# Numerical Investigation on the Principle of Energy Separation in the Vortex Tube
## 3 Results and Discussion
### Temperature Separation  
For a steady and axisymmetric swirl flow, the energy equation, Equation (3), can be simplified considerably. Further, the heat conduction term is not considered in the temperature separation discussion because we are concerned with temperature separation; however, heat conduction always plays a role in making the temperature uniform in space. Consequently, the energy equation can be written as  
\[u_{r}\frac{\partial T}{\partial r}+u_{z}\frac{\partial T}{\partial z}\propto \frac{1}{\rho c_{p}}\bigg{(}u_{r}\frac{\partial p}{\partial r}  
To compare the first and second terms on the left-hand side of Equation (4), we define  
\[\xi_{T}=\frac{|\partial T/\partial z|}{|\partial T/\partial z|+|\partial T/ \partial r|}. \tag{6}\]  
Figure 8 shows the \(\xi_{T}\) contours for the different cold mass fractions. Overall, \(\xi_{T}\) is not very small in most regions, and this implies that \(|\partial T/\partial z|\) is comparable to or higher than \(|\partial T/\partial r|\) in most regions. In addition, considering the strong swirl flow characteristics, \(|u_{z}|\) is considerably greater than \(|u_{r}|\) in most regions. Therefore, the first term on the left-hand side of Equation (4) can be safely neglected, and the temperature equation can be simplified further as:  
\[\frac{\partial T}{\partial z}\propto\frac{1}{\rho c_{p}}\frac{u_{r}}{u_{z}} \frac{\partial p}{\partial r}+\frac{1}{\rho c_{p}}\frac{\partial p}{\partial z }+\frac{1}{\rho c_{p}}\frac{\mu_{eff}}{u_{z}}\Phi. \tag{7}\]  
The axial temperature gradient depends on the radial pressure gradient, axial pressure gradient, and viscous dissipation. For comparing the three terms on the right-hand side of Equation (7), we define  
\[\psi_{1}=\frac{1}{\rho c_{p}}\frac{u_{r}}{u_{z}}\frac{\partial p}{\partial r}, \ \psi_{2}=\frac{1}{\rho c_{p}}\frac{\partial p}{\partial z},\ \psi_{3}=\frac{\mu_{eff}}{\rho c_{p}u_{z}}\Phi, \tag{8}\]  
\[\zeta_{n}=\frac{|\psi_{n}|}{|\psi_{1}|+|\psi_{2}|+|\psi_{3}|}\ \ \ \ \ (n=1,2,3), \tag{9}\]  
where \(\zeta_{1}\), \(\zeta_{2}\), and \(\zeta_{3}\) represent the measures of the relative importance of the first, second, and third terms on the right-hand side of Equation (7), respectively. Figures 9 and 10 show the \(\zeta_{1}\), \(\zeta_{2}\), and \(\zeta_{3}\) contours and streamlines for cmf = 0.2 and 0.8, respectively. \(\zeta_{2}\) is dominant in the inner core region; \(\zeta_{1}\) and \(\zeta_{3}\) are large in the periphery region and tend to have similar values, except in the region near the hot exit. Figure 9 shows that \(\zeta_{3}\) is very high at approximately \(z/R=15\), which is attributed to the increased turbulent viscosity there. As seen in the streamlines created using the radial and axial velocities, the flow is complicated because of the presence of the stagnation point. Figure 8: Contours for different cold mass fractions. As shown in Figure 6, the axial velocity \(u_{z}\) is positive and negative in the periphery and core regions, respectively. The calculations are performed along the axial direction lines at \(r/R=0.1\) and \(0.8\) to avoid the singularity in calculating \(\psi_{1}\) and \(\psi_{3}\) from Equation (8). The two horizontal lines belong to either the core or periphery region; the sign of \(u_{z}\) does not change in most regions of each horizontal line. It is desirable to interpret the positive and negative values of \(\psi_{n}\) as positive and negative contributions to \(\partial T/\partial z\), respectively, because the heat conduction effect was not included and some assumptions were used in Equation (7). Figure 11 shows the distributions of \(\psi_{1}\), \(\psi_{2}\), and \(\psi_{3}\) along the axial-direction line at \(r/R=0.1\) (core vortex region). The value of \(\psi_{2}\) is considerably higher than those of \(\psi_{1}\) and \(\psi_{3}\). This means that the axial pressure gradient in the near-axis region is the major contributor to an increase in temperature in the axial direction.