Design Improvement of a Vertically Oriented Thermal …
249
Continuity equation:
∂ρ
∂t
+
∂(ρu i )
∂ x i
= 0
( 1 )
Momentum equation:
∂(ρu i )
∂t
+
∂
ρu i u j
∂ x j
=
∂τ i j
∂ x j
+ ρβg i (T i − T ref ) + S i
(2)
where ρ is the density, u is the fluid velocity, τ is the shear stress, p is the pressure, g is
the gravitational acceleration and T ref is the reference temperature. The Boussinesq
approximation is considered to deal with the natural convection in the melted PCM
region. The source term S i derived from the Darcy’s law is expressed as,
S i = A mush (1 − f )
2
u i
f 3 + e
(3)
where f is the liquid volume fraction and A mush is mushy zone constant ranges from
10
4 to 10
7 . The term e is a small value (taken as 0.001) considered to avoid division
by zero.
Energy equation:
∂(ρ H )
∂t
+
∂(ρu i H )
∂ x i
=
∂
∂ x i
k
∂ T
∂ x i
(4)
where k is the thermal conductivity and T is temperature. H and h are the total and
sensible enthalpy, respectively, and are related as follows:
H = h + f L
(5)
The liquid volume fraction (f ) can be obtained with the help of the lever rule as
follows:
f =
⎧
⎪ ⎨
⎪ ⎩
0
if T < T sol
T −T sol
T liq −T sol
if T sol < T < T liq
1
if T > T liq
(6)
3.1 Initial and Boundary Conditions
The HTF enters into the tube from top at a constant mass flow rate of 0.008 kg/s and
leaves the other end at the atmospheric pressure. The inlet temperature of the HTF is