# Numerical Investigation on the Principle of Energy Separation in the Vortex Tube
## 1 Introduction
Because this study is focused on the principle of temperature separation, the previous works that simply presented the results of geometry/condition optimization were not included in Table 1. In most works listed in Table 1, the temperature separation was not explained or was partially discussed based on the temperature distribution in the radial direction; however, as we show later in this study, the temperature does not always increase in the radial direction. \begin{table}
\begin{tabular}{c c c} \hline
**Ref.** & **Dim.** & **Main Concern and Summary** \\ \hline
[32] & 2D & Transient flow behavior \\ \hline
[33] & 2D & Turbulence model, working fluid \\ \hline
[34] & 3D & Flow structure, inlet shape, and aspect ratio \\ \hline
[35] & 2D & Temperature separation analysis based on heat transfer and shear work \\ \hline
[36] & 2D & Turbulence mode, flow structure, and working fluid \\ \hline
[24] & 2D & Turbulence model \\  & & Temperature separation analysis based on heat transfer and shear work \\ \hline
[37] & 2D & Turbulence model \\ \hline
[38] & 2D & The increase in temperature in the radial direction was due to compression. \\ \hline
[39] & 3D & Turbulence model \\ \hline
[12] & 2D & Turbulence model \\ \hline
[40] & 3D & Flow structure based on LES \\ \hline
[9] & 2D & Flow structure based on LES \\ \hline
[41] & 3D & Flow structure based on LES \\ \hline
[42] & 3D & Flow structure based on LES \\ \hline
[30] & 3D & Flow structure based on LES \\ \hline \end{tabular}
\end{table} 
Table 1: Previous numerical simulation works. The flow structures were relatively well clarified [9; 30; 33; 40; 41; 42]. However, the principle of temperature separation was less investigated. [PERSON] and [PERSON] [8] claimed that the angular momentum transfer, pressure expansion, and transformation from kinetic energy resulted in energy separation. [PERSON] and [PERSON] [43] tested the effect of each term in the energy equation on temperature separation. The mean kinetic energy diffusion was the main influence on the temperature separation in the inlet region, while pressure work, stress generation, and stress gradient transport played also a significant effect. However, the results were simply reported without a detailed explanation of the reasons. In addition, only one cold mass fraction was considered. [PERSON] et al. [35] quantified the energy transfer between the core and periphery regions and attributed the temperature separation to the work transfer caused by a viscous shear between the regions. However, viscous heating increases the temperature in the core region as well as in the periphery region. Therefore, there is room for numerical simulations to explain the principle of temperature separation despite many previous works. Two-dimensional numerical simulations were carried out in this study. Two-dimensional simulations were enough to explain the principle of temperature separation and were a practical choice for considering many simulations with different cold mass fractions. In this study, the relative contributions of the radial pressure gradient, axial pressure gradient, and viscous dissipation to the temperature separation were discussed based on the simulation results and energy equation. Moreover, the effect of the cold mass fraction on the hot and cold exit temperatures were explained for the first time.