chemical engineering - trends and developments by miguel a. galan and eva martin del valle

The rate of reaction per unit volume of the reactor is a quantity associated with the averaging volume V illustrated in Figure 1.1.. are not drawn to scale and thus are not consistent wi

Chemical Engineering: Trends and Developments Edited by Miguel A Galán and Eva Martin del Valle

Copyright  2005 John Wiley & Sons, Inc., ISBN 0-470-02498-4 (HB)

Chemical Engineering

Trends and Developments

Editors

Miguel A Galán Eva Martin del Valle

Department of Chemical Engineering, University of Salamanca, Spain

Telephone ( +44) 1243 779777

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Library of Congress Cataloging-in-Publication Data

Chemical engineering : trends and developments / editors Miguel A Galán, Eva Martin del Valle.

p cm.

Includes bibliographical references and index.

ISBN-13 978-0-470-02498-0 (cloth : alk paper)

ISBN-10 0-470-02498-4(cloth : alk paper)

1 Chemical engineering I Galán, Miguel A., 1945– II Martín del Valle, Eva, 1973–

TP155.C37 2005

660—dc22

2005005184

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN-13 978-0-470-02498-0 (HB)

ISBN-10 0-470-02498-4 (HB)

Typeset in 10/12pt Times by Integra Software Services Pvt Ltd, Pondicherry, India

Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire

This book is printed on acid-free paper responsibly manufactured from sustainable forestry

in which at least two trees are planted for each one used for paper production.

1 The Art and Science of Upscaling 1

Pedro E Arce, Michel Quintard and Stephen Whitaker

2 Solubility of Gases in Polymeric Membranes 41

M Giacinti Baschetti, M.G De Angelis, F Doghieri and G.C Sarti

3 Small Peptide Ligands for Affinity Separations of Biological Molecules 63

Guangquan Wang, Jeffrey R Salm, Patrick V Gurgel and

Ruben G Carbonell

4 Bioprocess Scale-up: SMB as a Promising Technique for

Industrial Separations Using IMAC 85

E.M Del Valle, R Gutierrez and M.A Galán

5 Opportunities in Catalytic Reaction Engineering Examples

of Heterogeneous Catalysis in Water Remediation and

Preferential CO Oxidation 103

Janez Levec

6 Design and Analysis of Homogeneous and Heterogeneous

Alberto E Cassano and Orlando M Alfano

7 Development of Nano-Structured Micro-Porous Materials and

their Application in Bioprocess–Chemical Process

Intensification and Tissue Engineering 171

G Akay, M.A Bokhari, V.J Byron and M Dogru

8 The Encapsulation Art: Scale-up and Applications 199

M.A Galán, C.A Ruiz and E.M Del Valle

v

9 Fine–Structured Materials by Continuous Coating and Drying

or Curing of Liquid Precursors 229

L.E Skip Scriven

10 Langmuir–Blodgett Films: A Window to Nanotechnology 267

M Elena Diaz Martin and Ramon L Cerro

11 Advances in Logic-Based Optimization Approaches to Process

Integration and Supply Chain Management 299

Ignacio E Grossmann

12 Integration of Process Systems Engineering and Business

Decision Making Tools: Financial Risk Management and

Other Emerging Procedures 323

Miguel J Bagajewicz

List of Contributors

Engineering and Advanced Materials, (2) Institute for Nanoscale Science and Technology,Newcastle University, Newcastle upon Tyne NE1 7RU, UK

3450 (3000) Santa Fe, Argentina

Cookeville, TN 38505, USA

73019-1004, USA

(2) Process Intensification and Miniaturization Centre, School of Chemical Engineeringand Advanced Materials, (3) Institute for Nanoscale Science and Technology, NewcastleUniversity, Newcastle upon Tyne NE1 7RU, UK

Newcastle University, Newcastle upon Tyne NE1 7RU, UK, (2)Process Intensificationand Miniaturization Centre, School of Chemical Engineering and Advanced Materials

Carolina State University, Raleigh, NC 27695-7905, USA

3450 (3000) Santa Fe, Argentina

Alabama in Huntsville, Huntsville, AL 35899, USA

Ambientali, Università di Bologna, viale Risorgimento 2, 40136 Bologna, Italy

Caídos 1–5, 37008 Salamanca, Spain

vii

M Elena Diaz Martin Department of Chemical and Materials Engineering, University

of Alabama in Huntsville, Huntsville, AL 35899, USA

Ambientali, Università di Bologna, viale Risorgimento 2, 40136 Bologna, Italy

M Dogru Process Intensification and Miniaturization Centre, School of Chemical neering and Advanced Materials, Newcastle University, Newcastle upon Tyne NE17RU, UK

Caídos 1-5, 37008 Salamanca, Spain

Tecnolo-gie Ambientali, Università di Bologna, viale Risorgimento 2, 40136 Bologna, Italy

