Cambridge University Press 978-1-108-47374-3 — Dynamics of Multiphase Flows Chao Zhu , Liang-Shih Fan , Zhao Yu Index More Information

Index

ABF. See agglomerate bubbling fluidization Basset historic integral. See Basset force abrasive erosion, 508 Basset-Boussinesq-Oseen equation, 82, 105, 176 absorption, 99, 133, 185, 200, 535 BBO equation. See Basset-Boussinesq-Oseen AC. See alternating current equation acceleration number, 106 beam attenuation method, 331–334 acoustic field, 339, 438, 446 bend erosion, 477, 507 acoustic wave, 331, 337–339 Bernoulli equation, 43 added mass, 90, 128, 131, 232, 245, 567 Bernoulli principle, 313, See Bernoulli equation agglomerate bubbling fluidization, 444 binary collision, 62–64, 182 agglomerate particulate fluidization, 444 Biot number, 98 agglomerates, 10, 22, 444, 447 body forces. See field forces agglomeration, 321, 541 boiling, 481, 527 aggregate, 8, 444, 558 film boiling, 163, 528, 530 aggregation, 240, 560 flow boiling, 527, 532 alternating current, 397 flow regimes, 531 Amontons’ law of sliding friction, 138, 139 heat flux regimes, 527 angular momentum balance, 142 nucleate boiling, 528–530, 532, 540 anisokinetic sampling, 346 pool boiling, 527, 532 annular flow, 448, 482, 514, 532 sub-cooled boiling, 527 APF. See agglomerate particulate fluidization boiling heat transfer, 527, 532, 540 arbitrary Lagrangian-Eulerian technique, 275 Boltzmann equation, 62, 262, 294 Archimedes number, 131, 429 Boltzmann-Maxwell distribution, 296 arithmetic mean diameter, 330 bounce back scheme, 297, 298 Arrhenius law, 563 boundary conditions atomization, 534, 535 charged surface, 136 atomizers, 535, 540 fluid-solid interface, 37 attrition, 10, 15, 126, 154, 218, 320, 507 immiscible fluids, 40 averaged transport equations, 192, 212, 217, 221 of interfacial mass transfer, 40, 103, 563 averages, 212 permeable interface, 58 ensemble average, 63 types of, 272 intrinsic average, 213, 214, 219 boundary-fitted mesh technique, 274 phase average, 213, 216 boundary-layer theory, 484 time average, 222 Bourger-Beer relation, 134 volume average, 213, 214 Boussinesq formulation, 48, 51, 223 volume-time average, 222 Boussinesq history integral force. See Basset force averaging theorems, 212–216 Boussinesq’s gradient approach. See Boussinesq formulation baffles, 425, 551 Bragg cell, 326 Bagnold friction law, 62 breakage, 240, 242, 457, 560 Bagnold number, 61 Brinkman equation, 32, 55, 58–60 ball probe method, 346, 358 brittle erosion, 508, 509 Basset force, 19, 81, 83, 90 Brownian diffusion, 391, 393

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-47374-3 — Dynamics of Multiphase Flows Chao Zhu , Liang-Shih Fan , Zhao Yu Index More Information

