Experimental and Numerical Analysis of a PCM-Supported ...

S
(4.24)
Sáez and Carbonell [22] analyzed data for liquid holdup and pressure drop available
in the literature over a wide range of Reynolds and Galileo numbers in packed beds,
to determine the dependence of the relative permeability on the saturation for each
phase (Phase saturation refers to the phase volume per void volume). They
correlated the available information in the form:
2.43
K
rl
(4.25)
2.43
K
rg
S g
(4.26)
The quantity δ in Equation (4.25) is the reduced liquid saturation:
S S
(4.27)
0
l l
0
1
Sl
where
S is the reduced liquid saturation, Expressing krl in this form suggests that
0
l
the saturation corresponding to the static liquid holdup ( S
0 l
0
l
) consists of
l
essentially stagnant liquid. The use of a reduced liquid saturation in the formula for
the liquid phase relative permeability is common practice in two-phase flow analyses
[21].
In the first attempt to use the so called slit model (one of the porous media flow
models) to correlate experimental data, Holub et al. [23] analysis were identical in
form to equations (4.25) and (4.26), but with relative permeability functions for the
gas and liquid phases with a cubic dependence on saturation:
3
K
rl
S l
(4.28)
3
K
rg
S g
(4.29)
Holub et al. [23] compared the relative permeability functions in equations (4.31) and
(4.29) to the empirically determined results in equations (4.25), (4.26). They were
successful in fitting holdup and pressure drop data in the literature, provided that
Ergun’s constants were set to the values measured experimentally in each case [21].
For the laminar mixed or free convection flow induced by the combined thermal and
mass buoyancy forces, there is a buoyant force on the gas phase and also an
acceleration term, which should be included in Darcy’s law. Taking all of these flow
characteristics into consideration, we finally reach a form similar to Brinkman
equation. Brinkman equation extends Darcy’s law to include a term that accounts for
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