EFD_UA_Example - Experimental Uncertainty Assessment...

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Unformatted text preview: Experimental Uncertainty Assessment Methodology: Example for Measurement of Density and Kinematic Viscosity F. Stern, M. Muste, M-L 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. A-23): 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 = B-A 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) 1.4e-3 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) 1.2e-3 1.0e-3 8.0e-4 6.0e-4 4.0e-4 1.6e-3 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) 1.6e-3 1.4e-3 1.2e-3 1.0e-3 8.0e-4 6.0e-4 4.0e-4 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., M-L. 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, McGraw-Hill, Inc., White, Fluid New York, NY. New ...
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