Flow structure and mechanism of heat transfer in a Ranque–Hilsch vortex tube

Journal Pre-proofs Flow structure and mechanism of heat transfer in a Ranque–Hilsch vortex tube D.G. Akhmetov, T.D. Akhmetov PII: DOI: Reference:

S0894-1777(19)30589-8 https://doi.org/10.1016/j.expthermflusci.2019.110024 ETF 110024

To appear in:

Experimental Thermal and Fluid Science

Received Date: Revised Date: Accepted Date:

12 April 2019 24 November 2019 15 December 2019

Please cite this article as: D.G. Akhmetov, T.D. Akhmetov, Flow structure and mechanism of heat transfer in a Ranque–Hilsch vortex tube, Experimental Thermal and Fluid Science (2019), doi: https://doi.org/10.1016/ j.expthermflusci.2019.110024

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier Inc.

Flow structure and mechanism of heat transfer in a Ranque–Hilsch vortex tube D. G. Akhmetova,, T. D. Akhmetovb,c a Lavrentiev

Institute of Hydrodynamics, 15 Lavrentieva prospekt, Novosibirsk 630090, Russia Institute of Nuclear Physics, 11 Lavrentieva prospekt, Novosibirsk 630090, Russia c Novosibirsk State University, 2 Pirogova street, Novosibirsk 630090, Russia

b Budker

Abstract Flow structure of incompressible fluid in a Ranque–Hilsch vortex tube was studied experimentally. A pattern of streamlines in the whole volume of the vortex tube was constructed from the velocity distributions measured using laser Doppler anemometry. The radial distributions of the azimuthal velocity in the Ranque–Hilsch vortex tube are shown to be dierent from those in a vortex tube with a tangential inlet and a single central exit orifice. Based on the obtained structure of the flow, a simple qualitative model is proposed to explain the physical mechanism of the temperature separation eect. Keywords: Ranque–Hilsch vortex tube, laser Doppler anemometry, flow structure, Ranque eect, heat transfer

1. Introduction The Ranque–Hilsch vortex tube (RHVT) is a cylindrical device which separates tangentially injected compressed gas into hot and cold streams. The gas injected through a nozzle in a side wall of the tube rapidly rotates around the axis and exits through a central circular orifice at one end cap and through a narrow annular channel between the wall and the other cap [1]. Experimenting with such a tube Ranque found that the temperature in the stream escaping through the central orifice was well below, and the temperature in the stream escaping through the annular slit was well above the temperature of the initial compressed gas. Ranque’s discovery remained unnoticed until this eect was investigated in more detail by Hilsch [2], whose work started a series of studies of this paradoxical eect of separation of a gas flow supplied to the tube into two exhaust streams with dierent stagnation temperatures. Since then the Ranque eect (“temperature separation”) has been studied experimentally, theoretically and numerically in hundreds of papers. Some studies considered practical applications of 

Corresponding author Email address: [email protected] (D. G. Akhmetov)

Preprint submitted to Experimental Thermal and Fluid Science

December 20, 2019

Nomenclature v cp

fluid velocity vector specific heat at constant pressure D vortex tube diameter L vortex tube length N number of scattering particles p pressure q volume flow rate R vortex tube radius r radial coordinate T temperature V magnitude of fluid velocity x radial coordinate z axial coordinate Greek Symbols

specific heat ratio  kinematic viscosity

stream function Subscripts 0 tangential inlet nozzle 1 central outlet orifice 2 annular outlet orifice  location of zero axial flow rate on side wall a; b axial boundaries of inlet nozzle e external vessel i number of axial cross-section j number of radial location r; z;  radial, axial and azimuthal components Abbreviations LDA laser Doppler anemometer RHVT Ranque–Hilsch vortex tube

the Ranque eect such as spot cooling and determined optimal geometric parameters of the vortex tube and the ratio of gas flows entering and exiting the tube. Considerable interest in this eect is attracted by the search for the physical mechanism explaining temperature separation between the outlet gas flows. Details can be found in reviews and research papers [3–29]. This list is by no means exhaustive or complete. It includes some papers involving velocity measurement, construction of the flow pattern, and discussing the energy separation in the Ranque–Hilsch vortex tubes relevant to the present study. Eiamsa-ard and Promvonge reviewed current research on RHVTs especially regarding geometrical characteristics of the vortex tube and thermo-physical parameters such as inlet gas pressure, cold mass fraction, and type of gas [17]. Yilmaz et al. overviewed the investigations of the design criteria of vortex tubes in detail using experimental and theoretical results [21]. Gutsol reviewed the existing theories of the Ranque eect and suggested its explanation treating a gas flow as a collection of microvolumes with dierent forward velocities [7]. Subudhi and Sen reviewed the literature on RHVT experiments using air as the working fluid [27]. Besides the variables of interest and the important experimental results, they focused on curve-fitting equations using data from the literature which could provide a rough estimate of temperatures achieved, and which could be used in practice for preliminary vortex tube design. Experimental studies by Merkulov [3, 4] showed that the tube length could be made much shorter compared to traditional tubes while keeping the same cooling eect, when a multiblade cross was installed at the tube cap with the annular exit channel.

2

Gulyaev [5] demonstrated experimentally using helium as a working gas that cooling of the near-axial stream occurred due to heat transfer with the cold mass of gas in contact with it (see details in Sec.5). Balmer performed experiments with liquid water in a commercial counterflow RHVT designed for use with air and verified that significant temperature separation occurred when a suciently high inlet pressure was used [6]. He concluded that a temperature separation mechanism existed which did not depend upon the compressibility of the operating fluid. Ahlborn et al. studied the RHVTs experimentally [8–10]. They measured radial profiles of the azimuthal and axial velocities in the RHVT working with compressed air and found that at the center of the tube the total flow to the “cold end” was larger than the cold mass flow emerging out of the orifice [9]. So somewhere between the cold end and the center of the tube the excess flow turned back to the outer regions and moved there towards the hot end. This secondary circulation was imbedded into the primary vortex flow, but their measurements did not determine whether this secondary flow was open or closed. This secondary flow was used as a key agent in a refrigeration cycle in the vortex tube to explain the Ranque eect [10]. Arbuzov et al. [11] visualized large-scale structures in the form of a vortical double helix in a swirling Ranque flow. After analyzing and estimating possible heating mechanisms they concluded that the viscous heating of the gas in a thin boundary layer at the wall of the vortex chamber was the main reason of strong heating in RHVTs. Rebrov and Skovorodko showed by numerical simulation [13] that the Ranque eect was existent in rarefied gas as well and suggested that the dierence in stagnation temperatures of the cold and hot flows was mainly determined by the dierence in angular momentum of corresponding fluxes. Fr¨ohlingsdorf and Unger in their numerical simulation [14] suggested that the rotating cold gas could transfer its mechanical energy to the warm gas layers near the tube wall through friction caused by dierence in their angular velocities. However, to fit the measured data, artificial enhancement of this mechanical transfer rate was required. Dubnishchev et al. studied flow structure and energy separation in a RHVT with square cross-section [12]. Gao et al. experimentally studied flow structure in nitrogen in long RHVTs by measuring radial profiles of all three velocity components using a hot-wire anemometer probe, but did not construct a flow pattern [15, 16]. They measured temperature distributions and suggested a modified secondary circulation model of the temperature separation. Behera et al. [18] developed a three-dimensional numerical model of a RHVT, analyzed the flow parameters and energy separation mechanism inside the tube and discussed possible energy transfer mechanisms. Secchiaroli et al. [19] performed simulation of the turbulent, compressible, high swirling flow by both RANS and LES techniques. Results in dierent sections of the tube showed significant dierences in the velocity profiles, temperature profiles and secondary vortex structures, varying turbulence model. Their simulations proved capable of predicting secondary circulation flow inside a RHVT including a single re-circulating vortex structure in the twodimensional axisymmetric RANS model and more complex secondary vortex structure due to the three-dimensional and unsteady features of the flow field in the LES model. Farouk et al. [20] made numerical simulations of the flow field and temperature separation in a counter-flow RHVT. The time averaged profiles indicated a hot peripheral flow and a reversing cold inner core flow together with a small secondary circulation near the cold exit. Shtern and Borissov [22] studied the physical mechanism of elongated 3

