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Transport Phenomena in Multiphase Systems Transport Phenomena in Multiphase Systems Amir Faghri Dean and UTC Endowed Chair Professor in Thermal-Fluids Engineering School of Engineering University of Connecticut Storrs, Connecticut Yuwen Zhang Associate Professor Department of Mechanical and Aerospace Engineering University of Missouri-Columbia Columbia, Missouri Amsterdam • Boston • Heidelberg • London New York • Oxford • Paris • San Diego San Francisco • Singapore • Sydney • Tokyo To Our Families Pouran, Tanaz, and Ali Faghri Jennifer, Angela, and Joanna Zhang Whose Love and Support Make All Things Possible Table of Contents Preface Nomenclature Chapter 1Introduction to Transport Phenomena 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 1.3.4 Cohesion and Adhesion 1.3.5 Enthalpy and Energy 1.4 Review of Fundamentals of Transport Phenomena 1.4.1 Continuum Flow Limitations 1.4.2 Transport Phenomena 1.4.3 Microscale and Nanoscale Transport Phenomena 1.4.4 Dimensional Analysis 1.4.5 Scaling 1.5 Multiphase Systems and Phase Changes 1.5.1 Overview and Classifications 1.5.2 Solid-Liquid Phase Change Including Melting and Solidification 1.5.3 Solid-Vapor Phase Change Including Sublimation and Vapor Deposition 1.5.4 Interfacial Phenomena 1.5.5 Condensation 1.5.6 Evaporation and Boiling 1.5.7 Two-Phase Flow xv xxi 1 1 3 3 5 7 9 9 10 15 19 19 21 21 23 44 48 58 62 62 66 67 68 69 71 72 vii 1.6 Applications of Transport Phenomena in Multiphase Systems 1.6.1 Energy Systems, Including Fuel Cells and Combustors 1.6.2 Food and Biological Material Processing 1.6.3 Laser-Assisted Manufacturing 1.6.4 Heat Pipes 1.6.5 Electronics Cooling 1.6.6 Microscale Phase Change Heat Transfer References Problems 103 73 73 83 84 88 92 96 98 Chapter 2Thermodynamics of Multiphase Systems 2.1 Introduction 2.2 Fundamentals of Thermodynamics 2.2.1 Thermodynamic Laws 2.2.2 Thermodynamic Relations 2.2.3 Gibbs Phase Rule 2.3 Equilibrium and Stability of Single-Phase Systems 2.3.1 Equilibrium Criteria for Pure Substances 2.3.2 Maxwell Relations 2.3.3 Closed Systems with Compositional Change 2.3.4 Stability Criteria 2.3.5 System with Chemical Reactions 2.4 Thermodynamic Surfaces and Equations of State 2.4.1 Thermodynamic Surfaces of a Single-Component Substance 2.4.2 p-T, p-v and T-s Phase Diagrams for a Pure Substance 2.4.3 Equations of State for Pure Substances 2.4.4 Phase Diagrams for Multicomponent Systems 2.5 Equilibrium and Stability of Multiphase Systems 2.5.1 Two-Phase Single-Component Systems 2.5.2 Clapeyron Equation 2.5.3 Multiphase Multicomponent Systems 2.5.4 Metastable Equilibrium and Nucleation 2.6 Thermodynamics at the Interfaces 2.6.1 Equilibrium at the Interface 2.6.2 Surface Tension: Thermodynamic Definitions 2.6.3 Microscale Vapor Bubbles and Liquid Droplets 2.6.4 Disjoining Pressure: Thermodynamic and Hydrodynamic Definitions 2.6.5 Superheat-Thermodynamic and Kinetic Limit Definitions References 172 Problems 173 107 107 108 108 109 111 111 112 114 115 117 123 128 128 129 131 136 141 141 142 146 147 150 150 152 156 162 167 Chapter 3Generalized Governing Equations: Local Instance Formulations 3.1 Introduction viii Transport Phenomena in Multiphase Systems 177 177 3.2 Macroscopic (Integral) Formulation 3.2.1 Conservation of Mass 3.2.2 Momentum Equation 3.2.3 Energy Equation 3.2.4 The Second Law of Thermodynamics 3.2.5 Species 3.3 Microscopic (Differential) Formulation 3.3.1 Conservation of Mass 3.3.2 Momentum Equation 3.3.3 Energy Equation 3.3.4 The Second Law of Thermodynamics 3.3.5 Species 3.3.6 Jump Conditions at the Interfaces 3.3.7 Classification of PDEs and Boundary Conditions 3.3.8 Rarefied Vapor Self-Diffusion Model 3.3.9 An Extension: Combustion References 230 Problems 231 179 183 183 185 186 187 189 190 191 193 198 199 206 219 222 223 Chapter 4Generalized Governing Equations: Averaging Formulations 4.1 Introduction 4.2 Overview of Averaging Approaches 4.2.1 Eulerian Averaging 4.2.2 Lagrangian Averaging 4.2.3 Molecular Statistical Averaging 4.3 Volume-Averaged Multi-Fluid Models 4.3.1 Continuity Equation 4.3.2 Momentum Equation 4.3.3 Energy Equation 4.3.4 The Second Law of Thermodynamics 4.3.5 Species 4.4 Volume-Averaged Homogeneous Model 4.4.1 Continuity Equation 4.4.2 Momentum Equation 4.4.3 Energy Equation 4.4.4 The Second Law of Thermodynamics 4.4.5 Species 4.5 Area-Averaged Models for Channel