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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NOTATION 513
θ component in the θ-direction
φ component in the φ-direction
0 initial, upstream, reference, or infinite dilution value
∞ evaluated as the spatial or temporal variable approaches infinity
1 property of phase 1 or denotes a particular function
2 property of phase 2 or denotes a particular function
= tenor quantity
COMMONLY USED DIMENSIONLESS GROUPS
Bi
m
Biot number for mass transfer or solutal Biot number, ratio of the total mass
transfer external to the system to the diffusive mass transfer within the system
Bi
t
Biot number for heat transfer or thermal Biot number, ratio of the total heat
transfer external to the system to the conductive heat transfer within the system
Da
II
second Damkohler number, ratio of the time scale for radial diffusion to that for
heterogeneous reaction
Fo
m
Fourier number for mass transfer, ratio of the contact time to the characteristic
time for species diffusion
Fo
t
Fourier number for heat transfer, ratio of the contact time to the characteristic
time for heat conduction
Fr Froude number, ratio of the kinetic energy to the gravitational potential energy of
the flow
Gr
m
Grashof number for mass transfer, ratio of the solutally induced convective to
viscous transport of momentum
Gr
t
Grashof number for heat transfer, ratio of the thermally induced convective to
viscous transport of momentum
Gz Graetz number, a measure of the relative convection to diffusional transport of
species
Le Lewis number, ratio of heat conduction to species diffusion
Ma Mach number, ratio of the characteristic fluid velocity divided by the speed of
sound in the medium
Nu Nusselt number, a measure of the overall heat transfer to that by conduction alone
Pe
m
Peclet number for mass transfer or solutal Peclet number, ratio of convective to
diffusive transport of mass
Pe
t
Peclet number for heat transfer or thermal Peclet number, ratio of convective to
conductive transport of heat
Pe
V
Peclet number for mass transfer or solutal Peclet number based on the transverse
velocity
Pr Prandtl number, ratio of heat conduction to viscous transport of momentum
Ra
m
Rayleigh number for mass transfer, ratio of solutally induced convective to
diffusive transport of mass
Ra
t
Rayleigh number for heat transfer, ratio of thermally induced convective to
conductive transport of heat
Re Reynolds number, ratio of the kinetic energy per unit mass of the flow to the
principal viscous stress or equivalently, ratio of convective to viscous
transport of momentum
Re
V
Reynolds number based on the transverse velocity
Sc Schmidt number, ratio of viscous transport of momentum to species diffusion
514 NOTATION
Sh Sherwood number, a measure of the overall mass transfer to that by diffusion
alone
Sh
v
Sherwood number in the presence of axial oscillations or vibrations
Th Thiele modulus, ratio of the characteristic time for diffusion relative to that for
homogeneous reaction
MATHEMATICAL OPERATIONS
D
Dt
substantial time derivative or time derivative in a coordinate system convected at
the fluid velocity
d
dt
total time derivative allowing for the change in the quantity in time as well as
with spatial position
∂
∂t
partial time derivative allowing for the change in the quantity in time at a fixed
spatial position
erf(ϕ) error function of z, =
2
√
π
(
ϕ
0
e
−˜ϕ
2
d ˜ϕ
exp (ϕ) exponential function of ϕ,= e
ϕ
ln (ϕ) logarithm of ϕ to the base e
∇ del or nabla operator
INDEX
Absorption, see Gas absorption
Adsorption, see Pressure-swing adsorption
Adsorption time scale, see Time scale,
adsorption
Annulus:
conductive heat transfer in, 226
flow in hydraulic ram, 76
flow in rotating, 88
flow in tube with porous, 133
flow with fluid injection and withdrawal,
139
mass transfer between fixed and
rotating, 352
Approximation:
Boussinesq for free convection, 183
constant density, 56
constant diffusivity, 277, 349, 351
constant thermal conductivity, 246, 250
constant thermal diffusivity, 246
constant viscosity, 180
creeping flow, 26, 30, 50, 66, 75,
diffusional domain of slow reaction
regime, 381
fast reaction regime, 372
film theory for heat transfer, 153
film theory for mass transfer, 253, 357
film theory for mass transfer with
chemical reaction, 368, 406
fully developed flow, 62
hydrodynamic boundary layer, 32, 79,
84, 119, 120, 121
incompressible flow, 56, 135
inner domain of instantaneous reaction
regime, 405
Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach
to Model Building and the Art of Approximation, By William B. Krantz
Copyright © 2007 John Wiley & Sons, Inc.