counter-flows occurring in vortex tubes. Their numerical solution for vortex tubes with two outlets represented the flow as a sum of two parts: one part being the pipe flow that would develop with no swirl far downstream of the inlet and the rest part being the flow driven by swirl. They came to the flow pattern consisting of two through-flows, the global meridional circulation, and found that swirl decay due to fluid-wall friction induced both the U-shape through-flow and the circulation. Xue et al. reviewed the current research in RHVTs [23] and pointed in particular that clarification of the role of the secondary circulation in counter- and uni-flow vortex tubes was important. This group also performed experiments using water, found multicirculation inside the tube and suggested a schematic flow pattern [24]. Recently they measured radial distributions of the azimuthal and radial velocities in water in a vortex tube and found that in a longer tube the azimuthal velocity decays towards the hot end with the peripheral exit channel stronger than in a very short tube [25]. Liew et al. used phase Doppler particle analysis to measure three velocity components in a RHVT with addition of a dispersed phase (water droplets) to nitrogen [26]. Data analysis showed that high intensity velocity fluctuations in the core region of the main vortex were caused by wobbling of the vortex axis. Kolmes et al. [28] suggested a quantitative model for heat separation in a fluid due to motion along a pressure gradient and applied it to explanation for the temperature separation in a vortex tube. They discussed to what extent the concept of saturation could be used when the fluids temperature and pressure were related at its boundaries by an adiabatic law. Kobiela et al. [29] presented an analysis of the heat transfer mechanism in a RHVT, based on consideration of the exact equation governing the conservation of the turbulent heat fluxes. The experimental investigation of flow in a swirl chamber demonstrated that the new model yielded predictions that were distinctly better than those obtained using conventional closures. It was shown important to account for the eects of pressure gradients in the prediction of swirl-induced thermal energy separation. Various explanations exist why the gas stream from the central exit diaphragm is colder than at the inlet to the tube. However, these explanations cannot be assumed yet as fully conclusive, since they are not based on the real measured flow structure in the chamber. The physical mechanism of the Ranque eect can be established when a clear picture of the structure of a rotating flow in the chamber will be obtained, which itself is a dicult task. Some works present only sparse data on velocity and temperature distributions along a vortex tube. Some authors provide quite comprehensive measured velocity, temperature and pressure distributions [15, 16], but do not proceed further to constructing a pattern of streamlines and explaining the physical mechanism of heat transfer and temperature separation. Moreover, the majority of these measurements were performed by probes inserted in the flow. However, even miniature probes inserted in a swirled flow may significantly change its structure, which impairs validity of such experimental data. The use of optical methods to construct the velocity field of a gas flow by measuring velocities of small particles suspended in it, also faces challenges. First, optical beams deflect at a cylindrical chamber wall. Second, there are no particles with the density equal to the gas density, hence the particle velocity becomes dierent from the gas velocity. By contrast, in case of liquid media, particles with the density equal to the ambient fluid density can be easily found. Let us consider the conditions under which the fluid may be regarded as incom4

pressible. It is important, because in this study an incompressible fluid is used to model a flow pattern of a compressible fluid in a RHVT. In steady flow the fluid can be considered incompressible, if the fluid velocity is small compared with the velocity of sound [30]. It should be noted that a rotational motion of a gas in a cylindrical vortex chamber is the flow along a concave wall. Depending on the pressure of compressed gas in an external vessel, the velocity of the gas after its expansion into the chamber from an inlet nozzle can be greater or less than the velocity of sound. It is known that in a supersonic gas flow past a concave surface, a system of shock waves forms [30], and as the flow passes through these shocks, it becomes subsonic. Experimental data with air as a working fluid also proves that the flow inside the RHVT is always subsonic [8]. Although the velocity of a gas near the inlet is subsonic, it can be close to the velocity of sound. However, as follows from the experimental studies in long RHVTs with gases, the azimuthal velocity, which is dominant in RHVTs, rapidly decreases with distance from the inlet and falls below the half of the sound velocity at a distance of a few tube diameters from the inlet [16], whereas ecient RHVTs typically have a total length of about 50 diameters. So the required inequality v2 =c2  1 becomes approximately valid, and the assumption that the fluid is incompressible can be used everywhere in the vortex tube, except a relatively small region near the inlet, where the velocity can be high. Since the present work considers large-scale features of a time-averaged velocity field and consequences for the mechanism of the Ranque eect, it is reasonable to extend the approximate assumption of an incompressible fluid to the whole volume of the tube including the region near the inlet. So an incompressible fluid can be used to model the flow pattern of a gas. A liquid is a simpler object in terms of correct velocity measurements. Moreover, since the mean flow in a RHVT is two-dimensional, the continuity equation for an incompressible fluid, div v = 0, allows construction of a stream function and streamline pattern. Accordingly, in the present work the velocity field of water flow was measured using laser Doppler anemometry (LDA). Neutrally buoyant latex particles of diameter approximately 20 micrometers were used as seed tracers characterizing the fluid velocity. This technique does not perturb the original flow and yields correct local velocity of the fluid. Based on the measured velocity field, a time-averaged streamline pattern was constructed in the whole axial cross-section of the vortex chamber. Thereby suciently complete understanding of the structure of the mean swirled flow in the whole volume of the chamber was obtained. Here the experimental results briefly reported in Akhmetov et al. [31] are considered in detail with the emphasis on the specific features of the flow structure in the Ranque–Hilsch vortex tube and on explaining the mechanism of temperature separation. 2. Experimental setup The experimental setup is shown in Fig. 1. The vortex chamber is a transparent plexiglas cylinder with the inner radius R = 25 mm, length L = 0:47 m, and 1 mm thick wall. To measure the velocity components of a tracer particle in the fluid, the LDA beams were directed into the chamber through its side wall along direction normal to the cylinder axis. In order to limit beam refraction at the wall of the chamber, the latter was installed inside a water-filled square-prism container with transparent walls. A water tank placed 2:5 m above the tube inlet supplied a constant fluid flowrate. 5

