Index

Absorption

description, 216

film theory models

bulk concentration and bulk reactions, 670–672

first-order and pseudo-first-order reactions, 707

first-order reaction of dissolved gas, 668

two gases, 684–688

multistage cascades, 467–468

in slurries

instantaneous reaction case, 694–697

overview, 692–693

particle size effect, 693–694

Accumulation

evaporation in open containers, 197–198

flux form equations, 154

macroscopic models

batch reactors, 91

dissolved oxygen concentration in stirred tanks, 104

macroscopic balance, 90

reactor–separator combination, 97–98

single-phase systems, 173–174

sublimation of spherical particles, 101

silicon oxidation, 208

tracer response in two-phase systems, 505

Acetic acid in water diffusion rate, 257

Acetone in water, diffusion coefficient for, 245

Ackermann correction factor, 851–852

Activated carbon in adsorption, 920

Activation energy

interstitial diffusion, 246–247

porous catalyst temperature effects, 612

Active diffusion, 258

Active zones in backmixed assumption tests, 111–112

Activity coefficients

condensation unmixed model, 858

liquid-liquid interface, 19

models for non-ideal liquids, 244–245

Activity correction factor for non-ideal liquids, 243–244

Adiabatic dryers, 827

Adiponitrile (ADN) production, 807–808

Adsorption

applications and adsorbent properties, 920–921

batch slurry adsorbers collocation method, 929–931

complexities, 931

linear driving force model, 927–928

model equations, 925

overview, 924–925

particle-level model, 925–926

transients, 928–929

fixed bed adsorption

axial dispersion effects, 934–936

equilibrium model, 932–934

heterogeneous model, 936

Klinkenberg equation, 936–937

overview, 931–932

scale-up aspects, 937

isotherms, 921–924

overview, 919–920

problems, 941–943

review questions, 941

summary, 939–941

Adsorption constant in tracer response in two-phase systems, 505

Adsorption isotherm in fluid–solid interface, 19

Advection effects for environmental transport, 461

Aeration systems, 148

Age distribution functions in tracer studies, 441–442

Agitated tanks, 325–327

Air

diffusion volume, 232

heat balance, 819

Lennard-Jones constant, 229

Air and water mixture, condensation rate in, 853–854

Air phase in compartmental models fugacity of pollutants, 460

Ambipolar diffusion, 552

Amine solution

CO₂ in, 744

H₂S absorption in, 716–717

Amino acids, 956

Ammonia (NH₃)

Henry’s law constant, 16

molar volume in liquid, 241

Ammonia (NH₃)

