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Secondary flow in a vortex tube
FLUIDDYNAMICS RESEARCH ELSEVIER
Fluid Dynamics Research 21 (1997) 73-86
Secondary flow in a vortex tube Boye Ahlbom*, Stuart Groves Department of Physics, The University of British Columbia, 6224 Agricultural Road, Vancouver, B.C, Canada V6T 1Z1 Received 10 May 1996; revised 24 October 1996; accepted 13 January 1997
Abstract A novel Pitot probe was used to measure the axial and azimuthal velocities in a vortex tube. The probe has only a single measuring port and is hence smaller than standard devices. It monitors stagnation and reference pressure sequentially as the probe is rotated around its axis. From the measured velocity field in the 25 mm diameter vortex tube the local mass flux was determined and it was observed that the return flow at the center of the tube is much larger than the cold mass flow emerging out of the cold end. Therefore, the vortex tube must have a secondary circulation imbedded into the primary vortex, which moves fluid from the back flow core to the outer regions.
Keywords: Ranque Hilsch tube; Secondary circulation; Velocity measurements; Velocity vector angle
I. Introduction A vortex tube is a cylindrical pipe o f radius R and length L typically o f the size o f a bicycle pump. It has one or more tangentially oriented inlet holes arranged on the circumference close to one end, and two exits: a small hole on the axis close to the inlet plane with a typical radius Rc ~ 0.3R and a larger single or multiple port on the far end o f the pipe (Fig. 1). This simple mechanical device separates a stream o f gas entering at a rate j0 (kg/s) into a hot flow jh that emerges on the far end, and a cold component jc which issues out o f the small hole on the axis. In this "temperature separation" process some heat finds its way from the cold to the hot part o f the flow. Most vortex tubes have a valve at the hot end b y which the cold flow fraction Y =jc/jo can be adjusted in the range 0 ~< Y ~< 1. The effect depends on the inlet pressure P0 and on the mass flow ratio Y. Vortex tubes are used for specialized cooling applications and are commercially available. The vortex tube effect was discovered in 1933 by Ranque (1933), and examined in detail b y Hilsch (1946) and has since been the subject o f many studies. Several different qualitative explanations have been offered to explain the anomalous heat flow: (i) Internal friction with various assumptions about the radial and axial velocity distribution (van Deemter, 1952; Pengelly, 1957; Dreissler and Perlmutter, * Corresponding author. 0169-5983/97/$17.00 (~) 1997 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved. PIISO 1 6 9 - 5 9 8 3 ( 9 7 ) 0 0 0 0 3 - 8
74
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73-86
~
(b )
(a) pitot pipe
inlet
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.......~ z
hot
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Fig. 1. (a) Vortex tube with location of Pitot pipe. (b) Geometry of Pitot pipe.
1960; Sibulkin, 1962a,b, Fig. 12) which includes acoustic streaming (Kurosaka, 1982) whereby more momentum is pumped into the swirling flow; (ii) Turbulent heat transfer in a stratified flow (SchultzGrunow, 1951); (iii) Goertler vortices (Stephan et al., 1983); (iv) Compressibility of the working fluid (Amitani et al., 1983); (v) Turbulent transfer of thermal energy in an incompressible flow (Lindstrom Lang, 1971 ); and (vi) Effects associated with the particular working fluid (Erd61yi, 1962). While each of these explanations may capture certain aspects of the device, none of the models has sufficient predictive power to allow the optimization of the temperature separation (Balmer, 1988). This implies that none of these mechanisms altogether explains the Ranque Hilsch effect. Experimental investigations which only deal with the parameters at the entrance and the exit ports (Schultz-Grunow, 1951; Stephan, 1983; Ahlborn et al., 1994) are unlikely to yield new understanding of the temperature separation mechanism. More insight could be gained from detailed measurements of the internal flow structure, namely from a knowledge of the r, ~0 and z components of the velocity u. Lay (1959) mapped the fields of pressure, temperature, and speed as a function of radius and downstream position z. Lay's various probes did not allow him to measure any of these quantities at radii smaller than 0.3R. In addition, he only presented the velocity component uz at a single axial position, so that the full structure of velocity field could not be deducted. The complete pressure, temperature and velocity fields were measured by Bruun (1969) in a 90 mm diameter vortex tube, and by Takahama (1981) in various tapered vortex tubes. Bruun used a Pitot pipe (diameter 3 ram) with 3 holes located about 9 mm away from its tip to measure the stagnation pressure, and the free stream pressure. He employed a thermocouple mounted inside a 3 mm pipe with a fine measuring hole to measure the local temperature. The pressure probe could
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73~6
75
be rotated to determine the flow direction and hence the azimuthal and axial velocity components. A similar setup was used up by Takahama. His 3 mm diameter Pitot probe had only 2 holes mounted at an angle of 81 o. The vortex tubes used by Bruun and Takahama had relatively large diameters. Therefore, the perturbations caused by their 3 mm diameter probes could be neglected in first approximation. The mass flow ratios Y =j~/jo in both sets of experiments were chosen in the range 0.23 ~< Y ~<0.7 where the vortex tube produces a significant temperature separation effect. These experimental data depict a flow field with a core like a forced vortex surrounded by an external annular flow somewhat like a free vortex. The core shows a back flow region in which the flow direction is opposite to the hot fluid on the periphery. It was not known from these experiments if the back flow disappears when the cold gas flow is cut off entirely (Y = 0). O f interest here is the structure of the flow, and how it develops over the hole range of mass fractions, starting at Y = 0. The most accurate and non-invasive measurements of the local velocity components in flow fields are presently obtained with laser Doppler (LD) methods (see, for instance, Robertson and Crow, 1975). Unfortunately, it appears impossible to seed the flow with scattering particles, because due to the high centrifugal acceleration (typically 1069) all particles injected into the gaseous working fluid remain close to the wall so that no scattering centers reach the axial regions of the vortex tube. For this reason, apparently, Schlenz (1982) used his laser Doppler system only with water as working fluid and only in an unidirectional vortex tube. One must therefore stick to the tried and proven Pitot devices. The Pitot pipes deployed by Bruun and Takahama and Yokosawa with two or three holes, respectively, had a diameter of 3 mm. Such devices would present too much of an obstacle to the flow in our 25.4 mm diameter vortex tube. We have therefore developed a smaller Pitot pipe which is stretched across a diameter of the vortex tube at an arbitrary position z. The probe measures stagnation and reference pressure sequentially but otherwise functions like a Pitot tube. Using this probe we have detected in the middle of the vortex tube a secondary circulation pattern, somewhat like a vortex ring with superimposed toroidal flow, that may be open or closed. This secondary circulation is even present at Y = 0 when no gas exits the cold end so that no return flow is required and all the gas exits the vortex tube at the hot end.
2. Flow direction field measured with a Pitot pipe The basic component of our Pitot device is a thin pipe (diameter 1.6 mm) with a tiny hole (diameter 0.3 mm) on the side (Fig. lb). The pipe can rotate around its axis. It is closed at one end. The other (open) end protrudes through a sliding seal into a small chamber which contains a (differential) pressure gauge. The chamber is long enough to accommodate a length H of the pipe so that the sampling hole can be placed anywhere across the span H of the flow field to be measured. The data recording in this device differs from a standard Pitot arrangement, where stagnation and reference pressure are taken simultaneously at different measuring ports. Our Pitot pipe records the stagnation and reference pressures sequentially as the probe is rotated around its axis, using the same measuring port. This makes the probe unsuitable for measurements of rapidly fluctuating pressure fields. However, since the device has only one measuring hole and a single pressure duct, its diameter can be made considerably smaller than the 2-hole or 3-hole devices employed by Brutm
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73~86
76
e 99/0 0 Ps
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Fig. 2. (a) Pressure profile around a circular cylinder; (b) pressure curve measured with the Pitot pipe in vortex tube.