instantaneous reaction regime, 373
intermediate domain of slow reaction
regime, 403
intermediate reaction regime, 371
kinetic domain of slow reaction regime,
380
large Damk
¨
ohler number, 325, 340,
large Reynolds number, 32, 79, 84, 119,
120, 121
large solutal Biot number, 297
large solutal Grashof number, 281, 354
large solutal Peclet number, 269, 338
large thermal Biot number, 209
large thermal Grashof number, 218, 247
large thermal Peclet number, 167, 206,
236, 237, 238, 239, 240, 241
large Thiele modulus, 261, 339
lubrication flow, 26, 45, 72, 76, 101,
113, 114, 115, 116, 127, 141
lumped capacitance, 163
macroscale element, 362, 377
microscale element, 362
negligible curvature in fluid flow, 45,
52, 76, 94, 123, 127, 128, 134, 139
negligible curvature in heat transfer,
209, 229
negligible curvature in mass transfer,
308, 328, 338, 340, 342, 343, 344,
346, 352
negligible end effects in fluid flow, 43,
88, 133
negligible side-wall effects in fluid flow,
43, 52, 116
negligible viscous dissipation, 202, 235
515
516 INDEX
Approximation: (continued)
penetration theory for heat transfer, 153
penetration theory for mass transfer,
259, 336
penetration theory for mass transfer with
chemical reaction, 369, 400
physical absorption, 379
plug flow reactor, 266
quasi-parallel flow, 45, 94, 126, 128,
143
quasi-stationary-hypothesis, 380, 402
quasi-steady-state fluid flow, 38, 45, 72,
115, 117, 122, 142
quasi-steady-state heat transfer, 173,
242, 243, 244
quasi-steady-state mass transfer, 273,
297, 303, 308, 316, 321, 340, 342,
344, 346, 347
reaction boundary layer, 261, 328, 339,
340
slow reaction regime, 371
small Damk
¨
ohler number, 266
small Reynolds number, see
Approximation, creeping flow
small solutal Biot number, 297
small solutal Peclet number, 261, 340
small thermal Biot number, 159, 196,
231, 232, 233
small thermal Peclet number, 163, 202
small Thiele modulus, 328
solutal boundary layer, 259, 269, 277,
281, 308, 312, 325, 337, 338, 344,
345, 346, 347, 349, 350, 351, 354
steady-state fluid flow, 92, 116
steady-state heat transfer, 153
steady-state mass transfer, 253, 364
Stoke’s flow, see Approximation,
creeping flow
surface domain of instantaneous reaction
regime, 405
Taylor dispersion, 303
thermal boundary layer, 153, 167, 173,
206, 218, 236, 237, 238, 239, 240,
241, 247
uniformly accessible surface for mass
transfer, 79, 312
Auxiliary condition:
free surface flows, 47, 68, 96, 102, 103,
moving boundary in heat transfer, 174,
213, 242, 243, 246, 442
moving boundary in mass transfer, 275,
295, 299, 309, 323, 337, 342, 344,
345, 346, 347, 348, 351, 442
moving front for instantaneous reaction,
374
Axial dispersion time scale, see Time
scale, axial dispersion
Big oh of one, 2, 19, 145, 252
Biot number, see Dimensionless
groups, Biot number for heat transfer,
Biot number for mass transfer
Boundary condition:
free surface, 46, 95, 127, 128, 129
instantaneous reaction front, 373
moving boundary in heat transfer, 173,
211, 242, 243, 244
moving boundary in mass transfer, 275,
295, 299, 309, 337, 342, 344, 345,
346, 347, 348, 351
moving front for instantaneous reaction,
374
normal stress, 46, 96, 127, 128, 129
tangential stress, 46, 95, 127, 128, 129
Boundary layer, see Boundary-layer flow
Boundary-layer flow, 32, 79, 84, 119,
120, 121. See also Region of
influence
Brinkman term, see Porous media
Buckingham Pi theorem, see Pi theorem
Change of order one, see Big “oh” of one;
Little “oh” of one
Channel, gravity-driven flow with
sidewalls, 43
Chemical reaction:
heterogeneous, 266, 325, 340, 346, 347,
358
homogeneous, 261, 308, 328, 339, 340,
342, 343, 344, 349, 351, 364, 371,
372, 373, 380, 381, 390, 394, 401,
403, 404, 405, 406, 407, 408, 409,
411
See also Mass transfer with chemical
reaction
INDEX 517
Chemical reactor:
continuous stirred tank, 390, 408, 409
fluid-wall aerosol flow, 448, 475, 476,
477, 478, 479
