Chemical Engineering Science 61 (2006) 2643–2669www.elsevier.com/locate/ces

Nonlinear, multicomponent, mass transport in porous media

M. Quintarda,∗, L. Bletzackera, D. Chenua, S. Whitakerb

aInstitut de Mécanique des Fluides de Toulouse, Av. du Professeur Camille Soula, 31400 Toulouse, FrancebDepartment of Chemical Engineering and Material Science, University of California at Davis, Davis, CA 95616, USA

Received 13 May 2005; received in revised form 14 November 2005; accepted 16 November 2005Available online 9 January 2006

Abstract

In this paper we consider multicomponent mass transport in porous media for non-dilute solutions. This process is described by coupled,nonlinear transport equations that must be spatially smoothed in order to be useful. This spatial smoothing, or upscaling, is achieved by themethod of volume averaging for the case of negligible adsorption, desorption, and heterogeneous reaction. For pure diffusion, the resultsdemonstrate that a single tortuosity tensor applies to the transport of all species. When convective transport is important, the process becomesmuch more complex and it is difficult to generalize about the behavior of the various dispersion tensors.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Stefan–Maxwell; Convection; Diffusion; Porous media; Volume averaging

1. Introduction

The general problem of mass transport in porous mediainvolves a wide range of complications such as bulk andKnudsen diffusion, adsorption/desorption and heterogeneousreaction, slip and non-slip boundary conditions, deformableporous media, electrostatic and electrodynamic effects, etc.Often these transport processes are associated with dilutesolutions, and the analysis is based on a linear form of theconvective–diffusion equation. When the dilute solution ap-proximation is not valid, one is confronted with a set of cou-pled, nonlinear transport equations. In this paper, we examinethis transport process under passive conditions, i.e., in theabsence of adsorption/desorption and heterogeneous reaction,and in the presence of the no-slip condition. This provides animportant point of departure for future studies of the morecomplex processes involving active conditions associated withthe fluid–solid interface.

The general problem of mass transport in porous mediahas been the object of many studies, and monographs byJackson (1977) and by Mason and Malinauskas (1983) provide

∗ Corresponding author. Tel.: +33 5 61 28 59 21; fax: +33 5 61 28 58 99.E-mail address: quintard@imft.fr (M. Quintard).

0009-2509/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.11.034

extensive reviews of this subject. The central issue associatedwith multicomponent mass transport in porous media is the up-scaling of the species momentum equation (Whitaker, 1987),and this upscaling is generally accomplished in an intuitivemanner. The dusty gas model is a classic example of this intu-itive approach and recent studies by Kerkhof (1996, 1997) haveshed new light on the strengths and weaknesses of this model.Extensive calculations have been carried out using the dustygas model and recent examples are given by Suwanwarangkulet al. (2003) and by Fen and Abriola (2004). In this paper,we consider the formal upscaling of a special form of thespecies momentum equation, i.e., the Stefan–Maxwell equa-tions. The method of volume averaging is employed to predictboth the form of the spatially smoothed transport equations andthe effective transport coefficients that appear because of theupscaling. The theory is compared with numerical experimentsand good agreement is obtained.

The first formal upscaling of the linear problem was done byTaylor (1953) for the case of passive transport and by Golay(1958) for the case of active transport caused by linear adsorp-tion. Neither Taylor nor Golay presented their work in termsof the currently established procedures for upscaling; however,the essential elements were given, i.e., transport equationswere developed for averaged concentrations and a closurewas achieved that allowed for the prediction of the dispersion

2644 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

coefficient. In both studies, the linear convective–diffusionequation provided the basis for the analysis which includedthe no-slip condition. In addition, both studies made use of abundle of capillary tubes as a model of a porous medium.

Brenner (1980) extended the work of Taylor to the case ofa spatially periodic porous medium using the method of mo-ments, and Carbonell and Whitaker (1983) used the methodof volume averaging to achieve the upscaled transport equa-tion for disordered porous media. In this latter approach, onemust use a spatially periodic model to achieve closure and pre-dict the value of the effective diffusivity and the dispersioncoefficient. The relation between volume averaging for disor-dered porous media and the use of spatially periodic mod-els for closure has been discussed in detail by Quintard andWhitaker (1994a–e). Similar results for the explicit relation be-tween porous medium structure and effective diffusivity anddispersion coefficients were also obtained within the frame-work of homogenization theory (Mei, 1992; Auriault, 1995).Based on these theoretical analyses, many calculations havebeen performed for different types of unit cells in order toprovide effective diffusivities and dispersion coefficients (Kimet al., 1987; Edwards et al., 1991; Quintard, 1993; Sahraouiand Kaviany, 1994; Adler, 1994; Souto and Moyne, 1997; Namand Kaviany, 2003). Comparison with available experimentssupport the theoretical results (Eidsath et al., 1983; Kim et al.,1987; Quintard, 1993), provided that experimental conditionsare such that anomalous diffusion does not occur (Cushmanand Moroni, 2001). These upscaling studies were devoted tolinear systems, and the question of effective diffusion and dis-persion for general multicomponent systems has received lessattention.

Models for multicomponent transport in porous media havebeen used in many applications, such as petroleum engineer-ing, environmental hydrogeology, and chemical engineering.In general, these models assume that diffusion at the pore-scaleof the multicomponent mixture involves uncoupled transportequations with single diffusion coefficients, i.e., spherical ordiagonal generalized diffusion matrices, and, consequently,the same tortuosity and dispersion curve are used at themacro-scale for all components (Corapcioglu and Baehr, 1987;Abriola and Pinder, 1985; Wang and Cheng, 1996; Prins-Jansenet al., 1996; Gurau et al., 1998; Zheng and Wang, 1999;He et al., 2000; Douglas et al., 2003).

Some attempts to take into account full dispersion matri-ces have been undertaken; however, most models postulate thestructure of the macro-scale equations with multicomponenteffective diffusion coefficients involving the same “tortuosity”parameters for all components (Mason and Malinauskas, 1983;Bernardi and Verbrugge, 1991, 1992; Arnost and Schneider,1995; Bevers et al., 1997; Wohr et al., 1998; Berning et al.,2002; Djilali and Lu, 2002). In a recent study based on the pore-scale equations, Vynnycky and Birgersson (2003) presentedvolume averaged equations using a simplified closure schemeassuming that the tortuosity is the same for all components ofthe generalized diffusion matrix, and that a single dispersionterm may be added to the matrix diagonal. A similar approxi-mation was made within the framework of an averaging method

Fig. 1. Mass transport in porous media.

by Roos et al. (2003), and the main purpose of this paper is toexplore the validity of this approximation.

In this study, we consider the passive transport process ofdiffusion and convection in the porous medium illustrated inFig. 1. We identify the solid phase shown in Fig. 1 as the �-phase and the fluid phase as the �-phase so that the governingequations and interfacial flux boundary conditions for an N-component system are given by

��A�

�t+ ∇ · (�A�vA�) = 0, A = 1, 2, 3, . . . , N (1)

BC n�� · (�A�vA�) = 0,

A = 1, 2, 3, . . . , N, at the �.� interface (2)

Here it is apparent that there is no homogeneous reaction, noadsorption, and no heterogeneous reaction, thus we are treatingthe simplest possible non-dilute solution mass transfer process.In addition to Eqs. (1) and (2), we need N momentum equations(Whitaker, 1987) to determine the N species velocities repre-sented by vA�, A = 1, 2, . . . , N . The total, or mass average,momentum equation can always be expressed in the form

�

�t(��v�) + ∇ · (��v�v�) = ��b� + ∇ · T� (3)

in which �� and v� are defined by

�� =A=N∑A=1

�A�, v� =A=N∑A=1

�A�vA�, �A� = �A�/��. (4)

There are certain processes for which the N momentum equa-tions consist of the total momentum equation and the N − 1Stefan–Maxwell equations

0 = −∇xA� +B=N∑B=1B �=A

xA�xB�(vB� − vA�)

DAB

,

A = 1, 2, . . . , N − 1 (5)

in which the DAB are referred to as the binary diffusion co-efficients. This form of the species momentum equation is ac-ceptable when molecule–molecule collisions are much morefrequent than molecule–wall collisions, thus Eq. (5) is inap-propriate when Knudsen diffusion must be taken into account.

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2645

In addition, Eq. (5) requires that the �-phase be treated as anideal mixture, thus the driving force for diffusion becomesthe gradient of the mole fraction rather than the gradient ofthe chemical potential. For non-ideal mixtures, the general-ized Stefan–Maxwell equations are recommended (Taylor andKrishna, 1993) and these are given by

0 = −dA� +B=N∑B=1B �=A

xA�xB�(vB� − vA�)

DAB

,

A = 1, 2, . . . , N − 1, (6a)

dA� = xA�

RT∇�A�. (6b)

Here �A� is the chemical potential given by

�A� = �0A� + RT ln(�A�xA�) (7)

and �A� is the activity coefficient for species A in the �-phase.The species velocity in Eq. (6a) can be decomposed into

an average velocity and a diffusion velocity in more than oneway (Taylor and Krishna, 1993; Slattery, 1999; Bird et al.,2002), and arguments are often given to justify a particularchoice. In this work we prefer a decomposition in terms ofthe mass average velocity because governing equations, suchas the Navier–Stokes equations, are available to determine thisvelocity. The mass average velocity in Eq. (3) is defined by

v� =A=N∑A=1

�A�vA� (8)

and the associated mass diffusion velocity is defined by thedecomposition

vA� = v� + uA�. (9)

The mass diffusive flux has the attractive characteristic thatthe sum of the fluxes is zero, i.e.,

A=N∑A=1

�A�uA� = 0. (10)

Use of Eq. (9) in Eq. (6a) provides the generalized Stefan–Maxwellequations in terms of the mass diffusion velocity:

0 = −dA� +B=N∑B=1B �=A

xA�xB�(uB� − uA�)

DAB

,

A = 1, 2, . . . , N − 1. (11)

This form can be inverted to solve for the mass diffusive flux,�A�uA�, and the result is given in Appendix A where we showthat

�A�uA� = −E=N−1∑

E=1

��DAE∇��E ,

A = 1, 2, . . . , N − 1. (12)

The elements of the diffusivity matrix, DAE , are functions ofthe binary diffusion coefficients, the activity coefficients, andthe mass fractions.

One can make use of Eqs. (4) and (12) in Eq. (1) to ob-tain species continuity equations in terms of mass fractions.Since only N − 1 of the mass fractions are independent, the N

species continuity equations given originally by Eq. (1) are nowexpressed as

�(���A�)

�t+ ∇ · (���A�v�) = ∇ ·

[E=N−1∑

E=1

��DAE∇�E�

],

A = 1, 2, . . . , N − 1, (13a)

���

�t+ ∇ · (��v�) = 0. (13b)

Directing our attention to the boundary condition given byEq. (2), we note that the summation over all N species providesthe following boundary condition for Eq. (13b):

BC n�� · (��v�) = 0, at the �.� interface. (14)

In addition, this result can be used with Eq. (2) to obtain

n�� · (�A�uA�) = 0, A = 1, 2, 3, . . . , N − 1,

at the �.� interface (15)

in which we have noted that there are only N − 1 of theseconditions because of the constraint on the mass diffusive fluxesgiven by Eq. (10). Making use of the representation for thediffusive flux given by Eq. (12) leads to the following boundarycondition for use with Eq. (13a):

BC n�� ·E=N−1∑

E=1

��DAE∇�E� = 0,

A = 1, 2, . . . , N − 1, at the �.� interface. (16)

Eqs. (13), (14) and (16) provide the basis for the developmentof local volume-averaged mass transport equations.

There are a wide variety of flow processes for which themomentum transport problem can be simplified by neglectingthe inertial terms in Eq. (3) and treating that equation as quasi-steady. When the no-slip condition can be imposed, this leadsto Darcy’s law for the volume-averaged mass average velocity,under the condition that viscosity and density variations overthe averaging volume may be neglected. For these conditionswe need only develop the volume-averaged form of Eq. (13a).

