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Statistical model and prior knowledge for

the determination of calibration curves of

flow

Deliverable 1.1.4

of Work Package

WP 1 (Regression)

Authors:

Gertjan Kok Adriaan van der Veen VSL

Peter Harris Ian Smith National Physical Laboratory

A report of the EMRP joint research project NEW04 “Novel mathematical and statistical approaches to uncertainty evaluation”

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DELIVERABLE REPORT DOCUMENTATION PAGE

1. Work package

WP 1 (Regression)

2. Deliverable number

D1.1.4

3. Reporting date

November 2012

4. Title (and subtitle)

Statistical model and prior knowledge for the determination of calibration curves of flow

5. Author(s)

Gertjan Kok (VSL), Adriaan van der Veen (VSL), Peter Harris (NPL), Ian Smith (NPL)

6. Lead author (e-mail)

gkok@vsl.nl

7. Contributing researchers (institutes)

VSL, National Physical Laboratory

8. Other contributing work packages

None

9. Lead researchers in other WPs

None

10. Supplementary notes

None

11. Abstract

This report describes the problem of determining and using the calibration curve of a flow meter and the calculation of relevant uncertainty information. Based on available information obtained from VSL’s flow group a statistical model is formulated. Also prior knowledge concerning the result of the calibration is recorded. Some more general models are formulated as well. This report lays the foundation for further statistical analysis, which will be reported in subsequent deliverables.

12. Key words

Linear regression, flow meter, calibration, statistical model, prior knowledge

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TABLE OF CONTENTS

1 BACKGROUND .............................................................................................................. 1

2 INTRODUCTION ............................................................................................................. 1

3 TERMINOLOGY .............................................................................................................. 1

4 FLOW METER CALIBRATION ....................................................................................... 1

4.1 Overview ......................................................................................................................... 1

4.2 Literature on flow meter calibration .................................................................................. 2

5 USER QUESTIONS OF FLOW DEPARTMENT .............................................................. 2

6 STATISTICAL MODEL FOR FLOW METER CALIBRATION ......................................... 3

6.1 Basic model ..................................................................................................................... 3

6.2 Classification ................................................................................................................... 4

7 EXTENSIONS ................................................................................................................. 4

7.1 Heteroscedasticity ........................................................................................................... 4

7.2 Uncertainty in the reference and correlations .................................................................. 5

7.3 Temporal drift of the curve ............................................................................................... 5

7.4 Calibration curves with inverse usage ............................................................................. 7

8 PRIOR KNOWLEDGE .................................................................................................... 7

9 STATISTICAL APPROACH TO SOLVE THE MODEL ................................................... 8

10 CONCLUSION ................................................................................................................ 8

ACKNOWLEDGMENTS ............................................................................................................. 8

REFERENCES ........................................................................................................................... 8

APPENDIX A: UNCERTAINTY BUDGET FOR THE CALIBRATION OF A FLOW METER ....... 9

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LIST OF FIGURES

Figure 1: Example of flow meter calibration result. .......................................................................... 2

Figure 2: Calibration results of the same flow meter at several time points. .................................... 6

LIST OF TABLES

Table 1: Uncertainty budget for reference volume in flow meter calibration. .................................... 9

Table 2: Explanation of symbols used in Table 1. ......................................................................... 10

Table 3: Explanation of the uncertainty sources in the uncertainty budget of Table 1. ................... 10

Table 4: Uncertainty budget for pulse counting and interpolation of flow meter under test. ........... 11

Table 5: Uncertainty budget for flow meter pulse factor calibration. .............................................. 11

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1 BACKGROUND

This report covers deliverable D1.1.4: “Statistical model and prior knowledge for the determination of calibration curves of flow meters” of the EMRP project ‘Novel mathematical and statistical approaches to uncertainty evaluation’ (EMRP NEW04). It is part of the series of deliverables D1.1.4, D1.1.8, D1.1.12, D1.2.4, D1.2.8, D1.3.3, D1.3.5 and D1.3.6. Jointly the deliverables describe a case study on determining and using a flow meter calibration curve with the focus on the statistical analysis of the data. The methodology is more broadly applicable, however, than just for flow meters. The use of the case study enables a demonstration of how prior knowledge, and the context in which the measurement data are generated, can be utilized.

