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Introduction

Experimental methodology, experimental results and discussion, computational methodology, computational results and discussion, conflict of interest, data availability statement, nomenclature, appendix: mass and energy balances, an experimental and computational investigation of ranque–hilsch vortex tube heat transfer characteristics.

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Fuqua, M. N., and Rutledge, J. L. (November 16, 2023). "An Experimental and Computational Investigation of Ranque–Hilsch Vortex Tube Heat Transfer Characteristics." ASME. J. Thermal Sci. Eng. Appl . February 2024; 16(2): 021001. https://doi.org/10.1115/1.4063826

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Ranque–Hilsch vortex tubes have the extraordinary ability to split an incoming stream of fluid into two streams—one with a lower absolute total temperature than the incoming flow and the other with greater total temperature. The physical mechanism involves inducing an intense swirl of the flow down the length of the tube. The warmer flow exits around the periphery at the end of the tube, while the cooler central flow changes direction within the core and exits the opposite end. While much research has focused on the physical mechanisms of the energy separation, relatively little attention has been paid to the heat transfer behavior should a heat flux be applied to the walls. In the present work, experiments were performed using a vortex tube with air and varying levels of heat addition, up to approximately 15 kW/m 2 . Companion computational experiments were performed that allowed the determination of axially resolved Nusselt number distributions, the first of their kind for vortex tube flows. A notable finding is that the vast majority of heat added to the vortex tube flow remains within the hot stream; i.e., the cold stream experiences relatively little temperature rise due to the heat addition. For example, even when only 30% of the flow exits the hot side of the tube, it retains more than 80% of the heat added to the flow. Additionally, a modified swirl number was also defined that was found to scale the Nusselt number augmentation across the two different total flowrates examined presently.

It is well known that swirling flows in a circular pipe augment heat transfer compared to non-swirling flows. Swirl is typically induced by curved flow obstructions such as twisted tapes or by tangentially injected fluid streams. One classical example of a strongly swirling flow is that inside the Ranque – Hilsch vortex tube, which is a fluidic device commonly employed for its distinctive temperature separation characteristics. Georges Ranque obtained French and US patents for the device in 1933 [ 1 ] and 1934 [ 2 ], respectively, but the device was not well known until Hilsch [ 3 ] eventually wrote about the device in 1947. By tangentially injecting pressurized gas through a set of nozzles, a vortex tube creates an intense confined vortex that drives a transfer of energy internal to the flow. The outer layer of the vortex achieves a higher total temperature than the inlet and is exhausted from an annular exit at one end of the tube, while the inner core is brought to a lower total temperature than the inlet and is exhausted from an orifice at the opposite end of the tube.

The cold stream, m ˙ c ⁠ , is drawn from the cooler core of the vortex and the hot stream, m ˙ h ⁠ , is drawn from the hotter periphery. Only the hot stream is in contact with the vortex tube wall; therefore, any heat transferred from the wall to the cold stream must be conducted through the hot stream. A diagram approximating the flow paths of the hot and cold streams is shown in Fig. 1 . Both streams follow a vortical flow path initially in the direction of the hot exit, but the cold stream reverses its axial direction and exits via the cold exit.

Vortex tube inlet flow schematic. Note that both cold and hot flows have a strong swirling component around the axial direction that is not depicted in this two-dimensional schematic.

Current applications of vortex tubes focus primarily on industrial settings where compressed air is readily available, such as spot cooling for machining or for cabinet coolers. However, to understand how vortex tubes might be best incorporated into modern heat transfer systems, it is of great practical interest to gain new insights into the heat transfer characteristics of the tube walls containing the swirling flow. Efforts to date have been limited. Most vortex tube studies tend to be concerned with factors that affect the performance of vortex tubes. Besides parametric studies of geometric features, many have concerned themselves with the role of inlet-to-outlet pressure ratios and the cold fraction. Stephan et al. [ 4 ], for instance, did just that by examining the flow of air through a vortex tube. Among their findings was that the maximum temperature drop measured in the cold exit flow occurred with a cold fraction of μ C ≈ 0.3 and the highest inlet pressure they used (5.0 bar gauge), which provided a temperature drop of 37 K below the inlet temperature. Other more recent studies such as that by Cartlidge et al. [ 5 ] have expanded on the earlier studies by also examining the results in terms of the refrigeration coefficients of performance and the isentropic efficiencies of the devices under various conditions. Many other studies of vortex tube heat transfer, such as those by Fulton [ 6 ], Scheper [ 7 ], Scheller [ 8 ], Sibulkin [ 9 ], and Bruun [ 10 ], are dedicated to unraveling the mechanism of temperature separation and therefore are focused on heat transfer within the vortex. The aim of this paper, by contrast, is primarily to characterize heat transfer into the tube at the interface between the fluid and the solid wall and to investigate the practical consequences. To the authors' knowledge, only one paper in 1990 examined the heat transfer characteristics of vortex tubes. Lin et al. [ 11 ] cooled a vortex tube exterior with a water jacket and devised a correlation between the Nusselt number, Reynolds number, and ratio of mass flowrates at the tube exits. The analysis considered only the overall heat removed from the tube and thus yielded mean Nusselt numbers rather than local values, and no information was provided regarding the influence of tube cooling on the temperature separation characteristics of the vortex tube. Furthermore, the work on Lin et al. did not consider heat addition to the vortex tube, which is of importance for practical applications involving the use of embedded vortex tubes employed for thermal management. No previous research has been found regarding the effect of external heat addition on vortex tube performance, nor of local heat transfer characteristics of a vortex tube.

