fluent理论:ANSYS_Fluent_Theory_Guide.pdf

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Table of Contents
Using This Manual .................................................................................................................................. xxxvii
1. The Contents of This Manual ......................................................................................................... xxxvii
2.Typographical Conventions .......................................................................................................... xxxviii
3. Mathematical Conventions ................................................................................................................ xli
1. Basic Fluid Flow ....................................................................................................................................... 1
1.1. Overview of Physical Models in ANSYS Fluent .................................................................................... 1
1.2. Continuity and Momentum Equations ............................................................................................... 2
1.2.1. The Mass Conservation Equation .............................................................................................. 2
1.2.2. Momentum Conservation Equations ........................................................................................ 3
1.3. User-Defined Scalar (UDS) Transport Equations .................................................................................. 3
1.3.1. Single Phase Flow .................................................................................................................... 4
1.3.2. Multiphase Flow ....................................................................................................................... 5
1.4. Periodic Flows .................................................................................................................................. 5
1.4.1. Overview ................................................................................................................................. 6
1.4.2. Limitations ............................................................................................................................... 7
1.4.3. Physics of Periodic Flows .......................................................................................................... 7
1.4.3.1. Definition of the Periodic Velocity .................................................................................... 7
1.4.3.2. Definition of the Streamwise-Periodic Pressure ................................................................ 7
1.5. Swirling and Rotating Flows .............................................................................................................. 8
1.5.1. Overview of Swirling and Rotating Flows .................................................................................. 8
1.5.1.1. Axisymmetric Flows with Swirl or Rotation ....................................................................... 8
1.5.1.1.1. Momentum Conservation Equation for Swirl Velocity ............................................... 9
1.5.1.2.Three-Dimensional Swirling Flows .................................................................................. 10
1.5.1.3. Flows Requiring a Moving Reference Frame ................................................................... 10
1.5.2. Physics of Swirling and Rotating Flows .................................................................................... 10
1.6. Compressible Flows ........................................................................................................................ 11
1.6.1.When to Use the Compressible Flow Model ............................................................................ 12
1.6.2. Physics of Compressible Flows ................................................................................................ 13
1.6.2.1. Basic Equations for Compressible Flows ......................................................................... 13
1.6.2.2.The Compressible Form of the Gas Law .......................................................................... 14
1.7. Inviscid Flows ................................................................................................................................. 14
1.7.1. Euler Equations ...................................................................................................................... 14
1.7.1.1.The Mass Conservation Equation .................................................................................... 14
1.7.1.2. Momentum Conservation Equations .............................................................................. 15
1.7.1.3. Energy Conservation Equation ....................................................................................... 15
2. Flows with Moving Reference Frames ................................................................................................... 17
2.1. Introduction ................................................................................................................................... 17
2.2. Flow in a Moving Reference Frame .................................................................................................. 18
2.2.1. Equations for a Moving Reference Frame ................................................................................ 19
2.2.1.1. Relative Velocity Formulation ......................................................................................... 20
2.2.1.2. Absolute Velocity Formulation ....................................................................................... 21
2.2.1.3. Relative Specification of the Reference Frame Motion ..................................................... 21
2.3. Flow in Multiple Reference Frames .................................................................................................. 22
2.3.1.The Multiple Reference Frame Model ...................................................................................... 22
2.3.1.1. Overview ....................................................................................................................... 22
2.3.1.2. Examples ....................................................................................................................... 23
2.3.1.3. The MRF Interface Formulation ...................................................................................... 24
2.3.1.3.1. Interface Treatment: Relative Velocity Formulation ................................................. 24
2.3.1.3.2. Interface Treatment: Absolute Velocity Formulation ............................................... 25
2.3.2. The Mixing Plane Model ......................................................................................................... 25
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2.3.2.1. Overview ....................................................................................................................... 26
2.3.2.2. Rotor and Stator Domains .............................................................................................. 26
2.3.2.3. The Mixing Plane Concept ............................................................................................. 27
2.3.2.4. Choosing an Averaging Method ..................................................................................... 28
2.3.2.4.1. Area Averaging ..................................................................................................... 28
2.3.2.4.2. Mass Averaging .................................................................................................... 28
2.3.2.4.3. Mixed-Out Averaging ............................................................................................ 29
2.3.2.5. Mixing Plane Algorithm of ANSYS Fluent ........................................................................ 29
2.3.2.6. Mass Conservation ........................................................................................................ 30
2.3.2.7. Swirl Conservation ......................................................................................................... 30
2.3.2.8. Total Enthalpy Conservation .......................................................................................... 31
3. Flows Using Sliding and Dynamic Meshes ............................................................................................ 33
3.1. Introduction ................................................................................................................................... 33
3.2. Dynamic Mesh Theory .................................................................................................................... 34
3.2.1. Conservation Equations ......................................................................................................... 35
3.2.2. Six DOF Solver Theory ............................................................................................................ 36
3.3. Sliding Mesh Theory ....................................................................................................................... 37
4.Turbulence ............................................................................................................................................. 39
4.1. Underlying Principles of Turbulence Modeling ................................................................................. 39
4.1.1. Reynolds (Ensemble) Averaging .............................................................................................. 39
4.1.2. Filtered Navier-Stokes Equations ............................................................................................. 40
4.1.3. Hybrid RANS-LES Formulations ............................................................................................... 41
4.1.4. Boussinesq Approach vs. Reynolds Stress Transport Models ..................................................... 41
4.2. Spalart-Allmaras Model ................................................................................................................... 42
4.2.1. Overview ............................................................................................................................... 42
4.2.2. Transport Equation for the Spalart-Allmaras Model ................................................................. 43
4.2.3. Modeling the Turbulent Viscosity ............................................................................................ 43
4.2.4. Modeling the Turbulent Production ........................................................................................ 43
4.2.5. Modeling the Turbulent Destruction ....................................................................................... 44
4.2.6. Model Constants .................................................................................................................... 45
4.2.7. Wall Boundary Conditions ...................................................................................................... 45
4.2.7.1.Treatment of the Spalart-Allmaras Model for Icing Simulations ....................................... 45
4.2.8. Convective Heat and Mass Transfer Modeling .......................................................................... 46
4.3. Standard, RNG, and Realizable k-ε Models ........................................................................................ 46
4.3.1. Standard k-ε Model ................................................................................................................ 47
4.3.1.1. Overview ....................................................................................................................... 47
4.3.1.2.Transport Equations for the Standard k-ε Model ............................................................. 47
4.3.1.3. Modeling the Turbulent Viscosity ................................................................................... 47
4.3.1.4. Model Constants ........................................................................................................... 48
4.3.2. RNG k-ε Model ....................................................................................................................... 48
4.3.2.1. Overview ....................................................................................................................... 48
4.3.2.2.Transport Equations for the RNG k-ε Model ..................................................................... 48
4.3.2.3. Modeling the Effective Viscosity ..................................................................................... 49
4.3.2.4. RNG Swirl Modification .................................................................................................. 49
4.3.2.5. Calculating the Inverse Effective Prandtl Numbers .......................................................... 50
4.3.2.6.The R-ε Term in the ε Equation ........................................................................................ 50
4.3.2.7. Model Constants ........................................................................................................... 51
4.3.3. Realizable k-ε Model ............................................................................................................... 51
4.3.3.1. Overview ....................................................................................................................... 51
4.3.3.2.Transport Equations for the Realizable k-ε Model ............................................................ 52
4.3.3.3. Modeling the Turbulent Viscosity ................................................................................... 53
4.3.3.4. Model Constants ........................................................................................................... 54
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4.3.4. Modeling Turbulent Production in the k-ε Models ................................................................... 54
4.3.5. Effects of Buoyancy on Turbulence in the k-ε Models ............................................................... 54
4.3.6. Effects of Compressibility on Turbulence in the k-ε Models ...................................................... 55
4.3.7. Convective Heat and Mass Transfer Modeling in the k-ε Models ............................................... 56
4.4. Standard, BSL, and SST k-ω Models ................................................................................................... 57
4.4.1. Standard k-ω Model ............................................................................................................... 58
4.4.1.1. Overview ....................................................................................................................... 58
4.4.1.2.Transport Equations for the Standard k-ω Model ............................................................. 58
4.4.1.3. Modeling the Effective Diffusivity ................................................................................... 58
4.4.1.3.1. Low-Reynolds Number Correction ......................................................................... 58
4.4.1.4. Modeling the Turbulence Production ............................................................................. 59
4.4.1.4.1. Production of k ..................................................................................................... 59
4.4.1.4.2. Production of ω ..................................................................................................... 59
4.4.1.5. Modeling the Turbulence Dissipation ............................................................................. 59
4.4.1.5.1. Dissipation of k ..................................................................................................... 59
4.4.1.5.2. Dissipation of ω ..................................................................................................... 60
4.4.1.5.3. Compressibility Effects .......................................................................................... 60
4.4.1.6. Model Constants ........................................................................................................... 61
4.4.2. Baseline (BSL) k-ω Model ........................................................................................................ 61
4.4.2.1. Overview ....................................................................................................................... 61
4.4.2.2.Transport Equations for the BSL k-ω Model ..................................................................... 61
4.4.2.3. Modeling the Effective Diffusivity ................................................................................... 62
4.4.2.4. Modeling the Turbulence Production ............................................................................. 62
4.4.2.4.1. Production of k ..................................................................................................... 62
4.4.2.4.2. Production of ω ..................................................................................................... 62
4.4.2.5. Modeling the Turbulence Dissipation ............................................................................. 63
4.4.2.5.1. Dissipation of k ..................................................................................................... 63
4.4.2.5.2. Dissipation of ω ..................................................................................................... 63
4.4.2.6. Cross-Diffusion Modification .......................................................................................... 63
4.4.2.7. Model Constants ........................................................................................................... 63
4.4.3. Shear-Stress Transport (SST) k-ω Model ................................................................................... 64
4.4.3.1. Overview ....................................................................................................................... 64
4.4.3.2. Modeling the Turbulent Viscosity ................................................................................... 64
4.4.3.3. Model Constants ........................................................................................................... 64
4.4.3.4.Treatment of the SST Model for Icing Simulations ........................................................... 64
4.4.4.Turbulence Damping .............................................................................................................. 65
4.4.5. Wall Boundary Conditions ...................................................................................................... 66
4.5. Generalized k-ω (GEKO) Model ........................................................................................................ 66
4.5.1. Model Formulation ................................................................................................................. 67
4.5.2. Limitations ............................................................................................................................. 69
4.6. k-kl-ω Transition Model ................................................................................................................... 69
4.6.1. Overview ............................................................................................................................... 69
4.6.2.Transport Equations for the k-kl-ω Model ................................................................................ 69
4.6.2.1. Model Constants ........................................................................................................... 72
4.7.Transition SST Model ....................................................................................................................... 72
4.7.1. Overview ............................................................................................................................... 73
4.7.2.Transport Equations for the Transition SST Model .................................................................... 73
4.7.2.1. Separation-Induced Transition Correction ...................................................................... 75
4.7.2.2. Coupling the Transition Model and SST Transport Equations ........................................... 76
4.7.2.3.Transition SST and Rough Walls ...................................................................................... 76
4.7.3. Mesh Requirements ............................................................................................................... 77
4.7.4. Specifying Inlet Turbulence Levels .......................................................................................... 80
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4.8. Intermittency Transition Model ....................................................................................................... 81
4.8.1. Overview ............................................................................................................................... 81
4.8.2.Transport Equations for the Intermittency Transition Model ..................................................... 82
4.8.3. Coupling with the Other Models ............................................................................................. 86
4.8.4. Intermittency Transition Model and Rough Walls ..................................................................... 86
4.9.The V2F Model ................................................................................................................................ 86
4.10. Reynolds Stress Model (RSM) ......................................................................................................... 86
4.10.1. Overview ............................................................................................................................. 86
4.10.2. Reynolds Stress Transport Equations ..................................................................................... 87
4.10.3. Modeling Turbulent Diffusive Transport ................................................................................ 88
4.10.4. Modeling the Pressure-Strain Term ....................................................................................... 89
4.10.4.1. Linear Pressure-Strain Model ........................................................................................ 89
4.10.4.2. Low-Re Modifications to the Linear Pressure-Strain Model ............................................ 90
4.10.4.3. Quadratic Pressure-Strain Model .................................................................................. 90
4.10.4.4. Stress-Omega Model ................................................................................................... 91
4.10.4.5. Stress-BSL Model ......................................................................................................... 92
4.10.5. Effects of Buoyancy on Turbulence ........................................................................................ 92
4.10.6. Modeling the Turbulence Kinetic Energy ............................................................................... 93
4.10.7. Modeling the Dissipation Rate .............................................................................................. 93
4.10.8. Modeling the Turbulent Viscosity .......................................................................................... 94
4.10.9. Wall Boundary Conditions .................................................................................................... 94
4.10.10. Convective Heat and Mass Transfer Modeling ...................................................................... 95
4.11. Scale-Adaptive Simulation (SAS) Model ......................................................................................... 95
4.11.1. Overview ............................................................................................................................. 96
4.11.2.Transport Equations for the SST-SAS Model ........................................................................... 96
4.11.3. SAS with Other ω-Based Turbulence Models .......................................................................... 98
4.12. Detached Eddy Simulation (DES) ................................................................................................... 98
4.12.1. Overview ............................................................................................................................. 98
4.12.2. DES with the Spalart-Allmaras Model .................................................................................... 99
4.12.3. DES with the Realizable k-ε Model ....................................................................................... 100
4.12.4. DES with the BSL or SST k-ω Model ...................................................................................... 100
4.12.5. DES with the Transition SST Model ...................................................................................... 101
4.12.6. Improved Delayed Detached Eddy Simulation (IDDES) ........................................................ 101
4.12.6.1. Overview of IDDES ..................................................................................................... 101
4.12.6.2. IDDES Model Formulation .......................................................................................... 102
4.13. Shielded Detached Eddy Simulation (SDES) ................................................................................. 103
4.13.1. Shielding Function ............................................................................................................. 103
4.13.2. LES Mode of SDES .............................................................................................................. 105
4.14. Stress-Blended Eddy Simulation (SBES) ........................................................................................ 106
4.14.1. Stress Blending ................................................................................................................... 106
4.14.2. SDES and SBES Example ..................................................................................................... 106
4.15. Large Eddy Simulation (LES) Model .............................................................................................. 107
4.15.1. Overview ........................................................................................................................... 108
4.15.2. Subgrid-Scale Models ......................................................................................................... 108
4.15.2.1. Smagorinsky-Lilly Model ............................................................................................ 110
4.15.2.2. Dynamic Smagorinsky-Lilly Model .............................................................................. 110
4.15.2.3.Wall-Adapting Local Eddy-Viscosity (WALE) Model ...................................................... 111
