Krantz W.B. Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation
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INDEX 523
convective between parallel plates with
fluid injection, 202
convective between parallel plates with
temperature-dependent viscosity, 180
convective between parallel plates with
viscous dissipation, 163, 234, 235
convective in fluid-wall aerosol flow
reactor, 448, 475, 476, 477, 478,
479
evaporative cooling of nonflowing film,
211, 242
falling film flow, 206, 236, 237
free convection between parallel plates
at different temperatures, 183
free convection next to heated vertical
plate with suction, 247
free convection next to heated vertical
plate, 218, 247
freezing of water-saturated soil 173,
243, 244
home freezer performance, 250
hot wire anemometer, 249
melting of frozen soil, 242, 243
rectangular fin, 196
unsteady-state axial in solid cylinder,
228
unsteady-state conductive in cooking
turkey, 187
unsteady-state conductive through plane
wall with imposed temperatures, 153
unsteady-state convective from solid
sphere at high Biot number, 209
unsteady-state convective from solid
sphere at low Biot number, 159, 231
unsteady-state convective to plane wall,
232
unsteady-state convective to solid
cylinder, 233
unsteady-state in membrane thermal
casting, 438, 471, 472, 473, 474
unsteady-state in slab with heat
generation, 248
unsteady-state radial conduction in
sphere, 248
unsteady-state radial in solid cylinder,
228
unsteady-state radial in spherical shell,
229
unsteady-state resistance heating in
wire, 200, 223
unsteady-state to sphere with
temperature-dependent conductivity,
250
Heat transfer coefficient, 159, 193, 196,
209, 227, 229, 232, 233, 234, 244,
246, 248, 250
Heterogeneous chemical reaction, see
Chemical reaction, heterogeneous
High Reynolds number flow, see
Approximation, high Reynolds
number
Hollow fiber, 101, 287, 328, 349, 415
Homogeneous chemical reaction, see
Chemical reaction, homogeneous
Hydraulic ram, 76, 116
Hydrodynamic boundary layer, see
Approximation, hydrodynamic
boundary layer
Hydrogen fuel production, see Chemical
reactor, fluid-wall aerosol
Identity tensor, 47, 63, 483
Incompressible flow approximation, see
Approximation, incompressible flow
Integral balance:
auxiliary condition, 47, 68, 96, 102,
103, 119, 174, 175, 176, 195, 213,
214, 243, 244, 246, 374
conservation of energy, 174
conservation of mass, 47, 299, 323, 337,
342, 344, 345, 346, 351
species, 275, 298, 309, 347, 348, 374
Integral relationships:
Gauss-Ostrogradskii divergence
theorem, 504
Leibnitz’ formula, 504
Interphase mass transfer time scale, see
Time scale, interphase mass transfer
Isolating quantities in dimensional
analysis, 14, 16, 436, 461, 479
Jet flow, 94, 126
Kinematic surface condition, 47, 52, 68,
96, 127, 128, 129
524 INDEX
Leibnitz formula, 504
Lewis number, see Dimensionless groups,
Lewis number
Lie groups, 7
Little oh of one, 2, 19, 145, 252
Local scaling, 10
in fluid flow, 31, 32, 36, 47, 49, 67, 68,
83
in heat transfer, 166, 167, 170, 194,
in mass transfer, 264, 268, 281, 294
Low Reynold’s number approximation, see
Approximation, creeping flow
Lubrication flow, see Approximation,
lubrication flow
Mach number, see Dimensionless groups,
Mach number
Macroscale scaling analysis, see Scaling
analysis, macroscale
Mass balance, see Equation of continuity
for binary mixture
Mass transfer:
convective between rotating cylinders
with concentration-dependent
viscosity, 352
convective from sphere, 355
convective in falling film flow down a
vertical cylinder, 338
convective in falling film flow down a
vertical plane, 269, 338
convective in field-flow fractionation,
316
