INDEX

Absorption, see Gas absorptionAdsorption, see Pressure-swing adsorptionAdsorption time scale, see Time scale,

adsorptionAnnulus:

conductive heat transfer in, 226flow in hydraulic ram, 76flow in rotating, 88flow in tube with porous, 133flow with fluid injection and withdrawal,

139mass transfer between fixed and

rotating, 352Approximation:

Boussinesq for free convection, 183constant density, 56constant diffusivity, 277, 349, 351constant thermal conductivity, 246, 250constant thermal diffusivity, 246constant viscosity, 180creeping flow, 26, 30, 50, 66, 75,diffusional domain of slow reaction

regime, 381fast reaction regime, 372film theory for heat transfer, 153film theory for mass transfer, 253, 357film theory for mass transfer with

chemical reaction, 368, 406fully developed flow, 62hydrodynamic boundary layer, 32, 79,

84, 119, 120, 121incompressible flow, 56, 135inner domain of instantaneous reaction

regime, 405

Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approachto Model Building and the Art of Approximation, By William B. KrantzCopyright 2007 John Wiley & Sons, Inc.

instantaneous reaction regime, 373intermediate domain of slow reaction

regime, 403intermediate reaction regime, 371kinetic domain of slow reaction regime,

380large Damkohler number, 325, 340,large Reynolds number, 32, 79, 84, 119,

120, 121large solutal Biot number, 297large solutal Grashof number, 281, 354large solutal Peclet number, 269, 338large thermal Biot number, 209large thermal Grashof number, 218, 247large thermal Peclet number, 167, 206,

236, 237, 238, 239, 240, 241large Thiele modulus, 261, 339lubrication flow, 26, 45, 72, 76, 101,

113, 114, 115, 116, 127, 141lumped capacitance, 163macroscale element, 362, 377microscale element, 362negligible curvature in fluid flow, 45,

52, 76, 94, 123, 127, 128, 134, 139negligible curvature in heat transfer,

209, 229negligible 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, 153penetration theory for mass transfer,

259, 336penetration theory for mass transfer with

chemical reaction, 369, 400physical absorption, 379plug flow reactor, 266quasi-parallel flow, 45, 94, 126, 128,

143quasi-stationary-hypothesis, 380, 402quasi-steady-state fluid flow, 38, 45, 72,

115, 117, 122, 142quasi-steady-state heat transfer, 173,

242, 243, 244quasi-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, 371small Damkohler number, 266small Reynolds number, see

Approximation, creeping flowsmall solutal Biot number, 297small solutal Peclet number, 261, 340small thermal Biot number, 159, 196,

231, 232, 233small thermal Peclet number, 163, 202small Thiele modulus, 328solutal boundary layer, 259, 269, 277,

281, 308, 312, 325, 337, 338, 344,345, 346, 347, 349, 350, 351, 354

steady-state fluid flow, 92, 116steady-state heat transfer, 153steady-state mass transfer, 253, 364Stoke’s flow, see Approximation,

creeping flowsurface domain of instantaneous reaction

regime, 405Taylor dispersion, 303thermal boundary layer, 153, 167, 173,

206, 218, 236, 237, 238, 239, 240,241, 247

uniformly accessible surface for masstransfer, 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 Timescale, axial dispersion

Big oh of one, 2, 19, 145, 252Biot number, see Dimensionless

groups, Biot number for heat transfer,Biot number for mass transfer

Boundary condition:free surface, 46, 95, 127, 128, 129instantaneous reaction front, 373moving boundary in heat transfer, 173,

211, 242, 243, 244moving 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, 129tangential stress, 46, 95, 127, 128, 129

Boundary layer, see Boundary-layer flowBoundary-layer flow, 32, 79, 84, 119,

120, 121. See also Region ofinfluence

Brinkman term, see Porous mediaBuckingham Pi theorem, see Pi theorem

Change of order one, see Big “oh” of one;Little “oh” of one

Channel, gravity-driven flow withsidewalls, 43

Chemical reaction:heterogeneous, 266, 325, 340, 346, 347,

358homogeneous, 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 chemicalreaction

INDEX 517

Chemical reactor:continuous stirred tank, 390, 408, 409fluid-wall aerosol flow, 448, 475, 476,

