Facilitated transport in porous media

Chemical Engineenng Science, Punted m Great Britain

Vol. 46, No. 2, pp 477496,

1991.

FACILITATED J. ALBERT0

TRANSPORT

OCHOA-TAPIA,

Department

of Chemical

Doo9-zm9/91 53.00 + 0.00 0 1990 Pergamon Press plc

PIETER

Engineering,

(First receiued 31 May

IN POROUS

STROEVE

University

1988; accepted

and STEPHEN

of California, in revisedform

MEDIA WHITAKER’

Davis, CA 95616, U.S.A. 19 January

1990)

Abstract-The process of Lcilitated transport in porous media has been studied theoretically using the method of volume averaging, and experimentally in terms of the facilitated transport of carbon dioxide in an aqueous solution. For facilitated transport, the reaction rate leads to the appearance of interact& d~j$siwj?uxes in the spatially smoothed transport equations. It also requires that the closure problem be solved simultaneously with the macroscopic boundary value problem. This is a general characteristic of nonlinear, multiphase transport processes, and this work appears to be the first example in which a rigorous solution to the closure problem is given. For the system studied experimentally, the influence of the nonlinear reaction rate on the process of spatial smoothing was not particularly important and the interactive diffusive fluxes were small. Theoretical values of the several effective dNusivities necessary to describe our experimental system were determined; thus the comparison between theory and experiment was made in the absence of adjustable parameters. Good agreement between theory and experiment was obtained.

1. INTRODUCTION

The enhancement of mass transfer due to carrierfacilitated transport has been known for six decades (Hill, 1928), and Roughton (1932) speculated that the diffusion rate of oxygen in blood was enhanced by the presence of hemoglobin. The facilitated transport of dissolved gases has been studied extensively in biological systems (Kreuzer and Hoofd, 1987), and chemical engineers have explored the use of facilitated transport in separation processes (Schultz et al., 1974). A number of methods have been developed to solve the boundary value problems that describe this diffusion-reaction prgcess, and examples are available from the works of Goddard (1977), Smith et al. (1977) and Hoofd et al. (1986). Ward and Robb (1967) proposed a facilitated transport system for the separation of gases that was based on the use of a liquid membrane. For practical systems it has been suggested that the reactive liquid membrane be supported by a porous medium, and many feasibility studies of such systems have appeared in the literature (Kimura et al., 1979; Johnson et al., 1987). An example of transport in a porous medium containing a reactive liquid solution is shown in Fig. 1. Supported liquid membranes can also be used for the separation of solutes from liquid phases. The use of carrier facilitation in industrial membrane separation systems is of great interest because of the increased mass transfer rates and the improved selectivity for the permeate over the other species (Kimura et nl., 1979; Hughes et al., 1986; Lonsdale, 1982). Porous polymeric membranes are commonly used to support a liquid in which the carrier can be dissolved (Way et al., 1982), and in a recent review Way et al. (1985) discussed

‘Author

the

characteristics

to whom

correspondence

of an

ideal

should

support for separation processes. Design calculations for mass transfer in separation devices containing reactive, supported liquid membranes have been presented by Stroeve and Kim (1987) and Kim and Stroeve (1988). Most of the theoretical and experimental studies concerning carrier-facilitated transport have been devoted to homogeneous, single-phase systems. In experimental studies where the solution containing the carrier species is supported by a porous material, it is always assumed that the governing equations for facilitated diffusion in porous media have the sameform as the point equations in a homogeneous medium. In addition, it is always assumed that the effective diffusivities measured in passive systems can be used without modification in the analysis of reactive systems. To our knowledge, no fundamental theoretical and/or

Solid

- Liquid

dispersion

membrane

be addressed.

Fig. I. One-dimensional solution

diffusion through a reactive in a porous medium.

liquid

J. ALBERTA

478

OCHOA-TAPIA

experimental studies have been carried out in order to verify these assumptions. In this work we use the method of volume averaging (Carbonell and Whitaker, 1984) to analyze facilitated transport in porous media, and we follow the method of Crapiste et al. (1986) to predict the effective transport coefficients. In this analysis we assume that the solutes and the solvent diffuse only in the fluid phase, and that neither heterogeneous reaction nor adsorption takes place on the surface of the impermeable solid phase. Under these circumstances the diffusion of CO, in the P-phase is described by

3C.A= V.(BAVC,)

-

- k,

at

-k,

C, - &$,, 1 C,C,

and the chemical and physical parameters are well documented in the literature. 2.

THEORY: AVERAGING

In this section we derive the volume-averaged transport equations and we develop the closure problem that allows us to predict the effective transport coefficients for the diffusion of CO, through a reactive, aqueous bicarbonate solution containing impermeable polystyrene beads. The following reactions occur when CO, is present in an aqueous bicarbonate medium (Meldon et al., 1982): reaction 1: CO2 + PI,0 2 HCO;

>

-$CB

et al

+ H+

K2

(l-1)

(Z-la)

+ H’

(2-lb)

reaction 3: CO1 + OH- 2 HCO;

(2-lc)

reaction 4: H,O 2 OH-

(2-ld)

reaction 2: HCO;

+ CO:-

3

while the flux at the J!!-u interface is given by BC

- ‘a< LBAVC, = 0,

at

&,,

(l-2)

where A refers to CO*, B refers to HCO;, C to CO:-, while 0 and H indicate the OH- and H+ ions, respectively. Boundary conditions (BCs) of the form given by eq. (l-2) apply to all molecular species, and under these circumstances one would guess that the local volume-averaged equation (Slattery, 1981) for CO2 would be identical in form to eq. (l-l). However, analysis shows that the volume-averaged form of eq. (l-l) is given by

+

C V.(E~HA,.V(Cj>‘)

j=l

(l-3)

where N refers to the number of chemical species in the system, and the last term represents an interactive diffusive flux. The fact that the interactive diffusive flux is generated by the presence of the inert solid phase illustrated in Fig. 1 and the homogeneous reaction rate terms present in eq. (l-l) is indeed a curiosity that we have investigated in detail in a subsequent study (Ochoa-Tapia et al., 1990b). The effective transport coefficients in eq. (l-3) were determined for a spatially periodic porous medium using numerical methods, and they were determined analytically using Chang’s (1982) unit cell. Since the effective transport coefficients in eq. (l-3) depend on the volume-averaged concentrations, the macroscopic transport problem is solved iteratively along with the closure problem. The theoretical predictions are compared with our experimental data for the facilitation of CO, diffusion in aqueous bicarbonate layers containing impermeable polystyrene beads. This system has been extensively used with single-phase systems

+ H+.

The kinetic rate constants and equilibrium constants for these reactions have been tabulated by Hoofd et al. (1986). Reaction 1 is the combination of two successive reactions given by CO, + H,O + H&O,, H,CO,

it HCO;

slow

+ H+, rapid.

(2-2a) (Z-2b)

Because the second reaction is extremely rapid relative to the first, it can be treated as instantaneous and these two reactions can be replaced by eq. (2-1~~).The reaction represented by eq. (2-ld) is not independent since it can be obtained by a combination of the reactions represented by eqs (2-la) and (2-1~). In order to simplify the representation of the transport equations, such as eq. (l-l), we define four reaction rate parameters by (2-3a)

R, = CA - ;C&,

I R, = C+&

R, = C,C,

R,=C,-&Co.

(2-3b)

2 1 - -C, KS

(2-3~) (2-3d)

4

Once again we note that the chemical species have been identified according to A=CO,,

B=HCO,.

O=OH-,

H=H+,

C=CO;W = H,O

(2-4)

while Ki has been used to identify the equilibrium coefficient for reaction i as indicated in eqs (2-l). Reactions 2 and 4 are ionic recombination reactions which are essentially instantaneous and equi-

Facilitated

transpol ‘t in porous media

librium can be assumed. Because of this, it is convenient to follow the method used by Gallagher et al. (1986) to arrive at the following representation of the boundary value problem for diffusion and homogeneous reaction in a porous medium. Point

transport __

at

f(CB

+ C,)

equations

in the P-phase

= V.(9AVCA) - k, R, - k,R, = V.(B~VCB

(2-5)

+ s@=VC,) + k, R, + k,R,

(2-6)

+ k, R, + k, R,

(2-7)

~(C,-C,-C,)=v.~~“vC,-~~vC, -

Equilibrium

&,VC,)

relations C&,

Boundary

conditions

= ccc,

(2-8)

K, = cue,

(2-9)

Fig. 2. Averaging

at

dfl,,,

i=A,B

,...

