## 2 Computational Models
### Computational Model for Fine Inspection Conditions of a Multiport Hybrid Molecular Pump  
The main difference between the multiport hybrid molecular pump and the traditional hybrid molecular pump is that the gas is shunted inside the molecular pump, and the shunting position of the gas is at the turbine stage during the fine inspection condition. The pressure, flow rate and pumping speed of the gas at each part are shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Geometric Median** & & **Integral Mean Method** & **Test Value** \\  & **Method** & & & \\ \hline \(K_{\text{max}}\) & 17,104 & 11,900 & 11,280 \\ \hline \(S_{\text{max}}\) (L/s) & 55.980 & 55.662 & 49.796 \\ \hline \(H_{\text{max}}\) & 0.3039 & 0.2904 & 0.2899 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of the calculated and measured results.  
Figure 4: Calculation flow chart of the integral mean method.  
In Figure 5a, \(p_{1}\) indicates the pressure inside the inspected part, \(p_{2}\) indicates the pressure inside the mass spectrometer chamber, and \(p_{3}\) indicates the pressure at the opening of the molecular pump. In the working process of the helium mass spectrometer leak detector, the influence of the pipeline from the inspected part to the opening of the pump body is ignored. When the system reaches a stable state, \(p_{1}\) is equal to \(p_{3}\), and the pumping speed of the molecular pump is equal to the reflux speed. Therefore, by taking the fine inspection port as the boundary, the pumping speed above the opening is the reflux speed of helium, which is recorded as \(S_{2}\), and the pumping speed of the part below the opening and the front stage pump is \(S_{3}\), as shown in Figure 5b.  
After the gas molecules (air and helium) enter the vacuum system of the helium mass spectrometer leak detector through the leak hole, part of the gas diffuses in the reverse direction into the mass spectrometer chamber installed at the inlet of the multiport composite molecular pump, and the other part is pumped out by the combined action of the second half of the hybrid molecular pump and the front stage pump [4]. The flow relationship can be expressed as follows:  
\[Q_{1}=Q_{2}+Q_{3} \tag{8}\]  
Since \(Q=S\cdot p\), Equation (8) can be rewritten as:  
\[S_{1}\cdot p_{1}=S_{2}\cdot p_{2}+S_{3}\cdot p_{3} \tag{9}\]  
The compression ratio is:  
\[K=p_{1}/p_{2} \tag{10}\]  
The following three solution cases can be derived from Equations (8)-(10):  
When the flow rate of helium molecules into the inspected part \(Q_{1}\) is known, \(p_{1}\), \(p_{2}\), and \(Q_{2}\) can be calculated as follows:  
\[p_{1}=\frac{KQ_{1}}{KS_{3}+S_{2}},\ p_{2}=\frac{Q_{1}}{KS_{3}+S_{2}},\ Q_{2}=S _{2}\cdot p_{2}=\frac{S_{2}Q_{1}}{KS_{3}+S_{2}} \tag{11}\]  
When the minimum leakage rate \(Q_{2}\) that can be detected is known, \(p_{1}\), \(p_{2}\), and \(Q_{1}\) can be calculated as follows:  
\[p_{1}=\frac{KQ_{2}}{S_{2}},\ p_{2}=\frac{Q_{2}}{S_{2}},\ Q_{1}=Q_{2}+Q_{3}=Q_{ 2}+S_{3}\cdot\frac{KQ_{2}}{S_{2}} \tag{12}\]  
Figure 5: Computational model for fine inspection conditions: (a) flow, pressure distribution model, (b) pumping performance model.  
When the working pressure \(p_{1}\) in the inspected part is known, \(p_{2}\), \(Q_{2}\), and \(Q_{1}\) can be calculated as follows:  
\[p_{2}=\frac{p_{1}}{K},\ Q_{2}=\frac{S_{2}p_{1}}{K},\ Q_{1}=Q_{2}+Q_{3}=\frac{S_{2 }p_{1}}{K}+S_{3}p_{1} \tag{13}\]  
\(S_{2}\), \(S_{3}\), and \(K\) in these equations are the pumping characteristic parameters of the multiport hybrid molecular pump, which are related to the rotational speed, inclination angle and pitch chord ratio of the molecular pump but are affected by the position of the extraction port when the molecular pump is determined.