## 1 Introduction
The PCM cooling process was analytically analyzed under three different geometrical configurations: plate, cylinder, and sphere. The PCM was encapsulated in polyethylene containers and the dimensions for the plate thickness and for the diameters of the sphere and cylinder were equal to 2 cm. The model evaluated the speed of advancement of the solidification front: the shortest time for complete solidification was achieved with spherical capsules with high Jacob numbers and high thermal conductivity. In the above-mentioned studies, it was demonstrated that spherical capsules allow the best performance during the discharge process, and, as a matter of fact, this geometry finds major application in packed-bed TES configurations [9] and numerous works have been presented in the scientific literature about the melting and/or solidification of phase change materials inside a spherical container [13; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. [PERSON] and [PERSON] [15] were the first to report experimental and numerical studies related to \"unconstrained\" melting in a sphere (i.e., the solid portion falls and passes through the molten portion due to its higher density). It has been found that convective currents deriving from the fall of the solid mass cannot be neglected within the molten mass. [PERSON] et al. [13] investigated by numerical simulations the solidification process of erythritol in spheres with different diameters (from 10 to 40 mm) and different wall temperatures (from 10 to 40 K below the phase change temperature of the considered material). The initial fill volume of the studied capsules was 98.5% and the remaining volume was assumed to contain air. The charging and discharging processes, which consist of the melting and solidification of the PCM, represent a major concern for several LHS systems, because heat transmission depends on PCMs' thermal conductivity, which, in some applications and for most of the conventional materials, is lower than needed [25]. This is also true in macroencapsulation, in particular during energy release: the liquid solidifies in the correspondence of the exchange surface, thus creating a motionless layer of solid material with low thermal conductivity. Several studies were conducted focusing on the development of analytical models describing packed-bed systems with sphere-encapsulated PCMs and their optimization [5; 26; 27; 28; 29; 30]. [PERSON] et al. [21] developed a CFD model of a heat transfer fluid flowing past a single sphere containing PCM (neglecting buoyancy forces within the PCM). After experimental validation at various temperatures, this study established an empirical correlation that can be used to determine the effective thermal conductivity within a sphere during melting. This parameter can be applied for the representation of the average natural convection over the melting process and to refine its representation in different numerical models. In further studies, [PERSON] et al. [22; 23] studied the phase change time of PCM encapsulated in a sphere with pins and copper plating adopted to increase the performance of the heat transfer. The numerical model, validated against physical tests, demonstrated the improvement in heat transfer inside the PCM sphere, confirming the validity of this design option. [PERSON] et al. [24] numerically explored the solidification process of a phase change material in a sphere, where high thermal conductivity was obtained by the addition of expanded graphite. The simulations also took into account the volume change in paraffin during phase change and the distribution of cavities formed due to the manufacturing process. The authors concluded that the volume change in paraffin has a great impact on the heat transfer process during the later stages of solidification, and that the addition of a certain quantity of expanded graphite in the capsule can effectively improve heat transfer characteristics.