COMPUTATIONAL TWO-PHASE FLOWS IN CONDUITS

WITH AND WITHOUT HEAT ADDITION

by

JURIZAL JULIAN LUTHAN, B.E., M.S.M.E.

A DISSERTATION

IN

MECHANICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

May. 1992

T'-'

[He.

N 0 • 5 0 ACKNOWLEDGMENTS

During the course of this work several fine people have made contribution

toward its completion and for that I owe my gratitude to them. In particular, I

wish to express my gratitude to:

• Dr. Siva Parameswaran, my advisor, for his help and advice.

• Drs. T. T. Maxwell, H. J. Carper, Jr., F. A. Mohamed, and R. S. Narayan

that have served as my committee members.

• Michael Malin from CHAM Ltd. of England for providing the 1-D to-fluid

model program.

• My friends Ahmet Unal for helping me with the literatures; Steven Ekwaro,

Ghulam Mustafa, and many others for their encouragement.

The real burden of this work has been borne by my wife and my daughter.

For their patience, understanding, and love I dedicated this work to them.

My deepest appreciation goes to my parents and my brothers and sisters that

have stood by me all these years with their du^a and love. Finally, I'd like to

express my sincere gratitude to Bpk. Julius Tahija sekeluarga that have made

me believe that real friendship exists and have made me feel its warmth and that

without their help this endeavor will end up to be just another wild dream.

11

CONTENTS

ACKNOWLEDGMENTS ii

ABSTRACT v

LIST OF TABLES vi

LIST OF FIGURES vii

NOMENCLATURE ix

I. INTRODUCTION 1

II. LITERATURE REVIEW 6

2.1 Introduction 6 2.2 Local Instant Formulation 7 2.3 Averaging Techniques 7 2.4 Constitutive Equations 9

III. MATHEMATICAL MODELS 16 3.1 Introduction 16 3.2 Local Instantaneous Formulation of the General Balance

Equation 17 3.3 Time Averaging 23 3.4 Space Averaging 24 3.5 Covariance 25 3.6 Time-Averaged Two-Fluid Model General Balance

Equation 27 3.7 Two-Fluid Model Conservation Equations 28

3.7.1 Mass Balance 29 3.7.2 Momentum Bsdance 29 3.7.3 Energy Balance 29

3.8 One-Dimensional Two-Fluid Model Governing Equations . 32 3.8.1 Conservation of Mass 32 3.8.2 Conservation of Momentum 33 3.8.3 Conservation of Energy 34

3.9 Flow Regimes 37 3.9.1 Bubble to Slug Transition 37

111

3.9.2 Slug to Annular-Mist Transition 39 3.10 Interphase Drag Relations 39

3.10.1 Bubbly Flow Regime 40 3.10.2 Slug Flow Regime 41 3.10.3 Annular-Mist Flow Regime 42

3.11 Heat Transfer Modeling 43 3.11.1 Single-Phase Forced Convection 45 3.11.2 Two-Phase Heat Transfer Processes 46

3.11.2.1 Saturated Nucleate Boiling 47 3.11.2.2 Subcooled Nucleate Boiling 49

3.12 Interphase Mass Transfer ModeHng 49 3.12.1 Wall Mass Transfer 50

3.12.1.1 Subcooled and Saturated Nucleate Boiling Heat Transfer 50

3.12.1.2 Condensation Heat Transfer 51 3.12.2 Bulk Mass Transfer 51

3.12.2.1 Heat Transfer Process {TL < T') 51 3.12.2.2 Flashing Process {TL > T') 53 3.12.2.3 Condensation Heat Transfer Process . . . . 54

IV. NUMERICAL APPROXIMATIONS 56 4.1 Introduction 56 4.2 Finite-Difference Formulations 56

4.2.1 Conservation of Mass 58 4.2.2 Conservation of Momentum 62 4.2.3 Conservation of Energy 69

4.3 PEA and TDMA 74 4.3.1 PEA 74 4.3.2 TDMA 76

4.4 Guessed Pressure Field and Pressure Field Correction . . . 79 4.4.1 Guessed Pressure Field 79 4.4.2 Pressure Field Correction 80

4.5 Solution Procedure 93

V. RESULTS AND DISCUSSIONS 95 5.1 Introduction 95 5.2 One-Dimensional Stratified Flow 96

5.2.1 Some Specific Relations 96 5.2.2 Discussion of Results 99

IV

5.3 Simplified Two-Phase Flow With Heat Addition 103 5.3.1 Problem Description 106 5.3.2 Discussion of Results 110

5.4 Two-Phase Flow With Heat Addition 123 5.4.1 Experimental Setup and Problem Description . . . . 123 5.4.2 Discussion of Results 133

VI. CONCLUSIONS AND RECOMMENDATIONS 152 6.1 Conclusions 152 6.2 Recommendations 154

REFERENCES 156

ABSTRACT

The main objectives of this study of two-phase ga^-liquid flows are to reduce

the time and cost and to improve prediction capability of process development

in comparison with purely empirical design methods.

The problem associated with mathematical modeling of the detailed flow pat­

terns in two-phase flows involves the solution of strongly coupled, nonlinear par­

tial differential equations of the field equations. The solution of these equations

lies well beyond any existing analytical approach. Therefore finite-diflference ap­

proximations, based on IPSA (Inter-P,hase £lip A^ialyzer) algorithm, are used to

solve the problem.

Three cases are considered in this study. The first is the problem of two-phase

gas-liquid stratified flow with constant properties for both fluids. The second is

the problem of idealized boiling problem where, again, the properties of the two

fluids are taken to be constant. As the last one, the previous problem is revisited

by relaxing the simplifying assumptions.

The last two cases are treated as pseudo-transient problems. In addition, all

three problems are computed with one spatial dimension dependency. While the

flow model employed is two-fluid or six-equation model.

The results are then compared with the available analytical solution and

experimental data. It was found that they are satisfactorily comparable. The

methodology developed may be useful in future research with other fluid pairs

or components.

VI

LIST OF TABLES

3.1 Definition of terms used in the general balance equation 19

4.1 Variations of interfacial friction direction with velocities' directions

for \wGp\ > \wLp\ 64

4.2 Variations of interfacial friction direction with velocities' directions

for \wGp\ < \wLp\ 64

Vll

LIST OF FIGURES

3.1 Sketch of two-fluid material volume 18

3.2 Regions of heat transfer and flow patterns in convective boiHng . . 38

3.3 Variation of void fraction along a heated pipe 44

4.1 Sketch of main and velocity control volumes 57

4.2 Sketch of variation of wcp vs. P/y 86

4.3 Sketch of shifted control volumes 88

4.4 Sketch of variation of WG, VS. P5 91

5.1 Geometries in stratified flow 97

5.2 Grid independence study for stratified flow 100

5.3 Liquid surface plots for frictionless case 101

5.4 Velocity distributions at t = 2.5 5 102

5.5 Liquid surface plots for the case where the effects of friction are included 104

5.6 Liquid surface plots for the case where the effects of friction are included for circular and rectangular channels 105

5.7 Sketch of idealized annular flow 109

5.8 Grid independence study for idealized boiling I l l

5.9 Convergence history of a^ 112

5.10 EflFect of mass flux to void fraction distribution 113

5.11 Vapor void fraction distribution for rectangular duct 114

5.12 Liquid void fraction distribution for rectangular duct 115

5.13 Liquid stagnation enthalpy distribution for rectangular duct . . . . 117

5.14 Vapor stagnation enthalpy distribution for rectangular duct . . . . 118

5.15 Vapor velocity distribution for rectangular duct 120

5.16 Liquid velocity distribution for rectangular duct 121

5.17 Pressure distribution for rectangular duct 122

Vl l l

5.18 Comparison of vapor void fraction distributions for circular and rectangular ducts 124

5.19 Comparison of liquid void fraction distributions for circular and rectangular ducts 125

5.20 Comparison of vapor stagnation enthalpy distributions for circular and rectangular ducts 126

5.21 Comparison of liquid stagnation enthalpy distributions for circular and rectangular ducts 127

5.22 Comparison of vapor velocity distributions for circular and rectangular ducts 128

5.23 Comparison of liquid velocity distributions for circular and rectangular ducts 129

5.24 Comparison of pressure distributions for circular and rectangular

ducts 130

5.25 Sketch of experimental set-up of Schrock and Grossman 131

5.26 Grid independence study for E-260 experimental run 135

5.27 Wall temperature distribution for E-260 experimental run 136

5.28 Phasic temperature distributions for E-260 experimental run . . . 139

5.29 Pressure distribution profile for E-260 experimental run 140

5.30 Void fraction distribution for E-260 experimental run 142

5.31 Christensen experimental data 143

5.32 Velocity distributions for E-260 experimental run 145

5.33 Wall temperature profile for E-278 experimental run 147

5.34 Phasic temperature distributions for E-278 experimental run . . . 148

5.35 Pressure distribution for E-278 experimental run 149

5.36 Void fraction distribution for E-278 experimental run 150

5.37 Velocity profiles for E-278 experimental run 151

IX

NOMENCLATURE

a

A

A

b

B

Cp

C

CD

4

E

f

F

9

G

h

h-LG

H

I

J

k

M

N

Nu

P

finite-difference coefficients or constants

area or constant

interfacial area per unit volume

constant

body force or constant

constant pressure heat capacity

convective coefficients

interfacial drag

bubble diameter

hydrauHc diameter

energy source

coefficient of friction

Reynolds number factor in Chen's correlation

gravitational acceleration

mass flux

enthalpy

enthalpy of vaporization

heat transfer coefficient

interfacial source term

efflux of quantity ip

heat conductivity

momentum source

outward normal of Sm

Nusselt number

pressure

Pr

Q

r , R

Re

S

S :

^m '

t :

T

T

U

V

v,v w

We

X

X, y, z

Xu

Prandtl number

heat flux per unit volume

pipe radius

Reynolds number

finite-difference source terms

suppresion factor in Chen's correlation

material surface

time

temperature

stress tensor

velocity

velocity vector

volume

velocity

Weber number

mass quality

: coordinate directions

: Lockhart-Martinelli parameter

Greek

a '

/^

u

P

a

void fraction

kinematic viscosity

dynamic viscosity

: density

: surface tension

XI

r

r e

i>

4'

: wall shear stress

: mass source term

: volumetric error

: general variable

: source term of ij)

Subscript

6

c

DB

fd

G

L

m

n,N

NcB

V '

p,P :

5, S :

W :

: pertaining to bubble

: convective

: Dittus-Boelter

: fully developed

: gas or vapor

: liquid

: mean

North

nucleate boiling

pertaining to particle

pertaining to current grid node

South

wall

Superscript

: pertaining to interfacial

: averaged or time averaged value

Xll

771

M

o

s

mass conservation related quantity

momentum conservation related quantity

value of old time level

indicates saturation value

X l l l

CHAPTER I

INTRODUCTION

The simultaneous flow of two phases or the flow that consists of several compo­

nents occurs in a wide range of industrial applications as well as in many natural

phenomena. Examples of industrial applications include nuclear reactors, con­

ventional power generating plants, crude oil pipeline, as well as in air-conditioning

and refrigeration equipment. The type of flows studied in this investigation is the

simultaneous flow of gas and liquid as a subset of the whole family of multiphase

flows. Here, the words gas and vapor will be used interchangeably.

The complexity with a gas-liquid flow lies in the fact that the two phases can

distribute themselves in the conduit in a large variety of ways which are beyond

the control of the experimenter or designer. For example, the distribution is

susceptible to small changes of flow rates, fluid properties, conduit inclination,

or conduit shape. Furthermore, the velocities and shapes of the interfaces are

unknown. Therefore it is impossible to determine the fluid properties to be

used at a certain point in time when attempting to solve the differential balance

equations of conservation of momentum or energy for this kind of flow because

the spatial location of the phases is unknown. To aggravate the situation, the

boundary conditions related to the interface needed to solve the problem are also

unknown. Hence, at a glance it seems that the various multiphase systems and

phenomena have very little or nothing in common. Fortunately, this is not true.

It is known that all two-phase flow systems share the same singular characteristic

in the presence of several interfaces between the phases or components so that

many of the two-phase flows have a common structure via this interface. Now if

a single phase flow can be classified according to the dynamics of the flow into

laminar, transitional, or turbulent by virtue of its Reynolds number; two-phase

flows can be classified according to the geometry of the interface into three main

categories. They are separated, mixed or transitional, and dispersed flows.

Due to these complexities, a sound fundamental understanding of the process

is needed to support the development of rational methods in designing two-phase

systems. Therefore, the gas-liquid two-phase flows have become a subject of great

importance for researchers in various industries.

Major objectives of the analysis of two-phase flows are to reduce the time and

cost and to improve prediction capability of process development in comparison

to purely empirical design methods. The problem associated with mathemati­

cal modeling of the detailed flow patterns involves the solution of the strongly

coupled, nonlinear partial differential equations of the field equations of conser­

vations of mass, momentum, and energy. The solution lies beyond the existing

analytical approach. Hence, a numerical approach must be adopted. The solu­

tion procedure employed in this study is based on the IPSA (Inter-JPhase Slip

Analyzer) algorithm. The development of the mathematical basis of the general

procedure will be discussed in the following chapters.

To undertake this complex task, a step-by-step approach is desirable, and this

will be followed as the format of this report. The first problem to be studied is the

problem of two-phase gas-liquid stratified fiow. In this problem, the properties of

both fluids are taken to be constant. Also, there is no heat addition or substrac-

tion from the system, and it is assumed that there is no mass exchange between

the two fluids. Therefore the whole problem is reduced to solving the coupled

mass and momentum conservation equations. The results are then compared

with the analytical solution available for this type of problem with an additional

simplifying assumption in that the flow is frictionless. The completion of this

problem gives confidence in handling and developing the numerical scheme for

mass and momentum equations.

The next step is to study the problem of idealized boiling as suggested by

Spalding (1987). Here, the transport and thermodynamic properties of the two

fluids are again taken to be constant. Also, the geometry of the interface is

assumed to be constant, that is, from the inlet to the outlet annular flow pat­

tern, which is one type of the separated flow category, prevailed. This way the

problem solution can be simplified quite a bit because the code does not have to

include the capability to determine the properties of the fluids as well as the flow

pattern transitions. Thus, the previous problem is extended a little bit with the

addition of energy equation and the exchange of mass which cannot be neglected

any longer. The results obtained are then compared with experimental data. Of

course, it seems ridiculous to compare this highly idealized problem with experi­

mental data. However, the main objective in this comparison is to check that the

trend showed by this simplified problem is in conformity with the experimental

data as well as gaining experience in handling the complete set of conservation

equations.

As the last step, the above problem is revisited by relaxing the simplifying

assumptions. Here it is assumed that there is no dissolved gas in the system

which is also apphcable to the first two problems. Thus, a routine that handles

the transport and thermodynamic properties of the fluids as well as a capability

to determine the variations of the flow patterns together with their associated

flow parameters need to be included. This way, the problem closely approaches

a practical problem. And the computational results are then compared with the

experimental data of Schrock and Grossman (1959) for low quality flow boiling.

The last two problems are treated as pseudo-transient problems. That is,

they are treated as marching-in-time problems until the results obtained from

two consecutive time levels do not show any appreciable changes. In addition,

all the three problems are computed with one spatial dimension dependency.

Because most of the well established correlation equations for the constitutive

relations are developed for one spatial dimension. This preclude the idea to

extend the working balance equations to a higher degree of dimensionality. In

this last problem, it is assumed that neither phase can exist in a metastable form,

that is, the vapor can be either saturated or superheated but not subcooled

whereas the liquid can be either saturated or subcooled but not superheated.

This assumption follows the practice of Moeck and Hinds (1975).

There are many models developed to study the phenomena of two-phase flows.

Until recently various mixture models have been extensively used to study two-

phase flow problems. The reason is not only because of the simplicity they offer

in terms of the field equations but also because of the smaller number of closure

relations needed to specify the problems completely. In this environment where

the data base presently available is limited and the difficulties encountered in

the attempts of measuring two-phase flows in detail, an advanced mixture model

such as the drift-flux is perhaps the most favored and accurate theoretical ap­

proach for standard two-phase flow problems. However, a more detailed account

of two-phase flow problems is promised by the two-fluid model. In this model,

each of the fluids that makes up the flow is considered separately. Thus the num­

ber of field equations are doubled with one set for each fluid. The same thing is

true for the closure relations: the number is considerably higher for the two-fluid

model in comparison with that for drift-flux model. Consequently, much more

detailed experimental data are needed to develop satisfactory closure relations.

Unfortunately, this information is often not available. This is because the com­

paratively young age of this type of modeling. So that the present state-of-the-art

in two-phase flow measurement techniques implies considerable uncertciinties in

the closure relation expressions for the case of two-fluid model. Nevertheless,

this study is built exclusively on the ground of two-fluid modeling.

The remainder of this report consists of various topics of interest in the realm

of two-phase flows of liquid and gas and are discussed in the chapters outlined

below. The previous endeavors in the field of two-phase gas-liquid flows are

reviewed in the form of literature survey and are covered in Chapter 2. The

conceptual models for two-phase flows are formulated in terms of field equations

which describe the conservation laws of mass, momentum, and energy. These field

equations are complemented by the appropriate constitutive equations such as the

constitutive equations of state, heat transfer, and stress all of which are presented

in Chapter 3. Chapter 4 outlines the numerical formulations and algorithms

based on the conceptual models developed in the previous chapter for both the

gas and liquid phases. In Chapter 5, the theory is applied to stratified two-phase

gas-liquid flow, idealized (or more appropriately, highly simplified) two-phase

flow in a conduit with heat addition, and lastly a problem of constant heat flux

boiling in a vertical circular pipe is considered. In the end. Chapter 6 draws the

conclusions and is then followed by the recommendations for future work based

on the experience gained throughout the developments of the theoretical models

presented in this work.

CHAPTER II

LITERATURE REVIEW

2.1 Introduction

This chapter reviews the literature related to the problems under consider­

ation. After selecting the physical problems to be studied, together with de­

termining their initial and boundary conditions, then comes the mathematical

formulation of the problems. In this formulation stage, the problem can be di­

vided into three main areas. They are

1. Local instant formulation

2. Averaging to obtain working equations

3. Determination of constitutive relations.

Following which the numerical approach can be formulated to effect the solution

to the physical problems. This review will follow the above classification.

In spite of the papers and articles on specific aspects of two-phase fiows to

be cited shortly, several fine books are used as general references. They are

ColHer (1972), Tong (1965), Hsu and Graham (1976), Ishii (1975), Wallis (1969),

and Govier and Aziz (1972). The first three books devote themselves to the

problem of boiling and condensation with Collier extends the coverage to the flow

phenomena in boiling and condensation processes and Ishii exclusively discusses

the development of two-fluid modeling of two-phase flow systems while the last

two focus their attention on the discussions of the flow aspect of two-phase flow

systems.

2.2 Local Instant Formulation

This class is the most fundamental in the development of mathematical mod­

eling for two-phase flows. Microscopically, a two-phase flow system is formed

by several single phase regions which are bounded one another by moving inter­

faces. Therefore it is possible to formulate mathematical model for two-phase flow

problems by considering a field which is subdivided into single phase regions with

moving boundaries. In each of these subregions the standard differential balance

equations holds. To patch these individual subregions, appropriate boundary

and jump conditions at the phase interfaces are imposed so that the solutions

obtained match the solutions of the differential balance equations. It can be seen

that this kind of formulation is nothing but an extension of the formulation for

single phase flows in terms of local instantaneous variables. This type of treat­

ment for two-phase flow problems is called local instant formulation to emphasize

that it is based on microscopic rather than macroscopic treatment.

The derivations of the field formulations of conservation laws can be found

in the work of Ishii (1975, 1990), and Delhaye (1981). While rigorous basis for

the local instantaneous formulation is presented by Delhaye and Achard (1976).

Lastly, a sHghtly different approach of the formulation is discussed by Addessio

(1981).

2.3 Averaging Techniques

The set of equations obtained from local instantaneous formulation results

in solving a moving multi-boundary problem with the positions of the interfaces

being unknown. For most two-phase systems the mathematical complexities thus

introduced by the local instantaneous formulation can be abundant (consider the

problem of bubbly flow in a conduit) and are almost impossible to solve. This

makes direct applications of the local instantaneous formulation to practical two-

8

phase flow systems not appealing. However, there are two very important reasons

in performing the local instantaneous formulation. They are as follows

1. it can be applied directly to study basic phenomena in simple problems like adiabatic stratified two-phase flow or discrete bubbly flow

2. it is the raw material to be fed to an appropriate averaging technique to get the macroscopic two-phase flow model.

Because most of two-phase flow systems occur in practice have extremely

complicated interfacial motions and geometries, it is infeasible to solve for local

instantaneous motions of all the fluid particles that comprise the whole sys­

tem. Fortunately, the microscopic details of the fluid motions and the associated

variables are seldom needed in the solution of engineering problems. It is the

macroscopic aspects of the flow that play the important role. To achieve this,

the method based on averaging the local instantaneous formulation offers the

practical approach. Hence, the major objective in performing averaging is to

transform the set of equations from microscopic level to macroscopic one. By

averaging the respective fluid fields, part of the details of the local instantaneous

formulation is eliminated and this results in simplification of the problem. What

is left from averaging beside the macroscopic effects is the statistical effects. In

addition, collective interactions between the phases are the only thing needed to

be modeled in a macroscopic formulation rather than the individu2d interactions.

A detailed discussions of the averaging techniques can be found in Ishii (1975,

1981). Also, a rigorous derivation is presented by Delhaye and Achard (1976).

Recent expositions on the subject are contained in articles by Delhaye (1981,

1981a, 1981b) while the presentation of the subject by Addessio (1981) is partic­

ularly interesting.

2.4 Constitutive Equations

The mathematical model of a two-phase flow systems comprise of a num­

ber of differential equations complemented by initial and boundary conditions

equations. There are, basically, two sets of equations involved to completely

characterize the two-phase flow systems. The first set of equations results from

the application of the fundamental conservation laws, such as those for mass, mo­

mentum, and energy. While the second set of equations takes into account the

character of the fluids under consideration. It involves the intrinsic properties of

its mechanical and thermodynamic behaviors. These mathematical expressions

are known as the constitutive equations of the fluid following Truesdell (1969)

and Ishii (1975). The rest of the mathematical expressions needed to completely

describe the system are either the relevant thermodynamic relations—for exam­

ple, spatial derivative of fluid density—or definitions (for example, Reynolds and

Nusselt numbers).

Ishii (1975) mentioned that there are three fundamental bases in constructing

the constitutive laws. They are

1. the entropy inequaHty which should be satisfied by any constitutive laws,

2. constitutive zixioms which ideaUze the responses and behaviors of the fluids under consideration, and

3. the mathematical modeling of the responses and behaviors of the fluids

being studied.

Now, according to their physical significance, the constitutive equations can

be classified into three main classes. They are

1. Mechanical constitutive equations which specify the behaviors of the stress

tensor and the body force.

2. Energetic constitutive equations which supply the expressions for the heat

flux and the body heating.

