . v - q» . n] = er (43)

where Vi is the velocity of the interface, n is the unit normal, and [ ] denotes the jump across the interface between the gas and the liquid. Here mf is the traction associated with surface tension; and Ef is the surface energy associated with the interface.

The surface traction, which has the dimensions of stress, is defined as [4]

(44)

where tu is the hybrid tensor; aUU is the metric tensor in the surface; u is the surface tension; and ( ) .u denotes the covariant derivative. The surface energy

42

source term is given by

(45)

The generic conservation principle for an interface is expressed by the following jump condition:

(46)

where Vi is the velocity of the interface, n is the unit normal, and Mi is the interfacial source of 'iP 0

3.1.2 Summary of the Exact Equations of Motion The generic form of the equations of motion for the exact motions of the

materials involved have been shown to be

8p'iP lit + TV • p'iPv = TV 0 J + pi, (47)

with corresponding jump conditions

(48)

The usual values for q;, J, f and m are given in Table 1.

Conservation q; J f Mi Principle Mass 1 0 0 0 Momentum V T g mi Energy 1£ + T o v - q g · v + r e� ,

Table 1. Variables in Generic Conservation and Jump Equations

4. ENSEMBLE AVERAGING

A very elementary concept of averaging involves simply adding the observed values and dividing by the number of observations. Ensemble average is a gen­eralization of adding the values of the variable for each realization, and dividing by the number of observations. We shall refer to a "process" as the set of possible flows that could occur, given that the initial and boundary conditions are those

43

appropriate to the physical situation that we wish to describe. We refer to a "re­alization" of the flow as a possible motion that could have happened. Generally, we expect an infinite number of realizations of the flow, consisting of variations of position, attitude, and velocities of the discrete units and the fluid between them.

If f is some field (i.e. , a function of position x and time t) for some particular realization p, of the process, then the average of f is

f(x, t) = i f(x, t ; p,) dm(p,) (49)

where dm(p,) is the measure (i.e., probability) of observing realization p, and £ is the set of all realizations of the process of interest. We refer to £ as the ensemble. The ensemble average is an average that allows the interpretation of the phenomena in terms of the repeatabili ty of experiments. Any one exact experiment or realization will not be repeatable; however, any repetition of the experiment will lead to another realization, or member of the ensemble.

4.1 Other Averages The literature in fluid mechanics contains many different types of averages.

These are often motivated by the type of application. In this section we shall discuss some of these averaging processes.

The time average is defined by taking one realization, p,", integrating over the time interval from t - T to t, and normalizing:

� 1 f (x, t) == T f(x, t - T; p,*) dT. (50)

The use of this average relies on the ergodic hypothesis, which assumes that if a suming operation samples enough values of the random variable (in this case, f(x, t) , then the operation approximates the mean, or average of it. Use of the time average has the advantage that we need not sample the ensemble repeatedly but need make only one detailed observation to infer averaged values.

A similar interpretation can be given to the volume average. Thus, for a realization p, * ,

(51)

We note that in order for the time average or the volume average to be valid, the integration must sample values of the variable f in a way appropriate to be assigned to the space-time point (x, t). If, for example, the process is non­stationary, in the sense that the average is not steady (i.e. , changes in time), then the time average samples values of f that are contaminated with information from earlier times, and is inappropriate. A similar interpretation suggests that the

44

volume average is inappropriate near a macroscopic spatial inhomegeneity such as the volumetric wave that develops when a carbonated beverage is poured. and the head forms.

The ensemble average is the more fundamental averaging process. Both the time and volume average should be viewed as approximations to the ensemble average. which can be justified for steady flow or homogeneous flow. respectively.

4.2 Averaging.Procedures In order to apply the ensemble averaging procedure to the equations of mo­

tion, we shall need some results about the averaging procedure.

4.2.1 Gauss and Liebniz Rules

In order to average to the exact equations. we need expressions for 8! / fJt and V I. If I is "well behaved," then it is clear from the definition of the ensemble average that

al a7 at = at

and

(52)

(53)

Functions are generally discontinuous at the interface in most multiphase flows. They are well behaved within each phase, however. Thus, consider X1c V!. where Xle is the phase indicator function for phase k:

x { 1 , if x E k; Ie - 0, otherwise.

Then

and

= VX1c! - IVXIe•

Xle = aXlel _

at {}t at

(54)

(55)

= aX.f _ f

ax" lJt lJt .

45

(56)

The second term on the right hand side in both of these equations is related to the surface average of f, evaluated on the phase Ie side, over the interface.

4.2.2 The Topological Equation In the averaging process we use require the result

DiXie ax" -- = - + v· · Vx. = o

Dt at • (57)

This relation (57) has a reasonable physical explanation. Note that it is the "material" derivative of X. following the interlace. If we look at a point that is not on the interface, then either X" = 1 or XIa = O. In either case, the partial derivatives are both zero, and hence the expression (57) is zero. If we consider a point on the interface, if we move with the interface, we see the function X" as a constant jump. Thus, its material derivative is zero.

4.3 Averaged Equations The process of deriving the averaged equations involves multiplying the exact

conservation equation, eq. (40) by XIa and ensemble averaging. The result of this procedure can be seen by manipulating the phase function X •. We have:

(58)

Subtracting the average of p'iJ! times the result in eq. (57) reduces the right hand side of (58) to

(59)

This is the interfacial source of \Ii', and is due to phase change, if v · n =f Vi ' n, and to the molecular flux J.

Using 'iIi' = 1, J = 0, and f = 0 results in the equation of conservation of mass; using 'iJ! = v, J = T. and f = g results in the equation of conservation of mass; and using \Ii' = e = 'U. + �v2; J = T . v q; and f = g . v + r gives the equations of conservation of mass, momentum and energy for each component.

The resulting ensemble averaged equations are

46

Mass

8X,.p ---- + v . X,.pv = p(v - Vi) . V X,.

Momentum

8X,.pv -- -- ---- + V · X,.pvv = V · X.T + X,.pg + (pv(v - Vi) - T) . VX,..

Energy

1 + X,.p(g · V + r) + [P(u. + 2V2)(V - Vi) - (T . V - q)] . VX,.

4.4 Definition of Average Variables

(GO)

(61)

(62)

In this section, we define the appropriate average variables describing mul­ti phase flow.

First, the geometry of the exact, or microscopic situation is defined in terms of the phase indicator function X,.. The average of X,. is the average fraction of the occurrences of phase Ie at point x at time t.

(63)

It has become customary to call this variable the volume fraction of phase Ie, even though technically the volume fraction is the ratio of the volume of phase Ie in a small region, divided by the total volume. We note that this concept is inti­mately tied to volume averaging, and so is correct only for spatially homogeneous situations.

Another important geometric variable is the interfacial area density. The interfacial area density is defined by

(64)

where n,. is the unit external normal to material Ie.

47

All the remaining variables are defined in tenDS of weighted averages. The main variables are either weighted with the component function X" or mass­weighted averaged (weighted by X"p). Other variables are weighted with the interface variable V X".

The "conserved" variables are the appropriately weighted density.

(65)

velocity,

(66)

and internal energy,

(67)

The variables representing the averaged effects of the molecular fluxes are stress,

(68)

and heat flux,

(69)

The average body sources are body force,

(70)

and energy source,

(71)

The molecular fluxes to the interface act as sources of mass, momentum, energy or entropy to the interface. These terms are important in the theory of multi com­ponent flows since they represent the interactions between the materials. The interfacial momentum source is defined by

48

MAI = -T · VX" , (72)

the interfacial heat source is defined by

E" = q , VXAl 1 (73)

and the interfacial work is defined by

WI! = -T · v · VX" (74)

The relative motion of the various phases give rise to velocities that are not "lam­inar" in general. The effect of the fluctuations of a variable from its mean value is accounted for by introducing its fluctuating field (denoted by the prime super­script). One variable defined in this way is the so-called Reynolds stress

(75)

where

(76)

The pieces of the fluctuation energy flux are the fluctuation (i.e., Reynolds) kinetic energy flux

(77)

the fluctuation (i.e., Reynolds) pressure-velocity correlation

(78)

and the fluctuation (i.e., Reynolds) shear working

(79)

We can combine eqs. (78) and (79) as the fluctuation (i.e., Reynolds) shear work

(80)

49

The fluctuation (i.e., Reynolds) internal energy flux is defined by

(81)

In the energy equation, it is convenient to combine the fluctuation internal energy flux with the turbulent energy flux. Thus we may write

(82)

The fluctuation (i.e., Reynolds) kinetic energy is

(83)

Note that

(84)

The averaged interfacial pressure Plci and shear stress TIci are introduced to sep­arate mean field effects from local effects in the interfacial force. The interfacial pressure is defined by

(85)

and the interfacial shear stress is

(86)

Several tenns appear representing the actions of the convective and molecular fluxes at the interface. The convective flux terms are the mass generation rate

ric = p(v - Vi) ' VXIc I (87)

the interfacial momentum. flux

VZir. = pv(v - Vi) . V X. I (88)

50

the interfacial internal energy flux

(89)

and the interfacial kinetic energy flux

(90)

4.5 Averaged Equations We now present the averaged equations. The averaged equations governing

each phase are

Mass Balance

(91)

Momentum Balance

(92)

Energy Balance

(93)

4.5.1 Jump Conditions

The jump conditions are derived by multiplying the exact jump condition (46) by DI V Xl and averaging. This process yields the following conditions: Mass Jump

(94)

5 1

Momentum Jump

(95)

Energy Jump

(96)

where mi is the surface tension source, and "'if is the interfacial energy source.

The ensemble averaged equations and the postulated equations are identical if we interpret the various quantities properly. First, the densities of each phase are the z-weighted averaged density:

(97)

The velocity of phase k is the zp, or mass weighted average:

(98)

The mass source due to phase change is related to the microscopic fields in that it is the average of the mass flux through the interface. Thus,

(99)

The (total) stress is made up of the average of the exact stress, plus the Reynolds stress:

(100)

The interfacial force density is the interfacial average of the normal component of the stress at the interface:

(101)

The internal energy consists of two parts, one that corresponds to the local inter­nal energy, and one that corresponds to the fluctuation kinetic energy 1L�:

52

'ILl. = UZP + 'lLr- (102)

The energy flux has several parts, one corresponding the the local energy flux, one corresponding the the flux of internal energy by the fluctuating velocity, one corresponding the the flux of fluctuation kinetic energy by velocity fluctuations, and one that corresponds to stress workings with the velocity fluctuations. Thus

(l03)

The interfacial energy transfer is given by

(104)

In order to formulate the equations further, many terms must be constituted in order to achieve closure. Thus, let us now turn our attention to the closure laws for two-fluid models of two-phase flows.

5. CLOSURE CONDITIONS

5.1 Completeness of the Formulation

The three-dimensional, unsteady multiphase model given by (91), (92), and (93) must be supplemented with state equations, constitutive equations such as those discussed in Section 5.3 and boundary and initial conditions. Thus let us discuss a general approach by which closure can be achieved.

Initial conditions specify how the experiment or the multiphase flow starts. The boundary conditions specify how the flow interacts with its environment, specifically with the inlet and outlet flow devices and with the walls of conduits.

The constitutive equations generally specify how the materials (i.e., phases) interact with themselves and with each other. State equations specify the ther­modynamic state of the material as functions of those (state) variables that determine it. For example, in a single phase fluid, the internal energy is determined by the density and the entropy. Boundary and initial conditions are important,

in the sense that if one is specified incorrectly, the predicted flow will be in er­ror. However, the key to the accurate modeling of multiphase flows is to specify correctly the phase interaction terms (M/co ), and the self-interaction terms <T":, T�, ilk, qfe, . . . ) in terms of the state variables of the (o/e> yZP, . . .).

5.2 Constitutive Equations

The variables still to be specified involve the phasic interaction terms, and the self-interaction terms, which, in turn, depend on transport properties and on

53

turbulence effects. In view of the interfacial jump conditions, not all of the phase interaction and phase change terms are independent. The first step in completing the formulation, then, requires the choice of a set of dependent variables and a set of independent variables whose values determine them. Once this functional dependence is established, we must find the functions which describe the con­stitution of the materials. The following principles may be applied to guide the development of these constitutive relations.

5.2.1 Guiding Principles The principles of constitutive equations [5] provide a rational means for ob­

taining descriptions of classes of materials, without inadvertently neglecting an important dependence. We shall consider here the principles of equipresence, well-posedness, phase separation and frame indifference. These are the most fundamental and restrictive of the guiding principles provided by continuum me­chanics.

The principle of equipresence states that all variables should be included in each constitutive equation, unless another principle shows that a particular dependence cannot occur.

The principle of well-posedness states that the description of the motion should be such that a solution to the initial boundary value problem exists and depends continuously on the initial and boundary conditions. Moreover, it implies that all eigenvalues of the system of partial differential equations be real.

The principle of objectivity asserts that the constitutive equations cannot de­pend on the reference frame. The motivation is that reference frames are inven­tions of the observer and are not intrinsic to the materials being described. Thus, the functional dependence of the constituted variable on the variables describing the mechanical/thermal state must be independent of frame (i.e. , objective), and the variables used to describe the mechanical/thermal state must also be objec­tive, and thus independent of coordinate system. We shall discuss this principle in some detail next.

5.2.2 Objectivity

Consider two reference frames, so that x and x', are different representations of the same vector. The two representations are related by

x/ = Q . x + b , (105)

where Q(t) is an orthonormal transformation, and b is a translation. A scalar is objective ifits value is invariant to a change of coordinate system, that is, if " = ,. A vector is objective if it changes coordinates according to V' = Q . V. Tensors are objective if they transform each phase correctly. Specifically, a second order tensor is objective if T' = Q . T . QT. Velocity is not objective. If we differentiate eq. (l05) following material k we see that

54

V� = Q . VII + (Q . x + Ii) ( 106)

If eq. (106) is evaluated for two different values of k and the results subtracted, the velocity difference becomes

(107)

Thus, velocity differences are objective. Specifically, for two-phase gas-liquid flow,

V.,. = Vg - V, (108)

is objective, where the subscript 9 denotes the gas, and the subscript I denotes the liquid.

In multi phase flows there are reasons to believe that the interactions can depend on accelerations. However, in general, accelerations are not objective. Following phase m and diferentiating eq. (106) leads to

(l09)

Again subtracting, we have

(110)

Thus, particular acceleration differences are objective. In exactly the same manner as in classical continuum mechanics, since

we have

is objective, whereas

I T 1 ( . T T T) W,, = Q · W,, · Q + 2 Q . Q - )

(111)

(112)

55

(113)

Thus, the rotation tensor is not objective. However, there are other objective tensors in multiphase continuum mechanics. IT we add eq. (111) for different values of Ie, we have

(114)

where we have used the identity Q . QT = I to obtain Q . QT + Q . QT = O. Hence

is objective for any Ie and m. It is also clear from eq. (113) that

WA:m = Wj, - Wm is objective.

We also note that

Dj,v. Dmv!,. _ 2W' . (v� v' ) = Q . (DA:VIc

DmVm) Dt Dt p m Dt Dt

so that the quantity

Dlcvlc Dm.vm ( -- - -- - 2W . Vic - V ) Dt Dt p m

(115)

(1 16)

(117)

(118)

is objective. The objective acceleration difference that we shall choose for describ­ing two-phase gas-liquid flows is

(119)

56

One implication of the principle of frame indiference is that constituted vari­ables can depend on only objective variables. The second implication is that the function giving the dependence of the constituted variable on the objective vari­ables must be independent of the coordinate system, or frame indiferent. The complicated forms which are assumed for the interfacial interaction terms and the average molecular and Reynolds fluxes can be made objective by assuming that they depend on only objective quantities. This wil ensure that the total is objective. It is the total quantity (interfacial force, for example) which should be compared with physical observations.

5.3 Inviscid Flow Around a Sphere In order to motivate the constitutive assumptions given in the next Section,

let us compute the corresponding averaged quantities for the inviscid flow of an incompressible fluid around a sphere. This calculation should give valid values for the constituted quantities for dilute flows. The notation for this section is cho­sen to distinguish the variables specifying the flow and the averaged variables. We perform the averaging for spherical bubbles in a liquid, so that the averaged variables are weighted averages for the liquid (subscript l) and for the gas (sub­script g). The velocities needed to specify the flow field around the sphere are denoted by v. , for the velocity of the sphere, and v" for the velocity of the fluid.

Consider a sphere of radius a located at a point z in a flow field, moving with velocity v •. The sphere velocity is treated as a field. That is, if the sphere is at z, its velocity is v.(z, t). The velocity of the flow field is assumed to be such that it would be vJ(x, t) = Vo + X · L/ at point x, if the sphere were not present. The velocity potential for the flow of fluid around the sphere is given by [6]

1 (�)

+ 2 (v.(z, t) - vo(t) - z · Lt) · (x - z) r3

1 (a5 ) + -(x - z) . L J • ( x z) -

3 r5

5.3.1 Averaged Velocities

( 120)

The ensemble average for the flow of a dilute suspension of spherical bubbles can be approximated by the "cell" average [7], obtained by averaging the value of the variable to be averaged, f, calculated with a sphere at location z. The aver­aging is performed by integrating the z variable over al positions in a sphere of raduis R surrounding the field point x. See Figure 3. The radius of this "cell" is chosen so that

, , .. ..

Figure 3. A "cell."

- - - - - -

, , '

.. _ - - - - --

I , ,

\ , I I , ,

I

57

(121)

We shall also assume that the distribution of positions within a cell depends on the gradient of the volume fraction of the particles. This accounts for non­uniform. distributions of spherical particles in a natural way, in determining the density function. Thus, we assume that

(122)

is the probability of finding the sphere in a volume dV surrounding the point z, where x! = x - z.

Thus, the average velocity of the liquid phase is given by

v7P = 4 3) lRJ v(x, tlz) dO dr , i1r R - a Q

where the integration is over the z variable.

( 123)

In order to evaluate the integrals appearing in eq. ( 123), we must express the z dependence of the velocities in terms of x and x' = x - z. We have

(124)

58

and

v.(z) = v.(x) x' . L. I (125)

where L. is the velocity gradient tensor for the motion of the sphere. We shall assume that this tensor is constant.

It is convenient to have expressions for the integrals of powers of r over a spherical shell O(r) given by Ix'i = r. For these integrals, we note that

X/ . . . x/ dO = O if the factor x' appears an odd number of times, and

1 I I 4

4

x x dO = -3

11"r I , O(�)

1 I I I 4 X X X X = 15

7rr , O(�)

(126)

(127)

( 128)

(129)

where E is a fourth order isotropic tensor which is symmetric in all of its compo­nents. It is given in Cartesian co-ordinates by

Using these results in eq. (123) gives

The interfacial averaged velocity of the liquid is given by

V/i(x, t) = 41

2 v(x, t lz) dO 7r4

Substituting and performing the integrations lead to the result

(130)

(131)

(132)

59

v,,(x, t) = v,(x, t). (133)

5.3.2 Averaged Pressures Now let us compute averaged pressures using this formalism. The exact pres­

sure can be computed by Bernoulli's equation

(134)

Calculating the average liquid pressure and the average interfacial liquid pres­sure is tedious, but results in [7]

(135)

(136)

where we have ignored terms of order al R in addition to those ignored previously.

5.3.3 Interfacial Force The interfacial force density is calculated for the fluid phase first. M, is given

by M, = p'V X,. Thus,

M, = -� np(x, z, t) dO , a1rR (137)

where n is the nonnal pointing into the gas. Note that the integration is again over the z variable. this corresponds to letting the sphere be tangent to the point x, so that z is integrated over the spherical shell of radius a centered at x. If we substitute Bernoulli's equation for the pressure, and recognize that n = -ria, the result is that

60

agpJ [v,. . (L. + L: + tr L.) + a,PJv,. . LJ

-PiVag + �Pf(V" . v,.)Vag • Va,)v,. (138)

5.3.4 Dispersed Phase Stress IT a distribution of stresses is applied to the surface of a bubble, the bubble

deforms so that the surface tension can balance the pressure distribution over the interface. This results in a net interfacial force due to surface tension. This force is computed in this section. We shall assume that the surface tension is large enough that the bubble will remain essentially spherical. Then, the distribution of the pressure in the liquid is given by Bernoulli's equation (134). However, the pressure inside the bubble must remain approximately constant. The pressure is approximately constant because the accelerations that can be generated are not sufcient to produce a net force, since the density is small. Thus, the pressure in the gas will be approximately equal to the average interfacial pressure in the liquid, plus the contribution due to the surface tension. Thus

(139)

where (T is the surface tension, and is the mean curvature of the interface, which is given by

- 2 H = - , 14

where 14 is the average bubble radius.

5.3.5 Momentum Jump Condition The surface traction is given by eq. (44) as

(140)

(141)

It is difficult to compute this term directly. The flow of the gas and liquid cause the bubble to deform. If the surface tension is sufciently large, the deformation will be small. We shall calculate the surface traction force using a "ping-pong ball" model for the surface tension [8]. Assume that the surface is a thin elastic solid, having inner radius a- and outer radius a+, where a+ - a- is smal. The stress inside the shell satisfies the equation of motion V · T = o. Then the average momentum equation for this third "phase" is

6 1

( 142)

Thus, we see that

V . 'T. = mi (143)

We can compute this term by using the following insight [9]. Note that

-V . (T . x') = V . T . x' + T . VX' = T ( 144)

Then the average of T is

(145)

Here V. is the volume of the elastic shell. U sing the divergence theorem, and assuming that

-T · x' = T · na+ = -pna+ (146)

on the outside of the spherical shell, and

T . x' = T . na = -pna (147)

on the inside. Here n is normal exterior to the shell. If we assume that the pressure is uniform inside the shell, and given by Bernoulli's equation outside the shell, we can compute the average stress. The result, for a thin shell, is

- [_ _ ( 9 8 2 ) ] T. = 0:9 Pgi - P,i + PI -20 v,.v,. + 20 IV,. I I (148)

5.3.6 Reynolds Stresses We next tum to computation of the Reynolds stress in the fluid because of

the velocity fluctuations induced due to the inviscid flow around a sphere. Using the expression for the velocity potential, eq. (120), we see that

62

1 (43) 3 (

43) v/(x, t lz) = 2(V,(Z) - V.(Z» - 2(vAz) - V.(Z» . (X - Z) r6

(X - z) 2

(46) 5 (

45) + -(X - z) . L, - - - (X - z) . Lr (X - Z) - (X - z) 3 r5 3 r7

Therefore Tf" = -X1PvJvJ can be computed, and the result is

( 149)

( 150)

The fluid fluctuation kinetic energy is u� = �vJ . and can be computed by taking the negative of the trace of eq. (150) for T�. The result is

(151)

In addition, the interfacial averaged Reynolds stress can be computed by averag­ing -pvJvJ over the interface. The result is to be calculated in the form

(152)

All of the terms derived in this section are comparable in magnitude to the inertia of the liquid; consequently a model capable of describing effects that depend on inertia would be incomplete without closure terms of this type.

5.4 Constitutive Assumptions Some general properties of constitutive equations were discussed in the pre­

vious section. Additional guidance in formulating realistic constitutive equations can be obtained by assuming that the quantities computed from the "exact" solu­tions can be taken as forms for constitutive equations. For example, the dynamics of the irrotational flow of an inviscid fluid about a sphere (see Section 5.3) can be used to compute the average force on a sphere and the average interfacial pressure. Constitutive equations arrived at in this way should reduce to the appropriate limit for dilute flows.

As is evident from the state of our abilities to calculate the flow around as­semblages of bodies of arbitrary shape, we cannot hope to obtain solutions that can be averaged to give appropriate constitutive equations for general motions. The ultimate solution to the problem of obtaining constitutive equations rests in empiricism. The calculated forms from "exact" solutions provide a guide to the appropriate forms for empirical testing. In order to obtain a system of equations that will predict the flow of a dispersed multi phase material, it is necessary to find experiments that isolate appropriate phenomena and give (relatively) direct

63

measurements of unknown coefficients involved therein. For best results, the physical experiment should be simple, and should correspond to a simple solu­tion of the equations that depends mostly on the particular constitutive form being evaluated. These flows are sometimes called "viscometric," or "separate effects" experiments. It is clear that experiments which isolate effects, and the corresponding exact solutions are highly desirable.

In order for the mixture to flow, one of the materials present must be a fluid. We shall refer to this material as material c, for "continuous" phase. In particulate flows, the other phase is solid, while in gasliquid flows, both are fluids. In both of these cases, we refer to this phase as phase d, for dispersed. In order to simplify the notation throughout the rest of this Chapter, we shall drop all notation for averaging. Thus, for example, the average pressure of the dispersed phase is denoted by Pd, and the average velocity of the continuous phase is denoted by Ve.

5.4.1 Stress The stress in a multi phase mixture represents the force per unit area on

compomnent k across an element of area having nonnal n. When both materials "flow," fluid models seem appropriate, but the particle stress is due to a different mechanism than the fluid stress. We first discuss the averaged stress.

We shall assume that the micro scale stress can be written as a pressure plus a viscous stress

(153)

Similarly, the Reynolds stress is written as

(154)

Pressure The pressure in a compressible fluid is related to the temperature and density

of the fluid through a constitutive equation called the equation of state. When a phase is incompressible, then extra assumptions are reqUired to give closure. We treat these extra assumptions as constitutive equations. If one phase is finely dispersed throughout the other, the pressure in the dispersed phase cannot difer much from its value at the interface. Thus, for dispersed flows, one often assumes

Pdt = Pd ( 155)

With constant surface tension and spherical bubbles or drops, one normally uses the classical Laplace form:

64

Pei = Pdi - H u , (156)

where H is the mean curvature of the interface. If the dispersed phase is spher­ical, H = 2/ R.i, where R.i is the radius of the dispersed phase elements (bubbles or drops), and u is the surface tension.

Bernoulli's equation for the variation of pressure in a flowing (inviscid) fluid suggests that the interfacial pressure should be given by

( 157)

where v,. = Vd - Ve' The calculations for the averaged fields for the flow of an inviscid fluid around a single sphere given in Section 5.3 suggest that for dilute flows, we have e = 1/4 when the boundary layer remains attached to the spherical particle. For low Reynolds number flows, the calculation of the averaged fields indicates that e = -9/32. We speculate that since eq. ( 157) holds in both limits of the inviscid and very viscous flows, then we might expect a similar form to hold in somewhat more generality. Thus we use

(158)

where the dispersed phase Reynolds number is defined by

( 159)

where lie is the kinematic viscosity of the fluid.

Shear Stress In most boiling flows, viscous forces are unimportant, except in boundary

layers, and other microscopic flows. Thus, we shall assume that

(160)

Fairly complicated models for the turbulent, or Reynolds stresses have been used in particle/fluid suspensions [10]. For the Reynolds stresses in a gas-liquid flow, the situation will be similar if the surface tension forces are large enough to keep the bubbles spherical. The resulting Reynolds stress in the liquid is then given in the form

65

The calculations in Section 5.3 give a.� = -� , b� = �, and p.� = 'Yc = o. In general, we expect a.�, b�, 'Yc and p.� to be functions of ael and Reel.

We take

(162)

5.4.2 Interfacial Force The interfacial force MAo arises from stresses acting on the interface. It is the

crucial term in modeling two-phase flows, since it is the presence of the interface which makes the discipline of multiphase fluid mechanics so interesting.

We shall assume

(163)

where . . . includes scalar properties, such as the viscosities and the densities of the phases, as well as intrinsic parameters, such as distorted particle radius and interfacial area density. By the principle of objectivity, we see that

(164)

We represent Me as a sum of forces associated with drag, virtual mass, unsteady, and convective effects,

-Me = E Fn, (165)

Drag The drag force on a single particle, drop or bubble is the force felt by that

object as it moves steadily through the surrounding fluid. The concept is clouded when an array of particles moves through a fluid, but the drag force is still at­tributed to steady effects. The drag force is usually given in terms of a dimen­sionless drag coefficient CD, where the drag force is defined as

(166)

Here, the drag coefficient CD is assumed to be a function of ael and Reel. The dependence of CD on ael and on Reel has been studied for bubbly, droplet, and par­ticle flows [11]. The drag force for spherical particles or bubbles can be written

66

in terms of the bubble radius and the volume fraction using the relation

while

Thus,

Then, for bubbly flow,

FD = -� QdPeCD Iv Iv 8 Rd .. ..

Virtual Mass and Lift

(167)

(168)

(169)

(170)

The virtual mass force is the added force exerted on a moving object when it accelerates through a fluid. If an object is immersed in fluid and accelerated, it must accelerate some of the surrounding fluid. This results in an interaction force on the object. This force is calculated explicitly for a single sphere in Section 5.3. We shall take this force to be

(171)

The parameter Cvm is called the virtual volume coefficient For a spherical parti­cle, it is

1 Cum = '2 (172)

This is valid for dilute suspensions of spheres in a fluid. For general flows, the virtual volume coefficient Gvm is taken to be a function of ad and Red. For the high Reynolds number (nearly inviscid) flow of a dilute suspension of spheres we have [12]

(173)

67

If an object moves through a fluid that is in shearing motion then the particle experiences a force transverse to the direction of motion. Such a force is often called a "lift force." If the object is spherical, the force is given by [13]

(174)

The coefficient CL is called the lift coefficient. Note that the virtual mass and lift forces as proposed above are not frame- indiferent separately. However, if we take Cvm = eL, the sum F vm +FL is frame-indifferent and agrees with calculations for the force on a single sphere in an inviscid fluid undergoing strain. Thus, the principle of objectivity requires that

(175)

Other Forces Other forces can be calculated from the flow of an inviscid fluid around a

single sphere. These forces are not usually included in multiphase flow models, but should be, for completeness. We write

(176)

The values G,. = i, Cc = �, Ce = �, Cd = -fo, Gj = �, G,.ot = �, and Cn = -� can be calculated for irrotational flow around a single sphere [7,8] . For general flows, the coefficients are assumed to be functions of Qd and Red.

Other forces which are sometimes included in the interfacial force are the Faxen and Basset forces. Both forces arise from viscous effects in the continuous phase. The Faxen force is taken to be [14]

(177)

Single-sphere calculations indicate GF = 3/4, this value may be expected to be valid for low concentrations of the dispersed phase.

The Basset force is an unsteady force associated with a viscous wake and is due to boundary layer formation. If a flat plate is impulsively started from rest

68

moving parallel to itself, it experiences a force proportional to t-1/2 • The analogous force on a single sphere has been calculated [15,16], and can be expressed as

(178)

where the acceleration is here taken to be the frame-indifferent quantity a"" .

5.4.3 Momentum Source from Surface Tension The calculation in Section 5.3 suggests that the momentum source due to

surface tension can be related to the inertia of the fluid around the outside of the bubble. Thus, we take

(179)

The inviscid calculation gives 4 .. = -� and b .. = �. In general, we assume that 4 .. and b .. are functions of a" and Re".

5.4.4 Interfacial Stress It has become customary to write [17]

M", = T · VX"

= -T;. . Va. (T - T;.) . VXIo

(180)

where T;. is an appropriate averaged interfacial stress, and Mt is the reduced interfacial force density. The most natural interfacial averaged stress to use here is

(181)

6. SOME CONSEQUENCES OF THE FORMULATION

The equations of motion, together with the constitutive equations, derived in previous Sections describe the behavior of bubbly liquids. In order for the reader to gain confidence in these equations, this Section presents some discussion of the

69

mathematical nature of the equations. The ultimate test of equations designed to predict two-phase flows is in what predictions they are capable of making, and in what circumstances. There are several other instances of predictions made with these equations, or similar sets, in the remainder of this volume.

6.1 Discussion of the Force on a Sphere The force on a sphere at z in the straining flow of an inviscid fluid is computed

by integrating -pn with respect to x over the surface of the sphere. Therefore

(182)

where the integration is over the variable r, with the sphere centered at x, and Ir l = 4. Substituting the pressure from Bernoulli's equation (134) and performing the integration results in

Fp(z) = �1r43Pe + VJ VVJ + �

- + VJ · VVJ] ) ( 183)

Note that this force agrees with Taylor's [18] calculation of the force necessary to hold a sphere at rest in an accelerating stream.

Another force that has been calculated for a stationary sphere in a rotating fluid is given by [19]

-

(184)

U sing the relations between constitutive equations for the stresses and the inter­facial force density, the equations of motion for a dilute suspension of spherical bubbles in liquid become

8ae 0 7jt + V . aeVe = I

(185)

(186)

70

(187)

o.cPc + Vc · vvc) = -o.c Vpc + �Pclvc - VtlI2Vo.tI

-�o.tlPC + VC . VVC)

_ + VtI . VVd) ]

The equations of motion for the mixture can be obtained by adding eqs. (185) and ( 186) to obtain the equation of balance of mass for the mixture, and by adding .eqs. (187) and (188) to obtain the equation of balance of momentum. The results are

(189)

(190)

It is enlightening to consider the balance of momentum for the control volume pictured in Figure 4. Momentum balance for the mixture (192) suggests that the rate of change of momentum for the mixture should be balanced by stresses applied on the boundary of the control volume and the total body force applied to the interior of the control volume. The body forces and the pressure forces are

7 1

clearly appropriate for the control volume and i ts boundary. The extra tenn can be considered as a stress, and has its origin in two parts. First, the Reynolds stress in the fluid due to the relative motion between the phases contributes a stress. Second, the effect of the surface tension acting on the interface also acts as a stress, transmitting force across the boundary of the control volume.

T

o

Figure 4. Momentum Balance for Mixture

It is evident that letting ad -+ 0 in eqs. (189) and (190), that the mixture behaves as an inviscid fluid. Thus,

v ' Vc = 0 , ( 191)

( 192)

The dispersed component momentum equation then reduces to

1 1 -iPe(Vd - ve) x V X Vd - iPe(Vd - ve) x V x Ve ( 193)

This equation shows that for dilute flows of spheres, the equations reduce to Tay­lor's result for an accelerating sphere, or for a stationary sphere in an accelerating fluid. Moreover, it also includes a lift force that reduces to eq. (184) if the spheres

72

are stationary. It also includes a lift force that gives a transverse force if the spheres are translating and rotating as a unit. Note that this is not a rotation of each sphere.

6.2 Nature of the Equations

It should be recognized that any system of equations that is expected to de­scribe the behavior of a physical system is a model, and will, at best, describe the subset of phenomena that falls under the limitations of the model. These limitations are often unwritten and, unfortunately, unrecognized by even the most careful researcher. In a classical context, the equations of (linear) elastic­ity provide an excellent description of a large body of phenomena. However, the model fails to describe such common phenomena as permanent bending, crack propagation, and shear bands. Faced with the task of solving a formulation of governing equations, we must be assured that the formulation reproduces the essential physics of the problem; and the solution method does not introduce arti­facts which distort the character of the formulation. 'Ib this end we next consider what can be understood about the nature of a formulation before attempts are made to solve it. We then discuss a number of specific consequences of neglecting terms to form reduced models.

6.2. 1 Well Posedness A model that is not properly formulated mathematically canot describe phys­

ical phenomena. There are three prerequisites for a mathematical formulation to correspond with physical reality:

• The solution must exist;

• The solution must be uniquely determined;

• The solution must depend in a continuous fashion on the initial and bound­ary data.

If a formulation satisfies these prerequisites, it is said to be well posed [20]. However, because of the absence of general theorems, it is usually only possible to investigate existence and uniqueness for special cases of the equations. Exact solutions, when obtainable for limiting cases facilitate this investigation and help to remove the questions associated with an approximate (numerical) solution of the complete model.

The reader who is familiar with the large body of work in "ill-posed" problems will recognize that the physically meaningful ill-posed problems occur in a con­text different from that intended here. An ill-posed problem that occurs in the inviscid limit of some well-posed problem deserves study. In many situations, multiply valued steady states occur, but the initial-value problem that governs the evolution to them should be well-posed.

73

One aspect of the third prerequisite, a requirement of stability, can be quan­tified by examining the characteristic values of the governing equations in the following way.

6.2.2 Characteristic Values The one-dimensional model equations presented here can be written in the

matrix form

{}F (}F A- + B- = C at 8z

where

( 194)

(195)

and A('!P) and B(qi') are 4 x 4 square matrices; and C(qi') is a column vector, none of which contains derivatives ,of !P 'Ib investigate the behavior of this set of equations, suppose that arbitrary data for qi' are specified at all points (z,t) along a curve C, which is expressed as z = z(t) . A solution for " in the neighborhood ofC can be obtained by means of a Taylor series expansion about points on C provided that the first and higher derivatives of I' can be calculated. The calculation is simplified by a transformation to normal n(z, t) and tangential T(Z, t) variables (see Figure 5), whence eq. ( 194) becomes

n t

z

Figure 5. Normal and Tangential Coordinates

(196)

74

Since tp is given by the initial data as a function of r along the curve, the right-hand side of eq. (196) is known, and the derivatives 8tp/8n will be uniquely determined, if the coefficient matrix

( 197)

is non-singular, that is, if

(198)

Combining the solution from Eq. (196) with the results of successive differentia­tions of the governing equation (194) then provides all the derivatives needed for the Taylor series expansion; hence, the solution can be developed from the data given on C.

When the determinant of the coefficient matrix is zero, the curve C is called a characteristic curve. In this case, the existence of a solution for atp/8n requires that each detenninant, which appears as a numerator in the solution for each element of fJiP/{}n, must also be zero. The resulting equations become compati­bility conditions. Since terms from the right-hand side of (194) are involved in the compatibility conditions, an immediate consequence is that the initial data cannot be arbitrarily prescribed on a characteristic curve. One also finds that not all of the compatibility conditions are independent, so that there exist infinite combinations of 8lP/8n and 8fi/8r, or {}fij{}z and Eifij{}t, that will satisfy the prob­lem. Finally, it can be shown that higher order normal derivatives can no longer be determined from the differential equation; hence, the solution at points in the neighborhood of a characteristic curve cannot be obtained from data on the curve. For this reason, a solution is said to be "transported" along a characteristic curve, and the solution at a neighboring point "arrives" there on the characteristic curve that passes through that point.

The dependence of the solution on prescribed initial data can thus be reduced to an investigation of the equation

det (A(fi)� - B(fi)) = 0

in which we have introduced the characteristic value(s) � defined as

� = _ 8n/ 8n . at {}z

(199)

(200)

75

If the values of ..\ that satisfy (199) are complex, the coefficient matrix in (197) is non-singular and the solution is uniquely determined from the data given along the curve C. In this case, the system (194) is called elliptic. Although this seems satisfactory, we shall see that complex characteristics are not suited to initial value problems.

On the other hand, if the characteristic values of (199) are real, then C is a characteristic curve, and the solution acquires the special properties just de­scribed. In this case, the system (194) is called hyperbolic. Based on the orthog­onality of the local tangent vector dte, + dzez with the normal Bn/Bte, + Bn/8zez to curve C, it follows that

..\ _ _ Bn/ Bt _ dz -

8n/8z - dt ·

Thus, real characteristic values are the slopes of the characteristic curves.

(201)

It is significant to note that boundary value problems are associated with el­liptic equations, whose solutions depend on data on the boundary; whereas initial value problems are typically represented by hyperbolic equations, whose solutions depend on the initial and boundary data prescribed on restricted domains.

Hadamard's Example The classical example of the type of "ill-posedness" that arises from complex

characteristics can be seen in the Hadamard problem, where Laplace's equation is used for an initial value problem. The initial value problem for Laplace's equation can be written as

8u._

8v= O

at 8z '

8v

8t +

8z -,

for 00 < Z < 00, and 0 < t < 00, with initial conditions

u.(z, 0) = /(z)

v(z, O) = g(z ) ,

for -00 < z < 00 .

(202)

(203)

Suppose 'UQ(z, t) and vo(z, t) give a solution of this problem. Consider the problem given by

76

811.' 8v' - - - = 0 8t 8z '

(W' 81' - + - = 0 8t 8z '

with initial conditions

u'(Z, O) = ; Sin(Z/f)

v'(z, O) = -f COS(Z/E) .

for 00 < Z < 00. The solution of this second problem is easily seen to be

u'(Z, t) = l'et/I sin(z/E) ,

v'(z, t) = -Eec/, COS(Z/E) ,

Then the solution of the problem

811. _

(W = 0 at 8z '

8v

8t +

8z - ,

with initial conditions

u(Z, O) = f(z) + � sin(z/ E)

v(z,O) = g{z) f sin(z/f) ,

(204)

(205)

(206)

(207)

(208)

is u.o(z, t) + u'(z , t), vo(z , t) + v'(z , t), for any f. For small f, the error made in the initial conditions can be made as small as desired. However, the error in the solution can be made to exceed any given value at any time t > 0 by choosing f small enough. This is an unhealthy state of affairs; no matter how well the correct initial condition is approximated for numerical purposes, there is no assurance that the computed solution wil be close to the desired one.

The problem is generic to complex characteristics. If we consider the char­acteristics in one complex space variable z = Z + iy and one real time variable, we see that we need to extend the (real) initial values for f! from the real axis y = 0 to a neighborhood of the real axis. The most reasonable way to do this is to extend the initial conditions as analytic functions (i. e., having a power series in the complex variable z). If an error proportional to Sin(Z/f) is incured

77

in the initial conditions, this extends analytically as sin ( z / E) cosh(y / E) . This com­plex initial condition error grows exponentially away from the real axis. Complex characteristics entering a point (z, t) in the real plane come from both sides (i.e., y > 0 and y < 0 to the point (z, t). As t increases, these complex characteristics come from farther away from the real (z, t) plane, and consequently can pick up larger and larger errors from a smal error proportional to sine z / E).

6.2.3 Two-Phase Flow Characteristics A physically realizable two-phase flow evolves from some initial conditions,

subject to the physically imposed boundary conditions. Depending on the phe­nomena being modeled, the model should have either a parabolic or hyperbolic nature. Hyperbolic systems, those having real characteristic values, may be ex­pected to transport either continuous or discontinuous initial data as well as to develop embedded discontinuities, either stationary or moving. A parabolic model has the ability to describe the formation and evolution of regions of sharp change that provide a better model than the discontinuities inherent in hyperbolic mod­els.

Elliptic behavior, when one of the variables is time, is unacceptable. Nonethe­less, there are some subtle issues in requiring real characteristics. It seems clear that a formulation which produces complex characteristic values is physically unsound and in need of revision to incorporate missing physics. In subsequent sections, we consider whether, and in what sense, that the instabilities associated with "ill-posed" problems in multicomponent flow may sometimes be real and ob­servable. Whatever the case, further theoretical and experimental investigations will be needed to resolve this issue.

The Simplest Model The simplest model for multiphase flow is obtained by assuming that the

phasic pressures Pd and Pc are both equal and that Md is given as an algebraic combination of the averaged fields, but that it contains no derivatives of these averaged fields. The most commonly used form is Md = aGDPc/ Rdlvc - Vdl(vc - Vd). The equations of motion in one dimension then become

8( 1 - a) 8( 1 - a )vc _ 0 at + 8% -

(8Vd 8Vd) 8p apd - + Vd = -a- + Md 8t 8% 8%

(209)

(210)

(211)

78

(1 - alp (Bve + 11 8ve) = -(1 - a) 8p -Md e 8t e {Jz {Jz

where Md involves terms not having derivatives. The matrices A and B are

[ !1 A 0 o

[

v,

B = T

o o

a 0

o 0 ] o 0 o 0

(1 - a)Pe 0

0 I - a

apd'Vd 0 0 (1 a)Pel1e tL ]

The characteristics are given by

where

and

(212)

(213)

(214)

(215)

(216)

(217)

(218)

The characteristics are complex, unless l1e = l1d. Thus, the simplest model is not physically realistic to describe the evolution of multi component flows.

79

Effect of Viscosity A model including the effects of viscosity in the form of a higher order deriva­

tive of the ftuid velocity can be put in the form

8(1 - a) 8(1 - a)vc _ 0 at +

8z -

(aVd aVd) 8p apd - + Vd- = - a- + Md

8t 8z 8z

(219)

(220)

(221)

(222)

(223)

where the last equation gives the definition of the viscous stress in terms of the derivative of the continuous phase velocity.

The vector of unknowns becomes (a, Vd, Ve, p, Tc)tr, and the matrices of the for­mulation with viscosity (I'e > 0) become

A 0 0

A = 0 (224) 0

0 0 0 0 0

B 0 0

B = 0 (225) -1

0 0 (1 - a) 0 0

80

The characteristics are

(226)

they are always real. Thus, the characteristics of the viscous system (Pc > 0) and the characteristics of the inviscid system (lLc = 0) are unrelated. If the flow is sufciently viscous that the appropriate model should contain the viscosity terms, then no problems with complex characteristics will be evident.

On the other hand, when the flow is nearly inviscid, we expect to be able to use inviscid equations with correlations for wall boundary-layer effects. In this case, the roles of inertia, virtual mass and other first-derivative terms become dominant in determining whether the characteristics are real, that is, whether the problem is well posed.

If the characteristics of the inviscid system are complex, an infinitesimally small disturbance in the initial data will grow large in a short time. A small amount of viscosity can alleviate the instablility. To understand this, consider the linear stability of a steady-state solution Fa to the fonnulation represented by eq. (194). A perturbation qI of the fonn j;e"" eiloz will be a nontrivial solution, if

(227)

Here A and B are the same matrices encountered in the inviscid characteristics problem and D = ()C/{)F. As k __ 00 , u/ik __ �, which is the characteristic slope at Fo. Suppose that � = lL+ill and >; .. = IL ill are complex conjugate characteristic values, where " i O. The two corresponding growth rates of the perturbation are Re u = ±kll. Since II i 0, one of the growth rates must be positive; in fact, as Ie -- 00, the positive growth rate becomes infinite.

For the viscous system, direct substitution of the perturbation in F and lin­earization yield

as the condition for a nontrivial solution, where

[ 0 0 0 0 0 1

I 0 0 0 0 0 B = ILc(1 - Qa) 0 0 0 0 0

o 0 0 0 1

(228)

(229)

If ILck/pcvo « 1, the values of u are close to those for the inviscid system. Thus,

8 1

for small values of viscosity, there will exist disturbances whose wavenumbers Ie satisfy this criterion, and we conclude that complex characteristics can also taint viscous models in the inviscid limit. In this section we consider the conse­quences of several modeling assumptions to the characteristics of the associated formulations.

Inertial Effects The simplest model in this Section includes the inertia effects of the "left

hand side" terms, but, as derived in Section 5.3, there are many other inertial effects that can be included in the dilute multiphase flow equations. Using those constitutive equations, and specializing the equations or motion to one dimension, gives the system

8(1 a) 8(1 a)vc _ 0 {}t +

{}

z -

- C:mpc( Vd - vc)2 + Vd (a( "d + bd)pc( Vd Vc)2)

{} 28a + aCgdPc(Vd Vc)

8z(Vd Vc) + CodPc(Vd Vc) 8z

+ FD

(230)

(231)

(232)

(233)

82

where the inviscid calculation gives Own = i. e = 1. ae + be = i. ad + bd = fo. Cad = t\. Cgd = 10. Cal! = -/0 and CtIC = -10. Here. FD is the drag force. and involves terms not having derivatives.

The "simplest" inviscid two-fluid model is obtained by setting al the constants equal to zero. The characteristics for that system are complex, unless u = '1 again. If Own > O. but all the other constants are set equal to zero, the characteristics are again complex. Thus. virtual mass, alone, cannot make the characteristics real. The interfacial pressure coefficient, e, does have a significant role in making real characteristics. For the inviscid model calculated in Section 5.3, with the values of the constants given above. the characteristics are real for 0 < Q < 0.16. If we increase e to 0.35, the characteristics are real for 0 < Q < 0.39.

7. CONCLUSION

In this Chapter, we have presented the two-fluid model for two-phase flow. This model considers the equations of balance of mass. momentum, and energy for each phase. These equations are derived in two ways to give the reader an appreciation for (i) what the equations mean, from a continuum mechanics point of view. and (ii) how averaging works.

The averaging process was applied to the irrotational flow of an inviscid fluid around a single sphere in order to obtain closure conditions for dilute bubbly flow.

Constitutive equations were proposed to give closure for more general two­phase flows. The assumptions are reasonable, in that they reduce to the bubbly flow model. but they are untried.

Two important aspects of the model are discussed. Firat, the equations are shown to reduce to the force on a single sphere in the dilute limit. Second, the well-posedness of the model is discussed.

The problem of complex characteristics seems to arise from the coupling be­tween the two momentum equations, since it does not appear in the drift-flux model, which has only one momentum equation. The argument has been ad­vanced that viscosity will change the type of system of partial differential equa­tions, and the problem of complex characteristics will be irrelevant. While it is true that a viscous system has real characteristics. in the limit of vanishing vis­cosity, the complex characteristics of the inviscid system give rise to small-scale instabilities which are artifacts of the model, and not physically real.

Thus, effort should be devoted to understanding the inviscid, incompressible two-fluid momentum equations. The alternative of including differential terms to force real characteristics. either in the name of engineering or by invoking continuum mechanics principles, is unsatisfactory. We believe that the systematic inclusion of all terms arising in the averaged momentum equations, each soundly based on physics, will yield an appropriate working model.

83

8. REFERENCES

1. M. Ishi, Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, (1975).

2. J. M. Delhaye, Basic Equations for Two-Phase Flow Modeling, Two Phase Flow and Heat Transfer in the Power and Process Industries, A. E. Bergles, et al. , Eds. Hemisphere Publishing, Washington, (1981).

3. R. T. Lahey, Jr. and D. A. Drew, The Three Dimensional Time- and Volume Averaged Conservation Equations of Two-Phase Flow, Advances in Nuclear Science and Technology, vol. 20 (1988) 1-69.

4. C. Truesdell and R. Toupin, The Classical Field Theories. Handbuch der Physik, Vol. III, Pt. 1, Springer-Verlag, Berlin, (1960).

5. R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics,.Pren­tice Hall, Englewood Cliffs, New Jersey, (1962).

6. O. V. Voinov, Force Acting on a Sphere in an Inhomogeneous Flow of an Ideal Incompressible Fluid, Journal of Applied Mechanics and Technical Physics, vol. 14, (1973) pp. 592-594.

7. G. S. Arnold, D. A. Drew, and R. T. Lahey, Jr., Derivation of Constitu­tive Equations for Interfacial Force and Reynolds Stress for a Suspension of Spheres Using Ensemble Cell Averaging, Chemical Engineering Communi­cations, vol. 86 (1989) pp. 43-54.

8. J.-W. Park, Void Wave Propagation in Two Phase Flow, Ph. D. Thesis, Rensselaer Polytechnic Institute, Troy, NY (1992).

9. G. K. Batchelor, The Stress System in a Suspension of Force-Free Particles, J. Fluid Mech., 41, (1970) 545-570.

10. T. W. Abou-Arab and M. C. Roco, Solid Phase Contribution in the Two­phase 1Urbulence Kinetic Energy Equations, J. Fluids Engineering, Vol. 112, (1990) pp.351-361.

11. M. Ishii and N. Zuber, Relative Motion and Interfacial Drag Coefficient in Dispersed Two-Phase Flow of Bubbles, Drops and Particles, A. I. Ch. E. Journal, vol. 25 (1979) 843-855.

12. N. Zuber, On the Dispersed Two-Phase in the Laminar Flow Regime, Chern Eng. Sci., vol. 19, (1964) pp. 897-903.

13. D. A. Drew and R. T. Lahey, Jr., Some Supplemental Analysis Concerning the Virtual Mass and Lift Force on a Sphere in a Rotating and Straining Flow, Int. J. Multiphase Flow, voL 16, (1990) pp. 1127-1130.

84

14. H. Brenner. The Stokes Resistance of an Arbitrary Particle IV Arbitrary Fields of Flow. Chem. Eng. Sci . • vol. 19, (1964) pp. 703-727.

15. A. B. Basset. Treatise on Hydrodynamics. Deighton Bell (1888).

16. M. R. Maxey and J. J. Riley. Equations of Motion for a Small Rigid Sphere in a Nonumiform Flow. Physics of Fluids. Vol. 26 (1983) pp. 883-889.

17. M. Ishii and K Mishima. Two-Fluid Model and Hydrodynamic Constitutive Equations. Nuclear Engineering and Design. vol. 82. (1981) pp.l07-126.

18. G. I. Taylor. The Forces on a Body Placed in a Curved or Converging Stream of Fluid. Proc. Roy. Soc .• A vol. 120, (1928) pp. 260-283.

19. J. Proudman. On the motion of solids in a liquid possessing vorticity, Proc. Roy. Soc. A. vol. 92. (1916) pp. 408-424.

20. R. Courant and D. Hilbert. Methods of Mathematical Physics. Volume II. Wiley-Interscience ( 1962).

85

THE PREDICTION OF PHASE DISTRIBUTION AND

SEPARATION PHENOMENA USING TWO-FLUID MODELS

R.T. Lahey, Jr.

The Edward E. Hood, Jr. Professor of Engi neering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 USA

Abstract This chapter presents material on the use of multidimensional two-fluid

models for the analysis of phase distribution and phase separation phenomena. In particular, the analysis of lateral phase distribution for bubbly upflows and downflows in pipes and in more complex geometry conduits, is presented. In addition, the analysis of phase separation in a horizontal tee is discussed for stratified two-phase flow.

It is shown that accurate mechanistic predictions of two-phase flows can be performed using multidimensional two-fluid models and appropriate computational fluid dynamic (CFD) numerical techniques.

1. INTRODUCTION

Many interesting and challenging problems in two-phase flow are concerned with multidimensional phenomena. In particular, phase separation and phase distribution phenomena. Unfortunately, because of the complexity involved, limited progress has been made until recently in understanding how to perform accurate multidimensional analyses of flowing two-phase systems.

Most previous models have been primarily concerned with one-dimensional phenomena. The more sophisticated of these models were based on a two-fluid modelling approach in which the phasic conservation equations were used in conjunction with appropriate closure conditions. Closure of such systems was achieved by postulating interfacial and wall transfer laws which attempted to reintroduce some of the physics which was lost during the (space/time or ensemble) averaging process. The two-fluid modeling approach can be readily extended to multidimensional flows [1] and has been adopted by virtually all state-of-the-art multidimensional computer codes (eg, TRAC and PHOENICS). Unfortunately, there remains considerable uncertainty concerning the proper formulation of the interfacial transfer laws.

It is the purpose of this paper to investigate the ability of multidimensional two-fluid models to predict the observed phase separation and lateral phase distribution phenomena. It will be shown that such phenomena can be predicted once all significant physical mechanisms have been adequately modelled.

86

Let us begin the discussion by a consideration of lateral phase distribution phenomena.

2. DISCUSION · PHASE DlSTRIBurION DATA

It has been known for some time that pronounced lateral phase distribution can occur in two-phase conduit flow. In particular, MaInes [2] observed void peaking near the wal for bubbly air/water upflows in a pipe. In contrast, Qshinowa & Charles [3] found interior peaked void profiles (ie, void coring) for bubbly air/water downflows in a pipe.

In a pioneering investigation, Serizawa [4] measured the lateral void distribution as well as the turbulent axial liquid velocity fluctuations for bubbly air/water upflows in a vertical pipe. Some typical data are shown in Figure 1.

0.4

0.3

c 0 U � U. "C ·0 0.2 > Cti (.J 0 .. 1:

0. 1

j £ .1 .03 m/s Z I D ", 30

.. 0.0085 0 0.01 70 • 0.0258 .6 0.0341 A 0.0427

o 1 .0

Radial Position r \ R

Figure 1. Radial Void Distribution · Upflow [4]

87

Wal peaking can be seen for low quality (0:» bubbly flows, while void coring evolves as the quality is increased and slug flow occurs. Subsequently, Michiyoshi & Serizawa [5] used an X-probe to measure various components of the Reynolds stress tensor.

A detailed experimental investigation of phase distribution phenomena in a pipe was performed by Wang et.al. [6]. In this study, special single element cylindrical probes, and three element conical probes, were used to measure the lateral void distribution as well as all components of the Reynolds stress tensor for both bubbly air/water upflows and downflows. Typical data are shown in Figures 2 to 5. It can be seen that, as expected, lateral void distribution is

0.50

0.40

z o 0.30 � o « a: u. o 0.20

o >

0. 1 0

U PWAR D FLOW, j l = 0 . 7 1 mls

Reo = 34,000

C u rves: Fo rce B a l a n c e P red ict i o n

S y m b o l : Co rrected P robe M ea s u re m e n ts

jv = 0.40 mls

jv = 0. 1 0 mls

1 .00 0 . 80 0 .60 0 .40 0 . 20 0 .00

r/R Figure 2a. Void Fraction Profiles - Upflow [6]

88

z 0 � (J < a: u. 0 0 >

0.50

0.40

0 .30

0.20

0. 1 0

DOWNWAR D FLOW, j l = 0.94 mls

Reo = 44,000

C u rves: .Force B a l a n c e P red i ct i o n

S y m b o l : Corrected P robe Meas u re m e n ts

jv = 0.40 mls

jv = 0.27 mls

1 .00 0 .80 0 .60 0 .40 0 .20 0 .00

riA

Figure 2b. Void Fraction Profiles - Downflow [6]

strongly influenced by flow direction. Also, it can be seen in Figures 3 that, for two-phase flow, the mean liquid phase velocity can have a maximum off the centerline for both upflow and downflow. In addition, we note that the two­phase turbulent RMS liquid velocity fluctuations in the axial direction can be below those for corresponding single-phase flows at large liquid phase Reynolds numbers (Re,). This implies that for such conditions turbulence suppression mechanisms exceeded bubble-induced turbulence production mechanisms. Similar data trends were also observed by Serizawa [4].

89

u'.

1 .0

� � � -. 0.5 BOUNDARY LAYER PROBE '5 Z >" 0 � i= (5 c:( 0 � .. � W U > � 0 0.1 .. :; U.

0 � :: Z W

.. � al a: � � 0.0 1.0 0.5 0.0 RADIAL DISTANCE, rlR

Figure 3a. Liquid Velocity and Turbulent Fluctuations-Upflow, Ret = 34,000 [6]

1.5

1 E � -;. 1 .0

'5 >" Z � 0 (5 i= 0 c:( .. ::l w � > u 0 0.1 � 5 .,j u. 0 � :: z W .. � al a: � 0.0 0.0 �

1 .0 0.5 0.0 RADIAL DISTANCE, rIA

Figure 3b. Liquid Velocity and Turbulent Fluctuations-Upflow, Ret = 44,000 [6]

90

' .0

1 :3" 0.5 -> I-U o ..J W > o :5 o ::J

DOWNWARD FLOW, Ret " 3-4,00 LID " 35, if . 0.7' m/s

BOUNDARY LAYER PROBE

(Legend ume as Fig. 3.)

RADIAL DISTANCE, rlR

Fig. 3c. Liquid Velocity and Turbulent Fluctuations-Downflow, Ret = 34,000 [6]

DOWNWARD FLOW, R� " 44 00 LID " 35, I! . 0.94 mls '.5 BOUNDARY LAYER PROBE

(Legend same as Fig. 381

� � '5" ,.0 lC; -> Z I-U 0 0 � ..J < W ::J > I-0 0 ::J :5 0.5 0.' ..J 0 u.

::J I-Z W ..J ::J m a: ::J 0.0 0 0 I-' .0 0.5 0.0

RADI AL DISTANCE, rlR

Fig. 3d. Liquid Velocity and Turbulent Fluctuations-Downflow, Ret = 44,000 [6]

i 0.15 UPWARD FLOW, ReI · 23,000

� LID • 35, k · 0.43 mls. 10 - 0 Le\lend:

� • 4? 1 . ..j? 30 Conical

� . .;:. 0.10 I:. fl } 4So.Probe en Z o .j? ..j':) 0 � « ::J � 0 ::J ..J

0.05 u. � Z W ..J ::J m a: ::J � ..J 0.0 « 0.5 0.0 :: 1 .0 a: RADIAL DISTANCE, rlR 0 Z

Figure 4a. Normal Turbulent Fluctuations (Single-Phase) [6]

i � � tt; en Z 0 � « ::J � 0 ::J

..J u. � Z W ..J ::J m a: ::J � ..J « :: a: 0 z

0.15

0.10

0.05

0.0 1 .0

UPWARD FLOW, RIIf · 23.000

LID • 35. it · 0.43 mls. (Legend .ame 8S Fig . •• )

0.5

10 " 0.4 mls

RADIAL DISTANCE. rlR

0.0

Figure 4b. Normal Turbulent Fluctuations (Two-Phase) [6]

9 1

92

.004

.003

N en ;;-

.002

I�

.001

0.0 1 .0

UPWARD FLOW , Ret - 23,000

LID " 35, It · 0.43 m/s Legend: ia (m/s)

c. 0.00 o 0.10 IJ 0.27 o 0.40

Light Symbol: 3D Conical Probe Dark Symbol: 45- Probe

0.5

RADIAL D ISTANCE, rlR

0.0

Figure 5a. Reynolds Stress Measurements - Upflow [6] 0.006

0.005

0.004

N en ;;-.§. 0.003

I�

0.002

0.001

0.0

1 .0

DOWNWARD FLOW , Ret . 34,00 LID " 35, it " 0.71 m/s (Legend same as Fig. Sa)

0.5

RADIAL D ISTANCE, rlR

0.0

Figure 5b. Reynolds Stress Measurements - Downflow [6]

93

Let US next consider some typical Reynolds stress measurements which are shown in Figures 4 and 5. We see in Figures 4 that the turbulent structure is nonisotropic for both single and two-phase flow. In addition, Figures 5 show the shear stress trends for bubbly upflows and downflows. Moreover, it can be seen that redundant measurements using difl'erent hot film probes (ie, a 3-D conical probe and a 450 cylindrical probe) gave essentially the same results for the various Reynolds stress components. Finaly, there is consistency between Wang's [6] shear stress and void distribution measurements. That is, as can be seen in Figures 2, a fully developed force balance, given by,

r ( dp ) g(Pl - PV )Sr

)"

't (r ) = - ---P g - ( l- a r dr l 2 dz v r o

yields a predicted radial void distribution that agrees quite well with the appropriately corrected [6] local void fraction data.

Serizawa & Kataoka [7] have summarized the observed lateral void distributions in a "flow regime" map. This map is shown in Figure 6. It can be seen that as the liquid phase superficial velocity tiL) increases, the void peak moves further away from the wall. Moreover, as slug flow conditions are approached, void coring is observed.

Al the experiments that have been considered so far have been concerned with bubbly pipe flows. Even though most investigators have made an effort to control bubble size in their experiments, there was invariably a range of bubble sizes present. A completely different experimental approach has been adopted by other investigators in which single bubbles were investigated. These experiments have the advantage of being able to easily control the bubble size, however the effect of bubble interaction is absent. Sekoguchi et.aI. [8] observed that the motion of the bubbles appeared to be related to bubble size and distortion, the location of the injection point, and the liquid Reynolds number. Significantly, they found that all spherical bubbles, and distorted bubbles larger than about 3 m.m in diameter, did not collect near the wall for upflows. The results of similar experiments by Zun [9] and Kariyosaki [10] have also indicated that bubble size and shape may play an important role in lateral void distribution.

While the influence of bubble size and distortion on lateral phase distribution is not yet understood in two-phase flows of practical importance, it is clear that bubble size is important. Indeed, it is well known that microbubbles are widely used for flow visualization (eg, H2 bubbles) since they follow the turbulent flow pattern of the liquid phase. In contrast, large Taylor bubbles flow preferentially in the open regions of the conduits, and for such flows, the void fraction peaks away from the wall. Hence models which are to be valid over a wide range of conditions should include the effect of bubble size. This conclusion is underscored by the results of Valukina et.al. [1 1] who took "laminar flow" air/water bubbly flow data in a vertical tube. As can be seen in Figure 7, they found that bubble size had a noticeable effect on lateral void distribution.

94

.. 5 "-Ii .. . .., x => .. La.

or .. UJ ::I: :: 0 . 5 c >

CORE PEAl(

\/ALL PEAK

BUBl!lY F LOW S LUG F LOW

GAS VOLUMETR I C F LUX Jg Jl/s Figure 6. Lateral Phase Distribution Patterns [7]

c:x IrreMAX

(b)

o +1 -1 0 +1 rIA

Profiles of local void fraction (a) and liquid velocity ( b ) for � = 1 0%; (I) Re = 990; ( n ) Re = 2280; ( 1 ) d = 1 ; ( 2 ) d = 0.5 mm; 'ITe I ueMAX in m/sec; c:x in %.

Figure 7. Laminar Two-Phase Flow [11]

It should also be noted that similar phase distribution phenomena can be observed in '1aminar" solidliquid flows [ 12, 13]. Indeed, Figure 8 shows the observed lateral distribution when solid spheres of the same size were dispersed in liquid upflow in pipes. It can be seen that positive buoyant particles exhibit wall peaking while negative buoyant particles core into the center of the pipe. These trends are similar to the upflow and downflow trends

95

-J (J) (J) -J

< X ::J :: < -J

::J W W Z

J: J: Z f- f- < a: a: (!) < LlS a: w w z Z II Z Z a: 0 0 W � � II -J < Ci5 -J -J ::J ::J � � - UJ

::J ::J a: (.) (.) (!) (.) (.) UJ

< < C/) (a). Positive (b) . Negative (c) . Neutral

Bouyant Bouyant Bouyant Particles Particles Particles

Figure 8. The Lateral Distribution of Rigid Spheres

seen with (positive buoyant) vapor bubbles. Thus it appears that the effect of bubble distortion may not be too important in such flows, however bubble size may be important.

The configuration shown in Figure 8(c) is particularly interesting. It is known as the Segre-Silberberg anulus. It can be seen that the concentration always peaks at about rIR '2' 0.6. The Segre-Silberberg annulus occurs for neutral buoyant particles, thus the relative velocity is zero. This is an important observation since it implies that a balance of the lateral forces on rigid spheres, which are not related to buoyancy effects, control the lateral distribution. We will return to this point later when lateral lift forces are considered.

Let us next consider phase distribution phenomena in conduits of more complex geometry than pipes. Zigami [14] and Furukawa et.al. [15] have taken void distribution data with air/water flow in concentric annuli. They found that the voids peaked near both the inner and outer walls for low quality bubbly flows and peaked in the interior region of the conduit as slug flow conditions were approached. Shiralkar & Lahey [16] took void distribution data in an eccentric annulus using a two-phase flow of boiling freon. They found that the void fraction was the highest in the more open portion of the conduit. Similar observations were made in air/water by Ohkawa & Lahey [17]. It is obvious

96

that strong lateral void-drift forces exist which lead to the observed nonuniform void distributions.

Measurements of the phase distribution in isosceles triangular conduits have been obtained with air/water flows by Sekoguchi et.al. [18], Sadatomi et.al. [19], and Sim & Lahey [20]. Typical data trends are shown in Figure 9. It can be seen that for low quality bubbly flows the local void fraction peaks near the comers and walls of the conduit, while for higher quality (slug) flows void peaking occurs in the interior of the conduit. These data are extremely interesting since they show that along the apex bisector the local void fraction, and indeed the local "flow regime", can vary over a wide range.

A number of researchers have investigated phase distribution phenomena in rectangular chanels. In particular, Jones & Zuber [21], Ohba & Yhuara [22] and Sadatomi et.al. [19] took data in air/water flows. Typical data trends are shown in Figure 10. As in triangular conduits, wall and comer peaking was observed for low quality bubbly upflows, while the void fraction profUe peaked in the interior of the chanel for higher flow qualities. It is interesting to note in both Figures 9 and 10 that when wall peaking occurred, the highest local void fractions were in the comers of the conduit. Thus it appears that the secondary flow pattern may influence void distribution.

Finally, let us consider what is known about void distribution in boiling water nuclear reactor (BWR) fuel rod bundles. Lahey & Shiralkar [23] took diabatic steam/water subchannel data in a simulated 3x3 fuel rod bundle using an isokinetic sampling technique. The three subchannels sampled are shown in Figure 11. Typical data trends are given in Figure 12. It can be seen that even though the corner sub chanel (No. 1) had the highest power-to-flow ratio, its flow quality was the smallest. In contrast, the more open interior subchannel (No. 3) had the largest flow quality. These data imply that the

jJ . 1 .02 IT.,! j . 0 .221 m/s 9 <1» · 0 . 1 50

Figure 9. Phase Distribution in Isosceles Triangular Chanels [19]

1 .5

1 .5

E 5

jl - 1 . 00 m/s jg - 0 . 208 rr/s

<a> - 0 . 1 5 7 BUBBLY FLOW

0.5

10 15 20 U Y !Tom

jl - 1 .OO rnls j - 4 .40 m/s <�>= 0 . 63B SLUG FLOW

o 10 n 20 Y m

0.5

Figure 10. Phase Distribution in Vertical Rectangular Channels [19]

8

' . 8 8

SQUARE 'IWR TYPE) LATTIC E

f U E L "00 OUTER CHANNE L.

Figure 11. Typical Subchannels in a 3x3 Rod Bundle [23]

97

98

P = 100 PSIA G = 1.0 X 106 LS/HR -FTZ

� q. = 0.4 5 X 106 STU/HR -FT'

o CORNER SUBCHANNE L , G I C S IDE SUBCHANN£L , Gz. A CENTtR SUBCHANN£L , � 3

Figure 12a. Comparison o f Subchannel Flows for the Three Subchannels [23]

0.2 8

0.24

0.2 0

1\ � 0 . 1 6

V 0 . 1 2.

0.08

P = 1000 PSIA G = 1.0 X J06 Le/HR - FT' Q-= 0.4 5 X 106 BT U/HR-FT2.

o SUBOIANNEL I , <X I > o SUBOiANNEL , , <X, > o SUBCHANN£L 3 , <X1 >

0.1 0.2. 0.3 0 .4 < x > BUNDLE A V E R AG E QUALITY

Figure 12b. Variation of Subchanel Qualities with Average Quality for Three Subchanels [23]

99

global void fraction in subchanel-l is less than that in subchannel-2, which, in tum, is less than that in subchanel-3. Such data trends have been independently confirmed by Bayoumi [24], Herkenrath & Hufschmidt [25], and Yadigaroglu & Maganus [26]. These non-uniform lateral void distribution data are another clear demonstration of the strong lateral "void-drift" forces [27] which occur in two-phase flows.

8. DISCUSION · THE ANALYSIS OF PHASE DISI'RIBUTION

Let us now consider the physical modelling of the forces which produce the observed non-uniform void distributions. A two-fluid modelling approach will be adopted since this approach represents the current state-of-the-art. Since most of the detailed data is for adiabatic air/water flows, we will restrict the model to such flows.

The time-averaged three-dimensional continuity equations of adiabatic air/water flows in straight conduits are given as [28]:

Dkcxk + cxkV • u = 0 Dt -k (k = l,v) (1)

where the subscript-k refers to the liquid (l) or vapor (v) phase, the overbars indicate time-averaged quantities, the underbars indicate vector quantities, ak is the volumetric fraction of phase-k, .lUt is the corresponding phase velocity, and �/Dt represents the material derivative of phase-k.

Similarly, the phasic momentum equations for adiabatic air/water flows are [28]:

(2)

where,

The parameters Pk and J.lk are the density and viscosity of each phase, Pk is the static pressure of phase-k, g: is the gravitational acceleration, .M.ik is the volumetric interfacial force on phase-k, Mwk is the volumetric wall shear

force, �w is a volumetric lateral wall force (to be discussed later), and,

1k � JJ.k VWt - Pk( l1kl1k )· The term, - Pk( l1kl1k)' corresponds to the Reynolds

stress tensor. In the vapor phase momentum equation, the Reynolds stresses are normally very small in comparison to the pressure gradient and the

1 00

interfacial forces, and thus can be neglected. This assumption is quite good for low pressure air/water flows where the density ratio, P"/Pb is very small.

Detailed closure models for bubbly two-phase flows have been presented by Park [53] and Arnold [57]. In this chapter we have used a more simplified model [54]. In particular, we have assumed: constant transport properties, that the phasic pressures are equal (ie, PIt = p), that Jki = lit, and, Pvi = Pv·

Also, we have used the well known inviscid result [28],

- - 2 Pii - Pi = - Cp Pl W.V -ll) where we have assumed [54],

Cp = 1.0.

The interfacial terms in the momentum equations are related by the so-called jump condition:

Miv = - Mil = .M.i (3)

The forces at the interface can be decomposed into the drag (d) force and the non-drag (nd) forces,

Since the data of interest here are for the bubbly flow regime, the interfacial drag force is given by:

(d) (d) 1 1 - - 1 (-- - ) t O O

M ·L = -M. = -PL CD U - uL U - uL A. 1 -IV 8 -v - -v - 1 (4)

where A:" is the interfacial area density. For bubbly flows having only one 1 bubble size, this parameter is given by:

where Db is the bubble diameter. The drag coefficient, CD, was taken here to be the "dirty water" model given by Wallis [29]:

C 6.3

D = Re0.3S5 b (5)

A nondrag force of potential significance is the so-called lateral lift force. There has been considerable confusion concerning the form of this force. Early workers [30] assumed that the lateral lift force of the particles was due to

10 1

the Magnus effect. That is , they assumed that this force arose due to particle spin. However. Theodore [31] showed experimentally that eccentrically weighted solid particles (which did not rotate) exhibited the same lateral forces as those which could rotate. This implied that a Magnus-type formulation was not valid.

Subsequently Safan [32] showed that for laminar shear flows the lateral lift force is proportional to the vorticity of the liquid phase and the relative velocity between a dispersed sphere and the continuous fluid. This approach has been extended by Ho & Leal [33] and Vasseur & Cox [34] to handle wall effects. More recently. Drew & Lahey [35. 36] have derived the lateral lift force for inviscid flows. This force can be written in the form:

(6a)

which for axisymmetric pipe flow becomes.

(6b)

It can be shown that the lift coefficient (CL) for a single bubble is 0.5 for inviscid flows [35] but may be as low as 0.01 for highly viscous flows [37].

In this chapter. Eq. (6b) was used with a value of CL = 0.1 . This value is considered [54] to be the appropriate one to use for the bubbly flow data analyzed herein.

It should be noted that previous researchers [9, 38] have proposed the use of dispersion-type terms in the continuity equation. While such ad hoc models can apparently be made to work, they are not rigorous and involve empirical eddy-diffusivity parameters that are not directly measured. but rather are adjusted to produce agreement with the data. As a consequence, models of this type were not used here. Rather. a turbulent dispersion force was used in the momentum equation. By analogy with Einstein's work on Brownian difsion, this volumetric non-drag force is given by [54],

TD TD Miv = - �l = - CTD Pl kl Vln(a) (7)

where kl is the turbulent kinetic energy of the liquid phase and the turbulent diffusion parameter was taken to be. �D = 0.1. Interestingly, for lower flow

rates (ie, iil < 1.0 mls) the effect of this term is small. It has also been found [55] that for '1aminar" two-phase flows a lubrication­

like wall force becomes significant for bubbles very close to the wall. This force is given by [55]:

M!, = - MJ:, = [ CW1 + CW2RlIyJ � e lL­Rb (8)

1 02

where, Rb is the mean bubble radius, y is the distance between the wall and the center of the bubble, Yr is the local relative velocity, and [54], CW1 = 0. 1,

Cw2 = 0.12. Finally, the wall shear force (per unit volume) is given by [54],

1 f - 1 - 1 Mwt = "2 DH Pt!!t !!t (9a)

(9b)

where, a is the distance from the wall at which y+ = 5 (ie, a :- 100 uj<Yt» and CW}, = 1.0. Naturally, wall shear forces are only used in computational cells in which walls are present, and, as will be discussed in Section 3.2, the single-phase "law of the wall" often replaces the need for Eq. (9a).

For the adiabatic air/water flows under consideration, no energy equations are necessary and the thermodynamic and transport properties were assumed to be constant. This is not rigorously true because there is some internal dissipation, but this effect is normally negligible for adiabatic air/water flows.

The conservation equations, Eqs. (1) & (2), are closed once :l� = -Pi< is related to the state variables of the problem (ie, the velocity field). It should be noted that the turbulence structure of the liquid phase must be carefully modelled since it is known [39, 40] that two-phase turbulence is quite important. Indeed, the 3-D turbulence structure in the continuous phase induces a lateral pressure gradient and helps determine the local gradient of the mean liquid phase velocity. Since the gas bubbles have a relatively low axial inertia, they preferentially respond to the lateral pressure gradient, as well as the lateral lift force given by Eqs. (6). The net result is that these, and the other lateral forces in Eq. (2), determine the lateral void distribution.

3.1. Turbulence Modeling The total Reynolds stress for the continuous liquid phase is given by,

T T T It =�(BI) + �(SI)

where the bubble-induced shear stress is [41],

(1-a>.?�1) = PiCl [at Y.r� + bt<Ur ·�r) il and the bubble-induced parameters are given by [42],

(lOa)

(lOb)

1 al = - ID , and,

3 bl = - ID

1 03

The parameter �SI) is the two-phase shear-induced Reynolds stress. It can be approximated using a k-E model, and an algebraic stress law [58]. In particular, the turbulent kinetic energy, TKE (kL), and turbulence dissipation (EL) transport equations are given by [54]:

(l-a) = (l-a)( V.[UiVkL(SI)] + P-E) [ T ] 1

(I-a) Dt (l-a)V · \) L VkL(BI) + 'tb (k(BI)a - kL(BI»

= (l-a{ v.[�q VEL + P, - E, J where,

PrT = 1.3

'tb = Il)/ 1 U:r 1

�BI)a = a(l-a) Cp 1 � 1 2

kl = kl(SI) + kl(BI)

T � -\) L = C� El + Cllb Rbag 1 llr 1

where [58], CIl = 0.09, and [59], C).1h = 1.2.

(l1a)

(Ub)

(Uc)

(12a)

(12b)

(l2c)

(12d)

(12e)

U sing these results, the net shear-induced Reynolds shear stress is given by:

Re T [V - - V] Ji(SI) = \)l llL + III (13)

and the Reynolds normal stresses come from partitioning kL using a single­phase algebraic stress law [58].

It should be noted that the asymptotic bubble-induced turbulent kinetic energy of the liquid, �Bl)a' comes from an inviscid analysis of the relative

motion between dispersed spheres and the continuous liquid phase [57] , and

1 04

the last term in Eq. (12e) is the bubble-induced viscosity enhancement model of Sato [59].

3.2. BoUDdary Conditions The boundary conditions at the wal are an essential part of the k-£ model.

Because of computational complexity it is impractical to numericaly evaluate the flows in the buffer zone and the laminar sublayer. Instead the boundary conditions are normally placed at the inertial sublayer where the logarithmic "law of the wall" is assumed to be valid. This assumption is known to work well for single-phase flows and has also been used in this chapter for two­phase flow. While the validity of the single-phase "law of the wall" has not been completely verified for two-phase flows, it appears to be a reasonable assumption [45].

Thus the momentum equations' boundary conditions at the "wall" are zero normal velocities and axial velocities are given by [54]:

Ui + .- = 2. 3 1n y + 5. 4 u

where, y is the distance from the wall and,

(14)

(15)

All evaluations shown in this chapter were made assuming "wall" boundary conditions evaluated at y+ = 75.

These closure conditions comprise a state-of-the-art two-fluid model for the prediction of the observed phase distribution phenomena. For convenience, they were numerically evaluated using the PHOENICS code [46]. Unlike most previous analyses, the resultant model allows for the closed-loop prediction of all the state variables, including, the local void fraction and phase velocities.

Figures 13 and 14 show comparisons of the two-fluid model with some typical bubbly upflow and downflow data. It can be seen in Figures 13 that there is fairly good agreement between the model and the bubbly air/water upflow data of Serizawa et.al. [47]. Antal et.al. [55] used virtually the same two-fluid model and, as can be seen in Figure 14, was also able to predict the lateral void distribution for laminar two-phase bubbly upflows [11].

Similar good agreement is seen in Figures 15 between the model and the downflow data of Wang et.al. [47]. It is significant that the same mechanistic model is able to predict the phase distribution for both bubbly upflows and downflows in a pipe.

Even more significant is the fact that, as can be seen in Figures 16, the same multidimensional two-fluid model, which is given by Eqs. (1)-(15), can also predict bubbly upflow data taken in a more complex geometry conduit. In particular, an isosceles triangle [54]. While it should be obvious that more research is needed to predict all the phenomena seen in the data, it appears that the physics of lateral phase distribution is now reasonably well

__ mocel c e c c o Serizcwc's data

0 . 25 j 0 . 2e

l'= 1 .�� m/s J,= O .u t 7 m/s e . l ; l

� ] e . 1 e \

e . e5 �

e . ee e . 2e e . 48 e . 6Z e . ee 1 . ee r/2

105

Figure 13a. Comparisons with Serizawa's Upflow Void Frdction Data in a Vertical Pipe [47]: Ol = 1.36 m/s;jg = 0.077 m/s; Db = 3 mm.; CL = 0.1; Cp = 1.0)

. 60 - - - -'-_ _

";'1 . 20

'-.

E "-"'

0 . 40 0 0 0 0 Seri zO'l'lO ' s data : jg;O .077

-- mode ! : j g;O .077 m/s - - - mode! : jg= O .O m/s

j ,= 1 . 3 60 m/ s

m/s

e . e0 0 . 20 e . 40 0 . 60 r/R 0 . 80 1 . e0

Figure 13b. Comparisons with Serizawa's Upflow Mean Axial Liquid Velocity Data in a Vertical Pipe [47]: Gl = 1.36 mls; jg = 0.077 m/s; Db = 3 mm; CL = 0.1; Cp = 1.0)

1 06

,. '" (J) ..

0 . 01 0 0 .008

}, 0 .006 I� 0 .004

0 . 002

e c c e Serizaw�'s data: jg= O.077 m/s model : Jg=O.O m/s model : Jg= 0.077 mis, C�= 1 . 2 mode l : JIl=0 .077 mis, C�=O.O

j,= 1 .360 m/s

0 . 20 0 . 40 0 . 60 r/R

0 . 80 1 .00 Figure 13c. Comparisons with Serizawa's Upflow Liquid Phase Turbulent Velocity Fluctuation Data in a Vertical Pipe [47]: (jt = 1.36 m/s; jg = 0.077 m/s; Db = 3 mm; CL = 0.1; Cp = 1.0)

0 . 200

�0 . 1 50 S

c c c o o v' Serizawa's data t:. t:. t:. t:. t:. u ' Seriza wa's data -- mode l : 19= 0.077 m/s

mode l : J9=0.0 m/s

j,= 1 . 3 6 m/s

0. 000 0 .00 0 . 20 0 . 40 0. 60 r/R

0 . 80 1 . 00

Figure 13d. Comparisons with Serizawa's Upflow Liquid Phase Reynolds Stress Data in a Vertical Pipe [47]: (jt = 1.36 mls; jg = 0.077 m/s; Db = 3 mm; CL = 0.1; Cp = 1.0)

0. 10

0.09 - PREDICTED VO,I D

0.08 • EXPER I MENTFL DATA

0.07

a 0.06

0.03 •

• • . . . .. . • • • • • • •

0.0 0. 1 0 . 2 0. 3 0 . 4 0.5 0.6 0 .7 O.B 0.9 1 . 0

r/R L I FT COEFF I C I ENT 1 0

1 07

Figure 14. Prediction of Void Fraction Profile for Co curent Laminar, Bubbly Upflow in a Vertical Pipe; Rel = 1276, Po = 1 atm, UR = 0.1 m/s [55]

understood, at least for intermediate size bubbles (ie, 1 mm � Dt, � 6 mm), and that major breakthroughs in the way we analyze two-phase flows are possible.

Let us now turn our attention to other interesting applications of the two-, fluid model. In particular, let us use it to analyze phase separation phenomena in branching conduits.

4. DISCUSION - THE ANALYSIS OF PHASE SEPARATION PHENOMENA

A detailed summary of phase separation phenomena in branching conduits has been given previously [48] and will not be repeated here. Rather, only a few of the highlights of the phase separation research which has been published since this review article will be presented.

A detailed experimental and analytical study of phase separation in horizontal Wyes and Tees has been performed by Hwang et.al. [49]. Their data trends are shown schematically in Figure 17. It can be seen that at high mass extraction ratios (Ws/Wl) essentially complete phase separation occurs (ie, waXa = WIX1). As the extraction ratio decreases the phase separation ratio (Xs/Xl) peaks and falls to zero in the limit as Ws/Wl -+ O. The branch angle (0) is seen to have a pronounced effect at the lower mass extraction ratios.

1 08

0 . 1 3

Cl

" Cl � 0. 08

0 . 05 - - - CL = 0.05 -- CL = 0 . 1

�--"

-------- CL = 0 . 1 5 0. 03 Cl Cl Cl Cl Cl Wa ng's downflow data

j,= 1 .0 mis, jg=O. l m/s

\ \ \

0 . 00 0 . 20 0. 40 0 . 60 r/R 0 . 80 1 . 00

Figure 15a. Comparisons with Wang's Downflow Void Fraction Data in a Vertical Pipe [6]: The Effect of Lift (Cp = 1.0; Db = 3 mm)

1 . 50

- - - CL = 0 .05 -- CL = 0. 1 0 . 50 -- - ----- CL = 0. 1 5 ClClCCC Wang's downflow data

1= 1 .0 mis , jg=O. l m/s

0 . 20 0 . 40 0 . 60 r/R 0 . 80 1 . 00

Figure 15b. Comparisons with Wang's Downflow Mean Axial Liquid Velocity Data in a Vertical Pipe [6]: The Effect of Lift (Cp = 1.0; Db = 3 mm)

0 . 30

0 . 25

0 . 20

0 . 1 5 �

0 . 1 0

e . e5

0 . 00 I . 0 2'2 40 60

Y (mm) 8e

1 09

1 00

Figure 16a. Comparisons with Lopez de Bertodano's Bubbly Upflow Void Fraction Data in a Vertical Isosceles Triangle [54]: Ul = 1.0 m/s; jg = 0.1 m/s; CL = 0.1; Cp = 1.0)

1 . 512)

1 . 25

1 . 012)

,,-. II 0 . 75 '-.

E ..

IA 15 0 . 512)

0 . 25

12) . 00 I 0 20 612) 812) 1 12)0

Y (mm) Figure 16b. Mean Axial Liquid Velocity Comparisons with Lopez de Bertodano's Bubbly Upflow Data in an Isosceles Triangle [54]: Ul = 1.0 m/s; jg = 0.1 m/s; CL = 0.1, Cp = 1.0)

1 10

,,- �

E .. �

j=; 0: 1

3 . 121121

2 . 121121

. 121121

121 . 121121

. ee

- 2 . 1210

- 3 . 121121 I

m o d e l C�b= 1 . 2 model C�=O .O

22 40 63 Y (mm)

Figure 16c. Comparisons with Lopez de Bertodano's Reynolds Stress Data in a Vertical Isosceles Triangle [54]: (jL = 1.0 mls; jg = 0.1 mls; CL = 0.1; Cp = 1.0)

CD ®

Figure 17. Phase Separation Trends in Dividing Wyes and Tees

These data trends support the notion of "Zones of Influence" [48] from which the phases are extracted. Such zones can be quantified in terms of the dividing "streamlines" shown schematicaly in Figure 18.

Hwang et.al. [49] developed a phenomenological dividing "streamline" model which was able to predict the available data reasonably well.

Dividing Streamline for Gos

Streamline for liquid 0

II

Zones of Influence I I

Figure 18. Phase Separation Model Based on Dividing Streamlines [49]

1 1 2

Unfortunately, this model is semi-empirical and thus its extrapolation remains uncertain.

In principle, the multidimensional two-fluid model which was used for the analysis of phase distribution phenomena should also be applicable to the analysis of phase separation. That is, it should be able to automatically predict the 3-D "Zones of Influence" and the associated dividing "streamlines" shown in Figure 18.

A set of Karlsruhe (KfK) phase separation data [50] was selected as a benchmark data set against which to appraise analytical models for phase separation. These data were used as the basis for comparisons with the two­fluid model discussed herein. Since these data were for stratified flow, the closure conditions used in Eq. (2) were not the same as those discussed herein for bubbly flow.

When one considers typical two-phase dividing flows in Wye and Tee junctions it appears that relatively large spatial accelerations may occur. In order to accurately analyze such flows we must introduce another nondrag force into the two-fluid model. In particular, the virtual mass force, has been given as [35, 36]:

M�nd) = _M{nd) = C [Dv�v _ -11 -Iv apt vm Dt Dt

(16)

For a single spherical bubble the correct value of the virtual volume coefficient, Cvm, is 0.5. In general, this coefficient is a function of void fraction, however, for simplicity, a constant value (CYm = 0.0 or 0.5) was used in the numerical comparisons with the KfK data.

As noted previously, in this chapter the stratified flow KfK benchmark experiments [50] were analyzed. The branching conduit was a Tee junction. The inlet, run and side branch were of 55 mm ID, and had a 1.85 m long horizontal inlet section and a 3.09 m long horizontal run section. The side branch, on the other hand, could be horizontal, or in a vertical upward or downward orientation. For each orientation, the corresponding lengths of the side branch pipe were 3.2 m, 2.1 m, or 1.76 m, respectively.

A two-phase mixture of quality Xl was introduced into the inlet at flow rate W I . The mass withdrawal ratio (w3/w d was controlled by adjusting the pressures at the exits of both the side branch and the run. For each mass withdrawal ratio, the phase separation ratio (X3/Xl) was measured using appropriate phase separators and flow meters.

Also, pressure taps were used along the test section to measure the pressure profiles. The measured pressure distributions were used to determine the reversible and irreversible pressure changes in the junction. The inlet-to-branch pressure change data (6P l-3J) indicated a net pressure loss, while the inlet-to-run pressure change data (6Pl-2J) indicated a net

pressure rise. The complexity of the two-phase flow regimes greatly complicates the

analysis of phase separation phenomena. For example, for annular flow at low extraction ratios (Wa/Wl) the liquid film on the wall of the inlet pipe would

1 1 3

be preferentially extracted. Therefore, xS!ltl will be close to zero. AB the extraction ratio (Ws/Wl) increases, the gas phase in the core region of the inlet pipe will also be extracted and the phase separation ratio (Xs/Xl) will increase accordingly.

The influence of gravity and the orientation of the side branch can also be quite important. For a vertically-upward branch, a substantial amount of gas must be extracted before the relatively heavy liquid phase, which was stratified at the bottom of the pipe, is entrained due to the transverse pressure field. On the other hand, for a verticaly-downward branch. the stratified liquid on the bottom of the conduit is preferentially extracted through the side branch. especially at low mass extraction ratios (Ws/Wl). As more fluid is extracted through the side branch. vapor pul-through occured, thus increasing xs.

For stratified flow in a horizontally-oriented side branch the level of the liquid in the inlet pipe and the size of the "Zone of Influence" strongly effects the phase separation ratio (Xs/Xl). In order to predict these phenomena and other observed data trends, a detailed three-dimensional two-fluid model analysis is required.

5. PHASE SEPARATION ANALYSIS

Two-phase flow through a Tee junction was analyzed using the two-fluid model given by Eqs. (1)-(2) and the appropriate closure conditions. For simplicity it was assumed that 'tki = 0 and Pki = Pk. The drag force was expressed by Eq. (4), in which for the stratified flow conditions considered in the KfK study, a wavy-stratified interfacial drag coefficient [51] was used. In particular:

0.0112Re;>·2

CD =

r,ml8 r,ml8

(17)

where hi is the depth of the liquid phase and pvo is the gas phase density at STP.

In contrast, in the junction, the flow was well mixed and the bubbly flow drag law given by Eq. (5) was assumed.

To evaluate the interfacial area density (A:") in the junction, two limiting 1 flow regimes which may occur were considered. In the bubbly flow regime. the interfacial area density is directly related to the local void fraction. That is, as noted previously,

1 14

. . . 6a A. = _ (18) IB Db In contrast, for a stratified flow regime the global interfacial area density is given by,

A�'� = 2 R Sin (cp l 2) (19)

where R is the pipe radius and the angle defining the location of the interface, cp, is related to the global void fraction through,

< a >= 1-� (cp - Sincp) 2x

(20)

It should be noted that the interfacial area density determined using Eqs. (19) and (20) is the global value applicable to 1-D analysis. In 3-D analysis, the interfacial area in the grid cells which span the two-phase interface must be identified. If the flow is stratified and a particular cell includes an interface, then the interfacial area in each cell can be determined. These areas can then be divided by the cell volumes to give the local interfacial area density in each cell. Cells which are in either the liquid or gas phase only have no interfacial area and thus are not affected by the interfacial drag force.

In all branches of the KfK Tee, the flow regime was always stratified [50]. Therefore, the local 3-D interfacial area density corresponding to a stratified flow regime is appropriate. However, in the Tee junction, the flow regime was luite mixed and not well defined, thus, by default, a weighted-average of the ocal bubbly and the global stratified flow interfacial area density was used. rhe weighting factor was chosen to give the best agreement with the

experimental data [52]. The two-fluid model was evaluated using the general purpose flow

simulation code PHOENICS [46]. The Tee junction for which the simulation was performed was broken up into four regions: the inlet pipe, the run, the side branch and the junction. In order to efficiently compute the 3-D flow, the two-phase flow in the inlet pipe was simulated using the two-fluid conservation equations written in cylindrical coordinates. These results were used as inputs to the junction where a body-fitted cylindrical coordinate system was used to discretize the irregular flow domain. Lastly, the side branch and the run were simulated using body-fitted and cylindrical coordinates, respectively, and employed the results from the junction calculations as inputs.

In the junction, the grid was distorted in a prescribed manner such that the same topological character of the cells in the branches was retained (i.e., each cell had six faces and six adjacent neighbors). This distortion of the grid was done by specifying the coordinates of the cell comers. The body-fitted coordinate system for each of these cells was local and nonorthogonal . Figure 19 shows a 3-D view of the grid cells used in the region of the junction.

1 1 5

Figure 19. 3-D View of the Junction's Computational Grid

The inlet and the run were each divided into 100 axial nodes, 3 radial nodes, and 12 non-uniform azimuthal nodes. The junction also had 100 axial nodes but was divided, using body fitted coordinates, to match the grid structure of the interfacing inlet, run and side branch pipes.

A stratified two-phase mixture of given quality and flow rate was introduced into the inlet pipe. The inlet flow rate and quality (WI and Xl) were in accordance with experimental measurements [50]. The two-phase mixture in the inlet pipe flowed into the junction and the two exit pressures of the junction (P2 and Pa) were adjusted to obtain the measured mass extraction ratio (Wa/Wl). The simulation, in tum, gave the predicted phase separation ratio (Xa/Xl). The outlet conditions from the junction were used as the inlet conditions for the run and the side branch, and the static pressure profIles were evaluated.

8. COMPARISONS WITH MEASUREMENTS

The calculated results obtained in the two-fluid model are compared with the KfK experimental data [50] in Figures 20-23.

Figure 20 shows the phase separation ratio (Xa/Xl) for a horizontal side branch.

As can be seen in Figure 20, neglecting the virtual mass force (i.e., setting Cvm = 0) slightly increased phase separation and moved the predictions closer to the total phase separation line, and farther away from the data. Nevertheless, the spatial acceleration in these runs were relatively small and

1 1 6

6

5

4

X -C') X

2

CASE 2

Pl =O.6 M Pa

ie l =1 m1s � 1 =40 mls

-�- EXPERIM ENTAL

•

•

0

DATA

PRED ICTION CASE -1 Cvm=O.5

PREDICTION CASE -1 Cvm=O.O

P R EDICTION CASE 2

Cvm=0.5 TOTAL PHASE SEPARATION

0.8 1 .0

Figure 20. Phase Separation for a Horizontal Branch

thus the virtual mass force had relatively little effect on the phase separation results.

It should be noted that the 3-D simulation accurately predicts the low mass withdrawal ratio results for a horizontal side branch. This is apparently a consequence of the 3-D model adequately predicting the "Zone of Influence" in the junction. Such predictions are not possible with 2-D simulations.

U sing the same drag law, virtual volume coefficient (ie, Cvm = 0.5), and weighted-average interfacial area density model in the junction, Figure 21 shows good prediction of the data from experiments having a vertical side branch above junction branch. Figure 22 shows the corresponding liquid phase velocity vectors for this case. The position of the dividing streamline is easy to identify. Moreover, a recirculating flow at the upper right comer of the junction is evident.

Figure 23 shows two-fluid model predictions and data for the case of a vertical side branch below the junction. It can be seen that the observed vapor pull-through is well predicted.

5

4

x -M X a

2

-A- EXPERIMENTAL DATA

• PREDICTION

EQUAL PHASE SEPARATION (X1=x3)

0.0 0.2 0.4 0.6 0.8 1 .0

wa / W1 Figure 21. Phase Separation for a Vertical Branch Above Junction

, \ \ t i 1

f f / f

/I

/

-. -. -. -. --+ --+ --+ -+

-. --. -. --. -. -. --+ -+

-to --+ -+ --+ -to -to -. -.

1 1 7

Figure 22. Liquid Phase Velocity Vectors for Vertical Branch Above Junction

1 1 8

6

� EXPERIM ENTAL DATA

X • PREDICTION -CO) x

3

0.0 0.2 0.4 0.6 0.8 1 .0

W3 / w1 Figure 23. Phase Separation for a Vertical Branch Below Junction

7. ANALYSIS OF PRES DROP

The two-fluid model that was implemented into the PHOENICS code also predicted the static pressure profiles in the Tee. A typical prediction is shown in Figure 24. It can be seen that pressure recovery is predicted in the run, due to the Bernoulli effect, while there is a significant net pressure loss predicted in the side branch. That is, the irreversible hydraulic losses overpower the Bernoulli effect in the side branch. It is significant to note that in these predictions that no empirical distributed or local loss coefficients, nor two-phase multipliers (eg, CP�o)' were needed. These results are very encouraging and indicate the versatility and capability of a multidimensional two-fluid model.

8. SUMY AND CONCLUSIONS

It has been shown that a properly formulated three-dimensional two-fluid model is capable of predicting phase distribution and phase separation

0

-1

-2

-3

ra Q. � -4 Q. <l

-5

-6

-7

AP13JI

I ', I I

2 3 Z [m]

Figure 24. Pressure Drop in a Tee Junction

1 1 9

5

phenomena. Indeed, such a model has been found to predict the correct data trends for both bubbly upflows and downflows in both simple and complex geometry conduits. In addition, the same type model can predict the observed data trends for phase separation and pressure drop in a Tee.

These results are very promising and indicated that a major breakthrough in the way two-phase flows are analyzed may be near at hand.

It should be noted, however, that more R&D is needed before multidimensional two-phase flows can be predicted with confidence. In particular, further data and analysis are badly needed for other flow regimes.

It is hoped that this chapter will help stimulate researchers to give more serious attention to multidimensional two-phase flow phenomena and to the more widespread use of computational fluid dynamic (CFD) evaluation of

1 20

multidimensional two-fluid models. While there are many important problems which remain to be solved, the way to proceed is clear and the rewards are great. Indeed, it appears that we may be close to having the capability to perform mechanistic predictions of two-phase flow phenomena which are devoid of the global correlations which have characterized the field in the past.

1 M. Ishii, Thermofluid Dynamic Theory of Two-Phase Flow, Eyrolles, 1975.

2 D. MaInes, Report KR-110, Institutt for Atomenergi, Kjelles, Norway (1966).

3 T. Oshinowa and M.E. Charles, Can. J. Chem. Engng. 52 (1974) 25-35. 4 A. Serizawa, Fluid Dynamic Characteristics of Two-Phase Flow, Ph.D.

Thesis, Kyoto Univ., Japan (1974). 5 I. Michiyoshi and A. Serizawa, Nuc. Eng. & Des. 95 (1986) 253-267. 6 S.K. Wang, S.J. Lee, O.C. Jones, Jr. and R.T. Lahey, Jr. , Int. J.

Multiphase Flow 13(3) (1987) 327-343. 7 A. Serizawa, and I. Kataoka, Proceedings of ICHMT Conference on

Transport Phenomena in Multiphase Flow, Dubrovnik, Yugoslavia (1987).