Flows 4.5.1 Homogeneous Model 4.5.2 Separated Flow Model 4.6 An Extension: Porous Media 4.6.1 Conservation of Mass 4.6.2 Conservation of Momentum 4.6.3 Energy Equation 238 238 239 239 243 243 244 244 245 248 251 252 258 258 258 259 260 260 264 266 270 273 275 276 285 Table of Contents ix 4.6.4 The Second Law of Thermodynamics 4.6.5 Species 4.6.6 Multiphase Transport in Porous Media 4.7 Boltzmann Statistical Averaging 4.7.1 Boltzmann Equation 4.7.2 Lattice Boltzmann Model (LBM) 4.7.3 LBM for Multiphase Flows References Problems 320 287 288 289 303 303 311 314 318 Chapter 5Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer 5.1 Introduction 5.2 Surface Tension 5.2.1 Capillary Pressure: The Young-Laplace Equation 5.2.2 Interface Shapes at Equilibrium 5.2.3 Effects of Interfacial Tension Gradients 5.3 Wetting Phenomena and Contact Angles 5.3.1 Equilibrium and Apparent Contact Angles 5.3.2 Wettability and Adsorption 5.4 Phase Equilibrium in Microscale Interfacial Systems 5.4.1 Ultra-Thin Liquid Films, Disjoining Pressure 5.4.2 Change in Saturated Vapor Pressure over a Curved Interface 5.5 Transport Effects at the Interface 5.5.1 Interfacial Mass, Momentum, Energy, and Species Balances 5.5.2 Interfacial Resistance in Vaporization and Condensation 5.5.3 Formation of and Heat Transfer through Thin Liquid Films 5.5.4 Heat Transfer in the Thin-Film Region of an Axially-Grooved Structure 5.6 Dynamic Behaviors of Interfaces 5.6.1 Rayleigh-Taylor and Kelvin-Helmholtz Instabilities 5.6.2 Surface Waves on Liquid Film Flow 5.7 Numerical Simulation of Interfaces and Free Surfaces 5.7.1 Continuum Approach: Interface Tracking Techniques 5.7.2 Noncontinuum Approach: Molecular Dynamic Simulation References 406 Problems 410 331 331 332 332 334 337 342 342 344 347 347 350 351 351 366 368 373 382 382 386 392 393 402 Chapter 6Melting and Solidification 6.1 Introduction 6.2 Boundary Conditions at the Solid-Liquid Interface 6.3 Exact Solution 6.3.1 Governing Equations of the Solidification Problem 6.3.2 Dimensionless Form of the Governing Equations 6.3.3 Exact Solution of the One-Region Problem x Transport Phenomena in Multiphase Systems 421 421 424 427 427 429 430 6.3.4 Exact Solution of the Two-Region Problem 6.4 Integral Approximate Solution 6.4.1 Heat Conduction in a Semi-Infinite Body 6.4.2 One-Region Problem 6.4.3 Two-Region Problem 6.4.5 Solidification/Melting in Cylindrical Coordinate Systems 6.5 Numerical Simulation 6.5.1 Overview 6.5.2 Enthalpy Method 6.5.3 Equivalent Heat Capacity Method 6.5.4 Temperature-Transforming Model 6.6 Solidification of a Binary Solution System 6.6.1 Overview 6.6.2 Integral Approximate Method 6.6.3 Mixture Model 6.6.4 Volume-Averaging Model 6.7 Contact Melting in a Rectangular Cavity 6.7.1 Fixed Melting and Contact Melting 6.7.2 Contact Melting in a Rectangular Cavity 6.8 Melting and Solidification in Porous Media 6.8.1 Convection-Controlled One-Region Melting Problem 6.8.2 An Enthalpy Model for Two-Region Melting/Solidification 6.9 Applications of Solid-Liquid Phase Change 6.9.1 Latent Heat Thermal Energy Storage 6.9.2 Heat Pipe Startup from Frozen State 6.9.3 Thermal Protection from Intense Localized Heating Using PCM 6.9.4 Microwave Thawing of Food and Biological Materials 6.9.5 Laser Drilling 6.9.6 Selective Laser Sintering (SLS) of Metal Powders 6.10 Microscale Phase Change 6.10.1 Overview 6.10.2 Two-Step Model References 517 Problems 522 435 436 436 439 443 450 454 454 455 460 462 465 465 466 471 475 479 479 480 484 484 488 492 492 495 499 501 505 508 512 512 513 Chapter 7Sublimation and Vapor Deposition 7.1 Introduction 7.2 Sublimation 7.2.1 Sublimation over a Flat Plate 7.2.2 Sublimation inside an Adiabatic Tube 7.2.3 Sublimation inside a Tube Subjected to External Heating 7.2.4 Sublimation with Chemical Reaction 7.3 Chemical Vapor Deposition (CVD) 7.3.1 Introduction 531 531 534 534 538 543 550 554 554 Table of Contents xi 7.3.2 Governing Equations of CVD 7.3.3 Transport Properties 7.3.4 Typical Selected Applications References 574 Problems 577 557 559 563 Chapter 8Condensation 8.1 Introduction 8.2 Dropwise Condensation 8.2.1 Dropwise Condensation Formation Theories 8.2.2 Critical Droplet Radius for Spontaneous Growth and Destruction 8.2.3 Thermal Resistances in the Condensation Processes 8.2.4 Heat Transfer Coefficient for Dropwise Condensation 8.3 Filmwise Condensation 8.3.1 Regimes of Filmwise Condensation 