Principles of Heat and Mass Transfer

Symbols xxi


CHAPTER 1 Introduction 1


1.1 What and How? 2


1.2 Physical Origins and Rate Equations 3


1.2.1 Conduction 3


1.2.2 Convection 6


1.2.3 Radiation 8


1.2.4 The Thermal Resistance Concept 12


1.3 Relationship to Thermodynamics 12


1.3.1 Relationship to the First Law of Thermodynamics (Conservation of Energy) 13


1.3.2 Relationship to the Second Law of Thermodynamics and the Efficiency of Heat Engines 31


1.4 Units and Dimensions 36


1.5 Analysis of Heat Transfer Problems: Methodology 38


1.6 Relevance of Heat Transfer 41


1.7 Summary 45


References 48


Problems 49


CHAPTER 2 Introduction to Conduction 67


2.1 The Conduction Rate Equation 68


2.2 The Thermal Properties of Matter 70


2.2.1 Thermal Conductivity 70


2.2.2 Other Relevant Properties 78


2.3 The Heat Diffusion Equation 82


2.4 Boundary and Initial Conditions 90


2.5 Summary 94


References 95


Problems 95


CHAPTER 3 One–Dimensional, Steady–State Conduction 111


3.1 The Plane Wall 112


3.1.1 Temperature Distribution 112


3.1.2 Thermal Resistance 114


3.1.3 The Composite Wall 115


3.1.4 Contact Resistance 117


3.1.5 Porous Media 119


3.2 An Alternative Conduction Analysis 132


3.3 Radial Systems 136


3.3.1 The Cylinder 136


3.3.2 The Sphere 141


3.4 Summary of One–Dimensional Conduction Results 142


3.5 Conduction with Thermal Energy Generation 142


3.5.1 The Plane Wall 143


3.5.2 Radial Systems 149


3.5.3 Tabulated Solutions 150


3.5.4 Application of Resistance Concepts 150


3.6 Heat Transfer from Extended Surfaces 154


3.6.1 A General Conduction Analysis 156


3.6.2 Fins of Uniform Cross–Sectional Area 158


3.6.3 Fin Performance 164


3.6.4 Fins of Nonuniform Cross–Sectional Area 167


3.6.5 Overall Surface Efficiency 170


3.7 The Bioheat Equation 178


3.8 Thermoelectric Power Generation 182


3.9 Micro– and Nanoscale Conduction 189


3.9.1 Conduction Through Thin Gas Layers 189


3.9.2 Conduction Through Thin Solid Films 190


3.10 Summary 190


References 193


Problems 193


CHAPTER 4 Two–Dimensional, Steady–State Conduction 229


4.1 Alternative Approaches 230


4.2 The Method of Separation of Variables 231


4.3 The Conduction Shape Factor and the Dimensionless Conduction Heat Rate 235


4.4 Finite–Difference Equations 241


4.4.1 The Nodal Network 241


4.4.2 Finite–Difference Form of the Heat Equation 242


4.4.3 The Energy Balance Method 243


4.5 Solving the Finite–Difference Equations 250


4.5.1 Formulation as a Matrix Equation 250


4.5.2 Verifying the Accuracy of the Solution 251


4.6 Summary 256


References 257


Problems 257


4S.1 The Graphical Method W–1


4S.1.1 Methodology of Constructing a Flux Plot W–1


4S.1.2 Determination of the Heat Transfer Rate W–2


4S.1.3 The Conduction Shape Factor W–3


4S.2 The Gauss Seidel Method: Example of Usage W–5


References W–9


Problems W–10


CHAPTER 5 Transient Conduction 279


5.1 The Lumped Capacitance Method 280


5.2 Validity of the Lumped Capacitance Method 283


5.3 General Lumped Capacitance Analysis 287


5.3.1 Radiation Only 288


5.3.2 Negligible Radiation 288


5.3.3 Convection Only with Variable Convection Coefficient 289


5.3.4 Additional Considerations 289


5.4 Spatial Effects 298


5.5 The Plane Wall with Convection 299


5.5.1 Exact Solution 300


5.5.2 Approximate Solution 300


5.5.3 Total Energy Transfer 302


