## 2 Computational Models
### Turbine Stage Pumping Performance Calculation Model  
Expanding a single blade row of the turbine stage gives a simplified model, as shown in Figure 1. The thermal velocity of gas molecules conforms to the law of [PERSON]'s velocity distribution, and the adsorption and desorption on the blade row obey the cosine law [17]. As shown in Figure 1, the turbine blades divide the space into areas I and II; the pressure, molecular number density, and temperature of the gas molecules on both sides are represented by \(p\), \(n_{\mathrm{d}}\) and \(T\), respectively. Moreover, the blade thickness is \(t\), the pitch chord ratio is a/b, and the channel port areas are \(A_{1}\) and \(A_{2}\). The positive transmission probability is \(M_{12}\), and the reverse transmission probability is \(M_{21}\). Additionally, the pumping performance of a single-stage blade can be calculated [18]. The maximum extraction coefficient is as follows:  
\[H_{\mathrm{max}}=M_{12}-M_{21} \tag{1}\]  
The maximum compression ratio is:  
\[K_{\mathrm{max}}=(\frac{p_{2}}{p_{1}})=\frac{M_{12}}{M_{21}} \tag{2}\]  
The maximum pumping speed of a single blade is:  
\[S_{\mathrm{max}}=36.4H_{\mathrm{max}}F_{i}\sqrt{\frac{T}{M}} \tag{3}\]  
where \(M\) is the molecular weight of the gas and \(F_{i}\) is the sum of the area of the \(i\)-th working impeller row channel. The pumping characteristics of the turbine stage depend mainly on the transmission probability of gas molecules through blade rows \(M_{12}\) and \(M_{21}\). The turbine stage is usually composed of multistage dynamic blades and static blades in series. In the calculation process, the transmission probabilities of first blade and the second blade are calculated in series. Then, the results are used in series to calculate the transmission probabilities of third blade. Finally, the overall positive and reverse transmission probabilities of the turbine stage are obtained [19]. The positive and reverse transmission probabilities of gas molecules from the first-stage blade to the \(n-1\)-stage blade are defined as \(P_{n-1}\) and \(Q_{n-1}\), respectively. The positive and reverse transmission  
Figure 1: Expanded view of a single-stage turbine blade. probabilities of the \(n\)-th stage single blade are defined as \(P_{1}\) and \(Q_{1}\), respectively. The total positive and reverse transmission probabilities of \(n\)-stage blades are defined as \(P_{n}\) and \(Q_{n}\), respectively. \(P_{n-1}P_{n}\) represents the unit flow gas molecules that can enter the turbine blade row channel and directly reach the exhaust side through the n-stage blade row. \(P_{n-1}(1-P_{n})\) represents the flow of gas molecules reflected from the \(n\)-th stage. The calculation model of the combined blade row is shown in Figure 2.  
\(P_{n-1}P_{1}\) is the unit flow of gas molecules that pass through the \(n\)-stage turbine blades, and \(P_{n-1}(1-P_{1})\) is a gas molecule that reaches the nth stage blade row but is reflected instead of passing through the blade row. The total positive transmission probability of \(n\)-stage blades is as follows:  
\[P_{n}=\frac{P_{1}P_{n-1}}{1-(1-P_{1})(1-Q_{n-1})} \tag{4}\]  
The total reverse transmission probability of \(n\)-stage blades is as follows:  
\[Q_{n}=\frac{Q_{1}Q_{n-1}}{1-(1-P_{1})(1-Q_{n-1})} \tag{5}\]  
The maximum compression ratio is:  
\[K_{\max}=\frac{P_{n}}{Q_{n}}=\frac{P_{1}P_{n-1}}{Q_{1}Q_{n-1}}=\prod_{i-1}^{n} K_{i} \tag{6}\]  
The maximum extraction coefficient is:  
\[H_{\max}=P_{n}-Q_{n}=\frac{P_{1}P_{n-1}-Q_{1}Q_{n-1}}{1-(1-P_{1})(1-Q_{n})} \tag{7}\]  
The calculation of the turbine blade transmission probability is usually based on the Maxwell velocity distribution incidence condition and the wall diffuse reflection emission condition. The methods of gas molecular motion tracking and transmission statistics are used to simulate the movement of many molecules in the blade channel. Then, the number of molecules that move from one side of the turbine blade to the other side is obtained.