[S. Mostafa Ghiaasiaan] Two-Phase Flow, Boiling an(Bookos.org)

P1: KNP9780521882767pre CUFX170/Ghiaasiaan 978 0 521 88276 7 September 27, 2007 18:52

TWO-PHASE FLOW, BOILING AND CONDENSATION INCONVENTIONAL AND MINIATURE SYSTEMS

This text is an introduction to gasliquid two-phase flow, boiling, and condensation forgraduate students, professionals, and researchers in mechanical, nuclear, and chemicalengineering. The book provides a balanced coverage of two-phase flow and phase-changefundamentals, well-established art and science dealing with conventional systems, andthe rapidly developing areas of microchannel flow and heat transfer. It is based on theauthors more than fifteen years of teaching experience. Instructors teaching multiphaseflow have had to rely on a multitude of books and reference materials. This book remediesthat problem by covering all the topics that are essential for a first graduate course.Among the important areas discussed in the book that are not adequately covered by mostof the available textbooks are two-phase flow model conservation equations and theirnumerical solution for steady and one-dimensional flow; condensation with and withoutnoncondensables; and two-phase flow, boiling, and condensation in miniature systems.

S. Mostafa Ghiaasiaan is a Professor in the George W. Woodruff School of MechanicalEngineering at Georgia Tech. Before joining the faculty, Professor Ghiaasiaan worked inthe aerospace and nuclear power industry for eight years, conducting research and devel-opment activity on modeling and simulation of transport processes, multiphase flow, andnuclear reactor thermal-hydraulics and safety. His current research areas include nuclearreactor thermal-hydraulics, particle transport, cryogenics and cryocoolers, and multiphaseflow and change-of-phase heat transfer in microchannels. Professor Ghiaasiaan has morethan 150 publications, including 80 journal articles, on transport phenomena and multi-phase flow. He is a Fellow of the American Society of Mechanical Engineers (ASME)and has been a member of that organization and the American Nuclear Society (ANS)for more than twenty years. Currently, he serves as the Executive Editor for Asia, MiddleEast, and Australia of the journal Annals of Nuclear Energy.

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Two-Phase Flow, Boiling andCondensation

IN CONVENTIONAL ANDMINIATURE SYSTEMS

S. Mostafa GhiaasiaanGeorgia Institute of Technology

iii

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-88276-7

ISBN-13 978-0-511-48039-3

S. Mostafa Ghiaasiaan 2008

2008

Information on this title: www.cambridge.org/9780521882767

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provision of relevant collective licensing agreements, no reproduction of any part

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accurate or appropriate.

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

eBook (NetLibrary)

hardback


Contents

Preface page xi

Frequently Used Notation xiii

PART ONE. TWO-PHASE FLOW

1 Thermodynamic and Single-Phase Flow Fundamentals . . . . . . . . . . . . 3

1.1 States of Matter and Phase Diagrams for Pure Substances 31.1.1 Equilibrium States 31.1.2 Metastable States 5

1.2 Transport Equations and Closure Relations 71.3 Single-Phase Multicomponent Mixtures 101.4 Phase Diagrams for Binary Systems 151.5 Thermodynamic Properties of Vapor-Noncondensable Gas Mixtures 171.6 Transport Properties 21

1.6.1 Mixture Rules 211.6.2 Gaskinetic Theory 211.6.3 Diffusion in Liquids 25

1.7 Turbulent Boundary Layer Velocity and Temperature Profiles 261.8 Convective Heat and Mass Transfer 30

2 GasLiquid Interfacial Phenomena . . . . . . . . . . . . . . . . . . . . . . 38

2.1 Surface Tension and Contact Angle 382.1.1 Surface Tension 382.1.2 Contact Angle 412.1.3 Dynamic Contact Angle and Contact Angle Hysteresis 422.1.4 Surface Tension Nonuniformity 43

2.2 Effect of Surface-Active Impurities on Surface Tension 442.3 Thermocapillary Effect 462.4 Disjoining Pressure in Thin Films 492.5 LiquidVapor Interphase at Equilibrium 502.6 Attributes of Interfacial Mass Transfer 52

2.6.1 Evaporation and Condensation 522.6.2 Sparingly Soluble Gases 57

2.7 Semi-Empirical Treatment of Interfacial Transfer Processes 592.8 Interfacial Waves and the Linear Stability Analysis Method 642.9 Two-Dimensional Surface Waves on the Surface of an Inviscid

and Quiescent Liquid 662.10 RayleighTaylor and KelvinHelmholtz Instabilities 68

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vi Contents

2.11 RayleighTaylor Instability for a Viscous Liquid 742.12 Waves at the Surface of Small Bubbles and Droplets 762.13 Growth of a Vapor Bubble in Superheated Liquid 80

3 Two-Phase Mixtures, Fluid Dispersions, and Liquid Films . . . . . . . . . . 89

3.1 Introductory Remarks about Two-Phase Mixtures 893.2 Time, Volume, and Composite Averaging 90

3.2.1 Phase Volume Fractions 903.2.2 Averaged Properties 92

3.3 Flow-Area Averaging 933.4 Some Important Definitions for Two-Phase Mixture Flows 94

3.4.1 General Definitions 943.4.2 Definitions for Flow Area-Averaged one-Dimensional Flow 953.4.3 Homogeneous-Equilibrium Flow 97

3.5 Convention for the Remainder of This Book 973.6 Particles of One Phase Dispersed in a Turbulent Flow Field

of Another Phase 983.6.1 Turbulent Eddies and Their Interaction with Suspended Fluid

Particles 983.6.2 The Population Balance Equation 1033.6.3 Coalescence 1053.6.4 Breakup 106

3.7 Conventional, Mini-, and Microchannels 1073.7.1 Basic Phenomena and Size Classification for

Single-Phase Flow 1073.7.2 Size Classification for Two-Phase Flow 111

3.8 Laminar Falling Liquid Films 1123.9 Turbulent Falling Liquid Films 1143.10 Heat Transfer Correlations for Falling Liquid Films 1153.11 Mechanistic Modeling of Liquid Films 117

4 Two-Phase Flow Regimes I . . . . . . . . . . . . . . . . . . . . . . . . 121

4.1 Introductory Remarks 1214.2 Two-Phase Flow Regimes in Adiabatic Pipe Flow 122

4.2.1 Vertical, Cocurrent, Upward Flow 1224.2.2 Cocurrent Horizontal Flow 126

4.3 Flow Regime Maps for Pipe Flow 1294.4 Two-Phase Flow Regimes in Vertical Rod Bundles 1304.5 Comments on Empirical Flow Regime Maps 134

5 Two-Phase Flow Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.1 General Remarks 1375.2 Local Instantaneous Equations and Interphase Balance Relations 1385.3 Two-Phase Flow Models 1415.4 Flow-Area Averaging 1425.5 One-Dimensional Homogeneous-Equilibrium Model:

Single-Component Fluid 1445.6 One-Dimensional Homogeneous-Equilibrium Model:

Two-Component Mixture 1485.7 One-Dimensional Separated Flow Model: Single-Component Fluid 1495.8 One-Dimensional Separated-Flow Model: Two-Component Fluid 158


Contents vii

5.9 Multidimensional Two-Fluid Model 1605.10 Numerical Solution of Steady, One-Dimensional Conservation

Equations 1635.10.1 Casting the One-Dimensional ODE Model Equations

in a Standard Form 1635.10.2 Numerical Solution of the ODEs 169

6 The Drift Flux Model and VoidQuality Relations . . . . . . . . . . . . . 173

6.1 The Concept of Drift Flux 1736.2 Two-Phase Flow Model Equations Based on the DFM 1766.3 DFM Parameters for Pipe Flow 1776.4 DFM Parameters for Rod Bundles 1786.5 DFM in Minichannels 1796.6 VoidQuality Correlations 180

7 Two-Phase Flow Regimes II . . . . . . . . . . . . . . . . . . . . . . . . 186

7.1 Introductory Remarks 1867.2 Upward, Cocurrent Flow in Vertical Tubes 186

7.2.1 Flow Regime Transition Models of Taitel et al. 1867.2.2 Flow Regime Transition Models of Mishima and Ishii 189

7.3 Cocurrent Flow in a Near-Horizontal Tube 1937.4 Two-Phase Flow in an Inclined Tube 1977.5 Dynamic Flow Regime Models and Interfacial Surface Area

Transport Equations 1997.5.1 The Interfacial Area Transport Equation 1997.5.2 Simplification of the Interfacial Area Transport Equation 201

8 Pressure Drop in Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . 207

8.1 Introduction 2078.2 Two-Phase Frictional Pressure Drop in Homogeneous Flow and the

Concept of a Two-Phase Multiplier 2088.3 Empirical Two-Phase Frictional Pressure Drop Methods 2108.4 General Remarks about Local Pressure Drops 2148.5 SinglePhase Flow Pressure Drops Caused by Flow Disturbances 215

8.5.1 Single-Phase Flow Pressure Drop across a Sudden Expansion 2178.5.2 Single-Phase Flow Pressure Drop across a Sudden Contraction 2198.5.3 Pressure Change Caused by Other Flow Disturbances 219

8.6 Two-Phase Flow Local Pressure Drops 220

9 Countercurrent Flow Limitation . . . . . . . . . . . . . . . . . . . . . . . 228

9.1 General Description 2289.2 Flooding Correlations for Vertical Flow Passages 2339.3 Flooding in Horizontal, Perforated Plates and Porous Media 2369.4 Flooding in Vertical Annular or Rectangular Passages 2379.5 Flooding Correlations for Horizontal and Inclined Flow Passages 2409.6 Effect of Phase Change on CCFL 2409.7 Modeling of CCFL Based on the Separated-Flow Momentum

Equations 241

10 Two-Phase Flow in Small Flow Passages . . . . . . . . . . . . . . . . . . 245

10.1 Two-Phase Flow Regimes in Minichannels 24510.2 Void Fraction in Minichannels 252


viii Contents

10.3 Two-Phase Flow Regimes and Void Fraction in Microchannels 25410.4 Two-Phase Flow and Void Fraction in Thin Rectangular Channels

and Annuli 25710.4.1 Flow Regimes in Vertical and Inclined Channels 25810.4.2 Flow Regimes in Rectangular Channels and Annuli 259

