ASME 19.1 Test Uncertainty_05.pdf

ASME PTC 19.1-2005 (Revision of ASME PTC 19.1-1998)

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Test Uncertainty

A N A M E R I C A N N AT I O N A L STA N DA R D

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Date of Issuance: October 13, 2006

The 2005 edition of ASME PTC 19.1 will be revised when the Society approves the issuance of the next edition. There will be no Addenda issued to ASME PTC 19.1-2005. ASME issues written replies to inquiries as code cases and interpretations of technical aspects of this document. Code cases and interpretations are published on the ASME website under the Committee Pages at http://www.asme.org/codes/ as they are issued.

ASME is the registered trademark of The American Society of Mechanical Engineers.

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This code or standard was developed under procedures accredited as meeting the criteria for American National Standards. The Standards Committee that approved the code or standard was balanced to assure that individuals from competent and concerned interests have had an opportunity to participate. The proposed code or standard was made available for public review and comment that provides an opportunity for additional public input from industry, academia, regulatory agencies, and the public-at-large. ASME does not “approve,” “rate,” or “endorse” any item, construction, proprietary device, or activity. ASME does not take any position with respect to the validity of any patent rights asserted in connection with any items mentioned in this document, and does not undertake to insure anyone utilizing a standard against liability for infringement of any applicable letters patent, nor assumes any such liability. Users of a code or standard are expressly advised that determination of the validity of any such patent rights, and the risk of infringement of such rights, is entirely their own responsibility. Participation by federal agency representative(s) or person(s) affiliated with industry is not to be interpreted as government or industry endorsement of this code or standard. ASME accepts responsibility for only those interpretations of this document issued in accordance with the established ASME procedures and policies, which preclude the issuance of interpretations by individuals.

No part of this document may be reproduced in any form, in an electronic retrieval system or otherwise, without the prior written permission of the publisher. The American Society of Mechanical Engineers Three Park Avenue, New York, NY 10016-5990 Copyright © 2006 by THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS All rights reserved Printed in U.S.A.

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CONTENTS Notice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Committee Roster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii viii ix

Section 1 1-1 1-2 1-3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harmonization With International Standards . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1

Section 2 2-1 2-2

Object and Scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Object. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 2

Section 3 3-1 3-2

Nomenclature and Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3

Section 4 4-1 4-2 4-3 4-4

Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pretest and Posttest Uncertainty Analyses. . . . . . . . . . . . . . . . . . . . . .

5 5 5 5 11

Section 5 5-1 5-2 5-3 5-4 5-5

Defining the Measurement Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of the Appropriate “True Value” . . . . . . . . . . . . . . . . . . . . . Identification of Error Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Categorization of Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative Versus Absolute Testing. . . . . . . . . . . . . . . . . . . . . . . . .

13 13 13 13 15 16

Section 6 6-1 6-2 6-3 6-4

Uncertainty of a Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random Standard Uncertainty of the Mean . . . . . . . . . . . . . . . . . . . . Systematic Standard Uncertainty of a Measurement . . . . . . . . . . . . . Classification of Uncertainty Sources . . . . . . . . . . . . . . . . . . . . . . . . . . Combined Standard and Expanded Uncertainty of a Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 19

22 22 23 23 24

7-6

Uncertainty of a Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation of Measurement Uncertainties Into a Result. . . . . . . . . . Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random Standard Uncertainty of a Result . . . . . . . . . . . . . . . . . . . . . Systematic Standard Uncertainty of a Result. . . . . . . . . . . . . . . . . . . . Combined Standard Uncertainty and Expanded Uncertainty of a Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of Uncertainty Propagation . . . . . . . . . . . . . . . . . . . . . . . . .

Section 8 8-1 8-2 8-3 8-4

Additional Uncertainty Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . Correlated Systematic Standard Uncertainties. . . . . . . . . . . . . . . . . . . Nonsymmetric Systematic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . Fossilization of Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 28 31 35 36

Section 7 7-1 7-2 7-3 7-4 7-5

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19

24 24

8-5 8-6

Analysis of Redundant Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regression Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 38

Section 9 9-1 9-2

Step-by-Step Calculation Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 41

Section 10 10-1 10-2 10-3

43 43 47

10-4 10-5

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Measurement Using Pitot Tubes . . . . . . . . . . . . . . . . . . . . . . . . . Flow Rate Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Rate Uncertainty Including Nonsymmetrical Systematic Standard Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressor Performance Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . Periodic Comparative Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Section 11

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Section 12

Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

Illustration of Measurement Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Error Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distribution of Measured Values (Normal Distribution) . . . . . . . . . . Uncertainty Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generic Measurement Calibration Hierarchy . . . . . . . . . . . . . . . . . . . Difference Between “Within” and “Between” Sources of Data Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pareto Chart of Systematic and Random Uncertainty Component Contributions to Combined Standard Uncertainty . . . . . . . . . . . . . Schematic Relation Between Parameters Characterizing Nonsymmetric Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation Between Parameters Characterizing Nonsymmetric Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three Posttest Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traverse Points (Example 10-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of a 6 in. ⴛ 4 in. Venturi. . . . . . . . . . . . . . . . . . . . . . . . . . . Typical Pressure and Temperature Locations for Compressor Efficiency Determination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The h-s Diagram of the Actual and Isentropic Processes of an Adiabatic Compressor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Installed Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Design Curve With Factory and Field Test Data Shown . . . . . Comparison of Test Results With Independent Control Conditions . Comparison of Test Results Using the Initial Field Test as the Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 7 8 11 14

Figures 4-2-1 4-2-2 4-3.1 4-3.3 5-3.1 5-4.3 7-6.2 8-2.1 8-2.2 8-5.1 10-1.1 10-2.1 10-4.1 10-4.7 10-5.1-1 10-5.1-2 10-5.1-3 10-5.2 Tables 6-4-1 6-4-2 7-6.1-1 7-6.1-2 7-6.2-1 7-6.2-2 8-1

Circulating Water Bath Temperature Measurements (Example 6-4.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Systematic Uncertainty of Average Circulating Water Bath Temperature Measurements (Example 6-4.1) . . . . . . . . . . . . . . . . . . Table of Data (Example 7-6.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Data (Example 7-6.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Data (Example 7-6.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Data (Example 7-6.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burst Pressures (Example 8-1-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

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50 51 62

16 27 32 34 37 44 48 57 61 63 64 64 67

20 21 25 26 26 27 29

8-6.4.5 9-2-1 9-2-2 10-1.2 10-1.3-1 10-1.3-2 10-1.6 10-1.9 10-2.1-1 10-2.1-2 10-2.1-3 10-2.1-4 10-2.1-5 10-2.1.1-1 10-2.1.1-2 10-2.1.1-3 10-2.1.1-4 10-2.1.1-5 10-2.1.1-6 10-2.1.1-7 10-3-1

10-3-2 10-4.1-1 10-4.1-2 10-4.1-3 10-4.3.2-1 10-4.3.2-2 10-4.7 10-5.1-1 10-5.1-2 10-5.2-1 10-5.2-2 10-5.2-3

Systematic Standard Uncertainty Components for Yˆ Determined from Regression Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average Values (Example 10-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Deviations (Example 10-1) . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Average Velocity Calculation (Example 10-1) . . . . . . . . Standard Deviation of Average Velocity (Example 10-1) . . . . . . . . . . Uncertainty of Result (Example 10-1) . . . . . . . . . . . . . . . . . . . . . . . . . Uncalibrated Case (Example 10-2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absolute Sensitivity Coefficients in Example 10-2 (Calculated Numerically) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absolute Sensitivity Coefficients in Example 10-2 (Calculated Analytically) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absolute Contributions of Uncertainties of Independent Parameters (Example 10-2: Uncalibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . Summary: Uncertainties in Absolute Terms (Example 10-2: Uncalibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative Uncertainty of Measurement (Example 10-2: Uncalibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative Contributions of Uncertainties of Independent Parameters (Example 10-2: Uncalibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . Summary: Uncertainties in Relative Terms for the Uncalibrated Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative Uncertainties of Independent Parameters (Example 10-2: Calibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative Contributions of Uncertainties of Independent Parameters (Example 10-2: Calibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary: Uncertainties in Relative Terms for the Calibrated Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary: Comparison Between Calibrated and Uncalibrated Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absolute Contributions of Uncertainties of Independent Parameters (Example 10-3: Uncalibrated, Nonsymmetrical Systematic Uncertainty Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary: Uncertainties in Absolute Terms (Example 10-3: Uncalibrated, Nonsymmetrical Systematic Uncertainty Case) . . . . Elemental Random Standard Uncertainties Associated With Error Sources Identified in Para. 10-4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . Independent Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculated Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inlet and Exit Pressure Elemental Systematic Standard Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inlet and Exit Temperature Elemental Systematic Standard Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of Analysis Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Design Data (Tc p 20°C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncertainty Propagation for Comparison With Independent Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary: Uncertainties in Absolute Terms . . . . . . . . . . . . . . . . . . . . Summary of Results for Each Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

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40 42 42 44 45 45 46 48 48 50 51 52 52 52 53 53 53 54 54 54

55 56 56 57 57 58 59 62 63 63 66 66 66

10-5.3-1 10-5.3-2

Uncertainty Propagation for Comparative Uncertainty . . . . . . . . . . . Sensitivity Coefficient Estimates for Comparative Analysis . . . . . . . .

68 68

Nonmandatory Appendices A Statistical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Uncertainty Analysis Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C Propagation of Uncertainty Through Taylor Series . . . . . . . . . . . . . . . . . . . . . . . D The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 84 87 92

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NOTICE All Performance Test Codes must adhere to the requirements of ASME PTC 1, General Instructions. The following information is based on that document and is included here for emphasis and for the convenience of the user of the Supplement. It is expected that the Code user is fully cognizant of Sections 1 and 3 of ASME PTC 1 and has read them prior to applying this Supplement. ASME Performance Test Codes provide test procedures which yield results of the highest level of accuracy consistent with the best engineering knowledge and practice currently available. They were developed by balanced committees representing all concerned interests and specify procedures, instrumentation, equipment-operating requirements, calculation methods, and uncertainty analysis. When tests are run in accordance with a Code, the test results themselves, without adjustment for uncertainty, yield the best available indication of the actual performance of the tested equipment. ASME Performance Test Codes do not specify means to compare those results to contractual guarantees. Therefore, it is recommended that the parties to a commercial test agree before starting the test and preferably before signing the contract on the method to be used for comparing the test results to the contractual guarantees. It is beyond the scope of any Code to determine or interpret how such comparisons shall be made.

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FOREWORD In March 1979 the Performance Test Codes Supervisory Committee activated the PTC 19.1 Committee to revise a 1969 draft of a document entitled PTC 19.1 “General Considerations.” The PTC 19.1 Committee proceeded to develop a Performance Test Code Instruments and Apparatus Supplement which was published in 1985 as PTC 19.1-1985, “Measurement Uncertainty,” and which was intended—along with its subsequent editions—to provide a means of eventual standardization of nomenclature, symbols, and methodology of measurement uncertainty in ASME Performance Test Codes. Work on the revision of the original 1985 edition began in 1991. The two-fold objective was to improve the usefulness to the reader regarding clarity, conciseness, and technical treatment of the evolving subject matter, as well as harmonization with the ISO “Guide to the Expression of Uncertainty in Measurement.” That revision was published as PTC 19.11998, “Test Uncertainty,” the new title reflecting the appropriate orientation of the document. The effort to update the 1998 revision began immediately upon completion of that document. This 2005 revision is notable for the following significant departures from the 1998 text: (a) Nomenclature adopted for this revision is more consistent with the ISO Guide. Uncertainties remain conceptualized as “systematic” (estimate of the effects of fixed error not observed in the data), and “random” (estimate of the limits of the error observed from the scatter of the test data). The new aspect is that both types of uncertainty are defined at the standard-deviation level as “standard uncertainties.” The determination of an uncertainty at some level of confidence is based on the root-sum-square of the systematic and random standard uncertainties multiplied times the appropriate expansion factor for the desired level of confidence (usually “2” for 95%). This same approach was used in the 1998 revision but the characterization of uncertainties at the standard-uncertainty level (“standard deviation”) was not as explicitly stated. The new nomenclature is expected to render PTC 19.12005 more acceptable at the international level. (b) There is greater discussion of the determination of systematic uncertainties. (c) There is new text on a simplified approach to determine the uncertainty of straightline regression. ASME PTC 19.1-2005 was approved by the PTC Standards Committee on September 13, 2005, and was approved as an American National Standard by the ANSI Board of Standards Review on November 3, 2005.

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PERFORMANCE TEST CODE COMMITTEE 19.1 ON TEST UNCERTAINTY (The following is the roster of the Committee at the time of the approval of this Supplement.)

OFFICERS R. H. Dieck, Chair W. G. Steele, Vice Chair G. Osolsobe, Secretary

COMMITTEE PERSONNEL J. F. Bernardin, Pratt & Whitney D. A. Coutts, WSMS R. H. Dieck, Ron Dieck Associates, Inc. R. S. Figliola, Clemson University H. K. Iyer, Colorado State University J. Maveety, Intel Corp. J. A. Rabensteine, Environmental Systems Corp. M. Soltani, Bechtel National Corp. W. G. Steele, Mississippi State University

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PERFORMANCE TEST CODES STANDARDS COMMITTEE OFFICERS J. G. Yost, Chair J. R. Friedman, Vice Chair S. D. Weinman, Secretary

COMMITTEE PERSONNEL P. G. Albert R. P. Allen J. M. Burns W. C. Campbell M. J. Dooley A. J. Egli J. R. Friedman G. J. Gerber P. M. Gerhart T. C. Heil R. A. Johnson D. R. Keyser S. J. Korellis

P. M. McHale M. P. McHale J. W. Milton S. P. Nuspl A. L. Plumley R. R. Priestley J. A. Rabensteine J. W. Siegmund J. A. Silvaggio W. G. Steele J. C. Westcott W. C. Wood J. G. Yost

HONORARY MEMBERS W. O. Hays

F. H. Light

MEMBERS EMERITI R. L. Bannister R. Jorgensen

G. H. Mittendorf R. E. Sommerlad

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Section 1 Introduction 1-1 GENERAL

harmony with international guidelines and standards. In this Supplement, “standard” uncertainties are always equivalent to a single standard deviation of the average. The most common confidence level used in this Supplement is 95% although methods for employing alternate confidences are also given. The confidence level of 95% is applied to “expanded” uncertainty. This term, too, was included in this Supplement for improved harmony with international guidelines and standards. While this Supplement is in harmony with the ISO GUM, this Supplement emphasizes the effects of errors rather than the basis of the information utilized in the estimation of their limits. The ISO GUM utilizes two major classifications for errors and uncertainties. They are “Type A” and “Type B.” Type A uncertainties have data with which to calculate a standard deviation. Type B uncertainties do not have data to calculate a standard deviation and must be estimated by other means. This Supplement utilizes two major classifications for errors and uncertainties. They are “systematic” and “random.” Random errors (whose effects are estimated with “Random Standard Uncertainties”) cause scatter in test data. Systematic errors (whose effects are estimated with “Systematic Standard Uncertainties”) do not. Harmonization of this Supplement with the ISO GUM is achieved by encouraging subscripts with each uncertainty estimate to denote the ISO Type, i.e., using subscripts of either “A” or “B.”

This Supplement has significant additions and Sections that have been rewritten to both add to the available technology for uncertainty analysis and to make it easier for the practicing engineer. Throughout, the intent is to provide a Supplement that can be utilized easily by engineers and scientists whose interest is the objective assessment of data quality, using test uncertainty analysis.

1-2 HARMONIZATION WITH INTERNATIONAL STANDARDS It is recognized that this Supplement and promulgated international uncertainty standards and/ or guides must be in harmony. In rewriting this Supplement, great care was taken to assure continued harmony with the International Organization for Standardization (ISO) Guide to the Expression of Uncertainty in Measurement (GUM) [1]. For the practicing engineer, this harmonization means the elimination of such ambiguous terms as bias, precision, bias limit, and precision index. In addition, careful attention was paid to discriminating between errors, the effects of errors, and the estimation of their limits, which is the uncertainty. The term “bias” is not used in this Supplement. Instead, the combined terms of “systematic error” and “systematic uncertainty” are used. The former describes an error source whose effect is systematic or constant for the duration of a test. The latter describes the limits to which a systematic error may be expected to go with some confidence. The term “precision” also is not used in this Supplement. Instead the combined terms of “random error” and “random uncertainty” are used. The former describes an error source that causes scatter in test data. The latter describes the limits to which a random error may be expected to reach with some confidence. Throughout the Supplement, the term “standard” uncertainty has been introduced to improve

1-3 APPLICATIONS This Supplement is intended to serve as a reference to the various other ASME Instruments and Apparatus Supplements (PTC 19 Series) and to ASME Performance Test Codes and Standards in general. In addition, it is applicable for all known measurement and test uncertainty analyses. The paramater values and uncertainty levels used throughout the examples are for illustrative purposes only and are not intended to be typical of standard tests. 1

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Section 2 Object and Scope 2-1 OBJECT

(e) documents uncertainty for assessing compliance with agreements.

The object of this Supplement is to define, describe, and illustrate the various terms and methods used to provide meaningful estimates of the uncertainty in test parameters and methods, and the effects of those uncertainties on derived test results. Analysis of test measurement and result uncertainty is useful because it (a) facilitates communication regarding measurement and test results; (b) fosters an understanding of potential error sources in a measurement system and the effects of those potential error sources on test results; (c) guides the decision-making process for selecting appropriate and cost-effective measurement systems and methodologies; (d) reduces the risk of making erroneous decisions; and

2-2 SCOPE The scope of this Supplement is to specify procedures for evaluation of uncertainties in test parameters and methods, and for propagation of those uncertainties into the uncertainty of a test result. Depending on the application, uncertainty sources may be classified either by the presumed effect (systematic or random) on the measurement or test result, or by the process in which they may be quantified (Type A or Type B). The various statistical terms involved are defined in the Nomenclature (subsection 3-1) or Glossary (subsection 3-2). The end result of an uncertainty analysis is a numerical estimate of the test uncertainty with an appropriate confidence level.

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Section 3 Nomenclature and Glossary ␦p (unknown) total error; difference between the assigned value of a parameter or a test result and the true value ⑀p (unknown) true random error; random component of ␦ ␪p absolute sensitivity ␪ ′p relative sensitivity ␮p (unknown) true average of a population ␯p number of degrees of freedom ␴p (unknown) true standard deviation of a population ␴ 2p (unknown) true variance of a population Indices Ip total number of variables ip counter for variables jp counter for individual measurements Kp total number of sources of elemental errors and uncertainties kp counter for sources of elemental errors and uncertainties Lp total number of correlated sources of systematic error lp counter for correlated sources of systematic error Mp total number of multiple results mp counter for multiple results Np total number of measurements

3-1 NOMENCLATURE b Xp systematic standard uncertainty component of a parameter b Xkp systematic standard uncertainty associated with the kth elemental error source bRp systematic standard uncertainty component of a result bXYp covariance of the systematic errors in X and Y b+, b−p upper and lower values of nonsymmetrical systematic standard uncertainty Np number of measurements or sample points or observations available (sample size) Rp result sRp random standard uncertainty of a result sXp standard deviation of a data sample; estimate of the standard deviation of the population ␴x sXp random standard uncertainty of the mean of N measurements SEEp standard error of estimate of a leastsquares regression or curve fit tp Student’s t value at a specified confidence level with ␯ degrees of freedom, i.e., t95, ␯ up combined standard uncertainty Up expanded uncertainty U+, U−p upper and lower values of the nonsymmetrical expanded uncertainty Xp individual observation in a data sample of a parameter Xp sample mean; average of a set of N individual observations of a parameter ␤p (unknown) true systematic error; fixed or constant component of ␦

3-2 GLOSSARY calibration hierarchy: the chain of calibrations that links or traces a measuring instrument to a primary standard. calibration: the process of comparing the response of an instrument to a standard instrument over some measurement range. confidence level: the probability that the true value falls within the specified limits. 3

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degrees of freedom (␯ ): the number of independent observations used to calculate a standard deviation.

result (R): a value calculated from a number of parameters.

elemental random error source: an identifiable source of random error that is a subcomponent of total random error.

sample size (N): the number of individual values in a sample. sample standard deviation (sx): a value that quantifies the dispersion of a sample of measurements as given by eq. (4-3.2).

elemental random standard uncertainty (s Xk ): an estimate of the standard deviation of the mean of an elemental random error source.

sensitivity: the instantaneous rate of the change in a result due to a change in a parameter.

elemental systematic error source: an identifiable source of systematic error that is a subcomponent of the total systematic error.

standard error of estimate (SEE): the measure of dispersion of the dependent variable about a least squares regression or curve.

elemental systematic standard uncertainty (b Xk ): an estimate of standard deviation of an elemental systematic error source.

statistic: any numerical quantity derived from the sample data. X and s X are statistics. Student’s t: a value used to estimate the uncertainty for a given confidence level.

expanded uncertainty (UX or UR ): an estimate of the plus-or-minus limits of total error, with a defined level of confidence, (usually 95%).

systematic error (␤): the portion of total error that remains constant in repeated measurements of the true value throughout a test process.

influence coefficient: see sensitivity. mean (X): the arithmetic average of N readings.

systematic standard uncertainty (b X ): a value that quantifies the dispersion of a systematic error associated with the mean.

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

parameter: quantity that could be measured or taken from best available information, such as temperature, pressure, stress, or specific heat, used in determining a result. The value used is called the assigned value.

total error (␦): the true, unknown difference between the assigned value of a parameter or test result and the true value.

population mean (␮): average of the set of all population values of a parameter.

traceability: see calibration hierarchy. true value: the error-free value of a parameter or test result.

population standard deviation (␴): a value that quantifies the dispersion of a population.

Type A uncertainty: uncertainties are classified as Type A when data is used to calculate a standard deviation for use in estimating the uncertainty.

population: the set of all possible values of a parameter. random error (⑀): the portion of total error that varies randomly in repeated measurements of the true value throughout a test process.

Type B uncertainty: uncertainties are classified as Type B when data is not used to calculate a standard deviation, requiring the uncertainty to be estimated by other methods.

random standard uncertainty of the sample mean (s X ): a value that quantifies the dispersion of a sample mean as given by eq. (4-3.3).

uncertainty interval: an interval expressed about a parameter or test result that is expected to contain the true value with a prescribed level of confidence.

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Section 4 Fundamental Concepts 4-1 ASSUMPTIONS

contributions of several elemental random error sources. Elemental random errors may arise from uncontrolled test conditions and nonrepeatabilities in the measurement system, measurement methods, environmental conditions, data reduction techniques, etc.

The assumptions inherent in test uncertainty analysis include the following: (a) The test objectives are specified. (b) The test process, including the measurement process and the data reduction process, is defined. (c) The test process, with respect to the conditions of the item under test and the measurement system employed for the test, is controlled for the duration of the test. (d) The measurement system is calibrated and all appropriate calibration corrections are applied to the resulting test data. (e) All appropriate engineering corrections are applied to the test data as part of the data reduction and/or results analysis process. For expanded uncertainty, 95% confidence levels have been used throughout this document in accordance with accepted practice. Other confidence levels may be used, if required. (See Nonmandatory Appendix B.)

4-2.2 Systematic Error Systematic error, ␤, is the portion of the total error that remains constant in repeated measurements throughout the conduct of a test. The total systematic error in a measurement is usually the sum of the contributions of several elemental systematic errors. Elemental systematic errors may arise from imperfect calibration corrections, measurement methods, data reduction techniques, etc. 4-3 MEASUREMENT UNCERTAINTY There is an inherent uncertainty in the use of measurements to represent the true value. The total uncertainty in a measurement is the combination of uncertainty due to random error and uncertainty due to systematic error.

4-2 MEASUREMENT ERROR --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

Every measurement has error, which results in a difference between the measured value, X, and the true value. The difference between the measured value and the true value is the total error, ␦. Since the true value is unknown, total error cannot be known and therefore only its expected limits can be estimated. Total error consists of two components: random error and systematic error (see Fig. 4-2-1). Accurate measurement requires minimizing both random and systematic errors (see Fig. 4-2-2).

4-3.1 Random Standard Uncertainty Any single measurement of a parameter is influenced by several different elemental random error sources. In successive measurements of the parameter, the values of these elemental random error sources change resulting in the random scatter evident in the successive measurements. If an infinite number of measurements of a parameter were to be taken following the defined test process, the resulting population of measurements could be described statistically in terms of the population mean, ␮, the population standard deviation, ␴, and the frequency distribution of the population. These terms are illustrated in Fig. 4-3.1 for a population of measurements that is normally distributed. For measurements with zero systematic

4-2.1 Random Error Random error, ⑀, is the portion of the total error that varies randomly in repeated measurements throughout the conduct of a test. The total random error in a measurement is usually the sum of the 5 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

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Fig. 4-2-1 Illustration of Measurement Errors

error (refer to para. 4-2.2), the population mean is equal to the true value of the parameter being measured and the population standard deviation is a measure of the scatter of the individual measurements about the population mean. For a normal distribution, the interval ␮ ± ␴ will include approximately 68% of the population and the interval ␮ ± 2␴ will include approximately 95% of the population. Since only a finite number of measurements are acquired during a test, the true population mean and population standard deviation are unknown but can be estimated from sample statistics. The sample mean, X, is given by

兺

jp1

sX p

Xj

N

(4-3.1)

(4-3.2)

sX

冪N

(4-3.3)

For a normally distributed population and a large sample size (N > 30), the interval X ± sX is expected to contain the true population mean with 68% confidence and the interval X ± 2sX is expected to contain the true population mean with 95% confidence [where the value 2 represents the Student’s t value for 95% confidence and degrees of

where Xj represents the value of each individual measurement in the sample and N is the number of measurements in the sample. The sample standard deviation, sX, is given by 6 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

(Xj − X)2 N−1

Since the sample mean is only an estimate of the population mean, there is an inherent error in the use of the sample mean to estimate the population mean. For a defined frequency distribution, the random standard uncertainty of the sample mean, sX, can be used to define the probable interval about the sample mean that is expected to contain the population mean with a defined level of confidence. The random standard uncertainty of the sample mean is related to the sample standard deviation as follows:

N

Xp

冪jp1兺 N

sX p

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Fig. 4-2-2 Measurement Error Components

sources contributes a constant, but unknown, error, ␤ Xk, to the successive measurements of a parameter for the duration of the test (the subscript k is used to denote a specific elemental error source). As ␤ Xk is constant for the test, the error imparted to the average value of successive measurements, X [as given by eq. 4-3.1], is equivalent to the error imparted to each individual measurement. While ␤ Xk is unknown, it may be postulated to come from a population of possible error values from which a single sample (error value) is drawn and imparted to the average measurement for the test. Knowledge of the frequency distribution and standard deviation of this population permits describing the uncertainty in X due to this single sample elemental systematic error in terms of a confidence interval. The elemental systematic standard uncertainty, b Xk, is defined as a value

freedom of greater than or equal to 30 where the degrees freedom for the random standard uncertainty is N−1 (see subsection 6-1)]. In general, increasing the number of measurements collected during a test and used in the preceding formulas is beneficial as (a) it improves the sample mean as an estimator of the true population mean; (b) it improves the sample standard deviation as an estimator of the true population standard deviation; and (c) it typically reduces the value of the random standard uncertainty of the sample mean. 4-3.2 Systematic Standard Uncertainty Every measurement of a parameter is influenced by several different elemental systematic error sources. Each of these elemental systematic error 7 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

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Fig. 4-3.1 Distribution of Measured Values (Normal Distribution)

that quantifies the dispersion of the population of possible ␤ Xk values at the standard deviation level. All of the elemental systematic errors associated with a measurement combine to yield the total systematic error in the measurement, ␤ X. As with elemental systematic error, total systematic error is constant, unknown, and may be postulated to come from a population of possible error values from which a single sample (error value) is drawn and imparted to the average measurement for the test. Total systematic standard uncertainty, b X, is defined as a value that quantifies the dispersion of the population of possible ␤ X values at the standard deviation level. Typically, total systematic standard uncertainty is quantified by (a) identifying all elemental sources of systematic error for the measurement; (b) evaluating elemental systematic standard uncertainties as the standard deviations of the possible systematic error distributions; and (c) combining the elemental systematic standard uncertainties into an estimate of the total systematic standard uncertainty for the average measurement.

of test uncertainty. Attempting to identify all elemental sources of systematic error requires a thorough understanding of the test objectives and test process. For further discussion refer to subsection 5-4. 4-3.2.2 Evaluating Elemental Systematic Standard Uncertainties. Once all elemental sources of systematic error are identified, elemental systematic standard uncertainties for each source are evaluated. By definition, an elemental systematic standard uncertainty is a value that quantifies the dispersion of the population of possible ␤ Xk values at the standard deviation level. As ␤ Xk is both constant and unknown during a test, successive measurements of a parameter do not provide sufficient data for direct computation of a standard deviation as described in para. 4-3.1. Therefore, the evaluation of an elemental systematic standard uncertainty requires that a standard deviation be evaluated from engineering judgment, published information, or special data. 4-3.2.2.1 Engineering Judgment. When neither published information or special data is available, it is often necessary to rely upon engineering judgment to quantify the dispersion of errors associated with an elemental error source. In these situations, it is customary to use engineering analyses and experience to estimate the limits of the

4-3.2.1 Identifying Elemental Sources of Systematic Error. Attempting to identify all of the elemental sources of systematic error for a measurement is an important step of an uncertainty analysis, as failure to identify any significant source of systematic error will lead to an underestimation 8 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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ASME PTC 19.1-2005

elemental systematic error at 95% confidence. In other words, an interval is estimated which is expected to contain 95% of the population of possible ␤ Xk values. Not withstanding information to the contrary, the analyst typically assumes that the population of possible ␤ Xk values is normally distributed, that the estimation of the limits of the error is based upon large degrees of freedom, and that the limits of error are symmetric (equally spread in both the positive and negative directions). Based upon these assumptions, the elemental systematic standard uncertainty is estimated as follows:

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

b Xk p

BXk 2

uncertainty statement, or a multiple of a standard deviation. If the published information is presented as a confidence interval (limits of error at a defined level of confidence), then the elemental systematic standard uncertainty is estimated as the confidence interval divided by a statistic that is appropriate for the frequency distribution of the error population. The specific value of this statistic must be selected on the basis of the defined confidence level and degrees of freedom associated with the confidence interval. For a normal distribution, the Student’s t statistic is used. For a 95% confidence level and large degrees of freedom, the value of the Student’s t statistic is approximated as 2 and eq. (4-3.4) would apply (refer to Nonmandatory Appendix B for values of the Student’s t statistic at other confidence levels and degrees of freedom). For situations in which the frequency distribution and degrees of freedom are unspecified, a normal distribution and large degrees of freedom are often assumed. For situations involving other frequency distributions, refer to an appropriate statistics textbook. If the published information is presented as an ISO expanded uncertainty at a defined coverage factor (sometimes referred to as a “k factor”), then the elemental systematic standard uncertainty is estimated as the expanded uncertainty divided by the coverage factor. If the published information is presented as a multiple of a standard deviation, then the elemental systematic standard uncertainty is estimated as the multiple of the standard deviation divided by the multiplier.

