The Art and Science of Upscaling

The Art and Science of Upscaling

Pedro E. Arce Department of Chemical Engineering

Tennessee Tech University Cookeville, TN 38505 - USA

Michel Quintard

Institut de Mécanique des Fluides de Toulouse Av. du Professeur Camille Soula

31400 Toulouse, France

Stephen Whitaker Department of Chemical Engineering and Material Science

University of California at Davis Davis, CA 95616 – USA

February 3, 2005

I. 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 an 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. In order to use

information associated with the averaging volume to successfully design the reactor, the

averaging volume must be large enough to provide a representative average and it must

be small enough to accurately capture 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 2.

There we have identified the fluid as the γ-phase, the porous particles as the κ-phase, and

Art & Science of Upscaling Arce, Quintard & Whitaker

γ as the characteristic length associated with the γ-phase. In addition to the

characteristic length associated with the fluid, we have identified the radius of the

Figure 1. Design of a packed bed reactor averaging volume as . In order that the averaging volume be large enough to provide a

representative average we require that

or

or γ>> , and in order that the averaging volume

be small enough to accurately capture the variations of the rate of reaction we require that

. Here the choice of the length of the reactor, L, or the diameter of the reactor,

D, depends on the concentration gradients within the reactor. If the gradients in the radial

direction are comparable to or larger than those in the axial direction, the appropriate

constraint is . On the other hand, if the reactor is adiabatic and the non-uniform

flow near the walls of the reactor can be ignored, the gradients in the radial direction will

be negligible and the appropriate constraint is . These ideas suggest that the

length scales must be disparate or separated according to

o,L D r>>

oD >> r

oL >> r

page 2


o,L D r γ>> >> (1)

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 and Whitaker, 1994a-e; Whitaker, 1999). It is important to understand that

Figures 1 and 2 are not drawn to scale and thus are not consistent with the length scale

constraints contained in Eq. 1.

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

determine the rate of reaction in the porous catalyst identified as the κ-phase in Figure 2.

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

concentrations of the reacting species can be treated as constants within the averaging

volume. This allows one to specify the rate of reaction per unit volume of the reactor in

terms of the concentrations associated with the averaging volume illustrated in Figure 1.

A reactor 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, as a plug-flow tubular reactor

(Schmidt, 1998). In order to measure

Figure 2. Averaging volume

reaction rates and connect those rates to concentrations, one attempts to achieve the

approximation of a uniform concentration within an averaging volume. However, the

page 3


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. This motivates the construction of a second, smaller averaging

volume illustrated in Figure 3.

Figure 3. Transport in a micropore-macropore model of a porous catalyst.

Porous catalysts are often manufactured by compacting microporous particles

(Froment and Bischoff, 1979) and this leads to the micropore-macropore model of a

porous catalyst illustrated at level II in Figure 3. In this case, diffusion occurs in the

macropores while diffusion and reaction takes place in the micropores. Under these

circumstances, it is reasonable 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 regions make up the porous catalyst illustrated at level I in

Figure 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

page 4


constants within this second averaging 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 3. The constraints associated with the validity of a one-equation model for the

micropore-macropore system are given by Whitaker (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 4. Analysis at this level leads to a micropore effectiveness factor that is discussed

by Carberry (1976, Sec. 9-2) and by Froment and Bischoff (1979, Sec. 3.9).

Figure 4. Transport in the micropores

page 5


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 in

order to achieve a complete analysis of the process. This leads to yet another averaging

volume illustrated as level IV in Figure 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 catalytic

surface. It would appear that this aspect of the diffusion and reaction process has

received little attention and the required information associated with level IV is always

obtained by experiment based on the assumption that the experimental information can be

used directly at level III.

Figure 5. Reaction at a non-uniform catalytic surface

page 6


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

illustrated in Figure 6. There are several important observations that must be made 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

Figure 6. Train of information

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 provide

the 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

page 7


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 been

discussed by Whitaker (1999).

In Figures 1 through 6 we have provided a qualitative description of the process of

upscaling. In the remainder of this chapter we will focus our attention on level II with the

restriction that the diffusion and reaction process in the porous catalyst is dominated by a

single pore size. In addition, we will assume that the pore size is large enough so that

Knudsen diffusion does not play an important role in the transport process.

II. 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 porous

medium. We then direct our attention to the more complex problem of coupled, non-

linear diffusion and reaction in a real porous catalyst. We show how the information lost

in the upscaling process can be recovered by means of a closure problem that allows us to

predict 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 and reaction.

We consider a two-phase system consisting of a fluid phase and a solid phase as

illustrated in Figure 7. There we have identified the fluid phase as the γ−phase and the

solid phase as the κ−phase. The foundations for the analysis of diffusion and reaction in

this two-phase system consist of the species continuity equation in the γ−phase and the

species jump condition at the catalytic surface. The species continuity equation can be

expressed as

( ) , 1,2,...,AA A A

cc R A

tγ

γ γ γ∂

+ ∇ ⋅ = =∂

v N (2a)

or in terms of the molar flux as given by (Bird et al., 2002)

page 8


, 1,2,...,AA A

cR A

tγ

γ∂

+ ∇ ⋅ = =∂

N N (2b)

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

Figure 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

( ) , at the interface , 1, 2, ...,AsA A As

c c R At γ γ γκ

∂= ⋅ + γ − κ =

∂v n N (3a)

in which represents the unit normal vector directed from the γ-phase to the κ-phase.

In terms of the molar flux that appears in Eq. 2b, the jump condition is given by

γκn

, at the interface , 1, 2, ...,AsA As

c R At γκ

∂= ⋅ + γ − κ =

∂N n N (3b)

page 9


In Eqs. 2a through 3b, we have used Ac γ to represent the bulk concentration of species A

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

(moles per unit area). The nomenclature for the homogeneous reaction rate, AR γ , and

heterogeneous reaction rate, AsR , 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 shell

balance around the interfacial region. The jump condition can also be thought of as a

surface transport equation (Slattery, 1990) and it forms the basis for various mass transfer

boundary conditions that apply at a phase interface.

In addition to the continuity equation and the jump condition, we need a set of N

momentum 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, Asc , to the bulk concentration, Ac γ .

Before exploring the general problem in some detail, we consider the typical,

intuitive approach commonly used in text books on reactor design (Carberry, 1976;

Fogler, 1992; Froment and Bischoff, 1979; Schmidt, 1998; Levenspiel, 1999). In this

approach, the analysis consists of the application of a shell balance based on the word

statement given by

accumulation flow of species inof species to the control volume

rate of productionflow of species out

of species owingof the control volume

to chemical reaction

AA

AA

=

+ +

(4)

This result is applied to the cube illustrated in Figure 8 in order to obtain a balance

equation associated with the accumulation, the flux, and the reaction rate. This balance

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

involved and thus takes the form

page 10


( )

( )

( )

Ax Axx x x

AAy Ay Ay y y

Az Azz z z

N N y z

c x y z N N x z R x y zt

N N x y

+∆

+∆

+∆

− ∆ ∆

∆∆ ∆ ∆ = − ∆ ∆ + ∆ ∆ ∆

∆

− ∆ ∆

(5)

One divides this balance equation by x y z∆ ∆ ∆ and lets the cube shrink to zero to obtain

AyA Ax AzA

Nc N N Rt x y z

∂ ∂ ∂ ∂= − + + +∂ ∂ ∂ ∂

(6)

In compact vector notation this takes a form

AA A

c Rt

∂= − ∇ ⋅ +

∂N (7)

that can be easily confused with Eq. 2b. To be explicit about the confusion, we note that

Ac in Eq. 7 represents a volume averaged concentration, AN represents a volume

averaged molar flux, and AR represents a heterogeneous rate of reaction. Each one of

the three terms in Eq. 7 represents something different than the analogous term in Eq. 2b

and this leads to considerable confusion among chemical engineering students.

Figure 8. Use of a cube to construct a shell balance

page 11


The diffusion and reaction process illustrated in Figure 7 is typically treated as one-

dimensional (in the average sense) so that the transport equation given by Eq. 6 simplifies

to

A AzA

c N Rt z

∂ ∂= − +

∂ ∂ (8)

At this point, a vague reference to Fick’s law is usually made in order to obtain

eA A

Ac c Rt z z

∂ ∂ ∂ = ∂ ∂ ∂ D + (9)

in which D is identified as an effective diffusivity. Having dispensed with accumulation

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

e

Heterogeneous reaction: eA A

v Ac c a k ct z z

∂ ∂ ∂ = − ∂ ∂ ∂ D (10)

in which represents the surface area per unit volume. va

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

Homogeneous reaction: A AA

c c k ct z z

∂ ∂ ∂ = ∂ ∂ ∂ D − (11)

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 Eq. 2, is carried out in a purely intuitive manner

with no regard to the precise definition of the dependent variable, Ac . If the meaning of

the dependent variable in a governing differential equation is not well understood, trouble

is sure to follow.

