TRANSPORT MODELING FOR ENVIRONMENTAL ENGINEERS AND SCIENTISTS

TRANSPORT MODELING FOR ENVIRONMENTAL ENGINEERS AND SCIENTISTS

Second Edition

MARK M. CLARK

)WILEY A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Clark, Mark M. Transport modeling for environmental engineers and scientists / Mark M. Clark. - 2nd ed.

p. cm. Summary: "Transport Modeling for Environmental Engineers and Scientists, Second

Edition, builds on integrated transport courses in chemical engineering curricula, demonstrating the underlying unity of mass and momentum transport processes. It describes how these processes underlie the mechanics common to both pollutant transport and pollution control processes"-Provided by publisher.

Includes index. ISBN 978-0-470-26072-2

1. Environmental chemistry-Mathematical models. 2. Transport theory-Mathematical models. I. Title.

TD193.C55 2009 628.53-dc22

2009025905

Printed in Mexico

10 9 8 7 6 5 4

Consider how all events are interconnected. When we see the lightning, we listen for the thunder; when we hear the wind, we look for the waves on the sea; in the chill autumn, the leaves fall. Everywhere order reigns, so that when some circum-stances have been noted we can foresee that others will also be present. The progress of science consists in observing these interconnections and in showing with patient ingenuity that the events of this evershifting world are but examples of a few general connections or relations called laws. To see what is general in what is particular and what is permanent in what is transitory is the aim of sci-entific thought.

—Alfred North Whitehead, Introduction to Mathematics, Barnes and Noble Publishing, 2005 (originally published in 1911)

CONTENTS

Preface

Acknowledgments

List of Symbols

1 Conservation Laws and Continua

1.1. Introduction, 1 1.2. Conservation Laws: Systems Approach, 2 1.3. Conservation Laws: Control Volume Approach, 4 1.4. Conservation Laws: Differential Element Approach, 10 1.5. Continua, 14 1.6. Sources, Sinks, Reactions, and Box Models, 18 1.7. Summary, 20 Exercises, 21 References, 32 Bibliography, 32

2 Low-Concentration Particle Suspensions and Flows

2.1. Introduction, 33 2.2. Drag on a Sphere, 34 2.3. Drag Force on Nonspherical Particles, 37 2.4. Low Reynolds Number Particle Dynamics and Stokes'

Law, 39 2.5. Particle Motions in Electric Fields, 43

viii CONTENTS

2.6. Quiescent and Perfect-Mix Batch Sedimentation, 45 2.7. Continuous Sedimentation Processes, 48 2.8. Inertial Forces on Particles and Stopping Distance, 55 2.9. Inertial Forces in Particle Flows, 56 2.10. Rotating Flows, 59 2.11. Centrifugation, 62 2.12. Summary, 66 Exercises, 68 References, 81 Bibliography, 82

3 Interactions of Small Charged Particles 83

3.1. Introduction, 83 3.2. Importance of Surface, 85 3.3. Acquisition of Surface Charge, 88

3.3.1. Hydrosols, 89 3.3.2. Aerosols, 91

3.4. Particle Size, Shape, and Polydispersity, 91 3.5. The Double Layer and Colloidal Stability, 96

3.5.1. Debye-Hückel Model of the Double Layer, 97 3.5.2. Overlapping Double Layers and Interparticle

Repulsion, 101 3.5.3. Van der Waals Forces and the DLVO Theory, 103

3.6. The Schulze-Hardy Rule, 106 3.7. Electrophoresis and Zeta Potential, 108 3.8. Particle Collision and Fast Coagulation, 112

3.8.1. The General Dynamic Equation, 112 3.8.2. Collision-Frequency Functions, 114 3.8.3. Simplified Discrete Particle Dynamics, 119 3.8.4. Simple Kinetics with Mass Conservation, 120 3.8.5. Simple Kinetics with Mass Conservation and

Breakup, 123 3.9. Slow Coagulation, 125 3.10. Summary, 128 Exercises, 129 References, 133 Bibliography, 134

4 Adsorption, Partitioning, and Interfaces 135

4.1. Introduction, 135 4.2. Accumulation of Solutes at Interfaces, 136

4.2.1. Gibbs Adsorption Isotherm, 136 4.2.2. Orientation of Large Molecules at Interfaces, 141

4.3. Adsorption at Solid-Liquid and Solid-Gas Interfaces, 145

CONTENTS ix

4.4. Adsorption Isotherms, 147 4.5. Linear Equilibrium Partitioning Between Two

Phases, 157 4.5.1. Dalton's and Raoult's Laws, 157 4.5.2. Henry's Law and Partition Coefficients, 160 4.5.3. Mass Balances and the Partition Coefficient, 163

