Comparison techniques for small sonic nozzles

Flow Meas. lnstrum., Vol. 7, No. 2, pp. 109-114, 1996 © 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0955-5986/96 $15.00 + 0.00

ELSEVIER

PII: S0955-5986(96)00008-8

Comparison techniques for small sonic nozzles N. Bignell CSIRO Division of Applied Physics, Lindfield, Australia 2070

Received 1 August 1995; in revised form 21 March 1996 Two techniques are described that allow small sonic nozzles to be compared. The first allows one nozzle to be compared easily and accurately with another at the same inlet conditions. The second allows a nozzle with a particular inlet pressure to be compared with the same nozzle at a different inlet pressure. A description of the calibration of two sets of sonic nozzles is given in some detail. The first is a set of nominally equal nozzles and the second is a set of nozzles of varying sizes used to produce a range of closely spaced flows for the calibration of flow meters. The redundant measurement scheme with its design and variance--covariance matrices is described, and some results of actual measurements given. Results of the measurement of the change of discharge coefficients with Reynolds number for nozzles of throat diameters 0.7, 1.0 and 1.4 mm are given and compared with previous work. © 1997 Elsevier Science Ltd.

Keywords: sonic nozzle; discharge coefficient; calibration; comparison; Reynolds number dependence

1. Introduction

Qm = N(p) p

A sonic nozzle, sometimes known as a critical flow venturi, is so-called because under the conditions of operation the gas flow through the narrowest part, the throat, reaches sonic velocity. When this happens information about the downstream conditions cannot propagate upstream to affect the flow rate, which is thus independent of downstream conditions. This makes the sonic nozzle useful in flow metering. The mass flow rate Qm of a perfect gas through a sonic nozzle with upstream pressure p can be written simply as

where N(p) will be referred to as the nozzle coefficient. An international standard for sonic nozzles ~, ISO 9300, enables the coefficient of a sonic nozzle to be calculated with an uncertainty of 0.5% provided that the nozzle has the required standard form and that its dimensions have been measured to sufficient accuracy. These requirements can be met without too much difficulty when the nozzles are larger than ca 5 mm throat diameter, but meeting them becomes increasingly difficult as the diameter becomes smaller. For nozzles of ca 1 mm diameter the dimensional metroIogy task is very difficult and so is the manufacturing task. Even smaller nozzles are sometimes required to generate flows for the calibration of smaller flow meters and to obtain accurately-known small steps in larger flows 2. Even if the measurement and manufacturing problem could be overcome, the standard, ISO 9300, does not claim to give acceptable results for low Reynolds numbers such as are likely to apply to small diameter nozzles. Recent work 3, using nozzles operating at the low end of the range of Reynolds numbers allowed for in the standard, has shown a greater rate of change of nozzle coefficient with Reynolds number than that given in the standard. This paper gives some techniques that allow sonic nozzles to be compared. They allow the coefficient of one nozzle to be compared with that of another nozzle at the same inlet conditions. A variation of this technique allows the coefficient of a particular nozzle at a particular Reynolds number to be compared with the coefficient of the same nozzle at another Reynolds

Qm = pc c(t~, Pc) A = Np (1) where pc is the density of the gas at the throat conditions of temperature t~ and pressure Pc, c is the velocity of sound, A is the area of the throat and N is called the nozzle coefficient. For real gases this is usually written as~ Qm = CD Cr A f(p,T)

(3)

(2)

CD is called the discharge coefficient and depends on the Reynolds number of the gas in the throat. Cr is the coefficient that takes account of the fact that the gas is not perfect and is therefore called the real gas critical flow coefficient. The value of both of these will be of the order of unity. The actual geometrical area of the throat is A and f(p,T) which incorporates the density and velocity of sound in equation (1) is, for a particular gas, a function of the absolute pressure p and temperature, T. For a perfect gas it will have the form p/qT. In this work the temperature and nature of the inlet gas do not change so that equation (2) may be written as

109

110

N. Bignell

number. Thus the pressure dependence, or the Reynolds number dependence, of the nozzle coefficient can be determined without measuring the coefficient itself. These techniques have been applied to small nozzles but they should be applicable to nozzles of any size. The nozzles used in this work have throat diameters less than 2 mm. Some of them have been made from glass and some from brass. The brass nozzles were made in groups of three with closely matched specifications. An attempt was made to construct them with the toroidal form as specified by ISO 9300 but for nozzles of this size it is very difficult to quantify the degree to which this has been achieved.

