ME 322: InstrumentationLecture 23

March 13, 2015

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

– How did plotting and LabVIEW programing go?

• Midterm II, April 1, 2015– Next week is Spring Break!

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

Transient Instrument Response

• Can measurement devices follow rapidly changing measurands?– temperature– pressure – speed

Lab 9 Transient Thermocouple Response

• At time t = t0 a small thermocouple at initial temperature TI is dropped 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 the TC to reach TF?

T

tt = t0

TI

TF FasterSlower TC

Error = E = TF – T ≠ 0

T(t)

TI

TF

Environment Temperature

Initial Error EI = TF – TI

Heat Transfer from Water to TC

• 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

– Q = [TF – TS(t)]Ah• Units [h] =

• Can we predict TC temperature versus time?

Q [J/s = W]

Fluid (water) TempTF

T(t,r)

Surface TempTS(t)

D=2r

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)

• and change with time

For a Uniform Temperature TC• Assume:

– Good for a “small” and “high-conductivity” TC• Compared to what? (later)

• -)

– Let: • For a sphere:

• Units • TC time constant (assumes h is constant)

Solution

– ID: 1st order linear differential equation (separable)–

• Initial Temperature and time, and .

– –

• Error decays exponentially with time• Let be the dimensionless temperature error

0.37

0.140.05 0.011

Dimensionless Temperature Error

• To get (TF-T) ≤ 0.37(TF – TI)

• Wait for time t - tI ≥ t = – For fast response use

• small rc (volumetric specific heat, material properties)• Small D (use small diameter wire to make TC)• Large h

– Increase mixing– High conductivity fluid (air or water)

𝜃=𝑇 𝐹−𝑇𝑇 𝐹−𝑇 𝐼

=𝐸𝐸 𝐼

=𝑒−𝑡 −𝑡 𝐼𝜏

Prediction versus Measurement

• 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 the assumption exactly true? – Does the temperature at the thermocouple center responds as soon as it is placed

in the water– Does the TC lead wire affect the TC temperature?– How long will it take before the center starts to respond?

t-tI [sec]

T [

C]

tI

TI

TF

Semi-Infinite Body Transient Conduction

• Consider a very large body whose surface temperature changes at t = 0• Thermal penetration depth, which exhibits a temperature change, (accurate

within an “order of magnitude”) 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 > ~tt the TC center temperature starts to change.– Will measured TC temperature follow the “expected” time-dependent shape after that?

dTi

T1

t = 0 ∞

T

x

After t > tt, is TC temperature uniform?

• When is DTTC << DTCONV (uniform temp TC)?

• Balance conduction and convection–

• If BiD < 0.1, (small D or large kTC )– Then (lumped body)

DTCONV

DTTC

T

r

DTCONV

DTTC

T

r

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)

• Thermocouple temperature responds much more quickly in water than in air

• How to determine h all three environments?

Measured Thermocouple Temperature versus Time

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8

Time, t [sec]

Te

mp

era

ture

, T

[oC

]

tB = 0.78 sIn Boiling Water

tA = 3.36 sIn Air

tR = 5.78 sIn Room Temperature Water

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)

• 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)

• Best estimate: • Uncertainty:

Have a good spring break!

• 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.

Table 1 Thermocouple PropertiesEffective

Diameter D [in]

Density ρ [kg/m3]

Thermal Conductivity kTC [W/mK]

Specific Heat c [J/kgK]

Initial Transient

Time tT [sec]

Value 0.059 8400 45 421 0.183s Uncertainty 0.006 530 24 26 0.10

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

• 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?

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8

Time, t [sec]

Te

mp

era

ture

, T

[oC

]

tB = 0.78 sIn Boiling Water

tA = 3.36 sIn Air

tR = 5.78 sIn Room Temperature 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.

For t = 1.14 to 1.27 sq = 1.867E+06e-1.365E+01t

0.01

0.1

1

0.8 0.9 1 1.1 1.2 1.3 1.4

Time, t [sec]

q BO

IL =

(T

B-T

(t))

/(T

B-T

R)

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.

In AirFor t = 3.83 to 5.74 sec

q = 2.8268e-0.3697t

In Room Temp WaterFor t = 5.86 to 6.00 sec

q = 2E+19e-7.856t

0.01

0.1

1

3 3.5 4 4.5 5 5.5 6 6.5 7

Time t [sec]

q Ro

om

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

Environment b [1/s]

h

[W/m2C]

Wh

[W/m2C]kFluid

[W/mC]NuD

hD/kFluid

Bi hD/kTC

Lumped (Bi < 0.1?)

Boiling Water -13.7 12016 1603 0.680 26 0.403 noAir -0.37 325 43 0.026 19 0.011 yes

Room Temperature Water -7.86 6915 923 0.600 17 0.232 no

Lab 9Find h in: Boiling Water

Room Temper Air and water

Why does h vary so much in different environments? Water, AirWhat does h depend on?

T

D

T

r

TF

𝛿

h ≈𝑘𝐹

𝛿≈𝑘𝐹

𝐷

h=𝑘𝐹

𝐷𝑁𝑢𝐷

NuD ≡ Nusselt number

Lab 9Expect 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

Environment b [1/s]

h

[W/m2C]

Wh

[W/m2C]kFluid

[W/mC]NuD

hD/kFluid

Bi hD/kTC

Lumped (Bi < 0.1?)

Boiling Water -13.7 12016 1603 0.680 26 0.403 noAir -0.37 325 43 0.026 19 0.011 yes

Room Temperature Water -7.86 6915 923 0.600 17 0.232 no

Can we measure time-dependent heat transfer rate, Q vs. t, to/from the TC?

1st Law

Differential time step

Measurement Results

• Choice of dtD is a compromise between eliminating noise and responsiveness

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0.6 0.8 1 1.2 1.4 1.6

Time t [sec]

Hea

t T

ran

sfer

Q [

W]

Dt = 0.001 secDt = 0.01 secDt = 0.1 secDt = 0.05 sec

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