ME 322: Instrumentation Lecture 23

1. ME 322: InstrumentationLecture 23 March 16, 2012 Professor Miles Greiner

2. Announcements/Reminders • HW 8 is due now • Midterm II, April 2, 2014 • Next week is Spring Break!

3. So far in this course… • Quad Area Measurement • Multiple, independent measurements of the same quantity don’t give the same results (random and systematic errors, mean, standard deviation) • Steady Measurements • Pressure Transducer Static Calibration • Metal Elastic Modulus • Fluid Speed and Volume Flow Rate • Boiling Water Temperature • Discrete sampling of time varying signals using computer data acquisition (DAQ) systems • Allows us to acquire unsteady outputs versus time • LabVIEW, derivatives, spectral analysis

4. Transient Instrument Response • Can measurement devices follow rapidly changing measurands? • temperature • pressure • speed

5. Lab 9 Transient Thermocouple Response T Environment Temperature TF Faster Slower TC Initial Error EI = TF – TI Error = E = TF – T ≠ 0 TI TI t t = t0 • At time t = t0 a small thermocouple at initial temperature TI is put into boiling water at temperature TF. • How fast can the TC respond to this step change in its environment temperature? • What causes the TC temperature to change? • What affects the time it takes to reach TF? TF T(t)

6. Heat Transfer from Water to TC Surface Temp TS(t) Fluid Temp TF Q [J/s = W] • Convection heat transfer rate Q [W] is affected by • Temperature difference between water and thermocouple surface, TF– TS(t) • Assume TF is constant but TS(t) changes with time • TC Surface Area, A • Linear convection heat transfer coefficient, h • Affected by • Water thermal conductivity k, density r, specific heat c • Water motion • Units [h] = • Q = [TF – TS(t)]Ah D=2r T(t,r)

7. Energy Balance (1st law) • Internal energy of an incompressible TC • U = mcTA = rVcTA • r = density • c =specific heat • TA = Average TC temperature (may not be isothermal) • -) • TA and TS change with time t

8. For a Uniform Temperature TC • When is this a good assumption? (later) • -) • For a sphere: • Units • TC time constant (assumes h is constant)

9. Solution • ID: 1storder linear differential equation (separable) • Error decays exponentially with time • Let be the dimensionless temperature error

10. Dimensionless Temperature Error • To get (TF-T) ≤ 0.37(TF – TI) • Wait for time t - tI ≥ t = • For fast response use • small rc(material properties) • Small D (use small diameter wire to make TC) • Large h • Increase mixing • High conductivity fluid 0.37 0.14 0.05 0.011

11. Prediction versus Measurement TF • Theoretical Solution: • What is different between the theory (expectation) and the measurements? • Why doesn’t the measured temperature slope exhibit a step change at t = tI • Is exactly true? • Does the temperature at the thermocouple center responds as soon as it is placed in the water • How long will it take before the center responds? TI tI

12. Semi-Infinite Body Transient Conduction T1 t = 0 ∞ • Consider a very large body whose surface temperature changes at t = 0 • Thermal penetration depth, which exhibits a temperature change, grows with time • Thermal diffusivity: (material property) • How long will it take for thermal penetration depth to reach TC center? • D (order of magnitude) • D2/k • After t > ~ttthe TC center temperature starts to change. • Does the average temperature then follow the expected time-dependent shape? Ti d

13. After t > tt, is TC temperature uniform? DTCONV T T DTCONV DTTC • When is DTTC << DTCONV (uniform temp TC)? • Balance conduction and convection • If BiD < 0.1, (small D or large kTC ) • Then (lumped body) r r DTTC

14. Lab 9 Transient TC Response in Water and Air • Start with TC in air • Measure its temperature when it is plunged into boiling water, then room temperature air, then room temperature water • Determine the heat transfer coefficients in the three environments , hBoiling, hAir, and hRTWater • Compare each h to the thermal conductivity of the environment (kAir or kWater)

15. Measured Thermocouple Temperature versus Time • Thermocouple temperature responds much more quickly in water than in air • How to determine h all three environments?

16. Dimensionless Temperature Error • For boiling water environment, TF = TBoil, TI = TAir • During what time range t1 < t < t2 does decay exponentially with time? • Once we find that, how do we find t?

17. Data Transformation (trick) • Reformulate: • Where , and b = -1/t • Take natural log of both sides • Instead of plotting versus t, plot ln() vs t • Or, use log scale on y axis • During the time period when decays exponentially, this transformed data will look like a straight line • Use least-squares to fit a line to that data • Slope = b = -1/t, Intercept = ln(A) • Since t= , then calculate

18. TC Wire Properties (App. B) • Best estimate: • Uncertainty:

19. Table 1 Thermocouple Properties • The diameter uncertainty is estimated to be 10% of its value. • Thermocouple material properties values are the average of Iron and Constantan values. The uncertainty is half the difference between these values. The values were taken from [A.J. Wheeler and A.R. Gangi, Introduction to Engineering Experimentation, 2nd Ed., Pearson Education Inc., 2004, page 431] • The time for the effect of a temperature change at the thermocouple surface to cause a significant change at its center is tT = D2rc/kTC. Its likely uncertainty is calculated from the uncertainty in the input values.

20. Lab 9 Sample Data • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2009%20TransientTCResponse/LabIndex.htm • Plot T vs t • Find Tboil and Tair • Calculate q and plot q vs time on log scale • Select regions that exhibit exponential decay • Find decay constant for those regions • Calculate h and wh for each environment • Calculate NuD, BiD

21. Lab 9 • Place TC in (1) Boiling water TB, Room temperature air TA, and Room temperature water TW • Plot Temperature versus time • Why doesn’t TC temperature versus time slope exhibit a sudden change when it is placed in different environments?

22. Fig. 4 Dimensionless Temperature Error versus Time in Boiling Water • The dimensionless temperature error decreases with time and exhibits random variation when it is less than q < 0.05 • The q versus t curve is nearly straight on a log-linear scale during time t = 1.14 to 1.27 s. • The exponential decay constant during that time is b = -13.65 1/s.

23. Fig. 5 Dimensionless Temperature Error versus Time t for Room Temperature Air and Water • The dimensionless temperature error decays exponentially during two time periods: • In air: t = 3.83 to 5.74 s with decay constant b = -0.3697 1/s, and • In room temperature water: t = 5.86 to 6.00s with decay constant b = -7.856 1/s.

24. Table 2 Effective Mean Heat Transfer Coefficients • The effective heat transfer coefficient is h = -rcDb/6. Its uncertainty is 22% of its value, and is determined assuming the uncertainty in b is very small. • The dimensional heat transfer coefficients are orders of magnitude higher in water than air due to water’s higher thermal conductivity • The Nusselt numbers NuD (dimensionless heat transfer coefficient) in the three different environments are more nearly equal than the dimensional heat transfer coefficients, h. • The Biot Bi number indicates the thermocouple does not have a uniform temperature in the water environments

26. So far in this course… • Quad Area Measurement • Multiple, independent measurements of the same quantity don’t give the same results (random and systematic errors, mean, standard deviation) • Steady Measurements • Pressure Transducer Static Calibration • Transfer Functions, Linear regression, Standard Error of the Estimate • Metal Elastic Modulus • Strain Gage/Wheatstone Bridge, Propagation of Uncertainty • Fluid Speed and Volume Flow Rate • Pitot-Static Probes, Venturi Tubes • Boiling Water Temperature • Thermocouples • Discrete sampling of time varying signals using computer data acquisition (DAQ) systems • Allows us to acquire unsteady outputs versus time • LabVIEW, derivatives, spectral analysis