In addition to examining those aspects of vortex tube behavior, the present paper is concerned with the extent to which heat added to the flow is retained in the hot stream versus transferred to the cold stream. The mechanism of vortex tube temperature separation, while not central to the present paper, is nevertheless relevant here. Previous computational studies by Behera et al. [ 12 ] and Aljuwayhel et al. [ 13 ] have suggested that the phenomenon is driven by viscous work transfer from the vortex core to the periphery, while heat transfer simultaneously transfers energy from the hotter periphery to the core and therefore mitigates temperature separation. The contribution was substantial: Aljuwayhel et al. [ 13 ] estimated heat transfer to be equivalent to 62% of the net energy separation for μ C = 0.30, while Behera et al. [ 12 ] estimated it to be 13–126% of the net energy separation for 0.65 ≤ μ C ≤ 0.85. In essence, these researchers have posited that a strong mechanism for transferring energy from the periphery to the core in a vortex tube does exist, but under normal circumstances, it is outpaced by the mechanism for energy transfer from the core to the periphery. This is an important concept to keep in mind in this novel examination of vortex tube behavior when external heat is added to the fluid at the periphery and the question of how much of the added heat is transferred into the core flow that ultimately exits as the cooler of the two streams.

Bulk analysis will examine the fraction of added heat that is retained in the cold flow, α , and consider the extent to which it depends on flow and heat transfer parameters such as vortex tube cold fraction, heat flux, and vortex tube nozzle velocity.

In film cooling applications described by Bogard and Thole, heat transfer experiments are normally coupled with auxiliary experiments performed with an adiabatic surface for the purpose of directly measuring the adiabatic wall temperature which differs substantially from the freestream temperature. A similar approach is used herein where an adiabatic boundary condition is used to determine the adiabatic wall temperature distribution. Unlike traditional pipe flow, the energy separation occurring within the vortex tube along with the region of reverse flow causes the adiabatic wall temperature to vary with position along the length of the tube. A sample adiabatic wall temperature distribution is shown in Fig. 2 for m ˙ = 1.42 g / s and μ C = 30 demonstrating how the adiabatic wall temperature increases somewhat as the flow near the periphery of the tube gains heat from the flow closer to the center of the tube. Also shown in Fig. 2 is the companion non-adiabatic data with q″ = 5 kW/m 2 .

Sample T aw distribution ( ⁠ m ˙ = 1.42 g / s and μ C = 0.30) along with wall temperature distribution with q″ = 5 kW/m 2

The present investigation examines vortex tube performance under the condition of external heat addition using experimental and computational techniques. First, the bulk effects of heat addition on temperature separation were established experimentally. Next, a more detailed examination of local heat transfer characteristics was accomplished computationally.

