## 2 Materials and Methods
### Numerical Model  
The energy equation in enthalpy form is written as [35; 36]:  
\[\frac{\partial(\rho\mathrm{H})}{\partial t}\ =\ \
abla(\mathrm{k}\
abla T) \tag{1}\]  
where the total enthalpy (H) corresponds to the sum of the sensible enthalpy and the latent heat. According to [35], the enthalpy-temperature relationship (Figure 3) during the melting process can be described by four phases: solid-state PCM, solid-state PCM--phase change, semi-solid-state PCM--phase change, liquid-state PCM--liquid. Consequently, according to [35; 36; 37; 38; 39], the enthalpy is the sum of the enthalpy h and the enthalpy at phase change. \[\mathrm{H}\ =\ \mathrm{h}+\Delta\mathrm{H} \tag{2}\] \[\mathrm{H}\ =\ \mathrm{C_{ps}T+\frac{\mathrm{L_{p}}}{(a1+a2)}\Big{(}T-T _{(p-a1)}\Big{)}},\] \[\mathrm{T_{(p-a1)}}\ <T<T_{(p+a2)}\] \[\mathrm{H}\ =\ \mathrm{C_{ps}T+T_{p}},\] \[\mathrm{H}\ =\ \mathrm{C_{pl}T+L_{p}+\frac{L}{(a1+a2)}\Big{(}T-T_{(m-e1)} \Big{)}},\] (3) \[\mathrm{H}\ =\ \mathrm{C_{pl}T+L_{p}+L},\]  
Figure 2: Scheme of experimental setup. Subsequently, the thermal conductivity of the PCM, k, at the interface phase is expressed as follows:  
\[\rm{k}\ =\ \rm{k_{s}}+\left(\frac{k_{l}-k_{s}}{(\Delta T)_{m}}\right)(T-(T_{m-e1})),\ T_{(m-e1)}<T<T_{(m+e2)} \tag{4}\]  
The stored energy in a unit volume of PCM can be expressed as follows [35]:  
\[\rm{E}\ =\ \int_{T_{0}}^{T_{p-al}}\rho_{s}C_{ps}dT+\int_{T_{p-al}}^{T_{p+a2}} \rho_{s}C_{ps}dT+\rho_{s}L_{p}+\int_{T_{m-e1}}^{T_{m+a2}}\rho_{l}C_{pl}dT+\rho_{ l}L+\int_{T_{m+e2}}^{T_{f}}\rho_{l}C_{pl}dT \tag{5}\]  
Consequently, the sensible enthalpy h is given by [25; 26; 40; 41; 42]  
\[\mathrm{H}\ =\ \mathrm{h}+\Delta\mathrm{H} \tag{2}\]0  
\(\rm{h_{ref}}\)--reference enthalpy;  
\(\rm{T_{ref}}\)--reference temperature;  
\(\rm{C_{p}}\)--specific heat capacity at constant pressure. The enthalpy of latent  
H--enthalpy; P--density; \(\dot{\mathrm{v}}\)--flow velocity. During the phase change process, heat is exchanged between the heat transfer fluid and the phase change interface in the PCM at the phase change temperature [40; 41; 42]. This heat transfer is a function of the thermal resistance of the PCM that is in the phase change process. The result is a functional dependence of the outlet fluid temperature on the thermal resistance of the system and the constraints of the phase change temperature. Maximum heat transfer is achieved under the condition where the outlet temperature is equal to the phase change temperature of the PCM. When the phase change interface is reached, the PCM stack can be analyzed as an exchanger between the fluid (heat transfer fluid) and the cooler/source (constant-temperature PCM). Therefore, for the average efficiency of the system at phase change according to [42; 43], one can write the relation: \[\mathrm{H}\ =\ \mathrm{h}+\Delta\mathrm{H} \tag{2}\]1 (10) \(\varepsilon\)--efficiency; Qact--energy released from the reservoir during the phase change; Qth--maximum theoretical energy released from the tank during the phase change; \(\dot{\mathrm{m}}\)--mass flow rate of the heat transfer fluid; Cp--heat capacity of the heat transfer fluid; Ti--inlet temperature of the heat transfer fluid into the tank; Tout--outlet temperature of the heat transfer fluid from the storage tank; \(\Delta\mathrm{t}\)--time interval; TPCM--PCM phase change temperature. The actual usable amount of phase change energy stored in this way is defined by the charging and discharging efficiency. Since the efficiency is based on a constant outlet temperature, this efficiency provides information about the temperature reached at the outlet of the storage tank. The efficiency can thus be characterised as a function of the operating flow rate, which characterizes the energy capacity of the storage tank. Further, as [42; 43; 44] shows, efficiency increases as flow rates decrease because as the thermal resistance of the PCM changes, the amount of energy transferred to the heat transfer fluid also changes.