TR AE 6504

A COMPARATIVE STUDY OF STEADY ANDNONSTEADY-FLOW ENERGY SEPARATORS

by

Joseph S. Hashem

CONTRACT NO. DA-31-124-ARO-D-318

U. S. Army Research Office-Durham

A Thesis Submitted to the Faculty of the Department ofAeronautical Engineering and Astronautics in Partial Fulfillmentof the Requ4 rements for the Degree of Master of AeronauticalEngineering.

RENSSELAER POLYTECHNIC INSTITUTETROY, NEW YORK

OCTOBER 1965 4 d 0l

___ Best Available Copy

TABLE OF CONTENTS

Page

LIST OF SYMBOLS ................... .................... iv

ABSTRAC ............................................... vi

INTRODU CTION AND HISTORICAL REVIEW ..................... 1

PART I. THE RANQUE-HILSCH TUBE .............................

I-A. Flow in the Free Vortex ..................... 4I-B. Flow in the Forced Vortex ......... ... 6

1. Relationship between IJ and k ......... 82. Determination of the temperature near

the wall (Tw ) ........................ 93. Numerical Solution ............. 9

I- • Comparison with Experimental Results ........ 12

PART II. THE NO -STEADY FLOW ENERGY SEPARATOR ................... 14

I-A. General Equations.......................... 14II-B. Comparison with Vortex Tube ................. 16

PART III. COEFFICIENT OF PERFORMANCE ............................ 19

III-4. Conventional Gas Refrigerating Machine ...... 19III-B. Energy Separators ........................... 19III-. Comparison of Energy Separators with Standard

Refrigerating Machines ................... 20

DISCUSSION OF RESULTS AND CONCLUSIONS .................. 22

REFERENCES AND BIBLIOGRAPHY ............................. 24

iii

d - - .-...~mln

LIST OF SYMBOLS

A Tw-8rL

C flow velocity relative to frame Fs in which the flow field is

stationary

specific heat at constant pressure.

e,,eA, ej constants

gravitational acceleration

H ° total stagnation enthalpy of a mass occupying a unit length of

. tube at statio.n__r (see Fig. I-la)

static, stagnation enthalpy per unit mass

constant

1total angular momentum of a mass occupying a unit length of tube

at station r (see Fig. I-la)

m mass of gas which occupies a unit of length of the vortex tube

at station r

,L mass flow rate

static, stagnation pressure, respectively

rotor radius

R gas constant

r .radial distance from vortex-tube axis

S cross sectional area of vortex tube

T ,To static, stagnation temperature, respectively

L flow velocity relative to frame of reference Fu, in which energy

separation is utilized

LL whirl in frame Fu

V rotor peripheral speed

iv

injection velocity

itangential velocity of flow at station (i)

inclination of the flow to plane of rotation at runner inlet

e see Fig. Il-la, Il-lb and II-2b

ratio of specific heats

2

angular velocity of fluid in the forced vortex

IA' fluid density

Subscripts

a,b cold, hot stream, respectively

d discharge

i irrotational

r rotational

w wall

' 1 through 5 See Fig. II-2a

v

ABSTRACT

This paper analyzes and compares two methods of energy

separation -- a steady-flow method proposed by Ranque over thirty years

ago and a nonsteady-flow method recently proposed and studied by Foa.

Part I of this paper develops a simple theory for devices

utilizing the first method (Ranque tubes). Despite its simplicity and

extreme idealization, the theory appears to provide -better agreement with

experimental data than any of the available analytical treatments.

Part II is an extension of the Foa theory of the non-steady-flow

energy separator. The extension is more 1dealizedtha -the--original theory.

but covers a wider variety of embodiments of the concept. A comparison is

also given between the performance characteristics of steady-flow and non-

steady-flow energy separators.

Part III defines a coefficient of performance for the two devices

as refrigerators or air conditioners, and compares these coefficients with

those of standard refrigerating cycles.

vi

INTRODUCTION AND HISTORICAL REVIEW

What is known today as the Ranque-Hilsch effect was first dis-

covered in 1931 by George Joseph Ranque, a French metallurgist associated

with a steel company in the town of Montlouyon, Central France. It is not

known how Ranque made this discovery, but it is assumed that he noticed

such an effect in connection with cyclone separators, where the air drawn

from the center is slightly cooler than the air drawn from the periphery.

He sought to utilize this effect in the device which is now known as the

vortex tube, Ranque tube, or improperly, the iiilsch tube. For this device

he obtained a French patent in 1932 and a United States patent (No. 1,952,281)

in 1934.

The vortex tube is extremely simple and has no moving parts. It

consists of a straight tube provided with an orifice which allows a supply

of compressed gas to be injected tangentially into the tube. This stream of

compressed gas divides into Zwd streams at different temperatures -- the

cold stram leaving through the center and the hot stream through the peri-

phery.

Ranque was obviously hoping to accomplish significant changes in

refrigcration with h~s invention, but subsequent development showed this de-

vic. to be very inefficient as a refrigerator. Nothing more was heard about

it until 1945, when it was learnedthat during the war the vortex tube had

been studied in Germany by Rudolph Hilsch, a physicist of the University of

Erlangen. Publication of a paper by Hilsch (Ref. 1) aroused much interest

in the devi-e in the United States.

Since 1946, several attempts have been made to explain the curious

phenomenon taking place inside this tube. It is now understood that when a

:reami of compressed gas is injected tangentially into a tube of ,constantI

-oss section, a free vortex is formed due to the absence of any external

)rque. The velocity profile in this vortex ic that of an irrotational

low, with constant angular momentum. The total energy of each particle in

ie free vortex is also constant. Therefore there is a continuous drop in

emperature from the walls of the tube to the ay-is. A free vortex cannot,

..wevei, be maintained in a real gas, due to the presence of viscosity.

herefore, the flow cannot maintain a constant energy distribution as it pto-

eeds down.the tube. .'The fast moving inner layers lose part of their kinetic

nergy to the outer layers and are slowed down. The conversion from irrota-

ional to rotational flow is completed near the discharge end, where the

ortex becomes completely forced ( go = constant).

