## 5 Results and Discussion
### Sensitivity Analysis  
Sensitivity analysis is according to Equation (17) to calculate the \(\theta\) value of each parameter to evaluate the influence of \(\phi\), \(Re\), and \(H/d\) on \(\overline{Nu}\), as shown in Table 5. The larger the value of \(\theta\), the more significant the influence. The results show that the \(\theta\) value corresponding to \(Re\) is higher than the \(\theta\) value corresponding to \(\phi\) and \(H/d\). This means that \(Re\) affects the global heat transfer performance most obviously, and it also means that the inlet velocity of the PMIFJ is an important parameter, followed by \(\phi\), and \(H/d\) has the slightest effect. Figure 10: Response surface plots of the average Nusselt number (\(\overline{Nu}\)) with respect to \(\phi\) and \(H/d\) at different \(Re=\) (**a**) 600, (**b**) 800, (**c**) 1000, and (**d**) 1200. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\phi\) & \(Re\) & \(H/d\) \\ \hline \(\theta\) & 0.213 & 0.474 & 0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Each parameter corresponds to the \(\theta\) value of \(\overline{Nu}\). The Effect of Re and \(\phi\) on the Flow, Temperature, and Heat Transfer Characteristics of the PMIF  
By sensitivity analysis, it is known that \(Re\) and \(\phi\) are the crucial parameters affecting the heat transfer performance of the PMIF. The flow field, temperature field, and radial \(Nu\) distribution are presented to further explore the reasons for their influence. First, the effect of \(Re\) on the PMIF is discussed. At fixed \(\phi=1.2\) and \(H/d=1.5\), the contours of the velocity and temperature fields at different Reynolds numbers are shown in Figure 12. The premixed cone's height rises with \(Re\) because of the increase in inlet axial velocity. It is also observed that as \(Re\) increases, the velocity behind the reaction region also rises, causing the PMIMJ to spread more and the high-temperature region to become more extensive. Figure 13 provides the radial \(Nu\) distribution for different Reynolds numbers. Compared with the \(Nu\) distribution of \(Re=800\), the \(Nu\) distribution of \(Re=600\) in the stagnation region is much lower because the lower Reynolds number equals less fuel usage. Another reason is that Figure 12a shows the existence of a low-velocity region, which causes the stagnation region to expand and reduce its temperature. This is consistent with the results measured by [PERSON] and [PERSON] using non-invasive laser diagnostic technology [60]. It may be related to the velocity distribution at the burner exit. The velocity distribution of the top hat is produced by the nozzle burner used in this study and in Ref. [60]. It is different from the parabolic velocity distribution using a tube burner [4,8,26,45]. The stagnation point's Nusselt number (\(Nu_{0}\)) of \(Re=1000\) is reduced to 9 due to the low-temperature premixed cone (unburned mixture) slightly hitting the impingement plate. In addition, the peak Nu is far away from the stagnation point because of the shift of the peak axial velocity [26]. When the \(Re\) value increases to 1200, more low-temperature premixed cones are intercepted by the impingement plate, which means there is more unburned mixture in the stagnation region, further suppressing the heat flux at the stagnation point. Therefore, \(Nu_{0}\) has zero value in a small region. The unburned mixture burns a short distance from the stagnation point to obtain chemical energy, resulting in maximum heat flux in this region. Therefore,  
Figure 11: Response surface plots of the average Nusselt number (\(\overline{Nu}\)) with respect to \(\phi\) and \(Re\) at different \(H/d=\) (**a**) 1.5, (**b**) 2.0, (**c**) 2.5, and (**d**) 3.0.  
the peak \(Nu\) of \(Re=1200\) is greater than the peak \(Nu\) of \(Re=1000\). In the wall jet region, the radial \(Nu\) distribution significantly rises as \(Re\) increases. As mentioned, the PMIFJ with high \(Re\) has a more diffuse and extensive high-temperature region, promoting more convective heat transfer.