analytical and experimental studies of asme flow nozzles

R. P. BENEDICT Fellow ASME.

J. S. WYLER Mem. ASME.

Westinghouse Electric Corp., Steam Turbine Division,

Lester, Pa. 19113

Analytical and Experimental Studies of AS1E F l u Nozzles In this paper, we first present a state of the art review of the published work concerning theoretical nozzle discharge coefficients. Then, we develop a new nozzle discharge coefficient based on an axisymmetric boundary layer solution, which in turn is based on a new axisymmetric potential flow solution. These solutions apply to plenum inlet installations which offer major advantages over conventional ASME pipe inlet installations as to losses, and as to predictability of the discharge coefficient at the higher Reynolds numbers encountered in industry. Next, we present new correlations for static pressure tap errors and apply these to the theoretical (zero lap size) discharge coefficients. Finally, we present new experimental data showing how a laminar boundary layer is preserved and tap error is accordingly minimized for the case of a plenum inlet with ASME nozzle contour.

Introduction ASMK elliptical nozzles are often used to meter the flow of

water, steam and air (see Fig. 1). Discharge coefficients, based in part on empirical values, are prescribed by the ASME [1], and are almost always used to determine actual flow rates. However, when the discliarge coefficient must be extrapolated beyond the calibration Reynolds numbers to the higher Reynolds numbers encountered in industry, a more rational basis is required.

Several theoretical studies of the discharge coefficient have been published: Shapiro-Smith [2], Ilivas-Shapiro [3], Hall [4], Cotton-Westcott [5], Leutheuser [6], Soundranayagam [7], Cotton-Corcich-Sehofield [8], and Benedict-Wyler [9], to name a few; but they all show severe limitations in their formulations. For examples: parts of flat plate boundary layer theory have been used by some, although flow through a nozzle is clearly axisymmetric with important pressure gradient effects to be considered; most solutions to date are given in terms of either all laminar or all turbulent boundary layers, although often a combination of both prevails; and all solutions concern flow from inlet plenums, although some of these solutions have been applied to the more conventional pipe installations. The now-famous relations of Hall [4], i.e.,

CDL = 1 - 6.92 RD-o-»

and

CDT — 1 0.184 RD-o-s

(1)

(2)

include all of the above limitations.

This paper begins with the development of a new discharge coefficient based on a theoretical, stepwise, axisymmetric, laminar-turbulent boundary layer solution of Walz [10]. This is based on a new axisymmetric potential flow solution obtained for a plenum inlet installation of the standard ASME nozzle. We further present new correlations for static pressure tap errors and apply those to the theoretical (zero tap size) discharge coefficient to obtain values of Cn one would expect to measure if throat taps were used. Finally, when possible, our analytical results are compared with the analytical and experimental results of other published works as well as with new experimental data that we have recently obtained.

2/3 D

UJJ1 LL IIIIIIIIUIIIIII Ulllll I

Contributed by the Research Committee on Fluid Meters of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS and presented at the Winter Annual Meeting, Atlanta, Ga., November 27-December 2, 1977. Manuscript received at ASME Headquarters April 27, 1977. Paper No. 77-WA/FM-l.

-fc-

Fig. 1 Geometry of ASME low p elliptical nozzle

Journal of Fluids Engineering S E P T E M B E R 1978, Vol. 100 / 265 Copyright © 1978 by ASME

Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 05/07/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Potential Flow Studies Three primary inputs are required for a boundary layer so

lution. These are: the position parameter, S/D; the potential velocity ratio, U/U\; and the geometric parameter, 11/D. These represent respectively: distance S along the nozzle contour in terms of a reference length, taken to be the nozzle throat diameter D; the potential velocity U at the outer edge of the boundary layer in terms of a reference velocity, taken to be the potential velocity U\ at the nozzle throat in the absence of a boundary layer; and the radius R of the nozzle boundary, again with respect to D.

To obtain the initial potential velocity ratio, U/U'i, we at first used a conducting sheet analog [11]. However, this provided only rough estimates of the information needed since only two-dimensional flow could be treated in this manner. Then we considered the use of an electrolytic tank analog [12], since this could handle the axisymmetric aspects of the problem. We rejected this approach also because of the complexity of the tank bottom required. Next, we applied a computerized numerical relaxation method but abandoned this approach because of the long computer times involved to achieve convergence, and because of the difficulty of modeling the elliptical surface of the ASME nozzle. Finally, we turned to a powerful, proprietary, finite element, computer program (WECAN), and found it entirely satisfactory in providing potential solutions to the various ASME nozzle installations of interest.

In these potential flow studies, we made use of an analogy, well known since the times of Fourier, Ohm, and Kirchhoff. Briefly, for constant density fluid flow (which we are considering), for constant resistivity electric flux (which the conducting sheet analog offered), and for constant thermal conductivity heat transfer (which the WECAN approach utilized), the Laplace equation applies and expresses the variation of field potential. In two-dimensional cartesian coordinates, for example, this is simply

*<p f <>v dx1 By1

= 0

In electric fields, the governing equation is:

I =

In fluid fields, the analogous equation is:

1 dV

p dS

~ ~~ p A dS

where the potential gradient here is more familiarly the fluid velocity, i.e., U = dtp/dS.

In heat transfer fields, the analogous equation is:

kA dT

dS

I t is clear from the above three statements that temperature voltage, and velocity potential are analogous, and in particular that the required fluid velocity can be obtained either from the voltage gradient in the electric field, or from the temperature gradient in the heat transfer field.