W (2-10)

conditions

at the entrances

Here we have used the definition given by

and exits of the



C,=di(r,

t).

at

&,,,

i=A,B,.

.W

(2-11)

i = A, B,.

W.

(2-12)

Initial conditions C, = Ee,(r),

at

volume

at the /?-a interface

- ngo .@VC,=O,

Boundary b-phase

LAveraging Volume

t = 0,

In eq. (2-10) we have used d,, to represent the interfacial area, while in aq. (2.11) ,tp,, has been used to represent the bounding surfaces of the entrances and exits of the continuous phase illustrated in Fig. 1. In writing eqs (2-5)-(2-7) we have assumed that the diffusive transport of the ionic species is described by the Fickian part of the diffusive flux (Gallagher et al., 1986), and in eqs. (2-10) we have indicated that no adsorption or heterogeneous reaction takes place at the surface of the polystyrene beads. In many cases of aqueous solutions of CO,, the pH lies in the range from 5 to 9 and the concentrations of the OH- and H+ ions are negligible compared to the other species. We have included the transport equations for these species in the development of eqs (2-5)-(2-7) in order to present a complete theory; however, we have been willing to limit the theory to the case for which the concentration of the water, Cw. can be treated as constant. I,n order to obtain the local volume-averaged transport equations associated with the system shown in Fig. 1, we make use of the averaging volume shown in Fig. 2 and form the phase average of eq. (2-5) to obtain

=$

of the phase average

I

(2-14) C,dV V# volume illustrated in Fig. 2, of the b-phase contained

where V is the averaging and VP is the volume within -Y-. If we assume that the polystyrene beads are fixed in space, V, is independent of time and we can interchange differentiation and integration in the first term in eq. (2- 13). This leads to y

=
- (k, R,)

-


Interchanging differentiation and integration in the first term on the right-hand side of eq. (2-15) requires the use of the spatial averaging theorem (Anderson and Jackson, 1967; Marie, ‘1967; Slattery, 1967; Whitaker, 1967) and this provides

= V~(~*VC,)

Thus flux BC represented by eq. (2-10) leads to the obvious simplification and allows us to write eq. (2-15) as a = V.

at

-
-


(2-17)

=

- (k,R,)

- (2-13)

The spatial variation of the coefficients such as g* and kl can generally be neglected within the averaging

J. ALBERTO OCHOA-TAPIA et a[.

480

volume (Carbonell and Whitaker, (2-17) can be simplified to a -= dt

V.(S*(VC,))

1984); thus, eq.

- k,(R,)

- k,
A second application leads to

of the averaging theorem now

We avoid these difficulties, at least for the present, by foilowing the type of development encountered in the study of turbulent transport phenomena (Bradshaw et al., 1981). This leads us to the use of Gray’s (1975) decomposition, which can be expressed as ci = s

+ Ei,

i = A, B, C, H and 0.

The average of the product

takes the form

(C&Y

+ G
= <as

+
- k, - b
and at this point we must recognize that it is the intrinsic phase average concentration that represents the preferred dependent variable. This is defined by C,dV B J v.6 and it is related to the phase average concentration

Use of this result, along with similar representation for (R,) and , in eq.(2-19) yields

aB __ ~ = v. % at [

9*

%I

L

(C,C,)B

= p

ns,CAdA

=;

nB,eA dA ‘? s Aga

where we have required that Edbe independent of time since the u-phase is rigid; however, we have not imposed the restriction that the porous medium is homogeneous. We now return to eqs (2-3) and assume that the equilibrium coefficients are essentially constant within the averaging volume, thus allowing us to write eq. (2-22) as

+c,>fl 3 Here we are confronted

1

can

1

=i.J

(2-22)

-

(2-27)

nsc(C,)BdA

1

1

+ <&?‘,)b.

The length-scale constraint given by eq.(2-26) also be used to justify the approximation

E~V
1

is

i-1

(

:,-~cBc”,n

constraint

2

r0

by

(2-21)

= &@B.

length-scale

(2-25)

we can simplify eq. (2-25) to (Carbonell and Whitaker, 1984, Section 2)

(C,>S =:_



and when the following satisfied

+ Z&>B

(2-24)

T+‘. Ap.

n,,dA

s

1

s

(2-28) na,cA dA. 40 One can use the averaging theorem to show that

and when this result, along with eq. (2.27), is used in eq. (2-23) we obtain

@3
,

(2-23)

with two problems:

- E,&

"p

-&I)~ 3

(1) The governing equation for {CA)@ contains the point concentration CA. (2) The governing equation for s contains the average of the product of point concentrations instead of the product of average concentrations.

+

(C,E~>fl . 1

(2-30)

At this point we will make use of information available in the closure problem which is presented in Section 3. There we show that the boundary condition

Facilitated transport in porous media given

by eq. (2-10) provides the only source of the eifields, and this allows us to immediately estimate the magnitude of Ci. The result is that the spatial deviation concentrations are small compared to the average concentrations when I, is small compared to L. We express this idea as i = A, 8, C, H

ci 6 (CiY,

and

0

(2-31)

and it allows us to simplify eq. (2-30) to

aa

Eg-=V.

at

[

Ea9*

!

V
+~J‘,,?&Aj] (CAY -

1

;
(2-32) One can express the IocaI volume-averaged forms of eqs (2-6) and (2-7) in the same form as eq. (2-32) and we will do so in Section 4. To complete the process of volume averaging, we need to develop the averaged forms of the equilibrium relations given by eqs (2-8) and (2-9). The intrinsic phase average form of these algebraic relations can be expressed as

a
provided

the following

=

(2-33)

< (C,>fl(C”>fl.

(2-35)


{c,>a{c,>fl.

(2-36)

These inequalities are based on eqs (2-3 1) which we have not yet proved: however, we will do so in the course of developing the closure problem in the next section. Before going on to the closure problem, one might note that if C, were directly proportional to V@, i.e. (5, - V@

THE

CLOSURE

Bensoussan et al. (1978). Bourgeat et al. (1988) examined the two methods and have concluded

have that

they are essentially identical. To follow the method

of Crapiste

et (II. (1986)

one

uses eq. (2-24) in eq. (2-5) and separates the terms according to the two length scales. This leads to

+ k,

E,6, + (C,>flCo + E,(C,)fl - +

=-

a8 ___

3

at

-

V.@‘,Vs)

+

>

k,(
(2-37)

the firm of eq. (2-32) would be identical to that of eq. (l-l). This means that one could deduce the local volume-averaged transport equation by judiciously replacing point values of the concentration with averaged values and replacing the molecular diffusivity with an effective diffusivity. Whether eq. (2-37) is correct or not is the objective of the next section. 3. THEORY:

The existence of disparate length scales makes the use of asymptotic methods attractive, and this approach has been used by Chang (1982) to study diffusion and heat conduction in porous media. Rubinstein and Mauri (1986) have followed a similar line of analysis to study dispersion in porous media, and their results are identical to those produced by Carbonell and Whitaker (1983) using the method of volume averaging. The asymptotic method is often referred to as the method of spatial homogenization and the mathematical details are available in the monograph by

are satisfied

P

In order to develop

(3-l)

(2-34)

(Gi)B8

constraints

vious studies of Ochoa et al. (1986) and Nozad et al. (1985) which are based on the general method of Crapiste r?t al. (1986). To develop a boundary value problem for CA, we return to eq. (2-5) and make use of the decompositions given by eq. (2-24). It is important to recognize that eq. (2-24) represents a decomposition of length scales; the length scale associated with ( Ci)s being L and the length scale associated with ci being I,,. The macroscopic length scale, L, is illustrated in Fig. 1 while the pore length scale, I,, is shown in Fig. 2. Throughout this development we shall assume that the length scales are constrained by !fl Q L.

j+J6
(c*>~a -

481

PROBLEM

a predictive theory of facilitated transport in porous media, we need to determine the spatial deviation concentration defined by eq. (2-24). To develop the closure problem for c, and the other spatial deviation concentrations, we follow the pre-

When the length-scale constraint given by eq. (2-26) is valid, the right-hand side of this result can be treated as a constant with respect to integration over the averaging volume. This type of simplification is illustrated in eq. (2-28), and it allows us to express eq. (3-2) as (Crapiste et al., 1986)

482

J.ALBERTO OCHOA-TAPIA et al.

