Summary of Experimental Uncertainty Assessment Methodology with Example

1
Summary of Experimental Uncertainty Assessment
Methodology with Example

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

2
Table of Contents

• Introduction
• Terminology
• Uncertainty Propagation Equation
• UA for Multiple and Single Tests
• Recommendations for Implementation
• Example

3
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

4
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

5
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)

6
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

7
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

8
Terminology

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

9
Uncertainty propagation equation

• One variable, one measurement

DRE
10
Uncertainty propagation equation

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

The total error in the kth determination of r
(1)
11
Uncertainty propagation equation
(2)

• A measure of dr is

Substituting (2) in (1), and assuming that
bias/precision errors are correlated
(3)
ss 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)
12
Uncertainty propagation equation

• Generalizing (3) for J variables

sensitivity coefficients
Example
13
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

14
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

15
Precision limits (multiple tests)

• Precision limit of the result (end to end)

t coverage factor (t 2 for N gt 10)
standard deviation for M readings of the result

• The average result

16
Precision limits (single test)

• Precision limit of the result (end to end)

t coverage factor (t 2 for N gt 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.
17
EFD Validation

• Conduct uncertainty analysis for the results

• EFD result A UA
• Benchmark or EFD data B UB
• E B-A
• UE2 UA2UB2
• Validation
• E lt UE

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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

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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

20
Experimental Uncertainty Assessment Methodology
Example for Measurement of Density and Kinematic
Viscosity
21
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, viscosity m, and kinematic
  viscosity n ( m/r).
• Flow situations
• - Re VD/n ltlt1 (Stokes law)
• - Re gt 1 (asymmetric wake)
• - Re gt 20 (flow separates)

22
Test Design

• Assumption Re VD/n ltlt1
• Forces acting on the sphere

• Apparent weight

• Drag force (Stokes law)

23
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
  for n and r in terms of measurements of the
  individual variables Dt, Ds, tt, ts, l

24
Measurement Systems and Procedures

• Individual measurement systems
• Dt and Ds micrometer resolution 0.01mm
• l scale resolution 1/16 inch
• tt and 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

25
Block-diagram
26
Test results
27
Uncertainty assessment (multiple tests)

• Density r (DRE
  )

• Bias limit

Sensitivity coefficients e.g.,

• Precision limit

• Total uncertainty

28
Uncertainty assessment (multiple tests)

• Density r

29
Uncertainty assessment (multiple tests)

• Viscosity n (DRE
  )
• Calculations for teflon sphere

• Bias limit

• Precision limit

• Total uncertainty

30
Comparison with benchmark data

• Density r

E 4.9 (reference data) and E 5.4 (ErTco
hydrometer)
Neglecting correlated bias errors
Data not validated
31
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)
32
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.

33
References

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