Page 119 - gas transport in porous media

112
Second Problem for Species A
3 Whitaker
γ γ −1 γ
0 =∇ · D AB D AB D AA ∇b AB +∇b BB (6.219a)
4
γ −1 γ γ −1 γ

+ D AB D AC ∇b CB + ... . + D AB D A, N−1 ∇b N−1, B

γ −1 γ
BC. − n γκ ·∇b AA − n γκ · D AA D AB ∇b BA (6.219b)
γ −1 γ

−n γκ · D AA D AC ∇b CA − ... .
γ −1 γ

− n γκ · D AA D A, N−1 ∇b N−1, A = n γκ , at A γκ
Periodicity: b DB (r + i ) = b DB (r) , i = 1, 2, 3 , D = 1, 2, ... , N−1
(6.219c)
Third Problem for Species A
etc. (6.220)

N − 1 Problem for Species A
etc. (6.221)

Here it is convenient to define a new set of closure variables or mapping variables
according to

γ −1 γ γ −1 γ

d AA = b AA + D AA D AB b BA + D AA D AC b CA (6.222a)
γ −1 γ

+ ... . + D AA D A, N−1 b N−1, A
γ −1 γ γ −1 γ

d AB = D AB D AA b AB + b BB + D AB D AC b CB (6.222b)
γ −1 γ

+ ... . + D AB D A, N−1 b N−1, B
γ −1 γ γ −1 γ

d AC = D AC D AA b AC + D AC D AB b BC (6.222c)
γ −1 γ

+ b CC + ... . + D AC D A, N−1 b N−1,C
etc. (6.222n − 1)
With these definitions, the closure problems take the following simplified forms:

First Problem for Species A
2
0 =∇ d AA (6.223a)