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Guidelines for measurement errors caused by buckled orifice plates.

This article documents research on flow measurement errors caused by orifice plates that have undergone permanent, plastic deformation. Tests were performed on orifice plates in both 6- and 10-inch diameter line sizes. The plates were flow calibrated in senior orifice fittings flowing natural gas, both before and after being permanently deformed. Results were used to assess measurement error as a function of plate deflection angle, orifice bore diameter, and other critical dimensions. Estimates of the flow measurement error for the bent plates were also predicted using the models published by Jepson and Chipchase (1975), and by Mason, Wilson, and Birkhead (1975).

It was determined that models that include explicit measurements of the bent plate bore diameter can best predict the error in flow rate associated with bent plates. Typical disagreements between the Jepson and Chipchase (J&C) model and observed experimental errors were within [+ or -] 0.6% of flow rate. It was also found that adequate measurements of deflected plate dimensions may be made with hand tools and used with this model to predict flow measurement errors.

Previous Work

When a pressure differential is applied across an orifice plate, the plate deflects due to the applied force. AGA Report No. 3 (2000) specifies guidelines for minimum orifice plate thickness and maximum pressure differential. When these guidelines are followed, the orifice plate bending stress is less than the elastic limit, and the plate retains its original shape when the pressure differential is relieved. These guidelines also limit the error in the orifice discharge coefficient due to elastic plate bending to the order of 0.1%. If the bending stress exceeds the elastic limit, the orifice plate will become permanently bent or buckled.

Previous investigators have tested permanently bent orifice plates to determine the associated measurement error, but the amount of available test data is limited. For a 4-inch diameter orifice meter, Burgin (1971) reported that plate deflections of 1/8 inch gave an under-measurement error between 2.4% and 3.1%. Deflections of 1/4 inch increased the magnitude of the under-measurement error to 8-9%. J&C developed a theoretical model to predict the flow measurement error from elastic and/or plastically deformed plates. They confirmed the model using test data on 8-inch orifice plates, plastically deformed to give edge deflections ranging from 1/32-inch to 1/2 -inch. Good agreement was found between the theoretical calculations and experimental measurement error for plates with beta ratios of [beta] = 0.70. The agreement was worse for plates with [beta] = 0.20, however. They also concluded that permanent deformation in the flow direction, such that the leading edge of the bore is deflected by more than 1[degrees] from the unbent position, could result in under-measurement of the flow by several percent.

Yeyssandier and Chisman (1984) tested 8- and 10-inch bent plates with original [beta] ratios of 0.5 to 0.6 and deflections of up to 0.54 inch in the direction of flow. Their study also concluded that measurement errors become more negative with increasing deflection angle, but the largest under-measurement error observed in the study was 3.3% for a deflection of about 1/2 inch. Ting (1993) flow calibrated 4- and 6-inch orifice plates with beta ratios between 0.25 and 0.75, then recalibrated the same plates after they had been bent over a mandrel to deflection angles between 1[degrees] and 10[degrees]. Except for [beta] = 0.25, the measurement error increased with increasing [beta] and deflection angle. From his test results, Ting concluded that bent plates might cause under-measurement errors up to 4.5%, less than the errors reported by Burgin for similar deflection angles.

Unfortunately, the results of these studies are contradictory, and a need has been identified for more experimental data to verify the reported trends. More experimental data are particularly needed to validate calculational models for measurement errors created by bent orifice plates. Such information could provide natural gas companies with a potential method to more accurately estimate the errors associated with bent orifice plates discovered in the field. To this end, tests were performed at the Metering Research Facility (MRF) located at Southwest Research Institute (SwRI[R]), on orifice plates in both 6- and 10-inch diameter line sizes, before and after permanent deformation. This article describes the tests and, based on the results, presents recommendations for predicting measurement errors of buckled orifice plates.

Review Of Formulas

Formulas commonly used by the natural gas industry to predict measurement error caused by bent plates have been published by J&C (as mentioned above) and by Mason, Wilson, and Birkhead (MW&B). These formulas are derived from the basic equation for the flow rate through an orifice meter, and estimate errors in measured flow rate due to changes in physical dimensions of the plate and the downstream flow field. The J&C paper includes separate formulas to predict the effects of both elastic (temporary) and plastic (permanent) deformation. The latter is described here. Notably, the original derivation by MW&B assumes only elastic deformation, but it is commonly used for predicting errors of plastically deformed plates.