2 INTRODUCTION

This report describes the problem of determining and using the calibration curve of a flow meter and the calculation of relevant uncertainty information. Based on available information obtained from VSL’s flow group a statistical model is formulated. Also prior knowledge concerning the result of the calibration is recorded. This report lays the foundation for further statistical analysis, which will be reported in subsequent deliverables.

3 TERMINOLOGY

In regression analysis there are two (sets of) variables involved. Many different terms exist for the ‘x-coordinates’ and the ‘y-coordinates’. In this report the terms ‘explanatory variable’ (for x) and ‘explained variable’ (for y) are used throughout.

4 FLOW METER CALIBRATION

4.1 Overview

A flow meter is a device for measuring the quantity of fluid (e.g. water) transported through a pipeline, both measured as transported volume in a given time and as transported volume per unit of time (flow rate). A flow meter has typically (at least) two outputs: one analogue output for flow rate and one pulse or frequency output for (total) volume. For the latter output a conversion factor or pulse factor (often referred to as a “K-factor”) is specified, meaning that one output pulse is equivalent to a specified volume that has passed the meter (e.g., 10 pulses/l or alternatively 100 ml/pulse). In a flow meter calibration usually this pulse output is calibrated, i.e. the pulse factor is the explained variable in the regression model (and flow rate the explanatory variable). The manufacturer specifies a pulse factor that is independent of the flow rate, e.g. 10 pulses/l. For the calibration pulse factors are determined at 12 different flow rates, and the calibration is repeated 5 times. This is a typical choice used in VSL’s water flow laboratory, but may vary between calibrations and between different flow laboratories. For example, the calibration result could be that the calibrated (measured) pulse factor is 10.03 pulses/l at the flow rate 2000 l/min and 10.08 pulses/l at 4000 l/min, etc. For these calibrated pulse factors a regression curve is determined as a function of the (reference) flow rate. This curve can then be used to correct the flow meter reading at a measured flow rate. This use of the curve is not entirely mathematically correct (as the pulse factor is determined as a function of reference flow rate and not as a function of indicated flow rate by the meter), but the induced error is negligible. Often a polynomial curve for the pulse factor is determined using linear regression on certain fixed powers of the flow rate q (e.g., q-1, q, q2, q3) and a constant intercept. The reference in a flow meter calibration can be a volume prover (often a

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steel cylinder of known volume), a vessel on a weighing scale or another calibrated flow meter. An example of the result of a flow meter calibration is shown in Figure 1.

Figure 1: Example of flow meter calibration result.

4.2 Literature on flow meter calibration

More information concerning the particular flow meter calibration procedure followed at VSL can be found in VSL’s quality system [7]. Reference [8], ‘Flow Measurement Handbook’, gives an overview about flow measuring techniques and instruments and includes information on flow meter calibration and uncertainties. The International Organization for Standardization (ISO) has published two documents concerning the ‘Assessment of uncertainty in calibration and use of flow measurement devices’, ISO/TR 7066-1:1997 [5] and ISO 7066-2:1988 [6]. They deal with linear regression used in flow meter calibrations. Explicit use of these documents seems not to be widespread however. They are currently not under revision.

5 USER QUESTIONS OF FLOW DEPARTMENT

The questions raised by technicians of VSL’s flow group can be summarized as follows:

1. What is the best way to estimate the regression coefficients and evaluate the associated

uncertainties?

2. Sometimes the flow meter shows poorer repeatability at some flow rates in comparison to others.

How can this phenomenon be included in the regression and uncertainty calculations?

3. For consecutive calibrations undertaken at intervals of several months, often the regression

coefficients are approximately the same, but sometimes there can be a substantial shift of the

curve. How should account be taken for these time drift phenomena in the uncertainty evaluation?

