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Comparison of different approaches to calculate a final meter factor for rotary-type natural gas displacement meters
Flow Measurement and Instrumentation 30 (2013) 160–165
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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Comparison of different approaches to calculate a final meter factor for rotary-type natural gas displacement meters Elcio Cruz De Oliveira a,b,n, Tu´lio Campos Lourenc- o c a
Petrobras Transporte S.A., Project Management, 20091-060, Rio de Janeiro, RJ, Brazil Post-Graduation Metrology Programme, Metrology for Quality and Innovation, Pontifical Catholic University of Rio de Janeiro, 22453-900, RJ, Brazil c Petrobras Transporte S.A., Natural Gas Measurement Division, 20091-060, Rio de Janeiro, RJ, Brazil b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 March 2012 Received in revised form 8 January 2013 Accepted 20 February 2013 Available online 1 March 2013
The meter factor is the ratio between the reference volume and the indicated test meter volume for a particular flow rate. In some applications, a final and a single meter factor that covers all the flow meter rates is required, and there are several approaches to calculate it. However, none of them are specific to rotary-type natural gas displacement meters. In this paper, certain established approaches, such as AGA 7, AGA 9, non-weighted and weighted regression lines were applied to calibration data of this type of meter and their results were compared. An Excel spreadsheet was developed to calculate the final meter factors using all these approaches and to indicate users the one with the lowest uncertainty, based on input data in order to configure the flow computer. In the specific case studies shown, the approaches using linear regression were found to be more suitable. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Final meter factor Linear regressions Calibration curve Rotary-type natural gas displacement meter
1. Introduction The use of natural gas, an attractive fossil fuel is increasing. Gas pipeline transportation companies are demanding credibility and excellence in meter calibration as the principal parameter in ensuring accountability for the gas invoiced [1]. When a meter is a mechanical device, such as a displacement or turbine meter [2], it can be affected by slippage, drag, and wear. Since there is little literature concerning rotary-type displacement meters, and turbine meters have similar mechanical characteristics, concepts and references of these were broadened to comprise the MF (meter factor) applied to rotary-type meters. A turbine meter is a type of velocity flow meter comprising a turbine, a bearing and a preamplifier [3] that generates frequencies proportional to volumetric flow rates [4]. In the right situations, they offer a useful combination of simplicity, accuracy, and economy [5]. Since it is mechanical equipment, through years of use its MF can gradually change, which means that regular recalibration is needed to provide an updated MF and consequently an accuracy control [6]. Usually, an optimum meter factor and zero bias are chosen so that the meter error at any given flow rate lies within the manufacturer’s specified error band for a flowcalibrated meter [7].
n
Corresponding author. Tel.: þ55 21 3211 9223; fax: þ 55 21 3211 9300. E-mail address: [email protected] (E.C. De Oliveira).
0955-5986/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.flowmeasinst.2013.02.005
A meter factor corrects the indicated volume to the reference or actual metered throughput. It can be defined as a number by which the result of a measurement is multiplied to compensate for systematic error. It is a non-dimensional value determined for each flow rate at which the meter is calibrated, and is calculated by dividing the value from the reference meter by the indicated value of the meter under test (MUT). Though, in some cases it is not feasible to apply multiple meter factors, each one to an individual flow rate, given that many flow computer models require an average single factor (final meter factor – FMF). FMF is the number developed either by averaging the individual meter factors over the range of the meter or by weighting more heavily the meter factors over flow rates at which the meter is more likely to be used. In addition, multi-point linearization or polynomial curve fitting techniques may be used [8]. The simplest way to express the meter factor or to describe the flow rate difference between the working standard (reference meter) and the meter under test (MUT) is given by the Eqs. (1) [9,10]: MF ¼
qStandard qMUT
ð1Þ
Alternatively, meter factors can be calculated from the percentage error values provided at each calibration flow rate, by the Eq. (2) [8]: Meter f actor ¼
100 100 þ percent error
ð2Þ
E.C. De Oliveira, T. Campos Lourenc- o / Flow Measurement and Instrumentation 30 (2013) 160–165
Thus, the meter factor example of 1.005 would be the same as 0.5 percentage error. The error adjustments can be calculated offline manually or online in an electronic accessory device, and can be applied to each specific flow rate, using individual meter factor, or across the range of flow rates, using a single FMF [11]. The calibration facility may provide meter factors in addition to or in place of percentage error values for each test flow rate of a meter [8]. The FMF may be the arithmetic average of the meter factors or calculated by a non-weighted ordinary least square regression over the range of flow rates at which the meter is to be used. The FMF may also be weighted more heavily toward the individual meter factors at the higher flow rates at which the meter is to be used or by a weighted ordinary least square regression. Rotary-type natural gas displacement meters are commonly used in custody transfer applications, and reference [12] represents a basic standard for safe operation, substantial and durable construction, and acceptable performance for this type of meter. However, this reference does not mention how to establish a single meter factor. The flow rate literature principally mentions two approaches to calculate a final (single) meter factor, based on turbine and ultrasonic meter technologies, although they can be applied to other similar measurement systems. Alternatively, statistical techniques can also be used to provide a solution to this issue. The aim of this paper is to compare the results of FMF calculations using different approaches, applied to rotary-type natural gas displacement meters.
2.2. Different approaches to calculate the final meter factor 2.2.1. AGA 7 approach This approach calculates each meter factor based on Eq. (1). Afterwards, an arithmetic average of meter factors is taken, and then the error between each meter factor and the average one is calculated. The final meter factor is the one which has the least bias among all meter factors. Here, the standard uncertainty is simply considered as the standard deviation of the meter factor. 2.2.2. AGA 9 approach In this approach, it is necessary to have available both the meter under test (MUT) results and actual or reference meter results, the nominal test rate, or desired flow rate. The percentage error of each flow rate is calculated by Eq. (3). The next step is to calculate weighting factor values, wfi, using the relationship between each actual flow rate and the maximum desired flow rate, qmax, Eq. (5), [7,16]. wf i ¼
qActuali qmax
ð5Þ
The flow mean error (FME) is then found by Eq. (6), [7,16]. Pn i wf i Errorð%Þqi Pn ð6Þ FME ¼ i wf i Considering the uncorrelated quantities in Eq. (6), the combined standard uncertainties are shown in Eqs. (7) and (8): !2 2 @FME @FME uðwf i Þ þ uðErrorð%Þqi Þ ð7Þ u2c ðFMEÞ ¼ @wf i @Errorð%Þqi
2. Methodology The methodology comprises two topics. The first one is the uncertainty evaluation which is the criterion used to choose the final meter factor. The second one details the different approaches used to calculate the FMF.