Uni-versity, Pittsburgh, PA 15213, USA

Carolina State University, Raleigh, NC 27695-7905, USA

Caidos 1-5, 37008, Salamanca, Spain

National Institute of Chemistry, PO Box 537 SI-1000 Ljubljana, Slovenia

Camille Soula, 31400 Toulouse, France

Caídos 1–5, 37008 Salamanca, Spain

Carolina State University, Raleigh, NC 27695-7905, USA

Ambientali, Università di Bologna, viale Risorgimento 2, 40136 Bologna, Italy

Engineering and Materials Science and Industrial Partnership for Research in Interfacialand Materials Engineering, University of Minnesota, 421 Washington Avenue S E.,Minneapolis, Minnesota 55455, USA

Car-olina State University, Raleigh, NC 27695-7905, USA

Univer-sity of California at Davis, Davis, CA 95459, USA

Usually the preface of any book is written by a recognized professional who describesthe excellence of the book and the authors who are, of course, less well-known thanhimself In this case, however, the task is made very difficult by the excellence of theauthors, the large amount of topics treated in the book and the added difficulty of findingsomeone who is an expert in all of them For these reasons, I decided to write the prefacemyself, acknowledging that I am really less than qualified to do so

This book’s genesis was two meetings, held in Salamanca (Spain), with the old studentarmy of the University of California (Davis) from the late 1960s and early 1970s, togetherwith professors who were very close to us The idea was to exchange experiences aboutthe topics in our research and discuss the future for each of them In the end, conclusionswere collected and we decided that many of the ideas and much of the research donecould be of interest to the scientific community The result is a tidy re-compilation ofmany of the topics relevant to chemical engineering, written by experts from academiaand industry

We are conscious that certain topics are not considered and some readers will findfault, but we ask them to bear in mind that in a single book it is impossible to includeall experts and all topics connected to chemical engineering

We are sure that this book is interesting because it provides a detailed perspective

on technical innovations and the industrial application of each of the topics This is due

to the panel of experts who have broad experience as researchers and consultants forinternational industries

The book is structured according to the suggestions of Professor Scriven It starts bydescribing the scope and basic concepts of chemical engineering, and continues withseveral chapters that are related to separations processes, a bottleneck in many industrialprocesses After that, applications are covered in fields such as reaction engineering,particle manufacture, and encapsulation and coating The book finishes by coveringprocess integration, showing the advances and opportunities in this field

I would like to express my thanks to each one of the authors for their valuablesuggestions and for the gift to being my friends I am very proud and honoured by theirfriendship Finally, a special mention for Professor Martín del Valle for her patience,tenacity and endurance throughout the preparation of this book; to say thanks perhaps isnot enough

For all of them and for you reader: thank you very much

Miguel Angel Galán

ix

1 The Art and Science of Upscaling

Pedro E Arce, Michel Quintard and Stephen Whitaker

1.1 Introduction

The process of upscaling governing differential equations from one length scale to another

is ubiquitous in many engineering disciplines and chemical engineering is no exception.The classic packed bed catalytic reactor is an example of a hierarchical system (Cushman,1990) in which important phenomena occur at a variety of length scales To design such

a reactor, we need to predict the output conditions given the input conditions, and this

prediction is generally based on knowledge of the rate of reaction per unit volume of the reactor The rate of reaction per unit volume of the reactor is a quantity associated with the averaging volume V illustrated in Figure 1.1 In order to use information associated

with the averaging volume to design successfully the reactor, the averaging volume must

be large enough to provide a representative average and it must be small enough to

capture accurately the variations of the rate of reaction that occur throughout the reactor

To develop a qualitative idea about what is meant by large enough and small enough,

we consider a detailed version of the averaging volume shown in Figure 1.2 Here wehave identified the fluid as the
-phase, the porous particles as the -phase, and 
as thecharacteristic length associated with the
-phase In addition to the characteristic lengthassociated with the fluid, we have identified the radius of the averaging volume as r0

In order that the averaging volume be large enough to provide a representative average

we require that r0

, and in order that the averaging volume be small enough to

capture accurately the variations of the rate of reaction we require that L D
r0 Herethe choice of the length of the reactor, L, or the diameter of the reactor, D, depends onthe concentration gradients within the reactor If the gradients in the radial direction arecomparable to or larger than those in the axial direction, the appropriate constraint is

D
r0 On the other hand, if the reactor is adiabatic and the non-uniform flow near thewalls of the reactor can be ignored, the gradients in the radial direction will be negligible

Chemical Engineering: Trends and Developments Edited by Miguel A Galán and Eva Martin del Valle

Copyright  2005 John Wiley & Sons, Inc., ISBN 0-470-02498-4 (HB)

L

Packed bed reactor

Figure 1.1 Design of a packed bed reactor

r0γ-phase

V κ-phase

Figure 1.2 Averaging volume

and the appropriate constraint is L
r0 These ideas suggest that the length scales must

be disparate or separated according to

These constraints on the length scales are purely intuitive; however, they are characteristic

of the type of results obtained by careful analysis (Whitaker, 1986a; Quintard andWhitaker, 1994a–e; Whitaker, 1999) It is important to understand that Figures 1.1 and 1.2

are not drawn to scale and thus are not consistent with the length scale constraintscontained in equation 1.1.

