Fundamentals of Heat and Mass Transfer

Ted Bergman received his Ph.D. from Purdue University, and has been a faculty member at the University of Kansas (2012 - present), the University of Connecticut (1996 - 2012), and The University of Texas at Austin (1985 - 1996). He directed the Thermal Transport Processes Program at the U.S. National Science Foundation from 2008 to 2010. Early in his career, Dr. Bergman designed the cooling systems of large electric power generation stations.

Adrienne Lavine is Professor and past Department Chair (2006 - 2011) in the Mechanical and Aerospace Engineering Department at the University of California, Los Angeles. She began her academic career there in 1984 as an Assistant Professor after obtaining her Ph.D. in Mechanical Engineering from the University of California, Berkeley.

Symbols xix

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 ... 28
1.4 Units and Dimensions ... 33
1.5 Analysis of Heat Transfer Problems: Methodology ... 35
1.6 Relevance of Heat Transfer ... 38
1.7 Summary ... 42
References ... 45

Chapter 2 Introduction to Conduction ... 47
2.1 The Conduction Rate Equation ... 48
2.2 The Thermal Properties of Matter ... 50
2.2.1 Thermal Conductivity ... 51
2.2.2 Other Relevant Properties ... 58
2.3 The Heat Diffusion Equation ... 62
2.4 Boundary and Initial Conditions ... 70
2.5 Summary ... 74
References ... 75

Chapter 3 One-Dimensional, Steady-State Conduction ... 77
3.1 The Plane Wall ... 78
3.1.1 Temperature Distribution ... 78
3.1.2 Thermal Resistance ... 80
3.1.3 The Composite Wall ... 81
3.1.4 Contact Resistance ... 83
3.1.5 Porous Media ... 85
3.2 An Alternative Conduction Analysis ... 99
3.3 Radial Systems ... 103
3.3.1 The Cylinder ... 103
3.3.2 The Sphere ... 108
3.4 Summary of One-Dimensional Conduction Results ... 109
3.5 Conduction with Thermal Energy Generation ... 109
3.5.1 The Plane Wall ... 110
3.5.2 Radial Systems ... 116
3.5.3 Tabulated Solutions ... 117
3.5.4 Application of Resistance Concepts ... 117
3.6 Heat Transfer from Extended Surfaces ... 121
3.6.1 A General Conduction Analysis ... 123
3.6.2 Fins of Uniform Cross-Sectional Area ... 125
3.6.3 Fin Performance Parameters ... 131
3.6.4 Fins of Nonuniform Cross-Sectional Area ... 134
3.6.5 Overall Surface Efficiency ... 137
3.7 Other Applications of One-Dimensional, Steady-State Conduction ... 141
3.7.1 The Bioheat Equation ... 141
3.7.2 Thermoelectric Power Generation ... 145
3.7.3 Nanoscale Conduction ... 153
3.8 Summary ... 157
References ... 159

Chapter 4 Two-Dimensional, Steady-State Conduction ... 161
4.1 General Considerations and Solution Techniques ... 162
4.2 The Method of Separation of Variables ... 163
4.3 The Conduction Shape Factor and the Dimensionless Conduction Heat Rate ... 167
4.4 Finite-Difference Equations ... 173
4.4.1 The Nodal Network ... 173
4.4.2 Finite-Difference Form of the Heat Equation: No Generation and Constant Properties ... 174
4.4.3 Finite-Difference Form of the Heat Equation: The Energy Balance Method ... 175
4.5 Solving the Finite-Difference Equations ... 182
4.5.1 Formulation as a Matrix Equation ... 182
4.5.2 Verifying the Accuracy of the Solution ... 183
4.6 Summary ... 188
References ... 189

