## 3 Results and Discussion  
The measurement results on the temperature differences between the tube and the surroundings (\(T_{t}-T_{s}\)) for several tilt angles (\(\alpha\)), numbers of fins (\(N\)), and heat inputs (\(q\)) are shown in Table 2 and Figure 4. From the measurement results, the cooling performances of tilted finned tubes are calculated and presented in Table 2 and Figure 5. In the present study, the thermal conductance, which is a reciprocal of the thermal resistance \(R_{th}\), is used as an index of cooling performance. The thermal conductance \(1/R_{th}\) can be defined as:  
\[\frac{1}{R_{th}}\equiv\frac{q}{T_{t}-T_{s}} \tag{1}\]  
The thermal conductance \(1/R_{th}\) is the heat dissipation rate per unit temperature difference between the tube and the surroundings. As shown in Figure 5, the cooling performance of the horizontal tube with tilted rectangular fins is maximized when \(\alpha=60^{\circ}\) and \(N=36\). According to Figure 4 in [27], in the case of vertical configuration, the cooling performance is also maximized when \(\alpha=60^{\circ}\) and \(N=36\). Therefore, among eleven tested finned tubes, the tilted finned tube with \(\alpha=60^{\circ}\) and \(N=36\) is the best, regardless of configuration. The thermal conductance \(1/R_{th}\) can be expressed as a function of the effective surface area \(A_{eff}\), the unfinned surface area \(A_{b}\), the fin surface area \(A_{f}\), the heat transfer coefficient \(h\), and the fin efficiency \(\eta\):  
\[\frac{1}{R_{th}}=hA_{eff}=h(A_{b}+\eta NA_{f}) \tag{2}\]  
where  
\[A_{b}=\pi DL-NLt \tag{3}\]  
\[A_{f}=Lt+2H_{f}t+2H_{f}L \tag{4}\]  
\[h=Nu_{D}k_{f}/D \tag{5}\]  
\[\eta=\frac{\sqrt{hpk_{s}A_{c}}}{hA_{f}}\frac{\left(h/\sqrt{\frac{hp}{k_{s}A_{c }}}k_{s}\right)+tanh\left(\sqrt{\frac{hp}{k_{s}A_{c}}}H_{f}\right)}{\left(h/ \sqrt{\frac{hp}{k_{s}A_{c}}}k_{s}\right)tanh\left(\sqrt{\frac{hp}{k_{s}A_{c}}} H_{f}\right)+1} \tag{6}\]  
Here, \(k_{f}\) and \(k_{s}\) are the thermal conductivities of the fluid and the solid, respectively. In Equation (6), the convection heat transfer from the fin tip is considered. The fin height \(H_{f}\), the fin perimeter \(p\), and the fin cross-sectional area \(A_{c}\) are given as:  
\[H_{f}=\sqrt{HD+H^{2}+D^{2}\cos^{2}\alpha/4}-D\cos\alpha/2 \tag{7}\]  
\[p=2t+2L \tag{8}\]  
Figure 5.0 \begin{table}
\begin{tabular}{c c c c c c c} \hline
**No.** & \(N\) & **\(\alpha\)** & \(L\) & \(D\) & \(H\) & \(t\) \\ \hline
1 & 9 & & & & & \\
2 & 12 & 90\({}^{\circ}\) & & & & \\
3 & 18 & & & & & \\ \hline
4 & 9 & & & & & \\
5 & 12 & & & & & \\
6 & 18 & 60\({}^{\circ}\) & 50 mm & 60 mm & 30 mm & 1.0 mm \\
7 & 36 & & & & & \\ \hline
8 & 9 & & & & & \\
9 & 12 & & & & & \\
10 & 18 & & & & & \\
11 & 36 & & & & & \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensions of tested tubes with tilted fins. The Nusselt numbers \(Nu_{D}\) in Equation (5) are calculated from measured temperature differences and heat inputs by using Equations (1)-(9), and are tabulated in Table 2. In addition, the hydraulic diameter \(D_{h}\) of the yellow region in Figure 2b, i.e., the channel between two adjacent fins, is given as:  
\[D_{h}=\frac{\pi(D+2H)^{2}/N-\pi D^{2}/N-4H_{f}t}{2\pi R/N+2H_{f}-t} \tag{10}\]  
Therefore, the dimensionless hydraulic diameter \(D_{h}\)* can be defined as:  
\[D_{h}^{*}=\frac{D_{h}}{D} \tag{11}\]  
In the present study, the Nusselt number correlation for the horizontal tubes with tilted rectangular fins that best matches the measurement results is developed. The functional form of the Nusselt number correlation is proposed by modifying that for the horizontal tube without fins as follows:  
\[Nu_{D}=f\cdot Nu_{D,yll} \tag{12}\]  
Figure 4: Temperature differences for several heat inputs, tilt angles, and numbers of fins. Figure 5: Cooling performances for several tilt angles and numbers of fins.