477, 478, 479hollow fiber membrane, 328, 349laminar flow with heterogeneous

reaction, 266, 325, 340laminar flow with homogeneous

reaction, 261, 339packed column, 394, 410, 411photocatalytic, 358plug flow, 266

Chemical reactor design:continuous stirred tank, 390, 408, 409diffusional domain of slow reaction

regime, 384fast reaction regime, 385fluid-wall aerosol flow, 448, 475, 476,

477, 478, 479inner domain of instantaneous reaction

regime, 386intermediate reaction regime, 384kinetic domain of slow reaction regime,

383packed column, 394, 410, 411surface domain of instantaneous reaction

regime, 386Compressible flow, 56, 135Condensation, 127Conduction, see Heat transferConduction time scale, see Time scale,

conductionConservation of energy equation, see

Thermal energy equationConstitutive equation, 481, 483, 489, 491,

492, 496, 497, 498Contact time scale, see Time scale, contact

timeContinuity equation:

cylindrical coordinates, 487generalized notation, 482rectangular coordinates, 486spherical coordinates, 487

Continuous stirred tank reactor, seeChemical reactor, continuous stirredtank

Correlating experimental or numericaldata, 2, 415, 424, 436, 438, 448, 460

Creeping flow, see Approximation,creeping flow

Crystallization, 345Curtain coating, 128,143Curvature effects, see Approximation,

negligible curvatureCylinder:

falling needle viscometer, 117free convection mass transfer adjacent

to vertical, 281, 354heat conduction with

temperature-dependent thermaldiffusivity, 246

heat transfer for hot wire anemometer,249

heat transfer with resistance heating,200, 223

mass transfer in dissolutionof, 343, 344

mass transfer to film flow down vertical,338

steady-state heat conduction withexternal convection, 230

two-dimensional steady-state heatconduction, 225

unsteady-state axial heat conduction,228

unsteady-state heat conduction withexternal convection, 233

unsteady-state radial heat conduction,228

Cylindrical coordinates:continuity equation, 487equation of motion for porous media,

494equations of motion, 490species continuity equation, 500thermal energy equation, 497

Cylindrical tube:compressible gas flow in, 56, 135constant injection through in flow

between parallel disks, 114constant injection through in flow

between spinning parallel disks, 115countercurrent liquid-gas flow in, 123entry region flow with porous annulus,

134entry region flow, 121

518 INDEX

Cylindrical tube: (continued )falling head method for determining

permeability, 105falling needle viscometer, 117flow in hollow fiber, 101flow through porous media in, 52gravity- and pressure-driven flow in

vertical, 111impulsively initiated flow in, 92, 142mass transfer in membrane-lung

oxygenator, 287, 415mass transfer of evaporating liquid

from, 273, 337mass transfer via Taylor dispersion in,

303mass transfer with photocatalytic

reaction in, 358mass transfer with reaction for flow in,

266, 325, 340non-constant injection through for flow

between parallel disks, 115pressure-driven flow in oscillating, 122,

142pressure-driven flow in rotating, 112radial flow from porous, 133

d’Alembert’s paradox, 35Damkohler number, see Dimensionless

groups, Damkohler numberDarcy flow, see Porous media flow,

Darcy’s lawDerivative scaling, 9, 21

compressible gas flow in cylindricaltube, 59, 60, 69, 135

convective mass transfer betweenparallel membranes, 339

curtain-coating flow, 130dissolution of a spherical capsule, 308draining of liquid film, 47, 48, 68evaporation of a liquid, 273evaporative casting of polymer film, 297evaporative cooling of a liquid film, 211field-flow fractionation, 316flow between moving and stationary

plates, 110, 111flow between parallel permeable

membranes with homogeneousreaction, 339

fluid-wall aerosol flow reactor, 448

freezing of water-saturated soil overlaidby snow, 244

heat transfer with phase change, 173hollow fiber membrane reactor, 328jet flow, 96, 97, 126membrane lung oxygenator, 415membrane permeation cell, 321rusting of metal surface, 347steady-state conduction in cylinder, 230steady-state conduction in rectangular

fin, 196thermal casting of membrane, 438

Developing flow:between approaching parallel circular

plates, 72between closed-end parallel plate

membranes, 141between converging flat plates, 26, 113between diverging flat plates, 113between parallel disks with constant

radial injection, 114between parallel disks with

time-dependent radial injection, 115between permeable and impermeable

parallel flat plates, 137between spinning parallel disks with

constant radial injection, 115boundary layer over flat plate with

blowing, 120boundary layer over flat plate with

suction, 120boundary layer over flat plate, 32, 119curtain coating, 128, 143diverging nozzle, 114entry region between parallel

plates, 84entry region in a tube with porous

annulus, 134entry region in cylindrical tube, 121field-flow fractionation, 136gravity-driven draining film down

vertical wall, 45gravity-driven free surface film flow

with condensation, 127gravity-driven free surface of film over

horizontal filter, 127hydraulic ram, 116liquid jet, 94, 126permeable hollow fiber membrane, 101