8 K. Sekoguchi, T. Sato and T. Honda, Trans. Japan Soc. Mech. Engng. 40(33) (1974) 1395-1403.

9 I. Zun, Proceedings of the ICHMT Seminar on Transient Phenomena in Multiphase Flow, Dubrovnik, Yugoslavia (1987).

10 A. Kariyasaki, Proc. 7th Two-Phase Flow Symposium of Japan (1985). 11 N.V. Valukina. B .K. Koz'menko and O.N. Kashinskii, Inzhenemo-

Rizichenskii Zhurnal 36(4) (1979) 695-699. 12 D.F. Young. ASME Preprint 60-HYD-12 (1960). 13 R.C. Jeffrey and J.R.A. Pearson, J. Fluid Mech. 22 (1965) 721. 14 T. Zigami, Heat Transfer and Fluid-Flow in Air-Water Two-Phase Flow

in an Annular Channel, M.S. Thesis, Kyoto University (1976). 15 T. Furukawa, H. Fukuk and K. Sekoguchi, Proc. 17th Nat. Heat

Transfer Symposium of Japan (1980) 349-35l. 16 B. Shiralkar and R.T. Lahey, Jr., ANS Trans. 15(2) (1972) 880. 17 K. Ohkawa and R.T. Lahey, Jr., Int. J. Multiphase Flow 15(3) (1989) 447-

457. 18 K. Sekoguchi, O. Tanaka, S. Esaki, H. Sugi and O. Ueno, Proc. 17th Nat.

Heat Transfer Symposium of Japan (1980) 352-354. 19 M. Sadatomi, Y. Sato and S. Saruwatari, Int. J. Multiphase Flow 8(6)

(1982) 641-655. 2D S.K Sim and R.T. Lahey, Jr., Int. J. Multiphase Flow 12(3) (1986) 405-

425. 21 O.C. Jones, Jr. and N. Zuber, AIChE Symposium Series (1978). 22 K. Ohba and Y. Yhuara, Trans. Japan Soc. Mech. Engrs. 48(425) (1982)

78-85. 23 R.T. Lahey, Jr. and B. Shiralkar, J. Heat Transfer 93 (1971) 197-209.

1 2 1

24 M.A.A. Bayoumi, Etude des Repartitions de Debit et d'Enthalpie dans Les Sous-Canaux d'use Geometrie en Grappe, des Reacteurs Nucleaires en Ecoulements Monophasique et Diphasique, Doctorate Thesis, U. de Grenoble (1976).

25 H. Herkenrath and W. Hufschmidt, Experimental Investigation of the Enthalpy and Mass Flow Distribution Between Sub channels in a BWR Cluster Geometry (PELCO-S), ElJR.6585-EN (1979).

28 G. Yadigaroglu and A. Maganus. Quality and Mass Flux Distribution in an Adiabatic Three-Subchanel Test Section, UC-BINE 3342, 1978.

Zl R.T. Lahey, Jr. and F. J. Moody, The Thermal-Hydraulics of a Boiling Water Nuclear Reactor, ANS Monograph. 1977.

28 R.T. Lahey, Jr. and D.A. Drew, (Lewins & Becker - Eds.) Advances in Nuclear Science and Technology, 1988.

29 G.B. Wallis, One Dimensional Two-Phase Flow, McGraw-Hill, New York, 1969.

:J) S.I. Rubinow and J.K Keller, J. Fluid Mech. 11 (1961) 447. 31 L. Theodore, Sidewise Force Exerted on a Spherical Particle in Poiseuille

Flow, Eng. Sc.D. Dissertation, NYU, 1964. 32 P.G. Safman, J. Fluid Mech. 22, Part-2 (1965) 385. 33 B.P. Ho and L.G. Leal, L.G., J. Fluid Mech. 65, Part-2 (1974) 365. 34 P. Vasseur and R.G. Cox, J. Fluid Mech. 78, Part-2 (1976) 385. 35 D. Drew and R.T. Lahey, Jr., Int. J. Multiphase Flow 13(1) (1987) 113-

121. 36 D.A. Drew and R.T. Lahey, Jr., Int. J. Multiphase Flow 16(6) (1990) 1127-

1130. � R. Eichhorn and S. Smal, S., J. Fluid Mech. 20(3) (1964) 513. 38 S.W. Beyerlein, R.K. Cossmann and H.J. Richter, H.J., Int. J.

Multiphase Flow 11(5) (1985) 629-641. 39 D. Drew and R.T. Lahey, Jr., J. Fluid Engng. 203 (1981) 583-589. 40 D. Drew and R.T. Lahey, Jr., J. Fluid Mech. 117 (1982) 91-106. 41 R.I. Nigmatulin, Int. J. of Multiphase Flow 5 (1979) 353. 42 A. Biesheuvel and L. van Wijngaarden, J. Fluid Mech. 168 (1984) 301-

318. 43 B.E. Launder, et al., J. Fluid Mech. 68-3 (1975) 537-566. 44 H. Tennekes and J.L. Lumley, A First Course in Turbulence, MIT

Press, Boston, 1972. 45 J.L. Marie, Int. J. Multiphase Flow 13(3) (1987) 309-326. 46 D.B. Spalding, et al., PHOENICS - Beginner's Guide and User Manual,

CHAM TRl100, 1986. 47 A. Serizawa, I. Kataoka, and I. Michiyoshi, Int. J. Multiphase Flow 2(3)

(1975) 221-260. 4B R.T. Lahey, Jr., Nuc. Eng. & Des. 95 (1986) 145-161. 49 S.T. Hwang, H. Soliman and R.T. Lahey, Jr., Int. J. Multiphase Flow

14(4) (1988) 439. 50 R.T. Lahey, Jr., Multiphase Science and Technology 3, Hemisphere

Press, 1987. 51 N. AnClrltsos and T.J. Hanratty, AIChE Journal 33(3) (1987) 444-453. 52 R.T. Lahey, Jr., Nuc. Eng. & Des. 122 (1990) 17-40.

1 22

53 J-W. Park, Void Wave Propagation In Two-Phase Flow, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY (1992).

54 M. Lopez de Bertodano, Turbulent Bubbly Two-Phase Flow in a Triangular Duct, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY (1992).

55 S.P. Antal, R.T. Lahey, Jr. and J.E. Flaherty, Int. J. Multiphase Flow 17(5) (1991) 635-652.

56 M. Lopez de Bertodano, S-J. Lee, R.T. Lahey, Jr. and D.A. Drew, Journal of Fluids Engineering 112(1) (1990) 107-113.

57 G. Arnold, Entropy and Objectivity as Constraints Upon Constitutive Equations for Two-Fluid Modeling of Multiphase Flows, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY ( 1988).

58 W. Rodi, Turbulence Models and their Applications in Hydraulics, IAH&'AIHR Monograph (1981).

59 Y. Sato, M. Sadatomi and K Sekoguchi, Int. J. Multiphase Flow 7(6) (1981) 167-190.

123

WAVE PROPAGATION PHENOMENA IN TWO-PHASE FLOW

R.T. Lahey, Jr.

The Edward E. Hood, Jr. Professor of Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 USA

Absiract This chapter presents data and a one-dimensional two-fluid model analysis

of pressure and void waves. Both linear and nonlinear analysis are performed. The analytical linkage between standing and propagation pressure waves, as well as critical flow, is developed. The well-posedness of a two-fluid model is found to be determined by the void wave eigenvalues, which are, in turn, determined by the closure conditions. Moreover, the criterion for the formation of void shocks and solitons is given.

It is shown that wave propagation phenomena may be quite sensitive to the closure laws used in the two-fluid models. As a consequence pressure and void wave data are very useful for two-fluid model assessment.

1. INTRODUCTION

Wave propagation mechanisms in two-phase flow are of great importance. Indeed, many transient and steady-state phenomena are controlled by the propagation of pressure, enthalpy and void disturbances. Examples include, choking, flooding, shocks, density-wave instabilities and flow regime transition. Significantly, it has also been found that both sonic and density­waves strongly depend on the closure conditions used in two-fluid models. As a consequence, wave propagation data offers an excellent means of model assessment.

This paper will focus on sonic and void wave phenomena. Those readers specifically interested in the effect of enthalpy and density-waves on two-phase flow instabilities, are referred to previous work [1], and to the chapter in this book on Instabilities in Two-Phase Systems.

2. DISCUSION

Let us begin by considering propagating pressure perturbations (ie, sonic waves). Previous investigators have taken data on the speed of sound in a two­phase mixture (C2�) using both standing wave and pulse propagation (ie, time-of-flight) technIques (eg, [2-5]). In these studies both side mounted and traversing pressure transducers (ie, hydrophones) were used for the standing

1 24

wave experiments. Unfortunately, no data were taken using both of these techniques, thus it was not possible to determine what bias may have occurred. For example, what effect the acoustic impedance mismatch associated with an intrusive probe (ie, a traversing hydrophone) might have had on the data. Moreover, standing wave and pulse propagation was not taken in the same facility and the analytical linkage between the two-phase speed of sound measured in both ways was not previously demonstrated.

These shortcomings have been addressed by Ruggles et.aI. [6]. Thus, let us consider this study in some detail. The apparatus used for standing wave and pressure pulse measurements is shown schematically in Figure 1. The wave guide used to generate the standing wave pattern was constructed from a 63.5 mm ID, 76 .2 mm OD, stainless steel tube, which was two meters in length. This tube was fitted with three side mounted pressure transducers and a hydrophone mounted on a traversing mechanism. Sinusoidal pressure oscillations were introduced through a Wye fitting, using an electromechanical shaker and piston arrangement. An isolation system prevented sound energy from the shaker and piston from entering the waveguide walls and disturbing the side mounted transducers. In addition, an air cushion isolated the entire wave guide and lower plenum from laboratory floor vibrations.

All standing wave data were taken for conditions of no liquid flow, while some of the data for the pulse propagation experiments involved finite liquid flows (to verify that it had no effect). Air bubbles were introduced in the lower plenum using one of four banks of hypodermic needles. Bubble radii were measured directly using high speed strobe photographic techniques. The global void fraction, <CX>, was determined from the hydrostatic pressure drop, and by using quick closing valves.

In the standing wave experiments the global void fraction was varied from 0.5% to 18% and the mean bubble radii, Rb, varied from 0.55 mm to 2.5 mm.

The forcing frequency was varied from 20 Hz to 200 Hz to allow measurement of the dispersion and attenuation curves for each flow situation. The propagation speed and attenuation of the pressure perturbations associated with standing waves was measured using three independent techniques. In the first of these techniques a hydrophone was traversed through the waveguide from above, and the locations and amplitudes of the pressure nodes and anti-nodes were recorded. The propagation speed and attenuation could then be calculated since the distance between the nodes (LN) is half of the wavelength, A. Thus [7]:

(1)

(2)

where, n is the node number, counting from the top of the wave guide, and T\ is the spatial attenuation coefficient.

1 25

® TRANSDUCER STAT I O N A ® TRANSDUCER STAT I O N B © T R A N S DU C ER STAT I O N C @ WAV E I N LET ® M A N O M ETER P O RTS

® WAV E G U I D E @ LOWER TA N K ® B U B B LE G E N E R ATO R CD TRAVERS I N G H Y D R O P H O N E

Figure 1. Measurement system for wave propagation in a bubbly air/water mixture.

Another redundant measurement technique used the root mean square pressure amplitude readings, �PRMS ' from the three side mounted

transducers to infer the wavelength and attenuation of the standing wave. Thus [7]:

1 26

(3)

where, as can be seen in Figure 1, the distance from the free surface is,

l = L - z

and A + is the amplitude of the upward traveling wave at z = L. The third redundant measurement technique involved varying the induced

angular frequency (00) until a pressure node was situated over one of the side­mounted pressure transducers. This indicated an integer number of half wavelengths existed between that transducer and the bubbly air/water interface. The number of half wavelengths was then measured using the traversing hydrophone and the celerity was calculated using Eq. (1). These redundant measurement techniques were used to assure the accuracy of the data and showed that the presence of the traversing hydrophone did not significantly affect the data.

Representative standing wave data is shown in Figures 2 through 6. It can be seen in Figure 2 that the sonic speed (C2cp) is a strong function of both

frequency and bubble radius. In addition, Figures 3 & 4 show that the sonic speed decreased markedly as void fraction increases. Figures 5 and 6 show that the spatial attenuation coefficient (11) is a strong function of frequency but not void fraction.

C2� 1 74 (m/s) 1 72

<a> = 0.50% CVM = 0.5 (-)

1 5 60 20 40 60 80 100 120 1 40 160 180 200 220 FREQUENCY ( Hz)

Figure 2. Sound speed vs. frequency (effect of bubble size)

C .. 68 (m/s)

66

6 4

6 Z

6 0

5 8

5 6

5 4

5 0

4 8

�a> _ 4.020/0 Ii! - 1 .6 mm Cw .. - 0.5 (-)

�a> _ 5.00% RB - l .7 mm Cw .. - 0.5(-)

� <a> - 8.00"10 FiB - 1 .95 mm Cw .. - 0.5 (-)

FREaUENCY (HZI

1 27

Figure 3. Sound speed VB. frequency (effect of void fraction - low void fractions).

C .. 45 (m/s)

44

43

42

41

40

39

39

37

36

<a> - 10% REi � 2.20 mm Cw .. - 0.5 (-)

r!<a> _ 13% RS - 2.53 mm Cv .. - 0.55 (-)

- 0.6 (-)

Rs - 2.4B mm Cw .. - 0.7(-)

FREaUENCY (HZI

Figure 4. Sound speed VB . frequency (effect of void fraction - high void fractions).

1 28

2 <a> = 4 .02% = 1 .6 m m

CVM = 0 . 5 (-)

o 20 40 60 80 100 1 20 FREQUEN CY ( Hz)

Figure 5. Attenuation VB. frequency (low void fraction)

� (�) 5 4

o 20 FREQUEN CY ( Hz)

Figure 6. Attenuation VB. frequency (high void fraction)

<a> = 1 8 % R B = 2 .46 m m CVM 1& 0.7 (-)

1 29

The apparatus used by Ruggles et.al. [6] to take the pressure pulse propagation speed measurements was identical to that shown in Figure 1. The pressure pulses were introduced by driving the electromechanical shaker with a square-wave generator. For these experiments, the upper end of the waveguide was connected to a phase separation tank and the lower plenum fed by a metered water flow to facilitate variation of the superficial liquid velocity, <jj>. Flow experiments were run to verify that the sonic velocity was unafected by liquid phase velocity.

A typical pressure pulse is given in Figure 7, as it appeared at the lower side mounted transducer station (C) in Figure 1. It was noted that the pressure pulses exhibited some attenuation as they passed the upper transducer stations. However, no significant distortion of the pulse shape or steepening of the leading edge was observed.

Pressure pulse speed measurements were made using two independent techniques. The first of these was a time-of-flight method. This technique used a Tektronix 7854 digital oscilloscope with the peak positive pressure chosen as the discrete time feature of each pulse. Another independent pressure pulse speed measurement technique involved taking the slope of the phase vs. frequency plot of the cross power spectral density (CPSD) function [8] between a lower and upper side-mounted transducer pair. This slope can be related to the propagation speed of a pulse by noting that,

BV(t,z) = C2cj1

6P )( 1 0 3

(dynes/em)

4 2

- 2

.4

-6

- 8

6

T I M E )( 1 02 (sec)

Figure 7. Pressure pulse (typical)

(4)

1 30

If we Fourier transform Eq. (4) we obtain:

(5)

where e is the phase angle of frequency component, 00. The CPSD may be written as,

(6)

This indicates the phase angle of the CPSD of a propagating perturbation is locally linear with frequency. Thus the slope of the CPSD phase angle (11))

versus frequency plot, -!' yields,

C = (z - Z _ dIP ) 2, 2 1 doo

The CPSD's were calculated using a Hewlett-Packard 3562A dynamic signal analyzer. This technique was a useful independent verification that the time­of-flight measurements accurately indicated the sonic speed.

The pressure pulse propagation data taken in the study of Ruggles et.a!. [6] are presented in Figure 8. The sonic speed is seen to be a strong function of the global void fraction, <a>.

The standing wave dispersion data given in Figure 2 clearly shows the strong dependence of the propagation speed of pressure perturbations on mean

bubble radius (RB). This effect is due to the dependence of the propagation speed on the interfacial heat transfer between the two phases. For the same global void fraction, a few larger bubbles have less interfacial area available for heat transfer than more smaller bubbles, and thus exhibit a more-nearly adiabatic process, and thus a greater celerity. It can also be noted that, for essentially the same reason, the celerity increases with frequency. That is, more time exists for heat transfer at the lower frequencies thus promoting a more-nearly isothermal process and a lower celerity.

Let us now consider the analysis of wave propagation phenomena. The model used will be a one-dimensional two-fluid model of the two-phase flow.

3. ANALYSIS

The space-time averaged one-dimensional two-fluid conservation equations for adiabatic air/water flow in a constant area duct are given by [9]:

c2• 140 (m/s)

120

100

80

60

40

20

+ BAN K 2. jL = 0.0 X BANK 3. jL :: 0.0 o BANK 4. jL = 0 .0 C BANK 4, JL = 0.6 m/s � BANK 4, iL = 1 .0 m/s V BANK 5, k = 0.2 m/s

o o 2 3 4 5 6 7 8 9 10 I I 12 13 14 15 16 17 18

<a>, VOID FRACTION (%)

Figure 8. Pressure pulse propagation speed vs. global void fraction

CONSERVATION OF MASS

a a at « ak > Pk ) + az

« ak > Pk < Uk >k) = 0

1 3 1

(7)

1 32

CONSERVATION OF MOMENTUM

CONSERVATION OF ENERGY

i.« (lk > Pkhk )+i.« (lk > Pk < uk >k hk) at az

=< (lk > (aPk + < uk >k apk < Uk >k 'tk + qk·A. � w 1 1

(8)

(9)

where the subscript k denotes either the gas (k=g) or the liquid (k=l) phase. In this model it is assumed that both phases are compressible and that, we have an equation of state given by:

(10)

Appropriate constitutive e.quations must be used to model the interaction between the flow components. Detailed closure laws for bubbly two-phase flows have been given by Park [34] and Lopez de Bertodano [35] , however let us assume that the momentum. transfer between the gas and liquid phases can be written as the sum of three forces [10]:

(11)

where,

The interfacial drag force, FD, for bubbly flow of uniform bubble size, Rb, can be modeled as:

(12)

The drag coefficient, CD' for distorted bubbly flows is given by [11] as,

1 33

(13)

In contrast, the interfacial drag force can also be written as:

(14)

where, for example, the interfacial friction factor for undistorted bubbles is given by [12]:

[ Re

°·75J

DJI 1 + 0.1 2cp fi = 18 D}) Re 2cp

where,

The virtual mass force, FVM' is given by,

(15a)

(15b)

(l5c)

(16)

The virtual volume coefficient, Cvm ' can be expressed as an empirical function of the global void fraction [6] as,

« a» S 20%

and the virtual mass acceleration, Bvm' is given by [13]:

or, equivalently,

Bvm - Dt - Dt

(17)

(18a)

(l8b)

1 34

The axial reaction force due to bubble pulsation, FR, results from the interaction of a vibrating spherical bubble with the flow field around the bubble. This flow field is due to both bubble translation relative to the liquid phase and to radial bubble pulsations. This force is given by [10]:

3 FR = Rt, Cvm Pi « Ug>g - <ui>/.) Dt (19)

Since, for bubble flow, the gas phase is assumed to be dispersed within the liquid phase, the wall shear stress on the gas phase is,

(20a)

while the liquid phase wall shear stress is,

(20b)

For low pressure air/water flows, the interfacial pressure in the gas phase is often related to the average pressure of the gas phase by,

(21)

This is normally a good assumption and implies that as one is dealing with situations in which the bubble has essentially a uniform internal pressure.

In contast, the difference between the interfacial average pressure and the mean pressure in the liquid phase is, for a non-pulsating bubble, given by [14]:

� 2 Pli - Pi = i\Pli = - 4 (<ug>g - <u/.>/.) (22)

The Reynolds stress (�r) is negligible in the gas phase phase but not in the liquid phase. It is beyond the current state-of-the-art to predict this term in general, however, the Reynolds stress in the liquid phase due to "bubble­induced" turbulence has been given by Nigmatulin [15] as:

�i = - <a>p /. [ C1 1 <JIg> g - <lli> i l2 1 + C2 « llg> g - <lli> i) « l1g> g - <llr i) ] (23a)

It can be shown [16] that for a spherical bubble the coefficients C1 and C2 are given by,

(23b)

It should be noted that for one-dimensional pipe flow, Eqs. (23) reduce to:

T 1 2 tzzl = - 5 <a> Pl - <l1l>l)

Hence, the Reynolds stress gradient term in Eq. (8) can be written as,

1 35

(24)

(25)

This is the same expression as developed by Biesheuvel and van Wijngaarden [16] and used by Pauchon and Banerjee [17].

The wall-induced interfacial shear, tki' is often set to zero. However, one

may also assume that it is equal to t;z . When this is done the third and l fourth terms on the right hand side of Eq. (8) combine to yield,

(26)

For quasi-static conditions we note from Eq. (22), and the well-known Laplace equation, that:

(27)

Let us next develop a dynamic relationship between the phasic pressures, and Pl. This can be done in a manner similar to that of Prosperetti [18] and

et.al. [10]. To accomplish this, a single bubble is considered to be surrounded by an infinite liquid medium, and excited by sinusoidal pressure oscillations. The bubble response is assumed to be spherically symmetric and without translatory oscillations. The continuum assumption is valid for pressure excitations having wavelengths, A, which are much greater than the bubble radius, Rb.

Continuity of normal stress at the bubble surface gives,

The Bernoulli equation for the liquid phase can be written as,

(Rb,t) 1 2 acp(Rb) (t) + - [(Vcjl(Rb)] + -- = Pl 2 at Pl

where the velocity potential of a stationary pulsating bubble is given by [19]:

(28)

(29)

1 36

and the wave number is given by,

CJ) kb = C2cp(ro)

(30)

(31)

If we combine Eqs. (28), (29) and (30) we obtain for the bubble dynamics equation,

PlRbRb · 2 [ 2 1 [ k})Rb pg<t) - PI = (1 - ik})R� + PIRb (1 - ik� - 2 - iklJRb - 1

�[ 20 + 1% 1 - (1 - ik})Rb) Rb + Rb

(32)

where we have assumed that the average pressure of the liquid phase, Pb is the appropriate far field liquid pressure ( Pl_)'

It is interesting to note that if terms of order kbRb are neglected we recover the classical Rayleigh equation for a stationary pulsating bubble,

• where, R = Dt

(33)

A thermodynamic analysis of a bubble suspended in a liquid media can be used instead of the gas thermal energy equation, Eq. (9). This type of analysis has been done by previous investigators [18,20,7] and results in one being able to write the perturbed form of Eq. (34) as a damped harmonic oscillator.

1 37

3.1 The Dispersion Relation Equations (7), (8), (9), and (34) can be written in matrix form as,

(35)

where,

(36)

Equation (35) can be perturbed as follows,

[d'lo � + oAl dt, + � + oID dz + oz- = � + oQJ �o + �] (37)

The steady-state equation describing the unperturbed fully developed two­phase flow is given by,

(38)

Assuming that the spatial derivatives of the steady-state solution are of order 0, the linearized equation set describing the response of the system to small perturbations can be expressed as,

where,

Let us assume that the perturbation of the state variables is of the form,

o! = 'l' ei(kz-mt)

This transforms Eq. (39) into the following algebraic equation,

�<'D [-ico] + llo<'l> [ik] - ��)} 'l' = 0

(39)

(40)

(41)

Equation (41), in conjunction with the requirement that'l' be finite, implies a linear dispersion relationship of the form,

1 38

det[(�o (mJk) - (iJk)��) - �o] = 0 (42)

For standing waves, Eq. (42) gives a relationship between the angular frequency, (I), and wavenumber, k. The wavenumber is in general a complex number, with its real part corresponding to 27t divided by the wavelength, and its imaginary part corresponding to a spatial attenuation coefficient (1'\).

The dispersion relation normally gives seven roots (wlk) for each value of angular frequency, (I). Three of these roots yield celerities typical of the convective velocities of the liquid and vapor phase, <ur l and <ug> g. Two

other roots are associated with the celerity of void perturbations (C!>. The remaining two roots travel at celerities typical of the so-called speed of sound in the two-phase medium. One of these acoustic roots, [(I)/Re(k)t , is positive (ie,

it travels with the flow). The other, [co/Re(k)]�, is negative, and travels against the flow. These roots have velocities that vary in absolute value by a two-phase convective velocity. The difference of these two roots has been used to determine the standing wave speed previously discussed. That is, the speed of sound was given by,

(43)

It can be seen in Figures 2-6 that this model predicts the standing wave data of Ruggles et.al. [6] very well. Standing wave data near bubble resonance was also taken by Silberman [3] . It is interesting to note in Figure 9 that the two­phase speed of sound is nondispersive at both low and high frequencies, but varies dramatically with frequency near the bubble's resonant frequency. Moreover, it should be noted that C2cp is essentially Cl above the resonant frequency. As will be explained later, this is because the inertia of the liquid is so large that the bubbles can no longer respond to the imposed forcing function and thus the two-phase mixture behaves acoustically like a single-phase liquid. Finally, it is interesting to note in Figure 10 that damping of the acoustic wave is quite large near bubble resonance.

3.2 Prediction of Propagating Pres Pulses Let us now consider the analytical relationship between standing waves and

propagating pressure perturbations. A linear perturbation of the form � can be represented in terms of the

harmonic perturbations given in Eq. (40) through a Fourier superposition,

"i l o w

Model run lor: R" =2.5 mm

� C·.� en o z ::> o en MEAN BUBBLE RADIUS (R •• ). m

o 1.92 X 10·'

o 1 98 X 1 0 ' o 2.07 X 1 0·' I:l. 2.50 X 1 0·'

1 0' 1 02 1 03 1 04 1 05 FREQUENCY (Hz)

Figure 9. Sound speed vs. frequency (Data of Silberman [3])

Model run tor R,·2.5 mm

<a>' 0.5S4'1o

MEAN BUBBL.E RADIUS (R •• l.m o 1.92 X 10 ' o 1.9S X 10·· C 2.07 X 10 ' I:l. 2.50 X 10·'

1 0' 10' 10' 10'

FREQUENCY (Hz)

Figure 10. Attenuation VB. frequency (Data of Silberman, [3])

1 39

1 40

00

z, t) = f f g j (0) )a", .dO) j=1 -J 0)=--

00

f[j�l gi!m)!/(kj(O»z) }-imtdCl CIl=--

00 J[jt Q i ! Cl)e i(k j( O» z) ]--imtdm

CIl=-oo where,

7 7 [ ' J 1 00 ' t L Q:j(O)) = L gj(O))"" = - f&¥(0,t)e1Cl dt j=1 j=1 -J 2x t= oo

and,

00 g j( 0)) = J a'l /0, t) . ",�eiCJ)tdt

t=-oo

(44)

(45a)

(45b)

Normally, the pressure perturbation, api' is the dominant eigenmode. Thus, Eq. (44) may be simplified to give the approximate Fourier integral representation (ie, considering only the pressure mode) of the single state variable Pi in the propagating eigenmode associated with the traveling pressure wave,

(46a)

where,

(46b)

1 4 1

Note that the wave number of the pressure perturbation travelling in the negative z direction (ie, in the flow direction), k� , has been used. This technique has been employed to predict the propagation speed and dispersion of the pressure perturbation shown in Figure 7, thus relating the standing wave and propagating pressure perturbations. The time trace of the pressure perturbation (measured at the lower transducer) was used to generate the function gp(co) using a standard FFT algorithm.. The time trace of the pressure pulse at the upper transducer was then constructed through a numerical inversion of Eq. (46a) with the values for k�(co) taken from the model's dispersion relation, given in Eq. (42). The results are shown in Figure 11. It can be seen that the model predictions are in close agreement with the data.

It is also interesting to use the two-fluid model to predict the dispersion of a hypothetical square pulse. The results are shown in Figure 12. Notice that the pressure pulse rapidly separates into two distinct pulses, one that is traveling relatively slowly, at the low frequency two-phase speed of sound (C�cp)' and one that is traveling much more rapidly, at the speed of sound in liquid (Cl). This is easily understood by examining the predicted values of in Figure 9. All the frequency components of the pulse which have values than bubble resonance (oor) are traveling relatively slowly, at the low frequency two-phase

6P )1 1 0-' e (dynes/emf) 6

4

-- 10_' I,ansdue., 51111

-- - - - Ul'I'e, I'lnsduea, 1'"lIon

: o 0 o p'�"led vllues 2

0

- 2

_ 4

- 6

< a > - 5.00 %

A - 1 .7 mm b

- 8 2 4 6 8 TIME II 10' (sec)

Figure 11. Measured and predicted pressure traces.

1 42

speed of sound (C�cp)' In contrast, al the frequencies above bubble resonance are traveling much faster, at the speed of sound in liquid (Cl)' Thus, the high frequency components (CI) > Cl)r) which comprise the leading and trailing edge of a square pulse, rapidly separate from the low frequency components (CI) < CJ)r)' All of this is accompanied by strong attenuation of all components of frequency near bubble resonance.

The rapid separation of a square 'pulse into 'slow' and 'fast' portions indicates that the physical behavior that governs sound propagation for frequencies below bubble resonance is quite different from that which governs the propagation of sound for frequencies above bubble resonance.

Indeed, well below bubble resonance the generalized Rayleigh equation, Eq. (34), reduces to the quasi-static approximation,

p <a>-O.SO % Ro=2.2mm zsO.Om

0 .0

0 .0

<a>=0.50 0/ Ro 2.2mm z=0.1 m

1 .0 2.0 TIME (ms)

1 .0 TIME (ms)

Figure 12. Predicted dispersion of a step input.

20 £l 2 Pg - Pt = Rb - 4 « ug>g - <Ut>t)

143

(47)

It has been previously shown that the bubble's thermal response is essentially adiabatic, over a wide range of frequencies below bubble resonance [18]. Thus when the quasi-steady phasic pressure relationship, Eq. (47), replaces Eq. (34) in the two-fluid model we obtain:

(48)

where the subscript, s, denotes matrices appropriate for quasi-static (ie, slow, low frequency) disturbances.

It has been shown [6] that two of the eigenvalues of Eq. (48) yield the C2� shown in Figure 9. The linear dispersion relation, Eq. (42), which results from Eq. (48), also yields C2�' This is apparently because, for bubbly air/water flows, viscous effects are negligible for frequencies below bubble resonance [18], hence, £s : O.

The 'fast' portion of the original square wave pulse is composed of frequencies above bubble resonance. For perturbations of high frequency (ie, CI) » Cl)r) the Rayleigh equation yields, SRb == O. This implies that the bubbles are limited dynamically in their ability to follow the liquid pressure excitations, therefore, the gas phase pressure remains unperturbed. Thus, the liquid phase pressure perturbations do not sense the gas phase compressibility since the bubbles behave more or less as rigid spheres. Hence, as shown in Figure 9, when the high frequency pressure relationship between the gas and liquid phases is introduced into the dispersion relation, Eq. (42), the resulting prediction for the two-phase speed of sound is the speed of sound in liquid, Ct.

Let us now determine how C� is related to the eigenvalues ofEq. (48). The eigenvalues (�j) of this nonlinear system of equations can be determined

from:

detr�s � - �sl = 0 (49)

where, the celerity-j is given by:

Re[�j] = (�:)j (50)

The seven roots ofEq. (50) are similar to those ofEq. (42). However, they are not frequency dependent. Subtracting �/2 from ��/2 we obtain the speed of the

propagating pressure pulse, C2�' The pressure pulse data was compared to

1 44

the data of Ruggles et.al. [6] for a range ofCvm' As can be seen in Figure 8/the agreement was excellent when Cvm values corresponding to Eq. (17) were used.

3.3 The Be1atioDsbip to Critical Flow Let us now consider the relationship of C2, to the critical flow velocity. The

form of the two-fluid models commonly used for critical two-phase flow analysis is identical to Eq. (48). In particular, for steady choked flow, Eq. (48) reduces to:

(51)

A necessary condition for choking is [21]:

detQkl = 0 (52)

The choking condition can be determined by solving Eq. (51) as an initial value problem, while simultaneously testing for the condition of Eq. (52). A detailed discussion of how this type of analysis is performed has been given previously [21] and thus will not be repeated here.

The velocity associated with the choked state (ie, the critical flow velocity) is calculated from the characteristics (�) ofEq. (48) by [22]:

det�� -�} I choked = 0 (53)

Obviously Eq. (53) is just a special case ofEq. (49). One of the eigenvalues of Eq. (53), �c = 0, determines the local choking

condition. This eigenvalue is the only one capable of transmitting information on pressure variations downstream of the throad to positions upstream of the throat. Once the discharge velocity at the throat reaches sonic conditions, further lowering of downstream pressure is not capable of affecting conditions upstream of the choking plane (ie, the throat). Moreover, since Eq. (53) is valid for all eigenvalues, and in particular for the one that vanishes, this equation reduces to Eq. (52) when choking occurs.

It should be noted that Eq. (53) is somewhat different from the corresponding linear dispersion relation for the 'slow' pulses:

(54)

where, l;* = CIlI'k. However, Eq. (54) reduces to Eq. (53) in the limit as the wave number, k,

becomes large (ie, the high frequency limit):

1 45

(55)

where km is the maximum wavenumber for which the low frequency interfacial pressure relationship, Eq. (47), holds. In practice the celerities predicted by Eqs. (53), (54) and (55) are almost identical for all frequencies where the bubble behavior is essentially adiabatic. Thus we find that the wave propagation model associated with low frequency phenomena predicts the so­called two-phase choking velocity, C2 •.

The reader is cautioned that while simple adiabatic models may be adequate for predictions of the two-phase sound speed they are not adequate for predictions of the spatial attenuation. Fortunately, a complete two-fluid model appears to be adeq�ate.

It is also interesting to note that some of the terms in matrix �s are due to algebraic constitutive laws often used to model phenomena such as bubble drag, wall shear, and heat transfer. These phenomena are actually based on local gradients, but for convenience they are normally modeled algebraically. Hence, their appearance on the right hand side of Eq. (48), and ofEq. (35), is a result of pr<fedures used in averaging and modeling.

Finally, it is interesting to note that a general spatial pressure field can be constructed eitheJ; from frequencies above resonance (fast waves), or from frequencies below resonance (slow waves). This non-unique representation is possible because, as shown in Figure 13, a significant range of frequencies above bubble resonance, and below bubble resonance, produce an identical range of wave numbers.

The discussion so far has had to do with the prediciton of the propagation of pressure perturbations. Let us now turn our attention to the propagation of void perturbations.

4. THE LINEAR ANALYSIS OF VOID WAVE PROPAGATION

Linear void wave propagation phenomena has been extensively studied for the assessment of two-fluid model constitutive relations [23 ,24,17 ,25]. Moreover, it has been proposed that the void waves may be responsible for flow regime transition [26,27].

The void wave propagation data of Pauchon and Banerjee [28] and Bernier *

[29] are compared in Figure 14 with the root (�j ) of Eq. (54) associated with the

fast void wave (C�). For consistency with previous investigations [28, 17] the virtual volume coefficient, Cvm' was taken to be 0.5 in these evaluations.

The two interfacial drag coefficients given in Eq. (13) and Eqs. (15) were used in conjunction with the two-fluid model. Predictions are also shown on Figure 14 for the dispersion model with no interfacial drag (ie, CD = 0). It can be seen that the drag models for distorted bubbles, Eq. (13), and zero interfacial drag

1 46

300

<a>= 5.00 % RD "'1 .7mm

- - - Near Bubble Reso nance

c: 200 w

I 1 1 I

CD ::E :: Z w > « � .,;

l Oa 2" Slope " C2� = 0.1 0

27T Slope= c;- '" 0.0042

, - - OX1 0· lXl0·

CoJ R 27T

FREQUENCY (Hz)

Figure 13. Dispersion relation for pressure perturbations

(CD = 0) tend to bracket the data. It is evident that the void wave predictions are quite sensitive to the interfacial drag law used.

In order to better understand void wave phenomena, let us consider a one­dimensional two-fluid model for adiabatic and incompressible air/water flow in a constant area duct. Equations (7), (8) and ( 11 )-(27) comprise an appropriate two-fluid model. A more complete model has been derived by Park [34], but it will not be considered herein since it results in qualitatively similar void wave phenomena as the model presented herein. Also, in the study of void waves, the force due to bubble pulsation (FR) can be neglected since it is only important near bubble resonance.

It is convenient to use dimensionless fonns of Eqs. (7) and (8):

(56)

1 .00

0.75

0.50

0.25

Dispersion Model. Eq. (54). Using the Distorted Bubble Drag Law

Dispersion Model, Eq. (54). With CD ", 0.0

<j.> " 0.3 1 8 m/s <j.> = 0. 1 69 m/s <j.> " 0.013 m/s <j.> • 0.0 m/s <jo> " 0 . 1 0 m/s

} B.m'" (1 982)

Pauchon and Bane�ee (1 986)

o 1 0 20 30

<a> (%)

Figure 14. Void Wave Data, "A.* vs. <a>

where, the Lagrangian form of Eq. (7) has been used, and,

t* = �t, uRo

'" g z

uRo

ag =< a >

at = 1- < a >

147

(57)

1 48

Let us now first perform a linear analysis of void wave propagation. When the two-phase system is disturbed about a fully developed steady-state condition, the perturbed variables satisfy:

(58)

(59)

To achieve closure, the constitutive relations must be expressed in terms of the state variables, u;, u* and a.. Thus, we may assume that the right hand side g • of Eq. (59) can be expressed as, of* + of* , where (for Fn· , t Tz or tk• ) : vm Z k w

(60)

Equation (60) can only be used for the algebraic interfacial and wall transfer laws. Other "forces", such as those due to virtual mass (F* ), must be treated vm differently. Using Eqs. ( 16) and ( 18), the nondimensional virtual mass force can be written as:

(61)

Assuming Cvm is a constant, we obtain the perturbation in the virtual mass force as:

• [dOU; . dOU; dOU; • dOU;] of vrn=Cvrn +ug - -uL at 0 dz dt 0 dz (62)

149

If we neglect the interfacial pressure difference for the gas phase (ie, L\p; . >, 1

and the surface tension between phases is neglected (ie, P;. = Pl' >' we obtain, 1 1

* '" * Bpg - Bp l = BL\p li

(63)

If we subtract Eq. (59) for the gas phase from that for the liquid phase to eliminate pressure, and eliminate gradients in the velocity, and the phase average pressure perturbations by using Eqs. (58), and (60), we obtain [36]:

(64)

where,

(66)

(67)

1 50

(68)

o o

(69)

o

Equation (64) can he rewritten in more compact form as:

a&x • a&x • ( a • a )( a • a )0 -. + a+-. + T -. + r_ -. -. + r+ -. = 0 at az at az at az

(70)

where ,

(71a)

(71h)

r* = r±

=_ K4 ± ..!.(K4 )2 _(K5 )

± U Ro 2Ka 4 Ka Ka (71c)

* *

1 5 1

It can be shown [30] that a+ , r ± and T* are the dimensionless forms of the kinematic wave speed, the characteristics and the relaxation time, respectively. It is interesting to note that Eq. (70) is similar to, but more general than, the previous results of Pauchon and Banerjee [17].

Let us now consider the properties of Eq. (70). The dispersion relationship can be obtained by assuming a solution of the form:

I i(K*z*-co*t*) Ba = a e (72)

Inserting Eq. (72) into Eq. (70), we obtain the following linear dispersion relationship:

(73)

* If we consider the region where Eq. (70) is -hyperbolic (ie , where r± are real), the solution of Eq. (73) for traveling waves (ie, where K* is real) can be found by solving the following coupled equations:

where,

*

C. = coR a. •

K

- ( . *) r = r+ + r_ 1 2

(74a)

(74b)

(74c)

(74d)

(74e)

Equations (74b) and (74d) imply that void wave dispersion is pronounced for *

large values of the relaxation time (T*), since the wave speed (Ca) is strongly dependent on angular frequency when relaxation time " is large.

1 52

It is interesting to plot Eqs. (74). As shown in Figures 15 and 16, two speeds

of propagation are possible (ie, C� and C� waves) for a specified frequency.

The faster one (C: ) , is easily recognized as the predominant speed of propagation, and, in the limit as CI>R � 0, it is the so-called [31] kinematic wave

* -speed (a+). However, a complementary void wave (Ca) is also present. The

dispersion relationship implies that the C� wave is slower than C� and, as shown in Figures 15 and 16, has relatively high damping. As a consequence it is not easily measured. The complementary kinematic wave speed, the

counterpart of the classical kinematic wave speed, (ie, C� at zero frequency) can be found from Eq. (74b) as:

(75)

The well-known stability criteria [30] for C� waves can be easily seen, by examining the solid lines in Figures 15 and 16, to be:

(76)

As discussed previously, appropriate constitutive relations must be used to quantify the properties of void waves. Using the constitutive relations in Eqs. (13) - (15), and Eqs. (20) - (24), we can obtain the dimensionless form of the kinematic wave speed in a frame referenced to the liquid phase velocity:

(77)

where,

(78a)

Using Eqs. ( 15) we obtain:

c ; w a ve

--------

-Cex w a ve

•

o I i I i i

w a ve l i i • . r +

o i i i i ____ �_. ___ ___ L._. __ J ___ .1.. _______ _

I

I I ' I - ', \ i , ' , C o: w a ve I " ! i ', !

Figure 15. Plot ofEqs, (74) for r. < a+ < r+ (Stable)

1 54

C ; *

0+

* r + _

* r

• a

WI

+ C ex w a ve .. _-----_ .. _---

I I , , I , , I , , , \ , \

\ \

\ \

O . . . I -.l ._ . . - . -l---.----.1.--.--r-.-. - -t-/ 1 _ ,

I I " i i '\ i " i "

C - i \ ex w a ve i ;

Figure 16. Plot of Eqs. (74) for a+ > r+ (Unstable)

Similarly, Eq. (71c) yields,

where,

. A 2 [Cvm-l1-kao + P;(l-<lo)t u =(Hxo) •

ao {Hxo)�vm +Pg{Hxo)

+2(1 -ao)2(11-Cvm / 2) - p� (1 -aol -ka�(1 ao)

1 55

(78b)

(79)

(80a)

(BOb)

(80c)

It should be noted that except for the last term on the right hand side of Eq. (80c), Eqs. (79}-(80) reduce to the results of Pauchon & Banerjee [17] if we let: fw = 0, p; = 0, Cvm = V2, 11 = V4 and k = VS. The difference reflects the fact that Pauchon & Banerjee [17] assume 'tlj = 0 in their study, where here 'tlj = T

'tzz was assumed. I If we neglect the wall shear stress in Eq. (78b), we find that 1.71 S n S 3.0 for

all possible values of Re2«1>o. To bound the possibilities, the dimensionless kinematic wave speed given by Eq. (77) for n = 1.71 and n = 3.0 is shown in Figure 17 along with the characteristics. According to the stability criteria given by Eq. (76), the kinematic wave is stable (ie, it decays during propagation) over a limited range of Re2«1>o. However, other models may yield different results [Park, 1992]. Since Re2«1>o is proportional to relative velocity, n is increased as the relative velocity is decreased. Thus, the kinematic wave can be stabilized by reducing the phasic slip in the steady flow. It should be noted that since Re2«1>o is proportional to the size of bubbles, small bubbles can also stabilize the kinematic wave.

1 56

1 . 0

* A+t

0 . 5

o

- 0 . 5

A: ( n = 1 . 7 1 ) / A � ( n = 3 . 0 )

ex

A.: ( n = 3 . 0 )

Figure 17. Kinematic wave speeds and characteristics (Cvm = 0.5, 11 = 0.25, k = 0.2, Pg = 0, � = 0)

The dimensionless relaxation time can be found from Eq. (73) as:

T* _ gr _ Fro Re2+o [ao{l-ao}+Cvm+p;{l-ao)2] - u Ro - 18DH/Db [ 1+O.175Reg:: � ao 't ;wo Db 12. 25DH )ae2+0

(81)

As can be seen in Eq. (81), the Froude number (gDH/ui ) strongly influences o

the void wave relaxation time.

1 57

As mentioned earlier, the dispersion relationship presented herein yields a complementary kinematic wave <a). Using the constitutive relations previously discussed and Eq. (75), we find that the dimensionless speed of the complementary kinematic wave is:

A * = a_ - < Ut >10 = i.* + i.* - A * - + - + uRo (82)

.. As shown in Figure 17, the intersection of the kinematic wave speeds (ie: A+ =

A�) occurs at a lower void fraction than where the characteristics intersect (ie: i.: = i.�) when the corresponding a+ wave is stable.

If we use Eqs. (74a) and (82), we obtain the temporal damping of the C� wave as,

(83)

This implies that the C� wave has large temporal damping when the phasic slip is large. Thus, based upon the constitutive relations used herein, it appears to be possible to observe the C� wave only when phasic slip is small.

Naturally, once C�«(J) ) is known we may determine a_ since, lim C� = a_. COR-70

It is also interesting to relate these results to classical kinematic wave theory. A kinematic drift-flux model for void waves has been proposed by Wallis [31] . Significantly, this model is based entirely on steady-state considerations. The celerity of void perturbations is given by,

a _ _ . + - a«x> - J + a<a> (84)

since

we have,

(85)

1 58

Wallis [31] has also proposed an empirical drift-flux relation of the form,

(86)

Thus, Eqs. (84)-(86) yield, *

A + = 1 - n<a.> == 1 - nao (87)

Interestingly, this is exactly the same form as given in Eq. (77). The value of the drift-flux parameter (n) depends on the flow conditions. For

example, in steady bubbly two-phase flow Eqs. (8) & (12) yield:

u [� g(p I - P 1 - <a.> Ro - 3 PI CD

1/2 (88)

For the distorted bubble regime, Eq. (13) can be combined with Eq. (88) to yield,

That is,

uRo = uoo (1 - <a» 3/4 (89)

Hence Eqs. (86) and (89) imply n = 7/4 for distorted bubbly flow. Finally, it should be noted that, using an entirely different analytical

approach, Pauchon and Banerjee [17] have deduced a result for the case of a constant interfacial drag coefficient, which yields, n = 3/2. This result is consistent with the result proposed previously by Zuber and Hench [32].

Thus we find that the void wave celerity given by a drift-flux model (which is based only on steady-state considerations) is closely linked, via the interfacial drag law, to the corresponding linear dispersion results of the two-fluid model which is valid for low frequencies. This clearly shows the importance of interfacial drag on void wave propagation phenomena. Moreover, it implies that a lot of physics associated with interfacial momentum transfer (eg: virtual mass effects, turbulence, etc.) is implicit in the empirical drift-flux parameter, n.

It appears that the linear dispersion relation of a typical two-fluid model is capable of predicting small amplitude void wave propagation data. In contrast, the void eigenvalues generally underpredict such data since these results imply zero interfacial drag, among other things (ie, they imply setting £0 = 0). Moreover, it has also been shown that kinematic drift-flux models of void wave propagation are closely linked to the dispersion relation in which the corresponding interfacial drag law is used.

1 59

It can be noted in Figure 18, Ruggles et al [24] showed that the eigenvalues from Eq. (55), or Eqs. (79)-(80) exhibited a strong sensitivity to Cvm and models for 't i. ' In particular, the domain of hyperbolicity of the two-fluid model is 1 reduced when Cvm is greater than 0.5 and increased when it is less.

Moreover, including the interfacial shear stress, 'ti. ' also reduces the 1

domain of hyperbolicity. Indeed, a significant change can be noted when Cvm S 0 .5. It is also interesting to note that for Cvm < 0.5, the lower branch of A * can be negative (ie, r_ < <ui> io)'

In addition, we note increasing values for A * as the void fraction approaches unity. This occurs because the primary contribution to the liquid pressure gradient is the hydrostatic head due to the mixture density. Thus, the relative velocity, uR = <Ug>g <ui> i' approaches zero as the mixture density approaches that of the gas phase (ie, bubble buoyancy goes to zero). This result is somewhat artificial since the bubbly flow drag law used in the eigenvalue predictions shown in Figure 18 is not appropriate for very high void fractions, nor are some of the other closure laws which have been used. While the same comments apply, this behavior is not seen in the model of Pauchon and Banerjee [17] since no explicit expression for relative velocity was included.

1 .0

0 .8

" . = r-<ul>' <ug>g - <uPl 0.2

'0. 2

0.4

- - - - - - - - - - - - - - - - - - - � 1 a. Cy .. =0.3, TL,�O.O l b. Cy .. =0.3, Pauchon and Banerjee ( 1 988) 1 c. Cy .. -0.3, Te, .T�ZL 2a. Cy .. =0.5,TL;'0.0 2b. Cy .. -0.5, Pauehon and Baneriee ( 1 988) 2c. Cv .. aO.S,rL,=T1zl. 3a. Cy .. =0.7, TL,=0.0 3b. Cy .. �0.7, Pauehon and Banerjee ( 1 988) 3e. Cy .. =0.7,TLi=r'

_1 a

Figure 18. Sensitivity of eigenvalue model to changes in Cvrn (with Reynolds stress)

1 60

Let us next turn our attention to nonlinear phenomena in void wave propagation.

6. NONLINEAR ANALYSIS OF VOID WAVE PROPAGATION

While linear analysis is important, it is clear that the propagation of finite amplitude void waves canot be adequately modeled using linear theory. Thus, in this part of the chapter, the propagation of finite amplitude void waves wil be investigated.

The two-fluid model can be recast into a moving coordinate system using a Galilean transformation. It will be shown that solutions for nonlinear void wave profiles include, solitons, shocks and rarefactions.

If we introduce a new coordinate system which is moving at celerity Cs' we obtain the Galilean relationship between the physical and the moving coordinate variables as,

Thus, Eqs. (7) and (8) can be rewritten in the moving coordinate system as,

d<uk>k Pk( <Uk>k - Cs) d�

(90)

(91)

(92)

If we subtract Eq. (92) for the gas phase from that for the liquid phase, and eliminate the phasic velocities using Eq. (91), we obtain:

Pl - P gCos8 + � _ _ W

-{ ) Mt· Mg. 4 ( 'tt 'tgw ) g l- < a > < a > DH (l- < a » < a > (93)

1 6 1

where,

II <a> = <ag> = (1 - <at»

Using Eqs. (14), (16) and (18), the two-phase interfacial momentum transfer in the moving coordinate system is:

(94)

Inserting the closure laws, Eq. (94) and Eqs. (20) (24), we obtain a single equation which quantifies the void fraction gradient in the moving coordinate system:

where,

P;[( I- < a » +cvrn](

)2 + C - < u > < a > ( 1- < a > ) s g g

(95)

(96a)

1 62

* A Pg = pip, (96c)

5.1 Charactecs, Shocks and Wave Spe It is interesting to note that the two-phase system's characteristics (ie.

eigenvalues) satisfy the following equation:

(97)

Solving Eq. (97) for the wave celerity. CSt relative to the liquid phase velocity. we obtain the dimensionless characteristics as,

A. = V ± ± Ug - ul

where, V*, u* and 'C* have been previously defined in Eqs. (80).

(98)

The characteristics given by Eqs. (98) are the same as those in Eqs. (79). and reduce to those obtained by Pauchon & Banerjee [17] if we let Cvm = 112. 11 = 114.

* k = 115 and P = o. g If we assume that H« a>.<ul>l,<ug>g'Cs) � 0 in Eq. (95), the void fraction

gradient vanishes when,

(99)

Thus, the roots which satisfy Eq. (99) can be recognized as the steady-states. To identify the steady-states given by Eq. (99), we must consider phasic

continuity. If we add Eqs. (91) for each phase, we obtain, d

d� [«x><ug>g + (1-<a.» <ut>l] = 0 (100)

Integration of Eq. (100) directly yields the result that the total volumetric flux (j) is a conserved quantity in the �-a plane. That is,

(101)

We may obtain another conserved quantity, K, by noting that Cs is a constant, and integrating Eq. (100) for the liquid phase to obtain,

<a>Cs + (1-<CD)<Ul>l = K (102)

Using Eqs. (101) and (102), we obtain the phasic velocities in terms of the void fractions and the conserved quantitie�:

<Ut>l = (K - <a.>Cs)/(1 - <CD) (103a)

1 63

<ug>g = Cs + (j-K)/<a> (103b)

Inserting Eqs. (103) into Eq. (96b), we obtain:

G 1 f Cs - K J - K f K- < a > Cs { { . ]2 [ ]2}

< a > = . < a > + -- -( ) l ( (l- < a » < a > W (l- < a »

II< - (1 - Pg) g CosO (104)

Unfortunately, an analytic expression for the zeroes of G« a» is not possible. However, it has been found numerically that G« a» has two zeroes within the range of void fraction « a» from zero to unity, for all plausible values of j, K, Cs and interfacial drag laws. We designate these two zeroes by a1 and a2'

Since j and K have been found to be conserved quantities we have, from Eqs. (101) and (102),

(105a)

(105b)

Solving Eq. (l05b) for Cs, we obtain:

(106)

It is interesting to note that the celerity given by Eq. (l06) is the same as the continuity shock speed derived by Wallis [31]. Also, if we neglect wall friction, the shock speed referenced to the liquid phase velocity can be found from Eqs. (104), (105a) and (106) as:

(107)

Thus, Eq. (107) can be used to determine the shock speed (Gs) when a step change of the void fraction (eg, from a1 to a2) is specified.

As a special case, the linear kinematic wave speed can be found from Eq. (107) when the void fraction change is small enough. That is,

al (l - al) dfi l 2fi(al ) da «1

(l08)

1 64

Interestingly, this is the same result as given by Eqs. (77) & (78) for the special case of no wal shear, and an 4 which is only a fraction of a.

5.2 NonBnear Void Wave Proftles As mentioned previously, phasic velocities in Eqs. (96) can be eliminated by

using Eqs. (l03). Therefore, we may obtain another form of Eq. (95) where void fraction is the only dependent variable:

d<a> H« a» <IC = O« a» (109)

where, H( <a» can be obtained by inserting Eqs. (l03) into Eq. (96a) and 0( <a» is given by Eq. (104).

Integrating Eq. (l09) by separation of variables, we obtain an implicit expression for the void fraction profile in the �-a plane:

[a>

H(a') � = --, da' <ii> G(a )

where,

a1 < « a» < a2 and � is taken so that, � = 0 at <0.> =a.

(110)

Thus, the void fraction profile determined by Eq. (110) propagates at celerity Cs, given by Eqs. (107) or (112) when the void fraction is changed from a1 to a2 (or, from a2 to a1) without a variation in the total volumetric flux, j.

The kinematic wave speed is the first order celerity which controls the propagation of void fraction disturbances for nondispersive conditions [23]. As seen in Eq. (l08), the kinematic wave speed depends on the interfacial friction factor. As shown in Figure 19, when we plot the kinematic wave speeds along with the characteristics, we find that a distorted bubble drag law [11], n = 1.75, yields unstable kinematic wave propagation based upon the linear stability

* • criteria given by Whitham [11], A._ < A* < A.+. However, the kinematic wave is

stabilized when an undistorted drag law [33], n = 2.5, is used [23]. Some related properties between the stability of the kinematic wave and the nonlinear void wave will be discussed later.

5.3 Void Waves and Their Stability To determine the nonlinear void wave profile using Eq. (110), we need the

conserved quantities, j and K, and the speed of propagation, Cs' when the void fractions at the initial and the final states are specified.. Since the total volumetric flux is normally known, K and Cs remain to be determined. These

1 . 0

A * J.. . +

0 . 5

o

\

,

, ,

* A ( n = 2 . 5 )

;.* +

* A ( n = 1 . 7 5)

0 . 5

165

ex

Figure 19. Kinematic wave speeds and characteristics (Cvm = 0.5, 11 = 0.25, k = 0 .2 , p* = 0, 'tL* = 0, A* = I-no.) g w

quantities can be obtained if we obtain the steady-states , by solving G(a.l) = G(a.2) = 0 simultaneously, resulting in:

(111)

1 66

Before proceeding further, it should be noted that the nonlinear void wave profile breaks (ie, the solution is multivalued in the �-«l> plane) when,

(113)

where, the eigenvalues are given by:

H(<lol) = H(<lo2) = ... = 0

Possible nonlinear void wave profiles, which depend on the integrand of Eq. (110), are shown in Figure 20. When the zeroes of H« a» satisfy Eq. (113), the void wave profile breaks as Type B, C, E or F shocks. Wave breaking occurs when Cs is between the larger characteristic speed of the initial state and that of the final state [30], that is,

when a1 > a2. If we rearrange Eq. (114) by using Eqs. (103), (111) and (112), we obtain the condition for nonlinear void wave breaking (ie, shock formation) as,

Based upon the above wave breaking criteria, different types of void wave profiles are categorized in Figure 21. As can be seen, Cs decreases as a2 increases for a specified initial state (a1). This result can be easily understood if we realize that these shock (or rarefaction) solutions are based on a kinematic condition which implies a reduction in wave speed with void fraction (ie, note that dA * Ida < 0 in Figure 19).

If we consider the region, a1 > a2 and a1 > ac in Figure 21, we see that solitons, that is smooth void wave profiles (ie, type A or D) break when the void fraction change, a1 - a2, is large. The evolving void wave profiles for different values of the void fraction change are shown in Figure 22.

1 67

Typ e

Typ e B

Typ e C

Typ e 0

Typ e E

Typ e F

Figure 20. Types of nonlinear void wave profiles

In the region, (1 < ac in Figure 21, there exists a region where small amplitude void wave solutions are not possible based on the present technique. Interestingly, <Xc is the point where the kinematic wave speed and the faster characteristic coalesce, as shown in Figure 19. Since, based upon the linear stability criteria, A.: < A * < A.:, the linear void wave is unstable where 0 < a < ac, we find this result is more general. It should also be noted that the nonlinear void wave solutions are also not possible in the region where the initial and the final void fractions are large.

None of the solutions in the region, a 1 < a2 , are stable [30] since the upstream celerity (ie, the kinematic wave speed) (ie, A * for a = (Xl) is smaller

1 68

0 . 5

CXz

, , , , ' c ' '- Typ e " s = J

'- 8 or C '-, , , , " ,)+ 5b ,

, " " j+ 1 a " , .. "

Typ e ' .. .. E or F .. .. .. .. ..

"" ' l

" " "' , .. .. ..

..

I ' " o

<Xc 0 . 5

Figure 21. Different types of nonlinear void waves (Cvm = 0.5, 11 = 0.25, k = 0.2, P * = 0, n = 2.5) g a : No solution i s possible b: All units are in em/sec

than that for the downstream conditions (ie, A * for <a> = a2). Indeed the solutions are rarefactions, such that if void waves are generated by increasing the' void fraction, they will decay out.

As can be seen in Figure 23, if we use a distorted bubble drag model (eg, n = 1. 75), smooth wave profile solutions (ie, Types A or D) are not possible. This result suggests that the increased interfacial drag associated with bubble distortion promotes discontinuity in the two-fluid variables before and after the transition.

The effects of the virtual mass force, the interfacial pressure difference and the bubble-induced Reynolds stress are shown in Figure 24. From these results we see that the virtual mass force is crucial in determining the behavior of nonlinear void waves. In particular, the smooth wave profiles in the high void fraction region totally disappear when the virtual volume coefficient is increased from 0.5 to 0.6. In contrast, increasing the interfacial pressure difference and/or the Reynolds stress increases regions A and D monotonically, which means that these effects reduce the void fraction gradient between the two steady-states. It should be noted that when using more detailed two-fluid models, it has also been found [34] that void wave phenomena is very sensitive to the closure laws used, however the results are quantitatively different.

It has been shown that a linear dispersion model appears to be able to quantify void wave dispersion. According to this model, void wave dispersion is pronounced for large values of the relaxation time (T), which is determined

1 69

ex

CX2

0. 2 - 0 . 1 o 0 . 1

Figure 22. Breaking of void waves (Cvrn = 0.5, " = 0.25, k = 0.2, p* = 0 , n = 2.5) g

by the Froude number, the virtual mass coefficient and the void fraction. Since the relaxation time is small for large values of the relative velocity (ie, where the Froude number is small), void waves in a stagnant pool of liquid can be successfully described by a kinematic wave model. However, since relaxation time increases as relative velocity decreases (ie, the Froude number becomes large), void waves become more dispersive when slip is reduced. The virtual mass effect also promotes void wave dispersion since the relaxation time is increased as the virtual volume coefficient (Cvm) is increased.

It was found that nonlinear void wave speed increases as the amplitude of the wave increases. Moreover, the nonlinear void wave speed turns out to be a generalization of the linear void wave speed.

Different types of void wave profiles have been predicted. Unfortunately, the existing void wave propagation data exhibits significant scatter and does not always include the measurement of wave amplitude and attenuation coefficient. Furthermore, the existing data for C� is not always accompanied by sufficient supporting information on the flow state, or the nature of the void perturbation (ie : frequency, amplitude, etc.), thus, complete two-fluid model assessment is not possible at this time. Nevertheless, it appears that carefully

1 70

0 . 5

C(2

o

Typ e B o r C

Typ e E o r F

CX 1 0 . 5

Figure 23. Different types of void waves with the distorted bubble drag models (Cvm = 0.5, Tl = 0.25, k = 0.2, p; :: 0, n = 1.75)

taken void wave propagation data can be a powerful tool for assessing two-fluid interfacial momentum transfer laws due to the relatively large sensitivity in the void wave speed, C! , to the interfacial closure laws. Indeed, it appears

that the analysis of void wave phenomena is very valuable for the assessment of two-fluid models.

6. NOMENCLATURE

a Acceleration avm Virtual mass acceleration c Speed of propagation (celerity) Cvm Virtual volume coefficient D Diameter f

M 'al d ' . a( ) aten envatIve, at a( )

+ uk az Frequency; friction factor

1 7 1

0. 5 K t �ype . 5 . 25 .2

. 2 5 --e- . 2 -+- . 2 5

0( 2

I

N

Type E or F

o 0. 5

Figure 24. Dependency of void wave types on the two-fluid model's constitutive relations (p* = 0, n = 2.5) g

F Force CI}ri Interfacial heat transfer rate g Gravitational acceleration Eigenvalue Gravitational constant r gc R Radius h Enthalpy STP One atmosphere, 23°C

Imaginary number, {T T Temperature jk Superficial velocity of phase-k t time jgL Drift-flux u Velocity

z Axial location k Wavenumber Mki Interfacial momentum

transfer � n Drift-flux parameter a Void fraction llLS Interfacial area density S( ) Perturbed quantity p Pressure A. Wave length

Ii Dynamic viscosity

1 72

(Jl Angular frequency p Density IC Reynolds stress parameter � State vector (J Surface tension a Angle of inclination of flow

from vertical t Stress or parameter defined

in Eq. (BOb) '\) Kinematic viscosity or

parameter defined in Eq. (80c) � Interfacial pressure

distribution parameter

b 2$ 1$ g H

Bubble Two-phase One-phase Gas Hydraulic Interfacial

REFERENCES

k Phase indicator (g = gas, 1 =liquid)

1 Liquid o Equilibrium (steady-state)

value vm Virtual mass w Wall

Terminal rise (velocity)

( )T Transpose of a vector matrix

� Defined as v O( ) O( ) d( ) dO

Gradient

Partial derivative

Total derivative

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28 C. Pauchon and S. Banerjee, Int. J. Multiphase Flow, 12, No. 4 (1986) 559-573.

2} RN.J. Bernier, Report E200.4, Division of Engineering and Applied Science, California Institute of Technology, 1982.

3) G.B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, 1974. 31 G.B. Wallis, One-Dimensional Two-Phase Flows, McGraw-Hill, 1969. 32 N. Zuber and J. Hench, General Electric Report No. 62 GL 100 (1962). 33 M. Ishii and K. Mishima, Nuclear Eng. & Des., 82 ( 1984) 107-126. 34 J-W. Park, Void Wave Propagation in Two-Phase Flow, Ph.D. Thesis,

Rensselaer Polytechnic Institute, Troy, NY, 1992. :Ii M. Lopez de Bertodano, Turbulent Bubbly Two-Phase Flow in a

Triangular Duct, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY, 1992.

:J) RT. Lahey, Jr. , AIChE Journal, 37(1) (1991) 123-135.

1 75

CRITICAL FLOW: Basic Considerations and Limitations in the Homogeneous Equilibrium Model.

Owen C. Jones

Professor of Nuclear Engineering and Engineering Physics

Rensselaer Polytechnic Institute, Troy, NY 1 2 180-3590

Abstract The basic equations of homogeneous equilibrium flow are presented. The general equation for the sonic velocity is developed and presented for water in the two-phase region. Discontinuities at 0% and 100% quality are discussed. Limitations in the flow rates implicit in the homogeneous equilibrium model are presented in the context of a single-phase inlet to a decompressing zone for Fanno flow, Rayleigh flow, and isentropic flow. It is shown that with a subcooled inlet, the

maximum velocity in a straight pipe or c<?nverging pipe is the two-phase sonic velocity at vanish­ing quality and that this condition occurs with the saturation line placed at the minimum pressure location.

1. INTRODUCTION

Choking and critical flows are of importance in many instances in the process, chemical, and power industries. This is especially true where multiphase flows are possible since the formation of vapor in a liquid leads to significantly IQwer sonic velocities and larger frictional losses than in either phase alone. Resultant critical flow rates are thus substantially reduced from corresponding single-phase values.

When a liquid is decompressed, either temporally or spatially, fluid particles may reach a point where the local pressure decreases below the saturation pressure according to the local tempera­ture. Such a circumstance is depicted in Fig. l. Generally, this local temperature is sufficiently close to the inlet (stagnation) temperature that the diferences are ignored. Thermodynamic con­siderations would suggest that as the pressure is further reduced, there would be vapor formation which could be determined directly from the equation of state of the fluid coupled with a suitable assumption regarding the decompression process itself: isentropic, polytropic, isothermal, etc.

Furthermore, once the vapor is formed, some assumption must be made regarding the relative motion between the phases. Should the relative motion be ignored, not an urrealistic assumption considering the usually high velocities of flashing flows, and the phase change determined by

thermodynamic considerations alone, tle resultant model would be termed the homogeneous equilibrium model or HEM. Further, if tlle expansion is assumed to occur in an isentropic manner than the model would be termed the isentropic homogeneous equilibrium model, or IHE model.

1 76

w a: ::J en en w a: 0.

L

r- �a11 R2

z = o

- ct

I

il I �P I SUp. 0

Figure 1 . Schematic of flashing in a

converging-diverging geometry.

The subject of homogeneous eqUilibrium is the main topic of this chapter. There are some quite instructive things which can be learned through consideration of the equations which govern the flow in such

cases, things which bear directly upon the

actual behavior of decompressing liquids. For this reason, the first part of this chapter

shall deal with the limitations inherent in the

thermofluid mechanical field equations. This chapter will present the basic equations,

present the variations in sonic velocity from

the equation of state, and discuss limitations in the flow through pipes and nozzles for subcooled inlets.

2. HOMOGENEOUS EQUI­

LIBRIUM

There are several approaches to the

development of expressions for sonic

velocity in fluids. One enlightening method is to develop an expression for the pressure

gradient from the momentum equation, and

then find the conditions where the pressure gradient becomes singular--that is, infinite.

For homogeneous flow, this process is quite straightforward, and virtually identical to that for single-phase fluids. The only difference is to recognize that quality or void fraction plays a role.

The continuity and momentum equations for quasi-one-dimensional, steady flows may be

written as

d -(eAv) = 0 dz and

dp = -..!.�(eAv2) - iwf dz A dz A

( 1 )

(2)

where e is density, z is the flow direction, fw is wall shear stress, � is the wall perimeter, A the

cross sectional area, and v the velocity of the mixture. The mechanical and caloric equations of

state are

� = �(P, s) and i = i(P, s).

where p is pressure, s is entropy, and i is enthalpy. Equation (3) may be written as

dp ; �v2 dA dv2 de - - = Tw-- -- - e - - vl -dz A A dz dz dz

which. using the continuity equation may be rearanged to read

dp ; ev2 dA de - - = Tw- - -- - vl -dz A A dz dz.

Using the mechanical equation of state and differentiating using the chain rule yields

_ dp {l _ v2( iJ� ) } = Twt- �v2 dA

_ V2(�) ds dz up s A A dz iJs P dz

1 77

(3)

(4)

(5)

(6)

where the subscripts represent differentiation keeping the subscripted variable constant. Thus,

(7)

Note that if the denominator in (7) vanishes, the pressure gradient becomes singular. The sonic velocity and Mach number are thus defined as

and (8)

so that (7) becomes

(9)

Thus. if the Mach number becomes unity, the pressure gradient becomes infinite.

1 78

Up to this point, there has been no difference from standard, single-phase flow analysis. If now the density is considered a function of void fraction, a., and the specific volume, u, is considered a function of quality, x, then the sonic velocity c may be written as

(10)

where X is the quality. The sUbscriptsf and g represent saturated liquid and vapor respectively, andfg represents the change between saturated liquid and vapor. This equation can also be written as

1 { de! de g ( iJa ) } - = ( l - a) - + a - + {!!g -r? dp dp dp s ( 1 1 )

where i t i s understood that the saturation properties are functions of one variable only.

Byrecalling the expression relating quality and void fraction, Eq. (10) may be written in terms of void fraction and sonic velocity as

( 12)

For two-phase homogeneous flows, it is seen that the inverse of the sonic velocity can be con­sidered a void-weighted average of the inverse sonic velocities of liquid and vapor, reduced by a

compressibility factor {!fg(da / de)s . It is not uncommon to read of a "frozen sound speed." This is

the value obtained by neglecting the compressibility due to phase change as a result of pressure change.

The values of isentropic homogeneous sound speed are shown for water as a function of tem­perature in Fig. 2 for lines of constant quality, and in Fig. 3 versus pressure. Similarly, for lines of constant void fraction, the sonic velocity is shown versus pressure in Fig. 4, and as a function of void fraction for lines of constant pressure in Fig. 5.

Substantial differences are seen between the values in the quality region and single-phase val­ues, even as the limiting values of quality or void fraction are approached. This is because the compressibility resulting from the phase change processes results in large jumps in sonic velocity at the liquid or vapor saturation lines, especially at the former. These jumps are seen in Figs. 6 and 7.

All these curves were determined using the 1967 IFC international formulation for the equa­tion of state for water. Since there is an anomaly in the state equation at a temperature of 350 C

Figure 2. Sonic velocity for water: temperature-quality behavior.

where the slopes are discontinuous, non­physical results are obtained in this region. This is seen most dramatically in Fig. 4.

Consider that for single-phase fluids one can write

while for a two-phase mixture

(�) = e2( dP )-uS P dt In both cases,

( up) < 0 uS S and ( uv ) > 0 uS P

( 14)

(15)

except for liquid water of temperature be­low 4C which is not of concern here. We

can, therefore, assume that (ue / uS)p is al­

ways negative. Let

( 16)

and

( 17)

and

Bs = V2( iJe ) . uS P (18)

1 79

10- 1

50 100 150 200 250 300 350 Temperature (OC)

25 50 75 100 125 150 175 200 225 Pressure (bar)

FIGURE 3. Sonic velocity for water: pressure-quality behavior.

1 80

500

450

400

-. 350 Vol 8 '-" 300 � 1: =:l 0 250 C/.)

4-0 0 200 � 0 0 c. 1 50 C/.)

100

50

0 0 25 50 75 100 125 150 1 75 200 225

Pressure (bar)

1()3

= 220 bar 102

� SO

20 ;;- 101 10 1: =:l 0

C/.) 4-0 0 � 0.5 0 0 c. 0.2 C/.)

0.1 10-1 0.05

0.02 om

10-2 10 5 10-4 10 3 10 2 10- 1

Void Fraction FIGURE 5. Sonic velocity for water:

void fraction-pressure behavior.

Figure 4. Sonic velocity for water: pressure-void fraction behavior.

Then, Eq. (9) becomes

Equation (19) is a general equation re­lating pressure gradient to Mach number, area change, heat addition, and friction. It will be shown in what follows that this equation is useful in both two-phase criti­cal flow as well as single-phase gas dy­namics.

3. LIMITATIONS IN THE HO­

MOGENEOUS EQUILIBRIUM

MODEL

The balance of this chapter is taken from the thesis and paper of Yang ( 1984, 1985). Fano flow, also called flow with simple friction, is the case where wall fric­tion is the sale mechanisms which makes flow become choked. This flow can be realized by letting fluids flow through an adiabatic duct of constant cross section with wall friction. In this case, Eq. ( 1 9) be­comes

_ dp Bf+ Bsds/dz dz [l �]

(20)

According to the second law ofthenno­dynamics, ds > 0 when only wall fliction

Figure 6. Sonic velocity for water: discontinuous behavior at the

liquid-two-phase line.

is present. As a result, the numerator of the right-hand side ofEq. (20) is always posi­tive. From inspection of Eq. (20). it is ap­parent that

dp < 0 when M < 1 . dz

and

dp > 0 when M > 1 . dz

(2 1 )

(22)

Thompson ( 1972) demonstrated that in Fanno flow, dp and dv are always of

opposite signs. From the continuity equa­tion,

d«(}v) = 0

so that

dv v - = - > 0. dv v

(23)

(24)

Combining Thompson's result and Eq. (24) yields

"d C g CI) 4-0 o

1 8 1

1400

800

600

400

200

o 25 50 75 100 1 25 150 Pressure (bar)

1400

1200

800

dv 0 - <

dp . (25) "2 600

In what follows. it will be useful to con­sider two separate sonic velocities and

Mach numbers. Clq" c2.<p. M1qh . and

M2.<p The subscripts 1 cp and 2cp refer to the

single-phase liquid region and the two­phase region respectively. Due to phase­change compressibility, clifJ < ClIP at sat­

uration (Fig. 6).

8-CI)

FlGURE 7. Sonic velocity for water: discon­tinuous behavior at the two-phase-vapor line.

1 82

Now, examination of Eqs. (20). (21 ), (22), and (25), indicates that friction always drives the flow towards unity Mach number, an accepted realization in gas dynamics. Furthermore, consid­ering flow at the saturation line, if the two-phase Mach number exceeded unity, the pressure downstream would increase as shown by Eq. (22). and the velocity would decrease as shown in Eq. (25). However, the differential specific volume neglecting liquid compressibility is

Taking the limit on the two-phase side of the saturation line for vanishing quality indicates that for dv to be negative, dx must also be negative. causing condensation of any vapor formed.

From the preceding, then, two observations for Fanno flow including both single-phase and two-phase regions include:

1 . It is impossible for subsonic Fanno flow to become supersonic, or for supersonic Fanno flow to become subsonic through a continuous process. Friction in the absence of other effects will always drive the flow towards unity Mach number.

2. It is impossible in the case of simple friction in a duct for the liquid velocity at the satura­

tion line to exceed the two-phase sonic velocity, c2cfJ . at the same condition. Thus. for

flashing to occurvJ �(x=O ) � C2�(X=O +), . and the flow upstream must be everywhere sub­sonic in the constant area duct.

Since the pressure gradient upstream of flashing is linear, a decrease in stagnation conditions will cause the flashing front (saturation line) to move upstream. With fixed exit pressure, the flow rate must also decrease. Therefore, a third observation for Fanno flow is:

3. Maximum flow with simple friction will occur with flashing inception located at the exit of the constant area duct.

Rayleigh flow, also called flow with simple heat addition, can be realized by letting fluids flow through frictionless channels with heat transfer. Since the focus is on two-phase flows with sub­cooled inlet. only heat addition is of interest here.

Now heat addition is the sole mechanism that alters flow velocity and pressure. Equation ( 19) is simplified to

dp Bsds/dz - dz = [ 1 - M2J . (27)

In a heating process, ds/dz > O . As a result, the numerator of Eq. (27) is positive, and Eqs.

(2 1) and (22) also hold for Raleigh flow. If reversible heat addition is assumed,

Tds = di - vdp (28)

and the stagnation entropy can be written as

Tds + vdp + vdv = dio.

But in a reversible process,

Tds = dQ = dio

so we have

vdp + vdv = O. This gives

dv 0 - <

dp

for the Rayleigh flow process, identical to Fanno flow.

1 83

Now, if a subsonic liquid were to enter a constant area duct with a velocity greater than the

two phase sound speed at x = 0+ , heating, which leads to boiling would cause an increase in pressure immediately after saturation according to Eq. (27). Equation (32), then, requires a de­crease in the velocity while the continuity equation, Eq. (24), equally valid for Rayleigh flow, then requires a decrease in specific volume, as does the momentum equation. This again can only happen with a return to single-phase conditions . Using arguments similar to those for Fanno flow, similar conclusions are reached:

1 . It is impossible for subsonic Rayleigh flow to become supersonic or vice versa through a continuous process. Heating in the absence of other effects will always drive the flow to­ward unity Mach number.

2. It is impossible in the case of simple heating in a duct for the liquid velocity at the satura­tion line to exceed the two-phase sonic velocity, C2tP , at the same condition. Thus, for boil-

ing to occur, Vltp(x = 0-) S C2tjJ(X = 0+) , and the flow upstream will be everywhere sub­

sonic in a constant area duct.

3. Maximum flow with simple heating will occur with boiling onset located at the exit of the constant area duct.

Equation ( 19) for the case of simple area change has Bf = Bs = 0 and so

dp BadA/dz - dz = [ 1 - M2] .

Rearanging this equation results in

(33)

1:84

(34)

A simple area change case, (Le., isentropic flow of initially subcooled liquids through a con­verging nozzle will be discussed in what follows.

For the case of a simple area change, it seems useful to discuss the behavior in two stages: firs t considering only an incompressible, always subsonic liquid; second considering the possibility of liquid compressibility and possible supersonic liquid conditions.

Incompressible liquid.

(a) Converging duct, dAldz < 0

For the case of an initially subcooled liquid, being incompressible has required that dA/dp is

positive resulting in decompression. At the saturation line, if Mu/x=O+) is larger than unity, re­

compression would commence eliminating the possibility of further expansion and vaporization. Therefore, a liquid velocity exceeding the two-phase sonic velocity at saturation can not exist if flashing is to occur.

(b) Diverging duct, dAldz > 0

In this case, if a subcooled, incompressible liquid were to exist in the diverging section of the

nozzle, dA/dp would be again positive and pressure would increase in the flow direction. Satura­

tion could not be encountered under these circumstances.

(c) Constant area, dA/dz = 0

In this case, for M *" 1 , dp/dz = 0 and no change occurs. In the case where M=l, Eq. (34) indi­

cates dp = 0 also.

A controversy arises for the situation where the saturation line exists immediately at the throat. Can the liquid velocity exceed the two-phase sonic velocity at the saturation line in this case?

Since the case in question represents a singular situation, previous arguments are not valid.

Rather, the following logical process will be utilized:

1. Choose an arbitrary set of conditions;

2. Prove that an exclusion zone exists completely surrounding these conditions which are im­

possible to achieve;

3. Conclude that since the initial conditions were completely arbitrary, that they, too, are im­

possible to achieve.

For the purposes of this argument, and in the context of Eq. (34), consider the possibility of a liquid expanding in the converging portion of a converging-diverging nozzle. Let the stagnation

pressure and temperature, Po and To, be appropriate so that the throat pressure p, is just at satura­

tion according to the throat temperature, T,: i.e., letp, = Psa,(T,). Assume that the flow rate is suffi-

1 85

ciently large that the liquid-side throat velocity, Vlt(x=O ), is larger than the two-phase sonic ve­locity at x=O, C21j1(X=O+). but still smaller than the liquid sonic velocity at throat conditions. Clt(x=O ): i.e., assume the inequality C

Now imagine that small perturbations in stagnation pressure and temperature occur according to the following situations: C2,(X=O+) < v,dx=O-) < CII(X=O-) .

• Cases 1 & 2: Decreasing stagnation pressure by -5po or increasing stagnation temperature by 5To•

Both cases wold tend to cause the flashing location to be perturbed upstream of the throat into the converging section. Thus, v21j1(x=O+fu), the two-phase velocity located between the flashing front. zf. and the throat location, Zto would exceed the two-phase sonic velocity,

v2I/I(x = 0 + dx) > c2I/I(x = 0 + dx). According to Eq. (34), with M > 1 .0, the pressure would in­

crease. But increasing pressure would recompress any vapor formed keeping the flow liquid. A direct contradiction arises between the decreasing pressure requirement of subsonic single­phase flow and the increasing pressure requirement of supersonic two-phase flow, yielding an impossible situation.

• Cases 3 and 4: Increasing stagnation pressure by -fipo or decreasing stagnation tempera­ture by 5To.

Both cases would cause the throat pressure to exceed the saturation pressure, PI > PsarC.T I) •

The liquid, itself being subsonic at the throat would, therefore, recompress downstream of the throat with increasing pressure. The flashing condition would never be reached.

Compressible liquid. The restriction discussed above did not take into consideration the pos­sibility that supersonic liquid conditions could be encountered (see Jones and Shin. 1983). Con­sidering the liquid compressibility, it is shown in Fig. 8 that withpi greater thanpsadTo) (curve 1). liquid sonic condition can be achieved with a very high stagnation pressure. Continued liquid ex­pansion would occur downstream of the throat until saturation is reached. Beyond this point, flashing would continue with two-phase mixture hypersonic with regard to two phase sonic con­ditions. This is not a violation of the previously stated hypothesis since dAldp is negative and at the same time (1 - Ml) is also negative. Eq. (34).

A reduction in stagnation pressure would result in a reduction in PI until the saturation condi­tion is brought into the throat (curve 2). This condition is, in fact, a violation of the restriction previously mentioned and results from the inclusion of the liquid compressibility and the ultra­high pressures considered. It must be noted that in al cases the flow rate would be virtually con­stant due to liquid-sonic limitations at the tlu'oat.

A fwtler reduction in stagnation pressure yields a conundrum and the restrictions discussed in the preceding section would apply. If flashing moves upstream, converging, two-phase flow would result which is hypersonic with respect to two-phase sonic conditions. dA/dp would be negative since M21j1> > 1 . and the pressure would increase, recompressing the liquid. Hence, it is not possible for upstream flashing to occur under these circumstances given constant flow rate.

1 86

1 0

9 Limit of two-phase sonic throat condition - 2 m. m)< m1c I t l

7 m2< m. m2

5 m3

Limit of onset of

30 upstream flashing -.".,.

m< �lc"(lI = 0+)

20

Upstream flashing

10 Ar A* = 1 .35

0 10 5 5 10

Area Ratio - AlA * FIGURE 8. Pressure profiles for flow of compressible liquid

and two-phase mixture of steam and water in a nozzle.

On the other hand, a reduction inflow rate to keep Pt = PSaJ would result in a reduction in throat velocity below the liquid sonic velocity, but still much above the two-phase sonic velocity atx =

0 +. This condition is unstable according to the previous discussion. The only alternative is for the liquid to recompress downstream of the throat as a single-phase liquid.

Further reduction in Po causes commens urate reduction in flow rate until the single-phase liq­uid throat velocity is identical to the two-phase sonic velocity atx = 0 +, and the previously identi­fied conditions are matched (curve 3). It is obvious that further reduction in Po can move the flash­ing inception point upstream.

In summary, it is seen for the case where the stagnation pressure is lower than that required to achieve liquid-sonic throat conditions, there is a space surrounding the stagnation conditions for

1 87

which the possibility ofliquid velocity greater than the two-phase, zero quality, sonic velocity is excluded as long as the liquid itself remains subsonic. Thus. with flashing at the minimum area, the velocity of saturated liquid can not exceed the sonic velocity of a two-phase mixture at van­ishing qUality.

4.GENERAL SUMMARY

The concept of flashing from initially subcooled liquids is placed into context from the view­point of the homogeneous equilibrium model. It is shown that for isentropic flow, Fanno flow, and Rayleigh flow that the maximum flow rates possible in straight ducts are those where flashing begins immediately at the exist of the duct, and that the exit liquid velocity can never exceed the two-phase sound speed at zero quality.

5. NOMENCLATURE

See the nomenclature in the next chapter.

6. REFERENCES

1 . Yang. 1., ( 1984) . Homogeneous Equilibrium Considerations of lnitially Subcooled Two­

Phase Flows of Water. M.S. Thesis, Rensselaer Polytechnic Institute, Troy, NY. August.

2. Yang, J., Jones, D.C., and Shin, J.T., 1986. Critical flow of initially subcooled flashing liquids: limitations in the homogeneous equilibrium model. Nuel. Eng. Des. 95, pg. 197-206.

1 89

NONEQUILIBRIUM PHASE CIlANGE--l. Flashing In­ception, Critical Flow, and Void Development in Ducts

Owen c. Jones Professor of Nuclear Engineering and Engineering Physics Rensselaer Polytechnic Institute, Troy, NY 12180-3590

Abstract

This chapter will introduce the first concepts of nonequilibrium with emphasis on nucleation and flashing in nozzles, and subsequent void development. Both number and size of bubbles must be accurately determined for initial calculation of flashing void development downstream of re­strictions. Void development upstream of a restriction is negligible if the inlet is subcooled. A new method is presented of accurately determining number of nuclei and their size, and shows this results in accurate calculation of downstream void development.

The activation criterion developed for site nucleation, different from that developed for sub­cooled boiling, is one sided due to the uniform superheat. A priactical method of utilizing the acti­vation criterion is then identified. A defined figure-of-merit for the particular fluid solid combi­nation yields minimum nucleation surface energy per site, and allows characteristic site nucle­ation frequencies, and number densities of nucleation sites of given sizes to be obtained from data.

A bubble transport equation is used to predict the number density and size of bubbles at the throat. Throat superheats are calculated with a standard deviation of 1 . 9K for throat superheats up to -lOOK and expansion rates between 0.2 barfs to over 1 Mbars, extending previous correlations by more than three orders of magnitude. Throat void fractions for al data found in the literature are less than 1 % confirming earlier ass umptions and allowing nozzle critical flow rates to be cal­culated with an accuracy of -3%. Accurate calculation of throat superheats allows for correct cal­culation of throat pressures up to -70 bars below saturation, a critical factor in calculating choked flows.

A quasi-one-dimensional, five-equation model, developed on a microcomputer, was used to calculate the behavior of flowing, initially subcooled, flashing liquids. Equations for mixture and vapor mass conservation, mixtw"e momentum conservation, liquid energy conservation, and bubble transport were discretized and linearized semi-implicitly, and solved using a successive iteration Newton method. Closure was obtained through simple constitutive equations for fric­tion and spherical bubble growth, and a new nucleation model for wall nucleation in small

1 90

nozzles combined with an existing model for bulk nucleation in large geometries. Good qualita­tive and quantitative agreement with expeliment confirms the adequacy of the nucleation models in determining both initial size and number density of nuclei. It is shown that bulk nucleation be­comes important as the volume-to-surface ratio of the geometry is increased.

1. INTRODUCTION

There are a number of areas where thermal nonequilibrium play an important role in the de­sign, optimizing, safety, and operation of engineering equipment. Such cases include once­through boilers-superheaters, flash evaporators, condensing heat exchangers, and nuclear power systems, to name but a few. There are also numerous situations where differences in temperature between phase in juxtaposition can exist, and where the phase change rates are limited by the ther­mal processes rather than the thermodynamics involved. Such situations can include subcooled boiling of heated liquids, spray condensation, post-dry out heat transfer, and flashing of initially subcooled liquids. Choking and critical flows aTe of importance in many instances in the pTocess, chemical, and power industries. This is especially true where multiphase flows are possible since the formation of vapor in a liquid leads to significantly 10weT sonic velocities and larger frictional losses than in either phase alone. Resultant critical flow rates are thus substantially reduced from corresponding single-phase values.

This chapter will introduce the concepts of nonequilibrium phase change through consider­ation of the last situation--flashing of initially subcooled liquids in reducing pressuTe fields. This area is important in any situation where fluids are escaping from pressurized containers and are subjected to choked or critical flow conditions. While the previous chapter dealt with clitical flows in general, those with two-phase inlets tend to be more well understood and easier to deal with than those with subcooled, single-phase liquid inlets. This is because in the case of the for­mer the interfaces are generally well established with high area density and low thermal resis­tance leading to excellent coupling between the phases and little thermal nonequilibrium. In addi­tion' the void fractions tend to be considerably higher than those that are nearly all liquid so that velocities are generally considerably lower. Thus, relative velocity, that is, mechanical nonequili­brium, plays a larger Tole in controlling the critical flow rates with two-phase inlets.

When a liquid is decompressed, either temporally or spatially, fluid particles may reach a point where the local pressure decreases below the saturation pressure according to the local tempera­tUTe. Such a circumstance is depicted in Fig. 1 . GeneTally, this local temperature is sufficiently close to the inlet (stagnation) temperature that the differences are ignored. Thermodynamic con­siderations would suggest that as the pressure is further reduced, there would be vapoT formation which could be determined directly from the equation of state of the fluid coupled with a suitable assumption regarding the decompression process itself: isentropic, polytropic, isothermal, etc. Furthermore, once the vapor is formed, some assumption must be made regarding the Telative motion between the phases.

This chapter presents the state-of-the-art in the description of the overall flashing process start­ing with nucleation and continuing on into the region of net void development.

2. THERMOFLUID DYNAMICS

OF REAL FLUIDS IN THE NU­

CLEATION ZONE

Considering Fig. I, the initially-sub­cooled liquid must first be decompressed to the saturation line, then must begin nucleat­ing. Then the nuclei must grow producing net vaporization and void development. The concept of nucleation in initially subcooled, decompressing liquids has been given little attention. Most persons concerned with void development under such circumstances sim­ply make an assumption regarding the num­ber and size of initially nucleated bubbles, and then proceed to treat the growth of these bubbles in some fashion.

It is well known that the mass flow rates in critical flow are highly dependent on the va­por content of the flow. The vapor content can be computed from the equations of mass, mo­mentum, and energy conservation if both the interfacial area and the driving potential for mass transfer at interfaces is known, or if the

LlJ a: :: CI) CI) LlJ a: c.

Z

I I -, :- AZ I I I I

I

I I I

I Z Z+AZ

1 9 1

- -ct

l AP I sup, 0

z.

Figw'e 1 . Schematic of flashing in a converging-diverging geometry.

nucleation rate and size of nuclei are known. For pre-nucleated flows, the vaporization rate per unit volume, r, is given by

r dA v = qk ' Ok

ALl1lv Xi k=l,v dz ( 1 )

where ililv is the "equivalent" latent heat taking into account the subcooling of the liquid and the superheating of the vapor, qk" is the local heat flux vector and qk

" ' nk is the outward-directed

heat flux normal to phase-k at the interface. The sum of the normal heat flux on either side of the interface is the net heat flux which goes into phase change and mass transfer. Also, (JIA)dA/dz is the interfacial area density. Thus, the rate at which vapor is generated per unit volume is given by the net heat flux going into evaporation, multiplied by the total interfacial area in the volume and divided by the product of the latent heat and the volume, Adz.

It is seen from examination ofEq. (1) that both the total quantity of inteIfacial area and the net heat flux for evaporation are required to calculate the volumetric vapor generation rate from first principles. lbis calculation involves determining the initial distribution of vapor in a given vol­ume as well as the evolution of this vapor with phase change.

1 92

The problem in ''initial nucleation" is one which has received some attention. It can become clear the extent to which flashing inception is a consideration by examining the evolution of non­

equilibrium vapor generation. For steady, quasi-one-dimensional flows with phase change, the mass conservation equations for the vapor are

and (2)

where r v and r e are the actual and equilibrium-path volumetric rates of vapor generation. Re­

call that G is the mass flux. Putting Eq. (107) into dimensionless terms and subtracting the space

derivative for actual quality from that for equilibrium quality yields

(3)

Now, it is recognized that the actual vapor source, r v , is due to the difference between the actual

and equilibrium (saturation) temperatures, and recalling that this difference is directly related to

the difference between equilibrium and actual qualities, one may write (108) as

d(xe - x) N ( ) - 1 -- + , Xe - X - . dxe

where

(4)

(5)

Note that since rv is dependent on the temperature difference between the phases, and since this temperature difference is directly dependent on the quantity (Xe - x), the parameter N, is indepen­dent on the nonequilibrium potential (Xe - x).

Equation (109) shows that the quantity (xe - x), dimensionless nonequilibrium potential for

phase change, behaves as a first orderrelaxation process. The inhomogeneity is the forcing func­

tion dxeldxe equal to unity and is due to the equilibrium path the pressure takes causing the equi­

librium quality to change. The relaxation number, Nr, is related to the actual vaporization rate

relative to the equilibrium rate, and is responsible for allowing the phase change to occur and the

actual quality to approach the equilibrium value.

Of interest in this equation are several things:

1 . Development of nonequilibrium is an initial value problem;

2. Development of nonequilibrium is a first-order, forced relaxation process;

3. Development of nonequilibrium is patb- (history) dependent.

1 93

4. Only the relaxation number is a local variable, dependent on the interface area and thermal field distributions and on the resultant rate limitations for phase change.

For flashing of initially subcooled liquids, the initial value is the inception of the flashing process. lbis shall be considered in the next section.

Traditionally, researchers interested in void development of initially subcooled liquids, when faced with the necessity of providing initial conditions, simply chose a particular number density for vapor nuclei, and also arbitrarily chose a size for these nuclei. Typical values for the number density chosen ranged from 108 to 1013 m-3 . Similarly, values for initial sizes ranged down to 10-6 m. various equivalent alternatives chosen producing similar results include initial void fraction, initial superheat, and its equivalent, initial underpressure (the difference in pressure between the saturation value according to the stagnation temperature and the throat pressure). Note that these models are essentially instantaneous inception models and do not consider the nucleation devel­opment lengths.

The first attempt to do more than this was undertaken by Alimgir and Lienhard ( 1 979, 1981 ). They considered that nucleation would occur similar to that in bulk liquids except for a reduction in the work function. They then developed an expression for the underpressure in terms of the rate of decompression and the thermodynamic state of the fluid. This method was applied to transient decompression data with some degree of success. The correlation developed related static pres­sure "undershoot" in terms of the expansion rate 1:', and other parameters as

a3/2 T13.76 6.PFio = 0.258 j--( , IJI) j 1 + 13 .25l: '0.8

kTc I - -v, (6)

where k is Boltzman's constant, and where the subscripts on the temperatures represent reduced and critical conditions. For this dimensional equation, the expansion rate is in Mbar/s.

The description of Alimgir and Lienhard did not properly account for flashing inception pres­sure undershoot in flowing liquids. To take these conditions into account for both friction and acceleration dominated situations, Jones ( 1980), building on the work of Alimgir and Lienhard ( 1979), showed that the fluctuations in velocity associated with turbulence in flowing liquids pro­duced pressure fluctuations which must be considered to accurately predict the underpressure at flashing inception.

By considering the trace of the turbulent fluctuation intensity for isotropic turbulence, the rela­tionship

1 94

p'

G

(a) Envelope of pressure fluctuations

------J-/ /

/ - / � ./

G

(b) Qualitative effect of fluctuations on overexpansion at flashing inception

Figure 2. Qualitative behavior of pressure fluctuations as they affect flashing inception.

where !lpFi is the underpressure at flashing inception with turbulence compared with tlPFio , the value without turbulence. The mean flow velocity is v, the turbulent fluctuation V, and the subscript on the density I represents general liquid, not necessarily saturated. Thus, the kinetic energy of turbulent fluctuations was seen to decrease the inception "superheat" which is directly related to the pressure undershoot. Of course this equation is limited to zero undershoot and so a maxi­mum value was specified as either zero of that provided by Eq. (7).

The rationale for this behavior is shown in Fig. 2. The pressure fluctuations themselves in­crease with the square ofthe mass flux (Fig. 2a) and add to the actual pressure as seen in Fig. 2b. When compared with data for frictional decompression in straight pipes, the result is quite soik­ing as seen in Fig. 3 where the turbulence intensity was taken at the commonly accepted value of 0.072. When added to the static decompression correlation of Alimgir and Lienhard ( 1979), the result in dimensionless terms becomes

where !lE' is the convective expansion rate, and

The dimensionless undershoot is defined as

= �PFi Fi - --�PFio

The parameter 1/J is dependent on the factor causing the decompression:

(8)

(9)

( 10)

195

Friction : tp = [2d lQ", jfJ2/3 ( 1 1) � 24 � • & 20 A o Rcocreux T\ = 1 16°C [ A j ]213

Acceleration : tp = ( 12) <]

c 1210C A 1260C • 5cynhacve T\ = 1 160C

with/being the D' Arcy friction factor and d the duct diameter. This flashing incep­tion undershoot is shown in Fig. 4. While turbulence is important when friction is the predominant cause of decompression, in many cases with acceleration effects causing the decompression, the turbu­lence effects are not important (Abuaf, Jones, and Wu, 1980).

While this expression for the flashing inception superheat provided one part of the needed information, there still is still an infinite combination of initial number densities and bubble sizes which would

c: .9

� oS CI) c: :.E <12 CIS ii: � � 5! Ctl �

1 6 1230C 1340C

1 2 1390C

8

4 �PFi = �PFi 0

1� 1� 1�

•

:J Square of the Mass Velocity - kg2/m4 . 52

Figure 3. Comparison of the flashing inception data of Reocreux ( 1974) and of Seynhaeve et aL

( 1976) as reported by Jones, ( 1 980).

yield identical superheats . Since each combination would yield different interfacial area, differ­ent void development rates could be calculated for different combinations.

The problem with determining correct initial conditions for use in Eqs. ( 109) and ( 1 10) is that the nucleation rate is really continuous, and the size of the bubbles depend both on the conditions of the surface and on the local flow conditions adjacent to the surface during nucleation. That is, surface nucleation dominates most processes for which experimental evidence exists. Surface nucleation is known to occur at preexisting cavities. The nucleation superheatis dependent on the size of the nucleating cavities as is the "holding power" of the surface by surface tension at the base of the bubble. The forces attempting to remove the nucleus from the surface include buoyan­cy and drag, mostly the latter in critical and near critical flows. A model is required to describe the physics of this overall nucleation process.

The transition from liquid to a two-phase mixture through short pipes and nozzles by flashing usually takes place in several stages. Initially subcooled liquid encounters an acceleration-con­trolled pressure drop (Fig. 1 ) and is brought to a saturated state. A further decrease in pressure causes the liquid to become superheated. As liquid superheat is obtained, bubble nucleation starts. Nucleation is a strong function of the thermodynamic state of the superheated liquid.

Nucleation rates will first increase to a maximum value. Nucleation has been shown to occur on solid boundaries, at least in laboratory situations, and so heterogeneous wall nucleation shall be

1 96

, 0 0 Mbar

1: 0= . 1 sec

10 2 10-4 10 3 1 0 2 10 1 10° 101

100

10 1

l O Mbar

- . sec

1 0-2

10 4

10- 1

10 4 10 3 10 2 10 1 100 101

Convective Expansion Rate t:r: - Mbar/sec Convective Expansion Rate t:r: Mbar/sec

Figure 4. Dimensionless flashing inception correlation combining Alimgir and Lienhard ( 1979) with turbulence effects. 39

the mode considered herein. It is expected that the nucleation rate will subsequently decrease due to evaporative cooling of the liquid boundary layers adjacent to solid surfaces. In this respect it is similar to the formation of condensation nuclei in shock tubes (Wegener, 1969, 1 975). The net result is a limited-length nucleation zone wherein virtually all the bubbles which go into the sub­sequent void development process are formed.

Following the nucleation zone, bubbles will grow as they flow downstream. This growth is generally limited by conduction processes at liquid-vapor interfaces. It is expected that spherical bubble growth processes will dominate void development in the early stages, and perhaps up to quite high void fractions owing to the rapid transport times and the limited times available for bubble agglomeration in high speed flows.

The analysis of this overall problem, therefore, can thus be broken up into three stages:

1 . nucleation, growth, and departure at a single cavity (Jones and Shin, 1 986); 2. overall nucleation zone and bubble transport to the throat (Jones and Shin, 1987); 3. void development downstream of the throat (Blinkov et aI., 1990a).

1 97

. . . boiling . . . flashing

Tsup (z)

wall

Too (z)

x x

Figure 5. Stability criteria for cavity nuclei: (a) subcooled boiling; (b) flashing.

Single cavity nucleation. The first problem is to develop a criterion for determining when cavities are active. Consider the differences between flashing and boiling. One major difference was the way in which the thennal layers adjacent to the walls behaved. As shown in Fig. 5a, the wall layer in subcooled boiling is superheated and a bubble at a cavity would be stable and support growth if it were completely confined in the superheated layer (Hsu's criterion: Hsu, 1962).

