| |
| |
| Preface | |
| |
| |
| |
| Experimentation, Errors, and Uncertainty | |
| |
| |
| |
| Experimentation | |
| |
| |
| |
| Why Is Experimentation Necessary? | |
| |
| |
| |
| Degree of Goodness and Uncertainty Analysis | |
| |
| |
| |
| Experimentation and Validation of Simulations | |
| |
| |
| |
| Experimental Approach | |
| |
| |
| |
| Questions to Be Considered | |
| |
| |
| |
| Phases of Experimental Program | |
| |
| |
| |
| Basic Concepts and Definitions | |
| |
| |
| |
| Errors and Uncertainties | |
| |
| |
| |
| Degree of Confidence and Uncertainty Intervals | |
| |
| |
| |
| Expansion of Concept from "Measurement Uncertainty" to "Experimental Uncertainty," | |
| |
| |
| |
| Elemental Systematic Errors and Effects of Calibration | |
| |
| |
| |
| Repetition and Replication | |
| |
| |
| |
| Experimental Results Determined from Multiple Measured Variables | |
| |
| |
| |
| Guides and Standards | |
| |
| |
| |
| Experimental Uncertainty Analysis | |
| |
| |
| |
| Validation of Simulations | |
| |
| |
| |
| A Note on Nomenclature | |
| |
| |
| References | |
| |
| |
| Problems | |
| |
| |
| |
| Errors and Uncertainties in a Measured Variable | |
| |
| |
| |
| Statistical Distributions | |
| |
| |
| |
| Gaussian Distribution | |
| |
| |
| |
| Mathematical Description | |
| |
| |
| |
| Confidence Intervals in Gaussian Distribution | |
| |
| |
| |
| Samples from Gaussian Parent Population | |
| |
| |
| |
| Statistical Parameters of Sample Population | |
| |
| |
| |
| Confidence Intervals in Sample Populations | |
| |
| |
| |
| Tolerance and Prediction Intervals in Sample Populations | |
| |
| |
| |
| Statistical Rejection of Outliners from a Sample | |
| |
| |
| |
| Uncertainty of a Measured Variable | |
| |
| |
| |
| Systematic Standard Uncertainty Estimation | |
| |
| |
| |
| Overall Uncertainty of a Measured Variable | |
| |
| |
| |
| Large-Sample Uncertainty of a Measured Variable | |
| |
| |
| |
| Uncertainty of Measured Variable by Monte Carlo Method | |
| |
| |
| |
| Summary | |
| |
| |
| References | |
| |
| |
| Problems | |
| |
| |
| |
| Uncertainty in a Result Determined from Multiple Variables | |
| |
| |
| |
| Taylor Series Method for Propagation of Uncertainties | |
| |
| |
| |
| TSM for Function of Multiple Variables | |
| |
| |
| |
| Expanded Uncertainty of a Result | |
| |
| |
| |
| Large-Sample Approximation for Uncertainty of a Result | |
| |
| |
| |
| Example of TSM Uncertainty Propagation | |
| |
| |
| |
| Numerical Approximation for TSM Propagation of Uncertainties | |
| |
| |
| |
| Monte Carlo Method for Propagation of Uncertainties | |
| |
| |
| |
| General Approach for MCM | |
| |
| |
| |
| Example of MCM Uncertainty Propagation | |
| |
| |
| |
| Coverage Intervals for MCM Simulations | |
| |
| |
| |
| Example of Determination of MCM Coverage Interval | |
| |
| |
| References | |
| |
| |
| Problems | |
| |
| |
| |
| General Unvertainty Analysis: Planning an Experiment and Application in Validation | |
| |
| |
| |
| Overview: Using Uncertainty Propagation in Experiments and Validation | |
| |
| |
| |
| Application in Experimentation | |
| |
| |
| |
| General Uncertainty Analysis Using the Taylor Series Method (TSM) | |
| |
| |
| |
| Application to Experiment Planning (TSM) | |
| |
| |
| |
| Simple Case | |
| |
| |
| |
| Special Functional Form | |
| |
| |
| |
| Using TSM Uncertainty Analysis in Planning an Experiment | |
| |
| |
| |
| Example: Analysis of Proposed Particulate Measuring System | |
| |
| |
| |
| The Problem | |
| |
| |
| |
| Proposed Measurement Technique and System | |
| |
| |
| |
| Analysis of Proposed Experiment | |
