# Simulation-Based Design for Inlet Nozzle of Vortex Tube to Enhance Energy Separation Effect
## 2 Numerical Methods  
### Governing Equations  
For compressible turbulent thermal flows, the governing equations include the continuity, RANS, and energy transport equations, as given in Equations (1)-(3), where the bar over a variable indicates time averaging. The effect of gravity on the flow was neglected in the conservation of momentum equation (2018; 2019; 2020; 2020).  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{i})}{\partial x_{i}}=0 \tag{1}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Authors** & **Year** & **Research Focus** \\ \hline \hline   [PERSON] (2018) & 1933 & Invention \\   [PERSON] (2018) & 1947 & Improvement and principle description \\   [PERSON] and [PERSON] (2018) & 1966 & Length and shape of chamber \\   [PERSON] and [PERSON] (2018) & 2003 & Inlet pressure and geometrical parameters \\   [PERSON] et al. (2018) & 2004 & Nozzle shape design \\   [PERSON] et al. (2018) & 2009 & Numbers of nozzle, inlet pressure \\   [PERSON] et al. (2018) & 2009 & Diameter of cold outlet \\   [PERSON] (2018) & 2010 & Plug angle \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of research focus of experimental studies on vortex tube.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Authors** & **Year** & **Research Focus** & **Turbulence Model** \\ \hline   [PERSON] et al. (2018) & 2006 & area of nozzles & \(k-\epsilon\) and RNG \(k-\epsilon\) \\   [PERSON] et al. (2018) & 2009 & LES & RSM and \(k-\epsilon\) \\   [PERSON] et al. (2018) & 2007 & cold end shape & \(k-\epsilon\) \\   [PERSON] et al. (2018) & 2019 & shapes of vortex chambers & \(k-\epsilon\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of research focus and turbulence model of numerical studies on vortex tube.  
\[\frac{\frac{\partial(\rho u_{i})}{\partial t}}{\partial t}+\frac{ \frac{\partial\left(\rho u_{i}u_{j}\right)}{\partial x_{j}}}{\partial x_{j}}=- \frac{\partial p}{\partial x_{i}}+\frac{\partial\tau_{ij}}{\partial x_{j}}- \frac{\frac{\partial\left(\rho u_{i}^{\prime}u_{j}^{\prime}\right)}{\partial x _{j}}}{\partial x_{j}}, \tag{2}\] \[\tau_{ij}=\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{i}}{\partial x_{i}}\right)-\frac{2}{3}\mu\delta_{ij}\frac{\partial u _{k}}{\partial x_{k}} \tag{3}\]  
where \(\rho\) is the density; \(u_{i}\) and \(x_{i}\) are the velocity component and spatial coordinates in the \(i\)-direction; \(p\) is the pressure; \(\tau_{ij}\) is the viscous stress tensor; \(\mu\) is the dynamic viscosity; \(\delta_{ij}\) is the Kronecker delta, which is 1 when \(i=j\) and 0 otherwise; \(\overline{u_{i}^{\prime}u_{j}^{\prime}}\) is the Reynolds stress tensor representing turbulent fluctuations, which is replaced using a turbulence model; \(E\) is the total specific energy, defined as \(E=e+(u_{i}u_{i})/2\); \(e\) is the internal energy; \(k_{T}\) is the thermal conductivity; \(c_{p}\) is the specific heat; \(\mu_{t}\) is the eddy viscosity, defined as \(\mu_{t}=\rho C_{p}k^{2}/\varepsilon\), where \(C_{p}\), \(k\), and \(\varepsilon\) are model parameters (with \(C_{p}\) set to 0.09 in this study), the turbulent kinetic energy, and the turbulent dissipation rate, respectively; \(P_{t}\) is the turbulent Prandtl number; \(T\) is the temperature; \(\left(\tau_{ij}\right)_{eff}\) is the effective viscous stress tensor, where \(\mu\) in Equation (2) is replaced by the sum of \(\mu\) and \(\mu_{t}\); and \(\Phi\) is the viscous dissipation representing the conversion of mechanical energy into heat due to viscosity.