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Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Advanced Heat and Mass Transfer Amir Faghri United Technologies Endowed Chair Professor in Thermal-Fluids Engineering Department of Mechanical Engineering University of Connecticut Storrs, Connecticut, USA Yuwen Zhang Professor Department of Mechanical and Aerospace Engineering University of Missouri Columbia, Missouri, USA John Howell Baker-Hughes Centennial Professor Ernest Cockrell, Jr., Memorial Chair Department of Mechanical Engineering University of Texas at Austin Austin, Texas, USA Global Digital Press Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Published by Global Digital Press 601 Business Loop 70 W., Suite 134H Columbia, MO 65203, USA This book is printed on acid-free paper. Copyright © 2010 by Global Digital Press. All rights reserved. Except permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without writing permission from the publisher. Permissions may be sought directly from Global Digital Press, 601 Business Loop 70 W., Suite 134H Columbia, MO 65203, USA, Email: administrator@GlobalDigitalPress.org. ISBN: 978-0-9842760-0-4 Printed in the United States of America 06 07 08 09 10 9 8 7 6 5 4 3 2 1 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table of Contents Preface Nomenclature Chapter 1 Introduction 1.1 Introduction 1.2 Physical Concepts 1.2.1 Sensible Heat 1.2.2 Latent Heat 1.2.3 Phase Change 1.3 Molecular Level Presentation 1.3.1 Introduction 1.3.2 Kinetic Theory 1.3.3 Intermolecular Forces and Boltzmann Transport Equation 1.3.4 Cohesion and Adhesion 1.3.5 Enthalpy and Energy 1.4 Fundamentals of Momentum, Heat and Mass Transfer 1.4.1 Continuum Flow Limitations 1.4.2 Momentum, Heat and Mass Transfer 1.4.3 Microscale and Nanoscale Transport Phenomena 1.4.4 Dimensional Analysis 1.4.5 Scaling 1.5 Modern Applications of Heat and Mass Transfer 1.5.1 Energy Systems 1.5.2 Biological and Biomedical Systems 1.5.3 Security 1.5.4 Information Technology 1.5.5 Nanotechnology References Problems xii xv 1 1 3 3 5 7 9 9 10 16 20 21 23 23 25 43 48 59 61 62 66 69 71 74 78 83 Table of Contents v Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Chapter 2 Generalized Governing Equations 2.1 Introduction 2.2 Macroscopic (Integral) Local Instance Formulation 2.2.1 Conservation of Mass 2.2.2 Momentum Equation 2.2.3 Energy Equation 2.2.4 The Second Law of Thermodynamics 2.2.5 Species 2.3 Microscopic (Differential) Local Instance Formulation 2.3.1 Conservation of Mass 2.3.2 Momentum Equation 2.3.3 Energy Equation 2.3.4 The Second Law of Thermodynamics 2.3.5 Species 2.3.6 Classification of PDEs and Boundary Conditions 2.3.7 Jump and Boundary Conditions at the Interfaces 2.3.8 Rarefied Vapor Self-Diffusion Model 2.3.9 An Extension: Combustion 2.4 Volume Averaged Models 2.4.1 Overview of Averaging Approaches 2.4.2 Volume-Averaged Multi-Fluid Models 2.4.3 Volume-Averaged Homogeneous Model 2.4.4 An Extension: Porous Media 2.5 Fundamentals of Turbulence 2.5.1 Description of Turbulence 2.5.2 Time-Averaged Governing Equations References Problems 89 89 91 93 94 95 99 99 101 102 103 105 110 111 121 124 137 138 145 145 150 161 167 181 181 184 190 192 Chapter 3 Heat Conduction 3.1 Introduction 3.2 Steady State Heat Conduction 3.2.1 One Dimensional Heat Conduction 3.2.2 Multidimensional Heat Conduction 3.3 Unsteady State Heat Conduction 3.3.1 Lumped Analysis 3.3.2 One Dimensional Transient Heat Conduction 3.3.3 Multidimensional Transient Heat Conduction 3.4 Numerical Simulation of Heat Conduction Problems 3.4.1 One-Dimensional Steady-State Conduction 3.4.2 One-Dimensional Transient Conduction 3.4.3 Multidimensional Transient Conduction 3.5 Melting and Solidification 3.5.1 Introduction 209 209 212 212 227 238 238 240 261 264 265 