Page 92 - gas transport in porous media

Chapter 6: Conservation Equations
Use of this result with Eq. (6.82) allows us to express the species momentum equation
in the form 85

∂
−1
p (ρ A v A ) +∇ · (ρ A v A v A ) − ρ A b A + x A ∇p −∇ · τ A − r A v A (6.84)
∂t
B=N
x A x B (v B − v A )
=−∇x A + , A = 1, 2, ... , N
D AB
B=1
The left hand side of this result can be simplified by the use of Eqs. (6.11) and the
total momentum equation given by Eq. (6.49). The algebra is rather lengthy and the
result is given by

−1 ∂u A −1
p ρ A + v A ·∇u A + u A ·∇v − p (ω A − x A )∇p (6.85)
∂t
+ p −1 (ω A ∇· τ −∇ · τ A ) − p −1 ρ A (b − b A )+
B=N
x A x B (v B − v A )
=−∇x A + , A = 1, 2, ... , N
D AB
B=1
When the two terms on the right hand side are dominant, all of the terms on the left
hand side can be neglected. We express this idea as

p −1 ρ A ∂u A + v A ·∇u A + u A ·∇v − p −1 (ω A − x A )∇p (6.86)
∂t
+ p −1 (ω A ∇· τ −∇ · τ A ) − p −1 ρ A (b − b A )
∇x A , A = 1, 2, ... , N

which leads us to the N −1 independent Stefan-Maxwell equations that take the form
(Bird et al., 2002)

B=N
x A x B (v B − v A )
0 =−∇x A + , A = 1, 2, ... , N − 1 (6.87)
D AB
B=1
In this approach, one uses Eq. (6.49) to determine the mass average velocity, v, and
Eq. (6.87) to determine N − 1 species velocities. There are numerous problems of
mass transport in porous media for which the simplifications leading to Eq. (6.87)
are not valid. Some of these cases are described in the classic work of Jackson (1977)
on transport in porous catalyst and in the summary of the dusty gas model by Mason
and Malinauskas (1983). More recent studies are described in the work of Kerkhof
(1996, 1997).