Document Type : Full Lenght Research Article
Authors
^{1} Islamic Azad University, Semnan Branch
^{2} Semnan Branch, Islamic Azad University
^{3} Semnan University
Abstract
Keywords
1.
History: Received 31 May 2014 Received in revised form 27 July 2014 Accepted 08 September 2014
Keywords: MicroScale Vortex Tube Energy separation CFD Analysis Refrigeration Capacity 
A B S T R A C T In this study, fluid flow and energy separation in a microscale RanqueHilsch Vortex Tube are numerically investigated. The flow is assumed as 2D, steady, compressible ideal gas, and shearstresstransport is found to be a best choice for modeling of turbulence phenomena. The results are in a good agreement with the experimental results reported in the literature. The results show that fluid flow and energy separation inside the microscale vortex tube is quite similar to those of traditional ones. Moreover, it is found that nondimensional forms of coldtemperature difference and refrigerating capacity are only dependent on cold mass fraction. In addition, two correlations have been proposed to estimate nondimensional forms of cold temperature difference and refrigeration capacity in the microscale vortex tube.
© 2015 Published by Semnan University Press. All rights reserved. 
2D numerical simulation of a micro scale RanqueHilsch vortex tube
Nader Rahbar^{1,*}, Mostafa Shateri^{1}, Mohsen Taherian^{1}, Mohammad Sadegh Valipour^{2} ^{1}Department of Mechanical Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran ^{2}School of Mechanical Engineering, Semnan University, Semnan, Iran 


Journal of Heat and Mass Transfer Research
Journal homepage: http://jhmtr.journals.semnan.ac.ir 


