## 3 Results and Discussion
\(Nu_{D,cyl}\) is the Nusselt number for the horizontal tube without fins, and is given as the following equation [28]:  
\[Nu_{D,cyl}=\left(0.60+\frac{0.387 Ra_{D}^{1/6}}{\left(1+\left(0.559/\text{Pr} \right)^{9/16}\right)^{8/27}}\right)^{2} \tag{13}\]  
In Equation (13), the Rayleigh number \(Ra_{D}\) is defined as Equation (14) in terms of the thermal diffusivity of the fluid \(\alpha_{f}\), the kinematic viscosity of the fluid \(\
u_{f}\), the volume expansion coefficient of the fluid \(\beta_{f}\), and the gravitational acceleration \(g\):  
\[Ra_{D}=\frac{g\beta_{f}(T_{w}-T_{amb})D^{3}}{\
u_{f}\alpha_{f}} \tag{14}\]  
The function \(f\) in Equation (12) is related to the influence of the tilted fins on the heat dissipation. Function \(f\) should monotonically decrease as the hydraulic diameter \(D_{h}\) decreases. This decrease occurs because the heat dissipation from a fin is suppressed by adjacent fins owing to the overlap of the thermal boundary layers developing on the surfaces of the fins. As the hydraulic diameter \(D_{h}\) decreases, the fin-by-fin spacing decreases, and the overlap of the thermal boundary layers and the suppression of the heat dissipation from the fins increases. Several functional forms for function \(f\) that satisfy this condition are tested by using a least-squares fit on the measurement results, and it is found that the functional form:  
\[f=C_{1}-C_{2}\exp(-C_{3}D_{h}^{*}) \tag{15}\]  
best matches with the measurement results when the coefficients are given as:  
Table 2.0  
The coefficient \(C_{1}\) is the ratio of Nusselt numbers of the tubes with and without fins in the case when the hydraulic diameter of the space between two adjacent fins is much greater than the diameter of the tube. Therefore, the heat transfer coefficient of the tube with tilted fins is 1.08 times greater than that of the tube without tilted fins, when the hydraulic diameter is much greater than the tube diameter. Finally, the Nusselt number correlation for the horizontal tubes with tilted rectangular fins under free convection is given as:  
Table 2.1  
When the fin length is zero (\(H=0\), \({D_{h}}^{*}=0\)), the Nusselt number should be reduced to that for the horizontal tube without fins. To satisfy this limiting conduction, the correlation can be modified as follows:  
Table 2.2  
Equations (17) and (18) are identical when \(H\)/\(D\) is 0.5. In Figure 6a, the Nusselt numbers obtained from the measurement results for the tilt angles of 30\({}^{\circ}\) and 60\({}^{\circ}\) and those calculated from Equation (18) are presented as functions of the dimensionless hydraulic diameter. Figure 6a indicates that the Nusselt number is strongly dependent on the hydraulic diameter, and that the Nusselt number correlation is in good agreement with the measurement results within a \(\pm\)10% error. The Nusselt numbers obtained from the measurement results for the tubes with radial rectangular fins (\(\alpha=0\)) presented in [24] are also compared with those calculated from Equation (18). The correlation can predict the Nusselt numbers of the tubes with radial rectangular fins within a \(\pm\)10% error, in spite of the fact that the correlation is developed for the tubes with tilted fins. In Figure 6b, the Nusselt numbers obtained from the measurement results for the tilt angle of 90\({}^{\circ}\) are compared with those calculated from Equation (18). In this case, Equation (18) can predict the Nusselt number within a \(\pm\)20% error. In addition, another correlation for the Nusselt number is developed to improve the prediction regarding only the tilt angle of 90\({}^{\circ}\), as in the following:  
Table 2.3  
Table 2.4 The Nusselt numbers obtained from measurement results and from the correlations. (**a**) Nusselt numbers for tilt angles of 0\({}^{\circ}\), 30\({}^{\circ}\), and 60\({}^{\circ}\); (**b**) Nusselt numbers for the tilt angle of 90\({}^{\circ}\). When the fin length is zero (\(H=0\), \(D_{h}{}^{*}=0\)), the Nusselt number should be reduced to that for the horizontal tube without fins.