hollow fiber membrane, 328, 349
laminar flow with heterogeneous
reaction, 266, 325, 340
laminar flow with homogeneous
reaction, 261, 339
packed column, 394, 410, 411
photocatalytic, 358
plug flow, 266
Chemical reactor design:
continuous stirred tank, 390, 408, 409
diffusional domain of slow reaction
regime, 384
fast reaction regime, 385
fluid-wall aerosol flow, 448, 475, 476,
477, 478, 479
inner domain of instantaneous reaction
regime, 386
intermediate reaction regime, 384
kinetic domain of slow reaction regime,
383
packed column, 394, 410, 411
surface domain of instantaneous reaction
regime, 386
Compressible flow, 56, 135
Condensation, 127
Conduction, see Heat transfer
Conduction time scale, see Time scale,
conduction
Conservation of energy equation, see
Thermal energy equation
Constitutive equation, 481, 483, 489, 491,
492, 496, 497, 498
Contact time scale, see Time scale, contact
time
Continuity equation:
cylindrical coordinates, 487
generalized notation, 482
rectangular coordinates, 486
spherical coordinates, 487
Continuous stirred tank reactor, see
Chemical reactor, continuous stirred
tank
Correlating experimental or numerical
data, 2, 415, 424, 436, 438, 448, 460
Creeping flow, see Approximation,
creeping flow
Crystallization, 345
Curtain coating, 128,143
Curvature effects, see Approximation,
negligible curvature
Cylinder:
falling needle viscometer, 117
free convection mass transfer adjacent
to vertical, 281, 354
heat conduction with
temperature-dependent thermal
diffusivity, 246
heat transfer for hot wire anemometer,
249
heat transfer with resistance heating,
200, 223
mass transfer in dissolution
of, 343, 344
mass transfer to film flow down vertical,
338
steady-state heat conduction with
external convection, 230
two-dimensional steady-state heat
conduction, 225
unsteady-state axial heat conduction,
228
unsteady-state heat conduction with
external convection, 233
unsteady-state radial heat conduction,
228
Cylindrical coordinates:
continuity equation, 487
equation of motion for porous media,
494
equations of motion, 490
species continuity equation, 500
thermal energy equation, 497
Cylindrical tube:
compressible gas flow in, 56, 135
constant injection through in flow
between parallel disks, 114
constant injection through in flow
between spinning parallel disks, 115
countercurrent liquid-gas flow in, 123
entry region flow with porous annulus,
134
entry region flow, 121
518 INDEX
Cylindrical tube: (continued)
falling head method for determining
permeability, 105
falling needle viscometer, 117
flow in hollow fiber, 101
flow through porous media in, 52
gravity- and pressure-driven flow in
vertical, 111
impulsively initiated flow in, 92, 142
mass transfer in membrane-lung
oxygenator, 287, 415
mass transfer of evaporating liquid
from, 273, 337
mass transfer via Taylor dispersion in,
303
mass transfer with photocatalytic
reaction in, 358
mass transfer with reaction for flow in,
266, 325, 340
non-constant injection through for flow
between parallel disks, 115
pressure-driven flow in oscillating, 122,
142
pressure-driven flow in rotating, 112
radial flow from porous, 133
d’Alembert’s paradox, 35
Damk
¨
ohler number, see Dimensionless
groups, Damk
¨
ohler number
Darcy flow, see Porous media flow,
Darcy’s law
Derivative scaling, 9, 21
compressible gas flow in cylindrical
tube, 59, 60, 69, 135
convective mass transfer between
parallel membranes, 339
curtain-coating flow, 130
dissolution of a spherical capsule, 308
draining of liquid film, 47, 48, 68
evaporation of a liquid, 273
evaporative casting of polymer film, 297
evaporative cooling of a liquid film, 211
field-flow fractionation, 316
flow between moving and stationary
plates, 110, 111
flow between parallel permeable
membranes with homogeneous
reaction, 339
fluid-wall aerosol flow reactor, 448
freezing of water-saturated soil overlaid
by snow, 244
heat transfer with phase change, 173