(4-3.4)

The variable BXk in the preceding equation represents the 95% confidence level estimate of the symmetric limits of error associated with the kth elemental error source. In certain situations, knowledge of the physics of the measurement system will lead the analyst to believe that the limits of error are nonsymmetric (likely to be larger in either the positive or negative direction). For treatment of nonsymmetric systematic uncertainty see subsection 8-2. The value of 2 in the equation is based on the assumption that the population of possible systematic errors is normally distributed. If the analyst thinks that the error distribution might be other than normal, such as uniform (rectangular), then a different factor would be used to convert the 95% confidence level estimate of the systematic error limits to an elemental systematic standard uncertainty (see Nonmandatory Appendix B). Also, there is some level of uncertainty associated with the estimate of BXk. This uncertainty in the estimate can be converted into a degrees of freedom for the systematic standard uncertainty as shown in Nonmandatory Appendix B. Usually, the BXk estimates are made such that this degrees of freedom will be large (≥30). Using the recommendations in Nonmandatory Appendix B, it can be shown that this large degrees of freedom (≥30) corresponds to an uncertainty in the estimate of BXk of 13% or less.

4-3.2.2.3 Special Data. For some elemental systematic error sources, special data may be obtained that manifests the dispersion of the population of possible, unknown ␤ Xk values. Possible sources of this special data include (a) interlaboratory or interfacility tests; and (b) comparisons of independent measurements that depend on different principles or that have been made by independently calibrated instruments; for example, in a gas turbine test, airflow can be measured with an orifice or a bell mouth nozzle, or computed from compressor speed-flow rig data, turbine flow parameters, or jet nozzle calibrations. For these cases, the elemental systematic standard uncertainty may be evaluated as follows:

4-3.2.2.2 Published Information. For some elemental systematic error sources, published information from calibration reports, instrument specifications, and other technical references may provide quantitative information regarding the dispersion of errors for an elemental systematic error source in terms of a confidence interval, an ISO expanded

b Xk p

冪

1 N ␤X

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NX

k

兺

jp1 k

(X kj − X k )2 N Xk − 1

(4-3.5)

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where NXkp the number of special data values used in the computation of b Xk N␤ X p the number of independent samples from k the population of possible ␤ Xk values that are averaged together in the computation of the average measurement for the test (X) Xkp the average of the set of special data

of the total systematic standard uncertainty for the measurement, b X. Provided all elemental systematic standard uncertainties are evaluated in terms of their influence on the parameter being measured and in the units of the parameter being measured, these elemental systematic standard uncertainties are combined per subsection 6-2. Otherwise, these elemental systematic standard uncertainties are combined per subsection 7-4. In some cases, elemental systematic standard uncertainties may arise from the same elemental error source and are therefore correlated. See subsection 8-1 for a detailed discussion.

Xkjp the j th data point of the set of special data that manifests the dispersion of the population of possible ␤ Xk values associated with the k th elemental error source For most measurements (especially those made using a single instrument calibrated at a single laboratory and installed in a single location), only a single sample from the population of possible ␤ Xk values is included in the computation of the average measurement for the test (X) and hence N␤ X p 1. The following illustrate some k possible cases where N␤ X may be greater than one. k (a) Several independent measurement methods that depend on different principles are used to measure the same parameter. The results from each of the measurement methods (each determined as an average value over the duration of the test) are used as input to eq. (4-3.5) to evaluate the elemental systematic standard uncertainty associated with the error inherent to the various measurement methods. If the average measurement reported for the test is the average of the results from all of the measurement methods, then the value for N␤ X used in eq. k (4-3.5) is equal to the number of independent measurement methods employed. (b) An instrument is sent to multiple laboratories to obtain calibration data for the instrument prior to using the instrument in a test. The results from each of the independent laboratories (each determined as an offset to be applied to the instrument when measuring a specific input level) are used as input to eq. (4-3.5) to evaluate the elemental systematic standard uncertainty associated with the error inherent to the various laboratories. If the average measurement from the instrument reported for the test is based upon application of the average offset from all of the laboratories, then the value for N␤ X used in eq. (4-3.5) is equal to the number of k independent laboratories employed.

4-3.3 Combined Standard Uncertainty and Expanded Uncertainty As mentioned previously, the total uncertainty in a measurement is the combination of uncertainty due to random error and uncertainty due to systematic error. The combined standard uncertainty of the measurement mean, which is the total uncertainty at the standard deviation level, is calculated as follows: uX p

(4-3.6)

where b Xp the systematic standard uncertainty s Xp the random standard uncertainty of the mean The expanded uncertainty of the measurement mean is the total uncertainty at a defined level of confidence. For applications in which a 95% confidence level is appropriate, the expanded uncertainty is calculated as follows: U X p 2u X

(4-3.7)

where the assumptions required for this simple equation are presented in subsection 6-4. Expanded uncertainty is used to establish a confidence interval about the measurement mean which is expected to contain the true value. Thus, the interval X ± UX is expected to contain the true value with 95% confidence (see Fig. 4-3.3).

4-3.2.3 Combining Elemental Systematic Standard Uncertainties. Once evaluated, all of the elemental systematic standard uncertainties influencing a measurement are combined into an estimate 10 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

冪 (bX)2 + (sX)2

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--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

Fig. 4-3.3 Uncertainty Interval

4-4 PRETEST AND POSTTEST UNCERTAINTY ANALYSES

(2) selecting alternative measurement methods by varying test instrumentation, calibration techniques, installation methods, and/or measurement locations; and (3) increasing sample sizes by increasing sampling frequencies, increasing test duration, and/or conducting repeated testing. Additionally, a pretest uncertainty analysis facilitates communication between all parties to the test about the expected quality of the test. This can be essential to establishing agreement on any deviations from applicable test code requirements and can help reduce the risk that disagreements regarding the testing method will surface after conducting the test. (b) The objective of a posttest analysis is to establish the uncertainty interval for a test result, after conducting a test. In addition to the data and information used to conduct the pretest uncertainty analysis, a posttest uncertainty analysis is based upon the additional data and information gathered for the test including all test measurements, pretest and

(a) The objective of a pretest analysis is to establish the expected uncertainty interval for a test result, prior to the conduct of a test. A pretest uncertainty analysis is based on data and information that exist before the test, such as calibration histories, previous tests with similar instrumentation, prior measurement uncertainty analyses, expert opinions, and, if necessary, special tests. A pretest uncertainty analysis allows corrective action to be taken, prior to expending resources to conduct a test, either to decrease the expected uncertainty to a level consistent with the overall objectives of the test or to reduce the cost of the test while still attaining the objectives. Possible corrective actions include (1) selecting alternative testing methods that rely upon different analysis procedures, testing under different conditions, and/or measurement of different parameters; 11 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

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posttest instrument calibration data, etc. A posttest uncertainty analysis serves to (1) validate the quality of the test result by demonstrating compliance with test requirements;

(2) facilitate communication of the quality of the test result to all parties to the test; and (3) facilitate interpretation of the quality of the test by those using the test result.

12 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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Section 5 Defining the Measurement Process 5-1 OVERVIEW

represented as uncertainties. These uncertainties in the measurement process can be grouped by source (a) calibration uncertainty (b) uncertainty due to test article and/or instrumentation installation (c) data acquisition uncertainty (d) data reduction uncertainty (e) uncertainty due to methods and other effects

The first step in a measurement uncertainty analysis is to clearly define the basic measurement process. This simple step, often overlooked, is essential to successfully develop and apply the uncertainty information. Consideration must be given to the selection of the appropriate “true value” of the measurement and the time interval for classifying errors as systematic or random. This section provides an overview of how the measurement process should be defined.

5-3.1 Calibration Uncertainty Each measurement instrument may introduce random and systematic uncertainties. The main purpose of the calibration process is to eliminate large, known systematic errors and thus reduce the measurement uncertainty to some “acceptable” level. Having decided on the “acceptable” level, the calibration process achieves that goal by exchanging the large systematic uncertainty of an uncalibrated or poorly calibrated instrument for the smaller combination of systematic uncertainties of the standard instrument and the random uncertainties of the comparison. Calibrations are also used to provide traceability to known reference standards or physical constants, or both. Requirements of military and commercial contracts have led to the establishment of extensive hierarchies of standards laboratories. In some countries, a national standards laboratory is at the apex of these hierarchies, providing the ultimate reference for every standards laboratory. Each additional level in the calibration hierarchy adds uncertainty in the measurement process (see Fig. 5-3.1).

5-2 SELECTION OF THE APPROPRIATE “TRUE VALUE”

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

Depending on the user’s perspective, several measurement objectives or goals and hence corresponding “true values” (measurements with ideal zero error) may exist simultaneously in a measurement process. For example, when analyzing a thermocouple measurement in a gas stream, several starting points or “true values” can be selected. The starting point for the analysis could begin with the “true value” defined as the metal temperature of the thermocouple junction, the gas stagnation temperature or junction temperature corrected for probe effects, or the mass flow weighted average of the gas temperature at the plane of the instrumentation. Any of the aforementioned “true values” may be appropriate. The selection of the “true value” for the uncertainty analysis must be consistent with the goal of the measurement [3]. 5-3 IDENTIFICATION OF ERROR SOURCES

5-3.2 Uncertainty Due to Test Article and/or Instrumentation Installation

Once the true value has been defined, the errors associated with measuring the true value must be identified. Examples of error sources include imperfect calibration corrections, uncontrolled test conditions, measurement methods, environmental conditions, and data reduction techniques. Estimates to reflect the extent of these errors are

Measurement uncertainty can also exist from interactions between (a) the test instrumentation and the test media or (b) between the test article and test facility. Examples of these types of uncertainty are 13

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Fig. 5-3.1 Generic Measurement Calibration Hierarchy

(a) Interactions Between the Test Instrumentation and Test Media: (1) Installation of sensors in the test media may cause intrusive disturbance effects. An example could be the measurement of airflow in an air conditioning duct. Depending on the design of the pitot static probe, it may affect the measured total and static pressure and thus the calculated airflow. (2) Environmental effects on sensors/instrumentation may exist when the sensors experience environmental effects that are different from those observed during calibration. These may be such things as conduction, convection, and radiation on a sensor when installed in a gas turbine. (b) Interactions Between the Test Article and Test Facility: (1) Test-facility limitations for certification testing affects product measurement uncertainty. An example may be an air conditioner that was bench tested in a laboratory but used in an automotive mechanics shop. The effect of the oily air can influence the quoted rating of the unit. A second example is the testing of a gas turbine engine in an altitude facility. The facility simulates altitude by lowering the ambient pressure at the test article exhaust and raising the inlet pressure at the engine inlet. In appli-

cation the inlet pressure is elevated due to the ram drag effects of the aircraft. A correction factor must be applied that corrects between uninstalled to installed aircraft engine performance. (2) Facility limitations for testing may require extrapolations to other conditions. An example is the testing of an automotive engine. The fuel consumption of an automotive engine changes with altitude and speed. An automotive test facility may only be able to test at specified altitudes and speeds, and the effects at other altitude conditions may need to be extrapolated.

5-3.3 Data Acquisition Uncertainty Uncertainty in data acquisition systems can arise from errors in the signal conditioning, the sensors, the recording devices, etc. The best method to minimize the effects of many of these uncertainty sources is to perform overall system calibrations. By comparing known input values with their measured results, estimates of the data acquisition system uncertainty can be obtained. However, it is not always possible to do this. In these cases, it is necessary to evaluate each of the elemental 14

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uncertainties and to combine them to predict the overall uncertainty.

stated. The significance of this is discussed in para. 5-4.2. In addition, the objective of the test may affect the categorization as discussed in para. 5-4.3.

5-3.4 Data Reduction Uncertainty 5-4.1 Alternate Categorization Approach

Computations on raw data are done to produce output (data) in engineering units. Typical uncertainty sources in this category stem from curve fits and computational resolution. With the recent advances in computer systems, the computational resolution uncertainty sources are often negligible; however, curve-fit error uncertainty can be significant. Other examples of data reduction uncertainty include (a) the assumptions or constants contained in the calculation routines; (b) using approximating engineering relationships or violating their assumptions; and (c) using an empirically derived correlation such as empirical fluid properties. These additional uncertainties may be of either a systematic or random nature depending on their effect on the measurement.

An alternate approach, which is used in the ISO GUM, categorizes the uncertainties based on the method used to estimate uncertainty. Those evaluated with statistical methods are classified as Type A, while those, which are evaluated by other means, are classified as Type B. Depending on the selection of the defined measurement process, there may be no simple correspondence between random or systematic and Type A or Type B. 5-4.2 Time Interval Effects Errors that may be fixed over a short time period may be variable over a longer time period. For example, calibration corrections, which are assumed fixed over the life of the calibration interval, can be considered variable if the process consists of a time interval encompassing several different calibrations. The time interval must be clearly specified to classify an error, and it may not always be the same interval as the test duration. For example, when comparing results among various laboratories, it may be appropriate to classify an error as random rather than as systematic even though that error may have been constant for the duration of any single test. The effects of a time interval may also be important when considering the stability and control of a test process. The stability of a measurement method is a generic concept related to the closeness of agreement between test results. Process stability is estimated from observations of scatter within a data set and is treated as a random error. Variability in independent test results obtained under different test conditions, varying experimental setups, or configuration changes allow for additional between-test random errors.

5-3.5 Uncertainty Due to Methods and Other Effects Uncertainties due to methods are defined as those additional uncertainty sources that originate from the techniques or methods inherent in the measurement process. These uncertainty sources, beyond those contained in calibration, installation sources, data acquisition, and data reduction, may significantly affect the uncertainty of the final results. 5-4 CATEGORIZATION OF UNCERTAINTIES This Standard delineates uncertainties by the effect of the error (i.e., systematic and random). This categorization approach supports the identification, understanding, and managing of test uncertainties. If the nature of an elemental error is fixed over the duration of the defined measurement process, then the error contributes to the systematic uncertainty. If the error source tends to cause scatter in repeated observations of the defined measurement process, then the source contributes to the random uncertainty. Because measurement uncertainties are categorized by the effect of the error, the time interval and duration of the measurement process can be important considerations and so must be clearly

5-4.3 Test Objective The classification and number of error sources are often affected by the test objective. For example, if the test objective is to measure the average gas mileage of model “XYZ” cars, the variability among or between cars of the same model must be considered. Random error obtained in a test from a given car would not include car-to-car 15

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Fig. 5-4.3 Difference Between “Within” and “Between” Sources of Data Scatter

5-5 COMPARATIVE VERSUS ABSOLUTE TESTING

variations and thus would not represent all random error sources. To observe the random error associated with car-to-car variability, the experiment would need to be run again using a random selection of different cars within the same model (see Fig. 5-4.3). The total variation in the test result is greater than that observed from a test of a single given car. This variation would be more representative of the total random error associated with determining gas mileage for the fleet of model “XYZ” cars. Of course, if the data of interest is gas mileage of a given single car, then the estimated variation with testing the representative given car is an appropriate estimate for the random error. The same short-term and long-term effects must be applied for other variables affecting gas mileage (temperature, altitude, humidity, road conditions, driver variations, etc.).

The objective of a comparative test (also known as a back-to-back test) is to determine, with the smallest measurement uncertainty possible, the net effect of a design change. The first test is run with the standard or baseline configuration. The second test is then run in the same facility with the design change and hopefully with instruments, setups, and calibrations identical to those used in the first test. The difference between the results of these tests is an indication of the effect of the design change. Depending on whether common instrumentation, setups, and calibrations are used between comparative tests, the effects of correlated uncertainties (see Section 8) may cause the total uncertainty of the difference between the test results to be less than the uncertainty of each separate test result. An example of back-to-back uncertainty analysis is shown in Example 8.1 in subsection 8-1.

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Section 6 Uncertainty of a Measurement 6-1 RANDOM STANDARD UNCERTAINTY OF THE MEAN

in the variable. For example, taking multiple measurements as a function of time while holding all other conditions constant would identify the random variation associated with the measurement system and the unsteadiness of the test condition. If the sample standard deviation of the variable being measured is also expected to be representative of other possible random variations in the measurement, e.g., repeatability of test conditions, variation in test configuration, etc., then these additional error sources will have to be varied while the multiple data measurements are taken to determine the standard deviation. Another situation where previous values of a variable would be useful is when a small sample size (N) is used to calculate the mean value (X) of a measurement. If a much larger set of previous measurements for the same test conditions is available, then it could be used to calculate a more appropriate standard deviation for the current measurement [4]. Typically, these larger data sets are taken in the early phases of an experimental program. Once the random variation of the test variables is understood, then this information can be used to streamline the test procedure by reducing the number of measurements taken in the later phases of the test. When NP previous values (XPj) are known for the quantity being measured, the sample standard deviation for the variable can be calculated as

6-1.1 General Case For X that is determined as the average of N measurements, the appropriate random standard uncertainty of the mean (sX) is given by eq. (43.3). This type of estimate is an ISO Type A estimate. In a sample of measurements, the degrees of freedom is the sample size (N). When a statistic is calculated from the sample, the degrees of freedom associated with the statistic is reduced by one for every estimated parameter used in calculating the statistic. For example, from a sample of size N, X is calculated by eq. (4-3.1). The sample standard deviation (sX ) and the random standard uncertainty of the mean (sX ) are calculated from eqs. (4-3.2) and (4-3.3), respectively, and each has N − 1 degrees of freedom (␯ ) ␯pN−1

(6-1.1)

because X (based on the same sample of data) is used in the calculation of both quantities. 6-1.2 Using Previous Values of sX In some test situations, the measurement of a variable may be only a single measurement or an average of measurements taken over a short time frame, as with a computer-based data acquisition system. In this latter case, the time frame over which the measurements are taken may be on the order of milliseconds or less while the random variations in the process may be on the order of seconds, or minutes, or even days. This “short time frame averaged” value should then be handled in the same manner as a single measurement. Information about the possible variations in a single measurement must be obtained from previous measurements of the variable taken over the time frame and conditions that cover the variations

冤

1 s X p s X Pp NP − 1

P

兺 (XP jp1

j

− XP )

2

冥

1⁄

2

(6-1.2)

where

XP p

1 NP

N

P

兺 XP jp1

j

(6-1.3)

The appropriate random standard uncertainty of the mean for the current measurement (X) is then 17

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N

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sX p

sX

K

(6-1.4)

冪N

␯p

K

兺

where N is the number of current measurements averaged to determine X. The number of degrees of freedom for this random standard uncertainty of the mean sX is ␯ p NP − 1

兺 (sXk)2 冣 冢kp1 kp1

2

(6-1.7)

(s Xk) 4 ␯k

where ␯k is the appropriate degrees of freedom for sXk and is obtained from eq. (6-1.1) or (6-1.5) as appropriate. When all error sources have large sample sizes, the calculation of ␯ is unnecessary. However, for small samples, when combining elemental random standard uncertainties of the mean by the root-sum-square method [see eq. (6-1.6)], the degrees of freedom (␯) associated with the combined random standard uncertainty is calculated using the Welch-Satterthwaite formula [5] above [eq. (6-1.7)].

(6-1.5)

This estimate of the random standard uncertainty is an ISO Type A estimate since it is obtained from data. The case where the data sample is only a single measurement is handled above with N p 1. 6-1.3 Using Elemental Random Error Sources Another method of estimating the random standard uncertainty of the mean for a measurement is from information about the elemental random error sources in the entire measurement process. If all the random standard uncertainties are expressed in terms of their contribution to the measurement, then the random standard uncertainty for the measurement mean is the root-sum-square of the elemental random standard uncertainties of the mean from all sources divided by the square root of the number of current readings (N) averaged to determine X

sX p

1

冪N

冤 兺 (s K

kp1

) X k

2

冥

1⁄

6-1.4 Using Estimates of Sample Standard Deviation In a pretest uncertainty analysis, previous information might not be available to estimate the sample standard deviation as discussed in para. 6-1.2 or 6-1.3. In this case, an estimate of the sample standard deviation (sX) would be made using engineering judgment and the best available information. This type of uncertainty estimate would be an ISO Type B estimate.

2

6-2 SYSTEMATIC STANDARD UNCERTAINTY OF A MEASUREMENT

(6-1.6)

The systematic standard uncertainty b X of a measurement was defined in para. 4-3.2 as a value that quantifies the dispersion of the systematic error associated with the mean. The true systematic error (␤) is unknown, but b X is evaluated so that it represents an estimate of the standard deviation of the distribution for the possible ␤ values. It should be noted that while b X is an estimate of the dispersion of the systematic errors in a measurement, the systematic error that is present in a specific measurement is a fixed single value of ␤. The systematic standard uncertainty of the measurement is the root-sum-square of the elemental systematic standard uncertainties b Xk for all sources.

where K is the total number of random error (or uncertainty) sources. Each of the elemental random standard uncertainties of the mean sXk is calculated using the methods described in para. 6-1.1 or 6-1.2 depending upon which is appropriate, and each is assumed to be an ISO Type A estimate. If in each of the N measurements of the variable X the output of an elemental component is averaged Nk times to obtain Xk, then the method in para. 6-1.1 would be used. If instead previous information is used to obtain sXk, then the method in para. 6-1.2 would apply. The degrees of freedom for the estimated random standard uncertainty of the mean (sX) is dependent on the information used to determine each of the elemental random standard uncertainties of the mean and is calculated as

K

bX p

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兺 (b X )2冥 冤 kp1 k

1⁄

2

(6-2.1)

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ASME PTC 19.1-2005

where

There may be situations when it is convenient to classify elemental uncertainties by both effect and source. Such classifications would be useful in international test programs. This Supplement recommends the following nomenclature for dual classification:

Kp the total number of systematic error sources and each bXkp an estimate of the standard deviation of the kth elemental error source

b Xk,Ap elemental systematic standard uncertainty calculated from data, as in a calibration process b Xk,Bp elemental systematic standard uncertainty estimated from the best available information s Xk,Ap elemental random standard uncertainty calculated from data s Xk,Bp elemental random standard uncertainty estimated from best available information

Note that in eq. (6-2.1) all of the elemental systematic standard uncertainties are expressed in terms of their contribution to the measurement. For each systematic error source in the measurement, the elemental systematic standard uncertainty must be estimated from the best available information. Usually these estimates are made using engineering judgment (and are therefore ISO Type B estimates). Sometimes previous data are available to make estimates of uncertainties that remain fixed during a test (and are therefore ISO Type A estimates). If any of the elemental systematic uncertainties are nonsymmetrical, then the method given in subsection 8-2 should be used to determine the systematic standard uncertainty of the measurement. There can be many sources of systematic error in a measurement, such as the calibration process, instrument systematic errors, transducer errors, and fixed errors of method. Also, environmental effects, such as radiation effects in a temperature measurement, can cause systematic errors of method. There usually will be some elemental systematic standard uncertainties that will be dominant. Because of the resulting effect of combining the elemental uncertainties in a root-sum-square manner, the larger or dominant ones will control the systematic uncertainty in the measurement; however, one should be very careful to identity all sources of fixed error in the measurement.

6-4 COMBINED STANDARD AND EXPANDED UNCERTAINTY OF A MEASUREMENT For simplicity of presentation, a single value is often preferred to express the estimate of the error between the mean value (X) and the true value, with a defined level of confidence. The interval X ± UX

represents a band about X within which the true value is expected to lie with a given level of confidence (see Fig. 4-3.3). The uncertainty interval is composed of both the systematic and random uncertainty components. The general form of the expression for determining the uncertainty of a measurement is the rootsum-square of the systematic and random standard uncertainties of the measurement, with this quantity defined as the combined standard uncertainty (uX ) [1].

6-3 CLASSIFICATION OF UNCERTAINTY SOURCES As discussed in subsection 1-1, the ISO Guide [1] classifies uncertainties by source as either Type A or Type B. Type A uncertainties are the calculated standard deviations obtained from data sets. Type B uncertainties are those that are estimated or approximated rather than being calculated from data. Type B uncertainties are also given as standard deviation level estimates. In this Supplement, uncertainties are classified by their effect on the measurement, either random or systematic, rather than by their source. This effect classification is chosen since most test operators are concerned with how errors in the test will affect the measurements.

uX p

冪 (b X )2 + (sX )2

(6-4.2)

where b Xp the systematic standard uncertainty [eq. (6-2.1)] s Xp the random standard uncertainty of the mean [eq. (4-3.3), (6-1.4), or (6-1.6) as appropriate] In order to express the uncertainty at a specified confidence level, the combined standard uncer19

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Table 6-4-1 Circulating Water Bath Temperature Measurements (Example 6-4.1) Elapsed Time, min

Measured Temp., °C

Elapsed Time, min

Measured Temp., °C

Elapsed Time, min

Measured Temp., °C

0 1 2 3 4

85.11 84.89 85.07 84.77 85.24

11 12 13 14 15

85.28 85.11 84.80 84.79 85.22

21 22 23 24 25

85.23 85.12 85.43 84.50 85.22

5 6 7 8 9 10

84.72 85.00 85.39 84.72 85.50 85.18

16 17 18 19 20

85.05 84.58 85.20 85.14 85.05

26 27 28 29 30

85.39 84.74 85.35 84.75 84.56

tainty must be multiplied by an expansion factor taken as the appropriate Student’s t value for the required confidence level (see Nonmandatory Appendix B). Depending on the application, various confidence levels may be appropriate. The Student’s t is chosen on the basis of the level of confidence desired and the degrees of freedom. The degrees of freedom used is a combined degrees of freedom based on the separate degrees of freedom for the random standard uncertainty and the elemental systematic standard uncertainties (see Nonmandatory Appendix B). A t value of 1.96 (usually taken as 2) corresponds to large degrees of freedom and defines an interval with a level of confidence of approximately 95%. This expansion factor of 2 is used for most engineering applications. For other confidence levels see Nonmandatory Appendix B. The uncertainty for 95% confidence and large degrees of freedom (t p 2) is calculated by U X p 2u X

(a) Uncertainty Due to Random Error. The uncertainty due to the random error of the average temperature measurement is evaluated as follows. (1) The sample mean, or average value, of the temperature measurements is determined using eq. (4-3.1) as follows:

Xp

1 N

N

兺 Xj p 85.04°C jp1

(2) The sample standard deviation is determined using eq. (4-3.2) as follows:

冪jp1兺 N

sX p

(Xj − X)2 p 0.28°C N−1

(3) The random standard uncertainty of the sample mean is determined using eq. (4-3.3) as follows:

(6-4.3)

where uX p the combined standard uncertainty [eq. (6-4.2)]

sX p

Example 6-4.1. A digital thermometer was used to measure the average temperature of a circulating water bath that is being used in an experiment. The experiment lasted a total of 30 min. Temperature measurements were collected every minute resulting in a total of 31 data points as presented in Table 6-4-1.

冪N

p 0.05°C

(b) Uncertainty Due to Systematic Error. The uncertainty due to the systematic error of the average circulating water bath temperature measurement is evaluated by (1) identifying all elemental sources of systematic error for the measurement; 20

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Table 6-4-2 Systematic Uncertainty of Average Circulating Water Bath Temperature Measurements (Example 6-4.1) Description of Systematic Uncertainty Source

Systematic Standard Uncertainty, °C

Calibration of digital thermometer Environmental influences (ambient temperature, humidity, etc.) on digital thermometer Effects of conduction and radiation heat transfer between the digital thermometer and the surroundings Uniformity of circulating water bath (spatial uncertainty) Total systematic uncertainty

0.05 0.005

0.0005

0.05 0.07

(c) Expanded Uncertainty. The expanded uncertainty of the average circulating water bath temperature measurement is evaluated using eqs. (6-4.2) and (6-4.3) as follows:

(2) evaluating elemental systematic standard uncertainties as the standard deviations of the possible systematic error distributions; and (3) combining the elemental systematic standard uncertainties into an estimate of the total systematic standard uncertainty for the measurement. For the purposes of this example, a summary of this evaluation is presented in Table 6-4-2. Refer to para. 4-3.2 and subsections 6-2, 7-4, 8-1, and 8-2 for further discussion of the process for identifying, evaluating, and combining elemental systematic uncertainties.

