ME 322: Instrumentation Lecture 23 March 11, 2016 Professor Miles Greiner Transient TC response , Modeling, expected and observed behaviors, Lab 9, Plot transformation, Heat transfer measurement

Announcements/Reminders HW 8 is due now Any problem accessing LabVIEW? How did plotting and LabVIEW programing go? Midterm II, March 30, 2016 After Spring Break 3x5 Card Survey Please give me you input Constructive criticism and suggestions welcomed

So far in this course… Quad Area Measurement Steady Measurements 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 Material Strain and 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 Problems finding derivatives and spectral content

Transient Instrument Response Can measurement devices follow rapidly-changing measurands? temperature pressure speed

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

Heat Transfer from Water to TC Surface Temp TS(t) Fluid (water) Temp TF Q [J/s = W] D=2r T(t,r) 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 Convection heat transfer coefficient, h, is affected by fluid Properties: thermal conductivity k, density r, specific heat c Motion Q = [TF – TS(t)]Ah Units [h] = 𝑊 𝑚 2 𝐾 Can we predict TC temperature versus time?

Energy Balance (1st law) 𝑄−𝑊= 𝑑𝑈 𝑑𝑡 =ℎ𝐴( 𝑇 𝐹 − 𝑇 𝑆 ) Internal energy of an incompressible TC U = mcTA = rVcTA r = density c = specific heat TA = Average TC temperature (may not be isothermal) 𝑑(𝜌𝑉𝑐 𝑇 𝐴 ) 𝑑𝑡 =𝜌𝑉𝑐 𝑑 𝑇 𝐴 𝑑𝑡 =ℎ𝐴( 𝑇 𝐹 − 𝑇 𝑆 ) 𝑇 𝐴 and 𝑇 𝑆 change with time 𝑡

For a Uniform Temperature TC 𝑇 𝑠 = 𝑇 𝐴 =𝑇 Good assumption for a “small” and “high-conductivity” TC 𝜌𝑉𝑐 𝑑𝑇 𝑑𝑡 =𝐴ℎ( 𝑇 𝐹 -𝑇) 𝑑𝑇 𝑑𝑡 =− (𝑇− 𝑇 𝐹 ) 𝜌𝑉𝑐 𝐴ℎ =− (𝑇− 𝑇 𝐹 ) 𝜏 Let: 𝜏 = 𝜌𝑉𝑐 𝐴ℎ For a sphere: V= 4 3 𝜋 𝑟 3 ; A= 4𝜋𝑟 2 ; 𝜏 = 𝜌( 4 3 𝜋 𝑟 3 )𝑐 ( 4𝜋𝑟 2 )ℎ = 𝜌𝑐𝑟 3ℎ = 𝜌𝑐𝐷 6ℎ Units 𝑘𝑔 𝑚 3 𝐽 𝑘𝑔𝐾 𝑚 𝑚 2 𝐾 𝑊 =𝑠𝑒𝑐 𝜏 is TC time “constant” (assuming h is constant) “Close” to a sphere

Solution 𝑑𝑇 𝑑𝑡 =− (𝑇− 𝑇 𝐹 ) 𝜏 ID: 1st order linear differential equation (separable) 𝑇= 𝑇 𝐼 𝑇 𝑑𝑇 (𝑇− 𝑇 𝐹 ) =− 𝑡 𝐼 𝑡 𝑑𝑡 𝜏 Initial Temperature and time, 𝑇 𝐼 and 𝑡 𝐼 . ln 𝑇− 𝑇 𝐹 𝑇 𝐼 − 𝑇 𝐹 =− 1 𝜏 𝑡− 𝑡 𝐼 𝑇 𝐹 −𝑇 𝑇 𝐹 − 𝑇 𝐼 = 𝐸 𝐸 𝐼 = 𝑒 − 𝑡− 𝑡 𝐼 𝜏 Error E= 𝑇 𝐹 −𝑇 decays exponentially with time Let θ= 𝑇 𝐹 −𝑇 𝑇 𝐹 − 𝑇 𝐼 be the dimensionless temperature error

Dimensionless Temperature Error 0.37 0.14 0.05 0.011 𝜃= 𝑇 𝐹 −𝑇 𝑇 𝐹 − 𝑇 𝐼 = 𝐸 𝐸 𝐼 = 𝑒 − 𝑡− 𝑡 𝐼 𝜏 To get (TF-T) ≤ 0.37(TF – TI) Wait for time t - tI ≥ t = 𝜌𝑐𝐷 6ℎ For fast response use Small rc (material properties) Small D (use small diameter wire to make TC) Large h Increase mixing High conductivity fluid (air or water)

Prediction versus Measurement TF Convex Downward Convex Downward Convex Upward TI tI Theoretical Solution: 𝑇= 𝑇 𝐹 −( 𝑇 𝐹 − 𝑇 𝐼 ) 𝑒 − 𝑡− 𝑡 𝐼 𝜏 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 the assumption 𝑇 𝑠 = 𝑇 𝐴 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 starts to respond?

