I-curves for transient balance, 442

ICFUN function, 300

Ideal contactors for bimolecular reaction, 717–721

Ideal flow patterns

non-idealities, 499–503

overview, 495–496

plug-backmixed model, 496–499

Ideal gas law, partial pressure from, 10

Immobilized liquid membranes (ILMs), 744

In and out flow terms

batch reactors, 91

dispersion model, 478

macroscopic balance, 88–89

reactor–separator combination, 97

solid dissolutions from walls, 38–40, 126

stirred reactors, 33

surface reactions, 208

tubular flow reactors, 41, 130

Inclined flat plates

flow over inclined and curved surfaces, 389

similarity variable, 392

Inclined surfaces, flow over. See Flow over inclined and curved surfaces

Indirect dryers, 827

Inert gas flow rate in film theory models, 689

Inhibitors in tissues, 771

Initial value problems (IVP) matrix representation, 456–457

Instantaneous reaction

asymptote in single solute diffusion, 732–734

Instantaneous reaction case in film theory models

absorption in slurries, 694–697

bimolecular reactions, 677–678, 681–684, 714–717

Instantaneous velocity in turbulent flow, 406–408

Integral balance approach for laminar boundary layers

flow over inclined and curved surfaces, 390–391

high flux analysis, 384–385

homogeneous reactions, 381–382

momentum balance, 376–379

no reaction case, 379–381

overview, 376

species mass balance, 379

Integral method in semi-infinite slab analysis, 290–292

Intercalation in li-ion batteries, 800

Interconnected cells model for backmixed assumption tests, 111

Interface flux values in film theory models, 665–666

Interface temperature

condensation

binary gas mixtures, 860–861

unmixed model, 858

vapor with non-condensible gas, 852–853

cooling towers counterflow model, 819

Interface transfer term in macroscopic balance, 89

Interfacial across charged membranes, 553–554

Interfacial area for gas bubbles, 325

Interfacial mass transfer, 6

Interfacial partial pressures in film theory models, 664

Interfacial turbulence in turbulent flow, 430

Intermedia transport effects in environmental transport, 461–462

Internal age distribution in transient balance, 442

Internal diffusion

three-phase catalytic reactions, 610

tracer response in two-phase systems, 506

Internal laminar flow. See Convective mass transfer for internal laminar flow

Interstitial diffusion, 246–248

Interstitial velocity in tracer response in two-phase systems, 506

Intrinsic volume average in multiphase systems, 177

Invariance property for bimolecular reactions, 676–677

Invariant of system in single solute diffusion, 732

Ions

across charged membranes, 554

transport of charged species, 545

Isoelectric focusing, 957

Isotherms in adsorption, 921–924

j-factor form in convective mass transport, 318

Jacobi polynomial in orthogonal collocation method, 615

Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation for solid–solid reactions, 655

Joule heating in rotating annular beds, 961

Jump model for diffusion coefficients in liquids, 237

Kedem-Katchalski model, 903, 905–907

Key dimensionless groups

convective mass transport, 313–315

laminar flow reactors, 568–572

Kidneys, hemodialyzers for, 763–766

Kinetics

diffusion coefficients in gases based on, 225–232

dispersion model, 482

electrochemical reaction

Butler-Volmer equation, 788–791

overview, 786–788

plug flow idealization, 477–478

porous catalysts

diffusion-reaction model, 589

finite difference methods, 620–621

linking with reactor models, 623

Neumann and Robin conditions, 621

reacting solids, 645–646

shrinking core model, 640

topochemical reactions, 659

volume reaction model, 645–646

King expression for convective mass transfer in turbulent flow, 428

Klinkenberg equation for fixed bed adsorption, 936–937

Knudsen diffusion

fluids in porous solids, 248–250, 252–253

reacting systems, 525

Knudsen model for gas translation, 879–881

Kremser equation for multistage cascades, 467–468

Krogh cylinder problem for diffusion-reaction model, 600

Krogh model for transport of oxygen in tissues, 82, 757–760

Kummer functions for laminar flow, 340, 343

Lagrangian in transient balance, 441

Laguerre polynomials, 448

Laminar boundary layers

bubbles and drops, 396–399

flat plate with low flux mass transfer

concentration equation, 367–368

exact and Blasius analysis, 371–376

overview, 366–367

scaling results and analogies, 369–370

velocity equations, 368–369

flow over inclined and curved surfaces

Falkner-Skan equation, 393–394

integral balance method, 390–391

overview, 388

pressure variation term, 388–390

rotating disks, 395

similarity variable, 392

spheres, 395–396

stagnation point, 394–395

high flux analysis

Blasius approach, 386–388

film model, 383–384

integral balance method, 384–385

overview, 383

integral balance approach

homogeneous reactions, 381–382

momentum balance, 376–379

no reaction case, 379–381

overview, 376

species mass balance, 379

overview, 365–366

problems, 401–402

review questions, 401

summary, 399–400

Laminar flow

convective mass transport, 312–313, 315–316

flat plates

convective mass transport, 317

mass transfer coefficient, 201

internal. See Convective mass transfer for internal laminar flow

liquid separation membranes, 899

mesoscopic models

solid dissolutions from walls, 38

tubular flow reactors, 133

non-Newtonian fluids, 513

velocity profile in, 48

Laminar flow dispersion, Taylor model for, 488–491

Laminar flow reactors

examples

axial dispersion model for turbulent cases, 580–581

channel flow, 577

heat transfer effects, 578–579

non-Newtonian fluids, 577–578

turbulent flow reactor, 579–580

key dimensionless groups, 568–572

limiting cases

overview, 572

plug flow model, 574–575

pure convection model, 572–574

mesoscopic dispersion model, 575–577

model equations, 568–572

overview, 567–568

problems, 582–584

review questions, 582

summary, 581–582

Laminar jet apparatus in bimolecular reaction, 718

Laminar regime in condensation, 847–849

Laminar sublayer velocity profiles, 414, 416

Langmuir-Hinshelwood models for shrinking core model, 640

Langmuir isotherm

adsorption, 921–924

gas separation membranes, 879

nonlinear equilibrium models, 19

Langmuir kinetics

topochemical reactions, 659

volume reaction model, 646

Laplace domain in tracer response method, 484–485

Laplace transform

moments from, 446–449

N tanks in series, 447–448

in penetration theory, 707–710

Laplacian equation

electrolytic systems electric field, 549

steady state diffusion, 58, 189–191

Laplacian operator

batch slurry adsorbers, 930

diffusion and reaction, 75

porous catalysts

diffusion-reaction model, 593

orthogonal collocation method, 615

steady state mass transfer with reaction, 70–71

transient diffusion processes, 266–267

Laplacian terms

constant-density systems, 167

convective mass transfer, 348

laminar flow in pipes, 337

Large Thiele modulus in volume reaction model, 648

Law of mass action in oxygen-hemoglobin equilibrium, 751

Le Chatelier’s principle, 923

Leibnitz rule for differentiation, 183–184

transient diffusion processes, 292

Lennard-Jones constants for diffusion coefficients in gases, 229–230

Lethal corner in oxygen transport, 759

Level I model for environmental transport, 460–461

Level II model for environmental transport, 461

Level III model for environmental transport, 461–462

Level IV model for environmental transport, 462

Leveque solution for convective

mass transfer, 349

Lewis number

fog formation, 855

wet and dry bulb temperature, 813–815

Liebnitz rule in integral balance approach, 377

Limited bulk reactions, 671

Limiting cases in laminar flow reactors

overview, 572

plug flow model, 574–575

pure convection model, 572–574

Limiting current

copper electrowinning, 796

electrochemical reaction mass transfer effects, 791–793

electrodialyzer design, 954–955

mass balance for reacting systems, 550

Limiting reactants in porous catalysts multiple species, 606

Linear driving force model for batch slurry adsorbers, 927–928

Linear equilibrium case for NTU equation, 144

Linear isotherms in adsorption, 922–923

Linearized model for multicomponent diffusivity matrix, 539

LINSPACE function, 300

Linton-Sherwood correlation for mass transfer in pipe flow, 128

Liquid enthalpy equation for cross-flow cooling towers, 825

Liquid-filled pores, 250–251

Liquid-liquid extraction

backmixed–backmixed model, 115

equilibrium stage model, 115–116

overview, 112–114

unit operations, 22–23

Liquid-liquid interface, partition constant for, 18–19

Liquid-liquid reactions in film theory models, 697–698

Liquid mixtures, evaporation rate in, 532

Liquid phase balance

condenser model, 862–864

fluid–fluid systems, 463–464

ideal flow patterns, 500–501

plug-backmixed model, 498

Liquid-phase driving force in gas–liquid interface, 215

Liquid pools, oxygen profiles in, 59–62

Liquid separation membranes

classification, 898–899

forward osmosis, 907–908

overview, 897–898

pervaporation, 908–913

problems, 915–917

review questions, 914–915

semi-permeable membranes

concentration polarization effects, 903–905

equipment-level model, 907

Kedem-Katchalski model, 905–907

osmotic pressure, 900–901

overview, 900

reverse osmosis, 901–903

summary, 913–914

Liquid side coefficients for packed bed absorbers, 328

Liquids

diffusion coefficients in. See Diffusion coefficients

evaporation

in open containers, 197–198

in ternary mixtures, 528–530

Lithium-ion (Li ion) batteries

modeling, 800–802

transfer and reaction steps, 27

Local concentration in gas permeator models, 883–884

Local permeate composition in pervaporation, 911–913

Local volume average in multiphase systems, 176–178

Log mean driving force (LMDF)