U X p 2 冪 (bX ) 2 + (s X ) 2 p 0.17°C

Therefore, the true average temperature of the circulating water during the experiment is expected to lie within the following interval with 95% confidence: X ± UX p 85.04 ± 0.17°C

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Section 7 Uncertainty of a Result 7-1 PROPAGATION OF MEASUREMENT UNCERTAINTIES INTO A RESULT

mean values based on Ni repetitions. Ni can be different for each Xi. Repeated tests are those run under the same conditions to estimate a parameter or a set of parameters (Xi ). The statistics from repeated tests allow for quantifying the expected variation in a parameter or in a result derived from parameters. The estimation of the sample mean and standard deviation based on Ni measurements is calculated as described in para. 4-3.1. When tests are duplicated under similar but somehow changed operating conditions, these generate multiple data sets for the measured parameters. The statistics found by combining these duplicated data sets allow for a reasonable estimate of the variations possible in the result that might be due to the control of test operating conditions, or use of different test rigs, instrumentation, or test location. Whereas these influences might normally be considered systematic errors during repeated tests, the duplicated tests can randomize these systematic errors providing error estimates from the statistical variations in the combined data pool [6]. The overall reported result will usually be combined to provide the mean of the multiple results, R. Careful consideration should be given to designing the test series to average as many causes of variation as possible within cost constraints. The test design should be tailored to the specific situation. For example, if experience indicates that timeto-time and test apparatus-to-apparatus variations are significant, a test design that averages multiple test results on one rig or for only one day may produce optimistic random uncertainty estimates compared to testing several rigs, each monitored several times over a period of several days. The list of test variation causes are many and may include the above plus environmental and test crew variations. Historic data are invaluable for studying these effects. A statistical technique called analysis of variance (ANOVA) is useful for partitioning the total variance by source. If the pretest

Calculated results, such as in an estimate of efficiency, are not usually measured directly. Instead, more basic parameters, such as temperature and pressure, are either measured or assigned and the required result is calculated as a function of these parameters, including using unmeasured properties, such as tabulated coefficients. Uncertainties in these measurements or assigned values of the parameters are propagated to the result through the functional relationship between the result and the parameters. The effect of the propagation can be approximated by the Taylor series method (see Nonmandatory Appendix C). To estimate the uncertainty of a calculated result, the result, R, is expressed in terms of the average or assigned values of the independent parameters (Xi ) that enter into the result. That is, R p f( X1, X 2, . . ., X I )

(7-1.1)

where the subscript I signifies the total number of parameters involved in R, and the average values of the independent parameters are obtained as

Xi p

1 Ni

Ni

兺 Xi jp1

j

(7-1.2)

where Ni is the number of repeated measurements of Xi. Ni will be 1 for a single data point or assigned value of a parameter. 7-1.1 Single and Multiple Tests In some experimental situations, a set of parameters (Xi ) is measured and a single result, R, is calculated. This case is called a single test result. In this case, some of the parameters may be based on single measurements and others may be the 22 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

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uncertainty analysis (see subsection 4-4) identifies unacceptably large error sources, special tests to measure the effects of these sources should be considered.

Numerical differentiation is covered in various references [e.g., 8]. To best approximate the sensitivity that would be obtained analytically, the value of ⌬Xi used should be as small as practical (i.e., large enough to keep truncation errors from influencing the calculations).

7-2 SENSITIVITY Sensitivity is the instantaneous rate of change in a result to a change in a parameter. Two approaches to estimating the sensitivity coefficient of a parameter are discussed in the following text.

7-3 RANDOM STANDARD UNCERTAINTY OF A RESULT 7-3.1 Single Test

7-2.1 Analytically

The absolute random standard uncertainty of a single test result may be determined from the propagation equation (see Nonmandatory Appendix C) as

When there is a known mathematical relationship between the result, R, and its parameters (X1, X2, . . ., XI ), then the absolute (dimensional) sensitivity coefficient (␪i) of the parameter Xi may be obtained by partial differentiation.

sR p

Thus, if R p f(X1, X2, . . ., XI), then ␪i p

∂R

(7-2.1)

∂X i

␪ i′ p

∂X i

p

冢 冣

X i ∂R R ∂X i

sR p R

(7-2.2)

7-2.2 Numerically Finite increments in a parameter also may be used to evaluate sensitivity using the data reduction calculation procedure. In this case, ␪i is given by ⌬R

␪ i′ p

⌬X i Xi

p

冢 冣

X i ⌬R R ⌬X i

(7-3.1)

冤兺 冢 I

ipi

␪ ′i

sX i Xi

冣冥 2

1⁄

2

(7-2.4)

M

The result is calculated using Xi to obtain R [7]. The derivatives and values for ⌬R are best estimated by use of central differencing methods.

R p

兺 Rm mp1 M

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(7-3.2)

When more than one test is conducted with the same instrument package (i.e., repeated tests), the uncertainty of the average test result may be reduced from that for one test because of the reduction in the random uncertainty of the average. However, systematic uncertainty will remain the same as for a single test. The average result from more than one test is given by

and ␪i′ by ⌬R R

冥

2

7-3.2 Multiple Tests: Repeated Tests

(7-2.3)

⌬Xi

共 ␪ i s X i兲

1⁄

The symbols ␪i and ␪i′ are the absolute and relative sensitivity coefficients, respectively, of eqs. (7-2.1) or (7-2.3) and (7-2.2) or (7-2.4), and sX i is the random standard uncertainty of the measured parameter average (Xi ), determined according to the methods presented in subsection 6-1.

Xi

␪ i′ p

ipi

2

The relative random standard uncertainty of a result is

Analogously, the relative (nondimensional) sensitivity coefficient (␪i′) is ∂R R

冤兺 I

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(7-3.3)

ASME PTC 19.1-2005

TEST UNCERTAINTY

where M signifies the number of tests available. Following eq. (4-3.1), the estimate of the standard deviation of the distribution of the results is

sR p

冢

M

兺 (Rm − R)2 mp1 M−1

冣

1⁄

6-2). Equations 7-4.1 and 7-4.2 assume the systematic uncertainties for the different parameters are independent.

2

7-5 COMBINED STANDARD UNCERTAINTY AND EXPANDED UNCERTAINTY OF A RESULT

(7-3.4)

The general form of the expression for determining the combined standard uncertainty of a result is the root-sum-square of both the systematic and the random standard uncertainty of the result. The following simple expression for the combined standard uncertainty of a result applies in many cases:

where sR includes random errors within tests and variation between tests. The degrees of freedom associated with sR is determined by ␯ p M − 1. The random standard uncertainty of the result is estimated directly from the sample standard deviation of the mean result from multiple tests and is sR p

u R p 关 (bR) 2 + (S R)2兴

sR

1⁄

2

(7-5.1)

(7-3.5)

冪M

where bR is obtained from eq. (7-4.1) and sR is obtained from either eq. (7-3.1) for a single test result or from eq. (7-3.5) for a multiple test result. The expanded uncertainty in the result at approximately 95% confidence is given by

This random standard uncertainty of the result also has ␯R pM − 1 degrees of freedom. 7-3.3 Multiple Tests: Combined Tests

U R,95 p 2uR

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

When tests are duplicated under similar but somehow changed operating conditions, the average result and average standard deviations are given by combining the information of each test. The statistical equations used to assess variations between tests are similar to those discussed in para. 7-3.2 but discussion of their interpretation is beyond the intention of this document and is presented in detail in reference [6].

where the use of the factor of 2 assumes sufficiently large degrees of freedom for the 95% confidence level (i.e., t95 p 2). This factor can be modified as appropriate for other confidence levels and small degrees of freedom as discussed in Nonmandatory Appendix B. The interval within which the true result should lie with 95% confidence is given as R ± U R,95

7-4 SYSTEMATIC STANDARD UNCERTAINTY OF A RESULT

bR p

兺 (␪i bX ) 冥 冤 ip1

1⁄

2

2

(7-4.1)

i

The relative systematic standard uncertainty of a result is bR p R

I

冤兺 冢 ip1

␪i′

bXi

2

X 冣 冥

1⁄

2

7-6 EXAMPLES OF UNCERTAINTY PROPAGATION

(7-4.2)

7-6.1 The Magnitude of Uncertainty in the Test

i

The symbol b Xi is the systematic standard uncertainty of the measured parameter (see subsection

The magnitude of uncertainty in a test can be quantified by using the data reduction equations 24

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(7-5.3)

The special cases of correlated systematic uncertainties and nonsymmetric systematic uncertainties are covered in subsections 8-1 and 8-2, respectively. Treatment of uncertainty intervals with alternate confidence levels, error distributions, and alternate uncertainty equations is addressed in Nonmandatory Appendix B. With high-speed computing capabilities, Monte Carlo methods have become popular for determining the test result uncertainty using the test input variable values and their associated uncertainties.

The absolute systematic standard uncertainty of a result may be determined from the propagation equation (see Nonmandatory Appendix C) as I

(7-5.2)

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TEST UNCERTAINTY

ASME PTC 19.1-2005

Table 7-6.1-1 Table of Data (Example 7-6.1) Independent Parameters Uncertainty Contribution of Parameters to the Result (in Result Units Squared)

Parameter Information (in Parameter Units)

Absolute Absolute Absolute Systematic Absolute Systematic Random Standard Random Standard Standard Absolute Uncertainty Uncertainty Uncertainty, Uncertainty, Sensitivity, Contribution, Contribution, bX SX ␪i ␪i b X 2 ␪i sX 2

Symbol

Description

Units

Nominal Value

Standard Deviation

Ni

P L A

Power Length Crosssectional area Temperature difference

W m m2

18.4 0.025 0.018

0.65 0.0012 0.0013

50 32 32

0.1 6.4E-05 6.4E-05

0.092 2.12E-04 2.30E-04

°C

8.5

0.5

50

0.1

0.071

⌬T

i

and estimates for each of the test parameters. This is important and useful because it provides insight into whether a particular test or methodology is feasible based on the acceptable order of magnitude in uncertainty. If for example the uncertainty in the results needs to be 10% or less, and the pretest calculations show the best results will be on the order of 15%, then appropriate corrective action to reduce the uncertainty can be taken. Example 7-6.1: Tests are often conducted using the guarded hot plate technique to determine the thermal conductivity of a material. The guarded hot plate is used because it is relatively inexpensive and effective in providing the boundary conditions necessary to ensure one dimensional heat flow through the material. For steady state conditions, the one-dimensional form of Fourier’s law can be used to characterize the thermal conductivity of the test specimen. P L A ⌬T

i

2.67E-04 5.9E-05 1.14E-04

2.25E-04 6.50E-04 1.47E-03

−0.354

1.25E-03

6.26E-04

(7-6.1)

7-6.2 Ranking of Uncertainty Components

The result, R, which in this example is given by the thermal conductivity k, is determined as a function of the parameters using their nominal values and eq. (7-6.1), which itself is just the specific form of eq. (7-1.1). In eq. (7-6.1), P is the electrical power dissipated by the hot plate, ⌬T is the temperature difference across the hot and

Uncertainty analysis can be used to identify the largest contributors to the overall uncertainty. This enables designers and engineers to make pretest hardware and/or experimental methodology improvements. Example 7-6.2: A test is to be conducted to determine the pressure loss coefficient across a 25

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i

0.163 120.3 −167

cold surfaces, A is the cross-sectional area, and L is the thickness of the material, respectively. The uncertainty in conductivity will be directly dependent on the method used to measure each parameter in the computation. To estimate the expected conductivity uncertainty, we must know or estimate the measurement errors associated with the power, temperature, and geometry. Table 7-6.1-1 lists the independent parameters for this example and Table 7-6.1-2 the final results. From the previous analysis, the conductivity was determined to be k p 3.01 ± 0.14 W/mK at 95% confidence. Inspection of Table 7-6.1-1 lends insight into the magnitude of the contributions to the overall uncertainty. The uncertainty in power required to generate a constant heat flux will depend on the type of material being tested. Therefore, it is important to select the power setting such that measurements are at least at the midscale of the power supply range.

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

kp

i

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ASME PTC 19.1-2005

TEST UNCERTAINTY

Table 7-6.1-2 Summary of Data (Example 7-6.1)

Symbol

Description

Units

Calculated Value, R

K

Conductivity

W/mK

3.01

Absolute Systematic Standard Uncertainty, bR

Absolute Random Standard Uncertainty, sR

Combined Standard Uncertainty, uR

Expanded Uncertainty of the Result, UR, 95

0.041

0.055

0.068

0.14

Table 7-6.2-1 Table of Data (Example 7-6.2) Parameter Information Uncertainty Contribution of Parameters to the Result (in Result Units Squared)

Parameter Information (in Parameter Units)

Symbol Description ⌬P D Q

␳

Pressure drop Diameter Volumetric flow rate Air density

Nominal Value

Units Pa

230.0

Absolute Systematic Standard Uncertainty, bX

Standard Deviation

Ni

10.0

50

5.0

1.41 2.25E-05 2.12E-04

i

m m3/s

0.030 0.0116

1.27E-04 1.20E-03

32 32

6.35E-06 1.0E-04

kg/m3

1.192

...

...

0.013

valve at a specified operating condition. The apparatus consists of a closed loop tube containing a differential pressure gauge for measuring the pressure drop across the valve, a blower for cycling the working fluid (air), and a flow meter. The equation characterizing the loss coefficient is Kp

⌬P 1 2 ␳V 2

␲ 2 D4⌬P 8 ␳Q 2

i

...

Absolute Sensitivity, ␪i 0.0062 191.0 −247.0 −1.202

P p 101.3 kPa T p 296.2 K

(7-6.2)

Absolute Random Standard Uncertainty Contribution, ␪i S X 2

9.70E-04

7.64E-05

1.47E-06 6.10E-04

1.85E-05 2.74E-03

2.45E-04

...

i

i

sP p 8.0 kPa sT p 1.6 K

NP p 50 NT p 50

Following the procedure outlined in subsection 7-1, the density and expanded uncertainty in density were determined to be ␳ p 1.192 ± 0.027 kg/ m3 at 95% confidence, where the uncertainty contribution from the gas constant was assumed negligible (9). Data from the loss coefficient test are summarized in Tables 7-6.2-1 and 7-6.2-2. The density uncertainty is treated as simply a systematic contribution in the loss coefficient calculation. From the previous analysis, the loss coefficient was determined to be K p 1.433 ± 0.137 at 95%

(7-6.3) 26

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Absolute Systematic Standard Uncertainty Contribution, ␪i b X 2

The Mach number for this flow is well below 0.3, and the density is assumed to follow the ideal gas equation of state, ␳ p P/RT. Measurements of the ambient air were made, and the data are given as:

where ⌬P is the pressure drop across the valve, ␳ is the density of air and V is the average air velocity. From the definition of mass flow rate, the velocity can be replaced with the volumetric flowrate Q through the expression Q p AV, where A is the cross-sectional area of the tube. In this example we will consider a round tube so A p ␲D2/4. Substituting into eq. (7-6.2) gives Kp

Absolute Random Standard Uncertainty, SX

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TEST UNCERTAINTY

ASME PTC 19.1-2005

Table 7-6.2-2 Summary of Data (Example 7-6.2) Calculated Results

Symbol

Description

Units

Calculated Value, R

K

Loss coefficient

Dimensionless

1.433

Absolute Systematic Standard Uncertainty, bR

Absolute Random Standard Uncertainty, sR

Combined Standard Uncertainty of the Result, uR

Expanded Uncertainty of the Result, UR, 95

0.043

0.053

0.068

0.137

Fig. 7-6.2 Pareto Chart of Systematic and Random Uncertainty Component Contributions to Combined Standard Uncertainty confidence. A pareto chart (see subsection A-4) of the systematic and random uncertainty contributions to the loss coefficient is given in Fig. 76.2. The relative contributions of the systematic uncertainty, as given by 共␪ibXi 兲2冫(uK )2, and of the random uncertainty, as given by 共␪isXi 兲2冫(uK )2, are shown and the pareto is ranked (left to right)

in terms of the combined standard uncertainty contribution of each parameter to the result. The analysis shows that random errors in the volumetric flow rate measurement and systematic errors in the pressure drop measurement are the largest contributors to the overall uncertainty in loss coefficient.

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ASME PTC 19.1-2005

TEST UNCERTAINTY

Section 8 Additional Uncertainty Considerations 8-1 CORRELATED SYSTEMATIC STANDARD UNCERTAINTIES

source making them correlated, or thus their measurement errors are no longer independent. The units of the correlation terms (covariances), bXiXk, are the product of the units of Xi and Xk. The covariance terms in eq. (8-1.2) must be properly interpreted. Each bXiXk, term represents the sum of the products of the portions of bXi and bXk that arise from the same source and are therefore perfectly correlated [10]. For instance, if elemental systematic standard uncertainties 1 and 2 for parameters 2 and 3 were from a common source, then bX2X3, would be determined as

The expressions for the systematic standard uncertainty of the result in subsection 7-4 [eqs. (7-4.1) and (7-4.2)] assume that the systematic standard uncertainties of the measured parameters are all independent of each other. There are many situations where systematic errors for some of the parameters in a result are not independent. Examples would include using the same apparatus to measure different parameters or calibrating different parameters against the same standard. In these cases, some of the systematic errors are said to be correlated and these nonindependent errors must be considered in the determination of the systematic standard uncertainty of the result [9]. Consider an example where the result (R) is determined from three parameters (X 1 , X 2 , X 3 ) that have correlated systematic errors. The result is calculated as R p f( X 1 , X 2 , X 3 )

b X2 X 3 p b X 2 + b X2 b X 3 1

(8-1.1) I

bR p

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

+ 2␪ 2␪ 3 bX 2 X 3 兴

I-1

(8-1.2)

(8-1.4)

bip systematic standard uncertainty in parameter i bikp covariance between the systematic standard uncertainties for the ith and kth parameters, calculated as follows:

2

The first three terms under the square root in eq. (8-1.2) are the same as those obtained by using eq. (7-4.1), and the last three terms are those that account for the correlation among the systematic standard uncertainties in X 1 , X 2 , and X 3 . The terms bXiXk are the estimates of the covariances of the systematic errors in Xi and Xk (see Nonmandatory Appendix B). These terms must be included when systematic standard uncertainties for separate parameters, Xi and Xk, are from the same

L

b ik p

兺 bi bk lp1 l

l

(8-1.5)

Ip number of parameters i and kp indexes indicating the ith and kth parameters ␪p sensitivity coefficients 28

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I

兺 (␪i bi)2 + 2 ip1 兺 kpi+1 兺 ␪i␪k bik ip1

where

b R p 关共 ␪1 b X 1兲 2 + 共␪ 2 b X 2兲 2 + 共 ␪ 3 b X 3兲2

1⁄

(8-1.3) 2

The example in eq. (8-1.2) can be expanded to any number of parameters by including the term for each pair of parameters that has correlated systematic standard uncertainties. Therefore, another form of eq. (8-1.2) is

and the absolute systematic standard uncertainty of the result is given as

+2␪ 1␪ 2 bX 1 X 2 + 2␪ 1␪ 3 bX 1 X 3

2

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TEST UNCERTAINTY

ASME PTC 19.1-2005

R p 0.0250 (106 Pa)−1 Pn b 2Rp [(−0.0325)(106 Pa)−1 (0.2)(106 Pa)]2 + [(0.025)(106 Pa)−1 (0.2)(106 Pa)]2 bRp 0.0082

␪np

Table 8-1 Burst Pressures (Example 8-1-1)

Program 1 Meter #1 Meter #2 Program 2 Meter #3

Base Design, P b, 106 Pa

Improved Design, Pn, 106 Pa

Systematic Standard Uncertainty, bP, 106 Pa

40.0 ...

... 52.0

0.2 0.2

42.0

54.7

0.5

Program 2 (correlated systematic uncertainties): 54.7 p 1.30 42.0 ␪bp −0.0310 (106 Pa)−1 ␪np 0.0238 (106 Pa)−1 Rp

b 2Rp [(−0.0310)(106 Pa)−1 (0.5)(106 Pa)]2 + [(0.0238)(106 Pa)−1 (0.5)(106 Pa)]2 + 2(−0.0310)(106 Pa)−1 (0.0238)(106 Pa)−1 (0.5)(106 Pa)(0.5)(106 Pa) bRp 0.0036

where Lp number of common (correlated) error sources lp an index

This example demonstrates the strength of the back-to-back testing technique using the same instrumentation. Even though the pressure transducer in Program 2 had a systematic standard uncertainty of more than twice those of the transducers in Program 1, the systematic standard uncertainty of the result for Program 2 was less than half of that for Program 1. Example 8-1-2 (adapted from [9]): Consider the piping arrangement shown with the four flowmeters:

Example 8-1-1: The use of back-to-back tests is an excellent method to reduce the systematic standard uncertainty when comparing two or more designs. This method is a special case of correlated systematic standard uncertainties. Consider a burst test for an improved container design. The improvement in the design can be expressed as the fraction Rp

Pn Pb

where Pnp the burst pressure of the new design Pbp the burst pressure of the original or base design Table 8-1 provides burst tests for two different programs. In the first test program, different pressure transducers were used in the tests on the two designs. There were no correlated systematic standard uncertainties common between these two transducers. In the second program, the same pressure transducer was used for both tests; therefore, the systematic standard uncertainty was the same and was correlated for the two test measurements. Program 1 (no correlated systematic uncertainties):

From conservation of mass, a balance check would yield z p m 4 − m1 − m 2 − m 3 p 0

If the errors in the flow rate measurements are predominantly systematic, then for the balance check to be satisfied the absolute value of z must be less than or equal to the uncertainty in z

52.0 p 1.30 40.0 −R ␪bp p −0.0325 (106 Pa)−1 Pb Rp

|z| ≤ 2bz

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ASME PTC 19.1-2005

TEST UNCERTAINTY

Consider the case where the dominant systematic errors are from the calibration standard and from the calibration curve fit. The calibration standard systematic standard uncertainty for each flowmeter is ±1.5 kg/h for the three small meters and ±4.5 kg/h for the large meter. The curve fit systematic standard uncertainty for each meter is ±0.5 kg/h. Note that for this example the derivatives for eq. (8-1.2) are

because it is due to the scatter in the calibration line. The final uncertainty is obtained as follows:

␪ m1 p ␪m2 p ␪m3 p −1

b m1m2 p b m1m3 p b m2 m3 p (1.5 kg/h)(1.5 kg/h)

b m1 p bm2 p bm3 p 1.6 kg/h b m4 p 4.5 kg/h

and

and

Using eq. (8-1.4) for four parameters with three of them having correlated systematic standard uncertainties, the systematic standard uncertainty for z becomes

␪ m4 p 1

Case 1: Each Flowmeter Calibrated Against a Different Standard. In this case, all of the systematic standard uncertainties are uncorrelated with the systematic standard uncertainty for the three small flowmeters determined as 1⁄

b z p 关共 ␪m1 bm1 兲 2 + 共 ␪ m2 b m2 兲 2 + 共 ␪ m3 b m3 兲 2 + 共 ␪ m4 b m4 兲 2 + 2 ␪ m 1␪ m 2 b m 1 m2 + 2 ␪ m 1␪ m 3 b m 1 m3 + 2 ␪ m 2␪ m 3 b m 2m 3 兴

2

or b z p 关共 bm1 兲 2 + 共 bm2 兲 2 + 共 bm3 兲 2 + 共 b m4 兲 2

and the systematic standard uncertainty for the large flowmeter calculated as 1⁄

+ 2b m1m2 + 2b m1m3 + 2b m2m3 兴

2

2

b z p 6.4 kg/h

and the balance check will be satisfied if

Using eq. (7-4.1), the systematic standard uncertainty for z is b z p 关共 ␪m1 bm1 兲 2 + 共 ␪ m2 b m2 兲 2 + 共 ␪ m3 b m3 兲 2 + 共 ␪ m4 b m4 兲 2 兴

1⁄

|z| ≤ 2bz p 12.9 kg/h

Note that in this case the signs for all the correlated terms are positive because all of the derivatives of z with respect to m1, m2, and m3 are negative. If flowmeters 1, 2, and 3 are calibrated against the same standard, and flowmeter 4 is calibrated against a different standard, the systematic standard uncertainty for z is larger than if all the meters had been calibrated against different standards (Case 1). Case 3: Flowmeters 1, 2, 3, and 4 Calibrated Against the Same Standard. In this case, the systematic standard uncertainties are

2

or 1⁄

2

b z p 关共 bm1 兲2 + 共 bm2 兲 2 + 共 bm3 兲 2 + 共 b m4 兲 2 兴 p 5.3 kg/h

and the balance check will be satisfied if |z| ≤ 2bz p 10.6 kg/h

Case 2: Flowmeters 1, 2, and 3 Calibrated Against the Same Standard and Flowmeter 4 Calibrated Against a Different Standard. In this case, the systematic standard uncertainty for the three small flowmeters that originates from the calibration standard is correlated. The systematic standard uncertainty from their curve fits is not,

b m1 p bm2 p bm3 p 1.6 kg/h

and b m4 p 4.5 kg/h 30

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1⁄

2

b m4 p 关(4.5) + (0.5) 兴 p 4.5 kg/h 2

2

2

b mi (i p 1, 2, 3) p 关 (1.5) + (0.5) 兴 p 1.6 kg/h 2

1⁄

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ASME PTC 19.1-2005

8-2 NONSYMMETRIC SYSTEMATIC UNCERTAINTY

with

In some experiments, physical models can be used to essentially replace the asymmetric uncertainties with symmetric uncertainties in additional experimental variables. If this can be done then it should be, but if not then the method of para. 8-2.1 should be used. This paragraph presents a method for determining nonsymmetric uncertainty intervals in these cases [11].

b m1m2 p b m1m3 p b m2m3 p (1.5 kg/h)(1.5 kg/h)

and b m1m4 p b m2m4 p b m3m4 p (1.5 kg/h) (4.5 kg/h)

Using eq. (8-1.4) for four parameters, all with correlated systematic standard uncertainties, the systematic standard uncertainty for z is

8-2.1 Nonsymmetric Systematic Uncertainty Interval for a True Value

b z p 关共 ␪m1 bm1 兲 2 + 共 ␪ m2 b m2 兲 2 + 共 ␪ m3 b m3 兲 2

If the distribution of the systematic error associated with a measured variable is symmetrically distributed but not centered at zero, then the overall uncertainty interval for the unknown true value will not be centered on the measured value of the variable. In this case, the following procedure should be employed for constructing a nonsymmetric uncertainty interval for the unknown true value of the quantity being measured (see Fig. 8-2.1): (a) Specify an interval (X − B−, X + B+ ) relative to the measured value X within which one may expect the true value to fall with 95% confidence, in the absence of random errors. This interval accounts for systematic errors only. (b) Define the offset, q, as the difference between the center of the interval specified in (a) and the measured value. Thus

+ 共␪ m4 b m4 兲 2 + 2 ␪m1␪m2 bm1m2 + 2 ␪ m 1␪ m 3 b m 1 m3 + 2 ␪ m 1␪ m 4 b m 1 m4 + 2 ␪ m 2␪ m 3 b m 2 m3 + 2 ␪ m 2␪ m 4 b m 2 m4 + 2 ␪ m 3␪ m 4 b m 3 m4 兴

1⁄

2

or b z p 关共 bm1 兲 2 + 共 bm2 兲 2 + 共 bm3 兲 2 + 共 b m4 兲 2 + 2b m1m2 + 2b m1m3 − 2b m1m4 + 2b m2m3 − 2b m2m4 − 2b m3m4 兴

1⁄

2

b z p 1.0 kg/h

and the balance check will be satisfied if

qp

( X + B+ ) + ( X − B−) B+ − B − −Xp 2 2

|z| ≤ 2bz p 2.0 kg/h

(c) Define the quantity B as follows:

Note the signs for each of the correlated terms. For this case, calibrating all the flowmeters against the same standard will yield the minimum systematic standard uncertainty for z. It is interesting to note that the systematic standard uncertainty in z is less than the smallest estimate of systematic standard uncertainty for any of the flowmeters. In general, correlated systematic standard uncertainties can either decrease, increase, or have no effect on the systematic standard uncertainty of the result, depending on the form of the data reduction equation and on which parameters have correlated systematic errors.

