## 2 Thermal Characterization
### Experimental Measurement Method
This function represents the quadratic difference between the numerically simulated heat fluxes and the experimentally measured fluxes, to the left and right of the sample; it is calculated as the difference between the experimental and theoretical heat fluxes on both sides of the sample according to the following formula:  
\[f_{obj}=\sum\Bigl{(}\Phi_{left\;exp}-\Phi_{left\;calcalcalcalcalcalcalcalcalcal  
At the beginning of melting ( ), a nearly identical curve of thermal flux measured on both the ETMV 8 sample and the binary solution can be observed. Concerning the solidification phase, two common points between the two bodies essentially stand out, namely the levels at the beginning ( ) and end of solidification ( ). It should also be noted that during the solidification phase, except for the second peak of thermal flux identified for the ETMV 8 coating, the curves are on the whole similar. All these elements serve to reinforce the choice of a binary solution model to analyze the coating behavior.  
One of the most accurate existing models to describe the thermal behavior of PCMs with a binary solution behavior is detailed in [35, 36, 37, 41]. In particular, this model has been used to accurately characterize a composite material consisting of cement mortar and paraffin-based PCM. [PERSON] et al. [35] compared this model with the other main existing models, in deducing that it is the most precise to represent the thermal behavior of binary PCMs.  
In general, the thermal behavior of PCMs can be described by expressing the evolution of enthalpy (h) as a function of temperature [21]. According to the first principle of thermodynamics, the heat equation associated with a phase change problem can be described in terms of the derivative of enthalpy (Equation (2)). For a binary solution, this derivative can be expressed by Equation (3), and the liquid fraction \(f_{\text{liq\_bin}}\) (T) used to model the change of state is given by Equation (4) [42].  
\[\rho.\frac{\partial h}{\partial T}.\frac{\partial T}{\partial t}=\lambda. \frac{\partial 2T}{\partial x2} \tag{2}\]  
\[\frac{\partial h}{\partial T}=\frac{df_{\text{liq\_bin}}(T)}{dT}.L_{A}+C_{S^{ \ast}}\cdot\left(1-f_{\text{liq\_bin}}(T)\right)+C_{L}\cdot f_{\text{liq\_bin}} (T) \tag{3}\]  
\[\text{where }f_{\text{liq\_bin}}\cdot(T)=\left\{ \begin{array}{cc}\frac{(T_{A}-T_ {M})}{(T_{A}-T)}&T\leq T_{M}\\ 1&T>T_{M}\end{array} \right. \tag{4}\]  
Other existing models describe the thermal behavior of binary PCMs, although numerous authors [43, 44, 21, 45, 46] have shown that many of them do not satisfactorily model the behavior of a PCM composed of several materials with distinct thermal behavior. Some models fail to simulate a continuous evolution of the heat capacity over the entire temperature range, while others consider the heat capacity to be identical in the solid and liquid states or simply do not allow for asymmetric PCM behavior. However, the main problem with the model presented above (Equation (2) through to Equation (4)), which considers the PCM as a binary solution, is its inability to describe the phenomena caused by the presence of two oils in the PCM, resulting in the appearance of several heat flux peaks during solidification. Figure 5: Comparison of the experimental heat flux of the ETMV 8 coating with a binary solution. The use of this binary solution model, even if a different equation is considered for solidification and melting (Figure 6), is not satisfactory to accurately reproduce the thermal behavior of the ETMV 8 coating. Figure 7 shows the comparison between the experimental heat flux and the heat flux obtained numerically by the given model, for a thermal load varying between 10 \({}^{\circ}\)C and 30 \({}^{\circ}\)C over 10 h. Notable differences are visible between the two heat fluxes, such as the asymmetry between the melting/solidification phases and a 2 nd flux peak during solidification.