Effect Of Stagnation Point And Length Biology Essay


In this numerical study, performance of Ranque-Hilsch vortex tubes, with a length to diameter ratios (L/D) of 8, 9.3, 10.5, 20.2, 30.7, and 35 with six straight nozzles, on the basis of experimental results were investigated. Also, this study has been done to understanding the effects of stagnation point position on the performance of RHVT. It was found that the best performance was obtained when the ratio of vortex tube's length to the diameter was 9.3 and also at this model the stagnation point was found the farthest from the inlet. The results showed that the closer distance to the hot end will provide large amounts of temperature difference. Presented results in this paper shown good agreement with experimental results.


Ranque-Hilsch vortex tube is a device without any intricacy, which can produce temperature separation. When a vortex tube is injected with compressed air through tangential nozzles into its vortex chamber, a strong rotational flow field is established. Due to the wall friction, the velocity of gas near the tube wall is lower than the velocity at the tube center; as a result, gas in the center region transfers energy to the gas at the tube wall. 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(a) shows the schematic diagram of a vortex tube and its flow pattern. But, vortex tubes were discovered in 1932 by Ranque [1] and later Hilsch [2] performed the detailed examination of the Ranque effect. Since then, the vortex tube has been a subject of much interest. Harnett and Eckert [3] invoked turbulent eddies, Ahlborn and Gordon [4] described an embedded secondary circulation and Stephan et al. [5] proposed the formation of Gortler vortices on the inside wall of the vortex tube. Kurosaka [6] reported the temperature separation to be a result of acoustic streaming effect. Aljuwayhel et al. [7] utilized a fluid dynamics model of the vortex tube to study the flow behavior. Skye et al. [8] used a model similar to Aljuwayhel et al. [7]. Chang H Sohn et al .[12] conducted a experiment using surface tracing method to investigation the flow field and to indicate the stagnation position in a vortex tube.Ameri et al.[13] and Eisma et al.[14] performed a numerical study to research the temperature separation phenomenon.Volkan kirmaci [15] employed a different method to optimize the nozzles number .Recently some studies has been done to use vortex tube as a refrigeration system, instead of the conventional refrigeration systems [16],[17]. Vortex tubes generally are used as a cooling system for Industrial purposes.


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Diameter of vortex tube (mm)


Cold gas fraction


Turbulence kinetic energy (m2s-2)


Turbulence dissipation rate (m2s-3)


Length of vortex tube (mm)


Dynamic viscosity (kg m-1s-1)


Mass flow rate (g s-1)


Turbulent viscosity (kg m-1s-1)


Radial distance from axis(mm)


Density (kg m-3)


Temperature (K)


Stress (N m-2)


Axial length from nozzle cross section


Shear stress (N m-2)


Temperature difference between cold and hot end(K)


Stress tensor components


Temperature difference between inlet and cold end(K)



Temperature difference between hot end and inlet(K)


Cold gas


Hot gas

Greek Symbols


Inlet in vortex tube

Fig.1.(a) Flow pattern and schematic diagram of vortex tube (b) Schematic of an ExairTM 708 slpm vortex tube

2. Numerical modeling

The numerical simulation of the vortex tube has been created by using the FLUENTTM software package. The models are three dimensional, steady state, compressible and employ the standard k-epsilon turbulence model. 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:


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2.1 Turbulence model

The flow field inside the vortex tube is fully turbulent. Thus the turbulence kinetic energy, k and its rate of dissipation, ε are obtained from the following transport equations:



Where, 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 coefficients. σk and σε are the turbulent prandtl numbers for k and ε, respectively. The turbulent viscosity, μt is computed by combining k and ε as follows:


Where, Cμ is a constant. The model constants C1ε, C2ε, Cμ, σk and σε have the following default values: C1ε = 1.44, C2ε = 1.92, Cμ = 0.09, σk = 1.0, σε = 1.3.

