Page 87 - gas transport in porous media

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be expressed as
AXIOM I: Whitaker
B=N

d 1 2
(ρ A e A + ρ A v )dV =− q A · ndA + Q AB dV (6.53)
A
2
dt
B=1
V A (t) A A (t) V A (t)

+ v A · t A(n) dA + ρ A b A · v A dV
A A (t) V A (t)
B=N

+ v A · P AB dV
B=1
V A (t)

2
1
+ r A (ρ A e A + ρ A v )dV
2 A
V A (t)
AXIOM II:
A=N
1 2
r A (e A + v ) = 0 (6.54)
2 A
A=1
AXIOM III:
A=N B=N

Q AB = 0 (6.55)
A=1 B=1
AXIOM IV:
e A is a function of the state of the system (6.56)
Here we have used e A to represent the internal energy per unit mass of species A,we
have used q A to represent the heat flux (conductive and radiative) transferred from
the surroundings to species A, and we have used Q AB to represent the volumetric rate
of transfer of thermal energy from species B to species A. Determination of e A on
the basis of Eq. (6.53) would be extremely difficult; however, the axiom given by
Eq. (6.56) indicates that this is unnecessary and instead we need only determine the
temperature, pressure, and composition of the system in order to determine e A .
To produce a useful result from Eq. (6.53), we first derive the governing differential
2
1
equations for e A + ρ A v . The species mechanical energy equation is then subtracted
2 A
from the species total energy equation to obtain the species thermal energy equation.
This is a rather lengthy process (Whitaker, 1989) that leads to the governing equation
for the species energy.
B=N
∂
(ρ A e A ) +∇ · (ρ A e A v A ) =−∇ · q A + Q AB +∇v A : T A + r A e A (6.57)
∂t
B=1
There are a variety of paths that one can follow at this point; however, the development
of an enthalpy transport equation seems to be the most productive route. The species