hollow fiber membrane reactor, 328
jet flow, 96, 97, 126
membrane lung oxygenator, 415
membrane permeation cell, 321
rusting of metal surface, 347
steady-state conduction in cylinder, 230
steady-state conduction in rectangular
fin, 196
thermal casting of membrane, 438
Developing flow:
between approaching parallel circular
plates, 72
between closed-end parallel plate
membranes, 141
between converging flat plates, 26, 113
between diverging flat plates, 113
between parallel disks with constant
radial injection, 114
between parallel disks with
time-dependent radial injection, 115
between permeable and impermeable
parallel flat plates, 137
between spinning parallel disks with
constant radial injection, 115
boundary layer over flat plate with
blowing, 120
boundary layer over flat plate with
suction, 120
boundary layer over flat plate, 32, 119
curtain coating, 128, 143
diverging nozzle, 114
entry region between parallel
plates, 84
entry region in a tube with porous
annulus, 134
entry region in cylindrical tube, 121
field-flow fractionation, 136
gravity-driven draining film down
vertical wall, 45
gravity-driven free surface film flow
with condensation, 127
gravity-driven free surface of film over
horizontal filter, 127
hydraulic ram, 116
liquid jet, 94, 126
permeable hollow fiber membrane, 101
INDEX 519
permeable parallel membranes, 143
pressure-driven compressible in
cylindrical tube, 56, 135
Diffusion, see Mass transfer
Diffusion time scale, see Time scale,
diffusion
Dimensional analysis in fluid flow:
around sphere falling at terminal
velocity, 62, 142
between permeable membranes, 143
curtain-coating, 143
falling head method for determining
permeability, 105
hollow fiber membrane, 144
oscillating cylindrical tube, 142
tube with impulsively applied pressure,
142
Dimensional analysis in heat transfer:
cooking a turkey, 187
home freezer characterization, 250
hot wire anemometer performance, 249
resistance heating in electrical wire, 223
slab with heat generation, 248
sphere with temperature-dependent
thermal conductivity, 250
steady-state convective from sphere, 248
Dimensional analysis in mass transfer:
convective from sphere, 355
evaporating liquid, 357
film theory, 357
membrane-lung oxygenator, 287, 415,
468
pressure-swing adsorption, 424, 471
spherical red blood cell, 332
tubular photocatalytic reactor, 358
Dimensional analysis in mass transfer with
chemical reaction,
fluid-wall aerosol flow reactor, 448, 475,
476, 477, 478, 479
Dimensional constants, 13
Dimensionless groups:
Biot number for heat transfer, 162, 513
Biot number for mass transfer, 302, 513
Damk
¨
ohler number, 268, 513
Fourier number for heat transfer, 155,
513
Fourier number for mass transfer, 258,
513
Froude number, 30, 513
Graetz number, 292, 513
Grashof number for heat transfer, 221,
513
Grashof number for mass transfer, 285,
513
Lewis number, 445, 513
Mach number, 61, 513
Nusselt number for heat transfer, 159,
513
Nusselt number for mass transfer, 292
Peclet number for heat transfer, 166, 513
Peclet number for mass transfer, 263,
513
Prandtl number, 166, 513
Rayleigh number, 184, 221, 513
Reynolds number, 30, 513
Schmidt number, 263, 513
Sherwood number, 292, 514
Thiele modulus, 263, 514
Disk:
mass transfer from uniformly accessible
rotating, 312
oscillating viscometer, 117
rotating flow, 79
rotating viscometer, 116
Dissipation of energy, see Viscous
dissipation
Dissolution:
cylindrical capsule, 343, 344
spherical capsule, 308, 342
Drainage of liquid:
falling head method for determining
permeability, 105
unsteady down vertical wall, 45
Dynamic pressure, 63, 219, 283
End effects:
fluid flow, 43, 68, 77, 88, 118, 133
heat transfer, 146, 224, 225, 226
mass transfer, 343
Energy dissipation, see Viscous
dissipation; Heat transfer
Energy equation, see Thermal energy
equation
Entrance effects:
fluid flow, 84, 121, 134
heat transfer, 234, 236, 247
