## 3 Model Description  
A lumped parameter model was developed in the Modelica language, using the Dymola environment and the commercial TIL libraries for components and media [28]. The following assumptions were made:  
* the thermo-physical properties of the refrigerant, the HTF, and the PCM are independent of temperature; however, the properties in the solid phase are different from the properties in the liquid phase;
* HTF is incompressible;
* the temperature of the refrigerant, HTF, and the metal walls varies only in the flow direction (x-direction) along their lengths, which are divided into n nodes;
* radiative heat transfer is not considered;
* the heat loss to the surroundings is negligible;
* natural convection in the liquid PCM is ignored, as this effect is small due to the low fin height and pitch [19; 29]. A schematic view of the model structure is shown in Figure 2, together with the main boundary conditions. For each cell of refrigerant and HTF, momentum, energy, and mass balance equations are solved. For the refrigerant of the heat pump:  
\[p_{in_{ref}}-p_{out_{ref}}=\Delta p_{ref} \tag{1}\]  
\[\dot{m}_{in_{ref}}-\dot{m}_{out_{ref}}=\frac{dm_{ref}}{dt} \tag{2}\]  
\[m_{ref}\frac{dh_{ref}}{dt}=V_{ref}\frac{dp_{ref}}{dt}+\dot{m}_{in_{ref}}\Big{(} h_{in_{ref}}-h_{ref}\Big{)}+\dot{m}_{out_{ref}}\Big{(}h_{out_{ref}}-h_{ref} \Big{)}+\dot{Q_{ref}} \tag{3}\]  
where \(p\) (bar) is the pressure, \(\dot{m}\) (kg s\({}^{-1}\)) is the mass flow rate, \(h\) (J kg\({}^{-1}\)) is the specific enthalpy, and \(\dot{Q}\) (W) is the heat transfer term. The pressure drop \(\Delta p\) in the refrigerant cell is considered constant and evaluated according to experimental data. The heat transfer term is instead calculated as:  
\[\dot{Q}=U_{ref}A\Big{(}T_{refrigerant}-T_{wall}\Big{)} \tag{4}\]  
\[U_{ref}=\frac{Nu_{ref}\ k}{D_{herf}} \tag{5}\] \begin{table}
\begin{tabular}{c c} \hline Phase change temperature range (\({}^{\circ}\)C) & 2–4 \\ \hline Density (liquid) (kg m\({}^{-3}\)) & 770 \\ \hline Density (solid) (kg m\({}^{-3}\)) & 880 \\ \hline Specific heat (kJ kg\({}^{-1}\) K\({}^{-1}\)) & 2 \\ \hline Melting latent heat (kJ kg\({}^{-1}\)) & 175 \\ \hline Volume expansion (\%) & 12.5 \\ \hline Thermal conductivity (W m\({}^{-1}\) K\({}^{-1}\)) & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 2: Properties of the PCM. \[Nu=j\;Re^{\frac{1}{\text{ }}} \tag{6}\]  
where \(U_{nf}\) (W m\({}^{-2}\) K\({}^{-1}\)) is the heat transfer coefficient for the refrigerant cell, \(A\) (m\({}^{2}\)) is the heat transfer area for each cell, \(k\) (W m\({}^{-1}\) K\({}^{-1}\)) is the thermal conductivity of the wall material, \(D_{h}\) (m) is the hydraulic diameter of the channel, \(Re\) is the Reynolds number, \(Pr\) is the Prandtl number, and \(Nu\) is the Nusselt number. The friction factor \(j\) is calculated as:  
\[j_{ref}=A+B\;Re^{c}_{ref} \tag{7}\]  
with \(A\), \(B\), and \(C\) supplied by the heat exchanger manufacturer. For the HTF:  
\[p_{in_{HTF}}-p_{out_{HTF}}=\Delta p_{HTF} \tag{8}\]  
\[\dot{m}_{in_{HTF}}-\dot{m}_{out_{HTF}}=\frac{dm_{HTF}}{dt} \tag{9}\]  
\[m_{HTF}c_{p_{HTF}}\frac{dT_{HTF}}{dt}=\dot{m}_{in_{HTF}}\big{(}h_{in_{HTF}}-h _{HTF}\big{)}+\dot{m}_{out_{HTF}}(h_{out_{HTF}}-h_{HTF})+\dot{Q}_{HTF} \tag{10}\]  
where \(cp\) (J kg\({}^{-1}\) K\({}^{-1}\)) is the specific heat capacity of the heat transfer fluid. Similarly to the case of the refrigerant cell, a constant pressure drop was considered and the heat transfer was calculated as:  
\[\dot{Q}=U_{HTF}A(T_{HTF}-T_{wall}) \tag{11}\]  
The heat transfer coefficient for the HTF cell \(U_{HTF}\) was calculated according to a Nu correlation, as in Equation (6), with:  
\[j_{HTF}=D+E\;Re^{f}_{HTF} \tag{12}\]  
The parameters for the calculation of the friction factor were supplied by the heat exchanger manufacturer.