kechagia2002_

Chemical Engineering Science 57 (2002) 2565 – 2577
www.elsevier.com/locate/ces
On the upscaling of reaction-transport processes in porous media
with fast or #nite kinetics
Persefoni E. Kechagiaa , Ioannis N. Tsimpanogiannisa , Yanis C. Yortsosa; ∗ ,
Peter C. Lichtnerb
a Department
b Earth
of Chemical Engineering, University of Southern California, 925 West 37th Street, Los Angeles, CA 90089-1211, USA
& Environmental Sciences Division (EES-5) MS F-649, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Received 27 October 2000; received in revised form 17 July 2001; accepted 12 December 2001
Abstract
We show that for reaction-transport processes with fast kinetics (in the limit of thermodynamic equilibrium), conventional volume
averaging for determining e6ective kinetic parameters applies only when the macroscopic variable approaches its equilibrium value. Even
under such conditions, computing the e6ective mass transfer coe7cient requires solving an eigenvalue problem, which couples the local
microstructure with the global. Two examples, one involving a simple advection–dissolution problem and another a drying problem in
a pore network, illustrate the theoretical predictions. Similar considerations apply for the case of #nite kinetics, when the macroscale
concentration approaches an equilibrium value. In that case, the e6ective kinetic parameter is not equal to the local, as typically assumed,
but it becomes a function of the local Thiele modulus. ? 2002 Published by Elsevier Science Ltd.
Keywords: Porous media; Mass transfer; Dissolution; Kinetics; Scale-up; Drying
1. Introduction
Mass transport with heterogeneous chemical reaction in
porous media is important in a variety of applications in the
physical, chemical, and engineering sciences. Transport is
typically by advection and di6usion, with reaction occurring at the (usually) disordered pore surface. As a result of
reaction, the pore morphology may also evolve, with examples ranging from gas–solid reactions (Dogu, 1980) to
the acidizing of carbonate rocks (Hoefner & Fogler, 1988;
Daccord, Lenormand, & Lietard, 1993). A speci#c class of
problems of interest to this paper involves heterogeneous reactions with #nite or fast kinetics. A limiting case are problems, where thermodynamic equilibrium applies at the pore
surface (Lichtner, 1996).
The latter type of problem arises in various contexts,
for example in the solubilization of trapped non-aqueous
phase liquids (NAPLs) in soils (Imho6, Ja6e, & Pinder,
1993; Jia, Shing, & Yortsos, 1999), or in the drying of a
porous medium (Laurindo & Prat, 1998; Tsimpanogiannis,
Yortsos, Poulou, Kanellopoulos, & Stubos, 1999), where
∗ Corresponding author. Tel.: +1-213-74-00-317;
fax: +01-213-740-8053.
E-mail address: [email protected] (Y. C. Yortsos).
0009-2509/02/$ - see front matter ? 2002 Published by Elsevier Science Ltd.
PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 1 2 4 - 0
the concentration at the source=sink interface can be considered as constant. In many applications, mass transfer is
driven by the Fow of a liquid in a porous medium. For example, this is the case in the remediation by solvent injection of
ground-water aquifers contaminated by organic chemicals
(NAPLs). The NAPLs are trapped by capillarity in the pore
interstices, forming immobile, disconnected ganglia, which
act as contaminant sources of a generally disordered geometry. A similar situation arises in dissolution or precipitation
and involve mass transport between the pore surface and the
injected Fuid. If, in such problems, the kinetics at the interface are fast, and one can proceed with the assumption of
local thermodynamic equilibrium (for example, the use of
Henry’s law at the source interface).
In view of the complexity of the pore space and=or the
source-sink geometry, and the need for workable models, an
important topic pursued for several years has been the development of e6ective models to scale-up the process from
the micro- (pore) scale to the macro- (continuum) scale.
While the upscaling of conventional di6usion–reaction systems has been researched for some time (Whitaker, 1999),
the upscaling of dissolution-type processes has been rather
recent. Of particular relevance in this context is the work
of Quintard and Whitaker (1994), to be hereafter referred
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P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
to as QW, who applied volume averaging to derive e6ective mass transfer coe7cients for the dissolution of trapped
NAPLs in a porous medium with a periodic microstructure.
Implicit to their approach are two important assumptions,
scale separation at the microscale, and small gradients of the
volume-averaged variable at the pore-network scale. Under
conditions of fast kinetics, however, this requires that the
volume-averaged concentration must be close to its thermodynamic value. It will be shown in this paper that in this
case, the unit-cell problem for calculating e6ective coef#cients involves information both from the microstructure
and the from the gradients of the macroscopic concentration,
thereby resulting in the coupling of micro and macroscale
problems.
Volume averaging is one of two main methods for upscaling. It is closely related to homogenization (Bensoussan, Lions, & Papanicolaou, 1978), with which it shares the same
foundation. Pioneering work in volume averaging in porous
media has been done by Whitaker (1999). An alternative approach involving ensemble averages is based on the method
of moments, applied for example in Edwards, Shapiro, and
Brenner (1993) to derive Darcy-scale di6usion–reaction coe7cients. Stochastic theories for reactive transport in various contexts (Hu, Cushman, & Deng, 1997; Cvetkovic &
Dagan, 1994) have also been developed, where the emphasis is on the (macroscopic) permeability heterogeneity, thus
representing upscaling over a larger scale. In this paper, we
will restrict our interest to volume averaging as particularly
applied in the above-cited work of QW.
For volume averaging to be meaningful, the averages
should not depend on the averaging volume itself. In general, this requirement can be translated into two conditions:
that the length scale of the averaging volume (macroscopic
length L) be su7ciently larger than the correlation length of
the microstructure, ,
1 ≡ 1
(1)
L
and that the upscaled variable vary slowly at the small scale
2 ≡ |∇CD |1;
(2)
where CD represents a dimensionless upscaled variable.