~,~,-, + 1

=-

V$ S{v,

!G at

--&CR

(CA)a~o+ ~,
of the nature of the EA-field, the change of eA that occurs over the pore-scale distance, I,, is estimated by 2,. On the basis of the lengh-scale con_straint given by eq. (3-l) and estimates of Cs, C, and Co analogous to that given eq. (3-7), we can simplify eq. (3-5) to

aEA ~-w.%VC*) + k,

&-:_((C&)CH+ 1

[ +k,

~B‘)

(C,)p~o+&
[ + ~A(C,)b-$-~g

dV.

1

=_-

3

V.(g,VcA)dF’.

i=A,B,C,H

and

0

(3-4)

and it allows us to remove averaged concentrations from the integral on the right-hand side of eq. (3-3). This simplification leads to

In our own experimental studies, the macroscopic problem illustrated in Fig. 1 was a steady process; however, there are many practical problems in which the macroscopic mass transfer process described by eq. (2-32) may be unsteady. Even when the macroscopic process is unsteady, the closure problem may be quasi-steady. This occurs when sit

i = A, B,C, H

9 1,

7

and under these conditions V.(5@AVchA)-k, I + &(

C,)‘)

1

--

cB

K3 1 V I( b Yp

v .(gAVCY*) + ;tCC, 1 (3-5)

Returning to the BC given by eq. (l-2), or the generalized version given by eq. (2-lo), we use the decomposition given by eq. (2-24) to obtain BC

- npa . BAVEA = nsB. BA

vay

at

don-

(3-6) At this point it is important to recognize that: (1) all the deviation transport equations and BCs have the same form as eqs (3-5) and (3-6), and (2) the transport equation for CA is homogeneous in the spatial deviation concentrations. This means that the BC given by eq. (3-6) provides th_ only source of the e,-field; thus the magnitude of C, can be estimated in terms of eq. (3-6) according to

EA4[(~)A(c_,q

0

(3-9)

eq. (3-8) reduces to

-k,

1 I =’

(CA)“&

+ 6~(c,)”

[

V . (L%*Vc,) d V.

(3-10)

yav,

It may be intuitively appealing, and it has been demonstrated by Ochoa (1988), that the integral term on the right-hand side :f eq. (3-10) makes a negligible contribution to the CA-field. In addition, it should be clear that eq. (3-10) and the boundary condition given by eq. (3-6) are part of a local problem for ?A that is not influenced by the BCs at the surface of the macroscopic system except for a small zone close to that surface (Pratt, 1989). We are willing to use the model of a spatially periodic porous medium (Brenner, 1980) and to neglect the variation in the diffusivity in our solution of the closure problem. Under these circumstances, we have ProbIem

1

0,V2cA

= k, [

+k,

(3-7)

where A< C,)B represents the change of ( CA )j that occurs over the macroscopic distance L, and because

and

&-$((C.}‘c,, 1

I

=--

(3-8)

v, I v,

This represents the integraldifferential equation for C,, and it appears to be much more complex than the original equation for C, given by eq. (1-1). However, there are some simplifications to be made when the constraint given by eq. (2-26) is valid. To begin with, that constraint allows us to employ (eL)@=O,

1 1

CA - &(c,)$, 1

+ c”,( C,)@) I

1

(3-l 1)


BC npa.VcA

= - nga.V(

C,>@,

at A,,

(3-12)

Facilitated transport in porous media Periodicity

i = 1, 2, 3

C*(r + Ii) = C*(r), ( E* >o = 0.

(3-13) (3-14)

Here Ii represents the three, nonunique lattice vectors that are needed to describe a spatially periodic porous medium. An example of a unit cell in a spatially Periodic porous medium is illustrated in Fig. 3. It is convenient to attack the remaining closure problems in terms of combinations of eqs (2-5)-(2-7). This approach leads to

483

in which the diffusivities are considered to be constant within the averaging volume. In addition to these three closure problems, we also require the algebraic relations for the spatial deviation concentrations that result from the equilibrium relations given by eqs (2-8) and (2-9). These relations provide us with KZ&

= Ec< C”)fl

0 = &{

ci = 0, = 0

(3-15)

BC ns;VcD

= - nBa.V( C,)#,

at Age

(3-16)

Periodicity i = 1, 2, 3

ED@ + Ii) = ED(r), (E,)“=O

(3-17) (3- 18)

(3-24)

Co)fl + !Eo( C,)@.

(3-25)

A key feature of eqs (3-ll)-(3-25) is that the null solution results when naa. V (C, )@,n,, V ( C, >fl and ns,, . V < C, >@are zero. We can express this idea as

Problem 2 v’c,

+ E”{ Cc)@

when

V ( Ci >8 = 0,

i = A, B, C,H,

and

0.

(3-26)

S$e the governing differential equations are linear in Ci, and since they are couple_d through the reaction rate terms, the solution for Ci must be of the form (Ochoa, 1988) i=N ej

=

fji.V ( ci )fl,

c i=L

Problem 3

for i,j = A, B, C, H, v=Ee = 0

(3-19)

BC nsS.V&

= - n,;V(

C,)“,

at A,,

(3-20)

Periodicity LE,(r + ii) = E,(r), (c;)B

i = 1,2, 3

= 0.

Here Co and C, are modified fined by

(3-21) (3-22)

“concentrations”

CD = C, - (%/ S,)C,

- (%/~,)G

CE = C, - (%i/~*)CH

+ (%/~*)C,

+ (90/9,4)Co

dc(3-23a)

(3-2313)

and

0.

(3-27)

We refer to the fij as the closure variables and we note that there are 25 of these vector fields. It is of some importance to keep in mind that only the diagor.aZ terms of the matrix, f,, will belong to boundary value problems that contain a nonhomogeneous boundary condition [such as eq. (3-12)]. All the off-diagonaI term.9 will be coupled to the diagonal terms by the chemical reaction rate terms, and in the absence of homogeneous chemical reactions all the off-diagonal vector fields would be zero. Ochoa (1988) sh_awed that eq. (3-27) is a unique representation for Cj when the length-scale constraint given by eq. (2-26) is valid. From the representation given by eq. (3-27) and the three boundary value problems given by eqs (3-ll)-(3-22), we can construct the following problems for the vector fields: Problem

1

f*j-$(flij(Ce)P

v2fAj=> A [

+fBj‘)

1

fAjp+foj(c*)‘-~fgj 3

1 1 (3-28)

BC - 4c .Vf,,

A,,

(3-29)

i = 1,2, 3

(3-30)

= nBb6Aj,

at

Periodicity fAj(r + Ii) = fAj(r),

( fAj>@= 0, j = A, B, C, H

and

0

(3-31)

Problem 2 Fig. 3. Complex unit cell.

V2ADj = 0

(3-32)

_

484

J. ALBERTO OCHOA-TAPIA et al

BC - nfla VA,, = nSa(aAj - d,6,,

- LL,&~),

at A,, (3-33)

Periodicity

-------1

A,(r

+ Ii) = ADj(r),

i = 1, 2, 3

< Aoj >B = 0 A,j = f*j - dBf=j - d,fcj,

p - phase

(3-34) (3-35)

T !

j = A, B, C, H, and 0 (3-36)

d,

(3.37)

A, = gc/gA

= %l/g*,

Problem 3 V2A,j = 0 BC

(3-38) Fig.

- nga VA,, = na,(S,j ~ d,SHj + d,Scj

+ doSoj),

at A,,

4.

