May 19, 1987 - diffusional processes that take place in porous media, there exists an .... CA is a linear function of the gradient of the volume averaged .... ___ ~ _____ I ______ L __ ... In. Equation (1.27) we have used D xx , D xy , etc. to represent the .... Natural minerals such as clay and sandstone tend to be anisotropic.
Transport in Porous Media 2 (1987), 327-356
327
© 1987 by D. Reidel Publishing Company.
Diffusion in Anisotropic Porous Media JIN-HWAN KIM*, J. ALBERTO OCHOA, and STEPHEN WHIT AKER ** Department of Chemical Engineering, University of California at Davis, Davis, CA 95616, U.S.A. (Received: 2 September 1986; revised: 19 May 1987) Abstract. An experimental system was constructed in order to measure the two distinct components of the effective diffusivity tensor in transversely isotropic, unconsolidated porous media. Measurements were made for porous media consisting of glass spheres, mica particles, and disks made from mylar sheets. Both the particle geometry and the void fraction of the porous media were determined experimentally, and theoretical calculations for the two components of the effective diffusivity tensor were carried out. The comparison between theory and experiment clearly indicates that the void fraction and particle geometry are insufficient to characterize the process of diffusion in anisotropic porous media. Key words. Diffusion, anisotropy, diffusivity tensor, volume averaging.
o.
Nomenclature
Roman Letters d YK dye AYK a b CA
..
Co
(CA)Y CA
C 9lJ Deff
Dxx
Dyy
Deff
f h
interfacial area between 'Y- and K-phases for the macroscopic system, m 2 area of entrances and exits of the K-phase for the macroscopic system, m 2 interfacial area contained within the averaging volume, m 2 characteristic length of a particle, m average thickness of a particle, m concentration of species A, moles/m 3 reference concentration of species A, moles/m 3 intrinsic phase average concentration of species A, moles/m3 3 CA - (CAP, spatial deviation concentration of species A, moles/m (CA) Y / Co, dimensionless concentration of species A binary molecular diffusion coefficient, m 2/s effective diffusivity tensor, m 2/s component of the effective diffusivity tensor associated with diffusion parallel to the bedding plane, m 2/s component of the effective diffusivity tensor associated with diffusion perpendicular to the bedding plane, m2/s effective diffusivity for isotropic systems, m2/s vector field that maps V (CA) Y on to CA, m depth of the mixing chamber, m
*Current address: Department of Chemical Engineering, Chonnam National University, Korea. ** Author to whom correspondence should be addressed.
328 H
I Iy Ii
L
JIN-HWAN KIM ET AL.
EL / h, dimensionless parameter unit tensor characteristic length of the 'Y-phase, m lattice vectors (i = 1,2,3), m characteristic length for (CA)Y, m; and the depth of the sample chamber, m
ro
r t*
"if x X
length of the sides of a two-dimensional unit cell, m molar flux vector for species A, moles/m 2 s phase average of the molar flux vector for species A, moles/m 2s unit normal vector directed from the 'Y-phase toward the K-phase radius of the averaging volume, m' position vector, m time, s characteristic time, s; dimensionless time (tDxx / L 2) averaging volume, m 3 distance, m x / L, dimensionless distance
Greek Letters E
An 'T
Vy/"if, porosity nth eigenvalue tortuosity
1. Introduction Diffusion in porous media is a central issue in the subject of reactor design (Satterfield, 1970; Jackson, 1977; Luss, 1977) and in a wide variety of mass transfer operations (Cussler, 1984). The transport of nutrients to the roots of plants (Tinker, 1970) and the recovery of methane from coal beds (Smith and Williams, 1984) are processes which are intimately involved with diffusion in porous media. Under certain circumstances the rate of drying is controlled by diffusion of water vapor through a porous medium (Whitaker, 1977) and when drying involves biological materials the structure of the porous medium can be quite complex (Crapiste et al., 1986). Because of the complexity of the diffusional processes that take place in porous media, there exists an enormous body of literature dealing with both the experimental determination and the theoretical prediction of effective diffusivities. The complexities consist of simultaneous bulk, Knudsen, and surface diffusion in bidispersed systems that are difficult to characterize (Dullien, 1979). In realistic systems the diffusion takes place in the presence of adsorption, reaction, temperature gradients, and convective transport caused by mole number changes associated with chemical reaction or imposed pressure gradients. In addition, one must sometimes deal
.
DIFFUSION IN ANISOTROPIC POROUS MEDIA
329
with structural changes that occur during the diffusion process (Gavalas and Kim, 1981; Bhatia and Perlmutter, 1983). The theoretical approaches are as diverse as the problems and they include the 'dusty gas' (Mason and Malinauskas, 1983), random sphere models (Strieder and Aris, 1973), nonuniform capillary tube models (Foster and Butt, 1966), random capillary tube models (Johnson and Stewart, 1965; Gavalas and Kim, 1981), general statistical methods (Beran, 1968; Batchelor, 1974), effective medium theory (Webman, 1982) and multiscale methods (Bensoussan et al., 1978; Chang, 1982). The theory used in this study is based on the method of volume averaging (Anderson and Jackson, 1967; Marie, 1967; Slattery, 1967; Whitaker, 1967) and the closure scheme developed by Ryan et al. (1980, 1981). The general subject of heat and mass transport in porous media has been reviewed by Carbonell and Whitaker (1984) and the specific matter of diffusion has been considered by Whitaker (1986) with a special emphasis on micropore-macropore systems and the combined process of bulk diffusion, Knudsen diffusion and Darcy flow. In this work we are concerned with the simplest possible diffusion process, i.e., bulk diffusion of species A under dilute solution conditions with N B , N c , etc. = O(N A ,) or constrained by equi-molar counter diffusion in a binary system. The pressure and temperature are constant and there is neither adsorption nor reaction at the gas-solid interface. The system is illustrated in Figure 1 in which the solid is identified as the K-phase and the gas as the 'Y-phase. The governing
Averaging Volume V Fig. 1. A granular porous medium.
330
JIN-HWAN KIM ET AL.
point equation for the concentration of species A is (1.1 ) and the boundary conditions are (1.2) B.C.2
CA = @P(r, t),
(1.3)
on dye
Here d YK represents the entire interfacial area of the system and dye represents the area of entrances and exits of the y-phase for the macroscopic system. In general, the boundary condition at the entrances and exits is known only in terms of averaged quantities rather than the point concentration given in Equation (1.3), and this latter boundary condition serves to remind us of what we do not know rather than what we do know. The local volume averaged form of Equation (1.1), subject to Equation (1.2), is given by €
a(~)Y = V.[€rzlJ (V (CA)Y + ~y to'K DyKCA dA)].
(1.4)
Here € represents the void fraction, Vy is the volume of the y-phase contained in the averaging volume illustrated in Figure 1, and AyK represents the y-K interfacial area contained within the averaging volume. In deriving Equation (1.4), we have neglected variations of rzlJ within the averaging volume, and we have imposed the length scale constraint (Carbonell and Whitaker, 1984, Sec. 2)
(Lro)2 ~1
(1.5)
in which L represents the characteristic length scale for (CA)Y. The spatial deviation concentration is defined by CA
= CA -
(1.6)
(CA)Y
and a closure scheme is required in order to represent CA in terms of (CA)Y. One can follow the original development of Ryan et al. (1981) or the more compre~ hensive treatment of Crapiste et al. (1986) to find that the governing equations and boundary conditions for CA are given by (1.7) -D yK · rzlJV CA CA
=