gas absorption column design, 141–142

mass exchangers, 140–141

solid dissolutions from walls, 127–128

Long cylinders, diffusion in with first-order reactions in, 72

porous catalysts, 593–594

transient diffusion processes, 278–279

Long reactors in dispersion model, 484

Low flux mass transfer

film model, 204

transient diffusion processes, 266

Low flux model

diffusion-induced convection, 193

drift correction, 195–196

steady state diffusion, 189–191

surface reactions

first-order reaction, 206–207

nonlinear reactions, 209–210

Lungs, oxygen uptake in meso-model for capillaries, 756–757

overview, 751

oxygen-hemoglobin equilibrium, 751–753

transport steps, 753–756

Macrofluid models

continuous stirred tank reactors, 109–110

macroscopic and compartmental models, 450–453

Macropores

applications and adsorbent properties, 920

catalysts, 588

Macroscopic models

batch reactors, 90–96

continuous stirred tank reactors, 106–110

development, 29

dissolved oxygen

concentration in stirred tanks, 104–106

fluid–fluid systems, 462–465

liquid–liquid extraction, 112–117

macrofluid models, 450–453

macroscopic balance, 87–88

accumulation term, 90

in and out terms from flow, 88–89

interface transfer term, 89

rate term, 90

multistage cascades, 465–468

overview, 85–87, 435–436

problems, 119–122, 471–473

reactor–separator combination, 96–101

review questions, 118–119, 471

single-phase systems, 172–175

stirred reactors, 436–438

sublimation of spherical particles, 101–104

summary, 117–118, 469–470

tanks in series models, 449

tracer data moments, 444–449

tracer experiments, 110–112

transient balance

age distribution functions, 441–442

overview, 438

pulse input, 439–441

step input, 438–439

tanks in series model, 442–444

variance-based models for partial micromixing, 453–454

Macroscopic scale

equilibrium stage model, 36

mixer-settler models, 35–36

overview, 32–33

stirred tank reactors, 33–34

sublimation of solid spheres, 34–35

Maleic anhydride production, 587

MAPLE

diffusion-reaction model, 593

Laplace transform method, 708

matrix representation, 456–457

slab solutions, 271

Marangoni effect

gas bubbles, 322

gas–liquid interface, 429–430

Margules equation

condensation unmixed model, 858

non-ideal liquids, 244–245

Mass balance

adsorption and chromatography

batch slurry adsorbers, 929

model equations, 925

batch reactors, 92–93

batch slurry adsorbers, 925, 928–929

compartmental models, 455

concentration form, 166–168

condensation, 858

condenser model, 862–864

conservation principle, 29

control volume, 37

cooling towers counterflow model, 818–819, 821

differential models, 31

diffusion in porous solids, 248, 250

dispersion model, 478

dissolved gas, 666

electrodialyzer design, 951

evaporation, 197

film flow, 354

fixed bed adsorption, 932

fluid-fluid systems, 463, 465

flux form, 154

gas permeator models, 883–886

gas-solid reaction models, 652

hemodialyzers, 764

high flux analysis, 383

instantaneous reaction case, 714

integral balance method, 376, 379, 391

laminar boundary layers, 386

liquid-liquid extraction, 114–115

macroscopic models

stirred reactors, 437

transient balance, 439–441

mass exchangers, 135, 139–141, 144

multicomponent systems, 235–236, 519

oxygen uptake in lungs, 756

plug flow, 477, 496, 498, 500

porous catalysts

linking with reactor models, 622

temperature effects, 611–612

reacting systems electric field, 550–551

reactor–separator combination, 97–98

semibatch reactors, 688–689

shrinking core model, 638–639, 642

single solute diffusion, 730, 732

slab solutions, 278

surface reactions, 208

tanks in series model, 449

tracer studies, 507

velocity profiles, 415

volume reaction model, 646

Mass basis

concentration form, 166–167

flux form equations, 153–155

overall continuity, 168–169

Mass diffusion flux in equations of mass transfer, 158

Mass exchangers

NTU and HTU representation, 143–144

overview, 134

single stream, 134–136

two streams, 136–141

Mass flow ratio in cooling towers design considerations, 817

Mass flux

Blasius approach, 386

condensation, 852, 858, 861

convective mass transfer, 349, 426

definition, 10–12

diffusion coefficients in gases, 226

film model, 383

flat plates, 366–367

laminar flow, 341–344, 349

slab solutions, 273–274, 276, 292–293

stirred reactors, 437–438

thermal diffusion, 258

tracer studies, 439, 442, 445

transient diffusion processes, 292

turbulent, 409

Mass fractions

averaged velocity, 156–159, 161

cooling towers counterflow model, 818–819

definition, 10

diffusion coefficients in gases, 226

mole fractions conversions, 46

weighted velocity, 161

Mass transfer coefficients

convective mass transfer in laminar flow

channel flows with mass transfer, 353

film flow, 356, 358

convective mass transfer in turbulent flow

channels and pipes, 421

gas–liquid interface, 428–429

van Driest model, 426

convective mass transport

film flow, 319

gas bubbles, 323

internal and external flows, 310–313

cooling towers

counterflow model, 819

design considerations, 817

copper electrowinning, 796

electrodialyzer design, 955

evaporation, 813

film model, 202

fixed bed adsorption, 935

flat plates with laminar flow, 201

gas–liquid interface two-film model, 212–217

gas–liquid reactions in film theory models

average rate of mass transfer, 711

bimolecular reaction, 719, 721

first-order reaction of dissolved gas, 664–665, 667

hemodialyzers, 765

laminar boundary layers

bubbles and drops, 397

flat plate with low flux mass transfer, 370

laminar flow in pipes, 342–343

macroscopic models

dissolved oxygen concentration in stirred tanks, 105

fluid–fluid systems, 464

interface transfer term, 89

liquid–liquid extraction, 117

reactor–separator combination, 98

sublimation of spherical particles, 102–104

mass transfer from spheres to stagnant gas, 192

mesoscopic models

ideal flow patterns, 501

mass exchangers, 138

plug-backmixed model, 497

solid dissolutions from walls, 40–41, 126–127

oxygen uptake in lungs, 754–755

porous catalysts, 590

shrinking core model no solid product, 640

solid product, 643

solid spheres sublimation, 34–35

Mass transfer correlations in pipe flow, 128

Mass transfer equations

boundary conditions, 171–172

common simplifications, 170–171

concentration form, 166–171

diffusion flux, 163–165

flux form, 153–155

frame of reference, 156–163

laminar boundary layers

flat plate with low flux mass transfer, 374

flow over inclined and curved surfaces, 393

mass fraction averaged velocity, 156–158

multiphase systems local volume averaging, 175–178

overview, 151–152

problems, 181–184

pseudo-binary diffusivity, 165–166

review questions, 181

single-phase systems, 172–175

summary, 179–181

Mass transfer overview

application examples

biomedical, 25

bioseparations, 24

electrochemical processes, 26–27

environmental, 28

metallurgy and metal winning, 25–26

overview, 20

product development and engineering, 26

reacting systems, 20–21

semiconductor and solar devices, 24–25

unit operations, 21–24

causes, 7

concentration

mass units, 9–10

mole units, 8–9

partial pressure units, 10

concentration jump at interface, 15–20

conservation principle, 29–30

continuum assumption, 7

description, 3–6

flux vector

convection flux, 12

diffusion flux, 12–15

molar and mass flux, 10–12

interfacial, 6

model development, 28–29

problems, 46–49

review questions, 45–46

summary, 44–45

Mass transfer rate

condensation vapor with non-condensible gas, 851

electrochemical reactions, 792

evaporation, 812

macroscopic models

liquid–liquid extraction, 114–115

reactor–separator combination, 99

mass exchangers, 139

spherical cap bubbles, 398

Mass transfer resistance in shrinking core model, 642–643

Mass transfer–limiting regime for surface reactions, 207

Mass transport in electrolytic systems

charge neutrality, 547–548

charged species

mobility and diffusivity, 544–545

Nernst-Planck equation, 545–547

across charged membranes, 553–556

electric field, 548–551

overview, 543–544

problems, 562–564

review questions, 561–562

summary, 560–561

transfer rate in diffusion film near electrodes, 556–559

across uncharged membrane, 551–553

Mass units in concentration, 9–10

Mass velocities for cooling

towers

cross-flow, 825

design considerations, 817

Material balance

batch slurry adsorbers, 928–929

condenser model, 861

dispersion model, 480, 483

electrodialysis, 948

film theory models, 691

fluid–fluid systems, 465

gas permeator models, 883

liquid separation membranes, 907

packed column absorbers, 691

MATHEMATICA

Laplace transform method, 485, 708

slab solutions, 271

MATLAB program

backmixed–backmixed model, 115

batch reactors, 93–96, 100

bimolecular reactions, 679

compartmental models matrix representation, 456–457

condensation

condenser model, 864

vapor with non-condensible gas, 854–855

convective mass transfer

entry region analysis, 349

film flow, 355

diffusion coefficients in gases, 230–231

evaporation of binary liquid mixtures, 532

H₂S absorption in amine solution, 717

heterogeneous reactions, 527

laminar boundary layers, 372, 374

laminar flow in pipes, 340

laminar flow reactors, 579

macrofluid models, 451–452

plug flow closure in tubular flow reactors, 131

porous catalysts temperature effects, 614

reacting solids, 648

reacting systems, 520–522

slab solutions, 272

stirred reactors, 437

temporal evolution concentration profiles in slabs, 63

tracer response method, 488

transient diffusion processes

eigenvalue computations with, 297–298

semi-infinite slab analysis, 290

variable diffusivity, 296

Matrix form

compartmental models, 456–457

evaporation of binary liquid mixtures, 531–532

Maxwell-Boltzmann distribution, 261–262

Maxwell’s equation for electrolytic system charge neutrality, 547

McAdams correlation, 849

Mean area for diffusion across cylindrical shells, 70

Mean free path for diffusion coefficients in gases, 227

Mean molecular speed for diffusion coefficients in gases, 226–227

Mean residence time

laminar flow reactors, 570, 572–573

plug flow, 477

tanks in series, 448

tracer studies, 439–441, 484, 487

Measured vs. true kinetics in low flux model surface reactions, 207

Mechanically agitated tanks, convective mass transport in, 325–327

Megascopic model for environmental transport, 459

Membranes

aeration systems, 148

electrodialysis and electrophoresis, 948–949

gas transport in. See Gas transport in membranes

liquid separation. See Liquid separation membranes

mass transport in electrolytic systems

charged, 553–556

uncharged, 551–553

for oxygen separation, 79

Mercury porosimetry catalysts, 588

Merkel equation for cooling towers, 821–823

Mesh size in cross-flow cooling towers, 826

Meso-model

capillaries, 756–757, 760

copper electrowinning, 798

Mesoscopic models

development, 29

dispersion coefficient values, 484–488, 493–495

dispersion in laminar flow reactors, 575–577

dispersion model

boundary conditions, 480–481

CHEBFUN, 482–483

first-order reaction, 481

negligible dispersion, 483–484

nonlinear reactions, 482

overview, 478–480

hemodialyzers, 763–766

ideal flow patterns

non-idealities, 499–503

overview, 495–496

plug-backmixed model, 496–499

mass exchangers

NTU and HTU representation, 143–144

overview, 134

single stream, 134–136

two streams, 136–141

overview, 37, 123–124, 475–476

plug flow idealization, 476–478

problems, 146–149, 510–515

review questions, 146, 510

segregated flow model, 491–493

solid dissolutions from walls, 38–41, 124–128

summary, 145–146, 507–509

Taylor model, 488–491

tracer response, 484–488, 503–507

tubular flow reactors, 41–43, 129–134

Metabolism modeling, 761–762

Metal oxide semiconductor (MOS) devices, 25

Metal recovery

co-transport in, 737

emulsion liquid membranes, 743–744

from waste streams, 806

Metal winning, 25–26

Metallurgy

application examples, 25–26

fluid bed reactors, 660

Methane

adsorption on activated carbon, 941

Lennard-Jones constant, 229

synthesis gas from, 890

Methane-ethane pairs, diffusivity of, 230, 232

Methanol (ME)