porous catalysts in production of, 586

selective catalytic reduction of NO, 209–210

Amorphous solids, drying, 829

Anemia, 769

Anion exchange membranes (AEMs), 946–950

Anions in mass transport in electrolytic systems

across charged membranes, 554

of charged species, 545

across uncharged membrane, 551

Annular ring catalyst, 596–597

Anodic current and reactions definitions, 776–777

electrochemical reaction voltage balance, 793–795, 796

half, 777–779

hydrogen fuel cells, 798

Anoxic core in oxygen transport in tissues, 759

Antoine equation for vapor pressure, 18, 48

Apparent rate constant for porous catalysts

diffusion-reaction model, 597–599

linking with reactor models, 624

Applied potential in electrochemical reactions, 786–787

Arnold cells for evaporation, 197, 532–533

Arnold diffusion in film model, 204

Arrhenius law and equation batch reactors, 92

boron in silicon, 247

Butler-Volmer equation, 788–791

gas translation model, 880

laminar flow reactor heat transfer effects, 578

porous catalyst temperature effects, 612

Arsenic diffusion in silicon, 247–248, 296

Ash layer effects in shrinking core model, 641–644

Asymmetric membranes in gas separation, 873

Augmentation factor for condensation vapor with non-condensible gas, 851–852

Automobile catalytic converters, 21

Average absorption rate in first-order and pseudo-first-order reactions, 707

Average concentration batch slurry adsorbers, 925, 927–928, 930

capillary flow models, 756

cross-sectional, 490, 576

differential models, 5

dispersion model, 478

fixed bed adsorption, 936

hemodialyzers, 764

internal flows, 311

laminar flow reactors, 571

long cylinders, 279

mass exchangers, 134

multiphase systems, 177

partial micromixing, 453–454

pipe flows, 423

reactor-separation combination, 98, 100

single-phase systems, 174

slab solutions, 274–275

solid dissolutions from walls, 39, 126

stirred reactors, 437

transient diffusion processes, 280–282

tubular flow reactors, 41, 129–131

2-D problems, 287

wall transfer term, 89–90

Average cross-sectional rate in tubular flow reactors, 131

Average flux bimolecular reactions, 713

penetration theory, 711

transient diffusion processes, 290

Average moisture content in drying falling rate period, 836–837

Average molecular weight calculation, 9–10

constant-density systems, 167

from mole fractions, 10

Average pore radius in catalysts, 587–588

Average rate

condensation, 848

diffusion-reaction model, 592, 597–598

film theory models, 710–711

laminar boundary layers, 398

laminar flow, 356

macroscopic models continuous stirred tank reactors, 108

interface transfer term, 90

Average reaction rate

collocation method, 615

porous catalysts, 604

second-order reactions, 454

single-phase systems, 173

Average velocity diffusion coefficients in gases, 227

film flow, 353

flux form, 155

kinetic model, 227

laminar flow in pipes, 337

mixtures, 156–157, 164

plug flow, 477

solid dissolutions from walls, 39

system, 152

turbulent flow, 413

wall shear stress, 416

Avrami model for solid–solid reactions, 654–655

Axial diffusion time in laminar flow reactors, 570

Axial dispersion

biomedical applications, 764

in channel flow, 493

chromatography, 938

in dispersion model, 479, 482

fixed bed adsorption, 934–936

Hannig design, 960

mesoscopic models, 42–43, 575

packed beds, 494

tracer response, 506

tubular flow reactors, 132–133

turbulent cases, 580–581

in turbulent flow, 493–494

Axial distance

fixed bed adsorption, 937

laminar flow, 337–338, 568

Sherwood number, 344, 352–353

Axial flux values in mesoscopic models, 39

Axial length in laminar flow reactors, 569

Axial velocity profiles

mesoscopic models, 38

segregated flow model, 492

Backmixed assumption

stirred reactors, 436–438

test for, 110–112

Backmixed reactor transient balance, 440

Backmixed–backmixed model

fluid–fluid systems, 462–464

gas permeator models, 884–886

liquid–liquid extraction, 115

Backmixing model for multistage cascades, 466

Balance equations

batch slurry adsorbers, 925

condenser model, 861, 863

cooling towers, 818–819, 825–826

differential models, 31, 92

fixed bed adsorption, 936

flow over inclined plates, 389, 392

flux form, 154

gas permeator models, 883, 887, 889

gas-solid reactions, 652, 654

hydrogen fuel cells, 798

mass basis, 166

membranes, 740

mesoscopic models

mass exchangers, 136, 138, 140

tracer response, 504–505

mole basis, 168

momentum equation, 371, 377–378

multicomponent systems, 236, 519, 524

packed column absorbers, 691

porous catalysts, 611–612, 622

semibatch reactors, 688

single-phase systems, 174

species, 113–114

stirred reactors, 436–437

tracer studies, 438–439, 442

two-phase systems, 504

Balance of forces in diffusion coefficients in liquids, 238

Barrer units in gas separation membranes, 875–876

Barrier diffusion, 236

Batch dryers, 827

Batch reactors

differential equations, 91–93

electrochemical, 806–807

ODE45 with CHEBFUN, 93–96

overview, 90

with product removal, 99–100

Batch slurry adsorbers

collocation method, 929–931

complexities, 931

linear driving force model, 927–928

model equations, 925

overview, 924–925

particle-level model, 925–926

transients, 928–929

Batteries, li ion

modeling, 800–802

transfer and reaction steps, 27

BCFUN function, 300

Bed voidage in fixed bed adsorption, 933

Benzene

cracking cyclohexane to, 891

system velocity in presence of diffusion, 160–161

Benzoic acid–coated wall in contact with flowing NaOH solution, 381–382

Bernoulli equation for flow over inclined and curved surfaces, 388

Bessel functions steady

state mass transfer with reaction, 72

transient diffusion processes, 278–279

BET (Brunauer-Emmet-Teller) isotherms, 923–924

Bhavaraju correlation in gas bubbles, 324

Bidispered particles in batch slurry adsorbers, 931

Bimodal distribution catalysts, 588

fluids in porous solids, 253–254

Bimolecular reactions

gas–liquid reactions in film theory

dimensionless representation, 673–675

invariance property, 676–677

overview, 672–673

gas–liquid reactions penetration theory

dimensionless form, 712–713

illustrative results, 713–714

instantaneous reaction case, 714–717

overview, 712

surface reactions, 209–210

Binary diffusion

in gases, 230–231