convective in fluid-wall aerosol flow
reactor, 448
convective in membrane-lung
oxygenator, 287, 415
convective in oxygen transfer to red
blood cell, 332
convective in pressure-swing gas
absorption, 424
convective in Taylor dispersion of a
solute, 303
convective to uniformly accessible
rotating disk, 312
diffusive in crystallization from a
supersaturated solution, 345
diffusive in evaporation of liquid in
tube, 273, 337
diffusive in evaporative casting of
polymer film, 297, 350
diffusive in membrane permeation cell,
321
diffusive in membrane permeation with
concentration-dependent diffusivity,
277
diffusive in membrane thermal casting,
438, 471, 472, 473, 474
diffusive in stationary liquid film, 253,
336, 357
diffusive in swelling membrane with
nonconstant diffusivity, 349
diffusive in tapered pore, 306, 337
diffusive to nucleated water droplet,
344, 346
film theory, 253
free convection from horizontal
evaporating liquid, 357
free convection in evapotranspiration
from vertical cylinder, 281,
354
free convection next to permeable
vertical plate, 354
penetration theory, 259, 336
Mass-transfer coefficient, 212, 216, 259,
260, 289, 294, 297, 299, 332, 355,
356, 357, 358, 359, 387, 400, 407,
416
Mass transfer with chemical reaction:
continuous stirred tank reactor, 390,
408, 409
diffusional domain of slow reaction
regime, 381, 404
fast reaction regime, 372, 404, 405, 407
fluid-wall aerosol flow reactor, 448, 475,
476, 477, 478, 479
heterogeneous in growth of nucleated
water droplet, 346
heterogeneous in rusting of metal
surface, 347
heterogeneous in tube flow, 266, 325,
340
heterogeneous in tubular photocatalytic
reactor, 358
homogeneous in aeration of water
containing aerobic bacteria, 340
homogeneous in dissolution of
cylindrical capsule, 343, 344
INDEX 525
homogeneous in dissolution of spherical
capsule, 308, 342,
homogeneous in flow between parallel
membranes, 261, 339
homogeneous in hollow fiber membrane
reactor, 328, 349
homogeneous in liquid film with
concentration-dependent diffusivity,
351
inner domain of instantaneous reaction
regime, 373, 374, 405, 406
intermediate domain of slow reaction
regime, 403
intermediate reaction regime, 371, 404,
407
kinetic domain of slow reaction regime,
380, 405
packed column chemisorption, 394, 410,
411
slow reaction regime, 371, 400, 401, 408
surface domain of instantaneous reaction
regime, 373, 405
Membrane:
convective heat transfer between parallel
with flow injection, 202
flow between parallel with velocity
profile distortion, 136
flow between parallel, 143
flow between permeable and parallel flat
plate, 137
flow in annular region between
concentric, 139
flow in closed-end parallel, 141, 143
flow in hollow fiber with permeation,
101
heat transfer in boundary-layer flow
over with suction, 241
mass transfer in artificial lung
oxygenator, 287, 415, 467, 468
mass transfer in field-flow fractionation
between parallel, 316
mass transfer in hollow fiber, 328, 349
mass transfer in permeation cell, 321
mass transfer in swelling, 277, 349
mass transfer with nonconstant
diffusivity through, 277, 349
mass transfer with reaction in flow
between parallel, 261, 339
thermal casting, 438, 471, 472, 473,
474
Membrane-lung oxygenator, 287, 415,
467, 468
Microscale scaling analysis, see Scaling
analysis, microscale
Minimum parametric representation, 13,
35, 62, 64, 65, 69, 70, 97, 101, 104,
108, 109, 187, 190, 192, 196, 224,
287, 288, 291, 293, 296, 297, 335,
414, 415, 444, 456, 457, 464
Momentum balance, see Equations of
motion; Equations of motion for
porous media
Momentum boundary layer, see Region of
influence, fluid flow
Moving boundary:
dissolution of spherical capsule, 308
evaporation of liquid, 273
