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ME 322: Instrumentation Lecture 23. March 16, 2012 Professor Miles Greiner. Announcements/Reminders. HW 8 is due now Midterm II, April 2, 2014 Next week is Spring Break!. So far in this course…. Quad Area Measurement

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me 322 instrumentation lecture 23

ME 322: InstrumentationLecture 23

March 16, 2012

Professor Miles Greiner

announcements reminders
Announcements/Reminders
  • HW 8 is due now
  • Midterm II, April 2, 2014
    • Next week is Spring Break!
so far in this course
So far in this course…
  • Quad Area Measurement
    • 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
    • Metal 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
    • LabVIEW, derivatives, spectral analysis
transient instrument response
Transient Instrument Response
  • Can measurement devices follow rapidly changing measurands?
    • temperature
    • pressure
    • speed
lab 9 transient thermocouple response
Lab 9 Transient Thermocouple Response

T

Environment Temperature

TF

Faster

Slower TC

Initial Error

EI = TF – TI

Error = E = TF – T ≠ 0

TI

TI

t

t = t0

  • 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?

TF

T(t)

heat transfer from water to tc
Heat Transfer from Water to TC

Surface Temp

TS(t)

Fluid Temp

TF

Q [J/s = W]

  • 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
    • Linear convection heat transfer coefficient, h
      • Affected by
        • Water thermal conductivity k, density r, specific heat c
        • Water motion
      • Units [h] =
    • Q = [TF – TS(t)]Ah

D=2r

T(t,r)

energy balance 1 st law
Energy Balance (1st law)
  • Internal energy of an incompressible TC
    • U = mcTA = rVcTA
      • r = density
      • c =specific heat
      • TA = Average TC temperature (may not be isothermal)
  • -)
  • TA and TS change with time t
for a uniform temperature tc
For a Uniform Temperature TC
    • When is this a good assumption? (later)
  • -)
      • For a sphere:
      • Units
      • TC time constant (assumes h is constant)
solution
Solution
    • ID: 1storder linear differential equation (separable)
  • Error decays exponentially with time
  • Let be the dimensionless temperature error
dimensionless temperature error
Dimensionless Temperature Error
  • To get (TF-T) ≤ 0.37(TF – TI)
  • Wait for time t - tI ≥ t =
    • For fast response use
      • small rc(material properties)
      • Small D (use small diameter wire to make TC)
      • Large h
        • Increase mixing
        • High conductivity fluid

0.37

0.14

0.05

0.011

prediction versus measurement
Prediction versus Measurement

TF

  • Theoretical Solution:
  • 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 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 responds?

TI

tI

semi infinite body transient c onduction
Semi-Infinite Body Transient Conduction

T1

t = 0

  • Consider a very large body whose surface temperature changes at t = 0
  • Thermal penetration depth, which exhibits a temperature change, grows with time
    • Thermal diffusivity: (material property)
  • How long will it take for thermal penetration depth to reach TC center?
    • D (order of magnitude)
    • D2/k
  • After t > ~ttthe TC center temperature starts to change.
    • Does the average temperature then follow the expected time-dependent shape?

Ti

d

after t t t is tc temperature uniform
After t > tt, is TC temperature uniform?

DTCONV

T

T

DTCONV

DTTC

  • When is DTTC << DTCONV (uniform temp TC)?
  • Balance conduction and convection
  • If BiD < 0.1, (small D or large kTC )
    • Then (lumped body)

r

r

DTTC

lab 9 transient tc response in water and air
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
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 error1
Dimensionless Temperature Error
    • For boiling water environment, TF = TBoil, TI = TAir
  • During what time range t1 < t < t2 does decay exponentially with time?
    • Once we find that, how do we find t?
data transformation trick
Data Transformation (trick)
  • Reformulate:
    • Where , and b = -1/t
  • Take natural log of both sides
  • Instead of plotting versus t, plot ln() vs t
    • Or, use log scale on y axis
    • During the time period when decays exponentially, this transformed data will look like a straight line
    • Use least-squares to fit a line to that data
      • Slope = b = -1/t, Intercept = ln(A)
      • Since t= , then calculate
tc wire properties app b
TC Wire Properties (App. B)
  • Best estimate:
  • Uncertainty:
table 1 thermocouple properties
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
Lab 9 Sample Data
  • 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, BiD
lab 9
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
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
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
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
so far in this course1
So far in this course…
  • Quad Area Measurement
    • 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
      • Transfer Functions, Linear regression, Standard Error of the Estimate
    • Metal Elastic Modulus
      • Strain Gage/Wheatstone Bridge, Propagation of Uncertainty
    • Fluid Speed and Volume Flow Rate
      • Pitot-Static Probes, Venturi Tubes
    • Boiling Water Temperature
      • Thermocouples
  • Discrete sampling of time varying signals using computer data acquisition (DAQ) systems
    • Allows us to acquire unsteady outputs versus time
    • LabVIEW, derivatives, spectral analysis