biodiesel production, 704

and water mixture in rate of condensation, 859–860

Methanol poisoning, ethanol for, 736

Methanol synthesis, porous catalysts for

application, 587

three-phase catalytic reactions, 608

Michaelis-Menten kinetic model for pharmacokinetics, 761–762

Microfiltration for liquid separation membranes, 898–899

Micromixed reactors with complete backmixing, 109

Micromixing, partial, 453–454

Microphase

absorption in slurries, 693–694

description, 453

Micropores

applications and adsorbent properties, 920

catalysts, 588

Minimum liquid flow rate in mass exchangers, 141

Mixed condenser model, 863

Mixer-settler models, 35–36

Mixing assumption in matrix representation, 456

Mixing cell model for fluid–fluid systems, 465

Mixing concentration. See Cup-mixed concentration

Mixing model for stirred tank reactor, 33–34

Mixture average velocity, 157

Mixture density, 9–10

Mixture velocity, weighting factors for, 183

Mobility

electrical, 550

electrodialysis, 946

electrophoretic separation devices, 957–959

frictional interpretation, 234

transport of charged species, 544–546

Model development

basic methodology, 28–29

compartmental models, 43–44

conservation principle, 29–30

differential models, 30–32

macroscopic scale, 32–36

mesoscopic, 37–43

Model equations

adsorption in batch slurry adsorbers, 925

gas–liquid reactions in film theory, 685

laminar flow reactors, 568–572

porous catalysts multiple species, 606

reactive membranes, 729–730

Moisture. See Humidification and drying

Molar flux, definition, 10–12

Molar rate

dissolved oxygen concentration in stirred tanks, 105

gas in film theory models, 692

gas permeator model countercurrent flow, 887

macroscopic balance, 88

mass exchangers, 136

solid dissolutions from walls, 125–126

Molar units in macroscopic balance, 88

Molar volume

diffusion coefficients in liquids, 241–242

reacting solids, 651

Mole balance

condenser model, 862–863

solid dissolutions from walls, 127

Mole basis

concentration form, 168–169

flux form equations, 153–155

Mole fraction

in averaged velocity for equations of mass transfer, 158–159

condensation

binary gas mixtures, 857–858

condenser model, 864

ternary systems, 865

unmixed model, 858

vapor with non-condensible gas, 850

definition, 9

in fog formation, 856–857

gas permeator models, 883–885

gas–liquid interface two-film model, 215–216

gradients

equations of mass transfer, 159–160

multicomponent diffusivity matrix, 537

liquid separation membranes

osmotic pressure, 901

pervaporation, 912

semi-permeable membranes, 903

mass fractions conversions, 46

in UMD, 196–197

vapor in evaporation, 812

Mole ratio

liquid–liquid extraction, 113–116

mass exchangers, 138

Mole units in concentration, 8–9

Molecular sieves, 921

Molecular speed in Maxwell-Boltzmann distribution, 261–262

Molecular weight

definition, 9–10

diffusion coefficients in gases, 228–229

liquid separation membranes, 898

mass transfer equations, 153

spherical particles, 101

water, 8

Molten salts, immobilized liquid membranes for, 744

Moments

N tanks in series, 447–448

tracer data analysis, 444–449

tracer response, 485–487, 506

Momentum balance in laminar boundary layers

high flux analysis, 385

integral balance approach, 376–379

Momentum boundary in layer film model, 200–201

Momentum diffusivity in convective mass transfer in turbulent flow

constant wall flux, 418

van Driest model, 426

Momentum equation for flat plate with low flux mass transfer, 371–372

Momentum integral for high flux analysis, 384–385

Momentum transfer

convective mass transport

channels and pipes, 422–423

film flow, 318–319

flat plate with low flux mass transfer, 367, 369–370

laminar boundary layers high flux analysis, 387

Morton number for spherical cap bubbles, 397

Moving reference frame in Taylor model, 490

Mujumdar correlation for drying, 831–832

Multicomponent systems

constitutive model for, 518–520

diffusion coefficients in gases, 235–236

heterogeneous reactions, 525–527

multicomponent diffusivity matrix, 535–539

non-reacting systems, 528–534

overview, 519

problems, 541–542

reacting system, 520–525

review questions, 540–541

summary, 539–540

Multiphase systems, local volume averaging in, 175–178

Multiple reactions in batch reactors, 92

Multiple species in porous catalysts, 605–607

Multistage cascades

equilibrium model, 466–468

overview, 465–466

Murphree vapor efficiency, 514

Myoglobin, oxygen binding to, 753

N tanks in series, moments for, 447–448

Nanofiltration in liquid separation membranes, 898–899

Natural draft cooling towers, 816

Navier-Stokes equation, 409–410

Near wall region in turbulent flow

constant wall flux, 419–420

velocity profiles, 414

Necrosis, 758

Negligible dispersion, 483–484

Nernst equation

electrochemical reactions, 788

thermodynamic considerations, 781–786

Nernst-Planck equation

electrodialyzer design, 951

mass transport in electrolytic systems

across charged membranes, 554–555

electric field, 548

electrolyte transport across uncharged membrane, 551

transfer rate in diffusion film near electrodes, 557–558

transport of charged species, 545–547

Net efflux term in differential models, 31

Net mole efflux in flux form equations, 153–154

Neumann boundary conditions

description, 171

laminar flow reactors, 571

oxygen profiles in pools of liquid, 60

porous catalysts, 621

reacting systems, 523

Neumann problem in drying

falling rate period, 836–838

Newton-Raphson method for orthogonal collocation, 616

Newtonian fluid

channel flow reactors, 577

turbulent flow reactors, 580

Newtonian laminar flow in Taylor model, 489

Newton’s law of viscosity

laminar boundary layers wall stress, 373

turbulent flow closure models, 411

Nickel recovery, emulsion liquid membranes for, 743–744

Nitric oxide (NO)

production, 769

selective catalytic reduction of, 209–210

Nitrogen (N)

diffusion volume, 231

Henry’s law constant, 16

No reaction case for laminar boundary layers, 379–381

Non-condensible gas, vapor with, 850–855

Non-homogeneous problems in transient diffusion processes, 283–285

Non-hygroscopic solids, 829

Non-ideal liquids, diffusion in, 243–245

Non-idealities in flow patterns, 499–503

Non-isothermal batch reactor modeling, 120

Non-Newtonian fluids

flow reactors, 577–578

laminar flow, 513

pipe flow, 493

Non-reacting systems, evaporation in

binary liquid mixtures, 531–533

ternary liquid mixtures, 528–530

Nonadiabatic dryers, 827

Nonlinear effects in gas separation membranes, 878–879

Nonlinear equilibrium models, 19–20

Nonlinear kinetics in porous catalyst finite difference methods, 620–621

Nonlinear reactions

dispersion model, 482

low flux model surface reactions, 209–210

NRTL three-constant model for non-ideal liquids, 244

NTU (number of transfer units)