Jepson And Chipchase Formula

The formula developed by J&C to estimate errors in mass flow due to permanent, plastic deformation of an orifice plate is:

[DELTA][q.sub.m]/[q.sub.m] = -1/[1 - [C.sup.2.sub.c] [[beta].sup.4]] ([DELTA][C.sub.c]/[C.sub.c] + 2[DELTA]d/d) (1)

where

[q.sub.m] = mass flow rate through the orifice meter

[DELTA][q.sub.m] = error in mass flow rate caused by the deflected plate (same units as [q.sub.m])

[C.sub.c] = contraction coefficient, the ratio of the jet area at the vena contract a downstream of the bore to the area of the bore itself (dimensionless)

[beta] = d/D = beta ratio of undetected orifice plate bore diameter to meter tube diameter (dimensionless)

d = undetected orifice plate bore diameter (inches)

D = meter tube diameter (inches)

[DELTA]d = d' - d = change in orifice bore diameter due to plate deflection (inches)

[DELTA][C.sub.e] = [C.sub.c]'- [C.sub.c] = change in contraction coefficient due to plate deflection (dimensionless)

The dimensions of the undeflected and deflected orifice plates used in this equation are illustrated in Figure 1. Values of the contraction coefficient [C.sub.c] in Equation 1 are determined by the following correlation, where the parameters [K.sub.i] are given in Table 1 and [theta] is the angle (in radians) between the downstream face of the orifice plate and the meter tube wall. [C.sub.c]' is computed from Equation 2 using the angle [theta]' after plate deflection, while [C.sub.c] is computed before plate deflection using [theta] = [pi] / 2. Values of [K.sub.i] may be interpolated from the table as necessary.

[C.sub.c] = [K.sub.1] + [K.sub.2][theta] + [K.sub.3][[theta].sup.2] (2)

[FIGURE 1 OMITTED]

Mason, Wilson And Birkhead Formula

MW&B began with the same theory and approach as J&C to predict flow measurement errors caused by orifice plate deformation. Their published formula is different in two respects, however: (1) their approximation for the deflection angle of the bent plate assumes elastic, not plastic, plate deformation, and (2) a different correlation is used for the contraction coefficient, [C.sub.c]. The MW&B published formula takes the form:

[DELTA][q.sub.m]/[q.sub.m] = [phi] - [phi]'/[phi] + 2(1 - 1/[beta])(1 - sin [theta]'), (3)

where [phi] for the undeflected plate is determined by the expression

[phi] = [C.sub.c]/[square root of 1 - [C.sup.2.sub.c][[beta].sup.4]] (4)

and [phi]' for the deflected plate is determined using the [beta] ratio before deflection, but the contraction coefficient [C.sub.c]' after deflection:

[phi]' = [C'.sub.c]/[square root of 1 - [([C'.sub.c]).sup.2] [[beta].sup.4] (5)

The downstream plate angle (in radians) is given by the approximate geometric relationship:

[theta]' [approximately equal to] [pi]/2 - arctan [2[delta]/D - d] (6)

Finally, the contraction coefficient correlation, taken from von Mises (1917), is given by Equation 7, with values of Mi interpolated from Table 2. As with the J&C formula, [C.sub.c]' is computed from Equation 7 using the angle [theta]' after plate deflection.

[C.sub.c] = [M.sub.0] + [M.sub.1][theta] + [M.sub.2][[theta].sup.2] + [M.sub.3][[theta].sup.3] (7)

Test Conditions And Test Equipment

The purpose of this research was to systematically gather data on buckled orifice plates and use the data to evaluate methods of predicting measurement errors of bent plates. To this end, baseline tests were performed on a series of new, unbent plates before deformation to record their discharge coefficients. A series of plates of the same line diameter and [beta] ratio were then bent in successive amounts, and the deformed plates were retested to assess the change in discharge coefficient and the resulting measurement error. Tests were performed on each plate before and after deformation at up to three flow rates, at a single line pressure of 415 psia.

Plate dimensions were also chosen to produce data on a useful range of flow rate errors while avoiding extremely low or extremely high differential pressures across the plates. Measurements were made of critical plate dimensions before and after deformation, using both hand tools and a computer-controlled coordinate machine. This allowed a determination of whether common instruments carried by field personnel would be sufficient to estimate measurement errors by bent plates.