161
u2c ðFMEÞ ¼
n X Errorð%Þqi i
þ
!2 P wf i ni wf i Errorð%Þqi u2 ðwf i Þ Pn 2 i wf i
Pn i
2 n X wf Pn i u2 ðErrorð%Þqi Þ i wf i i
ð8Þ
Finally, the final meter factor is calculated by Eq. (9) and its uncertainty by Eq. (10). 2.1. Uncertainty evaluation FMF ¼ To metrologists, measurement results cannot be appropriately expressed and evaluated without knowing their uncertainty [13]. The uncertainty has a probabilistic basis, while the error is deterministic, and reflects incomplete knowledge of the quantity. All measurements are subject to uncertainty and it can be used to evaluate the quality of a result. It is not always possible to correct significant systematic effects from a calibration curve [14]. In these situations, the total uncertainty becomes increased by this source of uncertainty called here as Error. In this paper, the total uncertainty is evaluated as the algebraic sum of the expanded uncertainty U (k¼2; 95.45%), considering that there is no bias, and also the maximum absolute value of the Error, over the calibration range. The rotary-type displacement meter performance is usually expressed by giving the relative meter error as a function of flow rate. The flow rate relative meter error and the FMF error are defined by Eqs. (3) and (4), respectively [15]: Errorð%Þqi ¼
Errorð%ÞFMF
qMUT i qStandardi 100 qStandardi FMFMF i ¼ 100 MF i
ð3Þ
100 100 þFME
u2c ðFMFÞ ¼
ð9Þ
2 100 u2c ðFMEÞ ð100 þFMEÞ
2.2.3. Non-weighted ordinary least squares (OLS) In the classical univariate calibration, considering Z calibration points, the calibration curve is defined by y¼f(x), and the unknown quantity (x0) is determined by the solution to the Eq. y0 ¼f(x0), where y0 is the result for the unknown variable. The most simple and widely used case is the following linear model:(y ¼b0 þb1x) [17]. The unweighted linear regression is used to obtain estimates of the calibration parameters b0 and b1, derived from x0 ¼(y0 b0)/b1. In this paper, this latter equation becomes MUT 0 ¼ ðStandard0 b0 Þ=FMF. So, it is assumed that there is a linear relationship between the reference meter and the meter under test (MUT) and the error in the y-values is constant, having homoscedastic behavior. It can be shown that the least squares straight line is given by Eq. (11): Pn Slope of least squares line : FMF ¼
ð4Þ
where Standard is the reference or actual flow meter, and MUT is the meter under test.
ð10Þ
i
fðqMUT i qMUT ÞðqStandardi qStandard Þg Pn 2 i ðqMUT i qMUT Þ
ð11Þ The random error in value for the slope is thus significant and it must be now considered. Firstly, the value of sy/x is calculated.
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162
This estimates the standard deviation of the regression in the ydirection, given by Eq. (12) [18]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i ðyi Y i Þ ð12Þ sy=x ¼ n2
Table 1 Calibration results from CTGas [20].
where Yi values are the points on the calculated regression line corresponding to the individual x-values, i.e. the fitted y-values. After the sy/x value is calculated, it is possible to calculate uFMF, the standard uncertainty for the final meter factor, as shown in Eq. (13): sy=x sqStandard =qMUT ffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ub1 ¼ uFMF ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pn 2 2 i ðqMUT i qMUT Þ i ðxi xÞ
Nominal flow rate (m3/h)
Standard flow rate (m3/h)
3508.1 6916.9 21,910.1 37,681.9 60,190.3 86,183.4
3750 8750 21,875 35,000 61,250 87,500
3475 6890 21,980 37,801 60,415 86,500
Table 3 Calibration results from TCC [21]. MUT flow rate (m3/h)
Standard flow rate (m3/h)
80.30 225.85 362.88 725.36 1797.62 2883.82 5045.88 7124.42
81.65 227.11 364.11 727.10 1800.94 2888.63 5048.59 7129.52
ð14Þ
n
The weighted slope of the regression line, FMF, is given by Eq. (15):
i
9.98 24.75 42.08 53.91 68.90 82.66 96.63
MUT flow rate (m3/h)
i
FMF ¼
10.00 25.00 42.00 54.00 69.00 83.00 97.00
Table 2 Calibration results from CEESI [7,16].
bias
n P
Standard flow rate (m3/h)
ð13Þ
2.2.4. Weighted ordinary least squares (weighted OLS) In some calibration curves, the error in the y-values is not constant. These heteroscedastic data should be treated by weighted regression methods. The regression line must be calculated to give additional weighting to those points where the errors are smaller [18]. In this paper, it is achieved by giving each point a weighting inversely proportional to the square of the bias (difference between the reference meter and the MUT). The individual weightings, wi, are given by Eq. (14), considering n as the number of measurements for the calibration curve, and bias as the difference between MUT and Standard flow rates: bias2 wi ¼ P n 2
MUT flow rate (m3/h)
wi qMUT i qStandardi nqMUT w qStandardw n P i
ð15Þ wi q2MUT i nq2MUT w
P P where qMUT w ¼ ni wi qMUT i =n and qStandardw ¼ ni wi qStandardi =n . As in Section 2.2.3, the statistic S(y/x)w is calculated, which estimates the weighted standard deviation of the regression in the y-direction, Eq. (16) [18]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i wi ðyi Y i Þ ð16Þ sðy=xÞw ¼ n2 Finally, it is possible to calculate uFM, the standard uncertainty for the final meter factor, Eq. (17) [19]: sðy=xÞw sðqStandard =qMUT Þw ffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uFMF ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pw Pn 2 2 i wi ðqMUT i qMUT Þ i wi ðxi xÞ
ð17Þ
3. Experimental data Three calibration data sets are analyzed in this paper.
Table 4 Data from case study 1.