multicomponent transport, 518

Binary gas mixtures, condensation of

interface temperature, 860–861

overview, 857–858

unmixed model, 858–859

Binary liquid mixtures, evaporation of, 531–533

Binary mixture with UMD, Stefan-Maxwell model for, 237

Binary pair diffusivity in multicomponent diffusivity matrix, 536–539

Biodiesel production, 704

Biological systems counter-transport, 736

Biomedical applications and systems

compartmental models, 43–44

examples, 25

hemodialyzers, 763–766

membrane aeration systems, 148

overview, 749–751

oxygen uptake in lungs

meso-model for capillaries, 756–757

overview, 751

oxygen-hemoglobin equilibrium, 751–753

transport steps, 753–756

pharmacokinetics, 760–763

problems, 768–773

review questions, 767–768

summary, 766–767

transport in tissues, 757–760

Bioseparations, 24

Biot number

film theory models

bimolecular reactions, 674

simultaneous absorption of two gases, 685

fugacity of pollutants, 460

laminar flow in pipes, 345, 347

laminar flow reactors, 571

oxygen uptake in lungs, 755

porous catalysts

diffusion-reaction model, 591, 594

temperature effects, 612–613

Robin condition, 276–278

transient diffusion processes

cylinders and spheres, 279–282

problems in 2-D, 287

Bipolar membranes (BPMs) in electrodialysis, 950–951

Bischoff approximation, 605

Blasius approach

dispersion coefficient values, 494

laminar boundary layers

flat plate with low flux mass transfer, 371–376

flow over inclined and curved surfaces, 392

high flux analysis, 386–388

Blasius correlation for axial dispersion model, 581

Blood cells

glucose uptake, 747

hemodialyzers, 763–766

oxygen transport in tissues, 757–760

oxygen uptake in lungs, 753–757

Blowing factor

laminar boundary layer high flux analysis, 387

steady state diffusion across slabs, 54

transport rate in presence of convection, 67

vapor with non-condensible gas, 851

Bodenstein number in dispersion model, 479

Boltzmann constant

diffusion coefficients in gases, 227

interstitial diffusion, 247

Bolus input

pharmacokinetic analysis, 471

transient balance, 439–441

Boron in silicon interstitial diffusion, 247

Boundary conditions

batch slurry adsorbers, 926, 930

concentration profiles in film model, 203–204

condensation in ternary systems, 865

convective mass transfer

entry region analysis, 348–349

film flow, 355

turbulent flow, 417

wall boundaries, 362

cross-flow cooling towers, 826

D-D problem in slab geometry, 283–285

description, 171–172

diffusion

with convection, 65–66

with first-order reactions in long cylinders, 72

drying falling rate period, 835–837

in electrolytic systems, mass balance for reacting systems, 550

equations of mass transfer, 171–172

evaporation binary liquid mixtures, 531

ternary liquid mixtures, 530

flat plate with low flux mass transfer, 367

gas–liquid reactions in film theory

bimolecular reactions, 674–676, 715

first-order reaction of dissolved gas, 663–664

simultaneous absorption of two gases, 685–686

gas–liquid reactions in penetration theory, 707

hemodialyzers, 765

heterogeneous reactions, 526

laminar boundary layers flat plate with low flux mass transfer, 372, 374

flow over inclined and curved surfaces, 393

high flux analysis, 386–387

integral balance approach, 377, 381

laminar flow in pipes, 339–340, 344–348

laminar flow reactors, 571

mesoscopic dispersion model, 576

model equations, 571–572

liquid separation membranes, 904

mass transfer from spheres to stagnant gas, 192

mass transport in electrolytic systems, 550

mesoscopic models

dispersion model, 480–483

ideal flow patterns, 500–502

Taylor model, 490

tracer response method, 484, 486–487

oxygen profiles in pools of liquid, 59–60

oxygen transport in tissues, 758

porous catalysts

diffusion-reaction model, 589–590, 593, 597, 600

finite difference methods, 621

orthogonal collocation method, 616

temperature effects, 611–613

reacting systems, 520–525

single solute diffusion, 729

slab solutions

Dirichlet case, 269–270

Robin condition, 276–278

temporal evolution, 63

steady state mass transfer with reaction, 71–72

transient diffusion

integral method, 290–293

problems in 2-D, 286

with reaction, 301–302

semi-infinite slab analysis, 289

in slabs, 62

variable diffusivity, 296

Boundary layers

channels and pipes, 421–422

fluid–solid mass transfer

benefits, 201

film model, 198–201

laminar. See Laminar boundary layers

Breakthrough curve for fixed bed adsorption, 935

Bubble point in binary gas mixtures condensation, 860–861

Bubbles and drops

convective mass transport, 321–325

laminar boundary layers

overview, 396

rigid bubbles, 396–397

spherical cap bubbles, 397–399

penetration theory of mass transfer, 294–295

Buffer zones in turbulent flow

constant wall flux, 420

velocity profiles, 414–416

Bulk concentration

gas–liquid reactions in film theory models, 668–672

porous catalysts temperature effects, 611

Bulk density in catalysts, 587

Butler-Volmer equation

hydrogen fuel cells, 799

kinetic model, 788–791

mass transfer effects, 792

mass transport in electrolytic systems, 550

voltage balance, 793, 797

BVP4C method

Blasius approach, 387

condensation in ternary systems, 865–866

evaporation in liquid mixtures, 531–534

Falkner-Skan equation, 393

flat plate with low flux mass transfer, 374

heterogeneous reactions, 527

porous catalyst temperature effects, 614

reacting systems, 520–522

ternary systems, 865

transient diffusion processes, 296

Calcium carbide, nitrogen reaction with, 658

Calcium cynamide, nitrogen reaction for, 658

Calderbank correlation

gas bubbles, 324

mechanically agitated tanks, 326

Calderbank-Moo-Young correlation, 323

Capacity term in tracer response in two-phase systems, 505

Capillaries

drying flow models, 838–839

meso-model for, 756–757

transport in tissues, 757–760

Capsules, drug release rate from, 282–283

Carbon

adsorption, 920

diffusion volume, 231

Carbon dioxide (CO₂)

in amine solution, 744

diffusion volume, 232

Henry’s law constant, 16, 47

Lennard-Jones constant, 229

molar volume in liquid, 241

oxygen-hemoglobin equilibrium, 753

removal with NaOH solution, 680–681

sequestration

mass transfer in human body, 750

underground mines, 28

Carbon monoxide (CO)