10.5 Two-Phase Pressure Drop 26110.6 Semitheoretical Models for Pressure Drop in the Intermittent

Flow Regime 26810.7 Ideal, Laminar Annular Flow 27110.8 The Bubble Train (Taylor Flow) Regime 272

10.8.1 General Remarks 27210.8.2 Some Useful Correlations 275

10.9 Pressure Drop Caused by Flow-Area Changes 279

PART TWO. BOILING AND CONDENSATION

11 Pool Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

11.1 The Pool Boiling Curve 28711.2 Heterogeneous Bubble Nucleation and Ebullition 291

11.2.1 Heterogeneous Bubble Nucleation and ActiveNucleation Sites 291

11.2.2 Bubble Ebullition 29611.2.3 Heat Transfer Mechanisms in Nucleate Boiling 299

11.3 Nucleate Boiling Correlations 30011.4 The Hydrodynamic Theory of Boiling and Critical Heat Flux 30611.5 Film Boiling 309

11.5.1 Film Boiling on a Horizontal, Flat Surface 30911.5.2 Film Boiling on a Vertical, Flat Surface 31211.5.3 Film Boiling on Horizontal Tubes 31511.5.4 The Effect of Thermal Radiation in Film Boiling 315

11.6 Minimum Film Boiling 31611.7 Transition Boiling 318

12 Flow Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

12.1 Forced-Flow Boiling Regimes 32112.2 Flow Boiling Curves 32812.3 Flow Patterns and Temperature Variation in Subcooled Boiling 32912.4 Onset of Nucleate Boiling 33112.5 Empirical Correlations for the Onset of Significant Void 33612.6 Mechanistic Models for Hydrodynamically Controlled Onset

of Significant Void 33712.7 Transition from Partial Boiling to Fully Developed Subcooled Boiling 34012.8 Hydrodynamics of Subcooled Flow Boiling 34112.9 Pressure Drop in Subcooled Flow Boiling 34612.10 Partial Flow Boiling 34712.11 Fully Developed Subcooled Flow Boiling Heat Transfer Correlations 34712.12 Characteristics of Saturated Flow Boiling 34912.13 Saturated Flow Boiling Heat Transfer Correlations 35012.14 Flow-Regime-Dependent Correlations for Saturated Boiling

in Horizontal Channels 35812.15 Two-Phase Flow Instability 362


Contents ix

12.15.1 Static Instabilities 36212.15.2 Dynamic Instabilities 365

13 Critical Heat Flux and Post-CHF Heat Transfer in Flow Boiling . . . . . . 371

13.1 Critical Heat Flux Mechanisms 37113.2 Experiments and Parametric Trends 37413.3 Correlations for Upward Flow in Vertical Channels 37813.4 Correlations for Subcooled Upward Flow of Water in Vertical

Channels 38713.5 Mechanistic Models for DNB 38913.6 Mechanistic Models for Dryout 39213.7 CHF in Inclined and Horizontal Channels 39413.8 Post-Critical Heat Flux Heat Transfer 399

14 Flow Boiling and CHF in Small Passages . . . . . . . . . . . . . . . . . . 405

14.1 Minichannel- and Microchannel-Based Cooling Systems 40514.2 Boiling Two-Phase Flow Patterns and Flow Instability 407

14.2.1 Flow Regimes in Minichannels with Hard Inlet Conditions 41014.2.2 Flow Regimes in Arrays of Parallel Channels 411

14.3 Onset of Nucleate Boiling and Onset of Significant Void 41414.3.1 ONB and OSV in Channels with Hard Inlet Conditions 41414.3.2 Boiling Initiation and Evolution in Arrays of Parallel Mini-

and Microchannels 41714.4 Boiling Heat Transfer 419

14.4.1 Background and Experimental Data 41914.4.2 Boiling Heat Transfer Mechanisms 42014.4.3 Flow Boiling Correlations 423

14.5 Critical Heat Flux in Small Channels 42714.5.1 General Remarks and Parametric Trends in the Available Data 42714.5.2 Models and Correlations 430

15 Fundamentals of Condensation . . . . . . . . . . . . . . . . . . . . . . . 436

15.1 Basic Processes in Condensation 43615.2 Thermal Resistances in Condensation 43915.3 Laminar Condensation on Isothermal, Vertical, and Inclined

Flat Surfaces 44115.4 Empirical Correlations for Wavy-Laminar and Turbulent Film

Condensation on Vertical Flat Surfaces 44715.5 Interfacial Shear 44915.6 Laminar Film Condensation on Horizontal Tubes 45015.7 Condensation in the Presence of a Noncondensable 45415.8 Fog Formation 457

16 Internal-Flow Condensation and Condensation on Liquid Jetsand Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

16.1 Introduction 46216.2 Two-Phase Flow Regimes 46316.3 Condensation Heat Transfer Correlations for a Pure Saturated Vapor 467

16.3.1 Correlations for Vertical, Downward Flow 46716.3.2 Correlations for Horizontal Flow 46916.3.3 Semi-Analytical Models for Horizontal Flow 472


x Contents

16.4 Effect of Noncondensables on Condensation Heat Transfer 47716.5 Direct-Contact Condensation 47816.6 Mechanistic Models for Condensing Annular Flow 48316.7 Flow Condensation in Small Channels 48816.8 Condensation Flow Regimes and Pressure Drop in Small Channels 491

16.8.1 Flow Regimes in Minichannels 49116.8.2 Flow Regimes in Microchannels 49216.8.3 Pressure Drop in Condensing Two-Phase Flows 493

16.9 Flow Condensation Heat Transfer in Small Channels 493

17 Choking in Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . 499

17.1 Physics of Choking 49917.2 Velocity of Sound in Single-Phase Fluids 49917.3 Critical Discharge Rate in Single-Phase Flow 50117.4 Choking in Homogeneous Two-Phase Flow 50217.5 Choking in Two-Phase Flow with Interphase Slip 50417.6 Critical Two-Phase Flow Models 505

17.6.1 The Homogeneous-Equilibrium Isentropic Model 50517.6.2 Critical Flow Model of Moody 50717.6.3 Critical Flow Model of Henry and Fauski 509

17.7 RETRAN Curve Fits for Critical Discharge of Water and Steam 51217.8 Critical Flow Models of Leung and Grolmes 51417.9 Choked Two-Phase Flow in Small Passages 51917.10 Nonequilibrium Mechanistic Modeling of Choked Two-Phase Flow 523

APPENDIX A: Thermodynamic Properties of Saturated Water and Steam . . . . . 529

APPENDIX B: Transport Properties of Saturated Water and Steam . . . . . . . . . 531

APPENDIX C: Thermodynamic Properties of Saturated Liquid and Vaporfor Selected Refrigerants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

APPENDIX D: Properties of Selected Ideal Gases at 1 Atmosphere . . . . . . . . 543

APPENDIX E: Binary Diffusion Coefficients of Selected Gasesin Air at 1 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

APPENDIX F: Henrys Constant of Dilute Aqueous Solutionsof Selected Substances at Moderate Pressures . . . . . . . . . . . . . . . . . . . . 551

APPENDIX G: Diffusion Coefficients of Selected Substances in Waterat Infinite Dilution at 25C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

APPENDIX H: LennardJones Potential Model Constants for SelectedMolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

APPENDIX I: Collision Integrates for the LennardJones Potential Model . . . . . 557

APPENDIX J: Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

APPENDIX K: Unit Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

References 563

Index 601


Preface

This book is the outcome of more than fifteen years of teaching graduate courseson nuclear reactor thermal-hydraulics and two-phase flow, boiling, and condensa-tion to mechanical, and nuclear engineering students. It is targeted to be the basisof a semester-level graduate course for nuclear, mechanical, and possibly chemicalengineering students. It will also be a useful reference for practicing engineers.

The art and science of multiphase flow are indeed vast, and it is virtually impossibleto provide a comprehensive coverage of all of their major disciplines in a graduatetextbook, even at an introductory level. This textbook is therefore focused on gasliquid two-phase flow, with and without phase change. Even there, the arena is toovast for comprehensive and in-depth coverage of all major topics, and compromise isneeded to limit the number of topics as well as their depth and breadth of coverage.The topics that have been covered in this textbook are meant to familiarize thereader with a reasonably wide range of subjects, including well-established theoryand technique, as well as some rapidly developing areas of current interest.

Gasliquid two-phase flow and flows involving change-of-phase heat transferapparently did not receive much attention from researchers until around the middleof the twentieth century, and predictive models and correlations prior to that timewere primarily empirical. The advent of nuclear reactors around the middle of thetwentieth century, and the recognition of the importance of two-phase flow andboiling in relation to the safety of water-cooled reactors, attracted serious attention tothe field and led to much innovation, including the practice of first-principle modeling,in which two-phase conservation equations are derived based on first principles andare numerically solved. Today, the area of multiphase flow is undergoing acceleratingexpansion in a multitude of areas, including direct numerical simulation, flow andtransport phenomena at mini- and microscales, and flow and transport phenomena inreacting and biological systems, to name a few. Despite the rapid advances in theoryand computation, however, the area of gasliquid two-phase flow remains highlyempirical owing to the extreme complexity of processes involved.

In this book I have attempted to come up with a balanced coverage of funda-mentals, well-established as well as recent empirical methods, and rapidly developingtopics. Wherever possible and appropriate, derivations have been presented at leastat a heuristic level.

The book is divided into seventeen chapters. The first chapter gives a concisereview of the fundamentals of single-phase flow and heat and mass transfer. Chap-ter 2 discusses two-phase interfacial phenomena. The hydrodynamics and mathemat-ical modeling aspects of gasliquid two-phase flow are then discussed in Chapters 3

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xii Preface

through 9. Chapter 10 rounds out Part One of the book and is devoted to the hydro-dynamic aspects of two-phase flow in mini- and microchannels.

Part Two focuses on boiling and condensation. Chapters 11 through 14 aredevoted to boiling. The fundamentals of boiling and pool boiling predictive methodsare discussed in Chapter 11, followed by the discussion of flow boiling and criticaland postcritical heat flux in Chapters 12 and 13, respectively. Chapter 14 is devotedto the discussion of boiling in mini- and microchannels. External and flow conden-sation, with and without noncondensables, and condensation in small flow passagesare then discussed in Chapters 15 and 16. The last chapter is devoted to two-phasechoked flow. Various property tables are provided in several appendices.