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4. Does a model specified by a set of four or fewer powers of the flow rate exist that is both simple

(preferably comprising integer powers or integers + 1/2) and has particular good fitting properties?

Or maybe even has a physical meaning?

6 STATISTICAL MODEL FOR FLOW METER CALIBRATION

6.1 Basic model

In this section a flow meter calibration model is studied and a statistical model including the main error sources deduced. The full model for the uncertainty budget of the calibration is given in Appendix A, and only the main results are presented in this section. An even more detailed description can be found in VSL’s quality system [7]. In VSL’s flow group well-established uncertainty budgets for the reference volume, the counted pulses (including pulse fractions) and the pulse factor have been developed [1]. This budget includes uncertainty contributions from temperature and pressure measurements, geometrical measurement of the prover, some material constants, pulse counting and interpolation. The result is that the relative expanded uncertainty of the reference volume Vref equals 0.021 % (k = 2) [3]. The uncertainty contribution due to systematic effects amounts to 0.020 % and the contribution due to random effects during the measurement is 0.003 %. As time measurement is relatively accurate the uncertainty in the reference volume is also the uncertainty in flow rate, which is the explanatory variable of the regression. The explained variable is the pulse factor. The calibrated value has relative expanded uncertainty 0.024 % (k = 2). The random uncertainty associated with the mean of 5 calibrations of the pulse factor at a given flow rate varies between 0.004 % and 0.013 %. A contribution is due to the random behavior of the facility (0.003 %) and another is due to the flow meter under test. The next step is to compare the magnitudes of the uncertainties in the explanatory and explained variables on a comparable scale and to decide whether one is negligible compared to the other. The effect of the uncertainty in flow rate on the uncertainty in predicted pulse factor is assessed.

From the data leading to Figure 1, the maximum local slope of the regression curve is dk/dq = 2 10-5 (p/l)/(l/min) for the smallest flow rates. (This value for the slope is also similar in magnitude to those of the curves in Figure 2.) The relative uncertainty of the reference flow rate is 0.021 %, and so this uncertainty is largest in absolute value for the largest flow rate qmax = 6000 l/min. The effect

on the predicted pulse factor is at most Uq(k) = 0.021 % 6000 (2 10-5) 2 10-5 l/min. This value for the uncertainty has to be compared with the smallest random uncertainty in the explained

variable Uk(k) = 0.004 % 13.202 5 10-4 l/min. Thus at worst Uq(k)/Uk(k) 0.04, and so the uncertainty induced by the uncertainty in the explanatory variable (flow rate q) is negligible compared to the uncertainty in the explained variable (pulse factor k) and is therefore not taken into account in the modeling. Admittedly this reasoning is somewhat intuitive, but this approximation seems reasonable at this stage. The measurement uncertainty of the pulse factors, whether due to the flow meter itself or to the facility, is modeled as a nuisance variable, and for all measurements the errors in the factors are independent and identically normally distributed with mean zero. With the above assumptions, a basic calibration model is given by models 1a and 1b below. Extensions to these basic models based on fewer assumptions are considered in the following sections.

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MODEL 1a:

ki = βT·qi + εi (i = 1, ... , N) (superscript

T denotes transpose of a vector)

βT = (β0, β1, ..., βP), qi

T = (1, qi

r_1, ... , qi

r_P) (r_m denotes rm for all m)

εi ~ N(0, σ2), i.i.d. (normal distribution with mean 0 and known variance σ

2)

MODEL 1b:

As Model 1a, but with variance σ2 unknown.

The qi are assumed to be known without uncertainty and the ki are known from the measurement with uncertainty (due to the terms εi). The powers r1, ... , rP are assumed to be known fixed values. The goal is to estimate β (and σ as an intermediate quantity in case of model 1b) and to calculate the uncertainty associated with the predicted value of the pulse factor k at a new data point (flow rate) s.

