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In this numerical study, performance of Ranque-Hilsch vortex tubes, with a length to diameter ratios of 8, 9.3, 10.5, 20.2, 30.7, and 35 with six straight nozzles, on the basis of available experimental results were investigated. Also, this study has been done to understanding the physical behavior of the flow field in the vortex tube. CFD analysis is employed to achieve the highest temperature separation and optimum length to diameter (L/D) ratio of the Ranque-Hilsch vortex tubes. The temperature separation phenomenon in the vortex tube has been obtained by a 3D compressible turbulent CFD model. Also it was found that the best performance was obtained when the ratio of vortex tube's length to the diameter was 9.3. Moreover, it is found that increasing the cold mass fraction decreases the cold temperature difference and efficiency. Finally the computed results such as velocity and temperature variations are presented and discussed in more details. Presented results in this paper shown good agreement with experimental results.
Ranque-Hilsch vortex tube is a device with a simple geometry and without any intricacy, which can produce temperature separation. Generally it consists of nozzle, vortex chamber, separating cold plate, hot valve, hot and cold exit without any moving parts. When a vortex tube is injected with compressed air through some tangential nozzles into its vortex chamber, a strong rotational flow field is established. This vortex in the inlet area causes pressure distribution of the flow in radial direction. As a result a free vortex is produced as the peripheral warm stream and a forced vortex as the inner cold stream. Swirled flow near the wall of the tube tends to have higher velocity compared to those in the central region of the tube. After energy separation in the vortex tube, the inlet air stream was separated into two air streams: hot air stream and cold air stream, the hot air stream left the tube from one end and the cold air stream left from another end. Fig. 1 shows the schematic diagram of a vortex tube and its flow pattern.
Fig 1. Flow pattern and schematic diagram of vortex tube
But vortex tube history goes back to early in the twentieth century. In 1931, a French physics student George Ranque  occasionally found the phenomenon of energy separation in the vortex tube when he was studying processes in a dust separation cyclone. He noticed that the warm air would be drawn from one end, and the cold air from the other. Later it was discovered that the mechanism is closely related to the swirling flow of the air within the tube. In 1945, Rudolph Hilsch  published his systemic experimental results on the thermal performances of vortex tubes with different geometrical parameters and under different inlet pressures.
Since then, the vortex tube has been a subject of much interest. In the following years, many experimental studies and CFD investigations have been carried out in which attempts were concentrated on explaining the mechanism of energy separation in the vortex tube. Harnett and Eckert  invoked turbulent eddies, Ahlborn and Gordon  described an embedded secondary circulation and Stephan et al. proposed the formation of Gortler vortices on the inside wall of the vortex tube that drive the fluid motion. Kurosaka  reported the temperature separation to be a result of acoustic streaming effect that transfer energy from the cold core to the hot outer annulus. Despite all the proposed theories, none has been able to explain the temperature separation effect satisfactorily. Aljuwayhel et al.  utilized a fluid dynamics model of the vortex tube to understand the process that drives the temperature separation phenomena. Skye et al.  used a Model similar to that of Aljuwayhel et al .. In recent years, many numerical investigation has been carried out to simulate the flow field and energy separation[11,12]and. Volkan kirmaci  used Taguchi method to optimize the number of nozzle of vortex tube.While each of these explanations may capture certain aspects of vortex tube, none of these mechanisms altogether explains the vortex tube effect.Vortex tubes generally are used as a cooling system for Industrial purposes.
2. Governing Equation
The compressible turbulent flows in the vortex tube are governed by the conservation of mass, momentum and energy equations. The mass and momentum conservation and the state equation are solved as follows:
Flow in the vortex tube is highly turbulent. The steady state assumption and practical considerations indicate that a turbulence model must be employed to represent its effects. The turbulence kinetic energy, k and its rate of dissipation, ε is obtained from the following transport equations:
In these equations, Gk represents the generation of turbulence kinetic energy due to the mean velocity gradients, Gb is the generation of turbulence kinetic energy due to buoyancy, YM represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, C1ε, C2ε and C3ε are constants. σk and σε are the turbulent prandtl numbers for k and ε, respectively. The turbulent (or eddy) viscosity, μt is computed by combining k and ε as follows:
Where, Cμ is a constant.
4. CFD model
In this numerical investigation the FLUENT software package was used to create the CFD model of the vortex tube. The models are three-dimensional steady state, ax symmetric, and employ the standard k-epsilon turbulence model. Since the nozzle consists of 6 straight slots, the CFD model assumed to be a rotational periodic flow and only a sector of the flow domain with angle of 60°, needs to be considered which is shown in Fig2(b). The three-dimension model showing boundary regions is shown in Fig. 2(a) and (b).