In the forced vortex the flow is highly turbulent and the changes

n state properties associated with the convection of small masses of fluid

n this turbulent flow obey the isentrcpic law p/f - constant. Therefore

,oth the kinetic energy the and static temperature distribution which complies

rith this law are higher at the walls of the tube than in the center. If,

-hen, the flow is divided into two separate concentric flows along any desired

'adius, two streams at different stagnation temperatures are obtained.

Kassner and Knoernschild (Ref. 2) contributed much to the under-

itanding of this phenomenon, through analyses in which they calculated tempera-

:ure and velocity profiles in rotational and irrotational flows of fluids of

:onstarit density.

Part I of the present paper gives a simplified analytical study of

a compressible flow in a uniflow* vortex tube. In this theory the variation

* In the uniflow type tube the cold and hot streams leave the vortex

tube through the same end,'whereas in.the counterflow vortex tube the twostreams leave through opposite ends, See Fig. I-la.

in density, which was left out in the Kassner-Knoernschild analysis, is taken

into account. The results obtained from this theory agree very well with

Hilsch's experimental results given in Ref. 1.

Also treated cn a more general basis, in Part II, is an energy

separator proposed by Fop (Ref. 3), which represents the unsteady-flow counter-

part of the vortex tube. In this device, the process by which the original

homogeneous stream of compressed gas separates into two streams at different

temperatures is an essentially nondissipative "crypto-steady" process. A

crypto-steady process is one which is nonsteady in the frame of reference Fu

in which-it is observed and utilized but admits a frame of reference F. in

which it is steady. This process is easily analyzed as a steady-flow process

in this unique frame of reference, while retaining the advantages of nonsteady

flow-processes in che frame of reference in whichit is used.

The production of cold and hot streams in the nonsteady-flow energy

separater can be explained briefly by ieference to Fig. ll-la. Here (i)

represents the initial homogeneous jet, which divides into streams a and b

upon impinging on a wall. The surface of contact (s) between the two flows

a and b in the original stream is stationary in frame Fs, but is moving in any

other frame of reference Fu . The pressure forces acting on this surface are

therefore doing work in Fu . The work done by these pressure forces is equal

. . --to the energy transferred from the one flow to the other.

Part III of this paper is devoted to the definition and calculation

of suitable coefficients of performance for both the vortex tube and the non-

steady-flow energy separator, and to the comparison of these with the co-

efficicnt of performance of a standard refrigerating cycle (reversed Brayton

cycle).

PART I

THE RANQUE-HILSCH TUBE

I-A. Flow in the Free Vortex

Inrediately after injection the flow is irrotaticnal and free

from external torque. Therefore, at this injection end

vur (1)

where k is a constant.

For radial equilibrium,

r (2)

Therefore, at the injection station

, - dr (3)

Since the stagnation enthalpy is uniform at station (i), neglect-

ing the axial component of the flow vel.,Lity at that station one has

T T 0 IL1 (4)7..1 -(

Ccmbining equation (i) with the equation of state for a pe fect gas and

utilizing (3) one obtains

which when integrated yields

:. r-, V21 , T.° '

• ,1 .1 a J i Je -

or

;, ,,, (5)

Equation (5) gives the pressure distribution in the free vortex.

The density distribution is easily obtained from equation (4) and the equa-

tion of state, which give

(7* -7.- k %IFj

or

By virtue of the conservztion of angular momentum, the expression

for the total angular momentum integrated over the mass m which occupies a

unit length of tube at station r is given by

Similarly, because of the unif'rmity of the total energy distribution the

integrated stagnation enthalpy can be written in the form

. . .. .. ------ ---- ------ --

The velocity, according to the distribution given by equation (1)

and shown in Figure 1-2 would increase indefinitely as the radius tends to

zero. The fluid is however taken to be irrotational down to a nucleus radius

rl, which is arbitrarily specified to be that at which the temperature f the

gas would reach absolute zero. Furthermore it is assumed that the region

lying inside rI , and extending through the whole length of the tube, is re-

placed by a solid core of the same dimensions.

6

The expression for rj is easily obtained by letting Ti equal

zero in equation (4). Thus,

= ' (9)

I-B. Flow in the Forced Vortex

The flow in the forced vortex will now be studied on the basis of

the following simplifying assumptions:

(a) there is no exchange of energy between the

flow inside the tube and the surroundings.

(b) the friction at the walls is negligible,.and

(c) at the discharge end, the axial velocity is

uniform and the radial velocity zero.

As mentioned previously., the pressure and temperature distributions

at the discharge end obey the isentropic law

t~o: constant(1)

This, combined with the equation of state and the expression for

the centrifugal pressure gradient, gives the distribution of temperature

along the radius in terms of the two unknown parameters Tw and ta

To deteL-mine these two quantities, use will be made'of the fact

that the total angular momentum, the tota' energy, and the total mass crossing

this section in unit of time are the same as at the injection section (where

free vortex flow prevails).

Following the same procedure as before, the expression for radial

eqailibrium becomes

dr (11)

Usi .ng the equation of state, equation (1)can be rewritten as

~ (12)

or . -

-----Cmbiingequations (10) and (12) ---t---

dT t~e'rdr

which, when integrated, yields the desired exprcssion for the vari-at Iion --o -f

the temperature along the radius.

- *±~.& (r~I (13)2 4 Y

On-the other hand, by definition

21

-Therefore,

21~~ 2...I (14)

The expressions for the density and pressure are readily obtained

f rom equationS,(12) and (13), thus giving-

and (17)

Rem~emberinlg that m is the mass of gas in a unit length of

k 8

the tube at the rotational end:

Substituting for /_ from equation (15) and integrating, one obtains

Ir ell= ' [(A ar") -(A + r, 2 (7S

or

, _ (19)7res

where

8 -

1. Relation Between w& and k.

For the rotational flow, the expression for the angular momentum

is given by

Mr 27r j rC r

which when integrated by parts gives

11, - (20)

Equations (18), (19), and (20) give

r ICe I I

9

Equtin th toal nguar ometaM. and M rone finally obtains

2. 'Determination of the T-emperature at the Wall(TW

The total stagnation enthialp pe1 uileng th -f the tube- in-

rotational flow is-P.

or, a a.V~ rc7; /ai +. d

orr C Ar +

Substituting mk for M r in the above expression and integrating one obtains

1'w e3 Ine3 7 je

Equating the total stagnation enthalpies iii and H 0 and making user

df equation (18) and (20), one obtains

or 07.- -(2

3. NumeriLal Solution

Equation (21) cannot be solved explicitly for w It becomes

therefore necessary at this point to determine the temperature distribution

numerically for each specific case.