2. Volume averaging

In order to develop an upscaled version of Eqs. (13a)–(16),we associate with every point in space an averaging vol-ume, V, such as we have illustrated in Fig. 2. For anyquantity �� associated with the �-phase, this allows us to

2646 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

Fig. 2. Averaging volume.

define two averages, a superficial average defined by

〈��〉 = 1

V

∫V�

�� dV (17)

and an intrinsic average that takes the form

〈��〉� = 1

V�

∫V�

�� dV . (18)

Here we have used V� to represent the volume of the �-phasecontained within the averaging volume. These two averages arerelated by

〈��〉 = ��〈��〉� (19)

in which �� is the porosity or volume fraction of the void space.Throughout this analysis we will impose the length-scale con-straints indicated by

�� � r0 � L. (20)

In general, these length-scale constraints are overly severe, andone can often develop satisfactory volume-averaged equationsand closure schemes for processes that do not satisfy the con-straints indicated by Eq. (20) (Quintard and Whitaker, 1993;Goyeau et al., 1997). Given the definitions indicated by Eqs.(17) and (18), along with the length-scale constraints indicatedby Eq. (20), we can make use of the general transport theoremand the spatial-averaging theorem to upscale the point equa-tions and boundary conditions indicated in the previous sec-tion. The details are provided in Appendix B and the results

for species A continuity equation are given here as

�(��〈��〉�〈�A�〉�)�t

+ ∇ · (��〈��〉�〈�A�〉�〈v�〉�)

+ ∇ ·⎛⎜⎝〈��〉� 〈�A�v�〉︸ ︷︷ ︸

filter

⎞⎟⎠

= ∇ ·

⎡⎢⎢⎢⎣

E=N−1∑E=1

��〈��〉�〈DAE〉�∇〈�E�〉�

+ 〈��〉�〈DAE〉�V

∫A��

n���A� dA︸ ︷︷ ︸filter

⎤⎥⎥⎥⎦ ,

A = 1, 2, . . . , N − 1. (21)

This represents the volume-averaged form of Eq. (13a), andwe have identified the area integral of n���A� as a filter sincenot all the information available at the closure level will passthrough the area integral and contribute to the volume-averagedtransport equation. The dispersive transport can be expressedexplicitly as

〈�A�v�〉 = 1

V

∫V�

�A�v� dV (22)

and we have identified this volume integral as a filter sincenot all the information available at the closure level will passthrough this volume integral. The fact that information at theclosure level is filtered provides the justification for many ofthe simplifications made in the closure problem to be developedin subsequent paragraphs. In addition to the volume-averagedform of Eq. (13a) we also require the volume-averaged form ofEq. (13b). This is developed in Appendix B and given here as

�(��〈��〉�)�t

+ ∇ · (��〈��〉�〈v�〉�) = 0. (23)

Before moving on to the closure problem, we need to dealwith the relation between point and volume-averaged functionalrelations. For example, the diffusivity, DAE , depends on thebinary diffusion coefficients, the activity coefficients, and themass fractions. It would be an acceptable approximation totreat the binary diffusion coefficients as constants, and if weignore for the present the influence of the activity coefficients,the functional dependence of DAE could be expressed as

DAE = F(�A�, �B�, �C�, . . . ,�N−1�). (24)

In order to use this result in Eq. (21) we need to know how〈DAE〉� depends on 〈�A�〉�, 〈�B�〉�, . . . , 〈�N−1�〉�. This prob-lem is also important in order to understand the thermody-namic significance of the volume-averaged temperature, and ina separate study Hager and Whitaker (2002) have shown that

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2647

Eq. (24) leads to the functional relation for 〈DAE〉� given by

〈DAE〉� = F(〈�A�〉�, 〈�B�〉�, 〈�C�〉�, . . . , 〈�N−1�〉�) (25)

provided that constraints of the following type are satisfied:

r20

L2�

� 1. (26)

Here L� represents the distance over which significant varia-tions in 〈�A�〉� occur where A=1, 2, . . . , N −1. The constraintgiven by Eq. (26) will always be satisfied when the genericlength-scale constraints indicated by Eq. (20) are valid.

In order to make use of Eqs. (21) and (23) to predict themass fractions as a function of time and space, we need to de-velop closure problems for the spatial deviations, v� and �A�.The closure problem for v� is well known for the case of in-compressible flow and that result will serve as a reasonableapproximation for this study. The closure problem for �A� isdeveloped in the following section.

3. Closure

In this section, we wish to develop the boundary valueproblem for the spatial deviation density and mass fractionsdefined by

�� = �� − 〈��〉�, �A� = �A� − 〈�A�〉�,

A = 1, 2, . . . , N − 1. (27)

To obtain the governing differential equation for ��, we needto subtract the governing equation for 〈��〉� from the governingequation for ��. Noting that �� is treated as a constant, we cansubtract Eq. (23) from Eq. (13b) to obtain

�

�t(�� − 〈��〉�) + ∇ · (��v� − 〈��〉�〈v�〉�) = 0. (28)

On the basis of the decompositions expressed by Eqs. (B.7),this result can be rearranged in the form

���

�t+ ∇ · (��v� − 〈��〉�v�) = 0. (29)

The characteristic length for both ��v� and 〈��〉�v� is the smalllength scale, ��, thus the order of magnitude estimates as-sociated with the convective transport terms in Eq. (29) aregiven by

∇ · (��v�) = O(

��v�

��

), ∇ · (〈��〉�v�) = O

( 〈��〉�v�

��

).

(30)

On the basis of �� � 〈��〉�, these estimates lead to

∇ · (��v�) � ∇ · (〈��〉�v�) (31)

and this allows us to simplify Eq. (29) to the form

���

�t+ ∇ · (〈��〉�v�) = 0. (32)

The characteristic length associated with the average densityand the average mass fractions is the large length scale, L whilethe characteristic length associated with the spatial deviationdensity is the small length scale, ��. Both these length scalesare illustrated in Fig. 2, and the fact that �� � L indicates thatEq. (32) is quasi-steady and thus takes the form

∇ · (〈��〉�v�) = 0. (33)

At this point we are willing to ignore variations of 〈��〉� in theclosure problem so that Eq. (33) is given by

∇ · v� = 0. (34)

Ignoring variations of volume-averaged quantities in closureproblems is a routine simplification (Whitaker, 1999, Section1.4.3), and in this case it is achieved by assuming that 〈��〉� isa constant. However, one must keep in mind that quantities thatare treated as constants in a closure problem may not neces-sarily be treated as constants in the volume-averaged transportequation. For example, treating 〈��〉� as a constant in Eq. (33)is an acceptable approximation while this may not be the casein Eq. (23).

To obtain the governing equation for �A�, we subtractEq. (21) from Eq. (13a) leading to the result given by

�

�t(���A� − 〈��〉�〈�A�〉�)+ ∇ · (���A�v� − 〈��〉�〈�A�〉�〈v�〉�)− ∇ · (�−1

� 〈��〉�〈�A�v�〉)

= ∇ ·[

E=N−1∑E=1

��DAE∇�A� − 〈��〉�〈DAE〉�∇〈�E�〉�]

− ∇ ·[

E=N−1∑E=1

〈��〉�〈DAE〉�V�

∫A��

n���A� dA

],

A = 1, 2, . . . , N − 1. (35)

Directing our attention to the first term, we use Eqs. (B.7) toobtain

�

�t(���A� − 〈��〉�〈�A�〉�)

= �

�t(〈��〉��A� + ��〈�A�〉� + ���A�). (36)

This accumulation term can be simplified on the basis of theinequalities given by the first two of Eqs. (B.17) which provide

���A� � 〈��〉��A�, ���A� � ��〈�A�〉�. (37)

In addition, we restrict our analysis by the inequality

Restriction:��

〈��〉�� �A�

〈�A�〉� . (38)

2648 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

Under these circumstances, Eq. (36) simplifies to

�

�t(���A� − 〈��〉�〈�A�〉�) = �

�t(〈��〉��A�). (39)

Moving on to the second term in Eq. (35), we make use of thedecompositions given by Eqs. (B.7) in order to represent theconvective transport as

∇ · (���A�v� − 〈��〉�〈�A�〉�〈v�〉�)= ∇ · [〈��〉�(〈�A�〉�v� + �A�v�) + ���A�v�]. (40)

After carrying out the divergence operation on the right-handside, we can arrange this result in the form

∇ · (���A�v� − 〈��〉�〈�A�〉�〈v�〉�)= 〈�A�〉�∇ · (〈��〉�v�) + 〈��〉�v� · ∇〈�A�〉�

+ �A�∇ · (〈��〉�〈v�〉�) + �A�∇ · (〈��〉�v�)

+ 〈��〉�v� · ∇�A�. (41)

On the basis of Eq. (33) we simplify this result to

∇ · (���A�v� − 〈��〉�〈�A�〉�〈v�〉�)= 〈��〉�v� · ∇〈�A�〉�

+ �A�∇ · (〈��〉�〈v�〉�) + 〈��〉�v� · ∇�A� (42)

and on the basis of the disparate length scales associated with〈��〉�〈v�〉� and �A� we impose the inequality

�A�∇ · (〈��〉�〈v�〉�) � 〈��〉�v� · ∇�A�. (43)

This allows us to express Eq. (42) as

∇ · (���A�v� − 〈��〉�〈�A�〉�〈v�〉�)= 〈��〉�v� · ∇〈�A�〉� + 〈��〉�v� · ∇�A�. (44)

Use of this result and Eq. (39) in Eq. (35) provides a simplifiedform of the closure equation given by

�

�t(〈��〉��A�) + 〈��〉�v� · ∇〈�A�〉�

+ 〈��〉�v� · ∇�A� − ∇ · (�−1� 〈��〉�〈�A�v�〉)

= ∇ ·[

E=N−1∑E=1

��DAE∇�A� − 〈��〉�〈DAE〉�∇〈�E�〉�]

− ∇ ·[

E=N−1∑E=1

〈��〉�〈DAE〉�V�

∫A��

n���A� dA

],

A = 1, 2, . . . , N − 1. (45)

While the dispersive transport, 〈��〉�〈�A�v�〉, is of importancein the volume-averaged transport equation when the Pécletnumber is large compared to one, it can be neglected here onthe basis of the inequality

∇ · (�−1� 〈��〉�〈�A�v�〉) � 〈��〉�v� · ∇�A�. (46)

This result is based on the disparate length scales indicated byEq. (20) and it allows us to express Eq. (45) as

�

�t(〈��〉��A�) + 〈��〉�v� · ∇〈�A�〉� + 〈��〉�v� · ∇�A�

= ∇ ·[

E=N−1∑E=1

��DAE∇�A� − 〈��〉�〈DAE〉�∇〈�E�〉�]

− ∇ ·[

E=N−1∑E=1

〈��〉�〈DAE〉�V�

∫A��

n���A� dA

],

A = 1, 2, . . . , N − 1. (47)

The characteristic length for the closure problem is �� and thisleads to a characteristic time given by

{characteristic time forthe closure problem

}= �2

�

D(48)

in which D is representative of the binary diffusion coefficientsthat appear in Eq. (5). In contrast, the characteristic time forthe volume-averaged transport equation takes the form

{characteristic time forthe volume-averagetransport equation

}= L2

D. (49)

Under these circumstances, the closure problem is alwaysquasi-steady and Eq. (47) takes the form

〈��〉�v� · ∇〈�A�〉� + 〈��〉�v� · ∇�A�

= ∇ ·[

E=N−1∑E=1

��DAE∇�A� − 〈��〉�〈DAE〉�∇〈�E�〉�]

− ∇ ·[

E=N−1∑E=1

〈��〉�〈DAE〉�V�

∫A��

n���A� dA

],

A = 1, 2, . . . , N − 1. (50)

On the basis of the constraints given by (see Eq. (B.23) inAppendix B)

�� � 〈��〉�, DAE � 〈DAE〉� (51)

and the plausible inequality associated with the density and themass fractions

�A�∇�� � ��∇�A�, A = 1, 2, . . . , N − 1 (52)

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2649

the first term on the right-hand side of Eq. (50) can be simplifiedto obtain

v� · ∇〈�A�〉� + v� · ∇�A�

= ∇ ·[

E=N−1∑E=1

〈DAE〉�∇�A�

]

− ∇ ·[

E=N−1∑E=1

〈DAE〉�V�

∫A��

n���A� dA

],

A = 1, 2, . . . , N − 1. (53)

Since the area integral of n���A� is associated with the largelength scale, L, illustrated in Fig. 2, and �A� is associatedwith the small length scale, ��, the last term in Eq. (53) canbe discarded leading to the following governing differentialequation for �A�:

v� · ∇〈�A�〉� + v� · ∇�A� = ∇ ·[

E=N−1∑E=1

〈DAE〉�∇�A�

],

A = 1, 2, . . . , N − 1. (54)

