126 uncertainty analysis

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Uncertainty Analysis of Turbine and Ultrasonic Meter Volume

Measurements Part 2, Advanced Topics

THOMAS KEGEL, Senior Staff Engineer,

COLORADO ENGINEERING EXPERIMENT STATION, INC., NUNN, CO

ABSTRACT

This paper continues from the first paper1 in describing the process of estimating the

uncertainty of volume measurements made with turbine and ultrasonic flowmeters. Componentsthat contribute uncertainty include the pressure and temperature transducers, the gas

chromatograph, state equation and flow computer as well as the meter itself. In the first paper

each component was described and numerical uncertainty values were estimated based on a

hypothetical set of measurements. The individual component values were combined to provide

the uncertainty in the total volume.

The present paper discusses uncertainty issues associated with calibration, it is organized

based on two examples. The first example concerns pressure transducer calibration, the second

example discusses flowmeter calibration. Topics include short and long term random effects,

percent-of-reading effects, and full-scale effects Additional discussion covers proper

interpretation of flowmeter calibration results and guidance to replace manufacturer specifications

with calibration results.

STANDARD UNCERTAINTY AND STANDARD DEVIATION

A simplified five-step uncertainty analysis procedure was described in the first paper. One of

those steps is to identify and then classify the numerical values of uncertainty. In order to

combine the uncertainties in the individual components they have to be defined in a uniformmanner. The standard uncertainty2 is the term given to the uniform method of expressing

numerical values of uncertainty.

In order to estimate a standard uncertainty value, a component uncertainty is classified based

on how the numerical value is determined. An estimate is classified as Type Awhen statistical

data are available. The standard uncertainty is defined as the statistically determined standard

deviation3. An estimate is classified as Type B when statistical data are not available. The

standard uncertainty is defined as u= 0.58Uwhere Udefines the limits within which the true

value is expected to lie. In some cases the standard uncertainty of a Type B estimate is u =

0.50Uif it is known to be based on a statistical analysis. In the first paper most of the standarduncertainties were classified as Type B, in the present paper Type A uncertainty estimates are

discussed in greater detail.

Given a set of n data points, the mean is defined as:

=

=

n

1i

ixn

1x [1]

and the standard deviation is defined as:


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( )=

=

n

1i

2i xx

1n

1s [2]

The n data points are samples of a population, if n is sufficiently large then 95% of the data

points lie within 2s of the population mean. The sample mean, x , lies within 1n

s

of the

population mean.

PRESSURE TRANSDUCER CALIBRATION

A simple example has been formulated to demonstrate some important statistical principles.

The subject is a fictitious pressure transducer that is used over a 1001000 psi range. The output

is a voltage that varies over the 05 volt range; the nominal sensitivity is therefore 200 psi/volt.

The transducer is calibrated every 90 days, a history has been developed based on 16 calibrations

made between January 1999 and December 2002.

Long and Short Term Random Effects

Each calibration involves determining the transducer sensitivity based on values calculatedfrom pressure and voltage standards. Sensitivity data are obtained at ten pressure levels equally

spaced over the input range, typical calibration results contained in Figure 1. The ten sensitivity

values that make up a calibration are used to calculate a mean and standard deviation, values are

given in Figures 2 and 3. Each symbol represents one calibration; the solid lines are described

later in the paper.

Two values of standard deviation, with different interpretation, are calculated from the

calibration data to form the control limits. A pooled value of calibration standard deviations,

called swfor within, represents the average variation observed in the time required to perform a

ten-point calibration. The term pooled refers to a process for combining multiple standard

deviation values3. The second standard deviation, called sbfor between, represents random effects

that are only observed over longperiods of time. It is calculated as the standard deviation of themean values. The duration of longand shortperiods of time depends on the application. In the

present example short represents the time required for a single calibration and long represents

four years.

The reported standard deviation, sr, accounts for both short term and long term random

effects4. The measurement uncertainty will be underestimated both effects are not included. The

value for sris calculated from values of swand sbcombined in quadrature (root sum square):

08.103.132.0sss 222w2

br =+=+= [psi/v] [3]

The interpretation of sr is as follows: all of the random effects associated with the

measurement process amount to (21.08) = 2.16 [psi/v] with a confidence level of 95%. The

2 term means that two standard deviation values are required to achieve the 95% level ofconfidence. If the sbterm is not accounted for the uncertainty is underestimated, s r= 1.03 instead

of 1.08. While this omission results in a minor difference in the current example, it could be more

significant in other applications.

Statistical Process Control

In addition to providing estimated uncertainty values, the above analysis is used to develop

control charts that monitor process consistency. The application of this well developed technique


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to the measurement process is described in Reference 4, a pressure transducer applications in

Reference 5. The solid lines in Figs. 2 and 3 represent control limits; the calculation procedure is

described in the references. All the data points are between the lines in Fig. 2 and below the

single limit of Fig 3. Operation within the control limits indicates that the transducer calibration

process is operating in a state of statistical control. This means that the observed variations are

due only to the inherent random effects that are part of the process.

