measurement of density and kinematic viscosity s. ghosh, m. muste, f. stern

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Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

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Page 1: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Measurement of density and kinematic viscosity

S. Ghosh, M. Muste, F. Stern

Page 2: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Table of contentsTable of contents

PurposeExperimental designExperimental process

• Test Setup• Data acquisition• Data reduction• Uncertainty analysis • Data analysis

Page 3: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

PurposePurpose

Provide hands-on experience with simple table top facility and measurement systems.

Demonstrate fluids mechanics and experimental fluid dynamics concepts.

Implementing rigorous uncertainty analysis.

Compare experimental results with benchmark data.

Page 4: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Experimental designExperimental design

Viscosity is a thermodynamic property and varies with pressure and temperature.

Since the term /, where is the density

of the fluid, frequently appears in the equations of fluid mechanics, it is given a special name, Kinematic viscosity ().

We will measure the kinematic viscosity through its effect on a falling object.

FF

F

V

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

bd

g

The facility includes:• A transparent cylinder containing glycerin.• Teflon and steel spheres of different

diameters

• Stopwatch• Micrometer• Thermometer

Page 5: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Experimental processExperimental process

Page 6: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Test set-upTest set-up

Verify the vertical position for the cylinder.

Open the cylinder lid. Prepare 10 teflon and 10 steel

spheres. Clean the spheres. Test the functionality of stopwatch,

micrometer and thermometer.

Page 7: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Data AcquisitionData Acquisition

Experimental procedure:

1. Measure room temperature.

2. Measure λ.

3. Measure sphere diameter using micrometer.

4. Release sphere at fluid surface and then release gate handle.

5. Release teflon and steel spheres one by one.

6. Measure time for each sphere to travel λ.

7. Repeat steps 3-6 for all spheres. At least 10 measurements are required for each sphere.

Page 8: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Data reductionData reduction

Terminal velocity attained by an object in free fall is strongly affected by the viscosity of the fluid through which it is falling.

When terminal velocity is attained, the body experiences no acceleration, so the forces acting on the body are in equilibrium.

Resistance of the fluid to the motion of a body is defined as drag force and is given by Stokes expression (see above) for a sphere (valid for Reynolds numbers, Re = VD/n <<1),

where D is the sphere diameter, rfluid is the density of the fluid, rsphere is the density of the falling sphere, is the viscosity of the fluid, Fd, Fb, and Fg, denote the drag, buoyancy, and weight forces, respectively, V is the velocity of the sphere through the fluid (in this case, the terminal velocity), and g is the acceleration due to gravity (White 1994).

gD

mgGravity sphere 6

3

g 6

D gm = FBuoyancy f fluidb

3

:

D V 3 = ForceDrag fluid :

Page 9: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Data reduction (contd.)Data reduction (contd.) Once terminal velocity is achieved, a summation of the vertical forces

must balance. Equating the forces gives:

where t is the time for the sphere to fall a vertical distance .

Using this equation for two different balls, namely, teflon and steel spheres, the following relationship for the density of the fluid is obtained, where subscripts s and t refer to the steel and teflon balls, respectively.

18

t 1) - /( g D = fluidsphere

2

t D - tD

t D - t D =

s2stt

ss2stt

2t

fluid 2

Page 10: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Data reduction (contd.)Data reduction (contd.)Sheet 1 Sheet 2

Page 11: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Experimental Uncertainty Experimental Uncertainty AssessmentAssessment

• Uncertainty analysis (UA): rigorous methodology for uncertainty Uncertainty analysis (UA): rigorous methodology for uncertainty assessment using statistical and engineering concepts. assessment using statistical and engineering concepts.

• ASME (1998) and AIAA (1999) standards are the most recent updates of ASME (1998) and AIAA (1999) standards are the most recent updates of UA methodologies, which are internationally recognized as summarized in UA methodologies, which are internationally recognized as summarized in IIHR 1999. IIHR 1999.

