# Numerical Investigation on the Principle of Energy Separation in the Vortex Tube
## 2 Numerical Methods  
The flow and temperature fields inside the Ranque-Hilsch vortex tube were obtained by solving the mass, momentum, and energy equations, which are given respectively as:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{u})=0, \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho\mathbf{u})+\
abla\cdot(\rho\mathbf{u} \mathbf{u})=-\
abla p+\
abla\cdot\left\{\mu_{eff}(\
abla\mathbf{u}+\
abla \mathbf{u}^{T})-\frac{2}{3}\mu_{eff}(\
abla\cdot\mathbf{u})\mathbf{I}-\frac{2 }{3}\rho\mathbf{I}\right\}, \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho E)+\
abla\cdot(\rho\mathbf{u}E)=-\
abla\cdot(p \mathbf{u})+\
abla\cdot(\lambda_{eff}\
abla T)+\
abla\cdot(\tau_{eff}\cdot \mathbf{u}), \tag{3}\]  
where \(\rho\), \(\mathbf{u}\), \(p\), \(\mu_{eff}\), \(\mathbf{I}\), \(k\), \(E\), \(\lambda_{eff}\), \(T\), and \(\tau_{eff}\) denote the density, velocity vector, pressure, effective viscosity, identity matrix, turbulent kinetic energy, total energy, effective thermal conductivity, temperature, and effective viscous stress tensor, respectively [46]. In Equation (3), the last term is related to viscous dissipation and must be included because of the strong vortex flow inside a small tube. In some existing studies [34; 35; 36], \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ref.** & **Dim.** & & **Main Concern and Summary** \\ \hline
[43] & 2D & – & The effects of source terms in the energy equation on temperature separation was tested and simply reported. \\
[43] & 2D & – & 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 also played a significant effect. \\  & & – & Only one cold fraction case was considered. \\ \hline
[44] & 3D & & Variable fluid properties \\ \hline
[8] & 3D & The interaction of angular momentum transaction and kinetic energy transfer were found to be responsible for the temperature difference between the cold and hot fluids. \\ \hline
[45] & 3D & & Gas compressibility \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._the standard \(k-\varepsilon\) turbulence model yielded better predictions of the vortex tube flow. Therefore, the standard \(k-\varepsilon\) turbulence model was also used in the present study. Numerical simulations were conducted for the representative experimental data of [47], which is used in the literature on vortex tube. Figure 1 shows the computational domain for the axisymmetric swirl flow in the vortex tube. The tube length (\(L\)), tube radius (\(R\)), cold exit radius (\(R_{cold}\)), inlet length (\(L_{in}\)), and hot exit length (\(L_{hot}\)) values are 106 mm, 5.7 mm, 3.1 mm, 0.95 mm, and between 1-6 mm, respectively; air was used as the working fluid. The National Institute of Standards and Technology (NIST) air property model was used to consider more realistic thermodynamic properties [48]. Numerical simulations were conducted using the ANSYS Fluent solver [46]. Inlet conditions were specified according to the approach used by [PERSON] et al. [47]. At the inlet, the mass flow rate and total temperature were set to 0.0083 kg/s and 294 K, respectively. The radial and circumferential components of the inlet flow direction were set to \(-0.25\) and 0.97, respectively. The hot and cold exits were exposed to atmospheric pressure. The hot exit area was adjusted by the throttle to control the cold mass fraction (cmf), which is defined as the ratio of the cold exit mass flow rate to the inlet mass flow rate. For the numerical methods, pressure-based coupled algorithms were used for pressure-velocity coupling, and the QUICK scheme was applied to the spatial discretization of variables. The residual criterion for convergence was set to \(10^{-6}\). A grid test was conducted under the experimental conditions of cmf = 0.5. The quadrilateral grid system with finer grids near the inlet and two exits was used. Figure 2 shows the grid convergence test result.