The flow tests were performed in the High Pressure Loop (HPL) at the MRF located at SwRI. Orifice plates in both 6- and 10-inch senior orifice meter runs were installed as shown in Figure 2 and Figure 3, respectively. These sizes allowed both 1/4-inch thick plates and 1/8-inch thick plates to be included in the work to assess any scaling effects. The use of 10-inch meter testing also provided critical data on a line size common to the natural gas industry for which little bent plate data previously existed. The meter rims were inspected before tests for conformance to AGA-3 specifications of surface roughness, tap location, and geometry. Both rims were installed with flow conditioner arrangements to yield discharge coefficients for unbent plates in agreement with the Reader-Harris/Gallagher correlation of AGA-3. The 10-inch meter run was installed in the HPL well upstream of the 6-inch meter run to allow testing of plates in both meters simultaneously.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Kelley Instrument Machine provided six plates of each line size (D = 6 inch and 10 inch) and each nominal [beta] ratio (0.2, 0.35, and 0.5), for a total of 36 orifice plates. All plates were tested in the undeformed state to record their baseline discharge coefficients. Three plates of each diameter/[delta] ratio combination were then bent in amounts of 1/16 inch, 1/8 inch, and 3/16 inch, the deflection amount [delta] being measured in the direction of flow as shown in Figure 1. The original upper limit on [delta] of 3/16 inch was based on the width of the openings for inserting plates into senior fittings. After initial tests, it was found that plates with 1/4-inch and 5/16-inch deflections could be inserted into the 10-inch senior fitting, and additional tests were performed on 10-inch plates with [beta] ratios of 0.35 and 0.5 deflected by these larger amounts.

Deflection Of Bent Plates

Plates used in the tests were deformed into shapes similar to those of buckled orifice plates found in the field so that test results would be as useful as possible in predicting flow rate errors under field conditions. Orifice plates bent in service were provided by two of the project sponsors for inspection. The dimensions and shapes of the field-bent plates were measured and found to fall into three categories, as shown in Figure 4: (1) "conic" plates with flat surfaces, (2) "parabolic" plates with the radius of curvature facing upstream, and (3) "funnel-shaped" plates with the radius of curvature facing downstream. It was also found that a deflection angle of 4[degrees] marked the approximate transition between plates bent into conic shapes and plates buckled into parabolic shapes. Accordingly, it was decided that plates to be deflected at angles above 4[degrees] should be bent into a parabolic shape, and plates with smaller deflection angles should be bent into a conic shape.

Kelley Instrument Machine built a series of fixtures, similar to conical mandrels, over which the plates were bent manually with a machine press into the appropriate shapes. These fixtures were designed to avoid damage to the leading edge of the bore during the deflection process. The plates were first inspected for conformation to AGA-3 standards, and then calibrated in the MRF HPL in their undeformed condition. A series of 18 plates was then sent to Kelley to be deflected in controlled amounts. SwRI staff provided guidance to Kelley on the desired plate deflection angles and shapes. After bending, Kelley staff allowed the plates to rest for 24 hours and "spring back" to their final shape. The plates were then inspected again, and returned to SwRI for testing.

[FIGURE 4 OMITTED]

Inspection Of Plates

The models for predicting bent-plate measurement errors require the dimensions of the deformed plate as input. Plate dimensions were measured using both machinists' tools, to simulate common field practices, and using a computer-controlled coordinate machine owned by project participant Southern Star Central Gas Pipeline. Measurement errors estimated using both sets of dimensions were compared to assess the effectiveness of the methods.

Southern Star performed checks on the orifice plates before testing and after deformation to confirm that the plates conformed to AGA-3 specifications. Plate characteristics checked for conformity included orifice edge sharpness and integrity; plate surface roughness; bore diameter, d, and bore roundness; plate thickness, E, and bore thickness, e; departure from flatness, [delta]; and bevel dimensions. The coordinate machine used to measure many of the plate dimensions used a ruby-tipped stylus to measure coordinates of an orifice plate surface or edge in three-dimensional space, as illustrated in Figure 5.

Differences in vertical coordinates (shown in example in the figure) were used to calculate deflection angles and deflection depths. Any slope in the granite table was compensated for in determining plate deflection and flatness. The radius of the ruby tip itself was also subtracted from measurements of the plate outer edge and bore edge to obtain the plate and bore diameters, respectively.

Hand measurements of the deformed plates were deferred until after tests of the plates in the HPL were completed to avoid damaging the plates by handling. SwRI staff and project sponsors tried several hand methods to measure the required dimensions, and final recommendations for measurement methods were made based on ease of use and accuracy. AGA-3, Part 2 specifies three different methods of measuring the plate deflection, [delta], two of which were effectively used here. In one approach, a rigid straightedge was placed across the plate, and straight or angled feeler gauges were inserted under the straightedge to measure the plate deflection at the bore edge and the location of the meter tube wall. The difference in these two measurements was then taken as the value of [delta]. In the other approach, a Vernier caliper depth probe was used to measure the deflection at the bore edge and the meter tube wall location, and the difference was again taken as [delta]. Illustrations of both methods are shown above in Figure 6.

