1. F. Stern, M. Muste, M-L Beninati, W.E. Eichinger
Experimental Uncertainty Assessment Methodology: Example for Measurement of Density and Kinematic Viscosity
F. Stern, M. Muste, M-L Beninati, W.E. Eichinger
11/9/2018

2. Table of contents Introduction Test Design
Measurement Systems and Procedures
Test Results
Uncertainty Assessment for Multiple Tests
Uncertainty Assessment for Single Test
Discussion of Results
Comparison with Benchmark Data

3. Introduction
Purpose of experiment: to provide a relatively simple, yet comprehensive, tabletop measurement system for demonstrating fluid mechanics concepts, experimental procedures, and uncertainty analysis
More commonly, density is determined from specific weight measurements using hydrometers and viscosity is determined using capillary viscometers

4. 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 regimes:
- Re = VD/n <<1 (Stokes law)
- Re > 1 (asymmetric wake)
- Re > 20 (flow separates)

5. Test Design Assumption: Re = VD/n <<1
Forces acting on the sphere:
Apparent weight
Drag force (Stokes law)

6. Test Design Terminal velocity:
Solving for n and substituting l/t for V
(1)
Evaluating n for two different spheres (e.g., teflon and steel) and solving for r
(2)
Equations (1) and (2): data reduction equations for n and r in terms of measurements of the individual variables: Dt, Ds, tt, ts, l

7. Measurement Systems

8. 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 (2) and (3) by substituting the measurements for each test into the data reduction equation (2) for evaluation of r and then along with this result into the data reduction equation (1) for evaluation of n

9. Test Results

10. UA multiple tests - density
Data reduction equation for density r :
Total uncertainty for the average density:

11. UA multiple tests - density
Bias limit Br
Sensitivity coefficients

12. UA multiple tests - density
Precision limit
(Table 2)

13. UA multiple tests - density

14. UA single test - density

15. UA multiple tests - viscosity
Data reduction equation for density n :
Total uncertainty for the average viscosity (teflon sphere):

16. UA multiple tests - viscosity
Bias limit Bnt (teflon sphere)
Sensitivity coefficients:

17. UA multiple tests - viscosity
Precision limit (teflon sphere)
(Table 2)

18. UA multiple tests - viscosity
Teflon spheres

19. UA single test - viscosity
Teflon spheres

20. Discussion of the results
Values and trends for r and n in reasonable agreement with textbook values (e.g., Roberson and Crowe, 1997, pg. A-23): r = kg/m3 ; n = m2/s
Uncertainties for r and n are relatively small (< 2% for multiple tests)

21. Discussion of the results
Calibration against benchmark
EFD result: A ± UA
Benchmark data: B ± UB
E = B-A
UE2 = UA2+UB2
Data calibrated at Ue level if:
|E|  UE
Unaccounted for bias and precision limits if:
|E| > UE

22. Comparison with benchmark data
Density r (multiple tests)
E = 4.9% (benchmark data)
E = 5.4% (ErTco hydrometer)
Neglecting correlated bias errors:
r is not validated against benchmark data (Proctor & Gamble) and alternative measurement methods (ErTco hydrometer because
E~constant suggests unaccounted for bias errors

23. Comparison with benchmark data
Viscosity n (multiple tests)
E = 3.95% (benchmark data)
E = 40.6% (Cannon viscometer)
Neglecting correlated bias errors:
n is not validated against benchmark data (Proctor & Gamble) and alternative measurement methods (Cannon capillary viscometer) because
E~constant suggests unaccounted for bias errors

24. 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.