2. Comparison of the coefficients of two nozzles The basis for this intercomparison is the connection of two nozzles, with coefficients N~ and N~, in series with a vacuum pump as shown in Figure 1A. Air is drawn from the laboratory for this measurement. For both nozzles to be operating in the critical region it is obviously necessary for the second nozzle to be larger than the first. Indeed usually the area of the throat of the second will be twice that of the other nozzle. Clearly the mass flow rate is the same through both nozzles in Figure 1A, so that from equation (3) we have Qm =

NI(Po)

Po = NB(pl) Pl

(4)

Similarly, from Figure 1B, N2(po) Po = NB(p2) P2

(5)

Dividing equation (4) by equation (5) gives

Po

Po

N1 Pl

lll

P2

NB

) C

C)

A

B

Figure 1 The two configurations of nozzles used for the comparison of nozzles N1 and N2 using reference nozzle NB which is connected directly to a vacuum pump. The pressure Po is normally laboratory air

Nl(po) N2(po)

P~ NB(PT) Pl P2 NB(p2) P~ -

-

(6)

o

if we assume that NB(pl) = NB(p2)

(7)

Hence in order to compare N~(po) and N~(po) it is not necessary to know the value of the nozzle NB. This nozzle may, for convenience, be composed of two nozzles in parallel, each the size of the nozzles being compared, or one nozzle of approximately twice the throat area. For two nozzles in parallel the values of nozzle coefficient add, since the mass flow rates add. This is similar to the rule governing the addition of electrical conductances, but there is no rule for nozzles in series that corresponds to the addition of electrical resistors in series. For nozzles in series operating in the critical region, the effective nozzle coefficient of the combination is just the nozzle coefficient of the smaller nozzle. The assumption made in equation (7) is accurate if p~ is close to P2 which is obviously true if the nozzles being compared are closely matched. The degree of matching needed is quantitatively examined in the discussion. The transducer used in the measurements consists of a bellows acting on a quartz force transducer. Its output goes to a frequency counter used in period mode. A cubic equation was used to relate the period to the pressure. The measurement of the ratio of approximately equal pressures using the transducer is much more accurate than the measurement of a single pressure. For nozzles of nominally equal size the pressures p~ and P2 are nearly equal. Let the true pressures be pT and p~ so that pl_ P2

p[ + a p l pT + ap2

~

p~_ 1 + p~ \ pT

p~ ]

(8)

where 8p~ and ~P2 are the errors introduced by the calibration of the transducer. Now for p~ approximately equal to Pz, 8P~ and 8P2 will be very nearly equal and hence the error terms in brackets in equation (8) will largely cancel. For independent errors we could not assume that the signs of the terms were opposite; furthermore their magnitudes could have any value. But with a smooth function relating the pressure to the period measured, as here, the pressure errors are strongly correlated and we can be sure not only that they are of opposite sign but also that their magnitudes are not very different as long as pl and P2 are themselves not too different. This is true even if we do not know, very well, the smooth functional relationship between the period and pressure. Drift can be almost eliminated by the order and timing of measurements in the usual way, leaving only random uncertainties that can then be reduced by averaging repeated measurements. For the present work the uncertainty in the pressure ratio is estimated to be better than 0.01%.