Some results of these investigations are given in Figs. 2 and 3. In Fig. 2, we compare the two-dimensional results from the conducting sheet along with the axisymmetric results from the finite element program, for a nozzle of /3 = 0.43. From this work it is clear that we cannot use two-dimensional results for axisymmetric flows. In Fig. 3, we present axisymmetrie potential flow solutions from the finite element analysis for ASME low /3 profile nozzles for /3's of 0, 0.43, and 0.6. Also shown in this figure, for comparison purposes, is the Rivas-fihapiro potential flow solution for this same nozzle. The agreement is satisfying.

Boundary Layer Discharge Coefficient The discharge coefficient developed in this section is based

on the Walz [10] stepwise, integral method for approximating incompressible, axisymmetric boundary layers. Briefly, we require statements for momentum balance, energy balance, velocity profile shape factors, various boundary layer thickness parameters, and a boundary layer transition criterion.

Momentum Balance, turn equation:

dd ,_ + 0(2 + Hn) TJ

do U

We begin with the standard momen-

dU/dS

pU* = 0 (3)

where 9 is momentum thickness, Hu is a shape parameter = S*/9, and 5* is displacement thickness. Equation (3) is transformed, by Z = 6 RsN and, with the definition r/pU2

= a/R$N, becomes

d_Z dUJdS

ds + z - u {Fl} - Fl 0 (4)

where the laminar boundary layer is characterized by NL = 1, the turbulent boundary layer by JVT = 0.268, and all other

•Nomenclature-A — area Rd* —

CD = theoretical discharge coefficient RD = (zero t ap size) Re =

COM = measured discharge coefficient Re* = d - pressure tap diameter S =

1) — nozzle throat diameter e = error in static pressure measure- ' T —

ment u = H = shape parameter, &**/d </ =

Hu = shape parameter, 8*/0 I = electric current intensity V = k = thermal conductivity V* = m —~ mass flow rate x, y = N — characteristic exponent 1/ = q = dynamic pressure, rate of heat Z =

transfer (3 = R — radius

tap Reynolds number throat Reynolds number momentum Reynolds number critical value of Ro, defined by (8) distance along nozzle contour,

measured from leading edge temperature velocity in boundary layer potential velocity at outer edge

of boundary layer electric potential friction velocity cartesian coordinates radial distance from wall transformation variable diameter ratio, Dthroat/^pipo (ex

cept in Appendix 1)

5 5*

5** A

e V

p T

<P

= boundary layer thickness = displacement thickness = energy thickness = finite difference = momentum thickness = kinematic viscosity = fluid density, electrical resistivity = wall shear stress = potential

Subscripts

1 2

/, T

= inlet = throat = laminar = turbulent

Superscripts i = ideal or inviscid

266 / Vol. 100, S E P T E M B E R 1978 Transactions of the ASME


\:i-

0.8

0.6-j

0.4

n? -

n

T • 1 1 i

^ 2-D RESULTS

^ ^

~'"H

• i l 1 l

/ /

j L^ AXISYMMETRIC / / RESULTS

/ / 0 = 0.43

-1.6 -1.2 -0.8 - 0 . 4 0 0.4 0.8 1.2 1.6 2.0 S / D

S / D

Fig, 4 ¥arioys velocity distributions, ASME low 0 nozzle

Fig. 2 Potential solutions to low p ASME nozzle

Fig. 3 Axisymmetric potential solutions to low $ ASME nozzle

undefined quantities are given in Appendix 1. In a stepping form, for axisymmetric boundaries, (4) becomes

~zA Hi A, + B.Fi AS

"ft 1 + R7 (5)

Energy Balance. In similar fashion, the standard energy equation is:

(Ui + M dU'dS

( F ) F' - n as + H ~u~ (/a) ~ z - ° (6)

where if is another shape parameter = 8**/d, and 8** is energy thickness. In a stepping form, independent of the axisymmetric condition, (6) becomes

AS — i - = 4 + Bn F- , (7)

Method of Solution. Equations (5) and (7) are solved as a simultaneous system of equations in terms of the thickness parameter Z and the shape factor H, once S/D, U/U\. and R/D are specified for the solid boundaries of interest. A linear change between these specified points is used to give the required precision, and average values of H and Z are used in these small AS intervals.

Usually, one starts with a laminar boundary layer by setting H = 1.572 (the flat plate value), Ar = 1, and Z = 0. One then proceeds with the stepwise solution until either separation or transition of this boundary layer occurs. Separation is recog

nized when r/q = 0. Transition, by the Walz method, is recognized when log R$ exceeds a critical value defined by

(log Re*) = 2.42 + 24.2 (H - 1.572) (8)

In the first element that shows transition, the solution is changed to a turbulent boundary layer by simply redefining Z as ZT = 6L R$LNT. From then on, only turbulent parameters are used.

Iteration. I t is necessary to modify the initial velocities ( [ / ' x /U'i) obtained from potential flow theory, in accordance with the displacement thickness obtained from the boundary layer solution. The velocities are corrected according to the relation

V, W) 1

D%) / \Z>a/_

( t v ) (9)

which is valid in a rigorous sense only in the region where one-dimensional flow prevails. This procedure yields a new potential flow velocity distribution from which is obtained a new boundary layer solution. Such corrections, and such repeated computer runs are only important at the lower throat Reynolds numbers (say below 1Q6), as shown in Fig. 4. Even then, the effect of this iteration is to increase Co by only a slight amount (on the order of 0.1 percent at l i e = 105).