[FIGURE 6 OMITTED]

Bull-nosed calipers were found to be fastest for measuring plate diameters, bore diameters, and plate thicknesses, and are strongly recommended over sharp-edged calipers, which may damage the bore leading edges. A swing-arm protractor was used to measure all plate deflection angles. A machinists' square was laid across the upstream edges of the plate so that its vertical arm defined the plane of the pipe wall. The face of the protractor was then placed against the vertical arm, and the angle of the swing arm resting on the bore edge (as shown in Figure 7) was used to determine the downstream plate angle, [theta]'.

[FIGURE 7 OMITTED]

Comparisons of hand measurements to dimensions taken with the coordinate machine demonstrated that the two methods produced statistically similar results, and that hand measurements are suitable for predicting bent orifice plate measurement errors. A general recommendation is to use templates for centering the plates and defining the axes along which measurements are made; simple paper templates, shown in the figures above, are sufficient for this purpose.

Test Results

All plates were first flow-calibrated in the meter runs as received from the manufacturer. Calibrated values of the orifice coefficient, [C.sub.d], were compared to values calculated from the Reader-Harris/Gallagher (R-G) equation. While the goal was to study the relative effect of deflection on [C.sub.d] and measurement accuracy, it was felt that the use of orifice plates conforming to the R-G equation before deformation would improve confidence in the results. Next, a subset of 18 orifice plates conforming to the R-G equation was sent to Kelley to be deformed to successive bore deflection angles. These plates consisted of six combinations of line diameter and [beta] ratio (D = 10-inch Schedule 80 and 6-inch Schedule 40; [beta] = 0.2, 0.35, and 0.5), with three plates of each geometry deflected in successive amounts of [delta] = 1/16 inch, 1/8 inch, and 3/16 inch. In follow-on tests, four more 10-inch plates were deflected, with four combinations of [beta] ratio ([beta] = 0.35 and 0.5) and deflection amounts ([delta] = 1/4 inch and 5/16 inch). The bent orifice plates were again calibrated at the same line pressure and flow rates to determine their new discharge coefficients. Differences in [C.sub.d] were translated into corresponding errors in flow rate, assuming that the discharge coefficient from the unbent plate would be used with the differential pressure measured on the same plate after buckling to compute flow rate in the field.

The observed measurement errors for the 6-inch plates and for most test conditions with the 10-inch plates were negative, meaning that the bent plates would under-register the actual flow rate. The magnitude of the under-measurement increased with increasing deflection angle, as expected. Data for the 10-inch, [beta] = 0.2 plates showed small under-measurement errors, and in some cases, small over-measurement errors; this trend is likely due to the small differential pressures involved in the tests of these plates. Many of the observed shifts in [C.sub.d] were less than the uncertainty in the R-G correlation, so the Student's t-test was applied to the data to evaluate the significance of the changes. The majority of the changes in [C.sub.d] with plate deflection were statistically significant at the 95% confidence level; most of the statistically insignificant changes occurred with the 10-inch diameter, [beta] = 0.2 plates. Changes in [C.sub.d] were also found to be an order of magnitude more significant than changes expected from the reproducibility of conditions in the MRF HPL.

Each predictive model was used with three sets of plate dimensions as input. One set of calculations used the nominal dimensions of the plates, such as the dimensions specified during manufacture or the requested deflection angles, as input. The second set of calculations used the dimensions measured by hand with machinists' rules, T-squares, protractors and calipers. The third set of calculations (labeled "SSCGP") used measurements from the Southern Star coordinate machine.

[FIGURE 8 OMITTED]

Figure 8 compares the measurement errors observed in tests to the errors predicted by the original MW&B model. Predicted and observed measurement errors are plotted versus plate serial number. The serial number begins with "10" or "6" to indicate the line diameter, while the two digits before the dash indicate the bore diameter in tenths of an inch. Except for plates 1018-2 and 1018-3, the plate deflection increases with the digit after the dash. The blue vertical bars represent the range of measurement errors observed for a given plate over all Reynolds numbers at which it was tested.

