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ME 322: Instrumentation Lecture 40 April 30, 2014 Professor Miles Greiner • • • • Announcements/Reminders This week: Lab 12 Feedback Control HW 14 due now (Last HW assignment) Review Labs 9, 10, 11, and 12; Today and Friday Supervised Open-Lab Periods • May 2-4, 2014, 11-2 Friday through Sunday (will those times work?) • Extra Credit Lab 12.1 (due in class Monday, 5/5/2014) • See Lab 12 instructions (study effect of DT, DTi, TSP, heater and TC locations) • Check out Lab-in-a-Box for DeLaMare Library • Only 0.5% of grade • Lab Practicum Finals (May 6-14) – Guidelines, New Schedule • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Tests/Index.htm • How many of you will graduate this year (2014) or next year (2015). • If it will be later than 2015, was there something the ME Department did that delayed your graduation? Lab 9 Transient TC Response in Water and Air 100 90 Temperature, T [oC] 80 70 60 tR = 5.78 s In Room Temperature Water tA = 3.36 s In Air 50 40 30 20 tB = 0.78 s In Boiling Water 10 0 0 1 2 3 4 5 6 7 8 • Start with TC in room-temperature air • Measure its time-dependent 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 those environments (kAir or kWater) • Also calculate Biot number (dimensionless thermal size) and delay time for center to respond Time, t [sec] LabVIEW VI Dimensionless Temperature Error , , TI T Environment Temperature TF TF T(t) ℎ Initial Error EI = TF – TI Error = E = TF – T ≠ 0 TI t t = t0 • At time t = t0 a thermocouple at temperature TI is put into a fluid at temperature TF. • Theory for a uniform-temperature TC predicts: – Dimensionless Error: = = − − = − − 0 – Time Constant for a spherical thermocouple = 6ℎ Measured Thermocouple Temperature versus Time 100 90 Temperature, T [oC] 80 70 60 tR = 5.78 s In Room Temperature Water tA = 3.36 s In Air 50 40 30 20 tB = 0.78 s In Boiling Water 10 0 0 • From this chart, find 1 2 3 4 5 6 7 8 Time, t [sec] – Times when TC is placed in Boiling Water, Air and RT Air (tB, tA, tR) – Temperatures of Boiling water (maximum) and Room (minimum) (TB, TR) • Thermocouple temperature responds more quickly in water than in air • Slope does not exhibit a step change in each environment – Temperature of TC center does not response immediately • Transient time for TC center: tT ~ D2rc/kTC Type J Thermocouple Properties Effective Diameter D Density ρ [in] [kg/m3] Value 3s Uncertainty 0.059 0.006 Thermal Conductivity kTC [W/mK] Specific Heat c [J/kgK] Initial Transient Time tT [sec] 45 24 421 26 0.18 0.10 8400 530 • State estimated diameter uncertainty, 10% or 20% of D • Thermocouple material properties (next slide) – Citation: A.J. Wheeler and A.R. Gangi, Introduction to Engineering Experimentation, 2nd Ed., Pearson Education Inc., 2004, page 431. + 2 − 2 – Best estimate: = – Uncertainty: = • tT ~ D2rc/kTC; 2 =? TC Wire Properties (App. B) Dimensionless Temperature Error, 100 90 Temperature, T [oC] 80 70 60 tR = 5.78 s In Room Temperature Water tA = 3.36 s In Air 50 40 30 20 tB = 0.78 s In Boiling Water 10 0 0 1 2 3 4 5 6 7 8 Time, t [sec] • = − − = − − = −0 − – For boiling water environment, TF = TBoil, TI = TRoom – For room-temperature air and water, TF = TRoom, TI = TBoil • How can we find the time range t1 < t < t2 when decays exponentially with time? Data Transformation (trick) • = − − 0 = 0 – Where = , and b = -1/t are constants • Take natural log of both sides – ln = ln = ln + • Instead of plotting versus t, plot ln() versus t – Or, use log-scale on y-axis – During the time period when decays exponentially, this transformed data will look like a straight line Find decay constant