Frequently Used Notation

A Flow area (m2); atomic numberAC Flow area in the vena-contracta location (m2)Ad Frontal area of a dispersed phase particle (m2)a Speed of sound (m/s)aI Interfacial surface area concentration (surface area per unit mixture

volume; m1)Bd Bond number = l2/( g )Bh Mass-flux-based heat transfer driving forceBh Molar-flux-based heat transfer driving forceBm Mass-flux-based mass transfer driving forceBm Molar-flux-based mass transfer driving forceBi Biot number = hl/kBo Boiling number = qw/(G hfg)C Concentration (kmol/m3)C Constant in Walliss flooding correlation; various constantsc Wave propagation velocity (m/s)Ca Capillary number = LU/Cr Crispation number =

l (k

CP)

C2 Constant in TienKutateladze flooding correlationCC Contraction ratioCD Drag coefficientCHe Henrys coefficient (Pa; bar)Co Convection number = (g/ f )0.5[(1 x)/x]0.8CP Constant-pressure specific heat (J/kgK)CP Molar-based constant-pressure specific heat (J/kmolK)Csf Constant in the nucleate pool boiling correlation of RohsenowCv Constant-volume specific heat (J/kgK)Cv Molar-based constant-volume specific heat (J/kmolK)C0 Two-phase distribution coefficient in the drift flux modelD Tube or jet diameter (m)DH Hydraulic diameter (m)D Mass diffusivity (m2/s)Dij Binary mass diffusivity for species i and j (m2/s)DiG, DiL Mass diffusivity of species i in gas and liquid phases (m2/s)d Bubble or droplet diameter (m)dcr Critical diameter for spherical bubbles (m)

xiii


xiv Frequently Used Notation

dSm Sauter mean diameter of bubbles or droplets (m)E Eddy diffusivity (m2/s)E1, E One-dimensional and three-dimensional turbulence energy spectrum

functions based on wave number (m3/s2)E1 , E

One-dimensional and three-dimensional turbulence energy spectrumfunctions based on frequency (m2/s)

EB Bulk modulus of elasticity (N/m2)EH Eddy diffusivity for heat transfer (m2/s)Eo Eotvos number = g l2/e Total specific convected energy (J/kg)e Unit vectorF Degrees of freedom; force (N); Helmholtz free energy (J); correction

factorFI Interfacial Helmholtz free energy (J)FI Interfacial force, per unit mixture volume (N/m3)Fo Fourier number = ( k

CP) tl2

Fr Froude number = U2/(gD)FVM Virtual mass force, per unit mixture volume (N/m3)Fw Wall force, per unit mixture volume (N/m3)FwG, FwL Wall force, per unit mixture volume, exerted on the liquid and gas

phases (N/m3)F Surface tension force (N)f Fanning friction factor; frequency (Hz); distribution function (m1 or

m3); specific Helmholtz free energy (J/kg)f Darcy friction factorf I Specific interfacial Helmholtz free energy (J/m2)fcond Condensation efficiencyG Mass flux (kg/m2.s); Gibbs free energy (J)GI Interfacial Gibbs free energy (J)Ga Galileo number = L g l3

2L

Gr Grashof number = ( g l32L

)( LgL

)

Gz Graetz number= 4U l2z ( CPk )g Gravitational acceleration vector (m/s2)g Specific Gibbs free energy (J/kg); gravitational constant ( = 9.807

m/s2 at sea level); breakup frequency (s1)gI Specific interfacial Gibbs free energy (J/m2)H Heat transfer coefficient (W/m2K); height (m)Hr Radiative heat transfer coefficient (W/m2K)He Henry numberh Specific enthalpy (J/kg); mixed-cup specific enthalpy (J/kg); collision

frequency function (m3s)hL Liquid level height in stratified flow regime (m)hfg, hsf, hsg Latent heats of vaporization, fusion, and sublimation (J/kg)hfg, hsf, hsg Molar-based latent heats of vaporization, fusion, and sublimation

(J/kmol)Im Modified Bessels function of the first kind and mth orderJ Diffusive molar flux (kmol/m2s)Ja Jakob number = ( CP)L T/ghfg or CPL T/hfg


Frequently Used Notation xv

J Flux of a transported property in the generic conservation equations(Chapters 1 and 5)

J Dimensionless superficial velocity in Walliss flooding correlationJa Modified Jacob number =

LG

CPLThfg

j Diffusive mass flux (kg/m2s); molecular flux (m2s1); superficialvelocity (m/s)

k Thermal conductivity (W/mK); wave number (m1)K Loss coefficient; Armands flow parameter; mass transfer coefficient

(kg/m2s)K parameter in Kattos DNB correlation (Chapter 13)K Molar-based mass transfer coefficient (kmol/m2s)K* Kutateladze number; dimensionless superficial velocity in

TienKutateladze flooding correlationKa Kapitza number = 4L3Lg/ 3Khor Correction factor for critical heat flux in horizontal channelsLe Lewis number = /DLB Boiling length (m); bubble (vapor clot) length (m)Lheat Heated length (m)LS Liquid slug length (m)l Length (m); characteristic length (m)lD Kolmogorovs microscale (m)lE Churn flow entrance length before slug flow is

established (m)lF Length scale applied to liquid films (m)M Molar mass (kg/kmol); component of the generalized drag force (per

unit mixture volume) (N/m3)Ma Marangoni number = (

T )T l2

( CPk )

Mo Morton number = g 4L /(2L 3)MIk Generalized interfacial drag force (N/m3) exerted on phase kMID Interfacial drag force term (N/m3)MIV Virtual mass force term (N/m3)Mk Signal associated with phase kM2 Constant in TienKutateladze flooding correlationm Mass fraction; mass of a single molecule (kg); dimensionless constantm Mass (kg)m Mass flux (kg/m2s)N Molar flux (kmol/ m2s)N Unit normal vectorNAv Avogadros number ( = 6.022 1026 molecules/k mol)Ncon Confinement number =

/g/ l

Nu Nusselt number H l/kN Viscosity number = L/[L

/(g)]1/2

n Number density (m3); number of chemical species in a mixture;dimensionless constant; polytropic exponent

p Perimeter (m)P Pressure (N/m2); Legendre polynomialPP Pump (supply) pressure drop (N/m2)PC Channel (demand) pressure drop (N/m2)


xvi Frequently Used Notation

Pe Peclet number = U l ( CP/k)Pr Prandtl number = CP/kPr Reduced pressure = P/PcrPrturb Turbulent Prandtl numberpf Wetted perimeter (m)pheat Heated perimeter (m)Q Volumetric flow rate (m3/s); dimensionless wall heat fluxq Heat generation rate per unit length (W/m)q Heat flux (W/m2)qv Volumetric energy generation rate (W/m3)R Radius (m); gas constant (Nm/kgK)Rc Radius of curvature (m)RC Wall cavity radius (m)Re Reynolds number (U l/)ReF Liquid film Reynolds number = 4F/LRj Equilibrium radius of a jet (m)Rl Volumetric generation rate of species l (kmol/m3s)Ru Universal gas constant ( = 8,314 Nm/kmolK)r Distance between two molecules () (Chapter 1); radial

coordinate (m)rl Volumetric generation rate of species l (kg/m3s)S Sheltering coefficient; entropy (J/K); source and sink terms in

interfacial area transport equations (s1m6); distance definingintermittency (m)

Sc Schmidt number = /DSh Sherwood number = Kl/D or Kl/CDSo Soflata number = [(3 3)/(3g4)]1/5Su Suratman number = l/2Sr Slip ratios Specific entropy (J/kgK)T Unit tangent vectorT Temperature (K)t Time (s); thickness (m)tc Characteristic time (s)tc,D Kolmogorovs time scale (s)tgr Growth period in bubble ebullition cycle (s)tres Residence time (s)twt Waiting period in bubble ebullition cycle (s)U Internal energy (J)U Velocity (m/s); overall heat transfer coefficient (W/m2K)UB Bubble velocity (m/s)UB, Rise velocity of Taylor bubbles in stagnant liquid (m/s)Ur Slip velocity (m/s)U Friction velocity (m/s)u Specific internal energy (J/kg)u Velocity (m/s)uD Kolmogorovs velocity scale (m/s)V Volume (m3)Vd Volume of an average dispersed phase particle (m3)


Frequently Used Notation xvii

Vg j Gas drift velocity (m/s)Vg j Parameter defined as Vg j + (C0 1) j (m/s)v Specific volume (m3/kg)We Weber number = U2l/W Width (m)w Interpolation length in some flooding correlations (m)x Qualityxeq Equilibrium qualityX Mole fraction; Martinellis factor

Greek characters

Void fraction; wave growth parameter (s1); phase index Thermal diffusivity (m2/s)k In situ volume fraction occupied by phase k Volumetric quality; phase index; parameter defined in Eq. (1.75);

coefficient of volumetric thermal expansion (K1); dimensionlessparameter

(V, V) Probability of breakup events of particles with volume V that resultin the generation of a particle with volume V (m1)

Plate thickness (m) Kronecker delta; gap distance (m); thermal boundary layer

thickness (m)F Film thickness (m)m Thickness of the microlayer (m) Porosity; radiative emissivity; Bowrings pumping factor (Chapter 12);

turbulent dissipation rate (W/kg); perturbationD Surface roughness (m) Energy representing maximum attraction between two molecules (J) Parameter in Bakers flow regime map (Chapter 4) Cavity side angle (rad or degrees); transported property (Chapters 1

and 5); stream function (m2/s)2 Two-phase multiplier for frictional pressure drop Two-phase multiplier for minor pressure drops; dissipation

function (s2) Velocity potential (m2/s); pair potential energy (J) Transported property (Chapters 1 and 5); relative humidity Volumetric phase change rate (per unit mixture volume) (kg/m3s);

correction factor for the kinetic model for liquidvapor interfacialmass flux; surface concentration of surfactants (kmol/m2)