Bp

( X + B+ ) − ( X − B−) B + + B− p 2 2

Thus B is equal to one-half the length of the interval specified in (a). (d) Calculate b X, the systematic standard uncertainty for the measurement, as

bX p

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B 2

ASME PTC 19.1-2005

TEST UNCERTAINTY

Fig. 8-2.1 Schematic Relation Between Parameters Characterizing Nonsymmetric Uncertainty

This is based on the assumption that a Gaussian distribution is an appropriate model for the systematic error. (See Nonmandatory Appendix B for other distributional models.) (e) Calculate u X, the combined standard uncertainty for the measurement, using the standard formula uX p 冪b 2 + s 2 . X X (f) Calculate U95, the expanded uncertainty for the measurement, using U 95 p 2uX p 2 冪 b X2 + s X2

Example 8-2.1: Suppose a thermocouple is being used to measure the temperature of a gas stream, but the user of the thermocouple believes there may be a tendency for the thermocouple to provide a temperature reading that is lower than the actual gas temperature due to a radiative heat transfer mechanism. The user does not have enough information to properly correct the thermocouple reading for these effects, but wishes to account for them in an uncertainty analysis. From a sample of more than 30 readings using the thermocouple, the user finds that X p 534.7°C and sX p 2.4°C. If the user believes that the true gas temperature may be as much as 10°C higher than X due to radiation effects, then a nonsymmetric confidence interval accounting for this nonsymmetric systematic uncertainty may be computed as follows: (a) Specify an interval (corresponding to 95% confidence) for the systematic error in question. In this case, the user of the thermocouple believes that the true gas temperature falls between the average measured with the thermocouple, X p 534.7°C, and a value that is 10°C higher than X, i.e., 544.7°C. So B− p 0°C and B+ p 10°C. (b) Determine q, the difference between the center of the interval specified in (a) and the value measured with the thermocouple. In this case

(8-2.1)

This calculation is based on the assumption that the degrees of freedom for the combined standard uncertainty are large. (For small degrees of freedom see Nonmandatory Appendix B.) (g) Calculate an approximate 95% confidence interval for the true value using

关X + q兴 ± U95

(8-2.2)

(h) Express the final result as an asymmetric 95% confidence interval for the true value with the lower limit given by X lower limit p X + q − U 95 p X − U−

(8-2.3)

and the upper limit given by X upper limit p X + q + U 95 p X + U+

(8-2.4)

where U− p U95 − q and U+ p U95 + q.

q p (544.7°C + 534.7°C)/2 − 534.7°C p 5°C

32 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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ASME PTC 19.1-2005

(c) Calculate the quantity B as Bp

and the upper limit given by

B + + B− 10°C + 0°C p p 5°C 2 2

X upper limit p X + U+ p 534.7°C + 11.9°C p 546.6°C

(d) Calculate b X, the systematic standard uncertainty for the measurement, as bX p

8-2.2 Nonsymmetric Systematic Uncertainty Interval for a Derived Result

B p 2.5°C 2

A nonsymmetric systematic uncertainty in a measured variable may also result in a nonsymmetric uncertainty interval for a derived result. The following procedure may be employed for propagating the nonsymmetric uncertainties in a set of measured variables to a derived result (see Fig. 8-2.2): (a) Determine Xi , uXi , and qi for each average Xi that contributes to the determination of the derived result, r 共X1 , X2 , . . ., Xn 兲. (b) Determine the offset, qr , which is defined as

This is based on the assumption that a Gaussian distribution is an appropriate model for the systematic error. (e) Calculate uX, the combined standard uncertainty for the measurement, using the standard formula uX p

冪 b X2 + sX2 p 冪 (2.5°C)2 + (2.4°C)2 p 3.45°C

(f) Calculate U95, the expanded uncertainty for the measurement, using

q r p r 共 X1 + q 1 , X 2 + q2 , . . ., Xn + q n兲 − r共 X 1 , X 2 , . . ., X n兲 .

U 95 p 2u X p 6.9°C

(c) Determine the sensitivity coefficient, ␪i, for each average X; that contributes to the derived result following standard procedure. If a sensitivity coefficient depends on the values of any averages, i.e., ␪i p ␪i 共X1 , X2 , . . ., Xn兲, then it should be evaluated at the point 共X1 + q1 , X2 + q2, . . ., Xn + qn兲. (d) Calculate ur , the combined standard uncertainty for the derived result, using the standard formula:

This calculation is based on the assumption that the degrees of freedom for the combined standard uncertainty are large. (For small degrees of freedom see Nonmandatory Appendix B). (g) Calculate an approximate 95% confidence interval for the true value using

关X + q兴 ± U95 In this case, this 95% confidence interval is given by

ur p

冪关共␪1uX 兲2 + 共␪2uX 兲2 + . . . + 共␪nuX 兲2兴 1

2

n

(8-2.5)

[534.7°C + 5°C] ± 6.9°C

(e) Calculate U95,r the expanded uncertainty for the derived result at a 95% confidence level, as

(h) Calculate U − p U 95 − q p 6.9°C − 5°C p 1.9°C

U 95,r p 2u r

and (This is based on the assumption that the degrees of freedom are large. For small degrees of freedom, see Nonmandatory Appendix B.) (f) Calculate an approximate 95% confidence interval for the derived result using

U + p U 95 + q p 6.9°C + 5°C p 11.9°C

The final result may be expressed as an asymmetric 95% confidence interval for the true value using the lower limit given by

r 共X 1 + q1 , X 2 + q 2 , . . . , X n + qn兲 ± U95,r (8-2.6)

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TEST UNCERTAINTY

Fig. 8-2.2 Relation Between Parameters Characterizing Nonsymmetric Uncertainty

(g) Express this confidence interval as an asymmetric 95% confidence interval for the derived result as follows: r 共X 1 , X 2 , . . . , X n兲 ± 共U 95,r ± q r兲

negligible uncertainty and T is the measured value of the absolute temperature. In this case, T p (X + 273.2)K, where X is the average value of the temperature of the gas using the thermocouple. The uncertainty interval for c may be calculated as follows: (a) Determine T, uT , and qT for the measured variable T. In this case, T p 807.7K, uT p 3.45K, and qT p 5K. (b) Determine the offset, qc , as follows:

(8-2.7)

where the lower limit on this interval is given by r lower limit p r( X 1 , X 2 , . . . , X n) − (U 95,r − q r ) p r − U r−

(8-2.8) q c p c(T + q T) − c(T) p [kR(812.7K)]

and the upper limit on this interval is given by

− [kR(807.7K)]

r upper limit p r( X 1 , X 2 , . . . , X n) (8-2.9)

with Ur− p U95,r − qr and U +r p U95,r + qr. Example 8-2.2: Suppose the user of the thermocouple in the example in para. 8-2.1 wishes to use this gas temperature to estimate the speed of sound for the gas using the following relation: c p [kRT]

1⁄

p [kR]

␪ T p ( 1 ⁄2)[kR/(812.7K)]

1⁄

2

(0.0878K)

1⁄

2

1⁄

2

p [kR]

1⁄

2

(0.0175K) −

1⁄

2

(d) Estimate the combined standard uncertainty for the derived result, uc. In this case,

2

u c p [{(kR)

where k, the ratio of specific heats, and R, the gas constant for the gas, are taken to be constant with

1⁄

2

(0.0175K−

p [kR]

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2

2

(c) Determine the sensitivity coefficient, ␪T , for the measured variable T. In this case, ␪T p 1 (1/2)[kR/T] ⁄2. Since this sensitivity coefficient depends on T, it should be evaluated at T + qT p 812.7K, so that here

+ (U 95,r + q r) p r + U r+

1⁄

1⁄

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1⁄

2

1⁄

2

(3.45K)} 2]

(0.0604K)

1⁄

2

1⁄

2

TEST UNCERTAINTY

ASME PTC 19.1-2005

calibrations (i.e., calibrations performed prior to each of the multiple tests or periodically throughout a test program), the calibration process random error will introduce scatter in the test data set and therefore should remain classified as a random error source or random standard uncertainty in the uncertainty analysis. However, for test measurements involving a single calibration (i.e., calibrations performed only once during a test program, and all data samples processed using the same calibration constants), the calibration process random error does not have an opportunity to introduce scatter in the test data set and therefore should be reclassified (fossilized) as systematic error during the final uncertainty analysis. In this case, the calibration random standard uncertainty should be treated as another elemental systematic standard uncertainty and combined with the other calibration systematic standard uncertainties to obtain the total systematic standard uncertainty for the calibration process as:

(e) Calculate U95,c the expanded uncertainty for the derived result c, at a 95% confidence level, as U 95,c p 2uc p 2[kR]

1⁄

2

(0.0604K)

1⁄

2

p [kR]

1⁄

2

(0.1208K)

1⁄

2

(This is based on the assumption that the degrees of freedom are large. For small degrees of freedom, see Nonmandatory Appendix B.) (f) Compute a 95% confidence interval for the derived result using c(T + qT) ± U95,c. In this case, this 95% confidence interval is given by [kR]

1⁄

2

[812.7K]

1⁄

2

± [kR]

1⁄

2

1⁄

(0.1208K 2)

(g) Express the final result as an asymmetric 95% confidence interval using c(T) ± (U 95,c ± q c)

In this case, this 95% confidence interval is given by [kR]

1⁄

2

[807.7K]

1⁄

2

± {[kR]

1⁄

2

1⁄

(0.1208K 2)

1⁄

bc* p

1⁄

± [kR] 2 (0.0878K 2)}

1⁄

1⁄

2

[28.42K 2 ] − [kR]

1⁄

b*cp total systematic standard uncertainty of the single calibration process b1, b2, . . .p elemental systematic standard uncertainty components of the calibration process Ncp number of calibration points within a single calibration scp standard deviation of the calibration process; an estimated standard deviation of the random error components of the calibration process scp sc冫冪Nc; the fossilized elemental systematic standard uncertainty due to the random error of the calibration process A more general form of eq. (8-3.1) for a single calibration is

1⁄

2

(0.033K 2)

and whose upper limit is equal to c upper limit p [kR]

1⁄

2

1⁄

[28.42K 2 ] + [kR]

1⁄

2

1⁄

(0.2086K 2)

In this example, the uncertainty interval for the speed of sound of the gas extends from 0.12% below to 0.73% above the value for the speed of sound assessed using the measured value of the temperature. 8-3 FOSSILIZATION OF CALIBRATIONS Definition of the Measurement Process is a prerequisite for determining measurement uncertainty estimates. For example, different Defined Measurement Processes for the same test will result in different estimates of measurement uncertainty. Elemental errors are classified as random if they add scatter to a result. If they do not, they are systematic. Final classification is dependent on the Defined Measurement Process. Calibration errors are frequently reclassified as a result of the Defined Measurement Process. As an example, for test measurements involving multiple

bc* p 关 (b c) 2 + (s c)2兴

1⁄

2

(8-3.2)

where bcp calibration process initial systematic standard uncertainty before reclassifying (fossilizing) the calibration random standard uncertainty into systematic standard uncertainty 35 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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(8-3.1)

where

whose lower limit is equal to c lower limit p [kR]

冪 (b1)2 + (b2)2 + . . . (sc)2

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8-4 SPATIAL VARIATION [12]

To illustrate the fossilization process, consider a master flowmeter installed in line with a test flowmeter for use in establishing a calibration correction for the test meter. As a result of the calibration process, the systematic standard uncertainty of the test meter is replaced by that of the master meter. The calibration process random standard uncertainty is a function of the random standard uncertainties in both the master and test meters. When a data set of interest from the test meter involves multiple calibrations, the calibration process random standard uncertainty will cause scatter in the individual test meter data samples and thus should remain classified as calibration random standard uncertainty in the uncertainty analysis. However, when a data set from the test meter involves only a single calibration, the calibration process random standard uncertainty is common to all data samples and thus manifests itself in the data set as a systematic standard uncertainty (becomes fossilized). In this case, the random and systematic standard uncertainties of the calibration process should be combined and carried forward as a systematic standard uncertainty. Fossilization of calibration random standard uncertainty can occur at any or all levels of calibration hierarchy, from a national laboratory to the test application, depending on the defined calibration, or measurement processes, or both. Sometimes, multiple calibrations are performed but the results are averaged into a single set of calibration constants for use in processing all data samples (e.g., pre- and post-calibrations). In this case, a portion of the random standard uncertainty still becomes fossilized into systematic standard uncertainty. The magnitude has been reduced by having averaged multiple calibrations. The term sc in eq. (8-3.2) should be reduced by dividing by 冪Nrc. Thus a more general form of eq. (8-3.2) would be

冤

b c* p (b c) + 共 sc / 冪 N rc兲

2

冥

1⁄

Measurement requirements for a performance test are often such that an average measurement of individual parameters is required. Most instrumentation, however, yields a point measurement of a parameter rather than an average measurement. While this point characteristic may be useful for other purposes, it raises a problem in determining performance level. In many instances, the quantity measured varies in space, making the point measurement inadequate. Thus, it often is necessary to install several measurement sensors at different spatial locations to account for spatial variations of the parameter being measured. Spatial variation effects are considered errors of method (see para. 5-2.5). A simple illustration of the impact of spatial variation is the measurement of average velocity in fully developed flow of an incompressible fluid in a pipe. At low Reynolds numbers, the flow is laminar and has a parabolic velocity profile. One measurement will not give the true average value directly. For example, the velocity at the center of the pipe is twice the average velocity. This is in contrast to the situation at high Reynolds numbers, where the flow is turbulent. At higher Reynolds numbers, the profile approaches uniformity and any measurement will yield a reasonable estimate of the average velocity. Usually, test conditions vary between these two extremes and it is not possible to correct the readings in a simple manner. Circumferential variations may also be present. Therefore, other approaches are chosen, such as installing multiple sensors and averaging the outputs. An example of an averaged output is an averaging pressure probe often used to measure average velocity at a cross section of pipe. The uncertainty of this mean velocity would be calculated by considering it to be a determined result. An additional systematic standard uncertainty may need to be assigned to the mean result to account for the possible difference between the determined mean and the true mean.

2

(8-3.3)

where 8-5 ANALYSIS OF REDUNDANT MEANS

Nrcp number of repeated independent (single) calibrations averaged and used in obtaining a single set of calibration constants common to all samples within the test data set

When redundant instrumentation or calculation methods are available, the individual results and their uncertainties should be compared with each other and with the pretest uncertainty analysis. 36

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ASME PTC 19.1-2005

Fig. 8-5.1 Three Posttest Cases

When comparing redundant means (X1 and X2) and their uncertainty intervals, the three cases illustrated in Fig. 8-5.1 need to be considered. Case 1: A problem clearly exists when there is no overlap between uncertainty intervals. Either uncertainty intervals have been grossly underestimated, or the true value is not constant. Investigation to identify bad readings, overlooked or underestimated systematic uncertainty, etc., is necessary to resolve this discrepancy. Case 2: When the uncertainty intervals completely overlap, as in Case 2, one can be reasonably confident that there has been a proper accounting of all major uncertainty components. The smaller uncertainty interval X2 ± U2 is wholly contained in the larger interval X1 ± U1 Since the individual measurements are valid, the weighting method described in para. A-2 may be used to obtain a better estimate of the true value than either of the individual measurements. Case 3: Case 3, where a partial overlap of the uncertainty intervals exists, is the most difficult to analyze. For both measurements and both uncertainty intervals to be correct, the true value must lie in the region where the uncertainty intervals overlap. Consequently, the larger the overlap, the more confidence we have in the validity of the

measurements and the estimate of the uncertainty intervals. As the difference between the two measurements increases, the overlap region shrinks. Standard statistical hypothesis testing may be used to evaluate the significance of the difference observed. The process is outlined in the following paragraphs. Let s1 and b1 denote, respectively, the random and systematic standard uncertainties associated with X1. Likewise, let s2 and b2 denote, respectively, the random and systematic standard uncertainties associated with X2. Assuming that the degrees of freedom associated with the systematic and random uncertainty components are large, and that there are no correlated errors, the Z-statistic for testing the null hypothesis that the two measurements X1 and X2 have an expected difference of zero (i.e., the two measurements are consistent with one another) is given by Zp

X 1 − X2

关 + s22 + b21 + b22兴 s 21

2

If |Z| > 2, then the data indicate, at the 95% confidence level, that the two measurements are consistent with one another. Otherwise, there is 37

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1⁄

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TEST UNCERTAINTY

no compelling evidence to declare the two measurements to be inconsistent with one another.

where for N data pairs (Xj, Yj), the slope m is determined from

Caution: Lack of evidence for demonstrating inconsistency does not “prove” consistency.

N

N

N 兺 XjY j − mp

jp1

N

N 兺 (Xj2) −

8-6 REGRESSION UNCERTAINTY

jp1

8-6.1 Linear Regression Analysis

N

cp

冢兺 冣 jp1

(8-6.2)

2

Xj

N

N

N

兺 (X2j ) jp1 兺 Yj − jp1 兺 Xj jp1 兺 (XjYj) jp1 N

N兺

jp1

N

(X 2j )

−

冢兺 X冣 jp1

2

(8-6.3)

j

The least-squares process essentially provides an average for the data so that the regression expression in eq. (8-6.1) represents the relationship between the mean value of Y and X. This mean, Yˆ, is not the average of the Yj data but the mean Y response from the curve-fit for a given X. Once the slope and intercept are calculated from eqs. (8-6.2) and (8-6.3), these constants can be substituted into eq. (8-6.1) along with several values of X and the resulting straight line can be plotted over the (Xj, Yj) data. Since the Yˆ vs. X curve is a mean value for the data set, the curve should be a good representation of the data if the simple linear fit is appropriate. 8-6.3 Random Standard Uncertainty for Yˆ Determined From Regression Equation The statistic that defines the standard deviation for a straight-line curve-fit is the standard error of estimate

SEE p

冤

N

兺 (Yj − mXj − c)2 jp1 N−2

冥

1⁄

2

(8-6.4)

For a given value of X, the random standard uncertainty associated with the Yˆ obtained from the curve-fit [eq. (8-6.1)] is

8-6.2 Least-Squares For a straight-line, or a simple linear regression, the curve-fit expression is

冤

1 s Yˆ p SEE + N

(8-6.1) 38

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N

and the intercept c is determined from

Curve fitting often is used in the calibration process, in the data reduction program, and in the representation of the final test results. Leastsquares regression analysis is the most popular means of curve fitting. In many cases, the anticipated representation of the data is a straight line, or a simple (first-order) linear regression. In other cases, the data to be curve-fit often can be rectified, or transformed, into linear coordinates [9, 13, 14]. Subsection 8.6 will cover only straight-line regressions to estimate the relationship for Y versus X. For higher-order linear regressions and other regression methodologies, see Refs. [9, 15, 16, 17, 18]. For regression uncertainty when X and Y are functions of other variables, see Refs. [9, 17]. The random standard uncertainty for the curvefit will be determined using standard least squares analysis [9, 14, 16], where the assumption is made that there is no random standard uncertainty in the X values and the random standard uncertainty in the Y values is constant over the range of the curve-fit. In this section, only a special case is considered for the systematic standard uncertainty. This special case is where the systematic standard uncertainty for the Y values and/or the X values is a constant (i.e., percent of full scale) and there are no correlated elemental systematic standard uncertainties between the X and Y values. A more general approach to regression uncertainty is presented in [9] where the methodology applies for variable random standard uncertainties in X and Y, variable systematic standard uncertainties in X and Y, and correlated systematic standard uncertainties between X and Y.

Yˆ p mX + c

N

兺 Xj jp1 兺 Yj jp1

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(X − X)2 N

兺

jp1

(X j − X) 2

冥

1⁄

2

(8-6.5)

TEST UNCERTAINTY

ASME PTC 19.1-2005

where

b Yˆ1 p bY1

Xp

1 N

N

兺 Xj jp1

(8-6.6)

8-6.4.2 Systematic Standard Uncertainty in Xj Data With No Systematic Standard Uncertainty in Xnew. If each of the Xj data points has the same systematic standard uncertainty, bX, and Xnew has no systematic standard uncertainty, then the resulting elemental systematic standard uncertainty for the mean Yˆ from the curve-fit is determined as [9, 17]

If there is no random standard uncertainty in the Xj data or the new X values used in the regression equation, the random standard uncertainty sYˆ obtained from eq. (8-6.5) is combined with the systematic standard uncertainty (discussed in para. 8-6.4 using eq. (7-5.1) to obtain the total uncertainty for the Yˆ value from the curve-fit. For random standard uncertainty in the Xj or X values, the general approach in [9] should be used.

b Yˆ2 p mb x

(8-6.8)

This case would occur when the regression equation from a set of test data is used later in a design or analysis process where Xnew might be taken as a value that has no uncertainty.

8-6.4 Systematic Standard Uncertainty for Yˆ Determined From Regression Equation There can be systematic standard uncertainty, bYj and bXj respectively, in the Yj and Xj data. There also can be systematic standard uncertainty in the X value used in the curve-fit to find a Yˆ value. This curve-fit X will be called Xnew to distinguish it from the Xj data points, and the systematic standard uncertainty for Xnew is bXnew. It is very likely that most, and probably all, of the elemental systematic standard uncertainties for each of the Yj data points are from the same sources, and are, therefore, correlated. The same is true for the Xj data points. There is also the possibility that the Xnew values will have systematic standard uncertainties from the same sources as the Xj data causing these uncertainties to be correlated. In subsection 8-6, only constant systematic standard uncertainties for (Xj, Yj) and Xnew are considered. All of the bYj uncertainties are assumed to be completely correlated with each other, and all of the bXj uncertainties are assumed to be completely correlated with each other. The assumption is made that there are no common uncertainty sources between Yj and Xj (no correlation between the bYj and bXj systematic standard uncertainties). Cases are considered where Xnew has systematic standard uncertainty which is correlated with that in Xj and where Xnew has systematic standard uncertainty which is not correlated with that in Xj.

8-6.4.3 Systematic Standard Uncertainty in Xj Data With Correlated Systematic Standard Uncertainty in Xnew. If each of the Xj data points has the same systematic standard uncertainty, bX, and Xnew has the same systematic standard uncertainty (from the same sources), then the resulting elemental systematic standard uncertainty for the mean Yˆ from the curve-fit is zero [9]. This case would occur if the same instruments are used to measure Xnew as were used to measure Xj. Since all of the systematic standard uncertainties for Xj and Xnew are correlated, the systematic standard errors are all the same. The effect on the curve-fit is to shift it to the right or left depending on the sign of the errors (the signs and magnitudes of the errors are unknown). This shift has no effect on the value of Yˆ obtained from the curve since the shift in Xnew is the same as the shift in Xj. 8-6.4.4 Systematic Standard Uncertainty in Xj Data With Uncorrelated Systematic Standard Uncertainty in Xnew. If each of the Xi data points has the same systematic standard uncertainty, bX, but Xnew has a different (no common systematic error sources) systematic standard uncertainty, bXnew, then the resulting elemental systematic standard uncertainty for the mean Yˆ from the curve-fit is [9]

8-6.4.1 Systematic Standard Uncertainty in Yj Data. If each of the Yj data points has the same systematic standard uncertainty, bY1, then the resulting elemental systematic standard uncertainty for the mean Yˆ from the curve-fit is [9, 14, 17]

b Yˆ3 p 关(mbX) 2 + (mb Xnew) 2兴

1⁄

2

(8-6.9)

This case would occur if different instruments were used to measure the Xj values and Xnew. 39 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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(8-6.7)

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ASME PTC 19.1-2005

TEST UNCERTAINTY

Table 8-6.4.5 Systematic Standard Uncertainty Components for Yˆ Determined from Regression Equation Systematic standard uncertainty in Yj data

bYˆ p bY

Systematic standard uncertainty in Xj data with no systematic standard uncertainty in Xnew

bYˆ p mbX

1

2

1

3

关共mbX兲2 + 共mbXnew兲2兴

(8-6.13)

1⁄ 2

bYˆ p b Yˆ3

(8-6.14)

For no systematic standard uncertainty in the Yj data and correlated systematic standard uncertainty between Xj and Xnew bYˆ p 0

(8-6.15)

8-6.5 Uncertainty for Yˆ From Regression Equation The total uncertainty in the Yˆ obtained from the simple linear regression expression, eq. (8-6.1), is given by eq. (7-5.1) for the case where the degrees of freedom for Yˆ are sufficiently large so that t ≈ 2

(8-6.10) 2

2

UYˆ p 2 关b Yˆ + s Yˆ兴

For systematic standard uncertainty in the Yj data and the Xj data and no systematic standard uncertainty in Xnew 2

2

bYˆ p 关 bYˆ1 + b Yˆ2兴

1⁄

2

1⁄

2

(8-6.16)

Note that the degrees of freedom for Yˆ is based on the degrees of freedom for sYˆ , which is N − 2, and the degrees of freedom for bYˆ (see Nonmandatory Appendix B). The use of the factor t ≈ 2 will be appropriate in most cases. The uncertainty band UYˆ in eq. (8-6.16) will vary with X (i.e., Xnew) because of the expression for sYˆ from eq. (8-6.5). As noted earlier, the uncertainty expression in eq. (8-6.15) only applies if there is no random standard uncertainty in X and if the systematic standard uncertainties are percent of full-scale values.

(8-6.11)

For systematic standard uncertainty in the Yj data and the Xj data and uncorrelated systematic standard uncertainty in Xnew, the systematic standard uncertainty for the curve-fit value of Yˆ is

40

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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(8-6.12)

For no systematic standard uncertainty in the Yj data, systematic standard uncertainty in the Xj data, and uncorrelated systematic standard uncertainty in Xnew, the systematic standard uncertainty for the curve-fit value of Yˆ is

8-6.4.5 Systematic Standard Uncertainty for Yˆ . The systematic standard uncertainty for the mean Yˆ from the curve-fit will be the appropriate rootsum-square of the bYˆi elemental systematic standard uncertainties defined above and summarized in Table 8-6.4.5. For systematic standard uncertainty in Yj only or systematic standard uncertainty in Yj with correlated systematic standard uncertainty between Xj and Xnew bYˆ p b Yˆ1

2

bYˆ p b Yˆ2

0

Systematic standard uncertainty in Xj data with uncorrelated systematic standard bYˆ p uncertainty in Xnew

1⁄

For no systematic standard uncertainty in the Yj data, systematic standard uncertainty in the Xj data, and no systematic standard uncertainty in Xnew

2

Systematic standard uncertainty in Xj data with correlated systematic standard uncertainty in Xnew

2

bYˆ p 关 bYˆ1 + b Yˆ3兴

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ASME PTC 19.1-2005

Section 9 Step-by-Step Calculation Procedure 9-1 GENERAL CONSIDERATIONS

(1) The systematic uncertainty and random uncertainty (sample standard deviations of the means) of the independent parameters are propagated separately all the way to the final result. (2) Propagation of the standard deviations of the means is done, according to the functional relationship defined in step (a)(4), by using the Taylor series method (see section 7). This requires a calculation of the sensitivity factors, either by differentiation or by numerical analysis. (e) Calculate Uncertainty (see subsection 7-5). (1) Combine the systematic and random uncertainties 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 result level. For most uncertainty analyses, all measured parameters will have symmetric systematic uncertainties and large degrees of freedom. For some analyses, one or more of the systematic uncertainties may be nonsymmetric (see subsection 8-2), and for other analyses, the degrees of freedom may be small for some of the uncertainties (see Nonmandatory Appendix B). The detailed report table should include, as a minimum, the following information 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 measurements), X (e) systematic standard uncertainty, b i (f) sample random standard uncertainty, standard deviation of the mean, s x,i (g) sensitivity, ␪␫ (h) systematic standard uncertainty contribution to the combined uncertainty of the result, (␪i b i)2

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 instrumentation systems and test designs and to determine 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 results. 9-2 CALCULATION PROCEDURE (a) Define Measurement Process (see section 5). (1) Review test objectives and test duration. (2) List all independent measurement parameters 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 simultaneously (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 Random Uncertainty (Standard Deviation of the Mean) for Each Parameter (see subsections 6-1 and 6-2). (d) Propagate the Systematic and Random Standard Deviations (see subsections 7-1 through 7-4). 41 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

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Table 9-2-1 Table of Data Independent Parameters

Symbol

Description

Units

Nominal Value

C d D ␳

Discharge coefficient Throat diameter Inlet diameter Water density at 60°F Differential pressure head across venturi (at 68°F)

... in. in. lbm/ft3

0.984 3.999 6.001 62.4

h

in. H2O

100

Absolute Systematic Standard Uncertainty, bX

Absolute Random Standard Uncertainty, SX

0.00375 0.0005 0.001 0.002

0.0 0.0 0.0 0.002

0.15

0.4

i

Absolute Sensitivity, ␪i

i

140 86.2 −11.4 1.11 0.6919

Absolute Systematic Standard Uncertainty Contribution, ␪i b X 2

Absolute Random Standard Uncertainty Contribution, ␪i SX 2

0.276 0.00186 0.00013 0.0000049

0.0 0.0 0.0 0.0000049

0.0108

0.0766

i

i

Table 9-2-2 Summary of Data Calculated Result

Symbol

Description

Units

Calculated Result, R

˙ m

Mass flow rate

lbm/s

138.4

Absolute Systematic Standard Uncertainty, bR

Absolute Random Standard Uncertainty, sR

Absolute Combined Standard Uncertainty, uR

Absolute Expanded Uncertainty, UR

0.276

0.0766

0.286

0.572

(i) random standard uncertainty contribution to the combined uncertainty of the result, (␪i sX)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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ASME PTC 19.1-2005

Section 10 Examples 10-1 FLOW MEASUREMENT USING PITOT TUBES

appropriate calibration corrections and unit conversions; (e) the DAS automatically takes 60 readings at each traverse point and computes and records average values and standard deviations for the data collected at each traverse point; (f) the Pitot tube, differential pressure transmitter, 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.