5.5.4 Additional Considerations 302


5.6 Radial Systems with Convection 303


5.6.1 Exact Solutions 303


5.6.2 Approximate Solutions 304


5.6.3 Total Energy Transfer 304


5.6.4 Additional Considerations 305


5.7 The Semi–Infinite Solid 310


5.8 Objects with Constant Surface Temperatures or Surface Heat Fluxes 317


5.8.1 Constant Temperature Boundary Conditions 317


5.8.2 Constant Heat Flux Boundary Conditions 319


5.8.3 Approximate Solutions 320


5.9 Periodic Heating 327


5.10 Finite–Difference Methods 330


5.10.1 Discretization of the Heat Equation: The Explicit Method 330


5.10.2 Discretization of the Heat Equation: The Implicit Method 337


5.11 Summary 345


References 346


Problems 346


5S.1 Graphical Representation of One–Dimensional, Transient Conduction in the


Plane Wall, Long Cylinder, and Sphere W–12


5S.2 Analytical Solutions of Multidimensional Effects W–16


References W–22


Problems W–22


CHAPTER 6 Introduction to Convection 377


6.1 The Convection Boundary Layers 378


6.1.1 The Velocity Boundary Layer 378


6.1.2 The Thermal Boundary Layer 379


6.1.3 The Concentration Boundary Layer 380


6.1.4 Significance of the Boundary Layers 382


6.2 Local and Average Convection Coefficients 382


6.2.1 Heat Transfer 382


6.2.2 Mass Transfer 383


6.2.3 The Problem of Convection 385


6.3 Laminar and Turbulent Flow 389


6.3.1 Laminar and Turbulent Velocity Boundary Layers 389


6.3.2 Laminar and Turbulent Thermal and Species Concentration Boundary Layers 391


6.4 The Boundary Layer Equations 394


6.4.1 Boundary Layer Equations for Laminar Flow 394


6.4.2 Compressible Flow 397


6.5 Boundary Layer Similarity: The Normalized Boundary Layer Equations 398


6.5.1 Boundary Layer Similarity Parameters 398


6.5.2 Functional Form of the Solutions 400


6.6 Physical Interpretation of the Dimensionless Parameters 407


6.7 Boundary Layer Analogies 409


6.7.1 The Heat and Mass Transfer Analogy 410


6.7.2 Evaporative Cooling 413


6.7.3 The Reynolds Analogy 416


6.8 Summary 417


References 418


Problems 419


6S.1 Derivation of the Convection Transfer Equations W–25


6S.1.1 Conservation of Mass W–25


6S.1.2 Newton s Second Law of Motion W–26


6S.1.3 Conservation of Energy W–29


6S.1.4 Conservation of Species W–32


References W–36


Problems W–36


CHAPTER 7 External Flow 433


7.1 The Empirical Method 435


7.2 The Flat Plate in Parallel Flow 436


7.2.1 Laminar Flow over an Isothermal Plate: A Similarity Solution 437


7.2.2 Turbulent Flow over an Isothermal Plate 443


7.2.3 Mixed Boundary Layer Conditions 444


7.2.4 Unheated Starting Length 445


7.2.5 Flat Plates with Constant Heat Flux Conditions 446


7.2.6 Limitations on Use of Convection Coefficients 446


7.3 Methodology for a Convection Calculation 447


7.4 The Cylinder in Cross Flow 455


7.4.1 Flow Considerations 455


7.4.2 Convection Heat and Mass Transfer 457


7.5 The Sphere 465


7.6 Flow Across Banks of Tubes 468


7.7 Impinging Jets 477


7.7.1 Hydrodynamic and Geometric Considerations 477


7.7.2 Convection Heat and Mass Transfer 478


7.8 Packed Beds 482


7.9 Summary 483


References 486


Problems 486


CHAPTER 8 Internal Flow 517


8.1 Hydrodynamic Considerations 518


8.1.1 Flow Conditions 518


8.1.2 The Mean Velocity 519


8.1.3 Velocity Profile in the Fully Developed Region 520


8.1.4 Pressure Gradient and Friction Factor in Fully Developed Flow 522


