Table of Contents
Preface vii
1 Introduction 1
References 3
2 Differential Equations of the Two-Phase Flow System 5
2.1 Introduction 7
2.1.1 Governing Equations and Characteristics of the Homogeneous, Equilibrium Model 7
2.1.2 Invariance of Characteristics to Change of Dependent Variable and EOS 11
2.1.3 Eigenvectors and Compatibility Conditions 13
2.1.4 The Ill-Posed Problem 15
2.2 Mass and Momentum Equations 17
2.3 The Energy Equation 18
2.4 The Viscous Stress and Drag Function 19
2.5 Energy Equation of Enthalpy 22
References 22
3 Finite Differences of the Governing Equations 25
3.1 Introduction 27
3.2 The Continuity Equation 27
3.3 The Momentum Equations 37
3.4 The Viscous Stresses 45
3.5 The Internal Energy Equation 51
3.6 Finite Difference for the Enthalpy Equation 55
References 59
4 A Pressure Iteration Scheme for Two-Phase Flow Modeling 61
4.1 Introduction 63
4.2 Two-Fluid Model and Constitutive Correlations 63
4.3 Numerical Solution Procedures 66
4.4 Derivations of $$ and $$ 67
4.5 Simulation of Two-Phase Jet Impinged on Vertical Plate 78
4.6 Discussions 79
References 82
5 Pollution Particle Transport 83
5.1 Introduction 85
5.2 The Mathematical Formulation 87
5.2.1 Governing Equations of the Meso-Scale Meteorology 87
5.2.2 Plume Rise Model 96
5.2.3 Eddy Diffusivity 97
5.2.4 Quality Equations 100
5.3 A Simple Model for Wind Field Computation from Sparse Data 102
5.4 Downwind SO2 Concentration Calculations for the Four Corners Generating Station 104
5.5 Sulfur Dioxide Concentration Calculations for the Dickerson Power Plant 106
5.6 Behavior of Massive LNG Spills from Storage Tanks at Prince William Sound, Alaska 110
5.7 Conclusions 112
References 114
6 Nuclear Fission 117
6.1 Introduction 117
6.2 The Yield of Fission Fragments 118
6.3 The Fission Reaction Rate 118
6.4 Possible Fission Reactions 122
6.5 Calculations of the Released Energy Q 124
References 125
7 Radiation Hydrodynamics 127
7.1 Introduction 129
7.2 Inertial Confinement Fusion Problems 130
7.3 Computational Method for Radiation Hydrodynamics 135
7.3.1 Introduction 135
7.3.2 Finite Difference Equations 137
7.3.3 Boundary Conditions 145
7.3.4 Variable Eddington Factor 147
7.4 Sample Calculations 154
7.4.1 Introduction 154
7.4.2 Diffusion Wave in a Slab 155
7.4.3 Tests of the Variable Eddington Factors 156
7.4.4 Compton Scattering Test Problems 166
References 180
8 Natural Convection between Two Heated Vertical Plates 181
8.1 Introduction 183
8.2 Basic Equations and Boundary Conditions 184
8.2.1 Formulation of the Problem 184
8.2.2 Derivation of the Density Term 185
8.2.3 Derivation of Governing Equations 186
8.2.4 Boundary Conditions 190
8.2.5 Normalization of the Governing Equations 192
8.2.6 Laplace Operator, Vorticity and Stream Function 194
8.2.7 Non-Dimensional Forms of Governing Equations 196
8.3 Finite Difference Forms 197
8.3.1 Introduction 197
8.3.2 Taylor Series Expansions 197
8.3.3 Finite Difference Approximations for First and Second Derivatives 199
8.3.4 Alternating-Direct ion Implicit Method 201
8.3.5 Finite Difference Forms of Vorticity and Energy Equations 203
8.3.6 Finite Difference Forms of Stream Function Equation 209
8.3.7 Finite Difference Forms of Velocity Equations 211
8.3.8 Approximation Forms for those Points Near or on the Boundary 212
8.3.9 Finite Difference Forms of Nusselt Number 217
8.4 Stability 217
8.5 Results 222
8.5.1 General 222
8.5.2 Calculating Problems Where the Left Plate Temperature is Equal to One and Right Equal to Zero 223
8.5.3 Calculating Problems Where the Temperatures of both Plates are with Linear Distributions 227
8.6 Conclusions and Recommendations for Future Work 229
8.6.1 Conclusions 229
8.6.2 Recommendations for Future Work 233
References 239
Appendix A The Two-Phase Flow Computer Code 241
A.1 Definition of the Variables and Symbols Used in the Code 241
A.2 Input Data Descriptions 247
A.3 Sample of the Input Data 261
A.3.1 Two Phase Jet Impinged on Vertical Plate 261
A.3.2 One-Dimensional Fluidized Bed 266
A.4 Output Data Description 272
A.4.1 The Output for the Two-Phase Jet Impinged on Vertical Plate 272
A.5 Complete code listing 273
A.5.1 The Two-Phase Code Program 273
A.5.2 The Common Block for the Program Described in Section A.5.1 396
A.5.3 Equation of State of Water Including Liquid and Vapor 397
A.6 CD-Rom of the Computer Program 398
References 398
Appendix B Photon Transport 399
B.1 Introduction 400
B.2 The Variable Eddington Approximation Method 401
B.3 The Variable Eddington Factor 402
B.4 The Flux-limited Diffusion Description 403
B.5 Sn Radiative Transfer 404
B.6 Monte Carlo Method 413
B.6.1 Introduction 413
B.6.2 Discretization of the Diffusion Equation 416
B.6.3 Solving the Discretized Diffusion Equation by a Monte Carlo Technique 419
References 423
Appendix C Charge Particle Transport 425
C.1 Introduction 426
C.2 Fokker-Planck-Boltzmann Equation 426
C.3 Application of the Levermore Theory 430
C.4 Spatial Discretization 436
C.5 Time Integration 438
References 440
Appendix D Neutron Transport 441
D.1 Introduction 442
D.2 Criticality of the Diffusion Equation 442
D.3 Numerical Method for Solving the Diffusion Equation 443
D.4 The Method of Obtaining k 445
D.5 Solution Method for One-Dimensional, Single Group Diffusion Equations 446
D.6 Computational Method for a Two-Dimensional, Single Group Diffusion Equation 449
References 452
Index 453
