## 4 Finite Element Analysis of Temperature Distribution Characteristics in HSLLRBs
### Temperature Field Analysis Model
#### 4.2.2 Heat Transfer Model  
Under the conditions of high speed and light load, the heat convection between the viscous flow lubricant and bearing elements is mainly considered in rolling bearings. The internal heat radiation of the bearings can be neglected. In this study, considering that skidding takes place, thermal conduction through solid-solid contact is negligible in comparison with the thermal convection.  
Figure 7: The heat flux and action time of contact surface between the roller and ring raceway.  
Heat is transferred from the bearing element to the lubricant and, thereafter, to the rest of the bearing seat. There is also heat convection between the outer surface of the housing and the surrounding air. The heat flux \(\rm{H_{v}}\) on the solid surface can be expressed as:  
\[\rm{H_{v}}=\alpha A(T_{surface}-T_{ambient}) \tag{27}\]  
Here, \(\rm{T_{surface}}\) is the surface temperature and the \(\rm{T_{ambient}}\) is the environment temperature. The formula of heat convection coefficient \(\alpha\) proposed by [PERSON] is adopted [7]. This formula considers the heat convection of the inner (outer) raceway surface, cage surface, roller surface, and lubricant.  
(1) The heat convection coefficient of oil between the inner ring and the cage can be calculated as:  
\[\rm{\alpha_{1}}=0.175\frac{K}{R}\sqrt{\frac{R_{i}\omega_{i}C}{\eta}\sqrt{ \frac{C}{R}}}/\ln\left(1+\frac{C}{R}\right) \tag{28}\]  
(2) The heat convection coefficient of oil between the outer ring and the cage can be calculated as:  
\[\rm{\alpha_{2}}=0.175\frac{K}{R}\sqrt{\frac{R(\omega_{cm}-\omega_{cm}\times s _{f})C}{\eta}\sqrt{\frac{C}{R}}}/\ln\left(1+\frac{C}{R}\right) \tag{29}\]  
(3) The heat convection coefficient of air on the surface of the bearing seat can be obtained as follows:  
\[\rm{\alpha_{3}}=0.53\frac{k}{D_{h}}(G_{r}P_{r})^{1/4} \tag{30}\]  
Here, \(\rm{G_{r}}\) is the Grashof number and \(\rm{P_{r}}\) is the Prandtl constant of the air. The natural heat convection coefficient between the outer surface of the bearing seat and air can be calculated.  
(4) The loading analysis model of heat flux and the heat convection coefficient of the bearing is shown in Figure 8a, and the convection boundary conditions on each surface of the bearing are shown in Figure 8b. Loading is as follows: the heat flux of the outer ring raceway is \(\rm{q_{1}}\), the heat convection coefficient of the outer ring raceway is \(\rm{\alpha_{1}}\), the heat convection coefficient of the outer surface of the outer ring is \(\rm{\alpha_{5}}=\rm{\alpha_{1}}\)/3, and the natural convection coefficient of the outer surface of the outer ring is \(\rm{\alpha_{6}}\). The heat convection density of the inner ring raceway surface is \(\rm{q_{2}}\), the heat convection coefficient of the inner ring raceway is \(\rm{\alpha_{2}}\), and the heat convection coefficient of the outer end of the inner ring is \(\rm{\alpha_{4}}=\rm{\alpha_{2}}\)/2. The thermal convection coefficient between the cage and the guide surface of the outer ring is \(\rm{\alpha_{3}}=2\rm{\alpha_{1}}\). The physical properties of the bearing roller and inner (outer) ring material are the same or similar, and the friction heat between the bearing roller and the ring raceway proposed by [PERSON] [31] is distributed in a ratio of 1:1 on the contact surface.  
Figure 8: Temperature distribution analysis model. (**a**) Solid model; (**b**) boundary condition model.