III. ANALYSIS

To eliminate the confusion between homogeneous and heterogeneous reactions, and

to introduce the concept of upscaling in a rigorous manner, we need to illustrate the

general features of the process without dealing directly with all the complexities. To do

so, we consider a bundle of capillary tubes as a model of a porous medium. This model

page 12


is illustrated in Figure 9 where we have shown a bundle of capillary tubes of length 2L

and radius . The fluid in the capillary tubes is identified as the γ-phase and the solid as or

Figure 9. Bundle of capillary tubes as a model porous medium

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

{ } 22oporosity r b= π (12)

and we will use to represent the porosity. γε

Our model of diffusion and heterogeneous reaction in the one of the capillary tubes

illustrated in Figure 9 is given by the following boundary value problem:

2

21 , A A Ac c c

r in the phaset r r r z

γ γ γγ

∂ ∂ ∂ ∂= + ∂ ∂ ∂ ∂

D γ −

0

(13)

B.C.1 ,A Ac c zγ γ= = (14)

B.C.2 ,AA

ckc r r

rγ

γ γ∂

− =∂

D = (15)

page 13


B.C.3 0 ,Acz L

zγ∂

= =∂

(16)

I.C. unspecified (17)

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

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

1986b). Equations 13 through 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 Eq. 13 and that we determine

under what circumstances the concentration at r r= can be replaced by the area-

averaged concentration, . The area-averaged concentration is defined by Ac γγ⟨ ⟩

2

0

1 2

r r

A

r

cr

=

γγ

=

⟨ ⟩ = ππ ∫ Ar c drγ (18)

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

intrinsic area average of Eq. 13 to obtain

o

o o

2

0

2

2 2

0 0

1 2

1 1 12 2

r r

A

r

r r r r

A A

r r

crdr

tr

c cr rdr r

r r rr r

=

γ

=

= =

γ γγ

= =

∂ π = ∂π

∂ ∂ ∂ = π + ∂ ∂π π

∫

∫ ∫D 2 drz

π∂

(19)

The first and last terms in this result can be expressed as

2 2

0 0

1 12 2

r r r r

A AA

r r

c crdr c rdr

t tr r

= =γ

γ γγ

= =

∂ ∂ ∂ π = π = ∂ ∂π π

∫ ∫ t

⟨ ⟩

∂ (20)

2 22

2 2 2 2

0 0

1 12 2

r r r r

A AA

r r

c crdr c rdr

r z z r z

= =γ

γ γγ

= =

∂ ∂∂ π = π = π ∂ ∂ π ∂ ∫ ∫ 2

⟨ ⟩ (21)

page 14


so that Eq. 19 takes the form

2

2

0

2

r r

A A

r

c cr dr

t r rr z

=γ γ

γ γγ

=

∂⟨ ⟩ ∂ ∂ ⟨ ⟩ ∂

= ∂ ∂ ∂ ∂

∫D 2Ac γ

γ+ D (22)

Evaluation of the integral leads to

2

22A A

r r

c ct rz

γ γγγ γ

γAcr

γ

=

∂⟨ ⟩ ∂ ⟨ ⟩ ∂= +

∂ ∂

DD

∂ (23)

and we can make use of the boundary condition given by Eq. 15 to incorporate the

heterogeneous rate of reaction into the area averaged diffusion equation. This gives

2

22A A

A r r

c c k ct rz

Dγ γ

γ γγ γ =

∂⟨ ⟩ ∂ ⟨ ⟩= −

∂ ∂ (24)

Here we remark that the boundary condition is joined with the governing differential

equation, and that means that the heterogeneous reaction rate in Eq. 15 is now beginning

to “look like” a homogeneous reaction rate in Eq. 24. This process, in which a boundary

condition is joined to a governing differential equation, is inherent in all studies of

multiphase transport processes. The failure to explicitly identify this process often leads

to confusion concerning the difference between homogeneous and heterogeneous

chemical reactions.

Equation 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 average concentration, the area averaged transport equation will be of

little use to us and we will be forced to return to Eqs. 13 through 17 to solve the boundary

value 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

across the tube. We obtain this by using the flux boundary condition to construct the

following order of magnitude estimate

page 15


(0A Ar r rA r r

c ck c

rOD

γ γ= =γ =

− =

)γ (25)

which can be arranged in the form

0A Ar r r

A r r

c c k rc

γ γ= =

γγ =

− =

O

D

1

(26)

When it should be clear that we can use the approximation o /kr γ <<D

A r rc Ac γ

γ γ== ⟨ ⟩ (27)

which represents the closure for this particular process. This allows us to express Eq. 24

as

2

o2

2 ,A AA

c c k k rct rz

γ γγ γ γ

γ γγ

∂⟨ ⟩ ∂ ⟨ ⟩= − ⟨ ⟩

∂ ∂D

D1<< (28)

Here we see that the heterogeneous reaction rate expression that appears in the flux

boundary condition given by Eq. 15 now appears as a homogeneous reaction rate

expression in the area averaged transport equation. It should be clear that the

“homogeneous reaction rate coefficient” contains the geometrical parameter, r , and this

is a clear indication that

o

2 is something other than a true homogeneous reaction rate

coefficient. When the constraint,

k r

o /k r γ 1<<D , is not satisfied, the closure represented

by Eq. 27 becomes more complex and this condition has been explored by Paine et al.,

(1983).

IIIA. Porous Catalysts

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

average transport equation given by

page 16


2

2o

2A AA

accumulation reaction ratediffusive fluxper unit volume per unit volumeper unit volume of fluid of fluid of fluid

c c k ct rz

γ γγ γ γ

γ∂⟨ ⟩ ∂ ⟨ ⟩

= −∂ ∂

D γ⟨ ⟩ (29)

Here we have emphasized the intrinsic nature of our area-averaged transport equation,

and this is especially clear in terms of 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.,

{ }

volumeof the fluid

porosityvolume of theporous medium

γ

ε = =

(30)

the superficial averaged diffusion-reaction equation is expressed as

2

22A A

A

accululation per rate of reactiondiffusive flux per unit volume of per unit volume unit volume of porous media of porous media porous media

c cc

t rz

γ γγ γ kγ γ

γ γ γ∂⟨ ⟩ ∂ ⟨ ⟩ ε

ε = ε − ⟨∂ ∂

D γ ⟩ (31)

Here we see that the last term represents the rate of reaction per unit volume of the porous

medium and this is the traditional interpretation in reactor design literature. One can

show that orγε2 represents the surface area per unit volume of the porous medium, and

we denote this by so that Eq. 31 takes the form va

2

v2A A

Ac c

a k ct z

γ γγ γ γ

γ γ γ∂⟨ ⟩ ∂ ⟨ ⟩

ε = ε − ⟨∂ ∂

D γ ⟩ (32)

Sometimes the model illustrated in Figure 9 is extended to include tortuous pores such as

we have shown in the two-dimensional illustration in Figure 10. Under these

circumstances one often writes Eq. 32 in the form

page 17


2

v2A A

Ac c

a k ct z

Dγ γγ γ γ γ γ

γ∂⟨ ⟩ ε ∂ ⟨ ⟩

ε = − ⟨∂ τ ∂

γ ⟩ (33)

Here is a coefficient referred to as the tortuosity and the ratio, τ D , is called the

effective diffusivity which is represented by . This allows us to express Eq. 33 in the

traditional form given by

γ τ

effD

2

v2A A

eff Ac c

D at z

γ γγ γ k c γ

γ γ∂⟨ ⟩ ∂ ⟨ ⟩

ε = ε − ⟨∂ ∂

γ ⟩ (34)

The step from Eq. 32 for a bundle of capillary tubes to Eq. 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 Eqs. 13 through 17 to the upscaled

� - phase

� - phase

Figure 10. Tortuous capillary tube as a model porous medium result given by Eq. 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 paragraphs.

IV. COUPLED, NON-LINEAR DIFFUSION AND REACTION

Problems of isothermal mass transfer and reaction are best represented in terms of

the species continuity equation and the associated jump condition. We repeat these two

equations here as

page 18


( ) , 1,2,...,AA A A

cc R A

tγ

γ γ γ∂

+ ∇ ⋅ = =∂

v N (35)

( ) , at the interface , 1, 2, ...,AsA A As

c c R At γ γ γκ

∂= ⋅ + γ − κ =

∂v n N (36)

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

surface concentration, Asc , and the bulk concentration, Ac γ . One classic connection is

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

the form

(37a) , at the interface , 1, 2, ...,As A Ac K c Aγ= γ − κ = N

The condition of local mass equilibrium can exist even when adsorption and chemical

reaction 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)

( ) 1 1 , at the interface , 1, 2, ...,A A A A A Asc k c k c Aγ γ γκ γ −⋅ = − γ − κ =v n N (37b)

in which 1Ak and 1Ak− represent the adsorption and desorption rate coefficients for

species A.

