Preface ........................................................ ix
Nomenclature ................................................. xtii
Chapter 1 Transport Equations .................................. l
1.1 Basic Approach ............................................. 1
1.2 Mass Balances .............................................. 2
1.3 Momentum Balances .......................................... 3
1.4 Energy Balances ............................................ 7
1.5 Entropy Balance ........................................... 10
1.6 Mixture Equations ......................................... 11
1.7 Multicomponent Multiphase Flow ............................ 20
Exercises ...................................................... 21
Literature Cited ............................................... 28
Chapter 2 One-Dimensional Steady Gas-Solid Flow ............... 31
2.1 One-Dimensional, Steady Mixture Momentum Balance .......... 31
2.2 One-Dimensional, Steady Gas Momentum Balance: Pressure
Drop in Both Phases - Model A ............................. 33
2.3 Fluid Particle Drag ....................................... 35
2.4 Buoyancy .................................................. 37
2.5 Drag for Models В and С ................................... 39
2.6 Entrance Length in Pneumatic Transport .................... 40
2.7 Pressure Drop ............................................. 45
2.8 Pressure Drop Correlation in a Dilute Lift Line ........... 51
Exercises ...................................................... 52
Literature Cited ............................................... 58
Chapter 3 Drift Flux .......................................... 61
3.1 Introduction .............................................. 61
3.2 Hold-Up in Homogeneous and Slip Flow ...................... 61
3.3 Single Particle Analysis .................................. 62
3.4 Ergun Equation Prediction ................................. 63
3.5 Two Regimes in a Bubble Column ............................ 66
3.6 Other Applications ........................................ 68
Exercises ...................................................... 69
Literature Cited ............................................... 71
Chapter 4 Critical Granular Flow
4.1 One-Dimensional Granular Flow Momentum Balance ............ 73
4.2 One-Dimensional Statics: Jannssen Equation ................ 75
4.3 Incompressible, Frictionless Flow and Discharge ........... 77
4.4 Thermodynamics of Powders: Compressibility ................ 79
4.5 Critical Flow Theory for Granular Materials ............... 82
4.6 Example of Critical Granular Flow ......................... 86
4.7 "Sound Speed" from Kinetic Theory ......................... 88
Exercises ...................................................... 89
Literature Cited ............................................... 95
Chapter 5 The Fluidized State ................................. 97
5.1 Introduction: Fluidization Regimes ........................ 97
5.2 Minimum Fluidization Velocity ............................. 98
5.3 Geldart's Classification of Powders ...................... 104
5.4 Kinetic Energy Dissipation Analysis ...................... 106
Exercises ..................................................... 109
Literature Cited .............................................. 113
Chapter 6 On The Origin of Bubbles ........................... 115
6.1 Introduction ............................................. 115
6.2 Void Propagation in Incompressible Fluids ................ 116
6.3 Shocks and Dispersion with No Solids Stress .............. 119
6.4 Bubbling Criterion for Small Particles ................... 121
Acknowledgment ................................................ 123
Exercises ..................................................... 123
Literature Cited .............................................. 125
Chapter 7 Inviscid Multiphase Flows: Bubbling Beds ........... 129
7.1 Basic Equations .......................................... 129
7.2 Compressible Granular Flow ............................... 131
7.3 Well-Posedness of Two-Phase Models ....................... 134
7.4 Homogeneous Flow and Pressure Propagation ................ 139
7.5 Solids Vorticity and Void Propagation .................... 141
7.6 Davidson Bubble Model .................................... 146
7.7 Computer Fluidization Models ............................. 149
7.8 Comparison of Computations to Observations ............... 154
Exercises ..................................................... 183
Literature Cited .............................................. 192
Chapter 8 Viscous Flow and Circulating Fluidized Beds
8.1 Introduction ............................................. 197
8.2 Multiphase Navier-Stokes Equation Model .................. 198
8.3 Dimensional Analysis: Scale Factors ...................... 201
8.4 CFB or Riser Flow: Experimental .......................... 207
8.5 Need for Clusters in One-D Modeling ...................... 214
8.6 Computation of Cluster Flow .............................. 216
8.7 Computation of Core-Annular Regime ....................... 224
8.8 Radial Profiles and Turbulence ........................... 228
Exercises ..................................................... 233
Literature Cited .............................................. 235
Chapter 9 Kinetic Theory Approach ............................ 239
9.1 Introduction ............................................. 239
9.2 Maxwellian Distribution for Particles .................... 240
9.3 Properties of the Maxwellian State ....................... 242
9.4 Dynamics of an Encounter between Two Particles ........... 244
9.5 The Frequency of Collisions .............................. 247
9.6 Mean Free Path ........................................... 252
9.7 Elementary Treatment of Transport Coefficients ........... 253
9.8 Boltzmann Integral-Differential Equation ................. 256
9.9 Maxwell's Transport Equation ............................. 259
9.10 Conservation Laws with No Collisions ..................... 261
9.11 Second Approximation to the Frequency Distribution ....... 263
9.12 Integral Equation Solver Strategy ........................ 272
9.13 Viscous Kinetic Stress Tensor ............................ 274
9.14 Dense Transport Theorem .................................. 276
9.15 Particulate Momentum Equation ............................ 279
9.16 Fluctuating Kinetic Energy Equation ...................... 281
9.17 Viscosity — Collisional Momentum Transfer ................ 284
9.18 Granular Conductivity .................................... 290
Literature Cited .............................................. 294
Chapter 10 Applications of Kinetic Theory ..................... 297
10.1 Granular Shear Flow ...................................... 297
10.2 Flow down a Chute ........................................ 307
10.3 Bubbling Bed: Flow Patterns .............................. 311
10.4 Liquid-Solid Fluidization ................................ 318
10.5 Circulating Fluidized Bed Loop Simulation ................ 324
10.6 Maximum Solids Circulation in a CFB ...................... 330
Literature Cited .............................................. 334
Chapter 11 Kinetic Theory of Granular Mixtures ................ 337
11.1 Empirical Input: Restitution Coefficients ................ 337
11.2 Boltzmann Equations for a Mixture ........................ 339
11.3 Dense Transport Theorem .................................. 339
11.4 Granular Temperatures and Applications ................... 341
11.5 Particle-to-Particle Drag ................................ 343
11.6 Summary .................................................. 345
Exercises for Kinetic Theory (Chapters 9-11) .................. 348
Literature Cited .............................................. 353
Chapter 12 Sedimentation and Consolidation ................... 355
12.1 Conservation of Particles ................................ 355
12.2 Settling in a Sedimentation Column: Introduction ......... 357
12.3 Free Settling ............................................ 360
12.4 Compression Settling ..................................... 366
12.5 Consolidation: Relation to Osmotic Pressure .............. 370
12.6 Electrokinetic Phenomenon: Zeta Potential ................ 376
12.7 Effect of Zeta Potential on Sedimentation ................ 379
Exercises ..................................................... 382
Literature Cited .............................................. 388
Appendices
Formulation of Continuum Problems: Introduction ............... 391
Appendix A: Overall (Macroscopic Balances) ............... 393
Appendix B: Eulerian Approach: One-D Energy Balance ...... 405
Appendix C: Leibniz Formula and Relation to Transport .... 411
Appendix D: Lagrangian Approach: One-D Conservation of
Species and Population Balance ............... 413
Appendix E: Reynolds Transport Theorem ................... 423
Appendices
The Methods of Characteristics: Introduction .................. 429
Appendix F: First Order Partial Differential Equation .... 431
Appendix G: Solution of a Hyperbolic System of First
Order Partial Differential Equations ......... 441
Index ......................................................... 457
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