In the particular flow calibration of Figure 1, the number of measurements is N = 12 5 = 60, corresponding to 12 different flow rates each repeated M = 5 times (however they are not replicated exactly, therefore no double indexing is used in the formulation of the model). The flow rates qi lie in the interval [800 l/min, 5900 l/min]. Besides the intercept β0, P = 4 powers of the flow rates are fitted, with r1 = -4, r2 = -2, r3 = 0.5, r4 = 5. These powers were chosen by the technician by trial and error until a fit was obtained that was satisfactory for the technician in his subjective, though expert, opinion. As the repeatability of the flow meter was not known beforehand, the variance σ2 was unknown, and the flow meter calibration model is of type 1b.

6.2 Classification

The statistical models 1a-1b can be classified as multiple linear regression in a controlled calibration setting with independent and identically distributed zero mean normal errors [2]. ‘Multiple’ indicates that there is more than one explanatory variable, namely the rP-tuple (qr_1, ... , qr_P), which are not independent of course. Note that in the case of more than one explained variable, the problem is termed ‘multivariate’. The adjective ‘linear’ indicates that the model is linear in the parameters β that are to be estimated. A controlled calibration setting indicates that the values of the explanatory variable qi were fixed by the person performing the calibration and were not the result of a random sampling distribution, which is sometimes called ‘natural calibration’.

7 EXTENSIONS

In this section some possible extensions of model 1 are formulated.

7.1 Heteroscedasticity

It was noted by the technicians performing the calibrations that for some flow meters the repeatability at different flow rates is clearly different. Heteroscedasticity is described by the following model in which repeated measurements are assumed to be available at some fixed flow rates.

MODEL 2a:

kij = βT·qi + εij (i = 1, ... , N, j = 1, ... , M)

βT = (β0, β1, ..., βP), qi

T = (1, qi

r_1, ... , qi

r_P) (r_m denotes rm for all m)

εij ~ N(0, σi2), all independent, σi

2 known.

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MODEL 2b:

As Model 2a, but with variances σi2 unknown.

The qi are assumed to be known without uncertainty and the kij are known from the measurement with uncertainty (due to the terms εij). The powers r1, ... , rP are assumed to be known fixed values. The goal is to estimate β (and σi as intermediate quantities if unknown) and to calculate the uncertainty associated with the predicted value of the pulse factor k at a new data point (flow rate) s. The expected outcome is an uncertainty value that is clearly flow rate dependent. As the repeatability of the flow meter is not known beforehand, the variances σi

2 are unknown, and the heteroscedastic flow meter calibration model is of type 2b.

7.2 Uncertainty in the reference and correlations

In the justification of the statistical flow meter calibration model it was argued that the uncertainties in the explanatory variables are negligible. In some cases this might not be the case. There may also exist correlations associated with the explanatory variables, associated with the explained variables or even associated with both sets of variables. This leads to the following general models. These models can be simplified by adding assumptions on the structure of the covariance matrices, in which the models may simplify to e.g. Generalised Distance Regression or Orthogonal Regression. See reference [4] for more details. Observed are the pairs (ki, qi) and the powers r1, ..., rP are fixed.

MODEL 3a:

ki = βT·q

*i + εi (i = 1, ... , N)

qi = q*i + δi (i = 1, ... , N)

βT = (β0, β1, ..., βP), qi

T = (1, qi

r_1, ... , qi

r_P) (r_m denotes rm for all m)

δT = (δ1, ... , δN) ~ N(0, Vq) multivariate normal distribution with known covariance matrix Vq

εT = (ε1, ..., εN) ~ N(0, Vk) multivariate normal distribution with known covariance matrix Vk

MODEL 3b:

As Model 3a, but with unknown covariance matrices Vq and Vk.