In addition to Eqs. 35 and 36, we need N momentum equations (Whitaker, 1986a)

that are used to determine the N species velocities represented by Aγv , .

There are certain problems for which the N momentum equations consist of the total, or

mass average, momentum equation

1,2,...,A N=

( ) ( )t γ γ γ γ γ γ γ γ∂

ρ + ∇ ⋅ ρ = ρ + ∇ ⋅∂

v v v b T (38)

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

1

( )0 , 1, 2, ...., 1

E NA E E A

AAE

EE A

x xx A

=γ γ γ γ

γ

=≠

−= − ∇ + = −∑ v v

DN (39)

This form of the species momentum equation is acceptable when molecule-molecule

collisions are much more frequent than molecule-wall collisions, thus Eq. 39 is

page 19


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

in Eq. 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 mass

average velocity in Eq. 38 is defined by

1

A N

A A

A

=

γ γ γ

=

= ω∑v v (40)

and the associated mass diffusion velocity is defined by the decomposition

A Aγ γ γ= +v v u (41)

The mass diffusive flux has the attractive characteristic that the sum of the fluxes is zero,

i.e.,

(42) 1

0A N

A A

A

=

γ γ

=

ρ =∑ u

As an alternative to Eqs. 40 through 42, we can define a molar average velocity by

*

1

A N

A A

A

x=

γ γ γ

=

= ∑v v (43)

and the associated molar diffusion velocity is given by

*A A

∗γ γ= +v v γu (44)

In this case, the molar diffusive flux also has the attractive characteristic given by;

(45) 1

0A N

A A

A

c=

∗γ γ

=

=∑ u

however, the use of the molar average velocity defined by Eq. 43 presents problems when

Eq. 38 must be used as one of the N momentum equations.

page 20


If we make use of the mass average velocity and the mass diffusion velocity as

indicated by Eqs. 40 and 41, the molar flux in Eq. 35 takes the form

(46) total molar molar convective mixed-mode flux flux diffusive flux

A A A A Ac cγ γ γ γ γ γ= +v v c u

N

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

(47) , 1, 2, ...,A A Ac Aγ γ γ= =J u

so that Eq. 35 takes the form

( ) , 1,2,...,AA A A

cc R A

tγ

γ γ γ γ∂

+ ∇ ⋅ = − ∇ ⋅ + =∂

v J N (48)

The single drawback to this mixed-mode diffusive flux is that it does not satisfy a simple

relation such as that given by either Eq. 42 or Eq. 45. Instead, we find that the mixed

mode diffusive fluxes are constrained by

( )1

0A N

A A

A

M M=

γ

=

=∑J (49)

in which AM is the molecular mass of species A and M is the mean molecular mass defined by

1

A N

A A

A

M x M=

γ

=

= ∑ (50)

There are many problems for which we wish to know the concentration, Ac γ , and the

normal component of the molar flux of species A at a phase interface. The normal

component of the molar flux at an interface will be related to the adsorption process and

the heterogeneous reaction by means of the jump condition given by Eq. 36 and relations

of the type given by Eqs. 37, and this flux will be influenced by the convective, cAγ γv ,

and diffusive, AγJ , fluxes.

page 21


The governing equations for Ac γ and γv are available to us in terms of Eqs. 38 and

48, and here we consider the matter of determining AγJ . To determine the mixed-mode

diffusive flux, we return to the Stefan-Maxwell equations and make use of Eq. 41 to

obtain

1

( )0 , 1, 2, ...., 1

E NA E E A

AAE

EE A

x xx A

=γ γ γ γ

γ

=≠

−= − ∇ + = −∑ u u

DN (51)

This can be multiplied by the total molar concentration and rearranged in the form

1 1

0 , 1, 2, ...., 1E N E N

E E EA A A A

AE AEE EE A E A

c xc x x c A N

= =γ γ γ

γ γ γ γ γ

= =≠ ≠

= − ∇ + − = −

∑ ∑uu

D D (52)

which can then be expressed in terms of Eq. 47 to obtain

1 1

0 , 1, 2, ...., 1E N E N

E EA A A

AE AEE EE A E A

xc x x A N

= =γ γ

γ γ γ γ

= =≠ ≠

= − ∇ + − = −

∑ ∑JJ

D D (53)

Here we can use the classic definition of the mixture diffusivity

1

1E N

E

Am AEE A

E

x=

γ

=≠

= ∑D D (54)

in order to express Eq. 53 as

1

, 1, 2, ...., 1E N

AmA A E Am A

AEEE A

x c x A=

γ γ γ γ γ

=≠

− = − ∇ =∑J JD DD

N − (55)

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

solution representation for the diffusive flux

page 22


, 1 (56) A Am A Ac x xγ γ γ γ= − ∇ <<J D

and the transport equation for species A takes the form

( ) ( ) ,AA Am A A

cc c x R x

tγ

γ γ γ γ γ γ∂

+ ∇ ⋅ = ∇ ⋅ ∇ + <<∂

v D 1A (57)

Given the condition, , it is often plausible to impose the condition 1Ax γ <<

A Ax c c xγ γ γ γ∇ << ∇ (58)

and this leads to the following convective-diffusion equation that is ubiquitous in the

reactor design literature:

( ) ( ) ,AA Am A A A

cc c R

tγ

γ γ γ γ γ∂

+ ∇ ⋅ = ∇ ⋅ ∇ + <<∂

v D 1x (59)

When the mole fraction of species A is not small compared to one, the diffusive flux in

this transport equation will not be correct. If the diffusive flux plays an important role in

the rate of heterogeneous reaction, Eq. 59 will not lead to a correct representation for the

rate of reaction.

V. DIFFUSIVE FLUX

We begin our analysis of the diffusive flux with Eq. 55 in the form

1

, 1, 2, ...., 1E N

AmA Am A A E

AEEE A

c x x A N=

γ γ γ γ γ

=≠

= − ∇ + = −∑J JDDD

(60)

and make use of Eq. 49 in an alternate form

( )1

0A N

A A N

A

M M=

γ

=

=∑J (61)

in order to obtain N equations relating the N diffusive fluxes. At this point we define a

matrix [ ]R according to

page 23


1 ......

1 ......

[ ] 1 ......

. . . . . ...... .

. . . . . ...... .

...... 1

A Am A Am A Am

AB AC AN

B Bm B Bm B Bm

BA BC BN

C Cm C Cm C Cm

CA CB CN

A B C

N N N

x x x

x x

x x xR

M M MM M M

γ γ γ

γ γ

γ γ γ

− − − −

− + − − −

= − − + − −

− − − − + + + +

D DD D D

D DD D

D DD D D

x γ

D

DD

D (62)

and use Eqs. 60 and 61 to express the N diffusive fluxes according to

[ ]

1 1

... ...

...

0

Am AA

Bm BB

Cm CC

N m N

N

x

x

xR c

x

γγ

γγ

γγγ

− − γ

γ

∇ ∇ ∇ = −

∇

J

J

J

J

D

D

D

D

(63)

We assume that the inverse of [ ]R exists in order to express the column matrix of

diffusive flux vectors in the form

[ ] 1

1 1

... ...

...

0

Am AA

Bm BB

Cm CC

N m N

N

x

x

xc R

x

γγ

γγ

− γγγ

− − γ

γ

∇ ∇ ∇ = −

∇

J

J

J

J

D

D

D

D

(64)

in which the column matrix on the right hand side of this result can be expressed as

page 24


(65)

1 11 1

0 0 ...... 0 00 0 ...... 0 0

0 0 ...... 0 0

. . . ...... . . .....0 0 0 ...... 0

0 0 0 ...... 0 00

AmAm A A

BmBm B B

CmCm C C

N m NN m N

Nm

x x

x x

x x

xx

γ γ

γ γ

γ γ

− − γ− − γ

∇ ∇ ∇ ∇ ∇ ∇ =

∇∇

DDDD

DD

DDD

The diffusivity matrix is now defined by

1

1

0 0 ...... 0 00 0 ...... 0 0

0 0 ...... 0 0[ ] [ ]

. . . ...... . .0 0 0 ...... 0

0 0 0 ...... 0

Am

Bm

Cm

N m

Nm

D R −

−

=

DD

D

D

D

(66)

so that Eq. 64 Takes the form

1

[ ]... ......

