# Research on Temperature Field Analysis Method of High-Speed Bearing Chamber-Bearing System
## 2 Model and Method
### Numerical Method of Temperature Field of Bearing Chamber-Bearing System  
(1) Bearing Heat Generation  
Referring to the distributed heat generation law of roller bearing, this article takes 60% of the total ball bearing heat generation as oil churning loss and the remaining 40% as mechanical friction loss. The heat generated by oil churning is carried away by lubricating oil, and the heat generated by mechanical friction is imposed on the bearings. The total bearing heat generation is distributed in a ratio of 1:2:1 to the inner ring, rolling elements, and outer ring, respectively. The raceway surfaces on the inner and outer rings of the bearing, as well as the surfaces on rolling elements, are taken as heat loading surfaces. Based on the heat loading surface's area and a wall thickness of 2 mm, the heat generation rate (unit: W/m3) is calculated and applied to the inner ring, outer ring, and rolling elements. The specific data are shown in Table 5.  
(2) Convection Heat Transfer Conditions Inside Bearings  
The Crecelius convective heat transfer coefficient calculation model is used to obtain convective heat transfer coefficients within the bearing, which are applied to the surfaces of the bearing outer ring, rolling elements, and the bearing inner ring. The calculation formula is as follows [20]:  
\[h=0.0986\Big{(}\frac{n}{v}\Big{(}1\pm\Big{(}\frac{d_{0}\cos\beta}{d_{m}}\Big{)} \Big{)}\Big{)}^{\frac{1}{2}}k\text{Pr}\frac{1}{3} \tag{11}\]  
where \" + \" represents the rotation of the bearing outer ring, \" - \" represents the rotation of the bearing inner ring, \(n\) is the bearing rotational speed, \(d_{0}\) is rolling element diameter, \(d_{m}\) is bearing pitch diameter, \(v\) is oil kinematic viscosity, \(\beta\) is contact angle of ball bearing, \(k\) is oil thermal conductivity, Pr is Prandtl number. The oil physical parameters in the formula are qualitatively determined by the oil inlet temperature of the bearing. According to Formula (11), the convection heat transfer coefficient within ball bearings under Operating Conditions 1, 2, and 3 is 4970 W/(m2\(\cdot\)K). The temperature of the bearing inner wall is set to the oil supply temperature of 75 \({}^{\circ}\)C.  
(3) Boundary Condition of Oil Supply Pipeline Walls \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Operating**} & \multicolumn{2}{c}{**Heat Flux of Bearing 1, Bearing 4**} & \multicolumn{2}{c}{**Heat Flux of Bearing 2, Bearing 3**} \\  & \multicolumn{2}{c}{**(W/m3)**} & \multicolumn{2}{c}{**(W/m3)**} \\ \cline{2-7}  & **Inner Ring** & **Balls** & **Outer Ring** & **Inner Ring** & **Balls** & **Outer Ring** \\ \hline
1 & \(1.177\times 10^{9}\) & \(1.22\times 10^{9}\) & \(1.017\times 10^{9}\) & \(1.16\times 10^{9}\) & \(1.20\times 10^{9}\) & \(1.00\times 10^{9}\) \\
2 & \(1.176\times 10^{9}\) & \(1.22\times 10^{9}\) & \(1.017\times 10^{9}\) & \(1.16\times 10^{9}\) & \(1.20\times 10^{9}\) & \(9.99\times 10^{9}\) \\
3 & \(1.176\times 10^{9}\) & \(1.22\times 10^{9}\) & \(1.016\times 10^{9}\) & \(1.16\times 10^{9}\) & \(1.20\times 10^{9}\) & \(9.99\times 10^{9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Heat flux of bearings. In the steady state of the oil supply pipeline before the injection nozzle, the fluid temperature remains relatively constant. To simulate this condition, a constant wall temperature boundary condition is applied to the walls of the oil supply pipeline.