Given k, the corresponding values of u and Tw can be determined

ds follows: 1,

Assigning first the value zero to the quantity fj -

in equation (21), the corresponding values of k j and Twj can be obtained.

The quantity Pj is calculated using these values. By substitution back

in equation (21), new values 4j+l and Twjl are obtained. The process

is continued till successive values of Pn become sufficiently close to

one another.

-Two cases are andlyzed here,-pertaining to injection velocities - . . .

(Vil = k/rw) of 1000 and 750 fps. Tables 1-1 and I-/ list the results of

such computations. The ratio rl/rw in equation (21) is determined in each

case by equation (9).

I11

TABLE I-1: V 1000 f.p.s., rl/rw = 0.4, Ti° - 5280R and ? = 1.4

1st Iteration 2nd Iteration 3rd Iteration 4th Iteraton

P 0 PJ+- , 0.132 P J+2 0.181 P = 0.23

wrw - 1758 f.p.s. wrw a 1625 f.p.s. wrw * 1570 f.p.s. wrw = 155 f.p.s.

.. 494 R Tw a 479°R Tw =4720 R Tw - 470 R

PJ+I 0.132 Pj+ 2 a 0.181 PJ+ 3 = 0.203 Pj+4 = 0.209

TABLE 1-2: Vi 750 f.p.s., rl/r w 0.3,ii= 528°R and. 1.4

1st Iteration 2nd Iteration 3rd Iteration 4th Iteration

Pj -0 P J+ f 0.194 P J+2 0.275 p j+3 0.316

wrw = 1643 f.p.s. wrw = 1460 f.p.s. wrw = 1378 f.p.s. wrw = 1315 f.p.s.0 0 = 51( Tw

Tw 548oR Tw 5230 R Tw 514 R = O

Pj+I = 0.194 Pj+2 = 0.275 Pj+ 3 = 0.316 PJ+4 0.'41

5th Iteration 6th Iteration

P'+ 0.341 p -0.362

wrw f 1295 f.p.s. wrw = 1290 f.p.s.

Tw 506°R Tw = 5050R

Pj+ 5 0.362 Pj+6 0.365

- ~ - - t

I-C. Comparison with Experimental Results

Setting Pr = PdPmin

Furthermore, since the pressure at the wall remains substantially

uniform throughout (Ret. 4, Part 1), the ratio pic'/Pd can be written as

follows: . .

-, (23)

where -(f ) = ( r/.... ...

and Pr'Pr is obtained from equation (17) with r - r1 . Equation (23)

gives, for the two velocities of 1000 and 750 f.p.s. considered here, the

pressure ratios of 8.15 and 3.78 respectively.

Figure 1-3 shows the variation of the pressure along the radius in

the irrotational flow. Figures 1-4 and 1-5 show the variation in density and

pressure along the radius for the rotational case.

Now define

where barred quantities represent average densities given by

and computed using Figure 1-4.

From equation (14), (22), (24) and using the data given in the last

column of Tables I-I and 1-2, a final plot of T0 vs . can be made for the

two values of the parameter pi°/Pd of 8.15 and 3.78 and is shown in,,-

Figure 1-6. Also plotted on the same graph for comparison are the experi-

mental results obtained by Hilsch in Ref. 1 and the simplified theory due

to Kassner and Knoernschild (Ref. 2). The agreement between the theory pre-

sented above and Hilsch's result is surprisingly good tqpecially for large

values of ,i.e., in the range where this device would be expected to

be most useful as a refrigerator. ------

PART II/

I THE NONSTEADY FLOW ENERGY SEPAPATOR

II-A. General Equations

The Foa concept of non-steady-flow energy separation lends itself

to embodiment in a great variety of arrangements. Only the simplest of those

(shown in Fig. l-1b) is analyzed in Ref. 3, for the purpose of illustration.

The following treatment is an extension of Ref. 3. -It is less exact, since it

does not account for any dissipation or losses, but it is considerably more

general, in that it covers most of the conceivable embodiments of the Foa con-

cept.

In the simplest form of this separator, continuous flows exchange

mechanical energy directly through the action of mutually exerted pressure

forces at their interfaces. In more complex arrangements the energy transfer

function is performed, in part or in full, by turbines and compressors, but

the principle of operation is still the same, in that one portion of the

initially homogeneous flw is made to do work on the remaining portion.- The

study of non-steady flow separation as a turbine-compressor action has the

advantage of leading in a direct manner to the determination of. the effects

of the various controls that may be applied to the two flows.

The "generalized energy separator" considered here is diagram-

matically shown in Fig. II-2a. There C denotes a compressor and tI and t2

two impulse turbines which, together, drive the compressor. The two turbines

are taken to be of the same diameter, and it is stipulated that t2 is so de-

signed that the whirl at its exit is zero. Therefore, when t2 is present, the

system is designed to produce the maximum extraction of pcwer from flow a.

Thus, 1A = Co.S s = The assumption is now made that

O= LA . u and V

From inspection of Fig. II-2b, the following relaticns are obtained:

= , s .OS

+ v a u. s

(A V C S (,

a2

"U - C,.o -

C& C, + v

, ( -.

C 2 Vi

LA Chi. +COSO2v( b CS 6

., c 6 cos e&,, + V

Thc.reforc., one obtains

22

and

A. aV

The energy equation can bc written in the following form

h= (6)

and since the static enthalpy is the same at station 5 as at station 3

2 210 = I (" - LA

1.3 LS(7)

From equations (6) and (7) the following expressions can be deduced,

A ,. ~- ,.)(8)

o~ 0 2 2

and 01 Ih6 (9)- -- ---- -

,2. 2

By means of equations (3), (4) and (5), Ub3 and UXj. can be expressed

in terms of V ar j. only. V is obtained as a function of/A. by equating

the power output of the turbines to the work required to drive the compressor.