A schematic of the vortex tube laboratory employed is shown in Fig. 3 . Pressurized air was supplied through a pressure regulator and heat exchanger to establish desired inlet conditions. Pressure and temperature were measured at the inlet to the vortex tube with a NetScanner pressure transducer measured static pressure and an Omega grounded-junction K-type thermocouple, and compressible flow relations were applied to determine properties at the vortex tube nozzles in a process detailed in Fuqua and Rutledge [ 22 ]. The vortex tube was a modified Exair ™ model 3208 (shown in Fig. 4 ) with a model 8R vortex generator (shown in Fig. 5 ). The inside diameter of the stainless steel tube was 6.2 mm and inside length was 58.1 mm, so the internal surface area of the tube wall is 1.13 × 10 −3 m 2 . The vortex generator features six rectangular nozzles, each with dimensions of 0.7940 mm by 0.660 mm. An internal cruciform feature evidently used as a flow straightener was removed from the stock vortex tube resulting in a small deterioration in the performance of the vortex tube, but greatly simplifying computational modeling of the flow. Pressure and temperature were also measured at the exits, and the cold fraction was set by two Omega 2612A digital mass flow controllers. Heat exchangers were used to protect the mass flow controllers from elevated exit temperatures. The ratio of the Grashof number to the square of the Reynolds number is of order 0.01 indicating that free convection had no effect on the internal heat transfer behavior. As for external heat transfer, some heat from the heater cord is lost due to natural convection, but this was kept to a minimum through insulation and only the energy added to the vortex tube was used in the energy balances.

Vortex tube laboratory schematic

Exair ™ 3208 vortex tube

Solid model of Exair ™ 8R vortex generator

The vortex tube was heated externally by an Omega HTC-030 heater cord, powered by a Staco 3PN1510B variable transformer; approximately 51 cm of cord was coiled tightly around the tube and then foam rubber insulation was wrapped around the vortex tube and heater cord. Heat addition to the vortex tube was controlled by adjusting the voltage output by the variable transformer to the heater cord, specified as a percentage of the transformer's rated voltage of 120 VAC. Cases were run in increments of 20% of the rating, from 0% to 80%. The case of 0% voltage, i.e., with the heater off, was conducted using an inlet temperature that was closely matched to ambient conditions, nominally 302 K, to minimize heat transfer between the tube and the surroundings despite the insulation. The inlet temperature was then held approximately constant for all tests. The flowrate for the tests was nominally 1.420 × 10 −3 kg/s (72 standard liters per minute, (SLPM)) and inlet gauge pressure was nominally 3.0 bar. Actual nozzle conditions for all cases, averaged across the cold fractions, are shown in Table 1 .

Nozzle conditions for heat addition cases

Nozzle velocity was computed using standard compressible flow relations whereby the flow path and reduction in cross-sectional area between the pressure and temperature instrumentation upstream of the vortex tube and the vortex tube nozzles is modeled as an isentropic converging nozzle. The process of computing nozzle velocity is outlined in the following.

It must be emphasized, however, that although the acceleration of the gas through the nozzles is modeled as an isentropic converging nozzle and therefore it is assumed that the total pressure loss in the 7 cm stretch between the instrumentation section and the nozzles is negligible, the vortex tube as a whole is certainly not an isentropic device and incurs a substantial pressure drop through the tube itself (the discharge coefficient across the nozzle was determined to be 0.92 through the computational fluid dynamics (CFD) simulations performed in this work). Detailed information regarding the pressure drop across this device for the same configuration has been documented in Ref. [ 22 ].

Measurements for the heated cases were taken at cold fractions of μ C = 0.3, 0.4, 0.5, and 0.7. The heat added to the flow was computed using an energy balance outlined in Eq. (2) , using the mass flowrates and specific enthalpies at the inlet and hot and cold exits. Air properties were drawn from nist refprop 9.1 [ 24 ].

Inlet and exit properties for each data point were computed as the mean of ten successive measurements sampled at a rate of 1 Hz. The accuracy of each mass flow controller was equal to 0.2% of full scale (500 SLPM) plus 0.8% of the measured value (72 SLPM), corresponding to an uncertainty of 2.2% of the measured value for each mass flow controller. The nominal accuracy of the thermocouples was 0.5 K, though the standard error of the measured values for each data point—an indicator of precision uncertainty—was typically less than 0.1 K. The accuracy of the pressure measurements was 345 Pa (0.05 psi), though associated standard error was typically less than 138 Pa (0.02 psi). Using the method of Kline and McClintock [ 25 ], the uncertainty in nozzle velocities was found to be approximately 3.0 m/s or 2.9% of the measured value.

The temperature separation curves corresponding to the heat addition cases are presented in Fig. 6 . Most notably, the addition of heat to the vortex tube causes a dramatic increase to the temperature of the flow exiting via the hot exit, while only slightly increasing the temperature at the cold exit. For example, at a cold fraction of μ C = 0.50, the temperature change at the cold exit increases by 2.1 K from the 0% rating case (heater off) to the 80% case, while that of the hot exit increases by 23.2 K. Even with heat added to the vortex tube, the cold flow is still at least colder than the inlet for every case; in the most extreme case—with the heater setting at 80% of full scale at a cold fraction of μ C = 0.70—the hot flow increased in temperature by 44.7 K compared to the inlet while the cold flow temperature decreased by 4.4 K below the inlet temperature.