In order to determine the average rate of reaction in the volume V , one needs to

deter-mine the rate of reaction in the porous catalyst identified as the -phase in Figure 1.2

If the concentration gradients in both the
-phase and the -phase are small enough,

the concentrations of the reacting species can be treated as constants within the aging volume This allows one to specify the rate of reaction per unit volume of thereactor in terms of the concentrations associated with the averaging volume illustrated

aver-in Figure 1.1 A reactor aver-in which this condition is valid is often referred to as an ideal reactor (Butt, 1980, Chapter 4) or, for the reactor illustrated in Figure 1.1, as a Plug-flow

tubular reactor (PFTR) (Schmidt, 1998) In order to measure reaction rates and connectthose rates to concentrations, one attempts to achieve the approximation of a uniformconcentration within an averaging volume However, the approximation of a uniform

concentration is generally not valid in a real reactor (Butt, 1980, Chapter 5) and the

concentration gradients in the porous catalyst phase need to be taken into account Thismotivates the construction of a second, smaller averaging volume illustrated in Figure 1.3.Porous catalysts are often manufactured by compacting microporous particles (Fromentand Bischoff, 1979) and this leads to the micropore–macropore model of a porous catalystillustrated at level II in Figure 1.3 In this case, diffusion occurs in the macropores, whilediffusion and reaction take place in the micropores Under these circumstances, it is rea-sonable to analyze the transport process in terms of a two-region model (Whitaker, 1983),one region being the macropores and the other being the micropores These two regionsmake up the porous catalyst illustrated at level I in Figure 1.3 If the concentration

gradients in both the macropore region and the micropore region are small enough, the

concentrations of the reacting species can be treated as constants within this second aging volume, and one can proceed to analyze the process of diffusion and reaction with

a one-equation model This leads to the classic effectiveness factor analysis (Carberry,

1976) which provides information to be transported up the hierarchy of length scales

to the porous medium (level I) illustrated in Figure 1.3 The constraints associated withthe validity of a one-equation model for the micropore–macropore system are given byWhitaker (1983)

If the one-equation model of diffusion and reaction in a micropore–macropore system

is not valid, one needs to proceed down the hierarchy of length scales to develop an

analysis of the transport process in both the macropore region and the micropore region.This leads to yet another averaging volume that is illustrated as level III in Figure 1.4.Analysis at this level leads to a micropore effectiveness factor that is discussed byCarberry (1976, Sec 9.2) and by Froment and Bischoff (1979, Sec 3.9)

In the analysis of diffusion and reaction in the micropores, we are confronted with the

fact that catalysts are not uniformly distributed on the surface of the solid phase; thus

the so-called catalytic surface is highly non-uniform and spatial smoothing is required inorder to achieve a complete analysis of the process This leads to yet another averagingvolume illustrated as level IV in Figure 1.5 The analysis at this level should make use

of the method of area averaging (Ochoa-Tapia et al., 1993; Wood et al., 2000) in order

to obtain a spatially smoothed jump condition associated with the non-uniform catalyticsurface It would appear that this aspect of the diffusion and reaction process has receivedlittle attention and the required information associated with level IV is always obtained

by experiment based on the assumption that the experimental information can be useddirectly at level III

The train of information associated with the design of a packed bed catalytic reactor

is illustrated in Figure 1.6 There are several important observations that must be made

III IV

Non-uniform catalytic surface

Figure 2, p.xiv; with kind permission of Kluwer Academic Publishers

about this train First, we note that the train can be continued in the direction of decreasing length scales in search for more fundamental information Second, we note that one can board the train in the direction of increasing length scales at any level, provided that

appropriate experimental information is available This would be difficult to accomplish

at level I when there are significant concentration gradients in the porous catalyst Third,

we note that information is lost when one uses the calculus of integration to move up the

length scale This information can be recovered in three ways: (1) intuition can providethe lost information; (2) experiment can provide the lost information; and (3) closure can

provide the lost information Finally, we note that information is filtered as we move up

the length scales By filtered we mean that not all the information available at one level

is needed to provide a satisfactory description of the process at the next higher level

A quantitative theory of filtering does not yet exist; however, several examples have beendiscussed by Whitaker (1999)

In Figures 1.1–1.6 we have provided a qualitative description of the process of ing In the remainder of this chapter we will focus our attention on level II with therestriction that the diffusion and reaction process in the porous catalyst is dominated by

upscal-a single pore size In upscal-addition, we will upscal-assume thupscal-at the pore size is lupscal-arge enough so thupscal-atKnudsen diffusion does not play an important role in the transport process

