## 4 Conclusions  
A theoretical method MDMT for evaluating the MPP's hydrodynamic lubrication properties of the SSC is proposed in this paper. The main idea of this method is to calculate the micro deflecting motion trajectory of the gate-rotor and judge whether the trajectory crosses the limits. An algorithm is developed to realize the MDMT method. Three MPPs including SSLT, SCT and MCT are evaluated under different working conditions and in different machines. Based on the calculation results, the following conclusions may be made:  
1. The hydrodynamic water film in the MCT meshing pair usually has the highest pressure and action area at tooth flank. In the studied case, compared to the SSLT and SCT, the MCT increases the peak value of the water film pressure by 29.7% and 19.1%, respectively.
2. The total torque \(T_{\rm t}\) applied the on the gate-rotor decreases with the increment of the discharge pressure and increases with the machine size. The SCT has the minimum \(T_{\rm t}\) among the three MPPs and also the maximum load \(|T_{\rm t}|\) for the water film at leading sides.  
Figure 20: The fluid film lubrication ratio in a whole period: (**a**) the influence of discharge pressure, (**b**) the influence of velocity.  
3. The total torque \(T_{\rm t}\) increases with the deflecting angle. With the increment of the discharge pressure and the reduction of the machine size, the self-regulating ability of the gate-rotor declines. The MCT has the strongest self-regulating capability among the three MPPs.
4. The water film stiffness firstly decreases and then increases when \(\delta\) increases. The water film stiffness \(S\) at trailing side is always larger than that at leading side. The water film stiffness \(S\) raises with the discharge pressure and machine size. The relation between \(S\) and discharge capacity is unlinear. The MCT has the biggest water film stiffness at the leading side.
5. As the discharge pressure increases, the MDMT curves approach the leading flank, and the lubrication properties get worse. Under 0.8 MPa, the ratios of no contact region to the whole period are 12.2%, 0 and 100% for SSLT, SCT and MCT, respectively. Under 1.2 MPa, the ratios decrease to 10.4%, 0 and 100%. Under 1.6 MPa, the ratios change to 0, 0 and 54.1%. The MCT shows some high-pressure resistance.
6. With the machine size increasing, the MDMT curves move towards the trailing side and the lubrication performance is improved. In the 3 m\({}^{3}\cdot\)min\({}^{-1}\) SSCs, the ratios of no contact region to the whole period are 0, 0 and 84.8% for SSLT, SCT and MCT, respectively. In the 6 m\({}^{3}\cdot\)min\({}^{-1}\) SSCs, the ratios raise to 12.2%, 0 and 100%. In the 12 m\({}^{3}\cdot\)min\({}^{-1}\) SSCs, the ratios further increase to 73.51%, 83.75% and 100%. In small machines, the MCT can still keep relative high lubrication performance.
7. The MDMT method is an intuitive and effective method to evaluate the MPPs of SSCs comprehensively.  
To verify the MDMT method accurately by experiment is an interesting work in the future. In addition, this study can lay a foundation for improving the existing thermodynamic model since the gate-rotor is always set to no deflection when calculating the leakage through the gaps at tooth flanks.  
Project administration and conceptualization, D.C.; methodology, T.L.; formal analysis, W.J.; visualization and supervision, X.G.; data curation, T.L. and W.J.; software, R.H.; funding acquisition, T.L. and R.H.; resources, investigation and validation, Q.F.; writing--original draft, T.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Fundamental Research Funds for the Central Universities (GrantNo. 2452017319), Science and Technology Project of Shaami Provincial Water Resources Department (Grant No.20198 lkj-17), the National Natural Science Foundation of China (Grant No. 51706179).  
The authors declare no conflict of interest.