For flashing (Fig. 5b), the liquid is uniformly superheated away from the cavity. At the cavity during growth, the wall drops to saturation and then recovers as the wall and liquid start to equili­brate following departure of the vapor nucleus. The equivalent to Hsu's nucleation criterion for flashing is that a stable bubble will nucleate when the wall temperature increases to the saturation temperature inside a bubble of radius identical to that of the cavity. Assumptions for this develop­ment include:

1 . The problem is one dimensional and actual cavity geometry is ignored;

2. Properties of liquid and solid are constant.;

3. There is no contact resistance between solid and liquid;

4. Liquid and solid are semi-infinite and stationary;

5. Liquid has uniform superheat far from the solid;

6. Convection of liquid is negligible.

The cavity nucleation problem is broken up into two intervals: bubble growth period to depar­ture; dwell time between departure and nucleation. A detailed solution to this problem may be found in Jones and Shin ( 1 985) .

Stage-l , growth period:

During this time, the vapor temperature inside the bubble, and hence at the vapor-solid inter­face, is at saturation temperature according to the internal bubble pressure. It is assumed that this pressure decreases rapidly and has a value characteristic of the local liquid pressure. The wall temperature, then, is assumed to start at uniform superheat and have an instantaneous decrease to local saturation temperature. The solution to this problem is well known to be

1 98

Ts(y, t) - TSal Tsup - Tsat

( 13)

where the distance into the solid, y, is negative. Here, T is temperature with subscripts s, sat, and sup representing solid, saturation, and superheat, respectively, and any temperature difference is the positive value so !l.Tsup is the local superheat. Also, a is the thermal diffusivity and t is time after the step change in temperature from the superheat value to the saturation value. Of course the assumption implies the wall equilibrates to the superheat temperature during the dwell times and is strictly applicable only to the case where the dwell times are long compared to the growth times. Nevertheless, this temperature profile will be used where such is not the case

Stage-2, dwell period:

During this period, the precursor bubble departs, uniformly superheated liquid at a tempera­ture Tsup impacts the solid, and an interfacial contact temperature is established providing in both liquid and solid boundary layers. This layer then diffuses into both layers causing each to equili­brate towards the uniform superheat temperatw·e.

The solution to stage- l shall provide the initial conditions for the solution to this stage. In this case, both liquid and solid are considered with one dimensional conduction equations for each. The problem to be solved is thus,

and ( 14)

where the subscript I on the temperature and diffusivity represents the liquid. This field equation is subject to the initial conditions

At t = 0, for y < 0 ( 15)

where tg is the growth time of the precursor bubble. Also, the liquid temperature at the start of the dwell peliod will be that of the uniformly superheated outer fluid,

At t = 0, ( 1 6)

for y >0,

Boundary conditions require that the fluid and solid temperature and heat flux match at the interface so that

At y = 0, Ts(O, t) = T [(0, t)

and

( 17)

1 99

At y = 0, ( 18)

Furthermore, both liquid and solid temperatures must be well behaved everywhere so that

As y - 00 , Tl(Y, t) is finite and

As Y - - 00 , Tiy, t) is finite.

( 19)

(20)

The solution for the liquid temperature distribution in dimensionless terms with reference to the temperature of the superheated liquid is

where H2n is the Hermite polynomial, and where

fJ = - 2 jaltg

and

and

(2 1)

(22)

(23)

(24)

Note that k is the thermal conductivity and C the specific heat. The transient temperature distribu­tion is shown in Fig. 6a. By the choice of parameters, the different combinations of material p�op­erties are automatically accounted for without change in dimensionless temperature profIles.

The contact line (wall) temperature time behavior is obtained from (9) by letting the distance from the wall, y, vanish to obtain

(25)

It is readily seen that as the dwell time progresses, the contact line temperature (Fig. 7) increases back toward the superheat temperature. At some point, this temperature will be sufficiently large

200

u.i -0.1 a: :: � a: w 0. :! w � � W .. Z o CiS z w :! 25

-0.4

-0.5

-0.6

-0.7

't = 1 00 20 1 0 5 2.5 1 .67 1 .25

o 4

DIMENSIONLESS POSITION, "

Figure 6. Dimensionless liquid temperature profiles during the waiting period following wall contact of the cavity nucleation cycle.

to sustain stable nucleation at a given cavity according to the equivalent Hsu-criterion as described next. e -0.2

� E' -0.4 � � -0.6 � � -0.8 o .� � .S 0 o 20 40 60 80 100

Dimensionless Dwell Time, 't Figure 7. Dimensionless contact line temper­ature following contact during the waiting pe­riod of the cavity nucleation cycle.

Using the Clausius-Clapyron equation, the saturation temperature corresponding to the local inside pressure of a bubble with ara­dius Rc is given by

eRC = 1 ) (26)

• • • ( 1 + j(keC)I/(kec)s) where (J is the swface tension, Il.�g "is the la­tent heat, and Rc is the cavity radius. Follow­ing Hsu ( 1962), it is assumed that in order for a new bubble to nucleate at the cavity, the

temperature everywhere on the surface of the nucleus must equal or exceed the saturation temper­ature according to the pressure inside the bubble. This, of course, is slightly conservative since condensation on the colder surfaces could be balanced by evaporation on the hotter swfaces. Nevertheless, it is assumed that the wal-liquid interface temperature must rise following contact

20 c ·s *� 15 � iZi 10 � ·S u 5

-1 -0.8 -0.6 -0,4 -0.2

9 .. /[1 + o

Figure 8. Minimum active cayity size as a function of wall temperature.

20 1

to that temperature which just equals the satu­ration temperature inside the nucleus in order for the nucleus to be stable and grow. Tran­sient conduction in both wall and liquid, coupled with the local temperatures, thenno­dynamic and transport properties, will define the duration of the waiting period.

The expression for the minimum active cavity size comes directly from the Laplace equation for a metastable bubble combined with the Clausius-Clapyron equation to yield

(27)

where '1g is the vaporization specific volume

of expansion and R;,min is the dimensionless minimum value of the active cavity size. Combina­

tion with (26) yields

(28)

where 1d is the total dwell time following departure of the precursor bubble. This stability criteri­on is shown in Fig. 8. As the waiting time increases, E>w vanishes and the minimum active cavity size approaches that associated with the uniform superheat. As the dwell time vanishes, 8w ap­proaches its limiting value of negative unity and only large cavities can nucleate since the wall superheat would be a minimum.

The difficulty in this analysis, of course, is identical to the difficulty in previous stability crite­ria developed for boiling. While this procedure is useful in qualitatively explaining observations, it requires quantitative information about the nucleating surface characteristics to be useful. In. the next section, an approach shall be taken which circumvents this difficulty.

Figure of merit. Actual determination of nucleation site density. even if possible, is patently impractical. An alternate approach would be to find a reasonable "figure-of -merit" which would be definitive and which would allow analysis to proceed.

While there is no theoretical basis for a minimum energy principal for nucleation at cavities, it may be that preferential nucleation might take place at cavities in a way that the sUlface energy for production is minimized. If this were to be the case, nucleation would favor the maximum cavity size which will produce stable cavities. This, in fact, would allow nucleation to proceed with minimum superheat.

202

'5 � 'Ci Q) .. ;: til) ti:

20

10

A figure-of-merit for the system is thus obtained by considering active cavities on the surface to be only those producing the largest stable nuclei--those for which the dwell time vanishes and the wall temperature is at its minimum value.

Letting the waiting time vanish so that

Sw(rd) � -1 yields the maximum value of

minimum active cavity size from Eq. (28) as

Figure 9. "Figure-of-Merit" for nucleating which is shown in Fig. 9. Having assumed the surfaces. Maximum value of minimum active surface to be made up of cavities of this (ficti-cavity size. tious) size, one could proceed to calculate the

needed quantities such as growth times, de­parture sizes, and nucleation frequencies, and see if the results proved useful. By considering this figure-of-merit, dwell time between bubbles has been neglected. Therefore, calculation of growth rate through standard methods, coupled with a departure size criteria, will then yield the nucleation frequency at a given site, this frequency being a maximum value.

Bubble departure size. By balancing drag and surface tension forces, the departure of a bubble is given by,

( 4aRc ) 1/2 Rd = K 2 (30)

where CD is the drag coefficient and where K accounts for, among other things, the fraction of the surface tension forces acting in opposition to the drag. Also, V, is the average velocity acting over the bubble. It is assumed that all the bubbles grow entirely within the laminar sublayer. (This as­sumption has ben confined by the result.) The average velocity is given by

1 f 2Rd ( T Y) Ttfi. VI = 2Rd 0 dy = -;- . (3 1 )

where 'tw is the wall shear stress, 1.1. the viscosity, y the distance from the wall, and i?(} the departure radius of the bubble. The normal turbulent friction coefficient (Schlichting, 1979) is used to cal­culate the wall shear stress such that -1/4 Cf= 0.0791 Red (32)

203

where RI!,J is the local Reynolds number based on the diameter. This assumption will underestimate the wall shear in nozzles where the convergence suppresses the boundary layer thereby steepening the gradients. TIris will lead to late departure estimates and underestimated frequencies, tending to offset the vanishing dwell time estimate. The departure size thus becomes

The drag coefficient is taken to be (Schlichting, 1979) { 24/ReB CD = 18.5 Reilo.6

0.44

ReB < 1 1 s ReB < 500

500 s ReB < 2x1cP

(33)

(34)

where ReB is the bubble Reynolds number based on the bubble diameter and local duct-averaged liquid velocity. Thus, since all data fall within the range of bubble diameters between 1 and 500, the departure size becomes

7{( aRc) 1/2( f.l )7/10( 1 )3/10}5/7 Rd = 0.58XS/ - - -

(}t "l"w V (35)

K accounts for the relative balance between drag and surface tension at departure is arbitrarily taken as unity in what follows.

Available data on flashing in pipes and nozzles (Abuaf et al., 198 1 ; Reocreux, 1976; Brown, 196 1 ; Celata, 1982; Wu et al., 198 1 ; Sozzi and Sutherland, 1975; Bailey, 1 959; Ardron and Ack­erman, 1 978) were examined and departure sizes calculated using Eq. ( 1 9) with K= 1 . For all these data, departure sizes in the range of 1 1lm to 100 Ilm were obtained. The smallest values corre­spond to the highest flows and largest superheats near lOOC.

In al cases, the values of departure size calculated were approximately equal to or larger than the given value of Max {Rc,min } .

Bubble nucleation frequency. For wall nucleation conditions, the nucleation frequency per site and the site density are the quantities which describes the system nucleation behavior. The cavity size distribution in the nozzle should be characterized by that in the throat where the super­heat and nucleation frequency is at a maximum. Thus, the frequency in the throat will be deter­mined and then used to obtain the site density. The latter will then be used for the balance of the nozzle.

Data which exist in the literature (Abuaf et at, 1981 ; Reocreux, 1974; Brown, 1961 ; Celata, 1982; Wu et al., 1 98 1 ; Sozzi and Sutherland, 1975; Bailey, 195 1 ; Ardron and Ackerman, 1978; Reocreux, 1 976) were used to calculate the nucleation frequency for the figure-of-merit cavities (assumed size). Growth rates before departure were taken as the smaller of momentum controlled

204

or thel1l1al-difusion-controlled (Pies set and Zwick, 1954) rates. Results are shown in Fig. 10. The liquid superheats span the range from less than IC to close to 10 A reasonable fit close to the least squares condition of these data was found to yield a maximum departure frequency of

fmax = lQ4ill1up. (36)

. The least squares coefficient and exponent were 1 1 170 and 2.785 respectively. Frequencies close to 1010 s-l were calcu­lated at the largest values of superheat near l00C. These are un­expectedly large values characteristic of what one nOl1l1ally would consider at the lower bound for homogeneous nucle­ation. In spite of the fact that these are artificial values, it is seen

in Fig. 1 1 that the ratio of distance traveled in one period after �=N departure, ul!f, to the bubble size at departure, 2�, is generally O SOZ.SUTHERLAND above unity (only two cases, 0.9, and 0.95, is this ratio below REOCRBJX unity, well within the accuracy of the calculation. Note that

2 these calculations are based on bubble growth rates which are

10 . . 0.1 1.0 10 100 assumed controlled by thel1l1a1 hmlts . Momentum may actually UQUID SUPERHEAT, .6. T (0G) . . . sup,o be controllIng at higher superheats which would lead to lower

Figure 10. Throat bubble frequencies. 40 departure frequencies for 0

"Figure-of -Merit" cavities. The correlation for nucleation frequency is di­

mensional, a somewhat undesirable situation. However, at this time no satisfactory dimensionless method of correlating the data has been found. o 0

o QjJ Q

o o

Maximum local wall nucleation rate and nu­cleation site density. It is typical of rapidly nu­cleating systems that the process of nucleation it­self can tum off the nucleation process due to the energy absorbed or liberated by fOl1l1ation of the nuclei (Wegener, 1969, 1 975). Thus, it is expected that with sufficiently rapid nucleation, the available superheat in the nucleation layer will be consumed. In order to calculate a maximum wall nucleation rate for each set of data, the nucleation site density can be calculated as

(37)

w U Z � (J) a (J) (J) �

LEGEND o BNL o ARCIAON '" BAIlEY O SOZZl· SUTH� Q REOCRBJX OBAO'N O CElATA

o 0 0 0

Q 0 0 %0 � o o 0 °� '" o o '" '" :0

8 0 '" Q "' ", '"

z o en z w � 1 a "' ''

0.1 1 10

UQUID SUPERHEAT, Ll. T sup,O (0C)

Figure 1 1 . Distance traveled by one de­parting bubble in one nucleation period relative to departure diameter.

100

205

where Jwm is the maximum wall nucleation density which must be determined, A- the local cross

sectional area of the nozzle, and � the nozzle perimeter. Considering the nucleation wall layer, a convective energy balance yields

(38)

where B is the thickness of a thin cylinder ofliquid of radius equal to the cavity radius having the mass of the nucleus. Furthermore, RN(Z) is the nozzle radius as a function of axial location. It is

assumed that a disk of diameter 2Rc and thickness B is instantaneously evaporated to give

c5 = 3.Rc((!g ) . 3 (!l (39)

The velocity in the integral of (38) is obtained from the universal velocity profile over the

thickness B. Equation (38) must, of course, be evaluated for each given geometry. For pipe flow, R(z) is

fixed and the evaluation is relatively straightforward. For the

between RJ at the inlet and R2 at the throat, and where the acceleration pressure profile in the nu­cleation zone can be approximately linearized, it is found that

Jwm = 2.94xlO-3 plm (!g :rup,a H(z) ( C · 7/4 ) ( IlT )2 ap.3/4T:raJ (!ILn

(40)

where m is the flow rate, IlTsup,a is the superheat at the throat, Ln is the nucleation length, z is the

distance from saturation (0 < z < Ln) and where

H(z) = _z {I _ 2.75( 1lR) (2.)} RJiI\z) R LN

(41)

accounts for the nozzle geometry with IlR=RJ -R2 being the difference between inlet and throat diameters and LN the length of the inlet section of the nozzle. Of course, the more general nozzle shapes must be evaluated individually according to Eq. (38). Values for dimensionless nucleation

site density can be obtained from the experimental data previously cited. As seen in Fig. 12. it is

found that the dimensionless nucleation site density correlates approximately with the superheat­based cavity size Res (Laplace equation) as

(42)

where R; = Rc:r/Rd ' The least squares coefficient and exponent were 1 .039 x 10-7 and -3.808 respectively.

206

.. Z � (/) Z W C � en (/) (/) W -I Z 0 en z w � 0

10'"

10 '

10-1

10 '

O BNl. o ARORON t: 1IAJLEY o SOZZl-SUTHERlANO Q REOCREUX O 8ROWN O CElATA

Equation (42) shows the density of active nucleation sites to be proportional to the square of the departure size and inversely de­pendent on the fourth power of the super­heat-related cavity size. Thus, both thermo­dynamic and hydrodynamic states are impor­

tant. Increasing the local velocity decreases the departure size and snuffs out active sites

in accordance with the observations of

Bergles and Rohsenow ( 1 962) for subcooled boiling. An increase in frequency at remain­

ing active sites would thus occur with de­creasing pressure, increasing superheat, and increasing velocity.

A model for heterogeneous nucleation in the bulk fluid was developed by Soplenkov and Blinkov ( 1983) and further described in

DIMENSIONLESS CAVITY SIZE, R� detail by Nigmatulin et al. (1987) The as-

Figure 12 . Correlation of dimensionless sumption was that the liquid always carried

nucleation site density. suspended particles whose size distribution is fIp( d). At nucleation sites, only supercritical particles (d > d*) can be active where d* is the diame­ter of the critical spherical vapor nucleus and depends on the physical properties of the liquid and degree of metastability. Thus the total number of nucleation sites where evaporation and bubble formation can occur is

NB(Gi) = f Q) n(d)dd d"

(43)

where Gi is the Gibbs number. An empirical correlation for NB = NB(Gi) was obtained using ex­

perimental data on blowdown of initially subcooled water through short tubes (4 � Lid � 10, L � 0.3 m) with sharp entrances where Gi � 1500. The result was

10g(N B) = 12 .5 - O. 15 10g(Gi). (44)

The nucleation source term is thus obtained as

(45)

It is reasonable to expect that the nucleation site density at

the throat is characteristic of the rest of the nucleation zone. The bubble number density at the throat may thus be detennined by approximate integration of the bubble transport equation as

(46)

where Nns is the nucleation site density as a function of position. The results are shown in Fig. 13 in comparison with the esti­mates provided by Wu et al. (198 1) on the basis of an assumed inception void fraction. Furthermore, the values predicted by the subcooled boiling model of KocamustafaoguUari and Ishii (1983), while not explicitly valid for flashing are also shown several orders lower than those predicted herein.

The wall nucleation rate may be written in terms of the nucle­ation site density and site frequency as

lwm = NnifmBx� . Ac (47)

10"

A 000 �. � .

QJ O

•

o A Ao • •

• t:J JC1t:J t:J

o eN. • PRESENT C1 SLeCOOLED BOLING o T .. =149"C A T .. 121·C o T",al00C

207

1 2 3 4 5 7 10 20 LIQUID SUPERHEAT, � T sUP.O (0C)

Figure 13 . Comparison of calculated bubble number densities at the nozzle throat with other predictions.

Recalling the expression for departure size, and using the cavity size in terms of the superheat through the Oausius Clapyron equation, the throat superheat is obtained as

where

B == 8,41xlO-5 eg '1g � .! _ ( tli )23/7{( ) 1/2( )7/10( 1 ) 3110} 10/7 2aTsat el l'w V

(48)

(49)

This dimensional coefficient has units of ( l/sK3) because of the nature of the nucleation frequen­cy correlation.

A comparison between the calculated and observed values is shown in Fig. 14. The standard deviation is 1 .9C, indicating a reasonable accuracy in the calculation.

208

c: 0 � � 8

-0 ci il

�

10

o BNL C ARORON A BAlLEY o SOZZI-SUTHERLAND 9 REOCREUX o BROWN o CELATA

Figure 14. Comparison of calculated throat superheats with measured values.

The void fraction at the throat may be found by integrating the vapor continuity equation to obtain

(50)

where the integration is taken over the nu­cleation length from the saturatiohn line to the throat. The volumetric vapor source for variable area geometry may be ex­pressed as (Zuber et al., 1966)

(5 1)

ry =�{�f Z mB(Z, z l )JW(Z I�(Z I )dZ ' } dz Ac 0

where m'(z,z') is the mass ofa bubble atz which was nucleated atl' The mass of the bubble at any location may be determined through the departure size at the nucleation site given by Eq.(35) and an analysis for bubble growth in a variable pressure field (c.f. Jones and Zuber, 1978).

1 0

� � 8

� < fE 6 o (5 > :ci 4

� )(

UJ 2 o

o 0.2

DIMENSIONLESS DISTANCE, z*

Figure 15. Void Development for Run C35 of Ardron and Ackerman (1978)

In virtually all previous flashing models, the nucleation zone is treated as a single point of flashing inception. This has been justified since the zone of supersaturation in many cases is quite narrow. However, in this zone, the voids which develop from the nu­clei form the basis for interfacial mass trans­fer and subsequent growth downstream. It is, therefore, important that both the size and number be determined so that accurate cal­culations of void development may be un­dertaken.

Calculations were made for allruns in the previously referenced data sets and all had similar behavior. The smallest throat void fraction (Fig. 15) was computed for Ardron and Ackerman's Run C25 shown in , having a throat superheat of 1 .66K with 1 .6 bar inlet pressure. While the nucleation site density

209

2.0 4

5 2.0 � '? .s .s II! i z

-, 4 � W 1 .5 Ci5 � z c( w a: Cl z 3 w 0 � � 1 .0 Ci5 z w 0 .. 2 � 0 :> z w .. .. 0.5 0 .. :> � z

X mX � 0 0

00 0.2 0.4 0.6 0.8 1 .00

6 � 1 .5 l-iiI <1 f-.:' <: w � 1 .0 w a. ::> (J) o 5 0.5 o �

0.2 0.4 0.6

w a: 3 �N=f

(/) Q. w � � a (/) <1

2 � 1-" .. 0 z O O ::I: C; � z w W o � z 0 ::> x

o ,.

DIMENSIONLESS DISTANCE, Z· DIMENSIONLESS DISTANCE, z* (a) Wall nucleation rate and (b) Liquid superheat and

nucleation site density. pressure undershoot. Figure 16. Calculations for the nucleation zone for the conditions of Ardron and Ackerman

( 1 978), Run C35 . Conditions for this data set: 1in = 1 l 1 .5C; Pin = 1 .59 bar; G = 7740 kg/m2s; !1Tsup,o = 1 .66K.

shown in Fig. 1 6 increased to approximately 140 m-2 with overall wall nucleation rates of about 5 x 108 m 3s-1 (bulk equivalent), the throat void fraction was only 10-5.

The largest throat void fraction of 0.9% was calculated for Brown's run 39 shown in Fig. 17, having a throat superheat of 8 1 .6K, t:I

an inlet pressure of 68.4 bar, and a wall nucle � 8

ation rate at the throat of 3 x 1023 m-3s-1 t (Fig. 18 ), fifteen orders of magnitude larger � 6 than that found for Ardron and Ackerman. � Note that the throat superheat of 8 1 .6K repre- � sents a ;i 4

X Where measurements of void fraction ex- c(

..,x 2 o

ist, the agreement was within experimental accuracy of the experiment. This however, is no real test, since the data are usually accurate to within 1 -2% voids at best and the com­puted maximum throat void fraction when data existed was on the order of6xlO-5• Nev­ertheless, the calculations and existing data suppon the original hypothesis of Abuaf et aI.

o 0.2 . 0.4 0.6

DIMENSIONLESS DISTANCE, z*

Figure 17. Void development for Run 39 of Brown ( 1 961 ).

2 1 0

) ui � a: Z 0 � w .. U ::J z x

�

1 00 1 00

4.0 � 80 80 � )

60 � � 3.0 w � 60 60 c <C W W l- I en II: 2.0 40 z w 40 0 IL. 40 � :: en � 0 u :5 1 .0 20 � 0 20 20

x :J ";> �

0 0 1 .0 0 00 0.2 0.4 0.6 0.8 1 .0 0 0.2 0.4 0.6 0.8 DIMENSIONLESS DISTANCE, Z' DIMENSIONLESS DISTANCE, Z'

(a) Wall nucleation rate and (b) Liquid superheat and nucleation site density. pressure undershoot.

w II: :: en en N O w :: II: C-IL. s:! en -a en � � � Z 0 Q o en I z en w II: ::! w - 0 O z x :: <?

Figure 18 . Calculations for the nucleation zone for the conditions of Brown ( 1961 ), Run 39. Conditions for this data set: Tin = 280C; Pin = 68.4 bar; G = 303 kg/m2s; llTsup,o = 8 1 .6K.

( 1983) that flashing flows with subcooled inlet had viItually all single-phase flow upstream of the restriction or throat.

2.7. Critical Mass Flow Rates

� 1

· E

10

0.1 o ARORON 6 BAILEY o SO�-SUTHERLAND o REOCREUX o BROWN o CELATA

0.01 0.1 10 100 m measured (kg/s)

Figure 19. Comparison of calculated critical flow rates with measured values.

The work described above shows that negligible voids exist at the throat for flash­ing of initially subcooled liquids. Single­phase theory may thus be used to calculate critical flow rates under such conditions, where the correct throat pressure must be obtained from the calculation of the throat superheat as given by Eq. (48).

The result of these calculations is shown in Fig. 19 for the data cited previously. The standard deviation between predicted and measured critical flow rates is approxi­mately 3%, a 40% improvement over earli­er work (Abuaf et al, 1983) where 5% accu­racy was obtained using the pressure under­shoot correlation of Jones ( 1980).

2 1 1

This section has described a distributed model for nucleation in the superheated zone upstream of the throat in nozzles dUllng flashing which has the following features:

1. A new stability criterion for active cavities;

2. Identification of a figure-of-merit for a nucleating surface which ties the stability criteria to an obtainable nucleation site density and cavity nucleation frequency in flashing flows;

3. Identification of a method to determine the departure size of nuclei in the nucleation zone;

4. Correlation of nucleation frequencies at a given site and surface density of nucleation sites as determined f om existing data;

5. Detennination of the maximum, energy-limited rate of nucleation;

It was then shown that utilization of the nucleation model allows the throat superheat to be calcu­lated within 2% for existing data over the approximate range from less than l K to nearly lOOK.

The range of data include pressures to almost 70 bar, and expansion rates from 0.2 barfs to over 1 Mbarfs extending existing methods by four orders of magnitude.

Bubble sizes at departure upstream of the throat were determined to be in the range of 1 - 100

Ilm and were not constant as has been heretofore assumed by various investigators. Nucleation rates at the throat were also variable and were calculated to span the range of 108 to 1023 m -3 s - 1 . Resultant calculations of throat number densities in all cases ranged between -108 and 1011 m-3

Bubble transport calculations show that even for the cases with largest superheat near l OOK,

negligible « 1 %) voids exist at the throat, confirming the previous hypotheses of Abuaf et al. ( 1980, 1983) . A result of this confirmation is the calculation of critical flow rates by single phase methods within 3% accuracy once the correct throat superheat (and thus pressure) is obtained us­ing the methods identified herein.

It was noted that the differences accounted for by the methods described herein represent throat pressure undershoots up to the order of 60 bar! This large correction, coupled with the real­ization that the throat conditions are essentially single-phase liquid, was the major factor in de­veloping an ability to calculate the critical mass flow rates accurately for all data. As shall be seen in the next section, the ability to determine both the number density and the size of initial nuclei is also the key to accurately calculating void development downstream of the nucleation zone.

3. VOID DEVELOPMENT DOWNSTREAM OF THE NUCLEATION

ZONE

The common approach to determination offlow structure is the use of flow regime maps. They are typically constructed with superficial velocities, flow rates and quality, and/or void fraction as

2 1 2

(a) Flow regime #1

- - .. . . : . .. :

(b) Flow regime #2

Figure 20. Assumed flow diagrams of flashing liquid in nozzles.

coordinates. Among the variety of internal two-phase flow structures, the bubble, slug, chum, annular, dispersed-annular, and dis­persed regimes may be identified. These clas­sifications and transition criteria both have a qualitative nature so models constructed with the use of these flow maps may utilize exist­ing transition criteria for comparison with data.

Advanced, best-estimate computer pro­gram for transient analysis of two-phase sys­tems such as TRAC (LASL, 1979) and RE­LAP (Chow and Ransom, 1984) apply flow regime maps with void fraction as the main transition criterion. Indeed, while other

methods exist which are more detailed, and which are, perhaps, more accurate for specific transi­tions, the use of void fraction has achieved recommended general usage (Wu et al., 1981 , Ishii and Mashima, 1983) due to its simplicity. In what follows, two variants of this flow structure scheme are considered. A comparison of the resultant features of these two maps will be discussed fol­lowing identification of the equations given particular to the two.

Flow regime 1 (Fig 20a). Flashing inception is assumed to result in a bubbly mixture. The limits of this flow regime vary depending on the rate of void development. For slowly developing systems, the transition is generally taken to be at approximately a = 0 2. For rapidly expanding systems, the transition to slug or churn flows may be inhibited until quite large void fractions up to over 0.7. For the pwpose of this discussion, the bubbly mixture is assumed to exist up to a = 0.3, consistent with previous assumptions (Wu et aI., 198 1 , and Dobran, 1985) . Note that Ishii and Mashima ( 1983) showed that for a > 0.3 spherical bubbles must touch. For bubbly flows to exist at larger void fractions, obviously the bubbles must distOl1:. Such is the case observed in liquid metal MHO generators. In moderately accelerating systems, agglomeration begins to occur after bubbles begin to touch, thus leading to transition of the bubbly structure to the adjacent regime.

As void development continues past a > oj, it is assumed that this coalescence results in the formation of larger bubbles with the region between filled with small bubbles. This bubbly-slug regime is taken to exist throughout the region 0.3 < a < 0.8 whereupon the slugs have grown so long that they, in tum, coalesce to fOlm annular flow or mist-annular. Dispersed droplet flow is assumed to exist for void fractions over 0.9. The region between a = 0.8 and a = 0.95 is consid­ered herein to be a transitional zone coupling slug and dispersed liquid flows.

Flow regime 2 (Fig. 20b). In this case, the structure of slug flow is not developed and the zone between a = 0.3 and a = 0.7 is considered as a transitional zone. This zone may be characterized by intensive interaction or coalescence of bubbles and deviations from sphericity characteristic of the churn-turbulent regime (Solbrig et al., 1978). In this case dispersed flow is assumed to oc-

2 1 3

cur as a breakdown of the continuous liquid filaments resulting in continuous vapor dispersed

flows.

For the current case the interest is in the lower void fraction regions below dispersed flows. In such cases, the vapor is tightly coupled to the liquid from a mechanical viewpoint, relative veloci­ties are small and any variations are due more to disttibution than local slip. It is thus assumed that mechanical equilibrium exists and the phases have identical velocities and pressures. Vapor tem­peratures are assumed to be at local saturation conditions since the primary source of heat transfer is through the liquid continuum which may be subcooled, saturated, or superheated locally (ac­cording to the local pressure).

The model thus chosen is a quasi-one-dimensional, transient model which uses two continu­ity equations (mixture and vapor phase), one energy equation for the liquid, one momentum equa­tion for the mixture, and one bubble transport equation. These are expressed below as:

Continuity equation for the mixture:

iJem 1 iJ - + -- (emAw) = 0

iJt A iJz

Continuity equation for the vapor-phase:

iJaev 1 i) -- + --(ae,.w) = rv

iJt A iJz

Momentum equation for the mixture:

iJw + w

iJw =

_ "!"!J w2 iJt iJz em iJz em '2d

Energy equation for the liquid:

iJ [ j 1 iJ [ 1 p i) iJa "

- ( l a)etUt + -- (l a)etutAw + --[{ l a)Awj = p - rviJ A;q ; nel iJt A iJz A iJz iJt '

Conservation of bubble number density:

iJN 1 iJ -+ -- (NAw) = Jw + JB iJt A iJz

(52)

(53)

(54)

(55)

(56)

Note in these equations that w is taken as the z-direction velocity and that it is assumed that there is no slip. TIris is in accordance with the intention that this model be utilized to predict void development in critical or near critical flows of high velocity where slip velocities are expected to

2 1 4

be of negligible importance. Note too that,if is the friction factor and u is the internal energy with appropriate phase subscripts.

The unknowns in this equation set include void fraction. a, pressure, p, axial velocity, w. liq­uid temperature. 11. and bubble number density, N, where 11 is tied to the specific internal energy Ui through the caloric equation of state.

Constitutive equations - flow regime map 1. Constitutive equations for wall friction and interfacial heat and mass transfer must be provided for closure as well as those for nucleation.

Wall friction:

For single-phase flow (ex = 0) the shear stress is taken to be

where

Cf = CIRe) with Re = Qlwd

�I

(57)

(58)

where d is the (hydraulic) diameter of the duct. The Blassius friction coefficient q is used for turbulent flow. Wall friction force per unit volume of the mixture is

(59)

For two-phase flow, a friction multiplier, cj)2, is utilized so that

(60)

where the friction coefficient and Reynolds number relationships are given by Eq. (58). The mul­tipliers used were taken from Beattie (1973) as included in RETRAN (McFadden, 198 1 ), where a flow regime map similar to that utilized herein was used. The two-phase friction multiplier equa­tions adopted are:

for a < 0.3

(61)

for 0.3 :s a < 0.8

(62)

for 0.8 s a < 0.95

for a > 0.95

( )O'2( )O.8{ ( )} 1 .8 if>2

1 + x - 1

Interfacial heat and mass transfer:

2 1 5

(63)

(64)

The rate of vapor generation is limited by the heat transfer rate and interfacial area according to

the relation

(65)

where

"

-1 f

q i,net = A ' I Ai I cit" ' Ok dAi

Ic=l,v (66)

where Rk is the unit outward normal to phase-k. this relationship expresses the balance across the interface of the rate of energy loss due to vaporization and the rate of energy replenishment by heat transfer from the bulk liquid.

(a) For bubbly flow, a. < 0.3, the model assumes that the zone of intensive nucleation is va y narrow, Lnuc « L and is located very close to the minimum area portion of the nozzle, which is confirmed by calculations and is consistent with the distributed nucleation model presented in Sect. 2.2. In this case, uniform sized bubbles may be assumed to exist at any cross section. From sphericity, the interfacial area density A; is given by

(67)

where N is the bubble number density and rB the bubble radius. also,

(68)

so that

(69)

2 1 6

The growth of the bubbles is assumed to be controlled by transient conduction. An analytical solution of thermally-controlled bubble growth for constant values of liquid superheat due to Scriven ( 1959), expressed by an approximation given by Labuntsov et al. (1964) gives the heat input from the liquid to the bubble-liquid interface as:

q" i,net = h6.Tsup

where

with tne Nusselt number expressed in terms of the Jakob number as

Nu = -Ja 1 +- - +- . 12 { 1 ( n )2/3 n } 1T. 2 6Ja 6/a

The Jakob number is defined as

(70)

(7 1 )

(72)

(73)

While this expression in general is valid only for uniform superheat and not for variable pressure fields (Jones and Zuber, 1978), it has been shown approximately correct when the m1 superheat is used (Wu et al., 198 1 )

(b) For bubbly-slug flow i n the range o f 0.3 < a <0.8, i t i s assumed that some of the bubbles coalesce to form larger (Taylor) bubbles while others continue to grow according to Eqs. (70) to (73). Thus, two classes of bubbles are assumed to exist and grow at different rates.

Since vapor generation may take place on the surface of both kinds of bubbles, the total interfa­cial flux is given by

'I" i,net = q" sAs + 'I" BAB (74)

which is the sum of the net heat flux going into evaporation along the slugs plus that for the small, spherical bubbles. The Taylor bubbles are assumed to be cylinders which, at a=O.8 absorb small­er bubbles and merge with one another to form annular flow (Wu et al., 1981 ). The interfacial area density is thus given by

(75)

For the slugs, the area density is

(76)

2 1 7

and for the bubbles i t i s given by

(77)

Note that the total void fraction is the sum of that due separately to the slugs and the bubbles given by

(78)

with the void fraction for the slugs given by the relation

(79)

where aOrN1 = 0.3 and asma• = 0.8 The heat transfer coefficient appropriate for Taylor bubbles

is approximated by that given in the TRAC-PIA code (LASL, 1 979) for slug flows as

(80)

so that

(8 1)

(c) For the case of transitional and dispersed droplet flows where a> 0.8, the difference is in the friction multiplier as expressed in Eq. (61) to (64). The heat and mass transfer occur on liquid droplets fonned as a result of bubble coagulation and droplet entrainment from the lateral surface of the Taylor bubbles. In this case, the interfacial area density is given by

(82)

and the interfacial heat flux is

(83)

In this case, the Nusselt is assumed to be constant at a value of 16 as suggested by Solbrig et al (1978). the droplet diameter is assumed to be

R aWe d = 2 2ev(Wv - WI) (84)

where the Weber number We is assumed to be constant at 5 .0. Note that even though from the standpoint of mass conservation there is assumed to be no slip, the calculation of the droplet heat

2 1 8

and mass transfer requires a value for relative velocity and this is calculated based on solid par­ticle dynamics. A simple formulation which eliminates the transcendental nature of the drag­Reynolds number relationships was used as given by Jones ( 1 984). The terminal Reynolds num­ber for the droplets is taken as a function of the Archimedes number as:

{ (

4 )O.452}-

Re = 1 + 0.0487 3Ar

1 .74 {A,. where

A, < 3.227xlOS

Ar � 3.227x105

Re s 2xIOS

(85)

(86)

The Reynolds number is based on the droplet diameter and the terminal velocity, taken identically as the slip velocity.

Constitutive equations - flow regime map 2. In bubbly flows there are no differences from

flow regime map 1 . The surface area density is proportional to a2J3 assuming NI/3 is constant. If the void fraction exceeds 0.3, the surface area density is influenced by two opposing effects. On the one hand, the continuing bubble growth and distortion of their shape tend to increase the area density, Ai . On the other hand, the coalescence and formation of large bubbles tends to reduce Ai. The variation of Ai with void fraction here may be small so it can be assumed that Ai remains constant having the value obtained at a=O.3. The heat flux for bubbly flow is obtained from Eqs. (70) to (73).

For the case where a � 0.7 ' the nansition to dispersed droplet flow occurs and the interfacial

area density is given by

(87)

Thus the variation with a is assumed to be symmetrical about a = 0.5 (Aj -a213 if a < 0.3, and Ai -(l-a)213 if a > 0.7). The Nusselt number for the heat flux is determined by Eqs. (82) to (84).

Wall nucleation. The wall nucleation process is desclibed in previous section 2.2. The wall nucleation source term is taken from Eq. (47) to be

lwm = Nnsfmax5. . A

(88)

2 1 9

Bulk nucleation. The nucleation source term for the numerical model is thus obtained from

Eq. (45) as

dNB NB(Gi)w J8 = - = .

dt llz (89)

The model applies the bulk. nucleation zone to the mesh cell spanning the throat. consistent with maximum superheat at this location. Bulk nucleation thus vanishes everywhere except for the throat cell and in this cell is additive to the continuous wall nucleation determined from the model of Shin and Jones ( 1988). For the computational model the characteristic time for bulk nucleation is thus taken as lltn = &/w where Ilz is the mesh spacing.

The detailes of the numerical model are not included in this chapter. The interested reader is directed to the papers by Blinkov. Jones, and Nigmatulin ( l992a/b).

For solving the equations given by Eqs.(52) to (56), a semi-implicit method (EPR!. 1 983) is used having the stability criterion

llz llt s --Wmax (90)

where Wmax is the maximum axial velocity. This significantly increases the allowable time step compared with explicit methods and makes feasible the use of a microcomputer. The channel was divided into a number of cells with uniform spatial mesh spacing. All thermodynamic variables were taken to be cell-centered whereas velocities were computed at cell faces. Details are given in Blinkov et al. ( 1988a/b).

The experience with the computations were reasonably good from the viewpoint of using a microcomputer for the calculations. The entire program required approximately 80 KB of main memory on a Hewlett-Packard HP-9816S having floating point hardware based on a Motorola MC-6800 microprocessor. CeU lengths for typical calculations were 5 mm for the BNL geome­tty (Abuaf et al., 198 1 ) representing 1 10 cells, and -2.5 mm for the Marvikin geometry (EPRI, 1 982) requiring 40-70 cells, and for the geometry of Sozzi and Sutherland ( 1975) requiring 64 cells.

Computational times required to reach a steady state solution usually took 200-300 time steps with each time step requiring -4 internal iterations in the early stages of solution and 1 -2 itera­tions near steady state. A single iteration required approximately 40 seconds so that calculational times ran 8- 10 hours. These times would be significantly faster on more modern microproces­sor-based personal computers . this is a relatively old desk-top computer and it is expected that computational times would be considerably shortened using more modern systems.

220

550

co � a. Z :E a c: � u.i C,) c: < � c: (/) u. (/) 0 w 0.2 0.4 6 c: > a.

0.2

DISTANCE, z(m)

Figure 2 1 . Comparison between experimental

and calculated distributions of pressure and void

fraction for the BNL nozzle run 273.3 Flow rates: measured = 8.71 kg/s, calculated = 8.8 kg/so Po = 0573 MPa; To = 421 .85K; !lTo =

8.4K; pco = 0.442 MPa. Dot-dash lines,flow re­gime map 1; solid lines, map 2 ; dash lines, bubbly flow only.

_ 10'0 420 _1 � � lo' 418 �� � u.i� �� 10' �416 �� � It: 10' W 414 Q. .. � �� 10' 412 CD ::J CD

10"

Figure 22. Void development parameters for

BNL nozzle run 273.3

Void development downstream-­small nozzles. Calculations for the verti­

cal nozzle in the experiments of Abuaf et

al. ( 1 98 1 ) taken at Brookhaven National

Laboratory (BNL) are shown in Fig. 2 1 , where comparisons are shown between

calCUlated void fraction and pressure pro­

files with those measured in the experi­

ment.

In all these calculations, it was con­

firmed that the calculated effect of bulk

nucleation was negligible. This lends ad­

ditional support to the experimental evi­

dence that for nozzles of this small diame­

ter wall nucleation predominates. Note

also that the correlations whic h are used to

calculate the wall nucleation rate were

based on this assumption.

Three methods of calculation were

used to make these calculations shown in

Fig. 2 1 :

1 . Flow regime consisting of

bubbly, bubbly-slug, transitional,

and dispersed flows, dot-dash

lines;

2. Flow regime 2 conslstmg of

bubbly, transitional chum turbu­

lent. and dispersed flows, solid

lines;

3. Bubbly flow for any void frac-

tion, dash lines.

One can see that the three calculated

values of void fraction in Fig. 21 differ

slightly due to the differing interfacial

area density for phase change. Figure 22

221

shows curves of the liquid temperature, T" vapor temperature, Tv, interfacial area density, Ai, mixture velocity, w, bubble number density in the region up to 30% voids, 10g(N), and local fro­zen sonic velocity. Comparing with Fig. 2 1 , it is seen that bubbly flow gives the highest void cal­culated since the interfacial area density is the largest. Model flow regime 1 gives a smaller void fraction since the area density is lower and decreases after a. = 0.3 (Fig. 2 1 b). For the third model, flow regime 2, the interfacial area density is constant after a. = 0.3, but the void fraction is still lower. lIDs is because as seen in the pressure profiles, this third model results in increased super­heat offsetting the effect of reduced interfacial area density. While the void fraction in the latter case is closer to the data, the pressure profiles are further away. The reasons for these results are unknown but show both the need for careful modeling of surface area density and for more defini­tive experiments to delineate the type of behavior to be expected.

Table 2. Summary of conditions for BNL experiments (Abuaf et aI., 198 1 )3

BNL Run No.

148

288

309

Po (bar) 3 .05

5.30

5 .559

To (K) b.To (K) 394.35 12.80

422.35 4.80

422.25 6.75

poo (bar) 2.06

4.591

4.05

TnmefJS (kgts) 7 .50

7 .25

8 .80

IncfJlc (kgts) 7.80

7.27

8.23

Table 3. Comparison of calculated and experimental values of exit void fraction for the BNL data. Model l is bubbly flow only; Model 2 is for flow regime map 1 .

BNL Run No. Bubbly Flow Only Flow Regime Map 1 Measured Void Data

1 48 0.66 0.59 0.55

273 0.69 0.61 0.57

288 0.52 0.49 0.49

309 0.7 1 0.70

Figure 23 shows similar results for different BNL runs having conditions summarized in Table 2 (conditions for Run 273 are summarized in the caption to Fig. 21 ) . In these cases, only the bubbly flow regime results are shown. Table 3 compares the exit void fraction for the bubbly flow calculation compared with that for the flow regime 1 results in comparison with the data. It is seen that in all cases, the inclusion of a more realistic flow regime calculation which considers the re­duction in interfacial area density due to agglomeration produces results closer to the data.

Critical flow of saturated water through a Laval nozzle having a throat diameter of 3.84 mm is shown in Fig. 24 for the data of Karasev et aI. ( 1977). In this case, wall nucleation dominates by two orders of magnitude the bulk nucleation, again affinning that this is the predominate mode of void formation in the nucleation zone in small geometries. It is seen that the flow becomes over­expanded and goes supersonic downstream of the throat. Substantial expansive cooling of both liquid and vapor are seen.

Void development downstream - large nozzles. Figure 25 shows calculations for a round­entrance short pipe used in Marviken experiments (EPRI, 1982). The single point at the inlet rep-

222

5 -Ca 4 e w

� 3 C/) C/) w 2 0: CL

5 -Ca 4 e w 0: 3 � C/) C/) W 2 0: CL

5

'i 4 e w 0: 3 � C/) C/) W 2 0: CL

BNL RUN 1 .0

0.8 5 i= 0 0.6 « 0: LL

0.4 0 0

0.2 >

0

1 .0

0 .8 t5 �

0.6 � 0: LL

0.4 0 0

0.2 >

0

BNL RUN 309 1 .0

0 .8 5 i=

0.6 � 0: LL

0.4 0 0

0.2 >

0 0.1 0.2 0.3 0.4 0.5

LOCATION (m) Figure 23. Comparison with exper­

imental data for BNL nozzle runs.

resents the measured inlet pressure used to drive the flow calculations. In Fig. 25a,

results are shown with 18 = 0, no bulk nu­cleation. Figure 25b shows calculations where the bulk nucleation model is included. The number density is substantial­ly larger in the early stages of decompres­sion near the inlet when bulk nucleation is

included and the voids begin to grow earli­

er in the nozzle but little effect on the tem­

peratures or pressures is seen. However,

near the exit where void development be­comes significant, the number densities

approach each other and void growth be­

comes similar. This shows that in the larg­er geometries, as the volume to surface ra­tio increases, the role of bulk nucleation

can be expected to become increasingly important.

Figure 26 shows a comparison be­

tween calc ulated and experimental clitical

mass flow rates of the Malviken experi­

ments when the water temperature is near­

ly constant. It is obvious from this case

that the inclusion of bulk nucleation for

this larger geometry substantially im­proves the computation of void develop­

ment with resultant improved prediction

of the critical flows. In this case, accurate calculation is crucial since the flow rate is void dominated.

The transient flow calculations for

Marviken agree well with the experi­

mental values (Figs. 27 and 28) up to

about 30 ms after which the liquid temper­ature begins to become affected by the nonunifonnity in the vessel temperature.

Figure 29 shows that as the Lid ratio of the

nozzle increases, the flow and the model

approach that calculated by homogeneous equilibrium.

101•

;r- '0

.,� 800 of '0" g � 700 �ffi W

I:[ 1:[ 0 ,0' ::l �� i 800 §� W Q. � O W 'o' 1- 50 �� ::l II

,0" 40

.,:l 84

0.04 O.oe O.oe DISTANCE, z( )

1.0

t:I

.. i 0 4 0 �

Figure 24_ Calculated distributions of flashing flow parameters along the small scale Laval nozzle of Karasev et al. ( 1977).

Po = 8.5 MPa; To = 572K; I1To = OK. Mass flow rates: measured, 0535 kg/s; calcu­lated, 0575 kg/so

A computational framework for calculat­ing the behavior of flowing, initially sub­cooled liquids in pipes and nozzles has been described for use on a microcomputer. The model uses the distributed nucleation model of Shin and Jones (1988) coupled with a previous model for bulk nucleation developed by Sopolenkov and Blinkov

( 1983) to determine appropriate initial conditions for flashing and void development downstream of the tlroat. The model, a five­

equation, mechanical equilibrium, thermal nonequilibrium model incorporates the distributed nucleation model for wall

nucleation in small ducts, as well as a

previously developed model for bulk nucleation on suspended particles in larger geometries described in Section 2.3.

223

Figure 25. Calculated disnibutions of flashing flow parameters for the Marviken nozzle. 53 Po = 452 MPa; To = 507K; !lTo = 23.85K. Massfluxes: measured, 51 ,000 kg/mls; calculated, 55,280 kg/m)s.

10"

� 10'2

:;:-� '! z. 10'0 &� . CJ) � � lOS < 0 a: a: Z w Q aI 1 00 � :! < :J W z c3 w 1Q4 :: .. z gj

ffi w 100

10'4

� 1012

:;:-� '! � 10'0 �� .CJ) w z 10' � � � � 10'

� � .. z g � 104 z al

� 102

100

f- 1820 (a) Wall nucleation only

520

600

580 � Ii � Q. � 560 :! 4 c: :: ui � � 3 ffi 540 CJ) Q. CJ) :! w � 52O g: 2

50

480 0,4 0.8 1 _2

DISTANCE, z(m)

1 .0

es Z

0.5 � U < a:

0.4 U. 0 (5 > 0_2

(b) Combination of wall nucleation and bulk nucleation

620

500

580 1 .0 � � Ii � 560 0.8 es Q, 4 ::! Z � 15: 0 ffi 540 � 3 0,6 § Q" :: < ::! CJ) a: CJ) u. � 520 � 2 0,4 0

Q" (5 > 500 0 2

480 00 0_4 0,8 1 _2 DISTANCE, z(m)

224

"i N E � (5' x 3 u. U) U) c( � x

-·-------------1"

� -----------------6

o 0.4 0.8 DISTANCE, z(m)

Figure 26. COmp811S0n between calculated and experimental critical mass fluxes for the Marviken nozzle 181'ge tube (d=O.509 m) having a round entrance (EPR!, 1982). 1 . calculation with wall nucleation only; 2. calculation with both wall and bulk nucleation; 3.po = 452 MPa, To = 507K; 4.po = 4.52 MPa, To = 507K;5 .po = 356 MPa, To = 507K.

Semi-implicit difference methods were used with all properties computed at cell centers and a donor cell method used to calculate convective flux effects with velocities computed at cell boundaries. The sys-tem was solved by Newton iteration. Typical computational times on a Hewlett-Packard 98 16S microcomputer based on the MC-6800 processor chip at 8 MHz were 40-45 seconds per time step and a total of 8-10 hours for a converged solution with all variables within 1 part in 103 .

4.GENERAL SUMMARY

� C) .: E ui .. « e: � 0 .. u. x

C? 0 2

The model consisted of mixture and vapor mass conservation equations, a mixture momentum equation, the liquid energy equation, and a bubble transport equation. Spherical bubble growth was calculated by traditional thelmaUy-li­mited growth methods using local super­heat.

For closure, a relatively simple wall friction model was utilized represented by a friction multiplier having different for­mulations in four different regions of void fraction.

ctf a. �

0: ui 0:

4 ::l CJ)

8 CJ) w e: a.

TIM E, t(s)

It is shown that both number and size of bubbles must be accurately determined for initial calculation of flashing void developmentdownstream of restrictions. This chap­ter presents a new method of accurately determining both which results in accurate cal­culation of downstream void development.

Figure 27. Blowdown in the Marviken nozzle experiment: L = 0.955 m,D = 0.5 m.53 1 . experiment; 2 calculation; 3. pressure history (Exptl.)

First. the concept of flashing from initial­ly subcooled liquids is placed into context from the viewpoint of the homogeneous equilibrium model. It is shown that for isen­tropic flow, Fanno flow, and Rayleigh flow that the maximum flow rates possible in straight ducts are those where flashing be­

gins immediatel y at the exist of the duct, and that the exit liquid velocity can never exceed the two-phase sound speed at zero quality.

Forreal fluids, a wall cavity model is de­sClibed for use in the calculation of nucleation rates and bubble number .densities at flashing inception in nozzles, and subse­quently in the calculation of void development downstream of minimum area zones. The model is based on the physics of the nucleation phenomena in flashing and consid­ers transient conduction to be the sole means of heat transfer from the superheated liquid to the vapor bubble. 'This latter assumption

(i) 45 C\I

E C, C. 35 ·E X

25 => -.J u. CI) CI) 1 5 < � x

CO) 5 b

u.i 6 I-< a: � -.J LL x 2

M 6

Figure 29. Critical flow for different size Mar­viken nozzles . (EPRI, 1982). 1 . Model pres­ented, D = 0.5 m, L = 0.955 m; 2. model pres­ented, D = 0.3 m, L = 1 .266 m; 3. homogeneous equilibrium.

225

8

ca a. 6 �

c: u.i

a: 4 :J CI) CI) w 2

a: a.

0

TIME, t(s) Figure 28. B lowdown of the Marviken nozzle experiment: L = 1 .226 m, D = 0.3 m.53 1 . flow rate from experiment; 2 flow rate from calculation; 3. experimental pressure history.

is expected to over estimate the bubble departure frequencies at a given nucleation site, expecially under high superheat conditions.

The activation critelion developed for site nucleation is one sided due to the uni­form superheat rather than two sided as in boiling. A figure of-merit for the particular fluid solid combination is then determined which yields minimum nucleation surface energy� per site. Characteristic site nucleation frequencies, and number den­sities of nucleation sites of gi ven sizes are then obtained from data.

226

A bubble transport equation is used to predict the number density and size of bubbles at the

throat. Throat superheats are calculated with a standard deviation of 1 .9K for throat superheats up

to -lOOK and expansion rates between 0.2 bar/sec to over 1 Mbar/sec, extending previous corre­

lations by more than three orders of magnitude. Throat void fractions for all data found in the literature are less than 1 % confining earlier assumptions and allowing nozzle critical flow rates to be calc\.llated with an accuracy of -3%.

The last section of this chapter describes a quasi-one-dimensional, five-equation model which was used on a microcomputer to calculate the behavior of flowing, initially subcooled, flashing liquids. Equations for mixture and vapor mass conservation, mixture momentum conser­vation, liquid energy conseIVation, and bubble transport were discretized and linearized semi­

implicit! y, and solved using a successive iteration Newton method. Closure was obtained through simple constitutive equations for friction and spherical bubble growth, and a new nucleation model for wall nucleation in small nozzles combined with an existing model for bulk nucleation

in large geometries. Good qualitative and quantitative agreement with experiment confirms the adequacy of the nucleation models in determining both initial size and number density of nuclei.

It is shown that bulk nucleation becomes important as the volume-to-surface ratio of the geome­

try is increased.

5. NOMENCLATURE

English

A Area Ai Interfacial area density Ar Archimedes number B Coefficient

c Sonic velocity C Specific heat d Diameter or hydraulic diameter ff Friction factor g Acceleration of gravity G Mass flux Gi Gibbs number h Heat transfer coefficient

Enthalpy J Nucleation rate Ja Jacob number k Thermal conductivity K Surface force coefficient L Length m Mass flow rate M Mach number N Bubble number density

Nu Nusselt Number p Pressure Pr Prandtl number

II" Heat flux

rB Bubble radius T Radius Re Reynolds number s Entropy S Cross section area St Stanton number

t Time T Temperature AT Temperature difference u Specific internal energy

v GeneralVelocity w Velocity in z-direction We Weber number x Quality

y Distance from a surface z Axial coordinate

Greek

a Void fraction or thermal diffusivity r Volumetric vapor generation rate

5 Bubble diameter or boundary layer thickness A Positive difference

J.1 Dynamic viscosity v Kinematic viscosity

p Density a Surface tension r: Expansion rate , Shear stress or dimensionless time e Dimensionless temperature u specific volume

� perimeter

Subscripts

a Area B Bubble or bulk c Critical or cavity

227

228

d Droplet e Equilibrium f Saturated liquid fg Positive saturated liquid-vapor difference Fi Flashing inception Fio Flashing inception without convection g Saturated vapor I Liquid (not necessarily at saturation) m Mixture n Nucleation zone ns Nucleation site o Stagnation sat Saturation sup Superheated S Re Taylor bubble

Throat v Vapor (not necessarily at saturation) w Wall wm Maximum wall value

Terminal 1cp Single-phase 2cp Two-phase

6. REFERENCES

There are a considerable number of bibliobraphical citations given in this section to which no reference is made in the main body of this chapter. They ae provided for information. The inter­ested reader is directed to the paper by Shin and Jones ( 1992) for a detailed summary ofthe litera­ture.

1 . Abdollahian, D., Healzer, J., Jansssen, E. and Amos, c., 1982. Critical Flow Data Review and Analysis. EPRI report NP-2 1 92.

2 . Abuaf, N. , Jones. O.c., and Wu, B .J .c., 1 980. Critical flashing flows in nozzles with subcooled inlet conditions. Polyphase Flow and Transport Technology, presented at the Symposium on Polyphase Flow and Transport Technology, Century 2 - Emerging Technology Conferences, San Francisco, CA, Aug. 13-15 . , pg. 65-74.

3. Abuaf, N., Zimmer, G.A., and Wu, BJ.C., 198 1 . AStudy of Nonequilibrium Flashing of Water in a Converging-Diverging Nozzle. BNL Report NUREG/CR-1864, BNL-NUREG-5 13 17, Vol. 1 .

4 . Abuaf, N., Jones, O.C., Jr., and Wu, B .lC., 1983. Critical Flashing Flow in Nozzles with Sub­cooled Inlet Conditions. Trans. AS ME, J. Heat Transfer, 105, 379.

229

5. Aguilar, E, and Thompson, S., 198 1 . Non-Equilibrium Flashing Model for Rapid Pressure Transient. ASME Paper No. B I-HT-35.

6. Alimgir, M.D., and Lienhard, J.H., 1 979. Private communication.

7. Alimgir, M.D., and Lienhard, 1. H., 19B 1 . Correlation of pressue undershoot during hot-water depressurization. Trans. ASME, J. Heat Transfer, 103, pg. 52.

8. Ardron, K.H., 1978. A Two-Fluid Model for Critical Vapor-Liquid Flow. Int. J. Multiphase Flow, 4, 323.

9. Ardron, K.H., and Ackerman, M.C., 1978. Studies of the Critical Flow ofSubcooled Water in a Pipe. Proc. of 2nd CSNI Specialist Meeting, June, Paris.

10. Bailey, lE, 195 1 . Metastable Flow of Saturated Water. Trans. ASME, pg. 1 109.

1 1 . Bankoff, S.G., 1958. Entrapment of Gas in the Spreading of a Liquid Over a Rough Surface. AIChE Journal, 4, p. 24.

1 2. Bauer, E.G., Houdayer, G .R, and Sureau, H.M., 1976. A Non-Equilibrium Axial Flow Mod­el and Application to LOCA Analysis: the CLYSTERE System Code. Paper presented at the OECD/nea Specialists' Meeting on Transient Two-Phase Flow," Toronto, Canada, Aug.

13. Beattie, D.RH., 1 973. A Note on the Calculation of Two-Phase Pressure Losses. Nuel. Eng. Des., 25, pp. 395-402.

1 4. Bergles, A.E., and Roshenow, W.M., 1962. Forced Convective Surface B oiling Heat Transfer and Burnout in tubes of Small Diameter. MIT Report No. 8767-21 , May.

15 . Blinkov, Y.N., Jones, O.c., and Nigmatulin, B .I., 1988. Flashing ofinitially subcooled liquids in nozzles: 2. A 5-equation model for vapor void development. Proc. 3rdJapan-U.S. Seminar on Two Phase Flow Dynamics, Ohtsu, Japan, July 15-20, paper 0.4.

1 6. Blinkov, Y.N., Jones, O.c., and Nigmatulin, B .I., 1988. Flashing of initially subcooled liquids in nozzles: 3. Comparisons with Experiment. Proc. 3rd Japan-U.S. Seminar on Two-Phase Flow Dynamics, Ohtsu, Japan, July 15-20, paper 0.4.

17. Blinkov, Y.N., Jones, O.C., and Nigmatulin, B.I., 1 992a. Flashing of initially subcooled liq­uids in nozzles: 2. Comparisons with Experiment using a Homogeneous 5-Equation Model. In Two-Phase Flow Dynamics, Begell House Press, New York.

lB. B linkov, Y.N., Jones, O.c., and Nigmatulin, B .1. 1992b. On Nucleation and Flashing of FlOwing Liquids in Nozzles: 2. Comparison with Experiments using a 5-Equation Model for Va­por Void Development. Int. J. Heat and Mass Trans., in press.

1 9. Brown, RA., 1961 . Flashing Expansion of Water Through a Converging-Diverging Nozzle MS Thesis, Univ. of California, Berkeley, UKAEC Rep. UCRL-6665-T

230

20. Celata, C.P., Cumo, M., Farello, G.E., and Incalcaterra, P.C., 1982. CIitical Flow of Sub­cooled Liquid and Jet Forces. ENEA-RT/lNC(82 1 8) .

21 . Chow, H. , and Ransom, V.H., 1984. A Simple Interphase Drag Model for Numerical Two­Fluid Modeling of Two-Phase Flow Systems." Second Proc. Nucl. Thermal Hydraulics. Sum­mer Annual ANS Meeting, pp. 137- 145

22. Clark, H.B. , Strenge, P.S., and Westwater, J.W., 1959. Active Sites for Nucleate Boiling. Chern. Eng. Prog. Symposium Ser., 55, p. 103.

23. Dobran, F., 1 985. A Nonequilibrium Model for the Analysis of Two-Phase Critical Flows in Tubes. Presented at the 23rd National Heat Transfer Conference, Denver, Colorado, August 4-7.

24. Edwards, A.R. , 1968. Conduction Controlled Flashing of a Fluid and the Prediction of Criti­cal Flow Rates in One-Dimensional System. AHSB (S) R- 147, UKAEA, AERE.

25 . EPRI, 1982. The Marviken Full Scale Critical Flow Tests. Vol. 1 . Summary Report. EPRI report NP-2370.

26. Fincke, J .R., 198 1 . The Correlation of Nonequilibrium Effects with Choked Flow with Sub­cooled Upstream Conditions. Conference Paper, ANS Small Break Specialist Meeting, Califor­nia, 4- 1 - 4-30.

27. Fincke, J.R ., Collins, D.R., and Wilson, M.L., 1 98 1 . The Effects of Grid Turbulence on Non­Equilibrium Choked Nozzle Flow. EG&G report NUREG/CR-1997, EGG-2088, Apr.

28 . Forster, H.K., and Zuber, N., 1954. Growth of a Vapor Bubble in a Superheated Liquid. J. Applied Physics, 25, 474

29. Fritz, G., Riebold, W., and Sculze, W., 1976. Studies on Thermodynamic Non-Equilibrium in Flashing Flow. Paper presented at the OECD/NEA Specialists Meeting on Transient Two-Phase Flow, Toronto, Canada, Aug.

30. Griffith, P., and Wallis, J., The Role of Surface Conditions in Nucleate Boiling. Preprint 106, ASME-AIChE Heat Transfer Conf., Storrs, Conn., 1959.

3 1 . Han, D., and Griffith, P., 1965. The Mechanisms of Heat Transfer in Nucleate Pool Boiling. Int. J. Heat Mass Transfer, 8, p. 887.

32. Hendricks, R.C., Simoneau, R.J., and B arows, R.F., 1976. Two-Phase Choked Flow of Sub­cooled Oxygen and Nitrogen. NASA-TN-D-81 69.

33. Henry, R.E., Fauske, H.K., and McComas, S .T., 1970. Two-Phase Critical Flow at Low Qual­ities, Part I: Experimental. Nuclear Sci. and Eng., 41, 79.

34. Henry, R .E., and Fauske, H.K., 197 1 . The two-phase critical flow of one component mixtures in nozzles, orifices, and short tubes. Trans. ASME, J. Heat Trans., 93, pg 179.

23 1

35. Hsu, Y.Y., 1 962. On the Size Range of Active Nucleation Cavities on a Heating Surface. Trans. AS ME, J. Heat Trans., 94, pg. 207.

36. Hsu, Y. Y., 1 972. Review of Critical Flow, Propagation of Pressure Pulse, and Sonic Velocity. NASA report TND-6814.

37. Hsu, Y.Y., and Graham, RW., 1961 . An Analytical and Experimental Study of the Thermal

Boundary Layer and the Ebullition Cycle in Nucleate Boiling. NASA report TN-D-594.

38. Ishii, M. , and Mishima, K., 1983. Flow Regime Transition Criteria Consistent with Two­Fluid Model for Vertical Two-Phase Flow. Argonne Report NUREG/CR-3338, ANL-83-42.

39. Jones, a.c., 1980. Flashing inception in flowing liquids. Trans. ASME, J. Heat Trans., 102, pg. 439-444.

40. Jones, a.c., Jr., 1982. Toward a Unified Approach for Thermal NonequilibIium in Gas-Li­quid Systems. Nucl. Eng. Des., 69, 57.

4 1 . Jones, O.C., Jr., 1 984. Thermal Design Concepts for the Rotating Fluidized Bed Reactor. Nucl. Sci. Eng., 87, pp. 13-27.

42. Jones, a.c., and Shin, T.S. , 1983. Upstream flashing inception and critical flow: II. Effect of liquid compressibility. Trans. ANS, 46, pg 885.

43. Jones, O.C., Jr. , and Shin, T.S., 1986. An Active Cavity Model for Flashing. Nuc. Eng. Des., 95, pg. 185 .

44. Jones. O.c., Jr., and Saha, P., 1977. Non-Equilibrium Aspects of Water Reactor Safety. in Thermal Hydraulic Aspects of Nuclear Reactor Safety: Vol. 1. Light Water Reactors. O.c.

Jones, Jr. and S .G. B ankoff, Eds., ASME, New York.

45 . Jones, a.c., Jr., and Shin, T.S. , 1986. An Active Cavity Model for Flashing. Nuc. Eng. Des., 95, pg. 1 85- 196.

46. Jones, O.C. Jr., and Zuber, N., 1978. Bubble Growth in Variable Pressure Fields. Trans. ASME, J. Heat Transfer, 100, 453.

47. Karasev, E.K., Vasinger, V. v., Mingaleyeva, G.S. , Tru bkin, E.I., 1977 . Investigation of Water Adiabatic Expansion from the Saturation Line. Nuclear Energy, 42, 6, pp. 478-48 1 .

48. Kochamustafaogullari, G., and Ishii, M., 1982. Interfacial Area and Nucleation Site Density in Boiling Systems. lot. J. Heat Mass Trans., 26, 1377.

49. Labuntzov, D.A., Kolchugin, B .A., Golovin, V.S . , Zakharova, E.A., and Vladimirova, L.N.,

1964. High Speed Camera Investigation of Bubble Growth for Saturated Water Boiling in a Wide

Range of Pressure Variations. Thermophysics of High Temperature, 2, 2, pp. 446-453.

50. LASL, 1 979. ''TRAC-PIA. An Advanced Best-Estimate Computer Program for PWR LOCA Analysis. Los Alamos report NUREG/CR-0665, LA-7777-MS .

232

5 1 . Levy, S., and Abdollahian, D., 1982. Homogeneous Non-Equilibrium Critical Flows.

52. MaInes, D. , 1975. Critical Two-Phase Flow Based on Non-Equilibrium Model. In Non­Equilibrium Two-Phase Flow, RT. Lahey and G.B. Wallis, Eds., 1 1 .

53. Marviken, 1979. Marviken Full Scale Critical Flow Tests," Dec.

54. McFadden, I.H., et at, 198 1 . ''RETRAN-02. A Program for Transient Thermal-Hydraulic Analysis of Complex Fluid Flow Systems. EPRI Report NP- 1850 CCM, V. 1 .

55. Moody, FJ., 1966. Maximum Two-Phase Vessel Blowdown from Pipes. Trans. ASME, J. Heat Transfer, 88, 285.

56. Plesset, M.S., and Zwick, S.A, 1954. The Growth of Vapor Bubbles in Superheated Liquids. J. Appl. Physics, 25, 4.

57 . Powell, AW., 1961 . Flow ofSubcooled Water Through Nozzles. Westinghouse Electric Cor­poration, WAPD-PT -(V)-90, April.

58. Reocreux, M., 1974. Contribution a l 'Etude des Debits Critiques en Ecoulement Diphasique Eau-Vapeur. Ph.D. Thesis, Universite Scientifique et Medicale de Grenoble, France.

59. Reocreux, M., 1976. Experimental Study of Steam-Water Choked-Flow. Paper presented at the OECD/NEA Specialists ' Meeting on Transient Two-Phase Flow, Toronto, Canada, Aug.

60. Richter, HJ., 198 1 . Separated Two-Phase Flow Model: Application to Critical Two-Phase Flow. EPRI report NP-1800, April.

6 1 . Rivard, W.c., and Travis, 1.R, 1980. A Non-Equilibrium Vapor Production Model for Criti­cal Flow. Nuel. Sci. Eng., 74, 40.

62. Rohatgi , U.S ., and Reshotko, E., 1 975. Non-Equilibrium One-Dimensional Two-Phase Flow in Variable Area Channels. In Non-Equilibrium Two-Phase Flow, RT. Lahey and G.B. Wallis, Eds., 47.

63. Saha, P., 1978. A review of two-phase steam-water critical flow models with emphasis on thermal nonequilibrium. Brookhaven report BNL-NUREG-50907, Sept.

64. Saha, P., Abuaf, N., and Wu, B J.C., 198 1 . A Non-Equilibrium Vapor Generation Model for Flashing Flows. ASME Paper No. 8 1-HT-84.

65. Schlichting, H., Boundary Layer Theory, McGraw-Hill, New York, 1979.

66. Schrock, Y.E., Starkman, E.S. and Brown, R.A, 1977. Flashing Flow of Initially Subcooled Water Through Nozzles. Trans. ASME, J. Heat Transfer, 99, 263.

67. Scriven, L.E., 1959. On the Dynamics of Phase Growth. Chern. Eng. Sci., 1, pp. 1-13 .

68 . Seynhaeve, I .M., Giot, M.M., and Fritte, AA, 1976. Non-equilibrium effects on critical flow rates at low qUalities. Presented at the specialists meeting on transient two-phase flow, Toronto, Canada, Aug. 3-4.

233

69. Shin, T.S., and Jones, a.c., Jr., 1988. Flashing of initiaUy subcooled liquids in nozzles: 1. A distributed nucleation model for flashing inception and critical flow. Proc. 3rd Japan-U.S. Semi­nar on Two-Phase Flow Dynamics, Ohtsu, Japan, July 15-20, paper G.3.

70. Shin, T.S., and Jones, O.C., Jr., 1992a. Flashing of initially subcooled liquids in nozzles: 1. A distributed nucleation model for flashing inception and critical flow. In Two-Phase Flow Dy­namics, Begell House Publishers, New York, 1992.

7 1 . Shin, T.S ., and Jones, O.c., Jr., 1992b. On Nucleation and Flashing of Liquids in Nozzles: 1. A Distributed Nucleation Model. Int. J. Heat and Mass Trans., in press.

72. Shoukri, M.S .M., and Judd, R.L., A Theoretical Model for Bubble in Frequency in Nucleate Pool B oiling Including Swface Effects. 6th International Heat Transfer Conference, Toronto, Canada, 1978.

73. Simoneau, R.J., 1 975. Pressure Distribution in a Converging-Diverging Nozzle during Two­Phase Choked Flow of Subcooled Nitrogen. In Non-Equilibrium Two-Phase Flows, R.T. Lahey and G.B. Wallis, Eels., 37.

74. Simpson, H.C., and Silver, R.S . , 1962. Theory of One-Dimensional Two-Phase Homoge­neous Non equilibrium Flow. Proc. Inst. Mech. Eng. Symposium on Two-Phase Flow, 45.

75 . Solbrig, C.W . • McFadden, J.H., Lyczkowski, R.W., and Hughes. E.D. , 1978. Heat Transfer and Friction Correlations Required to Describe Steam-Water Behavior in Nuclear Safety Stu­dies. AIChE Sym. Ser. 74, No. 174, pp. 100-128.

76. Soplenkov, K.I., and B linkov, Y.N., 1 983. Heterogeneous Nucleation in the Flow of Super­heated Liquid. In Multi-Phase Systems Transient flows with Physical and Chemical Trans­formation, R.I. Nigmatulin and AI. Ivandayev, Eds., Moscow State University, pp. 105- 109.

77. Sozzi, G.L., and Sutherland, W.A, 1975. Critical Flow of Saturated and Subcooled Water at High Pressure. General Electric Report NEDO-13418.

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79. Wallis, G.B ., 1969. One-Dimensional lWo-Phase Flow. McGraw Hill, New York.

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8 1 . Wegener, P.P., 1975. Non-Equilibrium Flow with Condensation. Acta-Mechanica, 2 1 ; 65.

82. Winters, W.S., and MeTte, H., 1979. Experiments and Non-Equilibrium Analysis of Pipe Blowdown. Nuel. Sci. Eng., 69, 4 1 1 .

83. Wolfert, K., 1 976. The Simulation of Blowdown Processes with Condensation ofThennody­namic Non-Equilibrium Phenomena. Paper presented at the OECD/NEA Specialists Meeting on Transient Two-Phase Flow, Toronto, Canada, Aug.

234

84. Wu, B .lC., Abuaf, N., and Saha, P., 198 1 . A Study of Non equilibrium Flashing of Water in a Converging-Diverging Nozzle. BNL Report NUREG/CR- 1 864, BNL-NUREG-5 1 3 17, Vol. 2

85. Zimmer, G.A., Wu. B J.C., Leonhard, W.L., Abuaf, N. and Jones, O.c., Jr., 1979. Pressure and Void Distributions in a Converging-Diverging Nozzle with Non-Equilibrium Water Vapor Gen­eration. BNL repOlt 'BNL-NUREG-26003.

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235

TWO-PHASE FLOW DYNAMICS

M.Z. Podowski

Department of Nuclear Engineering & Engineering Physics, Rensselaer Polytechnic Institute, Troy, New York 12180-3590

1. INTRODUCTION

Applications of two- and multi-phase flows in modern technologies are very extensive. Among others , they include: power systems, chemical and processing plants, petroleum industry, and even electronic and computer industry. B ecause of the complexity of phenomena involved, our understanding of various important mechanisms governing two-phase flow and heat transfer accompanied by phase change (boiling or condensation) is still not fully satisfactory. The above applies to both steady-state processes and, in particular, to transients. Nevertheless, the state-of-the-art in the analysis of two-phase system dynamics has steadily been advancing during recent years, including both theory and experiments, substantially improving the predictive capabilities of analytical models. The purpose of the present materials is to discuss selected aspects of modeling and analysis of boiling system dynamics. First, unsteady-state processes in boiling channels and loops will be summarized, including the physics of governing phenomena, mathematical models, external perturbations and boundary conditions. Next, the analysis of selected transients will be presented. And finally, multi­dimensional aspects of two-phase flow and heat transfer will be discussed.

2. TRANSIENTS IN BOILING LOPS

2.1 Depending on the particular application, different design concepts of boiling

loops (and systems) can be used. Schematics of some typical boiling systems are shown in Figures 1 through 4. A basic boiling loop is shown in Fig. 1 . The loop consists of a multi-channel heaterlboiler and a condenser, as well as piping connecting them. The pump supplies liquid coolant to the boiler. It should be mentioned that in natural-convection-driven systems the loop does not contain the pump. Following partial or complete evaporation in the boiler, two-phase vaporniquid mixture or single phase (superheated) vapor flows to the condenser, where it is converted back into liquid phase. In particular, the schematic shown in Fig. 1 represents a typical experimental test facility used to study boiling system dynamics. It can also represent more complex industrial systems, such as once-through boilers in power plants (in

236

particular, once-through steam generators in PWR nuclear power plants). In this case, however, the section connecting the boiler to the condenser would include a turbine (or turbines) not shown in the figure.

KOUT

R ISER CONDENSER

r - - - - - -

I lpARALLEL I CHANNELS I I I I L _ _ _ _ _ _

LOWER PLENUM DOWNCOMER

Figure 1. A loop containing parales boiling chanels.

A boiling loop in which vapor is separated from partially evaporated liquid, passed to another part of the system, external to the loop (e.g. , turbine), and eventually returns as condensed liquid (feedwater), is shown in Fig. 2. In this case, only the unevaporated liquid phase recirculates inside the loop.

The loop in Fig. 2 can represent a simplified version of boiling water nuclear reactor (BWR) systems (with or without the pump), a U-tube steam generator (with the pump removed and the reactor core replaced by U-tubes, as shown in Fig. 3), or any industrial boiling system with internal recirculation. A more detailed schematic of the recirculation loop in a modern BWR (General Electric manufactured BWR-5 or -6) is shown in Fig. 4. In this case a part of the single-phase liquid (about 113) in the downcomer flows to the recirculation pumps and, then, drives the remaining portion of the coolant through the jet-pump diffusers to the lower plenum. In reality, the system is even more complex, since several (up to ten) jet pumps are connected to each recirculation pump.

Steam Separator A SW

SW

� W SW,in

Riser

__ L_ I Upper

plenum

Core

Lower plenum

Figure 2. A simplified schoomatic of a boiling water reactor (BMR) loop.

2.2

237

The analysis of transients in boiling loops and systems involves the modeling of several inter-related phenomena, such as: single-phase and two­phase flows in pipes/channels (see Fig. 5) and plena, boiling heat transfer

238

(including thermodynamic non-equilibrium), phase separation and mixing, unsteady-state heat conduction in solid structures, effect of void on power generation (e.g. due to neutronic feedback in boiling water reactors), and others. Combining the equations describing fluid flow and heat transfer phenomena in individual sections of the overall system with the appropriate boundary conditions and physical constraints (which depend on system geometry and particular class of transients under consideration), and including various possible external perturbations of the system operating conditions (input variables), yields a self-contained model of the system. Needless to say, a given system can be modeled in several different ways, depending on the complexity of mathematical description used and the class of transients to be analyzed.

f Steam Outlet

Steam Dryers

Steam Separators 4- Feedwater

Primary Side In let Plenum

U-tube

Downcomer

Tube Sheet

Primary Side Outlet Plenum

Figure 3. U-tube steam generator.

In let Plenum

Figure 4. A dual-loop boiling water reactor (BWR-5,6).

239

Several modeling concepts can be used to model fluid flow dynamics and heat transfer. For example, large computer codes, such as RELAP, TRAC, etc., use two- or even three-dimensional two-fluid models. Because of the complexity of such models, and the fact that their verification is usually very limited, several aspects of boiling system dynamics can be studied much more effectively by using one-dimensional models of fluid flow combined with empirical correlations for single-phase and boiling heat transfer. In fact, several qualitative system properties can be successfully investigated by using even simpler, lumped-parameter models. In any case, the correctness of theoretical analyses must be always checked against experimental data.

The most commonly used approach to fluid dynamics in boiling systems is based on the drift-flux model. The following basic conservation equations used by this model [1] are summarized in Tables 1 and 2

In the equations shown in Table 1, A is the chanel cross-section area, r is

the evaporation rate (per unit volume), < jl > and < jv > are the volumetric fluxes (superficial velocities) of the liquid and vapor phase, respectively, <p'> is the so-called momentum density, B<z> is the Dirac's impulse function, Q is the volumetric flow rate, and the remaining notation is conventional.

The area-averaged void fraction, <a>, is related to the volumetric flux of

vapor, < jv >, and the total volumetric flux, <j> = < jl >+ < jv > = QlA, via the

240

relationship given by Eq. (6), where Co is the concentration parameter and Vvj is the drift velocity. Also, < jv >, < ji >, and <a> can be expressed in terms of the flow quality, <X> = Gv/G, as shown in Eqs. (7) - (9).

Bulk Boiling

,- P�\ • 1

�

I , • J : �O .. :·e,,, ==> <= q (t) • 3 " , .

• 6 '.; � '

Subcooled ; ; ,. � I � �·t I

BOil ing , . �

t I A(t)Distance to Boiling

j in (t)

Figure 5 . Two-phase flow in a heated channel.

For single-phase flows, the above-described model simplifies to three equations (Eqs. (1), (3) and (5) for liquid flows, and Eqs. (2), (4) and (5) for vapor/gas flows). Also, by making some additional simplifying assumptions, the present general five equation model of two-phase flow can be simplified to four or even three equations only. In particular, assuming thermodynamic equilibrium between phases (i.e. , hi'= hf, hv = hg , Pi = Pf , Pv = Pg ) and ignoring phasic slip yields the homogeneous two-phase flow model given by the mixture conservation equations shown in Table 3.

A similar three-equation model can be obtained assuming slip between the phases. In this case, however, it is convenient to replace the original mass and energy conservation equations of the two-phase mixture by a modified continuity equation and the void propagation equation, both shown in Table 4 .

The presence of thermodynamic nonequilibrium conditions in boiling channels usually manifests itself as the interaction between subcooled liquid and saturated vapor (although, rigorously speaking, vapor is always slightly superheated, the effect of superheat can be neglected in most cases of practical interest). Consequently, the modeling of two-phase flow dynamics can still be

241

based on equations obtained from Eqs. (13) - (14) by replacing saturated liquid (subscript "f') by subcooled liquid (subscript «L"), supplemented by an additional equation - the energy balance for subcooled liquid, as shown in Table 5.

Table 1. One-dimensional of flow

(a) Liquid Phase

� [PI (1- < a » A] + � (PI < jl > A) = - rA at az (b) Vapor Phase

� (Pv < a > A) + �(Pv < jv > A) = rA � az

(a) Liquid Phase

a a " [(Plhl - p)(l < a » A] + -[Plhl < jl > A] = qlIH at az

(b) Vapor Phase

� [(p h - p) < a > A] + � [p h < j > A] = q PH at v v az v v v v

- + - - + - + CIio - + L KiCliO(Z-Zi ) - + < j5 > gSin9 = O ac 1 a ( G2 A J ap [ f n ] G2 at A az < P > az DH i=l 2Pl

where

(1)

(2)

(3)

(4)

(5)

242

Table 2. Drift. flux model

< }' > < a > = v Co < j >+Vvj

<jv> = G<x>/Pv

< jl > = G(l-<X» /Pl

< a > = {CO [1 + (Pv / Pi )(1- < x » / < x >] + Pv V vj / (c < x » }-1

Table 3, flow model

Ph >< 11 » a(C < h » q"PfI ap -- + -at dz A at

ac a ( c2 J ap [ f n ] c2 , - + - -- = - - - - + LKiB(Z - zd + < Ph > gSm8 at az < Ph > az DB i=l 2 < Ph >

where

< Ph >= Pf(l- < a » + Pg < a >

< 11 >= [Pfhf (l- < a » + Pghg < a >]/ < Ph >

<h>=hf( l-<x> )+hg<x>

� < 11 >=< h >

(6)

(7)

(8)

(9)

(10)

(11)

(12)

243

Table 4. Drift flux model - thermodynamic equilibrium. between the liquid and vapor

a < j > = r Vfj _ < a > dPf + < a > �)CJp _ dPf + > �)CJp

az g pf dp Pg dp at pf dp Pg dp az (13)

a < a. > a < a > r < a. > dpg ap < jg > dpg ap CJCK -- + CK -- = - - -- -- - ----- - < a > (14) at az Pg Pg dp at Pg dp az az

where the volumetric vapor generation rate can be obtained from,

and CK is the kinematic wave speed given by,

Table 5. between (subcooled

Additional Equation: Energy Balance for Subcooled Liquid,

ahl ahl PI(l- < a. » - + Pl < jl > - + r(hg - hl ) = at az

q"PrI ( dhg] ap . dhg ap -- + I - Pg < a. > Pg < Jg >A dp at dp az

NOm

(15)

(16)

(17)

The volumetric vapor generation rate, r, is given by an empirical relationship which accounts for wall heat flux partitioning between evaporation and single-phase convection.

244

It should be mentioned that in order to obtain the volumetric vapor generation rate, r, in Eq. (17), Eq. (15) must be replaced by another, usually not fully mechanistic, relationship accounting [2] for the fact that heat transferred from the channel wall is partially used to increase the temperature of the subcooled liquid.

The above-given equations can be used to model two-phase flow dynamics in a single boiling channel (see Fig. 5). In various practical applications, the boiler consists of several parallel channels, having common inlet and exit plena. In addition, the individual channels can be axially interconnected, including ventilations made in the channel walls , and flow across gaps formed between heated rods in a multi-rod array. In order to account for the effect of cross-flow between the channels, modified equations can be used [3].

The equations obtained for heated channels can be easily modified for use under adiabatic flow conditions, including both two-phase channels (chimney) and single-phase (liquid or vapor) sections of the loop.

An important aspect of model derivation for a boiling loop deals with the boundary conditions at the junctions between individual sections of the loop. Two typical problems are discussed next.

(a) Sudden change in the channel cross section area.

Two-phase flow parameters which maintain continuity along flow direction are: mass flow rate and quality, given by Eqs. (18) and (19) in Table 6, where Zc is the axial location of the junction.

If the channel area changes from Al to A2 , the new mass flux, G2, is given for Eq. (20).

Table 6. Sudden in the channel cross section area

In general (see Eq. (9»

It follows from Eqs. (7)-(9) that other parameters of the one­dimensional drift-flux model considered herein may also experience a discontinuity at Z=Zc (note that Eq. (9) implies that if V vj =0, then <a> is continuous at Z=Zc , otherwise it is not). Consequently, the integration of governing equations along a multi-sectional channel must be performed over each section

(18)

(19)

(20)

separately, and the inlet conditions for each section must be established based on the exit conditions for the preceding section.

(b) Mixing in large plena.

Since, regardless of a particular shape, the UA ratio for large plena is typically small, a perfect mixing homogeneous flow model can usually be applied. Specifically, the lumped parameter mass and energy conservation equations are given in Table 7, where Vp is the volume of the plenum, and <h>p= dl>p (perfect mixing assumption). A short flow distance between the inlet(s) and the exit{s) in the plenum makes pressure losses due to gravity, friction and acceleration small compared to the local losses at the inlet{s) and/or exit{s). Consequently. the momentum conservation can be ignored in this case.

Table 7. in

245

Using perfect mixing homogeneous flow model, lumped-parameter mass and energy conservation equations become, respectively,

d < Ph >p V P dt

where

LWin,i -Wout,j i

(21)

(22)

So far, the thermal energy added to a boiling loop system was considered in the form of a given heat flux, q", at the channels wall (see Eqs. (11) and (15), for example). In reality, the heater power, rather than the wall heat flux, should be used as a given (controlled) parameter. Any changes in the power generated in the heater will be transmitted to the coolant with a delay depending on the heater geometry and material properties. Two options of a heated wall dynamics model are given in Table 8. For heaters characterized by spatially-distributed heat sources (electrically-heated rods, nuclear reactor

246

fuel elements, etc.), the lateral heat transfer can be described by a one­dimensional heat conduction equation, Eq. (23), where T=T(y,t) is the temperature of the heater, q'" is the volumetric heat generation rate and ah is the thermal diffilsivity of the heater. The wall heat flux can be obtained from Eq. (24). For simplicity, the distributed-parameter model, given by Eq. (23), can be replaced by a lumped-parameter model, given by, Eq. (25), where Ah is the heater cross-section area, and Tw can be related to q" via Eq. (26), with H and Tb being, respectively, the effective (single or two-phase flow) heat transfer coefficient between the channel wall and the coolant, and the coolant bulk temperature.

Table 8. Heated wall

1. One-dimensional Heat Conduction Equation

aT = ah V2T + q" ' /(pcp)

at

with Wall Heat Flux given by

q -ay wall

2. Lumped-parameter Model

dTw PhCp,hAh � = q"'Ah - q"Ib

with Wall Heat Flux given by

q" = H(Tw - Tb)

(23)

(24)

(25)

(26)

If the heater power depends on some other system parameters, an additional relationship must be given to evaluate q"' . For example, in boiling water nuclear reactors the reactor power, P, (and, thus, q" ') varies with average void fraction in the core. A simple model which can be used in this case is given by Eq. (27) in Table 9, where � and A are constant parameters, and p (the reactivity) is a linear function of the core-average void fraction, <a>av (which can be obtained by integrating local void fraction in heated chanels over the entire coolant volume). The void-reactivity feedback is given by Eq. (28).

Table 9. in water nuclear reactors

Point Kinetics Equation

dP = pP - �A I t Exp[-A,(t - 't)][P(t) - P('t)]d't dt -00

where,

247

(27)

(28)

The final part of model derivation for the analysis of boiling loop (or a section thereof) dynamics concerns the formulation of boundary conditions and selection of external variables perturbing the system.

The fundamental boundary condition for a boiling loop is obtained by summing-up pressure drops over the closed flow path around the loop, as shown by Eq. (29) in Table 10, where �Ppump is the pump head (for a natural­convection-driven loop, �Ppump=O). Eq. (29) is complemented by other boundary conditions, such as those at the junctions (discussed before).

Table 10.

1. Pressure Drop over closed flow around the loop

�Ploop - 6ppump = 0

2. Parallel Channels

(29)

(30)

In some cases, it is interesting to study transient behavior of the boiler itself rather than of the entire loop. In particular, the above applies to multi­channel, non-uniformly heated assemblies, where highest temperatures and boiling rates are experienced by a single (or a few, at most) channel only. For channels having common inlet and exit plena, the same (given) pressure drop boundary condition can be used, given by Eq. (30).

Transients in boiling loops may be caused by several reasons. In general, depending on the nature of the initial events, the following three classes of transients can be defined:

(a) intended operational transients, (b) accidents,

248

(c) system response to random perturbations. In any of the above-mentioned transients, the initiating events (external

perturbations) are related to changes in several physical parameters of the system The most important external perturbations are listed in Table II. Other external perturbations may include: vibrations of solid structures (flow-structure interaction), valve manipulations, etc. Since the dynamic response of a boiling loop usually includes several parameters of interest (flows, temperatures, void fraction, power, etc.), boiling loops must be treated as multi-inputlmulti-output (MIIMO) systems.

Table II. External

(a) System Pressure

(b) Coolant Velocity (eg, variation in the feedwater flow rate or pump speed)

(c) Coolant Temperature (feedwater)

(d) Thennal Power of the Heater

3. EXAMPLES OF 'IWO-PHASE FLOW TRANSIEN

Boiling channel's response to a step change in inlet flow rate (velocity)

Assumptions: uniformly heated channel, saturated inlet conditions (hin=hf), homogeneous two-phase flow model, constant system pressure.

Governing equations are:

where,

q" PJ-IVf Q = rVfg = g

= cons tan t Ah£g

(31)

(32)

(33)

Trajectories of particles moving with velocity, <j>, satisfy the equation,

dz = < j > = jin (t) + nz dt Assuming that jin(t) is given by,

·in (t) = { �l for t�O J J2 for t>O

249

(34)

(35)

(36)

for t > 0, where to is the particle entrance time (z(to)=O, as shown in Fig. 6 (a) in the case when j2<.i 1).

Solving Eq. (31) along trajectories (i.e. , along the characteristic dz = < j » dt and combining with Eq. (36) yields,

nz j2 j2 - j1 -. + � - h h 11

� .Qz 1 +-j2

for 0 � t � � In(l + Q J2

for t > �ln(l + Q J2

(37)

Solutions for <j> and <Ph>, given by Eqs. (34) and (37), respectively, are shown in Fig. 6(b) and 6(c).

Boiling channel's response to a step change in inlet flow rate, using drift-flux model.

Using the remaining assumptions same as in Problem 1, and taking constant Co and Vgj, the governing equations can be obtained from Eqs. (13) and (14) as,

250

t*

< j >

z· (b)

• • I Z = [ Exp (nt ) - 1 ] j 2 / n I

z · (c)

I Z L

I j 1 + nL j 2 + nL I

I I I I

L z

Figure 6. Response of a boiling channel to a step change in the inlet flow rate (Example 1). Assumptions: saturated inlet conditions, homogeneous two-phase flow model.

a < j > � = n = rVfg

where,

(38)

(39)

CK = CO<j> + Vgj

and the boundary/initial conditions are:

<a(O,t» = 0

jin (t) = { �1 for t�O J2 for t>O

<j(Z,t» = j1 + Q Z for t S 0

Let,

dz - = CK = CK in + CoQz dt ' where,

Ck,in = Co,jin + V gj Integrating Eq. (44), yields,

for t > o.

25 1

(40)

(41)

(42)

(43)

(44)

(45)

(46)

if to > 0

Integrating Eq. (40) along trajectories z(t,to), and combining the resultant equation with Eq. (46), we obtain,

< a(z, t) >= (47)

Eq. (47) can be used to obtain the two-phase density,

252

< p(z, t) > = pI[ 1 - < a(z, t) >] + Pg < a(z, t) > (48)

As can be easily checked, assuming Co=1 and Vgj=O, Eq. (48) becomes Eq. (37) obtained under the homogeneous flow model assumption.

Boiling channel with subcooled liquid at the inlet at the inlet (same model and transient as in Example 1).

Integrating the single-phase energy conservation along the characteristic,

dz = jin (t) dt yields,

q"PH �hsub = hf - hin = -- v

PfA

(49)

(50)

where v is the particle residence time in the nonboiling section of the channel. Assuming hin=constant, v can be calculated as,

Apf�hsub v q"PH = constant (51)

Integrating Eq. (49) between the inlet (z=O) and the boiling boundary (z=A,(t» yields,

t A,{t) = lhn(t)dt

t-v

Now, Eq. (32) can be integrated for Z�A,(t) to obtain,

< j(z,t) > = jin (t) + O[z - A,(t)]

Again, solving along the characteristic, �� = < j >, we have,

t

(52)

(53)

z(t, to ) = [hv+ (h - h )to]Exp[n(t - v- to)] + JExp[n(t - 't)Ih -m.('t)] dt (54) YHo

253

forbO.

In the case of step change in the inlet velocity, given by Eq. (35), Eq. (54) yields,

[it ( ) h it ] p[ ( ] ( 1 ) aExp -ato + -0 Ex a t v) +h v a for t > v

for v + to S t < v (55)

Trajectories, z(t,to), for different to and for a step change in the inlet velocity, are shown in Fig. 7(a). Solving Eq. (31) along the trajectories z(t,to),

<Ph(t,to» = PfExp[-!l(t-to)] (56)

and eliminating to from Eqs. (55) and (56), yields the solution, <Ph(z,t» , shown in Fig. 7(c)(note that A,l=vh, A,2=vj2). The solution ofEq. (53) is shown in Fig. 7(b).

Uniformly heated boiling channel subject to a step change in the wall heat flux.

Under the assumptions of Example 1, Eqs. (31) and (32) can still be used with,

!l = {01 for t � 0 02 for t > 0

Consequently, integrating the equation of the characteristic,

dz . . - = < J > = Jin dt

yields,

(57)

(58)

254

for t>O.

t 1

h i 2 : I I

I I

1 0 < t < t 1

A. 2 A. 1 (C)

I J I t 1 = v+{qn [1 +Q ( L - A.2) ! j 2 ]} ! Q

; L Z I I

h + Q ( L - A. 1 ) .. j 2 + Q ( L - A. 2)

t � t 1

I 1 I

L Z

Z

(59)

Figure 7. Response of a boiling channel to a step change in the inlet flow rate (Example 3). Assumptions: subcooled inlet conditions, homogeneous two-phase flow model.

255

Now, integrating Eq. (31) along the trajectories z(t,to) and eliminating to, the density variation, <Ph(Z,t» , can be obtained as,

Plots for <j(z,t» and <Ph(z,t» are shown in Fig. 8.

< j >

P f

(a)

! O < t < t 1

t � t 1 = [qn ( 1 +n2 L/ j in )] /n2 I I

j in + n2L

i in + Q 1 L I I I I I I I L z

z = [ Exp (n2t)- 1 ] j in l n 2 L z (b)

Figure 8 . Response of a boiling channel to a step change in the wall heat flux (Example 4). Assumptions: saturated inlet conditions, homogeneous two-phase flow model.

256

Pool swelling due to depressurization.29

Consider a pool of saturated liquid. The initial pressure, Po , starts decreasing at t=O, and is a given function of time. As a result, vapor is generated inside the pool and gradually moves towards the surface. Assuming a constant velocity of the vapor bubbles, Y (this assumption is good as long as the void fraction remains small), and ignoring pressure variation with the height in Eq. (15), we have,

(61)

r = {[ I-Pf (l - < ex » - Pg < ex >-tJ Z} (62)

Now, Eqs. (61) and (62) can be substituted into Eq. (2), to obtain

a(pg < ex > ) a(pg < ex » 1 { dhr dhf �) }

� at +Y az - hfg I-Pfdp + pg dp - dp Pg < ex > dt (63)

Substi tuting,

y = Pg<Cl> Eq. (63) can be rewritten as,

ay ay - + Y - = - (Bl at az ) � B2Y dt

(64)

(65)

(66)

(67)

Let us consider a special case, when the system pressure decreases linearly with time, i.e.

pet) = Po - at (8.>0) for t � 0 (68)

Assuming that the parameters, Bl and B2, remain constant during the transient, the solution of Eq. (70) (i.e. obtained by using the Laplace­Transform method) with the boundary/initial conditions,

y(z,O) = 0 (69)

257

y(O,t) = 0 (70) can be written as,

Bl z z z y (z,t) = � [l-Exp (-B2 at) ]U(t) - (l-Exp [-B2a(t - ,,)] )Exp (-B2a,,)U(t -,,)} (71)

where U(t) is the unit step function. Eq. (71) yields the following expression for the transient axial distribution of

the pool void fraction,

1 (ptdhr/dp - l) - �)�t]} for t <..!. prdhr/dp - pgdhgldp pg dp dp hfg - V

< ex (z,t) > = (prdhr/dp -1) {1 Exp

[ dhf ..!.]} for t > .!. (72)

prdhr/dp - pgdhgldp - - Pg dp - dp hfg V V

The solution given by Eq. (72) is graphically shown in Fig. 9. As seen, a steady-state exponential axial distribution is established when t exceeds LmaxN, where Lmax is the final height of the swollen pool. The transient pool height, L(t), (Lmax, in particular) can be calculated from the following transcendental equation,

Pfg fo L(t) < ex(z,t) > dz = Pf [L(t) - Lo] (73)

where Lo is the collapsed (also initial) pool height.

<a >

z Vt Lo L(t) Lmax

Figure 9. Pool swelling due to depressurization. Assumption: saturated liquid at t=O, constant pressure reduction rate.

258

4. EFFECT OF LATERAL DISTRIBUTION OF FLOW PARAMETERS ON BOILING CHANL DYNAMICS

• I-D Drift - Flux model is useful, but not always accurate enough. • Possible problems due to:

- Use of non-mechanistic empirical correlations Oimited to certain conditions and geometries)

- These correlations refer to steady-state conditions • Alternative Approach

- 1-D Two-fluid model - Multi-dimensional two-fluid models

• Problems to be discussed Effect of lateral flow and heat transfer phenomena on subcooled boiling dynamics

The discussion in the preceding section was concerned with the modeling and analysis of boiling system dynamics based on a one-dimensional drift-flux model. Although such an approach has several advantages (in particular, relative simplicity as compared to other models), there are also considerable limitations inherently built into it. Specifically, the model requires several closure conditions which, as a rule, are obtained from steady-state experiments. Examples include: drift flux parameters (Co and Vvj), two-phase friction multipliers (<I>�o and <l>LOCAL) single- and two-phase heat transfer

coefficients (H 1cj1 and H2cj1), volumetric vapor generation rate under subcooled boiling conditions, etc. The closure laws are not only stationary, so that their application to transients may lead to inaccurate results, but, moreover, are usually non-mechanistic, so that they can be used only in a certain range of operating conditions. For example, using constant values of Co and V vj in a boiling channel undergoing flow regime change from sub cooled-boiling bubbly flow near the inlet to annular flow at the exit, constitutes a substantial simplification of the actual phenomena, and makes sense only if the channel­averaged values of the above-mentioned parameters are known for the particular geometry and operating conditions. Alternative approaches to the drift-flux (D-F) model include two-fluid models,

either one- or multi-dimensional. Whereas the 1-D two-fluid model can relax some of the non-mechanistic quasi-stationary assumptions of the D-F model, such as those concerning the interaction between component-phases in the axial direction, it is still unable to capture various important lateral effects, to mention only thermal i:nertia and thermodynamic nonequilibrium between phases in a heated channel. Such effects can only be included in a mechanistic way by using 2-D or 3-D models. Needless to say, such models also include several constitutive laws, which must be verified experimentally.

259

As an illustration, a comparison of transient subcool-boiling dynamics phenomena in a uniformly heated channel, obtained from a 1-D and a 2-D model, is presented below. The two-dimensional two-fluid model of two-phase flow used in this comparison is given in Table 12 [4].

