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Summary of Experimental Uncertainty Assessment Methodology with Example. F. Stern, M. Muste, M-L. Beninati, and W.E. Eichinger. Table of Contents. Introduction Terminology Uncertainty Propagation Equation UA for Multiple and Single Tests Recommendations for Implementation Example.

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Summary of experimental uncertainty assessment methodology with example l.jpg

Summary of Experimental Uncertainty Assessment Methodology with Example

F. Stern, M. Muste, M-L. Beninati, and W.E. Eichinger


Table of contents l.jpg
Table of Contents with Example

  • Introduction

  • Terminology

  • Uncertainty Propagation Equation

  • UA for Multiple and Single Tests

  • Recommendations for Implementation

  • Example


Introduction l.jpg
Introduction with Example

  • Experiments are an essential and integral tool for engineering and science

  • Uncertainty estimates are imperative for risk assessments in design both when using data directly or in calibrating and/or validating simulations methods

  • True values are seldom known and experiments have errors due to instruments, data acquisition, data reduction, and environmental effects

  • Determination of truth requires estimates for experimental errors, i.e., uncertainties


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Introduction with Example

  • Uncertainty analysis (UA): rigorous methodology for uncertainty assessment using statistical and engineering concepts

  • ASME and AIAA standards (e.g., ASME, 1998; AIAA, 1995) are the most recent updates of UA methodologies, which are internationally recognized

  • Presentation purpose: to provide summary of EFD UA methodology accessible and suitable for student and faculty use both in classroom and research laboratories


Terminology l.jpg
Terminology with Example

  • Accuracy: closeness of agreement between measured and true value

  • Error: difference between measured and true value

  • Uncertainties (U): estimate of errors in measurements of individual variables Xi (Uxi) or results (Ur)

  • Estimates of U made at 95% confidence level, on large data samples (at least 10/measurement)


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Terminology with Example

  • Bias error(b): fixed, systematic

  • Bias limit(B): estimate of b

  • Precision error(e): random

  • Precision limit (P): estimate of e

  • Total error: d = b + e


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Terminology with Example

  • Measurement systems for individual variables Xi: instrumentation, data acquisition and reduction procedures, and operational environment (laboratory, large-scale facility, in situ)

  • Results expressed through data-reduction equations (DRE)

    r = r(X1, X2, X3,…, Xj)

  • Estimates of errors are meaningful only when considered in the context of the process leading to the value of the quantity under consideration

  • Identification and quantification of error sources require considerations of:

    • steps used in the process to obtain the measurement of the quantity

    • the environment in which the steps were accomplished


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Terminology with Example

  • Block diagram: elemental error sources, individual measurement systems, measurement of individual variables, data reduction equations, and experimental results


Uncertainty propagation equation l.jpg

r(X) with Example

i

r(X)

i

dr

dX

i

r

d

r

r

t

r

u

e

d

X

i

X

i

X

X

i

i

true

Uncertainty propagation equation

  • One variable, one measurement

DRE =


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Uncertainty propagation equation with Example

  • Two variables, the kth set of measurements (xk, yk)

The total error in the kth determination of r

(1)


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Uncertainty propagation equation with Example

  • A measure of dr is

(2)

Substituting (2) in (1), and assuming that bias/precision errors are correlated

(3)

s’s are not known; use estimates for the variances and covariances of the distributions of the total, bias, and precision errors

The total uncertainty of the results at a specified level of confidence is

(K = 2 for 95% confidence level)


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Uncertainty propagation equation with Example

  • Generalizing (3) for J variables

sensitivity coefficients

Example:


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Single and multiple tests with Example

  • Single test: one set of measurements (X1, X2, …, Xj) for r

  • Multiple tests: many sets of measurements (X1, X2, …, Xj) for r

  • The total uncertainty of the result (single and multiple)

(4)

  • Br: determined in the same manner for single and multiple tests

  • Pr: determined differently for single and multiple tests


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Bias limits with Example(single and multiple tests)

  • Br given by:

  • Sensitivity coefficients

  • Bi: estimate of calibration, data acquisition, data reduction, and

    conceptual bias errors for Xi

  • Bik: estimate of correlated bias limits for Xi and Xk


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Precision limits with Example(multiple tests)

  • The average result:

  • Precision limit of the result (end to end):

t: coverage factor (t = 2 for N > 10)

: standard deviation for M readings of the result


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Precision limits with Example(single test)

  • Precision limit of the result (end to end):

t: coverage factor (t = 2 for N > 10)

Sr: the standard deviation for the N readings of the result. It is not

available for single test. Use of “best available information”

(literature, inter-laboratory comparison, etc.) needed.


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EFD Validation with Example

  • EFD result: A ±UA

  • Benchmark or EFD data: B ±UB

    E = B-A

    UE2 = UA2+UB2

  • Validation:

    |E| < UE

    • Conduct uncertainty analysis for the results:


    Recommendations for implementation l.jpg
    Recommendations for implementation with Example

    • Determine data reduction equation: r = r(X1, X2, …, Xj)

    • Construct the block diagram

    • Identify and estimate sources of errors

    • Establish relative significance of the bias limits for the individual variables

    • Estimate precision limits (end-to-end procedure recommended)

    • Calculate total uncertainty using equation (4)

    • Report total error, bias and precision limits for the final result


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    Recommendations for implementation with Example

    • Recognition of the uncertainty analysis (UA) importance

    • Full integration of UA into all phases of the testing process

    • Simplified UA:

      • dominant error sources only

      • use of previous data

      • end-to-end calibration and estimation of errors

    • Full documentation:

      • Test design, measurement systems, data-stream in block diagrams

      • Equipment and procedure

      • Error sources considered

      • Estimates for bias and precision limits and estimating procedures

      • Detailed UA methodology and actual data uncertainty estimates


    Slide20 l.jpg

    Experimental Uncertainty Assessment Methodology: Example for Measurement of Density and Kinematic Viscosity


    Test design l.jpg
    Test Design Measurement of Density and Kinematic Viscosity

    A sphere of diameter D falls a distance l at terminal velocity V (fall time t) through a cylinder filled with 99.7% aqueous glycerin solution of density r, viscositym, and kinematic viscosityn (= m/r).

