04/22/23 1

Experimental Uncertainty Assessment Methodology:

Example for Measurement of Density and Kinematic Viscosity

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

Table of contents

IntroductionIntroduction Test DesignTest Design Measurement Systems and ProceduresMeasurement Systems and Procedures Test ResultsTest Results Uncertainty Assessment for Multiple TestsUncertainty Assessment for Multiple Tests Uncertainty Assessment for Single TestUncertainty Assessment for Single Test Discussion of ResultsDiscussion of Results Comparison with Benchmark DataComparison with Benchmark Data

Introduction

Purpose of experiment: to provide a relatively Purpose of experiment: to provide a relatively simple, yet comprehensive, tabletop measurement simple, yet comprehensive, tabletop measurement system for demonstrating fluid mechanics system for demonstrating fluid mechanics concepts, experimental procedures, and concepts, experimental procedures, and uncertainty analysisuncertainty analysis

More commonly, density is determined from More commonly, density is determined from specific weight measurements using hydrometers specific weight measurements using hydrometers and viscosity is determined using capillary and viscosity is determined using capillary viscometersviscometers

Test Design

FF

F

V

S p h e refa ll in g a tte rm in a lv e lo c ity

bd

g

A sphere of diameter A sphere of diameter DD falls a falls a distance distance at terminal velocity at terminal velocity

VV (fall time (fall time tt) through a ) through a cylinder filled with 99.7% cylinder filled with 99.7%

aqueous glycerin solution of aqueous glycerin solution of densitydensityviscosityviscosityand and

kinematic kinematic viscosityviscosity

Flow regimes:Flow regimes:- Re- Re = = VDVD//<<1 (Stokes law)<<1 (Stokes law)- Re- Re > 1 (asymmetric wake) > 1 (asymmetric wake)- Re- Re > 20 (flow separates) > 20 (flow separates)

Test Design

FF

F

V

S p h e refa l l in g a tte rm in a lv e lo c ity

bd

g

Assumption: Assumption: ReRe = = VDVD//<<1<<1

Forces acting on the sphere:Forces acting on the sphere:

dbga FFFW

)1( SWa

VDFd 3

6/3D

Drag force (Stokes law)Drag force (Stokes law)

Apparent weightApparent weight

/;6/; 3sphereSDg

Test Design

Terminal velocity: Terminal velocity:

Solving for Solving for and substituting and substituting /t /t forfor V V

(1)(1)

Evaluating Evaluating for two different spheres (e.g., teflon and steel) and for two different spheres (e.g., teflon and steel) and solving for solving for

Equations (1) and (2): data reduction equations Equations (1) and (2): data reduction equations forforandandin terms of measurements of the individual variables: in terms of measurements of the individual variables: DDtt, , DDss, , tttt, , ttss, ,

)1(18

),,,(2

StgDtD

t D - tD t D - t D =tDtD

s2stt

ss2stt

2t

sstt 2),,,(

tVSgDV

);1(

18

2

Measurement Systems EXPERIM ENTAL ERROR SOURCES

EXPERIMENTALRESULTS

XB , P

SPHEREDIAMETER

FALLDISTANCE

FALLTIM E

XB , P

INDIVIDUALMEASUREMENT

SYSTEMS

MEASUREMENTOF INDIVID UAL

VARIABLES

DATA REDUCTIONEQUATIONS

XB , P

= (X , X ) =D t - D t

D t - D t

= (X , X , X , X ) =

D g( -1)t

18

B , P

B , P

D

D

DD

t

D t

t tt2 2

2

2

2tts s

s s s

sphere

s,t

s,t s,t

t

t

t

Measurement Systems and Procedures Individual measurement systems: Individual measurement systems:

DDtt and and DDss – micrometer; resolution 0.01mm – micrometer; resolution 0.01mm – – scale; resolution 1/16 inchscale; resolution 1/16 inch tttt and and tts s - stopwatch; last significant digit 0.01 sec. - stopwatch; last significant digit 0.01 sec. T T (temperature) – digital thermometer; last significant digit 0.1(temperature) – digital thermometer; last significant digit 0.1 FF