F Film mass flow rate per unit width (kg/ms) Specific heat ration (CP/Cv); perforation ratio c Convective enhancement factorK Curvature (m1) von Karmans constantB Boltzmanns constant (= 1.38 1023J/K) Interfacial pressure (N/m) Molecular mean free path (m); wavelength (m); coalescence

efficiency; parameter in Bakers flow regime map (Chapter 4)


xviii Frequently Used Notation

d Fastest growing wavelength (m)H Critical Rayleigh unstable wavelength (m)L Laplace length scale =

/g (m)

Viscosity (kg /ms) Kinematic viscosity (m2/s) Number of phases in a mixture; 3.1416 Azimuthal angle (rad); angle of inclination with respect to the

horizontal plane (rad or degrees); contact angle (rad or degrees)0, a, r Equilibrium (static), advancing, and receding contact angles

(rad or degrees) Density (kg/m3) Momentum density (kg/m3) Surface tension (N/m); smaller-to-larger flow area ratio in a flow area

change Smaller-to-lager flow area ratios in a flow-area contraction Molecular collision diameter ()A Molecular scattering cross section (m2)c, e Condensation and evaporation coefficients Molecular mean free time (s); shear stress (N/m2) Viscous stress tensor (N/m2) Azimuthal angle for film flow over horizontal cylinders (rad)k, D Collision integrals for thermal conductivity and mass diffusivity Angular frequency (rad/s); humidity ratio; dimensionless parameter

(Chapter 17) Chemical potential (J/kg); noncondensable volume fraction Interphase displacement from equilibrium (m) Tangential coordinate on the liquidgas interphase

Superscripts

r Relative+ In wall units In the presence of mass transfer Averaget Time averagedt k Time averaged for phase k= Tensor Dimensionless Molar based; dimensionless

Subscripts

avg AverageB BubbleBd Bubble departureb Boiling; bulkc Continuous phasech Choked (critical) flowcond Condensationcont Contraction


Frequently Used Notation xix

cr Criticald Dispersed phaseeq Equilibriumev Evaporationex Expansionexit Exitf Saturated liquidf0 All vaporliquid mixture assumed to be saturated liquidfr FrictionalFC Forced convectionF Liquid or vapor filmG Gas phaseg Saturated vapor; gravitationalg0 All liquidvapor mixture assumed to be saturated vaporGI At interphase on the gas sideG0 All mixture assumed to be gash Homogeneousheat HeatedI Gasliquid interface; irreversiblein Inletinc Inception of wavinessL Liquid phaseL0 All mixture assumed to be liquidLI At interphase on the liquid siden Sparingly soluble (noncondensable) inert speciesout OutletR Reversiblerad Radiationref Referenceres Associated with residence times s surface (gas-side interphase); isentropic; solid at melting or

sublimation temperaturesat SaturationSB Subcooled boilingslug Liquid or gas slugspin SpinodalTB Transition boilingTP Two-phaseturb TurbulentUC Unit cellu u surface (liquid-side interphase)V Virtual mass forcev Vapor when it is not at saturation; volumetricW Waterw WallwG Wallgas interfacewL Wallliquid interfacez Local quantity corresponding to location z0 Equilibrium state


xx Frequently Used Notation

Abbreviations

BWR Boiling water reactorCFD Computational fluid dynamicsCHF Critical heat fluxDC Direct-contactDFM Drift Flux ModelDNB Departure from nucleate boilingDNBR Departure from nucleate boiling ratioHEM Homogeneous-equilibrium mixtureHM Homogeneous mixtureMFB Minimum film boilingLOCA Loss of coolant accidentNVG Net vapor generationOFI Onset of flow instabilityONB Onset of nuclear boilingOSV Onset of significant voidPWR Pressurized water reactor


TWO-PHASE FLOW, BOILING AND CONDENSATION INCONVENTIONAL AND MINIATURE SYSTEMS

xxi


xxii

P1: KNP9780521882761c01 CUFX170/Ghiaasiaan 978 0 521 88276 7 August 29, 2007 23:15

PART ONE

TWO-PHASE FLOW

1


2


1 Thermodynamic and Single-PhaseFlow Fundamentals

1.1 States of Matter and Phase Diagrams for Pure Substances

1.1.1 Equilibrium States

Recall from thermodynamics that for a system containing a pure and isotropic sub-stance that is at equilibrium, without any chemical reaction, and not affected by anyexternal force field (also referred to as a PvT system), an equation of state of thefollowing form exists:

f (P, v, T) = 0. (1.1)This equation, plotted in the appropriate Cartesian coordinate system, leads to asurface similar to Fig. 1.1, the segments of which define the parameter ranges forthe solid, liquid, and gas phases. The substance can exist in a stable equilibriumstate only on points located on this surface. Using the three-dimensional plot isawkward, and we often use the phase diagrams that are the projections of the afore-mentioned surface on Pv (Fig. 1.2) and Tv (Fig. 1.3) planes. Figures 1.2 and 1.3also show where vapor and gas occur. The projection of the aforementioned surfaceon the PT diagram (Fig. 1.4) indicates that P and T are interdependent when twophases coexist under equilibrium conditions. All three phases can coexist at the triplepoint.

To derive the relation between P and T when two phases coexist at equilibrium,we note that equilibrium between any two phases and requires that

g = g, (1.2)where g = u + Pv Ts is the specific Gibbs free energy. For small changes simul-taneously in both P and T while the mixture remains at equilibrium, this equationgives

dg = dg. (1.3)From the definition of g one can write

dg = du + Pdv + vdP Tds sdT. (1.4)However, from the Gibbs equation (also referred to as the first Tds relation) wehave

Tds = du + Pdv. (1.5)3


4 Thermodynamic and Single-Phase Flow Fundamentals

Solid + liquid

Solid + Vapor

P

Solid Liquid

T = constant

P = constant

T = constant

T

v

T =Tcr

VaporLiquid + vapor Triple line

Critical point

Figure 1.1. The PvT surface for a substance that contracts upon freezing.

We can now combine Eqs. (1.3) and (1.4) and write for the two phases

dg = sdT + vdP (1.6)

and

dg = sdT + vdP. (1.7)

Vapor

T1 = constant

Solid

Psat T1

Solid + liquidP

Critical point

Solid + vapor

Triple line

Liquid + vapor

LiquidGas

Figure 1.2. The Pv phase diagram.


1.1 States of Matter and Phase Diagrams for Pure Substances 5

Liquid and Vapor

Liquid

P = Pcr

Supercritical FluidT

Tcr

Tsat(P1)

Gas

P1

P2 < P1

Vapor

v

Figure 1.3. The T phase diagram.

Solid

P

Fusio

n

Liquid

Vaporiz

ation

Triple point

Sublimation

T

Vapor

Critical State

Figure 1.4. The PT phase diagram.

Substitution from Eqs. (1.6) and (1.7) into Eq. (1.3) gives

dPdT

= s sv v . (1.8)

Now, for the reversible process of phase change of a unit mass at constant tempera-ture, one has q = T(s s) = (h h), where q is the heat needed for the process.Combining this with Eq. (1.8), the well-known Clapeyrons relations are obtained:

evaporation:

dPdT

=(

dPdT

)

sat= hfg

Tsat(vg vf) , (1.9)

sublimation:(

dPdT

)

sublim= hsg

Tsublim(vg vs) , (1.10)

melting:(

dPdT

)

melt= hsf

Tmelt(vf vs) . (1.11)

1.1.2 Metastable States

The surface in Fig. 1.1 defines the stable equilibrium conditions for a pure substance.Experience shows, however, that it is possible for a pure and unagitated substance toremain at equilibrium in superheated liquid (TL > Tsat) or subcooled (supercooled)vapor (TG < Tsat) states. Very slight deviations from the stable equilibrium diagramsare in fact common during some phase-change processes. Any significant deviationfrom the equilibrium states renders the system highly unstable and can lead to rapidand violent phase change in response to a minor agitation.

In the absence of agitation or impurity, spontaneous phase change in a metastablefluid (homogeneous nucleation) must occur because of the random molecular fluctu-ations. Statistical thermodynamics predicts that in a superheated liquid, for example,pockets of vapor covering a range of sizes are generated continuously while surfacetension attempts to bring about their collapse. The probability of the formation of



AT = constant line

P

B

CD

EF

spinodal linesG

v

Figure 1.5. Metastable states and thespinodal lines.

vapor embryos increases with increasing temperature and decreases with increasingembryo size. Spontaneous phase change (homogeneous boiling) will occur only whenvapor microbubbles that are large enough to resist surface tension and would becomeenergetically more stable upon growth are generated at sufficiently high rates.

One can also argue based on classical thermodynamics that a metastable state isin principle only possible as long as (Lienhard and Karimi, 1981)

(v

P

)

T 0. (1.12)

This condition implies that fluctuations in pressure are not followed by positivefeedback, where a slight increase in pressure would cause volumetric expansion ofthe fluid, itself causing a further increase in pressure. When the constant-T lines onthe Pv diagram are modified to permit unstable states, a figure similar to Fig. 1.5results. The spinodal lines represent the loci of points where Eq. (1.12) with equal signis satisfied. Lines AB and FG are constant-temperature lines for stable equilibriumstates. Line BC represents metastable, superheated liquid. Metastable subcooledvapor occurs on line EF, and line CDE represents impossible (unstable) states.

Using the spinodal line as a criterion for nucleation does not appear to agreewell with experimental data for homogeneous boiling. For pure water, the requiredliquid temperature for spontaneous boiling can be found from the following empiricalcorrelation (Lienhard, 1976):

TLTcr

= 0.905 + 0.095(

TsatTcr

)8, (1.13)

where

Tcr = critical temperature of water (647.15 K),TL = local liquid temperature (K),Tsat = Tsat(P) (K), andP = pressure of water.

EXAMPLE 1.1. Calculate the superheat needed for spontaneous boiling in pure atmo-spheric water.


1.2 Transport Equations and Closure Relations 7

Table 1.1. Summary of parameters for Eq. (1.14)

Conservation/transport law J

Mass 1 0 0Momentum U g P I Energy u + U22 g U + q/ q + (P I ) UThermal energy in terms of

enthalpyh 1

(qv + DPDt + : U

)q

Thermal energy in terms ofinternal energy

u 1

(qv + P U + : U) q

Species l, mass flux based mlrl

j lSpecies l, molar flux based Xl

RlC

J lIn Eq. (1.14), must be replaced with C.