8.2 Thermal Considerations 523


8.2.1 The Mean Temperature 524


8.2.2 Newton s Law of Cooling 525


8.2.3 Fully Developed Conditions 525


8.3 The Energy Balance 529


8.3.1 General Considerations 529


8.3.2 Constant Surface Heat Flux 530


8.3.3 Constant Surface Temperature 533


8.4 Laminar Flow in Circular Tubes: Thermal Analysis and Convection Correlations 537


8.4.1 The Fully Developed Region 537


8.4.2 The Entry Region 542


8.4.3 Temperature–Dependent Properties 544


8.5 Convection Correlations: Turbulent Flow in Circular Tubes 544


8.6 Convection Correlations: Noncircular Tubes and the Concentric Tube Annulus 552


8.7 Heat Transfer Enhancement 555


8.8 Flow in Small Channels 558


8.8.1 Microscale Convection in Gases 558


8.8.2 Microscale Convection in Liquids 559


8.8.3 Nanoscale Convection 560


8.9 Convection Mass Transfer 563


8.10 Summary 565


References 568


Problems 569


CHAPTER 9 Free Convection 593


9.1 Physical Considerations 594


9.2 The Governing Equations for Laminar Boundary Layers 597


9.3 Similarity Considerations 598


9.4 Laminar Free Convection on a Vertical Surface 599


9.5 The Effects of Turbulence 602


9.6 Empirical Correlations: External Free Convection Flows 604


9.6.1 The Vertical Plate 605


9.6.2 Inclined and Horizontal Plates 608


9.6.3 The Long Horizontal Cylinder 613


9.6.4 Spheres 617


9.7 Free Convection Within Parallel Plate Channels 618


9.7.1 Vertical Channels 619


9.7.2 Inclined Channels 621


9.8 Empirical Correlations: Enclosures 621


9.8.1 Rectangular Cavities 621


9.8.2 Concentric Cylinders 624


9.8.3 Concentric Spheres 625


9.9 Combined Free and Forced Convection 627


9.10 Convection Mass Transfer 628


9.11 Summary 629


References 630


Problems 631


CHAPTER 10 Boiling and Condensation 653


10.1 Dimensionless Parameters in Boiling and Condensation 654


10.2 Boiling Modes 655


10.3 Pool Boiling 656


10.3.1 The Boiling Curve 656


10.3.2 Modes of Pool Boiling 657


10.4 Pool Boiling Correlations 660


10.4.1 Nucleate Pool Boiling 660


10.4.2 Critical Heat Flux for Nucleate Pool Boiling 662


10.4.3 Minimum Heat Flux 663


10.4.4 Film Pool Boiling 663


10.4.5 Parametric Effects on Pool Boiling 664


10.5 Forced Convection Boiling 669


10.5.1 External Forced Convection Boiling 670


10.5.2 Two–Phase Flow 670


10.5.3 Two–Phase Flow in Microchannels 673


10.6 Condensation: Physical Mechanisms 673


10.7 Laminar Film Condensation on a Vertical Plate 675


10.8 Turbulent Film Condensation 679


10.9 Film Condensation on Radial Systems 684


10.10 Condensation in Horizontal Tubes 689


10.11 Dropwise Condensation 690


10.12 Summary 691


References 691


Problems 693


CHAPTER 11 Heat Exchangers 705


11.1 Heat Exchanger Types 706


11.2 The Overall Heat Transfer Coefficient 708


11.3 Heat Exchanger Analysis: Use of the Log Mean Temperature Difference 711


11.3.1 The Parallel–Flow Heat Exchanger 712


11.3.2 The Counterflow Heat Exchanger 714


11.3.3 Special Operating Conditions 715


11.4 Heat Exchanger Analysis: The Effectiveness NTU Method 722


11.4.1 Definitions 722


11.4.2 Effectiveness NTU Relations 723


11.5 Heat Exchanger Design and Performance Calculations 730


11.6 Additional Considerations 739


11.7 Summary 747


References 748


Problems 748


11S.1 Log Mean Temperature Difference Method for Multipass and Cross–Flow Heat Exchangers W–40