4.15.2.4. Algebraic Wall-Modeled LES Model (WMLES) .............................................................. 112
4.15.2.4.1. Algebraic WMLES Model Formulation ................................................................ 112
4.15.2.4.1.1. Reynolds Number Scaling ......................................................................... 113
4.15.2.4.2. Algebraic WMLES S-Omega Model Formulation ................................................. 114
4.15.2.5. Dynamic Kinetic Energy Subgrid-Scale Model ............................................................. 114
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4.15.3. Inlet Boundary Conditions for Scale Resolving Simulations .................................................. 115
4.15.3.1.Vortex Method ........................................................................................................... 115
4.15.3.2. Spectral Synthesizer ................................................................................................... 117
4.15.3.3. Synthetic Turbulence Generator ................................................................................. 117
4.15.3.3.1. Limitations ........................................................................................................ 119
4.16. Embedded Large Eddy Simulation (ELES) ..................................................................................... 119
4.16.1. Overview ........................................................................................................................... 119
4.16.2. Selecting a Model ............................................................................................................... 119
4.16.3. Interfaces Treatment ........................................................................................................... 120
4.16.3.1. RANS-LES Interface .................................................................................................... 120
4.16.3.2. LES-RANS Interface .................................................................................................... 120
4.16.3.3. Internal Interface Without LES Zone ........................................................................... 121
4.16.3.4. Grid Generation Guidelines ........................................................................................ 121
4.17. Near-Wall Treatments for Wall-Bounded Turbulent Flows .............................................................. 122
4.17.1. Overview ........................................................................................................................... 122
4.17.1.1.Wall Functions vs. Near-Wall Model ............................................................................. 123
4.17.1.2. Wall Functions ........................................................................................................... 125
4.17.2. Standard Wall Functions ..................................................................................................... 125
4.17.2.1. Momentum ............................................................................................................... 125
4.17.2.2. Energy ....................................................................................................................... 126
4.17.2.3. Species ...................................................................................................................... 128
4.17.2.4. Turbulence ................................................................................................................ 128
4.17.3. Scalable Wall Functions ....................................................................................................... 129
4.17.4. Non-Equilibrium Wall Functions .......................................................................................... 129
4.17.4.1. Standard Wall Functions vs. Non-Equilibrium Wall Functions ....................................... 131
4.17.4.2. Limitations of the Wall Function Approach ................................................................. 131
4.17.5. Enhanced Wall Treatment ε-Equation (EWT-ε) ...................................................................... 131
4.17.5.1.Two-Layer Model for Enhanced Wall Treatment ........................................................... 132
4.17.5.2. Enhanced Wall Treatment for Momentum and Energy Equations ................................. 133
4.17.6. Menter-Lechner ε-Equation (ML-ε) ...................................................................................... 135
4.17.6.1. Momentum Equations ............................................................................................... 137
4.17.6.2. k-ε Turbulence Models ............................................................................................... 137
4.17.6.3. Iteration Improvements ............................................................................................. 137
4.17.7. y -Insensitive Wall Treatment ω-Equation ........................................................................... 137
4.17.8. User-Defined Wall Functions ............................................................................................... 138
4.17.9. LES Near-Wall Treatment ..................................................................................................... 138
4.18. Curvature Correction for the Spalart-Allmaras and Two-Equation Models ..................................... 138
4.19. Production Limiters for Two-Equation Models .............................................................................. 140
4.20. Definition of Turbulence Scales .................................................................................................... 142
4.20.1. RANS and Hybrid (SAS, DES, and SDES) Turbulence Models .................................................. 142
4.20.2. Large Eddy Simulation (LES) Models .................................................................................... 143
4.20.3. Stress-Blended Eddy Simulation (SBES) Model ..................................................................... 143
5. Heat Transfer ....................................................................................................................................... 145
5.1. Introduction ................................................................................................................................. 145
5.2. Modeling Conductive and Convective Heat Transfer ...................................................................... 145
5.2.1. Heat Transfer Theory ............................................................................................................. 145
5.2.1.1.The Energy Equation .................................................................................................... 145
5.2.1.2.The Energy Equation in Moving Reference Frames ........................................................ 146
5.2.1.3.The Energy Equation for the Non-Premixed Combustion Model .................................... 146
5.2.1.4. Inclusion of Pressure Work and Kinetic Energy Terms .................................................... 147
5.2.1.5. Inclusion of the Viscous Dissipation Terms .................................................................... 147
5.2.1.6. Inclusion of the Species Diffusion Term ........................................................................ 147
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5.2.1.7. Energy Sources Due to Reaction ................................................................................... 148
5.2.1.8. Energy Sources Due To Radiation ................................................................................. 148
5.2.1.9. Energy Source Due To Joule Heating ............................................................................ 148
5.2.1.10. Interphase Energy Sources ......................................................................................... 148
5.2.1.11. Energy Equation in Solid Regions ............................................................................... 148
5.2.1.12. Anisotropic Conductivity in Solids .............................................................................. 149
5.2.1.13. Diffusion at Inlets ....................................................................................................... 149
5.2.2. Natural Convection and Buoyancy-Driven Flows Theory ........................................................ 149
5.3. Modeling Radiation ...................................................................................................................... 150
5.3.1. Overview and Limitations ..................................................................................................... 150
5.3.1.1. Advantages and Limitations of the DTRM ..................................................................... 151
5.3.1.2. Advantages and Limitations of the P-1 Model ............................................................... 151
5.3.1.3. Advantages and Limitations of the Rosseland Model .................................................... 152
5.3.1.4. Advantages and Limitations of the DO Model ............................................................... 152
5.3.1.5. Advantages and Limitations of the S2S Model .............................................................. 152
5.3.1.6. Advantages and Limitations of the MC Model ............................................................... 153
5.3.2. Radiative Transfer Equation .................................................................................................. 154
5.3.3. P-1 Radiation Model Theory .................................................................................................. 155
5.3.3.1. The P-1 Model Equations ............................................................................................. 155
5.3.3.2. Anisotropic Scattering ................................................................................................. 157
5.3.3.3. Particulate Effects in the P-1 Model .............................................................................. 157
5.3.3.4. Boundary Condition Treatment for the P-1 Model at Walls ............................................. 158
5.3.3.5. Boundary Condition Treatment for the P-1 Model at Flow Inlets and Exits ...................... 159
5.3.4. Rosseland Radiation Model Theory ....................................................................................... 159
5.3.4.1.The Rosseland Model Equations ................................................................................... 159
5.3.4.2. Anisotropic Scattering ................................................................................................. 160
5.3.4.3. Boundary Condition Treatment at Walls ........................................................................ 160
5.3.4.4. Boundary Condition Treatment at Flow Inlets and Exits ................................................. 160
5.3.5. Discrete Transfer Radiation Model (DTRM) Theory ................................................................. 160
5.3.5.1.The DTRM Equations .................................................................................................... 160
5.3.5.2. Ray Tracing .................................................................................................................. 161
5.3.5.3. Clustering .................................................................................................................... 161
5.3.5.4. Boundary Condition Treatment for the DTRM at Walls ................................................... 162
5.3.5.5. Boundary Condition Treatment for the DTRM at Flow Inlets and Exits ............................ 162
5.3.6. Discrete Ordinates (DO) Radiation Model Theory ................................................................... 163
5.3.6.1. The DO Model Equations ............................................................................................. 163
5.3.6.2. Energy Coupling and the DO Model ............................................................................. 164
5.3.6.2.1. Limitations of DO/Energy Coupling ..................................................................... 165
5.3.6.3. Angular Discretization and Pixelation ........................................................................... 165
5.3.6.4. Anisotropic Scattering ................................................................................................. 168
5.3.6.5. Particulate Effects in the DO Model .............................................................................. 169
5.3.6.6. Boundary and Cell Zone Condition Treatment at Opaque Walls ..................................... 169
5.3.6.6.1. Gray Diffuse Walls ............................................................................................... 171
5.3.6.6.2. Non-Gray Diffuse Walls ........................................................................................ 171
5.3.6.7. Cell Zone and Boundary Condition Treatment at Semi-Transparent Walls ...................... 171
5.3.6.7.1. Semi-Transparent Interior Walls ........................................................................... 172
5.3.6.7.2. Specular Semi-Transparent Walls ......................................................................... 173
5.3.6.7.3. Diffuse Semi-Transparent Walls ............................................................................ 175
5.3.6.7.4. Partially Diffuse Semi-Transparent Walls ............................................................... 176
5.3.6.7.5. Semi-Transparent Exterior Walls ........................................................................... 176
5.3.6.7.6. Limitations .......................................................................................................... 178
5.3.6.7.7. Solid Semi-Transparent Media ............................................................................. 179
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5.3.6.8. Boundary Condition Treatment at Specular Walls and Symmetry Boundaries ................. 179
5.3.6.9. Boundary Condition Treatment at Periodic Boundaries ................................................. 179
5.3.6.10. Boundary Condition Treatment at Flow Inlets and Exits ............................................... 179
5.3.7. Surface-to-Surface (S2S) Radiation Model Theory .................................................................. 179
5.3.7.1. Gray-Diffuse Radiation ................................................................................................. 179
5.3.7.2.The S2S Model Equations ............................................................................................. 180
5.3.7.3. Clustering .................................................................................................................... 181
5.3.7.3.1. Clustering and View Factors ................................................................................ 181
5.3.7.3.2. Clustering and Radiosity ...................................................................................... 181
5.3.8. Monte Carlo (MC) Radiation Model Theory ............................................................................ 182
5.3.8.1. The MC Model Equations ............................................................................................. 182
5.3.8.1.1. Monte Carlo Solution Accuracy ............................................................................ 182
5.3.8.2. Boundary Condition Treatment for the MC Model ......................................................... 183
5.3.9. Radiation in Combusting Flows ............................................................................................ 183
5.3.9.1. The Weighted-Sum-of-Gray-Gases Model ..................................................................... 183
5.3.9.1.1.When the Total (Static) Gas Pressure is Not Equal to 1 atm .................................... 184
5.3.9.2.The Effect of Soot on the Absorption Coefficient ........................................................... 185
5.3.9.3.The Effect of Particles on the Absorption Coefficient ..................................................... 185
5.3.10. Choosing a Radiation Model ............................................................................................... 185
5.3.10.1. External Radiation ...................................................................................................... 186
6. Heat Exchangers .................................................................................................................................. 187
6.1.The Macro Heat Exchanger Models ................................................................................................ 187
6.1.1. Overview of the Macro Heat Exchanger Models .................................................................... 187
6.1.2. Restrictions of the Macro Heat Exchanger Models ................................................................. 189
6.1.3. Macro Heat Exchanger Model Theory .................................................................................... 190
6.1.3.1. Streamwise Pressure Drop ........................................................................................... 191
6.1.3.2. Heat Transfer Effectiveness ........................................................................................... 192
6.1.3.3. Heat Rejection ............................................................................................................. 193
6.1.3.4. Macro Heat Exchanger Group Connectivity .................................................................. 194
6.2. The Dual Cell Model ...................................................................................................................... 195
6.2.1. Overview of the Dual Cell Model ........................................................................................... 195
6.2.2. Restrictions of the Dual Cell Model ........................................................................................ 196
6.2.3. Dual Cell Model Theory ......................................................................................................... 196
6.2.3.1. NTU Relations .............................................................................................................. 197
6.2.3.2. Heat Rejection ............................................................................................................. 197
7. Species Transport and Finite-Rate Chemistry ..................................................................................... 199
7.1. Volumetric Reactions .................................................................................................................... 199
7.1.1. Species Transport Equations ................................................................................................. 199
7.1.1.1. Mass Diffusion in Laminar Flows ................................................................................... 200
7.1.1.2. Mass Diffusion in Turbulent Flows ................................................................................ 200
7.1.1.3.Treatment of Species Transport in the Energy Equation ................................................. 200
7.1.1.4. Diffusion at Inlets ......................................................................................................... 200
7.1.2.The Generalized Finite-Rate Formulation for Reaction Modeling ............................................ 201
7.1.2.1. Direct Use of Finite-Rate Kinetics (no TCI) ...................................................................... 201
7.1.2.2. Pressure-Dependent Reactions .................................................................................... 203
7.1.2.3.The Eddy-Dissipation Model ......................................................................................... 205
7.1.2.4. The Eddy-Dissipation Model for LES ............................................................................. 206
7.1.2.5. The Eddy-Dissipation-Concept (EDC) Model ................................................................. 206
7.1.2.6.The Thickened Flame Model ......................................................................................... 207
7.1.2.7.The Relaxation to Chemical Equilibrium Model ............................................................. 210
7.2.Wall Surface Reactions and Chemical Vapor Deposition .................................................................. 211
7.2.1. Surface Coverage Reaction Rate Modification ....................................................................... 213
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7.2.2. Reaction-Diffusion Balance for Surface Chemistry ................................................................. 213
7.2.3. Slip Boundary Formulation for Low-Pressure Gas Systems ..................................................... 214
7.3. Particle Surface Reactions ............................................................................................................. 216
7.3.1. General Description .............................................................................................................. 216
7.3.2. ANSYS Fluent Model Formulation ......................................................................................... 217
7.3.3. Extension for Stoichiometries with Multiple Gas Phase Reactants .......................................... 218
7.3.4. Solid-Solid Reactions ............................................................................................................ 219
7.3.5. Solid Decomposition Reactions ............................................................................................ 219
7.3.6. Solid Deposition Reactions ................................................................................................... 219
7.3.7. Gaseous Solid Catalyzed Reactions on the Particle Surface .................................................... 219
7.4. Electrochemical Reactions ............................................................................................................. 220
7.4.1. Overview and Limitations ..................................................................................................... 220
7.4.2. Electrochemical Reaction Model Theory ................................................................................ 220
7.5. Reacting Channel Model ............................................................................................................... 223
7.5.1. Overview and Limitations ..................................................................................................... 223
7.5.2. Reacting Channel Model Theory ........................................................................................... 224
7.5.2.1. Flow Inside the Reacting Channel ................................................................................. 224
7.5.2.2. Surface Reactions in the Reacting Channel ................................................................... 225
7.5.2.3. Porous Medium Inside Reacting Channel ...................................................................... 226
7.5.2.4. Outer Flow in the Shell ................................................................................................. 226
7.6. Reactor Network Model ................................................................................................................ 227
7.6.1. Reactor Network Model Theory ............................................................................................ 227
7.6.1.1. Reactor network temperature solution ......................................................................... 228
8. Non-Premixed Combustion ................................................................................................................. 229
8.1. Introduction ................................................................................................................................. 229
8.2. Non-Premixed Combustion and Mixture Fraction Theory ............................................................... 229
8.2.1. Mixture Fraction Theory ....................................................................................................... 230
8.2.1.1. Definition of the Mixture Fraction ................................................................................ 230
8.2.1.2.Transport Equations for the Mixture Fraction ................................................................ 232
8.2.1.3. The Non-Premixed Model for LES ................................................................................. 233
8.2.1.4.The Non-Premixed Model with the SBES Turbulence Model ........................................... 233
8.2.1.5. Mixture Fraction vs. Equivalence Ratio .......................................................................... 233
8.2.1.6. Relationship of Mixture Fraction to Species Mass Fraction, Density, and Temperature ..... 234
8.2.2. Modeling of Turbulence-Chemistry Interaction ..................................................................... 235
8.2.2.1. Description of the Probability Density Function ............................................................ 235
8.2.2.2. Derivation of Mean Scalar Values from the Instantaneous Mixture Fraction ................... 236
8.2.2.3. The Assumed-Shape PDF ............................................................................................. 237
8.2.2.3.1.The Double Delta Function PDF ........................................................................... 237
8.2.2.3.2.The β-Function PDF ............................................................................................. 237
8.2.3. Non-Adiabatic Extensions of the Non-Premixed Model .......................................................... 238
8.2.4. Chemistry Tabulation ........................................................................................................... 241
8.2.4.1. Look-Up Tables for Adiabatic Systems ........................................................................... 241
8.2.4.2. 3D Look-Up Tables for Non-Adiabatic Systems .............................................................. 243
8.2.4.3. Generating Lookup Tables Through Automated Grid Refinement .................................. 245
8.3. Restrictions and Special Cases for Using the Non-Premixed Model ................................................. 247
8.3.1. Restrictions on the Mixture Fraction Approach ...................................................................... 247
8.3.2. Using the Non-Premixed Model for Liquid Fuel or Coal Combustion ...................................... 250
8.3.3. Using the Non-Premixed Model with Flue Gas Recycle .......................................................... 251
8.3.4. Using the Non-Premixed Model with the Inert Model ............................................................ 251
8.3.4.1. Mixture Composition ................................................................................................... 252
