CONVECTIVE HEAT AND MASS TRANSFER -...

CONVECTIVE HEAT AND MASS TRANSFER

This book was developed by Professor S. Mostafa Ghiaasiaan during10 years of teaching a graduate-level course on convection heat andmass transfer. The book is ideal for a graduate course dealing with the-ory and practice of convection heat and mass transfer. The book treatswell-established theory and practice on the one hand; on the otherhand, it is enriched by modern areas such as flow in microchannels andcomputational fluid dynamics–based design and analysis methods. Thebook is primarily concerned with convective heat transfer. Essentialsof mass transfer are also covered. The mass transfer material and prob-lems are presented such that they can be easily skipped, should that bepreferred. The book is richly enhanced by exercises and end-of-chapterproblems. Solutions are available for qualified instructors. The bookincludes 17 appendices providing compilations of most essential prop-erties and mathematical information for analysis of convective heatand mass transfer processes.

Professor S. Mostafa Ghiaasiaan has been a member of the WoodruffSchool of Mechanical Engineering at Georgia Institute of Technologysince 1991 after receiving a Ph.D. in Thermal Science from the Univer-sity of California, Los Angeles, in 1983 and working in the aerospaceand nuclear power industry for eight years. His industrial researchand development activity was on modeling and simulation of transportprocesses, multiphase flow, and nuclear reactor thermal hydraulicsand safety. His current research areas include nuclear reactor thermalhydraulics, particle transport, cryogenics and cryocoolers, and multi-phase flow and change-of-phase heat transfer in microchannels. Hehas more than 150 academic publications, including 90 journal arti-cles, on transport phenomena and multiphase flow. Among the hon-ors he has received for his publications are the Chemical EngineeringScience’s Most Cited Paper for 2003–2006 Award, the National HeatTransfer Conference Best Paper Award (1999), and the Science Appli-cations International Corporation Best Paper Award (1990 and 1988).He has been a member of American Society of Mechanical Engineers(ASME) and the American Nuclear Society for more than 20 yearsand was elected an ASME Fellow in 2004. Currently he is the Exec-utive Editor of Annals of Nuclear Energy for Asia, Africa, and Aus-tralia. This is his second book with Cambridge University Press—thefirst was Two-Phase Flow, Boiling, and Condensation, In Conventionaland Miniature Systems (2007).

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Convective Heat and Mass Transfer

S. Mostafa GhiaasiaanGeorgia Institute of Technology






cambridge university pressCambridge, New York, Melbourne, Madrid, Cape Town,Singapore, Sao Paulo, Delhi, Tokyo, Mexico City

Cambridge University Press32 Avenue of the Americas, New York, NY 10013-2473, USA

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c© S. Mostafa Ghiaasiaan 2011

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.

First published 2011

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication data

Ghiaasiaan, Seyed Mostafa, 1953–Convective heat and mass transfer / Mostafa Ghiaasiaan.

p. cm.Includes bibliographical references and index.ISBN 978-1-107-00350-7 (hardback)1. Heat – Convection. I. Title.QC327.G48 2011536′.25 – dc22 2011001977

ISBN 978-1-107-00350-7 Hardback

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To my wife Pari Fatemeh Shafiei, and my son Saam






CONTENTS

Preface page xv

Frequently Used Notation xvii

1 Thermophysical and Transport Fundamentals . . . . . . . . . . . . . . . . . . . 1

1.1 Conservation Principles 11.1.1 Lagrangian and Eulerian Frames 11.1.2 Mass Conservation 21.1.3 Conservation of Momentum 31.1.4 Conservation of Energy 6

1.2 Multicomponent Mixtures 111.2.1 Basic Definitions and Relations 111.2.2 Thermodynamic Properties 15

1.3 Fundamentals of Diffusive Mass Transfer 171.3.1 Species Mass Conservation 171.3.2 Diffusive Mass Flux and Fick’s Law 181.3.3 Species Mass Conservation When Fick’s Law Applies 191.3.4 Other Types of Diffusion 201.3.5 Diffusion in Multicomponent Mixtures 20