In order to develop boundary conditions for this governing dif-ferential equation, we return to Eq. (16) repeated here as

BC n�� ·E=N−1∑

E=1

��DAE∇�E� = 0,

A = 1, 2, . . . , N − 1 at the �.� interface. (55)

We use of the decompositions given by the first two ofEqs. (B.7), the single decomposition given by Eq. (B.22), andthe inequalities given by the first two of Eqs. (B.23) in orderto express this result as

BC n�� ·E=N−1∑

E=1

〈��〉�〈DAE〉�∇�E�

= −n�� ·E=N−1∑

E=1

〈��〉�〈DAE〉�∇〈�E�〉�,

A = 1, 2, . . . , N − 1 at the �.� interface. (56)

To complete our statement of the boundary value problem, weneed a condition at the boundary of the macroscopic systemillustrated in Fig. 2. We state this condition as

BC �A� = F(r, t), at A�e. (57)

Here we have used A�e to represent the area of the entrancesand exits of the �-phase associated with the macroscopic sys-tem. It should be clear that �A� is generally unknown at theboundary of the macroscopic system; however, this is not im-portant since the value of �A� at the boundary will influencethe �A�-field only in a region of thickness ��. In order to de-velop a local solution for �A� that can be used to evaluate theintegral in Eq. (21), we make use of a spatially periodic modelof the porous medium. A representative region of this modelis illustrated in Fig. 3 and we refer to that region as a unit

Fig. 3. Representative region in a spatially periodic model of a porous medium.

cell to which we apply the periodic condition given by

�A�(r + li ) = �A�(r), i = 1, 2, 3. (58)

Here we have use li (i = 1, 2, 3) to represent the three latticevectors needed to describe a spatially periodic system. Giventhis model, our closure problem takes the form

v� · ∇〈�A�〉� + v� · ∇�A� = ∇ ·[

E=N−1∑E=1

〈DAE〉�∇�A�

],

A = 1, 2, . . . , N − 1, (59)

BC n�� ·E=N−1∑

E=1

〈��〉�〈DAE〉�∇�E�

= −n�� ·E=N−1∑

E=1

〈��〉�〈DAE〉�∇〈�E�〉�,

A = 1, 2, . . . , N − 1 at the �.� interface, (60)

Periodicity: �A�(r + li ) = �A�(r), i = 1, 2, 3,

A = 1, 2, . . . , N − 1. (61)

Within the framework of the closure problem, we treat ∇〈�E�〉�as a constant, and we recognize that Eqs. (59)–(61) can be usedto determine �A� to within an arbitrary constant. This arbitraryconstant can be removed by imposition of the condition

Average: 〈�A�〉� = 0. (62)

Since any constant associated with �A� will not pass throughthe filter in Eq. (21), this condition on the average need not beimposed; however, we will retain Eq. (62) in order to clarifythe nature of the spatial deviation mass fractions. This meansthat �A� can be expressed in terms of the N − 1 independentgradients of the mass fractions, ∇〈�E�〉�, E =1, 2, . . . , N −1.

2650 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

It is convenient to express the closure problem in the compactform given by

v� · [∇〈��〉�] + v� · ∇[��] = ∇ · {[〈D〉�]∇[��]}, (63a)

BC − n�� · [〈D〉�]∇[��] = n�� · [〈D〉�][∇〈��〉�], at A��,

(63b)

Periodicity [��](r + li ) = [��](r), i = 1, 2, 3, (63c)

Average: [〈��〉] = 0. (63d)

Here the column matrices for the gradient of the volume-averaged mass fractions and spatial deviation mass fractionsare given by

[��] =

⎡⎢⎢⎢⎣

�A��B��C�· ·

�N−1�

⎤⎥⎥⎥⎦ , [∇〈��〉�] =

⎡⎢⎢⎢⎣

∇〈�A�〉�∇〈�B�〉�∇〈�C�〉�

· ·∇〈�N−1�〉�

⎤⎥⎥⎥⎦ (64)

while [〈D〉�] has been used to represent the diffusivity matrixgiven by

[〈D〉�]

=

⎡⎢⎢⎢⎣

〈DAA〉� 〈DAB〉� 〈DAC〉� · · · 〈DAN−1〉�〈DBA〉� · · · · · · · 〈DBN−1〉�〈DCA〉� · · · · · · · 〈DBN−1〉�

· · · · · · · · · · ·〈DN−1A〉� · · · · · · · 〈DN−1N−1〉�

⎤⎥⎥⎥⎦ .

(65)

The form given by Eqs. (63) represents a complex boundaryvalue problem for the spatial deviation mass fractions; how-ever, the problem can be greatly simplified if we make use ofthe diagonal form of the diffusivity matrix (Toor, 1964; Culli-nan, 1965; Stewart and Prober, 1964; Gupta and Cooper, 1971;Taylor, 1982). It is important to note that this procedure will beapplied only to the closure problem, and not to the final macro-scale equations that will remain fully non-diagonal. To obtainthis form, we denoted the nodal matrix by [P ] so that we have

[〈D〉�diag] = [P ]−1[〈D〉�][P ], (66)

[w�] = [P ]−1[��], (67)

[∇〈w�〉�] = [P ]−1[∇〈��〉�] (68)

in which the diagonal diffusivity matrix is given explicitly by(note the use of the non-italic D to distinguish from the originaldiffusion coefficients)

[〈D〉�diag]

=

⎡⎢⎢⎢⎣

〈DAA〉� 0 0 · · · · 00 〈DBB〉� · · · · · · 00 · · 〈DCC〉� · · · · 0· · · · · · · · · · · ·0 · · · · · · · · 〈DN−1N−1〉�

⎤⎥⎥⎥⎦ .

(69)

For convenience, we will call w� and 〈w�〉� pseudo-componentsand the resulting space the pseudo-composition space.

Operating on Eq. (63a) with [P ]−1 and making use ofEq. (66) provides the following form:

v� · ([P ]−1[∇〈��〉�]) + v� · ∇([P ]−1[��])− v� · (∇[P ]−1)([��])

= ∇ · {([〈D〉�diag])[P ]−1∇[��]}− (∇[P ]−1)[P ]([〈D〉�diag])[P ]−1∇[��]. (70)

At this point we need to recognize that the characteristic lengthassociated with [P ] is the same as the characteristic length as-sociated with 〈DAE〉�, and this is the large length scale, L, illus-trated in Fig. 2. In addition, the characteristic length associatedwith [��] is the small length scale, ��, that is also illustrated inFig. 2. We express these ideas as

∇[P ]−1 = O(P/L), ∇[��] = O(��/��),

∇∇[��] = O(��/�2�) (71)

and on the basis of the constraint on length scales indicated byEq. (20), we have the following inequalities:

(∇[P ]−1)([��]) � ∇([P ]−1[��]), (72a)

(∇[P ]−1)[P ][〈D〉�diag][P ]−1∇[��]� ∇ · {[〈D〉�diag][P ]−1∇[��]}. (72b)

These two restrictions allow us to simplify Eq. (70) to the form

v� · ([P ]−1[∇〈��〉�]) + v� · ∇([P ]−1[��])= ∇ · {[〈D〉�diag][P ]−1∇[��]}. (73)

We now make use of Eq. (72a) to simplify the diffusive termleading to

v� · ([P ]−1[∇〈��〉�]) + v� · ∇([P ]−1[��])= ∇ · {[〈D〉�diag]∇([P ]−1[��])}. (74)

Finally, we employ Eqs. (67) and (68) to express the closureproblem in terms of pseudo-components according to

v� · [∇〈w�〉�] + v� · ∇[w�] = ∇ · {[〈D〉�diag]∇[w�]} (75a)

BC n�� · ∇[w�] = −n�� · [∇〈w�〉�] at A��. (75b)

Periodicity: [w�](r + Ii ) = [w�](r), i = 1, 2, 3. (75c)

Average: [〈��〉�] = 0. (75d)

Often it is convenient to view this problem in the expandedform given by

v� ·

⎡⎢⎢⎢⎣

∇〈wA�〉�∇〈wB�〉�

· · ·· · ·

∇〈wN−1�〉�

⎤⎥⎥⎥⎦ + v� · ∇

⎡⎢⎢⎢⎣

wA�wB�· · ·· · ·

wN−1�

⎤⎥⎥⎥⎦

= ∇ ·

⎡⎢⎢⎢⎣

〈DAA〉�∇wA�〈DBB〉�∇wB�

· · ·· · ·

〈DN−1N−1〉�∇wN−1�

⎤⎥⎥⎥⎦ , (76a)

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2651

BC n�� · ∇

⎡⎢⎢⎢⎣

wA�wB�· · ·· · ·

wN−1�

⎤⎥⎥⎥⎦ = −n��

⎡⎢⎢⎢⎣

∇〈wA�〉�∇〈wB�〉�

· · ·· · ·

∇〈wN−1�〉�

⎤⎥⎥⎥⎦ at A��,

(76b)

Periodicity:

⎡⎢⎢⎢⎣

wA�wB�· · ·· · ·

wN−1�

⎤⎥⎥⎥⎦ (r + Ii ) =

⎡⎢⎢⎢⎣

wA�wB�· · ·· · ·

wN−1�

⎤⎥⎥⎥⎦ (r),

i = 1, 2, 3, (76c)

Average:

⎡⎢⎢⎢⎣

〈wA�〉�〈wB�〉�

· · ·· · ·

〈wN−1�〉�

⎤⎥⎥⎥⎦ = 0. (76d)

Here it becomes apparent that each of the N −1 boundary valueproblems represented by Eqs. (76) contains a single nonhomo-geneous term, i.e., ∇〈wA�〉�, ∇〈wB�〉�, etc. Because of this, thesolutions for wA�, wB�, etc., take the form

wA� = bA · ∇〈wA�〉�, wB� = bB · ∇〈wB�〉�, etc. (77)

This indicates that we can express the column vector of spatialdeviation variables as

[w�] = [b] · [∇〈w�〉�], (78)

where the operator [b] is the diagonal matrix given by

[b] =

⎡⎢⎢⎢⎣

bA 0 · · · · · · · · ·0 bB · · · · · · · · ·· · · · · · · · · · · · · · ·· · · · · · · · · · · · 0· · · · · · · · · 0 bN−1

⎤⎥⎥⎥⎦ . (79)

Substitution of the representation given by Eq. (78) intoEqs. (75) leads to

v�[1] + v� · ∇[b] = ∇ · ([〈D〉�diag]∇[b]) (80a)

BC n�� · ∇[b] = −n�� at A��, (80b)

Periodicity: [b](r + Ii ) = [b](r), i = 1, 2, 3, (80c)

Average: [〈b〉�] = 0 (80d)

in which we have used [1] to represent the (N − 1) × (N − 1)

unit matrix given by

[1] =

⎡⎢⎢⎢⎣

1 0 · · · · · ·0 1 · · · · · ·· · · · · · · · · ·· · · · · · · · 0· · · · · · 0 1

⎤⎥⎥⎥⎦ . (81)

Both the convective and diffusive terms have been simplifiedby imposition of the restriction

[b] · [∇∇〈w�〉�] � [∇b] · [∇〈w�〉�] (82)

that is consistent with the length scale restriction given byEq. (20). In addition, it is consistent with Eq. (20) to imposethe restriction given by

∇[〈D〉�diag] · ∇[b] � [〈D〉�diag]∇2[b] (83)

so that the general closure problem takes the form

v�[1] + v� · ∇[b] = [〈D〉�diag]∇2[b] (84a)

BC n�� · ∇[b] = −n�� at A�� (84b)

Periodicity: [b](r + Ii ) = [b](r), i = 1, 2, 3, (84c)

Average: [〈b〉�] = 0. (84d)

This represents a system of N − 1 independent closure prob-lems differing only by the value of the diffusion coefficient inEq. (84a). The form of the closure problem is analogous tothat obtained by numerous authors (Brenner, 1980; Carbonelland Whitaker, 1983; Mei, 1992; Auriault, 1995) for the caseof dilute solution mass transfer, i.e., for the species A continu-ity equation when xA� � 1. In this case the problem is morecomplex because the dilute solution mixture diffusivity hasbeen replaced with the diagonal diffusivity matrix, [〈D〉�diag].One should also remember that this simple form applies to thetransformed concentrations, or pseudo-components identifiedby [w�], and not to the original components identified by [��].When we return to the original concentrations, a more complexform will be encountered.