The correlation length expresses the scale over which microstructure variables are correlated. The #rst condition ensures that process-independent averages of the microstructure exist. For this, the averaging scale must be much larger
than the scale of correlation of the microstructure (which
is also known as the integral scale, e.g. see Dagan, 1989).
Upscaling, however, involves process-dependent variables,
as well, such as the upscaled concentration, for which constraint (2) must also be satis#ed. This will depend in general
on process parameters, and possibly on space and time.
In conventional reaction–di6usion problems, constraint
(2) can be enforced, if among other parameters, the local
Thiele modulus is small, namely if the kinetics are su7ciently slow (Whitaker, 1999). In dissolution–precipitation
processes of fast kinetics, which is one class of problems of
interest to this paper, however, the condition of local thermodynamic equilibrium precludes the existence of a small
dimensionless parameter, to ensure the validity of Eqs. (1)
and (2) (in such cases, the local Thiele modulus is in#nite).
It follows that Eq. (2) is satis#ed only in the limit when
the upscaled concentration approaches its thermodynamic
equilibrium value, in all other cases the upscaling being inherently non-local. We will show that even under the #rst
condition, however, micro and macroscale problems still remain coupled. A similar situation exists for conventional
advection–reaction processes, when the kinetics are #nite
and the process approaches an equilibrium state. In the latter
limit, constraint (2) is satis#ed, even though the local Thiele
modulus is not necessarily small. It will be shown that in
this case as well, the e6ective parameter arises from the solution of a problem coupling both local and global scales.
The paper is organized as follows: First, we provide a
brief review of the upscaling constraints of the type (2) for
typical transport-reaction systems. In this we will also draw
an analogy with two-phase Fow in porous media. Then, we
provide simple 1-D examples which illustrate the problem
and provide the motivation for the subsequent development.
Using the methodology of QW we will derive a problem for
the computation of the e6ective mass transfer coe7cient, in
processes where thermodynamic equilibrium applies, with
particular application to a dissolution problem. The closure
problem is found to be di6erent than in QW, as it depends
also on the gradients of the macroscale variable, and can
be cast in terms of an eigenvalue problem. We demonstrate
the theory with two examples, one involving advection–
dissolution and another involving drying in a pore network.
Similar ideas apply to reaction–di6usion systems with #nite
kinetics, when an equilibrium state is approached.
2. A brief background
Problems involving transport, reaction or other physicochemical processes in porous media are controlled by two
competing processes, one local and one non-local. The local process does not involve spatial gradients, and although
involving Fuctuating (possibly random) phenomena, such
as chemical reaction at disordered surfaces, it is generally
a stationary process. The non-local process involves spatial
gradients.
Consider, #rst, the example of di6usion with heterogeneous reaction. Here, the local process is the chemical reaction, while the non-local process is di6usion. The ratio in
the magnitude between local and non-local “forces” is expressed by a dimensionless parameter, which in a reaction–
di6usion system is the microscale Thiele modulus
ke
l =
:
(3)
aV D
Here, ke is the heterogeneous reaction kinetic parameter,
aV is the pore surface area per unit volume and D is the
P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
di6usion coe7cient. Recognizing that for relatively homogeneous media the speci#c surface area scales inversely with
the microscale length, aV ∼ 1=l, Eq. (3) expresses the ratio
in the speed of reaction to that of di6usion in the microscale.
For negligible microscale gradients in the upscaled variable,
which is the volume-average concentration, di6usion must
be fast at the small scale, compared to reaction. Then, the
homogenization constraint (2) becomes
l 1:
(4)
A measure of the competition between reaction and di6usion, but at the macroscale, is the conventional Thiele modulus
ke L
L
=
≈ l
:
(5)
D
l
This classical reaction engineering parameter is, thus, the
product of two terms, one of which must be small, according
to Eq. (4), and the other large, according to Eq. (1), for
conventional upscaling to apply.
As l increases, it becomes increasingly di7cult to satisfy Eq. (2). Then, an explicit coupling between the two
scales arises, leading in general, to a non-local description.
Nonetheless, gradients of the macroscale variable can be
small, regardless of the value of l , if the process approaches
an asymptotic equilibrium state (where reaction will cease).
For example, this is the case in dissolution processes, where
the solution eventually approaches thermodynamic equilibrium. Even then, it will be shown that the e6ective mass
transfer coe7cient is determined from a problem coupling
the two scales.
Before we proceed, it is instructive to draw a comparison
with immiscible, two-phase displacements in porous media,
also a well-studied topic (e.g. see Lake, 1989). Here, the
local “force” is capillarity, which controls Fuid–Fuid interfaces at the pore scale, while the non-local “forces” are viscous. The ratio between viscous and capillary forces at the
microscale is expressed through the capillary number
ql l
q
Ca = 2
=
;
(6)
l
where q is a typical Fow velocity, denotes viscosity and is the interfacial tension. Capillarity dominates at the small
scale and conventional upscaling is valid if
Ca1:
(7)
This is the immiscible Fow counterpart of (4). As in the corresponding reaction–di6usion example, a macroscopic capillary number can also be de#ned (Dullien, 1979)
L
L
;
(8)
∼ Ca
CaL = Ca √
l
k
where k ∼ l2 is the permeability of the porous medium, thus
providing the analog of the macroscopic Thiele modulus (5).
Porescale models, as well as experimental results from
two-phase displacements, indicate that viscous e6ects can
2567
be neglected at the small scale provided that Ca is small, e.g.
Ca ¡ O(10−4 −10−5 ). A manifestation of this is the amount
of the capillarity-trapped displaced phase (the “residual oil
saturation”, denoted by Sor ). As shown in Fig. 1, Sor is independent of Ca, suggesting that capillarity is dominant at the
small scale, for values satisfying the above constraint. It is
in this region that the conventional use of rate-independent
Fow and capillary parameters (and the associated upscaled
description in terms of relative permeabilities, e.g. see Lake,
1989) is valid. Increasing Ca leads to a decreasing Sor , indicating that viscous forces become progressively stronger
at the microscale to counteract the trapping capillary forces.
