Preface xv
1 Experimentation, Errors, and Uncertainty 1
1-1 Experimentation, 2
1-1.1 Why Is Experimentation Necessary?, 2
1-1.2 Degree of Goodness and Uncertainty Analysis, 3
1-1.3 Experimentation and Validation of Simulations, 5
1-2 Experimental Approach, 6
1-2.1 Questions to Be Considered, 7
1-2.2 Phases of Experimental Program, 8
1-3 Basic Concepts and Definitions, 8
1-3.1 Errors and Uncertainties, 9
1-3.2 Categorizing and Naming Errors and Uncertainties, 13
1-3.3 Estimating Standard Uncertainties, 15
1-3.4 Determining Combined Standard Uncertainties, 16
1-3.5 Elemental Systematic Errors and Effects of Calibration, 18
1-3.6 Expansion of Concept from "Measurement Uncertainty" to "Experimental Uncertainty", 20
1-3.7 Repetition and Replication, 22
1-3.8 Associating a Percentage Coverage or Confidence with Uncertainty Estimates, 24
1-4 Experimental Results Determined from a Data Reduction Equation Combining Multiple Measured Variables, 25
1-5 Guides and Standards, 27
1-5.1 Experimental Uncertainty Analysis, 27
1-5.2 Validation of Simulations, 29
1-6 A Note on Nomenclature, 31
References, 31
Problems, 32
2 Coverage and Confidence Intervals for an Individual Measured Variable 33
2-1 Coverage Intervals from the Monte Carlo Method for a Single Measured Variable, 34
2-2 Confidence Intervals from the Taylor Series Method for a Single Measured Variable, Only Random Errors Considered, 35
2-2.1 Statistical Distributions, 35
2-2.2 The Gaussian Distribution, 36
2-2.3 Confidence Intervals in Gaussian Parent Populations, 42
2-2.4 Confidence Intervals in Samples from Gaussian Parent Populations, 43
2-2.5 Tolerance and Prediction Intervals in Samples from Gaussian Parent Populations, 48
2-2.6 Statistical Rejection of Outliers from a Sample Assumed from a Gaussian Parent Population, 51
2-3 Confidence Intervals from the Taylor Series Method for a Single Measured Variable: Random and Systematic Errors Considered, 55
2-3.1 The Central Limit Theorem, 55
2-3.2 Systematic Standard Uncertainty Estimation, 56
2-3.3 The TSM Expanded Uncertainty of a Measured Variable, 58
2-3.4 The TSM Large-Sample Expanded Uncertainty of a Measured Variable, 61
2-4 Uncertainty of Uncertainty Estimates and Confidence Interval Limits for a Measured Variable, 63
2-4.1 Uncertainty of Uncertainty Estimates, 63
2-4.2 Implications of the Uncertainty in Limits of High Confidence Uncertainty Intervals Used in Analysis and Design, 65
References, 67
Problems, 68
3 Uncertainty in a Result Determined from Multiple Variables 71
3-1 General Uncertainty Analysis vs. Detailed Uncertainty Analysis, 72
3-2 Monte Carlo Method for Propagation of Uncertainties, 73
3-2.1 Using the MCM in General Uncertainty Analysis, 73
3-2.2 Using the MCM in Detailed Uncertainty Analysis, 75
3-3 Taylor Series Method for Propagation of Uncertainties, 78
3-3.1 General Uncertainty Analysis Using the Taylor Series Method (TSM), 79
3-3.2 Detailed Uncertainty Analysis Using the Taylor Series Method (TSM), 80
3-4 Determining MCM Coverage Intervals and TSM Expanded Uncertainty, 82
3-4.1 MCM Coverage Intervals for a Result, 82
3-4.2 TSM Expanded Uncertainty of a Result, 85
3-5 General Uncertainty Analysis Using the TSM and MSM Approaches for a Rough-walled Pipe Flow Experiment, 87
3-5.1 TSM General Uncertainty Analysis, 88
3-5.2 MCM General Uncertainty Analysis, 89
3-5.3 Implementation Using a Spreadsheet, 89
3-5.4 Results of the Analysis, 92
3-6 Comments on Implementing Detailed Uncertainty Analysis Using a Spreadsheet, 95
References, 96
Problems, 97
4 General Uncertainty Analysis Using the Taylor Series Method (TSM) 99
4-1 TSM Application to Experiment Planning, 100
4-2 TSM Application to Experiment Planning: Special Functional Form, 103
4-3 Using TSM Uncertainty Analysis in Planning an Experiment, 107
4-4 Example: Analysis of Proposed Particulate Measuring System, 109