Temperature separation curves with heat addition

It is not surprising that the hot exit flow is more sensitive to external heat addition than the cold exit flow; the hot exit flow is, after all, immediately adjacent to the tube wall. However, the temperature difference between the hot and cold exit flows across a 3.1 mm inside radius tube is nevertheless striking.

Figure 7 indicates that heat addition at a constant heater power setting is a function of cold fraction, where less heat is added at higher cold fractions. Since the heater power setting is constant along each curve in Fig. 7 , this indicates that the remaining heat output from the cord heater is lost to the surroundings, a consequence of a higher cord heater temperature and imperfect insulation around the heater. The mechanism for this is explained by the fact that the hot exit flow experiences a steep rise in temperature with increasing cold fraction—a classic vortex tube characteristic—and the tube wall likewise gets hotter. It follows that the radial temperature gradient decreases, which reduces the heat flux from the cord heater through the tube wall. Thus the remainder of the heat from the cord heater is conducted radially outward, away from the vortex tube and to the environment.

Heat transferred to the vortex tube for each case

The heater-off case experiences some heat loss from the tube wall in spite of the insulation, and the 20% rating case is nearest to an adiabatic condition: the elevated temperature of the vortex tube due to the hot flow is very nearly offset by the warm temperature of the heater, apparently minimizing the temperature gradients. At a cold fraction of μ C = 0.40, the heat transfer for the 20% case is calculated to be 0.22 W, which corresponds to a mean surface heat flux of 195.0 W/m 2 . This is clearly not zero, but it is small compared to all other cases and it will serve as the quasi-adiabatic reference case for higher heater settings at μ C = 0.40. Since the heat transfer into the vortex tube is computed from bulk transport properties, it is possible that the local heat flux is locally positive or negative at various axial positions along the tube, but instrumentation was not available to determine local values. For each case at a cold fraction of μ C = 0.40, the heat added to the vortex tube and the corresponding mean heat flux are compiled in Table 2 . Using the 20% case as the near-adiabatic reference case, the fractions of heat remaining in the cold flow and hot flow, m ˙ c o l d i t , c o l d − m ˙ c o l d , r e f i t , c o l d , r e f q − q r e f ⁠ , m ˙ h o t i t , h o t − m ˙ h o t , r e f i t , h o t , r e f q − q r e f ⁠ , and β are shown for the 40%, 60%, and 80% cases. The residual is the difference between unity and the combination of the three terms using Eq. (4) ; this represents the experimental error in the energy balance and is at most 0.063%.

Results for the cases ( μ C = 0.40)

The fractions of heat retained in the cold and hot flows were then renormalized using Eqs. (6) and (7) to yield values of α and 1 − α , as shown in Table 3 . For the cases available, only 1.0–7.5% of heat added to the vortex tube is conducted to the cold flow. The overwhelming majority of the heat added to the vortex tube, between 92.5% and 99.0%, is retained in the hot flow. This is the first demonstration of the effect of heat addition on temperature separation in a vortex tube in the known literature.

Results for the cases ( μ C = 0.40), continued

The computational geometry was based on the modified stock vortex tube without the cruciform flow straightener with the annular exit. As with the experimental hardware, six rectangular nozzles inject air tangentially into the vortex tube; the nozzle dimensions of 0.7940 mm by 0.660 mm yield a hydraulic diameter of 0.721 mm. The length of the tube, from the nozzles to the hot exit, was 58.1 mm. A cone with a 60 deg half-angle and a diameter of 2.2 mm was used to obstruct the center of the hot exit, the last 0.44 mm of which was tapered by 30 deg; this was done to closely mimic the experimental configuration. The origin with respect to axial position is located at the side of the nozzle closest to the cold exit; the cold exit plane is at an axial position of −1.0 mm. Figure 8 depicts the geometry implemented, with tube walls in gray, inlet faces in green, cold exit orifice face in blue, and hot exit annulus face in red; these are the faces at which nozzle and exit properties were computed. A three-dimensional unstructured mesh was created from the basic geometry using pointwise v18.0 via its automated hybrid mesh generation method known as anisotropic tetrahedral extrusion, or “T-Rex.” A close-up view of the mesh is shown in Fig. 9 ; however, for reasons to be described, the mesh required further modification at the exits to achieve satisfactory solutions.