11S.2 Compact Heat Exchangers W–44


References W–49


Problems W–50


CHAPTER 12 Radiation: Processes and Properties 767


12.1 Fundamental Concepts 768


12.2 Radiation Heat Fluxes 771


12.3 Radiation Intensity 773


12.3.1 Mathematical Definitions 773


12.3.2 Radiation Intensity and Its Relation to Emission 774


12.3.3 Relation to Irradiation 779


12.3.4 Relation to Radiosity for an Opaque Surface 781


12.3.5 Relation to the Net Radiative Flux for an Opaque Surface 782


12.4 Blackbody Radiation 782


12.4.1 The Planck Distribution 783


12.4.2 Wien s Displacement Law 784


12.4.3 The Stefan Boltzmann Law 784


12.4.4 Band Emission 785


12.5 Emission from Real Surfaces 792


12.6 Absorption, Reflection, and Transmission by Real Surfaces 801


12.6.1 Absorptivity 802


12.6.2 Reflectivity 803


12.6.3 Transmissivity 805


12.6.4 Special Considerations 805


12.7 Kirchhoff s Law 810


12.8 The Gray Surface 812


12.9 Environmental Radiation 818


12.9.1 Solar Radiation 819


12.9.2 The Atmospheric Radiation Balance 821


12.9.3 Terrestrial Solar Irradiation 823


12.10 Summary 826


References 830


Problems 830


CHAPTER 13 Radiation Exchange Between Surfaces 861


13.1 The View Factor 862


13.1.1 The View Factor Integral 862


13.1.2 View Factor Relations 863


13.2 Blackbody Radiation Exchange 872


13.3 Radiation Exchange Between Opaque, Diffuse, Gray Surfaces in an Enclosure 876


13.3.1 Net Radiation Exchange at a Surface 877


13.3.2 Radiation Exchange Between Surfaces 878


13.3.3 The Two–Surface Enclosure 884


13.3.4 Radiation Shields 886


13.3.5 The Reradiating Surface 888


13.4 Multimode Heat Transfer 893


13.5 Implications of the Simplifying Assumptions 896


13.6 Radiation Exchange with Participating Media 896


13.6.1 Volumetric Absorption 896


13.6.2 Gaseous Emission and Absorption 897


13.7 Summary 901


References 902


Problems 903


CHAPTER 14 Diffusion Mass Transfer 933


14.1 Physical Origins and Rate Equations 934


14.1.1 Physical Origins 934


14.1.2 Mixture Composition 935


14.1.3 Fick s Law of Diffusion 936


14.1.4 Mass Diffusivity 937


14.2 Mass Transfer in Nonstationary Media 939


14.2.1 Absolute and Diffusive Species Fluxes 939


14.2.2 Evaporation in a Column 942


14.3 The Stationary Medium Approximation 947


14.4 Conservation of Species for a Stationary Medium 947


14.4.1 Conservation of Species for a Control Volume 948


14.4.2 The Mass Diffusion Equation 948


14.4.3 Stationary Media with Specified Surface Concentrations 950


14.5 Boundary Conditions and Discontinuous Concentrations at Interfaces 954


14.5.1 Evaporation and Sublimation 955


14.5.2 Solubility of Gases in Liquids and Solids 955


14.5.3 Catalytic Surface Reactions 960


14.6 Mass Diffusion with Homogeneous Chemical Reactions 962


14.7 Transient Diffusion 965


14.8 Summary 971


References 972


Problems 972


APPENDIX A Thermophysical Properties of Matter 981


APPENDIX B Mathematical Relations and Functions 1013


APPENDIX C Thermal Conditions Associated with Uniform Energy Generation in One–Dimensional, Steady–State Systems 1019


APPENDIX D The Gauss Seidel Method 1025


APPENDIX E The Convection Transfer Equations 1027


E.1 Conservation of Mass 1028


E.2 Newton s Second Law of Motion 1028


E.3 Conservation of Energy 1029


E.4 Conservation of Species 1030


APPENDIX F Boundary Layer Equations for Turbulent Flow 1031


APPENDIX G An Integral Laminar Boundary Layer Solution for Parallel Flow over a Flat Plate 1035


Index 1039