catalytic oxidation of, 541

counter-transport, 736

diffusion volume, 232

Carbothermic reduction of ores, 654

Cartesian coordinates

diffusion with convection, 64–67

divergence operator, 155

Laplacian of concentration, 190

steady state diffusion

across slabs, 52–55

reactions in slabs, 56–62

transient diffusion in slabs, 62

Catalysts. See Porous catalysts

Catalytic converters, 20–21

Catalytic oxidation of CO, 541

Cathodic current and reactions

definitions, 776–777

electrochemical reaction voltage balance, 794, 797

half, 777–779

hydrogen fuel cells, 799

reduced product, 780

Cation exchange membranes (CEMs) in electrodialysis, 946–950, 953

Cations in electrolytic systems

transport across charged membranes, 554

transport across uncharged membrane, 551

transport of charged species, 545

Cauchy-Euler type for laminar boundary layers, 380

Cement production, 654

Center concentration

laminar flow, 341, 359

limiting cases, 573

porous catalysts

diffusion-reaction model, 600, 603

finite difference methods, 619

spherical catalysts, 595

temperature gradients, 614

spheres, 280, 282, 303

Stanton number, 424

Central difference equations for porous catalysts, 618–619

Channels

axial dispersion, 493

convective mass transfer, 350–353

description, 577

laminar flow, 315

overview, 316

turbulent flow

constant wall flux, 418–421

correlations, 315

momentum transfer analogy, 422–423

overview, 417–418

Stanton number for boundary layers, 421–422

Stanton number for pipe flows, 423–425

Chapman-Enskog model

diffusion coefficients in gases, 228–231

effective diffusivity, 253

Charge equality in electrolytic systems, 558

Charge neutrality in electrolytic systems, 547–548

Charged membranes in electrolytic systems, 553–556

Charged species diffusion, 258

Charging li-ion batteries, 801

Chebechev polynomials

batch reactors, 94

channel flows with mass transfer, 352

CHEBFUN solutions

batch reactors, 93–96

channel flows with mass transfer, 351–353

dispersion model, 482–483

Falkner-Skan equation, 393

laminar flow in pipes, 345–346

macrofluid models, 452

reacting systems, 521–522

simultaneous absorption of two gases, 684, 686–688

transient diffusion processes, 297–298

Chemical potential in Nernst equation, 781–782

Chilton-Colburn analogy for heat and mass transfer, 370

Chlorine formation, 948

Chloromethylation, 949

Chromatography, 938–939

Classification

cooling towers, 816–817

electrode reactions, 779–780

gas separation membranes, 873–874

liquid separation membranes, 898–899

Clift correlation for gas bubbles, 324–325

Closed-closed boundary

conditions in dispersion model, 480–481

Closure

axial dispersion model, 43

convective mass transfer in turbulent flow, 407, 411–413

Co- and counter-transport in reactive membranes

illustrative results, 741–742

overview, 736–739

Cobalt recovery, 806

Coefficient matrix for evaporation of binary liquid mixtures, 532

Collision integral for diffusion coefficients in gases, 230–231

Collocation methods

adsorption and chromatography

batch slurry adsorbers, 929–931

fixed bed adsorption, 936

porous catalysts, 615–617

Colloids in electrophoresis principle, 956

Column design in gas absorption, 141–142

Column extractors in multistage cascades, 466

Columns, bubble, 322

Combined flux equation for steady state diffusion, 187

Compartmental models

development, 43–44

drug distribution, 750

environmental transport

fugacity of pollutants, 460

Level I, 460–461

Level II, 461

Level III, 461–462

Level IV, 462

overview, 459

matrix representation, 456–457

overview, 436, 454–456

pharmacokinetics, 457–459, 760–763

Competitive adsorption isotherms, 923

Completely mixed film assumption in condensation, 858

Composite dielectric medium, 256

COMSOL-based solutions

laminar boundary layers, 390

laminar flow reactors, 579–580

Concentration and concentration profiles

adsorption and chromatography

batch slurry adsorbers, 925, 927–930

fixed bed adsorption, 933, 936–937

Butler-Volmer equation, 789

capillary flow models, 756

convective mass transfer

channel flow, 316

entry region analysis, 349

external and internal flows, 310–311

film flow, 354, 356–358

turbulent flow, 406–407, 419–421, 424

wall reaction, 347

cross-sectional, 490, 576

differential models, 5

diffusion

with convection, 66

across cylindrical shells, 69

in gases, 226

electrochemical reaction voltage balance, 797

electrodialyzer design, 952–953

film model, 199–200, 203–204

fixed bed adsorption, 936

fog formation, 856

gas permeator models, 883–884

gas separation membranes, 874–875, 878

gas–liquid reactions in film theory models

absorption in slurries, 693–694, 696

bimolecular reactions, 673, 675–676, 681, 714–715

bulk concentration and reactions, 668–672

coupling with reactor models, 688–690

first-order reaction of dissolved gas, 665–667

Laplace transform method, 709–710

liquid–liquid reactions, 698

hemodialyzers, 764

internal flows, 311

jump at interface

fluid–solid interface, 19

gas–liquid interface, 15–17

liquid–liquid interface, 18–19

nonlinear equilibrium models, 19–20

overview, 15

vapor–liquid interface, 17–18

laminar boundary layers

film model, 384

flat plate with low flux mass transfer, 367–369, 375–376

flow over inclined and curved surfaces, 393

high flux analysis, 386–388

integral balance approach, 379–380

wall stress, 373

laminar flow in pipes, 339–344

laminar flow reactors, 569, 571, 576

liquid separation membrane

osmotic pressure, 900–901

long cylinders, 279

macroscopic models

continuous stirred tank reactors, 107–108

fluid–fluid systems, 463–465

macrofluid models, 450–451

moment analysis, 445

stirred reactors, 437

sublimation of spherical particles, 102

tanks in series models, 449

transient balance, 439, 443–444

variance-based models, 453–454

mass exchangers, 134

mass transfer equations, 160

mass transfer from spheres to

stagnant gas, 192

mass units, 9–10

mesoscopic models

dispersion model, 478, 480

first-order reaction, 481

ideal flow patterns, 501–503

plug-backmixed model, 496–498

plug flow idealization, 477

solid dissolutions from walls, 38–39, 126

Taylor model, 488–491

tracer response method, 487

tubular flow reactors, 131–132

mole units, 8–9

multiphase systems, 177

oxygen-hemoglobin equilibrium, 751

oxygen profiles in pools of liquid, 61

oxygen transport in tissues, 758–760

oxygen uptake in lungs, 756

partial micromixing, 453–454

partial pressure units, 10

pipe flows, 423

polarization effects in semi-permeable membranes, 903–905

porous catalysts

diffusion-reaction model, 588–592, 600–602

finite difference methods, 619–621

linking with reactor models, 622

multiple species, 606–607

orthogonal collocation method, 616

spherical catalysts, 595, 598

reacting solids, 646, 650

reacting systems, 521

reactive membranes and facilitated transport, 739–740

reactor-separation combination, 98, 100

shrinking core model solid product, 641

single-phase systems, 174

single solute diffusion, 729–735

slabs

illustrative results, 272, 274–275

semi-infinite, 288–290

temporal evolution, 63–64

steady state diffusion

low flux model, 190

across slabs, 55

across spherical shells, 74

transfer rate in diffusion film

near electrodes, 558

transient diffusion processes

cylinders and spheres, 280–281

integral method, 293

problems in 2-D, 287

semi-infinite slab analysis, 288–291

spherical coordinates, 76–77

tubular flow reactors, 41

tumors, 770–771

2-D problems, 287

variable diffusivity, 296

wall transfer term, 89–90

Concentration-dependent diffusion, 247

Concentration form

constant-density systems, 167–168

mass basis, 166–167

mole basis, 168–169

overall continuity, 168

overview, 166

Concentration overpotential in electrochemical reaction, 793

Concentration ratio parameter for bimolecular reaction, 713

Concurrent flow in ideal flow patterns, 500–501

Condensation

binary gas mixtures

interface temperature, 860–861

overview, 857–858

unmixed model, 858–859

condenser model

liquid and vapor phase balances, 862–864

overview, 861

vapor with non-condensible gas, 854–855

fog formation, 855–857

overview, 845–846

problems, 868–869

pure vapor, 846–850

review questions, 867–868

summary, 866–867

ternary systems, 864–866

vapor with non-condensible gas

condenser model, 854–855

heat transfer rate and Ackermann correction factor, 851–852

interface temperature, 852–853

mass transfer rate, 851

overview, 850

Condensation rate

methanol and water mixture, 859–860

water and air mixture, 853–854

Conductivity

electrolytic systems electric field, 548

salt solution in electrodialyzer design, 953

Conservation laws and equations

diffusion with convection, 64–65

flux form equations, 154

macroscopic models balance, 87

continuous stirred tank reactors, 106

dissolved oxygen concentration in stirred tanks, 104–105

stirred reactors, 436–437

sublimation of spherical particles, 101

mass basis, 166

mesoscopic models, 38, 125–128

model development, 28–30

mole basis, 168

multiphase systems, 175

reacting systems, 520, 525

steady state diffusion radial, 68

with slabs, 53–54, 57

transient diffusion in slabs, 62

tubular flow reactors, 41–42

Constant and falling drying rates, 829–830

Constant-density systems

common simplifications, 170

concentration form, 167–168

Constant rate drying period, 830–833

Constant surface concentration in semi-infinite slab analysis, 288–291

Constant volume in batch reactors, 92–93

Constant wall concentration in laminar flow in pipes, 339–341, 347

Constant wall flux, 418–421

Constitutive model for multicomponent transport, 518–520

Constraint condition in single solute diffusion, 732

Contact point model in solid–solid reactions, 655–656

Contacting efficiency in fluid–fluid systems, 465

Contactors in bimolecular reactions, 684, 717–721

Contaminants in bubbles and drops, 397

Continuity equation for turbulent flow, 407–408

Continuous dryers, 827

Continuous stirred tank reactors (CSTRs)

first-order reaction, 107

overview, 106–107

second-order reaction, 108–110

Continuum assumption, 7

Contracted axial distance parameter for laminar flow in pipes, 338

Control surfaces in macroscopic models, 87, 174

Control volumes

differential models, 31

gas–liquid reactions in film theory models, 668

macroscopic models

balance, 87–89

liquid–liquid extraction, 113

mass transfer equations, 157–158

mesoscopic models

description, 37

mass exchangers, 136

plug-backmixed model, 496

tubular flow reactors, 129–130

model development, 28–29

multiphase systems, 176

porous catalysts, 622

steady state diffusion

radial, 68

across spherical shells, 73

Convection

diffusion-induced, 170, 187–188, 193–198

diffusion with, 64–67

limiting cases, 572–574

Taylor model, 488–489

transport rate in presence of, 66

Convection-diffusion equation

convective mass transfer in turbulent flow

channels and pipes, 417

gas–liquid interface, 427

time-averaged equation, 408

film flow, 357–358

laminar boundary layers film model, 383

flat plate with low flux mass transfer, 367–368, 374

laminar flow in pipes, 336–337

parallel plate electrolyzer, 807

Convection-diffusion-reaction (CDR) model for laminar flow reactors, 568–569

Convective flux

description, 12

diffusion coefficients in gases, 226

Convective heat transfer in cooling towers, 820

Convective mass transfer for internal laminar flow

channel flows with mass transfer, 350–353

entry region analysis, 348–350

film flow

gas absorption from interface, 357–358

overview, 353–354

solid dissolution at walls, 354–356

laminar flow in pipes

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

constant wall concentration, 339–341

dimensionless form, 337–339

overview, 336–337

overview, 335–336

PDEPE for, 358–359

problems, 361–364

review questions, 361

summary, 359–360

wall reaction, 344–348

Convective mass transfer in turbulent flow

channels and pipes

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

closure models, 411–413

gas–liquid interface, 427–430

overview, 403–404

problems, 433–434

review questions, 432

summary, 430–432

time-averaged equation, 408–411

time averaging, 406–408

turbulent flow properties, 404–406

van Driest model, 425–427

velocity and turbulent diffusivity profiles, 413–417

Convective mass transport

bed absorbers, 327–329

external and internal flows, 310–311

flows

in film, 318–320

in flat plates, 316–318

in pipes and channels, 315–316

gas bubbles, 321–325

key dimensionless groups, 313–315

mechanically agitated tanks, 325–327

overview, 309–310

problems, 331–333

relation to differential model, 311–313

review questions, 330–331

solid spheres, 320–321

summary, 329–330

Cooling towers, 815

classification, 816–817

counterflow model, 817–823

design considerations, 817

NTU calculation, 823–824

Copper (Cu)