Table 12. Two-dimensional two-fluid model

£.p . .!2.(U. ) = v-[£. (Il' + J.L!)VU. )] +gp.£. + v(£,p) + F.d + FY + FF 1 1 Dt 1 1 1 1 1 1 1 1 1 1 1

o Pi [ ( t ) ] £.p. -(h. --) = V- £. k. + k. VT. + Q. 1 1 Dt 1 Pi

1 1 . 1 1 1

FV = _Fv = E P C � - � (D U 0 U J L G G L vm Dt Dt

where

A - 4.5 £G- EGs &Gs I - Db 1-EG dB 1-EGs

dB = 10-4 (Tr., - T sat> +0.0014

(74)

(75)

(76)

(77)

(78)

(79)

(SO)

(81)

(82)

260

Table 12. Two-dimensional two-fluid model (continued)

Ct1$ = A1$PLCpuL,aSta (Tw-Ta)

where

2 2 L8 AQ = 1td Bm Nsd = 1td Bm [210 (Tw - Tsat) ]

3 PL

(83)

(84)

(85)

(86)

(87)

(88)

89)

In Eqs. (74) - (76) the index, i, indicates either the liquid or vapor phase, 'Fd is the drag force, FV is the virtual mass force, and Fr is the force due to phase change. These three interfacial forces, are given by Eqs. (77) - (79), with the assumption that the interface velocity is equal to the dispersed (vapor) phase velocity.

261

The expression for the drag coefficient, Eq. (80), was introduced by Ishii [5]. In this expression, dB is the bubble diameter and Red is the Reynolds number based on the bubble diameter and the magnitude of the vapor-liquid relative velocity. The interfacial area density, Ai, (i.e. the area per unit volume of the

mixture) in Eq. (77) is given by Eq (81) , where Dh is the pipe diameter, and £Gs=min(£G,0.25). The bubble diameter has been assumed to be a function of local liquid

subcooling. Specifically, a linear interpolation has been used between the diameter of bubbles detaching from the wall [6], and the diameter at low subcooling [7] . The resultant expression is given by Eq. (82), where T is in rOC] and dB is in em]. The virtual mass force showed little significance for the range of flow

conditions considered, however it was retained in the model in order to improve numerical stability. The virtual mass coefficient Cvm, was set to 0.5, which is a theoretical result for spherical bubbles. The turbulent viscosity, j.1.t, is calculated as a sum of the wall-generated

turbulent viscosity, obtained by a single phase solution of lC-£ turbulence model for the same flow conditions, and bubble-generated turbulent viscosity given by Sato et al. [8]. The phase change per unit volume, r, is obtained from the energy balance

at the interface, assuming that all the heat transferred from the interface to the liquid phase is associated with phase change. The heat transfer coefficient between the interface and the liquid phase is given by Ishii [5] . Thus the source term in Eq. (76) is given by Eq. (83) in Table 12, where the viscous dissipation was neglected as compared to the source due to phase change. For the momentum boundary conditions at the wall, the well known

logarithmic law is used. The boundary conditions for the energy equation at the wall require the knowledge of partitioning the wall heat flux between that transferred to the liquid phase and that used to form vapor phase. In the present model, the wall heat flux has been partitioned as shown in Eq. (84), where q 1 e!> is the heat transferred to the liquid phase outside the zone of influence of the bubbles, qQ is the heat transferred to (relatively) cold liquid that fills the volume vacated by detaching bubbles (the so-called quenching heat flux), and qe is the net heat to form vapor phase. The single phase heat flux, obtained from the Reynolds analogy, is given by Eq. (85), where the index, S, indicates a location within the buffer layer, and Ale!> is the fraction of the wall area not affected by the presence of bubbles. The quenching heat flux is given by Eq. (86) [9], where f is the bubble

detachment frequency, tQ is the waiting period, aL is the thermal diffusivity of the liquid, and AQ is the fraction of the wall area influenced by the detaching bubbles, which is given by Eq. (87).

262

Here, Nsd is the number of nucleation sites per unit area given as a function of wal superheat [9]. The bubble detachment frequency is given by Eq. (88) [10], where dBm is the bubble diameter at detachment from the wall. This diameter is given as a function of the wall heat flux, liquid sub cooling and liquid velocity [6].

The heat to form vapor is, given by Eq. (89). The 2-D model equations has been solved using the PHOENICS code [11] as

a solver of the governing equations. The calculations were performed based on an elliptic scheme. Time-dependent problems used a fully implicit scheme starting from initial steady state conditions.

The results of the comparison between the experiments of Bartolemei & Chanturia [12] and the calculations using the present model are shown in Fig. 10. In addition, 1-D calculations were performed for the same conditions with a drift-flux model. The resultant void fraction and temperature distributions are also given in Fig. 10. The diameter of the vertical test chanel was 0.0154 m, the wall heat flux was 5.7 105 W/m2, and the inlet mass flux was 900 kglm2-s at 4.5 MPa. The wall temperature in the drift-flux model was calculated from the Jens-Lottes correlation. The radial profiles of void fraction and superficial velocities are given in Fig. 11 , and the axial distributions of averaged drag and virtual mass forces are given in Fig. 12, both obtained from the 2-D calculations.

Temperature Void 20 (CO) Fraction

0.75

-5

o Bartolemei (1 967) 0.50 -- Drift Flux Model

30 - 2-D Model

0.25 -55

-80 0.00 0.0 0.5 1 .0 1 .5 2 .0

Distance from Inlet (m)

Figure 10. Axial steady-state distrubutions of temperature and void fraction in a vertical uniformly heated channel.

1 .0

Superficial velocity (m/s) and void fraction

0.0 0.00 0.20 0.40

Radius (cm) 0.60

263

Figure 1 L Radial distributions of superficial velocities and void fraction at 1.6 m from the inlet of the pipe (same conditions as in Fig. 10).

2000

Interfacial Force (N/m3)

0.50 I

0.75 I

1 .00

Drag Force

Virtual Mass

1 .25 1 .50 1 .75

Distance from Inlet (m)

0.5

Heat Flux x 1 0-6 (W/m2)

Total

0.0 0.00 0.25 0.50 0.75 1 .00 1 .25 1 .50 1 .75

Distance from Inlet (m) Figure 12. Distributions of selected two-phase parameters along the channel.

264

As can be seen, good agreement has been obtained between the 2-D calculations and the experimental data, whereas the 1-D drift-flux model yields a void distribution which is substantially lower than the measured values.

Following the testing and verification of both 2-D and 1-D models, described above, two particular aspects of the radial distributions of flow and heat transfer parameters on forced-convection subcooled boiling in heated chanels have been investigated. One of them is concerned with the void distribution in the developing flow region near the channel entrance, the other with the effect on flow conditions of an unheated section being an extension of a heated chanel. A commonly used approach to evaluate the void distribution along boiling

channels is the use of 1-D models which are based on experimental correlations for various flow parameters. One such correlation is the expression for the void detachment point, proposed by Saha & Zuber [13] . Whereas this correlation compares well with a wide range of experimental results, its application is limited to fully-developed flow conditions only.

Using the present 2-D model, the effect has been studied of decreasing inlet subcooling, on the location of the void detachment point in the developing flow region. The results obtained using a fixed value of void fraction at detachment (corresponding to that given by the Saha-Zuber correlation in fully developed flow) for each case have been correlated by the following expression,

hf - hd h (h ) = 1 - 0.044'1' e-O·0223 'I' + e-O·0505 'I' f - d S-Z (90)

where 'I' = G�hin/q. The correlation given by Eq. (90) is illustrated in Fig. 13. This correlation was subsequently used to verify the 1-D model described before. The results of calculations are shown in Fig. 14. As seen, whereas this improved 1-D model still cannot predict local boiling in the low void fraction region, good agreement has been obtained for the remaining part of the boiling section.

If subcooled boiling still exists at the exit of a heated channel which is followed by an unheated section, the two-phase mixture in the latter section will gradually return to the thermodynamic equilibrium conditions. In order to see how a 1-D drift-flux model (the validation of which is mainly based on flows in adiabatic or uniformly heated channels) compares against a mechanistic 2-D model for such flow and heat transfer conditions, calculations were performed using the previously discussed models. The results are shown in Fig. 15. As seen, the exit subcooling of 3°C was already high enough to cause the (2-D)-model-calculated void fraction to start diminishing quickly in the unheated section. In contrast, the 1-D model predicted only a very slow void fraction decrease, substa.n.tially overestimating the overall vapor concentration in this section.

265

400

h f - h in

200

200 400

( h f - h in )Glq"

Figure 13_ The effect of inlet subcooling on the enthalpy at the void detachment point in developing flows; hd is evaluated from (see Eq. (90)).

0. 1 5 Void Fraction

- - Original Saha-Zuber Carr.

0. 1 0 - - - Modified Saha-Zuber Carr. -- 2-D model

0.05

/' /'

o .00 0.0 0.5 1 .0

Distance from In let (m)

Figure 14. Entrance effect on void fraction. D=0.0254 m, q�=1.17 105 W/m2, Re=135000, Tsat-Tin=8°C.

266

5

-5

-1 5

o

Temperature (C)

---- Drift Flux Model -- 2-D Model

2 3 4

Distance from Inlet (m)

Void Fraction

5

0.50

0.25

Figure 15. Void fraction and temperature distributions in a partially heated pipe. D=0.0254 m, q� =3.885 106 W/m2, G=630.4 kg/m2s, heated length=2.8 m.

A very important practical question is concerned with the impact of lateral flow and heat transfer effects on the location of the void detachment point under unsteady-state conditions. 1-D transient calculations are inherently based on quasi-steady state correlations such as the Saha-Zuber correlation. Whereas this correlation proves very useful in determining the steady-state beginning of boiling from the hydrodynamic view point, the calculated void detachment point is usually considerably shifted downstream from the onset­of-nucleation point (i.e. the point where boiling starts affecting heat transfer, as shown by the wall temperature distribution in Fig. 10). The accuracy of this correlation under transient conditions has not been studies systematically before. In order to quantify the effect of unsteady-state heat transfer conditions on

the void detachment point location, the present 2-D model has been compared against the results of 1-D calculations. In this comparison, the transient void fraction at detachment point was the same as that obtained from the Saba­Zuber correlation for steady state conditions [14].

267

The calculations were performed for a sinusoidal change in the wall heat flux of varying frequency. The results are shown in Figures 16, 17 and 18. As can be seen in Fig. 17, the 1-D and 2-D results agree well only for low frequencies (quasi-steady state conditions) departing considerably from each other for increasing frequencies. In particular, for high frequencies, the 2-D value of zd tends to stabilize (due to liquid inertia) while the 1-D correlation indicates oscillations proportional to those in the wall heat flux.

0.20

0. 1 5

0. 1 0

Void Fraction

At 3 . 1 m from inlet i n let

3 Time (5)

4 5

Figure 16. Variation of average void fraction following a (20%) sinusoidal change in wall heat flux with a period of 2 sec.

268

2.3

2.0

Zd (m) __ 2-D Model ----- Drift Flux Model

0.0 (a) 0.2

Ti�e(5)

0.6

zd (m) 2.3

2.0

1·�.00 0.25 0.50 0.75 ' 1 .00 '1 .25 1 .50

(b) Time (s) zd (m)

2.3

2.0

1. 7 0.0 1 .0 2 .0 3.0 4.0 5.0 6.0 (c) Time (s)

2.4 Zd (m)

2 . 1

1 .8

1 .5 " ' , I

0.0 2.5

2.4

2.1

1 .8

(d)

1 .5 ' I

0.0 (e) 5.0

" 1 I I . I I I I I 1 ' 1

5.0 7.5 1 0.0 Time (5)

I • I ' I

1 0.0 1 5.0 20.0 Time (s)

Figure 17. Void detachment point trajectories as calculated by 1-D and 2-D models for various frequencies of the wall heat flux oscillations: a) 4Hz, b) 2Hz, c) 0.5 Hz, d) 0.25 Hz, e) 0.125 Hz.

90

60

30

1 0-1 1

Frequency ( 1 Is) 1 0

269

Figure 18. Phase shift (cj>UJ-C\l10) and relative amplitude (A20/AIO) of the location of void detachment point in a pipe with sinusoidal heat flux oscillations, calculated by I-D and 2-D models.

5. REFERENCES:

1 R.T. Lahey, Jr. and M.Z. Podowski, (eds. G.F. Hewitt, J.M. Delhaye and N. Zuber) Multiphase Science and Technology, 4, 1989, 183-371.

2 R.T. Lahey, Jr. and F.J. Moody, The Thermal-Hydraulics of a Boiling Water Nuclear Reacter, ANS, 1977.

3 R.P. Taleyarkhan, M.Z. Podowski and R.T. Lahey, Jr. , An Analysis of Density-Wave Oscillations in Ventilated Channels, NUREG/CR-2972, 1983.

4 N. Kurul and M.Z. Podowski, On the Modeling of Multidimensional Effects in Boiling Channels, ANS Proceedings, HTC, 5 (1991).

5 M. Ishii, Two-Fluid Model for Two-Phase Flow, 2nd International Workshop on Two-Phase Fundamentals, RPI, Troy, NY, 1987.

6 H.C. Unal,Int. J. Heat Mass Transfer, 19 (1976) 643-649. 7 R.M. Thomas, Int. J. Multiphase Flow, 7 (1981) 709-717. 8 Y. Sato, M. Sadatomi and K. Sekoguchi, Int. J. Multiphase Flow, 7

(1981) 167-190. 9 M. Del Valle, and D.B.R. Kenning, Int. J. Heat Mass Transfer, 28 (1985)

1907-1920. 10 W.C. Ceumern-Lindenstjerna, (eds. E. Hahne and U. Grigull) Heat

Transfer in Boiling, Academic Press and Hemisphere, 1977.

270

11 D.B. Spalding, Mathematics and Computers in Simulation, 23, North Holland Press. 1981

12 G.G. Bartolemei and V.M. Chanturiya, Thermal Engineering, 14(2) (1967) 123-128.

13 P. Saha and N. Zuber, Point of Net Vapor Generation and Vapor Void Fraction in Subcooled Boiling, Proc. Fifth International Heat Transfer Conference, 4, 1974.

14 N. Kurul and M.Z. Podowski, Multidimensional Effects in Forced Convection Subcooled Boiling, Proc. Ninth International Heat Transfer Conference, 2, 1990.

27 1

INSTABILITIES IN TWO-PHASE SYSTEMS

M. Z. Podowski

Department of Nuclear Engineering and Engineering Physics, Rensselaer Polytechnic Institute, Troy, New York 12180-3590

1. INSfABILITY MODES

The instability phenomena associated with two-phase flow dynamics may cause problems in the operation of energy systems. Various instability modes and mechanisms have been identified to date, depending on the geometry and operating conditions of particular boiling and condensing systems. A classification of two-phase flow instabilities is shown in Table 1. It is based [1, 2, 3, 4] on dividing the different instability modes into two classes: static instabilities and dynamic instabilities. Each class covers a broad spectrum of physical phenomena resulting in either fundamental (i.e . , a single dominant mechanism governs the flow dynamics) or compound instabilities. In general, a two-phase system is stable if its response to an external perturbation (e.g., in flow, temperature, pressure, power, etc.) converges asymptotically to the operating conditions of the unperturbed system. If certain perturbations of arbitrarily small magnitude result in a diverging­type response, the system is unstable. A common feature of static instabilities is that the perturbed system suddenly departs from the initial operating conditions to reach a new operating state, the parameters of which are significantly different from the original ones. This new state may be either stable or unstable. In the latter case another departure may occur so that after some time the system will temporarily return to the original state and the cycle may repeat. Dynamic instabilities usually manifest themselves as either self-sustained (periodic) or diverging oscillations associated with the propagation of kinematic (i.e. , density) or dynamic (i.e. , pressure) waves.

Because of its practical importance, the question of two-phase flow instabilities has been studied quite extensively. Various physical mechanisms have been identified and modeled analytically. However, knowledge of the subject is still far from satisfactory. As a consequence, predictions based on either measurements or calculations (or both) are usually associated with a substantial degree of uncertainty.

272

Table 1 modes

• Excursive (Ledinegg) Instabilities • Flow Regime Relaxation Instabilities • Nucleation Instabilities

• Density-Wave Oscillations • Pressure-Drop Oscillations • Acoustic Instabilities • Condensation-Induced Instabilities • Instabilities

One of the most important instability modes, which usually occurs at low or medium qualities and high-power-to-flow ratios, is associated with density­waves. The so-called density-wave oscillations may appear in systems consisting of combined sections of single-phase (liquid) flows and two-phase (vapor/liquid) flows. These oscillations have been observed mainly in boiling systems, such as boiling water nuclear reactors (where they are combined with neutronic effects), but may also occur in condensing systems. The nature of density-wave oscillations can be explained by taking into consideration the different speeds of propagation of flow velocity perturbations in the single-phase and two-phase regions of the flow channel. For example, in the liquid sections of a boiling system any such perturbations propagate at the speed of sound (i.e., they traverse these sections very rapidly compared to the residence time of the liquid). On the other hand, a velocity perturbation appearing in the boiling region leads to an increase (when the flow slows down) or decrease (when the flow speeds up) of the local void fraction. Such void fraction perturbations will propagate downstream at a kinematic velocity characteristic of density-waves which is much smaller than the two-phase speed of sound ( in fact, the speed of density-waves is often close to the speed of the vapor). Any changes in the flow velocity and/or void fraction in the two­phase region result in a pressure drop variation in this region. It is interesting to note that the hydrodynamic effect of a decreasing velocity is to reduce the pressure drop, while the thermal effect of it in a heated channel is to increase the amount of gas phase, which in turn leads to an increased pressure drop. Since the perturbations discussed above travel rather slowly along the two-phase region, a constant pressure drop boundary condition across the entire system may result in the two-phase pressure drop and the induced single-phase flow velocity (in particular, the velocity at the inlet to the two-phase region) oscillating out-of-phase with each other, even after the external perturbation, which originated the transient, has ceased. Such self­excited oscillations may either diverge or reach a self-sustained periodic mode. In the latter case, the magnitude of oscillations may be so high that the thermal limits of the system (such as those due to CHF) will be exceeded, causing damage to the heater.

273

In general, boiling and condensing systems can experience various modes of density-wave instabilities (see Table 2).

Table 2 instabilities in

• Loop Instabilities • Parallel-Channel Instabilities • Channel-To-Channel Instabilities • Instabilities

The two basic modes are: loop instabilities and parallel channel instabilities. It is important to note that a given system can experience either one or the other mode, depending on the operating conditions, and either can be the limiting one. The loop instabilities may occur in phase-change systems operating in a closed-loop fashion (e.g. , BWRs, U-tube steam generators, etc.). Parallel channel instabilities refer to systems consisting of multi-channels having a common inlet and common exit (e.g., BWRs, once-through steam generators, moisture separators, etc.). The thermal-hydraulic mechanisms associated with the loop instability mode are fairly well understood. It has been shown that such instabilities are governed by the constant (zero) pressure drop boundary condition around the loop. A similar boundary condition applies to parallel channel instabilities, provided the number of interconnected channels is very large (or if the two-phase channels interact with a large single-phase bypass). In this case oscillations are usually limited to one or a few channels operating at low flow and/or high power conditions, and the constant channel pressure drop is imposed by the remaining stable channels. The problem gets considerably more complicated when the number of parallel channels is small and their operating conditions are similar. In this case the so-called channel-to-channel oscillations have been observed. In the case of two identical channels, the phase of oscillations in one channel will be opposite to that in the other, while the oscillation patterns for three or more channels are more complicated and not clearly understood. Indeed, several questions still require further investigation, including the effect of an even/odd number of channels and the overall analysis of the phenomena controlling the magnitude and frequency of self­sustained nonlinear oscillations (i.e., the limit cycle). While density-wave oscillations constitute the dominant instability mode

encountered in many systems, they are frequently combined with other phenomena, such as neutronic feedback, compressibility effects (i.e. , pressure-drop oscillations), acoustic instabilities, and others. The neutronic feedback, which is particularly important in boiling water nuclear reactors, occurs through the effect of voids in the reactor coolant channels on the reactor power. Whereas neutronically-coupled density-wave oscillations have been rather extensively studied using point kinetics models of core neutronics, very little has been done so far to investigate the effect of spatial neutronics. An area which is particularly interesting concerns neutronically-coupled parallel channel oscillations. Such phenomena have been observed

274

experimentally and confirmed numerically [5], but no consistent theoretical studies have yet been undertaken to explain the basic mechanisms associated with them. Another example of compound instabilities concerns the superposition of

density-wave oscillations and pressure drop oscillations. Pressure drop oscillations may occur in boiling systems directly connected to compressible volumes, such as surge tanks or plena. As in the case of Ledinegg-type (static) instabilities, pressure-drop instabilities occur when the slope of the pressure drop-to-flow curve is negative. In this case, however, the effect of compressibility results in periodic oscillations the period of which can vary from a few to tens of seconds. The amplitude of such oscillations is usually high, so that the boiling crisis is likely to occur. If, due to pressure-drop oscillations, the threshold of density-wave instabilities is reached, the effect of such compound instability modes may be even more dramatic. Although acoustic instabilities usually occur at frequencies higher than the characteristic frequency of density-wave oscillations (e.g., 10-40 Hz vs 1 Hz), the possible interaction between these two modes (especially in low pressure systems) may lead to simultaneous overheating and mechanical vibrations which can substantially augment system damage.

2. LINEAR ANALYSIS OF TWO-PHASE FLOW INSTABILITIES

Because of the nonlinear mathematical form of the conservation equations governing two-phase flow dynamics, a rigorous stability analysis of a boiling channel is only possible if some simplifying assumptions are made. In particular (see Table 3), if the threshold of instability is of interest, linearized models are often used. Such models are obtained by perturbing the conservation and related constitutive equations around a given steady-state operating point. It should be noted that while linear stability analysis can be used to determine whether a steady-state solution of a nonlinear system is stable or unstable to small perturbations, this approach does not provide information concerning other characteristics of nonlinear systems, such as the magnitude and frequency of any limit cycle oscillations. For this purpose a non-linear stability analysis must be performed (see Table 3 and Section 3).

In order to demonstrate the analytical approach used in linear stability analysis, a simple thermal-hydraulic model will be used. The basic assumptions and fundamental equations for this model are summarized in Table 4.

Table 3 of

• Linearize Governing Equations Around Steady-State Operating

Parameters • Obtain Transfer Function(s) • Examine Properties Of Roots Of Characteristic Equation

Table 3 of (continued)

• Hopf s Bifurcation Method • Method Of Liapunov • Harmonic Quasi-Linearization (Describing Function Method) • Fractals As Measure Of Attractors (Chaotic Vibrations)

Table 4 Linear of flow instabilities

(A) Upflow In A Vertical Boiling Channel (B) Uniform Axial Power Distribution (C) No Subcooled Boiling (D) Homogeneous Two-Phase Flow (E) Constant System Pressure (F) Incompressibility Of Both Liquid And Vapor Phases

275

The total length of the channel is divided into two sections; the nonboiling single-phase liquid region, and the saturated bulk boiling region, and the pressure drop across each section is evaluated from Eq. (12)[in reference 6] as,

( )

LH{ac a ( c2 J f c2 } �p = peA) p = I -+ - - + + g < P dz 2cp H ex A at az < Ph > DH 2 < Ph > h>

+ K. 1 C2 (z. ) I 2 < P (z. ) > I

i tL241 h I

where A is the position of the boiling boundary.

(1)

(2)

Eqs. (1) and (2) can be combined with the appropriate mass and energy conservation equations (see [6]), then perturbed (linearized) around steady­state operating parameters and Laplace-transformed. A detailed derivation is given in Appendix A and the final results are summarized in Table 5.

276

Table 5

(3)

(4)

where and (i=l, 2, 3) are given by Eqs. (A.42) - (A.46) in Appendix A.

PARALCHAN INsrABILITIES

Let us now consider a multichannel system, composed of a number of parallel boiling channels, having common inlet and outlet plena. Such system is shown in Fig. 1 as a part of a boiling loop. It is clear that all the channels must satisfy an equal-pressure-drop boundary condition.

KOUT

CONDENSER

COOLING WATER

DOWNCOMER

Figure 1. A loop containing parallel boiling channels.

Because of possible differences in the operating conditions of the various chanels, one channel will often be the least stable (Le., the most unstable) one. If the number of channels composing the system is very large and only one channel reaches the instability threshold, the effect of oscillations in this single channel will not afect the operation of the other (stable) channels.

277

Therefore, the investigation of the onset of the so-called parallel channel instabilities can be done by considering the most unstable channel subject to a constant pressure drop boundary condition.

The conditions for neutral stability (onset of instability) may change in the case of an assembly containing a relatively small number (N) of parallel chanels. For this case, the flow rate in each chanel is not an independent variable, since the common pressure drop canot be considered to be constant. Indeed, the multi-channel system may be unstable, with variations in the channel flow rates, even though the total flow rate of the boiling loop is nearly constant. This mode of instability, called channel-to-channel instability, can be studied through the analysis of the response of the inlet flow rates of the individual chanels to an external perturbation in the total loop flow rate.

A. LARGE NUMBER OF CHANLS

Assuming a constant power of the heater and ignoring changes in the lower plenum temperature, the channel's pressure drop perturbation can be obtained from Eqs. (AAO) - (A.41) as,

o(�phi = O(�Plc1 )H +O(�P2c1)H = [rl,H(S) + n1,H(S)] OWin PfAx-s

or, (5)

(6)

If O(�P)H = 0 , Eq. (6) always has a solution for dWin (dWin = 0.0, the steady­state case). It may also have a nonzero periodic solution provided that,

(7)

for some S = jco '" 0 In this case, the channel is self-excited and will undergo self-sustained periodic oscillations at the angular frequency, co . Equation (7) is the characteristic equation of a boiling channel subjected to a constant pressure drop boundary condition. The appropriate block diagram is shown in Fig. 2.

B. SMAL NUMBER OF CHANLS

The transfer function for channel-i can be defined as,

OW · H· (s) = __ 1 1

(8)

278

/I. /I. /I. o j in ext + o j in t o j in t

+ /I. o j in fb

/I. /I. - 1

O (� P H)1 1j) -1

o (� PH)21j) IT I ,H fl ,H

Figure 2. Block diagram for a parallel channel model.

where ,

OWi is the Laplace-transformed perturbation in the inlet flow to channel-i,

oWT is the Laplace-transformed perturbation in the total loop flow rate. Using the following boundary conditions,

and, N

oWT = 2. Ow. i = 1 1

where ,

yields the following expression for Hi(s),

Hi (5) = N ] .n. n . Gk (5) J�l k:;l:J

(i, j = 1, . ,N) (9)

(10)

(11)

(12)

279

None of the transfer functions Gj(s), (j = I, . , N), can have poles with Re(s) � O. In fact, if this were not true, it follows from Eq. (11) that variations in chanel pressure drop could occur even if the inlet flow rate was constant. Since the other parameters characterizing the channel wall heat flux and inlet enthalpy are constant, this is physicaly impossible. The system given be Eq. (12) becomes unstable if the transfer function, Hi(s),

has a nonremovable singularity in the complex right half-plane. In other words, when the loop is completely stable (Le., SWT = 0) Eqs. (8) & (12) imply the characteristic equation,

(13)

If Zi is the number of zeros with positive real part, of the transfer function Gi(s), and Nz is the number of unstable channels, the multichannel system will be stable if the Nyquist diagram of IIH(jro) encircles the origin of the

Nz complex plane in the counter clockwise direction K times, where K = I,Zi If i=l

all the channels composing the system are identical, Eqs. (9) and (11) yield,

(14)

which is precisely Eq. (6), the classical parallel channel result.

TWO NON-IDENTICAL PARAL CIIAN

It is also interesting to notice in Eq. (13), that if that system consists of two non-identical parallel channels, the characteristic equation can be reduced to,

(15)

It is clear that the stability of this system depends on the location of the roots of(Gl + G2) in the complex plane. If the Nyquist diagram of (Gl + G2) encircles the origin, the system will be unstable (neither Gl nor G2 have poles in the right-half plane), otherwise it will be stable. In fact, if both channels are unstable, the function (Gl + G2) encircles the origin in the clockwise direction at least once, so that the two-channel system is unstable. On the other hand, if both channels are stable, their Nyquist loci do not encircle the origin, and neither does (Gl + G2), so the interconnected system is stable. In the same way, if channel-l is unstable and channel-2 stable, but (Gl + G2) encircles the origin, the system will be unstable.

280

EFCT OF THE NUMBER OF PARAL CI

It can be easily shown how the characteristic equation of a multi-channel system tends to that of a classical parallel channel when the number of chanels increases. For this purpose let us consider the case of an assembly composed of (N-l) identical chanels and one channel (io) which is different from the others. In this case Eq. (11) can be rewritten as,

where Gj(s) = Gl(s) for i * io.

One chanel unstable and remaining chanels stable.

(16)

If the i� channel is unstable and the other channels are stable, and we denote,

l!. l!. Gio = Gun' and Gl = Gsb Eq. (16) becomes,

BWT = [1 + (N - 1) Gun ] BWun Gst Equation (17) yields the following characteristic equation,

(17)

(18)

Since the transfer function of the stable channels, Gst, have neither poles nor zeros with positive real parts, Eq. (18) is equivalent to,

l!. (N-l) 1 G(s) = -,:.r- Gun (s) + N Gst(s) = 0 (19)

where the number of channels, N, was used as a normalizing factor (i.e., the . - 1 -mput, BWT' was replaced by N BWT)'

As the number of identical stable channels, N- l, increases, Eq. (19) asymptotically approaches,

0un(s) = 0 (20)

28 1

which is precisely the characteristic equation for the classical parallel­channel model, Eq. (7). The effect of increasing the number of stable channels is shown in the Nyquist plots for G(s), given in Figure 3.

-1 .00 -0.50

+ M = 1 (stable) 1 .40 ;Ie M :: 2 (stable)

o M = 5 (unstable) .. M .. 10 (unstable) 6 M :: :.o (same as for the single unstable channel)

1 .00

2.00

Figure 3. Nyquist plots for two separate channels, one stable (*), the other unstable (�), and for a bundle of parallel chanels consisting of one unstable. channel and M stable chanels.

One chanel stable, remaining chanels unstable

An interesting result can also be obtained by considering the opposite case, in which the single different channel is stable and the remaining (N-l) channels are unstable. In this case Eq. (16) yields,

- °un(s) -aWT =[(N-l) + Gst(s) ] awun.

The corresponding characteristic equation is,

CN-l) 1 O(s) = -w- Gst(s) + N Gun(s) = 0

(21)

(22)

282

If we now increase the number of identical chanels, N-l, Eq. (22) yields, Gst(s) = 0 (23)

This result means that if the number of unstable channels is sufciently large, channel-to-channel oscillations will not occur. Instead, the entire bundle will behave as a single unstable channel. Hence, oscillations of the entire boiling loop constitute the only possible mode of density-wave instabilities and will only occur if the loop is unstable.·

EFCT OF En'EBNAL LOP

Let us now consider the effect of the external loop shown in Figure 1 on the dynamics of boiling parallel channels. The different components of the external loop are the: upper plenum (UP),

riser (R), condenser (C), pump (P), downcomer (DC) and lower plenum (LP). The momentum equation integrated around the loop yields [7],

The basic assumptions implicit in the loop model are: (a) perfect mixing in the plena, (b) constant liquid level in the condenser.

The terms corresponding to the single-phase region of the external loop are related directly to the total flow. That is, S(.�p)p = Gp(s) S;'T (25)

S(6P1JC = GDC(s) SWT (26)

(27)

The other terms in Eq. (24) correspond to the two-phase region of the loop, and thus are related to the flows and the enthalpies in those regions,

(28)

• The present analysis assumes that the total inlet flow rate is the only external perturbation for the entire channel assembly. If external perturbations were imposed on the individual channels (e.g., by changing the thermal power or the inlet pressure loss in a given channe}), channel to-channel oscillations could occur.

283

(29)

(30)

where, in general, aWk and ahk mean the perturbations in the flow rate and enthalpy at the entrance to component-k. Each aWk �d ahk can be related to perturbations in the total flow (awex) and average enthalpy (ahex) of the two­phase mixture at the exit of the heated channels. In turn, Wex and hex can be expressed in terms of similar parameters referring to individual channels,

N wex = L wex J' j=l '

(31) 1 N

hex = L wex J' hex J' Wex j=l ' , (32)

Since the perturbations of flow and enthalpy at the exit of each channel can be expressed in terms of the flow perturbation at the inlet to the corresponding channel, the right hand side of Eq. (24) can be directly related to the flow perturbations at the inlet to the individual channels,

(33)

Using Eqs. (9) and (11), one can also express the loop pressure drop perturbation as a function of the total flow rate perturbation in the single­phase portion of the system,

(34)

The pressure drop across the parallel channels is balanced by the pressure drop through the external loop. Hence,

(35)

Using Eq. (34), we obtain,

284

N Li(s)

1: G'(S) + 1 = 0 i=l 1

(36)

Equation (36) is the characteristic equation of a system of parallel channels coupled with an external loop. Let us now rewrite the transfer functions, 4(s), as [8],

(37)

where Lo represents the frictional and local hydraulic losses in the single­phase portion of the external loop. In particular, Lo can be expressed as,

where Kl<1> is the effective pressure loss coefficient (frictional and local) of the

single-phase portion of the external loop, wT ,0 is the steady-state total flow rate, and Ax-s is the reference cross-section area of a channel. Similarly, let Gi(S) be given by,

G. (s) = G . + (s) (39) 1 0,1

where Go ' are constants representing single-phase/entrance losses in , 1 channel-i.

Using Eqs. (37) and (39), Eq. (36) becomes,

N [Lo + L�(5)] 1 + 2, , = 0 i=1 [GO/i + Gi (5)] (40)

If, due to a large loss coefficient (K1<1> » 1), the external loop is very stable,

Lo becomes the dominant term in Eq. (40). That is,

I L i (s) I /Lo < E (41)

Hence, the terms, L i (s) in Eq. (40) can be neglected, to obtain,

[ N 1 1 ] L 2, --+- = 0 o i=l Gi (5) Lo (42)

For a sufficiently large Lo ' Eq. (42) can be approximated by,

N 1 L - = 0 i=l Ci (s)

285

(43) Equation (43) is equivalent to the characteristic equation for the channel-to­

channel instability mode, Eq. (13). On the other hand, if al channels are very stable, the Gi(s) terms can be approximated by the constants, Go,i' so that Eq. (36) becomes,

N L. (s)

i=l Co,i (44)

In this case, the loop dynamics dominates, so that the channels may only oscillate in-phase and only if the loop is unstable. As seen, one can always stabilize the loop by increasing the magnitude of G . (e.g. , the channel inlet 0,1 loss coefficients).

Predictions of instabilities using the model presented herein have been compared [8] with the experimental data obtained for two interconnected parallel channels by Aritomi, et al [9] . Specifically, the geometry, system pressure and inlet temperature used in the experiments reported in the above­mentioned reference were applied to obtain the characteristic equation, Eq. (15). N ext, for specified values of the channel heat flux, the Nyquist loci of G(joo) = G1 (joo ) + G2 (joo ) were evaluated using the inlet velocity as a parameter. The values of heat flux and velocity for which the Nyquist locus intersected the origin in the (1m G(joo), Re G(joo)} plane, were then plotted against each other and compared with the measurements. Good agreement was obtained, as shown in Figure 4.

An interesting question in the stability analysis of parallel channels concerns the modes of oscillation in the case when all the channels are geometrically similar and operate at similar power, flow and inlet temperature conditions.

Theoretically, if a bundle of identical channels was externally perturbed due to a small change in the total flow rate at the inlet, the response of all the channels would be in-phase with one another and no channel-to-channel oscillations would be observed. Such oscilations would be possible only if the initial perturbations in the flow rates at the inlet to the individual channels varied, or if there were asymmetries in the channels, such as slightly different inlet loss coefficients. In addition, channel-to-channel oscillations may be caused by differences in the geometry and/or operating conditions between the individual chanels. Moreover, the modes of oscillation in such systems depend on which asymmetry effects are dominant.

286

N E 3 '

1 .00

0.80 Unstable

Region

Prediction

� 0. 0

X ::J -.J u. I- 0.40 Stable Region « w

:J: 0.20

AVERAGE INLET VELOCITY (m/s) Figure 4. Comparison between experimental data and prediction [9].

In order to study the effects of individual parameters in parallel channel arrays on their modes of oscillations, numerical tests were performed for a system of three parallel channels [8] . For ease of interpretation these evaluations were made in the time domain. For this purpose, the frequency­domain relationship between the channel inlet flow rate and the total flow rate was converted into the time-domain by applying the numerical inverse Laplace transform algorithm developed by Balaram et al (1983), and then used to calculate the system's impulse response.

Identical channels , each operating at their instability threshold, were modified by changing the power level in each channel. The results obtained by varying channel power level are shown in Figures 5 through 7.

As seen in Figures 5 and 6, for three channels operating at different heat fluxes (i.e. , 126.2, 116.7 and 107.3 W/cm2), the highest power (i.e., "hot") chanel oscillates out-of-phase with the other two channels. In contrast, the other two channels oscillate almost in-phase. In addition, the magnitude of oscillations of the lowest-power channel is always much smaller than that of the intermediate and high-power channels. These figures also show the stabilizing effect of inlet subcooling, however, the basic mode of oscillation does not appear to change. As expected, when two of the three chanels have identical thermal-hydraulic parameters (see Figure 7), these two channels oscillate in-phase.

0.40 _

>- 0.20 _ J-() o .. W > J­w .. -0.20 _ Z

0.40

(a) q" = 1 26 .2 w/cm2

(b) q" = 1 1 6.7 w/cm2

(c) q" = 1 07.3 w/cm2

Figure 5. Impulse response of a three-channel assembly; different power level in each chanel, .6T sub = 3.3°C.

� 0 -

() 0 .. W > J-W .. Z

0.60

0.40 -

0.20 -

-0.40 _

0.60

1 0.00 1 2.00 1 4.00 1 6.00

TIME (SEC)

(a) q" = 1 26.2 w/cm2

(b) q" = 1 1 6.7 w/cm2

(c) q" = 1 07.3 w/cm2

Figure 6. Impulse response of a three-channel assembly; different power level in each channel, .6T sub = 14.9°C.

288

0.45

_ 0.30 -

� � >-I- 0. 1 5 -

() o -1 W > I­W -1 Z

0.00 0.00

-0. 1 5

-0.30

-0.45

12.00 1 4.00 1 6.00

(a) q" = 1 26.2 w/cm2

(b) q" = 1 26.2 w/cm2

(c) q" = 85.2 w/cm2

Figure 7. Impulse response of a three-channel assembly; two channels identical, the third at a lower power level.

The influence of the hydraulic characteristics of an external loop on the channel-to-channel oscillations was also studied [8]. In particular, the case of two channels and loop, such as that shown in Figure 1 , was investigated. In Figure 8, the loop was very stable (i.e., KIN was large) but the two-channel assembly was marginally stable. As a result, both channels oscillated out-of­phase, practically without perturbing the loop flow, as predicted by Eq. (43). It should be pointed out that while the perturbations in the loop flow rate were very small (that is, dWT « wT,o)' the channel pressure drop oscillations may still be substantial; this conclusion can be directly drawn from the equation,

(45)

which is valid if KIN is large. Figure 9 shows the case when the inlet loss coefficient in the loop was

reduced. In this case, some phase shift can be noted but the loop was still stable and was driven into forced oscillations by the unstable channels. In Figure 10, the loop's inlet loss was reduced even more. It is interesting to note that the loop was still stable, but as its stability was reduced, the amplitude of the forced oscillations of the loop flow increased. In addition, we note that the

289

amplitude of oscillations of one of the paralel channels is greater than that of the other. It is clear that complex modes of oscillation can occur.

� 0.6 -

� 0.4 -en

� 0.2 -0

-I u.

0.0 --I W Z -0.2 -Z « I -0.4 -()

-0.6 0 0.2 0.4 0 .6 0 .8

TIME 0.6

� � 0.4 .

� 0.2 -0 -I 0.0 . u. -I « -0.2 .. 0 .. -0.4

-0.6 0 0.2 0.4 0.6 0.8

TIME

Figure 8. Interaction between channel-to-channel and loop instabilities, Channel Loss Coefficients Loop loss coefficients:

(i-inlet, e-exit) KIN = 1000; KoUT = 8

�l = �2 = 9; Kel = Ke2 = 1 Channel heat flux: q:1 = 85.2 w/cm2, q '2 = 88.3 w/cm2

290

1 5

f!. 1 0 en � -I u. -I � -5

� - 10 :c <.:> o 0.2 0.4 0.6 0.8 1

TIME

1 5

;g-� 1 0

� 5 0 -I 0 � u. -I -5 « � 0 -1 0 �

-1 5 0 0.2 0.4 0.6 0.8

TIME

Figure 9. Interaction between channel­to-channel and loop instabilities, KIN= 150 ,other parameters same as in Figure 8.

- 1 20 f!. ;; 80

� 40

u. 0 -I W � -40

� -80 <.:>

;g-�

� -I u. -I � 0 I-

o

1 20

80

40

0

-40

-80

- 1 20 0

0.2 0.4 0.6 0.8 TIME

J\M 0.2 0.4 0.6 0.8

TIME

Figure 10. Interaction between channel-to-channel and loop instabilities, KIN=20, other con­ditions same as in Figure 8.

In Figure 11, the loop's inlet loss coefficient was reduced even more while the inlet loss coefficients in each chanel were increased to stabilize them. In this case, the loop is only marginally stable but the channels are stable. Thus, the chanels oscillate almost in-phase. Note, however, that due to channel-to­chanel interaction, a slight component of the out-of-phase mode can be seen.

3. NONLINEAR PHENOMENA

The propagation of density-waves is governed by combined hydrodynamic and thermodynamic phenomena (which can also be coupled with other effects, such as thermal inertia of heaters, or neutron dynamics in nuclear reactors) which are highly nonlinear in nature. These nonlinear effects

29 1

determine various system characteristics, including the amplitude and frequency of oscilations. Since most boiling and condensing systems are only conditionally stable, the response of a perturbed system strongly depends on the magnitude of the initial (external) perturbations. This is illustrated in Fig. 12. In particular, while for a sufciently small perturbation the system may return to its original steady-state operating conditions, an increase in the perturbation's magnitude may lead to a divergent response. Furthermore, even if the perturbed system's response will eventually converge to the equilibrium operating level, its amplitude at some time may temporarily exceed the required thermal limits of the system (e.g. , a flow-reduction­induced CHF).

9

� 0 6 -

en � 3 0 .. U. o -.. w

-3 -Z Z « -6 :r: 0

-9 0.2 0 0.4 0.6 0.8

TIME 9

� 6

� � 3

0 o -.. U.

.. -3 «

I-0 -6 -I-

-9 0 0.2 0.4 0.6 0.8 1

TIME Figure 1 1 . Interaction between channel-to-channel and loop instabilities, Kil = lG2 = 11; Kel = Ke2 = 1.

292

P1

a x

P2 · _ · _ · - · - · - P 2

CASE (A). STABLE CHANNEL

LINEAR NONLINEAR \/-,

CASE (B) . UNSTABLE CHANNEL

Figure 12. Boiling channel response to step reduction in system pressure.

Whereas the linear system theory can be applied to nonlinear two-phase systems, it can be used to describe the behavior of these systems around their steady state operating points only as long as the perturbations remain sufficiently small. However, this approach is unable to predict other important system properties, such as the magnitude of perturbations in

293

various parameters external to the system (e.g., the temperature and flow rate of fluid entering the system, the system pressure, thermal power, etc.) required to maintain the stable operating mode of a conditionally stable system. Neither can it be used to predict the response of such systems when an unstable mode is reached.

When the operating parameters of a boiling system exceed the stability limits, the properties of the system response depend on which of two basic unstable modes will occur: subcritical bifurcations or supercritical bifurcations [7] (see Fig. 13).

linear stability I

OJ· . boundary I In � h !

STABLE UNSTABLE �

Ojin� linear stability I � boundary I

Oj in � - t t I

STABLE UNSTABLE �

(a) Supercritical bifurcation

(b) Subcritical bifurcation

Figure 13. Typical stability boundaries showing amplitude response for (a) limit cycle (b) finite amplitude boundary [7].

294

In the case of subcritical bifurcations, the unstable mode is reached when the amplitude of external perturbation is sufciently large even though the system is linearly stable. Clearly, this is a potentially dangerous situation. Supercritical bifurcations (Le., limit cycles) occur when the threshold of linear instability is exceeded. It has already been shown theoretically [10] that bifurcations of both kinds may occur in a boiling channel. Moreover, they have been measured in actual boiling systems, including both small-scale experiments [11] and operating nuclear power plants [5, 12] . The former were concerned with the channel-to-channel instabilities between two electrically heated parallel chanels. The latter data mainly refer to the neutronically­coupled density-wave instabilities, where the void reactivity feedback has a profound effect on the behavior of the perturbed system. For example, if a BWR is stable for a given negative void coefficient of reactivity, and then this coefficient changes (as a result of changing the operating conditions) to a positive value, even small oscillations in reactor parameters may eventually be large enough to cause power excursion which, in turn, may lead to a serious accident (the Chernobyl accident illustrates what may happen if the void coefficient of reactivity is positive). On the other hand, if the magnitude of the void coefficient of reactivity becomes too large on the negative side, supercritical bifurcations may occur, resulting in the onset of self-sustained oscillations of a finite, possibly large, magnitude. For example, in the stability tests performed on the CAORSO (Italy) boiling water nuclear reactor, such oscillations in power and flow were observed in selected channels of the reactor core These local neutronics-induced parallel-channel instabilities (see Fig. 14) may not cause oscillations in the recirculation loop, and still lead to the failure of fuel elements in certain core regions.

The development of nonlinear stability methods lags far behind those for linear systems. In most cases the analyses of nonlinear models of diabatic two-phase systems are based on numerical solutions of system equations in the time domain. Even though reasonable agreement with experimental data was obtained in some cases, especially when sophisticated computer codes were used, such as RETRAN [13] or RAMONA [5], both the complexity of the models and the nature of the method itself (direct numerical integration) do not allow for applying this approach to generic studies of the system's stability characteristics. Various theoretical methods of nonlinear stability analysis have also been applied to two-phase systems. Four major methods are listed in Table 6.

Table 6 Nonlinear methods

• Hopfs Bifurcation Method • Method of Liapunov • Harmonic Quasi-linearization (Describing Function Method) • Of Chaos (Fractals As Measure Of Attractors)

a: w

I I

A

FUEL ROD ASSEMBLIES

n. ..

z < U

1 0 1 2 1 4 1 6 1 8 20

TIME (SEC) 22 24 26

Figure 14. Typical local instabilities observed in a boiling water reactor.

295

The approach based on Hopfs theory was applied so far to rather simple models only [ 10, 14, 15] such as: a single boiling channel, uniform and constant heat flux, constant inlet temperature, etc. This method can be used, among others, to determine which of the instability modes (subcritical or supercritical bifurcations) will occur, and, as long as the amplitude of oscillations remains small, to evaluate this magnitude.

Whereas the original method of Lyapunov is also difficult to apply to advanced models of boiling systems, its recent extension [16, 17, 18] not only provides effective ;stability criteria for a large class of nonlinear systems, but also yields additional information about the properties of perturbed system trajectories . This new approach, which has mainly been used to study

296

nuclear reactor systems (including BWRs with void reactivity feedback [16, 7]), can be readily extended to other non-nuclear two-phase systems. The method itself can be used to study instabilities related to subcritical bifurcations (i.e., if a system is conditionally stable) and allows for determining regions of attraction of the steady state operating level and estimating the magnitude of perturbed trajectories as a function of the magnitude of various initial perturbations.

The method of harmonic quasi-linearization is an extension of the classical describing function method. It has been successfully used [19] to predict the magnitude and period of limit cycle oscillations in a boiling chanel.

One of the most interesting problems in the analysis of nonlinear effects in two-phase systems concerns the so-called period-doubling bifurcation phenomena, possibly leading to chaotic vibrations (see Fig. 15).

y(t)

y(t)

y(t)

Figure 15. Period-doubling bifurcation and the development of chaotic oscillations.

297

So far, only very limited theoretical studies have been undertaken in this area [20], and much more work is still necessary, including both experiments and theoretical/numerical analyses.

Each of the three methods described above (i .e., Hopfs bifurcation, Lyapunov method and describing function method) provides important, but partial only, information about properties of nonlinear systems. Usually, analytical studies are limited to one particular method. Since, in fact, these methods complement one another, it is interesting to compare the results obtained by applying each of them to a boiling channel model.

Let us consider for this purpose a uniformly heated boiling channel under constant-pres sure-drop boundary condition. U sing a homogeneous model of two-phase flow, the lumped-parameter mass and momentum conservation equations are given as Eqs. (46) and (47) in Table 7.

In Eqs. (46) and (47), the subscripts "1", "2" , and "24>" refer to the channel inlet (single-phase liquid), exit (two-phase mixture), and two-phase region,

Table 7 A model of uniformly heated boiling channel under constant-pressure-drop

condition

2 2 dO f G1 G2 f G2

L Tt = i\p + (1 - 2DH A.) - - 2DH ( ) (L - A.) Pf P2 P

f v 1 - 0\1 - [1 + (L - - G1)] (-- G1 + aL) G21 Pf Pf

(46)

(47)

(48)

(49)

298

Table 7 A model of uniformly heated boiling channel under constant-pressure-drop

condition (continued)

xl = J.lX1 + w2x2 + F(X1, x2)

X2 = -xl

• 3 Xl = J.lX1 + - 'Y X 1 X2 = -Xl

(50) (51)

(52a)

(52b)

respectively, p and G are the channel-length-averaged density and mass flux, A. is the nonboiling length, and the remaining notation is conventional.

v Assuming that A. = G1, where v=constant is the fluid residence time in

Pf the non-boiling region, and taking,

dp d Pf+P2 1 dp2 1 d� dt <It

2 2 G2 1 G1 G2 ( ) = ( - - -) P 2q, 2 Pf P2

- 1 G = 2 (G1 + G2)

where n = (q'Vfg)/Axshfg), Eqs. (46) - (47) can be rewritten as Eqs. (48) - (49) in Table 7.

Substituting, Xl = n (G2-G1) and X2 = G2 - Go, where Go is the steady-state flow rate, Eqs. (48) and (49) respectively, can be rewritten in the state-variable form as Eqs. (50) (51), where F (Xl X2 ) is a third order polynomial, containing quadratic and cubic term� with respect to Xl and X2 , the coefficients of which depend on the operating conditions and geometry of the channel. In particular, the above-mentioned coefficients can be chosen in such a way that Eqs. (50) - (51) become Eqs. (52) - (53).

As can be readily seen, Eq. (52) is linearly (i.e. with 'Y = 0) stable if J.I.<O and unstable if J.I.>O. Concerning the properties of solutions of the full nonlinear system, it can be shown by using the method based on Hopfs bifurcation theorem and its extensions that if J.l and 'Y are small and )1.1>0 then Eq. (52)

299

has a non-zero periodic solution, (limit cycle), x*(t) = (xi(t), x2(t)}. Moreover, if J.L>O (supercritical case), this periodic solution is stable and characterizes the asymptotic behavior (as t-+oo) of all trajectories with initial values, x(O), within a certain distance from x*(O). On the other hand, if J.L<O (subcritical case), the periodic solution is unstable, so that no solution, other than x* itself, will satisfy: lim [x(t) -x*(t)] =0. The above-given results are sumarized in Table 8.

t-+o

Table 8 Results of • Linear System (Eq. (52) with 'Y=O) is stable ifJ.L <0 and unstable ifJ.L >0 • Hopfs Bifurcation Theorem shows that if J.L and y are small and IlY <0,

then Eq. (52) has a nonzero periodic solution x * (t) = ( xi(t), xi (t) } Moreover, if Jl >0 (Supercritical Case) this periodic solution is stable (Stable Limit Cycle). If J.L <0 (Sub critical Case), x *(t) is unstable.

• Lyapunov Method based on the following function

where,

yields:

(53)

(54)

(1) ifJ.L>O, the solution x(t) = 0 of the nonlinear system given by Eq. (52) is unstable,

(2) if J.L<O, the solution x(t) = 0 is asymptotically stable, and: (2a) if "'(>0, all non-zero trajectories asymptotically approach the zero­

solution as t-+-, (2b) ify<O, stability is conditional only, and the set

Do = ( (x1(0),x2(0» : x� (0) + o>J. x �(O) �) y

belongs to the region of attraction of the origin,

300

Table 8 Results of (continued)

(2c) for any solution asymptotically approaching zero, the maximum departure from the zero-steady-state can be evaluated (estimated) as a function of the initial conditions from,

X� (t) + olJ. x �(t) < x� (0) + olJ. x �(O). • Harmonic Quasi-Linearization.

Transforming Eq. (52) into,

y - f.1Y + IDly + 'Y (y)3 = 0

and assuming,

y= A' sin (oo*t)

yields,

y = Xl = A'oo* cos(oo*t) = A cos(oo*t)

(y)3 = A3 [ � cos(oo*t) + i cos(3ro*t) ]

(55)

(56)

(57)

(58)

Ignoring the higher (3rd) order term in Eq. (58), a quasi-linear form of Eq. (55) can be written as,

• • 3 A2 'Y • Y - Il (1 - 4 ) y + = 0

f.1

Assuming f.1'Y > 0, and substituting Eq. (56) into Eq. (59), yields,

Xl (t) = 3 1 cos(oot)

X2(t) = 1. sin (oot) ro

which is the limit solution of (52).

(59)

(60)

(61)

Other aspects of the solution properties of Eq. (52) can be studied by using the method of Lyapunov. The results based on a particular form of the function of Lyapunov, are also given in Table 8.

301

If Eq. (52) has a periodic solution (limit cycle), it can be approximately evaluated by using the method of harmonic quasi-linearization. For this purpose, Eqs. (52a) - (52b) will be replaced by a single second-order equation with respect to y=-x2(t). The resuls are shown in Table 8. As can be seen, Eq. (55) is a generalization of the classical Van der Pol equation.

It is interesting to note that the limit cycle magnitude, given by Eq. (60),

(if 1l,,(>0)

depends on JJi'Y only (i.e. it remains constant for different (small) Il as long as 1.1 and 'Y change proportionally). Let us now compare the above-given theoretical predictions with the

numerical results, obtained for JJi'Y =3, 1.1 =± 0 . 1 and (1)=1, shown in Figs. 16 and 17. As seen in Fig. 16, if Il = 0.1 > 0, there exists a stable limit cycle, the theoretically predicted magnitude (A=2) and angular frequency «(1)*=1) of which agree very well with the values obtained from numerical calculations. It is also clear that in this case the zero-solution is unstable. If Il = -0.1 < 0, the unstable limit cycle, predicted theoretically, is not

explicitly shown in Fig. 17, but its presence can be readily deduced from the two oscillatory solutions (one diverging, the other converging) given there. In this case, the zero-solution is asymptotically stable and its theoretically-estimated region of attraction includes all initial conditions such that x� (0) + x� (0) <3. The result shown in Fig. 17 (upper part) has been obtained for Xl (0) = -0.8 and X2 = 1.5, i.e. x� (0) + x� (0) = 2.89. The maximum theoretically calculated initial value of Xl to guarantee

convergence ofx(t) to zero is, xl (0) = 1.73 (ifx2 (0) = 0) The actual permissible magnitude may be slightly higher, but by no more than 15%, since x1(0)=2, x2(0)=0 already yield the limit cycle solution. The maximum values of xl (t) in this case (the first peak in the upper part of Fig. 17), obtained from the theory and numerical calculations (X1,max = 1.7) are in perfect agreement with each other.

4. STABILITY MARGINS

Because of several modeling simplifications, theoretical predictions of instabilities in boiling systems always contain a substantial uncertainty level. In particular, the uncertainties associated with the evaluated instability thresholds can be quantified by using the so-called stability margins. VariOllS definitions of linear system stability margins have been used for this purpose, including the classical gain and phase margins. Another definition, which encompasses both the gain and phase margins combined together, can be obtained from the normalized phase diagrams [7], as shown in Fig. 18.

302

w --' � 2.00 C/) w � 0.00 --' w a: -c5 -2.00

I C\J C!J

TIME

-4.00

0.00 30.00 60.00 90.00 1 20.00 1 50.00

w --' « &5 2.00

� -

c5 -2.00 I C\J C!J

TIME

0.00 30.00 60.00 90.00 1 20.00 1 50.00

Figure 16. Flow rate oscillations in a boiling channel - supercritical case, � = 0.1, 'Y = 0.033, CJ) = 1.

W .. <C o 2.00 en il > � 0.00 .. il a: -c5 -2.00

I C\I (!J

0.00

W .. <C � 2.00

il > � 0.00 .. il a: -c5 -2.00

I C\I

(!J

30.00 60 .. 00 90.00 1 20.00 1 50.00 TIME

0 . 00 30.00 60.00 90.00 1 20.00 1 50.00 TIME

303

Figure 17. Flow rate oscillations in a boiling channel subcritical case, J.1 = -0.1, "( = 0.033, co = 1.

304

. I m

Re

Figure 18. Typical nyquist loci for various operational parameters.

All the above-mentioned definitions apply to single-input/single-output (SISO) systems. As discussed before, two-phase loops and channels are, in general, multi-inpuUmulti-output (MIMO) systems. It can be shown [7] that the calculated conditions of the onset of instabilities are the same, regardless of the particular input/output functions used in the analysis. Consequently, both SISO or MIMO models can be used for this purpose. A question arises, however, whether SISO and MIMO models of a given boiling system are fully equivalent in describing stability characteristics of the system. As has been shown by many authors, the SISO representation of multivariable systems may become inadequate in quantifying various types of uncertainties associated with mathematical modeling of complex physical phenomena.

In order to investigate the sensitivity of multivariable system properties to the uncertainties caused by the differences between models and reality, a method has been developed [21] relating MIMO stability margins to the nonn of the close-loop transfer function of a boiling system, given by the following equation,

i (s) = !I (s) i (s) (62)

In Eq. (62), the matrix, !! (s), is the system's closed loop transfer function, and its norm is defined as,-

11HI12 A max 11&112 = [lmax am·)] 112 � 5max (!!) - - IIxII 2 <1

- - (63)

305

where IIHII2 is the Euclidian norm of the vector, x, !!* is the matrix conjugate to !I, Amax is the largest eigenvalue of and 0max is the largest singular value ofH.

As it can be readily shown, if the system under consideration is stable, we have,

det �-1 (joo)] >0

and, consequently,

(Omax[ [!!(joo)]} -1 >0 for all w.

When the instability threshold is reached, there exists such 00* that, det [!-l (joo*) = {omax[ [!(j00*)]r1

= o.

Let us define the system stability margin as,

(64)

For a stable system, m is always a positive number, decreasing to zero when the system is approaching the instability threshold. Hence, the definition given by Eq. (64) can be indeed used to quantify the stability margins of multivariate system.

The fundamental practical problem concerning applications of the above­mentioned concept deals with establishing a minimum of the stability margin, below which system operation is unacceptable. Since the existing stability acceptance criteria are mainly based on the SISO analysis, it is natural to compare the value of m, evaluated for a multidimensional model, with similar results for individual one-dimensional components. Specifically, in the case of model given by Eq. (62), and assuming that H is a (2x2) matrix, we can calculate m for the three different cases: !!, HII, and H22. It should be mentioned at this point that, for a scalar systeIl, given by a transfer function, Hii (joo), Eq. (64), reduces to,

m . . = min [H . . (joo)] -l 11 w 11 (65)

In order to illustrate the relationship between m, m1 1 , m22 , results obtained for a model of the Peach Bottom nuclear reactor for various operating conditions, are shown in Fig. 19. The analysis of the results given there yields the following observations:

306

the stability margins, given by Eq. (65), for the reactivity (Hll) and the inlet velocity state variables (H22) are similar, the two-dimensional stability margin, m, is two orders of magnitude lower than the one-dimensional margins mll and m22, the frequency at which the minimum in Eqs. (64) and (65) is reached the same for al three cases under consideration, after the system becomes unstable, the values of m and mii (i=1,2) start increasing again.

It follows from the above that, at least for the case considered here, the MIMO stability margin, given by Eq. (64), can be directly related to the 8180 results, and consequently, a unique acceptable margin can be established for the entire two-dimensional system. On the other hand, however, having evaluated a given value of m>O from Eq. (64), we still cannot determine whether the system is stable or unstable. Hence, the stability margin concept discussed herein must always be used together with one of the existing stability criteria, such as the Nyquist criterion, for instance.

O.jOO 5 4 6

m22

0.48 m

Figure 19. Comparison between MIMO and 8180 stability margins.

REFERENCES

1 J.A. Boure, A.E. Bergles and L.S. Tong, Nucl. Eng. Des., 25 (1973).

307

2 A.E. Bergles, (eds. S. Kakac and F. Mayinger) Two-Phase Flow and Heat Transfer, 1, Hemisphere Publishing Corp., 1976.

3 M. Ishii and N. Zuber, Thermally Induced Flow Instabilities in Two­Phase Mixtures, Proc of the 4th Int. Heat Transfer Conf., 5, 1970.

4 R.T. Lahey, Jr. and D. Drew, NUREG/CR-144, (1980). 5 L. Moberg and K. Tangen, The Time Domain BWR Stability Analysis

Using 3-D Code RAMONA-3B, Proc. of the ENO 2986 International ENSIANS Conference, Geneva, Switzerland, 1986.

6 M.Z. Podowski, Chapter entitled "Two-Phase Flow Dynamics," in this same book.

7 R.T. Lahey, Jr. and M.Z. Podowski, (eds. G.F. Hewitt. J.M. Delhaye and N. Zuber) Multiphase Science and Technology. 4, Hemisphere Publishing Corporation. New York. 1989. 183-371.

8 M.Z. Podowski. R.T. Lahey. Jr .• A. Clausse and N. DeSanctis. Chern. Eng. Corom., 93, (1990).

9 M. Aritomi. S. Aoki and A. Inoue. Journal of Nuclear Science and Technology, 14(2), (1977).

10 J.L. Achard, D.A. Drew and R.T. Lahey. Jr., J. Fluid Mech., 155. (1985) 213-232.

11 T.N. Veziroglu and S.S. Lee, ASME Paper No. 71-H-12, 1971. 12 S.A. Sandoz and S.F. Chen, Trans, Am. Nucl. Soc. , 45, (1986). 13 K Hornyik, EPRI NP-2494-SR, 1982. 14 Rizwan-uddin and J.J. Dorning. Nuc!. Eng. Des., 91. (1986). 15 Rizwan-uddin, Trans. Am. Nucl. Soc .• 53, (1986). 16 M.Z. Podowski, Proc. of the ANS Topical Meeting on Reactor Physics

and Core Thermal-Hydraulics, Kiamesha Lake, NY, 1982. 17 M.Z. Podowski, IEEE Transactions on Automatic Control, V. AC31,

198&. 18 M.Z. Podowski, IEEE Transactions on Automatic Control, V. AC31.

1986b. 19 A. Clausse, A. Kerris and J. Conventi, Proc. of the 3rd International

Symposium on Multi-Phase Transport and Particle Phenomena, Miami Beach. FL, 1986.

ID J.J. Dorning, Trans. Am. Nuc!. Soc., 53, (1986). 21 J. Balaram, C.N. Shen, R.T. Lahey, and M. Becker, An Analysis of

Boiling Water Nuclear Reactor Stability Margin, NUREG/CR-3291, 1983.

APPENDIX A Derivation Of A Linear Frequency-Domain Model For Boiling Channel Stability Analysis

In order to perturb Eqs. (1) and (2) around a steady-state operating point, the steady-state parameters must be evaluated. For a given system pressure, p, and inlet velocity of the subcooled liquid, jin,o. the steady-state mass flux is given by,

308

Go = Pf jin ,o (A.l)

If the inlet subcooling is L\hin 0' and the assumed uniform axial wall heat flux is q � the steady-state le�gth of the liquid region (i.e., the boiling boundary), A.o• can be evaluated from Eq. (11) in [6] as,

(A.2)

Because of the assumed incompressibility of the liquid. the volumetric flux in the single-phase region is constant and equal to jin,o In the two-phase region. the volumetric flux. <jo(z» . can be obtained by integrating Eq. ( 13), in

[6], in which ro = (q� PH) / (hfgAx s) , and p are assumed to be constant. The resultant equation can be written as,

where,

Hence, the homogeneous density, < Pho >, can be obtained from,

Go Go < Pho (z) > = < jo (z) > Oo(Z-Ao) + jin,o

(A.3)

(AA)

(A.5)

For transient analysis, Eqs. (1) . (A.3), (AA), and (A.5), respectively, generalize to,

<j(t,z) > = O(t,z)[z-A(t)] + jin(t)

O(t,z) = �

Ax-s hfg

< Ph(t,z) > = < j(t,z) >

(A.6)

(A.7)

(A.B)

(A.9)

309

Using Eqs. (A.I) - (A.9), we can perturb and Laplace-transform Eqs. (1) and (2), to obtain,

(A. 11)

The next step in the analysis deals with relating the variables in Eqs. (A.IO) and (A. II), B�, B < J > , and B < Ph >, to the channel external perturbations.

The volumetric flux (i.e. , the superficial velocity) perturbation. B < J > . can be obtained from Eq. (A.7) as,

(A. 12)

The boiling boundary perturbation. o�. can be expressed in terms of the

local enthalpy perturbation. B < ii (S,Ao) > , by integrating the energy equation.

Eq. (A.9). in [6] , from Ao, to A(t), and then perturbing it. The resultant equation is,

Taking into account that < h (t,A) > = hf = ho (Ao) ' and Laplace­transforming Eq. (A. 13). yields.

- GoVfg - A -BA.(s) = - -- B < h(s,A.o) > = - 0 B < h(s,A.o > nohfg Llsub (A. 14)

3 1 0

The enthalpy perturbation at the boiling boundary, S < 1\ (5,""0) > , can be evaluated by perturbing and Lapace-transforming Eq. (11) in [6]. Under the assumptions mentioned earlier, we obtain,

d(Sh) 5 - PH q" Sq" (5/Z) Sj-__ + Sh(s,z) = 0 .. � dz Jin,o Go Ax-s qo Jin,o (A.15)

Before Eq. (A.l5) is integrated, we should notice that the perturbation in the surface heat flux, Sq"(s,z), is not an independent variable, and can be expressed in tenns of the perturbation in the heater internal heat generation rate, &1�' (5) . The specific relationship between &1" and sq' " depends on the model used to quantify the heated wall dynamics, as well as the heater geometry. In general, the following expression can be obtained,

Zl (5) &1" (s,z) + Z2 (5) sq� (5) = STw (s,z) (A. l6)

If a lumped-parameter model is used, given by Eq. (25) in [6], we have:

In the single-phase region, the wall temperature, Tw, is given by Newton's Law of Cooling, as,

(A. l7)

where Tb is the bulk fluid temperature. Perturbing Eq. (A. l 7 ) , and taking into account that,

H1cp(t) = H1cp,o [jin (t) / hn,o]a (where, normally, a = 0.8), yields, - -

" Sin Sci" = H1cp,o (STw - STb) + qoa Jin,o (A. l8)

Eqs. (A.16) and (A.l8) can be combined to obtain sq" (s,z) in the single-phase region as a function of sq�' (s) , sin (s) , and an(s,z). Substituting the resultant expression into Eq. (A.l5) and rearranging, we obtain,

' " d(�h) _ �Jm (5) &] (S) -- + 4>(S) �h(s,z) = 9(s) + 13(5) dz J . m,o qH,o where,

3 1 1

(A. 19)

(A.20)

(A.21)

(A.22)

Integrating Eq. (A.19) from z = 0 to z == A.o , with the boundary condition,

�h(s,O) = �hin<S), yields,

�h(s,A.o) = Exp [-4><s)A.o] �hin + {1 EXP[ 4>(S)A.oU[9<S) .��n + J3(s) ] 4>(5) lin 0 qH , ,0 (A.23)

Substituting Eq. (A.23) into Eq. (A.14), the expression for �):(s) becomes,

where,

A 9(s) { [ ]} Al (s) = - 0.. 1 Exp -4>(s)A.o MlsubJm,o

A. J3(s) { } A2 (s) = 0

II 1 - Exp[-q,(S)AO] Mlsubq 0

A. A3 (s) = 0 Exp [-4>(5)A.O] Llsub

(A.24)

(A.25)

(A.26)

(A.27)

3 12

(A.28)

Eq. (A.24) can be used to eliminate ai from Eq. (A.12). In order to eliminate an from the expression for a < } > , we can use the equation which describes the heated wall dynamics, Eq. (18) in [6], combined with Newton's Law of Cooling for the boiling heat transfer,

q" = C(p) (Tw - Tsat ) 11m

After perturbing, Eq. (A.29) becomes,

where, Z3 = [(Twto - Tsat) / q� / m] = Cons tan t .

Substituting Eq. (A.30) into Eq. (A.l6), and rearranging, yields,

an(s) = Z4 (5) &i�' (s) where ,

Now, substituting Eqs. (A.24) and (A.31) into Eq. (A.12), we obtain,

o < }(s,Z» = [l -Oo Al(S)] a1m(S) + [(Z- Ao) Z4 (S) - OoA2 (S)] &i�' (s)

(A.29)

(A.30)

(A.31)

(A.32)

- 0oA3 (s) ahin (s) (A.33)

Finally, in order to obtain an equation for 0 < PH(S,Z) > , we can combine Eqs. (10) and 13) in [6], and perturb the resultant equation,

where &l is given by Eq. (A.3l). Substituting Eqs. (A.3) and (A.5) into Eq. (A.34) yields,

d _ ) (s I 00 + 1) _ G [ a < > -(S < P > + a < p > =.. dz h (Z-AO + jm' 0 / °0) h 0 2 (Z- A + J'. , 0 0 m,o 0 SO

3 1 3

(A.35)

In order to integrate Eq. (A.35), a boundary condition must be established at Z = 1.0 ' For this purpose Eq. (10) in [6] can be integrated from Ao to A, to yield,

(A.36)

After perturbing and Laplace-transforming Eq. (A.36), and taking into account that, < Ph (t,A) > = Ph and < j(t,A) > = hn(t) , we obtain,

a < Ph (S,AO) > = � [S1n (t) - Sj(s,Ao) >] hn,o

Using Eq. (A.12), Eq. (A.37) can be rewritten as,

S < Ph (s,Ao ) > = SiCs) Jin,o

(A.37)

(A.38)

Now, Eq. (A.35) can be integrated with the boundary conditions given be Eq. (A.38), to obtain,

- S < Ph (s,z) > = {S[jin,J < jo(z) > ](sl°o·l) -Oo�Oo / (8-00)][ Go / < jo(z) >2] SX(s)

+ -[iin,ol < io(z» ](slilo - I)} [no I (s -no)][Gol < io(z) >2]6;" in { [ ](S/O -l)} 1 [ 2] -- 1 - hn,ol < jo(z) > 0 (s -Oo) Gojin,ol < jo(z) > aO(s)

(A.39)

Finally, Eqs. (A.24), (A.33), and (A.39) can be substituted into Eqs. (A.10) and (A.ll), and the spatial integrations performed. Assuming for simplicity that the only local losses are those at the channel inlet and exit, performing the integration in Eq. (A. ll), and rearranging, yields the following expressions for the pressure drop perturbations in the single-phase and two­phase regions, respectively,

(A.40)

3 14

where, [ sA. A. (fjin 0 g ) ] r (s) = G � + f--2. + Kin + Al (s) 1,H 0 J. DH 2DH Jin 0 mp ,

lll,H(s) = Go [F1(s)-F2 (s) A1 (S)]

llZ,H(s) = Go [FZ(s)AZ(s)-F3 (s)]

ll3,H(s) = Go F2(s)A3 (s)

(i =Z,3)

+ + ( ){EXP[(2no-S)tex] -1}+ J� ( (1-ExP( Ootex)

H 8-00 s-2Oo m,o 8-

+ no [Exp(-st ) - 1] } + K {I + no (1 -Exp [(00 - s) tex] )} s ex ex 2(s - no)

(A.41)

(A.42)

(A.43)

(A. 44)

(A.45)

(A.46)

(A.47)

- K 0 {.!.ExP[(O - S)'t ] - 1 - 00 [1 - Exp[(0 - S)'t ] J} ex 0 2 0 ex 2(s -00) 0 ex

{<S+2(0) 82 . [ ] F3 (s) = 00 -A.o) - (8-00)00 Jin,o 'tex - (8-00)2 Exp (00 -s}tex -1

f ( ( )2 iin,o

+ LH - 1.0 - (LH -1.0) H 0 0 0

- (S goo) [l -EXP(-Ootexl] - ; [l - ExP(-S<exl] 1

+ \(LH - "0) - (1 - Exp [(00 - S)tex])) Kex } Z 4 (s)

3 1 5

(A.48)

(A.49)

(A.50)

3 1 7

APPLICATIONS O F FRACTAL AND CHAOS THEORY IN THE FIELD OF MULTIPHASE FLOW & HEAT TRANSFER

R.T. Lahey, Jr.

The Edward E. Hood, Jr. Professor of Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 USA

Abs1ract The elements of fractals, static & dynamic bifurcations, and chaos theory

are presented and discussed. Engineering examples for the application of these analytical techniques are given for density-wave instabilities in a single­phase thermosyphon and a two-phase boiling channel.

1. INTRODUCTION

The study of chaotic phenomena in deterministic dynamical systems relies heavily on the concept of fractal geometry and the topological interpretation of phase space trajectories. This emerging science has already produced some dramatic breakthroughs in our understanding of nonlinear dynamics. It is the purpose of this tutorial chapter to summarize the essential concepts necessary for understanding how these ideas and analytical techniques may be applied to physical systems. In particular, applications in the area of single- and multi-phase flow & heat transfer will be stressed.

This paper will begin by reviewing some important concepts concerning fractals and fractal dimensions. Next, some of the essential elements of the theory of chaos will be presented. Examples of static and tlynamic bifurcations will be presented and discussed, and phase plane interpretations of both lineal' and nonlinear oscillations will be presented and generalized to higher order systems of equations.

Next, chaotic phenomena will be discussed and the concepts of a "strange attractor" and basins of attraction will be presented. Bifurcation diagrams, Poincare sections, so-called first return maps, and Lyapunov exponents, as well as other tests to verify a chaotic response, will be discussed.

Finally, these techniques will be applied to problems of density-wave instability in a single-phase natural circulation loop and in a boiling channel. It will be shown that chaotic phenomena may occur for some operating conditions.

3 1 8

2. FRACfALS

There are many good books on the theory of fractals in which fractals are carefuly defined [1,2,3,4]. For our purposes here it is sufficient to define a fractal as, a self-similar mathematical object which is produced by simple repetitive mathematical operations. In order to quantify what is meant, let us consider a few sets which exhibit important fractal properties.

As shown schematically in Figure 1, the Koch set is generated by dividing a line segment of length L into three equal segments (U3), removing the center segment and in its place forming an equilateral triangle having side length U3. As the process is repeated a rather fuzzy looking continuous curve, which is nowhere differentiable, is formed. Significantly, in the limit this set has infinite length.

The Cantor set is shown in Figure 2. One way to form it is by again dividing a line segment of length L into three equal segments (U3), except that in this case the center segment is removed and discarded. As the process is repeated, we obtain in the limit a set having zero length and an infinite number of points. The Cantor set is a very important one in the theory of chaos, since the phase space trajectories often form a pattern on so-called Poincare sections (to be discussed later) which have some properties similar to that of a Cantor set.

(n=O) (n= l ) (n= 2 )

(n=3 )

(n=4)

Figure 1. The Koch Set for four applications of the Koch algorithm

3 1 9

1 /3

1 /9 1 /9

Figure 2. The Cantor Set.

Other more complicated sets are also possible using simple iterative algorithms. One of the most famous is the Mandelbrot set. This set is generated from the iteration of the following complex nonlinear function:

(1)

where, z = x + iy, and, C = A + iB.

Equation ( 1) is equivalent to iteration of the following coupled real nonlinear functions:

(2a)

yn+1 = 2xnYn +B (2b)

As shown in Figure 3, astoundingly complex patterns can be generated with the relatively simple Mandelbrot set. Moreover, this set demonstrates the fractal property that it repeats itself over and over as we zoom in on a particular portion. This is an important property of fractal sets, and one which we will find significant when we discuss chaotic "first-return maps".

320

Figure 3. The Mandelbrot Set [4]

Let us now turn our attention to how to characterize the various sets. In particular, the fractal dimension of a set. There are a number of possible fractal dimensions which one can define. Unfortunately, the fact that they are not all equivalent has often let to confusion. In this chapter we will consider only a few of the most important fractal dimensions for practical applications.

The Hausdorff-Besicovitch dimension (DH-B) is defined below:

DJi = lim in N(e) -B

e�O in( 11e)

hence,

N(e) oc E-DH-B

(3a)

(3b)

32 1

As shown in Figure 4, £ is the length of a hypercube, and N(e) is the smallest number of hypercubes necessary to enclose the set of hyperspace points shown in Figure 4.

We note that when the set is a single point, N(e ) = 1, then Eq. (3b) implies the Hausdorft'-Besicovitch dimension is, DH-B = 0.0. In contrast, when the set is a line segment of length L, the number of hypercubes needed is, N(e) = Ue. Thus, Eq. (3a) implies:

DH-B = lim {Ln(L/e)} = lim {LnL-Lne} = 10 e-+O In(l/e) £--+0 -Lne

Similarly, when the set is a surface of area S, N(£) = 8/£2, thus:

DH-B = lim {LnS- 2ine} = 2.0 e-+O -ine

For a fractal set, such as the Koch set, which covers more hyperspace than a line but less than a surface, we expect, 1.0 < DH-B < 2.0. Indeed, for this set we obtain [5]:

ln4 DH-B = - .:. 126 Ln3

Figure 4. TIlustration of the covering of an object (a set of points) by cubes of linear dimension [5]

322

Similarly, for the Cantor set, which covers more hyperspace than a single point but less than a line, we expect, 0.0 < DH-B < 1.0. Since the Cantor algorithm implies, E = (l/3)m and N(e) = 2m, we obtain:

. ln2m ln2 DH-B = lim -- = -"='O.63 e�O ln3m ln3

Unfortunately, for many practical applications the limiting operation implicit in the definition of the Hausdorf-Besicovitch dimension converges very slowly. Thus it is not useful for most cases of interest.

Hence, let us next consider the so-called correlation dimension, DC. This dimension is particularly useful for the analysis of data. It consists of centering a hypersphere about point-i in hyperspace and then letting the radius (r) of the hypersphere grow until all n points are enclosed. In practice, since the number of points (n) is finite, many spheres are used (centered about different points) and the results are averaged. That is, the correlation function, N(r), is given by:

N(r) = lim .. r �H(r- I!i - ! · I) n2 . . . J l�J J

where, H(�) is a Heaviside step operation defined such that: {to H(�) =

0.0

It has been found that, in the limit as r�O [6],

Thus,

log N = DC log r + C2

(4)

(5)

(6)

As can be seen in Figure 5, if we plot N(r) versus r in a log-log plot then the slope of the line will be the correlation dimension, DC.

If we have an analytic function that is being numerically evaluated it is straightforward to compute the correlation dimension since the dimension of the hyperspace is known (ie, al the state variables are known). For example, the Cantor set has a correlation dimension of, DC = 0.63. Interestingly this is the same as the Hausdorff-Besicovitch dimension. Unfortunately this is not always true. In fact, it has been found [6] that, DC S DH-B.

l og N

o

D(5)=D(6)=

/ /

/ /'\ / WHITE NOISE

log r

(FRACTAL DIMENSION)

5 -(IMBEDDING ) DIMENSION

p

Figure 5. Correlation Dimension (DC).

323

Let us now consider evaluating the correlation dimension (DC) of data. If

we are processing experimental data we normally do not know how many state variables are needed to characterize the process. Fortunately, it has been shown [7] that a pseudo-phase-space can be constructed using time-delayed measurements of only one temporal measurement, and that the basic topology of the attractor being investigated will be unchanged. This was an important discovery since it allows one to calculate the fractal dimension of a wide range of experimental data.

Let us assume we are measuring some variable X (eg, pressure, temperature, etc.). We then have readings X(t), x(t+'t), X(t+2't), . . . , where the

324

time delay t is somewhat arbitrary, however it is often taken to be the period of the maximum energy peak in the power spectral density (PSD) function. To determine DC, we must first assume a dimension for the hyperspace. Let us start by assuming a 3-D space, and plot x(t), x(t+t) and X(t+2t) as shown in Figure 6. We then center a sphere of radius-r about point-i and apply Eq. (4). Next we make a log-log plot, as in Figure 5, and obtain a slope, DC(3). We then assume that four state variables are needed to describe the process and apply Eq. (4) to the 4-D hyperspace given by, x(t), x(t+t), X(t+2t) and X(t+3t). As before we make a log-log plot of the results and get a new (steeper) slope, DC(4). We continue this process for 5-D space, then 6-D space and so on, until, as shown in Figure 5, the slope (DC) no longer changes.

In the example shown in Figure 5, it has been found that the minimum number of state variables (p) to describe the process being measured is p = 5. This is called the imbedding dimension of the process. As shown in Figure 5, the imbedding dimension can be most easily recognized by plotting D versus p. The degree of freedom (p) at which D stops changing is the imbedding dimension and the slope for this value is the correlation dimension (DC) of the process. It should be noted that a truly random process (ie, one which has no underlying structure) has no finite imbedding dimension. That is, as can be seen in Figure 5, the slope never saturates for so-called "white noise" .

When random noise is superimposed on the chaotic (ie, deterministic) signal, one expects [8] the slope in the "large r" part of the plot to converge to D C , as discussed above. In contrast, for the "smaller rIO part, the slope is different and never converges. Indeed, this part of the curve reflects the white noise in the system.

• • • •

•

•• • •

. .. . : • •

• ••

• • • • • • • • •

: . . . . .

X (t + r)

Figure 6. 3-D Phase Space Representation of Signal X(t)

325

Another important and interesting fractal dimension has been discussed by Feder [1]. This dimension is sometimes called the Hurst dimension, DH-B. Hurst showed that many time varying chaotic processes, �(t), of record length t, can be correlated by:

where,

R(t) � Max X(t;t) - Min X(t;t) te t tet

t X(t;t) = J [�(t') - �(t)]dt'

t-t

t �(t) = .! g(t')dt' t t-t

and, { t }�2 S = � H�(t,) _ �]2

dt'

t-t

(7)

(Sa)

(Bb)

(Be)

(Bd)

Obviously, X(t;t) is the cumulative temporal variation of �(t) about its mean, �; R(t) is the difference between the maximum and minimum value of X(t;t) in interval t, and S is the standard deviation of �(t).

It has been found [1] that random processes are characterized by DH = 0.5, while chaotic processes are correlated by, DH = 0.7. Thus, processes having DH > 0.5, have some underlying structure. Thus the Hurst dimension, DH, is a relatively easy way of determining when a random-looking process may have a hidden structure.

While there is much more than can be said about fractals, the ideas which have been presented above should be sufcient to allow one to understand the fractal nature of chaos. Thus let us now tum our intention to the mathematics which underlies chaos theory. First, we will consider the elements of bifurcation theory.

326

3. BIFURCATION THEORY

In order to understand how the solutions of nonlinear equations may bifurcate, and what it means when a bifurcation takes place, let us consider some simple examples of static and dynamic bifurcations. This tutorial approach follows the work of Doming [9]. There are two main classifications of bifurcations; static and dynamic bifurcations.

3.1. S�c �aoDS The most common and important static bifurcations are the turning point

(ie, saddle point) bifurcation, the transcritical bifurcation and the pitchfork bifurcation. It should be noted that the solution of a single nonlinear differential equation, having only one state variable, may exhibit these static bifurcations. In contrast, at least two state variables are required for dynamic bifurcations.

Let us consider some nonlinear first order ordinary differential equations which represent the standard forms that the various static bifurcations can be reduced to. We begin by considering,

�(t) = fl (x(t),J,1) = J.1 - x2(t) (9)

The fixed (ie, steady-state) points of this differential equation are given by setting the time derivative in Eq. (9) equal to zero. Hence, fl(X;J,1) = 0, and thus,

(lOa)

or,

(lOb)

In order to examine the stability of these fixed points, we linearize Eq. (9), obtaining:

ax = ax • �fl et o thus,

where the perturbation of the state variable (x) is given by:

(Ua)

(Ub)

327

11 Sx = x(t) - Xo (Uc)

There are two possible values of the steady-state solution, xo' These are given in Eq. (lOb). Thus we have:

.(1) (1) Sx = - 2xo Sx =

and,

(12a)

(12b)

Assuming that the parameter 1.1 is a positive real number, the solution of Eq. (12a) is stable, and is given by:

(13a)

while Eq. (12b) is unstable, and is given by:

(13b)

The turning point (ie, saddle point) bifurcation and its solution flow are given

in Figure 7. It can be seen that the lower branch, x�), is shown dashed to

denote that it is unstable. Next we tum our attention to a somewhat more complicated equation given

by:

(14)

This equation exhibits what is called a transcritical bifurcation. As before the fixed points are given by:

(15a)

thus,

(15b)

328

.. .. .. -

Figure 7. Turning (ie. , saddle) point bifurcation

The stability of the fixed point is given by the solutions of the linearization ofEq. (14):

(16)

The solution ofEq. (16) is:

(i) (i) (�-2x�i))t 5x (t) = 5x(o)e

(17)

For i=l, Eqs. (15b) and (17) yield,

(18a)

329

while for i=2,

(l8b)

Equations (18) show that when J.1 > 0, the solution given by Eqs. (18a) is unstable while the solution in Eq. (18b) is stable. In contrast, when J.1 < 0, the solution in Eq. (18a) is stable while that in Eq. (18b) is unstable. This bifurcation and its solution flow are shown in Figure 8.

Let us next consider the so-called pitchfork bifurcation. This interesting and important bifurcation occurs in the solution of the following nonlinear ordinary differential equation:

�(t) = f3(X(t);J.1) = J.1X - x3 (19)

As before, the fixed points are determined by setting the time derivative to zero. Thus,

(2Oa)

hence,

x(l) = 0 o J.1 < 0 (2Ob)

x(l) = 0 o j.1 > 0 (2Oc)

The stability of these fixed points are determined from the solutions of the linearization of Eq. (19):

(21)

The solution of Eq. (21) is given by:

(22)

From Eqs. (20b) and (21) we have:

330

x

Figure 8. Transcritical Bifurcation

(1) (1) ax (t) = ax (o)eJ.1t

(2) Xo = �

- - - - - - - � Jl

(23a)

Thus, when J.1 > 0 the solution is unstable, while for J.1 < 0 the solution is stable. Next, from Eqs. (20c) and (22), we find stable solutions for both branches of

the pitchfork:

ax(2)(t) = Sx

(2)(o) e -2J.1t

Sx(3)(t) = Sx

(3)(o)e -2J.1t

(23b)

(23c)

The pitchfork bifurcation and its solution flow are given in Figure 9. It can be seen that this bifurcation looks similar to a saddle point bifurcation, except that both the upper and lower branch of the pitchfork bifurcation have stable fixed points. As a consequence the solution flow on the lower branch is different.

Finally, we note that Eq. (19) is a special case of the following nonlinear ordinary differential equation:

33 1

� (t) = f4(X(t);�,9) = 9 + � - x3 (24)

where 9 is often referred to as the imperfection parameter. Obviously when 9 = 0, Eq. (24) reduces to Eq. (19). When 9 :1= 0, the fixed points of Eq. (24) are given by:

which has three roots given by [10],

X�1) = (81 + 82)

(2) 1 i� Xo = -"2(81 +82)+2(81 -82)

Figure 9. Pitchfork bifurcation

(25)

(26a)

(26b)

(26c)

332

where,

(27a)

(27b)

As before, the stability of these fixed points is given by linearizing Eq. (24),

(28)

which is the same as Eq. (21). The solution ofEq. (28) is given by:

(29)

Thus the stability of the fixed points depends on the value of the three roots

. . E (26) (i) gIven 1D qs. , Xo •

It is interesting to consider the static bifurcation diagram of this two parameter, J.1 and e, one state variable, x(t), diferential equation. We see in

"Figure-IO that for e � 0 the pitchfork bifurcation unfolds. Indeed we have a so­called cusp catastrophe such that when I e I is large enough, hysteresis causes a sudden change from the one branch of fixed points to the other (eg, from the lower to the upper branch as e > 0 is increased beyond e·).

The physical significance of the unfolding of a pitchfork bifurcation has been studied previously in connection with the problem of instability in natural convection flows [11]. That is, for the onset of Benard cells in a square pool of fluid heated from below and inclined from the vertical by an angle e. For example, as can be seen in Figure 11, when the Rayleigh number (Ra) is increased for a horizontal pool (ie, e = 0°), we have a pitchfork bifurcation (at Ra·) giving rise to two roll cells of opposite polarity. It should be noted that the critical Rayleigh number (Ra·) is closely related to the parameter J.1; indeed, J.1 = Ra - Ra·.

As the heating of the lower wall is increased further, we have another pitchfork bifurcation (Ra •• ) which leads to the formation of two symetric roll cell pairs of opposite polarity. In contrast, when the heated cavity is tilted by e = 2° the pitchfork bifurcation unfolds and the roll cell pairs formed at the second bifurcation, Ra··(2°), are non-symmetric.

UNFOLDED PITCHFORK BIFURCATION

(SIDE VIEW, 9 > 0)

9

CUSP CATASTROPHE (TOP VIEW)

SOLUTION SURFACE OF FIXED POINTS

Figure 10. The unfolding of a pitchfork bifurcation.

3.2 Dynamic Bifurcations

UNFOLDED PITCHFORK BIFURCATION

(SIDE VIEW, 9 < 0)

HYSTERESIS CURVE (FRONT VIEW)

333

Let us now turn our attention to the analysis of dynamic, or Hopf, bifurcations [12]. As noted previously, at least two state variables are required for a Hopf bifurcation. A simple example of a system having a Hopf bifurcation is given by:

(30a)

(30b)

The fixed points of this system of equations for real J.L are given by, (xl ,x2 ) = o 0 (0,0). As usual the stability of these fixed points can be determined by linearizing the system of equations about the fixed points:

334

FLUID VELOCITY

FLUID VELOCITY

� � �

Ra -

Figure 11. Effect of Rayleigh Number on the Bifurcation for 9=0° and 9=2°.

(31a)

(31b)

or, equivalently, in matrix notation:

If we assume a modal solution of the form,

we find that the only non-trivial solution is for:

det[(J.1 -i.) -

1 ] _ 0 1 (J.1 - i.)

Hence,

thus the eigenvalues are:

Thus,

335

(32)

(33)

(34)

(35)

(36)

Equations (36) and (33) imply that we will have an oscillatory solution (with an angular frequency of (J) = 1.0 radls) which is damped (ie, stable) for J.1 < 0 (a stable focus), and unstable for J.1 > 0 (an unstable focus).

It is interesting to note that (x1 ,x2) = (0,0) is a solution to Eqs. (30), and the only one valid for small perturbations. However, Eqs. (30) also have another finite amplitude solution given by:

(37)

336

Equation (37) is a circular orbit in XI-X2 phase space. As shown schematically in Figures-12 for J.1 > 0, and the plus (+) sign, in Eqs. (30) and (37), we may have a supercritical bifurcation, while for J.1 < 0, and the minus (-) sign, we may have a subcritical bifurcation. As can be seen in Figure 12a, for a supercritical bifurcation, all phase plane (ie, xl-x2 plane) trajectories (ie, solution flows) converge to a stable limit cycle of finite amplitude for conditions on the unstable side of the linear stability boundary (ie, J.1 = 0). Moreover, as given by Eq. (37), the amplitude of this limit cycle is �. In contrast, Figure 12b

STABLE FIXED POINTS, fJ. < 0

STABLE LIMIT CYCLE, fJ. > O

- � fJ.

Figure 12a. Supercritical Hopf Bifurcation

STABLE FIXED POINTS, fJ. < 0

X 2

UNSTABLE LIMIT CYCLE, fJ. < O

UNSTABLE FIXED POINTS, fJ. > 0

Figure 12b. Subcritical Hopf Bifurcation

337

shows that for a subcritical bifurcation, an unstable limit cycle (of amplitude

{1l) exists, such that the phase plane trajectories either converge to the negative I.l axis, (0,0), for small perturbations, or diverge exponentially for large enough perturbations. Thus, in this problem there are two so-called basins of attraction which are separated by the unstable limit cycle.

The occurence of both subcritical and supercritical bifurcations have been predicted in boiling chanels [13] but only supercritical bifurcations (ie, limit cycles) have been measured to date. It is significant to note that subcritica1 bifurcations are potentially quite dangerous since they imply that divergent instability can occur in the region of linear stability if large enough amplitude perturbations occur in a boiling channel. This may result in a critical heat flux (CHF), and physical damage to the heated surface.

The Hopf theorem is an existence theorem which states in essence that a dynamic (ie, Hop£) bifurcation occurs when one has a complex conjugate pair of eigenvalues ) which crosses the imaginary axis with,

Im(A) � 0 (38a)

dl.l � O (38b)

We note that the eigenvalues given in Eq. (36) satisfy the Hopf criteria. The actual application of the Hopf analysis normally involves the use of

higher order perturbation theory and a Floquet analysis of the stability of the resultant limit cycle. The application of this analytical method to a boiling channel has been discussed in detail by Achard et.al. [13] and Rizwan-Uddin et.al. [14], and thus wil not be repeated here.

8.3 Self-Similarty and Mixed Bifurcations Many bifurcation diagrams exhibit self-similarity. In particular, let us

define:

(39a)

(39b)

As shown schematically in Figure-13, I.ln is the value of the parameter I.l at the onset of the nth bifurcation and An is the maximum amplitude of the oscillation at the end of the nth bifurcation (ie, at the onset of the (n+l)th bifurcation). A cascade of period doubling bifurcations in the phase plane orbits is normaly a precursor to the onset of chaos. Indeed, period doubling is often used as an indication of the approach to chaos.

It has been observed [15] that for many nonlinear systems,

338

Figure 13. A typical Hopf bifurcation followed by a cascade of period doubling bifurcations

lim an = 4.669202 . . . n�oo lim an = 2502907 n�oo

(40a)

(40b)

These results are called the Feigenbaum numbers (or Feigenvalues), and provide a means of assessing numerical results and of estimating where the next, that is, (n+l)th, bifurcation will occur.

Not all bifurcations are simple static and dynamic bifurcations of the type previously discussed. Figure-14a shows an example of a mixed sub critical bifurcation which, at large enough amplitudes, exhibits a stable limit cycle response. Such a system has the features of both a subcritical and a supercritical bifurcation. Moreover, significant hysteresis may occur as shown in Figure-14b.

Figure-15 shows a mixed supercritical bifurcation, however, in this case if the initial excitation is large enough (ie, larger than the amplitude of the unstable limit cycle), or Il > Il·, the solution will diverge with time. Thus, this system of equations has the features of both a supercritical and a subcritical bifurcation.

It is also possible to have both static and dynamic bifurcations in a particular system of equations. Figure-16 shows a pitchfork bifurcation (0 < Il < Il·) which subsequently experiences a supercritical Hopf bifurcation at Il =

STABLE AXED POINTS

Figure 14a. Mixed Subcritical Hopf Bifurcation �'�I I , I I

.. ,.1" 0 J.1

Figure 14b. Hysteresis in Mixed Subcritical Hopf Bifurcation

339

J.1*. Physically, this may correspond to the case of a natural circulation loop which begins to circulate at J.1 = 0 (a pitchfork bifurcation) and as J.1 is increased the loop becomes unstable due to a Hopf bifurcation. Naturally, more complicated situations are also possible and depend on the nonlinearities in the mathematical system of equations q,eing investigated.

Now that we have reviewed the elements of the mathematics associated with bifurcation theory, let us consider the application of these concepts to the study of chaos in engineering systems.

4. CHAOS THEORY

Let us begin with a review of a simple linear oscillator. In particular, a one-dimensional damped spring/mass system which is being excited with a forcing function, F(t). Such a system can be written as:

(41a)

Dividing through by the mass (m), we obtain:

(41b)

340

STABLE FIXED POINTS

I I I I Jl *

- - -t- \ - - - �-II UNSTABLE FIXED

POINTS I I

",/ ,,� UNSTABLE LIMIT

CYCLE

Figure 15. Mixed Supercritical Hopf Bifurcation.

· dx where, x = dt ' P = B/m, k = Kim, and, ftt) = F(t)/m.

Equation (4 1b) can be written as a system of first order coupled differential equations as:

x = y (42a)

(42b)

In matrix form, Eqs. (42) are:

. [0 � = -k (43a)

34 1

Figure 16. Combined Pitchfork and Supercritical Hopf Bifurcation.

where,

� = (x y)T (43b)

Now an autonomous (ie, unforced) oscillator has f=O, and is thus given by:

(44)

Let us assume a modal solution of the form,

(45)

Combining Eqs. (44) and (45) we obtain:

B eAt [M -p.] = 0 (46)

where J is a matrix having elements given by the Kronecker delta function, aij. The only non-trivial solution to Eq. (46) is given by:

det -JA.] = 0 (47)

That is,

[-A. 1 ]

det -k -(j} + A.) = 0 (48)

342

Obviously A. are the eigenvalues of the system matrix, M. Expanding out Eq. (48) we obtain the so-called characteristic equation:

-

Thus,

A. = (-� ±�)/2

We see that A. will be complex if, �2 - 4k < O. Recalling that,

Re [e(O'+ico)t] = eat Cos(cot)

(49)

(50)

(51)

we see in Figures-17 that there are four possible solutions. Namely, a stable elliptic attractor, an unstable elliptic repellor (Fig. 17a), and a stable point attractor, and an unstable saddle (ie, turning) point (Fig. 17b). It should be noted that these figures present the locus of the time varying trajectories in the

so-called phase plane. That is, a plane defined by the state variables, y(t) = i(t) and x(t).

Moreover, for the two stable cases, the origin (y = x = 0) is the stable fixed point of the solution.

The discussion given above for a linear damped spring-mass system may seem overly simple. However, it presents the ideas necessary to understand more complicated situations. Indeed, the stability analysis of a nonlinear system of differential equations is just a generalization of what has been discussed. This generalization depends on the Center Manifold Theorem, which states in essence that the linear stability analysis of an N-dimensional nonlinear system can be reduced to the study of an equivalent linear one­dimensional problem on the so-called center manifold [16].

To understand how the stability of a system of equations may be analyzed, let us consider the following N-dimensional non-linear system:

(52)

where the underbar denotes a matrix vector and Il is a parameter of the system.

As before, the stability of the fixed points, x.o(Il), of this system can be investigated by linearizing the system. Thus, using a Taylor series expansion:

Note that the fixed points <xc) are defined by,

(54)

Thus neglecting higher order terms in the Taylor series expansion, we can rewrite Eq. (53) as:

•

E l l i pt ic Attracto r ({3 > 0)

x x

E l l i pt ic Repel l o r ({3 < 0)

Figure 17a. Phase Plane Trajectories for Spring/Mass System (�2 - 4k < 0)

P o i nt Attract o r (fJ > 0)

Sad d l e P o i n t (fJ < 0)

Figure 17b. Phase Plane Trajectories for Spring/Mass System (�2 - 4k > 0)

where, the so-called Jacobian (go) matrix of the system is given by:

343

(55)

344

�F J (Jl) = --0 ax ( ) - �o Jl (56)

We note that since Xo(Jl) is a constant we can rewrite Eq. (55) as,

(57)

where, the perturbations are defined as:

(58)

As for the one-dimensional case, let us assume a modal solution of the form,

Equations (57) and (59) yield,

�At[A! -:0 (Jl)] = 0

The only non-trivial solution is when,

-J (Jl)] = 0 - -0 Obviously the A are the eigenvalues of the Jacobian, go(Jl).

(59)

(60)

(61)

Thus, from Eq. (59) we note that, as for one-dimensional systems, the N-dimensional system, is linearly stable if all the eigenvalues, lj, of go(Jl) have negative real parts, ReO"i) < O. In contrast, the system is linearly unstable when any eigenvalue has a positive real part and is said to be marginally stable when the real part of any eigenvalue is zero. For the latter case, the fixed point is normally referred to as a singular point.