4.6. Partitioning and Separation in Flow Systems, 165 4.7. Summary, 168 Exercises, 169 References, 172 Bibliography, 173

5 Basic Fluid Mechanics of Environmental Transport 175

5.1. Introduction, 175 5.2. The Joy of Fluid Mechanics, 176 5.3. The Navier-Stokes Equations, 179 5.4. Fluid Statics and the Buoyancy Force, 181 5.5. Capillarity and Interfacial Tension, 185 5.6. The Modified Pressure and Free-Surface Flows, 189 5.7. Steady Unidirectional Flows and Steady Circular Streamline

Flows, 190 5.8. Fluid Shear Stresses and the Viscosity of Newtonian

Fluids, 194 5.9. Slip Flow, 196 5.10. Field-Flow Fractionation, 198 5.11. Nonsteady Unidirectional Flows: Stokes' First Problem, 200 5.12. Low Reynolds Number Flows, 203

5.12.1. The Reynolds Number and the Stokes Equation, 203 5.12.2. Sphere in Axisymmetric Low Re Flow: Stokes' Drag

Law, 205 5.12.3. Flow Through Porous Media, 208

5.13. Ideal Fluids, Potential Flows, and Stream Functions, 212 5.13.1. Inviscid Flows and the Euler Equation, 212 5.13.2. Irrotational Flows and the Velocity-Potential

Function, 212 5.13.3. The Stream Function, 213

5.14. The Bernoulli Equation, 217 5.15. Steady Viscous Momentum Boundary Layers, 223 5.16. Turbulent Flows, 229

5.16.1. Characteristics of Turbulence, 230 5.16.2. Reynolds Averaging, 234 5.16.3. Reynolds Stresses and the Closure Problem, 237 5.16.4. Eddy Viscosity and the Mixing-Length Model, 239 5.16.5. Advanced Turbulence Modeling: the k-e Model, 241

X CONTENTS

5.16.6. Isotropie Turbulence and Kolmogoroff s Universal Equilibrium Range, 242

5.16.7. Flow Through Cylindrical Conduits, 244 5.16.8. Turbulent Boundary Layers and Universal Velocity

Laws, 247 5.17. Summary, 250 Exercises, 251 References, 265 Bibliography, 267

6 Diffusive Mass Transport 268

6.1. Introduction, 268 6.2. Thermodynamics of Diffusion, 269 6.3. Fick's First Law and General Diffusive Transport, 271 6.4. The Diffusion Coefficient, 275

6.4.1. Classical Approaches, 275 6.4.2. Effect of Solution Conditions on Diffusion

Coefficient, 280 6.4.3. Knudsen Diffusion, 282

6.5. Steady-State Diffusion Problems with No Overall Diffusive Mass Transfer, 283

6.6. Steady-State Mass Balances Over Differential Elements, 284 6.6.1. Diffusion of One Gas Through Another Stagnant

Gas, 284 6.6.2. Diffusion with Heterogeneous Reaction at the Particle

Surface, 289 6.6.3. Brownian Coagulation as a Steady, Diffusion-Limited,

Heterogeneous Reaction, 295 6.6.4. Diffusion with Homogeneous First-Order Reaction, 296

6.7. Fick's Second Law and Nonsteady-State Diffusion, 298 6.7.1. Fick's Second Law, 298 6.7.2. Diffusion from a Plane Source, 300 6.7.3. Diffusion in a Semi-Infinite Medium with Constant

Boundary Concentration, 304 6.7.4. Diffusion and Partitioning into a Homogeneous

Sphere, 306 6.8. Effective Diffusion Coefficients in Porous Media, 309 6.9. Hindered Diffusion, 313 6.10. When Chemicals Diffuse Against a Concentration

Gradient, 315 6.11. Summary, 317 Exercises, 318 References, 333 Bibliography, 335

CONTENTS xi

7 Convective Diffusion, Dispersion, and Mass Transfer 336

7.1. Introduction and Simple Example of Convective Diffusion, 336

7.2. The Convective-Diffusion Equation, 340 7.3. Mass Transport in Steady Laminar Flow in a Cylindrical

Tube, 342 7.4. Taylor-Aris Dispersion, 352 7.5. Turbulent Dispersion: The Lagrangian Approach, 353 7.6. Turbulent Dispersion: The Eulerian Approach, 359 7.7. Mass Transfer in Laminar Flow Along Reacting or Dissolving