1.4 Boundary and Interfacial Conditions 221.4.1 General Discussion 221.4.2 Gas–Liquid Interphase 241.4.3 Interfacial Temperature 241.4.4 Sparingly Soluble Gases 271.4.5 Convention for Thermal and Mass Transfer Boundary

Conditions 301.5 Transport Properties 31

1.5.1 Mixture Rules 311.5.2 Transport Properties of Gases and the Gas-Kinetic Theory 321.5.3 Diffusion of Mass in Liquids 37

1.6 The Continuum Flow Regime and Size Convention for FlowPassages 38

Problems 39

vii






viii Contents

2 Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.1 Boundary Layer on a Flat Plate 442.2 Laminar Boundary-Layer Conservation Equations 482.3 Laminar Boundary-Layer Thicknesses 512.4 Boundary-Layer Separation 532.5 Nondimensionalization of Conservation Equations and

Similitude 54Problems 58

3 External Laminar Flow: Similarity Solutions for Forced LaminarBoundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1 Hydrodynamics of Flow Parallel to a Flat Plate 613.2 Heat and Mass Transfer During Low-Velocity Laminar Flow

Parallel to a Flat Plate 653.3 Heat Transfer During Laminar Parallel Flow Over a Flat Plate

With Viscous Dissipation 713.4 Hydrodynamics of Laminar Flow Past a Wedge 733.5 Heat Transfer During Laminar Flow Past a Wedge 783.6 Effects of Compressibility and Property Variations 80Problems 85

4 Internal Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.1 Couette and Poiseuille Flows 904.2 The Development of Velocity, Temperature, and Concentration

Profiles 944.2.1 The Development of Boundary Layers 944.2.2 Hydrodynamic Parameters of Developing Flow 974.2.3 The Development of Temperature and Concentration

Profiles 1004.3 Hydrodynamics of Fully Developed Flow 1034.4 Fully Developed Hydrodynamics and Developed Temperature or

Concentration Distributions 1074.4.1 Circular Tube 1074.4.2 Flat Channel 1104.4.3 Rectangular Channel 1134.4.4 Triangular Channel 1134.4.5 Concentric Annular Duct 114

4.5 Fully Developed Hydrodynamics, Thermal or ConcentrationEntrance Regions 1174.5.1 Circular Duct With Uniform Wall Temperature Boundary

Conditions 1174.5.2 Circular Duct With Arbitrary Wall Temperature

Distribution in the Axial Direction 1244.5.3 Circular Duct With Uniform Wall Heat Flux 1264.5.4 Circular Duct With Arbitrary Wall Heat Flux Distribution

in the Axial Coordinate 129






Contents ix

4.5.5 Flat Channel With Uniform Heat Flux BoundaryConditions 130

4.5.6 Flat Channel With Uniform Wall Temperature BoundaryConditions 132

4.5.7 Rectangular Channel 1354.6 Combined Entrance Region 1354.7 Effect of Fluid Property Variations 137Appendix 4A: The Sturm–Liouville Boundary-Value Problems 141Problems 141

5 Integral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.1 Integral Momentum Equations 1515.2 Solutions to the Integral Momentum Equation 153

5.2.1 Laminar Flow of an Incompressible Fluid Parallel to a FlatPlate Without Wall Injection 153

5.2.2 Turbulent Flow of an Incompressible Fluid Parallel to aFlat Plate Without Wall Injection 156

5.2.3 Turbulent Flow of an Incompressible Fluid Over a Body ofRevolution 158

5.3 Energy Integral Equation 1595.4 Solutions to the Energy Integral Equation 161

5.4.1 Parallel Flow Past a Flat Surface 1615.4.2 Parallel Flow Past a Flat Surface With an Adiabatic

Segment 1635.4.3 Parallel Flow Past a Flat Surface With Arbitrary Wall

Surface Temperature or Heat Flux 1655.5 Approximate Solutions for Flow Over Axisymmetric Bodies 167Problems 173

6 Fundamentals of Turbulence and External Turbulent Flow . . . . . . . . 177

6.1 Laminar–Turbulent Transition and the Phenomenology ofTurbulence 177

6.2 Fluctuations and Time (Ensemble) Averaging 1806.3 Reynolds Averaging of Conservation Equations 1816.4 Eddy Viscosity and Eddy Diffusivity 1836.5 Universal Velocity Profiles 1856.6 The Mixing-Length Hypothesis and Eddy Diffusivity Models 1886.7 Temperature and Concentration Laws of the Wall 1926.8 Kolmogorov Theory of the Small Turbulence Scales 1966.9 Flow Past Blunt Bodies 200Problems 205