10-1.1 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 integratedaverage velocity may be estimated as the 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 positioned. The point velocity values at individual traverse points are treated as measurements (Section 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 pitot tube is measured by a differential pressure transmitter; (c) the output of the differential pressure transmitter is measured and recorded by a computerized data acquisition system (DAS); (d) the DAS computes velocity for each measurement by taking the square root of the output of the differential pressure transmitter, making the appropriate corrections for fluid density, and making the

10-1.2 Data Summary The computerized data acquisition system is used to compute average values from the 60 measurements 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 summarized in Table 10-1.3-1. The traverse points are located at the centroid of equal areas so that the integrated-average velocity 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 10

Vi (ft/sec) ≈

1

10

) X if (ft/sec)

(10-1.1)

Then, the average velocity in the pipe V is approximated as 4

V (ft/sec) ≈

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兺 (⁄ jp1

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兺 ( ⁄ ) Vi (ft/sec) ip1 1

4

(10-1.2)

ASME PTC 19.1-2005

TEST UNCERTAINTY

Fig. 10-1.1 Traverse Points (Example 10-1)

Table 10-1.2 Average Values (Example 10-1) Traverse Point, j

j j j j j j j j j j

p p p p p p p p p p

1 2 3 4 5 6 7 8 9 10

Radius 1, i ⴝ 1 X ij (ft/sec)

Radius 2, i ⴝ 2 X ij (ft/sec)

Radius 3, i ⴝ 3 X ij (ft/sec)

Radius 4, i ⴝ 4 X ij (ft/sec)

5.31 5.46 5.55 5.63 5.65 5.69 5.73 5.74 5.76 5.72

5.27 5.53 5.61 5.68 5.74 5.77 5.79 5.76 5.75 5.80

5.21 5.25 5.37 5.47 5.58 5.62 5.65 5.65 5.70 5.70

5.00 5.16 5.31 5.42 5.50 5.55 5.63 5.65 5.65 5.67

44 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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Table 10-1.3-1 Standard Deviations (Example 10-1) Radius 1, i ⴝ 1 SX Traverse Point, j 1 2 3 4 5 6 7 8 9 10

SX

Radius 2, i ⴝ 2

Radius 4, i ⴝ 4

SX

SX

(ft/sec)

(ft/sec)

(ft/sec)

(ft/sec)

(ft/sec)

(ft/sec)

(ft/sec)

(ft/sec)

1.21 1.06 1.03 1.21 1.29 1.09 0.81 1.00 1.15 0.81

0.156 0.157 0.133 0.156 0.167 0.141 0.105 0.129 0.148 0.105

1.31 1.61 1.36 1.31 1.06 1.26 1.03 0.93 1.34 1.45

0.169 0.208 0.176 0.169 0.137 0.163 0.133 0.120 0.173 0.187

1.61 1.78 1.89 1.84 1.65 1.09 1.43 1.18 1.36 1.00

0.208 0.230 0.244 0.238 0.213 0.141 0.185 0.152 0.176 0.129

1.41 1.65 1.26 1.80 2.04 1.74 1.61 2.14 1.43 1.54

0.182 0.213 0.163 0.232 0.263 0.225 0.208 0.276 0.185 0.199

ij

ij

SX

Radius 3, i ⴝ 3

ij

ij

SX

ij

SX

ij

Table 10-1.3-2 Summary of Average Velocity Calculation (Example 10-1) Parameter

Value (ft/sec)

V1 V2 V3 V4 V

5.62 5.67 5.52 5.45 5.57

S Xij p

SX

ij

ij

S Xij 冪 N ij

where values for SXij are shown in Table 10-1.3-1. Since there are 60 measurements at each traverse point, Nij p 60, the degrees of freedom at each traverse point is ␯ij p Nij − 1 p 59

The resulting values for SXij are also presented in Table 10-1.3-1.

The subscripts i and j are used in the previous equations to designate radius and traverse positions, respectively. The results of these calculations are summarized in Table 10-1.3-2.

10-1.6 Propagate Random Standard Uncertainty

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

The random standard uncertainty for each average velocity along a radius SVi is calculated from eq. (7-3.1) as

10-1.4 List Elemental Uncertainty Sources The sources of uncertainty which are considered random in this simplified example are those causing 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

SVi p

兺 关共∂Vi /∂Xij兲共sX 兲兴2冧 冦jp1 ij

1⁄

2

(10-1.4)

where 1 ∂ Vi p ∂ X ij 10

The random standard uncertainty of the average velocity in the pipe s V is then calculated as

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

4

sV p

兺 冢 ∂V 冤jp1

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(10-1.3)

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∂V i

s Vi

冣冥 2

1⁄

2

(10-1.5)

ASME PTC 19.1-2005

TEST UNCERTAINTY

Table 10-1.6 Standard Deviation of Average Velocity (Example 10-1) Parameter, ft/sec

Value

sV 1 sV 2 sV 3 sV 4 sV

0.0440 0.0523 0.0618 0.0687 0.0287

of spatial position. Therefore, there is an inherent uncertainty in the method used to approximate the integrated-average velocity. This uncertainty of method is sometimes referred to as uncertainty due to spatial variation. Since the velocity profile remains fixed for the duration of the test, the spatial variation is a source of systematic uncertainty for the test result. This uncertainty can be estimated in a variety of ways, including (a) special tests which provide independent knowledge of the velocity profile; (b) special tests which compare the measurement technique to other techniques which yield the desired integrated-average; (c) published reports which document the uncertainty of similar measurement techniques in similar measurement situations; and (d) evaluation of the variation in test measurements as a function of spatial position. For this example, the velocity profile is distorted due to the presence of flow disturbances upstream of the measurement location. Special tests had previously been conducted to determine if taking additional traverse points (20) along each of the 4 radii would result in a significant change in the measurement of the overall average velocity in the pipe, as computed using eq. (10-1.2). The results of these past tests were used to compute a standard deviation representing the dispersion of errors (differences) in average pipe velocities computed using 20 traverse points along each of the 4 radii versus those obtained using 10 traverse points. The standard deviation published from these past tests was assumed to be representative of the systematic standard uncertainty (for the present test) resulting from spatial variation in the radial direction. The resulting value is shown in the following equation. The published degrees of freedom for this value is 30.

where ∂V ∂Vi

p

1 4

The results are summarized in Table 10-1.6. While not shown here, the degrees of freedom associated with each s Vi as well as SV can be calculated using eq. (B-1.7). 10-1.7 Calculate Systematic Standard Uncertainties 10-1.7.1 Calibration. Velocity is measured at each point with the same Pitot tube, digital pressure transmitter, and DAS. The Pitot tube, digital pressure transmitter, and DAS are calibrated together as a system. For this example, the calibration uncertainty of the instruments, estimated as the limits of the elemental systematic error at 95% confidence, is 3% of measured velocity. Using eq. (4-3.4), the elemental systematic standard uncertainty associated with the calibration of the instruments, of the average pipe velocity measurement is estimated as bVc p (0.03/2)(5.57) p 0.0836 (ft/sec)

(10-1.6)

The degrees of freedom of bVC is assumed to be large (≥30).

b VSR p 0.0111 ft/sec

10-1.7.2 Spatial Variation. The true value being measured is the integrated average of the velocity over the cross-sectional flow area. Averaging the velocities at the centroid of equal area points is a numerical approximation of this integrated average. Even if exact values of velocity are known at each of the traverse points, the analyst must recognize that the numerical average may not equal the integrated-average value over the entire cross-sectional flow area. This is due to incomplete sampling of a profile which varies as a function --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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Since no additional pipe taps are available for testing, it is unknown whether the average of the traverses of the four radii sufficiently characterizes the integrated average around the pipe due to circumferential variations. Therefore, the uncertainty due to spatial variation must be inferred from available data. For this example, it will be estimated by evaluation of the variation in the radial averages. Assuming that the four measured 46 Licensee=Mott MacDonald Ltd/5956936002 Not for Resale, 08/17/2010 06:03:02 MDT

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radial averages come from a population of values which are normally distributed, the systematic standard uncertainty in the average velocity measurement due to spatial variation in the circumferential direction is estimated using eq. (4-3.5) as

analysis at the test result level is reported in Table 10-1-9. 10-2 FLOW RATE UNCERTAINTY [12] 10-2.1 General Description

bVSC p

冪

1 N␤ V

NV

i

i

兺 ip1

2

( V i − V) p 0.0494 ft/sec N Vi− 1

In this example, the test objective is to determine the flow of water using a 6 x 4 in. venturi (see Fig. 10-2.1) within an uncertainty of 0.5%. A pretest analysis is required to determine if an uncalibrated venturi could be used to satisfy the test objective, and, if not, whether calibration of the venturi would achieve the desired objective. This example outlines differences in the analysis for both the uncalibrated and calibrated cases. The example looks at each case in both absolute and relative (i.e., percent) value formats. The clearest way to present the results of the steps in the uncertainty analysis is to develop a table in which the names, definitions, values, uncertainties, sensitivities, etc., are displayed for each of the variables required for the analysis. The table associated with the uncalibrated case is shown here first, as Table 10-2.1-1. The table has been developed in accordance with the step-bystep procedure of Section 9. The steps in the development of the table are as follows: (a) Define Measurement Process. Flow rate can be calculated by making the measurements required to define the independent variables found in eq. (102.1)[21].

(10-1.7)

where NVip the number of independent radial average velocities used in the computation of bVSC N␤ V p the number of radial average velocities i averaged together in the computation of V. Both of these values are equal to 4. The degrees of freedom associated with b VSC is equal to NVˆi − 1 p 3. 10-1.8 Propagate Systematic Standard Uncertainties The systematic standard uncertainties due to calibration and spatial variation are combined using eq. (7-4.1) as follows: b V p [(b VC ) 2 +(b VSR ) 2 + (b VSC ) 2]

1⁄

2

p [(0.0836) 2 + (0.0111) 2 + (0.0494)2]

1⁄

2

(10-1.8)

p 0.0977 (ft/sec) ˙ p m

0.099702Cd 2 冪 ␳ h

10-1.9 Uncertainty of Result The combined standard uncertainty of the resulting average pipe velocity measurement is determined using eq. (7-5.1) as follows: uV p [(b V )2 + (s V ) 2]

1⁄

2

4

(10-2.1)

The definitions and values for each of the measurements used in the calculation of the mass flow rate are displayed in the first four columns of Table 10-2.1-1.

p 0.102 (ft/sec) (10-1.9)

10-2.2 Uncalibrated Venturi Case

The expanded uncertainty of the resulting average pipe velocity measurement is determined using eq. (7-5.2) as follows:

(b), (c), (d) List Elemental Systematic Uncertainty Sources; Estimate Elemental Uncertainties; Calculate the Systematic and Random Uncertainties. The systematic values were evaluated and are listed in Table 102.1-1. The degrees of freedom associated with each of the uncertainty estimates are assumed to be greater than 30. The systematic uncertainties for each input parameter are displayed on an absolute basis. The standard deviation of the mean of the

U V p 2u V p 0.204 (ft/sec)

Use of the 2 in the above equation is appropriate as it can be shown using eq. (B-1.7) that the combined degrees of freedom of the result is ≥30. A summary presentation of the uncertainty 47 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

冪 冢冣 d 1− D

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TEST UNCERTAINTY

Table 10-1.9 Uncertainty of Result (Example 10-1)

Symbol V

Description

Absolute Absolute Absolute Systematic Random Combined Absolute Standard Standard Standard Expanded Uncertainty, Uncertainty, Uncertainty, Uncertainty, Units Calculated Value bv sv uv Uv

Average ft/sec velocity in pipe

5.57

0.0977

0.0287

0.102

0.204

Fig. 10-2.1 Schematic of a 6 in. x 4 in. Venturi

Table 10-2.1-1 Uncalibrated Case (Example 10-2) Independent Parameters

Symbol

Description

Units

Nominal Value

Absolute Systematic Standard Uncertainty, bX

C d D ␳ h

Discharge coefficient Throat diameter Inlet diameter Water density at 60°F [25] Differential pressure head across venturi (at 68°F)

... in. in. lbm/ft3 in. H2O

0.984 3.999 6.001 62.37 100

3.75E–03 5.0E–04 1.0E–03 0.002 0.15

Random Standard Uncertainty of the Mean, sX 0 0 0 0.002 0.4

i

GENERAL NOTES: (a) The systematic and random estimates for density are based on water temperature measurements having systematic and random uncertainties of 0.2°F and 0.1°F, respectively. (b) The systematic uncertainty for the differential pressure head is assumed to be one-half the least count of the manometer scale.

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it is necessary only to divide the absolute uncertainty by the nominal value of the parameter. Multiplying by 100 would provide an expression of the uncertainty in terms of a percentage of the nominal value. The sensitivity coefficients were converted to relative terms using eq. (7-2.2). Multiplying by 100 yields the percentage change in the result for a 1% change in the measured parameter. Tables 102.1.1-1, 10-2.1.1-2, and 10-2.1.1-3 display the same parameters as Tables 10-2.1-1, 10-2.1-4, and 102.1-5, respectively, but with the sensitivities and uncertainties expressed in relative (percent) terms rather than in absolute terms.

measurements taken is shown for each of the independent variables. These values are also expressed in absolute terms, so the units are the same as the units for the measured parameter. (e) Propagate the Systematic and Random Uncertainties. The sensitivity of the result to each of the individual parameter uncertainties is calculated, either numerically or analytically, in accordance with subsection 7-2. A quick and easy way to numerically calculate the sensitivity coefficients of the independent parameters is to develop a table using a spreadsheet program on the personal computer. Table 10-2.1-2 shows the results obtained using such a spreadsheet, along with the formulas used in the spreadsheet. As can be seen by looking at the formulas shown in Table 10-2.1-2, the same basic equation for the ˙ was repeated seven times (once calculation of m for each independent parameter, X, with Xi being replaced with Xi + Xi Qx, where Qx is the quantity by which Xi is to be perturbed. A forward differencing scheme is used with a value of Qx equal ˙ is therefore to 0.1% (i.e., 0.001). A perturbed m calculated for each independent parameter and ˙ is subtracted from each the baseline value of m ˙ perturbed m. Dividing the difference by the amount by which Xi was perturbed (Xi + Xi Qx) provides the absolute sensitivity coefficient. As noted in para. 7-2.1, the sensitivity coefficient can also be determined analytically, by finding the partial derivatives of the result with respect to each of the measured parameters. Table 10-2.13 shows the formulas for the partial derivatives for each of the measured parameters with respect to the calculated mass flow rate, and the sensitivity coefficients found using these formulas. By looking at the numbers in the last two columns of Table 10-2.1-4, it can be seen at once which parameters contribute most to the uncertainty of the result. In this case, the largest contributor is the systematic uncertainty in the discharge coefficient, C. (f) Calculate Uncertainty. The total uncertainty of the result is then calculated by root-sum-squaring the systematic and random contributions. Table 102.1-5 shows the nominal value, and the systematic, ˙ , calculated as random, and total uncertainties for m described previously.

10-2.3 Calibrated Venturi Case (a) Define the Measurement Process (b), (c), (d) List Elemental Uncertainty Sources; Estimate Elemental Uncertainties; Calculate the Systematic and Random Uncertainties. In this case the systematic uncertainty of the discharge coefficient changes. In addition, the systematic uncertainties of the throat and pipe diameters are eliminated (set to zero in Table 10-2.1.1-4) because the entire flow section is calibrated as a unit, which is standard practice. The uncertainty estimate for discharge coefficient comes from calibration data, which has been fossilized, and is listed as a systematic uncertainty in Table 102.1.1-4 as Bc /C p 0.122%, which corresponds to an absolute systematic standard uncertainty of bc p Bc /2 p 0.061%. (e), (f) Propagate the Systematic and Random Uncertainties; Calculate Uncertainty. These new values can be inserted into the tables (spreadsheets) used for the uncalibrated case. As the formulas do not change, there is nothing more that needs to be done. The parameters and results are displayed using a relative basis for the uncertainties and sensitivities in Tables 10-2.1.1-4, Table 10-2.1.1-5, and 10-2.1.1-6. A summary of these uncertainties is given in Table 10-2.1.1-7. ˙ ± 0.5% is marginally The test objective of m satisfied by using a calibrated venturi (Table 10-2.111). The most promising path to take to obtaining additional reductions in the uncertainty of the mass flow rate determination would be through an increase in the number of differential head readings to reduce s X. (g) Report. Tables 10-2.1.1-1 through 10-2.1.1-7 would be included in the report, with text noting that calibration of the venturi is required to meet

10-2.2.1 Relative Sensitivities and Uncertainties. Sensitivities and uncertainties may also be stated in relative terms. To convert the uncertainty of any parameter from absolute to relative terms, 49 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

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Table 10-2.1-2 Absolute Sensitivity Coefficients in Example 10-2 (Calculated Numerically) Quantity by Which to Perturb Independent Parameters (Qx ): 0.1% Symbol, Xi C

Nominal Values 0.984

Formulas for Absolute Sensitivity ⵲m˙ ⵲Xi

冪␳h − m˙

140.6

冪␳h − m˙

86.3

0.099702 (C + CQx ) d 2

冪1 − 冢D冣 d

Absolute Sensitivity ⵲m˙ ⵲Xi

4

CQx

d

3.999

0.099702(d + dQx ) d2

冪1 − 冢

冣

d + dQx D

D

6.001

4

dQx

0.099702d 2

冪␳h

冪1 − 冢D + DQ 冣 d

4

−11.3 ˙ −m

x

DQx

␳

62.37

0.099702d 2

冪( ␳ + ␳Qx )h − m˙

冪1 − 冢D冣 d

1.11

4

␳ Qx

h

100

0.099702d 2

冪␳(h + hQx ) − m˙

冪1 − 冢D冣 d

0.692

4

hQx

Symbol

Result

˙ m

138.4

˙ Formula for m 0.099702Cd 2

冪1 − 冢D冣 d

the objective of 0.5% uncertainty or less in the mass flow rate determination.

冪␳h 4

(a) the discharge coefficient will have a nonsymmetrical absolute systematic uncertainty (95% confidence estimate) represented by B− p 0.0095 and B+ p 0.0055; and (b) the differential pressure head across the venturi (at 68°F) will have a nonsymmetrical absolute systematic uncertainty (95% confidence estimate) represented by B− p 0.1 in. H2O and B+ p 0.5 in. H2O. Using the methodology in subsection 8-2, the discharge coefficient will have a nonsymmetrical

10-3 FLOW RATE UNCERTAINTY INCLUDING NONSYMMETRICAL SYSTEMATIC STANDARD UNCERTAINTY This example is nearly identical to that presented in subsection 10-2 for the uncalibrated venturi case. The only differences are the following: 50

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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Table 10-2.1-3 Absolute Sensitivity Coefficients in Example 10-2 (Calculated Analytically)

Symbol, Xi

C

Nominal Values

0.099702d 2

0.984

冪␳h

冪 冢冣 d 1− D

冪1 − 冢D冣 d

d

D

3.999

6.001

4

(2)0.099702C

−

冪␳hd −

冢

冢D冣 d d 2 冪1 − 冢D冣 D冢1 − 冢D冣 冣 4

4(0.099702)Cd 6 d

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

absolute systematic standard uncertainty of b− p 0.00575 and b+ p 0.00175, and the differential pressure head across the venturi will have a nonsymmetrical absolute systematic standard uncertainty of b− p −0.05 in. H2O and b+ p 0.35 in. H2O. The results of this uncertainty analysis are presented in Tables 10-3-1 and 10-3-2 in which each symbol has the same description as in Table 10-2.1-1, and in which each corresponding number has the same units as those given in Table 10-2.1-1. Following the method given in subsection 8-2, the absolute systematic standard uncertainty, b, for each variable is estimated by b p (b+ + b−)/ 2 and the offset, q, is estimated by q p (b+ − b−)/ 2. The absolute sensitivity, ␪, for each variable is estimated with all variables set to their offset values, X + q. The apparent offset in the mass

4

−11.4

1.11

2 4

0.692

flow rate, qr, is determined by the difference between the mass flow rate evaluated when all variables are set to their offset values, RX + q, and the mass flow rate evaluated when all variables are set to their measured values, RX. The lower and upper limits on the uncertainty of the mass flow rate are given by U − p U95,R − qr, and U + p U95,R + qr, respectively.

10-4 COMPRESSOR PERFORMANCE UNCERTAINTY The following example follows the step-by-step procedure outlined in Section 9. This example highlights the following: (a) the identification and quantification of elemental sources of uncertainty; 51

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72.6

冪␳/h

冪 冢冣 d 1− D

冣

2

d

100

4

3

冪h/␳

冪1 − 冢D冣

62.37

d

冪␳h

4 3⁄2

冢 冢D冣 冣

2D5 1 −

0.099702Cd 2 h

140

4

共0.099702C冪 ␳h兲 d 2 (−4)

0.099702Cd 2

␳

Absolute Sensitivity ⵲m˙ ⵲Xi

Formulas for Absolute Sensitivity ⵲m˙ ⵲Xi

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ASME PTC 19.1-2005

TEST UNCERTAINTY

Table 10-2.1-4 Absolute Contributions of Uncertainties of Independent Parameters (Example 10-2: Uncalibrated Case) Independent Parameters Uncertainty Contribution of Parameters to the Result (in Results Units Squared)

Parameter Information (in Parameter Units)

Symbol Xi C d D ␳ h

Description

Units

Nominal Value

Discharge coefficient Throat diameter Inlet diameter Water density at 60°F [25] Differential pressure head across venturi (at 68°F)

... in. in. lbm/ft3

0.984 3.999 6.001 62.37

in. H2O

100

Absolute Systematic Standard Uncertainty, bX

Absolute Random Standard Uncertainty, sX

3.75E–03 5.0E–04 1.0E–03 0.002

0 0 0 0.002

0.15

0.4

i

Absolute Sensitivity, ␪i 140 86.2 −11.4 1.11 0.692

Absolute Systematic Standard Uncertainty Contribution, (b X ␪i )2

Absolute Random Standard Uncertainty Contribution, (s X ␪i )2

0.276 1.86 ⴛ 10−3 1.29 ⴛ 10−4 4.93 ⴛ 10−6

0 0 0 4.92 ⴛ 10−6

1.08 ⴛ 10−2

7.66 ⴛ 10−2

i

i

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

Table 10-2.1-5 Summary: Uncertainties in Absolute Terms (Example 10-2: Uncalibrated Case)

Symbol

Description

˙ m

Mass flow rate

Units

Calculated Value

Absolute Systematic Standard Uncertainty, bR

Absolute Random Standard Uncertainty, sR

Combined Standard Uncertainty of the Result, uR

Total Absolute Uncertainty, UR,95

138.4

0.537

0.276

0.604

1.21

lbm/sec

Table 10-2.1.1-1 Relative Uncertainty of Measurement (Example 10-2: Uncalibrated Case) Independent Parameters

Symbol

Description

Units

Nominal Value

Relative Systematic Standard Uncertainty, b′X ⴝ b X /Xi

C d D ␳ h

Discharge coefficient Throat diameter Inlet diameter Water density at 60°F [24] Differential pressure head across venturi (at 68°F)

... in. in. lbm/ft3 in. H2O

0.984 3.999 6.001 62.37 100

0.361% 0.0125% 0.0167% 0.0032% 0.150%

i

i

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Relative Random Standard Uncertainty, s′X ⴝ s X /Xi 0% 0% 0% 0.0032% 0.400%

i

i

TEST UNCERTAINTY

ASME PTC 19.1-2005

Table 10-2.1.1-2 Relative Contributions of Uncertainties of Independent Parameters (Example 10-2: Uncalibrated Case) Independent Parameters

Symbol

Description

Units

Nominal Value

C d D ␳ h

Discharge coefficient Throat diameter Inlet diameter Water density at 60°F [24] Differential pressure head across venturi (at 68°F)

... in. in. lbm/ft3 in. H2O

0.984 3.999 6.001 62.37 100

Relative Sensitivity, ␪i′ 1.0 2.4923 −0.4923 0.50 0.50

Relative Systematic Standard Uncertainty Contribution, (b′X ␪ i′)2

Relative Random Uncertainty Contribution, (s′X ␪ i′)2

ⴛ ⴛ ⴛ ⴛ ⴛ

0 0 0 2.57 ⴛ 10−10 4.0 ⴛ 10−6

i

1.45 9.71 6.73 2.57 5.62

i

10−5 10−8 10−9 10−10 10−7

Table 10-2.1.1-3 Summary: Uncertainties in Relative Terms for the Uncalibrated Case Calculated Result

Symbol

Description

Units

Calculated Value

Relative Systematic Uncertainty, bR /R

˙ m

Mass flow rate

lbm/sec

138.4

0.39%

Relative Random Uncertainty, sR /R

Relative Combined Standard Uncertainty, uR /R

Relative Expanded Uncertainty, UR,95 /R

0.20%

0.44%

0.88%

Table 10-2.1.1-4 Relative Uncertainties of Independent Parameters (Example 10-2: Calibrated Case) Independent Parameters

Symbol

Description

Units

Nominal Value

Relative Systematic Standard Uncertainty, b′X ⴝ bX / Xi

C d D ␳ h

Discharge coefficient Throat diameter Inlet diameter Water density at 60°F [24] Differential pressure head across venturi (at 68°F)

... in. in. lbm/ft3 in. H2O

0.984 3.999 6.001 62.37 100

0.061% 0% 0% 0.0032% 0.150%

i

i

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Relative Random Standard Uncertainty, s′X ⴝ sX / Xi i

i

0% 0% 0% 0.0032% 0.400%

ASME PTC 19.1-2005

TEST UNCERTAINTY

Table 10-2.1.1-5 Relative Contributions of Uncertainties of Independent Parameters (Example 10-2: Calibrated Case) Independent Parameters

Symbol

Description

Units

Nominal Value

C d D ␳ h

Discharge coefficient Throat diameter Inlet diameter Water density at 60°F [24] Differential pressure head across venturi (at 68°F)

... in. in. lbm/ft3 in. H2O

0.984 3.999 6.001 62.37 100

Relative Sensitivity, ␪i′

Relative Systematic Standard Uncertainty Contribution, (b′X ␪ i′ ) 2

1.0 2.4923 −0.4923 0.50 0.50

3.72 ⴛ 10−7 0 0 2.57 ⴛ 10−10 5.62 ⴛ 10−7

i

Relative Random Standard Uncertainty Contribution, (s′X ␪ i′ ) 2 0 0 0 2.57 ⴛ 10−10 4.0 ⴛ 10−6

i

Table 10-2.1.1-6 Summary: Uncertainties in Relative Terms for the Calibrated Case Calculated Results

Symbol

Description

˙ m

Mass flow rate

Units

Calculated Value

Relative Systematic Uncertainty, bR /R

Relative Random Uncertainty, sR /R

Relative Combined Standard Uncertainty, uR /R

Relative Expanded Uncertainty, UR,95 /R

138.4

0.097%

0.20%

0.22%

0.44%

lbm/sec

Table 10-2.1.1-7 Summary: Comparison Between Calibrated and Uncalibrated Cases Calibrated Case ˙ m bR /R sR /R uR /R UR,95 /R

138.4 lbm/sec 0.097% 0.20% 0.22% 0.44%

conditions. The compressor was operated at normal, steady-state conditions for two hours prior to the test and for one hour during the test. Measurements of the total pressure and total temperature at the inlet and exit of the compressor were collected at one minute intervals resulting in 60 discrete measurements of each parameter over the test period. The mean value for each measured parameter was calculated using eq. (4-3.1). The resulting averages are presented in Table 10-4.1-1. A simplified schematic depicting the test measurement locations is shown in Fig. 10-4.1. The compressor inlet and exit total pressures were measured using multiport impact pressure arrays which are permanently installed in the compressor inlet and exit. Each array was connected to a digital pressure indicator in a manner which yielded a spatially averaged pressure measurement. The compressor inlet and exit total temperatures were measured using multipoint thermocouple stagnation probe arrays which were permanently installed in the compressor inlet and

Uncalibrated Case 138.4 lbm/sec 0.39% 0.20% 0.44% 0.88%

(b) treatment of correlated sources of uncertainty in a practical manner; and (c) the importance of applying all known engineering corrections and using appropriate engineering relationships as part of the test results analysis process. 10-4.1 Define the Measurement Process A test was conducted to determine the adiabatic efficiency of an air compressor at normal operating --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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TEST UNCERTAINTY

ASME PTC 19.1-2005

Table 10-3-1 Absolute Contributions of Uncertainties of Independent Parameters (Example 10-3: Uncalibrated, Nonsymmetrical Systematic Uncertainty Case) Independent Parameters --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

Parameter Information (in Parameter Units)

Symbol Xi C d D ␳ h

Description

Units

Nominal Value

Absolute Negative Systematic Standard Uncertainty, − bX

Discharge coefficient Throat diameter Inlet diameter Water density at 60°F [24] Differential pressure head across venturi (at 68°F)

... in. in. lbm/ft3

0.984 3.999 6.001 62.37

5.75E–03 5.0E–04 1.0E–03 0.002

in. H2O

100

−0.05

Absolute Positive Systematic Standard Uncertainty, b X+

Absolute Systematic Standard Uncertainty, bX

Absolute Random Standard Uncertainty, sX

1.75E–03 5.0E–04 1.0E–03 0.002

3.75E–03 5.0E–04 1.0E–03 0.002

0 0 0 0.002

0.35

0.15

0.4

i

Independent Parameters Uncertainty Contribution of Parameters to the Result (in Result Units Squared)

Symbol, Xi C d D ␳ h

Description

Units

Absolute Sensitivity, ␪i

Discharge coefficient Throat diameter Inlet diameter Water density at 60°F [24] Differential pressure head across venturi (at 68°F)

... in. in. lbm/ft3

140.8 86.14 −11.34 1.108

in. H2O

exit. The thermocouples from each multiprobe array were connected in parallel in a manner which yielded a spatially averaged temperature measurement. A common digital temperature indicator, with built-in cold junction compensation, was used to measure both the inlet and the exit temperatures. Pretest and posttest calibrations were performed on all instrumentation. The adiabatic efficiency of the air compressor was initially calculated using the following simplified engineering relationship:

0.6898

Absolute Systematic Standard Uncertainty Contribution, (bX ␪i ) 2

Absolute Random Standard Uncertainty Contribution, (sX ␪i ) 2

0.279 1.86 ⴛ 10−3 1.29 ⴛ 10−4 4.9 ⴛ 10−6

0 0 0 4.9 ⴛ 10−6

1.07 ⴛ 10−2

7.61 ⴛ 10−2

i

␩p

(P2 /P 1)[(␥ −1)/␥] − 1 T2/T 1 − 1

(10-4.1)

where P1p measured P2p measured T1p measured ature T2p measured ature

compressor inlet total pressure compressor exit total pressure compressor inlet total tempercompressor exit total temper-

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i

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ASME PTC 19.1-2005

TEST UNCERTAINTY

Table 10-3-2 Summary: Uncertainties in Absolute Terms (Example 10-3: Uncalibrated, Nonsymmetrical Systematic Uncertainty Case)

Symbol

Description

Units

Calculated Value

Absolute Systematic Standard Uncertainty, bR

˙ m

Mass flow rate

lbm/sec

138.4

0.540

P1 P2 T1 T2

Combined Standard Uncertainty of the Result, uR

Expanded Absolute Uncertainty, UR,95

0.276

0.606

1.21

Symbol

Description

Units

Offset, qR

Expanded Absolute Negative Uncertainty, UR,95−

˙ m

Mass flow rate

lbm/sec

−0.14

1.35

Table 10-4.1-1 Elemental Random Standard Uncertainties Associated With Error Sources Identified in Para. 10-4.2

Symbol

Absolute Random Standard Uncertainty, sR

Description

i

14.70 95.52 520.0 960.0

1.07

the test result introduces an error that was not accounted for in the reported uncertainty, the test result was reevaluated using more rigorous engineering relationships to eliminate some of the assumptions noted. The difference between the two test results was compared to the previously reported uncertainty to illustrate the importance of applying all known engineering corrections and using appropriate engineering relationships as part of the results analysis process. The details of this comparison are discussed in para. 10-4.7.