MODEL 4a:

ki = βT·q

*i + εi (i = 1, ... , N)

qi = q*i + δi (i = 1, ... , N)

βT = (β0, β1, ..., βP), qi

T = (1, qi

r_1, ... , qi

r_P) (r_m denotes rm for all m)

(δT, ε

T) = (δ1, ... , δN, ε1, ..., εN) ~ N(0, V) multivariate normal distribution with known covariance matrix V

MODEL 4b:

As Model 4a, but with unknown covariance matrix V.

7.3 Temporal drift of the curve

A particular concern of the technicians in the flow department is the drift over time of the calibration curve. A reference flow meter is calibrated approximately twice a year. The new curve often lies close to the old curve, though outside the uncertainty bands around the old curve. An appreciable shift of the calibration curve has also been observed. In Figure 2 the calibration results of the same flow meter at different time points are shown. The deviation on the y-axis is the deviation found

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during the calibration with respect to the K-factor specified by the manufacturer (a constant). The powers of the flow rate used for the curve fits are different from the ones proposed in the last sections.

Figure 2: Calibration results of the same flow meter at several time points.

The question is: how should the behaviour of the calibration curve as a function of time be modeled? The model is now extended to a sequence of regression models depending on a parameter t for time described by parameters β(t) and other model parameters depending on t as well. The difference between consecutive β(t) is in general rather small, suggesting random walks or scaled Wiener processes with small scaling parameters b for the standard deviation. However with a small probability there may a big jump in the intercept β0 in an unknown direction. This suggests assigning a random distribution to the scaling parameter b0 such that it takes a value b0,1 with high probability (1-p), and an appreciably different value b0,2 with low probability p. The model is formalized below. The model is more involved than models 1 to 4 and its statistical treatment may be beyond the scope of the current research project.

MODEL 5: ki(t) = β

T(t)

·qi(t) + εi(t) (i = 1, ... , N)

βT(t) = (β0(t), β1(t), ..., βP(t)), qi

T(t) = (1, qi

r_1(t), ... , qi

r_P(t)) (r_m denotes rm for all m)

εi(t) ~ N(0, σ2(t)), i.i.d. for fixed t, all εi(t) are independent

t = t0, t1, t2, ..., tL L+1 fixed discrete time points with t0 = 0

β(t) = β(0) + b .* WP(t), WP(t) denotes a (P+1)-dimensional vector of independent Wiener processes and the symbol ‘.*’ denotes component wise multiplication.

b = (b0, b1, ..., bP)T with (b1, ..., bP) fixed values and b0 ~ b0,1 + (b0,2 – b0,1) · Bern(p) (Bern(p) denotes

the Bernoulli distribution with parameter p).

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Given will be time points t = t0, t1, t2, ..., tL, data vectors qi(t) and ki(t) for all time points t and the exponents r1, ..., rP. The goal is to estimate the regression parameters β(t), the scaling parameters b0,1, b0,2, b1, ..., bP and the probability p. In the case of an insufficient amount of data, b0,1, b0,2, b1, ..., bP and p may be given beforehand by expert knowledge. With the help of these estimated parameters, a prediction of the pulse factor k(tL+1) at a flow rate s has to be calculated, together with its uncertainty. The assumption that the (P+1) Wiener processes of WP(t) are independent may not be realistic, as the calibration curve is expected to have small slope which may dictate a relationship between the regression coefficients. In that case WP(t) should be modeled as a fully correlated (P+1)-dimensional Wiener process.

7.4 Calibration curves with inverse usage

In the case of flow meter calibration the use of the calibration curve involves the prediction of values of the explained variable corresponding to values of the explanatory variable. In other cases the user may have a value for the explained variable and want to predict the value of the explanatory variable and its uncertainty. The statistical models for this type of calibration are the same as models 1 to 5 with the difference being in the question that the model and the statistical solution approach should answer. An example is the calibration of a resistance thermometer. During the calibration the reference temperature is measured accurately and the electrical resistance of the thermometer is measured with some noise. When the user uses the instrument, an electrical resistance is measured and the user wants to determine the corresponding temperature and its uncertainty.