    Flow situations:

    - Re = VD/n <<1 (Stokes law)

    - Re > 1 (asymmetric wake)

    - Re > 20 (flow separates)


    Test design22 l.jpg
    Test Design Measurement of Density and Kinematic Viscosity

    • Assumption: Re = VD/n <<1

    • Forces acting on the sphere:

    • Apparent weight

    • Drag force (Stokes law)


    Test design23 l.jpg
    Test design Measurement of Density and Kinematic Viscosity

    • Terminal velocity:

    • Solving for n and substituting l/t for V

      (5)

    • Evaluating n for two different spheres (e.g., teflon and steel) and solving for r

      (6)

    • Equations (5) and (6): data reduction equations forn andrin terms of measurements of the individual variables: Dt, Ds, tt, ts, l


    Measurement systems and procedures l.jpg
    Measurement Systems and Procedures Measurement of Density and Kinematic Viscosity

    • Individual measurement systems:

      • Dtand Ds – micrometer; resolution 0.01mm

      • l – scale; resolution 1/16 inch

      • ttand ts - stopwatch; last significant digit 0.01 sec.

      • T (temperature) – digital thermometer; last significant digit 0.1F

    • Data acquisition procedure:

      • measure T and l

      • measure diameters Dt,and fall times tt for 10 teflon spheres

      • measure diameters Ds and fall times ts for 10 steel spheres

    • Data reduction is done at steps (5) and (6) by substituting the measurements for each test into the data reduction equation (6) for evaluation of r and then along with this result into the data reduction equation (5) for evaluation of n


    Block diagram l.jpg
    Block-diagram Measurement of Density and Kinematic Viscosity


    Test results l.jpg
    Test results Measurement of Density and Kinematic Viscosity


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    Uncertainty assessment Measurement of Density and Kinematic Viscosity(multiple tests)

    • Density r (DRE: )

    • Bias limit

    Sensitivity coefficients: e.g.,

    • Precision limit

    • Total uncertainty


    Uncertainty assessment multiple tests28 l.jpg
    Uncertainty assessment Measurement of Density and Kinematic Viscosity(multiple tests)

    • Density r


    Uncertainty assessment multiple tests29 l.jpg
    Uncertainty assessment Measurement of Density and Kinematic Viscosity(multiple tests)

    • Viscosity n (DRE : )

      • Calculations for teflon sphere

    • Bias limit

    • Precision limit

    • Total uncertainty


    Comparison with benchmark data l.jpg
    Comparison with benchmark data Measurement of Density and Kinematic Viscosity

    • Density r

    E = 4.9% (reference data) and E = 5.4% (ErTco hydrometer)

    Neglecting correlated bias errors:

    Data not validated:


    Comparison with benchmark data31 l.jpg
    Comparison with benchmark data Measurement of Density and Kinematic Viscosity

    • Viscosity n

    E = 3.95% (reference data) and E = 40.6% (Cannon capillary viscometer)

    Neglecting correlated bias errors:

    Data not validated (unaccounted bias error):


    References l.jpg
    References Measurement of Density and Kinematic Viscosity

    • AIAA, 1995, “Assessment of Wind Tunnel Data Uncertainty,” AIAA S-071-1995.

    • ASME, 1998, “Test Uncertainty,” ASME PTC 19.1-1998.

    • ANSI/ASME, 1985, “Measurement Uncertainty: Part 1, Instrument and Apparatus,” ANSI/ASME PTC 19.I-1985.

    • Coleman, H.W. and Steele, W.G., 1999, Experimentation and Uncertainty Analysis for Engineers, 2nd Edition, John Wiley & Sons, Inc., New York, NY.

    • Coleman, H.W. and Steele, W.G., 1995, “Engineering Application of Experimental Uncertainty Analysis,” AIAA Journal, Vol. 33, No.10, pp. 1888 – 1896.

    • ISO, 1993, “Guide to the Expression of Uncertainty in Measurement,", 1st edition, ISBN 92-67-10188-9.

    • ITTC, 1999, Proceedings 22nd International Towing Tank Conference, “Resistance Committee Report,” Seoul Korea and Shanghai China.


    References33 l.jpg
    References Measurement of Density and Kinematic Viscosity

    • Granger, R.A., 1988, Experiments in Fluid Mechanics, Holt, Rinehart and Winston, Inc., New York, NY.

    • Proctor&Gamble, 1995, private communication.

    • Roberson, J.A. and Crowe, C.T., 1997, Engineering Fluid Mechanics, 6th Edition, Houghton Mifflin Company, Boston, MA.

    • Small Part Inc., 1998, Product Catalog, Miami Lakes, FL.

    • Stern, F., Muste, M., M-L. Beninati, and Eichinger, W.E., 1999, “Summary of Experimental Uncertainty Assessment Methodology with Example,” IIHR Technical Report No. 406.

    • White, F.M., 1994, Fluid Mechanics, 3rd edition, McGraw-Hill, Inc., New York, NY.


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