INDEX 519

permeable parallel membranes, 143pressure-driven compressible in

cylindrical tube, 56, 135Diffusion, see Mass transferDiffusion time scale, see Time scale,

diffusionDimensional analysis in fluid flow:

around sphere falling at terminalvelocity, 62, 142

between permeable membranes, 143curtain-coating, 143falling head method for determining

permeability, 105hollow fiber membrane, 144oscillating cylindrical tube, 142tube with impulsively applied pressure,

142Dimensional analysis in heat transfer:

cooking a turkey, 187home freezer characterization, 250hot wire anemometer performance, 249resistance heating in electrical wire, 223slab with heat generation, 248sphere with temperature-dependent

thermal conductivity, 250steady-state convective from sphere, 248

Dimensional analysis in mass transfer:convective from sphere, 355evaporating liquid, 357film theory, 357membrane-lung oxygenator, 287, 415,

468pressure-swing adsorption, 424, 471spherical red blood cell, 332tubular photocatalytic reactor, 358

Dimensional analysis in mass transfer withchemical reaction,

fluid-wall aerosol flow reactor, 448, 475,476, 477, 478, 479

Dimensional constants, 13Dimensionless groups:

Biot number for heat transfer, 162, 513Biot number for mass transfer, 302, 513Damkohler number, 268, 513Fourier number for heat transfer, 155,

513Fourier number for mass transfer, 258,

513Froude number, 30, 513

Graetz number, 292, 513Grashof number for heat transfer, 221,

513Grashof number for mass transfer, 285,

513Lewis number, 445, 513Mach number, 61, 513Nusselt number for heat transfer, 159,

513Nusselt number for mass transfer, 292Peclet number for heat transfer, 166, 513Peclet number for mass transfer, 263,

513Prandtl number, 166, 513Rayleigh number, 184, 221, 513Reynolds number, 30, 513Schmidt number, 263, 513Sherwood number, 292, 514Thiele modulus, 263, 514

Disk:mass transfer from uniformly accessible

rotating, 312oscillating viscometer, 117rotating flow, 79rotating viscometer, 116

Dissipation of energy, see Viscousdissipation

Dissolution:cylindrical capsule, 343, 344spherical capsule, 308, 342

Drainage of liquid:falling head method for determining

permeability, 105unsteady down vertical wall, 45

Dynamic pressure, 63, 219, 283

End effects:fluid flow, 43, 68, 77, 88, 118, 133heat transfer, 146, 224, 225, 226mass transfer, 343

Energy dissipation, see Viscousdissipation; Heat transfer

Energy equation, see Thermal energyequation

Entrance effects:fluid flow, 84, 121, 134heat transfer, 234, 236, 247mass transfer, 340

520 INDEX

Entrance length, see Entrance effectsEntropy production, 126Equation of continuity, see Continuity

equationEquation of continuity for a binary

mixture:cylindrical coordinates, 500generalized notation, 484rectangular coordinates, 499spherical coordinates, 502

Equation of energy, see Thermal energyequation

Equations of motion:cylindrical coordinates, 490generalized notation, 482rectangular coordinates, 489spherical coordinates, 492

Equations of motion for porous media:cylindrical coordinates, 494generalized notation, 483rectangular coordinates, 494spherical coordinates, 495

Error estimate in scaling, see Scalinganalysis, estimated error

Evaporation:casting of polymer film, 297, 350cooling of stationary liquid film, 211,

242free convection mass transfer from

horizontal liquid layer, 357liquid in cylindrical tube, 273, 337polymeric membrane and film casting,