Obtaining the Discharge Coefficient. Solving the boundary layer equations yields: 8*/D, 6/D, 8**/D, R», and r/q (this latter being the dimensionless wall shear stress), at every point of interest along the nozzle contour. There are several formulations available to define the discharge coefficient from this information. Tha t given by Rivas-Shapiro is:

CD 8*

4 -D + \D) J.

y u 8U

(10)

However, most authors (e.g., [4, 6, 7] neglect the bracketed term in (10) as being negligibly small, and settle for

CD = 1 5*

4D (11)

Equations (10) and (11) do not admit that the potential core velocity must change in the presence of a boundary layer. When an energy balance as well as a mass balance is applied across the nozzle, a new formulation for the discharge coefficient results.

A general derivation is given in Appendix 2. Thus, for our plenum inlet boundary layer solution we use

Journal of Fluids Engineering S E P T E M B E R 1978, Vol. 100 / 267


c " - 0 - 4 S ) t l - 46V/ft

" - 4 52*/ft Z 4 82**/A! (12)

11 is clear from (12) that the discharge coefficient will be influenced by the energy thickness (5**) as well as by the displacement thickness (5*).

Boundary Layer Transition. A boundary layer, which is initially laminar for a plenum inlet nozzle installation, may, at some location in the nozzle and at some throat Reynolds number, undergo transition and become turbulent. The boundary layer solution is, of course, strongly dependent on the choice made from among the transition criteria, and these include:

1. No transition. That is, the laminar boundary layer may persist throughout the nozzle at all fiVs. In this regard, Schlich-ting [13] states: In a region of decreasing pressure (accelerating flow) the boundary layer remains, generally speaking, laminar."

3. Transition at the Indifference Point. This earliest of transitions occurs when Re exceeds Rg*, and is identified as the Toll-mien-Schlichting indifference point of (8). To his comment that the boundary layer in accelerating flow is generally laminar, Schlichting adds: "even a small increase in pressure causes almost immediate transition." The basic indifference point is known to move towards higher Ro*'s as the wall roughness and/or free stream turbulence decreases. Walz [10] indicates that Re* increases by as much as 750 at very low turbulence levels. The turbulent boundary layer tha t results from transition at the indifference point will cause a CDT which exceeds CDL for Rj> < 10« because such a boundary layer grows at a smaller rate and is thus thinner than the corresponding laminar layer at the nozzle throat.

3. Transition Because of Laminar Separation. Whenever r/q ~ 0, the laminar boundary layer can no longer prevail.

In its mildest form, the laminar separation will trigger transition to a turbulent boundary layer in much the same manner as indicated for the indifference point transition. However, transition by laminar separation usually takes place later than by indifference. In its most severe form, a separation bubble may grow and cause a thickening of the resulting turbulent boundary layer. This has the immediate effect of decreasing CDT with respect to CDL-

4. Flat Plate Transition. According to flat plate theory, transition occurs when R« = 470. However, this zero pressure gradient value has little to recommend it for the sharp pressure gradient flow found in ASME nozzles.

I t is the existence of these various transition possibilities that introduces such large uncertainties in theoretical discharge coefficients. Fig. 5 describes, by schematic curves, the effect of the various transition criteria on the discharge coefficient. One must rely on experimental data to determine which of the transition criteria applies for a particular nozzle profile.

Boundary Layer Results Figs. 6, 7, 8, 9, and 10 present our boundary layer results in

terms of §*, 6, §**, Hu, and r/q for the ASME nozzle. I n Fig. 6, we compare our results for displacement thickness

with those of Leutheuser [6]. This should be recognized as a generalized plot in the laminar boundary layer region in the sense that S*/D \/RD is independent of the throat Reynolds number. After transition, however, in the turbulent regime, such as often prevails at the throat tap, no such simple correlation holds.

In Fig, 7, we give a similar generalized plot for momentum thickness in terms of d/D VRD, and compare it with the results of Rivas-Shapiro [5].

•-"K IUKBULENT B.L. AFTER ^ ^ SEPARATION BUBBLE

TRANSITION BY SEVERE LAMINAR SEPARATION (BUBBLE FORMED)

lO5 l O 6

THROAT REYNOLDS NO., 10?

Fig. 5 Discharge coefficient of a plenum inlet ASME nozzle as a function of transition criterion

s / o

Fig. 6 ASME nozzle displacement thickness versus contour position

RIVAS-SHAPIRO LAMINAR

S/ D Fig. 7 ASME nozzle momentum thickness versus contour position

268 / Vol. 100, S E P T E M B E R 1978 Transactions of the ASME


S / 0

Fig. 8 ASME nozzle energy thickness versus contour position

^TURBULENT

_FLAT_ PLATE "TURBULENT

S / D

Fig. 9 ASME nozzle shape factor versus contour position

I0 5 I 0 6 I07 I0B

THROAT REYNOLDS NO., R0

Fig. 10 Wal l shear stress to dynamic pressure parameter for A S M E throat tap nozzle

AUTHORS' TURBULENT-2-;^-

A//X / /r^

—-~~~"~\^s^^AUTH0RS' LAMINAR

/"^ ^-^'f^ / RIVAS- ^ ^ ^ ^ " b , , - S H A P I R O ^ = ^ /

^ . - - ^ COTTON-WESTCOTT

-HALL(FLAT PLATE)

/ / ° LEUTHEUSER'S DATA

J /?