exit orifice at axis

water tank buffer reservoir

exhaust tube

q1

r

inlet slit nozzle za z*

q0

zb

vortex chamber container cross suppressing fluid rotation

z

exit annular channel

exhaust tube

buffer reservoir

q2

q0 inlet slit nozzle

vortex chamber container Figure 1: Experimental setup of the Ranque–Hilsch vortex tube showing the transparent cylindrical vortex tube, transparent square-prism container, tangential inlet nozzle with flow rate q0 , central exit orifice with flow rate q1 , and annular exit channel with flow rate q2 .

The flow entering the vortex tube through a rectangular slit nozzle of 3 mm width in the radial direction and 20 mm width in the axial direction, is perpendicular to the tube axis and tangential to its inner surface. Tangential fluid injection produced strong rotation around the tube axis, and finally the fluid exited through the central orifice of radius r0 = 8 mm in the tube cap near the inlet nozzle, and through an annular channel between the tube wall and the other cap. The exhaust fluid entered buer vessels and then moved through exhaust tubes. The majority of experimental studies suggest that the maximum gas cooling eect in the RHVT is achieved when the tube length exceeds about fifty diameters. However, eective temperature separation was achieved in much shorter tubes as well, with the length of just 9–10 diameters, if a special cross is installed inside the tube at the cap with the annular channel to stop the swirling motion of the gas [3]. Hence, to reduce the dimensions of the experimental

6

device, a relatively short tube with a four-bladed cross was used in the present study. All measurements were performed at the fluid inlet flow rate q0 = 5:05
10 5 m3 /s, which for the inlet area of 20  3 mm2 gives the inlet bulk velocity V0 = 0:84 m/s. The fluid flow rate from the central exit orifice was q1 = 1:58
10 5 m3 /s (the minus sign accounts for the direction of the axial velocity in the central orifice), and the flow rate from the annular exit slit was q2 = 3:47
10 5 m3 /s. The vortex tube and the flow in it are characterized by the following dimensionless parameters: relative tube length L=D = 9:4, radii ratio r0 =R = 0:35, ratio of the central to inlet flow rates (cold mass fraction) jq1 j=q0 = 0:31, and the Reynolds number Re = V0 R= = 2:1
104 , where  is the kinematic viscosity of water. The flow was considered in cylindrical coordinates (z; r), where the z-axis coincided with the tube axis and was directed from z = 0 at the cap with the central orifice to the other end of the tube. Figure 2 shows the LDA setup for measurement of the azimuthal and axial velocities. The LDA complex which was used in the experiments was described in [32]. Two laser beams of LDA were focused on the point of their intersection, and the velocity component lying in the plane of the beams and perpendicular to the beam angle bisector was measured. The ellipsoidal intersection region of the beam formed the measurement volume of length about 1 mm along the bisector and 0.05 mm in diameter. The length determined the spatial resolution of the velocity measurement. The beams were directed inside the chamber through its side wall and the radial coordinate of the intersection point inside the chamber was set by changing the distance between the experimental device and LDA. cold exit inlet z LDA

LDA x

laser beams

x

v

vz laser beams hot exit z

azimuthal velocity

axial velocity

Figure 2: Schematic setup of laser Doppler anemometer for measurement of the azimuthal and axial velocities.

3. Velocity field in Ranque–Hilsch vortex tube The radial distributions of the axial and azimuthal (swirl) velocity components were measured by displacing the laser intersection region along the diameter with a step of 3 mm in each of 28 cross-sections of the tube: zi =L = 0:011; 0.032, 0.043, 0.053, 0.074, 0.085, 0.096, 0.138, 0.160, 0.202, 0.245, 0.287, 0.383, 0.404, 0.447, 0.489, 0.532, 0.574, 0.617, 0.638, 0.702, 0.734, 0.777, 0.819, 0.862, 0.894, 0.904, and 0.915. 7

These locations are shown in Fig. 3. One can see that the whole volume of the chamber is covered with a suciently dense measurement grid allowing resolution of relatively small flow features.

z/L 0

0.2

0.4

0.6

0.8

1.0

Figure 3: All 28 axial locations zi along the vortex tube where velocity component distributions were measured. Thick blue lines show particular locations in which velocity profiles in Fig. 4 are plotted. z = 0 is at the end with the central orifice.

When the axial velocity is measured, the plane formed by the two LDA beams coincides with the axial plane of the chamber, and the laser beams enter the chamber through a plane wall of the square-prism container with water and through a generating line of the cylinder. Displacement of LDA relative to the chamber in the radial direction corresponds to the change in the radial coordinate of the velocity measurement point, and no distortions of the measured axial velocity component appear. However, when the azimuthal velocity component is measured, the LDA beam plane is normal to the chamber axis, and the laser beams enter the chamber at dierent distances from the axial plane of the chamber depending on the distance from LDA to the chamber. In this case the refraction angles of the beams passing through the cylindrical wall and hence the angle between the beams inside the chamber depend on the distance between LDA and chamber. This leads to small distortions in the measured values of the azimuthal velocity and in the radial coordinate of the beam intersection point, despite the small thickness of the chamber wall and small dierence of indices of refraction of the plexiglas wall and water. These aberrations were calculated analytically providing corrections to the measured values of the azimuthal velocity and radial coordinate of the observation point. The fluid velocity was measured during the time interval of 30 s in each point (zi ; x j ) of the axial cross-section of the tube; the x-axis was directed along the diameter of the tube, and the origin x = 0 coincided with the tube axis. During the measurement interval the region of the LDA beam intersection was crossed by Ni j latex particles scattering light, and a specified velocity component of each particle was recorded. The number Ni j was typically of several tens. The velocity component of a particle was equal to the instantaneous velocity component of the flow. Then the flow velocity component was calculated as an arithmetic mean of the values for all particles within the measurement time interval. It is known that when a number of individual realizations is recorded for particles having dierent velocities, the arithmetic mean tends to be biased towards higher velocity, because particles with higher than average velocities cross the measurement volume larger than average number of times. To avoid this biasing, so-called velocity-weighted average is often used with the weights equal to the inverse velocity magnitude of each particle. However, this procedure requires spatial uniformity of seed particle density, spherical measurement volume, and knowledge of