SOLUTION. We have P = 1.013 bar; therefore Tsat = 373.15 K. The solution ofEq. (1.13) then leads to TL = 586.4 K. The superheat needed is thus TL Tsat =213.3 K.

Example 1.1 shows that extremely large superheats are needed for homogeneousnucleation to occur in pure and unagitated water. The same is true for other commonliquids. Much lower superheats are typically needed in practice, owing to heteroge-neous nucleation. Subcooled (supercooled) vapors in particular undergo fast nucle-ation (fogging) with a supersaturation (defined as Psat(TG)PPsat(TG) ) of 1% or so (Friedlander,2000).

1.2 Transport Equations and Closure Relations

The local instantaneous conservation equations for a fluid can be presented in thefollowing shorthand form (Delhaye, 1969):

t+ ( U) = J + , (1.14)

where is the fluid density, U is the local instantaneous velocity, represents thetransported property, is the source term for , and J** is the flux of . Table 1.1summarizes the definitions of these parameters for various conservation laws. Allthese parameters represent the mass-averaged mixture properties when the fluid ismulticomponent. Other parameters used in the table are defined as follows:

g = acceleration due to all external body forces,qv = volumetric heat generation rate,rl = mass generation rate of species l in unit volume,Rl = mole generation rate of species l in unit volume,q = heat flux,u = specific internal energy,h = specific enthalpy, andml , Xl = mass and mole fractions of species l.



Angular momentum conservation only requires that the tensor (P I ) remainunchanged when it is transposed. The thermal energy equation, represented by eitherthe fourth or fifth row in Table 1.1, is derived simply by first applying U (i.e., the dotproduct of the velocity vector) on both sides of the momentum conservation equation,and then subtracting the resulting equation from the energy conservation equationrepresented by the third row of Table 1.1. The energy conservation represented bythe third row and the thermal energy equation are thus not independent from oneanother.

The equation set that is obtained by substituting from Table 1.1 into Eq. (1.14)of course contains too many unknowns and is not solvable without closure relations.The closure relations for single-phase fluids are either constitutive relations, meaningthat they deal with constitutive laws such as the equation of state and thermophysicalproperties, or transfer relations, meaning that they represent some transfer rate law.The most obvious constitutive relations are, for a pure substance,

= (u, P) (1.15)or

= (h, P). (1.16)For a multicomponent mixture these equations should be recast as

= (u, P, m1, m2, . . . , mn1) (1.17)or

= (h, P, m1, m2, . . . , mn1), (1.18)where n is the total number of species. For a single-phase fluid, the constitutiverelations providing for fluid temperature can be

T = T(u, P) (1.19)or

T = T(h, P); (1.20)For a multicomponent mixture,

T = T(u, P, m1, m2, . . . , mn1) (1.21)or

T = T(h, P, m1, m2, . . . , mn1). (1.22)In Eqs. (1.17) through (1.22), the mass fractions m1, m2, . . . , mn1 can be replacedwith mole fractions X1, X2, . . . , Xn1.

Let us assume that the fluid is Newtonian, and it obeys Fouriers law for heatdiffusion and Ficks law for the diffusion of mass. The transfer relations for the fluidwill then be

= i j ei e j ; i j = (

uixj

+ u jxi

) 2

3i j U, (1.23)


1.2 Transport Equations and Closure Relations 9

q = kT +

l

j lhl , (1.24)

j l = Dlmml , (1.25)J l = CDlm Xl , (1.26)

where ei and e j are unit vectors for i and j coordinates, respectively, and Dlm repre-sents the mass diffusivity of species l with respect to the mixture.

Mass-flux- and molar-flux-based diffusion will be briefly discussed in the nextsection. The second term on the right side of Eq. (1.24) accounts for energy transportfrom the diffusion of all of the species in the mixture. For a binary mixture, one canuse subscripts 1 and 2 for the two species, and the mass diffusivity will be D12. Thediffusive mass transfer is typically a slow process in comparison with the diffusionof heat, and certainly in comparison with even relatively slow convective transportrates. As a result, in most nonreacting flows the second term on the right side of Eq.(1.24) is negligibly small.

The last two rows of Table 1.1 are equivalent and represent the transport ofspecies l. The difference between them is that the sixth row is in terms of mass fluxand its rate equation is Eq. (1.25), whereas the last row is in terms of molar fluxand its rate equation is Eq. (1.26). A brief discussion of the relationships amongmass-faction-based and mole-fraction-based parameters will be given in the nextsection. A detailed and precise discussion can be found in Mills (2001). The choicebetween the two formulations is primarily a matter of convenience. The precise def-inition of the average mixture velocity in the mass-flux-based formulation is consis-tent with the way the mixture momentum conservation is formulated, however. Themass-flux-based formulation is therefore more convenient for problems where themomentum conservation equation is also solved. However, when constant-pressureor constant-temperature processes are dealt with, the molar-flux-based formulationis more convenient.

In this formulation, and everywhere in this book, we consider only one typeof mass diffusion, namely the ordinary diffusion that is caused by a concentrationgradient. We do this because in problems of interest to us concentration gradient-induced diffusion overwhelms other types of diffusion. Strictly speaking, however,diffusion of a species in a mixture can be caused by the cumulative effects of at leastfour different mechanisms, whereby (Bird et al., 2002)

j l = j l,d + j l,P + j l,g + j l,T. (1.27)The first term on the right side is the concentration gradient-induced diffusion

flux, the second term is caused by the pressure gradient in the flow field, the thirdterm is caused by the external body forces that may act unequally on various chemicalspecies, and the last term represents the diffusion caused by a temperature gradient,also called the Soret effect. A useful discussion of these diffusion terms and their ratelaws can be found in Bird et al. (2002).

The conservation equations for a Newtonian fluid, after implementing these trans-fer rate laws in them, can be written as follows:

Mass conservation:

t+ ( U) = 0 (1.28)



or

DDt

+ U = 0. (1.29)Momentum conservation, when the fluid is incompressible and viscosity is

constant:

D UDt

= D(U)

Dt= P + g + 2 U. (1.30)

Thermal energy conservation equation for a pure substance, in terms of specificinternal energy:

DuDt

= kT P U + . (1.31)Thermal energy conservation equation for a pure substance, in terms of specific

enthalpy:

DhDt

= (kT) + DPDt

+ , (1.32)

where the parameter is the dissipation function (and where representsthe viscous dissipation per unit volume). For a multicomponent mixture, theenergy transport caused by diffusion is sometimes significant and needs to beaccounted for in the mixture energy conservation. In terms of specific enthalpy,the thermal energy equation can be written as

DhDt

= kT + DPDt

+ n

l=1j lhl . (1.33)

Chemical species mass conservation, in terms of partial density and mass flux:

l

t+ (l U) = (D12ml) + rl . (1.34)

Chemical species mass conservation in terms of mass fraction and mass flux:

[mlt

+ (ml U)]

= (D12ml) + rl . (1.35)Chemical species mass conservation, in terms of concentration and molar flux:

Clt

+ (Cl U) = (CD12 Xl) + Rl . (1.36)Chemical species mass conservation, in terms of mole fraction and molar flux:

C[ Xlt

+ U Xl]

= (CD12 Xl) + Rl Xln

j=1Rj . (1.37)

1.3 Single-Phase Multicomponent Mixtures

By mixture in this chapter we mean a mixture of two or more chemical species in thesame phase. Ordinary dry air, for example, is a mixture of O2, N2, and several noblegases in small concentrations. Water vapor and CO2 are also present in air most ofthe time.


1.3 Single-Phase Multicomponent Mixtures 11

The partial density of species l, l , is simply the in situ mass of that species in a unitmixture volume. The mixture density is related to the partial densities accordingto

=n

l=1l , (1.38)

with the summation here and elsewhere performed on all the chemical species in themixture. The mass fraction of species l is defined as

ml = l

. (1.39)

The molar concentration of chemical species l, Cl , is defined as the number of molesof that species in a unit mixture volume. The forthcoming definitions for the mixturemolar concentration and the mole fraction of species l will then follow:

C =n

l=1Cl (1.40)

and

Xl = ClC . (1.41)

Clearly,n

l=1ml =

nl=1

Xl = 1. (1.42)

The following relations among mass-fraction-based and mole-fraction-basedparameters can be easily shown:

l = MlCl , (1.43)

ml = Xl Mlnj=1 Xj Mj

= Xl MlM

, (1.44)

Xl = ml/Mlnj=1

mjMj

= ml MMl

, (1.45)

where M and Ml represent the mixture and chemical specific l molar masses, respec-tively, with M defined according to

M =n

j=1Xj Mj (1.46a)

or

1M

=n

j=1

mjMj

. (1.46b)

When one component, say component j, constitutes the bulk of a mixture, then

M Mj (1.47)



and

ml XlMl M j . (1.48)

In a gas mixture, Daltons law requires that

P =n

l=1Pl , (1.49)

where Pl is the partial pressure of species l. In a gas mixture the components of themixture are at thermal equilibrium (the same temperature) at any location and anytime and conform to the forthcoming constitutive relation:

l = l (Pl , T) , (1.50)where T is the mixture temperature and Pl is the partial pressure of species l. Someor all of the components may be assumed ideal gases, in which case for the ideal gascomponent j, one has

j = PRuMj

T, (1.51)

where Ru is the universal gas constant. When all the components of a gas mixtureare ideal gases, then

Xl = Pl/P. (1.52)

EXAMPLE 1.2. The atmosphere of a laboratory during an experiment is at T = 25Cand P = 1.013 bar. Measurement shows that the relative humidity in the lab is 77%.Calculate the air and water partial densities, mass fractions, and mole fractions.

SOLUTION. Let us start from the definition of relative humidity, :

= Pv/Psat(T).Thus,

Pv = (0.77) (3.14 kPa) = 2.42 kPa.The partial density of air can be calculated by assuming air is an ideal gas at 25Cand pressure of Pa = P Pv = 98.91 kPa to be a = 1.156 kg/m3.

