## 4 Design of the Helical-Grooved Self-Cooling Case  
A motor generally requires a device for cooling the motor because the core and coil have a high possibility of failure due to heat generation. Therefore, a conventional water-cooled cooling motor is used to cool the motor, injecting cooling oil into the inside and outside of the housing and circulating the cooling oil. However, a conventional water-cooled cooling motor requires a separate external power source to circulate the cooling oil. In particular, the conventional water-cooled cooling motor cooled the motor using an external pump, circulating cooling oil through the power of the pump. Likewise, the installation cost and the maintenance cost for cooling the motor increase because the conventional water-cooled cooling motor requires a pump, which is a separate power source. In this study, technology for cooling the motor enabling the cooling oil to flow up the outer wall of the rotor was proposed. In particular, a case surrounding the outer wall of the rotor was fabricated, and a helix groove was formed on the outer wall of the rotor to form an upward swirling flow of cooling oil at the bottom of the outer housing when the outer rotor rotated. In other words, the motor can be cooled by circulating cooling oil without a separate power source. The intention was to determine the effect of the core and coil, which are the main causes of motor heat generation, and undergo cooling and heat dissipation due to the cooling oil. The diffusion rate, \(v\), of a gas in a one-dimensional flow field with a number density gradient is expressed as follows:  
\[v=-D\ \frac{1}{n}\ \frac{dn}{dx}, \tag{8}\]  
where \(D\) is the diffusion coefficient. For free molecular flow, the diffusion coefficient \(D_{m}\) in the helical groove can be derived as follows. First, the throughput \(Q\) through helical groove is expressed as shown in the following equation [20]. \[Q=\ C_{1}\frac{8}{3}\left(\frac{RT}{2\pi}\right)^{\frac{1}{2}}\frac{w^{2}h^{2 }}{w+h}\left(-\frac{dp}{dx}\right), \tag{9}\]  
where \(C_{1}\) denotes a correction factor according to the aspect ratio of the cross-section, \(R\) denotes the gas constant, \(T\) denotes the absolute temperature, and \(w\) and \(h\) are the width and height of the helical groove, respectively. The diffusion velocity \(v_{m}\) for the free  
Figure 3: Electromagnetic field analysis result of the IPM motor. molecular flow can be derived by dividing Equation (9) by the product of pressure and cross-sectional area, \(pA\), and using the equation of state, \(p=nkT\). \[\frac{Q}{pA}=v_{m}=-D_{m}\,\frac{1}{n}\,\frac{dn}{dx}, \tag{10}\]  
where the diffusion coefficient is \(D_{m}=2C_{1}v_{0}wh/\{3(w+h)\}\), and the molecular average velocity is \(v_{0}=\sqrt{8 RT/\pi}\). The conductance, \(C\), in the helical groove is as follows:  
\[C=3.54\times 10^{-2}C_{2}\frac{w^{2}h^{2}}{\mu L}\,\overline{p}, \tag{11}\]  
where \(\mu,\ L,\ p\), and \(C_{2}\) denote the viscosity coefficient, length of the pipe, average pressure at the inlet and outlet, and correction coefficient according to the aspect ratio of the section, respectively. \(C\) can be expressed as the product of the diffusion velocity \(v_{v}\) and the cross-sectional area \(A\) in the viscous region; therefore, Equation (14) can be ultimately summarized for \(v_{v}\) as follows:  
\[ \begin{split} v_{v}&=-C_{2}\frac{wh}{28\,240\mu} \left(-\frac{dp}{dx}\right)\\ &=-D_{v}\,\frac{1}{n}\,\frac{dn}{dx},\end{split}  \tag{12}\]  
where \(D_{v}=C_{2}v_{0}wh/\left(35.967\lambda\right)\). Figure 4 shows the structure where the helix structure of the helical-grooved self-cooling case is unfolded in the circumferential direction. It is simplified by rotating the motor at a speed \(u\) along the helix groove on the outer wall of the rotor.