Super-critical compressible flow through a square-edged orifice

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NAME AND ADDRESS

DATE

NORTHWESTERN UNIVERSITY

SUPER-CRITICAL COMPRESSIBLE FLOW THROUGH A SQUARE-EDGED ORIFICE

A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY

DEPARTMENT OF MECHANICAL ENGINEERING

By Richard Greenlaw Cunningham

Evanston, Illinois April 1950

ProQuest Num ber: 10101309

All rights re se rve d INFORMATION TO ALL USERS The q u a lity o f this re p ro d u c tio n is d e p e n d e n t u p o n th e q u a lity o f th e c o p y s u b m itte d . In th e unlikely e v e n t th a t th e a u th o r d id n o t se nd a c o m p le te m a n u s c rip t a n d th e re a re missing p a g e s , th e s e will b e n o te d . Also, if m a te ria l h a d to b e re m o v e d , a n o te will in d ic a te th e d e le tio n .

uest P ro Q u e st 10101309 P ublished b y P roQ uest LLC (2016). C o p y rig h t o f th e D issertation is h e ld b y th e A uth o r. All rights re served. This w o rk is p ro te c te d a g a in s t u n a u th o riz e d c o p y in g u n d e r Title 17, U n ite d States C o d e M icro fo rm E dition © P roQ uest LLC. P roQ uest LLC. 789 East E isenhow er P arkw ay P.O. Box 1346 A n n A rbor, Ml 48106 - 1346

ACKNOWLEDGMENTS

This work was made possible by a graduate fellowship granted by the Pure Oil Company, and aided by a research grant from the Technological Institute of Northwestern University« The author wishes to thank Professor Edward F* Obert for his valuable help and encouragement during the course of this investigation.

TABLE OF CONTENTS Fage LIST OF.............. T A B L E S ......................

iv

LIST OF........... ILLUSTRATIONS....................

v

NOMENCLATURE

viii

Chapter INTRODUCTION

............................

1

STATEMENT OF THE P R O B L E M .................. I II

III IV

V VI

THE METERING OF COMPRESSIBLE FLUIDS

2

.........

3

A STUDY OF THE ORIFICE JET Survey of the, Literature ......... Experimental Inves11gations.........

19 26

A THEORETICAL SOLUTION FOR SUPER-CRITICAL COMPRESSIBLE FLOW THROUGH AN O R I F I C E.......

H

LABORATORY INVESTIGATION OF ORIFICE FLOW Introduction .................... Extension of Expansion Factors To the Super-Critical Region 1. Air Flow Through an Orifice in a ........ Two Inch Pipe . 2. Steam Flow Through an Orifice in a Three Inch P ipe............ Super-Critical Flow With High Velocity of Approach .....................

63

67 85 95

COMPARISON OF THE EXPERIMENTAL RESULTS WITH THE THEORETICAL SOLUTION..............

122

SUMMARY AND DISCUSSION

..................

125

APPENDIX i

..........................

133

APPENDIX ii

..........................

HI

APPENDIX iii

............

151

BIBLIOGRAPHY

...........................

153

VITA

159

LIST OF TABLES Table

Page

1.

Air Flow Through an Orifice in a Two Inch Pipe . . . 78

2*

Steam Flow Through an Orifice in a Three Inch P i p e ........................................... 89

3.

Dimensions of Orifice Plates for a Meter with High Velocities of Approach......................101

4*

Air Flow Through an Orifice Meter with High Velocities of Approach............

103

5*

Experimental Discharge Coefficients for an Orifice Meter with High Velocities of Approach . . . 113

6*

Comparison of Expansion Factors at a Pressure Ratio of 0.63 .................

•* 117

LIST OF ILLUSTRATIONS Figure 1. 2a.

Page Apparatus Used by Stanton

....................

22

Radial Distributions of Pressures Over the Minimum Jet Section as Determined by Stanton

23

Shape of the Orifice Jet Plotted from the Data of Stanton .......................................

23

Orifice and Probe Arrangement for Free Discharge Into the Atmosphere •

28

4*

Axial Pressures in an Orifice J e t ..............

30

5*

Axial Pressure Traverses with Condensation Shocks . . .

32

6.

Effect of Orifice Pressure Ratio on the Axial Pressures in the Free J e t ......................

2b. 3.

7.

Orifice Discharge Chamber with Probe Guide

8.

Axial Pressure Traverses of Air Jets Discharging from an Orifice with Constant Initial Conditions

9o

•••••• ...

34 36 38

Effect of Pressure Ratio on the Axial Pressures Near the Orifice Plate.......................

39

10.

Orifice and Pressure Probe for Steam Jet Measurements .

4*1

11.

Static Pressure Traverses of Steam Jets Showing Shock Phenomena..............

4*2

Approximate Pressure and Velocity Profiles of the Orifice Jet ..........

4*8

Square-Edged Orifice

4*9

12. 13.

.......................

14a,

Super-Critical Flow Impulse Forces

...............

49

14b.

Constant Density Flow Impulse F o r c e s ..............

49

15a.

Orifice Discharge Coefficients from Bachmann*s Air Tests Compared with the Theoretical Solutions . . . . .

58

Figure 15b,

Page Values of computed from Schiller^ Data for Steam Flow Compared with the Theoretical Solutions • •

16a,

Borda Nozzle in a Plane Wall • • • • • .............

16b,

Super-Critical Jet Issuing from a Borda Nozzle •

17,

58 62

,

Contraction Coefficient for the Borda Nozzle Versus the Pressure Ratio

62 62

Apparatus for Metering Air F l o w ................

19*

Test Orifice Meter in Two Inch Pipe

20.

Experimental Expansion Factors Versus the Orifice Pressure Ratio for Air Flow in a Two Inch Pipe Measured with Corner and Flange Taps ............

81

Experimental Expansion Factors Versus the Orifice Pressure Ratio for Air Flow in a Two Inch Pipe Measured with Pipe Taps ........................

81

Apparatus for Measuring the Flow of Steam Through an Orifice Meter in a Three Inch Pipe ......

85

Expansion Factors Measured for the Flow of Super­ heated Steam Plotted Versus the Pressure Ratio . . . .

92

24-*

Orifice Meter for High Velocity of Approach Tests

. .

97

25.

Temperature Relations for High Velocity Flow Meter . .

99

26.

Orifice Plates Used for High Velocity of Approach ....................... Tests

101

Discharge Coefficients KY Versus the Pressure Ratio Determined from Tests on Seven Orifice Plates in a Smooth Copper Pipe .................

Ill

Incompressible Flow Coefficient K as a Function of the Orifice Diameter Ratio........

114

Expansion Factors Versus Orifice Pressure Ratios for Air Flow at High Approach Velocities . • • • . . . • •

116

Experimentally Determined Expansion Factors for Air Flow in a Two Inch Pipe Meter Compared with the Theoretical Solutions .......................

124

21.

22. 23.

27.

28. 29* 30,

.

68

18.

•

70

Figure 31.

32, 33*

Page Experimentally Determined Expansion Factors for Air Flow in a 0,527 inch Pipe Meter Compared with the Theoretical Solutions .....................

124-

Air and Steam Expansion Factors at a Diameter Ratio of p = 0.15 for Flange and Corner Taps .............

128

Expansion Factors for High Approach Velocities

• . . •

128

NOMENCLATURE Subscripts 1 2 o

position of undisturbed flow upstream from orifice discharge region orifice plate, circular opening Symbols

A abs Ca Ca Cp Cv cfm d dQ e E F ft G gc h ID J K lbf lbm M m mm p psi psia Pic Plw P2c P2w Q

area absolute adiabatic discharge coefficient adiabatic discharge coefficient, based upon total initial pressure and temperature at the orifice specific heat at constant pressure specific heat at constant volume cubic feet per minute diameter orifice diameter, inches internal energy ASME thermal expansion factor for orifice plates fahrenheit degrees feet mass flow rate, pounds per second gravitational constant, lbm ft/lbf sec^ enthalpy per pound of fluid inside diameter mechanical equivalent of heat orifice discharge coefficient for incompressible flow pound force pound mass Mach number orifice meter area ratio A q/A^ millimeters pressure pounds per square inch pounds per square inch absolute orifice initial pressure at the Corner Tap orifice initial pressure at the Wall Tap orifice discharge pressure at the Corner Tap orifice discharge pressure at the Wall Tap heat added

R r rc Re s sec t T u v W w I

gas constant, ft lbf/lbm F abs pressure ratio P2/PI theoretical critical pressure ratio Reynolds number entropy per pound of fluid seconds temperature absolute temperature internal energy per pound of fluid specific volume work velocity orifice meter expansion factor

Greek Letters p

orifice meter diameter ratio d^d^ ratio of specific heats, Cp/Cv

Fffeet of Ori/'‘ice Pressure Tntio on the Axial Pressures in the Free Jet.