Index 585

Brownian force, 199, 200, 414 chemical absorption, 526 Brownian motion, 178, 198, 230, 390, 399, 414 combustion, 526, 561–566 bubble column, 3, 306, 452, 457 polymerization, 526 gas-liquid, 248, 424, 458 CHF. See critical heat flux gas-liquid-solid, 424 choking, 447, 451 reactors, 549, 552 churn flow, 482 regimes, 458 churn-turbulent regime, 458 slurry, 10, 363, 424, 458, 552 circulating fluidized bed, 21, 446–452, 551 vortical-spiral flow structure, 459 compoents, 447 bubble condensing, 567 flow modeling, 451 bubble formation, 452–454, 527 flow regimes, 447 bubble-induced turbulence, 245, 248–249, 571, flow structure, 449 572 pressure balance, 463 bubbles clarification, in slurry separation, 379 acceleration, 19 Clausius-Clapeyron equation, 104 breakup, 434, 457 Clausius-Mossotti factor, 404–406 coalescence, 456–459 cluster interactions, 457, 533 bubbles, 459 motion, 432–434, 456, 567 formation and instability, 22, 447 bubbling velocity, 432 mass transfer, 134 bubbling, 107, 455, 456 clustering, 23, 238 stability, 432 coalescence, 155–157 wake, 433, 456 bubbles, 242, 434, 456 bubbling, 452 drainage time, 155 bubbling flows, 452 efficiency, 155 bubbling fluidization, 426, 432, 438 of fluid particles, 155–156 Buckingham Pi-theorem, 484 coefficient of restitution, 145, 183 buoyancy force, 19, 159, 454, 495 collection efficiency, 383 Burgers viscous vortex model, 385 collection mechanisms, 391 Burke-Plummer’s equation, 56 cyclone, 25, 387, 407 electrostatic precipitator, 401, 412 Cahn-Hilliard equation, 284 fractional, 407 cake filtration, 390, 396 multi-stage collector, 328 capacitance number, 452 of a fiber, 391–394 capacitance transducers, 334 of a fiber, size-averaged, 395 capillary force, 148–150 of a filter, 396, 410 Capillary number, 109 of a filter, overall, 395, 407–410 capillary tubes, 56 of a granular particle, 394 carried mass, 4, 19, 81, 83, 90, See added mass particle size distribution, 360 cascade impaction, 324 collision cascade impactors, 324, 378, 382 bubble-particle, 454, 457 catalytic cracking, 553 critical normal collision velocity, 145 cavitation, 527 droplet-droplet, 152 central differencing scheme, 266, 267 droplet-particle, 541 centrifugal force, 5, 377, 381, 385, 438, 506 droplet-surface, 150, 163 centrifugal sedimentation, 324 frequency, 155, 240 centrifugal separation, 4 head-on, 10 CFB. See circulating fluidized bed Hertzian, 144 CFD. See computational fluid dynamics non-equilibrium, 22 channeling, 428, 443 normal, 140–142, 145 charge generation, 560 oblique and rotational, 142–144 charge measurement, 358 particle-droplet, 7 charge transfer, 15, 147 particle-particle, 63 charge-induced electric field, 195 particle-wall, 5, 144, 477 charging probability, 400 collisional force, 107, 140, 141 chemical reactions, 15, 81, 526 collisional stress of solids, 32

586 Index

combustion, 6, 561–566 covariance, 222–226, 230, 231, 342 coal particle, 564 creeping flow, 43 droplet, 562 critical heat flux, 528, 530, 532 compressibility cross-correlation function, 349 bulk volume, 33 cross-correlation measurements, 349 flow, 41, 81 cross-correlation method, 348–350 thermodynamic, 41 cumulative particle size distribution, 323 computational fluid dynamics, 41, 182, 261 Cunningham slip, 81, 88, 392 computer aided design, 263 cut-off size, 406 condensation, 242, 481, 560, 567 cyclone, 388, 406 condensing bubble, 567 impactor, 324 confocal microscopy, 318 multi-stage collector, 328 conformal mesh methods, 273–276 cyclone, 4, 5, 14, 380, 381, 447 conservation equation collection efficiency, 25, 406–407 general form, 35 flow modeling, 385–388 constitutive relations particle trajectory, 25 fluid-solid flows, 238 tangential-inlet, 4 kinetic theory of granular flow, 31–67 cyclone separation mechanism, 385 multi-fluid, 226, 229–234 cyclone separator, 345, 386, 406, 554 turbulent flows, 48 viscous fluid, 33–34 d2-law, 82, 104, 116, 545 contact angle, 39, 149 Dalton’s law, 100 contact theory, 136, 145 damping coefficient, 186, 189, 190 contact time Darcy’s equation. See Darcy’s law inter-particle, 146, 188 Darcy’s law, 32, 55, 58, 390 particle-fluid, 13, 98 DC. See direct current continuity equation DeBroucker’s mean diameter, 330 fluid without phase change, 36 Debye shielding distance, 399 overall of all phases, 292 deconvolution, 328 stratified flow, 499 DEM. See discrete element method volume-averaged, 217 dense phase transport, 22, 126, 467 volume-time averaged, 223 dense-phase fluidization, 429–437, 448, 468 continuous-discrete phase coupling, 287 DEP. See dielectrophoresis continuum modeling. See Eulerian-Eulerian deposition velocity, 485 modeling depth filtration, 390 flow in porous media, 55–60 deterministic trajectory model, 176–177, 253 kinetic theory of granular flow, 60–68 Deutsch equation, 402 multiphase flows, 212–237 dielectric particle, 94, 177, 402–405 viscous fluid flow, 55, 234 dielectrophoresis, 95, 402 continuum-discrete model, 173, 262, See dielectrophoretic force, 94, 95, 397 Eulerian-Lagrangian modeling differencing schemes, 266 convection differential settling method, 379 effect on evaporation, 104 diffraction, 132, 318, 320 heat transfer, 99, 131, 194 diffuse interface method, 283 mass transfer, 135 diffusion charging, 399 phase motion, 493 diffusion equation, 545 volume fraction in VOF, 280 diffusive flux, 36, 223, 267, 545 cooling tower, 6 diffusive transport coefficient, 223 core-annular pipe flow, 496 dilute transport, 18, 426, 447, 495–497, 544 Coriolis force, 96, 505 dimensional analysis, 49, 50, 156 corona charging, 398, 399 direct current, 397 corona-charging-ball probe method, 348 direct numerical simulation, 46, 53, 200, 239, 273, corona-discharging method, 355–356 543 Coulomb force, 113, 397, 400 discrete element method, 17, 182–191, 568 Coulomb’s law, 135 hard-sphere model, 182–185 Coulter counting, 318 soft-sphere model, 185–191

