1 Introduction 1
1.1 Thermal Transport 1
1.2 Mass Transfer and Fluid Flow 3
1.3 An Example 4
1.4 Importance of Analytical and Experimental Methods 6
1.5 Numerical Approach 9
1.6 Basic Considerations in a Numerical Solution 12
1.7 Outline and Scope of the Book 15
References 17
Part 1 Mathematical Background 19
2 Governing Equations 21
2.1 Classification 21
2.2 Representative Differential Equations from Heat Transfer and Fluid Flow 23
2.3 Boundary and Initial Conditions 26
2.4 Integral Forms 28
2.5 Numerical Solution 31
2.5.1 Basic Equations 31
2.5.2 Different Approaches 33
References 34
Problems 35
3 Finite Differences 37
3.1 Basic Concepts 39
3.1.1 Direct Approximation Approach 39
3.1.2 Polynomial Representation 41
3.1.3 Taylor Series Approach and Accuracy 44
3.1.4 Control Volume Approach and Conservation 48
3.1.5 Numerical Considerations 50
3.1.5.1 Total Truncation Error 51
3.1.5.2 Discretization and Roundoff Errors 52
3.1.5.3 Convergence 53
3.1.5.4 Numerical Stability and the Equivalence Theorem 53
3.2 Steady-State Diffusion 55
3.2.1 Discretization 55
3.2.2 Solution of Simultaneous Equations 58
3.2.2.1 Iterative Methods 59
3.2.2.2 Direct Methods 65
3.3 Transient Diffusion 70
3.3.1 Two-Level Time Discretization 70
3.3.2 Matrix Stability Analysis 72
3.3.3 Fourier Series Stability Analysis 76
3.3.4 An Example of Numerical Instability 78
3.3.5 Other Explicit and Implicit Schemes 80
References 81
Problems 82
4 Finite Elements 85
4.1 Basic Concepts 86
4.1.1 Discretization 88
4.1.2 Interpolation Functions 88
4.1.3 Integral Representations and Galerkin’s Method 89
4.1.4 Assembly 90
4.1.5 Elements 90
4.1.6 Condensation and Substructuring 91
4.1.7 Practical Implementation 93
4.2 Steady-State Diffusion 94
4.2.1 Matrix Equations with Boundary Conditions 94
4.2.2 One-Dimensional Diffusion 97
4.2.3 Two-Dimensional Diffusion 99
4.2.4 Typical FEM Solutions 102
4.3 Transient Diffusion 108
4.3.1 The Matrix System 108
4.3.2 Finite Differences in Time 109
4.3.3 Diagonalization 111
4.3.4 Transient One-Dimensional Diffusion 111
4.3.5 Other Methods and Solutions 112
References 113
Problems 114
Part 2 Simulation of Transport Processes 119
5 Numerical Methods for Conduction Heat Transfer 121
5.1 Governing Equations 122
5.2 Numerical Solution of Steady-State Conduction 124
5.2.1 One-Dimensional Conduction 124
5.2.1.1 Basic Equations 124
5.2.1.2 Finite Difference Approximation of the Boundary Conditions 127
5.2.1.3 An Example：Numerical Solution of Heat Transfer in an Extended Surface 130
5.2.1.4 Runge-Kutta Methods 132
5.2.1.5 Finite Difference Method 135
5.2.2 Multidimensional Steady-State Conduction 137
5.2.2.1 Finite Difference Formulation 139
5.2.2.2 Solution：Iterative and Direct Methods 144
5.2.2.3 Improvement in Accuracy of Numerical Results 149
5.2.2.4 Finite Element Formulation 150
5.2.3 Variable Property and Other Considerations 152
5.3 Numerical Solution of Unsteady-State Conduction 166
5.3.1 One-Dimensional Unsteady-State Conduction 168
5.3.1.1 FTCS Explicit Method 169
5.3.1.2 Other Methods 178
5.3.2 Numerical Approximation of Lumped Mass and Semi-infinite Solids 180
5.3.3 Multidimensional Unsteady-State Conduction 184
5.3.4 Numerical Methods for Time-Varying Boundary Conditions 189
5.3.5 Property Variation 195
5.3.6 Finite Element Solution 198
5.4 Grid Generation 203
5.5 Summary 206
References 207
Problems 209
6 Numerical Methods for Convection Heat Transfer 215
6.1 Governing Equations 217
6.2 Computation of Forced Convection with Constant Fluid Properties 220
6.2.1 Inviscid Flow：Introduction to Stream Function and Vorticity 221
