## 2 Methodology  
### Force Analysis and Parameters of the HTB  
#### 2.1.1 Determination of Forces Acting on a Star Wheel  
The HTB force used for the star wheel shaft is related to the force of the star wheel. Thus, the force analysis of the star wheel is also the focus of the research on an HTB's working characteristics. Forces acting on a star wheel mainly come from the action of gas pressure in the cylinder and constricting forces exerted by bearings on them. Illustrations of the acting forces and torques are shown in Figure 4.  
As shown in the figure, the HTB mainly bears the gas force, \(\rm F_{g}\). According to the working principle of single-screw compressors, gas pressure acting on a star wheel tooth surface often varies with star wheel rotation angle, \(\rm\alpha\). This situation also causes the resultant force, \(\rm F_{g}\), from the gas pressure to change. Figure 4: Illustrations of forces acting on a star wheel shaft. Figure 3: Workflow of the methodology. The variable force acting on a star wheel can be calculated using the following equation and relationship:  
Gas force, \(F_{ig}\), of the \(i\)th star wheel tooth meshed into the screw groove is calculated as follows:  
\[F_{ig}(\alpha)=p_{i}(\alpha)A_{i}(\alpha) \tag{1}\]  
Gas pressure, \(p_{i}(\alpha)\), acting on the \(i\)th star wheel tooth meshing with the screw groove can be calculated using the formulas in the literature [23].The area, \(A_{i}(\alpha)\), of the \(i\)th star wheel tooth end face closed in the screw groove can be expressed as follows:  
\[A_{i}(\alpha)=\left\{ \begin{array}{ll}\int_{-\frac{k}{2}}^{\frac{k}{2}}\Big{(} \sqrt{R_{sw}^{2}-\eta^{2}}-\eta\tan\alpha-\frac{A-R_{sw}}{\cos\alpha}\Big{)} d\eta\ \ \alpha_{in}\leq\alpha\leq\alpha_{o1}\\ \int_{-\frac{k}{2}}^{R_{sw}\sin\left(\alpha_{out}-\alpha\right)}\Big{(}\sqrt {R_{sw}^{2}-\eta^{2}}-\eta\tan\alpha-\frac{A-R_{sw}}{\cos\alpha}\Big{)}d\eta \ \alpha_{o1}<\alpha\leq\alpha_{o2}\end{array} \right. \tag{2}\]  
where \(b\) is the tooth width of the star wheel, \(R_{sw}\) is the radium of the star wheel, \(d\eta\) is an area element parallel to the tooth centerline on the tooth surface of the star wheel, \(\eta\) is the distance between the area element and the tooth centerline, \(A\) is the center distance of the screw rotor and star wheel, \(R_{sw}\) is the radium of the star wheel, \(\alpha_{in}\) is the star wheel rotation angle at the end of suction, \(\alpha_{o1}\) is the star wheel rotation angle when the star wheel tooth begins to drop off the screw groove, \(\alpha_{out}\) is the star wheel rotation angle when the center of the star wheel begins to drop off the screw groove, and \(\alpha_{o2}\) is the star wheel rotation angle at the end of discharge. Thus, the resultant force that acts on the star wheel teeth engaging with their screw grooves can be obtained as follows:  
\[F_{g}(\alpha)=\sum_{i=1}^{3}F_{ig}(\alpha) \tag{3}\]  
This resultant force will be the key parameter to determine the water film thickness. On the basis of the preceding description, this study developed a WSSC prototype. The related technique parameters of the prototype are presented in Table 1.  
Under a common discharge pressure of 0.5, 0.8 and 1.2 MPa, the variation of the resultant force, \(F_{g}\), with the rotation angle of the star wheel shaft is shown in Figure 5.  
It is shown that the minimum value of gas force, \(F_{gmin}\), is 1203 N, and the maximum value of gas force, \(F_{gmax}\), is 2777 N. This largely fluctuating force, \(F_{g}\), will act on the HTB of the star wheel shaft and change the water film thickness between the two sliding surfaces. In general, the screw rotor of a single-screw compressor has six screw grooves. Each elementary volume will complete two gas transmission processes during every rotation of the screw rotor. Therefore, the resultant force, \(F_{g}\), has six periods of variation for each rotation of the screw rotor.