POWER UNDERSTANDING MEASUREMENT UNCERTAINTY IN DP FLOW DEVICES

Transcription

1 Proceedings of the ASME 2014 Power Conference POWER2014 July 28-31, 2014, Baltimore, Maryland, USA POWER UNDERSTANDING MEASUREMENT UNCERTAINTY IN DP FLOW DEVICES Michael S. Hering Rosemount DP Flow Emerson Corp. Boulder, CO, USA David R. Mesnard Rosemount DP Flow Emerson Corp. Boulder, CO, USA ABSTRACT This paper examines the concepts and methods involved in the uncertainty analysis of a differential pressure flow measurement, including which variables have the greatest influence on the uncertainty, and how to effectively manage the influence of error sources on the measurement. An uncertainty analysis will provide the user with a good measure of actual and/or potential system performance and can be instrumental in identifying potential areas of improvement in system performance. The wide availability of instruments of all types and levels of performance presents a difficult decision to those who are responsible for project success and budgets. Optimizing the investment in instrumentation is possible when the required performance is known based an appropriate uncertainty analysis. INTRODUCTION An uncertainty analysis can be a great tool to evaluate the performance of prospective measurement technologies and can be used both as a process optimization tool as well as a cost control measure. For measurements that involve several instruments, it is not always easy to determine the uncertainty of the resulting measurement. This paper will attempt to provide an introduction to the process of performing an uncertainty analysis and the resulting insight into the measurement it can provide. Differential pressure flow measurement is somewhat unique in that it involves the measurement of several independent variables, which together with the appropriate equation, calculate the flow rate. An uncertainty analysis can make it easier to predict the effect of uncertainties in the independent variables on the uncertainty of the measured flow. BACKGROUND & BASIC CONCEPTS (DEFINITIONS) To begin with, there are some basic uncertainty terms and concepts that the reader should be familiar with: - Systematic Uncertainty, sometimes referred to as Bias and often represented by the symbol b, represents an uncertainty that is inherent in the measuring instrument. It is considered fixed or unchanged due to random effects and is normally determined as part of an instrument calibration to a traceable standard. - Random uncertainty is represents the scatter of the measurements and is typically determined from the standard deviation of the measured values. For a normally distributed sample, a single standard deviation interval covers 68% of the data population. The random uncertainty is usually represented by the variable S. It should be noted that, for both the systematic and ransom uncertainties, the values are estimates based on a finite number of samples. - Absolute Uncertainty is uncertainty expressed in units of measure, such as cm for length. - Relative Uncertainty is uncertainty expressed as a percentage of the measured parameter. - Data Distribution: There are several basic types of data distribution; however, for purposes of this paper it will be assumed that all data is normally distributed. Figure 1 illustrates a sample set of 100 measurements. When the data in Fig. 1 are plotted in a histogram, a normal data 1 Copyright 2014 by ASME

2 distribution will appear as shown in the plot shown in Fig. 2. Fig. 1: Normally Distributed Data Fig. 2: Histogram of a Normal Data Distribution - Confidence Interval: Represents the percentage of the population that is enclosed by the upper and lower error limits. Common confidence intervals are shown in Table 1. Table 1 Confidence Interval (Sigma) Standard Deviations Confidence Interval 68.27% 1- Sigma % 95.45% 2- Sigma % 99.73% 3- Sigma 3 - Coverage Factor: A multiplier which is used to expand the standard uncertainty to include a higher confidence interval. The factor is determined from the Student s T distribution, based on the desired confidence and degrees of freedom (). The factor is represented by the variable k p, where p is a subscript use to represent the probability associated with the confidence interval, e.g. k 95 would represent the coverage factor for a 95% confidence interval. Figure 3 and Table 2 illustrate the relationship between the degrees of freedom and the coverage factor. Fig. 3: - Combined Uncertainty (u c ): To determine the actual uncertainty of the measurement, the random and systematic uncertainty components must be combined. This is done via the Root-Sum-Square method as follows: (1) The uncertainty represented by this calculation will enclose 68% of the data points. - Degrees of Freedom (): represents the number of independent pieces of information that go into the estimate of a parameter. When a parameter is calculated as a mean of N measurements, then the degrees of freedom is expressed as N-1. Table 2: Coverage Factors Confidence Degrees of Freedom (v) Interval =120 =60 =30 =15 = % % % % % Expanded Combined Uncertainty (U c ) represents the uncertainty limits required to enclose the desired percentage of the population (confidence interval). The expanded combined uncertainty is determined by multiplying the combined standard uncertainty (u c ) by 2 Copyright 2014 by ASME