297, 350Example problems:

fluid dynamics, 70heat transfer, 196mass transfer, 297

Falling film, see Film flowFalling head method, 105Falling needle viscometer, see Viscometer,

falling needleFick’s law, 314, 342, 353, 356, 358, 484,

499, 500, 501, 502, 503Field-flow fractionation, 136, 316Film flow:

countercurrent liquid-gas in cylindricaltube, 123

curtain coating, 128, 143fully developed falling film in presence

of viscous gas phase, 70gravity-driven draining film down a

vertical wall, 45gravity-driven free surface film flow

with condensation, 127gravity-driven free surface of film over

horizontal filter, 127gravity-driven in channel with side

walls, 43gravity-driven liquid film over porous

media, 99, 133mass transfer to gravity-driven down

vertical cylinder, 338mass transfer to gravity-driven down

vertical plate, 269, 338Film theory, see Approximation, film

theory for heat transfer;Approximation, film theory for masstransfer

Filter, 127Flat plate:

boundary layer flow over, 32, 119boundary-layer flow over with blowing,

120boundary-layer flow over with suction,

120flow down with condensation, 127flow over filter, 127free convection heat transfer adjacent to,

218, 247free convection mass transfer adjacent

to, 354gravity-driven flow down in presence of

viscous gas, 70gravity-driven flow over with side walls,

43heat transfer to falling film flow down,

206, 236, 237mass transfer to falling film flow down,

269, 338porous media flow bounded by, 131thermal boundary layer in flow over

with suction, 241thermal boundary layer in flow over,

167, 236, 238, 239, 240Flat plates:

flow between converging, 26

INDEX 521

flow between diverging, 113flow between parallel membrane, 143flow between parallel stationary and

moving, 20, 110, 111flow between stationary and oscillating,

38flow between with fluid injection and

withdrawal, 136flow in entry region between parallel, 84flow in porous media between parallel,

132flow of stratified liquid layers between,

125heat transfer in flow between with

entrance effects, 234heat transfer in flow between with

permeable walls, 202heat transfer in flow between with

temperature-dependent viscosity, 180heat transfer in flow between with

viscous dissipation, 163heat transfer in laminar flow between,

235heat transfer in thermal boundary layer,

236mass transfer with field-flow

fractionation between parallelmembrane, 316

mass transfer with reaction in flowbetween parallel membranes, 261, 339

Fluid dynamics, see Fluid flowFluid flow:

annulus with injection and withdrawal,139

axial in a rotating tube, 112between approaching parallel circular

plates, 72between converging flat plates, 26between cylinder and piston in hydraulic

ram, 76between oscillating and stationary

parallel circular plates, 38between diverging flat plates, 113between parallel disks with constant

radial injection, 114between parallel disks with

time-dependent radial injection, 115between parallel membranes, 143

between permeable and impermeableparallel flat plates, 137

between spinning parallel disks withconstant radial injection, 115

boundary layer over a flat plate, 32, 119boundary-layer over flat plate with

blowing, 120boundary-layer over flat plate with

suction, 120closed-end parallel plate membranes,

141countercurrent liquid-gas in cylindrical

tube, 123curtain coating, 128, 143diverging nozzle, 114entry region between parallel plates, 84entry region for cylindrical tube, 121entry region in a tube with porous

annulus, 134falling needle viscometer, 117field-flow fractionation, 136fully developed falling film in presence

of viscous gas phase, 70gravity and pressure-driven in vertical

tube, 111gravity-driven cylindrical jet, 94gravity-driven draining film down a

vertical wall, 45gravity-driven free surface film flow

with condensation, 127gravity-driven free surface of film over

horizontal filter, 127gravity-driven in channel with side

walls, 43gravity-driven liquid film over porous

media, 99, 133hydraulic ram, 76, 116liquid jet, 94, 126oscillating disk viscometer, 117over falling sphere, 62, 142permeable hollow fiber membrane, 101,

144porous media bounded by flat plate, 131porous media bounded by parallel flat

plates, 132porous media in cylindrical tube, 52pressure driven in oscillating cylindrical

tube, 122

522 INDEX

Fluid flow: (continued )pressure-driven between moving and

stationary parallel plates, 20pressure-driven compressible in

cylindrical tube, 56, 135pressure-driven of two stratified liquid

layers, 125radial from porous cylindrical tube, 133rotating disk viscometer, 116rotating disk, 79rotating in annulus with end effects, 88tube with impulsively applied pressure,

92, 142Fluid particle, 480Fluid-wall aerosol flow reactor, see

Chemical reactor, fluid-wall aerosolflow

Force on a fluid particle, 480Forgiving 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, 349Fourier 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, 128cylindrical jet, 94down plate with condensation, 127gravity-driven down a vertical wall, 45over horizontal filer, 127

