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

where β t , β c , ρ ref , c ref , and Т ref are the volumetric thermal expansion coefficient,
volumetric diffusion (concentration) expansion coefficient, reference density,
reference concentration, and reference temperature respectively. Single phase flow
in a dense porous medium with low flow velocity is typically governed by Darcy’s
law, which relates the fluid velocity v to the pressure drop ∆P over a distance ∆L,
using the viscosity μ and the absolute permeability K of the medium as proportional
constants.
K P
v
L
(4.20)
Ergun determined an empirical relationship between K, d, the pore size, and the void
fraction ε for spherical particles:
2 3
d
K
(4.21)
150
1
2
It can also be derived from Karman-Kozeny’s hydraulic radius theory [19]; in this
case the constant is 180 instead of 150 and d represents the particle (bead)
diameter. Darcy’s law can then be used to derive relations between the velocity and
the pressure gradient for each of the two fluid phases as follows, (Nield and Bejan
[20]):
v
v
l
g
K
rlK
P
l
g
(4.22)
l
K
rgK
P
g
g
(4.23)
g
For the case of two phase flow in equations (4.22) and (4.23), the concept of relative
permeability K r has been introduced by Sáez and Carbonell [22] in momentum
balance equations. The idea behind the concept of relative permeability is that an
expression for the drag force for single-phase flow is used. However, this has to be
altered (divided) by a certain parameter to take into account the presence of the
second phase. This parameter is known as the relative permeability of a certain
phase (krα). The relative permeability can in fact be viewed as the ratio between the
effective permeability of a certain phase within two-phase flow (keff,α) and the
permeability of the bed (k). The relative permeabilities of the gas and liquid phases
are normally assumed to be primarily a function of the saturation S of each phase
[21]:
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