# Thermal Characteristics Study of the Bump Foil Thrust Gas Bearing
## 2 Numerical Simulation Research
### Heat Transfer Model of Bearing Components  
#### 2.5.1 Heat Transfer Model of Top Foil Side  
In this paper, the equivalent thermal resistance is applied to calculate the overall heat transfer processes when cooling air is introduced or not.  
Figure 4 is the heat transfer model on the top foil side. The figure on the left shows that when cold air is not passed into the foil, the heat generated in the gas film passes through the top foil through conduction. Part of the heat is conducted to the base plate through the bump foil, and the other part is conducted to the base plate through the air in the gap between foils. As the base plate is in close contact with the bearing housing, the heat is transferred to the bearing housing through heat conduction, and finally spread out through natural convection. The figure on the right shows that when the cooling air flow is introduced into the foil, the heat generated in the gas film transferred through a different way. The heat passes through the top foil through conduction and part of it is transferred to the bump foil through heat conduction, and another part is taken away by the cold air directly. The heat transferred to the bump foil is conducted to the base plate partly, and the rest of it is taken away by the cold air which passed through the upper and lower surfaces of the bump foil. Then, part of the heat transferred to the base plate is taken away by the cooling air flow on the upper surface of the base plate through convection, the other part is transferred to the bearing housing through heat conduction, and finally spread out through natural convection. Correspondingly, the equivalent thermal resistance models of these two situations are established, respectively.  
Figure 5 is the schematic diagram of thermal resistance model established based on the ways of heat transfer in the foil structures, where \(T_{1}\) is the air film temperature on the top foil side; \(T_{0}\) is the ambient temperature. The remaining parameters are the corresponding equivalent thermal resistance.  
When there is no cooling air flow injected, the equivalent total thermal resistance is calculated as follows:  
\[R_{a}=R_{-}t+\frac{1}{\frac{1}{R_{-}bp}+\frac{1}{R_{-}lb}}+R_{-}b+R_{-}w+R_{-}e \tag{13}\]  
when the cooling air flow is introduced into the bump foil, the equivalent total thermal resistance is calculated as follows:  
\[R_{a}=R_{-}t+\frac{1}{\frac{1}{R_{-}bp+\frac{1}{R_{-}b}+\frac{1}{R_{-}c}}}+ \frac{1}{R_{-}c} \tag{14}\]  
where \(R_{-}i\) is calculated as follows:  
\[R_{-}i=R_{-}b+\frac{1}{\frac{1}{R_{-}lb}+\frac{1}{R_{-}w+R_{-}c}} \tag{15}\]  
Figure 4: Heat transfer paths near the top foil.  
the calculation formula for each thermal resistance is as follows:  
\[\begin{split} R\_t,R\_bp,R\_tb,R\_b,R\_w=t_{i}/k_{i}A_{i}\\ R\_c,R\_hbc,R\_hb,R\_e=1/A_{j}h_{j}\end{split} \tag{16}\]  
where _i,j_ are the serial number of the thermal resistance; \(t\) is the thickness on heat transmission direction (m); \(k\) is the thermal conductivity of the material (W/K\(\cdot\)m); \(h\) is the coefficient of heat convection(W/m\({}^{2}\cdot\)K); \(A\) is the heat transmission area (m\({}^{2}\)). Under normal circumstances, the parameters \(t\) and \(A\) are determined by the geometric features of the components, exceptionally, the _A_\({}_{i}\) of _R_bp_ is contact area between the bump foil and base plate, and the _A_\({}_{j}\) of the _R_hbc_ is approximated obtained by straight inclined segments of the bump foil.  
When the gas film temperature reaches equilibrium, the heat transferred into the top foil from the gas film will be in dynamic balance with the heat transferred from the top foil side to the outside. Therefore, the heat balance equation between \(T_{1}\) and \(T_{0}\) considering the total thermal resistance of the foil structures \(R_{a}\) will form the:  
\[-k_{a}A\frac{\partial T_{1}}{\partial_{Z}}=\frac{T_{1}-T_{0}}{R_{a}} \tag{17}\]