•Error: difference between measured and true value. Error: difference between measured and true value.

• Uncertainties (Uncertainties (UU): estimate of errors in measurements of individual ): estimate of errors in measurements of individual variables variables XXii ( (UUxxii) or results () or results (UUrr) obtained by combining ) obtained by combining UUxxii..

• Estimates of Estimates of UU made at 95% confidence level. made at 95% confidence level.

Page 12: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

• Bias errorBias error ::Fixed and systematic Fixed and systematic

•Precision errorPrecision error : : ± and random ± and random

Total error:Total error:

DefinitionsDefinitions

Page 13: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Propagation of errorsPropagation of errors

r = r (X , X ,......, X ) 1 2 J

1 2 J

MEASUREMENTOF INDIVIDUALVARIABLES

INDIVIDUALMEASUREMENTSYSTEMS

ELEMENTALERROR SOURCES

DATA REDUCTIONEQUATION

EXPERIMENTALRESULT

XB , P

1

1 1

XB , P

2

2 2

XB , P

J

J J

rB , P

r r

Block diagram showing elemental error sources, individual measurement systems measurement of individual variables, data reduction equations, and experimental results

EXPERIMENTAL ERROR SO URCES

EXPERIMENTALRESULTS

X

B , P

SPHEREDIAMETER

FALLDISTANCE

FALLTIME

XB , P

INDIVIDUALMEASUREMENT

SYSTEM S

MEASUREMENTOF INDIVIDUAL

VARIABLES

DATA REDUCTIONEQUATIONS

X

B , 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 tt

2 2

2

2

2tts s

s s s

sphere

s,t

s,t s ,t

t

t

t

Page 14: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Uncertainty equations for single Uncertainty equations for single and multiple testsand multiple tests

Measurements can be made in several ways:

Single test (for complex or expensive experiments): one set of measurements (X1, X2, …, Xj) for r

• According to the present methodology, a test is considered a single test if the entire test is performed only once, even if the measurements of one or more variables are made from many samples (e.g., LDV velocity measurements)

• Multiple tests (ideal situations): many sets of measurements (X1, X2, …, Xj) for r at a fixed test condition with the same measurement system

Page 15: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Uncertainty equations for single Uncertainty equations for single and multiple testsand multiple tests

• The total uncertainty of the result

P + B = U 2r

2rr

2

• Br : same estimation procedure for single and multiple tests• Pr : determined differently for single and multiple tests

Page 16: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Uncertainty equations for single Uncertainty equations for single and multiple tests: bias limitsand multiple tests: bias limits

• Br :

• Bi: estimate of calibration, data acquisition, data reduction, conceptual bias errors for Xi.. Within each category, there may be several elemental sources of bias. If for variable Xi there are J significant elemental bias errors [estimated as (Bi)1, (Bi)2, … (Bi)J], the bias limit for Xi is calculated as

• Bike: estimate of correlated bias limits for Xi and Xk

• Sensitivity coefficients

J

i

J

i

J

ikikkiiir BBB

1

1

1 1

222 2

ii X

r

L

kiik BBB1

2

1

2

k

J

kii BB

Page 17: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Precision limits for single testPrecision limits for single test

• Precision limit of the result (end to end):

t: coverage factor (t = 2 for N > 10)

Sr: the standard deviation for the N readings of the result. Sr must be determined from N readings over an appropriate/sufficient time interval

• Precision limit of the result (individual variables):

rr tSP

)P ( P 2ii

J

=1ir

iii StP the precision limits for Xi

Often is the case that the time interval for collecting the data is inappropriate/insufficient and Pi’s or Pr’s must be estimated based on previous readings or best available information

Page 18: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

• The average result:

t: coverage factor (t = 2 for N > 10)

: standard deviation for M readings of the result

• Precision limit of the result (end to end):

• The total uncertainty for the average result:

• Alternatively can be determined by RSS of the precision limits of the individual variables

M

kkrM

r1

1

M

tSP r

r 2/1

1

2

1

M

k

kr M

rrS

rS

rP

22222 2 MSBPBU rrrrr

Precision limits for multiple testPrecision limits for multiple test

Page 19: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

• 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

Uncertainty Analysis - densityUncertainty Analysis - density

Page 20: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Bias Limit for DensityBias Limit for Density

Bias limit Bias limit BB

ststststsssstttt ttttDDDDttDDttDD BBBBBBBBB 22222222222

4222

2

808,296m

kg

t s D - t D

)t - s( Dt t s tt D 2

Dstt

s

tDt

sm

kg

t D -t D

)t - s( t s DD

tsstt

ts

ttt

3222

22

60.30

4222

2

208,527m

kg

t s D - t D

)s - t( Ds t s tt D 2

Dstt

t

sDs

sm

kg

t D -t D

)s - t( tt DD

tsstt

ts

sts

3222

22

1.78

Sensitivity coefficientsSensitivity coefficients

Correlated Bias : two variables are measured with the

same instrument

Page 21: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Precision limit for densityPrecision limit for density

Precision limitPrecision limit

M

SP

2

2/1

1

2

1

M

k

k

MS

P

Page 22: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Typical Uncertainty resultsTypical Uncertainty results

Page 23: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Uncertainty Analysis - ViscosityUncertainty Analysis - 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

Page 24: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Calculating Bias Limit for Calculating Bias Limit for ViscosityViscosity

Bias limitBias limit B Btt(teflon sphere)(teflon sphere)

222222222 BBBBB ttDD ttt

Sensitivity coefficients:

s

m202.0

18tt1gtD2

tDtDt

skg

mttgD tt

5

62

2

1036.118

2

251027.2

18

12

s

mx

gt

D

ttt

t

s

mx

tgt

D tt 32

1015.118

12

No Correlated Bias errors contributing to viscosity

Page 25: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Precision limit for viscosityPrecision limit for viscosity

Precision limit P Precision limit P υυ (teflon sphere) (teflon sphere)

2/1

1

2

1

M

k

k

MS

M

SP t

t

2

Page 26: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Typical Uncertainty resultsTypical Uncertainty results

Teflon spheres

Page 27: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Presentation of experimental Presentation of experimental results: General Formatresults: General Format

• EFD result: EFD result: A ±A ± UUAA

• Benchmark data: Benchmark data: B ±B ± UUB B

• E E = = BB--A A

• UUEE2 2 = = UUAA

22++UUBB2 2

• Data calibrated at UData calibrated at UEE

level if: level if: • ||EE| | UUE E

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

• ||EE| > | > UUE E

Independent variable X i

20 25 30 35 40 45

Re

sult

R

1.3

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

Page 28: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Data analysisData analysis

Temperature (Degrees Celsius)

12 14 16 18 20 22 24 26 28 30 32

Den

sity

(kg

/m3 )

1254

1256

1258

1260

1262

1264 Reference data

Temperature (degrees Celsius)

18 20 22 24 26 28 30 32

Kin

emat

ic V

isco

sity

(m

2 /s)

4.0e-4

6.0e-4

8.0e-4

1.0e-3

1.2e-3

1.4e-3

Reference data

(Proctor & Gamble Co (1995))(Proctor & Gamble Co (1995))

UA bandsshowing % uncertainty

Compare results with manufacturer’s data

Page 29: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Flow Visualization using ePIVFlow Visualization using ePIV

• ePIV-(educational) ePIV-(educational) PParticle article IImage mage VVelocimetryelocimetry

• Detects motion of particles Detects motion of particles using a camerausing a camera

• Camera details: digital , Camera details: digital , 30 frames/second, 30 frames/second, 600×480 pixel 600×480 pixel resolution resolution