1.2 Intuition

We begin our study of diffusion and reaction in a porous medium with a classic, intuitive approach to upscaling that often leads to confusion concerning homogeneous

and heterogeneous reactions We follow the intuitive approach with a rigorous upscaling

of the problem of dilute solution diffusion and heterogeneous reaction in a model porousmedium We then direct our attention to the more complex problem of coupled, non-lineardiffusion and reaction in a real porous catalyst We show how the information lost inthe upscaling process can be recovered by means of a closure problem that allows us topredict the tortuosity tensor in a rigorous manner The analysis demonstrates the existence

of a single tortuosity tensor for all N species involved in the process of diffusion andreaction

We consider a two-phase system consisting of a fluid phase and a solid phase asillustrated in Figure 1.7 Here we have identified the fluid phase as the
-phase and thesolid phase as the -phase The foundations for the analysis of diffusion and reaction inthis two-phase system consist of the species continuity equation in the
-phase and thespecies jump condition at the catalytic surface The species continuity equation can beexpressed as

This latter form fails to identify the species velocity as a crucial part of the species

transport equation, and this often leads to confusion about the mechanical aspects of

Porous catalyst

Catalyst deposited

on the pore walls

κ-phase γ-phase

Figure 1.7 Diffusion and reaction in a porous medium

multi-component mass transfer When surface transport (Ochoa-Tapia et al., 1993) can

be neglected, the jump condition takes the form

cAs

t =
cA
vA


· n
+ RAs at the
− interface A = 1 2     N (1.3a)

where n
represents the unit normal vector directed from the
-phase to the -phase In

terms of the molar flux that appears in equation 1.2b, the jump condition is given by

cAs

t = NA· n
+ RAs at the
− interface A = 1 2     N (1.3b)

In equations 1.2a–1.3b, we have used cA
to represent the bulk concentration of species A

(moles per unit volume), and cAs to represent the surface concentration of species A

(moles per unit area) The nomenclature for the homogeneous reaction rate, RA
, andheterogeneous reaction rate, RAs, follows the same pattern The surface concentration is

sometimes referred to as the adsorbed concentration or the surface excess concentration,

and the derivation (Whitaker, 1992) of the jump condition essentially consists of a shellbalance around the interfacial region The jump condition can also be thought of as asurface transport equation (Slattery, 1990) and it forms the basis for various mass transferboundary conditions that apply at a phase interface

In addition to the continuity equation and the jump condition, we need a set of Nmomentum equations to determine the species velocities, and we need chemical kinetic

constitutive equations for the homogeneous and heterogeneous reactions We also need

a method of connecting the surface concentration, cAs, to the bulk concentration, cA

Before exploring the general problem in some detail, we consider the typical intuitive approach commonly used in textbooks on reactor design (Carberry, 1976; Fogler, 1992;

Froment and Bischoff, 1979; Levenspiel, 1999; Schmidt, 1998) In this approach, the

analysis consists of the application of a shell balance based on the word statement



−

flow of species A out

of the control volume

equation is usually written with no regard to the averaged or upscaled quantities that are

involved and thus takes the form

NAxx− NAxx+ x

cA

t x y z = NAyy− NAyy+ y + RA x y z (1.5)

NAzz− NAzz + zOne divides this balance equation by x y z and lets the cube shrink to zero to obtain

In compact vector notation this takes a form

cA

that can be easily confused with equation 1.2b To be explicit about the confusion, we

note that cAin equation 1.7 represents a volume averaged concentration, NArepresents a

of the three terms in equation 1.7 represents something different than the analogous term

in equation 1.2b and this leads to considerable confusion among chemical engineeringstudents

The diffusion and reaction process illustrated in Figure 1.7 is typically treated as dimensional (in the average sense) so that the transport equation given by equation 1.6simplifies to

where De is identified as an effective diffusivity Having dispensed with accumulation

and diffusion, one often considers the first-order consumption of species A leading to

where avrepresents the surface area per unit volume

Students often encounter diffusion and homogeneous reaction in a form given by

and it is not difficult to see why there is confusion about homogeneous and heterogeneous

reactions The essential difficulty results from the fact that the upscaling from an unstated

point equation, such as equation 1.2, is carried out in a purely intuitive manner with

no regard to the precise definition of the dependent variable, cA If the meaning of thedependent variable in a governing differential equation is not well understood, trouble issure to follow

1.3 Analysis

To eliminate the confusion between homogeneous and heterogeneous reactions, and tointroduce the concept of upscaling in a rigorous manner, we need to illustrate the generalfeatures of the process without dealing directly with all the complexities To do so, we

Figure 1.9 Bundle of capillary tubes as a model porous medium

consider a bundle of capillary tubes as a model of a porous medium This model isillustrated in Figure 1.9 where we have shown a bundle of capillary tubes of length 2Land radius r0 The fluid in the capillary tubes is identified as the
-phase and the solid

as the -phase The porosity of this model porous medium is given by

and we will use 
to represent the porosity

Our model of diffusion and heterogeneous reaction in one of the capillary tubesillustrated in Figure 1.9 is given by the following boundary value problem:

cA

t = D

1r

Here we have assumed that the catalytic surface at r= r0is quasi-steady even though the

diffusion process in the pore may be transient (Carbonell and Whitaker, 1984; Whitaker,