8.3.4.1.1. Property Evaluation ............................................................................................. 253
8.4.The Diffusion Flamelet Models Theory ........................................................................................... 253
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8.4.1. Restrictions and Assumptions ............................................................................................... 253
8.4.2.The Flamelet Concept ........................................................................................................... 253
8.4.2.1. Overview ..................................................................................................................... 253
8.4.2.2. Strain Rate and Scalar Dissipation ................................................................................. 255
8.4.2.3. Embedding Diffusion Flamelets in Turbulent Flames ..................................................... 255
8.4.3. Flamelet Generation ............................................................................................................. 256
8.4.4. Flamelet Import ................................................................................................................... 257
8.5. The Steady Diffusion Flamelet Model Theory ................................................................................. 258
8.5.1. Overview ............................................................................................................................. 259
8.5.2. Multiple Steady Flamelet Libraries ........................................................................................ 259
8.5.3. Steady Diffusion Flamelet Automated Grid Refinement ......................................................... 260
8.5.4. Non-Adiabatic Steady Diffusion Flamelets ............................................................................. 260
8.6. The Unsteady Diffusion Flamelet Model Theory ............................................................................. 261
8.6.1. The Eulerian Unsteady Laminar Flamelet Model .................................................................... 261
8.6.1.1. Liquid Reactions .......................................................................................................... 263
8.6.2. The Diesel Unsteady Laminar Flamelet Model ....................................................................... 264
8.6.3. Multiple Diesel Unsteady Flamelets ....................................................................................... 264
8.6.4. Multiple Diesel Unsteady Flamelets with Flamelet Reset ........................................................ 265
8.6.4.1. Resetting the Flamelets ................................................................................................ 265
9. Premixed Combustion ......................................................................................................................... 267
9.1. Overview and Limitations ............................................................................................................. 267
9.1.1. Overview ............................................................................................................................. 267
9.1.2. Limitations ........................................................................................................................... 268
9.2. C-Equation Model Theory .............................................................................................................. 268
9.2.1. Propagation of the Flame Front ............................................................................................ 268
9.3. G-Equation Model Theory ............................................................................................................. 270
9.3.1. Numerical Solution of the G-equation ................................................................................... 271
9.4. Turbulent Flame Speed Models ..................................................................................................... 271
9.4.1. Zimont Turbulent Flame Speed Closure Model ...................................................................... 271
9.4.1.1. Zimont Turbulent Flame Speed Closure for LES ............................................................. 272
9.4.1.2. Flame Stretch Effect ..................................................................................................... 273
9.4.1.3. Gradient Diffusion ....................................................................................................... 274
9.4.1.4.Wall Damping .............................................................................................................. 274
9.4.2. Peters Flame Speed Model .................................................................................................... 274
9.4.2.1. Peters Flame Speed Model for LES ................................................................................ 276
9.5. Extended Coherent Flamelet Model Theory ................................................................................... 276
9.5.1. Closure for ECFM Source Terms ............................................................................................. 278
9.5.2.Turbulent Flame Speed in ECFM ............................................................................................ 280
9.5.3. LES and ECFM ...................................................................................................................... 280
9.6. Calculation of Properties ............................................................................................................... 282
9.6.1. Calculation of Temperature ................................................................................................... 283
9.6.1.1. Adiabatic Temperature Calculation ............................................................................... 283
9.6.1.2. Non-Adiabatic Temperature Calculation ....................................................................... 283
9.6.2. Calculation of Density .......................................................................................................... 283
9.6.3. Laminar Flame Speed ........................................................................................................... 284
9.6.4. Unburnt Density and Thermal Diffusivity ............................................................................... 284
10. Partially Premixed Combustion ........................................................................................................ 285
10.1. Overview .................................................................................................................................... 285
10.2. Limitations .................................................................................................................................. 285
10.3. Partially Premixed Combustion Theory ........................................................................................ 286
10.3.1. Chemical Equilibrium and Steady Diffusion Flamelet Models ............................................... 286
10.3.2. Flamelet Generated Manifold (FGM) model ......................................................................... 287
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10.3.2.1. Premixed FGMs in Reaction Progress Variable Space ................................................... 287
10.3.2.2. Premixed FGMs in Physical Space ............................................................................... 289
10.3.2.3. Diffusion FGMs .......................................................................................................... 290
10.3.3. FGM Turbulent Closure ....................................................................................................... 290
10.3.3.1. Scalar Transport with FGM Closure ............................................................................. 292
10.3.4. Calculation of Mixture Properties ........................................................................................ 293
10.3.5. Calculation of Unburnt Properties ....................................................................................... 294
10.3.6. Laminar Flame Speed ......................................................................................................... 294
10.3.7. Generating PDF Lookup Tables Through Automated Grid Refinement .................................. 296
11. Composition PDF Transport .............................................................................................................. 297
11.1. Overview and Limitations ............................................................................................................ 297
11.2. Composition PDF Transport Theory ............................................................................................. 297
11.3.The Lagrangian Solution Method ................................................................................................. 298
11.3.1. Particle Convection ............................................................................................................ 299
11.3.2. Particle Mixing ................................................................................................................... 300
11.3.2.1.The Modified Curl Model ............................................................................................ 300
11.3.2.2.The IEM Model ........................................................................................................... 300
11.3.2.3. The EMST Model ........................................................................................................ 301
11.3.2.4. Liquid Reactions ........................................................................................................ 301
11.3.3. Particle Reaction ................................................................................................................. 301
11.4. The Eulerian Solution Method ..................................................................................................... 302
11.4.1. Reaction ............................................................................................................................. 303
11.4.2. Mixing ................................................................................................................................ 303
11.4.3. Correction .......................................................................................................................... 303
11.4.4. Calculation of Composition Mean and Variance ................................................................... 304
12. Chemistry Acceleration ..................................................................................................................... 305
12.1. Overview and Limitations ............................................................................................................ 305
12.2. In-Situ Adaptive Tabulation (ISAT) ................................................................................................ 305
12.3. Dynamic Mechanism Reduction .................................................................................................. 307
12.3.1. Directed Relation Graph (DRG) Method for Mechanism Reduction ....................................... 308
12.4. Chemistry Agglomeration ........................................................................................................... 309
12.4.1. Binning Algorithm .............................................................................................................. 310
12.5. Chemical Mechanism Dimension Reduction ................................................................................ 312
12.5.1. Selecting the Represented Species ...................................................................................... 312
12.6. Dynamic Cell Clustering with ANSYS Fluent CHEMKIN-CFD Solver ................................................ 313
12.7. Dynamic Adaptive Chemistry with ANSYS Fluent CHEMKIN-CFD Solver ........................................ 313
13. Engine Ignition .................................................................................................................................. 315
13.1. Spark Model ................................................................................................................................ 315
13.1.1. Overview and Limitations ................................................................................................... 315
13.1.2. Spark Model Theory ............................................................................................................ 315
13.1.3. ECFM Spark Model Variants ................................................................................................. 318
13.2. Autoignition Models ................................................................................................................... 319
13.2.1. Model Overview ................................................................................................................. 319
13.2.2. Model Limitations .............................................................................................................. 319
13.2.3. Ignition Model Theory ........................................................................................................ 320
13.2.3.1.Transport of Ignition Species ...................................................................................... 320
13.2.3.2. Knock Modeling ........................................................................................................ 320
13.2.3.2.1. Modeling of the Source Term ............................................................................. 321
13.2.3.2.2. Correlations ...................................................................................................... 321
13.2.3.2.3. Energy Release .................................................................................................. 322
13.2.3.3. Ignition Delay Modeling ............................................................................................. 322
13.2.3.3.1. Modeling of the Source Term ............................................................................. 322
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13.2.3.3.2. Correlations ...................................................................................................... 323
13.2.3.3.3. Energy Release .................................................................................................. 323
13.3. Crevice Model ............................................................................................................................. 323
13.3.1. Overview ........................................................................................................................... 323
13.3.1.1. Model Parameters ...................................................................................................... 324
13.3.2. Limitations ......................................................................................................................... 325
13.3.3. Crevice Model Theory ......................................................................................................... 326
14. Pollutant Formation .......................................................................................................................... 327
14.1. NOx Formation ........................................................................................................................... 327
14.1.1. Overview ........................................................................................................................... 327
14.1.1.1. NOx Modeling in ANSYS Fluent .................................................................................. 327
14.1.1.2. NOx Formation and Reduction in Flames .................................................................... 328
14.1.2. Governing Equations for NOx Transport .............................................................................. 328
14.1.3.Thermal NOx Formation ...................................................................................................... 329
14.1.3.1. Thermal NOx Reaction Rates ...................................................................................... 329
14.1.3.2. The Quasi-Steady Assumption for [N] ......................................................................... 329
14.1.3.3.Thermal NOx Temperature Sensitivity ......................................................................... 330
14.1.3.4. Decoupled Thermal NOx Calculations ......................................................................... 330
14.1.3.5. Approaches for Determining O Radical Concentration ................................................ 330
14.1.3.5.1. Method 1: Equilibrium Approach ....................................................................... 330
14.1.3.5.2. Method 2: Partial Equilibrium Approach ............................................................. 331
14.1.3.5.3. Method 3: Predicted O Approach ....................................................................... 331
14.1.3.6. Approaches for Determining OH Radical Concentration .............................................. 331
14.1.3.6.1. Method 1: Exclusion of OH Approach ................................................................. 331
14.1.3.6.2. Method 2: Partial Equilibrium Approach ............................................................. 331
14.1.3.6.3. Method 3: Predicted OH Approach ..................................................................... 332
14.1.3.7. Summary ................................................................................................................... 332
14.1.4. Prompt NOx Formation ....................................................................................................... 332
14.1.4.1. Prompt NOx Combustion Environments ..................................................................... 332
14.1.4.2. Prompt NOx Mechanism ............................................................................................ 332
14.1.4.3. Prompt NOx Formation Factors .................................................................................. 333
14.1.4.4. Primary Reaction ....................................................................................................... 333
14.1.4.5. Modeling Strategy ..................................................................................................... 333
14.1.4.6. Rate for Most Hydrocarbon Fuels ................................................................................ 334
14.1.4.7. Oxygen Reaction Order .............................................................................................. 334
14.1.5. Fuel NOx Formation ............................................................................................................ 335
14.1.5.1. Fuel-Bound Nitrogen ................................................................................................. 335
14.1.5.2. Reaction Pathways ..................................................................................................... 335
14.1.5.3. Fuel NOx from Gaseous and Liquid Fuels .................................................................... 335
14.1.5.3.1. Fuel NOx from Intermediate Hydrogen Cyanide (HCN) ....................................... 336
14.1.5.3.1.1. HCN Production in a Gaseous Fuel ............................................................ 336
14.1.5.3.1.2. HCN Production in a Liquid Fuel ................................................................ 336
14.1.5.3.1.3. HCN Consumption .................................................................................... 337
14.1.5.3.1.4. HCN Sources in the Transport Equation ..................................................... 337
14.1.5.3.1.5. NOx Sources in the Transport Equation ..................................................... 337
14.1.5.3.2. Fuel NOx from Intermediate Ammonia (NH3) ..................................................... 338
14.1.5.3.2.1. NH3 Production in a Gaseous Fuel ............................................................. 338
14.1.5.3.2.2. NH3 Production in a Liquid Fuel ................................................................ 338
14.1.5.3.2.3. NH3 Consumption .................................................................................... 339
14.1.5.3.2.4. NH3 Sources in the Transport Equation ..................................................... 339
14.1.5.3.2.5. NOx Sources in the Transport Equation ..................................................... 339
14.1.5.3.3. Fuel NOx from Coal ........................................................................................... 340
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14.1.5.3.3.1. Nitrogen in Char and in Volatiles ............................................................... 340
14.1.5.3.3.2. Coal Fuel NOx Scheme A ........................................................................... 340
14.1.5.3.3.3. Coal Fuel NOx Scheme B ........................................................................... 340
14.1.5.3.3.4. HCN Scheme Selection ............................................................................. 341
14.1.5.3.3.5. NOx Reduction on Char Surface ................................................................ 341
14.1.5.3.3.5.1. BET Surface Area .............................................................................. 342
14.1.5.3.3.5.2. HCN from Volatiles ........................................................................... 342
14.1.5.3.3.6. Coal Fuel NOx Scheme C ........................................................................... 342
14.1.5.3.3.7. Coal Fuel NOx Scheme D ........................................................................... 343
14.1.5.3.3.8. NH3 Scheme Selection ............................................................................. 344
14.1.5.3.3.8.1. NH3 from Volatiles ........................................................................... 344
14.1.5.3.4. Fuel Nitrogen Partitioning for HCN and NH3 Intermediates ................................ 344
14.1.6. NOx Formation from Intermediate N2O ............................................................................... 345
14.1.6.1. N2O - Intermediate NOx Mechanism .......................................................................... 345
14.1.7. NOx Reduction by Reburning ............................................................................................. 346
14.1.7.1. Instantaneous Approach ............................................................................................ 346
14.1.7.2. Partial Equilibrium Approach ..................................................................................... 347
14.1.7.2.1. NOx Reduction Mechanism ............................................................................... 347
14.1.8. NOx Reduction by SNCR ..................................................................................................... 349
14.1.8.1. Ammonia Injection .................................................................................................... 349
14.1.8.2. Urea Injection ............................................................................................................ 350
14.1.8.3. Transport Equations for Urea, HNCO, and NCO ............................................................ 351
14.1.8.4. Urea Production due to Reagent Injection .................................................................. 352
14.1.8.5. NH3 Production due to Reagent Injection ................................................................... 352
14.1.8.6. HNCO Production due to Reagent Injection ................................................................ 352
14.1.9. NOx Formation in Turbulent Flows ...................................................................................... 353
14.1.9.1. The Turbulence-Chemistry Interaction Model ............................................................. 353
14.1.9.2. The PDF Approach ..................................................................................................... 354
14.1.9.3.The General Expression for the Mean Reaction Rate .................................................... 354
14.1.9.4.The Mean Reaction Rate Used in ANSYS Fluent ........................................................... 354
14.1.9.5. Statistical Independence ............................................................................................ 354
14.1.9.6.The Beta PDF Option .................................................................................................. 355
14.1.9.7.The Gaussian PDF Option ........................................................................................... 355
14.1.9.8. The Calculation Method for the Variance .................................................................... 355
14.2. SOx Formation ............................................................................................................................ 356
14.2.1. Overview ........................................................................................................................... 356
14.2.1.1.The Formation of SOx ................................................................................................. 357
14.2.2. Governing Equations for SOx Transport ............................................................................... 357
14.2.3. Reaction Mechanisms for Sulfur Oxidation .......................................................................... 358
14.2.4. SO2 and H2S Production in a Gaseous Fuel ......................................................................... 359
14.2.5. SO2 and H2S Production in a Liquid Fuel ............................................................................. 360
14.2.6. SO2 and H2S Production from Coal ..................................................................................... 360
14.2.6.1. SO2 and H2S from Char .............................................................................................. 360
14.2.6.2. SO2 and H2S from Volatiles ........................................................................................ 360
14.2.7. SOx Formation in Turbulent Flows ....................................................................................... 361
14.2.7.1. The Turbulence-Chemistry Interaction Model ............................................................. 361
14.2.7.2. The PDF Approach ..................................................................................................... 361
14.2.7.3.The Mean Reaction Rate ............................................................................................. 361
14.2.7.4.The PDF Options ........................................................................................................ 361
