1. F. Stern, M. Muste, M-L. Beninati, and W.E. Eichinger
Summary of Experimental Uncertainty Assessment Methodology with Example
F. Stern, M. Muste, M-L. Beninati, and W.E. Eichinger
2/17/2019

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 Total error: d = b + e
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 d r X
r(X) i
true t u e dr dX
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
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)

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 > 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 > 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 UE2 = UA2+UB2 Validation:
Conduct uncertainty analysis for the results:
EFD result: A ± UA
Benchmark or EFD data: B ± UB
E = B-A
UE2 = UA2+UB2
Validation:
|E| < UE

18. 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

19. 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
2/17/2019

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 <<1 (Stokes law)
- Re > 1 (asymmetric wake)
- Re > 20 (flow separates)

22. Test Design Assumption: Re = VD/n <<1
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
ASME, 1998, “Test Uncertainty,” ASME PTC
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 – 1896.
ISO, 1993, “Guide to the Expression of Uncertainty in Measurement,", 1st edition, ISBN
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.
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.