Froude number, see Dimensionless groups,Froude number

Gas absorption:chemical in bubble column, 362, 364,

377continuous stirred tank reactor, 390,

408, 409packed column, 394, 410, 411

Generalized notation:continuity equation, 482continuity equation for binary mixture,

484equations of motion, 482equations of motion for porous media,

483thermal energy equation, 483

Graetz number, see Dimensionless groups,Graetz number

Grashof number, see Dimensionlessgroups, Grashof number for heattransfer, Grashof number for masstransfer

Group theory, 7

Heat conduction, see Heat transferHeat convection, see Heat transferHeat transfer:

boundary layer flow between parallelplates with unheated entrance length,236

boundary-layer flow over flat plate withunheated entrance length, 239

boundary-layer flow over flat plate, 167,238, 240

boundary-layer flow over plate withspecified heat flux, 240

boundary-layer flow over plate withsuction, 241

conductive in cylinder, 225, 230conductive film theory model, 153conductive in annulus with prescribed

temperatures, 226conductive in circular fin, 227conductive in cylinder with

temperature-dependent thermaldiffusivity, 246

conductive in rectangular parallelepiped,225

conductive penetration theory model,153

conductive through plane wall, 153, 224conductive two-dimensional with end

effects, 146convective between heated parallel

plates, 236

INDEX 523

convective between parallel plates withfluid injection, 202

convective between parallel plates withtemperature-dependent viscosity, 180

convective between parallel plates withviscous dissipation, 163, 234, 235

convective in fluid-wall aerosol flowreactor, 448, 475, 476, 477, 478,479

evaporative cooling of nonflowing film,211, 242

falling film flow, 206, 236, 237free convection between parallel plates

at different temperatures, 183free convection next to heated vertical

plate with suction, 247free convection next to heated vertical

plate, 218, 247freezing of water-saturated soil 173,

243, 244home freezer performance, 250hot wire anemometer, 249melting of frozen soil, 242, 243rectangular fin, 196unsteady-state axial in solid cylinder,

228unsteady-state conductive in cooking

turkey, 187unsteady-state conductive through plane

wall with imposed temperatures, 153unsteady-state convective from solid

sphere at high Biot number, 209unsteady-state convective from solid

sphere at low Biot number, 159, 231unsteady-state convective to plane wall,

232unsteady-state convective to solid

cylinder, 233unsteady-state in membrane thermal

casting, 438, 471, 472, 473, 474unsteady-state in slab with heat

generation, 248unsteady-state radial conduction in

sphere, 248unsteady-state radial in solid cylinder,

228unsteady-state radial in spherical shell,

229

unsteady-state resistance heating inwire, 200, 223

unsteady-state to sphere withtemperature-dependent conductivity,250

Heat transfer coefficient, 159, 193, 196,209, 227, 229, 232, 233, 234, 244,246, 248, 250

Heterogeneous chemical reaction, seeChemical reaction, heterogeneous

High Reynolds number flow, seeApproximation, high Reynoldsnumber

Hollow fiber, 101, 287, 328, 349, 415Homogeneous chemical reaction, see

Chemical reaction, homogeneousHydraulic ram, 76, 116Hydrodynamic boundary layer, see

Approximation, hydrodynamicboundary layer

Hydrogen fuel production, see Chemicalreactor, fluid-wall aerosol

Identity tensor, 47, 63, 483Incompressible flow approximation, see

Approximation, incompressible flowIntegral balance:

auxiliary condition, 47, 68, 96, 102,103, 119, 174, 175, 176, 195, 213,214, 243, 244, 246, 374

conservation of energy, 174conservation of mass, 47, 299, 323, 337,

342, 344, 345, 346, 351species, 275, 298, 309, 347, 348, 374

Integral relationships:Gauss-Ostrogradskii divergence

theorem, 504Leibnitz’ formula, 504

Interphase mass transfer time scale, seeTime scale, interphase mass transfer

Isolating quantities in dimensionalanalysis, 14, 16, 436, 461, 479

Jet flow, 94, 126

Kinematic surface condition, 47, 52, 68,96, 127, 128, 129

524 INDEX

Leibnitz formula, 504Lewis number, see Dimensionless groups,

Lewis numberLie groups, 7Little oh of one, 2, 19, 145, 252Local 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, seeApproximation, creeping flow

