Ind. Eng. Chem. Res. 1995,34, 3508-3513
3508
The Mass Flux Boundary Condition at a Moving Fluid-Fluid Interface Michel Quintard L.E.P.T.-ENSAM (UA CNRS), Esplanade des Arts et Mktiers, 33405 Talence Cedex, France
Stephen mitaker* Department of Chemical Engineering and Material Science, University of California at Davis, Davis, California 95616
The dissolution of organic chemicals into ground water aquifers represents a mass transport process of considerable complexity. The rate of dissolution can depend on the manner in which the organic material is distributed, on the nature of the mechanical and chemical heterogeneities associated with a particular aquifer, and on the equilibrium relations for the individual chemical species. One aspect of this problem that has not been considered in detail is the flux boundary condition at the organic phase-aqueous phase interface, and it is the nature of this boundary condition that is explored in this paper. At a moving fluid-fluid interface, one must make use of the species mass jump condition for a singular surface, the appropriate form of the StefanMaxwell equations, and the mass average momentum equation in order to develop a general form of the mass flux boundary condition. In this paper we show that the molar flux condition for species A a t the j3-y interface can be expressed as CA(VA - w)mvp = - ~ ~ ~ ~ - ‘ ~ A E c @ B m v-w H..wMN-le(v ~ y p - whyp in which HAB represents a matrix of coefficients that depend on the binary diffision coefficients and the concentration at the interface. We have used MN to represent the molecular weight of the Nth component, and it is the last term in the above relation that is ignored in the traditional models of the interfacial flux.
Introduction Contamination of aquifers by chemical solvents and petroleum products is a problem of increasing public concern. The chemicals migrate through the subsurface and eventually reach the water table where they contaminate municipal water supplies. For nonaqueous phase liquids (NAPL), capillary forces can cause the organic phase to become trapped as immobile blobs or ganglia which slowly dissolve into the aqueous phase. This situation is represented in Figure 1 where we have used y to identify the trapped hydrocarbon phase, j3 t o represent the aqueous phase, and o t o represent an impermeable solid phase. Contamination is due to the slow dissolution of chemicals from the NAPL into the groundwater with subsequent convection and dispersion into the aquifer. In a previous study (Quintard and Whitaker, 19941, we have examined the transport process in the /3-phase for the case in which the concentration at the /3- y interface could be specified and the interfacial flux could be inferred from a solution t o the @-phasetransport problem. In this paper, we are interested in the development of a precise description of the interfacial flux condition for the problem of multicomponent mass transfer at a moving interface. This is a matter of general concern for a variety of chemical engineering processes, and it is of particular importance for situations in which laboratory measurements cannot be easily used to develop reliable empiricisms for the interfacial flux. The mass transport process in an N-component system is described by a set of continuity equations
Figure 1. Trapped organic phase in contact with an aqueous phase.
