lab1_lecture - Measurement of density and kinematic...

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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 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 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 set-up 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 3-6 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 = B-A • 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 4.0e-4 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 • Visualization-a means of viewing fluid flow as a way of examining the Visualization-a relative motion of the fluid relative • Generally fluid motion is highlighted by smoke, die, tuff, particles, Generally shadowgraphs, Mach-Zehnder 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 Visualization-Flow 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 Visualization-Flow 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 Visualization-Flow around a circular cylinder con’t circular Re=26 Re=30 Re=55 Re=140 Flow Visualization-Flow around a circular cylinder con’t circular • Typical ePIV images Re=30 Re=60 Re=90 The End The ...
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