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ME 322: Instrumentation Lecture 24 March 24, 2014 Professor Miles Greiner Announcements/Reminders • This week: Lab 8 Discretely Sampled Signals – Next Week: Transient Temperature Measurements • HW 9 is due Monday • Midterm II, Wednesday, April 2, 2014 – Review Monday • Extra Credit Opportunity, Friday, April 18, 2014 – Introduction to LabVIEW and Computer-Based Measurements Hands-On Seminar • NI field engineer Glenn Manlongat will walk through the LabVIEW development environment • 1% of grade extra credit for actively attending • Time, place and sign-up “soon” Transient Thermocouple Measurements • Can a the temperature of a thermocouple (or other temperature measurement device) accurately follow the temperature of a rapidly changing environment? Lab 9 Transient TC Response in Water and Air • Start with TC in room-temperature air • Measure its time-dependent temperature when it is plunged into boiling water, then room temperature air, then roomtemperature 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) Dimensionless Temperature Error , , TI T Environment Temperature TF TF Initial Error EI = TF – T I T(t) ℎ 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. – Error: E = TF – T • Theory for a lumped (uniform temperature) TC predicts: – Dimensionless Error: = – = 6ℎ = (spherical thermocouple) − − = − − Lab 9 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 • However, 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 • = − − = − − = − − – For boiling water environment, TF = TBoil, TI = TRoom • During what time range t1<t<t2 does decay exponentially with time? – Once we find that, how do we find t? Data Transformation (trick) • = − − = − = – 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 To find decay constant b using Excel • Use curser to find beginning and end times for straight-line period • Add a new data set using those data • Use Excel to fit a y = Aebx to the selected data – This will give b = -1/t – Since t = 6ℎ – Calculate ℎ = 1 =− , − 6 (power product?), ℎ 2 ℎ =? • Assume uncertainty in b is small compared to other components • What does convection heat transfer coefficient depend on? Thermal Boundary Layer for Warm Sphere in Cool Fluid Thermal Boundary Layer TF T r D • = ℎ − = • ℎ≈ ≈ ≈ − Conduction in Fluid – h increases as k increase and object sized decreases • ℎ= – = ℎ = Dimensionless Nusselt Number Lab 9 Sample Data • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab %2009%20TransientTCResponse/LabIndex.htm • Plot T vs t – Find TB 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 and wh for each environment • For each environment calculate – NuD = – BiD = ℎ ℎ Fig. 4 Dimensionless Temperature Error versus Time in Boiling Water 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] • • 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. Fig. 5 Dimensionless Temperature Error versus Time t 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] • 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. 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 • Heat Transfer Coefficients vary by orders of magnitude – Water environments have much higher h than air – Similar to kFluid • Nusselt numbers are more dependent on flow conditions (steady versus moving) than environment composition Air and Water Thermal Conductivities Appendix B • kAir (TRoom) • kwater (TRoom, TBoiling) Lab 9 Extra Credit • Measure time-dependent heat transfer rate Q(t) to/from the TC (when TC is placed into boiling water) • 1st Law – – 2 − = = 6 +Δ −(−Δ ) t = 2Δ – Differentiation time step Δ = mΔ • Sampling time step Δ • Integer m – What is the best value of m? Measurement Results 3.5 Dt = 0.001 sec Dt = 0.01 sec Dt = 0.1 sec Dt = 0.05 sec 3 Heat Transfer Q [W] 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0.6 0.8 1 1.2 1.4 1.6 Time t [sec] • Choice of DtD is a compromise between eliminating noise and responsiveness What Do We Expect? Expected for Uniform Temperature TC T F = TB Ti = TR Measured t0 What do we measure? Expected for uniform temperature TC Q Measured tT t ti Sinusoidally-Varying Environment Temperature TENV TTC • For example, a TC in a car exhaust line • Eventually the TC will have – The same average temperature and unsteady frequency as the environment temperature – However, its unsteady amplitude will be less than the environment temperature’s. Heat Transfer from Fluid to TC Fluid Temp TF(t) Q =hA(T – T) T D=2r • Environment Temperature: TE = M + Asin(wt) • − = = = ℎ − – Divided by hA and – Let the TC time constant be = • ℎ = 6ℎ (for sphere) + = – 1st order, linear differential equation, non-homogeneous Solution • + = = + () • Solution has two parts – Homogeneous and non-homogeneous (particular): – T = TH+TP • Homogeneous solution – + = 0 − – Solution: = – Decays with time, not important as t∞ • Particular Solution to whole equation – Assume = + + () • = − () • Plug into non-homogeneous differential equation to find constants C, D and E Particular Solution • + = + () • Plug in assumed solution form: – ( − ()) + + + () = + () • Collect terms: – + + )(− + − + − = 0 =0 =0 • Find C, D and E in terms of A and M – C=M – E = − – 2 + = ; • = – E= 2 +1 − 2 +1 =0 Result • = + + () • = + 2 +1 • = + [ 2 +1 • = + 2 +1 + − () 2 +1 − () ] − – where tan() = Compare to Environment Temperature tD T • = + () • = + 2 +1 − ; tan() = • Same mean value • If 1 >> =2 = • ≫ 2 2 • Then = + – Minimal attenuation and phase lag • Otherwise • = = < 2 +1 arctan() 2 = arctan() 360 (since 0 = − ) Example • A car engine runs at f = 1000 rpm. A type J thermocouple with D = 0.1 mm is placed in one of its cylinders. How high must the convection coefficient be so that ATC = 0.5 AENV? • If the combustion gases may be assumed to have the properties of air at 600C, what is the required Nusselt number? Can we measure time-dependent heat transfer rate, Q vs. t, to/from the TC? 1st Law Differential time step Measurement Results 3.5 Dt = 0.001 sec Dt = 0.01 sec Dt = 0.1 sec Dt = 0.05 sec 3 Heat Transfer Q [W] 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0.6 0.8 1 1.2 1.4 1.6 Time t [sec] • Choice of dtD is a compromise between eliminating noise and responsiveness Lab 9 Transient Thermocouple Response T TI Environment Temperature TF TF T(t) Faster Initial Error EI = TF – T I Slower TC Error = E = TF – T ≠ 0 TI t = t0 t • 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? VI Lab 9 1) Setup TC-Signal Conditioner MyDAQ Boiling water, Room temp water, Air, Measure TC “diameter” D = ~(0.1 0.2), 2) VI = (volts) × 40°c Volt 3) Data Acquisition f = 1000Hz T1 = 8 s 2 sec in each a) Boiling water b) Air at room temp c) Water at room temp Save data Initial Transient Time Note: Slope does not exhibit a step change when TC enters new environment. Predict order of magnitude of initial transient time tT for TC center to begin to respond. ~ ~ 2 2 = = Pg 455 Properties for TC , , , , →average for Iron and const , , Transform Trick Straight Line Fit data 1 < < 2 1 = −13.67 1 −6ℎ = = ℎ ℎ 2 = 2 + 2 + Same for air & water 2 + 2 Small Lab 9 Find h in: Boiling Water Room Temper Air and water Why does h vary so much in different environments? Water, Air What does h depend on? = ℎ − = ≈ − T ℎ≈ ≈ ℎ= T D NuD ≡ Nusselt number TF r