Section 4

 

 Flow Acceleration and Centrifugal Force as a Possible Cause of the  Observed Temperature Rise Anomaly in the Ranque-Hilsch Vortex Tube  

 

Abstract    In strong vortices small temperature rise anomalies are observed. These are particularly evident and well studied in the so-called Ranque-Hilsch Vortex Tube, but the  reason for the anomalies has been a matter of controversy. Here, we present evidence that this effect is due to an increase in random  vibratory kinetic molecular motion in the radial dimension set up by imbalances in the strong centrifugal forces,  thereby slightly raising the gas temperature. 

 

Contents

4.1 Introduction

 

4.2 The Ranque-Hilsch Vortex Tube

 

4.3  Previous Explanations for the Anomalous Temperature Rise Phenomenon   

 

4.4  A New Approach: Centrifugal Force in Vortex Flow

 

4.5  Vortex Flow

 

4.6 The Temperature Rise Anomaly in the Ranque-Hilsch Tube

 

4.7   Additional Atmospheric Observational Evidence for this  Temperature Rise  Anomaly

 

4.8   Conclusions

 

References

 

 

 

 

4.0 Introduction

 

Following observations on a rare whirlwind which occurs over cool lakes in fine warm weather [1]  we have, since 2008, carried out  laboratory studies of various isentropic flow transformation models and devices   with a view  (1) to understanding and controlling transformations of internal pressure-volume energy into kinetic or flow energy, and eventually (2) into the possibility of extracting this kinetic energy into useful work at better efficiencies than with existing air motors such as turbo machines, jet engines, vacuum machines and compressors..

 

Two laboratory models studied have been, first, isentropic nozzle flow, both linear and swirled, and, second, the peculiar forced vortex called the Ranque–Hilsch Tube [2,3]. 

 

Here we report some preliminary findings on the latter phenomenon,

 

4.2  The Ranque-Hilsch Vortex Tube

Ranque [2] invented and patented a pressure or “push flow”  vortex tube consisting of a cylinder into which compressed air is injected tangentially to form a gas vortex, having, typically, at one end of the cylindrical tube, an axial exit port, and at the other end a peripheral exit annulus around a cone valve.  Astonishingly, while the axial port emits cold air as might be expected, the annular port simultaneously emits hot air. The total temperature spread can range up to 40 or 50 degrees C when the inlet pressure is high and the tube radius is small. The same general effect is observed with a so-called ‘uniflow’ model in which the cold and hot exits are at the same end of the tube.

 

 

 

 

The Ranque-Hilsch tube is diagrammed in Figure 1. Typical inlet operating air pressures are in the range of 1 to 6 atmospheres. Typical tube sizes are from 5 cm in diameter, down to 1 or 2 centimeters for maximum temperature separation.

 

Many detailed theoretical and experimental studies have been carried out since Ranque’s discovery of the temperature separation phenomenon  in 1933, and  subsequent findings and improvements by Hilsch in 1946.  However, while  it  might be expected that gas exiting at an axial port would naturally be colder than the inlet gas temperature because of isentropic expansion of the compressed gas into the  vortex core, the accompanying exit of hot air at the other end of the tube has made the phenomenon a  matter of continuing interest  and  controversy [2, 3,4,5,6}.

 

4.3   Previous Explanations for the Anomalous Temperature Rise Phenomenon  

 

Ranque [2] proposed that the observed cooling is due to adiabatic expansion of the gas into the vortex core and that the heating is due to adiabatic compression of the gas at the periphery or walls of the tube. Hilsch [3] investigated the added the effects of internal fluid viscosity.

 

Later work [4,5,6] has investigated the effects of turbulence, heat transfer processes,  internal secondary circulations and certain acoustic effects. Thus, the theoretical explanations have become quite complex.

 

There was even some early speculation that the phenomenon might involve a sort of “Maxwell’s  Demon”  effect in  which centrifugal  forces might somehow actually separate slow  moving ( cold ) molecules from fast moving  ( hot) molecules.  This approach was quickly dismissed, as it involved a violation of the second law of thermodynamics.

 

Here, we adopt the basic original Rankine-combined vortex models studied by Ranque and Hilsch, and then we analyze the effect of the very strong centrifugal forces to provide a possible  new explanation for the anomalous heating – the general cooling is considered to  be already reasonably well accounted for by adiabatic isentropic expansion of the air flowing from the compressor  into the vortex core.

 

 

4.4  A New Approach: Centrifugal Force in Vortex Flow

We turn first to the centrifugal force which enters into  linearly accelerated flow and in  ‘curved’ accelerated flow, such as occurs in a vortex.

The centrifugal force  Fc  at any radius r of the vortex is given by

Fc= VT  2 /r

where  VT    is the tangential velocity.

