Measurement of density and kinematic viscosity

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Measurement of density and kinematic viscosity

• S. Ghosh, M. Muste, F. Stern

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Table of contents

• Purpose
• Experimental design
• Experimental process
• Test Setup
• Data acquisition
• Data reduction
• Uncertainty analysis
• Data analysis

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Purpose

• 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.

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Experimental design

• Viscosity is a thermodynamic property and varies
  with pressure and temperature.
• Since the term m/r, where r is the density of the
  fluid, frequently appears in the equations of
  fluid mechanics, it is given a special name,
  Kinematic viscosity (n).
• We will measure the kinematic viscosity through
  its effect on a falling object.

• The facility includes
• A transparent cylinder containing
• glycerin.
• Teflon and steel spheres of different
• diameters
• Stopwatch
• Micrometer
• Thermometer

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Experimental process
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Test 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.

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Data Acquisition

• Experimental procedure
• Measure room temperature.
• Measure ?.
• Measure sphere diameter using micrometer.
• Release sphere at fluid surface and then release
  gate handle.
• Release teflon and steel spheres one by one.
• Measure time for each sphere to travel ?.
• Repeat steps 3-6 for all spheres. At least 10
  measurements are required for each sphere.

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Data 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 ltlt1),
• where D is the sphere diameter, rfluid
  is the density of the fluid, rsphere is the
  density of the falling sphere, n 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).

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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 l.
• 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.

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Data reduction (contd.)
Sheet 2
Sheet 1
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Experimental Uncertainty 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.
• Uncertainties (U) estimate of errors in
  measurements of individual variables Xi (Uxi)
  or results (Ur) obtained by combining Uxi.
• Estimates of U made at 95 confidence level.

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Definitions

• Bias error b
• Fixed and systematic
• Precision error e
• and random
• Total error d b e

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Propagation of errors
Block diagram showing elemental error
sources, individual measurement systems
measurement of individual variables, data
reduction equations, and experimental results
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Uncertainty equations for single and 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

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Uncertainty equations for single and multiple
tests

• The total uncertainty of the result

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

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Uncertainty equations for single and multiple
tests bias limits

• Br

• Sensitivity coefficients

• 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

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Precision limits for single test

• Precision limit of the result (end to end)

t coverage factor (t 2 for N gt 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)

the precision limits for Xi
Often is the case that the time interval for
collecting the data is inappropriate/insufficient
and Pis or Prs must be estimated based on
previous readings or best available information
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Precision limits for multiple test

• The average result

• Precision limit of the result (end to end)

t coverage factor (t 2 for N gt 10)
standard deviation for M readings of the result

• The total uncertainty for the average result
• Alternatively can be determined by RSS
  of the precision limits of the individual
  variables

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Uncertainty Analysis - density

• Data reduction equation for density r

• Total uncertainty for the average density

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Bias Limit for Density
Correlated Bias two variables are measured with
the same instrument

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Precision limit for density
Precision limit
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Typical Uncertainty results
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Uncertainty Analysis - Viscosity
Data reduction equation for density n

Total uncertainty for the average viscosity
(teflon sphere)
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Calculating Bias Limit for Viscosity
No Correlated Bias errors contributing to
viscosity
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Precision limit for viscosity
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Typical Uncertainty results
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Presentation of experimental results General
Format

• EFD result A UA
• Benchmark data B UB
• E B-A
• UE2 UA2UB2
• Data calibrated at UE level if
• E ? UE
• Unaccounted for bias and precision limits if
• E gt UE

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Data analysis
Compare results with manufacturers data
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Flow Visualization using ePIV

• ePIV-(educational) Particle Image Velocimetry
• Detects motion of particles using a camera
• Camera details digital , 30 frames/second,
  600480 pixel resolution
• Flash details 15mW green continuous diode laser

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Results of ePIV

• Identical particles are tracked in consecutive
  images to have quantitative estimate of fluid
  flow
• Particles have the follow specifications
• neutrally buoyant density of SG 1.0
• small enough to follow nearly all fluid motions
  diameter11µm
• Qualitative estimates of fluid flow can also be
  shown

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Flow Visualization

• Visualization-a means of viewing fluid flow as a
  way of examining the relative motion of the fluid
• Generally fluid motion is highlighted by smoke,
  die, tuff, particles, shadowgraphs, Mach-Zehnder
  interferometer, and many other methods

• Answer the following questions
• Where is the circular cylinder?
• In what direction is the fluid traveling?
• Where is separation occurring?
• Can you spot the separation bubbles?
• What are the dark regions in the left half of the
  image?

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Flow Visualization-Flow around a circular cylinder

• Flow around a sphere is approximated by a
  circular cylinder
• Flow in laboratory exercise has a Reynolds number
  less than 1.
• Flow with ePIV has a Reynolds number range from
  2 to 90.
• Reynolds number Re (VD)/? (? V D)/µ

Re 2
Re lt1

• Glycerine solution with aluminum
• powder, V1.5 mm/s, dia10 mm

• ePIV, water and 10µm polymer
• particels, V1.5 mm/s, dia4 mm

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Flow Visualization-Flow around a circular
cylinder cont

• Flow separation occurs at Re 5
• Standing eddies occur between 5 lt Re lt 9
• Length of separation bubble is found to grow
  linearly with Reynolds number until the flow
  becomes unstable about Re 40
• Sinusoidal wake develops at about Re 50
• Kármán vortex street develops around Re 100

Re1.54
Re9.6
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Flow Visualization-Flow around a circular
cylinder cont
Re26
Re55
Re140
Re30
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Flow Visualization-Flow around a circular
cylinder cont

• Typical ePIV images

Re30
Re90
Re60
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The End