Page 116 - gas transport in porous media

Chapter 6: Conservation Equations
109
condition imposed at the γ –κ interface, we have identified the diffusive flux, and a
non-homogeneous term referred to as the diffusion source. If the diffusion source in
Eq. (6.200) were zero, the ˜c Aγ field would be generated only by the non-homogeneous
terms that might appear in the boundary condition imposed at A γ e or in the initial
condition given by Eq. (6.202). One can easily develop arguments indicating that the
closure problem for ˜c Aγ is quasi-steady, thus the initial condition is of no importance
(Whitaker, 1999, Chapter 1). In addition, one can develop arguments indicating that
the boundary condition imposed at A γ e will influence the ˜c Aγ field over a negligibly
small portion of the field of interest. Because of this, any useful solution to the
closure problem must be developed for some representative region which is most
often conveniently described in terms of a unit cell in a spatially periodic system.
These ideas lead to a closure problem of the form
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢E=N−1 ⎥ ⎢ E=N−1 γ ⎥
⎢ D AE ⎥
⎢ γ ⎥
D AE ∇˜c Eγ ⎥ −∇ · ⎢ n γκ ˜c Eγ dA ⎥ (6.203)
0 =∇ · ⎢
⎢ ε γ V ⎥
⎢ ⎥
⎣ E=1 ⎦ ⎢ E=1 ⎥
⎣ A γκ ⎦

diffusion
non-local diffusion
E=N−1

γ
− n γκ · D AE ∇˜c Eγ
E=1

diffusive flux
E=N−1
γ γ
BC.1 = n γκ · D AE ∇ c Eγ , at the γ –κ interface (6.204)
E=1

diffusive source
BC.2 ˜ c Aγ (r + i ) =˜c Aγ (r), i = 1, 2, 3 (6.205)
Here we have used i to represent the three base vectors needed to characterize
a spatially periodic system. The use of a spatially periodic system does not limit
this analysis to simple systems since a periodic system can be arbitrary complex
(Quintard and Whitaker, 1994a–e). However, the periodicity condition imposed by
γ
γ
γ
Eq. (6.205) can only be strictly justified when D AE , c Aγ , and ∇ c Aγ are
constants and this does not occur for the types of systems under consideration. This
matter has been examined elsewhere (Whitaker, 1986a) and the analysis suggests
γ
γ
that the traditional separation of length scales allows one to treat D AE , c Aγ , and
γ
∇ c Aγ as constants within the framework of the closure problem.
One can show (Whitaker, 1999) that the non-local diffusion term is negligible
when the length scales are separated as indicated in Eq. (6.117), and under these