cooling towers, 823–824

mass exchangers, 143–144

Nusselt model for condensation, 847–849

Nutrient transport, 28

ODE45 solver

batch reactors, 93–96

evaporation of binary liquid mixtures, 531

flat plate with low flux mass transfer, 372

plug flow closure in tubular flow reactors, 131

porous catalysts, linking with reactor models, 622–624

Ohmic potential, 797

Ohm’s law

electrodialyzer design, 951–953

electrolytic systems electric field, 549

Onda correlations for packed bed absorbers, 328–329

One-compartment model for pharmacokinetics, 761

1-D (differential) balances. See Differential (1-D) balances

1-D problems, 266–267, 300–304

One-term approximation

concentration in laminar flow in pipes, 341

transient diffusion processes in cylinders and spheres, 280–282

Onsager reciprocal relations, 536

Open containers, liquid evaporating in, 197–198

Open-open boundary conditions in tracer response method, 486–487

Operating current density in copper electrowinning, 795–796

Organs, drug accumulation in, 762–763

Orthogonal collocation method

batch slurry adsorbers, 929

porous catalysts, 615–617

Orthogonality property of F function for laminar flow in pipes, 340

Oseen model for solid spheres, 321

Osmosis in liquid separation membranes, 907–908

Osmotic diffusion, 236

Osmotic pressure in semi-permeable membranes, 900–901

Osmotic transport in reactive membrane equilibrium model, 742

Overall continuity in mass basis, 168

Overall mass balance

batch reactors, 93

batch slurry adsorbers, 929

concentration form, 168

condensation of binary gas mixture, 858

fluid-fluid systems, 465

liquid–liquid extraction, 115

mass exchangers, 140–141

trace methods, 439, 507

Overall rate constant in shrinking core model

no solid product, 637–638

solid product, 642

Overall reaction, definitions, 777–779

Overpotential form of Butler-Volmer equation, 789–791

Oxidation

CO, 541

electrochemical reactions, 782, 784–788

li-ion batteries, 801

partial, 891

porous catalysts, 608

reduction, 776–777

silicon, 207–209

Oxygen

binding, in Hill’s equation, 768

concentration in water, 17

counter-transport, 736

diffusion volume, 231–232

dissolved concentration in stirred tanks, 104–106

Henry’s law constant, 16

mass transfer in human body, 750

molar volume in liquid, 241

in pools of liquid, 59–62

separation, 79, 256

silicon oxidation, 207–209

transient diffusion of, 275

transport in tissues, 82, 757–760

transport with fixed-site

carrier membranes, 744–745

uptake in lungs

meso-model for capillaries, 756–757

overview, 751

oxygen-hemoglobin equilibrium, 751–753

transport steps, 753–756

Oxygen-hemoglobin equilibrium, 751–753

Ozone, Henry’s law constant for, 16

p-substitution in diffusion-reaction model nth-order reaction, 603

Packed beds

dispersion coefficient, 494–495

gas–liquid mass transfer in, 327–329

Packed column absorbers

coupling with reactor models, 691–692

mass exchangers, 136–137

Palladium gas separation membranes, 873–874

Parallel plate electrolyzer modeling, 807

Partial micromixing, 453–454

Partial oxidation, 891

Partial pressure difference in gas separation membranes, 875, 877

Partial pressure units in concentration, 10

Partial solute rejection for semi-permeable membranes, 902–903

Particle-level model for batch slurry adsorbers, 925–926

Particle-Pellet model for gas–solid reaction, 653

Particle size effect for absorption in slurries, 692–693

Partition coefficient

gas separation membranes, 875

liquid–liquid extraction, 114

liquid–liquid interface, 18–19

mass exchangers, 138

Patient-dialyzer system in hemodialyzers, 765–766

PDEFUN function, 300

PDEPE solver

bimolecular reaction, 713–714

channel flow reactors, 577

convective mass transfer, 358–359

drying falling rate period, 837–838

laminar flow reactors

boundary conditions, 571

heat transfer effects, 579

non-Newtonian fluid flow reactors, 578

temporal evolution

concentration profiles in slabs, 63

tracer response method, 488

transient diffusion processes, 299–304

volume reaction model, 648

PDEVAL function, 301

Peclet number

defined, 314

film model, 383–384

laminar flow in pipes, 338–339

laminar flow reactors, 569–570, 576

mesoscopic models

dispersion coefficient, 494–495

dispersion model, 479

ideal flow patterns, 501–502

pipe flow, 493

tracer response method, 484, 486

non-Newtonian fluid flow reactors, 493, 578

solid spheres, 320

sublimation in pipes, 129

transport rate in presence of convection, 66–67

PEG (polyethylene glycol) in water splitting, 950

Pellets in gas–solid reaction models, 652–653

Penetration depth

doping silicon with phosphorus, 294

transient diffusion process integral method, 292

Penetration theory approach

film flow, 319–320

gas–liquid reactions. See Gas-liquid reactions in penetration theory

transient diffusion processes, 294–295

Performance equations

mass exchangers, 135–136

plug flow idealization, 477

stirred reactors, 437

Permeability

gas transport in membranes

gas separation membranes, 874–876

gas translation model, 879, 881

heterogeneous media, 254

oxygen uptake in lungs, 754–755

polymeric membranes, 256

semi-permeable membranes, 905

Permeance transport rate for gas separation membranes, 876–877

Permeate flux model for pervaporation, 910–911

Permittivity in electric field, 547

Pervaporation

application examples, 909–910

local permeate composition, 911–913

overview, 908–909

permeate flux model, 910–911

pH

electrophoresis principle, 956–957

oxygen-hemoglobin equilibrium, 753

Pharmacokinetics

bolus injection, 471

compartmental models, 457–459, 760–763

Philpot design, 957–959

Phosphorus

doping silicon with, 293

in silicon, interstitial diffusion, 248

Physiologically based

pharmocokinetic (PBPK) models, 762–763

Pipes and pipe flow

convective mass transfer in turbulent flow

constant wall flux, 418–421

momentum transfer analogy, 422–423

overview, 417–418

Stanton number for boundary layers, 421–422

Stanton number for pipe flows, 423–425

convective mass transport, 312–316

internal laminar flow

concentration, wall mass flux, and Sherwood number, 341–344

constant wall concentration, 339–341

dimensionless form, 337–339

overview, 336–337

mass transfer correlations in solid dissolutions from walls, 128

non-Newtonian liquid, 493

segregated flow model, 491–493

sublimation in, 129

tubular flow reactors, 133

Plasma in three-compartmental model, 772

Plate and frame design in electrodialysis, 946

Plug-backmixed model, 496–499

Plug flow

fixed bed adsorption, 933

laminar flow reactors

axial dispersion model for turbulent cases, 581

limiting cases, 574–575

mesoscopic models

closure in tubular flow reactors, 131

idealization, 476–478

solid dissolutions from walls, 38

tracer response method, 487

tubular flow reactors, 42

tracer studies, 440

Plug flow reactors for porous catalysts, 622

Poisson’s equation for electrolytic system charge neutrality, 547

Polarization effects for semi-permeable membranes, 903–905

Pollutants in environmental transport, 460

Polymeric membranes

diffusion, 256–257

gas separation, 873–874

hydrogen fuel cells, 798

Polystyrene, 949

Pools of liquid, oxygen profiles in, 59–62

Pores

batch slurry adsorbers, 925–926

fluids in porous solids

liquid-filled, 250–251

size, 248–250

gas separation membranes, 873

liquid separation membranes, 898–899

Porosity

gas–solid reaction models, 651–653

volume reaction model, 644

Porous catalysts

diffusion-reaction model

first-order reaction, 589–598

nth-order reaction, 603–605

overview, 588–589

zero-order reaction, 599–603

finite difference methods

central difference equations, 618–619

Neumann and Robin conditions, 621

nonlinear kinetics, 620–621

overview, 617–618

zero-order reaction, 619–620

fluids in porous solids, 251–254

linking with reactor models, 622–625

multiple species, 605–607

orthogonal collocation method, 615–617

overview, 585–586

problems, 628–633

properties and applications, 586–588

review questions, 627

summary, 625–627

temperature effects

dimensionless boundary conditions, 612–613

dimensionless representation, 611–612

equations for heat and mass transport, 610–611

temperature gradients, 613–614

three-phase catalytic reactions, 607–610

Potential field in electrolytic systems, 549, 558

Potentiostatic mode for electrochemical batch reactors, 806–807

Power consumption in mechanically agitated tanks, 326

Power input for gas bubbles, 324

Power-law kinetics

dispersion model, 482

plug flow idealization, 477–478

porous catalysts, linking with reactor models, 623

volume reaction model, 645–646

Power law model

non-Newtonian fluids flow reactors, 577–578

variable diffusivity, 296

Prandtl analogy for laminar boundary layers

flat plate with low flux mass transfer, 370, 375

flow over inclined and curved surfaces, 391, 395

Prandtl model for eddy viscosity, 412–413, 425

Prandtl number

channels and pipes, 422–423

evaporation, 813

in fog formation, 855

Prandtl universal velocity profile, 580

Pre-exponential factor in gas translation model, 880–881

Pressure

flow over inclined and curved surfaces, 388–390

gas permeator models, 882, 888–889

liquid separation membranes, 901

local permeate composition, 911–912

partial pressure units, 10

semi-permeable membranes, 907

in small pores, 250

Pressure diffusion, 258

Primary variables, definitions, 781

Probability factor in gas translation model, 880

Product counter-diffusion for surface reactions, 210–212

Product development and engineering application examples, 26

Product removal, batch reactor with, 99–100

Product solution method for transient diffusion processes, 285–287

Production rate

batch reactors, 91–92

reacting systems, 524–525

volume reaction model, 645–646

Proteins

electrophoresis principle, 956

electrophoretic separation devices, 958–960

Proton exchange membrane (PEM) in fuel cells, 779

Pseudo-binary diffusion coefficient in evaporation of liquids, 529

Pseudo-binary diffusivity, 165–166

Pseudo-first-order reactions

bimolecular reactions, 677

penetration theory approach, 706–707

reactive membrane single solute diffusion, 735–736

Pseudo-homogeneous models for fixed bed adsorption, 932

Pseudo-steady state hypothesis for volume reaction model, 647

Pulse injection in tracer response method, 487

Pulse input

fixed bed adsorption, 933

tracer studies, 439–441

Pulse response

integral method, 292–293

tracer studies, 441

Pure convection model for laminar flow reactors, 572–575, 578

Pure vapor in condensation, 846–850

Quadratic profile assumption for batch slurry adsorbers, 927