I 0 6

THROAT REYNOLDS NO., 107

Fig. 11 parison

Various boundary layer solutions and an exper imental com-for p lenum inlet ASME nozzles

In Fig. 8, the characteristics of energy thickness are given in terms of 8**/D VliD-

Fig. 9 shows the shape factor, Hu, from our results compared with Leutheuser's choice of a constant Hu = 2.554 in the laminar region, and compared with the Oat plate value of Hu = 1.286 in the turbulent region.

In Fig. 10, we present the parameter r/q in terms of the throat Reynolds number. This is a key factor, as will be shown in the next section, in determining static tap corrections. Given in this figure are: the turbulent boundary layer values resulting from the indifference transition; the laminar values resulting from no transition; and the values obtained by imposing the lie = 470 transition criterion.

Our primary results are summarized in Fig. 11, where we present the theoretical discharge coefficient for a plenum inlet ASME nozzle in terms of the throat Reynolds number. Here, we compare our two solutions (one based on the laminar separation criterion, and the other based on the indifference point transition) with the laminar boundary layer solution of Rivas-Shapiro, with the laminar and turbulent flat plate solutions of Hall, and with the turbulent solution of Cotton-Westcott.

For convenience, we have provided empirical fits for our solutions and for the Cotton-Westcott solution. These are:

CDL = L

Authors

5.25 RD~°- ;

j;ood for Reynolds numbers from 105 through 5 X 107,

CDT = 1 ~ 0.130 Ro-o-ass

Authors

(13)

(14)

good for Reynolds numbers from 106 through 5 X 10', and

CDT = 1 - 0.106 RD-°-w (15)

Cotton-Westcott

good for Reynolds numbers from 3 X 10« through 5 X 107. I t is clear from Fig. 11 that significant differences exist be

tween our new boundary layer solutions and previous theoretical solutions. These differences are to be accounted for mainly by two factors: the formulation of Co, and the choise of a transition criterion, both of which we have already discussed.

For the plenum inlet ASM E nozzle, only the data of Leutheuser [6] are available in the literature. This has therefore been added to Fig. 11. Since these data- were obtained by traversing the nozzle exit, without the use of throat taps, it can be compared directly with the theoretical curves. I t strongly suggests that a laminar boundary layer (such as described by our equa-



1.00

Q u 9 9

LLT UJ o o UJ

5.98-X CO

Q

.97-

1 AUTHORS' LAMINAR _____

THEORY —^-___——— ~

AUTHORS' SIMULATED BUBBLE______.

SOUNDRANAYAGAM * • 2" NOZZLE

0 4"NOZZLE

#

I05 10° THROAT REYNOLDS NO., Rp

Fig. 12 Theoretical and experimental comparisons for a plenum inlet ISA nozzle

tion (13) exists in the ASME nozzle at Reynolds numbers below 106.

We have further found, from our boundary layer studies, t h a t the nozzle profile shape does affect CD. This confirms what was found by Soundranayagam [7], and means that we should expect differences between ISA circular arc profiles and ASME elliptical profiles for example. Fig. 12 presents the Soundranayagam data compared with our laminar boundary layer results of (13). Since his data fall far below (13), it strongly suggests the presence of a laminar separation bubble in the ISA nozzle. Indeed, reference [7] indicates that such is the case, Soundranayagam having measured and visually observed a separation bubble in this nozzle.

As a matter of interest, we have simulated the separation bubble effect on CD by imposing on the boundary layer solution the condition that (d/D) at laminar separation is fixed at an arbitrary value larger than that just before separation. This is similar to a suggestion made by Hall [14] in his work on the orifice coefficient, where a separation bubble is always encountered.

We have also found that the length of the plenum inlet face of the nozzle does not affect the discharge coefficient. This confirms what was already found by Leutheuser [6],

As a final observation, we note that the discharge coefficient based on the boundary layer parameters at the throat tap location is essentially the same as that based on the boundary layer at the nozzle exit. This means that we should expect very little difference between the theoretical (zero tap size) discharge coefficients for throat and pipe wall t ap nozzle installations.

Static Pressyre Tap Errors

I t is possible to define a nozzle discharge coefficient in terms of measurements obtained by traversing across a nozzle throat according to (12). For examples of such applications, see [1, 6|, and [7], However, the more usual method is to form an ideal, incompressible flow rate in terms of a measured pressure drop across the nozzle (from inlet to throat, or from inlet to a downstream wall tap) ; to obtain an actual flow rate by a catching-weighing-timing procedure; and to ratio these two to form the discharge coefficient. The latter method invo'.ves the use of static pressure taps.

But, all pressure taps, even if they are constructed perpendicular to the flow direction, are square-edged, are in a region of essentially zero pressure gradient, have the proper length-to-diameter ratio, and do not protrude into the flow region, exhibit an error in static pressure measurement which depends on tap size.

Hence, it is important to consider here the effects of such tap errors on the discharge coefficient, especially so in the high Reynolds number region where CD must be extrapolated beyond calibration data.

Rayle [15], has chosen to express the pressure tap size error (e) in terms of the mean dynamic pressure (q). Others, including: Ray [16], Shaw [17], Jackson [18], Rajaratnam (19), Rainbird [20], Duffy-Norbury [21], Franklin and Wallace [22], and Zogg-Thomann [23], have chosen to express the pressure tap size error in terms of the wall shearing stress (r) .