3. Physical modeling

The present CFD model was created based on the model which used by Skye et al [8]. It is noteworthy that, an ExairTM 708 slpm vortex tube (The geometry summary is given in Table 1) was used by Skye et al [8] to collect all of the experimental data which is shown in Fig.1 (b). In addition to the model which was used by Skye et al [8], to study the effects of length on the performance of vortex tube, all geometrical properties of the Skye et al [8] model are kept constant, and only with changing the length of the model, five other models with different lengths and six numbers of straight nozzles were created. The radius of the vortex tubes fixed at 5.7 mm, and the length are set to 92, 106, 120, 230, 350, and 400 mm respectively. 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

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

considered which is shown in Fig2(b). The three-dimension model showing boundary regions is shown in Fig. 2(a) and (b) :

3.1.Boundary conditions for analysis

Boundary conditions were determined based on the experimental measurements by Skye et al [8] 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. 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.:

4. Results and Discussion

CFD analyses were carried out for a 11.4 mm diameter vortex tube with L/D of 8, 9.3, 10.5, 20.2, 30.7, and 35 to achieve the optimum length to diameter ratio (L/D).The results showed that, the best performance is obtained when the length to diameter ratio is 9.3(L=106 mm). Notice that this is the same model that was used by skye et al.[8].The obtained results for optimum case (106mm) were compared and validated with experimental and numerical results of skye et al. [8] (Fig.3 and 4).

4.1. Effect of Length to diameter ratio

As shown in Fig.3 and 4, the obtained temperature difference at present analysis for the optimum vortex tube (L=106 mm), were compared with the experimental and computational results of skye et al. [8], that both models have similar geometry and boundary conditions. The skye et al. [8] CFD model was developed using a two-dimensional, and the present CFD models are three dimensional .As shown in Fig. 4 the ΔT h,i ,predicted by the our model is in good agreement with the experimental results. Prediction of the ΔTi,c is found to lie between the experimental and computational result of skye et al. [8] which is shown in Fig. 3. The obtained ΔTh,i simulated at both models were close to the experimental results. Though both models get values less than experimentally results of ΔTi,c, but the predictions from the present model were found closer to experimental results. Fig.3 shows the

Table 1.Geometry summary of the vortex tube which was used in experiment (Model :ExairTM 708 slpm)



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Working tube length

106 mm

Nozzle height

0.97 mm

Nozzle width

1.41 mm

Nozzle total inlet area (An)

8.2 mm2

Cold exit diameter

6.2 mm

Cold exit area

30.3 mm2

Hot exit diameter

11 mm

Hot exit area

95 mm2

Fig. 3.Comparison of cold temperature difference Fig .4. Comparison of hot temperature difference

obtained at present CFD simulation and skye et al[8] obtained at present CFD simulation and skye et al[8]

simulation and experiments. simulation and experiments


maximum ΔTi,c is obtained at cold gas fraction of about 0.3 through the experiment and CFD simulation.The optimum vortex tube length (L=106mm) can produce hot gas temperature of 363.2 K at 0.8 of cold gas fraction, and a minimum cold gas temperature of 250.24 K at about 0.3 cold gas fraction. The results of analysis for all vortex tube lengths that were investigated, are given in Table 2.

Fig.5. Radial profiles of axial velocity at different axial locations for a=0.3: (a)z/l=0.1 (b)z/l=0.4 (c)z/l=0.7

Fig. 5 shows the radial profiles of the axial velocity at different axial locations (z/l = 0.1, 0.4 and 0.7) at specified cold gas fraction of 0.3 (a=0.3), for various length of vortex tubes. For the optimum vortex tube 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, therefore a maximum value of 83 m/s is seen at the tube axis near the inlet zone (z/l=0.1). Fig. 6 shows the radial profiles for the swirl velocity at z/l=0.1, 0.4 and0.7. Comparing the velocity components, it is observed that swirl velocity has the highest value. 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 [6], Gutsol [9]. Also the axial and swirl

Table 2.The results of CFD analysis for investigated tube lengths at a=0.3 (maximum cooling effect).