evaporative casting of polymer film, 297
evaporative cooling of liquid film, 211,
242
freezing of water-saturated soil, 243,
244
instantaneous reaction front, 373
melting of frozen soil, 173, 242, 243
thermal casting of membrane, 438, 471,
472, 473, 474
Navier Stokes equation, see Equations of
motion
Newton’s constitutive equation, 483, 489,
491, 492, 493
Nomenclature, see Notation
Normal stress boundary condition, see
Boundary condition, normal stress
No-slip condition, see Boundary condition,
no-slip
Notation, 506
Nozzle, 114
Nucleation, 344, 346
Nusselt number,
see Dimensionless groups,
Nusselt number for heat transfer,
Nusselt number for mass transfer
Observation time, see Time scale,
observation time
526 INDEX
Order of magnitude, see Big ”oh” of one;
Little ”oh” of one; Scaling analysis
Order-of-one scaling analysis, see Big
”oh” of one; Little ”oh” of one;
Scaling analysis
Oscillated plate, 38, 68, 110, 111
Oscillating disk viscometer, see
Viscometer, oscillating disk
Oscillating tube, 122, 142, 287, 415, 467,
468
Packed column, 124, 394, 410, 411
Peclet number, see Dimensionless groups,
Peclet number for heat transfer, Peclet
number for mass transfer
Penetration theory, see Approximation,
penetration theory in heat transfer;
Approximation, penetration theory in
mass transfer
Permeation:
cell for membrane characterization, 321
evapotranspiration through vertical
cylinder, 281, 354
hollow fiber membrane reactor, 328, 349
influence on heat transfer in
boundary-layer flow over flat
membrane, 241
influence on velocity profile in flow
between parallel membranes, 136
membrane-lung oxygenator, 287, 415,
467, 468
through closed-end hollow fiber causing
axial flow, 101
through closed-end parallel membranes
causing axial flow, 141, 143
through concentric membranes with
axial flow in annular region, 139
through membrane parallel to flat plate
with laminar flow, 137
through membrane with nonconstant
diffusivity, 277, 349
through parallel membranes in field-flow
fractionation, 316
through parallel membranes in flow with
homogeneous reaction, 261, 339
through parallel membranes with
convective heat transfer, 202
through parallel membranes with
laminar flow, 141, 143
through walls of rotating co-axial
cylinders with axial flow, 352
Phase transition:
freezing of water-saturated soil, 243,
244
melting of frozen soil, 173, 242, 243
thermally induced phase separation
process for membrane formation, 438,
471, 472, 473, 474
Pi theorem:
in fluid flow, 13, 62, 65, 66, 70, 109
in heat transfer, 187, 192, 193, 196,
224, 248, 250, 251
in mass transfer, 287, 292, 297, 335, 437
Plate, see Flat plate
Point quantities, 25, 26, 37, 43, 67, 111,
156, 173, 193
Porous media flow:
annulus with radial tube flow, 133
between parallel flat plates, 132
bounded by flat plate, 131
Brinkman term, 69, 106, 483, 494, 495,
cylindrical tube, 52
Darcy’s law, 53, 106, 426
determining soil permeability, 105
gravity-driven film flow over, 99, 133
liquid film flow over, 99
permeability, 53, 99, 105, 106, 109,
110, 137, 426
superficial velocity, 53, 55, 99, 426
Practice problems:
fluid flow, 110
heat transfer, 224
mass transfer, 336
mass transfer with chemical reaction,
399
process design, 467
Prandtl number, see Dimensionless groups,
Prandtl, number
Pressure-swing adsorption, 424, 468, 469,
470, 471
Primary quantity, 8
Process design:
chemisorption in diffusional domain of
slow reaction regime, 384
chemisorption in fast reaction regime,
385
chemisorption in inner domain of
instantaneous reaction regime, 386
INDEX 527
chemisorption in intermediate reaction
regime, 385
chemisorption in kinetic domain of slow
reaction regime, 383
chemisorption in surface domain of
