# Numerical Investigation on the Principle of Energy Separation in the Vortex Tube
## 3 Results and Discussion
### Temperature Separation
Overall, the value of \(\psi_{2}\) is higher for cmf = 0.8 than that for 0.2, which implies that the axial pressure gradient increases with the cold mass fraction and is consistent with the pressure contours shown in Figure 5. The higher the cold mass fraction, the higher the axial pressure gradient in the near-axis region. This is because as the pressure in the near-axis region at the hot exit section increases due to narrowing the hot exit area by the throttle valve, the return of flow from the hot exit section toward the cold exit section becomes large. Figure 10: \(\zeta_{2}\), \(\zeta_{3}\), and streamlines for cmf = 0.8. Figure 9: \(\zeta_{2}\), \(\zeta_{3}\), and streamlines for cmf = 0.2. Figure 12 plots the distributions of \(\psi_{1}\), \(\psi_{2}\), and \(\psi_{3}\) along the horizontal lines at \(r/R=0.8\) (periphery region). The viscous dissipation effect (\(\psi_{2}\)) is seen to contribute the most to the temperature separation in the peripheral region. However, that effect is expected to be reduced by the other effects. Figure 13 compares the distributions of \(\psi_{1}+\psi_{2}+\psi_{3}\) at \(r/R=0.1\) and 0.8. For both cmf = 0.2 and cmf = 0.8, the magnitudes of \(\psi_{1}+\psi_{2}+\psi_{3}\) at \(r/R=0.1\) are considerably greater than those at \(r/R=0.8\). Consequently, the increase in the hot exit temperature results from the axial pressure gradient in the core region; the higher the cold mass fraction, the larger the axial pressure gradient is. Thus, the hot exit temperature increases with the cold mass fraction. According to the adiabatic expansion and compression theory, the temperature inside the tube must be lower than the inlet temperature because the pressure at any point inside  
Figure 11: Distributions of \(\psi_{1}\), \(\psi_{2}\), and \(\psi_{3}\) along the axial-direction line at \(r/R=0.1\). Figure 12: Distributions of \(\psi_{1}\), \(\psi_{2}\), and \(\psi_{3}\) along the axial-direction line at \(r/R=0.8\). \[\frac{\partial T}{\partial z}\propto\frac{1}{\rho c_{p}}\frac{u_{r}}{u_{z}} \frac{\partial p}{\partial r}+\frac{1}{\rho c_{p}}\frac{\partial p}{\partial z }+\frac{1}{\rho c_{p}}\frac{\mu_{eff}}{u_{z}}\Phi. \tag{7}\]0 Comparison of the distributions of \(\psi_{1}+\psi_{2}+\psi_{3}\) at \(r/R=0.1\) and 0.8.  
the tube is lower than the inlet pressure, as shown in \[\frac{\partial T}{\partial z}\propto\frac{1}{\rho c_{p}}\frac{u_{r}}{u_{z}} \frac{\partial p}{\partial r}+\frac{1}{\rho c_{p}}\frac{\partial p}{\partial z }+\frac{1}{\rho c_{p}}\frac{\mu_{eff}}{u_{z}}\Phi. \tag{7}\]1 In this study, the axial temperature gradient was analyzed in relation to the axial pressure gradient. Therefore, although the pressure at any point inside the tube is lower than the inlet pressure, the temperature may increase significantly in the axial direction because of the axial pressure gradient in the core region. As the first approximation of Equation (7), the temperature difference between the hot and cold exits can be approximated as \(\Delta T\sim\Delta p/\rho c_{p}\), where \(\Delta p\) denotes the pressure difference between the center of the right end of the tube and the center of the left cold exit. This approximation is based on the fact that the axial pressure gradient in the core region is the major contributor to an increase in temperature in the axial direction. Consider the pressure contour for cmf = 0.8, as shown in Figure 5, where \(\Delta p\approx\) 80 kPa. Assuming that \(\rho\approx\) 1 kg/m\({}^{3}\) and \(c_{p}\approx\) 1000 J/kg \(\cdot\) K, the temperature difference \(\Delta T\) between the hot and cold exit is estimated to be 97 K, which is not very different from the temperature difference of 90 K for cmf = 0.8, as shown in \[\frac{\partial T}{\partial z}\propto\frac{1}{\rho c_{p}}\frac{u_{r}}{u_{z}} \frac{\partial p}{\partial r}+\frac{1}{\rho c_{p}}\frac{\partial p}{\partial z }+\frac{1}{\rho c_{p}}\frac{\mu_{eff}}{u_{z}}\Phi. \tag{7}\]2  
Meanwhile, [PERSON] et al.