Data acquisition procedure: Data acquisition procedure: 1.1. Measure Measure TT and and 2.2. Measure diameters Measure diameters DDtt,,and fall times and fall times tttt for 10 teflon spheres for 10 teflon spheres3.3. Measure diameters Measure diameters DDss and fall times and fall times ttss for 10 steel spheres for 10 steel spheres

Data reduction is done at steps (2) and (3) by substituting Data reduction is done at steps (2) and (3) by substituting the measurements for each test into the data reduction the measurements for each test into the data reduction equation (2) for evaluation of equation (2) for evaluation of and then along with this and then along with this result into the data reduction equation (1) for evaluation of result into the data reduction equation (1) for evaluation of

Test Results

UA multiple tests - density

Data reduction equation for density Data reduction equation for density

Total uncertainty for the average density:Total uncertainty for the average density:

t D - tD

t D - t D =s

2stt

ss2stt

2t

2

22 PBU

UA multiple tests - density Bias limit Bias limit BB

ststststsssstttt ttttDDDDttDDttDD BBBBBBBBB 22222222222

4222

2

808,296mkg

ts D - t D

)t - s( Dt t s tt D 2D

stt

s

tDt

smkg

t D -t D

)t - s( t s DD

tsstt

ts

ttt

3222

22

60.30

4222

2

208,527mkg

t s D - t D

)s - t( Ds t s tt D 2D

stt

t

sDs

sm

kgt D -t D

)s - t( tt DD

tsstt

ts

sts

3222

22

1.78

Sensitivity coefficientsSensitivity coefficients

UA multiple tests - density

Precision limitPrecision limit

M

SP

2

2/1

1

2

1

M

k

k

MS

(Table 2)

P

UA multiple tests - density

UA single test - density

UA multiple tests - viscosity

Data reduction equation for density Data reduction equation for density

Total uncertainty for the average viscosity Total uncertainty for the average viscosity (teflon sphere):(teflon sphere):

)1(

18

2

StgD

222

tttPBU

UA multiple tests - viscosity Bias limitBias limit B Btt(teflon sphere)(teflon sphere)

222222222 BBBBB ttDD ttt

Sensitivity coefficients:

sm202.0

18tt1gtD2

tDtDt

skg

mttgD tt

5

62

2

1036.118

2

251027.2

18

12

sm

xgtD

ttt

t

sm

xtgtD tt 3

2 1015.118

12

UA multiple tests - viscosity

Precision limit Precision limit (teflon sphere)(teflon sphere)

(Table 2)(Table 2) 2/1

1

2

1

M

k

k

MS

M

SP t

t

2

UA multiple tests - viscosity Teflon spheres

UA single test - viscosity Teflon spheres

Discussion of the results

Values and trends for Values and trends for andand in reasonable agreement in reasonable agreement with textbook values (e.g., Roberson and Crowe, 1997, pg. with textbook values (e.g., Roberson and Crowe, 1997, pg. A-23): A-23): = 1260 = 1260 kg/mkg/m33 ; ; = = 0.00051 m0.00051 m22/s/s

Uncertainties for Uncertainties for and and are relatively small (< 2% for are relatively small (< 2% for multiple tests)multiple tests)

EFD result: EFD result: A A ±± UUAA Benchmark data: Benchmark data: B B ±± UUBB

E E = = BB--AA

UUEE2 2 = = UUAA

22++UUBB22

Data calibrated at UData calibrated at Uee level if:level if:

||EE| | UUEE

Unaccounted for bias Unaccounted for bias and precision limits if:and precision limits if:

||EE| > | > UUEE

Independent variable X i

20 25 30 35 40 45

Res

ult

R1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1Experimental Result (UA= 3%)Benchmark data (UB = 1.5% )

Validated data Data not validated

Calibration against benchmarkCalibration against benchmark

Discussion of the results

Comparison with benchmark data Density Density (multiple tests) (multiple tests)

E E = 4.9% (benchmark data) = 4.9% (benchmark data)

E E = 5.4% (ErTco hydrometer)= 5.4% (ErTco hydrometer)

