2.3. Governing equations :
2.3.1. Equations used in the heat transfer in solid surface :
ρCp\(\frac{\text{dT}}{\text{dt}}\) + ∇.q = Q
q = - k.∆T
Where,
ρ = Density of the material
Cp = Heat capacity
q = Heat flux
k = Thermal conductivity
Q = Heat source
2.3.2. Fluid heat transfer equation used in the liquid metal model :
ρCp\(\frac{\text{dT}}{\text{dt}}\) + ∇.q +
ρCp u. ∆T = Q
Where,
ρ = \(\frac{P}{\text{RT}}\) in ideal gas domain
R = Universal gas constant
u = velocity of the fluid
2.3.3. Equation for thermal insulation :
- n . q = 0
Here,
n = Unit vector
2.3.4. Equations for heat flux :
- n .q = qo
qo = h * ( Text. - T )
Where ,
qo = Inward heat flux
h = Convection heat transfer co-efficient
2.3.5. Equations for the phase change materials :
ρ = θ1.ρ1θ2.ρ2
Cp = \(\frac{1}{\rho}\) (
θ1ρ1Cp,1 +
θ2ρ2Cp,2 ) +
L1→2\(\frac{\partial}{\partial T}\frac{\partial\alpha_{m}}{\partial T}\)
\begin{equation}
\alpha_{m}\ =\ \frac{1}{2}\ \frac{\theta_{1}\rho_{1}\text{\ .\ }\theta_{2}\rho_{2}}{\theta_{1}\rho_{1}+\ \theta_{2}\rho_{2}}\nonumber \\
\end{equation}k = θ1k1 + θ2k2
θ1 + θ2 = 1
Where,
Cp,1 = Constant pressure heat capacity of the solid
phase
Cp,2 = Constant pressure heat capacity of the liquid
phase
k1 = Thermal conductivity of the solid phase
k2 = Thermal conductivity of the liquid phase
θ1 = Volume fraction of the solid phase
θ2 = Volume fraction of the liquid phase
L1→2 = The latent heat of melting
2.4 : Numerical implementation :
A finite element method-based simulation software, ”COMSOL Multiphysics
5.6,” had been used to solve the governing equations. Constant heat was
applied to the base of the heat sink, and boundary walls were kept
insulated. Tetrahedral and triangular meshes were selected, with maximum
and minimum values of the element size of 0.0055 m and 0.00099 m,
respectively. Figure-2 shows the temperature profiles of the three types
of heat sinks after simulation.
[insert figure-2 here]