Chapter 5 Transient Conduction ... 191
5.1 The Lumped Capacitance Method ... 192
5.2 Validity of the Lumped Capacitance Method ... 195
5.3 General Lumped Capacitance Analysis ... 199
5.3.1 Radiation Only ... 200
5.3.2 Negligible Radiation ... 200
5.3.3 Convection Only with Variable Convection Coefficient ... 201
5.3.4 Additional Considerations ... 201
5.4 Spatial Effects ... 210
5.5 The Plane Wall with Convection ... 211
5.5.1 Exact Solution ... 212
5.5.2 Approximate Solution ... 212
5.5.3 Total Energy Transfer: Approximate Solution ... 214
5.5.4 Additional Considerations ... 214
5.6 Radial Systems with Convection ... 215
5.6.1 Exact Solutions ... 215
5.6.2 Approximate Solutions ... 216
5.6.3 Total Energy Transfer: Approximate Solutions ... 216
5.6.4 Additional Considerations ... 217
5.7 The Semi-Infinite Solid ... 222
5.8 Objects with Constant Surface Temperatures or Surface Heat Fluxes ... 229
5.8.1 Constant Temperature Boundary Conditions ... 229
5.8.2 Constant Heat Flux Boundary Conditions ... 231
5.8.3 Approximate Solutions ... 232
5.9 Periodic Heating ... 239
5.10 Finite-Difference Methods ... 242
5.10.1 Discretization of the Heat Equation: The Explicit Method ... 242
5.10.2 Discretization of the Heat Equation: The Implicit Method ... 249
5.11 Summary ... 256
References ... 257

Chapter 6 Introduction to Convection ... 259
6.1 The Convection Boundary Layers ... 260
6.1.1 The Velocity Boundary Layer ... 260
6.1.2 The Thermal Boundary Layer ... 261
6.1.3 The Concentration Boundary Layer ... 263
6.1.4 Significance of the Boundary Layers ... 264
6.2 Local and Average Convection Coefficients ... 264
6.2.1 Heat Transfer ... 264
6.2.2 Mass Transfer ... 265
6.3 Laminar and Turbulent Flow ... 271
6.3.1 Laminar and Turbulent Velocity Boundary Layers ... 271
6.3.2 Laminar and Turbulent Thermal and Species Concentration Boundary Layers ... 273
6.4 The Boundary Layer Equations ... 276
6.4.1 Boundary Layer Equations for Laminar Flow ... 277
6.4.2 Compressible Flow ... 280
6.5 Boundary Layer Similarity: The Normalized Boundary Layer Equations ... 280
6.5.1 Boundary Layer Similarity Parameters ... 281
6.5.2 Dependent Dimensionless Parameters ... 281
6.6 Physical Interpretation of the Dimensionless Parameters ... 290
6.7 Boundary Layer Analogies ... 292
6.7.1 The Heat and Mass Transfer Analogy ... 293
6.7.2 Evaporative Cooling ... 296
6.7.3 The Reynolds Analogy ... 299
6.8 Summary ... 300
References ... 301

Chapter 7 External Flow ... 303
7.1 The Empirical Method ... 305
7.2 The Flat Plate in Parallel Flow ... 306
7.2.1 Laminar Flow over an Isothermal Plate: A Similarity Solution ... 307
7.2.2 Turbulent Flow over an Isothermal Plate ... 313
7.2.3 Mixed Boundary Layer Conditions ... 314
7.2.4 Unheated Starting Length ... 315
7.2.5 Flat Plates with Constant Heat Flux Conditions ... 316
7.2.6 Limitations on Use of Convection Coefficients ... 317
7.3 Methodology for a Convection Calculation ... 317
7.4 The Cylinder in Cross Flow ... 325
7.4.1 Flow Considerations ... 325
7.4.2 Convection Heat and Mass Transfer ... 327
7.5 The Sphere ... 335
7.6 Flow Across Banks of Tubes ... 338
7.7 Impinging Jets ... 347
7.7.1 Hydrodynamic and Geometric Considerations ... 347
7.7.2 Convection Heat and Mass Transfer ... 348
7.8 Packed Beds ... 352
7.9 Summary ... 353
References ... 356

Chapter 8 Internal Flow ... 357
8.1 Hydrodynamic Considerations ... 358
8.1.1 Flow Conditions ... 358
8.1.2 The Mean Velocity ... 359
8.1.3 Velocity Profile in the Fully Developed Region ... 360
8.1.4 Pressure Gradient and Friction Factor in Fully Developed Flow ... 362
8.2 Thermal Considerations ... 363
8.2.1 The Mean Temperature ... 364
8.2.2 Newton’s Law of Cooling ... 365
8.2.3 Fully Developed Conditions ... 365
8.3 The Energy Balance ... 369
8.3.1 General Considerations ... 369
8.3.2 Constant Surface Heat Flux ... 370
8.3.3 Constant Surface Temperature ... 373
8.4 Laminar Flow in Circular Tubes: Thermal Analysis and Convection Correlations ... 377
8.4.1 The Fully Developed Region ... 377
8.4.2 The Entry Region ... 382
8.4.3 Temperature-Dependent Properties ... 384
8.5 Convection Correlations: Turbulent Flow in Circular Tubes ... 384
8.6 Convection Correlations: Noncircular Tubes and the Concentric Tube Annulus ... 392
8.7 Heat Transfer Enhancement ... 395
8.8 Forced Convection in Small Channels ... 398
8.8.1 Microscale Convection in Gases (0.1 μm ≤ Dh ≤ 100 μm) ... 398
8.8.2 Microscale Convection in Liquids ... 399
8.8.3 Nanoscale Convection (Dh ≤ 100 nm) ... 400
8.9 Convection Mass Transfer ... 403
8.10 Summary ... 405
References ... 408