Spatially periodic porous medium.

(3-39)

Periodicity

i = 1, 2, 3

AEj(C + Ii) = A,(r),

(3-40)

( AEj)fl = 0 A,

= f~j - dufuj f d,fcj j= d,

= L&Jg,+,

(3-41)

f d,foj,

A,B,C,

H

Here we have used the Kronecker ing sense Sij =

1,

i=j

1 0,

i#j,

and

0

a!, = G&/a,.

(3-42) (3-43)

delta in the follow-

i,j=A,B,C,HandO (3-44)

and the boundary conditions problems have been constructed BC

- nbO. Vf, = Sijnam,

for all three closure on the basis of at

A,,

(3-45)

In order to complete our specification of the vector fields fij> we make use of the equilibrium relations given by eqs (3-24) and (3-25) to obtain

and

0

(3.46)

0.

(3-47)

‘@

a( C, )p

and

FORM

The closed form of eq. (2-32) is obtained by the use of eq. (3-27) for c,. This leads to

o=f~j
square array as a model of a spatially periodic porous medium.

Two-dimensional

4. THEORY: THE CLOSED

fBjK2=fcj8, j = A, B, C, E

Fig. 5.

The computational difficulties associated with the solution of the closure problem represented by eqs (3-28)-(3-47) would appear to be overwhelming. However, all the differential equations and boundary conditions are of the same form, and for simple, twodimensional arrays, such as the ones shown in Figs 4 and 5, one need only solve for a single component of the vector fij. Even without solving the closure problem, we can use the results of our development to predict the form of the volume-averaged diffusion equations and the form of the volume-averaged diffusive flux.

___ at

= V-(E~D:~~.V( C,)B

j=N + c [email protected] j=I

- Epk

( Cj>@)

~
Here the effective diffusivity tensor is defined by np,f,,

dA

1

(4-2)

Facilitated transport in porous media

and the effective transport coefficients active difitsioe flux take the form 1

.Q HAj = cBaa

for the inter-

1.

(1 - 6Aj)na,fAjdA

L-S‘0

At.

(4-3)

Here we can see that the effective diffusive flux of species A depends on the concentration gradients of all the diffusing species. Thus, we could write eq. (4-l) as

a
p= Eb at

-V-
485

tion rate term that appears explicitly in eq. (1-l) for the point concentration and in eq. (3-11) for the spatial deviation concentrations. This reaction rate term is especially evident in eq. (3-28) where it clearly indicates the origin of the coupling between the vector fields, fij. It should be intuitively appealing that this coupling increases with increasing reaction rate and that the nonlinear nature of the rate expression plays an important role. The influence of this nonlinearity is evident both in eq. (3-28) and in eqs (3-46) and (3-47). To complete our discussion of the averaged transport equations, we need to list the equilibrium relations which were given earlier as

(4-4) where

(R,


># are dejined by

>” and (R,

(4-12)

= (C,Y
(4- 13)

K, = aP. < R, >’ = < C, >” - ;

and the intrinsic form

1

< G, >O< C, >@

(4-5)

phase average diffusive flux takes the


-E~D$,.V(C,)~ j=N -

j?l

E~H~~.V
(4-7)

In order to solve the macroscopic problem illustrated in Fig. 1, we need the volume-averaged forms of eqs (2-6) and (2-7), and we list these results without derivation as

These representations are based on eqs (2-35) and (2-36). At this time we can see that the order of magnitude estimate given by eq. (3-7) supports the inequality, eq. (2-31), that led to eqs (4-12) and (4-13). The interesting point here is that the nonlinearforrns represented by eqs (2-8) and (2-9) remain unchanged in form when subjected to the process of volume averaging. This results from the fact that ( ci >fl = 0,

Ei 4 ( ci )‘, i = A,B,C,H

j=N + C V.C(EaHBj j=t

+ EbHCj).V(

+ kleB(

+ k&

Cj)‘]

RI)’

and

0.

(4-8) 5. PREDICTION

OF

EFFECTIVE

TRANSPORT

COEFFICIENTS

= V.(E@Dzr .V(Cn)B-~CDZ;f.V~Cc)P j=N + C VC(+HHj j= L

- E~H~.~- E~H~~).V(

Cj)fl]

+ klEB(R1

>#

(4-9)

Y’.

The effective diffusivity tensors and the interactive diffusive tensors are given by generalization of eqs (4-2) and (4-3): (4-10)

(1 - G,)ne,fijdA It is not interactive

i = A, B, C,H

Because of this, one must proceed with caution in the treatment of nonlinear transport problems in porous media.

&~&((C”}B-(CC)B-(CO)I)

+ k,Eg CR,

(4-14)

(4-15)

’

- ~Dr)fr.V
0.

However, one mtrst keep in mind that the inequality associated with Ci and ( Ci)P does not apply to the gradients of these concentrations. In that case, one finds that VEi = O(V ( ci )P),

+ ~~D&.“‘)

and

1

difficult to prove that the origin diffusive fluxes is the homogeneous

(4-11) of the reac-

In this section we solve the closure problem for porous media that are isotropic with respect to the diffusion. This does not mean that the porous media are isotropic; it simply means that D&r and Hij are isotropic and under these circumstances we need only solve for a single component of the vector fields fij (Ochoa, 1988). This allows us to calculate the single distinct component associated with D:,, and Hjj. To be specific, we identify the x-components of fij, Anj and A, as Aj = i.fij,

A,

= i. A,,

AEj = i. AEj.

(5-l)

The closure problem in terms of these components is given by eqs (3-28)-(3-47) with nBo being replaced by nW .i. From eqs (3-32)-(3-44) we see that the boundary value problems for hnj and AEi are uncoupled and independent of the volume-averaged concentrations of the diffusing species. This means that these two

J. ALBERTO OCHOA-TAPIA et al.

486

closure problems can be solved for assigned values of d,, d,, d, and d, in exactly the same way as the closure problems associated with diffusion and heterogeneous reaction in porous media (Ryan et al., 1981) and pure diffusion in anisotropic porous media (Kim et al., 1987). Once the solutions for A, and A, are known, they can be used with eqs (3-46) and (3-47) to produce values offBj,fc-,fHj andfoj in terms OffAj. The problem at this point becomes more difficult since eqs (3-46) and (3-47) involve the local volumeaveraged concentrations as does the boundary value problem for fAj. These concentrations cannot be specified arbitrarily

+ YiAEj - Jiktis*j, i, j = A, B, C,H

The coefficients is given by

and

0.

(5-2)

gi and Yi are given in Table 1 and J& &=ac+Ypi, _,+$= - 1,

( ~,..~,,j 1

1

-k3

(C,>“I,-2 3)

i

(CK”,PyB+F&) 1

1 (5-10)

again we note that A, and hEj can bc determined independently by solving for the x-component of the vector fields A, and A,, . This means that eqs (3-32)-(3-43) are solved for specified values of d,, d,, d, and do. The solution of eqs (5-4))(5-7) requires values of the average concentrations in order to evaluate Fl , F, and F3. Because- of this, JAj is determined iteratively along with the solution of the macroscopic problem. In this work we have used two types of structures in our solution of the closure problem. The first of these is the two-dimensional array illustrated in Fig. 4. We have used this system in previous studies (Ryan et al., 1981: Nozad et al., 1985; Ochoa et al., 1986); thus, its use allows us to compare calculated effective ditTusiv-

Once

ities in the presence

of homogeneous

previous

for passive systems. In addition,

gAV2fAj=fAjFl - Aoj& - AE,F, - nsg. VfAj= i.nBa8Ai, at 4,

Here the parameters

>

(5-3b)

BC

(fAj)@=O,

(5-9)

i = A.

1

+ Ii) =_&j(r),

f54

(5-3a)

Problem

i = 1,293

j=A,B,C,HandO.

1.

reaction

with

the geometry allows us to use simple Anite-difference techniques and this is an important consideration

(5-5) (5-G)

putation

(5-4)

is restricted

to one quarter

of the unit cell.