Under steady flow conditions, there is  cyclostrophic balance  in which, at any point, the centrifugal force outward is balanced by the pressure gradient force acting radially inward: 

                                         Fc  = VT 2 /r =  −  1/ρ (dp/dr)   

The centrifugal forces arising in atmospheric vortices, such as in the cores of tornadoes, and even more so in intense  laboratory vortices such as the Ranque-Hilsch tube, are very large.  We shall now explore the possible role of these very large vortex forces in statistically modifying the kinetic motions and energy partition of the molecules of the compressible fluid  itself so as to produce the  observed Ranque-Hilsch tube  temperature anomaly. 

 

 

4.5  Vortex Flow

 

In a vortex, the gas or fluid  flow is roughly circular around the central axis of rotation of the vortex  (Figure 2, below ), and  the necessary  acceleration dV/dt arises from the change in the direction of flow dV circulating around the vortex core or axis, rather than from change in the magnitude of V as in the case of linear flow accelerations.. Thus, because of the acceleration,  the pressure  in the vortex decreases radially inwards towards the vortex core in both compressible and incompressible flow..

 

There are two main types of vortex flow.  First, there is the compressible potential vortex with ‘swirl’  or circulation of the  fluid around a central vortex axis,   but where the individual  fluid elements themselves do not rotate. Rather, their motion is analogous the motion of a compass needle when the  compass itself is moved around in a circular motion  but the compass, needle does not rotate on its own axis but always points  in the same direction even while the compass as a whole ‘circulates’.  

 

In a compressible potential vortex the relation between tangential velocity VT and radial distance r from the centre is given by

 

VT  r = Γ =constant

 

which is just a statement of the conservation of angular  momentum Vr, where Γ with dimensions of angular momentum is called the circulation, and VT is the  tangential velocity at any point of radius r. 

 

The potential vortex is so-called, because its  motion can also be described by a potential velocity function φ, such that dφ/dx = V.   It is also  often called a free vortex [7,8, 9,10].

 

The second main type of vortex motion  is solid or wheel-like rotation,  sometimes called  a forced  vortex or rectilinear vortex, which is described  mathematically as

V T/r = ω

 

were ω is the rotation with the dimensions of frequency (radians per second).

 

Natural atmospheric vortices such as tornadoes, waterspouts and dust devils are often modeled as a combination of these two vortex motions, that is, with  an outer potential  fluid motion and an inner core in solid rotation. [ 8, 9,10, 11 ].This combination is called a  Rankine combined vortex, or simply a Rankine vortex.(Fig. 2). 

 

For the radial pressure profile in a Rankine vortex, we have, [10 ] the following relationships;

 

 (a) In the outer potential vortex :

 

                                   p = po −( ρΓ2 / 4π2) ∫dr/r3 = ρΓ2/  8 π2 r2

                                                                                                                    

(b) In the inner forced vortex or core:

                 

                                   p = po [ρΓ2 (r1 2 – r2 ) /  8 π2 r14 ]                               

 

The combined pressure profile across the Rankine vortex  is shown in fig. 2 below.

-p
 

 

 

Figure 2.  Radial Profiles of Tangential Velocity and Pressure in a Rankine-combined Vortex

 

 

It is worth noting that there is an important difference is the relationship of velocity to the radius in a vortex versus that in the Equation of Continuity. In a vortex, the velocity V is proportional to 1/r; but in the Equation of Continuity ( ρVA = constant)  the velocity is proportional to 1/r2.  We shall have occasion to make use of this difference shortly.              

 

 

4.6  The Temperature Rise Anomaly in the Ranque-Hilsch Tube 

 

We shall use this basic Rankine-combined vortex model to describe the operation of the Ranque- Hilsch tube, and then propose some modifications to it that we feel are required  by the enormous centrifugal forces generated by high flow velocities and small radius of the vortex tube.

 

We now consider the equation relating average specific molecular energy ( c2 )  i.e. sound speed energy, to internal pressure volume energy in the ideal gas law:

 

c 2  = k pv = k RT

 

Here,  k, the ratio of specific heats cp/c­v ,  is also related to the number of degrees of energy partition n as

 

k = (n+2) /n = cp/c­v

 

Thus, for n equal to 5 as in air, k equals 1.4. Therefore, if  n were to increase from 5 to 6 with a gain of one whole degree of freedom,   k  decreases   to 1.33 . Then both c2 and the gas temperature T must increase by the factor 1.4/1.33 = 1.05. Therefore at a base temperature of 293 K this would mean a rise of 14.6 C in the air temperature  from the action of the centrifugal force.

 

Of course, at small tangential velocities and consequently small centrifugal forces, the effect on the motion of the gas molecules will be less, and only some fraction of a degree of freedom change would be expected,  giving  a smaller temperature rise.   

