## 3 Results and Discussion
To satisfy this limiting conduction, Equation (19) can be modified as follows:  
Table 2.5  
The functional forms of Equations (18) and (20) are identical; however, the values of the coefficients for function \(f\) are slightly different. As shown in Figure 6b, Equation (20) can predict the Nusselt number within a \(\pm 10\%\) error. As shown in Figure 6a,b, the ratio of the Nusselt number of the tube with tilted rectangular fins (\(Nu_{D}\)) to the Nusselt number of the tube without fins (\(Nu_{D,cyl}\)) is less than 1. However, that does not mean that the tube without fins has better cooling performance compared to the tube with tilted rectangular fins. To compare the resistances of the tubes with and without tilted rectangular fins, it is useful to consider the ratio of the thermal resistance of the tube without fins (\(R_{th,cyl}\)) to the thermal resistance of the tube with tilted rectangular fins (\(R_{th}\)). This ratio satisfies the following equation:  
Table 2.6  
In this equation, \(h_{cyl}\) is the heat transfer coefficient of the tube without fins and \(A_{cyl}\) is the surface area of the tube without fins. This ratio of the thermal resistances is above 1 when the tube with tilted rectangular fins is better. On the other hand, this ratio is below 1 when the tube without fins is better. The effective surface area of the tube with tilted rectangular fins (\(A_{cyl}\)) is much greater compared to the surface area of the tube without fins(\(A_{cyl}\)), because a lot of additional surface area is provided by the fins in the case of the tube with tilted rectangular fins. As a result, \(A_{cyl}\)/\(A_{cyl}\) is much greater than 1. Consequently, according to Equation (21), the ratio of the thermal resistances \(R_{th,cyl}\)/\(R_{th}\) is greater than 1, because \(A_{cyl}\)/\(A_{cyl}\) is much greater than 1 even though \(Nu_{D}\)/\(Nu_{D,cyl}\) is less than 1. The tube with tilted rectangular fins has much better cooling performance compared to the tube without fins in general. Figures 7 and 8 show the cooling performances for several numbers of fins and tilt angles, respectively. In Figure 7, the cooling performance increases as the number of fins increases. This increase occurs mainly because the fin surface area increases as the number of fins increases. The cooling performance also increases as the temperature difference increases because free convection is driven by the buoyancy force, which increases as the temperature difference increases. In Figure 8, the cooling performance increases as the tilt angle increases from \(0^{\circ}\) to \(60^{\circ}\), after which it decreases for \(\alpha>60^{\circ}\). Therefore, there exists a specific tilt angle  
Table 2.7 Cooling performances for several tilt angles (\(N=18\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Constraints**} & \multicolumn{2}{c}{**Properties**} \\ \hline \(T_{t}-T_{s}\) & 50 \({}^{\circ}\)C & \(\alpha_{f}\) & 2.23 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/s \\ \(L\) & 50 mm & \(\beta_{f}\) & 0.0033/K \\ \(D\) & 60 mm & \(\
u_{f}\) & 1.6 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/s \\ \(H\) & 30 mm & \(k_{f}\) & 0.026 W/m\(\cdot\)K \\ \(N\) & 9–36 & \(k_{s}\) & 220 W/m\(\cdot\)K \\ \(\alpha\) & 60\({}^{\circ}\) & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Constraints and properties for thermal optimization. Figure 7: Cooling performances for several numbers of fins (\(\alpha=60^{\circ}\)). However, as the thickness of fins increases, the heat transfer coefficient decreases due to a decrease in fin-by-fin spacing (Figure 11). As a result, for the fixed number of fins, there exists a specific thickness of fins for which the cooling performance is maximized in Figure 9. Finally, Figure 9 shows that the optimal cooling performance of the tube with tilted rectangular fins is 0.543 W/K.