Fig 2. (a)Three-dimensional model of vortex tube with six straight nozzles provided with refinement in mesh
(b) A part of sector that taken for analysis showing computational domain
A compressible form of the Navier-Stokes equation together with the standard k-ε model by second order upwind for momentum and turbulence equations and the quick numerical schemes for energy equation has been used to simulate the phenomenon of flow pattern and temperature separation in a vortex tube .The pressure and temperature data obtained from the experiments are supplied as input for the analysis. Boundary conditions for the model were determined based on the experimental measurements by Skye et al  for all case in this analysis. The inlet is modeled as a mass flow inlet; the total mass flow rate, stagnation temperature, were specified and fixed at 8.35 g sec−1, 294.2 K respectively. Also the backflow temperature value is set to 290 k. The static pressure at the cold exit boundary was fixed at experimental measurements pressure. The static pressure at the hot exit boundary was adjusted to vary the cold fraction. A no-slip boundary condition is enforced on all walls of the vortex tube.It is noteworthy that ,an ExairTM 708 slpm (25 scfm) vortex tube was used by Skye et al  to collect all of the experimental data .
In this numerical simulation the radius of the vortex tubes fixed at 5.7 mm for all models, and the length are set to 92, 106, 120, 230, 350, and 400 mm respectively. Thus, performance of vortex tubes, with a length to diameter ratio of 8, 9.3, 10.5, 20.2, 30.7, and 35, with 6 straight nozzles were investigated. The height of each slot is 0.97 mm and the width is 1.41mm. The cold and hot exits are axial orifices with areas of 30.2 (cold end diameter dc = 6.2 mm) and95 mm2, respectively. So following analysis is made for specific vortex tube with six numbers of straight nozzles and various L/D ratios.
5.Results and discussion
The analysis has been done to achieve the optimum length to diameter ratio (L/D), between six various lengths of vortex tubes. It was deduced that the best performance was obtained when the length to diameter ratio was 9.3(L=106 mm), which, the same model was investigated by skye et al. . Because of maximum temperature drop at cold exit of vortex tube achieved at cold mass fraction of 0.288 (Fig. 3), that obtained by experiments of skye et al. , and because of vortex tube mostly provided for cooling applications, and present investigation concentrated on cooling performance of RHVT, so further investigations were carried out for cold mass fraction equal to 0.288(ξ=0.288). Also it must be said that the skye et al.  CFD model was developed using a two-dimensional, steady axisymmetric model (with swirl), and the present CFD models are three-dimensional. The fraction for cold gas ξ was defined as the ratio of the mass flow rate of the cold stream to the mass flow rate of the inlet stream:
ξ took different values between 0 and 1, when the flow was controlled by the valve on the hot exhaust line. Performance was defined as the difference between the temperature of the hot stream and the temperature of the cold stream, âˆ†Tch= (Th-Tc) .Cold temperature difference or temperature separation is defined as the difference in temperature between inlet flow temperature and cold flow temperature:
Where Tin is the inlet flow temperature and Tc is the cold flow temperature. Similarly hot temperature difference is defined as:
Fig 3. Comparison of cold exit temperature Fig 4. Comparison of hot exit temperature
difference as a function of cold mass difference as a function of cold mass
fraction between optimum length of this fraction between optimum length of this
calculation and skye et al calculation calculation and skye et al calculation
and experiments. and experiments.
6.Length to diameter ratio
The investigators who studying vortex tube, have suggested different values for length to diameter ratios, but should be said that the L/D ratios, can be different for each specific project .In the present study the tube with L = 106 mm presented the optimum results for the highest possible temperatures (L/D =9.3).The obtained temperature separation at present calculations for optimum length of vortex tube (L=106 mm), were compared with the experimental and computational results of skye et al. ,that both models have similar geometry, inlet and boundary conditions. As shown in Fig. 4 the hot exit temperature difference, ΔTh,i ,predicted by the our model is in good agreement with the experimental results. Prediction of the cold exit temperature difference ΔTi,c for optimum model of present study is found to lie between the experimental and computational result of skye et al.  that is shown in
Fig. 5 Experimentally measured and CFD model predictions of hot and cold exit static pressure as a function of the cold fraction
Fig. 3. Compared to the present calculations k-ε model predictions with computational results of skye et al. , clearly observed that the hot exit temperature difference ΔTh,i simulated at both models were close to the experimental results. Though both models get values less than experimentally results of cold exit temperature difference ΔTi,c, but the predictions from the present model were found closer to experimental results.