8.3.2 Modeling of Laminar Film Condensation of a Binary Vapor Mixture 8.3.3 Filmwise Condensation in a Stagnant Pure Vapor Reservoir 8.3.4 Effects of Vapor Motion 8.3.5 Turbulent Film Condensation 8.3.6 Other Filmwise Condensation Configurations 8.3.7 Effects of Noncondensable Gas 8.3.8 Flooding or Entrainment Limit 8.4 Nongravitational Condensate Removal 8.4.1 Condensation in a Tube with Suction at the Porous Wall 8.4.2 Annular Condensation Heat Transfer in a Microgravity Environment 8.4.3 Condensation Removal by a Centrifugal Field via a Rotating Disk 8.4.4 Condensation by Capillary Action in a Heat Pipe 8.5 Film Condensation in Porous Media 8.5.1 Overview 8.5.2 Gravity-Dominated Film Condensation on an Inclined Wall 8.5.3 Effect of Surface Tension on Condensation in Porous Media References 667 Problems 670 581 581 587 587 588 594 597 599 599 600 605 614 619 625 627 633 638 638 643 649 652 658 658 660 663 Chapter 9Evaporation 9.1 Introduction 9.2 Classification and Criteria of Evaporation 9.3 Evaporation from an Adiabatic Wall 9.3.1 Evaporation from Horizontal Films 9.3.2 Evaporation from a Vertical Falling Film 678 678 681 685 685 691 xii Transport Phenomena in Multiphase Systems 9.4 Falling Film Evaporation on a Heated Wall 9.4.1 Classical Nusselt Evaporation 9.4.2 Laminar Falling Film with Surface Waves 9.4.3 Turbulent Falling Film 9.4.4 Surface Spray Cooling 9.4.5 Evaporation from a Wedge or Cone Embedded in a Porous Medium 9.5 Direct Contact Evaporation 9.5.1 Evaporation of a Liquid Droplet in a Hot Gas 9.5.2 Evaporation of a Liquid Jet in a Pure Vapor 9.6 Evaporation inside Pores and Slots/Microchannels 9.6.1 Evaporation from Cylindrical Pore under Low/Moderate Heat Flux 9.6.2 Fluid Flow Effect in Pore/Slots during Evaporation 9.6.3 Evaporation under High Heat Flux 9.6.4 Evaporation in an Inclined Microchannel 9.7 Evaporation from Inverted Meniscus in Porous Media References 755 Problems 758 701 702 706 714 716 718 724 724 727 729 730 734 738 744 747 Chapter 10Boiling 10.1 Introduction 10.2 Pool Boiling Regimes 10.3 Nucleate Boiling 10.3.1 Nucleation and Inception 10.3.2 Bubble Dynamics 10.3.3 Bubble Detachment 10.3.4 Nucleate Site Density 10.3.5 Bubble Growth and Merger 10.3.6 Heat Transfer in Nucleate Boiling 10.4 Critical Heat Flux 10.5 Transition Boiling and Minimum Heat Flux 10.5.1 Transition Boiling 10.5.2 Minimum Heat Flux 10.6 Film Boiling 10.6.1 Film Boiling Analysis 10.6.2 Direct Numerical Simulation of Film Boiling 10.6.3 Leidenfrost Phenomena 10.7 Boiling in Porous Media 10.7.1 Nucleate Boiling in a Wicked Surface 10.7.2 Boiling in Porous Media Heated from Below 10.7.3 Film Boiling Analysis in Porous Media References 843 Problems 849 765 765 767 770 770 775 786 791 792 796 803 806 806 809 811 811 821 824 833 833 837 840 Table of Contents xiii Chapter 11Two-Phase Flow and Heat Transfer 11.1 Introduction 11.2 Flow Patterns of Liquid-Vapor (Gas) Two-Phase Flow 11.2.1 Concepts and Notations 11.2.2 Flow Patterns in Vertical Tubes 11.2.3 Flow Patterns in Horizontal Tubes 11.3 Two-Phase Flow Models 11.3.1 Homogeneous Flow Model 11.3.2 Separated Flow Model 11.3.3 Frictional Pressure Drop 11.3.4 Void Fraction 11.4 Forced Convective Condensation in Tubes 11.4.1 Two-Phase Flow Regimes 11.4.2 Heat Transfer Predictions 11.5 Forced Convective Boiling in Tubes 11.5.1 Regimes in Horizontal and Vertical Tubes 11.5.2 Bubble Lift-Off Size in Forced Convective Boiling 11.5.3 Heat Transfer Predictions 11.6 Two-Phase Flow and Heat Transfer in Micro- and Minichannels 11.6.1 Two-Phase Flow Patterns 11.6.2 Flow Condensation 11.6.3 Flow Evaporation and Boiling References 937 Problems 944 853 853 854 854 857 861 864 864 866 870 878 883 883 885 889 889 895 899 904 904 908 920 Appendix A: Constants and Conversion Factors Appendix B: Thermophysical Properties Appendix C: Vectors and Tensors Index 950 954 1006 1013 xiv Transport Phenomena in Multiphase Systems Preface Transport phenomena in multiphase systems with phase change is of great interest to scientists and engineers working in the power, nuclear, chemical processes, environmental, microelectronics, biotechnology, nano-technology, polymer science, food processing, cryogenics, space, and many other industries, from the established to emerging multidisciplinary technologies. For example, almost two-thirds of industrial heat exchangers undergo phase change; therefore, physical understanding