instantaneous reaction regime, 387
continuous stirred tank reactor, 390
fluid-wall aerosol flow reactor, 448
membrane formation, 438
membrane-lung oxygenator, 415
packed column chemisorption, 394
physical absorption, 382
pressure-swing absorber, 424
Quasi-stationary hypothesis, see
Approximation, quasi-stationary
hypothesis
Quasi-steady-state, see Approximation,
quasi-steady-state
Rayleigh free convection, 183, 218, 247,
281, 354, 357
Rayleigh number, see Dimensionless
groups, Rayleigh number
Reaction domain:
diffusional of slow reaction, 381
inner of instantaneous, 373, 405, 406
intermediate of slow reaction, 403
kinetic of slow reaction, 380
surface of instantaneous, 373, 405
See also Mass transfer with chemical
reaction
Reaction processes, see Mass transfer with
chemical reaction
Reaction regime:
fast, 372
instantaneous, 373
intermediate, 371
slow, 371
See also Mass transfer with chemical
reaction
Reaction time scale, see Time scale,
reaction in macroscale element; Time
scale, reaction in microscale element
Reactions, see Mass transfer with chemical
reaction
Rectangular coordinates:
continuity equation for binary system, 499
continuity equation, 486
equations of motion, 489
equations of motion for porous media,
494
thermal energy equation, 496
Reference factor, 10
Region of influence scaling, 10
fluid flow, 20, 25, 32, 35, 36, 39, 40, 41,
42, 43, 45, 53, 55, 67, 68, 69, 80, 83,
85, 87, 111, 113, 120, 122, 132, 134,
135, 137, 139, 418, 435, 468, 471,
heat transfer, 146, 151, 152, 153, 156,
157, 158, 167, 170, 171, 193, 194,
205, 206, 208, 209, 216, 217, 220,
225, 228, 229, 230, 231, 235, 236,
237, 238, 239, 240, 243, 244, 247,
473, 476
mass transfer, 253, 260, 265, 271, 272,
279, 280, 281, 285, 288, 294, 295,
296, 301, 311, 312, 315, 325, 331,
337, 338, 339, 340, 341, 345, 346,
347, 348, 350, 351, 356, 415, 418,
422, 435
mass transfer with chemical reaction,
261, 266, 308, 325, 328, 339, 340,
342, 343, 344, 347, 351, 358, 364,
377
Reynolds number, see Dimensionless
groups, Reynolds number
Rotating disk viscometer, see Viscometer,
rotating disk
Rotating flow:
annulus with end effects, 88
between spinning parallel disks with
radial injection, 115
disk viscometer, 116
spinning disk, 79
tube with axial flow, 112
Rusting, 347
Scale-up analysis, 2, 143, 248, 250
Scaling analysis:
o(1) procedure, 8
dimensional analysis procedure, 14
estimated error, 3, 12, 24, 37, 94, 150,
156, 163, 193, 462
macroscale element, 361, 377, 402, 403,
404, 407, 426, 449
mathematical basis, 7
microscale element, 361, 362, 364, 401,
426, 449
528 INDEX
Scaling derivatives, see Derivative scaling
Schmidt number, see Dimensionless
groups, Schmidt number
Secondary quantity, 8
Sherwood number, see Dimensionless
groups, Sherwood number
Sidewall effects in fluid flow, 43, 116
Sign convention:
constitutive equation, 481, 483, 489,
491, 492
force on fluid particle, 480
Sizing of equipment, 3, 424, 448
Species continuity equation, see Equation
of continuity for a binary mixture
Sphere:
conductive heat transfer at low Biot
number, 159, 231
conductive heat transfer in at high Biot
number, 209
convective heat transfer from, 248
convective mass transfer from, 355
diffusional growth of nucleating, 344,
346
diffusive mass transfer from dissolving,
308, 342
diffusive mass transfer into red blood
cell, 332
falling at terminal velocity, 62, 142
unsteady-state with
temperature-dependent conductivity,
250
water treatment via aeration from, 340
Spherical coordinates:
continuity equation, 487
continuity equation for binary system,
502
equations of motion, 492
equations of motion for porous media,
495
thermal energy equation, 497
Stratified flow, 125
