9-1 GENERAL CONSIDERATIONS
It is recommended that an uncertainty analysis, following the methods of Sections 4 through 8 of this Supplement, be conducted before and after each test, according to the procedure that follows.
The pretest analysis (see subsection 4-4) is used to determine if the test result can be measured with sufficient accuracy, i.e., the predicted uncertainty should be smaller than the required uncertainty.
It may also be used to compare alternative instru-mentation systems and test designs and to deter-mine corrective action if the predicted uncertainty is unacceptably large. Furthermore, it may be used to evaluate the need for calibration. The posttest analysis (see subsection 4-4) validates the pretest analysis, provides data for validity checks, and provides a statistical basis for comparing test re-sults.
9-2 CALCULATION PROCEDURE
(a) Define Measurement Process (see section 5).
(1) Review test objectives and test duration.
(2) List all independent measurement parame-ters and their nominal levels.
(3) List all calibrations and instrument setups that will affect each parameter. Be sure to check for uncertainties in measurement system components that affect two or more measurements simultane-ously (correlated uncertainties).
(4) Define the functional relationship between the independent measurement parameters and the test result.
(b) List Elemental Error Sources (see subsection 5-3).
(1) Make a complete and exhaustive list of all possible test uncertainty sources for all parameters.
(c) Calculate the Systematic Uncertainty and Ran-dom Uncertainty (Standard Deviation of the Mean) for Each Parameter (see subsections 6-1 and 6-2).
(d) Propagate the Systematic and Random Stan-dard Deviations (see subsections 7-1 through 7-4).
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(1) The systematic uncertainty and random un-certainty (sample standard deviations of the means) of the independent parameters are propagated sepa-rately all the way to the final result.
(2) Propagation of the standard deviations of the means is done, according to the functional rela-tionship defined in step (a)(4), by using the Taylor series method (see section 7). This requires a calcula-tion of the sensitivity factors, either by differentia-tion or by numerical analysis.
(e) Calculate Uncertainty (see subsection 7-5).
(1) Combine the systematic and random uncer-tainties to obtain the total uncertainty.
(f) Report.
(1) The uncertainty analysis for each calculated result should be reported on two tables. The first is a detailed report that displays all the information used in the calculation of the nominal value and uncertainty of the result. The second is a table that summarizes the uncertainty information at the re-sult level. For most uncertainty analyses, all mea-sured parameters will have symmetric systematic uncertainties and large degrees of freedom. For some analyses, one or more of the systematic uncer-tainties may be nonsymmetric (see subsection 8-2), and for other analyses, the degrees of freedom may be small for some of the uncertainties (see Nonman-datory Appendix B). The detailed report table should include, as a minimum, the following infor-mation for each parameter used in the calculation of the result:
(a) symbol used in the calculations (b) description
(c) units
(d) nominal value (average of measure-ments), X
(e) systematic standard uncertainty, bi (f) sample random standard uncertainty, standard deviation of the mean, sx,i
(g) sensitivity,
(h) systematic standard uncertainty contri-bution to the combined uncertainty of the result, (ibi)2
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Table 9-2-1 Table of Data
Independent Parameters
Absolute Absolute
Absolute Absolute Systematic Random
Systematic Random Standard Standard
Standard Standard Absolute Uncertainty Uncertainty Nominal Uncertainty, Uncertainty, Sensitivity, Contribution, Contribution,
Symbol Description Units Value bX
i SX
i i ibX
i
2 iSX
i 2
C Discharge coefficient . . . 0.984 0.00375 0.0 140 0.276 0.0
d Throat diameter in. 3.999 0.0005 0.0 86.2 0.00186 0.0
D Inlet diameter in. 6.001 0.001 0.0 −11.4 0.00013 0.0
Water density at lbm/ft3 62.4 0.002 0.002 1.11 0.0000049 0.0000049
60°F
h Differential pressure in. H2O 100 0.15 0.4 0.6919 0.0108 0.0766
head across venturi (at 68°F)
Table 9-2-2 Summary of Data
Calculated Result
Absolute Absolute Absolute
Systematic Random Combined Absolute
Calculated Standard Standard Standard Expanded
Result, Uncertainty, Uncertainty, Uncertainty, Uncertainty,
Symbol Description Units R bR sR uR UR
m˙ Mass flow lbm/s 138.4 0.276 0.0766 0.286 0.572
rate
(i) random standard uncertainty contribu-tion to the combined uncertainty of the result, (isX)2
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The summary report table should display the information associated with the result as detailed in Tables 9-2-1 and 9-2-2 which are based on Example 10-2.
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Section 10 Examples
10-1 FLOW MEASUREMENT USING PITOT TUBES 10-1.1 Define the Measurement Process
The flow rate of an incompressible fluid in a pipe may be determined by multiplying the integrated-average velocity of the fluid by the cross-sectional flow area of the pipe. One technique for measuring the integrated-average velocity of the fluid is to traverse the cross-sectional flow area with a Pitot tube. Measurements at each traverse point can be used to determine local fluid velocity. Traverse points are typically specified at the centroid of equal areas so that the integrated-average velocity may be estimated as the integrated-average of the measured values for all traverse points.
This avoids the need to develop weighting factors for each sample area.
For this example, the velocity is measured at 40 unique traverse points (10 traverse positions along 4 equally spaced radii) corresponding to the centroid of equal areas as shown in Fig. 10-1.1.