10

3. Constitutive equation of state which gives the relationships between the well known thermodynamic variables.

Boure (1978) gives conceptual discussions in the development of the consti­

tutive laws. In this work he differentiates between intrinsic constitutive laws as

opposed to external constitutive laws. The intrinsic constitutive laws include the

equations of state which are generally well known for many single-phase fluids,

for example, the steam table. While the external constitutive laws are those that

often expressed by empirical correlations and usually depend both on the fluid

properties as well as on the initial and boundary conditions of the problem. The

example of this last type is the flow patterns. On the other hand, Ishii (1975)

covers the derivations of the relevant constitutive equations for two-phase flow

systems with a general overview is presented in Ishii (1990).

Attention is now focused on reviewing the relevant mathematical models or

empirical correlations for each of the three types of constitutive equations.

The study of Lockhart and Martinelli (1949) is one of the earliest attempts

to model the functional relations of pressure drop for two-phase gas-liquid flows.

It is one of the best and simplest procedures for calculating pressure and void

fraction in two-phase flow systems. In their study a definite portion of the flow

area is specified to each phase and they presumed that the conventional frictional

pressure drop can be applied to the flow of each phase. Thus interaction between

the two phases is neglected. The important contributions they made to the study

of two-phase flow are the ingenious inventions of the dimensionless pressure drop

parameter and the so called Lockhart-Martinelli parameter, X«. It took more

than 20 years later for Johannessen (1972) to develop a theoretical model that

explained the dependence of the pressure drop parameter with the Lockhart-

Martinelli parameter for stratified flow. However he made some simplifications

that were unnecessary like neglecting the shear stress in the interface and that

11

motivated Taitel and Dukler (1976) to relax those simplifying assumptions and

incorporated them in their investigation.

As one of the problems to be considered here is two-phase flow in a conduit

with heat addition then a review of pertaining correlations or mathematical re­

lations associated with heat transfer should be included. Now, at the inlet of the

pipe in flow boiling problems the liquid may still be in subcooled state. Thus,

before the liquid undergoes the subcooled boiling process, a single-phase con­

vective heat transfer will take place. Also, at a certain distance in the boiling

tube, for high quahty boiling, the liquid might have all transformed into vapor

so that there is a portion of the pipe in which the mode of heat transfer is again

single-phase convective heat transfer with steam as the working fluid. Therefore

a correlation for convective heat-transfer is needed. Molki and Sparrow (1986)

proposed an average value of heat transfer coefficient for turbulent flow in circular

tubes with simultaneous velocity and temperature development. They claimed

that it is the average values that are more often needed in practice. They gave

a least-square fit that corrects the local Nusselt number for fully developed flow.

There are various expressions for local Nusselt number for fully developed flows

and the one that is used in this report is that of Dittus and Boelter (1930) which

has been found satisfactory for turbulent flows. Another intresting account on

the developing flow in heated round tubes is given by McEhgot, et al. (1965).

As long as the the temperature of the heating surface is below the the satu­

ration temperature of the fluid at that particular location, no boiling can occur.

Collier (1972) reviewed the minimum limiting conditions for nucleation to begin

based on the suggestion of Bowring (1962). While Bergles and Rohsenow (1963)

obtained a graphical solution to that Hmiting conditions. Their equation is sim­

ple and is valid only for water over a wide range of operating pressure. Later on,

Davis and Anderson (1966) carried out the study to get the analytical solution.

12

Both results are in good agreement with each other and adequately predict the

onset of nucleation.

With the Bergles and Rohsenow equation being satisfied, the so-called sub­

cooled boiling process takes place. There are quite a number of empirical corre­

lations for heat transfer coeflicient and void fraction predictions for this boiling

regime. The subcooled boiling region is further subdivided into high and low sub-

cooling regions. In the high subcooHng region, the works of Rohsenow and Clark

(1951), Griffith et al. (1958), Bowring (1962), and Bergles and Rohsenow (1963)

are the important studies on the heat transfer aspects of this region. As far as

the flow's void fraction is concerned, just after the onset of nucleation the vapor

generated remains as discrete bubbles attached to the surface and is essentially

a wall effect. In this region, small bubbles grow and condense while they are still

attached to the wall so that they do not penetrate far into the bulk subcooled

stream. Therefore the void fraction in this region usually remains very low and

can be neglected according to Collier (1981). The works of Bowring (1962), Levy

(1967), and Saha and Zuber (1974) outline the procedures to estimate the void

fractions in low subcooling region with the procedure of Levy to be preferred as

being the simpler one to use.

Following these two subregions is the region of fully developed subcooled boil­

ing. In this region, the studies of Jens and Lottes (1951) and Thom et al. (1965)

are two of the most important ones. Jens and Lottes summarized experiments on

subcooled boiling of water flowing upwards in vertical electrically heated stainless

steel or nickel tubes and the data were correlated by a dimensional equation valid

for water only. While Thom et al. modifled the correlation given by Jens and

Lottes and also valid for water only. Thom et al., in the same publication, pro­

posed a procedure for predicting the void fraction for fully developed subcooled

boiling region based on the data of their experiment.

13

When the bulk liquid temperature flowing inside a heated tube reaches the

saturation temperature, nucleate boiling process takes over. There have been

many studies conducted related to this process, however, they are not considered

satisfactory so that Chen (1963) proposed a new correlation which proved very

successful in correlating all the forced convective boiling heat transfer data for

water and organic systems. He assumed that both nucleation and convective

mechanisms occur to some degree over the entire range of the correlation and that

the contributions of both mechanisms are additive. Hence, the local heat transfer

coefficient is the summation of the heat transfer coefficient due to nucleate boifing

and that is due to convection.

Numerous studies have been done to analyze void fraction in saturated nu­

cleate boiling regime. Marchaterre and Hoglund (1962) proposed the shp ratio

correlation for vertical two-phase flow. The acquired slip ratio value then can be

used to estimate the void fraction. A different empirical correlation for the slip

ratio in a variable density two-phase flow was suggested by Bankoff (1960). Later

on, Hughmark (1962) extended the application of that correlation to horizontal

and vertical flows of fluids other than steam-water mixture. Meanwhile the same

paper by Lockhart and Martinelli (1949) suggested an empirical void fraction

correlation mostly based on the data of horizontal adiabatic two-component flow

at low pressures. Subsequently, Martinelli and Nelson (1948) extended the corre­

lation to steam-water mixtures for various values of working pressures. All in all

the Martinelli-Nelson correlation gives better agreement with the experimental

data and it should be mentioned that their correlation was originally developed

for annular flow.

Consider a low quality flow boiling in which subcooled liquid flows in at the

inlet and a mixture of liquid and vapor comes out of the pipe, it is obvious

that the flow pattern will change along the pipe. Beginning with single-phase

14

subcooled liquid at the inlet, the flow becomes a bubbly flow as the fluid gets

into the subcooled boihng regime, and it becomes slug flow as more heat is

added to it, and lastly the flow takes on the annular flow near the outlet of the

pipe. This makes it necessary to be able to predict the changes in flow pattern

along the pipe. The earliest and possibly the most durable of flow regime maps

for two-phase gas-liquid flow was proposed by Baker (1954). Mandhane et al.

(1974) gives a new flow regime correlation for various flow pattern maps for two-

phase gas-liquid flow in horizontal pipes and it represents an extension to the

work done by Govier and Aziz (1972). In the work of Taitel and Dukler (1976) a

mechanistic model is developed for the analytical prediction of transition between

flow regimes for horizontal and near horizontal gas-liquid flow. While the Hewitt

and Roberts ' (1969) flow pattern map is the most widely used chart for air-

water and steam-water flows in vertical tubes. Taitel et al. (1980) also presented

models for predicting flow pattern transitions during steady gas-liquid flow in

vertical tubes based on physical mechanisms suggested for each transition. They

claim that the models incorporate the effect of fluid properties and pipe size so

that they are generally free from the limitations hampering the empirically based

transition maps or correlations. Quite recently, Dukler and Taitel (1986) gives a

review of the state-of-the-art in predicting flow pattern transitions in two-phase

flow systems.

The standard field conservation equations discussed above are, together with

the appropriate constitutive relations, valid within the region of each phase up

to a phase interface. Across the interface—for example, the boundary of gas and

liquid region or the wall and fluid boundary—the density, energy, and velocity

experience a jump discontinuity. Hence, a special form of the balance equations

should be used to take into account the singular nature of the interface. In order

to completely specify the balance equations at the interface, several pertaining

15

flow parameters need to be determined. It is obvious that each flow pattern has

its own relevant flow parameters, such as equivedent diameter. Also, the drag

or frictional correlations to be used are different for different flow patterns. The

works done by, among others, Ishii (1977), Ishii and Chawla (1979), Ishii and

Mishima (1980, 1984) contain the necessary information.

As the last problem to be considered in this report involved large changes

in thermodynamic and transport properties of the fluids, equation of state for

the fluids should be made available. There are several books that concentrate

on the discussions of the necessary transport and thermodynamic properties to

be used in solving the problem of boiling. They are, among others, by Schmidt

and Grigull (1981), Meyer et al. (1967) and Reynolds (1979). The last reference

is worth special mention since it not only contains a systematic presentation of

the equations to be used to calculate the thermodynamic properties but also an

example of program implementation. However, there is a shortcoming by not

containing any information on how to calculate the transport properties.

CHAPTER III

MATHEMATICAL MODELS

3.1 Introduction

It is well established that the continuum model for liquid or gas in a single

phase flow are assigned in terms of the conservation laws of mass, momentum, en­

ergy, chemical species, etc. These conservation laws are constructed on the basis

of integral balances. In these integral balances, if the integrands are continuously

differentiable and if the Jacobian of the transformation between the spatial and

material coordinates exists then the so-called Reynolds transport theorem can be

used to produce Eulerian-type differential balance equations—see for example,

Aris (1962) or Arpaci and Larsen (1984). These differential balance equations

are then complemented by appropriate constitutive equations specifying the ther­

modynamics and mechanical states as well as the chemical behavior of the fluids

under consideration at a particular point in space and at a certain time level.

The same approach is applicable in the case of multiphase flow systems. How­

ever, the derivation is considerably complicated due to the singular characteristic

of multiphase flows in the presence of interfaces separating the phases or com­

ponents involved. The fact that the variables are not continuously differentiable

in the domain of integration neccesitates a slightly different approach. Here, the

conservation equations are derived for each phase involved with jump conditions

patching up the discontinuity of variables on each side of the phase interface.

Theoretically, these equations together with appropriate inital and boundary

conditions could be solved to characterize the dynamics of each phase. However,

this methodology would result in a multiboundary problem with the positions of

the phase interface being unknown and hence should be computed. Unless the

16

17

interface geometry is simple—for example, that of separated flow category—such

an approach encounters overwhelming mathematical difficulties. Fortunately, for

the engineering analysis of systems and the development of constitutive models

from experimental measurements, one is interested in the space-time average be­

havior of each component not in the instantaneous formulation of each particle

in the flow. Therefore, multiphase flow analysis is usually performed using some

kind of averaged field equations. It is worth mentioning that this averaging pro­

cedure is shared even by single-phase flows. Consider the single-phase turbulent

flow without moving interfaces, so far it has not been possible to obtain exact

solutions expressing local instantaneous fluctuations in the flow.

The most commonly employed averaging techniques in continuum mechanics

is the so-called Eulerian averaging because it is closely related to human observa­

tions as well as instrumentation's measurement methods. Of particular interest

is the spatial-temporal Eulerian averaging technique where the averaging is taken

over an interval At that is large enough to smooth out the local fluctuation of

properties but small enough to preserve the overall unsteadiness of the flow. The

resulting time averaged equation can then be formulated in terms of either a

multi-fluid model or a diffusion (mixture) model, both of which have specific

advantages and disadvantages.

3.2 Local Instantaneous Formulation of

the General Balance Equation

So far, subjectively, the most concise and clearest formulation of the gen­

eral balance equations for two separated fluids is given by Addessio (1981). A

summary of his formulation is given below.

Consider a material volume Vm with material surface Sm that encloses two

separate fluids as shown in Fig. 3.2. This volume consists of three distinct regions.

18

Volumes Vi and V2 contain the individual fluids, while Vi includes the interfacial

region where the properties change continuously from those associated with one

fluid region to the other. A general integral balance, with the definition of the

terms summarized in Table 3.1, can be written on this material volume for the

total time rate of change for any quantity V* that varies continuously within Vm,

4- f PHV = I PHV - I J • ndS at JVrn -^Vm ^ Sm

w here

J <!>

n

efflux source term of quantity tj) outward normal of Sm-

(3.1)

Figure 3.1: Sketch of two-fluid material volume

Table 3.1: Definition of terms used in the general balance equation

Balance Eqn.

Mass

Momentum

Energy

Bal. Quantity {ip)

1.0

V

u^v^/2

Eflaux ( J )

0.0

P6ij - T

q — T ' V

Source (^)

0.0

9

9 v

19

Separating eqn. (3.1) into those applicable to the individual volume elements

and the interfacial region yields

-[[ piiPidV-\- f p2il^2dV+ f p4idV] = [f Pi(t>idV-\- I p24>2dV-^ dt Jvi M M -'^i •' 2

I pi(t>idV]-[l Ji'nidA-\- f J2'n2dA+ i JiUidA]. (3.2) JVi JAi JM -^^'^

According to Aris (1962), the Reynolds Transport Theorem is

'V(t)

where

f{x,t) : continuous function defined within V(t) and on S{t)

— I fix tWV = / —dV -H / fvA • riAdS dt Mi) ^ Mt) dt Js{t)

UA

VA ^A

outward normal of S(t) speed of displacement of point on S{t).

Applying to the above geometry, the following expression is obtained for region 1

± f fjy^ [ ^dV+ I hv.-n.dA^ / hvi-n,dA, (3.3)

dtJvx M dt J Ax JAi

So far the development of the general balance equation is stiU analogous to

that of single phase flows. However, as can be seen in the last term of the

20

above equation, for multi-fluid the presence of the interface manifests in the

general balance formulation because the Reynolds transport theorem requires

the integration to be performed over all surfaces bounding the fluid and this now

includes the interface. Applying the similar of eqn. (3.3) to both regions, it is

possible to transform eqn. (3.2) into, with grouping the same terms together

t.i L [ % r ^ - PkMdV + f {p,i^,v, . n,)dA}

+ / (pifpiVi • Tii + p2i}2Vi • ni)dA + y ] / Jk- TikdA

JAi 1^^^ J A,

+ T : / Pi'^idV - I pi(j)idV -hi Ji mdA = 0. (3.4) dt JVi JVi JAIC

The surface integral containing the phasic efflux term can be transformed into

volume integral plus an interfacial area integral by applying Gauss' theorem

/ Jk ' rikdA = f Jk- rikdS - Jk- UkdA JAt Jst JAi

= / V • JkdV - I Jk- UkdA, JVk JAi

Utilizing the above relation, eqn. (3.4) can be written as

E { / [ ^ ^ ^ + "^ ' iP'^<i>^^^) "rV-Jk- Pkcl>k]dV} ,tt V. dt

+ ( 4 / Pii^idV - I Pi(i>idV + i Ji TiidA dt JVi JVi JAlC

+ / y,[Pk'^k{vi - Vk) - Jk] • fikdA) = 0. J^i k=i

In the grouping above, it can be seen that there are two groups. The first group

is applicable for the fluid regions. While the second group is for the interface

region. Separating the groups, the following two integrations resulted

/ [ ^ ^ ^ + V . {pki^kVk) + V . J , - PkMdV = 0. (3.5) ^Vt dt

21

Because the volume of integration was arbitrary then the integrand must be zero.

This step results in an Eulerian differential balance equation identical to that for

single phase flows where the second set takes care of the balance at the interface

and couples the two fluid regions. If A/ -^ 0 the following integration is obtained . 2 ,

- / T.lPkM'"i-'^k)-Jk]-nkdA = - judA- I jedA+ I I-NdC (3.6) JAi ^^j at JAi JAi Jc

where the interfacial quantities on the right hand side are now defined as surface

properties (e.g., 7 is the mass per unit surface) with N is the unit normal to the

curve C in the plane of Ai and I is the analogous efflux.

Further manipulation is needed for the first and the third terms on the right

hand side of the above jump condition. First, the Reynolds' transport theorem

for the geometric surface A according to Aris (1962) can be written in tensor

notation as

in which F is a property of the surface, r " is the fluid velocity within the surface,

and a"^ is the surface metric tensor. Here, a is the determinant of the metric

tensor while a is the time derivative of a. Second, the surface form of Green's

theorem according to McConnell (1957) is of the following expression

f I'NdC = J I%dA. (3.8)

Substituting these last two expressions into the jump condition relation above,

the following is obtained

• ^ k=i

- f ^edA -\- I rjA. (3.9) JAi JAi '

Now, because of the integration domains of eqns. (3.5) and (3.9) are arbi­

trary, the following general differential balance equation may be obtained from

22

eqn. (3.5) for the bulk fluid while the interfacial jump condition is from eqn. (3.9):

dipki^k) g^— + V . (pki^kVk) -i-V -Jk- pk4>k = 0 (3.10)

and

2

illpkM^k - Vi) + Jk] . njfe = [ - ^ -f- V . (7a;«) -f 70;;^] - 7^ + V • / . (3.11) k=\ Ot 2a

In the last equation, the variables with subscript k are understood as the quanti­

ties in the bulk fluid evaluated at the interface. It is a common assumption that

the mass, momentum, energy, and body forces associated with the interface (the

first four terms on the right hand side of the above equation) are taken to be

negligible. Thus the general balance formulation for multifluid flows leads to 2

(two) balances to be satisfied for each of the conserved quantities. For example,

the mass conservation equation gives

^ + V . ( / , , r , ) = 0 (3.12)

and 2

Y.rnk = Q (3.13) k=\

where

rhk = pk{vk-Vi)-nk. (3.14)

In this last expression, it is stated that the mass crossing the interface from

one fluid region to another must also be conserved. There are analogous beil-

ances for the conservation of momentum and energy, for example, in Ishii (1975),

Kocamustafaogullari (1971), Delhaye (1981), and Stuhmiller (1976).

The above local instantaneous differential balance formulations for two-phase

flows are valid at any given time. However, the spatial position of the fluid

regions and the interfaces is varying with time. Therefore the differential bedance

equations m.ust be time averaged for the results to be of practical benefit.

23

3.3 Time Averaging

As has been said, the local instantaneous equations for general two-fluid prob­

lems are difficult to obtain mathematically. In fact, the microscopic details offered

by local instantaneous equations are unnecessary and unmeasurable. Hence, a set

of working equations that does not contain the high-frequency phenomena, insta­

bilities, and discontinuous variables as found in the local, microscopic equations

is necessary. To obtain a smooth set of equations, the local instantaneous equa­

tions must be averaged. The most commonly employed time averaging technique

is the Eulerian approach. Time average of variable F may be defined as

_ 1 ft+T/2 Fk = 7f F{x,T)dr.

1 Jt-T/2

With this averaging process, two consequences are resulted. They are

1. smoothing out of turbulent fluctuations 2. properly defining the local volume fraction of the i^^ phase.

Consider averaging over the time interval [t - T/2-, t + T/2] where Ti being the

cummulative residence time of phase i during the interval [T], then the precise

definition of the time fraction of phase i, ai, is obtained

a; = - = - / Xi(x,t)dt l[T]

where Xi{x,t) is the phase density function defined by

I 1 if point X pertains to phase i ^ • ( " ' ' ' ) = „ ... •

I 0 otherwise.

The term local volume fraction, or simply void fraction, is also applied to a^.

The time averaged value of any quantity fi is defined as

- _ l/TJ^T]Xifidi ^'~ 1/TJ^j^Xidt •

24

3.4 Space Averaging

Consider a scalar, vector, or higher order tensor quantity Tpi{x,t) of the i^^

phase with volume V, enclosing the i^^ cross-sectional plane. Then the volume-

averaged value of quantity V'i can be defined as

«<i,,»>{t) = -l- I ^,dV. Vi{t) Jvi

The area-averaged value of quantity V*., « V ' , » , can be obtained by expressing

the volume as V,- = AiAx where Ai is the cross-sectional area of the i"* phase

and by considering the limit of the above equation as Ax —> 0:

« V ' t » {x,t) = — [ ip{x,y,z,t)dA. Ai JAi

Note that since the integration is performed over the cross-sectional plane normal

to the main flow direction (x), the resultant area-averaged quantity « V'i must

be a function of x and time, t. The averaged value « V ' i » then applies to the

center of area of the i^^ phase. It is advantageous to formulate the averaged

values with respect to the center of mass instead of the center of area. The

mass-weighted, area-averaged value of quantity xj^i is defined as

Jx. PidA

Utilizing the definition for for area-averaged values, the relationship between the

two is «Piil^i»

In particular, for incompressible fiows, the following relation can be deduced

Pi «fJ^i » ,

25

3.5 Covariance

Area-averaged system variables are normally employed when one-dimensional

numerical methods are desired to solve the field equations. The introduction of

area-averaged system into the non-linear field equations increases the analytical

complexity of the problem because, in general, the average of a product is not

equcd to the product of averages. That is, for two variables V'i and 7,-

^ / ^ -^Ai Pii^ilidA

JAi PidA

In particular,

unless "^i is constant over the cross-sectional plane over which the averaging is

performed.

The difference between the average of a product and the product of averages

is given by the so-called covariance and takes on the form of

cov (^. . 7.) = <-0. . 7. > - <'0i > . <7- >

The value of the covariance of squared quantities, such as the fluid velocity, de­

pends upon the variation of the quantity over the cross-sectional area. If the value

is nearly constant as in turbulent flows, the covariance is small. However, for a

laminar flow of an incompressible fluid in a circular duct of radius R where the

velocity distribution is parabolic the covariance can be significant, for example,

using the expression that relates the local velocity distribution and maximum

local velocity for laminar fiow in a circular tube given by Bird et al. (1960)

7*

where

ApL

26

yields

< ^ > ^ So^'S^v^TdrdB ^ 2i:J^v,^ma.[^-(r/R)^]dr

"2^ rR 2

Therefore

Meanwhile

< V z > =

/o^/o '^drdS -KR

1 A P 2

«...»'=<^,>'=i(ii^fl'). 4 4//i

2_ /o" /o" "f'•rf'-'i* 2,r ;„« < „ „ . [ 1 - (r/iJ)^]'dr '27r rfl

Thus

S^^'S^vdrdO irR^

= IRW . — = ^ ^ - l(^R'\ 3 '•^°'= 7r/22 3 3M/iX ^ •

4 1 cou(i;^ -v^) =<v]> - <v,>^= ( - - 1 ) <v,>^= - <v,>^

For turbident fiows, a 1/7-power is assumed as the velocity profile to obtain

the value of the covariance for turbident flow in a duct. According to Schlichting

(1979) the following relation can be used for turbulent flows in duct

Now,

< ^ > = r2. rR

1 - (y.\''^

/o ' //* u{R - y)dyde 27r / ^ Uir/R)'f'dy

So''So{R-y)dyde ^R' dSirUR^ 98 1207ri?2 120

U.

