## 4 Design of the Helical-Grooved Self-Cooling Case
The following equation can be expressed if the leakage flow rate per area is \(q\), as gas leaks through the gap \(\delta\), as shown in Figure 4.  
\[u_{d}-\frac{q}{n}=(D_{m}f_{m}+D_{v}f_{v})\frac{1}{n}\frac{dn}{dx}, \tag{13}\]  
where \(u_{d}\) is as follows:  
\[u_{d}=u\frac{w}{2(w+h)}\cos\alpha. \tag{14}\]  
The leakage flow rate \(q\) in the equation is assumed as shown in Equation (15). \[q=n_{2}u_{d}\frac{A_{l}}{A_{p}}, \tag{15}\]  
where \(A_{l},A_{p}\), and \(n_{2}\) denote the leakage area, area of the flow path, and water density at the outlet, respectively. The leakage amount was calculated by considering that leakage is  
Figure 4: Helical-grooved self-cooling case. continuously found from the inlet to the outlet through the distance outlet \(\delta\) between the rotor and stator, as shown in Figure 2.  
\[q=\int_{0}^{L}nu\frac{\delta L\sin\alpha}{2A_{p}(L+\delta)}dx. \tag{16}\]  
Therefore, Equation (13) is as follows:  
\[u_{d}n\bigg{(}1-b\frac{uL}{2u_{d}(L+\delta)}\frac{\int_{0}^{L}n\ dx}{n}\bigg{)} =\frac{v_{0}}{\alpha h}\ \bigg{\{}C_{1}\frac{2 nuh}{3(w+h)}\bigg{\}}\times\bigg{\{} \frac{\lambda}{\lambda+h}+\frac{C_{2}\big{(}\frac{J}{4w}+1\big{)}}{23.978C_{1} }\ \frac{h}{\lambda+h}\bigg{\}}\ \frac{dy}{dx}, \tag{17}\]  
where \(AR\) denotes the aspect ratio \(\frac{h}{w}\) of the cross-section. Substituting \(y=h/\lambda\), Equation (17) can be summarized as follows:  
\[\frac{u_{d}}{v_{0}}=\frac{2C_{1}wh}{3(w+h)}\frac{\bigg{(}\frac{1}{1+y}+C_{3} \frac{y^{2}}{1+y}\bigg{)}\ \frac{dy}{dx}}{\sigma ln\bigg{(}1-b\frac{uL}{2u_{d}(L+\delta)}\frac{\int_{0}^ {L}n\ dx}{n}\bigg{)}}, \tag{18}\]  
where \(C_{3}\) is as follows:  
\[C_{3}=\frac{C_{2}\Big{(}\frac{1}{AR}+1\Big{)}}{23.978C_{1}}. \tag{19}\]  
The integral term included in the denominator on the right-hand side of Equation (18) is as follows:  
\[u_{d}-\frac{q}{n}=(D_{m}f_{m}+D_{v}f_{v})\frac{1}{n}\frac{dn}{dx}, \tag{13}\]0  
Using Equation (20), Equation (18) from 0 to \(L\) and from \(y_{1}\) to \(y_{2}\) can be integrated as follows:  
\[ \begin{array}{ll}\frac{u_{d}}{v_{0}}\frac{L}{h}\Big{\{}\frac{3(1+AR)}{2C_{1} }\Big{\}}&=C_{3}(y_{2}-y_{1})\\ &+\bigg{(}1+C_{3}\frac{1+C_{1}\overline{Z}^{2}}{2}\bigg{)}\ \bigg{\{}\frac{1}{1+C_{4} \overline{Z}}\ln\frac{(y_{1}+1)\big{(}y_{2}-C_{4}\overline{Z}\big{)}}{(y_{2}+ 1)\big{(}y_{1}-C_{4}\overline{Z}\big{)}}\bigg{\}}\\ &+C_{3}\frac{1-C_{4}\overline{Z}}{2}\ln\frac{(y_{1}+1)\big{(}y_{2}-C_{4} \overline{Z}\big{)}}{(y_{2}+1)\big{(}y_{1}-C_{4}\overline{Z}\big{)}}.\end{array}  \tag{21}\]  
where \(C_{4}\) and \(\overline{Z}\) are as follows:  
\[C_{4}=b\frac{uL}{2u_{d}(L+\delta)}, \tag{22}\]  
\[\overline{Z}=\int_{0}^{L}y\ dx, \tag{23}\]  
The helix angle, channel width, channel depth, and clearance, which can improve the suction performance of the helical-grooved self-cooling case, were selected based on the above equation. The selected values are listed in Table 2. The drawing design, based on the design specifications, is shown in Figure 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{1}{c}{**Unit**} & \multicolumn{1}{c}{**Value**} \\ \hline Helix angle & \(\alpha\) & \(\circ\) & 10 \\ Channel width & \(w\) & mm & 1.5 \\ Channel depth & \(h\) & mm & 0.6 \\ Clearance & \(\delta\) & mm & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Helical-grooved self-cooling case parameter.