AGA 7 AGA 9 OLS WEIGHTED OLS
FMF
Total uncertainty (%)
0.9962 0.9972 0.9964 0.9990
1.5 1.1 1.1 0.99
Table 5 Meter factor and its error. MUT flow rate (m3/h)
Standard flow rate (m3/h)
MFi (MUT/ Standard)
Error (%)FMF ¼ FMF MFi/ MFi 100 (%)
3508.1 6916.9 21,910.1 37,681.9 60,190.3 86,183.4
3475 6890 21,980 37,801 60,415 86,500
0.9906 0.9961 1.0032 1.0032 1.0037 1.0037 MFaverage ¼ 1.0001
1.26 0.703 0.003 0.000 0.057 0.051 Maximum Error (%)FMF ¼ 1.26
3.1. Case study 1 3.2. Case study 2 Data from a calibration certificate from ‘‘Centro de Tecnologias do Ga´s e Energias Renova´veis – CTGas’’, in Brazil, of a rotary-type natural gas displacement meter (certificate number 440/11LMVG), are reported in Table 1.
Data from a calibration certificate from Colorado Engineering Experimental Station – CEESI of an 8-inch multipath ultrasonic flowmeter, before adjustments, are available in Table 2.
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163
Table 6 Totaled values MUT flow rate (m3/h) 3,508.1 6,916.9 21,910.1 37,681.9 60,190.3 86,183.4
Nominal flow rate (m3/h)
Standard flow rate (m3/h)
3,750 8,750 21,875 35,000 61,250 87,500
Errorð%Þqi ¼
qMUT qStandard i i qStandard
wf i ¼ q
100
0.953 0.390 0.318 0.315 0.372 0.366
3,475 6,890 21,980 37,801 60,415 86,500
qStandard
wf i Errorð%Þqi
i
max nominal
i
0.0397 0.0787 0.2512 0.4320 0.6905 0.9886 Sum=2.4807
0.0378 0.0307 0.0799 0.1361 0.2568 0.3618 Sum=-0.7671 %
Table 10 MF systematic errors.
Table 7 Differences between values. MUT flow rate (m3/h)
Standard flow rate (m3/h)
qMUT i qMUT
qStan dardi qStan dard
3508.1 6916.9 21,910.1 37,681.9 60,190.3 86,183.4 qMUT ¼ 36,065
3475 6890 21,980 37,801 60,415 86,500 qStan dard ¼ 36,177
32,557 29,149 14,155 1,617 24,125 50,118
32,702 29,287 14197 1,624 24,238 50,323
MUT flow rate (m3/h)
AGA 7 (%)
AGA 9 (%)
OLS (%)
Weighted OLS (%)
3508.1 6916.9 21,910.1 37,681.9 60,190.3 86,183.4
1.3 0.69 0.00 0.00 0.06 0.05
1.3 0.68 0.01 0.01 0.06 0.06
1.4 0.80 0.11 0.11 0.06 0.06
1.4 0.84 0.15 0.16 0.10 0.11
Table 11 Summarized data from case study 2 Table 8 Totaled values
qMUT i qMUT qStandardi qStandard
FMF
Random component (%)
Systematic component (%)
Total uncertainty (%)
1.0032 1.0031 1.0043 1.0047
0.55 0.36 0.019 0.013
1.3 1.3 1.4 (1.41) 1.4 (1.44)
2.4 2.0 1.4 1.4
2 qMUT i qMUT AGA 7 AGA 9 OLS WEIGHTED OLS
1,064,674,133 853,658,963 200,956,412 2,625,926 584,750,214 2,522,110,725
1,059,959,334 849,618,535 200,364,497 2,613,988 582,024,471 2,511,842,324
Sum ¼ 5; 228; 776; 374
Sum ¼ 5; 206; 423; 149
Table 12 Data from case study 3.
(The sums are changed too. It is only an observation; it does not belong to the manuscript.) AGA 7 AGA 9 OLS Weighted OLS
Table 9 Bias and weightings MUT flow rate (m3/h)
Standard flow rate (m3/h)
bias (m3/h)
1=bias2
wi
3,508.1 6,916.9 21,910.1 37,681.9 60,190.3 86,183.4
3,475 6,890 21,980 37,801 60,415 86,500
33.1 26.9 69.9 119.1 224.7 316.6
0.000913 0.001382 0.000205 0.000070 0.000020 0.000010
2.11 3.19 0.47 0.16 0.05 0.02
Sum
Sum ¼ 0:0026
3.3. Case study 3 Data from a calibration certificate from TransCanada Calibrations – TCC of an ultrasonic flowmeter, before adjustments, are available in Table 3.
4. Results and discussion Computational approaches of three case studies were compared, and the criterion for the choice of one approach, based on the uncertainty evaluation, was determined.
FMF
Total uncertainty (%)
1.0034 1.0013 1.0005 1.0005
2.4 2.3 1.7 1.6
The values in bold are the recommended final meter factors, given that they presented the lowest uncertainty. In situations where the maximum desired flow rate, needed for the AGA 9 approach, is not available, the maximum reference flow rate is used, since these two values are almost the same. 4.1. Case study 1 The equations presented in the paper were applied and Table 4 shows the results of FMF and its uncertainty. 4.2. Case study 2 This case is detailed numerically in this paper, showing how the cited equations were used, including the weightings assigned to each calibration point. 4.2.1. AGA 7 approach Table 5 shows each meter factor and its error, where the largest one to be computed in the final uncertainty it is considered.
E.C. De Oliveira, T. Campos Lourenc- o / Flow Measurement and Instrumentation 30 (2013) 160–165
164
Table 13 Advantages and limitations of each approach.
AGA 7 AGA 9 OLS Weighted OLS
Advantages
Limitations
Simplicity. Considers a flow weighted mean error. Suitable when the error is constant over the calibration range. Considers a flow weighted error.
No statistical treatment and high uncertainties. Nominal flow rates need to be available in the calibration certificate. Does not take into consideration different errors over the calibration range. Not always, the weighting used in this paper (1/Error2) was not the most appropriate. Other weightings such as (1/Error/Error2, etcy), in other data sets, should be more robust, achieving better results. However, for each set of data, it is possible to have a different weighting, although the effort to find it does not compensate.
The FMF is selected by checking where the error is minimal: in bold, 1.0032. The random component of the FMF uncertainty is uFMF ¼ MF standarddeviation =MF average = 100 ¼ 0:55% and the systematic component is the Maximum Errorð%ÞFMF ¼ 1:3%. The total uncertainty is: Uð2 uFMF Þþ Maximum Errorð%ÞFMF ¼ 2:4%
Table 10 shows the systematic errors of each meter factor. It is clear that at lower flow rates, the systematic errors are the largest. In Table 11, it is observed that the systematic component is similar for all approaches. However, the regression approaches present random components much smaller than the others, which is reflected in the total uncertainty.
