Page 120 - gas transport in porous media

Chapter 6: Conservation Equations
113
BC.
− n γκ ·∇d AA = n γκ ,
at A γκ
(6.223c)
Periodicity: d AA (r + i ) = d AA (r) , i = 1, 2, 3 (6.223b)
Second Problem for Species A
2
0 =∇ d AB (6.224a)

BC. − n γκ ·∇d AB = n γκ , at A γκ (6.224b)

Periodicity: d AB (r + i ) = d AB (r) , i = 1, 2, 3 (6.224c)

Third Problem for Species A

etc. (6.225)

N − 1 Problem for Species A

etc. (6.226)

To obtain these simplified forms, one must make repeated use of inequalities of
the form given by Eq. (6.214). Each one of these closure problems is identical to that
obtained by Ryan et al. (1981) and solutions have been developed by several workers
(Ryan et al., 1981; Ochoa-Tapia et al., 1994; Chang, 1982, 1983; Quintard, 1993;
Quintard and Whitaker, 1993a,b). In each case, the closure problem determines the
closure variable to within an arbitrary constant, and this constant can be specified by
imposing the condition

5
G = 1, 2, ... , N − 1
γ
γ
˜c Dγ = 0, or d GD = 0, (6.227)
D = 1, 2, ... , N − 1
However, any constant associated with a closure variable will not pass through the
filter in Eq. (6.195), thus this constraint on the average is not necessary.

6.5.1 Closed Form for Non-Dilute Diffusion
The closed form of Eq. (6.195) can be obtained by use of the representation for ˜c Eγ
given by Eq. (6.209), along with the definitions represented by Eqs. (6.222). After