3 the appropriate coverage factor (k p ). The expanded uncertainty often includes a subscript indicative of the confidence interval, such as (U c,95 )., (2) - Uncertainty of a Mean Value: is the random uncertainty component that predicts the upper and lower error limits within which the mean value is expected to remain, assuming that future data follows the same trends as past data. Using mean values and assigning the uncertainty to the mean value can reduce the uncertainty attributed to that variable. If the average value of a variable is used to calculate a result, or if the average value is the actual parameter of interest, then the uncertainty of the mean value is applicable. The random uncertainty of a mean of X, where S x is the standard deviation of X, and N is the number of samples: Fig. 4: (3) EXAMPLE UNCERTAINTY OF A DIRECT MEASUREMENT As an example of an uncertainty calculation, consider the measurement of steel rods as they are cut by an automatic saw. The rods are measured with a caliper once they are cut, and the data is recorded for a quantity of 100 bars. We will use the data shown in Fig. 1. In this example the systematic uncertainty is taken from the calibration certificate of the caliper, and the random uncertainty is the standard deviation of the data set. - Mean Value = cm - Systematic Uncertainty (b i ) = cm (calibration uncertainty of caliper) - Random Uncertainty (S i ) = cm (standard deviation of measurements) - The combined standard uncertainty is calculated as: Since we have 100 samples, the degrees of freedom (n) is 100 1=99. If we wanted to know the uncertainty to a 95% confidence, then we would need to apply the coverage factor calculated for n=99 and 95% probability, which is Note that the spreadsheet (MS Excel) function to determine coverage factor is TINV. - The Expanded Combined Uncertainty is then calculated as:, UNCERTAINTY OF A RESULT To illustrate the principle involved in calculating the uncertainty of a result, the volume of a cone will be used as an example: - Uncertainty of a Calculated Result: In many applications the variable to be output is actually calculated from several measured parameters. In this case the uncertainties of all the parameters must be taken into account when calculating the overall uncertainty. This is further complicated by the fact that parameters often are associated with exponents, which affects their influence on the uncertainty of the result. - Sensitivity Coefficients ( i ): Represent the effect a change in a particular variable will have on the calculated result. Sensitivity Coefficients can be determined either from numerical analysis or from partial differentiation of equations. To perform a numerical uncertainty analysis, the following calculations are performed: - For a radius (r) = 100 cm and a height (h) of 100 cm, the volume of the cone is cm 3. - If r is changed by 1% to 101 (h remains unchanged), the volume is cm 3, which represents a change in volume of 2%. Therefore the relative sensitivity of V to r is 2. - If h is changed by 1% to 101 (and r remains at 100), the volume becomes cm 3, which represents a change in volume of 1%. Therefore the relative sensitivity of V to h is 1. 3 Copyright 2014 by ASME