b using Excel qBOIL = (TB-T(t))/(TB-TR) 1 0.1 For t = 1.14 to 1.27 s q = 1.867E+06e-1.365E+01t 0.01 0.8 0.9 1 1.1 1.2 1.3 1.4 Time, t [sec] • Use curser to find beginning and end times for straight-line period – q exhibits random variation when it is less than q < 0.05 • Add a new data set using those data • Use Excel to fit a y = Aebx to the selected data – For this data b = -13.65 1/s – Since b = -1/t, and t = – Calculate ℎ = − 6 6ℎ 1 =− , (power product?), ℎ 2 ℎ =? • Assume uncertainty in b is small compared to other components Dimensionless Temperature Error versus Time for Room-Temperature Air and Water 1 qRoom In Air For t = 3.83 to 5.74 sec q = 2.8268e-0.3697t In Room Temp Water For t = 5.86 to 6.00 sec q = 2E+19e-7.856t 0.1 0.01 3 3.5 4 4.5 5 5.5 6 6.5 7 Time t [sec] • Decays exponentially during two time periods: – In air: • t = 3.83 to 5.74 sec, b = -0.3697 1/s – In water: • t = 5.86 to 6.00 sec, b = -7.856 1/s. Lab 9 Results Environment Boiling Water Air Room Temperature Water h b [1/s] [W/m2C] Wh [W/m C] kFluid [W/mC] 2 NuD Lumped (Bi Bi < 0.1?) hD/kFluid hD/kTC -13.7 -0.37 12016 325 1603 43 0.680 0.026 26 19 0.403 0.011 no yes -7.86 6915 923 0.600 17 0.232 no • Water environments have orders of magnitude higher h (and b) than air – Similar to kFluid • Nusselt numbers = ℎ (power product) are more dependent on flow conditions (steady versus moving) than environment composition • Biot number = ℎ (dimensionless size) Air and Water Properties (bookmark) Lab 9 Sample Data • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab %2009%20TransientTCResponse/LabIndex.htm • Plot T vs t – Find TB, TR, tB, tA, and tR • Calculate q and plot vs time on log scale – In Boiling Water, TI = TR, TF = TB – In Room Temperature air and water, TI = TB, TF = TR – Select regions that exhibit exponential decay • Find decay constant for those regions • Calculate ℎ = − 6 and wh for each environment • For each environment calculate – NuD = – BiD = ℎ ℎ Lab 10 Vibration of a Weighted Cantilever Beam L Clamp LB E MW W T E (given) LT MT • Accelerometer Calibration Data – i.e. C = 616.7 mV/g – Use calibration constant for the issued accelerometer – Inverted Transfer function: a = V/C • During Final you will be given values and uncertainties of E, W, T • Measure (3s uncertainty) • MT, MW: Analytical balance, = 0.1 g – LB, LE, LT: Tape measure, = 1/16 in Accelerometer Table 1 Measured and Calculated Beam Properties LE LB Clamp MW W T E (given) Accelerometer LT MT Units Elastic Modulus, E [Pa] [GPa] Beam Width, W [inch] Beam Thickness, T [inch] Beam Total Length, LT [inch] End Length, LE [inch] Beam Length, LB [inch] Beam Mass, MT [g] Intermediate Mass, MI [g] Combined Mass, Mw [g] • Intermediate mass (later) Value 63 0.99 0.1832 24.00 0.38 10.00 196.8 21.9 741.2 3s Uncertainty 3 0.01 0.0008 0.06 0.06 0.06 0.1 1.5 0.1 Figure 2 VI Block Diagram Convert to Dynamic Data Convert to Dynamic Data Converts numeric, Boolean, waveform and array data types to the dynamic data type for use with Express VIs. Statistics This Express VI produces the following measurements: Time of Maximum Figure 1 VI Front Panel Disturb Beam and Measure a(t) Aluminum Steel • Use a sufficiently high sampling rate to capture the peaks – fS > 2fM – When plotting a versus t, use time increment Dt = 1/fS • Looks like = sin(2 + ) – Is b constant? • Measure f from spectral analysis ( fM ) • Find b from exponential fit to acceleration peaks Figure 4 Acceleration Oscillatory Amplitude Versus Frequency • The sampling period and frequency were T1 = 10 sec and fS = 200 Hz. – As a result the system is capable of detecting frequencies between 0.1 and 100 Hz, with a