## 5 Energy Balance and Error Analysis
### Evaluation of Errors Related to Uncertainties  
In the context of this study, uncertainties may arise in the measurement of heat flux and temperature of the sample during testing, as previously mentioned. These error sources can lead to misinterpretation of the results and thus reduce the accuracy of the characterization of thermophysical properties of the material. According to the law of propagation of uncertainties, the compound uncertainty (\(U_{c}\)) includes several components [62]. In the present case, it is essential to take into account errors related to the T-type thermocouples (\(U_{T}\)), the tangential gradient fluxmeters FGT (\(U_{\varphi}\)), and the data acquisition unit (\(U_{mod}\)). For heat flux measurements, this compound uncertainty is given by the following equation:  
\[U_{c,\Phi}=\left[{U_{\Phi}}^{2}+U_{b,\Phi}^{2}\right]^{\frac{1}{2}} \tag{20}\]  
Similarly, the compound uncertainty related to temperature measurements is expressed as:  
\[U_{c,\text{ T}}=\left[{U_{T}}^{2}+U_{b,T}\right]^{\frac{1}{2}} \tag{21}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Experimental**} \\ \hline
**Loading** & **Melting [J]** & **Solidification [J]** & **Difference [\%]** \\ \hline
3-h ramp & 136,233 & 135,727 & 0.37 \\
6-h ramp & 134,579 & 135,850 & 0.94 \\
10-h ramp & 135,770 & 135,937 & 0.12 \\ Max difference [\%] & 1.2 & 0.2 & \\ \hline \hline \end{tabular}
\end{table}
Table 6: Energy balance, based on the experimental heat flux.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Numerical**} \\ \hline
**Loading** & **Melting [J]** & **Solidification [J]** & **Difference [\%]** \\ \hline
3-h ramp & 137,232 & 133,968 & 2.38 \\
6-h ramp & 137,235 & 133,978 & 2.37 \\
10-h ramp & 137,225 & 133,922 & 2.41 \\ Difference [\%] & 1.0 10\({}^{-2}\) & 4.0 10\({}^{-2}\) & \\ \hline \hline \end{tabular}
\end{table}
Table 7: Energy balance, from the numerical heat flux.  
The error on fluxmeter sensitivity (114 \(\mathrm{\SIUnitSymbolMicro W\cdot W^{-1}\cdot m^{-2}}\)) has been estimated at 2% [63], with the uncertainty values being calculated for a heat flux of approx. 100 W\(\cdot\)m\({}^{-2}\) (Table 8). The standard uncertainty of the acquisition unit is provided by the manufacturer. Thus, the compound uncertainty for the heat flux measurement is evaluated here at 2.04 W\(\cdot\)m\({}^{-2}\). The temperature measurement errors are mainly correlated with both the accuracy of thermocouples provided by the manufacturer and the accuracy of the central acquisition unit. Temperature is therefore measured with a precision of plus or minus 0.5 \({}^{\circ}\)C. The error range related to uncertainties of heat flux measurements during tests conducted to characterize the ETMV 8 coating is shown in Figure 21.  
The actual heat flux through the coating may fall within the range of uncertainty shown in Figure 21. The accumulation of these uncertainties in heat flux and temperature measurements can affect the thermophysical characterization of the coating using inverse methods since such a characterization depends mainly on experimentally collected data. This aspect can also have an impact on the calculated energy balance during the melting and solidification phases of the material, as discussed in Section 5.1. However, these errors, equal to around 2% on the heat flux, as well as the inaccuracy of the temperature measurement (\(\pm\)0.5 \({}^{\circ}\)C), which generated about 4% error on the determination of the PCM liquid fraction, do remain quite acceptable, which allows validating the results proposed in this article.  
Figure 21: Uncertainties of heat flux measurement on the ETMV 8 coating.