To construct the closed form of the volume-averaged trans-port equation, we first express Eq. (21) in terms of the matricesgiven by Eqs. (64) and (65). This leads to

�(��〈��〉�[〈��〉�])�t

+ ∇ · (��〈��〉�〈v�〉�[〈��〉�])+ ∇ · (〈��〉�〈v�[��]〉)

= ∇ ·[��〈��〉�[〈D〉�]

{∇[〈��〉�]

+ 1

V�

∫A��

n��[��] dA

}]. (85)

At this point we can make use of Eq. (78) along with Eqs. (67)and (68) to express the spatial deviation mass fractions as

[��] = [P ][b][P ]−1 · [∇〈��〉�]. (86)

When this representation is used with the convective and dif-fusive terms in Eq. (85) we obtain

〈v�[��]〉 = 1

V

∫V�

v�[P ][b][P ]−1[∇〈��〉�] dV , (87a)

1

V�

∫A��

n��[��] dA

= 1

V�

∫A��

n��[P ][b][P ]−1[∇〈��〉�] dA. (87b)

2652 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

When the length scale constraints indicated by Eq. (20) aresatisfied, and the porous medium is disordered, Quintard andWhitaker (1994) have shown that [∇〈��〉�] can be removedfrom the volume and area integrals in Eqs. (87). This simplifi-cation allows us to express Eq. (85) in the form

�(��〈��〉�[〈��〉�])�t

+ ∇ · (��〈��〉�〈v�〉�[〈��〉�])+ ∇ · (〈��〉�〈v�[P ][b][P ]−1〉 · [∇〈��〉�])

= ∇ ·[��〈��〉�[〈D〉�]

×{[I] + 1

V�

∫A��

n��[P ][b][P ]−1 dA

}· [∇〈��〉�]

].

(88)

We now define the effective diffusivity tensor and the hydro-dynamic dispersion tensor according to

[Deff ] = [〈D〉�]{[I] + 1

V�

∫A��

n��[P ][b][P ]−1 dA

}, (89a)

[D�] = −〈v�[P ][b][P ]−1〉� (89b)

and this allows us to express Eq. (88) in the compact formgiven by

�(��〈��〉�[〈��〉�])�t

+ ∇ · (��〈��〉�〈v�〉�[〈��〉�])= ∇ · {〈��〉�(��[Deff ] + ��[D�]) · [∇〈��〉�]}. (90)

Here it is important to note that the effective diffusivity tensorand the hydrodynamic dispersion tensor are defined in termsof intrinsic average quantities and this leads to the use of �� asa pre-multiplier in Eq. (90). Alternate representations for thesetwo quantities are often encountered in the literature, thus onemust be aware of the definitions such as we have provided interms of Eqs. (89). For comparison with experiment, the diffu-sivity and dispersion tensors are generally grouped together sothat Eq. (90) takes the compact form

�(��〈��〉�[〈��〉�])�t

+ ∇ · (��〈��〉�〈v�〉�[〈�〉�])= ∇ · {��〈��〉�[D∗

�] · [∇〈��〉�]}. (91)

Here the total dispersion tensor is defined as

[D∗�] = [Deff ] + [D�], (92)

where the details of [D∗�] are given by

[D∗�] =

⎡⎢⎢⎢⎣

D∗AA� D∗

AB� D∗AC� · · · · D∗

AN−1�D∗

BA� · · · · · · · · D∗BN−1�

D∗CA� · · · · · · · · D∗

BN−1�· · · · · · · · · · · ·D∗

N−1A� · · · · · · · · D∗N−1N−1�

⎤⎥⎥⎥⎦ .

(93)

At this point we are in the position to discuss the kind ofapproximations used in the literature to construct effectivegeneralized diffusion and dispersion tensors.

3.1. Pure diffusion

When convective transport is negligible, the equations are ho-mogeneous with respect to the diagonal diffusion coefficients,and the closure problem given by Eqs. (84) can be expressed as

0 = ∇2[b], (94a)

BC n�� · ∇[b] = −n�� at A��, (94b)

Periodicity: [b](r + Ii ) = [b](r), i = 1, 2, 3, (94c)

Average: [〈b〉�] = 0. (94d)

In this case each closure variable, bA, bB , etc., satisfies thesame boundary value problem, and Eq. (79) simplifies to

[b] = b

⎡⎢⎢⎢⎣

1 0 · · · · · ·0 1 · · · · · ·· · · · · · · · · ·· · · · · · · · 0· · · · · · 0 1

⎤⎥⎥⎥⎦ . (95)

The single closure problem takes the form

0 = ∇2b, (96a)

BC n�� · ∇b = −n�� at A��, (96b)

Periodicity: b(r + Ii ) = b(r), i = 1, 2, 3, (96c)

Average: 〈b〉� = 0 (96d)

and Eq. (86) simplifies to

[��] = b · [∇〈��〉�]. (97)

This is the closure problem that has been abundantly discussedin the literature described in the introduction. For this caseof pure diffusion, the volume-averaged transport equation isgiven by

�(��〈��〉�[〈��〉�])�t

= ∇ · (��〈��〉�[Deff ] · [∇〈��〉�]) (98)

in which [Deff ] takes the form

[Deff ] = [〈D〉�]{I + 1

V�

∫A��

n��b dA

}. (99)

Here we see that only a single closure problem is required inorder to determine the tortuosity effects for all the diffusingspecies. In terms of the single tortuosity tensor defined by

� = I + 1

V�

∫A��

n��b dA (100)

we can express Eq. (99) as

[Deff ] = [〈D〉�]�. (101)

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2653

This result, along with Eq. (98), indicates that we can de-scribe multicomponent diffusion in porous media in terms ofthe porosity, ��, the single tortuosity tensor, �, and the diffusiv-ity matrix, [〈D〉�]. For the special case of an isotropic porousmedium, this result takes the form

[Deff ] = 1

�[〈D〉�] (102)

in which � is the classic scalar tortuosity given by

� = I�

. (103)

It should be emphasized that this result validates, for the caseof pure diffusion, the simplifying assumptions used by manyauthors.

3.2. Dispersion

For the general case, Eqs. (89) do not offer special simpli-fications, and the manner in which generalized dispersion ten-sors are constructed in the literature is not supported by thetheoretical results presented in this paper. However, a specialcase must be considered, i.e., the linear dispersion regime. Thisregime is characterized by expressions for the dispersion curveof the type a + b Pe. In this case, the dispersion coefficientdivided by a molecular diffusivity is a linear function of thePéclet number which is inversely proportional to the moleculardiffusivity. For this particular case, the dispersion curve in thepseudo-composition space may be approximated by (for sakeof simplicity we only consider longitudinal dispersion)

D∗AA�|pseudo-component = 〈DAA〉�

�+ L〈v�〉� (104)

in which the dispersive term, being linear with respect to ve-locity, does not depend on the diffusion coefficient. This latterpoint is the source of the important simplification describedbelow. It must be emphasized that this expression gives verygood results in the diffusive regime, i.e., small Péclet num-bers. In addition, it gives good results in the linear dispersiveregime, provided that dispersion is, of course, a valid descrip-tion as opposed to anomalous dispersion. As we shall see later,its accuracy in the transition regime may be questioned.

On the basis of Eqs. (89) and (92), the total dispersion tensorcan be expressed as

[D∗�] = [〈D〉�]

{[I] + 1

V�

∫A��

n��[P ][b][P ]−1 dA

}− 〈v�[P ][b][P ]−1〉� (105)

and the longitudinal or axial component takes the form

i · [D∗�] · i = [〈D〉�]i ·

{I + 1

V�

∫A��

n��[P ][b][P ]−1 dA

}· i

− i · 〈v�[P ][b][P ]−1〉� · i. (106)

If we restrict this discussion to isotropic porous media andapproximate the diffusive contribution by the representationgiven Eq. (101) for pure diffusion, the above result simplifies to

i · [D∗�] · i = 1

�[〈D〉�] − i · 〈v�[P ][b][P ]−1〉� · i. (107)

Making use of Eq. (79) and ignoring variations in [P ] leads to

i · [D∗� ] · i

= 1

�[〈D〉�] − [P ]

×

⎡⎢⎢⎢⎣

i · 〈v�bA〉� · i 0 · · · · · ·0 i · 〈v�bB 〉� · i · · · · · ·· · · · · · · · · ·· · · · · · · · 0· · · · · · 0 i · 〈v�bN−1〉� · i

⎤⎥⎥⎥⎦ [P ]−1.

(108)

If we adopt the model represented by Eq. (104), this ex-pression implies that the total dispersion coefficient will begiven by

i · [D∗� ] · i = 1

�[〈D〉�] + [P ]

×

⎡⎢⎢⎢⎣

L〈v�〉� 00 ...

...

... 00 L〈v�〉�

⎤⎥⎥⎥⎦ [P ]−1 (109)

in which we have assumed that the average velocity is ori-ented along the x-axis. The dispersive part has been modifiedusing the fact that in the linear dispersive regime all bA are in-dependent of the diffusion coefficients, i.e., beyond the transi-tion regime, dispersion is the same for all pseudo-components.Finally, the longitudinal component of the total dispersion ten-sor may be written

i · [D∗�] · i = 1

�[〈D〉�] + L〈v�〉�

⎡⎢⎢⎢⎣

1 0 · · · · 00 1 · · · · ·· · · · · · ·· · · · · · 00 · · · · 0 1

⎤⎥⎥⎥⎦ .

(110)

Here we see that, in this particular case, a linear dispersion termmay be added to the diagonal of the generalized diffusion ma-trix to estimate the effective properties. It should be emphasizedthat this dispersion effect does not apply to the off-diagonalterms.

4. Comparison with numerical experiment for a ternarysystem

In order to test the theory described in the previous sec-tion, we performed two-dimensional computations for non-linear ternary systems. We compared the resulting averagedconcentration fields to predictions using the one-dimensionalmacro-scale equations in which the effective diffusivities hadbeen obtained from the solution of the closure problem. Manydifferent systems have been investigated, and we chose to re-port the results obtained with the acetone–benzene–CCl4 liquid

2654 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

0

0.5

1 00.2

0.40.6

0.81

0

0.5

1

1.5

2

ωBωA

Den

sity

(g/c

m3 )

0

0.5

1 00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1x 10

-3

ωB

ωA

Dyn

amic

visc

osity

(Pa.

s)

Fig. 4. Density and viscosity.

system. We adopted this system because nonlinear effects arelarge enough so the test can be considered significant, and be-cause correlations for the various physical properties are avail-able in the literature (Taylor and Krishna, 1993).

The thermodynamic properties of this system were computedusing the software and database provided by BibPhy Add-InTM

using the NTRL thermodynamic model. The referencetemperature and pressure were 20 ◦C and 1 bar. The resultingcorrelation tables with the concentrations were replaced bycontinuous polynomial interpolations for ease of use in thenumerical computations. The density and viscosity variationswith concentrations are illustrated in Fig. 4, where we haveused the following nomenclature:

• �A� represents the acetone mass fraction.• �B� represents the benzene mass fraction.

The multicomponent diffusion coefficients were calculated us-ing the procedure outlined in Taylor and Krishna (1993, pp. 91and 545). The results were similarly smoothed using interpo-lating functions. The evolution of the various diffusion coeffi-cients with the concentrations is shown Figs. 5 and 6.

00.2

0.40.6

0.81

0

0.5

10

1

2

3

4x 10

-9

ωA

ωB

DA

A(m

2 /s)

00.2

0.40.6

0.81

0

0.5

1-10

-8

-6

-4

-2

0

2

x 10-10

ωA

ωB

DA

B(m

2 /s)

Fig. 5. Diffusion coefficients for Species A.

The micro- and macro-scale equations were solved usingthe finite element toolbox FEMLABTM with the partial deriva-tive equations being programmed using the general formula-tion provided by the software. Mesh sizes and other numericalparameters were adapted to reach sufficient accuracy.

The chosen micro-scale structure was an array of cylinders,as illustrated in Fig. 7. The number of unit cells (NUC) beingadapted to the problem under consideration.

For completeness, the pore-scale or micro-scale equations tobe solved are summarized here as

���

�t+ ∇ · (��v�) = 0, (111)

0 = −∇p� + ∇ · (��∇v�), (112)

(�� + �A�

���

��A�

)��A�

�t+

(�A�

���

��B�

)��B�

�t

+ ∇ · (���A�v�)

= ∇ · {��(DAA∇�A� + DAB∇�B�)}, (113)

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2655

00.2

0.40.6

0.81

0

0.5

1-1

0

1

2

3x 10

-8

ωAωB

DB

A(m

2 /s)

00.2

0.40.6

0.81

0

0.5

10

1

2

3

4

5

x 10-9

ωA

ωB

DB

B(m

2 /s)

Fig. 6. Diffusion coefficients for Species B.