The numerical method used to solve the boundary value problems for ADj, .Aej andfAi was a five-point block finite-difference scheme. The pentadiagonal system of equations resulting from the discretization was IMSL subroutine (Program solved using an LEQTBlB from IMSL, Inc., Houston, TX) as a direct

Coefficients in eq. (5-2)

4

calculations

when extensive numerical computation is required. By appropriate transformations (Ochoa, 1988) in the boundary value problems for hD,, h, and.fAj associated with the square unit cell shown in Fig. 4, it can be shown that these fields are symmetric about y = 0 and y = f l/2, and that they are skew-symmetric about x = 0 and x = f l/2. This means that numerical com-

(5-7)

F,, F2 and F, are given by

Table 1

3

i#A

Substitution of eq. (5-2) into eq. (3-28) allows us to represent the boundary value problem for the five fields represented byfAj as follows:

&(r

F,=k,

((CO)‘-(CA)‘&+?)

since they must satisfy the macro-

scopic boundary value problem, i.e. the solution of eqs (4-l), (4-8) and (4-9) for the system shown in Fig. 1. This means that the closure problem must be solved simultaneously with the macroscopic problem. This is a situation that arises naturally with nonlinear systems; however, it appears that this is the first time that this type of detailed analysis has been carried out for a nonlinear transport process in a porous medium. The algebra associated with the determination of fBj, fcj, fHj and foj can be expressed as Jj = &Iih,

+k,

that

relateJj

tofAj

%

Facilitated

solver.

When

all the values

single

distinct

components

transport

in porous

media

487

ofAj

are calculated, the and Hij can be calculated using eqs (4-10) and (4-11). It is important to keep in mind that, in order to computef;-i, one must first calculate A,, and A, for j = A, B, C, H and 0 and store these fields. One then solves eqs (5-4)-(5-7) in order to calculate fAj for j = A, B, C, H and 0 while simultaneously solving the macroscopic problem so that F,, F, and F3 can be determined. This is a computationally intensive and expensive procedure; thus, there is considerable motivation for developing analytical solutions for fii. of D&

5.1. Analytical solution In addition to the system shown in Fig. 4, we have also solved the closure problems associated with diffusion and heat conduction for the system shown in Fig. 5 (Ochoa, 1988). These closure problems are less complicated than the closure problem associated with facilitated transport; however, we can draw conclusions from our studies of these simpler processes that are extremely useful in our study of the facilitated transport problem. In the solution of a closure problem it is permissible, as a matter of convenience, to specify that the closure variable is zero at some point and ignore the constraint given by eq. (5-7). If we impose the condition that f = 0 at the corner of a unit cell, as illustrated in Fig. 5, periodicity requires that f be zero at all four corners of the cell. For this condition, our detailed numerical calculations indicate negligible oariations in the f-field in the shaded regions shown in Fig. 6. For the facilitated transport problem, this means that the value offij in the shaded regions is essentially zero and one is motivated to replace the unit cell shown in Fig. 5 with the system shown in Fig. 7. This idea was first utilized by Chang (1982), and it has been explored in considerable detail by Ochoa-Tapia et nl. (1990a) for the problem of diffusion with interfacial resistance in anisotropic systems. It should be intuitively appealing that the difference between results determined using Figs 5 and 7 would be small when E, is small compared to one. This is indeed the case and Chang (1982) found that use of the system shown in Fig. 7 was equivalent to Rayleigh’s model for diffusion and heat conduction in dilute suspensions. Following Ochoa-Tapia et al. (1990a) one can find that the solutions for Si, have the form fij = &(cos

0) rij(i)

(5-11)

where 8 is illustrated in Fig. 7, and [ is the dimensionless radial coordinate given by

Fig. 6. Unit cell.

ce

l/2

ch %

+

‘.h Fig. 7. Chang’s unit cell.

express

the results

as

ADj

=

Szch(cos

@I

%i

(i)

(5-13)

AEj

=

+z,,(cos

0)

x,

(&J.

(5-14)

When these expressions are used in the boundary value problems expressed by eqs (3-32)-(3-43) we obtain boundary value problems for ZDj and EEj that are almost identical to the problem solved originally by Chang (1982). The details are available in Ochoa (1988) and Ochoa-Tapia et al. (1990a), and the solutions can be expressed as

%h

the diameter of the unit cell where l,, represents shown in Fig. 7. For a valid comparison between the two systems, the porosity in the square unit celt should be the same as that in Chang’s unit cell. When eq. (5-11) is used in eqs (3-36) and (3-42), we can

ZEj = - (s,j + d,6,, - &S,)(

- d,6,j z)(

q.

(5-16)

488

J. ALBERTO

OCHCXA-TAPIA et al

Use of eqs (5-13)-(5-l@ and (5-11) in eq. (5-2) leads to solutions for rij in terms of rAj. These are given by

i,j = A, B, C, H The coefficients Ai are defined coefficients Mij are given by

+ where All closure priate lowing

Yi(6,j

+

d,6,j-

and

0.

(5.17)

by eqs (5-3) and the

d,h,j-

d,d,j)

(5-18)

one should note that .‘lrAj = 0. that remains to complete the solution of the problem is to determine IAj. Use of the approequations in eqs (5-4-(5-10) leads to the folboundary value problem for rAj:

Problem

tration of CO, at the upstream boundary shown in Fig. 1. The solution for rA, can be expressed in terms of Bessel functions, and the solution alIows us to determine rij(<) according to eq. (5-17). These results are then used in eq. (5-11) which is, in turn, substituted into eqs (4-10) and (4-l 1) in order to produce values of the effective transport coefficients. Since Chang’s unit cell represents an isotropic system, the single distinct value of DaFr takes the form

i = A,B,C,H while eq. (4-11) provides the following for the interactive diffusive coefficient:

(5-25)

representation

1-E&q -._qij ___ >I ( 2--cEg

(5-19) dr;\j __ = - aAj, dC

BCl BC2

rAj = 0,

at c = E:‘*

(5-20)

at [ = 1

s: = $;

1

fK

[

K,

(S-22)

8_Af,j

7 [

- fpj [ < c, >fl _Moj

( > I -

+

1

-

Ls!x _NHj Kl 1 2

L 3

1 (CX)_

I

&g

X2-Eg.

these representations

0.

(5-26)

we have used

(5-27) where Q is given by

Q=

K~ts,)b(Ef’~s~)+ ~,(SI)&~E:‘~S,) _ - r,(s,)K;(&ys,)

1

3

4:

In developing

and

1

1 (5-28)

8J&+~ (cx)-’ [ 1

+&

i,j = A, B, C, H

K,(S,)I;(&yS,)

(CHY AB+=)BH 1

’

(5-21)

Here we should note that eq. (5-19) is extracted from eq. (5-4); the first boundary condition given by eq. (5-20) is analogous to the interfacial flux condition given by eq. (5-6); and eq. (5-21) replaces the periodicity condition given earlier by eq. (5-6). The parameters S: and Szj take the forms

{

0

H.. ~=(‘-6,)[&(f$Jr,&::~)

1

s2j=

and

(5-23)

The macroscopic Damkahler numbers #1 and & are defined in terms of the diameter of the circular unit cell, I,,, according to (5-24a)

The concentration C: is a characteristic concentration of CO,, and in our analysis we used the concen-

where K,, K,, I, and I, represent the appropriate Bessel functions (Abramowitz and Stegun, 1964) and the prime represents differentiation with respect to [. In Section 8 of this paper we present a comparison between the approximate analytical solution represented by eqs (5-25) and (5-26) and results obtained from the numerical solution of eqs (3.28H3.43). In order to do this, we must compute the average concentrations on the basis of the macroscopic boundary value problem that describes the experimental conditions used in this work. Once again we remark that this requires a simultaneous solution of the macroscopic boundary value problem and the closure problem, and in the next section we describe the solution of the macroscopic problem.