0

A A

B B

C

N

N

x

x

xc D

x

γ γ

γ γ

γγ

Cγ

− γ

γ

∇

∇ ∇ = −

∇

J

J

J

J

N

(67)

This result can be expressed in a form analogous to that given by Eq. 60 leading to

(68) 1

1

, 1, 2, ....,E N

A AE E

E

c D x A= −

γ γ γ

=

= − ∇ =∑J

In the general case, the elements of the diffusivity matrix, AED , will depend on the mole

fractions in a non-trivial manner. When this result is used in Eq. 48 we obtain the non-

linear, coupled governing differential equation for Ac γ given by

page 25


( )1

1

, 1,2,...,E N

AA AE E A

E

cc c D x R A

t

= −γ

γ γ γ γ γ

=

∂ + ∇ ⋅ = ∇ ⋅ ∇ + = ∂

∑v N (69)

We seek a solution to this equation subject to the jump condition given by Eq. 36 and this

requires knowledge of the concentration dependence of the homogeneous and

heterogeneous reaction rates and information concerning the equilibrium adsorption

isotherm. In general, a solution of Eq. 69 for the system shown in Figure 7 requires

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

volume averaging (Whitaker, 1999).

V. VOLUME AVERAGING

To obtain the volume averaged form of Eq. 69, we first associate an averaging

volume with every point in the γ−κ system illustrated in Figure 7. One such averaging

volume is illustrated in Figure 11, and it can be represented in terms of the volumes of the

individual phases according to

V Vγ κ= +V (70)

The radius of the averaging volume is r and the characteristic length scale associated

with the γ-phase is indicated by

o

γ as indicated in Figure 11. In Figure 11 we have also

illustrated 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 length scales

are disparate, i.e., the length scales are constrained by

oL r γ>> >> (71)

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 Eq. 71 are

different than those listed in Eq. 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 important to

page 26


Figure 11 Averaging volume for a porous catalyst clearly define each one. We define the superficial average of some function γψ

according to

1

V

dV

γ

γ⟨ψ ⟩ = ψγ∫V (72)

and we define the intrinsic average by

1

V

dVV

γ

γγ

γ⟨ψ ⟩ = ψγ∫ (73)

These two averages are related according to

γγ γ γ⟨ψ ⟩ = ε ⟨ψ ⟩ (74)

page 27


in which is the volume fraction of the γ-phase defined explicitly as γε

Vγ γε = V (75)

In this notation for the volume averages, a Greek subscript is used to identify the

particular phase under consideration while a Greek superscript is used to identify an

intrinsic average. Since the intrinsic and superficial averages differ by a factor of γε , it is

essential to make use of a notation that clearly distinguishes between the two averages.

When we form the volume average of any transport equation, we are immediately

confronted with the average of a gradient (or divergence), and it is the gradient (or

divergence) 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 11 this theorem can be expressed as

1

A

dA

γκ

γ γ γκ⟨∇ψ ⟩ = ∇⟨ψ ⟩ + ψγ∫ nV (76)

in which ψ is any function associated with the γ-phase. Here represents the

interfacial area contained within the averaging volume, and we have used

γ Aγκ

γκn to

represent the unit normal vector pointing from the γ-phase toward the κ-phase.

Even though Eq. 69 is considered to be the preferred form of the species continuity

equation, it is best to begin the averaging procedure with Eq. 35 and we express the

superficial average of that form as

( ) , 1,2,...,AA A A

cc R A

tγ

γ γ γ∂

+ ∇ ⋅ = ⟨ ⟩ =∂

v N (77)

For a rigid porous medium, one can use the transport theorem and the averaging theorem

to express this result as

page 28


( )1AA A A A

A

cc c dA

tγκ

γγ γ γκ γ γ γ

∂⟨ ⟩+ ∇ ⋅ ⟨ ⟩ + ⋅ = ⟨ ⟩

∂ AR∫v n vV

(78)

where it is understood that this applies to all N species. Since we seek a transport

equation for the intrinsic average concentration, we make use of Eq. 74 to express Eq. 78

in the form

( )1AA A A A A

A

cc c dA

tγκ

γγ γ

γ γ γ γκ γ γ∂⟨ ⟩

ε + ∇ ⋅ ⟨ ⟩ + ⋅ = ε ⟨∂ ∫v n v

VRγ γ ⟩ (79)

At this point, it is convenient to make use of the jump condition given by Eq. 36 in order

to obtain

1 1A AsA A A As

A A

c cc R dAt t

γκ γκ

γγ γ

γ γ γ γ∂⟨ ⟩ ∂

ε + ∇ ⋅ ⟨ ⟩ = ⟨ ⟩ − +∂ ∂

R dA∫ ∫vV V

(80)

We now define the intrinsic interfacial area average according to

1

A

dAA

γκ

γ γκ γγκ

⟨ψ ⟩ = ψ∫ (81)

so that Eq. 80 takes the convenient form given by

v v

transport homogeneous heterogeneousaccumulation adsorption reaction reaction

A AA A A As

c cc R a a

t t

γγ γγ

γ γ γ γ∂⟨ ⟩ ∂⟨ ⟩

ε + ∇ ⋅ ⟨ ⟩ = ⟨ ⟩ − + ⟨∂ ∂

v s Rκγκ⟩ (82)

One must keep in mind that this is a general result based on Eqs. 35 and 36; however,

only the first term in Eq. 82 is in a form that is ready for applications.

VI. 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 Eq. 82. The chemical kinetic constitutive equation for the heterogeneous rate

of reaction can be expressed as

page 29


( ), , .... ,As As As Bs NsR R c c c= (83)

and here we see the need to relate the surface concentrations, Asc , Bsc , …, Nsc , to the

bulk concentrations, Ac γ , Bc γ , …, Nc γ , and subsequently to the local volume averaged

concentrations, Ac γγ⟨ ⟩ , ⟨ , …, B

γγ ⟩c Nc γ

γ⟨ ⟩ . In order for heterogeneous reaction to occur,

adsorption at the catalytic surface must also occur. However, there are many transient

processes 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 quasi-

steady, the local volume averaged transport equation simplifies to

v

transport heterogeneousaccumulation reaction

AA A As

cc a

t

γγ

γ γ γ∂⟨ ⟩

ε + ∇ ⋅ ⟨ ⟩ = ⟨∂

v R γκ⟩ (84)

and this result provides the basis for several special forms.

VII. CONVECTIVE AND DIFFUSIVE TRANSPORT

Before examining the heterogeneous reaction rate in Eq.84, we consider the

transport term, A Ac γ γ⟨ v ⟩

c ⟩u

γγv

. We begin with the mixed-mode decomposition given by

Eq. 46 in order to obtain

(85) total molar molar convective mixed-mode flux flux diffusive flux

A A A A Ac cγ γ γ γ γ γ⟨ ⟩ = ⟨ ⟩ + ⟨v v

Here the convective flux is given in terms of the average of a product, and we want to

express this flux in terms of the product of averages. As in the case of turbulent

transport, this suggests the use of decompositions given by

, (86) A A Ac c cγγ γ γ γ γ= ⟨ ⟩ + = ⟨ ⟩ +v v

At this point one can follow a detailed analysis (Whitaker, 1999, Chapter 3) of the

convective transport to arrive at

page 30


(87) total flux average convective mixed-mode

flux diffusive fluxdispersive flux

A A A A Ac c cγγ γ γ γ γ γ γ γ⟨ ⟩ = ε ⟨ ⟩ ⟨ ⟩ + ⟨ ⟩ + ⟨ ⟩v v v J

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 velocity

since this is the quantity that normally appears in Darcy’s law (Whitaker, 1999) or the

Forchheimer equation (Whitaker, 1996). Use of Eq. 87 in Eq. 84 leads to

( )

v

diffusive dispersive heterogeneoustransport transport reaction

AA

A A

cc

t

c a R

γγ γ

γ γ γ γ

γ γ γ

∂⟨ ⟩ε + ∇ ⋅ ε ⟨ ⟩ ⟨ ⟩

∂

= − ∇ ⋅ ⟨ ⟩ − ∇ ⋅ ⟨ ⟩ + ⟨ ⟩

v

J v As γκ

(88)

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

irreversible representation for the heterogeneous reaction, one can show that AsR is given

by (Whitaker, 1999, Sec. 1.1)

1

1, at the interfaceAs A

As As As AAs A

k kR k c ck k γ

−

= − = − γ − κ +

(89)

when species A is consumed at the catalytic surface. Here we have used Ask to represent

the intrinsic surface reaction rate coefficient, while 1Ak and 1Ak− are the adsorption and

desorption rate coefficients that appear in Eq. 37b. Other more complex reaction

mechanisms can be proposed; however, if a linear interfacial flux constitutive equation is

valid, the heterogeneous reaction rates can be expressed in terms of the bulk

concentration as indicated by Eq. 89. Under these circumstances the functional

dependence indicated in Eq. 83 can be simplified to

( ), , .... , , at the interfaceAs As A B NR R c c cγ γ γ= γ − κ (90)

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

( ), , .... , , at the interfaceAs As A B NR R c c cγκ γκ γ γκ γ γκ γ γκ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ γ − κ (91)

page 31


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 12. There

we have shown an averaging volume with the centroid located (arbitrarily) in the

κ−phase. In this case, the area average of the bulk concentration is given explicitly by

( )

1( )A

A

cA

γκ

γ γκ γ +γκ⟨ ⟩ = Ac dA∫x

xx x y

(92)

in which x locates the centroid of the averaging volume and y locates points on the

γ−κ interface. We have used to represent the area of the γ−κ interface contained

within the averaging volume.