4.2.2. AGA 9 approach Table 6 shows values using Eqs. (3) and (6).
4.3. Case study 3
FME ¼
0:7671 ¼ 0:3088% 2:4807
FMF ¼
100 ¼ 1:0031 100 þð0:3088Þ
The random component of the FMF uncertainty is mFM ¼0.36% and the systematic component is the Maximum Errorð%ÞFMF ¼ 1:3%. The total uncertainty is: Uð2 uFMF Þ þMaximum Errorð%ÞFMF ¼ 2:0% 4.2.3. OLS approach In this paper, the FMF is the slope of regression angular coefficient. Tables 7 and 8 show the intermediate computations. Pn fðqMUT i qMUT ÞðqStandardi qStandard Þg FMF ¼ i Pn 2 i ðqMUT i qMUT Þ ¼
5228780,184 ¼ 1:0043 5206430,713
The random component of the FMF uncertainty is uFMF ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qP n 2 and the systematic sqStandard =qMUT = i ðqMUT i qMUT Þ ¼ 0:019% component is the Maximum Errorð%ÞFMF ¼ 1:4%. The total uncertainty is: Uð2 uFMF Þ þ MaximumErrorð%ÞFMF ¼ 1:4%. 4.2.4. Weighted OLS approach Table 9 shows individual weightings, based on bias for each calibration point. n P
FMF ¼
i
wi qMUT i qStandardi nqMUT w qStandardw n P i
¼ 1:0047 wi q2MUT i nq2MUT w
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i wi ðyi Y i Þ ¼ 9:6294 sðy=xÞw ¼ n2 The random component of the FMF uncertainty is uFMF ¼ sðy=xÞw ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pw 2 2 = i wi ðxi xÞ ¼ sðqStandard =qMUT Þw = i wi ðqMUT i qMUT Þ ¼ 0:013% and the systematic component is the Maximum Errorð%Þ FMF ¼ 1:41% . The total uncertainty is: ðUð2 uFMF Þ þ Maximum Errorð%ÞFMF ¼ 1:4%Þ.
The equations presented in the paper were applied and Table 12 shows the results of FMF and its uncertainty. The results show that the AGA 7 approach always presents the greatest uncertainty. Table 13 compares all approaches that have been studied in this paper, showing advantages and limitations of each. Through these results, it is possible to assume that statistical approaches (OLS and Weighted OLS) in calculating the FMF respond better than the AGA approaches. However, it is not possible to determine the better one since each may be more suitable according to the calibration data set evaluated.
5. Conclusions In natural gas stations where volumetric or linear meters are used, the meter factor plays an important role. Hence, for rotarytype natural gas displacement meters, it is essential to calculate this correction factor properly, and also consider its contribution to the global uncertainty. Some techniques to calculate a final meter factor have been provided in more detail in this paper, allowing the reader to understand them more easily. Each approach presents different results. Considering that the specific standards related to rotarytype natural gas displacement meters do not mention how to calculate the FMF, it is recommended to choose the techniques that deal with regression techniques (rather than the simpler ones), as the total uncertainty is lower than the other approaches. It can be explained because systematic errors are relevant components in the total uncertainty and they can not be negligible. Moreover, the regression techniques minimize the random errors. References [1] Haner W. Ultrasonic flow meter calibration: considerations and benefits. Trans Canada Calibrations. Manitoba, Canada; 2006. [2] Cheesewright R, Bisset D, Clark C. Factors which influence the variability of turbine flowmeter signal characteristics. Flow Measurement and Instrumentation 1998;2(4):83–89. [3] Zheng G, et al. Gas–liquid two phase flow measurement method based on combination instrument of turbine flowmeter and conductance sensor. International Journal of Multiphase Flow 2008;34:1031–1047. [4] Mattingly GE. The characterization of a piston displacement-type flowmeter calibration facility and the calibration and use of pulsed output type flowmeters. Journal of Research of the National Institute of Standards and Technology 1992;97(5):509–531.
E.C. De Oliveira, T. Campos Lourenc- o / Flow Measurement and Instrumentation 30 (2013) 160–165 [5] Chen G. et al. Investigation of the meter factor of turbine meter with unsteady numerical simulation. The 4th International Symposium on Fluid Machinery and Fluid Engineering: Beijing, China; 2008. [6] Manual of Petroleum Measurement Standards, Chapter 4.8—Operation of Proving Systems, 1st edition, reaffirmed March; 2007. [7] AGA 9. Measurement of Gas by Multipath Ultrasonic Meters. Transmission measurement committee report no. 9, AGA catalog no. XQ0701. Washington, DC; 2007. [8] AGA 7. Measurement of Natural Gas by Turbine Meters. Transmission measurement committee report no. 7, AGA catalog no. XQ0601. Washington, DC; 2006. [9] Feng C, Lin W, Yang C. Laminar Flow Meter with Straight Glass Capillary. Journal of Metrology Society of India 2011;26(3):237–245. [10] Shaluf IM, et al. An overview on ADCO crude oil storage tanks. Disaster Prevention and Management 2010;19(3):370–383. [11] Bonner JA. Turbine meter measurement of natural gas. In: Proceedings of 2nd International Symposium on Fluid Flow Measurement : Calgary, Canada; 1990.
