Page 113 - gas transport in porous media

106
we can impose the simplification
γ
(6.186)
c γ = c γ = constant Whitaker
Imposition of this condition means that there are only N − 1 independent transport
equations of the form given by
E=N−1
∂c Aγ
=∇ · D AE ∇c Eγ , A = 1, 2, ... , N − 1 (6.187)
∂t
E=1
and for the case of passive transport the boundary conditions at the fluid-solid
boundary are given by

E=N−1

− n γκ · D AE ∇c Eγ = 0, at the γ –κ interface (6.188)
E=1
A = 1, 2, ... , N − 1,

One can make use of the averaging theorem and the no-flux boundary condition to
show that the volume averaged form of Eq. (6.187) is given by

γ E=N−1
∂ c Aγ
ε γ =∇ · D AE ∇c Eγ , A = 1, 2, ... , N − 1 (6.189)
∂t
E=1
At this point we decompose the elements of the diffusion matrix according to
γ ˜
D AE = D AE + D AE (6.190)
γ
˜
and neglect D AE relative to D AE so that Eq. (6.189) simplifies to
γ E=N−1
∂ c Aγ γ
ε γ =∇ · D AE ∇c Eγ , A = 1, 2, ... , N − 1 (6.191)
∂t
E=1
We can represent this simplification as,

˜
D AE D AE γ (6.192)
and when it is not satisfactory it may be possible to develop a correction based on
˜
the retention of the spatial deviation, D AE . The inequality represented by Eq. (6.192)
is equivalent to ignoring variations of D AE within the averaging volume.
The volume averaging theorem can be used with the average of the gradient in Eq.
(6.191) in order to obtain

1
∇c Eγ =∇ c Eγ + n γκ c Eγ dA (6.193)
V
A γκ