8 PRIOR KNOWLEDGE

In this section the prior knowledge concerning the flow meter to be calibrated is described. Such knowledge is available before a calibration is performed and can typically be used in a Bayesian solution approach. In a frequentist setting the use of such knowledge is rather unusual. The prior knowledge is here expressed in plain words. In deliverable D1.2.4 these words will be transformed to prior distributions where possible. The prior knowledge can be summarized as follows:

1. The calibrated pulse factor curve should deviate by less than 1 % from the value given by the

manufacturer. In particular the pulse factor must be positive.

2. With high probability the calibrated pulse factor curve should deviate by less than 0.1 % from the

values found at the last calibration. With small probability pulse factor shifts of up to 0.5 % are

possible.

3. The manufacturer specifies a typical flow meter repeatability which is prior knowledge for the

variance σ2. This is particularly useful when the repeatability of the calibration facility is known or

negligible.

4. The calibrated pulse factor curve is usually very flat. The main purpose of the calibration is to

determine the off-set (intercept) of the curve. The difference between maximal and minimal

relative deviation of the flow meter pulse factor compared to the pulse factor specified by the

manufacturer is usually less than 0.2 %.

5. At low flow rates the calibration curve may be less flat and therefore negative powers of the flow

rate q are included in the model.

6. The powers of the flow rate q used in the linear regression fit are often used for traditional reasons

and are not the requirements of a specific (documentary) norm. They can be freely chosen. For

example, another laboratory uses -1, 1, 2, 3.

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7. Finally, in many calibrations no prior knowledge at all is assumed. In this case non-informative prior

distributions can be used, which will be further specified in D1.2.4.

9 STATISTICAL APPROACH TO SOLVE THE MODEL

The problem of solving the model actually consists of two problems: the determination of the calibration curve (undertaken by an NMI) and the use of the calibration curve (undertaken by a customer). Determination of the calibration curve involves estimation of the parameters β and the evaluation of the associated uncertainties. The calibration curve can be used in two different ways. Firstly, to predict values of the explained variable at new data points of the explanatory variable, e.g. as is done with the flow meter calibration curve. The second type of use is when the relationship between the variables is inverted and the curve is used to predict the value of the explanatory variable from an observed explained variable, e.g. when using the measured electrical resistance to predict the temperature in case of a resistance thermometer. As the statistical approach to solve the model is not part of this deliverable, which concerns the specification of the model, the possible solution methods are not included in this report. There will be a separate informal report entitled ‘Statistical approaches for solving regression problems’ that describes possible solution approaches. The selected approaches including numerical methods will be presented in deliverable D1.1.12.

10 CONCLUSION

Five statistical models for the calibration of a flow meter have been presented. The models are of increasing complexity and therefore not all of them will be solved in subsequent stages of the project. In this project Model 1 will be solved in any case. The models have been formulated in a very general way, such that they are applicable to many other calibration and linear regression problems. Prior knowledge available before the flow meter calibration takes place has also been formulated.

ACKNOWLEDGMENTS

The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union.

REFERENCES

[1] F.M. Smits, Excel file with flow meter calibration uncertainty budget for VSL’s water flow facility (2010), 100125 Smith turbine 6 inch VLM001.xlsm

[2] P.J. Brown, Measurement, Regression, and Calibration, Oxford statistical science series, Clarendon Press Oxford (1993)

[3] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, 2008 Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement, Joint Committee for Guides in Metrology, JCGM 100:2008

[4] R M Barker, M G Cox, A B Forbes, P M Harris, Discrete Modelling and Experimental Data Analysis, NPL Report DEM-ES 018: SSfM, BPG 04, March 2007.

