## 2 Methodology
### Theoretical Evaluation Methodology  
In order to analyse and compare the charging and discharging of the PCM in the different modes, nine well distributed temperature sensors were used, thus dividing the container into nine respective sections, as shown in Figure 5. The amount of PCM at each section was evaluated as shown in Equation (1):  
\[m_{i}=\frac{l_{i}\text{-}w_{i}}{l\cdot w}\cdot p_{l}\cdot v_{T}[\text{kg}] \tag{1}\]  
where \(l_{i}\) and \(w_{i}\) are the length and the width of PCM section \"i\", \(l\) and \(w\) are the length and the width of the region occupied by PCM (Block A), \(v_{T}\) is the total volume of PCM inside the tank that is in direct  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Initial PCM** & **Compressor** & **Evaporation** & **Inlet HTF** & **HTF Flow Rate** \\
**Temperature** [\({}^{\circ}\)**C] & **Power [\%]** & **Set-Point** & **Temperature [\({}^{\circ}\)**C]** & **[L-h\({}^{-1}\)]** \\ \hline \multirow{4}{*}{12} & 15 & & & \\  & 30 & & & \\  & 45 & \(-\)10 & 12 & 150 \\  & 60 & & & \\  & 75 & & & \\ \hline \multirow{4}{*}{12} & 15 & & & \\  & 30 & & & \\  & 45 & \(-\)10 & 12 & 100 \\  & 60 & & & \\  & 75 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 6: Set of experiments in Mode 3a.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Initial PCM** & **Compressor** & **Evaporation** & **Inlet HTF** & **HTF Flow Rate** \\
**Temperature** [\({}^{\circ}\)**C] & **Power [\%]** & **Set-Point [\({}^{\circ}\)**C]** & **Temperature [\({}^{\circ}\)**C]** & **[L-h\({}^{-1}\)]** \\ \hline \multirow{4}{*}{\(-4\)} & 15 & & & \\  & 30 & & & \\ \cline{1-1}  & 45 & \(-\)10 & 12 & 150 \\ \cline{1-1}  & 45 & \(-\)10 & 12 & 150 \\ \cline{1-1}  & 60 & & & \\ \cline{1-1}  & 75 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 7: Set of experiments in Mode 3b.  
thermal contact with the refrigerant and HTTF channels, and \(\rho_{l}\) is the density of PCM in liquid state. It is assumed that the mass of PCM inside each section of the PCM remains constant regardless the state of the PCM during the phase change process.  
The specific enthalpy of the PCM was derived from DSC results shown in Figure 6 to relate the cold energy stored in the PCM with its temperature, as shown in Equation (2):  
\[\left\{\begin{array}{cc}&h_{i}=h_{T_{i}-4^{\circ}\mathrm{C}}+Cp_{s}(-4-T_{i} )\left[k!\cdot\!kg^{-1}\right],\ \mathrm{if}\ T_{i}<-4^{\circ}\mathrm{C}\\ h_{i}=-0.1139\cdot&T_{i}^{3}+1.1139\cdot T_{i}^{2}-8.5545\cdot T_{i}+131.97\left[ k!\cdot\!kg^{-1}\right],\ \mathrm{if}\ -4^{\circ}\mathrm{C}\leq T_{i}<6^{\circ}\mathrm{C}\\ h_{i}=-0.0985\cdot&T_{i}^{3}+2.8732\cdot T_{i}^{2}-28.629\cdot T_{i}+99.839\left[ k!\cdot\!kg^{-1}\right],\ \mathrm{if}\ 6^{\circ}\mathrm{C}\leq T_{i}\leq 12^{\circ} \mathrm{C}\\ &h_{i}=-Cp_{l}(T_{i}-12)\left[k!\cdot\!kg^{-1}\right],\ \mathrm{if}\ T_{i}>12^{\circ} \mathrm{C}\end{array}\right. \tag{2}\]  
where \(h_{i}\) and \(T_{i}\) are the specific enthalpy and instantaneous temperature of the PCM at section \"i\", respectively. The specific heat capacity values of the PCM at temperatures less than \(-4\ ^{\circ}\mathrm{C}\) and higher than \(12\ ^{\circ}\mathrm{C}\) were considered constant and equal to \(Cp_{s}=2.03\ \mathrm{kJ\cdot kg^{-1}\cdot K^{-1}}\) and \(Cp_{l}\ =1.10\ \mathrm{kJ\cdot kg^{-1}\cdot K^{-1}}\), respectively.  
The total cold energy stored in the PCM (\(E_{T}(t)\)) and the power of charging or discharging of the cold energy (\(\dot{E}_{T}(t)\)) at a given time t were evaluated as described in Equations (3) and (4), respectively:  
\[E_{T}(t)=\sum_{i=1}^{9}m_{i}\cdot h_{i}(t)\ [kJ] \tag{3}\]  
\[\dot{E}_{T}(t)=\frac{dE_{T}(t)}{dt}[\mathrm{kW}] \tag{4}\]