NATIONAL ADVISORY COMMITTEE

FOR AERONAUTICS

TECHNICAL NOTE 1947

INVESTIGATION OF FLOW COEFFICIENT OF CIRCULAR, SQUARE, AND

ELLIPTICAL ORIFICES AT HIGH PRESSURE RATTOS

By Edmund E. Callaghan and Dean T. Bowden

Lewis Flight Propulsion Laboratory Cleveland, Ohio

Washington September 1949

https://ntrs.nasa.gov/search.jsp?R=19810068720 2018-01-30T05:20:45+00:00Z

NATIONAL ADVISORY COMM1'rrJa FOR AERONAUTICS

TECHNICAL NOTE 1947

INVESTIGATION OF FLOW COEFFICIENT OF CIRCULAR, SQUARE, AND

ELLIPTICAL ORIFICES AT HIGH PRESSURE RATIOS

By Edmund E. Callaghan and Dean T. Bowden

SUMMARY

An experimental investigation has been conducted to determine the orifice coefficient of a jet directed perpendicularly to an air stream as a function of pressure ratio and jet Reynolds number for circular, square, and elliptical orifices. The effect of air-stream velocity on the jet flow was also determined for three tunnel-air velocities. Equations for the flow coefficients In terms of jet Reynolds number and pressure ratio were obtained for the various shapes. Excellent correlatIon was obtained between the results for a Jet discharging into still air and the results for a jet discharging into a moving air stream, provided that the correct outlet pressure was used.

INTRODUCTION

The introduction of a gas or a vapor into an air stream for purposes of heating or cooling the air stream may be easily accom-plished by the use of a high-velocity jet directed perpendicularly to the air stream, as reported in reference 1. In order to evaluate the heating or cooling effect of such a Jet, knowledge of the mass air flow of the jot or, more fundamentally, the discharge coeffi-cient of the orifice from which the jet is issuing is essential. In addition to the effects of jet Reynolds number and orifice shape on the discharge coefficient, the jet pressure ratio and the inter-action of the air stream and the jet must be considered.

An investigation to determine the orifice-discharge coefficient of a high-velocity air jet directed perpendicularly to an air stream was undertaken in a 2- by 20-inch duct tunnel at the NACA Lewis laboratory. The discharge coefficients of circular, square, and elliptical orifices were determined at tunnel-air velocities of 160, 275, and 380 feet per second for several jet (orifice) areas, a range of pressure ratios from 1.15 to 3.2, and a jet total tem-perature of approximately 4000 F.

2

NACA TN 1947

APPARATUS AND PROCEDURE

The experimental investigation consisted of two parts: (1) determination of the flow coefficients of the orifices in still air, and (2) determination of the effect of tunnel-air velocity on the jet flow.

Flow coefficients In still air. - The nine 1/16-inch-thick thin-plate orifices investigate of three circles, two ellipses with an axis ratio of 4:1 1 two ellipses with an axis ratio of 2:1, and two squares. The circular orifices had diameters of 0.375, 0.500, and 0.625 inch;, the ellipses and squares had areas equivalent to the 0.375- and 0.625-Inch-diameter circles. Each orifice was investigated over a range of pressure ratios from 1.15 to 3.2 and for jet total temperatures of approximately 700 and 4000 F. Air for the jets was obtained by passing high-pressure air through an electric heater and into a plenum chamber, the upper wall of which contained the orifice, as shown in figure 1. The mass flow was measured by a calibrated orifice in the ducting upstream of the heater. The calibration curve for the measuring orifice indicated an over-all accuracy of 2 percent.

The plenum chamber was constructed with an Inside diameter of 6 inches to minimize the effect of approach velocity and to insure that the static temperature and pressure measured in the plenum chamber would be equal to the total temperature and pressure of the jet at the vena contracta.

Static pressures were measured at the orifice and at a point upstream of the orifice by pressure taps in the tunnel walls.

Effect of tunnel-air velocity on jet. - The elliptical orifices were mounted with the major axis parallel to the air stream and the square orifices with two edges parallel to the air stream. Each of the nine orifices was investigated over a range of pressure ratios from 1.15 to 3,2 for a jet total temperature of 4000 F and at

tunnel-air'veloçitiee of 160, 275, an 380 feet per second. The static pressure was measured by the wall taps used In the static investigation and later by a static tap located in the tunnl floor at the point of maximum thickness of the jet and about 0.030 inch from the orifice.