and development of the first principal models are not only of interest in fundamental research, they also are greatly needed for a more accurate and reliable design of multiphase thermal systems. The subject of transport phenomena in a multiphase system with phase change is important, because a unified physical/mathematical treatment is essential for engineering practitioners in the 21 st century, who must cope with issues such as high heat flux and micro- or nanoscale systems for various applications. Our motive in preparing this new textbook was to address the challenges and opportunities facing graduate education and teaching in thermal sciences within the mechanical engineering discipline and/or advanced transport phenomena in chemical engineering, which have remained basically unchanged for five decades. For example, the convection and/or conduction courses offered by most mechanical engineering departments as core courses in thermal sciences focus almost exclusively on single-phase, single-component, simple geometry such as channel flows or flat plates with the goal of an analytical solution with the continuum approach. Similarly, advanced transport phenomena in chemical engineering are based mostly on the excellent classical book by Bird et al., which was originally published in 1960. In contrast with their educational training, practicing engineers working in the thermal sciences or scientists in academia and the private sector have in recent years focused mostly on multiphase, multicomponent, non-conventional geometries, with coupled heat and mass transfer and phase change, with the goal of developing a numerical simulation using a continuum or non-continuum approach. We therefore developed this new textbook with the intention of helping instructors to bridge classroom learning and engineering practice xv by offering them advanced fundamental and general course materials that can replace conventional, limited, approaches for teaching advanced heat and mass transfer or transport phenomena. The purpose of this textbook is to accurately present the basic principles for analyzing transport phenomena in multiphase systems and to demonstrate their wide variety of possible applications. Since it would take many book volumes to do justice to all aspects of multiphase systems, the scope of this book is limited to thermodynamics and momentum, heat and mass transfer fundamentals, with emphases on melting, solidification, sublimation, vapor deposition, condensation, evaporation, boiling, and two-phase flow. Several books over the last 20 years have summarized the state of the art in liquid vapor systems. No serious attempts were made to bring all three forms of phase change, i.e., liquid vapor, solid liquid, and solid vapor, into one volume and to describe them from one perspective (in this text, pairs of arrows, , are used to portray energy and mass exchange associated with multiphase transfer between the phases listed). Furthermore, most of the existing texts were developed as monographs rather than textbooks. In writing this textbook, our goal was to provide basic engineering fundamentals related to transport phenomena in multiphase systems with phase change, including microscale and porosity effects. In most cases, the basic physical phenomena are presented with different mathematical models. Historically, the field of transport phenomena has developed successful textbooks for momentum, heat and mass transfer in single-phase systems because these are straightforward and well developed concepts, in terms of physical and mathematical modeling. The same is not true for multiphase systems, which involve some components of the semi-empirical approach, are much more complex, and are thus less well understood. However, because of significant developments in transport phenomena in multiphase systems with phase change during the last two decades, we have much better physical, analytical, and numerical tools to model these types of problems: this is the purpose of our textbook. Furthermore, traditionally three approaches were used to present transport phenomena: microscopic (differential), macroscopic (integral) and molecular level. Most heat transfer textbooks place the emphasis on microscopic and/or macroscopic. With the importance of microscale heat transfer or transport phenomena in applications of nanotechnology and biotechnology, as well