In the CFD model, the cold exit pressure boundary condition was specified at the measured cold exit pressure and the hot exit pressure was iteratively specified until the experimentally measured cold fraction was achieved. As shown in Fig. 5, the present model values generally are much higher than the experimentally predicted at the hot exit pressure required for a given cold fraction, however, the general trend agrees well.
The magnitude of swirl velocity is one of most important factors that affect performance of vortex tube. So, increasing the swirl velocity will increase the performance of vortex tube. Comparing the swirl generation of different vortex tube lengths it is observed that swirl generation of vortex tube with L/D=9.3(L=106mm) has the highest value. The optimum length of present calculation(L=106mm) can produce maximum swirl generation of 428 m/s at inlet zone, and hot gas temperature of 363.2 K at 0.8 of cold mass fraction, and a minimum cold gas temperature of 250.24 K at about 0.288 cold mass fraction. The results of analysis for all vortex tube lengths that were investigated, are given in Table 1.
Fig. 6 Radial profiles of axial velocity at different axial Fig. 7 Radial profile of swirl velocity at different axial
locations: (a)z/l=0.1 (b)z/l=0.4 (c)z/l=0.7 locations: (a)z/l=0.1 (b)z/l=0.4 (c)z/l=0.7
The radial profiles of velocity components axial, and swirl (tangential) are shown in Fig. 6and7. Because of the effects of wall friction, the speed of fluid near the tube wall is lower than the speed at the center of tube. Fig. 6 shows the radial profiles of the axial velocity at different axial locations (z/l = 0.1, 0.4 and 0.7) at specified cold mass fraction equal to 0.288(ξ=0.288) for various length of vortex tubes that were investigated. Also Fig. 6 shows the highest value of axial velocity belongs to models with L=92and106mm. Furthermore, Fig. 6 shows the highest value of axial velocity, near of tube wall, and also in the core of tube and in all axial sections belongs to models with L=92 and 106 mm. Fig. 6 shows ,near of cold end exit (z/l=0.1), maximum axial velocity is in the core of tube, but in near the hot end exit, axial velocity near the wall of tube is maximum .As shown in Fig. 6 the variations of axial velocity shows the direction of flow near the wall of tube is towards the hot end, and the direction of flow in core of tube is towards the cold end. Also it was observed that the maximum value of the axial velocity decreased with increasing axial distance from the inlet zone.
For optimum vortex tube (L=106mm) at axial locations of z/l=0.1, 0.4 and 0.7 the maximum axial velocity was found 83, 63 and 57 m s−1 respectively. The axial velocity profiles (Fig. 6) show that the flow reversal takes place at less radial distance from center of the tube, at z/l=0.7 compared to the z/l=0.4 and z/l=0.1. It means that with increasing distance from the cold end exit, the flow reversal takes place at less radial distance from the center of tube. Also it was observed that the axial velocity in the tube core after stagnation point is directed towards the cold end exit. The axial velocity in the cold core was found to increase with a decrease in the axial distance from inlet zone.
Fig. 7 shows the radial profiles for the swirl velocity (tangential velocity) at different axial locations (z/l=0.1, 0.4 and 0.7). Comparing the velocity components, it is observed that swirl velocity has the highest value. It has a value almost equal to inlet tangential flow in the nozzle inlet zone, which rapidly decreases in amplitude towards the hot end discharge. The radial profile of the swirl velocity indicates a free vortex near the wall and the values become negligibly small at the core, which is in conformity with the observations of Kurosaka , Gutsol . Also the axial and swirl velocity profiles obtained at different axial locations of the vortex tube are in good conformity with observations of Gutsol  and Behera . According to variations of the swirl velocity which is shown in Fig. 7,
Fig. 8 Radial profiles of total temperature at different axial locations:(a) z/l=0.1 (b) z/l=0.4 (c) z/l=0.7
Table 1 The results of CFD analysis for all vortex tubes lengths that investigated
1L/D: length to diameter ratio 5Thm: Maximum temperature at hot end
2L: Length of tube 6âˆ†T i,c: Temperature separation between inlet and cold end
3Vsm: Maximum generated swirl velocity 7âˆ†T h,i: Temperature separation between inlet and hot end
4Tcm: Minimum temperature at cold end 8âˆ†Tch: Temperature separation between cold and hot end
in near of the inlet zone (z/l=0.1), compared to other models the highest swirl velocity belongs to model with L=106mm, but with increasing the distance from inlet zone towards the hot end, the swirl velocity magnitude decrease in all models, so that, in the models which is longer than the others, the magnitude of swirl velocity along
the tube, have more dropping. As shown in Fig. 7c in near of the tube end (z/l=0.7) the maximum and minimum swirl velocity belongs to models with L=92 mm and L= 400 mm respectively.