Superficial velocity, see Porous media,
superficial velocity
Tangential stress boundary condition, see
Boundary condition, tangential stress
Tangential unit vector, see Unit vector,
tangential
Taylor dispersion, 303
Taylor series expansion:
of concentration-dependent density, 283,
355
of diffusion coefficient, 279
of pressure-dependent density, 58
of small dimensionless group, 15, 66
of temperature-dependent density, 185,
219
of temperature-dependent viscosity, 181
Terminal velocity, 62
Thermal boundary layer, see Region of
influence, heat transfer
Thermal energy equation:
cylindrical coordinates, 497
generalized notation, 483
rectangular coordinates, 496
spherical coordinates, 497
Thermally induced phase-separation
process for membrane formation, 438,
471, 472, 473, 474
Thiele modulus, see Dimensionless
groups, Thiele modulus
Time scale:
adsorption, 431
axial dispersion, 431
conduction, 162, 193, 201
contact, 368, 379, 431
diffusion, 258, 268, 305, 368
interphase mass transfer, 379
observation, 38, 40, 49, 50, 68, 93, 94,
155, 157, 161, 162, 177, 193, 194,
201, 210, 215, 216, 217, 246, 257,
276, 301, 305, 310, 319, 324, 341
periodic motion, 40, 420
pressurization, 430, 431, 434
reaction in macroscale element, 263,
268, 379
reaction in microscale element, 368
viscous, 40
Time scaling, see Time scale
Transient phenomena, see Unsteady-state
Tube, see Cylindrical tube
Uniform magnifications and contractions, 7
Unsteady-state fluid flow:
between approaching parallel circular
plates, 72
INDEX 529
between stationary and oscillating
parallel plates, 38
draining of film down vertical plate, 45
falling head method for determining soil
permeability, 105
hydraulic ram, 116
impulsively initiated in tube, 142
membrane-lung oxygenator, 287, 415
oscillating disk viscometer, 117
oscillating tube, 122, 142, 287, 415
Unsteady-state heat transfer:
conductive in cooking turkey, 187
conductive in slab with heat generation,
248
conductive in solid cylinder, 228, 230,
233
conductive in spherical shell, 229
conductive in thermal casting of
membrane, 438, 471, 472, 473, 474
conductive through wall, 153, 224, 232
convective from solid sphere, 159, 209,
231, 250
evaporative cooling of nonflowing film,
211, 242
freezing of water-saturated soil, 173,
243, 244
melting of frozen soil, 242, 243
resistance heating in wire, 200, 223
Unsteady-state mass transfer:
convective in aeration of water, 340
convective in field-flow fractionation,
316
convective in membrane-lung
oxygenator, 287, 415
convective in pressure-swing adsorption,
424, 468, 469, 470, 471
convective in Taylor dispersion, 303
diffusive evaporation of liquid, 273, 337
diffusive in crystallization from
supersaturated liquid, 345
diffusive in dissolution of cylindrical
capsule, 343, 344
diffusive in dissolution of spherical
capsule, 308, 342
diffusive in evaporative polymer film
casting, 297, 350
diffusive in growth of nucleated water
droplet, 344, 346
diffusive in membrane permeation cell,
321
diffusive in rusting of planar surface,
347
diffusive in thermal casting of
membrane, 438, 471, 472, 473, 474
diffusive through stationary film, 253,
259, 336
Unsteady-state mass transfer with chemical
reaction:
macroscale element scaling, 377
microscale element scaling, 364
Vector-tensor notation, 482
Viscometer:
falling needle, 117
oscillating disk, 117
rotating disk, 116
Viscous dissipation 163, 180, 202, 235
Viscous time scale, see Time scale, viscous
Wall:
conduction through one-dimensional,
153, 224, 232
conduction through one-dimensional
with heat generation, 248
conduction through three-dimensional,
225
conduction through two-dimensional,
146
Wetted-wall column, see Falling flow