Hence

Meanwhile

2 2 9604 _,2

«^»=<^>=Iii5o^-

^ .^_CJo''u'{R-y)dyde ^ 2^5^U\rlRfl'dy

tS^{R-y)dyde 7ri?2 .2

27

_9S7rlPR^ _ 100 2 1447rit:2 ~ 98 ^""^ '

This results in

cov{u -u) =<u^> - <u>^= (-— - 1) <u>^= — <u>^ .

For most practical two-phase flow problems, there will be a large variation in

velocities over a cross-sectional plane normal to the principal flow direction owing

to the large difference in densit)'^ between the hquid and the vapor phases. The

situation can be worse for the important mass-weighted, area-averaged quantities

when there is appreciable droplet flow moving with the vapor in conjunction with

slower continuous liquid flow—see, for example, Wallis (1982).

Thus, the covariance terms can be expected to be important for most practical

two-phase flow analysis. However, Delhaye (1981) in discussing two-phase flow

modeling states that generally the covariance terms are neglected. This is due to

the fact that it is essentially impossible to specify the value of the covariance in

multiphase flows. It is, therefore, possible to obtain a more accurate description

by considering the total flow field as being composed of several phases (or fluids)

rather than as a mixture. This implies that the covariance over each phase is

assumed to be negligible rather than over the entire mixture. This is one of the

primary advantages of modeling multiphase flows with a multifluid formulation

in comprison with a mixture (diffusion) formulation. For an in-depth discussion

about the covariance, the work of Yadigaroglu and Lahey (1976) can be consulted.

3.6 Time-Averaged Two-Fluid Model

General Balance Equation

Ishii (1975) and others [Delhaye (1981), Stuhmiller (1976,1981), and Addessio

(1981)] have shown by application of the time averaging techniques discussed in

28

Section 3.3 to the instantaneous general balance eqn. (3.10), that it is possible to

obtain the following macroscopic, time averaged balance equation for each fluid

phase

dioLkPi^ib,) , _ 7,

dt + "^ • ( ^ ^ ^ ) = -"^'Wk^Jk^Jk)]

+ kPkh^^k (3.15)

where J^ and J^ represent the effluxes due to the average molecular diffusion

and the statistical effects of the two-phase and turbulent fluctuations while Ik

represents the interfacial source of property V* for the k^^ phase. The interfacial

transfer condition can be written as

2

Y^lk-Im = 0 (3.16)

where X^ is the total interfacial source of property ip for the two-phase mixture.

Thus, these two equations express the macroscopic balance of property tp for the

k^^ phase and at the interface, respectively.

The original purpose of the averaging has now been accomplished. That is,

the alternate occupying of a point by two separate phases has been transformed

into two coexisting continuum. Additionally, the comphcated two-pha^e and

turbulent fluctuations have been smoothed out and their statistical effects have

been taken into account by the covariance, or turbident flux terms.

3.7 Two-Fluid Model Conservation Equations

The macroscopic balance eqn. (3.15) and the interfacizd transfer condition

eqn. (3.16) which have been time averaged are applied to the conservation laws

of mass, momentum, and energy in this section. The variables to be used in

these equations follow the definitions of the local instantaneous formulations of

Section 3.2 (see Table 3.1).

29

3.7.1 Mass Balance

The mass conservation equation for each phase is

djakPk) , ^ f - - \ r — — — -h V . [akPkVk) = Tk

and the interfacial transfer relation is,

J f e = l

where Tk is the interfacial mass source due to the phase change.

3.7.2 Momentum Balance

The momentum balance for each phase is

^ ^ " ^ y ' ^ + V • {akPkVkVk) = -V{akPk) + ^-HTk^Tl)]

+ ock'Pk9k + ^k

and the its interfacial transfer relation is

^ M , - M ^ = 0 k=\

where r ^ and Mk denote the turbulent fiux and the k^^ phase momentum source,

respectively, and Mm is the mixture momentum source which is usually assumed

to be due to the surface tension effect.

3.7.3 Energy Balance

The energy balance for each phase is,

d[a,Uu, + vll2)] ^ v\a,Uu, + %)v,] = -V •[a.iq. + ql)] dt ^

-I- V • {aiJ" • rjt) + 0Lk'Pk9k -^k + Ek

30

and its respective interfacial transfer relation is

J2Ek-Em = 0 k=l

where Ek represents the interfacial supply of energy to the k^^ phase, and Em is

the energy source for the mixture. Thus, energy can be stored or released from

the interfaces. The apparent internal energy Uk consists of the standard thermal

energy and the turbulent kinetic energy. The turbulent heat flux q^ accounts for

the turbulent energy convection as well as for turbulent work.

The two-fluid model is characterized by two independent velocity fields, and

is based on the above six field equations, i.e., two mass, two momentum, and two

energy equations. The interfacial exchange relations for mass, momentum, and

energy couple the transport processes of each phase. These balances must be

supplemented by various constitutive equations or exchange correlations which

specify molecular diffusion, turbulent transport, and interfacial exchange mech­

anism as well as the thermodynamic state variables.

There are, see Ishii (1975), 33 (thirty-three) unkown variables appearing in

the conservation equations and the equations of state. In order for the problem to

be properly posed, it is therefore necessary to specify a total of 33 (thirty-three)

equations. These are:

Field Equations 6 Interfacial Transfer 3 Axiom of Continuity 1 Average Molecular Diffusion Fluxes 4 Turbulent Fluxes 4 Body Force Fields 2 Interfacial Transfer Equations 3 Interfacial Sources 2 Equations of State "* Turbulent Kinetic Energy 2

31

Phase Change Condition Specifying the Interfacial Temperature 1

Mechanical Conditions at the Interface Relating PL and PQ 1

This is the two-fluid formulation in its most general form. For most practi­

cal engineering analyses, assumptions are made which can simplify the problem

somewhat.

Restricting the investigation to one-dimensional spatial variable reduces the

number of variables involved considerably. Additional effects from this simplifi­

cation is that no turbulent related variables need to be considered. Their effects

are included in correlations to be employed as the external constitutive relations

for both conservations of momentum and energy.

The fact that the void fractions should sum up to one,

Q:G + a^ = 1,

gives additional advantage in reducing the number of variables involved. Also,

employing the assumption that no differentiation be made between the vapor

pressure and the liquid pressure

PG = PL = P (3.17)

reduces the number of variables even further. Finally, the following hypotheses

are generally admitted [Boure and Reocreux (1972)]

1. The time correlation coefficients are all equal to 1.

2. The equation of state valid for local quantities applies to averaged equa­

tions.

3. Longitudinal conduction in each phase together with their derivatives are

negligible.

32

4. The phase viscous stress derivatives and the power of these viscous stresses are negligible.

5. The pressure is constant over a cross section in a vertical flow.

3.8 One-Dimensional Two-Fluid Model Governing Equations

In all the field equations below the averaging signs are dropped for simplicity.

The derivation of the field equations is well established, for example, in Ishii

(1975), Delhaye and Achard (1976), with Delhaye (1981) discusses from local

instantaneous formulation up to the various averaging processes. In addition,

the correlation coefficients are asummed to be unity, for instance see Yadigaroglu

and Lahey (1976), so that the average of a product of variables is equal to the

product of averaged variables.

3.8.1 Conservation of Mass

Vapor: ^iPo^'o) + djpGaGWG) ^ p^^ (3^gj

dt dz

Liquid:

^!£l^ + ?i£i^i^ = T^a. (3.19) dt dz

Since the mass exchange terms on the right hand side of the above two equations

constitute the total mass exchange, then

TGL + TLG = 0

or, using the convention that the mass exchange due to evaporation is positive,

the following is resulted

TGL = - T L G = T.

Also, it is assumed that the net mass exchange is the result of two separate

mass exchange processes, one which occurs in the bulk of the fluid and the other

33

occurs at the wall. The phase change that occurs at the interface between the

two fluids is treated as a process in which the bulk fluid is heated or cooled at

the saturation temperature and the phase change takes place at the saturation

state. This means the total mass exchange

r = FG + r w-

3.8.2 Conservation of Momentum

Vapor:

d{pGOLGWG) , d[pGOLGWGWG) , dP „

dt— + dl + "^ aT - ^^"^ ' = TWG - AGLTGL{'^G - WL) - AWGTWGWG-

Liquid:

where

Tki

Twk

Wk

Aki

Awk

dipiaiwi) , dipLOCLWiWi) , dP -K: \ 5 ^ OLL-^ pLOtirit = ot oz oz

-TwL - AicTiGi^L - WG) - AWLTWL'^L

frictional coefficient between phase k and / frictional coefficient between wall and phase k body force in the z direction interfacial velocity of phase k surface area per unit volume between phases k and / surface area per unit volume of phase k in contact with the wall.

For both phases, the terms on the right hand side are, respectively, momen­

tum transfer due to mass-transfer, interphase friction, and wall-to-phase friction.

While the interphase jump condition requires that .

. \ ^

TWG - TwL 4- AGLTcLi-^G - I^L) + ALG(WL - WG) = 0.

34

3.8.3 Conservation of Energy

Vapor:

djpGO^GhG) d(pGaGhGWG) dP dP

dl + di = -^^"aT - "^^^:^

~^^GL + QGi + PGOCGB^WQ.

Liquid:

dipLaihi) , dipiaihiwi) dP dP dt + Fz = ""'-m ~ ^ ^ " ^ ^

-^^LG + Qii + PLOLLB^WL

where

hk : specific enthalpy of phase k E'ki : interface energy exchange between phases k and /.

Again, the terms on the right hand side, save for the pressure terms, are energy

transfer due to mass-transfer, interphase energy transfer due to heating, and the

effect of body force. It should be noted that the interphase energy transfer due

to heating consists of two components: the energy transfer due to wall heating

and energy transfer that occurs in the bulk fluid. That is

j ^ ^ *\

QGi = QGL + QwG

and ^ *\ ^

Qli = QLG + QwL-

While

Qw — QwG + QwL

is the total heat transfer rate to the fluids from the duct wall. Also, as is indicated

in the Conservation of Mass that vapor generation or disappearance is due to the

following

35

1. mass exchange due to the bulk energy exchange, TG

2. mass exchange due to heat transfer from wall, Tw

Thus the interphase heatings caused by the transfer of mass, following the dis­

cussion in Carlson et al. (1986), are

and

^LG = —^G^L — Twh'i-

By summing the two phasic energy equations, the mixture energy equation is

obtained in which it is required that the interface transfer terms to be identically

zero.

QGi + QLi + TGih'G - HL) + Twih'G - hi) = 0. (3.20)

Since each phase at the most is in contact with two other phases, for example the

vapor phase is in contact with the liquid and the duct wall, so that the interface

heat transfer rates can be written as

QGi = QGL + QwG = HG(T' - TG) + QwG (3.21)

and

QLi = QLG + QwL = HL{T' - TL) + QWL (3.22)

where, for both expressions, the first term on the right is the thermal energy

exchange between the bulk fluid and the interface. While the second term is due

to the heat transfer from the wall. This second term contributes to the overall

mass exchange either by boiling or condensation.

Substituting eqns. (3.21) and (3.22) into eqn. (3.20) gives

HG{T' - TG) + QwG + HL{T' - TL) + QWL + TGihh - HL) + Tw{h'G - H'L) = 0.

36

Gathering those terms associated with the interface and those with the wall and

requiring them to be identically zero results in

HG{T' - TG) + HL(T' - TL) + TGih'^ - hL) = 0, (3.23)

and

QwG + QWL + Twih'G - hi) = 0. (3.24)

The former expression takes care the transfer process between the bulk fluids and

their respective interfaces while the latter handles the transfer process between

the phase and the duct wall. Also, it is assumed that for boiling process the

vapor phase in contact with the wall is negligible in comparison with the liquid

phase. Because the vapor bubbles generated at the wall will detach from the wall

and flow downstream. This gives, for boiling process, QWG = 0 where Tw > 0.

That is, the liquid phase is being heated to produce vapor bubbles. Therefore,

the rate of vaporization at the wall is

Substituting the last two relations into eqns. (3.21) and (3.22) respectively,

the interfacial heat transfer for the liquid and gas phases are

QGi = HG{T' - TG) (3.26)

and

QLi = HL(T' - TL) - Twih'G - h'^^). (3.27)

Finally, with a little algebra, the interphase rate of vapor generation from

eqn. (3.23) by means of eqns. (3.26) and (3.27) is

HGiT' - TG) + HLiT' - TL)

^^"" ihi-ht) This gives the total rate of mass exchange to be

HGJT' - TG) + HLiT' - TL) ^ ^

37

3.9 Flow Regimes

The flow regime is determined using the method proposed by Taitel and

Dukler (1980, 1986). For the present, only vertical flows are considered for the

majority of the boiling experiments are done for vertical flows. Since the objective

is to simulate the experiment in which the heat transfer does not reach the critical

heat flux (CHF) condition then only three regimes will be considered. These three

flow regimes are idealization of the so many flow regimes that might occur in such

a flow—see Collier (1972) and the accompanying Fig. 3.9.1. The flow regimes

are the bubbly flow, slug flow, and annular-mist flow and the discussions here

follow closely that of Carlson et al. (1986).

3.9.1 Bubble to Slug Transition

Taitel and Dukler (1980) suggested that bubbly flow cannot occur when gas

bubble rise velocity greater than the velocity of Taylor bubble in small diameter

tubes. The rise velocity of relatively large bubbles is given by

while the rise velocity of the Taylor bubbles is given by

UG ^ 0 . 3 5 ^ ^ .

Solving for D using the two equations, the dimensionless critical diameter can

be found as

Dc > 19.11,

where

^^^ (3.28) \ (ripL- PG)

Meanwhile, for flows in tubes with diameters greater than 19.11, Taitel and Duk­

ler (1980) suggested that bubble-slug transition occurs at a void fraction ag^ =

38

WALL AND FLUID TEMP VARIATION

FLOW PATTERNS

Wall temp

HEAT TRANSFER REGIONS

Fluid temp

Sat temp

H

x-1

Vapour core temp

*Dryout'

'Fluid temp ^

D

Liquid 'Core temp

•0

Fluid temp

B

A

Single- Convective phase heat transfer vapour to vapour

Drop flow

__ V-:-

Liquid deficient region

Annular flow with

entrainment Forced

convective heat transfer thro'

liquid film

Annular flow

Slug flow

Bubbly flow

Single-phase liquid

Saturated nucleate boiling

Subcooted boiling

Cortvective heat transfer

to liquid

Figure 3.2: Regions of heat transfer and flow patterns in convective boiling

39

0.25 for low mass fluxes, that is, for G < 2000 kg/m^s. Thus, the competing

conditions between ag^ = 0.25 and Dc > 19.11 should be considered in bubble-

slug transition.

At high mass fluxes, in which G > 3000 kg/m^s, Taitel and Dukler (1980)

indicates that bubbly flow with finely-dispersed bubbles can exist up to a void

fraction of a § ^ = 0.52. In between these two mass flux brackets, a linear interpo­

lation can be used to determine the transitional void fraction between a bubbly

and a slug flows. Hence, if a^ < a § ^ then the flow is in the bubbly flow region

otherwise if a^ > OCQ^ then the flow regime is slug flow.

3.9.2 Slug to Annular-Mist Transition

Taitel and Dukler (1980) suggested that annular flow cannot exist unless the

gas velocity in the gas core is sufficient to suspend the entrained droplets. The

minimum gas superficial velocity, UGS^ required to lift a drop is given by

r . Mn, . = 3.1 (3.29) [crgipL - PG)]'-'' ^ ^

in which the slug flow regime exist if the gas superficial velocity is smaller than

UGS while the annular flow regime is the flow type if the gas superficial velocity

is greater than UGS-

3.10 Interphase Drag Relations

The interphase drag per unit volume between phase k and phase / in terms

of relative phasic velocity is given by

Fki = AkiTki

in which according to White (1979)

^ Cppcjwk - wi)^ •iki = ;;

40

where

CD

Pc

interfacial area per unit volume drag coefficient density of continuous phase.

The following discussion is aimed at determining the appropriate Aki and CD

for different flow regimes.

3.10.1 Bubbly Flow Regime

Following Wallis (1972) and Shapiro et al. (1957) the dispersed bubbles can

be assumed to take the form of spherical particles with size distribution being

determined by the Nukiyama-Tanasawa non-dimensional formulation. Also of

interest is the discussions presented in Brodkey (1967) and Kuo (1986).

where T> = D/D' with D' being the most probable particle diameter, and V is

the probability of occurence of particles with non-dimensional diameter T>. With

this distribution, it can be shown that the average particle diameter D = 1.5£)',

so that the surface area per unit volume is

_ QaG J V^V dV _ 2AaG _ 3.6QG

AGL - - ^ jj)3p dV~ D' ~ "D '

The average diameter, D, is obtained by assuming that

-D = ^ ^ (3.30)

where the maximum diameter, Dmax-, is related to the critical Weber number

given by

We = DmaxPci^G " WL)^aG

41

with pc being the density of the continuous phase which in this case is the liquid

density. The value of the critical Weber number is taken (Ishii, 1990) as 10 for

bubbly flow.

The drag coefficient for bubbly flow in the viscous regime, according to Ishii

and Chawla (1979), is given by

24(1 -f O.lile/ '^ ') Cn =

Rep

where the particle Reynolds number is calculated by using

PC\WG-WL\D Rep =

P'm

in which the mixture viscosity, Pmi for bubbly flow is given by

P'L Mm = — •

3.10.2 Slug Flow Regime

Slug flow is modeled as a series of Taylor bubbles separated by fiquid slugs

that contains small bubbles. Letting a c , be the average void fraction in the

liquid film and the slug region, the void fraction of a single Taylor bubble, Q J ,

in the total mixture is then

OLG — O^Gs ar = — »

1 - OCG,

where a c is the overall average void fraction. By approximating the ratio of the

Taylor bubble diameter to the tube diameter and the diameter to length ratio of

a Taylor bubble, Ishii and Mishima (1980), obtained the interfacial area per unit

volume for slug flow as

4.5 3.6aG«/- V AGL = -^ocT + - ^ — ( 1 - ^ ^ ) -

42

While the drag coefficient for Taylor bubbles is, according to Ishii and Chawla

(1979), given by

CD = 9.8(1 - Q r ) ^

3.10.3 Annular-Mist Flow Regime

This type of flow is characterized by a liquid film along the wall and a vapor

core containing entrained liquid droplets. Then, see Ishii (1990), the interfacial

area per unit volume is

AGL = —jY^V^ - ocLL + 3.6Q:LdZ)(l - a^^),

where Can is the roughness parameter due to waves in the film iCan ^ 1) and

aLd is the average liquid volume fraction in the vapor core which is given by

OtL — OtLL O^Ld = — •

1 - OCLL

The correlation for the average liquid film volume fraction is

aLL = a^C/exp[ -7 .5 x 1 0 - ^ ( ^ ^ ) « ] , UGS

where UGS is the expression in eqn. (3.29). While the term Cj is expressed as

D Cf = PLOLL'^L— X 10

ML

- 4

The interfacial friction factor, / i , to follow replaces CD in the interfacial friction

force per unit volume.

fi = 0.02 -i- AA8'^

where 4 _ -in-O.Se-l-S.OT/Dc

4.74 B = 1.63 4 - - y p

43

and

. DaL

Here, 6 is the film thickness and Dc is the dimensionless diameter in eqn. (3.28).

3.11 Heat Transfer Modeling

The correlations for heat transfer processes, by referring to Carlson et al.

(1986), will be discussed in this section. The set of correlations includes heat

transfer for single phase forced convection, subcooled nucleate boiling, saturated

boiHng, and condensation. Note that the boiling heat transfer processes are those

for pre-CHF boiling. Here, in order to determine the appropriate heat transfer

correlations reference is made to Fig. 3.9.1 and Fig. 3.11.

This model assumes that the duct wall is completely in contact with liq­

uid. Hence, the contribution due to the vapor in contact with the duct wall is

neglected. This is not such a bad assumption, for example see Collier (1972).

Therefore, the heat transfer from the wall to the vapor and vice versa is

QWG = 0.

While the heat transfer from wall to the liquid is

QWL = HwLAwLiTw - TL),

where HwL^ Tw, and TL are the heat transfer coefficient, wall temperature, and

liquid temperature, respectively.

The heat transfer coefficient used for single-phase forced convection is that of

Dittus and Boelter (1930) and the heat transfer coeflficient for nucleate boiUng is

that of Chen (1966).

44

Region A Convection to single-phase liquid

i B Subcooled boiling

C Bulk boiling

Flow

•o

5

H i w n M W i t i i w i t t i M "' 11M111 iipjy m^m y j u U^l^^y.^''Jjp^-^M^^ir' —

m

» I M M H f M I M f f t } f M f } f Uniform heat flux, 0 j

Based on .thernwdynamic equilibrium

Length (z)

Figure 3.3: Variation of void fraction along a heated pipe

45

3.11.1 Single-Phase Forced Convection

Because the fluid flow into the pipe may still be in subcooled state then the

mode of heat transfer is single-phase convective heat transfer. However, since

this mode of heat transfer is usually very close to the inlet where there might

be simultaneous velocity and temperature development then the effect of flow

development should be taken into account. For that, the suggestion made by

Molki and Sparrow (1986) is employed here. That is, the local Nusselt number

is corrected to include these two effects.

Nu a

= 1 + Nufd iz/Df

with

a = 23.99i2e-°-23o

and

b= -2 .08 X 10"®ile-f 0.815.

Here z is the distance from the inlet of the pipe and Nufd is fully developed

Nusselt number that, in this report, is the correlation developed by Dittus and

Boelter (1930) which is valid for circular geometry and turbulent flows

j^uf, = ^ ^ ^ ^ = 0.023Pr''Re'' kL

where the fluid properties are evaluated at fluid temperature and

H : heat transfer coefficient kL : fluid thermal conductivity Dh ' hydraulic diameter Pr : Prandtl number Re : Reynolds number.

Thus the wall to liquid heat flux can be calculated using

QWL = HDBAwLiTw - TL)

w here

46

BDB

Tw TL

AWL

Dittus-Boelter heat transfer coefficient correlation wall temperature liquid temperature contact area per unit volume.

3.11.2 Two-Pha^e Heat Transfer Processes

If heat is continuosly supplied to the flowing single-phase liquid at a certain

point along the pipe flow boiling process begins to take place, see Fig. 3.11.

The subcooled boiling region begins with the onset of nucleate boiling at ZA

while the mean or bulk temperature of the fluid is still below the saturation

temperature. However, for nucleation to occur the fluid temperature near the

wall must be somewhat higher than saturation temperature, T', so that vapor

bubbles can begin to form at the wall. Because the bulk of the fluid is still

subcooled the bubbles formed do not detach from the wall but grow and collapse

while still attached to the wall. This process gives a small nonzero void fraction

that usually neglected. Nevertheless, beginning at 2 = 2^ the correlation for

subcooled nucleate boiling heat transfer coeflficient can be used in this region.

When the temperature of the bulk fluid equals the saturation temperature the

heat transfer coeflficient correlation to be used is switched to the one that is

applicable for saturated nucleate boiling process.

Now, the criterion to be used for boiling inception is the one developed by

Bergles and Rohsenow (1963) which has the following form and valid for water

oidy

T- - T ' -f 0 556f ^ 10 .463P0-"

where q'w is the surface heat flux in W/m^ and P is the system pressure in bar.