| |
| |
| |
| Implications of Uncertainty Analysis Results | |
| |
| |
| |
| Design Changes Indicated by Uncertainty Analysis | |
| |
| |
| |
| Example: Analysis of Proposed Heat Transfer Experiment | |
| |
| |
| |
| The Problem | |
| |
| |
| |
| Two Proposed Experimental Techniques | |
| |
| |
| |
| General Uncertainty Analysis: Steady-State Technique | |
| |
| |
| |
| General Uncertainty Analysis: Transient Technique | |
| |
| |
| |
| Implications of Uncertainty Analysis Results | |
| |
| |
| |
| Examples of Presentation of Results from Actual Applications | |
| |
| |
| |
| Results from Analysis of a Turbine Test | |
| |
| |
| |
| Results from Analysis of a Solar Thermal Absorber/Thruster Test | |
| |
| |
| |
| Application in Validation: Estimating Uncertainty in Simulation Result due to Uncertainties in Inputs | |
| |
| |
| References | |
| |
| |
| Problems | |
| |
| |
| |
| Detailed Uncertainty Analysis: Designing, Debugging, and Executing an Experiment | |
| |
| |
| |
| Using Detailed Uncertainty Analysis | |
| |
| |
| |
| Detailed Uncertainty Analysis: Overview of Complete Methodology | |
| |
| |
| |
| Determining Random Uncertainty of Experimental Result | |
| |
| |
| |
| Example: Random Uncertainty Determination in Compressible Flow Venturi Meter Calibration Facility | |
| |
| |
| |
| Example: Random Uncertainty Determination in Laboratory-Scale Ambient Temperature Flow Test Facility | |
| |
| |
| |
| Example: Random Uncertainty Determination in Full-Scale Rocket Engine Ground Test Facility | |
| |
| |
| |
| Summary | |
| |
| |
| |
| Determining Systematic Uncertainty of Experimental Result | |
| |
| |
| |
| Systematic Uncertainty for Single Variable | |
| |
| |
| |
| Some Practical Considerations | |
| |
| |
| |
| Digital Data Acquisition Errors | |
| |
| |
| |
| Property Value Uncertainty | |
| |
| |
| |
| Systematic Uncertainty of a Result Including Correlated Systematic Error Effects | |
| |
| |
| |
| Example: Correlated Errors in a Temperature Difference | |
| |
| |
| |
| Example: Correlated Errors in an Average Velocity | |
| |
| |
| |
| Comparative Testing and Correlated Systematic Error Effects | |
| |
| |
| |
| Result Is a Difference of Test Results | |
| |
| |
| |
| Result Is a Ratio of Test Results | |
| |
| |
| |
| Comprehensive Example: Sample-to-Sample Experiment | |
| |
| |
| |
| Problem | |
| |
| |
| |
| Measurement System | |
| |
| |
| |
| Zeroth-Order Replication-Level Analysis | |
| |
| |
| |
| First-Order Replication-Level Analysis | |
| |
| |
| |
| Nth-Order Replication-Level Analysis | |
| |
| |
| |
| Comprehensive Example: Debugging and Qualification of a Timewise Experiment | |
| |
| |
| |
| Basic Ideas | |
| |
| |
| |
| Example | |
| |
| |
| |
| Some Additional Considerations in Experiment Execution | |
| |
| |
| |
| Choice of Test Points: Rectification | |
| |
| |
| |
| Example of Use of Rectification | |
| |
| |
| |
| Choice of Test Sequence | |
| |
| |
| |
| Relationship to Statistical Design of Experiments | |
| |
| |
| |
| Use of Balance Checks | |
| |
| |
| |
| Application to a Flow System | |
| |
| |
| |
| Use of a Jitter Program | |
| |
| |
| |
| Comments on Transient Testing | |
| |
| |
| References | |
| |
| |
| Problems | |
| |
| |
| |
| Validation Of Simulations | |
| |
| |
| |
| Introduction to Validation Methodology | |
| |
| |
| |
| Errors and Uncertainties | |
| |
| |
| |
| Validation Nomenclature | |
| |
| |
| |
| Validation Approach | |
| |
| |
| |
| Code and Solution Verification | |
| |
| |
| |
| Estimation of Validation Uncertainty uval | |
| |
| |
| |