Separation of Components

The ability to separate short and long term random effects is important in some applications.

For example, laboratory testing of flowmeter installation effects involves one test at baseline

conditions and a second test at distortedconditions. The installation effects associated with the

distorted conditions are calculated as the difference in flowmeter performance. The uncertainty

associated with the difference between two measurements made with the same instrument will

include only short term random effects provided the measurements are made during a short time.

Systematic Effects

The reported (sr) value of standard deviation includes all random effects present during

calibration. It is typically assumed that the same effects will be present when the transducer isused to make a measurement. The combined uncertainty in pressure measurement includes

systematic effects as well as the reported standard deviation. Systematic effects contrast with

random effects in that they do not contribute random variability in the final result. There are three

systematic effects present in a typical pressure measurement. The first is the calibration standard,

usually a deadweight tester. The second is the data acquisition system, usually a flow computer.

The third systematic effect accounts for ambient temperature effects present when the transducer

is used. All three effects are determined from manufacturer specifications.

FLOWMETER CALIBRATION

The previous example concerned a pressure transducer; this section contains an example based

on a flowmeter calibration. The previous discussion concerning long and short term effects are

fully applicable to flowmeter calibrations, a recent example is described in Reference 6.

There are a number of differences between pressure transducer and flowmeter calibration.

First, the flowmeter contributes more uncertainty than the pressure transducer. The uncertainty

associated with a good pressure transducer might be 0.075% while that associated with a flow

meter can be as high as 1%. Second, the pressure typically shows less variation than the flowrate

in a when installed in the field. It is more important to characterize the flowmeter over the

operating range. Finally, a flow calibration is more costly than a pressure calibration, especially

with a large meter. It is important that the flow calibration provides as much information as

possible.

Full Scale and Percent Reading Effects

The performance of any instrument is characterized by both full-scaleandpercent-of-readingeffects. A percent-of-reading effect acts equally over the entire range while a full-scale effect

dominates at the low end of the range. The manufacturers specifications in Fig. 1 of the firstpaper1contained both effects; they are also present in ultrasonic and turbine flowmeters. Typical

data contained in Figure 4, the solid lines represent the 95% confidence interval, which is the

statistical interval that contains 95% of the data. The interval width varies with flowrate due to

full-scale effects associated with the meter. A properly designed calibration process needs to

determine both the full-scale and percent-of-reading effects


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A typical flowmeter calibration begins with the selection of several velocity or flowrate

points. Multiple samples are obtained under steady conditions (flow, pressure, temperature). The

total sample time, the sample rate and the total number of samples are dependent upon cost and

uncertainty issues associated with the operation of the lab. These issues are discussed in this

section based on an example. Assume that calibration conditions are held steady and 512 samples

are obtained. In the present example a set of random numbers simulate the multiple samples. The

standard deviation of the 512 samples (random values) is calculated to be 0.128%. Theinterpretation of standard deviation is as follows: if the process continues to hold steady, 95% of

any additional samples will lie within 20.128 = 0.256% of the mean value (of the 512samples).

The example continues by grouping the samples, for example, 16 groups of 32 samples each.

A mean value is calculated for each group. A standard deviation is calculated for the 16 group

means, the value is sb= 0.061%. The interpretation of this standard deviation is as follows: if the

process continues to hold steady, 95% of any additional groups will lie within 20.061% =

0.122% of the mean value (of the 16 groups).

Each of the 16 groups of 32 samples is characterized by a standard deviation; they range in

value from 0.099% to 0.141%. An average value of sw= 0.116% is calculated for the current

example. The grouping of data within the present example results in the same two types of

standard deviation as with the pressure transducer example. The standard deviation, sw, accounts

for random effects within a group while sbaccounts for random effects between groups.

The present example continues further by organizing the 512 data points into different sized

groups and calculating standard deviation values. The results are summarized in Table 1. The first

two columns identify the quantity and size of sample groups. The final three columns contain

values for the three standard deviations. The lowest reported values (sr= 0.130%) are observed

when the data are not grouped. Slightly larger reported values are observed for the shaded rows

(6-16 groups, 32-85 samples per group). If either the group size or the samples per group

becomes too small, the reported value increases. This increase is a result of statistical

considerations beyond the scope of this paper.

Time Scales

Various time scales can be associated with the 512 samples; the same uncertainty

considerations apply regardless of the time frame. If the samples were obtained at a rate of

between one and five seconds per sample the total time would be between 500 and 2500 seconds

(between 8 and 40 minutes). This range of time scales would represent a typical calibration

process. In a calibration the termgroupis replaced by the term data point. If 512 samples were

obtained in ten minutes, eight data points based on 64 samples each would be reported.

A sample rate between 5 and 10 samples per second would result in a total time of between 50

and 100 seconds to obtain 512 samples. A group of samples might represent, for example,

samples leading to a single measurement made by the voltmeter that is part of the flow computer.

