# Simulation-Based Design for Inlet Nozzle of Vortex Tube to Enhance Energy Separation Effect
## 4 Energy Separation Efficiency Improvement
### Simulation Result  
Simulations were performed on the original and four modified shapes using the grid system, turbulence model, numerical schemes, and boundary conditions outlined in Section 3. The results were compared to assess the effect of the modified shapes on the energy separation efficiency of the vortex tube. The temperature distributions at the central cross-section for each shape, with the outlet pressure set to 1.7 atm, are shown in Figure 17. As illustrated, an increase in the expansion ratio leads to a rise in temperature inside the vortex tube, particularly toward the hot outlet relative to the vortex generator. In contrast, there is relatively little change in the temperature near the cold outlet. This observation is further confirmed in Figure 18, which displays the temperatures at both the cold and hot outlets. Figure 19 presents the temperature differences between the hot and cold outlets for the various shapes, demonstrating that the temperature difference increases as the expansion ratio becomes larger. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value or Definition** \\ \hline Radius of nozzle throat & \(R_{t}\) & 2 mm \\ Radius of nozzle exit & \(R_{e}\) & \(\sqrt{e}R_{t}\) \\ Expansion ratio & \(\epsilon\) & 4, 6, 8, 10 \\ Parabolic entry angle & \(\theta_{n}\) & \(15^{\circ}\) \\ Parabolic exit angle & \(\theta_{e}\) & \(15^{\circ}\) \\ Length of nozzle & \(L_{n}\) & \(K\big{(}\sqrt{e}-1\big{)}R_{t}/\tan(\theta_{e})\) \\ Percent of the conical nozzle length & \(K\) & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Geometrical parameters of the bell shape nozzle. Figure 16: Original and modified nozzle and internal shapes of the vortex generator. Figure 15: Parameters of parabolic bell shape nozzle design. Figure 19: Comparison of the temperature difference between cold and hot outlets due to expansion ratio \(\epsilon\). Figure 17: Comparison of temperature distribution in the center section of the vortex between the original shape and modified ones changing expansion ratio \(\epsilon\). Figure 20b,c show the velocity magnitude and the velocity change ratio, as defined in Equation (6), at the smallest cross-section indicated in Figure 20a, respectively. As observed in the figure, as the expansion ratio increases, although the average velocity at the inlet decreases, the velocity change ratio increases significantly, resulting in a higher velocity at the smallest cross-section. In addition, a comparison of the velocity change ratio in Figure 20c with the temperature difference in Figure 19 reveals a similar tendency. Therefore, it can be said that the primary reason for the improvement in energy separation performance with higher expansion ratios for the inlet nozzles is the higher velocity at the smallest cross-section. \[velocity\;change\;ratio\;(\%)=\frac{U_{sc}-U_{in}}{U_{in}}\times 100=\left(\frac{U_{ sc}}{U_{in}}-1\right)\times 100 \tag{6}\]  
where \(U_{sc}\) is the velocity magnitude at the smallest cross-section and \(U_{in}\) is the average velocity magnitude across the four inlets. The radial velocity distributions at the x-center planes of different vortex generators are compared in Figure 21. As shown in the figure, as the expansion ratio increases, the region with high radial velocity--one of which is highlighted by the red circle at the top right--expands, indicating a reduction in interference between the incoming and rotating flows. Figure 20: Comparison of velocity magnitudes at inlet and velocity change ratios at contraction against expansion ratio \(\epsilon\). Additionally, Figure 22 presents a comparison of temperature differences against cold fractions, with varying outlet pressures, for both the original and modified shapes. The results show that both the cold fraction and the temperature difference increase with a higher expansion ratio. Except for the case where the expansion ratio is 4, the simulations indicate an average improvement of about 15% in the energy separation effect.