## 3 Numerical Model
### Impact of Mushy-Zone Temperature Range  
The mushy-zone temperature range (\(\Delta T\)) defines how rapidly the PCM properties change from the liquid to solid values as a function of temperature, and over which range \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mesh** & \begin{tabular}{c} **Domain** \\ **Elements** \\ \end{tabular} & \begin{tabular}{c} **Boundary** \\ **Elements** \\ \end{tabular} & \begin{tabular}{c} **Total** \\ **Elements** \\ \end{tabular} & \
\begin{tabular}{c} **Average** \\ **Element Size** \\ \end{tabular} &
\begin{tabular}{c} **Simulation** \\ **Time** \\ \end{tabular} \\ \hline
1 & 5135 & 888 & 6023 & 0.00090 m & 17 s \\
2 & 8152 & 985 & 9137 & 0.00057 m & 16 s \\
3 & 15,419 & 1140 & 16,559 & 0.00046 m & 23 s \\
4 & 30,907 & 1805 & 32,712 & 0.00030 m & 46 s \\
5 & 130,768 & 3617 & 134,385 & 0.00013 m & 194 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistics of the meshes used in the mesh independence study. Figure 4: Mesh 2 used for the validation study. Figure 5: System equivalent power as a function of time for the 12 fins, 9 LPM, and 65 to 21 \({}^{\circ}\)C study for the five meshes used in the mesh independence study. of temperature latent heat is accounted for during the phase change. As presented by [PERSON] and [PERSON] [37], the mushy-zone temperature range (\(\Delta T\)) also has an impact on the simulated results when the solid-liquid phase change is accounted for. Previous studies have also shown that the smallest \(\Delta T\) values lead to more physically accurate results at the expense of increased computational cost [36]. Therefore, simulations were performed to look at the impact of \(\Delta T\) on the local results and on the extracted power results. The effect of modifying \(\Delta T\) was studied regarding its impact on the conductive heat transfer and phase change behavior through a parametric mushy-zone \(\Delta T\) study. The simulations used mesh 3, as defined in Section 3.5, and were otherwise identical to the simulations ran in Section 3.5 for the mesh independence study. Figure 6 presents the position of the solidification interface after 7020 s for the four values of \(\Delta T\) used (1, 4, 8, and 12 K). The results are shown at such a long time after the start of solidification because the impact of the mushy-zone temperature range (\(\Delta T\)) becomes more noticeable then. It can be observed that the position of the interface, therefore the amount of PCM solidified, varies slightly with values of \(\Delta T\). The position of the interface is mostly the same for \(\Delta T\) of 1 and 4 K, with the difference increasing with an increase in \(\Delta T\). Figure 7 shows the effect of changing \(\Delta T\) on the system equivalent power curve. In reality, there is no mushy range for the PCM used. Rather, the material goes through its phase transition at a well-defined melting/solidification temperature, and the impact of the latent heat of fusion is felt solely at this transition temperature. Therefore, numerically, increasing \(\Delta T\) yields a smoother, more continuous power curve with a slightly higher value for the system equivalent power. When \(\Delta T\) = 1 K, the power curve closer mimics the discontinuous behavior seen in the experimental data at approximately 1250 s.  
Also, as \(\Delta T\) decreases, the modified specific heat capacity function becomes much steeper and taller due to the latent heat correction having to be accounted for within a smaller temperature range. Numerically, this makes the simula  
Figure 6: Solidification interface position at 7020s for \(\Delta T\) = 1, 4, 8, and 12 K (red, blue, green, and orange lines, respectively). Figure 7: System equivalent power as a function of time for the 12 fins, 9 LPM, and 65 to 21 \({}^{\circ}\)C study for four mushy-zone temperature intervals: \(\Delta T\) = 1, 4, 8, and 12 K.  
and causes convergence issues when varying thermal conductivities individually. Therefore, a \(\Delta T=2\) K was chosen to perform the simulations for this study.