7 Internal Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7.1 General Remarks 2087.2 Hydrodynamics of Turbulent Duct Flow 211

7.2.1 Circular Duct 2117.2.2 Noncircular Ducts 217






x Contents

7.3 Heat Transfer: Fully Developed Flow 2187.3.1 Universal Temperature Profile in a Circular Duct 2187.3.2 Application of Eddy Diffusivity Models for Circular Ducts 2217.3.3 Noncircular Ducts 224

7.4 Heat Transfer: Fully Developed Hydrodynamics, ThermalEntrance Region 2247.4.1 Circular Duct With Uniform Wall Temperature or

Concentration 2247.4.2 Circular Duct With Uniform Wall Heat Flux 2267.4.3 Some Useful Correlations for Circular Ducts 2297.4.4 Noncircular Ducts 231

7.5 Combined Entrance Region 231Problems 238

8 Effect of Transpiration on Friction, Heat, and Mass Transfer . . . . . . . 243

8.1 Couette Flow Film Model 2438.2 Gas–Liquid Interphase 248Problems 256

9 Analogy Among Momentum, Heat, and Mass Transfer . . . . . . . . . . . 258

9.1 General Remarks 2589.2 Reynolds Analogy 2599.3 Prandtl–Taylor Analogy 2619.4 Von Karman Analogy 2639.5 The Martinelli Analogy 2659.6 The Analogy of Yu et al. 2659.7 Chilton–Colburn Analogy 267Problems 272

10 Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

10.1 Natural-Convection Boundary Layers on Flat Surfaces 27510.2 Phenomenology 27810.3 Scaling Analysis of Laminar Boundary Layers 28010.4 Similarity Solutions for a Semi-Infinite Vertical Surface 28510.5 Integral Analysis 28910.6 Some Widely Used Empirical Correlations for Flat Vertical

Surfaces 29410.7 Natural Convection on Horizontal Flat Surfaces 29510.8 Natural Convection on Inclined Surfaces 29710.9 Natural Convection on Submerged Bodies 29810.10 Natural Convection in Vertical Flow Passages 30010.11 Natural Convection in Enclosures 30410.12 Natural Convection in a Two-Dimensional Rectangle With

Heated Vertical Sides 30510.13 Natural Convection in Horizontal Rectangles 30710.14 Natural Convection in Inclined Rectangular Enclosures 309






Contents xi

10.15 Natural Convection Caused by the Combined Thermal andMass Diffusion Effects 31110.15.1 Conservation Equations and Scaling Analysis 31110.15.2 Heat and Mass Transfer Analogy 316

10.16 Solutions for Natural Convection Caused by CombinedThermal and Mass Diffusion Effects 317

Problems 327

11 Mixed Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

11.1 Laminar Boundary-Layer Equations and Scaling Analysis 33211.2 Solutions for Laminar Flow 33711.3 Stability of Laminar Flow and Laminar–Turbulent Transition 34111.4 Correlations for Laminar External Flow 34311.5 Correlations for Turbulent External Flow 34811.6 Internal Flow 349

11.6.1 General Remarks 34911.6.2 Flow Regime Maps 351

11.7 Some Empirical Correlations for Internal Flow 351Problems 358

12 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

12.1 Reynolds-Averaged Conservation Equations and the EddyDiffusivity Concept 362

12.2 One-Equation Turbulence Models 36412.3 Near-Wall Turbulence Modeling and Wall Functions 36712.4 The K–ε Model 371

12.4.1 General Formulation 37112.4.2 Near-Wall Treatment 37412.4.3 Turbulent Heat and Mass Fluxes 376

12.5 Other Two-Equation Turbulence Models 37612.6 The Reynolds Stress Transport Models 377

12.6.1 General Formulation 37712.6.2 Simplification for Heat and Mass Transfer 38012.6.3 Near-Wall Treatment of Turbulence 38012.6.4 Summary of Equations and Unknowns 381