It is also interesting to note that the Laplace transform of Eq. (57) yields,

(62)

or,

(63)

345

Thus, recalling the definition of the inverse of a matrix and for a transfer function, we find that the characteristic equation is given by,

(1)] = 0 - -0 (64)

Comparing Eqs. (61) and (64) we see that the roots, s, of the characteristic equation are just the eigenvalues, A, of the Jacobian, go(J.l). Hence, we confirm that the linear stability of the system of differential equations is determined by the sign of the real part of the most limiting root(s). Hence, analysis of the stability of the Nth order system gives the same result as if a one-dimensional stability analysis was performed on the center manifold for the most limiting eigenvalue(s).

Let us now extend some of the ideas that have been discussed for a linear oscillator to a non-linear oscillator. Historically, one of the most important nonlinear oscillators is the Van der Pol oscillator. This autonomous oscillator can be written in the form,

(65)

Comparing Eqs. (65) and (41b), we find that the most significant difference is that for the Van der Pol oscillator we have a nonlinear damping term, � = J.l(1-x2), which changes sign with the value of the state variable, x(t).

The response of the Van der Pol oscillator depends on the sign of the parameter J.l. As shown in Figure-19a for J.l < 0, one has a supercritical Hopf bifurcation. That is for this negative damping case, � < 0, when I x(t) I < 1, the system behaves as an elliptic repellor. In contrast for, � > 0, we have positive damping when I x(t) I > 1 , and the system behaves as an elliptic attractor. In general, all phase plane trajectories converge to a stable limit cycle for the case shown in Figure-1Sa.

In Figure- 1Sb we see the case in which J.l > o. This case is called a subcritical bifurcation, and is characterized by the response of an elliptic attractor for I x(t) I < 1, and an elliptic repellor for I x(t) I > 1. The limit cycle shown is clearly unstable and will not persist. The mechanical analogy for this case is a ball oscillating in a bowl. For small amplitude perturbations, the ball will oscillate and come to rest at the center of the bowl. For large enough perturbations, the ball will jump over the side of the bowl and will fall out. If the perturbations is such that the ball is perched on the rim of the bowl (the unstable limit cycle) it will not remain there since any minor disturbance will cause it either to fall out of, or into, the bowl. As noted previously, subcritical bifurcations are potentially very dangerous occurrences since the system is linearly stable and yet finite amplitude perturbations may cause it to become exponentially divergent.

Let us now turn our attention to the analysis of chaotic, or "strange", attractors . In order to understand chaos, let us again consider an autonomous second order oscillator of the form:

346

. X

x

FIGURE - 1 8a Su percrit ical Bifu rcation (/1 < 0)

FIGURE - 1 8b Bifurcat ion (/1 > 0)

Figure 18. Van der Pol Oscillator, x + � (1 - x2)i + kx = 0

x-ax+ x = 0 (66) This equation can also be rewritten in system form as:

x = -y (67a)

y = x + ay (67b)

If the damping parameter is positive (ie, a > 0), we find that Eqs. (66) and (67) will have negative damping and are thus unstable. Indeed, we have an elliptic repellor in the phase plane, (y,x).

347

In order to limit the amplitude of x(t), we can modify the system in Eqs. (67) by introducing a new state variable (z). The resultant system is:

x = -y - z (68a)

y = x+ ay (68b)

z = b+ z(x - c) (68c)

We see that if a, b and c are all positive when x(t) becomes greater than the

parameter c, then z > 0 and thus z(t) will increase, causing x to decrease. This can produce a limit cycle or even a strange attractor.

Equations (68) are a form of the equations which yield the R6ssler band attractor [17]. It is well known that the occurrence of strange attractors require at least three state variables, thus the R6ssler band attractor is a simple example of a far more general result.

It is instructive to analyze Eqs. (68) using a methodology which will apply to the most general case. The first thing that one must do is to find the fixed points (ie, the steady-state solution) of the system. Thus, if we set the time derivatives in Eqs. (68) to zero we obtain:

Yo = - Zo (69a)

(69b)

z01,2 = (c ± c2 - 4ab )l2a (69c)

where subscript-o denotes the steady-state and subscripts 1,2 denote the positive and negative root, respectively, of Eq. (69c). Obviously, we have two fixed points in the three-dimensional (x,y,z) phase space.

The next step is to linearize Eqs. (68):

Sx = -Sy - Sz (70a)

Sy = Sx + aSy (70b)

(70c)

These equations can be written in matrix form as:

348

(71)

where,

� = (ox oy oz)T (72a)

and,

(72b)

As before, if we assume a modal solution of the form,

(73)

Equations (71) and (73) yields a non-trivial solution for:

det[�"A

� - "A) �1

] = 0 z01,2 0 (Xo - c - "A)

(74a)

or,

(74b)

There are three roots ("A) to this equation for zOl' and three for z02 ' These roots are of the form:

"A�� = 0(1) ± ioo(1)

"A�) = - e(1) and,

(75a)

(75b)

(76a)

349

(76b)

We see in Figure-19 that the roots in Eqs. (75) yield the elliptic repellor and point attractor shown on the left of the figure, and the roots in Eqs. (76) yield the elliptic attractor and point repellor shown on the right of the figure. As we shall see shortly, the interaction between these two fixed points produce a phase space vortex which causes a folding of the orbits in the phase plane, and a resultant inability to predict the future response of the deterministic system. This property will lead to what is known as BelUlitivity to initial conditiolUl (SIC). That is, smaU differences in initial conditions will be exponentially magnified as the process continues.

Let us now consider the numerical evaluation of the nonlinear system in Eqs. (68). If we fix the value of two of the parameters to be, b = 2.0 and c = 4.0, and then vary the value of parameter-a, we obtain the phase plane response shown in Figure-20. It is significant to note the cascade of even bifurcations (ie, period doubling) as the parameter-a is increased from a = 0.3 to a = 0.3909. When a = 0.398 we have a chaotic response in which the stretched and folded orbits never repeat. This strange attractor is caled the ROssler band attractor [17]. It is also interesting to note that as the parameter-a is further increased that we have a reverse cascade of odd bifurcations.

Many strange attractors have been found in physical systems. Probably the most famous is one that was found first; the butterfly-shaped strange attractor of Lorenz [18]. This attractor resulted from a study by Lorenz of weather prediction using a simplified three state variable model of natural convection:

Figure 19. Fixed Points of ROssler's Band Attractor

350

Figure 20. Rossler's Band Attractor (b = 2.0, c = 4.0) [17]

� = o(y - x} (77a)

y = px - y - xz (77b)

(77c)

where, 0, p and � are parameters. The three fixed points of this system of equations are shown in Figure-21,

and the strange attractor in Figure-22. It can be seen that the strange attractor is basically comprised of two foci rotating in the opposite direction. Physically this 3-D phase space motion implies that the direction of motion of the Bernard roll cells is changing chaotically in time.

3 5 1

One of the most important challenges for the analyst is to determine when a chaotic response (ie, strange attractor) exists and when the response is something else, such as, random noise or the superposition of sinusoids. There are various tests for chaos which can, and should, be used.

When one is working with data, then the various fractal dimensions (DC and DH) should be computed to determine whether there is some underlying structure in the data. In addition, the power spectral density (PSD) function should be computed. The spectrum wil be broad band for chaotic data as well as noise. In contrast, if the signal is comprised of superimposed sinusoids, it will have spikes at the frequency of each sinusoid.

It is also often useful to plot the data in a 3-D phase plane. [ie: x(t). x(t+t), X(t+2't)]. to appraise the attractor. Other. more quantitative techniques involve the use of a Poincare section. This important technique will be described shortly.

When one is numerically evaluating an analytical model, the approach to chaos is often characterized by a cascade of bifurcations. Such bifurcations are relatively easy to detect since they yield period doubling in the phase plane orbits. In addition, a chaotic attractor will always exhibit sensitivity to the initial conditions (SIC). That is, two nearby points in phase space diverge rapidly as the orbits progress. Indeed. the points are known to diverge exponentially. and this divergence is characterized by a Lypunov exponent. A. This exponent may be defined in terms base-e, or base-2 . That is , if we imagine an initial condition filled hypersphere in phase space of diameter do, then as the points in this sphere move through phase space the hypersphere will distort into a hyperellipsoid of major axis, d(t). This distortion can be quantified by the Lypunov exponent:

Figure 21. Fixed Points of Lorenz Attractor [6]

Phase space

352

z Attractor

Figure 22. Lorenz Attractor [6]

(78a)

or, alternatively,

(78b)

In either case the Lypunov exponent (A or A) implies that the orbits become far apart after sufficiently long time. Indeed, even computer roundoff error will lead to nonpredictability due to SIC. Thus, to verify a chaotic response, one should always check the divergence of the orbits to make sure Eqs. (78) are satisfied.

Another excellent way of assessing whether there is a strange attractor or not is to analyze how the phase space orbits pass through a Poincare section. Thus, let us next discuss this valuable technique.

A typical Poincare section in phase space of order three is shown in Figure 23. The penetration of the Poincare plane by the phase space orbits (ie, the solution flow) creates a pattern in the plane which is the signature of the attractor under investigation.

353

y

Figure 23. Poincare Section [6]

Figure-24a shows the Rossler band attractor in 3-D state space. Figure-24b shows a Poincare section after the trajectories have been stretched and folded after completing one orbit in phase space. It can be seen that the original line segment is now a U-shaped segment. Next, as can be seen in Figure-24c, after a second orbit a double-U-shaped pattern emerges. Figure-24d indicates the third orbit produces a quadruple U pattern. Finally, after many orbits we achieve the Poincare map shown in Figure-24e. This map has a fractal pattern which is similar to a Cantor set. The appearance of a fractal pattern in the Poincare section is good confirmation of a strange attractor.

Other interesting information can be obtained from a Poincare section. Indeed, if we keep track of the solution flow and plot where it pierces the Poincare plane on successive orbits we can construct a first-return map, in which we plot state variable Xn' where the penetration of the plane is now, versus Xn+ l ' where the corresponding penetration will be at the end of the next orbit. Figure-25 gives the first return map for a Rossler band attractor. we note the non-uniqueness implied by the parabolic shape of the first-return map. That is, two different Xn can give the same Xn+ l ' This loss of uniqueness implies nonpredictability.

In order to appreciate the importance of a first-return map we note in Figure-25 that one can iterate the first-return map by taking an initial point (say Xn+ 1 = A) and note that on the next iteration this point will become Xn = A, thus Xn+1 = B, and so on. It is more convenient to just draw a 45° line and move on a horizontal line from each point, to the 45° line then vrtically to the first return map. This will give the next iterant (Xn+1)' It can be seen that, as expected, the Rossler band attractor is unstable and aperiodic. Such motion in a first return map is the signature of the strange attractor.

Figure 24a .

. . . . . . . . . . . . . .

Figure 24b. Figure 24c.

355

Poincare Section

.. .. .. .. .. .. .. .. . . . . . . . .. . .. .

. .. . . .

. . '

. ' . . "

.. .. .. .. .. ; � .. .. .. " .. ,-

�{f�� :':: '. :: , � : ':

Figure 24d. Figure 24e.

Figure 24a-e. The Stretching and Folding Process for a Rossler's Band Attractor [27]

It is also possible to interpret the first-return map in the time domain [9]. Figures-26 show how a record of analog data might appear in a first-return map for a period signal of period T which is sampled on interval T/4.

In order to more easily appreciate the iterative procedure in a first-return map, and the stability criterion that controls the motion in this plane, let us consider a simpler problem. In particular, let us consider the so-called Logistic map of population dynamics. The iterative equation for this problem is given by:

(79)

where the parameter J.1 quantifies the birth rate. The fixed point (Xo) of this problem is where the 450 line intersects Eq. (79).

That is,

356

• 'A

. •

F • • + • •

..

X n

. D + .

\ C \

• •

\. G

Figure 25. First-Return Map - ROssler's Band Attractor [17]

thus,

Xo = 1 - 1IJ,1

The condition for stability is known [17] to be given by,

, I f (xo) I < 1 .0

Now, from Eq. (79),

, f (x) = J,1(1-2x)

Combining Eqs. (80) and (82),

(SO)

(81)

(82)

(83)

357

Figure 26a. Data in the Time Domain for Signal of Period T Which is Sampled On Interval T/4 [9]

X n+1 (X3 , X4 ) •

(X 1 , X2 ) x2 •

X3 • (X2 , X3 ) x 1 • (X O , X 1 )

X1 X3 X2 Xo X n

Figure 26b. Data on a First Return Map [9]

358

We note that for � = 2.5, the system is stable, and, as can be seen in Figure-27a, the iteration converges to the fixed point (xo)' In contrast, for � = 3.1 the system is unstable, and results in the "limit cycle" response shown in Figure-27b. Interesting, when � = 3.8 the system becomes chaotic (ie, aperiodic), as shown in Figure-27c. This implies that if the birth rate (�) becomes too large the population dynamics can become completely unpredictable. A frightening prospect at best.

While there is much more than can be said about the analysis of chaos, we conclude with the observation that it is a very exciting and rapidly developing field.

FIGURE 27a

IJ. '

! I ! i I :

1/

, ,: i l , ,! , !

FIGURE - 27b

X.

FIGURE 27c

Figure 27. Behavior of iterated maps: (a) logistic map, showing a transient settling to an attracting equilibrium for � = 2.5, (b) attracting limiting cycle for J.L = 3.1, and (c) aperiodic behavior for J.L = 3.8.

359

Let us now turn our attention to the application of these analytical techniques to problems of interest in multiphase flow and heat transfer. In particular, let us consider the prediction of nonlinear density-wave instability phenomena in single-phase natural circulation loop"s and in boiling chanels.

5. THE ANALYSIS OF CHAOS IN SINGLE·PHASE NATURAL CONVEC1'ION LOPS

A thermo syphon is an excellent example with which to demonstrate some analytical procedures that can be used in the analysis of chaos. Thus, let us consider the analysis of nonlinear density-wave instabilities which may occur in a single-phase natural circulation loop. Such phenomena have been previously considered by Bau et.al. [19]. The discussion below is based on this interesting piece of work.

A quite general thermosyphon loop is shown in Figure-28. It can be seen that the bottom half of the loop is uniformly heated and the upper half is uniformly cooled. For the case of nonsymmetric heating (cp -:I: 0), the heated section would be rotated.

Let us begin by deriving the conservation equations which describe fluid motion during natural circulation conditions in the loop shown in Figure-28. Adopting Boussinesq's approximation, Newton's second law implies:

(84)

where,

and,

1t iit = J Ut (r) 2"r dr

o The corresponding loop momentum equation comes from integrating Eq. (84) around the loop (ie, from 9=0 to 21t) and noting that p(O) = p(21t). The result of this integration is:

2It dli g P � J

4tw Pl dt = - 21t Cos (9 + cp) d9 - %

o

(85)

360

HEATED

Figure 28. Schematic of a Toroidal Thermosyphon [19]

The corresponding energy equation comes from applying the first law of thermodynamics to a one-dimensional differential control volume which spans the loop:

aT _ 1 if k aaf (T l -T w ) , 0 E T w(O) specified P C � + p u c -� -- - H l l at l l I. � ao - � ao2 - �

DH ' 0 E q"(O) specified

(86)

In accordance with the Boussinesq approximation the liquid in the loop is assumed to be incompressible but varies (with temperature) in the density head term of the loop's momentum equation. As a consequence the continuity equation is automatically satisfied.

It is convenient to nondimensionalize Eqs. (85) and (86). To this end we define the loop's time scale as,

the loop's Prandtl number as,

2 PL = 8 PrlNu = 32 'U tlDH and the loop's Rayleigh number as,

(87a)

(87b)

(87c)

361

where, lir l is the instantaneous temperature difference in the liquid from one side of the loop to the other (actually, any two points around the loop may suffice). Similarly, the Biot number of the loop was defined as,

We can now define the following nondimensional quantities:

t* � th

Thus, Eq. (85) becomes:

2� * u �* .. * w d-* 1 f �

dt* == u = i Ra PL T Cos (9 + 4» d9 - (DH/1O o

It is interesting to note that for laminar flow:

hence,

* 4 'tw (DH/1O = PL u*

(87d)

(88)

(89)

Let us next derive the nondimensional form of the loop's energy equation. Recalling that,

362

Equation (86) can be written as,

aT· _. if· alf· _ {-(r* -T'; ) , a E T w(a) specified at· + u aa - B aa2 -

q"*(a) • a E q"(a) specified (90)

Equations (88), (89) and (90) comprise the conservation equations needed for the analysis of a single-phase thermosyphon undergoing laminar natural circulation flow.

In order to use the mathematical machinery which has been developed for the analysis of nonlinear deterministic systems, it is necessary to convert the partial differential equation, given in Eq. (90), into an equivalent ordinary differential equation. Because of the nature of this particular problem it is convenient to use spectral methods. In particular, to decompose the various temperature fields into Fourier series.

Thus we assume:

T: (a,t) = wo(t) + L wn(t) Sin(ne) n=l

-

T* (a,t) = L [Sn(t) Sin(n9) + Cn(t) Cos(na>] n=O

(91)

(92)

• where we have used the fact that the normalized wall temperature (Tw) is an odd function of e.

Introducing Eq. (92) into Eq. (88), and, for simplicity, assuming the validity ofEq. (89), we obtain:

21t

;';. = Ra PL f i [8,(t) Sin(nO) + Cn(t) Cos(nO)] Cos (0 + �) dO · PL u· (93) o n=O

Recalling the following orthogonality relations,

21t

f Sin (nO) Cos (mO) dO = { � ,n=m

o

21t f Sin (nB) Sin (mB) dB = { : o

21t

f Cos (nB) Cos (mB) dB = { : o

Equation (93) yields,

,n=m

,n=m

Next, we introduce Eqs. (91) and (92) into Eq. (90):

363

(94)

i [Sn(t) Sin (nB) + Cn(t) Cos (nB)] + ii' i [n Sn(t) Cos (nB) - n Cn Sin (nB)] n=O n=O

+ B L [n2 Sn(t) Sin (n9) + n2Cn(t) Cos (n9)] n=O

q"(9) ,9 E q"specified

00 00

wo(t) + I. wn(t}Sin(n9) - I.[Sn(t}Sin(n9) +Cn(t)Cos(n9)] ,9 E Twspecified n=l n=O

We now multiply Eq. (95) through by Cos(9) and integrate to obtain:

21t I q"(9)Cos(9)d9 ,9 E q"specified

• _* 0 C1+ BC1 + u Sl =

-C1 ,9 E Twspecified

Similarly, we multiply Eq. (95) through by Sin(9) and integrate to obtain:

(95)

(96)

364

21t I q"(9)Sin(9)d9 ,9 E q"specified

• _* 0 8l+ B8l + u Cl =

,9 E Twspecified (97)

For simplicity let us now consider the special case where the wal temperature (Tw) is specified for all 9. For this case, Eqs. (94), (96) and (97) reduce to:

�. = Ra PL [Cl Cose!> - 81 Sine!>] - PL u· (98)

Cl + (1+B)Cl +� 81 = 0

81 + (1+B)Sl -� Cl = WI

It is convenient to make a change of variables as follows:

.1 .. Ra = Ra ( 1+B2)

.1 A

8 = Ra 8l/(1+B)

u � U*/(1+B

t � t(l+B)

Making these transformations, Eqs. (98)-(100) become:

� = [C Cose!> - 8 8ine!>] P - Pu

C + C + u8 = 0

For the special case of a constant wall temperature, we may let:

(99)

(100)

(lOla)

(101b)

(lOlc)

(lOld)

(lOle)

(IOU)

(102)

(103)

(104)

(105)

365

thus, Eq. (104) becomes, • S + S - uC + RaA = 0 (106)

In order to reduce Eqs. ( 102), ( 103) and (106) to a more recognizable form we define,

II x = u

where, R � ARa.

Hence Eqs. ( 102), (103) and (106) can be rewritten as,

; = (y Cos(�) - [z - R] Sin (�») P - Px

y = - x[z - R] - y . z =o - z + xy

(107a)

(l07b)

(107c)

We note that for the special case of symmetric heating (ie, � = 0) that Eq. (107a) reduces to:

; = P[y - x] (107d)

The system of nonlinear ordinary differential equations given by Eqs. ( 107b), (107c) and (107d) is a special case (ie, (J = P, P = R, 13 = 1 .0) of the well-known Lorenz System [18] given in Eqs. (77).

Let us now investigate the system of equations given by Eqs. (107a), (107b) and (107c). First of al the fixed points of the system are given by setting the

time derivatives to zero (ie, ; = Y = ; =0). This yields,

Xo = Yo Cos(�) - [zo - R] Sin (�)

Combining Eqs. (108) we obtain,

x! + [1 - R Cos (�)]xo - R Sin (�) = 0

(10Ba)

(l08b)

(lOBe)

(109)

366

It is important to note that for the nonsymmetric heating case there are no non-motion (ie, Xo == Uo = 0) solutions. Indeed, the formula for a cubic equation implies that there is only one (motion) solution for Xo == Uo when R S; Rl' where Rl comes from the solution of:

3 27 · 2 [1 - Rl Cos (�)] + 4" Rl Sin2(�) = 0 (110)

In contrast, when R > Rl there are three (motion) solutions for xo' however, only two of these solutions are stable. A bifurcation diagram for the case in which � > 0 is shown in Figure-29. This type bifurcation represents an unfolded pitchfork bifurcation, in which the upper branch is for counterclockwise (CCW) loop flow and the lower branch is for clockwise (CW) loop flow. Clockwise motion implies downflow against the effect of buoyancy (due to nonsymmetric heating). Such steady flows do not occur unless special provisions are made to establish them. Moreover, above a certain generalized Rayleigh number (R2) the loop flow becomes unstable and reverses direction in going from the lower (CW) branch to the upper (CCW) branch. Such a transition is known as a "catastrophe".

It should also be noted in Figure-29 that the upper (CCW) branch will also become unstable as the generalized Rayleigh number is increased to Ra . Moreover, due to the hysteresis inherent in a subcritical Hopf bifurcation, a chaotic response may occur in the region R2 < R < Ra.

Let us next consider the case of symmetric heating (�=o). For this important case, Eqs. (l08) imply that the fixed points are given by,

Xo = Yo (111a)

(ll1b)

(1nc)

Thus we find that for symmetric heating we may have either nonmotion solutions (ie, Xo = Yo = Zo = 0) or motion solutions given by,

Xo = Yo (l12a)

(l12b)

Equation (112a) yields the pitchfork bifurcation diagram shown in Figure-aO. It can be seen that the upper branch implies counterclockwise (CCW) loop flow, while the lower branch is for clockwise (CW) loop flow. Naturally for the case of symmetric heating either flow is possible. Also, it should be noted that for both flow directions a subcritical bifurcation and a chaotic response may

x = u (t) ( <I> > 0)

CW Motion

", .. , \ , ' I ' I I - - - - -

\ , .. "" -

367

R

Figure 29. A Bifurcation Diagram for an Asymmetrically Heated Loop (Dashed Lines Represent Unstable Conditions).

x = u (t) - - - - -

Motion

R = 1 .0 R

- -- - -

Figure 30. A Bifurcation Diagram for a Symmetrically Heated Loop (Dashed Lines Represent Unstable Solutions).

368

occur for sufficiently high generalized Rayleigh number (ie, as Rb < R S Rc = Moreover for R > Rc a chaotic response with so-called "windows of

periodicity" is found. A typical time trace for R > Rc is shown in Figure-31a. It can be seen that deterministic chaos is quite complicated, involving occasional reversals in the direction of the flow. Also, Figure-31b shows that there are no distinctive frequencies in the corresponding Power Spectral Density function. This implies that the time signal shown in Figure-31a is aperiodic.

In order to gain more insight into the properties of Eqs. (107d), ( 107b) and ( 107c), let us perform a linear stability analysis. If we linearize these equations we obtain:

ai = p[ay - ax]

ay = - [zo - R]ax - xoaz ay

Equations ( 1 13) can be written in matrix form as,

a! =£ ay where, the state variable vector is given by,

and the so-called Jacobian matrix is,

[ -p p

£ = (R-zo) -1

Yo Xo

Let us now assume a modal solution of the form,

(113a)

(113b)

(113c)

(114)

(115)

(116)

(117)

Combining Eqs. ( 1 17) and ( 1 14) we find that the only nontrivial solution is when,

p

-(1+1.) (118)

369

u (t)

0 ---

o t

Figure 31a. The Velocity in the Thermosyphon Loop as a Function of Time (for R � He) [19]

PSD

{(Hz)

Figure 3lb. The Power Spectral Density (PSD) Function of the Time Series Shown in Fig. 31a [19].

370

For the nonmotion solutions (ie, Xo = Yo = Zo = 0), Eq. (118) implies,

A -(1+P) ± + 4PR 1,2 2 (119a)

(119b)

For convenience, we refer to this set of roots as those of fixed point Co ' Comparing Eqs. (117) and (119) we see that the system is stable (ie, Aj < 0) for R < 1.0. For R = 1.0 we have neutral stability (ie, 1.1 = 0.0), while for R > 1.0 the system is unstable (ie, 1.1 > 0.0).

Using the loop motion solutions given by Eqs. (112), Eq. (118) implies:

",3 + (2+P) ",2 + [(1+2P) + x� - P(R-zo)] 1. + P[1 + x� - (R-zo) + XoYo] = 0 (120) This cubic equation gives two sets of three eigenvalues depending on which value (ie, ±) is chosen for the Xo = Yo in Eq. (112a). Because these roots are algebraically complicated we shall not write out the set of roots for the fixed points C- and C+

Figure-32 shows the fixed points, Co' C- and C+ for the Lorenz attractor for R > 1.0. It can be seen that Co behaves as a point attractor (stable manifold)

and a point repellor (unstable manifold). In contrast, C- and C+ behave as

x

Unstable �:�Old

Stable c+ manifold

of C'"

Unstable manifold

Stable manifold of Ca

Figure 32. The fixed points of the Lorenz attractor for R > 1 in phase space. Also shown schematically are the eigenvectors associated with the linearized stability problem and the associated stable and unstable manifolds [19].

3 7 1

elliptic repellors (unstable manifold) and point attractors (stable manifold). A number of phase space trajectories are possible. One which starts at the unstable manifold of a given fixed point and returns to the stable manifold of the same fixed point is called a homoclinic orbit. In contrast, orbits which connect different fixed points are called heteroclinic orbits.

Figures-33 show the time response of the Lorenz system for Ra < R < Rc (see Figure-30). It can be seen that since the system can no longer be at rest (ie, with higher density fluid over lower density fluid) an oscillatory response from the unstable manifold of Co occurs. This solution may converge to steady flow

in the CW direction (C+ stable manifold) or in the CCW direction (C· unstable manifold). In contrast, at R = Rc a homoclinic explosion occurs. The resultant strange attractor is shown in Figures-34, where the famous butterfly-shaped Lorenz (strange) attractor can be noted.

It should be noted that hysteresis occurs when going into and out of chaos. For example, when the generalized Rayleigh number (R) is increased we have the onset of chaos when R = Rc (see Figure-3D). In contrast, when the generalized Rayleigh number is reduced chaos will persist until we have reduced it to Rb [5] .

Figure-35 shows a Poincare section of the Lorenz attractor. It can be seen to have a very distinctive fractal signature (and as an aside, has a correlation dimension of Dc = 2.06). If we zoom in on one of the "line segments" shown in Figure 35 we would find that it consists of a vast number of closely packed sheets, each of which also consist of a large number of sheets. Moreover, the well-known sensitivity to initial conditions (SIC) of the Lorenz attractor (actually for any strange attractor) is clearly shown in Figure-36.

Bau et.al . [ 19] have also shown that for a forced time-periodic wall temperature case rather than the time-independent wall temperature case just discussed, that chaos does not suddenly occur after a homoclinic explosion but rather at the end of a cascade of Hopf bifurcations (ie, period doubling bifurcation). As discussed in this chapter, this property is also characteristic of the onset of chaos in autonomous boiling natural circulation loops. A typical bifurcation diagram for a single-phase thermo syphon which is being forced by a periodic wall temperature (with period T) is given in Figure-37. It can be seen that there are bands of deterministic chaos for several different ranges of the generalized Rayleigh number (R). Interestingly the onset of chaos for the forced case is at a slightly larger generalized Rayleigh number than for the unforced case. That is, for heating from below, forcing the wall temperature has a stabilizing effect.

6. APPLICATIONS OF CHAOS THEORY - THE ANALYSIS OF NONLINEAR DENSITY-WAVE INSTABILITIES IN BOILING CHAN

The phenomena of density-wave instabilities in boiling channels is well­known [20] . These oscillations may be found for certain operating conditions of boiling systems which become unstable due to lags in the phasing of pressure-

372

10.0

5.0

x(t) E u(t)

0.0

-5.0

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Figure 33a. Time series exhibiting the approach to the steady-state solution (C+) for R = 7 and initial conditions on the RHS unstable manifold of Co [19].

10.0

5.0

x(t) iR u(t)

0.0

-5.0

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Figure 33b. Time series exhibiting the approach to the steady-state solution (C-) for R = 8 with initial conditions similar to those in Fig. 33a. Note that the change in the Rayleigh number (R) causes trajectories with similar initial data to end up at diferent fixed points.

373

Figure 34a. Lorenz attractor in phase space for R = 20 and P = 4. Sufficient time has been allowed for the initial transient to die out [19].

15.0

10.0

5.0

y 0.0

-5.0

- 10.0

-10.0 -8.0 6.0 4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0

x

Figure 34b. Pr.ojection of the Lorenz Attractor in x-y Phase Plane [19]

374

35.0

30.0

25.0

20.0

z 15.0

10.0

5.0

0.0

-10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0

x

Figure 34c. Projection of the Lorenz Attractor in the x-z Phase Plane [19] 20.0

16.0

12.0

8.0

4.0

Y 0.0

-4.0

-8.0

- 12.0

- 16.0

- 10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8 . .0 10.0

X Figure 35. Poincare Section Through the Plane z = R-l [19].

375

1 0 .

x(t) = u(t)

- 1 0 .

Figure 36. Sensitivity to initial conditions (SIC). Two solutions with slightly different initial conditions (black & gray lines) exhibit vastly different behavior after sufcient time [19].

8

6

2

x 0 h:> �:: :': .� I,

-2 -4 -6 -8

0 2 6 7 8 9 10

R

Figure 37. The upper half of a bifurcation diagram for a single-phase thermo syphon with a modulated wall temperature . The x values were stroboscopically sampled every period T [19].

376

drop feedback mechanisms. The most common manifestation of density-wave instabilities are self-excited oscillations of the flow variables.

The analytical tool which is often used to study the problem of density-wave instabilities is linear frequency-domain stability analysis. Presently rather accurate and reliable models are available for the linear stability analysis of complicated systems such as boiling water nuclear reactors (BWRs).

The study of the non-linear behavior of density-wave instabilities has attracted considerable interest recently. In particular, Hopf bifurcation techniques have been used to study the amplitude and frequency of the oscillations [2 1 ,22] . A numerical analysis of the nonlinear dynamics of a steam generator has been performed by LeCoq [23]. Similarly, a numerical analyses was also performed by Rizwan-Uddin & Doming [22], where a chaotic attractor was indicated for periodically forced flows.

Let us now consider a non-linear analysis of autonomous density-wave instabilities using a lumped parameter model. The model is based on a Galerkin nodal approximation of the conservation equations for a boiling channel.

We start the boiling channel model description by considering the following assumptions made concerning the flow:

• the flow is homogeneous (ie, no phasic slip) • the system pressure is constant • the heat flux is uniform • both phases are incompressible • the two phases are in thermodynamic equilibrium • viscous dissipation, kinetic energy, potential energy and flow work are

neglected in the energy equation • the channel inlet temperature is constant

For these assumptions, the one-dimensional conservation equations can be written as [24]:

o 0 q"PH at (ph) + (phu) = A x-s

(121b)

a 0 [ f l� � � (pu) + oz (pu2) = - DH + 2 - pg - oz

and the corresponding equations of state are:

p = Pf , for h � hr

(121a)

(121c)

(122a)

377

, for h > hf (l22b)

The single-phase region of the heated chanel extends from the channel inlet to the boiling boundary (ie, the location where bulk boiling begins). As can be seen in Figure-3B, this region is subdivided into N s nodes, having variable length. The partition of the single-phase volume was found necessary to properly describe the propagation of enthalpy waves. These waves are of importance in determining the dynamics of the boiling boundary.

The enthalpy increase from the inlet, hi, to saturation, hf, is divided into N. equal intervals, (hf-hi)/Ns. Therefore, the boundary, Ln, between subcooled node-n and node-Cn+l) is defined as the point where the fluid enthalpy is:

(123)

It should be noted that this enthalpy (hn) is a constant, while its spatial location is a function of time.

P ..... ,. - - - - -- -

LA/N. L� w I/)

PR ' - - - - - - -

P • . h. - - - ---L · L ...

Pf ·h , -- --- 0 qN N. - 1 � h. + ""b. h .. - - - - - - 0 w en

L ... = � C w - - - - -- � -< L .... , w

h. + b.h'.b N.

h.

u 1

Figure 3B. Schematic Diagram of the Boiling Channel.

378

The differential equations governing the evolution of the node boundaries. Ln. can be derived using a Galerkin technique by assuming a linear shape function (ie: enthalpy profile) inside each node. That is.

(124)

Integrating the energy equation. Eq. ( 121b), between Ln-1 and Ln, we have, using Liebnitz's rule:

(125)

It is convenient to choose the channel mass (Mch) as a state variable. Its corresponding conservation equation can be derived by integrating the continuity equation over the heated channel's length, which gives:

(126)

The two-phase mixture's exit velocity, \le, can be calculated by first combining Eqs. ( 121a), (121b) and (122b), which yields:

dU q" PH .!f.g � 0 dZ - Ax-s hfg - (127)

Then, integrating Eq. (127) between the boiling boundary (A = LN s) and some

location z in the two-phase part of the heater gives:

u = ui + O(z - A)

In particular, at the exit of the heated channel:

ue = ui + O(L-z)

(128)

(129)

The exit density, Pe, can be expressed in terms of the total heated channel's mass, Mch. by also assuming a linear enthalpy profile inside the two-phase region. Combining Eqs. (121a) and ( 122b), and integrating between the boiling boundary and the channel exit, yields:

M2, = Ax-s(L-A) Pf pf7'Pe - 1 (130)

379

and the total mass of the heated channel, Mch, is given by:

Mch = PfAx-s A. + M2q, (131)

An adiabatic riser was also included in the model. It was found that for low flow conditions the presence of the riser can effect the dynamic characteristics of the system.

The riser was divided into NR fixed axial nodes of equal length, as can be seen in Fig. 38. Integrating the continuity equation over node-r, gives:

(132)

The riser's node-r mass, Mr, can be expressed in terms of Pr- l and Pr using a procedure analogous to that used in the derivation of Eq. (130). This yields:

_ AR Mr - NR 1. Pr

- Pr-l

(133)

At this point we have Ns + NR + 1 equations (ie, Ns Eqs. ( 125), NR Eqs. ( 132) and one Eq. ( 126» , and Ns + NR + 2 unknowns (ie, Ln, Mch' Mr and �p). The model is closed by imposing the external pressure drop (�Pext) boundary condition on the boiling channel and riser. The momentum equation, Eq. (121c), was integrated using the assumption of linear enthalpy profiles inside the various nodes. The result is:

�Pext = �PI + �Pg + �Pf + �PR + �Pa

where:

L f a d L

�PI = -(pu)dz = - Jpudz at dt 0 o

thus,

Similarly,

(134)

(135)

3&0

L

&Pg = fpgdZ = g � Pf (� En - hfLN AX-S) Ax-s cpfAx-s n=l s

o

where,

Ln En = Ax-s Ihdz = Ax-s{Ln -Ln-1){hn + hn-1)/2

Ln 1 The irreversible hydraulic losses are given by:

L �Pf = n

dz

o

thus, considering only inlet and exit losses,

Next the spatial acceleration term is given by:

L J

opu2 2 2 &Pa = --dz = Peu - Pfu.

oz e 1 o

Finally, the riser pressure drop is given by:

(136)

(137)

(138)

(139)

(140)

where, MR = Nf Mr ' r=l

38 1

We now have derived the model. Let us now consider the results of its evaluation.

The boiling channel and associated riser are described using Ns + NR + 2 independent differential equations. The system of equations was numerically integrated by means of the IMSL library subroutine, DGEAR.

For certain operating conditions of boiling channels, it was found that the system evolves to limit cycles near the linear stability boundary. Indeed, this nodal model allows the simulation of self-sustained oscillations in excellent agreement with a more detailed distributed parameter HEM model [14] .

A particularly interesting behavior was found for low Froude (Fr) numbers. Physically, this means operating the boiling channel at low inlet flows. A number of runs were made for the parameters given in Table-I . In these runs only the phase change number (N pch) was varied.

Figure-39 shows a projection of the limit cycle in the aut -ap� plane, R

where, P�R = PNR/Pf' and, Uj+ = is the normalized inlet velocity. For

this condition the fluid velocity tends to drop as the boiling boundary approaches the end of the heated length, due to an increase of the density head in the channel and thus a decrease in the net driving head.

As can be seen in Figure-40 , by reduci ng the channel power a period doubling bifurcation occurs. On further reduction of the channel power a cascade of bifurcations takes place which leads to a chaotic response. This

1 . 0

aui 0. 00

-0. 10 0. 00 O. 10

Figure 39. Limit Cycle Projection in the aUj+

- ap�R plane

382

Table 1 Parameters Used in the Analysis of Chaos

NSUB = 100

KIN = 84

AR = O

+ zD = O

Ns = NR = O

Fr = 0.0016

KEXIT = O

Kr1 = Kr2 = 0

+ zR = 30

b = 0.OO2

A = O

Kr3 =30

+ AR = 4

interesting behavior has been encountered for a large variety of non-linear differential equations (Moon, 1987). The most common manifestations are 80-called strange attractors, which are asymptotic orbits of the system (ie, the solution flow) describing fractal trajectories in hyper-phase-space. One important property of strange attractors is the inability to predict future events due to the exponential magnification of any uncertainties. This feature, often known as sensitivity to initial conditions (SIC), may be a source of concern if an accurate knowledge of the system evolution is required, as in nuclear reactor safety problems for example.

A projection of the strange attractor which was found is shown in Figures-41. This attractor has a correlation dimension [6] of De = 1.8 and an imbedding dimension of six (6). Next, Figure-42 shows the corresponding temporal evolution of the inlet velocity. It can be seen that these aperiodic nonlinear oscillations are extremely irregular. Significantly, rather similar chaotic oscillations have been reported for experiments in natural circulating boiling loops having a riser [25,26].

A type of Poincare map was constructed in which strobed points are plotted in the ut - PNR plane each time a pre specified characteristic time is reached

[6]. In this case the time to lose subcooling, 'U, was chosen as the appropriate characteristic time. Figure-43 shows the resultant mapping. The fractal structure of the strange attractor is evident.

7. CLOSURE

It is hopefully clear to the reader that single-phase and boiling natural circulation systems may exhibit a chaotic response. Moreover, the occurrence of such nonlinear instabilities may be very detrimental to the operation of power production or process equipment (eg, CHF may occur).

Hopefully this brief introduction to the theory of fractals and chaos will be sufficient to allow some of the readers to begin to do work in this exciting and rapidly expanding field of scientific analysis.

NpCH .. 1 07.6

6ut 0. 00 -

-0. 1 0 0. 00 O. 10

6PN A

Figure 40. Period Doubling in the out - OP�R Plane

6ut 0. 00

- 1 . 0 -0.05 0. 00 0.05

Figure 41. The Strange Attractor for Density-Wave Instability

383

384

r I

1. 0

Su� 0. 00

6. 0 7. 0 B. O 9. 0 10. I+ . VU

Figure 42. Temporal Evolution of Inlet Velocity for Chaotic Density-Wave Oscillations.

Su�

-o.os � OO �

Figure 43. Poincare map for strange attractor

NOMENCLATURE Ax.s Heated channel cross sectional area AR Riser cross sectional area cpr Liquid specific heat

D Channel diameter Dc: Correlation dimension E Energy f Friction coefficient g Gravity h Specific enthalpy hrg Latent heat of vaporization hr Liquid specific heat K Loss coefficient L Channel length M Mass M2 Two-phase mass in the heated channel Ns Number of nodes in the subcooled region NR Number of nodes in the riser PH Heated perimeter p Pressure �p Pressure drop q I I Heat flux q Total power t Time u Velocity Vf Specific volume of the liquid Vfg Liquid to vapor specific volume difference w Mass flow rate z Space variable

� 13 Liquid thermal expansion coefficient, - � �

BX Perturbation, X(t) - Xo P Density Pf Liquid density n Characteristic frequency A Boiling boundary �p Channel pressure drop

a Acceleration head term c h Channel D Downcomer e Channel exit ext External f Friction head term i Channel inlet I Inertial head term g Gravity head term n nth subcooled node r rth riser node

385

386

ref Reference value R Riser 241> Two-phase o Steady-state

Qo � Npch = Wo hfgVf

Phase change number

N -(hf - hi) � Subcooling number sub - hfg Vf

u� 1 0 Fr = gLH

Froude number

Nsub L u = -- - = Aolui Npch uio 0

fL A = 2D

� b = Cpf Vfg

t+ = tiu

z+ = z/L

Friction number

Thermal expansion number

1 J. Feder, Fractals, Plenum Press, New York, 1988. 2 B.B. Mandelbrot, Fractals , Form, C hance and Dimension, W.H .

Freeman, San Francisco, 1977. 3 M. Bamsley, Fractals Everywhere, Academic Press Inc., 1988. 4 M.F. Barnsley, R.L. Devaney, B.B. Mandelbrot, H-O. Peitgen, D. Saupe

and R.F. Voss, The Science of Fractal Images, Springer-Verlag, 1988. 5 P. Berge, Y. Pomeau and C. Vidal, Order Within Chaos, John Wiley &

Sons, New York, 1984. 6 F.C. Moon, Chaotic Vibrations, John Wiley & Sons, New York, 1987.

387

7 N.H. Packard, J.P. Crutchfield, J .D. Farmer and R.S. Shaw, Philadelphia Review Letters, 45 ( 1980).

8 A. Ben-Mizrachi and I. Procaccia, Physical Review A, 29, No. 2 (1984). 9 J. Dorning, AIChE Symposum Series-269, 85 (1989). 10 M. Abramowitz and LA. Stegun (Editors), Handbook of Mathematical

Functions, NBS-AMS.55, 1968. 11 Y.Y. Azmy and J.J. Dorning, ( (Eds. - Taylor et al) , Proc. 3rd Int. Conf.

Num. Meth. for Nonlinear Problems, Pine ridge Press, Swansea, U.K (1986).

12 J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, App. Math. Series Vol-18, Springer-Verlag, 1976.

1.3 J.L. Achard, D.A. Drew and R.T. Lahey, Jr. , J. of Fluid Mech., 155 ( 1985). 15 M.J. Feigenbaum, J. Statistical Physics, 19( 1) (1978). 14 Rizwan-Uddin and J.J. Doming, Nuc. Eng. & Des. , 93 ( 1986). 16 B .D. Hassard, N.D. Kazarinoff and Y-H. Wau, London Mathematical

Society Lecture Note Series-41, Cambridge University Press, 1981. 17 J.M.T. Thompson and H.B. Stewart, Nonlinear Dynamics and Chaos,

John Wiley & Sons, New York, 1986. 18 E.N. Lorenz, J. Atmos. Sci . , 20 ( 1963). 19 H.H. Bau and Z.Y. Wang, (Ed. C.L. Tien), Annual Reviews of Heat

Transfer, IV, Hemisphere Publishing, 1991 . ID RT. Lahey & M.Z. Podowski (eds . G.F. Hewitt, J.M. Delhaye and N.

Zuber), Multiphase Science and Technology, Vol . IV, Hemisphere Publishing, 1989, 183-370.

21 J.L. Achard, D.A. Drew and R.T. Lahey, Jr. , J. of Fluid Mech., 155 ( 1986) 213-232.

22 Rizwan-Uddin and J.J. Dorning, Nucl. Sci. & Eng., 100 ( 1988) 393-404. Z3 G. LeCoq, Proc. 3rd. Int. Topical Meeting on Nuclear Power Plant

Thermal-Hydraulics and Operations, Seoul, Korea, 1988. 24 RT. Lahey, Jr. and F.J. Moody, The Thermal-Hydraulics of a Boiling

Water Nuclear Reactor, Chapter 7, ANS Monograph, LaGrange Park, 1977.

25 A. Clausse, Efectos no Lineales en Ondas de Densidad en Flujos Bifasicos, PhD thesis, Instituto Balseiro, 8400 Bariloche, Argentina, 1986.

20 D.G. Delmastro, Influencia de la Gravedad Sobra la Estabilidad de Canales en Ebullici6n, M.S. thesis, Instituto Balseiro, 8400 Bariloche, Argentina, 1988 .

'Zl RH. Abraham and C.D. Shaw, Dynamics, The Geometry of Behavior, Aerial Press, Inc. , 1984.

389

ELEMENTS OF BOILING HEAT TRANSFER

A.E. Bergles

School of Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590 USA

Abstract The fundamentals of pool boiling and forced convection boiling heat

transfer are described in this chapter. Typical correlations for the various boiling modes or flow regimes are presented. The emphasis is on simple geometries such as a single tube in a large pool and a single vertical circular tube. Mention is made of practical problems such as boiling curve hysteresis resulting from difficult nucleation, shifts in the boiling curve due to dissolved gas, and surface or fluid contamination. A section on two-phase flow and heat transfer under microgravity conditions is included.

1. INTRODUCTION

The phase-change heat transfer coefficients and pressure loss factors required for the design of boilers and evaporators involve some of the most complex thermo-fluid phenomena. Out of necessity, and additionally because of the intellectual challenge, research in this area has exploded during the past 50 years. The patron of the science and art of boiling heat transfer is confronted with an accumulated literature of about 30,000 publications, about 50 text and reference books, and an output of another 1000 papers each year. Clearly, it is no longer possible to digest or even summarize this information. Nevertheless, the designer must have predictive methods for heat transfer and pressure drop. The relations used need not always be theoretically based, but they must be reasonably accurate. It is important to have an understanding of the physical phenomena and the mechanisms so that the correlations can be used appropriately.

The emphasis in this chapter will be on the heat transfer characteristics of simple geometries such as a single tube in a large pool and a single vertical circular tube. In most cases, as will be seen in subsequent chapters, the correlations for complex geometries, e.g., horizontal tube bundles and multiple vertical channels, are based on the experience for the simpler configurations. The present discussion focuses on pure fluids; mixture boiling and fouling will be the subjects of other chapters in this volume.

Boiling processes respond differently to the traditional boundary conditions of constant heat flux or constant wall temperature. The former boundary condition is associated primarily with systems having essentially fixed heat dissipation such as an electric boiler or a nuclear reactor core. This situation also occurs with liquid cooling of high power density devices such as electron

390

accelerator targets or computer chips. A constant wall temperature is frequently encountered in two-fluid heat exchangers with phas� change,. but there are situations, e.g., fossil boilers, where the heat flux IS essentIally constant.

2. POL BOILING

Pool boiling represents the traditional starting point for discussion of heat transfer in boiling systems. With pool boiling, it is possible to minimize the number of variables that must be considered in an experimental apparatus or analytical formulation. Due to extensive research effort, the mechanism of pool boiling is relatively well understood. However, it is still not possible to predict the heat transfer characteristics for this simplest of boiling systems with the precision associated with single-phase systems. This was evident at the Engineering Foundation Conference on Pool and External Flow Boiling held in Santa Barbara, CA, 1992. An inherent difficulty is to quantitatively characterize all the important surface and fluid characteristics, and to describe vapor and liquid flows in the complex geometries found in practice.

2.1 The Boiling Curve The first complete characteristics of pool boiling were reported by

Nukiyama [1] . The popular version of his "boiling curve" is chosen here as the basis of discussion because it is still the most meaningful representation.

As shown in Figure 1, the boiling characteristics are generally represented as a log-log plot of heat flux versus wall superheat - the heater surface temperature minus the saturation temperature.

N = 1: "-.a OJ : C7

Melting b _ _ _ _ �oint

Electric Heating 10 :1

104

I 10 10 Tw -Tsot OF

Figure 1 . Typical boiling curve for saturated pool boiling of water at atmospheric pressure.

39 1

The data represent saturated pool boiling of degassed liquid, which provides a suitable reference case because it involves the least number of system variables. For saturated boiling, the heat transfer coefficient is given simply as the heat flux of interest divided by the corresponding wall superheat. The coefficient is obviously a strong function of the superheat. The boiling curve can be traced out entirely by heaters with constant wall temperature (high temperature fluid or condensing vapor as the heat source) or partially with constant heat flux heaters (electrical heating). The basic regions and points on the boiling curve are now described.

I. Conduction with convection occurs at the heated surface. There is then convective transport to the vicinity of the free liquid surface where the energy is transferred by conduction, with convection, and evaporation.

(a) Incipient boiling. The first bubbles form at the heated surface, depart, and rise to the free surface. In practice, the initial bubbles may consist largely of air trapped in surface cavities. It is also difficult to maintain a saturated pool at low heater power without the use of auxiliary heaters; thus, vapor bubbles may condense before reaching the surface.

II. Bubbles form at many favored sites on the surface. At lower heat flux, individual bubbles can be distinguished, and there is little interaction between bubbles generated from adjacent sites. At higher flux, the bubble generation rate is so high that nearly continuous columns of vapor appear, with apparent interaction among these columns. At still higher heat flux, vapor masses result from the coalescence of vapor columns, and boiling occurs in a thin liquid layer beneath a vapor mass. The surface may become partially and intermittently dry, whereupon the average surface temperature rises substantially (departure from nucleate boiling).

(b) Peak nucleate boiling heat flux condition. The vapor generation rate becomes so high that there is restricted liquid flow to the surface, whereupon the surface becomes essentially completely blanketed with a vapor film.

III. The vapor film is unstable, collapsing and reforming under the influence of convection currents and surface tension. Large vapor bubbles originate at the outer edge of the film and at the random locations where liquid contacts the surface. As the surface temperature is increased, the average wetted area of the surface decreases and a lower heat flux is obtained.

(c) Minimum film boiling condition. A continuous film just covers the heated surface at this condition, frequently referred to as the Leidenfrost point.

N. An orderly bubble discharge occurs from the edge of the vapor film covering the surface; however, the shape of the interface varies continuously. At higher surface temperatures, radiation supplements film conduction.

392

All regions of the boiling curve can be exhibited with an appropriate constant temperature heating system. For instance, condensing steam. at various pressures might be used to generate the curve of Figure 1 up to the lower region of film boiling. The transient calorimeter or quenching technique has also been used for some systems (Merte and Clark [2]); however, significant alterations in the curve due to transient effects have been noted (Bergles and Thompson [3]). With a nearly constant heat flux system, such as electric heating, it is not possible to operate in Region III. When the power is increased to point b, a first-order instability ensues, and the operating point shifts rapidly to the film boiling region. For many systems, this new operating point corresponds to a temperature greater than the melting temperature; hence, point b is commonly referred to as the "burnout point". If operation in the film boiling region is achieved, the power may be increased to actual burnout or reduced to the Leidenfrost point, where the system reverts back to operation in the nucleate region. Successful attempts have been made to develop electrically heated systems with appropriate feedback control so that Region III can be studied (Sakurai and Shiotsu [4]).

A summary of present concepts regarding the various regions of the pool boiling curve is now presented. The discussion is oriented toward providing representative data and correlations that might be used for design. No attempt is made to incorporate all of the available analytical models or experimental observations.

2.2 Natural Convection The initial portion of the boiling curve is predicted by standard correlations

for heat transfer with free convection. Once the heat-transfer coefficient is obtained, the curve for Region I can be obtained, because

q" = h(Tw - Tsat) (1)

The heat-transfer coefficient for natural convection thus governs the entry into boiling conditions. A typical graphical representation of the Nusselt number as a function of Rayleigh number is given in Figure 2.

2.3 Nucleation In nucleate boiling we observe two separate processes

- the formation of bubbles (nucleation) and the subsequent growth and motion of these bubbles. In general, nucleation may be either of the homogeneous or heterogeneous variety; both types involve superheated liquid, which is a metastable state. Figure 3 depicts the superheated liquid state A where the liquid is heated above the saturation temperature TI for the constant system pressure Pl. This is referred to as (T2 - TI) degrees of liquid superheat. The corresponding subcooled vapor state B, which is also a metastable state, has (TI - T3) degrees of vapor subcooling. Nucleation theory and experience are concerned with finding the superheat required to initiate vapor formation.

Several theories based on statistical mechanics have been proposed to account for homogeneous nucleation in a pure liquid. One approach using classical rate theory, e.g. , Volmer [6] , presumes that numerous molecules have the activation energy

393

required for existence in the vapor phase. These energetic molecules could combine through collisions to form a cluster, which is then a vapor bubble. Theories of this type yield extremely high liquid superheats for nucleation in a pure liquid, for example, 50°C for water at 1 atm. Such high superheats are contrary to experimental observations with most engineering systems.

1000

10

Horllonlal FacillQ Upward

Pial .. and Horizonlal

( L ' r 012 )

100

Nu

1 .0

0 1 10•2 102 104 10' 108 10 10 1012 ( G, · Pr J

Figure 2. Correlation of natural convection data for typical boiling surfaces [5] .

Figure 3. lllustration of metastable states for liquid and vapor.

394

In a real system, of course, the liquid contains foreign particles and dissolved gas that could act as nuclei. The predicted nucleation superheats would be considerably less in the presence of a preexisting gas phase. This form of homogeneous nucleation implies that vapor formation would be noted at random points where the nuclei happen to be located. In actual practice, however, bubbles form at specific locations associated with the heated surface, not the fluid. It has furthermore been found by microscopic observation that these locations are small imperfections or cavities on the heated surface (Clark et al. [7]). For practical boiling systems, then, one is forced to discard homogeneous nucleation theories and concentrate upon heterogeneous or cavity nucleation.

Consider an idealized conical cavity with a pre­existing gas phase, presumed to be pure vapor for the moment, as shown in Figure 4. A wetting situation, or contact angle � � 90° is presumed. For a segment of a spherical bubble, mechanical equilibrium requires to a close approximation that

(2)

Thermodynamic equilibrium requires that Tl - Tv. At equilibrium, then, it is evident that the liquid temperature is superheated with respect to the liquid pressure, for a finite radius of curvature. In other words, the traditional property tables and charts for liquid and vapor are applicable only when a flat interface separates the phases.

In order to arrive at a general expression for the superheat required for nucleation, it is necessary to relate T and p along the saturation line. It is most convenient to employ the Clapeyron relation:

!!L � VfgT = dT (3)

One popular set of approximations assumes that the quantity hfg/vfgT is constant, with T = T sat. Integration then yields

hfg P - Pl = (Tv - Tsat) (4)

Substituting Eq. (4) in Eq. (2), the following relation for the equilibrium superheat is obtained:

Tsat Tv - Tsat = h = Tl - Tsat fg r (5)

As the superheat is raised, the radius of curvature decreases, from (1) to (2) in Figure 4, for example. It is noted, however, that the minimum radius corresponds to the hemispherical state (3), with the radius equal to the cavity mouth radius. With further increases in superheat (4), the bubble is no longer

395

stable, because its radius must increase in violation of the requirement for mechanical equilibrium. The bubble "nucleates" and grows at a rate dependent primarily upon the rate of heat transfer from the surrounding liquid, shortly becoming visible to the naked eye. It is particularly significant that the superheat required for incipient boiling is dependent on a single dimension of the cavity, the mouth radius. The validity of Eq. (5) has been established by Griffith and Wallis [8] , who manufactured cavities of a known size on a copper surface and produced nucleation by uniformly superheating the water pool.

L iq u id

Figure 4. Vapor bubble in a conical cavity with � = 90°.

It is evident that Eq. (5) will not be generally valid for predicting the incipience of boiling in the usual pool with a submerged heater where a temperature gradient exists in the liquid adjacent to the heated surface. Nucleation cannot take place from a cavity unless the environment surrounding the bubble is sufficiently high in temperature. Basic assumptions must be made regarding the liquid temperature profile and the relation of the liquid temperature to the vapor temperature. The following illustrates how this problem might be handled.

Assume that the liquid temperature profile is linear:

(6)

Further, assume that the liquid at all points adjacent to the bubble must be at a higher temperature than the required vapor temperature. Referring now

396

T,e Eq .

Figure 5. Conditions required for nucleation in a temperature gradient.

to Figure 5, it is seen that the liquid and vapor temperature are related as follows:

dTl dTl Tl = Tv and dy = "'Ci'r at y = rc

Utilizing Eqs. (5), (6), and (7), the cavity size at incipient boiling is

_ sat 20 r - hfg q"

and the heat flux - wall superheat relation is

,, _ kl (T q - w - 8Tsat Vfg O

(7)

(8)

(9)

To apply this criterion, it is convenient to plot Eq. (9) using the coordinates of Figure 1; the intersection of this equation with the convection relation, Eq. (2), defines the point of incipient boiling. In practice, however, the large cavities called for may not be available as nucleation sites. They may simply not be present or may not contain the necessary vapor nucleus. The appropriate equation is then

q" = T T sat 20 k rmax - hfg(rmax) (10)

397

where rmax is the radius of the largest initially active nucleation site. Figure 6 schematically depicts the incipient boiling loci for a range of maximum active cavity radii. For typical free convection behavior, the incipient boiling point would occur at a superheat corresponding to the intersection with Eq. ( 10) rather than Eq. (9), i .e . , (lBP)real rather than (ffiP)ideal.

"a-01 o

I I I I I I I , I I I

I I " rc • ( rmax ) . (T -T ) • 'w sat q " � o

Figure 6 . Determination o f incipient boiling point for various values o f rmax.

Inert gas is frequently present in boiling systems, either in solution or trapped in surface cavities. It is quite possible that homogeneous nucleation can occur with a large dissolved gas concentration. The gas is essentially driven out of solution near the heated surface, presenting the appearance of normal boiling. The previous analysis can be modified to account for the presence of an inert gas. Equation (7) becomes

(11)

where Pg is the partial pressure of the inert gas in the homogeneous or heterogeneous nucleation site. Following a similar development, the equilibrium superheat is obtained:

398

Tsat 20' Tv - T sat = h (- - Pg) fg r (12)

The inert gas thus reduces the superheat required for nucleation; in fact, nucleation can occur for temperatures below saturation. Unfortunately, it is virtually impossible to determine Pg, so only a qualitative assessment can be made.

It is appropriate to examine under what conditions a surface imperfection becomes a nucleation site. When the pool is filled, air can be trapped in a cavity when the advancing liquid touches the far side of the groove or pit before touching the base. The criterion can be stated in terms of the contact and cavity angles shown in Figure 4 as p > cp. Nucleation proceeds according to the previous analysis; however, the point of incipient boiling shifts to higher wall superheat as the air becomes depleted. Eventually a steady-state condition is reached where there is only vapor trapped in the cavity.

Boiling systems are normally operated so that there are repeated cycles of heating and cooling. If the air in a conical cavity has been exhausted, and the temperature is reduced below the saturation temperature, the requirements for equilibrium can no longer be satisfied and the interface recedes steadily into the cavity until there is no vapor left. The nucleation site has then been deactivated or "snuffed out". Re-entrant cavities, either natural or artificial, can remain active for considerable sub cooling. As indicated in Figure 7, the square re-entrant cavity retains vapor for a subcooling which depends on the inner mouth radius in accordance with Eq. (5). Note, however, that the degree of sub cooling is reduced as the contact angle becomes smaller. For the alkali liquid metals or refrigerants, where p- 0, the cavity of Figure 7a will flood for any sub cooling. The doubly re-entrant cavity shown in Figure 7b has been suggested to remedy the notorious nucleation instability of these fluids because the cavity will sustain sub cooling to the indicated radius of curvature. The stability of cavities containing non-condensable gases is discussed by Mizukami [9].

Liquid L iquid

a. Square re-entrant. b. Doubly re-entrant.

Figure 7. Re-entrant cavities used to improve boiling stability.

399

Cavity stability is much enhanced if nonwetting conditions are present. From Figure 8, it is seen that the cavity can retain vapor for virtually any amount of subcooling ( 1), (2).

4

Figure 8. Vapor bubbles in a conical cavity with � > 90°.

It is noted, however, that the cavity radius no longer defines the point of incipient boiling, because the bubble becomes unstable (3) before reaching the hemispherical state. The boiling process is generally erratic due to the tendency of the bubble to cover a large portion of the surface (4). Such behavior is encountered with mercury or when boiling water from teflon or oily surfaces.

As noted in the following section, all theoretical models of nucleate boiling need input data on the size distribution of active cavities. Visual observation of the surface during boiling is not reliable because the bubbles obscure the surface, except at very low heat flux. Post boiling observations of deposits surrounding nucleation sites, e.g., Heled and Orell [10] , are likely to be suspect due to modification of the nucleation sites by the deposits. It is difficult to obtain quantitative information by analyzing surface roughness parameters or by studying photographs such as those obtained by a scanning electron microscope (Nail et a1. [ 11]), as it is not clear which "pits" represent potential active nucleation sites. Gas diffusion experiments patterned after that of Brown [ 12] have also been attempted, but also with limited success (Lorenz et a1. [13], Eddington et al. [14]).

400

2.4 Saturated Nucleate Pool Boiling Upon reaching

the incipient boiling condition, further heat transfer promotes bubble growth, due to the excess vapor pressure that is no longer balanced by the surface tension forces. The growth of bubbles is a dynamics problem coupled with a heat transfer problem. Considerable effort has been devoted to examining the behavior of single bubbles. These isolated bubbles, which are amenable to analysis, are found at low heat flux.

In general, heat is transferred from the wall to the liquid so as to establish a superheated layer. This layer involves free convection heat transfer, perhaps augmented by "bubble agitation", over that portion of the surface not directly affected by the bubbles. Transient free convection is observed prior to nucleation or in the vicinity of the bubble after nucleation.

After nucleation, bubble growth is promoted by heat transfer from the superheated layer and, especially in the case of highly wetting liquids, vaporization of a "microlayer" of liquid between the bubble and the wall. There may also be evaporation at or near the base of the bubble and condensation at the top of the bubble ("latent heat transport"). Eventually, the bubble grows to the point where it departs due to buoyancy, carrying with it a substantial portion of the superheated liquid layer ( "bubble pumping" or "microconvection") .

A miniature thermocouple or thin film sensor, e.g. , Cooper and Lloyd [ 15], installed near a nucleation site, records temperature fluctuations, T w(t), that would not be noted by the usual instrumentation, which records merely Tw. These temperature fluctuations and associated bubble states are shown schematically in Figure 9. We start with the condition subsequent to incipient boiling where the bubble is growing rapidly outside the cavity. The microlayer vaporizes, producing a drop in surface temperature (1) . When the layer has evaporated, the surface rises in temperature (2) because heat transfer to the vapor is poor. During this period,. the bubble is also growing due to evaporation at the interface between the bubble and the superheated liquid layer. When the bubble departs, the surface is quenched by cooler bulk liquid and there is a sharp drop in temperature (3). The thermal boundary layer is then re-established and the surface rises in temperature until conditions are again suitable for the incipience of boiling (4). The temperature oscillations depend strongly upon the system; for instance, a glass heater would produce substantial fluctuations, while the fluctuations would be small for a high conductivity metallic heater.

The relative importance of these energy transport mechanisms varies according to the boiling system. Graham and Hendricks [ 16], for example, estimated that microlayer evaporation accounted for about 50% of the wall heat transfer in the case of methanol but only about 25% of the wall heat transfer in the case of water.

Numerous attempts have been made to formulate and solve reasonable mathematical models for the growth of vapor bubbles. The classic formulation of Rayleigh, originally applied to cavitation bubbles, has been utilized to examine the initial stage of bubble growth that is controlled by surface tension and inertia. During the latter stage of growth, the problem can be formulated relatively simply in terms of heat transfer,

40 1

be�use �e growth rate is controlled by the rate of evaporation at the interface, whIch, In tu�n, . depends on heat conduction from the surrounding su�erheat�d hqUI� layer . . Expressions derived by Mikic et a1. [ 17, 18] satisfactonly .de�cnbe expenmental results for bubble growth in a uniformly superheated bqwd or at a heated wall. The predictions and data are shown in nondimensional form in Figure 10.

( I ) S uperheated Liquid

M i cro layer

Figure 9. Surface temperature variation in vicinity of a nucleation site, with corresponding bubble states.

The bubble departure size is an important parameter in the study of pool boiling, because it has a direct bearing on the heat transfer characteristics. Fritz [19] utilized the theory of capillarity to get an equilibrium bubble shape, and formulated a differential equation representing a balance of gravity and surface-tension forces. An approximate solution to this equation yielded the bubble diameter at departure:

Db = 0.0148J3[g( 2a

PI - Pv) (13)

where � is in degrees. This equation has been verified for numerous systems, including steam and hydrogen bubbles in water. The contact angle presents a problem in applying this equation, because a dynamic contact angle seems to be required rather than the static contact angle that is normally measured.

402

• { Experimental data

o Water, pressure range

0.18 to 5.6 psia

Figure 10. Predicted bubble growth curves compared with experimental data. (Mikic, et al. , [17, 18]).

With the preceding information, it is possible to combine the individual processes of bubble inception, growth, and departure to predict the heat transfer performance of a boiling surface. It is instructive to outline the method of Han and Griffith [20, 21] , even though it will become apparent that the procedure cannot be used for engineering calcuations.

The model is formulated for the isolated bubble regime, as indicated in Figure 11. An idealized grid of nucleation sites is postulated that defines the so-called bulk convection area. The mechanism of heat removal from the surface consists of two parts: 1) the enthalpy transport represented by the repeated removal (bulk convection) of the superheated layer in the vicinity of the bubbles, and 2) continuous removal of heat by the usual free convective process in the area uninfluenced by the bubbles. Referring to the earlier discussion of the mechanism of nucleate pool boiling, it is seen that only Regions 3 and 4 of Figure 9 are considered.

At Stage 1 the bubble departs, carrying with it a section of the superheated transient thermal layer. Colder liquid from the bulk of the pool quenches the heated surface, and the transient thermal layer is reformed. A waiting period is required before the layer is superheated sufficiently to activate the cavity

403

(Stage 2). The bubble then grows (Stage 3) until the departure diameter is reached (Stage 4), at which point the cycle is repeated. A simple experiment indicated that the area from which the superheated liquid is pumped away corresponds to about twice the bubble departure diameter. Assuming only pure conduction to the superheated liquid layer in the area of influence, the problem was modelled as conduction to a semi-infinite body with a step change in temperature at the surface. The superheated layer is replaced at a rate corresponding to the frequency of bubble departure.

No. of slaQI. I

r/ WoilinQ period j

�BUbble Bulk convection

l ayer -- ,

Staqe I�

�ral conv4ct'jon

I'.

5109e :3 �1Pld'di

2 3 • �1" ' 4� j rj 'od

p e r iod

Figure 11 . Model for nucleate boiling in the isolated bubble region (Han and Griffith, [21]).

In somewhat simplified form, the final expression for the heat flux is

(14)

(15)

2 [x(kpchf ]O.5Db 2n(T w - 1b) (16)

where an average departure frequency is presumed to be valid for the bubbles issuing from the n active sites per unit area of the heating surface. To use this equation, it is necessary to specify f, Db, and n(rc). Han and Griffith [20]

404

obtained the frequency from transient conduction calculations, in essence, getting the time required for nucleation of a specified cavity (tw) and the time required for the bubble to grow to departure size (tci). The departure diameter was obtained from the Fritz relation, Eq. (13); the contact angle was taken as the dynamic value (measured from motion pictures) at the average bubble growth rate. The cavity size distribution was inferred from the wall heat flux and wall superheat at which incipient boiling was observed. The formulation thus represents a "two adjustable parameter" model. The agreement between measured and predicted heat transfer rates was quite satisfactory.

Modifications of this basic model were employed by Mikic and Rohsenow [22] as shown in Figure 12, and Lorenz et al. [23], with similar success. Judd and Hwang [24] extended the model to include microlayer evaporation, as shown schematically in Figure 13 and given by Eq. (17).

0' Irf

10 10

IIEE CONVECTION �IIIEDICTED :

.. .. - 1.5 I T;T •• , )

102 T.- T •• , 0 ,

(17)

Figure 12. Comparisons of measured (Gaertner and Westwater [25]) and predicted (Mikic and Rohsenow, [22]) data for nucleate boiling of water.

This term introduces still another experimentally determined parameter, the average microlayer volume. The total heat flux was predicted quite well if the area of influence was reduced from that corresponding to twice the diameter of the bubble at departure to 1.3 times the diameter.

405

It is evident that the prediction of the boiling performance is an elusive undertaking. The models proposed to date provide insight into the physics; however, the requirement that microscopic data be available for specific systems renders these models useless for engineering calculations.

The preceding discussion has given some idea of the variables that have an effect on nucleate boiling. A more complete summary is given below.

Fluid state - the fluid properties p, Vfg, hfg, cr, k, p, c, and the fluid-surface property � are important parameters for the semi-analytical treatment of nucleate boiling. In addition, the dissolved-gas concentration affects nucleation, as noted earlier.

Surface condition The material properties k, p, and c are important in certain cases. Mechanical properties have a bearing on how a particular finishing operation produces the characteristic cavity size distribution. The amount of gas trapped in the cavities also affects nucleation, particularly when the system is first started up. Surface coating, oxidation, or fouling can markedly affect the surface wettability and, therefore, the effective cavity sizes.

Heater and pool geometry - These variables determine the pool convective conditions, which have an important bearing on the natural convection heat transfer coefficient as well as the bubble motion.

Body forces - Significant differences in boiling behavior have been noted in fractional-gravity and multi-gravity situations. The former is discussed in Section 4.

Method of heating - The heating method is immaterial except for the case of a.c. resistance heating, which may cause bubble generation in phase with the power supply.

TRANSFER SlR"ACE AREA AT

BY

Figure 13. Schematic representation of boiling heat transfer model including microlayer evaporation (Judd and Hwang, [24]).

406

History - Due to the peculiarities of nucleation, hysteresis behavior may be noted where the boiling curve is different when the heat flux is increased than when it is reduced. An aging phenomena can also occur as the surface becomes degassed or contaminated.

Several sets of data illustrate certain of these effects. Figure 14 demonstrates the effect of relatively large-scale surface roughness (5), oxidation (2), and aging due to surface degassing (4,6,7).

C'I -+-oJ "H s. ..c: -. ::I -+-oJ l:Q

-0"

104

103

W a l l ' . I A''' HOfl lontal S u , ' a c .

I ACkl9h.".d by .and bla,1

2 So", • • o.,diud

3 Sand blast.d avain

4 So"' •• 10"9.' u •• , RCklqh" ... o.cn.n added

6 So", • • 4 h, boolinq . 24 ." loak inq

7 So". •• a ." bai l i nq .

24 h, laakinQ

T - T , OF w sat

Figure 14. Boiling curves for surfaces with various surface treatments and operating history (Jakob and Fritz, [26]).

Additional roughness effects are illustrated in Figure 15, where it is seen that a four-fold increase in the boiling heat transfer coefficient for this system (evaluated at constant heat flux) can be obtained by choosing a lap over a mirror finish.

C'I +> <t-o � ..

• R u n 31 : E m ery 3 20

x R u n 32 : E m e ry 6 0

o R u n s 1 7 So 22 ' Lap E D R un s 2 a 3 : Mirrar Finish

� +> CQ

407

Figure 15. Effect of surface finish on boiling curves for copper-pentane (Berenson, [27]).

Data that demonstrate hysteresis are presented in Figure 16. These data were taken from a small vertical copper plate that was indirectly electrically heated. The temperature overshoot is due to the deactivation of large nucleation sites, which 'was discussed earlier. Between runs, the power was shut off so that the deactivation reoccurred. The boiling curve hysteresis with a highly wetting liquid is evidently large and repeatable.

It is apparent that neither a precise description nor a universal correlation will be possible for nucleate boiling due to the number of variables, in particular, those that might be termed nuisance variables.

Numerous attempts have been made to correlate the portion of the boiling curve that is log-linear. The most successful models are semi-analytical and involve a form of dimensional analysis. The most widely used correlation, developed by Rohsenow [29], is described here. Adopting the argument that the major portion of the heat is transferred directly from the surface to the liquid, a conventional combination of dimensionless groups is sought:

Nu = f(Re, Pr) (18)

The forced-convection turbulent flow groups are pertinent because the bubbles are "pumping" the liquid.

408

The Reynolds number represents the ratio of bubble inertia to frictional forces on the bubble:

(19)

Use of Gb is permitted because Gb = Gl, by continuity. The bubble superficial mass flux is given by

A bubble Nusselt number is given by

The latent heat transfer of the bubbles is represented as

qb = hfg 1;Db3 f Pv n

RUN INCREASING

1 6 2 v 3 0

PLAI N COPPER

DECREASI NG • • •

H • 5 . 0 nvn \I = 4 . 9 nm

Tsat 46 . 4 °c

I I S

I :� • .9

• v • 4 S � .' 6

.J9 1.! v

� ." :-

10 ] o · 0 . 1 1 0

(20)

(21)

(22)

1 00

Figure 16. lllustration of boiling curve hysteresis with boiling of Refrigerant 113 from a copper surface (Kim and Bergles, [28]).

Experiments indicate that for a wide range of conditions,

q" - n

Thus,

q" = Cq qb

The Prandtl number of the liquid is appropriate:

Introducing Eq. (13) for Db, and formulating the functional relationship as

the final correlation can be obtained as

cl(Tw - Test) _ C [ 2 X 0.0148P]n [--L 0" ]n [ C� Jm hfg - q Cq hfgll1 g(Pl - Pv) k 1

409

(23)

(24)

(25)

(26)

(27)

The collection of constants preceding the slightly redefined Reynolds number function is designated as Csf, a constant reflecting the condition of particular fluid and surface combination. Because all properties are evaluated at the saturation temperature, this expression is analogous to the general expression for fully established boiling:

q" = f (p, fluid, surface) (T w - T sat)lln (28) It was possible to correlate data for a wide range of fluids at different pressures utilizing n = 0.33 and m = 1.7, where Csf has a different value for each fluid-surface combination, as listed in Table 1. An example of the extent to which the data are correlated is shown in Figure 17. This "one adjustable constant" correlation with a value of Csf from the table is useful in preliminary design. It is, of course, desirable to have some data for the actual system at hand to establish the appropriate Csf. This testing might be done in a simple pool at atmospheric pressure; the correlation then permits extrapolation to high pressure. Note that a subsequent re-evaluation of the data indicated that m = 1.0 is more accurate over the entire pressure range for water. This is a consequence of the Prandtl number for water not being a monotonic function of pressure.

4 1 0

100

r- -> C- 10 t» I

a: bo

- I � -0' .! 1.0

L.

0.1

ali & - a l 4 a -

/ 4 • • ,"4 .7 0 &· 4 Dala of :

o J Addoms • Pool BOiling i Plotlnum

o 4 Wire - Woter _ 0.024· dlom. 0 1 4.7 PSIA

0/ 6 �;� :�:: -

A 1 2 05 PSIA x 1 60 2 PSIA

a 2465 PSIA

0.01 0.1

� u ..

lit

0' .. (5 �

'i .--�

U I:

C f I

- Tsot) PrL7 ,g

Figure 17. Correlation of pool boiling data for water (Rohsenow. [29]).

TABLE 1. Constants in the Rohsenow Nucleate Boiling Correlation, Eq. (27) [30].

Surface-Fluid Combination

Water-nickel Water-platinum Water-copper Water-brass Carbon tetrachloride - copper Benzene-chromi urn n-Pentane-chromium Ethyl alcohol - chromium Isopropyl alcohol - copper 35% Potassium carbonate - copper 50% Potassium carbonate - copper n-Butyl alcohol - copper

Csf

0.006 0.013 0.013 0.006 0.013 0.010 0.015 0.0027 0.0025 0.0054 0.0027 0.0030

41 1

It should be noted that the slope of the log-linear portion of the boiling curve may be different from that assumed in the correlation. In actual practice, a range of n = 0.04 to 1.0 has been reported. However, in order to use the correlation's pressure prediction capability, which is really the only reason for using the correlation, the exponent n = 0.33 must be retained, because the property groups are afected by this exponent.

At this point, it is apparent that the designer is still at a loss to predict nucleate boiling without direct information on the fluid and surface of interest. A natural question is: In view of the vast amount of data accumulated on nucleate boiling, isn't it possible to get a useful correlation by purely statistical means? The early study of Armstrong [31] was not v�ry encouraging, as shown in Figure 18, there is a large variation in the wall superheat for a given heat flux for organic liquids. However, according to the study of Stephan and Abdelsalam [32], the answer is a qualified "yes". They identified the physical properties and variables characterizing the process and developed a possible set of 13 independent dimensionless groups. A product relation among these groups was postulated, and a linear regression analysis was used to obtain the lead constant and the exponents of the most important groups. The fluids were divided into four categories: water, hydrocarbons, cryogenic fluids, and refrigerants; about 5000 data points were considered. While only 8 of the groups were ultimately utilized, and only 3-5 groups appeared in the correlation for a given category, the correlations were quite involved.

Tw-Tsat = 1 1 .48 (q"/lOOO) O . 2 9 3 _ c I � _--80 80

� 40 o

o o

2 4 6 8 105 q" , Btu/hr ft 2

Figure 18. Compilation of nucleate pool boiling data of various organics at 1 atm together with a statistical correlation (Armstrong. [31]).

4 12

Accordingly, a simple set of heat transfer coefficients were proposed as

h = c(p) (q")n (29) where the exponent was fIxed for each category and the pressure function is given graphically for each fluid or group of fluids. For water, for example, the expression is

h = Cl(p) (q")O.673

q" = [ cl(p) (Tw - Tsat)]3·06

(30)

(31)

which is in excellent agreement with Rohsenow's recommendation of an exponent of 3.03 in Eq. (27). The pressure function Cl(P) is shown in Figure 19. It is important to note, however, that the surface roughness appears in only the correlation for cryogenic fluids, even though surface finish and material are known to afect boiling of all fluids.

Cooper [33] has proposed an alternative correlation procedure for saturated nucleate boiling utilizing reduced properties. While the pressure dependence is taken care of quite satisfactorily, there is considerable scatter in thl� data due to surface effects.

1 02 8 6 4

-

c,

I

100 2 4 6 8 10 ' 2 4 6 8 102 bor 2

10 2 2 4 6 B 10 -' 2 bor 4 6 B 100

- p Figure 19. Pressure factor for water (Stephan and Abdelsalam, [32]).

4 1 3

2.5 Peak Nucleate Boiling Heat Flux In spite of intensive research effort, there is still disagreement regarding

the actual mechanism of the peak nucleate or critical heat flux. Two interpretations of this condition are

The number of nucleation sites becomes so numerous that neighboring bubbles or vapor columns coalesce, causing a vapor blanketing of the surface (Figure 20), e.g., Rohsenow and Griffith [34]. Their correlation is given by:

<kit = 143 hCg Pv (iY·25 Btulhrft2 (32)

(hCg in Btulbm and Pv in Ibmlft3)

where a/g is a term added later for fractional gravity and multigravity situations. The comparison with data is shown in Figure 21.

Figure 20. Bubble·packing model of critical heat flux in pool boiling (Rohsenow and Griffith, [34]).

. At high heat flux, the number of sites is so

large and the vapor generation rate so high that the area between the bubble columns for liquid flow to the surface is reduced, as shown in Figure 22. The relative velocity is so large that the liquid-vapor interface becomes unstable, thus essentially starving the surface of liquid and causing the formation of a vapor blanket, e.g. , Kutateladze [35], Zuber [36], Chang and Snyder [37], and Moissis and Berenson [38]. This has also been visualized as a flow pattern transition where the liquid filaments break up into drops that are suspended or "fluidized" by the vapor stream (Wallis [39]). The Zuber [36] correlation is given by

4 14

Figure 21 . Comparison of bubble-packing correlation with data (Rohsenow and Griffith, [34]).

Vv

t � I

Figure 22. Simple visualization of jets for hydrodynamic instability model of critical heat flux.

" . - 0 131 h [O(PI - qcnt - . fg Pv 2 Pv

(33)

The constant in Eq. (33) is generally considered to be low; the value of 0. 18 has been recommended by Rohsenow [30], as shown in Figure 23.

• ,Q

ITMAMOL. C'CHIL.LI AMO aO.ILL.A.

... .. ' .. '.MI. . . . IlfIlI"I. · • • .. n ..... O L • • 'IT."" ' _ ", ,. 0 •• "T ..... ILO

w .. Tta. AODOln .. f._. oa.,.

4 1 5

Figure 23 . Comparison of hydrodynamic instability correlation with data (Zuber, [36]).

If the heaters are small in a hydrodynamic sense , i .e . , a tube with R' = R[g(Pl · pv)/cr]O.5 < 0.5, the critical heat flux is increased, e.g. , Sun and Lienhard [40] . For saturated water at 1 atm boiling under standard gravity, the value of R' at the point where the critical heat flux increases corresponds to a 0.5 in. diameter tube. This is depicted in Figure 24.

L' = L (g(Pt - pv)/a)o . 5

Figure 24. Adjustments to flat plate correlation for various geometries (Lienhard and Dhir, [41]).

4 1 6

As shown in Figure 25 , the vapor removal patterns are di�erent for vari�us geometries. Accordingly, the critical heat flux correlations change wIth geometry, as shown in Figure 24.

Small sphere

Infinite flat plate

Sphere or cyl inder cross -sec' ion

R i bbon with one side insula ted

Figure 25. Vapor removal patterns postulated near critical heat flux (Lienhard and Dhir, [41]).

The hydrodynamic theory breaks down at small values of the dimensionless size. Figure 26 illustrates this for a cylinder, where the critical heat flux is poorly defined for R < 0.1 . The rather random behavior is associated with the way in which large bubbles behave shortly after nucleation occurs. This region has been studied by Bakhru and Lienhard [42]. A recent, very comprehensive review of critical heat flux behavior on cylinders is given by Lienhard [43]. A growing number of challenges to the hydrodynamic theory are noted. From a practical point of view, however, the various mechanistic corrleations, such as Eqs. (32) and (33) are quite similar. This is because the empirical constants have been adjusted to fit the experimental data.

Additional effects not accounted for in the critical heat flux correlations items that been observed are usually a reduction in q�rit with

vertical orientation of long heaters non-wetted surfaces restriction of the liquid circulation to the heater mechanical or metallurgical defects in the heater thin, low thermal conductivity heaters

and an increase in q�rit with small amounts of certain additives use of d.c. power instead of a.c. power rapid increases in power certain types of surface fouling

2 0

1·5 .., ..,

0-r.:I 10

-0" :r

'"'

-0"

v v

• I • t /

13 Ethanol data} G) Water data 'V'.�.O.� data

g /g� � I .O for R'� 0.07

OimenslOl1less r adius. R'

4 1 7

20

Figure 26. Breakdown of hydrodynamic theory at low R' according to Sun and Lienhard [40].

These factors do not seem to be involved in the aforementioned studies carried on by Lienhard and co-workers; however, they are expected to be reflected in data from a wide variety of sources.

Figure 27 presents a composite of data for tubes from 47 separate investigations that demonstrate the actual variability of the critical heat flux. The generally accepted baseline of Eq. (33) and the correlation of Sun and Lienhard [40] are close to the averages for R' > 0.1; however, the variation in the data is seen to be quite large. The spread is due to inherent statistical fluctuations in the phenomenon itself (±15%) as well as the factors mentioned above. Accepting this variability and adopting a statistical approach, the following equation is recommended (Park and Bergles [44]):

41 8

= 1.235 - 0.687X - 0.590X2 -H>.987X3 + 0.673X4

q"crit, Eq. ( 33)

- 0.296X5 - 0.330X6 - 0.090X'7 - 0.OO8X8 (34)

where X = log R'.

Z.S

Co z.o !AI

_U 1 .5 -0' �

10 u -

0' 1.0

0.5

Eq. (33) •

I . 10 1 R '

• S • "' 1(1

+ ISOPIPAI XII[TIWIl 1' fTIWC. • liz· Oz . R·m . R· I I

x

I • 101

Figure 27. Critical heat flux data for cylindrical heaters [2377 points] (Park and Bergles, [44]).

2.6 Transition and film boiling Film boiling is the most tractable regime analytically,

because the flow pattern is relatively simple. Exact solutions can be obtained for laminar film boiling in a saturated pool by solving the momentum and energy equations separately, and then relating the solutions by means of a heat balance.

The relation of Breen and Westwater [45], which accounts for interfacial shear and curvative effects, is

h = (0.59 + 0.0069 Ac [kv3 g (PI - Pv) D �v (Tw - Tsat) Ac

where the minimum wavelength for Taylor instability, Ac, is given by

(35)

[ a Ac = 21t g(Pi - Pv)

4 1 9

(36)

A change in film boiling behavior, apparently analogous to the transition to turbulence, has been noted at a critical value of a modified Rayleigh number according to Frederking and Clark [46]:

(37)

The correlation for the turbulent regime, Ra*> 5 x 107, is

Nu = 0.15 (Ra*)113 (38)

which is apparently valid for vertical plates, horizontal tubes, and spheres. The temperature level is generally quite high in film boiling, and it is

necessary to account for radiation. A simple superposition of heat transfer coefficients is not adequate, however, because the radiant heat transfer results in an increase in the film thickness. The following equation is recommended for the resultant heat transfer coefficient:

(39)

where

(40)

and the interchange factor F LO. The lower limit of stable film boiling corresponds to the

breakdown of the continuous insulating film and the onset of liquid-solid contact. Numerous analyses have been made to predict this condition, generally based on hydrodynamic stability theory similar to that employed in determining the critical heat flux. The result for large tubes in terms of the minimum heat flux is (Zuber [36], Zuber, et a!. [47], Berenson [48])

" [ag (Pi - qmin = C hfg Pv ( )2 Pi + Pv

(41)

where C ranges from 0.09 to 0.177. The higher value is recommended because the usual liquid impurities and surface contamination tend to raise the minimum heat flux.

Due to the transient character and ill-defined flow pattern of transition boiling, no theory has been formulated for this regime. An accurate representation of the average heat transfer characteristics can be

420

obtained by linearly interpolating between the peak nucleate heat flux point and the Leidenfrost point on the boiling coordinates Oog-log).

2.7 Influence ofSubcollng on the Boiling Curve Subcooled pool boiling is generally a transient condition observed as the

pool is heating up. In certain cases, however, the system heat loss may be high or some cooling means is employed so as to achieve a subcooled bulk condition. Subcooling may be visualized as a perturbation of the saturated pool boiling curve discussed in detail in the preceding sections.

In the natural convection region, the procedure of Section 2 . 1 can be utilized where now

(42)

Fully established boiling may deviate from saturated boiling with a similar system. The boiling curve for a horizontal cylindrical heater shifted to the right (Bergles and Rohsenow [49]), as shown in Figure 28. However, the opposite trend was observed for a horizontal flat heater (Duke and Schrock [50]). In the absence of definite guidelines, it is suggested that this effect be ignored.

..

106 9 8 7 6

4

.;: 2 � -= is = - lOS C' 9

8 7 6 5 4

Pol Boiling 16 90 55 P '" 29 Ib,. /in�obs a (Tsat - T},) 3 OF D 71 , 149

2 10 20 30 40 50 60 80 100

Tw - Tsat, of

Figure 28. Reported trends of subcooling on nucleate boiling curve for water (Bergles and Rohsenow, [49]).

42 1

The critical heat flux is strongly dependent upon the degree of subcooling. The data are usually represented as a linear function of the subcooling, as exemplified by the simple expression recommended by Ivey [51]:

sub = [1 + 0.1 {ClPl (Tsat - (!.)0.273 qcnt, sat hfg Pv g (43)

Ponter and Haigh [52] found that the following equation is accurate for water at subatmospheric pressures:

sub = 1.06 + 0.015 (T 0.474

qcrit, sat sat P (44)

where (Tsat-Tb is in °C, p is in torr. Eq. (33) is used as the reference in both Eq. (44) and Eq. (43).

More recent work by Elkassabgi and Lienhard [53] indicates that the influence of subcooling is negligible at very high subcooling. As shown in Figure 29, they found that the critical heat flux ratio is essentially linear up to the point where the highly subcooled boiling region begins.

! '6

-0' --.0

� ..

-0'

3.0

2.5

2.0

1.5

1 .0

0.5

reoion 01 mod.r.,e

aubcoollng

iloproplnol d.,.

o 0.11 1 3 mm dl •. " •• ,.rl

e 1 .042 mm dl •• h •• ,.r. (I 1.215 mm dla. h •• ,.r.

• 1.524 mm dlL " •• '.r.

Figure 29. The effect of subcooling as the critical heat flux for heaters of various size in isopropanol (Elkassabgi and Lienhard, [53]).

Subcooled film boiling has not been investigated extensively; however, general experimental evidence indicates that heat transfer coefficients are increased in the stable film boiling region as the subcooling increases. Quantitative predictions can be obtained from the analysis of Sparrow and Cess [54] for an isothermal vertical plate. The analysis predicts that the heat

422

transfer coefficient for large subcooling will be identical to that obtained for free convection of the liquid alone.

Subcooling increases q;in and transition boiling heat fluxes; however, no reliable data appear to be available to quantitatively establish this region of the boiling curve.

2.8 Comtruciion of the Complete Bolling Curve In recapitulation. it is appropriate to develop the complete boiling curve

from the preceding equations. The result is shown in Figure 30 for a stainless steel tube in water. This is the approach that would be taken to predict the performance without performing tests on the actual equipment. As emphasized in the preceding discussion, significant deviations are possible in all regions of the boiling curve. The generated curve, however, represents a typical estimate based on the current state of the art.

2 x 1 06

1 06

1 05

..

.. .t:.

.= II

1 0'

b Eq . ( 4 3 )

b Eq . ( 3 2 )

n Eq . ( 2 7 )

TABLE 1

I Eq . ( 4 2 )

FIG. 2 - -

IY

DEGASSED WATE R 1 1 at m HOR I ZONTAL S TA I NLESS TUBE 0 . 5 I N . D I A .

-- SATU R ATE D Tb = 2 1 2 O F - - - S U S COOLED Tb= 1 00 O F

1 03

3 1 0 1 0 0 0 3 000

Figure 30. Prediction of boiling curve for saturated and subcooled pool boiling.

423

2.9 Crosow Efects on Boiling from Cylinders It has been demonstrated experimentally that pool boiling heat transfer is

influenced by any of the schemes that impart a velocity to the bulk liquid, including propeller-type stirrers, pumps, and injected vapor (e.g. , Pramuk and Westwater [55]). As might be expected, heat transfer in the nonboiling convective region is improved, there is little effect on fully developed nucleate boiling, the peak heat flux is elevated, and coefficients in the transition and film boiling regions are increased. Quantitative prediction of these effects is difficult, however, due to the inability to describe the velocity field in the vicinity of the heater.

Numerous experiments have been devised to study boiling heat transfer from single horizontal cylinders with an established crossflow velocity. The data are useful for anticipating the heat transfer characteristics of tubes in a bundle when the velocity arises from the gross natural circulation. The single-phase portion of the boiling curve can be obtained from well-established relations for flow over cylinders. The transition region between incipient boiling and fully established boiling can usually be ignored, especially given the usual uncertainty of quantifying the established boiling.

700 500

300

200 <='>e --­�

t 100 -

>< 70 -0"

50

30

20

o 20 40 60 80 100 120 140

T - T , K w sat

Figure 31. Effect of crossflow velocity on nucleate, transition, and film boiling (R-113 subcooled 4.5 K at 1.12 atm flowing normal to a 6.4 mm OD cylinder). Adapted from Yilmaz and Westwater ([56]).

424

There is some disagreement how the imposed velocity, which is higher than the natural circulation rate in free convection from a single tube, affects the boiling curve. Yilmaz and Westwater [56] suggest that there is a substantial shift in the nucleate boiling curve with velocity, as shown in Figure 31 . However, close examination of the actual data o n log-log coordinates indicates that the shift is pronounced only at low heat fluxes and is due to the accentuated knee of the boiling curve in the presence of both velocity and subcooling. Thus, the data are in accordance with most observations of forced convection subcooled boiling.

The peak nucleate heat flux, seen in Figure 31 to increase with increasing velocity, was correlated by Yilmaz and Westwater [56]:

q�rit = PvhfgU cr (45)

Gravity has a significant effect at low flow velocities, as demonstrated for low velocities in the horizontal plane by Sadasivan and Lienhard [57]. As shown in Figure 32, the critical heat flux ratio, where the denominator is a term. similar to Eq. (45), tends to unity at low values of the parameter involving gravity.

The film boiling heat transfer coefficient increases with velocity and subcooling, and is further elevated by radiation effects at higher temperature. Correlations are discussed by Yilmaz and Westwater [56].

2.6 2.5

. 2.0 'tl Q) � 0. � 1 .5 -rt � 0

=0" "

"" 1 .0 'M � U

'0" 0.5

0 1 00

Data:

o isogroganol ; Ungar

CJ isogroganol ; Ha.an et al. • methanol : Huan et al. o Freon : Yllmaz & We.t.ater

Brou •• ara & We.twe.er

.. Freon ; Cochran & Andrecchio

o Water ; Ungar

• Water ; Vlie' & Leggert

o I I d

,�o I

• 1 • 1 0 0 0 I

I I

1 + 0.000020 (PI/ Pv) 1. 7/Fr

Fi�re 32. Critical . heat flux �atio shOwing regions of gravity influence and nonmfluence (Sadaslvan and LIenhard, [57]).

425

3. FLOW INSIDE TUBES When flow occurs in a channel, as in in-tube boilers and evaporators, the

hydrodynamic and heat transfer behavior becomes, in effect, three dimensional. For instance, in low velocity horizontal evaporative flows, the heat transfer coefficient varies around the channel periphery as well as along the length of the tube. The designer requires circumferentially average, but axially local, coefficients in order to accurately size or rate heat exchangers.

The emphasis will be on the internal flow in circular tubes that is so common in heat exchanger equipment. It is recognized, however, that heat exchangers with other configurations, e.g., plate-and-frame, spiral, and tube­and-plate fin, are accounting for an increasing percentage of the market.

3.1 Flow Patterns The local hydrodynamic and heat transfer behavior is related to the

distribution of liquid and vapor, referred to as flow pattern or flow regime. It is helpful to briefly discuss flow patterns even though experience has been that reasonably accurate correlations for pressure drop and heat transfer coefficient can be obtained without consideration of the flow pattern.

The traditional picture of a once-through boiler is shown in Figure 33. The flow enters as subcooled liquid and exits as superheated vapor. Subcooled boiling is observed before the fluid reaches a bulk saturated condition; the flow pattern is The "bubble boundary layer" thickens because of the accumulation of uncondensed vapor, which is promoted by the decreasing condensation potential. The fluid is in a non-equilibrium state, with superheated liquid near the wall and subcooled liquid in the core, and the vapor would condense if the flow were brought to rest and mixed. At some point, the bulk enthalpy is at the saturated liquid condition (x = 0).

As the vapor volumetric fraction increases, the bubbles agglomerate and is observed. (Slug flow may also be observed in the subcooled region.)

Agglomeration of the slug flow bubbles leads to a transition regime termed where the nominally liquid film and the nominally vapor core are

in a highly agitated state. The subsequent flow pattern, has the phases more clearly separated spacewise. However, the film may contain some vapor bubbles and the core may contain liquid in the form of drops or streamers. The film thickness usually varies with time, with a distinct wave motion, and there is an interchange of liquid between the film and core.

There is a gradual depletion of the liquid due to evaporation. At some point before complete evaporation, the wall becomes nominally dry due to net entrainment of the liquid or abrupt disruption of the film. Beyond this dryout condition, prevails. A non-equilibrium condition again occurs, but in this case the vapor becomes superheated to provide the temperature difference required to evaporate the vapor. Eventually, beyond the point where the bulk enthalpy is at the saturated vapor condition (x = 1), the liquid evaporates and normal superheated vapor is obtained.

The fluid and wall temperature profiles shown in Figure 33 pertain to uniform heat flux, as might be approximated in a fired boiler with complete vaporization. The temperature difference between the wall and the fluid is inversely proportional to the heat transfer coefficient. The normal single-

426

Wall temp

x- I I Vapour \ core temp

' Oryout '

Liquid temp x - o

Fluid temp

+

t D

f

Figure 33. Flow patterns and temperature profiles in a vertical evaporator tube (Collier, [58]).

phase coefficient is observed near the entrance of the tube, perhaps with an increase right at the inlet due to flow and/or thermal development. The

427

coefficient increases rapidly as subcooled boiling is initiated, because of the intense agitation of the bubbles, but levels off in established boiling in the subcooled and low quality regions. The coefficient is usually expected to increase in annular flow because of the thinning of the liquid film. At the dryout point, the coefficient decreases rapidly as a result of the transition from a basically solid-liquid heat transfer to a solid-gas heat transfer. Droplets striking the surface elevate the heat transfer coefficient above what it would be for pure vapor.

Flow patterns in horizontal tubes have the Bame general characteristics; however, the phase distributions are asymmetric because of gravity. There is the possibility of intermittent drying and rewetting of the upper surface of the tube in slug or churn flow and dryout of the upper surface in stratified annular flow. At high mass fluxes, the gravity is less effective, and the flow patterns are reasonably close to those shown in Figure 33.

Because the single-phase coefficients are generally known quite accurately, the design or rating calculation hinges on knowledge of the nucleate boiling coefficient (or boiling curve) and convective vaporization coefficients together with incipient boiling and dryout information. The problem may be reduced in complexity by ignoring the knee of the boiling curve at incipient boiling.

3.2 Subcoled Boiling A typical set of boiling curves for forced convection subcooled boiling of

water in an annulus is shown in Figure 34. It is seen that there is a rather pronounced transition from incipient boiling to fully established boiling. the nonboiling region responds to the usual change in heat transfer coefficient with velocity and the effect of subcooling according to Eq. (412.

Established forced convection subcooled boiling is subject to the same surface and fluid variables as pool boiling. However, it was shown by Brown [12] that the surface effects become less pronounced as the levels of velocity and subcooling increase. A correlation of the form of Eq. (27) is usually sought. For example, Jens and Lottes [60] suggested the following equation for water boiling from stainless steel or nickel surfaces:

q" = 3.91 x 105eO.065p (T w - T saV"4 (46)

where p is the absolute pressure in bar, Tw - Tsat is in K and q" is in MW/m2• No correlations based on a variety of surfaces are available for other fluids. The pressure effect has not been established for other fluids; however, Eq. (27) has occasionally been used for this purpose.

In view of the rather large transition region or knee between single-phase force convection and established subcooled nucleate boiling, procedures have been developed to estimate this region. The most accurate procedure involves the establishment of nucleate boiling, essentially using Eq. (9), and the use of the following interpolation formula (Bergles and Rohsenow, [49]), as shown in Figure 35.

q" = qf{l +{�l -�)T (47)

428

3

2

10" 8 6

S 4

3

2 "' ..

,9","

��

I I 8

6 S

4 I I Annulul 1.1 I 3 p ' 60 p.'.

a V ' I III ... . 4

2 0 12 ---- - T -T • 20'f ___ s b SO -- 1 00 --- I SO

10 20 30 40 50 60 eo 100

T - T , OF w sa.t

Figure 34. Typical data for forced convection subcoiled boiling (McAdams et al., [57]).

The boiling curve knee is likely to be important for high heat flux systems cooled by subcooled nucleate boiling; however, the knee is much less important for evaporators, because that region usually covers only a small part of the tube length.

'

4 Sleel in. 10 :3 p . 22 IbF/in,2 obs

• V • 1.7 flsee 2 a V • 11.0 flsee • V • 6.8 flsee 'lb- 1 25 OF

106 8 Eq. (47)

: Eq. (9) I � a5 I • -c:r 2

5 4 3

Tw - Tsat, of

429

Figure 35. Interpolation formula for knee of boiling curve (Bergles and Rohsenow, [49]).