8

all three velocity components of each particle. We use one dimensional LDA setup which measures only one chosen velocity component. The degree of uniformity of the seed particle density is not known. The measurement volume is not a sphere, but a prolate ellipsoid with length-to-diameter ratio of 20. In this situation the velocity-weighted average is not recommended [33], therefore we used the arithmetic mean. The instantaneous velocity component of a single particle was measured with 1 % accuracy determined by the LDA hardware used in these experiments [32]. The standard deviation of the velocity determined by the dierence in velocities of individual particles recorded during the measurement time interval was up to 30 % of the mean velocity. A typical number of particles that crossed the measurement volume within a 30 s interval was in the range of Ni j = 20–50 at each (zi ; x j ) location. Relatively large standard deviation means that the flow in the vortex tube is highly turbulent, although the Reynolds number defined earlier from the inlet fluid velocity is moderate. However, since the dierence between turbulent and laminar flows occurs mainly within boundary layers, the turbulence should not lead to fundamental change in the global pattern of the flow which is the goal of this experimental study. Distributions of the axial velocity vz and azimuthal velocity v along the tube diameter in seven particular axial locations z specified in Fig. 3 are shown in Fig. 4 by filled and empty circles, where the symbols change from plot to plot only to avoid confusion between the points from neighboring plots. The radial velocity in strongly swirled flows is negligible compared with the axial and azimuthal velocities, and is of no interest. In each cross-section zi the experimental profiles were used to construct a 5-th degree polynomial approximation using the weights Ni j , with the requirement that the axial velocity should be an even function of x and the azimuthal velocity should be an uneven function of x, which produced smoothed profiles of velocity components vz (zi ; x) and v (zi ; x). The smoothed profiles of velocities are shown in Fig. 4 by solid lines. The approximations agree well with the experimental data in the same cross-sections, which supports the assumption that the time-averaged flow in the tube is axially symmetric. The azimuthal velocity in each cross-section is a monotonically increasing function of radius and decreases with the axial distance from the inlet nozzle. Hence the flow inside the chamber does not have any concentrations of vorticity in particular regions of the chamber. 4. Flow structure in Ranque–Hilsch vortex tube The flow structure is most fully represented by a pattern of streamlines. Since the mean flow in the vortex chamber is axisymmetric, a stream function can be introduced, which satisfies the equation of continuity for an incompressible fluid and determines the axial and radial velocities, vz = (1=r)@ =@r and vr = (1=r)@ =@z respectively. The stream function (z; r) is obtained from the axial velocity distribution vz (z; r) by integration, and then a pattern of streamlines in a plane R r through the tube axis is constructed. Hereafter an axial fluid flow rate q(z; r) = 2 0 vz (z; r)r dr through a circle of radius r is used instead of , which diers only by a factor of 2. The flow rate must satisfy the boundary conditions at the tube surface and at all orifices. At the tube axis and at the end cap with the annular slit q = 0. Then the fluid velocity is assumed to be constant over the area of each orifice, because the local details of 9

0.15

vz (m/s)

1.0

0.10

z/L 0.05 0 1.0 0.5

0

0.5

0.05

0.5

0.915 1.0  0.5 x/R

0.777

0.10

v (m/s)

0 0

0.5

0.5 1.0

0.617

0.447

0.287

0.138

0.011 1.0

0.5

0 0.5

1.0

1.0 0.5

0

0.5

1.0

Figure 4: Radial profiles of axial velocity vz and azimuthal velocity v in seven cross-sections specified in Fig. 3. Positive axial velocity is directed towards the annular exit slit.

10

1.0

x/R

the velocity distribution at the orifices have little eect on the global flow field in the chamber. Along the inlet slit nozzle at the side wall from za = 18 mm to zb = 38 mm the flow rate linearly increases from q1 to q2 = q0 jq1 j, passing through the value q = 0 at z = z . At the surfaces of the tube the flow rate has constant values equal to its values at the orifice edges. Given these boundary conditions, the distributions q(zi ; r) were calculated in each of the 28 cross-sections of the tube, zi . Then two-dimensional smoothing third degree B-splines [34] were used to obtain a continuous function q(z; r) over the whole (z; r) domain, which allowed plotting a pattern of streamlines shown in Fig. 5, where neighbor streamlines are separated by an increment of 4:0
10 6 m3 /s.

r/R

za

z  q0 zb

q2

3

1.0 0.8

2

0.6

1

0.4 0.2 q1

0 0

0.2

0.4

0.6

0.8

Figure 5: Experimental flow field in the Ranque–Hilsch vortex tube. Streamline 1 with flow rate q1 is the last closed streamline. Streamline 2 starting from the inlet nozzle at (z ; R) corresponds to zero axial flow rate and separates the inlet flow into two parts, proceeds to the annular channel and to the central orifice respectively. Streamline 3 coincides with the side wall of the tube.

It is clear from this flow field that inside the tube there is a large closed toroidal volume bounded by streamline 1. The flow of the fluid entering the tube is separated into two flows by streamline 2 starting at the inlet nozzle from the point z = z . The part of the flow between streamline 2 and the side wall equal to q2 = 3:47
10 5 m3 /s moves along the tube wall to the annular exit slit at the other end of the tube. The second part of the inlet flow between streamlines 1 and 2 equal to q1 = 1:58
10 5 m3 /s travels around the region with closed streamlines and comes to the central exit orifice. Knowledge of the smoothed distributions of the velocity components in the r-z plane q allows calculation of the velocity magnitude V = jvj  v2z + v2 at any point inside the vortex tube (the radial velocity is neglected). Figure 6 shows the velocity magnitude along the characteristic streamlines specified in Fig. 5. It is clear that the velocity magnitude in the RHVT is maximum near the inlet nozzle and decreases with distance from this region. The above velocity distributions in the tube and the pattern of streamlines can be compared with other studies. The existence of the “secondary circulation” (equivalent to the toroidal volume inside streamline 1 in Fig. 5) was suggested by Ahlborn and Groves [9] who measured the radial profiles of the azimuthal and axial velocities in one cross-section in the middle of the RHVT working with compressed air. They found 11

z/L

1.0

1.0

V (m/s)

0.8 0.6

2 3

0.4 0.2 0

1 0 za zb 0.2

0.4

0.6

0.8

1.0 z/L

Figure 6: Velocity magnitude on streamlines from Fig. 5: streamline 1 bounds the region with closed streamlines, streamline 2 corresponds to zero axial flow rate, and streamline 3 coincides with the side wall of the tube. Arrows show direction from the inlet nozzle towards the exit orifices.

