# A computational and experimental study of thermal energy separation by swirl

Abstract: When compressed air is introduced into a tube in such a way as to generate a strong axial vortex, an interesting phenomenon is observed wherein the fluid temperature at the vortex core drops below the inlet value, while in the outer part of the vortex, the temperature is higher than at inlet. The most familiar manifestation of this phenomenon is known as the Ranque-Hilsch effect, and several alternative explanations for it have been proposed. In this study, we present an analysis of the heat transfer mechanism underlying this phenomenon, based on consideration of the exact equation governing the conservation of the turbulent heat fluxes. The outcome is a model that explicitly accounts for the dependence of the heat fluxes on the mean rates of strain, and on the gradients of mean pressure. These dependencies, which are absent from conventional closures, are required by the exact equation. To verify the model, an experimental investigation of flow in a swirl chamber was conducted, and the measurements were used to check the model’s performance as obtained by three-dimensional numerical simulations. Comparisons between predictions and measurements demonstrate that the new model yields predictions that are distinctly better than those obtained using conventional closures.

## Summary (2 min read)

### 1. Introduction

- Vortices that influence the local temperature distribution are frequently encountered in nature and in engineering practice.
- When flow occurred in a vortex tube with an open outlet, measurements showed that the temperature in the vortex core was lowered, vortex breakdown occurred and pressure fluctuations with descent frequencies.
- Thus, while the phenomenon is clearly evident when the working fluid is air [8–10], the situation is far less clear when the working fluid is water.
- From consideration of the equation governing the conservation of energy in a rotating fluid under adiabatic conditions, they derive an expression for the total temperature that shows this quantity to depend on both the axial and angular velocities and hence vary in the radial direction leading to temperature separation.
- Taken together, these results strongly suggest that the simple model for the turbulent heat flux is not adequate in this case.

### 2. Analysis and model development

- The exact equations that govern the conservation of the turbulent heat fluxes in compressible flows are obtained from the Navier-Stokes and energy equations by replacing the instantaneous variables by the sum of mean and fluctuating parts, and by time-averaging after some manipulation.
- When the mean pressure gradient is finite, the following functional relationship is obtained: uit ¼ f i uiuj; @Ui @xj ; @T @xj ; @P @xj ð3Þ Smith [18] gives the general representation for uit, a first-order tensor, in terms of the first- and second-order tensors in the functional relationship of Eq. (3).
- When the heat transport is accomplished by fluid particles that are moving along a pressure gradient and work is done, they can maximally change their temperature according to an isentropic change of state.
- This results in the same value for C5 as the one given above.

### 3.1. Geometry

- The outlet from the tube is open and the pressure there is atmospheric.
- For the flow inside a Ranque-Hilsch tube, the axial flow is bi-directional in the sense that the flows in the central core and the periphery move in opposite directions and thus the swirl number as defined in Eq. (10) would not be an appropriate indicator of the strength of swirl at a given streamwise section.
- At inlet to the swirl chamber, however, the axial flow is uniformly directed across the entire section and hence the swirl number as defined in Eq. (10) is a meaningful indicator of the strength of swirl at that location.
- This being the case, the inlet swirl number in the experiments is obtained as SI ¼ 5:30.

### 3.2. Instrumentation

- A schematic representation of the test rig used for the present experiments is shown in Fig.
- From the mass flow element, the flow passed through a plenum chamber followed by a honeycomb flow straightener before entering the swirl chamber via the tangential slots.
- Temperature TW was measured using ten thermocouples that were placed directly below the surface at different axial positions.
- The air flow was seeded with small oil droplets with a diameter of about 0:25 lm.
- Each time an image was captured with a camera orientated perpendicularly to the laser sheet.

### 4.1. Computational details

- The computations were performed using the compressible flow form of the Ansys CFX (v. 11sp1) software in which the governing equations are discretized by second-order accurate finite-volume methodology.
- Implementation of the latter into the computations software was fairly straightforward and was accomplished via user defined subroutines.
- For comparison, a model with a constant turbulent Prandtl number Prt ¼ 0:9 was also used.
- This was done to ensure that the computations accurately captured the steep temperature gradients that occurred there.
- The refinement factor for the thickness of the grid cells from the wall is 1:20.