Introduction
A RanqueHilsch Vortex Tube (RHVT) is a simple device without any moving parts. When a highpressure gas is entered tangentially in a vortex tube, it is separated into two lowpressure hot and cold temperature streams. This phenomenon which is known as energy separation is first discovered by Ranque [1, 2] and after that it is developed by Hilsch [3].
To better understanding of the energy separation inside traditional RHVTs, several numerical investigations have been done during last decade. Fröhlingsdorf and Unger [4] numerically evaluated the fluid flow in a traditional RHVT. They used an axisymmetric 2D model to evaluate the flow structure inside the RHVT. They identify a secondary circulating zone inside the vortex tube and conclude that this zone receives energy from a cold stream and transmits it to the hot gas. In a numerical and experimental study, Behera et al. [5] investigated the effect of different configuration of nozzles on energy separation in a traditional vortex tube. They report that for attaining a maximum cold temperature difference, the values of length to diameter ratio and nondimensional orifice diameter should be 20< L/D<30 and dc/D=0.5, respectively. Aljuwayhel et al. [6] studied the energy separation and flow structure in a RHVT by using Standard Kε and RNG Kε turbulence models. They concluded that RNG Kε model provided better prediction than Standard Kε model. In a similar study, Skye et al. [7] reported the opposite results of Aljuwayhel for simulating of a commercial vortex tube. In another numerical study, Eiamsaard and Promvonge [8] simulated a uniflow vortex tube. They used CFX commercial CFD code and axisymmetric 2D model to simulate the energy separation inside the tube. Farouk and Farouk [9] reported that LES turbulence model was better than Kε model for simulating of a vortex tube. Behera et al. [10] evaluated the effects of fluid flow properties, secondary circulation and tube length on the energy separation of a 3D RHVT. They report that, in a largescale vortex tube, the static temperature radially decreases near the entrance region.
In another study, Farouk et al. [11] simulated gas species and temperature separation inside a counter flow RHVT. They reported that, only a very minimal gas separation occurred inside the vortex tube as a result of diffusion effects. They also conclude that inner core flow has large values of eddy heat flux and Reynolds stresses.
Ameri and Behnia [12] used 2D and 3D RSM turbulence models to investigate the energy separation in a vortex tube. They find an optimum inletpressure for maximum efficiency. They also suggested the optimum dimensional values for their vortex tube. Dutta et al. [13] compare several different turbulence models and conclude that the value of hot and cold outlet temperatures obtained by the standard Kε and SST Kω models is very close to the experimental data. Hossein Nezhad and Shamsoddini [14] compared 2D and 3D numerical modeling of the flow stream in a vortex tube. They conclude that the results of a threedimensional model are more accurate than 2D one. Moreover, in another study Shamsoddini and Hossein Nejad [15]reportthat the number of nozzles has direct effect on the power of cooling and opposite effect on the cold outlettemperature. Gas separation at atmospheric and cryogenic temperatures was numerically studied by Dutta et al. [16]. They reported an enhancement in predicting of the energy separation by using of NIST real gas model for accurately computing of the air properties. Baghdad et al. [17] investigated the energy separation mechanism and flow phenomena within a vortex tube by using four different turbulence models. They reported that advanced RSM model is the most accurate model to estimate cold and hot outlet temperatures. Khazaei et al. [18] reported that Spalart–Allmaras turbulence model has also good ability in estimating of flow field and energy separation in a typical vortex tube.
Typical vortex tubes have fairly large tube diameters (i.e. 1025 mm) which limit their application specially in smal scale devices. A microscale vortex tube has a good potential for smallscale cooling applications such as cooling of electronic chips, cutter blades,plastic injection molds, and setting solders and adhesives. To the best knowledge of the authors of the present paper, there are not enough investigations on microscale vortex tubes. Dyskin and Kramarenko [19] were the first researchers that conducted some experimental procedures to determine the performance characteristics of a microscale vortex tube. Their vortex tube had an operating pressure ratio of 6 and diameters of 1 mm, 2 mm, and 3 mm. They reportthat, by decreasing of the flow rate, the cooling effect decreases. Hamoudi et al. [20] experimentally investigated the performance of a microscale vortex tube. They conducted some experiments over a wide range of working pressure, different cold air mass ratio, different tube length, and orifice diameters. The results of their experiments at low Reynolds numbers show that by increasing of Reynolds number, dimensionless temperature increases in both hot and cold air flows. They also conclude that the optimum cold air mass fraction is not constant at high inlet pressure and it is higher than that of conventional vortex tubes. However, the effect of and ratios are similar to those of conventional devices. Rahbar et al. [21] numerically investigated the flow behavior and energy separation inside a microscale vortex tube. They show that both 2D and 3D CFD simulation have a good potential to estimate the performance of a microscale vortex tube. They also reported that in a microscale vortex tube, the expansioneffect on static temperature is more than that of largescale vortex tubes.
The energy separation inside a microscale vortex tube is a significant phenomenon and all operational and performance characteristics are dependent on it. As mentioned before, there are few works to estimate flow characteristics of microscale vortex tubes. So, it requires more work to find insight of this process and to obtain some accurate correlations to estimate the performance of a microscale vortex tube. The aim of this work is to investigate the energy separation phenomenon and flow structure inside a 2D microscale vortex tube by using of computational fluid dynamic. In addition, some correlations are proposed to estimate nondimensional forms of cold temperature difference and refrigeration capacity in the microscale vortex tube.
2. Numerical Model Formulation
In this study, numerical analysis of the fluid regime in a microscale vortex tube is performed. Because of small size of inlet nozzles, it is essential to check the validation of continuum model in the microscale vortex tube. The continuum model is not valid, when the characteristic dimension is comparable with the mean free path of molecules. The ratio of the mean free path to the characteristic length, defines an important dimensionless parameter, called the Knudsen number, and it is given by [22, 23]:
(1) 
For values of , the flowregime is continuum. In the microscale vortex tube, the value of Knudsen number is equal to at inlet nozzles and it is possible to apply NavierStokes and energy equations in the flow simulation.
In this study, the flow is assumed as steady, turbulent, compressible and, the governing equations for fluid flow and heat transfer are as follows [13]:
Continuity equation:
(2) 
Momentum equations:
(3) 
Energy equation:
(4) 
State equation for an ideal gas:
(5) 
The term of is called as Reynolds stress and must be modelled to close the Eq. (3). Boussinesq hypothesis is a common method for modelling of Reynolds stresses, and it is given as follows:
(6) 