Radial averaging in laminar flow reactors, 575

Radial concentration gradient for tubular flow reactors, 132

Radial diffusion

in cylindrical geometry, 191

laminar flow reactors, 570

steady state, 68–70

Taylor model, 488–489

tubular flow reactors, 132–134

Radial direction in mesoscopic models, 37

Radial position in laminar flow reactors, 569

Radius dependency in shrinking core model, 640

Random flow field in fully turbulent flow, 405

RANS (Reynolds-averaged Navier-Stokes) equation, 410

Ranz-Marshall equation, 640

Raoult’s law and model

condensation unmixed model, 858

vapor–liquid interface, 17–18

Rate constants

drying falling rate period, 835

Laplace transform method, 708–709

porous catalysts, 597–599

shrinking core model

no solid product, 637–638

solid product, 642

Rate equations for bimolecular reactions, 682

Rate model for pharmacokinetics, 761–762

Rate term

bimolecular reaction, 712

macroscopic balance, 90

Reacting solids

gas–solid reaction models, 651–654

overview, 635–636

problems, 658–660

review questions, 658

shrinking core model

no solid product, 637–640

overview, 636

solid product, 641–644

solid–solid reactions, 654–656

summary, 656–658

volume reaction model

concentration profile for gas and solid, 646

first-order reaction, 646–649

kinetic model, 645–646

overview, 644–645

zero-order reaction, 649–651

Reacting systems

examples, 20–21

flat plate with low flux mass transfer, 375–376

multicomponent systems, 520–525

Reaction contribution in time-averaged equation, 410–411

Reaction kinetics in matrix representation, 456

Reaction-limiting regime in surface reactions, 207

Reaction planes

absorption in slurries, 694–697

bimolecular reactions, 678, 681–682

shrinking core model solid product, 642

Reaction rate in porous catalysts

diffusion-reaction model, 589

linking with reactor models, 622

Reaction time in laminar flow reactors, 570

Reactive membranes and facilitated transport

application examples

emulsion liquid membranes, 743–744

fixed-site carrier membranes, 744–745

immobilized liquid membranes, 744

overview, 742–743

co- and counter-transport

illustrative results, 741–742

overview, 736–739

equilibrium model, 739–742

overview, 727–728

problems, 747–748

review questions, 746

single solute diffusion

dimensionless representation, 730–732

model equations, 729–730

overview, 729

summary, 745–746

Reactor coupled with membrane separators, 890–891

Reactor models

gas–liquid reactions in film theory models, 688–692

porous catalysts, 622–625

Reactor performance for tanks in series models, 449

Reactor types in three-phase catalytic reactions, 608–609

Redlich-Kister equation for non-ideal liquids, 245

Reduction in oxidation, 777

Reference temperature for cooling towers enthalpy balance equations, 820

Regions for constant wall flux, 419

Relative humidity, 815

Relative velocity

defined, 157

diffusion coefficients in liquids, 238–239

gas bubbles, 323–325

multicomponent diffusion, 235

multiphase systems, 176–178

Resistances

batch slurry adsorbers, 928

diffusion coefficients in liquids, 239–240

electrochemical reactions, 792

electrodialyzer design, 952–953

gas–liquid interface in two-film model, 216–217

hemodialyzers, 765

heterogeneous media, 254

mass exchangers, 136, 138

oxygen uptake in lungs, 754–756

porous catalysts diffusion-reaction model, 597

shrinking core model, 642–643

surface reactions, 206–207

Response curves

tracer response method, 485–487

transient balance, 440, 443–444

Retention time

chromatography, 938

fixed bed adsorption, 933–935

Hannig design, 959–960

Reverse diffusion, 236

Reverse osmosis

salt water purification, 898

semi-permeable membranes, 901–902

Reverse reaction rate in Butler-Volmer equation, 788

Reynolds analogy

channels and pipes, 422–423

flat plate with low flux mass transfer, 369

Reynolds number

channel flows with mass transfer, 351

condensation, 847–849

convective mass transfer in turbulent flow

channels and pipes, 423–424

constant wall flux, 419

gas–liquid interface, 427–428

transition criteria, 404–405

velocity profiles, 416–417

convective mass transport

dimensionless representation, 313–314

film flow, 319

gas bubbles, 322

gas–liquid mass transfer in packed bed absorbers, 328

mechanically agitated tanks, 326

packed bed reactors, 333

solid spheres, 321

dispersion coefficient, 495

drying constant rate period, 832

fixed bed adsorption, 937

laminar flow in pipes, 338

mass transfer coefficient for sublimation of spherical particles, 102–103

mass transfer correlations in pipe flow, 128

spherical cap bubbles, 397

sublimation in pipes, 129

Reynolds stresses in time-averaged equation, 409–410

Richardson-Zaki correction for gas bubbles, 325

Rigid bubbles, 396–397

Robin boundary conditions

batch slurry adsorbers, 926

description, 171–172

in electrolytic systems, mass balance for reacting systems, 550

film theory models

bimolecular reactions, 674

first-order reaction of dissolved gas, 664

simultaneous absorption of two gases, 685

heterogeneous reactions, 526

laminar flow in pipes, 344–348

laminar flow reactors, 571

porous catalysts diffusion-reaction model, 590, 593

finite difference methods, 621

orthogonal collocation method, 616

slab solutions, 276–278

Rotary dryers, 827

Rotating annular beds, 960–962

Rotating disks, flow over inclined and curved surfaces, 395

Runge-Kutta method for batch reactors, 93

Salt

decomposition, 785–786

electrolysis, 794–795

osmotic transport in membranes, 742

separation

electrodialyzer design, 952–953

voltage balance, 953–954

solutions

electrodialysis, 948

freezing point, 915

splitting, 950

Salt water purification, 898

Scale-up aspects in fixed bed adsorption, 937

Scaled rate of reaction in porous catalyst temperature effects, 611

Scaling results and analogies in flat plate with low flux mass transfer, 369–370

Schmidt number

convective mass transfer in turbulent flow

channels and pipes, 423–425

closure models, 411–412

constant wall flux, 418

velocity profiles, 416–417

convective mass transport

dimensionless representation, 313–314

solid spheres, 320

evaporation, 813

film model, 200

fog formation, 855

laminar boundary layers

flat plate with low flux mass transfer, 370, 374–375

high flux analysis, 387

laminar flow in pipes, 338

laminar flow reactors, 580

mass transfer coefficient in sublimation of spherical particles, 102

mass transfer correlations in pipe flow, 128

sublimation in pipes, 129

Second-order reactions

bimolecular reactions, 677–679

continuous stirred tank reactors, 108–110

low flux model surface reactions, 209

macrofluid models, 451

plug flow idealization, 478

porous catalysts, linking with reactor models, 624

tubular flow reactors, 131

Segregated model

continuous stirred tank reactors, 109

laminar flow, 492–493

Selectivity

gas permeator models, 882

gas separation membranes, 877

liquid separation membranes, 899

pervaporation, 911

Self-diffusion coefficient in gases, 227

Semi-infinite slab analysis in transient diffusion processes

constant surface concentration, 288–291

integral method, 290–292

overview, 287

pulse response, 292–293

Semi-permeable membranes

concentration polarization effects, 903–905

equipment-level model, 907

Kedem-Katchalski model, 905–907

osmotic pressure, 900–901

overview, 900

reverse osmosis, 901–903

Semibatch reactors, 688–691

Semiconductor devices

diffusion-reaction analysis, 24–25

power law model, 296

Sensible heat correction in vapor with non-condensible gas, 853

Separation

ethanol–water mixture in liquid separation membranes, 909

multistage cascades, 465

VOCs, 909–910

Series coefficient slab solutions, Dirichlet case, 271–272

Series reaction simulation for batch reactors, 95

Series solution in slab solutions, Dirichlet case, 269–271

Shape normalization in porous catalysts, 595–596, 605

Shear stress

total, 433

turbulent flow closure models, 412

Shell balance approach for transient diffusion in spherical coordinates, 76

Sherwood number

absorption in slurries, 696

channel flow, 316

convective mass transfer

channel flows with mass transfer, 352–353

entry region analysis, 349

convective mass transport

dimensionless representation, 313–314

flow in flat plate, 317

copper electrowinning, 796

laminar boundary layers

bubbles and drops, 397

flat plate with low flux mass transfer, 374–375

high flux analysis, 388

integral balance approach, 381

laminar flow in pipes, 315, 341–344, 346–347

mass transfer coefficient for sublimation of spherical particles, 102–103

mass transfer correlations in pipe flow, 128

oxygen uptake in lungs, 754

turbulent flow, 315

Shrinkage rate of subliming spheres, 103–104

Shrinking core model for reacting solids

no solid product, 637–640

overview, 636

solid product, 641–644

Sievert’s Law for gas separation membranes, 877–878

Silica gel, 920

Silicon

doping with phosphorus, 293

interstitial diffusion, 247–248

Silicon oxide (Sio₂) in surface reactions, 207–209

Similarity variable for flow over inclined and curved surfaces, 392

Simultaneous absorption of two gases, 684–688

Single flowing phase in tracer response in two-phase systems, 504–506

Single-phase systems, 172–175

Single-pore gas diffusion, 248–250

Single pore model for reacting solids, 652–653

Single solute diffusion

dimensionless representation, 730–732

instantaneous reaction asymptote, 732–734

invariant of system, 732

model equations, 729–730

overview, 729

pseudo-first-order reaction asymptote, 735–736

Single stage in multistage cascades, 467

Single stream in mass exchangers, 134–136

Slabs

concentration profiles in temporal evolution, 63–64

D-D problem in transient diffusion processes, 283–285

Dirichlet case

average concentration, 274–275

dimensionless representation, 268–269

illustrative results, 272–274

overview, 267–268

series coefficient, 271–272

series solution, 269–271

Slabs

porous catalysts

diffusion-reaction model, 591

external transport effects, 592–593

temperature effects, 610–611

Robin condition, 276–278

steady state diffusion across, 52–55

steady state diffusion with reaction in, 56–62

transient diffusion in, 62, 287–293

Slow reactions

dispersion model, 483

high flux model surface reactions, 211

Slurries

absorption in film theory models, 692–697

three-phase catalytic reactions, 608

Slurry adsorbers. See Batch slurry adsorbers

Sodium chloride (NaCl)

decomposition, 785–786

electrolysis, 793–795

osmotic transport in membranes, 742

Sodium hydroxide (NaOH)