14.3. Soot Formation ........................................................................................................................... 362
14.3.1. Overview and Limitations ................................................................................................... 362
14.3.1.1. Predicting Soot Formation ......................................................................................... 362
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14.3.1.2. Restrictions on Soot Modeling ................................................................................... 363
14.3.2. Soot Model Theory ............................................................................................................. 363
14.3.2.1.The One-Step Soot Formation Model .......................................................................... 363
14.3.2.2.The Two-Step Soot Formation Model .......................................................................... 364
14.3.2.2.1. Soot Generation Rate ........................................................................................ 364
14.3.2.2.2. Nuclei Generation Rate ...................................................................................... 365
14.3.2.3. The Moss-Brookes Model ........................................................................................... 365
14.3.2.3.1.The Moss-Brookes-Hall Model ............................................................................ 367
14.3.2.3.2. Soot Formation in Turbulent Flows .................................................................... 368
14.3.2.3.2.1.The Turbulence-Chemistry Interaction Model ............................................ 369
14.3.2.3.2.2.The PDF Approach .................................................................................... 369
14.3.2.3.2.3. The Mean Reaction Rate ........................................................................... 369
14.3.2.3.2.4.The PDF Options ....................................................................................... 369
14.3.2.3.3.The Effect of Soot on the Radiation Absorption Coefficient ................................. 369
14.3.2.4.The Method of Moments Model ................................................................................. 369
14.3.2.4.1. Soot Particle Population Balance ....................................................................... 369
14.3.2.4.2. Moment Transport Equations ............................................................................ 371
14.3.2.4.3. Nucleation ........................................................................................................ 371
14.3.2.4.4. Coagulation ...................................................................................................... 373
14.3.2.4.5. Surface Growth and Oxidation ........................................................................... 376
14.3.2.4.6. Soot Aggregation .............................................................................................. 379
14.4. Decoupled Detailed Chemistry Model ......................................................................................... 383
14.4.1. Overview ........................................................................................................................... 383
14.4.1.1. Limitations ................................................................................................................ 383
14.4.2. Decoupled Detailed Chemistry Model Theory ..................................................................... 384
15. Aerodynamically Generated Noise ................................................................................................... 385
15.1. Overview .................................................................................................................................... 385
15.1.1. Direct Method .................................................................................................................... 385
15.1.2. Integral Method by Ffowcs Williams and Hawkings .............................................................. 386
15.1.3. Method Based on Wave Equation ........................................................................................ 387
15.1.4. Broadband Noise Source Models ........................................................................................ 387
15.2. Acoustics Model Theory .............................................................................................................. 387
15.2.1.The Ffowcs Williams and Hawkings Model ........................................................................... 388
15.2.2.Wave Equation Model ......................................................................................................... 390
15.2.2.1. Limitations ................................................................................................................ 390
15.2.2.2. Governing Equations and Boundary Conditions .......................................................... 391
15.2.2.3. Method of Numerical Solution ................................................................................... 391
15.2.2.4. Preventing Non-Physical Reflections of Sound Waves .................................................. 392
15.2.2.4.1. Mesh Quality ..................................................................................................... 392
15.2.2.4.2. Filtering of the Sound Source Term .................................................................... 392
15.2.2.4.3. Ramping in Time and Limiting in Space (Masking) of the Sound Source Term ..... 392
15.2.2.4.4. Damping of Solution in a Sponge Region Using Artificial Viscosity ...................... 393
15.2.3. Broadband Noise Source Models ........................................................................................ 393
15.2.3.1. Proudman’s Formula .................................................................................................. 393
15.2.3.2.The Jet Noise Source Model ........................................................................................ 394
15.2.3.3.The Boundary Layer Noise Source Model .................................................................... 395
15.2.3.4. Source Terms in the Linearized Euler Equations ........................................................... 396
15.2.3.5. Source Terms in Lilley’s Equation ................................................................................ 396
16. Discrete Phase ................................................................................................................................... 399
16.1. Introduction ............................................................................................................................... 399
16.1.1.The Euler-Lagrange Approach ............................................................................................. 399
16.2. Particle Motion Theory ................................................................................................................ 400
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16.2.1. Equations of Motion for Particles ........................................................................................ 400
16.2.1.1. Particle Force Balance ................................................................................................ 400
16.2.1.2. Particle Torque Balance .............................................................................................. 400
16.2.1.3. Inclusion of the Gravity Term ...................................................................................... 401
16.2.1.4. Other Forces .............................................................................................................. 401
16.2.1.5. Forces in Moving Reference Frames ............................................................................ 401
16.2.1.6.Thermophoretic Force ................................................................................................ 402
16.2.1.7. Brownian Force .......................................................................................................... 402
16.2.1.8. Saffman’s Lift Force .................................................................................................... 403
16.2.1.9. Magnus Lift Force ...................................................................................................... 403
16.2.2.Turbulent Dispersion of Particles ......................................................................................... 404
16.2.2.1. Stochastic Tracking .................................................................................................... 404
16.2.2.1.1. The Integral Time .............................................................................................. 405
16.2.2.1.2.The Discrete Random Walk Model ...................................................................... 405
16.2.2.1.3. Using the DRW Model ....................................................................................... 406
16.2.2.2. Particle Cloud Tracking ............................................................................................... 407
16.2.2.2.1. Using the Cloud Model ...................................................................................... 409
16.2.3. Integration of Particle Equation of Motion ........................................................................... 409
16.3. Laws for Drag Coefficients ........................................................................................................... 411
16.3.1. Spherical Drag Law ............................................................................................................. 412
16.3.2. Non-spherical Drag Law ..................................................................................................... 412
16.3.3. Stokes-Cunningham Drag Law ............................................................................................ 412
16.3.4. High-Mach-Number Drag Law ............................................................................................ 413
16.3.5. Dynamic Drag Model Theory .............................................................................................. 413
16.3.6. Dense Discrete Phase Model Drag Laws .............................................................................. 413
16.3.7. Bubbly Flow Drag Laws ...................................................................................................... 414
16.3.7.1. Ishii-Zuber Drag Model .............................................................................................. 414
16.3.7.2. Grace Drag Model ...................................................................................................... 415
16.3.8. Rotational Drag Law ........................................................................................................... 415
16.4. Laws for Heat and Mass Exchange ............................................................................................... 416
16.4.1. Inert Heating or Cooling (Law 1/Law 6) ............................................................................... 416
16.4.2. Droplet Vaporization (Law 2) ............................................................................................... 418
16.4.2.1. Mass Transfer During Law 2—Diffusion Controlled Model ........................................... 419
16.4.2.2. Mass Transfer During Law 2—Convection/Diffusion Controlled Model ........................ 420
16.4.2.3. Mass Transfer During Law 2—Thermolysis ................................................................. 420
16.4.2.4. Defining the Saturation Vapor Pressure and Diffusion Coefficient (or Binary Diffusivity) ......................................................................................................................................... 421
16.4.2.5. Defining the Boiling Point and Latent Heat ................................................................. 422
16.4.2.6. Heat Transfer to the Droplet ....................................................................................... 422
16.4.3. Droplet Boiling (Law 3) ....................................................................................................... 424
16.4.4. Devolatilization (Law 4) ...................................................................................................... 425
16.4.4.1. Choosing the Devolatilization Model .......................................................................... 426
16.4.4.2.The Constant Rate Devolatilization Model ................................................................... 426
16.4.4.3. The Single Kinetic Rate Model .................................................................................... 426
16.4.4.4.The Two Competing Rates (Kobayashi) Model ............................................................. 427
16.4.4.5. The CPD Model .......................................................................................................... 428
16.4.4.5.1. General Description .......................................................................................... 428
16.4.4.5.2. Reaction Rates .................................................................................................. 428
16.4.4.5.3. Mass Conservation ............................................................................................ 429
16.4.4.5.4. Fractional Change in the Coal Mass .................................................................... 429
16.4.4.5.5. CPD Inputs ........................................................................................................ 430
16.4.4.6. Particle Swelling During Devolatilization .................................................................... 432
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16.4.4.7. Heat Transfer to the Particle During Devolatilization ................................................... 432
16.4.5. Surface Combustion (Law 5) ............................................................................................... 433
16.4.5.1.The Diffusion-Limited Surface Reaction Rate Model .................................................... 433
16.4.5.2.The Kinetic/Diffusion Surface Reaction Rate Model ..................................................... 434
16.4.5.3. The Intrinsic Model .................................................................................................... 434
16.4.5.4.The Multiple Surface Reactions Model ........................................................................ 436
16.4.5.4.1. Limitations ........................................................................................................ 436
16.4.5.5. Heat and Mass Transfer During Char Combustion ....................................................... 436
16.4.6. Multicomponent Particle Definition (Law 7) ........................................................................ 437
16.4.6.1. Raoult’s Law .............................................................................................................. 439
16.4.6.2. Peng-Robinson Real Gas Model .................................................................................. 439
16.5.Vapor Liquid Equilibrium Theory .................................................................................................. 440
16.6. Physical Property Averaging ........................................................................................................ 442
16.7.Wall-Particle Reflection Model Theory .......................................................................................... 443
16.7.1. Rough Wall Model .............................................................................................................. 446
16.8.Wall-Jet Model Theory ................................................................................................................. 447
16.9.Wall-Film Model Theory ............................................................................................................... 448
16.9.1. Introduction ....................................................................................................................... 448
16.9.2. Interaction During Impact with a Boundary ......................................................................... 450
16.9.2.1. The Stanton-Rutland Model ....................................................................................... 451
16.9.2.1.1. Regime Definition ............................................................................................. 451
16.9.2.1.2. Rebound ........................................................................................................... 452
16.9.2.1.3. Splashing .......................................................................................................... 452
16.9.2.2.The Kuhnke Model ..................................................................................................... 457
16.9.2.2.1. Regime definition ............................................................................................. 457
16.9.2.2.2. Rebound ........................................................................................................... 460
16.9.2.2.3. Splashing .......................................................................................................... 460
16.9.3. Separation and Stripping Submodels .................................................................................. 463
16.9.4. Conservation Equations for Wall-Film Particles .................................................................... 463
16.9.4.1. Momentum ............................................................................................................... 463
16.9.4.2. Mass Transfer from the Film ........................................................................................ 464
16.9.4.2.1. Film Vaporization and Boiling ............................................................................ 464
16.9.4.2.2. Film Condensation ............................................................................................ 466
16.9.4.3. Energy Transfer from the Film ..................................................................................... 467
16.10. Wall Erosion .............................................................................................................................. 470
16.10.1. Finnie Erosion Model ........................................................................................................ 470
16.10.2. Oka Erosion Model ........................................................................................................... 471
16.10.3. McLaury Erosion Model .................................................................................................... 472
16.10.4. Modeling Erosion Rates in Dense Flows ............................................................................. 472
16.10.4.1. Abrasive Erosion Caused by Solid Particles ................................................................ 473
16.10.4.2.Wall Shielding Effect in Dense Flow Regimes ............................................................. 473
16.10.5. Accretion ......................................................................................................................... 474
16.11. Particle–Wall Impingement Heat Transfer Theory ....................................................................... 475
16.12. Atomizer Model Theory ............................................................................................................. 477
16.12.1.The Plain-Orifice Atomizer Model ...................................................................................... 477
16.12.1.1. Internal Nozzle State ................................................................................................ 479
16.12.1.2. Coefficient of Discharge ........................................................................................... 480
16.12.1.3. Exit Velocity ............................................................................................................. 481
16.12.1.4. Spray Angle ............................................................................................................. 482
16.12.1.5. Droplet Diameter Distribution .................................................................................. 482
16.12.2. The Pressure-Swirl Atomizer Model ................................................................................... 483
16.12.2.1. Film Formation ........................................................................................................ 484
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16.12.2.2. Sheet Breakup and Atomization ............................................................................... 485
16.12.3.The Air-Blast/Air-Assist Atomizer Model ............................................................................. 487
16.12.4.The Flat-Fan Atomizer Model ............................................................................................. 488
16.12.5.The Effervescent Atomizer Model ...................................................................................... 489
16.13. Secondary Breakup Model Theory ............................................................................................. 490
16.13.1.Taylor Analogy Breakup (TAB) Model ................................................................................. 491
16.13.1.1. Introduction ............................................................................................................ 491
16.13.1.2. Use and Limitations ................................................................................................. 491
16.13.1.3. Droplet Distortion .................................................................................................... 491
16.13.1.4. Size of Child Droplets ............................................................................................... 493
16.13.1.5.Velocity of Child Droplets ......................................................................................... 493
16.13.1.6. Droplet Breakup ...................................................................................................... 493
16.13.2.Wave Breakup Model ........................................................................................................ 495
16.13.2.1. Introduction ............................................................................................................ 495
16.13.2.2. Use and Limitations ................................................................................................. 495
16.13.2.3. Jet Stability Analysis ................................................................................................. 495
16.13.2.4. Droplet Breakup ...................................................................................................... 496
16.13.3. KHRT Breakup Model ........................................................................................................ 497
16.13.3.1. Introduction ............................................................................................................ 497
16.13.3.2. Use and Limitations ................................................................................................. 497
16.13.3.3. Liquid Core Length .................................................................................................. 498
16.13.3.4. Rayleigh-Taylor Breakup ........................................................................................... 498
16.13.3.5. Droplet Breakup Within the Liquid Core .................................................................... 499
16.13.3.6. Droplet Breakup Outside the Liquid Core .................................................................. 499
16.13.4. Stochastic Secondary Droplet (SSD) Model ........................................................................ 499
16.13.5. Madabhushi Breakup Model ............................................................................................. 500
16.14. Collision and Droplet Coalescence Model Theory ....................................................................... 504
16.14.1. Introduction ..................................................................................................................... 505
16.14.2. Use and Limitations .......................................................................................................... 505
16.14.3.Theory .............................................................................................................................. 506
16.14.3.1. Probability of Collision ............................................................................................. 506
16.14.3.2. Collision Outcomes .................................................................................................. 507
16.15. Discrete Element Method Collision Model .................................................................................. 507
16.15.1.Theory .............................................................................................................................. 508
16.15.1.1. The Spring Collision Law .......................................................................................... 509
16.15.1.2. The Spring-Dashpot Collision Law ............................................................................ 509
16.15.1.3. The Hertzian Collision Law ....................................................................................... 510
16.15.1.4. The Hertzian-Dashpot Collision Law ......................................................................... 510
16.15.1.5.The Friction Collision Law ......................................................................................... 510
16.15.1.6. Rolling Friction Collision Law for DEM ....................................................................... 512
16.15.1.7. DEM Parcels ............................................................................................................. 512
16.15.1.8. Cartesian Collision Mesh .......................................................................................... 513
16.16. One-Way and Two-Way Coupling ............................................................................................... 513
16.16.1. Coupling Between the Discrete and Continuous Phases .................................................... 514
16.16.2. Momentum Exchange ...................................................................................................... 514
16.16.3. Heat Exchange ................................................................................................................. 515
16.16.4. Mass Exchange ................................................................................................................. 516
16.16.5. Under-Relaxation of the Interphase Exchange Terms ......................................................... 516
16.16.6. Interphase Exchange During Stochastic Tracking ............................................................... 516