4-4.1 The Problem, 109
4-4.2 Proposed Measurement Technique and System, 109
4-4.3 Analysis of Proposed Experiment, 110
4-4.4 Implications of Uncertainty Analysis Results, 112
4-4.5 Design Changes Indicated by Uncertainty Analysis, 113
4-5 Example: Analysis of Proposed Heat Transfer Experiment, 114
4-5.1 The Problem, 114
4-5.2 Two Proposed Experimental Techniques, 115
4-5.3 General Uncertainty Analysis: Steady-State Technique, 117
4-5.4 General Uncertainty Analysis: Transient Technique, 121
4-5.5 Implications of Uncertainty Analysis Results, 123
4-6 Examples of Presentation of Results from Actual Applications, 124
4-6.1 Results from Analysis of a Turbine Test, 124
4-6.2 Results from Analysis of a Solar Thermal Absorber/Thruster Test, 125
References, 126
Problems, 127
5 Detailed Uncertainty Analysis: Overview and Determining Random Uncertainties in Results 131
5-1 Using Detailed Uncertainty Analysis, 131
5-2 Detailed Uncertainty Analysis: Overview of Complete Methodology, 134
5-3 Determining Random Uncertainty of Experimental Result, 137
5-3.1 Example: Random Uncertainty Determination in Compressible Flow Venturi Meter Calibration Facility, 139
5-3.2 Example: Random Uncertainty Determination in Laboratory-Scale Ambient Temperature Flow Test Facility, 141
5-3.3 Example: Random Uncertainty Determination in Full-Scale Rocket Engine Ground Test Facility, 143
5-3.4 Summary, 146
References, 146
6 Detailed Uncertainty Analysis: Determining Systematic Uncertainties in Results 147
6-1 Estimating Systematic Uncertainties, 149
6-1.1 Example: Estimating Uncertainty in Property Values, 152
6-1.2 Example: Estimating Systematic Uncertainties in the Turbulent Heat Transfer Test Facility (THTTF), 153
6-1.3 Example: An "Optimum" Calibration Approach Used in a Test to Determine Turbine Efficiency, 163
6-2 Determining Systematic Uncertainty of Experimental Result Including Correlated Systematic Error Effects, 165
6-2.1 Example: Correlated Systematic Error Effects with "% of Full Scale" (%FS) Systematic Uncertainties, 168
6-2.2 Example: Correlated Systematic Error Effects with "% of Reading" Systematic Uncertainties, 170
6-2.3 Example: Correlated Systematic Error Effects with Systematic Uncertainties that Vary with Set Point, 171
6-2.4 Example: Correlated Systematic Error Effects When Only Some Elemental Sources Are Correlated, 172
6-2.5 Example: Correlated Systematic Error Effects When Determining Average Velocity of a Fluid Flow, 176
6-3 Comparative Testing, 177
6-3.1 Result Is a Difference of Test Results, 178
6-3.2 Result Is a Ratio of Test Results, 181
6-4 Some Additional Considerations in Experiment Execution, 183
6-4.1 Choice of Test Points: Rectification, 183
6-4.2 Choice of Test Sequence, 188
6-4.3 Relationship to Statistical Design of Experiments, 189
6-4.4 Use of a Jitter Program, 191
6-4.5 Comments on Transient Testing, 193
6-4.6 Comments on Digital Data Acquisition Errors, 193
References, 194
Problems, 195
7 Detailed Uncertainty Analysis: Comprehensive Examples 199
7-1 TSM Comprehensive Example: Sample-to-Sample Experiment, 199
7-1.1 The Problem, 199
7-1.2 Measurement System, 200
7-1.3 Zeroth-Order Replication-Level Analysis, 201
7-1.4 First-Order Replication-Level Analysis, 205
7-1.5 Nth-Order Replication-Level Analysis, 206
7-2 TSM Comprehensive Example: Use of Balance Checks, 207
7-3 Comprehensive Example: Debugging and Qualification of a Timewise Experiment, 210
7-3.1 Orders of Replication Level in Timewise Experiments, 211
7-3.2 Example, 212
7-4 Comprehensive Example: Heat Exchanger Test Facility for Single and Comparative Tests, 216
7-4.1 Determination of the Uncertainty in q for a Single Core Design, 219
7-4.2 Determination of the Uncertainty in ¿q for Two Core Designs Tested Sequentially Using the Same Facility and Instrumentation, 224