H₂S absorption in, 683–684

production, 775

Soil, fugacity of pollutants in, 460

Solar devices, 24–25

Solid concentration profiles in volume reaction model, 646

Solid density of catalysts, 587

Solid dissolutions at walls

film flow, 354–356

mesoscopic models, 38–41, 124–128

Solid reactant mass balance in shrinking core model, 638–639

Solid spheres

convective mass transport, 320–321

sublimation of, 34–35

Solid to liquid in film flow, 318–319

Solids

drying, 828–829

reacting. See Reacting solids

Solid–solid diffusion

interstitial diffusion, 246–248

vacancy diffusion, 246

Solid–solid reactions in reacting solids, 654–656

Solutes

in electrophoresis principle, 956–957

transport rate in semi-permeable membranes, 906

Solvents in multicomponent

diffusivity matrix, 537

Species balance equations

gas permeator models, 889

reacting systems, 524–525

single-phase systems, 174

tracer studies, 438

Species mass balance

compartmental models, 455

gases, 236

integral balance approach, 379

multicomponent systems, 519

plug-backmixed model, 496

porous catalysts

linking with reactor models, 622

temperature effects, 612

reacting systems, 550

tanks in series models, 449

two-phase systems, 500

Species mole fraction

condensation of vapor with non-condensible gas, 850

mass exchangers, 136

reacting systems, 520

Species velocity in mass transfer equations, 156–157

Speed, molecular, in Maxwell-Boltzmann distribution, 261–262

Spheres

bimolecular reaction, 720

convective mass transport, 320–321

dispersion coefficient, 495

flow over inclined and curved surfaces, 395

mass transfer to stagnant gas, 191–192

porous catalysts diffusion-reaction model, 594–595

sublimation of, 34–35

transient diffusion, 279–280, 303–304

two-point collocation, 616–617

Spherical cap bubbles in laminar boundary layers, 397–399

Spherical coordinates

diffusion and reaction, 75–76

divergence operator, 155

Laplacian of concentration, 190

steady state diffusion across

spherical shells, 73–74

transient diffusion in, 76–77

Spherical particles, sublimation of, 101–104

Spherical shells, steady state

diffusion across, 73–74

Stage efficiency

fluid–fluid systems, 465

liquid–liquid extraction, 116–117

Stages in condenser model, 861

Stagnant gas in mass transfer from spheres, 191–192

Stagnation point in flow over inclined and curved surfaces, 394–395

Stanton number

convective mass transfer in turbulent flow

boundary layers, 421–422

pipe flows, 423–425

van Driest model, 426

convective mass transport dimensionless representation, 314

flat plate with low flux mass transfer, 369

Starling equation for semi-permeable membranes, 902–903

Staverman constant for semi-permeable membranes, 903

Steady-on-average flows in turbulent flow, 405–406

Steady state case

porous catalysts diffusion-reaction model, 588

radial diffusion, 68–70

Steady state diffusion, 186

combined flux equation, 187

determinacy condition, 188–189

diffusion-induced convection, 187–188

low flux model, 189–191

radial, 68–70

with reaction in slabs, 56–62

across slabs, 52–55

across spherical shells, 73–74

Steady state mass transfer with reaction, 70–72

Stefan cells, 197

Stefan-Maxwell model

binary mixture with UMD, 237

condensation in ternary systems, 865–866

constitutive model for multicomponent transport, 519–520

diffusion coefficients in gases

frictional interpretation, 234

multicomponent diffusion, 235–236

electrodialyzer design, 955

equimolar counter-diffusion, 533–534

evaporation

binary liquid mixtures, 531

ternary liquid mixtures, 528–530

heterogeneous reactions, 526–527

multicomponent diffusivity matrix, 535–538

reacting systems, 520, 525

Step input in transient balance, 438–439

Steric partition constant for fluids in porous solids, 251

Stimulus-response studies for transient balance, 438

Stirred cells in bimolecular reaction, 720–721

Stirred reactors

backmixing assumption, 436–438

dissolved oxygen concentration in, 104–106

mixing model, 33–34

Stochastic nature in turbulent flow, 405–406

Stokes-Einstein model

diffusion coefficients in liquids, 239–240

Wilke-Chang extension, 241

Stokes law

convective mass transport in solid spheres, 321

diffusion coefficients in liquids, 240

Stream function for flat plate with low flux mass transfer, 371–372

Stress

total shear, 433

turbulent flow closure models, 412–413

turbulent flow reactors, 580

walls, 373

Stress discontinuity in turbulent flow gas–liquid interface, 429

Stripping factors in multistage cascades, 467–468

Structural changes in gas–solid reaction models, 651–654

Subcooling in condensation, 849

Sublimation

in pipes, 129

solid spheres

mass transfer coefficient, 34–35

rate of shrinkage, 103–104

spherical particles, 101–104

Substitution diffusion, 247

Sucking factor in laminar boundary layer high flux analysis, 387

Sulfur (S)

diffusion volume, 231

from hydrogen sulfide bearing gas, 702–703

Sulfur compounds, removal by porous catalysts, 630

Sulfur dioxide (SO₂)

