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Unformatted text preview: Measurement of density and
kinematic viscosity S. Ghosh, M. Muste, F. Stern Table of contents
Table Purpose Experimental design Experimental process
•
•
•
•
• Test Setup
Data acquisition
Data reduction
Uncertainty analysis
Data analysis Purpose
Purpose Provide handson 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. Experimental design
Experimental
Fd F S p h ere
fa llin g a t
te rm in a l
v e lo c ity b λ Fg Viscosity is a thermodynamic property
and varies with pressure and temperature. Since the term µ /ρ , where ρ is the density
of the fluid, frequently appears in the
equations of fluid mechanics, it is given a
special name, Kinematic viscosity (ν ). We will measure the kinematic viscosity
through its effect on a falling object. V The facility includes:
• A transparent cylinder containing
glycerin.
• Teflon and steel spheres of different
diameters • Stopwatch
• Micrometer
• Thermometer Experimental process
Experimental Test setup
Test 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. Data Acquisition
Data
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 36 for all spheres.
At least 10 measurements are
required for each sphere. Data reduction
Data
D3
Gravity = mg = ρ sphereπ
g
6
Buoyancy : D3
g
F b = m f g = ρ fluid π
6 Drag Force : = 3 ρ fluid π ν V D 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, ν 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). Data reduction (contd.)
Data Once terminal velocity is achieved, a summation of the vertical forces
must balance. Equating the forces gives: D g ( ρ sphere / ρ fluid  1) t
ν=
18 λ
2 where t is the time for the sphere to fall a vertical distance λ . 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. ρ t  D2 t s ρ s
Dt
s
ρ fluid =
2
Dt t t  D 2 t s
s
2
tt Data reduction (contd.)
Data
Sheet 1 Sheet 2 Experimental Uncertainty
Assessment
Assessment • Uncertainty analysis (UA): rigorous methodology for uncertainty
assessment using statistical and engineering concepts.
• ASME (1998) and AIAA (1999) standards are the most recent updates of
UA methodologies, which are internationally recognized as summarized in
IIHR 1999.
•Error: difference between measured and true value.
Error:
• Uncertainties (U): estimate of errors in measurements of individual
Uncertainties ):
variables Xi (Uxi) or results (Ur) obtained by combining Uxi.
obtained
• Estimates of U made at 95% confidence level. Definitions
• Bias error β:
Bias
Fixed and systematic
Fixed
•Precision error ε:
± and random
and
• Total error: δ = β + ε
Total Propagation of errors
Propagation
Block diagram showing elemental error sources, individual measurement
systems measurement of individual variables, data reduction equations, and
experimental results E X P E R IM E N T A L E R R O R S O U R C E S E LE M EN TA L
ERROR SOURCES 1 X
1
B ,P
1 2 1 IN D IV ID U A L
M EASUREM ENT
SYSTEMS J X
2
B ,P
2 X
J
B ,P 2 J J M EASUREM ENT
O F IN D IV ID U A L
V A R IA B L E S SPHERE
D IA M E T E R FA LL
D IS T A N C E X X X B D D λ B λ, Pλ , PD D 1 2 r
B ,P
r J r D A T A R E D U C T IO N
E Q U A T IO N E X P E R IM E N T A L
R E S U LT t 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 MEASUREMENT
O F IN D IV ID U A L
V A R IA B L E S t Bt,P ρ = ρ (X , X ) = r = r ( X , X ,......, X ) IN D IV ID U A L
MEASUREMENT
SYSTEM S FA LL
T IM E 2
t
2
t t t ρt
t 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 S U LT S Uncertainty equations for single
and multiple tests
and
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 Uncertainty equations for single
and multiple tests
and
• The total uncertainty of the result U r2 = B 2 + P 2
r
r
• Br : same estimation procedure for single and multiple tests
• Pr : determined differently for single and multiple tests Uncertainty equations for single
and multiple tests: bias limits
and
• Br : J J −1 B = ∑ θ B + 2∑
2
r i =1 2
i • Sensitivity coefficients 2
i J ∑θ θ i =1 k =i +1 ik Bik ∂r
∂X i θi = • 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
J 2 k =1 k B = ∑ ( Bi )
2
i L • Bike: estimate of correlated bias (limitsBfor Xi and Xk
B =∑ B) ( )
ik α =1 iα kα Precision limits for single test
Precision
• Precision limit of the result (end to end):
Pr = tS r 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):
J Pr = ∑ ( θ i Pi )2
i=1 Pi = ti Si 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 Precision limits for multiple test
1
r=
M • The average result: M ∑r
k =1 k • Precision limit of the result (end to end): M ( rk − r ) 2 S r = ∑ k =1 M − 1 tS r
Pr =
M 1/ 2 t: coverage factor (t = 2 for N > 10) • Sr : standard deviation for M readings of the result
The total uncertainty for the average result: ( U = B + P = B + 2 Sr