12.7 Algebraic Stress Models 38112.8 Turbulent Models for Buoyant Flows 38212.9 Direct Numerical Simulation 38512.10 Large Eddy Simulation 39012.11 Computational Fluid Dynamics 394Problems 395

13 Flow and Heat Transfer in Miniature Flow Passages . . . . . . . . . . . . . 397

13.1 Size Classification of Miniature Flow Passages 39713.2 Regimes in Gas-Carrying Vessels 39913.3 The Slip Flow and Temperature-Jump Regime 40213.4 Slip Couette Flow 406






xii Contents

13.5 Slip Flow in a Flat Channel 40813.5.1 Hydrodynamics of Fully Developed Flow 40813.5.2 Thermally Developed Heat Transfer, UHF 41013.5.3 Thermally Developed Heat Transfer, UWT 413

13.6 Slip Flow in Circular Microtubes 41513.6.1 Hydrodynamics of Fully Developed Flow 41513.6.2 Thermally Developed Flow Heat Transfer, UHF 41613.6.3 Thermally Developed Flow Heat Transfer, UWT 41813.6.4 Thermally Developing Flow 420

13.7 Slip Flow in Rectangular Channels 42213.7.1 Hydrodynamics of Fully Developed Flow 42213.7.2 Heat Transfer 424

13.8 Slip Flow in Other Noncircular Channels 42613.9 Compressible Flow in Microchannels with Negligible

Rarefaction 42713.9.1 General Remarks 42713.9.2 One-Dimensional Compressible Flow of an Ideal Gas

in a Constant-Cross-Section Channel 42813.10 Continuum Flow in Miniature Flow Passages 431Problems 441

APPENDIX A: Constitutive Relations in Polar Cylindrical and SphericalCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

APPENDIX B: Mass Continuity and Newtonian Incompressible FluidEquations of Motion in Polar Cylindrical and Spherical Coordinates . . . . . . . 451

APPENDIX C: Energy Conservation Equations in Polar Cylindrical andSpherical Coordinates for Incompressible Fluids With Constant ThermalConductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

APPENDIX D: Mass-Species Conservation Equations in PolarCylindrical and Spherical Coordinates for Incompressible Fluids . . . . . . . . . . 454

APPENDIX E: Thermodynamic Properties of Saturated Water and Steam . . . 456

APPENDIX F: Transport Properties of Saturated Water and Steam . . . . . . . 458

APPENDIX G: Properties of Selected Ideal Gases at 1 Atmosphere . . . . . . . 459

APPENDIX H: Binary Diffusion Coefficients of Selected Gases in Air at1 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

APPENDIX I: Henry’s Constant, in bars, of Dilute Aqueous Solutions ofSelected Substances at Moderate Pressures . . . . . . . . . . . . . . . . . . . . . . . . . 466

APPENDIX J: Diffusion Coefficients of Selected Substances in Water atInfinite Dilution at 25 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467






Contents xiii

APPENDIX K: Lennard–Jones Potential Model Constants for SelectedMolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

APPENDIX L: Collision Integrals for the Lennard–Jones Potential Model . . 469

APPENDIX M: Some RANS-Type Turbulence Models . . . . . . . . . . . . . . . . 470M.1 The Spalart–Allmaras Model 470M.2 The K–ω Model 472M.3 The K–ε Nonlinear Reynolds Stress Model 475M.4 The RNG K–ε Model 477M.5 The Low-Re RSM of Launder and Shima 478

APPENDIX N: Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

APPENDIX O: Unit Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

APPENDIX P: Summary of Important Dimensionless Numbers . . . . . . . . . . 485

APPENDIX Q: Summary of Some Useful Heat Transfer andFriction-Factor Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

References 501

Index 517






Preface

We live in an era of unprecedented transition in science and technology educationcaused by the proliferation of computing power and information. Like most otherscience and technology fields, convective heat and mass transfer is already too vastto be covered in a semester-level course even at an outline level and is yet under-going exponential expansion. The expansion is both quantitative and qualitative.On the quantitative side, novel and hitherto unexplored areas are now subject toinvestigation, not just by virtue of their intellectual challenge and our curiosity,but because of their current and potential technological applications. And on thequalitative side, massive sources of Internet-based information, powerful personalcomputers, and robust and flexible software and other computational tools are noweasily accessible to even novice engineers and engineering students. This makesthe designing of a syllabus for courses such as convection heat and mass transferall the more challenging. Perhaps the two biggest challenges for an instructor of agraduate-level course in convection are defining a scope for the course and strikinga reasonable balance between the now-classical analytic methods and the recentlydeveloping modern areas. Although the importance of modern topics and methodsis evident, the coverage of these topics should not be at the expense of basics andclassical methods.