270 273 276 276 vi Advanced Heat and Mass Transfer by Faghri, Zhang and Howell Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 3.5.2 Exact Solution 3.5.3 Integral Approximate Solution 3.5.4 Numerical Simulation 3.6 Microscale Heat Conduction 3.6.1 Extensions of Classic Model 3.6.2 Two-Step Model 3.6.3 Microscale Phase Change References Problems 281 289 304 314 314 316 319 323 325 Chapter 4 External Convective Heat and Mass Transfer 4.1 Introduction 4.2 Concepts of the Boundary Layer Theory 4.3 Boundary Layer Approximation 4.4 Governing Equations for Boundary Layer Approximation 4.5 Laminar Boundary Layer Solutions for Momentum, Heat, and Mass Transfer 4.6 Similarity Solutions 4.6.1 Uncoupled Mass, Momentum, and Heat Transfer Problems 4.6.2 Coupled Mass, Momentum, and Heat Transfer Problems 4.7 Integral Methods 4.8 Computational Methodologies for Forced Convection 4.8.1 One-Dimensional Steady-State Convection and Diffusion 4.8.2 Multidimensional Convection and Diffusion Problems 4.8.3 Numerical Solution of Flow Field 4.8.4 Numerical Simulation of Interfaces and Free Surfaces 4.9 Application of Computational Methods 4.10 Analogies and Differences in Different Transport Phenomena 4.11 Turbulence 4.11.1 Turbulent Boundary Layer Equations 4.11.2 Algebraic Models for Eddy Diffusivity 4.11.3 K-ε Model 4.11.4 Momentum and Heat Transfer for Turbulent Flow over a Flat Plate References Problems 339 339 341 343 344 350 351 352 362 369 375 376 385 388 395 400 406 412 412 414 422 424 430 433 Chapter 5 Internal Convective Heat Transfer 5.1 Introduction 5.2 Basic Definitions, Terminology and Governing Equations 5.3 Hydrodynamically and Thermally Fully Developed Laminar Flow 5.4 Hydrodynamically Fully Developed and Thermally Developing Laminar Flow 5.4.1 Constant Wall Temperature 438 438 439 447 453 454 Table of Contents vii Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 5.4.2 Constant Heat Flux at the Wall 456 5.5 Hydrodynamically Fully Developed Flow with Coupled Thermal and Concentration Entry Effects 458 5.5.1 Sublimation inside an Adiabatic Tube 458 5.5.2 Sublimation inside a Tube Subjected to External Heating 463 5.6 Developing Flow, Thermal and Concentration Effects 470 5.7 Full Numerical Solutions 475 5.8 Forced Convection in Microchannels 482 5.8.1 Introduction 482 5.8.2 Fully Developed Laminar Flow and Temperature Profile 485 5.8.3 Fully Developed Flow with Developing Temperature Profile 493 5.9 Turbulence 499 5.9.1 Time-Averaged Governing Equations 499 5.9.2 Velocity Profile and Friction Coefficient for Fully Developed Flow 500 5.9.3 Heat Transfer in Fully Developed Turbulent Flow 503 References 509 Problems 511 Chapter 6 Natural Convection 6.1 Introduction 6.2 Governing Equations for Natural Convection 6.2.1 Generalized Governing Equations 6.2.2 External Natural Convection from Heated Vertical Plate 6.2.3 Dimensionless Parameters 6.3 Scale Analysis 6.3.1 High Prandtl Number Fluids (Pr ≫1) 6.3.2 Low Prandtl Number Fluids (Pr ≪1) 6.4 External Natural Convection 6.4.1 Similarity Solution for Natural Convection on a Vertical Surface 6.4.2 Integral Solution for Laminar and Turbulent Natural Convection 6.4.3 Natural Convection over Inclined and Horizontal Surfaces 6.4.4 Natural Convection over Cylinders and Spheres 6.4.5 Free Boundary Flow 6.5 Natural Convection in Enclosures 6.5.1 Scale Analysis 6.5.2 Rectangular Enclosures 6.5.3 Annular Space between Concentric Cylinders and Spheres 6.6 Natural Convection in Melting and Solidification 6.6.1 Solidification around Horizontal Cylinder 6.6.2 Melting in a Rectangular Enclosure Heated from the Side 6.7 Instability Analysis of Natural Convection References Problems 515 515 517 517 519 520 521 523 525 526 526 533 540 543 551 555 556 560 569 572 572 575 580 583 587 viii Advanced Heat and Mass