Index 587

discrete phase, 126, 172, 217 scanning, 319 discretization schemes, 266–267 transmission, 319 disengagement method, 364 electro-osmotic flow, 71 dispersed bubble flow, 481 electrophoresis, 107, 173 dissipation function, 36 electrostatic precipitation, 13, 378, 397–402 dissipation rate electrostatic precipitator, 13, 397, 400, 412 heat, 37 elongated bubble flow, 481 specific kientic energy, 199 elutriation, 435 sub-grid energy, 54 elutriators, 378 turbulent kinetic energy, 45, 49 emulsion, 426, 432–434, 446 DNS. See direct numerical simulation energy equation Doppler effect, 325 enthalpy, 37, 220 Doppler frequency shift, 326, 352–354 internal energy, 36, 220 Doppler phase shift, 318, 326 temperature, 37, 221 downcomer, 447 total energy, 36, 219, 226 drag coefficient, 84, 87, 88, 127, 129, 244, 247 energy transfer, 15 drag force, 19, 83, 84, 107, 129 ensemble average, 63 drag reduction, 159, 478, 491 ensemble averaging, 213 drift flux model, 533 Enskog’s equation, 63 drift velocity, 533 entrainment drying air, by hydraulic jump, 305 coal particle, 564 circulating fluidized bed, 446, 497 fluidized bed, 10 dense-phase fluidized bed, 423, 434 spray, 526, 537–539 jet, 10, 536, 542, 547 spray coating, 7 rotation-induced, 93 dryout, 532 wake, 155 DTM. See deterministic trajectory model EOS. See equation of state dual-beam cross-correlation method, 349 Eötvös number, 119, 510 ductile erosion, 508 equation of motion, 82, 105–107, 128, 159, 162, dynamic diameter, 321 173, 176, 180, 191, 199 equation of state, 268, 297 ECT. See electrical capacitance tomography general form, 37 eddy diffusivity, 223, 230, 231 ideal gas, 37 eddy existence time, 178, 236 mixure, 533 eddy generation non-ideal fluid, 300 mechanisms, 44 non-ideal gas, 299 phenomenological causes, 44 equivalent diameter, 109, 242, 330 effective viscosity, 59, 249, 494 Ergun equation, 32, 56–57, 390, 431, 445, 465 Einstein model, of slurry viscosity, 488 erosion, 4, 477, See bend erosion ejector spray, 342 brittle, 509 elastic collision, 141, 187 ductile, 508 elastic contact theory, 124, 136–138, See Hertzian ERT. See electrical resistance tomography contact Euler backward scheme, 267 elastic modulus, 235, 277 Euler forward scheme, 267 elastic-plastic model, 146 Eulerian coordinates, 172, 181, 192, 221 electric ball-probe method, 342 Eulerian-Eulerian algorithm, 289–294 electric force, 13, 14, 71, 93, 94, 177, 397, 413 Eulerian-Eulerian modeling, 18, 212–238 electric wind, 401 fluidized bed, with kinetic theory, 467–470 electrical capacitance tomography, 331, 339, 359, gas-liquid bubble column, 248–250 458 gas-solid fluidized bed, 250–251 electrical capacitance volume tomography, 342 mechanically agitated slurry tank, 246–248 electrical impedance tomography, 331, 339–342 slurry flow in F-T bubble reactor, 570–572 electrical resistance tomography, 331, 339 slurry over a bend, 512–513 electrification, 15, 135, 136 slurry pipe flow, 515–517 electromagnetic wave, 331, 339 Eulerian-Eulerian-Lagrangian modeling electron microscopy spray in gas-solid flow, 251–254