Chapter 9 Free Convection ... 409
9.1 Physical Considerations ... 410
9.2 The Governing Equations for Laminar Boundary Layers ... 412
9.3 Similarity Considerations ... 414
9.4 Laminar Free Convection on a Vertical Surface ... 415
9.5 The Effects of Turbulence ... 418
9.6 Empirical Correlations: External Free Convection Flows ... 420
9.6.1 The Vertical Plate ... 421
9.6.2 Inclined and Horizontal Plates ... 424
9.6.3 The Long Horizontal Cylinder ... 429
9.6.4 Spheres ... 433
9.7 Free Convection Within Parallel Plate Channels ... 434
9.7.1 Vertical Channels ... 435
9.7.2 Inclined Channels ... 437
9.8 Empirical Correlations: Enclosures ... 437
9.8.1 Rectangular Cavities ... 437
9.8.2 Concentric Cylinders ... 440
9.8.3 Concentric Spheres ... 441
9.9 Combined Free and Forced Convection ... 443
9.10 Convection Mass Transfer ... 444
9.11 Summary ... 445
References ... 446

Chapter 10 Boiling and Condensation ... 449
10.1 Dimensionless Parameters in Boiling and Condensation ... 450
10.2 Boiling Modes ... 451
10.3 Pool Boiling ... 452
10.3.1 The Boiling Curve ... 452
10.3.2 Modes of Pool Boiling ... 453
10.4 Pool Boiling Correlations ... 456
10.4.1 Nucleate Pool Boiling ... 456
10.4.2 Critical Heat Flux for Nucleate Pool Boiling ... 458
10.4.3 Minimum Heat Flux ... 459
10.4.4 Film Pool Boiling ... 459
10.4.5 Parametric Effects on Pool Boiling ... 460
10.5 Forced Convection Boiling ... 465
10.5.1 External Forced Convection Boiling ... 466
10.5.2 Two-Phase Flow ... 466
10.5.3 Two-Phase Flow in Microchannels ... 469
10.6 Condensation: Physical Mechanisms ... 469
10.7 Laminar Film Condensation on a Vertical Plate ... 471
10.8 Turbulent Film Condensation ... 475
10.9 Film Condensation on Radial Systems ... 480
10.10 Condensation in Horizontal Tubes ... 485
10.11 Dropwise Condensation ... 486
10.12 Summary ... 487
References ... 487

Chapter 11 Heat Exchangers ... 491
11.1 Heat Exchanger Types ... 492
11.2 The Overall Heat Transfer Coefficient ... 494
11.3 Heat Exchanger Analysis: Use of the Log Mean Temperature Difference ... 497
11.3.1 The Parallel-Flow Heat Exchanger ... 498
11.3.2 The Counterflow Heat Exchanger ... 500
11.3.3 Special Operating Conditions ... 501
11.4 Heat Exchanger Analysis: The Effectiveness–NTU Method ... 508
11.4.1 Definitions ... 508
11.4.2 Effectiveness–NTU Relations ... 509
11.5 Heat Exchanger Design and Performance Calculations ... 516
11.6 Additional Considerations ... 525
11.7 Summary ... 533
References ... 534

