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The orifice plate discharge coefficient equation
Flow Meas. Instrum. Vol 1 January 1990
67
The orifice plate discharge coefficient equation M. J. READER-HARRIS and J. A. SATTARY
The orifice plate discharge coefficient equation described in this paper is the result of the analysis of the data collected in the last ten years in Europe and the United States. It was derived at NEL and accepted by an international meeting of flow measurement experts in New Orleans in November 1988. It is based on an understanding of the physics, and consists of C= term for corner tappings at infinite Reynolds number, slope term consisting of throat Reynolds number term and velocity profile term, and upstream and downstream tapping terms. Keywords: orifice plate flowmeter, discharge coefficient equation
Introduction Although the orifice plate is the recognised flowmeter for the measurement of natural gas and light hydrocarbon liquids, the orifice discharge coefficient equations in current use are based on data collected more than 50 years ago. Moreover, for almost 20 years the United States and Europe have used different equations, a discrepancy with serious consequences for the oil and gas industry since many companies are multinational. For the past ten years data on orifice plate discharge coefficients have been collected in Europe and the United States in order to provide a new database from which an improved discharge coefficient equation could be obtained which would receive international acceptance. In November 1988 a joint meeting of US and EEC flow measurement experts in New Orleans unanimously accepted the equation derived by NEL. This paper describes this equation (with one small amendment agreed at the meeting) and the physical understanding on which it is based. Anyone deriving a new equation benefits from the work of Stolz 1 who made a vital contribution to the subject. He realized that discharge coefficients obtained with different sets of pressure tappings must be related to one another by equations based on the physics. He therefore derived a discharge coefficient equation whose terms had known physical significance. This is the equation included in the present international standard ISO 51672. The new equation is made up of several terms of known physical significance; the tapping terms are described first because it was necessary to determine them before the other terms could be calculated. Throughout this work the best-fit terms were obtained by fitting the values in the database; this consists of 11 346 points; the diameter ratios range from 0.10.75, throat Reynolds numbers from 1700 to 5 x 107, and pipe diameters from 50-250 mm. The data were collected in nine laboratories 3-12 in four fluids: water, air, natural gas and oil. The data points for which the orifice diameter was less than 12.5 mm (I/2 in) are The authors are with the National Engineering Laboratory, East Kilbride, Glasgow, Scotland
0955-5986/90/020067-10
very scattered and were excluded. The 600 mm (24 in) data are still being collected and only the Gasunie tapping term data 13 were available (no other laboratory had then made 600 mm measurements and the Gasunie discharge coefficients for 600 mm pipe were not available). However, in obtaining the form of each of the terms, data collected by many other experiments were also used.
The tapping terms The tapping terms are equal to the difference between the discharge coefficient using flange or D and D/2 tappings and those using corner tappings. The value of the tapping terms was determined from the EEC data which included all three standard pairs of tappings on each orifice run. The American data were collected with only one pair of tappings (flange) and so the value of the tapping terms could only be calculated approximately for comparison.
Value of the tapping terms in the EEC database The data in the EEC database were analysed to determine the values of the tapping terms for 100 mm (4 in) and 250 mm (10 in) flange tappings and for D and D/2 tappings. First it was established that for the data in the EEC database the tapping terms are independent of Reynolds number; then the values of the tapping terms were obtained for each diameter ratio ,6. The EEC database does not contain any oil data, and so only 5% of the data were obtained with throat Reynolds number less than 80000, and most of that 5% were with ,8 = 0.2 (see later section on fitting for low Reynolds numbers). After the tapping term work described in this section was done, a small number of additional files were added to the database; inclusion of these would only have a very small effect on the calculations in this section.
Establishing that the tapping terms are independent of Reynolds number for the data in the EEC database For this purpose only data which had been collected in water at NEL or in air at CEAT (Centre d'Etudes Aerodynamiques et Thermiques, Poitiers) were used, because only these data consisted of simultaneous
(~ 1990 Butterworth & Co (Publishers)Ltd
M. J. Reader-Harris and J. A. 5attary - The orifice p/ate discharge coefficient equation
68
measurements of discharge coefficients using different sets of tappings. The tapping terms were calculated for each value of ,8 and plotted against IogReo, where ReD is the pipe Reynolds number. The best-fit straight line and 95% confidence limits were obtained, and for each value of ,8 the horizontal line representing the mean difference lay within the confidence limits. For some values of ,8 the slope of the best-fit straight line was small and positive and for other values small and negative. It was established that the tapping terms are independent of Reynolds number for the data in the EEC database.
Finding the value of the tapping terms for the diameter ratios and pipe diameters available in the EEC database For the NEL water and the CEAT data the value of the tapping terms for each set of data can be obtained by taking differences between discharge coefficients and averaging them. For the other sets of data, average values were obtained by fitting
C = acgc + a~gf + adgd + bReD -°'75,
(1)
where C is the discharge coefficient, ac, at and ad are the constant terms for corner, flange and D and D/2 tappings respectively, and b is the slope. If a discharge coefficient was measured with corner tappings, gc = 1 and gt = gd = O, and similarly for the other tappings. This fit, in which the slope is assumed to be the same for each tapping, is the appropriate one to use since it has been assumed that the tapping terms are independent of Reynolds number. Thus for a given set of data the difference between, for example, the discharge coefficients using D and D/2 and those using corner tappings is given by ad a~. The method just described was also used for the NEL water and the CEAT data and in most sets of data the differences obtained by fitting and those obtained by averaging differed by less than 0.00001; in no case did they differ by more than 0.00005. These very small differences suggest that the effect of using different exponents of ReD in the fit in equation (1) will be very small. For each diameter ratio the mean value of each of the three dimensionally different tapping terms (D and D/2, flange in 100 mm pipe, and flange in 250 mm pipe) was obtained by averaging the data-set-means (the mean values of each set of data). The estimated standard deviation of the data-set-means depends on tapping position but is generally in the range 0.00020.0010. Since the D and D/2 tapping terms for the -
-
two sets of NEL air data with diameter ratio 0.75 differ from the mean value by 0.005 70 and 0.00696, these data were excluded from subsequent calculations as outliers since they both lie more than six standard deviations from the mean. With these outliers removed the mean values of the tapping terms are as given in Table 1. The work of other experimenters
Upstream data It is natural to begin by analysing the upstream data since previous workers have found them easier to fit. The change in discharge coefficient, AC, when the downstream tapping is fixed in the corner and the upstream tapping is moved from the corner to a distance one pipe diameter (D) upstream, is shown in Figure 1 as a function of ,84/(1- ~4). Data from Herning and Bellenberg 14 (A), Witte ~S (B), Herning 16 (C), Sens17 (D), Witte and SchrOder (E) (in reference 18), Teyssandier and Husain ~9 (F), Stuttgart (G) (in reference 20), Darrow 2~ (H), Bean et, a122 (I), BHRA (J) (in reference 23), Enge124 (K), NEL 25 (L) and Gasunie 13 (M) are plotted. The linear dependence on ,84/(1 - ,84) is confirmed. For the calculations for New Orleans, provisional data from Gasunie were used; in this paper their final data are used, but the changes are very small. The data from Bean et al should perhaps not be included since their upstream tap closest to the corner was about 0.04D away from it. The line on the figure is A C = 0.043/64/(1 -/64); the constant was chosen to make it possible to obtain a good fit to the EEC data means in Table 1. One reason why the line lies a little below the mean of the data on the figure is that the tapping term formula should include not only an upstream and a downstream term but also a product term because the discharge coefficient depends on the reciprocal of the square root of the pressure difference 26. When this term is removed to produce a simpler formula, the upstream tapping term is reduced to compensate. The dependence of the upstream tapping term on L~ is also required, where L I is the distance between the upstream tapping and the upstream face of the orifice plate divided by the pipe diameter. To determine this dependence, points were taken from most of the experimenters listed above and ,8-4(1- ,84)AC was plotted against L~. Since it is much more important to obtain a good fit for large /3, the symbols 0.030
o.o25~0.020
Table I Mean measured tapping terms for the EEC Database
B~
K
- -
0,015M
,8
Flange 250 mm
Flange 100 mm
D and D/2
0.20 0.375 0.50 0.57 0.66 0.75
-0.000 46 -0.00086 -0.001 08 -0.000 95 -0.000 51 0.00057
-0.000 57 -0.001 73 -0.001 34 -0.000 95 0.001 32 0.00710
-0.000 92 -0.001 2 7 -0.000 28 0.000 94 0.004 79 0.011 65
c y
~,~'F"IAH
0.010 H IA
0.005
~
Equation
15)
E/
0.000 ~I 0.00
I 0.10
I 0.20
I 0.30
I 0.q0
I 0.50
0.60
Figure I Upstream tapping term one diameter upstream
Flow Meas. Instrum.