| Estimating uval When Experimental Value D of Validation Variable Is Directly Measured (Case 1) | |
| |
| |
| |
| Estimating uval When Experimental Value D of Validation Variable Is Determined from Data Reduction Equation (Case 2 and 3) | |
| |
| |
| |
| No Measured Variables Share Identical Error Sources (Case 2) | |
| |
| |
| |
| Measured Variables Share Identical Error Sources (Case 3) | |
| |
| |
| |
| Estimating Uval When Experimental Value D of Validation Variable Is Determined from Data Reduction Equation That Itself Is a Model (Case 4) | |
| |
| |
| |
| Interpretation of Validation Results Using E and uval | |
| |
| |
| |
| Interpretation with No Assumptions Made about Error Distributions | |
| |
| |
| |
| Interpretation with Assumptions Made about Error Distributions | |
| |
| |
| |
| Some Practical Points | |
| |
| |
| References | |
| |
| |
| |
| Data Analysis, Regression, and Reporting of Results | |
| |
| |
| |
| Overview of Regression Analysis and Its Uncertainty | |
| |
| |
| |
| Categories of Regression Uncertainty | |
| |
| |
| |
| Uncertainty in Coefficients | |
| |
| |
| |
| Uncertainty in Y From Regression Model | |
| |
| |
| |
| (Xi, Yi) Variables Are Functions | |
| |
| |
| |
| Least-Squares Estimation | |
| |
| |
| |
| Classical Linear Regression Uncertainty: Random Uncertainty | |
| |
| |
| |
| Comprehensive Approach to Linear Regression Uncertainty | |
| |
| |
| |
| Uncertainty in Coefficients: First-Order Regression | |
| |
| |
| |
| Uncertainty in Y from Regression Model: First-Order Regression | |
| |
| |
| |
| Higher Order Regressions | |
| |
| |
| |
| Reporting Regression Uncertainties | |
| |
| |
| |
| Regressions in Which X and Y Are Functional Relations | |
| |
| |
| |
| Examples of Determining Regressions and Their Uncertainties | |
| |
| |
| |
| Experimental Apparatus | |
| |
| |
| |
| Pressure Transducer Calibration and Uncertainty | |
| |
| |
| |
| Venturi Discharge Coefficient and Its Uncertainty | |
| |
| |
| |
| Flow Rate and Its Uncertainty in a Test | |
| |
| |
| |
| Multiple Linear Regression | |
| |
| |
| References | |
| |
| |
| Problems | |
| |
| |
| |
| Useful Statistics | |
| |
| |
| |
| Taylor Series Method (TSM) for Uncertainty Propagation | |
| |
| |
| |
| Derivation of Uncertainty Propagation Equation | |
| |
| |
| |
| Comparison with Previous Approaches | |
| |
| |
| |
| Abernethy et al. Approach | |
| |
| |
| |
| Coleman and Steele Approach | |
| |
| |
| |
| ISO Guide Approach | |
| |
| |
| |
| AIAA Standard [11], AGARD [12], and ANSI/ASME [13] Approach | |
| |
| |
| |
| NIST Approach | |
| |
| |
| |
| Additional Assumptions for Engineering Applications | |
| |
| |
| |
| Approximating the Coverage Factor | |
| |
| |
| References | |
| |
| |
| |
| Comparison of Models for Calculation of Uncertainty | |
| |
| |
| |
| Monte Carlo Simulations | |
| |
| |
| |
| Simulation Results | |
| |
| |
| References | |
| |
| |
| |
| Shortest Coverage Interval for Monte Carlo Method | |
| |
| |
| Reference | |
| |
| |
| |
| Asymmetric Systematic Uncertainties | |
| |
| |
| |
| Procedure for Asymmetric Systematic Uncertainties Using TSM Propagation | |
| |
| |
| |
| Procedure for Asymmetric Systematic Uncertainties Using MCM Propagation | |
| |
| |
| |
| Example: Biases in a Gas Temperature Measurement System | |
| |
| |
| References | |
| |
| |
| |
| Dynamic Response of Instrument Systems | |
| |
| |
| |
| General Instrument Response | |
| |
| |
| |
| Response of Zero-Order Instruments | |
| |
| |
| |
| Response of First-Order Instruments | |
| |
| |
| |
| Response of Second-Order Instruments | |
| |
| |
| |
| Summary | |
| |
| |
| References | |
| |
| |
| Index | |