EML 4141L Lecture Uncertainty Analysis There’s no such thing as a perfect measurement!! Uncertainty Estimation When we measure some physical quantity with an instrument and obtain a numerical value, we want to know how close this value is to the true value. The difference between the true value and the measured value is the error. Unfortunately, the true value is in general unknown and unknowable. Since this is the case, the exact error is never known. We can only estimate error. Types of Errors Difference between measured result and “true” value. Illegitimate errors Blunders result from mistakes in procedure. You must be careful. Computational or calculation errors after the experiment. Bias or Systematic errors An error that persists and cannot be considered to exist entirely by chance. This type of error tends to stay constant from trial to trial. (e.g. zero offset) Systematic errors can be corrected through calibration Faulty equipment--Instrument always reads 3% high or low Consistent or recurring human errors-- observer bias This type of error cannot be studied theoretically but can be determined by comparison to theory or by alternate measurements. Types of Errors (cont.) Random or Precision errors: The deviation of the measurement from the true value resulting from the finite precision of the measurement method being used. Instrument friction or hysteresis Errors from calibration drift Variation of procedure or interpretation of experimenters Test condition variations or environmental effects Reduce random errors by conducting more experiments/take more data. Grouping & Categorizing Error Sources Calibration Laboratory certification of equipment Data Acquisition Errors in data acquisition equipment Data Reduction Errors in computers and calculators Errors of Method Personal errors/blunders How to combine bias and precision error? Rules for combining independent uncertainties for measurements: Both uncertainties MUST be at the same CI RSS-Root-sum-square Method Provides 95% CI coverage Most commonly used/we will use this method throughout course U x Bx Px2 or U x Bx Px2 2 2 ADD-Addition Method Provides 99 % CI coverage Used in aerospace applications/more conservative U x, ADD Bx Px or U x , ADD Bx Px How to Estimate Bias Error Manufacturers Specifications Assume manufacturer is giving max. error Accuracy - %FS, %reading, offset, or some combination (e.g., 0.1% reading+0.15 counts) These are generally given at a 95% confidence interval Independent Calibration Device is calibrated to known accuracy Regression techniques and accuracy of standards Use smallest readable division Typically ± 1/2 or ± 1/4 smallest division (judgment call) Summing Bias Error Btotal ( Bi ) 2 12 General Uncertainty Analysis The estimate of possible error is called uncertainty. Includes both bias and precision errors. Need to identify all errors for the instrument(s). All measurements should be given in three parts Best value/average value ± Confidence limits or uncertainty interval Specified probability/confidence interval (typically 95% C.I.) Uncertainty can be expressed in either absolute terms (i.e., 5 Volts ±0.5 Volts) or in percentage terms (i.e., 5 Volts ±10%) (relative uncertainty= V/V) **Always use a 95 % confidence interval in throughout this course Propagation of Error Used to determine uncertainty of a quantity that requires measurement of several independent variables. Volume of a cylinder = f(D,L) Volume of a block = f(L,W,H) Density of a gas = f(P,T) Again, all variables must have the same confidence interval to use this method and be in proper dimensions. RSS Method (Root Sum Squares) For a function y(x1,x2,...,xN), the RSS uncertainty is given by: R 2 R 2 R 2 x U x1 x U x2 ... x U xN U R 1 2 N Rules Rule 1 - Always solve the data reduction equation for the experimental results before doing the uncertainty analysis. Rule 2 – Always try to divide the uncertainty analysis expression by the experimental result to see if it can be simplified. Determine uncertainty in each independent variable in the form ( xN ± xN) Use previously established methods including bias and precision error. RSS Method (Special Function Form) For relationships that are pure products or quotients a simple shortcut can be used to estimate propagation of error. R=k X1a X2b X3c… 2 2 2 2 2 UR U x1 2 U x2 U x3 x b x a c x ...... R 1 2 3 Example Problem: Propagation of Error Ideal gas law: P RT Temperature T±T How do we estimate the error Pressure P±P in the density? R=Constant Apply RSS Formula to density relationship: 2 2 2 1 P2 RSS p T P T p T RT RT 2 Apply a little algebra: P RT 2 p T 2 p T Uncertainty Analysis in EES Uncertainty Calculation in EES Experimental Data Analysis References ASHRAE, 1996. Engineering Analysis of Experimental Data, ASHRAE Guideline 2- 1996 Deick, R.H., 1992. Measurement Uncertainty, Methods and Applications, ISA. Coleman, H.W. and Steele, G.W., 1989. Experimentation and Uncertainty Analysis for Engineers. Plotting and Data Analysis with MicroSoft Excel Outline Basic Plotting with Excel Regression Analysis Example Basic Plotting with Excel 97 Plotting Experimental Data X-Y Plots RULE: Data points are discreet; therefore they should be represented by symbols. Do not connect symbols with lines. Functions, on the other hand, are continuous hence they should be represented by lines. Basic Plotting with Excel 97 Create the basic plot. Format the axis and titles Axes should have clear labels and units e.g., Pressure, P (Pa) Adjust the scale to maximize the amount of plot space occupied by the data. Tick marks should be used Add Greek letters. Basic Plotting with Excel 97 Format the data series Use open symbols before solid symbols Add legend if needed Add error bars linked to the worksheet. Add additional data sets. Plotting Common Sense Colors and Font Do not use Excel Chart Defaults Black points are difficult to see on a gray background. Remove unnecessary borders and headers like “Sheet 1” Prepare the plot in Black & White only. Color plots look nice in presentations and reports, but office copiers and publishers are still B&W only. To a copier red and yellow both appear gray. Format text for clarity Superscript Greek Symbols Plotting Common Sense Trend Line do’s and don’ts Avoid using “Insert Trend Line” because it only gives, slope, intercept, and R2. Use Analysis Tool Pack instead. Use “Insert Trend Line” to obtain polynomial fits only when a curve fit for the data is required and one is not concerned with the underlying physics. DO NOT insert trend lines for cosmetic reasons. Measurements Lab Reporting Requirements Present the plot, clearly labeled, error bars, etc. If the plot is included directly in the body of a report, do not insert a title. Use figure captions to describe the plot. Present the original worksheet used to analyze and plot the data that we can spot mistakes and give partial credit. Also, neatly format and annotated so that we can follow your analysis. Sample calculations (longhand or computer generated) of the data and uncertainty analysis so that we can give partial credit.