 

If we were to express the relative centrifugal force strength in a vortex by the Mach V/c, at a base temperature of 293C we could calculate the temperature rise as

 

+ ΔT  = [M x 1/6] /5] x 293˚C                                                                                                         

 

with the results shown in Table 1.

 

 

Table 1

 

Temperature Rise Anomaly  Due to Vortex Centrifugal Force

 

Maximum                Mach Number    Portion of Molecules  Temperature Rise + ΔT                                                                                                          

Tangential Velocity                                    [ M x 1/6] /5         [ M x 1/6] /5] x 293˚C                          

      (m/s)                                                                                         

                         

30                                     0.09                        0.003                        0.9˚ C               

40                                     0.12                        0.004                        1.2

50                                     0.15                        0.005                        1.5

60                                     0.19                        0.006                        1.9

70                                     0.22                        0.007                        2.1   

80                                     0.25                        0.008                        2.4

90                                     0.29                        0.01                         2.8

100                                   0.32                        0.011                       3.1  

300                                   1.00                        0.033                       9.8

 

These temperature rises are in the same general range as those observed in the Ranque-Hilsch tube experiments.

 

We must naturally now address a further relevant key question : What possible physical mechanism might introduce such a postulated additional degree of energy partition into the vortex?

 

As we have pointed out, in a vortex there is cyclostrophic balance between the pressure gradient force directed radially inward and the centrifugal force directed radially outward.

 

Fc  = Vt2 /r =  −  1/ρ (dp/dr)

 

 

However, it would be unreasonable to expect a perfect and continuous balance between these opposing forces  in any real flow, and  so  we may reasonably expect that there will be substantial vibratory oscillations set up as the cyclostrophic balance is momentarily disturbed and then restored . 

 

We now propose that the vibratory oscillations in the vortex flow will constitute an additional source  of energy partition, so that  the value of n, ( in air for example ) will tend to increase from 5  towards the value of 6.  The values set out in Table 1 above would then apply and this mechanism  would  provide an explanation for the observed temperature increase in strong vortex flows such as the Ranque-Hilsch Tube.

 

Of course, the above formulation of the proof is preliminary; it can undoubtedly be made more quantitative and rigorous, but the essential proposed process should be  clear, and appears to be  supported by the preliminary results.. There can be either one or two outlet ports  or nozzles as described earlier. The  temperature and pressure changes are quasi-isentropic.

 

The expansion of the pressurized inflowing  air in the tube takes place in two ways: (a) by  the usual isentropic expansion in the duct  with the pressure drop ( and cooling)  determined by the ratio of inlet orifice area to outlet orifice or nozzle area and  (b) the additional expansion ( and cooling) which takes place in the superimposed  vortex  because of the accelerated or curved  flow.

 

To repeat, the new  proposed flow effect which may help explain the anomalous temperature rise in  the R-H Tube  is that the very high centrifugal forces induced in some portions of the  flow introduce vibratory motions which constitute  an additional degree of energy partition  and hence are a main cause of  the   temperature rise anomalies that are observed  the tube flow.

 

 

[Another possible xplanation is that the centrifugal force suppresses the   small scale turbulence in the flow and converts it into molecular vibrations or heat motion and so that this accounts for he heat rise.[   Check this thought.

 

(Insert May 16/11) : Usual explanation of the Ranque-Hilsch effect is via conservation of angular momentum, etc,  Here we seek an explanation at the energy level. The internal energy given by the sound speed mc2  = m pv = mRT involves the temperature T via the  5 degrees of freedom of random motions of the gas. However, there is also the linear flow aspect where the isentropic relationship predict the temperature dependence on the flow velocity.  

 

We now consider the additional energy  (kinetic) in the random small scale turbulence which is suppressed by the centrifugal force of the vortex circular motion. And we attempt to quantify its effect on the observed temperatures.

 

We know that  streamline flow velocity affects the temperature via the isentropic relationships. So why would not small scale random turbulence packets of kineticc energy not also affect the temperature when theeeeey arise and disappear?. We know that if turbulence sets in in a previously smooth linear flow there is a pressure rise and so also a temperature rise. So, a linear velocity drop is a temperature rise. Q. Is then a random velocity drop ( i.e  turbulence suppression) also  a temperature rise ( as observed in the R-H effect.?

 

 

 

4.7   Additional Atmospheric  Observational Evidence for the Temperature Rise Anomaly 

 

Because of the flight hazard from swirling airborne debris in land-based tornadoes, detailed aircraft temperature profiles through them are not available. However, waterspouts, even severe ones, are relatively debris-free, and they can be penetrated with suitable aircraft, so that data from a series of waterspout temperature and pressure soundings are available [12,13,14]. 

