Page 88 - gas transport in porous media

81
Chapter 6: Conservation Equations
enthalpy per unit mass is defined by
ρ A h A = ρ A e A + p A (6.58)
in which p A is the partial pressure of species A. This is defined according to

2 ∂e A
p A = ρ (6.59)
A
∂ρ A s, ρ B , ρ C , ...
in which s is the total entropy per unit mass of the mixture. Following the approach
used for a Stokesian fluid, the species stress tensor is decomposed according to
T A =−I p A + τ A (6.60)
and Eqs. (6.58) through (6.60) can be used in Eq. (6.57) to obtain
B=N
∂
(ρ A h A ) +∇ · (ρ A h A v A ) =−∇ · q A + Q AB (6.61)
∂t
B=1
+∇v A : (−I p A ) +∇v A : τ A + r A e A
∂p A
+ +∇ · (p A v A )
∂t
We now make use of the definitions,
A=N A=N A=N

ρh = ρ A h A , q = q A , p = p A (6.62)
A=1 A=1 A=1
along with Axiom III, and sum Eq. (6.61) over all species to obtain
A=N A=N
∂
(ρh) +∇ · ρ A h A v A =−∇ · q + ∇v A : (− I p A ) (6.63)
∂t
A=1 A=1
A=N A=N

+ ∇v A : τ A + r A e A
A=1 A=1
A=N
∂p
+ +∇ · p A v A
∂t
A=1
On the basis of Eqs. (6.6), (6.26), and (6.54), and the velocity decomposition given
by Eq. (6.37) we can prove that
A=N A=N

1 2
r A e A + r A u = 0 (6.64)
2 A
A=1 A=1
In addition, we can use the velocity decomposition given by Eq. (6.37) along with
the stress decomposition given by Eq. (6.60) to obtain the following results:
A=N A=N

ρ A h A v A = ρhv + ρ A h A u A (6.65)
A=1 A=1