Lubrication flow, see Approximation,lubrication flow

Mach number, see Dimensionless groups,Mach number

Macroscale scaling analysis, see Scalinganalysis, macroscale

Mass balance, see Equation of continuityfor binary mixture

Mass transfer:convective between rotating cylinders

with concentration-dependentviscosity, 352

convective from sphere, 355convective in falling film flow down a

vertical cylinder, 338convective in falling film flow down a

vertical plane, 269, 338convective in field-flow fractionation,

316convective in fluid-wall aerosol flow

reactor, 448convective in membrane-lung

oxygenator, 287, 415convective in oxygen transfer to red

blood cell, 332convective in pressure-swing gas

absorption, 424convective in Taylor dispersion of a

solute, 303convective to uniformly accessible

rotating disk, 312diffusive in crystallization from a

supersaturated solution, 345diffusive in evaporation of liquid in

tube, 273, 337

diffusive in evaporative casting ofpolymer film, 297, 350

diffusive in membrane permeation cell,321

diffusive in membrane permeation withconcentration-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 withnonconstant diffusivity, 349

diffusive in tapered pore, 306, 337diffusive to nucleated water droplet,

344, 346film theory, 253free convection from horizontal

evaporating liquid, 357free convection in evapotranspiration

from vertical cylinder, 281,354

free convection next to permeablevertical plate, 354

penetration theory, 259, 336Mass-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, 409diffusional domain of slow reaction

regime, 381, 404fast reaction regime, 372, 404, 405, 407fluid-wall aerosol flow reactor, 448, 475,

476, 477, 478, 479heterogeneous in growth of nucleated

water droplet, 346heterogeneous in rusting of metal

surface, 347heterogeneous in tube flow, 266, 325,

340heterogeneous in tubular photocatalytic

reactor, 358homogeneous in aeration of water

containing aerobic bacteria, 340homogeneous in dissolution of

cylindrical capsule, 343, 344

INDEX 525

homogeneous in dissolution of sphericalcapsule, 308, 342,

homogeneous in flow between parallelmembranes, 261, 339

homogeneous in hollow fiber membranereactor, 328, 349

homogeneous in liquid film withconcentration-dependent diffusivity,351

inner domain of instantaneous reactionregime, 373, 374, 405, 406

intermediate domain of slow reactionregime, 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, 408surface domain of instantaneous reaction

regime, 373, 405Membrane:

convective heat transfer between parallelwith flow injection, 202

flow between parallel with velocityprofile distortion, 136

flow between parallel, 143flow between permeable and parallel flat

plate, 137flow in annular region between

concentric, 139flow in closed-end parallel, 141, 143flow in hollow fiber with permeation,

101heat transfer in boundary-layer flow

over with suction, 241mass transfer in artificial lung

oxygenator, 287, 415, 467, 468mass transfer in field-flow fractionation

between parallel, 316mass transfer in hollow fiber, 328, 349mass transfer in permeation cell, 321mass transfer in swelling, 277, 349mass transfer with nonconstant

diffusivity through, 277, 349mass 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 Scalinganalysis, 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 ofmotion; Equations of motion forporous media

Momentum boundary layer, see Region ofinfluence, fluid flow

Moving boundary:dissolution of spherical capsule, 308evaporation of liquid, 273evaporative casting of polymer film, 297evaporative cooling of liquid film, 211,

242freezing of water-saturated soil, 243,

244instantaneous reaction front, 373melting of frozen soil, 173, 242, 243thermal casting of membrane, 438, 471,

472, 473, 474

Navier Stokes equation, see Equations ofmotion

Newton’s constitutive equation, 483, 489,491, 492, 493

Nomenclature, see NotationNormal stress boundary condition, see

Boundary condition, normal stressNo-slip condition, see Boundary condition,

no-slipNotation, 506Nozzle, 114Nucleation, 344, 346Nusselt 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, 111Oscillating disk viscometer, see

Viscometer, oscillating diskOscillating tube, 122, 142, 287, 415, 467,

468

Packed column, 124, 394, 410, 411Peclet number, see Dimensionless groups,

Peclet number for heat transfer, Pecletnumber for mass transfer

Penetration theory, see Approximation,penetration theory in heat transfer;Approximation, penetration theory inmass transfer