Hysteresis effects are present in subcooled boiling of highly wetting liquids, as illustrated by the data of Murphy and Bergles [61] shown in Figure 36. For this low velocity boiling of R-113, the temperature "overshoot" at incipient boiling is large for degassed liquid, but becomes even larger as the gas content is increased. The outgassing has a substantial effect at lower heat fluxes; however, at high heat flux the effect of gas is negligible. The enlarged boiling curve knee has erroneously led to the impression that established subcooled boiling is significantly afected by gas content.

The critical heat flux for subcooled boiling has been the subject of numerous investigations and the parametric trends have been well established. Some reasonably accurate correlations are available [63]. Thistopic will not be pursued further here because of the emphasis in this chapter on vaporization.

430

e·_ · - ·

z/D = 23

K E Y

X a . 0.06 )1 - 10-4 moles/mole

X a a 8.57 )1 10-4 moles/mole

10 T - T , OF

w sat

Figure 36. Hysteresis and dissolved gas effects on forced convection subcooled boiling (Murphy and Bergles, [61]).

8.3 Foro Convection Vaporimtion

The more popular correlations for vaporization heat transfer incorporate both low quality and higher quality behavior in additive fashion. The widely used Chen [63] correlation, for example, is

htp = hmic + hmac (48)

where the microscopic contribution is obtained from a pool boiling correlation and the macroscopic contribution involves a single-phase convective correlation. The former term has a correction factor that suppresses the nucleation at higher quality and the latter term is corrected by another factor that incorporates two-phase effects.

Much effort has been directed recently toward the development of more accurate correlations for specific orientations and specific classes of fluids. The equations of Kandlikar [64] are given for illustration:

43 1

Vertical flow

(49)

Horizontal flow

(50) where

(51)

and for Frl > 0.04, D5 = 0 and D6 = O. The convection number, boiling number, Froude number, and Prandtl number are defined in the Nomenclature. The constants D 1-D6 are optimized for high and low ranges of Co. The final factor Fn is a fluid-dependent correction factor that takes care of the vagaries of nucleate boiling. The empirical parameters for Eqs. (48) and (49) are given in Table 2. Kandlikar's correlation is more accurate than other correlations for a wide range of data for water, refrigerants, and cryogens.

TABLE 2 Constants in the Kanclikar Correlation, Eqs. (49-51)

D1 D2 D3 D4 D5 D6

Fluid

Water R-11 R114 R-12 Nitrogen Neon

1.091 -0.948

887.46 0.726 0.333 0.182

Fn

1.0 1.35 2.15 2.10 3.0 3.0

0.809 -0.891

387.53 0.587 0.096 0.203

432

A true test of any correlation, of course, involves comparison with data not used in the development of the correlation. Such a comparison was prepared by Reid et a!. [65]. As shown in Figure 37, the data are characterized by high coefficients in the entrance region and a slowly increasing coefficient in the downstream high quality region. None of the correlations noted follow this trend but several of the correlations, including Eq. 50, predict the average behavior quite well. In general, the prediction of heat transfer in the quality region is an elusive business that requires much more attention.

1 0000 o Experimental date 8.11 mm Smooth Tube

Mass Velocity - 248 to/m2 s Heal Flux - H!383 W/m2 Inlet Pressure - 346 kPa

S!n _ _ _ _ £!1a!do,£k-.!ru!!.e,,!n!:!

8000 _ _

�-.. - . . . -��!,�!i�!,!_ . _ _ _ _ _ • • _ .

�';I!!9.�!:���!!'!.t.�� • • • . .

6000

- '" 4000 0 o . - �-

. • • •••••• • •• . • • • • • • • • • • • • • • • • • • • •••••• • • • • • • • • • • • • . . • • • . • . . . . . �.::-: . • . . . • • • • t> o 0 • • -G- . - • • a. _ • . . -

2 000

o 0.0

. -- - - -- - ­ - - --- --

0.1 0.2

0.3 0.4 x

0.5 0.6

Figure 37. Evaporation of Refrigerant 113 (Reid et aI. , [65]).

3.4 Critical Heat Flux or Dryout

0.7 0.8

The problem of critical heat flux, also referred to as dryout, in the quality region has received a great deal of study. As noted by Collier [58] and Bergles [62, 66] , the technical importance of the dryout condition has led to a bewildering variety of correlations. The majority of the work has been empirical, and, with few exceptions, the correlations apply only to water. As an example, the Bowring [67] expression for water flowing vertically upward in circular tubes is

" A' - DH xol4 qcrit = C '

where

(52)

A' 2.31{�JFl

1 .0 + 0.143 F21l0.5 G

I 0.077 F300 C = 1.0 + 0.347 F4 (G11356)n

n = 2.0 - 0.00725p

433

(53)

(54)

(55)

Here D is in m, G in kglm2s, hfg in J/kg, and p in bar. The constants Fl - F 4 are presented as functions of system pressure from 1 - 200 bar.

There are many geometrical and flow effects that influence the critical heat flux, as summarized in [58] , [62] and [66]. Collier [58] provides a comprehensive list of references to studies of fluids other than water. Very little work has been done with horizontal or inclined tubes, but premature dryout due to stratification has been documented.

3.5 Transition and Film Boiling Transition and film boiling have been widely studied, and many elaborate

procedures are suggested to account for the very complex phase change behavior in these regimes. The treatment of drop of mist flow is particularly difficult due to the problems in determining the drop size spectrum, the drop­wall interactions, and the degree of thermal nonequilibrium. Further information can be found in Collier [58] . For purposes of preliminary (and conservative) design of high quality evaporators and once-through steam generators, the heat transfer coefficient can be presumed to be single-phase vapor immediately after the dryout point.

4. TWO-PHASE FLOW AND HEAT TRANSFER UNDER MICROGRA VITY CONDITIONS

4.1. Introduction Body forces are often critical to the success of systems involving boiling

heat transfer and two-phase flow. For example, gravity drives boiling in simple pools, horizontal shell-and-tube heat exchangers with shellside boiling, thermosyphon reboilers, etc. Also gravity can affect forced flow by causing stratification in in-tube condensers and evaporators (Figure 38). On earth, gravity is exploited or accommodated with design rules developed from extensive experience. These rules may not hold in the "microgravity" of outer space. This is usually considered to be aJg = 10 4•

Various space vehicle systems are shown in Figure 39. It is evident that these systems will have fluid management problems in a weightless environment. For example, in Figure 39a, if liquid is not available at the pump inlet, the engine will not restart. Also, venting will be required during prolonged storage, and the frequency of the venting will depend on the heat transfer mechanism.

434

Singl�Bubbl1-phas� flow liquid

I X = o

Plug flow

Slug f low

Figure 38. Flow patterns in a horizontal tube evaporator (Collier, [58]).

Ot ..

ATTITUM COIlTIIOl

' OUTFLOW

"EAT GAIIII

.. nOAD S""�OfITS

( a) Propellant tank system.

( c) Life support system.

EVA,.A"", COIIIKII

"UT

( b ) Space power system. 'OWE. TO

1.0. �" _Ttll MCEIVlII 1.11. SWPLY ( d ) Fuel cell system.

Figure 39. Fluid Management problems in space vehicle systems (Otto, [68]).

435

The problems encountered in microgravity environments encompass four areas:

Interface configurations, i.e., shape and location of the interface, and liquid-vapor separation,

Interface dynamics, Pool boiling mechanisms, and Forced convection evaporation and condensation phenomena.

4.2 Interface Configuration and Dynamics Surface tension forces are important in determining the interface

configuration and interface dynamics. This leads to predominance of the dimensionless groups Bond number and Weber number:

accleration forces 2 surface tension forces = Bo = P L aJa

inertia forces _ _ 2 surface tension forces - We - p LU /a

(56)

(57)

It is possible to use these groups to define hydrodynamic regions as shown in Figure 40. Here, the Froude number, Fr = (We/Bo)O.5, is introduced to define the boundary between the inertia-dominated regime and the acceleration­dominated regime.

We = pLU2/a

1 0 0 0

1 00

1 0

. 1

.0 1

I I NERTIA DOMINAT� D

U/.. = 1

ACC E L E RATION

OR GRAV I TY --

CAPILLARY

DOMINATED

DOMI NATED

100 1 000

Bo =

Figure 40. Hydrodynamic regions for gas-liquid flow (Otto, 1966).

In the capillary-dominated regime, the interface configuration can be predicted analytically for the simpler geometries. For instance, Figure 41

436

shows the weightless (Bo = 0) configura�ion fo: c�linders and spheres as a function of contact angle and volwne fraction of liqwd.

� DEG

180

160

1 40

1 20

100

80

60

o

20

o 0

o 20 40 60 80 100 PERCENT VOLUME

Figure 41. Weightless configurations in spheres (Otto, [68]).

Because the location of liquid and vapor is often undesirable from the standpoint of pumping and venting, much effort has been directed toward obtaining reliable phase separation. Referring to Figure 42, capillary control is a possibility, and one embodiment is shown. The pump inlet would be located at the bottom of the tank and the vent at the top of the tank. Screens with small mesh are another possibility to position the liquid.

The dynamic behavior of the interface as it undergoes transition from high acceleration to microgravity, or in response to altitude control accelerations or overflow disturbances, is also of importance to pumping and venting. For large acceleration disturbances, the stability limit of the interface may be exceeded.

Several types of facilities have been utilized to create a microgravity environment to confirm theoretical expectations and uncover problems with fluid management: drop towers, airplanes flown on a ballistic trajectory, and ballistic rockets. Respective test times are about 2s, 158, and 1205.

437

(a) l - g con,igu rat ion. (bl Zero-g configuration.

Figure 42. Capillary tube for fluid control in microgravity (Otto. [68]).

4.3 Pool Boiling Pool boiling has been subject of much attention because of its occurrence in

several of the systems shown in Figure 39. Earth-based correlations suggest difficul ty:

Free convection h _ g1l2 (laminar theory)

Established nucleate boiling q" _ g1l6 (Rohsenow model)

Peak nucleate or critical heat flux q;'rit _ g1l4 (Zuber model)

Our intuition does suggest that if there is no buoyancy to remove bubbles. the surface will quickly be blanketed with vapor. On the other hand, early experiments with drop towers suggested that nucleate boiling is unchanged with microgravity. as shown in Figure 43. The explanation is that bubbles are removed from the surface by acceleration of the surrounding liquid during the growth of the bubbles. This is certainly a possibility for subcooled conditions. It seems. however. that even quasi-steady equilibrium can be questioned when small transient calorimeters (and test times of only about 1s) are used.

In general. the dependence of critical heat flux on local acceleration does seem to follow the 1/4-power dependence. as shown in Figure 44. Because ck'rit � 0 as a/g � 0, the concept of a saturated nucleate boiling curve seems to lose meaning under microgravity.

With subcooling, a steady boiling curve can be established, provided, of course. that there is some way to keep the pool subcooled. This is illustrated by the more recent experiments conducted during rocket flight by Weinzierl and Straub [69]. The relatively long test period allowed setting the heat flux to the wire at three different levels. as shown in Figure 45. Although established

438

, I 1 • ," D'A . • " 0 '

• l i Z " O'A. OI,o, • , " D'A. 0. 01 ( .,, ( O.DS D 'I2"OIA .O.OI ( . , < O . O S .. OATA 0' I_UY

• DATA 0' I!UZICIIA 6 DATA 0' MSU • WlIT.AlIl!

• ," O'A . • ,, 0 0.20

• ," OIA • i. - o. " • ' · D .... • ,, - 0. 10

•

Figure 43. Boiling curves for saturated nitrogen boiling on a copper surface (transient calorimeter at standard and near-zero gravity) [2] .

boiling corresponded only to the highest heat flux, it is evident that the microgravity data are identical to the standard gravity data.

There is still a general air of uncertainty surrounding the nucleate boiling mechanism; however, it is agreed that the effective utilization of pool boiling is sustained microgravity would require a means to remove the vapor from the vicinity of the heating surface. Pool boiling experiments are being planned for a space shuttle mission. It is intended to investigate the onset of saturated and subcooled nucleate boiling and confirm model(s) for bubble growth [70].

4.4. Forced Convection Phase Change Forced convection phase change is the time-honored way to compensate for

lack of gravity, as suggested in Figure 40. Reasonable velocities and pressure drops will impose shear control, permitting operation in microgravity and the resisting of adverse accelerations. Early solutions to potential problems with evaporators noted were tapered tubes, swirl flow devices, helical coils, and even rotating boilers. Activity in this area has intensified because future spacecraft will have greater power requirements and correspondingly greater heat dissipation needs. Two-phase systems are still highly desirable because they can transfer large amounts of energy over useful distances with a small weight penalty.

439

o. X X

o. X T I

0 .7 q"crit/q"crit,e = (a/ge)O . 25

INVE5TIGA TION FACILITY FI.UlD DURATION 0 .6 I UaiakJn &Dd Siegel Drop Tower Water

I 1 sec

� I 0 Lyon et al. Magnel 1.02 55

'6 0 .5 I 6 Merte aDd Clark Drop Tower LN2 1 . 4 sec

.1 0 Sherley Drop Tower LI� 1 sec

-0" �

X Papell aDd Faber Magnet Colloid 58 0 Clodfelter Drop Tower Water 1 . 8 sec

.. :y 0 .4 I Siegel aDd Howell Drop Tower Water 1 aec C" Ethyl

Alcohol Sucro8e Solution

0 . 3

0.2

0. 1 0 0

-0. 1 0. 1 0.2 0.3 0.4 0. 5 0. 6 0. 7 0. 8 0 .9 1 . 0

a/ge

Figure 44. Dependence of critical heat flux on local acceleration [2] .

10. 0 a/ge = 10-4

c a/ge = 1 I

-

.r:r" /

.

-0"

Tw Tsat Figure 45. Boiling curves for subcooled R-113 at nonnal and microgravity [69].

440

A typical example of recent results with aircraft is shown in Figure 46. Flow regimes for in-tube condensation of water were visually observed and depicted in a simple flow regime map. The flow regime transitions are different; flows that were stratified under normal gravity immediately reverted to slug flow upon inception of near-zero gravity. The modeling of flow pattern transitions, with data obtained from drop tower and aircraft experiments, is discussed by Dukler et al. [76].

I Dispersed Flow " Annular Flow III : S lug Flow

1 0000 .0

1 00 0 . 0 :f .! 1 00 . 0 bO � c5

1 0 . 0

1 . 0

0 . 1 0 . 00 0 1

X

0 .001 0 0 .0 1 00 x

0 . 1 000

• ANNULAR

6 SLUG

X DISPERSED

1 . 0000

Figure 46. Flow regimes for condensing water at standard and low gravity [71].

5. CONCLUDING REMARKS

This chapter has presented the basic understanding of pool boiling and forced convection boiling that is required for the design of boilers and evaporators. While the emphasis has been on simple geometries, the material presented provides the background for the more complex geometries encountered in actual heat exchangers. It is shown that variations in surface and fluid conditions may result in large changes in boiling performance; accordingly, the prediction of boiling and evaporation is subject to considerable uncertainty. Various aspects of boiling heat transfer are treated in subsequent chapters.

6. NOMENCLATURE A', C' parameters in Eq. (52) a acceleration, m/s C, Cq, Csf constants, -

441

c, Cl pressure functions (see Eqs. (29) - (31» c constant pressure specific heat, J/kgK D tube diameter, m DI, D2, etc. constants in Eqs. (49) and (50) Db bubble departure diameter, m De equivalent diameter. m F radiation interchange factor, -FI, F2, etc. factors in Eqs. (52) and (53), -Ffl fluid dependent correction factor in Eqs. (49) and (50), -f frequency of bubble formation, s-l G mass flux, kg/m2s g acceleration of gravity. m/s2 ge earth gravitational acceleration, rn/s2 H height of heater, m h heat transfer coefficient, W/m2K hmae convective contribution to two-phase heat transfer

coefficient, W/m2K hmie nucleate boiling contribution to two-phase heat transfer

coefficient, W/m2K hn e heat transfer coefficient for natural convection,

W/m2K hfg latent heat of vaporization. J/kg hfg' adjusted heat of vaporization, J/kg hr radiation heat transfer coefficient, W/m2K hT resultant film boiling heat-transfer coefficient, W/m2K htp two-phase heat transfer coefficient, W Im2K k thermal conductivity, W/mK. L characteristic length - cylinder radius. plate height,

m, n n p Pg q" qbe ck'rit ch;.e ch;.in �e R R '

heater length, m constants, -

etc. , m

number of nucleation sites per unit surface area, m l pressure, N/m2 partial pressure of noncondensable gas in bubble, N/m2 heat flux, W/m2 heat flux due to bubble-induced bulk convection. W/m2 critical heat flux, W/m2 heat flux due to microlayer evaporation, W/m2 minimum heat flux for stable film boiling, W/m2 heat flux due to natural convection, W/m2 tube or sphere radius. m characteristic length. R[g(Pl - Pl)/cr]O.5, -

442

Rj r rc rmax T .1T 'rb Tsat Tw t 11, tw U v

Vfg W X x y z Bo

Co Fr

Gr Nu Pr R+ Ra* Re t+ We �

a A.c A.d,A.H 11 cp P 0'

jet dimension in Figure 25, m bubble radius, m cavity radius, m radius of largest cavity available for nucleation, m temperature, °C wall-fluid temperature difference, °C bulk fluid temperature, °C saturation temperature, °C temperature of heated surface, °C time, s characteristic bubble times (noted in Figure 11), s external flow velocity, mls internal flow velocity, m/s specific volume, m3/kg specific volume change for vaporization, m3/kg width of heater, m log R', -flowing equilibrium mass quality, -nonnal distance from heating surface, m distance along heated channel, m boiling number, Eqs. (49-50), q"/Ghfg, -Bond number, pL2a/0', -convection number, [(1 - x)/x]O.8(Pv/Pl)o.5 , -Froude number, U/.., -Froude number, Eq. (50), G2/P1gD, -Grashof number, g�L3.1T/v2, -Nusselt number, hD/k, or hIlk, -Prandtl number, clJ.lk, -dimensionless radius in Figure 10, -

Rayleigh number, defined by Eq. (37), -Reynolds number, GD/Il, ­dimensionless time in Figure 10, -

Weber number, pLU2/0', -contact angle, deg bulk modulus of expansion, K-l vapor film thickness in Figure 26, m wavelength defined by Eq. (36) jet spacing in Figure 26, m dynamic viscosity, Ns/m2 angle of conical cavity, deg density, kg/m3 surface tension, N/m Stefan-Boltzmann constant, W/m2K4

Subscripts b e i n o sat sub v

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443

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Mechanics Inst., Stanford University Press, Stanford (1972) 400. 62 A.E. Bergles, Nuclear Safety, 18 (1977) 154. 03 J.C. Chen, Industrial and Engineering Chemistry, Process Design and

Development, 5 (1966) 322. 6l S.S. Kandlikar, Heat Exchangers for Two-Phase Flow Applications,

ASME, N.Y. ( 1983) 3. m RS. Reid, M.B. Pate and A.E. Bergles, ASME Paper No. 87-HT-51 (1987). ffi A.E. Bergles, Nuclear Safety, 20 (1979) 671. 01 RW. Bowring, Atomic Energy Establishment Report AEEW-R-789 (1972). ffi E.W. Otto, Chem. Engng. Progress Symposium Series, 62, 61 (1966) 158. m A. Weinzierl and J. Straub, Proc. 7th Int. Heat Transfer Conf. , 4 ,

Hemisphere, Washington, D.C. (1982) 21. 70 NASA Pool Boiling Experiment Conceptual Design Review, NASA Lewis

Research Center (1988). 71 L. Kachnik, D. Lee, F. Best and N. Faget, ASME 87-WAJHT-12 (1987). 72 A.E. Dukler, J.A. Fabre, J.B. McQuillen and R. Vernon, Int. Symp. on

Thermal Problems in Space-Based Systems, HTD-83, ASME, New York, N.Y. (1987) 85.

NONEQUILIBRIUM PHASE CHANGE--2. Relaxation Models, General Applications, and Post-Dryout Heat Transfer

Owen C. Jones Professor of Nuclear Engineering and Engineering Physics Rensselaer Polytechnic Institute, Troy, NY 12 180-3590

Abstract

447

A generalized model for phase change in steady, flowing, quasi-one-dimensional, two-phase mixtures is developed. Based on the vapor conselVation equations for actual and equilibrium conditions, a first order, inhomogeneous differential equation having one parameter is developed to describe the behavior of the nonequilibrium potential. This equation has one parameter, the relaxation number, which itself is a local variable and represents the ratio of actual vapor genera­tion rate to the equilibrium value per unit potential. It is thus shown that nonequilibrium phase change is an initial value problem which itself is path dependent and which is also strongly depen­dent on the initial conditions.

The concepts developed for nonequilibrium phase change are then applied to the case of post­dryout heat transfer. It is shown that the use of the relaxation equation describing the behavior of the nonequilibrium potential provides an accurate method to calculate the actual rates of phase change quality and temperature for equilibrium qualities between 0. 1 3 to over 3.0.

1. INTRODUCTION

This chapter expands on the original nonequilibrium concepts first introduced in Nonequili­brium Phase Change--l. A general relaxation model for nonequilibrium phase change is developed. It is shown that phase change in real fluids occurs as a first-order, forced relaxation process. The forcing function is the equilibrium path taken by the process. That is, the path which the fluid would take if heated or cooled, or if pressurized or decompressed, if in equilibrium. The actual path taken is restricted by the rate processes for nucleation and interfacial heat and mass transfer. These processes are local variables but the overall path taken, being described by an initial value problem, is path-dependent. The actual conditions achieved during a noneqiuli­brium phase-change process, then, is very dependent on the initial conditions, as was seen in the previous chapter on nonequilibrium.

The general theory for nonequilibrium phase change is disc used in relation to both subcooled boiling and for post-dryout.

448

2.GENERAL NONEQUILIBRIUM RELAXATION THEORY

The starting point for this development will be a finite vol­ume as shown in Fig. 1 within which there may be interfaces and more than one phase. This is different from the approach usually encountered using a differential volume, since no assumptions need be made as to the size of the discreteness. The only requirement is that each phase itself can be n'eated as a con­tinuum. Averaging takes care of the rest. It will be shown that the nature of the phase-change process evolves naturally, rather than being assumed. Figure 1 . Multiphase

flow region. At the expense of some rigor, the multiphase conservation laws

are written in averaged form without detailed delivation (cf. Truesdel and Toupin 1960, Vernier and Delhaye 1 968, Ishii 197 1 , 1975, and Kocamustafaogullari 197 1 among others). For the cur­rent case of steady state heat transfer, the results are reasonably well accepted so that considerable effort is saved on this assumption. As a reference, all the one-dimensional field equations may be taken for the area-averaged local phase balance equation

( 1 )

where Cl.k. i s the volume fraction for phase-k, 'Vk is the property being conserved for phase-k at a point in space-time (x,t), volume ( I), mass (Pk) , momentum (PkVk), energy (Pkek ), or entropy (PkSk), Vk is the velocity of phase-k and Vs is the velocity of the interface, and cj)k is the flux of 'Vk . The density of phase-k is Pko the entropy is Sko and the energy is ek . Table 1 gives appropriate val­ues for 'V, cj), and s. All properties are intensive. Vector English characters are identified in bold­face whereas vector Greek characters are identified with the "hat. II The shear stress tensor is iden-

Table 1 . Identification of equivalent volume intensive quantity, its flux and production terms for conserved quantities in equation ( 1 ). Note: e = u + 1/] v2

Quantity Conserved 'V cj) s

mass P 0 0

-momentum pv i

-'t pg

energy pe

-1 ' v + q" pg ' V + qlll

449

tified with an overbar. Also V is the volume under consideration and over which the field equa­tions have been averaged as,

<f(x, t) > == � f f(x' , t)dV. V v

(2)

Also, z represents the flow direction and Aok is the outer wall area of phase-k in V. If, on the other hand, the balance equation is averaged over the interface in dV=Adz the result is

(3)

The generation of one phase at the expense of another is, in a single-component system, tradi­tionally considered as the result of a quasistatic, thermodynamic equilibrium process wherein rate processes are not considered. The traditional thermodynamic definition used to define a ther­modynamic equilibrium process in a system requires equilibrium of all potentials during the pro­cess--chemical, mechanical, thermal, etc. Thus no temperature differences are allowed between points in an equilibrium thermodynamic system. In what follows, it will be shown that this repre­sents a paradoxical point of view when contrasted with the phenomenological description of the phase-change process.

From a local viewpoint, phase change can only occur at an interface or with the production or elimination of interfaces such as occurs in the nucleation process. The intelfacial balance equa­tion, Eq. (3), when applied to a local interface yields the Kochine relation (Truesdel and Toupin, 1960),

L 1/Jknk ' (Vk vs) + nk 'rpk = 0 k=1.v

(4)

Equation (4), then, represents the interfacial conseIVation equations for ", which must be applied in any complete description of a discontinuous field having intelfacial structure: i.e., two-phase flow fields. They are necessary to provide closure by coupling the phases at their inter­facial boundaries. The surface formulation of Ishii ( 1 975) is used including sulface tension ef­fects but ignoring surface storage.

The assumption oflocal thermodynamic equilibrium shaU be adopted so that all properties are logically defined by only two state variables. This implies that molecular relaxation times are much shorter than any encountered in the specification of system time constants and resultant derivation of any phenomenological time constants. Thus, i{ = � and iv = ig , the respective saturation values.

Since the subject of this chapter is phase change, the focus will be on this quantity throughout. The mass flux going as phase change into phase-k, is due solely to the difference between the

nonnal velocities of the phase itself and the interface, directed inward to the phase region. Thus, if nk is the unit � nonnal to phase-k at an interface, the phase-change rnass flux producing phase-k is given by

(5)

Equation (4) rnay thus be used in conjunction with (5) to express the evaporative rnass flux, given by Glv, in tenns of rnass, rnornenturn and energy conservation for the interface, to achieve

I q{ 'nk +aVs 'Vs - I (ik · nk)(Vk - V�) G1v =

k=1.v k=l.v �ifg + tv, . I (v, - Vs) k=1.v

(6)

where Qk" is the heat flux in phase-k, :; is the shear sn'ess tensor, and the r-subscript on the veloc­ity is the relative velocity between the vapor and the liquid. v, = (vv - VI) Note that "rnechanical energy" tenns were obtained by scalar product of the interfacial velocity with the rnornenturn equation. Also, with (j being the surface tension,

(7)

and is the surface-divergence of the interfacial velocity field, ta being the surface unit vector in the a-direction and aa� being the rnetric tensor for the surface (Aris, 1962). Thus, the surface tension tenn in the denominator is simply the energy which shows up due to stretching of the in­terface.

The kinetic energy transfer shows up in the small effect due to relative velocities in the denorn­inator of Eq. (6) in corn paris on with the latent heat. For sorne low latent heat fluids such as hydro­carbons or fluorocarbons, this relative kinetic energy transfer effect may be as rnuch as approxi­rnately 10% of the latent heat. For water, however, this effect is generally srnall, less than 1 %.

The preceding paragraphs and Eq. (6) express the local concept of the fundamental paradox. That is, phase change generally can not occur without net interfacial heat flux which requires the phases to be at different temperatures away from the interface. In the next section, this concept will be extended to the case of a quasi-one-dimensional. heterogeneous rnixture of phases and interfaces.

If we also neglect dilatational surface tension and shear work. the evaporative rnass flux is then given by

(8)

which is the usual approxirnation of interest in thennal systerns.

45 1

Since heat transfer, 'It", can only occur due to a temperature diference, Eq. (8), then, is a state­

ment of the fundamental paradox:

• phase change can occur only due to temperature gradients, the net result of which is a non­

vanishing discontinuity in the interfacial heat flux, and cannot occur when a system is in

thermodynamic equilibrium.

Since the description of heat exchange is by means of the rate processes involved in the trans­

fer of thermal energy, we see that equilibrium thermodynamics occupies no place in this concept

except through the local definition of the fluid state. Instead, the local phase balance equations or

suitable approximations must instead be used to determine the energy flux to the interface form

each phase. Furthermore, since the net heat flux implied by the summation in (8) requires a tem­

perature difference, and the entire equation was developed from first primciples by applying con­

servation laws to an interface, two additional restJlctions apply to phase change in real systems:

• Phase change can only occur when the two phases in contact are at different mean temper­

atures.

• Phase change in the absense of nucleation can only occur when there are two phases sepa­

rated by an interface.

These principles state that continuous phase change (to discriminate between evaporation and

nucleation) can not take place in a single-phase fluid region regardless of the degree of supersatu­

()V v + -dz (/z

A + ()A dz ()z

Figure 2. Sketch of a differential stream tube.

ration. Similarly, phase change can

not take place in a uniform tempera­

ture, two-phase mixture, regardless

of the extent of the interfacial area

density. Both are required simulta­

neously. It shall be shown that the

degree of coupling between the two

phases, i .e., the degree of thermal

nonequilibrium which must exist

for phase change of a given amount to occur, is controlled by the combi­

nation of the energy transfer rates

and the interfacial area density.

Take V as an incremental-length, stream-tube volume, Adz, within which there are interfaces,

(Fig. 2). Also. take the case of mass conservation where Eq. ( 1) is written for this stream-tube as

< > () dinA 1 f dA < ()tVtz » = - (at < ()tVkz » -- - - !(Vt - vs) +tPkf O Ot-ol OZ dz A �'�o.I. dz (9)

452

where � and J;ot are the interfacial and outer boundary perimeter associated with phase-k in A . Note that � accounts for the mass transfer due to phase change while l;., k accounts for mass addi­tion at the boundaries. The latter shall be dispensed with in what follows. Note that in this case, where the focus is on a differential volume of finite area nonnal to the stream direction, the defini­tion of the space-average becomes

<f(x, t) > == ..!. f f(x' , t)dA. A A

( to)

Consider now the integration of Gj: over all interfaces in V. This integral shall be defined as

( 1 1 )

where Gk is the net mass flux into phase-k. Note that ( 1 1) defines 1k as a volume-average of a more locally defined, spatially-dependent n (x,l) through the integral over all interface Ai. Ob­viously, rk is the rate of generation ofphase-k per unit volume: the volumetric source of phase-k. B ut, as shown in Eq. (5), the mass flux at the intelface is simply the difference between the nonnal components of the interface velocity and the phase-k velocity at the interface so that

( 12)

where the negative sign accounts for the fact that if the normal phase velocity is in the same direction and exceeds the interface velocity there is net efflux of matter from the region. Combination with (8) now yields

r v(t) = f ( I qk" OOt)dA. V6.ZJg AjE V k=l,v

( 1 3)

Now since dAj = dsd�, where ds and d� are incremental interfacial arc-length and perimeter, a change of variables can be performed from oS' to zresulting in dA/ = (dsldz )d�dz, where ds/ dz is the reciprocal of the direction cosine of the interface in the plane A. Thus, since dz is independent of �, integration inA and resultant cancellation of dz yields the quasi-one-dimensional equivalents of Eqs. ( 12) and ( 13) as

1

f

dA rk(z, t) = - - �t(Vk v�.) onk-

A �i dz

and

( 14)

( 15)

453

where l;t is the entire intelfacial perimeter in A. Thus, the mass conservation equation for quasi­one-dimensional flow of phase-k can be written as

( 16)

For the purpose of use in models of two-phase flows, it is thus desirable to obtain an expres­sion for rk .. Equation ( 14) simply identifies the source tenn in tenns of the mass fluxes whereas Eq. ( 15) identifies the fact that there are energy transfers involved which must be specified and provides the key to modeling the source term for vallous geometries. In general, Eq. ( 15) verifies the fundamental paradox for the area-average case.

It is seen from examination of Eq. (15) that both the total quantity of interracial area and the net heat flux for evaporation are required to calculate the volumetric vapor generation rate from first principles. This calculation involves detennining the initial distribution of vapor in a given vol­ume as well as the evolution of this vapor with phase change.

If two phases having initially different temperatures are brought into intimate thermal contact with each other and allowed to coexist for an infinite time, heat exchange would occur between the two at a rate governed by the laws of heat transfer which would occur at interfaces. The two phases would eventually come into thermal equilibrium with each other. A basic description of the process can be seen by examining the mass conservation equation for the vapor

Equations ( 1 ) and (3) with Table 1 allow generation of all the field equations germane to this discussion. Equation ( 16) is the resultant quasi-one-dimensional mass conservation equation. The following assumptions are now made:

4. The area of the cross section is time invaIlant;

5. There is no mass addition at the boundaIies;

6. There are no covarient interactions;

7. Processes are steady state;

8. Properties including pressure are constant.

Since the mixture continuity equation shows that dG/dz = -Gd(lnA)/dz, where G is the mass flux through the sO eam-tube, the actual and equilibrium vapor continuity equations become

( 17)

rv and re are the actual and equilibrium-path volumetric rates of vapor generation. Putting Eq. ( 17) into dimensionless terms and subtracting the space derivative for actual quality from that for equilibrium quality yields

454

( 18)

A nonequilibrium potential is defined as Q = (Xe - x). As shall be shown below, (and also shown in Chapter 1) , this potential is directly related to the mean temperature difference between the phases in the diferential volume under consideration. Thus, Q may be considered as the dimensionless temperature difference between the phases.

Now, it is recognized that the actual vapor source. rv, is due to the diference between the actual and equilibrium (saturation) temperatures. Since this difference is directly related to the dif­ference between equilibrium and actual qualities. one may write ( 18) as

dQ -+ N,Q = 1 dxe

where

( 1 9)

(20)

Note that since r" is dependent on the temperature difference between the phases, and since this temperature difference is directly dependent on the quantity difference Q. the parameter N, is independent on the nonequilibrium potential Q.

The inhomogeneity is the forcing function dxeldxe equal to unity and is due to the equilibrium path the pressure takes causing the equilibrium quality to change. The relaxation number, NT. is related to the actual vaporization rate relative to the equilibrium rate, and is responsible for allowing the phase change to occur and the actual quality to approach the equilibrium value.

Equation ( 19) shows that the behavior of the quantity Q, dimensionless nonequilibrium poten­tial for phase-k. is similar to that of a first order relaxation process. H the relaxation number. N,. were to be constant, the solution to this equation. subject to the ini.tial that atXe = Xeo. Q = Qo, becomes

(2 1 )

'This result shows that the initial condition effect dies out as the difference between the equilibliurn quality and its initial value increases. as to be expected. The dimensionless phase change parameter Nr is responsible for relaxing the nonequilibrium potential towards zero. Thus, as the re­

laxation number increases indicative of an increase in the actual vapor generation rate, rv , Q would tend towards the equilibrium value of zero; if the vapor generation rate diminishes relative to the equilibrium value, the nonequilibrium potential would become quite large, with Eq. ( 1 9) indicating that dQldxe tends toward unity.

455

Eq. (19) shows several important things relative to nonequilibrium phase change:

1 . Development of nonequilibrium is an initial value problem;

2. Development of non equilibrium is path (history) dependent with history effects dying out

as the process moves away from the initial conditions.

3. The relaxation number is a local variable, dependent on the interface area and thermal field

distributions and on the resultant rate limitations for phase change.

It is important to realize that there are no assumptions in the development, and thus going into these three points beyond those inherent in the description of conservation of mass by itself.

Changes in the energy content of a mixture are governed by the first law of thermodynamics.

Reviewing the results of the introductory chapter, the mixture enthalpy is given by

(22)

where the subscript e on the quality indicates the equilibrium value under which circumstances

both phases have the same temperature, the saturation temperature, and the f- and g-subscripts indicate saturation values for the liquid and vapor enthalpies.

Considering that the actual bulk liquid and vapor temperatures differ from saturation, then there will be a difference between the actual quality, X, and the equilibrium value 'h , given by

rewriting Eq. (22) as

xCiv - ig) - (1 - x)(ij - i1) (xe - x) = 6, ' . l[g (23)

Thus, only if there is a difference between the actual temperature of the liquid and/or vapor, and

saturation temperature, can there be a difference between actual and equilibrium qualities. In fact, even if there are differences, the actual and equilibrium qualities may be identical ifthe effeCts of

vapor superheat and liquid subcooling cancel each other.

From (23), if the vapor is superheated and the liquid is at saturation, the vapor temperature is given by

(24)

On the other hand, if the vapor is at saturation and the liquid is subcooled, the liquid temperature is

(25)

456

In the fonner case, vapor superheat would result in the equilibrium quality exceeding the ac­

tual quality while in the latter case, liquid subcooling would result in the actual qUality exceeding

the equilibrium value.

In all cases, it is generally assumed that the phases have identical temperatures at an interface.

Furthermore, energy continuity is generally assumed at an interface since, without the ability to

store mass, an interrace can not store energy if surrace tension is ignored. The assumption of iden­

tical interracial phasic temperatures, then, simultaneously with nonequilibrium, means that tem­perature gradients must occur in one or both phases. This is, of course, a dynamic situation which

would result in relaxation of both phases to a mutual equilibrium condition without the addition or

rejection of heat from the mixture.

3. APPLICATION TO POST-DRY OUT HEAT TRANSFER

Studies of liquid-deficient cooling in two-phase heat transfer equipment assumed a role of

major importance due to the increasing use of equipment where conditions Beyond the Critical

Heat Flux (BCHF also telmed "post dry out") are encountered. For instance. sub-critical once­

tlrrough steam generators may require operation in this region intermittently or continuously

(Baily, 1973) . In fact up to one-third of the evaporation surface may operate in the liquid-defici­

ent region in design of Fog-Cooled reactors (Collier, 1962) has been considered which would

operate almost exclusively in this region. Regenerative cooling of liquid-propellant rockets stim­

ulated considerable interest in this area as has the increasing use of cryogens in other fields. Stu­

dies of off-normal, hypothetical accident situations recent renewed interest in this area both from

the standpoint of understanding the Loss of Coolant Accident (LOCA) sequences and the Emer­

gency Core Cooling System (ECCS) perrormance.

To place the overall topic of post-dry out into perspective, consider that early attempts to pre­

dict thermal performance were based on the general understanding that liquid in the beyond criti­

cal heat flux (BCHF) region could no longer yield a primary heat transfer mechanism through

contact with the wall. As a result, a number of correlative attempts were made to predict heat

transfer rates with various modifications of existing single-phase methods as summarized by

Groeneveld ( 1 968). All were of the form

(26)

where NUv. Rev . and Pry were the vapor Nusselt, Reynolds, and Prandtl numbers and a, b, and c were correlation coefficients. At the time these were reviewed, the best appeared to be that of

Miropolskii. ( 1963). Based on these models, Groeneveld proposed a form similar to (26) but add­

ed terms to get

(27)

457

where the subscript w represents properties evaluated at the wall temperature and where Y was due to Miropolskii ( 1963) given by

(28)

Using computer regression techniques, values were determined for the coefficients a - e.

Plummer et al. ( 1973) showed, however, that this correlation did not accurately predict the trends because it was not developed as a local correlation where equilibrium qualities much exceeded

100%.

Additive approaches were attempted for instance by Stein et al. ( 1962) and by Parker and

Grosh ( 1961 ) including droplet impingement effects, but these methods were generally un­

successful. It was later shown that droplet impingement is of negligible importance as a heat

transfer mechanism in all but minor circumstances (Iloeje et aI. , 1975) since even when droplets

can wet the wall the contact time and areas are comparatively small.

The two-step approach to dispersed for heat transfer, developed in this country mainly by

workers at the Massachusetts Institute of Technology (Laverty and Rohsenow, 1964, Forslund

and Rohsenow, 1966, 1968, Hynek et aI., 1 969) and in England by workers at Harwell (Bennett et

aI. , 1967, 1968, and Bailey et al., 1973a/b) has been shown to accurately predict the trends of ex­

isting data. While Laverty and Rohsenow ( 1964) appear to have initiated this method, it is inter

esting that the effects in non-equilibrium vapor superheat were not accounted for in this study,

even though the work of Parker and Grosh ( 1961) had earlier indicated the importance of this

effect on the overall heat transfer process.

By the late 1 960's and early 1970's, however, it was recognized that the loss of coupling be­

tween the heat input and the direct vaporization process that occurs in post-dryout flow necessi­

tate significant degrees of vapor superheat in order that the liquid evaporate by convection. (For­

slund and Rohsenow, 1966, Benett et al., 1967, Bailey et ai, 1 973) The two limits of frozen flow, (low mass velocities), and equilibrium flow, (high mass velocities), have been shown to bracket

the heat transfer data. (Bailey et aI ., 1973a/b) Unfortunately, this two-step method generally re

quires the simultaneous solution of several differential equations expressing axial quality and

temperature gradients, droplet acceleration, and droplet size gradient as well as several additional

empirical correlation equations in order to obtain the desired results. While this method can be

readily applied by itself, its incorporation into larger design calculation sys terns is cum bersome at

best.

Several more recent works recognized that the nonequilibrium component of the total energy

can be attacked separately. Plummer et al. ( 1974) assumed a linear relationship between the ac­

tual-vs-dryout quality difference expressed as

(29)

458

and found that K appeared to vary directly as the logarithm of the group G( dlpvrJ )112 but did not correlate the three fluids tested on a single line. As will be shown later in this chapter, Eq. (29) is not accurate in that it implies a constant derivative d(xe - x)ldxe = (1 - K) for a given set of condi­tions, a linear, first-order, nonrelaxation process. Groeneveld and Delonne ( 1975), on the other hand, assumed that the quantity (x) - x) was given by

(30)

where

Xl = min(xe, 1 ). (3 1 )

They used computer regression techniques to eliminate insignificant vaIiables from (30) and ob­tain

(Xl - X) = e tamJI

where

or { 0 1/J < 0 1/J - for

11:/2 1/J > 11:/2

and the homogeneous flow Reynolds number is defined as

Gd Xl Rehorn ==

--­I-lv ahorn

(32)

(33)

(34)

(35)

The void fraction anom was that calculated for the actual quality neglecting slip. The advantage ofEq. (33) was that it correctly predicted the asymptotic trends of quality and non-equilibrium as Xe is increased above 100%. Also the inverse mass velocity effect on the nonequilibrium was in­cluded. The disadvantage was similar to that for any empirical correlation in that care must be exercised in its utilization outside of its developed range of conditions. Unlike phenomenological models, its utility, for instance, in transient conditions is questionable.

Considerable effort was undertaken by Chen and his coworkers (Chen et aI . , 1979; Webb, et al., 1982) in the development of correlations for the local heat transfer coefficient based on the momentum analogy. A parallel effort in development of methods for measurement of actual va-

459

por temperatures in post-dryout (Nijhawan et al . • 1979. 1980; Evans et aI. , 1983) has resulted in definitive new data which is invaluable in development of theoretical models.

Saha et al. ( 1977) developed two separate cOlTelations for the actual vapor generation rate in stearn-water mixtures which were similar to that of Jones and Zuber ( 1977) but did not recognize the implications in the relaxation approach. The first, termed the K I-correlation, was expressed as

(36)

where Pr is the reduced pressure (relative to critical). In the second correlation, i.e., their oDo-correlation. the average droplet diameter at the dryout location was given by the relationship ·2 j-j = 1 .47 Qs/g,DO (37)

where �Pfg is the density difference between liquid and vapor at saturation. and where the sub­script DO means "at dryout." This equation. with the assumption of no droplet breakup. was then used in their expression for the vapor source

rv = 6h(Tv - Ts)( 1 - a) lJ�ifg (38)

to calculate the rate of vapor generation, and resultant degree of thermal nonequilibrium. They used the drift-flux model to determine the void fraction assuming no distlibution effects. Combi­nation with the Heineman correlation fOT steam heat transfer allowed temperatures on the wall to be determined with reasonable accuracy.

Building on the two-step model. Yoder ( 1983) utilized differential equations for the axial gra­dients of liquid velocity, droplet diameter, actual quality, and vapor temperatures along with nu­merous best estimate constitutive relationships and empirical relations required to close the mod­el. Results were of the correct magnitude but the detailed behavior indicated that substantial effort relative to the details of the processes need to be undertaken. Differences of over 50C between the predicted and experimentally measured wall temperatures were noted and the trends were not well predicted when the axial wall temperatures were not monotonically changing.

More recently, Varone and Rohsenow ( 1986) examined the details of Yoder's model and con­cluded that the presence of droplets altered the tW bulent structure of the flow much in the same was as particles afect the ttrbulence in fluid-particle flows. The result, they hypothesized, was to alter the wall heat transfer from that which occurs simply due to vapor heat transfer, a key assump­tion in many of the previous methods. They detelmined the ration of the Nusselt number with droplets to that without droplets in Yoder's model by comparison wiht the data and found that this

460

ratio correlated with the viscosity ratio between the bulk vapor and that at the wall. By using these correction factors, excellent predictions of the data were obtained.

In summary, existing, physically-based models are qualitatively accurate if various factors are evaluated empirically bur can be exceedingly complex for use in large computer systems. Em­pirical correlations have thus found prominence in design verification applications but canot be expected to perform outside their domain of applicability. A different approach appeared to be needed: one which would be simpler to apply and yet maintain a working relationship with sound physical principles.

The balance of this chapter will present an alternative approach to the problem, and to show that this approach can work well when applied to systems having large degrees of nonequili­brium. It is not intended herein to develop or even demonstrate the overall capability of a devel­oped correlation. Instead, it is proposed to show that nonequilibrium, two-phase, dispersed-flow beyond the critical heat flux can be physically described as a first-o der, inhomogeneous relax­ation process which can be applied to accurately describe the relaxation of the thermal non-equi­librium at equilibrium qualities over 3.0.

In the curent case, equation ( 15) shall be applied to a dispersed liquid, the droplets assumed to

be spherical in shape, and at saturation temperature. The point of view similar to the two-step process (Bennett et all, 1 967; Forslund and Rohsenow, 1968) will be taken. Since the concentra­tion of droplets in dispersed flow has been shown to be uniformly distIibuted in space (Cumo, 1973), the one-dimensional approximation appears valid. The implication is that the two dimen­sionalities in the vapor heat transfer can be properl y accounted for by standard heat transfer corre­lations but neglecting the laminarization effects of the droplets on the vapor velocity profile (Gill et aI, 1963) . Momentum transfer transients are ignored thereby ignoring droplet acceleration and flashing due to pressure loss.

Since the liquid is assumed at saturation, the only heat flux is through the vapor phase to the relative velocity. In terms of a droplet heat transfer coefficient, hEl' Eq. (15) yields

r y = < ho(Ty - Ts) >i 1 f dA

!l.itg A �i dz

where <�>i indicates an interface-averaged value and where

1.f dA = 6( 1 - a) A �i dz 6

(39)

(40)

is the interfacial area density assuming uniformly-shaped spherical droplets. Here, 5 is the drop­let diameter. The volumetric vaporization rate thus becomes

r y = 6ho!l.Tys( l - a) 6!l.ifg (41)

46 1

where IlTys is the superheat.

The application of (41) will occur through the calculation of variations in actual quality under

post-dryout conditions. Considering Eq. (22), since the liquid is at saturation, the result for (41 ) is

r _ 6h6(1 - a) ( _ ) y - Xe X

(42)

where the specific heat of the vapor must be averaged over the superheat. Recalling (24), this

equation confins the phenomenological notion that vapor cannot be generated under equilibri­um conditions and that the relaxation number is truly independent of the nonequilibrium poten­

tial, Q.

Considering again Eq. ( l ), for the previous assumptions the actual and equilibrium energy

equations are

d {(I )

' ' \ qw" � - - x el + xey = --dz GA and

where

, . e = , 2

(43)

(44)

(45)

For two-phase flows of interest, velocities are generally less than 100 mls so kinetic energies

al"e less than 5 kJ/kg compared with latent heats for the vapor of over l OOKJ/Kg and similar for

liquid enthalpies near saturation. Similarly, velocities for the cryogenic data curently considered yield negligible kinetic effects. Thus, kinetic energies shall be neglected henceforth. Comparison

of ( 17) and (44) for equilibrium gives

r _ qw" � e - Allifg (46)

which assumes that all the energy goes directly tlu'ough the liquid from the wall to the interface

without resistance. Comparison with (15) shows this must be a limiting case for vanishing ther­

mal resistance. From a non-equilibrium viewpoint we have shown by (42) that ry -7 0 as X -7 Xe•

This, however, indicates that dx/dz -7 0 also as x -7 XI." strongly supporting the concept of a relax­

ation type process.

462

Finally, considering Eqs. (40), (42) and (46), the expression for the superheat relaxation num­ber is easily seen to be

N _ �(M)2/3 kytlfl.ifg ( 1 - )1/3 sr - a . 2 6 pv'/w x (47)

Note that if the local value ofthe relaxation number can be determined, then the local volumetric vapor source tenn can be calculated from

(48)

At this point, no real departure has been made from previously defined concepts utilized in the two-step model such as proposed by Bennett et al. ( 1967, 1968) or Forslund and Rohsenow ( 1 966, 1 967). The arangement of the variables has been acomplished in a fOlm more easil y interpreted. Of course, the difficulty now comes on calculating the superheat relaxation number Nsr. Standard procedures as previously defined would have us follow the droplets accounting for the drag, relative velocity, droplet splitting when the accelerative retardation causes the Weber num­ber to exceed the critical value, etc. On the other hand. it would be much simpler to use local v81iabIes and to assume that droplets would remain in mechanical equilibrium with the vapor specified completely through the fluid properties and the drag coefficient. In addition, the effect of increased droplet residence tending to increase the effective Nusselt number would have to be accounted for separately. Bennettet al. ( 1967, 1968) and subsequent workers, have correlated this in their "ventilation factor" incorporated into their droplet evaporation equation. It might be ex­pected that neglecting this effect, the values of NSf calculated from (47) might be lower that the actual values, and that an additional mass velocity effect would be omitted.

The extensive data of Forslund and Rohsenow ( 1966) and Bennett et al. ( 1967) have been chosen since equilibrium qualities range fonn a low of 0. 13, to a maximum of over 3.0. Unlike Groeneveld and Delorme ( 1 975) i > ig is represented by equilibrium qualities greater than unity since the thennal disequilibrium and the relaxation process are Duly driven by the difference between Xe and (i - �)/Il.itv and not between x and 1 .0. Also, these two sets of data includes a number of different heat fluxes, three tube sizes, a wide range of mass velocities, two fluids, and two widely separated reduced pressures.

Vapor heat transfer. In obtaining values for x vs Xe from actual data in the absence of direct measurements, an assumption must be made as to the actual mode of vapor heat transfer. Laverty and Rohsenow ( 1 964) used the Dittus-Boelter correlation based on the vapor velocity and satu­

rated vapor properties. Kearsey (1965) used the Dittus-Boelter equation based on the steam mass velocity and the bulk vapor temperature, Quinn (1965) used the Seider-Tate equation based on bulk steam phase except for the thennal conductivity of the vapor which was taken to be at the

wall. In addition, he also used the steam velocity Gx/a,p., and the "steam diameter," ad where the

463

void fraction was that due to Polomik (1966) with the slip ratio taken to be the density ratio to the

1/3 power. Bennett ( 1968) et al. chose to use a modification of Heineman's ( 1 960) equation for steam based on film properties and the actual steam velocity. Hynek. Rohsenow, and Bergles

( 1971 ) used the value determined by Forslund and Rohsenow (1966) in their experiments with bulk. vapor properties and actual steam velocity which do not vary significantly from those ob­tained from the Dittus-Boelter relation on the same basis. Groeneveld ( 1 972) on the other had used a modified McAdams correlation similar to Quinn ( 1965) but all properties were taken to be at the bulk vapor temperature.

Different correlations will yield somewhat different results and in the absence of actual mea­surements of the local quality, arguments of individual differences appear moot. It does apear,

however. that Quinn's ( 1 965) argument regarding the behavior of the specific heat for near-criti­

cal temperatures is valid so that in the present case, a bulk vapor property correlation was utilized. Since it is desired to eventually achieve some sort of uniformity between different fluid systems,

the bulk Dittus-Boelter equation was chosen where it is assumed that heat transfer between the

wall and the vapor occurs as if liquid droplets did not exist. This con elation has been shown valid

for a wide range of single-phase fluids and geomeoies. Thus. the vapor velocity is given as Vv =

GxlOfJv and the correlation becomes'

( Gd )0.8 Nuv = 0.023 -� PrS·4 I'-v a

(49)

This choice has the additional advantage that the reverse procedure of calculating the wall

temperature from a predicted actual quality will not depend strongly on film properties so that

errors should be similar to the errors encountered in predicting the vapor temperature itself. The

vapor fraction was related to the quality in the standard manner through the density ratio and slip

ration which is evaluated independently.

The procedure of obtaining a vapor temperature and thus the quality based on a measured wall

temperature is to some extent an iterative one. The vapor temperature can be assumed which al­

lows the quality x to be calculated which then allows the heat transfer coefficient to be calculated

from

(50)

and the process is repeated until compatibility is achieved.

Droplet size and slip. A similar aray of choices are available for calculation of droplet size

and slip, as for vapor heat transfer. A major simplifying assumption is to allow the drops to always

achieve mechanical equilibrium with the vapor. By balancing drag and buoyancy forces one

achieves the result that

464

d = 3CDI2v� ag�

(5 1 )

where the mixture density w as used in calculating the buoyancy i n a manner similar t o that of Zuber and Hench ( 1962) for bubbles. It is seen that two items must be specified: CD and v,. Wallis ( 1 974) has summarized the. relationship between drop size and terminal velocity and indicated that for most real systems containing some impullties, drops behave similar to solid spheres. Ishii ( 1968) developed an approximate equation representing the drag coefficient up to the velocity-li­mited region given by

24 CD =-- ( 1 + 0. 1 R�. cc )O.7S

Re6,cc (52)

which has been adopted here. A simple formulation which eliminates the transcendental nature of the drag-Reynolds number relationships was given by Jones ( 1984). The terminal Reynolds number for the droplets is taken as a function of the Archimedes number as:

R..,.� = 1 + 0.0487 JAr I A

{ (4

)O.4S2}-

1 .74 ,fA;

where

Ar < 3.227 x 10 5

Ar � 3.227 x 105 Re � 2 x 105

(53)

(54)

The Reynolds number is based on the droplet diameter and the terminal velocity, taken identi­cally as the slip velocity. This fonnulation is a direct one which allows the terminal Reynolds number to be immediately calculated from a knowledge of the droplet size and the thermodynam­ic state of the mixture.

In order now to tie down a definitive relationship between the velocity and the size we must make some assumption regarding an initial condition. Both Groeneveld ( 1972) and Hynek and Rohsenow ( 1 969) used a critical Weber Number criterion, these being taken as 6.5 and 7.5 respec­

tively. Forslund and Rohsenow ( 1968) showed that a value of 7.5 represented their visual mea­surements quite well, and this value is choosen herein. Since droplet acceleration is neglected, the droplets can only change size due to evaporation. The number density, n. is thus assumed to re­main constant. The initial condition chosen is that the droplet volume at the CHF location will be

just equal to half of that given by Wee = 75 so that

(55)

and thereafter

{ 6(I _ a) } 1/3

{ I - a } 1/3

o = Oc -- = Oc --7lnc I - ac

465

(56)

In virtually all cases, the void fraction is very close to unity. Thus, once compatibility is achieved by iteration at the critical quality condition. the void fraction is a lagged quantity. That is, the void fraction for the previous step is used to calculate the valiabies for the current step. Therefore, given the void fraction, the droplet diameter is calculated from (56) and the Archimedes number is then calculated from (54). Next the terminal Reynolds number is computed fol­lowed by the drag coefficient from (52) and the relative velocity from (5 1 ). Knowing the relative velocity the current value of the void fraction is then determined from basic quantities. That is, from the definition of the relative velocity, the slip ratio can be determined as

S = 1 + ( l -a)etV,

( l - x)G (57)

which then allows the void fraction to be determined from the void-quality-slip relationship.

Droplet heat transfer.

Since no definitive work on droplet evaporation in a mist cunently exists, the simple relation­ship given by Lee and Ryley ( 1 968) was used herein where

1/2 1/3 NU6 = 2 + 0.74 Red,.,., Prv (58)

Knowing the droplet heat transfer and the other parameters required by Eq. (47), the local value of the superheat relaxation number can be calculated.

Calculation of superheat relaxation number from experiment. The actual quality at any location where the heat flux and wall temperature are known may be calculated using the pre­viously described procedure. Once the actual qualities have been calculated, the superheat relax­ation numbers may be calculated from (2 1). That is, if the spatial step along the test section is small enough, the relaxation number from one location (initial conditions) to another (current location) can be assumed constant and given through solution for Nj in Eq. (2 1 ) written as

(59)

The values obtained are, of course, the center-weighted values for each cell. Each ceU-j has inlet node j-l and outlet node j. Experimental data were examined which had a profusion of ther­mocouples so that the steam temperature at any thelmocouple location could be obtained in con-

466

junction with the calculation of local nonequilibrium. That is, the thermocouple measurement provided the outside wall temperature for the test section. From the power measurement, knowledge of the geometry of the test section, and calibration for heat loss, the local heat flux at the thermocouple location could be obtained. Knowing the material type and thickness, the inside wall temperature and heat flux could be determined. Then, using Eq. (49) the superheated steam temperature was calculated, yielding the local nonequilibrium. Q. Since the value for both Q and eqilibrium quality for the current thermocouple location node and the previous are thus deter­mined, the value of the average relaxation number between the two thermocouple locations can be determined. 'This can then be compared with the values calculated from Eq. (47). The starting point for the computation was the critical heat flux point. determined by an intersection of the best fit straight lines through thermocouple data immediately below and above the CHF locatiop., but ingnoring the slight dip usually obtained in the temperatures immediately preceding the CHF point as well as any hysteresis noted in increasing and decreasing power data.

In applying the method in the suggested manner two things were fround. First. upon examina­tion of the extensive data of Forslund and Rohsenow ( 1966), a mass velocity effect not accounted for in (47) was encountered. The second thing found was that there were differences between the nitrogen data and the water data. These two findings will be discussed as the data are examined in detail below.

(a) Data of Forslund and Rohsenow (1966)

It can be argued that the mass velocity effect found was due to liquid holdup yielding effective Nusslet numbers higher than that given by (58). similar to the "ventilation" effect postulated by Bennet et al. ( 1967). A ratio of inertial forces tending to carry the drops out of the duct, to the gravitational forces tending to increase liquid holdup would seem to be appropriate to compen­sate for these effects. The Boussinesque number Bo represents such a ratio

(60)

so that the experimental values of the superheat relaxation number. NSf, were all divided by this parameter for correlation purposes.

The results of comparisons for the MIT 1 .7 4-bar nitrogen data are shown in detail Fig. 3 which include 67 1 data points covering the fonowing range of conditions:

95 :5 G :5 260 kgj� 5 .79 :5 d :5 1 1 .73 mm 16 :5 q" W :5 94 kWjm

0. 13 :5 Xc :5 3.2 0.6 :5 q" wi q" W,l/vg :5 1 .5

o

� z

o

� z

10-1

G-95kg/m2 s -'--- diO. ' 5.79 mm } -- dio ' 8. 2 mm REGION <t) .-------- dio : 1 1 .73mm

I

S == �(mr)2/3 NU6(1 - a)1/3 2 6 Cpvlw x

o G-95 kg/m2 5 o G - 175 kg/m2 s 6 G- 260 kglln2 s

Figure 3. Correlation of the data of Forslund and Rohsenow ( 1966) for nitrogen. p=1.74 bar; G-95 kglm2s.

467

10

The range in the ratio oflocal-to-average heat flux was due entirely to variations in the electrical resistivity of the test section rather than intentionally obtained. Data for all mass fluxes are shown together in the lower right of the figure.

Several things can be seen from these data and comparisons. First, the data seem quite scat­tered. This is, of course, due to the differential natw'e of the results obtained from the experiment. Note that the procedure previously outlined is equivalent to solving the diferential relaxation equation, Eq. ( 19), for the superheat relaxation number if the step-size is sufficiently small. That is, the equivalent calculation is

468

NOMENCLATURE G-kg/m2 s

dio. mm 95 1 75 260 5.79 0 � • 8.20 0 .. • 1 1 .73 b. .. ..

Figure 4. COlTelation of the low quality, near­

critical quality region for the nitrogen data of

Forslund and Rohsenow ( 1966). p=1.74 bar.

where the nonsubsclipted value of Q is the average of the lagged value and the CUlTent value.

Further, it can be noted that the data

progress from upper light to lower left in the figures as the distance from the CHF

location increases downstream. As the ac­

tual quality increases, the flows become more dilute and the actual relaxation num­

bers reduce more rapidly than those calcu­lated. This may be due to the "ventilla­tion" effect mentioned previously, and the

lower relative velocity and thus lower heat

transfer rates due to the neglect of droplet

acceleration effects. Finally, it should be

noted that since NSf is a differential param­

eter, effects of uncertaintly in its determination are integrated or smoothed during application to

determine the local noneqiulibrium.

A reasonable correlation of the data is given which separates the data into three regions. In the first two, droplet flow regimes, the correlation obtained is given by

Nsr { 170S2 for S :5 0.595 Bo

= O.855SI/8 for S > 0.0595

where

3 (rm)2/3 kvd6.iJg 1/3 S == - - NU6 ( l - a) 2 6 Cpvflw" x

(62)

(63)

In the third, low quality region shown in Fig. 4, it is thought that a different flow regime exists,

perhaps rivulet flow or a type of transition boiling region between annular tlow and mist flow, or maybe even inverse annular flow such as considered by Bromley ( 1950), Doughall and Rohse­

now ( 1963) or Hsu and Westwater ( 1959), among others. A simple cOlTelation in this regime,

which really contains very few points and is confined to a very narow region immediately adja­cent to the critical heat flux location is given by

-2!. = 2.l5x lO 2 - ..! exp[- 40(0.24 - xc)]S3 N { G ( ) I/3} 7/2 Bo dey (}vg

(64)

where Xc is the critical quality at CHF. No attempt to optimize these equations was made since this was only an interim step on the overall process. Note that value of Nsr/Bo to be used is the maximum of those given by Eqs. (62) to (64).

The ability of the present method to ad­equately describe both the magnitude and the trends of data for a single fluid is verified in Figs. 5a-d. In Figs. 5a-c, data for different mass fluxes are shown, each mass flux having a different clitical quali­ty. It is seen that the effect of initial condi­tions is quite pronounced, and persists for a considerable distance in equilibrium quality, especially at the lowest value of mass flux. It is only for the higher mass fluxes that the history effects diminish fairly rapidly. At constant clitical quality, the effects of different mass flux are shown in Fig. 5d wherein the inverse mass flux effect is clear. In addition, the diame­ter effect is also seen in Fig. 5d wherein the middle set of data having a significantly larger diameter and thus larger relax­ation parameter tends toward equilibrium faster .than the other two sets of data.

All data of Forslund and Rohsenow ( 1 966) were calculated and compared as shown in Fig. 6. All data are predicted with excellent accuracy except, perhaps for the higher quality regions for the lowest mass flux wherein the actual quality is slightly higher than the actual value.

(b) Extension to the Data of Bennett et al. (1967)

It would be surprising if the "cook book" methods used herein for the various calculations would include all the effects between thermodynamic systems. Indeed they don't. It was found that the two fluids

469

1 .0

O.B )-.. � <1 06 a oJ <I ;= 04 �

-- CALCULATED

0.4 D B 1 .2 1 .6 2.0 2.4 2.B 3.2

1 .0

O.B

� 0.6 � 0 oJ � 04 I-u

CALCULATED

0.5 1.0 1.5 2.0 2.5 3.0

1 .0

O.B

0.6 G- 260 kg/m2 s

04

CALCULATED

0.5 1 .0 1 5 2.0 2. 5 3.0

O B

0.6

0 4

G dio. 95 kg/m2 s I I 7mm

1 75 kg/m2 s S.2mm 256 kg/m2 s S.2 mm

-- CALCULATED

0.5 1 .0 1 5 2.0 2 5 3.0 EQUI L I BRIUM QUALI T Y

Figure 5 . Comparison of rutrogen correlation with data of Forslund and Rohsenow; p=1.77.

470

08

1 :; 0.6 o c; u & tQ 0.4 ::I o

c; a 0 0.2 -<

0.2 0.4 0.& 0.8 1.0 Actual Quality from Data

as Q8 10

Figure 6. Comparison of calculated and actual quality for the data of Forslund and Rohsenow using the nitrogen correlation given in Eq. (62) to (64).

10

Nsr

B 1 .0 o

0. 1

Nsr = 1 .23 (S//Pr) 311 Bo

0.01 0. 1 1 .0 10 100

s == �(M)2/3 NUo(l -a)1/3 2 6 Cpvl/w x

Figure 7_ Superheat relaxation numbers for both nitrogen data (1 .74 bar) and water data (70-bar) .

behaved differently when compared on the basis of the previous development. Fur­thennore, when a correspon­ding-states correction was tried, it was found that the re­duced pressure, introduced empirically, was able to ac­commodate these differences. Also, upon further examina­tion, a slightly higher power on S in the middle region, 3/8 rather than 1/8, was required to bring both sets of data into reasonable compromise.

The results are shown in Fig. 7 for the 1 .74-bar nitro­gen data of Forslund and Roh­senow (1966) and the 70-bar

water data of Bennett et a1.

(1967). The resulting correla­tion based on both the water and nitrogen data is:

Nsr =

)

'

Bo

( S )'" 1 .23 jp; where

and where

P = .! r -Pc

for

for

47 1

S j- � 0.22 Pr

(65)

S j- > 0.22 Pr

(66)

(67)

is the reduced pressure, based on the critical pressure, Pc . Futhennore, as before, the value of relaxation parameter to be used is the maximum value of those given by Eqs . (65) and that given by

� = 2. 15x lO 2 - � exp[- 40(0 .24 xc)] N { G ( ) 1/3}7/2 ( S )3 Bo �v e�

Again, with respect to Fig. 7, two things are noted:

(68)

1 . The superheat relaxation number, Nsr, is a differential parameter, obtained by differentiating experimental data. As such, it is expected to have considerable scatter, and it does.

2. The calculation of the actual quality and the nonequilibrium potential using the superheat relaxation number is a forward integration process from the critical heat flux location wherein smoothing of the scatter in the relaxation number occurs .

Thus, the test of the method � in the "goodness" or lack thereof for the Nsr correlation, but rather in the accuracy with which calculations are accomplished for actual nonequilibrium quality and ultimately wall temperature.

Typical calculations of actual quality compared with the data are shown in Fig. 8 for each of the three mass velocities tested by Forslund and Rohsenow, ( 1966). Complete comparisons of the calculated actual quality with that obtained from the 67 1 experimental data points are shown in­Fig. 9. It is noted that the comparisons are slightly worse than those shown previously based on a correlation of these data alone. Furthermore. there is still a slight mass flux effect which is not accounted for in the correlation.

Compa.Iisons of calculated actual quality with those derived from the 1084 data points at 70-bar in water by Bennett et aI. ( 1 967) are shown in Fig. 10. Note that the lower the mass flux,

472

.� O.B -; 0 °

6

� 0.4

� 0. 2

q 0.8 (;

00.6

(; B 0.4

< 0.2

0 0.8

� 0 °.

6

(; E 0.4 o -< 0.2

1 .0

'c SYMBOl. 289 0.229 • 260 0.288 259 0.353 206 0.393 •

RUN 268 293

- 175 kg/m2 , 'c SYMBOL CACULATION 0.134 • 0.273 • 0 343 •

2 .0 3.0

G-260 ka/m2,

;.�78 SYM

:OL CALCULATION

0.254 • 258 0.308 •

Equilibrium Quality

the higher the quality. Each overall quality range is separated from the adjacent range in this figure by altemating the filled circules with the open circular symbols. It should be noted that while these comparisons appear quite good, the higher mass velocities en­countered in these expeliments resulted in much less deviation between actual and equi­librium quality than for the nitrogen data. In addition, the higher heat fluxes resulted in significant temperature limitations in post­dryout condition to avoid test section failure.

A tabulation of the standard deviations in these comparisons is given in Table 2. The overall standard deviation for the MIT data ( 1966) is 0.035, and for the Harwell data ( 1 967) is 0.020. The overall rms deviation of 0.027 is quite remarkable in view of the ob­vious coarseness exhibited in Fig. 7, and the relatively wide range of parameters covered as shown in the table. This is indicative of the smoothing action of the integration process and confums previously stated expectations.

Once the nonequilibrium potential has been determined providing the local bulk va­por temperature, it is straightforward to de-

Figure 8. Selected comparisons of nonequili- termine the wall temperature using Eq. (49). brium qualities calculated from Eq. (65)-(68) Typical results are shown for the develop-for the 1 .74-bar nitrogen data of Forslund and mental data base in Figs. l l a and l lb, and Rohsenow ( 1966). for the data of Janssen and Kervinen in

Fig. l lc. In these cases, the comparisons are typical of all for the given data base: neither the best nor the worst.

The most severe test is probably a comparison of the data of Swenson et a1. ( 1967), with the predictions of ( 19) and (65). These data, taken at a pressure of 207 bar, represent an extrapolation by a factor of three over the range included in Fig. 7 and the development ofEq. (65) . It is found that the rms deviation in the quality was 0.041 and in the wall temperature was 13 . 1C, somewhat worse than that previously encountered. The predicted wall temperatures are shown in Fig. 12 .

The comparisons of Fig. 12 represent a very severe test case because of the very low critical quality and resultant low void fractions encountered even at moderate qualities. Consequently,

true droplet flow probably does not occur

Conditions 95 ::5 G ::5 5200 kg/nil s

5.79 " d " 12.37 mm

16 ::5 q" w ::5 1 836

0. 1 3 ::5 Xe ::5 3.2

0.6 :5 q" w/ q" w.avg :5 1 .5

0. 1 2 1 ::5 L s 0.553m

0.05 S Ze S 0.553m

0.05 S Pr S 0.3 1

Set

a1. B B et aL B et a1. B et al. B et a1. B et a1.

Range Data Runs kg/m2S

650 16 10 12

17 1970 16 2550 16 3850 14 5200 6

473

Mean Square Data Deviation

Points

1.!L.

1 24 0.0229 171 0.0254 1 3 1 0.02 1 1 140 0.0140 1 57 0.0084 6 1 0.0077

Table 2. Summary of experimental conditions and nos deviations in the calculation of actual quality when compared with 1 .74-bar nitrogen data of Forslund and Rohsenow ( 1966. F&R) and 70-bar water data of Bennett et al. ( 1 967. B et al.).

0.2 0" OA OA

Actual Quality from Data Figure 9. Comparison of calculated and actual quality for the data of Forslund and Rohsenow using the nitrogen correlation given in Eq. (65)-(68).

474

1.0

0.8

� as 0.6 "3 o � u £ � ::I o � 0.4 .a �

0.2

SYM BOL MASS VELOCITY

kQ Iml , 390 � 6� U'I 1000 �

ILI Z 1 350 Q "' 2000j : 2550 ffi � 3800 ffi � 5200

0.2 0.4 0.6 0.8 Actual Quality from Data

1 .0

Figure 10. Complete compalison of calculated actual quality with those obtained from the 70-bar water data of Bennett et al. (1967).

except at high qualities. It is not surprising that the low quality comparisons are poor­er than those obtained at higher qualities. The one high heat flux, high temperature run showing rather poor comparisons stands apalt from the rest of the data. Cal­culations of thermal conductivity and spe­cific heat near the critical point proved troublesome and is believed to be largely responsible for the discrepancies noted. It is emphasized that the rod-flow film boil­ing regime must be addressed in future work within the framework of the relax­ation mode.

The effect of an uncertainty in the ac­tual quality of 2% represents -8-1 2°C in the wall temperature if the vapor tempera­tures are in the range of 60-1200°C. It should be noted that the rapidly diminish­ing effects of critical quality along with the inverse mass velocity effect appear to be correctly accounted for in the proce­dure. The reader is reminded that no at­tempt at optimization of the method has been attempted at this point since it was desired only to demonstrate the adequacy of the correct physics, when combined with a minimum amount of empiricism, to correctly predict the magnitude and trends of the non-equiliblium.

Comparison of the methods ofvalious investigators in calculation of post-dryout heat transfer was given by Richlen et al. ( 1976). It is noted here, however, that, to the author's knowledge no one, including the originators of the data themselves, has demonstrated the ability to perform even circular calculations within the accuracies herein achieved.

It is important to recognize that the analysis described herein conclusively shows that post­dryout heat transfer behaves as an initial value problem, as do all relaxation phenomena. Thus, past history is important in determining local changes since it is the history that determines the local degree of non-equilibrium, Q. It is for this reason that single data points providing only local values of wall temperatures, equilibrium quality, mass velocity, heat flux, and pressure are useless for the purpose of providing comparative data for the procedures outlined herein. Rather, complete data describing the entire experiment, and definitively specifying the CHF-Iocation

G � � .. ::I � .. 8-E � E-

�

G � CI)

475

900 60 800

BOO 700

700

600 600 300

500 200 SOO ° 0 •

400 100 · 0 0 0

0

400

300 00000000.00 0

300 0

0a«ldloO 0 0 200 -100

1 00 -200 200 -O.S 0.0 O.S 1 .0 1 .5 2.0 0.0 O.S 1.0 1 .S 2.0 2.S -0.4 0.0 0.4 0.8 1 .2 1 .6 2.0

Equilibrium Quality-x (a) Bennett et al. (b) Forslund and Rohsenow (c) Janssen and KeTvinen

Figure 1 1 . Comparison with post-dryout data with Eq.(65)-(68) .

G - 1 356 kg /m2 S

o q:. 561 kw/m2

D �. 303 kw / m2

o o o

G= 950 kg/m2s o q:, a302 kW/m2

D q:, = 495 kWI m2

9 4 50 tU o

Ii3 0-S �

400 D � D DGJ D D a: D D D

a oo o o o ax

-0.2 0 0.2 0.4 0.6 0.8 1 .0 -02 00 0.2 0.4 0.6 08 1.0 L2

Local Equilibrium Quality Figure 1 2. Comparison of calculated wall temperatures with those measw'ed by Swensen and

Carver ( 1 967). Water, p=207 bar. along with inlet enthalpy, heat flux, and wall temperature profiles are required to adequately describe the conditions observed.

Finally, it is stressed that the success of the methods described herein is due to a reasonable inclusion of correct physics coupled with a separation of the critical heat flux from the prediction of the post-dryout conditions. That is, the critical heat flux is specified only as the initial condi­tons wherein the actual and equilibrium qualities were taken as identical. The reader is pointedly reminded that to do otherwise, such as many previous researchers, is to attempt to con-elate the

critical heat flux condition simultaneously with the post-dryout condition--a patently unlikely possibility. It is no wonder that virtually all previous post-dryout prediction methods fail so mis­

erably.

4.GENERAL SUMMARY

A general approach for the computation of nonequilibrium phase change in gas-liquid sys­tems was described. This method, derived from first principles, shows that the generation of va­

por under thermal nonequilibrium conditions is a first-order, inhomogeneous, initial-value prob­

lem As such, initial conditions are of overriding importance in the detelmination of the actual

vapor content of the flow. This method is termed the nonequiliblium relaxation description of

phase change.

Application of the relaxation phase change method to calculation of conditions beyond the

critical heat flux in liquid-deficient, dispersed flow heat transfer accounts for the non-equili­

brium effects using a combination of first principles and semi-empirical application.

Although the effects of history on post -dryout heat transfer appear to diminish with increasing

mass velocity, it has been clearly shown that the proper expression of the non-equilibrium behav­

ior involves an initial value problem. History effects become more pronounced at lower mass

velocities and may be quite important in natural circulation systems or in systems involving

emergency fe-flood. In addition, the reporting of experimental post dryout data from experi­

ments should include inlet, dryout, and profile data, as well as local variables.

s. NOMENCLATURE

English

a Metric tensor A Area Ai Interfacial area density Ar Archimedes number Bo Boussinesque number

Co Drag coefficient D Hydraulic diameter e Energy g Acceleration of gravity G Mass flux h Heat transfer coefficient

Enthalpy L Length m Mass flow rate n Droplet number density Nu Nusselt Number p Pressure Pr Prandtl number

q" Heat flux

Re Reynolds number s Entropy or generalized source S Slip ratio t Time T Temperature L\T Temperature difference u Specific internal energy v Generalvelocity V Volume w Velocity in z-direction We Weber number x Quality z Axial coordinate

Greek

ex Void fraction or thermal diffusivity

cP Flux

r Volumetric vapor generation rate

477

478

5 Bubble diameter or boundary layer thickness � Positive difference

IJ. Dynamic viscosity p Density (J Surlace tension 1: Shear stress or dimensionless time u specific volume

� perimeter

Subscripts

c Critical e Equilibrium f Saturated liquid fg Positive saturated liquid-vapor difference g Saturated vapor hom Homogeneous

Interlace k Phase-k

1 Liquid (not necessarily at saturation) m Mixture o Initial p Constant pressure r Relative or relaxation s Swface or saturated v Vapor (not necessarily at saturation) w Wall

Terminal 1, Single-phase 2, Two-phase

479

6. REFERENCES

1 . Aris, R.t 1962. Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Prentice Hall, New Jersey.

2. Bailey, N. A., 1. O. Collier, and J. e. Ralph, 1973a. Post Dryout Heat Transfer in Nuclear and Cryogenic Equipment. ASME pre print 73-HT - 16.

3 . Bailey, N. A., J. O. Collier, and 1. C. Ralph, 1973b. Post Dryout Heat Transfer in Nuclear and Cryogenic Equipment," AERE-M-75 19, August.

4. Bennett, A. W., ll.al., 1968. Heat Transfer to Steam-Water Mixture Flowing in Uniformly Heated Tubes in which the Critical Heat Flux has been Exceed. Paper #27 presented at the Ther­modynamics and Fluid Mechanics Convention, Bristol, March.

5. Bennett, A. W., O. F. Hewitt, H. A. Kea.rsey, and R. F. K. Keeys, 1967 . Heat Transfer to Steam­Water Mixture Flowing in Uniformly Heated Tubes in Which the Critical Heat Flux has been Ex­ceeded. AERE-R 5373, October.

6. Bromley, L. A., 1950. Heat Transfer in Film Boiling," Chemical Engineering Progress, 46, 221 .

7 . Chen, lC., Ozkaynak, F.T., Sundaram, R.K., 1979. Vapor Heat Transfer i n Post-CHF Region Including the Effect of Thermodynamic Equilibrium. Nucl. Eng. Des. 51, pg. 143.

8 . Collier, 1. 0., 1962. Heat Transfer and Fluid Dynamic Research as Applied Fog-Cooled Power Reactors. AECL-Chalk River report, CRARE-1 108, June.

9. Cumo, M., �., 1973. On Two-Phase Highly Dispersed Flows. ASME preprint 73-HT -18.

10. Doughall, R . S. , and W. M. Rohsenow, 1963. Film Boiling on the Inside of Vertical Tubes with Upward Flow of the Fluid at Low Qualities. MIT Report 9079-26, September.

1 1 . Evans, D., Webb, S.W., and Chen, J.C., 1983. Experimental Measurement of Axially Varying Vapor Superheats in Convective Film Boiling. In Interfacial Transport Phenomena, Pg. 85, HTD-23, le. Chen and S.O. B ankoft', Eds. , ASME, New York.

12 . Forslund, R. P. and W. M. Rohsenow, 1 966. Thennal Non-Equilibrium in Dispersed Flow Film Boiling in a Vertical Tube," MIT Report 753 12-44, November.

13 . Forslund, R. P. and W. M. Rohsenow, 1968. Dispersed Flow Film Boiling. Trans., ASME Ser. C. J. Heat Trans., 90, 399-407, ( 1968).

14. Gill, L.. E., G. F. Hewitt, and P. M. C. Lacey, 1963. Sampling Probe Studies of the Gas Core in Annular Two-Phase Flow, Park II, Studies of the Efect of Flow Rates on Phase and Velocity Dis­nibutions. Harwell report AERE-R 3955.

15 . Oroeneveld, D. C., 1 968. An Investigation of Heat Transfer in the Liquid Deficient Regime. AECL-3281 , December.

480

16. Groeneveld, D. C., 1972. The Thermal Behavior of a Heated Surface at and Beyond Dryout. AECL-4309, PhD Thesis, University of Western Ontario, November.

17 . Groeneveld, D. C. and G. G. l DelOlme, 1975. Prediction of Thermal Non - equilibrium in Post-Dryout Regime. (private communication).

18 . Heineman, J. B . , 1960. An Experimental Investigation of Heat transfer to Superheat Steam in Round and Tectangular Ducts. Argonne report ANL-62 13.

1 9. Hynek, S. l, W. M. Rohsenow, and A. E. Bergles, 1969. Forced Convection Dispersed Flow Film Boiling. MIT Report DSR 70586-63, April.

20. 11oeje, O. c., W. M. Rohsenow. P. Griffith, 1975. TIu'ee Step Model of Dispersed Flow Heat Transfer (POST CHF Vertical Flow). AS ME preprint 75-W A/HT - 1 .

2 1 . Ishii, M . • 1975. PIivate communication.

22. Ishii. M., 1 97 1 . Thermally Induced Flow Instabilities in Two-Phase Mixture in Thermal Equilibrium. PhD Thesis, Georgia Institute of Technology.

23. Ishii, M., 1975 Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolle, Paris.

24. Janssen, E., and Kervinen, lA., 1975. Film Boiling and Rewetting. General Electric Report NEDO-20975.

25 . Jones, O.C., 1982. Toward a Unified Approach for Thermal Nonequilibrium in Gas-Liquid Systems. Nuel. Eng. Des., 69, Pg. 57.

26. Jones, O.C., Jr., 1984. Thermal Design Concepts for the Rotating Fluidized Bed Reactor. Nuel. Sci. Eng., 87, pp. 13-27.

27. Jones, O.C., 199 1 . A Nonequilibrium Relaxation Model forPost-Dryout HeatTransfer. To be published in the Int. J. Heat and Mass Trans.

28. Jones, O.C., and Saha, P., 1977. Non-Equilibrium Aspects of Water Reactor Safety. In Ther­mal and Hydraulic Aspects of Nuclear Reactor Safety. Vol. 1: Light Water REactors. O.C. Jones and S .G. Bankoff, Eels., AS ME press, New York.

29. Jones, O.C., and Zuber, N., 1977. Post-Dryout Heat Transfer: A Nonequilibrium, Relaxation Model. ASME Preprint 77-HT -79.

30. Kocamustafaogullari, G., 197 1 . Thermo-Fluid Dynamics of Separated Two-Phase Flow. PhD Thesis, Georgia Institute of Technology.

3 1 . Hsu, Y. Y. and J. W. Westwater, 1959. Approximate Theory for Film Boiling on Vertical Sur­faces. Chern. Eng. Prog. Syrn. Ser #30, 56, 15-24. 32. Kearsey, H. A." 1 965 . Steam Water Heat Transfer - Post Burnout Conditions. Chern. & Proc. Energy., 46, 455-459.

48 1

33. Laverty, W.F., and W. M. Rohsenow, 1 964. Film Boiling of Saturated Liquid Flowing Upward Through a Heated Tube and High Vapor quality Range. MIT Report 9857-32, September.

34. Le, K. and D. Ryley, 1968. The Evaporation of Water Droplets in Superheated Steam. Trans. ASME Ser. C. J. Heat Trans., 90, 455-45 1 .

35. Miripolskii, Z. L., 1 963. Heat Transfer in Film Boiling of a Steam-Water Mixture in Steam Generating Tubes. Teploenergetika, 10, 49-53.

36. Nijhawan, W., Chen, J.c., Sundaram, R.K., and London, EJ. , 1 979. Measurement of Vapor Superheat in Post-Critical Heat Flux Boiling. In Nonequilibrium Interfacial Transport Pro­cesses, pg. 45., J.C. Chen and S .G. Bankoff, Eds., ASME, New York.

37. Nijhawan, W., Chen, J.c., Sundaram, R.K., and London, EJ., 1980. Measurement of Vapor Superheat in Post-Critical Heat Flux Boiling. Trans. ASME, J. Heat Trans, 102, pg. 465.

38. Parker, 1. D., and R. J . Grosh, 1961 . Heat Transfer to a Mist Flow. AND-629 1 , January.

39. Plummer, D. N., O. C. Iloeje, P. Griffith and W. N. Rohsenow, 1973. A Study of Post Critical Heat Flux Heat Transfer in a Forced Convective System. MIT Report DSR -73645-80, March.

40. Plummer, D. N., O. C. Iloeje, W. M. Rohsenow, P. Griffith, and E. Ganic, 1974. Post Critical Heat Transfer to Flowing Liquid in a Vertical Tube. MIT RepOlt 727 1 8-9 1 , September.

4 1 . Polomik, E. E., Phase Velocities in Boiling Flow Systems by Total Energy and by Diffusion. Trans ASME, Ser. C. J. Heat Trans, 88, 1 -9, 1966.

42. Quinn, E. P., 1 965 . Forced Flow Heat Transfer to Water Beyond the Critical Point. ASME Paper No. 65-WA/HT-36, December.

43. Stein, R., H. Firstenberg, S . Israel, R. Hankel, and M. Crane, 1 963. Investigations of Wet Steam as a Reactor Coolant (CAN-2). UNC-5008- 1 , August.

44. Truesdell, C. and R. Toupin, 1 960. The Classical Field Theories. from Handbuch der Phy­sik, by S. Flugge, Bank Ill/I .

45. Varone, A.F., and Rohsenow, W.M., 1986. Post Dryout Heat Transfer Prediction. Nuel. Eng. Des., 95, pg. 3 15 .

46. Vernier, P., and 1 . M. Delhaye, 1968. General Two-Phase Flow Equations Applied to the Thennodynamics of Boiling Nuclear Reactors . Energie Premaire, IV, 1-2.

47. Wallis, G. B ., 1 974. The Tenninal Speed of Single Drops or Bubbles in an Infinite Medium. Int. J. Multiphase Flow, 1, 49 1-5 1 1 .

48. Webb, S .W., Chen, J.c., and Sundaram, R.K. , 1982. Vapor Generation Rate i n Nonequili­

brium Convective Film Boiling. Proc. 7th lot. Heat Trans. Conf., Munich.

482

49. Yoder, J .L., and Rohsenow, W.M., 1983. A solution for Dispersed Flow Heat Transfer Using Equilibrium Fluid Conditions . Trans. ASME, J. Heat Trans., 105, pg. 10.

50. Zuber, N. and J. Hench, 1 962. Steady S tate and Transient Void Fraction of Bubbling Systems on Their Operating Limits, Part I: Steady State Operation. General Electric Report 62GLlOO, July.

SHELLSIDE BOILING AND TWO-PHASE FLOW

Michael K. Jensen

Rensselaer Polytechnic Institute Troy, NY 12180-3590

Abstract

483

The cw-rent state of knowledge on the prediction of two-phase flow patterns, pressure drop, heat transfer coefficients, and the critical heat flux condition in crossflow boiling on the shellside of multi-tube bundles is described in this chapter. The mechanisms' governing the processes are discussed. When available , correlations for the various phenomena are presented. Enhanced tube bundles and the modeling of the two-phase flow and heat transfer in large tube bundles are also discussed.

1. INTRODUCTION

Shellside boiling with crossflow in horizontal multi-tube bundles is used extensively in a variety of applications such as in kettle and thermo syphon reboilers in the chemical process industry, submerged evaporators in the refrigeration industry, waste heat boilers, and horizontal natural circulation steam generators in nuclear power plants in the USSR (Collier [1] ). Even though crossflow boiling is widely used and design techniques have been developed which can predict overall or average tube bundle performance, what is occurring at a particular point within the bundle (i.e . , locally) is unknown because our knowledge of crossflow boiling heat transfer and two-phase pressure drop characteristics is insufficient. This lack of knowledge inhibits the development of a more systematic and detailed design approach via computer simulation which could lead to better designs and more efficient heat exchangers.

An example of a heat exchanger with shellside boiling for which more information on local characteristics could lead to better designs and improved performance is a kettle reboiler, such as shown in Fig. 1A. An early view of the heat transfer processes in kettle reboilers was that nucleate pool boiling was the dominant mechanism; these results were based on inexact plant data

• Some of this material has been previously presented by the author in H.e.a.t Ed. R.K. Shah, ASME, New York, 1989.

484

FIGURE 1. Schematic of Cross-sectional View of (A) Kettle Reboiler and (B) Full Fundle Boiler (Payvar f4BD.

(Palen and Taborek [2], Palen and Small [3]). Later research (Palen et a1. [4]) indicated that heat transfer mechanisms other than nucleate pool boiling were involved, the early design methods 'Were verry conservative, and that the use of a nucleate pool boiling curve from a single 'heated tube did not represent the bundle heat transfer processes very well. (See Fig. 2) Not only was the bundle­average heat transfer coefficient in the nucleate boiling range much higher than that of the single tube, the bundle-iaverage critical heat flux (CHF) condition was significantly lower. The bundle CHF is defined as that condition when the integrated effect df all the tubes (not all the tubes dryout at the same time ) begin to have a significant adverse affect on the overall bundle performance. At this time for kettle reboNers, such as described above, and other multi-tube bundles with shellside boiling we do not have the necessary information to predict which tubes have dried out, what the heat transfer coefficient distribution is throughout the bWldle, what the effect of the bundle geometry is on the heat transfer processes, etc. If that information was available and simulation programs were developed, then we could tailor the bundle design to take advantage of favorable characteristics and to minimize unfavorable ones. However, it has been only in the past few years that research interest has been directed toward the determination of local conditions throughout the tube bundle. Once the flow behavior, the local heat transfer characteristics and the local pressure gradients for crossflow in tube bundles are quantified with respect to fluid conditions and geometry, then better designs and more efficient heat exchangers can be developed. It is the objective of this paper to briefly review the state of the art on forced and natural-convection induced crossflow

485

boiling in tube bundles and to suggest areas which need additional research. More detailed reviews can be found in Palen [5, 6] and Jensen [7].

.: .c .. :=l I- C" x :=l -I u-I-« L.U :J:

2

1 0 5 8 6 4

2

1 04 8 6 4

2

2

CALCULATED S INGLE TUBE M AXIM UM HEAT FLUX -

4 6 8 1 0 2 4 6 8 1 02 2

OVERALL AT, ( F )

6 8 1 03 2 4

FIGURE 2. Typical Tube Bundle Boiling Data Compared to Single Tube (Palen et a1. [4]).

2. FLOW PATI'ERNS

Flow patterns in two-phase crossflow have received much less attention than has intube two-phase flow. (See, for example, Diehl and Unruh [8], Nakajima [9] , Grant and Murray [10, 1 1], Grant and Chisholm [12], Kondo and Nakajima [13], Chisholm [14, 15].) Because it has been shown in several studies (e.g. Chisholm [14], Schrage et al. [16]) that the flow pattern can have an effect on the two-phase friction multiplier and, undoubtedly, the heat transfer, it is important that we can predict the flow pattern as a function of flow conditions, fluid properties and bundle geometry.

Several distinct flow patterns have been observed, but it appears from examination of the studies on two-phase flow patterns in tube bundles that, unlike intube flows, there are only a limited number of flow patterns present in tube bundles; there are not enough data to determine what affect the tube layout has on the flow patterns. Shown in Fig. 3 are the most prevalent flow patterns observed in the above investigations; bubbly and spray flow are common to both vertical and horizontal flows while slug or intermittent flow generally occurs only in horizontal flows. These flow patterns are analogous to those occurring in in-tube flow.

486

Gas bubbles LiQu:cl droplets Liquid lilm in liquid in gas on wans

and lubes

Bubbly llow Slug llow Spray llow

" 0 0 0 Gas bubbles __ o 0 0 0 0 0 00 0 Gas 0 ° in liquid

0 0 0 0 ° 0 0 0 o 0 0 0 ., 0 0 Liquid- Bubbly

LiQuid drOpJeIS _" j ' ,

m gas � e , & 0 I •

Liquid -- Siratilied· Spray

Slra�tied

LiquiJ � 6 , 4 g g • 6

droplelS -l. I I 0 ° , ,

Spray

FIGURE 3. Flow Patterns in Tube Bundles (Chisholm [14]).

To generalize the data and to quantify at what fluid conditions the transition occurs between flow patterns, flow pattern maps have been presented by Grant and Murray [10, 11] and Kondo and Nakajima [13]. Those by Grant and Murray, as given in Grant and Chisholm [12], are shown in Fig; 4. Note, however, that these maps were developed using low pressure air­water flows for a tube bundle with a p/D = 1.25 and an equilateral triangular layout, so they are not completely general.

Chisholm [15] presents equations with which the quality can be determined at the three transitions in horizontal crossflows.

Stratified:

Bubbly:

Spray:

I-xs =

Xs Bs

1-xb =

xb B) l-xf

=

Xf Bf (1)

In these equations , xs, Xb, and Xf are the transition qualities for the stratified, bubbly, and spray transition points, respectively; the quantities Bs, Bb, and Bf are

Bs = (22-m-2) I (r +1) ; Bb = {PH I Pg)1I2 ; Bf = (Ilfl llg)ml2

R = 1.3 + 0.59 Frfo N2 (Ilf I J.lg)-m r2 = (dP/dz)go I (dP/dz}fo = (pfl Pg) (J.lfl J.lg)-m

487

(2)

and m is the exponent in a Blasius-type single-phase friction factor equation. The quantity Frfo is a Froude number for the total flow as liquid with the velocity based on the minimum cross-sectional area in the tube bundle normal to the flow direction.

HORIZONTAL SIOE·TO·SIOE FLOW 0.01

0.1 1 0

FIGURE 4. Shellside Flow Pattern Maps (Grant and Chisholm [12]).

To date no theoretical or semi-empirical analyses of the flow pattern transitions have been attempted for shellside flow; all the work has been experimental. Generally, the data bank for flow pattern information is very sparse. Only one flow pattern map has been developed for one tube bundle geometry. With renewed interest in shellside boiling, this could be a fruitful area for research.

3. PRESSURE DROP

Evaluation of the pressure drop in crossflow boiling is important for a number of reasons. In kettle reboilers, for example, an estimate of the recirculating flow is determined by a balance between the driving hydrostatic

488

head in the liquid outside of the bundle and the total pressure drop (hydrostatic, acceleration and friction) across the bundle. In submerged evaporators in refrigeration service, the pressure drop through the bundle needs to be calculated so that the decrease in the saturation temperature of the refrigerant which will lower the system capacity can be estimated. In a variety of other applications the estimation of the pressure drop through a bundle is important so that circulation pumps can be sized more accurately. To calculate the hydrostatic and acceleration pressure drops, the primary quantity needed is the void fraction. For the friction pressure drop, using a separated flow model, a two-phase friction multiplier is required.

3 .1 . Void Fraction In vertical two-phase crossflow, there have been five studies that have

addressed void fractions in tube bundles. Kondo and Nakajima [13] have taken indirect void fraction measurements in vertical upflow across horizontal staggered tube bundles using air-water mixtures using a quick-closing valve technique. The void fraction increased with superficial gas velocity; the superficial liquid velocity had no effect. The results also showed that the number of tube rows affected the void fraction. The void fraction data were correlated but the correlation cannot be generally applied because of the very low mass velocities used to generate the data.

Schrage et al. [16] tested air-water mixtures in a square, inline tube bundle with p/D = 1.3 using quick-closing plate valves to isolate the bundle. At a fixed quality the void fraction increased with increasing mass velocity and was significantly lower than the homogeneous value. The data were correlated by applying a mass velocity correction factor in terms of a Froude number to the homogeneous void fraction:

a I aH = 1 + 0. I23Fr-O.191lnx (3)

Dowlati et al. [17] , using gamma densitometry, measured void fractions in a square, inline 10 row tube bundle with p/D = 1.26. They also found that the void fraction was a function of the mass flux and were able to correlate all their data using the Wallis parameter, jg*:

a = 0.88 jg*O.57 (4)

Note that the Schrage et al. correlation has poorer accuracy at low qualities than at high qualities (their experiment went to qualities as high as 68%) and that the Dowlati et aI. experiment was limited to qualities less than about 10%. The Schrage et al. [16] correlation tended to overpredict the low mass velocity data of Dowlati et al. [17]. In a later paper Dowlati et al. [18] measured void fractions in two 20 row inline tube bundles (p/D = 1.30 and 1 .75) and found no pitch-to-diameter ratio effect on the void fraction. (See Fig. 5.) They correlated their data with:

(5)

1 . 0

0 . 8 HOMOG ENEOUS VOID FRACTIO N ", 0 0'; ..

,. . 0 . 6 � t .

. ,.0"p CS • . � crt • 0

0 . 4 : . .. • 6 0

.. •

0 . 2 •

I .. • 0 0 6

1 0-4 1 0-3 1 0-2

Q UALITY

o 6 • 0 6 .t .. . 6 0 •

0

0 6 0 • 0 0

G kg/m2s o 27 • 348 • 77 v 503 6 9 6 ,. 599 .. 1 5 1 � 6 9 6 o 2 6 2 • 8 1 8

1 0- 1

489

FIGURE 5. Shellside Void Fraction in Vertical Upflow in p/D = 1.30, Inline Bundle (Dowlati et al. [18]).

Robinson et al. [19] studied void distributions in bubbly flows through yawed rod arrays. For an inline square array with the rods yawed up toward the left, large bubbles drifted to the right wall and small bubbles migrated toward the left. For the rotated square array no bubble migration along the rod axis was observed. This behavior is attributed to vortices forming behind the rods. Reinke and Jensen [20] compared the pressure drop across an inline and a staggered tube bundle with the same p/D ratio when operating at identical conditions. As shown on Fig. 6 for a low mass flux condition, the much lower pressure drop in the staggered tU:be bundle is directly attributable to a greater void fraction in the staggered tube bundle than in the inline tube bundle. Figure 7 is for another condition and shows the significantly different behavior between the staggered and inline bundles. The differences can be attributed to both the void fraction and friction factor characteristics of the two bundles.

In horizontal crossflow, Grant and Chisholm [12] have developed an expression for the void fraction in stratified flow. The correlation used to represent the data was

(1 - a) = 1

1 +

(6)

where k2 was a velocity evaluated by

(7)

490

-110. �

.n -a. <J

Xo .00 .05 . 1 0 . 1 5 .20 .25 .30 .35 .40 .45 .50

.045

o

o o

o STAGGERED ARRAY + IN-LINE ARRAY

.020 -

0 0 0 -

.0 2.5 5 . 0 7 . 5 1 0.0 1 2.5 1 5.0 1 7.5 20.0 22.5

q" ( kW/m2) FIGURE 6. Comparison of Total Pressure Drops Between Staggered and Inline Tube Bundles with p/D = 1.30 at P = 500 kPa and G = 100 kg/m2s using R-113 (Reinke and Jensen [20]). Xo

.20

�

.0 -

a. <J

.00 .05

.050

. 1 0 . 1 5 .25

o STAGGERED ARRAY + IN-LINE ARRAY

o

o o

O. 5. 1 0 . 1 5 . 20. 25. 30. 35.

q" (kW/m2)

.30 .35 .40

o

o

FIGURE 7. Comparison of Total Pressure Drops Between Staggered and Inline Tube Bundles with pm = 1.30 at P = 500 kPa and G = 300 kg/m2s using R-113 (Reinke and Jensen [20]).

49 1

The authors stated that this correlation represented the data well at low qualities and underestimated the data at higher qualities. Grant et al. [21] also have developed expressions for the volume fraction occupied by the separated liquid in stratified flow.

Studies on void fractions in tube bundles are rare, and the correlations which have developed have not been widely tested. In particular, the effect of bundle geometry has not been quantified. Hence, care must be exercised when using the available correlations, particularly when the tube bundle geometry is different than that used in development of the correlations. Likewise, since it has been shown that the homogeneous model poorly represents the void fraction in crossflow, the use of the homogeneous void fraction cannot be justified.

3.2. Two-Phase Friction Multiplier In comparison to the work that has been done on shellside void fraction,

there has been more attention given to the two-phase friction multiplier. However, the main emphasis of these experiments has been directed to horizontal crossflow over horizontal tube bundles; the lack of a suitable void fraction model for vertical crossflow, which would be required to properly reduce the experimental data to obtain the two-phase friction multiplier, is the reason for fewer vertical flow studies.

Chisholm and co-workers (e.g. Grant and Chisholm [12] , Chisholm [14, 22] have developed a model for the two-phase friction multiplier. Data were obtained from two bundles with a equilateral triangular layout with p/D = 1.25 and were fit using

'" = Bx(2-m)/2( 1_x)(2-m)/2 + x(2-m)

where 'V is a normalized two-phase friction multiplier defined as

As shown on Fig. 8, '" is a function of mass flux and flow pattern; early expressions for B are given by Eq. 2 for horizontal crossflow.