• Flash details: 15mW Flash details: 15mW green continuous diode green continuous diode laserlaser

Page 30: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Results of ePIVResults of ePIV• Identical particles are tracked in consecutive images to have Identical particles are tracked in consecutive images to have

quantitative quantitative estimate of fluid flowestimate of fluid flow

• Particles have the follow specifications: Particles have the follow specifications:

• neutrally buoyant : density of SG ~ 1.0 neutrally buoyant : density of SG ~ 1.0

• small enough to follow nearly all fluid motions: small enough to follow nearly all fluid motions: diameter~11diameter~11μμmm

• QualitativeQualitative estimates of fluid flow can also be shownestimates of fluid flow can also be shown

Page 31: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Flow VisualizationFlow Visualization

• Visualization-a means of viewing fluid flow as a way of examining the Visualization-a means of viewing fluid flow as a way of examining the relative motion of the fluidrelative motion of the fluid

• Generally fluid motion is highlighted by smoke, die, tuff, particles, Generally fluid motion is highlighted by smoke, die, tuff, particles, shadowgraphs, Mach-Zehnder interferometer, and many other methodsshadowgraphs, Mach-Zehnder interferometer, and many other methods

• Answer the following Answer the following questions:questions:

1.1. Where is the circular cylinder?Where is the circular cylinder?

2.2. In what direction is the fluid In what direction is the fluid traveling?traveling?

3.3. Where is separation occurring?Where is separation occurring?

4.4. Can you spot the separation Can you spot the separation bubbles? bubbles?

5.5. What are the dark regions in the What are the dark regions in the left half of the image?left half of the image?

Page 32: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Flow Visualization-Flow around a Flow Visualization-Flow around a circular cylindercircular cylinder

• Flow around a sphere is approximated by a circular cylinderFlow around a sphere is approximated by a circular cylinder

• Flow in laboratory exercise has a Reynolds number less than 1. Flow in laboratory exercise has a Reynolds number less than 1.

• Flow with ePIV has a Reynolds number range from ~2 to 90. Flow with ePIV has a Reynolds number range from ~2 to 90.

• Reynolds number = Re = (V∙D)/Reynolds number = Re = (V∙D)/υυ = ( = (ρρ ∙ V ∙ D)/ ∙ V ∙ D)/μμ

Glycerine solution with aluminumGlycerine solution with aluminum

powder, V=1.5 mm/s, dia=10 mmpowder, V=1.5 mm/s, dia=10 mm

ePIV, water and 10ePIV, water and 10μμm polymerm polymer

particels, V=1.5 mm/s, dia=4 mmparticels, V=1.5 mm/s, dia=4 mm

Re= <1 Re= ~2

Page 33: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Flow Visualization-Flow around a Flow Visualization-Flow around a circular cylinder con’tcircular cylinder con’t

• Flow separation occurs at Re ~ 5Flow separation occurs at Re ~ 5

• Standing eddies occur between Standing eddies occur between 5 < Re < 95 < Re < 9

• Length of separation bubble is Length of separation bubble is found to grow linearly with found to grow linearly with Reynolds number until the flow Reynolds number until the flow becomes unstable about Re = 40becomes unstable about Re = 40

• Sinusoidal wake develops at Sinusoidal wake develops at about Re = 50 about Re = 50

• Kármán vortex street develops Kármán vortex street develops around Re = 100around Re = 100

Re=9.6

Re=1.54

Page 34: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Flow Visualization-Flow around a Flow Visualization-Flow around a circular cylinder con’tcircular cylinder con’t

Re=26 Re=55

Re=140Re=30

Page 35: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

Flow Visualization-Flow around a Flow Visualization-Flow around a circular cylinder con’tcircular cylinder con’t

• Typical ePIV imagesTypical ePIV images

Re=30

Re=90Re=60

Page 36: Measurement of density and kinematic viscosity S. Ghosh, M. Muste, F. Stern

The EndThe End