3. Ratios of the coefficients of a set of nozzles The method just described can be applied to find the ratio of just two nozzles but, as for much of metrology,

Comparison techniques for small sonic nozzles it is better to perform a series of redundant measurements and to calculate the best values by a least squares technique. This is done here by taking a group of nozzles and measuring the many possible ratios between them, not all of which will be independent. The best set of values for the independent ratios can then be found by a calculation using a minimization of the sum of the squares of the residuals (least squares) technique 4. In this calculation the variance-covariance matrix can also be calculated to give a measure of the uncertainties of the calculated ratios as a result of the random uncertainties in the measurements. Systematic uncertainties show up as an increase in the variances. As an example, consider the simplest possible set, just three nozzles denoted NI, N2 and N3. If we measure all the possible ratios N,/N2, N2/N3 and N1/N3 and they are, for some reason, systematically high, then this will show in the comparison of the product of (NI/N2) and (N2/N3) with the third ratio N1/N3. This is in effect what is done in the least squares solution. For this particularly simple system there are only two independent ratios, which can be chosen to be N2/N3 and N~/N~. If we denote the measurements of these three ratios by r~2, r23 and r,3, then the three equations resulting from the measurements can be written in matrix form as

[i !21IR131 [°1

.ol 71s 1 .01568 .99903 .99778 .98796 .97764 0 0 0 0

D=

-I

0

0

0

0

0

0

I

0

0

0

0

io

o

1

0

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

-1 .O4O72

0

0

0

0

0

-1.02967

0

0

0

-1.01972

0

0

0

-1.01818

0

0

0

-1.00173

0

0

0

0

1

0

0

0

-1.03907

1

0

0

-1.02807

0

1

0

-1.01776

0

0

0

0

0 - I .02237

1

-1.0171

0

I

0

0

0

I

0

-I .01152

0

0

I

-1.00152

9

0

0

0

I

0

- I .02097

0

0

1

-I .01005

0

0

0

0

I

-I ,01063

(1 O)

0 0 0 0 0 0 0 0 0 0

j

The solution matrix b was then found (from b = (AT A) -I A T D) to be 73

1.0157

LR2~J

A=

o

1.01

(9) r13 Lr23J where RI~ and R23 are the values, to be found by the least squares solution of the three equations, of the ratios chosen to be independent. The first matrix in equation (9) is called the design matrix and is denoted by A, the second is the solution matrix, called b, and the third is the observation matrix, D. As an example of an actual measurement take a set of seven nozzles of ca 0.7 mm diameter with nozzle coefficients denoted by oq, o~2 ... ozT. There are six independent ratios that were taken to be al/OtT, cz2/cz7 ... cZ~aT. The design matrix and observation matrix were:

111

0.99911 b =

(11)

0.99779 0.98789

0.97745 The vanance-covariance matrix is a measure of the random errors present in the original data. The variances (the squares of the standard deviations) are the best estimate of the magnitudes of these random errors carried through into the solutions. They are not generally independent. An error in a particular original measurement effects a number of solutions to varying degrees. The degree of this dependency is expressed by the covariances. The variance--covariance matrix is, in this case, a 6 x 6 matrix with the diagonal elements (the variances) about 1.05 x 10-8 and the offdiagonal elements (the covariances) about 5.4 x 10-9. Thus the standard deviation of the values of the ratios is ca 0.0001. Nozzles of more than one size may be included in the scheme, e.g. in the calibration of nozzles for use in a flow generator 2. This is a device that consists of an array of nozzles of varying sizes so that a wide range of flows may be obtained by using combinations of nozzles in parallel to generate the flow. The nozzles are automatically selected according to the flow required, the prevailing pressure and temperature, and a knowledge of the nozzle coefficients. In one such calibration, a set of ten nozzles was calibrated ranging in diameter from 0.18 to 2 mm. The calibration was broken into two parts, those nozzles smaller than the reference nozzle and those nozzles larger than the reference. Of course the reference nozzle appeared in both calibrations. The design matrix for the smaller nozzles was an 11 x 24 matrix and that for the larger nozzles was 6 x 13. These could be combined to form one large matrix of 1 7 x 3 7 . There is little point in

112

N. Bignell

doing that in this case as the interaction between the two parts is only through the reference nozzle. With a different set of measurements this need not be so. The design of the calibration should, in fact, try to compare as many nozzles of different sizes as possible. This was done by treating up to four nozzles in parallel as a single nozzle in the measurements. The entire design matrix of the smaller nozzles is too large to be reproduced here but several rows are shown below: A =

I

(}

0

l

0

I

(I 096054 --U.c}(~71 ~

I!