6. THE MACROSCOPIC

PRORI.EM

In this section we outline the solution of the boundary value problem for the steady-state diffusion of CO, through a reactive, aqueous bicarbonate solution in a porous medium. This situation is illustrated in Fig. 1 where the flux of CO, is generated by the imposition of a partial pressure difference, pAo - pAL,

Facilitated transport in porous media

489 (6-12)

across the film of thickness L. Here we have used the subscript A to designate carbon dioxide, and we remind the reader that the general notation for identifying the various chemical species is given by A = CO,,

B = HCO;,

O=OH-,

H=H+,

C = CO:1

W=H,O.

+ @):

(6-l)

Since all species are present in small amounts with respect to the solvent, the concentration of water can be considered constant and the reaction network is given by eqs (2-l). The volume-averaged transport equations are represented by eqs (4-l), (4-8) and (4-9) while the volume-averaged equilibrium relations are given by eqs (4-12) and (4-13). One should keep in mind that the intrinsic phase average flux given by eq. (4-7) can be generalized for all species simply by replacing A with i. It is convenient to represent the steady-state, macroscopic problem in terms of dimensionless variables. This leads to the following problem statement: Volume-averaged

(6-13)

transport equations

PAP0

-

____

where the two macroscopic defined according to

(6-14)

PB >

K3aApAo

DamkGhler numbers are

(6-15) One should note that D$ represents the effective diffusivity at x = 0 for species A in rhe absence of chemical reaction. It is necessary to use this parameter to scale the equations since it is a constant for any given porous medium as opposed to D& which depends on the rate of chemical reaction. Goddard et al. (1970) have pointed out that only one of the zero flux conditions given by eq. (6-9) is independent. For this reason it is necessary to introduce the following constraint on the total sodium ion

L PM =

s0

(6-16)

(PB + 2MdX

The derivation of this result (Meldon et al., 1982) involves the use of local global electroneutrality conditions along with the constraint +~[j~~(h.i+h,)~]+R=O j-1

(6-3)

~(ds&)-&(dc~)-~(do~) +&

1

+I?=0

‘kN(h,,-hcj-h,)g [’ j=i

Volume-averaged

equilibrium

(64)

relations:

65) K4

(6-6)

(a,p,,)2 = pHpo Volume-averaged BCl BC2

boundary conditions: C(A= I,

PA

=

at

PAL/PAo,

X=0 at

BCs 3 and 4

x

(6-7) =

1

(6-8)

It is important to note that the boundary value problem given by eqs (6-2x6-9), along with eq. (6-16), also describes the CO, diffusion problem in a homogeneous bicarbonate solution. In that case the volumeaveraged concentrations become point concentrations and the value of Ed is one. The effective diffusivities are replaced by the molecular diffusivities and all the hij are zero. The boundary value problem given by eqs (6-2t (6-9) along with eq. (6-16) was solved with a finitedifference method using a variable grid to deal with the reaction boundary layers. The solution required an iterative method due to the nonlinearity of the equations. The details of the discretization, the variable grid, and the algorithm based on the method of quasi-linearization (Suchdeo and Schultz, 1971) are given by Ochoa (1988). Integration of the equations produced by adding eq. (6-2) to eqs (6-3) and (6-4) yields

= = 0; X

0, 1; i # A. (6-9)

The dimensionless given by

(6- 17)

5
variables

~, =

in eqs (6-2H6-9)

(CiY

I

are

x/L

C (hAj + hBj + hcj)z = -j,

j= 1

(6-18)

d,$+d,+$-d$-dog (6-10)

cLAPAo

x =

j=N +

(6-11)

j=N + C (hAj + h, i=l

- h,

- hoj)g

= -j,

(6-19)

490

J.

ALBERTO OCHOA-TAPIA ez al.

where the constant of integration, -j,, is the dimensionless total flux at X = 0 and 1. The presence of the interactive diffusive terms changes the form of the total flux. Usually the total flux equation includes only the terms involving the effective ditfusivities, but in this case it also contains the terms with the coefficients hIj. The CO2 facilitation factor is defined as the ratio of the total CO, flux to the purely diffusive flux of CO,. This factor is given by Y*

JAL

=

sfl @u

aA (PAo

-

(6-20)

1.

PAL)

We have compared our algorithm for the solution of the macroscopic problem to the combined Damkiihler number technique used by Hoofd et al. (1986) for the cuse of homogeneous bicarhonnte soZutions (i.e. cB = 1). The agreement between our method and that of Hoofd et ul. (1986) is very good, and the results are reported in detail by Ochoa (1988). The comparison has provided two important conclusions: (1) the numerical method is adequate, and (2) the mass transfer due to the diffusion potential can be neglected for the conditions studied in this work. 7. EXPERIMENTAL

METHOD

The experimental set-up is identical to the one described by Stroeve and Ziegler (1980) and is shown in Fig. 8. The thin liquid film of interest is supported horizontally in a sample holder on top of a synthetic, nonwettable porous membrane (Celgard) 25 pm thick. The liquid film is 1.OOmm thick. Tn comparison to the liquid film, the mass transfer resistance of the porous membrane to CO, transport is negligible. The sample holder consists of a center section which holds the liquid film, and upper and lower gas chambers.

PERFUSION PUMP \

Gas mixtures of known CO, concentration are fully humidified by passage through a series of saturators. The upstream gas chamber is flushed continuously with a humidified gas mixture of CO, in N, (8.04% CO,). The downstream chamber is perfused with humidified N, gas at a preset and much lower flow rate which is maintained by a syringe pump. The change in CO, concentration of the gases entering and leaving the gas chambers is measured by gas chromatography. The CO, mass transfer through the film is obtained from the difference in the exiting and entering CO, concentration of the downstream gas and the volumetric flow rate. All experiments were conducted at 23°C. 7.1. Materials Suspensions of surfactant-free monodisperse polystyrene beads in deionized water were prepared following the method proposed by Juang and Krieger (1976). The suspensions were prepared and characterized by Lewis (1987). Monodisperse spheres were synthesized at solid contents up to 14 vol% and then purified by filtration and ion exchange. The particle size analysis was performed by electron microscopy. Surface charge of the spheres was measured by conductometric titration. The diameter of the polystyrene particles was 0.320 pm. The uniformity ratio, defined as the ratio of the weight average diameter divided by the number average diameter, was found to be 1.00. The surface charge of the particles was 9.93 &/cm’ and the suspension conductivity was 22.5 pmho (Lewis, 1987). Suspensions were diluted by addition of distilled water or concentrated by evaporation at room temperature. Sodium bicarbonate was added as required. Solutions were stored at 4°C before use in the mass transfer experiments. Additional details are available elsewhere (Ochoa, 1988). UPSTREAM SATURATORS

DISPERSION LAYER

b

d

LET

DOWSTREAM SATURATOR

II

+

DRYING COLUMN

/SAMPLE

INJECTOR

AS DETECTOR CONTROL UNIT

(THERMOSTA1TED) DETECTOR

Fig. 8. Schematic diagram of experimental apparatus.

Facilitated transport in porous media 8. RESULTS AND DISCUSSION

The numerical method described above was used to solve the macroscopic problem for CO, diffusion in heterogeneous systems. In order to perform the iterative calculations, the effective diffusivities were assumed to be those encountered for purely passive diffusion in the porous medium. In addition, the initial values of Hij were taken to be zero. When the CO, concentration profile was determined, the effective diffusivities and the coefficients Hij were calculated using both the analytical and numerical methods described in the theory section. The new values for the effective diffusivities for CO,, HCO; and CO i - were always in agreement with the initial values, but the values for H+ and OH- were found to be equal to their molecular diffusivities. This gives a tortuosity factor of one for the difiusion of these two species. The definition of the tortuosity is given by r = %$/l&.

