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

the viscous transport and introduces velocities in special directions as dependent
variables. The flow field can be determined by solving the Brinkman equation for the
momentum balance in combination with the continuity equation. Applying Brinkman
equation on two phase flow by using the relative permeability K r , the gas phase
momentum balance equation reads:
vg
)
t
g
( p
g vg
gvg
(4.30)
KK
rg
where μ is the dynamic viscosity, K is the permeability, p
is the pressure gradient,
and ρ is the varying gas density. The buoyant force appears automatically when an
equation of state is inserted into the equation of motion (4.30) in the (ρg) term (but
not into the acceleration term ( v t
)).
v
t
rel,
g
T
T
c
c
g
( p
ref
g)
vg
gvg
ref
g t
ref c ref
(4.31)
KK
The form of equation (4.31) is similar to the Boussinesq equation for forced and free
convection in non-isothermal flow in free (non-porous media) flow when the second
and third term on the right hand side can be omitted. Therefore, this form of the
equation of motion is very useful for heat and mass transfer analysis. For forced
convection the buoyant term g
T
T
c
c
ref
t
ref
c
ref
, which represents the
momentum transfer caused by the combined buoyancy forces, can be neglected
while for free convection the term ( p
g)
is small and omitting it is usually
appropriate [19]. It is also customary [19] to replace ρ on the left side of equation
(4.31) by the gas density at inlet conditions. This substitution has been successful for
free convection at moderate temperature differences. Under these conditions the
fluid motion is slow, and the acceleration term v t
is small compared to g [19].
4.2.2.2 Flow in the condenser
Figure 4.5a shows the physical system of direct contact condensation in the PCM
condenser. Figure 4.5b shows a schematic illustration of energy and mass flow in a
rectangular finite element of the packed bed. Comparing figure (4.5) and figure (4.4),
due to similarity between the evaporator and condenser designs, a similar set of
coupled partial differential equations (4.11) – (4.18) as well as other state variable
equations are found to represent heat and mass balances for the condenser.
Altering the signs of the last two terms in both equations (4.12) and (4.13), the same
form of the energy and mass balance equations could be applied on the
condensation bed, with some modifications to take into account the condensation on
the solid surface that might take place in parallel with the absorption of water vapor
ref
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