4 The conclusion that can be reached from the above data is that the volume is more sensitive to changes in radius than it is to changes in height, which means that any uncertainties in the measurement of the radius would have twice the impact on the uncertainty of the volume than uncertainties in the height. To account for these effects in an uncertainty calculation is the function of the sensitivity coefficient, which is a method of weighting the impacts of various parameter uncertainties on the uncertainty of the calculated result. As can be seen in the example above, the sensitivities are generally related to the exponent associated with the variable. SENSITIVITY COEFFICIENTS In the example above, it was fairly straightforward to numerically evaluate the sensitivity of volume to changes in radius and height; however, it is often more convenient to mathematically determine a constant or equation that defines the sensitivity of the result to each parameter. This allows the sensitivities to more easily be evaluated over a wider range than when derived numerically (where in some cases they are only valid at a single point). The sensitivity coefficients are calculated by partially differentiating the equation with respect to each variable. The following example also illustrates the difference in the equations between the absolute sensitivity coefficient (result units/variable units) and the relative sensitivity coefficient (dimensionless - percent). - For an Equation in the form (4) - Absolute Sensitivity Coefficient of R to x is: The uncertainty of a result is determined from the component uncertainties using the following equation: (7) If it is assumed that the uncertainty of both the radius and the height is 1%, then the uncertainty of the volume would be calculated as follows: 1 1% 2 1% % Based on the above example, it is evident that sensitivity coefficients can have a significant impact on the overall uncertainty of a measurement, and that without determining the sensitivity coefficients; one may be unaware of a significant source of error. - Relative Sensitivity Coefficient of R to x is: (5) (6) DIFFERENTIAL PRESSURE FLOW APPLICATION - ORIFICE PLATE EXAMPLE Now it is time to put the previous information to practical use in analyzing some DP Flow installations. The first example will look at an Orifice Plate installation. The standard differential pressure flow equation for mass flow through an orifice [1] is: - From the cone example above, the relative sensitivity coefficient of the radius is: Similarly, it can be shown that the relative sensitivity coefficient of the height is: 2 (SI Units) (8) Where; Q m C E d = Mass flow Rate (kg/s) = Meter Discharge Coefficient (dimensionless) = Gas Expansion Factor (dimensionless) = Velocity of Approach Factor 1 1 = primary element throat diameter (m) 4 Copyright 2014 by ASME

5 f = fluid density (kg/m 3 ) p D = differential pressure (Pa) = Beta Ratio (d/d) (dimensionless) = Pipe inside Diameter (m) For liquid applications, 1, and, so for volumetric liquid measurement the above equation can be written as: 2 (9) From [1], the relative uncertainty of the Volumetric Flow as measured by an orifice is as follows: (10) In the above equation the values such as D/D represent the uncertainty of the various parameters, and are equivalent to the terms presented earlier. Note that for pipe diameter (D) and orifice bore diameter (d), the sensitivity coefficients are functions of the beta ratio (). This is a consequence of the velocity of approach term (E), which includes both d and D. Refer to the appendix for the mathematical derivation of the sensitivity coefficients. Before beginning the analysis there are some items to take into consideration: - Ensure that all uncertainties have been converted to a common confidence interval. If the various uncertainties are all known to differing confidence intervals, it is usually best to convert them to standard uncertainties (e.g. divide a 95% confidence uncertainty by 2 to obtain a standard uncertainty). - The methods in this paper use the relative uncertainties, so ensure that all uncertainties are expressed in relative terms (%). - The source of uncertainty information can come from a variety of sources, including published standards, manufacturer s specifications, and existing test data. For this example, the following uncertainties will be assumed: - It will be assumed that the orifice has been installed per ASME MFC-3M recommendations and that the C uncertainty specified in this document can be used. This standard specifies a +/- 0.75% uncertainty to a confidence of 95%, so it should be divided by 2 to get the standard uncertainty of ± 0.375%. - Pipe Diameter (D) is ± 0.20% standard uncertainty. (based on ASME MFC-3M requirements) - Bore Diameter is ± 0.10% standard uncertainty. (based on ASME MFC-3M requirements) - A water temperature range of 13⁰C to 30⁰C results in a density standard uncertainty of ± 0.20%. - The DP transmitter will require a different approach, as the uncertainty is generally given as a percent of span. This means that the relative uncertainty should be evaluated over a range of operating conditions that cover the anticipated operating conditions. Table 3: DP Transmitter Standard Uncertainty % Flow Diff Uncertainty Uncertainty (Pa) (Pa) (%) 100% ± 0.01% 25% ± 0.19% 12.5% ± 0.78% 10% ± 1.22% The next step is to evaluate the sensitivity coefficients. In the case of the orifice plate, the sensitivity coefficients for D and d are functions of beta (), so depending on the value of beta, the sensitivities will differ. The values in Table 4 below were evaluated at a beta of 0.5. Table 4: Orifice Sensitivity Coefficients Error Source Sensitivity Coefficient Description Symbol Equation Value Pipe Diameter D Bore Diameter d Diff. Pressure Density Discharge Coeff. C - 1 If the sensitivity coefficients for bore and pipe diameter are evaluated at various betas, it can be seen that both pipe diameter and bore diameter uncertainties will have a greater impact on the combined uncertainty at higher values of beta, as shown in Fig. 5 on the following page. 5 Copyright 2014 by ASME