ac, at accumulation
+
V.(CAVA)
totalflux molar
=
RA
9
homogeneous reaction A = 1,2, ..., N (1)
that determine the concentrations, CA, CB, ..., CN, and a set of momentum equations (Whitaker, 1991)
a
geAVA) +
=
v.(@AvAvA)
local acceleration @AbA
body force
+
convective acceleration
VTA
surface force
+
B=N
Pm ~ = i
diffusive force
* Author t o whom correspondence should be addressed. 0888-588519512634-3508$09.00/0 0 1995 American Chemical Society
+
?-Av*A
3
source of momentum owing to reaction A = 1,2, ...,N (2)
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3609 that determine the velocities, VA, VB,...,VN. Both eqs 1 and 2 apply to the ,4- and y-phases, and at the interface they are coupled by species jump conditions (Whitalter, 1992) and equilibrium relations. The jump conditions associated with eqs 1can be expressed as cAY(vAr
- wb,p = c @ ( v A p - w b Y p at Asy, A = 1 , 2 , ...,N (3)
provided there is negligible adsorption, surface transport, and heterogeneous reaction. In eq 3 we have used wmpVto represent the speed of displacement (Slattery, 1990) of the phase interface, and this quantity must, in the general case, be determined as part of the solution to the problem. The equilibrium relations can be expressed in terms of the chemical potential according to PAP
= PA^, A = 1 , 2 , -., N
(4)
where , U A ~and , U A ~ depend on the temperature, the pressure, and N - 1 of the mole fractions. In order t o complete our statement of the interfacial conditions, we should develop the jump conditions associated with the species momentum equations; however, those equations normally undergo significant simplificationbefore being used in the solution of mass transfer problems and we need t o consider those simplifications before returning to eqs 2. In general, there are two routes that are followed in order to obtain solutions to eqs 1and 2, and these same two routes can be followed in order t o develop the interfacial flux boundary conditions that are based on eqs 3. We identlfy these two approaches as the di.’f)cclsion solution and the mechanical solution. In both cases one avoids the direct application of the N species momentum equations given by eqs 2, and instead one makes use of the N - 1 Stefan-Maxwell equations that can be extracted from those relations. To develop the StefanMaxwell equations, one represents the diffusive force as (Maxwell, 1860; Chapman and Cowling, 1970, p 109) and one makes some reasonable simplifications (Whi-
constraint has not been imposed on the species continuity equation given by eq 1 or on the total momentum equation given by eq 7. It has been pointed out that this is not valid for small times; however, one needs to be clear about what is meant by “small’ before dismissing the quasi-steady form of the Stefan-Maxwell equations. The constraint associated with the quasi-steady form of eq 6 is given by (Whitaker, 1986)
t* >> u*L, IC2
(8)
in which t* is the characteristic process time, u* is the characteristic velocity, Lu is the diffusive length, and C is the speed of sound. A little thought will indicate that it is very difficult to violate this constraint for problems of practical importance.
Diffusion Solutions There is a class of problems which have the important characteristic that the Nth equation is discarded in favor of some plausible restriction. Taylor and Krishna (1993, p 145) have listed three examples of such problems as. 1. Equimolar Counterdiffusion. This imposes a constraint on the velocities in a binary system that is given by
and this relation eliminates one of the species momentum equations indicated by eqs 2. In eq 9 we have used e to represent a unit base vector that is parallel to the one-dimensional diffusion process in order to make it clear that this type of restriction can only be imposed on one-dimensional processes. 2. Stefan Diffision. This is actually a restriction on the solubility of gaseous species B in a liquid composed primarily of species A. In the analysis of the Stefan diffusion tube this restriction leads to (Whitaker, 1991)
(5) taker, 1986) to eventually arrive at
One could also use the generalized Stefan-Maxwell equations (Bird et al., 1960, section 18.4; Slattery, 1981, section 8.4.5; Taylor and Krishna, 1993, section 2.2); however, we will make use of eq 6 to keep this analysis as simple as possible. To determine the N species velocities in eqs 1,we need N momentum equations, and the Nth momentum equation is traditionally taken to be the sum of eqs 2. This is given by
a $ev)
+ V Q w ) = gb + V*T, N t h equation
(7)
Equations 6 and 7, along with eqs 1,3,4, and the jump conditions associated with eqs 2, form the basig for the solution of mass transfer processes in multicomponent, multiphase systems. The species momentum equation given by eq 6 is obviously quasi-steady, whereas this
and this condition eliminates one of the species momentum equations indicated by eqs 2. Equation 10 is valid when species B is insoluble in the liquid phase. 3. Flux Ratios Fixed. This condition results from imposing stoichiometric constraints on molar fluxes. Most oRen this condition is associated with diffusion and reaction in porous catalysts, but it also occurs during the process of diffusion to a catalytic surface (Bird et al., 1960, section 17.3). Given a reaction of the type
2A-B
(11)
and a steady, one-dimensional process, one can argue that the fluxes are constrained by e N A
= -2e.N~
(12)
Under these circumstances the velocities must conform to
yx,)”
ev, = -2 (1 -
(13)
3510 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
and this eliminates the need for one of the species momentum equations given by eqs 2. There is a fourth case in which not only is the Nth equation avoided, but N - 2 of the Stefan-Maxwell equations are also avoided. This is the classic dilute solution diffusion process. 4. Dilute Solution Diffusion. In this case one considers the diffusion of species A in an N-component system under the special conditions
Given these restrictions one can use a single StefanMaxwell equation and the continuity equation for species A t o obtain (Whitaker, 1986)
The remaining N - 1momentum equations are satisfied by
which is the mathematical consequence of the physical restrictions given by eqs 14. Unlike the first three methods of avoiding the Nth equation, this special case is not restricted to one-dimensional processes.
