## 5 Results and Discussions
### Numerical and Experimental Comparison  
The results presented hereafter are obtained in the forced convection regime, with the fan turned on (v\({}_{\text{0 air}}\) = 0.69 m s\({}^{-1}\)), using a current rate of 70 A for charge/discharge, which is the most unfavorable case studied during the experiment phase (maximum current rate and supercapacitor temperatures). Steady state is assumed for the numerical simulation. For experimental results, the temperature is obtained for t = 1 h, and at that moment, the evolution is asymptotic (Figure 11). The numerical and the experimental results are in good agreement (Table 3). Numerical simulation results of the temperature at the same location as the experimental measurements were compared (see Figure 1 for temperature sensor location for experimental measurements). The deviation for the numerical and experimental values for temperature is constantly less than 10%, for all of the compared zones. We concluded that the validation is satisfactory, and we proceeded for the use of the numerical simulation results to complete the understanding of physical phenomena in the supercapacitor module stack. The maximum temperature augmentation (\(\Delta T_{rise,max}\)--Equation (8)) and the average of the temperature rises (\(\Delta T_{rise,avg}\)--Equation (12)) of all of the SCs are approximately 3.1 \({}^{\circ}\)C higher inside compared to those on the surface. This increase is explained by the maximum temperature value located in the middle of the SCs. These parameters are important since they are linked to the maximum operating temperature which should stay less than 65 \({}^{\circ}\)C for safety working conditions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Experimental**} & **Numerical** & **Deviation [\%]** \\ \hline \(l\) & A & \multicolumn{2}{c}{70} & \\ \hline \(s\) & Wm\({}^{-3}\) & Boundary & & 9461 & \\ \hline \(\text{v}_{\text{0 air}}\) & ms\({}^{-1}\) & conditions & 0.69 & 0.69 & \\ \hline \(\text{T}_{\text{0 air}}\) & \({}^{\circ}\)C & & 26.48 & 26.48 & \\ \hline T3 & \({}^{\circ}\)C & & 34.84 & 36.5 & \The operating stability and lifetime of supercapacitor modules are likewise affected by its non-uniform temperature distribution. However, the \(\Delta T_{max}\) (Equation (10)), the difference between the highest and the lowest temperature augmentation SCs, is approximately 1 \({}^{\circ}\)C smaller inside compared to the value on the surface. This parameter is important since it should stay less than 5 \({}^{\circ}\)C [9] to prevent the premature aging of SCs. The obtained value, 10.5 \({}^{\circ}\)C, lets predict a premature aging of the supercapacitor module with this thermal management. Figure 13 presents the temperature distribution of the supercapacitors on their external surface (Figure 13a) and in the middle longitudinal plan (symmetry plane). The perpendicular plan, which divides each supercapacitor into two equal cylindrical volumes, is chosen since the most important overheating is recorded there. The maximum temperature is located in the middle of SCs around the axis. For the studied case the maximum internal temperature is 49.34 \({}^{\circ}\)C. For a full lifespan, SCs are required to maintain temperature below 50 \({}^{\circ}\)C [9]. As expected, similar with the experimental study, the temperature increases from the first to the last row. The contours of temperature and velocity in the whole symmetry plane are presented respectively in Figures 14 and 15. It can be seen on Figure 15 that the velocity field changes between the third and the fourth row. For the SCs aligned distribution, the main flow is tangential on the SCs lateral surface. After the modification of the arrangement distribution, which occurs between the thirds and fourth row, a significant flow is oriented perpendicular to the center of the SCs of the fourth row. Local velocity increase in this place explains the lower temperature rise between the third and fourth rows (from Figure 14).