588 Index

Eulerian-Lagrangian algorithm, 262, 285–289 fluid catalytic cracking, 553 Eulerian-Lagrangian model coupling fluid pressure force, 127, 128 charge, 195 fluid tracer, 26 energy, 194 fluidization, 423–463 mass, 192 bubbling, 14, 426, 429, 432 momentum, 193 centrifugal, 438 Eulerian-Lagrangian modeling, 17, 172–196 channeling flow, 426, 429 electrostatic precipitation, 412–414 circulating, 446–452 FCC reacting flow, 568–570 dense-phase, 423, 426, 429–437 flow transport of charged particles, 202–203 external field modulated, 438–443 gas-solid flow in cyclone, 410–412 fast, 14, 426, 446 gas-solid flow over a bend, 510–512 gas, 9 inhalation of aerosols, 198–200 gas-liquid bubble column, 452–459 rain flow over airfoil, 196–198 gas-liquid-solid, 423, 452, 460–463 reacting flow in coal gasifier, 572–574 gas-solid, 423, 430 solar-absorbing particulate-laden flow, 200–201 Geldart classification, 428–429 explicit scheme, 267 homogeneous, 423 extinction coefficient, 332–334 lean-phase, 423 liquid-solid, 423 Fanning friction factor. See friction factor magnetic-assisted, 442 Faraday cup, 358, 359 minimum, 430 fast fluidization nanoparticles, 444–446 clustering, 23 onset of bubbling, 432 flow modeling, 451–452 particle classification, 428 flow regimes, 447–448 particulate, 423, 426, 429 flow structure, 449–451 regimes for nanoparticles, 444 FCC. See fluid catalytic cracking slugging flow, 426, 429 Fick’s law of diffusion, 100, 545 spouting flow, 426, 429 field charging, 358, 399 turbulent, 426, 429, 433 field flow fractionation, 405 vibration-assisted, 443 field forces, 93–95, 219, 397, 426, 429 fluidization index, 443 field imaging techniques, 348 fluidized bed, 3, See fluidization field-diffusion charging, 398 application examples, 424, 551 filtration, 390–397 bed contraction, 463 by diffusion, 391 combustor, 14 by impaction, 392 components, 425 by interception, 392 distributors, 425 collection by fiber, 391–393 dynamic pressure drop, 430, 461 collection by granular particle, 393–394 flow regimes, 462 fibrous filter, 407–410 gas-liquid-solid, 460 mechanisms, 390 gas-solid, 9 filtration techniques, 390 internals, 425 finite difference method, 265 magnetic-assisted, 441 finite element method, 265, 276 of field force other than gravity, 426 finite volume method, 265, 287 phase holdup, 461 Fischer-Tropsch synthesis, 424, 551–553 rotating, 438 fixed bed, 57, 423, 426, 430, 442, 479, See packed three-phase, 9, 424, 460–463 bed fluidized bed. See fluidization flashing, 527 Fourier number, 98, 132 flow assurance, 477 four-lump kinetic model, 556 flow compressibility, 41, 83, 88 four-way coupling, 174, 285 flow regimes, in stirred tanks, 551 friction factor, 56, 489 flows in pipe bends, 477, 503–510 friction velocity, 484, 486 bend erosion, 507–509 frictional factor, 490 particle flow, 505–507 front tracking method, 278 single-phase flow, 503–505 Froude number, 488