I!

II

(/

i!

il

U

U1

1

I).9bl)54

I

0.~lh()S4

i

0 (t(ff)r~4

I)

()

I

I).9671

I

I

II t ) h T l

II

()

~

09671

/

/

Pl

Pi

N2)

N,) Pal

N.)

N'/

(I

(12)

()

Each row in one of these matrices stems from a particular measurement of a ratio of nozzles or nozzle combinations. A zero in a row means that the nozzle to which that column refers has not been used in that particular measurement. A negative number as opposed to a positive number means that the nozzle was used first or second in the measurement, respectively. If the nozzles associated with the columns are simply called A, B, C and so on the first row means that only nozzle A was used in a comparison with the standard nozzle and the result was that nozzle A equals 0.98846 times the standard. The second row means that A, J and K were not used, but B, D, F and H were used together in parallel and compared with C, E, G, and I also in parallel. It means that the combined sizes of B, D, F and H were measured to be equal to 0.96054 times the combined sizes of C, E, G and I. The zero for this row in the matrix D means that the reference nozzle was not used in this comparison.

I

N3)/ P~2

N')/

()

/ A

D=

Pl

P~

()

B

C

Figure 2 The first set of configurations of nozzles N t, N2 and N ~, used in the measurement of the pressure dependence of the nozzle coefficients

P0

N1)i/ Pl N3

I

4. Measurement of the pressure dependence of nozzle coefficients

A development of the technique described above enables the change in nozzle coefficient with pressure to be measured. In this technique the air is not drawn from the laboratory but from a supply whose pressure can be varied. Three closely matched nozzles N~, N2 and N~ are placed between a variable pressure supply p~ and a third nozzle as shown in Figure 2. Then by individually measuring p,,~, p,,2 and p,,~ we (an calculate, from equation (6), P,2 __ N2(Pl) . P,~ __ N~(pl) P.~I NI(pl) ' P,1 NI(pl)

(131

The assumption of equation (7) is valid because the nozzles have been closely matched to make p,~, p,,, and p.~ as equal as possible. The nozzles are then connected as shown in Figure 3 and the supply pressure Po adjusted to make the pressure at the junction of the nozzles equal to p~, the same as it was in Figure 2. Then from equation (4) poN1 (Po) = Pl [N~(pl) + N~(pl)]

and hence dividing by poN~(p~)

(14)

Hgure ~ The second configuration using nozzles N,, N, and N~ tor the measurement of the pressure dependence of N

N, tp,i p, tN,-,,i * N'I 'It P~e [P.,1

P.,lJ

I he quantities in the curly brackets are the pressure ratios in equation (1 3). They are approximately unity and will be able to be quite accurately measured because of the error cancellation discussed above. The pressure ratio Pl/P~, is, from equation (4), equal to NIl(N2 + N~) and thus has a value of ca 0.5 and so the error cancellation will not be nearly so effective. there m a y still be s o m e cancellation but it is not easy to estimate the extent of it. An uncertainty estimate based on the assumption of independence leads to an uncertainty in Pl/P~) of 0.035%, whereas if they were

Comparison techniques for small sonic nozzles

correlated, as are the ratios in would be ca 0.01%. Taking including the uncertainties in brackets gives 0.033% as the N1 (po)/ N1 (PT).

equation (13), the value a value of 0.03% and the ratios in the curly uncertainty in the ratio

5. Pressure-dependence results By absolute flow calibration methods nozzle coefficients have been measured at various values of the inlet pressure and it has been found from both these experiments and theory that there is a variation that has the form 1 N(p) = N(po) [a0 - al Re-"]

(16)

where Re is the Reynolds number at the throat, N(po) is the nozzle coefficient at the pressure Po and n, ao and a~ are constants. For toroidal throat nozzles the value of n is 0.5. Figure 4 shows the value of N(p)/N (100 kPa) for three sizes of nozzles, 0.7 mm, 1.0 mm and 1.4 mm throat diameter. The ratio is plotted against throat Reynolds number and as expected the smaller nozzles change more than the larger ones. When the type of gas and the temperature are constant the ratio of the nozzle coefficients is equal to the ratio of the discharge coefficients, or, from equation (16), N(p)

