## 2 Internal Leakage Minimization Based on Sealing Technology
### Constant Gap Sealing  
Other alternative solutions are based on gap sealing of the hydraulic cylinder, and the tiny gap between the piston and the cylinder is used for sealing. As the sealing interface does not contact, the friction between the cylinder and the piston is greatly reduced. Therefore, the response speed of the piston is effectively improved. We call it the second generation hydraulic cylinder, as shown in Figure 5.  
In 1983, [PERSON] and others improved the gap sealing technology and invented the annular gap seal (Servofloat) [23]. The floating ring was adopted to make the annular gap seal automatically centered against radial friction. The gap value was constant.  
Figure 4: Schematic diagram of relevant parameters in friction calculation.  
Figure 5: Schematic diagram of the constant gap sealing: (**a**) support ring of constant gap sealing; (**b**) structure of constant gap sealing.  
At the beginning of the 21 st century, [PERSON] et al. [24] proposed a concept of gap sealing with end-face modification. It could change the lubrication condition between the end faces to achieve long-lasting operation of the mechanical seal by cutting various grooves on the seal's end faces. In 2006, [PERSON] [25] and others further improved the gap sealing technology, which prevented the gap amount change caused by the deformation of the annular gap seal. By processing multiple pressure-equalizing grooves on the surface of the piston, the lubrication effect was increased, and the speed of the piston was effectively improved. This can achieve the drag reduction lubrication (for example, the movement speed of the piston was increased from 0.1 m/s to 0.7 m/s).  
However, there is serious internal leakage with the increasing pressure, which can result in low volumetric efficiency and also reduce the working efficiency. The internal leakage \(q\) is related to parameters such as the sealing gap (\(h_{0}\)), the pressure difference between the two chambers of the hydraulic cylinder (the high-pressure chamber \(p_{1}\) and the low-pressure chamber \(p_{2}\)), structure parameters (the length \(L\) and diameter \(d\) of the piston), and the properties of the oil (density \(\rho\) and viscosity \(\eta\)), shown in Figure 6. Then, the internal leakage can be obtained by Equation (2) [26].  
\[q=\frac{\pi dh_{0}{}^{3}(p_{1}-p_{2})}{12\rho\eta L} \tag{2}\]  
where the internal leakage is defined as \(q\), and the term \(h_{0}\) is defined as the sealing gap between the hydraulic cylinder and the piston. The symbols \(p_{1}\) and \(p_{2}\) are used to express, respectively, the two pressure of the high-pressure chamber and the low-pressure chamber. The length and the diameter of the piston are defined as \(L\) and \(d\), respectively. Meanwhile, the density \(\rho\) and the viscosity \(\eta\) are the physical properties of the hydraulic oil, which are shown in Figure 6 and Equation (2).  
The above calculation formula (Equation (2)) is obtained when the hydraulic cylinder and the piston are coaxial. While the piston is off-center, and the relative eccentricity is defined as the term \(\varepsilon\); then, its leakage will increase significantly and can be expressed by Equation (3) [26].  
\[q=\frac{\pi dh_{0}{}^{3}(p_{1}-p_{2})}{12\rho\eta L}(1+1.5\varepsilon^{2}) \tag{3}\]  
where the term \(\varepsilon\) is defined as the relative eccentricity, which is equal to \(e/h_{0}\). The term \(e\) is the eccentricity of the piston and the cylinder. The other parameters are the same as Equation (2).  
Then, it can be concluded that the leakage of the annular gap is proportional to the relative eccentricity when the other parameters are not changed.