Absolute Random Standard Mean Value, Uncertainty, Units Xi sX

Inlet pressure psia Exit pressure psia Inlet temperature R Exit temperature R

Expanded Absolute Positive Uncertainty, UR,95+

0.030 0.170 0.300 0.600

10-4.2 List Elemental Error Sources

␥p ratio of the specific heats, assumed to be 1.40 ␩p adiabatic compressor efficiency

Based upon a review of the measurement methods and instruments employed for the test, the following lists of elemental error sources were compiled for each of the measurements: (a) Compressor Inlet and Exit Pressure Measurements (1) random error associated with incomplete sampling of the average pressure over the duration of the test and random variability in the pressure measurement instrumentation (2) systematic error resulting from imperfect calibration and drift of the digital pressure indicator (3) systematic error resulting from environmental influences on the digital pressure indicator (4) systematic error resulting from imperfect spatial averaging (5) systematic error due to the inability of the impact pressure arrays to fully realize total pressure

Simplifying assumptions for this relationship include (a) use of ideal gas properties for dry air (b) negligible potential energy change (c) negligible heat loss to surroundings (d) constant specific heats and a specific heat ratio of 1.4 Using the previous simplified engineering relationship, a test result and associated uncertainty were calculated and are presented in Tables 104.1-2 and 10-4.1-3. The details of the uncertainty analysis are discussed in paras. 10-4.2 through 104.6. Recognizing that the use of the simplified engineering relationship in the computation of 56

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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TEST UNCERTAINTY

ASME PTC 19.1-2005

Fig. 10-4.1 Typical Pressure and Temperature Locations for Compressor Efficiency Determination

Table 10-4.1-2 Independent Parameters Uncertainty Contribution of Parameters to the Result (in Result Units2)

Parameter Information (in Parameter Units)

Symbol

Description

Units

Nominal Value, Xi

P1 P2 T1 T2

Inlet pressure Exit pressure Inlet temperature Exit temperature

psia psia R R

14.70 95.52 520.0 960.0

Absolute Systematic Standard Uncertainty, bX

Absolute Random Standard Uncertainty, sX

0.021 0.198 0.646 [Note (1)] 0.792 [Note (1)]

0.030 0.170 0.300 0.600

i

i

Absolute Systematic Standard Uncertainty Contribution, (␪i b X )2

Absolute Random Standard Uncertainty Contribution, (␪i s )2

7.53E–7 1.54E–6 5.62E–6 [Note (1)] 2.59E–6 [Note (1)]

1.51E–6 1.14E–6 1.21E–6 1.48E–6

Absolute Sensitivity, ␪i −0.0409 +0.00629 +0.00367 −0.00203

i

X

i

NOTE: (1) These systematic standard uncertainties have some components that are correlated. The correlated terms are not shown in the table.

Table 10-4.1-3 Calculated Result

Symbol

Description

Units

Calculated Value, R

Absolute Systematic Standard Uncertainty, bR

␩

Computed adiabatic compressor efficiency

Nondimensional

0.8355

0.00309

Absolute Random Standard Uncertainty, sR

Absolute Combined Standard Uncertainty, uR

Absolute Expanded Uncertainty, UR

0.00231

0.00366

0.00772

57 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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ASME PTC 19.1-2005

TEST UNCERTAINTY

Table 10-4.3.2-1 Inlet and Exit Pressure Elemental Systematic Standard Uncertainties

Error Source Calibration and drift of indicator [Note (3)]

Environmental influences on indicator [Note (3)] Spatial averaging [Note (3)]

Realization of total pressure [Note (3)] Total

Limits of Error for Inlet [Note (1)] psia, B P1

Absolute Systematic Standard Uncertainty for Inlet [Note (2)] psia, b P1

Limits of Error for Exit [Note (1)] psia, B P2

Absolute Systematic Standard Uncertainty for Exit [Note (2)] psia, b P2

Calibration and drift uncertainty reported by calibration laboratory based upon pretest and posttest calibration Published information provided by indicator vendor

0.04

0.02

0.25

0.125

0.01

0.005

0.06

0.03

Reported uncertainty provided by compressor vendor based upon test rig data Engineering judgment

0.01

0.005

0.3

0.15

Negligible

Negligible

Negligible

Negligible

N/A

0.021

N/A

0.198

Source of Information for Estimation of Limits of Error

Eq. 6.8

NOTES: (1) All limits of error are estimated at 95% confidence. It is assumed that these estimates are based on large degrees of freedom and that the population of possible error values associated with each elemental systematic error source is normally distributed. (2) Elemental systematic standard uncertainties are calculated using eq. (4-3.5). (3) It is assumed that the indicated elemental systematic standard uncertainties for the inlet and exit pressure measurements are not correlated because the inlet and exit measurements are substantially different in magnitude. For this particular example, assuming that these are not correlated will elevate the estimated uncertainty in the test result.

(b) Compressor Inlet and Exit Temperature Measurements (1) random error associated with incomplete sampling of the average temperature over the duration of the test and random variability in the temperature measurement instrumentation (2) systematic error resulting from imperfect calibration and drift of the thermocouple probes (3) systematic error resulting from imperfect calibration and drift of the digital temperature indicator (4) systematic error resulting from imperfect calibration and drift of the cold junction reference (5) systematic error resulting from environmental influences on the digital temperature indicator (6) systematic error resulting from imperfect spatial averaging

(7) systematic error due to the inability of the stagnation probes to fully realize total temperature 10-4.3 Calculate Random and Systematic Standard Uncertainties 10-4.3.1 Random Standard Uncertainty. The elemental random standard uncertainties associated with the error sources identified in para. 10-4.2 were evaluated by calculating the absolute standard deviation of the mean for each measured parameter [see eq. (4-3.2)]. The results are summarized in Table 10-4.1-1. 10-4.3.2 Systematic Standard Uncertainty. The elemental systematic standard uncertainties associated with the error sources identified in para. 104.2 were evaluated and totaled for each measured 58

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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TEST UNCERTAINTY

ASME PTC 19.1-2005

Table 10-4.3.2-2 Inlet and Exit Temperature Elemental Systematic Standard Uncertainties

Error Source Calibration and drift of thermocouple probes [Note (3)] Calibration and drift of indicator [Note (3)]

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

Calibration and drift of cold junction reference [Note (4)] Environmental influences on indicator [Note (3)] Spatial averaging [Note (3)]

Realization of total temperature [Note (3)] Total

Source of Information for Estimation of Limits of Error

Limits of Error for Inlet [Note (1)] °R, B T1

Calibration and drift uncertainty reported by calibration laboratory based upon pretest and posttest calibration Calibration and drift uncertainty reported by calibration laboratory based upon pretest and posttest calibration Calibration and drift uncertainty reported by calibration laboratory based upon pretest and posttest calibration Published information provided by indicator vendor Reported uncertainty provided by compressor vendor based upon test rig data Engineering judgment

Absolute Systematic Standard Uncertainty for Inlet [Note (2)] °R, b T1

Limits of Error for Exit [Note (1)] °R, B T2

Absolute Systematic Standard Uncertainty for Exit [Note (2)] °R, b T2

1.1

0.55

1.1

0.55

0.4

0.2

0.2

0.1

0.5

0.25

0.5

0.25

0.2

0.1

0.1

0.05

0.1

0.05

1.0

0.5

Negligible

Negligible

Negligible

Negligible

N/A

0.0646

N/A

0.792

Eq. (6-2.1)

NOTES: (1) All limits of error are estimated at 95% confidence. It is assumed that these estimates are based on large degrees of freedom and that the population of possible error values associated with each elemental systematic error source is normally distributed. (2) Elemental systematic standard uncertainties are calculated using eq. (4-3.4). (3) It is assumed that the indicated elemental systematic standard uncertainties for the inlet and exit temperature measurements are not correlated because the inlet and exit measurements are substantially different in magnitude. For this particular example, assuming that these are not correlated will elevate the estimated uncertainty in the test result. (4) It is assumed that the indicated elemental systematic standard uncertainties for the inlet and exit temperature measurements are correlated as an error in the common cold junction reference will cause equivalent errors in the inlet and exit measurements.

parameter as shown in Tables 10-4.3.2-1 and 104.3.2-2.

Appendix C. The absolute random standard uncertainty for the test result is [see eq. (7-3.1)]

10-4.4 Propagate Random and Systematic Standard Uncertainties

sR p [(␪ P1s P1 ) 2 + (␪ P2s P2 )2 + ( ␪T1sT1 )2 + ( ␪T2sT2 )2] p 0.00231

The individual parameter standard uncertainties are propagated into terms of the test result by a Taylor series expansion as given in Nonmandatory

2

(10-4.2)

where the absolute sensitivities are [see eq. (7-2.1)] 59

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1⁄

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ASME PTC 19.1-2005

␪P1 p

TEST UNCERTAINTY

−[ ␥ − 1)/␥ ][(P 2/P 1) −1/␥][P2 /(P1 ) 2] [(T 2 /T 1 ) − 1]

The uncertainty interval for the test result is [see eq. (7-5.3)]

(10-4.3)

p −0.0409(psia −1) R ± UR p 0.8355 ± 0.00772

␪P2 p

[( ␥ − 1)/␥ ][(P 2/P 1) −1/␥][1/P 1] [(T2 /T 1 ) − 1]

10-4.6 Report (10-4.4)

A summary presentation of the uncertainty analysis at the test result level is reported in Tables 10-4.1-2 and 10-4.1-3.

p +0.00629(psia −1)

␪T1 p

[T 2]{[(P 2 /P1 ) (␥−1)/␥] − 1}

(10-4.5)

[(T 2 − T1 ) 2] p +0.00367(°R−1)

␪T2 p

[−T1]{[(P 2 /P 1 )

( ␥ −1)/ ␥

10-4.7 Importance of Applying Known Engineering Corrections

] − 1}

As discussed in subsection 4-1, it is important that all known engineering corrections are applied and appropriate engineering relationships are used as part of the results analysis process. Failure to do so leads to additional errors in the reported test result which are not accounted for in the test uncertainty analysis. This could, in turn, lead to expression of an uncertainty interval for a test result that does not encompass the true value. For this particular example, the assumptions of the analysis method used to compute the adiabatic efficiency of the compressor were not identified as sources of error. To illustrate the potential significance of having overlooked these sources of error, the test result will be recalculated using more exact relationships that do not require the assumption of constant specific heats and the assumed specific heat ratio of 1.4. By definition, the adiabatic efficiency of a compressor is the ratio of the work input required to raise the pressure of a gas to a specified value in an isentropic manner to the actual work input.

(10-4.6)

2

[(T 2 − T 1 ) ] p −0.00203(°R−1)

Note: In computing the preceeding sensitivity coefficients, it was assumed that the specific heat ratio is independent of air temperature.

The absolute systematic standard uncertainty for the test result is [see eq. (8-1.2)] bR p [(␪ P1 b P1 ) 2 + (␪ P2 b P2 )2 + ( ␪T1 bT1 )2 + (␪ T2 b T2 )2 +2 ␪T1␪T2bT1T2]

1⁄

2

p 0.00309

(10-4.7)

where bT1T2p the covariance of the error sources common to T1 and T2 and is determined as (see subsection 8-1) bT1T2 p 0.25(°R) · 0.25(°R) p 0.0625(°R 2)

(10-4.8)

␩p

10-4.5 Calculate Uncertainty

uR p 关共b R兲 2 + 共 S R兲 2兴

1⁄

2

p 0.00386

(10-4.12)

wsp entropic compressor work wap actual compressor work

(10-4.9)

Assuming negligible change in the potential energy of the gas being compressed and negligible heat loss to the surroundings, the adiabatic efficiency can be expressed as a function of stagnation enthalpies as follows:

and the expanded uncertainty of the result is [see eq. (7-5.2)] U R p 2u R p 0.00772

ws wa

where

The combined standard uncertainty in the test result is [see eq. (7-5.1)]

(10-4.10)

The assumptions required for using this equation are presented in subsection 1-3.

␩p 60

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(10-4.11)

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h2s − h 1 h 2 − h1

(10-4.13)

TEST UNCERTAINTY

ASME PTC 19.1-2005

Fig. 10-4.7 The h-s Diagram of the Actual and Isentropic Processes of an Adiabatic Compressor (Used by Permission From McGraw-Hill Co.) where

where

h1p stagnation enthalpy at measured inlet conditions h2p stagnation enthalpy at measured exit conditions h2sp stagnation enthalpy assuming an isentropic compression process

P1p measured compressor inlet total pressure P2p measured compressor exit total pressure Pr1p the relative pressure determined from ideal gas property tables for air at the measured compressor inlet total temperature Pr2p the relative pressure at the exit corresponding to isentropic compression

An h-s diagram of the actual and isentropic processes of an adiabatic compressor is illustrated in Fig. 10-4.7. Assuming ideal gas properties for dry air, ideal gas property tables for air are used to evaluate h1, h2, and h2s. The values for h1 and h2 are evaluated at the measured inlet and exit conditions. The value for h2s, however, must be evaluated at a state corresponding to the measured exit pressure but assuming isentropic compression has occurred. This is done by first using the following relationship, which accounts for variable specific heats, to determine the relative pressure corresponding to isentropic compression. P r2 p P r1

冢P 冣 P2

Next, the value for h2s is evaluated from air property tables at a state corresponding to the value of Pr2 determined above. The results of this analysis are summarized in Table 10-4.7 and compared with the previously reported test result. Comparison of the previously reported test result with the more exact value determined by eq. (10-4.13) indicates an error in the previously reported test result that is greater than the magnitude of the previously reported expanded uncertainty. In this case, failure to use the more correct engineering relationships would lead to presentation of an uncertainty interval for the test result that does not encompass the true value. This illustrates

(10-4.14)

1 s p const

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TEST UNCERTAINTY

Table 10-4.7 Evaluation of Analysis Error Symbol h1 h2 Pr 1 Pr2

h2s

␩ ␩

UR ...

Parameter Description

Units

The performance of a hydraulic system can be evaluated using Bernoulli’s equation.

Average Value

Stagnation enthalpy at measured Btu/lbm inlet total temperature Stagnation enthalpy at measured Btu/lbm exit total temperature Relative pressure at measured ... inlet total temperature Relative pressure at exit ... corresponding to isentropic compression Stagnation enthalpy at exit Btu/lbm corresponding to isentropic compression Computed adiabatic compressor ... efficiency using eq. (10-4.13) Previously reported adiabatic ... compressor efficiency using eq. (10-4.1) Previously reported expanded ... uncertainty in the test result Difference between results from ... eq. (10-4.13) and eq. (10-4.1)

124.27

Hp 231.06

P v2 + +z ␳ 2g

where

1.2147

Hp total head, m Pp system pressure, Pa ␳p fluid density, kg/m3 vp fluid velocity, m/s gp gravitational constant, 9.81 m/s2 z1p elevation, m

7.8931

212.35

0.8248 0.8355

The effective head produced at a specified pressure is a measure of pump performance. The effective head in terms of the measured test parameters and physical property data is

0.00772 −0.0107

⌬P p P 2 +

8Q 2␳

␲ 2 d 42

+ (z 2 − z 1) ␳ g

where d 2p inside pipe diameter, m ⌬Pp pressure change between taps 1 and 2, Pa P 2p pressure at tap 2, Pa Qp fluid flow rate, m3/s z 2p elevation of pressure tap 2, m z 1p elevation of tap 1 or free-surface elevation, m

the importance of applying all known engineering corrections and using appropriate engineering relationships as part of the test results analysis process. 10-5 PERIODIC COMPARATIVE TESTING 10-5.1 Problem Definition Periodic testing of equipment is a common situation where measurement uncertainty must be considered. For this example, a pump is considered to perform consistently if the supply pressure measured at 100% of rated flow is consistent with prior test results. The pump design data is presented in Table 10-5.1-1 and the test data is presented in Table 10-5.1-2 After each test, it is necessary to evaluate the test results. As part of this effort the effect of measurement uncertainty must be considered. For the first test in Table 10-5.1-2 the available information is limited since it was a factory test. (See Figs. 10-5.1-1 and 10-5.1-2.) The conclusions that can be drawn from Fig. 10-5.1-3 are as follows: (a) The pump is operating consistently when compared to the factory test results since the uncertainty bands overlap. (b) The pump is operating better than the minimum required design condition. The confidence for this conclusion is better than 95%.

The density can be estimated numerically using the relationship [6], ␳ p 766.17 + 1.80396 T K −

(10-5.1)

where TK p TC + 273.15 p the absolute temperature and TC is the fluid temperature in Celsius Estimates of the density, ␳, using eq. (10-5.1) are reported to have a systematic standard uncertainty of 0.587 kg/m3. The random error in the curve fit was judged to be negligible and so the random uncertainty is set to zero. During testing it is not feasible to operate exactly at a specified operating condition. For a pump test, the applied flow might be slightly different for each test. This will result in a slight change in the resultant differential pressure. The variation 62

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3.4589 T 2K 1000

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TEST UNCERTAINTY

ASME PTC 19.1-2005

Table 10-5.1-1 Pump Design Data (TC ⴝ 20°C ) Flow

Differential Pressure

3

m /s

gpm

Percent Rated Flow

kPa

psi

0.000 0.126 0.189

0 2000 3000

0 100 150

827 689 552

119.9 99.9 80.1

Table 10-5.1-2 Summary of Test Results Factory Test [Note (1)] Raw Data Flow, m3/s P exit, Pa d exit, m z exit, m z inlet, m T, °C Resultants ␳, kg/m3 ⌬P, Pa ⌬P, psi

. . . .

0.126 .. .. .. .. 20.15

997.7 712000 103.3

Field Tests A

B

C

D

E

0.125 840000 0.254 2.40 15.00 19.71

0.126 845000 0.254 2.40 15.00 20.01

0.130 820000 0.254 2.40 15.00 20.31

0.123 836000 0.254 2.40 15.00 20.78

0.129 841000 0.254 2.40 15.00 21.10

997.8 718000 104.1

997.7 724800 105.1

997.7 707300 102.6

997.6 710200 103.0

997.5 726400 105.4

NOTE: (1) The factory test data only provides resultant information.

Fig. 10-5.1-1 Installed Arrangement

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ASME PTC 19.1-2005

TEST UNCERTAINTY

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

Fig. 10-5.1-2 Pump Design Curve With Factory and Field Test Data Shown

Fig. 10-5.1-3 Comparison of Test Results With Independent Control Conditions

in flow may be handled by normalizing the test results. This is accomplished by adding an additional random term. A normalization coefficient can be estimated from the factory test data in Table 10-5.1-1. A best fit correlation of the data is ⌬P p 827,000 − 376,000Q − 5,710,000Q 2

⌬P N p b(Q N − Q) + ⌬P

where ⌬PNp expected pressure change based on the nominal (specified) test conditions, Pa ⌬Pp measured pressure change, Pa QNp nominal (specified) test flow rate, m3/s Qp measured test flow rate, m3/s bp slope of eq. (10-5.2), Pa · s/m3

(10-5.2)

The normalizing coefficient is the slope of eq. (10.5.2) at the specified test conditions. 64 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

(10-5.3)

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TEST UNCERTAINTY

ASME PTC 19.1-2005

The slope of eq. (10-5.2) is ∂⌬P N ∂Q

冨

QN

The uncertainty for each test can be calculated as shown in Table 10-5.2-1. Table 10-5.2-2 shows the nominal value, and the systematic, random, and total uncertainties for ⌬P. The uncertainties for each test are presented in Table 10-5.2-3. The results are plotted in Fig. 10-5.2.

p b p −376,000 − 11,420,000 Q N

At the 100% rated flow condition (see Table 105.1-1), QN p 0.126 m3/s, and the value for the slope is b p −1 810 000 Pa · s/m3. The final data reduction equation becomes: ⌬P p P 2 +

8Q 2␳

␲ 2 d 42

10-5.3 Comparative Uncertainties Correlation of terms is an important consideration in comparison testing [23, 24] where two different operating conditions or constructions are being compared by use of a ratio

+ (z 2 − z 1) ␳ g + b(Q N − Q)

␩p

10-5.2 Comparison With Independent Control

␩p ratio of two resultants ⌿altp alternate (or variable) resultant ⌿controlp resultant used for the baseline or control For the pump test C, with test A considered the control test, eq. (10-5.4) becomes

The test results for the comparative analysis are summarized in Table 10-5.2-3. The uncertainty for the comparative analysis can be computed using the method from subsection 8-1. The partial derivatives for eq. (10-5.4) are

∂⌬P p1 ∂P 2 16Q␳

Q

−b

␲ 2d 42

∂␩ ␩ p ∂⌿a ⌿a

−32Q ␳ 2

d2

␲ 2d 52 8Q

␲ 2d 42

+ (z2 − z1 )g

z1

␳g

z2

− ␳g

b

QN − Q

TK

1.80396 −

␪ P2 p

␩ ∂⌬Pa ⌬Pa ∂P2a

␪Qa p

␩ ∂⌬Pa ⌬Pa ∂Qa

␪ d2 p

␩ ∂⌬Pa ⌬Pa ∂d2a

␪␳a p

␩ ∂⌬Pa ⌬Pa ∂␳a

␪z2 p

␩ ∂⌬Pa ⌬Pa ∂z2a

␪z1 p

␩ ∂⌬Pa ⌬Pa ∂z1a

␪b a p

␩ ∂⌬Pa ⌬Pa ∂ba

␪ TC p

␩ ∂⌬Pa ⌬Pa ∂Tca

␪ p2 p

−␩ ∂⌬Pc ⌬Pc ∂P2c

␪Qc p

−␩ ∂⌬Pc ⌬Pc ∂Qc

␪d2 p

−␩ ∂⌬Pc ⌬Pc ∂d2c

a

6.9178TK 1000

The temperature sensitivity coefficient is computed using the chain rule:

冢 冣 6.9178 T 冢1.80396 − 1000 冣

∂⌬P N ∂ ␳ ∂⌬P N 8Q 2 p p + (z 2 − z1) g ∂TK ∂ ␳ ∂T K ␲ 2d 42

K

c

a

a

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−␩ ∂␩ p ∂ ⌿c ⌿c

These may be combined with the partial derivatives presented earlier to estimate the sensitivity coefficients for the comparative analysis.

2

␳

707 kPa p 0.985 718 kPa

␩A p

Formulas for Absolute Sensitivity

P2

(10-5.4)

where

The field test data for the pump can be compared with the factory test results or the minimum rated pressure output. For this type of evaluation, the uncertainties are independent and a simple comparison of the test results with the benchmark value is adequate. The uncertainty is calculated using the method in subsections 7-1 through 7-4. The partial derivatives necessary to estimate the sensitivity coefficients for this problem are Symbol, Xi

⌿ alt ⌿ control

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a

a

a

ASME PTC 19.1-2005

TEST UNCERTAINTY

Table 10-5.2-1 Uncertainty Propagation for Comparison With Independent Control Independent Parameters Uncertainty Contribution of Parameters to the Result (in Result Unit Squared)

Parameter Information (in Parameter Units)

Symbol Q P2 d2 z2 z1 Tc ␳ b

Description

Nominal Value

Units

Flow m3 Exit pressure Pa Exit diameter m Exit elevation m Inlet elevation m Fluid temperature °C Fluid density kg/m3 Correlation Pa·s/m3 coefficient

Absolute Random Standard Uncertainty, sX

Absolute Sensitivity, ␪i

3.0ⴛ10−3 3500 1.0ⴛ10−3 0.0125 0.0125 0.25 0.6 5000

1.0ⴛ10−3 3000 0 0 0 0.10 0 0

1.86ⴛ106 1.0 −47810 9786 −9786 26.76 −120.5 1.0ⴛ10−3

i

0.125 840.0ⴛ103 0.254 2.40 15.00 19.71 997.8 −1.82ⴛ106

Absolute Absolute Systematic Random Standard Standard Uncertainty Uncertainty Contribution, Contribution, (b X ␪i )2 (s X ␪i )2

Absolute Systematic Standard Uncertainty, bX

i

i

i

3.11ⴛ107 1.23ⴛ107 2.29ⴛ103 1.50ⴛ104 1.50ⴛ104 44.8 5.23ⴛ103 25.0

3.46ⴛ106 9.0ⴛ106 0 0 0 7.16 0 0

Table 10-5.2-2 Summary: Uncertainties in Absolute Terms

Symbol

Description

Units

Calculated Value

Absolute Systematic Standard Uncertainty, bR

⌬P

Differential pressure

Pa

718,000

6,590

Absolute Random Standard Uncertainty, sR

Combined Standard Uncertainty of the Result, uR

Total Absolute Uncertainty, UR,95

3,530

7,476

14,951

Table 10-5.2-3 Summary of Results for Each Test Test

⌬P, kPa

b X, kPa

s X, kPa

U⌬P,95, kPa

␩

b␩

s␩

U␩,95

Factory Test A B C D E

712 718 725 707 710 726

... 6.5 6.5 6.5 6.5 6.5

... 4 4 4 4 4

... 15 15 15 15 15

0.9916 1.0000 1.0097 0.9847 0.9889 1.0111

0.0091 0.0048 0.0049 0.0049 0.0048 0.0049

0.0049 0.0069 0.0070 0.0069 0.0069 0.0070

0.0207 0.0168 0.0170 0.0169 0.0168 0.0170

66 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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ASME PTC 19.1-2005

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

Fig. 10-5.2 Comparison of Test Results Using the Initial Field Test as the Control

The uncertainty of the comparison ratio, ␩, when all of the systematic standard uncertainty terms are correlated and the corresponding sensitivity coefficients are equal (i.e., ␪i,alt p ␪i,control) is zero. For eq. (10-5.4) the systematic standard uncertainty summation based on eq. (8-1.2).