0.400

35 2. Probe Studies with Variable Discharge Pressure The apparatus shown in Figure 7 was designed to permit an in­ vestigation of the orifice jet under conditions of controlled discharge pressure.

The device is a discharge chamber containing an axial probe

guide that can be positioned transversely.

The chamber was bolted

across the orifice plate (d0 3 0.4-80 inch) in place of the retaining ring used in Part 1,

Note that, for a distance of about two pipe

diameters downstream, the orifice configuration is identical to that of a standard orifice meter.

The four equally spaced discharge con*

nections were joined and connected to the downstream pressure control valve.

Operation of this valve along with the upstream valve controlled

the initial pressure and the pressure ratio across the orifice.

The

discharge pressure P2 (which was atmospheric pressure for Stanton*s tests) was measured at the Corner Taps. tion will be made clear in Chapter IV,

The reason for this tap loca­ By means of the four adjusting

screws S the probe guide G could be moved in a plane perpendicular to the jet axis.

The centering of the probe tube in the orifice was

accomplished by. adjusting the probe guide position to yield a maximum jet pressure reading.

(The probe tube was similar in dimensions to

that used in Part 1 of this investigation.) Jet pressures with constant initial conditions. - For a constant up­ stream pressure of 82 psia and an initial temperature of about 75F, the pressure drop along the jet axis is shown in Figure 8 for nine orifice pressure ratios.

The critical pressure has the same value for all of

the curves but not the same position, and the shift of pc toward the

y poo

^^o

v/y//////A Y///7777Z7,

'/ / / / / / / / / / A

orifice plate, as the pressure ratio is lowered, is readily apparent. The super-critical curves further reveal that the free .jet develops shock waves for an orifice Measure ratio of 0.49. but not for a ratio of

* X-bys, like the convergent nozzle, shock waves first originate

IS

£o£ & pressure ratio near the critical value. A cross

plot of the data contained in Figure 8 reveals an interesting change in the orifice flow regime in the region of the critical pressure ratio* Figure 9 shows the behavior of the jet pressure at several locations upstream from the critical pressure, plotted versus the orifice pres­ sure ratio.

The pressure at, say, the plane of the orifice, responds

to a decrease in downstream pressure in the linear manner shown by curve AB.

The linearity fails in the super-critical range of pressure

ratios, thereby demonstrating that the flow regime changes near the critical ratio.

Note, however, that the family of curves all depart

from the sub-critical linearity at a pressure ratio that is slightly higher than the critical value (about 0.6). Reference to this fact will be made later. Examination of the chamber wall pressure. - In Figure 7 note the aux­ iliary probe B.

This auxiliary probe was coupled to the axial probe,

and measured the pressure along the chamber wall.

One number 73

(0.024 inch) hole was drilled in the side of the tube facing the center of the chamber*

A mercury differential manometer was connected between

the discharge pressure tap (P2) and this wall probe.

The wall pressure

was investigated throughout the length of the chamber, and exhibited only a slight decrease in value in the direction away from the plate, regardless of the orifice pressure ratio.

For the one diameter ratio

0.90

pnj=64.9---

0.80

0.70

©0.60 .—

^—

FLCW Relativje position ' and size of orifice' plate. | =0.434 0.40

Initial) pressure 8] .4 psia Initial) temperature 75 - 79 \

I

'

Orifice d0- 0.480 inch (See Figure 7 for details) 0.30

0.20

0.10 __ -0.400

-

0.600 0.400 0.200 0 Distance Downstream from Face of Orifice Plate, Inches

0.200

Figure 8.

Axial Pressure Traverses of Air Jets Discharging from an Orifice with Constant Initial Conditions.

0.800

39 1.00 of Static Pressure Tap j Holes Probe Tube Relative bo Ofrifice Plate A. 0.023 inch upstream i P. At the Orifice Fece C* 0.007 inch downstream

0 .9 6

Pressure at Axis of Orifice Initial Pressure

Jet

0 .9 2

0.88

0.84

lO.80

initial Presgure 81.4psis Initial temperature | 7$ - 79F Orifice Diameter 1 di= 0.480 incl ([See Figure 7 for details

0 .7 6

0.68

0.20

Figure 9*

0.40 rc 0.60 0.80 Orifice "Pressure Patio r = P 2/PI

1.00

Effect of Pressure ^ntio on the Axial Pressures TTear the Orifice Plate

0*232 investigated, no indication was found that the shock fronts in the .jet (as evidenced by the axial probe readings) affected the re_gion ad.iacent to the pipe wall. Shock waves in steam jets. - The availability of the steam nozzle de­ vice shown in Figure 10 suggested its use for super-critical orifice work.

The converging-diverging nozzle was replaced with a square-edged

orifice.

The probe control mechanism proved to be unsuitable for very

accurate work, but the outlines of the violent shock waves were easily traced, and it is interesting to examine the progressive change of the shock patterns.

The orifice was operated with saturated steam at

a constant initial pressure of 116 psia for a wide range of pressure ratios.

The axial pressure data have been plotted in Figure 11.

Three

important facts are to be noted: 1. The orifice jet first develops shock discontinuities at, or near, the critical ratio (about 0.54 for steam). 2. The wavelength of the shock discontinuities is a function of the pressure ratio, and is quite regular for any one curve. 3. As the pressure ratio is lowered, the orifice jet successively exhibits three types of shock formations; the curves labeled A, B, C, are representative. The above facts have long been well established for supersonic jets issuing from nozzles, but these curves demonstrate that the orifice _biet develops shock phenomena in a similar manner.

Probe — Traverse Screw

Probe Pressure

r - ° ----- ---- “ -1 -- Guide Bars • TT Probe Tube,0.125 inch Diameter Packing Gland

Thermometer in hell

P>poroach Diameter dU -3.2£incnes

3 inch Flanged Tee (Bolts not shown)

Probe Tube Guide Four holes, 0.016 inch Square-Edged Orifice dQ= 0 .b06 inch

[2 i-V

Steam Inlet

Discharge to Control Valve

Figure 10. Orifice and Pressure Probe for Steam Jet Measurements.

42

c;

-tfi ov

• KJ

UT.

•

"tf4Q CM

«tD H DO •'d4fij

P. 1C

•O 0 )

"d4»•'p U

P a: K

C

ShocF

CO P *4- O 0)

Phenol^on

CM

I— I

-P in E

Fhmv jry

4"r— l Q>*r-

c_: c

i r. ^rnssnre

^rsver^ps

of

ffea'n

Jofs

c c3a

F? rure

11,

CO

io

o B is a ^ 'q .9 ^

c, J ° S f x y q.Bk © a tu ,so aj

43 Summary. - The results of this experimental work may be summarized as follows: 1. Jet pressure measurements have been made on the discharge stream of an orifice plate installed with an unrestricted approach.

Similar results were obtained to those of

Stanton for an orifice with restricted approach. 2. The orifice jet develops shock waves at, or quite near, the critical ratio. 3. Probe investigation failed to detect any evidence of shock discontinuities upstream of the minimum section. 4. The characteristics of the flow, upstream from the jet minimum section, are different for super-critical flow than for sub-critical flow. 5. An orifice pressure ratio of about 0.6 marks the be­ ginning of a transition from the sub-critical into the super-critical type of flow.

CHAPTER III A THEORETICAL SOLUTION FOR SUPER-CRITICAL COMPRESSIBLE FLOW THROUGH AN ORIFICE An equation can be derived that, it will be shown, predicts the contraction behavior of the gas jet.

The contraction coefficient, in

conjunction with certain one-dimensional gas-dynamics relations, renders possible the prediction of the mass flow rate for a square-edged orifice.

This new solution, plus the solution of Buckingham for the

sub-critical zone, enables the gas flow rate of an orifice meter to be predicted quite accurately over the entire range of pressure ratios from unity to zero, A compressible flow contraction coefficient can be established by correcting the incompressible or water coefficient for the effects of compressibility.

Similitude studies

38 51 , of orifice flow have

established that for a given installation the incompressible flow co­ efficient is a function only of the Reynolds number.

Moreover, beyond

a certain high limiting value of Reynolds number, the coefficient at­ tains constancy and becomes independent of fluid viscous forces.

The

discharge coefficient for gas flow near unity pressure ratio approaches the value of the incompressible coefficient.

With the exception of

low flow rates through very small orifices, most meters operate at very high Reynolds numbers - well beyond the limiting value.

Hence

4-5 the value of the gas flow coefficient at unity pressure ratio is simply the limiting value of the incompressible flow coefficient* ■Oi® £?®® jet. - Unlike the nozzle, the contracting gas jet issuing from a square-edged orifice is unconfined and hence free to expand radially as well as longitudinally.

Because of this radial expansion,

the minimum area of the stream is not constant, but increases as the downstream pressure is lowered. By combining the isentropic one-dimensional energy equation, the continuity relation, and the definition of Mach number (M) an equation may be derived which relates the velocity (w) and cross sectional area (A) for the isentropic steady flow of a gas in a stream tube:

w

- 1) - dA A

This relation at once demonstrates that for subsonic flow (M < 1.0) the increase of velocity requires a diminishing area, ie., a convergent shape.