Geometry used in computational investigations: walls in gray, cold exit face in blue (top), hot exit face in red (bottom), and nozzle faces in green

Unstructured mesh used for CFD modeling (cold exit face highlighted)

The ansys fluent v17.2.0 computational fluid dynamics software was used to model the flow. The intrinsic complexity of the flow field inside a vortex tube greatly complicated the computational study, and considerable effort was exerted to identify the most appropriate turbulence model, solver settings, gas model, and boundary conditions which yielded reliable, stable temperature separation profiles. The standard k – ɛ model was implemented with a standard wall function; viscous heating and compressibility effects were enabled. A pressure-based solver was adopted with a semi-implicit method for pressure-linked equations pressure-velocity coupling scheme. Spatial discretization was accomplished with the Green–Gauss node-based method for gradients, the fluent v17.2 “standard” method for pressure, second-order upwinding for density and momentum, and quadratic upstream interpolation for convective kinematics for turbulent kinetic energy, turbulent dissipation rate, and energy. A real-gas model was applied based on the nist refprop database resident in fluent . fluent is capable of computing either relative or absolute total temperature, depending on the reference frame selected. In the present work, the solid boundary was stationary and the reference frame was set to “absolute”; therefore, all total temperatures are reported in absolute terms.

The standard wall function in fluent 17.2 [ 26 ] uses the methodology described by Launder and Spalding [ 27 ], including the “law of the wall” approach. This drives mesh requirements to ensure that y + wall units remain within bounds wherein the law of the wall remains valid; experimental findings summarized by Pope [ 28 ] suggest this range is approximately 30 < y + < 300. Figure 10 illustrates the actual y + values for the mesh that was actually utilized in the study, which generally lie in the desired range with minor departures in the region closest to the hot exit.

Dimensionless wall distance, y + , along the vortex model tube walls

Boundary conditions represented a challenging aspect of matching the experimental characteristics. One of the most fundamental variables to control when modeling vortex tube operation is the cold fraction, and building a characteristic set of performance curves requires precise prescription of both the mass flowrate entering the vortex tube and that exiting from the cold side. Because the mesh inlet faces were actually the nozzles, the terms “inlet” and “nozzles” are synonymous here; however, the term “nozzle” will be used because of the significance of nozzle properties as demonstrated in recent experimental studies [ 22 , 29 ].

The inlet boundary condition was best modeled using a mass flowrate condition with a specified total temperature. The pressure at the inlet (or nozzles), however, is not known a priori and emerges from the flow solution. The exit boundary conditions present a challenge since the software prohibits the use of “outflow” boundary conditions—in which the mass flowrate at the exits is defined—for its compressible flow solver. Instead, fluent v17.2.0 includes a solver tool to automatically adjust the exit static pressure at a pressure outlet to achieve a target mass flowrate across that boundary. In practice, the inlet mass flowrate was set to the desired value and the cold exit was set to a pressure outlet with a target mass flowrate corresponding to the desired cold fraction. As the computational simulation progressed, the necessary adjustments were applied iteratively to achieve this cold fraction until the solution converged. The cold outlet pressure adjustments were performed computationally without a human in the loop to ensure appropriate convergence. Where precise control of the inlet pressure was desired, this required run-by-run manipulation of the hot exit pressure outlet boundary condition.

One final boundary condition also required close attention: the reverse flow temperature. This was due to a swirling flow phenomenon described by Gupta et al. [ 30 ] as a “central toroidal recirculation zone (CTRZ),” which is responsible for reverse axial flow in regions of high swirl ( S ≥ 0.6). A region of reverse axial flow was indeed identified in the region surrounded by the cold exit and the inlets (nozzles). The reverse flow temperature defines the temperature of fluid brought into the mesh via an exit face in the event of local reverse flow. While this reverse flow is certainly accounted for in the overall energy balance of the flow solution, a CTRZ has the potential to distort the understanding of vortex temperature separation, since it is otherwise expected that the specific enthalpy of the flow at the exit faces is only a function of the temperature separation phenomenon in the vortex tube as well as any heat addition.

Two methods were applied to mitigate the effect of the CTRZ on the computational results. First, two extensions were added to the mesh: a 5 mm extension on the cold exit and 3 mm on the hot exit. These were found to move the flow feature out of the region of interest and away from the exit planes where the exit temperatures were computed, thereby minimizing any effects on observed temperature separation, although at an added computational cost. This is shown in Figs. 11 and 12 , which plot contours of axial velocity for the original, non-extended geometry, and the exit extensions, respectively. Second, and in addition to mitigating the CTRZ distortion through the extended fluid domain, the reverse flow temperature boundary conditions at the extended exit planes were specified such that the temperature of any reverse flow would be matched to that of the outgoing flow.