Index 589

fully suspended pipe flows, 477, 487, 493–497 immiscibility, 283, 299 electrostatic effect, 495 impact regimes, of droplet, 541 phase distributions, 493 implicit schemes, 267 slurry flow modeling, 515 independent scattering, 133 induction probe, 359 gas-liquid pipe flows, 481–483 inertia-based separation, 377–389 gas-liquid separators, 380 by flow rotation, 380–382 Gauss rule, 216, 217 by gravity settling, 378–380 Gauss theorem, 35, 216, 265, 499 by impaction, 382–383 Gauss’s law, 71, 400 in-situ measurement, 316 Gaussian distribution, 178, 179, 408, 409 integral length scale, 242 Geldart’s classification, 428–429, 440 interaction potential, 148, 300, 301 Group A, 428, 429, 432–434, 440, 443 interaction potential model, 300–301 Group B, 428, 429, 433 interactions Group C, 428, 429, 440, 443 adhesive, 444 Group D, 429, 434, 436, 440 bubble-bubble, 459 general transport equation, 265, 268 charge, 16, 135 gradient-related forces, 81, 91–93 contact, 16, 126 granular filtration, 394, 410 fluid-particle, 16, 82, 172 granular temperature, 65, 66, 68, 235, 251, 469 hydrodynamic, 16, 126, 451 Grashof number, 194 inter-phase, 18, 173 gravitational sedimentation, 324, 378–380, 390 non-hydrodynamic, 126 gravitational settling. See sedimentation particle-eddy, 245 gravity settlers, 378–380 particle-particle, 16, 61, 126, 182, 288 gravity settling, 378, See gravitational particle-turbulence, 23 sedimentation wake-induced, 129 interfacial area concentration, 233, 239 Hadamard-Rybcyznski equation, 86 interfacial area transport model, 241, 242 Hamaker constant, 149 interfacial boundary condition, 217, 218, 220 Happel cell model, 129, 391, 396 interfacial instability, 154, 514 hard sphere model, 174, 182–185, 288 interfacial transport, 192, 227, 231–233, 238, 273, hard-field techniques, 341 291 heat exchanger, 59, 425, 477 intrinsic averages, 213–214, 218, 219 heat transfer, 96–99 invasive methods, 317 Heaviside function, 283 inverse problem, 337, 339–341 Hertzian contact. See Hertzian theory isokinetic sampling, 317, 342–345, 359 Hertzian theory, 137–138, 140, 142, 185, 186 Hertz-Knudsen equation, 545 jet Hinze-Tchen equation. See Hinze-Tchen model cross-flow, 536 Hinze-Tchen model, 233, 235–237, 243, 492, 516 entrainment, 10, 536 holography, 319 gas, 536 hopper, 4 milling, 10 horizontal pipe flows, 477–482 similarity, 536 flow patterns, 478 spray, 535 flow regimes, 478 jet impingement, 382 gas-liquid flow regimes, 481 saltation, 483 Kelvin-Helmholtz instability, 154, 457, 482, 500 slurry flow regimes, 480 k-equation, 50, 199, 243, 245, 247 stratified, 499–503 kinetic energy of turbulence, 53 hot-wire anemometry, 348 kinetic theory Howard settling chamber, 379 of gases, 18, 48, 62 hydraulic conveying, 477 of granular flows, 18, 32, 60–67, 467 hydraulic jump, 302, 305–306 Kirchoff’s law, 134 Klinkenberg correction, 58 immersed boundary method, 276–278, 298 Knudsen number, 41, 88 immersed boundary techniques, 278 Kolmogorov scale, 32, 44, 45, 53, 247

590 Index

Kolmogorov time scale, 173, 174 mass transfer, 15, 100–105, 526 Kozeny theory, 55, 69, 70 mass-based size density distribution function, 324 Kuwabara factor, 392 matrix inversion, 267, 291 k-ε model, 25, See turbulence model Maxwellian equilibrium distribution, 66 k-ω model, 49–50 May cascade impactor, 382 measurement techniques, 316–359 Lagrangian coordinates, 17, 105, 172, 192, 193, charge, 358–359 276, 412, 510 list of typical methods, 318 Lagrangian correlation coefficient, 236, 237 mass flow, 342–348 Lagrangian model. See Lagrangian trajectory particle size and morphology, 318–330 modeling velocity, 348–358 Lagrangian trajectory modeling, 175, 346, 390, volume fraction, 331–342 410, See Eulerian-Largrangian modeling microscopy methods, 319 deterministic trajectory model, 17, 176–177 microwave transducers, 335 discrete element method, 17, 182 Mindlin’s theory, 136, 142, 185, 187 probability density function method, 17, minimum bubbling velocity, 431, 432 179–181 minimum fluidization velocity, 423, 430, 431, stochastic trajectory model, 17, 177–179 439, 445, 461 Lambert-Beer law, 331 minimum transport velocity, 483, 484 laminar flow, 16, 31, 43 mixing Langevin dispersion model, 511 blade-stirring, 9 Laplace equation, 529 bubble column, 458 Laplace transformation, 112 concrete, 8 Laplace-Young equation, 39 dry, 8 large eddy simulation, 32, 53–55, 543 jet, 536, 542 laser Doppler anemometry. See laser Doppler slurry, 8, 246 velocimetry sparged stirred tank, 550 laser Doppler velocimetry, 348, 352–354 turbulent fluidization, 426 laser marked shadowgraphy, 362 mixing length model, 48–49 latent heat, 87, 526, 527, 541 molecular diffusion, 31, 51, 81, 231 lattice Boltzmann method, 262, 294–301 momentum equation LBM. See lattice Boltzmann method discretized, 270 LDV. See laser Doppler velocimetry integral form, bend flow, 503 Leibnitz rule, 216, 217 integral form, jet flow, 549 Leidenfrost phenomenon, 163 RANS equation, 47 Leidenfrost temperature, 541 single-phase fluid, 36 LES. See large eddy simulation time-averaged, 47 level set method, 164, 281–283 volume-averaged, 219 Lewis number, 103, 116, 563, 564 volume-time-averaged, 224 lift force, 92, 93, 232, 485 momentum transfer, 15 Lockhart-Martinelli approach, 490 interfacial, 231 log-law velocity distribution, 484 of a particle, 172 lognormal distribution, 361, 410 particle-field, 93–96 particle-fluid, 82–94 Mach number, 41, 81, 88 momentum transfer coefficient, 193, 232, 469 magnetic force, 93, 107, 176, 443 Monte Carlo method, 175, 179, 389 magnetofluidized bed, 441 morphology, 318, 331 magnetophoretic force, 95 Morton number, 109, 455 Magnus effect, 93 moving bed, 14, 436, 479 Magnus force, 81, 83, 93–94, 107 multi-component single-phase flow, 15 mass conservation equation, 177, 193, 248, multi-continuum, 18 571–573 multi-fluid model, 250, 512 mass flow measurement, 342–348 multiphase flow modeling, 31, 261 ball probe method, 346–348 multiphase flows isokinetic sampling, 344–346 definition of, 15 mass flow rate, 342 dense-phase, 15, 348