For supersonic velocities (M >■ 1.0), the area must increase

for increasing velocities and hence a divergent shape is necessary. Furthermore, the stream area is a minimum for M = 1.0, and it may be shown that the ratio of pressure at this throat to the initial pressure is the so-called critical and has a value of 0*528 for diatomic gases such as air. Stanton*investigation of the gas jet issuing from a circular orifice, and further work by the author, established that the free jet behaves as a one-dimensional stream tube in two important respects:

4.6 1. The free jet is convergent only for sub-critical flow, that is, flow at pressure ratios greater than the critical (r >. rc). For super-critical flow (r

rQ),

the jet converges to a minimum section and then diverges. Furthermore, the pressure at the minimum section is essentially equal to the critical pressure. 2. Velocities of sonic magnitude, as signified by the occurrence of shock phenomena in the jet downstream from the minimum section, first appear at orifice pressure ratios quite near the critical. In the following development, impulse momentum relations are applied to the super-critical gas jet.

Unlike the Borda nozzle, the

pressure on the upstream face of the orifice plate is not uniform, but diminishes near the edge of the opening.

To overcome this problem,

resort has been made to a relation devised by Buckingham^ in his approximate theoretical solution for the sub-critical case.

(The force

exerted on the face of the plate by a compressible fluid and by an incompressible fluid are related.

The validity of the method is

evidenced by the excellent agreement between experimental [sub-critica]^ data and BuckinghamTs solution.) Super-critical flow hypothesis. - At the critical pressure ratio the average pressure at the minimum section of the jet equals the down­ stream pressure.

Now, as the downstream pressure is lowered, the

pressure and velocity at the throat section remain unaffected, but since a greater differential pressure exists across the jet boundary, t i U

it will stretch in a radial direction and thus increase the effective stream area.

Unlike the nozzle with its fixed dimensions, the lower­

ing of the orifice discharge pressure in the super-critical region en­ larges the throat area, and thus the mass flow rate increases.

Based

on this concept, certain impulse-momentum relations may be employed find a jet contraction coefficient which take cognizance of the phenomenon of limiting throat velocity and pressure. Assumed Conditions s 1. The orifice consists of a round hold in a thin plate which is fastened across the end of a pipe of circular cross-section.

The space into which the jet discharges

is large, so that the pressure surrounding the jet boundary is uniform.

The flow is steady and adiabatic.

2. The boundary of the gas jet is considered to be sharply defined, at least as far as the minimum section.

The

frictional drag and boundary layer effects will be dis­ regarded. 3. The velocity and pressure at the minimum jet area are averaged values of the true distributions (approximated in Figure 12).

48

m lli o m

PRESSURE DISTRIBUTION IN A FREE JET

VELOCITY DISTRIBUTION IN A FREE JET

FIG. ©.APPROXIMATE PRESSURE AND VELOCITY THE ORIFICE JET

PROFILES

4* The fluid is an ideal gas; a condition which is , broached by gases such as air. The above conditions may be summarized by noting that the flow upstream of the orifice plate, and at the minimum section will be treated as one-dimensional, and the expansion as isentropic.

(Obvious­

ly the flow in the plane of the orifice, for example., is not one dimen­ sional*) Nomenclature. - In Figure 13 subscripts 1 and 2 refer respectively to sections upstream of the orifice plate, and at the point of complete ex­ pansion of the free jet.

Subscript 0 refers to the plane of the orifice,

and c refers to the minimum section of the free jet.

The pressure at 2

OF

49

FIG. 13. SQUARE-EDGED ORIFICE

FIG. 14a. SUPER-C R ITIC A L IMPULSE FORCES

FIG. 14b. CONSTANT DENSITY FLOW IMPULSE FORCES

identical for the case of* sub—critical flow only# area absolute pressure pressure ratio p/px critical pressure ratio for isentropic one dimensional flow#

(if approach velocity is zero)

density linear average velocity gas jet contraction coefficient Ac/A0 Area Ratio Aq/A^ = p Mass rate of flow dimensional constant Absolute temperature Fahrenheit Gas constant Ratio of Specific Heats Discharge pressure that would produce the same mass flow from the same initial conditions (Aj Pi/5 i) if the flow occurred at constant density, Incompressible flow pressure ratio

^

4

P^/Pi

Velocity of the incompressible fluid at the minimum section (after reaching py) The minimum area of the incompressible jet A3/A0 The contraction coefficient, constant density expansion.

51 Fo

The force exerted by the fluid on the orifice plate, in the direction of flow*.

The super-critical contraction coefficient, - In Figure H a consider the fluid in the shaded area.

The increase of momentum of the fluid

element is equal to the impulse, CFdt =* d(mw)

or

£F — G A w

Summing the forces acting on the shaded area, SF

» F2 - F2 - F0 =

- p2 (A0 - A0) - P0A 0 - F ^

(1)

By combining the first law, the isentropic relation, and the perfect gas law, the velocity equation is obtained for a one-dimensional gas streams 2 w 2SC

^2

T -1 =

Pi

2gc

(1

-

*

)

(2)

From continuity, a-y» ^

*

1 -/ w A

Hence

w *

2gcy pi V- 1 p ^

tf-1 1 - r

(3 )

l-(i-)2 r2/T A1

*It is assumed (following Buckingham^) that the force F0 is the same for similar initial conditions and mass flow rates, whether the flow is compressible or incompressible. The effect of any local pressure dif­ ferences on F0 arising from differences between liquid and isentropic flow will approach zero for small m values. Furthermore, note that as r approaches unity, any effect of these differences on FQ will be minimized. !

i

52 If the overall pressure ratio is equal to or less than the critical ratio, r

r^, the velocity at the minimum section is the

critical, w = wc. By definition, jxg = Ac/aq, and in equation 3, A/Ai = (— ) • (^) =

Thus in the super-critical case the throat

velocity iss

2g0T

T-l 1 - rc r If

Pi

U)

, 2 2 2/rf T - l y o ! ul - /ig m r0 From continuity}

P c A< - r* W1 * ^ l A / c - c

V-r (5)

*

The increase of momentum from 1 to c is G A w = G(wc - W]_) Upon substituting equation U and equation 5 in the above relation, it follows that

TT .1 ^{1 - rc ^ )I m 2g c T |*c T- 1 1 + jig m r(i/tt

(6)

Equating impulse (equation l) and momentum change (equation 6), 1 - mr - — 7- 3 jig m L P1A 1

I

_ P) +

c_________ ^ ( 1 -rc^__r l "1 1 + /ig m rc ^

Similar steps may be carried out for the case of constant density flow (Figure 14b):

(7)

53 ^ F =» Ft1 -- F2 - F0 = [P1AX - P3A0 - F01

z L - A L » a d . x)

2gc

2gc

(10

(2 ')

f>

From continuity, G =

W1 A1 3 f* w3 a3 “ P w3 yu m Ai

Hence

U') ,3" (For the constant density flow, the minimum area will occur where the internal jet pressure has fallen to equality with the downstream or receiver pressure.) The increase of momentum from 1 to 3 is G A w =* G(w3 - wj) Substituting equation 4* &n

(6c

Setting this relation equal to equation (4.) and solving for

r

- JLt_l

(17)

If the velocity of approach is zero,

r° "

(18)

Since the solution of equation 17 requires the numerical value of jig, (a function of re) an approach by successive approximation is apparently necessary*

However, p.g for r = 0.53 may be found from the sub-critical

equation 13 and employed in equation 17*

Since the value of rc is

comparatively insensitive to changes in jig, one approximation usually suffices. Theoretical-experimental comparisons * - Few data on super-critical flow IS can be found in the literature; the work of Baehmann on air flow, and Schiller^ on superheated steam appear to be the best examples. The sub-critical portion of BaehmannTs data was employed by Dr. Buckingham in his analysis of orifice flow expansion factors for the ASME; he cited the consistent nature of the data as evidencing good ex­ perimental technique.

Baehmann^ coefficients are for free discharge

into the atmosphere, and probably constitute the closest approach to

57 the assumptions made in this development.

The other data are for flow

tests on German standard orifices3**. Figure 15a, b, shows the comparison between ;ag values computed from the experimental data and equations 12 and 13, in the super­ critical and sub-critical zones, respectively. The experimental values for air, Figure 15a, are the individual test results as tabulated by Baehmann*

Equations 12 and 13 evidently

predict the experimental results to within about 1% (with the exception of the slightly greater deviation of the points at the highest and lowest test pressure ratios)• For steam flow (Figure 15b) a similar (l%) correlation exists down to a pressure ratio of about 0.3, the difference increasing to 2.0% at r = 0.16.