(7) 
There are several models to calculate the turbulence viscosity . In turbulence model is calculated as a function of turbulent kinetic energy, , and turbulence dissipation rate, . In turbulence model, is calculated as a function of turbulent kinetic energy, , and specific dissipation rate, . There are also other RANS turbulence models such as , and discussed in more details in the literature [24].
The flow regime in a vortex tube is mainly classified into two categories: nearwall and central core regimes. The flow in central region has a higher level of turbulence and mixing, so using of turbulence model is preferred. On the other hand, as a result of viscosity effect, the flow near the walls has low velocity and turbulence, so turbulence model is the best choice for simulating of flow field near the walls. We show later that is the best choice for simulating of the turbulence regime in a microscale vortex tube. The shearstress transport , developed by Menter [25], effectively and accurately combines the formulation of and models by using a blending function. The blending function activates model near the wall, and model in the central region of the vortex tube. This modification improves the flow prediction with strong adverse pressure gradients and separation[26]. In using of , and satisfy two following equations:
(8) 

(9) 
More details about terms of , , , , , , , and , can be found in the work of Cebeci [24].
3. Theoretical Background
In order to evaluate the performance of a microscale vortextube, some operational parameters should be calculated which are as follows:
Total temperature difference:
(10) 
Cold air temperature difference:
(11) 
Nondimensional Cold air temperature:
(12) 
Refrigeration Capacity:
(13) 
Isentropic Efficiency:
(14) 
Isentropic temperature difference:
(15) 
Cold mass ratio:
(16) 
4. Solution Procedures
In this study, the experimental report of Hamoudi et al. [20] was used to validate the numerical results. As shown in Fig. 1, to investigate the flow structure and energy separation phenomena, an axisymmetric two dimensional microscale vortex tube was simulated by using Fluent 6.3.26 software. The inlet nozzle was modelled as a continuous annular opening with a crosssectional area equal to the total area of four inlet nozzles of the vortex tube reported in the study of Hamoudi et al. [20]. Moreover, the diameter of the cold outlet is chosen so that its area is the same as the coldoutlet area of 3D RHVT. Table 1 shows the other dimensions of 2D geometry. The boundary conditions are as follows:
In fluent 6.3.26, governing equations of fluid flow are solved by finitevolume method. For convectiondiffusion formulation, PRESTO was used as a pressure interpolation scheme and secondorder upwind was used for others. The pressurevelocity coupling was also handled by