1-(al]Re-°5 \ao/ 1 [a~°s

N(100 kPa)

- \aoY'

(17)

10o

where Re~oo is the throat Reynolds number at 100 kPa inlet pressure. Equation (17) has only one adjustable parameter, al/ao, so it is easy to fit the results of Figure 4 to this equation by varying the parameter to minimise the sum of the squares of the residuals. The curves obtained from this fit are shown as the solid lines in Figure4. The quality of the fit is good especially remembering that there is only one adjust-

able parameter. For nozzles of 0.7, 1.0 and 1.4 mm diameter the values for a~/ao are 3.73, 3.46 and 3.67, respectively, with an estimated standard deviation of 0.16. If these are taken as estimates of the same number, i.e. the nozzles are all assumed to fit the same law with the same parameter, then the mean value may be taken to give 3.62 with a standard deviation of 0.12. The value obtained by Watanabe and Komiya s is 3.53 for a nozzle of 0.821 mm throat diameter, which is in good agreement. Stratford 6 calculated the value 3.04. For a 1.26mm nozzle, at a higher Reynolds number, Brain and Macdonald 7 obtained a set of discharge coefficients that is best fitted to C = 0.9982 - 4.097Re -°-s

6. Discussion We can use the result just found for the pressure dependence of the nozzle coefficients to investigate the assumption made in equation (7). For a given nozzle throat diameter with a particular gas at a particular temperature the Reynolds number is proportional to the pressure so we can write (19)

Re = Fp

where F is a constant, which for air and a 1 mm diameter nozzle would be ca 0.126. Thus equation (17) becomes N(p)

1 - a! F-o'Sp-°5 ao

N(100 kPa)

1 - a! F-°s(100 000) -o.5 a0

1 - Dp -°5

~

1.02

~

1.01

(2O)

B

which defines D and equation (4) we have

1.03

B. Using equation(20)

N1 (Po) Po = NB(100 kPa)

in

(1 - DpT°s) B - - Pl

1.00

(21)

o 0.99

Using equation (20) in equation (5) similarly gives (1 - Dp~°'5) Nz(po) P0 = NB(100 kPa) ~ P2

0.98

.~ 0.97

(22)

0.96 0.95

(18)

giving a value for al/ao of 4.1. For an uncertainty in discharge coefficient of 0.2%, a~/ao can vary up to ca 0.34 so that all these experimental results could be said to be in agreement.

-

~ --

113

The division of equation (21) by equation (22) leads to

0

10

20

30

40

Reynolds

50

60 70 Number (/103)

80

90

100

Figure 4 The curves show the ratio of the nozzle coefficient at an inlet pressure P to that for the same nozzle with an inlet pressure of 100 kPa (all pressures are absolute) as a function of the Reynolds number for three nozzles. The diameters of the nozzles are 0.7ram (top curve), 1.0 mm (middle curve) and 1.4 mm (bottom curve)

Nl(Po)

[ I - DpT°sl Pl

(23)

If we write P2 = Pl + 8p and since D p-oS is small, this can be written as N~(po)

F _ D 8pl p~

cl 2p .Sj

(24)

114

N. Bignell

Thus, the expression in square brackets in equation (23) or (24) is a correction to the expression found previously in equation (6). For 8p equal to 1 kPa and for Pl equal to 50 kPa the correction term for a 1 mm diameter nozzle used as reference is 0.99953. For these numbers the measured ratio before correction would be 0.98 and after correction 0.9795. If the correction is not applied it becomes a systematic uncertainty. To verify this correction, two composite nozzles that generate a large difference between p~ and P2, when used in the manner of Section 3, were used. They were made from members of two pairs of nozzles. The first pair, with coefficients al and a2, were half the size (area) of the second, with coefficients b~ and b2. The members of each pair were closely matched and the ratios of their coefficients, R2 and R3, both close to unity, were measured by the pressure ratio method of Section 3. Similarly, the ratio of the combined coefficient of al plus bl to that of b2 alone, i.e. one small plus one large nozzle to one large nozzle, was measured. This, of course, has the value of ca 1.5. The ratio of the coefficients of the two small nozzles used together (i.e. in parallel) to b2 was also measured with the result R1, also close to unity. Thus, the measurements of ratios close to unity are a~ + a 2 a, b~ b2 =R1;a-1 =R2;b,~=R~