Schiller presented his test results as discharge co­

efficients of a form different from jp.g. The points shown in Figure 15b are values computed from a smoothed curve of Schiller*s coefficients. (The sub-critical portion of the data taken by Schiller was published separately by Witte

.)

As might be expected, the actual discharge falls below the theoretical flow at very high jet velocities (low pressure ratios). Extrapolation of these data (as well as several other sources) to zero pressure ratio, that is, flow into a vacuum, indicates the probable actual flows to be about 4% below the predicted values.

In view of

the fact that previous attempts35'39 at predicting orifice behavior have failed completely for pressure ratios below about 0.25, this agreement between theory and actual results is quite good.

Obviously,

a velocity coefficient could now be incorporated, as a simple function

0 .9 0 AIR FLOW

0.80 LlJ

o A. BUCKINGHAM EQUATION ui

B. EQUATION

O

0.70

12

BACHMANN

T E S T RESULTS

APPROACH D IA M ETER , 8 2 . 5 m m .— O RIFICE D IAM ETER, 2 0 .0 3 mm. m * 0 .0 5 9 0 U = 0 .5 9 7

rc = 0 .5 2 8 T = 1 .4 0

UJ

!t

0.60

0.50

1.0

0.8

0.6 ORIFIC E

PRESSURE

0

0.2

0.4 RATIO, r =

FIG.15a. ORIFICE DISCHARGE COEFFICIENTS FROM BACHMANN'S AIR TESTS WITH THE THEORETICAL SOLUTIONS.

COMPARED

o>0 .9 0 STEAM

FLOW

0 .8 0 A. BUCKINGHAM

B. EQUATION o

EQUATION

12

SCHILLER T E S T RESULTS

PIPE D IA M E TE R , 100 mm. O R IF IC E D IA M E T E R , 5 8 m m . m =0336

Hi 0 .7 0

i t « 0 .6 2 8 2

rc= 0 .5 5 4 5 If = 1.3 0

U.

0 .6 0

0.6

0.8 ORIFICE

i___________ I | I

0 .4

PRESSURE RATIO, r =

FIG.15b. VALUES OF jig COMPUTED FROM SCHILLER'S DATA FOR S T EAM FwQW C O M P A N D WITH T H E T H E O R E T IC A L

SOLUTIO NS

of pressure ratio, which would eliminate the difference between ex­ periment and theory evident at very high jet velocities (in Figure 15), However, such empiricism contributes little to the theory and has been omitted, It is interesting to note that low pressure ratios might be expected to cause greater departures from the idealized case so far as pressure distribution in the jet is concerned.

Opposed to this,

however, is the fact that a supersonic free jet, compared with a sub­ sonic jet, is more stable at the boundaries.

This may be demonstrated

theoretically and it has been recently confirmed experimentally by the MCA,

This latter effect may be in part responsible for the

success of both the Buckingham equation near r = rQ, and the new equa­ tion, Extrapolation of sub-critical solutions. - Several

3/5 39 16

attempts

have been made to apply sub-critical gas jet solutions to predict super-critical mass flow rates,

(Such extrapolations cannot, of course,

consider the change of flow regime that occurs at the critical pres3/ sure ratio,) Use of the various sub-critical solutions (Buckingham , Witte2^, Nusselt3^, Chaplygin1^) results in a gas jet contraction co­ efficient of unity at a pressure ratio of about 0.25*

The theoretical

mass flow rate, when based on such a contraction coefficient, declines from a maximum at r

0*25 to zero at r - 0. The (unjustified)

suggestion35 has been made that the behavior of the solution be ignored, and that the flow rate be considered to be constant, from r ^ r * 0o

0,25 to

Experiment has shown, however, that the mass flow rate con­

tinues to Increase in this range of pressure ratios.

t

Compressible flow in a Borda nozzle * - Impulse-momentum relations are conveniently applied to the Borda or re-entrant nozzle (Figure 16a), because the pressure on the face of the tank or container, in which the nozzle is mounted, is uniform (p^)« The incompressible flow coefficient is found to be 0*5 in this manner; and Busemann-^ found the following expression for the (sub-critical) gas jet contraction coefficients nr - i

1 - r ♦Mr -

(19) r ^

rc

The concept of the convergent-divergent jet, developed earlier, may be utilized to permit a solution of the super-critical contraction coefficient.

i®hen the ratio of the discharge pressure to the initial

pressure is less than the critical (r0 * 0.52S for air) the jet pres­ sure and velocity at the minimum jet section are independent of the discharge pressure P2* The jet pressure is equal to the critical, or pc * rc pi, and the velocity is sonic; the critical or sonic velocity is expressed by

(20)

W° = 1

From continuity,

G » Ac f* c wc

and since the initial velocity is zero, the increase of momentum is GA

w = Ac f ^

(21)

61 In Figure 16b the summation of the forces acting on the shaded element is

(22)

- pi Aq - pc Ac - P2^ o “ ^c^

Upon equating impulse (equation 22) and momentum increase (equation 21), the following expression for the super-critical contraction coefficient resuitss 1 - r /*g

(23)

2JSL L T + 1

As was the case with the orifice solution, equation 23 may be solved at r * 0, yielding a value of jig = 0.790 for the sub-critical (Busemann) jig - r ^lg - r

relation have been drawn.

= 1.A0.

In Figure 17

relation and the super-critical Note that these solutions for p.g

are exact in that they do not depend on an assumption concerning the force F0 necessary to solution of the orifice contraction coefficient.

62

A 2 ^2 W 2

Ac

FIG. 16a BORDA NOZZLE IN A PLANE WALL

FIG. 16b. SUPER-CRITICAL JET ISSUING FROM A BORDA NOZZLE

o* 0.8 A. BUSEMANN EQUATION B. EQUATION 23 0 .7 Li.

U.

0.6

0.5

1.0

0.8

0.6

0 .4

PRESSURE RATIO FIG 17

0.2

0

r= *“

CONTRACTION COEFFICIENT FOR THE BORDA NOZZLE VERSUS THE PRESSURE RATIO

CHAPTER IV

LABORATORY INVESTIGATION OF ORIFICE FLOW Introduction Orifice coefficients for incompressible flow must be determined experimentally if the exactness demanded by industry is to be satisfied* Moreover, theoretical analyses of the compressibility effects (Chapter III) are complex even when based upon the simplest boundary condi­ tions*

(Attempting to meet actual boundary conditions, such as the

complex question

of pressure tap location, would render the possi­

bility of a theoretical solution quite remote.)

Consequently, dis­

charge coefficients for compressible flow must be measured under actual flow conditions, if precision results are to be expected in the metering of compressible fluids. The initial and discharge pressures at an orifice meter may be measured in several different manners. taps are in general use: Flange Taps*.

Four locations of pressure

Corner Taps, Throat Taps, Pipe Taps, and

In this country, Flange Taps and Pipe Taps are the most

*The tap locations are as follows: Corner Taps - At the corner formed by the pipe wall and the orifice plate. Throat Taps - One diameter upstream and one—half diameter downstream from the respective faces of the orifice plate. Pipe Taps - Two and one-half diameters upstream and eight diameters downstream from the orifice plate. Flange Taps One inch upstream and one inch downstream from the respective faces of the orifice plate.

widely used, although the International Federation of National Standardizing Associations (ISA) has adopted (1934.) the German Standard Orifice; a meter with Comer Taps. The laboratory investigation of orifice flow was divided into separate phases: A. An extension of the present range (sub-critical) of the ASME orifice discharge coefficients.

At pressure ratios

as low as r - 0.13, coefficients were experimentally determined for the flow of air in a 2 inch pipe meter, and for the flow of superheated steam in a 3 inch pipe meter, the orifice-pipe diameter ratio

(p) being 0.15 in

both cases. B. An investigation of the effects of high velocity of ap­ proach on the super-critical flow characteristics of an orifice meter. Equation for orifice flow. - Appendix i contains a summary of the more important equations and discharge coefficients applicable to orifice meters.

The equation G = KY A0

pgcf>!

A p

hereafter referred to as the A S M equation, was adopted as the basic experimental flow equation.

It differs from the usual incompressible

flow equation only by the expansion factor Y«

The product of the

water coefficient K and the expansion factor Y will be referred to as the (compressible flow) discharge coefficient.

The ASM equation

6$ was selected for three reasons: 1* The complexity of the isentropic flow equation detracts from its usefulness in practice*

2* Two principal authorities on industrial flow metering, the ASME Fluid Meters^ Regeln flow*

(1937) and the German counterpart,

, both utilize this treatment for compressible Hence the equation is well established, and exten­

sion of available data is facilitated* 3* An analysis of Schiller *s^ super-critical data showed that the coefficient defined by the ASME equation could be correlated with the pressure ratio in a linear manner* The discharge coefficient KY consists of the product of the in­ compressible or water coefficient K and the expansion factor Y*

The

coefficient K is correlated in the ASME Tables by means of the Reynolds number, pipe size, diameter ratio and pressure tap location.

