Page 102 - gas transport in porous media

Chapter 6: Conservation Equations
95
The analysis of the convective transport is not simple; however, reasonable argu-
ments are available (Whitaker, 1999, Sec. 3.2.3) indicating that when the length-scale
constraint indicated by Eq. (6.117) is valid Eq. (6.132) simplifies to
γ γ
v γ c Aγ = ε γ v γ c Aγ + ˜v γ ˜c Aγ (6.133)
Use of this result in Eq. (6.130) leads to
γ
∂ c Aγ γ γ
ε γ +∇ · v γ c Aγ = ∇ · J Aγ +∇ · ˜v γ ˜c Aγ + R Aγ (6.134)
∂t
dispersive
transport
∂ c As γκ
− a v + a v R As γκ
∂t
where the volume average velocity has been expressed in terms of the superficial
average rather than the intrinsic average since it is the superficial average that appears
in the traditional form of Darcy’s law. In the following section we indicate how v γ
can be determined and we then return to the problem of expressing the diffusive and
dispersive flux in some useful form.

6.3 MOMENTUM TRANSFER
An analysis of momentum transfer is based on the continuity equation and the linear
momentum equation. The former was given earlier as Eq. (6.17) and represented here
in the form

∂ρ γ
+∇ · (ρ γ v γ ) = 0 (6.135)
∂t
When the density can be treated as a constant, this reduces to

∇· v γ = 0 (6.136)

and the local volume averaged form is given by
∇· v γ = 0 (6.137)

This form of the continuity equation is an attractive approximation for the case of
dilute solution mass transfer described by Eqs. (6.88) through (6.98). For non-dilute
solutions, the approximation given by Eq. (6.136) may not be acceptable.
In the total momentum equation given by Eq. (6.49), it is often reasonable to
ignore variations in the density and viscosity in the viscous term so that the governing
equation for v γ is given by

2
∂
ρ γ v γ +∇ · ρ γ v γ v γ = ρ γ g −∇p γ + µ∇ v γ (6.138)
∂t