Fig. 7. Spatially periodic porous medium.

(�B�

���

��A�

)��A�

�t+

(�� + �B�

���

��B�

)��B�

�t

+ ∇ · (���B�v�)

= ∇ · {��(DBA∇�A� + DBB∇�B�)}. (114)

Here we remind the reader that density, viscosity and diffusiv-ities all vary with the mass fractions. The boundary conditionsare such that:

• Velocity was taken to be equal to zero (no-slip) on everysurface except at the entrance and exit boundaries. Therefore,steep velocity gradients were also observed on the two upperand lower confining surfaces, in addition to the effect of no-slip on cylinder surfaces. This choice was made in order to

increase velocity fluctuations within the unit cell, and thusthe dispersion effects.

• Pressure was imposed at the entrance and boundary surfaces:the same pressure in the case of diffusion and a small headloss of a few Pa in the case of forced convection.

• Mass fluxes were taken to be zero on all fluid–solid surfaces.• Dirichlet conditions were imposed for the mass fractions at

the entrance boundary.• Convective condition (Danckwerts, 1953) was used for the

mass fractions at the exit boundary in the case of forcedconvection in the tube, while a Dirichlet condition was usedfor the diffusive case.

Since the theory is being compared with liquid-phase masstransport, the use of the no-slip condition should be an ac-ceptable approximation. However, one must recognize that theno-slip boundary condition violates the key result associatedwith Graham’s law (Kramers and Kistemaker, 1943). Recenttheoretical and experimental studies (Altevogt et al., 2003a,b)of gas-phase mass transport in porous media indicate that slipphenomena needs to be considered to obtain an accurate rep-resentation of gas phase, diffusion-dominated transport. In thiswork we compare theory with numerical experiment, as op-posed to laboratory experiment. Under these circumstances, werequire only that the simplifications imposed on the theoreticaldevelopment are the same as those imposed on the numericalcalculations carried out at the pore-scale. In the present form,the theory does not take into account the effect of slip; how-ever, future work will include all of the classic characteristicsof active surfaces, i.e., adsorption/desorption, heterogeneousreaction, slip phenomena, etc.

The closure problems were solved using specifically de-signed numerical models, developed for similar problems(Quintard, 1993; Quintard and Whitaker, 1993, 1994; Quintardet al., 1997). The appropriate unit cell used in the calculationsis represented in Fig. 8.

Fig. 8. Unit Cell geometry.

2656 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

We list below the averaged or macro-scale equations to besolved:

�(��〈��〉�)�t

+ ∇ · (〈��〉�〈v�〉) = 0, (115)

〈v�〉 = −K�

��· ∇〈p�〉�, (116)

��

(〈��〉� + 〈�A�〉�

�〈��〉��〈�A�〉�

)�〈�A�〉�

�t

+ ��

(〈�A�〉�

�〈��〉��〈�B�〉�

)�〈�B�〉�

�t+ ∇ · (〈��〉�〈�A�〉�〈v�〉)

= ∇ · {〈��〉���(D∗AA�∇〈�A�〉� + D∗

AB�∇〈�B�〉�)}, (117)

��

(〈�B�〉�

�〈��〉��〈�A�〉�

)�〈�A�〉�

�t

+ ��

(〈��〉� + 〈�B�〉�

�〈��〉��〈�B�〉�

)�〈�B�〉�

�t

+ ∇ · (〈��〉�〈�B�〉�〈v�〉)= ∇ · {〈��〉���(D∗

BA�∇〈�A�〉� + D∗BB�∇〈�B�〉�)}. (118)

For the array of cylinders used as our model porous medium,only the one-dimensional versions of Eqs. (115)–(118) wereused in the calculations. The intrinsic permeability has beenestimated from the pressure drop calculated in the two-dimensional case, for which we found K� ≈ 1.91 × 10−11 m2.The boundary conditions at the exit and entrance surfaces weretaken as similar to the pore-scale expressions but in terms ofthe averaged variables. It must be emphasized that, in theseequations, the density, the viscosity, and the dispersion coef-ficients depend nonlinearly on the averaged mass fractions,with correlations similar to the pore-scale versions. This isconsistent with Eqs. (24) and (25).

4.1. Diffusion

In this section, steady-state concentration fields are obtainedfor pure diffusion problems with Dirichlet boundary condi-tions on the left and right of the porous medium illustrated inFig. 7, and the same value of the pressure at the entrance andexit. The resulting calculated steady-state velocity is equal tozero, thus indicating a process of pure diffusion. Fig. 9 shows acomparison between direct numerical calculations and predic-tions from the averaged equation (single tortuosity coefficientequal to 1.34, as obtained from the solution of the closure prob-lem). The solid lines correspond to the intrinsic average concen-trations [denoted �A� (theor.) and �B� ( theor.) in the legend]obtained from the one-dimensional averaged equations witheffective parameters, while the symbols correspond to intrin-sic average concentrations obtained from the two-dimensionalmicro-scale computations [denoted �A� (num.) and �B� (num.)in the legend]. The results are in very good agreement. How-ever, the nonlinearities in this case may be considered as small,and this motivated the calculations for a complete problem in-cluding convection induced by density variations, with viscosity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m)

Vol

. Ave

r. M

ass

frac

tion

intr. veloc. 0 m/sωA

(num.)

ωB(num.)

ωA (theor.)

ωB (theor.)

Fig. 9. Comparison between theory and numerical experiment (NUC = 10).

Fig. 10. Results for two-dimensional computational model.

dependence on composition, as well as the nonlinear diffusioncoefficients already included in the diffusive computations.

4.2. Diffusion and convection (diffusive regime)

In this example, the intrinsic average velocity is smallenough so the pore-scale Péclet number is less than one and thedispersion mechanisms can be neglected. Therefore, effectivediffusion coefficients can be estimated with a single tortuositycoefficient (� = 1.34). An example of pore-scale numericalcomputations is given in Figs. 10 and 11. Fig. 10 shows themass fraction of species A at a given time, while Fig. 11 showsthe mass fraction of species B at the same time. The legend onthis figure indicates the value of an intrinsic average velocity.This is an average indication since, as will be shown later,

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2657

Fig. 11. Results for two-dimensional computational model.

the intrinsic average velocity is not constant in space and timebecause of the density variations with the concentrations. Tobe precise, the value for the velocity shown in Fig. 12 is theintrinsic average velocity associated with the first unit cell.

The pore-scale fields were used to calculate intrinsic aver-aged mass fractions, and the resulting values indicated by thesymbols O and � are plotted in Fig. 12 for two different times.The macro-scale equation was also solved with the effectivecoefficients, and the resulting fields are plotted in the same fig-ure. One sees a very good agreement between the numericalexperiments and the macro-scale predictions.

It should be noted that the time evolution is rather complexif one considers the effect of the density variations on the ve-locity field. This is illustrated in Fig. 13 which represents thespatial variations of the average intrinsic velocity at varioustimes, obtained from the two-dimensional calculations or theone-dimensional macro-scale model. Despite all the transientand nonlinear effects, the agreement between numerical exper-iments and theoretical predictions is remarkable.

4.3. Calculations in the dispersive regime

The closure problems have been solved and dispersioncurves have been constructed in the pseudo-composition space.The closure problem for bA may be written in dimensionlessform as

v′� + v′

� · ∇′b′A = 1

PeA

∇′2b′A, (119a)

BC n′�� · ∇′b′

A + −n′�� at A��, (84b)

Periodicity: b′A(r + li ) = b′

A(r), i = 1, 2, 3, (84c)

Average: 〈b′A〉� = 0, (84d)

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m)

Vol

. Ave

r. M

ass

frac

tion

t = 25.0 s

intr. aver. veloc. 0.72 10-5 m/s

ωA(num.)

ωB(num.)

ωA (theor.)

ωB (theor.)

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m)

Vol

. Ave

r. M

ass

frac

tion

t = 85.0 s

intr. aver. veloc. 1.04 10-5 m/s

ωA (num.)

ωB (num.)

ωA(theor.)

ωB(theor.)

Fig. 12. Comparison between theoretical and experimental volume-averagemass fractions (NUC = 13; pressure difference = 0.2 Pa).

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x 10-3

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4x 10

-5

x (m)

intr

insi

c ve

loci

ty (

m/s

)

15 s (num.)85 s (num.)

300 s (num.)

15 s (theor.)

85 s (theor.)300 s (theor.)

Fig. 13. Comparison between theoretical and experimental intrinsic averagevelocity.

2658 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

where the dimensionless variables are defined as

x′ = x

�UC

; v′� = v�

〈v�〉� ; b′A = bA

�UC

;

PeA = 〈v�〉��UC

〈DAA〉� . (120)

Here �UC is the characteristic length of the unit cell representedin Fig. 8. The effective diffusion and dispersion coefficients arecalculated by the following formulas:

[Deff ] = [〈D〉�][P ]{

[I] + 1

V ′�

∫A′

��

n��[b′] dA

}[P ]−1,

(121a)

[D�] = − [P ]

×⎡⎢⎣

PeA〈DAA〉�〈v′�b′

A〉� 0

0 PeB 〈DBB 〉�〈v′�b′

B〉�

. . . 00 . . .

⎤⎥⎦ [P ]−1.

(121b)

From the closure problem in dimensionless form, it is con-venient to construct dispersion curves in terms of a pseudo-concentration total dispersion coefficient defined as

D∗�,A

∣∣∣pseudo-comp.

〈DAA〉�= [I] + 1

V ′�

∫A′

��

n��b′A dA − PeA〈v′

�b′A〉�. (122a)

Such a dispersion curve is shown in Fig. 14. The symbolscorrespond to numerical solution of the closure problem, andthe solid line corresponds to the estimate given by

D∗�,A

∣∣∣pseudo-comp.

〈DAA〉� = 1

1.34+ 0.0128 Pe1.65

A . (123)

These data have the classical shape of dispersion curves, witha diffusive regime at small Péclet numbers followed first by atransition regime and then by a dispersive regime governed bya power law expression. We see that the estimate is very goodin the two limiting regimes, while a small discrepancy may beseen in the transition regime where the values are larger thanthose predicted by Eq. (123). This will be important later inthis paper, when comparing theoretical results with numericalexperiments.

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

Peclet Number

a + b Pen

num.

Fig. 14. Dimensionless longitudinal dispersion coefficient in the pseudo-composition space as a function of the cell Péclet number.

It is important to remember, when calculating the total dis-persion coefficient, that the dispersive part depends on concen-tration in a nonlinear manner, through the dependency on thediffusion coefficients, i.e., we have

D∗�,A

∣∣∣pseudo-component

= 〈DAA〉� 1

1.34+ 0.0128(〈DAA〉�)−0.65(〈v�〉��UC)1.65.

(124)

The only simplification comes when the exponent on the Pécletnumber is equal to one, and this leads to the results presentedbefore for the linear dispersive regime. When it is differentfrom one, this produces in the normal concentrations a com-plex structure of all the coefficients of the effective generalizedFick’s law. We have

[D∗�] = [〈D〉�]

[[I] + 1

V�

∫A��

n��[P ][b][P ]−1 dA

]− 〈v�[P ][b][P ]−1〉�, (125)

where, in the binary case, the matrix [P ] can be calculated from

[P ] =[

1 (〈DAA〉� − 〈DAA〉�)/〈DAB 〉�(〈DBB 〉� − 〈DBB 〉�)/〈DBA〉� 1

](126)

and the eigenvalues being given by

〈DAA〉� = 1

2(〈DAA〉� + 〈DBB〉�) + 1

2

√(〈DBB〉�)2 − 2〈DBB〉�〈DAA〉� + (〈DAA〉�)2 + 4〈DBA〉�〈DAB〉� (127)

〈DBB〉� = 1

2(〈DAA〉� + 〈DBB〉�) − 1

2

√(〈DBB〉�)2 − 2〈DBB〉�〈DAA〉� + (〈DAA〉�)2 + 4〈DBA〉�〈DAB〉�. (128)

Eq. (125) may be used in conjunction with the discussion aboveto calculate the total longitudinal dispersion coefficient as

[i] · [D∗�] · i = 1

1.34[〈D〉�]

+ [P ][

0.0128(〈DAA〉�)−0.65(〈�〉��UC)1.65 00 0.0128(〈DBB〉�)−0.65(〈�〉��UC)1.65

][P −1]. (129)

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2659

Numerical computations for the one-dimensional-averagedequations have been carried out with these formulas. At eachtime step this requires that we carry out the following calcula-tions:

• compute the values of the micro-scale generalized Fick’smatrix,

• compute the eigenvalues,• compute the effective dispersion coefficients using Eq. (129).