electrochemical reaction voltage balance, 797

electrowinning, 795–798

from leach solutions, 795

Correction factor

diffusion flux, 67

drift flux in diffusion-induced convection, 195–196

evaporation in open containers, 198

film model, 202, 204–205

gas–liquid mass transfer in packed bed absorbers, 327

liquid-filled pores, 251

non-ideal liquids, 243–244

tubular flow reactors, 132

Correlations

condensation, 849

constant rate drying period, 831–832

convective mass transport

gas bubbles, 323–325

gas–liquid mass transfer in packed bed absorbers, 328–329

low flux value, 312

mechanically agitated tanks, 325–327

dispersion coefficient values, 493–495

mass transfer coefficient sublimation of spherical particles, 102–104

solid dissolutions from walls in pipe flow, 128

Counter-diffusion

equimass, 162–163

equimolar, 161–162

high flux model surface reactions, 210–212

Countercurrent cascade in multistage cascades, 467

Countercurrent flow

gas permeator models, 886–889

ideal flow patterns, 502

Counterflow model for cooling towers

enthalpy balance equations, 819–821

mass balance equations, 818–819, 821

Merkel equation, 821–823

overview, 817–818

Coupled computational scheme for fixed bed adsorption, 936

Coupling with reactor models, 688–692

Cross-correlation in turbulent flow, 407

Cross-flow

cooling towers, 825–827

gas permeator models, 889–890

Cross-section averaged models

overview, 37

solid dissolutions from walls, 38–41

Cross-sectional area in gas translation model, 880

Cross-sectional average

dispersion model, 478, 482

plug flow idealization, 476

Taylor model, 490–491

tubular flow reactors, 130–131

Crosscurrent cascade in multistage cascades, 467–468

Crosslinked polystyrene, 949

Crystalline solids, drying, 828–829

Cubic profile in integral balance approach, 378

Cumulative concentration for reacting solids, 659–660

Cup-mixed concentration

description, 39

example, 40

hemodialyzers, 764

laminar flow in pipes, 342–343

laminar flow reactors, 571

mesoscopic models

mass exchangers, 135

solid dissolutions from walls, 126–127

tubular flow reactors, 130–131

oxygen uptake in lungs, 756

plug flow idealization, 476

pure convection model limiting cases, 573

Taylor model, 490

Current

diffusion film near electrodes, 559

electrochemical processes

Butler-Volmer equation, 789–790

description, 781

mass transfer effects, 792

voltage balance, 793–794

electrodialysis, 946

electrodialyzer design, 952–955

hydrogen fuel cells, 799–800

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

Cyclohexane to benzene cracking, 891

Cylinders

diffusion in first-order reactions, 72

porous catalysts in diffusion-reaction model, 593–594

transient diffusion in, 73, 278–279

Cylindrical coordinates

divergence operator, 154–155

Laplacian of concentration, 190

overview, 67–68

radial diffusion in, 68–70, 191

steady state mass transfer with reaction, 70–72

transient diffusion in, 73

Cylindrical pore filled with methane

diffusion, 249–250

fluids in porous solids, 249

Cylindrical shells, diffusion across, 69–70

D-D problem in slab geometry, 283–285

Dalvi-Suresh contact point model for solid–solid reactions, 655–656

Damkohler number

laminar boundary layers, 376

laminar flow reactors

dispersion model, 576

model equations, 569

plug flow model, 574–575

macroscopic models

continuous stirred tank reactors, 107–108

macrofluid models, 452–453

stirred reactors, 437

tanks in series models, 449

mesoscopic models

dispersion model, 479

plug flow idealization, 477

tubular flow reactors, 133

porous catalysts, 623

Damping of turbulence in gas–liquid interface, 428–429

Danckwerts boundary conditions

hemodialyzers, 765

laminar flow reactors, 576

mesoscopic models

dispersion model, 480, 482–483

ideal flow patterns, 500–501

tracer response method, 484

Danckwerts surface renewal model, 725

Darcy permeability, 906

Darcy’s law, 898–899

Darken relation in non-ideal liquids, 244

Dead zones

backmixed assumption tests, 111–112

porous catalysts, 630

Debye layer in electrolytic systems, 548

Density

constant-density systems, 167–168

mixture, 9–10

Desorption, 216

Determinacy condition

equimolar counter-diffusion, 533

evaporation in ternary liquid mixtures, 529

steady state diffusion, 188–189

Determinacy correction factor for film model, 204–205

Dew point for binary gas mixtures, 860

Diameters of bubble, 323–324

Differential (1-D) balances

Cartesian coordinates

diffusion with convection, 64–67

overview, 52

steady state diffusion across slabs, 52–55

steady state diffusion with reaction in slabs, 56–62

transient diffusion in slabs, 62

cylindrical coordinates

overview, 67–68

steady state mass transfer with reaction, 70–72

steady state radial diffusion, 68–70

transient diffusion in cylinders, 73

overview, 51–52

problems, 79–83

review questions, 78–79

spherical coordinates, 73

diffusion and reaction, 75–76

steady state diffusion across spherical shells, 73–74

transient diffusion in spherical coordinates, 76–77

summary, 77–78

Differential equations and models

development, 29

laminar boundary layers, 379

macroscopic models

batch reactors, 91–93

reactor–separator combination, 97

multicomponent systems, 525

overview, 30–32

relation to convective mass transport, 311–313

two streams, 139

Differential species mass balance equation, 168–169

Differential volume in fixed bed adsorption, 933

Diffusion

batch slurry adsorbers, 926

common simplifications, 170

complex effects, 257–258

with convection, 64–67

in cylinders, 73

cylindrical pore filled with methane, 249–250

across cylindrical shells, 69–70

diffusion coefficients. See Diffusion coefficients

electrophoretic separation devices

Hannig design, 960

Philpot design, 959

equimolar counter-diffusion, 161–163

with first-order reactions

example, 59

in long cylinders, 72

fluids in porous solids, 248–254

gas separation membranes, 877–878

heterogeneous media, 254–256

liquid separation membranes, 899

multicomponent diffusivity matrix, 535–539

non-ideal liquids, 243–245

overview, 223–224

polymeric membranes, 256–257

problems, 261–264

review questions, 260–261

single solute diffusion, 729

solid–solid diffusion, 246–248

in spherical coordinates, 75–77

steady state radial, 68–70

summary, 258–260

system velocity in presence of, 160–161

time-averaged equation in turbulent flow, 408

transient. See Transient

diffusion

with zero-order reactions, 58

Diffusion balances in integral balance approach, 382

Diffusion coefficients

electrolyte transport across uncharged membrane, 552

fixed bed adsorption, 937

gas separation membranes, 874

gas-solid reaction models, 651–652

in gases

frictional interpretation, 232–234

model based on kinetic theory, 225–232

multicomponent diffusion, 235–236

overview, 224–225

in liquids

ethanol in water, 242

overview, 237–239

Stokes-Einstein model, 239–240

Wilke-Chang equation, 241–242

values, 14

Diffusion-dominated processes

diffusion-induced convection, 193–198

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

overview, 185–186

problems, 219–221

review questions, 219

steady state diffusion, 186–192

summary, 217–218

surface reactions, 206–212

Diffusion film near electrodes, transfer rate in, 556–559

Diffusion flux

description, 12–13

diffusivity values, 13–14

dispersion, 15

laminar boundary layers in film model, 383–384

mass transfer equations, 156–157

properties, 163–165

steady state diffusion with slabs, 54, 57

turbulent diffusion, 14–15

Diffusion-induced convection

common simplifications, 170

condensation in unmixed model, 859

conditions for validity of low flux model, 193

description, 187–188

drift flux correction factor, 195–196

mole fraction profiles in UMD, 196–197

UMD analysis, 193–195

Diffusion limiting current (DLC) calculation in copper electrowinning, 795–798

Diffusion-reaction model

first-order and pseudo-first-order reactions, 706–707

porous catalysts

first-order reaction, 589–598

nth-order reaction, 603–605

overview, 588–589

zero-order reaction, 599–603

reactive membranes and facilitated transport equilibrium, 739

volume reaction model, 646

Diffusion type of models in drying falling rate period, 835–838

Diffusional effects for porous catalysts, 598–599

Diffusive flow field for fully turbulent flow, 405

Diffusivity

batch slurry adsorbers, 931

bimolecular reaction, 713

convective mass transfer in turbulent flow, 413–417

gas separation membranes, 878

pseudo-binary, 165–166

transport of charged species, 544–545

Dilute solutions in electrodialysis and electrophoresis, 948

Dilute system assumption for fluid–fluid systems, 463

Dimensionless boundary conditions in porous catalysts temperature effects, 612–613

