1、AN AM.ERICAN NATIONAL STANDARD Measurement Uncertainty for Fluid Flow in Closed Conduits ANSI/ASME MFC-2M-1983 SPONSORED AND PUBLISHED BY THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS United Engineering Center 345 East 47th Street New York, N. Y. 1 O01 7 ASME MFC-E“ 83 llsl 07-57b70 noLl7272 5 m Date
2、 of Issuance: August 31,1984 This Standard will be revised when the Society approves the issuance of a new edition. There will be no addenda or written interpretations of the requirements of this Standard issued to this Edition. This code or standard was developed under procedures accredited as meet
3、ing the criteria for Ameri- can National Standards. The Consensus Committee that approved the code or standard was balanced to assure that individuals from competent and concerned interests have had an opportunity to partici- pate. The proposed code or standard was made available for public review a
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8、opyright O 1984 by THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS All Rights Resewed Printed in U.S.A. ASME MFC-2M 83 B! 0757670 0047273 7 W c FOREWORD (This Foreword is not part of American National Standard, Measurement Uncer- tainty for Fluid Flow in Closed Conduits, ANSI/ASME MFC-2M-1983.) This St
9、andard was prepared by Subcommittee 1 of the American Society of Mechanical Engineers The methodology is consistent with that described in: Standards Committee on Measurement of Fluid Flow in Closed Conduits. Joint Army, Navy, NASA, Air Force Propulsion Committee (JANNAF). ICRPG Handbook for Esti- m
10、ating the Uncertainty in Measurements Made with Liquid Propellant Rocket Engine Systems. CPIA Publication 180. AD 851 127. Available from NTIS, 5285 Port Royal Road, Springfield, VA 22161. U.S. Dept. of the Air Force. Arnold Engineering Development Center. Handbook: Uncertainty in Gas Turbine Measur
11、ements. USAF AEDC-TR-73-5. AD 755356. Available from NTIS, 5285 Port Royal Road, Springfield, VA 22161. The Committee is indebted to the many engineers and statisticians who contributed to this work. Most noteworthy are J. Rosenblatt and H. Ku of the National Bureau of Standards for their helpful di
12、scussions and comments. The measurement uncertainty model is based on recommendations by the National Bureau of Standards. D. R. Keyser suggested the alternate model and other changes. B. Rinhser programmed the Monte Carlo simulations for uncertainty intervals and outliers. Encouragement and constru
13、ctive criticism were provided by: G. Adams, Chairman, The Society of Automotive Engineers, Committee E33C, USAF, WPAFB, ASD R. P. Benedict, Chairman, The American Society of Mechanical Engineers, Committee PTC19.1, Westinghouse J. W. Thompson, Jr., ARO, Inc. R. H. Dieck, Pratt Symmetrical Bias . 16
14、9 Measurement Uncertainty; Nonsymmetrical Bias 17 1 O Run-to-Run Difference . 18 11 Flow Through a Choked Venturi . 20 12 Schematic of Critical Venturi Flowmeter Installation Upstream of a Turbine Engine . 27 13 Typical Calibration Hierarchy . 27 14 Calibration Process Uncertainty Parameter U1 . +(B
15、1 t fg5S) 29 15 Temperature Measurement Calibration Hierarchy 34- 16 Typical Thermocouple Channel 36 17 Graph of0 vsB 49 6 Data Acquisition System 10 Vi Al Bias in a Random Process . 52 A2 Correlation Coefficients . 52 Cl Outliers Outside the Range of Acceptable Data . 64 C2 a. 0 Error in Thompsons
16、Outlier Test (Based on 1 Outlier in Each of 100 Samples C3 a. P Error in Grubbs Outlier Test (Based on 1 Outlier in Each of 100 Samples of C4 Results of Outlier Tests . 69 of Sizes 5. . 10. and 40) i 67 Sizes 5.10. and 40) 68 Tables 1 Values Associated With the Distribution of the Average Range . 6
17、2 Nonsymmetrical Bias Limits 8 3 Calibration Hierarchy Error Sources 10 4 Data Acquisition Error Sources 11 5 Data Reduction Error Sources . 11 6 Uncertainty Intervals Defined by Nonsymmetrical Bias Limits . 17 7 FlowData 21 8 Elemental Error Sources 23 9 Calibration Hierarchy Error Sources . 27 10
18、Pressure Transducer Data Acquisition Error Sources 29 1 1 Pressure Measurement Data Reduction Error Sources . 31 12 Temperature Calibration Hierarchy Elemental Errors 34 13 Airflow Measurement Error Sources . 42 14 . Error Comparisons of Examples One and Two 47 15 Values of0 andB . 49 16 Resultsford
19、= 14in.andB=O.667 . 50 B1 Results of Monte Carlo Simulation for Theoretical Input (ux2. cc,. cry2. P, ) 61 B2 Results of Monte Carlo Simulation for Theoretical Input pxi. uxi2 . 61 B3 Error Propagation Formulas 62 C2 Rejection Values for Grubbs Method 66 C4 Results of Applying Thompsons T and Grubbs