6 Fig.5: Table 6: Error Source Uncertainty (%Flow) Name & Symbol 100% 25% 12.5% 10% Combined Uncertainty Coverage Factor (95%) Expanded Combined Uncertainty 0.40% 0.41% 0.56% 0.73% % 0.82% 1.12% 1.46% Once the sensitivity coefficients have been calculated, the uncertainty contributions and overall uncertainties can be evaluated. This can be done on a spreadsheet such as the one shown below. If the uncertainty contributions are plotted on a bar chart as shown below, the significant contributors to the uncertainty can be identified. Fig. 6: Table 5: Error Source Uncertainty Contribution (%Flow) Name & Symbol 100% 25% 12.5% 10% Pipe Diameter Bore Diameter Differential Pressure 7.08E E E E E E E E E E E E-05 Density 1.00E E E E-06 Discharge Coefficient 1.41E E E E-05 Fig. 7: Sum of Squares Combined Uncertainty 1.63E E E E % 0.41% 0.56% 0.73% The combined uncertainty shown above is a standard uncertainty (68% confidence interval). To determine the uncertainty for a larger confidence interval, the standard uncertainty must be multiplied by the appropriate coverage factor as shown in Table 6. In the figures above, it can be seen that the uncertainty 6 Copyright 2014 by ASME

7 contribution of pipe diameter, which is relatively insignificant at 0.5 beta, becomes more significant at the higher beta of In the charts above, it is also obvious that the differential pressure uncertainty contribution becomes increasingly more significant at higher flow turndown (i.e. the ratio of max flow to min flow, such as 10:1). In the case of high flow turndown, the overall measurement uncertainty can be reduced at lower flows by utilizing a DP transmitter which has an uncertainty specification in % of reading rather than % of span. two transmitters of different ranges (a high range and a low range), however this adds additional complexity in terms of installation and electrical connections. It also illustrates that for small flow turndowns, the selection of the DP transmitter is not as critical as for larger turndowns. Fig. 10: Fig. 8: Although the transmitter with a % reading uncertainty specification still has a slight dependency on the DP as a percentage of URL, it is greatly reduced. The following chart illustrates the difference over a 14:1 flow turndown. Fig. 9: DIFFERENTIAL PRESSURE FLOW APPLICATION - AVERAGING PITOT TUBE EXAMPLE The following example will look at an Averaging Pitot Tube installation and how it will differ from an orifice plate installation. The standard differential pressure flow equation for mass flow through an averaging pitot [2] is: 2 (SI Units) (11) Where; Q m K D = Mass flow Rate (kg/s) = Meter Flow Coefficient (dimensionless) = Gas Expansion Factor (dimensionless) = Pipe Diameter (m) f = fluid density (kg/m 3 ) p = differential pressure (Pa) When this comparison is extended to the overall measurement uncertainty, it illustrates (Fig. 10) the large reduction in measurement uncertainty that can be achieved in large turndown flow measurement applications by careful selection of the DP transmitter. Similar results can be achieved by using For liquid applications, 1, and, so for volumetric liquid measurement the above equation can be written as: 2 (12) This equation is much more straightforward to differentiate than the orifice equation, as it does not contain the velocity of 7 Copyright 2014 by ASME

