# Characteristic Analysis of Bump Foil Gas Bearing Under Multi-Physical Field Coupling
## 3 Case Study
### Thermal Characteristic Analysis  
The fluid energy equation describes the energy conservation of fluid motion. Although in air bearings energy generation and dissipation are much smaller than with liquid lubrication, the influence of temperature cannot be ignored when running at high speed, and the distribution of temperature and the transfer of energy still need to be considered. In the process of gas lubrication, there are two main aspects of energy transfer: heat conduction and heat transfer. Figure 10 shows the temperature distribution of the middle section with a rotation speed of 40,000 rpm and an eccentricity of 0.84. From the diagram, it can be seen that the temperature change law in the thickness direction of the gas film is as follows: the gas enters the inlet, is heated by viscous shear along the circumference, and the temperature rises rapidly. After reaching the vicinity of the maximum load of the gas film, the temperature decreases to a certain extent and then rises with air pressure. The temperature rises again, and then mixes with the inlet temperature again to re-enter the bearing. Because the rotor rotates continuously, the shaft surface continuously passes through the high-temperature and low-temperature regions--and most of the shaft is a hollow shaft with cooling air flowing through--so the temperature of the shaft surface remains constant. Figure 9: The curve of the friction torque of the air bearing varies with the rotational speed and eccentricity rate. Figure 11 shows the temperature of the middle surface. It can be seen from Figure 11a,b that the coordinate line (1, 0, z) is the air inlet, the air inlet is clockwise, and the initial temperature is the ambient temperature. After the initial gas enters the bearing, the temperature rises rapidly. When the gas film pressure reaches the maximum near the load, it can be seen that this is a high-temperature gathering area. Then, due to the decrease in air pressure, the gas film temperature decreases, because at this time, considering the existence of negative pressure, the temperature decreases rapidly (when there is negative pressure, there is no contact between the flat foil and the wave foil, the conditions change, negative pressure occurs, and external environment gas enters quickly), and the temperature of the two free-end faces of the bearing is the ambient temperature. It can be seen from Figure 11c that the temperature of the gas film on the middle surface is distributed in the circumferential direction: after the gas enters, the temperature gradually rises to the maximum temperature of the circumference of this layer and then drops to the ambient temperature. When the pressure returns to positive pressure, the gas film temperature rises again until it returns to the air inlet. It can be seen from Figure 11d that the temperature of the gas film in the middle section is distributed in the axial direction: the temperature on both sides of the bearing presents an axisymmetric law. First, the ambient temperature increases to reach this layer's maximum axial temperature, and then it decreases to the ambient temperature. As can be seen from Figure 12, when the speed increases from 30,000 rpm to 55,000 rpm, the maximum gas film pressure increases continuously and the maximum gas film temperature in the bearing also increases continuously. When the rotational speed increases, the gas flow velocity inside the bearing increases, and the gas is subjected to centrifugal force in the bearing gap, causing the gas to collide with the bearing surface, thereby increasing the pressure of the gas near the bearing surface. Although the air bearing is in normal operation, there is no direct contact between the shaft and the flat foil. Although this 'non-contact' condition is achieved by the gas film pressure, the collision between the air molecules is more intense and the temperature increases due to the increase in rotational speed. The influence of the maximum pressure and the maximum temperature on the relative speed change is close.