## 4 Results and Discussion
### Validation Study
#### 4.1.1 Validation of the Heat Transfer Process in a Shell and Coil Tube Heat Exchanger
Because of the larger temperature gradients near the inner tube containing ethylene glycol channel, 10 prismatic layers are used near the inner tube on both sides (tube and shell). Initially, temperatures in the inner and outer tubes (ethylene glycol and ice) are fixed to the melting temperature of ice. The inlet mass flowrate and temperature of the heat transfer fluid (ethylene glycol) are 2 L/min and +5 \({}^{\circ}\)C, respectively. The thermo-physical properties of water, ice and ethylene glycol [9] are listed in Table 2. The buoyancy effect is modeled by using the Boussinesq approximation. As shown in Figure 5, the numerical results of the ice melting process are compared with the experimental results reported by [PERSON] et al. [7] with respect to the total rejected energy versus time. The total energy of the system for a given time can be defined as the integration of specific energies for each control volume, over the whole domain. The total rejected energy can be calculated as  
\[\mathrm{E}(t)=\mathrm{E}_{\mathrm{Sensible}}\left(t\right)+\mathrm{E}_{ \mathrm{Latent}}\left(t\right) \tag{11}\]  
Sensible and latent components can be defined for the melting process as  
\[\mathrm{E}_{\mathrm{Sensible}}(t)\ =\ \left\{ \begin{array}{cl}\mathrm{C}_{ \mathrm{Water}}T\ :\ \mathrm{Liquid\ Phase}\\ \mathrm{C}_{\mathrm{Ice}}T\ :\ \mathrm{Solid\ Phase}\end{array} \right. \tag{12}\]  
\[\mathrm{E}_{\mathrm{Latent}}(t)\ =\ \rho_{\mathrm{Ice}}\ \times\mathrm{f}(t)\ \times L \tag{13}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Density of Fluid (kg/m\({}^{3}\)) & \(\rho_{\mathrm{f}}\) & 999.833 \\ Specific Heat of the Fluid (J/(kg-K)) & \(\mathrm{C}_{\mathrm{I}}\) & 4180 \\ Specific Heat of the Solid (J/(kg-K)) & \(\mathrm{C}_{\mathrm{S}}\) & 2217 \\ Thermal conductivity of liquid water (W/(m-K)) & \(\mathrm{K}_{\mathrm{S}}\) & 1.918 \\ Thermal conductivity of ice (W/(m-K)) & \(\mathrm{K}_{\mathrm{I}}\) & 0.578 \\ Viscosity of the fluid (kg/(m-s)) & \(\mu_{\mathrm{f}}\) & 0.00162 \\ Latent Heat of Phase Change (J/kg) & L & 334.000 \\ Melting Temperature (K) & \(\mathrm{T}_{\mathrm{f}}\) & 273.15 \\ Density of Ethylene Glycol (kg/m\({}^{3}\)) & \(\mathrm{p}_{\mathrm{EG}}\) & 1111.4 \\ Specific Heat of Ethylene Glycol (J/(kg-K)) & \(\mathrm{C}_{\mathrm{EG}}\) & 2415 \\ Thermal conductivity of Ethylene Glycol (W/(m-K)) & \(\mathrm{K}_{\mathrm{EG}}\) & 0.252 \\ Viscosity of Ethylene Glycol (kg/(m-s)) & \(\mu_{\mathrm{EG}}\) & 0.0157 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermo-physical properties of water, ice, and ethylene glycol [9]. Figure 4: Schematic of validation model of the melting process of the ice storage system. where \(L\) is the latent heat and f(f) is the fraction of ice volume in each control volume. As show in Figure 5, the present solver predicts fairly well the temporal evolution of the total rejected energy measured by [PERSON] et al. [7]. The maximum difference with the experimental data reaches 7.1% at \(t\) = 440 min. For the mesh independent study, the results of the validated model were investigated for four different grids composed of 436,400; 584,300; 1,391,200; and 1,631,500 cells. The obtained results are displayed in Figure 6. There is no significant difference between the 584,300; 1,391,200; and 1,631,500 grids. To save computational resources, the grid composed of 584,300 cells associated with a 0.5 s time step is selected in the following.