and Takahama. An added advantage is the fact that the probe presents a constant obstruction to the whole flow field, which does not change as the measuring hole traverses the vortex tube. In contrast, Pitot tubes which have their measuring holes closed to the tip increase the obstructed cross section as the probe moves deeper into the flow field. The pressure gauge in our experiment measured Apw, the difference of the pressures at the measuring hole and at some suitable reference point. The pressure at the wall Pw was used as reference in this experiment. The Pitot pipe was mounted so that it could rotate around its axis and translate in its axial direction. The angular position q~ of the measuring hole was recorded with a suitable encoder. Angle ~b and pressure Apw ----p - P w were monitored by a computer. When the pipe was mounted in span direction across a flow field and then rotated it picked up the typical pressure distribution of a cylinder embedded in a flow (Fig. 2a). The maximum pressure on this trace is the stagnation pressure Ps, located at the angle ~bs. From this maximum value the pressure decays rapidly in both angular directions. On the back side in a direction ~/~b = ~)s = -4- 180 ° from the stagnation point the pressure p(~bb) = Pb is attained. In the vicinity of ~)b the pressure trace has a shallow extremum, which can be easily identified. The local
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73~86
77
free stream pressure pf appears at an angular position ~bf where the pressure changes rapidly, so that pf does not stand out on the pressure trace. By simply detecting the position of maximum pressure the direction of the local velocity vector is found. Before a measurement starts one must calibrate the angular encoder and align the measuring hole relative to a chosen reference direction. In order to examine the flow field of the vortex tube one begins by mounting the stand at some axial position z orienting the pipe across a diameter of the vortex tube. Initially, the measuring hole is set at a height h,~0 (radial position r,.~R). The Pitot pipe is then rotated around its axis to find the flow angle at this setting. Then the probe is shifted in radial direction by an arbitrary but constant amount Ah and the flow angle is again found. When this procedure has been repeated across the whole span H - - 2 R , the flow angle is known at N - - 2 R / A h locations across the vortex tube diameter. Instead of rotating the probe at a fixed elevation and then stepwise translating it after each rotation, one can connect the Pitot pipe to a screw and combine rotation and translation into a single continuous motion. We defined ~b--0 as the positive z direction. The measuring hole was then at the bottom of the horizontally mounted vortex tube. The Pitot pipe was fitted through two sliding seals across the diameter of the vortex tube. The closed end of the Pitot tube was connected to a screw which could be rotated by a gear head motor through a sliding clutch. Each full revolution of 360 ° shifted the measuring hole by the pitch height Ah---- 1.1 mm, producing pressure traces like in Fig. 2b with N = 21 measured points across the diameter of the vortex tube. A 10 turn resistor connected via a pulley to the motor shaft, was used as encoder. During each revolution the probe picked up one pressure maximum and hence one flow direction within the radial interval Ah. The radial position of this measured point was precisely known. For instance, if a maximum pressure was detected at the measured angle q~m = 543 ° the height of the measurement above the starting point was h - - ( 5 4 3 / 3 6 0 °) × 1.1 m m - - 1.66mm, and the angle against the positive z-axis was 543-360 °-- 183 °. In general, the height above the starting point is h = (~bm/360)Ah,
(1)
and the angle q~s between the direction of maximum pressure and the positive z-axis is ~bs = (/)m - - n X
360 °,
(2)
where n is the nearest number of full revolutions. Since the measuring hole sees a maximum pressure at ~bs when it faces into the flow, 180 ° must be added to get the angle ~bu between the velocity vector u and the z-axis: ~bu = qSs + 180 °.
(3)
Note that qbu is the angle of the velocity vector component in the plane normal to the Pitot pipe axis. For simplicity, the flow angle ~bu is called ~b from here on. By arranging the Pitot pipe across a diameter of the vortex tube the probe axis is in radial direction so that one can find the velocity vector angle in the z-q~ plane but nothing is known about the azimuth of the velocity vector.
78
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73-86
3. Velocity magnitude The flow direction can be determined with the Pitot pipe without any complicated calibrations or assumptions. The velocity magnitude can also be found. For standard Pitot tubes the flow velocity u is derived from the Bernoulli equation u = V/2(ps - p f ) / p .
(4)
This equation also serves as the basis for deriving the local flow velocity for a Pitot pipe; Ps stands clearly out as the maximum pressure. However, in our device the local free stream pressure pf is an unsuitable reference, because it is attained in a direction where the pressure changes rapidly with angle so that pf does not stand out as a distinct feature. Fortunately, the base pressure Pb = P(q~b = ~bs -5 180 °) is quite easily observed as a shallow secondary maximum on the pressure trace in a direction opposite to the principal maximum Ps. The base pressure Pb can be expressed as a function of pf: The stagnation pressure Ps exceeds the free stream pressure pf exactly by the dynamical pressure (½)pu 2. The difference between free stream pressure pf and base pressure Pb is related to the backpressure coefficient Cp(qSb) associated with the angular position ~bb = qSs+ 180 ° as 1
2
Pb -- Pf =: -~Cppu .
(5)
Therefore, one can express the required difference P s - P f in terms of the easily measurable difference Ap---- Ps - Pb, namely 1
2
1
2
~pu = Ps - Pf = Ps - Pb -k (Pb -- Pf) = A p + ~Cppu .
This yields Ap = (ps - pb) = ½PU2(1 -- Cp).
(6)
For the circular cylinder representing the Pitot pipe we approximate the backpressure coefficient Cp(q~b) with its value for laminar flow, namely Cp ,.~ - 1 so that 1 - Cp ~ 2. The free stream velocity is then
u=
(1 - C p ) p - ~
V
Pff
(7a)
where p has been eliminated with the ideal gas law. Rg is the gas constant, and T the local free stream temperature, which can be estimated in the vortex tube as the average between the hot and the cold exit temperatures. The local velocity u is a function of the pressure difference Ap between the principal maximum Ps and the secondary maximum Pb. The absolute value of the free stream pressure is known from traces such as Fig. 2b when the pressure at the wall Pw has been measured. Since generally Ap << pf the absolute value of pf is sufficiently well approximated as the average of Ps and Pb. The Pitot pipe measures the direction angle ~b of the velocity component u . in the plane normal to the Pitot pipe axis. The velocity component Ull parallel to the pipe axis does not contribute to the stagnation pressure, and is hence not determined. However, it is known from the flow field measurements of Bruun (1969) and Takahama (1981) that the velocity component Ur in a vortex
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73~86
79
tube is small compared to uz and u~. In arranging the Pitot pipe along a diameter of the vortex tube one has u± = ~2z22 _~_ uq~2 ~ u , since ull =
Ur << U.
From the measured flow angle q5 in the q)-z plane
the axial velocity component is obtained as uz = u± cos qS,
(7b)
and the azimuthal (toroidal) velocity component as u~ = u± sin ~b.
(7c)
The Pitot pipe was initially tried out on a straight pipe flow. The pipe had the same diameter (25.4 mm) as the vortex tube. The mass flow through the pipe was determined with a flow meter. In a first experiment the flow was injected straight into the pipe; in a second experiment a spiral twister was placed into the inlet area to generate a swirling flow with a significant azimuthal velocity component. Since the Pitot probe yielded direction fields and velocity magnitudes as expected for this pipe flow, the probe was then applied to the vortex tube.
4. The flow field of a vortex tube
Our vortex tube with an inner diameter 2R = 25.4 mm consisted of two Pyrex tube sections each 30.4 cm long mounted between brass ends. Compressed air entered the tube at the cold end through four tangential nozzles which were fed from a ring-shaped plenum chamber (more details are given in Camire (1995)). The cold exit orifice of R e = 4 . 1 mm was located in the inlet nozzle plane at z = 0. The mass flow ratio Y = j c / j o was adjusted with a valve at the hot end. The Pitot probe could be mounted near the inlet section at an axial distance z = 4.5 mm = 0.007L, or it could be placed at z ~ 0.5L between the two Pyrex tube sections. The vortex tube had standard probes to measure temperatures, pressures and mass flows at the entry and the exit ports. Flow angles and velocities were determined with our Pitot pipe for several settings of the mass flow ratio Y at the plenum chamber pressure PN 20 psig. Fig. 3a shows the flow angles at the location z = 0.007L for the two conditions: (i) Y = 0 when all gas exits at the hot end and (ii) Y = 1 where all gas exits the cold end. When no fluid escapes out of the cold end ( Y = 0 ) the flow angle is ~b~90% at small radial positions r ~<4 mm. This implies that the inner section of the flow field rotates like a forced vortex. When all fluid leaves through the cold end (Y = 1 ) the innermost layer of the working medium moves precisely in axial direction (~b= 180°). The flow direction changes continuously with increasing radius. At r = R t ~ 4 m m the flow is in azimuthal direction only (q~=90°), and at larger radii the flow goes towards the hot end, similar as in the case Y = 0. Fig. 3b shows Uz and u~ for Y = 1, and illustrates the flow structure on concentric shells, which have been pulled apart to see how the velocity direction changes with radius. From layer to layer the velocity field changes its pitch, reminding strongly of the twisted magnetic field lines in force free magnetic fields in plasmas, where the local plasma current Jpl is parallel to the magnetic field B, so that the Lorentz force FL = j × B vanishes. Such fields are, for instance, discussed by Richter (1960). Azimuthal velocity components are displayed in Fig. 4a for large and small mass flow ratios at two axial positions: z = 0.07L and z = 0.5L. The error bars indicate the uncertainties of individual =
80
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73~6 20o °
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m e a s u r e m e n t s discussed later. Axial velocities are s h o w n in Fig. 4b. There are five curves for the axial position z = 0.5L with the parameters Y = 0, 0.2, 0.4, 0.6, and 0.8. F o r all these m a s s fractions the axial v e l o c i t y Uz is positive at large radii. A t s o m e intermediate radius Rt the axial v e l o c i t y is zero, indicating that the flow is entirely azimuthal, u = u~. A l t h o u g h the curves are quite close
B. Ahlborn, S. Groves~Fluid D y n a m i c s Research 21 (1997) 73~86
0.8
.......................