The value

of K, for any one installation, is a function only of the Reynolds number*

It decreases with increase in Reynolds numbers up to a cer­

tain high value, beyond which K is independent of the viscous forces represented by the Reynolds criterion*

Except for very small orifices

at low flow rates, the Reynolds number for compressible flow is usually quite large - well beyond the limiting value; as a result, K may be assumed to be a constant. determined by water tests*

(See page 76).

The value of K

may

be

(The values tabulated in the ASME Tables

were largely determined by this method.)

The value of K may also be

established in quite another manner:

At low velocities the compress­

ibility effects approach zero, and therefore the flow of a gas through an orifice approaches that of an incompressible fluid.

Since

the product KY is conveniently linear when plotted versus the orifice pressure ratio, it is readily extrapolated.

The intersection of the

KY curve with the ordinate (at r = 1*0) defines the water coefficient K.

Values determined in this manner agree well with water flow

measurements*

(Bean^ has noted that the effects of compressibility

and viscosity are probably inter-related in a complex fashion*

But

the Reynolds number effect customarily is confined to K and the compressibility effects to Y, and this division offers a practical and successful method of data correlation.)

The experimental deter­

minations of the expansion factors (Y) in this work utilized both methods of finding K for the meter.

In the first part, the meters

were installed in strict conformance to ASME specifications, and K values were therefore available.

The second part concerned flow in

small pipes; the coefficients for the orifice meter were completely unknown*

The K coefficients were found by extrapolating the sub-

critical KY - r curves to unity pressure ratio*

The expansion factor

Y could then be found by dividing the KY values by K*

67

Extension of Expansion Factors to the Super-CriticaL Region 1* Air Flow Through an Orifice in a Two Inch Pipe Apparatus* - Air at pressures up to 100 psi was supplied by a reciprocating-piston compressor (24-0 cfm) equipped with a five pass aftercooler.

The compressor and auxiliaries were located on the lower level

of the laboratory.

The two metering orifices, used to measure the mass

flow rate, and the test orifice were located on the upper level.

The

orifices were separated from the air receiver at the compressor by approximately 60 feet of 3 inch pipe, thus minimizing pulsations. Mass flow. - The mass flow rate for each run was measured by two standard orifice meters placed in series with the experimental orifice (Figure 18).

The upstream meter (A) was made from (new) 2 inch steel pipe and

a pair of commercial orifice flanges (Flange Taps), while the down­ stream meter (C) was similar but with 3 inch pipe.

Such installation

details as the lengths of approach and discharge piping, the location and method of pressure and temperature measurements, and the orifice plate proportions were all determined in accordance with ASME specifi­ cations.

Screwed orifice flanges were installed, and the pipe ends

were faced flush with the flange faces.

To eliminate air leaks, the

Joints between flange and pipe were sweated.

The pipe diameters at

the flanges were measured with telescoping gage and micrometer; each pipe size was established as the average of eight measurements.

The

OC

to

4-

I—I Figure

18,

CO

Apparatus

for

o

■—i Fetering

Air

68

-p

•H

+5

M

orifice pressure taps (Flange) were duplicated to permit two independ­ ent measurements of the orifice pressure differential.

The edges of

the pressure tap holes (1/-4 inch) were slightly rounded to remove burrs The initial orifice pressures were measured with single-leg mercury manometers for pressures up to 30 psij for greater pressures Bourdontube test gages, calibrated daily, were used.

The pressure differen­

tials across the orifice were measured (in duplicate) by cistern-type water manometers.

The value of the pressure differential for each run

was taken as the average of the two readings.

The temperature of the

air-stream at each meter was measured, at a location ten pipe diameters downstream from the orifice, with bare-bulb thermometers held in pack­ ing glands.

The thermometers were precision grade with scale divisions

of 1/2 degree Fahrenheit.

Each meter was equipped with a set of five

graduated orifice plates, covering a diameter ratio to about 0,6.

(p>) range of 0.15

These plates were made from 1/16 inch stainless steel

by the Merriam Instrument Company.

The indicated diameters were re­

checked in the laboratory, using tapered gage blocks and micrometer. The density of the air was corrected for departures from the perfect gas relation, and the effect of humidity was taken into account (as explained in the next section).

The coefficients and expansion factors

for the orifice meters were taken from ASME tables.

The analysis in

Appendix iii shows that the tolerance on the mass flow rate G is of the order of - 0.9$. Test orifice meter. - In Figure 19 are shown the arrangement and detail of the experimental orifice.

The text meter mas made from (new) 2 inch

70

to Differential Manometer 1.

pressure gage -Two inch pipe d^=2.077 inches (thermometer

F-T

C

L inlet

s-22d| s>(

72 inche-ss^dj-

Key to pressure taos: C Corner Taps F Flange Taos P Pioe Taos T Throat Taps Dif ferential iianometer 2. ■Stainless steel orifice plafe, 1/16 inch, do=0.3122 inch,

Note: The thermometer is located 15 pipe diameters upstream from the orifice. Corner Tap holes— .__ drill 1/8 inch v \ \ \

y v

^

\

' v W s w U

n! \ \

\

2.077 inches K < \ ~ \ v v v~x x \ s. "St \

\

\

\ i x vw

V ////// Sweated joints

Diameter Ratio,^=0.l503

Four flange bolts omitted 1/16 inch gasket

Figure 19.

Test Orifice Meter in Two Inch Pioe

71 steel pipe and a pair of commercial orifice flanges.

The pipe ends

were faced flush with the flange faces, and the pipe-flange threaded joints were sweated to ensure air-tightness.

The meter was modified

to include Corner Tapp, Throat Taps, and Pipe Taps, as well Flange Taps. an angle of

& the

Corner Taps were formed by drilling a 1/8 inch hole at

U5 degrees to the orifice plate in the c o m e r formed by

the pipe and orifice plate.

The gasket between the orifice plate and

flange was carefully cut to prevent interference with the pressure tap holes.

The edges of the drilled Flange and Corner Tap holes were

carefully smoothed to remove burrs.

Brass (SAE) tubing fittings were

inserted into the wall to form the pipe wall taps (Pipe Taps and up­ stream Throat Taps)*

The ends of the fittings were filed flush with

the inner walls of the pipe, and burrs were removed from the edges of the holes.

This method resulted in a tap hole havings an inside diameter

of about l/4 inch.

All pressure connections were made with 5/l6 inch

copper tubing and (SAE) flared fittings. (Hoke lines.

High pressure needle valves

l/U inch) were installed in the manometer and pressure gage As indicated in Figure 19 each of the four sets of pressure

taps was duplicated on opposite sides of the pipe. lines were valved into four manifolds.

The pressure tap

Independent pressure measure­

ments were made between each pair of corresponding manifolds.

The

value of pressure or pressure differential for each run was taken as the average of the two readings; the method used to measure the pres­ sure differential ( d p ) for each run depended upon the pressure ratio. For runs at high pressure ratios the differential pressure

A p

was

72 measured with cistern-type mercury manometers.

Low pressure ratios re­

sulted in discharge pressures near atmospheric and p2 was measured directly with mercury manometers, having one leg open to the atmosphere. Intermediate pressure ratios necessitated the use of series manometers. Four 50 inch mercury manometers were connected in series, the spaces above the mercury and the connecting tubing being filled with water. Air bubbles were carefully eliminated and each of the two sets of manometers was checked against a single-leg manometer at 30 psi to en­ sure accuracy.

The differential pressure for each run was calculated

by summing the displacements of each of the four manometers, making proper allowance, of course, for the water legs.

The initial or up­

stream pressure was measured with a Bourdon-tube test gage which was calibrated for every operating period.

The two meters and the test

section were frequently tested for air leakage by painting all piping and pressure tubing joints with soap solution.

The initial temperature

of the air stream was measured with a bare bulb thermometer inserted into the line through a packing gland located 15 pipe diameters up­ stream from the orifice.

The line was not lagged, as the stream

temperature never differed from that of the room by more than a few degrees, thus effectively eliminating any error caused by heat transfer through the pipe wall. made from

The orifice plate used in the test meter was

l/lS inch stainless steel plate.

It was designed to ASME

specifications as regards diameter ratio (^), downstream beveling, etc. The finished diameter d0 was 0.3122 inch, resulting in a diameter ratio £ of 0.1503.

The upstream edge of the orifice plate was formed

73 by making the final boring cut toward the upstream face, and then carefully removing the resultant burr by honing with a fine-grade oil stone held flat against the face of the plate. square edge that did not

The result was a

reflect light*

Tegt procedure. - At the start of each run, the mass flow rate was estimated and the orifice plates for the two meters accordingly selected.

(The plates were changed whenever the pressure differential

became less than 10 inches of water or greater than 50 inches of water.) The compressor was started and caused to run continuously by controlled bleeding of excess air over that flowing through the meter run.