Vol 1 January 1990
0.055 0.050
m
v
0.045
_
0.040
m
0 vx
v E
X X
0 X
0.035 %
v 0.030
i _
V
0.025
Equation 0.020
0.015
(5)
841 (Im~ 4 )
-
_
V
0.010 ~
V
0.1
Z~ v +
0.05 o.1 0.2
X []
0.3 0.4
-
0.4
-
0.5
0
0.5
-
0.6
-
- 0.2 - 0.3
0.005 0.000 / 0.00
Y
I
t
I
0."~0
0.20
0.30
0.40
I 0.50
I 0.60
I 0.70
I 0.80
I 0.90
I 1.00
1.10
LI Figure 2 Upstream tapping term as a function of L i
used relate to the value of /~ rather than the experimenter. The fit marked in Figure 2 is one that comes from the fit to the EEC data means (see equation (5)). Downstream data Many experimenters have also measured the pressure profile downstream of an orifice plate. It is less obvious what formula will fit these data. However, Teyssandier and Husainl'~ non-dimensionalised downstream distances with the dam height rather than the pipe diameter, and this proves to be very useful Given the definition that M2 is the distance between a downstream position and the upstream face of the orifice plate divided by the dam height 2L2 M2 - - (2) 1-/~ where L2 is the distance between a downstream position and the upstream face of the orifice divided by the pipe diameter. M~ and L~, are defined identically except that in each case the distance from the downstream face of the orifice plate is used. The pressure downstream of the orifice plate decreases from its corner tap value to a minimum and thereafter it increases rapidly. In Figure 3 the value of L2 at the pressure minimum is plotted as a function of /~. The same notation as in Figure I is used with the addition of data from Beitler (N) (quoted in reference 27). This figure is similar to one in Teyssandier and Miller 27, except that this one includes a simple linear fit: at the pressure minimum
L2 = 1.65(1 - ~ , ie
(3) /~2
=
3.3.
It is worthwhile also to attempt to determine the greatest change in discharge coefficient, ACre,n, as the downstream tap is moved from the corner to the minimum pressure location and the upstream tap is kept in the corner. Whereas the difference between the distance to the pressure minimum from the upstream and that from the downstream face is nearly always less than 10%, one significant feature of much of the data is that the pressure changes rapidly in the vicinity of the downstream corner; so there is a substantial difference between the value of ACmin when the downstream corner is at NI~ = 0 and that with it at M2 = 0, where values at M2 = 0 are obtained by linear extrapolation from values for small M2. Substantial work has been done 26 to examine the advantages and disadvantages of using L2 and M2 rather than L~ and M~. Using M2 has theoretical advantages 2e.29 but is more complicated. It has been 1.80 1.60 1.40 1.2o-
1.00 A 0.600.40
C
0.20~ 0.00 I I I 0.00 0.10 0.20 0.30
k
N _ N
C t ~L
~
C I ~ ~ I I l I l I 0.40 0.50 0.60 0.70 0.00 0.90 1.00 B
Figure 3 Distance to the pressure minimum
70
M. J. Reader-Harris and J. A. 5attary - The orifice p/ate discharge coefficient equation piecewise linear fits or continuous functions. The fit which is superimposed on the data is that arising from the final fit to the EEC data means (see equation (5)).
0.000
Equation (4) -0.002 ,
Fitting the data
C
-0.004 _.CE
~
I
A ~
-0.006
As previously stated, the data outside the EEC database were only used to give the correct forms of the equations. The best fit equations were obtained by fitting the mean tapping terms from all the sets of data in the 100 mm (4 in) and 250 mm (10 in) EEC database together with the Gasunie 600 mm (24 in) flange tapping terms. The 600 mm tapping terms were available but the discharge coefficients were not, because the tapping terms were measured directly and final checking would make negligible difference to the tapping terms. In fact, the 600 mm flange tap terms do not have a very large effect on the final fit. The 600 mm D and D/2 tapping terms were in excellent agreement with the 100 and 250 mm data (and so it was not necessary to include them in the fitting: the maximum difference between the 100 and 250 mm and the 600mm D and D/2 tapping terms was 0.000 53). The tapping terms derived from the 600 mm data are given in Table 2. It was decided also to attempt to determine the values of the tapping terms required to fit the US data. Only the water data were used since for the oil data the Reynolds numbers are sufficiently low that there is an effect of Reynolds number (see the following section). The differences in the values of the tap terms between those for the 250 mm (10 in) data and those for each of the other diameters have been calculated and are listed in Tables 3 and 4. The fact that the actual values of /~ and D are different from the
I ~
C
I
-0. 008
B ~
-0.010
l
I
0.00 0.10
I
I
I
0.20 0.30 o.qo
I
I
C
l
I
o.so 0.60 0.70 0.80 0.90 1.00
Figure 4 Z~Cmin(from M'2 = 0) shown that, provided appropriate restrictions are placed on plate thickness, M~ can be used without introducing significant errors. These plate thickness restrictions are satisfied by the EEC and US plates. ACmin is expected to be proportional to/~k where k is some unknown exponent. To test this theory ACm~n was plotted against/~ and the results are shown in Figure4 (from M~ = 0). It is not necessary to determine the best fit, but the following, from equation (5), gives a good fit: ACmin =
-0.0099,8 TM (from M~ = 0).
(4)
Given possible vertical and horizontal scales it is now reasonable to plot /~-~.~AC against M~ in Figure 5. Although there is some scatter it is possible to discern the shape. It is appropriate to fit the data with either
0.0001~ ÷k
V
I ~ _o.o02F~,~,x~
I
-0.004-
[]
Equation
x **
A
8o~.~ ~ ~
-0
v +
-~x
,~
d;7
+A x
'ely
~AOXA
~V
~
El
(5)
Z~ V
0.25 - 0.35 0.35 - 0.45
+ x
o.4s - o.5s o55-o65
[]
o.65
0
0
[]
o.75
-
0.75 - 0.80 0
0
-o 0 0 6 -
u
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v
_ x
i+
-.
~
~ X []
v
A
+o+
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~o
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0 z~
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^
,~ ,.-,
^~
[]
x
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,,
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"-...~. + x
o
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"
rm _
o []
x
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x
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-0.014 0.00
I
I
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I
I
I
0.50
1.00
1.50
2.00
2.50
3.00
M~ Figure 5 Downstream tapping term as a function of M',
I
3.50
I 4.00
4.50
Flow Meas. Instrum. Vol 1 January 1990
71
nominal ones has been allowed for in the calculations using equation (8) to estimate changes in tapping term due to small changes in ,6 and D. In Table 4 the fact that the relative roughness varied between the pipes has also been allowed for (using equation (15)), whereas in Table 3 the effect of relative roughness has not been included (equation (18) was used to allow for the effect of RED). Slight discrepancies between equations (15) and (18) lead to the difference in the tapping term for ,6 = 0.2 and D = 50 mm being greater in Table 4 than in Table 3. The differences between 250 and 100 mm tapping terms are very similar to those from the EEC data. The 50 mm flange tapping terms differ significantly from the EEC D and D/2 tapping terms: the difference between them can be as much as about 0.0024 (about 0.4% of C) whereas at least for small ,6 they should be nearly equal. Since the values of the tapping terms for the American data were obtained by comparing data from one orifice plate and tube against those from another plate and tube, whereas those for the EEC data were all obtained by comparing data taken (sometimes simultaneously) from the same plate and tube, the American values were not used in the final fit. Upstream and downstream several types of fits were tried: the results are described in reference 26. It was decided that the upstream and downstream tapping terms should have continuous first derivatives. Table 2 Mean measured tapping terms for the 600 mm (24 in) Gasunie Data
/~
Flange 600 mm
D and D/2
0.2 0.375 0.5 0.57 0.66 0.75
-0.000 10 -0.000 18 0.000 07 0.000 23 0.000 93 0.001 95
-0.001 04 -0.001 47 - 0.000 09 0.001 30 0.005 32 0.011 75
By prescribing the upstream tapping term and optimizing the downstream one the following equation was obtained: A C = (0.0433 + 0.0712e -8sL, - 0.1145e -6L~) - 1
-0.0116(M~ - 0.52M~ 13),611.
