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Unformatted text preview: Experimental Uncertainty
Assessment Methodology:
Example for Measurement of
Density and Kinematic Viscosity
F. Stern, M. Muste, ML Beninati, W.E. Eichinger 12/09/11 1 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
Purpose
simple, yet comprehensive, tabletop measurement
system for demonstrating fluid mechanics
concepts, experimental procedures, and
uncertainty analysis
uncertainty More commonly, density is determined from
More
specific weight measurements using hydrometers
and viscosity is determined using capillary
viscometers
viscometers Test Design
A sphere of diameter D falls a
sphere
distance λ at terminal velocity
V (fall time t) through a
through
cylinder filled with 99.7%
Fd F b
aqueous glycerin solution of
density ρ, viscosity µ, and
density
viscosity
kinematic
viscosity ν ( = µ/ρ).
viscosity Fg
Flow regimes:
Flow
 Re = VD/ν <<1 (Stokes law)
VD <<1
 Re > 1 (asymmetric wake)
 Re > 20 (flow separates) S p h ere
fa llin g a t
te rm in a l
v e lo c ity λ
V Test Design ∇ = πD 3 / 6 R
VD <<1 Assumption: Ree = VD/ν <<1 Fd F b Forces Forces acting on the sphere:
Fg Wa = Fg − Fb = Fd Apparent weight
Apparent Wa = γ∇( S − 1)
γ = ρg ; Drag force (Stokes law)
Drag ∇ = πD 3 / 6; S = ρ sphere / ρ Fd = 3πµVD Test Design Terminal velocity: gD 2
V=
( S − 1);
18ν V= λ
t for Solving for ν and substituting λ//tt for V
gD 2t
ν = ν ( D, t , λ , ρ ) =
( S − 1)
18λ (1) Evaluating ν for two different spheres (e.g., teflon and steel) and
solving for ρ 2
2
Dt t t ρ t  D s t s ρ s
ρ = ρ ( Dt , tt , Ds , t s ) =
Dt2 t t  D 2 t s
s (2 ) Equations (1) and (2): data reduction equations
for ν and ρ in terms of measurements of the individual variables:
for
and Measurement Systems
E X P E R IM E N T A L E R R O R S O U R C E S SPHERE
D IA M E T E R FA LL
D IS T A N C E FA L L
T IM E X X X IN D IV ID U A L
M EASUREMENT
SYSTEM S B D D λ B , Pλ , PD B t,P λ ρ = ρ (X , X ) =
t D D 2
s D t sρ s D
2
s ts D 2 ν = ν (X D , X t , X ρ, X λ) = ν s ,t
B ν , Pν
s ,t s ,t M EASUREMENT
O F IN D IV ID U A L
V A R IA B L E S t 2
t
2
t t t tρt
t t D g ( ρs p h e r e / ρ  1 ) t ρ
B ρ , Pρ D A T A R E D U C T IO N
E Q U A T IO N S 18λ E X P E R IM E N T A L
R E SU LTS Measurement Systems and Procedures Individual measurement systems:
Individual Dt and Ds – micrometer; resolution 0.01mm
λ – scale; resolution 1/16 inch
scale;
tt and ts  stopwatch; last significant digit 0.01 sec.
T (temperature) – digital thermometer; last significant digit 0.1 ° F
(temperature) Data acquisition procedure:
Data 1. Measure T and λ
1. Measure
2. Measure diameters Dt,and fall times tt for 10 teflon spheres
Measure
and
3. Measure diameters Ds and fall times ts for 10 steel spheres
Measure Data reduction is done at steps (2) and (3) by substituting
Data
the measurements for each test into the data reduction
equation (2) for evaluation of ρ and then along with this
result into the data reduction equation (1) for evaluation
of ν Test Results UA multiple tests  density Data reduction equation for density ρ :
Data
2
2
Dt t t ρ t  D s t s ρ s
ρ=
Dt2 t t  D 2 t s
s Total Total uncertainty for the average density:
2
U ρ = ± Bρ + Pρ2 UA multiple tests  density Bias limit Bρ
Bias
2
2
2
2
2
Bρ = θ Dt BDt + θ t2 Bt2 + θ Ds BDs + θ t2 Bt2s + 2θ Dt θ Ds BDt BDs + 2θ tt θ t s Btt Bt s
t
t
s Sensitivity coefficients
θ Dt 2 Ds2 t t t s D t ( ρ s  ρ t )
∂ρ
kg
=
=
= 296,808 4
2
∂Dt
m
Dt2 tt  Ds2 t s Ds2 Dt2 t s ( ρ s  ρ t )
∂ρ
kg
θ tt =
=
= 30.60 3
2
∂tt
m ⋅s
Dt2 tt  Ds2 t s θ Ds 2 Dt2 t t t s D s ( ρ t  ρ s )
∂ρ
kg
=
=
= −527,208 4
2
∂Ds
m
Dt2 tt  Ds2 t s Ds2 Dt2 t t ( ρ t  ρ s )
∂ρ
kg
θ ts =
=
= −78.1 3
2
∂t s
m ⋅s
Dt2 tt  Ds2 t s [ [ [ [ UA multiple tests  density Precision limit Pρ
Pρ = 2 ⋅ Sρ
M M ( ρk − ρ ) 2 S ρ = ∑ k =1 M − 1 1/ 2 (Table 2) UA multiple tests  density UA single test  density UA multiple tests  viscosity Data reduction equation for density ν :
Data
gD 2t
ν=
( S − 1)
18λ Total uncertainty for the average viscosity
(teflon sphere):
(teflon Uν2t = Bν2t + P 2
νt UA multiple tests  viscosity Bias limit Bνt (teflon sphere)
(teflon 2
2
2
2
22
Bν2t = θ Dt BD + θ ρ Bρ + θ t2 Bt2 + θ λ Bλ
t Sensitivity coefficients:
( ) 2 Dt g ρ t ρ − 1 tt
∂ν
θD =
=
= 0.202
18 λ
t ∂Dt 5
Dt2 gρ t t
∂ν
t = 1.36 × 10 − 6 m
θ=
=
ρ ∂ρ
18λρ 2
kg ⋅ s m
s 2
∂ν Dt g ( ρ t ρ − 1)
θ=
=
= 2.27 x10 − 5
tt
∂t
18λ m2
s2 D 2 g ( ρ t ρ − 1) tt
∂ν
θλ =
=− t
= −1.15 x10 − 3
2
∂λ
18λ m
s UA multiple tests  viscosity Precision limit P
Precision
ν t (teflon sphere)
(teflon Pt =
ν 2 ⋅ Sν t
M M (ν k − ν ) 2 Sν = ∑ M −1 k =1 1/ 2 (Table 2)
(Table UA multiple tests  viscosity
Teflon spheres UA single test  viscosity
Teflon spheres Discussion of the results Values and trends for ρ and ν in reasonable agreement
Values
and
with textbook values (e.g., Roberson and Crowe, 1997, pg.
