## 2 Materials and Methods
### Theoretical Evaluation Methodology  
The overall cold energy transferred to the module during the charging process, \(E_{ch}\) (in kJ), was evaluated as shown in Equation (2). \[E_{ch}=\sum_{i=1}^{n}\bigl{[}m_{PCM,i}\bigr{(}T_{i,t=t_{ch}}\bigr{)}-h_{PCM} \bigr{(}T_{i,t=0}\bigr{)}\bigr{)}+(m_{AI,i}\cdot Cp_{AI}+m_{HTF,i}\cdot Cp_{HTF })\cdot\bigl{(}T_{i,t=0}-T_{i,t=t_{ch}}\bigr{)}\bigr{]} \tag{2}\]  
where the sub-index \(i\) refers to each of the volume elements, \(n=9\) is the total number of volume elements considered (equal to the number of PCM temperature sensors), \(m_{PCM,i}\) is the mass of PCM, \(m_{HTF,i}\) is the mass of HTF, and \(m_{AI,i}\) is the mass of aluminium in volume element \(i\), a sub-index \(t\) refers to the time instant, with \(t_{ch}\) being the time taken to complete the charging process. The enthalpy or the specific heat capacity, the measured temperature and the mass of each material were incorporated into the finite element method detailed in [15; 28] to obtain the energy level in the module. Thereafter, basic equations of thermodynamics [Equations (3)-(9)] on heat transfer and energy balance adapted from [29] were used to complete the evaluation. The average power during the charging process, \(\overline{\dot{E}}_{ch}\) (in kW), was evaluated as in Equation (3). \[\overline{\dot{E}}_{ch}=\frac{E_{ch}}{t_{ch}} \tag{3}\]  
The energy discharged from the systems during the discharging process, \(E_{dis}\) (in kJ), was evaluated as shown in Equation (4). \[E_{dis}=\sum_{t=0}^{t_{dis}}\bigl{[}\dot{V}_{HTF}\cdot p_{HTF}\cdot Cp_{HTF} \cdot(T_{HTF,in}-T_{HTF,out})\bigr{]}_{t}\cdot\Delta t+\sum_{t=0}^{t_{dis}} \bigl{[}UA_{loss}\cdot\left(T_{a}-T_{av}\right)_{t}\bigr{]}\cdot\Delta t \tag{4}\]  
where \(\dot{V}_{HTF}\) is the volume flow rate of the HTF, \(T_{HTF,in}\) and \(T_{HTF,out}\) are the temperatures of the HTF at the inlet and outlet of the module, respectively, \(t_{dis}\) is the time taken to complete the discharging process, \(UA_{loss}\) is the overall heat transfer coefficient for heat losses, \(T_{a}\) is the ambient air temperature, \(T_{av}\) is the average PCM temperature, and \(\Delta t=10\) s is the time interval between two consecutive measurements. The average power during the discharging process, \(\overline{\dot{E}}_{dis}\) (in kW), was evaluated as in Equation (5). \[\overline{\dot{E}}_{dis}=\frac{E_{dis}}{t_{disc}} \tag{5}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & **RT4 PCM [23]** & **HTF [25]** \\ \hline Phase change range [\({}^{\circ}\)C] & 2–4 & \(-18/127\) \\ Specific heat capacity [k]\(\cdot\)kg\({}^{-1}\).K\({}^{-1}\)] & 2 & 3.602 \\ Density [kg\(\cdot\)L\({}^{-1}\)] & 0.88 (solid) & 1.05045 \\ Thermal conductivity [W\(\cdot\)m\({}^{-1}\).K\({}^{-1}\)] & 0.2 & 0.418 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the PCM and the HTF. The effectiveness \(\varepsilon\) of the TES module was evaluated using Equation (6). \[\varepsilon=\frac{\dot{E}_{HTF_{actual}}}{\dot{E}_{HTF_{max}}} \tag{6}\]  
The actual HTF power, \(\dot{E}_{HTF_{actual}}\), was evaluated using Equation (7), whereas the maximum HTF power, \(\dot{E}_{HTF_{max}}\), was evaluated using Equation (8). \[\dot{E}_{HTF_{actual}}=\dot{V}_{HTF}\cdotp_{HTF}\cdotp Cp_{HTF}\cdot\left(T_{ HTF,in}-T_{HTF,out}\right) \tag{7}\]  
\[\dot{E}_{HTF_{max}}=\dot{V}_{HTF}\cdotp P_{HTF}\cdotp Cp_{HTF}\cdot\left(T_{ HTF,in}-T_{ref,in}\right) \tag{8}\]  
where \(T_{ref,in}\) is the refrigerant inlet temperature.