Power outp, r of turbines = m V(u, - tL + 4tv(u u., )

Power required to drive the compressor = Cfl'V (4A,- "',)

Therefore

b3 *

B. Comparison with Vortex Tube

Equations (8) and (9) were solved, for each of the following four

cases:

-P

17

Case 1.This is the only case involving the use of a turbine t2, formaximum power extraction from the cold flow. This turginehelps to drive the rotor, thereby increasing its rotationalspeed and the stagnation temperature of the hot flow.

Case 2.Prerotation is imparted to the entire flow at the entranceto the runner passages. This can be done eitaer by means ofstator vanes or by iniprting the gas tangentially inside therotor, as is done in the Ranque tube'.

Case 3.Prerotation is imparted to the cold flow only, by means of

-stator vanes at the entrance of the cold flow to the runnerpassages.

Case 4.This is the case treated in Ref. 3. It involves no pre-rotation, no turbine t2 , and no Ftator vanes.

In each case, the performance .was calculated for different values

of the angles ID , cc , and Ok Table II-1 lists the differ-

ent values given to these angles in each case. For simplicity, the angle @

.33was stipulated to be equal to 180°-G in all cases.

The results are plotted in Figs. 11-3 through 11-26. The results

of the analysis of Part I are also plotted, for comparison, on the same

graphs, as dctted curves. The numbers next to these curves refer to the pres-

sure ratio (pi0 /Pd) to which they correspond, whereas the numbers next-to the

solid curves refer to the particular case at hand.

c a a N k a C (' 4 9 3 ..

Cae19 0 0 goo8Case 1 900 90° #90 ° 180 ° -eO

Case 2 900 a 900 1800 -

Case 3 # 900 900 90 180 0 e,

Case 4 90 90 ° 90 180 -

TABLE II-1

Looking at Figs. I-3 through 11-26, it is notic-d th,!t th

curves corresponding to Case i, above, give the highest tempcrature differ-

ence. In fact, this difference increases indefinitely as/& approaches

zero. This, of course, is physically impossiblt. However, a closer look

reveals that the prerotation -in-thef-l ow-ahead of turbine t, cannot be

completely eliminated by expansion through this turbine if the magnitude of

the velocity C , is less than V. Therefore,

:----.--.------

From equation (1),

( s) - 2VCO,., (VC)

The coirresponding value of AA (i.e., the minimum value of1AL cOM-

potible with the stipulation of zero exit whirl in flow a) is cal.ulated by

means of equation (10), with the value of V determihed througi simult inecou,

solution of the second and third of equations (I), and equation (11).

Therefore, in igs. 11-3 through 11-26, only the parts of the

zurves beyond /'min ar physically meaningful.

[

19

PART III

THE COEFFICIENT OF PERFORMANCE

III-A. Conventional Gas Refrigerating Machine

The coefficient of performance of the energy separators will be

......-- calculated and compared with the coefficient of performance of a standard

gas refrigerating machine, using a conventional compressor-turbine arrange-

ment. The cycle normally used in such machines is the reversed Brayton

-. cycle,--(see* Ref.--5) 'where -- ideally --- the gas. is compressed-isentropically ......

in a compressor from a state a' to a ctate bl (sce Figure Ill-la), cooled at

constant pressure from b' to c', then expanded through a turbine from c' to

d' (where the pressure is the same as at a'),while doing work. This work

is utilized to help drive the compressor. The cold gas is then discharged

producing a continuous flow of cold gas.

The coefficient of performance for such a device may be defined

as the heat removed from the cold flow divided by the work of isentropic

compression. Therefore

(h - h'C.P.

hc j

Since "f'' = " the expression for C.P. reduces to

C.P. (1)

III-B. Energy Separators

In the case of the energy separators, the gas is first compressed

20

isentropically through a compressor, then cooled at constant pressure in

a heat exchanger before being allowed to separate into cold and hot streams.

Therefore, using the same definition as before, the expression for the

coefficient of performance of steady and nonsteady-flow energy sepa.rators

becomes:

C.P A - :)

If the energy of the hot stream can be regained, then

.=h 1 - h:

0 07Z-7

T, [,_ (, .)l-,_ (T _) -,..' (2)

III-C. Comparison of Energy Separators With

Standard Refrigerating Machines

The coefficient of performance of a standard refrigerating

machine operating in a closed cycle is independent of,.A , whereas that

of the energy separators is not, as can be seen from equation (2). The

quantity (T7 - Ta) and therefore the coefficient of performance,vary with

/AC . By inspection of Figs. 11-3 through !1-26 it becomes evident that

this quantity is largest for small values of,/c , high values of the ratio

Pi°/Pd and for the smallest possible angles t9 O3 0 0(a and X44

Fig. III-lb compares the coefficient of performance of the non-

steady-flow energy separator (Case 2, with e, - 150 and *eA - - 300),

21

//

with,'those of the vortex tube and of the standard refrigerating machine.

The coefficient of performance of the nonsteady flow energy

separator is far superior to that of the vortex tube although still lower

than that of the standard refrigerating machine. This difference, howevr,

decreases as the pressure ratio pi°/Pd is increased.

DISCUSSION OF RESULTS AND CONCI 'SIONS

The results of the steady-flow-energy separator theory (Part I

of this paper) are summarized in Fig. 1-6, which shows how well the new

theory developed here agrees with the experimental results obtained by

Hilach over the whole range of mass flow ratios. The agreement is best

for thE smaller mass flow ratios, which are the ones of greatest interest

when the device is uset" as a refrigerator. The simplified theory of Kassner

and Knoernschild agrees very well with the experimental data for small mass

flow ratios but not at all for the larger ratios.