in which Yrepresents the averaging volume shown in Figure 1 and A,&) represents the interfacial area contained within the averaging volume. In this work we are only concerned with the interfacial transport that we identify explicitly as
Iinterfacial flux 1 [to the y-phase
which clearly indicates the need t o understand and to develop useful representations for the term c ~ ( v 4 w)qY.While the formalism leading to eq 18 and the representation given by eq 19 is not often employed in chemical engineering texts, this detailed description of the physics is inherent in any analysis of mass transfer in a fluid-fluid system (Walker et al., 1937, Chapter XV). General Boundary Condition. Often the flux boundary condition at a /3-y interface is based upon the equality of the diffusive fluxes and is expressed as
Mechanical Solution When none of the four simplifications listed above are available, one simply makes use of the N momentum equations, as given by eqs 6 and 7, to develop solutions to multicomponent mass transfer problems. There are many examples of such solutions in the literature, and one of the early treatments is given by Dickson-Lewis (1968). More recently, Dandy and Coltrin (1994) and Meeks et al. (1994) have solved problems of this type; however, their studies do not include multiphase processes in which there is a moving interface. In such systems, the tradition appears t o be that one borrows one of the special cases given by the diffusion solutions in order to extract a flux boundary condition, and in this paper we wish to explore the general boundary condition that is consistent with the use of the Nth equation. This flux boundary condition represents a key aspect of the contaminant transport problem and other multiphase processes, and we demonstrate this in the following paragraphs. Volume-AveragedTransport Equation. To analyze the transport of chemicals in a system such as we have illustrated in Figure 1, one needs to develop volume-averaged transport equations (Anderson and Jackson, 1967; Marle, 1967, Slattery, 1967; Whitaker, 1967). As an example, we consider the continuity equation for species A in the /3-phase that can be expressed as
The volume-averaged form of this equation is given by (Whitaker, 1985)
and sometimes this flux relation is supplemented with a mass transfer constitutive equation of the form
Here kp represents a /3-phase mass transfer coefficient and e; is some suitable reference concentration. While ihese results might be applicable to certain special cases, they are not generally valid representations for the interfacial flux. To develop a general form of the boundary condition a t a moving interface, we begin with eq 6 written as B=NxAcB(vB o=-vxA+
B=l BtA
- W)
c@m
-c
B=Nx+A(vA B=l BtA
-W)
Cgm
(22)
in which w represents the velocity that provides the speed of displacement, w q Y . Since the flux, CA(VA w), is not involved in the second sum, we can express eq 22 as 0 = -cvxA
+ B=NxA[cB(vB
- w)] -
B=l
BtA
At this point it is convenient, but not necessary, to define the mixture diffisivity for species A by
Ind. Eng. Chem. Res., Vol. 34, No. 10,1995 3611 (24)
so that the momentum equation given by eq 23 can be expressed as
average momentum equation, it is not unreasonable t o neglect the jump conditions associated with eqs 2. In order to simplify the representations of eqs 26 and 29, we will make use of the nomenclature fA gA
= cA(vA - w).npy,A = 1 , 2 , ...,N
= c@AmVXA*npy, gN
(30a)
A = 1,2, ...,N - 1 (30b)
= -MN-'@(v
- w)
(30~)
so that our N momentum equations can be expressed as The normal component of this expression a t the P-y interface takes the form
[cA(vA
- w).nayl, a t Aay A = 1,2, ...,N - 1 (26)
We can use this result to determine N - 1 velocities (relative to w)a t the /3-y interface; however, we need to determine N velocities and this requires that we make use of the Nth equation. Here we simply note that eq 7, and the mass average momentum jump condition, will produce a value for the mass average velocity which we can express as B=N
v=
B=N
WBVB
or v
- w = C OJB(VB - W)
B=l
(27)
We now define the gradient vector as
B=l
Since the mass fraction of species B is given by OB
