1. ME 322: Instrumentation Lecture 25
March 16, 2016
Professor Miles Greiner
Thermocouple response to sinusoidally varying temperature, radiation error, 2 examples
Ran out of time during last example

2. Undergraduate Research Awards
An opportunity for undergraduates to apply for funding to support research, scholarship or creative activity.
Proposals Deadline April 11, 2016.
The first application writing workshop will be this Friday, March 18 at 3:00 PM in 104 MIKC.
I’m interested in working with students, especially ones who are doing well in this class
NASA Satellite Thermal Management
Used Nuclear Fuel Packaging Safety

3. Announcements/Reminders
This week in lab: Discretely Sampled Signals
How is the lab going?
Next Week: Spring Break
After break: Transient Temperature Measurements
HW 9 is due Monday after break
Midterm II, Wednesday, March 30, 2016
Review Monday after break

4. TC Response to Sinusoidally-Varying Environment TemperaturetD
TENV
TTC
For example, a TC in an engine cylinder or exhaust
“Eventually” the TC will have
The same average temperature and unsteady frequency as the environment temperature. However
Its unsteady amplitude may be less than the environment temperature’s.
The TC temperature peak may be delayed by time tD

5. Heat Transfer from Environment to TC
Environment Temp
TE(t)
D=2r T
Q = hA(TE–T)
Heat Transfer to TC
Environment Temperature: TE = M + Asin(wt)
w= 2pf = 2p/T (Assume M, A, and w are known)
𝑄−𝑊= 𝑑𝑈 𝑑𝑡 =𝜌𝑐𝑉 𝑑𝑇 𝑑𝑡 =ℎ𝐴 𝑇 𝐸 −𝑇
Divide by hA, and
Let the TC time constant be 𝜏= 𝜌𝑐𝑉 ℎ𝐴 = 𝜌𝑐𝐷 6ℎ (for sphere)
𝜏 𝑑𝑇 𝑑𝑡 +𝑇= 𝑇 𝐸 𝑡 Identify this equation
1st order, linear differential equation, non-homogeneous
For steady fluid temperature 𝑇 𝐹 we got, 𝜏 𝑑𝑇 𝑑𝑡 +𝑇= 𝑇 𝐹

6. Solution 𝜏 𝑑𝑇 𝑑𝑡 +𝑇= 𝑇 𝐸 𝑡 =𝑀+𝐴𝑠𝑖𝑛(𝜔𝑡) Solution has two parts
𝜏 𝑑𝑇 𝑑𝑡 +𝑇= 𝑇 𝐸 𝑡 =𝑀+𝐴𝑠𝑖𝑛(𝜔𝑡)
Solution has two parts
Homogeneous and Particular (non-homogeneous)
T = TH + TP
Homogeneous solution
𝜏 𝑑 𝑇 𝐻 𝑑𝑡 + 𝑇 𝐻 =0
Solution: 𝑇 𝐻 =𝐴 𝑒 −𝑡 𝜏
Decays with time, so not important after initial transient
Particular Solution (to whole equation)
Assume 𝑇 𝑃 =𝐶+𝐷𝑠𝑖𝑛 𝜔𝑡 +𝐸𝑐𝑜𝑠(𝜔𝑡) (but 𝐶, 𝐷, 𝐸=?)
𝑑 𝑇 𝑃 𝑑𝑡 =𝜔𝐷𝑐𝑜𝑠 𝜔𝑡 −𝜔𝐸𝑠𝑖𝑛(𝜔𝑡)
Plug into non-homogeneous differential equation
Find constants C, D and E in terms of M, A and 𝜔

7. Particular Solution 𝜏 𝑑 𝑇 𝑃 𝑑𝑡 + 𝑇 𝑃 =𝑀+𝐴𝑠𝑖𝑛(𝜔𝑡)
𝜏 𝑑 𝑇 𝑃 𝑑𝑡 + 𝑇 𝑃 =𝑀+𝐴𝑠𝑖𝑛(𝜔𝑡)
Plug in assumed solution form: 𝑇 𝑃 =𝐶+𝐷𝑠𝑖𝑛 𝜔𝑡 +𝐸𝑐𝑜𝑠(𝜔𝑡)
𝜏[𝜔𝐷𝑐𝑜𝑠 𝜔𝑡 −𝜔𝐸𝑠𝑖𝑛(𝜔𝑡)] +[𝐶+𝐷𝑠𝑖𝑛 𝜔𝑡 +𝐸𝑐𝑜𝑠(𝜔𝑡)]=𝑀+𝐴𝑠𝑖𝑛(𝜔𝑡)
Collect terms:
𝑐𝑜𝑠 𝜔𝑡 𝜏𝜔𝐷+𝐸 +𝑠𝑖𝑛 𝜔𝑡)(−𝜏𝜔𝐸+𝐷−𝐴 + 𝐶−𝑀 =0
Find C, D and E in terms of A, M and 𝜔
C = M
E=−𝜏𝜔𝐷
−𝜏𝜔 2 𝐷+𝐷=𝐴
D = 𝐴 𝜏𝜔 2 +1
E = −𝜏𝜔𝐴 𝜏𝜔 2 +1 =0 =0 =0
For all times