Consequently, if Tw is greater than Tw then it is said that the fluid is undergoing

subcooled nucleate boihng process.

47

Although logically and physically subcooled nucleate boiHng precedes satu­

rated nucleate boiling, the latter process is discussed first. Because subcooled

nucleate boiling heat transfer coefficient can be determined by modifying the

correlation for saturated nucleate boiling.

3.11.2.1. Saturated Nucleate Boiling. Here, the Chen's (1966) correlation is

used which proved to be very successful in correlating all the forced convective

boihng heat transfer data for water and organic system—see Colher (1972). The

proposed correlation covers both the saturated nucleate boiling region and the

two-phase forced convective region. Chen assumed that both mechanisms occur

to some degree over the entire range of the correlation and that the superposition

of the contribution made by each mechanism formed the two-phase heat transfer

coefficient.

In an evaporator tube, heat is transferred from the wall to the fluid by three

means: (a) nucleate boihng in the hquid in contact with the wall, (b) convection

from the wall to the liquid followed by surface evaporation at the hquid-vapor

interface, and (c) convection from the wall to the vapor. The third term is

usually quite small in comparison with the first two except in the post-burnout

region. The heat transfer is thus dominated by the first two mechanisms: nucleate

boiHng and forced convection to Hquid, both of which are present to a varying

degree depending on the flow conditions. Thus, the additive mechanism originally

proposed by Chen is a plausible one.

Hchen = HNCB + Be

where HNCB is the heat transfer coeflficient due to boiHng and He is heat transfer

coefficient due to the forced convection effect. The two are correlated by the

48

following expressions

1 , 0 . 7 9 0.45 0.49

^ PL ^LG PG

and

He = 0.023!^ Prl'Rel^F. Dl ^ ^ 'h

Here F is the so-called Reynolds number factor—see Bennet and Chen (1980)—

and has the following form

^ ^ 1.0 for X „ - ' < 0.1

2.35(X«"' + 0.213)°-^3^ for A ^ r ' > 0.1

and

V -1 _ r ^GPGWG 10.9 PLfP'G.o.i

(1 - ocGJpL^L V PG P'L

Also, S is the suppression factor [see Bennet et al. (1980)], and takes on the

following expression

S = I

in which

(1 + 0.12i?erp^-'^)-i for RCTP < 32.5

(1 + 0A2ReTP°'^^)-'^ for 32.5 < RCTP < 70.0

0.1 for RCTP > 70.0

R,^^ = .^LPLaLDH^^^.2S ^ 10-4

P'L

The meaning of variables involved above are

AT' : Tw-T' T' AP

saturation temperature difference in vapor pressure corresponding to A T ' .

Therefore the rate of heat transfer can be found by using

QWL = HchcTxAwLiTw - TL)

49

while

QWG = 0.

3.11.2.2. Subcooled Nucleate BoiHng. A modified Chen's correlation is

used for this region by foHowing the suggestion of Butterworth (1972). He ex­

tended the use of Chen's correlation to cover the subcooled region by setting the

Reynolds number factor, F , to unity. Although this may cause some underes­

timate of heat transfer coefficient at low subcooling where substantial amounts

of subcooled voids are present but tests with water, n-butanol, and ammonia

showed that this method can give satisfactory results.

Then the rate of heat flux is

QwL = AwL[HNcBiTw - T') + HciTw - TL)]

while

QwG = ^

with all the properties are evaluated at temperature TL-

3.12 Interphase Mass Transfer Modeling

Interphase mass transfer modeling depends on the flow and heat transfer

regimes. As is indicated previously that the net mass transfer consists of two

processes; they are the mass transfer rate in the bulk of the fluid and the mass

transfer rate at the wall:

r = FG + Tw-

Since there is no interphase mass exchange when the bulk of the fluid is still

undergoing single-phase heat transfer, attention is then focused on the subcooled

and saturated nucleate boiling processes only. For this, the Chen's correlation

proves to be very convenient.

50

It has been indicated that the Chen's correlation assumes that the total wall

to fluid heat transfer rate consists of boiHng and convective heat transfer contri­

butions. While the interfacial mass transfer at the wall is mainly due to boiHng

heat transfer mechanism. Therefore, to calculate the interfacial mass transfer at

the wall, the portion of heat transfer rate associated with the convective heat

transfer mechanism must be substracted from the total heat transfer rate.

The interfacial mass transfer in the bulk of the fluids is modeled according to

the flow regime. In the bubbly flow regime, for the Hquid side, interfacial mass

transfer is the larger of either the model for bubble growth developed by Plesset

and Zwick (1954) or the model for convective heat transfer for spherical bubble

(Kreith, 1973) and for the vapor side, an interphase heat transfer coefficient is

assumed that is high enough to drive the vapor temperature toward saturation.

The following discusses the correlations to be used in each heat transfer regime.

3.12.1 WaU Mass Transfer

3.12.1.1. Subcooled and Saturated Boiling Heat Transfer. The general form

of Chen's correlation is ^ ^ ^

QWL = QNCB + Qc

where QNCB is the heat transfer rate due to boiHng while Qc is due to convective mechanism. Therefore

T QWL - Qc lw = '

in which

and where

QWL

He hLG

hLi

Qc = HcAwLiTw - TL)

see previous section for corresponding regime convective part of Chen's correlation latent heat of vaporization.

51

3.12.1.2. Condensation Heat Transfer.

•p QWL — Qcon iw = /ILG[1 + 0.375cpGiTG - T')/hLG]

in which

Qcon = (1 - OCG)HDBAwLiTw " TL)

and where HDB is Dittus-Boelter coefficient of heat transfer.

3.12.2 Bulk Mass Transfer

As is outHned previously, the mass transfer in the bulk of the fluids are

evaluated from the following expressions

QGi = QGL + QWG

and

QLi = QLG + QWL-

Below, the relations to calculate the values of QGL and QLG are given.

3.12.2.1. Heat Transfer Process iTL < T').

1. Bubbly Flow Regime

• Liquid

QLG = HLiT' - TL)

where HL is according to Unal (1976)

. _ 3(t>ChLG(^G

^"-'IIPG-IIPL

52

in which

I 1.0 ifwL < 0.61 m/5

I 1.262wl^'^ iiwL >0.61m/s

and

I 65 - 5.69 X 10-^(P - 1.0 X 10^) if 0.1 < P MPa < 1.0

I 2.5 X 10VP^-^^« if 1.0 < P MPa < 1.7.7

here, P is in N/m^.

• Vapor

QGL = HGiT' - TG)

where

dl This is based on the correlation proposed by Tong and Young (1974)

with the Nusselt number of a single bubble Nu\, = 1.0 x 10^. For a

complete correlation, see the expression for HL in Flashing Process.

While the value of db can be found using the relation of eqn. (3.30).

2. Annular-Mist Flow Regime

• Liquid

QLG = HLiT' - TL)

with

HL = —jr^LNud. dd

Rather than bubbles, droplet diameter is more appropriate here. The

value of Nud is 1.0 x 10^ while droplet diameter can be estimated using

eqn. (3.30).

53

• Vapor

QGL = HGiT' - TG)

where, see Groeneveld and Snoek (1986)

6 Q ! L , . « . ^ 1 / 2 ^ 1 / 7 . . . . a°^^ HG = -irkGi2 -f OMRe'J'Pry') + 0.0023it:e°«Jfcc-^

b D^

and droplet Reynolds number is

pGdd \WG - WL Red =

P'G

and, again, the droplet diameter can be estimated using eqn. (3.30).

3.12.2.2. Flashing Process (T^ > T').

1. Bubbly Flow Regime

• Liquid

QLG = HLiT' - TL)

in which, see Tong and Young (1974)

^aGkLi2 + 0.74i2e°-^Pri/^) HL =

dl

and bubbble Reynolds number is

pLdb \WG -WL Reb =

P'L

while df, is the bubble diameter according to eqn. (3.30).

Vapor

QGL = HGiT' - TG)

in which 6<^G ,, ,^ T n4

5 b

HG = -if^G X 10 df

54

2. Annular-Mist Regime

• Liquid

where

• Vapor

where

HG = 60CL

QLG = HLiT' - TL)

HL = ^kL X 10^ dl

QGL = HGiT' - TG)

1/2 D„l/3> 0.5

-jfkGi2 -f O.lARe'J'Pry^) + 0.0023Re^ckG-^

pGdd \'WG -WL\ Red =

f^G

3.12.2.3. Condensation Heat Transfer Process.

1. Bubbly Flow Regime

• Liquid

QLG = HLiT' - TL)

where HL is according to Unal (1976)

3</>C/IX,GQ:G HL =

in w hich

<!> =

1.0

1/PG - IIPL'

if WL ^ 0.61 mis

1.262^;^^ if U;L > 0.61 m/5

and

65 - 5.69 X 1 0 - ' ( P - 1.0 X 10^) if 0.1 < P MPa < 1.0

2.5 X lOVP^-^^® if 1.0 < P MPa < 1.7.7

here, P is in NIvn}.

55

• Vapor

QGL = HGiT' - TG)

where

dl

with the Nusselt number Nub = 1.0 x 10"*.

2. Annidar-Mist Flow Regime

• Liquid

QLG = HLiT' - TL)

where

HL = [12 -f ^^^-:^]kL^ + PLWLCpLAf X 1 0 - ^ •Im - -LL "d

where Af is the area of liquid film per unit volume and

- T' -TL

" ^ " l+CpGiTG-T')lhLG

The heat transfer coefficient is the superposition of the one according

to Brown (1951) for the condensation of a single bubble in superheated

vapor and Theofanous (1979) for the film condensation.

• Vapor

QGL = HGiT' - TG)

where

HG = ^kG X 10^ dd

CHAPTER IV

NUMERICAL APPROXIMATIONS

4.1 Introduction

As has been mentioned above that on mathematical level the problem of

detailed analysis of two-phase flows reduces to the solution of strongly coupled,

nonlinear partial differential equations of the conservation equations. Because of

these peculiarities, the solution is well beyond any foreseeable analytical method

so that a numerical approach should be adopted. For that, this chapter is devoted

to building the numerical approximations to the relevant conservation equations.

In this chapter the work by Baghdadi et al. (1979) is referenced exhaustively and

is the main source for the various derivations of the finite-difference formulations.

The discussions begin with Section 4.2 containing the finite-difference approx­

imations to all the conservation laws involved. Section 4.3 outlines the methods

to solve the residting finite-difference expressions, while Section 4.4 discusses the

pressure correction method suitable for the two-phase flow problems to be stud­

ied. Lastly, Section 4.5 summarizes the major steps in solving the two-phase

problems.

4.2 Finite-Difference Formulations

Basically, before the finite-diflference approximation can be made, the flow

domain should be subdivided into discrete, small regions by constructing the

so-called finite difference grids. In between two finite difference grids a control

volume is defined with a grid node in the middle as is shown in Fig. 4.1. These

grid nodes are not necessarily midway between the grid fines; it can be made non­

uniform in which larger number of grids are provided in the regions where the

56

57

gradients of the flow parameters expected to be large. However, once selected,

the positions of the grid nodes are fixed during the entire calculation processes.

These grid nodes play the role as the reference locations for the flow parameters.

Main C.V.

X

^

Velocity C.V.

i>

1

Nl I

+ve distance n

Figure 4.1: Sketch of main and velocity control volumes

Fig. 4.1 also indicates the locations where the flow variables are "stored": all

fluids properties and the pressure are stored at the grid nodes whereas the phasic

velocities are stored midway between two adjacent grid nodes. This arrangement

is known as the "stagerred-grid" configuration. The differential equations are

then integrated over these discrete regions, also known as control volumes, to

give the finite-difference formulations. There are two kinds of control volumes,

they are, the "main" or grid-node control volumes to faciHtate the integration

of the variables stored in the grid nodes and the "velocity" control volumes for

integrating the momentum equations. These two control volumes do not have to

be of same size and so are the areas associated with them. In order to emphasize

this notion, the main control volume and its associated area are denoted by V'

58

and A^, respectively. While V and A are appHed to velocity control volume and

its area.

4.2.1 Conservation of Mass

• Vapor:

The conservation of mass for the vapor phase is copied and each term is

identified for ease of discretization

dipGO^G) dipGOLG^G) _ p

dt ^ dz ~ <^ ^ ^ ' ' V " source term

transient term convective term

Integrating this equation term by term with respect to the main control volume

results in

1. Transient term: To obtain finite-difference formulation of this time-depend­

ent term, it is assumed that the values of OLG and pc at point P prevails

throughout the main control volume. Thus

Transient = ^^^P^^^^P-P^P-IP)

At

The superscript o used above denotes the values of the variables at "old"

time level.

2. Convective term: When the scalar variables are assumed to vary Hnearly

between the grid nodes, then

Convective = A4,[iCGWG)p - (C'G^'^G)*]

where A^iCGy}G)p is the convective flux. The manner in which CGP is eval­

uated determines the type of "diflferencing" employed for the convective

term. The most straight forward way to calculate CGP is by linearly in­

terpolating the values CGP and CGN- This method is commonly known as

59

the central-differencing scheme. However, see the discussion in Patankar

(1983), the "upwind-differencing" formulation wiU be used here. Therefore

the convective coefficients

PGPOCGP if WGp > 0

PGNOCGN if WGp < 0

CGP =

and

Cn, = PGSOCGS if WGS > 0

pGPOtGP if U)Gi < 0.

Defining an operator [yl, 5 | to denote the greater of A and B, these last two

conditional statements can be incorporated compactly into the equation as

Convective = A^ipGpctcplwGp.O^ - PGNOCGNI - IUGP,0 | | -

pGSOtGsiwG,, 0] -f PGPOCGPI -WGS, Ol).

Collecting terms and rearrange the coefficients, the following expression is

obtained

Convective = a'^G^GP — O,^G^GN — O^^SG^GS

with

0'NG= ^<f>PGNi -1^Gp,0[

«?G = A^PGsiwGs.Ol

dpG = A^pGpilwGp^Ol + B - ^ G . , 0 I ) .

The values of pGp and PGI are obtained by Hnear interpolation of the two

nodal values that enclose each of them

pGS + pGP PG.= ^

while pGP + PGN

PGp = 2 •

And, lastly

60

3. Source te rm:

Source = TpV^

Collecting terms and rearranging the coefficients, the discretization of the

mass conservation for vapor is

ia^G ~ S^G)^GP = O'^NG^GN + 0,^G^GS + - J/G i^-^)

where, besides the definitions of the a"^'s coefficients above

f,m _ PGpy4,

^ ^ ~ " " A T

and

PGP^Gpy<t> At SuG = ^ ^ ^ ; ^ ^ ^ + TpV^'

• Liquid:

Similarly for the conservation of mass for the liquid phase

dipL^L) dipLOCLWL) _ _ p

dt , dz , ^^ ^ s/ '' ^ V " source term

transient term convective term

FoUowing the discretization steps for the vapor phase, the te rm by term dis­

cretization is

1. Transient te rm:

^ . , V4>ipLP<^LP - PLP^LP) Transient = -rr •

2. Convective term:

Convective = A^[iCLWL)p - ( C ' L ^ ' L ) . ]

with the convective coefficients

pLPOCLP if "^Lp ^ 0 CLP = ,

PLNOtLN if "^Lp < 0

61

and

^ J PLSOLLS if WLt > 0 ^LB — <

[ PLPOCLP if WL, < 0.

Utilizing the greater-of-the-two operator, these two conditional statements

becomes

Convective = A^^ipLpotLplwLpM - PLNOCLN\ -WLPM "

PLS0CLS\WL„ 0 | -f PLP0LLP\ -WLS, 0 | ) .

Collecting terms and rearrange the coefficients

Convective = a^^a^p - oJ^j^aLN - (I'SL^LS

with

OjVL = A4>PLN\ -WLp,0

C 5L =A^PLS\WL.M

a^L = A^PLpiiwLp,Ol + [ -i£;jr.,0

And

3. Source term:

Source = — FpF^.

Hence, the finite difference formulation to determine the value of liquid void

fraction is

io'^PL - S'PL)OCLP = CL'NL^LN + O,7L^LS + S^L (4.2)

with

and

f^m _ PLPy<t>

'^^^~~ At

cm P°LP^°Lpy4> p 1/

62

4.2.2 Conservation of Momentum

• Vapor:

Rewriting the vapor conservation of momentum as

dipGOLGWG) , dipGOCGWGWG) , dP ^. 1 ^ h OLG-^- -PGOLGBZ =

ot oz oz ' V ' ' "• ' ' ' r ' ' '" ' body force

transient term convective term pressure term

TWG - AGLTGLi'^G - WL) - AWGTWG'^G mass transfer interface friction yfelH friction

and integrating this equation term by term with respect to the velocity control

volume encloses by point P and N results in

1. Transient term:

. ^ VipGpO^GpWGp - Php^hp'^Gp) Iransient =

At

again the superscript o denotes the values of the "old" time level.

2. Convective term:

Convective = A[iCGWG)N - iGGWG)p]

with the convective coefficients

{ pGpOtGp'^Gp if "^Gp > 0

pGnOCGnWGn if ^Gp < 0

and J PG,0LG,'^G, if ^G* > 0

CGP = { -r c\ \ pGpOLGp'^Gp if ^ G J < 0-

These two conditional statements can be incorporated compactly into the

equation as

Convec t ive = AipGpOtCp'^Gpl'^GpM- PGnOiGnWGn\-WGpM

- /JG*aG-^yG,l^G*,0|l 4-PGpQ^Gp^Gpl -WG,M)-

63

3. Pressure term:

Pressure = aGpAiP^ - Pp).

4. Body force term:

Body force = pGpocGpB^V.

The relevant body force in this case is assumed to be due to the gravity

acceleration only.

5. Mass transfer term:

Mass transfer = T^WGpV

where WGp is the interfacial velocity. Assuming that it has the average

value of the two phasic velocities at that particular point, then

WGp + WLp ^Gp =

and the momentum transfer due to mass exchange becomes

Mass transfer = — —. 2

6. Interfacial friction term:

Interfacial friction = / . A G L ^ G L ^ ^ = ^GLp = AGLPTGLP^-

Now, there are eight combinations that can be formed from the directions of

each phasic velocity involved in the interphase friction. Each combination

gives a different direction for the interfacial friction to work. Therefore the

directions of the velocities need to be incorporated in the frictional force

formulation. The following arrangement will take care of the directions of

the phasic velocities and will result in the correct direction of the interfacial

frictional force.

^GLp Interfacial friction = XT — —

WGp - "^Lp iwGp-WLp).

64

Taking the —> direction to be positive direction and <— as negative, then

the first four combinations for which the vapor velocity is dominating is

given in Table 4.1 while the last four is covered by Table 4.2.

Table 4.1: Variations of interfacial friction direction

with velocities' directions for \wGp\ > \wLp\

WGp

-^

-*

• < —

^

WGp > WLp case

WLp

-^

^

- ^

^

WGp - WLp

- ^

-^

^

< -

IT —ve

—ve

+ve

+ve

Table 4.2: Variations of interfacial friction direction

with velocities' directions for \wGp\ < \wLp\

WGp

-^

^

^

WGp < WLp case

WLp

-*

- ^

^

WGp - WLp

*-

^

— >

IT -\-ve

—ve

-\-ve

—ve

By substituting the expression for TGL

AGLpCDPLpiwGp - WLp)^V Interfacial friction = — iwGp - WLp)

2iwGp - WLp)

AGLpCpPLpyi-^Gp - WLpliwGp - WLp)

2

65

7. Wall friction term:

Wall fnction = / AwGTwGdV = TwGp = AwGpTwGpV-

However, instead of eight there are two possibilities that can be formed

from the directions of vapor velocity in this case. Clearly, each possibility

gives a different direction for the wall friction to work. This makes it that

the direction of the vapor velocity needs to be accounted for in the friction

force. The following arrangement will take care of the direction of the vapor

velocity and will result in the correct direction of the wall frictional force.

TwGp

WGp

Again by taking the —> direction to be positive direction and <— as negative,

then

WGp -^' y^T - ve

WGp <—: y^T -f-ve.

Therefore, by substituting the expression for TWG

AwGpfpGpWcpV

Wall friction = WT = - WGI

Wall friction = — SwGp

_ _ AwGpfPGpy\wGp\wGp

8

where / is the Darcy friction factor and can be determined from the usual

friction factor correlation such as the Moody chart for pipe friction.

By employing the assumption made in the discussion of heat transfer pro­

cess above, the waH friction term for the vapor phase is identicaUy zero.

Because it is assumed that the only fluid that is in contact with the conduit

wall is the Hquid phase, hence AwGp = 0. Here, the product of .AH^GP^ is

nothing but the perimeter area for that control volume, App.

66

Collecting terms and rearrange coefficients, the finite difference formulation for

the vapor velocity is

ia^G - S^G)'^GP = a^G^Gn + a^G^Gs + S^G (4.3)

where

^NG = ApGnOCGni "I^Gp, 0|

^¥G = ApGsaG,iwG,,Ol

dpG = >i^GpaGp(l^Gp,Ol + 1 -WGs,Ol)

while

M _ PGpO^GpV , TpF AGLpCDPLpy\wGp-WLp\

and

SM^ ^ Ph,<£GpV_ ^ aGpAiPp-Pr,) + pGpaGpB.V-^^WLp^

AGLPCDPLPV\WGP - WLp\ ^^ . ^ ^ .

• Liquid:

dt ^ dz ^ ^ dz^ -—w—' ^ s r ^ ^ •^ ^ body force

transient term convective term pressure term

-TWL -ALGTLGi'WL-'^G)-AwLTwLWL-> ^ ' ' ^ " ' .;— '

mass transfer interface friction wall friction

Again, integrat ing this equation te rm by term with respect to the velocity

control volume results in

1. Transient te rm:

VipLpOtLpWLp - PlpO^LpKp) Transient =

At

67

2. Convective term:

Convective = AliCLWL)N - ( C I U ; L ) P ]

with the convective coefficients

^ J PLpOCLpWLp if WLp > 0 ^LN = <

[ pLn(^LnWLn if WLp < 0

and

^ I PL,OtL,WL, if WL, > 0 ^LP = <

[ pLpOCLpWLp if WL, < 0.

These two conditional statements can be incorporated compactly into the

equation as

Convective = AipLpaLpWLpjwLp, Ol - pLnOtLnWLnl -WLp,Ol

- PL,OCL,WL,IWL„01 -\- pLpOCLpWLpl -WL„01).

3. Pressure term:

Pressure = aLpAiPj^ — Pp)-

4. Body force term:

Body force = pLpOtLpB^V.

5. Mass transfer term:

Mass transfer = TpWLpV

where WLp is the interfacial velocity. Assuming that it has the average value

of the two phasic velocities

WLp + WGp WLp = ^

the momentum transfer due to mass exchange becomes

Mass transfer = ^ - ^ ' " ^ ! + " ° - ) .

68

6. Interfacial friction term:

Interfacial friction = / ; ALGlLGdV = Tror. = A LGp ^LGpJ- LGp V.

To take care of the phasic velocities combination, the foHowing arrangement

is used

TLGP Interfacial friction = WLp - WGp

Thus, by substituting the expression for TLG

iwLp - WGp).