as molecular dynamic simulations, it is important to discuss the molecular approach and the connection between the molecular and microscopic approaches. In this textbook, an attempt is made to better describe this relationship. For example, the generalized conservation equations in Chapter 3 have been developed not only microscopically and macroscopically using the continuum approach, but also using the Boltzmann equation. There are three types of information available in the area of transport phenomena in multiphase systems that can be covered in a textbook of this nature: xvi Transport Phenomena in Multiphase Systems 1. Significant existing experimental work and correlations 2. Analytical and physical models 3. Numerical simulation modeling due to recent significant advances in digital computers and computational methodologies We have not presented much in the way of item 1 except well established semi-empirical correlations that have been accepted in practice. The emphasis in this book is on the last two items. With respect to the final item, note that this is not a numerical method book; however, we have set up the framework so that students who wish to pursue this approach are equipped with the basic background material necessary to use existing commercial computer codes. Numerical methodologies and approaches are presented if they are specific to multiphase systems with phase change. Analytical and numerical physical models of transport phenomena in multiphase systems are the main focus in this textbook. Chapters 1 through 4 present materials that are fundamental to the entire text. These chapters should be considered before proceeding to other chapters. Chapter 1 begins with a review of the concept of phases of matter and a discussion of the role of phases in systems that include, simultaneously, more than one phase. This is followed by a review of transport phenomena with detailed emphasis in multicomponent systems, microscale heat transfer, dimensional analysis, and scaling. The processes of phase change between solid, liquid, and vapor are also reviewed, and the classification of multiphase systems is presented. Finally, some typical practical applications are described, which require students to understand the operational principles of these multiphase devices for further understanding and application in homeworks and examples in future chapters. The thermodynamics of multiphase systems is presented in Chapter 2, which begins with a review of single-phase thermodynamics, including thermodynamic laws and relations, and proceeds to the concepts of equilibrium and stability. This is followed by discussion of thermodynamic surfaces and phase diagrams for single- and multicomponent systems. Also discussed are equilibrium criteria for single and multicomponent multiphase systems and the metastable equilibrium that exists in a multiphase system. Chapter 2 concludes with a discussion of thermodynamics at the interface and the effects of surface tension and disjoining pressure, including the superheat effect. Chapter 3 presents the generalized macroscopic (integral) and microscopic (differential) governing equations for multiphase systems in local-instance formulations. The instantaneous formulation requires a differential balance for each phase, combined with appropriate jump and boundary conditions to match the solution of these differential equations at the interfaces. Also discussed in Chapter 3 are a rarefied vapor self-diffusion model and the application of the differential formulations to combustion. The generalized governing equations for multiphase systems in averaged formulations are presented in Chapter 4. The averaged formulations are obtained by averaging the govern- Preface xvii ing equations within a small time interval (time average) or a small control volume (spatial average). The governing equations for the multidimensional multi-fluid and homogeneous models, as well as area-averaged governing equations for one-dimensional flows, are also discussed. Chapter 4 also covers single- and multiphase transport phenomena in porous media, including multi-fluid and mixture models. Finally, Boltzmann statistical averaging, including a detailed discussion of the Boltzmann equation and the Lattice Boltzmann method for modeling both single and multiphase systems, is presented. Vector and tensor notations have been used in the