Tap size error is expressed as a function of r since it is believed to arise because of a local disturbance of the boundary layer. I t is well-documented that near the boundary wall, there is a region where conditions are determined solely by r, p, and v (the lat ter two being the fluid density and kinematic viscosity respectively), and a characteristic length, y. Dimensional analysis leads at once to a velocity distribution in the boundary layer given by

y. = / (A*) (16)

where u is the velocity at any distance y from the wall, and V* is the friction velocity defined by Vr /p , and R„* is the friction Reynolds number defined by V*y/v.

Equation (16) leads to the universal velocity profile, the so-called "law of the wall." I t is universal in the sense that the same distribution will be obtained in a developing turbulent boundary layer as in a fully developed turbulent pipe flow.

I t follows, from what has been said concerning the universal velocity profile, that the static pressure size error is also a function of r, p, c, and a characteristic length only. In tap analysis, the tap diameter (..) is taken as the characteristic length.

Dimensional analysis leads to two acceptable pairs of non-dimensional terms, depending on the grouping of variables chosen. For one grouping, used by Shaw, there results;

- = / ( R _ * ) r

(17)

For another grouping, first proposed by Preston [24], we obtain:

p*> = g (T*) (18)

where

R_* = V*d/v

etf/pv* = ( - jCR,.*)2

and

= T < 2 2 / > 2 = (R<i*)2

and Rayle provided a third correlation:

= h (d) (19)

Results of Static Tap Analysis. All the available experimental data are compared in Figs. 13, 14, 15. Fig. 13 is Rayle's plot based on (19). I t shows the utter failure of e/q to correlate static tap data, even including Rayle's own work. Fig. 14 is Shaw's plot based on (17). I t shows the very reasonable correlation of static tap data in terms of wall shear stress. However, note tha t there is a lack of definition of the faired-in curve at the higher Reynolds numbers, and this is just where our interest lies because of the necessity for extrapolating the nozzle discharge coefficient. Fig. 15 is a modified Preston plot based on (18). There are seen to be two distinct regions of pressure tap performance. One might be considered the laminar region, and the other might be taken as the fully turbulent region.

2 7 0 / Vol. 100, SEPTEMBER 1978 Transactions of the ASME


/ •

& R A Y L E ( I 9 4 9 ) , L / D = 4 . 2 A , " , L / D = 15.6 A " , L / D = 2 7 0 - - RAYLEI I959 ) X SHAW ( I 9 6 0 ) D FRANKLIN a WALLACE -

(1970)

Fig. 13 size

0 .04 .08 .12 .16 .20 .24 .28

HOLE S I Z E , J (INCHES)

Static tap error as a function of dynamic pressure and tap

These log-log plots yield straight line correlations, and in the fully turbulent region the straight line allows extrapolations to be made. Rajaratnam made vise of (18) in his work in connection with static tap pairs, but this was confined to the lower Rd* region only. He gave the expression

T * = 129 ( p * ) 0.555

which we have expressed in the convenient form

- = 0.000157 (Rd*) i-w* r

(20)

to describe static tap error in the range, 0 < Rd* < 385. In the fully turbulent region, we have correlated all the available data by the new expression.

- = 0.269 (Rd*) o-sss T

(21)

to describe static tap error, whether such taps are operating in developing turbulent boundary layers (as found in nozzles) or in fully developed turbulent pipe flow (as in the usual pipe wall taps) at tap Reynolds numbers greater than 385.

Note, there is necessarily a transition region between these two fits where neither is entirely adequate. We suggest that this region extends from 280 < Rd* < 500.

I t is important to note that Rainbird's data, the only da ta to extend into the higher Reynolds number region, does not contradict our straight line correlation of (21). I t is further important to note that Rayle's data correlates far better on our straight line plot Of Fig. 15 than on his own coordinate system of Fig. 13. This indicates that e/q is not the valid correlation parameter.

Effect of Static Tap Error on CD. The authors [25] have derived a relation between a measured discharge coefficient (CW) and a zero tap size theoretical discharge coefficient (CD) which can be given as

CDM — CD + Co3

1 - /3< g ^ 4 ) ' (22)

For most practical cases, et/qi, < <C2/<J2 and can be neglected-For a plenum inlet, (22) reduces to

CDM = CD + CD 3 (23)

4 0 0 8 0 0 1200 600 2000 2400 2800

Fig. 14 Static tap error as a function of wail shear stress and tap Reynolds number

7.0-

6.5-

6.0-

5.5-

5.0-

4 . 5 -

4 . 0 -

3 .5-

RAJARATNAM, (20) ,

1/

i i ' «

AUTHORS ,(21) j g

A

v j *

lc

A RAYLE

X SHAW

O RAINBIRO

D FRANKLIN a WALLACE _

-5.0 5.5

LOG T

Fig. 15 Static tap error as a function of wall shear stress and tap Reynolds number

Here, CD and (r/q\ are specified by the boundary layer analyses, (13), (14), and Fig. 10 respectively, and the quantity (e/rh is defined by the correlations of (20) and (21), depending on the tap Reynolds number,

Rd* = jr/q ( d '°P \ R B (24)

I t is significant to note that the static throat tap corrections which apply to a nozzle operating with a laminar boundary layer are negligibly small since the wall shear stresses are so small. However, there will be large and variable effects on CDM cavised by static throat taps when operating in a turbulent boundary layer where the wall shear stresses are high.