The

inlet and discharge pressures at the test section were adjusted to approximately the desired values and then sufficient time was allowed for constant pressures and temperatures to obtain.

(The most sensitive

indicators of steady flow were the differential water manometers.) operators were necessary to record the test data.

When steady flow

was established, one

of the operators recorded the pressures and

temperatures for the

two meters while the other recorded the test

section data.

Two

If the differential pressures across the standard meters

changed during a run, the data were discarded. The data recorded for the determination of the mass flow rate consisted of the following quantities, (for the upstream meter, and also for the downstream u»ter)# 1. Initial pressure, p^ 2. Air stream temperature, t^ Pressure differentials

A p (two readings)

4* Manometer temperatures 5# Orifice plate size, dQ At the test section the following data were recorded for each run: 1* Initial pressure Pi at each tap location 2. The initial temperature tl 3* The differential pressures across the four pairs of pressure taps: 4.

Flange, Throat, C o m e r and Pipe

Manometer temperatures

The barometer reading and barometer temperature were recorded (for use in calculating absolute pressures) at the start and at the end of each operating period in the laboratory*

It soon became apparent that

little or no difference existed between the pressures at the four up­ stream tap locations (Flange, Throat, Corner and Pipe)*

Since for a

2 inch pipe, the downstream Throat Tap and the Flange Tap coincide, the readings of the Throat Taps were discontinued* Range of variables* - One orifice plate (d0 3 0*3122 inch, £ * 0.1503) was tested with air ( V

* 1*4-0) over a range of pressure ratios from

unity to about 0*13 at nominal upstream pressures of 112, 100, 80 and 60 psia*

A majority of the runs were made with the orifice operating

at super-critical pressure ratios*

Sufficient sub—critical runs were

made to check the experimental coefficients against the ASME values. Air density. - The initial density of the air at the orifice was computed from the relation -

Ply ^ Ti

where y is a correction for departures from the perfect gas law.

The

value of y was obtained from the ASME Fluid Meters chart51, and, in addition, the data of Sage and Lacey5^ were used as a check. sources agreed well in the range encountered in this work.) constant

(Both The gas

was computed from Rjg * 1545.4/M, where M is the molecular

weight of the mixture of air and water vapor.

The air was assumed to

be saturated with water vapor at the pressure and temperature in the air receiver (see Appendix ii).

The temperature of the air at the re­

ceiver was essentially constant at 75F for all of the tests. temperature at the orifices was 10 to 15 degrees higher.)

(The

Since the

air was not throttled for these runs, the humidity calculations were based on 75F and the pressure at the orifice (thus neglecting the slight pressure drop in the piping between the receiver and the orifice). The value of the mixture molecular weight M, and hence Km , was calcu­ lated at each of the four nominal initial pressures of 113, 100, 80 and 60 psia.

(The effect of the water vapor is to increase slightly

the value of the gas constant over its value for dry air:

53.35.)

The same value of Rm was used, of course, for density calculation for the two metering orifices and the test orifice. Mass flow rate (G). - A sample determination of G is included in Appendix ii.

The values of G from each meter were averaged to estab­

lish the mass flow rate for each test.

The agreement between the two

meters was used as a criterion in accepting or rejecting a test:

If

the values differed by more than 1.0 .per cent, that particular test was r e je c t e d *

Reynolds number effect

gn K. - It was stated, in the introduction to

this chapter, that the water coefficient K for the test orifice was a constant and independent of the Reynolds number.

Since the orifice

conforms to ASME specifications, and very nearly to Regeln38 specifica­ tions (for corner taps), these authorities may be consulted for the limiting value of Reynolds number.

The orifice Reynolds number may

be computed from the relation

Re 3 — iffi?— .

Vd0 ^ where

is the viscosity of the fluid.

changes affect

Although the temperature

at high velocities, near unity pressure ratio

may be based on the initial orifice temperature (t^)•

xj

With a value of

at 60F of 12.2 x 10"^ lbj/ft sec and dQ = 0.3122 inch,

Re 3 4 x 106G The mass flow rate G varied from a low of about 0.037 Ibjj/sec to a high of 0.160 for this series of tests.

For the lower value,

Re * 15 x loA

pd Upon consulting the ASME tables

for the value of K for Flange Taps

at p 3 0.15 (2 inch pipe), it is found that the value of K at this Reynolds number is 0.6001. 0.5993.

The tabulated value of K at Re - a ®

is

The difference, due to a change in Reynolds number yalue

from 15 x lcA to infinity, is 0.13 per cent, and this is a maximum

effect for comparison has been made at the lowest flow rate.

Further­

more, the Reynolds number of the lowest flow rate is greater than the

38 SSBSla

limiting value for corner taps.

It is further noted that the

extensive analyses of compressible flow data by Buckingham3^ and Bean28 were similarly founded on the assumption that K was a constant. Experimental fllfichapge coefficients _(KY}. - The values of KY were computed for each test from the relationship KY = --------------- S-----------

0.525 d2 Forty-eight tests on the 0*3122 inch orifice were calculated, and two of these tests were discarded as obviously containing errors (sample calculations are included in Appendix ii)*

Three values of KY repre­

senting C o m e r Taps, Flange and Throat Taps and Pipe Taps were calcu­ lated for each test (tabulated in Table l). for each pair of taps*

The KY-r curve was plotted

The points were concentrated in narrow bands

without exhibiting any differences arising from the initial pressures (112, 110, 80, 60 psia)*

The best curve that could be drawn for the

Corner and Flange Taps consisted of two straight lines, one extending from r * 1*0 to r * 0*63, and a second line of greater slope extending from r ■ 0*63 to r * 0.0*

A similar linear correlation resulted for

Pipe Taps; however; the inflection point occurred not at r = 0*63 but at r * 0.77*.

The unity pressure ratio intercepts of the KY curves

pft agreed well with the proper K values tabulated*

by the ASME for Flange

taps and Pipe Taps, and with the value of "«( 11 given by Regeln^ C o m e r Taps*

for

The limiting value of K for Flange Taps differed from

*This stands in contradiction to Perry's work with Pipe T a p s ^ • How­ ever Perry smoothed his data by plotting certain "flow factors" for whibh he presented equations. Relations for KY as a function of r were based on these (empirical) equations.

78 TABLE 1 FLOW OF AIK THROUGH AN ORIFICE IN A TWO INCH PIPE

dQ = 0.3122; f& - 0*1503 Corner Tape K - 0.5993 tun

Pi psia

45 111.19 44 111.93 55 111*32 43 111.43 54 111.45 29 110o21 52 111.50 38 111o99 53 111.70 39 110.75 40 111.85 41 111.38 42 111.63 20 21 22 23 24 48 11 46 47 19 18 12 13 17

*1 F

93.4 90.6 89.2 92.1 88.0 91.0 89.1 90.9 87.3 93.0 91.4 92.7 91.4

38.5 89.0 90.0 89.2 89*0 90.5 80.5 89*3 88.6 86.5 87.0 98086 77.7 98.46 79.5 99*84 86.7

99.45 99*95 99.75 99.74 99.95 99.65 98.30 99*80 99*65 100.54 100 .14

G

r

KY

Flange Taps K - 0.5993

Pipe Taps

K = 0.6068

Y

r

KY

Y

r

KY

Y

lb/sec

.1596 .1225 .4286 .7152 *1483 .4338 *7149 .1216 .4284 .7148 .1599 ol815 .4407 .7354 .2021 *4463 .7355 .1805 .4404 .7348 .1580 .2363 .4525 .7550 o2523 *4573 .7536 .2342 o451S *7539 01559 2828 .4616 *7702 .2980 .4666 .7690 .2806 .4610 o7692 *1547 o3445 .4772 .7963 .3590 ,4826 .7953 .3411 o4760 *7943 .1506 .3885 .4877 .8138 *4052 .4945 .8149 .3904 .4885 .8151 oL4S9 .4410 .4977 .8305 *4535 *5034 .8296 *4379 .4964 .8283 .1461 .4879 ,5088 o8490 *4996 .5147 .8482 ,4868 .5082 .8480 O1420 .5299 .5159 .8608 .5422 .5227 .8614 .5306 .5162 .8613 .1327 .6039 .5323 .8882 .6124 *5381 08868 .6008 .5302 .8847 .1182 *7162 .5538 o924l .7227 *5602 .9232 .7146 *5522 .9214 .1004 .8043 o5696 .9504 .8089 o5766 *9502 .8034 .5683 .9483 .07008.9107 ,5868 .9791 *9133 .5954 .9812 .9105 .5879 o9810 o