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[12] American National Standard, ANSI B109.3—Rotary-type gas displacement meters, 4th edition, April; 2000. [13] Analytical Methods Committee. Uncertainty of measurement: implications of its use in analytical science, The Analyst, v.120, n.9, p. 2303-2308; 1995. [14] Guide to the expression of uncertainty in measurement (GUM). ISO; 1995. [15] Van Der Kam PMA, Dam AM. Large turbine meters for custody transfer measurements: the renovation of the Gasunie export stations. Flow Measurement and Instrumentation 1993;4(2):91–98. [16] Johansen W, Clancy J. Calibration of ultrasonic flowmeters. Nunn, CO, USA: Colorado Engineering Experimental Station, Inc; 2003. [17] Oliveira EC. Critical metrological evaluation of fuel analyses by measurement uncertainty. Metrology & Measurement Systems 2011;XVIII(2):235–248. [18] Miller JN, Miller JC. Statistics and chemometrics for analytical chemistry: New York; 2005. [19] Brownlee KA. Statistical theory and methodology in science and engineering: New York; 1960. [20] CTGas Calibration Certificate N 440/11-LMVG. ˚ [21] TransCanada Calibrations Certificate of Calibration N 2595. ˚
Contents lists available at SciVerse ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Comparison of different approaches to calculate a final meter factor for rotary-type natural gas displacement meters Elcio Cruz De Oliveira a,b,n, Tu´lio Campos Lourenc- o c a
Petrobras Transporte S.A., Project Management, 20091-060, Rio de Janeiro, RJ, Brazil Post-Graduation Metrology Programme, Metrology for Quality and Innovation, Pontifical Catholic University of Rio de Janeiro, 22453-900, RJ, Brazil c Petrobras Transporte S.A., Natural Gas Measurement Division, 20091-060, Rio de Janeiro, RJ, Brazil b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 March 2012 Received in revised form 8 January 2013 Accepted 20 February 2013 Available online 1 March 2013
The meter factor is the ratio between the reference volume and the indicated test meter volume for a particular flow rate. In some applications, a final and a single meter factor that covers all the flow meter rates is required, and there are several approaches to calculate it. However, none of them are specific to rotary-type natural gas displacement meters. In this paper, certain established approaches, such as AGA 7, AGA 9, non-weighted and weighted regression lines were applied to calibration data of this type of meter and their results were compared. An Excel spreadsheet was developed to calculate the final meter factors using all these approaches and to indicate users the one with the lowest uncertainty, based on input data in order to configure the flow computer. In the specific case studies shown, the approaches using linear regression were found to be more suitable. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Final meter factor Linear regressions Calibration curve Rotary-type natural gas displacement meter
1. Introduction The use of natural gas, an attractive fossil fuel is increasing. Gas pipeline transportation companies are demanding credibility and excellence in meter calibration as the principal parameter in ensuring accountability for the gas invoiced [1]. When a meter is a mechanical device, such as a displacement or turbine meter [2], it can be affected by slippage, drag, and wear. Since there is little literature concerning rotary-type displacement meters, and turbine meters have similar mechanical characteristics, concepts and references of these were broadened to comprise the MF (meter factor) applied to rotary-type meters. A turbine meter is a type of velocity flow meter comprising a turbine, a bearing and a preamplifier [3] that generates frequencies proportional to volumetric flow rates [4]. In the right situations, they offer a useful combination of simplicity, accuracy, and economy [5]. Since it is mechanical equipment, through years of use its MF can gradually change, which means that regular recalibration is needed to provide an updated MF and consequently an accuracy control [6]. Usually, an optimum meter factor and zero bias are chosen so that the meter error at any given flow rate lies within the manufacturer’s specified error band for a flowcalibrated meter [7].
n
Corresponding author. Tel.: þ55 21 3211 9223; fax: þ 55 21 3211 9300. E-mail address: [email protected] (E.C. De Oliveira).
0955-5986/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.flowmeasinst.2013.02.005
A meter factor corrects the indicated volume to the reference or actual metered throughput. It can be defined as a number by which the result of a measurement is multiplied to compensate for systematic error. It is a non-dimensional value determined for each flow rate at which the meter is calibrated, and is calculated by dividing the value from the reference meter by the indicated value of the meter under test (MUT). Though, in some cases it is not feasible to apply multiple meter factors, each one to an individual flow rate, given that many flow computer models require an average single factor (final meter factor – FMF). FMF is the number developed either by averaging the individual meter factors over the range of the meter or by weighting more heavily the meter factors over flow rates at which the meter is more likely to be used. In addition, multi-point linearization or polynomial curve fitting techniques may be used [8]. The simplest way to express the meter factor or to describe the flow rate difference between the working standard (reference meter) and the meter under test (MUT) is given by the Eqs. (1) [9,10]: MF ¼
qStandard qMUT
ð1Þ
Alternatively, meter factors can be calculated from the percentage error values provided at each calibration flow rate, by the Eq. (2) [8]: Meter f actor ¼
100 100 þ percent error
ð2Þ
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Thus, the meter factor example of 1.005 would be the same as 0.5 percentage error. The error adjustments can be calculated offline manually or online in an electronic accessory device, and can be applied to each specific flow rate, using individual meter factor, or across the range of flow rates, using a single FMF [11]. The calibration facility may provide meter factors in addition to or in place of percentage error values for each test flow rate of a meter [8]. The FMF may be the arithmetic average of the meter factors or calculated by a non-weighted ordinary least square regression over the range of flow rates at which the meter is to be used. The FMF may also be weighted more heavily toward the individual meter factors at the higher flow rates at which the meter is to be used or by a weighted ordinary least square regression. Rotary-type natural gas displacement meters are commonly used in custody transfer applications, and reference [12] represents a basic standard for safe operation, substantial and durable construction, and acceptable performance for this type of meter. However, this reference does not mention how to establish a single meter factor. The flow rate literature principally mentions two approaches to calculate a final (single) meter factor, based on turbine and ultrasonic meter technologies, although they can be applied to other similar measurement systems. Alternatively, statistical techniques can also be used to provide a solution to this issue. The aim of this paper is to compare the results of FMF calculations using different approaches, applied to rotary-type natural gas displacement meters.
2.2. Different approaches to calculate the final meter factor 2.2.1. AGA 7 approach This approach calculates each meter factor based on Eq. (1). Afterwards, an arithmetic average of meter factors is taken, and then the error between each meter factor and the average one is calculated. The final meter factor is the one which has the least bias among all meter factors. Here, the standard uncertainty is simply considered as the standard deviation of the meter factor. 2.2.2. AGA 9 approach In this approach, it is necessary to have available both the meter under test (MUT) results and actual or reference meter results, the nominal test rate, or desired flow rate. The percentage error of each flow rate is calculated by Eq. (3). The next step is to calculate weighting factor values, wfi, using the relationship between each actual flow rate and the maximum desired flow rate, qmax, Eq. (5), [7,16]. wf i ¼
qActuali qmax
ð5Þ
The flow mean error (FME) is then found by Eq. (6), [7,16]. Pn i wf i Errorð%Þqi Pn ð6Þ FME ¼ i wf i Considering the uncorrelated quantities in Eq. (6), the combined standard uncertainties are shown in Eqs. (7) and (8): !2 2 @FME @FME uðwf i Þ þ uðErrorð%Þqi Þ ð7Þ u2c ðFMEÞ ¼ @wf i @Errorð%Þqi
2. Methodology The methodology comprises two topics. The first one is the uncertainty evaluation which is the criterion used to choose the final meter factor. The second one details the different approaches used to calculate the FMF.
