# Pattern and Analysis of Corrugation-Sand Retaining Seals for
## 2 New Structure Sealing Theory and Model
### Sealing Model  
In Figure 1b, the a process represents the lubricating medium leakage process of the corrugated seal ring on the right half when it passes through the middle area, and b represents the process of the lubricating medium returning to the gap between the sealing ring and the shaft hole when the corrugated sealing ring of the left half passes through the middle area. Assume that the sealing ring in Figure 1b slides to the right at the speed of \(v\), and \(v_{1}\) and \(v_{2}\) can be obtained by orthogonal decomposition of \(v\), where \(v_{1}\) is the speed perpendicular to the edge of the corrugated belt, which represents the axial sliding speed of the reciprocating shaft seal. Additionally, \(v_{2}\) is tangent to the edge of the corrugated belt, representing the linear velocity of the rotary shaft seal rotating in the circumferential direction. Figure 2: Fluid flow in sealing gap: (**a**) There is pressure difference in the seal chamber, and the seal ring and shaft hole are relatively non-slip, (**b**) There is no pressure difference in the seal chamber, and the seal ring and shaft hole slide relatively, (**c**) There is pressure difference in the seal chamber, and the sealing ring slides in the direction of decreasing pressure, (**d**) There is pressure difference in the seal chamber, and the sealing ring slides in the direction of increasing pressure. According to [PERSON] theory, the leakage is calculated as follows:  
\[q=\frac{1}{2}vh+\frac{h^{3}(P_{1}-P_{2})}{12\eta L} \tag{7}\]  
where \(\frac{1}{2}vh\) is the grease leakage flow caused by the sliding of the seal ring relative to the shaft hole, and \(\frac{h^{3}(P_{1}-P_{2})}{12\eta L}\) is the influence of fluid pressure on the leakage. When the pressure inside the cavity is greater than the external pressure, the second term is positive, and leakage will be caused in this case. Under normal circumstances, changing the pre-compression of the seal ring can change the leakage. The fundamental reason is that the gap height \(h\) is changed. Then, the leakage \(q_{a}\) in a process is shown in Equation (8), where \(\theta\) is the angle between the velocity component perpendicular to the corrugated surface and the moving speed of the corrugated surface:  
\[q_{a}=\frac{1}{2}vh\cos\theta+\frac{h^{3}(P_{1}-P_{2})}{12\eta L} \tag{8}\]  
The process \(b\) can be regarded as the height of the sealing gap \(h\) becomes \(k\) times of the original gap, and the retention coefficient \(k\) represents the percentage of the total leakage that the lubricating medium enters the sealing gap again after entering the external mud environment. The leakage calculation formula is as follows; in this case, the angle \(\theta\) is greater than \(\frac{\pi}{2}\). \[q_{b}=\frac{1}{2}khv\cos\theta-\frac{k^{3}h^{3}(P_{1}-P_{2})}{12\eta L} \tag{9}\]  
Integrate the leakage at all points on the edge of the corrugation to obtain the leakage. Assuming that the corrugation is in the shape of a cosine, the number of cosine periods is \(N\), the length of the amplitude is \(A\), and the diameter of the sealing oil film is \(D\), then:  
\[ \begin{split} q=\sum\limits_{i=0}^{N-1}\Big{\{}\int_{2i\pi}^{(2i+ 1)\pi}&\Big{(}\frac{h^{3}(P_{1}-P_{2})}{12\eta L}+\frac{1}{2}hv \sin\Bigl{(}\arctan\Bigl{(}A\sin\Bigl{(}\frac{2 Nx}{D}\Bigr{)}\Bigr{)}\Bigr{)} \Bigr{)}dx\\ &+\int_{(2i+1)\pi}^{2(i+1)\pi}\Bigl{(}\frac{h^{3}h^{3}(P_{1}-P_{ 2})}{12\eta L}+\frac{1}{2}khv\sin\Bigl{(}\arctan\Bigl{(}A\sin\Bigl{(}\frac{2 Nx}{D}\Bigr{)}\Bigr{)}\Bigr{)}\Bigr{)}dx\Big{\}}\end{split}  \tag{10}\]  
Assuming that the leakage of each cosine period is consistent, the above equation is simplified to obtain:  
\[q=\frac{1-k}{2}NAhv\int_{0}^{\pi}\frac{\sin\Bigl{(}\frac{2 Nx}{D}\Bigr{)}}{ \sqrt{A^{2}\sin\Bigl{(}\frac{2 Nx}{D}\Bigr{)}}+1}dx+\frac{\bigl{(}1+k^{3}\bigr{)} h^{3}\pi D(P_{1}-P_{2})}{12\eta L} \tag{11}\]  
The above formula is the calculation formula for the leakage amount of the main seal.