In every comparison, the MW&B model predicts a more negative error in flow rate than was observed in the MRF experiments. Except for the 1018-X plates (D = 10 inch, [beta] = 0.2), the discrepancy between observed and predicted flow rate errors consistently becomes worse as the deflection amount increases. In the worst cases, the MW&B model predicts a flow measurement error of about -3.5%, while the observed error was -1.0% to -1.5%. Many of the predicted errors from the MW&B model were about twice the magnitude of the observed errors. This is consistent with comparisons of the MW&B model to data collected by Ting (1993), as reported by Morrow et al. (2002). The incorrect predictions are attributed to an approximation in the MW&B model for determining the deflected bore diameter, d', from the undeflected bore diameter, d, and the deflection depth, [delta].

During the research, a proprietary version of the MW&B model was made available to SwRI by a third party for testing. This version eliminates assumptions about the shape of the bent plate, and uses direct measurements of the bent plate dimensions. SwRI was allowed to evaluate the approach alongside the J&C method and the original MW&B method. The comparisons are shown in Figure 9. As with the original MW&B model, use of the nominal dimensions predicts flow rate errors more negative than those observed in tests.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Measured dimensions (by hand or by coordinate machine) lead to predicted errors that either fall within the range of observed values, or in most cases, deviate from the largest observed values by no more than [+ or -] 0.7%. This agreement is attributed to the proprietary changes to the original MW&B model that eliminate the assumption of elastic plate deformation and use actual plate measurements. For plates 1033-4, 1033-5, 1047-4, and 1047-6, the computer analysis of coordinate machine data was unavailable, and the raw coordinate data were processed using an analytic geometry method that may be responsible for the significant disagreement of the SSCGP data with the experiment.

Finally, Figure 10 compares the measurement errors from the SwRI tests to the predictions by the J&C model. The trends are very similar to those for the modified MW&B model. Using actual measurements of the bent plate dimensions again leads to the best predictions of observed measurement errors. When measured dimensions are used, predicted errors deviate from most observed values by no more than [+ or -] 0.3% of flow rate. In particular, the use of hand-measured dimensions with the J&C model yields predicted flow rate errors in very good agreement ([+ or -] 0.3%) with the most negative experimental flow rate errors (the lower ends of the error bars). Some users consider measurement errors of -0.5% or larger in magnitude to be significant for custody transfer applications. Using hand-measured dimensions with the formula also correctly predicts the deflection amounts at which flow errors exceed this limit, namely, [delta] = 3/16 inch for D = 10 inch, and [delta] = 1/8 inch for D = 6 inch. Clearly, the advantage of the J&C model (and of the proprietary version of MW&B) is in its use of measured dimensions of the deformed plate, particularly the deflection angle, deflection depth, and deformed bore diameter.

Note that these tests included bent orifice plates of different thicknesses: the standard 6-inch orifice plate is 1/8-inch thick, while 1/4-inch thick plates are used in 10-inch orifice meters. However, the J&C model and the proprietary MW&B model predicted flow rate errors for both line sizes with similar accuracy, and both predict the correct error trends with line diameter and deflection distance. It is concluded that the use of measured dimensions accounts for scaling effects and for the lack of geometric similarity between the 6- and 10-inch orifice plates.

A brief analysis concluded that the difference in predicted flow rate errors when hand tools are used to measure bent plate dimensions instead of a coordinate machine is acceptably small (no more than [+ or -] 0.24% of flow rate at the 95% confidence level). Measurements of bent plates by field personnel with hand instruments will lead to acceptable accuracy in predicting measurement errors of bent plates.

Conclusions And Recommendations

Based on the findings of this work, two recommendations are made for predicting flow rate errors caused by buckled orifice plates:

First, measurements of bent orifice plates may be made with hand tools, using the methods described above, to accurately predict flow rate errors made by the plates in service. Machine-assisted measurements may be used if practical, and will produce similar results.

Second, models used to predict bent plate flow rate errors should include measured dimensions of the bore diameter (and deflection angle, if required) of the bent plate to be accurate. The original model published by MW&B does not allow for this, and is not recommended. The model published by J&C more accurately models changes in bore diameter and flow conditions downstream of the bent plate, and is recommended for general use. It was found to predict all flow rate errors observed on 6- and 10-inch orifice plates in this work to within [+ or -] 0.6% of flow rate, and it predicted the most negative flow rate errors to within [+ or -] 0.3% of flow rate.