Interfacial friction = — ALGpCDpLpiwLp - WGp)^V

iwLp - WGp) 2iwLp - WGp)

ALGPCDPLPV\WLP - WGpliwLp - WGp)

7. Wall friction term:

Wall friction = / AwLTwLdV = TwLp — AwLpTwLp^-Jp

Also, to take into account the directional possibilities of the liquid velocity,

the following formulation is used

TwLp Wall friction = —

WLp WLp-

Then by substituting the expression for TWL

AwLpfpLpwlpV Wall friction = — fi ^ ^ P

SWLp

_ AwLpfPLpV\wLp\wLp

8

Hke before, / is the Darcy friction factor. Unlike vapor, this term survives

under the assumption made in the heat transfer section.

Collecting terms and rearrange coefficients, the finite difference formulation for

the Hquid velocity is

(^PL - S^L)y^Lp = a^L^Ln + a^L^L. + S^L. (4-5)

where

69

«ivL = ApLnOtLnl --^^Lp, 0[

('fL = ApL,aL,lwL„Ol

dpL = ApLpaLpliiwLpM + 1 - ^ L „ O B ) ]

while

^ M ^ PLpO^LpV _ TpV _ ^LGp^DPLp^l^^Lp " WGf

^^ At 2 2 -^WLfPLpVlwLpl

8

and

•^t/L = ^ ^ + OiLpAiPp - PN) -h /OLpCtLp^.l^ - - | - ^ G p +

ALGPCDPLPV\WLP - WGp\

2

WGp' (4.6)

4.2.3 Conservation of Energy

• Vapor:

dipGOLG^G) dipGOLGhGWG) dP dP m + ¥z = -c^G-^ - aawg—

V ^ ' V ^ ' N , /

transient term convective term pressure term

+ E'^lj^ + QGi -{-PGCCGB.WG.

mass transfer interfacial heat body force

Integrating this equation term by term with respect to the control volume results

in

1. Transient term:

rr, . ^ V^ipGpOLGphGP - PGP^GP^GP) Iransient = :

At 2. Convective term:

Convective = A^[iCGWG)p - (CGI^G) , ]

70

with the convective coefficients

j pGpOCGphGP if WGp > 0 UGP = <

[ PGNOCGN^GN if WGp < 0

and

^ J pGsocGshGs if WG, > 0 ^Ga = <

[ pGpo^Gph-GP if WG, < 0

or in a compact form

Convective = A^ipcpaGphGP^wcp^Ol - pGNOCGNhGN^-WGp^O^-

PGSOCGS^GSIWG,,01 -^ pGpo^GphGpl -WG„Ol).

Collecting terms and rearrange the coefficients

Convective = apGhGp - CLNG^GN — O^SG^GS (4.7)

with

a^G = A^pGNOtGN^ - I^Gp, 0 |

dsG = A^PGSOCGS^WG,,01

apG = A^pGpaGpi\iWGp,Ol + 1 -WG„0^).

3. Pressure term:

Pressure = — ocGpWGpAff,iPp — P,).

4. Mass transfer term:

Mass transfer = EGLpy<t>

However, the enthalpy associated with vapor appearance and dissapearance

takes on diflTerent value. This condition should be taken into account by

defining

0 for vapor appearance (evaporation)

1 for vapor dissappeaxance (condensation).

71

5. Interfacial heat transfer term

Interfacial heat = OG^^^-

6. Body force term:

Body force = PGPOCGPWGPB^V^.

Collecting terms and rearrange the coefficients, the finite difference formulation

for calculating the vapor energy is then

iapG — SpG)hGP = O'NGhcN + O^SGhGS + ^UG

with

a^G = A^PGNOCGNI - i ^ G p , 0 |

asG = A^pGsocGsiwG„Oi

apG = A^pGpOcGp[iiwGp,0l -f [I -WG„0^)]

while C, pGPO^GpVcf, fj. ,r jpG = ^ ^ c^ Gpy,i>

and

^ At At

[(1 - O^GP + Twp]h'GpV4, + QGiV^ + pGP^GPWGpB.V^.

Liquid:

dipLC^LhL) dipLCCLhLWL) dP_ _ dP dt dz dt dz - V V " ^ " v ^

transient term convective term pressure term

+ E'I'G + QLi ^ PLO^LBZWL-

mass transfer interfacial heat body force

72

Integrating this equation term by term with respect to the grid node control

volume results in

1. Transient term:

Transient = ^ ^ ( ^ ^ ^ ^ ^ ^ ^ ^ ^ - PJP^Lphlp) At

2. Convective term:

Convective = A^liCGWL)p - (CG^/^L),]

with the convective coefficients

^ I PLPOLLP^LP if WLp > 0

[ pLNOCLNhLN if WLp < 0

and

_, J pLSOLLsh-LS if WL, > 0 ^Ls = <

[ pLpocLphLP if WL, < 0

or in a compact form

Convective = A^ipLpaLphLp\wLpM - PLNOLLN^LNI -WLp,0\ -

pLSOLLshLs\wL,,0\ -\- pLPOiLphLp\ -WL„0\).

Collecting terms and rearrange the coefficients

Convective = apLhLp — CLNL^LN ~ O^SL^LS

with

a^L = A^PLNOLLNI -WLp.Ol

O'SL = A^pLSOLLslwL,,0\

O'PL = A^PLPOLLpi\wLpM + 1 -'^L,M)'

3. Pressure term:

Pressure = ^ ^ I ^ i ^ ^ ^ _ a.p.„.4,(P, - P.).

4. Mass transfer term:

Mass transfer = E'I'GPV^.

Substituting the expression for E'I'Q to the above equation

Mass transfer = -TGphLpV^ - Twph'LpV^.

5. Interfacial heat transfer term

Interfacial heat = (jLi^^-

6. Body force term:

Body force = PLPOCLPWLPB^V^.

Collecting terms and rearrange the coefficients, the finite difference formulation

for calculating the Hquid energy is then

iapL — SpL)hLP = CLNL^LN + O'SLhLS + SuL

where

CLNL = A^PLNO^LNI -WLp,0^

asL = A^pLSOtLsi'WL;^

apL = A^pLpaLpliiwLp.Oi + [ -WL^O^)]

w hile ^ PLPOCLPV^ p .r SpL = ^ ^ 1 GPV"

and

A I ^t

Twph'LpV^ + QLiy<l> + pLPOtLP'^LpBrV^.

74

4.3 PEA and TDMA

4.3.1 PEA

Consider the formulations to obtain the velocities of vapor and liquid, eqns. (4.3)

and (4.5), both of them contain the velocity that is sought for. That is WGp in

the case of Hquid momentum and WLp in vapor momentum. UnHke the finite-

difference equations for mass and energy which are solved using TDMA (Tri-

jDiagonal Matrix Algorithm) directly, the equations for WG and WL are treated

by the so-called PEA (partial Elimination Algorithm) from which they are then

fed to the TDMA solver. This algorithm takes special account of the interlink-

ages between the two momentum equations. The use of PEA leads to a fast and

stable solution of the equations. The following discussion gives the derivation of

the PEA method for the relevant equations set. It is based on the note provided

by MaxweU (1990).

Writing eqns. (4.3) and (4.5) in sHghtly different manner

M M I M I cM apGWGp = O'NG'^Gn + CLSG'^G, + ^UG

where now a^G — ^¥G ~ ^¥G ^^^

M M , M , cM dpLWLp = O'NL'^Ln + ^SL'^L, + ^UL

which does not change the formulation and yet shortened the writing. Also,

consider eqns. (4.4) and (4.6) where both equations contain operations involving

the other phasic velocity, that is, the last two terms on the right hand side of

each equation. Rearrange these two equations as

SifG = S^6 + SGLWLP

where

S^6 = ^°""°^"]"°' ' ' ' + '^o.AiPp - PN) + Pc^GpB.V

75

while

o _ ^P^ , -^GLpCDpLpVlwGp - WLJ

2 2

This gives the source term for vapor phase as

apGWGp = a^^WGn + af^wG, + S^G + SGLWLP- (4.8)

The velocity at point p is

_ O'NG'^Gn + afG'^Gs + 5 ^ ^ + SGLWLJ, WGp = - ^ , M ^ ^ ^ (4.9)

"PG

The same thing for the Hquid velocity source term, that is

5M QM" I c UL — ^UL + ^LGWGp

with CjM^ _ Plp^lp'^lpy .fr, n ^ , D rr

~ 'At otLpA[Pp - PN) + pLpOcLpB^V

and

C _ ^P^ , -^LGp^DPLp^kLp - It Gpl

SLG - — y - + ^ . Thus it becomes

O'PL'^Lp = a^^WLn + afi^WL, + S^£ -\- SGLWGP (4.10)

with the liquid velocity at point p as

y^Lp = -M • (4.11) ^PL

Substituting eqn. (4.11) into eqn. (4.8), the foUowing is obtained

O'PG'^Gp = aj^QWGn + O'SG'^G, + SJJG +

SGLi^NL'^Ln + CL^L'^L, + S^£ -\- SLGWGP)

^PL

76

or, simplifying

with

,Af neu»„.. ^ M ^.. I _M , I cM „iw neu;_.. M , M , nM new /A I ON O'PG ^Gp = O'NG'^Gn + O'SG'^G, + SuQ (4-12)

^Mneu; _ M SGLSLG

apQ — apG -g ^PL

and

^Mnew _ cM" , ^GlX°^NL^n + O.^L'^LB + •S' X ) >C/G — -^UG "I M

«PL

Similarly for the Hquid phase, by substituting eqn. (4.9) to eqn. (4.10) yields

O'PL'^Lp = a^L'^Ln + CL^L'^L. + S^£ +

SLGiO'NG'^Gn + afp'^Ga + S^Q + ^GLWLp)

apG

or, by simplifying the expression appearance

dpL ^Lp = O'NL^Ln + ^LsL^L, + -^f/L

in which Mneiu _ ^M ^GL^LG

apL — dpL -j^ apG

and cMnew _ c ^ . , 'gLG(QivG^Gn + d^G^G. + .^^0) • C/L — ^UL "t" ^M

4.3.2 TDMA

When the finite-diflference formulations are written for all the main and ve­

locity control volumes, the results are sets of simultaneous, algebraic equations

pertaining to the mass, momentum, and energy conservations and pressure cor­

rection equation. To solve these sets, any general matrix solver can be used.

However, careful observation wiU reveal that the matrix obtained from one-

dimensional finite-difference formulation is tridiagonal. That is, each control

77

volume under consideration has only two neighbors so that only the main diag­

onal of the matrix and two other diagonals that enclose the main diagonal that

hold non-zero values.

The TDMA (Tri-Diagonal Matrix Algorithm) is a matrix solver that tailored

to take advantage of the zeros in the tridiagonal matrix. It executes much faster

and requires less computer memory than the general matrix solver. Because

only those non-zero values are stored in the memory. The other advantage of

the TDMA solver is that the truncation error is less significant since it involves

only a fraction of the computational steps in comparison with other solvers.

The derivation of the TDMA solver here follows closely the discussion given by

MaxweU (1986).

All the finite-difference equations for ceU P have the following functional form

and generalizing the variables sought for by X

ApXp = BpXp+i -f CpXp-i + Sp.

Here Ap is the right hand side of each finite-difference formulation, while the

rest of the coefficients are those associated with respective equations minus the

subscript quaHfiers. Now, for control volume at grid node P = 2

A2X2 = B2X3 -H C2X1 + 52

but with A''! as the known boundary condition yields

X2 = B2X3 -f 52

with

and , _ C2X1 4- ^2

•^2 - :; •

78

For the control volume at grid node P = 3 the equation becomes

A3X3 = ^3X4 -f- C3A' 2 + S3

which, upon subt i tu t ing the value of A''2 gives

A' 3 = B^X^ -f ^3

where

B' = Bz

Ar, — C^B!, L3 — «-/3^2

and , _ C3S2 + ^3

AQ — C3B2

By continuing this process, a recurrence relation for solving the tridiagonal matr ix

systems can be formed in which the relation is

Xp = BpXp^i -\- Sp

where Bp

B' =

and

Ap — CpBp_-^

CpSp_i -f Sp S' = 'p

with

Ap — CpGp_i

C[=0

and

Si = Xi.

The values of variables Xp can then be calculated by back-substi tut ion process.

79

4.4 Guessed Pressure Field and Pressure Field Correction

Closely inspecting the momentum equations, it can be seen that they can be

solved only when the pressure field is given or is somehow estimated. Unless the

pressure field is the correct one, the resulting velocity fields wiH not satisfy the

continuity equations. The guessed velocities obtained from the guessed pressure

field, P% will be denoted by w^ and w'^. If a means is provided to guess the

pressure field close to the correct one, the number of iterations to reach the

correct solutions wiU theoreticafiy be less. Below, the discussion on guessing the

pressure field is presented and wiD be followed with discussion on the pressure

correction algorithm.

4.4.1 Guessed Pressure Field

Consider the momentum equations and recover the pressure term that im­

plicitly contained in the source term as

SlfG^'''" = aGpAiPp - Pj,) + rjG

and

S^L^'"" = aLpAiPp - Pr,) + VL

where TJG and TJL contain the rest of the terms that comprise the source terms.

Substituting these two expressions into the phasic momentum equations

a^^'^^WGp = a^G-^Gn + a^G^G. + cxGpAiPp -PN)-\-VG

while for liquid phase

^L'^^-^LP = O^L'^Ln + O^L-^L. + OLLpAiPp - P^) + 7)L

are obtained. These last two equations provide a way to calculate the pressure

of the current control volume, Pp. However, it can be seen that the they are a

80

function of Pj, which is not available as yet. So that although the two expressions

are correct to determine the value of Pp but they lead to more problems in

addition to the ones that need to be taken care of. As this step is aimed at

guessing the pressure Pp then the expressions to be used are not necessEirily the

consistent one. For that , replacing the term P^ with Ps and rearranging, the

following equations are resulted

Mnew^., „M _„ , _M a^C^'^^WGp = a^G'^Gn + a^G'^Gs + CiGpAiPs - Pp) + VG

w hil(

a^L^^^WLp = a^^WLn + a^^WL, + aLpAiPs - Pp) + VL-

It is known that in a heated channel, somewhere along the line vapor will be

produced. Because of the big density difference, there will be a big difference in

the contribution to the pressure drop, and hence the final pressure field, from

each of the phase. Then if aGp < 0.01 the guessed pressure will be of the form

P^ = Ps ^-^ (4.13)

while if aGp > 0.01 the guessed-pressure formidation is

p ^ p ^PG'^'^'^GP - ia^G'^Gn + g fG^G, + VG) ^ ^ . ^ ^ J

OLGpA

Because the vapor contribution to the pressure drop is greater as the vapor

fraction increases due to the higher vapor velocity and so is the frictional force.

The value 0.01 used as the deHmiter is an arbitrary one.

4.4.2 Pressure Field Correction

As it can be seen in the phasic momentum equations above that the pressure

term is buried in the source terms S/Sfj*** and S^L"""". Because the pressure

field is also ultimately calculated, it would be incovenient to proceed with these

81

formulations. Therefore, with the pressure term written expHcitly, the phasic

momentum equations become

and

(^PG^'^WGp = a^^wGn + afc^G. + 5/^^"^- + aGpAiPp - P^)

a^L^^^WLp = a^^WLn + a^^WL, + S^j;"'"" + aLpAiPp - P^,)

where now S{f(P'''^ and 5'5 "="' are the old definitions minus their respective pres­

sure term.

Writing the "starred" velocity fields to denote the results of the guessed pres­

sure field

^PG ^Gp - ^NG^Gn + ^SG^GB + -^t/G + acpA^Pp - PN)

and

^Mnew^,' „M „..» i „M .. » i nMnew , „ A / ID ID \ ^PL ^Lp = O'NL^Ln + ^SL^LB + ^UL + OLLpA[Pp - PN)-

Following the discussion in Patankar (1980), subtract these two equations

from the previous two equations, the following relations are obtained

„Mnew„,J „M „l , _M ,„/ i_ - A f r» T>' \ dpG ^Gp = ^NG^Gn + ^SG^GB + OCGpA[I'p - / ^ j

and

^PL ^Lp = ^NL^Ln + O'SL^LB + OLLpA{Fp - I^j,)

where the primed pressures are the so-called pressure correction. As is the prac­

tice in SIMPLE algorithm, the terms associated with the neighboring points to

P are dropped from the above two equations to yield

a^G^^^w'Gp = aGpAiP'p - P ; )

and

82

or

and

»Gp = dc„{p'p - p ; )

<, = di^iP'p - PN),

in which

whil(

, _ ^GpA °'GP - Mnew

apG

_ CLLPA "'^P ~ „Mnew-

^PL

The two equations for the primed velocities are caHed the velocity-correction

formulations. Thus the corrected phasic velocities are

'^Gp = W'GP + W'GP = W'G^ -h dGpiPp - P^)

and

^Lp = wlp -{- W'LP = wlp + dLpiP'p - P'N).

Judging from the above correction formulas, a discretization equation for P'

needs to be established. Now, assume that before attaining the correct values of

aGP and aLp the volumetric error is given by

e = i^GP - ^Lp) - 1-0

where the * quantities are the values before any pressure adjustment. The ad­

justed (or corrected) values of the void fractions are

O^GP = OC'GP + ^GP

and

"LP = «LP + ^'LP

83

in which the ' quantities are the adjustment values, that is, daGP = a'Gp. Now,

consider

1 + e = a'Gp -\- alp

1 = aGP + aLp

Substracting the second from the first yields

- e = a'Gp + a'j^p.

Assuming that

aGP and aLP = fiPs^Pp^PN^,

then differentiating the equations of continuity with respect to the pressures

(do^GP daLp\ , (daGP daLp\ „, ^

\dPp ^-dP7)^^^[~dP7^'dP;i)^''^ (doGP daLp\

An examination on the phasic conservation of momentum equations reveals that

both of them are a function of pressures. Therefore the contribution of each pha­

sic momentum equations should be included. However, both momentum equa­

tions are independent of the pressure at point S. But the momentum equations

are a function of the Ps if the attention is shifted to the left node. Consequently

rather than aGp and aLp that are being differentiated, a^p and a^p are more

appropriate. Therefore, the third term on the right hand side should be adjusted

to yield

daGP daLp\ , (docp . daLp\ ,

'm^'dPir^^x'dp;^ 'm '''^ daGP da

dPs dPs GP^2^]p'^^_,

84

with the understanding that the variables with superscript - 1 are those of the

left control volume. However, the superscripted variables are really of the 5

grid node because the left node of the main control volume was denoted by the

moving pointer P before it shifted to the next control volume. Therefore

daGP

dP, daLp\ , (daGP daLp\ ,

'' dPpj^^^y-dp;;^^'^'^ dagp daLs\ „ , _ dPs ^ ~dPl) ' " ~' S

Below, term by term derivation will be presented.

daGP ^

Taking into account the point just stated above, the following differentiation

maybe made daGP _ daGP dwGp

dP^ dwGp dPs

To determine the first partial differentiation on the right hand side, rewrite the

vapor mass conservation discretization, eqn. (4.1), in sHghtly different manner

„'mnew ^ „m ^ . „m ^ , c^^ O'PG ^GP = CLNG^GN + ^SG^GS + ^UG

to make the derivation easier. Here

mnew m c"^ apG — O'PG ~ '^PG-

Differentiating the the above equation with respect to WGp

mnewdocGP da^S'^ ^ daGN . ^ da^G , m daGs ^ apG ^ -\-OCGP-^ = Oj^G-E +OCGN-^ -^O^sG-^ +

dwGp dwGp dwGp OWGp dWGp da'sG . dS^G dwGp OWGp

85

Going back to definitions of the coeflicients and closely examining them, the

above relation can be reduced to the following

mnewdoGP da^S"" , da'l^c O'PG -^ = -OtGP r, + CtGN-^-^

OWGp OWGp dwGp

because the other terms are identically zero, that is, they are independent of WGp.

Since there are two possibiHties that can be attached to the direction of velocity

WGp then it needs to identify the value associated with each of them.

1. For WGp > 0 : a^c = 0 while a^J ' "' = pGP^l^^Gp

daGP _ PGpo^GpA^

OWGp O'PG

2. For WGp<0 -. a^G = PGNA^WGP while a^^'"" = 0

dcxGP _ PGNO^GNA4> / ^ » , , _ „ninew OWGp apG

While to determine the second partial differentiation on the right hand side,

the modified momentum eqn. (4.12) is used.

„Mnew„„ „M . . . I _Af ,„ _, QMnew OpG ^Gp = O'NG^Gn + O-SG^GB + ^UG

which, when differentiated with respect to Pjq yields

"^° Ipi; + '"°''~dpr - " " " ^ ^ +'"°" dP^ + "*° dP^ +

Checking with the definition of the coefficients involved, the above relation re­

duces to _MnewdwGp _ dS^^^"" _

86

because the rest of the coefficients are not a function of pressure PN and only

SUG"""" which is dependent on P^. Thus

dwGp _ aGpA

~dP^ ~ a^G"»- •

Again, realizing the fact the phasic momentum formulation was derived by taking

into account the velocity direction, the same thing should be appHed to this

derivation. Assuming that the velocity of the cell on the right hand side of the

cell under consideration is of the same direction and greater in magnitude then

with reference to Fig. 4.2 the partials with respect to the pressure P^ can be

found.

wr:„ +ve 'Gp

'N n n+1

V^Gp -ve

Figure 4.2: Sketch of variation of WGp vs. P^

8'

1. For WGp > 0 :

Hence

dwGp

dPN

_ -^GV - < p +ve pn+l _ pn ~ -,-^N -Ov ~ve

dwGp _ aGpA

2. For WGP < 0

Therefore

Then for WGp > 0

= —ve.

dP,

dwGp

N

n + 1 _ '^Gp W

a

n Gp

M PG new

dP N pn+l pn ^N ~ ^N

—ve

— ve = -fve.

dwGp _ aGpA

dP N a M new PG

daGP _ PGpo^GpA^ aGpA

dPN

while for WGp < 0, it becomes

a mnew PG a

M new PG

daGP PGNOCGNA^ aGpA

dP. N a 771 neu» PG a

M new PG

daLP , • -r-r— te rm

dP N

Similar approach can be used for the liquid phase, by omitt ing the derivation

steps and by replacing the subscript G with Z, the set of expressions applicable

for the liquid phase can be obtained. They are, for WLp > 0

daLP PLPOiLpA^ aLpA

dPN

hile for WLp < 0, it becomes

daLP

-.mnew "-PL

PLNOCLNA^

„M new ^PL

aLp A

dP N „mnew O'PL

„M new OpL

88

dacs ^

As has been pointed above, the attention is shifted one ceH to the left when

considering the contribution of the conservations of mass and momentum on this

particular term, see Fig. 4.3.

T ^ i> ^ P I

+ve distance

Figure 4.3: Sketch of shifted control volumes

Therefore, the following differentiation may be made

daGP _ daGs dwGB

dPs ~ dwGB ' dPs '

To determine the first partial differentiation on the right hand side, rewrite the

vapor mass conservation discretization, eqn. (4.1), in sHghtly different manner

aJ^'^^O^GS = aJ^G^^GP + d^G^GR + 5[?G

to make the derivation easier. Here

a .m neiu SG

m c* Ucn. '^SG' — "'SG

89

Differentiating the the above equation with respect to WG,

mnewdocGs,^ da^S'''" mdaGP^ da^G ^ m daGR ^

da^G^dS^G OWG, OWG.