[5] ISO/TR 7066-1:1997, Assessment of uncertainty in calibration and use of flow measurement devices -- Part 1: Linear calibration relationships

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[6] ISO 7066-2:1988, Assessment of uncertainty in the calibration and use of flow measurement devices -- Part 2: Non-linear calibration relationships

[7] VSL, Quality System, section flow meter calibrations. [8] R. C. Baker, Flow Measurement Handbook, Cambridge University Press (2000)

APPENDIX A: UNCERTAINTY BUDGET FOR THE CALIBRATION OF A FLOW METER

In this appendix the uncertainty budgets for calibrating the pulse factor of a water flow meter used in VSL’s flow group are described [1]. The budgets are reproduced essentially unchanged to reflect current practice in evaluating and reporting the uncertainties associated with flow meter calibration data. In these budgets it is assumed that all effects are normally distributed. This assumption justifies the use of the expanded uncertainty with a coverage factor of k = 2 throughout [3]. The values in the tables represent these expanded uncertainties. In view of current discussions in the Work Group Fluid Flow (WGFF), the form and contents of the uncertainty budgets may change in the near future. The following measurement model is used to determine the uncertainty of the reference volume when calibrating a flow meter against a reference volume prover:

The uncertainty budget for the reference volume is shown in Table 1. Symbols are explained in Table 2 and Table 3 explains the different uncertainty sources. Only the uncertainty sources due to variation of measurements during the calibration are random effects (i.e. sources 3(j), 7(j), 10(j) and 12(j)), all other uncertainty sources are systematic effects.

Quantity Value Sensitivity coefficient Ci Ci Source U(Xi) Ui(Vref) Ui(Vref) /

Vref

[Xi] [Xi] dm3 /[Xi] [Xi] [dm

3] [%]

Vprt [dm3] 120.220 (1/Vprt)Vref 1.00E+00 1(g) 2.40E-02 2.40E-02 0.0200%

p [g/dm3] 999.19 (1/p)Vref 1.20E-01

2(g) 1.68E-02 2.02E-03 0.0017%

3(j) 1.68E-02 2.02E-03 0.0017%

4(g) 2.00E-02 2.41E-03 0.0020%

[bar-1

] 4.60E-05 (-pm+pp)/((1+*pm)*(1+*pp))*Vref -8.05E+00 5(h) 2.00E-06 1.61E-05 0.0000%

pp [bar] 4.3 (D+*d*E+2*b*D*pp)/((1+*pp)*(d*E+D*pp))*Vref 6.38E-03 6(g) 2.00E-01 1.28E-03 0.0011%

7(j) 2.00E-01 1.28E-03 0.0011%

p [°C-1

] 3.19E-05 (tp-trp)/(1+p*(tp-trp))*Vref -4.15E+02 5(h) 2.00E-06 8.30E-04 0.0007%

tp [°C] 16.6 p/(1+p*(tp-trp))*Vref 3.84E-03 2(g) 1.00E-01 3.84E-04 0.0003%

3(j) 1.00E-01 3.84E-04 0.0003%

trp [°C] 20.0 -p/(1+p*(tp-trp))*Vref -3.84E-03 -- 0.00E+00 0.00E+00 0.0000%

D [mm] 444.5 pp/(d*E*(1+(D*pp)/(d*E)))*Vref 8.26E-06 8(g) 4.00E+00 3.30E-05 0.0000%

d [mm] 31.75 -(D*pp)/(d2*E*(1+(D*pp)/(d*E)))*Vref -1.16E-04 8(g) 5.00E-01 5.78E-05 0.0000%

E [bar] 2.0E+06 -(D*pp)/(d*E2*(1+(D*pp)/(d*E)))*Vref -1.87E-09 5(h) 1.00E+04 1.87E-05 0.0000%

m [g/dm3] 999.19 -(1/m)*Vref -1.20E-01

9(g) 1.68E-02 2.02E-03 0.0017%

10(j) 1.68E-02 2.02E-03 0.0017%

4(g) 1.00E-02 1.20E-03 0.0010%

pm [bar] 4.4 -/(1+*pm)*Vref -5.53E-03 11(g) 2.00E-01 1.11E-03 0.0009%

12(j) 2.00E-01 1.11E-03 0.0009%

Vref [dm3] 120.2097 UB [dm

3] / [%] 2.5E-02 0.0205%

Table 1: Uncertainty budget for reference volume in flow meter calibration.

