Sharing information about uncertainty budgets for flow meters could lead to a better understanding of measurement systems, says NEL’s Callum Hardie.
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Photo from NEL. |
While flow meters are calibrated under ideal laboratory conditions, the environments into which they are installed vary greatly. Uncertainty analyses are therefore essential to determine whether measurement systems, once installed, are capable of meeting required performance targets.
However, when developing uncertainty budgets for new measurement systems, it is often difficult to obtain reliable data to provide such evidence. Two main methods are used, the analytical method and the Monte Carlo method.
The analytical method
Calculating uncertainty using the analytical method is described in detail in the Guide to the expression of Uncertainty of measurement (GUM), produced by Working Group One of the Joint Committee for Guides in Metrology.
The technique involves a series of steps:
1. Define the relationship between all the inputs to the measurement and the final result.
2. For each input, draw up a list of all the factors that contribute to the uncertainty in that input.
3. For each of the uncertainty sources make an estimate of the magnitude of the uncertainty.
4. Convert the uncertainties to standard uncertainties by assigning a probability distribution to each uncertainty source.
5. From the relationship defined in step 1, estimate the effect that each input has on the measured result. This is usually achieved by calculating sensitivity coefficients.
6. Combine all the input uncertainties using the root sum squared technique to obtain the overall uncertainty in the final result.
7. Express the overall uncertainty as the interval about the measured value, within which the true value is expected to lie with the required level of confidence.
The uncertainty budgets created using the analytical method are very useful tools for optimizing measurement systems, as the effect of changes in input uncertainties on the output uncertainty can be seen very quickly. The input uncertainty sources can also be ranked to determine which sources have the most significant effect on the overall uncertainty. The process of developing uncertainty budgets can also be beneficial in that it helps to gain a full understanding of how the measurement system works.
The Monte Carlo method
The Monte Carlo method is an alternative way to estimate measurement uncertainties and is described in supplement 1 to GUM.
The method also involves a series of steps:
1. Define the relationship between all the inputs to the measurement and the final result.
2. For each input, draw up a list of all the factors that contribute to the uncertainty in that input.
3. For each of the uncertainty sources make an estimate of the magnitude of the uncertainty.
4. Assign a probability distribution to each of the uncertainty sources.
5. Use a random number generator to assign a “measured value” for each input variable based on its uncertainty value and probability distribution.
6. Calculate the final result using the “measured values” as inputs.
This process is repeated many times, until there is enough data to analyze the output distribution. The uncertainty in the final result can then be estimated by calculating the standard deviation of the output data.
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Comparison between analytical and |
The Monte Carlo method has some advantages, for example, it shows the distribution in the output which can be used to view whether the distribution is skewed or rectangular in shape. This technique is also particularly useful when the uncertainties are large compared with the measured values.
It has been shown that the combined use of the Analytical and Monte Carlo methods can be useful. The advantages of carrying out both methods on the same system are:
The table above right, shows the comparison between the analytical and Monte Carlo methods to calculate the uncertainty of a turbine meter. It can be seen that the agreement is within 0.015%. This agreement helps to increase confidence in the uncertainty calculation.
Uncertainty analysis with no historical data
Whatever method is used to calculate uncertainty, one of the most important stages is to estimate the magnitude of uncertainty sources. These estimations can be made from various different sources of information, including historical data from calibrations and verifications. Where a measurement system is newly installed, or if the historical data has not been well documented or is missing, then estimates of the magnitude of uncertainty sources have to be made using other sources of information, such as manufacturers’ specifications, engineering judgment or data from similar measurement systems.
Using historical data to carry out uncertainty analyses leads to many advantages. It will allow the operator to determine a more accurate estimation of uncertainty on measurement systems. While other methods of estimating uncertainty sources, such as using a manufacturer’s specifications, engineering judgment or data from similar systems, are all acceptable in the GUM, the assumptions made using these methods will lead to a less accurate estimation of uncertainty.
Having an accurate, evidence-based estimation of uncertainty for a measurement system will allow cost-effective improvements to be made to the system in the future. The uncertainty sources can also be ranked to determine which sources contribute most to the overall uncertainty in the system. The uncertainty of these sources can then be improved first ensuring that time and money is not wasted on improving the uncertainty of less significant sources.
Analyzing historical data will also bring benefits to maintenance teams. Typically, maintenance (including verifications and calibrations) is carried out very frequently, especially when a new system is installed, with the frequency reduced over time if the instrument passes verification checks.
However, by analyzing the stability of instruments over a period of time, calibration schedules can be determined based on evidence, rather than choosing arbitrary time periods. This means that instruments that are proving to be less stable can be calibrated more frequently, while more stable instruments can be calibrated less frequently, leading to improved safety procedures as it will avoid the need to break into the line, which involves isolation and depressurization.
The use of historical data also means that if new systems are installed with identical equipment, then evidence will be immediately available for determining initial calibration schedules. If the stability of instruments is likely to reduce over time, then such data will also help determine when these instruments need to be replaced.
Using historical data to carry out uncertainty analyses will also benefit the industry as a whole as increased knowledge of the uncertainty of measurement systems will lead to more effective allocation principles in shared pipelines. It will also help regulators to set regulations which are suitable and achievable based on current industry best practice.
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NEL’s flow measurement facilities, at East Kilbride, Glasgow. |
Effective analysis
The estimation of uncertainty in measurement systems can be greatly improved when historical data is available and should be used whenever possible to estimate uncertainty sources. Uncertainty analysis should also be seen as an iterative process, with uncertainty budgets being updated whenever new calibration data is available or changes are made to measurement system.
This means that historical records of calibrations should be kept in good order so that they can be analyzed at regular intervals, which should already be the case if the system is audited. If a system is new or calibration data is not available, uncertainty sources can be estimated by other methods. However the uncertainty values should be updated over time as more historical data becomes available, and it is recommended that as a minimum uncertainty budgets should be reviewed annually to ensure they are still relevant and accurate.
While performance data is generally available on the manufacturer’s datasheet, different manufacturers present the data in diverse forms. For example, a coverage factor and/or the stability of the instrument over time is not always given. Manufacturers should therefore make available more data on the performance and stability of instruments over time. This will allow more accurate estimates of uncertainty when historical data is not available.
There should also be more open sharing of data across industry, as this will lead to better understanding of measurement systems which will be beneficial for buyers, sellers, and pipeline users. This could involve the development of a calibration database, which would be very beneficial to the industry as a whole, especially when specifying and selecting new equipment.
Calum Hardie is a flow measurement engineer at TÜV SÜD Group company NEL, which provides technical consultancy, research, testing, and program management services.
Calum graduated with a degree in Mechanical Engineering from the University of Dundee and has been at NEL since 2008. Since then he has gained experience in a number of technical areas including flowmetering, measurement uncertainty, allocation, valves and erosion.
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