L (mm)




∆T i,c(K)

∆T h,i(K)


















































L/D: length to diameter ratio L: Length of tube

Tcm: Minimum temperature at cold end ∆Tch: Temperature separation between cold and hot end

∆Th,i: Temperature separation between inlet and hot end Thm: Maximum temperature at hot end

∆Ti,c: Temperature separation between inlet and cold end Vsm: Maximum generated swirl velocity

Fig.6. Radial profiles of swirl velocity at different Fig.7.Radial profiles of total temperature at different

Axial locations for a=0.3: (a) z/l=0.1 (b)z/l=0.4 axial locations for a=0.3(maximum cooling effect)

(c)z/l=0.7 :(a) z/l=0.1 (b) z/l=0.4 (c) z/l=0.7

Radial profiles of the total temperature for different length of vortex tube are presented in Fig.7. The maximum total temperature was observed to exist near the periphery of the tube wall in all vortex tubes. Comparing the total temperature and the swirl velocity profiles (Fig. 6and 7) show that the low temperature zone in the core coincides with the negligible swirl velocity zone. The total temperature profiles (Fig.7) shows an increase of the temperature values towards the periphery. As seen in Fig.7 (a) and(c), the model with L=106mm shows minimum cold gas temperature at cold exit, and maximum hot gas temperature at hot exit.

The total temperature distribution for the optimum vortex tube (L=106mm) is displayed in Fig. 8. Clearly can be seen that peripheral flow is warm and core flow is cold, furthermore increasing of temperature is seen in radial direction. The optimum length of this study, for a cold gas fraction of about 0.3, gives the maximum hot gas temperature of 311.5 K and minimum cold gas temperature of 250.24 K . Also, the three dimensional path lines of flow inside the vortex tube and rotational flow pattern is presented in Fig.9.

Fig.8.Contours of total temperature at: Fig.9.Streamlines inside of vortex tube colored

cold exit, z/l=0.1,z/l=0.4,z/l=0.7, hot exit . by total temperature

Fig. 10 shows the CFD analysis data on temperature difference between hot and cold end (∆Tch) for different L/D ratios. As seen that the peak value in ∆Tch is obtained for L/D ratio of 9.3(L=106mm) that investigated byskye et al. [8] experimentally and present numerical study. Fig. 11 shows the temperature difference at cold exit end for various lengths of vortex tube,which were investigated . 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.

Fig. 10 Temperature difference between hot Fig. 11.Temperature difference at cold exit

and cold gas for different L/D ratios for different lengths of vortex tubes

4.2 Effect of Stagnation Point

The results of present study shows that the performance of vortex tubes related to stagnation point location and also, study on the length effect, required to exploration of the stagnation point location along the tube to achieve the highest energy separation

The stagnation point position within the vortex tube can be determined by two ways: on the basis of velocity profile along the tube length at the point where axial velocity cease to have a negative value and according to maximum wall temperature, which this point represents the stagnation point determined by Fulton [18]. Also it was assumed that the wall temperature was representative of the gas temperature that reported by Frohlingsdorf [19].

Fig.12. Streamlines of optimum vortex tube in r-z plane. Fig.13. Schematic flow pattern of vortex tube

Fig. 12 shows the stagnation point and streamlines in the r-z plane associated with the flow inside the optimum vortex tube. Notice that in the CFD model the majority of energy separation occurs before stagnation point as reported by Aljuwayhel et al. [7]. Fig. 13 shows the flow within the vortex tube has forced and free vortex region up to stagnation point .The forced vortex usually is seen near the inlet region and the free vortex is seen near the wall, which is in accordance with the observations of Kurosaka [6], Gutsol [9]. As a result a free vortex is produced as the peripheral hot stream and a forced vortex as the inner cold stream.

