Page 79 - gas transport in porous media

Whitaker
72
in which r represents the time-dependent position of a material element whose
reference position is R. The velocity of any material element can be expressed as

dr Dr
v = = (6.2)
dt Dt
R
in which Dr/Dt is referred to as the material derivative. The motion of a species
body is described in an analogous manner, thus the motion of a material element of
species A is represented by (Slattery, 1999)
r A = r A (R A , t) (6.3)

and the velocity is given by

dr A
v A = (6.4)
dt
R A
In terms of the concept of a species body, we state the two axioms for the mass of
multicomponent systems as

d
AXIOM I: ρ A dV = r A dV, A = 1, 2, ... , N (6.5)
dt
V A (t) V A (t)
A=N

AXIOM II: r A = 0 (6.6)
A=1

Here ρ A represents the mass density of species A while r A represents the mass rate
of production per unit volume of species A owing to chemical reaction. In order to
extract a governing differential equation from Eq. (6.5), we first recall the general
transport theorem (Whitaker, 1981) for some scalar ψ and an arbitrary, continuous
velocity w

d ∂ψ
ψdV = dV + ψw · ndA (6.7)
dt ∂t
V a (t) V a (t) A a (t)

Here V a (t) represents an arbitrary volume, the surface of which has a speed of dis-
placement given by w · n. Use of the divergence theorem allows us to express the
general transport theorem in the form
d ∂ψ
ψdV = +∇ · (ψw) dV (6.8)
dt ∂t
V a (t) V a (t)