mass transfer, 340
520 INDEX
Entrance length, see Entrance effects
Entropy production, 126
Equation of continuity, see Continuity
equation
Equation of continuity for a binary
mixture:
cylindrical coordinates, 500
generalized notation, 484
rectangular coordinates, 499
spherical coordinates, 502
Equation of energy, see Thermal energy
equation
Equations of motion:
cylindrical coordinates, 490
generalized notation, 482
rectangular coordinates, 489
spherical coordinates, 492
Equations of motion for porous media:
cylindrical coordinates, 494
generalized notation, 483
rectangular coordinates, 494
spherical coordinates, 495
Error estimate in scaling, see Scaling
analysis, estimated error
Evaporation:
casting of polymer film, 297, 350
cooling of stationary liquid film, 211,
242
free convection mass transfer from
horizontal liquid layer, 357
liquid in cylindrical tube, 273, 337
polymeric membrane and film casting,
297, 350
Example problems:
fluid dynamics, 70
heat transfer, 196
mass transfer, 297
Falling film, see Film flow
Falling head method, 105
Falling needle viscometer, see Viscometer,
falling needle
Fick’s law, 314, 342, 353, 356, 358, 484,
499, 500, 501, 502, 503
Field-flow fractionation, 136, 316
Film flow:
countercurrent liquid-gas in cylindrical
tube, 123
curtain coating, 128, 143
fully developed falling film in presence
of viscous gas phase, 70
gravity-driven draining film down a
vertical wall, 45
gravity-driven free surface film flow
with condensation, 127
gravity-driven free surface of film over
horizontal filter, 127
gravity-driven in channel with side
walls, 43
gravity-driven liquid film over porous
media, 99, 133
mass transfer to gravity-driven down
vertical cylinder, 338
mass transfer to gravity-driven down
vertical plate, 269, 338
Film theory, see Approximation, film
theory for heat transfer;
Approximation, film theory for mass
transfer
Filter, 127
Flat plate:
boundary layer flow over, 32, 119
boundary-layer flow over with blowing,
120
boundary-layer flow over with suction,
120
flow down with condensation, 127
flow over filter, 127
free convection heat transfer adjacent to,
218, 247
free convection mass transfer adjacent
to, 354
gravity-driven flow down in presence of
viscous gas, 70
gravity-driven flow over with side walls,
43
heat transfer to falling film flow down,
206, 236, 237
mass transfer to falling film flow down,
269, 338
porous media flow bounded by, 131
thermal boundary layer in flow over
with suction, 241
thermal boundary layer in flow over,
167, 236, 238, 239, 240
Flat plates:
flow between converging, 26
INDEX 521
flow between diverging, 113
flow between parallel membrane, 143
flow between parallel stationary and
moving, 20, 110, 111
flow between stationary and oscillating,
38
flow between with fluid injection and
withdrawal, 136
flow in entry region between parallel, 84
flow in porous media between parallel,
132
flow of stratified liquid layers between,
125
heat transfer in flow between with
entrance effects, 234
heat transfer in flow between with
permeable walls, 202
heat transfer in flow between with
temperature-dependent viscosity, 180
heat transfer in flow between with
viscous dissipation, 163
heat transfer in laminar flow between,
235
heat transfer in thermal boundary layer,
236
mass transfer with field-flow
fractionation between parallel
membrane, 316
mass transfer with reaction in flow
between parallel membranes, 261, 339
Fluid dynamics, see Fluid flow
Fluid flow:
annulus with injection and withdrawal,
139
axial in a rotating tube, 112
between approaching parallel circular
plates, 72
between converging flat plates, 26
between cylinder and piston in hydraulic
ram, 76
between oscillating and stationary
parallel circular plates, 38
between diverging flat plates, 113
between parallel disks with constant
radial injection, 114
between parallel disks with