.1428 .1389 .4310 .7192 *1571 .4356 .1428 .1802 .4398 *7338 .1983 *4448 .1415 .2292 *4506 *7519 .2466 .4557 oUOO *2813 .4616 .7702 *2981 .4671 *1385 .3321 .4725 .7884 *3478 *4782 *1350 *3905 .4840 .8076 *4042 .4896 .1320 o4428 .4974 .8300 .4569 .5038 .1293 .4961 *5089 .8492 O5085 .5153 ol252 *5423 .5190 *8660 .5565 .5255 .1223 .5893 .5276 088Q4 .5989 *5338 .1066 .7116 *5513 .9199 *7193 .5588 .09339.7878 .5654 *9434 .7934 .5731 .06769.8944 *5843 .9750 .8970 *5918 o04280.9596 .5931 ,9896 .9605 *6000

79

TABLE 1 —

PI psia

tl

G

Continued

Corner Tape K * 0.5993 r KY Y

Pipe Taps

Flange Taps

K - 0.6068 r KY Y

K » 0.5993 r KY Y

F lb/sec 80.05 88e7 .1148 .1770 .4403 o7347 25 80o40 87.2 o!137 .2315 .4489 o7490 51 26 79.95 88.6 .1124 .2685 o4579 .7641 27 79c82 87o9 .1101 O3320 .4701 .7844 79.92 87.5 .1081 .3985

that the pressure differential, for the intermediate range of orifice pressure ratios, was determined by measuring the discharge pressures directly with a gage* The entry to the test section (at A, Figure 24) consisted of a polished convergent nozzle having a throat diameter equal to that the tubing inside diameter*

The inlet end of the nozzle (2 inches

in diameter) was coupled to the 2 inch discharge pipe from the up­ stream metering orifice*

The approach (A ** B) of the test meter was

long in length to ensure uniform flow at the orifice.

The discharge

length (B - C) terminated in a !■£• inch globe valve which was used to regulate the downstream pressure*

The air was finally discharged to

the atmosphere through the downstream metering orifice.

All piping

joints and pressure tap lines, from the upstream meter inlet to the downstream meter outlet, were frequently checked for leaks (with soap solution)*

The initial pressures at the test orifice, as well as the

discharge pressures at pressure ratios near 0.5> were measured with a Wallace and Tiernan Dial Manometer.

For the range encountered, the

gage could be read to about 1/10 per cent accuracy; it was specially calibrated by the manufacturer and checks in the laboratory indicated that, for the range in which it was used, no corrections were neces­ sary* T e m p e ra tn re measurement * — In addition to the fact that high fluid velocities produce thermometer errors, the tube size (0.527 inch) was such that a thermometer or thermocouple inserted through the tube wall would seriously restrict and disturb the flow.

As an alternative, the

temperature of the air was measured in the 2 inch pipe a few diameters

99 upstream from the convergent entry nozzle (at A in Figure 24).

The

assumption was made that the air upstream from the nozzle possessed zero velocity - a valid assumption, since the maximum velocity in the 2 inch pipe for any of the tests was less than 40 ft/sec.

The flow

from the nozzle entry to the orifice was assumed to be adiabatic be­ cause 1, The stream temperature (ti) in the 2 inch pipe was es­ sentially equal to the temperature of the room* 2* Froessel used a similar nozzle and tube arrangement in his determinations of high velocity friction factors* Froessel found that n.,..as a result of friction, the gas in the neighborhood of the wall acquired approximately the temperature of the outside air so that no heat exchange takes place through the pipe wall"

57

•

By thus assuming the total energy of the flow to be constant, the total temperature at the orifice is equal to the (total) temperature at the tube nozzle entrance and the stream temperature t^ at the orifice may be calculated*

0.527 inch

Figure 25.

100

For the locations illustrated in Figure 25,

2 m-i ^ “ hl * I T* 2g
x ^ p

(25)

shows that

T_ T- i

KY

r2/4r - r^TT*1

(26) (l m^ r^* )(1 - r\ * g

From equation 26, a theoretical orifice expansion factor Y may be calculated.

The values of jig may be found at all pressure ratios by

means of the relations presented in Chapter III. Orifice metering with high approach velocities. - A relation enabling the use of the total initial temperature temperature

in place of the static

may be developed as follows s

From the isentropic relations developed in the first part of this appendix, wi Z i Pl

li = T,

(27) 2gcT

By the continuity relation, the approach velocity in terms of ytig, m,

f)

may be expressed

an(* r:

rZk w! =

fil A1

H

m

(28)

139 Inserting equation 28 into equation 27, it follows that

_ 2 2 V+il 1 -ji* m r r

l± T,

i 2 2 . " ^g m r24-

-z

(29)

a relation that reduces to unity for zero approach velocity. For an ideal gas the mass flow may be expressed as

G =

A(

2goT Pi „ T _ i RTi "

r2/ T - * V

(30)

1 - u2 m2 r2''*' I rg

J

Thus the flow may be found from the initial and discharge static pres­ sures^ and from the initial total temperature T^.

Static pipe pressures

even at high velocities are conveniently measured by flush wall taps; measurement of the total temperature is usually accomplished by means of a thermocouple specially calibrated at high fluid velocities* Another approach to orifice metering with high approach velocities involves the use of the total initial pressure p^ as well as the total temperature T^.

The "total head” equation for an ideal gas is

1

g = c1 a a o II where

= Pa/Pi*

£L (rfr _ r1 r ) i

x RTi

(31)

Care must be taken in comparing coefficients (C^)

based upon equation 29 with the usual form of orifice coefficients* By means of the continuity relation the ’’total head” data may be con­ verted to conventional formst but the relations are rather complex* Prediction of high velocity temperature error* - In practice the ASME equation is most convenient to use.

An expression can be developed

which relates the ratio of the total to the static approach stream temperature (Ti/T1) to r, T

, and the product KI. The mass flow is

given by

G = H Ao | 2g0 / > i ^ p

(25)

By continuity, and the ideal gas relation,

Wi =

= n

02)

m 'f T T 7

Inserting this relation into equation 27 above, it follows that (KY)2 m2(l - r) -

T -1

(Tj/Ti - 1)

By means of equation 33* the magnitude of the difference may be evaluated at any pressure ratio.

(33) - T1)

APPENDIX ii

SAMPLE CALCULATIONS A. Humidity Relations for Chapter III The approximate relative humidity of the air entering the orifice may be calculated as follows* 0 p Ph Ps i% % ^

relative humidity total pressure, water vapor plus air water vapor partial pressure saturated partial pressure mass of water vapor in a unit volume of mixture mass of air in a unit volume of mixture specific humidity

mh “a

Eh ps

and

tu

V,T

0.622 ph

(See reference 61)

(3)

P - Ph Combining equations 1 and 3, (A/ = 0*622 fS ps

U)

p - 0 Ps Atmospheric air at temperature t0 is compressed and cooled to tr

t0 at the receiver tank pressure pp.

Normally this results in

*The subscripts indicate constant volume and constant temperature.

water removal at the water trap following the cooler. The atmospheric humidity necessary to produce saturated air at Pj. may be found from equation 4.

The initial specific humidity U0o equals the

specific humidity of the air in the receiver tanks

Thus

0*622 Po “

0 q

p g

_

0 * 6 2 2 ( l » 0 ) p g

Ps

" (1#°)Ps (5)

For example, if the receiver pressure is 100 psia and the atmospheric pressure is 14-#7 psia,

0O = 14*7 per cent Air was throttled from the receiver pressure pp to the orifice initial pressure Pi*

It was assumed that the cooled air in the re­

ceiver tank was saturated at the temperature tP# Experiment showed that the initial orifice temperature was essentially equal to the reBeivertewperflfftar^ and since no change in the specific humidity occurred, aquation 4 may be used to find the relative humidity at the orifices

0*622(1*0)ps = 0*622-01 Ps Pr - (l.0)ps

Pi - 01 Ps

(6)

H3

B a m f e . - For curve B, Figure 5, the conditions are Pj* ~ 59*0 psia tr = 79*0 F Pi = 4-1*4 psia ti = 80 F and neglecting the small temperature difference,

^1 "* 59#o * or 70*1 per cent relative humidity*

B* Determination of the Mass Flow Hate (G) for Air Tests The following calculations are for run 4*7, Table 1* Upstream orifice meter data, (This standard meter data is not included in Table 1,) Initial pressure Initial temperature

85*35 psi (including gage correction) ti = 88.5 F

Differential pressure manometer 1 manometer 2 Barometer Pipe diameter Orifice diameter Diameter ratio p

^ p = 24*8 inches water at 88 F A p = 24-*8 inches water at 88 F 29*4-65 inches at 84.F (14*40 psia) 2*087 inches 0.770 inch 0.369

Calculations: Pi

= 85.85 + 14.4-0 = 100.25 psia

A P = (24.8 » 24*8) 0.03595 = 0.8916 psia (average) 2 P2 - 99*36

U4

r * 1 -

A P - 0,9911,

and from

P1 reference 51, with T

38 1,40 Y = 0,9975 (expansion factor)

Gas constant for the air-vaoor mixture. - The air is assumed saturated at the air receiver.