20、 Method 68 C1 Rejection Values for Thompsons Tau . 65 C3 Samplevalues 68 Dl Two-Tailed Students t Table . 71 Appendices A Glossary . 51 B Propagation of Errors by Taylor Series . 57 C Outlier Detection . 63 D Students t Table . 71 Viii ASME MFC-2M 83 W 0757b70 0047278 b AN AMERICAN NATIONAL STANDARD
21、 MEASUREMENT UNCERTAINTY FOR F.UID FLOW IN CLOSED CONDUITS Section 1 - Introduction 1.1 OBJECTIVE The objective of this Standard is to present a method of treating measurement error or uncertainty for the measurement of fluid flow. The need for a common method is obvious to those who have reviewed t
22、he numerous methods currently used. The subject is complex and involves both engineering and statistics. A common standard method is required to produce a well-defined, consistent estimate of the magnitude of uncertainty and to make comparisons between experiments and between facilities. However, it
23、 must be recognized that no single method will give a rigorous, scientifically correct answer for all situations. Further, even for a single set of data, the task of finding and proving one method to be correct is almost impossible. 1.2 SCOPE 1.2.1 General This Standard presents a working outline de
24、tailing and illustrating the techniques for estimating measure- ment uncertainty for fluid flow in closed conduits. The statistical techniques and analytical concepts ap- plied herein are applicable in most measurement processes. Section 2 provides examples of the mathematical model applied to the m
25、easurement of fluid flow. Each example includes a discussion of the elemental errors and examples of the statistical techniques. An effort has been made to use simple prose with a minimum of jargon. The notation and definitions are given in Appendix A and are consistent with IS0 3534, Statistics - V
26、ocabulary and Symbols (1977). 1.2.2 The Problem All measurements have errors. The errors may be positive or negative and may be of a variable magnitude. Many errors vary with time. Some have very short periods and some vary daily, weekly, seasonally, or yearly. Those which can be observed to vary du
27、ring the test are called random errors. Those which remain constant or apparently constant during the test are called biases, or systematic errors. The actual errors are rarely known; however, uncertainty intervals can be estimated or inferred as upper bounds on the errors. The problem is to constru
28、ct an uncertainty interval which models these errors. 1.3 NOMENCLATURE 1.3.1 Statistical Nomenclature P = true bias error, i.e., the fEed, systematic, or constant component of the total error S. The S = total error, Le., the difference between the observed measurement and the true value prime () is
29、added to avoid confusion with engineering notation. 1 ANSl/ASME MFC-PM-1983 AN AMERICAN NATIONAL STANDARD MEASUREMENT UNCERTAINTY FOR FLUID FLOW IN CLOSED CONDUITS e = the random component of error, sometimes called repeatability error or sampling error p = the true, unknown average v = degrees of f
30、reedom (see Appendix A) u = the true standard deviation of repeated values of the measurement; also, the standard devia- (Note: 6 = 0 t e) tion of the error S. This variation is due to the random error e. u2 = the true variance, Le., the square of the standard deviation B = the estimate of the upper
31、 limit of the bias error 0 Bi = an estimate of the upper limit of an elemental bias error. The j subscript indicates the pro- cess, i.e.: j = (I) calibration = (2) data acquisition = (3) data reduction The i subscript is the number of the error source within the process. If i is more than a single d
32、igit, a comma is used between i and j. N = the number of samples or the sample size S = an estimate of the standard deviation u obtained by taking the square root of S2, It is the Si = the estimate of the precision index from one elemental source. The subscripts are the same precision index. as defi
33、ned under Bii above. S2 = an unbiased estimate of the variance u2 tg5 = Students t = statistical parameter at the 95% confidence level. The degrees of freedom v of the sample estimate of the standard deviation is needed to obtain the t value from Table D l. U= an estimate of the error band, centered
34、 about the measurement, within which the true value will fall; an upper limit of S. The interval defined as the measurement plus and minus U should include the true value with high probability. Xi = an individual measurement X= sample average of measurements 1.3.2 Engineering Nomenclature The follow