Transfer by Faghri, Zhang and Howell Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Chapter 7 Condensation and Evaporation 7.1 Introduction 7.2 Dropwise Condensation 7.2.1 Surface Tension and Capillary Pressure 7.2.2 Thermal Resistances in the Condensation Processes 7.2.3 Heat Transfer Coefficient for Dropwise Condensation 7.3 Filmwise Condensation 7.3.1 Regimes of Filmwise Condensation 7.3.2 Modeling for Laminar Film Condensation of a Binary Vapor Mixture 7.3.3 Filmwise Condensation in a Stagnant Pure Vapor Reservoir 7.3.4 Effects of Vapor Motion 7.3.5 Turbulent Film Condensation 7.3.6 Other Filmwise Condensation Configurations 7.3.7 Effects of Noncondensable Gas 7.4 Falling Film Evaporation on a Heated Wall and Spray Cooling 7.4.1 Classical Nusselt Evaporation 7.4.2 Laminar Falling Film with Surface Waves 7.4.3 Turbulent Falling Film 7.4.4 Surface Spray Cooling References Problems 590 590 599 599 603 607 609 609 610 615 623 629 634 636 642 642 646 649 649 652 655 Chapter 8 Boiling 8.1 Introduction 8.2 Pool Boiling Regimes 8.3 Nucleate Boiling 8.3.1 Nucleation and Inception 8.3.2 Bubble Dynamics 8.3.3 Bubble Detachment 8.3.4 Nucleate Site Density 8.3.5 Bubble Growth and Merger 8.3.6 Heat Transfer in Nucleate Boiling 8.4 Critical Heat Flux 8.5 Transition Boiling and Minimum Heat Flux 8.5.1 Transition Boiling 8.5.2 Minimum Heat Flux 8.6 Film Boiling 8.6.1 Film Boiling Analysis 8.6.2 Direct Numerical Simulation of Film Boiling 8.6.3 Leidenfrost Phenomena References Problems 665 665 666 669 670 675 685 690 691 695 701 705 705 707 709 709 719 722 730 735 Table of Contents ix Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Chapter 9 Fundamentals of Thermal Radiation 9.1 Electromagnetic Waves and Thermal Radiation 9.2 The Blackbody as the Ideal Radiator 9.2.1 The Planck Distribution and its Consequences 9.2.2 The Blackbody Fraction 9.3 Properties of Real Surfaces: Definitions, Measurements and Prediction 9.3.1 Opaque Surface Property Definitions 9.3.2 EM Theory Predictions of Properties 9.4 Application and Exploitation of Radiative Properties 9.4.1 Spacecraft Thermal Design 9.4.2 Solar Thermal Energy Collectors 9.4.3 Other Property Choices for Radiation/Surface Interactions 9.5 High-energy Radiation-Surface Interactions 9.5.1 Nanoscale Surface Modification for Tailoring Properties 9.5.2 Macroscale Laser-Surface Interactions 9.6 Light Pipes and Fiber Optics 9.7 Infrared Sensing, Cameras and Photography 9.8 Other Contemporary Applications and Research References Problems 739 739 741 742 748 750 750 759 766 766 771 777 778 780 781 783 785 786 787 788 CHAPTER 10 Heat Transfer by Radiation 794 10.1 Radiative Transfer through Transparent Media 794 10.1.1 Transfer between Two Areas 794 10.1.2 Diffuse Surfaces: The Configuration Factor 796 10.1.3 Configuration Factor Algebra 800 10.2 The Enclosure; The Net Radiation Method for Diffuse Surfaces 804 10.2.1 Radiosity, Irradiation, and Net Energy Transfer 805 10.2.2 Gray Surfaces 807 10.2.3 Nongray Surfaces 814 10.2.4 Surfaces with Varying Temperature, Radiative Flux, or Properties 817 10.3 Multimode Heat Transfer with Radiation 821 10.3.1 Numerical Methods 824 10.3.2 Conduction Dominated Problems 826 10.3.3 Radiation Dominated Problems 826 10.3.4 Problems with Both Modes Significant 827 10.4 Inverse Problems 828 10.4.1 An Inverse Design Problem 830 10.4.2 Regularization 835 10.4.3 Unresolved Problems in Inverse Cases 837 10.5 The Effect of Participating Media 838 10.5.1 Absorption, Emission and Scattering from a Medium 838 10.5.2 Properties of Participating Media 839 10.5.3 The Radiative Transfer Equation 844 10.5.4 Some Limiting Solutions for Radiative Transfer 848 x Advanced Heat and Mass Transfer by Faghri, Zhang and Howell Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 10.6 Applications