The results of the nonsteady-flow energy separator theory, pre-

sented in Figs. 11-3 through 11-26, point to the possibility of substantial

performance improvements over the statorless device which is analyzed as an

illustrative example in Ref. 3. The statorless device, identified here as

case 4, operates only for/A < 1. Some of the modifications covered by the

extended theory developed in Part II -- more specifically, those identi'ied

as cases 1, 2, and 3 -- produce better performance and are capable of operating

over wider ranges of the mass flow ratio. In general, the performance can be

improved by increasing the pressure ratio pi°/Pd and also by making the angles

45 , ea' and 6'6 as small as possible.

The coefficient of performarce of the nonsteady-flow energy separa-

tor is shown to be much higher than ihat ofthe vortex- tube.--The former is

comparable to that of the standard refrigerating machine, and may even exceed

it over a certain range of/a and for pressure ratios Pi°/Pd above 10, as

can be seen from Fig. Ill-lb.

In surmmary the mechanism by which cold and hot streams are obtained

in the Ranque tube is a highly dissipative one, and the only advantage of

23

this device as a refrigerator is in its extreme simplicity.

The performance of the non-steady flow energy separator is much

better than that of the Ranque tube, due to the fact that essentially non-

dissipative processes are responsible for the separation of energy. The

the retical coefficient of performance of this device is comparable to that!

of a standard refrigerating machine for low values of and for high values

of the ratio Pi°/Pd .

'REFERENCES

1. Hilsch, R., "The Use of the Expansion of a Gas in a Centrifugal Fieldas a Cooling Process," Review of Scientific Instruments, Vol. 18,No. 2, February 1947, pp. 108 (unabridged translation).

2. Knoernschild, E., Kassner, R., "Friction Laws and Energy Transfer inCircular Flow," Technical Report N-FTR-2198-ND. GS-USAF, WrightPatterson Air Force Base N-78, March 1948.

3. Foa, J.V., "Energy Separator," Rensselaer Polytechnic Institute,Technical Report, TR tVe 6401, January 1964.

4. Lay, J.E., "An Experimental and Analytical Study of Vortex-Flow-Tempera-ture Separation by Superposition of Spiral and Axial Flows,"Journal of Heat Transfer, August 1959.

5. Sears, F.W., Lee, J.F., "Thermodynamics," Addison-Wesley Publishing Co.,

Inc., 1955.

BIBLIOGRAPHY

6. Ranque, G.J., "Experience sur la Detente Girateire avec ProductionsSimultanees d'un Echappement d'air Chaud et dair Froid," Journalde Physique et de Radium, 1933.

7. Fulton, C.D., "Ranque's Tube," Journal of the ASRE, RefrigeratingEngineering, Vol. 58, 1950, pp. 473-479.

8. Webster, D.S., "An Analysis of the Hilsch Vortex Tube," RefrigeratingEngineering, Vol. 58, 1950, pp. 163-170.

9. Grunow-Schultz, F., "How the Ranque-Hilsch Vortex Tube Operates,"Refrigerating Engineering, Vol. 59, 1951, pp. 52-53.

10. Foa, J.V., "Crypto-Steady Energy Exchange," Renssel-er PolytechnicInstitute, Technical Report TR AE 6202, March 1962; also "A Methodof Energy Exchange," ARS Journal, 32, pp. 1396-1498, September19 62 . -. . . . . . ..

11. Sibulkin, M., "Unsteady, Viscous, Circular Flow, Part 3. Applicationto the Ranque-Hilsch Vortex Tube," Journal of Fluid Mechanics,12, pp. 269-293, 1962.

cold pipe cold pipei r Ir

counterflow type uniflow type

VORTEX TUBE

FIG.I-1

2 -

.2 4 a6 10 n

rFREE VORTE X

F IG. -2

FIG. 1-3,

1.0-

.2 .4.6 .8 1.0

r/r

FIG.I-4-

1.0 - - ---

.21 -

r/rn,

FIG.I-5

.( .O - - -O

r/ r

140 - -- - - - - - - - -

120--(8.15

,3.78

oc

0~~~. ----- - - - - --0

0001:;THIS THEORY___--EXPERIMENT (weltsco)

-THEORYOFREF2

.2 .4 .5 1.0

I ma

1*4ma~mb

FIG. 1-6

FIGIf- lb

b aBEARING

3 2 1 SUPPORT 2345

HOT_________ COLD

tEAMc t2STREAM

COMPRESSED

FIG. 1 -2a GAS

aUa, Ua V

li U a aa.,a

C aa

C b a0

Ub

FIG.KI-2b

FIG.!- 3

z 1200- - IiTM I I VLd - 4@di

104- 12 -72

400

0 .2 .4 .8 . 1.0 1.2 1.4 1*.1.8 2.@/U 0 .2 .4 .6 .6 :1.0 1.2 1.4 1.6 . .0Uk

14k0 140-------I

120PYPd 3 1200 -- - - -3 -

CASE. N, u3 CA3.74 20

2

CASE4uasab~za0 : CAS 3~&71~

FIG.R- 4

1400 1400 - -- v-l JI L - TOR \__

- -- - -, 25 - 1200- - - P/:

0000

.. ....... ..... . ... ..... . .. b OG\ - -\

20- - --

,° 1 1 - - 0 .2 .4, . 4 . . .2 1 .4 1 .... 2.0,40 .2 .4 .6 .3 1.0 1.2 1.4 1. 1.8 2 .oi, o .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.o,{Jjk

400 1400 ---

O°" 72 \ !a 0

12CC- -10 2CC--- .

10OC0 low-_ - ~P~~l

LI§7 200 1 2 4.4 i-- f -- -- 7- - , ---- -.

0 .2 .4 .6 .8 1.0 1.2 1.4 IA6 1.4 2.0/0J .2 .4 .6 .6 1.0 1.2 1.4 1.6 1,.1 2.0,

CASE 1 = -'a. 30* CASE 2 = als b=- 300

CASE 4*m&au*'b3=-a 4 q0 ° CASE 3:a,z 30•

FIG.- 5

14 1400 rrr

* IITR P I II

1200 --- Py~d 2 120( -- -

40 -- - 40U--

200 0 .2 .4 .6 .8 1.0 1.2 1.4 1.4 1.3 2.0,4U 1 * 0. . . 1.0 1.2 1. 4 .16 .1a2.

