## 4 Results and Discussion
### Influence of the Geometrical Parameters
#### 4.3.1 Effect of Coil Pitch (P)  
In this section, the effect of coil pitch is investigated numerically. Accordingly, two cases including case A and case B are considered for numerical simulation. The geometrical parameters of these cases (coils and shell) are listed in Tables 3 and 4. Two different coil pitches including 0.04 and 0.12 m and two calculated helix diameters as free parameter (in order to keep constant the heat transfer area) including 0.17 and 0.48 m are selected for cases A and B, respectively. The height of spiral coil and the coil diameter are considered 0.36 and 0.016 m, respectively. The dimensions of the shell are kept constant as listed in Table 4. The configurations of the two investigated cases (A and B) are illustrated in Figure 11. The liquid fraction (LF) of two cases of A and B are depicted in Figure 12. The LF is drawn for 280 min. These cases are compared in four points including 50, 100, 180, and 250 min.  
As illustrated in Figure 12, at time = 50 min, case B has the best performance with LF improvement of 16.3%. At time = 100 min, case B has a better thermal performance in comparison with the case A with LF improvement of 15.1%. Case B has a better thermal performance compared to the case A until time = 180 min. After that, the melting rate changes and case A shows a better performance. For instance, at time = 250 min, case A has a better thermal performance with melting rate improvement of 4.3%. The liquid fraction and temperature contours for these two cases at different time are illustrated in Figure 13. According to Figure 13, until time = 180 min, as the case A is placed at the center of the shell, the melting rate of the ice is more than the case B. However, after that, because of the coil geometry of the case B, case B covers the bottom of the shell more than case A. Also, because of the buoyancy effect, the melting rate of the bottom region of the shell is lower than that of the top region.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Value (mm)** \\ \hline Height of the shell & \(L\) & 700 \\ Shell diameter & \(D_{St}\) & 620 \\ Shell inlet and outlet diameter & \(d_{St}\) & 25 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Shell geometrical parameters.  
Figure 11: Geometries of the coils used in Cases A to F. (**a**) Case A; (**b**) Case B; (**c**) Case C; (**d**) Case D; (**e**) Case E; (**f**) Case F.  
Figure 12: Temporal evolution of the liquid fraction (LF) for cases A and B.  
Figure 13: Contours of the liquid fraction and temperature for cases A and B at four different times in the plane \(x=0\).  
Considering the melting rate in the top region, the results for cases A and B exhibit the same trend due to buoyancy effects. The melting rate is higher for case A compared to case B, but the results are opposite in the bottom region of the shell. Finally, at time = 180 min, the LF of both cases are equal. Although the melting region of the shell bottom region for case B is more than case A, the melting rate of the top region for case A is significantly more than case B, which can be attributed to the fact that the coil pitch of the case A is lower than case B. The melting rate in the shell for case A is more important than for case B but the difference gets low after 250 min (about 4.72%).