.
..........................
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Fig. 5. Torus radius Rt as function o f Y.
together and the points scatter somewhat, it is noticeable that Rt increases with the mass flow ratio Y; see Fig. 5. The measurements of u~(r) and uz(r) are in good qualitative agreement with Bruun's and Takahama's results. The axial velocity curves in Fig. 4b shift systematically upwards as Y is increased. The axial velocity uz is negative for radii smaller than Rt. This "backflow core" has been noted by Bruun (1969) and Takahama (1981). It must occur for conditions where all or most of the gas escapes out of the cold end, namely when Y is large. Unexpected is the presence of a backflow core at Y = 0, when the cold end is closed.
5. Errors and perturbations caused by the probe Any probe immersed into a fluid will perturb the flow field to some extent. Our Pitot pipe is smaller than conventional Pitot tubes. Its diameter of d = 1.6 mm is small compared to the diameter of the vortex tube 2R = 25.4 mm. The probe obstructs an area of 1.6 × 25.4 = 40 mm 2, blocking only 8% of the vortex tube cross section. Therefore, we assume that it does not alter the major structure of the flow field. Preliminary experiments with the probe in a pipe flow had convinced us that our device was capable of determining velocity fields with straight and spinning flow; however the device might influence the vortex tube flow in a different way. It has been suggested by Kurosaka (1982) that the temperature splitting in a vortex tubes arises from acoustic streaming. The Pitot pipe could cause strong acoustic perturbations initiated through vortex shedding. These should appear at the vortex shedding frequency f = Su/d. The local flow velocity u is typically ~ 102m/s and the pipe diameter is d = 0.0016 m, yielding a Reynolds numbers of about 104, and a Strouhal number S ~ 0.2. Therefore, perturbations in the flow caused by vortex shedding should appear at the frequency f ~ 12.5 kHz. To search for evidence of such perturbations we mounted a microphone on the hot side of the vortex tube and recorded the acoustic spectrum. The measurements had the following preliminary results: the sound field in a vortex tube is quite intense. The acoustic spectrum in our vortex tube is a continuum stretching from below 500 Hz to above 25 kHz with some very strong features near
B. Ahlborn, S. Groves~FluidDynamics Research 21 (1997) 73-86
82
19 kHz. The Pitot tube induces some minor peaks at frequencies where axial standing wave modes of high order have a node at the location of the probe. However, no new features at the expected vortex shedding frequency were seen, and the total intensity contained in the sound spectrum does not appear to increase. Therefore, we concluded that no major acoustic perturbations were introduced by the Pitot pipe. The accuracy of a single velocity measurement as determined from Eq. (7a) suffers for three reasons: errors in pf, errors in Ap and uncertainties in Cp, namely 5u/u ~ ½(Spf/pr + 5 A p / A p + ~T/T + 5Cp/[1 ÷ Cp]). pf is known with a typical accuracy of about 1%. The temperature T must lie between the limits Th and Tc. By linearly interpolating over the radius between these limits a typical uncertainty 8T,,~ a1( T h - T~),,~ 1 0 ° K is encountered giving 5 T / T ~ 1 0 / 2 9 0 ~ 3%. An individual measurement of Ap as taken from traces like Fig. 2b can be measured to about 10% where the velocity is large and to about 20% where u is small. The assumed value for the pressure coefficient on the backsurface C p ( ¢ = 1 8 0 ° ) ~ - 1 is valid for laminar flow. It is possible that the flow inside the vortex tube may be turbulent on a small scale. Therefore, Cp might differ from 1. If Cp differs from the assumed value by 30% its uncertainty contributes a maximum percentage error of 18%. Adding up these relative errors and taking into account that the sum has to be divided by 2 due to the square root in Eq. (7a) a maximum error of about 21% is found for small velocities, and about 16% for large velocities. Typical error bars for individual measurements are shown in Fig. 4. The relative error of velocities which are obtained at adjacent points will be smaller, since the uncertainties of temperature, pressure and Cp do not change abruptly from point to point in the flow. Furthermore, our measurements of the axial and radial velocity components are in good qualitative agreements with similar measurements taken in larger vortex tubes by Bruun (1969) and Takahama (1981). Hence, we conclude that our singlehole Pitot tube is a useful addition to the arsenal of velocity probes for twisted flow fields. The measurements obtained here can be further used to describe the features of the flow in a vortex tube.
6. A secondary circulation in the vortex tube The fluid in the middle of the vortex tube travels in the negative z-direction when the cold end is closed (Y = 0). There must be another turnaround for the fluid between z = 0 and z = 0.5L. The medium is forced to circulate in r-z planes. It is therefore of interest to find out how much mass is involved in this circulation. From the measured velocity distribution u(r) one can obtain the axial mass fluxes j ( r ) by integration, assuming cylindrical geometry:
j(r) =
2rtp(r')uz(r')r' dr'.
(8)
The mass fluxes were determined at the position z = L/2 for Y = 0.4 and pp! 146 kPa. At this setting mass was injected into the vortex tube at the rate j0 = 8.9 g/s, the hot stream contained jh = 5.2 g/s, and the cold stream emerged at the rate jc = 3.7 g/s. The temperatures at the exits were Th = 306 K and Tc = 270 K. =
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73-86
g~s 4 J(r) 2
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Ij<--~o
Fig. 6. Accumulated mass flow j(r/R) and number of revolutions N per AL = 0.3 m ( ~ L/2) as function of r/R.
To carry out the integration we approximated the measured velocity uz(r) as a third-order polynomial, and approximated p(r) as a linear function varying with a density between Pc and Ph obtained from measurements at the exit ports. The integration yielded a function j(r) that is negative in the backflow core, it has a minimum at r -- Rt and rises back up to zero at the radial position Rh = 1 1 mm. Fig. 6 gives j as a function of the reduced radius r/R. Rt is the center of circulation of the secondary flow. As previously observed the size and position o f Rt depends somewhat on the cold flow fraction Y; see Fig. 5. If one approximates the secondary circulation as a flat toms, its major radius equals Rt. At the wall of the tube, the mass flux j(R) ----jh should be attained. The integration gave j ( R ) = 5.1 g/s, which is close enough to the externally measured value jh = 5.2 g/s considering the approximation made for the density, and the uncertainties in the velocity profile measurements. At the minimum j ( R t ) = 7.2 g/s was found. Since the cold mass flow rate as measured by the external flow meter is only jc -- 3.7 g/s, there must be a forced convection or "secondary circulation" that contains a mass flow j2 = 7.2 - 3.7 = 3.5 g/s. The curve j(r/R) crosses the line j = j c at two places, labeled R2c and R2. The first crossing at R2c/R~ 2.6, or R2c ~ 3.3 mm is the average radius of the cold gas flow that exits the cold end. R2c is slightly smaller than the radius of the cold gas orifice Rc =4.1 ram. The second crossing at R2/R ,.~ 7.8, or R2 ~ 9.8 mm indicates the radius outside of which the entire throughput j0 flows towards the hot end. The ranges of flow are indicated in Fig. 6. The secondary circulation, j2, on average, propagates towards the hot end in the range Rt < r < R2. It moves towards the cold, and forms part of the backflow in the range Re2 < r < Rt. It is not clear from the data if the secondary circulation is closed by itself: a flattened vortex ring of separated fluid that never mixes with the rest, or if it is open so that fluid continually enters this region, is spun around and then leaves. As gas elements are moved by this secondary circulation in axial direction they are also participating in the azimuthal motion, because the secondary circulation is imbedded into the primary vortex flow, so that the fluid elements will spiral on corkscrew lines with small angles of pitch.
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 7346
84
A figure of merit for this convoluted motion is the number of revolutions N a fluid particle makes at the radius r while it traverses a certain axial distance zS~L: N-
ALu~
(9)
2ztrUz "
We assume that the secondary circulation covers a substantial part of the vortex tube, and arbitrarily chose the length AL = 0.30 m ,,~ L / 2 to calculate a figure of merit. N is plotted in Fig. 6. Near the r a d i u s Rt where the flow is predominantly in the azimuthal direction, N becomes very large. A mass element will travel a long distance As at the radius r before it passes through the length AL. The travel distance (10)
As= 2rtrN
is shown in Fig. 7 together with N (given with a larger scale). The large values of N and As indicate that mass elements only move relatively slowly on the path of the secondary circulation, and they are essentially "trapped" in the vortex tube by the primary vortex flow. Therefore, the different streams of fluid: the "hot flow", the "cold flow", and the "secondary circulation" can stay in contact over a much longer path than the trace of j2 in the r - z plane would suggest. The secondary circulation may well be the engine of the Ranque Hilsch effect, since it brings fluid elements at one time in close contact with the cold regions at R2c and moves them at a later time to R2 where they get into contact (or are possibly mixed up) with fluid elements in the hot regions. The contact extends over a long pathway As, before the secondary circulation convects them back to the other region of the vortex tube. This extended contact could facilitate an efficient exchange of energy by ordinary thermal conduction and/or by mixing between different layers of the flow. In short, the forced convection of the secondary circulation could carry energy from the cold to the hot regions, and the long contact distance between different layers allows conduction and/or mixing to take place so that energy can be exchanged. Regardless of whether the secondary circulation should be described as an open or closed vortex ring, the flow is quite unlike a normal smoke ring vortex, where the vorticity vector points along the toms axis and the fluid has no velocity component along the ring. The vortex ring in a vortex tube is part of the main swirling motion introduced by the injection nozzles. The vortex ring must AS
2000 ,
,
'
,
,
~
[I,
,
,
~1
N
3.5
1500
....................................