Solid Surfaces, 364 7.7.1. Flat Plate with Fast Heterogeneous Reaction or

Constant Surface Concentration, 365 7.7.2. Laminar Tube Flow with Fast Heterogeneous Reaction

or Constant Surface Concentration, 372 7.7.3. Laminar Flow Past Sphere with Fast

Heterogeneous Reaction or Constant Surface Concentration, 377

7.8. Mass-Transfer Coefficients, Models, and Correlations for Laminar and Turbulent Flows, 381 7.8.1. Mass-Transfer Coefficients and the Sherwood

Number, 382 7.8.2. Mass-Transfer Models and Analogies, 384 7.8.3. Mass-Transfer Correlations and Calculations, 394

7.9. Interphase Mass Transport and Resistance Models, 403 7.9.1. Nonsteady-State Mass Transfer to Particle, 404 7.9.2. Steady-State Interphase Mass Transfer with

Homogeneous Reaction in One Phase, 407 7.9.3. Steady-State Interphase Mass Transfer with

Heterogeneous Reaction, 410 7.9.4. Two-Film Models, Limiting Forms, and Gas

Absorption, 416 7.10. Summary, 424 Exercises, 424 References, 434

8 Filtration and Mass Transport in Porous Media 437

8.1. Introduction, 437 8.2. Porosity, Velocity, and Porous Media Continua, 440 8.3. Coefficients of Mechanical, Molecular, and Hydrodynamic

Dispersion, 443 8.4. Porous Media Dispersion Equation in a Homogeneous

Isotropie Medium, 446

CONTENTS

8.5. Solution of the Dispersion Equation in an Infinite One-Dimensional Medium, 448

8.6. Analytical Chromatography, 453 8.7. Filtration, 460

8.7.1. Classic Capture Mechanisms for Isolated Collectors, 460 8.7.2. Effect of Adjacent Collectors on Local Flow

Field, 472 8.7.3. Hydrodynamic Retardation, 473 8.7.4. Collection Efficiencies for Fibrous and Granular

Filters, 474 8.7.5. Attachment Probability, 476 8.7.6. Head Loss Through Filter Beds and Mats, 477

8.8. Osmotic Pressure and Reverse Osmosis, 478 8.8.1. Osmosis, 478 8.8.2. Reverse Osmosis, 485

8.9. Summary, 488 Exercises, 489 References, 493 Bibliography, 494

Reaction Kinetics 495

9.1. Introduction, 495 9.2. First-Order Reactions, 496 9.3. Second-Order Reactions, 503 9.4. Pseudo-First-Order Reactions, 507 9.5. Zero-Order Reactions, 508 9.6. Elementary and Nonelementary Reactions, 509 9.7. Simple Series and Parallel Reactions, 512 9.8. Reversible Reactions, 517 9.9. Characteristic Reaction Times, 523 9.10. Arrhenius' Law and the Effect of Temperature on Reaction

Rate, 525 9.11. The Fastest Reactions: Diffusion-Controlled Reactions, 527 9.12. Summary, 528 Exercises, 529 References, 536 Bibliography, 537

Mixing and Reactor Modeling 538

10.1. Introduction, 538 10.2. Simple Closed-Reactor and Residence-Time Distributions, 539 10.3. Measurement of Residence-Time Distributions, 546 10.4. Residence-Time Distributions from Discrete Data, 550

CONTENTS xiii

10.5. Perfect Mixing and Ideal Plug Flow, 555 10.6. F, W, and Disinfection, 561 10.7. Moments of Residence-Time Distributions, 563 10.8. Other Residence-Time Models, 567

10.8.1. Tanks in Series, 567 10.8.2. Gamma-Function Extension to Tanks-in-Series

Model, 569 10.8.3. Fractional Tubularity, 571 10.8.4. Crossflow and Stagnancy, 574 10.8.5. Recycle Models, 575

10.9. Axial-Dispersion Model, 580 10.10. Fitting Residence-Time Distributions to Data, 581 10.11. Mixing and Reactions, 583

10.11.1. Reaction Probabilities and Conversion for First-Order Reactions, 583

10.11.2. Conversion for Second-Order Reactions, 590 10.11.3. Complete Segregation and Micromixing, 592 10.11.4. Plug Flow Versus Perfect Mixing, 596

10.12. Summary, 597 Exercises, 598 References, 605 Bibliography, 605

Appendix I. SI Units and Physical Constants 606

Bibliography, 607

Appendix II. Review of Vectors 608

Bibliography, 615

Appendix III. Equations of Fluid Mechanics and Convective Diffusion in Rectangular, Cylindrical, and Spherical Coordinates 616