Permeation:cell for membrane characterization, 321evapotranspiration through vertical

cylinder, 281, 354hollow fiber membrane reactor, 328, 349influence on heat transfer in

boundary-layer flow over flatmembrane, 241

influence on velocity profile in flowbetween parallel membranes, 136

membrane-lung oxygenator, 287, 415,467, 468

through closed-end hollow fiber causingaxial flow, 101

through closed-end parallel membranescausing axial flow, 141, 143

through concentric membranes withaxial flow in annular region, 139

through membrane parallel to flat platewith laminar flow, 137

through membrane with nonconstantdiffusivity, 277, 349

through parallel membranes in field-flowfractionation, 316

through parallel membranes in flow withhomogeneous reaction, 261, 339

through parallel membranes withconvective heat transfer, 202

through parallel membranes withlaminar flow, 141, 143

through walls of rotating co-axialcylinders with axial flow, 352

Phase transition:freezing of water-saturated soil, 243,

244melting of frozen soil, 173, 242, 243thermally induced phase separation

process for membrane formation, 438,471, 472, 473, 474

Pi theorem:in fluid flow, 13, 62, 65, 66, 70, 109in heat transfer, 187, 192, 193, 196,

224, 248, 250, 251in mass transfer, 287, 292, 297, 335, 437

Plate, see Flat platePoint quantities, 25, 26, 37, 43, 67, 111,

156, 173, 193Porous media flow:

annulus with radial tube flow, 133between parallel flat plates, 132bounded by flat plate, 131Brinkman term, 69, 106, 483, 494, 495,cylindrical tube, 52Darcy’s law, 53, 106, 426determining soil permeability, 105gravity-driven film flow over, 99, 133liquid film flow over, 99permeability, 53, 99, 105, 106, 109,

110, 137, 426superficial velocity, 53, 55, 99, 426

Practice problems:fluid flow, 110heat transfer, 224mass transfer, 336mass transfer with chemical reaction,

399process design, 467

Prandtl number, see Dimensionless groups,Prandtl, number

Pressure-swing adsorption, 424, 468, 469,470, 471

Primary quantity, 8Process design:

chemisorption in diffusional domain ofslow reaction regime, 384

chemisorption in fast reaction regime,385

chemisorption in inner domain ofinstantaneous reaction regime, 386

INDEX 527

chemisorption in intermediate reactionregime, 385

chemisorption in kinetic domain of slowreaction regime, 383

chemisorption in surface domain ofinstantaneous reaction regime, 387

continuous stirred tank reactor, 390fluid-wall aerosol flow reactor, 448membrane formation, 438membrane-lung oxygenator, 415packed column chemisorption, 394physical absorption, 382pressure-swing absorber, 424

Quasi-stationary hypothesis, seeApproximation, quasi-stationaryhypothesis

Quasi-steady-state, see Approximation,quasi-steady-state

Rayleigh free convection, 183, 218, 247,281, 354, 357

Rayleigh number, see Dimensionlessgroups, Rayleigh number

Reaction domain:diffusional of slow reaction, 381inner of instantaneous, 373, 405, 406intermediate of slow reaction, 403kinetic of slow reaction, 380surface of instantaneous, 373, 405See also Mass transfer with chemical

reactionReaction processes, see Mass transfer with

chemical reactionReaction regime:

fast, 372instantaneous, 373intermediate, 371slow, 371See also Mass transfer with chemical

reactionReaction time scale, see Time scale,

reaction in macroscale element; Timescale, reaction in microscale element

Reactions, see Mass transfer with chemicalreaction

Rectangular coordinates:continuity equation for binary system, 499continuity equation, 486equations of motion, 489

equations of motion for porous media,494

thermal energy equation, 496Reference factor, 10Region 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 Dimensionlessgroups, Reynolds number

Rotating disk viscometer, see Viscometer,rotating disk

Rotating flow:annulus with end effects, 88between spinning parallel disks with

radial injection, 115disk viscometer, 116spinning disk, 79tube with axial flow, 112

Rusting, 347

Scale-up analysis, 2, 143, 248, 250Scaling analysis:

o(1) procedure, 8dimensional analysis procedure, 14estimated error, 3, 12, 24, 37, 94, 150,