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3
........................
\ i
2.5 m
1000
1.5 500
1
0.5 0 0
0.2
0.4
0,6
0.8 dR
1
Fig. 7. Path As travelled and numbers of revolution N at radius r/R while propagating the distance AL = 0.3 m ( ~ L/2) in z-direction.
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 7346
85
%o ~.. V~ >
co~d
o.d
/ \ ~ ~ . / . " ) . : ~
~
J
Fig. 8. Cartoon of the primary and secondary circulation system in a vortex tube.
share this azimuthal flow. Then in addition to the normal poloidal circulation of an ordinary "smoke ring vortex" there is motion along the torus axis of the vortex ring. This toroidal motion is induced by friction from the outer, or primary circulation. The lines of vorticity in this convoluted flow must spiral through the vortex ring like the ribbons around a wreath. Fig.8 is a cartoon illustrating the flow. The vorticity field ~--curlu, of a vortex tube with the components (toroidal,(poloidal,(z, is quite similar to the geometry of the magnetic field lines B in the extensively studied Tokamak fusion reactor (see, for instance, Chen, 1977). The field with its components Btonodal,Bpoloidal,Oz is made by three different sets of coils for the purpose of producing magneto-hydro-dynamical (MHD) stability. The sheared field lines in the Tokamak have B parallel to curl B so that the Lorentz force B x curl B vanishes and MHD stability is achieved. In the vortex tube (= arises from the tangential injection of the fluid, (toroidal, the "smoke ring vorticity", is induced because the fluid escapes out of the cold gas orifice or merely because the azimuthal velocity decays in the z-direction and fluid is sucked towards the axis where the pressure is lower - like the motion of tea leaves at the bottom of the cup when the tea is stirred with a spoon. ~po~oid=l is associated with the streaming of fluid in azimuthal direction. Of course, all three components are intimately connected. Observing how similar the flow field in a vortex tube is to the geometry of B-lines in a Tokamak it is very tempting to speculate that the flow in a vortex tube adjusts itself locally to ullcurlu, so that the Magnus forces vanish and hydrodynamical stability is attained. In summary, the novel Pitot tube has enabled us to measure the velocity components Uz and u~o in a vortex tube and permitted to derive the magnitude of a secondary circulation j2 which has the potential to convect energy from the cold to the hot flow. These observations evoke thoughts about the origin of the Ranque Hilsch effect and about fluid dynamics similarities between the vorticity structure in a vortex tube and the magnetic field structure in a Tokamak, which we are presently pursuing.
Acknowledgements We like to thank E. Rebhan, Diisseldorf for suggesting the similarity of magnetic fields in Tokamaks and the vorticity fields in a vortex tube. We like to thank Mr. W. Neill who measured
86
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73-86
the s o u n d spectrum, and Mr. A r a s z e w s k i and Mr. O l d k n o w w h o a s s e m b l e d part o f the apparatus in the E n g i n e e r i n g P h y s i c s u n d e r g r a d u a t e project #9513 U B C , M a r c h 1995.
References Ahlbom, B., J.U. Keller, R. Staudt, G. Treitz and E. Rebhan (1994) Limits of temperature separation in vortex tubes, J. Phys. D. Appl. Phys. 27, 480-488. Amitani, T., T. Adachi and T. Kato (1983) A study on temperature separation in a large vortex tube, JSME 49, 877-884. Balmer, R.T. (1988) Pressure driven Ranque Hilsch temperature separation in liquids, J. Fluid Eng. 110, 161-164. Bruun, H.H. (1969) Experimental investigation of the energy separation in vortex tubes, J. Mech. Eng. Sci. 11, 567-582. Camire, J. (1995) Experimental investigation of vortex tube concepts, M.Sc. Thesis, The University of British Columbia, Vancouver. Chen, F. (1977) Introduction to Plasma Physics (Plenum Press, New York, 1977) p. 289. Dreissler, R.G. and M. Perlmutter (1960) Analysis of the flow and energy separation in a turbulent vortex, Int. J. Heat Mass Transfer 1, 173-191. Erd~lyi, J. (1962) Wirkung des Zentrifugalfeldes auf den W~irmezustand der Gase, Erklgrung des Ranque Effekts, Forsch. Inv. Wes. 28, 181-186. Hilsch, R. (1946) Die Expansion von Gasen im Zentrifugalfeld als K/iRe prozess, Z. Naturforsch. 1, 208-214. Kurosaka, M. (1982) Acoustic streaming in a swirling flow and the Ranque Hilsch vortextube effect, J. Fluid Mech. 124, 139-172. Lay, J.E. (1959) An experimental and analytical study of vortex flow temperature separation by superposition of spiral and axial flows, Trans. Heat Transfer 81, 202-212. Lindstrom Lang, C.U. (1971) The three dimensional distribution of tangential velocity and total temperature in vortex tubes, J. Fluid. Mech. 45, 161-187. Pengelly, C.D. (1957) Flow in viscous vortex, J. Appl. Phys. 28, 86-92. Ranque, G.J. (1933) Exp~riencesd sur la detente giratoire avec productions simultanes d'un echappment d'air chand et d'un echappment d'air froid, J. Phys. Radium IV, 112-114. Richter, E. (1960) Zur Theorie kraftfreier Magnetfelder, Z. Physik. 159, 194-211. Robertson, J.A. and C.L. Crow (1975) Engineering Fluid Mechanics (Houghton Mifflin Comp. Boston, 1975) ISBN: 0-395-18607-2: Flow velocities, pp. 123 125. Laser Doppler systems, pp. 415-416. Schlenz, D. (1982) Kompressible Strahlgetriebene Drallstr6hmung in. rotationssymmetrischen Kan/ilen, Ph.D. Thesis, Univ. Erlangen. Schultz-Grunow, F. (1951) Turbulenter W/irmedurchgang im Zentrifugalfeld, Forsch. Inv. Wes. 17, 65-76. Sibulkin, M. (1962a) Unsteady viscous circular flow - application to the Ranque Hilsch vortex tube, J. Fluid Mech. 12, 269-293. Sibulkin, M. (1962b) Unsteady viscous circular flow - application to the Ranque Hilsch vortex tube, J. Fluid Mech. 12, 269-293. Stephan, K., S. Lin, M. Durst, F. Huang and D. Seher (1983) An investigation of energy separation in a vortex tube, Int. J. Heat Mass Transfer 26, 341-348. Takahama, H. and H. Yokosawa (1981) Energy separation in a vortex tube, ASME J. Heat Transfer 103, 196-203. van Deemter, J.J. (1952) On the theory of the Ranque Hilsch cooling effect, Appl. Sci. Res. A 3, 174-196.
Fluid Dynamics Research 21 (1997) 73-86
Secondary flow in a vortex tube Boye Ahlbom*, Stuart Groves Department of Physics, The University of British Columbia, 6224 Agricultural Road, Vancouver, B.C, Canada V6T 1Z1 Received 10 May 1996; revised 24 October 1996; accepted 13 January 1997
Abstract A novel Pitot probe was used to measure the axial and azimuthal velocities in a vortex tube. The probe has only a single measuring port and is hence smaller than standard devices. It monitors stagnation and reference pressure sequentially as the probe is rotated around its axis. From the measured velocity field in the 25 mm diameter vortex tube the local mass flux was determined and it was observed that the return flow at the center of the tube is much larger than the cold mass flow emerging out of the cold end. Therefore, the vortex tube must have a secondary circulation imbedded into the primary vortex, which moves fluid from the back flow core to the outer regions.