The water vapor is at 25C and 2.42 kPa and is therefore superheated. Its densitycan be found from steam property tables to be v = 0.0176 kg/m3. Using Eqs. (1.38)and (1.39), one gets mv = 0.015. Equation (1.45) gives Xv = 0.0183.

EXAMPLE 1.3. A sample of pure water is brought into equilibrium with a large mixtureof O2 and N2 gases at 1 bar pressure and 300 K temperature. The volume fractions ofO2 and N2 in the gas mixture before it was brought into contact with the water samplewere 22% and 78%, respectively. Solubility data indicate that the mole fractions ofO2 and N2 in water for the given conditions are approximately 5.58 106 and9.9 106, respectively. Find the mass fractions of O2 and N2 in both liquid and gas


1.3 Single-Phase Multicomponent Mixtures 13

phases. Also, calculate the molar concentrations of all the involved species in theliquid phase.

SOLUTION. Before the O2 + N2 mixture is brought in contact with water, we havePO2,initial/Ptot = XO2,G,initial = 0.22,PN2,initial/Ptot = XN2,G,initial = 0.78,

where Ptot = 1 bar. The gas phase after it reaches equilibrium with water will be amixture of O2, N2, and water vapor. Since the original gas mixture volume was large,and given that the solubilities of oxygen and nitrogen in water are very low, we canwrite for the equilibrium conditions

PO2,final/(Ptot Pv) = XO2,G,initial = 0.22, (a-1)PN2,final/(Ptot Pv) = XN2,G,initial = 0.78. (a-2)

Now, under equilibrium, we have

XO2,G,final PO2,final/Ptot, (b-1)Xg,N2,G,final PN2,final/Ptot. (b-2)

We have used the approximately equal signs in these equations because it wasassumed that water vapor acts as an ideal gas. The vapor partial pressure will beequal to vapor saturation pressure at 300 K, namely, Pv = 0.0354 bar. Equations(a-1) and (a-2) can then be solved to get PO2,final = 0.2122 bar and PN2,final =0.7524 bar. Equations (b-1) and (b-2) then give XO2,G,final 0.2122 and XN2,G,final 0.7524, and the mole fraction of water vapor will be

XG,v = 1 (XO2,G,final + XN2,G,final) 0.0354.To find the gas-side mass fractions, first apply Eq. (1.46a), and then Eq. (1.44):

MG = 0.2122 32 + 0.7524 28 + 0.0354 18 MG = 28.49,

mO2,G,final =XO2,G,final MO2

MG= (0.2122)(32)

28.49 0.238,

mN2,G,final =(0.7524)(28)

28.49. 0.739.

For the liquid side, first get ML, the mixture molecular mass number fromEq. (1.46a):

ML = 5.58 106 32 + 9.9 106 28+ [1 (5.58 106 + 9.9 106)] 18 18.

Therefore, from Eq. (1.44),

mO2,L,final =5.58 106

18(32) = 9.92 106,

mN2,L,final =9.9 106

18(28) = 15.4 106.



To calculate the concentrations, we note that the liquid side is now made up ofthree species, all with unknown concentrations. Equation (1.41) should be written outfor every species, while Eq. (1.40) is also satisfied. These give four equations in termsof the four unknowns CL, CO2,L,final, CN2,L,final, and CL,W, where CL and CL,W standfor the total molar concentrations of the liquid mixture and the molar concentrationof water substance, respectively. This calculation, however, will clearly show that,owing to the very small mole fractions (and hence small concentrations) of O2 andN2,

CL CL,W = L/ML = 996.6 kg/m3

18 kg/kmol= 55.36 kmol/m3.

The concentrations of O2 and N2 could therefore be found from Eq. (1.41) to be

CO2,L,final 3.09 104 kmol/m3,CN2,L,final 5.48 104 kmol/m3.

The extensive thermodynamic properties of a single-phase mixture, when repre-sented as per unit mass (in which case they actually become intensive properties) canall be calculated from

= 1

nl=1

ll =n

l=1mll , (1.53)

with

l = l (Pl , T) , (1.54)where can be any mixture property such as , u, h, or s, and l is the same propertyfor pure substance l. Similarly, the following expression can be used when specificproperties are defined per unit mole

= 1C

nl=1

Cl l =n

l=1Xl l . (1.55)

With respect to diffusion, Ficks law for a binary mixture can be formulated asfollows. First, consider the mass-flux-based formulation. The total mass flux of speciesl is

ml = l U + j l = ml( U) + j l , (1.56)where Ficks law for the diffusive mass flux is represented by Eq. (1.25), and themixture velocity is defined as

U =I

i=1ml Ul = G/. (1.57)

Consider now the molar-flux-based formulation. The total molar flux of speciesl can be written as

Nl = Cl U + J l . (1.58)


1.4 Phase Diagrams for Binary Systems 15

Ficks law is represented by Eq. (1.26), and the molar-average mixture velocity isdefined as

U =I

l=1Xl Ul = G/(CM). (1.59)

1.4 Phase Diagrams for Binary Systems

The phase diagrams discussed in Section 1.1 dealt with systems containing a singlechemical species. In some applications, however, we deal with phase-change phenom-ena of mixtures of two or more chemical species. Examples include air liquefactionand separation and refrigerant mixtures such as waterammonia and R-410A.

For a nonreacting PvT system composed of n chemical species, Gibbs phaserule states that

F = 2 + n , (1.60)where is the number of phases and F is the number of degrees of freedom. Fora single-phase binary mixture, n = 2, = 1, and therefore F = 3, meaning that thenumber of independent and intensive thermodynamic properties needed for specify-ing the state of the system is three. All the equilibrium states of the system can then berepresented in a three-dimensional coordinate system with P, T, and composition.We can use the mole fraction of one of the species (e.g., X1) to specify the composition,in which case the (P, T, X1) will be the coordinate system. When two phases are con-sidered in the binary system (say, liquid and vapor), then n = 2, = 2, and thereforeF = 2. The number of independent and intensive thermodynamic properties neededfor specifying the state of the system will then be two, meaning that only two of thethree coordinates in the (P, T, X1) space can be independent. The two-phase equi-librium state will then form a two-dimensional surface in the (P, T, X1) space. Whenall three phases at equilibrium are considered, F = 1, and the equilibrium states willbe represented by a space curve.

Let us now focus on the equilibrium vaporliquid system. We are interested inthe two-dimensional surface in the (P, T, X1) space representing this equilibrium.Rather than working with the three-dimensional space, it is easier to work withthe projection of the two-dimensional surface on (P, X1) or (T, X1) planes, and thisleads to the PX and TX diagrams, displayed qualitatively in Figs. 1.6 and 1.7,respectively, for a zeotropic (also referred to as nonazeotropic) mixture. A binarymixture is called zeotropic when the concentration makeup of the liquid and vaporphases are never equal. A mixture of water and ammonia is a good example of azeotropic binary system.

The behavior of a zeotropic binary system during evaporation can be better under-stood by following what happens to a mixture that is initially at state Z (subcooledliquid) that is heated at constant pressure. The process is displayed in Fig. 1.6. Themixture remains at the original concentration as long as it is in the subcooled liq-uid state, until it reaches the state B1. With further heating of the mixture, the liq-uid and vapor phases will have different concentrations. The concentration of theliquid phase moves along the B1 B2 curve, whereas the concentration of the vaporphase follows the D1 D2 curve. When evaporation is complete, the liquid will havethe state corresponding to point B2, and the vapor phase will correspond to point



Superheated Vapor

A

B

B2

B1

X1, E

Subcooled Liquid

Tem

pera

ture

, T

Saturated Vapor (dew point line)

Saturated liquid (bubble point line)

D2

D1

D

E

Z

Z

C

0.0 1.0

Mole Fraction of Species 1, X1

Figure 1.6. Constant-pressure phase diagram for a zeotropic (nonazeotropic) binary mixture.

D2. The line ABC is often referred to as the bubble point line, and the line ADC iscalled the dew point line. For refrigerants, the difference between the dew and bubbletemperatures is called the temperature glide.

In Fig. 1.7, a process is displayed where an initially subcooled mixture with con-ditions corresponding to the point z is slowly depressurized while its temperature ismaintained constant. Here as well, the concentration remains unchanged until point

Pre

ssur

e, P

a

0.0

b2

b1

d2

d1

db

z

z

c

eSuperheated Vapor

Saturated Vapor (dew point line)

Saturated Liquid(bubble point line)

Subcooled Liquid


1.0

Figure 1.7. Constant-temperature phase diagram for a zeotropic (nonazeotropic) binarymixture.


1.5 Thermodynamic Properties of Vapor-Noncondensable Gas Mixtures 17

0.0 1.0


Azeotrope

Tem

pera

ture

, T

Figure 1.8. Constant-pressure phase diagramfor a binary mixture that forms a singleazeotrope.

b1 is reached. With further depressurization the liquid phase will move on the b1b2curve, while the vapor moves along the d1d2 curve. Complete evaporation of themixture ends at point d2, where the mole fraction of species 1 will remain constantwith further depressurization.

For cooling and condensation of a binary system, the processes are similar tothose displayed in Figs. 1.6 and 1.7, only in reverse. The straight lines such as BD inFig. 1.6 and bd in Fig. 1.7 are referred to as tie lines. Tie lines have a useful geometricinterpretation. It can be proved that

NfNg

= Z D

Z B= z

dzb

, (1.61)

where Nf and Ng are the total numbers of liquid and vapor moles in the mixture.An azeotrope is a point at which the concentrations of the liquid and the vapor

phases are identical. Some binary mixtures form one or more azeotropes at interme-diate concentrations. A single azeotrope is more common and leads to TX and PXdiagrams similar to Figs. 1.8 and 1.9. A mixture that is at an azeotrope behaves like asaturated single-component species and has no temperature glide. Azeotropic mix-tures suitable for use as refrigerants are uncommon, however, because it is difficultto find one that satisfies other necessary properties for application as a refrigerant.