In this domain, rigorous upscaling methods are yet to be
developed.
For reaction–di6usion systems, an analogous quantity is
the e6ectiveness factor, , which is the ratio of the reaction
rate including di6usion to the rate if di6usion were in#nitely
fast. Actually, the analogy between the two quantities (Sor
and ) is not exact, in that in immiscible displacement increasing the non-local forces improves the process e7ciency
(at least for the case of favorable mobility), while the opposite is true for reaction–di6usion. However, they both measure the relative importance of “local” to “non-local” forces,
and more precisely the small-scale gradient of pressure or
concentration, respectively. Typical experimental results for
the variation of with the macroscale Thiele modulus are
shown in Fig. 2 (note that, compared to Fig. 1, the plot is
with respect to the macroscale parameter). For the same reasons as in the immiscible displacement problem, we expect
that conventional upscaling is valid only in the region where
is constant, which from Fig. 2 is the region ¡ 0:2. Taking a typical ratio of characteristic scales l=L ∼ O(10−2 ) Eq.
(5) provides the constraint l ¡ O(10−2 ), which is similar
(although not identical) to the constraint on the capillary
number (note, however, the logarithmic dependence on the
capillary number or the Thiele modulus in both #gures). For
larger values of , the validity of the conventional description should be questioned, although such a description is
routinely practiced. The analogy between reaction–di6usion
and immiscible Fow in porous media was exploited in earlier
work by Yortsos and Sharma (1986) and was #rst pointed
out in Yortsos (1990).
The problem of mass transfer by advection and di6usion
in the presence of sources (or sinks), embedded in a porous
medium, is analogous (where the Thiele modulus can also
be replaced by a DamkQohler number). In addition, if the interface kinetics are fast, the local Thiele modulus (or the
local DamkQohler number) is in#nitely large. By necessity,
therefore, the condition for homogenization in this case is
that the average concentration must be close to its thermodynamic equilibrium value. Enforcing this condition requires
knowledge of the actual value of the macroscopic average,
the evolution of which is unknown, however. As a result,
the two scales cannot be separated a priori and determining e6ective parameters requires the solution of a coupled
problem.
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P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
Fig. 1. The variation of the residual (trapped) saturation, Sor , in two-phase displacements in porous media vs. the capillary number of the displacement.
The value is normalized with its limiting value at low Ca (reprinted from Lake, 1989). Below a value of the order of 10−5 ; Sor is independent of Ca,
indicating that “local” equilibrium applies. It is in this region, where classical upscaling approaches are valid.
Fig. 2. The e6ectiveness factor, , vs. the macroscale Thiele modulus, , corresponding to a #rst-order reaction. The dots represent values for pellets of
various shapes (reprinted from Lapidus & Amundson, 1977). Conventional upscaling is rigorous valid in the region where is constant.
In the sections to follow, we will show how one can modify the volume-averaging methodology to derive rigorously
valid closure problems, even for #nite or fast kinetics, in the
limit where scale separation exists. The new unit-cell problem di6ers from that of QW, in that it involves macroscale
gradients of the upscaled variable and will be shown to
lead to an eigenvalue problem coupling both scales. For
an illustration, we will proceed #rst, with a simple 2-D
example.
3. An illustrative example
Consider a structureless, two-dimensional “porous”
medium, consisting of the strip (0; ∞) and (0; l) in the coordinate directions X and Y , respectively (Fig. 3). In this
example, the microstructure consists of the line 0 ¡ Y ¡ l,
hence volume averaging means integrating over Y in (0; l).
A Fuid phase is assumed to Fow at a constant velocity
vX = U . The solid interface is impermeable and located at
P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
equation
Y
U
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solid impermeable phase, σ
Y=l
aqueous
phase, β
dissolving interface, γ
Y=0
X=0
X
→
∞
X
Fig. 3. A simple, two-dimensional, structureless “porous” medium for
two illustrative examples. In advection–reaction, the injected Fuid reacts
with the interface at Y = 0, which acts as a sink with #nite kinetics. In
the advection–dissolution example, the interface is a dissolving interface.
Y = l. For further simplicity, we consider only transverse
di6usion and neglect longitudinal dispersion. In appropriate
dimensionless notation, where all lengths are scaled with l,
the steady-state advection–di6usion equation reads
@c
1 @2 c
;
=
@x Pe @y2
(9)
ke "
@c
=−
c;
(15)
@x
U
where "(!) = c|y=0 =c. Given the analytical expression for
c, one can determine the e6ective kinetic coe7cient, ke6 ,
hence the e6ective DamkQohler number, Da(!) ≡ ke6 =U =
ke "(!)=U . We are speci#cally
interested
on the e6ective
√
Thiele modulus = ke6 l=D = Da Pe. We #nd
∞ −a2 ! 4
e n ( + a2n )=((4l + a2n ) + 2l )sin2 an
2
= ∞n=1−a2 ! 4 l
2
n ( + a2 )=((4 + a2 ) + 2 ) sin a =a2
n n
n
n
n=1 e
l
l
l
(16)
the solution of which is plotted in Fig. 4 as a function of !.
Clearly, ; ke6 and Da are non-local. However, at large !,
and at a rate which depends on Pe, an equilibrium state is
reached, where
→ ∞ = a1
(17)
and a1 is the smallest positive root of Eq. (14). The latter
depends on l , thus
∞ tan ∞ = 2l :
(18)
where we introduced the Peclet number, Pe=Ul=D. We will
discuss two di6erent problems, one in which the surface at
Y = 0 is a sink reacting with the injected Fuid according to
a #rst-order reaction, and another in which it is a dissolving
interface. In both cases, the injected Fuid approaches an
equilibrium concentration su7ciently far downstream.