1986b) Equations 1.13–1.17 represent the physical situation in the pore domain and we need equations that represent the physical situation in the porous medium domain This

requires that we develop the area-averaged form of equation 1.13 and that we determine

under what circumstances the concentration at r= r0can be replaced by the area-averagedconcentration,cA

The area-averaged concentration is defined by

and in order to develop an area-averaged or upscaled diffusion equation, we form the

intrinsic area average of equation 1.13 to obtain

r=0

1r

+ 120

which a boundary condition is joined to a governing differential equation, is inherent

in all studies of multiphase transport processes The failure to identify explicitly this

process often leads to confusion concerning the difference between homogeneous andheterogeneous chemical reactions.

Equation 1.24 poses a problem in that it represents a single equation containing two concentrations If we cannot express the concentration at the wall of the capillary tube in

terms of the area-averaged concentration, the area-averaged transport equation will be oflittle use to us and we will be forced to return to equations 1.13–1.17 to solve the boundaryvalue problem by classical methods In other words, the upscaling procedure would fail

without what is known as a method of closure In order to complete the upscaling process

in a simple manner, we need an estimate of the variation of the concentration acrossthe tube We obtain this by using the flux boundary condition to construct the followingorder of magnitude estimate:

‘homoge-becomes more complex and this condition has been explored by Paine et al (1983).

1.3.1 Porous Catalysts

When dealing with porous catalysts, one generally does not work with the intrinsic

average transport equation given by

cA


t

  accumulation per unit volume

of fluid

− 2k

r0cA


  reaction rate per unit volume

(1.29)

Here we have emphasized the intrinsic nature of our area-averaged transport equation,and this is especially clear with respect to the last term which represents the rate of

reaction per unit volume of the fluid phase In the study of diffusion and reaction in real

porous media (Whitaker, 1986a, 1987), it is traditional to work with the rate of reaction

per unit volume of the porous medium Since the ratio of the fluid volume to the volume

of the porous medium is the porosity, i.e


= porosity =

volume

of the fluid



volume of theporous medium

−2
k

r0 cA


rate of reaction per unit volume

Sometimes the model illustrated in Figure 1.9 is extended to include tortuous pores such

as shown in the two-dimensional illustration in Figure 1.10 Under these circumstancesone often writes equation 1.32 in the form

Figure 1.10 Tortuous capillary tube as a model porous medium

Here  is a coefficient referred to as the tortuosity and the ratio, D
/, is called theeffective diffusivity which is represented by Deff This allows us to express equation 1.33

in the traditional form given by


cA


t = 
Deff

2cA


The step from equation 1.32 for a bundle of capillary tubes to equation 1.34 for a porous

medium is intuitive, and for undergraduate courses in reactor design one might accept

this level of intuition However, the development leading from equations 1.13 through

1.17 to the upscaled result given by equation 1.32 is analytical and this level of analysis

is necessary for an undergraduate course in reactor design The more practical problem deals with non-dilute solution diffusion and reaction in porous catalysts, and a rigorous

analysis of that case is given in the following sections

1.4 Coupled, Non-linear Diffusion and Reaction

Problems of isothermal mass transfer and reaction are best represented in terms of thespecies continuity equation and the associated jump condition We repeat these twoequations here as

A complete description of the mass transfer process requires a connection between the

surface concentration, cAs, and the bulk concentration, cA
One classic connection is

based on local mass equilibrium, and for a linear equilibrium relation this concept takes

the form

cAs= KAcA
 at the
− interface A = 1 2     N (1.37a)The condition of local mass equilibrium can exist even when adsorption and chemicalreaction are taking place (Whitaker, 1999, Problem 1-3) When local mass equilibrium

is not valid, one must propose an interfacial flux constitutive equation The classic linear

form is given by (Langmuir, 1916, 1917)

cA
vA


· n
= kA1cA
− k−A1cAs at the
− interface A = 1 2     N (1.37b)where kA1and k−A1represent the adsorption and desorption rate coefficients for species A

In addition to equations 1.35 and 1.36, we need N momentum equations (Whitaker,

1986a) that are used to determine the N species velocities represented by vA
,

A= 1 2     N There are certain problems for which the N momentum equationsconsist of the total, or mass average, momentum equation



t


v
+  ·

v
v


along with N− 1 Stefan–Maxwell equations that take the form

0= −xA
+

E=N E=1

inappropriate when Knudsen diffusion must be taken into account The species velocity

in equation 1.39 can be decomposed into an average velocity and a diffusion velocity

in more than one way (Taylor and Krishna, 1993; Slattery, 1999; Bird et al., 2002),

and arguments are often given to justify a particular choice In this work we prefer

a decomposition in terms of the mass average velocity because governing equations,

such as the Navier–Stokes equations, are available to determine this velocity The massaverage velocity in equation 1.38 is defined by