The sw value would form part of the uncertainty associated with the voltmeter and sb wouldrepresent the variation in the voltage being measured.

Finally, the samples could be obtained over a long period of time, perhaps many months. Each

group of samples would represent a complete calibration. This scenario is similar to the pressure

transducer example previously described.

There are significant differences between the calibration of a flowmeter and the use of that

flowmeter in the field. Calibration data are obtained for a limited time involving a very small

volume of gas. The flowrate is the parameter of interest and it is controlled to a steady value as


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data are obtained. Operation in the field is based on total volume measured over a long time. The

flowrate is less important a parameter than the total volume, and it can vary depending on the

application.

The correct interpretation of the calibration sample time depends on the application of the

flowmeter. The optimum sample time is a value that represents the shortest time period under

which the meter will measure steady flow. In general larger meters exhibit slower changes thansmaller meters and custody transfer applications exhibit smaller changes than production

applications. When the calibration sample time is appropriate to the meter application, the

contribution of sw can be reduced by the term1n

1

where n is the number of samples that

make up the data point. This reduced value of uncertainty is associated with the mean value of the

n samples.

This interpretation of sample time is illustrated in Figure 5. The thick line represents flowrate

changing over time. The initial slope is steeper than the final slope indicating a larger initial

change in flowrate over time. Two boxes are superimposed on the flowrate vs. time curve. The

height of each box represents a small range of flowrate that is considered steady. The width of

each box is subdivided into equal intervals, each representing one calibration sample timeinterval. The initial slope is considered steady for four calibration samples while the final slope is

considered steady for thirteen calibration samples. The swvalue resulting from the calibration of

the flowmeter can be reduced by the following values:

577.014

1=

if the initial slope represents typical flowmeter field operation

289.0113

1=

if the final slope represents typical operation

SUMMARY

The paper used two numerical examples to illustrate uncertainty concepts associated withcalibration. The mean and standard deviation of a series of samples were defined. Several

interpretations of the standard deviation were discussed. The standard deviation of the mean

differs from that of a single sample. Long and short term random effects are associated with

varying time scales. Full-scale effects vary over the range of an instrument. Percent-of-reading

effects are consistent over the range. The paper concluded with a discussion of how to interpret

the uncertainty of calibration results.

REFERENCES

1. Kegel, T. M., Uncertainty Analysis of Turbine and Ultrasonic Meter Volume

Measurements, AGA Operations Conference, Orlando, FL, May, 2003.

2. ___, Guide to the Expression of Uncertainty in Measurement, International Organization for

Standardization, 1994.

3. Wadsworth, H. M., Handbook of Statistical Methods for Engineers and Scientists, McGraw-

Hill, 1990.

4. Croarkin, Carroll, Measurement Assurance Programs, Part 2: Development and

Implementation, NBS Special Publication 676-2, 1985.


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5. Kegel, T. M., Statistical Control of a Differential Pressure Instrument Calibration Process,

45thInternational Instrumentation Symposium, Albuquerque, New Mexico, May, 1999.

6. Kegel, T. M., Quality Control Program of the CEESI Ventura Calibration Facility,

FLOMEKO, Gronigen, The Netherlands, May 2003.

197

0 200 400 600 800 1000

Applied Pressure [psia]

Figure 1: Typical Pressure Transducer Calibration Data

199

201

203

Sensitivity[psi/volt]


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197

199

201

203

Nov-98 Nov-99 Nov-00 Nov-01 Nov-02

Date

Mean[psi/v]

Figure 2: Pressure Transducer Calibration History Mean Values

1.6

si/v

0.4

0.8

1.2

Nov-98 Nov-99 Nov-00 Nov-01 Nov-02

Date

StandardDeviation[p

Figure 3: Pressure Transducer Calibration History Standard Deviation Values


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Table 1: Standard Deviations of Grouped Samples

QuantityGroups

Samples perGroup

Between

StandardDeviation

Within

StandardDeviation

Reported

StandardDeviation

512 1 0.130

256 2 0.098 0.120 0.155

128 4 0.077 0.120 0.143

64 8 0.066 0.119 0.136

32 16 0.060 0.119 0.133

16 32 0.057 0.119 0.132

10 51,52 0.057 0.119 0.132

8 64 0.057 0.119 0.132

6 85,86 0.059 0.118 0.132

5 102,103 0.063 0.118 0.134

4 128 0.058 0.120 0.133

3 170,171 0.064 0.119 0.135

1 512 0.130

0.4

-0.4

-0.2

0.0

0.2

0 20 40 60 80 100Velocity [ft/s]

Residual[%]

Figure 4: Typical Ultrasonic Flowmeter Calibration Data


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13 CALIBRATION SAMPLES

FLOWRATE VS TIME

CURVE

4 CALIBRATION

SAMPLES

Figure 5: Interpreting Calibration Sample Rate

126 uncertainty analysis

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