## 4 Results and Discussion
### Influence of the Geometrical Parameters  
In this section, a parametric study is performed to determine the effects of geometrical parameters including coil pitch (P), coil diameter (d), and coil height (\(H_{\text{C}}\)) on melting process. In each model, two cases are considered. For instance, to study the effect of coil pitch (P), cases A and B are considered to simulate the melting process of ice in the shell and coil tube ice storage system. To compare logically the cases, the heat transfer area that is the lateral area of the coil, is kept constant in the cases which should be compared with together (A with B, C with D, and E with F). Accordingly, in order to keep the heat transfer area constant, one of the geometrical parameters is considered as a free parameter. Here, helix diameter (\(D_{C}\)) is the free parameter. On the other hand, to study the effect one parameter, the helix diameter (\(D_{\text{C}}\)) changes for keeping constant the heat transfer.  
The coil geometrical parameters for all investigated models are listed in Table 3. Number of six models are studied which accordingly evaluates the effects of three different coil geometries. The helix diameter (\(D_{S}\)) and the heat transfer area (A) are kept constant to compare all models. The shell geometrical parameters are listed in Table 4. The dimensions of the shell are kept constant in all simulation. Also, \(L\) is the spiral coil length and \(A\) is the heat transfer area which are defined by the Equations (14) and (15), respectively. \(H_{C}\) is helix height, \(P\) is coil pitch, \(D_{C}\) is helix diameter, and d is the coil diameter. The schematic view of all considered models (case A-F) are illustrated in Figure 11.  
\[L = \frac{H_{\text{C}}}{P}\ \sqrt{\left(\pi D_{\text{C}}\right)^{2}+P^{2}} \tag{14}\] \[A = L\ \pi\ d \tag{15}\]  
Figure 10: Temporal evolution of the liquid fraction for different inlet flowrates of ethylene glycol.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multirow{2}{*}{\begin{tabular}{c} **Height of** \\ **Spiral Coil** \\ \end{tabular} } & \begin{tabular}{c} **Coil** \\ **Pitch** \\ \end{tabular} & \begin{tabular}{c} **Helix** \\ **Diameter** \\ \end{tabular} & \begin{tabular}{c} **Spiral Coil** \\ **Length** \\ \end{tabular} & \begin{tabular}{c} **Coil** \\ **Diameter** \\ \end{tabular} & \begin{tabular}{c} **Heat Transfer** \\ **Area** \\ \end{tabular} \\ \cline{2-7}  & & \(H_{\text{C}}\) **(m)** & \(P\) **(m)** & \(D_{C}\) **(m)** & \(L\) **(m)** & \(d\) **(m)** & \(A\) **(m\({}^{2}\))** \\ \hline \multirow{2}{*}{\begin{tabular}{c} Effect of \(P\) \\ Effect of \(d\) \\ Effect of \(H_{\text{C}}\) \\ \end{tabular} } & Case A & 0.36 & 0.04 & 0.17 & 4.83 & 0.016 & \(9.7\times 10^{-4}\) \\  & Case B & 0.36 & 0.12 & 0.48 & 4.55 & 0.016 & \(9.7\times 10^{-4}\) \\ \multirow{2}{*}{\
\begin{tabular}{c} Effect of \(d\) \\ Effect of \(H_{\text{C}}\) \\ \end{tabular} } & Case C & 0.36 & 0.09 & 0.48 & 6.05 & 0.016 & \(1.2\times 10^{-3}\) \\  & Case D & 0.36 & 0.09 & 0.25 & 3.17 & 0.022 & \(1.2\times 10^{-3}\) \\ \multirow{2}{*}{
\begin{tabular}{c} Effect of \(H_{\text{C}}\) \\ \end{tabular} } & Case E & 0.24 & 0.04 & 0.43 & 8.13 & 0.016 & \(1.6\times 10^{-3}\) \\  & Case F & 0.42 & 0.04 & 0.25 & 8.27 & 0.016 & \(1.6\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Coil geometrical parameters of the investigated models.