Index 591

dilute-phase, 15 particle-eddy interaction time, 244 examples of, 3 PDF. See probability density function separation, 377 PDPA. See phase Doppler particle analyzer multiple scattering, 133, 134, 339, 347 Peclet number multi-scaled modeling, 238 fiber, 392 granular, 393 nanoparticles, 321, 444–446, 478 permeability, 55, 56, 86 natural convection, 194, 528 permittivity measurement, 331, 334–336 Navier-Stokes equations, 32, 46, 389 phase change Newton’s law, 140, 172 chemical, 526 non-conformal mesh methods, 276–285 hydrodynamic phases, 526 non-contact mechanisms, 16, 126 physical, 526 non-invasive methods, 317 phase Doppler anemometry. See phase Doppler non-mechanical valve, 447 particle analyzer non-mechanical valves, 495 phase Doppler method, 325–328 non-Newtonian fluids phase Doppler particle analyzer, 325–328, 354 shear thickening fluids, 33 phase field function, 284 shear thinning fluids, 33 phase-coupled SIMPLE (PC-SIMPLE) algorithm, viscoplastic fluids, 33 291–293 nozzle photophoresis, 92, 107, 173 jet, 325, 389, 428 physical phase changes, 81, 218, 526 spray, 534, 540, 544, 555 cavitation, 526, 527 nuclear magnetic resonance imaging, 318, 331 condensation, 242, 481, 526, 560, 567 nucleate boiling, 528–532, 540 crystallization, 424, 526 number density function, 240, 328, 330 dissolution, 240, 482, 526 numerical algorithms, 262, 264 evaporation, 102, 526, 545 numerical simulation, 261, 262 pick-up velocity, 385, 477, 483–486 numerical solutions, 264–272 pipe flows, 477–510 Nusselt number, 97, 98, 132 flow patterns, 477 flow regimes, 477–483 oblique contact, 138, 139 flows in bends, 503–510 Ohm’s law, 341 fully suspended, 493–497 Ohnesorge number, 534 pressure drop, 486–493 one-way coupling, 174, 176, 203, 285, 304, 385, saltation and pick-up velocities, 483–486 387 slurry, 12 optical microscopy, 319 stratified flows, 497–503 optical visualization methods, 319 Pitot tube, 348 PIV. See particle image velocimetry packed bed, 56–57, 131, 430, 439, 463 Planck intensity function, 134 partial elimination algorithm, 291 plug flow, 478, 479, 514 particle charging methods, 398–400 pneumatic conveying, 3, 11, 342, 477 particle cloud tracking, 175, 179–181 flow regimes, 479 particle image velocimetry, 317, 348, 356–358, pressure drop, 478 458 Poisson equation, 311, 340 particle relaxation time, 173, 174, 494 Poisson’s ratio, 162, 188 particle residence time, 178, 384, 387 polarization, 94, 135, 173, 397, 402, 403, 406 particle roping, 505 polymerization, 553, 557–561 particle settling, 130, 157–158, 324, 384, 389, 483 pool boiling, 527 particle size density function, 328 population balance, 150, 240–241, 307, 560, 572 particle size distribution, 240, 241 porous media flow, 55–60 particle size measurement, 318–330 positron emission tomography, 331 particle stopping distance, 388 post-processing, 262, 264 particle terminal velocity, 385, 445, 485 potential flow, 43, 88 particle tracking velocimetry, 348, 357 Prandtl number, 99, 194, 494 particle trajectory, 110–114, 177 pre-processing, 262–264 particle wetting, 542, 546 pressure correction, 269, 289–291