(8-l)

The coefficients Hij were found to be negligible relative to the effective diflusivities (Hij smaller than Da,, by a factor of 10m4). Further iterations for updating the effective diffusivities did not change their values from those found after the first iteration. A range of values for the parameters, upstream and downstream CO* partial pressures, total global sodium concentration, fluid fraction, and lilm thickness, was explored in calculating the effective diffus-

ivities for the macroscopic problem. These values are given in Table 2. The effective diffusivity Dtfr was normalized by the molecular diffusivity in homogeneous media Qi. In any combination of the values in Table 2, the dimensionless effective diffusivities for CO,, HCO; and CO:were always found to be equal to the dimensionless effective diffusivities predicted for inert porous media (no reactions), while

Table 2. Range of values explored for the macroscopic problem and effective diffusivity calculations Global concentration

of Na+ (M)

0.07,0.15,0.35,0.50,

1.0

Upstream and downstream CO, partial pressures (mmHg): PA0

PAL

760 507 253 76 60 36.5

7.6 5.0 2.5 0.76 3.0 2.0

491

those of H* and OHwere equal to 1. For the reactive porous media, both our numerical method and the analytical method, based on the Chang’s approach, were used. For the prediction of diffusivities in inert suspensions we used the results of Ochoa-Tapia et al. (1990a) and the theory of Chang (1982). Both techniques predicted the dimensionless effective diffusivities of CO,, HCO; and CO:- to be equal to the values for the inert suspensions, and the dimensionless effective diffusivities for I-I+ and OHtobeequalto 1. Dimensionless effective diffusivities for CO,, HCO:-, CO;-, H+ and OH- diffusion in reactive porous media are invariant with position. The dimensionless effective diffusivities as a function of the fluid fraction are also shown in Fig. 9 for both the numerical and the analytical method. Again, both procedures show that the H+ and OH- dimensionless effective diffusivities are equal to 1. The fact that both the numerical technique and the analytical technique yield essentially the same values gives considerable confidence to the results obtained from the numerical solution. The values for the effective diffusivities of H+ and OH _ have a negligible impact on the prediction of the CO, flux and on the determination of the species concentration profiles across the heterogeneous layer. The arbitrary use of effective diffusivity values for H+ and OH- almost an order of magnitude larger or smaller than those predicted by the program did not significantly change the CO2 transport. If the presence of H+ and OH- ions are ignored, the calculations for the dimensionless effective diffusivities for CO,, HCO; and CO:are not significantly affected. The range of values explored for the macroscopic problem (Table 2) are such that 5 < pH c 9, and in these cases the H + and OH- concentrations are very small. Calculations for the H+ and OH- concentration profiles appear to be determined solely by the equilibrium relationships and not by diffusive transport. It is of interest to note that our calculations for thick films (100cm) do not change the effective diffusivities for CO,, HCO; and CO;-. For thick films, the chemical reactions are near equilibrium throughout the film, except at the boundaries.

- ---co*, Hco;,co; alan.gs “nilcell 08 - - co*, HCOj co; tdmmca

Fluid fraction: EB= 0.4, 0.81 Film thickness (cm):

L = 0.1, 1, 10, 100 Ratio of particle size to system size: d/L = 3.0 x 10-4, 3 x 10-3, 3 x lo-*

ow 0

’ 0.2

’

I n 0.4 9

’

0.6

I

’ 0.8

’

1

1 .o

Fig. 9. Normalized effective diffusivity vs fluid fraction of the fluid-solid dispersion (for conditions reported in Table 2).

J. ALBERTO OCHOA-TAPIA et al.

492

Ochoa-Tapia et al. (1990b) have analyzed a simpler problem (only the reaction A ti B occurs) and it was found that the effective diffusivity for A and B may exhibit the same behavior as that for the H+ and OH- ions. The study of Ochoa-Tapia et al. (1990b) shows that two conditions are necessary for this behavior: a macroscopic Damkijhler number much larger than 1, and a concentration of the solute much smaller than the solutes showing normal behavior. Therefore, in the present study, the much lower concentrations of H+ and OHrelative to the other species (two orders of magnitude) is probably responsible for the curious behavior of the effective diffusivities. Note that when one expresses the boundary value problem using eqs (2-8) and (2-9), the large Damkiihier number condition mentioned above is satisfied. Because of the complexity of the theoretical relationships for the effective diffusivities we do not have a simpler expIanation for the behavior of Hf and OH-. 8.1.

Comparison

with experimental

data

The diffusivity of CO, in water at 23°C was measured in this work to be 1.80 + 0.09 x 10m5 cm’/s, in good agreement with the value of 1.83 x 10m5 cm2/s found by Stroeve and Ziegler (1980). These measure-

ments were conducted on 0.1 cm thick films of distilled water, and represent over 10 separate measurements. In Table 3, the experimental data for the facilitation factors, Y,, for homogeneous bicarbonate solutions are compared with the theoretical predictions from the numerical technique. The initial NaHCO, concentrations were either 150 or 350 mM. Facilitation of CO, for the 150-mM solution is considerably lower

than for the 350-mM solution, and the facilitation factor is more difficult to determine accurately. If the standard deviation in the experimental results is taken into consideration, the agreement between the experimental and theoretical facilitation factors is very good. The experimental facilitation factors were calculated from eq. (6-20). In order to obtain the facilitation factors, the CO, diffusion coefficients in 150- and 350mM NaHCO, solutions are needed. Since the CO, in pure water was measured, we corrected this experimental value for the presence of salt using the approach described by Hoofd et al. (1986). This correction is small and leads to diffusion coeficients only a few percent lower than the value found in water. For the 150-mM NaHCO, solution the inert diffusion coefficient is 1.75 x 1O-5 cm’/s and for the 350-mM NaHCO, solution the inert diffusion coefficient is 1.69 x lo-’ cm’/s. Experiments were also conducted for facilitated CO2 transport in polystyrene suspensions with and without added NaHCO,. Without the presence of NaHCO, the polystyrene suspensions are inert, i.e. nonreactive. The experimental results for both inert and reactive suspensions are reported in Tables 4 and 5. For the results obtained in inert suspensions, the dimensionless effective CO, diffusivity is compared to the theoretical predictions obtained from the approximate analytical method and the numerical calculations (Ochoa, 1988). The molecular diffusivity of CO,, as measured in distilled water and then corrected for the total amount of salt in the system, was used to obtain the dimensionless effective diffusivity. For both suspensions the agreement between experiment and theory is good provided the standard devi-

of experimental (exp.) and theoretical (theor.) results for diffusion of CO, in aqueous bicarbonate layers, (T = 23°C)’

Table 3. Comparison

YA

NaHCO, @Ml

PAo

(mmW

6%

150 350

PAL (mmW

60.6 60.6

0.10 0.10

3.5 5.1

Exp.

Theor.

0.49 + 0.13 1.26 + 0.14

0.59 1.32

‘L, pAOand pAL are averaged values from the data reported by Ochoa (1988).

Table 4. Comparison

of experimental (exp.) and theoretical (theor.) results for diffusion of CO, in inert layers of water with polystyrene particles (T = 23°C)’

+D!$ ( x lo-’ L

(cm)

PA0 (mmW

0.875

0.10

60.7

0.789

0.10

60.9

PAL (mmW

cm2/s)

@%I~,

Exp.

Theor.

Exp.

Theor.

2.2

1.30 f 0.07

0.72 + 0.05

1.5

1.06 + 0.09

1.40~ 1.380 1.17’ 1.148

0.78’ 0.77’ 0.65’ 0.64’

‘L, pAo and pAL are averaged values from the data reported by Ochoa (1988). *From analytical method. $From numerical method.

0.59 + 0.06

Facilitated transport Table

in porous

493

media

5. Comparison

of experimental (exp.) and theoretical (theor.) results for diffusion of CO, in aqueous bicarbonate layers with polystyrene particles (T = 23°C)’ y’,

NaHCO, bW

%

PA0 &,

(mmH!s)

PAL (mm&9

Exp.

