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

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

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

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)

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

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

Measurement Systems

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

Test Results

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

UA multiple tests - density Bias limit Br Sensitivity coefficients

UA multiple tests - density Precision limit (Table 2)

UA multiple tests - density

UA single test - density

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

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

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

UA multiple tests - viscosity Teflon spheres

UA single test - viscosity Teflon spheres

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 = 1260 kg/m3 ; n = 0.00051 m2/s Uncertainties for r and n are relatively small (< 2% for multiple tests)

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

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

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

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.