## 2 Thermal Characterization
### New Hybrid Model  
Since the binary model is not satisfactory to model this material, which presents a very distinct behavior in melting and solidification. The hybrid model considers that the equations characterizing the thermal behavior of PCM differ in melting and in solidification. In melting, for instance in solidification, the heat equation of a binary solution (Equation  
Figure 6: Evolution of the derivate of enthalpy as a function of time. Figure 7: Evolution of the heat flux exchanged between the plate and the sample: binary solution and experimental. (3)) without eutectic [46] is used while being coupled to Gumbel's law [47] in order to reproduce the effects related to dissymmetry and supercooling, as generated by the presence of several materials in the PCM. Natural convection in PCM microcapsules is neglected due to their small diameter, varying from 5 to 25 \(\upmu\)m. The boundary conditions are of the DIRCHLET type, the imposed temperatures are \(T_{LP}\) on the left and \(T_{RP}\) on the right. The numerical model depends mainly on \(C_{S}\), \(C_{L}\), \(\lambda_{S}\), \(\lambda_{L}\), \(L_{A}\), \(T_{A}\), \(T_{M}\), \(R_{c\_right}\), and \(R_{c\_left}\). \(R_{c\_right}\) and \(R_{c\_left}\) are the thermal contact resistances between the heat fluxmeter and the sample. Other easily measurable parameters, such as density and thickness, are also needed to complete the input dataset. The operation of the model implemented (simplified to a cell), i.e., homogeneous, tri-layer unidirectional conduction (2 gypsum boards, 1 ETMV 8), is presented in Figure 8, where the depiction has been simplified yet each layer features several meshes, each containing two thermal resistances and one thermal capacity. Thus, the equations derived from this model are as follows:  
\[m=\rho\frac{e}{n}S\quad n:\text{ number of meshes, and }\] \[S:\text{ surface area (}m^{2}\text{)} \tag{5}\]  
\[Q=m.C.\frac{dT}{dt}\quad dT=T_{i}-T_{i-1},\text{ with }dt\text{ the interval }\] \[\text{ of the calculation time step (s) } \tag{6}\]  
\[\Phi_{i,\ \ i-1}=\frac{1}{R}.(T_{i}-T_{i-1})\quad\text{where R }=\frac{(e/n)}{\lambda} \tag{7}\]  
Contact resistances are integrated into the heat flux calculation at the material boundaries (left and right):  
\[\Phi_{left}=\frac{1}{R+R_{c-left}}.(T_{LP}-T_{i}) \tag{8}\]  
\[\Phi_{right}=\frac{1}{R+R_{c-right}}.(T_{i}-T_{RP}) \tag{9}\]  
The derivative of enthalpy is calculated by integrating the Gumbel distribution (\(F_{G}\)) and is represented in Figure 9. This distribution is a continuous probability function generally used to fit extreme data. It can be found in several fields, e.g., finance, hydrology, geology, and meteorology [48]. In this paper, we use the Gumbel probability density function (\(f_{G}\)), defined as the derivative of the Gumbel function:  
Figure 7:0  
Figure 7:1 Simplified model of the gypsum board (Gb) and ETMV8 (E8) complex. Its derivative is given by:  
Figure 7:2  
with \(x\in\mathbb{R}\), \(a\in\mathbb{R}\) is a position parameter, and \(b>0\) is a spread parameter. The advantage of this Gumbel derivative lies in its ability to easily obtain an asymmetric and adjustable form thanks to its position and spread parameters. Thus, the derivative of enthalpy is calculated according to the following equations. During solidification (s):  
\[\frac{dh_{\rm S}}{dT}=\left\{ \begin{array}{ll}c_{\rm S}\cdot\left(1-\frac{T_{ \rm As}-T_{\rm M_{\rm s}}}{T_{A_{\rm s}}-T}\right)+c_{L}\cdot\left(\frac{T_{\rm As }-T_{\rm M_{\rm s}}}{T_{A_{\rm s}}-T}\right)+\left(\frac{T_{\rm As}-T_{\rm M_{ \rm s}}}{T_{A_{\rm s}}-T}\right)\cdot\left(\frac{y_{\rm m}.L_{\rm A}}{T_{A_{ \rm s}}-T}\right)\\ c_{L}&T\geq T_{\rm M_{\rm s}}\end{array} \right. \tag{12}\]  
\[f_{\rm G_{s}}(T)=\left(\frac{1}{b_{\rm s}}\cdot\exp\left(-\exp\left(-\frac{T-a _{\rm s}}{b_{\rm s}}\right)\right)\cdot\exp\left(-\frac{T-a_{\rm s}}{b_{\rm s} }\right)\right)\cdot(1-\psi_{s})\cdot L_{A} \tag{13}\]  
During melting (m):  
\[\frac{dh_{\rm S}}{dT}=\left\{