T 4 T.10

4 - I --100PF -

Obs -5 -

600- "a -

---------- -40

200L 20 .2 .4 .4 .8. 1-0 12 1.4 1.4 1.4 2 .0/J A -0 .2 .4 .6 .4 1.0 1.2 1.4- 1.4 1.3

CASE 1uma 45 , 0 CASE 2 2a,caIb, 4 5 *

CASE 4 O'A320-iNK4 CASE 3 xa,a 450

.//

/./

1400 - G %R 00

1200 yd120F(

-' 4* - ° -- ,\eh ° -

'W0

.2 .4 .4 .0 1.0 1.2 1.4 1.4 2. J. 0 .2 .4 .4 .6 1.0 1.2 1.4 1.4 1.8 2.0/

T1400 .. . .o -rc- TY OR II - IT I, I rt 2 o 1200- Py,o,,

.. ,.. - _ b-, 1 5 0 _ Sb1\5 0

oot t\l+t 1 ' oo[-• I 8oe!

2WI - -- 2-o I-

0 2 .6 .8 . . 4 16 142 0 .2 .4 .4 S81.0 1.2 1.4 1.6 1.82.

,mo+ + .-- +.. --.- _ - -- m - - -- -.

CASE i =*ca. - 4? ° CASE 2 z cma~wbjn!S °

• C A S E 4 : °0la: z° b ," a. : 9 0* C A S E 3 =f, %: 4 5

12 0 i-..4~ PYJld 2 12001- P.

lool oo~

6002

400 K- - 400-- j -

0 2 .4 .4 .8 1.0 1.2 1.4 1.6 1.8 2.0 0 .2 . .6 8 101.2 14 16 1.8 2.,_

m P. =3 _ u

YF d.V 120 ,et

6 00

CASE I a.~~ @ CASE 2 m1 b~S

CASE 4*-a~xu -8b900.9 CASE 3 zola 13 S0

FIG.E- I

1400 100-ii I- 3Q1--10 12 1~ 14.8 20/i

200 ..' ~ ~ 00Kj - - ~-----

140-- 1400'-

1200 Pd -A 100 - P~

_._ 1.1 2.

T~ ~ ORI-j -

p 12~~00-- PiPd-l

eb, LI3 10 i5---. -

40 40W- i

200o 2 1.2 1 7 71 -. 1. 2 4 /T '0 2 .4 .2 1.0 1.2 I.A 1.4 1.8 .e t 2 4 . 9 .

CASE 1 2 Maz 6e CASE 2 =ws2 2%ib 600

CAS E 4la 2&z xb .0 CASE 3=gx&2-60

FIG1- 9

-40 HO4 - -i

120-K

420 1200 -/

4CC- - ----- 800- -

0 .2 .4 .6 .6 1.0 1.2 1.4 1A .8 0./C .2 .4 .6 .0 1.0 1.2 1.4 1.4 1.3 2.0/

0 00

CASE4UaUb,3m.W C~f 3too.

FIGI- 10

2 o 2 1200 - _

S - . - y 30* - , o-

1o - Iwo

a"" g oo E7. 1 l"T

6,0

o .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0/, 0 .2 .4 . . 1.0 1.2 1.4 1.6* 1.8 2.0/.4

1400 1400 _OR 3

o300I I YY .200 - 7Pd"'

1 ~ -01--01000 g-~---ooj-

60: -+ -- 6-00 ::-,i.1 _ .00 i

200 CSE0 200Hiit IIL0 .2 .4 .6 .3 1.0 1.2 1,4 1.6 1.8 2.0 . 0 .2 .4 .6 .6 1.0 1.2 1.4 1.8 1.3

CASE 1 z20(a s30° CASE 2 :O.a, (bt 300

CASE 4 t" b " La. O .b le' ° CASE 3 : a, :30

FIGL.-1i

1400- 1400

1200 ' o---4- -4 V120 - - ... - . ,,

Ob0 Pd 3

t°O°P--'+" .... + - 4 6-"L -

' 20\

0 '1 .4 .6 .6 1.01.2 1.4 1. 1. a-- .. .4 .0 1.2 1.4 1. 1.6 2.0t

.100 1do

1200 -Y 3--. --- 1 2 WO-i- yd3

4.30,----- ..... .. t0 -

W - .\. .. .

I.. ' \ i '.* - \ . .,oo . -+ .... ++.--- --. + -,+ * ...--....... +.-+ - . .....m ' 800 - 4..+

40400.1 .. ... . ..... ..

200 7t70 . .4 .6 1. 1. 1, 1A 14 .0 --0 .2 .4 .6 .4 1.0 1.2 1.4 1.4 1.6 3.0,4-t

CASE I=1a 4 CASE 2 =4Xa~zb,450

CASE 4. . . .CASE 3 = a ,.

F IG.U - 12

1400,

OR ILL120- -- Pd 251200' Ycl Pd:

- I -- --

400 ---. ~aO

400 200 ---

o ..2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0/,U 0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.82.d

1400 v v 1400,

U0 *. 101200 -. Py 210

6"o~3 goL - -- ' -41 --

I F, 2

200 11 --H FJLL J0I0 '.2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0~. 0 .2 .4 .6 . . . .

CASE I st,.=:450 CASE 2:z0aObx5

CAE44a,z- b,20a 4 §O CASE 3 zta:z450

FIGI- 13

1 1 4 00- - - .. . . . --E.... . .. ..fT %

1200 Pd 2 200/-- z 2I ~t 3( -eb-3s -.-

S o-.- g-----oot--i-l --

40.0t L -J 4__h_

200 P 1 1 210 iI'1' I0 .2 .4 .6 .i 1.0 1.2 1.4 1.6 1.3 2.0/J 0 .2 .4 .6 .1 1.0 1.2 1.4 1.6 1.8 2.0 JL

TI-I H r T - 1400 £J -,200 d = 3 120 -. Pypd 3

1001000 . ..

too 00-- L~mo&Pi -1

0 .2 .4 .6 . .0 .2 1.4 1.6 1. . 200 .2 .4 .6 .3 1.0 1.2 1.4 1., 1.8 2.0, -

CASE I a Ot a6 0 CASE 2 =0(adkb,,60CASE 4 sO u Gbb.ubKa.= 900 CASE 3 =Ca= 6C0

FIG. a- 14

140 -' 1400- I-1 R

hoc. .-. - it 1200 - -. - jS

Sb,23 0 - - b '30 "

1000-- 16-01

200 400v2 - -±±

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.3 2.0,44 0 ~2 .4 .6 .8 1.0 1.2 1.4 1.4 1.8 2.0/i.