( )Aγκ x

Figure 12. Position vectors associated with the area average over the γ−κ interface. To complete our analysis of Eq. 91, we need to know how the area averaged

concentration, , is related to the volume averaged concentration, ⟨ . When

convective transport is important, relating

Ac γ γκ⟨ ⟩ Ac γγ ⟩

Ac γ γκ⟨ ⟩ to Ac γγ⟨ ⟩ requires some analysis;

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

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

page 32


analysis indicated by Eqs. 24 through 28 is applicable. Under these circumstances,

Eq. 91 can be expressed as

(93) ( ), , .... , , at the interfaceAs As A B NR R c c cγ γ γγκ γκ γ γ γ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ γ − κ

and Eq. 88 takes the form

( )

( )v , , ...,

AA A

As A B N

cc c

t

a R c c c

γγ γ

γ γ γ γ γ

γ γ γγκ γ γ γ

∂⟨ ⟩ε + ∇ ⋅ ε ⟨ ⟩ ⟨ ⟩ = − ∇ ⋅ ⟨ ⟩ − ∇ ⋅ ⟨ ⟩

∂

+ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

v J Aγ γv

(94)

In porous catalysts one often neglects convective transport indicated by

A Ac c γγ γ γ γ γ γ⟨ ⟩ << ε ⟨ ⟩ ⟨ ⟩ << ⟨v v A ⟩J (95)

and this leads to a transport equation that takes the form

( )v , , ..., , 1, 2, ...,

AA

As A B N

ct

a R c c c A N

γγ

γ γ


∂⟨ ⟩ε = − ∇ ⋅ ⟨ ⟩

∂

+ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ =

J (96)

This result forms the basis for the classic problem of diffusion and reaction in a porous

catalyst such as we have illustrated in Figure 5. It is extremely important to recognize

that the mathematical consequence of Eqs. 95 and 96 is that the mass average velocity

has been set equal to zero, thus our substitute for Eq. 38 is given by

Assumption: 0γ =v (97)

This assumption requires that we discard the momentum equation given by Eq. 38 and

proceed to develop a solution to our mass transfer process in terms of the 1N −

momentum equations represented by Eqs. 39. The inequalities contained in Eq. 95 are

quite appealing when one is dealing with a diffusion process; however, Eq. 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 Eq. 95 need to be developed. When

convective transport is retained, some results are available from Quintard and Whitaker

page 33


(2004); however, a detailed analysis of the coupled, non-linear process with convective

transport remains to be done. At this point we leave those problems for a subsequent

study and explore the diffusion and reaction process described by Eq. 96.

VIII. NON-DILUTE DIFFUSION

We begin this part of our study with the use of Eq. 68 in Eq. 96 to obtain

( )

1

1

v , , ..., , 1, 2, ...,

E NA

AE E

E

As A B N

cc D x

t

a R c c c A N

= −γγ

γ γ γ

=


∂⟨ ⟩ε = ∇ ⋅ ∇

∂

+ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ =

∑ (98)

in which the diffusive flux is non-linear because AED depends on the mole

fractions. This transport equation must be solved subject to the auxiliary conditions

given by

1N −

1 1

, 1A N A N

A A

A A

c c= =

γ γ

= =

= =∑ x γ∑ (99)

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

be arranged in terms of volume averaged quantities before Eq. 98 can be solved, and any

reasonable simplifications that can be made should be imposed on the analysis.

VIIIA. Constant Total Molar Concentration

Some non-dilute solutions can be treated as having a constant total molar

concentration and this simplification allows us to express Eq. 98 as

( )

1

1

v , , ..., , 1, 2, ...,

E NA

AE E

E

As A B N

cD c

t

a R c c c A N

= −γγ

γ γ

=


∂⟨ ⟩ε = ∇ ⋅ ∇

∂

+ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ =

∑ (100)

The restriction associated with this simplification is given by

, 1, 2, ...,A Ax c c x Aγ γ γ γ∇ << ∇ = N (101)

page 34


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

given by

Assumption: (102) c c constantγγ γ= ⟨ ⟩ =

Imposition of this condition means that there are only 1N − independent transport

equations of the form given by Eq. 100, and we shall impose this condition throughout

the remainder of this study. The constraints associated with Eq. 102 need to be

developed and the more general case represented by Eqs. 98 and 99 should be explored.

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

AE AED D γ= ⟨ ⟩ + AED (103)

and if we can neglect AED relative to AED γ⟨ ⟩ , the transport equation given by Eq. 100

simplifies to

( )

1

1

v 1, , ..., , 1, 2, ..., 1

E NA

AE E

E

As A B N

cD c

t

a R c c c A N

= −γγ γ

γ γ

=

γ γ γγκ γ γ − γ

∂⟨ ⟩ε = ∇ ⋅ ⟨ ⟩ ⟨∇ ⟩

∂

+ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ = −

∑ (104)

We can represent this simplification as,

AE AED D γ<< ⟨ ⟩ (105)

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

retention of the spatial deviation, AED . However, it is not clear how this type of analysis

would evolve and further study of this aspect of the diffusion process is in order.

VIIIB. Volume Average of the Diffusive Flux

The volume averaging theorem can be used with the average of the gradient in

Eq. 104 in order to obtain

page 35


1E E

A

c c c

γκ

γ γ γκ⟨∇ ⟩ = ∇⟨ ⟩ + E dAγ∫ nV (106)

and one can follow an established analysis (Whitaker, 1999, Chapter 1) in order to

express this result as

1E E

A

c c

γκ

γγ γ γ γκ γ⟨∇ ⟩ = ε ∇⟨ ⟩ + Ec dA∫ nV

(107)

Use of this result in Eq. 104 provides

1

v

1

filter

1E N

AAE E E As

E A

cD c c dA a R

tγκ

= −γγ γ γ

γ γ γ γκ γ

=

∂⟨ ⟩ ε = ∇ ⋅ ⟨ ⟩ ε ∇⟨ ⟩ + + ⟨

∂

∑ ∫ nV γκ⟩ (108)

in which the area integral of Ecγκn γ has been identified as a filter. Not all the

information available at the length scale associated with Ec γ will pass through this filter

to influence the transport equation for Ac γγ⟨ ⟩ , and the existence of filters of this type is a

recurring theme in the method of volume averaging (Whitaker, 1999).

IX. CLOSURE

In order to obtain a closed form of Eq. 108, we need a representation for the spatial

deviation concentration, Ac , 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, Eq. 48 takes the form

γ

, 1,2,..., 1AA

cA N

tγ

γ∂

= − ∇ ⋅ = −∂

J (109)

Here one must remember that the total molar concentration is a specified constant, thus

there are only independent species continuity equations. Use of Eq. 68 along with

the restriction given by Eq. 101 allows us to express this result as

1N −

page 36


1

1

, 1,2,..., 1E N

AAE E

E

cD c A N

t

= −γ

γ

=

∂= ∇ ⋅ ∇ = −

∂ ∑ (110)

and on the basis of Eq. 103 and 105 this takes the form

1

1

, 1,2,..., 1E N

AAE E

E

cD c A N

t

= −γ γ

γ

=

∂= ∇ ⋅ ⟨ ⟩ ∇ = −

∂ ∑ (111)