time-dependent radial injection, 115
between parallel membranes, 143
between permeable and impermeable
parallel flat plates, 137
between spinning parallel disks with
constant radial injection, 115
boundary layer over a flat plate, 32, 119
boundary-layer over flat plate with
blowing, 120
boundary-layer over flat plate with
suction, 120
closed-end parallel plate membranes,
141
countercurrent liquid-gas in cylindrical
tube, 123
curtain coating, 128, 143
diverging nozzle, 114
entry region between parallel plates, 84
entry region for cylindrical tube, 121
entry region in a tube with porous
annulus, 134
falling needle viscometer, 117
field-flow fractionation, 136
fully developed falling film in presence
of viscous gas phase, 70
gravity and pressure-driven in vertical
tube, 111
gravity-driven cylindrical jet, 94
gravity-driven draining film down a
vertical wall, 45
gravity-driven free surface film flow
with condensation, 127
gravity-driven free surface of film over
horizontal filter, 127
gravity-driven in channel with side
walls, 43
gravity-driven liquid film over porous
media, 99, 133
hydraulic ram, 76, 116
liquid jet, 94, 126
oscillating disk viscometer, 117
over falling sphere, 62, 142
permeable hollow fiber membrane, 101,
144
porous media bounded by flat plate, 131
porous media bounded by parallel flat
plates, 132
porous media in cylindrical tube, 52
pressure driven in oscillating cylindrical
tube, 122
522 INDEX
Fluid flow: (continued)
pressure-driven between moving and
stationary parallel plates, 20
pressure-driven compressible in
cylindrical tube, 56, 135
pressure-driven of two stratified liquid
layers, 125
radial from porous cylindrical tube, 133
rotating disk viscometer, 116
rotating disk, 79
rotating in annulus with end effects, 88
tube with impulsively applied pressure,
92, 142
Fluid particle, 480
Fluid-wall aerosol flow reactor, see
Chemical reactor, fluid-wall aerosol
flow
Force on a fluid particle, 480
Forgiving nature of scaling, 12
in fluid flow, 25, 31, 49, 59, 96, 126,
133, 136
in heat transfer, 173, 177, 195, 456,
464, 466
in mass transfer, 329, 349
Fourier number, see Dimensionless groups,
Fourier number for heat transfer,
Fourier number for mass transfer
Fractional conversion, 449, 460, 462, 463,
478, 479
Free convection:
heat transfer, 183, 218, 247,
mass transfer, 281, 354, 357
Free surface flow:
curtain coating, 128
cylindrical jet, 94
down plate with condensation, 127
gravity-driven down a vertical wall, 45
over horizontal filer, 127
Froude number, see Dimensionless groups,
Froude number
Gas absorption:
chemical in bubble column, 362, 364,
377
continuous stirred tank reactor, 390,
408, 409
packed column, 394, 410, 411
Generalized notation:
continuity equation, 482
continuity equation for binary mixture,
484
equations of motion, 482
equations of motion for porous media,
483
thermal energy equation, 483
Graetz number, see Dimensionless groups,
Graetz number
Grashof number, see Dimensionless
groups, Grashof number for heat
transfer, Grashof number for mass
transfer
Group theory, 7
Heat conduction, see Heat transfer
Heat convection, see Heat transfer
Heat transfer:
boundary layer flow between parallel
plates with unheated entrance length,
236
boundary-layer flow over flat plate with
unheated entrance length, 239
boundary-layer flow over flat plate, 167,
238, 240
boundary-layer flow over plate with
specified heat flux, 240
boundary-layer flow over plate with
suction, 241
conductive in cylinder, 225, 230
conductive film theory model, 153
conductive in annulus with prescribed
temperatures, 226
conductive in circular fin, 227
conductive in cylinder with
temperature-dependent thermal
diffusivity, 246
conductive in rectangular parallelepiped,
225
conductive penetration theory model,
153
conductive through plane wall, 153, 224
conductive two-dimensional with end
effects, 146
convective between heated parallel
plates, 236