Keywords: Ranque Hilsch tube; Secondary circulation; Velocity measurements; Velocity vector angle
I. Introduction A vortex tube is a cylindrical pipe o f radius R and length L typically o f the size o f a bicycle pump. It has one or more tangentially oriented inlet holes arranged on the circumference close to one end, and two exits: a small hole on the axis close to the inlet plane with a typical radius Rc ~ 0.3R and a larger single or multiple port on the far end o f the pipe (Fig. 1). This simple mechanical device separates a stream o f gas entering at a rate j0 (kg/s) into a hot flow jh that emerges on the far end, and a cold component jc which issues out o f the small hole on the axis. In this "temperature separation" process some heat finds its way from the cold to the hot part o f the flow. Most vortex tubes have a valve at the hot end b y which the cold flow fraction Y =jc/jo can be adjusted in the range 0 ~< Y ~< 1. The effect depends on the inlet pressure P0 and on the mass flow ratio Y. Vortex tubes are used for specialized cooling applications and are commercially available. The vortex tube effect was discovered in 1933 by Ranque (1933), and examined in detail b y Hilsch (1946) and has since been the subject o f many studies. Several different qualitative explanations have been offered to explain the anomalous heat flow: (i) Internal friction with various assumptions about the radial and axial velocity distribution (van Deemter, 1952; Pengelly, 1957; Dreissler and Perlmutter, * Corresponding author. 0169-5983/97/$17.00 (~) 1997 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved. PIISO 1 6 9 - 5 9 8 3 ( 9 7 ) 0 0 0 0 3 - 8
74
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73-86
~
(b )
(a) pitot pipe
inlet
~ _
,,~h ~ ~ ~ ; ,
.......~ z
hot
-. pressure gauge ... pressure measuring plenum chamber t-" sliding seal ./ measuring hole jf
/.. 1 . 6 m m . ~ ~ control ...angle controll
resisto}" ~
.
~
-
sliding connection
stepping "~...~4,,~ motor - ~ - ~ _ ~.,===~
Fig. 1. (a) Vortex tube with location of Pitot pipe. (b) Geometry of Pitot pipe.
1960; Sibulkin, 1962a,b, Fig. 12) which includes acoustic streaming (Kurosaka, 1982) whereby more momentum is pumped into the swirling flow; (ii) Turbulent heat transfer in a stratified flow (SchultzGrunow, 1951); (iii) Goertler vortices (Stephan et al., 1983); (iv) Compressibility of the working fluid (Amitani et al., 1983); (v) Turbulent transfer of thermal energy in an incompressible flow (Lindstrom Lang, 1971 ); and (vi) Effects associated with the particular working fluid (Erd61yi, 1962). While each of these explanations may capture certain aspects of the device, none of the models has sufficient predictive power to allow the optimization of the temperature separation (Balmer, 1988). This implies that none of these mechanisms altogether explains the Ranque Hilsch effect. Experimental investigations which only deal with the parameters at the entrance and the exit ports (Schultz-Grunow, 1951; Stephan, 1983; Ahlborn et al., 1994) are unlikely to yield new understanding of the temperature separation mechanism. More insight could be gained from detailed measurements of the internal flow structure, namely from a knowledge of the r, ~0 and z components of the velocity u. Lay (1959) mapped the fields of pressure, temperature, and speed as a function of radius and downstream position z. Lay's various probes did not allow him to measure any of these quantities at radii smaller than 0.3R. In addition, he only presented the velocity component uz at a single axial position, so that the full structure of velocity field could not be deducted. The complete pressure, temperature and velocity fields were measured by Bruun (1969) in a 90 mm diameter vortex tube, and by Takahama (1981) in various tapered vortex tubes. Bruun used a Pitot pipe (diameter 3 ram) with 3 holes located about 9 mm away from its tip to measure the stagnation pressure, and the free stream pressure. He employed a thermocouple mounted inside a 3 mm pipe with a fine measuring hole to measure the local temperature. The pressure probe could
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73~6
75
be rotated to determine the flow direction and hence the azimuthal and axial velocity components. A similar setup was used up by Takahama. His 3 mm diameter Pitot probe had only 2 holes mounted at an angle of 81 o. The vortex tubes used by Bruun and Takahama had relatively large diameters. Therefore, the perturbations caused by their 3 mm diameter probes could be neglected in first approximation. The mass flow ratios Y =j~/jo in both sets of experiments were chosen in the range 0.23 ~< Y ~<0.7 where the vortex tube produces a significant temperature separation effect. These experimental data depict a flow field with a core like a forced vortex surrounded by an external annular flow somewhat like a free vortex. The core shows a back flow region in which the flow direction is opposite to the hot fluid on the periphery. It was not known from these experiments if the back flow disappears when the cold gas flow is cut off entirely (Y = 0). O f interest here is the structure of the flow, and how it develops over the hole range of mass fractions, starting at Y = 0. The most accurate and non-invasive measurements of the local velocity components in flow fields are presently obtained with laser Doppler (LD) methods (see, for instance, Robertson and Crow, 1975). Unfortunately, it appears impossible to seed the flow with scattering particles, because due to the high centrifugal acceleration (typically 1069) all particles injected into the gaseous working fluid remain close to the wall so that no scattering centers reach the axial regions of the vortex tube. For this reason, apparently, Schlenz (1982) used his laser Doppler system only with water as working fluid and only in an unidirectional vortex tube. One must therefore stick to the tried and proven Pitot devices. The Pitot pipes deployed by Bruun and Takahama and Yokosawa with two or three holes, respectively, had a diameter of 3 mm. Such devices would present too much of an obstacle to the flow in our 25.4 mm diameter vortex tube. We have therefore developed a smaller Pitot pipe which is stretched across a diameter of the vortex tube at an arbitrary position z. The probe measures stagnation and reference pressure sequentially but otherwise functions like a Pitot tube. Using this probe we have detected in the middle of the vortex tube a secondary circulation pattern, somewhat like a vortex ring with superimposed toroidal flow, that may be open or closed. This secondary circulation is even present at Y = 0 when no gas exits the cold end so that no return flow is required and all the gas exits the vortex tube at the hot end.
2. Flow direction field measured with a Pitot pipe The basic component of our Pitot device is a thin pipe (diameter 1.6 mm) with a tiny hole (diameter 0.3 mm) on the side (Fig. lb). The pipe can rotate around its axis. It is closed at one end. The other (open) end protrudes through a sliding seal into a small chamber which contains a (differential) pressure gauge. The chamber is long enough to accommodate a length H of the pipe so that the sampling hole can be placed anywhere across the span H of the flow field to be measured. The data recording in this device differs from a standard Pitot arrangement, where stagnation and reference pressure are taken simultaneously at different measuring ports. Our Pitot pipe records the stagnation and reference pressures sequentially as the probe is rotated around its axis, using the same measuring port. This makes the probe unsuitable for measurements of rapidly fluctuating pressure fields. However, since the device has only one measuring hole and a single pressure duct, its diameter can be made considerably smaller than the 2-hole or 3-hole devices employed by Brutm
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73~86
76
e 99/0 0 Ps
~b
Pf Pt
I
I
, 180 ° *
(a)
I~
=P'Pw
./Ps
t d
360 °
pf
ps~ -1
=2R I/-
0 (b)
I
I
1440
~
I
~
2880
I
4320 Angle
I
I
5760
i
I
I
7200
(degrees)
Fig. 2. (a) Pressure profile around a circular cylinder; (b) pressure curve measured with the Pitot pipe in vortex tube.
and Takahama. An added advantage is the fact that the probe presents a constant obstruction to the whole flow field, which does not change as the measuring hole traverses the vortex tube. In contrast, Pitot tubes which have their measuring holes closed to the tip increase the obstructed cross section as the probe moves deeper into the flow field. The pressure gauge in our experiment measured Apw, the difference of the pressures at the measuring hole and at some suitable reference point. The pressure at the wall Pw was used as reference in this experiment. The Pitot pipe was mounted so that it could rotate around its axis and translate in its axial direction. The angular position q~ of the measuring hole was recorded with a suitable encoder. Angle ~b and pressure Apw ----p - P w were monitored by a computer. When the pipe was mounted in span direction across a flow field and then rotated it picked up the typical pressure distribution of a cylinder embedded in a flow (Fig. 2a). The maximum pressure on this trace is the stagnation pressure Ps, located at the angle ~bs. From this maximum value the pressure decays rapidly in both angular directions. On the back side in a direction ~/~b = ~)s = -4- 180 ° from the stagnation point the pressure p(~bb) = Pb is attained. In the vicinity of ~)b the pressure trace has a shallow extremum, which can be easily identified. The local
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73~86
77
free stream pressure pf appears at an angular position ~bf where the pressure changes rapidly, so that pf does not stand out on the pressure trace. By simply detecting the position of maximum pressure the direction of the local velocity vector is found. Before a measurement starts one must calibrate the angular encoder and align the measuring hole relative to a chosen reference direction. In order to examine the flow field of the vortex tube one begins by mounting the stand at some axial position z orienting the pipe across a diameter of the vortex tube. Initially, the measuring hole is set at a height h,~0 (radial position r,.~R). The Pitot pipe is then rotated around its axis to find the flow angle at this setting. Then the probe is shifted in radial direction by an arbitrary but constant amount Ah and the flow angle is again found. When this procedure has been repeated across the whole span H - - 2 R , the flow angle is known at N - - 2 R / A h locations across the vortex tube diameter. Instead of rotating the probe at a fixed elevation and then stepwise translating it after each rotation, one can connect the Pitot pipe to a screw and combine rotation and translation into a single continuous motion. We defined ~b--0 as the positive z direction. The measuring hole was then at the bottom of the horizontally mounted vortex tube. The Pitot pipe was fitted through two sliding seals across the diameter of the vortex tube. The closed end of the Pitot tube was connected to a screw which could be rotated by a gear head motor through a sliding clutch. Each full revolution of 360 ° shifted the measuring hole by the pitch height Ah---- 1.1 mm, producing pressure traces like in Fig. 2b with N = 21 measured points across the diameter of the vortex tube. A 10 turn resistor connected via a pulley to the motor shaft, was used as encoder. During each revolution the probe picked up one pressure maximum and hence one flow direction within the radial interval Ah. The radial position of this measured point was precisely known. For instance, if a maximum pressure was detected at the measured angle q~m = 543 ° the height of the measurement above the starting point was h - - ( 5 4 3 / 3 6 0 °) × 1.1 m m - - 1.66mm, and the angle against the positive z-axis was 543-360 °-- 183 °. In general, the height above the starting point is h = (~bm/360)Ah,
(1)
and the angle q~s between the direction of maximum pressure and the positive z-axis is ~bs = (/)m - - n X
360 °,
(2)
where n is the nearest number of full revolutions. Since the measuring hole sees a maximum pressure at ~bs when it faces into the flow, 180 ° must be added to get the angle ~bu between the velocity vector u and the z-axis: ~bu = qSs + 180 °.
