Page 114 - gas transport in porous media

Chapter 6: Conservation Equations
and one can follow an established analysis (Whitaker, 1999, Chapter 1) in order to
express this result as 107
1
γ
∇c Eγ = ε γ ∇ c Eγ + n γκ ˜c Eγ dA (6.194)
V
A γκ
Use of this result in Eq. (6.191) provides
⎡ ⎛ ⎞⎤
⎢ ⎜ ⎟⎥
γ ⎢ E=N−1 ⎜ ⎟⎥
∂ c Aγ ⎢ ⎜ γ 1 ⎟⎥
⎜ γ ∇ c Eγ +
ε γ =∇ · ⎢ D AE γ ⎜ ε n γκ ˜c Eγ dA ⎟⎥ (6.195)
∂t ⎢ V ⎟⎥
⎢ E=1 ⎜ ⎟⎥
A γκ
⎣ ⎝ ⎠⎦

filter
in which the area integral of n γκ ˜c Eγ has been identified as a filter. Not all the infor-
mation available at the length scale associated with ˜c Eγ will pass through this filter
γ
to influence the transport equation for c Aγ , and the existence of filters of this type
is a recurring theme in the method of volume averaging (Whitaker, 1999).
In order to obtain a closed form of Eq. (6.195), we need a representation for the
spatial deviation concentration, ˜c Aγ , and this requires the development of the closure
problem. To develop this closure problem, we return to Eq. (6.187) and make use of
Eqs. (6.190) and (6.192) to obtain
E=N−1
∂c Aγ γ
=∇ · D AE ∇c Eγ , A = 1, 2, ... , N − 1 (6.196)
∂t
E=1
If we ignore variations in ε γ and subtract Eq. (6.195) from Eq. (6.196), we can arrange
the result as
⎡ ⎤
⎡ ⎤
E=N−1 E=N−1 γ
∂˜c Aγ D AE 1
γ ⎢ ⎥
=∇ · ⎣ D AE ∇˜c Eγ ⎦ −∇ · ⎣ n γκ ˜c Eγ dA⎦ (6.197)
∂t ε γ V
E=1 E=1
A γκ
in which it is understood that this result applies to all N − 1 species. In order to
develop a boundary condition for the spatial deviation concentration, we again make
use of the concentration decomposition given by the first of Eqs. (6.131) along with
Eqs. (6.190) and (6.192) to obtain
E=N−1

γ
− n γκ · D AE ∇˜c Eγ (6.198)
E=1
E=N−1
γ γ
= n γκ · D AE ∇ c Eγ , at the γ –κ interface
E=1