16.16.7. Interphase Exchange During Cloud Tracking ..................................................................... 517
16.17. Node Based Averaging .............................................................................................................. 517
17. Modeling Macroscopic Particles ....................................................................................................... 519
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17.1. Momentum Transfer to Fluid Flow ............................................................................................... 519
17.2. Fluid Forces and Torques on Particle ............................................................................................ 519
17.3. Particle/Particle and Particle/Wall Collisions ................................................................................. 520
17.4. Field Forces ................................................................................................................................. 521
17.5. Particle Deposition and Buildup .................................................................................................. 522
18. Multiphase Flows .............................................................................................................................. 523
18.1. Introduction ............................................................................................................................... 523
18.1.1. Multiphase Flow Regimes ................................................................................................... 523
18.1.1.1. Gas-Liquid or Liquid-Liquid Flows .............................................................................. 523
18.1.1.2. Gas-Solid Flows .......................................................................................................... 524
18.1.1.3. Liquid-Solid Flows ...................................................................................................... 524
18.1.1.4. Three-Phase Flows ..................................................................................................... 524
18.1.2. Examples of Multiphase Systems ........................................................................................ 525
18.2. Choosing a General Multiphase Model ........................................................................................ 526
18.2.1. Approaches to Multiphase Modeling .................................................................................. 526
18.2.1.1.The Euler-Euler Approach ........................................................................................... 526
18.2.1.1.1.The VOF Model .................................................................................................. 526
18.2.1.1.2. The Mixture Model ............................................................................................ 527
18.2.1.1.3.The Eulerian Model ............................................................................................ 527
18.2.2. Model Comparisons ........................................................................................................... 527
18.2.2.1. Detailed Guidelines ................................................................................................... 528
18.2.2.1.1.The Effect of Particulate Loading ........................................................................ 528
18.2.2.1.2.The Significance of the Stokes Number .............................................................. 529
18.2.2.1.2.1. Examples .................................................................................................. 529
18.2.2.1.3. Other Considerations ........................................................................................ 530
18.2.3.Time Schemes in Multiphase Flow ....................................................................................... 530
18.2.4. Stability and Convergence .................................................................................................. 531
18.3.Volume of Fluid (VOF) Model Theory ............................................................................................ 532
18.3.1. Overview of the VOF Model ................................................................................................ 532
18.3.2. Limitations of the VOF Model .............................................................................................. 532
18.3.3. Steady-State and Transient VOF Calculations ....................................................................... 532
18.3.4.Volume Fraction Equation ................................................................................................... 533
18.3.4.1. The Implicit Formulation ............................................................................................ 533
18.3.4.2.The Explicit Formulation ............................................................................................. 534
18.3.4.3. Interpolation Near the Interface ................................................................................. 535
18.3.4.3.1. The Geometric Reconstruction Scheme ............................................................. 536
18.3.4.3.2.The Donor-Acceptor Scheme ............................................................................. 537
18.3.4.3.3.The Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) ..... 537
18.3.4.3.4.The Compressive Scheme and Interface-Model-based Variants ........................... 538
18.3.4.3.5. Bounded Gradient Maximization (BGM) ............................................................. 538
18.3.5. Material Properties ............................................................................................................. 539
18.3.6. Momentum Equation ......................................................................................................... 539
18.3.7. Energy Equation ................................................................................................................. 539
18.3.8. Additional Scalar Equations ................................................................................................ 540
18.3.9. Surface Tension and Adhesion ............................................................................................ 540
18.3.9.1. Surface Tension ......................................................................................................... 540
18.3.9.1.1. The Continuum Surface Force Model ................................................................. 541
18.3.9.1.2.The Continuum Surface Stress Model ................................................................. 542
18.3.9.1.3. Comparing the CSS and CSF Methods ................................................................ 542
18.3.9.1.4.When Surface Tension Effects Are Important ...................................................... 543
18.3.9.2.Wall Adhesion ............................................................................................................ 543
18.3.9.3. Jump Adhesion .......................................................................................................... 543
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18.3.10. Open Channel Flow .......................................................................................................... 544
18.3.10.1. Upstream Boundary Conditions ............................................................................... 545
18.3.10.1.1. Pressure Inlet .................................................................................................. 545
18.3.10.1.2. Mass Flow Rate ................................................................................................ 545
18.3.10.1.3.Volume Fraction Specification .......................................................................... 545
18.3.10.2. Downstream Boundary Conditions ........................................................................... 546
18.3.10.2.1. Pressure Outlet ................................................................................................ 546
18.3.10.2.2. Outflow Boundary ........................................................................................... 546
18.3.10.2.3. Backflow Volume Fraction Specification ........................................................... 546
18.3.10.3. Numerical Beach Treatment ..................................................................................... 547
18.3.11. Open Channel Wave Boundary Conditions ........................................................................ 548
18.3.11.1. Airy Wave Theory ..................................................................................................... 549
18.3.11.2. Stokes Wave Theories ............................................................................................... 550
18.3.11.3. Cnoidal/Solitary Wave Theory ................................................................................... 551
18.3.11.4. Choosing a Wave Theory .......................................................................................... 552
18.3.11.5. Superposition of Waves ............................................................................................ 554
18.3.11.6. Modeling of Random Waves Using Wave Spectrum ................................................... 555
18.3.11.6.1. Definitions ...................................................................................................... 555
18.3.11.6.2.Wave Spectrum Implementation Theory .......................................................... 556
18.3.11.6.2.1. Long-Crested Random Waves (Unidirectional) ......................................... 556
18.3.11.6.2.1.1. Pierson-Moskowitz Spectrum ......................................................... 556
18.3.11.6.2.1.2. JONSWAP Spectrum ....................................................................... 556
18.3.11.6.2.1.3. TMA Spectrum ............................................................................... 557
18.3.11.6.2.2. Short-Crested Random Waves (Multi-Directional) .................................... 557
18.3.11.6.2.2.1. Cosine-2s Power Function (Frequency Independent) ....................... 558
18.3.11.6.2.2.2. Hyperbolic Function (Frequency Dependent) ................................. 558
18.3.11.6.2.3. Superposition of Individual Wave Components Using the Wave Spectrum ........................................................................................................................... 558
18.3.11.6.3. Choosing a Wave Spectrum and Inputs ............................................................ 559
18.3.11.7. Nomenclature for Open Channel Waves .................................................................... 561
18.3.12. Coupled Level-Set and VOF Model .................................................................................... 562
18.3.12.1. Theory ..................................................................................................................... 563
18.3.12.1.1. Surface Tension Force ...................................................................................... 563
18.3.12.1.2. Re-initialization of the Level-set Function via the Geometrical Method ............. 564
18.3.12.2. Limitations .............................................................................................................. 566
18.4. Mixture Model Theory ................................................................................................................. 566
18.4.1. Overview ........................................................................................................................... 566
18.4.2. Limitations of the Mixture Model ........................................................................................ 567
18.4.3. Continuity Equation ........................................................................................................... 568
18.4.4. Momentum Equation ......................................................................................................... 568
18.4.5. Energy Equation ................................................................................................................. 568
18.4.6. Relative (Slip) Velocity and the Drift Velocity ........................................................................ 569
18.4.7.Volume Fraction Equation for the Secondary Phases ............................................................ 571
18.4.8. Granular Properties ............................................................................................................ 571
18.4.8.1. Collisional Viscosity .................................................................................................... 571
18.4.8.2. Kinetic Viscosity ......................................................................................................... 571
18.4.9. Granular Temperature ......................................................................................................... 571
18.4.10. Solids Pressure ................................................................................................................. 572
18.4.11. Interfacial Area Concentration .......................................................................................... 572
18.4.11.1.Transport Equation Based Models ............................................................................. 573
18.4.11.1.1. Hibiki-Ishii Model ............................................................................................ 574
18.4.11.1.2. Ishii-Kim Model ............................................................................................... 575
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18.4.11.1.3.Yao-Morel Model ............................................................................................. 575
18.4.11.2. Algebraic Models ..................................................................................................... 577
18.5. Eulerian Model Theory ................................................................................................................ 578
18.5.1. Overview of the Eulerian Model .......................................................................................... 578
18.5.2. Limitations of the Eulerian Model ........................................................................................ 579
18.5.3.Volume Fraction Equation ................................................................................................... 579
18.5.4. Conservation Equations ...................................................................................................... 580
18.5.4.1. Equations in General Form ......................................................................................... 580
18.5.4.1.1. Conservation of Mass ........................................................................................ 580
18.5.4.1.2. Conservation of Momentum .............................................................................. 580
18.5.4.1.3. Conservation of Energy ..................................................................................... 581
18.5.4.2. Equations Solved by ANSYS Fluent ............................................................................. 582
18.5.4.2.1. Continuity Equation .......................................................................................... 582
18.5.4.2.2. Fluid-Fluid Momentum Equations ...................................................................... 582
18.5.4.2.3. Fluid-Solid Momentum Equations ...................................................................... 582
18.5.4.2.4. Conservation of Energy ..................................................................................... 583
18.5.5. Interfacial Area Concentration ............................................................................................ 583
18.5.6. Interphase Exchange Coefficients ....................................................................................... 584
18.5.6.1. Fluid-Fluid Exchange Coefficient ................................................................................ 585
18.5.6.1.1. Schiller and Naumann Model ............................................................................. 585
18.5.6.1.2. Morsi and Alexander Model ............................................................................... 586
18.5.6.1.3. Symmetric Model .............................................................................................. 586
18.5.6.1.4. Grace et al. Model .............................................................................................. 587
18.5.6.1.5.Tomiyama et al. Model ....................................................................................... 588
18.5.6.1.6. Ishii Model ........................................................................................................ 589
18.5.6.1.7. Ishii-Zuber Drag Model ...................................................................................... 589
18.5.6.1.8. Universal Drag Laws for Bubble-Liquid and Droplet-Gas Flows ........................... 590
18.5.6.1.8.1. Bubble-Liquid Flow .................................................................................. 591
18.5.6.1.8.2. Droplet-Gas Flow ...................................................................................... 592
18.5.6.2. Fluid-Solid Exchange Coefficient ................................................................................ 592
18.5.6.3. Solid-Solid Exchange Coefficient ................................................................................ 595
18.5.6.4. Drag Modification ...................................................................................................... 595
18.5.6.4.1. Brucato et al. Correlation ................................................................................... 596
18.5.7. Lift Force ............................................................................................................................ 596
18.5.7.1. Lift Coefficient Models ............................................................................................... 597
18.5.7.1.1. Moraga Lift Force Model .................................................................................... 597
18.5.7.1.2. Saffman-Mei Lift Force Model ............................................................................ 598
18.5.7.1.3. Legendre-Magnaudet Lift Force Model .............................................................. 598
18.5.7.1.4.Tomiyama Lift Force Model ................................................................................ 599
18.5.8. Wall Lubrication Force ........................................................................................................ 599
18.5.8.1.Wall Lubrication Models ............................................................................................. 599
18.5.8.1.1. Antal et al. Model .............................................................................................. 600
18.5.8.1.2.Tomiyama Model ............................................................................................... 600
18.5.8.1.3. Frank Model ...................................................................................................... 601
18.5.8.1.4. Hosokawa Model .............................................................................................. 601
18.5.9. Turbulent Dispersion Force ................................................................................................. 601
18.5.9.1. Models for Turbulent Dispersion Force ....................................................................... 602
18.5.9.1.1. Lopez de Bertodano Model ............................................................................... 602
18.5.9.1.2. Simonin Model .................................................................................................. 602
18.5.9.1.3. Burns et al. Model .............................................................................................. 603
18.5.9.1.4. Diffusion in VOF Model ...................................................................................... 603
18.5.9.2. Limiting Functions for the Turbulent Dispersion Force ................................................ 603
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18.5.10.Virtual Mass Force ............................................................................................................. 604
18.5.11. Solids Pressure ................................................................................................................. 605
18.5.11.1. Radial Distribution Function ..................................................................................... 606
18.5.12. Maximum Packing Limit in Binary Mixtures ....................................................................... 607
18.5.13. Solids Shear Stresses ......................................................................................................... 608
18.5.13.1. Collisional Viscosity .................................................................................................. 608
18.5.13.2. Kinetic Viscosity ....................................................................................................... 608
18.5.13.3. Bulk Viscosity ........................................................................................................... 608
18.5.13.4. Frictional Viscosity ................................................................................................... 609
18.5.14. Granular Temperature ....................................................................................................... 610
18.5.15. Description of Heat Transfer .............................................................................................. 612
18.5.15.1. The Heat Exchange Coefficient ................................................................................. 612
18.5.15.1.1. Constant ......................................................................................................... 613
18.5.15.1.2. Nusselt Number .............................................................................................. 613
18.5.15.1.3. Ranz-Marshall Model ....................................................................................... 613
18.5.15.1.4.Tomiyama Model ............................................................................................. 613
18.5.15.1.5. Hughmark Model ............................................................................................ 613
18.5.15.1.6. Gunn Model .................................................................................................... 613
18.5.15.1.7. Two-Resistance Model ..................................................................................... 614
18.5.15.1.8. Fixed To Saturation Temperature ...................................................................... 614
18.5.15.1.9. User Defined ................................................................................................... 615
18.5.16. Turbulence Models ........................................................................................................... 615
18.5.16.1. k- ε Turbulence Models ............................................................................................. 616
18.5.16.1.1. k- ε Mixture Turbulence Model ......................................................................... 616
18.5.16.1.2. k- ε Dispersed Turbulence Model ..................................................................... 617
18.5.16.1.2.1. Assumptions .......................................................................................... 617
18.5.16.1.2.2. Turbulence in the Continuous Phase ....................................................... 617
18.5.16.1.2.3.Turbulence in the Dispersed Phase .......................................................... 618
18.5.16.1.3. k- ε Turbulence Model for Each Phase ............................................................... 618
18.5.16.1.3.1.Transport Equations ................................................................................ 619
18.5.16.2. RSM Turbulence Models ........................................................................................... 619
18.5.16.2.1. RSM Dispersed Turbulence Model .................................................................... 620
18.5.16.2.2. RSM Mixture Turbulence Model ....................................................................... 621
18.5.16.3. Turbulence Interaction Models ................................................................................. 621
18.5.16.3.1. Simonin et al. .................................................................................................. 622
18.5.16.3.1.1. Formulation in Dispersed Turbulence Models .......................................... 622
18.5.16.3.1.1.1. Continuous Phase .......................................................................... 622
18.5.16.3.1.1.2. Dispersed Phases ........................................................................... 623
18.5.16.3.1.2. Formulation in Per Phase Turbulence Models ........................................... 624
18.5.16.3.2. Troshko-Hassan ............................................................................................... 624
18.5.16.3.2.1.Troshko-Hassan Formulation in Mixture Turbulence Models ..................... 624
18.5.16.3.2.2. Troshko-Hassan Formulation in Dispersed Turbulence Models ................. 625
18.5.16.3.2.2.1. Continuous Phase .......................................................................... 625
18.5.16.3.2.2.2. Dispersed Phases ........................................................................... 625
18.5.16.3.2.3.Troshko-Hassan Formulation in Per-Phase Turbulence Models .................. 625
18.5.16.3.2.3.1. Continuous Phase .......................................................................... 625
18.5.16.3.2.3.2. Dispersed Phases ........................................................................... 625
18.5.16.3.3. Sato ................................................................................................................ 625
18.5.16.3.4. None ............................................................................................................... 626
18.5.17. Solution Method in ANSYS Fluent ..................................................................................... 626
18.5.17.1.The Pressure-Correction Equation ............................................................................. 626
18.5.17.2. Volume Fractions ..................................................................................................... 626
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18.5.18. Dense Discrete Phase Model ............................................................................................. 627