NACA TN 1947

3

SYMBOLS

The following symbols are used in this report:

aj speed of sound at jet vena contracta, feet per second

b minor axis of elliptical orifice, feet

C flow coefficient, ratio of measured to theoretical flow

D diameter of circular orifice, feet

d length of side of square orifice, feet

g acceleration due to gravity, 32.2 feet per second per second

total pressure of jet, pounds per square foot absolute

P i outlet static pressure immediately adjacent to orifice, pounds per square foot absolute

p static pressure of air in plenum chamber, pounds per square foot absolute

B gas constant, 53.3 foot-pounds per pound per

Re jet Reynolds number

0 Tj total temperature of jet, F

t static temperature of air in plenum chamber, OR

Vj velocity of jet at vena contracta, feet per second

VP velocity of air in plenum chamber, feet per second

0, 13 flow parameters

y ratio of specific heats of air, 1.400

viscosity of jet at vena contracta, slugs per foot per second

Pi mass density of jet at vena contracta, slugs per cubic foot

PP mass density of air in plenum chamber, slugs per cubic foot

4

NACA TN 1947

METHODS OF CALCUlATION

-The jet velocity V and the jet density P j at the vena

contracta, or minimum section, were determined with the compressible-flow relations for the subsonic or unchoked conditions from the. following equations:

2 2

- -].P 2 Y-lPj 2

1

Pp

Because is very near zero, p = Pj and t = T + 460 and

therefore. p

V2.646 j t\117_1 \JPp

YJ

P j =P Pepp

When the static-pressure ratio across the jet exceeded that necessary for choking, the jet velocity at the vena contracta equaled the local speed of sound In the air. The jet veloo4ty for this case was determined from the following equation:

Vj = aj =t\V[igR(T + 460) = 44.8/\jr j +4 = 44.8

The jet density for the choked condition was determined from the total temperature and pressure by use of the following equation:

1

(Z2+1)1

Pj________ = 0.000369 = 0.000369

=o.000

T + 460 + 460 t

NACA TN 1947

5

The theoretical mass flow of the jet was calculated as the product of the Jet density P j , the jet velocity Vi., and the

orifice area. The flow coefficient C was calculated as the ratio of measured flow to theoretical flow.

RESULTS AND DISCUSSION

• The problem of orifice-discharge coefficients at high pressure ratios was first investigated by de Saint-Venant (reference 2) and later in references 3 to 5. A theoretical analysis was presented in 1902 by Chaplygin (reference 6) that showed good agreement with the results of references 3 to 5. All the experimental investigations, however, were conducted with circular orifices in still air and no attempt was made to obtain a jet Reynolds number correlation or shape effect. In addition, when'a jet is discharged into a moving air stream, an interaction between the stream and the jet may be expected.

Jets discharging into still air. - The first part of the inves- tigat ion was conducted to determine the orifice coefficients of a jet discharging into still air. The variation of flow coefficient as a function of pressure ratio Pj/Pj is shown in figure 2 for

the 0.375- and 0.625-inch-diameter circular orifices at a constant jet total temperature of 700 F. The slight offset of the similar curves obtained indicates a jet Reynolds number effect. This effect may also be seen in figure 3, where flow coefficients for the 0.625-inch-diameter orifice are plotted as a function of pres-sure ratio for Jet total temperatures of 700 and 4000 F.

In both figures 2 and 3, the effect of increasing the Reynolds number, either by using a larger orifice diameter or a lower Jet temperature, yields a lower flow coefficient.

The curves of figures 2 and 3 show that a linear variation of flow coefficient was obtained with pressure ratio at subsonic flows. A transition region occurred at pressure ratios slightly higher than choking flow (1.87) and a definite decrease in slope of the curve was evident, A linear variation of flow coefficient with pressure ratio was again obtained at high values of pressure ratio.

A cross plot of all the circular-orifice data is shown in figure 4 where the flow coefficient is plotted as a function of jet Reynolds number (PVD)/IL for constant Values of pressure ratio.

