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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

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slide1

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
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
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
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 design1
Test Design
  • Assumption: Re = VD/n <<1
  • Forces acting on the sphere:
  • Apparent weight
  • Drag force (Stokes law)
test design2
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 and procedures
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 0.1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
ua multiple tests density
UA multiple tests - density
  • Data reduction equation for density r :
  • Total uncertainty for the average density:
ua multiple tests density1
UA multiple tests - density
  • Bias limit Br

Sensitivity coefficients

ua multiple tests density2
UA multiple tests - density
  • Precision limit

(Table 2)

ua multiple tests viscosity
UA multiple tests - viscosity
  • Data reduction equation for density n :
  • Total uncertainty for the average viscosity (teflon sphere):
ua multiple tests viscosity1
UA multiple tests - viscosity
  • Bias limit Bnt(teflon sphere)

Sensitivity coefficients:

ua multiple tests viscosity2
UA multiple tests - viscosity
  • Precision limit (teflon sphere)

(Table 2)

discussion of the results
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)
slide21

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
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 data1
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
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.