Journal of Fluids Engineering S E P T E M B E R 1978, V o l . 100 / 2 7 1


1/8" INLET TAPS Table 1 Summary of statistical analysis

1/8" THROAT TAPS

A S M E NOZZLE

PLENUM PERFORATED PLATE

INLET « 5 0 % SOLIDITY

I I N C H ^ 2 . 5 4 CM

Fig. IS Schematic of plenum inlet nozzle installation

AUTHORS' DATA

- BOUNDARY LAYER THEORY THEORY ADJUSTED FOR STATIC TAP EFFECTS(d /D=0 .06 )

THROAT REYNOLDS NO. ,

Fig. 17 Theoretical and experimental comparisons for a plenum inlet ASME nozzle

The /3 Effect

Actually, there are two general problems of interest in applying boundary layer solutions to flow nozzles. The one, plenum inlet installations, has been dealt with at some length in this paper. The pipe inlet installation, on the other hand, has been all but ignored in this presentation, even though it is the most conventional of applications. This is because no rigorous boundary layer analysis is available at this time. For one thing, the inlet flow is rotational, and this is counter to the boundary layer concept of an irrotational core flow. For another, and possibly more important reason, there is a flow separation in the vicinity of the corner formed between the inlet pipe and the nozzle face. This separation is to be expected, but it has so far precluded further boundary layer solution. The existence of this separation strongly suggests the use of a cubic or double cubic nozzle in place of the elliptical ASME nozzle, as has often been suggested in the literature, [26, 27, 28, 29].

Several supposed analytical /3 solutions are available in the literature, [8, 30, 31, 32], but these are in direct opposition to each other. Although we believe that our CD equation (12A) is applicable to both pipe and plenum installations, the pertinent values of S* and 8** must be available before theoretical predictions can bo made. As we have seen, boundary layer solutions are available for plenum inlets, but we know of no such solutions for (3 flows.

New Experimental Studies A plenum inlet ASME nozzle installation was designed, built,

and tested to check out some of the ideas presented in the analytical portion of this paper (see Fig. 16). The plenum inlet retards the inlet flow, reduces losses across the straightening plate, presents essentially irrotational flow to the nozzle,

R D

(xio-S)

1.75

3.01

4.6

5.2

5.6

6.6

7.0

7.4

7.9

8.4

8.9

9.6

C D

.9888

.9914

.9920

.9912

.9934

.9955

.9958

.9944

.9946

.9948

.9940

,9950

S

.0007

-

.0018

.0013

.0016

.0008

.0002

.0005

.0012

.0006

.0014

-

fcN-l,p

4.303

-

2.770

4.303

2.447

4.303

4.303

12.706

12.706

12.706

2.571

.

CI (%)

.17

_

.23

.32

.15

.20

.05

.45

1.08

.54

.15

_

and introduces a step at the nozzle inlet edge which initiates a developing laminar boundary layer along the nozzle profile.

The plenum nozzle installation was tested in a continuous flow water loop which included a weigh tank system to provide the actual flow rate. The ideal flow rate was determined via

= A2 2g„ pAp

1 - /3* Vi (25)

where the significant pressure drop across the nozzle was determined from about 25 readings taken during each weighing. A special 120 inch Statham U-tube manometer, with digital readout to 0.001 inch_mercury under water, was used to provide Ap. A typical Ap had a standard deviation of 0.023 and a 2 sigma confidence interval of ±0 .01 inch Hg under H 2 0.

Fig. 17 presents our experimental results in terms of CD, the best value of CD at a given R j . The confidence interval defined by

CI = (26)

(see for example [33], where the probability p is taken as 95 percent to correspond to the two sigma interval, is summarized in Table 1.

The test points in Fig. 17 show that the initial laminar boundary layer neither undergoes transition nor separates, but remains laminar up to R D = 106. These data, obtained with water (along with the air data of Leutheuser [6] presented in Fig. 11) confirm both our new formulation of CD, as defined by (12), and our theoretical laminar boundary layer solution. Finally, these data indicate how small is the static tap effect when operating in a laminar boundary layer.

Summary 1. We have given potential flow velocity distributions for

ASME nozzles for /3's of 0, 0.43, and 0.6, which all show an over-acceleration and then a diffusion near the intersection of the contraction section and the cylindrical throat section of the nozzle.

2. We have obtained and described in some detail new boundary layer solutions to an ASME nozzle with a plenum

2 7 2 / Vol. 100, S E P T E M B E R 1978 Transactions of the ASME


inlet. These show the possibility of wide variations in Ci> depending on whether the boundary layer remains laminar, undergoes an indifference point transition to turbulent, or develops a laminar separation bubble.

3. We have derived a new Co formulation which accounts for energy thickness in the boundary layer as well as the conventional displacement thickness. Significant differences in Co are to lie expected when using the more complete analysis as compared with the conventional.

4. We have obtained new static tap error correlations-for several ranges of tap Reynolds numbers, and used these in conjunction with our boundary layer solutions of r/q to predict these effects on measured discharge coellicients.

5. Our work has been compared favorably with that of others, where possible, concerning: potential flow velocities, boundary layer parameters, and static tap errors.