Dimensionless groups

convective mass transport, 313–315

laminar flow reactors, 568–572

Dimensionless representation

bimolecular reaction, 712–713

Dirichlet case in slabs, 268–269

dispersion model, 479–480

gas–liquid reactions in film theory models

bimolecular reactions, 673

first-order reaction of dissolved gas, 664–665

simultaneous absorption of two gases, 685–686

internal laminar flow in, 337–339

porous catalysts temperature effects, 611–612

reactive membranes and facilitated transport

instantaneous reaction asymptote, 732–734

invariant of system, 732

pseudo-first-order reaction asymptote, 735–736

single solute diffusion, 730–732

Dirac delta function in transient diffusion processes, 292–293

Direct dryers, 827

Direction of transfer in two-film model, 216–217

Dirichlet boundary conditions, 171–172

Dirichlet case and problem

batch slurry adsorbers, 926, 930

falling rate period, 836–837

gas–liquid reactions in film theory models

bimolecular reactions, 674

first-order reaction of dissolved gas, 663–664

heterogeneous reactions, 526

laminar flow in pipes, 339–341

laminar flow reactors, 571

porous catalysts

diffusion-reaction model, 589, 591–594, 597, 600

orthogonal collocation method, 616–617

reacting systems, 523

slab solutions

average concentration, 274–275

dimensionless representation, 268–269

illustrative results, 272–274

overview, 267–268

series coefficient, 271–272

series solution, 269–271

Discharging li-ion batteries, 802

Discretization formula for porous catalysts, 615

Dispersed phase in multiphase systems, 176–177

Dispersion

axial. See Axial dispersion

diffusion, 15

fixed bed adsorption, 934–936

Dispersion closure in tubular flow reactors, 132–134

Dispersion coefficients

common cases, 493–495

Laplace domain, 485

moments of response curve, 485–487

non-Newtonian fluid flow reactors, 578

overview, 484–485

time domain solution, 487–488

Dispersion model

boundary conditions, 480–481

CHEBFUN, 482–483

first-order reaction, 481

negligible dispersion, 483–484

nonlinear reactions, 482

overview, 478–480

Dispersion number

dispersion model, 479

laminar flow reactors, 575

Dissipative flow field in fully turbulent flow, 405

Dissociation diffusion

description, 257

gas separation membranes, 877–878

Dissolved CO₂ in oxygen-hemoglobin equilibrium, 753

Dissolved oxygen concentration in stirred tanks, 104–106

Distillation

film model, 205

ternary mixtures, 534–535

Distributed pore models in gas–solid reaction models, 654

Dittus-Boelter correlation and equation

channel flow in convective mass transport, 315–316

drying constant rate period, 831

Divergence

flux form equations, 154–155

multiphase systems, 177

Domain statements in ODE45 with CHEBFUN, 94

Donnan equation, 553–556

Dopants

interstitial diffusion, 247–248

semiconductor and solar devices, 24–25

silicon with phosphorus, 293

Double collocation in fixed bed adsorption, 936

Drag coefficients

convective mass transfer in turbulent flow, 417

flat plate flow friction, 318

wall stress, 373

Drift flux correction

convective mass transport, 312

diffusion-induced convection, 195–196

Dropwise condensation, 846

Drug release from capsules, 26, 282–283

Dry bulb temperature, 812–815

Drying. See Humidification and drying

Dual mode transport in gas separation membranes, 878–879

E-curves in transient balance, 441

Eddy diffusivity

convective mass transfer in turbulent flow

gas–liquid interface, 428

velocity profiles, 415–416

description, 14

Eddy viscosity

convective mass transfer in turbulent flow

closure models, 412–413

gas–liquid interface, 427–428

van Driest model, 426

turbulent flow reactors, 580

Effective diffusivity of fluids in porous solids, 251–254

Effectiveness factor

oxygen transport in tissues, 759

oxygen uptake in lungs, 755

porous catalysts

diffusion-reaction model, 591–594, 601, 604–605

finite difference methods, 619

linking with reactor models, 622

multiple species, 607

spherical catalysts, 594–595

temperature effects, 613

Eigenvalues and eigenfunctions

channel flows with mass transfer, 351–352

Dirichlet case in slab solutions, 269–272

laminar flow in pipes, 340–341

matrix representation, 456

Neumann problem, 837

Robin condition in slabs, 276–278

tracer response method, 487

transient diffusion processes

CHEBFUN calculations, 297–298

cylinders and spheres, 278–280

wall reaction, 344–345

Einstein equation for transport of charged species, 545

Einstein relation for electric field, 548

Electric field in electrolytic systems

charge neutrality, 547

across charged membranes, 552

general expression, 548–549

Laplace equation for potential, 549

mass balance for reacting systems, 550–551

Nernst-Planck equation, 546

transfer rate in diffusion film

near electrodes, 558–559

transference number, 550

transport of charged species, 545–546

Electrochemical batch reactors, 806–807

Electrochemical processes applications, 26–27

Electrochemical reaction engineering

copper electrowinning, 795–798

definitions

anodic and cathodic reactions, 776–777

classification of electrode reactions, 779–780

half reactions and overall reaction, 777–779

overview, 776

primary variables, 780–781

hydrogen fuel cells, 798–800

kinetic model

Butler-Volmer equation, 788–791

overview, 786–788

li-ion batteries, 800–802

mass transfer effects, 791–793

overview, 775–776

problems, 805–808

review questions, 804–805

summary, 802–804

thermodynamic considerations, 781–786

voltage balance, 793–795

Electrochemist’s dream, 777

Electrodes

reaction classification, 779–780

transfer rate in diffusion film near, 556–559

Electrodialysis and electrophoresis

electrodialyzer design

current and voltage, 952–953

detailed models, 955

limiting current, 954–955

overview, 951

electrophoresis principle, 955–957

electrophoretic separation devices

Hannig design, 959–960

overview, 957

Philpot design, 957–959

rotating annular beds, 960–962

overview, 945–946

problems, 963–964

review questions, 963

summary, 962–963

technological aspects

bipolar membranes, 950–951

electrodialysis reversal process, 949

guidelines, 948

membranes, 948–949

overview, 946–948

Electrodialysis reversal (EDR) process, 949

Electrolyte transport across uncharged membrane, 551–553

Electrolytic systems. See Mass transport in electrolytic systems

Electroneutrality across charged membranes, 554–555

Electrons in semiconductor and solar devices, 24–25

Empirical models for drying falling rate period, 834–835

Emulsion liquid membranes (ELMs), 743–744

Energy balances in condenser model, 864

Enhancement factor

gas–liquid reactions in film theory models

absorption in slurries, 697

average rate of mass transfer, 711

bimolecular reactions, 677, 679, 682–683, 713, 716

coupling with reactor models, 691–692

first-order reaction of dissolved gas, 666–668

reactive membranes and facilitated transport

co- and counter-transport, 738

single solute diffusion, 731–732, 736

Enthalpy balance equations in cooling towers, 819–821

Entry regions

convective mass transfer, 348–350

laminar flow in pipes, 337

Environmental applications, 28

Environmental transport in compartmental models

fugacity of pollutants, 460

Level I, 460–461

Level II, 461

Level III, 461–462

Level IV, 462

overview, 459

Eotvos number for spherical cap bubbles, 397

Equilibrium constants

adsorption, 19

nonlinear models, 19–20

reactive membranes and facilitated transport

co- and counter-transport, 737

single solute diffusion, 734

single solute diffusion, 729, 731

Equilibrium curves for evaporation, 814

Equilibrium models

adsorption and chromatography

batch slurry adsorbers, 929

fixed bed adsorption, 932–934

Equilibrium models

environmental transport, 460–461

macroscopic models

fluid–fluid systems, 464–465

multistage cascades, 466–468

reactor–separator combination, 97

nonlinear, 19–20

plug-backmixed model, 499

reactive membranes and facilitated transport

co- and counter-transport, 738–739

computational scheme, 739–742

Equilibrium mole fraction in evaporation, 812

Equilibrium potential in electrochemical reactions, 781–786

Equilibrium stage model description, 36

liquid–liquid extraction, 115–116

Equimass counter-diffusion, 162–163

Equimolar counter-diffusion (EMD), 161–162

convective mass transport, 312

diffusion-induced convection, 193

film model, 204–205

non-reacting systems, 533–534

steady state diffusion, 188–189

Equipment-level model for semi-permeable membranes, 907

erfc function for semi-infinite slab analysis, 290

Erythrocyte, glucose uptake in, 747

Ethane, Lennard-Jones constant for, 229

Ethanol

diffusivity in water, 242

distillation of, 534–535

for methanol poisoning, 736

vapor phase dehydrogenation of, 527

in water

mole fraction, 912

separation of, 909

Wilke-Chang equation, 241

Ethylene oxide production, 586–587

Euler schemes in reactor models, 690

Evaporation

diffusion-induced convection, 187–188, 194

film model, 204

multicomponent systems

binary liquid mixtures, 531–533

ternary liquid mixtures, 528–530

in open containers, 197–198

wet and dry bulb temperature, 812

Exact analysis for flat plate with low flux mass transfer, 371–376

Excess reactants in porous catalysts multiple species, 606

Exchange current in Butler-Volmer equation, 788–791

Exit stream in macrofluid models, 450

expint function, 451

expm function, 527

Exponential matrix concept, 532

External flows in convective mass transport, 310–311

External mass transfer in porous catalysts, 591

External resistance in porous catalysts, 597

External transport effects in porous catalysts, 592–593

Extraction factor

liquid–liquid extraction, 116

multistage cascades, 467

Eyring theory, 237–238

F-curves in transient balance, 441–442

Facilitated diffusion, 257–258

Facilitated transport and reactive membranes. See Reactive membranes and facilitated transport