 

Golden [12]  found that waterspouts typically have temperature profiles which show an unusual temperature rise anomaly in an annular  zone  of about +3 degrees at moderate levels above the sea surface. C.  Later work with an aerobatic aircraft showed a smaller temperature rise of 0.3 ˚C  near the core near the cloud base [13,14].

 

 The cause of this waterspout temperature rise anomaly has apparently not been analyzed in detail, and it is ordinarily ascribed to  some such effect as compression and warming of subsiding air in the vortex  Now, however, we suggest that it is, in fact, caused by  the same basic phenomenon producing the ( much larger ) temperature rises which occur in the Ranque-Hilsch tube, and that it is therefore  to be explained as a result of an energy transformation involving an increase in internal kinetic vibratory motion arising from oscillations or vibrations in the imperfect cyclostrophic balance of the strong  centrifugal and pressure gradient forces in the  vortex..

 

 

  4.8  Conclusions

 

We have outlined the effect of very strong centrifugal force in vortices acting to distort  the average kinetic speed of molecules in a flow of fluid, and thus of slightly  raising the average temperature in the affected annular ring of the vortex. This we propose is the explanation for the observed temperature rise anomaly in the Ranque-Hilsch tube.  Such a temperature rise anomaly is also observed in atmospheric waterspouts;  presumably it also exists in tornados where the centrifugal forces are even stronger.

 

Many peculiarities remain to be studied in the Ranque-Hilsch tube vortex. Hopefully, the present proposed solution to the  problem of the temperature rise anomaly may assist in their solution. 

 

 

References

 

1. Power, Bernard A., Tornado-genesis by an Isentropic Energy Transformation.  http//www.energycompressibility.info   2008. 

 

2. Ranque, G. J.,  Experiences sur la detente giratoire avec production simultanee d’un echappement d’air chaud et d’un echappement froid. J. de Physique et de Radium, 4, h112-114, 1933.

 

3.  Hilsch, R.  The Use of expansion of gases in a centrifugal field as a cooling process. Review of Scientific Instruments,  18, 2 108-113, 1947.

 

4. Gao, Chengming. Experimental Study on the Ranque-Hilsch vortex tube. Doctoral thesis.   Technische Universitet Eindhoven.  pp 151.  2005.

 

5. Albhorn, B, and S. Groves. Secondary Flow in a Vortex Tube. Fluid Dyn. Res. 21, 73-86, 1997.

                                                                                                              

6.  Colgate, S. A., Vortex Gas Accelerator.  AIAA J., 2, No. 12, 2138-214.1964.           

 

7.  A. H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow. 2 Vols.  John Wiley & Sons, New York, 1953.

 

8.. Brunt, David, Physical and Dynamical Meteorology. Cambridge University Press, 1941.

  

9.  Lamb, Sir Horace. Hydrodynamics. Dover Publications Inc., New York, 1931.

 

10. Prandtl, L. and  O. G. Tietjens,  Applied Hydro- and Aeromechanics,  Dover Publications Inc., New York, 1957.

 

11. Rankine, W. J.,  A Manual of Applied Mechanics, Charles Griffin, London, 1882. 

 

12. Golden, Joseph H., The Life Cycle of Florida Keys’ Waterspouts: I.  J. Appl. Meteorology , pp 676-692, Vol 13, No. 6, Sept. 1974. 

13. Golden, Joseph, H,  and Howard  B. Bluestein,   “A Review of Tornado Observations”.  In . The Tornado: Its Structure, Dynamics, Prediction and Hazards. Church, C., Burgess, D., Doswell, C., Davies-Jones, R., (eds.). AGU  Monograph 79,  pp. 319-352, American Geophysical Union.  Washington, D.C. 1993.

 

14. Leverson, V.H., P.C. Sinclair and J.H. Golden. “Waterspout wind, temperature and pressure structure deduced  from aircraft measurements .”  Mon. Weather Rev.  105, 725-733, 1977.

                                                                                                                                                    

15. Standard Handbook for Mechanical Engineers, Sect. 9-8 to-9-13 by E. N. Fales, T. Baumeister and L.S. Marks, Editors.  McGraw-Hill Book Company, New York. 7th ed. 1967.

 

16. Also see http://www.reuk.co.uk/Betz-Limit.htm  

 

  

 

 Copyright: Bernard A. Power 2009      

 

Section Links:

 

Section 1:  Linear  ( streamline) Flow and Flow Power Amplification

 

Section 2:  Invention No.1:  A New Isentropic Air Motor and Clean Energy Source

 

Section 3:  To be posted in near future

 

Section 5:  A Note on Isentropic flow ‘ Perpetual  Motion’

 

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