6. Finally, we have presented results of now experimental studies of a plenum inlet ASMF. nozzle in water, and these confirm our theoretical predictions.

Acknowledgments We would like to express appreciation to our colleagues:

Kddie Phelts, for help in resolving problems encountered in developing the boundary layer computer program; Dr. Steve Bennett, for providing the WECAN potential solutions; and to Norm Deming and Les Southall, for suggesting and supporting this study. Vic Head, Chairman of Fluid Meters Research Committee SCS-WG1, offered many helpful suggestions during the review of this paper.

References 1 Fluid Meters - Their Theory and Application, Report of

the ASME Research Committee on Fluid Meters, o'th edition, 1971, Bean, II. S., ed.

2 Shapiro, A. H., and Smith, R. 1)., "Friction Coellicients in the Length of Smooth, Round Tubes," NACA TN 1785, Nov. 1948

3 Rivas, M. A., and Shapiro, A. II., "On the Theory of Discharge Coellicients for Rounded-Entrance Flowmeters and Venturis," TRANS. ASME, Apr. 1956, p. 489.

4 Hall, G. W., "Application of Boundary Layer Theory to Explain Some Nozzle and Venturi Flow Peculiarities," Proc. Instn. Mech. Engrs., Vol. 173, No. 36, 1959, p. 837.

5 Cotton, K. C., and Westcott, J. C , "Throat Tap Nozzles Used for Accurate Flow Measurements," ASME / . of Engrq. or Power, Oct, 1960, p. 247.

6 Leutheuser, II . J., "Flow Nozzles with Zero Beta Rat io ," ASME J. Basic Enqrg., Sept, 1964, p. 538.

7 Soundranayagam, S., "An Investigation into the Performance of Two ISA Metering Nozzles of Finite and Zero Area Ratio," ASME J. Basic Enqrg., June 1965, p. 525.

8 Cotton, K. C , Carcich, J. A., and Schofield, P., "Experience with Throat-Tap Nozzles for Accurate Flow Measurement," ASME J. Engrg. Power, Apr. 1972, p. 133.

9 Benedict, R. P., and Wyler, J. S., "A Generalized Discharge Coefficient for Differential Pressure Type Fluid Meters," ASME J. Engrg. Power, Oct. 1974, p . 440.

10 Walz, A., Boundary Layers of Flow and Temperature, (translated from the German by H. J. Oser), the M.I.T. Press, 1969.

11 Benedict, R. P., "Analog Simulation," Electro-Technology Science and Engineering Series No. 60, Dec. 1963, p. 73.

12 Benedict, R. P., and Meyer, C. A., "Electrolytic Tank Analog for Studying Fluid Flow Fields Within Turbomachinery," ASME Paper 57-A-120, Dec. 1957.

13 Sehlichting, II., Boundary Layer Theory (English translation by J. Kestin) McGraw-Hill, N. Y., 1955.

14 Hall, G. W., "Analytical Determination of the Discharge Characteristics of Cvlindrical-Tube Orifices," •/. Mech. Engrq. Science, Vol. 5, No. 1, 1963, p. 91.

15 Itayle, It. E., "An Investigation of the Influence of Orifice Geometry on Static Pressure Measurement," MS thesis, M.I.T., 1949. See also, "Influence of Orifice Geometry on Static Pressure Measurement," ASME Paper 59-A-234, Dec! 1959.

16 Ray, A. K., "On the Effect of Orifice Size on Static Pressure Reading at Different Reynolds Numbers," Translation in ARC Rep. T P 498, Nov. 1956. (See also ARC F M 2479).

17 Shaw, R., "The Influence of Hole Dimensions on Static Pressure Measurements," / . Fluid Mech., Vol. 7, Par t 4, Apr. 1960, p. 550.

18 Jackson, J. D., "A Note on the Relationship Between Static Hole Error and Velocity Distribution in the Boundary Layer," App. Sci. Res., Section A, Vol. II , 1962, p. 218.

19 Rajaratnain, N., "A Note on the Static Hole Error Problems," J. 'Roy. Aero. Soc, Vol. 70, Feb. 1960, p. 370.

20 Rainbird, W. J., "Errors in Measurement of Mean Static Pressure of a Moving Fluid Due to Pressure Holes," Quart. Bull., Div. Mech. Engrg., Nat, Aero. Est., NRC Rept. D M E / N A E , 1907 (3).

21 Duffy, J., and Norbury, J. F., "The Measurement of Skin Friction in Pressure Gradients using a Static Hole Pair," Proc. Inst. Mech. Enqrg., Vol. 182, Part 3H, 1967-1968, p. 76.

22 Franklin, It. E., and Wallace, J. M., "Absolute Measurement of Static-Hole Error using Flush Tranducers," / . Fluid Mech., Vol. 42, Part 1, 1970, p. 33.

23 Zogg, H., and Thomann, H., "Errors in Static Pressure Measurements Due to Protruding Pressure Taps," J. Fluid Mech., Vol. 54, Part 3, 1972, p. 489.

24 Preston, J. II., "The Determination of Turbulent Skin Friction by Means of Pitot Tubes," / . Roy. Aero. Soc, Vol. 58, 1954 p. 109.

25 Wyler, J. S., and Benedict, R. P., "Comparisons Between Throat and Pipe Wall Tap Nozzles," ASME ./. Engrg. Power, Oct, 1975, p. 569.