resolution of 0.1 Hz. – To plot aRMS vs t, use frequency increment Df = 1/T1 • The frequency with the peak oscillatory amplitude is fM = 8.70 ± 0.05 Hz. This frequency is easily detected from this plot. Fig. 5 Peak Acceleration versus Time Aluminum Steel • For aluminum, exponential decay changed at t = 2.46 s • During the first and second periods the decay rates are – b1 = -0.292 1/s – b2 = -0.196 1/s • Decay “constant” b was not constant Equivalent Endpoint Mass LE LB Clamp MW LT MT ME Beam Mass MB • Beam is not massless, – Its mass affects its motion and natural frequency • = + + 0.23 = + (linear sum) – = mass of weight, accelerometer, pin, nut – = + 0.23 • 2 = 2 + 2 (contribution form beam mass) + 2 + 0.23 +0.23 2 2 Beam Equivalent Spring Constant, KEQ F LB d • = = 3 3 – Power product? = 3 3 3 12 = 3 4 Predicted Frequencies • Undamped – 0 = 0 2 = 1 2 – Power Product? • Damped – = 2 = 1 2 − 2 2 = 1 2 − 2 – Power product? – If 2 ≪ , then ≈ 0 , and ≈ 0 • Measured Damping Coefficient – = −2 Table 2 Calculated Values and Uncertainties Equivalent Mass, MEQ Units Value [kg] 0.7631 3s Uncertainty 0.0005 2445 124 Equivalent Beam Spring [N/m] Constant, kEQ Predicted Undamped Frequency, foP [Hz] 9.0 0.2 Measured Damped Frequency, fM [Hz] 8.70 0.05 Decay Constant, b1 [1/sec] -0.292 - 0.45 0.00 9.0 0.2 3.5% - -0.196 - 0.30 0.00 9.0 0.2 3.5% - Damping Coefficient, lM [Ns/m] Damped Frequency, fp [Hz] Percent Difference (fP/fM-1)*100% Decay Constant, b2 [1/sec] Damping Coefficient, lM [Ns/m] Damped Frequency, fp Percent Difference (fP/fM-1)*100% [Hz] • The equivalent mass is not strongly affected by the intermediate mass • The predicted undamped and damped frequencies, fOP and fP, are essentially the same (frequency is unaffected by damping). • The confidence interval for the predicted damped frequency fP = 9.0 ± 0.2 Hz does not include the measure value fM = 8.70 ± 0.05 Hz. Time and Frequency Dependent Data • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%201 0%20Vibrating%20Beam/Lab%20Index.htm • Plot a versus t – Time increment Dt = 1/fS • Plot aRMS versus f – Frequency increment Df = 1/T1 • Measured Damped (natural) Frequency, fM – Frequency with peak aRMS 1 2 – Uncertainty = Δ = 1 21 • Exponential Decay Constant b (Is it constant?) – Show how to find acceleration peaks versus time • Use AND statements to find accelerations that are larger than the ones before and after it • Use If statements to select those accelerations and times • Sort the results by time • Plot and create new data sets before and after 2.46 sec – Fit data to y = Aebx to find b Thermal Boundary Layer for Warm Sphere in Cool Fluid Thermal Boundary Layer T TF r D • = ℎ − = • ℎ≈ ≈ ≈ − Conduction in Fluid – h increases as k increase and object sized decreases • ℎ= – = ℎ = Dimensionless Nusselt Number (power product?) Lab 9 Transient TC Response in Air & Water Wire yourself TC → Conditioner (+)TC → White Wire (-)TC → Red Stripe Conditioner to → MyDAQ Com → (-) Vout → (+) Write VI Easy Fig 1 & 2 will not be given Acquire Data Fs = 1000 Hz Ti = 8 sec At least 2 seconds in each environment. •Room temp water •Boiling water •Room temp air •Room temp water Fig 3 Plot T Vs. t ID time tB , tA , tR ID Temp TRoom = Tmin TBoil = Tmax Fig 4 For boiling water vs. t Identify: Start & end times of exponential decay period (looks linear) •Select exp decay data y •Add data to plot to that data •Fit Show results on the plot •Find b Units s-1 Fig 5 Room Temp Air & Water vs. t Find Find Table 2 Lab 10: Vibrating Beam You will be given beam and its E and WE VI fig 1 &2 Table 2 Undamped Predicted Frequency if b = 0, λ = 0 Measured Damping Coeff If Then Wfp ≈ Wfop Is