35、ing symbols are used in describing the primary elements and in the equations given for com- puting rates of flow. Letters used to represent special factors in some equations are defined at the place of use, as are special subscripts. 2 i- ASME MFC-2M 83 m 0757670 0047300 O m MEASUREMENT UNCERTAINTY
36、FOR FLUID FLOW IN CLOSED CONDUITS ANSI/ASME MFC-2M-1983 AN AMERICAN NATIONAL STANDARD 0 (beta) = ratio of diameters = d/D, ratio I(gamma) = isentropic exponent of a real gas, a function of pl, pz, and T, number 7 (gamma) = ratio of specific heats of a gas (ideal) = cp/cu, ratio Ap (delta p) = differ
37、ential pressure = pl - pz, psi or pascals (pa) p (rho) = density, lb,/ft3 or kg/m3 4* (phi) = sonic-flow function of a real gas, number q+* (phi) = sonic-flow function of an ideal gas, number a = area of an orifice, flow nozzle, or venturi throat, in. or mz C= coefficient of discharge, ratio cp = sp
38、ecific heat of a fluid at constant pressure, Btu/lb, * R or J/kg K c, = specific heat of a fluid at constant volume, Btu/lb,n * OR or J/kg - K D = diameter of pipe or meter tube, in. or m d = diameter of orifice, flow nozzle throat, or venturi throat in. or m E = velocity of approach factor = l/dm,
39、number F = isentropic expansion function of a real gas, ratio F, = area thermal expansion factor, ratio Fi = isentropic expansion function of an ideal gas c - i ) , ratio g, = proportionally constant in the force-mass-acceleration equation = 32.174, number (not re- g = acceleration due to gravity, l
40、ocal, ft/sec2 (not required in SI units) quired in SI units) h = effective differential pressure, ft of fluid (SI units not applicable) h, = effective differential pressure, in. of water at 68F (SI units not applicable) MW = molecular weight of a fluid, number m = mass rate of flow, lb,/sec or kg/s
41、p = pressure, absolute, psia (English units) Pa = pressure, pascal /m2; SI units) pt = total or stagnation pressure, psia or Pa R = gas constant in pu = R T (here p is lbf/ftz), ft X Ibf/lb, X R or J/(mol K) RD = Reynolds number based on D, ratio Rd = Reynolds number based on d, ratio T = absolute t
42、emperature, R or K V= velocity, ft/sec or m/s V, = velocity of sound (acoustic velocity), ft/sec or m/s u = specific volume = l/p, ft3/lb or m3/kg Y = expansion factor for a gas, ratio 2 = compressibility factor for a real gas, ratio 1.4 MEASUREMENT ERROR 1.4.1 General All measurements have errors.
43、These errors are the differences between the measurements and the true value, as shown in Fig. 1. In some cases, the true value may be arbitrarily defined as the value that would be obtained by the National Bureau of Standards (NBS). Uncertainty is an estimate of the test error which in most cases w
44、ould not be exceeded. Measurement error 6 has two components: a Tied error P and a random error e. 1.4.2 Precision (Random Error) Random error is seen in repeated measurements of the same thing. Measurements do not and are not expected to agree exactly. There are numerous small effects which cause d
45、isagreements. The precision of a measurement process is determined by the variation between repeated measurements. The standard devia- 3 ANSI/ASME MFC-2M-1983 AN AMERICAN NATIONAL STANDARD tion MEASUREMENT UNCERTAINTY FOR FLUID FLOW IN CLOSED CONDUITS Average Measured Value True (NBS) Value I 0.980
46、used a u in Fig. 2 is IS I I- l 0.985 0.990 0.995 Parameter Measurement Value FIG. 1 MEASUREMENT ERROR am I 1 .o mure of the precision error e. A large standard deviation me .ans large scatt .er in ea the measurements. The statistic S is calculated to estimate the standard deviation u and is called
47、the pre- cision index where N is the number of measurements made and 8 is the average value of individual measurements Xi. simultaneous observations and averaging. Averages wiU have a smaller precision index. The effect of the precision error of the measurement can often be reduced by taking several
48、 repeated or - uindividuals S uaverage - fi x- fi and S-“?- Throughout this document, the precision index is the sample standard deviation of the measurement, whether it is a single reading or the average of several readings. There are many ways to calculate the precision index. 4 ASME MFC-2M 83 8 0
49、757670 OOY7302 4 m MEASUREMENT UNCERTAINTY FOR FLUID FLOW IN CLOSED CONDUITS ANSI/ASME MFC-2M-1983 AN AMERICAN NATIONAL STANDARD Average Measurement 0.985 1 .o 1.015 Parameter Measurement Value FIG. 2 PRECISION ERROR (a) If the variable to be measured can be held constant, a number of repeated measurements can be used to evaluate Eq. (1). (b) If there are k redundant instruments and the variable to be measured can be held constant to take i repeated readings on each of k instruments, then the following pooled estimate of the precision index should be use