(5)
This equation has a minimum downstream at M~ = 3.7 and a finite derivative at the origin. The curves on Figures 1, 2, 4 and 5 have been extracted from this equation. The standard deviation of the mean tapping terms about this equation is 0.0007. Fitting for low Reynolds numbers
In analysing the EEC data, no effect of Reynolds number on the tapping terms was found. However, that is not true of the US data. Figure 6 is a plot of the US data for ,6 = 0.73. The tapping terms required to bring the data for high Reynolds numbers on to a common curve are clearly inappropriate for lower Reynolds number. A reduction in the upstream pressure rise at low Reynolds numbers has been seen previously in the work of Johansen 3° and of Witte and Schr6der (quoted in References 18 and 31). A reduction in the upstream pressure rise for extremely rough pipes (4 = 0.06) at moderate Reynolds numbers was also observed by Witte Is, but this effect will be extremely small with the pipe roughnesses encountered in accurate flow measurement. The change in the maximum decrease in downstream pressure has also been measured by Johansen. Compared with the data for high Reynolds numbers, these data are very sparse. It is not possible to produce perfect low Reynolds number tapping terms, but since these terms are significant it is necessary to produce the best possible ones. Both upstream and downstream it appears reasonable to consider the ratio of the low Reynolds number tapping terms to the high Reynolds number ones. Only the maximum changes in pressure (and thus C) are
Table 3 Differences in the values of the tapping terms between those for the 250 mm US data and those for each of the other diameters (with no allowance for the effect of pipe roughness)
Other diameters
/~ 0.2
50 mm 75 mm 1O0 mm 150 mm
0.001 8 -0.000 4 -0.000 8 0.000 8
0.375 0.001 -0.000 -0.000 -0.000
4 9 3 3
0.5
0.57
0.66
0.73
0.001 6 0.000 7 0.000 2 0.000 6
0.002 2 0.001 3 0.000 5 0.000 2
0.004 8 0.004 0 0.003 2 0.000 6
0.0094 0.007 2 0.006 3 0.002 8
Table 4 Differences in the values of the tap terms between those for the 250 mm US data and those for each of the other diameters (making allowance for the effect of pipe roughness) Other /~ diameters 0.2 0.375 0.5 0.57 0.66 0.73
50 mm 75 mm 1O0 mm 150 mm
0.002 0 -0.0004 -0.000 8 0.000 8
0.001 2 -0.001 0 -0.000 4 -0.000 3
0.001 2 0.000 5 0.000 2 0.000 5
0.001 6 0.001 0 0.000 0 0.000 1
0.003 6 0.003 5 0.002 2 0.000 5
0.0070 0.006 1 0.004 5 0.002 4
M. J. Reader-Harris and J. A. 5attary- The orifice plate discharge coefficient equation
72 0.68
Pipe diameter 0.67
A + {3 0 %7
0.66
50mm 75 mm I O0 mm 150 mm 250 mm
0.65 --
u
0.64 --
O u
0.63
--
o.62
-
L
~
0.611
0.60~
•
0.59 3.50
I
I
I
I
I
4.00
4.50
5.00
5.50
6.00
6.50
Logub Rot)
Figure 6 US data for/3 = 0.73 available and so it is necessary to assume that the same ratio applies for other values of L1 and M ; besides those at the maxima. From the work of Witte and Schr6der the upstream tapping term becomes effectively constant above an orifice Reynolds number, Red, of about 80000. It is assumed that the downstream tapping term becomes constant at the same orifice Reynolds number. The ratio of the upstream tapping term deduced from the data of Witte and Schroder (E) and of Johansen (O) to the standard tapping term is then plotted in Figure 7 for Red > 3700. A reasonable fit is ACI 119 0 0 0 1 °8
ACI
- 1 -
0 14( 19 0001o8 . I~-~ed /
(7)
1.20
There is a slight suggestion that the downstream low Red term may be a function of/3 since A C I / A C h is a little closer to 1 for /3 = 0.209, but it is probably not necessary to allow for this. For Red < 3700 it is not clear what the correct formulae are. Possible formulae are included in Reader-Harris-'% but none is in fact necessary to fit the database because all the data in the database for Red < 3700 are for [3 <-- 0.375 and the tapping terms are quite small.
1.00
Complete tapping term
=,
- 0.23/--E-
)
(0)
where ACI is the value of the tapping term for low Red and ACh is the high Red tapping term for L1 = 1 in equation (5).
Putting equations (5), (6) and (7) together, the complete tapping term is
ErE ~
O. 80
L) <
Downstream it is even harder to produce an accurate tapping term for low Reynolds numbers. A C J A C h is plotted for Red > 3700 for the data of Johansen (excluding that for /3 = 0.209 for which the tapping term is small and thus the value of the ratio is both less certain and less important) in Figure 8. A reasonable fit is
/o
0.60
A C = (0.0433 + 0.0712e -8.sL, - 0.1145e -6z,)
O. 40
0.20 0.0
/ I
3,00
3.50
x (1 - 0 . 2 3 A ) -
o o"
I
4100
4.50
,64
- 0.011 6(M~. - 0.52M~, 1~)
Equation (6) I
/34 1 -
x (1 - 0.14A)/3 TM,
5.00
Log Re d
Figure 7 Effect of low Reynolds number on upstream tapping term
where
A =
1,9000g° ~ ReD I
(8)
2L~ and M~ - - -
1-/3
Flow Meas. Instrum.
73
Vol 1 January 1990
data. In all cases the flange tapping data were convetted to corner tapping data using equation (8). In Figure 9, the data for ,6 = 0.1 and those for/3 = 0.2 lie on top of each other, and those for ,6 = 0.375 are slightly higher. So for very low /3 it appears that C depends only on Red. It then seems reasonable to suppose that for very low/3
1.20
0
1.00
I
o./o°-
0.80 ,,,¢
<] 0.60
J
~o
o
0.40
C = al + b(lO6/Red) n
0.20 0.00 3.00
= al + b(lO6fl/Reo) n.
Equation (7) I 3.50
I
4.00 Log Re d
I
4.50
5.00
Figure 8 Effect o f l o w Reynolds number on downstream tapping term
The C~ term C:~ is the discharge coefficient for corner tappings for infinite Reynolds number; it increases with ,6 to a maximum near /3= 0.55 and then decreases increasingly rapidly. Thus an appropriate form for C~ is C~ = a l + a2/~ mt + a3,6 m:.
(9)
With the slope term in either equation (14) or equation (17) it is possible to obtain a good fit with many pairs of values of ml and m2. Taking m~ = 2 and m2 = 8, while not necessarily optimum, enables a good fit to be obtained.
The slope term Reader-Harris and Keegans 32 found that, for a fixed pipe Reynolds number, as the pipe roughness changed so the change in discharge coefficient was approximately proportional to/~4A;~, where A). is the change in the pipe friction factor. This led Reader-Harris to suppose that the form of the equation for the discharge coefficient (omitting tapping terms) might be C = C~ + cl/~l;~,
(10)
where / is approximately 4. Rapier 33 made experimental measurements in very smooth pipes and as a result of his work also proposed that C was best expressed as a function of friction factor. However, equations of the form of equation (10) do not give good agreement with experiment for small ,6, since they then give C almost constant. The problem is due to the need for an additional term in equation (10). To establish what that term is it is necessary to consider what type of equation would be appropriate for small ,6 since it is for small ,8 that equation (10) seems to be most inappropriate. It seems reasonable that for low /~ the discharge coefficient should depend only on throat Reynolds number, Red C = al + fiRed)
(11)
To check this theory, C was plotted against log Red (Figure 9) for ,6 less than or equal to 0.375 using all the EEC and American flange tapping data. So that all the data included in this analysis had been collected with the same type of tappings, flange tapping data from the EEC were used rather than corner tapping
(12)
Since approximate equations for low and high /3 have now been obtained the most obvious general equation for C is given by C = C~ + b(lO6,6/Reo) ~ + cl,61;(.
(13)
Reader-Harris 34 tested this equation against flange tap data using a tapping term with no dependence on Red, and used data taken by several experimenters in rough pipe to establish the truth of the friction factor term. One problem is that when the low Red tapping terms in equation (8) are included, the fit against ~. is not as good as it was without them. But it is not surprising that only a poor fit should be obtained for Red < 80000 since the physical causes of the change in the tapping terms may also result in a change in the friction factor term. Such a change could not be detected in the rough pipe data in reference 34 since all these data were for Red > 80000. So it seems reasonable to suppose that the friction factor term should include a low Red term of the same form as that in the tapping terms. It is also necessary to make provision for transition from turbulent to laminar flow below ReD = 3500 since, except for very low /3, the slope of the data for ReD < 3500 is very different from that for ReD > 3500. This is because the velocity profile changes very rapidly as the flow changes from turbulent to laminar. Such a change has to be included in the friction factor term since this represents the effect of velocity profile. The slope term then becomes Ccr - C~ = b(lO°i~/ReD) ~ + (C! + c2A),6 I x m a x {A., 0 . 1 6 4
-
35(Reo/10°)}
(14) where A is as defined in equation (8), and Ccr is the discharge coefficient for corner tappings. Performing a least-squares fit of all the data in the database, the following equation for C= and slope is obtained: Ccr = 0.5966 + 0.0200,62 - 0.279,68 + 0.000 522(106/~/ReD) °'z + (3.974 + 1.044A),64 x max {~., 0.164 - 35(ReL)/10°)}.
(15)
The tapping terms in equation (8) were used to convert flange and D and D / 2 tapping data to corner tapping data. There is a problem with this equation: the use of friction factor as an extra variable would be unpopular. However, although ~. may appear to be a rather
M. J. Reader-Harris and J. A. Sattary - The orifice plate discharge coefficient equation
74 0.65
0.64
O 0.I Z~ 0.2 -I,. 0.375
0.63
tO
"~ 0 . 6 2 -
~2 u
P t-
0.61
--
0.60
--
0.59
--
I
0.58
3.00
3.50
1
4.00
I
4.50
I
I
I
I
5.00
5.50
6.00
6.50
7.00
LOglo Re d
Figure 9 Discharge coefficient for small fi inconvenient variable, it is possible to approximate it by a simple function of ReD if a value of relative pipe roughness, kin~D, is assumed. For instance,, if km/D = 0.00002, Z is always within 0.0005 of ~.' for ReD > 4000 if ,,1" = 0.008 59 + 0.809 26/Ren°~'L
C = 0.5960 + 0.0313,6 =
0.225,65
C= term
1
+ (0.0166 + 0.0035A),64 term max {(10"/ReD) °-4, 37.6 -- 8000(Ren/10~')}I Sl°pe + (0.0433 + 0.0712e -~L' - 0.1145e -~'L')
(17) (1 -
Including an appropriate velocity profile term for ReD < 3500 and fitting over the complete database gave the following equation for C= and slope:
0.23A)fi4/(1 - ,64)
-0.0116(M',
-
0.52M!,
x 1 - 0.14A),6 TM,
Ccr = 0.5960 + 0.0313/32 - 0.225,65
1.;i)
Upstream tap term
Downstream tap term (19)
+ 0.000 526(10('fl/Reo) °.7
where
+ (0.0166 + 0.003 5A)fl 4 x max {(106/Ren) °4, 37.6 - 8000(RED~10~)}.
