Page 118 - gas transport in porous media

111
Chapter 6: Conservation Equations
i = 1, 2, 3 ,
A = 1, 2, ... , N − 1 (6.212)
BC.2
b AE (r + i ) = b AE (r) ,
The derivation of Eqs. (6.210) and (6.211) requires the use of simplifications of
the form
γ γ
∇ b EA ·∇ c Aγ =∇b EA ·∇ c Aγ (6.213)
which result from the inequality
γ γ
b EA · ∇∇ c Aγ ∇b EA ·∇ c Aγ (6.214)
The basis for this inequality is the separation of length scales indicated by Eq. (6.117),
and a detailed discussion is available elsewhere (Whitaker, 1999). One should keep in
mind that the boundary value problem given by Eqs. (6.210) through (6.212) applies
to all N − 1 species and that the N − 1 concentration gradients are independent. This
latter condition allows us to obtain
E=N−1

γ
0 =∇ · D AE ∇b ED , D = 1, 2, ... , N − 1 (6.215)
E=1
E=N−1

γ γ
− n γκ · D AE ∇b ED = n γκ D AD ,at A γκ (6.216)
E=1
BC.1 D = 1, 2, ... , N − 1,

Periodicity: b AD (r + i ) = b AD (r), i = 1, 2, 3, D = 1, 2, ... , N − 1 (6.217)
At this point it is convenient to expand the closure problem for species A in order to
obtain
First Problem for Species A
3
γ γ −1 γ

0 =∇ · D AA ∇b AA + D AA D AB ∇b BA (6.218a)
4

γ −1 γ γ −1 γ

+ D AA D AC ∇b CA + ... + D AA D A, N−1 ∇b N−1, A
γ −1 γ

BC. − n γκ ·∇b AA − n γκ · D AA D AB ∇b BA (6.218b)
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 DA (r + i ) = b DA (r) , i − 1, 2, 3 , D = 1, 2, ... , N − 1
(6.218c)