that at the center of the tube the return flow to the cold end of the tube was much larger than the cold mass flow emerging out of the cold end orifice, meaning that somewhere between the cold end and the center of the tube the excess flow turned back to the outer regions and moved there towards the hot end. This secondary circulation was imbedded into the primary vortex flow, but their measurements did not determine whether this secondary flow was open or closed. Later they also suggested a model in which this secondary flow played a key role in a refrigeration cycle in the vortex tube [10]. Xue et al. [23] discussed the important areas of current research in RHVTs and concluded that clarification of the secondary circulation in dierent vortex tubes and an understanding of the influence of the secondary flow, when it was formed, were both required. A qualitative flow pattern was previously suggested by Xue et al. [24] from experiments in water in a relatively short tube (L=D = 6:9), where the existence of “multi-circulation” was deduced from axial variation of integral flow rates between certain fixed radii. Recently Xue et al. [25] measured radial distributions of the azimuthal and radial velocities in water in a vortex tube with dierent exit orifice geometries. They found that in a medium-length tube (L=D = 10) the azimuthal velocity decayed towards the hot end with the peripheral exit channel stronger than in a very short tube (L=D = 2:6). The flow pattern in the present work has one large circulation region. This may result from the concrete geometry of the vortex tube or from the time-averaged velocity distributions. For example, simulation of the gas flow within a counter-flow Ranque–Hilsch vortex tube by Farouk et al. [20] showed that while the small vortices were present in the instantaneous streamlines, they ceased to exist in the time-averaged streamline profiles. Flow structure in nitrogen was studied by Gao [15] in a long RHVT (L=D = 65) which was optimal for marked temperature separation eect. Azimuthal, axial and radial velocity profiles were measured by a hot-wire anemometer probe in five cross-sections of the tube, but a flow pattern was not constructed. The azimuthal velocity decayed towards the hot end and increased with radius in each cross-section,

12

and existence of the secondary circulation was demonstrated for any cold fraction. For small cold fractions the secondary circulation was observed even near the very hot end of the tube. Liew et al. [26] obtained radial profiles of the mean velocity components close to the inlet nozzle using LDA measurement of water droplet velocities in nitrogen gas in a long tube (L=D = 50). The mean swirl velocity increased with radius almost up to a tube wall in agreement with the present velocity profiles in Fig. 4, and there was no evidence of a “free vortex” with the swirl velocity decreasing with radius as r 1 . Swirl velocity increasing with radius everywhere in the tube was also obtained in numerical simulations by Behera et al. [18]. Thus a “free vortex” often postulated to exist in the peripheral stream of the RHVT, probably due to incorrect analogy with a simple vortex tube having a single exit orifice discussed further, does not seem to be a valid concept for the Ranque–Hilsch vortex tube. Some of the existing hypotheses attempting to explain the Ranque eect are based on the unsubstantiated assumption that the flow in the tube consists of a near-axial core with vorticity (so-called “forced vortex”) surrounded by a practically irrotational external flow (so-called “free vortex” mentioned earlier). However, such a flow structure is realized only in relatively short vortex chambers with a single central orifice at one tube cap, whereas the other end of the tube is closed by a solid cap. The flow structure in such a vortex tube was established experimentally in a recent work [35]. The aggregate of all azimuthal velocity data points measured in many cross-sections along the chamber axis except only boundary layers at the end caps is shown as a function of radius in Fig. 7. Since the scatter of the data points around a trend curve is relatively small, the azimuthal velocity in the whole volume of the vortex chamber is almost independent of the axial coordinate. The velocity distribution has a maximum at a radius of r=R  0:16 unlike the azimuthal velocity distributions in the Ranque–Hilsch vortex tube. This point of velocity maximum indeed divides the tube volume into two zones. In the near-axial zone the fluid rotates round the tube axis as a solid body, and in the outer zone the azimuthal velocity and vorticity rapidly decrease with radius. In such a chamber the near-axial core with vorticity is generated from the fluid arriving to the axis along the boundary layer at the solid cap of the tube under the action of the radial pressure gradient caused by the rotational fluid motion [36, 37]. The core with vorticity visualized by dye introduced into the end boundary layer is clearly seen in Fig. 8. Similar azimuthal velocity distributions with a forced vortex near the axis and a dominating free vortex in the rest volume of the tube were obtained by Xue et al. [25] in a short vortex tube (L=D = 2:6) with central orifices at both ends. The flow structure in a Ranque–Hilsch vortex tube is fundamentally dierent from the described flow in a short tube with a single exit orifice at the axis. This is related in particular with design features of the RHVT. As mentioned above, significant temperature separation in the RHVT is achieved only when the tube length-to-diameter ratio is very large, L=D  50. However, the optimal Ranque eect was obtained in a much shorter tube with L=D  10 as well, when a multiblade cross was installed inside the tube at the end cap with the annular channel [3]. These RHVT design requirements were established empirically, but it is clear that in a suciently long tube the rotational motion would decay from the inlet nozzle towards the tube cap with the annular exit slit due to viscous friction. Installation of the multiblade cross also leads to deceleration of the rotational motion of gas near this cap. Both of these results indicate that 13

2

v (m/s)

1.5

1

0.5

0 0

0.2

0.4

0.6

0.8

1.0

r/R Figure 7: Radial profile of the azimuthal velocity in a simple vortex tube with a central exit orifice only. Data points show the velocity maximum at r=R  0:16, rigidly rotating vortex core (approximated by solid red line) and decaying velocity outside the core [35].

Figure 8: Photograph of a vortex core in water flow in a simple vortex tube with a central exit orifice only. Visualization by skim milk [35].

14

deceleration of the rotational motion of gas near the tube end with the annular channel is a prerequisite for the Ranque eect. When the radial pressure gradient is present near the tube cap, it causes the inflow of the fluid with vorticity along the end boundary layer to the tube axis and formation of the near-axial vortex core. But in the absence of the rotational fluid motion near the cap, the radial pressure gradient disappears. It follows from Fig. 4 that the azimuthal velocity really decays with distance from the inlet nozzle and that in each cross-section it is a monotonically increasing function of radius. There is no particular near-axial core with vorticity inside the RHVT, hence the flow in the whole volume of the tube from the axis to the wall has vorticity. Thus the often used assumption that the flow in the RHVT consists of a near-axial core with vorticity surrounded by an external irrotational flow, does not agree with the real flow structure in the RHVT. So the hypotheses of the mechanism of the Ranque eect based on this assumption can be considered ill-founded. 5. Mechanism of Ranque eect The agreement on the basic physical mechanisms on which the RHVT operates is still lacking, despite decades of extensive studies. It is generally accepted that gas cooling in the vortex tube is produced by sudden expansion of the gas from the inlet nozzle into the tube. Heating of the gas stream leaving the tube from the annular channel causes controversy, and several possible reasons of gas temperature increase above the temperature of the compressed gas in the external tank have been suggested in the literature, from adiabatic compression and turbulent transfer to internal and wall viscous friction or combinations of the above [17, 23, 27]. However, since the full flow field has remained unclear, one could have only hypothesized, in particular, how the cold gas near the inlet nozzle is carried towards the central opening, whether a region with closed streamlines exists in the RHVT and what is its role in heat transfer. Here a mechanism of the temperature separation is suggested based on the experimentally obtained flow field in the RHVT. It is reasonable to expect that the obtained streamline pattern of the incompressible fluid shown in Fig. 5 represents the structure of a gas flow in the RHVT as well. Hence a physically consistent explanation of the Ranque eect can be proposed, namely the reason why two gas flows exit the tube with quite dierent stagnation temperatures. The Ranque eect is often considered as some process of energy separation, and hypotheses are suggested to explain its mechanism, although there is no such a thermodynamic process as energy separation at all. Two gas streams with significantly dierent temperatures flow out from the RHVT likely as a result of several simple and clear physical processes that take place inside the vortex tube. The mechanism of the Ranque eect will be explained, when the following questions will be answered: (1) What process in the vortex tube leads to cooling of the gas to the lowest temperature? (2) What mechanism cools the gas stream exiting from the central orifice of the tube? (3) What process leads to heating of the gas stream exiting from the annular slit at the other end of the tube? 15

