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

Table of contents Purpose Experimental design Experimental process Test Setup Data acquisition Data reduction Uncertainty analysis Data analysis

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.

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

Experimental process

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.

Data 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.

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 <<1), where D is the sphere diameter, r fluid is the density of the fluid, r sphere is the density of the falling sphere, is the viscosity of the fluid, F d, F b, and F g, 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.) 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.

Data reduction (contd.) Sheet 1 Sheet 2

Experimental Uncertainty Assessment Uncertainty analysis (UA): rigorous methodology for uncertainty assessment using statistical and engineering concepts. 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 ASME (1998) and AIAA (1999) standards are the most recent updates of UA methodologies, which are internationally recognized as summarized in IIHR Error: difference between measured and true value.Error: difference between measured and true value. Uncertainties (U): estimate of errors in measurements of individual variables X i (U x i ) or results (U r ) obtained by combining U x i. Uncertainties (U): estimate of errors in measurements of individual variables X i (U x i ) or results (U r ) obtained by combining U x i. Estimates of U made at 95% confidence level. Estimates of U made at 95% confidence level.

Bias error  :  Fixed and systematic Precision error  :Precision error  :  ± and random   Total error:   Definitions

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

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 (X 1, X 2, …, X j ) 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 (X 1, X 2, …, X j ) for r at a fixed test condition with the same measurement system

Uncertainty equations for single and multiple tests The total uncertainty of the result B r : same estimation procedure for single and multiple tests P r : determined differently for single and multiple tests

Uncertainty equations for single and multiple tests: bias limits B r : B i : estimate of calibration, data acquisition, data reduction, conceptual bias errors for X i.. Within each category, there may be several elemental sources of bias. If for variable X i there are J significant elemental bias errors [estimated as (B i ) 1, (B i ) 2, … (B i ) J ], the bias limit for X i is calculated as Bike: estimate of correlated bias limits for X i and X k Sensitivity coefficients

Precision limits for single test Precision limit of the result (end to end): t: coverage factor (t = 2 for N > 10) S r : the standard deviation for the N readings of the result. S r must be determined from N readings over an appropriate/sufficient time interval Precision limit of the result (individual variables): the precision limits for X i Often is the case that the time interval for collecting the data is inappropriate/insufficient and P i ’s or P r ’s must be estimated based on previous readings or best available information

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 Precision limits for multiple test

Data reduction equation for density  Data reduction equation for density  Total uncertainty for the average density: Uncertainty Analysis - density

Bias Limit for Density Bias limit B  Bias limit B  Sensitivity coefficients Correlated Bias : two variables are measured with the same instrument

Precision limit for density Precision limit

Typical Uncertainty results

Uncertainty Analysis - Viscosity Data reduction equation for density  Total uncertainty for the average viscosity (teflon sphere):

Calculating Bias Limit for Viscosity Bias limit B t  (teflon sphere) Sensitivity coefficients: No Correlated Bias errors contributing to viscosity

Precision limit for viscosity Precision limit P υ (teflon sphere)

Typical Uncertainty results Teflon spheres

Presentation of experimental results: General Format EFD result: A ± U AEFD result: A ± U A Benchmark data: B ± U BBenchmark data: B ± U B E = B-A E = B-A U E 2 = U A 2 +U B 2 U E 2 = U A 2 +U B 2 Data calibrated at U E level if: Data calibrated at U E level if: |E|  U E |E|  U E Unaccounted for bias and precision limits if: Unaccounted for bias and precision limits if: |E| > U E |E| > U E

Data analysis (Proctor & Gamble Co (1995)) UA bands showing % uncertainty Compare results with manufacturer’s data

Flow Visualization using ePIV ePIV-(educational) Particle Image VelocimetryePIV-(educational) Particle Image Velocimetry Detects motion of particles using a cameraDetects motion of particles using a camera Camera details: digital, 30 frames/second, 600×480 pixel resolutionCamera details: digital, 30 frames/second, 600×480 pixel resolution Flash details: 15mW green continuous diode laserFlash details: 15mW green continuous diode laser

Results of ePIV Identical particles are tracked in consecutive images to have quantitative estimate of fluid flowIdentical particles are tracked in consecutive images to have quantitative estimate of fluid flow Particles have the follow specifications:Particles have the follow specifications: neutrally buoyant : density of SG ~ 1.0neutrally buoyant : density of SG ~ 1.0 small enough to follow nearly all fluid motions: diameter~11μmsmall enough to follow nearly all fluid motions: diameter~11μm Qualitative estimates of fluid flow can also be shown Qualitative estimates of fluid flow can also be shown

Flow Visualization Visualization-a means of viewing fluid flow as a way of examining the relative motion of the fluidVisualization-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 methodsGenerally fluid motion is highlighted by smoke, die, tuff, particles, shadowgraphs, Mach-Zehnder interferometer, and many other methods Answer the following questions:Answer the following questions: 1.Where is the circular cylinder? 2.In what direction is the fluid traveling? 3.Where is separation occurring? 4.Can you spot the separation bubbles? 5.What are the dark regions in the left half of the image?

Flow Visualization-Flow around a circular 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)/υ = (ρ ∙ V ∙ D)/μReynolds number = Re = (V∙D)/υ = (ρ ∙ V ∙ D)/μ Glycerine solution with aluminum powder, V=1.5 mm/s, dia=10 mm ePIV, water and 10μm polymer particels, V=1.5 mm/s, dia=4 mm Re= <1 Re= ~2

Flow Visualization-Flow around a circular cylinder con’t Flow separation occurs at Re ~ 5Flow separation occurs at Re ~ 5 Standing eddies occur between 5 < Re < 9Standing eddies occur between 5 < Re < 9 Length of separation bubble is found to grow linearly with Reynolds number until the flow becomes unstable about Re = 40Length 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 = 50Sinusoidal wake develops at about Re = 50 Kármán vortex street develops around Re = 100Kármán vortex street develops around Re = 100 Re=9.6 Re=1.54

Flow Visualization-Flow around a circular cylinder con’t Re=26Re=55 Re=140Re=30

Flow Visualization-Flow around a circular cylinder con’t Typical ePIV imagesTypical ePIV images Re=30 Re=90Re=60

The End