Bibliography, 619

Appendix IV. Physical Properties of Water and Air 620

Bibliography, 621

Index 622

PREFACE

The second edition of Transport Modeling for Environmental Engineers and Scientists continues to be an effort to provide students an introduction to mod-eling of mass- and momentum-transport processes in the environment and pollution control equipment. The book strives to present material relevant to transport processes in air, water, and soil, and several areas have been updated in the second edition. However, since its original publication in 1996, our field has increasingly incorporated techniques from biotechnology and nanotech-nology; therefore, the second edition has an increased focus on the physical chemistry of selected biological analyses and nano-scale phenomena.

The following areas have been expanded on in the second edition: Cascade impactors; ultracentrifugation and biomolecules; coagulation modeling and breakup; zeta-potential, streaming-potential, and surface forces; octanol-water partition coefficients; Langmuir adsorption; hydrogen bonds and surfactants; chromatography and flow-field fractionation; nanofluidics and slip flows; capillarity, contact angle, and hydrophobicty; Donnan, hindered, and Knudsen diffusion; osmotic pressure and reverse osmosis; disinfection and disinfection byproducts; catalysis and enzymes; diffusion-controlled reactions; and tracers. More than 30 new exercises have been added in the second edition. For professors who have adopted this book and desire access to the solutions manual, the mannal can be accessed via the Wiley website for this title: http:// www.wiley.com/WileyCDA/WileyTitle/productCd-0470260726.html

SI units are used almost exclusively throughout the book, but other systems of units are occasionally used when they are more traditional. Many approxi-mations, such as » or « , and >» or <«, are also used. In this book, these symbols have the following meanings:

XV

xvi PREFACE

< or > Less than or greater than. <c or » Much less than, or much greater than. The compared quantities differ

by at least a factor of 10. <SK or »> Much, much less than, or much, much greater than. The compared

quantities differ by at least a factor of 100.

A good deal of algebra and simple differential equations are used in the book, so anyone with an engineering, physics, or chemistry background should find the mathematics quite accessible. Previous course work in dynamics, fluid mechanics, mass or heat transfer, and physical chemistry would be useful, but are not required. The book has been used as an introduction to mathematical modeling in several graduate environmental engineering programs, but I have found that undergraduates in these classes have always fared well.

Finally, a reviewer of the first edition suggested that the book could help students understand how things work. That is indeed the greatest contribution I would wish for this book.

Evanston, Illinois 2009

MARK M. CLARK

ACKNOWLEDGMENTS

With completion of the Second edition of this book, and looking back at my career, I would like to acknowledge the following teachers, colleagues, students and staff members who greatly influenced my training and growth as an envi-ronmental engineer and scientist, as well as those who made specific contribu-tions to the book: Shankha Banerji, Mriganka Ghosh, Louis Hemphiil, Henry Liu, John Novak, John T. O'Connor, Donald Fancisco, Donald Lauria, Stanley Corrsin, Martin Maxey, Charles O'Melia, M. Gorden Wolman, Georges Belfort, Jean-Luc Bersillon, François Fiessinger, Joël Mallevialle, Jean-Yves Bottero, René David, Bruce Nauman, Rhodes Trussell, Philippe Aptel, Jeffrey Collett, Claudia Cook, Wayland Eheart, David Freedman, Marcelo Garcia, Edwin Herricks, Susan Larson, Valentina Lazarova, Helen Mardis, Paul Newton, Gary Peyton, Massoud Rostam-Abadi, Eberhard Morgenroth, Thanh H. (Helen) Nguyen, Bruce Rittmann, Mark Rood, Vernon Snoeyink, Timm Strathmann, Joan Stolz, Albert Valocchi, Charley Werth, Ron Winburn, Julie Zilles, Bill Batchelor, Jean-Francois Gaillard, Mark N. Goltz, Mark Wiesner, Betsy Andrews, Mesenia Atenas, Richard Bernard, Fred Cannon, Joseph Flora, Catherine Jucker, Jaehong Kim, Mary Jo Kirisits, Detlef Knappe, Jean-Michel Laîné, Li Liu, Tze-Ling Ng, Mark Rhodes, Samer Adham, Ravindra Srivastava, Timothy Kramer, Joel Ducoste, Kerry Howe, Yonghun Lee, Adri-enne Menniti, Won-Young Ahn, David Ladner, and Manish Kumar.