It is only near this limit, where upscaling can be obtained
using conventional volume averaging. As shown below, this
involves coupling of both micro and macroscales. Fig. 5
shows the variation of ∞ with l . At small l , we have the
linear relationship ∞ = l , as expected from the conventional theory. In this limit Eq. (18) gives ∞ ≈ l , leading
to
3.1. Advection-reaction with 6nite kinetics
ke6 = ke :
In this problem, a reacting Fuid with concentration C0 is
injected at x = 0. The boundary conditions are
c = 1 at x = 0;
(10)
@c
=0
@y
(11)
at y = 1
and
@c
= 2l c
@y
at y = 0;
(12)
where c is normalized with respect to C0 , and we recognize
the small-scale Thiele modulus 2l = ke l=D. Straightforward
calculations show that the solution of Eq. (9) for this problem is
∞
4
2
2 ( + a )cos(an (1 − y)) sin an
n
c=2
e−an ! l
;
(13)
an
(4l + a2n ) + 2l
n=1
where ! = x=Pe and an is the nth positive root of the transcendental equation
a tan a = 2l :
(14)
The volume-averaged macroscopic equation is derived by
integration along the y-direction. We obtain the large-scale
(19)
This is the same with the result obtained from conventional
upscaling. As l increases, the e6ective Thiele modulus
grows less than linearly with l , and eventually saturates at
#=2, as l → ∞. The e6ective kinetic constant eventually
becomes independent of the local value.
In a section to follow, we will show that these predictions
can also be reached by solving a corresponding eigenvalue
problem resulting from upscaling. It is worthwhile to note,
that a behavior qualitatively similar to Fig. 5, is obtained if
one were to follow the simpler approach (e.g. Lapidus &
Amundson, 1977), where mass transport to the reacting surface is through steady-state di6usion across a boundary layer
of thickness l. By equating di6usive and reaction Fuxes, one
#nds
2l
ke6 l
2 ≡
(20)
=
D
1 + 2l
that relates the e6ective macroscopic parameter to the microscale Thiele modulus. Equation (20) has a behavior qualitatively similar to ∞ , namely a linear scaling = l , at
small l , and an approach to an asymptotic value (here equal
to 1) at large l . However, both the variation with l and
the asymptotic value are di6erent (see Fig. 5), suggesting
that a careful approach is necessary in the general upscaling
of such problems.
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P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
1
0.95
2
φ = 1
l
0.9
φ
0.85
0.8
φ2= 0.7
l
0.75
0.7
φ2= 0.5
l
0.65
0
0.1
0.2
0.3
0.4
0.5
0.6
ξ
Fig. 4. The variation of the e6ective Thiele modulus with distance and for di6erent values of the local Thiele modulus, for the advection–reaction
problem of Fig. 3. The (non-local) e6ective Thiele modulus eventually saturates to a l -dependent value.
3.2. Advection–dissolution
1.6
Analogous results apply for the case of dissolution. In
the schematic of Fig. 3, an injected (solvent) Fuid leads
to the slow dissolution of an interface at Y = 0, where the
equilibrium dimensionless concentration is 1. The solution
of this problem is readily obtained from the previous by
making the substitution c → 1−c and taking the limit l →
∞. We read
1.4
∞
c(y; !) = 1 − 2
e
n=0
−a2n !
an
sin(an y);
(21)
φ∞
1
0.8
0.6
0.4
0.2
where an = (2n + 1)#=2. The corresponding transverselyaveraged macroscopic equation is similar to the previous
@c "l
= (1 − c)
@x
U
1.2
(22)
the e6ective mass transfer coe7cient of which can be evaluated using the analytical solution. We #nd
∞ −a2 !
e n
"l2
"D (!) ≡
(23)
= ∞n=0 −a2 ! 2 :
n =a
D
n
n=0 e
The result is plotted in Fig. 6. As before, the e6ective mass
transfer coe7cient is in general non-local, depending on the
distance from the point of injection, !. At large !, which, depending on Pe can be equal to several microstructure lengths,
0
0
5
φl
10
15
Fig. 5. The limiting value of the e6ective Thiele modulus, ∞ , as a
function of the local Thiele modulus, l . Note the linear variation at small
l and the approach to the constant #=2 at large l . For comparison,
plotted also is the prediction from (20) denoted by a dashed line.
"D approaches an asymptotic value, however,
#2
(24)
"D → "D; ∞ = :
4
Again, upscaling to a constant mass-transfer coe7cient is
possible but only when the concentration approaches its
P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
2571
10
9
8
αD(ξ)
7
6
5
αD,∞=π2/4
4
3
2
0
0.1
0.2
0.3
0.4
0.5
0.6
ξ
Fig. 6. The variation of the dimensionless e6ective mass transfer coe7cient with distance, for the geometry of Fig. 3 and the advection–dissolution problem. The (non-local) coe7cient eventually saturates
to an asymptotic value.
equilibrium value. Even then, the result is obtained by solving a problem coupling both scales, as will be demonstrated
below. By contrast, by following the conventional closure
problem, the e6ective mass transfer coe7cient is di6erent
and found to be equal to "D; ∞ = 3 (see below), a value
which is larger than the actual by about 25%.
In the sections to follow, we will present a general approach for the upscaling of such problems. For convenience
in presentation, we will describe #rst the problem of dissolution (with in#nitely fast kinetics) followed by the problem
involving reaction with #nite kinetics.
4. Dissolution
equal to its equilibrium value
C$ = Ceq
on A$ ;
(26)
where A$ is the interface between trapped and Fowing
phases. The other boundary condition is zero Fux at the interface between aqueous and solid phases
n$% · ∇C$ = 0
on A$% ;
(27)
where n$% is the unit normal vector directed from the $-phase
toward the %-phase.