161
u2c ðFMEÞ ¼
n X Errorð%Þqi i
þ
!2 P wf i ni wf i Errorð%Þqi u2 ðwf i Þ Pn 2 i wf i
Pn i
2 n X wf Pn i u2 ðErrorð%Þqi Þ i wf i i
ð8Þ
Finally, the final meter factor is calculated by Eq. (9) and its uncertainty by Eq. (10). 2.1. Uncertainty evaluation FMF ¼ To metrologists, measurement results cannot be appropriately expressed and evaluated without knowing their uncertainty [13]. The uncertainty has a probabilistic basis, while the error is deterministic, and reflects incomplete knowledge of the quantity. All measurements are subject to uncertainty and it can be used to evaluate the quality of a result. It is not always possible to correct significant systematic effects from a calibration curve [14]. In these situations, the total uncertainty becomes increased by this source of uncertainty called here as Error. In this paper, the total uncertainty is evaluated as the algebraic sum of the expanded uncertainty U (k¼2; 95.45%), considering that there is no bias, and also the maximum absolute value of the Error, over the calibration range. The rotary-type displacement meter performance is usually expressed by giving the relative meter error as a function of flow rate. The flow rate relative meter error and the FMF error are defined by Eqs. (3) and (4), respectively [15]: Errorð%Þqi ¼
Errorð%ÞFMF
qMUT i qStandardi 100 qStandardi FMFMF i ¼ 100 MF i
ð3Þ
100 100 þFME
u2c ðFMFÞ ¼
ð9Þ
2 100 u2c ðFMEÞ ð100 þFMEÞ
2.2.3. Non-weighted ordinary least squares (OLS) In the classical univariate calibration, considering Z calibration points, the calibration curve is defined by y¼f(x), and the unknown quantity (x0) is determined by the solution to the Eq. y0 ¼f(x0), where y0 is the result for the unknown variable. The most simple and widely used case is the following linear model:(y ¼b0 þb1x) [17]. The unweighted linear regression is used to obtain estimates of the calibration parameters b0 and b1, derived from x0 ¼(y0 b0)/b1. In this paper, this latter equation becomes MUT 0 ¼ ðStandard0 b0 Þ=FMF. So, it is assumed that there is a linear relationship between the reference meter and the meter under test (MUT) and the error in the y-values is constant, having homoscedastic behavior. It can be shown that the least squares straight line is given by Eq. (11): Pn Slope of least squares line : FMF ¼
ð4Þ
where Standard is the reference or actual flow meter, and MUT is the meter under test.
ð10Þ
i
fðqMUT i qMUT ÞðqStandardi qStandard Þg Pn 2 i ðqMUT i qMUT Þ
ð11Þ The random error in value for the slope is thus significant and it must be now considered. Firstly, the value of sy/x is calculated.
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This estimates the standard deviation of the regression in the ydirection, given by Eq. (12) [18]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i ðyi Y i Þ ð12Þ sy=x ¼ n2
Table 1 Calibration results from CTGas [20].
where Yi values are the points on the calculated regression line corresponding to the individual x-values, i.e. the fitted y-values. After the sy/x value is calculated, it is possible to calculate uFMF, the standard uncertainty for the final meter factor, as shown in Eq. (13): sy=x sqStandard =qMUT ffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ub1 ¼ uFMF ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pn 2 2 i ðqMUT i qMUT Þ i ðxi xÞ
Nominal flow rate (m3/h)
Standard flow rate (m3/h)
3508.1 6916.9 21,910.1 37,681.9 60,190.3 86,183.4
3750 8750 21,875 35,000 61,250 87,500
3475 6890 21,980 37,801 60,415 86,500
Table 3 Calibration results from TCC [21]. MUT flow rate (m3/h)
Standard flow rate (m3/h)
80.30 225.85 362.88 725.36 1797.62 2883.82 5045.88 7124.42
81.65 227.11 364.11 727.10 1800.94 2888.63 5048.59 7129.52
ð14Þ
n
The weighted slope of the regression line, FMF, is given by Eq. (15):
i
9.98 24.75 42.08 53.91 68.90 82.66 96.63
MUT flow rate (m3/h)
i
FMF ¼
10.00 25.00 42.00 54.00 69.00 83.00 97.00
Table 2 Calibration results from CEESI [7,16].
bias
n P
Standard flow rate (m3/h)
ð13Þ
2.2.4. Weighted ordinary least squares (weighted OLS) In some calibration curves, the error in the y-values is not constant. These heteroscedastic data should be treated by weighted regression methods. The regression line must be calculated to give additional weighting to those points where the errors are smaller [18]. In this paper, it is achieved by giving each point a weighting inversely proportional to the square of the bias (difference between the reference meter and the MUT). The individual weightings, wi, are given by Eq. (14), considering n as the number of measurements for the calibration curve, and bias as the difference between MUT and Standard flow rates: bias2 wi ¼ P n 2
MUT flow rate (m3/h)
wi qMUT i qStandardi nqMUT w qStandardw n P i
ð15Þ wi q2MUT i nq2MUT w
P P where qMUT w ¼ ni wi qMUT i =n and qStandardw ¼ ni wi qStandardi =n . As in Section 2.2.3, the statistic S(y/x)w is calculated, which estimates the weighted standard deviation of the regression in the y-direction, Eq. (16) [18]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i wi ðyi Y i Þ ð16Þ sðy=xÞw ¼ n2 Finally, it is possible to calculate uFM, the standard uncertainty for the final meter factor, Eq. (17) [19]: sðy=xÞw sðqStandard =qMUT Þw ffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uFMF ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pw Pn 2 2 i wi ðqMUT i qMUT Þ i wi ðxi xÞ
ð17Þ
3. Experimental data Three calibration data sets are analyzed in this paper.
Table 4 Data from case study 1.
