ME 322: Instrumentation Lecture 13: Exam Review

1. ME 322: InstrumentationLecture 13: Exam Review February 19, 2014 Professor Miles Greiner

2. Announcements/Reminders • Labs • This week: Lab 6 Elastic Modulus Measurement • Next Week: No Lab • In two weeks: Lab 6 Wind Tunnel Flow Rate and Speed • Only 4 wind tunnels • Sign-up for 1.5 hour slots with your partner this week in lab

3. Midterm 1 Friday • Open book, plus bookmarks, plus one page of notes • If you have an e-book, you must turn off internet • 4 problems, some have parts • each part like HW or Lab calculations • Be able to use your calculator • sample average and sample standard deviation • linear regression YFIT = aX + b • Review Session • Josh McGuire, DMSC 103, Thursday 2/20/14 from 5pm – 6:30pm • Handout: last year’s midterm problems • These problems will not be on the exam • Neither Josh nor I will not provide answers or solutions for this • See me after class today regarding special needs

4. Multiple Measurements of a Quantity • Do not always give the same results. • Affected by Uncontrolled (random) and Calibration (systematic) errors. • Patterns are observed if enough measurements are acquired • Bell-shaped probability distribution function • The sample may exhibit a center (mean) and spread (standard deviation) • Statistical analysis can be applied to this “randomly varying” process Quad Area [m2]

5. Statistics • Find properties of an entire population of size N (which can be ∞) using a smaller sample of size n < N. • Sample Mean • Sample Standard Deviation • How can we use these statistics? • The standard deviation characterizes the measurement imprecision (repeatability) • The mean characterizes the best estimate of the measurand

6. Example Problem • Find the probability that the next sample will be within the range x1 ≤ x ≤ x2 • Let (# of SDs from mean) • I(z) on Page 146 • Useful facts: I(0) = 0, I(∞) = 0.5, I(-z) = -I(z)

7. “Typical” Problems • Find the probability the next value is within a certain amount of the mean (symmetric) • Find the probability the next value is below (or above) a certain value • If one more value is acquired, what is the likelihood it is above the mean? • How much must be added to that value to a specified-likelihood above the mean?

8. Instrument Calibration • Experimental determination of instrument transfer function • Record instrument reading y for a range of measurands x (determined by a standard) • Use the least squares method to fit a line yF = ax + b (or some other function) to the data. • Hint: Use calculator to find a and b unless told (remember Units) • Determine the standard error of the estimate of the Reading for a given Measurand • Hint: Lean to calculate this efficiently (use table format)

9. To use the calibration • Make a measurement and record instrument reading, • Invert the transfer function to find the best estimate of the measurand • Determine standard error of the estimate of the Measurand for a given Reading • sx,y = sy,x/a (Units!) • Confidence interval • (Units!) • Or • Calibration • Removes calibration (bias, systematic) error • Quantifies imprecision (random error) but does not remove it

10. Stand. Dev. of Best-Fit Slope and Intercept • = (68%) • = (68%) • Not in the textbook • wa= ?sa (95%)

11. Propagation of Uncertainty • Consider a calculation based on uncertain inputs • R = fn(x1, x2, x3, …, xn) • For each input xi find the best estimate for its value , and its uncertainty with a certainty-level (probability) of pi • Note: pi increases with wi • The best estimate for the results is: • …,) • The confidence interval for the result is • Find

12. Statistical Analysis Shows • In this expression • Confidence-level for all the wi’s, pi (i = 1, 2,…, n) must be the same • Confidence level of wR,Likely, pR = pi is the same at the wi’s

13. General Power Product Uncertainty • If where a and ei are constants • The likelyfractional uncertainty in the result is • Square of fractional error in the result is the sum of the squares of fractional errors in inputs, multiplied by their exponent. • The maximum fractional uncertainty in the result is • (100%) • We don’t use maximum errors much in this class

14. Instruments

15. U-Tube Manometer • Power product? Measurand Reading DP = 0 Fluid Air (1 ATM, 27°C) Water (30°C) Hg (27°C) 1.774 995.7 13,565

16. Inclined-Well Manometer R If and

17. Strain Gages • Electrical resistance changes by small amounts when • They are strained (desired sensitivity) • Strain Gage Factor: • Their temperature changes (undesired sensitivity) • Solution: • Subject “identical” gages to the same environment so they experience the same temperature change and the same temperature-associated resistance change. • Incorporate gages into a Wheatstone bridge circuit that cancels-out the temperature effect

18. Wheatstone Bridge Output Voltage, VO - + • When R1 R3 R2 R4, then • Small changes in Ri cause small changes in • If gages are in all 4 legs • with (S and ST same) + - R3

19. Quarter Bridge + - - + R3 • Only one leg (R3) has a strain gauge • Other legs are fixed resistors Undesired Sensitivity

20. Half Bridge + - • Wire gages at R2 (-) and R3 (+) • Place R3 on deform specimen; ε3, ΔT3 • Place R2 on identical but un-deformed; ε2=0, ΔT2=ΔT3 Automatic temperature compensation - + R3

21. Beam in Bending: Half Bridge ε3 • Twice the output amplitude as quarter bring, with temperature compensation ε2 = -ε3 ε2 = -ε3

22. Beam in Bending: Full Bridge 3 1 + - • V0 is 4 times larger than quarter bridge • And has temperature compensation. - + R3 2 4 = -e3 = e3 = -e3 = DT3 = DT3 = DT3

23. Tension Configuration (HW) ε1 = ε3 ε4 = ε2 = -υ ε3 + - 2 3 R3 - + 4 1

24. Beam Surface Strain W T • Bending: L y F Neutral Axis σ • Tension: • Could be used for force-measuring devices F

25. Fluid Speed V (Pressure Method) PT > PS PT > PS PS PS • Pitot Tube Transfer function: • To use: • C accounts for viscous effects, which are small • Assume C = 1 unless told otherwise • Less uncertainty for larger V than for small ones V

26. How to Find Density • Ideal Gases • P = PS = Static Pressure • R = Gas Constant = RU/MM • Ru = Universal Gas Constant = 8.314 kJ/kmol K • MM = Molar Mass of the flowing Gas • T = Absolute Temperature = T[°C] + 273.15 • Can plug this into speed formula • Liquids • Tables

27. Water Properties (Appendix B of Text)

28. Volume Flow Rate, Q Variable-Area Meters Nozzle Venturi Tube Orifice Plate • Measure pressure drop at specified locations • Diameter in pipe D, at throat d • Diameter Ratio: b = d/D < 1 • Ideal (inviscid) transfer function: • Less uncertainty for larger Q than for small ones

29. To use • Invert the transfer function: • C = Discharge Coefficient • C = fn(ReD, b = d/D, exact geometry and port locations) • Need to know Q to find Q, so iterate • Assume C ~ 1, find Q, then Re, then C and check…

30. Discharge Coefficient Data from Text • Nozzle: page 344, Eqn. 10.10 • C = 0.9975 – 0.00653 (see restrictions in Text) • Orifice: page 349, Eqn. 10.13 • C = 0.5959 + 0.0312b2.1 - 0.184b8+ (0.3 < b < 0.7)

32. Correlation Coefficient Student T If N >30 use student t