Experiment showed that the receiver temperature

was essentially constant at 75 E,

Neglecting the small pressure drop

between the receiver and the upstream orifice meter, the molecular weight of the air is calculated as that of saturated air at 75 E and the initial pressure at the orifice M = I%Xa + MhXh where X is the mole fraction and the subscripts a and h denote air and water vapor*

But _ £a X. "a = p 9

Xv h

a Ph = Ps p p

and pg = 0.4298 psia at 75 F (reference 56) M = 28.967 P ~ P

+ 18.015 2 * ^ P

For a mixture pressure p 31 Pi 53 100.25 psia 28.842 + 0.0774 = 28.919 ft Ibf lbm F abs Air density. - The mixture density was calculated from the ideal gas Relation by including a correction for supercompressibility (y)

1 !

i H5

>o

pi y RT^

px= m

Tl “ ^59*7 + 88*5 = 54-8.2 F abs

2? x I44 x 3,0015 ^ 53.44- x 548.2

.3 lbn/ft

Assuming a Reynolds number Re = 206,000 K = 0.6071

(reference 28)

G = 0.525 KT dj; || ^ A p

- 0.525 x .6071 x .9975 x (0.770)2 fc.4935 x 0.8916 G = 0.12505 lbm/seo Re =

4.8 G 7rd0 \

The viscosity at 88 F is

ij

-6 lb^ft sec » 12.6 x 10“°

4-8 x 0.12505 Re = — ; --— --- — = 200,000 TT x 0.770 x 12.6 x 10"6 The computed Re value is not sufficiently different from the assumed value to affect K, hence G« = 0.12505

lb jj/s e c

The above steps are repeated for the data collected for the downstream meter; it follows that G« * 0.12526

r G* + G» _ Q.12505 + 0.12526 b = 2 2 G * 0.1252 lbju/aec

146 he deviation of the two mass flow determinations is

71

G" - G*

5-----12516 = °*17 *** oent The value G - 0.1252 is tabulated for run 47 and is used in computing the discharge coefficient for the experimental orifice (which is con­ nected in series with the above two orifice meters).

C. Experimental Expansion Factors for the Two Inch Pipe Meter (Air) The following calculations are based upon the data in Table 1 for run 47* orifice diameter pipe diameter diameter ratio initial pressure initial temperature Corner Taps Flange Taps Pipe Taps mass flow rate

PI -

ti AP = AP = Ap = G = (as

0.3122 inch 2.077 inches 0.1503 99*65 psia 88.6 F 45.311 psi 45.314 psi 44.186 psi 0.1252 lbm/sec calculated above)

From the equation G - 0.525 KT d2

O

A ^

- M . P 1 RT-L

0*525 d^

12 px

y(l - r)

For Corner Taps = (x . _AJ2) - 0.5423 Pi

R => 53 LL ^ -fof lbjjj F aba aS ca^cu^a^e(^ above in B, y =» 1,0015 (references 51, 53) KI -

OI*IL__

0.1252 V54.8.3__________

0.525(0.3122)^ 12 x 99>65 jf1>0015(1 . 0.5423) KI « 0.5190 From references 28 and 38 the value of K for dx = 2 inches, p = 0.15, and Re «

oo / is found to be 0,5993,

* =

= °-8660

Similarly for Pipe Taps r « 0,5565 K = 0,6068

y = Qxl252 = 0,8663 0,6068 and for Flange Taps r = 0.5453 K - 0.5993

Y = &5A2Q = 0.8660 0,5993

D. Experimental Expansion Factors for the Three Inch Pipe Meter (Steam) The following calculation is based upon the data in Table 2, run 9.

orifice diameter pipe diameter initial pressure discharge pressure initial temperature injection water weight condensate weight time of run pressure ratio

d0 -0.4,630 inch dJ_ = 3>091 inchea pp = 113.0 psia P2 = 5 5 . 9 7 psia = 530.8 F 250 lbm 643.7 lb 34 minutes, 21 seconds r = 0,4950

•■- r "» ■

The mean temperature for the group of tests at pi = 113.0 psia was ts = 529 F .

To correct G for the departure from this

GS = G

= 0.1912 lbj/sec

From the steam tables-™ at 529 F and at 113.0 psia vs - 5.096 ft3/lbm ( - l//o ) The value of KY is computed from G = 0.525 KEY dj;

where £ is a

i *p

correction factor*^ for the thermal expansion of the

orifice plate; for ti = 530.8 F, E * 1,0060. KY = 0.525 E d^

V"a 7 0.1912 f5.096

0.525 x 1.0060 x (0.4680)2 |fll3.0 KI = 0.4940

i

149 For a three inch pipe, ft = 0.15, and at Re »

oo

, K = 0.5952

(reference 28)* Thus

y = Qt494-0 = 0.8300 0.5952

E. Experimental Expansion Factors for Orifice Meter fith High Approach Velocities The following calculations are based upon the data in Table 4 for run 35. orifice diameter pipe diameter diameter ratio initial pressure discharge pressure pressure ratio initial temperature mass flow rate

d0 &l p Plw P2c r t± G

= = = = = = = =

0.37150 inch 0.527 inch 0.7049 88.24 psia 4-6.13 psia 0.5228 70.4 F 0.1872 lbjj/sec determined as in example B.

The gas constant R = 53.45 ft lbf/lbm F abs is determined by the method illustrated in example B above. From example C. KY =

o T?T

f r

0.525

12 Pi Y y U " r)

hence KY =

V 53.4-5

0.1872

0.525(.37150)2 22 x 88.24. V1-0019^1 ' °-5228^" (KY)j * 0.594.0 based upon the total initial temperature Ti.

150

The value of K was determined as the unity pressure ratio intercept of the KY - r plot*

^ * Static Temperature Determination (Run 35) The static temperature

may be computed from the mass flow

rate and the total temperature Ti from

T1 + f T1 " f Ti = 0 (see derivation in Chapter IV). where f - 2£c Al Cp /Pl^ R2

G

=. 0.20013 (|b2 G

Accordingly, T-l = 523*9 F abs and

T ± - T j. - 6 . 2 F;

•^1.01183

The value of (Kl)i may now be divided by (KY). - 0.594.0/1.00592 - 0.5905

Both values of KY are and fi = 0,8.

tabulated in Table 4 for the runs with p - 0.7

APPENDIX iii ESTIMATES OF EXPERIMENTAL ACCURACY (See Reference 51) A. Mass Flow Rates for Air Tests The estimation of the tolerance on the mass flow rate is based upon the equation G = 0.525 KY d2

where

Quantity

JO

^1

\fpx Zip

«

RTi

Weight

Square

Pi

t 0.10 t 0.20

1/2 1/2

0.0025 0.0100

T1

t 0*20

0.0100

* 0*50

1/2 1/2

0.0625

t 0.10

2

0.04.00

± 0.50 * 0.50

1 1

0.5000 0.2500

Sum

0.8750

*P

Pi do K Y

Assumed Tolerance per cent

G overall tolerance = - Vo.8750 = t 0.94 per cent

152 B* Value of Experimental Expansion Factors for Air

Ap Pi

Weight

± 0.5 * 0.2 * 0.2

*0

K G

0.0625 0.0100 0.0100

± 0.06

2

0.0144

* 0.50 (asme) - 0.9

1 1

0.2500 0.8100

Sum

1.2194

Y overall tolerance ■ - V 1.22

C.

Square

1/2 1/2 1/2 1/2

vn

Ti

Assumed Tolerance per cent

i+ o •

Quantity

- -

i.i

0.0625

per cen-

Value of Experimental Expansion Factors for Steam

Quantity AP pi

Ti

P\ d 0 K G

Assumed Tolerance per cent

Weight

Square

1*0 0.2

1/2 1/2

0.2500 0.0100

0.2

1/2

0.0100

1.0

0.2500

0.06

1/2 2

0.0144

0.5 1.0

1 1

0.2500 1.0000

Sum

1.7844

Y overall tolerance - ■ )|1.7844

s - 1*34 per cent

153

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2*

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11*

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14«

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16.

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17*

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18.

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20.

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23*

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24.

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25.

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26.

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27.

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28.

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159

VITA Names

Richard Greenlaw Cunningham

Borns

September 23, 1921 at Olney, Illinois

Educations

Elementary and High Schools at Flora, Illinois Northwestern University: Bachelor of Science, Mechanical Engineering, 194-3 Master of Science, Mechanical Engineering, 194-7

Services

Engineering Officer (Lieutenant, Junior Grade) aboard a U, S. Navy Destroyer Escort type vessel, World War II.

Positions:

Research Assistant, Mechanical Engineering Department, Technological Institute, Northwestern University, April 194-6 to April 194-8, and September 194-8 to April 1950. Research Engineer, The Pure Oil Company, April 194-8 to September 194-8#

Societies:

Tau Beta Pi, Pi Tau Sigma, Sigma Xi, Society of Automotive Engineers.