AGA 7 AGA 9 OLS WEIGHTED OLS
FMF
Total uncertainty (%)
0.9962 0.9972 0.9964 0.9990
1.5 1.1 1.1 0.99
Table 5 Meter factor and its error. MUT flow rate (m3/h)
Standard flow rate (m3/h)
MFi (MUT/ Standard)
Error (%)FMF ¼ FMF MFi/ MFi 100 (%)
3508.1 6916.9 21,910.1 37,681.9 60,190.3 86,183.4
3475 6890 21,980 37,801 60,415 86,500
0.9906 0.9961 1.0032 1.0032 1.0037 1.0037 MFaverage ¼ 1.0001
1.26 0.703 0.003 0.000 0.057 0.051 Maximum Error (%)FMF ¼ 1.26
3.1. Case study 1 3.2. Case study 2 Data from a calibration certificate from ‘‘Centro de Tecnologias do Ga´s e Energias Renova´veis – CTGas’’, in Brazil, of a rotary-type natural gas displacement meter (certificate number 440/11LMVG), are reported in Table 1.
Data from a calibration certificate from Colorado Engineering Experimental Station – CEESI of an 8-inch multipath ultrasonic flowmeter, before adjustments, are available in Table 2.
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Table 6 Totaled values MUT flow rate (m3/h) 3,508.1 6,916.9 21,910.1 37,681.9 60,190.3 86,183.4
Nominal flow rate (m3/h)
Standard flow rate (m3/h)
3,750 8,750 21,875 35,000 61,250 87,500
Errorð%Þqi ¼
qMUT qStandard i i qStandard
wf i ¼ q
100
0.953 0.390 0.318 0.315 0.372 0.366
3,475 6,890 21,980 37,801 60,415 86,500
qStandard
wf i Errorð%Þqi
i
max nominal
i
0.0397 0.0787 0.2512 0.4320 0.6905 0.9886 Sum=2.4807
0.0378 0.0307 0.0799 0.1361 0.2568 0.3618 Sum=-0.7671 %
Table 10 MF systematic errors.
Table 7 Differences between values. MUT flow rate (m3/h)
Standard flow rate (m3/h)
qMUT i qMUT
qStan dardi qStan dard
3508.1 6916.9 21,910.1 37,681.9 60,190.3 86,183.4 qMUT ¼ 36,065
3475 6890 21,980 37,801 60,415 86,500 qStan dard ¼ 36,177
32,557 29,149 14,155 1,617 24,125 50,118
32,702 29,287 14197 1,624 24,238 50,323
MUT flow rate (m3/h)
AGA 7 (%)
AGA 9 (%)
OLS (%)
Weighted OLS (%)
3508.1 6916.9 21,910.1 37,681.9 60,190.3 86,183.4
1.3 0.69 0.00 0.00 0.06 0.05
1.3 0.68 0.01 0.01 0.06 0.06
1.4 0.80 0.11 0.11 0.06 0.06
1.4 0.84 0.15 0.16 0.10 0.11
Table 11 Summarized data from case study 2 Table 8 Totaled values
qMUT i qMUT qStandardi qStandard
FMF
Random component (%)
Systematic component (%)
Total uncertainty (%)
1.0032 1.0031 1.0043 1.0047
0.55 0.36 0.019 0.013
1.3 1.3 1.4 (1.41) 1.4 (1.44)
2.4 2.0 1.4 1.4
2 qMUT i qMUT AGA 7 AGA 9 OLS WEIGHTED OLS
1,064,674,133 853,658,963 200,956,412 2,625,926 584,750,214 2,522,110,725
1,059,959,334 849,618,535 200,364,497 2,613,988 582,024,471 2,511,842,324
Sum ¼ 5; 228; 776; 374
Sum ¼ 5; 206; 423; 149
Table 12 Data from case study 3.
(The sums are changed too. It is only an observation; it does not belong to the manuscript.) AGA 7 AGA 9 OLS Weighted OLS
Table 9 Bias and weightings MUT flow rate (m3/h)
Standard flow rate (m3/h)
bias (m3/h)
1=bias2
wi
3,508.1 6,916.9 21,910.1 37,681.9 60,190.3 86,183.4
3,475 6,890 21,980 37,801 60,415 86,500
33.1 26.9 69.9 119.1 224.7 316.6
0.000913 0.001382 0.000205 0.000070 0.000020 0.000010
2.11 3.19 0.47 0.16 0.05 0.02
Sum
Sum ¼ 0:0026
3.3. Case study 3 Data from a calibration certificate from TransCanada Calibrations – TCC of an ultrasonic flowmeter, before adjustments, are available in Table 3.
4. Results and discussion Computational approaches of three case studies were compared, and the criterion for the choice of one approach, based on the uncertainty evaluation, was determined.
FMF
Total uncertainty (%)
1.0034 1.0013 1.0005 1.0005
2.4 2.3 1.7 1.6
The values in bold are the recommended final meter factors, given that they presented the lowest uncertainty. In situations where the maximum desired flow rate, needed for the AGA 9 approach, is not available, the maximum reference flow rate is used, since these two values are almost the same. 4.1. Case study 1 The equations presented in the paper were applied and Table 4 shows the results of FMF and its uncertainty. 4.2. Case study 2 This case is detailed numerically in this paper, showing how the cited equations were used, including the weightings assigned to each calibration point. 4.2.1. AGA 7 approach Table 5 shows each meter factor and its error, where the largest one to be computed in the final uncertainty it is considered.
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Table 13 Advantages and limitations of each approach.
AGA 7 AGA 9 OLS Weighted OLS
Advantages
Limitations
Simplicity. Considers a flow weighted mean error. Suitable when the error is constant over the calibration range. Considers a flow weighted error.
No statistical treatment and high uncertainties. Nominal flow rates need to be available in the calibration certificate. Does not take into consideration different errors over the calibration range. Not always, the weighting used in this paper (1/Error2) was not the most appropriate. Other weightings such as (1/Error/Error2, etcy), in other data sets, should be more robust, achieving better results. However, for each set of data, it is possible to have a different weighting, although the effort to find it does not compensate.
The FMF is selected by checking where the error is minimal: in bold, 1.0032. The random component of the FMF uncertainty is uFMF ¼ MF standarddeviation =MF average = 100 ¼ 0:55% and the systematic component is the Maximum Errorð%ÞFMF ¼ 1:3%. The total uncertainty is: Uð2 uFMF Þþ Maximum Errorð%ÞFMF ¼ 2:4%
Table 10 shows the systematic errors of each meter factor. It is clear that at lower flow rates, the systematic errors are the largest. In Table 11, it is observed that the systematic component is similar for all approaches. However, the regression approaches present random components much smaller than the others, which is reflected in the total uncertainty.