Here we see that it is crucial to have a simple estimate for thedispersion curve in the pseudo-composition space, such as theone given by Eq. (124).

Unfortunately, it was not possible to perform the micro-scalecomputations at very large Péeclet numbers, and we stayed inthe transition regime (the actual Péclet number, while variablebecause of the non-linearities, was about 7). This is a rangewhere the estimate of the dispersion coefficient with the power-law expression is not highly accurate, as seen in Fig. 14. Theresults of the comparison are presented in Fig. 15. To showthat dispersion effects are present, a macro-scale calculationwas also performed with no dispersion effects (i.e., only termswith the tortuosity factor of 1/1.34), and they are reported asresults with no-dispersion. The comparison between the aver-aged concentrations obtained from the two-dimensional directsimulations are in very good agreement with the theoreticalpredictions including the dispersion effects, while the resultswith no dispersion clearly differ. Unfortunately, it was not pos-sible to perform direct numerical computations in the well-established dispersive regime (this would require a very longsystem with many unit cells). However, since no assumptionswere made in the comparison, the results are a good confirma-tion of the theory. The small discrepancy observed in the down-stream region is probably due to the fact that the dispersioncurve is not well fitted by the power law expression in this in-termediate regime. It also may be attributed to various effects,including:

• numerical dispersion,• the impact of pore-scale nonlinearities such that the assump-

tions made in developing the closure problem would fail tobe valid,

• scale effects, due to the relative small number of unit cellsinvolved in the numerical experiments.

Finally, we want to emphasize that the complex construction ofthe effective dispersion curve must be used in order to correctlyrepresent the averaged behavior. To illustrate that, we computedthe one-dimensional solution with heuristic dispersion curvesfollowing the literature proposal such that

[i] · [D∗�] · i = 1

1.34[〈D〉�]

+[

0.0128(〈DAA〉�)−0.65(〈�〉��UC)1.65 00 0.0128(〈DBB〉�)−0.65(〈�〉��UC)1.65

]. (130)

0 0.5 1 1.5 2 2.5

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m)

Mas

sfr

actio

n

t = 5.0 s

intr. aver. veloc. 15.15 10-5 m/s

ωA (num.)

ωB(num.)

ωA (theor. nodisp)

ωB(theor. nodisp)

ωA(theor. + disp)

ωB(theor. + disp)

0 0.5 1 1.5 2 2.5

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m)

Mas

sfr

actio

n

t = 10.0 s

intr. aver. veloc. 18.53 10-5 m/s

ωA(num.)

ωB (num.)

ωA(theor. nodisp)

ωB(theor. nodisp)

ωA(theor. + disp)

ωB(theor. + disp)

Fig. 15. Results for moderate dispersive regime (NUC = 26; pressuredifference = 8 Pa).

The result, illustrated in Fig. 16, will be called “specialdispersion”. This figure represents the one-dimensional resultsobtained previously and the results obtained with this “spe-cial dispersion” coefficient. We see that the results with the“special dispersion” coefficients are close to the predictionsobtained with the complete formulation. However, we haveto remember that dispersion and diffusion are about the sameorder at this relatively low Péclet number. A more thoroughexamination of the dispersion coefficients is needed beforeconcluding that it is possible to use the special dispersion coef-ficients instead of the whole theory. Fig. 17 shows that, for thisparticular case, the special dispersion coefficients are close

2660 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m)

Mas

sfr

actio

n

t = 10.0 s

intr. aver. veloc. 18.53 10-5 m/s

ωA (theor. nodisp)

ωB (theor. nodisp)

ωA (theor. + disp)

ωB (theor. + disp)

ωA (spec. disp)

ωB (spec. disp)

Fig. 16. Comparison between the different expressions for the effective dis-persion coefficients (NUC = 26; pressure difference = 8 Pa, t = 10 s).

0 0.1 0.2 0.3 0.4 0.5 0.6-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

ωA

Dis

pers

ion

ratio D

AA(spec. disp.)/D

AA

DBB

/DAA

DBB

(spec. disp.)/DAA

DBA

/DAA

DAB

/DAA

Fig. 17. Dispersion coefficients (i.e., D�) divided by DAA� as a function of�A�, for the composition characterized by �B� = 0.4.

to the diagonal terms of the full effective dispersion matrix.However, we see that the off-diagonal terms are not negligiblecompared to the diagonal terms, and they might play a role inother conditions, which may preclude the use of a simplifieddispersion matrix. We did not explore all possible conditions.However, to illustrate the impact of the off-diagonal terms un-der the conditions of our numerical experiments, we performedcalculations with only the dispersion terms, i.e., with a zero ef-fective diffusion matrix. One example of the obtained results isshown in Fig. 18, which represents the mass fractions plottedfor the two types of dispersion matrix, at a given time. Therewe see a small but observable difference between the two re-sults, especially for the second component, which indicates thatone must be careful when employing the simplified dispersionmatrix.

0.24 0.242 0.244 0.246 0.248 0.250.1

0.15

0.2

0.25

0.3

0.35

x (m)

Mas

sfr

actio

n

0. 24 0.242 0.244 0.246 0.248 0.25

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

ωA(theor. + disp)

ωA (spec. disp)

ωB(theor. + disp)

ωB(spec. disp)t = 1500 s

Fig. 18. Comparison between the different expressions for the effective dis-persion coefficients (intrinsic average velocity: 1.79 × 10−4 m/s).

5. Conclusions

In this paper, we have investigated the problem of multicom-ponent mass transfer in porous media. Using the method ofvolume averaging applied to the pore-scale equations, we ob-tained macro-scale equations involving a generalized total dis-persion matrix. The coefficients in this matrix are determinedby the solution of special closure problems which have theform of the classical closure problems for dispersion in thepseudo-composition space. Returning to the normal composi-tions results in a complex relationship between the pore-scalediffusion matrix and the effective total dispersion coefficients.It should be emphasized that this result applies provided thatnonlinearities may be neglected at the closure level over a unitcell, and this is generally the case when the classic length-scalerequirements are satisfied. The theory has been successfullytested by comparing averaged concentrations obtained fromtwo-dimensional micro-scale computations to theoretical pre-dictions without adjustable parameters.

From the theoretical results we deduce that the typical chem-ical engineering practice of applying dispersion correlations forbinary systems to the components of a non-dilute, multicompo-nent system is only strictly valid under the following circum-stances:

• in the purely diffusive case, effective coefficients are thepore-scale diffusion coefficients to which is applied the sametortuosity term,

• in the linear dispersive regime, and this only for the diagonaleffective coefficients.

Notation

A�� area of the �–� surface contained in the averagingvolume, m2

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2661

A�e area of the entrances and exits of the �-phase asso-ciated with the macroscopic system, m2

b closure variable that maps [∇〈��〉�] onto [��], m[b] diagonal matrix of closure variables composed of

bA, bB, bC , etc. that maps ∇〈wA�〉� onto wA�,

∇〈wB�〉� onto wB�, ∇〈wC�〉� onto wC�, etc., mb� body force vector for the �-phase, m/s2

cA� molar concentration of species A in the �-phase,k mol/m3

c� total molar concentration in the �-phase, k mol/m3

dA� (xA�/RT )∇�A�, m1

DAB binary diffusion coefficient, m2/sDAE element of the full diffusivity matrix, m2/s〈DAE〉� intrinsic average of an element of the full diffusivity

matrix, m2/sDAE DAE − 〈DAE〉� spatial deviation of an element of

the diffusivity matrix, m2/s[〈D〉�] full diffusivity matrix composed of 〈DAA〉�,

〈DAB〉�, 〈DBA〉�, etc., m2/s[〈D〉�diag] diagonal diffusivity matrix composed of 〈DAA〉�,

〈DBB〉�, 〈DCC〉�, etc.[Deff ] full tensor effective diffusivity matrix, m2/s[D�] full tensor dispersion coefficient matrix, m2/s[D∗

�] [Deff ] + [D�], full total dispersion tensor matrix,m2/s

[D∗] (N − 1) × (N − 1) diffusivity matrix that maps−c�[∇x] onto [J∗], m2/s

[D] (N − 1) × (N − 1) diffusivity matrix that maps−�y[∇�] onto [j], m2/s

[F ] (N − 1) × (N − 1) matrix that maps [j] onto [j∗][G] (N − 1) × (N − 1) matrix that maps [j∗] onto [j]I unit tensorli i = 1, 2, 3, lattice vectors for a spatially periodic

porous medium, m[I] (N − 1) × (N − 1) matrix of unit tensors[j] column matrix of mass diffusive fluxes, kg/m2 s[j∗] column matrix of mixed-mode diffusive fluxes,

kg/m2 sjA� ���A�uA�, mass diffusive flux of species A in the

�-phase, kg/m2 sj∗A� ���A�u∗

A�, mixed-mode diffusive flux of species A

in the �-phase, kg/m2 s[J∗] column matrix of molar diffusive fluxes,

kgmole/m2 sJ∗A� cA�u∗

A�, molar diffusive flux of species A in the �-

phase, kgmole/m2 sK� Darcy’s law permeability tensor, m2

�� small length scale (pore diameter) associated withthe �-phase, m

�� small length scale (particle diameter) associatedwith the �-phase, m

�UC characteristic length scale associated with a unitcell, m

L characteristic length scale associated with averagedquantities, m

L� characteristic length associated with 〈��〉�, mL� characteristic length associated with 〈�A�〉�, mn�� unit normal vector directed from the �-phase toward

the �-phasep� pressure in the �-phase, N/m2

[P ] nodal matrix that diagonalizes the full matrix,[〈D〉�]

r0 radius of the averaging volume, mr position vector, mR universal gas constant, N m/K kgmolet time, sT temperature,KT� total stress tensor for the �-phase, N/m2

uA� vA� − v�, mass diffusion velocity of species A inthe �-phase, m/s

u∗A� vA� − v∗

� , molar diffusion velocity of species A inthe �-phase, m/s

vA� velocity of species A in the �-phase, m/sv�

∑A=NA=1 �A�vA�, mass average velocity in the

�-phase, m/s〈v�〉� intrinsic average mass average velocity in the

�-phase, m/sv� v� −〈v�〉�, spatial deviation mass average velocity,

m/sv∗�

∑A=NA=1 xA�vA�, molar average velocity in the

�-phase, m/sV averaging volume, m3

V� volume of the �-phase contained within the aver-aging volume, m3

[w�] column matrix composed of wA�, wB�, wC�, etc.[∇〈w�〉�] column matrix composed of ∇〈wA�〉�, ∇〈wB�〉�,

∇〈wC�〉�, etc., m−1

xA� cA�/c�, mole fraction of species A in the �-phasey� position vector locating points in the �-phase rela-

tive to the centroid of the averaging volume, m

Greek letters

�A� activity coefficient for species A in the �-phase�AB elements of the unit matrix�� V�/V, volume fraction of the �-phase or the poros-

ity�A� chemical potential for species A in the �-phase,

N m/kgmole�A� mass density of species A in the �-phase, kg/m3

�� total mass density in the �-phase, kg/m3

〈��〉� intrinsic average total density, kg/m3

�� �� − 〈��〉�, spatial deviation total mass density,kg/m3

� tortuosity tensor� scalar tortuosity factor�A� �A�/��〈�A�〉� intrinsic average mass fraction of species A

�A� �A� − 〈�A�〉�, spatial deviation mass fraction for-species A

2662 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

[〈��〉�] mass fraction column matrix composed of 〈�A�〉�,〈�B�〉�, 〈�C�〉�, etc.

[��] spatial deviation mass fraction column matrix com-posed of �A�, �B�, etc.

Acknowledgements

Financial support from European program FEBUSS is grate-fully acknowledged.