7-5 Case Study: Examples of Single and Comparative Tests of Nuclear Power Plant Residual Heat Removal Heat Exchanger, 230
7-5.1 Single Test Results for an RHR Heat Exchanger (HX1), 231
7-5.2 Comparative Test Approach for the Decrease in Fouling Resistance and Its Uncertainty, 234
References, 235
Problems, 235
8 The Uncertainty Associated with the Use of Regressions 239
8-1 Overview of Linear Regression Analysis and Its Uncertainty, 240
8-1.1 Uncertainty in Coefficients, 241
8-1.2 Uncertainty in Y from Regression Model, 241
8-1.3 (Xi, Yi) Variables Are Functions, 243
8-2 Determining and Reporting Regression Uncertainty, 243
8-2.1 MCM Regression Uncertainty Determination, 244
8-2.2 TSM Regression Uncertainty Determination, 244
8-2.3 Reporting Regression Uncertainties, 244
8-3 Method of Least Squares Regression, 246
8-4 First-Order Regression Example: MCM Approach to Determine Regression Uncertainty, 249
8-5 Regression Examples: TSM Approach to Determine Regression Uncertainty, 252
8-5.1 Uncertainty in First-Order Coefficients, 252
8-5.2 Uncertainty in Y from First-Order Regression, 253
8-5.3 Uncertainty in Y from Higher-Order Regressions, 255
8-5.4 Uncertainty in Y from Regressions in Which X and Y Are Functional Relations, 255
8-5.5 Uncertainty Associated with Multivariate Linear Regression, 257
8-6 Comprehensive TSM Example: Regressions and Their Uncertainties in a Flow Test, 259
8-6.1 Experimental Apparatus, 261
8-6.2 Pressure Transducer Calibration and Uncertainty, 261
8-6.3 Venturi Discharge Coefficient and Its Uncertainty, 265
8-6.4 Flow Rate and Its Uncertainty in a Test, 269
References, 273
Problems, 273
9 Validation of Simulations 277
9-1 Introduction to Validation Methodology, 277
9-2 Errors and Uncertainties, 278
9-3 Validation Nomenclature, 279
9-4 Validation Approach, 280
9-5 Code and Solution Verification, 284
9-6 Interpretation of Validation Results Using E and uval, 284
9-6.1 Interpretation with No Assumptions Made about Error Distributions, 285
9-6.2 Interpretation with Assumptions Made about Error Distributions, 285
9-7 Estimation of Validation Uncertainty uval, 286
9-7.1 Case 1: Estimating uval When Experimental Value D of Validation Variable Is Directly Measured, 287
9-7.2 Cases 2 and 3: Estimating uval When Experimental Value D of Validation Variable Is Determined from Data Reduction Equation, 290
9-7.3 Case 4: Estimating uval When Experimental Value D of Validation Variable Is Determined from Data Reduction Equation That Itself Is a Model, 295
9-8 Some Practical Points, 297
References, 299
Answers to Selected Problems 301
Appendix A Useful Statistics 305
Appendix B Taylor Series Method (TSM) for Uncertainty Propagation 311
B-1 Derivation of Uncertainty Propagation Equation, 312
B-2 Comparison with Previous Approaches, 316
B-2.1 Abernethy et al. Approach, 316
B-2.2 Coleman and Steele Approach, 317
B-2.3 ISO Guide 1993 GUM Approach, 318
B-2.4 AIAA Standard, AGARD, and ANSI/ASME Approach, 319
B-2.5 NIST Approach, 319
B-3 Additional Assumptions for Engineering Applications, 319
B-3.1 Approximating the Coverage Factor, 320
References, 322
Appendix C Comparison of Models for Calculation of Uncertainty 325
C-1 Monte Carlo Simulations, 325
C-2 Simulation Results, 328
References, 336
Appendix D Shortest Coverage Interval for Monte Carlo Method 337
Reference, 338
Appendix E Asymmetric Systematic Uncertainties 339
E-1 Procedure for Asymmetric Systematic Uncertainties Using TSM Propagation, 340
E-2 Procedure for Asymmetric Systematic Uncertainties Using MCM Propagation, 344
E-3 Example: Biases in a Gas Temperature Measurement System, 344
References, 351
Appendix F Dynamic Response of Instrument Systems 353
F-1 General Instrument Response, 353
F-2 Response of Zero-Order Instruments, 355
F-3 Response of First-Order Instruments, 356
F-4 Response of Second-Order Instruments, 359
F-5 Summary, 362
References, 362
Index 363
