Experimental Uncertainty Assessment Methodology: Example for Measurement of Density and Kinematic Viscosity F. Stern, M. Muste, M-L Beninati, W.E. Eichinger
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, viscositym, and kinematic viscosityn (= 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 forn andrin terms of measurements of the individual variables: Dt, Ds, tt, ts, l
Measurement Systems
Measurement Systems and Procedures • 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 (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 randn 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 • 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 • Calibration against benchmark
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.