%30.1 DE UU

is not validated against is not validated against benchmark data (Proctor & benchmark data (Proctor & Gamble) and alternative Gamble) and alternative measurement methods (ErTco measurement methods (ErTco hydrometer because hydrometer because

EUE

Neglecting correlated bias Neglecting correlated bias errors:errors:

Den

sity

(kg/

m3 )

1100

1150

1200

1250

1300

1350

1400

Reference data (Procter & Gamble)Single test methodErTco hydrometerRoberson & Crowe (1997)

Temperature (Degrees Celsius)

18 20 22 24 26 28 30 32

Den

sity

(kg/

m3 )

1100

1150

1200

1250

1300

1350

1400

Reference data (Procter & Gamble)Multiple test methodErTco hydrometerRoberson & Crowe (1997)

E~constant suggests E~constant suggests unaccounted for bias errors unaccounted for bias errors

Comparison with benchmark data Viscosity Viscosity (multiple tests)(multiple tests)

E E = 3.95% (benchmark data)= 3.95% (benchmark data)

E E = 40.6% (Cannon viscometer)= 40.6% (Cannon viscometer)

)%(57.1 teflonUU DE

EUE

Neglecting correlated bias errors: Neglecting correlated bias errors:

)%(49.1 steelUU DE

Kin

emat

ic V

isco

sity

(m2

/s)

4.0e-4

6.0e-4

8.0e-4

1.0e-3

1.2e-3

1.4e-3

1.6e-3

Temperature (degrees Celsius)

10 15 20 25 30 35 40

Kin

emat

ic V

isco

sity

(m2/s

)

4.0e-4

6.0e-4

8.0e-4

1.0e-3

1.2e-3

1.4e-3

1.6e-3

Temperature (degrees Celsius)

10 15 20 25 30 35 40

Reference data (Procter & Gamble)Single test method (Steel)Cannon viscometerRoberson & Crowe (1997)

Reference data (Procter & Gamble)Single test method (Teflon)Cannon viscometerRoberson & Crowe (1997)

Reference data (Procter & Gamble)Multiple test method (Steel)Cannon viscometerRoberson & Crowe (1997)

Reference data (Procter & Gamble)Multiple test method (Teflon)Cannon viscometerRoberson & Crowe (1997)

is not validated against is not validated against benchmark data (Proctor & benchmark data (Proctor & Gamble) and alternative Gamble) and alternative measurement methods measurement methods (Cannon capillary viscometer) (Cannon capillary viscometer) because because

E~constant suggests E~constant suggests unaccounted for bias errors unaccounted for bias errors

References

Granger, R.A., 1988, Granger, R.A., 1988, Experiments in Fluid MechanicsExperiments in Fluid Mechanics, Holt, Rinehart , Holt, Rinehart and Winston, Inc., New York, NY.and Winston, Inc., New York, NY.

Proctor&Gamble, 1995, private communication.Proctor&Gamble, 1995, private communication. Roberson, J.A. and Crowe, C.T., 1997, Roberson, J.A. and Crowe, C.T., 1997, Engineering Fluid MechanicsEngineering Fluid Mechanics, ,

6th Edition, Houghton Mifflin Company, Boston, MA.6th Edition, Houghton Mifflin Company, Boston, MA. Small Part Inc., 1998, Small Part Inc., 1998, Product CatalogProduct Catalog, Miami Lakes, FL., Miami Lakes, FL. Stern, F., Muste, M., M-L. Beninati, and Eichinger, W.E., 1999, Stern, F., Muste, M., M-L. Beninati, and Eichinger, W.E., 1999,

“Summary of Experimental Uncertainty Assessment Methodology “Summary of Experimental Uncertainty Assessment Methodology with Example,” IIHR Technical Report No. 406.with Example,” IIHR Technical Report No. 406.

White, F.M., 1994, White, F.M., 1994, Fluid MechanicsFluid Mechanics, , 3rd edition, McGraw-Hill, Inc., 3rd edition, McGraw-Hill, Inc., New York, NY.New York, NY.