This book is the outcome of more than 10 years of teaching a graduate-levelcourse on convective heat and mass transfer. It also benefits from my more than20 years of experience of teaching undergraduate heat transfer and other thermalfluid science courses to mechanical and nuclear engineering students. The book isdesigned to serve as the basis for a semester-level graduate course dealing withtheory and practice of convection heat and mass transfer. My incentive in writingthe book is to strike a balance between well-established theory and practice on theone hand, and modern areas such as flow in microchannels and computational fluiddynamics (CFD)–based design and analysis methods on the other. I have had muchdifficulty finding such a balance in the existing textbooks while teaching convectionto graduate students and had to rely on my own class notes and recent issues ofjournals for much of the syllabi of my classes.

The book is primarily concerned with convective heat transfer. Essentials ofmass transfer are also covered, although only briefly. The mass transfer material

xv






xvi Preface

and problems are presented such that they can be easily skipped, should that bepreferred.

The book consists of 13 chapters. Chapter 1 reviews general and introductorymaterial that is meant to refresh the student’s memory about the material that he orshe will need to understand the remainder of the book. Chapters 2 and 3 deal withboundary layers and the transport processes that they control. Chapter 4 discusseslaminar internal flow, in considerably more detail than most similar textbooks, inrecognition of the importance of laminar flow in the now-ubiquitous miniature flowpassages. Chapter 5 discusses the integral method, a classical technique for theapproximate solution of boundary-layer transport equations. The fundamentals ofturbulence and classical models for equilibrium turbulence are discussed in Chap-ter 6, followed by the discussion of internal turbulent flow in Chapter 7. Chapter 8is a short discussion of the effect of transpiration on convective transport processes,and Chapter 9 deals with analogy among heat, momentum, and mass transfer pro-cesses. Buoyancy-dominated flows are discussed in Chapters 10 and 11.

Chapter 12 is on turbulence models. These models are the bases of the now-ubiquitous CFD tools. The chapter is primarily focused on the most widely usedReynolds-averaged Navier-Stokes (RANS)–type turbulent transport models in cur-rent convective heat transfer research and analysis. The discussions are meant toshow the students where these models have come from, with an emphasis on howthey treat not just the fluid mechanics aspects of turbulent flow but also the trans-port of heat and mass. Although access to and practice with CFD tools are help-ful for understanding these turbulence models, the chapter is written in a way thataccess to and application of CFD tools are not necessary. Only some of the problemsat the end of this chapter are meant to be solved with a CFD tool. These problems,furthermore, are quite simple and mostly deal with entrance-dominated internal tur-bulent flows. Finally, Chapter 13 is a rather detailed discussion of flow in microchan-nels. The importance of flow in microchannels can hardly be overemphasized. Thischapter discusses in some detail the internal gas flow situations for which significantvelocity slip and temperature jump do occur.

The book also includes 17 appendices (Appendices A–Q), which provide briefcompilations of some of the most essential properties and mathematical informationneeded for analysis of convective heat and mass transfer processes.

S. Mostafa Ghiaasiaan






Frequently Used Notation

A Flow or surface area (m2); atomic number�a Acceleration (m/s2)a Speed of sound (m/s); one-half of the longer cross-sectional

dimension (m)a′′

I Interfacial surface area concentration (surface area per unit)mixture volume (m−1)