Going back to definitions of the coefficients and closely examining them, the

above relation can be reduced to the following

mnewdaGS ^ d ^ M Z . r . ^ ^ ^ ^

OWG, OWG, OWG,

because the other terms are identically zero, that is, they are independent of WG,-

Since there are two possibilities that can be attached to the direction of velocity

WG, then it needs to identify the value associated with each of them.

1. For WGB>0 : ajf^ = 0 while a^J '="' = PGSA4,WG,

daGs pGso^GsA^, .mnew dwG, a^c"'

2. For WGB<0 : a^G = PGPA^^WQ, while afj*^'" = 0

dacs pGpOLGpA^

^«.. „mnew OWG, OSG

While to determine the second partial differentiation on the right hand side,

the modified momentum eqn. (4.12) is used

Mnew^,, „M ,„ 1 _M ... I cMnew ^SG ^GB = OpGWGp + O'RG^GT -T JUG

which, when differentiated with respect to Ps yields

"^^ TP; + """"^PT = '"^P^ °' dPs + ""^ dPs +

'"^'dK^'dpr-

90

Checking with the definition of the coefficients involved, the above relation re­

duces to

a M new SG

dwG, dS'frr^ = OCGBA

dPs dPs

because the rest of the coefficients are not a function of pressure Ps and only

Slfj"^"^ which dependent on Ps. Therefore

dwG, OtG,A

dP, a M new -

S "^SG

Again, realizing the fact the phasic momentum formulation was derived by taking

into account the velocity direction, the same thing should be appHed to this

derivation. Assuming that the velocity of the cell on the right hand side of the

cell under consideration is of the same direction and greater in magnitude then

with reference to Fig. 4.4 accompanying schematic the partials with respect to

the pressure Ps can be found.

1. For WG, > 0 : dwG,

dPs

."+1 .n

pr' - Pi -fve

—ve = —ve.

Therefore dwG,

dPs „M new ^SG

2. For WGB < 0 dwG,

dPs w

n+1 — W n GB

p n + l ^S

pn ^S

—ve

—ve = +ve.

Hence

Then for WGB > 0

dwGB _ O^GBA

art -Mncu; Ol^S 0,SG

daGs _ pGS<^GsA^ O^GBA

dPs ~ ~ SG _M neu; O-SG

91

while for WGB < 0, it becomes

daGs ~dP7

PGpocGpA^ aG,A

a mnew SG a M new SG

A^Gs+^c

n n+1

\JWQ5-VC

Figure 4.4: Sketch of variation of WG, VS. PS

daLS , • term dPs

Similar approach can be used for Hquid phase and by omitting the derivation

steps and by replacing the subscript G with L the set of expressions for the Hquid

phase can be obtained. They are, for WL, > 0

daLS pLso^LsA^ ocL,A

dPs -mnew aps a

M new SL

92

while for WL, < 0, it becomes

daLS pLpOLLpA^ OLL,A

fiP„ „mnew „M new O^S OsL Og^

daGP , daLP ^ • and -^-— terms

dPp dPp

Consider that when the correct values of aGP and aLp are found then when

they are summed up the result should be unity. That is

aGP + aLP = aGP + (1 - OLGP) = 1-

Now, employing the assumption made above, that is

aGP and aLP = /-f P5,Pp, Pjv^,

then the following partial differentiation can be made

(daGP daLp\ (daGP daLp\ (dacs daLs\ ^ ^

[ dPp " dPp ) ^ \ dPN dPNj [ dPs dPs J

Equating those terms that associated with the vapor and Hquid phases, these

two equations are obtained.

daGP _ daGP _ daGS

~dP7 ~~ dPN ~ dPs

and daLP _ daLP _ daLs

~dP^ ~~ dPN ~ dPs '

Substituting these two expressions into eqn. (4.15) gives

(daGP , daLP , daGS , daLs\ „, (daGP daLp\ , -[-dP;7^'dP;^^^P7^^K)^''^[dPj, ^ dPs)^^-^

(doGs.doLs.

93

It can be seen that the coeflicient in front of P^ is nothing but the sum of the

previous two term that have just been derived. By naming the coefficient in front

of P ^ as a^ and that in front of P^ as as then

-(aAT -f as)Pp -h a^P;^ -f asPs = -e

or

whe re

and

apPp = aNP'N + asP's -f e (4.16)

daGP , daLP

""^-dPi^^-m;' _ daGS daLS

""'' dPs ~dP7'

ap = apf -{• as-

while the source term is

e = ictGP -\-OiLp) - 1.

4.5 Solution Procedure

As the last section in this chapter, a general procedure to obtain the solution

to the problems under consideration will be outlined here. Asumming that the

initial guesses and the initial as well as the boundary values have been incorpo­

rated, the following calculational steps are then in order.

1. Guess the pressure field P* using eqn. (4.13) or (4.14).

2. Cadculate the energy equations by first determining the interfacial heat

transfer contributions. This is accompHshed by the help of the heat transfer

regime selection chart outlined in Chapter 3.

94

3. Calculate the mass exchanges due to vapor evolution on the waU of the

conduit and evaporation or condensation in the bulk fluid.

4. Calculate the continuity equations using eqns. (4.1) and (4.2) for vapor and

liquid, respectively.

5. Determine the type of flow at the point of interest, calculate the frictional

forces, and then calculate the momentum equations using eqns. (4.3) for

vapor and (4.5) for liquid.

6. Calculate the amount of pressure correction via eqn. (4.16).

7. Correct the velocities and void fractions.

8. Go back to step 1 for new time level if the time has not reached the end

time step, otherwise terminate the program.

CHAPTER V

RESULTS AND DISCUSSIONS

5.1 Introduction

The finite difference approximations developed in the previous chapter were

solved using the algorithm outHned earHer. Effects of number of grid nodes

variation were studied and the results obtained showed that fairly independence

of grid nodes used was typical.

The effectiveness of the mathematical model together with its numerical ap­

proximation can be demonstrated by comparing the computational results with

experimental data and, whenever available, with analytical solution.

The presentation and analysis of the results follow the approach delineated

in Chapter 1. That is, the results pertaining to the easiest problem (in this case

reflected by the number of conservation equations to be solved and whether or

not heat and mass exchanges are involved) is discussed first while the problem

employing the least simplifying assumptions as the last. Thus, Section 5.2 deals

with the problem of stratified flow with no heat and mass exchanges followed

by the discussions on the problem of simplified two-phase flow with heat ad­

dition with the assuminption that all thermodynamic and transport properties

are constant in Section 5.3. Lastly, the problem of two-phase flow with heat

addition in which thermodynamic and transport properties are no longer treated

as constants is analyzed in Section 5.4. For aU of these problems, the primary

variables of interest are the void fractions, phasic velocities, pressure, and phasic

temperatures wherever appHcable. While in the case of constant properties, the

phasic temperatures are replaced by phasic enthalpies.

95

96

5.2 One-Dimensional Stratified Flow

Generafiy, the equations derived in Chapter 3 and their discretizations are

appHcable for this flow. However, because of the assumption that there is no

heat addition to or extraction from the system, the conservation of energy equa­

tions can be dropped from the set and thus simpHfying the task of numerical

approximations with aU the fluids properties are then constant. The other con­

venient simplifying assumption is that there is no mass exchanges between the

two fluids that results in the famiHar form of continuity equations, in terms of

the void fractions, where the right hand side is now identicaHy zero.

The working fluids are water and air inside an enclosed rectangular and circu­

lar ducts. AU of the ducts are horizontally oriented with 10.0 m channel length

and 1.0 TTi sides for the rectangular duct while it is of 1.0 m in diameter for

the circular one with the same length. For the case where the frictional effects

are included the viscosity of the water is taken to be 8.960 x 10"^ Pa/s and

1.9632 X 10"^ Pa/s for the air.

5.2.1 Some Specific Relations

A careful examination of the flow regime discussions in Chapter 3 reveals

that there is no discussion on the stratified flow regime which is applicable here.

Because there is no heat transfer in this particidar problem, the stratified fiow

regime prevails throughout the whole length of the duct. The approach adopted

to obtain the flow related parameters for this problem is essentially those pro­

posed by Russel and EtcheUs (1969) which subsequently used by Agrawal et al.

(1973) and Taitel and Dukler (1976). They suggested that for stratified gas and

Hquid systems, the interface can be considered to act as a free surface with respect

to the Hquid phase. Whereas it is treated as an imaginary stationary surface or

soHd boundary with respect to the gas phase. Thus, referring to Fig. 5.1, the

97

equivalent (hydrauHc) diameters are

Dr.= AAi

for liquid phase and where

While

for the gas phase with

AL = TTB? X aL

SL = R<i>-

Dn = 4Ac

SG-^Si

Figure 5.1: Geometries in stratified flow

AG = T^B^OLG

SG = 2'KR — R(f>

98

5. = 2 P s i n - . 2

Since the position of the Hquid-gas interface in the channel is of interest, the

following relations can be shown to apply for the problem from geometry consid­

erations. A^i(l)-s\n4>)

^' = 2^

AG = A^ - AL

and

h = - ^ ( l - c o s - ) .

From the knowledge of the void fraction distribution along the channel, the

height of the interface can be calculated.

A simple code, based on IPSA, incorporating the finite difference approxima­

tions of mass and momentum conservations is developed to handle this problem.

The code is then used to predict wave propagation in confined plane and circular

channels. In these confined channels both the liquid and the gas are initially sta­

tionary and the interface experiences a discontinuity at the middle of the channel.

The relevant boundary conditions for these problems are zero velocities at the

closed ends for both fluids. Also, a computational case where the frictional ef­

fects are neglected is performed to enable the comparison of the approximation

results with a known analytical solution. Now, it is expected that the interface

discontinuity will form disturbances to the system and physically these distur­

bances will propagate in opposite directions from the middle of the channel and

wiU subsequently be reflected from the closed ends.

The analytical solution for the speed of these interfacial waves in the case of

frictionless flow in rectangidar channel, WalHs (1969), is given by

V —

\

ipL-PG)9H (5.1)

PL/O^L + PG/OLG

99

where H is the height of the channel. However, changing the problem sHghtly

by including the frictional effects leaves the problem without analytical solution

to compare to.

5.2.2 Discussion of Results

Results of the finite-difference approximation scheme are presented here for

the case of rectangular and circular channels with and without frictional effects.

Here, the results of grid independence study is presented first as is shown in

Fig. 5.2. It is found that as the spacings between two adjacent grid nodes are

decreased, with the number of grid nodes increases as the consequence, the error

is reduced substantiaUy.

Next on the line, the results for frictionless case is given in the form of Hquid

levels versus time plot. The plots. Fig. 5.3, depict the positions of the liquid-gas

interface as the disturbances travel in opposite directions as time is increased by

0.5 second increment. The calculation was performed using pL = 996.3 kg/m^

for water density while the density of the air is pG = 1.187 kg/m^. Plugging

in these numbers into eqn. (5.1) gives the waves speed to be r = 2.265 m/s.

With linear interpolation, the computational result indicates that the velocity

V = 2.249 m/s. This gives the computational error to be around 0.74% which

was found using the relation

_ Theoretical value — Computed value % Error = — —-. f x 100.

1 heoretical value

Following the liquid-gas interface is the plot of velocities variations across the

channel length, Fig. 5.4, at time level t = 2.h s. There it is seen that the bulk of

the air is stiU travelHng to the left while the water at the far end is just started to

move to the left again to begin a new cycle of back and forth movement. Another

100

0.55

O CO

^

13 cr

0.53

0.51

0.49

HJ 0.47

0.45 0

Legends: 95CVS 30 CVs

"iOCVs""

/

, . / . - ^ .

2 4 6 8

Channel Length [m] 10

Figure 5.2: Grid independence study for stratified flow

101

0.55

6 8

0.0 sec

10

6 8

0.5 sec 10

1.0 sec

1.5 sec

0.55

2.0 sec

0.55

0.50

0.45 2 4 6 8

Channel Length [m]

2.5 sec

10

Figure 5.3: Liquid surface plots for frictionless case

102

Gas iO-

6 \-^

Cd

o

Liquid

0.15 -0.10 -0 .05 0.00

w [m/s] 0.05 0.10 0.15

Figure 5.4: Velocity distributions at t = 2.5 s

103

interesting fact in this velocity plot is the magnitude of both fluids velocities are

relatively very smaU.

Fig. 5.5 shows the sequence of liquid-gas interface positions for the case where

frictions are taken into account and where, again, time is increased by 0.5 s. It

indicates that qualitatively there is no difference between it and the friction­

less case except for small differences in numerical values that the computational

scheme produced. This is due to the relatively smaU velocity distributions across

the channel even though those velocity distributions are for the frictionless case

and are supposed to be greater than that for the viscid case. Therefore the shear

stress that works on the liquid-gas interface can not impose more and bigger

ripples on the liquid surface, also, the retarding effects on the velocities because

of the frictions on the channel boundaries are not that significant.

Lastly, a plot of the interface positions for a circular channel of 1.0 m in

diameter in which the efltects of friction included is shown in Fig. 5.6. This figure

has the interface positions for the viscid rectangular case, shown as dashed fines,

superimposed to it to give a comparison. Clearly, the wave speed for the circular

channel case is smaUer, that is sHghtly below 2.249 m/s, than that of rectangular

channel for the same initial void fraction distributions. This slower wave speed

may be attributed to the fact that the driving force, in this case the Hquid height

at the discontinuity, is smaller in comparison with that for rectangular channel

so that the disturbances spread faster.

5.3 Simplified Two-Phase Flow With Heat Addition

A highly idealized problem of boiHng in pipe which is suggested by Spalding

(1987) is the attempt in this section. The problem is geared toward the approx­

imation of the complete laws governing the physical phenomena by numerical

means. As it is, by virtue of various simplifying assumptions to be discussed later.

104

0.55

0.0 sec

0.55

0.50

0.45 1 1 ~^v

1 . 1 1

1 •

6 8

0.5 sec

10

^ 0.55 0) > 0}

o

:3 cr

0.50 h

0.45 - "- X :

' • I 1 I 1 I — I —

0.55

0.50

0.45

6

6

8

8

1.0 sec

10

L

^*v^

I 1 • .J 1 V

1

1 •

1.5 sec

10

0.55

2.0 sec

2 4 6 8

Channel Length [m]

2.5 sec

Figure 5.5: Liquid surface plots for the case where the effects of friction are

included

105

0.55

6 8

0.0 sec

10

0.55

6 8

0.5 sec

10

0.55

1.0 sec

= 1.5 sec

0.55

0.50 h

0.45

—

1 1 1 1 1 1 1

6 8

= 2.0 sec 10

0.55

0.50

0.45 0

1 1 1 1 " X

2 4 6 8

Channel Length [m]

2.5 sec

10

Fieure 5.6: Liquid surface plots for the case where the effects of friction are included for circular and rectangidar channels

106

the governing equations solved do not represent reaHty faithfully as the focus of

this endeavor is more of numerical implementation rather than trying to solve a

faithful set of governing equations that characterizes the phenomena.

The brief description of the problem solved wiH be given next together with

the relevant initial and boundary conditions. FinaHy, the results in the form of

various plots are presented.

5.3.1 Problem Description

Saturated, entirely free of vapor, water enters a duct of uniform rectangular

cross-section which is 1.0 m long and 10.0 mm x 10.0 mm sides. The two-fluid

model is used, with full allowance for heat and mass transfer between the phases,

to solve the variables involved. The values of the variables are obtained by solv­

ing the six differential equations governing the conservation of mass, momentum,

and energy for the phases together with the constraint that the volume fractions

aL and a o must sum up to unity. Thus, this problem serves as the next logi­

cal extension to the previous one: the stratified flow problem where the set of

equations for conservation of energy are not included.

The effects of gravity are to be accounted for by way of the appropriate source

terms in the momentum equations for the two fluids. Its value is 9.81 m/s^.

In order that attention can be concentrated upon the numerical aspects, the

thermodynamic and transport properties of the liquid and vapor are represented

in an idealized manner. Specificcdly, the foHowing values are to be used in the

computation:

1. the densities of both fluids are assumed to be constant with pL = 1000.0 kg/m^

for water while />G = 1-0 kg/m^ for its vapor, respectively.

2. the saturation enthalpies of the two fluids are regarded as independent

107

of working pressure and therefore having constant values; they are h'^ =

4.0 X 10' J/kg and /i^ = 2.5 x 10« J/kg.

3. the effects of viscosity and thermal conductivity, other than those intro­

duced indirectly by way of the constitutive models, are neglected. The

viscosities of the two fluids are UL = 3.095 x 10"^ m^s and I^G = 2.355 x

10" ' m^/s respectively.

Because the problem is a steady-state one, initial conditions are not part of

the problem specifications. However, since this problem is solved as a pseudo-

transient in which the steady-state solution is reached by marching through time,

initial guesses of the variables are needed. These guess values are those at the

inlet of the duct, that is, the entering momentum per unit mass is 0.01 m/s.

This gives the mass velocity G =^pw = 1000.0 x 0.01 = 10.0 kg/im^s) while the

entering mass flow rate is m = pA^w = 1.0 x 10"^ kg/s.

At the outlet from the duct, the pressure is maintaned at the reference value.

This reference value is set to zero because the absolute value of pressure has no

influence on the calculation whatsoever. Meanwhile, to boil the saturated water,

a constant heat-flux of q'^ = 1250.0 W/rn^ is suppHed from the pipe waU. This

heat-flux is presumed to enter the liquid only. Thus enthalpy rise across the pipe

length is q''A, 1250.0x0.04 ^^ ^^^ ^,,

Ah = ^ ^ ^ = ^ ^ ^ , 3 = 50,000 J kg. m 1.0 X 10-2

Based on mixture model, the thermodynamic quality at the outlet of the channel

Ah 50,000

' ' °" ' " hLG ~ 2.5 X 10« - 4.0 X 10' ~

Although this calculation is unimportant in the context of two-fluid model but

it will prove to be useful for discussion purposes later.

108

It is noticed in Chapter 3 that to determine the appropriate flow regime,

the thermodynamic and transport properties should be available. Now, since

this problem treats any property as a constant then that determination is not

possible. However, relying on the piece of information given above that the heat

flux enters the system through the liquid phase only suggesting that the annular

flow pattern may be used as the flow regime for the entire length. It will be shown

later that this assumption is not that bad. Because annular flow is a particularly

important flow pattern from the fact that for a wide range of pressure and flow

conditions it is the flow pattern over the major part of the mass quality range:

from X less than 0.1 to unity so that in a vertical tube evaporator as much as

90% of the tube length may be in annular flow.

Below, the sketch of an ideafized flow field structure for annular flow regime

is given which is foUowed by various relevant flow parameters.

From the figure. Fig. 5.7, the interphase surface area per unit volume is

_ 27ri?.Aa _ 2Ri.

^ ^ ~ TTR^AX ~ R^'

However, a way to relate Ri and J? to a more useful variable should be available.

Now, for the case of Hquid on the waH, the radii Ri and R and the void fraction,

a c , are related by _ TTR^AX _ (Ri\^

""""-I^RFA^'KR) ' so that the interface surface area per unit volume becomes

^y/ctG AGL D

These last formulas are to be used in place of the whole flow regime corre­

lations presented in Chapter 3 so that, cleariy, this problem is a very simpHfied

version of boiHng in pipe.

109

Figure 5.7: Sketch of idealized annular flow

To complete the problem specification, a number of constitutive models are

provided. Here, the interphase friction is assumed, according to the specification

stated by Spalding (1987), to obey the following law

fki = Cakaipiiwi - Wk)

where fki is the force per unit volume exerted by phase / on phase k and the

constant C is 50.0.

The interphase heat transfer is calculated from the presumption that the

phase-to-interface heat-transfer rates are equal to the corresponding enthzdpy

diff'erences times CC\akaipi where Ci is equal to 0.01 for the vapor-to-interface

transfer and to 1.0 for the Hquid-to-interface transfer. The interface enthalpies

are taken to have the saturation values of the phase in question. While the

interphase mass-transfer rate is deduced from a heat balance over the interface.

no

that is, the net heat transfer to the system is being balanced by the enthalpy

increase experienced by the vaporizing water.

5.3.2 Discussion of Results

Computational predictions for verticaHy oriented heated channels is first

tested by comparing the results for a number of grid nodes as in Fig. 5.8. Except

at both ends of the channel, the results almost faU on one fine. As has been men­

tioned previously that this problem is treated as pseudo-transient one so that a

way to terminate the program execution should be suppHed. A convenient way

of stopping the execution was to check the computed variables at the end of the

channels in which an arbitrary difference of 1.0% was assigned against two time

levels for each variable. Thus, if any two computed variables of consecutive time

levels fall within this range then the problem is said to have attained steady-

state condition. The following figure. Fig. 5.9, was constructed using the vapor

velocity at the third control volume from the entrance of the rectangular pipe

in which it was divided into 95 control volumes. Taking the value at the end

time step as the correct one, it is seen that the manner in which convergence is

attained almost quadratic.

To accomodate the discussions for the following results, the plot, Fig. 5.10, of

influence of mass velocity on void fraction for saturated nucleate boiling due to

Zuber (1967) is included. This plot is followed by the plots of vapor and Hquid

void fractions, Figs. 5.11 and 5.12, with outlet thermodynamic quality x = 0.024.

Although the numerical values depicted by the plots are not important, because

of the constant fluids properties, but the trend pursued by the curve should be

able to teU whether the computational scheme foHowed the right track. Now, the

vapor void fraction experiencing a drastic jump at the beginning of the channel

length. This may be attributed to the fact that the void fraction, Fig. 5.10, does

Ill

0.125

0.100 -

o

o CO

^

0.075

0.050

0.025

0.000 0

Legends: 80 CVs 40 CVs

"20 CVs"

0.2 0.4 0.6 0.8 Channel Length [m]

1

Figure 5.8: Grid independence study for idealized boiHng

0.125

0.100

o

O

o CO

^

0.075

0.050

0.025 L

0.000

Legends: 80 CVs 40 CVs

"20 CVs"

0.2 0.4 0.6 0.6

Channel Length [m]

111

1

Figure 5.8: Grid independence study for idealized boiHng

112

0 .10

0.08 -

o CO

^

0.06 -

0.04

0.02 -

0.00 0 0.2 0.4 0.6

Time [sec] 0.8 1

Figure 5.9: Convergence history of a^

113

8

- 0 2 •01 0 01 02 03 Thermodynamic vapour quality x

Figure 5.10: Effect of mass flux to void fraction distribution

1 1 4

O CO

o >

o

0.125

0.100 -

0.075

0.050

0.025 -

0.000 0 0.2 0.4 0.6 0.8

Channel Length [m]

Figure 5.11: Vapor void fraction distribution for rectangular duct

115

1.00

G 0.97 -

O CO

« 4 ^

o >

13 cr

0.94

0.91

0.88 -

0.85 0.2 0.4 0.6 0.8

Channel Length [m]

Figure 5.12: Liquid void fraction distribution for rectangular duct

116

not start from zero when the mass quafity, x, equals to zero. There is a slight

diflficulty here, that is, the abscissa in Fig. 5.10 is not the length of the pipe as

it is in Fig. 5.11, however, since it is very much true that for a heated pipe the

thermodynamic vapor quaHty x increases with the pipe length then Fig. 5.10

can be used to make a comparison with Fig. 5.11. Therefore, it is as if the void

fraction for the vapor is forced to start from zero with the resulting eff'ect that

it increases drastically to catch up with the actual trend. The same is true for

the Hquid void fraction because the relation whereby the sum of the two void

fractions should be equal to 1.0 has to be satisfied.

