# 3D Numerical Simulation and Performance Analysis of CO2 Vortex Tubes
## 3 Results and Discussion
### Effect of Inlet Pressure on Radial Velocity  
Figure 16 illustrates a three-dimensional model based on a physical drawing of the swirl flow pipe. The 3-dimensional model is defined by the \(Z\)-axis in the direction of the hot-end tube, the \(Y\)-axis in the direction of the swirl flow pipe inlet, and the \(X\)-axis perpendicular to the direction of the hot-end tube, with the origin coordinates placed in the middle of the swirl flow pipe.  
The high-pressure flow passes through the inlet nozzle of the swirl flow pipe into the main tube. As the swirl flow pipe has no external source of work or heat, the pressure at the inlet is the only energy source available. The bulk of the pressure drop occurs in the nozzle, where the velocity increases dramatically, and a cyclonic flow is established. Figure 17 shows that as the swirl flow pipe inlet pressure continues to increase, the inlet radial cyclonic velocity rises, but the magnitude of the change decreases. The tangential velocity of the core flow is lower than that of the peripheral flow. It can be seen that the tangential velocity increases in the positive and negative Y-direction along the radius of the tube. The maximum velocity in the negative direction of the \(X\)-axis is greater than the maximum velocity in the positive direction, mainly because the former velocity is the velocity at the inlet of the swirl flow pipe into the vortex chamber, as shown in Figure 10. The pressure potential energy of the gas at the inlet is converted into kinetic energy, and the velocity rises sharply; this is the maximum velocity inside the swirl flow pipe. Figure 17 shows a gradual increase in tangential velocity from the center of the shaft to the outer edge of the swirl flow pipe, the low-velocity region at the center of the shaft gradually increases in the direction of the hot-end tube, indicating a gradual decrease in pressure along the hot-end tube.  
Figure 17 also shows that the tangential velocity increases from the center of the axis to the outer edge of the swirl flow pipe in the same axial section; the tangential velocity also  
Figure 16: Schematic diagram of 3-dimensional coordinates of the swirl flow pipe.  
Figure 17: Radial velocity distribution along the X-position at different inlet pressures.  
rises gradually with increasing pressure, but the increment decreases. There are significant differences in the tangential velocity in different axial sections. Along the hot-end tube, the tangential velocity becomes progressively weaker due to the fluid pressure difference and kinetic energy loss. As the fluid enters the vortex chamber tangentially, the fluid flows tangentially along the outer edge, so the pressure at the outer edge is higher than the pressure in the axial region. At different axial cross-sections, the exchange of work, kinetic energy, and heat (during the flow process) result in a reduction in pressure and, therefore, a reduction in tangential velocity. Figure 18 indicates that the effect of swirl flow pipe length on radial velocity is smaller for the same inlet pressure, as seen in the three plots, where the velocity of the swirl flow pipe with a tube length of 175 mm is slightly higher than that of the swirl flow pipe with a tube length of 125 mm. This leads to the conclusion that it is the number of swirl flow pipe inlets and the vortex chamber diameter, rather than the length of the swirl flow pipe hot end, that affects the radial velocity of the vortex chamber.