## 3 Results and Discussion
These profiles show the initial degradation of Nu at the smaller \(S/D\) and then the enhancement starts as the \(S/D\) increases, as mentioned earlier due to the velocity effect of the raising plume overcoming that of the accumulated thermal boundary layer. It is also noted that as Rayleigh's number increases, the Nusselt number profiles increase too, and the minimum degradation and maximum enhancement in Nu occur at \(S/D\) = 1.75 and 4.25, respectively. It should be noted that \(S/D\) = 0 in Figure 11 corresponds to that of natural convection from a single tube alone, and the rest is related to that of the tube in the array at different \(S/D\). Figure 12a-c shows the percent of degradation and enhancement in local Nu\({}_{\rm x}\) as \(S/D\) increases for different Rayleigh numbers compared to that of a single tube (at \(S/D\) = 0). These results show that the percent of degradation increases as the Ra\({}_{\rm x}\) increases, as shown for \(S/D\) = 1.75; however, Ra\({}_{\rm x}\) has a low effect on Nu\({}_{\rm x}\) at large \(S/D\) = 4.25. Table 5 shows the percentage of degradation or enhancement over the single tube for the three tubes in the array at different \(S/D\) as shown in Figure 12a-c.  
\begin{table}
\begin{tabular}{c c c c} \hline  & \multicolumn{2}{c}{Tube Number 1, (\(\copyright\), 2, 3)} \\ \hline S/D & A & B & R\({}^{2}\), \% \\ \hline Single tube & 0.299 & 0.240 & 98.3 \\ \hline
1.75 & 0.364 & 0.225 & 99.1 \\ \hline
2.75 & 0.346 & 0.231 & 97.8 \\ \hline
3.25 & 0.385 & 0.227 & 96.9 \\ \hline
3.75 & 0.336 & 0.235 & 97 Figure 11: _Cont._  
Figure 10: Local Nusselt numbers versus the modified Rayleigh numbers for different \(S/D\) ratios corresponding to the middle and right tube compared to that of a single tube; number 2 in (**a**), and number 3 in (**b**). Figure 11: Local Nusselt numbers versus \(S/D\) ratios for different modified Rayleigh numbers; tube number 1 in (**a**), tube number 2 in (**b**), and tube number 3 in (**c**). Figure 12: The percentage of degradation and enhancement of \(\rm{Nu_{x}}\) versus \(S/D\) for different Rayleigh numbers for the different tubes in the array; (**a**) tube number 1, (**b**) tube number 2, and (**c**) tube number 3. General correlations for each tube using the aspect ratio \(S/D\) as a parameter are obtained in the form:  
Figure 6:0  
\(C1\), \(C2\), \(C3\), and the coefficient of determination R\({}^{2}\) are shown in Figure 6:1 Figure 13a-c shows differences between the predicted and the experimental Nusselt numbers with \(S/D\) as a parameter using a correlation number (13). The maximum deviation, which appears at high Rayleigh numbers, is shown in Figure 6:2 Figure 14 shows a maximum difference of 17.9% between the predicted and the experimental Nusselt numbers using both \(S/D\) and the tube's number (n) as parameters. The overall correlation for the array is obtained as:  
Figure 6:3  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{Tube Number 1, (\(\mathbb{1}\), \(2\), \(3\))} \\ \hline S/D & Ra = \(5\times 10^{11}\) & Ra = \(1\times 10^{11}\) & Ra = \(1\times 10^{10}\) & Ra = \(1\times 10^{9}\) \\ \hline Single tube & \(0\) & \(0\) & \(0\) & \(0\) \\ \hline
1.75 & \(-17.74\) & \(-15.78\) & \(12.90\) & \(-9.92\) \\ \hline
2.75 & \(-8.06\) & \(-6.78\) & \(-4.94\) & \(-3.05\) \\ \hline
3.25 & \(-7.91\) & \(-6.05\) & \(-3.31\) & \(-0.51\) \\ \hline
3.75 & \(-0.70\) & \(0.05\) & \(1.11\) & \(2.19\) \\ \hline
4.25 & \(16.05\) & \(16.60\) & \(17.37\) & \(18.15\) \\ \hline \multicolumn{5}{c}{Tube number 2, (1, \(\mathbb{2}\), \(3\))} \\ \hline S/D & Ra = \(5\times 10^{11}\) & Ra = \(1\times 10^{11}\) & Ra = \(1\times 10^{10}\) & Ra = \(1\times 10^{9}\) \\ \hline Single tube & \(0\) & \(0\) & \(0\) & \(0\) \\ \hline
1.75 & \(-23.92\) & \(-21.23\) & \(-17.21\) & \(-1 Figure 6:4 Differences between the predicted and the experimental Nusselt numbers using \(S/D\) and tube number (\(n\)) as parameters for the array.