Original vortex tube fluid domain, shown with contours of axial velocity

Vortex tube fluid domain with exit extensions, shown with contours of axial velocity

A mesh sensitivity study was conducted, with cell counts of 1.6 × 10 6 , 2.5 × 10 6 , 5.2 × 10 6 , 9.4 × 10 6 , and 16 × 10 6 , respectively. Inlet boundary conditions were specified with a mass flowrate of 0.00142 kg/s (72 SLPM) and a total temperature of 293.4 K. The cold exit pressure outlet was set to achieve a cold fraction of μ C = 0.40, which typically yields the greatest cold exit temperature drop. The hot exit pressure outlet used a static pressure of 345 kPa. The meshes were compared in terms of temperature separation, quantified as the difference in mass-averaged total temperature between the exit measurement planes and the inlet plane. An additional measurement was included to assess net changes in the flow field: a volume-integrated x -vorticity. This was selected because the x -vorticity is one of the most prominent indicators of the swirl of the vortex; in the present configuration, the x -axis spans the length of the vortex tube. By integrating the x -vorticity across the volume of the mesh, a single indicator could aggregate large-scale changes in the flow pattern.

The results of the sensitivity study are presented in Fig. 13 . Hot and cold exit temperature separation are scaled with the left vertical axis. The x -vorticity is scaled according to the right vertical axis. Temperature separation and integrated x -vorticity are fairly sensitive to mesh size from 1.6 × 10 6 to 5.2 × 10 6 cells, but much less sensitive from 5.2 × 10 6 to 16 × 10 6 cells. Hot exit and cold exit temperature separation increase by 32% and 28% from 1.6 × 10 6 to 5.2 × 10 6 cells, while they increase by only 5.7% and 5.9%, respectively, from 5.2 × 10 6 to 16 × 10 6 cells. Similar trends occur in x -vorticity, though the maximum observed value is at 9.4 × 10 6 cells. Since the purpose of the mesh sensitivity is to balance concerns of accuracy with computational cost, it was decided to use the mesh with 5.2 × 10 6 cells for all subsequent investigations. The average cell size for this grid was 5.31 × 10 −4 mm 3 .

Mesh sensitivity study results ( μ C = 0.40)

Finally, verification of the computational model was conducted, comparing the results against an experimental benchmark case. The experimental work, which is described earlier in this paper, was performed, and the verification exercise involved a thorough comparison of the results as presented in Table 4 and in Fig. 14 . The boundary conditions were carefully matched between the experimental and computational cases. Specifically, mass flowrates and total temperatures were matched at the nozzles, hot exit static pressures were matched, and the cold exit static pressure was permitted to goal-seek to achieve the desired cold fractions. The computational solutions underestimated the pressure drop between the nozzle and hot exit by approximately 5%, with corresponding deviations in nozzle density, velocity, and Mach number. The computational simulations nevertheless yielded very similar temperature separation curves, as shown in Fig. 14 : the temperature separation curves track each other closely, although the CFD consistently overpredicted the temperature increase at the hot exit by approximately 0.4 K, and overpredicted the temperature drop at the cold exit by as much as 0.7 K for μ C = 0.30 to only 0.15 K for μ C = 0.70. This favorable comparison gives confidence that the CFD model faithfully represents the fundamental physics in the vortex tube.

Comparison of CFD temperature separation to experiment

Nozzle properties for verification case, experiment versus CFD ( μ C = 0.40)

Forty cases were run for two mass flowrates (1.42 g/s and 2.00 g/s), three heat flux conditions (5, 10, and 15 kW/m 2 ), and an adiabatic condition, and covering a range of cold fractions from μ C = 0.3 to μ C = 0.7. The heat flux was applied to the entire exposed surface area of 12.41 cm 2 , which included both tube walls and nozzle walls; therefore, the heat added for the cases with q ″ = 5 kW/m 2 , q ″ = 10 kW/m 2 , and q ″ = 15 kW/m 2 was 6.205 W, 12.410 W, and 18.615 W, respectively. Six of the heat flux cases did not achieve a converged solution, leaving 24 successful heat flux cases for analysis in addition to all 10 of the adiabatic cases. The cases are listed in Table 5 , in which non-converged solutions are indicated in italics and omitted from the presented results.