4.2.2. AGA 9 approach Table 6 shows values using Eqs. (3) and (6).
4.3. Case study 3
FME ¼
0:7671 ¼ 0:3088% 2:4807
FMF ¼
100 ¼ 1:0031 100 þð0:3088Þ
The random component of the FMF uncertainty is mFM ¼0.36% and the systematic component is the Maximum Errorð%ÞFMF ¼ 1:3%. The total uncertainty is: Uð2 uFMF Þ þMaximum Errorð%ÞFMF ¼ 2:0% 4.2.3. OLS approach In this paper, the FMF is the slope of regression angular coefficient. Tables 7 and 8 show the intermediate computations. Pn fðqMUT i qMUT ÞðqStandardi qStandard Þg FMF ¼ i Pn 2 i ðqMUT i qMUT Þ ¼
5228780,184 ¼ 1:0043 5206430,713
The random component of the FMF uncertainty is uFMF ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qP n 2 and the systematic sqStandard =qMUT = i ðqMUT i qMUT Þ ¼ 0:019% component is the Maximum Errorð%ÞFMF ¼ 1:4%. The total uncertainty is: Uð2 uFMF Þ þ MaximumErrorð%ÞFMF ¼ 1:4%. 4.2.4. Weighted OLS approach Table 9 shows individual weightings, based on bias for each calibration point. n P
FMF ¼
i
wi qMUT i qStandardi nqMUT w qStandardw n P i
¼ 1:0047 wi q2MUT i nq2MUT w
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i wi ðyi Y i Þ ¼ 9:6294 sðy=xÞw ¼ n2 The random component of the FMF uncertainty is uFMF ¼ sðy=xÞw ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pw 2 2 = i wi ðxi xÞ ¼ sðqStandard =qMUT Þw = i wi ðqMUT i qMUT Þ ¼ 0:013% and the systematic component is the Maximum Errorð%Þ FMF ¼ 1:41% . The total uncertainty is: ðUð2 uFMF Þ þ Maximum Errorð%ÞFMF ¼ 1:4%Þ.
The equations presented in the paper were applied and Table 12 shows the results of FMF and its uncertainty. The results show that the AGA 7 approach always presents the greatest uncertainty. Table 13 compares all approaches that have been studied in this paper, showing advantages and limitations of each. Through these results, it is possible to assume that statistical approaches (OLS and Weighted OLS) in calculating the FMF respond better than the AGA approaches. However, it is not possible to determine the better one since each may be more suitable according to the calibration data set evaluated.
5. Conclusions In natural gas stations where volumetric or linear meters are used, the meter factor plays an important role. Hence, for rotarytype natural gas displacement meters, it is essential to calculate this correction factor properly, and also consider its contribution to the global uncertainty. Some techniques to calculate a final meter factor have been provided in more detail in this paper, allowing the reader to understand them more easily. Each approach presents different results. Considering that the specific standards related to rotarytype natural gas displacement meters do not mention how to calculate the FMF, it is recommended to choose the techniques that deal with regression techniques (rather than the simpler ones), as the total uncertainty is lower than the other approaches. It can be explained because systematic errors are relevant components in the total uncertainty and they can not be negligible. Moreover, the regression techniques minimize the random errors. References [1] Haner W. Ultrasonic flow meter calibration: considerations and benefits. Trans Canada Calibrations. Manitoba, Canada; 2006. [2] Cheesewright R, Bisset D, Clark C. Factors which influence the variability of turbine flowmeter signal characteristics. Flow Measurement and Instrumentation 1998;2(4):83–89. [3] Zheng G, et al. Gas–liquid two phase flow measurement method based on combination instrument of turbine flowmeter and conductance sensor. International Journal of Multiphase Flow 2008;34:1031–1047. [4] Mattingly GE. The characterization of a piston displacement-type flowmeter calibration facility and the calibration and use of pulsed output type flowmeters. Journal of Research of the National Institute of Standards and Technology 1992;97(5):509–531.
E.C. De Oliveira, T. Campos Lourenc- o / Flow Measurement and Instrumentation 30 (2013) 160–165 [5] Chen G. et al. Investigation of the meter factor of turbine meter with unsteady numerical simulation. The 4th International Symposium on Fluid Machinery and Fluid Engineering: Beijing, China; 2008. [6] Manual of Petroleum Measurement Standards, Chapter 4.8—Operation of Proving Systems, 1st edition, reaffirmed March; 2007. [7] AGA 9. Measurement of Gas by Multipath Ultrasonic Meters. Transmission measurement committee report no. 9, AGA catalog no. XQ0701. Washington, DC; 2007. [8] AGA 7. Measurement of Natural Gas by Turbine Meters. Transmission measurement committee report no. 7, AGA catalog no. XQ0601. Washington, DC; 2006. [9] Feng C, Lin W, Yang C. Laminar Flow Meter with Straight Glass Capillary. Journal of Metrology Society of India 2011;26(3):237–245. [10] Shaluf IM, et al. An overview on ADCO crude oil storage tanks. Disaster Prevention and Management 2010;19(3):370–383. [11] Bonner JA. Turbine meter measurement of natural gas. In: Proceedings of 2nd International Symposium on Fluid Flow Measurement : Calgary, Canada; 1990.
165
[12] American National Standard, ANSI B109.3—Rotary-type gas displacement meters, 4th edition, April; 2000. [13] Analytical Methods Committee. Uncertainty of measurement: implications of its use in analytical science, The Analyst, v.120, n.9, p. 2303-2308; 1995. [14] Guide to the expression of uncertainty in measurement (GUM). ISO; 1995. [15] Van Der Kam PMA, Dam AM. Large turbine meters for custody transfer measurements: the renovation of the Gasunie export stations. Flow Measurement and Instrumentation 1993;4(2):91–98. [16] Johansen W, Clancy J. Calibration of ultrasonic flowmeters. Nunn, CO, USA: Colorado Engineering Experimental Station, Inc; 2003. [17] Oliveira EC. Critical metrological evaluation of fuel analyses by measurement uncertainty. Metrology & Measurement Systems 2011;XVIII(2):235–248. [18] Miller JN, Miller JC. Statistics and chemometrics for analytical chemistry: New York; 2005. [19] Brownlee KA. Statistical theory and methodology in science and engineering: New York; 1960. [20] CTGas Calibration Certificate N 440/11-LMVG. ˚ [21] TransCanada Calibrations Certificate of Calibration N 2595. ˚