Further down the channel, the vapor void fraction is increasing but in a very

mild manner. The reason for this behavior is that because the suppHed heat flux

to the channel is not sufficient to produce more vapor. This resulted in that the

supplied heat is used toward maintaining the existing conditions upstream and

hence the slow increase in vapor void fraction while the opposite effect is taking

place in the case of the liquid phase.

The plots of stagnation enthalpies are presented next, Figs. 5.13 and 5.14.

Now, because the phasic velocities are small then the values indicated by the

stagnation enthalpy plots are practically equal to the values of the enthalpies

and for that reason the enthalpy plots are not included here.

In actuality, the liquid enthalpy should be sloping down because the satura­

tion enthalpy is a function of saturation pressure. That is, since the pressure

should decrease due to frictional losses and others then the saturation enthalpy

should too. However, one of the assumptions employed in this problem is that

the working pressure does not have any influence whatsoever on the enthalpies so

that rather than decreasing the saturation enthalpy takes its own course—that

is, by going up—to alleviate for this assumption as weU as for the fact that the

channel is being heated. The same is true in the case of vapor enthalpy. Reaching

117

45.0 o ^

I: CO

o • i -H

CO

CO

C/D

44.0 -

43.0 -

42.0

41.0

40.0 0 0.2 0.4 0.6 0.8

Channel Length [m]

Figure 5.13: Liquid stagnation enthalpy distribution for rectangular duct

118

-^ 65.0

PL,

'cO Xi

fi

fi o

^ - >

CO c CO

^ - >

O

60.0

55.0

50.0

45.0

40.0 0.2 0.4 0.6 0.8

Channel Length [m]

Figure 5.14: Vapor stagnation enthalpy distribution for rectangular duct

119

the section near the outlet the enthalpies increase sHghtly. Again, this may be

due to the reasons stated in the discussion on the void fractions. Because there is

a direct relationship between the amount of vapor production with the diff'erence

in vapor and liquid enthalpies.

In the velocity plots. Figs. 5.15 and 5.16, the velocities are increasing with

the channel length. This is mainly because of more and more vapor produced

along the line so that , logicaUy, the vapor velocity increases. In the case of the

liquid phase, although the reverse is true (that is, Hquid is being depleted) but

stiU the velocity is sloping up. The reasons for this are twofold. First, there is

acceleration induced by the vapor phase and second, the area occupied by the

liquid phase is getting smaller. The last reason is a direct consequence of the

definition of void fraction. However, still the numerical values of both phasic

velocities are smaller than they should be. It is due to the assumption that the

densities are constant. If it were the other way around, in which the densities

are allowed to vary as the pressure, the velocities will be a lot higher because

the phasic densities should be decreasing as more heat is supplied to the system.

Therefore, for a constant mass flow rate and cross sectional area with decreasing

densities, the velocities should be higher for compensation.

As the last string of plots for vertical rectangular channel. Fig. 5.17 shows

the variation of pressure along the channel length. Now, due to the assumption

that the pressure has no influence on the thermodynamic properties, it can conve­

niently set to zero at the entrance of the channel. Then, given a value of absolute

pressure at the inlet, the pressure along the pipe can be found by substracting

the reference pressure at the inlet with local values. Also, it is seen in this plot

that the total pressure drop is small and almost straight line in appearance. This

is because the phasic velocities are small that they faH well within laminar regime

120

0.15

1 O

>

o

0.12

0.09

0.06 -

0.03 -

0.00 0 0.2 0.4 0.6 0.8

Channel Length [m]

Figure 5.15: Vapor velocity distribution for rectangular duct

121

cn

o JO

>

:3 cr

U . U i U

0.014

0.013

0.012

0.011

r 1 • 1

0.010 0 0.2 0.4 0.6 0.8

Channel Length [m] 1

Figure 5.16: Liquid velocity distribution for rectangular duct

122

0.0

- 0 . 8 -

CM

I :^ CO CO

u a.

-1 .6 -

-2 .4 -

- 3 . 2 -

- 4 .0 0 0.2 0.4 0.6 0.8

Channel Length [m]

Figure 5.17: Pressure distribution for rectangular duct

123

in addition to the contribution made by the assumption that the densities are

constant. Like the rest of the plots, this plot was obtained using 95 control

volumes dividing the length of the channel.

The computation was also done for vertical circular pipe of 10.0 mm in di­

ameter. The results in form of plots—Figs. 5.18, 5.19, 5.20, 5.21, 5.22, 5.23, and

5.24—are given below where the above discussions are appHcable in this case. It

is noticed that invariably the values for circular channel are higher than those

for rectangular channel. This is because the cross sectional area through which

the fluids flow is smedler in comparison with that of rectangular channel. These

plots were obtained by dividing the length of the pipe into 95 control volumes.

5.4 Two-Phase Flow With Heat Addition

Extending the above problems to include the various flow regimes and prop­

erty variations makes this last problem an accommodating one to work with

and serves as the ultimate goal in this report. For the purpose of determining

thermodynamic and transport properties, the methods contained in Reynolds

(1968), Meyer et al., and Schmidt and Grigull (1981) play an important role and

upon which the subroutine to facilitate the properties calculations was written.

To provide a means of comparison the computational results thus obtained, the

experiments conducted by Schrock and Grossman (1959) will be used.

5.4.1 Experimental Setup and Problem Description

The schematic diagram showing the essential components used in the exper­

iment of Schrock and Grossman (1959) is shown in Fig. 5.25. These components

include a hot-water storage tank, heaters, condenser, pumps, stainless steel type

347 test section, and the associated vales and piping. Heating of the test section

was accompHshed electrically using the test section as resistance heaters. The AC

o CO

o >

o

0.20

0.16

0.12

0.08 - I

0.04

0.00 0

Legends: Round Tube R^Qts^AgyJA^ Tube_

124

0.2 0.4 0.6 0.8 Channel Length [m]

Figure 5.18: Comparison of vapor void fraction distributions for circular and rectangular ducts

Ci

o o CO

o >

7i cr

1.00

0.96

0.92

0.88

0.84

0.80 0

Legends: Round Tube RQctangul_a_r Tube_

0.2 0.4 0.6 0.8

Channel Length [m]

125

1

Figure 5.19: Comparison of Hquid void fraction distributions for circular and rectangular ducts

126

T-H 65 .0

Xi

I: CO

Xi

c o

• ^ CO fi CO

CO o

60.0 -

55.0

50.0

45.0

40.0 0

Legends: Round Tube Rectangu.l_a_r Tube_

0.2 0.4 0.6 0.8

Channel Length [m]

Figure 5.20: Comparison of vapor stagnation enthalpy distributions for circular and rectangular ducts

127

4 5 . 0

o ^

Xi

'cO Xi ^ - >

fi

o -^

CO fi CO

C O

cr

44.0

43.0

42.0

41.0

40.0

Legends: Round Tube Rectangular Tube_

0.2 0.4 0.6 0.8

Channel Length [m] 1

Figure 5.21: Comparison of liquid stagnation enthalpy distributions for circular

and rectangular ducts

128

O O^

>

o p.

0.75

iP 0.60

0.45

0.30 -

0.15 -

0.00 0

Legends: Round Tube RQQtangul_a_r Tube_

0.2 0.4 0.6 0.8

Channel Length [m]

Figure 5.22: Comparison of vapor velocity distributions for circular and rectan­gular ducts

0.0140

t o

>

0.0132

0.0124

0.0116

:3

^ 0.0108

0.0100

Legends: Round Tube R?ctangul_ar Tube.

0.2 0.4 0.6 0.8 Channel Length [m]

129

1

Figure 5.23: Comparison of Hquid velocity distributions for circular and rectan­gular ducts

130

0.0

-1 .0 -

CM

I CO

u

- 2 .0

- 3 . 0 -

-4 .0

- 5 . 0

Legends: Round Tube R.QQtangRlAr Tubp_

0.2 0.4 0.6 0.8 Channel Length [m]

Figure 5.24: Comparison of pressure distributions for circular and rectangular ducts

131

TAP WATER

CONDENSER

WATER LEVEL

CONTROL TANK

ELECTRICAL HEATER

119 V •»- VENT

CONDENSER

CONTROL VALVE

SPRAY- 6 • n VENT

' / »^

HEATER

AND

STORAGE

TANK

?

SIGHT GLASS

TEST SECTION

I

SIGHT dLASS

DEGASSING LINE

• C ^

PRESSUREi^^ GAUGE ( )

SMALL PRE HEATER

LARGE PRE HEATER

ORIFICE

FILTER •cJo-

DRAIN

BOOSTER • PUMPS

• M J ( RELIEF VALVE

Figure 5.25: Sketch of experimental set-up of Schrock and Grossman

132

power suppHed was controHed by variable transformer. While the test sections

were furnished with a number of pressure taps in order to obtain the pressure

distribution along the test section length to be used in the pressure-gradient

correlation and in determining the local value of the saturation temperature.

Iron-constant an thermocouples were welded to the test sections for the purpose

of measuring the temperature of the outside surface of the tube wall. And an

inductance-capacitance was provided to eHminate the influence of the heating

alternating current on the thermocouple emf.

The experimental procedure was first done by bringing up the temperature

of the system by means of gas-fired water heater and electrical preheaters while

circulating the water through the degassing loop. This was done prior to each

series of runs to remove dissolved gas from the water. During the series of

experimental runs the water was kept boiHng in the storage tank and a small

amount of vapor was vented to the atmosphere continously. To establish the test

conditions, the water was circulated at a desired rate and the electrical preheaters

were used to bring the water near the saturation temperature at the entrance of

the test section. Following this step, heat was applied to the test section and the

electical power, pump speed, and throttle were adjusted simidtaneously until the

desired combination of heat flux, flow rate, and pressure level were achieved.

The experimental data were recorded for a sufficiently long period of time to

ensure that steady state conditions were reached. In all series of experimental

runs the instruments were read at least twice with the wall inside temperature

distribution was calculated based on the outer wall temperature readings.

For the purpose of comparison with computational results, two arbitrary ex­

perimental series E runs were selected. The reason for selecting the series E runs

was because these test data were considered dependable since the major prob­

lems associated with the instrumentation and fabrication of the test sections were

133

solved during the previous series of the tests. These problems include the unsta-

biHty of the system due to the pumps, thermocouples instalation method that

caused questionable tube-waH temperature readings, and the problem of Teflon

gasket flowing at higher temperature that caused flow Hne obstruction.

In the computational runs, the test sections were divided axially into a number

of control volumes. The control volumes were of equal geometric size. While

the initial and boundary conditions were obtained from the experimental data.

At the inlet, the boundary conditions fixed the fluid velocity, the fluid bulk

temperature and hence the fluid enthalpies as well as the inner wall temperature.

At the outlet, the exit pressure and the inner wall temperature are determined

using the experimental data. While at the outer wall of the pipe, a constant heat

flux supplies the energy to boil the liquid flowing inside the test section.

5.4.2 Discussion of Results

The first experimental run to be approximated is the E-260 test. In this ex­

perimental run, subcooled water flows into a 0.118 in. (2.9972 x 10"^ m) inside

diameter tube which has an electrically heated test section of 15 in. (0.381 m)

length and 35 mils thick. The inlet pressure is 2.075 MPa, the mass velocity is

3,200 kg/im^s), the uniform heat flux is 2.828 x 10^ W/m^, and the inlet tem­

perature is 207.78°C. This inlet temperature corresponds to a sHght subcooHng

at the inlet of 6.51°C.

In all the results wiU be presented shortly except for the grid independence

study, the pipe length is divided into 80 control volumes which are of equal geo­

metric size. The time step size. At, chosen for each experimental run was deter­

mined, following the suggestion of Fujita and Hughes (1979), from the following

134

relation

At = . Wmax, in

In most cases, the number of time steps aHowed is about 1000 with test E-260

requiring sHghtly less because at the inlet the Hquid is almost saturated. The

time increment used in these two tests are 0.00125 s for E-260 and 0.007 s for

E-278.

The result of grid independence study for experimental run E-260 using 20, 40,

and 80 control volumes is summarized in Fig. 5.26. It is seen that the resulting

curves representing each numbers of control volumes used fcdl on a relatively

narrow range.

The tube wall temperature profile is shown next in Fig. 5.27. Comparing with

the experimental value, it is found that the difference between the two are rela­

tively small. This temperature distribution was calculated by specifying the two

end values as boundary conditions because only the inside temperature distri­

bution given in the experimental report so that a one-dimensonal computational

scheme was used as the consequence.

The plot shows that, in comparison with the experimental vzdues, the com­

puted subcooled heat transfer coefficient using the modified Chen correlation

is a Httle high. Therefore the experimental values are sHghtly higher thzin the

computational results. Now, it is seen in this plot, rather thcin increasing—as is

anticipated with single-phase flow heat transfer—the wall temperature profile is

sloping down for boiling regions. It is due to the established fact that for single-

phase convective heat transfer, the heat transfer coefficient is relatively constant.

The magnitude changes only slightly because of the influence of temperature on

the Hquid physical properties. Thus, with constant suppHed heat flux and con­

stant heat transfer coefficient, the temperature difference between the wall and

1.0

135

0.8

o

O 0.6

O CO

^

0.4

0.2

0.0 0.2

Legend: 8a CVs 40 CVs '20 CVs"

0.4

z/L 0.6 0.8 1

Figure 5.26: Grid independence study for E-260 experimental run

2 5 0 . 0

136

O o

Cd u a.

^

^

242.0

234.0

226.0

218.0

210.0

Legend:

Prediction

Exp. Data

0.2 0.4 0.6

z/L 0.8

Figure 5.27: Wall temperature distribution for E-260 experimental run

137

and the bulk fluid temperatures increases. In the subcooled nucleate boiHng

region, the heat transfer coefficient increases in Hnearly fashion with pipe length

up to the point where x = 0. Therefore, the difference between the bulk fluid

and the waH temperatures decreases linearly with the length as weH up to the

point where a; = 0.

In the saturated nucleate boiHng region the temperature and therefore the

heat transfer coefficient remain constant. However, sHghtly beyond the saturated

nucleate boiHng region, due to the decreasing thickness of the liquid film in the

two-phase forced convective region, the heat transfer coefficient is increasing with

increasing pipe length or mass quality. Thus, the temperature difference between

the wall and the bulk fluid decreases with increasing pipe length.

All in all, the overall wall temperature profile follows a decreasing trend with

increasing pipe length. Amplified by the saturation temperature drop due to

the pressure drop as the result of various losses taking place in flow, the waU

temperature may decrease considerably from the start of the subcooled boiHng

regime to the end of the pipe.

Now, as was mentioned before that the inside waH temperature distribution

was obtained through calculation based on the measured outer wall temperatures.

It is stated in the report of Schrock and Grossman (1959) that they coidd not

measure the inside waH temperature so it was necessary to calculate this quantity

from the measured outside surface temperature and the known internal heat

generation. Many errors were involved in the calculation process which were

additive and they were

1. uncertainty in the thermal conductivity of the test section

2. asymmetry in test-section cross section

3. thermocouple error due to location of the junction, and

4. error in power input.

138

Based on this knowledge, the comparison made above was actually a comparison

between two calculational methods with their respective heat transfer coeflicient

correlations.

Fig. 5.28 shows the phasic temperature distributions. It is seen that, together

with the plot of void fraction, the magnitude of heat flux suppHed as weH as the

irdet mass flux caused the Bergles-Rohsenow's subcooled boiling inception cri­

terion is satisfied very close to the inlet of the tube so that beginning from the

inlet of the pipe the mode of heat transfer is already subcooled boiHng. While

the length of subcooled boiHng heat transfer regime is a function of, among oth­

ers, the heat supplied; it is seen that as early as around z/L = 0.15 saturated

nucleate boiHng heat transfer has taken place. Based on this result, the approxi­

mate inlet subcooling, with reference to the pressure distribution plot, is 7.266°C

which corresponds to around 11.6% absolute error. Further down the line, the

vapor and liquid temperature merge into one curve. This is because the liquid

enthalpy is at its saturation value while the vapor enthalpy is higher than that

of saturation value. However, the values obtained are still within the saturation

dome that correspond to the same saturation temperatures. Now, this difference

in enthalpies that causes the production of more and more vapor as the pipe

length increases.

It is noted that the temperatures are sloping down. This fact is closely tied

to the trend followed by the pressure: it is dropping due to the pressure loss as

the fluids get accelerated and hence causing more frictional losses. This pressure

plot, Fig. 5.29, is obtained by first specifying the values of all the control volumes

to be that of the outlet as the initial condition. Now, as time marches on, the

calculated pressure correction for each control volume elevates the pressure, see

the pressure correction relation, at that respective control volume except for the

pressure at the outlet because the boundary condition employed to obtain the

139

220.0

216.0 -

CJ

o

CO

CO u a.

^

212.0 -

208.0

204.0 -

200.0 0.2 0.4 0.6 0.8 1

z/L

Figure 5.28: Phasic temperature distributions for E-260 experimental run

2 . 2 5

140

Cd OH

P tw

2.10

1.95

1.80 -

1.65

1.50

Legend:

Prediction

Exp. Data

0.2 0.4 0.6

z/L 0.8

Figure 5.29: Pressure distribution profile for E-260 experimental run

141

pressure distribution is by specifying the outlet experimental value at the end

of the pipe. The steady-state solution to the pressure profile is rather flat in

the region of subcooled boiHng and it drops fairiy fast as the fluids reach the

saturation temperature which pertains to the saturated nucleate boiHng and the

two-phase forced convective heat transfer regimes. The reason for this behavior

may be attributed to the fact that the amount of vapor produced during the

subcooled boiHng regime is relatively small in comparison with the production

during the saturated nucleate boiHng and in turn this is reflected in the velocities

of both fluids.

The plot of void fraction distribution is shown next. Fig. 5.30. Now, with

reference to the fluid temperatures plot, it is noticed that the void fraction is

not zero during the length in which the heat transfer mode is subcooled nucleate

boiling. This is in conformity with the findings of Christensen (1961) which are

included as Fig. 5.31 and the findings of Zuber (1965) included here as Fig. 5.10.

Although the conditions in which the experiment is conducted do not exactly

match the problem under consideration, however, it was concluded that even

before saturated nucleate boiling takes place some amount of vapor has been

produced which result in the nonzero void fraction prior to the flow condition

reaches x = 0. So that it is seen that the void fraction increases sharply as fully

developed boiling begins at approximately z/L = 1.5. When the liquid pha^e

energy becomes equal to the saturation value, the rate of void formation slows

down.

The flow regimes covered include bubbly flow to about z/L = 0.05 from the

inlet, slug flow to z/L = 0.1 , and annular flow regime in the remainder of the

test section. This means that annular flow regime accounted for about 90% of

the test section length which makes the assumption employed in the previous

142

1.0

0.8

0.6

O CO

0.4 -

^

0.2 -

0.0 0 0.2 0.4

z/L 0.6 0.8 1

Figure 5.30: Void fraction distribution for E-260 experimental run

143

05

c o ^ •• 0-4 u o

u 0-3

E

o >

— o

-O02

400 psio 0-47xl0^lb/h-ft2 67xlO^Btu/h-ft2 Inlet chonnel subcooling I5*6*F

Prediction

-OOi Thernnal equilibrium steom quality

Figure 5.31: Christensen experimental data

144

p r o b l e m - t h a t annular flow is the only flow pattern to be considered-not that

far from reality for the case of low inlet subcooHng and saturated nucleate boiHng.

Finally, the velocity profiles for the Hquid and vapor phases are shown in

Fig. 5.32. As is expected that as the amount of vapor being produced increases-

indicated by the rise in void fraction profile—and so is its velocity. Now, although

the Hquid phase is being depleted but its velocity experiences a moderate increase.

This IS due to the acceleration induced by the vapor phase and also to the fact

that the area through which it flows decreases. It seems that the vapor velocity

plays a dominant role in characterizing the overall velocity distribution because

based on a simple mixture model calculation model, it was found that the bulk

velocity is in the order of 100.0 m/s which is relatively close to the terminal

vapor velocity.

Next comes the second experimental run to be discussed: E-278 test. Again,

in this experimental run, subcooled water flows into a 0.118 in. (2.9972 x 10"^ m)

inside diameter tube which has an electrically heated test section of 40 in.

(1.016 m) length. The inlet pressure is 1.422 MPa, the mass velocity is 1,600

kg/im}s), the uniform heat flux is 0.6 x 10® W/m^, and the inlet temperature is

172.78°C. This inlet temperature corresponds to a relatively higher subcooling

at the inlet of the pipe: 23.03°C

Although, generally, the discussions made above are also applicable in this

case; however, there are some pecuHarities that come with the fact that this

test was conducted with greater degree of subcooHng at the inlet. Now, com­

paring with the wall temperature distribution for experimental run E-260, the

temperature profile for this test—Fig. 5.33—has a greater portion of increasing

145

150.0

CO

CO 0)

o o

I CO "a

120.0 -

90.0 -

60.0 -

30.0 -

0.0

Figure 5.32: Velocity distributions for E-260 experimental run

146

ow trend. This is because, see also void fraction plot, the portion in which the fl

is essentially single-phase extends up to around z/L = 0.15.

UnHke the results for test E-260, Fig. 5.34 shows that the Hquid needs to be

heated for a considerable longer length to reach saturated nucleate boiHng regime

which is a logical consequence of the higher degree of subcooHng in addition to

the lower rate of supplied heat flux. The heat transfer regime changes from purely

single-phase to a subcooled nucleate boiHng at approximately z/L = 0.15 which

is foHowed by saturated nucleate boiHng process at around z/L = 0.26. Here, the

computational result indicates that the inlet subcooHng is sHghtly higher than

that of experimental value which is 23.92°C and it correponds to around 4.0%

absolute error.

Next is the plot of pressure profile. Fig. 5.35. It is noticed that the pressure

drop between the inlet and the outlet is relatively smaller in comparison with

that of E-260 test. This is due to smaller mass flux this test has to handle so

that the contribution of frictional and acceleration losses are also smaller.

Following the pressure plot is the void fraction distribution, Fig. 5.36. Be­

ginning from about z/L = 0.175 up to around z/L = 0.2 the flow pattern is

bubbly flow. And from z/L = 0.2 to z/L = 0.25 the flow pattern is slug flow.

Further down this latter point, the flow pattern is annular flow up to the pipe

outlet so that it may be deducted that about 75% of the test section length is in

annular flow regime. Therefore it can be concluded that the higher the degree of

subcooHng the shorter the length in which annular flow is found.

Lastly, the velocity profiles plot. Fig. 5.37, is in order. It is seen that, as a

consequence of being in single-phase flow regime, the Hquid velocity rises only

slightly up to the point where the subcooled nucleate boiling regime takes over.