A total of 60 measurements are taken in succession at each traverse point once the Pitot tube is posi-tioned. The point velocity values at individual traverse points are treated as measurements (Sec-tion 6); the average velocities calculated from these point velocities are treated as results (Section 7).
Several simplifying assumptions are made for this example:
(a) the pipe diameter is large compared to the Pitot tube diameter such that blockage effects and wall interference effects can be neglected;
(b) the velocity pressure developed across the pi-tot tube is measured by a differential pressure trans-mitter;
(c) the output of the differential pressure trans-mitter is measured and recorded by a computerized data acquisition system (DAS);
(d) the DAS computes velocity for each measure-ment by taking the square root of the output of the differential pressure transmitter, making the appro-priate corrections for fluid density, and making the
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appropriate calibration corrections and unit conver-sions;
(e) the DAS automatically takes 60 readings at each traverse point and computes and records aver-age values and standard deviations for the data col-lected at each traverse point;
(f) the Pitot tube, differential pressure transmit-ter, and DAS are calibrated together as a system; and (g) the flow rate and the velocity profile remain constant for the duration of the test.
10-1.2 Data Summary
The computerized data acquisition system is used to compute average values from the 60 mea-surements at each traverse point using eq. (4-3.1).
The resulting average values at each traverse point Xij are summarized in Table 10-1.2.
10-1.3 Velocity Results
The DAS is also programmed to output the sample standard deviation of the 60 measurements at each traverse point based on eq. (4-3.2). The sample standard deviation at each point is summa-rized in Table 10-1.3-1.
The traverse points are located at the centroid of equal areas so that the integrated-average veloc-ity in the pipe is approximated by the average of the velocities determined at the traverse points.
First, the average velocity along each radius, Vi, is approximated as
Vi(ft/sec)≈jp1
兺
10 (1⁄10) Xif(ft/sec) (10-1.1)Then, the average velocity in the pipe V is approximated as
V (ft/sec)≈ip1
兺
4 (1⁄4) Vi(ft/sec) (10-1.2)Copyright ASME International
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Fig. 10-1.1 Traverse Points (Example 10-1)
Table 10-1.2 Average Values (Example 10-1)
Radius 1, i ⴝ 1 Radius 2, i ⴝ 2 Radius 3, i ⴝ 3 Radius 4, i ⴝ 4 Traverse Point, j Xij(ft/sec) Xij(ft/sec) Xij(ft/sec) Xij(ft/sec)
jp1 5.31 5.27 5.21 5.00
jp2 5.46 5.53 5.25 5.16
jp3 5.55 5.61 5.37 5.31
jp4 5.63 5.68 5.47 5.42
jp5 5.65 5.74 5.58 5.50
jp6 5.69 5.77 5.62 5.55
jp7 5.73 5.79 5.65 5.63
jp8 5.74 5.76 5.65 5.65
jp9 5.76 5.75 5.70 5.65
jp10 5.72 5.80 5.70 5.67
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Table 10-1.3-1 Standard Deviations (Example 10-1)
Radius 1, iⴝ 1 Radius 2, iⴝ 2 Radius 3, iⴝ 3 Radius 4, iⴝ 4
Traverse Point, j (ft/sec) (ft/sec) (ft/sec) (ft/sec) (ft/sec) (ft/sec) (ft/sec) (ft/sec)
1 1.21 0.156 1.31 0.169 1.61 0.208 1.41 0.182
2 1.06 0.157 1.61 0.208 1.78 0.230 1.65 0.213
3 1.03 0.133 1.36 0.176 1.89 0.244 1.26 0.163
4 1.21 0.156 1.31 0.169 1.84 0.238 1.80 0.232
5 1.29 0.167 1.06 0.137 1.65 0.213 2.04 0.263
6 1.09 0.141 1.26 0.163 1.09 0.141 1.74 0.225
7 0.81 0.105 1.03 0.133 1.43 0.185 1.61 0.208
8 1.00 0.129 0.93 0.120 1.18 0.152 2.14 0.276
9 1.15 0.148 1.34 0.173 1.36 0.176 1.43 0.185
10 0.81 0.105 1.45 0.187 1.00 0.129 1.54 0.199
Table 10-1.3-2 Summary of Average Velocity Calculation (Example 10-1)
The subscripts i and j are used in the previous equations to designate radius and traverse posi-tions, respectively.
The results of these calculations are summarized in Table 10-1.3-2.
10-1.4 List Elemental Uncertainty Sources The sources of uncertainty which are considered random in this simplified example are those caus-ing variation in the 60 repeated measurements of velocity at each traverse point. The sources of uncertainty which are considered systematic in this simplified example are the uncertainty of the calibration of the instruments used to measure and record velocity at each traverse point and the uncertainty of the integrated-average velocity due to spatial variation.
10-1.5 Calculate Random Standard Uncertainty The random standard uncertainty of the mean value at each traverse point presented in Table 10-1.2 is calculated from eq. (4-3.3) as
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SXij 冪Nij
where values for SXij are shown in Table 10-1.3-1.
Since there are 60 measurements at each traverse point, Nijp60, the degrees of freedom at each traverse point is
ijpNij− 1 p 59 (10-1.3) The resulting values for SX
ij are also presented in Table 10-1.3-1.
10-1.6 Propagate Random Standard Uncertainty The random standard uncertainty for each aver-age velocity along a radius SV
i is calculated from eq. (7-3.1) as
The random standard uncertainty of the average velocity in the pipe sV is then calculated as
sVp