Computational models for polydisperse particulate and multiphase systems

Cover......Page 1
Contents......Page 9
Preface......Page 15
Notation......Page 19
1.1 Disperse multiphase flows......Page 41
1.2.1 The population-balance equation for fine particles......Page 43
1.2.2 The kinetic equation for gas--particle flow......Page 48
1.3 The mesoscale modeling approach......Page 54
1.3.1 Relation to microscale models......Page 56
1.3.2 Number-density functions......Page 58
1.3.3 The kinetic equation for the disperse phase......Page 59
1.3.5 Relation to macroscale models......Page 60
1.4.1 Hydrodynamic models......Page 63
1.4.2 Moment methods......Page 65
1.5 A road map to Chapters 2--8......Page 67
2.1 Number-density functions (NDF)......Page 70
2.1.1 Length-based NDF......Page 72
2.1.3 Mass-based NDF......Page 73
2.1.4 Velocity-based NDF......Page 74
2.2.1 The population-balance equation (PBE)......Page 75
2.2.3 The closure problem......Page 77
2.3.1 Moment-transport equations for a PBE......Page 78
2.3.2 Moment-transport equations for a GPBE......Page 80
2.4.1 Laminar PBE......Page 83
2.4.2 Turbulent PBE......Page 84
2.5 The moment-closure problem......Page 85
3.1 Univariate distributions......Page 87
3.1.1 Gaussian quadrature......Page 89
3.1.2 The product--difference (PD) algorithm......Page 91
3.1.3 The Wheeler algorithm......Page 93
3.1.4 Consistency of a moment set......Page 95
3.2 Multivariate distributions......Page 102
3.2.1 Brute-force QMOM......Page 103
3.2.2 Tensor-product QMOM......Page 108
3.2.3 Conditional QMOM......Page 114
3.3 The extended quadrature method of moments (EQMOM)......Page 122
3.3.1 Relationship to orthogonal polynomials......Page 123
3.3.2 Univariate EQMOM......Page 124
3.3.3 Evaluation of integrals with the EQMOM......Page 131
3.3.4 Multivariate EQMOM......Page 133
3.4 The direct quadrature method of moments (DQMOM)......Page 139
4.1.1 Definition of the NDF for granular systems......Page 142
4.1.2 NDF estimation methods......Page 145
4.1.3 Definition of the NDF for fluid--particle systems......Page 147
4.2 From the multi-particle--fluid joint PDF to the GPBE......Page 150
4.2.1 The transport equation for the multi-particle joint PDF......Page 151
4.2.3 The transport equation for the NDF......Page 152
4.2.4 The closure problem......Page 153
4.3.1 A few words about phase-space integration......Page 154
4.3.3 Disperse-phase volume transport......Page 156
4.3.4 Fluid-phase volume transport......Page 157
4.3.5 Disperse-phase mass transport......Page 158
4.3.6 Fluid-phase mass transport......Page 161
4.3.7 Disperse-phase momentum transport......Page 163
4.3.8 Fluid-phase momentum transport......Page 164
4.3.9 Higher-order moment transport......Page 167
4.4 Moment closures for the GPBE......Page 170
5.1 An overview of mesoscale modeling......Page 176
5.1.1 Mesoscale models in the GPBE......Page 177
5.1.2 Formulation of mesoscale models......Page 181
5.1.3 Relation to macroscale models......Page 185
5.2 Phase-space advection: mass and heat transfer......Page 187
5.2.1 Mesoscale variables for particle size......Page 189
5.2.2 Size change for crystalline and amorphous particles......Page 192
5.2.3 Non-isothermal systems......Page 195
5.2.4 Mass transfer to gas bubbles......Page 196
5.2.5 Heat/mass transfer to liquid droplets......Page 198
5.2.6 Momentum change due to mass transfer......Page 200
5.3 Phase-space advection: momentum transfer......Page 201
5.3.1 Buoyancy and drag forces......Page 202
5.3.2 Virtual-mass and lift forces......Page 211
5.3.3 Boussinesq--Basset, Brownian, and thermophoretic forces......Page 213
5.3.4 Final expressions for the mesoscale acceleration models......Page 215
5.4 Real-space advection......Page 217
5.4.1 The pseudo-homogeneous or dusty-gas model......Page 219
5.4.2 The equilibrium or algebraic Eulerian model......Page 220
5.4.3 The Eulerian two-fluid model......Page 221
5.4.4 Guidelines for real-space advection......Page 222
5.5 Diffusion processes......Page 223
5.5.1 Phase-space diffusion......Page 224
5.5.2 Physical-space diffusion......Page 227
5.5.3 Mixed phase- and physical-space diffusion......Page 228
5.6.1 Formation of the disperse phase......Page 229
5.6.3 Nucleation of vapor bubbles in a boiling liquid......Page 231
5.7 First-order point processes......Page 232
5.7.1 Particle filtration and deposition......Page 233
