## 4 Numerical Results
### Surface Heat Flux Distribution  
Figures 5-8 present contour plots of surface heat flux entering the ducts from the topside representing the back sheet of the PV module. The results have been plotted using an identical scale to distinguish the amount of heat flux through visual reference, i.e., colors on the lower end of the legend represent lower heat transfer to the back sheet leading to cooler back sheet temperatures and greater heat transfer to the cooling fluid. However, these contour plots must be viewed with the fact in mind that, except for the case of the duct flush with the bottom side of the back sheet, i.e., case (a) in Figures 5-8, all other cases also allow for heat transfer throughout the perimeter or other sides of the duct, which is not obvious from these contour plots. The case that allows for maximum heat transfer to the cooling fluid can be gleaned from Table 2, which lists the summary of area-weighted temperatures and heat transfer rates at the outlet of the cooling ducts. Table 2 indicates that the case (c), representing the duct embedded in the middle of the back sheet in all the cases, results in the highest heat transfer rate \(\dot{q}_{out}\) as well as the highest cooling fluid temperature \(T_{out}\) at the exit while case (a), representing the duct flush with the bottom side of the back sheet in all the cases, results in the lowest heat transfer rate \(\dot{q}_{out}\) as well as the lowest cooling fluid temperature \(T_{out}\) at the exit. Therefore, results shown in Figures 5-8 should be viewed together with the results summary in Table 2 to determine which of the cases result in enhanced heat transfer to the cooling fluid and lowest back sheet temperatures. Hence, for the case of the rectangular duct, Figure 5a-d, case (c) of the duct embedded in the middle of the back sheet represents the case that results in the highest heat transfer rate \(\dot{q}_{out}\) as well as the highest cooling fluid temperature \(T_{out}\) at the exit while case (a), representing the duct flush with the bottom side of the back sheet, results in the lowest heat transfer rate \(\dot{q}_{out}\) as well as the lowest cooling fluid temperature \(T_{out}\) at the exit of the duct. Similar results were obtained for all four different cooling duct cross-sections (shown in Figures 6-8), where case (c) of the embedded duct in all cases resulted in the highest heat transfer rate \(\dot{q}_{out}\) as well as the highest cooling fluid temperature \(T_{out}\) at the exit while case (a) of the duct flush at the bottom of back sheet in all cases resulted in the lowest heat transfer rate \(\dot{q}_{out}\) as well as the lowest cooling fluid temperature \(T_{out}\) at the exit of the duct. Ducts located flush on the bottom or top inside the side of the back sheet, cases (b) and (d) in Figures 5-8, exhibit similar performance, with the duct flush on the top inside of the back sheet being slightly superior regarding cooling performance compared to the duct flush on the bottom inside of the back sheet. Figure 5: Surface heat flux distribution received by the rectangular duct from top for the four different positions of the duct relative to the back sheet. (**a**) Rectangular duct flush on the outside bottom; (**b**) rectangular duct flush on the inside bottom; (**c**) rectangular duct embedded in the middle; (**d**) rectangular duct flush on the inside top. Figure 6: Surface heat flux distribution received by the semi-circular duct from top for the four different positions of the duct relative to the back sheet. (**a**) Semi-circular duct flush on the outside bottom; (**b**) semi-circular duct flush on the inside bottom; (**c**) semi-circular duct embedded in the middle; (**d**) semi-circular duct flush on the inside top. sections investigated in this study, numerical simulation results indicate that the lowest back sheet bottom-side temperatures and highest cooling fluid temperatures result when the ducts are placed in the middle of the back sheet, i.e., case (c) of the embedded duct in Figures 9-12.