(8)

(9)

While these expressions gave reasonable results, there was a problem with them since the expressions indicate that the two-phase friction multiplier was a function of the number of tube rows. However, Chisholm [22] has corrected this in a more recent paper and has presented the expression for B:

B = {.. +

(10)

where c = 1I(580000[kg!m�])(2-m)/2 and Bf is as in Eq. 2 Other models have been developed for horizontal flows but this one presented by Chisholm and co­workers is the best developed of the group.

492

eroao . .. v../CkI(.' i» . 17.5 e ns 0 532 D .. 1.lI '" J 5. 0 1;0 A S'," .. 265 • J130

I 0-3 1 0-4 10-� 10-1 100

Mus Drynes Fn�D, x

FIGURE 8. Normalized Two-Phase Multiplier (Chisholm [22]).

Schrage et al. [16] obtained two-phase friction multiplier data for vertical crossflow. There were strong mass velocity trends in the two-phase friction multiplier data and, depending on the value of the Martinelli parameter, <1>1 could either increase or decrease with increasing mass velocity. (See Fig. 9) This behavior was attributed to changing flow patterns. Although it was inappropriate, the Grant and Chisholm [12] flow pattern map for two-phase flow in bundles was used to classify the flows as either bubbly, slug, or spray. It was shown that in bubbly flow <1>1 decreased with increasing mass velocity,

but in both slug and spray flow <I>� decreased with increasing mass velocity. The data were fit to Eq. 11:

where

(11)

200 1><1

1 00

50

N -' 20 -s. 1 0

1 0 .01

*

1><1

G KG/M**2 S • 683 N 1 49

� 459 )( 1 1 7 o 390 lI( 78

1><1* Nk 254 � 59 � � <I 274 x 54

MI( f!- >-N I>( 1 95 X I><1 ��

O x * * � +(;� (N � � � � >-

� �� ° �t><+ + � � t><<t-

0 . 1 1 1 0 MART IN ELLI PARAM ETER

493

+0

1 00

FIGURE 9. Two-Phase Friction Multiplier for Inline Bundle with p/D = 1 .30 (Schrage et a1. [16]).

Table 1 Coefficients in Nondimensional Two-Phase Friction Multiplier Correlation (Eq. ll).

Flow Pattern t C1 C2 C3 C4 C5

0.036 1.51 7.79 -0.057 0.774 2.18 -0.643 11.6 0.233 1.09

0.253 -1.50 12.4 0.207 0.205

*If Fr S; 0.15, use Ishihara et a1. [23] correlation tFlow pattern evaluated with Grant and Chisholm [12] flow pattern map

The values of the coefficients are given in Table 1 as well as a restriction on the use of Eq. 11 . (Note that the use of C5 * 1 is different than what other investigators have used and thus restricts the use of this correlation to qualities less than about 50%.) Dowlati et a1. [17] also measured the void fraction in their experiment so that they could accurately calculate the two-phase friction multiplier. They concluded that '1 was not afected by the mass velocity and were able to satisfactorily correlate their data using Eq. 11 with C = 4.0 and C5 = 1.0.

494

Dowlati et a1. [18] successfully correlated their data for both test sections using Eq. 1 1 using m = 0.2 in Xtt. For p/D = 1.30, C = 8 and C5 = 1 .0 for G > 260 kg/m2s; for G < 260 kg/m2s a strong mass flux effect was noted. (See Fig. 10.) For p/D = 1.75, C = 50 and C5 = 1.0 for G > 200 kg/m2s fit the data reasonably well; again, at lower mass fluxes, strong mass flux effects were observed. Generally, for a given value of Xtt, the larger p/D bundle had the larger two-phase friction multiplier.

o 0 0 0

1 00 0 • • • 0 A •

6 . 6

1 0 • • N • -- kg/m29 8- G

o 27 • 348 • 77 " 503 A 96 " 599 • 1 5 1 o 696 o 262 • 8 1 8 Eq .8 (C=8)

0 . 1 0 . 1 1 0 1 00

X tt FIGURE 10. Liquid-Only Two-Phase Friction Multiplier Data and Martinelli Parameter, p/D = 1.30 (Dowlati et a1. [18]).

Other correlations have been developed for <1>1 in vertical crossflow but since the void fraction used to reduce the data were incorrect or were not given, the accuracy and validity of those expressions is questionable. Ishihara et a1. [23] reviewed several correlations and the data in the literature. In general, they concluded that all the correlations predicted the shear-controlled or high pressure drop data better than the low pressure drop data where the influence of the void fraction would be more evident. One of the major conclusions drawn by Ishihara et a1. , which is still valid, was that since the various models were developed under specific flow orientations and geometries, application of a model outside its intended limits is not recommended due to the empirical nature of the correlations. Equation 1 1 was then used to correlate the data bank with C = 8.0 and C5 = 1 .0. This predicted the shear-controlled flow data for Xtt < 0.2 with good results; however for

495

Xtt > 0.2 deviations were quite large, exceeding 60 percent. To improve the correlation in this range, Ishihara et al. concluded that by categorizing the data according to flow pattern and then obtaining the best curve fit for each pattern, an improvement in predictions could be made. However, this was not done by the authors.

As noted at the beginning of this section, the two-phase friction multiplier for crossflow in tube bundles has been addressed in more detail then has the void fraction. However, for vertical flows, only the Schrage et al. [ 16] and Dowlati et al. [17, 18] correlations can be recommended for general use since they are the only ones for which the true void fraction was available to reduce the experimental data. These correlations have not been tested extensively so their limits of applicability are unknown. The Grant and Chisholm [12] equation is recommended for horizontal flows. As with the void fraction, more research is required.

4. HEAT TRANSFER COEFFICIENTS

Two approaches have been used to obtain heat transfer coefficient data for shellside boiling: natural circulation and forced convection experimental set­ups. Palen and co-workers [2-4] used natural circulation bundles and found that the heat transfer coefficients in the bundles were much greater than those for a single tube. (See Fig. 2). Circulation and turbulence caused by the rising vapor bubbles were credited with increasing the coefficients. Clearance between tubes was found to be a significant factor affecting the heat transfer coefficient; tube layout angle was not. However, the coupled processes of the recirculating flow and the heat transfer obscure the effect of one parameter versus another. Because of proprietary considerations, no quantitative information was presented.

Tests on much smaller natural circulation tube bundles, e.g. , Mednikova [24], Wallner [25], Wall and Park [26], Fujita et a1. [27, 28], Kawai et al. [29], Hahne and Mueller [30], have confirmed the increased heat transfer coefficients (compared to a single tube in pool boiling) for the tubes higher in the bundle. The effect was most pronounced at lower wall superheats and decreased with increasing heat flux. Differences between the heat transfer coefficients for pool boiling from a single tube and in a bundle can be large at lower heat fluxes but decrease and eventually disappear as the heat flux increases. (See, for example, Fig. 11.)

More recently, the local heat transfer coefficients throughout a large (241-tube) kettle reboiler have been measured (Leong and Cornwell [31], Cornwell and Schuller [32]). Fluid recirculation (Fig. 12) in the shell of the simulated slice of the reboiler was caused by rising vapor. Heat transfer coefficients (Fig. 13) increased with increasing height in the bundle. This work was extended by Cornwell et al. [33] by rotating the inline tube bundle by 45° thereby making a staggered tube bundle. Comparison of the inline and staggered tube bundle data indicated that the heat transfer coefficients were about the same but with the staggered tube data slightly larger; this was found at each heat flux. A later paper (Andrews and Cornwell [34]) indicated that the small width of the bundle in Leong and Cornwell [31] affected the heat transfer seriously and that

496

� E .. �

c .c

C Q) ·0 ;;: Qi 0 e)

I. Q) en c: CIS !: Ci Q) I

51 03 4

f1 6 2 • • 5

• • • 4 • • • 3 1 03 • • • 2 8 • • • 1

6

4 R 1 1 p. 1 bar s 2d. 37 8 mm • haaled lube

2

1 02 1 02 2 4 6 8 1 03 2 4 6 8 1 04

Heat f low den sity qn

2 4 6 8 1 05

FIGURE 11. Heat Transfer Coefficients for Low Finned Tube Bundle (Hahne and Mueller [30]).

FIGURE 12. Streamlines in Tube Bundle (Cornwell et al. 1980).

q = 20 kW 1m 2

_ - - _ - -e

·5

2·

497

FIGURE 13. Heat Transfer Results of Leong and Cornwell [31]. Lines are Contours of Constant Heat Transfer Coefficient (kW/m2K).

the heat transfer coefficients should be at least 25% smaller. Other tests, e.g. Grant et al. [35], Nakajima and co-workers [36, 37], showed the same trends. One of the main drawbacks in using natural circulation through a bundle (from which an evaluation of how the heat transfer coefficient varies with local conditions is to be made) is that the local mass velocity and quality throughout the bundle are unknown; hence, the effects of flow rate, quality and bundle geometry on the heat transfer are not separable. To avoid this problem several studies have used forced flow through a modeled section of a tube bundle. Polley et al. [38] tested a 36-tube, 6-row bundle in forced convection with known fluid conditions. The two-phase forced flow was found to significantly enhance the boiling heat transfer coefficient, particularly at low wall superheats. However, this enhancement decreased as the wall superheat increased. They also found a large increase in heat transfer coefficient with increasing quality. Hwang and Yao [39, 40], in a similar study, obtained comparable results.

Jensen and co-workers [20,41-44] studied forced flow of R-113 in three tube bundles. They found that except when both the heat flux and mass velocity were low, there was only a modest increase in the heat transfer coefficient from the bottom tube row to the top tube row. As shown in Fig. 14, mass velocity and heat flux had strong influences on the heat transfer coefficient while the effect of quality was less. Tests using two square, inline bundles (pil = 1.30 and 1.70) and one equilateral triangular bundle (pil = 1.30) showed that bundle layout can affect the heat transfer coefficient but only at lower heat fluxes and mass fluxes. At higher heat fluxes, there were little differences in the heat transfer coefficients among the three bundles. (See, for example, Fig. 15.)

498

5.0

4.0

3.0 Z Z Z Z Q" z Z Z Z Z Z z Z

N E

.. 2.0

� "" "" "" "" "" "" "" "" * � * • � ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ .c 1 .0

0 0 0 0 0 0 0 0

0 0 0 0

o 2 . 1 7 kW/m' G s G, Xo =.082 + 8.30 kW/m' G . G, XO ".021

+ 4.40 kW/m' G - G. Xo ".206 Z 1 8.9 kW/m' G . G. XO ".079

X 5 S8 kW/m' G = G. Xo �.273 Y 25.2 kW/m' G s G, Xo = . 1 1 8

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 22 24 26 28

ROW N U M B ER FIGURE 14. Effect of Heat Flux on the Heat Transfer Coefficient at P = 517 kPa and at G1 = 50 kg/m2s, G2 = 460 kg/m2s (Jensen and Hsu [41]).

o o d o o

+ 0 +0 � 0 0

. 1 2

o

1 4 . 1 6

FIGURE 15. Comparison of Heat Transfer Coefficients Between the !nline and Staggered Tube Bundles (p/D = 1.30) P = 500 kPa, G = 500 kg/m2s , q" = 31.5 kW/m2 (Jensen et al. [44]).

499

The mechanisms governing the heat transfer behavior of boiling in a tube bundle appear to be nucleation, convection and, perhaps, thin film evaporation, e.g. , Fujita et a1. [27, 28], Hwang and Yao [39, 40] , Jensen and Hau [41] , Cornwell and Schuller [32]. Depending on the flow conditions and heat flux, there is a trade-off among these three mechanisms.

There are combinations of conditions when it appears that one or another of the mechanisms dominates and for different conditions a different mechanism will dominate. For example, as shown on Fig. 14 at a low mass flux and heat flux the increase in the heat transfer coefficient from the first to the last row is in the range of 300% while at higher mass and heat fluxes the increase is minimal. Measurements of the circumferential variation in the heat transfer coefficient (Jensen and Hsu [41], Reinke and Jensen [20]) tend to support the trade-off among the mechanisms. In an inline tube, Jensen and Hsu [41] found that generally the coefficients were highest at the two sides exposed to the highest fluid velocities, next highest at the bottom of the tube where the bubbles rising from the lower tube swept the surface and lowest at the top of the tube (Fig. 16). As the heat flux increased, the circumferential variations tended to level out. Comparable behavior was also obtained with flow through a staggered bundle (Reinke and Jensen [20]).

1 .20 0 o 6.30 kW/m2

1 . 1 5 -900 90 + 1 8.9 kW/m2

1 . 1 0 ±1 80 o 31 .5 kW/m2

0 CI �. 1 .05 a + + < 0 + ¢ 0

..c ¢ .. 1 .00 0 ¢ ..c +

¢ ¢ ¢

0.95 + 0 0 +

a 0 0.90

0.85 -1 80 -1 35 -90 -45 0 45 90 1 35

Theta ( Degrees)

FIGURE 16. Effect of Heat Flux on the Variation of Circumferential Heat Transfer Coefficient at P = 206 kPa and G = 190 kglm2s (Reinke and Jensen [20]).

Because of the trade-offs among the mechanisms evident in the data, many researchers (e.g. , Hwang and Yao [39, 40], Jensen and Hsu [41], Bennett et a1.

500

[45], Polley et aI. [38]) have used a Chen-type correlation to try to predict the heat transfer coefficients in tube bundles. The form of this correlation is

(12)

The convective coefficient, hconv' and the nucleate boiling coefficient, hNB, have been calculated using correlations from the literature or have been experimentally measured on the bundle under study. The suppression factor, S, frequently has been evaluated with an expression developed by Bennett et al. [45] in which the influence of bundle geometry on S is incorporated. Various researchers have applied the momentum analogy (similar to that developed by Chen [46] for intube flow) to flow across a tube bundle and have developed the following expression for F:

(13)

where n is the Reynolds number exponent in the single-phase convection heat transfer correlation of the form Nu = C RenprO •34 and m is the Reynolds number exponent in the single-phase friction factor in a Blasius-type correlation for the tube bundle of the form f = C Re-m . A variety of approximations have been used for <P� (e.g. , Fair and Klip [47], Payvar [48], Palen and Yang [49], Hwang and Yao [39, 40]); generally. a two-phase friction multiplier was used which was not appropriate for the bundle under study. However, reasonable results in predicting the heat transfer coefficients were obtained. Jensen and Hsu [41] did have an accurate expression for <P1 but their results using that expression were no better than the results using approximate expressions. They concluded that the Chen approach may not be sufficiently accurate for crossflow in tube bundles and recommended that different approaches may be necessary to obtain accurate predictions.

Other researchers have attempted to develop phenomonological models of the heat transfer process. Both Cornwell and co-workers and Fujita and co­workers have proposed models for low heat flux, low quality flows. Cornwell [50-52] has taken a very promising direction in explaining the governing heat transfer mechanisms and developing a method for predicting the heat transfer coefficient using sliding bubbles. He attributes the heat transfer to nucleation, convection and to thin film evaporation under bubbles which slide up the side of the tube. In an experiment in which the nucleation component was separated out from the total heat transfer coefficient, Cornwell showed that at lower heat fluxes the heat transfer could be attributed entirely to convection and to the sliding bubbles (see Fig. 17). He developed a model of the sliding bubble heat transfer process which was in reasonably good agreement with the experimental data.

The results of another study provided additional support for this approach. Kawai et a1. [29] made point measurements of void fractions and bubble-rise velocities near one tube in a seven tube bundle. They found that both the void fraction and bubble rise velocity should be taken into account when evaluating bundle effects at low wall superheats and assumed that thin film evaporation

JO

20

1 0

Sinel. Tub. 1n Sundle at qs • 15 kW/m2

Bundle

501

FIGURE 17. Sliding Bubble Analysis of the Q-6T Curves at a Bundle Heat Flux of 15 kW/m2 (Cornwell [50]).

under the rising bubbles caused the higher heat transfer coefficients compared to single tubes in pool boiling.

Fujita et al. [53] have developed a two-mechanism-model for bundle boiling by adapting the Mikic-Rohsenow [54] pool boiling model to include the effects of convection caused by bubbles rising from lower tubes. As with the Cornwell model, this model is for lower heat fluxes, qualities and mass fluxes. In this model, the heated surface is divided into three parts (Fig. 18): for the bottom 20% of a tube, bubbles flowing directly from the tube immediately lower in the bubble set this heat transfer coefficient; for the sides (60% of the tube), the convective effects are set by all the vapor flowing from lower tubes; the top 20% is assumed only to have nucleation heat transfer present. The tube average heat transfer coefficient is the area-weighted average of the three parts. Their correlation is based on data from their national circulation bundle. Hence, the correlation is not generally applicable, but does show a possible approach to the development of a general correlation.

5. CRITICAL HEAT FLUX CONDITION

Few studies have addressed the CHF condition in tube bundles. Palen and Taborek [2] and Palen and Small [3] studied full size tube bundles. They concluded that Kern's [55] recommendation for the bundle average CHF was very conservative and proposed their own correlation which was a

502

FIGURE 18. Variation in Boiling Behavior with Heat Flux and Pressure (Fujita et al. [28]).

modification of Zuber's [56] single tube pool boiling CHF correlation. Later work by HTRI (Palen et al. [4]) indicated that this, too, was very conservative. Generally, it was concluded that for kettle reboilers the average maximum heat flux was considerably less than that for a single tube in pool boiling (see Fig. 2) and was a strong function of bundle geometry. Grant et al. [35] also tested a kettle reboiler and obtained a boiling curve similar to that shown in Fig. 2. The maximum heat flux was about one-third less than the CHF calculated using the Leinhard and Dhir [57] correlation for a single tube and was about 2.5 times the value calculated with the tube bundle correlation of Palen and Small. Chan and Shoukri [58] tested a very small bundle and showed that the system hydrodynamics played an important role in affecting the CHF condition.

Schuller and Cornwell [59] tested a 241-tube electrically heated bundle. Each tube was instrumented with a thermocouple so that the heat transfer distribution throughout the bundle could be determined. Their results indicated a very complex and confusing behavior at higher heat fluxes (see Fig. 19); they suggested that some of the tubes in the bundle operated in the partial film boiling regime. They also found horizontal bands of tubes over which the heat transfer coefficients were alternately high and low. Generally, they concluded that the CHF decreases with increasing quality so that the most likely location for the CHF condition to initiate would be at the top of the bundle; however, they also concluded that the CHF condition may occur anywhere in the bundle. As these authors noted "These results do not simplify the task of the designer endeavoring to maximize the heat transfer in a boiler while avoiding dryout and vapor blanketing."

TIbe height In bundle

15

10

• 100 �wAol . ]0 �w/ol .. 10 . -0 %0 • • 60 . a 10 � , ,. .

503

10

FIGURE 19. Variation of Heat Transfer Coefficient with Tube Row in Reboiler Rig (Schuller and Cornwell [59]).

There have been only a few studies in which a known forced flow has been used for CHF studies in small arrays of tubes. Hasan et al. [60] investigated the CHF in saturated liquids on small diameter wires with adjacent unheated wires. An unheated cylinder any place except directly upstream of the heated upstream cylinder more than about four diameters away from the heated cylinder had only a minor influence; closer than four diameters resulted in a significant reduction (up to 90%) in the CHF. Above a certain velocity, there was a sharp drop in the CHF. Rahmani [61] found a decreasing CHF with increasing velocity at low qualities, but quality level was not given so that interpretation of these data is questionable. Cumo et al. [62] tested a 3x3 staggered bundle with a triangular pitch; inlet quality was varied. At zero quality, the mass velocity variation had only a minor effect on the CHF. Increasing the inlet quality caused an increase in the CHF such that at about 10-20% quality the CHF was 15 to 20% higher than the saturated liquid CHF at the same mass velocity. (See Fig. 20.) On the other hand, Schuller and Cornwell [59] for an inline bundle found a decreasing CHF with increasing quality; the effect of mass velocity was not given. Yao and Hwang [63] also tested an inline bundle and found that the CHF for a slightly subcooled liquid flow was much lower than that for a single tube in saturated pool boiling. Effect of quality was not given. They presented a CHF correlation based on a single tube in an equivalent channel.

Dykas and Jensen [64] and Leroux and Jensen [65] have measured the CHF on a single tube in 5 x 27 unheated inline and staggered tube bundles. The shape of the CHF-quality curves display three distinct patterns which progress from one to another as mass flux increased (Fig. 21). At low mass fluxes, the CHF data monotonically decreased with increasing quality. At intermediate

504

G - 4 1 Ik"m'·.1 J4

32 ��

.� J6 G • 91 Ikg/m' .,

30

28 •

o 10 20 JO K 1'\'

G · 6S lkg/m'·sl

G - 1 16 Ikglm'·"

10 10 JO K 1'\1

FIGURE 20. DNB Heat Fluxes Versus Quality in a Staggered Tube Bundle (Cumo et aI. [62]).

349 329 3aB 289 269 249 N ( 229 E , � 289 �

1.0. la9 J: U 169

149 129 lea a9 -. 1

1 . 3 INLINE 1 . 5 b a r LEGEND

MASS FLUX 0 S ekg/m"2*s 0 l eekg/mA2*s

00 ° ° X 2eekg/mA2*s A 3eekg/mA2*s 0 4e0kg/mA2*s

° Sbo ° S00kg/mA2*s

D O � O O� A � O 0 AD A A A. ° D °D A. i� ,, " X X X . A A. X X 00 X

� 0 00 a X � >b � � o O� 2SC o 0 0 0 0

., . 1 . 2 . 3 . 4 . S QUAL I T Y

({) a

0 <&' 0 0 . 6 . 7 . a

FIGURE 21. CHF data with R-113 for an Inline Tube Bundle with p/D = 1.30 (Dykas and Jensen [64]).

505

mass fluxes with increasing qualities, the CHF data initially decreased to a relative minimum, then increased to a relative maximum, and finally began to decrease again as higher qualities were reached. At high mass fluxes, as quality increased, the CHF rose gradually from the zero quality value to a maximum and then began to decrease. For all mass fluxes, the zero-quality CHF points clustered around an average value which varied slightly with test section geometry. The inline bundle with p/D = 1.30 had high CHF values (for the same conditions) than for the inline bundle with p/D = 1.70 on the staggered bundle with p/D = 1.30. These two studies show all the CHF trends that have been found individually in previous investigations. No correlation was presented.

While possible mechanisms for the CHF condition in tube bundles have been discussed (e.g. , Hewitt [66]) and correlations used for tube bundle performance predictions suggested (e.g., Yilmaz [67] , Palen [6]), too few data have been obtained for the CHF condition in crossflow to be considered quantified in any sense. Leroux and Jensen [65] and others have suggested that the various CHF mechanisms are probably due to the flow patterns present in the two-phase flow. (Fig. 22) At low mass fluxes, a departure from nucleate boiling phenomenon similar to that which occurs in pool boiling may bring on the CHF condition. The accompanying flow pattern may be described as bubbly. High quality CHF behavior is believed to result from a dry out process. A spray-annular flow pattern usually occurs at high qualities. At intermediate qualities, where the slope of the CHF-quality curve is upward, the CHF trend may reflect a transition from the low quality to the high quality mechanism. The flow pattern in this region also maybe transitional. Because no general flow pattern map for cross-flow in multi-tube bundles is available, it is not possible to confirm this speculation.

CHF

QUll l l ty

FIGURE 22. Basic curve patterns observed in the data and postulated flow patterns and CHF Mechanisms (Leroux and Jensen [65]).

506

6. SIMULATION OF CROSSFLOW BOILING

The design of heat exchangers with shellside boiling has progressed from approaches based on overall bundle performance predictions (ignoring local conditions) to local conditions predictions being integrated over the whole bundle. Fair and Klip [47] point out that early design relied extensively on empirical approaches and bundle average quantities were used for design. Kettle reb oiler type and full bundle boiler type heat exchangers (Fig. 1) were designed by this method.

However, after the appearance of the work of Leong and Cornwell [31] which graphically showed the recirculation flow patterns in a kettle reboiler, the recirculation in the shell and the local convective heat transfer effects in the tube bundle began to be addressed. Brisbane et al. [68] were the first to develop a recirculation model; others (Palen and Yang [49] , Whalley and Butterworth [69] , Yilmaz [67]) presented similar analyses. All these approaches used a simplified one-dimensional model of the bundle which assumed the same number of rows in each column in the bundle and uniform flow through the bundle. From this model, the total recirculating flow, the outlet quality and the variation in heat transfer coefficient with tube row could be estimated.

The results of these analyses were compared to the experimental results obtained by Leong and Cornwell [31] in their simulated slice of a kettle reboiler. Generally, there was reasonable agreement between the predicted and experimental circulation flow rates. For a given bundle diameter, the flow first increased with heat flux, reached a peak and then decreased with further increases in heat flux. The predicted heat transfer coefficients did show increases at tube rows higher in the bundle and tended to show the same trends as the experimental data. The transverse variations in flow and heat transfer coefficients across the bundle could not be shown because of the type of model chosen. Palen and Yang [49] and Fair and Klip [47] compared their results to proprietary data with reasonable agreement (+20 to -39%) between predicted and experimental overall heat flux for a given overall temperature difference; however, they adjusted numerous constants to obtain the best overall prediction. Jensen [70] also developed a one-dimensional geometric model but allowed for a different number of tubes and for a varying driving pressure head in each column. The effects of heat flux, weir height, geometric model , pressure drop model, bundle size and pressure level on the recirculating flow and exit quality were examined. It was concluded that the geometric model used does have a significant effect on the magnitude of the calculated flow as does the pressure drop model (up to a 100% variation in predicted recirculating flow) and that the simplified model used in previous investigations can obscure significant details of the recirculating flow.

Full bundle boilers, such as are used in submerged evaporators in the refrigeration industry, have been analyzed in Payvar [48] and Webb et al. [71, 72]. These studies have also developed one-dimensional models of the process and are similar to the kettle reboiler analyses. However, there are two main differences. As shown in Fig. 1B, the small shell to bundle clearance permits little liquid recirculation to occur; in addition, rather than the natural convection dominated situation as in kettle reboilers, the submerged evaporator is forced convection dominated.

507

Two-dimensional models for crossflow boiling on the shellside of horizontal multi-tube bundles have been presented by Carlucci et al. [73] and Edwards and Jensen [74]. Carlucci et al. developed a finite difference solution for the complete homgeneous flow field in a kettle reb oiler. The method used false porosity to account for the volume reduction in the shell due to the presence of the tube bundle. Unfortunately, the purpose of this investigation was only to show that the computational method can give a reasonable representation of the flow field; no discussion of the results was given and the model's accuracy was not verified.

Edwards and Jensen [74] performed a more complete study using a two­fluid model that accounted for slip between the phases (unlike Carlucci et a1.). Their results demonstrated that the two-dimensional simulations do provide more information and model reboilers better than one-dimensional simulations. These results accurately predicted the location of the recirculation center from Cornwell 's experimental reboiler rig and demonstrated that large transverse flows do exist in reboilers, but only over a limited extent of the bundle (Fig. 23). Also noted was the movement of the recirculation center down and away from the reboiler center as the bundle­average heat flux increased. This contributed to a significant decrease in the recirculation rate through the bundle. Additional work on the interfacial friction model and pressure drop correlation are required to improve predictions with this simulation. There are no data for local flow parameters with sufficient detail throughout a bundle to fully validate the accuracy of the two-dimensional models.

FIGURE 23. Mass Flux VectorNoid Fraction Contour Plots for Constant Wall Superheat: A) 4C; B) 15C (Edwards and Jensen [74].

508

7. TUBE BUNDLES WITH ENHANCED TUBES

Research on the use of augmented surfaces in tube bundles has also been performed, although not to the same extent as with single tubes. Generally, three types of enhanced surfaces have been used in tube bundle studies: finned tubes, finned tubes which have been cold-worked, and surfaces with a porous, sintered metallic matrix bQnded to the base tube. The heat transfer behavior of these surfaces in a bundle depends on the type of enhancement and the operating conditions.

Finned tubes have been studied in a variety of configurations, fluids and operating conditions (e.g., Myers and Katz [75], Wall and Park [26], Danilova and Dyundin [76], Hahne and Mueller [30], Yilmaz and Palen [77], Cornwell and Scoones [78], Hahne et a1. [79]). Generally, these investigations showed the finned tubes exhibit the same behavior as previously discussed for smooth tube bundles. The heat transfer coefficients varied with position and heat flux. At low heat fluxes, the heat transfer coefficient increased significantly at rows higher in the bundle; as the heat flux increased, the separate curves for each tube row converged to a single curve which was representative of a single finned tube. The induced convection of the vapor rising from the lower tubes and sweeping the upper tubes was credited with the enhanced heat transfer performance, just as it was in a plain tube bundle. Cornwell and Scoones [78] presented data that show the same behavior and suggest a correlation for the heat transfer coefficient. This correlation is based on the observation that nucleate boiling completely dominates for some conditions and for other conditions convection dominates. Their two-part correlation reflects the existence of the two regimes of heat transfer.

Hahne et al. [79] measured void fractions around a finned tube in a bundle and found that vapor jets form in the inter-tube space; this channelling has been observed by others. However, they did not find any slug or annular flow occuring. Likewise, they noted that visual observations of vapor distribution around a tube are misleading; the void measurements showed that rather than a vapor cloud surrounding a tube, the mixture generally has a normal distribution of liquid and vapor.

The second type of enhanced tubes in tube bundles (e.g., Arai et a1. [80], Yilmaz et al [81] , Stephan and Mitrovic [82]) to be discussed are finned tubes which have been cold worked. These tubes (e.g., Gewa-T and Thermoexcel-E) have had their fins bent over to form reentrant type cavities which are interconnected below the surface. While there are some inconsistencies among the results from the different investigations, several conclusions can be drawn. A single enhanced tube of this type out performs both a single plain tube or a plain tube bundle (Fig. 24). However, the enhanced tube's performance relative to a single enhanced tube was much smaller than that for a plain tube bundle relative to a single plain tube. Hence, it appears that the "bundle effect" evident in plain tube bundles is not as significant for this type of enhanced surface (Fig. 25). We can speculate as to the heat transfer mechanisms that may be responsible for the smaller or negligible enhancement of the enhanced tubes in a bundle compared to single tubes. In the plain tube bundle at lower wall superheats, convective effects are large compared to the nucleation effects. However, the nucleation characteristics o f the augmented tub e s are

.c

CD 0 Evaporator lutle bundle (I .. 2S"- LIOt' 20--)

4 (D c Evaporat r lube bundle(LII'I 21-' L"" '7.S"''")

10'

<!) (> Evaporalor lube bundle (Ll" 2'-·J... 17 5 .. )

(non un form heal laid dialrl ion)

-?""-LOw lln� (�48 !���:)

5 1 0'

Heat flux q (W/m2)

509

FIGURE 24. Bundle Boiling with a Finned and Thermoexcel-E Tube (Arai et. aI. [80]).

FIGURE 25. Bundle Boiling with GEWA-T Tube (Yilmaz et aI. [81]).

5 1 0

significantly enhanced because of the reentrant cavities. Due to the internal geometry of the cavities and the heat transfer processes which occur inside the cavities, convective effects outside the cavity will have little effect on these internal processes since the convective flow cannot intrude into these small cavities. The overall heat transfer process generally is dominated by the internal heat transfer processes. Convective effects will influence the process only at low heat fluxes and will become negligible at much lower heat fluxes than plain tubes.

Another enhancement technique which is directed toward improving the nucleation characteristics of tubes is the bonding of a porous, sintered metallic matrix to the base tube (e.g. , Union Carbide's High Flux tubing). This augmentation technique has been investigated in a variety of tube bundles (e.g., Fujita et al. [27 , 28] , Fujita [83] , Czikk et al. [84], O'Neill et al. [85], Koyama and Hashizume [86]). In a comparison between a bundle with this enhanced surface and a plain tube bundle, about a factor of ten increase in the heat flux at a given overall temperature difference was obtained. The most significant aspect of this surface is that there was essentially no difference between its heat transfer characteristics in a bundle or on a single tube in a pool. The heat transfer coefficient of a single enhanced tube was much higher than that of a single smooth tube, but there was no additional heat transfer augmentation of the enhanced tube in a bundle due to convective effects. The explanation for this behavior would be similar to that offered above for the Gewa-T and Thermoexcel-E surfaces.

8. CONCLUSIONS

In the past several years research efforts on shellside crossflow boiling and heat transfer have been directed more toward the details of the processes rather than just the overall performance of the bundle. Apparently, it has been realized that to develop more efficient or innovative heat exchangers the local processes need to be understood so that the integrated effects of all the tubes can be used to determine the overall bundle performance. Information is being developed in a number of investigations but there still are few predictive schemes or correlations which can be recommended with confidence. More experimental data on all facets of shellside crossflow boiling covering a wider range of fluids, flow conditions. and geometries are needed before accurate simulation/design programs can be developed.

9. NOMENCLATURE

parameter used in Eq. 7 (-) parameters defined in Eq. 2 (-) parameters in Eq. 10 (-) tube diameter (m) two-phase Reynolds number factor (-)

Froude number = G/pr'Jgfj (-)

.* Jg k K2 m n N p �P dP/dz q R S x xs,Xb,xf a f2 Il P

<Pr Xtt 'If

Subscripts conv f fo g go H NB T P

gravity (9.806 mls2) mass velocity based on minimum flow area (kg/m2s) heat transfer coefficient (W/m2K) superficial vapor velocity based on minimum flow area

Wallis parameter, Pg 112 jg I thermal conductivity (W/mK) parameter defined in Eq. 6 (-) exponent in Blasius equation Reynolds number exponent in Eq. 12 number of tube rows tube pitch (m) pressure drop (kPa) pressure gradient (kPalm) heat flux (W/m2) parameter given in Eq. 1 (-) suppression factor (-) mass quality (-) transition qualities given by Eq. 1 (-) void fraction (-) parameter defined in Eq. 2 (-) dynamic viscosity (kg/ms) density (kg/m3)

two-phase friction multiplier for liquid flowing alone (-) «1-x)/x)2-m (Pg/Pf) (Ilty'llg)m

parameter defined in Eq. 8

convective liquid total flow assumed liquid vapor total flow assumed vapor homogeneous nucleate boiling two phase

10. REFERENCES

5 1 1

1 J.G. Collier, in Two-Phase Flow Heat Exchangers Thermal-Hydraulic Fundamentals and Design, S. Kakac, A.E. Bergles and E.O. Fernandes (eds), Kluwer, Doedrecht (1988) 659.

2 J.W. Palen and J.J. Taborek, Chern. Engng. Prog. , 58, No. 7 (1962) 37.

5 1 2

3 J.W. Palen and W.M. Small, Hydrocarbon Processing, 43, No. 1 1 ( 1963) 199.

4 J.W. Palen, A. Yard en, and J. Taborek, AIChE Symp.Ser. 68, No. 118 (1972) 50.

5 J.W. Palen, Heat Exchanger Design Handbook, Hemisphere, Washington (1983).

6 J.W. Palen, Mechanical Engineering Handbook, Chap. 67, Wiley, New York (1986) 1893.

7 M.K. Jensen, in Two-Phase Flow Heat Exchangers Thermal-Hydraulic Fundamentals and Design, S. Kakac, A.E. Bergles and E.O. Fernandes (eds), Kluwer, Dordrecht (1988) 707.

8 J.E . Diehl and C.H. Unruh, ASME Paper No. 58-HT-20 (1958). 9 K.-I. Nakajima, Heat Trans.-Jap. Res. , 7, No. 2, (1978) l.

10 LD.R. Grant and I. Murray, NEL Report No. 500 (1972). 11 LD.R. Grant and 1. Murray, NEL Report No. 560 (1974). 12 I.D.R. Grant and Chisholm, J. Heat Trans. , 101 (1979) 38. 13 M. Kondo and K.-L Nakajima, Bulletin of the JSME, 23, No. 177 (1980) 385. 14 D . Chisholm, Two-Phase Flow in Pipelines and Heat Exchangers, George

Godwin, London (1983). 15 D. Chisholm, Heat Trans. Engng., 6 (1985) 48. 16 D.S. Schrage, J.-T. Hsu, and M.K. Jensen, AIChE Journal, 34 (1988) 107. 17 R. Dowlati, M. Kawaji and A.M.C. Chen, AIChE Symposium Series No.

263, 84 (1988) 126. 18 R. Dowlati, M. Kawaji and A.M.C. Chen, AIChE Journal, 36 ( 1990) 765. 19 J.T. Robinson, N.E. Todreas, and D. Ebeling-Koning, Int. J. Multiphase

Flow, 14 (1988) 645. m M.J. Reinke and M.K. Jensen, Boiling and Condensation in Heat

Transfer Equipment, E. G. Ragi et al. (eds), ASME, HTD-85 (1987) 41. 21 LD.R. Grant, C.D. Cotchin and D. Chisholm, Heat and Mass Transfer

Conference, Dubrovnik (1981). 22 D. Chisholm, AIChE Symposium Series No. 269, 85 (1989) 60. Z3 K. Ishihara, J.W. Palen and J. Taborek, Heat Transfer Engng. , 1, No. 3

(1979) 1. 24 N.M. Mednikova, Heat Trans.--Sov. Res., 6, No. 2 (1973) 30. 25 R. Wallner, Proc., 13th Int. Congress of Refrigeration, Paper 2.19 (1971)

185. aJ K.W. Wall and E.L. Park, Jr., Int. J. Heat Mass Trans., 21 (1978) 73. 27 Y. Fujita, H. Ohta, S. Hidaka and K. Nishikawa, Memoirs of the Faculty

of Engineering, Kyushu University, 44, No. 4, 427. 28 Y. Fujita, H. Ohta, S. Hidaka and K Nishikawa, 8th Int. Heat Transfer

Conf., San Francisco, 5 (1986) 2131 . 29 S. Kawai, N. Kawamura, T. Furukawa, T. Kitamoto and T. Machiyama,

Heat Transfer-Japanese Research, 18, No. 4 (1989) 52. 3) E. Hahne and J. Mueller, Int. J. Heat Mass Transfer, 26 ( 1983) 849. 31 L.S. Leong, and K.Cornwell, The Chemical Engineer, No. 343 (1979) 219. 32 K. Cornwell and R.B. Schuller, Int. J. Heat Mass Trans., 25 (1982) 683. � K. Cornwell, J.G. Einarsson and P.R. Andrews, 8th Int. Heat Transfer

Conf., San Francisco, 5 (1986) 2137. 34 P.R. Andrews and KJ. Cornwell, Chern. Eng. Res. Des. , 65 (1987) 127. 35 LD.R. Grant, C.D. Cotchin and J.A.R. Henry, Heat Exchangers for Two­

Phase Applications, HTD- 27, ASME, NY (1980) 41.

5 1 3

36 K.I. Nakajima and K Morimoto, Refrigeration (Japanese), 44, No. 495 (1969) 3.

:rl K.I. Nakajima and A. Shiozawa, Heat Trans. - Jap. Res. ,4, No. 4 (1975) 49.

38 G.T. Polley, T. Ralston and I.D.R. Grant, ASME Paper 80-HT-46, (1980). ffi T.H. Hwang and S.C. Yao, Int. J. Heat Mass Transfer, 29 (1986) 785. 40 T.H. Hwang and S.C. Yao, Int. Comm. Heat Mass Transfer, 13 (1986) 493. 41 M.K. Jensen and J.-T. Hsu, J. Heat Transfer, 110 (1988) 976. 42 J.-T. Hsu and M.K. Jensen, Collected Papers in Heat Transfer, K.T.

Yang (ed), ASME, NY, HTD-104 (1988) 239. 43 J.-T. Hsu, G. Kocamustafaogullari and M.K. Jensen, in "Experimental

Heat Transfer, Fluid Mechanics, and Thermodynamics 1988, RK. Shah, E.N. Ganic, and K.T. Yang (eds), Elsevier, NY (1988) 1634.

44 M.K. Jensen, M.J. Reinke and J.-T. Hsu, Experimental Thermal and Fluid Science, 2 (1989) 465.

45 D.L. Bennett, M.W. Davis and B.L. Hertzler, AIChE Symp. Ser. No. 199, 76 (1980) 91.

46 J. Chen, I&EC Proc. Des. Dev., 5, No. 3 ( 1966) 322. 47 J.R Fair and A. Klip, Chern. Engng. Prog., 79, No. 8 ( 1983) 86. 48 P. Payvar, Two-Phase Heat Exchanger Symposium, HTD-44, ASME, NY

(1983) 11. 49 J.W. Palen and C .C. Yang, Heat Exchangers for Two-Phase

Applications, HTD-27, ASME, NY (1983) 55. 50 K. Cornwell, HTFS Res. Symp., City University, London, RS695 (1987). 51 K. Cornwell, Int. J. Heat and Mass Transfer, 33 (1990) 2579. 52 K. Cornwell," 9th Int. Heat Transfer Conf., Jerusalem, 3 (1990) 455. 53 Y. Fujita, H.Ohta, K. Hoshida and S. Hidaka Heat Transfer-Japanese

Research, 19, No. 2 (1990) 25. M B.B. Mikic and W.M. Rohsenow, J. Heat Transfer, 91 (1969) 245. 55 D.Q. Kern, Process Heat Transfer, McGraw-Hill ( 1950). 56 N. Zuber, J. Heat Transfer, 80 (1958) 711. 57 J.H. Lienhard and V.K. Dhir, J. Heat Transfer, 95 (1973) 152. 58 A.M.C. Chan and M. Shoukri, Fundamentals of Phase Change: Boiling

and Condensation ASME, HTD-38 (1984) 1. 59 RB. Schuller and K. Cornwell, Inst. Chern. Engn. Ser. No. 86, 2 ( 1984)

795. m M.M. Hasan, R. Eichhorn and J.H. Leinhard, 7th Int. Heat Trans. Conf. ,

Munich, 4 (1982) 285. 61 R Rahmani, Ph.D Dissertation, University of California, Berkeley (1983). 62 M. Cumo, G.E. Farello, J. Gasiorowski, G. Iovino and A. Naviglio,

Nuclear Technology, 49 (1980) 337. 63 S.C. Yao and T.H. Hwang, Int. J. of Heat and Mass Transfer, 32 (1989) 95. 64 S. Dykas and M.K. Jensen, Experimental Thermal and Fluid Science, 4

(1991). ffi K.M. Leroux and M.K Jensen, J. Heat Transfer, 114 (1992). 00 G.F. Hewitt, in Two-Phase Flows and Heat Transfer, S. Kakac and T.N.

Veziroglu (eds), Hemisphere, Washington, D.C. (1976). ffI S.B. Yilmaz, Chemical Engineering Progress, 83, No. 11 (1987) 64. (:i3 T.W.C. Brisbane, I.D.R. Grant and P.B. Whalley, ASME Paper No . .80-

HT-42 (1980).

5 14

m P .B. Whalley and D. Butterworth, Heat Exchanger for Two-Phase Applications, ASME, NY, HTD-27 (1983) 47.

70 M.K Jensen, AIChE Symposium Series No. 263, 84 (1988) 114. 71 R.L. Webb, T.R Apparao and K.-D. Choi, ASHRAE Winter Meeting,

Chicago (1989). 72 R.L. Webb, K.-D. Choi and T.R. Apparao, ASHRAE Winter Meeting,

Chicago (1989). 73 L.N. Carlucci, P.F. Galpin and J.D. Brown, in A Reappraisal of Shellside

Flow in Heat Exchangers, W.J. Marner and J.M. Chenoweth (eds), ASME, NY, HTD-36 (1984) 19.

74 D.P. Edwards and M.K. Jensen, in Phase Change Heat Transfer - 1991, E. Hensel, V.K. Dhir, R Grief and J. Fillo (eds), ASME, NY, HTD-159 (1991) 9.

75 J.E. Myers and D.L. Katz, Chern. Engng. Prog. Symp. Ser. No 5., 49 (1953) 107.

76 G.N. Danilova and V.A. Dyundin, Heat Transfer Sov. Res. , 4, No. 4 (1972) 48.

71 S. Yilmaz and J.W. Palen, ASME Paper No. 84-HT-91 (1984). 78 K. Cornwell and D.J. Scoones, 2nd United Kingdom Conference on Heat

Transfer, Glasgow, 1 (1988) 21-32. 79 E. Hahne, J. Spindler, Q. Chen and R Windisch, 9th Int. Heat Transfer

Conference, Jerusalum, 6 (1990) 41. 8) N. Arai, T. Fukushima, A. Arai, T . Nakajima, K. Fujie and Y.

Nakayama, ASHRAE Trans., 83, Part 2 (1977) 58. 81 S. Yilmaz, J.W. Palen and J. Taborek, in Advances in Enhanced Heat

Transfer-1981, RL. Webb et al. (eds), ASME, NY, HTD-18 (1981) 123. 82 K. Stephan and J. Mitrovic, in Advanced in Enhanced Heat Transfer-

1981, RL. Webb et al. (eds), ASME, NY, HTD-18 (1981) 13l. 83 Y. Fujita, Research on Efficient Use of Thermal Energy, Reports of

Special Project Research on Energy under grant in aid of Scientific Research of the Ministry of Education, Science and Culture, Japan (1987) 27.

84 A.M. Czikk, C.F. Gottzmann, E .G. Ragi, J.G. Withers and E.P.Habdas, ASHRAE Trans., 76 ( 1970) 96.

85 P.S. O'Neill, R.C. King and E.G. Ragi, AIChE Symp. Ser. No. 199, 76 (1980) 289.

86 Y. Koyama and K. Hashizume, 16th Int. Congo of Refrigeration, Proc. , 2 (1983) 171.

f5l K. Cornwell, N.W. Duffin and RB. Schuller, ASME Paper No. 80-HT-45 (1980).

5 1 5

THE EFFECT OF FOULING ON BOILING HEAT TRANSFER

Euan F.e. Somers cales

Department of Mechanical Engineering, Aeronautical Engineering & Mechanics, Rensselaer Polytechnic Institute, Troy, New York 12180-3590 USA

Abstract The published information on the fouling of heat transfer surfaces in the

presence of boiling is revi�wed. The fundamental processes of fouling are discussed, and the terminology and nomenclature that is commonly used is described. The interaction between fouling and boiling is then considered for precipitation fouling, corrosion fouling, particulate fouling and chemical reaction fouling. The complexity of the topic and the difficulties of satisfactorily predicting the fouling performance of heat transfer surfaces under boiling conditions are pointed out. Quantitative data and theoretical predictions, where they are available, are presented.

1 . INTRODUCTION

1 . 1 �tion ofFouling Fouling can be defined as the formation of deposits on heat transfer surfaces

which impede heat transfer and increase the resistance to fluid flow. The growth of these deposits causes the thermal and hydraulic performance of heat transfer equipment to decline with time. Fouling affects the energy consumption of industrial processes and it can also decide the amount of material employed in the construction of heat transfer equipment, because it may be necessary to provide extra heat transfer area to compensate for the effects of fouling. In addition, where the heat flux is high, as in steam generators, fouling can lead to local hot spots and ultimately it may result in mechanical failure of the heat transfer surface, and hence an unscheduled shut down of the equipment. The designers and operators of heat transfer equipment must be able to predict the variation of its performance as fouling proceeds. The designer needs this information to ensure that the users' requirements with regard to cleaning schedules can be met and maintained for a heat transfer device designed for a predetermined first cost. The users of heat transfer equipment subject to fouling must be able to formulate rational operating schedules, both for equipment management purposes and in order to obtain from the manufacturer equipment that will meet the desired operating schedule.

5 1 6

1.2 Objectives offbe Lecture The objective of the lecture is to introduce the reader to fouling, particularly

fouling in the presence of boiling. It has been assumed that the reader is interested in the topic for one or more of the following reasons:

a . To obtain an overview of the state of the art of designing and operating heat transfer equipment subject to fouling in the presence of boiling.

b. To apply available data to predicting the performance of heat transfer equipment subject to fouling for the purpose of designing and/or operating specific items of such equipment.

It is not intended to present a comprehensive review of fouling; however, the lectures will contain references to sources of further information on various aspects of fouling.

1.3 Cost of Foulingt The fouling of heat transfer equipment introduces an

additional cost to the industrial sector of the national economy. This added cost is in the form of increased capital expenditure, and increased energy consumption and increased payments for labor.

a. Oversizing or redundant equipment

In order to compensate for the effects of fouling the heat transfer area of a heat exchanger is increased. Duplicate heat exchangers may be installed so as to ensure uninterrupted production while a fouled heat exchanger is cleaned.

b. Specialty materials and geometric configurations

Fouling, particularly that associated with corrosion, can be reduced by the careful selection of the materials used in the construction of the heat exchanger. Typically these are high cost materials, such as, titanium, stainless steel, glass, and graphite. Also, various special design features, reducing the size and number of crevices , and eliminating eddies and dead zones, help to decrease the effects of fouling. These special features add to the cost of an item of heat transfer equipment.

c. Additional downtime

Fouling increases the normally scheduled time incurred in maintaining and repairing equipment.

• This is based on Garrett Price et a1. [1].

5 1 7

d. Lost production

Downtime for cleaning fouled heat transfer equipment, when a plant is operating at or near capacity, represents a loss of valuable production. In addition, plant shut-down and start-up may result in production that does not meet normal product specifications. Such products must be discarded or sold at a discount.

e. Cleaning equipment and sources

The incorporation of fixed items of equipment for cleaning heat exchangers, such as soot blowers and vibrators, increases costs. In addition, chemical treatment of the process stream handled by the exchanger increases costs.

f. Additional energy requirements and loss of waste heat

Fouling reduces the transfer of energy between flowing streams, and it can result in higher pumping costs because of the reduction in the stream cross section. In certain cases waste heat is not captured and re-used because of fouling.

Estimates of the cost of fouling have been made by Thackery [2] for the British industrial economy.

Thackery based his estimate on the following cost components: (a) increased capital costs; (b) energy losses; (c) maintenance costs; (d) lost production. Using "considered guesses" and knowledge of fouling problems in Britain, Thackery estimated that the anual costs of fouling in Britain were $320 to $800 million in 1978.

L4 Observed Efects of Fouling The effect of fouling is to add a thermal resistance (Rr), commonly called the

��, •• to the convective thermal resistance (Rc) at the heat transfer surface, so that the � thermal resistance (R) is given by

(la)

or

(lb)

In equation (lb) it has been assumed, for simplicity that the surface is plane. The overall thermal resistance is related to the heat flux (q) by

•• It is probably advisable to restrict the terminology fouling factor to the maximum or design value of the fouling thermal resistance that is used to determine the size of a heat exchanger. The fouling thermal resistance is then a more general term, which includes the fouling factor, that recognizes the time dependent nature of fouling.

5 1 8

ITs - 1\,) q - R (2)

Fouling can also affect the fluid friction characteristics of the heat transfer surface. This means that for flow in a closed duct the pressure drop (&p) in a length L of the duct can increase because the Fanning friction factor (0 has increased, where

where

2'ti f:: -

pv2

(3a)

(3b)

The pressure drop may also be affected by the reduction in the flow cross section caused by fouling deposits on the duct wall. In the past the pressure drop effects of fouling have not been given much attention, because in most circumstances the heat transfer resistance is considered to have a much more important influence on the performance of heat transfer surfaces. The emphasis in this lecture will be on the heat transfer effects.

Numerous measurements covering all categories of fouling, have been made of the fouling thermal resistance, and one of the first things that is noticed is its variation with time. Broadly, graphs of the fouling thermal resistance as a function of time fall in one of three categories, as illustrated in Figure 1. Curve 1 can be called lin.e.n �, curve 2 can be called � IJl.k and curve 3 is However, it is sometimes suggested that the falling rate fouling (curve 2) is really asymptotic fouling that is taking a long time to reach the asymptotic case. In practice the curves obtained are not usually as smooth as curves 1, 2, and 3 shown in Figure 1 , and the final curve (curve 4) shows a more realistic form for the asymptotic case. Various explanations have been advanced to explain the form of these curves and these will be addressed later.

Curves 1, 2, 3 and 4 are shown as originating at time zero but quite often there is no measurable fouling thermal resistance for some extended period after the measurements begin and this initial period is usually called the

Fouling processes that exhibit an induction time are shown by curves lA, 2A and 3A in Figure 1.

1.5 Importance of Fouling Because fouling deposits introduce an added thermal resistance at a heat

transfer surface there is a reduction in the performance of a given item of heat transfer equipment. To demonstrate the technical importance of fouling the effect can be estimated of a given fouling resistance in a given heat exchanger on the required surface area and the resulting overal temperature difference in the two fluid streams. The starting point is the basic equation for a heat exchanger, which can be written

5 1 9

3A

I N D U CTI O N T I M E

Figure 1. Models of fouling: 1 ,lA linear fouling; 2 ,2A falling rate fouling; 3,3A asymptotic fouling

(4)

If we represent the performance of a clean heat exchanger by the following modified form of equation (4)

(5a)

and for the same heat exchanger when it is fouled

(5b) then we can write

(6)

where He is the fouling thermal resistance.

520

If equations (5a) and (5b) apply to the same heat exchanger, then Ac = AF = A. FC = FF = F, and

Qc = A Uc F 6TmC

QF = A UF F 6TmF

(7a)

(7b)

Suppose the heat exchanger is to operate at al times with a constant rate of heat transfer Q (= QC = QF), then from equations (7) we have the following relation between the temperatures (represented by the log mean temperature differences ATmC and 6TmF) in the clean and fouled heat transfer equipment

(8)

That is, as fouling proceeds and Rf increases then the log mean temperature temperature (6 T m ) must be increased relative to the clean value. Furthermore, this effect increases in importance as the value of the clean overall heat transfer coefficient (UC) increases (see Fig. 2).

0 0 1 00 ,.

.P ! a: 0 0 0

I 1 0 r-,O � � S

Rt Figure 2. Percent increase in the temperature driving force (or area) as a function of the clean overall coefficient (U c) and the fouling resistance (He) (from Knudsen [3])

521

If the heat exchanger is to operate at a constant rate of heat transfer Q (= QC = QF) and with a constant change in temperature (�TmC and �TmF)' then it can be shown from equations (5) that

(9)

In this case the effect of fouling is to increase that required heat transfer in the heat exchanger. From a comparison of equations (8) and (9), it is clear that the percentage change in the temperature difference and the percentage change in the heat transfer area is the same in both cases.

If the heat exchanger is to operate with a constant overall temperature difference, then from equations (7)

1 - (QlA)F (Q/A)C

(10)

This shows that as the fouling thermal resistance increases, then the heat transfer rate decreases. Again, this effect is more important with larger values of the clean overall hat transfer coefficient (UC) (see Fig. 2).

The aggravating effect of increasing the overall coefficient on the performance of a heat exchanger subject to fouling is of considerable importance. Knudsen [3] has pointed out that with increasing energy and material costs, there will be a strong incentive to design heat exchangers with high overall coefficients of heat transfer. This means that the prevention or control of fouling is a matter of increasing importance.

1.6 Categories ofFouJing Fouling can be classified in several ways [4]:

a. Type of heat transfer service being provided, e.g. , change of phase (boiling, condensation), or sensible (heating, cooling), or chemical reaction (endothermic, exothermic) heat transfer.

b. Type of fluid causing the fouling, e.g. , aqueous solutions, petroleum fractions, flue gases, etc.

c . Type of equipment experiencing the fouling, e .g. , plain surface, extended surface, enhanced surface, tubular heat exchanger, plate heat exchanger, etc.

d. Type of industry in which it occurs, e.g. , power generation, desalination, chemical processing (with numerous sub-categories), etc.

However, rather than these categorizations, the very fundamental classification that was proposed by Epstein [4] at the 1979 International Conference on the Fouling of Heat Transfer Equipment has been widely

522

adopted by engineers and scientists concerned with the fouling of heat transfer surfaces. This scheme classifies fouling according to the principle process that gives rise to the phenomenon. In this way the following six categories have been identified:

a. the precipitation of dissolved substances on the heat transfer surface.

b. the deposition of finely divided solids, suspended in the fluid, on the heat transfer surface.

c . deposits formed on the heat transfer surface by chemical reactions in which the surface material itself is not a reactant.

d . the heat transfer surface itself reacts to produce corrosion products that foul the surface.

e. the attachment of macro-organisms (macro­biofouling) and/or micro-organisms (micro-biofouling or microbial fouling) to the heat transfer surface, together with the adherent slimes generated by the microbial fouling.

f. solidification of a liquid, or some of its higher melting constituents onto a sub-cooled heat transfer surface.

Epstein [4] points out that each of the above six categories identifies a process that is essential to the particular observed fouling phenomenon. However, the process appearing in the title is not necessarily that which governs the rate of deposition of the fouling material. For example, in almost all of the categories of fouling, mass transfer from the bulk of the fluid to the heat transfer surface is important, and, furthermore, is often the process that governs the rate of fouling. This classification of fouling by the fundamental processes involved wil be considered further below.

An important limitation of the above classification is that it does not recognize the importance of combined modes of fouling e.g. , both precipitation fouling and corrosion fouling are frequently observed together on a heat transfer surface. Until the fundamental fouling processes are better understood, it seems more fruitful to concentrate our attention on single categories of fouling, rather than fouling situations that involve several categories.

The term fouling has been used in this lecture to describe deposits that form on heat transfer surfaces and inhibit the transfer of heat. This is, perhaps, not a generally accepted or universal terminology, because heat transfer inhibiting deposits are also known as scaling, crud, sludge, and so on. These different descriptive terms, which all refer to essentially the same situation, are probably associated with the different industries in which the problem of fouling is important. A lack of communication between the practitioners in

523

various areas of technology has, no doubt, been the cause of this wide variety of descriptive terms.

2. FUNDAMENAL PROCES OF FOULING

2.1 Introduction In view of the current state of our knowledge of heat transfer, mass

transfer, chemical kinetics, and fluid mechanics, and considering the success that has been experienced in applying these fundamental sciences to a wide variety of technical problems, it is natural to anticipate that they wil provide a means for the development of methods for predicting the fouling performance of heat transfer equipment, and, perhaps more importantly, suggest ways in which fouling can be minimized. As will be seen this expectation is overly optimistic because fouling is an extremely complex phenomenon, even if consideration is restricted to a single category of fouling without the complications arising from interaction between two or more categories of fouling. However, in spite of the limitations of approaching fouling from a fundamental viewpoint, it is a useful way to organize thinking about the topic and can lead to profound insights that will assist the designer and user of heat transfer equipment subject to fouling, not to say the research worker who is concerned with obtaining a complete understanding of this phenomenon.

In this lecture it is proposed to explore the possibility of constructing a theoretical model of fouling. This will be based on the idea that the net effect of fouling is the consequence of competition between growth and removal processes. The growth processes can be classified according to the transport processes involved. The formation of the fouling deposit from the transported material depends on 'reactions' occurring in the deposit. These reactions can vary during the life of the deposit and can affect the processes that act to remove the deposit. Knowledge of the removal processes is limited, but the information that is available will be reviewed.

2.2 Kern-Seaton Model The basic equation of fouling is due to Kern and Seaton [5,6] who

hypothesized that the observed rate of fouling on a heat transfer surface is the result of competition between the rate of growth processes and the rate at which the deposit is removed. That is, if dmlde is the observed rate of fouling, where mf is the mass of deposit on unit area of the surface at any instant of time (e), then

dmf • •

de = mg - mr (11)

where mg = rate of growth of the deposit and mr = rate of removal of the deposit (the units of both quantities are mass per unit area per time).

524

Equation (11) gives the fouling effect in terms of the mass (mf) of the deposit formed on unit area of the heat transfer surface, to relate this to the fouling thermal resistance (Rr) of the deposit per unit area we have

(12)

where Pf is the density and kf is the thermal conductivity of the deposit. These quantities can change during the course of the fouling process, as discussed below.

Fundamental studies of fouling are concerned with formulating suitable expressions for the growth (mg) and removal (mr) tenDS in equation (1 1). This involves introducing concepts connected with the transport of fouling materials both to and from the heat transfer surface and the rates of processes (e.g. , crystallization, polymerization, etc.) affecting the fouling deposit on the surface. These processes can vary over the time of exposure of the surface to the fouling stream. This is indicated by the observed character of the fouling thermal resistance at different times. Thus, when the surface is first exposed to the fouling stream it is frequently observed that there is no measurable magnitude for the fouling thermal resistance (Rr). The period during which these conditions persist is usually known as the .Q.[ dW tim& (en).

The irregular variation of the fouling thermal resistance is another characteristic that suggests that different fundamental processes are active at different times during the course of the fouling process. Typically a fairly steady increase in the fouling thermal resistance is followed by a decrease, and so OD. These "excursions" in the thermal resistance are irregular in magnitude and irregular in their time of occurrence, however, a general trend can be identified.

Clearly there are "phases" assOciated with the fouling process, such that different fundamental processes are active at different times during the "life" of the fouling deposit. This aspect of fouling will be considered in the next section.

2.3 The Phases of Fouling The discussion of the preceding section has suggested that

there are different phases in the fouling process. It is proposed to classify these as the ll and the � �. These are discussed in the following two subsections.

The mechanisms that operate during the induction time at the initiation of fouling are not completely understood; however, the following represents a summary, based on the discussion of Epstein [7,8], of current knowledge as it applies to different categories of fouling.

Precipitation fouling: The induction period is closely associated with the crystal nucleation processes, such that the induction time (en ) tends to decrease as the concentration in the fluid of the precipitating material increases relative to the concentration that would saturate the fluid. Crystal

525

nucleation processes and their relation to the induction time have been comprehensively reviewed by Troup and Richardson [9].

Chemical reaction fouling: The induction time appears to decrease as the surface temperature is increased which is presumably a consequence of the rates of the chemical reactions involved in the induction process increasing with temperature.

Biological fouling: According to Baier [10], the initiation of biological fouling is dependent on the adsorption of polymeric glycoproteins and proteoglycans on the heat transfer surface. Such materials are present in natural waters and act to condition the surface by laying down a film to which the micro­organisms subsequently attach. The adhesive characteristics of the film depend on the material of the heat transfer surface which can be characterized by the critical wetting tension (oc)' According to Baier if Oc is between 20 and 30 dyne s/cm , then bio-adhesion is minimized. The original reference should be consulted for more details.

All categories of fouling: It has been widely reported that for all categories of fouling the induction time (On ) decreases as the surface roughness increases. Surface roughness probably acts at the initiation of fouling in three ways:

i . roughness projections provide additional sites for crystal nucleation;

ii. grooves provide regions for deposition that are sheltered from the mainstream velocity;

iii. decreases the thickness of the viscous sublayer and hence increases eddy transport to the wall.

AJdu. Aging of the fouling deposit occurs as soon as it starts to form on the heat transfer surface. It can manifest itself as

i . change in crystal structure; ii. polymerization of the deposit material; iii. developing thermal stresses; iv. dissolution processes occurring at the deposit-surface

interface; v. poisoning of micro-organisms by corrosion products released

from the surface; vi. death of micro-organisms due to starvation.

The aging process causes the deposit to change its character so that removal processes change. Thus, where the removal of a microbial deposit initially occurs by erosion (deposit leaves as microscopic particles), it may at some later time occur by spalling (deposit leaves as a gross mass) because the micro­organisms have died and are no longer attached to the heat transfer surface.

2.4 Growth Proces The growth of the fouling deposit on a heat transfer surface

can be seen as a combination of processes that transport the fouling material to

526

the surface and some "reaction" that results in the attachment of the material to the surface. It is usual to represent such processes mathematicaly by

IDg = kt (Cb - Ci) (13)

where kt is a transport coefficient, and for the "reaction"

IDg = � (Ci - Cs)D (14)

where � is the reaction rate constant for the n-th order reaction. In these equations Cb is the bulk concentration of the transported species, Ci

is its concentration at the deposit-fluid interface, and Cs is its concentration at the deposit-surface interface.

Under steady state conditions··· the growth processes represented by equations ( 13) and ( 14) occur at the same rate. In that case the interface concentration Ci can be eliminated between equations (13) and (14) in favor of the observable quantities Cb and Cs. On doing this we obtain the result

. � mg = kr (Cb - Cs - kt)n (15)

This is a somewhat awkward expression in that the desired rate of growth

(fig) appears on both sides of the equation. However, for the special case of n =

1 we obtain the explicit relation

(16)

The use of equations (15) and (16) to determine the rate of growth of the deposit, for given values of the concentrations Cb and Cs of the fouling material, depends on knowledge of the transport coefficient (kt) and the reaction rate constant (kr). This depends, in its turn, on the type of material being transported to and attached, by means of the 'reaction', to the heat transfer surface. The materials can be either the fouling material itself, as is actually implied in equations ( 13) through (16), or it can be a 'nutrient', that is, a material that is essential to the formation of the fouling deposit. t The division

• • • In actual fact, steady state cannot exist in fouling, the fouling rate is continually varying with time. However, an instantaneous steady state can be imagined, and it is that situation that is implied here. t Where nutrient transport is under consideration it is necessary to relate rates for the nutrient and the fouling material. Typically, this could be through some chemical reaction involving both kinds of materials.

527

of transported materials into these two classes suggests that categories of fouling growth processes can be identified and related to the categories of fouling discussed earlier, as follows:

i . Growth due to material transport Particulate fouling Precipitation fouling Chemical reaction fouling

ii. Growth due to nutrient transport Corrosion fouling (oxygen being the nutrient)

iii. Growth due to both material and nutrient transport Biological fouling Combined categories of fouling

In the following three subsections these growth processes will be considered in greater detail.

If the transported material is in the form of ions, molecules or very small particles (smaller than 1J.Lm or O. lJ.Lm , typically) then the transport is diffusional in nature and the transport coefficient (kt) can be identified with the mass transfer coefficient (km). The latter can be obtained from the published mass transfer correlations of experimental data or from theoretical equations, provided the mass diffusivity (D) of the transported material can be determined. tt

Fouling which grows due to the transport of ions, molecules or fine particles can be classified in the categories of �, or �.

Experimental data on mass transfer in binary systems (i.e . , involving two chemical species) in forced convection is usually presented in the form of an equation involving dimensionless quantities, thus,

Sh = C Scl1l Ren (17)

where Sh = Sherwood number, which is a dimensionless mass transfer coefficient, and is defined in the NOMENCLATURE, Sc = Schmidt number, which is a dimensionless ratio of the momentum (u) and mass transport difusivities (D), and Re = Reynolds number, which is a dimensionless fluid velocity, and is defined in the NOMENCLATURE. The constant C and exponents m and n are determined by measurement, and depend on the geometry of the system and the fluid, chemical and thermal conditions in the system. It is worth noting that, provided the mass flux of the depositing component and its concentration, are not too large, equations of the form. of equation (17) can be deduced from the usually more widely available equations for heat transfer by substituting the Sherwood number (Sh) for the Nusselt number (Nu), and the Schmidt number (Sc) for the Prandtl number (Pr).

tt The determination of the mass diffilsivity is very complicated and is hindered by the limited infonnation that is available. A comprehensive discussion will be found in Skelland [11].

528

The deposit growth rate (mg) has been computed using the mass transfer model by

Preci pitation fouling: Hasson [12]

Chemical reaction fouling: Watkinson and Epstein [13] and Crittenden and Kolachowski [14]

Particulate fouling: Newson, Bott and Hussain [15]

The application of the simple mass transfer concepts to particulate fouling must be made with caution, because

(a) for larger size particles (diameters in excess of O. I�m to l�m) the diffusion mechanism for transport is replaced by mechanisms based on particle fluid mechanics:

(b) for even larger particles (diameter in excess of 10Jl.ID) gravity effects can control particle motion;

(c) in the presence of a temperature gradient, particles can move under the influence of that gradient, this is called and acts so as to increase the rate of transport of particles to the heat transfer surface.

Beal [16] has devised a comprehensive model of particulate fouling that recognizes the relative contributions of diffusion and particle momentum. He assumed that the conventional mass transfer techniques could be applied to all points in the fluid, except for a certain region immediately adjacent to the heat transfer surface. In this region fluid motions are dominated by viscous effects, and, accordingly, mass transport is dominated by diffusion effects. Such a transport mechanism is not appropriate for the larger sizes of particles, so Friedlander and Johnstone [17] assumed that a particle, rather than being transported directly to the surface by diffusion and turbulent eddies, is only transported by these mechanisms to within a certain distance of the surface, called the (S). From that point particles are imagined to coast to the wall by virtue of their momentum. Since the stopping distance is the distance a particle, with some initial velocity V po will coast in a stagnant or slowly moving fluid under the influence of viscous drag forces. This means that

d2p p p S = --Vpo 18�

(18)

529

Beal [16] modified this result by noting that the particle center would be at a distance dy'2 when it contacted the surface. That being so

d� d S = --Vpo + �

18J.1 2

The initial particle velocity V po would be

Vpo = O.9 < U > �

(19)

(20)

which is equal to the magnitude of the average velocity at its point of origin in the fluid where y = 80u/« u> fl2).

To determine the rate of transfer of particles to the fouled surface Beal [16] and Friedlander and Johnstone [17] used the expression

km fl2 <u> = U + U + (21)

Jc

JC d +

+ du+ + (Sc - 1) + 1 + Us Us where u/ and uc+ are, respectively determined at y = S and at the duct center line. For S > 30u/<u> with a Schmidt number (Sc) of unity, equation (21) takes on the simple form

km fl2 <u> =

1 - 13.73 ..JV2 where us+ = 13.73.

(22)

To relate the mass transfer coefficient (�) defined by equations (21) and (22) to the concentration (Cs) of particles at the heat transfer surface, Beal assumed that the particle concentration at points between the surface (y=O) and the stopping distance (y=S) is uniform and equal to the concentration at the surface.

The rate of growth of fouling due to the formation of corrosion products on the heat transfer surface depends, at least in part, on the rate of transport of oxygen to the surface. This can be viewed as a 'nutrient' for fouling, in the same way that microbial fouling depends in part on the rate of transport of true nutrients to the deposit. Because microbial fouling involves a combination of transport processes involving the fouling material (microbes can be viewed as very small particles) and the nutrient, consideration of that case is postponed until the next subsection.

The corrosion fouling of a metallic (metal M) heat transfer surface exposed to flowing oxygenated water involves the following overall chemical reaction

530

4 4 02 + 2H� + Z M --+ z M(OH)z

where z is the valence of metal M.

(23)

On removal from the water, and drying, the metal hydroxide loses its constituent water and the deposit consists of metal oxides. These oxides and hydroxides constitute the fouling deposit.

It is clear that the rate of growth of the fouling deposit depends on the rate of transport of the oxygen dissolved in the water to the heat transfer surface (assuming the cathodic reaction controls corrosion) , and its rate of consumption by the corrosion reaction. So we can write for the rate of growth

(mg) of the fouling deposit on the surface

• dml mg = KI de

where from equation (23)

and dml/de is the rate of oxygen transport and consumption.

(24)

(25)

Oxygen transport to the heat transfer surface can be viewed as involving two steps:

(a) convective transport from the bulk of the water to the interface between the corrosion products (fouling deposit) and the flowing water

(b) diffusion through the corrosion products.

Hence, we can write

dml _ Cbl - CsI dO" - 1 -- + ­km1 D1

(26)

Assuming for the moment that CsI is known, we will proceed to investigate the character of the mass transfer terms in the denominator of equation (26). The most realistic discussion of oxygen transport is due to Mahato et al [18-20]. From their calculations and measurements it can be shown that equation (26) can be written

(27a)

where kmo is an overall mass transfer coefficient

53 1

(27b)

To use equations (27) it is necessary to know Sno, kl, k2, and k3. These must be obtained by experiment. In addition a value must be assigned to the mass transfer eddy diffusivity (Ec).

An experimental investigation of corrosion on the inside of pipe under isothermal conditions was cared out by Mahato et.al. [20] and they reported the following values: k1 = 3.517 x 10"" d Re-0.371 [dm7 mg-2] k2 = 2.834 x 10"" Re-0.785 [m31kg]

k3 = 1.189 x 10-3 Re-O·256 [mg-l]

(28a)

(28b)

(28c)

Fouling due to microbial growths on a heat transfer surface involves transport of the fouling material, in this case the microbes which may be treated as small particles in the size range 0.51lm to 20llm [21], and the transport of nutrients to keep the microbial growth alive. So two parallel mass transfer processes are involved. Presumably, a similar situation exists where two or more categories of fouling are present on the heat transfer surface. From a theoretical point of view, fouling in these circumstances involves multicomponent mass transfer. This is a topic of current research, and only a few theoretical results are available for very simple flow situations. Certainly no attempts have been made, as yet, to introduce these concepts into the study of fouling. In these circumstances it does not seem appropriate to consider a strictly fundamental approach. Instead the overall empirical model proposed by Bryers and Characklis [22] for microbial fouling will be briefly described.

Bryers and Characklis [22] have proposed that the rate of formation (dmtld9) of the fouling deposit be given by

dmf d9 = kmf (29)

where k is the rate constant for the rate of formation of the deposit. This quantity is assumed, on the basis of empirical evidence, to be given by

(30)

where kl is an empirical constant (called the specific biofilm accumulation rate constant), Cb is the bulk concentration of the biomass, Re is the Reynolds number, and Il is the dispersed biomass growth rate. Analysis of

532

experimental data showed that a = 1, b = -1, c = 1, hence equation (29) can be written

dmf CbJl de = kl Re mC (31)

where kl = 125.0 ±25.0 l/mg. Note the biomass growth rate (Jl.) is obtained in a 'static' ('beaker' or batch) test, but applied to fouling under flow conditions. The convenience of this technique is obvious, but its validity is uncertain.

This approach appears to have had reasonable success with microbial fouling, so it might form an excellent starting point for studying the growth of combined categories of fouling that involve two or more parallel mass transport processes in the growth process.

2.5 in the deposit When the fouling material arrives at the heat transfer

surface it must be attached to that surface. There is considerable uncertainty about this process, but the available information will be summarized. It is convenient to divide the discussion according to the categories of fouling.

The mass transfer coefficients (km) deduced by Beal [16], and described above, can be used to compute the flux of particles (Ns) transported to the heat transfer surface using

(32)

Observation suggests that only a fraction of the particles transported to the heat transfer surface remain there and form a fouling deposit. Beal [16] assumed that the flux of particles that stick on the surface is proportional to the concentration (Cs) of particles at the surface and is also proportional to the average velocity (v) of the particles normal to the heat transfer surface. Hence

(33)

where p is the fraction of particles in the region next to the heat transfer surface that stick on the heat transfer, sometimes called the

Eliminating Cs between equations (32) and (33)

(34)

In estimating the normal component of velocity (v), Beal [16] assumed this was the sum of the normal component of the turbulent fluid velocity and the average diffusion velocity (the original reference should be consulted for details).

533

A major limitation of the theory of Beal is that he assumed. because of lack. of information. that the sticking probability (p) was unity. That is. all the particles transported to the wall stick on the wall. Subsequently two alternative expressions for the sticking probability have been developed. one due to Bea1 [23] and the other by Watkinson and Epstein [24].

Beal [23]:

( Cu (35a)

where C and n are constants depending on the fluid and on the value of S (see the original reference for details).

Watkinson and Epstein [24]:

a ( E ) p = exp - -< u >

2 f RT

(35b)

where a is an empirical constant, E is an activation energy (also determined empirically). R is the universal gas constant, and T is the absolute temperature. Clearly this expression difers from that proposed by Beal [23] in that it includes a temperature effect in the form of an Arrhenius type expression. In addition the velocity dependence is different. Thus, in Beal's expression p - 1I<u>5 . whereas in the Watkinson and Epstein formula p -1I< u > 2 . In view of the empirical nature of these expressions it is not surprising these differences exist.

In precipitation fouling the material attaches to the surface by a process known as surface integration, which can be represented by the expression

(36)

The attachment rate constant is assumed to depend on the temperature (Ts) of the heat transfer surface through an equation of the Arrhenius type

(37)

In chemical reaction fouling it is usual to assume that the fouling material forms on and attaches to the heat transfer surface according to

534

(38)

where it is assumed that the fouling material is completely consumed at the heat transfer surface. so that Cs = O.

It will be assumed. following Somerscales [25]. that the oxygen consumption reaction is first order. then

. mg = 1 � -- + ­kml Dl

(39)

where Kl is given by equation (25), and kml and �IDI are given by equation (27b).

According to empirical evidence gathered by Fletcher [26]. the rate of attachment of micro-organisms to a surface is given by

(40)

where A. is the fraction of the surface covered by micro-organisms. This particular expression does not appear to have been used in work on the microbial fouling of heat transfer surfaces, presumably because no suitable model for the transport of the organisms has been deduced.

2.6 Removal Proces Removal of deposit from the surface involves a release

"reaction" together with transport into the flowing fluid at the deposit-liquid interface. Very little is known about the removal of fouling deposits, but three plausible mechanisms have been proposed, they are

(a) dissolution - material leaves in ionic fonn; (b) erosion - material leaves in particulate fonn; (c) spalling - material leaves as a large mass.

It is proposed in this section to review each of these three mechanisms separately.

Assuming, for the purposes of illustration, that the fouling deposit consists of the metal hydroxide M(OH)z' that is. corrosion fouling is under consideration, then removal of the fouling deposit involves the dissolution of this substance according to

(41)

535

Assuming that the rate of the dissolution reaction is much greater than the rate of transport, it can be shown for [M+Z] « [ORe] that

(42)

The mass transfer coefficient k would be provided by the usual techniques discussed above.

Models for deposit removal by erosion have been proposed by Bartlett [27], Charlesworth [28], and Beal [16].

Bartlett proposed that

(43)

where � = constant, F" = effective applied force per unit area exerted by the flowing fluid on a particle, q, = effectiveness factor for the projected area (Rp) of the particle, ap = projected area of individual particles forming the deposit, Pp = particle density, and Vp = volume of the particles. The quantities �, F", • and ap have to be determined either theoretically or experimentally.

Bartlett's equation is based on a model of erosion release that assumes that the shearing force exerted by the flowing fluid causes weak bonds, where it is assumed that the particles are bonded to their neighbors, to break. The weak bonds are seen as developing as a result of an accumulation of small particles which combine to form a large particle. Eventually sufcient small particles have accumulated so that the resulting force exerted on the particle agglomeration by the fluid flow causes the weak bond joining the particles to its neighbors to break.

Charlesworth [28] supposes that the rate of removal of the deposit increases as the amount of deposit increases. The simplest model of this type would assume that the rate of removal is directly proportional to the amount of deposit on the surface and inversely proportional to the residence time (9R) of the deposit on the surface. Thus

dmr mf d9 = 9R

In the case cited by Charlesworth, 9R = 100 hours.

(44)

The only model that is clearly related to actual fouling conditions is due to Beal [ 16], who was able to provide numerical values for the constants in his removal rate expression. However, so far as can be ascertained, this model has never been tested.

Beal assumes that

536

(45)

where F is an empirically determined force [29] which acts on individual particles given by

(46) where vpc = the fluid velocity at the center of the particle. In equation (45) ke is the transport coefficient for eroded particles.

Consideration of the preceding discussion suggests that erosion and spal1ing differ by virtue of the factors that are important in the two processes. For erosion these are: fluid velocity, particle size, surface roughness, and bonding of the deposit. Spalling, however, depends on thermal stresses set up in the deposit by the heat transfer process, stresses induced by the addition of material within the deposit structure, changes in the deposit structure induced by thermal effects, and poor bonding of the deposit to the wall. The latter may b� caused, as pointed out by Ross [30], by changes in the solubility of the deposit resulting from changes in the temperature at the deposit-heat transfer surface interface that occur as the deposit grows on the transfer surface.

Models that attempt to quantitatively characterize spalling have been proposed by Kern and Seaton [5,6] , Taborek et.aI. [31] and by Loo and Bridgwater [32]. The first of these, due to Kern and Seaton, assumed that the spalling was caused by the shear stress ('ti) at the deposit-fluid interface. They further assumed that the rate of removal increased in proportion to the mass (mf) of the fouling deposit. Hence, they said

where C is an empirical constant.

Taborek et.al. [31] proposed that the rate of removal be given by

dmr 'ti a CI8 = C'V" mf

(47)

(48)

where 'I' is a function of the deposit structure and must be determined experimentally together with C and a. Information on 'I' can be obtained from the data reported by Morse and Knudsen [33].

Loa and Bridgwater [32] investigated the spalling of fouling deposits by considering the stresses set up in a deposit as a result of the accumulation of the deposit, and as a result of temperature gradients due to the heat transfer through the deposit. They show that the maximum stress in the deposit formed on the outer surface of the cylinder is given by

537

(49)

This stress will induce fracture in the deposit if it reaches a magnitude given by

(50)

where KIc = material constant depending on the physical processes occurring at the crack tip during crack propagation, al = geometric factor, a = initial flaw size. KIc can be determined experimentally by the introduction of a large artificial flaw and then loading the specimen until fracture.

2.7 Limitations of Fouling Models Attempting to understand fouling by considering the fundamental

processes that contribute to the overall fouling effect has the following limitations:

a . There i s a lack of information on many of the transport and chemical kinetic processes. This is particularly true as far as the removal processes are concerned.

b. Even if a satisfactory model can be deduced, its application is limited by the availability of suitable information on the properties of the fouling deposits, such as, the density (Pf), the thermal conductivity (kf)' nutrient diffusivity, and so on.

For these reasons the engineer concerned with the practical aspects of fouling is likely to find empirical data much more useful. Nevertheless, the utility of this data can be enhanced if simplified models, suggested by the theoretical considerations discussed in this lecture, can be used to put the data in a suitable mathematical form. The discussion of such simplified, quasi­empirical models is considered below.

3. EMPmICAL FOULING MODElS

3.1 Introduction Basing the design of heat transfer equipment subject to fouling on the

assumption that fouling effects are independent of time, as is implied in the conventional fouling factor, can lead to substantial errors in design. There is a need for simple, practical methods for estimating the effect of fouling on the performance of heat transfer equipment. Such methods are based on mathematical expressions that represent a combination of theoretical ideas

538

and experimental measurements. This lecture will review a representative selection of models for fouling.

The models will be classified by the observed temporal character of the fouling thermal resistance, i.e., (a) falling rate, (b) linear fouling, (c) asymptotic fouling. These modes of fouling are sketched in Figure 1.

The various models are based on the fundamental model of Kern and Seaton [5] that views fouling as a competitive process between the growth of the fouling deposit and its removal. Mathematical expressions are then derived for the three modes of fouling. Applications of these models in practical situations are described. It should be noted that for mathematical representations of the removal processes, the information given in the section entitled FUNDAMENTAL PROCESS OF FOULING wil be drawn upon.

An important feature of fouling is the so-called induction time, which is the time, when the surface is first exposed to the fouling stream, during which no detectable fouling is observed. The limited amount of information available on this phenomenon is reviewed.

8.2 Fal Rate Fouling The earliest mathematical model of fouling was proposed by McCabe and

Robinson [34] to describe the fouling of evaporators. Although this predated the Kern-Seaton model, it actually represents a special case in which the rate of growth (mg) of the deposit was proportional to the temperature difference between the deposit-liquid interface (Ti) and the bulk of the liquid (Tb). Hence

(51)

where lCr is a combined mass transfer coefficient and reaction rate constant (which, presumably, could be evaluated by the methods described above), and B is a temperature coefficient of solubility.

Combining equation (51) and the Kern and Seaton model with the assumption of no deposit removal we have

dRf =

lCrB (T. - Tb) dO Pfkf 1 (52)

To eliminate the interface temperature (Ti) in favor of the observable temperature (T s) of the heat transfer surface, we have

T. - 'E Rr(Ts - 'Ib) 1 - s Rc + Rf

where Rc is the convective thermal resistance at the deposit-fluid interface. Combining equations (52) and (53) we have

(53)

where K2 = x:rB/pptr.

539

(54)

If it is assumed that the temperature difference T s - Tb is constant (a reasonable assumption in the steam heated evaporators under consideration by McCabe and Robinson), we have on integrating equation (54)

Rr=Rr 9=9 I (Rc + Rf)dRf = K2 (Ts - 'Ib )Rc Id9 (55)

Rr=O 9=0

On integrating, assuming the convective resistance Rc at the deposit-fluid interface remains constant,

(56)

where K3 = 2K2(Ts - Tb)Rc' and R = Rc + Rr = total measured thermal resistance at the heat transfer surface.

This model predicts "falling rate" fouling. If the model is valid, then plotting the square of the total thermal resistance (R) measured in some item of heat transfer equipment against the elapsed time since it was exposed to the fouling stream will give a straight line.

In 1964 Reitzer [35] published a more general model than that used by McCabe and Robinson. It was assumed that the growth of the deposit from saturated solutions depends on an undefined power (n) of the temperature difference Ti - Tb' Then equation (51) becomes

mg = � Bn (Ti - Tb)D (57) Again, assuming the temperature difference T s - Tb is constant, we have on

following the procedure used to determine equation (56)

Rn+1 = R�+1 + Ka9 (58)

n where Ka = (n+1)K2(Ts - Tb)D Rc ' Clearly, equations (57) and (58) contain the

McCabe and Robinson model as a special case where n=1 . Epstein [4] suggests that equation (58) can be used to represent data from the

fouling of heat transfer surfaces on which boiling is occurring. In particular, he proposes

n '!:" 1: relatively low velocity non-stirred systems, in which the fouling process is mass transfer controlled

540

n ':' 2: high velocity and/or well agitated systems where the fouling process is controlled by the crystallization process on the heat transfer surface.

Hasson [12] showed that equation (58) can be used to represent experimental data on the fouling of the inner surface of steam heated double pipe heat exchangers carrying a calcium carbonate (CaC03) solution in the inner pipe. The exponent n was found to be dependent on the fluid velocity as follows:

v < 50 cmJs: n = 2.5 (duration: 400 hours)

v > 50 crnls: n = 2.5 (duration less than 100 hours)

n+1 n+1 . R - Rc --+ constant (duratIon greater than 100 hours).

Watkinson and Martinez [36] , who used higher fluid velocities (v) than Hasson, also used the Reitzer model with n = 2 to describe the growth process for calcium carbonate fouling in a double pipe heat exchanger. They also took into account removal processes which had been ignored by previous investiga tors.

In deriving the falling rate fouling model it was assumed that the

temperature difference T s - Tb between the heat transfer surface and the bulk of the fluid remained constant, if, instead of that assumption, it is assumed that the heat flux at the surface remains constant then a linear relation between the deposit thermal resistance (Rr) and time (0) is obtained. Thus, in equation (54) put T s - Tb = q (Rc + Rr); then

dRf d9 =K.t

Integrating

(59)

(60)

If the Reitzer [35] model for growth from a saturated solution [equation (57)] is used, then we obtain for the fouling thermal resistance

(61)

541

where K4, = ��n R: qn/pftf. The linear fouling model has been found to represent the fouling of a Kraft

liquor heater [24], data from a seawater desalination plant [37], precipitation fouling with pure crystalline substances [38]. In spite of this apparent occurence of linear rate fouling in engineering practice, care must be taken in applying this model, especially on the basis of data taken from observation periods of limited duration. There is a distinct possibility that if an item of heat transfer equipment is observed for a sufciently long time, the linear behavior might turn to falling rate fouling or even asymptotic fouling (see below).

A very widely observed form of fouling is the so-called asymptotic fouling.

In order to model this type of fouling, Kern and Seaton [5] assumed that m g was a constant and that IDr was propo.rtional to the mass (mf) of the fouling deposit. With this assumption the desired asymptotic form for the dependence of the fouling on time was obtained, as will be seen below. Accordingly,

dmf •

dO = mg - b mf (62)

where b is a constant. On integrating

mf = mr [ 1 - exp (-O/Oc)] (63)

where m'f = mgfb (remember IDg is assumed constant), and 0c = lib. The quantity mf is the asymptotic value of the mass of the fouling deposit and ac is a time constant. The latter quantity represents the average residence time of an "element" of the fouling deposit on the fouled surface. Alternatively 0c can be viewed as the time it would take to accumulate the asymptotic fouling

deposit if the fouling proceeded linearly at the constant growth rate mg . Making use of the relation between the fouling thermal resistance (Rr) and

the mass (mf) of the fouling deposit we can write equation (63) as

Rr = Rf [1 - exp(-O/Oc)] (64)

where R'f is the asymptotic magnitude of the deposit fouling resistance (Rr). Equation (64) has been used to represent the fouling behavior of many items

of heat transfer equipment. For example: Watkinson and Martinez [36], Morse and Knudsen [33], and Watkinson [39], where the latter two cases referred to hard water of various mixed salt compositions, and the first case referred to a pure calcium carbonate solution.

The designer and user of heat transfer equipment subject to fouling would require values for R'f and 0c. if equation (61) were to be used to predict the

542

fouling performance of heat transfer equipment. Very little theoretical information is available that could allow the Jl m:i.m:i determination of Rf and ac' but empirical data is beginning to be accumulated relating to the values of the parameters. The work reported by Knudsen and his associates [40-42] represents a valuable body of empirical data relating to precipitation fouling from water flowing through heated tubes.

It is frequently observed that no reasonable fouling effect can be detected on

a heat transfer surface for some time after it is exposed to a fluid. This time is known as the induction or delay time (see Figure 1). The models discussed in the preceding sections do not include any effects for the induction time, so some empirical procedure must be used. Knudsen and Story [43] modified the asymptotic fouling expression, equation (61), as follows

(65)

where aD is the induction time. No information was provided on methods for determining aD a :wi2ri in a given fouling situation.

Ritter [38] studied the fouling in an electrically heated annular heat exchanger carrying saturated solutions of either calcium. sulfate (CaS04) or lithium sulfate (Li2S04). He reported the fouling empirical expressions for the induction time (aD).

(66a)

(66b)

where le, Cb and Cs are measured in SI units and an is given in hours.

4. DESIGN OF HEAT TRANSFER EQUIPMENT SUBJECT TO FOULING

4.1 Introduction A heat exchanger subject to fouling may be designed on the assumption that

fouling introduces a time independent additional heat transfer resistance (Rf), or, recognizing that fouling is really a time dependent phenomenon, a schedule for cleaning may be devised based on operating and/or economic

543

considerations. The latter approach may, or may not, involve the provision of additional heat transfer area over and above that required on the assumption that the heat exchanger is clean.

This section will consider ways of specifying the allowance for fouling effects. In addition a brief review wil be given of forms of heat exchanger for fouling service, and of practical design considerations for shell-and-tube heat exchangers.

4.2 Fouling Thermal Resistances and FouJing Factors The design equation for a heat exchanger can be written

(67)

If we represent the performance of a clean heat exchanger by the following modified form of equation (67)

Qc = Ac Uc FC L\TmC , and for a fouled heat exchanger

where

(68a)

(68b)

(69)

and U F and U C are the overall heat transfer coefficients of the fouled and clean heat exchangers, respectively. It should be noted that equation (69) implies that the fouling thermal resistance associated with all the fluid streams can be consolidated into a single fouling thermal resistance He.

On the assumption that fluid stream temperatures at inlet and outlet, and the flow geometry are the same for both fouled and unfouled heat exchangers, i.e. , FF = FC and L\TmF = L\TmC' then equations (68) clearly indicate that if QF = QC then the heat transfer area of the fouled surface (AF) must be greater than that of the clean surface. This is the basis of the design of a heat exchanger for fouling service assuming that fouling is a time independent process.

To design heat transfer equipment assuming that fouling does not change with time requires the specification of a fouling resistance (He) either by the designer or by the user (mostly by the latter). The commonest procedure is to make use of the well known fouling factors originally proposed by the Tubular Exchanger Manufacturers Association (TEMA). The origin of these fouling factors (actually fouling thermal resistances) is uncertain, but they can be viewed as values based on the experience of a representative group of engineers with an extensive background in the design of heat transfer equipment.

544

The fouling resistances published by TEMA should not be viewed as asymptotic fouling resistances, because that would suggest that heat transfer equipment designed on the basis of these values need never be shut down for cleaning. This does not seem to be implied in the information published by TEMA. Accordingly it is better to assume that a heat exchanger with dimensions given by the TEMA values will meet performance requirements with a "reasonable" time between shutdowns for cleaning. The interval between cleaning would, presumably, be based on the analysis of the performance of the heat exchanger while in service.

In using the TEMA resistance values it must be recognized that they are subject to a number of limitations: (a) they do not take into account the time dependent nature of fouling; (b) they are not related to the specific design features and operational characteristics of particular heat exchangers; (c) information is available only for a limited number of fluids.

4.8 The Factor and Percent Oversurf In addition to the fouling factor, or time independent fouling thermal

resistance. fouling requirements can be given in terms of the so-called cleanliness factor (widely used in the electric power industry for steam condenser design) and in terms of the percent oversurface. These are related to the fouling deposit thermal resistance (Rr) and it is proposed to demonstrate that relationship.

The cleanliness factor (CF) is defined as

CF = UF"Uc (70)

where UF and Uc are determined under identical conditions of flow and temperature. Because the cleanliness factor involves convective heat transfer conditions, as well as the thermal characte.ristics of the fouling deposit. it confuses rather than clarifies the understanding of fouling.

From equation (70) it is possible to obtain an explicit relation between the fouling factor and the cleanliness factor

1 - (CF) Rr= UC(CF) (71)

The requirements that must be met in order to accommodate the effects of fouling can be expressed in terms of the heat transfer area that must be supplied. Thus

Percent oversurface = -1) 100 (72)

Equations (68) can be used to relate the fouling factor and the excess heat transfer area. H qF = qC. FF = FC' and 6TmC = 6TmF. then

Percent oversurface = 100 U cRr (73)

545

Equation (73) is given in graphical form in Figure 2.

In order to make the preceding discussion concrete, a simple example due

to Knudsen [44] wil be considered. A typical distillation column water-cooled shell-and-tube overhead

condenser is to be designed for fouling service with the fouling restricted to the surfaces exposed to the cooling water. Determine the overall heat transfer coefficient (UF)' based on the tube outside surface area, when operating with a specified fouling thermal resistance (Rr) of 0.00035 m2K1W. It is to be assumed that there is no fouling on the condensing side.

It is probably simplest to clean the interior of fouled tubes, so the cooling water wil be in the tubes with the condensing vapor in the shell.

The required overall coefficient of heat transfer is given by

1 1 ( x

J

(Aw

J

( 1

J

Al - = - + Rfl + - - + - + Rf2 -UF hI kW A2 h2 A2

(74)

where x is the thickness of the tube wall, and subscript 1 refers to conditions inside the tube, subscript 2 to conditions outside the tube, AW is the average of the inside and outside areas for thin wall tube.

It will be assumed: hI = 2840 W/m2 K = h2; tube O.D. = 0.091m; tube wall thickness = 0.00165m; kW = 45 W/mK (mild steel).

Equation (74) becomes

= 121

U F 2840 45 2840

So UF = 804 W/m2K. This is divided among the various resistances as follows: outside tube convective: 28.3%; wall: 3.3%; fouling deposit: 34.2%; inside tube convective: 34.2%.

4.4 The Efect of Fouling on Pres Drop The fouling deposit effectively roughens a heat transfer surface thereby

affecting fluid friction. In the case of flow through ducts, a sufciently heavy deposit of fouling will decrease the hydraulic diameter of the duct, thereby increasing the pressure drop over a length of the duct if the fluid flow rate is maintained constant. The effects of fouling on the fluid flow in the shell side of the shell-and-tube heat exchanger must be important, but they do not appear to have been investigated.

Marner and Suitor [45] show that if the fouling deposit is treated as a layer of uniform thickness on the inside surface of a duct of circular cross section then

546

(75)

where df and dc are the diameters of the available flow cross sections in the fouled and cleaned ducts, respectively. The application of equation (75) requires knowledge of the deposit thermal conductivity (kf) and the fouling deposit thermal resistance. Experience suggests that the former lies between 0.03 and 10 W/mK.

4.5 Design Features that Fouling Selection of an appropriate type of heat exchanger and its careful design can

mitigate the effects of fouling [46]. Firstly consider the form of heat exchanger:

Direct contact heat exchangers: heavily fouling liquids, such as geothermal brines.

Fluidized bed heat exchangers: scouring motion of fluidized particles scours away the fouling deposit.

Plate and frame heat exchangers: easily disassembled for cleaning (widely used in food processing for this reason).

Scraped heat exchangers: fouling deposit removed by the scrapers.

Compact heat exchangers: not advised for fouling service, because they are difficult to clean.

If the heat exchanger form has been selected consideration should be given to design features that will reduce the effects of fouling. Chenoweth has reviewed this aspect of design as it applies to shell-and-tube heat exchangers. Thus [46]:

Tube-side fluid: should be the more rapidly fouling fluid, because the tube interior is easier to clean than the external surfaces.

Orientation: horizontal heat exchangers are easier to clean than vertical ones.

Elimination of stagnant and low-velocity regions on the shell-side: baming arrangements are important and close axial spacing between the bafiles usually decreases the size of stagnant fluid regions.

Enhanced heat transfer surfaces: may be subject to fouling because they provide regions in which the fouling deposit accumulates, however evidence is accumulating that certain surfaces have fouling minimization properties.

Tube bundle layout: it is easier to mechanically clean bundles with square or rotated square tubefield layouts.

547

Tube spacing: increasing the tube pitch will provide wider lanes for hydraulic cleaning.

5. FOULING AND BOll.JNG

5.1 Introduction Fouling under boiling conditions involves the following categories of fouling:

(a) precipitation; (b) corrosion; (c) particulate; (d) chemical reaction. Each of these categories is reviewed in the following subsections. Because of its technical importance, a substantial body of literature is available on this topic, but space will only permit the highlights to be considered. In spite of the work that has been done on fouling on surfaces with boiling heat transfer, the complexity of the topic means that the available knowledge is still very incomplete and our understanding of the subject is in a confused state.

5.2 Precipitation Fouling The fouling of steam generator heat transfer surfaces by

insoluble salts present in the feedwater was the earliest manifestation of fouling reported in the technical Ii tera ture [47]. However, this particular occurrence of fouling is no longer a serious problem l;>ecause chemical means of controlling the phenomenon have now been developed. Nevertheless, continual vigilance is maintained because of the adoption of new materials for the heat transfer surfaces and because of steady advances in steam pressures and temperatures. This is particularly important in nuclear reactors where corrosion products, released from other parts of the steam circuit, have been found to deposit on the heat transfer surfaces. These corrosion products are sometimes in particulate form (magnetite, FeaO 4) or sometimes in dissolved form (hematite, a-Fe20a). The latter when it deposits on the heat transfer surface is a manifestation of precipitation fouling. Precipitation fouling under boiling conditions also occurs in evaporators used for sea water purification, sugar refining and crystallization.

No comprehensive review of precipitation fouling in the presence of boiling appears to have been published since that prepared by Partridge [48]. This deals with the relevant literature up to 1929, with a particUlar emphasis on fouling in steam generators. According to Partridge, the earliest investigation on the mechanism of precipitation fouling with boiling is due to Couste [49], who observed that calcium sulfate (a frequent contributor to fouling in boilers using untreated water) is an inverse solubility salt. He further proposed that local Bupersaturation of the solution in the vicinity of the heat transfer surface could be the cause of fouling. No further fundamental work on fouling under boiling conditions was reported until Hall [50] presented the results of an investigation that appeared to support the hypothesis of Couste.

No measurements of the fouling thermal resistance (He) due to precipitation fouling in the presence of boiling heat transfer have been made. This is a consequence of the difficulty in separating the fouling thermal resistance (Rr) from the convective thermal resistance (Re) and the directly measurable overall thermal resistance (R), as shown in

548

equation (1). However, measurements of the total thermal resistance (R) have been reported by Schmidt and Snodgrass [51], Reutlinger [52], Croft [53] and Partridge [48] (see also Partridge and White [54,55]). Of these, the most accurate are probably those of Reutlinger and of Partridge. If it can be assumed that the convective thermal resistance (Rc) is the same on the clean and fouled surfaces (a dubious assumption), then the fouling thermal resistance can be obtained from Rc = R - Rc.

