# CFD Analysis on the Heat Dissipation of a Dry-Lubricated Gear Stage
## 3 Results
### Analysis of the Heat Transfer Coefficient
#### 3.2.3 Surface-Averaged Heat Transfer Coefficient  
Based on the results in Section 3.2.1, surface-averaged heat transfer coefficients \(\overline{h}\) for the investigated gear geometries and circumferential speeds are evaluated. Figure 10 shows the surface-averaged HTC for the shaft, side, and tooth surfaces of the wheel of gear-C and gear-LL for the circumferential speeds \(\mathrm{v_{t}}\) = {2.0; 5.0; 10.0} m/s. They show an increase with \(\mathrm{v_{t}}\). The tooth surfaces exhibit the highest values, followed by the side surfaces and the shaft surfaces. Thus, higher heat transfer coefficients are present for surfaces that lie further out on the gear in the radial direction. The comparison of gear-C and gear-LL shows, for the shaft and side surfaces, comparable surface-averaged HTC values for all circumferential speeds. In contrast, the surface-averaged HTC values for the tooth surfaces of gear-LL are significantly higher. This is in agreement with Section 3.2.2, according to which the helical gear-LL shows a more pronounced incident flow of the teeth.  
Figure 9: Detailed view on axial flow velocity (z-direction) and HTC between gear surface and air for the wheels of gear-C (**a**) and gear-LL (**b**) for \(\mathrm{v_{t}}\) = 10.0 m/s.  
In the following, the derived surface-averaged HTC values are classified in the state of the art. Figure 11 illustrates the overall surface-averaged HTC of the pinion and wheel for gear-C and gear-LL for the circumferential speeds vt = {2.0; 5.0; 10.0} m/s. For gear-LL, higher surface-averaged HTC values result for a given circumferential speed compared to gear-C. Figure 11 also shows the calculated HTC values based on the Nusselt correlation according to [PERSON] [15], see Equations (2) and (3). The tip diameter was chosen as the reference length in accordance with [PERSON] et al. [17]. The underlying Reynolds number is within the given validity range of \(10^{3}<Re<10^{5}\). The analytical results predict the magnitude and trends over the circumferential speed in good agreement with the simulation results. With increasing circumferential speed, larger deviations can be noticed. Based on the Nusselt correlation, comparable HTC values are calculated for gear-C and gear-LL: The surface-averaged HTC value of gear-LL for the different circumferential speeds is only about 2.5% lower than that of gear-C, due to the smaller tip diameters of gear-LL. This demonstrates that the influence of the gear geometry and fluid flow on the HTC cannot be easily addressed by simple correlation analysis.  
The results show that for circumferential speeds vt = 2.0...10.0 m/s, the surface-averaged HTC for the considered dry-lubricated gears can be classified in the range of \(\overline{h}\) = 20...50 W/(m\({}^{2}\)K). This correlates with the magnitude of the HTC for solids and forced convection in gases specified by the VDI Heat Atlas [38]. The results of [PERSON] et al. [14] showed that the HTC between gears and oil for a circumferential velocity of about vt = 30 m/s is of the magnitude of \(\overline{h}=400\) W/m\({}^{2}\)K.  
Figure 11: Surface-averaged HTC \(\overline{h}\) for gear-C and gear-LL for circumferential speeds vt = {2.0; 5.0; 10.0} m/s in comparison to HTC based on Nusselt correlation according to [PERSON] [15].  
Figure 10: Surface-averaged HTC \(h\) for shaft, side, and tooth surfaces for the wheel of gear-C and gear-LL for circumferential speeds vt = {2.0; 5.0; 10.0} m/s.