1400T1 1400

1400 T*-R 2_

1200 - -p 2 0=10 PyPdUIO

0.10:

0 .2 .4 .6 .8 1.0 1.2 1.4 1.4 1.8 2.011j- 0 .2 .4 . . 1.0 1.2 1.4 1.4 1.8 .

CASE I &a 6e AS 2 =da~z Kb,%6e

CASE 42 dKA2 *Cb,,'oa 90 CASE 3 = (a26O0

1400 - ~1400--- TO.R J

1200-"Vd 1200 -yd2

1000 . . 1C

43 4

"0C 600 -4 C- ---- - -- - --------~

0 2 4 .6 . 10 1. .416 . 10,U. . 8 . -. 1.4 1.1.5 2.014,L

J400- J 1400 -- ~- 11 0

10C00 -- 10CC-

3.7

__ 3.3.

200 200t0 .2 .4 .6 .8 1.0 1.2 1.4 1.4 1.8 2.0/4L . . . 8 . 1.0 1.2 1.4 1.4 1.8 2.

CASE I =3e CASE 2 -- tdt,3

CASE 420(a3 . fbwG *,49O CA SE 3 zda 3 1O

FIG.1- 16

100g- T% 1200 -- I ~ 5 J TR - - -]I -

H 1100 Z

f2I ~N --- P)- ---.-

____________ G b ,2 45____________ _________

200 ~ ~~j~j. 7j 00o .!1 .4 .4 .6 1.0 1.2 1.4 1.6 4 la20 2. 410 1. 2 1 .L

T I 11400 TT T

IR OR1000

Mo 04- iP~*'

400 -4-00iV i7

gi00 20 I I-

.2 .4 .4 .8 1.0 1.2 1.4 1.6 1.4 2.0 0 . .4 . 1.0 1.2 1.4 1.4 1.5 .

CASE I=1a: 0 CASE 2 =O(a=*(b:3O

CASE 4 :a=~,oax0 CASE 3 a .F0

FIG.A-1?

T:, , , , -R

1r 111 IIE 1] 1400 1 YILL.- i .... b,1245* .b0 - 4-

o - - - ---4

. ....

" ,.* 1.3......* 4 .4 .3 1 1. 1 .16 0 .2 .4 . .1. 0 1.

-7_102 I 1 200

0 .2 .4 .6 .3 1.0 1.2 1.4 1.6 1.1 2.o/. 0 2 .4 ., .8 1.0 1.2 1.4 1.6 1.3 01, k

. ...... 1400 . . . '

.1 4...0 0 ' • -

00

CASE I ,Oa4. CASE 2 e:OaOb "45°CASE 43zla~z b , 9 o CASE 3,a-5 4 5• . ll , .. _ _IIIIII II II ll I lll I I 10 0 0 -I

Ia" - Soonli III I I I I

]//

FIG .1r-11,

, 0 140 0 l

120- PPd x 1200 - Pd

.- .- _ _,.-.

\ I

600--

0 .2 .4 .6 . 1.0 1.2 T.4 1.6 1.3 2.0/4, .2 .4 .6 .8 1.0 1.2 1.4 1.4 1.3 3.0/(.

F10- .140 1 Tfo - 1 0 12 0 - - -l P

IOOOI

Ob 3 ~ 4 5- - 4.~-450

400 400.

200 - goo* .2 .4 . , g 1.0 1.2 1.4 1.4 1.8 2 .O/ 0 .2 .4 .6 .. 1.0 1.2 1.4 1. 1. 2.014-t

0 '°- - + 0 + [ ' +°! l F - I- - i

CASE I = a 4S CASE 2= = OLsb 4 5

0

CASE 4 -. ta,db,'tla9 CASE 3: OLa,-45* i -,i iw.1-LI !-. V

FIG,1- 1

1400- 40I . .1400 .

,. i!!l. , - -° - - b4 -00d Pd

'00-

0 .2 .4 .6 .3 1.0 1.2 1.4 1.6 1.6 2.0/,41 0 .2 .4 .6 .8 1.0 1.2 1.4 i.6 1.0 2.o/J,

T R 4002ILL 1400

0

120d -4 -3- 1200-- PyPd-3eb 4 b =450

80-- ' SOeK z'5U

1 IN - I I40 4 00

200 E 2000 .2 .4 .6 .6 1.0 1.2 1.4 IA 1. 2.O/L, 0 .2 .4 .6 .6 1.0 1.2 1.4 1.6 1.6 2,0/

CASE 1. 0,a. 6 0 CASE 2 ='K&,-Ob,-60CASE 4 It*,"GLb,-d~a, " 0 CASE 3 a. &a-6O0

FIG.11-20

31400 140W

.001 ~ ~12001 m5

- bU4 0 bW

x 1L

200 ~r h L 71 2000 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.1 2.0/4. 0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.04.

1400 1400--TOR.. T! -- 00

1120- - 1200- P/Pd 10

8009b324-- -- + -0

-"

So, c .. .--- - -.- - .- , 0o'"-

200 LLA - [ 200 _

0 .2 4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0/LL 0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2A1A4

CASE I0a ea 650 CASE 2:2 CKa,2 O(b,2 60o

CASE 4 Ot a,=I =900 CASE 3 =O*a, 6 00

FIG 1- 21

140-- 1400-- ...-- -- i1I "T -' OO- -

1200 - " 1 y/d 2 1200 0 Pd

too too--O

600 6 .

4 I 000'-

. I+ ,---- -- , .. . .. ...

__ ._ 2 .. __+ _,,-

4001- - i- - 400om i-i.&1-.