Fig. 1. Geometry of the microscale vortex tube used in 2D simulation 

Table 1. Dimensions of the microscale vortex tubes used for CFD modelling 



2D simulation 
20 
2 
0.55 
0.141 

using of SIMPLE algorithm (Semi Implicit Method for Pressure Linked Equations), described by Patankar [27, 28]. Air is also considered as an ideal gas with constant specific heat and variable viscosity and thermal conductivity. The solution is considered to be fully converged when the values of scaledresiduals from iteration to iteration are smaller than a prescribed value, 10^{−7} for energy equation and 10^{−5} for others.
Grid dependency tests have been done for all configurations investigated. The grid independency is attained when the percent changes of total temperature difference and tangential velocity are smaller than a given accuracy value 1%. Total number of nodal point is 14675. Figure 2 shows griddependency plots for 2D microscale vortex tube, while Fig. 3 shows typical grids used for the CFD simulation.
5. Results and Discussions
To find the best turbulence model to simulate the microscale vortex tube, the results of five RANS turbulence models, standard , , , and are compared with experimental results of Hamoudi et al. [20]. Table 2 shows deviations of these models from experimental results. Moreover, the prediction of cold and hot outlet temperatures by different turbulence models are shown in Fig. 4 and Fig. 5. It is concluded that and
Fig. 2. Grid dependency check for 2D microscale vortex tube 
Fig. 3. Typical twodimensional grid 
are the best choices for prediction of cold outlet temperature and optimum coldmass ratio. On the other hand, in spite of very good behaviour of and , and have also good agreement with experimental results in prediction of hotoutlet temperature. As a result, in this study, is chosen for CFDsimulation of flow behaviours in a microscale vortex tube.
Variations of nondimensional cold temperature, refrigeration capacity and isentropic efficiency versus cold mass ratio for different inlet pressures and nondimensional tube length ( ) are shown in Figs. 611. It can be concluded that the numerical simulation has a reasonable agreement with experimental results. However, CFD simulation overpredicts the values of nondimensional cold temperature and refrigeration capacity. Moreover, the results show that optimum. values of , refrigeration capacity and isentropic efficiency are achieved at , and , respectively. These results are in accordance with experimental results of Skye et al. [7], and Valipour and Niazi [29] for a typical vortextube
Table 2. Deviations of different turbulence models from experimental results 

Model 
Deviation from experimental results 

ColdOutlet Temperature 
HotOutlet Temperature 
Optimum cold mass ratio 

standard 
3.2% 
8% 
2% 
2.46% 
1.41% 
55% 

2.16% 
1.76% 
55.5% 

1.86% 
4.4% 
0.9% 

1.84% 
4% 
0.5% 
Fig. 4. Comparison between different turbulence models in prediction of coldoutlet temperature 
Figures 12 13 show the variation of nondimensional cold temperature difference, , and refrigeration capacity, , versus cold mass fraction for different values of nondimensional length and inlet pressure. It is observed that they are independent of geometry and inlet pressure (for both experimental and CFD results) and they are only a function of cold mass ratio.
Figures 1415 show flow streamlines and contour of static temperature for cold mass ratios of 0.37 and 0.05, respectively. It is concluded that in a microscale RHVT, the back flow area for low values of cold mass flow rates is similar to that of conventional vortex tube reported by Skye et al. [7].
The numerical and experimental results of Figs. 1213 can be correlated as polynomials by the best fit of data as follows:
(17) 

(18) 

(19) 

(20) 
Variation of tangential velocity along radialdirection for different cross sections of the micro
Fig. 5. Comparison between different turbulence models in prediction of hotoutlet temperature 
Fig. 6. Variation of nondimensional cold temperature versus different cold mass ratio, 
Fig. 7. Variation of nondimensional cold temperature versus different cold mass ratio, 
Fig. 8. Variation of Refrigeration Capacity versus different cold mass ratio, 
Fig. 9. Variation of Refrigeration Capacity versus different cold mass ratio, 
Fig. 10 – Variation of isentropic efficiency versus different cold mass ratio,

Fig. 11. Variation of isentropic efficienc yversus different cold mass ratio, 
Fig. 12. Nondimensional cold temperature difference versus cold mass ratio 
Fig. 13. Nondimensional Refrigeration Capacity versus cold mass ratio 
scale vortex tube is shown in Fig. 16. It is concluded that tangential velocity is maximum in the vicinity of inletzone. When the fluid moves helically toward the hot exit, the tangential velocity decreases alongside the vortex tube as a result of wall friction and friction between the fluid layers [10].
Variation of axial velocity along radial direction is shown in Fig. 17. The axial velocity is zero near , which is the separatingline between hot





Fig. 14. Streamlines with back flow region and the distribution of static temperature, 




Fig. 15. Streamlines with back flow region and the distribution of static temperature, 




Fig. 16. Variation of tangential velocity at different cross sections of micro RHVT, 