-
+ 0.000 526(10~,6/ReD) °:
(16)
So in equation (14),). was replaced by an inverse power of Ren, and the slope term for ReD > 3500 becomes
CCT- C~ = b(lO~,6/Reo) " + (cl + c2A),6t(lO~'/Ren) n'.
and fitting the EEC/US database, led to the equation accepted in New Orleans in November 1988 and is given in equations (8) and (18):
M',--(18)
The standard deviation of the data about this equation is 0.285%. The standard deviation increases for small D, large fl and low ReD. When the 600 mm (24 in) data have all been collected and included in the database the equation will be refitted, but those data will not change the form of the equation. Conclusions
The work reported here, which consisted of both investigating the physics of flow through orifice plates
2L~ 1-,6
A = ( l ~' .000fl/°~ ~9 e o L1 is the distance between the upstream tapping and the upstream face of the orifice plate divided by the pipe diameter, and L~ is the distance between the downstream tapping and the downstream face of the orifice plate divided by the pipe diameter. The terms in A are only significant for small throat Reynolds number; M~ is in fact the distance between
Flow Meas. Instrum.
Vol I January 1990
the downstream tapping and the downstream face of the plate divided by the dam height.
75
13
Acknowledgement
14
This paper is published by permission of the Director, National Engineering Laboratory, and is Crown copyright. The work reported here was supported by DGXll of the European Economic Community and by the Standards, Quality and Measurement Advisory Committee of the Department of Trade and Industry.
15
References 1 SIolz, J. 'A universal equation for the calculation of discharge coefficient of orifice plates'. In: FIomeko 1978 - Flow Measurement of Fluids, H. H. Dijstelbergen and E. A. Spencer (Eds). North-Holland Publishing Co, Amsterdam (1978) 519-534 2 International Organisation for Standardization. ISO 5167 'Measurement of fluid flow by means of orifice plates, nozzles and venturi tubes inserted in circular crosssection conduits running full'. International Organization for Standardization, Geneva (1980) 3 Hobbs, J. M. 'Experimental data for the determination of basic 100 mm orifice meter discharge coefficients (European programme)'. Report EUR ]0027, Commission of the European Communities, Brussels, Belgium (1985) 4 Hobbs, J. M. and Satlary, I. A. 'Experimental data for the determination of 100 mm orifice meter discharge coefficients under different installation conditions (European programme)'. Report EUR 10074, Commission of the European Communities, Brussels, Belgium (1986) 5 Hobbs, I. M., Sattary, I. A. and Maxwell, A. D. 'Experimental data for the determination of basic 250 mm orifice meter discharge coefficients (European programme)'. Report EUR 10979, Commission of the European Communities, Brussels, Belgium (1987) 6 Hobbs, J. M., Satlary, I. A. and Maxwell, A. D. 'Experimental data for the determination of 250 mm orifice meter discharge coefficients under different installation conditions (European programme)'. Report EUR 10980, Commission of the European Communities, Brussels, Belgium (1987) 7 Hobbs, I. M. 'The EEC orifice plate project. Part I Traceabilities of facilities used and calculation methods employed'. Progress Report No PR5: EUEC/17. National Engineering Laboratory, East Kilbride, Glasgow (1987) 8 Hobbs, J. M. 'The EEC orifice plate project. Part II: Critical evaluation of data obtained during EEC orifice plate tests'. Progress Report No PR5: EUEC/17. National Engineering Laboratory, East Kilbride, Glasgow (1987) 9 Hobbs, J. M. 'The EEC orifice plate project: tables of valid data for EEC orifice analysis'. Progress Report No PR7: EUEC/17. National Engineering Laboratory, East Kilbride, Glasgow (1987) 10 'Coefficients of discharge for concentric, square-edged, flange-tapped orifice meters: equation data set - supporting documentation for floppy diskettes'. American Petroleum Institute, Washington, DC (1988) 11 Brillon, C. L., Caldwell, S., and Seidl, W. 'Measurements of coefficients of discharge for concentric, flange-tapped, square-edged orifice meters in white mineral oil over a low Reynolds number range'. American Petroleum Institute, Washington, DC (1988) 12 Whetstone, J. R., Cleveland, W. G., Baumgarten, G. P. and Woo, S. 'Measurements of coefficients of discharge for concentric, flange-tapped, square-edged orifice meters in water over a Reynolds number range of
16 17
18 19 20 21 22
23
24 25
26
27
28 29
30 31
32
1000-2 700000'. NIST Technical Note (TN-1264). American Petroleum Institute, Washington, DC (1988) '600 mm orifice plate project. Vol Xll. Discharge coefficients'. TP/T 88.R.2052, Gasunie, Groningen (December 1988) Herning, F. and Bellenberg, H. Neue Versuche mit Normblenden. Brennstoff-W~rme-Kraft 12 (1960)89-96 Witte, R. Die StrOmung durch Diisen und Blenden. Forschung auf dem Gebiet des Ingenieurwesens 2 7 (1931) Herning, F. Untersuchung des Druckverlaufs an Normblenden grosser Rohrweiten. Brennstoff-W~rme-Kraft 7 (1955) 303-307 Sens, M. Etude Exp~rimentale des Variations de la Pression au Voisinage d'un Diaphragme Normalis~ selon ISO/DIS/5167. ISO/TC30/SC2 (France-l l) 85 (March 1977) Engel, F. V. A. Durchflu,Smessung in Rohrleitungen. Bericht i.iber ein Symposium in East Kilbride (Schottland). Brennstoff-W~rme-Kraft 13 (1961) 125-133 Teyssandier, R. G. and Husain, Z. D. Experimental investigation of an orifice meter pressure gradient. Trans. ASME. J. Fluids Engng 109 (1987) 144-148 Private communication from J. Stolz regarding data collected in Stuttgart (1988) Darrow, R. R. (Ed). 'Investigation of orifice meter installation requirement'. Interim Research Report No 2, American Gas Association, (1954) Bean, H. S., Benesh, M. E. and Buckingham, E. Experiments on the metering of large volumes of air. Bureau of Standards, Washington. Journal of Research 7 (1931) 93-145 Marlin, C. N. B. 'Effects of upstream bends and valves on orifice plate pressure distributions and discharge coefficients'. NEL Report No 702. National Engineering Laboratory, East Kilbride, Glasgow (1986) Engel, F. V. A. Fluid-flow measuring devices. Engineering 176 (1953) 6-10 and 35-37 Certificate of Calibration on determination of the longitudinal pressure profile through six thin orifice plates mounted in the EEC 100 mm assembly using AGA tappings throughout for the European Economic Community DGXll Research, Science and Education. National Engineering Laboratory, East Kilbride, Glasgow (1988) Reader-Harris, M. J. 'The tapping terms in the orifice plate discharge coefficient formula'. Report to be submitted to the European Economic Community DGXII Research, Science and Education. National Engineering Laboratory, East Kilbride, Glasgow (1990) Teyssandier, R. G. and Miller, R. W. 'A new coefficient equation for concentric sharp-edged orifice: Part I Fundamental considerations'. ASME paper 79-WA/FM-3. American Society of Mechanical Engineers, New York (1979) Pfeffer, W. Untersuchungen an Blenden mit Eckanbohrungen. Brennstoff-W~rme-Kraft 20(1968) 108-115 Husain, Z. D. and Teyssandier, R. G. 'The effects of plate thickness and bevel angle in a 150 rnm line size orifice meter'. International Conference on Flow Measurement in the mid-80s. National Engineering Laboratory, East Kilbride, Glasgow (1986) Johansen, F. C. 'Flow through pipe orifices at low Reynolds numbers'. Aeron. Res. Committee, Report and Memo No 1252. HM Stationery Office, London (1930) Engel, F. V. A. 'New interpretations of the discharge characteristics of measuring orifices - an approach to improved accuracy'. ASME paper 62-WA-237. Amer. Soc. of Mechanical Engrs. New York (1962) Reader-Harris, M. I. and Keegans, W. 'Comparison of computation and LDV measurement of flow through orifice and perforated plates, and computation of the effect of rough pipework on orifice plates'. In: Proc. of
76
M. J. Reader-Harris and J. A. Sattary - The orifice plate discharge coefficient equation
the Int. Symp. on Fluid Flow Measurement, Washington, DC, Paper 12.1 (November 1986) 33 Rapier, A. C. 'The extrapolation of orifice calibrations to high Reynolds numbers'. In: Proc. of the Fluid Mechanics Silver Jubilee Conf. National Engineering Labora-
tory, East Kilbride, Glasgow, Paper 4.3 (1979) 34 Reader-Harris, M. J. 'A new type of formula for the orifice-plate discharge coefficient'. In: Proc. of the North Sea Flow Metering Workshop, National Engineering Laboratory, East Kilbride, Glasgow (1988)
67
The orifice plate discharge coefficient equation M. J. READER-HARRIS and J. A. SATTARY
The orifice plate discharge coefficient equation described in this paper is the result of the analysis of the data collected in the last ten years in Europe and the United States. It was derived at NEL and accepted by an international meeting of flow measurement experts in New Orleans in November 1988. It is based on an understanding of the physics, and consists of C= term for corner tappings at infinite Reynolds number, slope term consisting of throat Reynolds number term and velocity profile term, and upstream and downstream tapping terms. Keywords: orifice plate flowmeter, discharge coefficient equation
Introduction Although the orifice plate is the recognised flowmeter for the measurement of natural gas and light hydrocarbon liquids, the orifice discharge coefficient equations in current use are based on data collected more than 50 years ago. Moreover, for almost 20 years the United States and Europe have used different equations, a discrepancy with serious consequences for the oil and gas industry since many companies are multinational. For the past ten years data on orifice plate discharge coefficients have been collected in Europe and the United States in order to provide a new database from which an improved discharge coefficient equation could be obtained which would receive international acceptance. In November 1988 a joint meeting of US and EEC flow measurement experts in New Orleans unanimously accepted the equation derived by NEL. This paper describes this equation (with one small amendment agreed at the meeting) and the physical understanding on which it is based. Anyone deriving a new equation benefits from the work of Stolz 1 who made a vital contribution to the subject. He realized that discharge coefficients obtained with different sets of pressure tappings must be related to one another by equations based on the physics. He therefore derived a discharge coefficient equation whose terms had known physical significance. This is the equation included in the present international standard ISO 51672. The new equation is made up of several terms of known physical significance; the tapping terms are described first because it was necessary to determine them before the other terms could be calculated. Throughout this work the best-fit terms were obtained by fitting the values in the database; this consists of 11 346 points; the diameter ratios range from 0.10.75, throat Reynolds numbers from 1700 to 5 x 107, and pipe diameters from 50-250 mm. The data were collected in nine laboratories 3-12 in four fluids: water, air, natural gas and oil. The data points for which the orifice diameter was less than 12.5 mm (I/2 in) are The authors are with the National Engineering Laboratory, East Kilbride, Glasgow, Scotland
0955-5986/90/020067-10
very scattered and were excluded. The 600 mm (24 in) data are still being collected and only the Gasunie tapping term data 13 were available (no other laboratory had then made 600 mm measurements and the Gasunie discharge coefficients for 600 mm pipe were not available). However, in obtaining the form of each of the terms, data collected by many other experiments were also used.