De#ne, next, intrinsic and super#cial volume averages
1
1
C$ $ =
C$ dV and C$ =
C$ dV;
V$ V $
V V$
(28)
where V$ is the $-phase averaging volume and V is the averaging volume. Standard upscaling proceeds by decomposing the point concentration into its intrinsic average, which
is the macroscale quantity of interest, and a Fuctuation
C$ = C$ $ + C$
We brieFy summarize the problem of dissolution of a stationary phase in porous media using the volume-averaging
approach. We follow closely the procedure outlined in QW.
Denote by subscript $ a Fowing aqueous phase, by the
trapped hydrocarbon (NAPL) phase, and by % the solid
porous matrix (Fig. 7). Under the assumption of slow dissolution rates, phases and % are of #xed and stationary geometry. The pore-scale boundary value problem is described
by an advection–di6usion equation
@C$
+ ∇ · (vC$ ) = D∇2 C$ ;
@t
Fig. 7. Schematic of the porous medium containing a stationary dissolving
phase (NAPL). The aqueuous (or solvent) phase is mobile.
(25)
where C$ is the concentration of the dissolving species, t is
time, v is the Fuid velocity in the $-phase, and D is molecular
di6usivity, considered constant. The solute is assumed at a
su7ciently small concentration to not a6ect properties, such
as density and viscosity.
Under the condition of thermodynamic equilibrium at the
source interface, the solute concentration is constant and
(29)
and likewise
v = v$ + v :
(30)
Following QW, and based on assumptions of isotropy, uniform porosity and constant volume fractions, the governing
equations for the average and the spatial deviation concentration eventually become as follows:
$
@C$ $
+ v · ∇C$ $ + ∇ · v C$ @t
D
2
$
∇·
=$ D∇ C$ +
n$ C$ dA
V
A$
+
A$%
+
A$%
n$% C$ dA
+
n$% · ∇C$ dA
A$
n$ · ∇C$ dA
(31)
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P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
and
$
@C$
@t
Introducing Eq. (37) into Eq. (32) and noting that the transient term is small, we then obtain to #rst order the following
problem for the closure variable s$ :
∇C$ $
· ∇s$
v + 2D
(Ceq − C$ $ )
∇2 C$ $
∇C$ $
−D
s$
− v·
(Ceq − C$ $ )
(Ceq − C$ $ )
+ v · ∇C$ + $ v · ∇C$ $ − ∇ · v C$ =$ D∇2 C$ −
+
A$%
+
A$%
D
∇·
V
n$% C$ dA
A$
+
A$
n$ C$ dA
n$ · ∇C$ dA
n$% · ∇C$ dA ;
=D∇2 s$ − $−1 " − v ·
(32)
where $ is the volume fraction of the $-phase. The respective boundary conditions for the Fuctuation become
C$ = Ceq − C$ $
on A$
(33)
and
n$% · ∇C$ = −n$% · ∇C$ $
on A$% :
(34)
Up to this point, the analysis is identical to QW. Now, for
the solution of Eqs. (32) – (34), QW take the decomposition
C$ = bf · ∇C$ $ + s$ (Ceq − C$ $ )
(35)
with the understanding that the macroscopic concentration
and its gradient are linearly independent. Implicit in this
and their subsequent derivation, which allows to simplify
the equations considerably, is the important assumption that
the intrinsic average varies little over the scale of the microstructure. In other words, that constraint (2) is applicable. As pointed out above, in the particular problem, this is
tantamount to assuming C$ $ → Ceq . In this limit, however, the macroscopic concentration and its gradient are not
independent. Furthermore, higher-order gradients, such as
|∇2 C$ $ |, are not negligible compared to Ceq − C$ $ , an
assumption that allows QW to decouple bf from s$ in Eq.
(35). To demonstrate this point consider the averaged equation in this limit and at steady state. Then, for su7ciently
large distances, C$ $ will approach at an exponential rate
its equilibrium value, Ceq , e.g. as
C$ $ = Ceq − A exp(−ax);
(36)
where x is the coordinate aligned with the main Fow direction. Here, A is a positive constant, and a ¿ 0 is the non-zero
positive root of a quadratic, involving among other parameters, the e6ective mass-transfer coe7cient ", the determination of which is sought in the #rst place. Eq. (36) shows
that in the limit where homogenization is expected to apply,
the ratios |∇C$ $ |=(Ceq −C$ $ ), |∇2 C$ $ |=(Ceq −C$ $ ),
etc. are non-negligible constants, indicating that the variables are not independent.
In view of this observation, we will proceed with the
di6erent substitution
C$ = s$ (Ceq − C$ $ ):
(37)
∇C$ $
;
(Ceq − C$ $ )
(38)
where parameter " is given by the following expression:
D
"=
n$ · ∇s$ dA + ∇ ·
n$% s$ dA
V A$
A$%
∇C$ $
−
+
(Ceq − C$ $ )
A$%
n$% s$ dA :
(39)
Note the term for the interfacial transport between the dissolving ()-phase and the Fowing aqueous ($)-phase. In deriving the above expressions we assumed constant volume
fractions, isotropy and took account of the following boundary conditions:
s$ = 1
on A$ ;
n$% · ∇s$ + n$% ·
(40)
∇C$ $
(1 − s$ ) = 0
Ceq − C$ $
on A$% : (41)
These are supplemented with periodicity and self-consistency
conditions
s$ $ = 0:
(42)
Both the boundary value problem (38) and boundary condition (41) di6er from those presented in QW. In their corresponding closure problem for s$ , QW neglect all terms associated with the gradients of C$ $ , as being small within
the microstructure. For this to be true, however, requires the
condition
2
$ ∇C$ $ 1 and ∇ C$ 1:
(43)
Ceq − C$ $ Ceq − C$ $ A comparison with Eq. (36) shows that Eq. (43) does not in
general apply, when macroscopic equilibrium is approached,
even though this is the very limit where volume averaging
in the conventional sense is to be valid.