Based on the flow structure in the RHVT revealed in the present work, the answers to each of these questions are suggested. (1) Gas cooling process. The analysis of the obtained flow structure in the RHVT shown in Fig. 4 and Fig. 5, and of the character of the distributions of the velocity magnitude along the streamlines in Fig. 6 leads to the conclusion that the strongest gas cooling in the tube can be caused only by the mechanism of adiabatic expansion, when a compressed gas from the external vessel enters the tube through the inlet nozzle. When the gas at rest with the pressure pe and temperature T e expands adiabatically from the external vessel into the vortex chamber, the enthalpy conservation along a streamline gives c p T e = c p T 0 + V02 =2, where T 0 is the gas temperature immediately after the inlet nozzle, V0 is the velocity of gas outflow into the tube, and c p is the specific heat capacity at constant pressure. Hence, the temperature dierence of the gas before and after the inlet to the tube is a function of the inflow velocity, T e T 0 = V02 =2c p . Figure 6 implies that maximum of the velocity magnitude V in the vortex tube occurs in the region near the inlet nozzle, so the temperature dierence should also have maximum in this region of the vortex chamber. The pressure p0 and temperature T 0 of the gas after the entrance to the tube are related by T 0 =T e = (p0 = pe )( 1)= , where is the adiabatic index. When the air is used as a working gas, for the ratio of pressures is p0 = pe = 0:528, the velocity of outflow is equal to the sound velocity, and the temperature ratio becomes T 0 =T e = 0:833 implying that the temperature of the air supplied to the tube, for example, at T e = 300  K, decreases after the inlet into the vortex tube by (T e T 0 )  50  K. Such a temperature dierence is comparable with the experimental values for the temperature dierence between the air entering the tube and the air exiting the central orifice. However, if the pressure of the compressed air in the vessel is within a typical experimental range pe  (3–10)
105 Pa, the ratio p0 = pe can be less than the above estimate. Then the velocity of the air outflow from the inlet nozzle into the tube will exceed the sound velocity, and the temperature dierence (T e T 0 ) will be significantly greater than the above estimate. Thus, there is a real physical mechanism of gas cooling to very low temperatures inside the vortex tube, namely, the adiabatic expansion of the compressed gas from the external vessel through the inlet nozzle into the vortex tube. Such a strong gas cooling is hardly possible in any other region of the tube. (2) Mechanism of gas cooling in the central stream. The process of gas cooling in the flow from the central orifice of the vortex tube can be also explained from the analysis of the pattern of streamlines. Figure 5 implies that a part of the high-velocity cooled gas flow with the flow rate q = q1 entering the tube at the portion of the slit nozzle from za to z goes round the region of closed streamlines and proceeds to the central exit orifice. Streamline 2 is the outer boundary of this stream passing round the closed region. At the periphery near the inlet this low-temperature stream is in contact with the closed region. So in the absence of other heat exchange processes through its surface, the whole mass of the gas inside the closed region would be cooled due to heat transfer. The temperature of the gas at each point of a thin stream-tube around the closed region is determined by the balance of the two processes. Without heat transfer, the total enthalpy conservation implies that as the gas moves from the inlet nozzle to

16

the exit central orifice, its temperature should increase, since the velocity along this stream-tube decreases. On the other hand, due to heat transfer to the cold mass of the gas in the enclosed region across streamlines, the static temperature in this stream-tube would decrease as it proceeds to the exit. Finally, the stagnation temperature of the gas at the central exit orifice is lower than the temperature of the compressed gas in the vessel, as observed in experiments. The fact that the cooling of the near-axial stream occurs due to heat transfer to the cold mass of the gas in contact with it, was demonstrated experimentally by Gulyaev [5]. In that study helium was used as a working gas, and operation of a vortex tube was tuned by increasing the area of the annular outlet channel in such a way that the gas would flow out only through this channel. Then an additional flow of a gas without swirl was supplied through a tube inserted into the chamber along the axis through a cap with the annular channel. This gas flow passed directly along the chamber axis and left through the central orifice at the other end of the chamber. The temperature of the flow from the central orifice in these experiments occurred to be even lower than at standard operation of RHVT. This experimental result clearly confirms that the gas in the near-axial stream is cooled due to heat transfer to the surrounding mass of the cold gas. Thus, the cooling of the gas flow from the central outlet of RHVT occurs in two stages. The first stage is the cooling of the mass of gas within the closed streamlines due to heat transfer to the passing stream cooled at the inlet to the tube and moving round the enclosed region. In stage two, this cold mass of the gas within the closed streamlines in turn cools, also due to heat transfer, the near-axial gas stream which subsequently leaves the tube through the central orifice. (3) Mechanism of gas heating in the flow from the annular channel. From the flow structure in the vortex tube, it is also possible to explain the mechanism of gas heating in the flow from the annular channel at the other end of the tube. This stream is formed from the gas flow which enters the tube at a portion of the inlet nozzle from z to zb in Fig. 5 and then moves along the side wall to the annular exit slit. In an isentropic flow with or without viscous heating, the enthalpy and so the stagnation temperature of the gas would remain constant along a thin stream-tube from the inlet nozzle to the hot annular exit. Since the stagnation temperature of the stream from the annular channel is higher than that in the external vessel with compressed gas, the peripheral flow is not isentropic, and it is the heat transfer across the surfaces of constant flow that increases the static temperature of the peripheral flow and finally increases the stagnation temperature of the gas escaping from the annular channel. Near the inlet nozzle where the primary flow is the coldest, the azimuthal velocity at the boundary of the closed flow is much larger than the axial velocity. It provides long contact between the primary and secondary flows and ecient heat transfer, leading to heating of the primary flow and to cooling of the secondary flow. This heat transfer increases the enthalpy of the peripheral gas stream and finally leads to the increase in the stagnation temperature of the gas escaping from the annular channel. Figure 4 implies that the axial and azimuthal velocities reach their largest values near the tube wall, hence the radial gradient of the velocity magnitude V in a thin boundary layer near the wall is also very large. The rate of heat production by viscosity is proportional to 17

