Page 99 - gas transport in porous media

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Since the intrinsic and superficial averages differ by a factor of ε γ , it is essential to
make use of a notation that clearly distinguishes between the two averages. Whitaker
When we form the volume average of any transport equation, we are immediately
confronted with the average of a gradient (or divergence) whereas it is the gradient (or
divergence) of the average that we are seeking. In order to interchange integration and
differentiation, we will make use of the spatial averaging theorem (Whitaker, 1999).
For the two-phase system illustrated in Figure 6.3 this theorem can be expressed as

1
∇ψ γ =∇ ψ γ + n γκ ψ γ dA (6.116)
V
A γκ
in which ψ γ is any function associated with the γ -phase. Here A γκ represents the
interfacial area contained within the averaging volume, and we have used n γκ to
represent the unit normal vector pointing from the γ -phase toward the κ-phase. In
order for the method of volume averaging to lead to a successful result, the length
scales illustrated in Figure 6.3 must be disparate as indicated by
γ r o L (6.117)
When these constraints are satisfied, Eq. (6.116) can be used to derive an alternate
form given by
1
γ
˜
∇ψ γ = ε γ ∇ ψ γ + n γκ ψ γ dA (6.118)
V
A γκ
in which ˜ ψ γ is the spatial deviation defined according to
γ
˜ ψ γ = ψ γ − ψ γ (6.119)
To avoid errors in the development of closed-form volume averaged transport
equations, Gray (1975) has pointed out that one should define the spatial deviation
in terms of the intrinsic average as indicated by Eq. (6.119).

6.2.1 Mass Transfer
In order to explore the transport of species A in the porous medium illustrated in
Figure 6.3, we begin with Eq. (6.19) in the form
∂c Aγ
+∇ · (c Aγ v Aγ ) = R Aγ , A = 1, 2, ... , N (6.120)
∂t
If the γ –κ system is rigid and surface transport can be neglected, one can use Eq.
(6.120) to derive a jump condition (Slattery, 1990; Edwards et al., 1991; Whitaker,
1992) of the form
∂c As
= c Aγ v Aγ · n γκ + R As , at the γ − κ interface, A = 1, 2, ... , N (6.121)
∂t