of Radiative Transfer 10.6.1 Radiation Measurement and Sensing: IR Cameras, Optical Pyrometers and Remote Sensing 10.6.2 Atmospheric Phenomena Caused by Scattering 10.6.3 Pollution, Greenhouse Gases and the Greenhouse Effect, Atmospheric Radiation and the Global Energy Balance References Problems 862 862 863 865 866 868 List of Appendices Appendix A Constants, Units and Conversion Factors Appendix B Transport Properties of Solids Appendix C Transport Properties of Gases and Liquids at Atmospheric Pressure Appendix D Transport Properties for Phase Change Appendix E Mass Transfer Properties Appendix F Configuration Factors and Surface Properties for Radiation Appendix G Mathematical Relations 875 876 880 888 895 899 911 916 Index 925 Table of Contents xi Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Preface Numerous heat and mass transfer textbooks have been published over the last several decades. The field of heat and mass transfer has advanced many-fold during its evolution due to the development of new knowledge, techniques, and applications. If one examines the evolution of our understanding of heat and mass transfer, it can be seen that the first phase was dominated by the development of experimental results and correlations, as well as techniques for conventional problems related to momentum, heat and mass transfer, with primary emphasis on non-dimensional analysis. The second phase primarily involved the development of simple theoretical tools to predict these classical results using concepts such as boundary layer theory and kinetic theory, which provide some very useful physical insights and effective design tools for practicing thermal engineers. Phase three began with the development of large digital computers and more efficient computational techniques. We were able to develop much better physical insight into complex heat and mass transfer problems via the utilization of such tools. Furthermore, state-of-the-art experimental measurement techniques – including optics and data acquisition systems – produced vast amounts of experimental data in the field of heat and mass transfer. As we proceed into the fourth phase, one should realize the challenges and opportunities, such as the instant and ready availability of resources at one’s fingertips through the Internet from any point in the world. Secondly, we cannot continue business as usual; i.e., the cookbook approach that was employed in the development of some of the existing textbooks. We must integrate innovation, critical thought, and modern relevance into our textbooks so as to educate the engineers of the 21st century in a globally competitive market. Furthermore, most of the applications in the field of heat and mass transfer so far have pertained to traditional power and thermal engineering and conventional energy systems, as well as space activities. Modern technologies and new applications, e.g., nanotechnology, biotechnology, energy, material processing and information technology, will likely play a major role in the development of curricula in future years. Globally, universities need to better align heat and mass transfer in the engineering curricula, as werll as the nature of academic experiences, with the challenges and opportunities that engineers will face in this digital environment. Traditionally, heat and mass transfer at the graduate level is taught in four separate courses: heat conduction, convective heat transfer, mass transfer and xii Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press radiation. The materials covered in these courses are rather extensive while some of them are even irrelevant. Graduate students are not given appropriate exposure to topics related to modern emerging technologies. There is currently no advanced-level