## 2 Mathematical Model of Vortex Compressor Profile
### Performance Analysis of the Working Chamber  
In the meshing area of the dynamic and static vortex teeth of the compressor, there is always a gap to reduce friction and wear between parts [22]. For new volumetric vortex compressors, axial gaps are usually sealed using sealing strips and coatings. Therefore, considering only radial gaps and not axial gaps, the size of the radial gap determines the degree of gas leakage between compression chambers, significantly affecting the compressor's efficiency and performance. An appropriate control of the vortex disk's size and gaps during the design and manufacturing stages can optimize the compressor's performance, hence the introduction of a comparative analysis of important parameters such as radial leakage line length, stroke volume, unit rotation angle, and radial gas leakage amount. However, if the radial gap is too small or nonexistent, the compressor cannot operate smoothly; if it is too large, it causes adverse effects such as gas tangential leakage, chamber air pulsation, and rapid temperature rise. Therefore, important parameters like radial leakage line length, stroke volume, unit rotation angle, and gas radial leakage amount are introduced for comparative analysis. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Base circle radius \(\text{R}_{1}/\text{mm}\) & 3.2 \\ Base circle radius \(\text{R}_{2}/\text{mm}\) & 6.4 \\ Number of turns \(\text{n}_{1}\) & 2 \\ Number of turns \(\text{n}_{1}\) & 1 \\ Override Start involute Angle \(\varphi_{1}\) & 2\(\pi\) \\ Override Start involute Angle \(\varphi_{2}\) & \(\pi\) \\ Rotation angle \(\theta\) & \(\pi\) \\ Involute termination angle \(\varphi_{\text{e}}\) & 8\(\pi\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Related parameters of variable diameter double arc combined profile. Figure 1: Forming process of variable diameter double arc combined profile. The general theoretical equation for the radial leakage line length of the combined profile vortex compressor is shown in Equation (5):  
\[ \begin{array}{ll}I_{b}&=\int_{0}^{\varphi_{1}}\Big{(}{x^{\prime}}_{1}^{2}+y^{ \prime}_{1}\Big{)}^{1/2}d\varphi\\ &+\int_{\varphi_{1}}^{\varphi_{1}+2n_{2}\tau}\Big{(}{x^{\prime}}_{2}^{2}+y^{ \prime}_{2}\Big{)}^{1/2}d\varphi\\ &+\int_{\varphi_{1}+2n_{2}\tau}^{\varphi_{2}}\Big{(}{x^{\prime}}_{3}^{2}+y^{ \prime}_{3}\Big{)}^{1/2}d\varphi=\int_{0}^{\varphi_{1}}\big{(}R_{1\varphi} \big{)}d\varphi\\ &+\int_{\varphi_{1}}^{\varphi_{1}+2n_{2}\tau}\big{[}R_{2}(\varphi-\theta)]d \varphi+\int_{\varphi_{1}+2n_{2}\tau}^{\varphi_{\text{v}}}\big{(}R_{1\varphi} \big{)}d\varphi\end{array}  \tag{5}\]  
where \(x\) and \(y\) are the equations for the horizontal and vertical coordinates of the baseline. The general theoretical equation for the stroke volume \(V_{\text{b}}\) of the combined profile vortex compressor is shown in Equation (6):  
\[V_{b}=2S_{3}(\varphi)h \tag{6}\]  
\[S_{3}(\varphi)=R_{or}l+\xi=\bigg{\{}\int_{6\pi}^{8\pi}\Big{(}{x^{\prime}}_{3} ^{2}+{y^{\prime}}_{3}^{2}\Big{)}^{\frac{1}{2}}d\varphi+\big{[}R(6*\pi)-R(8* \pi)\big{]}\bigg{\}}R_{or} \tag{7}\]  
where S(\(\varphi\)) is the cross-sectional area of the suction chamber, \(l\) is the corresponding length of the vortex generatrix of the chamber, \(\zeta\) is the area reduced by the movement of the moving disk. The theoretical results are shown in Table 2. It can be observed that the stroke volumes of the uniform thickness and combined profile vortex compressors are equal. However, the leakage line length of the optimized new variable diameter dual circular arc combined profile is significantly less than that of the uniform thickness involute profile, with the radial leakage line length being reduced by 25% in comparison. When different working fluids in the chambers sequentially and continuously complete the compression process as the main shaft rotates, the radial engagement points along the leakage line length start from the involute development angle to the starting angle, completing the entire engagement process.