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Summary of Experimental Uncertainty Assessment Methodology with ExamplePowerPoint Presentation

Summary of Experimental Uncertainty Assessment Methodology with Example

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### Summary of Experimental Uncertainty Assessment Methodology with Example

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

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

Table of Contents with Example

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

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

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 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)

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

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

Terminology with Example

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

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 =

Uncertainty propagation equation with Example

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

The total error in the kth determination of r

(1)

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)

Uncertainty propagation equation with Example

- Generalizing (3) for J variables

sensitivity coefficients

Example:

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

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

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

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.

EFD Validation with Example Validation:

- EFD result: A ±UA
- Benchmark or EFD data: B ±UB
E = B-A

UE2 = UA2+UB2

|E| < UE

- Conduct uncertainty analysis for the results:

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

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

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 Design Measurement of Density and Kinematic Viscosity

- Assumption: Re = VD/n <<1
- Forces acting on the sphere:

- Apparent weight

- Drag force (Stokes law)

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 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 0.1F

- 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 Measurement of Density and Kinematic Viscosity

Test results Measurement of Density and Kinematic Viscosity

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 Measurement of Density and Kinematic Viscosity(multiple tests)

- Density r

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 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 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 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.

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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