From then on, the velocity increases fast until the heat transfer regime becomes

saturated nucleate boiHng in which the Hquid velocity increase slows down. As is

147

o o

a; u

CO u

^

^

225.0

217.0

209.0

201.0

193.0 -

185.0

Legend:

Prediction

Exp. Data

J I.

0.2 0.4 0.6

z/L 0.8 1

Figure 5.33: Wall temperature profile for E-278 experimental run

148

2 1 0 . 0

200.0 -

o

CO

o

CO

^

190.0 -

180.0 -

170.0 -

160.0 0.2 0.4 0.6

z/L 0.8 1

Figure 5.34: Phasic temperature distributions for E-278 experimental run

1.55

1.45

149

CO a. 1.35

0)

CO CO

OH

1.25

1.15

1.05

Legend:

Prediction

Exp. Data

0.2 0.4 0.6

z/L 0.8

Figure 5.35: Pressure distribution for E-278 experimental run

150

1.0

0.8

0.6 -

CO

0.4 -

^

0.2 -

0.0

Figure 5.36: Void fraction distribution for E-278 experimental run

151

7 5 . 0

CO

CO a;

o o

I

60.0 -

45.0 -

30.0 -

15.0 -

0.0

Figure 5.37: Velocity profiles for E-278 experimental run

expected, the vapor velocity is steadily increasing beginning with the first for­

mation of vapor as the result of heat being suppHed.

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions

The mathematical models characterizing two-phase flows in conduits with

and without heat addition derived from the fundamental conservation laws and

their predictions were described and discussed in the previous chapters. The

two-phase modeHng used in this work is exclusively two-fluid model. This model

is a relatively new model so that many of the required constitutive relations are

either not available or can be obtained by first manipulating those appHcable for

mixture models.

The major contributions and conclusions of this study can be summarized as

follows.

1. Mathematical models for general two-phase flow in a conduit with heat and

mass transfers were determined. The terms formed a certain conservation

law were identified and later were individually discretized so as to render

easy understanding. Flow patterns and flow pattern transitions were deter­

mined and calculated from appropriate correlations and/or formulations.

2. Finite difference methods were developed to predict two-phase stratified

flow in rectangidar and circular channels with and without the inclusion of

frictional effects. The computational result for rectangular channel without

friction was compared with its analytical solution. The comparison indi­

cated that the numerical approximation was quite accurate signifying the

scheme developed correctly modeled the mass and momentum conservation

equations of the problem.

152

153

3. A means to predict the pressure distribution to be performed at the be­

ginning of each time level was developed in addition to pressure correction

scheme suitable for two-phase flows. It was beHeved that the application

of pressure distribution prediction resulted in less number of iterations to

convergence.

4. An extension of the above numerical approximation was done and was used

to predict an idealized boiHng problem in which all the fluid properties were

assumed to be constant. The numerical scheme presumed that the flow was

annular throughout the pipe length. This assumption was later checked and

was concluded that it was a reasonable one except for a small percentage

of the pipe length at the inlet section. The results obtained were then

compared with the experimental findings of Zuber (1965) and it was found

that the scheme yielded the right trends for the void fraction distributions.

5. A stiU comprehensive extension was done to eliminate the assumptions

of constant properties and annular flow pattern as the only regime. The

modified code was then used to predict flow boiling problems. The results

were then compared with the experimental data of Schrock and Grossman

(1959). It was found that the numerical scheme approximated the exper­

imental runs fairly accurate. The computational results threw some Hght

on the otherwise mixture model based experiment where, for example, the

enthalpy of the fluids were not differentiated but weighted against the mass

quality, x.

6. A subprogram to handle the transport and thermodynamic properties of

water was developed and tested. This subprogram was introduced to the

main computational scheme in such a way as to ease its replacement with

other properties subprogram for other fluids.

154

7. In general this work served as a means to test and verify the constitu­

tive equations and other pertaining correlations. It was found that they

characterized the phenomena they represented adequately accurate.

Besides the many potential appHcations of the work described in this research,

it can also be extended and/or modified to address several practical problems.

For instance, the methodology can be used to tackle the problem associated with

water tube boilers by extending the number of flow patterns correlations and

their respective transitions as weH as the heat transfer correlations to include

Hquid deficient region and the determination of the point of critical heat flux.

To predict the phenomena occuring in the evaporator and/or condensor of a

refrigeration system with suitable replacement of the subprogram to determine

the refrigerant properties. To calculate the simultaneous flow of oil and gas and

to estimate the required pumping power to transport viscous crude oil by the

addition of a less viscous immiscible liquid.

6.2 Recommendations

The three stages with increasing complexities from the point of view of mathe­

matical modeHng and numerical approximations have been sufficiently presented

and discussed. The results have been compared with existing analytical solution

as well as experimental data with the indication that d l of them are in reason­

able agreement. Based on the above findings and the experience gained through

the development stages of all these modelings, the foUowing recommendations

for future work can be made.

1. Of all the primary variables of interest only pressure that did not undergo

the diflferentiation between the two phasic pressures. The extension of the

computer code to include the effects of two-pressure model wiU be the next

155

logical step. The two-pressure model wiU then be able to approximate

additional physical features and should be a viable approach for the case

of separated flow for there is a clear boundary between the two phases that

they may maintain different value of pressures.

2. Increase the number of spatial dimension by making the approach to be

two-dimensional in space. Of course, this stage is subject to the availability

of applicable constitutive correlations and/or formulations so that it won't

have to resort back to correlations developed based on a one-dimensional

measurement making it not more accurate than its one-dimensional coun­

terpart.

3. Conduct more experiments with two-fluid model in mind so that the mod­

elers do not have to make assumptions and/or extrapolations when he or

she try to dupHcate the experiment numerically.

REFERENCES

1. Addessio, F. L. (1981). "A Review of the Development of Two-Fluid Mod­els," Los Alamos Natl. Lab. Report No. LA-8852.

2. Agrawal, S. S.; Gregory, G. A.; and Govier, G. W. (1973). "An Analysis of Horizontal Stratified Two Phase Flow in Pipes." Can. J. Chem. Engng 5 1 , 280-286.

3. American Society for Metals (1961). Metals Handbook. Volume 1: Proper­ties and Selection of Metals. 8th. ed.. Metal Park, Ohio.

4. Aris, Rutherford (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice Hall, Inc., Englewood CHff"s, New Jersey.

5. Arpaci, Vedat S. and Larsen, Poul S. (1984). Convection Heat Transfer. Prentice Hall, Inc., Englewood CHffs, New Jersey.

6. Baghdadi, A. H. A.; Rosten, H. I.; Singhed, A. K.; Spalding, D. B.; and Tatchell, D. G. (1979). "Finite-Difference Predictions of Waves in Stratified Gas-Liquid Flows." in Durst, F.; Tsiklauri, G. V.; and Afgan, N. H. (eds.). Two-Phase Momentum, Heat and Mass Transfer. Hemisphere Publsh. Co., Washington.

7. Baker, 0 . (1954). "Simultaneous Flow of Oil and Gas." Oil Gas J., 53, 185-190.

8. Bankoff, S. G. (1960). "A Variable Density Single-Fluid Model for Two-Phase Flow with Particular Reference to Steam-Water Flow." Trans. ASME, Ser. C, J. Heat Transfer, 82, 265-272.

9. Bennet, Douglas L. and Chen, John C. (1980). "Forced Convective BoiHng in Vertical Tubes for Saturated Pure Components and Binary Mixtures." AIChE Journal, 26, 454-471.

10. Bennet, D. L.; Davis, M. W.; and Hertzler, B. L. (1980). "The Suppres­sion of Saturated Nucleate Boiling by Forced Convective Flow." AIChE Symposium Series, 76, 91-103, Oriando, Florida.

11. Bergles, A. E. and Rohsenow, W. M. (1963). "The Determination of Forced Convection Surface BoiHng Heat Transfer." Paper 63-HT-22 presented at the 6th. National Heat Transfer Conference of the ASME-AIChE, Boston.

156

157

12. Bird, R. Byron; Stewart, Warren E.; and Lightfoot, Edwin N. (1960). Transport Phenomena. John Wiley & Sons, Inc., New York.

13. Boure, J. A. (1978). "Constitutive Equations for Two-Phase Flows," in Gi-noux, Jean J. (ed.). Two-Phase Flows and Heat Transfer with .Application to Nuclear Reactor Design Problems. Hemisphere Publsh. Co., Washing­ton.

14. Boure, J. A. and Reeocreux, M. (1972). "General Equations of Two-Phase Flow. AppHcation to Critical Flows and to Non-Steady Flows," Fourth AU-Union Heat and Mass Transfer Conference, Minsk, U. S. S. R.

15. Bowring, R. W. (1962). "Physical Model Based on Bubble Detachment and Calculation of Steam Voidage in the Subcooled Region of a Heated Channel." OECD Halden Reactor Project Report HPR-10.

16. Brodkey, Robert S. (1967). The Phenomena of Fluid Motions. Addison-Wesley Series in Chemical Engineering.

17. Brown, G. (1951). "Heat Transmission by Condensation of Steam on a Spray of Water Drops." Institute of Mechanical Engineers, 49-52.

18. Butterworth, D. (1972). Private communication to Moles and Shaw as cited in Butterworth, D. and Hewitt, G. F. (eds.). Two-Phase Flow and Heat Transfer. Oxford University Press, 1977, p. 219.

19. Carison, K. E.; Roth, P. A.; and Ransom, V. H. (1986). "ATHENA Code Manual Volume 1. Code Structure, System Models and Solution Methods," EGG-RTH-7397-Vol.1, EG and G Idaho, Inc., Idaho Falls.

20. Chen, John C. (1963). "A Correlation for BoiHng Heat Transfer to Satu­rated Fluids in Convective Flow." ASME Preprint 63-HT-34 presented to 6th. National Heat Transfer Conference, Boston.

21. Chen, John C. (1966). "Correlation for BoiHng Heat Transfer to Saturated Fluids in Convective Flow." / & EC Process Design and Development, 5, 322-329.

22. Christensen, H. (1961). "Power-to-Void Transfer Functions," Argonne Natl. Lab Report No. ANL-6385.

23. ColHer, John G. (1972). Convective Boiling and Condensation. McGraw-

HiU Book Co. (UK), London.

158

24. Collier, John G. (1981). "Forced Convective BoiHng," in Bergles, A. E.; ColHer, J . G.; Delhaye, J. M.; Hewitt, G. F.; and Mayinger, F. (eds.). TweO'Phase Flow and Heat Transfer in the Power and Process Industries. Hemisphere Publsh. Co., Washington.

25. Davis, E. J . and Anderson, G. H. (1966). "The Incipience of Nucleate BoiHng in Forced Convection Flow." AIChE Journal, 12, 774-780.

26. Delhaye, J. M. (1981). "Local Instantaneous Equations," in Delhaye, J. M.; Giot, M.; and Riethmuller, M. L. (eds.). Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering. Hemisphere Publsh . Co., Washington.

27. Delhaye, J . M. (1981a). "Instantaneous Space-Averaged Equations," in Delhaye, J. M.; Giot, M.; and RiethmuHer, M. L. (eds.). Thermohy­draulics of Two-Phase Systems for Industrial Design and Nuclear Engi­neering. Hemisphere Publsh. Co., Washington.

28. Delhaye, J . M. (1981b). "Local Time-Averaged Equations," in Delhaye, J. M.; Giot, M.; and Riethmuller, M. L. (eds.). Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering. Hemisphere Publsh . Co., Washington.

29. Delhaye, J . M. and Achard, J. L. (1976). "On the Averaging Operators Introduced in Two-Phase Flow ModeHng," in Banerjee, S. and Weaver, K. R. (eds.). "Transient Two Phase Flow." Proceedings of a Specialists Meeting, Ontario, Canada.

30. Di t tus , F . W. and Boelter, L. M. K. (1930). "Heat Transfer in Automobile Radia tors of the Tubular Type." PubHcations in Engineering 2, 443-461, University of California, Berkeley.

31 . Dukler, A. E. and Taitel, Y. (1986). "Flow Pat tern Transitions in Gas-Liquid Systems: Measurement and ModelHng," in Hewitt, G. F.; Delhaye, J . M.- and Zuber, N (eds.). Multiphase Science and Technology Vol 2. Hemisphere Publsh. Co., Washington.

32. Fujita, R. K. and Hughes, E. D. (1979). "Comparison of a Two-Velocity Two-Phase Flow Model with BoiHng Two-Phase Flow Data," in Veziroglu, Nejat T . (ed.). Multiphase Transport. Fundamentals, Reactor Safety, Ap­plications. Vol. 2. Hemisphere Publsh. Co., Washington.

33. Govier, G. W . and Aziz, K. (1972). Flow of Complex Mixtures in Pipes.

Van Nostrand-Reinhold Co., New York.

159

34. Griflith P.; Clark, J. A.; and Rohsenow, W. M. (1958). "Void Volumes in Subcooled BoiHng Systems." Paper 58-HT-19 presented at ASME-AlChE Heat Transfer Conference, Chicago.

35. Groeneveld, D. C. and Snoek, C. W. (1986). "A Comprehensive Examina­tion of Heat Transfer Correlations Suitable for Reactor Safety Analvsis," in Hewitt, G. F.; Delhaye, J. M.; and Zuber, N (eds.). Multiphase Science and Technology Vol. 2. Hemisphere Publsh. Co., Washington.

36. Hewitt, G. F. and Roberts, D. N. (1969). "Studies of Two-Phase Flow Patterns by Simultaneous X-ray and Flash Photography." AERE-M 2159, H.M.S.O., London.

37. Hsu, Yih-Yun and Graham, Robert W. (1976). Transport Processes in Boil­ing and Two-Phase Systems—Including Near-Critical Fluids. Hemisphere Publsh. Co., Washington.

38. Hughmark, G. A. (1962). "Holdup in Gas-Liquid Flow." Chem. Eng. Progr., 58, 62-65.

39. Ishn, Mamoru (1975). Thermo-fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris.

40. Ishii, Mamoru (1977). "One-Dimensional Drift-Flux Model and Consti­tutive Equations for Relative Motion Between Phases in Various Flow Regimes," Argonne Natl. Lab. Report No. ANL-77-47.

41. Ishn, Mamoru (1981). "Foundation of Various Two-Phase Flow Models and Their Limitations." Paper No. EPRIWS-81-212. NRC-EPRI Conference on Simulation Methods for Nuclear Power Systems, 3-47.

42. Ishn, Mamoru (1990). "Two Fluid Model for Two-Phase Flow," in He­witt, G. F.; Delhaye, J. M.; and Zuber, N (eds.). Multiphase Science and Technology Vol. 5. Hemisphere Publsh. Co., Washington.

43. Ishn, M. and Chawla, T. C. (1979). "Local Drag Laws in Dispersed Two-Phase Flow," Argonne Natl. Lab. Report No. ANL-79-105 also NUREG/CR-1230.

44. Ishn, M. and Mishima, K. (1980). "Study of Two-Fluid Model and Inter­facial Area," Argonne Natl. Lab. Report No. ANL-80-111.

45. Ishn, M. and Mishima, K. (1984). "Two-Fluid Model and Hydrodynamic Constitutive Relations." Nucl. Engng. and Design, 82, 107-126.

160

46. Jens, W. H. and Lottes, P. A. (1951). "Analysis of Heat Transfer Burnout, Pressure Drop and Density Data for High Pressure Water." Argonne Natl. Lab. Report No. ANL-4627.

47. Johannessen, Thorbj6rn (1972). "A Theoretical Solution of the Lockhart and MartinelH Flow Model for Calculating Two-Phase Flow Pressure Drop and Hold-Up." Int. J. Heat Mass Transfer, 15, 1443-1449.

48. Kreith, F. (1973). Principles of Heat Transfer. Intertext Press, Inc., New York.

49. Kuo, Kenneth K. (1986). Principles of Combustion. John Wiley &; Sons, Inc., New York.

50. Levy, S. (1967). "Forced Convection Subcooled BoiHng Prediction of Vapour Volumetric Fraction." Int. J. Heat Mass Transfer, 10, 951-965.

51. Lockhart, R. W. and MartinelH, R. C. (1949). "Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes." Chem. Engng. Prog., 45, 39-48.

52. Mandhane, J. M.; Gregory, G. A.; and Aziz, K. (1974). "A Flow Pattern Map for Gas-Liquid Flow in Horizontal Pipes." Int. J. Multiphase Flow, 1, 537-553.

53. Marchaterre, J. F. and Hoglund, B. M. (1962). "Correlation for Two-Phase Flow." Nucleonics, 20.

54. MartinelH, R. C. and Nelson, D. B. (1948). "Prediction of Pressure Drops During Forced Circulation Boiling of Water." Trans. ASME, 70, 695-702.

55. MaxweH, Timothy T. (1986). "Numerical Methods in Conduction Heat Transfer," unpubHshed note, Dept. of Mechanical Engineering, Texas Tech University, Lubbock, Texas.

56. MaxweU, Timothy T. (1990). "Calculation Methods for the Temperature Fields in a SheU and Tube Heat Exchanger," unpubHshed note, Dept. of Mechanical Engineering, Texas Tech University, Lubbock, Texas.

57. McConnel, A. J. (1957). Applications of Tensor Analysis. Dover PubHca­

tions, Inc., New York.

58. McEligot, D. M.; Magee, P. M.; and Leppert, G. (1965). "Effect of Large Temperature Gradients on Convective Heat Transfer: The Downstream Region." Journal of Heat Transfer, 87, 67-76.

161

59. Moeck, E. 0 . and Hinds, H. W. (1975). "A Mathematical Model of Steam-Drum Analysis." Paper presented at the 1975 Summer Computer Simula­tion Conference, San Fransisco.

60. Molki, M. and Sparrow, E. M. (1986). "An Empirical Correlation for the Average Heat Transfer Coeflicient in Circular Tubes." Journal of Heat Transfer, 108, 482-484.

61. Patankar, S. V. (1980). Numerical Heat Transfer and Fluid Flow. Hemi­sphere Publsh. Co., Washington.

62. Plesset, M. S. and Zwick, S. A. (1954). "The Growth of Vapor Bubbles in Superheated Liquids." Journal of Applied Physics, 25, 493-500.

63. Reynolds, W. C. (1979). Thermodynamic Properties in SI—Graphs, Ta­bles, and Computational Equations for Forty Substances. Department of Mechanical Engineering, Stanford University.

64. Rohsenow, W. M. and Clark, J. A. (1951). "Heat Transfer and Pres­sure Drop Data for High Heat Flux Densities to Water at High Subcritical Pressure." 1951 Heat Transfer and Fluid Mechanics Institute, Stanford University Press.

65. Russel, T. W. F. and EtcheHs, A. W. (1969). Paper presented at the 19th Canadian Chemical Engineering Conference and 3rd Catalysis Symposium, Edmonton, ALberta.

66. Saha, P. and Zuber, N. (1974). "Point of Net Vapor Generation and Va­por Void Fraction in Subcooled BoiHng." Paper B4.7, Proceedings 5th. International Heat Transfer Conference, Tokyo.

67. SchHchting, Hermann (1979). Boundary-Layer Theory. McGraw-HiH Book

Co., New York.

68. Schmidt, Ernst and GriguU, Ullrich (1981). Properties of Water and Steam in Sl-Units. Springer-Veriag Inc., New York.

69. Schrock, V. E. and Grossman, L. M. (1959). "Forced Convection BoiHng Studies." Forced Convection Vaporization Project, Final Report, Series 73308 - UCX 2182, University of California, Berkeley.

70. Shapiro, A. H. and Erickson, A. J. (1957). "On the Changing Size Spectrum of Particle Clouds Undergoing Evaporation, Combustion, or Acceleration." Trans of the ASME, May.

162

71. Spalding, D. B. (1987). "BoiHng in Pipe," in Hewitt, G. F.; Delhaye, J. M.; and Zuber, N (eds.). Multiphase Science and Technology Vol. S. Hemisphere Publsh. Co., Washington.

72. Stuhmiller, J. H. (1976). "A Review of the Rational Approach to Two-Phase Flow ModeHng," Electric Power Research Institute Report No. EPRI-NP-147.

73. Stuhmiller, J. H.; Ferguson, R. E.; and Wang, S. S. (1981). "Two-Phase Flow Regime Modeling," CSNI Specialists Meeting on Transient Two-Phase Flow, Pasadena, California.

74. Taitel, Y. and Dukler, A. E. (1976a). "A Theoretical Approach to the Lockhart-MartinelH Correlation for Stratified Flow." Int. J. Multiphase Flow, 2, 591-595.

75. Taitel, Yemada and Dukler, A. E. (1976b). "A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal gas-Liquid Flow." AIChE Journal, 22, 47-55.

76. Taitel, Yehuda; Dvora, Bornea; and Dukler, A. E. (1980). "ModelHng Flow Pattern Transitions for Steady Upward Gas-Liquid Flow in Vertical Tubes." AIChE Journal, 26, 345-354.

77. Theofanous, T. G. (1979). "ModeHng of Basic Condensation Processes." Paper presented in the Water Reactor Safety Research Workshop on Con­densation, Silver Spring, Maryland.

78. Thom, J. R. S.; Walker, W. W.; Fallon, T. A.; and Reising, G. F. S. (1965). "BoiHng in Subcooled Water During Flow Up Heated Tubes or Annuli." Paper 6 presented at the Symposium on BoiHng Heat Transfer in Steam Generating Units and Heat Exchangers held by Inst. Mech. Engrs., Manchester, England.

79. Tong, L. S. (1965). Boiling Heat Transfer and Two-Phase Flow. John Wiley & Sons, Inc. New York.

80. Tong, L. S. and Young, J. D. (1974). "A Phenomenological Transition and Film BoiHng Correlation." Proceedings of the 5th. International Heat Transfer Conference, vol. IV, B 3.9, Tokyo.

81. TruesdeU, C. (1969). Rational Thermodynamics. McGraw-HiH Book Co.,

New York.

163

82. Unal, H. C. (1976). "Maximum Bubble Diameter, Maximum Bubble-Growth Time and Bubble-Growth Rate During the Subcooled Nucleate Flow BoiHng of Water Up To 17.7 MN/m\'' Int. J. Heat Mass Transfer, 19, 643-649.

83. WalHs, G. B. (1969). One Dimensional Two-Phase Flow. McGraw-HiU Book Co., New York.

84. WalHs, G. B. (1982). "Review—Theoretical Models of Gas-Liquid Flows." Journal of Fluids Engineering, 104, 279-283.

85. White, Frank M. (1979). Fluid Mechanics. McGraw-HiU Book Co., New York.

86. Yadigaroglu, G. and Lahey, Jr., R. T. (1976). "On the Various Forms of the Conservation Equations in Two-Phase Flow." Int. J. Multiphase Flow, 2, 477-494.

87. Zuber, N.; Staub, F. W.; Bijwaard, G.; and Kroeger, P. G. (1967). "Steady State and Transient Void Fraction in Two-Phase Flow Systems," as cited in ColHer, John G. (1972). Convective Boiling and Condensation. McGraw-HiU Book Co. (UK), London.