## 2 Materials and Methods
### Finite Element Analysis
#### 2.2.2 Results of the Thermal-Structural Analysis  
High contact stress improves sealing performance, while it does not allow the smooth driving of the disc. It is thus necessary to find an effective trade-off between needs for good operation and prevention of leakage. Contact stress, which is a little higher than the working pressure, could guarantee not only airtightness, but also smooth rotation of the disc. Tables 5 and 6 represent the maximum contact stresses for the different seal designs, temperatures (\(25\,^{\circ}\)C and \(350\,^{\circ}\)C) and slope angles (\(66.1^{\circ}\) and \(86.1^{\circ}\)). SS and G indicate stainless steel and graphite, and min and max mean the minimum (\(66.1^{\circ}\)) and maximum (\(86.1^{\circ}\)) slope angles. Figure 15: Analysis conditions of thermal-structural coupled simulation. For both stainless steel and graphite, the models with the slope angle of 66.1\({}^{\circ}\) showed lower contact stresses than those of 86.1\({}^{\circ}\), which proves that the increases of the contact surface led to a reduction of the peak contact stress, and sealing performance was improved with increasing temperature as a result of thermal expansion. In respect to the graphite layers, leakages were expected for the all models at 25 \({}^{\circ}\)C (G\({}_{\rm min}\); 0.19-0.28 MPa, G\({}_{\rm max}\): 0.14-0.18 MPa), and there were small changes in contact stress by varying the amount of the graphite, whereas at 350 \({}^{\circ}\)C, the larger contact stresses (G\({}_{\rm min}\): 12.33-4.73 MPa, G\({}_{\rm max}\): 5.03-0.79 MPa) beyond the working pressure were observed. This means that the graphite displayed a sealing effect at high temperature rather than at room temperature, and the contact stress decreases with increment of the number of laminated layer. In respect to the stainless steel layers, the graphite did not withstand the contact force at 25 \({}^{\circ}\)C, so the contact stress at the stainless steel increased with more amount of the graphite (SS\({}_{\rm min}\): 2.0 MPa \(\rightarrow\) 6.3 MPa, SS\({}_{\rm max}\): 1.2 MPa \(\rightarrow\) 4.9 MPa) because the contact force was intensively applied on the stainless steel. On the other hand, at 350 \({}^{\circ}\)C, the contact force was distributed to the graphite, so more graphite leads to a reduction of the peak contact stress at the stainless steel (SS\({}_{\rm min}\): 169.2 MPa \(\rightarrow\) 74.13 MPa, SS\({}_{\rm max}\): 129.9 MPa \(\rightarrow\) 63.0 MPa). The 1-layer model with the slope angle of 86.1\({}^{\circ}\) showed leakage (SS\({}_{\rm max}\): 0.61 MPa) at 25 \({}^{\circ}\)C, while the excessive contact stresses (SS\({}_{\rm min}\): 282.17 MPa, SS\({}_{\rm max}\): 199.71 MPa) at 350 \({}^{\circ}\)C would disturb smooth operation of the disc. In the 3-layer and 5-layer models, the contact stresses of the stainless steel (SS\({}_{\rm min}\): 169.22-115.37 MPa and SS\({}_{\rm max}\): 129.86-95.33 MPa for 3-layer, SS\({}_{\rm min}\): 118.94-94.82 MPa and SS\({}_{\rm max}\): 104.14-76.53 MPa for 5-layer) decreased compared with those of the single layered model \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Graphite** & \multicolumn{2}{c}{**G\({}_{\rm min}\) (66.1\({}^{\circ}\))**} & \multicolumn{2}{c}{**G\({}_{\rm max}\) (86.1\({}^{\circ}\))**} \\ \cline{2-6}  & **Thickness (mm)** & **22 \({}^{\circ}\)C** & **350 \({}^{\circ}\)C** & **22 \({}^{\circ}\)C** & **350 \({}^{\circ}\)C** \\ \hline \multirow{4}{*}{3-layer} & 0.6 & 0.26 & 12.33 & 0.16 & 5.03 \\  & 0.8 & 0.28 & 13.01 & 0.17 & 5.67 \\  & 1 & 0.22 & 8.71 & 0.16 & 4.03 \\  & 1.2 & 0.24 & 7.07 & 0.15 & 3.75 \\ \hline \multirow{4}{*}{5-layer} & 0.6 & 0.26 & 8.12 & 0.16 & 3.11 \\  & 0.8 & 0.19 & 7.20 & 0.15 & 2.56 \\  & 1 & 0.22 & 5.93 & 0.15 & 2.14 \\  & 1.2 & 0.21 & 5.72 & 0.13 & 2.10 \\ \hline \multirow{4}{*}{7-layer} & 0.6 & 0.24 & 7.52 & 0.18 & 2.40 \\  & 0.8 & 0.21 & 4.90 & 0.18 & 1.74 \\ \cline{1-1}  & 1 & 0.17 & 3.60 & 0.15 & 0.81 \\ \cline{1-1}  & 1.2 & 0.19 & 2.73 & 0.14 & 0.79 \\ \hline \hline \end{tabular}
\end{table}