## 2 Model Establishment
### Convective Heat Transfer of Bearings  
Bearing-induced heat is released through conduction, convection and radiation [24]. Given the modest temperature difference between the bearing components, heat from thermal radiation is ignored for its secondary position. Bearing heat is mainly dissipated through conduction and convection, among which the conduction heat between inner/outer rings and rollers is distributed in a proportion of 1:1 [25]. The bearing in the traction motor of a high-speed EMU is lubricated with grease; therefore, the bearing's convective heat transfer includes the convective heat transfer between the grease and the inner/outer raceway surfaces, the convective heat transfer between grease and rolling body/cage surfaces, and the convective heat transfer between outer surface of the bearing seat and the outer air.  
The fluid state can be divided into laminar flow and turbulent flow according to flow characteristics [26]. When calculating the convective heat transfer coefficient, the difference in material state will produce the Nusselt criterion of different flow states, and the convective heat transfer coefficient is also different. The lubricant moves in a laminar flow inside the bearing, which makes the description of heat convection rather difficult. The convective heat transfer coefficient can be approximately expressed by the surface heat transfer coefficient for a plate in a laminar flow [21].  
\[h_{g}=0.0332k_{g}P_{g}^{1/3}\bigg{(}\frac{u_{g}}{v_{g}D}\bigg{)}^{1/2} \tag{5}\]  
In the formula, when considering the heat transfer between the bearing and the lubricant, the surface speed of the bearing cage is \(u_{g}\); \(v_{g}\) is the lubricant grease kinematic viscosity; \(P_{g}\) is the lubricant grease Prandtl number, and \(k_{g}\) is the lubricant grease thermal conductivity, as shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Oil-Mist Lubrication** & **Oil-Bath/Grease** & **Vertical** \\  & **Lubrication** & **Oil-Bath/Spray-Grease** \\  & & **Lubrication** & **Lubrication** \\ \hline Ball bearing & 0.7–1 & 1.5–2 & 3–4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Value of coefficients related to bearing type and lubrication method with different lubrication types.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & \(f_{1}\) & \(P_{1}\) \\ \hline Ball bearing & \(0.0009(P_{0}/C_{0})^{0.55}\) & \(3F_{a}-0.1F_{r}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Calculation method of coefficient related to bearing type/load and calculated load value.  
Since the surface velocity of the bearing cage is related to angular velocity and the bearing pitch diameter, the above formula is simplified as  
\[h_{g}=0.0332k_{g}P_{g}\frac{1}{\left(\frac{\pi n_{c}}{60v_{g}}\right)^{\frac{1}{ 2}}} \tag{6}\]  
where \(n_{c}\) is the speed of the bearing cage. It can be seen from the above formula that the convective heat transfer coefficient between the lubricant and the bearing components is constant in the approximate calculation, which is only related to the cage speed and lubricant performance.  
When the traction motor is in operation, the bearing chamber can achieve a marked cooling effect, since forced ventilation cooling is at work internally and the external wind is at high velocity. The forced convection model is used to calculate the convection heat transfer coefficient between the bearing chamber outer surface and the air, and the calculation formula is as follows [27]:  
\[\alpha=0.3\frac{k_{air}}{D_{ch}}\bigg{(}\frac{u_{air}D_{ch}}{v_{air}}\bigg{)}^ {0.57} \tag{7}\]  
where \(k_{air}\) is the thermal conductivity of air; \(D_{ch}\) is the diameter of the bearing chamber; \(v_{air}\) is the kinematic viscosity of air; \(u_{air}\) is the flow rate of air.