EML 4141L Lecture Uncertainty Analysis

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					      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  RT 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.

				
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