(25)

For a set of nominally equal nozzles a scheme that uses a number of redundant measurements of nozzle coefficient ratios allows the least-squares best values of the independent ratios to be found. For a set of unequal nozzles a similar measurement scheme that uses the nozzles in various combinations, some of them in parallel, allows the set of best ratios between the nozzles to be found. A particular example of such a measurement has shown that standard deviations of 0.0001 can be obtained. Techniques, also using pressure ratios, have shown that the pressure dependence of the coefficient of a sonic nozzle can be measured. The results of a series of measurements on 0.7, 1.0 and 1.4 mm diameter nozzles compare favourably with previous work and show that the technique is useful for small nozzles of uncertain geometry. The pressure dependence work also allows a correction factor to be calculated that improves the measurement, using a pressure transducer, of nozzle coefficient ratios that differ substantially from unity.

Ac k nowle dge m e nt The difficult construction of the sets of closely matched small nozzles used in this work was done by Mr I. A. Groves. He also made the large number of measurements needed in the comparison of sets of sonic nozzles and in establishing pressure dependencies.

from which it is easy to show that

a~ + bL be

=

R~ 1 + R2

+ R~

(26)

Here the measured values of the ratios on the righthand-side (RHS) are all close to unity and no significant correction applies. However, the directly measured ratio on the left-hand-side (LHS) is ca 1.5 so that a significant correction should be applied. The value for the RHS calculated from R1, Re and R~, was 1.508, and that for the LHS was measured directly as 1.494 before correction and 1.507 afterwards, proving the worth of the correction. If, instead of using three closely matched nozzles for the measurements of Section 5, three nozzles only roughly equal, could have been used. It would then be necessary to apply the correction deduced in this section to the results. If the magnitude of the correction were to be derived from these results themselves, then self-consistency could be achieved by using an iterative calculation procedure. Though this has not been done, it is felt that this would rapidly converge due to the relative smallness of the correction.

7. Conclusions The ratios of sonic nozzle coefficients can be measured easily and accurately using a pressure transducer.

References 1 IS() 9300, Measurement of gas flow by means of critical flow Venturi nozzles (1990)

2 Bignell, N., Collings, A. F., Hews-Taylor, K. J., Martin, B. J., Braathen, C. W., Petersen, M. and Welsh, C. Calibration of ultrasonic domestic gas meters. Proceedings 6th Int. Conf. Flow Meas. (Flomeko '93), Seoul (1993) (ed. S. D. Park & F. C. Kinghorn) 385-390 Takamoto, M., Ishibashi, M., Watanabe, N., Aschenbrenner, A. and Caldwell, S. Intercomparison tests of gas flow rate standards. Proceedings 6th Int. Conf. Flow Meas. (Flomeko '93), Seoul (ed. S. D. Park and F. C. Kinghorn) 1993, 533-540 4 Martin, B. R. Statistics for Physicists. Academic Press, 1971 5 Watanabe, N. and Komiya, K. On the discharge coefficient of critical venturis. Bull. NRLM No. 41, 20 (1980) 20-25 6 Stratford, B. S. The calculation of the discharge coefficient of profiled choked nozzles and the optimum profile for absolute air flow measurement. I. Roy. Soc. 68 (]964) 237-245 7 Brain, T. J. S. and Macdonald, L. M. Evaluation of the performance of small scale flow venturis using the NEL gravimetric gas flow standard test facility. Fluid Flow Measurement in the mid-1970s Edinburgh: HMSO Vol. 1, 1977, 107-125