26 Rouse, II., and Hassan, M. M., "Cavitation-Free Inlets and Contractions," Mechanical Engineering, Mar. 1949, p. 213.

27 Redding, T. H., "Flow Characteristics of Metering Nozzles," The Engineer, July 1963, p. 129.

28 -Morel, T., "Comprehensive Design of Axisymmetric Wind Tunnel Contractions," ASME J. FLUIDS ENQRG., June 1975, p . 225.

29 Hussain, A. K. M. F., and Ramjee, V., "Effects of the Axisymmetric Contraction Shape of Incompressible Turbulent Flow," ASME JOURNAL OF FLUIDS ENGINEERING, Mar. 1976, p . 58.

30 Hall, G. W., Discussion of "An Experimental Investigation of the Flow in a Classical Venturimeter," by D. Lindley, Proc. Inst. Mech. Engrs., Vol. 184, Part 1, No. 8, 1969-70, p. 147.

31 Au, S. B., "The Prediction of Axisymmetric Turbulent Boundary Layer in Conical Nozzles," ASME J. App. Mech., Mar. 1974, p. 20.

32 "Steam Turbines," ANSI/ASME PTC 6-1976, An ASME Performance Test Code, 1976, p. 30.

33 Benedict, It, P., Fundamentals of Temperature, Pressure, and Flow Measurements, 2nd Edition, Wiley-Interscience, April 1977, p. 191.

A P P E N D I X 1

Boundary Layer Relations [10]

AZ = (Ui/Ui + iy\

Bz = (1 - Az (Ui/Ui + 1))/(1 + Fi) (1 - Ui/Ui + 1)

An = (Ui/U + i)F3

Bn = (1 - Au (Ui/Ui + 1))/(1 + F3) (1 - Ui/Ui + 1)

Fi = 2 + Ar + (1 + Ar) ff«

F, = (1 + A) a

F3 = 1 - Ha

F< = 2/3 Re*-™ - aH

NNL = 1

NL = 1

NNT = 0.2317ff - 0.2644 - 87000 (2 - ff)*>

NT = 0.268

aL = 1.7261 (H - 1.51.5)0-7158

aT = 0.03894 (H - 1.515)0-7

PL = 0.1564 + 2.1921 (H - 1.515)17



(ST = 0.00481 + 0.0822 (H - 1.5)*81

Hnh = 4.0306 - 4.2845 (II - 1.515)°-«

HuT = 1 + 1-48 (2 - ff) + 104 (2 - fl>7

2 = 0 2?^

"" 1/(1 + A')

i?« =

D D \RD u)

l / ( l + <v)

ir# = PTTRHP 5*

where

.symmetric — /UK) Tidr-

Hence, energy flux, applied between inlet and throat, yields!

Pl ~ P ^ = KE, - KE,

/'Pi - p A = /

\ P Actual \

pTrIWUjs/2g, 1 - 2

52*

A P P E N D I X 2

Discharge Coefficient

ideally

Continuity, applied between inlet and throat, yields

CD = ( 1 - 2 - -

1 - 2 ^ - 2 ^

1 - 2 5 2 * /%

Equating ideal and actual pressure drop heads yields,U%/U%x.

Discharge Coefficient

CD = -. = —

(i - /?*) ( 1 - 2 s.yii,) _

I *« ( i - 2 ^ v (~-~&•^_-i l - 2 S i* /BJ

(12a)

?«.' = p i i [/i1 = p AtUi

where A is geometric area = TTR2,

U' is ideal potential flow velocity,

and

0» = (fl,/«i)» = U\/U\

Energy, under these same conditions, yields

/ P i - p A (EV)2

\ P / idea!

Aetually

2(7= -(1 - 0«)

With the boundaries displaced by 5*, there results for continuity

m = pA1*U1 = pA2*U, = P7ri?22 ( 1 - 2 d,*/B2)U2

where vi* is area within displaced boundaries = ir(R — 5*)2. U is potential velocity in the core

/32 '1 - 2 W '

i - 2 5i*//ei r/i

and

o a: f •J R-S

1 _ ._- j ; . ._ d .

I t is necessary to use energy flux, in the actual case, to allow use of the core velocities, U.

KE =

KE = ( (KE)dm = 27r- f* J A ^9c J 0

/2TrpRm\

(KE)drh = — •A 29'

• R-i

tihdr

\ 2(7c ) f ->+ r [ / / i? :- dr

which, for a plenum inlet, reduces to

CD

Plenum

1 - 2 h* 1 - 2 d2*/Ri

1 - 2 52*/«2 - 2 52**/-R„

D I S C U S S I O N

D, R. Keyset

(12)

The authors are to be congratulated on taking yet another step forward in improving the understanding of the flow coefficients of flow nozzles generally. Especially enlightening is their inclusion and treatment of the effect of the pressure taps on the flow measurement. I t has been a recurring weakness in the field of head class fluid metering to concentrate on the nozzle or orifice itself, and ignore or circumvent the effects of other parts -of the measurement system, such as the pressure taps. In this paper

'Note that the loss term has been neglected in this formulation. If it is included, the resulting discharge coefficient becomes

CD =

' \i-2hvru)

The derivation of this equation is given in an ASME paper to be presented at the 1978 WAM by one of the authors.

3Naval Ship Engineering Center, Philadelphia, Pa. 19112. Mem. ASME.

274 / Vol . 100, S E P T E M B E R 1978 Transactions of the ASME


analytical and experimental studies of asme flow nozzles

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