CFD cases for heated vortex tube analysis

Note: Standard values: converged solutions; italicized values: non-converged solutions.

Nozzle conditions for the two mass flowrates are shown in Table 6 , corresponding to the adiabatic cases for μ C = 0.3. These nominal values apply to all low- and high-flow cases, both adiabatic and non-adiabatic; nozzle velocity varied by approximately 0.2 m/s (0.2% of nominal) across cold fractions for low-flow cases and 0.6 m/s (0.4% of nominal) for high-flow cases. Mass and energy balance analyses were conducted for each case and the detailed residuals are listed in   Appendix .

Nozzle properties for low-flow and high-flow cases

Temperature separation curves for the low- and high-flow cases are plotted in Figs. 15 and 16 , respectively. Figure 15 also shows the experimentally obtained temperature separation curves in green for the low flow condition and 80% heater power for which q m e a n ″ = 14948 W / m 2 ⁠ ; the heat fluxes and inlet conditions for computational and experimental cases are very similar but not identical, and the results compare favorably.

Temperature separation curves for low-flow cases

Temperature separation curves for high-flow cases

Fractions of added heat retained in the hot and cold flows

Figure 17 shows that the overwhelming majority of added heat is retained in the hot flow. Values of 1 − α range from 0.938 to 0.945 at a cold fraction of 0.30, and from 0.814 to 0.827 at a cold fraction of 0.70. In other words, even when the hot stream accounts for only 30% of the inlet mass flow, it retains more than 80% of the heat added to the flow. This is particularly striking when it is considered that the radius of the vortex tube is only 3.1 mm and therefore the hot and cold streams flow adjacent to each other in quite a compact space. The resulting temperature gradients within the flow are therefore quite large, but result in remarkably little heat transfer into the cold stream. These computational results compare favorably with experimental results: CFD results indicate that at a cold fraction of 0.40, 92.1–92.3% is retained in the hot flow, whereas the experimental results found that 0.925 ≤ 1 − α ≤ 0.990.

Swirl within the vortex tube flow and heat transfer characteristics on the inner surface of the vortex tube were closely examined. A cross section of the tube plotting tangential velocity at an axial location of 0.3 mm, i.e., near the middle of the nozzles, is shown in Fig. 18 and indicates the high degree of swirling flow in the tube.

Vortex tube tangential velocity at the axial position of 0.3 mm, inlet condition 1

The flow pattern is highly dependent on cold fraction because it alters the path of the hot and cold streams considerably; the flow pattern depends to a lesser extent on the nozzle condition and is virtually independent of the heat flux. Consider, for instance, how the radial location at which flow reversal occurs, i.e., the radial location corresponding to zero axial velocity at a particular station, varies between cases as shown in Fig. 19 .

Normalized radial location at zero axial velocity along the tube length

The resulting swirl number is plotted in Fig. 20 . For a given cold fraction, the progression of swirl along the length of the tube is virtually identical for all other conditions, demonstrating that swirl is largely independent of heat flux and nozzle condition when using the methodology shown here. Interestingly, the results indicate that swirl is slightly higher for the lower mass flowrate when within one diameter of the nozzles.

Swirl number as a function of axial location

This correlation is also plotted in Fig. 20 . The swirl in the vortex tube is found to be approximately two to three times as intense at any given axial location compared to pipe flow, though the vortex tube swirl decays at a similar rate. It is acknowledged that this is not entirely an apples-to-apples comparison: whereas the mass flow accounted for in Hay and West's pipe flow correlation is uniform with respect to axial position, the same cannot be said for the vortex tube. As the axial direction of the cold stream reverses, the mass flowrate accounted for by the swirl number decreases steadily with axial position.

Vortex tube local heat transfer augmentation along the vortex tube length

Curves of Nu D /Nu ∞ collapse nicely for heat flux and inlet conditions, indicating that local heat transfer augmentation is not a strong function of nozzle velocity; it is apparent, however, that cold fraction has a prominent effect on local heat transfer augmentation beyond x / D ≈ 1.5, where smaller cold fractions yield greater augmentation values. Axially-averaged heat transfer augmentation values span 3.95 ≥ Nu ¯ D / N u ∞ ≥ 4.89 ⁠ , bounded by cases with Inlet state 2 (high flow) and μ C = 0.70 on the low end, and Inlet state 1 (low flow) with μ C = 0.30 on the high end.