0.875

105

0.10

61.0

2.6

0.38 + 0.14

0.875

150

0.10

61 .O

2.9

0.71 f 0.12

0.789

105

0.10

61.0

1.9

0.31 f 0.14

0.789

150

0.10

61.0

2.3

0.67 f 0.17

Theor. 0.391 0.40~ 0.64’ 0.644 0.41 r 0.42’ 0~56~ 0.67’

‘L, pa. and pALare averaged values from the data reported by Ochoa (1988). *From analytical approximate method QFrom numerical method.

ation of the experimental results is taken into account. For each suspension several separate ditfusion measurements were made. For the reactive suspensions we report the facilitation factors in Table 5. In this case the experimental facilitation factors were calculated from eq. (6-20) using the experimental total flux of CO, and the experimental CO, flux in the dispersion with pure water for the same value of Q. A small correction, using the procedure of Hoofd ef al. (1986) clue to the presence of total salt, is included in this last term. The theoretical results were obtained using the analytical method and the numerical method for the square unit cell derived in Section 2. The comparison shows good agreement if the experimental uncertainty is considered. Further the experimental results show consistent trends with a decrease in fluid volume fraction and an increase in the concentration of the added NaHCO,. Suspensions of lower fluid fractions, i.e. E@= 0.70, were too viscous to handle experimentally; thus, data for more concentrated media could not be obtained. We did not conduct experiments with larger amounts of added NaHCO, to investigate higher levels of facilitated transport because the larger salt concentrations caused the polystyrene suspension to become unstable and flocculate.

9. CONCLUSlONS

In this paper we have considered diffusion with reversible chemical reaction in porous media. A typical example in chemical engineering is the separation of a solute from other species using reactive liquid membranes. In the biological sciences there are many other examples of reactive porous media, e.g. the transport of nutrients or metabollic waste products in cellular media, in which the solute undergoes reversible chemical complexation [e.g. Stroeve et al. (1976a, b)]. The design of a reactive liquid membrane system involves the determination of an appropriate carrierfacilitation transport system for a solute diffusing through the liquid membrane. The reactive liquid is

supported in a porous membrane, and this system may be well-characterized in terms of its diffusion and reaction characteristics prior to its introduction into the porous support. However, once the liquid is introduced into the support, the diffusion and reaction parameters of the porous medium are different. Volume averaging techniques can be used to accurately predict the effective transport properties and the facilitation factor of the reactive porous medium. Starting from the point transport equations, the equilibrium relations, and the boundary conditions, we have derived the volume-averaged transport equations for facilitated transport in porous media. patially periodic media are used in conjuction with a closure problem to determine the effective diffusivities. The use of a spatially periodic model does not impose any severe theoretical limitations, and the unit cell can be as complicated as desired in order to accurately represent the system of interest. We compare the theoretical predictions to experimental results for CO, facilitated transport in an aqueous solution of NaHCO, containing uniform, spherical latex particles. Thus the choice of the unit cells shown in Figs 4 and 7 for the numerical and the analytical method, respectively, are reasonable and appropriate. For inert porous media, the effective diffusivities for a unit cell in which the dispersed phase is a square particle arranged in a square pattern gives nearly identical results compared to the effective diffusivities obtained for a unit cell where the dispersed phase is a cylindrical particle in a square pattern (Ochoa, 1988). The use of the unit cell proposed by Chang (1982) is advantageous in that an analytical solution can be obtained for the effective diffusivities. For diffusion in inert porous media, we have found that the use of Chang’s unit cell gives accurate predictions of the effective diksivities when compared to results obtained from numerical methods (Ochoa-Tapia er al., 1990a). The macroscopic boundary value problem is not linear as expected from the form of the original boundary value problem in terms of the point transport equations. However, the closure problem is linear if the length restrictions indicated by eqs (2-26)

494

J. ALBERTO OCHOA-TAPIA et al.

and (3-1) are satisfied. The existence of nonlinear terms in the point equations gives rise to a closure problem dependent on the local averaged concentrations. The form of the closure problem yields additional diffusive terms in the averaged transport equations. For the conditions in this work, i.e. the absence of concentration boundary layers, the new diffusive interactive coeffcients were negligible. However, a recent work by Ochoa-Tapia et al. (1990b) shows that interactive coefficients might be important under certain conditions, i.e. large Damkohler numbers and a much smaller concentration of one solute with respect to the other solutes. The effective diffusivities depend on the solution of the macroscopic problem which provides the volumeaveraged concentrations. However, the solution of the macroscopic problem depends in turn on the effective diffusivities. Consequently, an iterative method is required to predict the effective diffusivities and the total transport of the solute. Effective diffusivities and effective interaction coefficients obtained from both the numerical method and the approximate analytical method give almost identical results. The effective diffusivities of the species which are responsible in determining the total transport of the solute are found to be equal to the effective diffusivities for the medium without any chemical reaction. The OH- and Ht ions, which are present in very low concentrations, have dimensionless effective diffusivities equal to one. This result agrees with the findings of Ochoa-Tapia er al. (1990b), because the large Damkohler number condition mentioned in Section 8 is implicitly satisfied in the use of eqs (2-8) and (2-9). Acknowledgement-This work was supported by NSF Grant CBT-85 13956. We also received computer time on the CRAY X-MP/48 from the San Diego Supercomputer Center (Grant 863UCD). We thank C. Shoemaker and J. I. Lewis for the gift of polystyrene latex suspensions. NOTATION

macroscopic area of entrances and exits for the b-phase, m* interracial area between /II- and a-phases for the system, m2 interfacial area between fi- and u-phases for the averaging volume, m2 coefficient in eq. (5-2) reported in Table 1 concentration for species A, B, C, H, 0 or W, kmol/m3 intrinsic phase average concentration for species i, kmol/m3 spatial local deviation concentration for species i, kmol/m3 effective diffusivity tensor of species i [eq. (4-lo)], m’/s x-component of effective diffusivity tensor for speces i, m’js effective diffusivity at x = 0 for species A in the absence of chemical reaction

4 &i

fij Aj

F,, & F3 Hij 4i hij JA JA

molecular diffusivity for species A, B, C, H or 0, m2/s dimensionless effective diffusivity coefficient Ceq. (6-1211 ratio of diffusivity or effective diffusivity of species i over the diffusivity or effective diffusivity of species A function that gives boundary conditions in entrances and exits for Ci vector that maps V ( C, )B to ei, m x-component of fij, m coefficients of eq. (5-4) defined by eqs (5-8)-(5-10) effective transport coefficient for interaction diffusive terms defined by eq. (4-11) x-component of Hjj dimensionless interactive effective coefficient [es. 613)l total flux of CO,, kmol/m’-s dimensionless flux of CO, (j, = J,L/

s2j

a,~,oD%) forward reaction rate constant for reaction i, s-r equilibrium constant for reaction i coefficient in eq. (S-2) defined by eq. (5-3) coefficient in eq. (5-17) given by eq. (5-18) outwardly directed unit normal vector for the p-phase at the B-a interface macroscopic length scale, m diameter of Chang’s unit cell, m lattice vector, m characteristic length for the P-phase, m partial pressure of CO,, mmHg term in D& equation given by eq. (5-28) reaction term for reactions 1 4 defined by eqs (2-3) position vector, m radial coordinate, m coefficient in eq. (5-2) reported in Table I dimensionless coefficient in eq. (5-19) defined by eq. (5-22) dimensionless coefficient in eq. (5-19) de-

t

fined by eq. (5-23) time, s

ki

L

1

If” ;, PA Q

Ri r I yi

Sl

v

VfJ x7 Y X

averaging volume, m3 volume of B-phase contained within averaging volume, m3 rectangular, Cartesian coordinates, m rectangular dimensionless, Cartesian ordinate

the

co-

Greek letters

solubility of CO,, kmol/m3 mmHg radial contribution offij for Chang’s solution [eq. (5-1 l)] volume fraction of w-phase dimensionless radial coordinate angular coordinate combination off,, defined by eqs (5-13) dimensionless concentration for species i defined by eq. (6-10) (i = A, B, C, H, 0)

Facilitated

global sodium radial tion

dimensionless ion [eq. (6-16)] contribution

[eqs

(5-13)

concentration

to A,,,, in Chang’s and

transport

of solu-

(5-14)]

tortuosity facilitation f6- 20) macroscopic

factor

of CO,

by eq.

number

of reaction

microscopic Damkiibler 1 defined by eq. (524a)

number

of reaction

macroscopic

number

of reaction

number

of reaction

1 defined

3 defined

Damkiihler

defined

by eq. (6-15)

Damkiihler

by eq. (6-15)

microscopic DamkBhler 3 defined by eq. (5-24b)

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