8. Result 𝑇 𝑃 =𝐶+ 𝐷𝑠𝑖𝑛 𝜔𝑡 + 𝐸𝑐𝑜𝑠(𝜔𝑡)
𝑇 𝑃 =𝐶+ 𝐷𝑠𝑖𝑛 𝜔𝑡 𝐸𝑐𝑜𝑠(𝜔𝑡)
𝑇 𝑃 =𝑀+ 𝐴 𝜏𝜔 𝑠𝑖𝑛 𝜔𝑡 + −𝜏𝜔𝐴 𝜏𝜔 𝑐𝑜𝑠(𝜔𝑡)
𝑇 𝑃 =𝑀+ 𝐴 𝜏𝜔 [𝑠𝑖𝑛 𝜔𝑡 −(𝜏𝜔)𝑐𝑜𝑠 𝜔𝑡 ]
𝑇 𝑃 =𝑀+ 𝐴 𝜏𝜔 𝑠𝑖𝑛 𝜔𝑡−𝜙
where tan(𝜙)=𝜏𝜔
Same frequency and mean as environment temperature
TE = M + Asin(wt)
But delayed and attenuated

9. Compare to Environment TemperaturetD
𝜏𝜔 = 2𝜋 𝜏 𝑇
𝑇 𝐸 𝑡 =𝑀+𝐴𝑠𝑖𝑛(𝜔𝑡)
𝑇 𝑃 = 𝑇 𝑇𝐶 =𝑀+ 𝐴 𝜏𝜔 𝑠𝑖𝑛 𝜔𝑡−𝜙 ; tan(𝜙)=𝜏𝜔
Same mean value and frequency
𝑇 𝑇𝐶 =𝑀+ 𝐴 𝑇𝐶 𝑠𝑖𝑛 𝜔𝑡−𝜙 ; 𝐴 𝑇𝐶 = 𝐴 𝜏𝜔 2 +1
If 1 >> 𝜏𝜔 = 𝜏2𝜋𝑓= 2𝜋𝜏 𝑇
𝑇≫2𝜋𝜏
Then 𝑇 𝑇𝐶 =𝑀+𝐴𝑠𝑖𝑛 𝜔𝑡
Minimal attenuation and phase lag
Otherwise 𝐴 𝑇𝐶 = 𝐴 𝜏𝜔 <𝐴
if 𝜏𝜔 < 0.1 (𝜏 < T/20p) , then 𝐴 𝑇𝐶 /A > (1/2% attenuation)
Delay time: when 0=𝜔 𝑡 𝐷 −𝜙
𝑡 𝐷 = 𝜙 𝜔 =𝑇 arctan(𝜏𝜔) 2𝜋 ; 𝑡 𝐷 𝑇 = arctan(𝜏𝜔) 2𝜋 = arctan(𝜏𝜔) 360 ≤0.25
𝜏𝜔 = 2𝜋 𝜏 𝑇

10. 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 the amplitude of the thermocouple temperature variations is 90% as large as the environment temperature variations? If the combustion gases may be assumed to have the properties of air at 600°C, what is the required Nusselt number?
ID: Steady or Unsteady?

11. Material Properties

12. Common Temperature Measurement Errors
Even for steady temperatures
Lead wires act like a fin, cooling the surface compared to the case when the sensor is not there
The temperature of a sensor on a post will be between the fluid and duct surface temperature

13. High Temperature (combustion) Gas Measurements
QConv=Ah(Tgas– TS)
Tgas TW
Sensor
h, TS, A, e TS
QRad=Ase(TS4 -TW4)
Radiation heat transfer is important and can cause errors
TC temperature changes until convection heat transfer to sensor equals radiation heat transfer from sensor
Q = Ah(Tgas – TS) = Ase(TS4 -TW4)
s = Stefan-Boltzmann constant = 5.67x10-8W/m2K4
e = Sensor emissivity (surface property ≤ 1)
T[K] = T[C]
Measurement Error = Tgas – TS = (se/h)(TS4 -TW4)

14. Problem 9.39 (p. 335)
Calculate the actual temperature of exhaust gas from a diesel engine in a pipe, if the measuring thermocouple reads 500°C and the exhaust pipe is 350°C. The emissivity of the thermocouple is 0.7 and the convection heat-transfer coefficient of the flow over the thermocouple is 200 W/m2-C.
ID: Steady or Unsteady?
What if there is uncertainty in emissivity?