## 4 Numerical Computational Method for Sealing Performance
### Method for Calculating the Bearing Capacity of Gas Film
#### 4.1.2 Gas Film Force  
The dynamic pressure gas film force calculated using the gas lubrication equation can be solved using the average Reynolds equation[21] in an isothermal state, as shown in Formula (14). Meanwhile, the following assumptions should be satisfied[22]:  
1. The fluid in the gap is fully developed laminar flow;
2. The fluid is viscous and conforms to [PERSON]'s law of viscosity;
3. The gas film thickness is very small compared with other geometric sizes, and the ratio is generally less than \(10^{-4}\);
4. The fluid is an isothermal ideal gas and
5. The fluid satisfies the no-slip boundary condition at the gas-solid boundary.  
Considering an isothermal ideal gas, the state equation for the gas is given by Formula (13):  
\[\frac{p}{\rho_{3}}=g\Re T \tag{13}\]  
\[\frac{\partial}{\partial x}(\phi_{x}\frac{h^{3}}{12\eta}\frac{\partial\overline {p}}{\partial x})+\frac{\partial}{\partial y}(\phi_{y}\frac{h^{3}}{12\eta} \frac{\partial\overline{p}}{\partial y})=\frac{(n_{1}r_{2}+n_{2}R_{1})}{2} \frac{\partial\overline{h}_{T}}{\partial x}+\frac{(n_{1}r_{2}-n_{2}R_{1})}{2} \sigma\frac{\partial\phi_{s}}{\partial x}+\frac{\partial\overline{h}_{T}}{ \partial t} \tag{14}\]  
In Equation (14):  
\[\phi_{x}=1-C_{0}e^{-r(\
icefrac{{h}}{{e}})},\gamma\leq 1,\ \phi_{x}=1+C_{0}( \
icefrac{{h}}{{\sigma}})^{-r},\gamma>1,\ \phi_{y}(\
icefrac{{h}}{{\sigma}},\gamma)=\phi_{x}( \
icefrac{{h}}{{\sigma}},\
icefrac{{1}}{{\gamma}})\]  
\(\gamma=\frac{l_{\theta}}{l_{y}}\), \(l_{\theta}\) is the circumferential direction of the sealing ring; \(l_{y}\) is the width direction of the sealing ring.  
\[\phi_{s}=(2V_{1}-1)\Phi_{S} \tag{15}\]  
In Equation (15):  
\(\Phi_{s}=A_{1}(\
icefrac{{h}}{{\sigma}})^{\alpha_{1}}e^{-2\
icefrac{{h}}{{ 2}}/\varepsilon+\alpha_{3}(\
icefrac{{h}}{{\sigma}})^{2}},\
icefrac{{h}}{{ \sigma}}\leq 5\),  
\(\Phi_{s}=A_{2}e^{-0.25\
icefrac{{h}}{{\sigma}},\
icefrac{{h}}{{\sigma}}>5}\),  
\(V_{1}=\
icefrac{{(\
icefrac{{h}}{{\sigma}})^{2}}}{{\sigma}},\)  
\(\sigma=\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}\),  
where \(\rho_{3}\) is gas density; \(C_{0}\), \(r\) are constant, respectively (available in reference[21]); \(\sigma_{1}\) is the root mean square value of the inner rotor surface roughness; \(\sigma_{2}\) is the root mean square value of the sealing ring roughness; \(\sigma\) is the comprehensive surface roughness (the value in this paper is 0.8); \(p\) is gas pressure; \(g\) is the gravitational acceleration; \(T\) is the absolute temperature; \(\phi_{x}\) and \(\phi_{y}\) are the pressure flow coefficient; \(\phi_{s}\) is the shear flow coefficient; \(\overline{p}\) is the mean fluid pressure and \(\overline{h}_{T}\) is the average film thickness. Let the resultant force in the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & \begin{tabular}{c} **Elasticity Modulus** \\ **\(\mathbf{E}\) (GPa)** \\ \end{tabular} & \
\begin{tabular}{c} **Poisson’s Ratio** \\ **\(\mathbf{\
u}\)** \\ \end{tabular} &
\begin{tabular}{c} **Density** \\ **\(\mathbf{\rho}\) (g/cm\({}^{3}\))** \\ \end{tabular} \\ \hline Graphite & 18 & 0.3 & 1.8 \\ Structural steel & 91 & 0.34 & 4.44 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Performance parameters of the inner rotor and the sealing ring.  
eccentric direction of the centre of the rotor and the centre of the floating sealing ring be \(W_{n}\), the resultant force in the vertical eccentric direction is \(W_{t}\), and the value is as follows [23]:  
\[\left\{\begin{array}{l}W_{n}=\int_{0}^{L}\int_{0}^{2\pi}\left(p-p_{a}\right)r_ {2}\cos\theta d\theta dy\\ W_{t}=\int_{0}^{L}\int_{0}^{2\pi}\left(p-p_{a}\right)r_{2}\sin\theta d\theta dy \end{array}\right. \tag{16}\]  
The dynamic pressure of the gas film is as follows:  
\[F_{p}=p_{a}r_{2}{}^{2}\sqrt{{W_{n}}^{2}+{W_{t}}^{2}} \tag{17}\]