Fig. 17. Variation of axial velocity at different cross sections of micro RHVT, 


and cold streams. For the values of , the direction of flow is toward the hotoutlet, and for the values of , its direction is toward the coldoutlet.
Distribution of static pressure along radial direction for different crosssections of the tube is shown in Fig. 18. The results show that for values of , the static pressure increases by moving from inlet nozzles toward hotoutlet. This means that there exists a flowing stream from hotoutlet to coldoutlet in the central part of the tube. However, for the values of , the static pressure decreases along axial direction and the direction of flow is toward hotoutlet. Moreover, it is observed that the static pressure is constant on the separatingline between hot and cold streams.
Fig. 19 shows streamlines inside the microscale RHVT. It is indicated that there exist a free vortex near the wall and forced vortex in the central region.
Distribution of static temperature as a function of radius, along the microscale vortex tube is shown in Fig. 20. It is concluded that the static temperature increases from inlet section toward hot
Fig. 18. Variation of static pressure at different cross sections of micro RHVT, 
Fig. 19 – Streamlines of flow inside the microscale vortex tube, 
exit as a result of decreasing of tangential velocity due to friction.
Moreover, it is observed that, except the entrance region, the static temperature is radially constant in the centralzone of the microscale vortex tube. Contours of total temperature are shown in Fig. 21. The separation of energy inside the microscale RHVT in radial and axial direction can be seen in this figure.
6. Conclusion
In this study, energy separation phenomenon inside a microscale vortex tube was investigated by using of computational fluid dynamic. For this purpose, a 2D axisymmetric model in Fluent 6.3.26 software has been used and its results have been compared with the experimental results reported in the literature. The main results obtained may be summarized as follows:
Fig. 20. Radial distribution of static temperature at different cross sections, 
Fig. 21. Contour of total temperature (K) inside the microscale vortex tube, 
the mechanism of energy separation and flow field are similar to those of conventional vortex tube reported in the literature.
Acknowledgments
This work was supported by the Office of the Vice Chancellor for Research, Islamic Azad University, Semnan Branch, with Grant No.1108  21/05/1389. The authors would like to express their grateful thanks to Islamic Azad University, Semnan Branch, for providing information, experimental facilities and their close cooperation.
Nomenclature 

Area, m^{2} 

Specific heat at constant Pressure, Jkg^{1}K^{1} 

, 
Diameter,m 
Nondimensional orifice diameter, (dc/D) 

E 
Total energy,Jkg^{1} 
h 
Mass average enthalpy, Jkg^{1} 
k 
Turbulence kinetic energy, m^{2}s^{2} 
K 
Thermal conductivity,Wm^{1}K^{1} 
Kn 
Knudsen number 
L 
Length, m 
L^{*} 
Nondimensional tube length to the diameter ratio, (L/D) 
M 
Mach number 
Mass flow rate, kgs^{1} 

P 
Pressure, Pa 
Pr 
Prandtle number 
Refrigeration capacity,W 

r 
Change in tube radius along y, m 
r^{*} 
Nondimensional radius (r/R) 
R 
Specific constant of an ideal gas, Jkg^{1}K^{1} 
Re 
Reynolds number 
T 
Temperature, K 
Mass averaged velocity, ms^{1} 

Fluctuating velocity component, ms^{1} 

x 
Axial distance from cold exit, m 
y 
Radial distance from tube axis, m 
y_{c} 
Cold mass fraction 
Greek symbols 

Kronecker delta 

Specific heat ratio, c_{p}/c_{v} 

Viscosity, N.s.m^{2} 

Eddy viscosity, N.s.m^{2} 

Kinematics viscosity, m^{2}.s^{1} 

Density,kg.m^{3} 

Stress tensor 

Specific dissipation rate 

Isentropic efficiency 

Stress tensor 

Subscripts 

a 
Atmospheric 
c 
Cold exit 
cs 
Isentropic 
eff 
Effective 
h 
Hot exit 
in 
Inlet 
is 
Isentropic 
I, j, k 
Cartesian indices 
t 
Turbulent 
o 
Overall 
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