## 2 Materials and Methods
### D Detailed Model  
Figure 1 shows the computational domain of the initial detailed model. Due to the simultaneous presence of a solid and liquid phase in the PCM, both heat conduction and natural convection are relevant heat transfer phenomena in LHTS systems. Thus, a full 3D numerical domain is generally recommended. However, the characteristics of the configuration analyzed in this study allow a substantial domain reduction. In fact, as claimed by [PERSON] et al. [18], compact LHTS designs hinder the development of natural convection phenomena, which are the main aspects responsible for problem asymmetries. According to [18], the Rayleigh number (\(Ra_{W}\)) and LHTS aspect ratio (A) can be used to determine the relevance of the wall heat flux due to natural convection in the liquid PCM. This correlation asserts that the flow is dominated by natural convection only if \(\frac{Ra_{W}}{A}\geq 500\). Regarding the LHTS configuration proposed in this analysis, the above-mentioned correlation yields \(\frac{Ra_{W}}{A}=374\). Therefore, the heat flux due to natural convection is likely to be negligible both during charging and discharging phases if compared to conduction. Hence, the computational domain is reduced to the horizontal cross-section plane. Here, the temperature gradient is expected to be larger compared to the axial direction [19], and heat conduction is dominant. The numerical domain is also further reduced, taking advantage of the symmetry created by the fins. Then, the physical problem is modeled using the finite volume method implemented in the commercial code Ansys Fluent. Regarding the materials, the fins and tubes are made of aluminum, while the PCM is assumed to be RT70 HC manufactured by Rubitherm GmbH [20]. A summary of the PCM properties is reported in Table 1. The enthalpy-porosity technique is used for modeling the melting and solidification process [21]. Combining this method with the hypothesis of negligible natural convection, the only governing equation to be solved is the energy equation, as expressed in Equation (1). \[\frac{\partial}{\partial t}(\rho H)-\
abla\cdot(k\
abla T)=0 \tag{1}\]  
where \(H\) is the sum of the sensible enthalpy and latent heat of the material. Adiabatic boundary conditions are applied on the external face (\(\Gamma_{3}\) in Figure 1), while symmetry is considered on \(\Gamma_{2}\). The heat transfer with the water in the pipes is modeled with an implicit Robin boundary condition on \(\Gamma_{1}\), as shown by Equations (2)-(4). \(T_{ref}\) represents the average temperature between the water inlet (\(T_{in}\)) and outlet (\(T_{out}\)) temperatures. However, \(T_{out}\) depends on the efficacy of the heat transfer between the flowing HTF and the rest of the LHTS (i.e., the ensemble of fins and PCM). Therefore, if \(T_{out}\) is expressed as a function of the heat flux at the wall (\(\text{q}_{\text{wall}}\)), Fluent solves Equation (2)iteratively. The heat transfer coefficient \(h_{conv}\) is computed using the Dittus-Boelter correlation (5) and the Nusselt number (6). In (5), n = 0.4 for discharge or n = 0.3 for charge. This correlation was preferred over the implicit one proposed by [PERSON] and [PERSON] due to its simplicity and acceptable uncertainty. Substantially, the simulated domain represents an average cross-section of the whole LHTS unit. \[\dot{\text{q}}_{\text{wall}}=h_{conv}\big{(}T-T_{ref}\big{)} \tag{2}\]  
\[T_{ref}=\frac{(T_{in}+T_{out})}{2} \tag{3}\]  
\[T_{out}=T_{in}+\frac{L_{tube}\pi d_{i}}{Gc_{p_{w}}}\text{q}_{\text{wall}} \tag{4}\]  
\[Nu=\ 0.023R\frac{e}{5}Pr^{n} \tag{5}\]  
\[h_{conv}=\ \frac{Nu*k_{w}}{d_{i}} \tag{6}\]  
The numerical model follows the same approach adopted by [PERSON] et al. [6], who validated their 2D cross-section model for PCM solidification against experimental data. For further details, the interested reader is referred to [6].