In this limit, the steady-state averaged equation is expressed in terms of the following expression:
v · ∇C$ $ = $ D∇2 C$ $ − ∇ · (s$ v (Ceq − C$ $ ))
+ "(Ceq − C$ $ ):
(44)
With the exception of the second term on the RHS, the other
three terms are conventional. The term containing sv is the
P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
average product of velocity and concentration Fuctuations
and represents hydrodynamic dispersion, although here it
has the unconventional form shown. In fact, this term resembles, but is not the same with, the velocity-like coe7cient d$
of QW. The equation can be expressed, if necessary, in terms
of a conventional advection–dispersion–reaction equation
v · ∇C$ $ = ∇ · (D$∗ ∇C$ $ ) + "e6 (Ceq − C$ $ );
(45)
D$∗
where
is the e6ective macroscopic dispersion coe7cient
for a passive tracer, which can be obtained independently
(for example using QW). In this formulation, however, the
e6ective reaction rate coe7cient "e6 is di6erent
"e6 = " − ∇ · s$ v +
$ D∇2 C$ $ − ∇ · (D$∗ ∇C$ $ ) + s$ v · ∇C$ $
Ceq − C$ $
:
(46)
In the particular case of the absence of a Fow #eld, where
the dispersion tensor is a scalar, we #nd, from this approach
D$∗ "
"e6 =
:
(47)
$ D
In general, it is apparent that the closure problem (38) –
(42) couples local and global scales, as the evaluation of
" requires the simultaneous solution of the global problem.
In general, the closure problem (38) – (45) is an eigenvalue
problem for ". In the absence of the new terms identi#ed here
the problem can be directly solved using the transformation
s$ = 1 + "
(48)
as shown by QW. Here,
satis#es the steady-state
advection–di6usion equation with a constant source term.
The solution for " is simply " = −1= $ . Local and global
problems are uncoupled in this approach. However, this is
not the case when the gradient terms in Eq. (38) are retained, which result in the coupling between the two scales.
To illustrate this consider the limit v = 0 (for example, as in
a drying application) and when the closure variable s$ does
not vary in the direction of the gradient of the concentration
(namely when macroscopic equilibrium is approached).
Using Eqs. (38), (44) and (48) leads now to the eigenvalue
problem
D∇2 = −$−1 "
(49)
with the homogeneous boundary conditions
(50)
= 0 on A$
and
n$% · ∇ − n$% ·
∇C$ $
Ceq − C$ $
= 0 on A$% :
(51)
Note also the implicit dependence on " in the second term in
Eq. (51). The solution of this eigenvalue problem generally
gives a discrete spectrum, the smallest eigenvalue of which
is the desired value. Speci#c examples will be shown below.
2573
We are, thus, led to the conclusion that in the very
limit where it is expected to be valid, the volume-average
methodology for processes with fast reactions gives rise
to additional terms, such as ∇C$ $ =(Ceq − C$ $ ) and
∇2 C$ $ =Ceq − C$ $ , which cannot be a priori discarded,
thus coupling micro and macroscales.
5. Examples
In this section, we will consider two illustrative examples, one involving the simple geometry described before,
where advection, di6usion and dissolution occur, and another involving drying in a pore network, where transport is
by di6usion alone.
5.1. Advection–dissolution
Consider the advection-dissolution problem described
previously (Fig. 3). Here, we will derive the exact result
(24) from the solution of the new closure problem (38)
– (42). In the previous dimensionless notation, Eq. (38)
reads as follows, in the limit !1, which is the limit that
must be taken in the volume-averaging approach, for the
microstructure variable s to be independent of !,
−
@c=@!
@2 s
s = 2 − "D :
(1 − c)
@y
(52)
Here we dropped subscript $ for simplicity and taken $ =1.
The dimensionless mass transfer coe7cient is obtained from
Eq. (39)
@s "D = − :
(53)
@y y=0
The corresponding boundary conditions become
s=1
at y = 0;
@s
=0
@y
(54)
at y = 1:
(55)
Taking into account the macroscale relation @c=@! =
−(1 − c)"D , and using the substitution s = 1 + "D , we
obtain the eigenvalue problem
= −"D
(56)
with boundary conditions
= 0 at y = 0
and
= 0 at y = 1;
(57)
n = 0; ±1; ±2; : : : :
(58)
which has the eigenvalues
"D(n) =
(2n + 1)2 #2
4
The smallest of these (n = 0) gives the exact previous result
for the dimensionless mass transfer coe7cient, "D0 = "D; ∞ =
#2 =4. The expression for s is obtained from Eq. (53), or the
2574
P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
self-consistency condition, and reads
# #y :
s = 1 − sin
2
2
(59)
A simple calculation shows that this solution coincides with
the exact result, as indeed expected. By contrast, in the corresponding closure problem of QW, the left-hand side of
Eq. (52) is neglected. The substitution s = 1 + "D gives
the inhomogeneous problem
=1
(60)
the solution of which subject to the previous boundary conditions is
y2
=
− y:
(61)
2
Use of the self-consistency condition, then, yields "D; ∞ = 3,
and
3y2
(62)
s = 1 − 3y +
2
both of which are di6erent from the actual.
5.2. Drying in a pore network
A second example to illustrate the above is the drying of a liquid in a pore network. In recent years, several
pore-network studies of drying have been conducted, where
the e6ect of various factors has been addressed (e.g. Laurindo & Prat, 1998; Yiotis, Stubos, Boudouvis, & Yortsos,
2000). In the present application, we are interested in
computing the e6ective drying rate, which is a variable of
interest to e6ective continuum models (Tsimpanogiannis et
al., 1999).
