Page 80 - gas transport in porous media

Chapter 6: Conservation Equations
We can choose the arbitrary velocity w to be equal to the species velocity v A so
that this result takes the form of a Reynolds transport theorem for species A 73
d ∂ψ A
ψ A dV = +∇ · (ψ A v A ) dV (6.9)
dt ∂t
V A (t) V A (t)
Here ψ A is any function associated with species A and if ψ A = ρ A this result can be
used with the axiom given by Eq. (6.5) to obtain

∂ρ A
+∇ · (ρ A v A ) − r A dV = 0, A = 1, 2, ... , N (6.10)
∂t
V A (t)
Here we note that the volume V A (t) is arbitrary in the sense that the Euler cut principle
(Truesdell, 1968) suggests that we can identify any region in space as the species body.
If we assume that the integrand in Eq (6.10) is continuous, the arbitrary nature of V A (t)
leads us to conclude that the integrand must be zero. Requiring that Eq. (6.10) be
satisfied leads to the species continuity equation
∂ρ A
+∇ · (ρ A v A ) = r A , A = 1, 2, ... , N (6.11)
∂t
which forms the basis for much of the analysis presented in this monograph.

6.1.1 Continuity Equation

To derive the continuity equation from the species continuity equation, we sum
Eq. (6.11) over all species and impose the axiom given by Eq. (6.6) to obtain

A=N A=N
∂
ρ A +∇ · ρ A v A = 0 (6.12)
∂t
A=1 A=1
We define the total mass density as
A=N

ρ = ρ A (6.13)
A=1
and note that the mass fraction is given by
ρ A
ω A = (6.14)
ρ
In terms of the total mass density Eq. (6.12) takes the form
A=N
∂ρ
+∇ · ρ A v A = 0 (6.15)
∂t
A=1