Another approach is to determine the mass (mf) of fouling deposit formed on unit area of the heat transfer surface then Rc can be determined from equation (12), provided the thermal conductivity (kf) and density (Pf) of the deposit are known. Appropriate measurements, in pool boiling in a saturated calcium sulfate solution, of the mass of fouling deposit were first reported by

Partridge [48], who proposed that the net rate of formation (mf) of the fouling deposit be given by

·

mf= - K5 dT qn (76)

Partridge made no attempt to evaluate the empirical constants Ks, m and n. The data on which equation (76) was based were obtained on the assumption

that the net rate of formation (�f) of the deposit was constant. Similar pool boiling measurements with saturated calcium sulfate solutions using this assumption have been reported by Palen and Westwater [56] and by Curcio [57]. The latter measurements involved plain and enhanced heat transfer surfaces.

Palen and Westwater represented their results as (assuming mg = 0)

mf= Ksq2 (77)

It can be shown (see Palen and Westwater [56]) that this implies that the exponent n=2 in Reitzger's model for faling rate fouling [equation (57)].

Experiments reported by Asakura et.al. [58,59] and by Mizuno et.al. [60] using hematite (a-Fe203) solutions have considered in addition to the heat flux (q), the latent heat of vaporization (hfg)' the bulk concentration of the solution (Cb)' and the time of exposure (9) of the surface to fouling conditions. They represented their results by

m m

f= K7 (�

) Cben hfg

with n = 0

(78)

549

Unfortunately, so far as the author is aware, no data is available on the properties (kf, PC> of the hematite deposit, so at this time equation (78) cannot be used to determine the fouling thermal resistance Re.

gf * Partridge [48] (see also Partridge and White [55]) studied the formation of the calcium sulfate deposit on the heat transfer surface and noted that at an early stage in the growth of the deposit it was seen to be in the form of rings. On the basis of this observation they hypothesized that the deposit was formed at the triple interface between the liquid, the vapor in the bubble and the heat transfer surface (see Fig. 3). Similar observations for calcium sulfate have been reported by Schmid-Schonbein [38] and for hematite deposits by Asakura et.al. [58,59]. Further work on the deposition of calcium sulfate, subsequent to that of Partridge [48], by Hospeti and Mesler [61] showed that the formation of deposits of calcium sulfate during nucleate boiling in saturated solutions was due to the evaporation of a microlayer (about 0.5 j.LDl to 2.6 J.1m thickness) of liquid beneath the bubble. Asakura et.al. [58] made use of this information to determine the constant K7 in equation (78), which was found to decrease with increasing flow rate of the solution over the heat transfer surface. Independent calculations by Schmid-Schonbein [37], using essentially the same model, have been carried out for saturated calcium sulfate solution.

3ENERAL SOLUTION

HEATING SURFACE AT ELEVATED TEMPERATURE

H IGH TEMPERATURE/AREA

CAUSING CRYSTALLIZATION OF SALTS WITH NEGATIVE

SOLUBILITY SLOPE

Figure 3. Dynamic mechanism of precipitation fouling deposit formation under boiling conditions proposed by Freeborn and Lewis [62].

550

llI The time dependence of the net rate of formation (�f) of the fouling deposit was observed by Mizuno et.al. [60], who found that it was constant up to some critical time, which depended on the concentration of hematite in the solution. Measurements of the heat transfer surface temperature by Palen and Westwater [56] indicated the temperature initially increased at a constant rate, then decreased, followed by an increasing trend. It seems reasonable to assume that these observations are related, but the exact nature of this relationsip requires further investigation.

Similar tests to those reported by Palen and Westwater have been carried out by Curcio [57] using enhanced heat transfer surfaces. He found that the surface temperature initially increased, but after about two to five hours (longer than Palen and Westwater, but shorter than Mizuno et.al.) the temperature did not increase significantly. This appeared to confirm results obtained by Gottzman, O'Neill and Minton [63] with UOP High Flux Q) enhanced heat transfer surface exposed to an aqueous solution.

As mentioned in the introduction to this section, the most important current application of the preceding is to evaporators handling seawater and certain other fluids, such as sugar solutions. However, these practical situations are complicated by the fact that the solutions handled involve a number of different salts that can precipitate on the heat transfer surface. Seawater, which is the most widely investigated evaporator fouling fluid typically contains several saltsttt that can precipitate on a heat transfer surface. Typically, but the composition is variable (surface temperature has an important effect), the deposits consist of calcium carbonate (CaC03)' magnesium hydroxide [Mg(OH)2]' and calcium sulfate (CaS04)'

Work on scaling in seawater evaporators up to 1958 has been comprehensively reviewed by Badger and Associates [64]. Probably the most important studies on this occurrence of fouling are due to Langelier et.al. [65], Hillier [66], Dooly and Glater [67], and Rankin and Adamson [68].

The investigation of Langelier et.al. [65] was concerned with the chemical mechanisms of fouling, and showed that the effect of carbon dioxide, as it influences the carbonate ion concentration is important. Hillier's study involved the use of a small evaporator in which heat was supplied by the condensation of steam in tubes submerged in the brine. He concluded from his tests that a minimum fouling occurred at 180°F, and that this temperature was a transition temperature such that calcium carhonate predominated in the deposit at lower temperatures and magnesium hydroxide was the major constituent at higher temperatures. The amount of deposit formed was proportional to the amount of seawater processed and the rate of

formation (�� of the fouling deposit increased with decreasing heat transfer surface temperature. These findings were questioned at the time of their publication and subsequently by Badger and Associates [64], and by Dooly and

ttt Typical analysis (parts per million by mass): calcium bicarbonate, Ca(HCOa)2: 180; calcium sulfate, CaS04: 1220; magnesium sulfate, MgS04 : 1960; magnesium chloride, MgC12: 3300; sodium chloride, N aCI: 25,620.

55 1

Glater [67]. However Rankin and Adamson [68] did find good agreement

between the rate of formation (mf) of the deposit as measured by Hilier and by themselves.

It is probably true to say that our understanding of fouling in seawater evaporators is very incomplete at this time, but this is not surprising considering the complexity of the fouling solutions.

Precipitation fouling in the presence of boiling is stil not fully understood, but the success in controlling fouling in steam generators suggests that where the economic incentive exists even a fouling situation as complicated as this is capable of resolution.

5.3 Corrosion Fouling Corrosion under boiling conditions has not been extensively

studied, and its effects on heat transfer have received even less attention. This type of fouling could be important in steam generators and engine cooling systems, for example, however corrosion inhibitors are universally used in these applications. Probably corrosion fouling in the presence of boiling heat transfer is only of importance when the use of inhibitors is impossible, such as, in chemical processing. The author is not aware of specific examples in the latter situation, so this discussion is restricted to some observations of corrosion fouling in laboratory boiling experiments where the metallic test surface has been exposed, on purpose or inadvertently, to oxygenated water.

EfmdiI m mllmW l: I:wat transfer. Bui and Dhir [69] have reported the effects of an oxide deposit on pool boiling heat transfer in water. Typical results are shown in Fig. 4(a). The nature of the oxide is not given, but it was probably a tarnish on the copper test surface, since the surface is described as clean with a mirror finish. It will be seen that this oxide has no effect on nucleate boiling (the left hand portion of the figure where &T is increasing with increasing heat flux, q). However there is a definite effect in the transition regime between nucleate and film boiling. No data was provided in the original reference on the change in heat transfer with time as the oxide film develops (presumably the oxide film was developed by exposing a clean surface with a mirror finish to the laboratory atmosphere for several hours). It is unlikely that oxide in the form of a tarnish will have much effect on the heat transfer and fluid flow in the vicinity of the surface, other than affecting the number of bubble nucleation sites. The observed shift of the transition boiling curve to higher values of & T is therefore probably an indication of the magnitude of the conductive thermal resistance (&lkf) of the oxide film.

Some other data provided by Bui and Dhir demonstrate the effects of a so­called deposit of dirt on the heat transfer surface. Since this was formed, according to the authors, when it operated "in nucleate boiling for several hours in the liquid pool exposed to the laboratory environment," and since the water was normally kept under nitrogen pressure in the test chamber it is likely that in the usual course of events little or no corrosion occurred while the apparatus was being operated. This suggests that what Bui and Dhir cal dirt may in fact be, to a greater or lesser extent, a heavy oxide deposit. As will be seen from Fig. 4(b), curves are shown for "Some Deposit of Dirt" and "Heavy

552

\ \ .

SURFACES ARE CLEAN

AND MIRROR F IN ISH

o . \ / \ �

\ \ WITH OXIDE 15 == 4 5

NO O X I D E == 90

o S T E A D Y S T A T E I RU G> T R A N S I E N T C O O L I N G N 4 9

V S T E A D Y S T A T E I � T R A N S I E N T C O O L I N G R U N 4 8

1 0

A T ( K ) 1 0 0 200

Figure 4(a). Effect of oxide film (tarnish) on pool boiling, according to Bui and Dhir [69]

Deposit of Dirt". presumably this is some indication of the progressive effect of fouling with time of exposure. As in the case of the oxide film [Fig. 4(a)], the effect of the deposit is to shift the transition boiling curve to higher values of AT for a given heat flux (q). It is possible, depending on the nature of the deposit (see the section below on particulate fouling), that this is a manifestation of both the conductive resistance of the deposit (Slkf) and its effect on the convective thermal resistance (Rc)'

There is a clear need for more investigations of the interaction between corrosion and heat transfer under boiling conditions.

5.4 Particulate Fouling Important examples of particulate fouling under boiling

conditions occur in fossil fuel fired and nuclear steam generators. In the former case, this is usually combined with corrosion products formed at the

.. '0

� S T E A D Y S T A T E 1 .. T R A N S I E N T C O O L I N G I R U N 5 0

• T R A N S I E N T H E A T I N G 0 S T U D Y S T A T E I 0 T R A N S I E N T C OO L I N G j RUN 5 1

• T R A N S I E N T H E A T I N G

0 S T E A D Y S T A T E 'I [3 T R A N S I E N T C O O L I N G RUN "6 • T R A N S I E N T H E A T I N G

\ \ DEPOSIT

.. ; 10 \ D E P O SIT

i A OF DIRT

CT

NO 0/ 6 \ . � / "8 $.A-J.,A '-A

ALL S U R F A C E S OF SAME

R OUGHNESS E.800

1 0 1 0 0

�T ( K )

2 0 0

553

Figure 4(b). Effect of surface deposit (corrosion product?) on pool boiling according to Bui and Dhir [69]

steel heat transfer surface (the extent depending on the effectiveness of the corrosion inhibition of the boiler feedwater). Nuclear power steam generator heat transfer sufaces, on the other hand, are usually made from zirconium alloys which corrode very little under heat transfer conditions. However, such surfaces are subject to fouling by corrosion products, produced elsewhere in the flow system, and transported to the heat transfer surface in dissolved and particulate form. The case of precipitation fouling, typicaly by dissolved hematite (a-Fe 20 3)' has been discussed above in the section devoted to precipitation fouling and boiling.

The fouling of fossil fuel fired boiler heat transfer surfaces can lead to excessive surface temperatures followed by damage and failure. This is a consequence of the normally high temperatures and heat fluxes at which these

554

operate. In nuclear reactors. which operate at comparatively low temperatures and heat fluxes. fouling can result in the transport of radioactivity outside of the reactor and possible damage to the heat transfer surface is a secondary consideration. * Such transport of radioactivity is undesirable from both the safety and the convenience points of view.

Particulate fouling in such industrial devices as evaporators and kettle reboilers does not appear to have been considered. This may be an indication of the relatively greater importance of precipitation fouling in these cases.

A good review of particulate and precipitation fouling under boiling conditions. with special reference to nuclear steam generators is due to Lister [70.71].

gf The only attempt to formulate a complete theory of particulate fouling under boiling conditions is due to Charlesworth [28]. although there have been a number of theoretical studies of certain aspects of the fouling process (see. for example. Gasparini et.al. [72]. Iwahori et.aI. [73]. and Styricovich et.al. [74]).

Charlesworth [28] made use of the Kern and Seaton [5] model for asymptotic fouling discussed above. In terms of the net amount (mf) of deposit formed on the surface we have from equation (63)

mf = mr [1 - exp (-O/Oc)] (79)

On the basis of a number of tests on heat transfer surfaces located both in the nuclear reactor and outside the reactor. Charlesworth obtained the following empirical values for the constants in equation (79)

m r = 25 kg/cm2

and

0c : = 100 hours

On the basis of earlier work by Mankina [75]. it is frequently stated that the net

rate of formation (�f) of the fouling deposit on a surface with boiling heat transfer is proportional to the square of the heat flux at the surface.** so the above values of mr and 0c can only be assumed to apply at one heat flux.

which. according to Charlesworth. was between 80 and 100 W/cm2 . Charlesworth's model has been used to calculate particle transport rates in nuclear reactor heat transfer systems (see. for example. Burrill [76-78]).

Simpler models for particulate fouling under boiling conditions. which owe their origin to the work of Mankina [75]. cited above. assume a dependence on

* However the so-called denting of fuel elements as a consequence of the accumulation of transported corrosion products in flow stagnation regions and their possible consequent effect on local temperatures should be noted. it Se further discussion below.

555

time that is different from the asymptotic form of Charlesworth. Examples are due to Asakura et.al. [58,59], Mizuno et.al. [60], and to Picone and Fletcher [79]. The first three references have already been discussed under precipitation fouling*** where the following expression was given

(SO)

Picone and Fletcher proposed

(81)

Equation (81) was not able to explain the measurements of fouling deposits in the Saxton reactor. Kabanov [80] sugested that a form like equation (80) would be better in which m=l and n=O.

An important investigation by Thomas and Grigull [81] considered the deposition of magnetite on heat transfer surfaces under both boiling and non­boiling conditions. In the absence of boiling, heat flux had little effect on the net rate of formation of the fouling deposit, but the turbulence level was very important. In nucleate boiling the extra turbulence induced by bubble motion led to an increase in the rate of deposition. Under film boiling conditions, the film of vapor at the surface impeded the transport of particles to the surface, so a decrease in the net rate of formation of the deposit was observed. The net rate of formation of the deposit was given in the form of a mass transfer Stanton number relation, namely

Stro = Stro 0 exp (-mth) I

(82)

where

(83)

and Stro,o is the initial (clean surface) mass transfer Stanton number. The initial net rate of formation of the deposit was given by

:ri:tf,o = K8 Re1.073 (84) Detailed discussions of the physical and chemical mechanisms of particulate fouling in the presence of boiling have been made by Gasparini et.al. [72] , Iwahori et.al. [73], and by Styricovich et.al. [74]. The speculative nature of

*H The work of these authors involved hematite (a-Fe203) which was injected into flowing water in a finely divided form (partide diameters -- 3.5 J,1m - 0.1 J,1m) part of which probably dissolved in the water and was deposited on the heat transfer surface by precipitation.

556

these theories, which is a consequence of the difficulty of making the associated measurements, does no warrant space being devoted to their consideration at this time.

.lAd IwU transfer. Fouling deposits, particularly those formed in the presence of boiling, are frequently porous in nature and this bas an important effect on the heat transfer at the surface. Macbeth [82] has proposed that 'chimneys' in the deposit play a role in the bubble formation and heat transfer. A conceptual diagram of such a chimney is shown in Fig. 5. Macbeth used this model to compute the effect of the chimneys on boiling heat transfer and showed (Fig. 6) that increasing size of the liquid capillaries alL) and of the vapor capillaries (Dy) increased the rate of heat transfer. Cohen [83] has used the chimney system to compute the rate of transport of foulant to the deposit. Particularly imporant in this respect are non-ferrous materials which substantially decrease the thermal conductivity (kf) of the deposit thereby increasing the fouling thermal resistance (Rr).

After precipitation fouling in the presence of boiling, particulate fouling has been studied the most extensively. In spite of this our understanding of this topic is very limited. There is assuredly insufficient information to allow the design of boiling heat transfer equipment subject to particulate fouling.

CAPILLARY CHANNELS

DRAWIN G UOUID TO THE BASE OF THE STEAM CHIMNEY

FLOW OF

STEAM ESCAPING FROM MOUTH OF

STEM� CHIMNEY BY SUCCESSIVE

FORMATION A ND RE LEASE OF STEAM BUB&LES.

__ = _

HEAT FLOW

CRUD DEPOSIT

THICK N E S S

Figure 5 . Proposed model of 'wick' boiling in a magnetite fouling deposit, according to Macbeth [82].

L � JX < L

EFFECTIVE CRUD VOIDAGE = 0'] CRUD THICKNESS 5O)MI

I !

10 20 )0 40 DIAMETER OF VAPOUR HOLES - ).1m

557

Figure 6. Maximum heat flux rates attainable in a porous fouling deposit with water at 1.0 bar, according to Macbeth [82].

5.5 Chemical Reaction Fouling Chemical reaction fouling is defined by Watkinson [84] "as a

deposition process in which a chemical reaction either forms the deposit directly, or is involved in forming deposit precursors (or foulants) which subsequently cause the deposition". The reaction does not involve the material of the wall, this would only occur in corrosion fouling. Precursors can be produced on the wall, in the wall region, or in the bulk of the fluid. The precursor, if formed in the fluid, may be soluble in the fluid and only precipitate on the wal. Alternatively, the precipitate may be in finely divided solid form in the bulk. of the fluid and deposit on the heat transfer surface by particulate fouling mechanisms. Clearly chemical reaction fouling involves features of precipitation fouling and/or particulate fouling. However it has not been possible to separate the processes of precursor formation from the

558

deposition process. In the presence oC boiling, Watkinson [84] suggests that the higher boiling point Coulant precursors will be concentrated in the liquid.

Chemical reaction Couling is most usualy observed in the heating of organic liquids, particularly petroleum refinery and petrochemical processing feedstocks. This type oC fouling also occurs in the fuel lines and combustor nozzles of gas turbines.

U seCul reviews oC chemical reaction Couing have recently been prepared by Crittenden [85] and Watkinson [84].

Thermal The most complete investigation oC chemical reaction fouling with boiling has been reported by Crittenden and Khater [86,87] who carried out experiments with kerosene. They demonstrated the importance oC liquid phase processes. This conclusion was based on two observations from their experiments, viz.,

(a) Fouling was greatest just before bulk vaporization commenced in the fluid, i.e., nucleate boiling was occurring when the rate of fouling was a maximum.

(b) Fouling was most significant in portions of the heat transfer surface that were known to be covered with liquid.

Crittenden and Khater also investigated the effect of pressure on the rate of fouling. This is a parameter of interest because raising the pressure could suppress boiling, however it was Cound that Couling was in fact enhanced. This finding also suggests that reducing the amount of vaporization by increasing the fluid velocity, which could also increase the rate oC deposit

removal (Iilr), may not be the best approach to reducing Couling in practical situations. Likewise the observation that fouling is less in the vapor phase, which corresponds to higher surface temperature, is contrary to conventional fouling wisdom. This serves to demonstrate that fouling is such a complex phenomena that the identification of operating strategies that will reduce fouling requires much more than a superficial acquaintance with the subject.

Some interesting experiments on fouling under boiling conditions with styrene dissolved in heptane have been reported by Fetissoff et.al. [88]. This results in the formation of a deposit of polymerized styrene on the heat transfer surface. The data (Fig. 7) showed that the time variation of the fouling thermal resistance had an asymptotic Corm [equation (64)]. Tests were carried out with two different concentrations of styrene (3.08% and 11.8%) and from a comparison of the results the authors concluded that the induction period generally decreased with increasing styrene concentration and that the fouling rate generally increased with the amount of styrene.

There have been fairly extensive studies of Couling in kerosene and jet fuels because of the tendency for deposits to form in fuel lines and combuster nozzles of turbojet engines. Particularly important contributions have been made by Taylor (see, for example. Taylor [89]). This body of work has identified the important role of oxygen in promoting fouling. The formation of the fouling deposit is believed to be related to a so-called auto­oxidation reaction. as follows:

,. � ..:

.. �

" e --

0:

0·24

0-1 6

0 ·08

S Y M B O L S H W P 6 ' F R U .. A 0 P f R u - a �

R U N 1 2

559

o 5 1 0 2 0 2 5

T i m e ( h o u r s ) 3 0 3 5 4 0

Figure 7. Fouling thermal resistance a s a function o f time for 3.1% styrene in n-heptane. A and B represent two separate runs using an HTRI Portable Fouling Research Unit (PFRU), according to Fetissoff, Watkinson and Epstein [88].

(a) Hydrogen absorption from the parent hydrocarbon R-H with a free .

radical X . .

R-H + X � R + XH .

(b) Reaction ofR with molecular oxygen . .

R + 02 � ROO

(85a)

(85b)

560

(c) Further hydrogen abstraction . .

ROO + RH -+ ROOH + R

(d) Homolysis of the weak 0-0 bond in ROOH to form more radicals . .

ROOH -+ RO + OH

(85c)

(85d)

The reaction chain breeds since equations (b). (c) and (d) result in one free radical giving rise to three.

Our current understanding of chemical reaction fouling is so limited that the addition of boiling to an already very complex problem makes it likely that progress on this type of fouling will be slow. Nevertheless. it deserves much more attention than it has received in the past because of its great technical importance.

6. SUMY AND CONCLUSIONS This lecture has reviewed the physical nature of fouling and its effect on

heat transfer and fluid flow. Fouling models based on fundamental processes have been discussed. but our present incomplete understanding of fouling limits the usefulness of these in the design and operation of heat transfer equipment. Empirical data. combined with simplified models, is currently a more suitable approach to the problems faced by a heat transfer engineer concerned with equipment exposed to fouling conditions. However design methods employing such data have not so far been developed and consequently fixed values of the fouling thermal resistance. known as fouling factors. based on a body of practical knowldge are widely used to design heat transfer equipment. The combination of such crude methods with the refined techniques now available for the design of heat transfer equipment is illogical. Clearly there is a need to develop methods for estimating fouling that are of comparable quality to the methods used to evaluate the performance of clean heat exchangers.

The interaction of fouling and boiling was also reviewed, but this is a topic that has not been very extensively studied. Because of this and because of the complexity of the process, involving the interaction between two complex processes - fouling and boiling, it is unlikely that significant progress toward understanding this topic will be made in the near future. Nevertheless the technical importance of this type of fouling suggests that it would be worthy of more study.

NOMENCLATURE A heat transfer area. m2

AC heat transfer area in clean heat exchanger, m2

AF heat transfer area in fouled heat exchanger, m2

a initial flaw size, m ap projected area of particles. m2

B temperature coefficient of solubility, kg/(m3oC)

ke kf km kmo km1 kr kro

m

. mf

p Q Qc

constant in equation (61). m2°C/(W s)

constant in equation (76). kg1 m m3m+2n-2 }{2m w-n s-l

constant in equation (77). kg m2 W 2 s-l

constant in equation (80). m2m+1 sm-n-1 kg-m

constant in equation (84). kg/m2 s material constant in Lo and Bridgwater theory. equation (50) solubility product. kg2/m6

56 1

rate constant for rate of formation of microbial fouling deposit. equation (29). s-l transport coefficient for eroded particles. mls thermal conductivity of material of the fouling deposit. W/m K mass transfer coefficient in equations (32). (34). (36). (38). mls overall mass transfer coefficient. defined in equation (27b). mls value of km for oxygen. m/s reaction rate constant. kgn-1 m3n 2 s l

Arrhenius constant for the temperature dependence of kr• same units as kr transport coefficient. mls thermal conductivity of material of the heat transfer surface. W/m K constant in corrosion model of Mahato et.al. [18-20]. dm 7mg-2

constant in corrosion model of Mahato et.al. [18-20]. m3/kg constant in corrosion model of Mahato et.al. [18-20]. mg-1 length of duct. m molar mass of oxygen. kglkg mole molar mass of fouling deposit. kg/kg mole exponent in equations (17). (76). (78). (80). dimensionless mass of oxygen per unit area of the heat transfer surface. kg/m2

mass of the fouling deposit per unit area of the heat transfer surface. kg/m2

net rate of formation of the fouling deposit. kglm2s

rate of growth of the fouling deposit. kglm2s

rate of removal of the fouling deposit. kg/m2s number flux of particles to the heat transfer surface. m-2s-1

exponent in equations (14). (15). (17). (35a). (38). (57). (58). (76). (78). (80). dimensionless sticking probability. dimensionless rate of heat transfer in the heat exchanger. W rate of heat transfer in the clean heat exchanger. W

562

b removal rate constant in equation (62), s-l; constant in equation (82), kg-I

C constant in equations (47) and (48) (units defined by respective equations)

Cb bulk concentration of fouling deposit material, kg/m3

Cbl bulk concentration of oxygen, kg/m3

Ci concentration of fouling deposit material at the interface between the deposit and the flowing fluid, kglm3

Cp specific heat of liquid, kJ/kgK Cs concentration of fouling deposit material at the interface between the

deposit and the heat transfer surface, kg/m3 Csi value of Cs for oxygen, kg/m3

Csat concentration of fouling deposit material at the saturation conditions, kg/m3

Dl mass diffusivity of oxygen in the deposit, m2/s Di mass diffusivity of oxygen in water, m2/s

d duct diameter, m dc diameter of available flow cross section in clean heat exchanger, m df diameter of available flow cross section in fouled heat exchanger, m dh duct hydraulic diameter, m dp particle diameter, m E activation energy, kJ/kg mole; elastic modulus in equation (49), N/m2 F correction factor for adjusting the actual heat exchanger to a double

pipe counter-flow heat exchanger, dimensionless; force, N F " effective applied force per unit area exerted by the fluid on the

particle, N/m2 FC correction factor F for clean heat exchanger, dimensionless FF correction factor F for fouled heat exchanger, dimensionless f Fanning friction factor, dimensionless fI activity coefficient of species I, dimensionless h coefficient of heat transfer, W/m2K hfg latent heat of vaporization, kJ/kg [ I ] concentration of chemical species I, kg/m3

(subscripts i and s indicate values at the liquid-deposit and deposit­surface interfaces, respectively)

Kl stoichiometric constant relating the rate of growth of the corrosion fouling deposit to the rate of consumption of oxygen [defined in equation (25)], dimensionless

� constant in equation (54), m2/W s K3 constant in equation (56), m2oC/(W2s); constant in falling rate fouling

model, equations (56), (58), m2oC/(W hr)

l<.i constant in equation (59), m2°C/(W s)

� q R

v

rate of heat transfer in the fouled heat exchanger, W heat flux, W/m2

563

total thermal resistance, m2K1W; universal gas constant, equations (35b), (37), kJlkmole K convective thermal resistance at the heat transfer surface, m2KJW fouling deposit thermal resistance, m2KIW asymptotic fouling resistance, see equation (64), mZKIW

Reynolds number = <u>dlu, dimensionless outside radius of the deposit formed on a cylindrical surface, m inside radius of the deposit formed on a cylindrical surface, m stopping distance, m Schmidt number = ulD, dimensionless Sherwood number = kmdHID, dimensionless

mass transfer Stanton number = xitf/P<U>(Cb - Cs)' dimensionless value of S1m at 9 = 0, dimensionless temperature, K bulk fluid temperature, K temperature at the interface between the fouling deposit and the flowing fluid, K; temperatures on the inside radius of the deposit formed on a cylindrical surface, K temperature on the outside radius of the deposit formed on a cylindrical surface, K temperature of the heat transfer surface, K overall heat transfer coefficient, W/rnZK. value of U on a clean surface, W/m2K value of U on a fouled surface,

·W/m2K

_ - <U> fluid velocity averaged over the duct cross section, mls particle volume, m3

initial velocity of a particle coasting toward the heat transfer surface, m/s average component of velocity of particles normal to the heat transfer surface, mls fluid velocity at the location of the center of the particle, mls

average fluid velocity, mls thickness of heat transfer surface normal to the direction of heat flow, m distance measured normal to the heat transfer surface, m . valence of the metal molecule, dimensionless geometric factor, dimensionless

564

pressure drop over the length L of the duct, m temperature diference between the heat transfer surface and the fluid, K log mean temperature difference, K value of L\Tm in a clean heat excbanger, K value of L\Tm in a fouled heat exchanger, K thickness of the deposit, m thickness of the damped turbulence layer on the clean heat transfer surface, m mass transfer eddy diffiJ.sivity, m2/s effectiveness factor in equation (43), dimensionless mass transfer coefficient in equation (66a), kg/m2s combined mass transfer coefficient and reaction rate constant, m1s fraction of surface covered by micro-organisms, dimensionless dynamic viscosity, kglm s kinematic viscosity, m2/s; Poisson's ratio in equation (49), dimensionless fluid density, kg/m3 density of the fouling deposit material, kg/m3

density of the particle material, kg/m3

deposit strength factor, N/m2 shear stress at the deposit-fluid interface, N/m2

time, hr or s time constant in asymptotic fouling model = 1Jb, s or hr or days induction time, s or hours or days residence time, s or brs critical wetting tension, Baier [10], dynes/cm; fracture stress, Loo and Bridgwater [31], N/m constant in equation (43), dimensionless

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565

4 N. Epstein, Fouling of Heat Transfer Equipment, eds. E.F.C. Somerscales and J.G. Knudsen (Hemisphere Publishing Corp., Washington, D.C., 1981) pp. 31-55.

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566

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41 RW. Morse and J.G. Knudsen, Can. J. Chem. Eng. 55 (1977) 272-278. 42 S.H. Lee and J.G. Knudsen, ASHRAE Trans. 85(1) (1979) 281-302. 43 J.G. Knudsen and M. Story, AIChE Symp. Series 74( 124) (Amer. Inst.

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de Chaleur, eds. J.G. Knudsen and E.F. C. Somerscales (L'lnstitut Algerien du Petrole, Centre d'Arzew, April 27-28, 1985) pp. 9.1-9.8.

45 W.J. Marner and J.W. Suitor, Handbook of Single Phase Heat Transfer, eds. S. Kaka�, RK. Shah, and Win Aung (John Wiley & Sons, New York, 1987).

46 J.M. Chenoweth,Fouling Science and Technology, NATO AS! Series E: Applied Sciences - vol. 145, eds. L.F� Melo, T.R. Bott, and C.A. Bernardo (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988) pp. 477-494.

47 E.F.C. Sornerscales, Heat Transfer Engineering 11(1) (1990) 19-36. 48 E.P. Partridge, Formation and Properties of Boiler Scale, University of

Michigan Engineeering Research Bulletin, No. 15 (June 1930). 49 E. Couste, An Mines 5(5) (1854) 69-164. 50 RE. Hall, Ind. Eng. Chem. 17 (1925) 283-290. 51 E.C. Schmidt and J.M. Snodgrass, Effect of Scale on the Transmission of

Heat Through Locomotive Boiler Tubes, University of Dlinois Engineering Experiment Station Buletin No. 11 (1907).

52 E. Reutlinger, Z. Ver. Deutscher Ing. 54 (1910) pp. 545-553, 596-601, 638-642, 676-681.

53 H.O. Croft, Power Plant Eng. 31 (1927) 1001-1002.

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Eng. Prog. Symposium Series 62(4), ed. J.G. Knudsen (American Institute of Chemical Engineers, New York, 1966) pp. 77-86.

57 L.A. Curcio, Jr., Pool Boiling of Enhanced Heat Transfer Surfaces in Refrigerant - Oil Mixtures and Aqueous Calcium Sulfate Solutions, M. Eng. Thesis (Rensselaer Polytechnic Institute, Troy, NY, 1989).

58 Y. Asakura, M. Kikuchi, S. Uchida and H. Yusa, Nucl. Sci. Eng. 67 (1978) 1-7.

59 Y. Asakura, M. Kikuchi, S. Uchida and H. Yusa, Nucl. Sci. Eng. 72 ( 1979) 117-120.

ro T. Mizuno, K. Wada and T. Iwahori, Corrosion 38 (1982) 15-19. 61 N.B. Hospetti and RB. Mesler, AIChE J. 11 (1965) 662-665. 62 G. Freeborn and D. Lewis, J. Mech. Eng. Sci. 4 (1962) 46-52. m C.F. Gottzmann, P.S. O'Neill and P.E. Minton, Chern. Eng. Prog. 69(7)

(1973) 69-75. 64 W.L. Badger and Associates, Inc., U.S. Department of the Interior, Office

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Eng. Chern. 42 (1950) 126-130. 66 H. Hillier, Proc. Instn. Mech. Engrs. I(B) (1952) 295-322. fj'{ R. Dooley and J. Glater, Desalination 11 (1972) 1-16. 68 B.H. Rankin and W.L. Adamson, Desalination 13 (1973) 63-87. m T.D. Bui and V.K Dhir, J. Heat Transfer 107 (1985) 756-763. 70 D.H. Lister, Corrosion Products in Power Generating Stations, Atomic

Energy of Canada Limited, Report No. AECL-6877 0,980). 71 D.H. Lister, Fouling in Heat Exchange Equipment, eds. E.F.C.

Somerscales and J.G. Knudsen (Hemisphere Publishing Corp . , Washington, DC, 1981) pp. 135-200.

72 R. Gasparini, C. Della Rocca and E. Ioanilli, Combustion 51(5) (1969) 12-18.

73 T. Iwahori, T. Mizuno and H. KOyama, Corrosion 35 (1979) 345-350. 74 M.A. Styricovich, 0.1. Martynova, V.S. Protopopov and M.G. Lyskov,

Heat Exchangers: Theory and Practice, eds. J. Taborek, G.F. Hewitt and N. Afgan (Hemisphere Publishing Corp., Washington, D.C., 1983) pp. 833-840.

75 N.N. Mankina, Teploenergetika 7(3) (1960) 8-12. 76 KA. Burrill, Can. J. Chern. Eng. 55 (1977) 54-61. Tl K.A. Burrill, Can. J. Chern. Eng. 56 (1978) 79-86. 78 K.A. Burll, Can. J. Chern. Eng. 57 (1979) 211-224. 79 L.F. Picone and W.D. Fletcher, Post-irradiation Examination of Saxton

Fuel Cladding, US Atomic Energy Commission Report no. WCAP-3269-57 (1980).

8) L. Kabanov, Energia Nucleaire 18(5) (1971) 285-289. 81 D. Thomas and U. Grigull, Brennstoff - Warme - Kraft 26(3) (1978) 167-180.

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85 B. Crittenden. Fouling Science and Technology. NATO ASI Series E: Applied Sciences - vol. 145. eds. L.F. Melo, T.R. Bott, and C.A. Bernardo (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988) pp. 293-313.

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87 B.D. Crittenden and E.M.H. Khater, J. Heat Transfer 109 (1987) 583-509. 88 P.E. Fetissoff. A.P. Watkinson and N. Epstein, Heat Transfer 1982, Vol. 6.

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89 W.F. Taylor, Ind. Eng. Chem. (Product R&D) 13 (1974) 133-138.

INTERMOLECULAR AND SURFACE FORCES WITH APPLICATIONS IN CHANGE-OF-PHASE HEAT TRANSFER

Peter C. Wayner, Jr.

569

The Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590

1 . INTRODUCTION

The successful application of many technologies depend on the characteristics of thin liquid films. This is discussed in BankofI's review of the importance of change-of-phase heat transfer in ultra-thin liquid films [ 1 J . Unfortunately, from the perspective of interfacial forces, we find that a thin liquid film is not a simple system. For example, ultra-thin liquid films are not macroscopic in nature because their intensive properties are a function of their thickness. Large density gradients are present near the liquid-solid and liquid-vapor interfaces where the transition from the intermolecular force field associated with a less dense substance to that associated with a denser substance occurs over a short distance. As a result of this density gradient, the stress tensor is anisotropic near interfaces. In ultra-thin films, these interfaces can overlap. The internal pressure in a bulk liquid due to cohesion can be of the order of thousands of atmospheres, whereas it is small in the vapor. In addition, adhesion changes the internal pressure near the solid­liquid interface. Therefore, we expect large interfacial effects on transport processes near interfaces.

Because of the nature of the transition from one phase to another, there is no simple way to model the interface. However, to overcome some of the theoretical difficulties, useful models have been developed. For example, Gibbs [2] developed the highly successful formalism of a two-dimensional dividing surface. At times, we find this convention and the resulting surface tension of a flat interface between two phases useful. In other situations, we find it necessary to use other concepts which are not easily described within the framework of the two dimensional dividing surface model. The thickness of the interface is also important. This requires the use of the surface excess convention. Both the Gibbs convention and the surface excess convention are used herein. A good discussion of the details and the comparison of these two conventions are given by deFeijter [3] .

The concept of thick interfaces is not new. For example, van der Waals (see Rowlinson, [4]) analyzed the extremely large density gradient located at the liquid-vapor interface. Thick interfaces are also associated with resistances to

570

transport processes. Using kinetic theory, Schrage [5], Nabavian and Bromley [6], Umur and Griffith [7], and Sukhatme and Rohsenow [8] demonstrated that extremely large interfacial heat and mass transfer coefficients are possible at the liquid-vapor interface. Large density gradients are also present near the liquid-solid interface. Derjaguin et a1. [9] discussed the effect of these interfacial intermolecular force gradients on fluid flow and the effective pressure in the liquid film using the concept of a disjoining pressure. Potash and Wayner [10] and Wayner et a1. [11] demonstrated that the disjoining pressure has a profound effect on the resistance to change-of­phase heat transfer in ultra-thin liquid films. Wayner [12] generalized these concepts by showing that the non-equilibrium processes of change-of-phase heat transfer and fluid flow are intrinsically connected because of their common dependence on the intermolecular force field and gravity. Moosman and Homsy [13], Stephan and Busse [14], Wayner, et a1. [15], and Schonberg and Wayner [16] demonstrated that the conductive resistance across the thin liquid film also has a significant influence on the evaporative process. As the thickness of a completely wetting system decreases, the stabilizing effect of interfacial forces increases. However, shear stresses become very large in small systems. In general, transport processes depend on the intermolecular force field which is a complicated function of temperature and pressure near interfaces. This will be discussed after a short introduction concerning classical interfacial phenomena.

When a liquid comes into ccmtact with a solid substrate, one of three conditions must exist at equilibrium: complete wetting, partial wetting, or non-wetting. Complete wetting occurs when the liquid spreads on the solid substrate to form an equilibrium film with a uniform thickness and a contact angle equal to zero (9 = 0). For 0° < 9 < 90° the liquid partially wets the solid. At equilibrium , the total interfacial free energy is a minimum, i.e.

(1)

where asv ' asp and a]v are the interfacial free energies of the solid-vapor, solid-liquid and liquid-vapor interfaces respectively and where A . . are the IJ areas. The solid-vapor interfacial free energy is used because the solid is in equilibrium with a vapor phase. The solid surface free energy is a function of the vapor pressure because the vapor can adsorb on the solid surface. As described herein, this leads to extremely important effects. Equation (1) can be used to describe the contact angle formed by a liquid in contact with a solid substrate which is shown in Figure ( 1). However, using a simple force balance, Young (see Rowlinson and Widom, [17]) derived the following equation for the apparent contact angle:

(2)

The modifier apparent is used to remind us that we cannot see the real contact angle, which is of molecular dimensions, and that the surface tensions were assumed constant in the derivation of Equation (2). Additional

FIGURE 1. Drop of liquid with a finite apparent contact angle on a solid substrate.

57 1

details concerning the microscopic details of this contact region are given in [ 18 and 19]. Since the interfaces are boundaries between phases, large stress gradients are present in the contact line region. Therefore, the resulting interfacial free energies are a function of the local liquid film thickness profile. At the molecular level, the interfaces are dynamic with large anisotropic density gradients. These differences vanish at the critical point. Obviously, the above equations are simple but successful macroscopic models of the intermolecular force field. However, we will find herein that additional detail is needed to optimize the use of interfacial forces in the modeling of transport processes.

We can also relate the wetting characteristics in terms of the final spreading coefficient, S, introduced by Cooper and Nuttal [20] .

Thus the condition for complete wetting is S = 0; while the condition for partial wetting and nonwetting is S < O.

(3)

Although the apparent contact angle and the surface tensions give considerable information concerning the general wetting characteristics of a system, the disjoining pressure concept, n, discussed next is more useful . It allows /Zuid /Zow and vapor pressure concepts to be easily introduced and evaluated because it can be viewed as an effective pressure resulting from a body force acting between the substrate and the mobile liquid film ( e.g. Derjaguin et al. [9,21], Wayner et. al. [ 1 1] , Wayner [ 12]) . Briefly, the disjoining pressure is (minus) the potential energy per unit volume due to intermolecular forces, F, and is a function of the film thickness, 0:

[1 (B) = - F (B) (4)

572

Disjoining describes the physical process whereby a completely spreading liquid naturally tends to "disjoin" a solid from the vapor by spreading. The concept is shown in Figure (2) for a completely wetting fluid which is evaporating in a gravitational field. To avoid inconsistencies, it is necessary to include gravity in some of the equations even though the other forces are usually substantially larger. In addition, it allows the use of the fundamental concept of hydrostatics to describe the phenomena. The relationship between surface tension and disjoining pressure is discussed in Section (2). At equilibrium, the gravitational force is equal and opposite to the surface forces. We note that the thickness of the thin film at its junction with the classical

capillary meniscus can be substantial ( of the order 10-7 m).

VAPOR

LIQUID Y

FIGURE 2. Completely wetting fluid in contact with a vertical plate. Evaporation and fluid flow is maintained by an external heat source which is not shown.

Adding the "pressure jumps" due to capillarity and disjoining pressure at the liquid-vapor interface, the following extended Young-Laplace equation for the hydrostatic pressure change in a non-evaporating equilibrium liquid film can be written:

(5)

in which PI is the density of the liquid in the gravitational field g, P vx and PIx are the pressures in the vapor and liquid at x respectively, and K is the

573

curvature of the liquid-vapor interface. The hydrostatic pressure change is approximate because the effect of the density of the vapor on buoyancy has been neglected. Neglecting n, Equation (5) is the classical equation of capillarity. An equivalent change in energy per unit surface area, Es' with

liquid film thickness could be given as

(6)

Therefore , for a completely wetting system, the "surface energy" decreases with an increase in the film thickness. This leads to a positive disjoining pressure, n, and a negative potential energy per unit volume, F, and demonstrates that fluid naturally flows from the thicker to the thinner region as a result of intermolecular forces. For a simple fluid, the film thickness due to van der Waals forces at the height reached by the classical capillary meniscus is of the order of 100 nm. Therefore, the weak van der Waals force has a profound effect on an equilibrium system. As demonstrated below, extremely small temperature differences also have a large effect on the film profile. Therefore, isothermal equilibrium is difficult to maintain ( see e.g. Sujanani and Wayner, [22] , and Truong, [23]). Although the disjoining pressure concept can be applied to more complicated systems and the finite contact angle case, we will restrict its use for convenience herein to completely wetting films interacting by only dispersion forces. We caution that intermolecular force fields are complicated and that the results for a simple system, which can be modeled using only van der Waals dispersion forces, can be misleading if applied to more complex systems. Nevertheless, the general objective of obtaining the heat transfer characteristics of a small system from the intermolecular force field is attainable. Some recent books that review and cover these concepts in detail are those by Israelachvili [24], Ivanov [25], Rowlinson and Widom [17], and Slattery [26] . Additional details concerning the contact line are given in articles by Wayner [18] and Brochard­Wyart et a1. [19] . Chandra and Avedisian [27] addressed the contact line region in their study of the collision dynamics of a liquid droplet on a solid substrate, which is important to processes like mist cooling.

2. THEORETICAL BACKGROUND

2.1 Equilibrium Vapor Pressure of a Liquid Film In this section, interfacial thermodynamics are used to relate the vapor

pressure of a curved film of liquid on a solid substrate in a gravitational field to the temperature and the effective pressure in the liquid film. The resulting vapor pressure will be subsequently used to model change of phase heat transfer.

The surface excess convention is used in which the single component liquid film can be thought to consist of a volume part, bounded by dividing surfaces at the liquid-vapor and liquid-solid interfaces (e.g. , see deFeijter [3]. and Wayner [12]). The properties of the volume phase are those of a bulk liquid phase in a gravitational field in equilibrium with the film. In the limit of a

574

thick film, the surface tensions of the two dividing surfaces are those of a bulk fluid in contact with the solid and with the vapor. When the film thickness is of the order of 10-9 m and less, bulk properties in the film have to be modified. When the film is very thin, the surface tensions of the dividing surfaces and other properties are a function of the film thickness. The Gibbs-Duhem equations for the two bulk phases a, liquid and v, vapor) in Fig. (2) are

dPl = sldT - Plgdx + nldlllg (7)

(8)

where s is the entropy/volume and n is the molar density. The chemical potential in a gravitational field is

Ilig = Ili + Mgx (9)

where Ili is the chemical potential/mole, and M is the molecular weight. The disjoining pressure for a flat film (curvature effects are included below) is

(10)

Therefore, a pressure jump model is used for the interfacial body force. The film tension, which is the excess tangential force relative to the surrounding bulk vapor, is

0+ -{ == J [PT(Y) - Pv]dy 0-

(11)

(12)

The first term on the right-hand side of Eq. ( 1 1) is the sum of both surface tensions at the dividing surfaces, CJ}v and crls. Near the interfaces the pressure tensor is nonisotropic and PT(y) is the component of the pressure in the direction tangent to the interface. Equation ( 1 1) is important because it relates the disjoining pressure to the more classical surface tensions. Another useful equation is

(13)

in which sl is the surface excess entropy. Using the above equations, dS = Sv -

Sl - SO-I, dp = Pv - Pl, L\n = nv - nl with local equilibrium between the vapor and liquid, J.1vg = J.1lg' gives

dyf L\s dllg = - + Mgdx - - d'I'

aL\n L\n

This can be rewritten in terms of the fugacity, f,

dyf dlnf = RTa6n

� dT RT6n

575

(14)

(15)

DeIjaguin and Zorin [21] essentially used Eq. (15) to study adsorption of a vapor in the form of a thin liquid film on a superheated solid surface placed close to but slightly above a pool of liquid. Using Eq. (14) with dJ,lg = 0 and Eq. ( 13) for dyf gives

dIT = (Sy - Sl)dT + pgdx (16)

Integration of Eq. (16) over a small temperature change (from T y, IT = 0 to Tly, IT) during which (sv - Sl) is approximately constant and x = 0 leads to

IT = (Sy - Sl) (Tly - Ty) (17)

Using IT = -Al67ta3 for film thicknesses less than 20 nm and (Sy - Sl) = L\h!Tj, where Ti = (Tly + T y)/2, gives

a = ( - ATi

)113 67t(Tly - T y)L\h

(18)

The constant A is the Hamaker constant and represents the relative importance of adhesion to cohesion which depend on the intermolecular force field. Eq. (18) can be used to calculate the thickness of an adsorbed layer of liquid on a superheated. (Tly - Ty), solid surface from the value of the volumetric heat of vaporization of the liquid film, £\h, and the Hamaker constant, A. Conversely, the Hamaker constant can be obtained by measuring, S(Tly, Ty). A conceptual drawing of an experimental cell to study adsorption on a superheated flat surface in a gravitational field is presented in Figure 3 .

For small changes i n the fugacity i n a range where the fugacity can be replaced by the vapor pressure, Eqs. (13 and 15) give on integration with (L\n)- 1 == -Vl

(19)

where Ply is the vapor pressure at the liquid-vapor interface. In this equation,

Py is a reference vapor pressure of a thick, flat, liquid film, where IT � 0, with

576

VAPOR, T v

LIQUID, T /

FIGURE 3. Experimental closed cell showing adsorption of a thin liquid film on a superheated solid substrate located at a height x above the reference bulk liquid. At equilibrium for mass transfer, the vapor pressure of the thin film is equal to the surrounding vapor pressure. However, Ts = Tlv > Tv = TI. As indicated, the film thickness can be measured optically using an ellipsometer.

The sensitivity of the film thickness to height and temperature difference is shown in Table 1. These results are based on Eq. (22) which is derived below.

TABLE 1. Sensitivity of the adsorbed film thickness to temperature difference and hydrostatic head using Plgx = - �h (Ts-Tv)/T + no for octane on Si02 at T =

298 K. no == -NO� is an important characteristic pressure difference based on

the characteristic thickness, 00,

�T, K x, mm 00, nm

a 1.7 30.0 0.001 1.7 7.2 0.01 1.7 3.3

small �T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0. 1 1.7 1.6 equivalent to large �p 0 12.4 m 1.6

1 1.7 0.7

a surface temperature Tlv = Tv and �h is the ideal heat of vaporization per unit volume of the film. Equation (13) can be modified to include curvature at

the liquid vapor interface by replacing n by n + O'lvK in which K is the curvature and O'lv is the surface tension of the liquid-vapor interface:

(20)

Equation (20) also represents the change in equilibrium vapor pressure relative to the pool surface of the extended meniscus presented in Figure (2) for a change in temperature of (Tlv - Tv) and hydrostatic pressure equal to

577

Plgx :: (n + O'lvK). The right hand side can be viewed as representing both the Kelvin and Clapeyron equations. This is approximate since buoyancy has been neglected. An integration in the vapor space at constant Tv gives for small changes in the vapor pressure

MgPvx Pyx - Pv � - RTv

Combining Eqs. (20 and 21) with Pyx = Ply gives for the non-isothennal equilibrium case

VlPv MgPv PvVl6h Ply - Pyx = - RTlv (n + O'lvK) + RTv x + RTlvTv (Tlv - Tv) = 0

For the isothermal case, Tlv = Tv, this reduces to the identity

Vlcn + O'lvK) = VZPZgx = (Mgx)l (Mgx)v

(21)

(22)

(23)

If Pyx 1t Ply, Eq. (22) can be used along with kinetic theory to calculate the rate of evaporation from (or condensation on) a curved thin film. This is demonstrated in the next section.

Equation ( 19) can be rewritten for constant temperature while retaining the natural logarithm to obtain Equation (24).

n � RT Z - Vl - - Vl n Pv (24)

.R n = -Bn

(25)

Therefore, the adsorbed film thickness is easily related to the static head in the liquid and vapor. Experimentally, it is observed that the film thickness decreases as the height above the pool increases (Sabisky and Anderson [28]; Dzyaloshinskii, et a1. [29]) . As an approximation in the thickness range 1 nm < 5 < 10 nm, B in Equation (25) (negative for a spreading system) can be viewed

as a constant, A, with n = 3 (Restrictions on the use of this approximation are

578

discussed by Truong and Wayner [30]). A is related to the classical Hamaker

constant, A, and intermolecular forces by A = 61tA. From Equation (25) it can be seen that B is negative for a spreading system

and that the effective pressure ( chemical potential per unit volume ), F(o) = - n, in the thin film decreases as the film thickness decreases. These results can be easily connected to statics and experimentally measured. If a drop of a spreading liquid is placed on a horizontal surface, fluid flows from the thicker portion of the ultra-thin film near the contact line to the thinner portion because of this gradient in the disjoining pressure which is represented by the slope of the thin film and which is due to the gradient in the surface force field. Gravitational effects are relatively small near the contact line of a horizontal film and the effective pressure gradient (F') for flow in a horizontal film is represented by

dF _ F' _ 3A dx -

- -04 (26)

An example of the variation of chemical potential with film thickness for a thin film is given in Figure (4a) for a simple spreading case (6 = 0) and for a simple finite contact angle case. In the limit 0 � 0, � � - 00 for both cases but is not shown. The variation for a complicated system which includes a fluid like water is given in Figure (4b). A discussion of various isotherms is given by Dzyaloskinskii et aI. [29] and Derjaguin et a1. [31].

9 > 0

jJ. (5) 5 jJ. (5)

9 = 0

(a)

FIGURE 4. Variation of Chemical Potential with Film Thickness: (a) simple spreading and non-spreading systems; (b) complicated system (e.g. , water)

2.2 Interfacial mass flux. In 1953, Schrage [5] reviewed the literature, presented and discussed the

following equation based on kinetic theory relating the net mass flux of matter

579

crossing a liquid-vapor interface to a jump change in interfacial conditions at the interface:

• _ C (..)IJ2 (PLv Pyx ) m - 1 21tR TIJ2 - TIJ2 Lv v

(27)

Herein, we presume that the net mass flux crossing the interface (e.g. , evaporation) results from a small vapor pressure drop across an imaginary plane at the interface in which Ply, is the quasi-equilibrium vapor pressure of the liquid film at (TZv, K, n, x) and Pyx is the equilibrium vapor pressure of a reference bulk liquid (K = 0, n = 0, x) at a temperature Tv. As discussed by Maa [32], the coefficient C1 accounts for a molecular exchange resistance at the liquid-vapor interface and has a maximum value of 2 if the exchange is ideal. Neglecting resistances in the bulk vapor space, Pyx and Tv can exist at a short distance from the interface, and a resistance to evaporation at the interface can be defined using Eq. (27). This resistance is associated with the effect of pressure (due to changes in the intermolec1ar forces near interfaces) and temperature on the vapor pressure. Conceivably, other resistances are associated with Cl . Using T}� "" T�2, this can be rewritten as

(28)

Wayner, et a1. [11] used an extended Clapeyron equation for the variation of equilibrium vapor pressure with temperature and disjoining pressure in a horizontal thin film to obtain Eq. (29) for the vapor pressure difference in Eq. (28).

(29)

We note that, when interfacial effects are important, the effective pressure in the liquid, Pl, is not necessarily equal to the pressure in the vapor or to its normal vapor pressure. For example, the equilibrium vapor pressure of a small droplet is not the same as that of a flat pool of bulk liquid at the same temperature, and the pressure inside the droplet is greater than the surrounding pressure because of surface tension. The concept of pressure is further clouded by the concept of internal pressure due to cohesion which is extended in thin films to include adhesion. Therefore, Ply is used to designate the equilibrium vapor pressure of an interface at Tlv changed by interfacial forces and (PZ - Pv) designates the pressure change in the liquid due to interfacial forces. For example, the above equation can be used to calculate the required subcooling needed to achieve equilibrium between a curved interface (droplet) and a flat interface when the vapor pressure is changed by the surface pressure jump due to surface tension.

Combining Eqs. (28 & 22) for the non-equilibrium case gives

580

• M PvMflhm VZPv MgPv m = Cl (--)112 T (TZv - Tv) - RT (n + crlvK) + -RT x)} (30)

21tRTi v Lv Zv v

which can be used to calculate the interfacial mass flux. Herein, there are two interfacial effects that can cause an effective pressure

jump at the liquid-vapor interface: capillarity, crLvK, and disjoining pressure, n. The significance of the disjoining pressure herein is that the vapor pressure of an adsorbed completely wetting liquid film is reduced by interfacial forces and therefore a superheated adsorbed liquid film can exist in ( vapor pressure) equilibrium with a bulk liquid at a lower temperature. For convenience, we will emphasize spreading (zero contact angle) fluids and will use the following sign convention

B A n == - - = - -on 03

(31)

(32)

The approximation, n = 3, in Eq. (32) restricts its use to thicknesses 0 � 20 nm. However, in the numerical examples presented below this is acceptable for either 0 � 20 nm or n :: O. The equations can be easily modified for general use andlor for other cases. It is useful to make the mass flux dimensionless using an ideal mass flux based on the temperature change only:

Iit VZTi M == = 1 -h T

(crZvK + n - PZgx) m Id M� m�

(33)

(34)

with �T = (TZy - Ty). Taking M = 0, n = 3, B = A = Al61t, K = 0 in Eq. (34), Eq. ( 18) for the thickness of a flat adsorbed liquid film, 00, on a superheated surface with �To = (TZv - Ty) can be recovered.

Restricting the following material to a constant interfacial temperature difference, �To = (TZy - Tv), and defining a reference disjoining pressure,

no = - Alo�, Eqs. (18 and 34) become

M = 1 - (crl� + n - PZgx) no; (35)

A constant temperature difference simplifies the analysis and should not alter the significance of the results. This is relaxed in one of the applications

581

given below. It is useful to keep in mind that no could be replaced by the equivalent temperature difference using Eq. (18). Therefore. the temperature difference increases the value of the disjoining pressure which causes the evaporating fluid to flow towards the contact line.

Using 1'\ = 6100 and � = x/xo • the curvature, K, for small interfacial slopes, can be written as

0" [1 + (0')2]312

(36)

Defining the parameter 3 Nc == crlv0olTIox� with no = 1'\3n and Plgx = TIoX gives

M = 1 - 3NCT\" - 1'\-3 + X (37)

With dimensionless terms the prime refers to differentiation with respect to �, otherwise x. Eq. (37) gives the change in the dimensionless evaporation rate resulting from the effects of gravity and interfacial forces on the vapor pressure. Using oAT 0 = �hm, this can also be written as a dimensionless liquid-vapor interfacial heat transfer coefficient:

(l -. = 1 - 3NCT\" - 1'\-3 + X (lid

(38)

Therefore, an interfacial heat transfer coefficient can be obtained from first principles. The dependence of the Hamaker constant on the "optical" properties of the liquid and solid is discussed in Section (4.4). Considerable insight can be obtained from Eq. (37). We find that at the leading edge of an evaporating horizontal (X = 0, due to g = 0) flat (1'\" = 0) thin film, 1'\ = 1, the evaporation rate can be zero even through the superheat can be substantial. This condition is required to define a stationary interline for an evaporating thin liquid film with varying thickness which has an equilibrium contact angle equal to zero. However, when the dimensionless film thickness is 1'\ = 4.6 (X = 0, 1'\" = 0), the evaporation rate will be 99% ofits substantial ideal value. When 1'\ < 1, condensation occurs. These equations allow the interfacial forces and thermal effects to be compared and combined, and have many uses in evaporating, condensing and/or equilibrium systems. For curvature to have the same effect, use ofEqs. ( 18 and 35) with n = 0 and g = 0 gives

(39)

582

Using Eqs. (18 & 39) and I10 = Plgx, the equivalents presented in Table 2 are

obtained for octane on Si02 at 298.16K with AJ61t = -3. 18 x 10-22 J. For the hydrostatic "x" equivalent, Ko and 6T are taken to be equal to zero. For the Ko equivalent, x and ITo are equated to zero.

TABLE 2. Equivalents calculated using Equations (18 & 39).

x, m

0. 1 3.94 x 106 8.49 x 1()4 1.55 12.4

This demonstrates that the decrease in film thickness associated with a 6T = 0 .1 K is equivalent to the pressure difference associated with a static liquid head of 12.4 m. This is a pressure drop available for fluid flow from the thicker region of a film. Extremely large pressure drops are available for monolayers.

2.3 Fluid Mechanics. Fluid flow in a curved thin film is also controlled by the "pressure" gradient.

Using the definitions for Tt, �, xo , no and the small slope assumption given above leads to Equation (40) for the pressure gradient.

d ( K ) 3no eN III Tt' X')

dx CJZv + IT - Plgx = Xo CTl - Tt4 - 3 (40)

Using the lubrication approximation for flow in a slightly tapered thin film while neglecting surface shear stress, the mass flow rate per unit width in a liquid film can be approximated by

B3 d r = - 3v dx [Pl + Plgx] (41)

Using Eqs. (3 1, 40 and41) while assuming Pv « PI gives

(42)

and the following equation for the dimensionless mass flow rates in the film

r · 3X' - xoV N Tl Tl r == = - - -

aollo Tl (43)

Using Eq. (44) for a steady state, stationary thin film with phase change

dr •

dx = - m

and Eq. (45), in which x� == -Nv �id, with Eq. (37) gives Eq. (46)

dr :nx� __ • -M d; - A

, 3X' (!L + Y _ NCT\3Tl"')' = 1 - 3Ncn" - Tl 3 + X

T\

(44)

(45)

(46)

583

Using four boundary conditions, Eq. (46) can be numerically solved to determine the he·at transfer characteristics of a stationary, steady state. thin film. When the right-hand side and the left-hand side of Eq. (46) are not equal, the above material can be used with the equation of continuity to analyze the transient case.

An evaluation of Eq. (30) demonstrates that, in an isothermal horizontal ultra-thin film, evaporation or condensation occurs unless (n + CJlvK) = O. Using simple models, this case without phase change has been previously discussed for the leading edge of a film (n > 0, K < 0) by Joanny and deGennes [33] and for the partial wetting case (n < 0, K > 0) by Wayner [18] . The results demonstrate that the interface becomes planar very rapidly. However, the phenomena controls the motion and phase change processes near the contact line.

For motion to occur in an isothermal, ultra-thin, horizontal film, Eq. (41) demonstrates that (n + CJlvK) "¢ O . This indicates that fluid flow due to interfacial forces in an "isothermal" thin film must be associated with change-of-phase heat transfer (Eq. (30» . Therefore, at the leading edge of a spreading ultra-thin film a small temperature gradient due to change-of­phase heat transfer is present. The size of the temperature gradient would be a function of the volatility of the fluid.

3. APPLICATIONS

3.1 An Evaporating Ultra·Thin Film Next we present a simple application of the above for a quasi-equilibrium

spreading system (an evaporating zero contact angle system with a fixed contact line). The thin film portion of a steady state evaporating film in the

584

contact line region consists of evaporating and non-evaporating regions as shown in Figure (5).

NON-EVAPORATING ADSORBED FILM

" "

,

" , - - - - - - -

dB dx

SOLID

FIGURE 5. Generic Contact Line Region for Spreading System, ac = 0 (not drawn to scale)

We note that the contact line is not moving and that the adjective spreading describes the equilibrium contact angle (a = 0). The apparent contact angle at the location represented by the thickness B. as. is larger in this case. The film and substrate are at a temperature above the reference saturation temperature set by condensation in a thick film in a region not shown in the figure. If the film is sufficiently thin. S = So. it is kept from evaporating by the additional van der Waals force acting between the solid and the liquid. n(S). as given in Eqs. ( 18 & 32) above. For the spreading case. the liquid-solid interaction is stronger than the liquid-liquid interaction. In the limit. the film thickness can be of the order of a monolayer. If a section of the film is slightly tapered. portions of the film may be sufficiently thick to evaporate since the force of attraction between the solid and the surface molecules decrease with an increase in the film thickness. Fluid flows towards the thinner region as a result of the thickness gradient. Gradients in the temperature. composition. and/or the curvature can either enhance or impede the flow. As a convenient example. it is possible to focus on a plane in the film where the thickness is approximately 10-8 m so that a continuum approach is acceptable. For developed laminar flow in a slightly tapered thin liquid film. the mass flow rate per unit width of the film can be represented by (Wayner. et a1. . [11])

@ dPl r = - (-d ) 3v x (47)

As a result of the van der Waals forces between the molecules, an "effective pressure" in the liquid can be represented by Equation (48). We note that the effective pressure can have a negative value because it is related to the internal pressure and A < 0 for spreading systems.

A Pz = (48)

Since there is almost a random inconsistency in the sign convention used in the literature for this equation, we re-emphasize that we are analyzing spreading systems with A < O. This is opposite the sign convention used in some articles but consistent with the Hamaker constant convention. In addition, the force concept is viewed two ways in the literature: Derjaguin pioneered the disjoining pressure concept (see, e.g. , Derjaguin, et at , [9 and 31]); an insight into the body force approach can be obtained in Huh and Scriven, [34], Miller and Ruckenstein, [35], or in Lopez, et at , [36]. Combining these two equations gives the mass flow rate per unit width at a given location"x" , nx), in terms of the film geometry (thickness and thickness gradient) and a flow coefficient.

Ao' r = ­

vo (49)

In turn, the slope of the liquid-vapor interface is a dimensionless mass flow rate in the liquid film due to van der Waals forces. Assuming that all the liquid which flows through a plane perpendicular to the substrate at a given location (represented by thickness 0) evaporates in the thinner portion of the film, a contact line heat sink can be defined as

(50)

In this case the slope is a dimensionless contact line heat sink:

(51)

Using r = -pVo in Equation (49) leads to considerable insight:

586

(52)

The slope equals the ratio of a representative shear stress in the flow field to the van der Waals force per unit area causing the flow. At equilibrium the slope is zero. Therefore, the slope represents a measure of the departure from equilibrium at a given thickness. Viewing the dimensionless contact line heat sink defined in Equation (51) in the same vein, the product of 0' and the

parameter (LlhmA/vo) gives the contact line heat sink. Numerical Example. There is an extensive literature concerning the

calculation of the van der Waals force. A recent book by Israelachvili [24] allows relatively easy entry into this literature. In Table 3, we use the approximate values for the Hamaker constant presented by Wayner [37] to calculate the interline heat flow parameter and a reference heat sink. In the third column of the following table, 0 = 10 8 m. (Unfortunately, the approximation for Teflon in this reference was based on an incorrect source and is not used here as an example of a surface with a low surface energy.)

TABLE 3. Estimated values of the interline heat flow parameter and reference heat sink.

System 6hmA LlhmA

- -- (watts) - � (watts/m) v

Pentane-Gold (293 K) 5.3 x 10 9 5.3 x 10 1 Octane-Gold (293 K) 2.6 x 10-9 2.6 x 10 1 Octane-Quartz (293 K) 0.42 x 10 9 0.42 x 10-1

For an angle of one degree, the slope is 0 .0 17 . Therefore, the contact line heat sink in the region 0 S 10-8 m for the Octane-Gold system discussed above is Q = 4 .5 x 10-3 w/m when the slope is 0.017 at this thickness. This value might appear small and we would not expect a much larger value for many non-polar systems operating within these assumptions. (The most important one being passive flow due to a disjoining pressure gradient with a small slope.) On the other hand, this analysis represents the ideal limiting case of a stationary contact line with a steady state low evaporation rate for a small temperature difference and it fixes a stable boundary condition at the contact line. At higher fluxes, the contact line is known to oscillate and "boiling" in

587

the thin film is possible. In addition, a shear stress due to a composition gradient can enhance flow and polar fluids have more impressive properties. Unfortunately, a curvature gradient impeding flow is also present. The value can be estimated by taking the derivative of the mass flow rate. Therefore, the limit given by Eq. (51) is only accurate at very low fluxes but demonstrates the phenomena. The above analysis can also be used to approximate an upper limit to the heat flux in the region 0 < 10-8 m which is only the tip of the evaporating meniscus. Assuming a triangle as an approximate (albeit poor) shape for the curved film, the minimum possible area under the film is 0/0' Therefore the maximum possible evaporative flux per unit width in the above case of a small slope is estimated to be

.

(53)

A better estimate of the flux can be obtained, with more effort, using the more exact profile given in the original article by Wayner, et a1. [11] . Such a comparison would demonstrate that although significant physical insight is provided by this example, the problem is more complicated because of additional effects like curvature and surface shear. These effects are discussed below. We note that surface shear can be significant even in a very pure fluid because some distillation occurs as 0 -? o. Therefore, the use of a properly selected additive which also evaporates could enhance the heat sink.

It is interesting to note that in the above ideal example it is possible to start with one of the four distinct forces of nature (the electromagnetic force between molecules as represented by the Hamaker constant or by recently improved models) and derive a dimensionless number that represents in the limit of very small slope the contact line heat sink of an evaporating pure liquid film using simplified transport equations. The observed change in interfacial slope in the contact line region is a dimensionless evaporation rate. A recent broad review of the details of the modeling of the intermolecular forces which makes this possible is given by Israelachvili [24]. Experimental data in [15, 22, 23] confirm that the liquid-vapor interfacial slope is a function of the evaporation rate. At a much thicker location outside the range of this analysis, the measured apparent contact was also found to be a function of the evaporation rate by Cook, et a1 . [38] and by Hirasawa and Hauptarnan [39].

To place the above work in perspective, we can discuss the following four related publications concerning experiments which had different emphases. Wu and Peterson [40] studied a wickless micro heat pipe which had dimensions of 1 x 1 x 10- 100 mm. Since the fluid had a finite contact angle, they successfully used the classical equation of capillarity (without disjoining pressure) to describe the internal fluid dynamics of this integral device. Xu and Carey [41] used a completely wetting fluid to study film evaporation from a micro-grooved surface with grooves 64 11m wide and 190 J.lm deep. The classical equation of capillarity was used in the apex region of the groove and the disjoining pressure equation was used in the thin film region on the side walls to describe fluid flow. Although these two studies did not completely

588

validate the models, the models appeared to provide a realistic treatment of the integral data. In order to study the microscopic details of the augmented Young-Laplace model, Cook, et a1. [38] used a scanning microphotometer to determine the heat transfer characteristics of the evaporating contact line region of a film on an inclined flat plate partially immersed in a pool of liquid. Since the reflectivity of the liquid film on the substrate was a function of the film thickness, the tapered film was a natural interferometer. A completely wetting fluid was studied by measuring the film thickness profile in the thickness range 10-7 < �(x) <10 5 m. The results supported the hypothesis that fluid flow and evaporation in the contact line region of a thin film results from a change in the thin film thickness profile. Truong and Wayner [30] used a photos canning ellipsometer with an attached interferometer to simultaneously measure both regions of an equilibrium extended meniscus down to thickness of 25 nm. The accuracy of the augmented Young-Laplace equation for describing the isothermal equilibrium meniscus was confirmed. The use of the augmented Young-Laplace equation for the non-equilibrium case was confirmed in [15, 22, and 42].

3.2 Nucleation Equation (20) can be related to classical material in nucleate boiling as

follows. Using only the second term on the right hand side which is the Clapeyron part, the differential precursor to Equation (20) becomes

�H d In Pv = RT2 dT (54)

Assuming that �H, which is the molar heat of vaporization, is a constant and that PvVv = RT gives

Using Equations (55 & 56) gives Equation (57)

20-Pv = Plv + r

(55)

(56)

(57)

which is a simple form of the activation equation for boiling nucleation at a cavity of radius r. A vapor bubble of radius fir" with a superheat greater than the value given by Eq. (57) will grow.

589

3.3 Efect of DUQoining Pres on Difion in an Arnold Cel In Figure (6) an Arnold cell is presented in which a very thin liquid pool of

thickness II at the bottom evaporates into a stream of air flowing over the top of the cell.

Ps

air + vapor

Pq ·

I . ·d VI IqUi

FIGURE 6. Arnold Cell

In this example the following two resistances to evaporation are compared:

1) At the liquid-vapor interface kinetic theory gives Equation (58) for the mass flux as a function of a small vapor pressure difference at the liquid-vapor interface (Schrage, [5]).

2) The diffusion equation for the air plus vapor mixture above the interface is

P DM P-PB rn = RT (H-ll)

In

(58)

(59)

in which P is the total pressure, PB is the external vapor pressure and Pvi is the vapor pressure in the vapor near the interface. Due to disjoining pressure the equilibrium vapor pressure at constant temperature over the liquid film is a function of the film thickness as given by Equation (60).

Plvi = psat exp lA RT�

(GO)

590

psat is the equilibrium bulk vapor pressure at temperature T. In this problem. we want the evaporation of the film at thickness So to stop because Plvi = PB even though the bulk saturation pressure psat > PB. Therefore

VlA -PB = psat exp . S = So . A < 0 • • . (61)

We note that PB < psat because of the disjoining pressure. Alsg, exerted by

the solid on the liquid when S = So.

The pressure on the vapor side of the liquid-vapor interface. Pvi. is found as a function of S by equating Equations (58) and (59) and using Equations (60) and (61)

2M VlA P DM P-PB (-)112 (psat exp - P . ) = - -- In (--) 1tRT VI RT (R-o) P-Pvi

Using Eqs. (62 & 60), PYi and PZvi are now known as functions of S.

(62)

The mass transfer rate is solved as a function of 0 using Equations (59 and 62). The thickness as a function of time is then obtained using

do �(S) =

- rl (63)

The results are presented in Figs. (7a-7c). In Fig. (7a)the following two curves are presented: one assuming Plvi is a function of 0 as given by Equation (60); the other assuming Plvi = psat At very large time, S -7 00 if adsorption occurs (curve 2). Fig. (7b) shows Plvi ,Pvi , and PB. (The difference between PIvi and Pvi is very small and cannot be distinguished on this scale.) Fig. (7c) shows (Plvi - Pvi) as a function of o.

Although these effects seem very small they are extremely important in small passive heat exchangers and other technologies like semi-conductor processing and adhesion. For example, in drying a small adsorbed film can remain on the surface. In the example presented in Fig. (7) the following values for the hexane/air mixture were used: So = 10 A; H = 1 cm; Sint = 20

A P . ; P = 1 atm; T = 298 K; RT = 40.87 moles/m3; M = 86.18; PI = 659 kg/m3,

VI = 1.30 x 10-4 rn3/mole; Dhexane = 7.7 x 10-6 m2/sec; A :: -4 x 10-21 J;

psat (25C) = 0 .2 atm.

2.0

x c.o en 1 .5 en

N E Z

w z � Q I �

-N 0 x

"> a. , 0:;

.. e:.

2.5

2.0

1 .5

1 .0

0.5

(a)

0.005

TIME, sec

1 .5

THICKNESS, 3 x 1 09, m

(c)

0.01

2.0

N .E z ..,.­o x a. u.i � 1 .75 en en w II: a. .. «

59 1

« 1 1 .5 2.0 a.

THICKNESS, 3 x 1 09, m

(b)

FIGURE 7. Results of calculation for Arnold cell described in text.

592

3.4 Marangoni Flows In the following section, an expanded continuum model which includes a

surface tension gradient is used to discuss the physicochemical phenomena of importance to the evaporating meniscus presented in Figure (2). However, in this case, we presume that the solid substrate is inclined at an angle of e relative to the horizontal.

Neglecting inertia terms and the "y" component of velocity, the Navier Stokes equation for velocity in the x direction becomes

in which

PI = Pv - O'K + pg(o-y) cos e

B <1> = - + <1>B

on

-ax

(64)

(65)

(66)

The Young-Laplace equation of capillarity, Eq. (65), is used to model the decrease in pressure on passing from the vapor to the liquid, caused by the curved interface. A potential energy function per unit volume, Eq. (66), is used to model the difference in behavior of a thin liquid film between two bulk phases relative to the same liquid in the bulk phases (this accounts for the London-van der Waals dispersion force effect using the model presented by Miller and Ruckenstein, [35]). This approach is equivalent to using a disjoining pressure (cited above) but circumvents the need to discuss disjoining pressure explicitly.

B B 0 � 40 nm - -

on 04 (67)

B A 0 � 20 nm

on =&3 (68)

Limitation on the use of these two approximate models are given in [30]. Equation (64) is solved for the velocity, u, using the following boundary conditions:

y = O,

y = o,

u = O

au dO' tyx = 11 ay

= dx

(69)

(70)

593

The boundary condition in Eq. (70) equates the surface shear at the liquid· vapor interface to the surface tension gradient. Since surface tension is a function of temperature and concentration, this boundary condition can have a large effect on the velocity distribution and therefore the flow. The resulting velocity distribution is

1 dP cr'y u(y) = - (-) oy) + -

� dx 2 �

The mass flow rate in the film is

o 03 dP 0'02 r = P J u dy = . - ( ) + -

3v dx 2v

03 [(

1.5 K)

' K

' nBo' B' . ] r = - - + cr + 0 + pg cos a . pg sm a

3v 0

In Equation (73), the following specific forces can be identified:

(oK') capillary pressure gradient

a 'o surface shear force/volume

disjoining pressure gradient

pg sin a gravitational force/volume

(71)

(72)

(73)

Equation (73) can be nondimensionalized using the flow rate in a uniform liquid film flowing down a vertical flat plate due to gravity, f vp:

pg OJ r - -­vp - 3 v

3vf 1.5a' Ka' oK' nBo' , -- = -- + - + - + -- . 0 cos a · sin a pg03 pgo pg pg pg&1+1

I II III N v

(74)

(75)

VI

The term including B' which is very small is neglected in the above. Terms III, IV, and V depend on the film profile which can be determined experimentally. Terms I and II depend on the surface tension gradient due to concentration and temperature gradients, and must be obtained by modeling the coupled transport equations. Term I accounts for Marangoni flow. We

594

find that, although the overall size of the region is very small, the physicochemical phenomena occurring in this region are complex and control many processes (e.g., the rewetting of a hot surface and evaporating multicomponent liquid films).

The evaporative mass flux, �, leaving the film surface is obtained from Eq. (73) as

• dr m = - dx

The evaporative heat flux, q, is

LlT and, therefore, U is left unspecified at this point. However, we can envision in a very approximate way at least three resistances, e.g.,

a) resistance in the solid, Slks

b) resistance in the liquid, BIkJ c) resistance at the liquid-vapor interface, l/hlv.

(76)

(77)

(78)

The van der Waals dispersion force accounts for the resistance at the liquid­vapor interface. There could also be a resistance to diffusion in the vapor. Equation (75) was used by Parks and Wayner [43] to analyze an evaporating binary meniscus which had been previously studied experimentally. The controlling process was found to change from region to region. However, for fairly thick films (0 > 10-6 m) and the conditions studied, cr' was found to be the most important term because of a composition gradient due to distillation.

3.5 Eft of Conduction Resistance The above equations can be expanded to include the effect of conduction in

the liquid on the resulting profile of the evaporating liquid film. In this case the constant temperature difference is between the substrate and the vapor. In this vein, Moosman and Homsy [13], Stephan and Busse [14], and Schonberg and Wayner, et al. [15, 16, 42] demonstrated that the conductive resistance across the thin liquid film has a significant influence on the evaporative process. Excluding the gravitational term, the resulting dimensionless form of Equation (46) with the addition of the resistance to conduction is

- - Tl3- = -- (1 + 4» 1 d ( d4» 1 3 d� d� 1 + 10'\

and 1 q, = -

Tl3

where the parameters x: and E = 3 N c are

in which

a - Cl M )1/2

- RTvTZv

595

(79)

(BO)

(B1)

(B2)

(B3)

The parameter x: is a measure of the importance in the film of the resistance to thermal conduction. The parameter E is a measure of the importance of capillary pressure effects relative to disjoining pressure effects. This can be solved numerically using two far field conditions The first condition is

(84)

The film thickness is assumed to asymptotically approach the non­evaporating thin film thickness mentioned previously. The second condition is

" � 00 , 4> � -4>m with � -+ - 00 (B5)

Using ellipsometry and microcomputer enhanced video microscopy (interferometry) with the experimental design presented in Fig. (B), we have recently found that the film thickness profile, S(x), of a completely wetting evaporating meniscus is as presented in Fig. (9) [15, 42].

596

FIGURE 8. Cross-sectional view of circular capillary feeder.

\\

-- lio .. 1 5nm, a .. O.OW

lio = 6.2nm, a = 0.2W

---- lio '" 5.6nm, a = 0.5W

\ \, \ \ \ '\ \ \ 0

0 50 1 00 1 50 200

RELATIVE DISTANCE (X) Ilm

FIGURE 9. Square root of the heptane film thickness versus relative distance,

A constant curvature would be represented by a straight line in this figure. The uppermost curve (Q=O) represents a system very close to equilibrium. We find that the thickness at the leading edge, °0, decreases and that the curvature profile changes with evaporation (Q>O) as predicted by the models

597

presented herein. Because of the scale used, the change in slope in the thicker film is hot apparent. However, there is a definite change. Since a liquid can be easily deformed under stress, the thickness profile represents the pressure field in the meniscus. For this purpose, the following .dimensionless interfacial pressure difference model is used:

(86)

where TJ = 0/00 is the dimensionless thickness, TJ" is the dimensionless

curvature, no = -A °0-3 > 0 is the disjoining pressure at the leading edge, and I: is the ratio of the characteristic capillary pressure to the characteristic disjoining pressure. We note that we have neglected in the approximate models presented herein the effect of film thickness on the surface tension and the Hamaker constant which we believe to be small for the thickness studied.

Since the non-equilibrium processes of change-of-phase heat transfer and fluid flow are intrinsically connected because of their common dependence on the intermolecular force field , Equation (79) for the pressure field in the film applies. The numerical solution of this equation for the case Q = 0.2 W with E = 1.5 and J( = 0.27, which is presented in Fig. (10) in terms of the pressure, was found to agree with the experimental results presented in Fig. (9). Therefore, the experimental results confirmed the model.

400

1 � 300 (.) z w a: w � 20 Ci w a: ::l 13 100 w a: a.

30 40 50

FILM THICKNESS ( l)) nm

60

FIGURE. 10 Interfacial pressure difference versus film thickness. 'Y = crLv in these calculations.

We find that the gradient in the disjoining pressure, n', associated with flow near the leading edge leads to a build-up in capillary pressure, crK, which

598

subsequently becomes the major cause of fluid flow. These characteristics for low evaporation rates and relatively large systems would also represent the phenomena in smaller systems with higher evaporation rates.

Although both the capillary and the disjoining pressures lead to fluid flow and a reduction in vapor pressure, the causes of these stresses are different. The disjoining pressure represents an increase in the average intermolecular force of attraction on the molecules at the liquid-vapor interface from solid molecules replacing liquid molecules as the film thickness decreases. Whereas, capillary pressure represents a change in the surface area and the replacement of vapor molecules by liquid molecules as the radius of curvature decreases. Since an increase in the solid-liquid intermolecular force should not directly lead to an increase in the propensity for cavitation in a flat film, we have proposed that the maximum tendency for cavitation occurs near the location where the capillary pressure is a maximum In addition, an increase in no leads to an increase in the maximum value of the capillary pressure where the interface is curved very near the leading edge of the meniscus. Using Eq. (87) for heptane with RTNI = 1.7 x 107 N/m2, which represents the change in chemical potential with film thickness, the value of no can be shown to be extremely large when the surrounding vapor pressure, P v' is kept substantially below the saturation vapor pressure, P sat. This can be easily obtained by pulling a partial vacuum at constant T.

Pv nov! = -RT ln -p sat (87)

When vapor is removed from the liquid-vapor interface, evaporation causes the intermolecular force field to change as described by the nonisothennal thickness profile presented in Fig. (10) in terms of the pressure difference and in Fig. (9). The maximum size of an isothermal change can be estimated using Eq. (87). For PvlPsat = 0.81, 00 = 0.5 run, and no = 3.6 x 106 N/m2. We note that a substantial change in the stress field can be obtained in a thin film by simply changing the equilibrium vapor pressure. Equation (88) represents the dimensionless interfacial mass flux as a function of the dimensionless interfacial pressure difference .

•

M = (1 + <1» I ( 1 + ICTl ) (88)

A plot of Equation (88) is presented in Figure (11) for the data given above.

o

1. HEPTANE - 0.2W

2. HEPTANE - 0.5W

o

ci 0 50 100 150 200 250 DIMENSIONLESS THICKNESS

DIMENSIONLESS MASS OR HEAT FLUX 1: 00 = 6.2 nm; qid = 24 1 w/m2

2: 00 = 5.6 nm; qid = 4 1 7 w/m2

FIGURE 11 . Dimensionless evaporative mass flux versus dimensionless thickness. ( 1: mid = 6.63 x 10-4 kg/m2 s,; 2: mid = 1 .15 x 10-3 kg/m2 s)

599

Thus we experimentally demonstrate that the resistance to conduction in the liquid and to evaporation at the interface leads to a maximum value of the evaporative heat flux in an evaporating meniscus. We note that the maximum ideal value of the heat flux based on kinetic theory cannot be reached. An increase in the evaporation rate leads to a large increase in the stress gradient and the capillary pressure very near the leading edge of the meniscus. The capillary pressure at the base of the meniscus would be less because of the pressure drop associated with flow in the meniscus. Since the stress has to be balanced, the stress field in the whole system is both a function of the size and shape of the system and the evaporation rate. The experimental results and theoretical modeling enhances our understanding of the deatails of the evaporative process as follows. First, the contact line thickness, 00' and temperature are measured as a function of energy input. This allows the parameter K ( Eq. 81) to be calculated. Then a best fit of the data is obtained using the model Equation (79) by varying the dimensionless parameter e. The parameter E can then be used to obtain the Hamaker constant using Equation (81). The values of the contact line thickness and the Hamaker constant can be used to calculate the characteristic pressure TIo . Using the previous information, the temperature difference fl T and the ideal heat flux, qid, can be calculated. The heat flux distribution can also be calculated using Eq. (88). Additional numerical results are given in [15].

600

3.6 Cavitation A recent discussion of cavitation and stress in a small system presented the

following material on cavitation [44]. Although both the capillary and the disjoining pressures lead to fluid flow and a reduction in vapor pressure, the causes of these stresses are different. The disjoining pressure represents an increase in the average intermolecular force of attraction on the molecules at the liquid-vapor interfac"e from solid molecules replacing liquid molecules as the film thickness decreases. Whereas, capillary pressure represents a change in the surface area and the replacement of vapor molecules by liquid molecules as the radius of curvature decreases. Since an increase in the solid-liquid intermolecular force should not directly lead to an increase in the propensity for cavitation in a flat film with a liquid-vapor interface, we propose that the maximum tendency for cavitation occurs nearer the location where the capillary pressure is a maximum. In addition, an increase in the drying rate would be associated with an increase in TIo and therefore the maximum value of the capillary pressure where the interface is curved very near the leading edge of the meniscus. The capillary pressure at the base of the meniscus would be less because of the pressure drop associated with flow in the meniscus. Since the stress has to be balanced, the stress field is both a function of the size and shape of the system and the local evaporation rate as observed in drying. As outlined by Hurd and Brinker [45] and others, an estimate of the absolute limit of the expansion of the liquid with the capillary pressure is given by the negative portion of the van der Waals loop. However, cavitation can occur at a lower value of the expansion and is a function of the maximum value of the capillary pressure (such as that given in Fig. (10», and other characteristics of the system. The dimensionless pressure difference, $, is a general measure of changes in the internal pressure and can be used to model fluid flow and evaporation.

A schematic diagram representing the change of phase process in the evaporating meniscus is given in Fig.(l2) where the Gibbs energy is presented as a function of the pressure. On the saturation line for a bulk liquid, the vapor pressure increases and the Gibbs energy decreases with an increase in temperature. Evaporation naturally occurs from the higher temperature liquid to the lower temperature liquid. This changes in the meniscus under tension. At the base of a relatively large meniscus with a fluid temperature of Tlv the vapor pressure, Plv' would be close to that of a bulk liquid. If a sink at Tv with P v existed evaporation and vapor flow between locations "2" and "I" would occur. As fluid flows towards the interline along line 2-31-41, the effective pressure in the liquid decreases. The equilibrium pressure jump across the liquid-vapor interface is represented by a horizontal tie line of constant chemical potential, e.g. , 31-31v. As long as the vapor pressure at the liquid-vapor interface is greater than the vapor pressure of the sink, evaporation occurs.

saturation line

� C!) cc w z w (J) £Xl II a liquid

�

constant temperature increase in film

3, 3fv vapor

P, PRESSURE

FIGURE 12. Gibbs energy verses pressure.

601

Since the liquid is under tension, cavitation can also occur. An increase in the tension in the liquid leads to an increase in the propensity for cavitation because of the increase in the length of the horizontal tie line. if a bubble were to form in the liquid with a vapor pressure lower than the vapor pressure on the tie line (i.e. , a bubble larger than the equilibrium one) it would grow because the liquid with a higher chemical potential would tend to evaporate and form a vapor with a lower chemical potential. Conversely, a smaller bubble would collapse. The longer the tie line, the larger the range of sizes that would grow when randomly formed. However, we feel that So represents the most stable location in the thin film due to the large liquid­solid intermolecular force field. Therefore, we propose that cavitation occurs at a location somewhere between the base of the meniscus and the interline. Rhetorically, can we further presume that it occurs at the peak value of the evaporation rate as given by Eq. (88)? The distribution for the simpler evaporation process is a measure of this non-equilibrium process.

4. THE VAN DER WAAlS DISPERSION FORCE

A considerable portion of the above material depends on the intermolecular force commonly known as the van der Waals dispersion torce. A good discussion of intermolecular and surface forces is given in the book by Israelachvili [24] . Some of the material in this section follows the direction of this book.

4.1. Derivation of the nonretarded van der Waals interaction fre energy (per unit area) between two flat surfaces across a vacuum, W= ·Al12 1tS2: A = Aij

a) First we analyze the molecule-surface interaction for a molecule at z = 0 interacting with the surface of a solid located at z = D as presented in Figure 13 .

602

ll x molecule at z = 0 solid at z = D

p = P i number dens ity of molecules

ring volume = 2 1t x dx dz

FIGURE 13. Molecule-Surface Interaction.

Using w(r) = � , the hard sphere pair interactive potential between two

molecules, the net interactive energy for a single molecule at a distance D away from the surface is

z=oo z=oo WeD) = J pw(r)dv = -21t Cp J dz J

z=D x=O and for n = 6,

WeD) = -1t C p/6D3 energy/molecule

4.2. Surface-surface interaction with number densities, Pi = Pj = P

(89)

(90)

From Eq. (90) the interaction energy of this sheet of volume number pdz with

the infinite surface is (pdz). Thus for the two surfaces separated by the

distance D,

Using the Hamaker Constant, A12 = 1t2CP1P2, and Pl = P2 = p, D = o.

A W(O) ( energy/area)

(91)

(92)

603

A typical value of A is A "" 10 19 J. In this case A > 0 and there is a decrease in the potential energy as the two slabs move closer together thereby naturally decreasing the vacuum space.

molecules : p dx dy dz = P 1 dz o (dx dy) = 1

z

p . J

d z

FIGURE 14. Surface-Surface Interaction.

4.3 The Force Law for Two Flat Surfaces Separated by a Vacuum using the Hamaker Constant Concept. (In this case, F is the van der Waa1s force per unit area.)

t

vacuum

(2)

aw A - = F = - = - TI ali 61tl)3

Since A>O, the two solids are attracted to each other. The disjoining pressure used in the other sections was for a liquid separating a solid and a vapor, therefore the liquid disjoined the solid from the vapor. Example in Adhesion: For two planar surfaces in contact (i.e . , 5 == 0.2 run) the sticking pressure is F = Ai61t53 = 7 x 108 N/m2 == 7,000 atm. Therefore the sticking pressure is large and decreases as li3 . For a liquid, this concept is related to the internal pressure.

604

4.4 Calculation of van der Waals Forces from the DLP Theory The following material from Truong's review of the literature [23] outlines

both the approximate and more accurate models for the evaluation of F(d). There are various inaccuracies associated with the classical Hamaker constant approach [46], e.g., the assumption of pairwise additivity. These problems are avoided in the DLP theory in which the forces are derived in terms of the dielectric properties of the materials [29, 47, 48]. The Dzyaloskinskii Lifshitz-Pitaevskii (DLP) theory enables one to calculate the

van der Waals forces (per unit area), F(B), between a solid (1) and a vapor (2) across a liquid film (3), from the temperature and the optical properties of the materials via the following equation:

where

Qj(X) = (Si + PX)/(Si - px) ,

and

Sj = (Ej/E3 - 1 + p2)112 , i = 1,2

The quantities kB,T,c,B are the Boltzmann's constant, the absolute temperature (K), the speed of light in vacuum, and the film thickness,

(93)

respectively. EI, E2 and E3 are the frequency-dependent dielectric susceptibilities of the three media evaluated on the imaginary frequency axis

at m = i�n. Here, the eigen frequencies �n are defined as 21tkTnIYl where 21th is Planck's constant. The prime on the summation sign indicates that the n = 0 term is to be given half weight.

F(B) can either be positive or negative, If there is a repulsion between the

solid and the vapor with a liquid film in between, F(B) is negative. This is the so-called repulsive van der Waals force and it tends to thicken the film (spreading case). In this case, one can also say that there is a stronger attraction between the liquid molecules and th� solid molecules than between

two liquid molecules. On the other hand, F(B) is positive when there is an "attraction between the solid and the vapor" (finite contact angle case). This attractive van der Waals force tends to thin the film. Therefore the film is unstable in this case, which is important for dropwise condensation.

To fully appreciate the usefulness of the DLP theory in the calculations of van der Waals forces, one must understand its limitations. First, the DLP theory assumes that the dielectric susceptibility of a continuum is the same throughout the material. This is not true since the dielectric characteristics

605

of a surface layer are different from those of the bulk (Jackson, [49]). Therefore, the theory is valid only when the separation between the surfaces - ­

i.e. , the thickness of the film -- is large compared to their interatomic distance. Second, the surface roughness is not taken into account in this theory. Third, if solids are brought into atomic contact, other short range forces may develop which are much larger than the van der Waals interactions. For example, the contribution of van der Waals interactions to the surface energy of metals is only about one-tenth of that due to metallic bonds (Buckley, [50]).

In the limit of a very thin film (non-retarded regime, B � 20 nun), F(B) is given in the following asymptotic form, (Dzyaloskinskii, et aI. , [29]),

A h 00 [El(i�)-E3(i�)] [E2(i�)-E3(i�)] F(B) = = 81t2@ J [E1(i�)+E3(i�)] [E2(i�)+£3(i�)]

while for thick film (retarded regime, B � 40 nm),

£10 £20 (SlO-P ) (S20-P ) }

£30 E30 � + £10 £20 p2

(SlO+P -) (S2o+P -) £30 £30

where SlO = (E1IYE30 - 1 + p2)112 and S20 = (£201£30 - 1 + p2)1l2.

(94)

(95)

£10, £20, £30 are the static dielectric constants of the solid, vapor and film, respectively. In Equation (94), A is the well-known Hamaker constant: while B is defined as the retarded dispersion force constant in Equation (95). Note that only the dispersion part of the van der Waals forces is present in non-polar liquids. For stable completely wetting films, both A and B (i .e. , F(B» must be negative.

The above asymptotic equations are valid only when T -+ O. At room temperature, calculations of the van der Waals forces using the asymptotic equations can differ as much as 25% from the results obtained from the complete equation (Rabinovich and Churaev [5 1], Truong and Wayner [30]). Furthermore, a comparison between the asymptotic equations and the complete equation showed that A and B are not constant (Gingell and Parsegian [52] , Christenson [53]). They are, however, a weak function of film thickness in their respective regimes. Although we have listed many limitations, the important basic connection between the heat transfer coefficient and the dielectric properties has been formulated.

606

Dielectric Susceptibility £(i�): The van der Waals forces can be calculated exactly from the DLP theory if the frequency-dependent dielectric functions of the participating media are known. These functions provide information about the strength and location of the energy absorption spectra at all frequencies. The dielectric function is a complex quantity and it can be expressed as a function of the real frequency w via the following equations [29],

£(CI)) = £'(00) + i£"(CI)

= [n(co) + ik(co)]2

where n is the real part of the refractive index and k is the absorption coefficient.

Comparing Equations (96) and (97) leads to the following,

£'(00) = n2(co) - k2(co)

e"(co) = 2n(co)k(co)

(96)

(97)

(98)

(99)

£"(00) is the measure of the energy dissipation of the electromagnetic field propagated in a medium. Through the Kramers-Kronig relation, we can relate £(i�) to £"(00) as follows, (Landau and Lifshitz [53]),

2 00 00 e"(co) E(i�) = 1 + - -- dco

1t 00 + �2 (100)

Therefore, a complete knowledge of £"(00) via Equation (99) will yield the dielectric function evaluated on the imaginary axis (i.e., £(i�». However, optical data over a wide range of frequencies for most materials are often limited. Fortunately, Parsegian and Ninham [55-58] have introduced a new approach to determine £(i�) from scanty spectroscopic optical data in the literature. The above material is presented to give an outline of the most accurate approach to determine function F(8). Examples of its use are in Truong and Wayner [30].

4.5 Approximate Model Instead of using the complete equations developed by Lifshitz we shall use as

an example the following simplified form for the interaction between two bodies ( 1,2) across another medium which was discussed in [24].

i

Liquid 3 B --L-

Solid 1

In this case we are calculating the force of attraction between vapor slab. 2. and solid slab. 1 . across liquid slab 3.

A = Av=o + Av>o

3hve (n�-n:) (n�-n:) Av>o =

8--/2 [(n�+n�)1J2 (n�+n�)1/2 {(nt+n:)1J2 +

(101)

(102)

(103)

For spreading we need A < O. Therefore. E2 < E3 < E1. That is Evapor (usually taken as £2 = 1) < £ liquid < £ solid. This gives a repulsive force F = (AlmB3) acting between the vapor slab and the solid slab for a spreading system.

4.6 Numerical Example: Hamaker Constant

kT = 4.112 x 10 21 J at 298 K

El = 3.8 (dielectric constant quartz)

= 1 (vapor) £3 = 1.84 (pentane)

Using Equations (102 and 103) we find that

( 0-21 J) 3.8-1.84) (

1-1.84 Av=O = (0.75) 4 .12 x l (3.8+1.84 1+1.84)

= -(0.75) (4.12 x 10-21 J) (0.102) = -3.18 x 10-21 J

(104)

(105)

607

608

Ve == main electronic absorption frequency in uv

= 3.1 x 1015 8-1

n�,3 index of refraction = E 2,3

nl = 1.448

Av>O =

8-. (2.096+ 1.84)1/2 (1+ 1.84) 1/2 [(2.096+ 1.84)112+(1 + 1.84)112+(1 + 1.84)112]

= -0.958 x 10-20 J (106)

Therefore, A = Av>o + Av=o = -1 .276 x 10-20 J

4.7 Combining Rules: Hamaker Constant (Approximate equations for simple systems are useful because data on Aii

are available.)

(107)

interaction of media with media "i" across a vacuum

o for air

or

Aslv = An - Als (108)

Therefore, the Hamaker constant for the liquid film represents the difference between cohesion and adhesion. If adhesion is stronger than cohesion, the liquid forms a completely wetting film. This can be theoreticaly determined from the dielectric susceptibilities of the liquid and substrate. These approximate equations were used by Wayner [37] to demonstrate the effect of the macroscopic optical properties of the system on the heat sink capability of an evaporating thin film.

4.8 Surface Energy The surface energy, 0", of a liquid can be calculated from the Hamaker

constant using

609

A 0 - ---24n: D5 D = 0.165 nm for hydrocarbons. (109)

Equation ( 109) gives the relationship between the DLP theory and the classical surface tension.

5. . SUMY

The above material outlines recent developments in interfacial phenomena and how they can be utilized in change of phase heat transfer. Evaporating thin liquid films were emphasized. The experimental and theoretical results demonstrate that it is possible to start with the intermolecular force concept (as described by the response of the materials to a fluctuating electromagnetic field) and detenrune the heat transfer coefficient and heat sink of an evaporating ultra-thin film from first principles. The successful application of the material to a relatively simple example of an evaporating spreading system with a non-polar fluid should lead to the use of this material in more complicated systems. Experimentally and theoretically, we find that the obtainable heat flux is less than that calculated using simpler models because of resistances in the liquid, at the liquid-vapor interface, and in the vapor space. Higher fluxes are probably possible with extremely small or turbulent and/or transient systems. On the other hand, small systems tend to be laminar. An understanding of these resistances and how they depend on the intermolecular force field is important to the proper utilization of small systems in improving technology. The basic principles have many applications in engineering.

NOMENCLATURE

A = Hamaker constant, area

A = N(6n:) a = defined by Eq. (82) b = defined by Eq. (83) B = constant in Eq. (67) C l = constant (evaporation coefficient) D = diffusion coefficient, separation distance d = differential change e = internal energy per molecule E = surface energy f = fugacity F(5) = surface force/area, potential energy/ volume g = gravitational force per unit mass H = molar enthalpy, distance in Arnold cell h = enthalpy/mass or volume, heat transfer coefficient, Planck's constant H. = Planck's constantl2n:

6 1 0

K = curvature k = Boltzman's constant, thermal conductivity, absorption coefficient M = molecular weight m = mass of a molecule, interfacial mass flux N c = dimensionless group, see Eq. (37) n = index of refraction, molar density P = pressure PI = "effective pressure" in liquid Pv = pressure in vapor Q = contact line heat sink. Q* = dimensionless contact line heat sink q = heat flux R = gas constant r = radius of curvature, distance between molecules S = distance in solid, spreading coefficient s = entropy T = temperature t = time U = overall heat transfer coefficient u = velocity V = average velocity, molar volume v = volume per molecule x = parallel to flow direction y - axis a = interfacial heat transfer coefficient

� = difference o = liquid film thickness 0' = slope of liquid-vapor interface E = dielectric constant, dimensionless group E = 3N c

cp = van der Waals potential energy/volume. dimensionless pressure difference

r = mass flow rate per unit width of film

'Y = film tension 11 = dimensionless film thickness � = dimensionless position

� = chemical potential per molecule or mole. dynamic viscosity

n = disjoining pressure 0' = surface free energy per unit area 't = shear stress a = angle of inclination, contact angle p = fluid density, number of density of molecules v = kinematic viscosity (j) = frequency of electromagnetic wave

� = imaginary axis

Subscripm and Superscripts

B = external partial pressure in Arnold cell, bulk g = includes gravity i = interface, average value id = ideal 1 = liquid Iv = liquid-vapor interface m = unit mass o = reference s = solid sl = solid-liquid interface dsv = solid-vapor interface sat = saturated v = vapor

6 1 1

vp = vertical plate x = evaluated at x

= derivative

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