18.5.18.1. Limitations .............................................................................................................. 628
18.5.18.2. Granular Temperature .............................................................................................. 628
18.5.19. Multi-Fluid VOF Model ...................................................................................................... 629
18.5.20. Wall Boiling Models .......................................................................................................... 630
18.5.20.1. Overview ................................................................................................................. 630
18.5.20.2. RPI Model ................................................................................................................ 631
18.5.20.3. Non-equilibrium Subcooled Boiling .......................................................................... 633
18.5.20.4. Critical Heat Flux ...................................................................................................... 634
18.5.20.4.1.Wall Heat Flux Partition .................................................................................... 634
18.5.20.4.2. Flow Regime Transition ................................................................................... 635
18.5.20.5. Interfacial Momentum Transfer ................................................................................. 636
18.5.20.5.1. Interfacial Area ................................................................................................ 636
18.5.20.5.2. Bubble and Droplet Diameters ........................................................................ 636
18.5.20.5.2.1. Bubble Diameter .................................................................................... 636
18.5.20.5.2.2. Droplet Diameter .................................................................................... 636
18.5.20.5.3. Interfacial Drag Force ...................................................................................... 637
18.5.20.5.4. Interfacial Lift Force ......................................................................................... 637
18.5.20.5.5.Turbulent Dispersion Force .............................................................................. 637
18.5.20.5.6. Wall Lubrication Force ..................................................................................... 637
18.5.20.5.7. Virtual Mass Force ........................................................................................... 637
18.5.20.6. Interfacial Heat Transfer ............................................................................................ 637
18.5.20.6.1. Interface to Liquid Heat Transfer ...................................................................... 637
18.5.20.6.2. Interface to Vapor Heat Transfer ....................................................................... 638
18.5.20.7. Mass Transfer ........................................................................................................... 638
18.5.20.7.1. Mass Transfer From the Wall to Vapor ............................................................... 638
18.5.20.7.2. Interfacial Mass Transfer .................................................................................. 638
18.5.20.8.Turbulence Interactions ............................................................................................ 638
18.6. Wet Steam Model Theory ............................................................................................................ 638
18.6.1. Overview of the Wet Steam Model ...................................................................................... 638
18.6.2. Limitations of the Wet Steam Model .................................................................................... 639
18.6.3.Wet Steam Flow Equations .................................................................................................. 639
18.6.4. Phase Change Model .......................................................................................................... 640
18.6.5. Built-in Thermodynamic Wet Steam Properties .................................................................... 642
18.6.5.1. Equation of State ....................................................................................................... 642
18.6.5.2. Saturated Vapor Line .................................................................................................. 643
18.6.5.3. Saturated Liquid Line ................................................................................................. 643
18.6.5.4. Mixture Properties ..................................................................................................... 643
18.7. Modeling Mass Transfer in Multiphase Flows ................................................................................ 643
18.7.1. Source Terms due to Mass Transfer ...................................................................................... 644
18.7.1.1. Mass Equation ........................................................................................................... 644
18.7.1.2. Momentum Equation ................................................................................................. 644
18.7.1.3. Energy Equation ........................................................................................................ 644
18.7.1.4. Species Equation ....................................................................................................... 644
18.7.1.5. Other Scalar Equations ............................................................................................... 645
18.7.2. Unidirectional Constant Rate Mass Transfer ......................................................................... 645
18.7.3. UDF-Prescribed Mass Transfer ............................................................................................. 645
18.7.4. Cavitation Models .............................................................................................................. 645
18.7.4.1. Limitations of the Cavitation Models .......................................................................... 646
18.7.4.1.1. Limitations of Cavitation with the VOF Model ..................................................... 647
18.7.4.2.Vapor Transport Equation ........................................................................................... 647
18.7.4.3. Bubble Dynamics Consideration ................................................................................ 648
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18.7.4.4. Singhal et al. Model .................................................................................................... 648
18.7.4.5. Zwart-Gerber-Belamri Model ..................................................................................... 650
18.7.4.6. Schnerr and Sauer Model ........................................................................................... 651
18.7.4.7. Turbulence Factor ...................................................................................................... 652
18.7.4.8. Additional Guidelines for the Cavitation Models ......................................................... 653
18.7.4.9. Extended Cavitation Model Capabilities ..................................................................... 655
18.7.4.9.1. Multiphase Cavitation Models ........................................................................... 655
18.7.4.9.2. Multiphase Species Transport Cavitation Model ................................................. 655
18.7.5. Evaporation-Condensation Model ....................................................................................... 656
18.7.5.1. Lee Model ................................................................................................................. 656
18.7.5.2.Thermal Phase Change Model .................................................................................... 658
18.7.6. Interphase Species Mass Transfer ........................................................................................ 659
18.7.6.1. Modeling Approach ................................................................................................... 660
18.7.6.2. Equilibrium Models .................................................................................................... 662
18.7.6.2.1. Raoult’s Law ...................................................................................................... 663
18.7.6.2.2. Henry’s Law ...................................................................................................... 664
18.7.6.2.3. Equilibrium Ratio .............................................................................................. 665
18.7.6.3. Mass Transfer Coefficient Models ................................................................................ 665
18.7.6.3.1. Constant ........................................................................................................... 666
18.7.6.3.2. Sherwood Number ............................................................................................ 666
18.7.6.3.3. Ranz-Marshall Model ......................................................................................... 666
18.7.6.3.4. Hughmark Model .............................................................................................. 666
18.7.6.3.5. User-Defined ..................................................................................................... 666
18.8. Modeling Species Transport in Multiphase Flows ......................................................................... 666
18.8.1. Limitations ......................................................................................................................... 668
18.8.2. Mass and Momentum Transfer with Multiphase Species Transport ....................................... 668
18.8.2.1. Source Terms Due to Heterogeneous Reactions .......................................................... 668
18.8.2.1.1. Mass Transfer .................................................................................................... 668
18.8.2.1.2. Momentum Transfer .......................................................................................... 669
18.8.2.1.3. Species Transfer ................................................................................................ 669
18.8.2.1.4. Heat Transfer ..................................................................................................... 669
18.8.3. The Stiff Chemistry Solver ................................................................................................... 670
18.8.4. Heterogeneous Phase Interaction ....................................................................................... 670
19. Population Balance Model ............................................................................................................... 673
19.1. Introduction ............................................................................................................................... 673
19.1.1. The Discrete Method .......................................................................................................... 673
19.1.2. The Inhomogeneous Discrete Method ................................................................................ 673
19.1.3.The Standard Method of Moments ...................................................................................... 675
19.1.4.The Quadrature Method of Moments .................................................................................. 676
19.2. Population Balance Model Theory ............................................................................................... 676
19.2.1. The Particle State Vector ..................................................................................................... 676
19.2.2. The Population Balance Equation (PBE) ............................................................................... 676
19.2.2.1. Particle Growth .......................................................................................................... 677
19.2.2.2. Particle Birth and Death Due to Breakage and Aggregation ........................................ 678
19.2.2.2.1. Breakage ........................................................................................................... 678
19.2.2.2.2. Luo and Lehr Breakage Kernels .......................................................................... 679
19.2.2.2.3. Ghadiri Breakage Kernels ................................................................................... 680
19.2.2.2.4. Laakkonen Breakage Kernels ............................................................................. 680
19.2.2.2.5. Parabolic PDF .................................................................................................... 681
19.2.2.2.6. Generalized PDF ................................................................................................ 681
19.2.2.2.7. Aggregation ..................................................................................................... 684
19.2.2.2.8. Luo Aggregation Kernel .................................................................................... 685
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19.2.2.2.9. Free Molecular Aggregation Kernel .................................................................... 685
19.2.2.2.10.Turbulent Aggregation Kernel .......................................................................... 686
19.2.2.3. Particle Birth by Nucleation ........................................................................................ 687
19.2.3. Solution Methods ............................................................................................................... 687
19.2.3.1.The Discrete Method and the Inhomogeneous Discrete Method ................................. 687
19.2.3.1.1. Numerical Method ............................................................................................ 687
19.2.3.1.2. Breakage Formulations for the Discrete Method ................................................. 689
19.2.3.2. The Standard Method of Moments (SMM) .................................................................. 690
19.2.3.2.1. Numerical Method ............................................................................................ 690
19.2.3.3.The Quadrature Method of Moments (QMOM) ............................................................ 691
19.2.3.3.1. Numerical Method ............................................................................................ 691
19.2.3.4.The Direct Quadrature Method of Moments (DQMOM) ............................................... 692
19.2.3.4.1. Numerical Method ............................................................................................ 692
19.2.4. Population Balance Statistics .............................................................................................. 694
19.2.4.1. Reconstructing the Particle Size Distribution from Moments ....................................... 694
19.2.4.2.The Log-Normal Distribution ...................................................................................... 695
20. Solidification and Melting ................................................................................................................. 697
20.1. Overview .................................................................................................................................... 697
20.2. Limitations .................................................................................................................................. 698
20.3. Introduction ............................................................................................................................... 698
20.4. Energy Equation ......................................................................................................................... 698
20.5. Momentum Equations ................................................................................................................ 699
20.6.Turbulence Equations .................................................................................................................. 700
20.7. Species Equations ....................................................................................................................... 700
20.8. Back Diffusion ............................................................................................................................. 702
20.9. Pull Velocity for Continuous Casting ............................................................................................ 702
20.10. Contact Resistance at Walls ........................................................................................................ 704
20.11.Thermal and Solutal Buoyancy ................................................................................................... 704
21.The Structural Model for Intrinsic Fluid-Structure Interaction (FSI) ................................................. 707
21.1. Equations ................................................................................................................................... 707
21.1.1. Linear Isotropic and Isothermal Elasticity ............................................................................. 707
21.1.2.The FSI Model ..................................................................................................................... 708
21.2. Finite Element Representation ..................................................................................................... 708
21.2.1. Construction of the Matrix of the System ............................................................................ 708
21.2.2. Intrinsic FSI ......................................................................................................................... 709
21.2.3. Dynamic Structural Systems ............................................................................................... 710
21.2.3.1. The Newmark Method ............................................................................................... 710
21.2.4. Limitations ......................................................................................................................... 711
22. Eulerian Wall Films ............................................................................................................................ 713
22.1. Introduction ............................................................................................................................... 713
22.2. Mass, Momentum, and Energy Conservation Equations for Wall Film ............................................. 714
22.2.1. Film Sub-Models ................................................................................................................. 715
22.2.1.1. DPM Collection .......................................................................................................... 715
22.2.1.2. Particle-Wall Interaction ............................................................................................. 715
22.2.1.3. Film Separation .......................................................................................................... 715
22.2.1.3.1. Separation Criteria ............................................................................................ 715
22.2.1.3.1.1. Foucart Separation ................................................................................... 716
22.2.1.3.1.2. O’Rourke Separation ................................................................................. 716
22.2.1.3.1.3. Friedrich Separation ................................................................................. 716
22.2.1.4. Film Stripping ............................................................................................................ 717
22.2.1.5. Secondary Phase Accretion ........................................................................................ 718
22.2.1.6. Coupling of Wall Film with Mixture Species Transport ................................................. 719
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22.2.2. Partial Wetting Effect .......................................................................................................... 720
22.2.3. Boundary Conditions .......................................................................................................... 720
22.2.4. Obtaining Film Velocity Without Solving the Momentum Equations .................................... 720
22.2.4.1. Shear-Driven Film Velocity ......................................................................................... 720
22.2.4.2. Gravity-Driven Film Velocity ....................................................................................... 721
22.3. Passive Scalar Equation for Wall Film ............................................................................................ 722
22.4. Numerical Schemes and Solution Algorithm ................................................................................ 722
22.4.1.Temporal Differencing Schemes .......................................................................................... 723
22.4.1.1. First-Order Explicit Method ........................................................................................ 723
22.4.1.2. First-Order Implicit Method ........................................................................................ 723
22.4.1.3. Second-Order Implicit Method ................................................................................... 724
22.4.2. Spatial Differencing Schemes .............................................................................................. 725
22.4.3. Solution Algorithm ............................................................................................................. 726
22.4.3.1. Steady Flow ............................................................................................................... 726
22.4.3.2. Transient Flow ........................................................................................................... 726
23. Electric Potential ............................................................................................................................... 727
23.1. Overview and Limitations ............................................................................................................ 727
23.2. Electric Potential Equation ........................................................................................................... 727
23.3. Energy Equation Source Term ...................................................................................................... 728
24. Modeling Batteries ............................................................................................................................ 729
24.1. Single-Potential Empirical Battery Model Theory .......................................................................... 729
24.1.1. Introduction ....................................................................................................................... 729
24.1.2. Computation of the Electric Potential and Current Density .................................................. 729
24.1.3.Thermal and Electrical Coupling .......................................................................................... 731
24.2. Dual-Potential MSMD Battery Model ............................................................................................ 731
24.2.1. MSMD approach ................................................................................................................ 731
24.2.2. NTGK Model ....................................................................................................................... 732
24.2.3. ECM Model ......................................................................................................................... 733
24.2.4. Newman’s P2D Model ......................................................................................................... 735
24.2.5. Coupling Between CFD and Submodels .............................................................................. 739
24.2.6. Battery Pack Simulation ...................................................................................................... 739
24.2.7. Reduced Order Solution Method (ROM) .............................................................................. 741
24.2.8. External and Internal Electric Short-Circuit Treatment .......................................................... 741
24.2.9.Thermal Abuse Model ......................................................................................................... 742
25. Modeling Fuel Cells ........................................................................................................................... 745
25.1. PEMFC Model Theory .................................................................................................................. 745
25.1.1. Introduction ....................................................................................................................... 745
25.1.2. Electrochemistry Modeling ................................................................................................. 747
25.1.2.1. The Cathode Particle Model ....................................................................................... 750
25.1.3. Current and Mass Conservation .......................................................................................... 750
25.1.4.Water Transport and Mass Transfer in PEMFC ....................................................................... 751
25.1.4.1.The Dissolved Phase Model ........................................................................................ 751
25.1.4.2. The Liquid Phase Model ............................................................................................. 752
25.1.4.2.1. Liquid Water Transport Equation in the Porous Electrode and the Membrane ..... 752
25.1.4.2.2. Liquid Water Transport Equation in Gas Channels ............................................... 754
25.1.5. Heat Source ........................................................................................................................ 754
25.1.6. Properties .......................................................................................................................... 755
25.1.7. Transient Simulations ......................................................................................................... 757
25.1.8. Leakage Current (Cross-Over Current) ................................................................................. 757
25.1.9. Zones where User-Defined Scalars are Solved ..................................................................... 758
25.2. Fuel Cell and Electrolysis Model Theory ........................................................................................ 758