A mixture is called near azeotropic if during evaporation or condensation theliquid and vapor concentrations differ only slightly. In other words, the temperatureglide during phase-change processes is very small for near-azeotropic mixtures. Agood example is the refrigerant R-410A, which is a fiftyfifty percent mass mixtureof refrigerants R-32 and R-125, and its temperature glide for standard compressorpressure and temperatures is less than about 0.1C.

1.5 Thermodynamic Properties of Vapor-Noncondensable Gas Mixtures

Vapor-noncondensable mixtures are often encountered in evaporation and con-densation systems. Properties of vapor-noncondensable mixtures are discussedin this section by treating the noncondensable as a single species. Although the



Azeotrope

Pre

ssur

e, P

0.0 1.0Mole Fraction of Species 1, X1

Figure 1.9. Constant-temperature phase dia-gram for a binary mixture that forms a singleazeotrope.

noncondensable may be composed of a number of different gaseous constituents,average properties can be defined such that the noncondensables can be treated asa single species, as is commonly done for air. Subscripts v and n in the followingdiscussion will represent the vapor and noncondensable species, respectively.

Airwater vapor mixture properties are discussed in standard thermodynamictextbooks. For a mixture with pressure PG, temperature TG, and vapor mass fractionmv, the relative humidity and humidity ratio are defined as

= PvPsat(TG)

XvXv,sat

(1.62)

and

= mvmn

= mv1 mv , (1.63)

where Xv,sat is the vapor mole fraction when the mixture is saturated. In the last partof Eq. (1.62) it is evidently assumed that the noncondensable as well as the vapor areideal gases. A mixture is saturated when Pv = Psat(TG). When < 1, the vapor is in asuperheated state, because Pv < Psat(TG). In this case the thermodynamic propertiesand their derivatives follow the gas mixture rules.

EXAMPLE 1.4. Find (hG/ PG)TG,mv for a binary vapor-noncondensable mixtureassuming that the mixture does not reach saturation.

SOLUTION. The mixture specific enthalpy is defined according to Eq. (1.53):

hG = mvhv + (1 mv)hn.From Eq. (1.60), the number of degrees of freedom for the system is three; thereforethe three properties PG, TG, and mv uniquely specify the state of the mixture. WithTG and mv kept constant, one can write(

hG PG

)

TG,mv

= mv{(

hv Pv

)( Pv PG

)}

TG

+ (1 mv){(

hn Pn

)( Pn PG

)}

TG

. (a)


1.5 Thermodynamic Properties of Vapor-Noncondensable Gas Mixtures 19

Since Pv PG Xv and Pn PG(1 Xv), and using the relation between mv and Xv,one then has (

Pv PG

)

mv

= Xv = mv/MvmvMv

+ 1mvMn,

( Pn PG

)

mv

= (1 Xv) = (1 mv)/MnmvMv

+ 1mvMn.

The specific enthalpy of an ideal gas is a function of temperature only. The non-condensable is assumed to be an ideal gas, therefore the second term on the rightof Eq. (a) will be zero. The term (hv/ Pv)TG on the right side of Eq. (a) can becalculated using vapor property tables.

The vapor-noncondensable mixtures encountered in evaporators and condensersare often saturated. For saturated mixtures, the following must be added to the othermixture rules:

TG = Tsat(Pv), (1.64)v = g(TG) = g(Pv), (1.65)hv = hg(TG) = hg(Pv). (1.66)

Using the identity mv = vn+v , and assuming that the noncondensable is an idealgas, one can show that

PG PvRuMn

Tsat(Pv)(1 mn) g(Pv)mn = 0. (1.67)

Equation (1.67) indicates that PG, TG and mv are not independent. This is of courseexpected, because now the mixture has only two degrees of freedom. By knowingtwo parameters (e.g., TG and mv), Eq (1.67) can be iteratively solved for the thirdunknown parameter (e.g., the vapor partial pressure when TG and mv are known).The variations of the mixture temperature and the vapor pressure are related by theClapeyrom relation, Eq. (1.9):

TG Pv

= Tsat(Pv) Pv

= TGvfghfg

. (1.68)

EXAMPLE 1.5. For a saturated vapor-noncondensable binary mixture, derive expres-sions of the forms (

G

PG

)

Xn

= f (PG, Xn)

and (G

Xn

)

PG

= f (PG, Xn)

SOLUTION. Let us approximately write

G = PGRuM TG

= MPGRuTsat(Pv)

,



where M = Xn Mn + (1 Xn)Mv, TG = Tsat(Pv), and Pv = (1 Xn)PG. The argumentof Tsat(Pv) is meant to remind us that Tsat corresponds to Pv = PG(1 Xn). Then(

G

PG

)

Xn

= MRuTsat

PG MRuT2sat

(Tsat PG

)

Also, using the Clapeyron relation

Tsat PG

= Tsat Pv

Pv PG

= vfgTsathfg

(1 Xn)

gives the result(

G

PG

)

Xn

= MRuTsat

Pvvfg MRuTGhfg

It can also be proved that(

G

Xn

)

PG

= PGRuTG

(Mn Mv) +P2Gvfg MRuTGhfg

, (1.69)

where vfg and hfg correspond to Tsat = TG.

EXAMPLE 1.6. For a saturated vapor-noncondensable mixture, derive an expressionof the form (

hGmn

)

PG

= f (PG, mn).

SOLUTION. Let us start with

hG = (1 mn)hg + mnhn (a)where hg is the saturated vapor enthalpy at Pv = Xv PG, with Xv = (mv M)/Mv, andwith M defined as in Eq. (1.46). Treating the noncondensable gas as ideal, one canwrite

hn = hn,ref +TG

Tref

Cp,ndT

where subscript ref represents a reference temperature for the noncondensableenthalpy. Noting that hg = hg(Pv) and Pv = PG(1 Xn), we have(

hGmn

)

PG

= hg + (1 mn) hg Pv

Pv Xn

Xnmn

+ hn + mn hnTG

TGmn

. (b)

By manipulation of this equation, one can derive(

hGmn

)

PG

= hg PG(1 mn) Xnmn

(hg Pv

)+ hn mnCP,n

(TGvfg

hfg

)PG

( Xnmn

),

(c)

where, again, vfg and hfg correspond to Tsat = TG. If, for simplicity, it is assumedthat CP,n = const. (a good assumption when temperature variations in the problemof interest are relatively small), then hn hn,ref = CP,n(TG Tref). The problem is


1.6 Transport Properties 21

solved by substituting from Eq. (1.45) for Xnmn

. Note that Clapeyrons relation hasbeen used for the derivation of the last term on the right side of this expression.

1.6 Transport Properties

1.6.1 Mixture Rules

The viscosity and thermal conductivity of a gas mixture can be calculated from thefollowing expressions (Wilke, 1950):

=n

j=1

Xj jni=1 Xi j i

, (1.70)

k =n

j=1

Xj kjni=1 Xi j i

, (1.71)

j i =[1 + ( j/i )1/2(Mi/Mj )1/4

]2

8[1 + (Mj/Mi )]1/2. (1.72)

These rules have been deduced from gas kinetic theory and have proven to be quiteadequate (Mills, 2001).

For liquid mixtures the property calculation rules are complicated and are notwell established. However, for most dilute solutions of inert gases, which are themain subject of interest in this book, the viscosity and thermal conductivity of theliquid are similar to the properties of pure liquid.

With respect to mass diffusivity, everywhere in this book, unless otherwise stated,we will assume that the mixture is binary; namely, only two different species arepresent. For example, in dealing with an airwater vapor mixture (as it pertains toevaporation and condensation processes in air), we follow the common practice oftreating dry air as a single species. Furthermore, we assume that the liquid onlycontains dissolved species at very low concentrations.

For the thermophysical and transport properties, including mass diffusivity, werely primarily on experimental data. Mass diffusivities of gaseous pairs are approxi-mately independent of their concentrations in normal pressures but are sensitive totemperature. The mass diffusion coefficients are sensitive to both concentration andtemperature in liquids, however.

1.6.2 Gaskinetic Theory

Gaskinetic theory (GKT) provides for the estimation of the thermophysical andtransport properties in gases. These methods become particularly useful when empir-ical data are not available. Simple GKT models the gas molecules as rigid and elasticspheres (hard spheres) that influence one another only by impact (Gombosi, 1994).When two molecules impact, furthermore, their directions of motion after collisionare isotropic, and following a large number of intermolecular collisions the orthogo-nal components of the molecular velocities are independent of each other. It is alsoassumed that the distribution function of molecules under equilibrium is isotropic.



These assumptions, along with the ideal gas law, lead to the well-known MaxwellBoltzmann distribution, whereby the fraction of molecules with speeds in the | U| to| U + d U| range is given by f (U)dU, and

f (U) =(

M2 RuT

) 32

eMU22RuT . (1.73)

If the magnitude (absolute value) of velocity is of interest, the number fraction ofmolecules with speeds in the |U| to |U + dU| range will be equal to F(U)dU, where

F(U) = 4U2 f (U). (1.74)Let us define, for convenience,

= m2BT

= M2RuT

, (1.75)

where m is the mass of a single molecule and B is Boltzmanns constant. (Note thatBm = RuM .) In Cartesian coordinates, we will have for each coordinate i

eU

2i dUi = 1. (1.76)

Various moments of the MaxwellBoltzmann distribution can be found. Forexample, using Eq. (1.74), we get the mean molecular speed by writing

|U| = 4(

)3/2

0

eU2U3dU =

8BTm

. (1.77)

Likewise, the average molecular kinetic energy can be found as

Ekin = 12mU2 = 2

(

)3/2m

0

eU2U4dU = 3

2BT. (1.78)

The average speed of molecules in a particular direction (e.g., in the positive xdirection in a Cartesian coordinate system) can be found by first noting that accord-ing to Eq. (1.73) the number fraction of molecules that have velocities along the xcoordinate in the range Ux and Ux + dUx is

(M

2 RuT

) 32

+

dUy

+

dUze

M(U2x +U2y +U2z )2RuT dUx. (1.79)

The average velocity in the positive direction will then will follow:

Ux+ =(

M2 RuT

) 32

+

dUy

+

dUz

+

0

eM(U2x +U2y +U2z )

2RuT UxdUx. (1.80)

Using Eq. (1.76), one can then easily show that

Ux+ =

0

eU2x+Ux+dUx+ =

BT2m

. (1.81)



For an ideal gas, furthermore, the number density of gas molecules is

n = NAV/M = PBT

, (1.82)

where NAV is Avagadros number. The flux of gas molecules passing, per unit time, inany particular direction (e.g., in the positive x direction in a Cartesian coordinate sys-tem), through a surface element oriented perpendicularly to the direction of interest,will be

jmolec,x+ = nUx+ = P2BmT

=(

M2 Ru

)1/2 Pm

T

. (1.83)

This expression, when multiplied by A, the surface area of a very small openingin the wall of a vessel containing an ideal gas, will provide the rate of moleculesleaking out of the vessel (molecular effusion) and is valid as long as the characteristicdimension of Ais smaller than the mean free path of the gas molecules. This expres-sion is also used in the simplest interpretation of the molecular processes associatedwith evaporation and condensation, as will be seen in Chapter 2.