The tapping terms The tapping terms are equal to the difference between the discharge coefficient using flange or D and D/2 tappings and those using corner tappings. The value of the tapping terms was determined from the EEC data which included all three standard pairs of tappings on each orifice run. The American data were collected with only one pair of tappings (flange) and so the value of the tapping terms could only be calculated approximately for comparison.
Value of the tapping terms in the EEC database The data in the EEC database were analysed to determine the values of the tapping terms for 100 mm (4 in) and 250 mm (10 in) flange tappings and for D and D/2 tappings. First it was established that for the data in the EEC database the tapping terms are independent of Reynolds number; then the values of the tapping terms were obtained for each diameter ratio ,6. The EEC database does not contain any oil data, and so only 5% of the data were obtained with throat Reynolds number less than 80000, and most of that 5% were with ,8 = 0.2 (see later section on fitting for low Reynolds numbers). After the tapping term work described in this section was done, a small number of additional files were added to the database; inclusion of these would only have a very small effect on the calculations in this section.
Establishing that the tapping terms are independent of Reynolds number for the data in the EEC database For this purpose only data which had been collected in water at NEL or in air at CEAT (Centre d'Etudes Aerodynamiques et Thermiques, Poitiers) were used, because only these data consisted of simultaneous
(~ 1990 Butterworth & Co (Publishers)Ltd
M. J. Reader-Harris and J. A. 5attary - The orifice p/ate discharge coefficient equation
68
measurements of discharge coefficients using different sets of tappings. The tapping terms were calculated for each value of ,8 and plotted against IogReo, where ReD is the pipe Reynolds number. The best-fit straight line and 95% confidence limits were obtained, and for each value of ,8 the horizontal line representing the mean difference lay within the confidence limits. For some values of ,8 the slope of the best-fit straight line was small and positive and for other values small and negative. It was established that the tapping terms are independent of Reynolds number for the data in the EEC database.
Finding the value of the tapping terms for the diameter ratios and pipe diameters available in the EEC database For the NEL water and the CEAT data the value of the tapping terms for each set of data can be obtained by taking differences between discharge coefficients and averaging them. For the other sets of data, average values were obtained by fitting
C = acgc + a~gf + adgd + bReD -°'75,
(1)
where C is the discharge coefficient, ac, at and ad are the constant terms for corner, flange and D and D/2 tappings respectively, and b is the slope. If a discharge coefficient was measured with corner tappings, gc = 1 and gt = gd = O, and similarly for the other tappings. This fit, in which the slope is assumed to be the same for each tapping, is the appropriate one to use since it has been assumed that the tapping terms are independent of Reynolds number. Thus for a given set of data the difference between, for example, the discharge coefficients using D and D/2 and those using corner tappings is given by ad a~. The method just described was also used for the NEL water and the CEAT data and in most sets of data the differences obtained by fitting and those obtained by averaging differed by less than 0.00001; in no case did they differ by more than 0.00005. These very small differences suggest that the effect of using different exponents of ReD in the fit in equation (1) will be very small. For each diameter ratio the mean value of each of the three dimensionally different tapping terms (D and D/2, flange in 100 mm pipe, and flange in 250 mm pipe) was obtained by averaging the data-set-means (the mean values of each set of data). The estimated standard deviation of the data-set-means depends on tapping position but is generally in the range 0.00020.0010. Since the D and D/2 tapping terms for the -
-
two sets of NEL air data with diameter ratio 0.75 differ from the mean value by 0.005 70 and 0.00696, these data were excluded from subsequent calculations as outliers since they both lie more than six standard deviations from the mean. With these outliers removed the mean values of the tapping terms are as given in Table 1. The work of other experimenters
Upstream data It is natural to begin by analysing the upstream data since previous workers have found them easier to fit. The change in discharge coefficient, AC, when the downstream tapping is fixed in the corner and the upstream tapping is moved from the corner to a distance one pipe diameter (D) upstream, is shown in Figure 1 as a function of ,84/(1- ~4). Data from Herning and Bellenberg 14 (A), Witte ~S (B), Herning 16 (C), Sens17 (D), Witte and SchrOder (E) (in reference 18), Teyssandier and Husain ~9 (F), Stuttgart (G) (in reference 20), Darrow 2~ (H), Bean et, a122 (I), BHRA (J) (in reference 23), Enge124 (K), NEL 25 (L) and Gasunie 13 (M) are plotted. The linear dependence on ,84/(1 - ,84) is confirmed. For the calculations for New Orleans, provisional data from Gasunie were used; in this paper their final data are used, but the changes are very small. The data from Bean et al should perhaps not be included since their upstream tap closest to the corner was about 0.04D away from it. The line on the figure is A C = 0.043/64/(1 -/64); the constant was chosen to make it possible to obtain a good fit to the EEC data means in Table 1. One reason why the line lies a little below the mean of the data on the figure is that the tapping term formula should include not only an upstream and a downstream term but also a product term because the discharge coefficient depends on the reciprocal of the square root of the pressure difference 26. When this term is removed to produce a simpler formula, the upstream tapping term is reduced to compensate. The dependence of the upstream tapping term on L~ is also required, where L I is the distance between the upstream tapping and the upstream face of the orifice plate divided by the pipe diameter. To determine this dependence, points were taken from most of the experimenters listed above and ,8-4(1- ,84)AC was plotted against L~. Since it is much more important to obtain a good fit for large /3, the symbols 0.030
o.o25~0.020
Table I Mean measured tapping terms for the EEC Database
B~
K
- -
0,015M
,8
Flange 250 mm
Flange 100 mm
D and D/2
0.20 0.375 0.50 0.57 0.66 0.75
-0.000 46 -0.00086 -0.001 08 -0.000 95 -0.000 51 0.00057
-0.000 57 -0.001 73 -0.001 34 -0.000 95 0.001 32 0.00710
-0.000 92 -0.001 2 7 -0.000 28 0.000 94 0.004 79 0.011 65
c y
~,~'F"IAH
0.010 H IA
0.005
~
Equation
15)
E/
0.000 ~I 0.00
I 0.10
I 0.20
I 0.30
I 0.q0
I 0.50
0.60
Figure I Upstream tapping term one diameter upstream
Flow Meas. Instrum.