xvii

LIST OF SYMBOLS

a particle radius, L a initial position in Lagrangian coordinate system, L aL longitudinal dispersivity coefficient, L aT transverse dispersivity coefficient, L A Hamaker constant, J A surface area or projected area, L2

Af preexponential facter Arrhenius equation, mol-L"3-?"1

A F constant, dimensionless Ap surface area of particle, L2

At total interfacial area of system of particles, L2

b ratio of rate constants in Langmuir isotherm, dimensionless Bi Biot number, dimensionless cA concentration of component A, mol/L3

CA,R dimensionless concentration cAji concentration of A at surface, mol/L3

cA „ concentration of component A in solution phase at equilibrium, mol/L3

cA „ concentration of component A at r —> °° (or very far from surface), mol/L3

C/i,ave average concentrat ion of A , mol /L 3

ct concentrat ion of ith component , mol /L 3

csoi solute molar concentrat ion, mol /L 3

c'A concentrat ion of componen t A in interstitial fluid, mol /L 3

cA dimensionless concentra t ion c'A fluctuating componen t of A , mol /L 3

xix

XX LIST OF SYMBOLS

c* equi l ibr ium concent ra t ion of A , mol /L 3

C concent ra t ion , mol /L 3

CA m e a n concent ra t ion of A , mol /L 3

cAJ inlet concent ra t ion of A , mol /L 3

c* characterist ic concent ra t ion scale or difference, mol /L 3

Cc Cunn ingham slip coefficient, dimensionless Cc ca rbon concentra t ion, MIL? CD d rag coefficient, d imensionless CfL average skin-friction coefficient, dimensionless Cfx local skin-friction coefficient, dimensionless Cn infinite series of constants , dimensionless Cp specific heat , energy/(M-°C) Cp part iculate-associated pesticide concent ra t ion , MIL3

d d iameter , L d decay term, mol-L^-T1

dc cylindrical collector diameter, L dw differential work, M-L~2-T~2

D system or box depth , L DAB diffusion coefficient of componen t A in componen t B, L2/T DA% diffusion coefficient in aerobic layer, L2/T DA

nB diffusion coefficient in anaerobic layer, L2IT

DA second D a m k ö h l e r number , dimensionless Dd i s dispersion coefficient, L2/T Deff effective diffusion coefficient, L2/T Dsol solvent diffusion coefficient, L2IT Djj relative diffusion coefficient class i and j particles, L2IT Djj mechanical dispersion tensor, LIT Dj diffusion coefficient of class /' particles, L2IT Diy hydrodynamic dispersion tensor, L2IT Djy principal componen t form of hydrodynamic dispersion tensor,

L2IT D'dis turbulent dispersion coefficient, L2IT Dk Knudson diffusion coefficient, L2IT D dep th of uncleared fluid, L er unit vector in radial direction, dimensionless e e unit vector in 6 direction, dimensionless E electric field strength, V/L EA activation energy, M-L2-T~2-mo\A

f friction factor, dimensionless / function of f(t) residence-time density function, 7"1

/ ( T ) dimensionless residence-time density /data('i) residence-time density function based on tracer data, T~l

/modei('i) residence-time density function based on model, 7"1

ff Fanning friction factor, dimensionless

LIST OF SYMBOLS xxi

F force, M-L-T2

F Lagrangian integral time scale, T Fe force on particle due to electric field, M-L-T"2

FD drag force, M-L-T'1

F(t) cumulative residence-time distribution function, dimensionless Fg net force of gravity on particle, M-L-T2

F, ith component of force acting on particle, M-L-T2

F„ nonsteady particle drag force, M-L-T"2

g acceleration of gravity, LIT2

g dimensionless function ge ö-component of acceleration of gravity, LIT2

G Gibbs function, M-L2-T2

G potential energy, L2-7* Gn infinite series of constants, dimensionless Gr Grashof number, dimensionless G„ Gravitational parameter, dimensionless h heat transfer coefficient, energy/(L2-T-°C) H Henry's law constant, atm/mole fraction or M-L2-vaotl-T2

HA attraction or adhesion group, dimensionless i unit vector along x axis, dimensionless / current, A j unit vector along v axis, dimensionless j A mass flux of component A relative to moving coordinates,

