# Numerical Evaluation of Rotordynamic Coefficients for Compliant Foil Gas Seal
## 3 Results
### Rotordynamic Coefficients with Eccentricity Ratio  
Considering the initial condition, the equations of leakage, gas force, and rotordynamic coefficients are numerically solved. The parameters for a typical turbine used in calculations are given in Table 1.  
The FDM is employed to solve the obtained equation. In this section, 2D FDM analysis of the compliant foil gas seal for different operating conditions is conducted. The steady performance of the eccentric ratios is shown in Figure 3. It is still extremely significant to analyze the steady results of the leakage, gas force, eccentric angle, and friction because these results are the basic performance of the gas seal, and they can be guaranteed to study the stiffness and damping coefficients of the compliant foil gas seal. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline
**R(mm)** & 25 \\ \(C\) **(mm)** & 0.005 \\ \(N\) & 16 \\
**Gas medium** & air \\
**The ratio of groove** & 0.6 \\
**Outlet pressure (Pa)** & 101,325 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The analysis layout of the gas seal [14,16]. Figure 2: (**a**) Description of non-dimensional gas thickness with non-dimensional length. (**b**) Description of non-dimensional pressure with nondimensional length. (**c**) Comparison of pressure calculated results. Figure 3a shows that leakage and gas force increases with the rise of eccentric ratio at approximately 10,000 rpm. The leakage changes at approximately 36.7%, and the gas force increases at approximately 83%, but the amplitude of the leakage is still relatively smaller than that of other seals, so the larger eccentric ratio can be selected without the influence of the dynamic factor. As a result, the larger eccentric ratio can be adjusted in the same conditions, including groove and operation parameters. The trend can be nearly discussed by Equations (6)-(10) [20, 21, 22]:  
\[dQ=\frac{C^{3}(1+\varepsilon\cos\theta)^{3}}{12\mu}\frac{\partial p}{\partial x }Rd\theta\pm\frac{U}{2}C(1+\varepsilon\cos\theta). \tag{6}\]  
The integral results are obtained by the short bearing approximation, and \(\partial p/\partial x\) is neglected as presented in Equation (7):  
\[Q=ULC\varepsilon. \tag{7}\]  
\(Q\) is the leakage, and \(U\) represents the mean velocity. The symbols '+' and '-' indicate the velocity direction. The gas flows from the high area to the lower area, which leads the leakage flow away, and the mean velocity is positive. Therefore, the leakage will increase gradually with the eccentric ratios increasing. The gas force, which is concluded by Equation (8), increases gradually and is consistent with the regular short-bearing approximation. This is helpful to improve the rotordynamic coefficients. \[F=\mu UL^{3}c\Big{(}\pi^{2}\Big{(}1-\varepsilon^{2}\Big{)}+16\varepsilon^{2} \Big{)}^{\frac{1}{2}}/[4c^{2}\Big{(}1-\varepsilon^{2}\Big{)}^{2}] \tag{8}\]  
Figure 3b shows the attitude angle \(\beta\), and friction F\({}_{\text{f}}\) changes with the increase of eccentric ratios for speed of 10,000 rpm. The attitude angle \(\beta\) decreases at approximately 55.6% as eccentric ratios increase, which is beneficial to rotordynamic stability. Friction increases at approximately 77% with increasing eccentric ratios. These are almost verified by Equations (9) and (10) of short-bearing approximations:  
\[\beta=\tan^{-1}\Big{[}\pi\Big{(}1-\varepsilon^{2}\Big{)}^{1/2}/4\varepsilon \Big{]} \tag{9}\]  
\[F_{f}=\mu ULR\pi(1+(1+\varepsilon))^{1/2}/C\Big{(}1-\varepsilon^{2}\Big{)}^{1 /2} \tag{10}\]  
As can be obtained in Figure 4, the impact of eccentric ratio on the rotordynamic coefficients can be seen clearly from 0.6. The dynamic coefficients are relatively insensitive to changes in eccentric ratio when the eccentricity ratio is smaller than 0.5. The absolute value of direct stiffness \(K_{xx}\) and \(K_{yy}\) shows almost the same trend, which arrives at a peak value as the eccentricity ratio is approximately 0.7. Similarly, the change of cross-coupled stiffness shows the same results.