## 4 Result and Discussion
### Influence of Solid Foam Insertion on the Response Rate of the System  
As already highlighted in the discussion above, despite the theoretical possibility of storing much larger amounts of energy than with conventional heat transfer systems, PCM are often characterized by relatively low thermal conductivity, which hinders a fast energy transfer between the PCM and the tank liquid. This markedly increases the process time, making the method unsuited for applications where the heat exchange time plays a significant role. Given the difficulty in finding melting materials with the required properties (from the thermal, safety, economic, etc., point of views), a possible solution is to include some high-thermal-conductivity materials in the capsule containing the basic PCM. Solid foams can be used to increase the conductivity of the phase change material encapsulated in the sphere. [PERSON] et al. [36] examined the solidification of PCM integrated with metal foam in a spherical capsule by applying a one-dimensional model, and they concluded that such materials could represent a reasonable approach for enhancing the heat transfer in TES systems. However, convective heat exchange inside the sphere was not taken into account even for PCM without embedded foam. In the present study, the influence of the inclusion of a higher-thermal-conductivity material in the sphere containing the PCM, was also assessed. For its evaluation, in addition the base case of the pure A46 paraffin, with k = 0.22 W/mK, two additional systems have been simulated, characterized by:  
* k = 1 W/mK, corresponding to A46 paraffin immersed in a composite SiC porous foam matrix (k = 200 W/mK), already presented elsewhere [37; 38];k = 10 W/mK, arbitrarily set to investigate the effect of \(k\), and representative of average values reported in the literature for the effective thermal conductivity of paraffin wax in metal foams [39]. In both cases, the composite structure was simulated as a pseudo-homogeneous material [38,39] but, due to the presence of the solid foam which prevents the movement of the PCM mass, the convective motions were discarded during simulation. The graphs below show a comparison between the temperature profiles obtained simulating the sphere containing pure A46 paraffin (with the convective motions allowed in the simulation) and those obtained for the two composite configurations. During the melting phase (Figure 10), a marked beneficial effect can be observed after a change in thermal conductivity from 1 W/mK to 10 W/mK. This might be expected, and it is especially true for the inner thermocouples (see for example Figure 10b). Overall, this increased efficiency reduces the time to reach thermal equilibrium from approximately 1700 s down to approximately 400 s. However, if we compare the results for k = 1 W/mK and k = 0.22 W/mK (pure A46), it is found that the difference is not significant and, in the upper part of the sphere, the efficiency is even higher in the case of the pure PCM rather than with the composite structure. This effect is likely due to the influence of the convective motions which have been already discussed above, and it has an important practical importance: in fact, since introducing a solid foam inside a spherical capsule is not a trivial procedure and would also increase the overall cost of the system, its adoption should be carefully considered and adopted only in those cases where it is expected to actually provide significant advantages. Figure 10: Comparison between the temperatures simulated corresponding to thermocouple (**a**) T1, (**b**) T2, (**c**) T3, and (**d**) T4 during the melting phase of the PCM, k = 0.22 W/mK, and the melting phase of composite materials (PCM+foam) with different thermal conductivities, 1 and 10 W/mK (T\({}_{\text{wall}}\) = 71 \({}^{\circ}\)C).