Page 117 - gas transport in porous media

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circumstances the boundary value problem for the spatial deviation concentration
takes the form Whitaker
E=N−1

γ
0 =∇ · D AE ∇˜c Eγ (6.206)
E=1
E=N−1

γ
BC.1 − n γκ · D AE ∇˜c Eγ (6.207)
E=1
E=N−1

γ γ
= n γκ · D AE ∇ c Eγ ,at A γκ
E=1
BC.2 ˜ c Aγ (r + i ) =˜c Aγ (r), i = 1, 2, 3 (6.208)
Here one must remember that the subscript A represents species A, B, C, ... , N − 1.
In this boundary value problem, there is only a single non-homogeneous term
γ
in the boundary condition imposed at the γ –κ interface. If
represented by ∇ c E γ
this source term were zero, the solution to this boundary value problem would be
given by ˜c Aγ = constant. Any constant associated with ˜c Aγ will not pass through the
filter in Eq. (6.195), and this suggests that a solution can be expressed in terms of
the gradients of the volume averaged concentration. Since the system is linear in the
N − 1 independent gradients of the average concentration, we are led to a solution of
the form
γ γ
˜ c Eγ = b EA ·∇ c Aγ + b EB ·∇ c Bγ (6.209)
γ γ
+ b EC ·∇ c Cγ + ..... + b E, N−1 ·∇ c N−1γ
Here the vectors, b EA , b EB , etc., are referred to as the closure variables or the mapping
variables since they map the gradients of the volume averaged concentrations onto
the spatial deviation concentrations. In this representation for ˜c Aγ , we can ignore the
γ
γ
spatial variations of ∇ c Aγ , ∇ c Bγ etc., within the framework of a local closure
problem, and we can use Eq. (6.209) in Eq. (6.206) to obtain

E=N−1 D=N−1

γ γ
0 =∇ · D AE ∇b ED ·∇ c Dγ (6.210)
E=1 D=1
E=N−1 D=N−1
γ γ
BC.1 − n γκ · D AE ∇b ED ·∇ c Dγ (6.211)
E=1 D=1
E=N−1
γ γ
= n γκ · D AE ∇ c Eγ , at A γκ
E=1