M-L2-Tl

j , mass flux of /th component relative to moving coordinates, M-L~2-Tl

J mas flux, M-L~2-Tl or mol -L^- r 1

JA flux of component A in transformed coordinate systems, mol-L^-r1

•^4,ave average flux over plate of length L, mol-L"2-!"1

J* relative molar flux of component A, mo\-L2-Tx

if molar flux of /th component relative to moving coordinates, mol-L- ' -r1

Jk molar gas flux knudson diffusion, mol-L2-?"1

/Soi solvent molar flux, mol-L"2-?"1

k reaction rate constant, moP~n-L3~3n-T (where n is reaction order)

k Boltzmann constant, J-K"1

k permeability, L2

k thermal conductivity, energy/(L-T-°C) ka adsorption rate constant, L3-T1-M~1

kc mass-transfer coefficient, LIT kd partition coefficient, L3/M kd desorption rate constant, 7"1

ke constant in Ergun equation, L

xxii LIST OF SYMBOLS

ks sedimentation rate constant, T~]

k„ constant, dimensionless kA first-order homogeneous rate constant, T~x

kA?s first-order heterogeneous reaction rate constant, LIT kA zero-order homogeneous rate constant, mol-L"3-!"1

kc, gas-phase mass-transfer resistance, rao\-T-M~x-L~x

kL liquid-phase mass-transfer resistance, LIT k* sum of k] and k2, T~x

k unit vector along z axis, dimensionless K Kozeny constant, dimensionless KAW air-water partition coefficient, dimensionless KB consolidated breakup rate constant, T~x

Keq equilibrium constant for reversible first-order reaction, dimensionless

Kf coefficient in Freundlich equation, {MIM)(LIM)Vn where n is Freundilich parameter

Kc, (overall) gas mass-transfer coefficient, mo\-T-M~x-L~x

Kj (overall) liquid mass-transfer coefficient, LIT Kja liquid mass-transfer coefficient, L/T /C o w octanol-water partition coefficient, dimensionless Kn Knudson number, dimensionless / system or box length, L I mixing length, L L characteristic length scale LA Avogadro's number, mol"1

Lc length required for fully developed concentration boundary layer, L

Le tube length in ideal porous medium, L Le length required for fully developed flow, L Lf fiber length, L L length dimension L liter, L3

Lo van der Waals force number, dimensionless m hydraulic radius, L m source strength, L2/T ma molar mass, M/mol mA molar mass of gas A, M/mol M mass dimension M mass tracer injected MAJ mass component A adsorbed at time /, M A/4 „ mass component A adsorbed at equilibrium, M Mf mass of fluid displaced by particle, M Mp particle mass, M Ms suspended solids concentration, MIL3

Mr total mass in multiple system, M

LIST OF SYMBOLS xxiii

MT total mass of tracer M molarity, mol /L n coefficient in Freundlich equat ion, dimensionless nAf x componen t of flux of componen t A , M-L~2-T~l

nt amoun t of /th type ions, mol itj n u m b e r concentrat ion of /th type ions or particles, L"3

nm n u m b e r moles in mobil phase, dimensionless ns n u m b e r moles in stat ionary phase, dimensionless «SOL n u m b e r solvent molecules, mol «SOi n u m b e r solute molecules, mol n«, particle concentrat ion outside control volume, L"3

n outward unit normal vector, dimensionless n,, mass flux, M-L~2-T~x

N n u m b e r pe r unit volume, L"3

Ny ra te of collisions be tween particles, T~x-L~3

NL particle concentrat ion at bed exit, L"3

Nmax asymptotic max imum particle concentrat ion, L"3

N0 particle concentration at bed entrance, L"3

NT total particle number concentration, L"3

NA molar flux of component A, mol-L"2-^1

NAiZ molar flux component A at position z, mol-L~2-T*~l

N, molar flux of /th component, mol-L^-T1

Nu Nusselt number, dimensionless p modified pressure, M-L^-V1

p production term, mol-L"3-^1

p partial pressure, M-L~l-T~2

PBM log mean concentration, dimensionless p0 local atmospheric pressure, M-L~x-T~2

p„ reference pressure, M-L~x-T~2

P average pressure, M-L~x-T~2

pA(x, t) probability density function at position x and time t, L"1

p\ equilibrium partial pressure of A, M-L_1-7"2

P pressure, M-L~x-T~2

PA density of constituent A in porous medium fluid phase, M/L3

Pr reaction probability, dimensionless Pe Peclet number, dimensionless Pr Prandtl number, dimensionless q particle charge, C q recycle flow, L3/T q heat flux, energy/(L2-T) qe equilibrium adsorbent capacity, mol/M qn roots of generating function, dimensionless qr unit flow in r direction, L2/T Q flow, ÜIT Qp permeate flow, L3/T