For the purposes of this example, we will simplify
the physical description by considering the following,
often-used, pore-network approximation. We take the
porous medium to be represented by a regular network
of bonds (pore throats) and sites (pore bodies). A liquid
phase is randomly distributed in the sites of the pore network. All volume is assigned to sites, hence the liquid
undergoing evaporation resides on sites only, which it fully
occupies, for the sake of this example. Transport of vapor
occurs by di6usion in the gas phase, across adjacent sites
connected by bonds. For convenience, all bonds are taken
with the same transport properties. E6ects of convection,
heat transfer, countercurrent di6usion, etc., are neglected.
As is conventional in these types of problems, variables are
de#ned in the sites of the lattice, over which the governing
equations are also discretized. Due to the limited accessibility of the network, percolation phenomena are possible,
and for the gas phase to be macroscopically connected, the
fraction of sites occupied by the liquid must be below the
value 1 − pc , where pc is the site-percolation threshold.
Because of the assumed thermodynamic equilibrium at
the liquid interface, the drying problem belongs to the same
class of in#nitely fast interfacial kinetics, described above.
Hence, a volume-averaged description would generally be
non-local, except in places where the macroscopic vapor
concentration approaches its equilibrium value. In dimensionless notation, the corresponding macroscale problem at
steady state reads from Eq. (44) as
d 2 c
= "D (c − 1);
(63)
d x2
where the dimensionless mass transfer coe7cient is
"l2
"D =
:
(64)
$ D
Here, it is understood that the porosity $ corresponds to
the accessible porosity of the gas phase (excluding trapped
regions) and all lengths have been scaled with the microscale
l. The solution of Eq. (63) at large x has an exponential
√
behavior similar to Eq. (36), with a = "D . Substituting in
Eq. (38) leads to
− "D s$ = ∇2D s$ − "D ;
(65)
where we have assumed that near the equilibrium limit, s$
is independent of x. For the solution of this problem, we use
again the transformation s$ = 1 + "D , to obtain
∇2D = −"D :
(66)
Solving on the pore network requires discretizing over the
lattice sites. In this context, therefore, only the boundary condition at the liquid–gas interface, Eq. (40), will be utilized.
Thus, the eigenvalue problem is given by Eq. (66), valid
over the gas-occupied sites, accompanied with the boundary
condition =0 in liquid-occupied sites and periodicity at the
boundaries of the lattice. This is a novel eigenvalue problem
for the Laplace operator, as the region involves boundaries
randomly distributed. It has some similarities with the classical problem of Anderson localization in statistical physics
(Elsner, Mehrmann, Milde Romer, & Schreiber, 1999), but
it is not the same. The problem was solved over a square
lattice, which represents a cross-section of the pore-network
in a direction perpendicular to x. The numerical procedure
used is somewhat elaborate and will not be described here.
Numerical results for a lattice of dimensions 40 × 40 are
shown in Fig. 8, where we plot the smallest eigenvalue as a
function of the liquid saturation Sl (fraction of sites occupied
by the liquid), for di6erent realizations of the microstructure.
The small lattice size used reFects the computational cost in
#nding the eigenvalues in this random lattice problem, and
it is itself reFected in the scatter shown in the #gure. We expect a decrease of the scatter and the approach to a limiting
curve as the lattice size increases. Plotted also are the results
corresponding to the problem for s$ as derived by QW, for
the same liquid con#gurations. These results have smaller
scatter, and also much smaller computational cost. The approach of the QW results to a limiting curve as the lattice
size increases was veri#ed in simulations using a 400 × 400
lattice. Both sets of data show that "D approaches zero at
Sl = 0 and that it increases as Sl increases. In view of Eq.
(64), however, the actual dimensional coe7cient " eventually decreases, as Sl increases, since the accessible fraction
P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
2575
1
0.9
0.8
0.7
α
D
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Sl
Fig. 8. The variation of the dimensionless e6ective mass transfer coe7cient as a function of the liquid saturation Sl , for a model problem of drying in
a pore-network. Values of "D obtained from the present theory are denoted by dots, those from QW by open circles. The scatter reFects the di6erent
realizations taken and the small size of the network (40 × 40).
of the gas phase $ decreases with Sl and vanishes at the
percolation threshold. Both these results are to be expected.
The di6erence between the two solutions is apparent, and
suggests that the coupling between the two scales must be
considered in the upscaling of such problems. We must remark, however, that Fig. 8 does not necessarily reFect the
exact di6erence between the present theory and that of QW.
For this purpose, it is necessary to evaluate "e6 which enters
in Eq. (45). Although rather straightforward for our problem, which requires only the evaluation of the e6ective diffusion coe7cient, this is a bit more elaborate for the QW
formulation, for which the two additional vectors u and d,
entering in their theory, must be computed. This was not
attempted here.
6. Advection-reaction with #nite kinetics
In the previous section we investigated the application
of volume averaging to the case of advection–reaction
processes with fast kinetics, where the reactant (solute)
concentration at the source remains constant and equal to
its thermodynamic equilibrium value. We showed the following results: that in general a non-local description is
necessary, that volume averaging in the conventional sense
is possible only when the macroscopic variable approaches
an equilibrium state, but that even under such conditions,
the microscale problem to be solved is coupled with the
macroscale. In the section to follow, we will extend these
concepts to the case where the kinetics at the interface are
not necessarily fast, but the macroscopic concentration does
approach an equilibrium state.