B Blowing parameterBh 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/kBr Brinkman number = µU2

k|�T|Bo Buoyancy number = Gr/Rem

b One-half of the shorter cross-sectional dimension (m)C Concentration (kmol/m3)Cf Fanning friction factor (skin-friction coefficient)CD Drag coefficientCHe Henry’s coefficient (Pa; bars)Cµ Constant in the k–ε turbulence modelCP Constant-pressure specific heat (J/kg K)CP Molar-based constant-pressure specific heat (J/kmol K)Cv Constant-volume specific heat (J/kg K)Cv Molar-based constant-volume specific heat (J/kmol K)D Tube or jet diameter (m)DH Hydraulic diameter (m)Dij Multicomponent Maxwell-Stefan diffusivities for species i and j

(m2/s)Dij Binary mass diffusivity for species i and j (m2/s)Dij Multicomponent Fick’s diffusivity for species i and j (m2/s)�dj Diffusion driving force for species j (m−1)E Eddy diffusivity (m2/s); gas molecule energy flux (W/m2)

xvii






xviii Frequently Used Notation

Ec Eckert number = U2

CP�TE1, E 1D and 3D turbulence energy spectrum functions based on wave

number (m3/s2)E∗

1, E∗ 1D and 3D turbulence energy spectrum functions based onfrequency (m2/s)

EB Bulk modulus of elasticity (N/m2)Ema Eddy diffusivity for mass transfer (m2/s)Eth Eddy diffusivity for heat transfer (m2/s)e Total specific advected energy (J/kg)�e Unit vector�F Force (N)F EigenfunctionF Dependent variable in momentum mixed-convection similarity

solutionsFo Fourier number = ( k

ρCP) t

l2

Foma Mass transfer Fourier number = D tl2

Fr Froude number = U2/ (gD)f Dependent variable in momentum similarity solutionsf Darcy friction factor; frequency (Hz); distribution function (m−1

or m−3); specific Helmholtz free energy (J/kg)G Mass flux (kg/m2s); Gibbs free energy (J); production rate of

turbulent kinetic energy (kg/m s3); filter kernel in LES methodGa Galileo number = ρL �ρ g l3

µ2L

Grl Grashof number = g βl3�Tν2

Gr∗l Modified Grashof number = g β q′′ l4

v2k

Grma,l Concentration-based Grashof number = g βmal3�m1ν2 or g β∗

mal3�x1

ν2

Gz Graetz number = 4U l2

x

(ρ CP

k

)g Specific Gibbs free energy (J/kg); gravitational constant

(= 9.807 m/s2 at sea level)�g Gravitational acceleration vector (m/s2)H Boundary-layer shape factor (= δ1/δ2); channel height (m)He Henry numberh Specific enthalpy (J/kg)h Heat transfer coefficient (W/m2 K); height (m)hr Radiative heat transfer coefficient (W/m2 K)h fg , hs f , hsg Latent heats of vaporization, fusion, and sublimation (J/kg)h fg , hs f , hsg Molar-based latent heats of vaporization, fusion, and sublimation

(J/kmol)Im Modified Bessel’s function of the first kind and mth orderJ Diffusive molar flux (k mol/m2 s)j Diffusive mass flux (kg/m2 s); molecular flux (m−2 s−1)K Turbulence kinetic energy (J/kg)K Loss coefficient; incremental pressure-drop numberK Mass transfer coefficient (kg/m2 s)K Molar-based mass transfer coefficient (kmol/m2 s )






Frequently Used Notation xix

k Thermal conductivity (W/m K); wave number (m−1)L Length (m)Le Lewis number = α

Dl Length (m)lc Characteristic length (m)lD Kolmogorov’s microscale (m)lent,hy Hydrodynamic entrance length (m)lent,ma Mass transfer entrance length (m)lent,th Thermal (heat transfer) entrance length (m)lM Turbulence mixing length (m)lM, ma Turbulence mixing length for mass transfer (m)lheat Heated length (m)lth Turbulence mixing length for heat transfer (m)M Molar mass (kg/kmol)Ma Mach numberm Mass fraction; dimensionless constantm Mass (kg); mass of a single molecule (kg)m′′ Mass flux (kg/m2s)N Ratio between concentration-based and thermal-based Grashof

numbers = Grl,ma/Grl�N Unit normal vectorN′′ Molar flux (kmol/ m2 s)NAv Avogadro’s number (= 6.024 × 1026 molecules/kmol)NS Navier-Stokes equationNul Nusselt number h l/k�n Total mass flux (kg/m2 s)n Component of the total mass flux vector (kg/m2 s); number density

