## 2 Materials and Methods
### Operating Principle and Theoretical Analysis of Triple Offset Butterfly Valve
#### 2.1.3 Mathematical Modeling of TOBV Seal Pair  
Conventional single offset valve with center shaft, which penetrates a seal, is suitable for low temperature and low pressure services only, and double offset valve with eccentric shaft results in an uninterrupted seal that can be used at higher pressures and temperatures, but the friction and seal wear deteriorate sealing performance. In the triple offset valve, both the seal and the disc are surfaces of a cone that is sectioned at an angle. The valve shaft is located slightly to one side of the seal center and above the plane of the seat, and its rotation center is also offset from the axis of the imaginary cone that extends from the surface of the seat. When the valve is closed, the surfaces of the seal and the disc are in full contact at all points, and opening the valve results in the disc moving away from the seal at all points, eliminating galling and minimizing seal wear due to non-rubbing sealing surfaces for long life and tight shut-off. The three separate offsets are designed in the butterfly valve, as shown in Figure 5.  
Offset 1 (\(H\)): The shaft is offset from the seal plane providing an uninterrupted sealing surface. Offset 2 (\(E\)): The centerline of disc is offset from the centerline of the shaft allowing the disc to freely lift off and away from the seal during opening. Offset 3 (\(\gamma\)): The centerline of seal cone with conical angle of 2\(\alpha\) is offset by \(\gamma^{\circ}\) from the centerline of the body to eliminate rubbing on the contact surface during opening and closing, thereby rotating the disc without interference [21]. Figure 4: Existing dual seal and new laminated seal. PTFE, polytetrafluoroethylene. Figure 5: Scheme of the triple-offset butterfly valve. A coordinate system is established by taking the apex of the cone, and the geometry of the triple offset seal pair is shown in Figure 6.  
The functions of cone where the disc is situated about \(y\)-axis are expressed in Equation (1), and the sections of the disc (\(S_{i}\)) in Equation (2) depend on \(H_{i}\), which are the distances between the coordinate origin (\(O\)) and \(N_{i}\), given by Equation (3) (\(i\) = 1, 2, and 3). \[y^{2}=k^{2}\cdot z^{2}-x^{2}, \tag{1}\]  
\[z=k_{0}\cdot y+H_{0}\text{ for }S_{0},\ z=k_{0}\cdot y+H_{1}\text{ for }S_{1},\ z=k_{0}\cdot y+H_{2}\text{ for }S_{2}, \tag{2}\]  
\[H_{i}=\left(H_{0}-\frac{T}{2\cdot\cos(\gamma)}\right)+\frac{T\cdot i}{\cos( \gamma)}, \tag{3}\]  
where k = tan (\(\alpha\)) and k\({}_{0}\) = tan (\(\gamma\)). When the sealing surface is truncated by an arbitrary plane that is in parallel with \(yOz\) plane with its \(x\)-coordinate as \(x_{0}\), the equation of the hyperbola (_ajbckd_) shown in Figure 7 is obtained by Equation (4), and the coordinate of two intersection points (\(x,y_{i},z_{i}\)) between hyperbola and disc surface (\(S_{i}\)) is obtained by Equation (5). \[\frac{z^{2}}{\left(\frac{x_{0}}{k}\right)^{2}}-\frac{y^{2}}{{x_{0}}^{2}}=1 \tag{4}\]  
\[\left\{ \begin{array}{l}x=x_{0}\\ y_{i}=\frac{-k^{2}\cdot k_{0}\cdot H_{i}\pm\sqrt{(k^{2}\cdot k_{0}^{2}\cdot x_{ 0}^{2}+k^{2}\cdot H_{1}^{2}-x_{0}^{2}}}{k^{2}\cdot k_{0}^{2}-1}\\ z_{i}=k_{0}\cdot y_{i}+H_{i}\end{array} \right. \tag{5}\]  
Figure 6: Geometry of triple offset seal pair. Figure 7: Projection diagram of disc section. The elliptical equations for the intersections between the disc cutting planes and the curved surface of cone are expressed by Equation (6). \[x^{2}+y^{2}=\left(k\cdot k_{0}\cdot y+H_{0}\right)^{2},\;x^{2}+y^{2}=\left(k\cdot k _{0}\cdot y+H_{1}\right)^{2},\;x^{2}+y^{2}=\left(k\cdot k_{0}\cdot y+H_{2} \right)^{2} \tag{6}\]  
The intersections between the vertical lines of \(N_{0}\), \(N_{1}\), and \(N_{2}\) with the plane \(x=x_{0}\) are the center point of \(jk\), \(bc\), and \(ad\), and apexes \(N_{i}\) of minor axis of the intersection constitutes the line \(\overline{N_{i}O}\).