(3)
Note that qbu is the angle of the velocity vector component in the plane normal to the Pitot pipe axis. For simplicity, the flow angle ~bu is called ~b from here on. By arranging the Pitot pipe across a diameter of the vortex tube the probe axis is in radial direction so that one can find the velocity vector angle in the z-q~ plane but nothing is known about the azimuth of the velocity vector.
78
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73-86
3. Velocity magnitude The flow direction can be determined with the Pitot pipe without any complicated calibrations or assumptions. The velocity magnitude can also be found. For standard Pitot tubes the flow velocity u is derived from the Bernoulli equation u = V/2(ps - p f ) / p .
(4)
This equation also serves as the basis for deriving the local flow velocity for a Pitot pipe; Ps stands clearly out as the maximum pressure. However, in our device the local free stream pressure pf is an unsuitable reference, because it is attained in a direction where the pressure changes rapidly with angle so that pf does not stand out as a distinct feature. Fortunately, the base pressure Pb = P(q~b = ~bs -5 180 °) is quite easily observed as a shallow secondary maximum on the pressure trace in a direction opposite to the principal maximum Ps. The base pressure Pb can be expressed as a function of pf: The stagnation pressure Ps exceeds the free stream pressure pf exactly by the dynamical pressure (½)pu 2. The difference between free stream pressure pf and base pressure Pb is related to the backpressure coefficient Cp(qSb) associated with the angular position ~bb = qSs+ 180 ° as 1
2
Pb -- Pf =: -~Cppu .
(5)
Therefore, one can express the required difference P s - P f in terms of the easily measurable difference Ap---- Ps - Pb, namely 1
2
1
2
~pu = Ps - Pf = Ps - Pb -k (Pb -- Pf) = A p + ~Cppu .
This yields Ap = (ps - pb) = ½PU2(1 -- Cp).
(6)
For the circular cylinder representing the Pitot pipe we approximate the backpressure coefficient Cp(q~b) with its value for laminar flow, namely Cp ,.~ - 1 so that 1 - Cp ~ 2. The free stream velocity is then
u=
(1 - C p ) p - ~
V
Pff
(7a)
where p has been eliminated with the ideal gas law. Rg is the gas constant, and T the local free stream temperature, which can be estimated in the vortex tube as the average between the hot and the cold exit temperatures. The local velocity u is a function of the pressure difference Ap between the principal maximum Ps and the secondary maximum Pb. The absolute value of the free stream pressure is known from traces such as Fig. 2b when the pressure at the wall Pw has been measured. Since generally Ap << pf the absolute value of pf is sufficiently well approximated as the average of Ps and Pb. The Pitot pipe measures the direction angle ~b of the velocity component u . in the plane normal to the Pitot pipe axis. The velocity component Ull parallel to the pipe axis does not contribute to the stagnation pressure, and is hence not determined. However, it is known from the flow field measurements of Bruun (1969) and Takahama (1981) that the velocity component Ur in a vortex
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73~86
79
tube is small compared to uz and u~. In arranging the Pitot pipe along a diameter of the vortex tube one has u± = ~2z22 _~_ uq~2 ~ u , since ull =
Ur << U.
From the measured flow angle q5 in the q)-z plane
the axial velocity component is obtained as uz = u± cos qS,
(7b)
and the azimuthal (toroidal) velocity component as u~ = u± sin ~b.
(7c)
The Pitot pipe was initially tried out on a straight pipe flow. The pipe had the same diameter (25.4 mm) as the vortex tube. The mass flow through the pipe was determined with a flow meter. In a first experiment the flow was injected straight into the pipe; in a second experiment a spiral twister was placed into the inlet area to generate a swirling flow with a significant azimuthal velocity component. Since the Pitot probe yielded direction fields and velocity magnitudes as expected for this pipe flow, the probe was then applied to the vortex tube.
4. The flow field of a vortex tube
Our vortex tube with an inner diameter 2R = 25.4 mm consisted of two Pyrex tube sections each 30.4 cm long mounted between brass ends. Compressed air entered the tube at the cold end through four tangential nozzles which were fed from a ring-shaped plenum chamber (more details are given in Camire (1995)). The cold exit orifice of R e = 4 . 1 mm was located in the inlet nozzle plane at z = 0. The mass flow ratio Y = j c / j o was adjusted with a valve at the hot end. The Pitot probe could be mounted near the inlet section at an axial distance z = 4.5 mm = 0.007L, or it could be placed at z ~ 0.5L between the two Pyrex tube sections. The vortex tube had standard probes to measure temperatures, pressures and mass flows at the entry and the exit ports. Flow angles and velocities were determined with our Pitot pipe for several settings of the mass flow ratio Y at the plenum chamber pressure PN 20 psig. Fig. 3a shows the flow angles at the location z = 0.007L for the two conditions: (i) Y = 0 when all gas exits at the hot end and (ii) Y = 1 where all gas exits the cold end. When no fluid escapes out of the cold end ( Y = 0 ) the flow angle is ~b~90% at small radial positions r ~<4 mm. This implies that the inner section of the flow field rotates like a forced vortex. When all fluid leaves through the cold end (Y = 1 ) the innermost layer of the working medium moves precisely in axial direction (~b= 180°). The flow direction changes continuously with increasing radius. At r = R t ~ 4 m m the flow is in azimuthal direction only (q~=90°), and at larger radii the flow goes towards the hot end, similar as in the case Y = 0. Fig. 3b shows Uz and u~ for Y = 1, and illustrates the flow structure on concentric shells, which have been pulled apart to see how the velocity direction changes with radius. From layer to layer the velocity field changes its pitch, reminding strongly of the twisted magnetic field lines in force free magnetic fields in plasmas, where the local plasma current Jpl is parallel to the magnetic field B, so that the Lorentz force FL = j × B vanishes. Such fields are, for instance, discussed by Richter (1960). Azimuthal velocity components are displayed in Fig. 4a for large and small mass flow ratios at two axial positions: z = 0.07L and z = 0.5L. The error bars indicate the uncertainties of individual =
80
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73~6 20o °
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shell model -~
r
,
II
.... ................ .
.
-__12_
I
=
Fig. 3. (a) Flow angle and (b) cartoon of the velocity field for Y ~ 0 and z = 0.007L.
Azimuthal
velocity
Axial velocity uz (r)
u e (r) 150
150 := ¥ - - 0 i
z= o.s t-.~ ~ :
m/s
p ! ...............
Ioo
.
................-
.
.
.
.
.
x
i
,~
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.
.
.
.
.
.
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.
.
.
.
.
.
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,
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50
. . . . . . . . . . . . . . . . . . . . . .
i
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~_. . . . . .
100
0 /
~
-.;......y.~.1.~T...,-7 T
50
..............
I
O~v~'× (a)
~XX
0
7
O.O07L
...............
'~
E~
\ ..........
]
-1 O 0
4
...... 6
8
i,,, 10 rnm 12
F•
o-'L~.-i
i
0.6
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-50
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i.~.,-i,,,i 2
z
' 14
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- 150
F,,,i
(b)
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"i."
,x
ii ' o 2"
ii "'° ii
. . . . . . . . . . . .
