Page 91 - gas transport in porous media

Whitaker
84
Stefan-Maxwell Equations
6.1.7
The species momentum equation given by Eq. (6.29) and repeated here as Eq. (6.79),
would appear to be overwhelmingly complex; however, some reasonable simplifica-
tions will lead us from that complex result to the relatively simple form given by the
Stefan-Maxwell equations.
B=N
∂
(ρ A v A ) +∇ · (ρ A v A v A ) = ρ A b A +∇ ·T A + P AB (6.79)
∂t
B=1
convective body surface
local
acceleration force force
acceleration diffusive
force
+ r A v A , A = 1, 2, ... , N

source of momentum
owing to reaction
We begin our analysis of the species momentum equation by making use of the
following representation for the species stress tensor

T A =−p A I + τ A , viscous fluid (6.80)

Here ρ A is the partial pressure of species A and this representation is only valid for
an isotropic, inelastic fluid. In addition to this limitation, we restrict our analysis to
ideal solutions so that the partial pressure can be expressed as

p A = x A p , ideal solution (6.81)

in which x A is the mole fraction of species A and p is the total pressure. Use
of Eqs. (6.80) and (6.81) in the species momentum equation given by Eq. (6.29)
leads to

∂
(ρ A v A ) +∇ · (ρ A v A v A ) = ρ A b A − x A ∇p − p∇x A +∇ · τ A (6.82)
∂t
B=N

+ P AB + r A v A , A = 1, 2, ... , N
B=1
Our next step in the analysis of the species momentum equation is the use of Maxwell’s
representation for the force P AB which we express as (Chapman and Cowling, 1970,
pg. 109)

px A x B (v B − v A )
P AB = , A, B = 1, 2, ... , N (6.83)
D AB