## 5 Results and Discussions
### Experimental Results  
#### 5.1.1 Natural Convection  
Effects of Position  
The graph of Figure 8 shows the temporal evolution of the temperatures T2, T8, T5, and T19 located in rows one, two, three, and five, respectively, and at the second line. As we can see, the temperature of the SC in the center of the fourth row (T5) is higher. Its central position within the module is unfavorable to cooling since it is not very exposed to the natural circulation of the ambient air. Moreover, the radiation exchange with the cold surfaces of the box is reduced since T5 SC it is surrounded by other hot SCs. The temperature recorded in T5 is higher than in the case of T8 SC since the upward air circulation is more complicated (less available space and more winding path) between rows four and five than between rows two and three. The supercapacitors located in the first (T2) and fifth (T19) rows are the best cooled since they are located at the ends of the SCs module. We have illustrated the results for a current rate of 40 A but these results have been verified for other current rates.  
Effect of the Current  
It is more appropriate to use relative temperature rather than absolute temperatures to compare the results from different tests since the ambient temperature varies from one test to another and sometimes even during the same test. For safety reasons, in the natural convection, the permanent regime is reached with a current less than 50 A.  
Figure 8: Evolution of T2, T5, T8, and T19 as a function of time at 40 A.  
The graph of Figure 9 shows the evolution of \(\Delta T_{rise,max}\) (the maximum temperature augmentation of all of the SCs--Equation (8)) as a function of time for five charges/discharge currents. The higher the charge/discharge current, the higher the maximum temperature reached. The relative temperature reaches 5.5 \({}^{\circ}\)C for 20 A and 27 \({}^{\circ}\)C for 50 A. The significant thickness of the lines is due to the repetition of the charge/discharge cycles and to the 1 Hz value of acquisition frequency. The time of charging/discharging cycle depends on current rate and varies between 202.5 s for 10 A and 40.5 s for 50 A. The temperature increases during the charge steep and decreasing during the discharge steep. However, the temperature increases everywhere over time during the transient period due to the fact that SCs accumulates heat during each charging/discharging cycle.  
The three relative temperatures and deviation values considered before (Equations (8)-(13)) are illustrated in Figure 10. As mentioned before, relative temperatures values increase with the increase of current. This evolution is not a linear curve.  
When the current rate is higher than 30 A, \(\Delta T_{max}\) becomes higher than 5 \({}^{\circ}\)C. Thus, the aging of SCs will be faster. These predicted curves can help to estimate the maximal relative surface temperature relative to current rates close to the limits of use. Thus, the projection of \(\Delta T_{rise,max}\) (the maximum temperature augmentation of all of the SCs) at 84 A  
Figure 10: Relative temperatures and deviation in the natural convection.  
Figure 9: Evolution of \(\Delta T_{rise,max}\) of each current as a function of time.  
is around 65 \({}^{\circ}\)C. It means that, if the ambient air temperature exceeds 0 \({}^{\circ}\)C, the maximum operating temperature (65 \({}^{\circ}\)C) is reached for the maximum continuous current (84 A).  
The results of this study will be compared with the results for the forced convection.