25.2.1. Introduction ....................................................................................................................... 758
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25.2.1.1. Introduction to PEMFC ............................................................................................... 759
25.2.1.2. Introduction to SOFC ................................................................................................. 760
25.2.1.3. Introduction to Electrolysis ........................................................................................ 760
25.2.2. Electrochemistry Modeling ................................................................................................. 761
25.2.3. Current and Mass Conservation .......................................................................................... 763
25.2.4. Heat Source ........................................................................................................................ 764
25.2.5. Liquid Water Formation,Transport, and its Effects (PEMFC Only) ........................................... 764
25.2.6. Properties .......................................................................................................................... 765
25.2.7. Transient Simulations ......................................................................................................... 767
25.2.8. Leakage Current (Cross-Over Current) ................................................................................. 767
25.3. SOFC Fuel Cell With Unresolved Electrolyte Model Theory ............................................................ 768
25.3.1. Introduction ....................................................................................................................... 768
25.3.2. The SOFC With Unresolved Electrolyte Modeling Strategy ................................................... 769
25.3.3. Modeling Fluid Flow, Heat Transfer, and Mass Transfer .......................................................... 769
25.3.4. Modeling Current Transport and the Potential Field ............................................................. 770
25.3.4.1. Cell Potential ............................................................................................................. 770
25.3.4.2. Activation Overpotential ............................................................................................ 771
25.3.4.3.Treatment of the Energy Equation at the Electrolyte Interface ..................................... 772
25.3.4.4.Treatment of the Energy Equation in the Conducting Regions ..................................... 774
25.3.5. Modeling Reactions ............................................................................................................ 774
25.3.5.1. Modeling Electrochemical Reactions .......................................................................... 774
25.3.5.2. Modeling CO Electrochemistry ................................................................................... 775
26. Modeling Magnetohydrodynamics .................................................................................................. 777
26.1. Introduction ............................................................................................................................... 777
26.2. Magnetic Induction Method ........................................................................................................ 778
26.2.1. Case 1: Externally Imposed Magnetic Field Generated in Non-conducting Media .................. 778
26.2.2. Case 2: Externally Imposed Magnetic Field Generated in Conducting Media ......................... 779
26.3. Electric Potential Method ............................................................................................................ 779
27. Modeling Continuous Fibers ............................................................................................................. 781
27.1. Introduction ............................................................................................................................... 781
27.2. Governing Equations of Fiber Flow .............................................................................................. 781
27.3. Discretization of the Fiber Equations ............................................................................................ 784
27.3.1. Under-Relaxation ............................................................................................................... 784
27.4. Numerical Solution Algorithm of Fiber Equations ......................................................................... 785
27.5. Residuals of Fiber Equations ........................................................................................................ 785
27.6. Coupling Between Fibers and the Surrounding Fluid .................................................................... 786
27.6.1. Momentum Exchange ........................................................................................................ 786
27.6.2. Mass Exchange ................................................................................................................... 787
27.6.3. Heat Exchange ................................................................................................................... 787
27.6.4. Radiation Exchange ............................................................................................................ 788
27.6.5. Under-Relaxation of the Fiber Exchange Terms .................................................................... 788
27.7. Fiber Grid Generation .................................................................................................................. 788
27.8. Correlations for Momentum, Heat and Mass Transfer .................................................................... 789
27.8.1. Drag Coefficient ................................................................................................................. 790
27.8.2. Heat Transfer Coefficient ..................................................................................................... 791
27.8.3. Mass Transfer Coefficient .................................................................................................... 792
27.9. Fiber Properties ........................................................................................................................... 792
27.9.1. Fiber Viscosity .................................................................................................................... 792
27.9.1.1. Melt Spinning ............................................................................................................ 793
27.9.1.2. Dry Spinning ............................................................................................................. 793
27.9.2.Vapor-Liquid Equilibrium .................................................................................................... 793
27.9.3. Latent Heat of Vaporization ................................................................................................ 794
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27.9.4. Emissivity ........................................................................................................................... 794
27.10. Solution Strategies .................................................................................................................... 794
28. Solver Theory .................................................................................................................................... 797
28.1. Overview of Flow Solvers ............................................................................................................ 797
28.1.1. Pressure-Based Solver ......................................................................................................... 798
28.1.1.1. The Pressure-Based Segregated Algorithm ................................................................. 798
28.1.1.2.The Pressure-Based Coupled Algorithm ...................................................................... 799
28.1.2. Density-Based Solver .......................................................................................................... 800
28.2. General Scalar Transport Equation: Discretization and Solution ..................................................... 802
28.2.1. Solving the Linear System ................................................................................................... 804
28.3. Discretization .............................................................................................................................. 804
28.3.1. Spatial Discretization .......................................................................................................... 804
28.3.1.1. First-Order Upwind Scheme ....................................................................................... 805
28.3.1.2. Power-Law Scheme .................................................................................................... 805
28.3.1.3. Second-Order Upwind Scheme .................................................................................. 806
28.3.1.4. First-to-Higher Order Blending ................................................................................... 807
28.3.1.5. Central-Differencing Scheme ..................................................................................... 807
28.3.1.6. Bounded Central Differencing Scheme ....................................................................... 808
28.3.1.7. QUICK Scheme .......................................................................................................... 808
28.3.1.8.Third-Order MUSCL Scheme ....................................................................................... 809
28.3.1.9. Modified HRIC Scheme .............................................................................................. 809
28.3.1.10. High Order Term Relaxation ..................................................................................... 811
28.3.2. Temporal Discretization ...................................................................................................... 811
28.3.2.1. Implicit Time Integration ............................................................................................ 811
28.3.2.2. Bounded Second-Order Implicit Time Integration ....................................................... 812
28.3.2.2.1. Limitations ........................................................................................................ 812
28.3.2.3. Second-Order Time Integration Using a Variable Time Step Size .................................. 812
28.3.2.4. Explicit Time Integration ............................................................................................ 814
28.3.3. Evaluation of Gradients and Derivatives .............................................................................. 814
28.3.3.1. Green-Gauss Theorem ............................................................................................... 814
28.3.3.2. Green-Gauss Cell-Based Gradient Evaluation .............................................................. 815
28.3.3.3. Green-Gauss Node-Based Gradient Evaluation ............................................................ 815
28.3.3.4. Least Squares Cell-Based Gradient Evaluation ............................................................. 815
28.3.4. Gradient Limiters ................................................................................................................ 817
28.3.4.1. Standard Limiter ........................................................................................................ 817
28.3.4.2. Multidimensional Limiter ........................................................................................... 818
28.3.4.3. Differentiable Limiter ................................................................................................. 818
28.4. Pressure-Based Solver ................................................................................................................. 818
28.4.1. Discretization of the Momentum Equation .......................................................................... 819
28.4.1.1. Pressure Interpolation Schemes ................................................................................. 819
28.4.2. Discretization of the Continuity Equation ............................................................................ 820
28.4.2.1. Density Interpolation Schemes ................................................................................... 821
28.4.3. Pressure-Velocity Coupling ................................................................................................. 822
28.4.3.1. Segregated Algorithms .............................................................................................. 822
28.4.3.1.1. SIMPLE .............................................................................................................. 822
28.4.3.1.2. SIMPLEC ........................................................................................................... 823
28.4.3.1.2.1. Skewness Correction ................................................................................ 823
28.4.3.1.3. PISO .................................................................................................................. 823
28.4.3.1.3.1. Neighbor Correction ................................................................................. 824
28.4.3.1.3.2. Skewness Correction ................................................................................ 824
28.4.3.1.3.3. Skewness - Neighbor Coupling ................................................................. 824
28.4.3.2. Fractional-Step Method (FSM) .................................................................................... 824
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28.4.3.3. Coupled Algorithm .................................................................................................... 824
28.4.3.3.1. Limitation ......................................................................................................... 826
28.4.4. Steady-State Iterative Algorithm ......................................................................................... 826
28.4.4.1. Under-Relaxation of Variables .................................................................................... 826
28.4.4.2. Under-Relaxation of Equations ................................................................................... 826
28.4.5.Time-Advancement Algorithm ............................................................................................ 826
28.4.5.1. Iterative Time-Advancement Scheme ......................................................................... 827
28.4.5.1.1.The Frozen Flux Formulation .............................................................................. 828
28.4.5.2. Non-Iterative Time-Advancement Scheme .................................................................. 829
28.5. Density-Based Solver ................................................................................................................... 831
28.5.1. Governing Equations in Vector Form ................................................................................... 831
28.5.2. Preconditioning ................................................................................................................. 832
28.5.3. Convective Fluxes ............................................................................................................... 834
28.5.3.1. Roe Flux-Difference Splitting Scheme ......................................................................... 834
28.5.3.2. AUSM  Scheme ......................................................................................................... 834
28.5.3.3. Low Diffusion Roe Flux Difference Splitting Scheme ................................................... 835
28.5.4. Steady-State Flow Solution Methods ................................................................................... 835
28.5.4.1. Explicit Formulation ................................................................................................... 836
28.5.4.1.1. Implicit Residual Smoothing .............................................................................. 836
28.5.4.2. Implicit Formulation .................................................................................................. 837
28.5.4.2.1. Convergence Acceleration for Stretched Meshes ................................................ 837
28.5.5. Unsteady Flows Solution Methods ...................................................................................... 838
28.5.5.1. Explicit Time Stepping ............................................................................................... 838
28.5.5.2. Implicit Time Stepping (Dual-Time Formulation) ......................................................... 838
28.6. Pseudo Transient Under-Relaxation ............................................................................................. 840
28.6.1. Automatic Pseudo Transient Time Step ............................................................................... 840
28.7. Multigrid Method ........................................................................................................................ 842
28.7.1. Approach ........................................................................................................................... 842
28.7.1.1.The Need for Multigrid ............................................................................................... 842
28.7.1.2.The Basic Concept in Multigrid ................................................................................... 843
28.7.1.3. Restriction and Prolongation ...................................................................................... 843
28.7.1.4. Unstructured Multigrid .............................................................................................. 844
28.7.2. Multigrid Cycles .................................................................................................................. 844
28.7.2.1. The V and W Cycles .................................................................................................... 844
28.7.3. Algebraic Multigrid (AMG) .................................................................................................. 848
28.7.3.1. AMG Restriction and Prolongation Operators ............................................................. 848
28.7.3.2. AMG Coarse Level Operator ....................................................................................... 849
28.7.3.3. The F Cycle ................................................................................................................ 849
28.7.3.4. The Flexible Cycle ...................................................................................................... 849
28.7.3.4.1.The Residual Reduction Rate Criteria .................................................................. 850
28.7.3.4.2. The Termination Criteria .................................................................................... 851
28.7.3.5.The Coupled and Scalar AMG Solvers .......................................................................... 851
28.7.3.5.1. Gauss-Seidel ..................................................................................................... 852
28.7.3.5.2. Incomplete Lower Upper (ILU) ........................................................................... 852
28.7.4. Full-Approximation Storage (FAS) Multigrid ......................................................................... 853
28.7.4.1. FAS Restriction and Prolongation Operators ............................................................... 854
28.7.4.2. FAS Coarse Level Operator ......................................................................................... 854
28.8. Hybrid Initialization ..................................................................................................................... 854
28.9. Full Multigrid (FMG) Initialization ................................................................................................. 856
28.9.1. Overview of FMG Initialization ............................................................................................ 856
28.9.2. Limitations of FMG Initialization .......................................................................................... 858
29. Adapting the Mesh ............................................................................................................................ 859
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29.1. Adaption Process ........................................................................................................................ 859
29.1.1. Hanging Node Adaption ..................................................................................................... 860
29.1.2. Polyhedral Unstructured Mesh Adaption ............................................................................. 861
29.2. Anisotropic Adaption .................................................................................................................. 862
29.3. Geometry-Based Adaption .......................................................................................................... 863
29.3.1. Geometry-Based Adaption Approach .................................................................................. 863
29.3.1.1. Node Projection ......................................................................................................... 863
29.3.1.2. Example of Geometry-Based Adaption ....................................................................... 866
30. Reporting Alphanumeric Data .......................................................................................................... 869
30.1. Fluxes Through Boundaries ......................................................................................................... 869
30.2. Forces on Boundaries .................................................................................................................. 870
30.2.1. Computing Forces, Moments, and the Center of Pressure ..................................................... 870
30.3. Surface Integration ..................................................................................................................... 873
30.3.1. Computing Surface Integrals .............................................................................................. 874
30.3.1.1. Area .......................................................................................................................... 874
30.3.1.2. Integral ...................................................................................................................... 874
30.3.1.3. Area-Weighted Average ............................................................................................. 874
30.3.1.4. Custom Vector Based Flux .......................................................................................... 874
30.3.1.5. Custom Vector Flux .................................................................................................... 874
30.3.1.6. Custom Vector Weighted Average .............................................................................. 875
30.3.1.7. Flow Rate ................................................................................................................... 875
30.3.1.8. Mass Flow Rate .......................................................................................................... 875
30.3.1.9. Mass-Weighted Average ............................................................................................ 875
30.3.1.10. Sum of Field Variable ................................................................................................ 876
30.3.1.11. Facet Average .......................................................................................................... 876
30.3.1.12. Facet Minimum ........................................................................................................ 876
30.3.1.13. Facet Maximum ....................................................................................................... 876
30.3.1.14.Vertex Average ......................................................................................................... 876
30.3.1.15. Vertex Minimum ...................................................................................................... 877
30.3.1.16.Vertex Maximum ...................................................................................................... 877
30.3.1.17. Standard-Deviation .................................................................................................. 877
30.3.1.18. Uniformity Index ...................................................................................................... 877
30.3.1.19. Volume Flow Rate .................................................................................................... 878
30.4. Volume Integration ..................................................................................................................... 878
30.4.1. Computing Volume Integrals .............................................................................................. 879
30.4.1.1.Volume ...................................................................................................................... 879
30.4.1.2. Sum .......................................................................................................................... 879
30.4.1.3. Sum*2Pi .................................................................................................................... 879
30.4.1.4. Volume Integral ......................................................................................................... 879
30.4.1.5.Volume-Weighted Average ......................................................................................... 880
30.4.1.6. Mass-Weighted Integral ............................................................................................. 880
30.4.1.7. Mass .......................................................................................................................... 880
30.4.1.8. Mass-Weighted Average ............................................................................................ 880
A. Nomenclature ....................................................................................................................................... 881
Bibliography ............................................................................................................................................. 885
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Theory Guide
List of Figures