Fig.15.Variation of axial velocity along the center line of the vortex tube :( a) L=92mm (b) L=106mm (c) L=120mm (d) L=230mm (e) L=350mm (f) L=400mm

The variations of axial velocity along the center line of the vortex tube are shown in Fig.15, where the Z/L represented the dimensionless length of vortex tube. As seen in Fig.15 at an axial distance between cold and hot end the velocity magnitude comes to zero, which this point shows the stagnation point position. According to Fig.15 it can be seen that in the vortex tubes with lengths of 230,350and400mm the point that axial velocity comes to zero (stagnation point) is further from hot end compared to vortex tube with L =92,106and120 mm. Fig.15 shows that the nearest stagnation point to the hot end belongs to vortex tube with L=106 mm(optimum case of present study).The results of analysis which are given in table.3 shows that the nearest stagnation point to the hot end gives the highest temperature difference which is an important point to get the best performance of vortex tubes. Also it can be seen that with increasing the length of vortex tube the stagnation point moves towards the cold exit end

The variations of total temperature along the wall of the vortex tube for different lengths of vortex tube are given in Fig.16. The point of maximum temperature in Fig .16 represents the stagnation point that described by Fulton et al [18]. As seen from the Fig.16 the point of maximum wall temperature (stagnation point) of vortex tubes with lengths of 92,106and120 mm is closer to the hot end compared to three other lengths of vortex tube, and also notice that the obtained temperature difference in these models is greater than the other models. The results of present

Table 3. Stagnation point position along the vortex tube length for various length of tube.











































L/D: length to diameter ratio Lsc: Stagnation point Location from cold end

Dsh: Distance between hot end and stagnation point (Z/L): Dimensionless tube length which is measured from the cold exit end

study about the position of stagnation point and its influence on the performance of vortex tube clearly confirms observation of Behera et al.[10]

The obtained results by two methods to determine the position of the stagnation point are given and compared in table .4. As seen in table.4 the average differnce between obtained values of these two methods is about 14.8%. Comparison of two methods shows the obtained results by both methods presents similar result about the effects of stagnation position, so that, in the both methods, the farthest stagnation point from the inlet, presents the highest temperature difference. The optimum model of this study (vortex tube with L=106 mm) has the farthest stagnation point from the inlet at both methods.

Fig.16.The variations of wall temperature along the vortex tube length: (a) L=92 mm (b) L=106 mm (c) L=120mm (d) L=230 mm (e) L=350 mm (f) L=400 mm

5. Conclusion

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 results showed that, the best performance is obtained when the length to diameter ratio is 9.3(L=106 mm). Comparison of present numerical model and skye et al.[8] experiments, shows the obtained temperature difference at hot and cold exit, predicted by the present CFD analysis is in good agreement with the experimental results of skye et al. [8], and is closer to the experimental results compared to the skye et al.[8] CFD model. The results showed that increasing the length to diameter ratio beyond 9.3 has no effect on the performance of vortex tube. By the optimum length of this research it is possible to get a temperature difference between hot and cold streams as high as 61.26 K. The maximum cold temperature difference was obtained at 0.3 of cold gas fraction. It was concluded that to use vortex tube as a cooling system lower cold gas fraction is required.

The analysis showed that the temperature difference between hot and cold gas flow can be improved by increasing the length of vortex tube such that stagnation point is farthest from the nozzle inlet and within the tube. In the optimum case of present study (vortex tube with length of 106mm) the stagnation point was found the farthest from the inlet. It is observed that in the long length of vortex tubes the stagnation point is far from the hot end and this affects the vortex tubes performance negatively.

Also both investigated methods to determine the stagnation point position, showed similar and reasonable results. So, to attain the large amounts of temperature separation in a vortex tube we recommended that the stagnation point must be in a minimum distance from the hot outlet.

Table 4.Comparison of the stagnation point location using two methods











































1L: Length of vortex tube 2L/D: length to diameter ratio 3T wmax: Maximum wall temperature

4(Z/L) w: Dimensionless distance of stagnation point based on measurement of wall temperature which is measured from the cold exit.

5(Z/L) v: Dimensionless distance of stagnation point based on variation of axial velocity which is measured from the cold exit

6Diff (%): Difference between two measurement methods of stagnation point location