### 4.2. Comparisons with measurements

- The computed and measured cross-stream profiles of the axial component of velocity are compared in Fig.
- The computed and measured circumferential velocity at four streamwise locations along the vortex chamber are compared in Fig.
- It is thus the case that the low velocity within the axial backflow in this region was also subject to high experimental uncertainty.
- Further downstream, the degree of temperature separation is reduced as the swirl weakens and with it the radial gradients of static pressure.
- The result correlates well with the numerically calculated static temperature distribution.

### 5. Conclusions

- The results presented in this paper demonstrate the importance of accounting for the effects of pressure gradients in the prediction of swirl-induced thermal energy separation.
- An algebraic model for the turbulent heat fluxes was thus developed to explicitly include the pressure-gradient effects.
- It was found that at the entry region to the chamber, where the swirl effects aremost pronounced, the predictions obtainedwith the newmodel matched quite closely the experimental results to within the estimated accuracy in the latter.
- It should be noted that the temperature variations in the experiment were not very large and hence the close agreement obtained here does not necessarily mean that the model would be equally successful in predicting the RanqueHilsch regime of parameters where the temperature differences are much larger.
- The swirl-induced temperature separation was clearly evident with a cold vortex core and a temperature distribution that looks almost like an adiabatic change of state compared to the pressure.

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698 citations

### "A computational and experimental st..." refers methods in this paper

...The unknown Reynolds stresses were obtained using an explicit algebraic Reynolds stress model [21,22] incorporating the modification proposed byWallin and Johansson [23] to account for streamline curvature....

[...]

621 citations

### "A computational and experimental st..." refers background in this paper

...where DT ij is the turbulent diffusivity tensor [17] which is unknown and in need of approximation, and Dpij is a second-order tensor that is a function of the Reynolds-stress tensor and the turbulence time scale but not a function of the velocity or temperature gradients....

[...]

411 citations

### "A computational and experimental st..." refers background in this paper

...B. Kobiela a, B.A. Younis b,⇑, B. Weigand a, O. Neumann c a Institut für Thermodynamik der Luft- und Raumfahrt, Universität Stuttgart, 70569 Stuttgart, Germany bDepartment of Civil & Environmental Engineering, University of California, Davis, CA 95616, USA cDepartment of Mechanical Engineering, University of Applied Sciences, 24149 Kiel, Germany a r t i c l e i n f o Article history: Received 22 December 2017 Received in revised form 14 March 2018 Accepted 16 March 2018 Available online 21 March 2018 Keywords: Energy separation by swirl Turbulent heat fluxes Ranque-Hilsch effect a b s t r a c t When compressed air is introduced into a tube in such a way as to generate a strong axial vortex, an interesting phenomenon is observed wherein the fluid temperature at the vortex core drops below the inlet value, while in the outer part of the vortex, the temperature is higher than at inlet....

[...]

...Their analytical solutions for the temperature distribution in a Ranque-Hilsch tube showed a static temperature distribution corresponding to an adiabatic change of state over the radial pressure distribution....

[...]

...For the flow inside a Ranque-Hilsch tube, the axial flow is bi-directional in the sense that the flows in the central core and the periphery move in opposite directions and thus the swirl number as defined in Eq....

[...]

...It should be noted that the temperature variations in the experiment were not very large and hence the close agreement obtained here does not necessarily mean that the model would be equally successful in predicting the RanqueHilsch regime of parameters where the temperature differences are much larger....

[...]

...The most familiar manifestation of this phenomenon of ‘‘thermal energy separation” is the Ranque-Hilsch effect (Fig....

[...]

202 citations

### Additional excerpts

...Kurosaka [5] attributed the effect to ‘‘acoustic streaming”....

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189 citations

### "A computational and experimental st..." refers methods in this paper

...The unknown Reynolds stresses were obtained using an explicit algebraic Reynolds stress model [21,22] incorporating the modification proposed byWallin and Johansson [23] to account for streamline curvature....

[...]