## 4 Result and Discussion
Figure 7 shows the distribution of temperatures inside the sphere at four instants, i.e., after: 150 s, 1200 s, 3600 s and 6000 s. Through the snapshots of the simulation, it is possible to gain an idea of the distribution of temperatures and masses of solid and liquid PCM in the spherical capsule during the solidification process: due to the external cooling, the solidification starts on the outer surface, with the formation of a solid shell, which then acts as a nucleus that gradually increases its thickness inward. From the color maps it is possible to see that the hotter liquid mass tends to remain in the inner and upper part of the sphere: the movements of the mass are, in any case, much slower and more regular than those occurring during melting (see for comparison Figures 7b and 9d both at 1200 s). Figure 8 shows the comparison between experimental and simulated data during the charge phase (melting of the PCM) for a wall temperature of 71 \({}^{\circ}\)C, while Figure 9 shows the calculated temperature distribution inside the sphere at four instants, namely 150 s, 300 s, 600 s and 1200 s from the start. The time sequence reported in Figure 9, clearly represents the free movement of the heavier block of solid paraffin towards the bottom of the control volume and the upwards movement of the liquid paraffin, which cause a markedly uneven distribution of the temperature inside the sphere: after 300 s, the distribution of the solid and liquid volumes in the sphere appears already strongly influenced by the effect of gravity. If we look at the temperature comparison reported in Figure 8, some significant differences can be noticed. The temperature at thermocouple T1 is predicted by the model rather well (Figure 8a), while in the correspondence of thermocouple T2, the model predicts  
Figure 9: Colormaps of the temperature during the melting phase with T1 = 71 \({}^{\circ}\)C and Twall = 22.5 \({}^{\circ}\)C after: (**a**) 150 s; (**b**) 300 s; (**c**) 600; (**d**) 1200 s.  
a much slower heating of the solid mass, compared to the experimental trend, up to 40 \({}^{\circ}\)C, followed by a steep increase up to 71 \({}^{\circ}\)C, after approximately 1200s since the beginning; experimentally, however, the equilibrium temperature is reached after 2200 s (Figure 8b). Similar, or even larger, discrepancies are observed in the correspondence of T3 and T4, with the model predicting a much faster temperature rise, compared to the experimental one (Figure 8c,d), followed by some waving trend. Besides those significant differences, overall, the model predicts a faster melting of the PCM mass in the sphere: thermal equilibrium is predicted after approximately 1700 s, instead of 2500 s, as measured experimentally. Modeling the melting of a phase change material in a sphere is a rather complex problem, hence the difficulty in obtaining quantitatively reliable results. [PERSON] et al. [34] developed a model to simulate the melting process of a PCM into a sphere with an internal diameter of 101 mm. The results were compared against experimental data and the model always predicted a higher melting speed. Similarly, [PERSON] et al. [35] carried out a CFD analysis of the constrained melting of n-octadecane paraffin wax in a spherical container at different operating conditions and various container diameters and validated the results of the simulations with experimental data. The results showed the existence of thermally stable layers on the top half of the sphere due to the increasing influence of natural convection; at the same time, the waviness behavior of the PCM in the bottom part of the container was observed. As far as the melting rate is concerned, this study highlighted the major influence of the surface temperature rather than other operational and geometrical parameters. These results agree with those reported in the present study.