Page 121 - gas transport in porous media

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some algebraic manipulation, one obtains
⎡ ⎛ ⎞ Whitaker
γ
∂ c Aγ 1
⎢ γ ⎜ ⎟ γ
ε γ =∇ · ⎣ε γ D AA ⎝I + n γκ d AA dA⎠ ·∇ c Aγ (6.228)
∂t V γ
A γκ
⎛ ⎞

1
γ ⎜ ⎟ γ
+ ε γ D AB ⎝I + n γκ d AB dA⎠ ·∇ c Bγ
V γ
A γκ
⎛ ⎞

1
γ ⎜ ⎟ γ
+ ε γ D AC ⎝I + n γκ d AC dA⎠ ·∇ c Cγ
V γ
A γκ
⎛ ⎞ ⎤

1
γ ⎜ ⎟ γ ⎥
+ ε γ D A, N−1 ⎝I + n γκ d A, N−1 dA⎠ ·∇ c N−1γ ⎦
V γ
A γκ
On the basis of the closure problems given by Eqs. (6.223a) through (6.226), we
conclude that there is a single tensor that describes the tortuosity for species A. This
means that Eq. (6.228) can be expressed as
γ
∂ c Aγ eff γ eff γ
ε γ =∇ · ε γ D AA ·∇ c Aγ + ε γ D AB ·∇ c Bγ (6.229)
∂t

+ ε γ D eff ·∇ c Cγ + ...... + ε γ D eff ·∇ c N−1γ γ
γ
AC
A, N−1
in which the effective diffusivity tensors are related according to
D eff D eff D eff D eff
A, N−1
AC
AA
AB
γ = γ = γ = ..... = γ (6.230)
D AA D AB D AC D A, N−1
The remaining diffusion equations for species B, C, ... , N − 1 have precisely the
same form as Eq. (6.229), and the various effective diffusivity tensors are related to
each other in the manner indicated by Eqs. (6.230). The generic closure problem can
be expressed as
Generic Closure Problem
2
0 =∇ d (6.231a)
BC. − n γκ ·∇d = n γκ , at A γκ (6.231b)

Periodicity: d(r + i ) = d(r) , i = 1, 2, 3 (6.231c)