ACKNOWLEDGMENTS:

This article is based on a presentation at the AGA Operations Conference May 14-16, 2008, in Phoenix, AZ. The author wishes to thank the sponsors of this Joint Industry Project for their guidance and participation: Atmos Energy, CenterPoint Energy, ConocoPhillips, El Paso Energy, Enbridge Energy, Enterprise Products, Kinder Morgan, National Fuel Gas, and Southern Star Central Gas Pipeline. A key contribution was made to the project by Tom Kelley and the staff of Kelley Instrument Machine, who donated the orifice plates used in the project and provided the equipment and manpower to bend the plates. Brad Massey and the staff of Southern Star Central Gas Pipeline are thanked for their support in the use of their coordinate machine to measure plate dimensions before and after deflection. Daniel Industries is gratefully acknowledged for providing its historical data on bent orifice plate errors. Last, but not least, the author would like to thank the Metering Research Facility technical staff of Roland Martinez, Michael Robertson, and John Sullenger, without whose efforts and advice this project would not have been possible.

REFERENCES:

American Gas Association (2000), AGA Report No. 3, Orifice Metering of Natural Gas and Other Related Hydrocarbon Fluids, Part 2: Specification and Installation Requirements, Fourth Edition, Arlington, VA, April 2000.

Burgin, E. J. (1971), "Factors Affecting Accuracy of Orifice Measurement (Primary Element)," Proceedings of the 1971 International School of Hydrocarbon Measurement.

Jepson, J, and R. Chipchase (1975), "Effect of Plate Buckling on Orifice Meter Accuracy," Journal of Mechanical Engineering Science, Vol. 17, No. 6, pp. 330-337.

Mason, D., M. P. Wilson, Jr., and W. G. Birkhead (1975), "Measurement Error Due to the Bending of Orifice Plates," Proceedings of the 1975 ASME Winter Annual Meeting, American Society of Mechanical Engineers, Houston, TX, ASME paper 75-WA/FM-6.

Morrow, T B., D. L. George, and M. Nored (2002), Metering Research Facility Program: Operational Factors that Affect Orifice Meter Accuracy, Gas Research Institute Topical Report GRI-00/0141, Gas Technology Institute, Des Plaines, IL, April 2002.

Teyssandier, R. G., and W. E. Chisman (1984), "Experimental Test Results Presented for Field-Damaged Orifice Meter Plates," Oil & Gas Journal, April 30, 1984, pp. 70-73.

Ting, V. C. (1993), "Effects of Nonstandard Operating Conditions on the Accuracy of Orifice Meters," SPE Production and Facilities, February 1993, pp. 58-62.

von Mises, R. (1917), "Berechnung von Ausfluss and Uberfallyalilem," Zeitschrift VDI, Vol. 61, p. 447.

By D. L. George, Ph.D., Senior Research Engineer, Southwest Research Institute[R], San Antonio, TX
Table 1: Parameters for the contraction
coefficient correlation used in the model of
Jepson and Chipchase.

[beta] [K.sub.1] [K.sub.2] [K.sub.3]

0.7 0.895 -0.190986 0.0372862
0.6 0.908 -0.225363 0.0437708
0.5 0.923 -0.257831 0.0510659
0.4 0.934 -0.278203 0.0543082
0.3 0.944 -0.294118 0.0567399
0.2 0.948 -0.300485 0.0567400

Table 2: Parameters for the contraction coefficient
correlation used in the model of MW&B.

[beta] [M.sub.0] [M.sub.1] [M.sub.2] [M.sub.3]

0 0.95402 -0.31646 0.06894 -0.00414
0.1 0.97151 -0.35154 0.03918 -0.00706
0.2 0.96840 -0.34810 0.08981 -0.00702
0.3 0.96440 -0.34174 0.08981 -0.00702
0.4 0.95161 -0.31931 0.08286 -0.00606
0.5 0.94027 -0.29814 0.07906 -0.00594
0.6 0.91687 -0.24612 0.05819 -0.00306
0.7 0.90321 -0.21015 0.05060 -0.00283
0.8 0.91456 -0.20376 0.06009 -0.00536
0.9 0.90895 -0.12380 0.02909 -0.00136

Figure 5: Example of the coordinate machine method for
measuring plate dimensions. Distances are in inches.

Granite Surface of Table

Reference 15.00583
A1 14.70666
A2 14.76286
A3 14.87172
A4 14.87310
A5 14.88521
A6 14.81782
A7 14.76023
A8 14.71553
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Comment:Guidelines for measurement errors caused by buckled orifice plates.
Author:George, D.L.
Publication:Pipeline & Gas Journal
Geographic Code:1USA
Date:Jul 1, 2008
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