6 NACA TN 1947

The curves obtained are similar to those normally given for orifice-discharge coefficients at pressure ratios near 1. The flow coef-ficient decreased with Increasing jet Reynolds number.

A cross plot of the data in figure 4 showed that at a constant Reynolds number two straight lines almost exactly represented the variation of flow coefflcient . with pressure ratio (fig. 5)..

These flow-coefficient relations at constant Reynolds number were utilized as a parameter e. From a replot of the data in figure 4 as the variation of flow parameter C/O with Reynolds number, a single curve was obtained (fig. 6). An equation for this curve has been obtained; It should be noted, however, that e rep-resents two functions that are valid only in their defined limits.

= 0.948 + 4.83 U

Re x 10 + 53.8

where for P j/pj from 1.15 to 2.09

9 = 0.151 - P.' + 0.44

Pj

and for P j/pj from 2.09 to 3.2

P.' U = 0.060 - Pj + 0.636

and

Re= PiV1D1

Aj

Two curves of flow coefficient as a function of pressure ratio have been constructed from this equation for Reynolds numbers of 60,000 and'350,000 and plotted with the theoretical results pre-sented by Chaplygin (fig. 7). The analysis of reference 6 is applicable only for two-dimensional flow in the subsonic region. The agreement between the theoretical calculations for two-dimensional flow and the experimental results for three-dimensional subsonic flow is evident in figure 7.

NACA TN 1947. 7

Analysis of the elliptical data showed that it was possible to correlate the flow coefficients of the ellipses with axis ratios of 4:1 and 2:1 as a function of jet Reynolds number for constant pressure ratios, provided that the jet Reynolds number was based on the minor axis. The variation of flow coefficient with Reynolds number at constant pressure ratios is shown in figure 8 for all four ellipses at jet total temperatures of 700 and 4000 F.

A comparison of the curves in figure 8 at a constant value of Reynolds number showed that each curve had a slightly different slope. It was therefore necessary to obtain a parameter 3 in terms of the pressure ratio based on the Reynolds number at which identical slopes were obtained. A plot of the variation of flow coefficient with Re/13 was made and curves similar to those for circular orifices were obtained. The parameter 0 could then be obtained so that do plotted as a function of Re/n yielded a single curve (fig. 9). The following equation for this experimental curve was obtained:

-0964+0.130

Re x 10

for Pj/Pj from 1.15 to 2.09

P1 o = 0.149 - + 0.469 Pj

= 5.97 Lj - 4.17 Pj

and for Pj/Pj from 2.09 to 3.2

Pa e = 0.054 - + 0.667

Pi

Pi 13 = 2.00 + 4.20

and

8

NACA TN 1947

P1 V1b Re =

I.LJ

The flow coefficients obtained with the square orifices were correlated with Reynolds number in the same manner as for the cir-cular orifices (fig. 10). The Reynolds number was based on the length of a side d.

It was again possible to plot C/e as a function of Reynolds number and obtain a single curve (fig. ii). The following equation for this experimental curve was obtained:

= 0.916-i-7.5

8Re x 10 -i- 74.0

where for P j/p j from 1.15 to 2.09

P. 8 = 0.150 _a + 0.467

pi

for PS/p. from 2.09 to 3.2P.

0 0.061 11 + 0.655 P

and

Pj V1d Re =

I-Li

A comparison of the various curves calculated from the orifice equations (fig. 12) shows that for a given orifice area, jet tempera-ture, and outlet pressure, the flow coefficients in descending order were the ellipse with an axis ratio of 4:1, the ellipse with an axis ratio of 2:1, the square, and the circle. This relation of the various flow coefficients logically follows from considerations of the types of flow involved. With the circular orifice, the lines of flow converge from all azimuths and hence result in the lowest flow coefficients.. The square-orifice flow lines converge in approximately four directions and thus should produce a higher flow

NACA TN 1947 9

coefficient than a circular orifice. The ellipses, particularly those with large axis ratio, approach a slot in shape and therefore yield the highest flow coefficients because the flow lines from a slot converge in only two directions.