xxiv LIST OF SYMBOLS

Q° ultimate adsorbent capacity, mol/M r radial distance, L r reaction rate, M-L~3-T~* or mot-L^-T x

rA reaction rate of component A , moVT ra adsorption rate, M-L3-Tl

r,i reaction rate of component B, mol/T rb boundary layer resistance, TIL rc surface reaction resistance, TIL ra(z) aerodynamic resistance, TIL rAz reaction rate of component A at position z, m o l - L 3 - T 4

rd desorption rate, M-L~3-T~x

rf forward reaction rate, mol/T r, radius of ith class particle, L r, radial coordinate of leading edge, L rr reverse reaction rate, mol/T rs sedimentation rate, M-L~3-T~l

r, radial coordinate of trailing edge, L r dimensionless radial coordinate R interception parameter, dimensionless R recycle ratio, dimensionless R retardation coefficient, dimensionless R fractional removal, dimensionless R universal gas constant, J -mor^-K 1

R ventilation rate, 7"1

Rh biofilm mass-transfer resistance, TIL Rd diffusion mass-transfer resistance, TIL Rf liquid film mass-transfer resistance, TIL Rfpb mass transfer resistance for fully penetrated biofilm, TIL RG gas-side resistance, M-L-motl-T~l

Rt inner dimension (radius) of couette device, L RL liquid-side resistance, M-L-maV-T~x

Rn infinite series of constants, dimensionless R0 outer dimension (radius) of couette device, L Rmn reaction mass-transfer resistance, TIL Rs surface reaction resistance, TIL Rr Lagrangian autocorrelation coefficient, dimensionless Re Reynolds number, dimensionless R e t boundary layer Reynolds number at x = L, dimensionless Rep particle Reynolds number, dimensionless Repb Reynolds number for packed bed, dimensionless Rex boundary layer Reynolds number, dimensionless 0Î retention, dimensionless s standard deviation, L s solid fraction, dimensionless 5 entropy (J/K)

LIST OF SYMBOLS

S total number of adsorpt ion sites, dimensionless S source of mater ia l , M-L~3-T~x

S s topping distance, L SA mass of A adsorbed per unit mass solids, dimensionless SAj„ equil ibrium concentra t ion of A in sphere, mol /L 3

Sd sedimentat ion coefficient, T 50 specific surface, L"1

51 grain surface a rea per unit of grain volume, L ' Si n u m b e r of adsorpt ion sites occupied, dimensionless Sc Schmidt number , dimensionless Sh Sherwood number , dimensionless St Stokes number , dimensionless T m e a n residence or de tent ion time, T tän arrival t ime, T tK re tent ion time, T fv viscous d is turbance t ime, T te exit t ime of nonadsorb ing solutes, T t0 starting time, T tw residence t ime exceeded by 9 0 % tracer molecules, T t dimensionless t ime tp m e a n residence t ime for plug-flow reactor, T fRTD mean residence t ime computed from the residence t ime

density, T Ts mean residence time for perfect mixer, T U time parameter in recycle model, T T time dimension T averaging period, T T temperature, K or °C Ts temperature at surface, K or °C T„ temperature far from surface, K or °C T dimensionless temperature fN time parameter for recycle model, T us velocity at surface, LIT ux x component of velocity, LIT uz streamwise velocity, LIT "?,ave average velocity in tube, LIT uz velocity in transformed coordinate system, LIT uB 6 c o m p o n e n t of velocity, LIT u vector velocity, LIT U{ component / of velocity, LIT U\ X\ component of velocity, LIT u2 x2 component of velocity, LIT «3 JC3 component of velocity, LIT u\ X] component of fluctuating velocity, LIT «2 x2 component of fluctuating velocity, LIT

xxvi LIST OF SYMBOLS

«3 JC3 c o m p o n e n t of fluctuating velocity, LIT we 9 c o m p o n e n t of velocity, LIT u* friction velocity, LIT u* mola r average velocity, LIT ü d imensionless velocity £/ap app roach velocity, LIT Umax bound a ry velocity, LIT £ 7 m a x centerline turbulent average velocity, LIT £/sup superficial bed velocity, LIT U0 characteristic velocity scale, LIT 0\ xx component of average velocity, LIT 02 x2 component of average velocity, LIT Ui x3 component of average velocity, LIT U* dimensionless boundary layer velocity U characteristic velocity, LIT Ut component / of velocity, LIT v particle velocity, LIT v average linear velocity in porous medium, LIT v v component of velocity, LIT ve streaming potential, L 2 -V ' -7" 1

vt settling velocity of the /th size class particle, LIT vt /th component of average linear velocity in porous medium,

