## 2 Thermal Characterization
### New Hybrid Model
\begin{array}{ll}c_{\rm S}\cdot\left(1-\frac{T_{ \rm M_{\rm m}}-T_{\rm M_{\rm m}}}{T_{A_{\rm m}}-T}\right)+c_{L}\cdot\left(\frac {T_{\rm M_{\rm m}}-T_{\rm M_{\rm m}}}{T_{A_{\rm m}}-T}\right)+\left(\frac{T_{ \rm M_{\rm m}}-T_{\rm M_{\rm m}}}{T_{A_{\rm m}}-T}\right)\cdot\left(\frac{y_{ \rm m}.L_{\rm A}}{T_{A_{\rm m}}-T}\right)\\ c_{L}&T\geq T_{\rm M_{\rm m}}\end{array} \right. \tag{14}\]  
\[f_{\rm G_{m}}(T)=\left(\frac{1}{b_{m}}\cdot\exp\left(-\exp\left(\frac{T-a_{m} }{b_{m}}\right)\right)\cdot\exp\left(\frac{T-a_{m}}{b_{m}}\right)\right)\cdot( 1-\psi_{m})L_{A} \tag{15}\]  
\(\psi_{s}\)_et_\(\psi_{m}\) denote the latent heat distribution coefficients between the binary and Gumbel functions during the solidification and melting phase of PCM, respectively. Figures 9 and 10 present the evolution of the full enthalpy derivative, the enthalpy derivative without the addition of the Gumbel function, as well as with the Gumbel function. Let's observe that the Gumbel function plays a more significant role during the melting phase in reproducing the effects of progressive melting of the material. During the solidification phase, the Gumbel function is mainly used to reproduce the effects related to supercooling, which occurs at approx. 17 \({}^{\circ}\)C.  
Figure 9: Evolution of the enthalpy derivative vs. temperature (hybrid model). This undercooling stems from the fact that the PCM used in this study is a mixture of two fatty acid-based PCMs [49], each with distinct thermal behavior. A difference of about 4% exists between the liquid fraction variation at the beginning of heating and at the end of cooling (Figure 11), as observed by [PERSON] [49] on several PCMs as well. The author assumed that this phenomenon was due to either incomplete melting of the material or measurement errors. An error analysis related to experimental uncertainties is carried out and detailed in Section 5.2. Moreover, the choice of using an inverse Gumbel function in the melting phase is justified by the fact that this function reflects a slow and progressive increase, followed by an abrupt decrease. It presented the best results in comparison with other mathematical laws, e.g., [PERSON]'s law. The opposite behavior was observed during the solidification phase, thus explaining the applicability of the Gumbel function, which shows a rapid rise with a steep slope followed by a gradual decrease. To optimize the parameters of this model and reduce the objective function to characterize the ETMV 8 sample, a differential evolution algorithm [50] available in the optimization module of Python, called Sippy, has been used. Due to its stochastic nature (i.e., the gradient method is not employed), this algorithm can search within a large candidate space area for the minimum of the objective function; its operation is detailed in [51]. Figure 11: Evolution of the liquid fraction vs. temperature. Figure 10: Evolution of the enthalpy derivate of the binary and Gumbel’s functions used to create the hybrid model. To reduce the number of parameters to be optimized, the thermal conductivities and capacities, as well as the contact resistances, are determined beforehand during experiments where the PCM is in either the solid or liquid state. Yet despite this, the optimization step still requires the identification of 11 parameters (L\({}_{\text{A}}\), T\({}_{\text{A}}\)\({}_{\text{m}}\), T\({}_{\text{Mn}}\), T\({}_{\text{A}}\)\({}_{\text{s}}\), T\({}_{\text{M}_{s}}\), \(\psi\)\({}_{\text{s}}\), \(\psi\)\({}_{\text{m}}\), a\({}_{\text{m}}\), b\({}_{\text{m}}\), a\({}_{\text{s}}\) and b\({}_{\text{s}}\)).