1.1. Example of Periodic Flow in a 2D Heat Exchanger Geometry .................................................................... 6
1.2. Example of a Periodic Geometry ............................................................................................................. 7
1.3. Rotating Flow in a Cavity ......................................................................................................................... 9
1.4. Swirling Flow in a Gas Burner .................................................................................................................. 9
1.5.Typical Radial Distribution of Circumferential Velocity in a Free Vortex .................................................... 10
1.6. Stream Function Contours for Rotating Flow in a Cavity ......................................................................... 11
1.7.Transonic Flow in a Converging-Diverging Nozzle .................................................................................. 12
1.8. Mach 0.675 Flow Over a Bump in a 2D Channel ...................................................................................... 12
2.1. Single Component (Blower Wheel Blade Passage) .................................................................................. 18
2.2. Multiple Component (Blower Wheel and Casing) ................................................................................... 18
2.3. Stationary and Moving Reference Frames .............................................................................................. 19
2.4. Geometry with One Rotating Impeller ................................................................................................... 23
2.5. Geometry with Two Rotating Impellers .................................................................................................. 24
2.6. Interface Treatment for the MRF Model .................................................................................................. 25
2.7. Axial Rotor-Stator Interaction (Schematic Illustrating the Mixing Plane Concept) .................................... 26
2.8. Radial Rotor-Stator Interaction (Schematic Illustrating the Mixing Plane Concept) .................................. 27
3.1. A Mesh Associated With Moving Pistons ................................................................................................ 33
3.2. Blower .................................................................................................................................................. 34
4.1. Effect of Increasing y  for the Flat Plate T3A Test Case ............................................................................ 77
4.2. Effect of Decreasing y  for the Flat Plate T3A Test Case ........................................................................... 78
4.3. Effect of Wall Normal Expansion Ratio for the Flat Plate T3A Test Case ..................................................... 79
4.4. Effect of Streamwise Mesh Density for the Flat Plate T3A Test Case ......................................................... 79
4.5. Exemplary Decay of Turbulence Intensity (Tu) as a Function of Streamwise Distance (x) .......................... 81
4.6. Resolved Structures for Cylinder in Cross Flow (top: URANS; bottom: SST-SAS) ........................................ 96
4.7. Eddy Viscosity Profiles ......................................................................................................................... 104
4.8.The Computational Domain and Mesh for the Subsonic Jet Flow .......................................................... 107
4.9. Iso-Surfaces of the Q-Criterion Colored with the Velocity Magnitude .................................................... 107
4.10. Distribution of the Mean (Left) and RMS (Right) Velocity along the Jet Centerline ................................ 107
4.11. Backward Facing Step Flow Using ELES .............................................................................................. 120
4.12.Typical Grid for ELES for Backward Facing Step ................................................................................... 122
4.13. Subdivisions of the Near-Wall Region ................................................................................................. 123
4.14. Near-Wall Treatments in ANSYS Fluent ............................................................................................... 124
5.1. Radiative Heat Transfer ........................................................................................................................ 155
5.2. Angles θ and φ Defining the Hemispherical Solid Angle About a Point P ............................................... 161
5.3. Angular Coordinate System ................................................................................................................. 166
5.4. Face with No Control Angle Overhang ................................................................................................. 166
5.5. Face with Control Angle Overhang ...................................................................................................... 167
5.6. Face with Control Angle Overhang (3D) ............................................................................................... 167
5.7. Pixelation of Control Angle .................................................................................................................. 168
5.8. DO Radiation on Opaque Wall ............................................................................................................. 170
5.9. DO Radiation on Interior Semi-Transparent Wall ................................................................................... 172
5.10. Reflection and Refraction of Radiation at the Interface Between Two Semi-Transparent Media ............ 174
5.11. Critical Angle θc ................................................................................................................................ 175
5.12. DO Irradiation on External Semi-Transparent Wall .............................................................................. 177
5.13. Beam Width and Direction for External Irradiation Beam .................................................................... 178
6.1. An Example of a Four-Pass Heat Exchanger .......................................................................................... 188
6.2. Core Discretized into 3x4x2 Macros ..................................................................................................... 189
6.3. Core with Matching Quad Meshes for Primary and Auxiliary Zones in a Crossflow Pattern ..................... 197
6.4. Core with Primary and Auxiliary Zones with Overlap of Cells ................................................................ 198
7.1. A Reacting Particle in the Multiple Surface Reactions Model ................................................................. 217
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7.2. Cross-section of a Channel and Outer Shell Around It. .......................................................................... 224
8.1. Relationship of Mixture Fractions (Fuel, Secondary Stream, and Oxidizer) .............................................. 231
8.2. Relationship of Mixture Fractions (Fuel, Secondary Stream, and Normalized Secondary Mixture Fraction) ......................................................................................................................................................... 231
8.3. Graphical Description of the Probability Density Function .................................................................... 236
8.4. Example of the Double Delta Function PDF Shape ............................................................................... 237
8.5. Logical Dependence of Averaged Scalars on Mean Mixture Fraction, the Mixture Fraction Variance, and
the Chemistry Model (Adiabatic, Single-Mixture-Fraction Systems) ............................................................. 238
8.6. Logical Dependence of Averaged Scalars on Mean Mixture Fraction, the Mixture Fraction Variance, Mean
Enthalpy, and the Chemistry Model (Non-Adiabatic, Single-Mixture-Fraction Systems) ................................ 240
8.7. Reacting Systems Requiring Non-Adiabatic Non-Premixed Model Approach ........................................ 241
8.8.Visual Representation of a Look-Up Table for the Scalar (Mean Value of Mass Fractions, Density, or Temperature) as a Function of Mean Mixture Fraction and Mixture Fraction Variance in Adiabatic Single-MixtureFraction Systems ....................................................................................................................................... 242
8.9.Visual Representation of a Look-Up Table for the Scalar φ_I as a Function of Fuel Mixture Fraction and
Secondary Partial Fraction in Adiabatic Two-Mixture-Fraction Systems ....................................................... 243
8.10.Visual Representation of a Look-Up Table for the Scalar as a Function of Mean Mixture Fraction and
Mixture Fraction Variance and Normalized Heat Loss/Gain in Non-Adiabatic Single-Mixture-Fraction Systems ......................................................................................................................................................... 244
8.11.Visual Representation of a Look-Up Table for the Scalar φ_I as a Function of Fuel Mixture Fraction and
Secondary Partial Fraction, and Normalized Heat Loss/Gain in Non-Adiabatic Two-Mixture-Fraction Systems ......................................................................................................................................................... 245
8.12. Chemical Systems That Can Be Modeled Using a Single Mixture Fraction ............................................ 249
8.13. Chemical System Configurations That Can Be Modeled Using Two Mixture Fractions .......................... 250
8.14. Premixed Systems That Cannot Be Modeled Using the Non-Premixed Model ...................................... 250
8.15. Using the Non-Premixed Model with Flue Gas Recycle ....................................................................... 251
8.16. Laminar Opposed-Flow Diffusion Flamelet ........................................................................................ 254
9.1. Borghi Diagram for Turbulent Combustion .......................................................................................... 277
10.1.The Scalar Dissipation Rate Along The Normalized Reaction Progress Variable .................................... 290
13.1. Flame Front Showing Accumulation of Source Terms for the Knock Model .......................................... 321
13.2. Propagating Fuel Cloud Showing Accumulation of Source Terms for the Ignition Delay Model ............ 322
13.3. Crevice Model Geometry (Piston) ...................................................................................................... 324
13.4. Crevice Model Geometry (Ring) ......................................................................................................... 324
13.5. Crevice Model “Network” Representation ........................................................................................... 324
14.1. De Soete’s Global NOx Mechanism with Additional Reduction Path .................................................... 347
14.2. Simplified Reaction Mechanism for the SNCR Process ........................................................................ 349
15.1. Schematic of the Convective Effect on the Retarded Time Calculation ................................................ 390
16.1. Coal Bridge ....................................................................................................................................... 428
16.2. Particle Reflection at Wall .................................................................................................................. 443
16.3. Particle-Wall Collision Forces ............................................................................................................. 444
16.4. Particle-Wall Interaction at a Rough Wall ............................................................................................ 446
16.5.Wall Roughness Parameters ............................................................................................................... 446
16.6.“Wall Jet” Boundary Condition for the Discrete Phase ......................................................................... 448
16.7. Mechanisms of Splashing, Momentum, Heat and Mass Transfer for the Wall-Film ................................. 449
16.8. Simplified Decision Chart for Wall Interaction Criterion ...................................................................... 451
16.9.The Stanton-Rutland Model: Impinging and Splashing ....................................................................... 453
16.10.The Kuhnke Impingement Model Regimes ....................................................................................... 457
16.11.The Kuhnke Model: Impinging Drop ................................................................................................. 461
16.12. Assumption of a Bilinear Temperature Profile in the Film .................................................................. 468
16.13. Geometric Parameters of Deformed Impinging Droplet in Heat Transfer Calculations ........................ 476
16.14. Single-Phase Nozzle Flow (Liquid Completely Fills the Orifice) .......................................................... 478
16.15. Cavitating Nozzle Flow (Vapor Pockets Form Just After the Inlet Corners) .......................................... 478
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Theory Guide
16.16. Flipped Nozzle Flow (Downstream Gas Surrounds the Liquid Jet Inside the Nozzle) .......................... 479
16.17. Decision Tree for the State of the Cavitating Nozzle .......................................................................... 481
16.18.Theoretical Progression from the Internal Atomizer Flow to the External Spray ................................. 484
16.19. Flat Fan Viewed from Above and from the Side ................................................................................ 489
16.20. Liquid Core Approximation ............................................................................................................. 498
16.21. Madabhushi Breakup Model ............................................................................................................ 501
16.22. Child Droplet Velocity ...................................................................................................................... 503
16.23. Madabhushi Diameter Distribution .................................................................................................. 504
16.24. Particles Represented by Spheres .................................................................................................... 508
16.25. An Example of a Friction Coefficient Plot .......................................................................................... 512
16.26. Force Evaluation for Parcels ............................................................................................................. 513
16.27. Heat, Mass, and Momentum Transfer Between the Discrete and Continuous Phases .......................... 514
17.1. Fixing Velocities in Fluid Cells Touched by the Particle ........................................................................ 519
18.1. Multiphase Flow Regimes .................................................................................................................. 525
18.2. Interface Calculations ........................................................................................................................ 536
18.3.Typical Wave Spectrum ...................................................................................................................... 559
18.4. Schematic View of the Interface Cut Through the Front Cell ................................................................ 565
18.5. Distance to the Interface Segment ..................................................................................................... 566
18.6. The Stability Phase Diagram .............................................................................................................. 657
18.7. Distribution of Molar Concentration in the Two-Resistance Model ...................................................... 661
19.1. Homogeneous Discrete Method ........................................................................................................ 675
19.2. Inhomogeneous Discrete Method ..................................................................................................... 675
19.3. A Particle Size Distribution as Represented by the Discrete Method .................................................... 687
19.4. Reconstruction of a Particle Size Distribution ..................................................................................... 694
20.1.“Pulling” a Solid in Continuous Casting ............................................................................................... 703
20.2. Circuit for Contact Resistance ............................................................................................................ 704
22.1. Subgrid Processes That Require a Wall Film Model .............................................................................. 713
22.2. Separation Criteria ............................................................................................................................ 716
22.3. Shear-Driven Film Velocity ................................................................................................................. 721
22.4. Gravity-Driven Film Velocity ............................................................................................................... 721
22.5. Spatial Gradient ................................................................................................................................ 725
24.1. Electric Circuits Used in the ECM Model ............................................................................................. 734
24.2. Electrode and Particle Domains in the Newman’s Model .................................................................... 736
24.3. Solution Domain for Two Potential Equations in a Battery Pack System ............................................... 739
25.1. Schematic of a PEM Fuel Cell ............................................................................................................. 746
25.2. Boundary Conditions for the Electric Potentials (Solid and Membrane) — PEM Fuel Cell ...................... 747
25.3. Schematic of a PEM Fuel Cell ............................................................................................................. 759
25.4. Boundary Conditions for the Electric Potential (Solid and Membrane) — PEM Fuel Cell ....................... 761
25.5. Schematic of a Solid Oxide Fuel Cell ................................................................................................... 768
25.6. How the SOFC With Unresolved Electrolyte Model Works in ANSYS Fluent .......................................... 769
25.7. Energy Balance at the Electrolyte Interface ........................................................................................ 773
27.1. Fiber Grid Penetrating Grid of the Gas Flow ........................................................................................ 786
27.2. Dimensionless Groups of Drag Coefficient and Nusselt Number ......................................................... 790
28.1. Overview of the Pressure-Based Solution Methods ............................................................................ 799
28.2. Overview of the Density-Based Solution Method ............................................................................... 801
28.3. Control Volume Used to Illustrate Discretization of a Scalar Transport Equation ................................... 804
28.4.Variation of a Variable Phi Between x=0 and x=L ................................................................................ 806
28.5. One-Dimensional Control Volume ..................................................................................................... 808
28.6. Cell Representation for Modified HRIC Scheme .................................................................................. 810
28.7. Cell Centroid Evaluation .................................................................................................................... 816
28.8. Overview of the Iterative Time Advancement Solution Method For the Segregate Solver .................... 828
28.9. Overview of the Non-Iterative Time Advancement Solution Method ................................................... 830
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28.10.V-Cycle Multigrid ............................................................................................................................. 845
28.11.W-Cycle Multigrid ............................................................................................................................ 846
28.12. Logic Controlling the Flex Multigrid Cycle ........................................................................................ 850
28.13. Node Agglomeration to Form Coarse Grid Cells ................................................................................ 853
28.14.The FMG Initialization ...................................................................................................................... 857
29.1. Example of a Hanging Node .............................................................................................................. 860
29.2. Hanging Node Adaption of 2D Cell Types ........................................................................................... 861
29.3. Hanging Node Adaption of 3D Cell Types ........................................................................................... 861
29.4. PUMA Refinement of a Polyhedral Cell ............................................................................................... 862
29.5. Mesh Before Adaption ....................................................................................................................... 864
29.6. Projection of Nodes ........................................................................................................................... 864
29.7. Levels Projection Propagation and Magnitude ................................................................................... 865
29.8. Coarse Mesh of a Sphere ................................................................................................................... 866
29.9. Adapted Mesh Without Geometry Reconstruction ............................................................................. 867
29.10. Mesh after Geometry-Based Adaption ............................................................................................. 868
30.1. Moment About a Specified Moment Center ....................................................................................... 871
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Theory Guide
List of Tables
1. Mini Flow Chart Symbol Descriptions ................................................................................................... xxxix
4.1.Wall-Resolved Grid Size as a Function of Reynolds Number .................................................................. 113
4.2.WMLES Grid Size as a Function of Reynolds Number ............................................................................ 113
9.1. Source Terms for ECFM Models ............................................................................................................ 278
9.2.Values of Constants for ECFM Model Source Terms ............................................................................... 278
13.1. Default Values of the Variables in the Hardenburg Correlation ............................................................ 323
14.1. Rate Constants for Different Reburn Fuels .......................................................................................... 349
14.2. Seven-Step Reduced Mechanism for SNCR with Urea ......................................................................... 351
14.3. Two-Step Urea Breakdown Process .................................................................................................... 351
14.4. Eight-Step Reduced Mechanism ........................................................................................................ 358
14.5. Sticking Coefficient for Different PAH Species ..................................................................................... 372
14.6. Arrhenius rate parameters for HACA mechanism ................................................................................ 377
16.1. Chemical Structure Parameters for C NMR for 13 Coals ....................................................................... 431
16.2. Example of the Oka Erosion Model Constants ................................................................................... 471
16.3. Example of the McLaury Erosion Model Constants ............................................................................. 472
16.4. List of Governing Parameters for Internal Nozzle Flow ........................................................................ 479
16.5.Values of Spread Parameter for Different Nozzle States ....................................................................... 483
16.6. Comparison of a Spring-Mass System to a Distorting Droplet ............................................................. 491
18.1. Slope Limiter Values and Their Discretization Schemes ....................................................................... 538
19.1. Luo Model Parameters ...................................................................................................................... 680
19.2. Lehr Model Parameters ..................................................................................................................... 680
19.3. Daughter Distributions ...................................................................................................................... 682
19.4. Daughter Distributions (cont.) ........................................................................................................... 683
19.5.Values for Daughter Distributions in General Form ............................................................................. 683
25.1.Volumetric Heat Source Terms ........................................................................................................... 754
25.2. Zones where UDSs are Solved in PEMFC ............................................................................................ 758
28.1. Summary of the Density-Based Solver ............................................................................................... 839