## 3 Construction of LPTN-Based VML
### Calculation and Classification of TRs
#### 3.3.2 Variable TRs
According to the expression of convective TRs [49] \[R_{cv}=\frac{1}{S_{cv}\;h_{cv}}=\frac{L}{S_{cv}\;k\;Nu}\] (9) where \(L\), \(k\), \(S_{cv}\) and \(Nu\) represent the characteristic length, fluid thermal conductivity, heat dissipation area, and Nusselt coefficient, respectively. The first three parameters remain basically unchanged, and the calculation of the Nusselt coefficient requires approximating each part of the bearing as a typical heat transfer model [39]. According to the calculation method summarized in [39], the convection model of balls and oil-grease-air mixture is equivalent to a roller immersed in an infinite medium, such as 3-6 in Figure 4. In the case, for the Nusselt number during convection between balls and oil-grease-air mixture, there are  
\[Nu_{1}=\left(1.2+0.53\left(\frac{\rho_{eff}\upsilon D}{\eta_{eff}}\right)^{0.54 }\right)\left(\frac{\eta_{eff}}{\alpha\rho_{eff}}\right)^{0.3}\left(\frac{\mu\; \mid\;T_{bulk}}{\mu\;\mid\;T_{wall}}\right) \tag{10}\]  
where \(\rho_{eff}\), \(\eta_{eff}\), \(\alpha\), \(\mu\) and \(D\) represent the density, viscosity pressure, viscosity coefficient, dynamic viscosity of the oil-grease-air mixture, and balls diameter, respectively. Theseparameters remain relatively constant during the stage operation. \(v\) is the race linear velocity of the rolling element, which is positively correlated with the speed of the bearing. Let the parameters to be identified as \(K_{11}\), \(K_{12}\), \(K_{13}\), where the physical meaning of \(K_{11}\) can be regarded as the natural convection heat transfer coefficient when the speed is 0. The simple relationship between the Nusselt number of rolling elements and the bearing speed can be expressed as:  
\[Nu_{cv1}(n_{s})=K_{11}+K_{12}\cdot{n_{s}}^{K_{13}} \tag{11}\]  
Such as the node pair 2-7 in Figure 4, the Nusselt number for convection between the outer ring and the oil-grease-air mixture is equivalent in the form of two concentric rotating cylinders and calculated using the following equation:  
\[Nu_{cv23}= \begin{cases}2&\text{Ta}<41\\ 0.167(\rho g_{r}(r_{in}\omega)\mu^{-1}\sqrt{g_{r}/r_{in}}^{0.69}\text{Pr}^{0.4 })&41<\text{Ta}<100\\ 0.401(\rho g_{r}(r_{in}\omega)\mu^{-1}\sqrt{g_{r}/r_{in}}^{0.5}\text{Pr}^{0.4 })&\text{Ta}>100\end{cases}  \tag{12}\]  
The Nusselt number in the above equation is also related to the bearing speed, and the simplified calculation formula for the convective Nusselt number of the inner/outer rings can also be used in a similar way to Equation (11)  
\[Nu_{cv2}(n_{s})=K_{21}+K_{22}\cdot{n_{s}}^{K_{23}} \tag{13}\]  
\[Nu_{cv3}(n_{s})=K_{31}+K_{32}\cdot{n_{s}}^{K_{33}} \tag{14}\]  
For the node pairs 4-7 and 6-7 in Figure 4, we simplify it into a convection model of a rotating disk and a shaft end to calculate the Nusselt number between inner ring/cage and oil-grease-air mixture using Equation (15). \[Nu_{cv4}= \begin{cases}0.4\sqrt{\text{Re}\text{Pr}}&\text{Re}<2.5\times 10^{5} \\ 0.0238\text{Re}^{0.8}\text{Pr}&\text{Re}>3.2\times 10^{5}\end{cases}  \tag{15}\]  
Similarly, the Nusselt number in this equation is also related to the rotational speed and is expressed in a similar form as:  
\[Nu_{cv4}(n_{s})=K_{41}+K_{42}\cdot{n_{s}}^{K_{43}} \tag{16}\]  
Based on the above analysis, 15 unknown parameters that need to be identified are stored in the set as follows:  
\[p=[f_{2},K_{c1},K_{c2},K_{11},K_{12},K_{13},K_{21},K_{22},K_{23},K_{31},K_{32},K_{33},K_{41},K_{42},K_{43}]\]