冢 冣 + 冢␪ b 冣 + 冢␪ b 冣 + 冢␪ b 冣 + 冢␪ b 冣 + 冢␪ b 冣 + 冢␪ b 冣 + 冢␪ b 冣 + 冢␪ b 冣 + 冢␪ b 冣 + 冢␪ b 冣 + 冢␪ b 冣 + 冢 ␪ b 冣 + 冢␪ b 冣 + 冢␪ b 冣

b2␩ p ␪P2 bP2 a

␳a

Tc

Tc

Qc

c

z2

2

2

z1

d2

2

z2

a

c

d2

2

z1

a

BN

d2

c

TC

c

2

Qa

a

BN

a

Qc

z2

Qa

2

␳a

a

2

a

P2

a

2

␳c

c

TC

z1

a

2

d2

a

2

2

a

2

a

P2

c

2

c

2

␳c

BN

c

BN

c

2

c

+ 2 ␪ P2 ␪ P2 b P2 P2 + 2 ␪ Qa␪ Qcb QaQc + 2 ␪ d2 ␪ d2 b d2 d2 a

c

a

c

a

c

+ 2 ␪ ␳ a ␪ ␳ c b ␳ a␳ c + 2 ␪ z 2 ␪ z 2 b z 2 z 2 + 2 ␪ z 1 ␪ z 1 b z 1 z 1 a

c

a

c

a

+ 2 ␪ Tc ␪ Tc bTc Tc + 2 ␪ bN ␪ bN b bN bN a

c

ca

c

a

c

a

For this example, all of the systematic standard uncertainties are considered fully correlated except for the exit pressure that is only partially correlated. The partial correlation for the pressure measurement must be derived from the elemental uncertainties. For this program the test procedure requires that the pressure gage be calibrated just prior to conducting the test. Standard protocol is to bring a replacement gage to the test location on the day of the test and replace the gage just prior to the test. Thus, each test is conducted with a different gage. The absolute systematic standard uncertainty, bP, associated with each gage is 3,500 Pa. Since the gage is randomly selected and several gage brands are used, much of the systematic error is not correlated. The exception is the calibration standard uncertainty, which is known to be 2,500 Pa. This calibration standard uncertainty is treated as a fully correlated standard uncertainty associated with the pressure gage. For test C, with test A considered the control test, the absolute random standard uncertainty component, SR, from Table 10-5.3-1 is 0.0069. The absolute systematic standard uncertainty of the result (from Tables 10-5.2-3 and 10-5.3-1) would be

c

a

a

c

c

c

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ASME PTC 19.1-2005

TEST UNCERTAINTY

Table 10-5.3-1 Uncertainty Propagation for Comparative Uncertainty ⵲␩ ⵲⌬p

⵲⌬P ⵲ xi

␪i

bi

(bi␪i )2

sX

1.39ⴛ10−6 1.39ⴛ10−6 1.39ⴛ10−6 1.39ⴛ10−6 1.39ⴛ10−6 1.39ⴛ10−6 1.39ⴛ10−6 1.39ⴛ10−6

1.86ⴛ106 1 −47,810 9,786 −9,786 26.76 −120.5 0.0010

2.590 1.39ⴛ10−6 −0.0665 0.0136 −0.0136 0.0372 −1.68ⴛ10−4 1.39ⴛ10−9

0.003 3,500 0.001 0.0125 0.0125 0.25 0.6 5,000

6.04ⴛ10−5 2.37ⴛ10−5 4.42ⴛ10−9 2.89ⴛ10−8 2.89ⴛ10−8 0.0 1.02ⴛ10−8 4.83ⴛ10−11

0.001 3,000 0.0 0.0 0.0 0.1 0.0 0.0

6.71ⴛ10−6 1.74ⴛ10−5 0.0 0.0 0.0 0.0 0.0 0.0

−1.37ⴛ10−6 −1.37ⴛ10−6 −1.37ⴛ10−6 −1.37ⴛ10−6 −1.37ⴛ10−6 −1.37ⴛ10−6 −1.37ⴛ10−6 −1.37ⴛ10−6

1.86ⴛ106 1 −47,810 9,786 −9,786 26.76 −120.5 0.0010

−2.55 −1.37ⴛ10−6 0.0655 −0.0134 0.0134 −3.66ⴛ10−5 1.65ⴛ10−4 −1.39ⴛ10−9

0.003 3,500 0.001 0.0125 0.0125 0.25 0.6 5,000

5.85ⴛ10−5 2.30ⴛ10−5 4.29ⴛ10−9 2.81ⴛ10−8 2.81ⴛ10−8 0.0 9.80ⴛ10−9 4.69ⴛ10−11

0.001 3,000 0.0 0.0 0.0 0.1 0.0 0.0

6.50ⴛ10−6 1.69ⴛ10−5 0.0 0.0 0.0 0.0 0.0 0.0

i

(s X ␪i )2 i

Variable Test C Q P2 d2 z2 z1 TC ␳ b Control Test A Q P2 d2 z2 z1 TC ␳ b

␩ (RSS value)

0.0129

0.0069

Table 10-5.3-2 Sensitivity Coefficient Estimates for Comparative Analysis Q P2 d2 z2 z1 TC ␳ b

␪ia

bi

␪ic

bi

2␪i ␪i bi i

2.593 1.39ⴛ10−6 −0.0719 0.0136 −0.0136 3.78ⴛ10−5 −1.68ⴛ10−4 5.56ⴛ10−9

0.003 2500 0.001 0.0125 0.0125 0.25 0.6 5000

−2.552 −1.37ⴛ10−6 0.0655 −0.0134 0.0134 −3.67ⴛ10−5 1.65ⴛ10−4 −1.37ⴛ10−9

0.003 2500 0.001 0.0125 0.0125 0.25 0.6 5000

−1.19ⴛ10−4 −2.38ⴛ10−5 0.0 0.0 0.0 0.0 0.0 0.0

a

c

␩ (RSS sum)

bR p

冪 (0.0129)2 + (−1.43 ⴛ 104) p 4.84 ⴛ 10−3

ac

The uncertainty for each test can be calculated as shown in Tables 1-5.3-1 and 10-5.3-2. The uncertainties for each test are presented in Table 1-5.22. The results are plotted in Fig. 10-5.2. The test conclusion that can be drawn from this figure is the pump is operating consistently when compared since the uncertainty bands overlap.

冪 (4.84 ⴛ 10−3)2 + (6.9 ⴛ 10−3)2 p 8.43 ⴛ 10−3

The total comparative uncertainty U R p 2u R p 0.0169

68 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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c

−1.43ⴛ10−4

The combined standard uncertainty of the result is uR p

a

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TEST UNCERTAINTY

ASME PTC 19.1-2005

Section 11 References [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8] --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

[9]

[10]

[11]

ISO “Guide to the Expression of Uncertainty in Measurement.” Geneva: International Organization for Standardization; 1995. ASME PTC 19.1, Measurement Uncertainty. New York: The American Society of Mechanical Engineers; 1985. Moffat, R. J. “Identifying the True Value — The First Step in Uncertainty Analysis.” ISA paper 88-0729. Steele, W. G., et al. Use of Previous Experience to Estimate Precision Uncertainty of Small Sample Experiments. AIAA Journal, 31: 1891–1896; October 1993. Brownlee, A. K. Statistical Theory and Methodology in Science and Engineering, 2nd edition. New York: John Wiley and Sons; 1967. Figliola, R. S., and D. E. Beasley. Theory and Design for Mechanical Measurements. 3rd edition. New York: John Wiley and Sons; 2000. Moffat, R. J. “Contributions to the Theory of a Single-sample Uncertainty Analysis.” Transactions of the ASME: Journal of Fluids Engineering, 104: 250–260; June 1982. New York: The American Society of Mechanical Engineers. James, M. L. Applied Numerical Methods for Digital Computation, 3rd edition. New York: Harper Collins; 1992. Coleman, H. W. and W. G. Steele. Experimentation and Uncertainty: Experimentation Uncertainty Analysis for Engineers, 2nd edition. New York: John Wiley & Sons; 1999. Brown, K. K., et al. Evaluation of Correlated Bias Approximations in Experimental Uncertainty Analysis. AIAA Journal, 34: 1013–1018; May 1996. Steele, W. G., et al. Asymmetric Systematic Uncertainties in the Determination of Experimental Uncertainty. AIAA Journal, 34: 1458– 1463; July 1996.

[12]

[13]

[14]

[15]

[16]

[17] [18] [19]

[20]

[21]

[22]

Wyler, J. S. Estimating the Uncertainty of Spatial and Time Average Measurements. Transactions of the ASME: Journal of Engineering for Power: 473–476; October 1975. Holman, J. P. Experimental Methods for Engineers, 6th edition. New York: McGraw-Hill; 1994. ISO/TR 7066-1: 1997 (E). “Assessment of Uncertainty in Calibration and Use of Flow Measurement Devices — Part 1: Linear Calibration Relationships.” Geneva: International Organization for Standardization; 1997. ISO 7066-2: 1988(E). “Assessment of Uncertainty in the Calibration and Use of Flow Measuring Devices — Part 2: Non-linear Calibration Relationships.” Geneva: International Organization for Standardization; 1988. Montgomery, D. C., and E. A. Peck. Introduction to Linear Regression Analysis. 2nd edition. New York: John Wiley and Sons; 1992. Price, M. L. “Uncertainty of Derived Results on X-Y Plots,” ISA paper No. 93-107. Fuller, W. A. Measurement Error Models. New York: John Wiley and Sons; 1987. ASME PTC 19.1, Test Uncertainty. New York: The American Society of Mechanical Engineers; 1998. NIST Technical Note 1297. Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurements. 1994. Fluid Meters — Their Theory and Application. 6th edition. New York: American Society of Mechanical Engineers; 1971. Chakroun, W., et al. “Bias Error Reduction Using Ratios to a Baseline Experiment — Heat Transfer Case Study. Journal of Thermophysics and Heat Transfer, 7: 754–757; October–December 1993.

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ASME PTC 19.1-2005

[23]

[24]

[25]

[26]

TEST UNCERTAINTY

Coleman, H. W., W. G. Steele, and R. P. Taylor. “Implications of Correlated Bias Uncertainties in Single and Comparative Tests.” Transactions of the ASME: Journal of Fluids Engineering, 117: 552–556; New York: The American Society of Mechanical Engineers. December 1995. Hahn, G. “Understanding Statistical Intervals.” Industrial Engineering, 45–48; December 1970. Thompson, R. W. “On a Criterion for Rejection of Observations and the Distribution of the Ratio of the Deviation to Sample Standard Deviation.” Annals of Mathematical Statistics, 6: 214–219; 1935. Grubbs, F. E. “Procedures for Detecting Outlying Observations in Samples.” Technome-

[27]

[28]

[29]

[30]

trics, 11: 1; February 1969. Steele, W. G., et al. “Computer-Assisted Uncertainty Analysis.” Computer Applications in Engineering Education, 1997. Steele, W. G., et al. “Comparison of ANSI/ ASME and ISO Models for Calculation of Uncertainty.” ISA Transactions, 33: 339–352; 1994. Strike, W. T., and R. H. Dieck. “Rocket Impulse Uncertainty.” Proceedings of the 41st ISA International Instrumentation Symposium, Denver, 1995. Coleman, H. W., and W. G. Steele. “Engineering Application of Experimental Uncertainty Analysis.” AIAA Journal, 33: 1888–1896; October 1995.

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Section 12 Bibliography [1]

[2]

[3]

[4]

[5] [6] [7]

Benedict, R. P. Fundamentals of Temperature, Pressure, and Flow Measurements. 3rd. edition. New York: John Wiley and Sons, 1984. ASME MFC-2M, Measurement Uncertainty for Fluid Flow in Closed Conduits. New York: The American Society of Mechanical Engineers; 1983. Hayward, A. T. J. “Repeatability and Accuracy.” Mechanical Engineering Publication, Ltd., 1977. Draper, N. R., and H. Smith. Applied Regression Analysis. 2nd. edition. New York: John Wiley and Sons; 1981. Williams, E. J. Regression Analysis. New York: John Wiley and Sons; 1959. Natrella, M. G. “Experimental Statistics.” National Bureau of Standards Handbook 91, 1963. Benedict, R. P. “Engineering Analysis of Experimental Data.” Transactions of the ASME:

[8]

[9]

[10]

[11]

[12]

Journal of Engineering for Power. New York: The American Society of Mechanical Engineers; January 1969, p. 21. Berkson, J. “Estimation of Linear Function for a Calibration Line.” Technometrics, 11: (4); November 1969. Mandel, J. “Fitting Straight Lines When Both Variables are Subject to Error.” Journal of Quality Technology, 15: (1); January 1984. AGARD Report AG-237. “Guide to In-flight Thrust Measurement of Turbojets and Fan Engines.” Davies, O. L. Design and Analysis of Industrial Experiments. 2nd edition, New York: Hafner, 1967. Hald, A. Statistical Theory With Engineering Applications. New York: John Wiley and Sons, 1952.

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Nonmandatory Appendix A Statistical Considerations A-1 UNDERSTANDING STATISTICAL INTERVALS

is no more difficult. Table A.1 lists the information needed to construct all three intervals.

It is often desirable to use the terms confidence, tolerance, and prediction interval. (The use of tolerance in this instance is different from that of PTC 1.) An eloquent treatment of these terms is given by Gerald Hahn [24] in his 1970 paper, which is reproduced here by permission. Statistical intervals are frequently misunderstood and misused. Here is an explanation of when to use confidence, tolerance, and prediction intervals. Engineers have come to appreciate that few things in life are known exactly. The most they can do is obtain an estimate and construct an interval which, with a high probability, contains the quantity of interest. This article describes three different types of statistical intervals and shows where each should be used. The three intervals are: (1) a confidence interval to contain a population mean, (2) a tolerance interval to contain a specified proportion of the population, and (3) a prediction interval to contain all of a specified number of future observations. Many nonstatistical users of statistics are well acquainted with confidence intervals. Some are also aware of tolerance intervals, but most nonstatisticians know very little about prediction intervals despite their practical importance. A frequent mistake is to calculate a confidence interval on the population mean when the actual problem calls for a tolerance interval or a prediction interval. At other times, a tolerance interval is used when a prediction interval is needed. This confusion is understandable since most texts on statistics devote extensive space to confidence intervals on population parameters, make limited reference to tolerance intervals, and almost never talk about prediction intervals. This is unfortunate because tolerance intervals or prediction intervals are needed as frequently in industrial applications as confidence intervals, and given the required tabulations, the procedure for constructing them

CONFIDENCE INTERVAL FOR THE POPULATION MEAN The sample mean y is an estimate of the unknown mean ␮, but differs from it because of sampling fluctuations. However, it is possible to construct a statistical interval known as a confidence interval for the population mean ␮. This interval contains ␮ with a specific probability. This probability is known as the associated confidence level. Thus a 95 percent confidence interval on the population mean is an interval which contains ␮ with a probability of 0.95. It is calculated as: y ± c M (n)s,

where cM (n) is obtained from the first column of Table A-1.1 as a function of n, the sample size. For the example shown in the box cM (5) p 1.24 and the 95 percent confidence interval for ␮ is: 50.10 ± (1.24)(1.31).

Consequently, one can be 95 percent confident that the interval 48.48 to 51.72 contains the unknown value of ␮. More precisely, over a large number of samples, the interval calculated in this manner will contain the unknown mean 95 percent of the time.

TOLERANCE INTERVAL TO CONTAIN A SPECIFIC PROPORTION OF THE POPULATION. Instead of, or in addition to, a confidence interval to contain ␮, many applications require an interval to enclose a specific proportion of the population. For a normal distribution, if ␮ and ␴ are known exactly, it can be stated that 90 percent of the population is located in the interval. 73

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Table A-1.1 Factors for Calculating the Two-Sided 95% Probability Intervals for A Normal Distribution Number of Given Observations

Factors for Confidence Interval to Contain the Population Mean ␮

Factors for Tolerance Interval to Contain at Least 90%, 95% and 99% of the Population

n

cM (n)

cT,90 (n)

cT,95 (n)

cT,99 (n)

cP,1 (n)

cP,2 (n)

cP,5 (n)

cP,10 (n)

cP,20 (n)

4 5 6 7 8 9 10 11 12 15 20 25 30 40 60 ⬁

1.59 1.24 1.05 0.92 0.84 0.77 0.72 0.67 0.64 0.55 0.47 0.41 0.37 0.32 0.26 0

5.37 4.28 3.71 3.37 3.14 2.97 2.84 2.74 2.66 2.48 2.31 2.21 2.14 2.05 1.96 1.64

6.37 5.08 4.41 4.01 3.73 3.53 3.38 3.26 3.16 2.95 2.75 2.63 2.55 2.45 2.33 1.96

8.30 6.63 5.78 5.25 4.89 4.63 4.43 4.28 4.15 3.88 3.62 3.46 3.35 3.21 3.07 2.58

3.56 3.04 2.78 2.62 2.51 2.43 2.37 2.33 2.29 2.22 2.14 2.10 2.08 2.05 2.02 1.96

4.41 3.70 3.33 3.11 2.97 2.86 2.79 2.72 2.68 2.57 2.48 2.43 2.39 2.35 2.31 2.24

5.56 4.58 4.08 3.77 3.57 3.43 3.32 3.24 3.17 3.03 2.90 2.83 2.78 2.73 2.67 2.57

6.41 5.23 4.63 4.26 4.02 3.85 3.72 3.62 3.53 3.36 3.21 3.12 3.06 2.99 2.93 2.80

7.21 5.85 5.16 4.74 4.46 4.26 4.10 3.98 3.89 3.69 3.50 3.40 3.33 3.25 3.17 3.02

Factors for Prediction Interval to Contain the Values of All of 1, 2, 5, 10, and 20 Future Observations

A two-sided 95 percent interval is y ± c(n)s, where c(n) is the appropriate tabulated value and y and s are the mean and the standard deviation of the given sample of size n.

The fact that both a population proportion (or percentage) and a statistical probability (also a percentage) are associated with a tolerance interval is sometimes confusing to the engineer. The first of these numbers refers to the proportion (or percentage) of the population that the interval is to contain. The second number specifies the probability that the calculated interval really contains at least the specified proportion of the population. When ␮ and ␴ are known exactly, an interval to contain a specified proportion of the population may still be of interest, but, in this case, there is no longer any uncertainty associated with the proportion of the population contained in the interval.

␮ ± 1.64 ␴

However, if only sample estimates y and s of the population values ␮ and ␴ are given, the best that can be stated is that with a chosen probability (say 0.95) the interval contains at least 90, 95, or 99 percent of the population. Such an interval is called a tolerance interval and can be calculated for a normal population with the help of the factors cT,90(n), cT,95(n), and cT,99(n), shown in columns 2, 3, and 4 of Table A-1.1. For example, it can be stated with 95 percent confidence that the interval: y ± c T,90 (n)s

contains at least 90 percent of a normal population The tolerance interval for the example in the box may be calculated as:

PREDICTION INTERVAL TO CONTAIN ALL OF A SPECIFIED NUMBER OF FUTURE OBSERVATIONS. Another type of interval is one that will contain all the values of one or more future observations. This is known as a prediction interval. The last five columns of Table A-1.1 provide values of the factor cP,k(n) such that all of k future observations from the same normal population will be located in the interval:

50.10 ± (4.28)(1.31)

or 44.49 to 55.71 where cT,90(n)s p 4.28. Thus, one may be 95 percent confident that the preceding interval contains at least 90 percent of the sampled population. 74 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

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Fig. A-1 How the Lengths of the Statistical Intervals for the Example Compare

interval approaches zero as the sample size increases (the interval converging to the point ␮).

y ± c P,k (n)s,

with a probability of 0.95. For example, if two additional readings are taken from the example in the box, k p 2 and n p 5. From Table A-1.1 the factor cP,2(5) p 3.70. Thus two future units from the sampled population will be located in the interval:

HOW TO SELECT THE RIGHT INTERVAL. The statistician’s job is to develop correct procedures for answering relevant questions. The engineer must decide upon the relevant questions. Once the questions to be answered have been clearly stated, it should be easy to decide upon the correct intervals. The following comments are offered to serve as a guide to the engineer in this process. The mean is the most commonly used single value to describe a population. For the normal distribution, the mean ␮ is one of the two parameters which uniquely defines the distribution. It is identical to the median (50 percent point) and mode (most common value) of the distribution. The population mean is therefore of great interest in characterizing product performance, and is often used as a standard by which competing processes are compared. Its use for such comparisons is especially appropriate when it is reasonable to assume that each of the competing processes has the same statistical variability (as measured by the process standard deviation) and, therefore, the differences between processes can be described

50.10 ± (3.70)(1.31)

or 45.25 to 54.95, with a probability of 0.95. The relative lengths of the three intervals obtained in the preceding examples are compared in Fig. A-1. It is seen that for the given sample of 5, the confidence interval to contain the population mean is appreciably smaller than both the tolerance interval and the prediction interval. Also a tolerance interval to include at least 90 percent of the population with a probability of 0.95 is somewhat larger than a prediction interval to contain both of two future observations. Inspection of the tabulations indicates that a confidence interval on the mean is always smaller than the other two intervals, but that the relative sizes of the tolerance and prediction intervals depend upon the proportion of the population to be contained in the prediction interval. Also, unlike the other two intervals, the length of a confidence 75 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

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completely by differences in their means. The assumption of equal standard deviations is frequently made. Because of random fluctuations, a sample does not provide perfect information about the population mean ␮. Thus, a confidence interval is established which contains the unknown value of ␮ with a specified degree of confidence. If, instead of characterizing typical process performance, you are interested in estimating the range of variation of the underlying population or of the observations in a future sample, then a tolerance interval or a prediction interval is needed. Specifically, a tolerance interval is applicable if limits are needed that contain most of the sampled population, while a prediction interval would be used to obtain limits to contain all of a small number of future units from the population. Thus, an engineer who is concerned with the performance of a mass-produced item, such as a transistor or a lamp, would generally be interested in a tolerance interval to enclose a high proportion of the sampled population. In contrast, a prediction interval to contain all of k future observations may be thought of as the astronaut’s interval. A typical astronaut, who has been assigned to a specific number of flights, is generally not very interested in what will happen on the average in the population of all space flights, of which his happen to be a random sample (confidence interval on the mean), or even what will happen in at least 99 percent of such flights (tolerance interval). His main concern is the worst that will happen in the one, three, or five flights in which he will be personally involved. Similarly, a turbine engineer who is bidding on an order of three units based upon his past experience on five units of the same type, would use a prediction interval to obtain specification limits to contain the performance parameter for all three units with a high probability. Prediction intervals are also required by the typical customer who purchases one or a small number of units of a given product and is concerned with predicting the performance of the particular units he has purchased (in contrast to the long-run performance of the process from which the sample has been selected).

discuss tolerance intervals, but make no mention of prediction intervals except in a regression context. Such intervals, however, are discussed in References 1, 2, 3, and 5. Further, Reference 3 provides a comprehensive comparison of statistical intervals for a normal population (including more detailed tabulations than are given here) and a discussion of methods for constructing the various intervals. This article also considers additional types of statistical intervals such as: A prediction interval to contain a future sample mean, A prediction interval to contain a future sample standard deviation, A confidence interval for the population standard deviation, A confidence interval for a population percentile. Finally, a new time-sharing computer program calculates a wide variety of statistical intervals, including confidence, tolerance, and prediction intervals, Reference 6. THE EXAMPLE PROBLEM The calculation of the three intervals are illustrated here by the following numerical example. Assume that readings obtained on a normally distributed performance parameter based on a random sample of five units are: 51.4, 49.5, 48.7, 49.3, and 51.6. From this information, the sample mean y¯ and the sample standard deviation s are calculated by well-known expressions: n

yp

兺 yi /n p (51.4 + . . . + 51.6)/5 p 50.10 ip 1

冤兺 n

sp

p

(yi − y¯)2

ip 1

冦

n−1

冥

1⁄

2

关(51.4 − 50.10) 2 + . . . + (51.6 − 50.10)2兴

冧

2

p 1.31,

where y1 , . . ., yn are the values of n given observations.

REFERENCES 1. Hahn, G. J., “Additional Factors for Calculating Prediction Intervals for Samples from a Normal Distribution.” Journal of the American Statistical Association, 65, December, 1970. 2. Hahn, G. J., “Factors for Calculating Two-Sided Prediction Intervals for Samples from a Normal Distribution.” Journal of the American Statistical Association, 64, September, 1969.

WHERE TO GET MORE INFORMATION. Standard books on elementary engineering statistics, Reference 4, give prime space to the concept of confidence intervals and, in many cases, also 76 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

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3. Hahn, G. J., “Statistical Intervals for a Normal Population,” Journal of Quality Technology, Volume 2, Number 3, pages 115–125, July 1970; Volume 2, Number 4, pages 195–206, October 1970. 4. Natrella, Mary Gibbons, Experimental Statistics, National Bureau of Standards Handbook 91, US Government Printing Office. 5. Nelson, W. B., “Two-Sample Prediction,” General Electric Company TIS Report 68-C-404, November 1968. (Available from Distribution Unit, PO Box 43, Building 5, Room 237, Schenectady, New York 12305). 6. “Summary Statistics Package — ONE-SAMS***” Document 003401, General Electric Information Service Department, 7735 Old Georgetown Road, Bethesda, Maryland.

where Wi are the required weighting factors. A weighting principle which is statistically valid and is based on weighting by variances is applicable in this case [5]. This is true since the systematic uncertainty component of a parameter BXi is assumed to equal two times the standard deviation (square root of the variance) of the possible distribution of systematic uncertainty. The variance of this distribution is (BXi/2)2. This is combined with (SXi)2, which is the variance of the average measurement. Therefore, UXi is a combination of variances:

A-2 WEIGHTING METHOD

Therefore,

N

U Xi p 2

Whenever the value of a test result is determined by several independent methods, then the test result and its associated uncertainty may be determined by weighting the means, random uncertainties, and systematic uncertainties of the various methods. The advantages of utilizing the weighting techniques described in this section are that the uncertainty associated with a weighted test result will usually be less than the uncertainty of the results determined from each of the independent methods by themselves. Prior to using the weighting techniques described in this section, the means and their associated uncertainty intervals from each of the independent methods should be compared as discussed in subsection 8-5. The weighting techniques described in this section should not be employed if the uncertainty intervals do not overlap to a significant degree as this is a possible indication of unaccounted for uncertainties. The weighting methods presented in this section are based on the following assumptions: (a) The various methods used to determine the test result are independent to the extent that there is no appreciable correlation between the sources of uncertainty of the various methods. (b) The assumptions presented in subsection 1-3 are valid for each of the independent methods such that the uncertainty model presented in subsection 1-3 may be used. Let Xi represent best estimates of a parameter by N measurement methods. Then X, the weighted mean of the measurements, can be given by

BXi

2

冤冢 2 冣

+ (S X i) 2

冢 冣 1 兺 冢U 冣 N

ip1

冥

1⁄

2

2

1 U Xi

Wi p

(A-1)

2

Xi

where UXi represents the uncertainties of X i. For two measurement methods with the means X1 and X2, eq. (A-1) yields

共 U X 2兲 2

W1 p

共 U X 1兲 2 + 共 U X 2兲 2

and

W2 p

共 U X 1兲 2 共 U X 1兲 2 + 共 U X 2兲 2

p 1 − W1

Using these same weighting factors, the systematic and random uncertainties of the weighted mean are given by the root-sum-square relations N

BX p

兺 共 W iBX 兲 冥 冤 ip1

SX p

兺 共 W iSX 兲2 冥 冤 ip1

1⁄

2

2

i

N

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X p

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1⁄

2

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These values are combined as UX to obtain the weighted uncertainty of X according to eq. (4-3.5) as follows: U X p 2 关共B X /2兲 2 + 共S X 兲 2兴

1⁄

All data should be inspected for spurious data points as a continuing check on the measurement process. To ease the burden of scanning large masses of data, computerized routines are available to scan steady state data and flag suspected outliers. The suspected outliers should then be subjected to an engineering analysis. The effect of outliers is to increase the standard deviation of the system. Tests are available to determine if a particular point from a sample is an outlier. In most of these tests, the probability for rejecting a good point is set at 5%. This means that the odds against rejecting a good point are 20 to 1 (or less). The odds could be increased by setting the probability of rejecting a good data point lower. However, this practice decreases the probability of rejecting bad data points. For small samples, bad data points are hard to identify. Two tests are in common usage for determining whether or not spurious data are outliers. These are the Thompson ␶ Technique [25] and the Grubbs Method [26]. The Thompson ␶ Technique is excellent for rejecting outliers, but also may reject some

2

As usual, X, SX , BX , and UX should be reported.

A-3 OUTLIER TREATMENT A-3.1 General All measurement systems may produce spurious data points. These points may be caused by temporary or intermittent malfunctions of the measurement system. Errors of this type should not be included as part of the uncertainty of the measurement. Such points are considered to be meaningless as steady-state test data, and should be discarded. Figure A-3.1 shows a spurious data point called an outlier.

Fig. A-3.1 Outlier Outside the Range of Acceptable Data 78 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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probe readings were average ␦ and ␦ was calculated from the average for each probe.

good values. The Grubbs Method does not reject as many outliers but the number of good points rejected is smaller. In this Supplement, the Modified Thompson ␶ Technique1 described below is recommended for identifying suspected outliers. The suspected outliers should then be subjected to an engineering evaluation to determine the cause of the outliers. The engineering evaluation should include an analysis of the instrumentation, the physics of the measurement methods employed, the expected temporal and spatial variation/profiles of the parameters being measured, data from similar tests, etc. If there is valid engineering justification, the suspected outliers may be removed from the analysis of the test result and its associated uncertainty. The removal of these outliers should be documented within the test report. If there is not valid engineering justification to remove the suspected outliers and if removal of the outliers will significantly change the test result and its associated uncertainty, the validity of the test should be questioned.

26 −11 148 −107 126

79 −137 −52 20 −72

58 120 −216 9 179

1 129 −56 40 127

−103 −38 89 2 −35

−121 25 8 10 334

−220 −60 −29 166 −555

334 and −555 are suspected outliers.

To illustrate the calculations for determining whether −555 is an outlier, the following steps are taken. Mean (X) p 1.125 Sample standard deviation (SX) p 140.8 Sample size (N) p 40 By the above equation for ␦, ␦ p |−55 − 1.125| p 556.125

From Table A-3.1, ␶ SX p 1.924 ⴛ 140.8 p 270.9

A-3.2 Thompson ␶ Technique (Modified)1

Since ␦ > ␶ SX, we conclude that −555 is a possible outlier. Repeating the above procedure for 334,

Consider a sample (Xi) of N measurements. The sample standard deviation (SX) and the mean (X) of the sample are calculated. Suppose Xj, the jth observation, is the suspected outlier. Then, the absolute difference of Xj from the mean (X) is calculated as

␦ p 冨334 − 1.125冨 p 332.875

Since ␦ exceeds ␶ SX for the suspected point, 334, we conclude that 334 also is a possible outlier. This procedure should be repeated for all remaining data points.