If we ignore variations in and subtract Eq. 108 from Eq. 111, we can arrange the

result as

γε

1

1

1v

1

1

E NA

AE E

E

E NAE

E A

E A

cD c

t

D ac dA R

γκ

= −γ γ

γ

=

= −γ

sγκ γ γκγ γ

=

∂ = ∇ ⋅ ⟨ ⟩ ∇ ∂

⟨ ⟩

− ∇ ⋅ − ⟨ ⟩ ε ε

∑

∑ ∫ nV

(112)

in which it is understood that this result applies to all 1N − species. Equation 112

represents the governing differential equation for the spatial deviation concentration, and

in order to keep the analysis relatively simple we consider only the first order,

irreversible reaction described by Eq. 89 and expressed here in the form

(113) , at the interfaceAs A AR k c γ= − γ − κ

One must remember that this is a severe restriction in terms of realistic systems and more

general forms for the heterogeneous rate of reaction need to be examined. Use of Eq. 113

in Eq. 112 leads to the following form

page 37


1

1

1v

1

1

E NA

AE E

E

E NAE A

E A

E A

cD c

t

D ac dA c

γκ

= −γ γ

γ

=

= −γ k γ

γκ γ γγ γ

=

∂ = ∇ ⋅ ⟨ ⟩ ∇ ∂

⟨ ⟩

− ∇ ⋅ + ⟨ ⟩ ε ε

∑

∑ ∫ nV

(114)

Here we have made use of the simplification

Ac Ac γγ γκ γ⟨ ⟩ = ⟨ ⟩ (115)

and the justification is given elsewhere (Whitaker, 1999, Sec. 1.3.3). In order to

complete the problem statement for cEγ , we need a boundary condition for c at the

γ−κ interface. To develop this boundary condition, we again make use of the quasi-

steady form of Eq. 36 to obtain

Eγ

(116) , at the interfaceA AsRγ γκ⋅ = − γ − κJ n

where we have imposed the restriction given by

(117) , at the interfaceAγ γκ γ γκ⋅ << ⋅ γ − κv n u n

This is certainly consistent with the inequalities given by Eq. 95; however, the neglect of

relative to u is generally based on the dilute solution condition and the

validity of Eq. 117 is another matter that needs to be carefully considered in a future

study. On the basis of Eqs.68, 101, 103 and 105 along with Eq. 113, the jump condition

takes the form

γ γ⋅v n κ κAγ γ⋅n

(118) 1

1

, at the interfaceE N

AE E A A

E

D c k c= −

γγκ γ γ

=

− ⋅ ⟨ ⟩ ∇ = γ − κ∑ n

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

concentration, we make use of the decomposition given by the first of Eqs. 86 to obtain

page 38


(119)

1

1

1

1

, at the interface

E N

AE E A A

E

E N

AE E A A

E

D c k c

D c k c

= −γ

γκ γ γ

=

= −γ γ γ

γκ γ γ

=

− ⋅ ⟨ ⟩ ∇ −

= ⋅ ⟨ ⟩ ∇⟨ ⟩ + ⟨ ⟩ γ − κ

∑

∑

n

n

With this result we can construct the following boundary value problem for Ac γ :

1

1

1v

1

accumulationdiffusion

reaction sourcenon-local diffusion

1

E NA

AE E

E

E NAE A

E A

E A

cD c

t

D ac dA c

γκ

= −γ γ

γ

=

= −γ

γkγκ γ γ

γ γ=

∂ = ∇ ⋅ ⟨ ⟩ ∇ ∂

⟨ ⟩

− ∇ ⋅ + ⟨ ⟩ ε ε

∑

∑ ∫ nV

(120)

BC.1 (121)

1

1

1

1

heterogeneous reactiondiffusive flux

reaction sourcediffusive source

, at the interface

E N

AE E A A

E

E N

AE E A A

E

D c k c

D c k c

= −γ

γκ γ γ

=

= −γ γ γ

γκ γ γ

=

− ⋅ ⟨ ⟩ ∇ −

= ⋅ ⟨ ⟩ ∇⟨ ⟩ + ⟨ ⟩ γ − κ

∑

∑

n

n

BC.2 e( , ) , at Ac tγ γ= rF A

t

(122)

IC. ( ) , at 0Ac γ = =rF (123)

In addition to the flux boundary condition given by Eq. 121, we have added an unknown

condition at the macroscopic boundary of the γ-phase, eγA , and an unknown initial

condition. Neither of these is important when the separation of length scales indicated by

Eq. 71 is valid. Under these circumstances, the boundary condition imposed at eγA

page 39


influences the Ac -field only over a negligibly small region, and the initial condition

given by Eq. 123 can be discarded because the closure problem is quasi-steady. Under

these circumstances, the closure problem can be solved in some representative, local

region (Quintard and Whitaker, 1994a-e).

γ

diffus

= −

⟨

In the governing differential equation for Ac γ , 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 γ−κ 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 Eqs. 120 and 121 were zero, the Ac γ -field would be generated only

by the non-homogeneous terms that might appear in the boundary condition imposed at

or in the initial condition given by Eq. 123. One can easily develop arguments

indicating that the closure problem for

eγA

Ac γ is quasi-steady, thus the initial condition is of

no importance (Whitaker, 1999, Chapter 1). In addition, one can develop arguments

indicating that the boundary condition imposed at eγA will influence the Ac -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. These

ideas lead to a closure problem of the form

γ

1 1v

1 1reactionion sourcenon-local diffusion

0E N E N

AE AAE E E A

E E A

D aD c c dA c

γκ

= −γ

γ γγ γκ γ

γ γ= =

⟨ ⟩ = ∇ ⋅ ⟩ ∇ − ∇ ⋅ + ⟨ ⟩ ε ε

∑ ∑ ∫ nV k

γ

(124)

page 40


BC.1 (125)

1

1

1

1

heterogeneous reactiondiffusive flux

reaction sourcediffusive source

, at the interface

E N

AE E A A

E

E N

AE E A A

E

D c k c

D c k c

= −γ

γκ γ γ

=

= −γ γ γ

γκ γ γ

=

− ⋅ ⟨ ⟩ ∇ −

= ⋅ ⟨ ⟩ ∇⟨ ⟩ + ⟨ ⟩ γ − κ

∑

∑

n

n

BC.2 (126) ( ) ( ) , 1,2,3A i Ac c iγ γ+ = =r r

Here we have used to represent the three base vectors needed to characterize a

spatially periodic system. The use of a spatially periodic system does not limit this

analysis to simple systems since a periodic system can be arbitrary complex (Quintard

and Whitaker, 1994a-e). However, the periodicity condition imposed by Eq. 126 can

only be strictly justified when

i

AED γ⟨ ⟩ , Ac γγ⟨ ⟩ , and Ac γ

γ∇⟨ ⟩ are constants and this does

not occur for the types of systems under consideration. This matter has been examined

elsewhere (Whitaker, 1986b) and the analysis suggests that the traditional separation of

length scales allows one to treat AED γ⟨ ⟩ Ac, γγ⟨ ⟩ , and Ac γ

γ∇⟨ ⟩ as constants within the

framework of the closure problem.

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

source in Eqs. 124 and 125 makes a negligible contribution to Ac γ . In addition, one can

demonstrate (Whitaker, 1999) that the heterogeneous reaction, A Ak c γ , can be neglected

for all practical problems of diffusion and reaction in porous catalysts. Furthermore, the

non-local diffusion term is negligible for traditional systems, and under these

circumstances the boundary value problem for the spatial deviation concentration takes

the form

1

1

0E N

AE E

E

D c= −

γγ

=

= ∇ ⋅ ⟨ ⟩ ∇ ∑ (127)

page 41


BC.1 (128) 1 1

1 1

, at E N E N

AE E AE E

E E

D c D c A= − = −

γ γγκ γ γκ γ γκ

= =

− ⋅ ⟨ ⟩ ∇ = ⋅ ⟨ ⟩ ∇⟨ ⟩∑ ∑n n γ

BC.2 (129) ( ) ( ) , 1,2,3A i Ac c iγ γ+ = =r r

Here one must remember that the subscript A represents species A, B, C, , . 1N −

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

represented by ∇⟨ in the boundary condition imposed at the γ−κ interface. If this

source term were zero, the solution to this boundary value problem would be given by

. Any constant associated with

Ec γγ ⟩

Ac constantγ = Ac γ will not pass through the filter in

Eq. 108, and this suggests that a solution can be expressed in terms of the gradients of the

volume averaged concentration. Since the system is linear in the independent

gradients of the average concentration, we are led to a solution of the form

1N −

, 1 1.....E EA A EB B EC C E N Nc c c cγ γ γγ γ γ γ −= ⋅∇⟨ ⟩ + ⋅∇⟨ ⟩ + ⋅∇⟨ ⟩ + + ⋅∇⟨ ⟩b b b b c γ

− γ

(130)

Here the vectors, b , , etc., are referred to as the closure variables or the mapping

variables since they map the gradients of the volume averaged concentrations onto the

spatial deviation concentrations. In this representation for

EA EBb

Ac γ , we can ignore the spatial

variations of , , etc., within the framework of a local closure problem,

and we can use Eq. 130 in Eq. 127 to obtain

Aγ

γ ⟩c∇⟨ Bc∇⟨ γγ ⟩

1 1

1 1

0E N D N

AE ED D

E D

D= − = −

cγ γγ

= =

= ∇ ⋅ ⟨ ⟩ ∇ ⋅∇⟨ ⟩ ∑ ∑ b

Aγ

N −

(131)

