Page 264 - gas transport in porous media

261
Chapter 14: Experimental Determination of Transport Parameters
o
amount (usually 500–900 Pa), P , and the gas inlet is closed. The time-change of
the pressure difference between compartments, P(t), is followed by the differential
pressure transducer.
p
The constitutive law for molar permeation flux density, N (t, x), of a pure gas in
a porous medium under isothermal conditions is given by the single component form
of the Darcy equation (14.10) (Dullien, 1979).
∂p/R g T
p
N =−B (14.32)
∂x
where p(t, x) is the gas pressure inside the porous medium. For single component
permeation the effective permeability coefficient (Eq. (14.11)) simplifies to Weber
equation (14.33)
ω + Kn 2 p
B = r ψK + r ψ (14.33)
1 + Kn 8η
for both MTPM and DGM models. x the pellets length co-ordinate, T the temperature,
R g the gas constant and t the time. In the majority of cases it appears that ω = 1
is a good approximation. Hence, in the following the simplified form of Eq. (14.33)
is used
p
2
B = r ψK + r ψ (14.34)
8η
Cellbalance. Becausetheporevolumeofporouspelletsmountedintheimpermeable
disk is much smaller than the volume of compartments the gas accumulation in the
p
pores can be neglected, that is, dN /dx = 0.
After sufficiently long time the pressure difference between cell compartment tends
L
U
to zero and pressures in the upper and lower compartments, P , P , are obviously
o
o
L
U
P (t →∞) = P (t →∞) = p = P + P /2 (14.35)
p
Using the assumption of negligible gas accumulation in pores (dN /dx = 0) the
gas conservation for cell compartments of identical volume can be expressed as
U
L
dP /dt =−dP /dt. The material balance then reads
1 d P SN p
=−2 (14.36)
R g T dt V
where V is the compartment volume (V = V L = V U ). Using the Darcy equation
(14.32) and the simplified Weber law (14.34), integration of Eq. (14.36) (from x = 0,
L
U
where p = P to x = L, where p = P ) yields
2
r ψ
p
N R g TL = r ψK P + P P (14.37)
8µ