textbook available that covers all of the pertinent topics (heat conduction, convective heat transfer, mass transfer, radiation, and multiphase phenomena) in a single volume. Therefore, we are hoping that this text will help to fill this gap. There are, of course, excellent generalized undergraduate textbooks, as well as advanced graduate-level books on single topics. However, due to curriculum limitations or small faculty sizes, a number of universities are offering single courses that cover all of these intermediate and advanced heat and mass transfer topics. The purpose of this textbook is to present the subject of heat and mass transfer with a focus on the significant advances in the field during the last decade, while emphasizing the basic, fundamental principles. Thus, we hope to provide a single textbook that these courses may use, so as to avoid requiring the use of several textbooks. A copyrighted solutions manual and PowerPoint presentation package are provided only to those instructors who adopt the book for the course. As the authors, we wish to express our appreciation to our following postdoctoral fellows and graduate students who generously reviewed individual chapters based on their expertise, and provided suggestions for improvements and necessary corrections: H. Bahrami, J. Huang, H. Shabgard, N. Sharifi, S. Wang, and J. Zhou. In addition, we would like to acknowledge the contributions of students over the last several years, who were taught from the manuscripts out of which this book evolved. This textbook is developed for use as an advanced-level undergraduate or graduate textbook on heat and mass transfer for various disciplines. We recognize a new trend at a number of universities to offer a single course in advanced heat and mass transfer, and therefore we have tried to cover the materials that various disciplines might wish to include in such a course. The authors were fortunate to develop previous heat transfer textbooks on various subjects; thus, some of the materials are taken from these sources. Your suggestions, comments, and criticisms are appreciated. Amir Faghri Yuwen Zhang John Howell Preface xiii Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Nomenclature A Bi Bo Br Brq c c cf ci cp cv C CD Cf d D D Dh Dij Dij area, m2 Biot number, hL / k (k is thermal conductivity of solid) Bond number, ( ρ − ρv ) gL2 / σ 2 Brinkman number, μ um /[k (Ts − Tc )] 2 ′′ Brinkman number constant heat flux, μ um /(qw D ) specific heat, J/(kg-K); velocity of the molecular random motion, m/s; speed of sound, m/s particle velocity (m/s) friction factor molar concentration of the ith species, kmol/m3 specific heat at constant pressure, J/kg-K specific heat at constant volume, J/kg-K heat capacity, J/K drag coefficient friction coefficient coefficient of pressure-diffusion term diameter, m; self diffusivity, m2/s; diffusion conductance rate of strain tensor, 1/s hydraulic diameter, m binary diffusivity, m2/s Maxwell-Stefan diffusivity, m2/s  ij DiT e E E Ê f multicomponent Fick diffusivity, m2/s multicomponent thermal diffusivity, m2/s specific internal energy, J/kg; kinetic energy of molecules, J internal energy or surface free energy, J; emissive power, W/m2; electric field electric field vector total energy, J friction coefficient; wave frequency, 1/s , molecular velocity distribution function. force, N; Helmholtz free energy, J/kg-K; flow rate through a control volume surface force vector, N F F xiv Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press F0-λT FA-B Fo Fr g G Gr Gz h h  h hv h′v hm hm ,G blackbody fraction configuration factor between surfaces A and B Fourier number, α t / L2 Froude number, U / gL or U 2 /( gL) gravitational acceleration, m/s2; specific Gibbs free energy, J/kg Gibbs free energy, J; coupling