# Numerical Investigation on the Principle of Energy Separation in the Vortex Tube
## 3 Results and Discussion
### Total Temperature Separation  
We discuss the simulation results of each of the exemplary cold mass fractions of 0.2, 0.5, and 0.8 to understand the total temperature separation. Figure 4 shows the total temperature difference contours for different cold mass fractions, where \(\Delta T_{0}\) denotes the local total temperature minus the inlet total temperature.  
The tendency of the total temperature to be higher in the periphery (near wall) region than in the core (near-axis) region is attributed to the strong vortex flow inside the tube. Figure 5 shows the pressure contours for the different cold mass fractions. The pressure in the periphery region is higher than that in the core region. Figure 6 shows the velocity contours for cmf = 0.2. The black line indicates the stagnation-stream surface at which the  
Figure 4: Total temperature difference contours for different cold mass fractions.  
Figure 3: Total temperature separation: (**a**) hot exit, (**b**) cold exit.  
core and periphery vortex regions are separated. The velocity magnitude is very large near the wall but small near the axis, and the circumferential velocity accounts for most of the velocity magnitude. The higher the flow velocity, the higher the total temperature is.  
The tendency of the gradual increase in total temperature in the axial direction is attributed to static temperature (unless otherwise stated, temperature refers to static temperature). Figure 7 shows the temperature difference contours for different cold mass fractions, where \(\Delta T\) denotes the difference between the local temperature and inlet temperature. Overall, the temperature increases in the axial direction. The temperature distribution shape is similar to the existing result [43]. It follows from Figures 6 and 7 that the hot exit temperature is more affected by the axial temperature increase rather than the radial temperature increase. The increase in temperature at the hot exit appears to be less related to the energy transfer between the core and periphery regions. If the energy transfer between the two regions is the main cause of temperature separation and the core (near-axis) region continues to lose energy, the temperature in the core region would decrease in the axial direction. Rather, the temperature increases in the axial direction. [PERSON] et al. [35] attempted to explain the temperature separation in terms of heat transfer and viscous work transfer through the stagnation-stream surface. They attributed the temperature increase at the hot exit to the work transfer due to viscous shear. However, viscous heating increases  
Figure 5: Pressure contours for different cold mass fractions.  
Figure 6: Velocity contours for cmf = 0.2.  
the temperature in the core region as well as in the periphery region. For these reasons, the present analysis was not made on the stagnation-stream surface.  
Notice that, for cmf = 0.8, the temperature increases significantly. This rapid increase in temperature contributes to a large total temperature at the hot exit, as shown in Figure 3a. Meanwhile, it is shown in Figure 7 that the decrease in the cold exit temperature is larger for cmf = 0.5 than it is for cmf = 0.8. Thus, the lower the cold mass fraction, the lower the total temperature is at the cold exit, except when the cold mass fraction is very low, as shown in Figure 3b. In the region where the cold mass fraction is very low, a partial backflow of atmospheric air at 294 K through the cold exit decreases the total temperature separation at the cold exit.  
In summary, the gradual increase in total temperature in the radial direction is mainly due to the gradual increase in velocity magnitude in the radial direction; \(\partial T_{0}/\partial r\sim\partial|V_{\theta}|/\partial r\). The gradual increase in total temperature in the axial direction is due to the gradual increase in temperature in the axial direction; \(\partial T_{0}/\partial z\sim\partial T/\partial z\). In the next section, the gradual increase in temperature in the axial direction is discussed.