␦ p |X j − X|

Using Table A-3.1, a value of ␶ is obtained for the sample size (N) at the 5% significance level. This limits the probability of rejecting a good point to 5%. (The probability of not rejecting a bad data point is not fixed. It will vary as a function of sample size.) The test for the outlier is to compare the difference (␦) with the product ␶ SX. If ␦ is larger than or equal to ␶ SX, we say Xj is an outlier. If ␦ is smaller than ␶ SX, we say Xj is not an outlier.

A-4 PARETO DIAGRAMS A-4.1 General It is often useful to display the relative sizes of the components of a whole with a bar chart. One particular type of bar chart is called a Pareto diagram after Vilfredo Pareto, an Italian economist who used this type of diagram in his studies of the unequal distribution of wealth. Most of the uses today, with extensive activity in the area of quality control, are attributed to Joseph Juran who defined the general principle known as the “Pareto Principle” — the “Vital Few, Trivial Many.” Mathematics were developed that described the distribution, but for the purposes illustrated here, the diagram can be defined as a bar chart with the bars arranged in descending order of size.

A-3.3 Example There were 40 temperature probes installed in one stage of the turbine of a jet engine. The 40 1

Thompson used a different equation for S. The Modified Thompson ␶ Technique uses the equation for S as defined in this Supplement.

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Table A-3.1 Modified Thompson ␶ (At the 5% Significance Level) N

␶

N

␶

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

1.151 1.425 1.571 1.656 1.711 1.749 1.777 1.798 1.815 1.829 1.840 1.850 1.858 1.865 1.871 1.876 1.881 1.885 1.889 1.893 1.896 1.899 1.901 1.904 1.906 1.908 1.910 1.911 1.913 1.915 1.916 1.917 1.919 1.920 1.921 1.922 1.923 1.924 1.925 1.926 1.927 1.927 1.928 1.929 1.929 1.930 1.931 1.931 1.932

52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 ⬁

1.932 1.933 1.934 1.934 1.935 1.935 1.935 1.936 1.936 1.937 1.937 1.937 1.938 1.938 1.938 1.939 1.939 1.939 1.940 1.940 1.940 1.941 1.941 1.941 1.941 1.942 1.942 1.942 1.942 1.942 1.943 1.943 1.943 1.943 1.944 1.944 1.944 1.944 1.944 1.944 1.945 1.945 1.945 1.945 1.945 1.945 1.946 1.946 1.946 1.96

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ASME PTC 19.1-2005

To apply this to a test uncertainty example, the first step is to define the individual systematic and random standard uncertainties of the mean in terms of their relative individual percentage contributions to the combined standard uncertainty of a test result, uR. The second step is to create a bar chart which depicts the percentage contributions of individual systematic and random standard uncertainties to the combined standard uncertainty in descending order of size. As elemental sources of standard uncertainty are not combined as arithmetic sums but are instead combined as described in subsection 4-8, derivation of the percentage contribution of an elemental source of standard uncertainty to the combined uncertainty is computed as the ratio of the square of the combined standard uncertainty that would be computed if the elemental source were the only source of standard uncertainty to the square of the combined standard uncertainty computed when accounting for all sources of uncertainty. Using the uncertainty model and associated assumptions presented in subsection 4-8, expressions for the percentage contributions from elemental sources of standard uncertainty were derived and are presented below. The percentage contribution of each elemental source of random standard uncertainty (sXi ) to the combined standard uncertainty (uR) is

共 ␪ i s X i兲 2 u 2R

of Pareto diagrams. The necessary values for parameter sensitivity, systematic standard uncertainties, random standard uncertainties, and combined standard uncertainty are summarized in the following table. (Note that these values were obtained from subsection 10-4). Symbol bP 1 bP 2 bT 1 bT 2 bT T 1 2 sP 1 sP 2 sT 1 sT 2 ␪P1 ␪P2 ␪T1 ␪T2 uR

u 2R

b P1 % contribution ~ to uR p p

u2R

ⴛ 100

(−0.0409 * 0.021)2 ⴛ 100 (0.00386)2

Similarly: ⴛ 100

b P2 p 10.4%, b T1 p 37.7%, b T2 p 17.3%

For bT1T2: b T1 T2 % contribution ~ to uR p

2

ⴛ 100

The percentage contribution of each correlated source of systematic standard uncertainty (bX1X2), to the combined standard uncertainty is 2 ␪ 1 ␪ 2 b X1 X 2

冤

2 ␪T1 ␪T2 bT1 T2 u 2R

2(0.00367)(−0.00203)(0.0625) (0.00386) 2

ⴛ 100

冥 ⴛ 100

p −6.3%

for sP1: ⴛ 100 s P1 % contribution ~ to u R p (−0.0409 * 0.030)2

A-4.2 Example

p

The compressor performance example in subsection 10-4 will be used to illustrate the application

(0.00386)2 p 10.1%

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共␪P1bP1兲2

p 5.0%

p

u 2R

0.021 0.198 0.646 0.792 0.0625 0.030 0.170 0.300 0.600 −0.0409 0.00629 0.00367 −0.00203 0.00386

For b P1:

The percentage contribution of each elemental source of systematic standard uncertainty (bXi ) to the combined standard uncertainty is

共 ␪ i b X i兲

Value

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共 ␪ P 1 s P 1兲 2 u 2R ⴛ 100

ⴛ 100

ASME PTC 19.1-2005

TEST UNCERTAINTY

Similarly:

As can be seen from this diagram, the combination of bT1, bT2, and bT1T2 is the largest contributor to the combined standard uncertainty, uR, and bP1 the smallest. The ideal end result of an analysis such as this would be to take corrective action, if possible, to reduce the contribution of major factors in the combined standard uncertainty through changes in methods, instrumentation, or both. Although this application of Pareto diagrams has been used to determine the relative contributions to combined standard uncertainty of systematic and random standard uncertainties, the method can be applied just as easily to the individual estimates of the elemental errors that contribute to systematic and random standard uncertainties.

s P2 p 7.7%, s T1 p 8.1%, sT2 p 10.0%

Figure A-4.2.1 illustrates the relative contributions to combined standard uncertainty of individual-parameter systematic and random standard uncertainties in terms of a Pareto chart. However, since the systematic standard uncertainties for T1 and T2 are correlated, the relative percentages for bT1, bT2, and bT1T2 should be combined as shown in Fig. A-4.2-2.

Fig. A-4.2.1 Pareto Chart for Random and Systematic Standard Uncertainties

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ASME PTC 19.1-2005

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Fig. A-4.2.2 Pareto Chart for Random and Some Combined Systematic Standard Uncertainties (Example 10-4)

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Nonmandatory Appendix B Uncertainty Analysis Models Ki

B-1 ISO UNCERTAINTY ANALYSIS MODEL Nonmandatory Appendix B is adapted from “Computer-Assisted Uncertainty Analysis” [27]. The uncertainty model given in the ISO Guide to the Expression of Uncertainty in Measurement (1995)[1] is presented in this paragraph. The uncertainty model requires estimates of the uncertainties for each of the elemental error sources for each parameter in the data reduction equation. These estimates are combined through use of the ISO model to calculate the band about the experimental result where the true result is thought to lie with C% confidence. The estimates of the elemental errors fall into two categories: systematic standard uncertainties, bik for the systematic errors, and random standard uncertainties of the mean, si for the random errors where i represents each parameter. The systematic standard uncertainties are related to those systematic errors, that remain after all calibration corrections are made. Systematic uncertainties can be estimated through manufacturer information, calibrations, and, in most cases, through sound engineering judgment. There will usually be a set, Ki, of elemental systematic standard uncertainties for each parameter, i. As discussed in para. 4-3.2, each elemental estimate, bik, is taken to be the prediction of the standard deviation for a particular distribution of possible errors for that particular error source. Typically, these error distributions are assumed Gaussian (normally distributed) or rectangular (uniformly distributed). For most engineering predictions of systematic uncertainty, the 95% limits of the possible distribution are estimated rather than the standard deviation of the distribution. Obtaining the standard uncertainty from the 95% estimate is simply a matter of dividing the estimate by the appropriate distribution factor, D [i.e., 2.0 for Gaussian, 1.65 p (1.73)(0.95) for rectangular]. The systematic standard uncertainty for parameter i, bi, is determined from the estimates for the Ki elemental error sources for that parameter as

kp1

b2ik

(B-1.1)

The estimate of the random error for a parameter is the random standard uncertainty of the mean, or the estimate of the error associated with repeated measurements of a particular parameter. The random standard uncertainty of the mean for each parameter is determined from Ni measurements as si p

1

冪 Ni

冤

Ni

1 Ni − 1

兺 (Xi jp1

j

− Xi)

2

冥

1⁄

2

(B-1.2)

where Ni

Xi p

兺 Xi jp1

j

(B-1.3)

Ni

For the case of a single measurement (Ni p 1), previous information must be used to calculate si [4]. Consider an experimental result that is determined from I measured variables as R p R 共 X1, X 2, . . ., X I兲

(B-1.4)

The ISO Guide defines the combined standard uncertainty of the result as I

uR2 p

I−1

I

兺 (␪i bi)2 + 2 ip1 兺 kpi+1 兺 ␪i ␪k bik ip1 I

+

兺

ip1

(B-1.5)

( ␪ i si ) 2

where ␪i p

∂R ∂X i

(B-1.6)

The first two terms on the right side of eq. (B-1.5) represent the systematic standard uncertainty of the result bR [eq. (7-4.1) including the correlated 84

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兺

bi2 p

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ASME PTC 19.1-2005

Table B-1 Values for Two-Sided Confidence Interval Student’s t Distribution [9]

terms], and the third term is the random standard uncertainty of the result sR eq. (7-3.1). The covariance of the random errors is assumed to be zero. The covariance of the systematic errors, or the correlated systematic standard uncertainty, bik, is determined by summing the products of the elemental systematic standard uncertainties for parameters i and k that arise from the same source and are therefore perfectly correlated [10] (see subsection 8-1). In order to obtain the overall uncertainty in the result, UR, at a specified confidence level, the ISO Guide recommends that the combined standard uncertainty of the result be multiplied by a coverage factor. The coverage factor is the value from the t distribution for the required confidence level corresponding to the effective degrees of freedom in the result, ␯R. The values for t are given in Table B-1. To find ␯R, the Welch-Satterthwaite formula is adapted as:

冦兺 兺冤 I

ip1

␯R p

I

ip1

关共␪i bi 兲 + 共␪isi兲 兴 2

共 ␪ i s i兲

K

4

␯ si

2

+

kp1

␯ bi

k

冥

␯ bi p 1⁄ 2 k

⌬ bik

冢b 冣

0.995

0.999

63.657 9.925 5.841 4.604 4.032

127.321 14.089 7.453 5.598 4.773

636.619 31.598 12.924 8.610 6.869

6 7 8 9 10

1.943 1.895 1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

3.707 3.499 3.355 3.250 3.169

4.317 4.029 3.833 3.690 3.581

5.959 5.408 5.041 4.781 4.587

11 12 13 14 15

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

3.106 3.055 3.012 2.977 2.947

3.497 3.428 3.372 3.326 3.286

4.437 4.318 4.221 4.140 4.073

16 17 18 19 20

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.921 2.898 2.878 2.861 2.845

3.252 3.223 3.197 3.174 3.153

4.015 3.965 3.922 3.883 3.850

21 22 23 24 25

1.721 1.717 1.714 1.711 1.708

2.080 2.074 2.069 2.064 2.060

2.831 2.819 2.807 2.797 2.787

3.135 3.119 3.104 3.090 3.078

3.819 3.792 3.768 3.745 3.725

26 27 28 29 30

1.706 1.703 1.701 1.699 1.697

2.056 2.052 2.048 2.045 2.042

2.779 2.771 2.763 2.756 2.750

3.067 3.057 3.047 3.038 3.030

3.707 3.690 3.674 3.659 3.646

40 60 120 ⬁

1.684 1.671 1.658 1.645

2.021 2.000 1.980 1.960

2.704 2.660 2.617 2.576

2.971 2.915 2.860 2.807

3.551 3.460 3.373 3.291

(B-1.8)

GENERAL NOTES: (a) See [9]. (b) Given are the values of t for a confidence level C and number of degrees of freedom ␯.

−2

(B-1.9)

ik

where the quantity in parentheses is an estimate of the relative variability of the estimate of bik. For instance, if one thought that the estimate of bik was reliable to within ±25%, then ␯bi p 1⁄ 2 (0.25) −2 p 8

With ␯R known, the proper t value is obtained from Table B-1 for C% confidence and multiplied by uR from eq. (B-1.5) to obtain the overall uncertainty in the result, UR, at a C% confidence level I

UR,C p tC I−1

(B-1.10)

+2

k

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0.990

12.706 4.303 3.182 2.776 2.571

(B-1.7)

or the degrees of freedom of the previous information if si is estimated [4]. The degrees of freedom of the elemental systematic standard uncertainties ␯bi may be known from previous information or k estimated. The ISO Guide recommends the approximation

0.950

6.314 2.920 2.353 2.132 2.015

where ␯si is either ␯ si p N i − 1

0.900

1 2 3 4 5

2

共 ␪ i bik兲 4

i

兺

冧

C ␯

I

冤 兺 (␪ b ) ip1

i i

I

2

兺 兺 ␪i ␪k bik + ip1 兺 ( ␪ i s i ) 2冥 ip1 kpi+1

(B-1.11) 1⁄

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2

ASME PTC 19.1-2005

TEST UNCERTAINTY

B-2 LARGE SAMPLE UNCERTAINTY ANALYSIS APPROXIMATION

The first two terms in the brackets in eq. (B-2.1) are the systematic standard uncertainty of the result bR [eq. (7-4.1) including the correlated systematic standard uncertainty terms], and the third term in the brackets is the random standard uncertainty of the result sR eq. (7-3.1). The large sample uncertainty expression given in eq. (7-5.1) is then obtained as (including the correlated systematic uncertainty terms)

The method described in subsection B-1 is the strict ISO method. It has been shown [28-30] that for most engineering applications, when the degrees of freedom for the result from eq. (B-1.7) is 9 or greater, t95 can be taken as 2 to a good approximation (93% to 95% coverage). Therefore, for large degrees of freedom in the result I

U R,95 p 2 I−1

+2

I

兺 兺 ip1 kpi+1

冤 兺 (␪ b ) ip1

␪ i ␪k b ik +

i i

I

兺 ip1

2

(␪i si )2

(B-2.1)

冥

1⁄

2

U R,95 p 2 共b 2R + s 2R兲

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1⁄

2

(B-2.2)

ASME PTC 19.1-2005

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Nonmandatory Appendix C Propagation of Uncertainty Through Taylor Series

C-1 INTRODUCTION E[X] p

Experimental results are not always directly measured. It is quite common for an experimental result, r(X1 , X2 , . . ., Xn ), to be defined as a function of certain variables, X1 , X2 , . . ., Xn , that are directly measured. The aim of this Appendix is to provide a method by which the variance of an experimental result that is not directly measured, r(X1 , X2 , . . ., Xn ), can be expressed in terms of the variances and covariances of its arguments, X1 , X2 , . . ., Xn , which are directly measured. The approach will be to relate the deviations in r(X1 , X2 , . . ., Xn ) to deviations in (X1 , X2 , . . ., Xn ) by means of a first order approximation to the Taylor series expansion for r(x1 , x2 , . . ., xn ) in the neighborhood of the point (␮X1 , ␮X2 , . . ., ␮Xn ), where ␮Xi is the true value of the measured variable Xi . In order to facilitate this project, the function r(x1 , x2 , . . ., xn ) will be assumed to be continuous with continuous partial derivatives in the neighborhood of the point (␮X1 , ␮X2 , . . ., ␮Xn ).

E[(X − ␮X ) 2] p

冕

-⬁

冕

x p(x) dx p ␮ X

(x − ␮ X ) 2 p(x) dx p ␴ X2

冕冕 ⬁

⬁

-⬁

-⬁

...

⬁

-⬁

f(x1 , x 2 , . . ., x n )

p(x 1 , x2 , . . ., xn ) dx1 dx 2 . . . dx n

where p(x1 , x2 , . . ., xn ) is the joint probability density function for X1 , X2 , . . ., Xn . The covariance ␴X1X2 of the random variables X1 , X2 presents the special case where f(X1 , X2 ) p (X1 − ␮X1 )(X2 − ␮X2 ), i.e.,

E[f(X 1 , X 2 )] p

冕冕 ⬁

⬁

-⬁

-⬁

(x1 − ␮ X1 )

(x2 − ␮ X2 ) p(x1 , x 2 ) dx 1 dx 2 p ␴ X1X2

C-3 PRELIMINARY CONSIDERATIONS If any random variable, Y, can be expressed as a linear combination of random variables, Xi , then the mean and variance of Y can be expressed in terms of the means, variances, and covariances of the variables Xi . Suppose

f(x) p(x) dx

where p(x) is the probability density function for X. The mean (or expected value) ␮X of the random variable X presents the special case where f(X) p X, i.e.,

Y p a 0 + a1 X 1 + a2 X 2 + . . . + a n X n 87

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⬁

E[f(X 1 , X 2 , . . ., X n )] p

⬁

-⬁

-⬁

The expected value of a function f(X1 , X2 , . . ., Xn ) of the random variables X1 , X2 , . . ., Xn is given by

The primary goal of Nonmandatory Appendix C is to present an expression for the variance in r in terms of the variances and covariances of X1 , X2 , . . ., Xn . The definitions below for “mean,” “variance,” and “covariance” will be used throughout this Appendix. The expected value of a function f(X) of a random variable X is given by

冕

⬁

The variance ␴ 2X of the random variable X presents the special case where f(X) p (X − ␮X )2, i.e.,

C-2 DEFINITIONS

E[f(X)] p

冕

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Then

using the results in para. C-3 we will be able to express the mean and variance of r(X1 , X2 , . . ., Xn ) in terms of the means, variances, and covariances of the variables X1 , X2 , . . ., Xn .

E[Y] p E[a 0 + a 1 X1 + a 2 X2 + . . . + an X n ]

or, using the definition of the mean value of a random variable,

C-4.1 The First Order Approximation

␮ Y p a 0 + a1 ␮ X1 + a 2 ␮ X2 + . . . + a n ␮Xn

Suppose r p r(x1 , x2 , . . ., xn ). If we expand r through a Taylor series in the neighborhood of ␮X1 , ␮X2 , . . ., ␮Xn we get

The variance of Y will be given as follows: ␴ 2Y p E[(Y − ␮Y ) 2]

r(x 1 , x2 , . . ., xn ) p r( ␮ X1 , ␮ X2 , . . ., ␮ Xn ) + ␪ X1 (x 1

p E{[a 1 (X 1 − ␮ X1) + a 2 (X2 − ␮ X2 )

− ␮X1 ) + ␪X2 (x2 − ␮ X2 ) + . . . + ␪Xn (x n − ␮ Xn )

+ . . . + an (Xn − ␮ Xn ) ]} 2

+ higher order terms

For example, if n p 3, then where ␪xi are the sensitivity coefficients given by

␴ 2Y p E{[a 1 (X1 − ␮ X1 ) + a2 (X 2 − ␮ X2 ) + a 3 (X 3 − ␮X3 ) 2]}

␴ 2Y

p

E[a21 +

a23

(X1 − ␮ X1 ) + a 2 (X 2 − ␮ X2 ) 2

2

␪ Xi p

2

(X 3 − ␮ X3 )2

Now, suppose that the arguments of the function r(X1 , X2 , . . ., Xn ) are the random variables X1 , X2 , . . ., Xn . Furthermore, assume that the higher order terms in the Taylor series expansion for r are negligible compared to the first order terms. Then in the neighborhood of ␮X1 , ␮X2 , . . ., ␮Xn we have

+ 2a 1 a 2 (X 1 − ␮X1 )(X 2 − ␮X2 ) + 2a 1 a 3 (X 1 − ␮X1 )(X 3 − ␮X3 ) + 2a 2 a 3 (X 2 − ␮X2 )(X 3 − ␮X3 )]

or ␴ 2Y p a 12 ␴X12 + a22 ␴ X22 + a 32 ␴X32

r(X 1 , X2 , . . ., Xn ) ≈ r( ␮ X1 , ␮ X2 , . . ., ␮ Xn ) + ␪ X1 (X 1

+ 2a 1 a 2 ␴X1X2 + 2a 1 a 3 ␴ X1X3 + 2a 2 a3 ␴ X2X3

− ␮X1) + ␪X2 (X2 − ␮ X2 ) + . . . + ␪Xn (X n − ␮ Xn )

or more generally, the variance of Y will be given by ␴ 2Y p

n

Consequently ␮ r ≈ r( ␮X1 , ␮ X2 , . . ., ␮ Xn )

n

兺 ai2 ␴Xi2 + 2 jpi+1 兺 ai aj ␴XiXj 冣 ip1 冢

and

C-4 PROPAGATION OF UNCERTAINTY/ERROR THROUGH TAYLOR SERIES

␴ 2r ≈

In subsection C-3 we found that if a random variable, Y, could be expressed as a linear combination of the random variables, Xi , the mean and variance of Y can be expressed in terms of the means, variances, and covariances of the variables Xi . Now suppose an experimental result, r, is defined as a function of certain measured variables, X1 , X2 , . . ., Xn . If the function r(X1 , X2 , . . ., Xn ) can be expressed as a linear combination of the measured variables X1 , X2 , . . ., Xn , by means of a Taylor series approximation to r(X1 , X2 , . . ., Xn ) in the neighborhood of ␮X1 , ␮X2 , . . ., ␮Xn , then

n

n

兺 ␪Xi2 ␴Xi2 + 2 jpi+1 兺 ␪Xi ␪Xj ␴XiXj 冣 ip1 冢

For the special case where the random variables X1 , X2 , . . ., Xn are all independent we get ␴ 2r ≈

n

兺 ␪Xi2 ␴Xi2 ip1

Table C-4 presents some useful formulas for propagating variance through the first order approximation to the Taylor series for an experimental result r. 88 --`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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∂r ∂x i

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TEST UNCERTAINTY

ASME PTC 19.1-2005

Table C-4 Taylor Series Variance Propagation Formulas Variance (in Absolute Units) and Absolute Sensitivities

Function r p f(x,y)

2

S2r p

冢∂x S 冣 + 冢∂y S 冣

␪x p

∂r ∂r ;␪ p ∂x y ∂y

r p Ax + By

∂r

∂r

x

Variance (Dimensionless) and Relative Sensitivities

2

y

S2r p A2S2x + B2S2y

冢∂x/x V 冣 + 冢∂y/y V 冣

␪x′ p

∂r/r ∂r/r ;␪′p ∂x/x y ∂y/y

V2r p

␪x p A; ␪y p B

␪x′ p rp

1 y

x x+y

S2r p

␪x p

rp

x 1+x

S2r p

␪x p r p xy

␪y′ p −1

y2 xSy

2

V2r p

2

; ␪y p −

x

␪x′ p

(x + y)2

(1 + x)

S2r p

␪x′ p 1; ␪y′ p 1 V2r p 4V2x

␪x′ p 2

S2x 4x

V2r p

1

␪x p

V2x

V2r p V2x + V2y

␪x p 2x rpx

y y ;␪′p− x+y y x+y

(1 + x)2 1 ␪x′ p 1+x

2

S2r p 4x2S2x

1⁄ 2

(x + y)2

V2r p

(1 + x)4 1

␪x p y; ␪y p x rpx

y2(V2x + V2y )

S2x

S2r p (ySx )2 + (xSy )2

2

Ax By ;␪′p Ax + By y Ax + By V2r p V2y

2

(x + y)2

V2x 4

␪x′ p 1⁄ 2

1⁄ 2

2x r p ln x

S2r p

S2x

x2 1 ␪x p x

r p kxayb

GENERAL NOTE: Vx p

Sx x

冢ln x冣

␪x′ p

1 ln x

Vx

2

V2r p (aVx )2 + (bVy )2

␪x p akybxa−1; ␪y p bkxayb−1

␪x′ p a, ␪y′ p b

Sy y

Vr p

Sr r

89

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V2r p

S2r p (akybxa−1Sx )2 + (bkxayb−1Sy )2

Vy p

2

y

(Ax + By)2

1

2

y

∂r/r

x

A2x2V2x + B2y2V2y

y4

冤(x + y) 冥 + 冤(x + y) 冥 ySx

∂r/r

S2y

Sr2 p

␪y p −

rp

2

V2r p

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TEST UNCERTAINTY

C-4.2 Assessing the Validity of the First Order Approximation

C-4.3 The Limitation of the Present Approach If the higher order terms in the Taylor series expansion are not small compared to the first order terms, then there is no way to express the variance in r(X1 , X2 , . . ., Xn ) directly in terms of the variances and covariances for X1 , X2 , . . ., Xn for an arbitrary joint probability density function p(x1 , x2 , . . ., xn ).

In para. C-4.1 we assumed that the Taylor series expansion for r could be reasonably approximated through the first order terms. In this paragraph we will briefly assess the conditions under which this approximation is meaningful. Let’s first consider the simplest case where r(x). Then

C-5 PROPAGATION OF SYSTEMATIC AND RANDOM COMPONENTS OF UNCERTAINTY

r(x) p r( ␮ x ) + ␪ x (x − ␮x ) + (1/2!) ␥ xx (x − ␮ x ) 2 + higher order terms

The total variance associated with a measured variable, Xi , can be expressed as a combination of the variance associated with a fixed component and the variance associated with a random component of the total error in the measurement. In this paragraph we will relate these two sources of variance in the measured variables, Xj , to the variance in an experimental result r(X1 , X2 , . . ., Xn ). In subsection 4-2, the total error in a measurement was given by

where ␥ xx p

∂ 2r ∂x 2

In this case the ratio of the second order term in the series to the first order term in the series is given by Rp

␥ xx (x − ␮ x ) ␪x

␦ p ␤ + ⑀

So that for this case the assumption that R Ⰶ1 reduces to the condition that ␥xx (x − ␮x ) Ⰶ␪x . More generally, if the second order terms in the Taylor series expansion are retained, the Taylor series expansion for r in the neighborhood of ␮x1 , ␮x2 , . . ., ␮xn becomes

so that E[␦ ] p E[ ␤ + ⑀ ]

␮␦ p ␮␤

also

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

r(x 1 , x2 , . . ., xn ) p r( ␮ x1 , ␮ x2 , ␮xn ) n

+

兺

ip1

E[( ␦ − ␮␦) 2] p E{[( ␤ − ␮ ␤ ) + ⑀ 2]}

␴ 2␦ p ␴ 2␤ + ␴ 2⑀ + 2␴ ␤⑀

␪ xi (x i − ␮xi ) n

+ (1⁄ 2!)

Assuming the systematic error to be independent of the random component of error this becomes

n

兺 兺 ␥xjxk jp1 kp1

␴ 2␦ p ␴ 2␤ + ␴ 2⑀

(xj − ␮ xj )(x k − ␮ xk )

Now, if we define the random variable Xi as follows:

+ higher order terms

where ␥ xjxk

X i p ( ␦ i − ␮ ␤i ) + ␮ Xi

∂ 2r p ∂x j ∂x k

then

The second order terms in this expansion may be compared directly to the first order terms in order to assess their significance.

E[X i] p ␮ Xi E[(X i − ␮ Xi ) 2] p ␴ ␦i2 p ␴ ␤i2 + ␴ ⑀i2 p ␴ Xi2 90

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TEST UNCERTAINTY

ASME PTC 19.1-2005

Assuming fixed errors to be independent of random errors, it can also be shown that

For no correlation among the random errors, with the use of sample statistics, and for a 95% confidence level, this equation leads to eq. (B-1.3).

E[(X i − ␮ Xi )(X j − ␮Xj )] p ␴ XiXj p ␴ ␤i␤j + ␴ ⑀i⑀j

C-6 THE PROBABILITY DENSITY FUNCTION OF A RESULT

These results can be combined with those in para. C-3.1 to yield

␴ r2 ≈

n

The preceding paragraphs make no assumptions about the joint probability density function of the measured variables Xi . Assuming that the first order approximation to the Taylor series expansion for r is adequate and that the measured variables Xi are jointly normally distributed, the experimental result r will also be normally distributed with mean ␮r and variance ␴2r .

n

兺 ␪Xi2 ␴⑀i2 + 2 jpi+1 兺 ␪Xi ␪Xj ␴⑀i⑀j 冣 ip1 冢 n

+

n

兺 ␪Xi2 ␴␤i2 + 2 jpi+1 兺 ␪Xi ␪Xj ␴␤i␤j 冣 ip1 冢

--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

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ASME PTC 19.1-2005

Nonmandatory Appendix D The Central Limit Theorem Paragraph G.2.1 in [1] states “The Central Limit Theorem:

Suppose Y p

N

c1 X1 + c2 X2 + . . . + cN XN p ∑ ci Xi. Then the disip1

tribution of Y will be approximately normal with N

expectation N

E(Y) p ∑ ci E(Xi) ip1

and

variance

␴ 2(X) p ∑ ci2 ␴ 2 (Xi), where E(Xi) is the expectaip1

tion of Xi and ␴ 2 (Xi) is the variance of Xi, provided that Xi are independent and ␴ 2 (Y) is much larger than any single component ci2 ␴ 2 (Xi) from a nonnormally distributed Xi.” Although not mathematically precise, this is a reasonable statement of the central limit theorem for application purposes. It is a more general version of the central limit theorem than what one would find in elementary statistics textbooks.

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ASME PTC 19.1-2005

D04505 Copyright ASME International Provided by IHS under license with ASME No reproduction or networking permitted without license from IHS

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