Radial profiles of the total temperature at various axial location (z/l = 0.1, 0.4 and 0.7) for different length of vortex tube are presented in Fig. 8. The maximum total temperature was observed to exist near the periphery of the tube wall. At the tube wall the total temperature is found to decrease, this is due to the no slip boundary condition at
the tube wall. The predicted temperature profiles are a result of the kinetic energy distribution in the vortex tube. The fluid at the core of the vortex tube has very low kinetic energy due to the minimum swirl fluid velocity at the central zone of the tube. From the swirl velocity profiles Fig. 7 it was observed that the swirl velocity had almost negligible value at the core of the vortex tube. Thus, the swirl velocity being the major component. Comparing the total temperature and the swirl velocity profiles (Fig. 7and 8) show that the low temperature zone in the core coincides with the negligible swirl velocity zone. The total temperature profiles (Fig. 8) shows an increase of the temperature values towards the periphery. The radial profiles of total temperature in Fig. 8 shows that the maximum temperature at the tube axis near the hot end belongs to vortex tube with length of 106mm and also, minimum temperature near the cold end belongs to L=106mm model that was investigated. Thus, the model with L=106mm shows minimum cold gas temperature at cold exit, and maximum hot gas temperature at hot exit.
Fig. 9 shows the CFD analysis data on temperature difference between hot and cold end (âˆ†Tch) for different L/D ratios. It can be noted that the peak value in âˆ†Tch is obtained for L/D ratio of 9.3(L=106mm) that investigated by skye et al.  and present study. The studies highlight that CFD has reasonable accuracy in predicting an optimum L/D ratio and is a suitable tool for the design and investigation of the vortex tube.
Fig. 9 Temperature difference between hot Fig. 10 Temperature difference at cold exit
and cold gas for different L/D ratios for different lengths of vortex tubes
Fig. 11 Temperature distribution in axial direction of optimum vortex tube in sections
Fig. 10 shows the temperature difference at cold exit end for various lengths of vortex tube,which were studied . Comparing the various lengths of vortex tube it is observed that the model with length of 106 mm has the maximum temperature separation about 43.96 K, at cold exit.
The total temperature distribution from CFD analysis along the length of tube, for optimum length of vortex tube with length to diameter ratio of 9.3 (L=106mm) is displayed in Fig. 11. Clearly can be seen that peripheral flow is warm and core flow is cold, furthermore temperature growth is seen in the radial direction. The optimum length of this study, for a cold mass fraction of equal to 0.288, gives the maximum hot gas temperature of 311.5 K and minimum cold gas temperature of 250.24 K. This means that temperature separation at cold exit is: âˆ†Ti,c =43.96 K, which is shown in Fig. 10.
Fig. 12 3D Streamlines inside of vortex tube colored by total temperature
Fig.12 shows the streamlines in three dimensional space associated with the flow inside the vortex tube. The highly rotational flow pattern inside of the optimum geometry of vortex tube can be seen in Fig.12.
A numerical investigation is performed to examine the performance of six vortex tubes which have an inner diameter of 11.4 mm and L/D ratio of 8, 9.3, 10.5, 20.2, 30.7 and 35. The vortex tube with L/D=9.3 (L=106mm) presented optimum results, with the highest temperatures at hot exit and lowest in the cold exit equal to 311.5 and 250.25 K respectively.
Comparison of present numerical model and skye et al. experiments, the obtained total temperature separations in hot and cold exit, predicted by the present CFD analysis have good conformity with the experimental results of skye et al. . As a result of present study it can be said that the length of vortex tube is one of the most important factors that affects its performance. The results show, depending on operating factors, the optimum L/D ratio can be different, and optimum length of vortex tube is function of geometrical and operating parameters such as inlet pressure and flow rate, so that, according to inlet condition of present CFD analysis, increasing the length to diameter ratio of vortex tube beyond 9.3 has no effect on performance of vortex tube.
The results of the numerical simulation show that it is possible to get a temperature difference between hot and cold streams as high as 61.26 K for optimum length of this study (L=106 mm). Because of maximum cold exit temperature separation achieved at 0.288 of cold mass fraction, it can be said that, if cooling is desired then lower cold mass fraction is required. It was deduced that, in the optimum case of present study, for the cooling purpose, the cold mass fraction should be fixed at 0.288.