As an alternative to equations 1.40–1.42, we can define a molar average velocity by

v
∗=

A=N A=1

= cA
v

  molar convective

+ cA
uA

  mixed-mode

(1.46)

Here we have decomposed the total molar flux into what we want, the molar convective

flux, and what remains, i.e a mixed-mode diffusive flux Following Bird et al (2002),

we indicate the mixed-mode diffusive flux as

A=NA=1

where MAis the molecular mass of species A and M is the mean molecular mass definedby

M=A=NA=1

There are many problems for which we wish to know the concentration, cA
, andthe normal component of the molar flux of species A at a phase interface The normalcomponent of the molar flux at an interface will be related to the adsorption processand the heterogeneous reaction by means of the jump condition given by equation 1.36and relations of the type given by equation 1.37, and this flux will be influenced by theconvective, cA
v
, and diffusive, JA
, fluxes

The governing equations for cA
and v
are available to us in terms of equations 1.38

and 1.48, and here we consider the matter of determining JA
To determine the mode diffusive flux, we return to the Stefan–Maxwell equations and make use ofequation 1.41 to obtain

which can then be expressed in terms of equation 1.47 to obtain

When the mole fraction of species A is small compared to 1, we obtain the dilute solution

representation for the diffusive flux

1.5 Diffusive Flux

We begin our analysis of the diffusive flux with equation 1.55 in the form

JA
= −c
DAmxA
+ xA

E=NE=1 E=A

where the column matrix on the right-hand side of this result can be expressed as

DAExE

+ RA
 A= 1 2     N (1.69)

We seek a solution to this equation subject to the jump condition given by equation 1.36and this requires knowledge of the concentration dependence of the homogeneous andheterogeneous reaction rates and information concerning the equilibrium adsorptionisotherm In general, a solution of equation 1.69 for the system shown in Figure 1.7

requires upscaling from the point scale to the pore scale and this can be done by the

method of volume averaging (Whitaker, 1999)

1.6 Volume Averaging

To obtain the volume-averaged form of equation 1.69, we first associate an averagingvolume with every point in the
− system illustrated in Figure 1.7 One such averagingvolume is illustrated in Figure 1.11, and it can be represented in terms of the volumes ofthe individual phases according to

The radius of the averaging volume is r0 and the characteristic length scale associatedwith the
-phase is indicated by 
as shown in Figure 1.11 In this figure we have alsoillustrated a length L that is associated with the distance over which significant changes

in averaged quantities occur Throughout this analysis we will assume that the lengthscales are disparate, i.e the length scales are constrained by

Here the length scale, L, is a generic length scale (Whitaker, 1999, Sec 1.3.2) determined

by the gradient of the average concentration, and all three quantities in equation 1.71

are different to those listed in equation 1.1 We will use the averaging volume V to define two averages: the superficial average and the intrinsic average Each of these

averages is routinely used in the description of multiphase transport processes, and it is

γ-phase κ-phase

important to define clearly each one We define the superficial average of some function

of the average that we are seeking In order to interchange integration and differentiation,

we will make use of the spatial averaging theorem (Anderson and Jackson, 1967; Marle,1967; Slattery, 1967; Whitaker, 1967) For the two-phase system illustrated in Figure 1.11this theorem can be expressed as

where 
is any function associated with the
-phase Here A
represents the interfacial

area contained within the averaging volume, and we have used n
to represent the unit

normal vector pointing from the
-phase toward the -phase.

Even though equation 1.69 is considered to be the preferred form of the speciescontinuity equation, it is best to begin the averaging procedure with equation 1.35, and

we express the superficial average of that form as

to express this result as

cA


t +  · cA
vA
 + 1

V

A

n
·
cA
vA


where it is understood that this applies to all N species Since we seek a transportequation for the intrinsic average concentration, we make use of equation 1.74 to expressequation 1.78 in the form

(1.82)

One must keep in mind that this is a general result based on equations 1.35 and 1.36;

however, only the first term in equation 1.82 is in a form that is ready for applications

1.7 Chemical Reactions

In general, the homogeneous reaction will be of no consequence in a porous catalyst and

we need only direct our attention to the heterogeneous reaction represented by the last

term in equation 1.82 The chemical kinetic constitutive equation for the heterogeneous

rate of reaction can be expressed as

RAs= RAscAs cBs     cN s (1.83)and here we see the need to relate the surface concentrations, cAs cBs     cN s, to thebulk concentrations, cA
 cB
     cN
, and subsequently to the local volume-averagedconcentrations,cA

cB

    cN

In order for heterogeneous reaction to occur,adsorption at the catalytic surface must also occur However, there are many transientprocesses of mass transfer with heterogeneous reaction for which the catalytic surface

can be treated as quasi-steady (Carbonell and Whitaker, 1984; Whitaker, 1986b) When

homogeneous reactions can be ignored and the catalytic surface can be treated as steady, the local volume-averaged transport equation simplifies to

quasi-
cA


t

  accumulation

(1.84)

and this result provides the basis for several special forms

1.8 Convective and Diffusive Transport

Before examining the heterogeneous reaction rate in equation 1.84, we consider thetransport term,cA
vA
 We begin with the mixed-mode decomposition given by equa-tion 1.46 in order to obtain

cA
vA


  total molar flux

= cA
v


  molar convective flux

+ cA
uA


  mixed-mode diffusive flux

(1.85)