A23): ρ = 1260 kg/m3 ; ν = 0.00051 m2/s
kg/m
0.00051 Uncertainties for ρ and ν are relatively small (< 2% for
Uncertainties
multiple tests)
multiple Discussion of the results
Calibration Calibration against benchmark EFD result: A ± UA
EFD
Benchmark data: B ± UB
Benchmark
E = BA 2.1
Experimental Result (UA= 3%) UE2 = UA2+UB2 Data calibrated at Ue
level if:
level
E ≤ UE Unaccounted for bias
Unaccounted
and precision limits if:
and
E > UE 1.9 Result R Benchmark data (UB = 1.5% ) 2.0 1.8
1.7
1.6
1.5
1.4 Data not validated Validated data 1.3
20 25 30 35 Independent variable X i 40 45 Comparison with benchmark data Density ρ (multiple tests)
Density E = 4.9% (benchmark data)
E = 5.4% (ErTco hydrometer)
5.4% 1400 U E ≈ U D = 1.30% ρ is not validated against
benchmark data (Proctor &
Gamble) and alternative
measurement methods (ErTco
hydrometer because
E ≥ UE E~constant suggests
E~constant
unaccounted for bias errors 1300
1250
1200
1150 Reference data (Procter & Gamble)
Single test method
ErTco hydrometer
Roberson & Crowe (1997) 1100
1400
1350
Density (kg/m3) Neglecting correlated bias errors: Density (kg/m3) 1350 1300
1250
1200
1150
1100
18 Reference data (Procter & Gamble)
Multiple test method
ErTco hydrometer
Roberson & Crowe (1997) 20 22 24 26 28 Temperature (Degrees Celsius) 30 32 Comparison with benchmark data Viscosity ν (multiple tests)
Viscosity (multiple E = 3.95% (benchmark data)
3.95%
E = 40.6% (Cannon viscometer)
40.6% ν is not validated against
benchmark data (Proctor &
Gamble) and alternative
measurement methods
(Cannon capillary viscometer)
because
E ≥ UE E~constant suggests
unaccounted for bias errors 2 / s)
Kinematic Viscosity (m U E ≈ U D = 1.49%( steel ) Reference data (Procter & Gamble)
Multiple test method (Teflon)
Cannon viscometer
Roberson & Crowe (1997) 1.4e3 Reference data (Procter & Gamble)
Single test method (Steel)
Cannon viscometer
Roberson & Crowe (1997) Reference data (Procter & Gamble)
Multiple test method (Steel)
Cannon viscometer
Roberson & Crowe (1997) 1.2e3
1.0e3
8.0e4
6.0e4
4.0e4
1.6e3 2 /s) U E ≈ U D = 1.57%(teflon) Kinematic Viscosity (m Neglecting correlated bias errors:
Neglecting Reference data (Procter & Gamble)
Single test method (Teflon)
Cannon viscometer
Roberson & Crowe (1997) 1.6e3 1.4e3
1.2e3
1.0e3
8.0e4
6.0e4
4.0e4
10 15 20 25 30 35 Temperature (degrees Celsius) 40
10 15 20 25 30 35 Temperature (degrees Celsius) 40 References Granger, R.A., 1988, Experiments in Fluid Mechanics, Holt, Rinehart
Granger,
Experiments
Holt,
and Winston, Inc., New York, NY.
and
Proctor&Gamble, 1995, private communication.
Roberson, J.A. and Crowe, C.T., 1997, Engineering Fluid Mechanics,
Roberson,
Engineering
6th Edition, Houghton Mifflin Company, Boston, MA.
6th
Small Part Inc., 1998, Product Catalog, Miami Lakes, FL.
Small
Product
Stern, F., Muste, M., ML. Beninati, and Eichinger, W.E., 1999,
Stern,
“Summary of Experimental Uncertainty Assessment Methodology
with Example,” IIHR Technical Report No. 406.
with
White, F.M., 1994, Fluid Mechanics, 3rd edition, McGrawHill, Inc.,
White,
Fluid
New York, NY.
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 Fall '10
 FredrickStern

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