factor, W/m3-K; irradiance, W/m2 Grashof number, g β ΔTL3 /ν 2 Graetz number heat transfer coefficient, W/(m2-K); specific enthalpy, J/kg average heat transfer coefficient, W/m2-K average enthalpy of the multiphase mixture, J/kg latent heat of vaporization, J/kg modified latent heat of vaporization, J/kg convective mass transfer coefficient, m/s mass transfer coefficient in noncondensable gas section, m/s hs hsv H H I I0 I1 I J J0 J1 Ji J* i Ja latent heat of fusion, J/kg latent heat of sublimation, J/kg enthalpy, J; height, m; Henry’s constant; magnetic field magnetic field vector intensity, W/m2-sr; electrical current, A modified Bessel function of the first kind of the zeroth order modified Bessel function of the first kind of the first order identity tensor total (diffusion + convection) flux; radiosity, (W/m2) Bessel function of the first kind of the zeroth order Bessel function of the first kind of the first order mass flux of the ith species relative to mass-averaged velocity, kg/m2-s molar flux of the ith species relative to molar-averaged velocity, kmol/m2-s Jakob number, c p ΔT / hv thermal conductivity, W /(m-K) thermal conductivity tensor, W /(m-K) modified Bessel function of the second kind of the zeroth order modified Bessel function of the second kind of the first order Boltzmann constant, J/K interface curvature, 1/m; Permeability, m2; dielectric constant Arrhenius constant momentum exchange coefficient between phases j and k, kg/(m3-s) Kapitza number, μ4 g /[( ρ − ρv )σ 3 ] Knudsen number, λ / L (characteristic) length, m k k kb K K0′ Kjk Ka Kn L K0 K1 Nomenclature xv Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Le Lb Lewis number, α / D bubble or capillary scale, σ /[ g ( ρ − ρ g )] , m Le m  m  m′′  m′′  m′′′ M Ma n n n ′′ nb NCR ′′ nD ni  ni′′ N N NA ′′ Na Nu Nu x p p′ Pe Pr Prt q q′ q′′ q′′ ′′ qmax ′′ qmin mean beam length mass, kg; exponent in flow over a wedge mass flow rate, kg/s absolute mass flux relative to stationary coordinate system, kg/m2-s mass flux vector, kg/m2-s mass source per unit volume, kg/m3-s molecular mass, kg/kmol; number of concave surfaces in enclosure Mach number, U / c number of moles; number of horizontal tubes in an array; refractive index complex refractive index, n − iκ unit normal vector number of vapor bubbles released per unit area and release cycle, 1/m2 conduction/radiation parameter, kδ/σTref3W2 liquid droplet size distribution, 1/m3 number of moles for the ith component in a multicomponent system absolute molar flux of the ith component relative to stationary coordinate system, kmol/s number of components, number of molecules number density of the molecules Avogadro’s number (1/mol) number density of nucleation sites Nusselt number, hL / k local Nusselt number, hx x / k pressure, Pa; number of retained singular values pressure correction Peclet number, UL / α Prandtl number, ν / α turbulent Prandtl number heat rate, W heat rate per unit length, W/m heat flux, W/m2 heat flux vector, W/m2 maximum (critical) heat flux in boiling, W/m2 minimum heat flux in boiling, W/m2 internal heat generation per unit volume, W/m3 total heat transfer, J; TDMA coefficient; volume flow rate, m3/s dimensionless radiative flux, q"/σTref4 ′′ qsolar solar constant, 1368 W/m2 q′′′ Q Q" xvi Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press r r reff R Rb Rc′′t re Rg ℜj Rlaser Rmen Rr Ru Rv Rδ Ra Re s gen s′′′ S S Sc radial coordinate, m residual vector; vector location on surface effective pore radius, m radius, m; radius of curvature, m; dimensionless radius, r/ri; electrical resistance, Ohm; thermal resistance, K/W; residual bubble radius, m contact thermal resistance (m2-K/W) electrical resistivity gas constant, Ru / M , kJ/kg-K net reaction rate of the jth chemical reaction radius