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Quantity [Xi] Explanation

Vprt [dm3] Calibrated prover volume

p [g/dm3] Water density in prover

[bar-1

] Pressure expansion coefficient of water

pp [bar] Pressure in prover

p [°C-1

] Temperature expansion coefficient of water

tp [°C] Temperature at prover

trp [°C] Reference temperature of water density

D [mm] Diameter of prover

d [mm] Wall thickness of prover

E [bar] Young’s modulus of prover

m [g/dm3] Water density at meter under test

pm [bar] Pressure at meter under test

Vref [dm3] Reference volume

Table 2: Explanation of symbols used in Table 1.

Uncertainty sources

1(g) = Uncertainty in the volume of the prover vessel

2(g) = Uncertainty in the calibration of the temperatures sensor in the prover

3(j) = Uncertainty due to temperature variation in the prover during the calibration

4(g) = Uncertainty in the calibration of the density of the used water

5(h) = Uncertainty in the value of tabulated constants

6(g) = Uncertainty in the calibration of the pressure transducer in the prover

7(j) = Uncertainty due to pressure variation in the prover during the calibration

8(g) = Uncertainty in dimensional measurements

9(g) = Uncertainty in the calibration of the temperature sensor at the meter under test

10(j) = Uncertainty due to temperature variation at the meter under test during the calibration

11(g) = Uncertainty in the calibration of the pressure transducer at the meter under test

12(j) = Uncertainty due to pressure variation at the meter under test during the calibration

Table 3: Explanation of the uncertainty sources in the uncertainty budget of Table 1.

NEW04 Uncertainty

- 11 -

As the measurement time may not correspond exactly to an integer number of pulses, the fractional part of pulses has to be estimated by means of the measurement time. The equation for pulse counting interpolation is shown below. The corresponding uncertainty budget is shown in Table 4. All uncertainty sources are mainly systematic effects.

Quantity Value

Sensitivity coefficient

Ci

Ci Source U(Xi) Ui(PI) Ui(PI) / PI

[Xi] Explanation [Xi] [P] /[Xi] [Xi] [P] [%]

Paanw [P]

Integral number of pulses counted 1587 1/Paanw*PI 1.00E+00 Pulse counter

uncertainty 0.00E+00 0.00E+00 0.0000%

Tref [sec] Time duration of calibration 1.230 1/Tref*PI 1.29E+03 Time base calibration 1.00E-04 1.29E-01 0.0081%

Tm [sec] Measuring time for collecting the

pulses 1.229 -1/Tm*PI -1.3E+03 Time base calibration 1.00E-04 1.29E-01 0.0081%

PI [P] Corrected number of pulses

including fractional part 1587.15 UB [P] / [%] 1.8E-01 0.0115%

Table 4: Uncertainty budget for pulse counting and interpolation of flow meter under test.

With the help of the number of pulses of the flow meter and the reference volume the pulse factor expressed as pulses per volume can be calculated as in the equation below. The corresponding uncertainty budget is shown in Table 5.

Quantity Explanation Value Sensitivity

coefficient Ci Ci Source U(Xi) Ui(K) Ui(K) / K

[Xi] [Xi] (p/l)/[Xi]

-1 [Xi] [p/l] [%]

PI [P] Corrected number of pulses of flow meter under test

1587.15 1/Vref 8.32E-03 (1)f 1.83E-01 0.0015 0.0115%

Vref [dm3] Reference volume 120.21 -p/Vref

2 -1.10E-01 (2)g 2.47E-02 0.0027 0.0205%

K [p/l] Pulse factor 13.203 Total UB [p/l] / [%]

0.0031 0.0235%

UA 2S [p/l] / [%] Standard deviation of the mean of the measured pulse factors in the calibration

0.0005 0.0039%

UT [p/l] / [%] Total uncertainty of the pulse factor 0.0031 0.0238%

Table 5: Uncertainty budget for flow meter pulse factor calibration.