Page 84 - gas transport in porous media

Chapter 6: Conservation Equations
the total momentum equation. This is given by
A=N A=N A=N A=N 77
∂
(ρ A v A ) +∇ · (ρ A v A v A ) = ρ A b A +∇ · T A (6.31)
∂t
A=1 A=1 A=1 A=1
in which Axioms III and IV have been used to eliminate the sum of the last two terms
in Eq. (6.29). We can define the following quantities
A=N

ρ = ρ A , total density (6.32)
A=1
ρ A
ω A = , mass fraction (6.33)
ρ
A=N

v = ω A v A , mass average velocity (6.34)
A=1
A=N

b = ω A b A , mass average body force (6.35)
A=1
so that Eq. (6.31) takes the form
A=N A=N
∂
(ρv) +∇ · ρ A v A v A = ρb +∇ · T A (6.36)
∂t
A=1 A=1
In order to extract a simplified form of the convective acceleration, we need to
introduce the important concept of a diffusion velocity.

6.1.5 Diffusion Velocity
There are two diffusion velocities that are commonly used; the mass diffusion veloc-
ity and the molar diffusion velocity. The first of these is defined according to the
decomposition

v A = v + u A (6.37)
and one can use this to show that that the mass diffusion velocities are constrained by
A=N

ρ A u A = 0 (6.38)
A=1
One can use the representation given by Eq. (6.37) in order to express the convective
inertial term in Eq. (6.36) as
A=N A=N

ρ A v A v A = ρ A v A (v + u A ) (6.39)
A=1 A=1