6.2.2 Equations for Viscous Flow：Primitive and Derived Variables 228
6.2.3 Linear Viscous Flow（Creeping Flow） 229
6.2.4 Computation of Boundary Layer Flows 233
6.2.4.1 Similarity Solution：Ordinary Differential Equations 234
6.2.4.2 Finite Difference Approach 238
6.2.5 Numerical Solution of the Full Equations 250
6.2.5.1 Central Differencing 252
6.2.5.2 Upwind，Hybrid and Other Lower-Order Differencing Schemes 253
6.2.5.3 Higher-Order Differencing Schemes for Convection 256
6.2.5.4 Other Numerical Methods and Considerations 259
6.2.5.5 Steady State Solution 265
6.2.5.6 Primitive Variables Approach 266
6.2.5.7 Simpler Algorithm 269
6.2.6 Finite Difference Considerations of the Conservative Form 274
6.2.7 Concluding Remarks on Flow Calculations 279
6.2.8 Energy Equation 280
6.2.8.1 Numerical Formulation 280
6.2.8.2 Boundary Conditions 284
6.2.8.3 Numerical Solution 289
6.2.9 Numerical Solution of Turbulent Flows 297
6.3 Computation of Natural Convection Flow and Transport 308
6.3.1 Similarity Solutions 310
6.3.2 Finite Difference Methods 315
6.3.3 Additional Considerations 324
6.4 Convection with Variable Fluid Properties 325
6.5 Finite Element Methods 330
6.5.1 Discretization and Interpolation Functions 331
6.5.2 Integral Representation 331
6.5.3 Element Equations and Assembly 333
6.5.4 Solution 335
6.5.5 Examples and Other Considerations 335
6.5.6 Comparison of Finite Element and Finite Difference Methods 337
6.6 Summary 338
References 339
Problems 345
7 Numerical Methods for Radiation Heat Transfer 353
7.1 Basic Concepts 354
7.2 Numerical Techniques for Enclosures with Diffuse-Gray Surfaces 359
7.2.1 Radiosity Method 359
7.2.2 Absorption Factor Method 364
7.2.3 Additional Considerations 365
7.2.3.1 Computation of View Factors 365
7.2.3.2 Temperature Dependence of Surface Properties 366
7.2.3.3 Spectral Variation 369
7.3 Nonuniform Irradiation and Emission：Discrete Integral Equations 370
7.4 Numerical Solution of Radiation in the Presence of Other Modes 379
7.4.1 Combined Modes at Boundaries：Nonparticipating Media 380
7.4.2 Participating Media 385
7.5 Other Methods For Participating Media 394
7.6 Monte Carlo Method 403
7.7 Summary 406
References 407
Problems 409
Part 3 Combined Modes and Process Applications 415
8 Applications of Computational Heat Transfer 417
8.1 Numerical Simulation of Thermal Systems in Manufacturing 418
8.1.1 Heat Treatment：Temperature Regulation 418
8.1.2 Surface Treatment：Semi-infinite Approximation 422
8.1.3 Continuously Moving Materials：Moving Boundary Effects 424
8.1.4 Melting and Solidification：Phase Change Considerations 427
8.1.5 Other Processes 436
8.2 Numerical Simulation of Environmental Heat Transfer Problems 442
8.2.1 Cooling Ponds：Periodic Processes 442
8.2.2 Recirculating Flows in Enclosed Spaces 447
8.2.3 Fire-Induced Flows in Partial Enclosures 454
8.2.4 Free Boundary Flows and Other Problems 457
8.2.5 Summary 466
8.3 Computer Simulation and Computer-Aided Design of Thermal Systems 468
8.3.1 General Approach 468
8.3.2 Example of Computer Simulation of a Thermal System 470
References 475
Problems 479
Appendices 483
A Finite Difference Approximations 483
B Sample Computer Programs 487
B.1 Successive Over-Relaxation（SOR）Method 488
B.2 Tridiagonal Matrix Algorithm（TDMA）or Thomas Algorithm 491
B.3 Gauss-Jordan Elimination Method 492
B.4 Forward-Time-Central-Space（FTCS）Method 495
B.5 Crank-Nicolson Method 497
B.6 Newton-Raphson Method 503
B.7 Finite Difference Method for ODEs 504
B.8 Runge-Kutta Method 507
B.9 Alternating-Direction-Implicit（ADI）Method 512
C Material Properties 523
Nomenclature 533
Index 537