BC.1

1 1 1

1 1 1

, at E N D N E N

AE ED D AE E

E D E

D c D c= − = − = −

γ γ γγκ γ γκ γ γκ

= = =

− ⋅ ⟨ ⟩ ∇ ⋅∇⟨ ⟩ = ⋅ ⟨ ⟩ ∇⟨ ⟩∑ ∑ ∑n b n

(132)

BC.2 (133) ( ) ( ) , 1,2,3 , 1, 2, ,..., 1AE i AE i A+ = = =b r b r

page 42


The derivation of Eqs. 131 and 132 requires the use of simplifications of the form

( )EA A EA Ac cγ γγ γ∇ ⋅ ∇⟨ ⟩ = ∇ ⋅ ∇⟨ ⟩b b (134)

which result from the inequality

EA A EA Ac cγ γγ γ⋅∇∇⟨ ⟩ << ∇ ⋅∇⟨ ⟩b b (135)

The basis for this inequality is the separation of length scales indicated by Eq. 71, and a

detailed discussion is available elsewhere (Whitaker, 1999). One should keep in mind

that the boundary value problem given by Eqs. 131 through 133 applies to all 1N −

species and that the concentration gradients are independent. This latter condition

allows us to obtain

1N −

(136) 1

1

0 , 1, 2, ... , 1E N

AE ED

E

D D= −

γ

=

= ∇ ⋅ ⟨ ⟩ ∇ = − ∑ b N

A

N

BC.1 (137) 1

1

, 1, 2, ... , 1 , at E N

AE ED AD

E

D D D N= −

γ γγκ γκ γκ

=

− ⋅ ⟨ ⟩ ∇ = ⟨ ⟩ = −∑ n b n

Periodicity: (138) ( ) ( ) , 1,2,3 , 1, 2, ,..., 1AD i AD i D+ = = = −b r b r

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

obtain

First Problem for Species A

(139a) ( )

( ) ( )

1

1 1, 1 1,

0

....

AA AA AA AB BA

AA AC CA AA A N N A

D D D

D D D D

−γ γ γ

− −γ γ γ γ− −

= ∇ ⋅ ⟨ ⟩ ∇ + ⟨ ⟩ ⟨ ⟩ ∇ + ⟨ ⟩ ⟨ ⟩ ∇ + + ⟨ ⟩ ⟨ ⟩ ∇

b b

b b

BC.

( )

( )

( )

1

1

1, 1 1,

.....

, at

AA AA AB BA

AA AC CA

AA A N N A

D D

D D

D D

−γ γγκ γκ

−γ γγκ

−γ γγκ − − γκ γκ

− ⋅∇ − ⋅ ⟨ ⟩ ⟨ ⟩ ∇

− ⋅ ⟨ ⟩ ⟨ ⟩ ∇ −

− ⋅ ⟨ ⟩ ⟨ ⟩ ∇ =

n b n b

n b

n b An

(139b)

page 43


Periodicity: (139c) ( ) ( ) , 1,2,3 , 1,2,..., 1DA i DA i D+ = − = −b r b r N

Second Problem for Species A

(140a) ( )

( ) ( )

1

1 1, 1 1,

0

....

AB AB AA AB BB

AB AC CB AB A N N B

D D D

D D D D

−γ γ γ

− −γ γ γ γ− −

= ∇ ⋅ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ∇ + ∇ +

⟨ ⟩ ⟨ ⟩ ∇ + + ⟨ ⟩ ⟨ ⟩ ∇

b b

b b

BC.

( )

( )

( )

1

1

1, 1 1,

.....

, at

AA AA AB BA

AA AC CA

AA A N N A

D D

D D

D D

−γ γγκ γκ

−γ γγκ

−γ γγκ − − γκ γκ

− ⋅∇ − ⋅ ⟨ ⟩ ⟨ ⟩ ∇

− ⋅ ⟨ ⟩ ⟨ ⟩ ∇ −

− ⋅ ⟨ ⟩ ⟨ ⟩ ∇ =

n b n b

n b

n b An

N

(140b)

Periodicity: (140c) ( ) ( ) , 1,2,3 , 1,2,..., 1DB i DB i D+ = = = −b r b r

Third Problem for Species A

etc. (141)

N-1 Problem for Species A

etc. (142)

Here it is convenient to define a new set of closure variables or mapping variables

according to

( )

( ) ( )

1

1 1, 1 1,....

AA AA AA AB BA

AA AC CA AA A N N

D D

D D D D

−γ γ

− −γ γ γ γ− −

= + ⟨ ⟩ ⟨ ⟩

+ ⟨ ⟩ ⟨ ⟩ + + ⟨ ⟩ ⟨ ⟩

d b b

b b A

(143a)

( )

( ) ( )

1

1 1, 1 1,....

AB AB AA AB BB

AB AC CB AB A N N

D D

D D D D

−γ γ

− −γ γ γ γ− −

= ⟨ ⟩ ⟨ ⟩ + +

⟨ ⟩ ⟨ ⟩ + + ⟨ ⟩ ⟨ ⟩

d b b

b b B

(143b)

( ) ( )

( )

1 1

1, 1 1,....

AC AC AA AC AC AB BC

CC AC A N N C

D D D D

D D

− −γ γ γ γ

−γ γ− −

= ⟨ ⟩ ⟨ ⟩ + ⟨ ⟩ ⟨ ⟩

+ + ⟨ ⟩ ⟨ ⟩

d b

b b

+b (143c)

etc. (143n-1)

page 44


With these definitions, the closure problems take the following simplified forms:

First Problem for Species A

20 AA= ∇ d (144a)

BC. (144b) , atAA Aγκ γκ γκ− ⋅∇ =n d n

Periodicity: (144c) ( ) ( ) , 1, 2, 3AA i AA i+ = =d r d r

Second Problem for Species A

20 AB= ∇ d (145a)

BC. (145b) , atAB Aγκ γκ γκ− ⋅∇ =n d n

Periodicity: (145c) ( ) ( ) , 1,2,3AB i AB i+ = =d r d r

Third Problem for Species A

etc. (146)

N-1 Problem for Species A

etc. (147)

To obtain these simplified forms, one must make repeated use of inequalities of the form

given by Eq. 135. Each one of these closure problems is identical to that obtained by

Ryan et al. (1981) and solutions have been developed by several workers (Ryan et al.,

1981; Ochoa-Tapia et al., 1994; Chang, 1982 and 1983; Quintard, 1993; Quintard and

Whitaker, 1993a-b). In each case, the closure problem determines the closure variable to

within an arbitrary constant, and this constant can be specified by imposing the condition

1,2,..., 1

0 , or 0 ,1,2,..., 1D GD

G Nc

D Nγ γ

γ= −

⟨ ⟩ = ⟨ ⟩ = = −d (148)

However, any constant associated with a closure variable will not pass through the filter

in Eq. 108, thus this constraint on the average is not necessary.

IXA. Closed Form

The closed form of Eq. 108 can be obtained by use of the representation for Ec γ

given by Eq. 130, along with the definitions represented by Eqs. 143. After some

algebraic manipulation, one obtains

page 45


, 1 , 1

1

1

1

1

AAA AA A

A

AB AB B

A

AC AC C

A

A N A N

A

cD dA

t V

D dA cV

D dA cV

D dAV

γκ

γκ

γκ

γκ

γγ γ γ

γ γ γκγ

γ γγ γκ γ

γ

γ γγ γκ γ

γ

γγ − γκ −

γ

∂⟨ ⟩

ε = ∇ ⋅ ε ⟨ ⟩ + ⋅ ∇⟨ ⟩ ∂

+ ε ⟨ ⟩ + ⋅ ∇⟨ ⟩ +

+ ε ⟨ ⟩ + ⋅ ∇⟨ ⟩ +

+ ε ⟨ ⟩ +

∫

∫

∫

∫

n d

n d

n d

n d

I

I

I

I 1 vN Ac a k

c γ +

Acγ γ− γ

γ⋅∇⟨ ⟩ + ⟨ ⟩

(149)

Here one must remember that we have restricted the analysis to the simple, linear

reaction rate expression given by Eq. 113, and one normally must work with more

complex representations for AsR .

On the basis of the closure problems given by Eqs. 144a through 147, we conclude

that there is a single tensor that describes the tortuosity for species A. This means that

Eq. 149 can be expressed as

1 v, 1......