Appendix A. Mass diffusive flux

In this appendix, in order to be clear about all definitionsand equations used in the main development, we give all for-mulas that have been used to manipulate and calculate the dif-fusive fluxes. We begin with the generalized Stefan–Maxwellequations in a form appropriate for non-ideal solutions

0 = −dA� +B=N∑B=1B �=A

xA�xB�(vB� − vA�)

DAB

,

A = 1, 2, . . . , N − 1, (A.1a)

dA� = xA�

RT∇�A�. (A.1b)

Here the chemical potential is given by

�A� = �oA� + RT ln

(�A�xA�

)(A.2)

in which we have used �A� to represent the activity coefficientfor species A in the �-phase. Use of this result in Eq. (A.1b)allows us to express dA� in the form

dA� = xA�

RT∇�A�

= xA�

RT

[B=N−1∑

B=1

RT� ln(�A�xA�)

�xB�

∣∣∣∣T ,p,xE�,E �=B

∇xB�

].

(A.3)

Simplification of this results leads to

dA� =B=N−1∑

B=1

[�AB + xA�

� ln(�A�)

�xB�

∣∣∣∣T ,p,xE�,E �=B

]∇xB�

(A.4)

in which �AB represents the components of the unit matrix.These can be expressed as

�AB ={

1, A = B

0, A �= BA, B = 1, 2, . . . , N − 1. (A.5)

To develop a more compact form of Eq. (A.4), we make use of

�AB = �AB + xA�� ln(�A�)

�xB�

∣∣∣∣T ,p,xE�,E �=B

,

A, B = 1, 2, . . . , N − 1, (A.6)

so that dA� can be expressed as

dA� =B=N−1∑

B=1

�AB∇xB� (A.7)

and we are ready to direct our attention to the right-hand sideof Eq. (A.1).

The species velocity can be decomposed in terms of themolar average velocity, v∗

� , and the molar diffusion velocity,u∗

A�, according to

vA� = v∗� + u∗

A� (A.8)

in which v∗� is defined by

v∗� =

N∑j=1

xA�vA�. (A.9)

Use of Eq. (A.8) in Eq. (A.1) leads to

0 = − c�dA� −E=N∑E=1E �=A

xE�J∗A�

DAE

+B=N∑B=1B �=A

xA�J∗B�

DAB

, A = 1, 2, . . . , N − 1, (A.10)

where J∗A� represents the molar diffusive flux defined explicitly

by Bird et al. (2002, p. 537)

J∗A� = cA�u∗

A�. (A.11)

The molar diffusive fluxes satisfy the relation

B=N∑B=1

J∗B� = 0 (A.12)

and this allows us to express Eq. (A.10) in the form

0 = − c�dA� − J∗A�

E=N∑E=1E �=A

xE�

DAE

− J∗A�xA�

DAN

+B=N−1∑

B=1B �=A

xA�J∗B�

DAB

− xA�

DAN

B=N−1∑B=1B �=A

J∗B�,

A = 1, 2, . . . , N − 1. (A.13)

This result can be compacted somewhat to obtain

0 = − c�dA� −

⎡⎢⎢⎣ xA�

DAN

+E=N∑E=1E �=A

xE�

DAE

⎤⎥⎥⎦ J∗

A�

+B=N−1∑

B=1B �=A

xA�J∗B�

(1

DAB

− 1

DAN

),

A = 1, 2, . . . , N − 1. (A.14)

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2663

In matrix form this can be expressed as

−c�[d] = [B][J∗] (A.15)

in which the column matrices are given by

[∇x] =

⎡⎢⎢⎢⎣

dA�dB�··

dN−1�

⎤⎥⎥⎥⎦ , [J∗] =

⎡⎢⎢⎢⎣

J∗A�

J∗B�··

J∗N−1�

⎤⎥⎥⎥⎦ (A.16)

and the (N − 1) × (N − 1) matrix is represented as

[B] =

⎡⎢⎢⎢⎣

BAA BAB · . . . BAN−1BBA BBB · . . . ·

· · BCC . . . ·· · · . . . ·

BN−1A · · . . . BN−1N−1

⎤⎥⎥⎥⎦ . (A.17)

The components of [B] are given explicitly by

BAA = xA�

DAN

+E=N∑E=1E �=A

xE�

DAE

, A = 1, 2, . . . , N − 1, (A.18a)

BAB = −xA�

(1

DAB

− 1

DAN

), A, B = 1, 2, . . . , N − 1,

A �= B. (A.18b)

From Eq. (A.15) we obtain the molar diffusive flux in the form

[J∗] = −c�[B]−1[d]. (A.19)

We can make use of Eq. (A.7) to express this result as

[J∗] = −c�[B]−1[�][∇x], (A.20)

where [�] represents the matrix having the coefficients givenby

�AB = �AB + xA�� ln(�A�)

�xB�

∣∣∣∣T ,p,xE�,E �=B

,

A, B = 1, 2, . . . , N − 1. (A.21)

Eq. (A.20) represents the generalized Fick’s law

[J∗] = −c�[D∗][∇x] (A.22)

in which the molar-based diffusivity matrix is given by

[D∗] = [B]−1[�]. (A.23)

Even for ideal solutions, this representation for the molar dif-fusive fluxes is nonlinear since the coefficients in [D∗] dependon the mole fractions of all the components.

We now turn our attention to the mass-based diffusion co-efficients. In this case, the species velocity is decomposed ac-cording to

vA� = v� + uA�. (A.24)

Here uA� represents the mass diffusion velocity and v� repre-sents the mass average velocity defined by

v� =N∑

j=1

�A�vA� (A.25)

in which the mass fraction is given by

�A� = �A�

��. (A.26)

We express the mass diffusive flux according to

jA� = ���A�uA� (A.27)

and note that these fluxes are constrained by

B=N∑B=1

jB� = 0. (A.28)

The mass fractions can be expressed in terms of the mole frac-tions according to

�A� = xA�MA

M(A.29)

in which MA is the molecular mass of species A and the meanmolecular mass is defined by

M =B=N∑B=1

xB�MB . (A.30)

At this point we introduce the mixed-mode diffusive flux (Birdet al., 2002, p. 537) defined by1

j∗A� = ���A�

(vA� − v∗

�

)= ���A�u∗

A� (A.31)

that has the characteristic given by

A=N∑A=1

xA�

�A�j∗A� = 0. (A.32)

We now seek the transformation that links j∗A� to the massdiffusive flux, jA�, defined by

jA� = ���A�(vA� − v�

) = ���A�uA�. (A.33)

We begin by extracting the mass diffusive flux from the mixed-mode diffusive flux with a decomposition of the flux to obtain

j∗A� = ���A�

(vA� − v∗

�

)= ���A�

(vA� − v�

)+ ���A�

(v� − v∗

�

). (A.34)

This can be expressed as

j∗A� = jA� + ���A�

(v� − v∗

�

)(A.35)

1 The notation for diffusive fluxes follows that of Bird et al. (2002).

2664 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

indicating that the last term needs to be expressed in terms ofeither j∗A� or jA� or both. A series of algebraic steps leads to

j∗A� = jA� + ���A�

(v� −

B=N∑B=1

xB�vB�

)

= jA� + ���A�

(B=N∑B=1

xB�(v� − vB�)

)

= jA� + ���A�

(B=N∑B=1

xB�(v� − vB�)

)

= jA� − �A�

(B=N∑B=1

xB����B�

�B�(vB� − v�)

)

= jA� − �A�

(B=N∑B=1

xB�

�B�jB�

). (A.36)

The last term in the summation can be isolated to obtain

j∗A� = jA� − �A�

(B=N−1∑

B=1

xB�

�B�jB�

)− �A�

xN�

�N�jN� (A.37)

and use of Eq. (A.28) allows us to express this result in the form

j∗A� = jA� − �A�

(B=N−1∑

B=1

xB�

�B�jB�

)

+ �A�xN�

�N�

B=N−1∑B=1

jB�. (A.38)

A compact representation of this result is given by

j∗A� = jA� − �A�

[B=N−1∑

B=1

(xB�

�B�− xN�

�N�

)jB�

](A.39)

which provides the connection of the mixed-mode diffusiveflux, j∗A�, to the mass diffusive flux, jA�. In matrix form thisrelation is expressed as

[j∗] = [F ][j]. (A.40a)

Here the column matrices are given by

[j∗] =

⎡⎢⎢⎢⎣

j∗A�j∗B�··j∗N−1�

⎤⎥⎥⎥⎦ , [j] =

⎡⎢⎢⎢⎣

jA�jB�··jN−1�

⎤⎥⎥⎥⎦ (A.41)

and the mixed-mode to mass transformation matrix is an (N −1) × (N − 1) matrix indicated by

[F ] =

⎡⎢⎢⎢⎣

FAA FAB · · · · FAN−1FBA FBB · · · · ·

· · FCC · · · ·· · · · · · ·

FN−1A · · · · · FN−1N−1

⎤⎥⎥⎥⎦ . (A.42)

The components of the mixed-mode to mass transformationmatrix are given by

FAB = �AB − �A�

(xB�

�B�− xN�

�N�

). (A.43)

The reverse transformation (mass to mixed-mode transfor-mation) can be obtained following the same methodology.A similar series of algebraic steps leads to

jA� = ���A�(vA� − v�)

= ���A�(vA� − v∗�) + ���A�(v∗

� − v�)

= j∗A� + ���A�

B=N∑B=1

�B�(v∗� − vB�)

= j∗A� − �A�

B=N∑B=1

j∗B�

= j∗A� − �A�

B=N−1∑B=1

j∗B� − �A�j∗N�

= j∗A� − �A�

B=N−1∑B=1

j∗B� + �A��N�

xN�

B=N−1∑B=1

xB�

�B�j∗B�

(A.44)

and a compact representation of this result is given by

jA� = j∗A� − �A�

×[

B=N−1∑B=1

(1 − �N�xB�

xN��B�

)j∗B�

]. (A.45)

In matrix form we express Eq. (A.45) as

[j] = [G][j∗] (A.46)

in which the details of the mass to mixed-mode transformationmatrix can be expressed as

GAB = �AB − �A�

(1 − �N�xB�

xN��B�

),

A, B = 1, 2, . . . , N − 1. (A.47)

On the basis of Eqs. (A.40) and (A.46) the transformation ma-trices are related by

[G] = [F ]−1. (A.48)

To obtain a relation between the mass diffusive flux and themass fraction gradients, we begin by representing the mixed-mode diffusive flux in terms of the molar diffusive flux accord-ing to

j∗A� = ���A�(vA� − v∗�) = ���A�

c�xA�J∗A�. (A.49)

In matrix form, this can be represented as

[j∗] = ��

c�[�][x]−1[J∗], (A.50)

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2665

where [�] and [x] are the diagonal matrices given by

[�] =

⎡⎢⎢⎢⎣

�A� 0 · · · · 00 �B� · · · · ·· · �C� · · ·· · · · · ·0 · · · · · �N−1�

⎤⎥⎥⎥⎦ ,

[x] =

⎡⎢⎢⎢⎣

xA� 0 · · · · 00 xB� · · · · ·· · xC� · · · ·· · · · · · ·0 · · · · · xN−1�

⎤⎥⎥⎥⎦ . (A.51)

We now make use of Eqs. (A.22) and (A.40) in order to expressEq. (A.50) as

[j] = −��[G][�][x]−1[D∗][∇x] (A.52)

and we are left with only the problem of determining the relationbetween [∇x] and [∇�]. In order to develop this relation, webegin with Eqs. (A.29) and (A.30) in the form

xA� = �A�M

MA

, M =B=N∑B=1

xB�MB (A.53)

and take the gradient of the mole fraction to obtain

∇xA� = M∇�A�

MA

+ �A�

MA

B=N∑B=1

MB∇xB�. (A.54)

The first of Eqs. (A.53) can be used for MA and MB in orderto express this result as

∇xA� = xA�

�A�∇�A� + xA�

B=N∑B=1

�B�

xB�∇xB� (A.55)

and the last term in the sum can be separated to obtain

∇xA� − xA�

B=N−1∑B=1

�B�

xB�∇xB� − xA�

�N�

xN�∇xN�

= xA�

�A�∇�A�. (A.56)

Since the sum of the mole fractions is one, we have

∇xN� = −B=N−1∑

B=1

∇xB�. (A.57)

Use of this result in Eq. (A.56) leads to the following relationbetween ∇xA�, ∇xB�, ∇xC�, etc. and ∇�A�

∇xA� − xA�

B=N−1∑B=1

�B�

xB�∇xB� + xA�

�N�

xN�

B=N−1∑B=1

∇xB�

= xA�

�A�∇�A�. (A.58)

A more compact form relating the gradient of the mass fractionto the gradients of the mole fractions is given by

∇�A� = �A�

xA�∇xA� − �A�

×B=N−1∑

B=1

(�B�

xB�− �N�

xN�

)∇xB�. (A.59)

This can be expressed in matrix form according to

[∇�] = [H ][∇x], (A.60)

where the components of [H ] are given by

HAB = �A�

xA��AB − �A�

(�B�

xB�− �N�

xN�

),

A, B = 1, 2, . . . , N − 1. (A.61)

At this point it is convenient to recognize that

�A�

xA��AB = �B�

xB��AB, A, B = 1, 2, . . . , N − 1, (A.62)

so that Eq. (A.61) takes the form

HAB = �B�

xB�

[�AB − �A�

(1 − �N�xB�

xN��B�

)],

A, B = 1, 2, . . . , N − 1. (A.63)

Comparison with Eq. (A.47a) indicates that

HAB = �B�

xB�GAB, A, B = 1, 2, . . . , N − 1. (A.64)

In terms of the appropriate matrices, this leads to

[H ] = [G][�][x]−1 (A.65)

and this allows us to express Eq. (A.60) in the form given by

[∇�] = [G][�][x]−1[∇x]. (A.66)

From this we obtain the desired representation of the gradientof the mole fractions given by

[∇x] = [x][�]−1[G]−1[∇�] (A.67)

and substitution of this result into Eq. (A.52) yields

[j] = −��[G][�][x]−1[D∗][x][�]−1[G]−1[∇�]. (A.68)

This can be expressed as

[j] = −��[D][∇�], (A.69)

where

[D] = [G][�][x]−1[D∗][x][�]−1[G]−1. (A.70)

The components of [D] are designated as DAE and appear inEq. (12) in the body of the paper.