(m−3); dimensionless constant; polytropic exponentP PropertyP Pressure (N/m2); Legendre polynomialPel Peclet number = U l (ρ CP/k)Pel, ma Mass transfer Peclet number = U l /DPo Poiseuille number = 2τs DH

µ UPr Prandtl number = µ CP/kPr Reduced pressure = P/Pcr

Prtu Turbulent Prandtl numberpf Wetted perimeter (m)pheat Heated perimeter (m)p Perimeter (m)Q Volumetric flow rate (m3/s); dimensionless wall

heat fluxq Volumetric energy generation rate (W/m3)q′′ Heat flux (W/m2)R Radius (m); gas constant (Nm/kg K)R EigenfunctionRal Rayleigh number = g β l3�T

ν α






xx Frequently Used Notation

Ra∗l Modified Rayleigh number = g β l4q′′

ν αkRc Radius of curvature (m)Re Reynolds number = ρU l/µReF Liquid film Reynolds number = 4F/µL

Rey Reynolds number in low-Re turbulence models = ρ K1/2 y/µ

Ri Richardson number = Gr/Re2

Rl Volumetric generation of species l (kmol/m3 s)Ru Universal gas constant (= 8314 Nm/kmol K)r Distance between two molecules (A) (Chapter 1); radial

coordinate (m)�r Position vector (m)rl Volumetric generation rate of species l (kg/m3 s)S Entropy (J/K); distance defining intermittency (m)S Channel width (m)Sc Schmidt number v/DShl Sherwood number = K l

ρ D or K lCD

Sij Component of mean strain rate tensor (s−1)St Stanton number = h

ρ CPU = hC CPU

= NulRel Pr

Stma Mass transfer Stanton number = Kρ U = K

C U = ShlRel Sc

s Specific entropy (J/kg K)s Coordinate on the surface of a body of revolution (m)T Temperature (K)T Turbulence intensity�T Unit tangent vectort Time (s); thickness (m)tc Characteristic time (s)tc,D Kolmogorov’s time scale (s)tres Residence time (s)U Internal energy (J)�U Velocity vector (m/s)U Overall heat transfer coefficient (W/m2 K); velocity (m/s)Uτ Friction velocity (m/s)u Specific internal energy (J/kg)u Velocity in axial direction, in x direction in Cartesian coordinates,

or in r direction in spherical coordinates (m/s)uD Kolmogorov’s velocity scale (m/s)V Volume (m3)Vd Volume of an average dispersed phase particle (m3)v Velocity in y direction in Cartesian coordinates, r direction in

cylindrical and spherical coordinates, or θ direction in sphericalcoordinates (m/s)

v Specific volume (m3/kg)W Work (J); width (m)W Power (W)We Weber number = ρ U2 l

σ






Frequently Used Notation xxi

w Velocity in z direction in Cartesian coordinates, in θ direction incylindrical coordinates, or in ϕ direction in spherical coordinates(m/s); work per unit mass (W/kg)

X Mole fractionY Parameter represents the effect of fluid compressibility in

turbulence models (kg/m s3); height of a control volume (m)y Normal distance from the nearest wall (m)

Greek Characters

α Thermal (energy) accommodation coefficientα Thermal diffusivity (m2/s)α∗ Aspect ratioβ Wedge or cone angle (rad); coefficient of volumetric thermal

expansion (1/K)βma Coefficient of volumetric expansion with respect to mass fractionβ∗

ma Coefficient of volumetric expansion with respect to concentration(kg/m3)−1

βma Coefficient of volumetric expansion with respect to mole fraction Correction factor for the kinetic model for liquid-vapor interfacial

mass flux; gamma functionF Film mass flow rate per unit width (kg/m)γ Specific heat ration (CP/Cv); shape factor [(Eq. 4.6.5)]δ Kronecker delta; gap distance (m); boundary-layer thickness (m)δF Film thickness (m)δ1, δ2, δ3, δh Boundary-layer displacement, momentum, energy, and enthalpy

thicknesses (m)ε Porosity; radiative emissivity; turbulent dissipation rate (W/kg)ε Energy representing maximum attraction between two molecules