Semi-Infinite Body Transient Conduction ∞ T x Consider a very large body whose surface temperature changes at t = 0 Thermal penetration depth, which exhibits a temperature change, 𝛿≅ 𝛼𝑡 (accurate within an “order of magnitude”) grows with time Thermal diffusivity: 𝛼= 𝑘 𝜌𝑐 (material property) How long will it take for thermal penetration depth to reach TC center? 𝛿≅ 𝛼 𝑡 𝑡 ≅ D (order of magnitude) 𝑡 𝑡 ≅ 𝐷 2 /𝛼 ≅ D2𝜌𝑐/k After t > ~tt the TC center temperature starts to change. Will measured TC temperature follow the “expected” time-dependent shape (concave downward) after that?

After t > tt, is TC temperature uniform? DTCONV T T DTCONV DTTC r r DTTC When is DTTC << DTCONV (uniform temp TC)? Balance conduction and convection Heat Transfer to TC (Order of magnitude analysis) 𝑄~ ℎ 𝐴 𝑆 Δ 𝑇 𝐶𝑂𝑁𝑉 ~ 𝑘 𝑇𝐶 𝐴 𝑆 𝑑𝑇 𝑑𝑟 ~ 𝑘 𝑇𝐶 𝐴 𝑆 Δ 𝑇 𝑇𝐶 𝐷 Δ 𝑇 𝑇𝐶 Δ 𝑇 𝐶𝑂𝑁𝑉 ~ ℎ𝐷 𝑘 𝑇𝐶 = 𝐵𝑖 𝐷 =𝐵𝑖𝑜𝑡 𝑁𝑢𝑚𝑏𝑒𝑟 If BiD < 0.1 (small D or large kTC ) Then Δ 𝑇 𝑇𝐶 ≪Δ 𝑇 𝐶𝑂𝑁𝑉 (lumped body)

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)

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

Dimensionless Temperature Error 𝑇 𝐵 𝑇 𝐴 𝜃 𝑡 = 𝑇 𝐹 −𝑇 𝑇 𝐹 − 𝑇 𝐼 = 𝑇 𝐵 −𝑇 𝑇 𝐵 − 𝑇 𝐴 = 𝐸 𝐸 𝐼 = 𝑒 − 𝑡− 𝑡 𝐼 𝜏 For boiling water environment, TF = TBoil, TI = TAir During what time range t1 < t < t2 does 𝜃 𝑡 decay exponentially with time? During that time we can t and h.

Data Transformation (trick) Reformulate: 𝜃 𝑡 = 𝑒 − 𝑡− 𝑡 𝐼 𝜏 = 𝑒 𝑡 𝐼 𝜏 𝑒 − 𝑡 𝜏 =𝐴 𝑒 𝑏𝑡 Where 𝐴= 𝑒 𝑡 𝐼 𝜏 , and b = -1/t are “constants” Take natural log of both sides ln 𝜃 = ln 𝐴 𝑒 𝑏𝑡 = ln 𝐴 +𝑏𝑡 Instead of plotting 𝜃 versus t, plot ln(𝜃) vs t Or, use log scale on y axis During the time period when 𝜃 decays exponentially, this transformed data will look like a straight line Use least-squares to fit a line to that data ln 𝜃 =𝑏𝑡+ ln 𝐴 (Excel) Slope = b = -1/t, Intercept = ln(A) Since t = 𝜌𝑐𝐷 6ℎ =− 1 𝑏 , then calculate ℎ=− 𝜌𝑐𝐷𝑏 6

TC Wire Properties (App. B) Best estimate: 𝑘 𝐽 = 𝑘 𝐼𝑟𝑜𝑛 + 𝑘 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛 2 Uncertainty: 𝑤 𝑘 𝐽 = 𝑘 𝐼𝑟𝑜𝑛 − 𝑘 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛 2 (𝑃= ?)

Type J Thermocouple Properties Not a sphere 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. Best estimate: 𝑘 𝐽 = 𝑘 𝐼𝑟𝑜𝑛 + 𝑘 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛 2 Uncertainty: 𝑤 𝑘 𝐽 = 𝑘 𝐼𝑟𝑜𝑛 − 𝑘 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛 2 tT ~ D2rc/kTC; 𝑊 𝑡 𝑇 𝑡 𝑇 2 = ? (68%, is that appropriately small?)

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.

Lab 9 Sample Data Plot T vs t 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 = ℎ𝐷 𝑘 𝐹𝑙𝑢𝑖𝑑 , BiD = ℎ𝐷 𝑘 𝑇𝐶

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?

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.

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.

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

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? 𝑄=𝐴ℎ 𝑇 𝐹 −𝑇 =𝐴 𝑘 𝐹 𝑑𝑡 𝑑𝑟 𝐹𝑙𝑢𝑖𝑑 ≈ 𝐴 𝑘 𝐹 𝑇 𝐹 −𝑇 𝛿 T ℎ≈ 𝑘 𝐹 𝛿 ≈ 𝑘 𝐹 𝐷 ℎ= 𝑘 𝐹 𝐷 𝑁𝑢 𝐷 TF T 𝛿 D r NuD ≡ Nusselt number

Lab 9 Expect h to increase as k increases and D decreases. k in appendix: kAir pg 454 → TRoom kwater pg 453 → TRoom , TBoiling

Lab 9 Results 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

Measurement Results Choice of dtD is a compromise between eliminating noise and responsiveness