592 Index

pressure drop of filter, 396–397 saturated flow boiling, 532 pressure-gradient force, 92 Sauter’s mean diameter, 330 probability density function, 389, 573 scanning electron microscopy, 319 PTV. See particle tracking velocimetry scanning modes, 337 scattering, 132–134, 318, 327, 331, 337 quality Schmidt number, 104 flow, 532 secondary-flow effects, 506, 537 thermodynamic, 532 sedimentation, 3, 480 sedimentation methods, 321–324, 328 radiation transport equation, 133, 331 sedimentation thickener, 379 radiometric forces, 92 segregation, 68, 284, 299, 301, 561 See RANS equation, 48 SEM. scanning electron microscopy Ranz-Marshall correlation, 99, 104, 116, 194 separation, 377–406 rapid granular flow, 62, 67 by external electric field, 397–406 rapidly dispersing flows, 350 by filtration, 390–397 Rayleigh regime, 534 by phase inertia, 377–389 See Rayleigh-Taylor instability, 154, 457 SGS model. sub-grid scale model reaction heat, 526, 559, 563, 569 shear modulus, 189 reconstruction algorithms, 341 shear thickening, 489 re-entrainment, 382, 385, 394 Sherwood number, 104 reflection, 132, 327, 362 shrinking sphere model, 565 refraction, 132, 327, 333 sieve series, 320 regimes of fluid-solids systems, 426 sieving, 317, 318, 320–321, 328 regional-averaged equations, 498 sieving mesh size. See sieve series regional-averaged modeling, 498–499 signal filtration, 350, 363 relaxation time SIMPLE algorithm, 269–271 particle, 173, 494 single scattering, 133, 134, 346 particle charging, 413 single-phase flows Stokes, 91 flows in porous media, 31, 55–60 restitution coefficient, 146, 185, 188 inertia-dominated granular flows, 31, 60–67 Reynolds averaged Navier-Stokes equation. See laminar flows, 31, 35–43 RANS equation turbulent flows, 31, 43–55 Reynolds decomposition, 46 sink-and-float method, 379 Reynolds number size distribution, 16, 328–329 droplet, 116, 197, 544 cumulative, 323–324 fluid flow, 84 deconvolution method, 360–362 minimum transport, 484 size distribution particle, 84–90 population balance model, 239–242 shear, 92 size of nanoparticle agglomerates, 445 Reynolds stress, 25, 47, 49, 51, 224 sliding contact, 139 Reynolds stress model, 32, 51–52 slug flow, 481 rheological models, 489 slugging, 428, 435 Richardson-Zaki equation, 21, 157, 158, 445, 461 smooth stratified flow, 481 Richardson-Zaki index, 131, 158, 445 soft sphere model, 174, 182, 185–191, 288, 410 riser, 12, 14, 447, 495 spatial deformation rate of fluid. See strain rate acceleration length of solids, 21 spatial resolution, 331 energy partitions, 465 species equation, 36, 556, 566 flow regimes, 447 speed of sound, 42, 88, 339 flow structure, 449 sphericity, 83, 85, 431 riser reactor, 553 spouted bed, 436–437 RSM. See Reynolds stress model spouting, 436 Runge-Kutta algorithms, 17, 287 spouting bed. See fluidization:spouting spray, 3, 6 Saffman force, 81, 83, 91–92, 107 abrasive sands, 7 saltation, 477, 479, 483 atomized, 6, 534 saltation velocity. See minimum transport velocity coating, 7, 541

