A vortex tube works by feeding high‑pressure air into a chamber, which causes the air to swirl and split into hot and cold streams. The tube is considered adiabatic so that:
\(\dot{m}_h\,H_h - \dot{m}_c\,H_c - \dot{m}_f\,H_f = 0\)
which simplifies to:
\(\dot{m}_c\,C_p\,(T_f - T_c) + \dot{m}_h\,C_p\,(T_h - T_f) = 0\)
The subscripts h, c and f represent the hot, cold and feed streams, respectively; ṁ is the mass flow rate (kg/s); H is specific enthalpy (J/kg); Cp is constant‑pressure specific heat capacity of air (J/(kg K)); and T is temperature (K). The hot and cold stream temperatures are functions of tube geometry, inlet pressure and temperature, and the mass fraction in the cold stream [1]. Vortex tube efficiency can be characterized by the isentropic efficiency (ηis),
\(\eta_{is} = \frac{T_f - T_c}{(T_f - T_c)_{rev}}\)
which is the ratio of actual temperature drop to the temperature change by reversible (isentropic) adiabatic expansion. ΔT of reversible expansion is given by:
\((T_f - T_c)_{rev} = T_f \left[\left(\frac{P_f}{P_{atm}}\right)^{(\gamma-1)/\gamma} - 1\right]\)
where Patm is ambient pressure and γ is the heat‑capacity ratio of the gas, Cp/Cv[1]. Isentropic efficiency does not consider the mass flow rate of the cold stream so the coefficient of performance (COP) may also be used to characterize efficiency. COP is the ratio of cooling rate to work input:
\(\mathrm{COP} = \frac{\dot{Q}_c}{W}\)
where Q̇c (J/s) is the cooling rate:
\(\dot{Q}_c = \dot{m}_c\,C_p\,(T_f - T_c)\)
and W (J/s) is the work input to the system, defined as the maximum possible cooling energy per time via reversible adiabatic expansion:
\(W = \dot{m}_f\,C_p\,T_f\left[\left(\frac{P_f}{P_{atm}}\right)^{(\gamma-1)/\gamma} - 1\right]\)
Note that the COP is equivalent to isentropic efficiency multiplied by the mass fraction of air in the cold stream ṁc/ṁf
The Ranque–Hilsch vortex tube produces a positive entropy change, ΔS:
\(\Delta S = \Delta S_h + \Delta S_c\)
The entropy change ΔṠ of the cold and hot streams is:
\(\Delta S_i = \dot{m}_i\left(C_p\ln\tfrac{T_f}{T_i} + R\ln\tfrac{P_f}{P_i}\right)\)
where subscript i may be replaced by either c or h.
A vortex tube is a mechanical device used to separate a compressed gas into hot and cold streams. High‑pressure air is fed into a Ranque–Hilsch vortex tube and air exits at 1‑bar pressure: cold air exits the left side and hot air exits the right side. Use the “fraction of feed in cold stream” slider to adjust the fraction of feed in the cold stream (left side); this adjusts the throttle valve on the right side (triangle). Use the “feed pressure (bar, absolute)” slider to modify the inlet pressure. The color of the flowing gas indicates its temperature; blue is cold and red is hot. The coefficient of performance and isentropic efficiency are shown in the top‑right corner. The coefficient of performance equals the isentropic efficiency multiplied by the fraction of feed gas in the cold stream.
This simulation was created in the Department of Chemical and Biological Engineering, at University of Colorado Boulder for LearnChemE.com by Sneha Nagaraju under the direction of Professor John L. Falconer and Michelle Medlin, with the assistance of Neil Hendren. It is a JavaScript/HTML5 implementation of a Mathematica simulation by Adam J. Johnston, Neil Hendren, and Rachael L. Baumann. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988) in collaboration with Washington State University. Address any questions or comments to LearnChemE@gmail.com.