= MBcBIQ
(32) (28)
we can use the second of eqs 27 to obtain
O = M N- '
Q(v
- wpnpy-
and the flux vector as
B=N
C
B=l
MN-~MB[cB(vB
[
- w>npYl (29)
Here we have used MN t o represent the molecular weight of the Nth species. This result provides us with the Nth momentum equation to be used with eqs 26 t o construct a useful form of the jump conditions given by eqs 3. If one of the simplifications identified by the diffusion solution is justifiable, the use of eq 29 can be avoided and one of the classic representations for the interfacial flux will result. In the formulation of the mass flux boundary condition given by eqs 26 and 29, we neglected all surface excess functions associated with the species mass jump conditions, and this led to the special form of the jump conditions given by eqs 3. The jump condition associated with the mass average momentum equation given by eq 7 is available in Slattery (1990, section 2.1.6), and it will be used in the determination of the mass average velocities, vp and vy. At this point, it is important to recognize that we have completely ignored the species momentum jump conditions associated with eqs 2. This means that we have ignored any surface excess associated with PAB, and it means that the jump in the species momentum flux, @AVAVA, has been ignored. Given the fact that the essential features of the species momentum equation are very different than those for the mass
f = fc
(33)
so that eqs 31 can be expressed in compact matrix notation according to
o=-g-D.f
(34)
Here we have defined the difusiuity matrix in Table 1. The inverse of eq 34 provides us with the desired flux vector which is given by
f = -Hog
(36)
where H can be written explicitly as H = D-'
(37)
Because the form of the last entry of the g-vector is different from the first N - 1 entries, it is worthwhile to write out the expression for f A to obtain BEN-1
HABgB-HANgN
fA=B=l
(38)
3512 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 Table 1. Diffusivity Matrix
+
1
-
+
-MB MN
+
I
I+2
in which HABhas been used to represent the elements of the matrix, H. If we now use the definitions given by eq 30, we see that eq 38 takes the form
of a simultaneous solution of both the fluid mechanical and mass transfer problems.
Binary Systems For a binary system, one can easily solve the system of two equations to obtain the following representation for the flux of species A
B=N-l
cA(vA
- w h s v=
This result indicates how one can evaluate the interfacial flux contained in the volume-averaged transport equation given by eq 18 and identified explicitly in eq 19. Use of eq 39 in the jump condition given by eq 3 leads to the following flux boundary condition that describes the conditions at a moving interface. When the following two conditions are satisfied
Flux Boundary Condition
This result is considerably more complex than the flux B=N-1
we recover the classic diffusive flux given by CA(VA
(HM),&fN-'ey(vy - w b S y (40) relations suggested by eqs 20 and 21, and certainly the most distinctive difference is the convective transport term that involves g(v - w>.n,qy.This contribution results from the use of the mass average momentum equation to predict the mass average velocity, and if one can justify the use of any of the diffusion solution approximations given by eqs 9-16, this last term can be eliminated. To make use of these results for the flux at a moving fluid-fluid interface, one must be able to determine the fluid velocity, v, and the speed of displacement, wqy. The former is determined, for example, by the NavierStokes equations, while the latter must be determined as part of the problem solution (Crank, 1984). In the absence of mass transfer, the speed of displacement is directly tied to the mass average velocity by the relation
vpPy = wmpy= v,,*npy at Apy
(41)
and techniques for determining the speed of displacement are discussed by Stone and Leal (1989, 1990). When mass transfer occurs, eq 41 is no longer valid and one must determine the speed of displacement as part
- w h p y= -@AmCvxA*np,,
at Apy
(44)
One generally thinks of this as the dilute solution case that is valid when x~