2666 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

Appendix B. Volume averaging

In order to develop an upscaled version of Eqs. (13a)–(16),we associate with every point in space an averaging volume, V,such as we have illustrated in Fig. 2, and we form the volumeaverage of Eq. (13a) to obtain⟨

�(���A�)

�t

⟩+ 〈∇ · (���A�v�)〉

=⟨∇ ·

[��

E=N−1∑E=1

DAE∇�E�

]⟩,

A = 1, 2, . . . , N − 1. (B.1)

For the case of passive transport in a rigid porous medium, thespeed of displacement of the �.� interface is zero and we caninterchange integration and differentiation in the first term toobtain⟨�(���A�)

�t

⟩= �〈���A�〉

�t. (B.2)

In addition, the volume averaging theorem given by

〈∇��〉 = ∇〈��〉 + 1

V

∫A��

n���� dA (B.3)

can be used to express the convective and diffusive terms as

〈∇ · (���A�v�)〉 = ∇ · 〈���A�v�〉+ 1

V

∫A��

n�� · ���A�v� dA, (B.4)

⟨∇ ·

[��

E=N−1∑E=1

DAE∇�E�

]⟩

= ∇ ·E=N−1∑

E=1

〈��DAE∇�E�〉

+ 1

V

E=N−1∑E=1

∫A��

n�� · ��DAE∇�E� dA. (B.5)

The boundary conditions represented by Eqs. (14) and (16)require that the area integrals of the fluxes in Eqs. (B.4) and(B.5) be zero and those two results, along with Eq. (B.2), canbe used in Eq. (B.1) to obtain

�〈���A�〉�t

+ ∇ · 〈���A�v�〉

= ∇ ·[

E=N−1∑E=1

〈��DAE∇�E�〉]

,

A = 1, 2, . . . , N − 1. (B.6)

Here we are confronted with the average of double and tripleproducts, and the analysis of these terms can become quitecomplex from the algebraic point of view. At this point weintroduce the spatial deviations, ��, ��, and v� that are definedby the decompositions

�� = 〈��〉� + ��, �A� = 〈�A�〉� + �A�,

v� = 〈v�〉� + v�. (B.7)

Fig. B1. Position vectors associated with the averaging volume.

We begin our analysis of the average of products with the term〈���A�〉 and make use of the first two decompositions givenby Eqs. (B.7) in order to obtain

〈���A�〉 = 〈〈��〉�〈�A�〉� + ��〈�A�〉� + 〈��〉��A� + ���A�〉.(B.8)

At this point we need to remove the intrinsic averages, 〈��〉�and 〈�A�〉�, from the superficial average indicated by〈 〉. To ac-complish this, we expand 〈��〉� and 〈�A�〉� in Taylor series ex-pansions about the centroid of the averaging volume illustratedin Fig. B1 . This has been done by Whitaker (1999, Section3.2.3), and after some algebra one obtains the following repre-sentation for 〈���A�〉

〈���A�〉 = ��〈��〉�〈�A�〉� + 〈���A�〉+ ∇〈��〉� · 〈y�y�〉 · ∇〈�A�〉� + 〈��y�〉 · ∇〈�A�〉�+ 1

2 〈��y�y�〉 : ∇∇〈�A�〉� + ∇〈��〉� · 〈y��A�〉+ 1

2 〈y�y��A�〉 : ∇∇〈��〉� + · · · (B.9)

To see how this expression can be simplified, we consider thethird term on the right-hand side and construct the estimategiven by Whitaker (1999, Section 1.3.2)

∇〈��〉� · 〈y�y�〉 · ∇〈�A�〉� = O[ 〈��〉�

L�r2

0 〈�A�〉�

L�

](B.10)

in which L� and L� represent the distances over which signif-icant variations in 〈��〉� and 〈�A�〉� occur. Our representationfor 〈���A�〉 can be simplified on the basis of the restrictiongiven by

∇〈��〉� · 〈y�y�〉 · ∇〈�A�〉� � ��〈��〉�〈�A�〉� (B.11)

M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669 2667

along with the assumption that small causes give rise to smalleffects. On the basis of the estimate given by Eq. (B.10) and theinequality given by Eq. (B.11), we can simplify Eq. (B.9) to

〈���A�〉 = ��〈��〉�〈�A�〉� + 〈���A�〉 + 〈��y�〉 · ∇〈�A�〉�+ 1

2 〈��y�y�〉 : ∇∇〈�A�〉� + ∇〈��〉� · 〈y��A�〉+ 1

2 〈y�y��A�〉 : ∇∇〈��〉� + · · · (B.12)

provided that the following length-scale constraint is satisfied

r20

L�L�� 1. (B.13)

This constraint will always be satisfied whenever the generallength-scale constraints given by Eq. (20) are valid. One canconstruct estimates of the higher-order terms in Eq. (B.12) andimpose constraints consistent with Eq. (20) in order to simplifyEq. (B.12) to the form

〈�y�A�〉 = ��〈��〉�〈�A�〉� + 〈���A�〉. (B.14)

In addition, it is well established that whenever the length-scaleconstraints given by Eq. (20) are valid the spatial deviations,�� and �A�, are small compared to the average values, 〈��〉�and 〈�A�〉�. We express this idea as

�� � 〈��〉�, �A� � 〈�A�〉� (B.15)

and note that it leads to

〈���A�〉 = ��〈��〉�〈�A�〉�. (B.16)

Directing our attention to the convective transport term,〈���A�v�〉, we first note that the spatial deviation quantitiesare constrained by

�� � 〈��〉�, �A� � 〈�A�〉�, v� = O(〈v�〉�). (B.17)

If we repeat the type of analysis that led to Eq. (B.12) for theconvective transport term we obtain

〈���A�v�〉 = ��〈��〉�〈�A�〉�〈v�〉�+ 〈��〉�〈�A�v�〉 + 〈�A�〉�〈��v�〉 + · · · (B.18)

in which the higher-order terms resulting from the Taylor seriesexpansion are not shown. Neglecting these terms is consistentwith the traditional treatment of dispersion phenomena; how-ever, further theoretical work is needed to establish this sim-plification. In addition to discarding the higher-order terms, weassume that

〈�A�〉�〈��v�〉 � 〈��〉�〈�A�v�〉. (B.19)

This is based on the idea that the total density, ��, undergoesmuch smaller variations than the species density, �A� =���A�.The motivation for this simplification is based on the differentforms of the governing equations represented by Eqs. (13a)and (13b); however, a detailed investigation of the validity ofEq. (B.19) remains to be carried out.

Use of Eq. (B.19) in Eq. (B.18) and neglecting the higher-order terms allows us to express the convective transport as

〈���A�v�〉 = ��〈��〉�〈�A�〉�〈v�〉� + 〈��〉�〈�A�v�〉. (B.20)

We now return to Eq. (B.6) and make use of Eqs. (B.16) and(B.20) to obtain

�(��〈��〉�〈�A�〉�)�t

+ ∇ · (��〈��〉�〈�A�〉�〈v�〉�)

+ ∇ · (〈��〉�〈�A�v�〉) = ∇ ·[

E=N−1∑E=1

〈��DAE∇�E�〉]

,

A = 1, 2, . . . , N − 1. (B.21)

In our treatment of the diffusive term, we introduce anotherspatial deviation given by

DAE = 〈DAE〉� + DAE (B.22)

and we assume that the spatial deviations associated with thediffusive flux satisfy the conditions

�� � 〈��〉�, DAE � 〈DAE〉�, ∇�A� = O(∇〈�A�〉�).(B.23)

The third of these conditions is consistent with the second ofEqs. (B.17) because of the disparity of length scales associatedwith �A� and 〈�A�〉�, i.e., �� � L�. At this point one canrepeat the analysis leading from Eq. (B.18) to Eq. (B.20) sothat the superficial average diffusive flux takes the form

〈��DAE∇�E�〉 = ��〈��〉�〈DAE〉�∇〈�E�〉�+ 〈��〉�〈DAE〉�〈∇�E�〉. (B.24)

We can use the averaging theorem with the last term in thisresult to obtain

〈∇�E�〉 = ∇〈�A�〉 + 1

V

∫A��

n���A� dA (B.25)

and it is consistent with the length-scale constraint given byEq. (20) to impose the condition

〈�A�〉 = 0. (B.26)

This allows us to use Eq. (B.24) in Eq. (B.21) to obtain

�(�〈��〉�〈�A�〉�)�t

+ ∇ · (��〈��〉�〈�A�〉�〈v�〉�)

+ ∇ ·⎛⎜⎝〈��〉� 〈�A�v�〉︸ ︷︷ ︸

filter

⎞⎟⎠

= ∇ ·

⎡⎢⎢⎢⎣

E=N−1∑E=1

��〈��〉�〈DAE〉�∇〈�E�〉�

+ 〈��〉�〈DAE〉�V

∫A��

n���A� dA︸ ︷︷ ︸filter

⎤⎥⎥⎥⎦ ,

A = 1, 2, . . . , N − 1 (B.27)

2668 M. Quintard et al. / Chemical Engineering Science 61 (2006) 2643–2669

which was given earlier as Eq. (21). In addition to the volume-averaged form of Eq. (13a), we also require the volume-averaged form of Eq. (13b). This is given by⟨���

�t

⟩+ 〈∇ · (��v�)〉 = 0. (B.28)

Use of the general transport theorem for a rigid porousmedium leads to

�〈��〉�t

+ 〈∇ · (��v�)〉 = 0 (B.29)

and the spatial averaging theorem can be used along with theboundary condition given by Eq. (14) to obtain

�(��〈��〉�)�t

+ ∇ · 〈��v�〉 = 0. (B.30)

Here we have replaced 〈��〉 with ��〈��〉�, and we can used Eq.(B.9) to express the convective transport term as

〈��v�〉 = ��〈��〉�〈v�〉� + 〈��v�〉 + ∇〈��〉� · 〈y�y�〉 · ∇〈v�〉�+ 〈��y�〉 · ∇〈v�〉� + 1

2 〈��y�y�〉: ∇∇〈v�〉�+ ∇〈��〉� · 〈y�v�〉 + 1

2 〈y�y�v�〉: ∇∇〈��〉� + · · · .

(B.31)

Traditionally, the higher-order terms in this result are discardedand the dispersive transport is neglected on the basis of

〈��v�〉 � ��〈��〉�〈v�〉�. (B.32)

This leads to a total continuity equation of the form

�(��〈��〉�)�t

+ ∇ · (��〈��〉�〈v�〉�) = 0 (B.33)

which is ready to use in the solution of mass transfer prob-lems; however, Eq. (B.27) requires the development of a clo-sure problem for �A� which is done in Section 3.

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