Page 122 - gas transport in porous media

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Chapter 6: Conservation Equations
and solution of this boundary value problem is relatively straightforward. The exis-
tence of a single, generic closure problem that allows for the determination of all the
effective diffusivity tensors represents the principle result of this section. On the basis
of this single closure problem, the tortuosity tensor is defined according to

1
τ = I + n γκ d dA (6.232)
V γ
A γκ
and we can express Eqs. (6.230) in the form

γ
D eff = τ D AA , D eff = τ D AB , ...., D eff = τ D A, N−1 γ (6.233)
γ
AA
A, N−1
AB
Substitution of these results into Eq. (6.229) allows us to represent the local volume
averaged diffusion equations as

γ E=N−1
∂ c Aγ γ γ
ε γ =∇· ε γ τ D AE ·∇ c Eγ , A = 1, 2, ... , N − 1 (6.234)
∂t
E=1
It is important to remember that this analysis has been simplified on the basis of
Eq. (6.186) and the problem becomes more complex when the total molar concen-
tration cannot be treated as a constant. For a porous medium that is isotropic in the
volume averaged sense, the tortuosity tensor takes the classical form

τ = I τ −1 (6.235)

in which I is the unit tensor and τ is the tortuosity. For isotropic porous media, we
can express Eq. (6.234) as

γ
∂ c Aγ E=N−1
ε γ γ γ

ε γ =∇· D AE ∇ c Eγ , (6.236)
∂t τ
E=1
A = 1, 2, ... , N − 1
Often ε γ and τ can be treated as constants; however, the diffusion coefficients in this
transport equation will be functions of the local volume averaged mole fractions, and
this means that we are faced with a coupled, non-linear diffusion problem. Future
studies of non-dilute solutions need to include convective effects along with adsorp-
tion and heterogeneous reaction. In addition, the restriction imposed by Eq. (6.186)
needs to be removed.