development of generalized governing equations in Chapters 3 and 4. The neatness, generality, and compactness of vector and tensor notations are considered sufficient to overcome the criticism of those who may consider the subject too sophisticated. Examples in Chapters 3 and 4 and applications of these in non-vectorial one-, two-, or three-dimensional forms for various geometries in following chapters will provide adequate experience. In many examples, equations for simple one-dimensional processes are also developed based on actual physical mass, momentum, and energy balance, so that students appreciate the physical significance of various terms. Chapter 5 introduces the concepts of surface tension, wetting phenomena, and contact angle, which are followed by a discussion on motion induced by capillarity. Additional detailed descriptions are presented for interfacial balances and boundary conditions for mass, momentum, energy, and species for multicomponent and multiphase interface. Also considered in Chapter 5 are heat and mass transfer through the thin film region during evaporation and condensation, including the effect of interfacial resistance and disjoining pressure. The dynamics of interfaces, including stability and wave effects, are presented. Finally, numerical simulations of interfaces and free surfaces using both continuum and non-continuum approaches are provided. Solid-liquid phase change, including melting and solidification, is treated in Chapter 6, starting with the classification of solid-liquid phase changes and generalized boundary conditions at the interface. Different approaches to the solution of melting and solidification problems, including exact, integral approximate, and numerical solutions, are introduced. Solidification in binary solution systems, contact melting, melting and solidification in porous media, applications of solid-liquid phase change, and microscale solid-liquid phase change are also presented. Solid-vapor phase change, including sublimation and vapor deposition, is introduced in Chapter 7. The discussion begins with a brief overview of solid-vapor phase change and proceeds to detailed analyses on sublimation without and with chemical reaction, as well as physical and chemical vapor deposition. Chapter 8 begins with a discussion of two main modes of liquid droplet embryo formation in condensation: homogeneous and heterogeneous, followed by a detailed examination of dropwise and filmwise condensation at both macro- and microscale levels. Applications of condensation in micro- xviii Transport Phenomena in Multiphase Systems gravity and condensation in porous media are also discussed. Chapter 9 presents criteria and classification of evaporation, evaporation from an adiabatic wall, evaporation from a heated wall, evaporation in porous media, evaporation in micro/miniature channels, as well as direct-contact evaporation. Chapter 10 introduces the pool boiling curve and characterizes the various boiling regimes (free convection, nucleate, transition, and film boiling), followed by detailed discussions of each of the four pool boiling regimes, critical heat flux, minimum heat flux, and direct numerical simulation. Also discussed in Chapter 10 are the Leidenfrost phenomena as well as physical phenomena of boiling in porous media. Chapter 11 starts with definitions of various parameters for two-phase flow and flow patterns in vertical and horizontal tubes. This is followed by two-phase flow models as well as prediction of pressure drops and void fractions. Finally, the two-phase flow regimes and heat transfer characteristics for forced convective condensation and boiling at both macro- and microscale levels are presented. The International System of Units (SI) is used throughout the book, and the conversion factors for different unit systems are provided in Appendix A. The complete thermophysical properties for all phases of various substances, along with empirical correlations of thermal properties as functions of temperature, are provided in Appendix B. Appendix C provides a brief review of vector and tensor operations. We have used consistent symbols throughout the book. However, we have used some symbols for more than one purpose in a number of cases. We believe the context, as well as the nomenclature section, will clarify the meaning of the symbols used in these cases. This textbook is designed