Here the convective flux is given in terms of the average of a product, and we want toexpress this flux in terms of the product of averages As in the case of turbulent transport,this suggests the use of decompositions given by

= 
cA

v


average convective flux

+ ˜cA
˜v


  dispersive flux

+ JA


  mixed-mode diffusive flux

(1.87)

Here we have used the intrinsic average concentration since this is most closely related

to the concentration in the fluid phase, and we have used the superficial average velocitysince this is the quantity that normally appears in Darcy’s law (Whitaker, 1999) or theForchheimer equation (Whitaker, 1996) Use of equation 1.87 in equation 1.84 leads to


cA


t +  ·

cA

v
= −  · JA


  diffusive

−  · ˜cA
˜v


dispersive transport

+ avRAs


  heterogeneous reaction

(1.88)

If we treat the catalytic surface as quasi-steady and make use of a simple first-order,

irreversible representation for the heterogeneous reaction, we can show that RAs is given

as indicated by equation 1.89 Under these circumstances the functional dependenceindicated in equation 1.83 can be simplified to

R = R
c  c      c 

Given the type of constraints developed elsewhere (Wood and Whitaker, 1998, 2000),the interfacial area average of the heterogeneous rate of reaction can be expressed as

Sometimes confusion exists concerning the idea of an area-averaged bulk concentration,

and to clarify this idea we consider the averaging volume illustrated in Figure 1.12 In thisfigure we have shown an averaging volume with the centroid located (arbitrarily) in the

-phase In this case, the area average of the bulk concentration is given explicitly by

where x locates the centroid of the averaging volume and y locates points on the
–

interface We have used A


within the averaging volume

To complete our analysis of equation 1.91, we need to know how the area-averagedconcentration, cA

, is related to the volume-averaged concentration, cA

Whenconvective transport is important, relating cA

 to cA

requires some analysis;

however, when diffusive transport dominates in a porous catalyst the area-averaged

concentration is essentially equal to the volume-averaged concentration This occurs

because the pore Thiele modulus is generally small compared to one and the type

of analysis indicated by equations 1.24–1.28 is applicable Under these circumstances,equation 1.91 can be expressed as

Figure 1.12 Position vectors associated with the area average over the
– interface

and equation 1.88 takes the form

that the mathematical consequence of equations 1.95 and 1.96 is that the mass average

velocity has been set equal to zero; thus our substitute for equation 1.38 is given bythe assumption

This assumption requires that we discard the momentum equation given by equation 1.38

and proceed to develop a solution to our mass transfer process in terms of the N− 1momentum equations represented by equation 1.39 The inequalities contained in equa-

tion 1.95 are quite appealing when one is dealing with a diffusion process; however,

equation 1.97 is not satisfied by the Stefan diffusion tube process (Whitaker, 1991), nor

is it satisfied by the Graham’s law counter-diffusion process (Jackson, 1977) It should

be clear that the constraints associated with the equalities given by equation 1.95 need

to be developed When convective transport is retained, some results are available fromQuintard and Whitaker (2005); however, a detailed analysis of the coupled, non-linearprocess with convective transport remains to be done At this point we leave those prob-lems for a subsequent study and explore the diffusion and reaction process described byequation 1.96

DAExE

&

+a R 
c 
c 
    c 


 A= 1 2     N (1.98)

where the diffusive flux is non-linear because DAEdepends on the N−1 mole fractions.This transport equation must be solved subject to the auxiliary conditions given by

c
=A=NA=1

cA
 1=A=N

A=1

and this suggests that numerical methods must be used However, the diffusive flux must

be arranged in terms of volume-averaged quantities before equation 1.98 can be solved,and any reasonable simplifications that can be made should be imposed on the analysis

1.9.1 Constant Total Molar Concentration

Some non-dilute solutions can be treated as having a constant total molar concentrationand this simplification allows us to express equation 1.98 as


cA


%E=N−1E=1

and it is important to understand that the mathematical consequence of this restriction is

given by the assumption

Imposition of this condition means that there are only N− 1 independent transportequations of the form given by equation 1.100, and we shall impose this conditionthroughout the remainder of this study The constraints associated with equation 1.102need to be developed and the more general case represented by equations 1.98 and 1.99should be explored

At this point we decompose the elements of the diffusion matrix according to

and when it is not satisfactory it may be possible to develop a correction based on the

retention of the spatial deviation, ˜DAE However, it is not clear how this type of analysiswould evolve and further study of this aspect of the diffusion process is in order

1.9.2 Volume Average of the Diffusive Flux

The volume-averaging theorem can be used with the average of the gradient in tion 1.104 in order to obtain

where the area integral of n
˜cE
has been identified as a filter Not all the information

available at the length scale associated with˜cE
will pass through this filter to influencethe transport equation forcA

, and the existence of filters of this type is a recurringtheme in the method of volume averaging (Whitaker, 1999)

1.10 Closure

In order to obtain a closed form of equation 1.108, we need a representation for the spatialdeviation concentration,˜cA
, and this requires the development of the closure problem.