the sum of squares of the velocity component gradients, so in flows with large velocity gradients, a significant part of mechanical energy of the flow is converted into heat due to the internal friction in a thermal boundary layer [38]. Arbuzov et al. [11] estimated possible heating mechanisms in the RHVT and concluded that viscous dissipation in a thin boundary layer near the side wall can also provide suciently large heating rate to explain high temperature of the gas leaving the tube from the annular channel. The above qualitative analysis of gas cooling and heating in the Ranque–Hilsch vortex tube uses the secondary circulation flow as a key intermediate part in the heat transfer mechanism leading to cooling of the central exit stream and to heating of the annular exit stream. This model is similar to the “heat pump” model suggested by Ahlborn et al. [10] in that both explanations use the idea that purely adiabatic processes cannot change the enthalpies of the exit gas streams and provide temperature separation, and it is the secondary circulation flow that plays the key role in heat exchange. The current explanation of the Ranque eect is based on the real pattern of streamlines determined from the measured distribution of the velocity components in the whole volume of the Ranque–Hilsch vortex tube. 6. Conclusion The velocity field in the whole volume of the Ranque–Hilsch vortex tube has been measured and the time-averaged pattern of streamlines has been constructed providing better insight into the structure of this flow. It is shown that unlike the flow in the vortex tube with a single central exit orifice, the azimuthal velocity in the RHVT increases with radius at any axial location. Based on the analysis of the flow structure, a simple physically consistent qualitative model of temperature separation in the Ranque–Hilsch vortex tube is suggested. The toroidal volume of secondary circulation is considered to be the key factor determining the Ranque eect. References [1] G. J. Ranque, Exp´eriences sur la d´etente giratoire avec productions simultan´ees d’un echappement d’air chaud et d’un echappement d’air froid, Journal de Physique et Le Radium IV(VII) (6) (1933) 112–115. [2] R. Hilsch, The use of the expansion of gases in a centrifugal field as cooling process, Review of Scientific Instruments 18 (2) (1947) 108–113. doi:10.1063/ 1.1740893. URL https://aip.scitation.org/doi/10.1063/1.1740893 [3] A. P. Merkulov, Investigation of a vortex tube, Soviet Physics: Technical Physics 1 (6) (1957) 1243–1248. [4] A. P. Merkulov, Vortex Eect and Its Application in Engineering, Mashinostroenie, Moscow (in Russian), 1969. [5] A. I. Gulyaev, Vortex tubes and the vortex eect (Ranque eect), Soviet Physics: Technical Physics 10 (10) (1966) 1441–1449. 18

[6] R. T. Balmer, Pressure-driven Ranque–Hilsch temperature separation in liquids, Journal of Fluids Engineering 110 (2) (1988) 161–164. doi:10.1115/ 1.3243529. URL https://doi.org/10.1115/1.3243529 [7] A. F. Gutsol, The Ranque eect, Physics-Uspekhi 40 (6) (1997) 639–658. doi: 10.1070/PU1997v040n06ABEH000248. URL http://stacks.iop.org/1063-7869/40/i=6/a=R06 [8] B. Ahlborn, J. U. Keller, R. Staudt, G. Treitz, E. Rebhan, Limits of temperature separation in a vortex tube, Journal of Physics D: Applied Physics 27 (3) (1994) 480–488. doi:10.1088/0022-3727/27/3/009. URL http://iopscience.iop.org/article/10.1088/0022-3727/27/3/ 009 [9] B. Ahlborn, S. Groves, Secondary flow in a vortex tube, Fluid Dynamics Research 21 (2) (1997) 73. doi:10.1016/S0169-5983(97)00003-8. URL http://stacks.iop.org/1873-7005/21/i=2/a=A01 [10] B. K. Ahlborn, J. U. Keller, E. Rebhan, The heat pump in a vortex tube, Journal of Non-Equilibrium Thermodynamics 23 (2) (1998) 159–165. doi:10.1515/jnet.1998.23.2.159. URL https://www.degruyter.com/view/j/jnet.1998.23.issue-2/ jnet.1998.23.2.159/jnet.1998.23.2.159.xml [11] V. A. Arbuzov, Y. N. Dubnishchev, A. V. Lebedev, M. K. Pravdina, N. I. Yavorski, Observation of large-scale hydrodynamic structures in a vortex tube and the Ranque eect, Technical Physics Letters 23 (12) (1997) 938–940. doi: 10.1134/1.1261939. URL https://doi.org/10.1134/1.1261939 [12] Y. N. Dubnishchev, V. G. Meledin, V. A. Pavlov, N. I. Yavorsky, Study of flow structure and energy separation in vortex tube with square section, Thermophysics and Aeromechanics 10 (4) (2003) 567–578. [13] A. K. Rebrov, P. A. Skovorodko, The Ranque eect in rarefied gas, in: Proceedings of the 21st International Symposium on Rarefied Gas Dynamics, Vol. 2, Marseille, France, 1998, pp. 221–228. [14] W. Fr¨ohlingsdorf, H. Unger, Numerical investigations of the compressible flow and the energy separation in the Ranque–Hilsch vortex tube, International Journal of Heat and Mass Transfer 42 (3) (1999) 415–422. doi:10.1016/S0017-9310(98)00191-4. URL http://www.sciencedirect.com/science/article/pii/ S0017931098001914 [15] C. Gao, Experimental study on the Ranque–Hilsch vortex tube, Ph.D. thesis, Department of Applied Physics (2005). URL https://research.tue.nl/en/publications/ experimental-study-on-the-ranque-hilsch-vortex-tube 19

