## 3 Results and Discussion
### Lubrication Performance under Different Discharge Pressure
#### 3.2.2 Water Film Stiffness  
Total torque \(T_{\rm t}\) is calculated in the range of \(-\delta_{\rm lim}\) to \(\delta_{\rm lim}\) under the gate-rotor rotating angle \(\varphi_{\rm gr0}\) when one of the teeth just closes the screw groove. The numerical results are shown in Figure 13.  
\(T_{\rm t}\) increases with the increase of \(\delta\), because the water films at leading side get thinner and generate bigger torque \(T_{\rm a}\) while the films at trailing side become thicker and generate less \(|T_{\rm b}|\). As \(\delta\) approaches \(\delta_{\rm lim}=0.02^{\circ}\), \(T_{\rm t}\) rises rapidly to prevent contact at leading side. When \(\delta\) approaches \(-\delta_{\rm lim}=-0.02^{\circ}\), \(T_{\rm t}\) falls rapidly to prevent contact at trailing side. If \(T_{\rm t}>0\) under \(\delta_{\rm lim}\) and \(T_{\rm t}<0\) under under \(-\delta_{\rm lim}\), the gate-rotor is viewed to have self-regulating capability.  
It can be seen that the variation ranges of \(T_{\rm t}\) under the discharge pressure 0.8 MPa are \(-\)1.773\(-\)0.028N-m, \(-\)6.03\(-\)\(-\)0.681N-m and \(-\)3.997\(-\)3.136N-m for SSLT, SCT and MCT, respectively. Only SCT cannot get a positive value even when \(\delta\) equals \(\delta_{\rm lim}\), which means wear occurs in the SCT meshing pair.  
When the discharge pressure increases to 1.2 MPa and 1.6 MPa, all the \(T_{\rm t}\) curves drop and only the MCT keeps the \(T_{\rm t}\) positive under \(\delta_{\rm lim}\), which represents that the MCT may have the best self-regulating ability under \(\varphi_{\rm gr0}\).  
By derivation the \(T_{\rm t}\) curves in Figure 13, the curves of the water film stiffness \(S\) are obtained as shown in Figure 14. The stiffness of the water film in the meshing pair can be understood as follows: the water film in the infinitesimal \(dl\) can be seen as a micro spring, the micro springs at different tooth length \(l\) in a tooth flank constitute a spring combination. All the spring combinations at different tooth flanks form a spring system. The water film stiffness can be regarded as the stiffness of the spring system.  
It can be observed that the water film stiffness firstly decreases and then increases as \(\delta\) increases (See Figure 14). In the range of -0.01-0.01\({}^{\circ}\), the water film stiffness almost remains a constant (Figure 13). The water film stiffness \(S\) at \(\delta=-\delta_{\rm lim}\) (trailing side) is often larger than that at \(\delta=+\delta_{\rm lim}\) (leading side).  
The water film stiffness rises when discharge pressure increases. Taking MCT as an example, the \(S\) values under \(\delta=-\delta_{\rm lim}\) are 391.01 N-m-deg\({}^{-1}\), 477.57 N-m-deg\({}^{-1}\) and 515.09 N-m-deg\({}^{-1}\) under 0.8 MPa, 1.2 MPa and 1.6 MPa.  
Since contact often occurs at leading side, the water film stiffness at \(+\delta_{\rm lim}\) is very important, which represents ability of resistance to wear. It is found that the MCT has much higher water film stiffness at \(+\delta_{\rm lim}\) when compared to the SSLT or SCT. Under discharge pressure of 0.8 MPa, the \(S\) values under \(\delta=+\delta_{\rm lim}\) are 157.69 N-m-deg\({}^{-1}\), 391.00 N-m-deg\({}^{-1}\) and 433 N-m-deg\({}^{-1}\) for SSLT, SCT and MCT, respectively.  
The results shown in Figures 13 and 14 are only under a certain gate-rotor rotating angle \(\varphi_{\text{g}\text{g}0}\); nevertheless, the evaluation of the MPP needs to be proceed in a whole period. In addition, calculations for loads and water film stiffness are indirect for hydrodynamic properties evaluation. Therefore, the MDMT evaluation method is proposed for this purpose.  
The gate-rotor tooth or the three teeth together can be viewed as a combined double slider. It can bear load in both directions and can even be unloaded. The gaps value is coupled with the load generated by water films and they are influenced by each other.  
Figure 13: The total torque at various deflecting angles under different discharge pressures.