Vol 1 January 1990
0.055 0.050
m
v
0.045
_
0.040
m
0 vx
v E
X X
0 X
0.035 %
v 0.030
i _
V
0.025
Equation 0.020
0.015
(5)
841 (Im~ 4 )
-
_
V
0.010 ~
V
0.1
Z~ v +
0.05 o.1 0.2
X []
0.3 0.4
-
0.4
-
0.5
0
0.5
-
0.6
-
- 0.2 - 0.3
0.005 0.000 / 0.00
Y
I
t
I
0."~0
0.20
0.30
0.40
I 0.50
I 0.60
I 0.70
I 0.80
I 0.90
I 1.00
1.10
LI Figure 2 Upstream tapping term as a function of L i
used relate to the value of /~ rather than the experimenter. The fit marked in Figure 2 is one that comes from the fit to the EEC data means (see equation (5)). Downstream data Many experimenters have also measured the pressure profile downstream of an orifice plate. It is less obvious what formula will fit these data. However, Teyssandier and Husainl'~ non-dimensionalised downstream distances with the dam height rather than the pipe diameter, and this proves to be very useful Given the definition that M2 is the distance between a downstream position and the upstream face of the orifice plate divided by the dam height 2L2 M2 - - (2) 1-/~ where L2 is the distance between a downstream position and the upstream face of the orifice divided by the pipe diameter. M~ and L~, are defined identically except that in each case the distance from the downstream face of the orifice plate is used. The pressure downstream of the orifice plate decreases from its corner tap value to a minimum and thereafter it increases rapidly. In Figure 3 the value of L2 at the pressure minimum is plotted as a function of /~. The same notation as in Figure I is used with the addition of data from Beitler (N) (quoted in reference 27). This figure is similar to one in Teyssandier and Miller 27, except that this one includes a simple linear fit: at the pressure minimum
L2 = 1.65(1 - ~ , ie
(3) /~2
=
3.3.
It is worthwhile also to attempt to determine the greatest change in discharge coefficient, ACre,n, as the downstream tap is moved from the corner to the minimum pressure location and the upstream tap is kept in the corner. Whereas the difference between the distance to the pressure minimum from the upstream and that from the downstream face is nearly always less than 10%, one significant feature of much of the data is that the pressure changes rapidly in the vicinity of the downstream corner; so there is a substantial difference between the value of ACmin when the downstream corner is at NI~ = 0 and that with it at M2 = 0, where values at M2 = 0 are obtained by linear extrapolation from values for small M2. Substantial work has been done 26 to examine the advantages and disadvantages of using L2 and M2 rather than L~ and M~. Using M2 has theoretical advantages 2e.29 but is more complicated. It has been 1.80 1.60 1.40 1.2o-
1.00 A 0.600.40
C
0.20~ 0.00 I I I 0.00 0.10 0.20 0.30
k
N _ N
C t ~L
~
C I ~ ~ I I l I l I 0.40 0.50 0.60 0.70 0.00 0.90 1.00 B
Figure 3 Distance to the pressure minimum
70
M. J. Reader-Harris and J. A. 5attary - The orifice p/ate discharge coefficient equation piecewise linear fits or continuous functions. The fit which is superimposed on the data is that arising from the final fit to the EEC data means (see equation (5)).
0.000
Equation (4) -0.002 ,
Fitting the data
C
-0.004 _.CE
~
I
A ~
-0.006
As previously stated, the data outside the EEC database were only used to give the correct forms of the equations. The best fit equations were obtained by fitting the mean tapping terms from all the sets of data in the 100 mm (4 in) and 250 mm (10 in) EEC database together with the Gasunie 600 mm (24 in) flange tapping terms. The 600 mm tapping terms were available but the discharge coefficients were not, because the tapping terms were measured directly and final checking would make negligible difference to the tapping terms. In fact, the 600 mm flange tap terms do not have a very large effect on the final fit. The 600 mm D and D/2 tapping terms were in excellent agreement with the 100 and 250 mm data (and so it was not necessary to include them in the fitting: the maximum difference between the 100 and 250 mm and the 600mm D and D/2 tapping terms was 0.000 53). The tapping terms derived from the 600 mm data are given in Table 2. It was decided also to attempt to determine the values of the tapping terms required to fit the US data. Only the water data were used since for the oil data the Reynolds numbers are sufficiently low that there is an effect of Reynolds number (see the following section). The differences in the values of the tap terms between those for the 250 mm (10 in) data and those for each of the other diameters have been calculated and are listed in Tables 3 and 4. The fact that the actual values of /~ and D are different from the
I ~
C
I
-0. 008
B ~
-0.010
l
I
0.00 0.10
I
I
I
0.20 0.30 o.qo
I
I
C
l
I
o.so 0.60 0.70 0.80 0.90 1.00
Figure 4 Z~Cmin(from M'2 = 0) shown that, provided appropriate restrictions are placed on plate thickness, M~ can be used without introducing significant errors. These plate thickness restrictions are satisfied by the EEC and US plates. ACmin is expected to be proportional to/~k where k is some unknown exponent. To test this theory ACm~n was plotted against/~ and the results are shown in Figure4 (from M~ = 0). It is not necessary to determine the best fit, but the following, from equation (5), gives a good fit: ACmin =
-0.0099,8 TM (from M~ = 0).
(4)
Given possible vertical and horizontal scales it is now reasonable to plot /~-~.~AC against M~ in Figure 5. Although there is some scatter it is possible to discern the shape. It is appropriate to fit the data with either
0.0001~ ÷k
V
I ~ _o.o02F~,~,x~
I
-0.004-
[]
Equation
x **
A
8o~.~ ~ ~
-0
v +
-~x
,~
d;7
+A x
'ely
~AOXA
~V
~
El
(5)
Z~ V
0.25 - 0.35 0.35 - 0.45
+ x
o.4s - o.5s o55-o65
[]
o.65
0
0
[]
o.75
-
0.75 - 0.80 0
0
-o 0 0 6 -
u
"
v
_ x
i+
-.
~
~ X []
v
A
+o+
"I#4" ~
"
~o
o-"~
0
+ ~
•
0 z~
-0.010
^
,~ ,.-,
^~
[]
x
[]
x
[]
[]
,,
[]
x
x
[]
[]
[]
x
a
0
x
0
Oa
+
[] ~ . I.- d r + Ox~ x3 ~
"-...~. + x
o
z~',"
"
rm _
o []
x
-o.oo0-
x
",~',-
0
D
[]
x 0
[]
x
0
[]
0 -
0
[] i"1
4-
[]
0
-0.012 +
-0.014 0.00
I
I
I
I
I
I
0.50
1.00
1.50
2.00
2.50
3.00
M~ Figure 5 Downstream tapping term as a function of M',
I
3.50
I 4.00
4.50
Flow Meas. Instrum. Vol 1 January 1990
71
nominal ones has been allowed for in the calculations using equation (8) to estimate changes in tapping term due to small changes in ,6 and D. In Table 4 the fact that the relative roughness varied between the pipes has also been allowed for (using equation (15)), whereas in Table 3 the effect of relative roughness has not been included (equation (18) was used to allow for the effect of RED). Slight discrepancies between equations (15) and (18) lead to the difference in the tapping term for ,6 = 0.2 and D = 50 mm being greater in Table 4 than in Table 3. The differences between 250 and 100 mm tapping terms are very similar to those from the EEC data. The 50 mm flange tapping terms differ significantly from the EEC D and D/2 tapping terms: the difference between them can be as much as about 0.0024 (about 0.4% of C) whereas at least for small ,6 they should be nearly equal. Since the values of the tapping terms for the American data were obtained by comparing data from one orifice plate and tube against those from another plate and tube, whereas those for the EEC data were all obtained by comparing data taken (sometimes simultaneously) from the same plate and tube, the American values were not used in the final fit. Upstream and downstream several types of fits were tried: the results are described in reference 26. It was decided that the upstream and downstream tapping terms should have continuous first derivatives. Table 2 Mean measured tapping terms for the 600 mm (24 in) Gasunie Data
/~
Flange 600 mm
D and D/2
0.2 0.375 0.5 0.57 0.66 0.75
-0.000 10 -0.000 18 0.000 07 0.000 23 0.000 93 0.001 95
-0.001 04 -0.001 47 - 0.000 09 0.001 30 0.005 32 0.011 75
By prescribing the upstream tapping term and optimizing the downstream one the following equation was obtained: A C = (0.0433 + 0.0712e -8sL, - 0.1145e -6L~) - 1
-0.0116(M~ - 0.52M~ 13),611.