Henry’s law constant, 16

in water, 20

Sulfuric acid (H₂SO₄) manufacture, 586

Superheat in condensation, 849

Supersaturation in fog formation, 855–857

Supporting electrolytes

copper electrowinning, 796

mass transfer effects, 793

Surface concentration

batch slurry adsorbers, 930

electrochemical reaction voltage balance, 797

porous catalysts

diffusion-reaction model, 600–601

finite difference methods, 621

semi-infinite slab analysis, 288–290

Surface contaminants in bubbles and drops, 397

Surface flux

diffusion-reaction model nth-order reaction, 603–604

transient diffusion processes

cylinders and spheres, 280

integral method, 292

semi-infinite slab analysis, 290

Surface reactions

electrochemical processes, 781

high flux model with product counter-diffusion, 210–212

low flux model

first-order reaction, 206–207

nonlinear reactions, 209–210

overview, 206

shrinking core model

no solid product, 638–640

solid product, 642

Surface tension in turbulent flow

at gas–liquid interface, 428–429

Surface term in differential models, 31

Surface velocity in film flow, 320

Swarms, bubble, 322

Swelling in polymeric membranes, 256

Symmetry factor in Butler-Volmer equation, 788

Synthesis gas, 890–891

System velocity in presence of diffusion, 160–161

Tafel equation

Butler-Volmer equation, 791

copper electrowinning, 796

mass balance for reacting systems, 550

Tanks in series model

ideal flow patterns, 502–503

reactor performance, 449

transient balance, 442–444

Taylor dispersion model for laminar flow, 488–491, 575

Temperature and temperature effects

Butler-Volmer equation, 791

condensation

binary gas mixtures, 860–861

vapor with non-condensible gas, 850

conversion effects, 659

diffusion coefficients in gases, 231

fog formation, 855–856

gas separation membranes, 877

laminar flow reactors, 578–580

polymeric membranes, 256

porous catalysts

dimensionless boundary conditions, 612–613

dimensionless representation, 611–612

equations for heat and mass transport, 610–611

temperature gradients, 613–614

slab solutions Robin condition, 277

wet and dry bulb, 812–815

Temporal evolution of concentration profile in slabs, 63–64

Ternary diffusion in heterogeneous reactions, 527

Ternary mixtures

distillation of, 534–535

liquid evaporation in, 528–530

pseudo-binary diffusivity in, 166

Ternary systems

condensation, 864–866

multicomponent diffusivity matrix, 536–538

Test for backmixed assumption, 110–112

Thermal boundary layer in evaporation, 813

Thermal diffusion, 258

Thermicity group for porous catalysts, 612

Thermodynamic considerations

in electrochemical reactions, 781–786

Thermodynamic equilibrium in liquid–liquid extraction, 114

Thermodynamic potential in hydrogen fuel cells, 798

Thickness

film, 354, 373

laminar boundary layers

flow over inclined and curved surfaces, 395

integral balance approach, 378–380

Thiele modulus

oxygen profiles in pools of liquid, 61

oxygen transport in tissues, 758

oxygen uptake in lungs, 755

porous catalysts

diffusion-reaction model, 590, 592, 600–603, 605

finite difference methods, 620

linking with reactor models, 623–625

spherical catalysts, 595–596, 599

temperature effects, 611

transient diffusion processes, 285

volume reaction model, 647–648

Three-compartmental model for plasma, 772

Three dimensional flow field for fully turbulent flow, 405

Three-phase catalytic reactions, 607–610

Time averaging in turbulent flow, 406–411, 417

Time-concentration plots for slab solution Robin condition, 277

Time constants in laminar flow reactor model equations, 570–571

Time-dependent effect for gas separation membranes, 878

Time domain fitting in transient balance, 444

Time domain solution in tracer response method, 487–488

Time for conversion in shrinking core model solid product, 643–644

Time in temporal evolution concentration profiles in slabs, 63

Time marching in film theory models, 690

Time scales in laminar flow reactor model equations, 570

Tissues

inhibitors in, 771

transport of oxygen in, 82, 757–760

Topochemical reactions in Langmuir kinetics, 659

Tortuosity factor for fluids in porous solids, 253

Total concentration

definition, 8

single solute diffusion, 734

Total flow balance in gas permeator model countercurrent flow, 887

Total heat released in condensation, 852

Total mass balance in condenser model, 862–864

Total mass flux in turbulent flow, 418

Total molar concentration, 8

Total molar flow rate in macroscopic models, 88

Total shear stress, 433

Tracer response method

Laplace domain, 485

overview, 484–485

response curve moments, 485–487

time domain solution, 487–488

two-phase systems, 503–507

Tracer studies

age distribution functions, 441–442

moment analysis, 444–449

overview, 438

pulse input, 439–441

segregated flow model, 492

step input, 438–439

tanks in series model, 442–444

test for backmixed assumption, 110–112

Track-etched membranes, 906

Transfer area

gas permeator models, 886

packed bed absorbers, 328–329

Transference number for electric field, 550

Transformation in environmental transport, 459

Transient balance in macroscopic models

age distribution functions, 441–442

overview, 438

pulse input, 439–441

step input, 438–439

tanks in series model, 442–444

Transient calculations of batch slurry adsorbers, 928–929

Transient diffusion

batch slurry adsorbers, 926

bimolecular reaction, 714–715

cylinders and spheres, 73, 278–282

drying falling rate period, 835

eigenvalue computations, 297–298

non-homogeneous problems, 283–285

1-D problems, 266–267

overview, 265–266

oxygen, 275

PDEPE Solver computations, 299–304

penetration theory of mass transfer, 294–295

problems, 306–308

with reaction, 284–285

review questions, 305–306

slab solutions. See Slabs

spherical coordinates, 76–77

summary, 304–305

2-D problems, 285–287

variable diffusivity, 295–297

Transient effects in environmental transport, 462

Transient species balance equation, 438

Transition criteria for turbulent flow, 404–405

Translocation in environmental transport, 459, 461

Transpiration velocity in diffusion with convection, 65

Transport of charged species in electrolytic systems

mobility and diffusivity, 544–545

Nernst-Planck equation, 545–547

Transport of oxygen in tissues, 82, 757–760

Transport rate

across charged membranes, 554–556

gas separation membranes, 874–876

in presence of convection, 66

Trickle bed reactors, 21–22

Triglycerides (TG) in biodiesel production, 704

True kinetics vs. measured

kinetics in surface reactions, 207

TSPAN function, 300

Tubular flow reactors

dispersion closure, 132–134

overview, 41–43, 129–131

plug flow closure, 131

Tumors, nutrient limitation for, 770

Turbulent core in constant wall flux, 420–421

Turbulent diffusion, 14–15

Turbulent flow

axial dispersion, 493–494

convective mass transfer in. See Convective mass transfer in turbulent flow

description, 579–580

dispersion model, 483

flat plates, 317

pipes and channels, 313, 315–316

solid dissolutions from walls, 38

velocity profile, 49

Turbulent mass flux in time-averaged equation, 409

Turbulent regime in condensation, 849–850

Turbulent Schmidt number, 411–412