200 200 1.0 1.2 14 16 1.8 2.0 tL

0 .2 .4 . 8 1.0 1.2 1.4 1.6 1.8 2. 0 .2 .4 .6 . . 0 . 2 --

-: ' 1400 . . 1

,o o -+.. .. 1 1 -: - - .:- --, - ! ,o -- r I 1 .

I I T TL ORrdoo, -F =~ 3 12000 1

- 6 ' 0 -0 _ .I

4.. ...... _.--... . .. . --- . +o -- - 4----: -+-

37

C i: 4----- --

A 20CASE 4xOa,2*3U j~w0 CASE 3 --@.,U-b3e

FIG-.1-22

1400. -t 1400 F F

1200-I pd 2 1200 - Pd-

_____ - 000~- Ob2. 0 __0_ ___ ______

-- 4001 -- j00

T .I IIT O

1200L. 10 1 I I200-1 1.11.*dzlIt 'i0 2 4 4 .8 1. 1 .4 1.6 . . ~ ~ 0 . 4 . 10 1.2 4=60 182Oj

400i-- ,---4.--

.2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0/0 .2 4 .0 . 10 12 1. 1. . 2.0.,4Lt

CAEI= t 0 CASE 2 =Oa2 ~d~tg3OI.CASE 4xOL&3UClb,-Q(&a*9 CASE 3 =cVa3o

FIGIU-23

1400 14 ,roo

- 1200 . .. 1--

1 -I ] i "1 a i 0 200

c .2 .4 . 9 10 12 1 16 18 2a ..4. .o .8 1.0_1.2 1.4_1o6., 0 2./4),~100o l 140", t 0 LJ L _! : L

• • {-..... i , -' I

4 0 0 '.. . .. -- ... ... + + + .. .. . . ... 0 - " " -* . ...*--...... .- - '- ---. ..-- I

. I , goo,

201.fi - 200111 d l

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.3 2.0 0 .2 .4 .6 . 1.0 1.2 1.4 1.8 1. 2.0,,L

120 1 I = t a 4 50 CAS 7 =046, Oyd..

1 C AS . ..... L b. ... .. CASE 3 , _ . 45,

3.7, \1,, ,a ' I !

1i - I" I T + " i 204f4Ii 12 1. ,~

0 .2 4 .81 ,I 1.0 1.2 1.41 .8 1.3 2.0O 2 4 ,1 . 2 • jII.,.,

C A SE I : m .,.45 ': C A SE ' : , d b -1 5 °

CASE 4sO a,'bZ t~a.2 9O ° CASE 3 "-lap4t . .

FIG f- 24

4 F°-y27 I fIj T ,+Rr° ....... I _ L !!120 i iy.5 __ lo' _ _

I 1i, i _ _

100 t - cl X 1200 m -

Nf 100C5(F goo--

S-0M 2 .... . ..

400.

Li0 I

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 .0 . 0 .2 .4 .6 .3 1.0 1.2 1.4 1.6 1.0 2.0,()

1400 0~ - -40

317 1 KF /

------ -

1000.... .. + ... 000 +- +-+ 400L-

I - -00.-

200 -. + + Lk-

0 .2 .4:. .8 1.0 1.2 1.4 1.6 1.3 2.0/,L 0 .2 .4 .6 . .0 1.2 1.4 1.6 1.8 2.0/2

CASE I COa.,450 CASE 2 =(aOCKb,=45,

CASE 41 09a,-tb,=:OCa,:900 CASE 3 :OKa&:45°

FIG U -25

140071-1 1TV 7] T4 T~~jJ 7000FT2 1200 - -- -

........ .... ....T. .. -T - ,- 2" °° ...!0CI-- -!-..-J- !1,,6 .-. 10c -

-3 - - I 40 -01 ,! ______I -v .____ i~

2 ,00 - 2-0

40 .. . . ! 4400'- -!- -, __ 2 i . .. -

I 91 ..[IL f -

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 0 .2 .4 .o . 1.0 1.2 1.4 1.6 1.8 2.0,4_L

.1400_ . - 71 , __0

l ; -I---1-- eb =6Ou V_-J b : u , l--

1 i i 2 ,#j" 1-H' ;1| ! iiT 2

o 1 - P ;t ' 4 : i 3

600. 600- -

.. .... -i-~ -... . -. ---...... ... + t - 1..

10 0 2 { -

-00 600.---38I i 1

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.1 2.0/0 .20 .4 ,. 1.0 1.2 1.4 1.6 1.1 2.0/4.t

CASE I = CC,.- 60' CASE 2 =O S,:=Cb,06 OCASE 4 z Oia iC .b,nOCa.- 9O CASE 3 =OkIa?600

FIG fl-26

1 - -y d1 M -PP

400 -400 I200 2001 -E tLft

0 .2 .4 .0 .8 1.0 1.2 1. . 5 2Oa 1 0 .2 .4 .6 .8 1.0 1.2 1.4 L60 1.3 2.0,t

1400 14 0 F T

I J ii0 1400 -

t20~' PdlO10 - - - - - Py,d 10

/ 10001 ---- i* 3 O -*,0

L 1..1

400 J--i L1

200 ~L 1117K 200 1~ z0 .2 .4 .6 .4 -1.0 1.2 1.4 1.0 1.8 20 0 .2 .. 0. 1.0 1.2 1.4 1.6 1.8 2.0 Al.

CASE I = Ota4 '600 CASE 2 z Ca~mOb,mSoo

CASE 4Zo= a 1 'k&**a4 CASE 329,-0

pressure

P.0 b

Pd -

Vol urneFIG. rn-lo STANDARD GAS REFRIGERATION CYCLE

C.R 7

5.0 . _-_- -

Standard Gas Refrtgerotton Cycle

4.0 -- -Foo Separator 'Case 21

IRanque Separat or

2.0-

103

11.0

FIG.EE-lb +I

pressure

Cb

Pd--0

VolumeFIG. N-la STANDARD GAS REFRIGERATION C.YCLE

5.0 &

Standard Gas Refrigeration Cycle

4.0 - - Foo Separator ICOSO 2

Ronque Separator

2-