A eff eff effA BAA AB AC

effN A AA N

cc c

t

c a k c

γγ

Ccγ γ γγ γ γ γ γ γ

γ γγ − γ γ−

∂⟨ ⟩ ε = ∇ ⋅ ε ⋅∇⟨ ⟩ + ε ⋅∇⟨ ⟩ + ε ⋅∇⟨ ⟩∂

+ + ε ⋅∇⟨ ⟩ + ⟨ ⟩

D D D

D

γ

(150)

in which the effective diffusivity tensors are related according to

page 46


, 1

, 1.....

effeffeff effA NACAA AB

AA AB AC A ND D D D−

γ γ γ−

= = = =⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

DDD Dγ (151)

The remaining diffusion equations for species B, C, … 1N − have precisely the same

form as Eq. 150, and the various effective diffusivity tensors are related to each other in

the manner indicated by Eqs. 151. The generic closure problem can be expressed as

Generic Closure Problem

20 = ∇ d (152a)

BC. , at A γκ γκ− ⋅∇ =n d n γκ (152b)

Periodicity: (152c) ( ) ( ) , 1, 2, 3i i+ = =d r d r

and solution of this boundary value problem is relatively straightforward. The existence

of a single, generic closure problem that allows for the determination of all the effective

diffusivity tensors represents the main finding of this work. On the basis of this single

closure problem, the tortuosity tensor is defined according to

1

A

dAV

γκ

γκγ

= + ∫ n dIτ (153)

and we can express Eqs. 151 in the form

(154) , 1, 1, , ..... ,eff eff effAA AB A NAA AB A ND Dγ γ

−−= ⟨ ⟩ = ⟨ ⟩ = ⟨ ⟩D D Dτ τ τ D γ

Substitution of these results into Eq. 150 allows us to represent the local volume averaged

diffusion-reaction equations as

1

v

1

, 1,2,.., 1E N

AAE E A A

E

cD c a k c A N

t

= −γγ γ γ γ

γ γ γ γ

=

∂⟨ ⟩ ε = ∇ ⋅ ε ⟨ ⟩ ⋅ ∇⟨ ⟩ + ⟨ ⟩ = − ∂ ∑ τ (155)

It is important to remember that this analysis has been simplified on the basis of Eq. 101

which is equivalent to treating cγ as a constant as indicated in Eq. 102. For a porous

medium that is isotropic in the volume averaged sense, the tortuosity tensor takes the

classical form

page 47


−1= τIτ (156)

in which is the unit tensor and I τ is the tortuosity. For isotropic porous media, we can

express Eq. 155 as

( )1

1

v , 1,2,.., 1

E NA

AE E

E

A A

cD c

t

a k c A N

= −γγ γ γ

γ γ

=

γγ

γ

∂⟨ ⟩ ε = ∇ ⋅ ε τ ⟨ ⟩ ∇⟨ ⟩ ∂

+ ⟨ ⟩ = −

∑ (157)

Often and can be treated as constants; however, the diffusion coefficients in this

transport equation will be functions of the local volume averaged mole fractions and we

area faced with a coupled, non-linear diffusion and reaction problem.

γε τ

X. CONCLUSIONS

In this paper we have first shown how an intuitive upscaling procedure can lead to

confusion regarding homogeneous and heterogeneous reactions, and in a more formal

development we have shown how the coupled, non-linear diffusion problem can be

analyzed to produce volume averaged transport equations containing effective diffusivity

tensors. The original diffusion-reaction problem is described by

1

1

, 1,2,..., 1E N

AAE E

E

cD c A N

t

= −γ

γ

=

∂= ∇ ⋅ ∇ = −

∂ ∑ (158a)

BC. (158b) 1

1

, at the interfaceE N

AE E A A

E

D c k c= −

γκ γ γ

=

− ⋅ ∇ = γ − κ∑ n

(158c) c c constantγγ γ= ⟨ ⟩ =

in which the AED are functions of the mole fractions. For a porous medium that is

isotropic in the volume averaged sense, the upscaled version of the diffusion-reaction

problem takes the form

page 48


( )1

1

v , 1,2,.., 1

E NA

AE E

E

A A

cD c

t

a k c A N

= −γγ γ γ

γ γ

=

γγ

γ

∂⟨ ⟩ ε = ∇ ⋅ ε τ ⟨ ⟩ ∇⟨ ⟩ ∂

+ ⟨ ⟩ = −

∑ (159)

Here we have used the approximation that AED can be replaced by AED γ⟨ ⟩ and that

variations of AED γ⟨ ⟩ can be ignored within the averaging volume. The fact that only a

single tortuosity needs to be determined by Eqs. 152 and 153 represents the key

contribution of this study. It is important to remember that this development is

constrained by the linear chemical kinetic constitutive equation given by Eq. 113. The

process of diffusion in porous catalysts is normally associated with slow reactions and

Eq. 93 is satisfactory; however, the first order, irreversible reaction represented by

Eq. 113 is the exception rather than the rule, and this aspect of the analysis requires

further investigation. The influence of a non-zero mass average velocity needs to be

considered in future studies so that the constraint given by Eq. 97 can be removed. An

analysis of that case is reserved for a future study which will also include a careful

examination of the simplification indicated by Eq. 117.

Nomenclature

eγA area of entrances and exits of the γ-phase contained in the macroscopic region, m2

Aγκ area of the γ−κ interface contained within the averaging volume, m2

va / , area per unit volume, m-1 Aγκ V

γb body force vector, m/s2

Ac γ bulk concentration of species A in the γ-phase, moles/m3

Ac γ⟨ ⟩ superficial average bulk concentration of species A in the γ-phase, moles/m3

Ac γγ⟨ ⟩ intrinsic average bulk concentration of species A in the γ-phase, moles/m3

Ac γ γκ⟨ ⟩ intrinsic area average bulk concentration of species A at the γ−κ -interface, moles/m3

page 49


Ac γ , spatial deviation concentration of species A, moles/m3 A Ac c γγ γ− ⟨ ⟩

cγ 1

A N

A

A

c=

γ

=∑ , total molar concentration, moles/m3

Asc surface concentration of species A associated with the γ−κ interface, moles/m2

ABD binary diffusion coefficient for species A and B, m2/s

AmD 1

1

E N

Am E

EE A

x=

−γ

=≠

= ∑D AED

]

, mixture diffusivity, m2/s

[D diffusivity matrix, m2/s

AED element of the diffusivity matrix, m2/s

AED γ⟨ ⟩ intrinsic average element of the diffusivity matrix, m2/s

AED , spatial deviation of an element of the diffusivity matrix, m2/s AE AED D γ− ⟨ ⟩

AγJ A Ac γu γ , mixed-mode diffusive flux, mole/m2s

AK adsorption equilibrium coefficient for species A, m

1Ak adsorption rate coefficient for species A, m/s

1Ak− desorption rate coefficient for species A, s-1

Ask surface reaction rate coefficient, s-1

γ small length scale associated with the γ-phase, m

or radius of the averaging volume, m

L large length scale associated with the porous medium, m

AM molecular mass of species A, kg/kg mole

M 1

A N

A A

A

x M=

γ

=∑ , mean molecular mass, kg/kg mole

γκn unit normal vector directed from the γ-phase to the κ-phase

r position vector, m

AR γ rate of homogeneous reaction in the γ-phase, moles/m3s

page 50


AsR rate of heterogeneous reaction associated with the γ−κ interface, moles/m2s

AsR γκ⟨ ⟩ area average heterogeneous reaction rate for species A, moles/m2s

t time, s

γT stress tensor for the γ-phase, N/m2

Aγu , mass diffusion velocity, m/s Aγ −v vγ

A∗

γu , molar diffusion velocity, m/s A∗

γ −v vγ

Aγv velocity of species A in the γ-phase, m/s

γv 1

A N

A A

A

=

γ γ

=

ω∑ v , mass average velocity in the γ-phase, m/s

∗γv

1

A N

A A

A

x=

γ γ

=∑ v , molar average velocity in the γ-phase, m/s

γγ⟨ ⟩v intrinsic mass average velocity in the γ-phase, m/s

γ⟨ ⟩v superficial mass average velocity in the γ-phase, m/s

γv , spatial deviation velocity, m/s γγ γ− ⟨ ⟩v v

V averaging volume, m3

Vγ volume of the γ-phase contained within the averaging volume, m3

Vκ volume of the κ-phase contained within the averaging volume, m3

Ax γ / , mole fraction of species A in the γ-phase Ac cγ γ

x position vector locating the center of the averaging volume, m

y position vector locating points on the γ−κ interface relative to the center of the averaging volume, m

Greek Letters

γε volume fraction of the γ-phase (porosity)

Aγρ mass density of species A in the γ-phase, kg/m3

γρ mass density for the γ-phase, kg/m3

Aγω / , mass fraction of species A in the γ-phase Aγρ ργ

page 51


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