One such problem is steady-state advection with a #rst
order, heterogeneous reaction at the pore surface, which
we will take to consume the injected species. The homogenization of a related problem but with transient di6usion
(instead of advection) and reaction has been discussed in
detail in Whitaker (1999). Under the two conditions of negligible transients and small local Thiele modulus (which is
the earlier postulated condition (4)), the e6ective, macroscopic equation is the standard reaction–di6usion equation,
where the e6ective kinetic parameter is identical to the
local. As a result, for obtaining the e6ective kinetic coe7cient, there is no need to solve a boundary-value problem at
the microscale. Now, consider the corresponding problem
in which an equilibrium state is reached, for example, in the
case when an injected reactant is ultimately fully consumed,
under #nite kinetics. Under these conditions, and for the
same reasons as in the above, the gradients of the macroscale
concentration are not necessarily small compared to the
concentration, hence the conventional approach does not
apply.
2576
P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
To proceed, we follow the previous approach, except that
phase is now absent, and derive equations for the average
and the Fuctuation. The analysis parallels closely Whitaker
(1999), thus details will be omitted. The closure problem is
obtained by taking
C$ = s$ C$ $ ;
(67)
which again di6ers from the conventional (e.g. see Whitaker,
1999). This substitution leads, to #rst order, to the following
closure problem for s$ :
∇C$ $
∇2 C$ $
∇C$ $
· ∇s$ + v ·
s$
−D
v − 2D
C$ $
C$ $
C$ $
2
= D∇ s$ +
$−1 "
∇C$ $
−v ·
:
C$ $
(68)
Here, we have de#ned
∇C$ $
D
∇·
n$% s$ dA + 2
·
n$% s$ dA
"=−
V
C$ $
A$%
A$%
+
A$%
n$% · ∇s$ dA :
(69)
Again, the problem is to be solved subject to periodicity,
self-consistency and the boundary condition
∇C$ $
ke
(1 + s$ ) = 0 on A$% :
+
n$% · ∇s$ + n$% ·
C$ $
D
(70)
The corresponding e6ective equation, under conditions of
steady-state advection and reaction reads
v · ∇C$ $
=$ D∇2 C$ $ − ∇ · (s$ v C$ $ ) − "C$ $ :
(71)
Given that the solution approaches an equilibrium state, we
have a similar exponential decay as previously, C$ $ ∼
A exp(−ax). Then, the ratios involving the gradient terms in
Eq. (68) cannot be neglected, unless a1, which however,
coincides with the condition (4) of small Thiele modulus.
Hence, in the general problem, even under the condition of
slow variation of the macroscopic concentration, the e6ective kinetic parameter must be determined from a problem
coupling local and global problems, as above. The solution
to the latter is demonstrated below using the previous simple example.
6.1. Advection over a reacting interface
Consider the same geometry as in the dissolution problem (Fig. 3). We will derive expression (17) by solving
the eigenvalue problem (68) instead. With the same assumptions as in the dissolution problem, variable s$ satis#es
the equation
@2 s
@ lnc
s = 2 + "D ;
@!
@y
(72)
where we dropped subscript $ for simplicity. The dimensionless mass transfer coe7cient "D is also equal to 2 , obtained from the macroscale equation, @c=@!=−2 c. The
corresponding boundary conditions are
@s
(73)
−
+ 2l (1 + s) = 0 at y = 0;
@y
@s
= 0 at y = 1:
(74)
@y
Taking into account the macroscale relation and using the
substitution s=−1+"D , we obtain the eigenvalue problem
= −"D
(75)
with boundary conditions
= 2l
at y = 0
and
= 0 at y = 1:
The eigenvalues of this problem are
√
√
"D tan "D = 2l :
(76)
(77)
The smallest of which gives the exact previous result, as
expected.
7. Concluding remarks
In this paper, we showed that for advection–reaction systems with fast kinetics (at the limit of thermodynamic equilibrium), as, for example, in dissolution-type processes, conventional volume averaging does not apply, except in the
limit where the macroscopic variable approaches its equilibrium value. While this is not unexpected, and in fact,
it is implied in various previous works, the limit where
the macroscale variable approaches thermodynamic equilibrium can still be homogenized by volume averaging. What
is interesting is that the solution for the e6ective parameters in this case results from a closure problem, where the
microscale and the macroscale are coupled. In this limit,
the coupling is only algebraic and results into a generally
non-linear eigenvalue problem for the e6ective kinetic parameter. This result is di6erent from the conventional, as for
example described in QW. Two examples, one involving a
simple advection–dissolution problem and another a drying
problem in a pore network, verify the theoretical predictions.
We note that with simple modi#cations, the above also apply directly to relevant heat transfer problems as well.
We also showed that systems with #nite kinetics can also
be homogenized in the limit where an equilibrium state is
approached, even though the local Thiele modulus is not
small. The analysis of a simple example presented reveals,
in fact, that the e6ective kinetic parameter is not the same
as the kinetic constant for the heterogeneous reaction, but it
is a function of the local Thiele modulus itself, eventually
P. E. Kechagia et al. / Chemical Engineering Science 57 (2002) 2565–2577
saturating at large values of the latter. Similar results are expected for more complex microstructures. The limiting behavior mimics that of a simple advection–reaction problem,
where di6usion is across a constant boundary layer. While
qualitatively similar, however, the precise dependence requires the solution of an eigenvalue problem, which was
derived above.
Given that the process under consideration is very near
equilibrium, the utility of the above results can be questioned. In response we claim that these do yield the correct results for the case when homogenization is expected to
apply. For tackling the more general problem, where the
process is far from equilibrium and the homogenization constraints do not apply, di6erent, non-local approaches are
needed. Some have already appeared in the literature involving integro-di6erential equations, moment expansions
and=or non-local parameters. An alternative approach of
some promise is to use transverse averaging across the main
direction where the macroscale variable changes. This alternative is currently under consideration.
Acknowledgements
This work was partly supported by Los Alamos National
Laboratory in the form of a student scholarship to the #rst
author. The research of the second and third authors was
partly supported by DOE Contract DE-AC2699BC15211.
Both contributions are gratefully acknowledged.
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