Index 593

dispersion, 535 transmission electron microscopy, 319 drying, 537 transmission tomography, 331, 336–339 evaporation effect, 536 transport theorem impingement, 6, 8 granular flow, 62–64 ligament, 534 viscous fluid flow, 32–43 modeling, 543 transport velocity, 447 phase interactions, 542 travelling wave dielectrophoresis, 405–406 quenching, 541 triboelectric charging, 358, 398, 494 sprinkler, 6 turbulence damping. See turbulence modulation spray atomization, 535 turbulence kinetic energy, 45 spray drying, 18, 526, 535, 537, 539, 545 bubble-induced, 245 spring-dashpot-slider model, 186, 190 definition of, 49 standpipe, 447, 463–465 equation of, 50 Stefan flux, 82, 101 equation of dissipation rate of, 50 stereomechanical impact model. See hard sphere particles, 237 model turbulence model stochastic Lagrangian model. See stochastic k-ǫ model, 26, 31, 43, 49–50 trajectory modeling large eddy simulation, 31, 43, 53–55 stochastic trajectory modeling, 175, 177–179 mixing length model, 31, 43, 48–49 Stokes drag, 91, 92, 105, 236, 385 Reynolds stress model, 31, 43, 51–52 Stokes flow. See creeping flow turbulence modulation, 233, 242, 243, 245 Stokes number, 392, 394 turbulent diffusion Stokes relaxation time, 91, 111, 173, 178, 180, kinetic energy, 50, 516 199, 201, 232, 236, 244, 387 particles, 177, 235–238, 496, 497 Stokes-Einstein equation, 392 Reynolds stress, 51 strain rate, 33 turbulent transport coefficients, 237 stratified pipe flows, 497–503 turbulent viscosity, 48, 199, 249 stress tensor twDEP. See travelling wave dielectrophoresis Reynolds, 51–52 two-way coupling, 174, 181, 195, 196, 285, 510, shear, of particles, 67 535, 560 time-averaged, 47 total, of fluid, 33 ultrasonic tomography, 336, 337, 339 total, of particles, 65, 66, 234, 469 uniformly dispersed flows, 350 viscous, 33, 230 UNIPOL process, 557 Strouhal number, 109 unstructured mesh, 263, 269, 275 structured mesh, 263, 265 upwind scheme, 266 sub-grid scale model, 54–55 sub-grid scale stress, 54 van der Waals force, 107, 148 surface mean diameter, 330 vaporization, 6 surface tension, 38–39, 279, 285 boiling, 527 by droplet-particle collision, 542, 546 tangential restitution coefficient, 184 flashing, 527 tangential-inlet cyclone, 387 spray, 535, 553, 555 Taylor bubbles, 482 volumetric rate, 252 Taylor’s expansion, 408 vapor-liquid pipe flows, 430. See gas-liquid pipe TEM. See transmission electron microscopy flows temporal discretization schemes, 267 velocity measurement techniques, 348–358 temporal resolution, 331, 339 corona-discharging method, 355–356 terminal velocity. See particle terminal velocity cross-correlation method, 348–350 thermal radiation, 96, 132, 194, 221 laser Doppler velocimetry, 352–354 thermophoresis, 92, 107, 173, 176 particle image velocimetry, 356–358 thickening, 379 venturimeter, 350–352 three-phase separators, 380 venturimeter, 342, 348, 350–351 time averaging, 18, 47, 213, 222–223 vertical pipe flows, 12, 477, 495 tomography techniques, 331 gas-liquid flow regimes, 482 tortuosity, 56 vapor-liquid flow regimes, with boiling, 531

594 Index

viscosity volume-time-averaged equations, 212, 221–228 bulk, 33, 235 vortex shedding, 174, 243, 456 dynamic, 33 mixture, 234 wake, 3, 24, 107 slurry mixture, 488 wake shedding, 9, 44, 84, 93, 108, 234, 433, See solid shear, 67 vortex shedding viscous dissipation, 42, 43, 45, 52, 56, 151, 189, wall friction, 20, 430, 451, 467, 483, 491, 492 195, 439 wall superheat, 527, 528 See VOF method. volume of fluid method waves volume averaging, 18, 59, 212–214, 217, 533 acoustic, 339 volume fraction liquid, 3, 481, 514 particle, 16 shock, 42 path-averaged, 332 wavy stratified pipe flows, 477, 481 phase, 18 wear. See erosion volume fraction function, 280–281 Weber number, 108, 152 volume fraction measurement techniques, wet scrubbing, 7 331–342 beam attenuation method, 331–334 X-ray tomography, 331, 341 electrical impedance tomography, 339–342 permittivity measurement method, 334–336 transmission tomography, 336–339 yield stress, 144, 185, 187, 508 volume mean diameter, 330 Young’s modulus, 162, 185, 188 volume of fluid method, 281 volume-averaged equations, 212, 216–219 γ -ray densitometer, 333, 349, 351 volume-averaging theorems, 18, 214–216 γ -ray tomography, 331 volume-time averaging, 213, 222 ε-equation, 50, 243, 245, 247