156, 163, 193, 462macroscale element, 361, 377, 402, 403,

404, 407, 426, 449mathematical basis, 7microscale element, 361, 362, 364, 401,

426, 449

528 INDEX

Scaling derivatives, see Derivative scalingSchmidt number, see Dimensionless

groups, Schmidt numberSecondary quantity, 8Sherwood number, see Dimensionless

groups, Sherwood numberSidewall effects in fluid flow, 43, 116Sign convention:

constitutive equation, 481, 483, 489,491, 492

force on fluid particle, 480Sizing of equipment, 3, 424, 448Species continuity equation, see Equation

of continuity for a binary mixtureSphere:

conductive heat transfer at low Biotnumber, 159, 231

conductive heat transfer in at high Biotnumber, 209

convective heat transfer from, 248convective mass transfer from, 355diffusional growth of nucleating, 344,

346diffusive mass transfer from dissolving,

308, 342diffusive mass transfer into red blood

cell, 332falling at terminal velocity, 62, 142unsteady-state with

temperature-dependent conductivity,250

water treatment via aeration from, 340Spherical coordinates:

continuity equation, 487continuity equation for binary system,

502equations of motion, 492equations of motion for porous media,

495thermal energy equation, 497

Stratified flow, 125Superficial velocity, see Porous media,

superficial velocity

Tangential stress boundary condition, seeBoundary condition, tangential stress

Tangential unit vector, see Unit vector,tangential

Taylor dispersion, 303Taylor series expansion:

of concentration-dependent density, 283,355

of diffusion coefficient, 279of pressure-dependent density, 58of small dimensionless group, 15, 66of temperature-dependent density, 185,

219of temperature-dependent viscosity, 181

Terminal velocity, 62Thermal boundary layer, see Region of

influence, heat transferThermal energy equation:

cylindrical coordinates, 497generalized notation, 483rectangular coordinates, 496spherical coordinates, 497

Thermally induced phase-separationprocess for membrane formation, 438,471, 472, 473, 474

Thiele modulus, see Dimensionlessgroups, Thiele modulus

Time scale:adsorption, 431axial dispersion, 431conduction, 162, 193, 201contact, 368, 379, 431diffusion, 258, 268, 305, 368interphase mass transfer, 379observation, 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, 420pressurization, 430, 431, 434reaction in macroscale element, 263,

268, 379reaction in microscale element, 368viscous, 40

Time scaling, see Time scaleTransient phenomena, see Unsteady-stateTube, see Cylindrical tube

Uniform magnifications and contractions, 7Unsteady-state fluid flow:

between approaching parallel circularplates, 72

INDEX 529

between stationary and oscillatingparallel plates, 38

draining of film down vertical plate, 45falling head method for determining soil

permeability, 105hydraulic ram, 116impulsively initiated in tube, 142membrane-lung oxygenator, 287, 415oscillating disk viscometer, 117oscillating tube, 122, 142, 287, 415

Unsteady-state heat transfer:conductive in cooking turkey, 187conductive in slab with heat generation,

248conductive in solid cylinder, 228, 230,

233conductive in spherical shell, 229conductive in thermal casting of

membrane, 438, 471, 472, 473, 474conductive through wall, 153, 224, 232convective from solid sphere, 159, 209,

231, 250evaporative cooling of nonflowing film,

211, 242freezing of water-saturated soil, 173,

243, 244melting of frozen soil, 242, 243resistance heating in wire, 200, 223

Unsteady-state mass transfer:convective in aeration of water, 340convective in field-flow fractionation,

316convective in membrane-lung

oxygenator, 287, 415convective in pressure-swing adsorption,

424, 468, 469, 470, 471convective in Taylor dispersion, 303diffusive evaporation of liquid, 273, 337diffusive in crystallization from

supersaturated liquid, 345

diffusive in dissolution of cylindricalcapsule, 343, 344

diffusive in dissolution of sphericalcapsule, 308, 342

diffusive in evaporative polymer filmcasting, 297, 350

diffusive in growth of nucleated waterdroplet, 344, 346

diffusive in membrane permeation cell,321

diffusive in rusting of planar surface,347

diffusive in thermal casting ofmembrane, 438, 471, 472, 473, 474

diffusive through stationary film, 253,259, 336

Unsteady-state mass transfer with chemicalreaction:

macroscale element scaling, 377microscale element scaling, 364

Vector-tensor notation, 482Viscometer:

falling needle, 117oscillating disk, 117rotating disk, 116

Viscous dissipation 163, 180, 202, 235Viscous time scale, see Time scale, viscous

Wall:conduction through one-dimensional,

153, 224, 232conduction through one-dimensional

with heat generation, 248conduction through three-dimensional,

225conduction through two-dimensional,

146Wetted-wall column, see Falling flow