Effect of tunnel-air velocity on jet flow. - For the initial study of the interaction of the jet and the air stream, the static pressure into which the jet was discharging was assumed equal to the free-stream static pressure of the air stream. The variation of flow coefficient with pressure ratio is shown in figure 13 for a 0.625-inch-diameter circular orifice at a jet total temperature of 4000 F and at tunnel-air velocities of 0, 160, 275, and 380 feet per second. Because the curves were separated by a function that was dependent on the free-stream velocity and because of the rela-tion of static pressure to velocity (Bernoulli's equation), a fur-ther investigation of the static pressure into which the jet was discharging was undertaken.

Measurements were made of the static pressure immediately adJacent to the jet at the point of maximum thickness by a tap located in the tunnel 0.030 inch from the orifice. These data indi-cated a static pressure considerably lower than free stream and somewhat lower than that indicated.by a static tap located on the tunnel wall at the same longitudinal position. Analysis of these data showed that the free-stream static pressure was related to the outlet static pressure in accordance with the Bernoulli and mass-flow equations, provided that the free area at the orifice was based on the tunnel width minus the jet width at the point of maximum thickness. When the data of figure 13 were replotted using the corrected values of flow coefficient and pressure ratio based on the corrected outlet pressure, data were obtained (fig. 14) that were in excellent agreement with the calculated curve for a jet discharging into still air. When this correction was applied to all the data for circular, square, and elliptical orifices, excellent agreement was obtained with the results for a Jet discharging into still air. Calculation of the flow coefficient for a Jet discharging into an air stream i thus possible when the equations for a Jet discharging into still air are employed, provided that the correct outlet pres-sure is used.

SU1'1MABY OF RESULTS

The following results were obtained In the investigation of the flow coefficients of circular, square, and elliptical orifices discharging into both a still and a moving air stream.

10

NACA TN 1947

1. Equations were obtained for the flow coefficients for the various shapesin terms of pressure ratio and Jet Reynolds number.

2. Agreement was obtained between the experimental results for three-dimensional flow with circular orifices and the theoretical results for two-dimensional flow.

3. The flow coefficient increased linearly with increasing pressure ratio for pressure ratios up to choking. A transition region was obtained at pressure ratios slightly higher than choking and the flow-coefficient variation again became linear with pressure ratio for the high values of pressure ratio.

4. The flow coefficient Increased more rapidly at the low pressure ratios than at high pressure ratios.

5. The flow coefficient decreased with Increasing jet Reynolds number at . constant pressure ratios.

6. For a particular value of pressure ratio and jet.total temperature, the ellipses yielded the highest flow coefficients.

7. No effect of stream velocity on flow coefficient was obtained, provided that the proper outlet static pressure was utilized.

Lewis Flight Propulsion Laboratory, National Advisory Committee for Aeronautics,

Cleveland, Ohio, May 13, 1949.

REFERENCES

1. Callaghan, Edmund E., and .Ruggeri, Robert S.: Investigation of the Penetration of an Air Jet Directed Perpendicularly to an Air Stream. NtCA TN 1615, 1948.

2. de Saint-Venant, Barre", et Wsntzel, Laurent:, Mmoire et exp4ri-ences sur l'couement de l'air. Jour. L'Ecole Roy. Poly., T. XVI, Cahier 27 1 1839, pp. 85-122.

NACA TN 1947 1_i

3. Him: Rflexions eur une critique de M. Hugoniot, parue aux "Coxnptee rendus" du 26 Juin. Comptee Rendus 1 T. 103, Juil.-Dc. 1886, PP. 109-113; Response relative la note de M. Hugoniot: "Sur la pression q .ui existe dane la section contracte'e d'une veine gazeuee", p. 371; Rernarques au sujet des notes de M. Hugonlot, inare'es -aux "Comptes rendue" des 15 et 22 Nov., pp. 1232-1236.

4. Parenty, H.: Sur le debit des gaz parfaits et de la vapeur d'eau sous preselon a travers lea orifices. Ann. Chiiu. et Phys., T. VIII, Sir. VII, 1896, pp. 5-79.

5. Parenty, H.: Sur lee vitesses, lea teznpe'ratures.et lee poide specifiq.ues des gaz parfaits et de la vapeur d'eau s'coulant a travere lee orifices. Ann. Chim. et Phye., T. XII, Ser. 7, 1897, pp. 289-373.

1.6. Chaplygin, S.: Gas Jets. NACA TM 1063, 1944.

12 NACA TN 1947

Figure 1. - Arrangement of orifice In plane parallel to air stream.