of laser beam radius of curvature of the meniscus, m characteristic micro roughness size, m universal gas constant, 8.3144 kJ/kmol-K vapor space radius, m interfacial thermal resistance, m2-K/W Rayleigh number, g β ΔTL3 /(να ) Reynolds number, UL /ν ; 4Γ / μ (for film condensation or evaporation) specific entropy, J/(kg-K); space variable; m; interface location, m entropy generation rate per unit volume, W/kg-K-m3 entropy, J/K; general source term in numerical solution; nondimensional interface location, s/L; solubility, kmol/Pa-m3 ; position along path Poynting vector; matrix of singular values Schmidt number, ν / D ; subcooling parameter, c ps (Tm − Ti ) / hs Sherwood number, hm L / D12 Stanton number, h /( ρ c pU ) Stefan number, c p Tw − Tm / hs Strouhal number, Lf / U time, s unit tangential vector laser pulse duration, s temperature, K melting point, K saturation temperature, K wall temperature, K temperature of environment, K velocity in the x- direction, m/s velocity, m/s; number of unknown configuration factors orthogonal matrices mean velocity, m/s Sh St Ste Sr t t tp T Tm Tsat Tw T∞ u U U U Nomenclature xvii Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ˆ u u* ub uc uf pseudovelocity in the x-direction, m/s velocity based on guessed pressure, m/s laser beam scanning velocity, m/s critical Helmholtz velocity, m/s 1/ 2 frictional velocity, (τ w / ρ  ) mean velocity, m/s specific volume, m3/kg; velocity in the y-direction, m/s; radial velocity, m/s; or vapor velocity along the η-coordinate, m/s volume, m3 velocity vector, m/s; orthogonal matrices Π k k mass-averaged velocity vector,   ε k ρ k Vk  ρ , m/s  k =1  pseudovelocity in the y-direction, m/s velocity based on guessed pressure, m/s velocity in the z-direction or axial velocity, m/s, specific work, J/kg pseudovelocity in the z-direction, m/s velocity based on guessed pressure, m/s liquid phase axial velocity, m/s vapor phase axial velocity, m/s wave velocity, m/s work, J; width of the cavity, m; width of a capillary groove, m Weber number, ρU 2 L / σ Womersley number, ω R 2 /ν Cartesian coordinate, m molar fraction of the ith species material coordinate, m; dimensionless coordinate, x/L body force per unit mass acting on the ith species in the kth phase, m/s2 Cartesian coordinate, m material coordinate, m Bessel function of the second kind of the zeroth order Bessel function of the second kind of the first order Cartesian coordinate, axial coordinate, m material coordinate, m um v V V  V ˆ v v* w ˆ w w* w wv ww W We Wr x xi X Xk,i y Y Y0 Y1 z Z GREEK SYMBOLS α thermal diffusivity, m2/s; relaxation factor; accommodation coefficient; absorptivity of a surface α(S) absorptance of participating medium β wedge angle, rad; coefficient of thermal expansion, 1/K; contact angle measured in degrees; attenuation coefficient, κ + σs βm composition coefficient of volume expansion xviii Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press γ χ δ Γ δ+ δ* Δ general diffusion coefficient; liquid mass flow rate per unit width, kg/m-s electric permittivity angle of refraction velocity boundary layer thickness, liquid or vapor film thickness, m; thermal penetration depth, m; laser irradiation penetration depth, m; thickness of the deposited film, m; thickness of surface element, m; Kronecker delta nondimensional film thickness, δ u f /ν  nondimensional film thickness, δ / LF , or δ (ν 2 / g ) −1/ 3 δt Δt ΔT εM εH ε(S) ΔV ε η ηf θ ′′ dimensionless thermal penetration depth, δ /(α1 ρ1hs /q0 ) ; dimensionless liquid layer thickness in contact melting, δ / W thermal boundary layer thickness, m time interval for time average; time step in numerical solution, s half of the width of phase change temperature range, K; temperature difference, K volume element for volume average, m3 