Facilitation factor in counter-transport, 738

Falkner-Skan equation for flow over inclined and curved surfaces, 390, 393–394

Falling rate period in drying

capillary flow models, 838

diffusion type of models, 835–838

empirical models, 834–835

model selection, 838–839

overview, 833–834

Faraday constant for transport of charged species, 545

Faraday’s law in electrodialyzer design, 951–952

Fast reactions

diffusion film near electrodes, 559

dissolved gas, 667

film theory models, 690–691

heterogeneous reactions, 172, 526

high flux model surface reactions, 211

homogeneous reactions, 382

reactive membranes, 738

silicon oxidation, 207–208

Fate and contaminant transport, 28

Fermi level in electrochemical reaction engineering, 787–788

Fibrous solids, drying, 829

Fick’s law

boundary conditions, 171–172

combined flux equation, 187

convective mass transfer, 356, 358

convective mass transport, 312

differential models, 31

diffusion coefficients

in gases, 226, 234

in liquids, 239

diffusion-induced convection problems, 170

diffusion modeling, 12–14

film model, 202

gas separation membranes, 874

gas–liquid reactions in film theory models

bimolecular reaction, 716

first-order reaction of dissolved gas, 665

flux, 710

laminar flow in pipes, 342

mass basis, 167

mass transfer equations, 158–160

mass transfer from spheres to stagnant gas, 192

mass transport in electrolytic systems

electrolyte transport across

uncharged membrane, 552

transfer rate in diffusion film near electrodes, 558–560

transport of charged species, 545–546

mole basis, 169–170

multicomponent diffusivity matrix, 536–539

multicomponent transport, 518–519

non-ideal liquids, 243

oxygen profiles in pools of liquid, 61

porous catalysts, 591

pseudo-binary diffusivity, 165

semi-permeable membranes, 903

slab solutions, 273

steady state diffusion

radial, 69

with reaction in slabs, 57

across slabs, 54

across spherical shells, 74

Taylor model, 488, 491

transient diffusion, 62, 266

Film concept in mass transfer analysis

boundary layer concept for fluid–solid mass transfer, 198–201

concentration profiles, 203–204

determinacy correction factor, 204–205

model approximation, 201–202

overview, 198

porous catalysts, 590

problems, 219–221

review questions, 219

summary, 217–218

Film flow in convective mass transfer

gas absorption from interface, 357–358

overview, 318–320, 353–354

solid dissolution at walls, 354–356

Film model

condensation, 850

flux transfer rate in diffusion film near electrodes, 556–559

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

laminar boundary layer high flux analysis, 383–384

overview, 185–186

Film thickness in condensation, 847–848

Filmwise condensation, 846–847

Filters in liquid separation membranes, 898

Finite difference methods for porous catalysts

central difference equations, 618–619

Neumann and Robin conditions, 621

nonlinear kinetics, 620–621

overview, 617–618

zero-order reaction, 619–620

First-order reactions

axial dispersion model for turbulent cases, 581

continuous stirred tank reactors, 107

diffusion with, 59, 76

dissolved gas

boundary conditions, 663–664

dimensionless version, 664–665

enhancement factor, 666–668

flux values, 665–666

overview, 662–663

gas–liquid reactions in film theory, 72

gas–liquid reactions in penetration theory, 706–707

low flux model in surface reactions, 206–207

mesoscopic models

dispersion model, 479, 481

plug flow idealization, 478

tubular flow reactors, 131, 133

porous catalyst linking with reactor models, 623–624

porous catalysts

diffusion-reaction model

apparent rate constant, 597–599

effectiveness factor, 591–592

external resistance, 597

long cylinders, 593–594

overview, 589–591

shape normalization, 595–596

slab external transport effects, 592–593

slab solution for Dirichlet condition, 591

spheres, 594–595

steady state mass transfer with reaction, 71–72

surface reactions in shrinking

core model, 638

volume reaction model, 646–649

Fisher-Tropsh synthesis in three-phase catalytic reactions, 608

Fixed bed adsorption

axial dispersion effects, 934–936

equilibrium model, 932–934

heterogeneous model, 936

Klinkenberg equation, 936–937

overview, 931–932

scale-up aspects, 937

Fixed bed type in three-phase catalytic reactions, 608

Fixed-site carrier membranes, 744–745

Flat plate flow

convective mass transport, 316–318

flow over inclined and curved surfaces, 389

laminar, 201

Flat plate with low flux mass transfer

concentration equation, 367–368

exact or Blasius analysis, 371–376

overview, 366–367

scaling results and analogies, 369–370

velocity equations, 368–369

Flow-averaged concentration in laminar flow in pipes, 342–343

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, 395–396

similarity variable, 392

stagnation point, 394–395

Flow rate in liquid separation membranes, 898–899

Flow ratio parameter

ideal flow patterns, 501

plug-backmixed model, 498

Flow regimes in condensation, 847

Flow weighted concentration in mesoscopic models, 39, 126

Flowing phases in tracer response in two-phase systems, 506–507

Fluid bed dryers, 827

Fluid bed reactors, 660

Fluid–fluid systems

backmixed–backmixed model, 462–464

equilibrium model, 464–465

mixing cell model, 465

overview, 462

Fluids in porous solids, diffusion with

liquid-filled pores, 250–251

overview, 248

porous catalysts, 251–254

single-pore gas diffusion, 248–250

Fluid–solid interface, adsorption isotherm at, 19

Flux and flux models

Blasius approach, 386

condensation, 852, 858–859, 861

convective mass transfer for laminar flow

entry regions, 349

internal and external flows, 310–311

convective mass transfer in turbulent flow

channels and pipes, 418–421

time-averaged equation, 409

van Driest model, 426

definition, 10–12

differential models, 31

diffusion

with convection, 65–66

properties, 163–165

steady state, 53, 187, 189–191

transient processes, 266

diffusion coefficients in gases, 226

diffusion-induced convection, 193–196

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

electrodialyzer design, 955

equations

mass basis, 155

mole basis, 153–155

evaporation of liquids in ternary mixtures, 529–530

film model, 202, 383

flat plates, 366–367

gas permeator models, 882

Flux and flux models

gas–liquid reactions in film theory models, 665–666, 710–711, 716

heterogeneous reactions, 525–526

laminar flow, 341–344, 349

liquid separation membranes pervaporation, 910–911

pore size, 899

semi-permeable membranes, 902–903, 906

mass transport in electrolytic systems

across charged membranes, 555

transport of charged species, 546

across uncharged membrane, 551–552

multicomponent transport, 518–519

porous catalysts in diffusion-reaction model, 590, 594

reacting systems, 520–522, 525

reactive membranes and facilitated transport

co- and counter-transport, 738–739

equilibrium model, 742

slab solutions, 273–274, 276, 292–293

stirred reactors, 437–438

surface reactions

first-order, 206–207

nonlinear, 209–210

product counter-diffusion, 210–212

thermal diffusion, 258

tracer studies, 439, 442, 445

transient diffusion processes

cylinders and spheres, 280

integral method, 292

semi-infinite slab analysis, 290

turbulent, 409

Flux of species in reactive membranes, 733–734

Flux vector

convection flux, 12

diffusion flux, 12–15

molar and mass flux, 10–12

solid spheres sublimation, 34–35

Fog formation, 855–857

Force balance model for gas bubbles, 323

Forced draft cooling towers, 816

Forward osmosis in liquid separation membranes, 907–908

Fractional extent of reaction in film, 666

Fractional order kinetics in stirred reactors, 437

Fractional order reactions in porous catalysts, 630

Frames of reference in mass transfer equations, 156–163

Free volume theory for polymeric membranes, 256–257

Freundlich isotherms in adsorption, 923

Frictional interpretation in diffusion coefficients in gases, 232–234

Frictional resistance in diffusion coefficients in liquids, 239–240

Froessling-Marshall correlation in solid spheres, 320

Froude number in packed bed absorbers, 329

FSOLVE

backmixed–backmixed model, 115

evaporation of binary liquid mixtures, 532