When convective transport is negligible and homogeneous reactions are ignored as being

a trivial part of the analysis, equation 1.48 takes the form

cA

t = − · JA
 A= 1 2     N − 1 (1.109)Here one must remember that the total molar concentration is a specified constant; thusthere are only N−1 independent species continuity equations Use of equation 1.68 alongwith the restriction given by equation 1.101 allows us to express this result as

cA

t =  ·E=N−1

E=1

DAE
cE
 A= 1 2     N − 1 (1.111)

If we ignore variations in 
and subtract equation 1.108 from equation 1.111, we canarrange the result as

˜cA

t =  ·

E=N−1E=1

in order to keep the analysis relatively simple we consider only the first-order, irreversiblereaction described by equation 1.89 and expressed here in the form

One must remember that this is a severe restriction in terms of realistic systems andmore general forms for the heterogeneous rate of reaction need to be examined Use ofequation 1.113 in equation 1.112 leads to the following form:

˜cA

t =  ·

-E=N−1

E=1

v
· n
 uA
· n
 at the
− interface (1.117)

This is certainly consistent with the inequalities given by equation 1.95; however, the

neglect of v
· n
relative to uA
· n
 is generally based on the dilute solution condition

and the validity of equation 1.117 is another matter that needs to be carefully considered

in a future study On the basis of equations 1.68, 1.101, 1.103, and 1.105 along withequation 1.113, the jump condition takes the form

−E=N−1

E=1

n
· DAE
cE
= kAcA
 at the
− interface (1.118)

In order to express this boundary condition in terms of the spatial deviation concentration,

we make use of the decomposition given by the first part of equation 1.86 to obtain

=E=N−1E=1

n
· DAE
cE


diffusive source+ kAcA


  reaction source

is quasi-steady Under these circumstances, the closure problem can be solved in somerepresentative, local region (Quintard and Whitaker, 1994a– e)

In the governing differential equation for˜cA
, we have identified the accumulation term,the diffusion term, the so-called non-local diffusion term, and the non-homogeneous term

referred to as the reaction source In the boundary condition imposed at the
– interface,

we have identified the diffusive flux, the reaction term, and two non-homogeneous terms

that are referred to as the diffusion source and the reaction source If the source terms

in equations 1.120 and 1.121 were zero, the ˜cA
-field would be generated only by thenon-homogeneous terms that might appear in the boundary condition imposed at A
e

or in the initial condition given by equation 1.123 One can easily develop argumentsindicating that the closure problem for ˜c is quasi-steady, thus the initial condition is

of no importance (Whitaker, 1999, Chapter 1) In addition, one can develop argumentsindicating that the boundary condition imposed at A
e will influence the˜cA
field over

a negligibly small portion of the field of interest Because of this, any useful solution

to the closure problem must be developed for some representative region which is most

often conveniently described in terms of a unit cell in a spatially periodic system Theseideas lead to a closure problem of the form

  reaction source

It is not obvious, but other studies (Ryan et al., 1981) have shown that the reaction

source in equations 1.124 and 1.125 makes a negligible contribution to˜cA
In addition, onecan demonstrate (Whitaker, 1999) that the heterogeneous reaction, kA˜cA
, can be neglectedfor all practical problems of diffusion and reaction in porous catalysts Furthermore, thenon-local diffusion term is negligible for traditional systems, and under these circumstancesthe boundary value problem for the spatial deviation concentration takes the form

0=  ·

E=N−1E=1

In this boundary value problem, there is only a single non-homogeneous term resented by cE

in the boundary condition imposed at the
– interface If thissource term were zero, the solution to this boundary value problem would be given by

rep-˜cA
= constant Any constant associated with ˜cA
will not pass through the filter in tion 1.108, and this suggests that a solution can be expressed in terms of the gradients ofthe volume-averaged concentration Since the system is linear in the N− 1 independentgradients of the average concentration, we are led to a solution of the form

N− 1 species and that the N − 1 concentration gradients are independent This lattercondition allows us to obtain

0= ·

-E=N−1

E=1

(1.138)

At this point it is convenient to expand the closure problem for species A in order toobtain

First problem for species A

0= ·/DAA
0

 bAA+ DAA
−1DAB
 bBA

+ DAA
−1DAC
 bCA+ · · · + DAA
−1DAN−1
 bN−1A

12(1.139a)

12(1.140a)

analysis of that case is given in the following sections

1.4 Coupled, Non-linear Diffusion and Reaction... cA
v
, and diffusive, JA
, fluxes

The governing equations for cA
and v
are available to us in terms of equations 1.38

and 1.48, and