ME 322: InstrumentationLecture 25

March 25, 2014

Professor Miles Greiner

Thermocouple response to sinusoidally varying temperature, radiation and conduction errors

Announcements/Reminders• This week: Lab 8 Discretely Sampled Signals– How is the lab going?– Next Week: Transient Temperature Measurements

• HW 9 is due Monday• Midterm II, Wednesday, April 1, 2015– Review Monday

• Trying to schedule an extra-credit opportunity – LabVIEW Computer-Based Measurements – NI field engineer will walk through the LabVIEW

development environment– Time TBA– 1% of grade extra credit for actively attending

TC Response to Sinusoidally-Varying Environment Temperature

• 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 will be less than the environment

temperature’s.

– TC temperature peak will be delayed by time tD

TENV TTC

T

tD

Heat Transfer from Environment to TC

• Environment Temperature: TE = M + Asin(wt)– w= 2pf = 2 /p T (assume M, A, and w are known)– Divide by hA, and – Let the TC time constant be (for sphere)

• Identify this equation – 1st order, linear differential equation, non-homogeneous

Q = hA(TE–T)

Environment TempTE(t)

TD=2r

Heat Transfer to TC

Solution

• Solution has two parts– Homogeneous and Particular (non-homogeneous)

– T = TH + TP

• Homogeneous solution

– Solution:

– Decays with time, so not important as t ∞• Particular Solution (to whole equation)– Assume ) (C, D, E=?)

• )• Plug into non-homogeneous differential equation

– Find constants C, D and E in terms of M, A and

Particular Solution

• Plug in assumed solution form: )

• Collect terms: +

• Find C, D and E in terms of A, M and C = M

•

–

=0 =0 =0 For all times

Result• )

– where

• Same frequency and mean as environment temperature– TE = M + Asin(wt)

Compare to Environment Temperature

– Same mean value and frequency– ;

• If 1 >> =

• Then – Minimal attenuation and phase lag

• Otherwise – if < 0.1 (< T/20p) , then /A > 0.995 (half percent

attenuation)

• Delay time: when • ;

T

tD

=

=

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?

Material Properties

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

High Temperature (combustion) Gas Measurements

• 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(TS

4 -TW4)

• s = Stefan-Boltzmann constant = 5.67x10-8W/m2K4

• = e Sensor emissivity (surface property ≤ 1)• T[K] = T[C] + 273.15

• Measurement Error = Tgas – TS = (se/h)(TS4 -TW

4)

QConv=Ah(Tgas– TS)

TS

QRad=Ase(TS4 -TW

4)

Tgas

TW

Sensorh, TS, A, e

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?

Conduction through Support (Fin Configuration)

• Sensor temperature TS will be between those of the fluid T∞ and duct surface T0

– Support: cross sectional area A, parameter circumference P, conductivity k– Convection heat transfer coefficient between gas and support h

• Fin Temperature Profile (from conduction heat transfer analysis):– – (dimensionless length)

• Dimensionless Tip Temperature Error from conduction– , (want this to be small)– Decreases as

• L, h and P increase• k and A decrease

T∞

h xLA, P, k

T0

TS

Example

• A 1-cm-long, 1-mm-diameter stainless steel support (k = 20 W/mK) is mounted inside a pipe whose temperature is 200°C. The heat transfer coefficient between gas in the pipe and the support is 100 W/m2K, and a sensor at the end of the support reads 350°C. What is the gas temperature? Assume esensor = 0

• Steady or unsteady• Radiation or Conduction errors

Solution

• Sensor temperature: •

• What is given and what must be found?

• What if esensor = 0.2?