8 approach factor, which has a dependency on d and D. to The relative uncertainty of the Volumetric Flow as measured by an averaging pitot is as follows: Error Source Table 8: Uncertainty Contribution (%Flow) 2 (13) In the above equation the values such as D/D represent the uncertainty of the various parameters, and are equivalent to the terms presented earlier. Note that compared to the orifice example, the sensitivity coefficient of pipe diameter (D) is no longer a function of another parameter. For this example, the following uncertainties will be assumed: - For the Flow Coefficient K, a common uncertainty (95% confidence) is ± 1.0%, or ± 0.50% standard uncertainty. - Pipe ID is ± 0.20% standard uncertainty. On a 6 pipe, this represents a measurement uncertainty of ± 0.30 mm. - A water temperature range of 13⁰C to 30⁰C results in a density standard uncertainty of ± 0.20%. - The flow rates will be set to give the same differential pressures as shown in Table 3 of the orifice example. - Sensitivity Coefficients are as follows: Table 7: Averaging Pitot Sensitivity Coefficients Error Source Sensitivity Coefficient Description Symbol Value Pipe Diameter D 2 Diff. Pressure 0.5 Density -0.5 Flow Coeff. K 1 Similar to the orifice example, the uncertainty contributions and overall uncertainties are evaluated. Name & Symbol 100% 25% 12.5% 10% Pipe Diameter Differential Pressure 1.60E E E E E E E E-05 Density 1.00E E E E-06 Flow Coefficient Sum of Squares Combined Uncertainty 2.50E E E E E E E E % 0.66% 0.76% 0.89% The combined uncertainty shown above is a standard uncertainty. To determine the uncertainty for a larger confidence interval, the standard uncertainty must be multiplied by the appropriate coverage factor as shown in Table 9. Error Source Table 9: Uncertainty (%Flow) Name & Symbol 100% 25% 12.5% 10% Combined Uncertainty Coverage Factor (95%) Expanded Combined Uncertainty 0.65% 0.66% 0.76% 0.89% % 1.32% 1.52% 1.78% If the uncertainty contributions are plotted on a bar chart as shown below, the significant contributors to the uncertainty can be identified. 8 Copyright 2014 by ASME

9 Fig. 11: The Pipe ID uncertainty contribution in the unmeasured Pipe ID case dwarfs all of the other uncertainty contributions. It is easy to see that in this case, not measuring the pipe inside diameter has caused a huge increase in overall uncertainty that the user may not be aware of if they have not done an analysis. In Fig. 13 it is shown that the pipe ID uncertainty must be reduced to about ±.25% to bring its uncertainty contribution in line with that of the other parameters. Fig. 13: When compared to the orifice case, it can be seen that the pipe diameter contributes to a much larger degree to the overall uncertainty of the measurement. The sensitivity coefficient of pipe diameter is 2 for the averaging pitot, compared to 1 for the 0.75 beta orifice, and for the 0.50 beta orifice. The reason for this is that on an averaging pitot, the pipe is the throat of the meter, whereas for an orifice, the machined bore is the throat. For averaging pitot tubes that are field-installed in existing piping, this is very significant in that the actual pipe inside diameter needs to be carefully measured during the installation process to avoid potentially high bias type error. As an illustration, let s compare an averaging pitot installation where the pipe inside diameter has been carefully measured to an installation where the pipe inside diameter is assumed to match the nominal ID of that pipe schedule. On a 6 S40 pipe, the nominal wall thickness of 7.1 mm is generally subject to a ±12.5% mill tolerance. The result is a range of potential pipe ID s of mm to mm, which represents a standard uncertainty of the nominal ID (154.1 mm) of ±1.1%. Fig. 12: The chart in Fig. 14 illustrates the comparison of the expanded combined uncertainty of the averaging pitot installation for the case of different pipe diameter uncertainties, and illustrates that careful measurement of the diameter can produce large reductions in uncertainty. In Fig. 14 the difference between the 1.1% and 0.2% pipe diameter uncertainty curves represents only a 1.4 mm difference in measurement uncertainty in a 6 pipe but results in a 3-fold increase in flow measurement uncertainty. The.5% standard pipe diameter uncertainty only represents a 0.5 mm increase in uncertainty compared to the 0.20% uncertainty, yet it almost doubles the flow measurement uncertainty. Fig. 14: 9 Copyright 2014 by ASME

