## Materials and Methods  
### PCM-Water Hybrid Thermal Storage  
#### 2.1.1 Dynamic Simulation Model  
A simulation model was developed in Trnsys based on type 860 proposed by [PERSON] and [PERSON] [18] to simulate water storage tanks including PCM modules (vertical cylinders, plates, or spheres bed). The model is a useful tool to predict the possible improvement of thermal energy storage capacity of PCM with Al-foam with respect to PCM. A detailed validation with experimental data was done in previous works [21; 23] by comparing the temperature along the section of the tubes at fixed heights. Here, for the sake of clarity, only the main aspects of the analysis and validation of the type 860 are reported and discussed. This model is based on an enthalpy approach which means that, for a given volume and material, a continuous and reversible function can be evaluated to obtain the local temperature, depending on the calculated enthalpy. This temperature is used to determine the node temperature, according to the enthalpy of the system at a given time. The type takes into the account both the conduction and convection into PCM as well as at the interface between PCM and water. The tank was modeled to be made of stainless steel, 700 mm height, 350 mm diameter, 1 mm thick, water inlet at 700 mm height, and outlet at the bottom base, in order to validate it with previously measured experimental data [9; 10; 11]. The tank was vertically divided into 35 water nodes (derivatives) and 30 PCM nodes, so axial nodes are 2 cm apart each other (Figure 1a). The simulations were made at an imposed inlet temperature in order to make them comparable to the experimental tests. This condition was obtained by fixing tank inlet water flow rate and temperature at 500 L h\({}^{-1}\) and 50 \({}^{\circ}\)C, respectively. Type 860 was set up with many parameters, among those:  
* the temperature-enthalpy characteristic of the PCM (Figure 1b reports the curve of the paraffin wax considered in the simulations);
* type and dimensions of the encapsulation: the PCM was considered inserted into two Al tubes (height 600 mm, inner diameter 48.6 mm, outer diameter 50.8 mm). One tube was supposed to be filled with the PCM, the other one with the PCM embedded in the Al-foam;
* values of solid and liquid thermal conductivity, specific heat capacity, and latent heat of fusion of the PCM as described in Table 1;
* the hysteresis parameter of type 860 was set up at 2.5 \({}^{\circ}\)C.  
As type 860 cannot directly simulate metal foams inside the PCM encapsulation, the parameters related to radial and axial thermal conductivity of PCM in the liquid phase were simulated by the simplified model by [PERSON] et al. [24]. The present authors estimated an apparent thermal conductivity of paraffin/Al foam composite phase change material, calculated as the equivalent thermal conductivity of an ideal homogeneous material exchanging the same heat as the real paraffin/Al foam composite. As the thermal conductivity of the Al foam alloy is equal to \(\lambda_{Al}\) = 170 W m\({}^{-1}\) K\({}^{-1}\), and that of the paraffin \(\lambda_{PCM}\) = 0.21 W m\({}^{-1}\) K\({}^{-1}\), the model estimates the apparent thermal conductivity of the composite PCM (\(\lambda_{a}\)) by a series-parallel model of metal material and filler material as by [PERSON] et al. [24]. The apparent thermal conductivity is within two limits, \(\lambda_{min}\) and \(\lambda_{max}\), defined when Al and paraffin are arranged in series and in parallel, respectively (Equation (1)):  
\[\lambda_{min}=\frac{\lambda_{Al}\lambda_{PCM}}{\varepsilon\lambda_{Al}+(1- \varepsilon)\lambda_{PCM}}\;\lambda_{max}=\varepsilon\lambda_{PCM}+(1- \varepsilon)\lambda_{Al} \tag{1}\]  
where \(\lambda_{Al}\) and \(\lambda_{PCM}\) are the thermal conductivities of Al foam and paraffin respectively, and \(\varepsilon\) is the porosity of Al foam.