An intermediate relationship between local heat transfer augmentation and swirl number is presented in Fig. 22 , which again indicates that cold fraction dominates the differences in heat transfer characteristics between cases.

Vortex tube local heat transfer augmentation as a function of swirl number

The temperature separation characteristics of the Ranque – Hilsch vortex tube are conventionally examined in the context of an adiabatic vortex tube; i.e., with regard for the internal fluid mechanics that motivate the separation of an inlet stream into two streams of different total temperatures. The present study expands the concept of vortex tube temperature separation to also include the extent to which the temperatures of the two streams remain separate under the condition of external heat addition.

The study demonstrates that the Ranque – Hilsch vortex tube has rather large internal heat transfer coefficient distributions. In the mid-length region of the tube, for instance, the Nusselt number is approximately four times that which would be expected of fully-developed turbulent axial flow in the same size tube. In that respect, vortex tubes show promise for novel heat transfer applications, such as cooling devices within certain hot structures. Remarkably, the cold flow exiting its respective end of the tube is relatively unheated; instead, the vast majority of the added heat is retained in the hot flow: even when the hot flow represents only 30% of the inlet flowrate, it retains over 80% of the added heat. This has the tendency to increase the temperature difference between the hot and cold exit flows.

Graphical relations were also presented between swirl number, axial location, and local heat transfer augmentation in the vortex tube. The definitions of axial momentum fluxes that appear in the swirl number were modified in such a way as to eliminate the region of reverse flow occurring within the central core of the vortex tube. This modified swirl number was shown to scale the Nusselt number augmentation across both flowrates of 1.42 g/s and 2.00 g/s examined presently.

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.

specific enthalpy, J/kg

thermal conductivity, W/(m · K)

arbitrary length scale, m

heat addition, W

radial position, m

axial location downstream of injection site

cross-sectional area, m 2

vortex tube inside diameter, m

pressure, Pa

tube inside radius, m

swirl number, Eq. (8) , dimensionless

temperature, K

velocity, m/s

mass flowrate, kg/s

axial component of velocity, m/s

tangential component of velocity, m/s

specific heat at constant pressure, J/(kg · K)

hydraulic diameter, m

axial flux of angular momentum, Eq. (9) , kg · m 2 /s 2

axial flux of axial momentum, Eq. (10) , kg · m/s 2

heat flux, W/m 2

dimensionless wall distance

Nusselt number, hD / k

Prandtl number, C p µ/k

Reynolds number, ρVD / µ

fraction of heat added to cold flow, Eq. (6)

last term in Eq. (4)

density, kg/m 3

dynamic viscosity, Pa · s

cold fraction

measured at the vortex tube cold exit

measured at the vortex tube hot exit

total property, i.e., stagnation value

adiabatic wall

measured at either vortex tube exit

measured at the instrumented inlet upstream of the vortex tube

measured at the vortex tube nozzles

static property

fully developed

Mass and energy balance analyses were conducted for each of the cases. The control volume of interest represented a subset of the entire mesh, bounded by the mesh inlet face and tube walls, but by arbitrary surfaces at the hot and cold exits to avoid the extension regions and reverse flow. A consequence of using this arbitrary volume was that the mass and energy balance residuals were larger than would be expected from the entire mesh. The mass balances for the low and high-flow cases are presented in Tables 7 and 8 ; residuals ranged from 0.14% to 0.51% of the inlet flowrate. Energy balances are shown in Tables 9 and 10 . It follows that the energy balance residuals closely track the mass balance residuals and range from 0.16% to 0.53% of the inlet advection. Moreover, it can be seen that the energy balance residuals are essentially independent of heat flux.

Mass balance analysis, low-flow cases

Mass balance analysis, high-flow cases

Energy balance analysis, low-flow cases

Energy balance analysis, high-flow cases

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Introduction to the Ranque–Hilsch Vortex Tube

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• André Kaufmann 2  

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Since the process of temperature separation by the vortex tube was first patented and published by Georges J. Ranque in 1933, the underlying physics have intrigued scientists and engineers. The device was continuously refined and found use in a variety of applications ranging from the cooling of milling tools to the cooling of clothing for firefighters. The vortex tube bifurcates a high pressure gas flow between a cold and a hot mass gas fraction. Effects of mixing are well known, but the effect of separating flow into cold and and hot is unique to the device.

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Kaufmann, A. (2022). Introduction to the Ranque–Hilsch Vortex Tube. In: The Ranque Hilsch Vortex Tube Demystified. Springer, Cham. https://doi.org/10.1007/978-3-030-89766-6_1

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