LIT vr r component of particle velocity, LIT v, terminal sedimentation velocity, LIT v, Lagrangian velocity at time /, LIT vx x component of particle velocity, LIT v- fluctuation in /th component of average linear velocity in

porous medium, LIT v' Lagrangian velocity fluctuation at time t, LIT v velocity in Lagrangian coordinate system, LIT v vector form of average linear velocity in porous medium,

LIT vs average migration velocity of adsorbing species, LIT v0 average migration velocity of nonadsorbing species, LIT V voltage, V V vo lume , L 3

VA mo la r vo lume of gas A , L3/mol Vj vo lume of /th size class particle, L 3

Vt / th c o m p o n e n t of l inear velocity in po rous med ium, LIT Vm vo lume of a monolayer , L 3

VM mobi le phase vo lume, L 3

VR r e ten t ion volume, L 3

Vs s ta t ionary phase volume, L 3

Vs p o r o u s m e d i u m solid vo lume, L 3

LIST OF SYMBOLS xxvii

>^SOL,m

v, v, w vv v0 w w W

wA W(t)

w X

X

X

xA

xc

-^SOL

xi X

X

y y y+

z z> Zo

z z

solvent partial molar volume, L3/mol total volume of system of particles, L3

Lagrangian mean velocity at time t, LIT porous medium total volume, L3

porous medium void volume, L3

overflow velocity, LIT system or box width, L z component of velocity, LIT inverse of stability ratio, dimensionless mass transfer of component A, MIT washout function, dimensionless unit width mass-transfer rate, M-L~x-T~x

liquid-phase mole fraction, dimensionless amount of unreacted reactant, mol/L3

rectangular coordinate, L mole fraction component A, dimensionless position of peak in concentration, L mole fraction solvent, dimensionless x coordinate in moving coordinate system, L dimensionless distance position in Lagrangian coordinate system, L gas-phase mole fraction, dimensionless rectangular coordinate, L dimensionless boundary layer distance rectangular coordinate, L valence /th type ion, dimensionless roughness length, L transformed coordinate system, L dimensionless axial coordinate

Greek

a thermal diffusivity, L2/T a equilibrium ratio of solute in solution phase to mass of solute

in sphere, dimensionless a collision efficiency function, dimensionless ad attachment efficiency, dimensionless ßn roots of equation, dimensionless )3(r„ /j) collision frequency for class i and ; particles, L'3

T, surface excess of ith type ion, mol/L2

7 parameter in recycle model, T y potential parameter in Gouy-Chapman theory, dimensionless A del operator 4//ads heat of adsorption, J/mol Are immobile double layer thickness, L

xxviii LIST OF SYMBOLS

Af/vap heat of vaporization, J/mol AmijM incremental mass of tracer, M 8 boundary layer thickness, L Ô Dirac delta function 5d concentration boundary layer thickness, L Sf film thickness e relative permittivity, dimensionless e porosity, dimensionless e parameter in recycle model, dimensionless e unit mass energy dissipation rate, L2/T3

Ejj c o m p o n e n t of eddy diffusivity, L2IT £o permittivity of vacuum, C2-L~l-i'x

Ç dummy integration variable, dimensionless Ç zeta potential, V 77 dimensionless variable rj Kolmogoroff microscale, L T] dimensionless independent variable in boundary layer 77 single-fiber efficiency, d imens ionless 77/ccd cylinder (fiber) collection efficiency for convective diffusion,

dimensionless i]FA{ cylinder (fiber) collection efficiency for direct interception,

dimensionless rj/ji cylinder (fiber) collection efficiency for inertial impaction,

dimensionless 77S c d sphere collection efficiency for convective diffusion,

dimensionless r]Säi sphere collection efficiency for direct interception,

dimensionless %gd sphere collection efficiency for gravitational deposition,

dimensionless 8 angle in cylindrical or spherical coordinates, radians 9 contact angle, degrees 9 fractional surface coverage, dimensionless K inverse of double layer thickness, L"1

K von Karman constant, dimensionless X mean free path, L Xn infinite series of eigenvalues, dimensionless \i absolute or dynamic viscosity, M-L~x-T~x

Pi chemical potential species i, J/mol pn nth moment about the origin, L" ji'„ nth moment about the mean, L" MSOL chemical potential solvent, J/mol H, eddy viscosity, M-L~x-T~x

v k inemat ic viscosity, L2/T v, e d d y diffusivity of m o m e n t u m , L2IT