(J)ε′ Parameter defined in Eq. (12.4.5) (W/kg)εs Surface roughness (m); a small numberζ Parameter defined in Eq. (3.1.26); dimensionless coordinateη Independent variable in similarity solution equations;

dimensionless coordinateη c Convective enhancement factorθ Nondimensional temperature; azimuthal angle (rad); angular

coordinate (rad); angle of inclination with respect to thehorizontal plane (rad or ◦)

K Curvature (m−1); coefficient of isothermal compressibility (Pa−1)κ von Karman’s constantκB Boltzmann’s constant ( = 1.38 × 10−23J/K molecule)λ Wavelength (m); second coefficient of viscosity (− 2

3µ) (kg/m s);eigenvalue

λmol Molecular mean free path (m)µ Viscosity (kg /m s)ν Kinematic viscosity (m2/s)






xxii Frequently Used Notation

ξ Parameter defined in Eq. (3.2.41); variableρ Density (kg/m3)σ Normal stress (N/m2); Prandtl number for various turbulent

properties; tangential momentum accommodation coefficientσ Molecular collision diameter (A)σA Molecular-scattering cross section (m2)σc, σe Condensation and evaporation coefficientsσ ′′′

gen Entropy generation rate, per unit volume (J/K m3)τ Molecular mean free time (s); viscous stress (N/m2)τ Stress tensor (N/m2)τ ′ Viscous stress tensor (N/m2)� Dissipation function (s−2); pressure strain term (W/kg)φ Velocity potential (m2/s); pair potential energy (J); inclination

angle with respect to vertical direction (rad or ◦); normalized massfraction

φ′ Inclination angle with respect to the horizontal plane (rad or ◦)ϕ Relative humidity; nondimensional temperature for mixed

convectionψ Stream function (m2/s)� Specific potential energy associated with gravitation (J/kg);

momentum flux of gas molecules (kg/m s)�k , �D Collision integrals for thermal conductivity and mass diffusivity�i j Component of vorticity tensor (s−1)ω Humidity ratio� Complex velocity potential (m2/s)

Superscripts

r Relative+ Dimensionless; in wall units. Time rate− Average; in the presence of mass transfer

–t Time averaged∗ Dimensionless or normalized; modified for velocity slip or

temperature jump∼ Molar based; dimensionless

Subscripts

ad Adiabaticavg Averageb Body forcec Center, centerlinecr Criticald Dispersed phasedf Downflowent Entrance region or entrance effect






Frequently Used Notation xxiii

eq Equilibriumev Evaporationex ExitF Forced convectionf Saturated liquidfd Fully developedfilm Filmfr FrictionalG Gas phaseg Saturated vapor; gravitationalH Hartree’s (1937) similarity solution�H1 Boundary conditions in which the temperature is circumferentially

constant while the heat flux is axially constantheat Heatedhy HydrodynamicI Irreversible; gas–liquid interphasein InletL Liquid phaselam Laminarm Mean, bulkma Mass transfermax Maximummol MolecularN Natural convectionn Sparingly soluble (noncondensable) inert speciesopt Optimizedout OutletR Reversiblerad Radiationref Referencerefl Reflectedres Associated with residence times Wall surface; s surface (gas-side interphase); isentropicsat Saturation©T Uniform wall temperatureth Thermaltu TurbulentUC Unit cellUHF Uniform heat fluxUMF Uniform mass fluxUWM Uniform wall mass or mole fractionUWT Uniform wall temperaturew Wallx, z Local quantity corresponding to location x or z∞ Ambient; fully developed= Tensor






xxiv Frequently Used Notation

Abbreviations

CFD Computational fluid dynamicsDDES Delayed detached eddy simulationDES Detached eddy simulationDNS Direct numerical simulationDSMC Direct simulation Monte CarloGKT Gas-kinetic theoryLES Large-eddy simulationMMFP Molecular mean free pathODE Ordinary differential equationRANS Reynolds-averaged Navier-StokesRNG Renormalized groupRSM Reynolds stress modelSGS Subgrid scaleUHF Uniform heat fluxUMF Uniform mass fluxUWM Uniform wall mass or mole fractionUWT Uniform wall temperature1D, 2D, 3D One-, two-, and three-dimensional






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