# An Experimental Study of the Leakage Mechanism in Static Seals
## 3 Experimental Design for Sealing Surface Leak-Rate Measurement
### Experimental Samples  
In this study, three common seal materials were selected, including stainless steel 316L, hardened SKD11 steel, and PTFE. The materials were processed and surface treated through the four different processes of turning, fine turning, electrical discharge machining (EDM), and polishing to obtain four different test pieces with obvious surface differences (as depicted in Figure 3). The material parameters of the four test pieces are listed in Table 1.  
Figure 3: Photographs of the four test pieces: (**a**) Test piece 1; (**b**) Test piece 2; (**c**) Test piece 3; and (**d**) Test piece 4. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multirow{2}{*}{**No. 2**} & \multicolumn{2}{c}{**Material Parameters**} & \multicolumn{2}{c}{**Measurement Parameters of Surface Topography**} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\ \cline{2-9}  & & **Elastic Modulus** & **Poisson’s Ratio** & **Hardness** & **Process** & **D** & **G** & **Ra** \\ \hline \multirow{2}{*}{Mold steel skd11} & 1 & 210 GPa & 0.3 & 850 MPa & Forging & 1.51 & 6.59 \(\times\) 10\({}^{-10}\) & 0.3 \\  & 2 & 142 GPa & 0.4 & 454 MPa & Turning & 1.46 & 8.28 \(\times\) 10\({}^{-9}\) & 1.3 \\ \cline{1-1} Stainless steel & 3 & 195 GPa & 0.3 & 187 MPa & EDM & 1.35 & 1.87 \(\times\) 10\({}^{-8}\) & 3.7 \\ Stainless steel & 4 & 195 GPa & 0.3 & 187 MPa & Fine & \(\times\) 1.42 \(\times\) 1.47 & \(\times\) 2.98 \(\times\) 10\({}^{-8}\) Y.381 \(\times\) 10\({}^{-8}\) & 2.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material and surface topography parameters of the test pieces. The cross-sectional profile data of the four test pieces were collected using an optical profilometer Veeco Wyko NT9100 (Veeco Corporate Headquarters, 1 Terminal Drive, Plainview, NY 11803, US) (vertical resolution: 0.1 nm, horizontal resolution: 400 nm), as shown in Figure 4.  
The structure function \(S(\beta)\) of the \(2D\) profile function of the fractal surface can be expressed as:  
\[S(\beta)=\left\langle\left[z(x+\beta)-z(x)\right]^{2}\right\rangle=\frac{ \Gamma(2D-3)sin[(2D-3)\pi/2]}{(4-2D)\ln\gamma}G^{2(D-1)}\beta^{4-2D}, \tag{11}\]  
where \(\beta\) denotes the scale of the structure function. By calculating the logarithms of both sides of Equation (11), we have  
\[\lg[S(\beta)]=\lg C+(4-2D)\lg\beta. \tag{12}\]  
As seen from the equation above, the cross-sectional profile curves of the rough surface with fractal characteristics, \(\lg(S(\beta))\) and \(\lg\beta\), present a linear relationship. The correlation between the slope \(k\) and the intercept \(B\), and the fractal parameters, can be expressed as  
\[D=(4-k)/2, \tag{13}\]  
\[G=\left\{\frac{C(4-2D)\ln\gamma}{\Gamma(2D-3)\sin[(2D-3)\pi/2]}\right\}^{\frac {1}{2(D-1)}}. \tag{14}\]  
As an illustration, one set of cross-sectional profile data was substituted into Equation (11) to demonstrate the linear relationship between \(\lg(S(\beta))\) and \(\lg\beta\). As demonstrated in Figure 5a, the slope \(k=0.9784\) of the trend line and the intercept \(B=-9.2844\) were substituted into Equations (13) and (14) to obtain fractal parameters \(D=1.51\) and \(G=2.1494\times 10^{-10}\). The fractal parameters were then substituted into the cross-sectional profile function of the AB model [18]. The results are shown in Figure 5b, where the blue line and the red line denote the fitted profile and the measured profile, respectively. The fitted three-dimensional surface is shown in Figure 5c. The fractal parameters of the sealing surfaces of the test pieces are shown in Table 1. The measurement and calculation results showed that the surfaces of test pieces No. 1 and 2 were smooth, and followed the fractal characteristics (\(\lg(S(\beta))\) and \(\lg\beta\) were in a linear relationship); the surface of test piece No. 3 was rough (Ra = 3.7), but still followed fractal characteristics; the surface of test piece No.