(5)
This equation has a minimum downstream at M~ = 3.7 and a finite derivative at the origin. The curves on Figures 1, 2, 4 and 5 have been extracted from this equation. The standard deviation of the mean tapping terms about this equation is 0.0007. Fitting for low Reynolds numbers
In analysing the EEC data, no effect of Reynolds number on the tapping terms was found. However, that is not true of the US data. Figure 6 is a plot of the US data for ,6 = 0.73. The tapping terms required to bring the data for high Reynolds numbers on to a common curve are clearly inappropriate for lower Reynolds number. A reduction in the upstream pressure rise at low Reynolds numbers has been seen previously in the work of Johansen 3° and of Witte and Schr6der (quoted in References 18 and 31). A reduction in the upstream pressure rise for extremely rough pipes (4 = 0.06) at moderate Reynolds numbers was also observed by Witte Is, but this effect will be extremely small with the pipe roughnesses encountered in accurate flow measurement. The change in the maximum decrease in downstream pressure has also been measured by Johansen. Compared with the data for high Reynolds numbers, these data are very sparse. It is not possible to produce perfect low Reynolds number tapping terms, but since these terms are significant it is necessary to produce the best possible ones. Both upstream and downstream it appears reasonable to consider the ratio of the low Reynolds number tapping terms to the high Reynolds number ones. Only the maximum changes in pressure (and thus C) are
Table 3 Differences in the values of the tapping terms between those for the 250 mm US data and those for each of the other diameters (with no allowance for the effect of pipe roughness)
Other diameters
/~ 0.2
50 mm 75 mm 1O0 mm 150 mm
0.001 8 -0.000 4 -0.000 8 0.000 8
0.375 0.001 -0.000 -0.000 -0.000
4 9 3 3
0.5
0.57
0.66
0.73
0.001 6 0.000 7 0.000 2 0.000 6
0.002 2 0.001 3 0.000 5 0.000 2
0.004 8 0.004 0 0.003 2 0.000 6
0.0094 0.007 2 0.006 3 0.002 8
Table 4 Differences in the values of the tap terms between those for the 250 mm US data and those for each of the other diameters (making allowance for the effect of pipe roughness) Other /~ diameters 0.2 0.375 0.5 0.57 0.66 0.73
50 mm 75 mm 1O0 mm 150 mm
0.002 0 -0.0004 -0.000 8 0.000 8
0.001 2 -0.001 0 -0.000 4 -0.000 3
0.001 2 0.000 5 0.000 2 0.000 5
0.001 6 0.001 0 0.000 0 0.000 1
0.003 6 0.003 5 0.002 2 0.000 5
0.0070 0.006 1 0.004 5 0.002 4
M. J. Reader-Harris and J. A. 5attary- The orifice plate discharge coefficient equation
72 0.68
Pipe diameter 0.67
A + {3 0 %7
0.66
50mm 75 mm I O0 mm 150 mm 250 mm
0.65 --
u
0.64 --
O u
0.63
--
o.62
-
L
~
0.611
0.60~
•
0.59 3.50
I
I
I
I
I
4.00
4.50
5.00
5.50
6.00
6.50
Logub Rot)
Figure 6 US data for/3 = 0.73 available and so it is necessary to assume that the same ratio applies for other values of L1 and M ; besides those at the maxima. From the work of Witte and Schr6der the upstream tapping term becomes effectively constant above an orifice Reynolds number, Red, of about 80000. It is assumed that the downstream tapping term becomes constant at the same orifice Reynolds number. The ratio of the upstream tapping term deduced from the data of Witte and Schroder (E) and of Johansen (O) to the standard tapping term is then plotted in Figure 7 for Red > 3700. A reasonable fit is ACI 119 0 0 0 1 °8
ACI
- 1 -
0 14( 19 0001o8 . I~-~ed /
(7)
1.20
There is a slight suggestion that the downstream low Red term may be a function of/3 since A C I / A C h is a little closer to 1 for /3 = 0.209, but it is probably not necessary to allow for this. For Red < 3700 it is not clear what the correct formulae are. Possible formulae are included in Reader-Harris-'% but none is in fact necessary to fit the database because all the data in the database for Red < 3700 are for [3 <-- 0.375 and the tapping terms are quite small.
1.00
Complete tapping term
=,
- 0.23/--E-
)
(0)
where ACI is the value of the tapping term for low Red and ACh is the high Red tapping term for L1 = 1 in equation (5).
Putting equations (5), (6) and (7) together, the complete tapping term is
ErE ~
O. 80
L) <
Downstream it is even harder to produce an accurate tapping term for low Reynolds numbers. A C J A C h is plotted for Red > 3700 for the data of Johansen (excluding that for /3 = 0.209 for which the tapping term is small and thus the value of the ratio is both less certain and less important) in Figure 8. A reasonable fit is
/o
0.60
A C = (0.0433 + 0.0712e -8.sL, - 0.1145e -6z,)
O. 40
0.20 0.0
/ I
3,00
3.50
x (1 - 0 . 2 3 A ) -
o o"
I
4100
4.50
,64
- 0.011 6(M~. - 0.52M~, 1~)
Equation (6) I
/34 1 -
x (1 - 0.14A)/3 TM,
5.00
Log Re d
Figure 7 Effect of low Reynolds number on upstream tapping term
where
A =
1,9000g° ~ ReD I
(8)
2L~ and M~ - - -
1-/3
Flow Meas. Instrum.
73
Vol 1 January 1990
data. In all cases the flange tapping data were convetted to corner tapping data using equation (8). In Figure 9, the data for ,6 = 0.1 and those for/3 = 0.2 lie on top of each other, and those for ,6 = 0.375 are slightly higher. So for very low /3 it appears that C depends only on Red. It then seems reasonable to suppose that for very low/3
1.20
0
1.00
I
o./o°-
0.80 ,,,¢
<] 0.60
J
~o
o
0.40
C = al + b(lO6/Red) n
0.20 0.00 3.00
= al + b(lO6fl/Reo) n.
Equation (7) I 3.50
I
4.00 Log Re d
I
4.50
5.00
Figure 8 Effect o f l o w Reynolds number on downstream tapping term
The C~ term C:~ is the discharge coefficient for corner tappings for infinite Reynolds number; it increases with ,6 to a maximum near /3= 0.55 and then decreases increasingly rapidly. Thus an appropriate form for C~ is C~ = a l + a2/~ mt + a3,6 m:.
(9)
With the slope term in either equation (14) or equation (17) it is possible to obtain a good fit with many pairs of values of ml and m2. Taking m~ = 2 and m2 = 8, while not necessarily optimum, enables a good fit to be obtained.
The slope term Reader-Harris and Keegans 32 found that, for a fixed pipe Reynolds number, as the pipe roughness changed so the change in discharge coefficient was approximately proportional to/~4A;~, where A). is the change in the pipe friction factor. This led Reader-Harris to suppose that the form of the equation for the discharge coefficient (omitting tapping terms) might be C = C~ + cl/~l;~,
(10)
where / is approximately 4. Rapier 33 made experimental measurements in very smooth pipes and as a result of his work also proposed that C was best expressed as a function of friction factor. However, equations of the form of equation (10) do not give good agreement with experiment for small ,6, since they then give C almost constant. The problem is due to the need for an additional term in equation (10). To establish what that term is it is necessary to consider what type of equation would be appropriate for small ,6 since it is for small ,8 that equation (10) seems to be most inappropriate. It seems reasonable that for low /~ the discharge coefficient should depend only on throat Reynolds number, Red C = al + fiRed)
(11)
To check this theory, C was plotted against log Red (Figure 9) for ,6 less than or equal to 0.375 using all the EEC and American flange tapping data. So that all the data included in this analysis had been collected with the same type of tappings, flange tapping data from the EEC were used rather than corner tapping
(12)
Since approximate equations for low and high /3 have now been obtained the most obvious general equation for C is given by C = C~ + b(lO6,6/Reo) ~ + cl,61;(.
(13)
Reader-Harris 34 tested this equation against flange tap data using a tapping term with no dependence on Red, and used data taken by several experimenters in rough pipe to establish the truth of the friction factor term. One problem is that when the low Red tapping terms in equation (8) are included, the fit against ~. is not as good as it was without them. But it is not surprising that only a poor fit should be obtained for Red < 80000 since the physical causes of the change in the tapping terms may also result in a change in the friction factor term. Such a change could not be detected in the rough pipe data in reference 34 since all these data were for Red > 80000. So it seems reasonable to suppose that the friction factor term should include a low Red term of the same form as that in the tapping terms. It is also necessary to make provision for transition from turbulent to laminar flow below ReD = 3500 since, except for very low /3, the slope of the data for ReD < 3500 is very different from that for ReD > 3500. This is because the velocity profile changes very rapidly as the flow changes from turbulent to laminar. Such a change has to be included in the friction factor term since this represents the effect of velocity profile. The slope term then becomes Ccr - C~ = b(lO°i~/ReD) ~ + (C! + c2A),6 I x m a x {A., 0 . 1 6 4
-
35(Reo/10°)}
(14) where A is as defined in equation (8), and Ccr is the discharge coefficient for corner tappings. Performing a least-squares fit of all the data in the database, the following equation for C= and slope is obtained: Ccr = 0.5966 + 0.0200,62 - 0.279,68 + 0.000 522(106/~/ReD) °'z + (3.974 + 1.044A),64 x max {~., 0.164 - 35(ReL)/10°)}.
(15)
The tapping terms in equation (8) were used to convert flange and D and D / 2 tapping data to corner tapping data. There is a problem with this equation: the use of friction factor as an extra variable would be unpopular. However, although ~. may appear to be a rather
M. J. Reader-Harris and J. A. Sattary - The orifice plate discharge coefficient equation
74 0.65
0.64
O 0.I Z~ 0.2 -I,. 0.375
0.63
tO
"~ 0 . 6 2 -
~2 u
P t-
0.61
--
0.60
--
0.59
--
I
0.58
3.00
3.50
1
4.00
I
4.50
I
I
I
I
5.00
5.50
6.00
6.50
7.00
LOglo Re d
Figure 9 Discharge coefficient for small fi inconvenient variable, it is possible to approximate it by a simple function of ReD if a value of relative pipe roughness, kin~D, is assumed. For instance,, if km/D = 0.00002, Z is always within 0.0005 of ~.' for ReD > 4000 if ,,1" = 0.008 59 + 0.809 26/Ren°~'L
C = 0.5960 + 0.0313,6 =
0.225,65
C= term
1
+ (0.0166 + 0.0035A),64 term max {(10"/ReD) °-4, 37.6 -- 8000(Ren/10~')}I Sl°pe + (0.0433 + 0.0712e -~L' - 0.1145e -~'L')
(17) (1 -
Including an appropriate velocity profile term for ReD < 3500 and fitting over the complete database gave the following equation for C= and slope:
0.23A)fi4/(1 - ,64)
-0.0116(M',
-
0.52M!,
x 1 - 0.14A),6 TM,
Ccr = 0.5960 + 0.0313/32 - 0.225,65
1.;i)
Upstream tap term
Downstream tap term (19)
+ 0.000 526(10('fl/Reo) °.7
where
+ (0.0166 + 0.003 5A)fl 4 x max {(106/Ren) °4, 37.6 - 8000(RED~10~)}.