Two-compartment models for pharmacokinetics, 456–457, 761

2-D models

transient diffusion processes, 285–287

turbulent flow reactors, 579–580

Two-film model for gas–liquid interface, 212–217

Two-phase flow

non-idealities, 499–503

overview, 495–496

plug-backmixed model, 496–499

Two-phase systems, tracer response in, 503–507

Two-point collocation in orthogonal collocation method, 616–617

Two streams in mass exchangers, 136–141

Two-way coupled problems for flat plate with low flux mass transfer, 367

Ultrafiltration in liquid separation membranes, 898–899

Uncharged membranes, electrolyte transport across, 551–553

Uniform reaction model for reacting solids, 648

Unimolecular diffusion (UMD) binary mixture with, 237

condensation vapor with non-condensible gas, 851

convective mass transport, 312

diffusion-induced convection, 193–195

film model, 204

mole fraction profiles in, 196–197

steady state diffusion, 188

UNIQUAC model for non-ideal liquids, 244

Unit operations, 21–24

Universal velocity profiles in turbulent flow, 414–415

Unmixed model for condensation

binary gas mixtures, 857–859

condenser model, 863

Vacancy diffusion in solid–solid diffusion, 246

Valency of ions in transport of charged species, 545

Validity of low flux model in diffusion-induced convection, 193

van Driest model for convective mass transfer in turbulent flow, 425–427

van Laar equation for non-ideal liquids, 244

vant Hoff equation for osmotic pressure, 901

van’t Riet correlation for mechanically agitated tanks, 325

Vapor in fog formation, 855–857

Vapor phase balance in condenser model, 862–864

Vapor phase dehydrogenation of ethanol, 527

Vapor pressure in Antoine equation, 18, 48

Vapor with non-condensible gas in condensation condenser model, 854–855

heat transfer rate and Ackermann correction factor, 851–852

interface temperature calculations, 852–853

mass transfer rate, 851

overview, 850

Vapor–liquid interface in Raoult’s law, 17–18

Variable diffusivity

batch slurry adsorbers, 931

transient diffusion with, 295–297

Variance-based models for partial micromixing, 453–454

Vector-matrix form for multicomponent diffusivity matrix, 537

Velocity and velocity profiles

bimolecular reaction, 718–719

channel flow reactors, 577

channel flows with mass transfer, 351

constitutive model for multicomponent transport, 518–519

convective mass transfer

entry region analysis, 348

film flow, 353–354, 357

convective mass transfer in turbulent flow, 405–406, 413–417

constant wall flux, 420

time-averaged equation, 408–410

convective mass transport in gas bubbles, 324–325

cooling towers

cross-flow, 825

design considerations, 817

diffusion coefficients in gases, 227

frictional interpretation, 233

multicomponent diffusion, 235

film flow surface, 320

flux form, 155

kinetic model, 227

laminar boundary layers

bubbles and drops, 396–397

flat plate with low flux mass transfer, 368–369, 372, 375

flow over inclined and curved surfaces, 393

high flux analysis, 385–387

integral balance approach, 378

laminar flow

formula, 48

pipes, 337

mesoscopic models

plug flow idealization, 477

solid dissolutions from walls, 125

tracer response in two-phase systems, 506

tubular flow reactors, 132–134

mixtures, 156–157, 164

non-Newtonian fluids flow reactors, 577–578

solid dissolutions from walls, 39

system, 152

transport of charged species in electrolytic systems, 544–545

turbulent flow

closure models, 412

model equation, 49

turbulent flow reactors, 579–580

Velocity defects in turbulent flow, 415

Vignes equation for diffusion coefficients in liquids, 242

Viscosity

convective mass transfer in turbulent flow

closure models, 411

constant wall flux, 419–420

velocity profiles, 414

laminar boundary layer wall stress, 373

Volatile organic compound (VOC)

recovery of, 895

separation of, 909–910

Voltage

electrochemical processes

calculations, 785–786

description, 781

electrodialyzer design, 952–953

hydrogen fuel cells, 800

Voltage balance

electrochemical reactions

copper electrowinning, 796–797

overview, 793–795

hydrogen fuel cells, 798

salt separation, 953–954

Volume averages in multiphase systems, 177

Volume reaction model for reacting solids

concentration profile for gas and solid, 646

first-order reaction, 646–649

kinetic model, 645–646

overview, 644–645

zero-order reaction, 649–651

Volumes

environmental transport, 461

tanks in series models, 449

transient balance, 442–443

Volumetric flow rate

hemodialyzers, 764

in and out flow terms, 88–89

mass exchangers, 135–136

solid dissolutions from walls, 126

Volumetric flux in liquid

separation membranes, 899, 906

Volumetric liquid flow rate in bimolecular reaction, 718

Volumetric rate of absorption

bimolecular reactions, 677

first-order reaction of dissolved gas, 668

Volumetric transfer rate in bulk concentration and reactions, 669

von Karman analogy for channels and pipes, 423

von Karman approximation for film model, 199–200

Wakao-Smith model

catalysts, 588

fluids in porous solids, 253–254

Wall flux

boundary conditions, 362

laminar flow in pipes, 341–344

turbulent flow in channels and pipes, 418–421

Wall shear stress relations, 416–417

Wall term in macroscopic balance, 89

Walls

helium leakage rate across, 55–56

laminar boundary layer stress, 373

Robin problem, 344–348

solid dissolutions from, 38–41, 124–128

Water

acetic acid in, 257

acetone in, 245

enthalpy balance, 819–820

fugacity of pollutants, 460

molar volume in liquid, 241

oxygen concentration in, 17

SO₂ in, 20

splitting, 950

Water and air mixture, condensation rate in, 853–854

Water and ethanol mixture, separation of, 909

Water and methanol mixture, rate of condensation in, 859–860

Water vapor in cooling tower

counterflow model, 819

Wave speed in fixed bed adsorption, 933

Wavy regime in condensation, 849–850

Weber number in convective mass transport

bed absorbers, 329

gas bubbles, 323–324

Wedge flow over inclined and curved surfaces, 393–394

Weighting factors in mixture velocity, 183

Weisz modulus, 629

Wet bulb temperature, 812–815, 817

Wetted spheres, 720

Wetted wall columns, 718–719

Wilke-Chang equation for diffusion coefficients in liquids, 241–242

Wilke equation for evaporation of liquids in ternary mixtures, 528–529

Wilson equation for non-ideal liquids, 244

x-momentum balance for laminar boundary layers

flat plate with low flux mass transfer, 368–369, 371

flow over inclined and curved surfaces, 392

integral balance approach, 376–377

XMESH function, 300

Xylene isomers, separation of, 938

Yamashita-Inoue correlation for gas bubbles, 325

Zeolites, 920–921

Zero-order behavior in oxygen transport in tissues, 759

Zero-order reactions

diffusion-reaction model, 75

diffusion with, 58

macrofluid models, 451–453

plug flow idealization, 478

porous catalysts

diffusion-reaction model, 589, 599–603

finite difference methods, 619–620

linking with reactor models, 624–625

steady state mass transfer with reaction, 71

stirred reactors, 437

volume reaction model, 649–651

Zero total velocity in electrolytic systems, 544–545

Zeroth moment in tracer response in two-phase systems, 507

Zwitterion, 956