2
4
6 mm 8
ii
!i I
I,,
i,,,
10
12
R
14
Fig. 4. (a) Azimuthal and (b) axial velocity components at z = 0.007L and z = 0.5L for various mass flow ratios.
m e a s u r e m e n t s discussed later. Axial velocities are s h o w n in Fig. 4b. There are five curves for the axial position z = 0.5L with the parameters Y = 0, 0.2, 0.4, 0.6, and 0.8. F o r all these m a s s fractions the axial v e l o c i t y Uz is positive at large radii. A t s o m e intermediate radius Rt the axial v e l o c i t y is zero, indicating that the flow is entirely azimuthal, u = u~. A l t h o u g h the curves are quite close
B. Ahlborn, S. Groves~Fluid D y n a m i c s Research 21 (1997) 73~86
0.8
.......................
.
..........................
i .........................
~ .........................
81
~ ...................
0.6 R/R t ,~
0.4
I
0.2
....................................................................... -.....................................
Y= Jc/Jo
r~ 0
,
0
,
,
0.2
,
,
,
0.4
,
,
,
0.6
,
,
I
I
I
=
0.8
Fig. 5. Torus radius Rt as function o f Y.
together and the points scatter somewhat, it is noticeable that Rt increases with the mass flow ratio Y; see Fig. 5. The measurements of u~(r) and uz(r) are in good qualitative agreement with Bruun's and Takahama's results. The axial velocity curves in Fig. 4b shift systematically upwards as Y is increased. The axial velocity uz is negative for radii smaller than Rt. This "backflow core" has been noted by Bruun (1969) and Takahama (1981). It must occur for conditions where all or most of the gas escapes out of the cold end, namely when Y is large. Unexpected is the presence of a backflow core at Y = 0, when the cold end is closed.
5. Errors and perturbations caused by the probe Any probe immersed into a fluid will perturb the flow field to some extent. Our Pitot pipe is smaller than conventional Pitot tubes. Its diameter of d = 1.6 mm is small compared to the diameter of the vortex tube 2R = 25.4 mm. The probe obstructs an area of 1.6 × 25.4 = 40 mm 2, blocking only 8% of the vortex tube cross section. Therefore, we assume that it does not alter the major structure of the flow field. Preliminary experiments with the probe in a pipe flow had convinced us that our device was capable of determining velocity fields with straight and spinning flow; however the device might influence the vortex tube flow in a different way. It has been suggested by Kurosaka (1982) that the temperature splitting in a vortex tubes arises from acoustic streaming. The Pitot pipe could cause strong acoustic perturbations initiated through vortex shedding. These should appear at the vortex shedding frequency f = Su/d. The local flow velocity u is typically ~ 102m/s and the pipe diameter is d = 0.0016 m, yielding a Reynolds numbers of about 104, and a Strouhal number S ~ 0.2. Therefore, perturbations in the flow caused by vortex shedding should appear at the frequency f ~ 12.5 kHz. To search for evidence of such perturbations we mounted a microphone on the hot side of the vortex tube and recorded the acoustic spectrum. The measurements had the following preliminary results: the sound field in a vortex tube is quite intense. The acoustic spectrum in our vortex tube is a continuum stretching from below 500 Hz to above 25 kHz with some very strong features near
B. Ahlborn, S. Groves~FluidDynamics Research 21 (1997) 73-86
82
19 kHz. The Pitot tube induces some minor peaks at frequencies where axial standing wave modes of high order have a node at the location of the probe. However, no new features at the expected vortex shedding frequency were seen, and the total intensity contained in the sound spectrum does not appear to increase. Therefore, we concluded that no major acoustic perturbations were introduced by the Pitot pipe. The accuracy of a single velocity measurement as determined from Eq. (7a) suffers for three reasons: errors in pf, errors in Ap and uncertainties in Cp, namely 5u/u ~ ½(Spf/pr + 5 A p / A p + ~T/T + 5Cp/[1 ÷ Cp]). pf is known with a typical accuracy of about 1%. The temperature T must lie between the limits Th and Tc. By linearly interpolating over the radius between these limits a typical uncertainty 8T,,~ a1( T h - T~),,~ 1 0 ° K is encountered giving 5 T / T ~ 1 0 / 2 9 0 ~ 3%. An individual measurement of Ap as taken from traces like Fig. 2b can be measured to about 10% where the velocity is large and to about 20% where u is small. The assumed value for the pressure coefficient on the backsurface C p ( ¢ = 1 8 0 ° ) ~ - 1 is valid for laminar flow. It is possible that the flow inside the vortex tube may be turbulent on a small scale. Therefore, Cp might differ from 1. If Cp differs from the assumed value by 30% its uncertainty contributes a maximum percentage error of 18%. Adding up these relative errors and taking into account that the sum has to be divided by 2 due to the square root in Eq. (7a) a maximum error of about 21% is found for small velocities, and about 16% for large velocities. Typical error bars for individual measurements are shown in Fig. 4. The relative error of velocities which are obtained at adjacent points will be smaller, since the uncertainties of temperature, pressure and Cp do not change abruptly from point to point in the flow. Furthermore, our measurements of the axial and radial velocity components are in good qualitative agreements with similar measurements taken in larger vortex tubes by Bruun (1969) and Takahama (1981). Hence, we conclude that our singlehole Pitot tube is a useful addition to the arsenal of velocity probes for twisted flow fields. The measurements obtained here can be further used to describe the features of the flow in a vortex tube.
6. A secondary circulation in the vortex tube The fluid in the middle of the vortex tube travels in the negative z-direction when the cold end is closed (Y = 0). There must be another turnaround for the fluid between z = 0 and z = 0.5L. The medium is forced to circulate in r-z planes. It is therefore of interest to find out how much mass is involved in this circulation. From the measured velocity distribution u(r) one can obtain the axial mass fluxes j ( r ) by integration, assuming cylindrical geometry:
j(r) =
2rtp(r')uz(r')r' dr'.
(8)
The mass fluxes were determined at the position z = L/2 for Y = 0.4 and pp! 146 kPa. At this setting mass was injected into the vortex tube at the rate j0 = 8.9 g/s, the hot stream contained jh = 5.2 g/s, and the cold stream emerged at the rate jc = 3.7 g/s. The temperatures at the exits were Th = 306 K and Tc = 270 K. =
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73-86
g~s 4 J(r) 2
. . . . . . .
~ '
'
!
'
1
'
~
' j'
' ~
83
200
.......................................................... -f.----N'"--i........................... i..............~-----./q N / [ i .....¢.~ 150
o .......................i ......................... .....................ii ...............i -2 ..............¢ ...................... ............... ..!£ Jo ,-I- .... ..... -8 r , , , 0 r/R
i ,i,, i,, 0.21 0.4
I
R
jo
100 50
I, i , , ,'i I, ,. 10.6 m J ~ . 8 I'R
R
2c
<
.............i
'12
I
I h
t
J2
I
J2
Ij<--~o
Fig. 6. Accumulated mass flow j(r/R) and number of revolutions N per AL = 0.3 m ( ~ L/2) as function of r/R.
To carry out the integration we approximated the measured velocity uz(r) as a third-order polynomial, and approximated p(r) as a linear function varying with a density between Pc and Ph obtained from measurements at the exit ports. The integration yielded a function j(r) that is negative in the backflow core, it has a minimum at r -- Rt and rises back up to zero at the radial position Rh = 1 1 mm. Fig. 6 gives j as a function of the reduced radius r/R. Rt is the center of circulation of the secondary flow. As previously observed the size and position o f Rt depends somewhat on the cold flow fraction Y; see Fig. 5. If one approximates the secondary circulation as a flat toms, its major radius equals Rt. At the wall of the tube, the mass flux j(R) ----jh should be attained. The integration gave j ( R ) = 5.1 g/s, which is close enough to the externally measured value jh = 5.2 g/s considering the approximation made for the density, and the uncertainties in the velocity profile measurements. At the minimum j ( R t ) = 7.2 g/s was found. Since the cold mass flow rate as measured by the external flow meter is only jc -- 3.7 g/s, there must be a forced convection or "secondary circulation" that contains a mass flow j2 = 7.2 - 3.7 = 3.5 g/s. The curve j(r/R) crosses the line j = j c at two places, labeled R2c and R2. The first crossing at R2c/R~ 2.6, or R2c ~ 3.3 mm is the average radius of the cold gas flow that exits the cold end. R2c is slightly smaller than the radius of the cold gas orifice Rc =4.1 ram. The second crossing at R2/R ,.~ 7.8, or R2 ~ 9.8 mm indicates the radius outside of which the entire throughput j0 flows towards the hot end. The ranges of flow are indicated in Fig. 6. The secondary circulation, j2, on average, propagates towards the hot end in the range Rt < r < R2. It moves towards the cold, and forms part of the backflow in the range Re2 < r < Rt. It is not clear from the data if the secondary circulation is closed by itself: a flattened vortex ring of separated fluid that never mixes with the rest, or if it is open so that fluid continually enters this region, is spun around and then leaves. As gas elements are moved by this secondary circulation in axial direction they are also participating in the azimuthal motion, because the secondary circulation is imbedded into the primary vortex flow, so that the fluid elements will spiral on corkscrew lines with small angles of pitch.
