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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

Summary of Experimental Uncertainty Assessment Methodology with Example

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

table of contents
Table of Contents
  • Introduction
  • Terminology
  • Uncertainty Propagation Equation
  • UA for Multiple and Single Tests
  • Recommendations for Implementation
  • Example
introduction
Introduction
  • 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
introduction4
Introduction
  • 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
Terminology
  • 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)
terminology6
Terminology
  • 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
terminology7
Terminology
  • 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
terminology8
Terminology
  • Block diagram: elemental error sources, individual measurement systems, measurement of individual variables, data reduction equations, and experimental results
uncertainty propagation equation

r(X)

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 =

uncertainty propagation equation10
Uncertainty propagation equation
  • Two variables, the kth set of measurements (xk, yk)

The total error in the kth determination of r

(1)

uncertainty propagation equation11
Uncertainty propagation equation
  • 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)

uncertainty propagation equation12
Uncertainty propagation equation
  • Generalizing (3) for J variables

sensitivity coefficients

Example:

single and multiple tests
Single and multiple tests
  • 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
bias limits single and multiple tests
Bias limits (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
precision limits multiple tests
Precision limits (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

precision limits single test
Precision limits (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.

efd validation
EFD Validation
    • 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
Recommendations for implementation
  • 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
recommendations for implementation19
Recommendations for implementation
  • 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

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

test design
Test Design

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
Test Design
  • Assumption: Re = VD/n <<1
  • Forces acting on the sphere:
  • Apparent weight
  • Drag force (Stokes law)
test design23
Test design
  • 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
Measurement Systems and Procedures
  • 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
uncertainty assessment multiple tests
Uncertainty assessment (multiple tests)
  • Density r (DRE: )
  • Bias limit

Sensitivity coefficients: e.g.,

  • Precision limit
  • Total uncertainty
uncertainty assessment multiple tests29
Uncertainty assessment (multiple tests)
  • Viscosity n (DRE : )
    • Calculations for teflon sphere
  • Bias limit
  • Precision limit
  • Total uncertainty
comparison with benchmark data
Comparison with benchmark data
  • Density r

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

Neglecting correlated bias errors:

Data not validated:

comparison with benchmark data31
Comparison with benchmark data
  • 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
References
  • 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
References
  • 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.