[16] C. Gao, K. Bosschaart, J. Zeegers, A. de Waele, Experimental study on a simple Ranque–Hilsch vortex tube, Cryogenics 45 (3) (2005) 173–183. doi:10.1016/j.cryogenics.2004.09.004. URL http://www.sciencedirect.com/science/article/pii/ S0011227504001900 [17] S. Eiamsa-ard, P. Promvonge, Review of Ranque–Hilsch eects in vortex tubes, Renewable and Sustainable Energy Reviews 12 (7) (2008) 1822–1842. doi:10.1016/j.rser.2007.03.006. URL http://www.sciencedirect.com/science/article/pii/ S1364032107000500 [18] U. Behera, P. Paul, K. Dinesh, S. Jacob, Numerical investigations on flow behaviour and energy separation in Ranque–Hilsch vortex tube, International Journal of Heat and Mass Transfer 51 (25) (2008) 6077–6089. doi:10.1016/j.ijheatmasstransfer.2008.03.029. URL http://www.sciencedirect.com/science/article/pii/ S0017931008002470 [19] A. Secchiaroli, R. Ricci, S. Montelpare, V. D’Alessandro, Numerical simulation of turbulent flow in a Ranque–Hilsch vortex tube, International Journal of Heat and Mass Transfer 52 (23) (2009) 5496–5511. doi:10.1016/j.ijheatmasstransfer.2009.05.031. URL http://www.sciencedirect.com/science/article/pii/ S0017931009003950 [20] T. Farouk, B. Farouk, A. Gutsol, Simulation of gas species and temperature separation in the counter-flow Ranque–Hilsch vortex tube using the large eddy simulation technique, International Journal of Heat and Mass Transfer 52 (13) (2009) 3320–3333. doi:10.1016/j.ijheatmasstransfer.2009.01.016. URL http://www.sciencedirect.com/science/article/pii/ S0017931009000647 [21] M. Yilmaz, M. Kaya, S. Karagoz, S. Erdogan, A review on design criteria for vortex tubes, Heat and Mass Transfer 45 (5) (2009) 613–632. doi:10.1007/ s00231-008-0447-8. URL https://doi.org/10.1007/s00231-008-0447-8 [22] V. N. Shtern, A. A. Borissov, Nature of counterflow and circulation in vortex separators, Physics of Fluids 22 (8) (2010) 083601. doi:10.1063/1.3475818. URL https://aip.scitation.org/doi/10.1063/1.3475818 [23] Y. Xue, M. Arjomandi, R. Kelso, A critical review of temperature separation in a vortex tube, Experimental Thermal and Fluid Science 34 (8) (2010) 1367–1374. doi:10.1016/j.expthermflusci.2010.06.010. URL http://www.sciencedirect.com/science/article/pii/ S0894177710001226

20

[24] Y. Xue, M. Arjomandi, R. Kelso, Visualization of the flow structure in a vortex tube, Experimental Thermal and Fluid Science 35 (8) (2011) 1514–1521. doi:10.1016/j.expthermflusci.2011.07.001. URL http://www.sciencedirect.com/science/article/pii/ S0894177711001361 [25] Y. Xue, J. R. Binns, M. Arjomandi, H. Yan, Experimental investigation of the flow characteristics within a vortex tube with dierent configurations, International Journal of Heat and Fluid Flow 75 (2019) 195–208. doi:10.1016/j.ijheatfluidflow.2019.01.005. URL http://www.sciencedirect.com/science/article/pii/ S0142727X18303369 [26] R. Liew, J. C. H. Zeegers, J. G. M. Kuerten, W. R. Michałek, 3D Velocimetry and droplet sizing in the Ranque–Hilsch vortex tube, Experiments in Fluids 54 (1) (2012) 1416. doi:10.1007/s00348-012-1416-z. URL https://link.springer.com/article/10.1007/ s00348-012-1416-z [27] S. Subudhi, M. Sen, Review of Ranque–Hilsch vortex tube experiments using air, Renewable and Sustainable Energy Reviews 52 (2015) 172–178. doi:10.1016/j.rser.2015.07.103. URL http://www.sciencedirect.com/science/article/pii/ S1364032115007509 [28] E. J. Kolmes, V. I. Geyko, N. J. Fisch, Heat pump model for Ranque–Hilsch vortex tubes, International Journal of Heat and Mass Transfer 107 (2017) 771–777. doi:10.1016/j.ijheatmasstransfer.2016.11.072. URL http://www.sciencedirect.com/science/article/pii/ S0017931016328095 [29] B. Kobiela, B. A. Younis, B. Weigand, O. Neumann, A computational and experimental study of thermal energy separation by swirl, International Journal of Heat and Mass Transfer 124 (2018) 11–19. doi:10.1016/j.ijheatmasstransfer.2018.03.058. URL http://www.sciencedirect.com/science/article/pii/ S0017931017357368 [30] L. D. Landau, E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, Vol. 6, Pergamon, Oxford, 1987. URL https://books.google.ru/books?id=CeBbAwAAQBAJ [31] D. G. Akhmetov, T. D. Akhmetov, V. A. Pavlov, Flow structure in a Ranque– Hilsch vortex tube, Doklady Physics 63 (6) (2018) 235–238. doi:10.1134/ S1028335818060010. URL https://doi.org/10.1134/S1028335818060010 [32] I. K. Kabardin, V. G. Meledin, N. I. Yavorsky, M. R. Gordienko, M. K. Pravdina, D. V. Kulikov, V. I. Polyakova, V. A. Pavlov, LDA diagnostics of velocity 21

fields inside the Ranque tube, Journal of Physics: Conference Series 980 (2018) 012043. doi:10.1088/1742-6596/980/1/012043. URL https://iopscience.iop.org/article/10.1088/1742-6596/980/ 1/012043 [33] R. V. Edwards, Report of the special panel on statistical particle bias problems in laser anemometry, Journal of Fluids Engineering 109 (2) (1987) 89–93. doi:10.1115/1.3242646. URL http://fluidsengineering.asmedigitalcollection.asme.org/ article.aspx?articleid=1426546 [34] C. de Boor, A practical guide to splines, Vol. 27 of Applied mathematical sciences, Springer Verlag, New York, 1978. [35] D. G. Akhmetov, T. D. Akhmetov, Flow structure in a vortex chamber, Journal of Applied Mechanics and Technical Physics 57 (5) (2016) 879–887. doi:10.1134/S0021894416050151. URL https://link.springer.com/article/10.1134/ S0021894416050151 [36] D. N. Wormley, An analytical model for the incompressible flow in short vortex chambers, ASME Journal of Basic Engineering 91 (2) (1969) 264–272. doi:10.1115/1.3571091. URL http://fluidsengineering.asmedigitalcollection.asme.org/ article.aspx?articleid=1434403 ´ P. Volchkov, S. V. Semenov, V. I. Terekhov, Aerodynamics of end-wall bound[37] E. ary layer in a vortex chamber, Journal of Applied Mechanics and Technical Physics 27 (5) (1986) 740–748. doi:10.1007/BF00916149. URL https://link.springer.com/article/10.1007/BF00916149 [38] H. Schlichting, Boundary-layer Theory, McGraw-Hill, New York, 1975. URL https://books.google.ru/books?id=hYaLmgEACAAJ

22

Authors:

D.G. Akhmetov and T.D. Akhmetov

Title:

Flow structure and mechanism of heat transfer in a Ranque–Hilsch vortex tube

Highlights: 

Fluid velocity was measured in the whole volume of a Ranque–Hilsch vortex tube



Pattern of streamlines was constructed from the velocity distributions



The azimuthal velocity distribution is shown to be different from that in a simple vortex tube



Physical model of heat transfer and temperature separation in the tube is suggested

CRediT author statement

Darvin Akhmetov: Conceptualization, Methodology, Investigation, Validation, Writing - Original Draft. Timur Akhmetov: Conceptualization, Methodology, Formal analysis, Visualization, Writing - Original Draft.