5.7.2 Particle breakage......Page 235
5.8 Second-order point processes......Page 242
5.8.1 Derivation of the source term......Page 243
5.8.2 Source terms for aggregation and coalescence......Page 245
5.8.3 Aggregation kernels for fine particles......Page 246
5.8.4 Coalescence kernels for droplets and bubbles......Page 252
6 Hard-sphere collision models......Page 254
6.1 Monodisperse hard-sphere collisions......Page 255
6.1.1 The Boltzmann collision model......Page 257
6.1.2 The collision term for arbitrary moments......Page 258
6.1.3 Collision angles and the transformation matrix......Page 261
6.1.4 Integrals over collision angles......Page 263
6.1.5 The collision term for integer moments......Page 270
6.2 Polydisperse hard-sphere collisions......Page 276
6.2.1 Collision terms for arbitrary moments......Page 277
6.2.2 The third integral over collision angles......Page 282
6.2.3 Collision terms for integer moments......Page 283
6.3.1 Monodisperse particles......Page 286
6.3.2 Polydisperse particles......Page 288
6.4 Moment-transport equations......Page 290
6.4.1 Monodisperse particles......Page 291
6.4.2 Polydisperse particles......Page 295
6.5.1 Flux terms......Page 301
6.5.2 Source terms......Page 303
7.1 Overview of methods......Page 306
7.2.1 Univariate PBE......Page 309
7.2.2 Bivariate and multivariate PBE......Page 319
7.2.3 Collisional KE......Page 323
7.3 The method of moments......Page 329
7.3.1 Univariate PBE......Page 330
7.3.2 Bivariate and multivariate PBE......Page 336
7.3.3 Collisional KE......Page 337
7.4 Quadrature-based moment methods......Page 340
7.4.1 Univariate PBE......Page 341
7.4.2 Bivariate and multivariate PBE......Page 347
7.4.3 Collisional KE......Page 354
7.5 Monte Carlo methods......Page 355
7.6.1 A few words on the spatially homogeneous PBE......Page 359
7.6.2 Comparison between the QMOM and the DQMOM......Page 363
7.6.3 Comparison between the CQMOM and Monte Carlo......Page 364
8.1 Overview of spatial modeling issues......Page 369
8.1.1 Realizability......Page 370
8.1.2 Particle trajectory crossing......Page 372
8.1.3 Coupling between active and passive internal coordinates......Page 375
8.1.4 The QMOM versus the DQMOM......Page 377
8.2 Kinetics-based finite-volume methods......Page 380
8.2.1 Application to PBE......Page 381
8.2.2 Application to KE......Page 385
8.2.3 Application to GPBE......Page 387
8.3.1 Moment-transport equations......Page 389
8.3.2 Standard finite-volume schemes for moments......Page 390
8.3.3 Realizable finite-volume schemes for moments......Page 393
8.3.4 Example results for an inhomogeneous PBE......Page 398
8.4 Inhomogeneous KE......Page 402
8.4.2 Operator splitting for moment equations......Page 403
8.4.3 A realizable finite-volume scheme for bivariatevelocity moments......Page 404
8.4.4 Example results for an inhomogeneous KE......Page 406
8.5.1 Classes of GPBE......Page 413
8.5.2 Spatial transport with known scalar-dependent velocity......Page 416
8.5.3 Example results with known scalar-dependent velocity......Page 417
8.5.4 Spatial transport with scalar-conditioned velocity......Page 421
8.5.5 Example results with scalar-conditioned velocity......Page 428
8.5.6 Spatial transport of the velocity-scalar NDF......Page 436
8.6 Concluding remarks......Page 441
A.1.1 The PD algorithm......Page 443
A.1.2 The adaptive Wheeler algorithm......Page 444
A.2.1 The correction algorithm of McGraw......Page 445
A.2.2 The correction algorithm of Wright......Page 447
A.3.1 Brute-force QMOM......Page 448
A.3.2 Tensor-product QMOM......Page 450
A.3.3 The CQMOM......Page 452
A.4.1 Beta EQMOM......Page 453
A.4.2 Gamma EQMOM......Page 456
A.4.3 Gaussian EQMOM......Page 458
B.1 Spatial dependence of GPBE......Page 461
B.2 Realizable FVM......Page 463
B.3 Advection......Page 467
B.4 Free transport......Page 469
B.5 Mixed advection......Page 474
B.6 Diffusion......Page 477
C.1 A model kinetic equation......Page 481
C.2.1 Segregating solution......Page 482
C.2.2 Mixing solution......Page 483
C.3.1 Moments of segregating solution......Page 484
C.3.2 Moments of mixing solution......Page 486
C.4.1 The moment-transport equation......Page 487
C.4.2 Transport equations for weights and abscissas......Page 488
D.2 Standard DQMOM......Page 490
D.3 DQMOM-FC......Page 493
D.4 Time integration......Page 495
References......Page 499
Index......Page 528