Chapter 12 Radiation: Processes and Properties ... 535
12.1 Fundamental Concepts ... 536
12.2 Radiation Heat Fluxes ... 539
12.3 Radiation Intensity ... 541
12.3.1 Mathematical Definitions ... 541
12.3.2 Radiation Intensity and Its Relation to Emission ... 542
12.3.3 Relation to Irradiation ... 547
12.3.4 Relation to Radiosity for an Opaque Surface ... 549
12.3.5 Relation to the Net Radiative Flux for an Opaque Surface ... 550
12.4 Blackbody Radiation ... 550
12.4.1 The Planck Distribution ... 551
12.4.2 Wien’s Displacement Law ... 552
12.4.3 The Stefan–Boltzmann Law ... 552
12.4.4 Band Emission ... 553
12.5 Emission from Real Surfaces ... 560
12.6 Absorption, Reflection, and Transmission by Real Surfaces ... 569
12.6.1 Absorptivity ... 570
12.6.2 Reflectivity ... 571
12.6.3 Transmissivity ... 573
12.6.4 Special Considerations ... 573
12.7 Kirchhoff’s Law ... 578
12.8 The Gray Surface ... 580
12.9 Environmental Radiation ... 586
12.9.1 Solar Radiation ... 587
12.9.2 The Atmospheric Radiation Balance ... 589
12.9.3 Terrestrial Solar Irradiation ... 591
12.10 Summary ... 594
References ... 598

Chapter 13 Radiation Exchange Between Surfaces ... 599
13.1 The View Factor ... 600
13.1.1 The View Factor Integral ... 600
13.1.2 View Factor Relations ... 601
13.2 Blackbody Radiation Exchange ... 610
13.3 Radiation Exchange Between Opaque, Diffuse, Gray Surfaces in an Enclosure ... 614
13.3.1 Net Radiation Exchange at a Surface ... 615
13.3.2 Radiation Exchange Between Surfaces ... 616
13.3.3 The Two-Surface Enclosure ... 622
13.3.4 Two-Surface Enclosures in Series and Radiation Shields ... 624
13.3.5 The Reradiating Surface ... 626
13.4 Multimode Heat Transfer ... 631
13.5 Implications of the Simplifying Assumptions ... 634
13.6 Radiation Exchange with Participating Media ... 634
13.6.1 Volumetric Absorption ... 634
13.6.2 Gaseous Emission and Absorption ... 635
13.7 Summary ... 639
References ... 640

Chapter 14 Diffusion Mass Transfer ... 641
14.1 Physical Origins and Rate Equations ... 642
14.1.1 Physical Origins ... 642
14.1.2 Mixture Composition ... 643
14.1.3 Fick’s Law of Diffusion ... 644
14.1.4 Mass Diffusivity ... 645
14.2 Mass Transfer in Nonstationary Media ... 647
14.2.1 Absolute and Diffusive Species Fluxes ... 647
14.2.2 Evaporation in a Column ... 650
14.3 The Stationary Medium Approximation ... 655
14.4 Conservation of Species for a Stationary Medium ... 655
14.4.1 Conservation of Species for a Control Volume ... 656
14.4.2 The Mass Diffusion Equation ... 656
14.4.3 Stationary Media with Specified Surface Concentrations ... 658
14.5 Boundary Conditions and Discontinuous Concentrations at Interfaces ... 662
14.5.1 Evaporation and Sublimation ... 663
14.5.2 Solubility of Gases in Liquids and Solids ... 663
14.5.3 Catalytic Surface Reactions ... 668
14.6 Mass Diffusion with Homogeneous Chemical Reactions ... 670
14.7 Transient Diffusion ... 673
14.8 Summary ... 679

References ... 680
Appendix A Thermophysical Properties of Matter ... 681
Appendix B Mathematical Relations and Functions ... 713
Appendix C Thermal Conditions Associated with Uniform Energy Generation in One-Dimensional, Steady-State Systems ... 719
APPENDIX D The Gauss–Seidel Method ... 725
APPENDIX E The Convection Transfer Equations ... 727
E.1 Conservation of Mass ... 728
E.2 Newton’s Second Law of Motion ... 728
E.3 Conservation of Energy ... 729
E.4 Conservation of Species ... 730
APPENDIX F Boundary Layer Equations for Turbulent Flow ... 731
APPENDIX G An Integral Laminar Boundary Layer Solution for Parallel Flow over a Flat Plate ... 735
Conversion Factors ... 739
Physical Constants ... 740
Index ... 741

Problems P-1
Chapter 1 Problems P-1
Chapter 2 Problems P-13
Chapter 3 Problems P-24
Chapter 4 Problems P-49
Chapter 5 Problems P-63
Chapter 6 Problems P-85
Chapter 7 Problems P-95
Chapter 8 Problems P-115
Chapter 9 Problems P-133
Chapter 10 Problems P-149
Chapter 11 Problems P-157
Chapter 12 Problems P-168
Chapter 13 Problems P-189
Chapter 14 Problems P-210