-
+ 0.000 526(10~,6/ReD) °:
(16)
So in equation (14),). was replaced by an inverse power of Ren, and the slope term for ReD > 3500 becomes
CCT- C~ = b(lO~,6/Reo) " + (cl + c2A),6t(lO~'/Ren) n'.
and fitting the EEC/US database, led to the equation accepted in New Orleans in November 1988 and is given in equations (8) and (18):
M',--(18)
The standard deviation of the data about this equation is 0.285%. The standard deviation increases for small D, large fl and low ReD. When the 600 mm (24 in) data have all been collected and included in the database the equation will be refitted, but those data will not change the form of the equation. Conclusions
The work reported here, which consisted of both investigating the physics of flow through orifice plates
2L~ 1-,6
A = ( l ~' .000fl/°~ ~9 e o L1 is the distance between the upstream tapping and the upstream face of the orifice plate divided by the pipe diameter, and L~ is the distance between the downstream tapping and the downstream face of the orifice plate divided by the pipe diameter. The terms in A are only significant for small throat Reynolds number; M~ is in fact the distance between
Flow Meas. Instrum.
Vol I January 1990
the downstream tapping and the downstream face of the plate divided by the dam height.
75
13
Acknowledgement
14
This paper is published by permission of the Director, National Engineering Laboratory, and is Crown copyright. The work reported here was supported by DGXll of the European Economic Community and by the Standards, Quality and Measurement Advisory Committee of the Department of Trade and Industry.
15
References 1 SIolz, J. 'A universal equation for the calculation of discharge coefficient of orifice plates'. In: FIomeko 1978 - Flow Measurement of Fluids, H. H. Dijstelbergen and E. A. Spencer (Eds). North-Holland Publishing Co, Amsterdam (1978) 519-534 2 International Organisation for Standardization. ISO 5167 'Measurement of fluid flow by means of orifice plates, nozzles and venturi tubes inserted in circular crosssection conduits running full'. International Organization for Standardization, Geneva (1980) 3 Hobbs, J. M. 'Experimental data for the determination of basic 100 mm orifice meter discharge coefficients (European programme)'. Report EUR ]0027, Commission of the European Communities, Brussels, Belgium (1985) 4 Hobbs, J. M. and Satlary, I. A. 'Experimental data for the determination of 100 mm orifice meter discharge coefficients under different installation conditions (European programme)'. Report EUR 10074, Commission of the European Communities, Brussels, Belgium (1986) 5 Hobbs, I. M., Sattary, I. A. and Maxwell, A. D. 'Experimental data for the determination of basic 250 mm orifice meter discharge coefficients (European programme)'. Report EUR 10979, Commission of the European Communities, Brussels, Belgium (1987) 6 Hobbs, J. M., Satlary, I. A. and Maxwell, A. D. 'Experimental data for the determination of 250 mm orifice meter discharge coefficients under different installation conditions (European programme)'. Report EUR 10980, Commission of the European Communities, Brussels, Belgium (1987) 7 Hobbs, I. M. 'The EEC orifice plate project. Part I Traceabilities of facilities used and calculation methods employed'. Progress Report No PR5: EUEC/17. National Engineering Laboratory, East Kilbride, Glasgow (1987) 8 Hobbs, J. M. 'The EEC orifice plate project. Part II: Critical evaluation of data obtained during EEC orifice plate tests'. Progress Report No PR5: EUEC/17. National Engineering Laboratory, East Kilbride, Glasgow (1987) 9 Hobbs, J. M. 'The EEC orifice plate project: tables of valid data for EEC orifice analysis'. Progress Report No PR7: EUEC/17. National Engineering Laboratory, East Kilbride, Glasgow (1987) 10 'Coefficients of discharge for concentric, square-edged, flange-tapped orifice meters: equation data set - supporting documentation for floppy diskettes'. American Petroleum Institute, Washington, DC (1988) 11 Brillon, C. L., Caldwell, S., and Seidl, W. 'Measurements of coefficients of discharge for concentric, flange-tapped, square-edged orifice meters in white mineral oil over a low Reynolds number range'. American Petroleum Institute, Washington, DC (1988) 12 Whetstone, J. R., Cleveland, W. G., Baumgarten, G. P. and Woo, S. 'Measurements of coefficients of discharge for concentric, flange-tapped, square-edged orifice meters in water over a Reynolds number range of
16 17
18 19 20 21 22
23
24 25
26
27
28 29
30 31
32
1000-2 700000'. NIST Technical Note (TN-1264). American Petroleum Institute, Washington, DC (1988) '600 mm orifice plate project. Vol Xll. Discharge coefficients'. TP/T 88.R.2052, Gasunie, Groningen (December 1988) Herning, F. and Bellenberg, H. Neue Versuche mit Normblenden. Brennstoff-W~rme-Kraft 12 (1960)89-96 Witte, R. Die StrOmung durch Diisen und Blenden. Forschung auf dem Gebiet des Ingenieurwesens 2 7 (1931) Herning, F. Untersuchung des Druckverlaufs an Normblenden grosser Rohrweiten. Brennstoff-W~rme-Kraft 7 (1955) 303-307 Sens, M. Etude Exp~rimentale des Variations de la Pression au Voisinage d'un Diaphragme Normalis~ selon ISO/DIS/5167. ISO/TC30/SC2 (France-l l) 85 (March 1977) Engel, F. V. A. Durchflu,Smessung in Rohrleitungen. Bericht i.iber ein Symposium in East Kilbride (Schottland). Brennstoff-W~rme-Kraft 13 (1961) 125-133 Teyssandier, R. G. and Husain, Z. D. Experimental investigation of an orifice meter pressure gradient. Trans. ASME. J. Fluids Engng 109 (1987) 144-148 Private communication from J. Stolz regarding data collected in Stuttgart (1988) Darrow, R. R. (Ed). 'Investigation of orifice meter installation requirement'. Interim Research Report No 2, American Gas Association, (1954) Bean, H. S., Benesh, M. E. and Buckingham, E. Experiments on the metering of large volumes of air. Bureau of Standards, Washington. Journal of Research 7 (1931) 93-145 Marlin, C. N. B. 'Effects of upstream bends and valves on orifice plate pressure distributions and discharge coefficients'. NEL Report No 702. National Engineering Laboratory, East Kilbride, Glasgow (1986) Engel, F. V. A. Fluid-flow measuring devices. Engineering 176 (1953) 6-10 and 35-37 Certificate of Calibration on determination of the longitudinal pressure profile through six thin orifice plates mounted in the EEC 100 mm assembly using AGA tappings throughout for the European Economic Community DGXll Research, Science and Education. National Engineering Laboratory, East Kilbride, Glasgow (1988) Reader-Harris, M. J. 'The tapping terms in the orifice plate discharge coefficient formula'. Report to be submitted to the European Economic Community DGXII Research, Science and Education. National Engineering Laboratory, East Kilbride, Glasgow (1990) Teyssandier, R. G. and Miller, R. W. 'A new coefficient equation for concentric sharp-edged orifice: Part I Fundamental considerations'. ASME paper 79-WA/FM-3. American Society of Mechanical Engineers, New York (1979) Pfeffer, W. Untersuchungen an Blenden mit Eckanbohrungen. Brennstoff-W~rme-Kraft 20(1968) 108-115 Husain, Z. D. and Teyssandier, R. G. 'The effects of plate thickness and bevel angle in a 150 rnm line size orifice meter'. International Conference on Flow Measurement in the mid-80s. National Engineering Laboratory, East Kilbride, Glasgow (1986) Johansen, F. C. 'Flow through pipe orifices at low Reynolds numbers'. Aeron. Res. Committee, Report and Memo No 1252. HM Stationery Office, London (1930) Engel, F. V. A. 'New interpretations of the discharge characteristics of measuring orifices - an approach to improved accuracy'. ASME paper 62-WA-237. Amer. Soc. of Mechanical Engrs. New York (1962) Reader-Harris, M. I. and Keegans, W. 'Comparison of computation and LDV measurement of flow through orifice and perforated plates, and computation of the effect of rough pipework on orifice plates'. In: Proc. of
76
M. J. Reader-Harris and J. A. Sattary - The orifice plate discharge coefficient equation
the Int. Symp. on Fluid Flow Measurement, Washington, DC, Paper 12.1 (November 1986) 33 Rapier, A. C. 'The extrapolation of orifice calibrations to high Reynolds numbers'. In: Proc. of the Fluid Mechanics Silver Jubilee Conf. National Engineering Labora-
tory, East Kilbride, Glasgow, Paper 4.3 (1979) 34 Reader-Harris, M. J. 'A new type of formula for the orifice-plate discharge coefficient'. In: Proc. of the North Sea Flow Metering Workshop, National Engineering Laboratory, East Kilbride, Glasgow (1988)