# Study the Effect of Operating Temperature on Performance in Dry Scroll Vacuum Pump
## 2 Numerical Model
### Numerical Simulation  
The commercial software is used to simulate the unsteady compressible flow inside the multi-stage dry scroll vacuum pump. The boundary conditions are set, and the parameters of the model are controlled according to the operating and experimental conditions.  
The discharges of the model are set as pressure boundaries and kept at 101,325 Pa (1 atmospheric pressure).  
The Knudsen number (\(Kn\)) [37] is a dimensionless number that allows for characterizing the gas flow regimes. It is defined as:  
\[Kn=\frac{\lambda}{d} \tag{10}\]  
where \(\lambda\) is the mean free path, \(d\) is a characteristic dimension of the system. Based on our previous work [29], the maximum Knudsen number in the flow field is calculated to be 0.133 when the suction pressure is 1 kPa. In fact, the transitional flow begins to appear at the radial clearance (0.05 mm) closing to the suction. Considering that, the flow field with a large Knudsen number is limited to the small local space proximity to the suction part, while in other areas, there is significant viscous flow, even turbulent flow. Therefore, it is appropriate to set the pressure boundary of the suction port to 1 kPa.  
The wall boundaries are set as no-slip smooth walls. According to the experience, the temperature on the surface of the scroll pump casing could reach 313.15 K-323.15 K in general while it operates, and a temperature gradient exists in the pump body. Therefore, the wall temperature of the actual situation is set at 338.15 K (65 \({}^{\circ}\)C).  
As shown in Figure 1a, the fixed scroll and orbiting scroll of the scroll vacuum pump have different heat dissipation structures. The fixed scroll can be cooled directly by the fan, while the orbiting scroll cannot. Thus, changing the heat dissipation capacity of the scroll pump, the temperature of the fixed scroll has a large change, and the temperature of orbiting scroll changes little. Therefore, in the numerical model, when the heat dissipation capacity is improved (e.g., changing fan airflow), the wall temperature of the fixed scroll will change, and the wall temperature of the orbiting scroll will remain the same. To analyze the effect of operating temperature on performance in the dry scroll vacuum pump, two other sets of wall temperature conditions are selected to simulate different heat dissipation capacities, as shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Operating Temperature** & **1 (Actual Situation)** & **2** & **3** \\ \hline \hline Fixer wall temperature/K & 338.15 K (65 \({}^{\circ}\)C) & 318.15 K (45 \({}^{\circ}\)C) & 298.15 K (25 \({}^{\circ}\)C) \\ Rotor wall temperature/K & 338.15 K & 338.15 K & 338.15 K \\ \hline \hline \end{tabular}
\end{table}
Table 2: Boundary conditions of wall temperature.  
Figure 5: Computational mesh.  
The rotating speed is taken at 1425 rpm (50 Hz), and the time step is equal to the time orbiting scroll rotate turns 1\({}^{\circ}\).  
The gas is controlled by the governing equation and the gas state equation. The gas is ideal air.  
Spatial discretization and time discretization adopt a hybrid format and implicit second-order accuracy format, respectively. The turbulence model is the RNG \(k\)-\(\epsilon\) two-equation model. The residual in the calculation is controlled to less than \(10^{-4}\). During the calculation, the pumping speed, the torque of the orbiting scroll caused by gas compression, and the pressure of two points (P1 and P2) are monitored.