1.4 108 202 2.6 310 3.4 Pressure ratio,

4)

41)

0

•1•1

.72

It

0

H.68

. 6 2.O

.84

.80

.76

.64

Orifice Reynolds number diameter, Dj range, Re

(in.)

o 0.375 10.5-41.6x104 o .625 18.1 - 70.0 x

NACA TN 1947

13

Figure 2. - Variation of flow coefficient with pressure ratio for two circular orifices at jet total tenpez'ature of 70 0 F.

1.4 1.8 2.2 2.6 310 Pressure ratio, Pj/Pj

.84

• 8C

o •' lb

.1•1 C)

q-4

Ilk 'I •'

0

I—I I4

•6

.6

.6C. - 1.0

Jet total Reynolds number temperature range, Re

o 400 9.9-39.3x104 o 70 18.1 -70 • 0 x

4

MACA TN 1947

Figure 3. — Variation of flow coefficient with pressure ratio for 0.625-inch circular orifice.

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co

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NACA TN 1947

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16

NACA TN 1947

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43

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0

.72 4) 0 C)

'-4

.6E

.64

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Pressure ratio, Pj/Pj

Figure 5. - Variation of flow coefficient with pressure ratio at constant Reynolds number of 400,000 for circular orifices.

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;uoo;;eoo £Otj

0

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MACA TN 1947

17

NACA TN 1947

/

//

Reynolds number, Re

----6.Ox iø ---35.0x

Theoretical (reference 6)

/

//__________

• _

8

• 74

.72

.70

o

4,

0

U •11

.66

0 0 '-I

.64

.62

.60

.58 1.0 1.2 1.4 11.6 1.8 2.0

Pressure ratio, PJ/Pj

Figure 7. - Comparison of flow-coefficient variation with pressure. ratio for two Reynolds numbers with results of reference 6.

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to

i)

44 IV4

I of

43

0 -4

0

U)

U)

0

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43

o

to 0

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loi

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co

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NACA TN 1947

9

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20 NACA TN 1947

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4.43 to to tD CDttDU) •il .rI • S S S • • S S

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Z,

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W 0 0 0

S S S

H H

x3q.9ure4'ed *o-

o i&Otd 14

NACA TN 1947 21

I VI

0

4)

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VI 0

•1•1

C)

VI VI 4)

S 43 VI

a) (d 0

w q- . 'ri'

0t 00

434) cdvi

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4-i 0

0 q-4 43

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S S S S

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C)l

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22

NACA TN 1947

/0 •

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4)

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00 0 Or-14)' U)U)LOtt) .rlcdO- • CC\ rI.r.r1 SS SS p

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Orifice configuration

Circular Square

---Elliptical; axis ratio, 2:1

----Elliptical; axis ratio, 4:1

- ---

7

Af1Z 4 1 1 11/7

I

I

.88

.84

.8c

0

I.

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NACA TN 1947

23

Figure 12. - Comparison of variation of calculated flow coefficient with pressure ratio for circular, square, and elliptical orifices. Orifice area, 0.307 square inch; jet total temperature, 400 0 F; outlet pressure, 2116 pounds per square foot.

24

NACA TN 1947

Tunnel-air velocity (ft/seo)

o 0 o 160.

275 380

•0

. 8C

, .76

43

U

.72 rj) 0 0

'-4

.68

.64

.60 .1.0 1.4 1.8 2.2 2.6 . 3.0

Pressure ratio,

Figure 13. - Variation of uncorrected flow coefficient with uncorrected pressure ratio for 0.625-inch circular orifice at several tunnel-air velocities. Jet total temperature, 400

0 F.

.84

NACA TN 1947 25.

Tunnel-air velocit (ft/sec

o 0 O 160

275 380

- - Calculated curve

e^O

1.4 1.8 2.2 2.6 3.0 3.4 Pressure ratio, PJ/PJ

Figure 14. - Variation of corrected flow coefficient with corrected pressure ratio for 0.625-inch circular orif ice at several tunnel-air velocities. Jet total temperature, 400 0 F.

.84

.80

.76

4)

C)

4-4 4-i

0 0

I4.68

.64

.60 1.0

NACA-Langley - 9-8-49 - 1100