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 7346
84
A figure of merit for this convoluted motion is the number of revolutions N a fluid particle makes at the radius r while it traverses a certain axial distance zS~L: N-
ALu~
(9)
2ztrUz "
We assume that the secondary circulation covers a substantial part of the vortex tube, and arbitrarily chose the length AL = 0.30 m ,,~ L / 2 to calculate a figure of merit. N is plotted in Fig. 6. Near the r a d i u s Rt where the flow is predominantly in the azimuthal direction, N becomes very large. A mass element will travel a long distance As at the radius r before it passes through the length AL. The travel distance (10)
As= 2rtrN
is shown in Fig. 7 together with N (given with a larger scale). The large values of N and As indicate that mass elements only move relatively slowly on the path of the secondary circulation, and they are essentially "trapped" in the vortex tube by the primary vortex flow. Therefore, the different streams of fluid: the "hot flow", the "cold flow", and the "secondary circulation" can stay in contact over a much longer path than the trace of j2 in the r - z plane would suggest. The secondary circulation may well be the engine of the Ranque Hilsch effect, since it brings fluid elements at one time in close contact with the cold regions at R2c and moves them at a later time to R2 where they get into contact (or are possibly mixed up) with fluid elements in the hot regions. The contact extends over a long pathway As, before the secondary circulation convects them back to the other region of the vortex tube. This extended contact could facilitate an efficient exchange of energy by ordinary thermal conduction and/or by mixing between different layers of the flow. In short, the forced convection of the secondary circulation could carry energy from the cold to the hot regions, and the long contact distance between different layers allows conduction and/or mixing to take place so that energy can be exchanged. Regardless of whether the secondary circulation should be described as an open or closed vortex ring, the flow is quite unlike a normal smoke ring vortex, where the vorticity vector points along the toms axis and the fluid has no velocity component along the ring. The vortex ring in a vortex tube is part of the main swirling motion introduced by the injection nozzles. The vortex ring must AS
2000 ,
,
'
,
,
~
[I,
,
,
~1
N
3.5
1500
....................................
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3
........................
\ i
2.5 m
1000
1.5 500
1
0.5 0 0
0.2
0.4
0,6
0.8 dR
1
Fig. 7. Path As travelled and numbers of revolution N at radius r/R while propagating the distance AL = 0.3 m ( ~ L/2) in z-direction.
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 7346
85
%o ~.. V~ >
co~d
o.d
/ \ ~ ~ . / . " ) . : ~
~
J
Fig. 8. Cartoon of the primary and secondary circulation system in a vortex tube.
share this azimuthal flow. Then in addition to the normal poloidal circulation of an ordinary "smoke ring vortex" there is motion along the torus axis of the vortex ring. This toroidal motion is induced by friction from the outer, or primary circulation. The lines of vorticity in this convoluted flow must spiral through the vortex ring like the ribbons around a wreath. Fig.8 is a cartoon illustrating the flow. The vorticity field ~--curlu, of a vortex tube with the components (toroidal,(poloidal,(z, is quite similar to the geometry of the magnetic field lines B in the extensively studied Tokamak fusion reactor (see, for instance, Chen, 1977). The field with its components Btonodal,Bpoloidal,Oz is made by three different sets of coils for the purpose of producing magneto-hydro-dynamical (MHD) stability. The sheared field lines in the Tokamak have B parallel to curl B so that the Lorentz force B x curl B vanishes and MHD stability is achieved. In the vortex tube (= arises from the tangential injection of the fluid, (toroidal, the "smoke ring vorticity", is induced because the fluid escapes out of the cold gas orifice or merely because the azimuthal velocity decays in the z-direction and fluid is sucked towards the axis where the pressure is lower - like the motion of tea leaves at the bottom of the cup when the tea is stirred with a spoon. ~po~oid=l is associated with the streaming of fluid in azimuthal direction. Of course, all three components are intimately connected. Observing how similar the flow field in a vortex tube is to the geometry of B-lines in a Tokamak it is very tempting to speculate that the flow in a vortex tube adjusts itself locally to ullcurlu, so that the Magnus forces vanish and hydrodynamical stability is attained. In summary, the novel Pitot tube has enabled us to measure the velocity components Uz and u~o in a vortex tube and permitted to derive the magnitude of a secondary circulation j2 which has the potential to convect energy from the cold to the hot flow. These observations evoke thoughts about the origin of the Ranque Hilsch effect and about fluid dynamics similarities between the vorticity structure in a vortex tube and the magnetic field structure in a Tokamak, which we are presently pursuing.
Acknowledgements We like to thank E. Rebhan, Diisseldorf for suggesting the similarity of magnetic fields in Tokamaks and the vorticity fields in a vortex tube. We like to thank Mr. W. Neill who measured
86
B. Ahlborn, S. Groves~Fluid Dynamics Research 21 (1997) 73-86
the s o u n d spectrum, and Mr. A r a s z e w s k i and Mr. O l d k n o w w h o a s s e m b l e d part o f the apparatus in the E n g i n e e r i n g P h y s i c s u n d e r g r a d u a t e project #9513 U B C , M a r c h 1995.
References Ahlbom, B., J.U. Keller, R. Staudt, G. Treitz and E. Rebhan (1994) Limits of temperature separation in vortex tubes, J. Phys. D. Appl. Phys. 27, 480-488. Amitani, T., T. Adachi and T. Kato (1983) A study on temperature separation in a large vortex tube, JSME 49, 877-884. Balmer, R.T. (1988) Pressure driven Ranque Hilsch temperature separation in liquids, J. Fluid Eng. 110, 161-164. Bruun, H.H. (1969) Experimental investigation of the energy separation in vortex tubes, J. Mech. Eng. Sci. 11, 567-582. Camire, J. (1995) Experimental investigation of vortex tube concepts, M.Sc. Thesis, The University of British Columbia, Vancouver. Chen, F. (1977) Introduction to Plasma Physics (Plenum Press, New York, 1977) p. 289. Dreissler, R.G. and M. Perlmutter (1960) Analysis of the flow and energy separation in a turbulent vortex, Int. J. Heat Mass Transfer 1, 173-191. Erd~lyi, J. (1962) Wirkung des Zentrifugalfeldes auf den W~irmezustand der Gase, Erklgrung des Ranque Effekts, Forsch. Inv. Wes. 28, 181-186. Hilsch, R. (1946) Die Expansion von Gasen im Zentrifugalfeld als K/iRe prozess, Z. Naturforsch. 1, 208-214. Kurosaka, M. (1982) Acoustic streaming in a swirling flow and the Ranque Hilsch vortextube effect, J. Fluid Mech. 124, 139-172. Lay, J.E. (1959) An experimental and analytical study of vortex flow temperature separation by superposition of spiral and axial flows, Trans. Heat Transfer 81, 202-212. Lindstrom Lang, C.U. (1971) The three dimensional distribution of tangential velocity and total temperature in vortex tubes, J. Fluid. Mech. 45, 161-187. Pengelly, C.D. (1957) Flow in viscous vortex, J. Appl. Phys. 28, 86-92. Ranque, G.J. (1933) Exp~riencesd sur la detente giratoire avec productions simultanes d'un echappment d'air chand et d'un echappment d'air froid, J. Phys. Radium IV, 112-114. Richter, E. (1960) Zur Theorie kraftfreier Magnetfelder, Z. Physik. 159, 194-211. Robertson, J.A. and C.L. Crow (1975) Engineering Fluid Mechanics (Houghton Mifflin Comp. Boston, 1975) ISBN: 0-395-18607-2: Flow velocities, pp. 123 125. Laser Doppler systems, pp. 415-416. Schlenz, D. (1982) Kompressible Strahlgetriebene Drallstr6hmung in. rotationssymmetrischen Kan/ilen, Ph.D. Thesis, Univ. Erlangen. Schultz-Grunow, F. (1951) Turbulenter W/irmedurchgang im Zentrifugalfeld, Forsch. Inv. Wes. 17, 65-76. Sibulkin, M. (1962a) Unsteady viscous circular flow - application to the Ranque Hilsch vortex tube, J. Fluid Mech. 12, 269-293. Sibulkin, M. (1962b) Unsteady viscous circular flow - application to the Ranque Hilsch vortex tube, J. Fluid Mech. 12, 269-293. Stephan, K., S. Lin, M. Durst, F. Huang and D. Seher (1983) An investigation of energy separation in a vortex tube, Int. J. Heat Mass Transfer 26, 341-348. Takahama, H. and H. Yokosawa (1981) Energy separation in a vortex tube, ASME J. Heat Transfer 103, 196-203. van Deemter, J.J. (1952) On the theory of the Ranque Hilsch cooling effect, Appl. Sci. Res. A 3, 174-196.