1. 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

2. 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

3. 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

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

5. 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?

6. 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?

7. 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

8. 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

9. 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

10. 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)

11. Prediction versus MeasurementTF
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?

12. 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?

13. 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)

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
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.

17. 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

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

20. 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?)

21. 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.

22. Lab 9 Sample Data Plot T vs t
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

23. 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?

24. 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 = /s.

25. 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 = /s, and
In room temperature water: t = 5.86 to 6.00s with decay constant b = /s.

26. 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

27. 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

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

29. 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

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