porosity; volume fraction; emissivity; eddy diffusivity, m2/s momentum eddy diffusivity, m2/s thermal eddy diffusivity, m2/s emittance of participating medium dimensionless variable for binary solidification, X /(2 τ ) ; dimensionless coordinate, y / δ ; coordinate normal to the solid-liquid interface, m fin efficiency inclination angle, rad; contact or wetting angle, rad; dimensionless temperature; cone angle in spherical coordinates excess temperature, K eigen function contact angle obtained from the smooth-surface model meniscus contact angle surface roughness, m; imaginary component of complex refractive index; absorption coefficient (m-1) Planck mean absorption coefficient (m-1) mean free path, m; wavelength, m; constant in solid-liquid phase change, S / (2τ 1/ 2 ) critical wavelength, m most dangerous wavelength, m eigenvalue dynamic viscosity, kg/(m-s); chemical potential, J/mol; magnetic permeability ϑ Θn θf θ men κ κP λ λc λD λn μ Nomenclature xix Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ν Π πs ζ kinematic viscosity, m2/s; frequency (Hz); C2 / λT number of phases surface pressure, N/m2 dimensionless stream function; pH 2O /( pH 2O + pCO2 ) density, kg/m3; reflectivity of a surface mass concentration of species i, kg/m3 surface tension force, N/m; collision diameter, Å scattering coefficient, (m-1) Stefan-Boltzmann constant, 5.67 × 10−8 W/m 2 -K 4 ρi ρ shear stress, N/m2; thermal relaxation time, s; dimensionless time, α t / L2 ; optical thickness τ(S) transmittance of participating medium τ' stress tensor, N/m2 viscous stress tensor, N/m2 τ τw shear stress at wall, N/m2 φ Lennard-Jones potential, J; specific value of general property, Φ; inclination angle; circumferential angle in spherical coordinates ϕ similarity variable; dependent variable in combustion of spherical droplet Φ general property Φ scattering phase function ψ stream function, m2/s ω blood perfusion rate; angular frequency (s-1); albedo, [κ/(σs + κ)] ωi mass fraction of species i Ω general vector quantity; surface tension parameter; solid angle, steradians ∇ Laplace operator vector Subscripts 0 reference variables; initial condition; reservoir conditions; in vacuum B bottom neighbor grid point b control volume face at bottom; blood; blackbody app apparent value c critical point, condenser, cutoff value cap capillary E east neighbor point e equilibrium; evaporator; control volume face at east eff effective value f final; fuel; thin film i ith component; initial; inner I interface k kth phase in a multiphase system  liquid L left; characteristic length σ σs σ SB τ xx Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press m men N n nb o P p q r R rad ref rel S s sat stag T t tr v W w mean; melting point or mushy zone; metabolic meniscus north neighbor grid point normal to surface; control volume face at bottom general neighbor grid points outer central grid point under consideration product; particle; pulse heat flux reduced; reactant right radiation reference frame relative south neighbor grid point shaft; solid; surface; control volume face at south saturation stagnation value top neighbor grid point; temperature turbulent; control volume face at top transition vapor phase west neighbor grid point wall; control volume face at west δ liquid-vapor interface λ wavelength dependent ∞ ambient or bulk property Superscripts k phase index n normal component t tangential component, turbulent ´ fluctuation * dimensionless + dimensionless Others ~ same order of magnitude < > volume averaged < >k phase average ¯ time averaged, mean  mass-averaged  parallel component of EM wave ⊥ perpendicular component of EM wave   maximum function Nomenclature xxi Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press xxii Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press