Page 112 - gas transport in porous media

Chapter 6: Conservation Equations
This clearly indicates that the pulse velocity is given by
γ 105
v γ
(6.181)
u p =
−1
1 + ε γ a v K eq
and that the dispersion tensor in this intrinsic averaged mass transport equation is
a complex function of the equilibrium coefficient K eq . It is convenient to express
Eq. (6.180) as
γ γ
∂ c Aγ v γ γ γ
·∇ c Aγ = D ∗∗
+
γ : ∇∇ c Aγ (6.182)
∂t −1
1 + ε γ a v K eq
in which the dispersion tensor takes the form

D ∗ γ 1 + ε γ −1 a v K eq + K eq d γ v γ γ
D ∗∗ =
(6.183)
γ
−1 2
1 + ε γ a v K eq
In order to determine D ∗∗ one must solve the two closure problems given by
γ
Whitaker (1997), and methods of solution are described by various authors (Eidsath
et al., 1983; Sahraoui and Kaviany, 1994; Quintard and Whitaker, 1994f, 1995).
In the previous sections we have given several examples of mass and momentum
transport for dilute solutions, and there are many other processes involving hetero-
geneous reaction, non-linear adsorption, slightly compressible flow, etc., that can be
analyzed in the same manner. The more general mass transfer problem based on Eq.
(6.111) is much more complex, and in the following section we treat the simplest case
of non-dilute solution transport in porous media.
6.5 CLOSURE FOR NON-DILUTE DIFFUSION

In this section we consider the simplest possible case of non-dilute mass transfer in
porous media. This occurs when there is no homogeneous reaction, no heterogeneous
reaction, no adsorption, negligible convective transport, and the conditions are such
that the total molar concentration can be treated as a specified constant. Even with
these severe simplifications, the analysis is complex; however, the result is relatively
simple, i.e., there is a single tortuosity tensor that can be used to determine all the
effective diffusivity tensors. For negligible convective transport and no homogeneous
reaction, Eq. (6.111) simplifies to

E=N−1
∂c Aγ
=∇ · c γ D AE ∇x Eγ , A = 1, 2, ... , N (6.184)
∂t
E=1
and we are confronted with a purely diffusive problem. Including heterogeneous
reaction is straightforward and the details are given by Arce et al. (2005). When the
total molar concentration is constrained by
x Aγ ∇c γ c γ ∇x Aγ , A = 1, 2, ... , N (6.185)