2
r 2
r 2
r 2
r M ) 2 Pr
• Alternatively can be determined by RSS of the precision limits of the individual variables Uncertainty Analysis  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 uncertainty for the average density:
Total
2
U ρ = ± Bρ + Pρ2 Bias Limit for Density
Bias
Correlated Bias : two variables are measured with the
same instrument Bias limit Bρ 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
∂ρ 2 Ds t t t s D t ( ρ s  ρ t )
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
∂t t
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 [ [ [ [ Precision limit for density
Precision
Precision limit Pρ
Precision
Pρ = 2 ⋅ Sρ
M ( ρk − ρ )
S ρ = ∑ k =1 M − 1 M 2 1/ 2 Typical Uncertainty results
Typical Uncertainty Analysis  Viscosity
Uncertainty
Data reduction equation for density ν :
gD 2t
ν=
( S − 1)
18λ Total uncertainty for the average viscosity
Total
(teflon sphere):
(teflon
Uν2t = Bν2t + P 2
νt Calculating Bias Limit for
Viscosity
Viscosity
Bias limit Bν t (teflon sphere)
(teflon
2
2
2
2
22
Bν2t = θ Dt BD + θ ρ Bρ + θ t2 Bt2 + θ λ Bλ
t No Correlated Bias errors
contributing to viscosity Sensitivity coefficients: ( ) 2 Dt g ρ t ρ − 1 tt
∂ν
θD =
=
= 0.202
18 λ
t ∂Dt 2
∂ν Dt g ( ρ t ρ − 1)
θ=
=
= 2.27 x10 − 5
tt
∂t
18λ 5
Dt2 gρ t t
∂ν
t = 1.36 × 10 − 6 m
θ=
=
ρ ∂ρ
18λρ 2
kg ⋅ s m
s
2 m
s2 D 2 g ( ρ t ρ − 1) tt
∂ν
θλ =
=− t
= − 1.15 x10 − 3
∂λ
18λ 2 m
s Precision limit for viscosity
Precision Precision limit P υ (teflon sphere)
Precision Pt =
ν 2 ⋅ Sν t
M (ν k − ν )
Sν = ∑ k =1 M − 1 M 2 1/ 2 Typical Uncertainty results
Typical Teflon spheres Presentation of experimental
results: General Format
results:
EFD result: A ± UA
EFD
Benchmark data:
Benchmark • E = BA • UE2 = UA2+UB2 B ± UB • Data calibrated at UE level if:
• E ≤ UE • Unaccounted for bias
Unaccounted and precision limits if:
• E > UE 2.1
Experimental Result (UA= 3%)
Benchmark data (UB = 1.5% ) 2.0
1.9 Result R •
• 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 4.0e4 Reference data Data analysis
Data
18 20 22 24 26 28 30 32 Temperature (degrees Celsius) Compare results with
manufacturer’s data UA bands
showing %
uncertainty (Proctor & Gamble Co (1995)) Flow Visualization using ePIV
Flow
• ePIV(educational) Particle
ePIV(educational) article
Image Velocimetry
mage
• Detects motion of particles
Detects
using a camera
using
• Camera details: digital ,
Camera
30 frames/second,
600×480 pixel
resolution
• Flash details: 15mW
Flash
green continuous diode
laser
laser Results of ePIV
Results
• Identical particles are tracked in consecutive images to have
Identical
quantitative estimate of fluid flow
estimate
• Particles have the follow specifications:
Particles
• neutrally buoyant : density of SG ~ 1.0
neutrally
• small enough to follow nearly all fluid motions:
small
diameter~11μm
diameter~11
• Qualitative estimates of fluid flow can also be shown
Qualitative estimates Flow Visualization
Flow
• Visualizationa means of viewing fluid flow as a way of examining the
Visualizationa
relative motion of the fluid
relative
• Generally fluid motion is highlighted by smoke, die, tuff, particles,
Generally
shadowgraphs, MachZehnder interferometer, and many other methods
shadowgraphs, • Answer the following
Answer
questions:
questions:
1. Where is the circular cylinder?
2. In what direction is the fluid
In
traveling?
traveling?
3. Where is separation occurring?
4. Can you spot the separation
Can
bubbles?
5. What are the dark regions in the
5. What
left half of the image?
left •
•
•
• Flow VisualizationFlow around a
circular cylinder
circular Flow around a sphere is approximated by a circular cylinder
Flow in laboratory exercise has a Reynolds number less than 1.
Flow
Flow with ePIV has a Reynolds number range from ~2 to 90.
Flow
Reynolds number = Re = (V∙D)/υ = (ρ ∙ V ∙ D)/μ Re= <1
Glycerine solution with aluminum
powder, V=1.5 mm/s, dia=10 mm Re= ~2
ePIV, water and 10μm polymer
particels, V=1.5 mm/s, dia=4 mm Flow VisualizationFlow around a
circular cylinder con’t
circular Re=1.54 Re=9.6 • Flow separation occurs at Re ~ 5
• Standing eddies occur between
Standing
5 < Re < 9
Re
• Length of separation bubble is
Length
found to grow linearly with
Reynolds number until the flow
becomes unstable about Re = 40
becomes
• Sinusoidal wake develops at
Sinusoidal
about Re = 50
• Kármán vortex street develops
Kármán
around Re = 100
around Flow VisualizationFlow around a
circular cylinder con’t
circular Re=26 Re=30 Re=55 Re=140 Flow VisualizationFlow around a
circular cylinder con’t
circular
• Typical ePIV images
Re=30 Re=60 Re=90 The End
The ...
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 Fall '10
 FredrickStern

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