# Research on Temperature Field Analysis Method of High-Speed Bearing Chamber-Bearing System
## 2 Model and Method
### Numerical Method of Oil-Air Two-Phase Flow in Bearing Chamber
In numerical calculations in Bearing Chambers 1, 2, and 5, the axial and circumferential oil inlet velocities are calculated using Equation (9) and Equation (10), respectively. The axial oil inlet velocity is determined by flow rate and inlet cross-sectional area:  
\[q=\rho vA \tag{9}\]  
The circumferential oil inlet velocity is the angular velocity of the bearing cage, in the case where the bearing outer ring does not rotate and the motion relationship inside the bearing occurs under ideal conditions. For deep groove ball bearings, the revolution speed of rolling elements and cage is [19]  
\[n_{m}=\frac{1}{2}n_{i}\bigg{(}1-\frac{d_{0}}{d_{m}}\bigg{)} \tag{10}\]  
where \(n_{m}\) is the revolution speed of rolling elements and cage, \(n_{i}\) is the rotation speed of the bearing inner ring, \(d_{0}\) is the rolling element diameter, \(d_{m}\) is the bearing pitch diameter. The oil inlet flow rate of the chamber is directly related to the flow rate of oil that is splashed by the bearing during operation. The oil inlet temperature of the chamber corresponds to the temperature of the oil as it exits the bearing, which is the oil outlet temperature of the bearing. In the computational model, the flow rate of oil splashed by the bearing is distributed evenly between the bearing chambers on either side of the bearing, with a 1:1 ratio. Given that the oil flow rates for each bearing can vary due to differences in bearing design, load, or operating conditions, the oil outlet temperature of each bearing will also differ. This variation in outlet temperature directly affects the oil inlet temperatures for each bearing chamber, as the oil entering the chambers will have different temperatures based on the specific bearing it originates from. The calculated oil inlet conditions for the bearing chambers are presented in Table 4, which provides details on the flow rates and temperatures of the oil entering each chamber. This table is essential for accurately simulating the thermal and hydrodynamic behavior of the oil within the bearing chambers and for ensuring that the simulation reflects the real-world operating conditions of the system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Bearing Chamber** & **Operating Condition** & \
\begin{tabular}{c} **Oil Flow Rate of** \\ **Chamber (L/min)** \\ \end{tabular} &
\begin{tabular}{c} **Oil Inlet** \\ **Temperature (°C)** \\ \end{tabular} & **Axial Speed (m/s)** & **Radical Speed (m/s)** \\ \hline \multirow{3}{*}{Chamber 1} & 1 & & 160.41 & \\  & 2 & 0.66 & 160.37 & 0.266 & 59.8 \\  & 3 & & 160.35 & \\ \hline \multirow{3}{*}{Chamber 2} & 1 & & left 160.41, right & \\  & 1 & & left 214.34 & \\  & 2 & & left 160.37, right & left 0.266, \\  & 2 & & right 0.4 & 214.26 & right 0.125 & 59.8 \\  & 3 & & left 160.35, right & \\  & 1 & & 214.22 & \\ \hline \multirow{3}{*}{Chamber 5} & 1 & & 160.41 & \\  & 2 & 0.66 & 160.37 & 0.266 & 59.8 \\ \cline{1-1}  & 3 & & 160.35 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Oil inlet condition of bearing chambers. As the rotational speeds are consistent across the different operating conditions, the oil outlet flow rates are also the same. Additionally, the differences in oil outlet temperatures are less than 1% under the three operating conditions considered. Therefore, this article focuses on the oil-air two-phase flow calculations for Bearing Chambers 1, 2, and 5 specifically under Operating Condition 1. By doing so, the convective heat transfer coefficients of the chamber walls, which are obtained from these calculations, are used as the thermal boundary conditions for the temperature field analysis of the entire bearing chamber-bearing system under all three operating conditions. This approach leverages the consistency in flow rates and temperature differences to simplify the analysis, ensuring that the thermal boundary conditions are representative of the overall system's behavior across the various operating scenarios.