# Thermal Characteristics Study of the Bump Foil Thrust Gas Bearing
## 2 Numerical Simulation Research
### Heat Transfer Model of Bearing Components
#### 2.5.2 Heat Transfer Model of the Thrust Plate  
Figure 6 is the assembly diagram of a single side thrust bearing. Considering that the thrust plate has high rotating speed, it is assumed that the surface temperature of the thrust plate is consistent along the circumferential direction in the calculation process.  
The governing equation of the one-dimensional thrust plate heat conduction model in the radial direction is formulated as:  
\[\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}k_{disc}r\frac{\partial T_{disc }}{\partial r}\bigg{)}+\frac{h_{conv\_disc}}{t_{disc}}(T_{\text{disc}}-T_{0})+ \overset{\bullet}{q}_{film\_disc}+\overset{\bullet}{q}_{disc\_sur}=0 \tag{18}\]  
Figure 5: Equivalent thermal resistance model diagram.  
where \(k_{disc}\) is thermal conductivity of thrust plate (W/K\(\cdot\)m); \(T_{disc}\) is the thrust plate surface temperature (K); \(h_{conv\_disc}\) is the convection coefficient of air and thrust plate (W/K\(\cdot\)m); \(t_{disc}\) is the thrust plate thickness (m).  
The last two items represent the heat flux density from the gas film to the thrust plate and from the thrust plate to the ambient environment through the inner and outer radial, respectively. The heat transferred from the air film to the thrust plate is expressed as:  
\[\dot{q}_{film\_disc}=\frac{k_{air}}{t_{disc}}\frac{dT}{dz} \tag{19}\]  
In this paper, when considering the heat transfer condition on the other side of the thrust plate, the heat dissipation model of the backside of the rotor thrust plate is regarded as the convection heat dissipation of the rotating plate in static air. When calculating the heat dissipation coefficient of convection between the back of the thrust plate and the environment, the calculation formula in reference [19] is used in this article:  
\[Nu=0.015\text{Re}_{\omega}{}^{0.8} \tag{20}\]  
\[h_{conv\_disc}=N_{u}\times k/L \tag{21}\]  
where \(Nu\) represents the Nusselt number of the fluid, which is used to describe the convective heat transfer intensity, and \(\text{Re}_{\omega}\) represents the Reynolds number, and \(k\) is the thermal conductivity of thrust plate (W/K\(\cdot\)m), and \(L\) is the characteristic length (m).  
In order the calculate the convective heat dissipation from the thrust plate outer surface to the surrounding air, the convection coefficient on the outer surface of a rotating cylinder is used in this paper [20]:  
\[h_{conv\_out}=0.133\text{Re}_{2}{}^{2/3}\text{Pr}^{1/3}(k_{air}/D_{R}) \tag{22}\]  
where \(\text{Re}_{2}\) is the Reynolds number with \(D_{R}\) being the characteristic length, and \(D_{R}\) is equal to \(2r_{2}\); \(r_{2}\) is the outer radius of the thrust bearing (m); Pr is the Prandtl number of air; \(k_{air}\) is the thermal conductivity of gas (W/K\(\cdot\)m).  
Figure 6: Single side gas thrust bearing configuration.  
\(\dot{q}_{\text{disc\_sur}}\) is the heat flux density transferred from the thrust plate to the environment in the radial direction, which is calculated as follows:  
\[\dot{q}_{\text{disc\_sur}}=\frac{h_{\text{conv\_out}}}{\Delta r}(T_{\text{disc}}- T_{0}) \tag{23}\]