reactive membranes equilibrium model, 741

stirred reactors, 438

Fuel cells

half reactions, 778–779

overview, 798–800

Fugacity of pollutants, 460

Fuller correlation for diffusion coefficients in gases, 231–232

Fully turbulent flow in convective mass transfer, 405

FZERO function, 717

Galvanostatic mode in electrochemical batch reactors, 807

Gamma function in convective mass transfer, 349

Gas, diffusion coefficients in. See Diffusion coefficients

Gas absorption column design, 141–142

from interface in convective mass transfer film flow, 357–358

Gas balance equation in fixed bed adsorption, 935–936

Gas bubbles in convective mass transport, 321–325

Gas concentration profiles in volume reaction model, 646, 650

Gas film diffusion in shrinking core model, 642

Gas film resistance in bimolecular reactions, 681–684

Gas holdup in gas bubbles, 325

Gas-liquid interface

convective mass transfer in turbulent flow, 427–430

Henry’s law, 15–17

two-film model, 212–217

Gas-liquid mass transfer

convective mass transport film flow, 319–320

packed bed absorbers, 327–329

Gas-liquid reactions in film theory

absorption in slurries

instantaneous reaction case, 694–697

overview, 692–693

particle size effect, 693–694

bimolecular reactions, 714

contactor choice, 684

dimensionless representation, 673–675

instantaneous asymptote, 678

instantaneous case, 681–684

invariance property, 676–677

overview, 672–673

pseudo-first-order case, 677

second-order reactions, 678–680

bulk concentration and bulk reactions, 668–672

coupling with reactor models, 688–692

first-order reaction of dissolved gas

boundary conditions, 663–664

dimensionless version, 664–665

enhancement factor, 666–668

flux values, 665–666

overview, 662–663

liquid–liquid reactions, 697–698

overview, 661–662

problems, 701–704

review questions, 701

simultaneous absorption of two gases, 684–688

summary, 698–700

Gas-liquid reactions in penetration theory

bimolecular reaction

dimensionless form, 712–713

ideal contactors, 717–721

illustrative results, 713–714

instantaneous reaction case, 714–717

overview, 712

concepts

average rate of mass transfer, 710–711

film theory and penetration theory, 711–712

first-order and pseudo-first-order reactions, 706–707

Laplace transform method, 707–710

overview, 706

overview, 705–706

problems, 723–726

review questions, 722–723

summary, 721–722

Gas phase balance in macroscopic models

fluid–fluid systems, 463–464

ideal flow patterns, 500–501

plug-backmixed model, 497

reactor–separator combination, 98

Gas-phase driving force in two-film model, 214–215

Gas-solid reaction models, 651–654

Gas transport in membranes

dissociative diffusion, 877–878

gas permeator models

backmixed-backmixed model, 884–886

countercurrent flow, 886–889

cross-flow pattern, 889–890

flux relations, 882

local concentration, 883–884

overview, 881–882

gas separation membranes

classification, 873–874

nonlinear effects, 878–879

overview, 872–873

permeability transport rate, 874–876

permeance transport rate, 876–877

selectivity, 877

gas translation model, 879–881

overview, 871–872

problems, 894–895

reactor coupled with membrane separators, 890–891

review questions, 893–894

summary, 892–893

Gaussian distribution

chromatography, 938

electrophoretic separation devices, 959

tracer response method, 487

Gel solids, drying, 829

Generation term

concentration, 30

plug flow model, 476

porous catalysts, linking with reactor models, 622

silicon oxidation, 208

stirred reactors, 33–34

tubular flow reactors, 41, 130

Glass transition temperature for polymeric membranes, 256

Glass walls, helium leakage rate across, 55–56

Glucose uptake in erythrocyte, 747

Gradients, temperature, in porous catalysts, 613–614

Graetz number and problem

convective mass transfer

entry region analysis, 349–350

turbulent flow, 417

laminar flow in pipes, 339–342

PDEPE solutions, 358–359

Grain model for gas–solid reaction models, 653

Granular solids, drying, 828–829

Graphite and li-ion batteries, 800

Grashof number for convective mass transport in solid spheres, 321

Gray’s theorem

and Leibnitz rule, 183–184

multiphase systems, 177–178

Green-Gauss theorem, 173

Grotthuss effect for transport of charged species, 545

Groundwater transport, 28

Gupta-Thodos correlation for packed bed reactors, 333

Hadamard-Rybczynski regime in gas bubbles, 322

Half reactions

copper electrowinning, 795

definitions, 777–779

electrochemical reaction, 784

plug flow idealization, 478

Hannig design for electrophoretic separation devices, 959–960

Hatta number in gas–liquid reactions

in film theory, 711–712

bimolecular reactions, 674, 679–680, 714

bulk concentration and bulk reactions, 669

first-order reaction of dissolved gas, 664–665

simultaneous absorption of two gases, 686

in penetration theory, 713

Hatta parameters

co- and counter-transport, 737

single solute diffusion, 730–731

Heat balance

in cooling towers, Merkel equation for, 821–823

porous catalysts temperature effects, 612

Heat transfer

condensation

binary gas mixtures, 860

vapor with non-condensible gas, 851–852

drying constant rate period, 831

flat plate with low flux mass transfer, 370

laminar flow reactors, 578–579

Heat transfer coefficient

condensation, 846–848

cooling towers, 822

drying falling rate period, 835

evaporation, 812

Heat transfer rate in cooling towers, 820–821

Heat transport equations, temperature effects in, 613–614

Heaviside step function for fixed bed adsorption, 933

Helium

diffusion volume, 232

leakage rate across glass walls, 55–56

tracer response in two-phase systems, 506

Hematocrit, 768

Hemodialyzer models, 763–766

Henry parameter for fluid–fluid systems, 463, 465

Henry’s law

bimolecular reactions, 673–674

boundary conditions, 171–172

for CO₂, 47

fugacity of pollutants, 460

gas separation membranes, 878

gas–liquid interface, 15–17

oxygen-hemoglobin equilibrium, 751

reactor–separator combination, 98–99

two-film model, 216–217

Heterogeneous media in diffusion, 254–256

Heterogeneous membranes in electrodialysis, 949

Heterogeneous models in fixed bed adsorption, 932, 936

Heterogeneous reactions

multicomponent systems, 525–527

ternary diffusion, 527

Hexachlorobiphenyl, 473

Hiemenz flow, 394–395

High flux analysis for laminar boundary layers

Blasius approach, 386–388

film model, 383–384

integral balance method, 384–385

overview, 383

High flux model for surface reactions product counter-diffusion, 210–212

Hikita-Asai model for reactor model coupling, 689

Hill’s equation

oxygen binding, 768

oxygen-hemoglobin equilibrium, 752–753

Hindered diffusion of fluids in porous solids, 250–251

Homogeneous membranes in electrodialysis, 949

Homogeneous reactions in integral balance approach, 381–382

Homogeneous solution in matrix representation, 456

HTU (height of a transfer units)

mass exchangers, 143–144

Merkel equation, 823

Humidification and drying

cooling towers

counterflow model, 817–823

cross-flow, 825–827

overview, 815–817

drying

constant and falling rates, 829–830

constant rate period, 830–833

dryer types, 827–828

falling rate period, 833–839

overview, 827

solid types, 828–829

overview, 811–812

problems, 841–843

review questions, 841

summary, 839–841

wet and dry bulb temperature, 812–815

Hydrocracking of porous catalysts, 608

Hydrodesulfurization of diesel, 608

Hydrodynamic hindrance factor of fluids in porous solids, 251

Hydrogen

diffusion volume, 231–232

Henry’s law constant, 16

Lennard-Jones constant, 229

molar volume in liquid, 241

transport in palladium membranes, 873–874

transport rate in presence of convection, 66–67

trickle bed reactors, 21–22

Hydrogen fuel cells

half reactions, 779

overview, 798–800

Hydrogen ion concentration, 956

Hydrogen sulfide (H₂S)

absorption

in amine solution, 716–717

in NaOH solution, 683–684

bearing gas, sulfur from, 702–703

Hydrogenation reactions in porous catalysts

three-phase catalytic reactions, 608

zero-order reaction, 599

Hygroscopic solids, 829

Hyperboloid design for cooling towers, 816

Hypergeometric function for laminar flow in pipes, 340

Hysteresis phenomena in adsorption, 924