10 Figure 14 also shows that the benefit of a higher performance transmitter can be completely negated by an inadequate pipe ID measurement, and that as the uncertainty of the pipe ID increases, the benefit of the lower uncertainty transmitter is reduced, as well as possibly any benefit of calibrating the averaging pitot tube. CONCLUSIONS In the preceding examples, uncertainty analyses were used to provide information on the system that may not have been evident otherwise. - In orifice flow measurement, the beta ratio has an effect on the sensitivity coefficient for bore diameter and pipe diameter, such that higher beta ratios result in bore and pipe diameter uncertainties having a more significant effect on overall uncertainty. REFERENCES [1] ASME MFC-3M (2004): Measurement of Fluid Flow in Pipes using Orifice, Nozzle, and Venturi [2] ASME MFC-12M (2004): Measurement of Fluid Flow in Closed Conduits Using Multiport Averaging Pitot Primary Elements [3] ASME PTC 19.1 (2005): Test Uncertainty [4] NIST TN 1297 (1994): Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results [5] ASME PTC 19.5 (2004): Flow Measurement [6] ISO Guide to the Expression of Uncertainty in Measurement - In orifice flow measurement at high turndowns, the DP uncertainty grows rapidly as flow is reduced. A high performance transmitter (with uncertainty as percent of reading) can be an excellent way to reduce uncertainty at low flows. - In averaging pitot flow measurement, the pipe inside diameter uncertainty has the greatest effect on the overall measurement uncertainty. Especially in smaller line sizes, very accurate measurement of the inside diameter is essential if absolute minimum error to a standard is required. If the pipe inside diameter is not measured (nominal diameter is used), the resulting flow measurement uncertainty can be several multiples of the flow meter reference accuracy. The measurement will still have good repeatability, however there is likely to be a bias to the measurement. - For liquid flow measurement, density uncertainties contribute very little to overall uncertainty, so there is typically not much to be gained by adding temperature compensation to liquid flow measurement. 10 Copyright 2014 by ASME

11 ANNEX A DERIVATION OF SENSITIVITY COEFFICIENTS FOR ORIFICE PLATE LIQUID VOLUMETRIC FLOW EQUATION Derivation of Sensitivity Coefficients for Orifice Plate Liquid Volumetric Flow Equation: 4 2 1) Relative Sensitivity Coefficient of to D: Since D is a component of Beta, and 1 1, must be partially differentiated with respect to D to obtain the sensitivity coefficient. Using equation (6) in the text to determine the relative sensitivity coefficient: 4 2 The term 2 can be simplified to : A. Determine by substituting 1 1 for (E): B. Determine derivative of 1 with respect to D: 1 4 C. Determine derivative of with respect to D: D. Substitute for in B: E. Substitute 1 for in A: F. Substitute for in A: Since 1 1 4, the above can be re-written as This agrees with the sensitivity coefficient in ASME MFC- 3M, except that ASME does not include the (-) sign since the value is squared during the root sum squares operation. 2) Relative Sensitivity Coefficient of to d: Since d is directly in the equation as well as being a component of Beta, and 1 1, the term must be partially differentiated with respect to d to obtain the sensitivity coefficient. Using equation (6) in the text to determine the relative sensitivity coefficient: 4 2 The term 2 can be simplified to : 1 A. Differentiate 2 using the product rule: B. Determine 2 by substituting 1 1 for (E): Copyright 2014 by ASME

12 C. Determine derivative of 1 with respect to d: 1 4 D. Determine derivative of with respect to D: E. Substitute F. Substitute G. Substitute for in C: for in B: for in A: H. Substitute for in 2: Since 1 1, the above can be re-written as: This agrees with the sensitivity coefficient in ASME MFC-3M. 3) Relative Sensitivity Coefficient of to C: Using equation (6) in the text to determine the relative sensitivity coefficient: This agrees with the sensitivity coefficient in ASME MFC- 3M. 4) Relative Sensitivity Coefficient of to : Using equation (6) in the text to determine the relative sensitivity coefficient: 4 2 The term 2 can be simplified to : This agrees with the sensitivity coefficient in ASME MFC- 3M. 5) Relative Sensitivity Coefficient of to : Using equation (6) in the text to determine the relative sensitivity coefficient: The term 2 can be simplified to : This agrees with the sensitivity coefficient in ASME MFC- 3M. The term 2 can be simplified to : 12 Copyright 2014 by ASME