for use as an advanced-level undergraduate or graduate textbook in mechanical engineering, chemical engineering, material science and engineering, nuclear engineering, biomedical engineering, or environmental engineering. It offers examples and homework problems as well as references from engineering and research applications related to multiphase systems. The only prerequisite courses necessary for the material are undergraduate thermodynamics, and heat transfer or transport phenomena. No graduate course in convection, conduction, or transport phenomena is required. In fact, convection, conduction, and/or transport phenomena are special cases of the general material presented here, if taught properly. We recognize a new trend at a number of universities to offer a single course in transport phenomena of multiphase system for all disciplines, and therefore we have tried to cover the materials that various departments might wish to have included in such a course. The materials included in this text may require more than one semester of instruction depending on the desired level of completeness. Therefore, it is recommended that the instructor choose the materials to be covered based on the background and needs of the students. This text is not intended as a reference tool or handbook summarizing the state-of-the-art, nor does it to detail the history of multiphase systems with phase change. Part of the text was developed originally from lecture notes pre- Preface xix pared by one of the authors (AF) who was teaching a graduate-level course at the University of Connecticut. Materials have been considerably rewritten by both authors and used as lecture notes for senior elective and/or graduate-level courses taught by the authors at the University of Connecticut, New Mexico State University, and the University of Missouri-Columbia. This textbook is suitable for students from a wide variety of backgrounds. The examples and homework problems were added to provide students a better physical understanding of theoretical concepts and uses for various applications. While the examples are designed to confer a better physical understanding, including mathematical modeling and a feeling for the order of magnitude of variables, end-of-chapter homework problems will help students appreciate fundamental concepts. There are three types of problems we have developed for this textbook: (1) simple numerical manipulation, (2) detailed physical and analytical models, and (3) open-ended problems. It is important that students gain experience in solving all three types of problems. A copyrighted solution manual and Microsoft PowerPoint presentation package are provided only to those instructors who adopt the book for the course. The authors would like to express their deep thanks to a number of distinguished members of the heat transfer community who shared their expertise and time in reviewing this book: Thomas Avedisian, Christopher Beckermann, Arthur Bergles, F.B. Cheung, John Howell, Raymond Viskanta, and Ralph Webb. In addition, we wish to thank the following individuals who generously reviewed individual chapters or part of the book: Yutaka Asako, Theodore Bergman, Yiding Cao, Baki Cetegen, Wilson Chiu, Emily Green, Hongbin Ma, Robert McGurgan, Dmitry Khrustalev, Roop Mahajan, Gregory Jewett, Ugur Pasaogullari, Ranga Pitchumani, Jeremy Rice, Scott Thomas, and Kambiz Vafai. We are grateful to these dedicated professionals for their support, sage advice, improvements, and additions, which resulted in a superior and more comprehensive text than we envisioned. It is important to acknowledge the contributions of students over the last several years who were taught from the manuscripts out of which this book evolved. Our special thanks to Nan Cooper and Emily Jerome for their expert editing of the manuscripts. This textbook provides an opportunity to cover fundamentals of transport phenomena in multiphase systems with all forms of phase change from one perspective. It is our hope that this textbook will influence some engineering colleges to treat transport phenomena in multiphase systems as a core requirement of the graduate curriculum in mechanical, chemical, environmental, nuclear, biomedical, and materials science disciplines. Your recommendations, comments, and criticisms are appreciated. Amir Faghri Yuwen Zhang xx Transport Phenomena in Multiphase Systems