According to simple GKT, the gas molecules have a mean free path of [seeGombosi (1994) for detailed derivations]:

= 12nA

, (1.84)

where A is the molecular scattering cross section. The molecular mean free timecan then be found from

= |U| =1

2nA|U|. (1.85)

Given that random molecular motions and intermolecular collisions are respon-sible for diffusion in fluids, expressions for , k, and D can be found based on themolecular mean free path and free time. The simplest formulas derived in this wayare based on the MaxwellBoltzmann distribution, which assumes equilibrium. Moreaccurate formulas can be derived by taking into consideration that all diffusion phe-nomena actually occur as a result of nonequilibrium. The transport of the molecu-lar energy distribution under nonequilibrium conditions is described by an integro-differential equation, known as the Boltzmann transport equation. The aforemen-tioned MaxwellBoltzmann distribution [Eq. (1.73) or (1.74)] is in fact the solution ofthe Boltzmann transport equation under equilibrium conditions. Boltzmanns equa-tion cannot be analytically solved in its original form, but approximate solutionsrepresenting relatively slight deviations from equilibrium have been derived, andthese nonequilibrium solutions lead to useful formulas for the gas transport proper-ties. One of the most well known approximate solutions to the Boltzmann equationfor near-equilibrium conditions was derived by Chapman, in 1916, and Enskog, in1917 (Chapman and Cowling, 1970). The solution leads to widely used expressionsfor gas transport properties that are only briefly presented and discussed in the fol-lowing. More detailed discussions about these expressions can be found in Bird et al.(2002), Skelland (1974), and Mills (2001).



0 r

Figure 1.10. The pair potential energy distribution according tothe LennardJones 612 intermolecular potential model.

The interaction between two molecules as they approach one another can bemodeled only when intermolecular forces are known. The force between two identi-cal molecules, F , defined to be positive when repulsive, can be represented in termsof a pair potential energy, , where

F = (r), (1.86)with r being the distance separating the two molecules. Several models have beenproposed for [see Rowley (1994) for a concise review]; the most widely used amongthem is the empirical LennardJones 612 model (Rowley, 1994):

(r) = 4[(

r

)12

(

r

)6]. (1.87)

Figure 1.10 depicts Eq. (1.87). The LennardJones model, like all similar models,accounts for the fact that intermolecular forces are attractive at large distances andbecome repulsive when the molecules are very close to one another. The function(r)in LennardJoness model is fully characterized by two parameters: , the collisiondiameter, and , the energy representing the maximum attraction. Values of and for some selected molecules are listed in Appendix H. The force constants for alarge number of molecules can be found in Svehla (1962). When tabulated valuesare not known, they can be estimated by using empirical correlations based on themolecules properties at its critical point, liquid at normal boiling point, or the solidstate at melting point (Bird et al., 2002). In terms of the substances critical state, forexample,

2.44(Tcr/Pcr)1/3 (1.88)and

/B 0.77Tcr, (1.89)where Tcr and /B are in degrees kelvin Pcr is in atmospheres, and calculated in thisway is in angstroms. The LennardJones model is used quite extensively in moleculardynamic simulations.

According to the ChapmanEnskog model, the gas viscosity can be found from

= 2.669 106

MT 2k

(kg/ms), (1.90)

where T is in kelvins is in angstroms, and k is a collision integral for thermalconductivity or viscosity. (Collision integrals for viscosity and thermal conductivityare equal.) For monatomic gases the ChapmanEnskog model predicts

k = ktrans = 2.5Cv = 154(

RuM

). (1.91)



For a polyatomic gas, the molecules internal degrees of freedom contribute to thegas thermal conductivity, and

k = ktrans + 1.32(

CP 52RuM

). (1.92)

The binary mass diffusivity of species 1 and 2 can be found from

D12 = D21 = 1.858 107

T3

(1

M1+ 1M2

)

212D P(m2/s), (1.93)

where P is in atmospheres, D represents the collision integral for the two moleculesfor mass diffusively, and

12 = 12(1 + 2), (1.94)

12 =

12. (1.95)

Appendix I can be used for the calculation of collision integrals for a number ofselected species (Hirschfelder et al., 1954).

1.6.3 Diffusion in Liquids

The binary diffusivities of solutions of several nondissociated chemical species inwater are given in Appendix G. The diffusion of a dilute species 1 (solute) in a liquid2 (solvent) follows Ficks law with a diffusion coefficient that is approximately equalto the binary diffusivity D12, even when other diffusing species are also present in theliquid, provided that all diffusing species are present in very small concentrations.

Theories dealing with molecular structure and kinetics of liquids are not suffi-ciently advanced to provide for reasonably accurate predictions of liquid transportproperties. A simple method for the estimation of the diffusivity of a dilute solutionis the StokesEinstein expression

D12 = BT32d1 , (1.96)

where subscripts 1 and 2 refer to the solute and solvent, respectively, and d1 is thediameter of a single solute molecule, and can be estimated from d1 , namely, theLennardJones collision diameter. Alternatively, it can be estimated from

d1 (

6

M11 NAv

)1/3. (1.97)

The StokesEinstein expression in fact represents the Brownian motion of spher-ical particles (solute molecules in this case) in a fluid, under the assumption of creepflow without slip around the particles. It is accurate when the spherical particle ismuch larger than intermolecular distances. It is good for estimation of the diffusivitywhen the solute molecule is approximately spherical and is at least five times largerthan the solvent molecule (Cussler, 1997).



Table 1.2. Specific molar volume at boiling pointfor selected substances

SubstanceVb1 103(m3/kmol) Tb(K)

Air 29.9 79Hydrogen 14.3 21Oxygen 25.6 90Nitrogen 31.2 77Ammonia 25.8 240Hydrogen sulfide 32.9 212Carbon monoxide 30.7 82Carbon dioxide 34.0 195Chlorine 48.4 239Hydrochloric acid 30.6 188Benzene 96.5 353Water 18.9 373Acetone 77.5 329Methane 37.7 112Propane 74.5 229Heptane 162 372

Note: After Mills (2001).

A widely used empirical correlation for binary diffusivity of a dilute and nondis-sociating chemical species (species 1) in a liquid (solvent, species 2) is (Wilke andChang, 1954)

D12 = 1.17 1016 (2 M2)1/2T

V0.6b1(m2s), (1.98)

where D12 is in square meters per second; Vb1 is the specific molar volume, in cubicmeters per kilomole, of species 1 as liquid at its normal boiling point; is the mixtureliquid viscosity in kg/ms; T is the temperature in kelvins; and 2 is an associationparameter for the solvent: 2 = 2.26 for water and 1 for unassociated solvents (Mills,2001). Values of Vb1 for several species are given in Table 1.2.

1.7 Turbulent Boundary Layer Velocity and Temperature Profiles

Near-wall hydrodynamic and heat transfer phenomena are crucial to many boilingand condensation processes. Examples include bubble nucleation, growth and releaseduring flow boiling, and flow condensation.

The universal velocity profile in a two-dimensional, incompressible turbulentboundary layer can be represented as (Schlichting, 1968)

a viscous sublayer:

u+ = y+ y+ < 5, (1.99)a buffer sublayer:

u+ = 5 ln y+ 3.05 5 < y+ < 30, (1.100)and an inertial sublayer:

u+ = 1

ln y+ + B 30 < y+ 400, (1.101)


1.7 Turbulent Boundary Layer Velocity and Temperature Profiles 27

where = 0.40, B = 5.5, y is the distance from the wall, u is the velocity parallel tothe wall, and

y+ = yU

, (1.102)

u+ = u/U , (1.103)U =

w/. (1.104)

This universal velocity profile can be utilized for determining the turbulent prop-erties in the boundary layer. For example, according to the definition of the turbulentmixing length, lm, one can write

w = lam + turb = [ + l2m

uy

]

uy

(1.105)

The mixing length is related to the turbulent eddy diffusivity by noting that

w = (E + )dudy . (1.106)

In a turbulent boundary layer near the wall, w = const., and as a resultEq. (1.106) can be manipulated to derive the following two useful relations:

Ev

=(

du+

dy+

)1 1, (1.107)

(1 + l+2m

du+

dy+

)

du+

dy+= 1, (1.108)

where l+m = lmU . Equation (1.108) can be rewritten as(

dy+

du+

)2 dy

+

du+ l+2m = 0. (1.109)

Equations (1.107) and (1.109), along with Eqs. (1.99)(1.101) can evidently beused for calculating the eddy diffusivity distribution in the boundary layer.

Turbulent boundary layers support a near-wall temperature distribution whenheat transfer takes place, which has a peculiar form when it is presented in appropriatedimensionless form. This temperature law of the wall is very useful and has beenapplied in many phenomenological models, as well as to the development of heattransfer correlations. The temperature law of the wall can be derived by noting that ina steady and incompressible two-dimensional boundary layer, when the heat transferboundary condition at the wall is a constant heat flux, one can write

qy qw = Cp(+ EH)Ty

= CP(

Pr+ E

Prturb

)Ty

, (1.110)

where y is the distance from the wall, EH is the eddy diffusivity for heat transfer, qyis the heat flux in the y direction, Prturb is the turbulent Prandtl number (which istypically 1 for common fluids), and T is the local time-averaged fluid temperature.Equation (1.110) can now be manipulated to get

T+ = Tw T(y)qwCpU

=y+

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