ME 322: InstrumentationLecture 24

March 23, 2015Professor Miles Greiner

Lab 9 calculations

Announcements/Reminders• This week: Lab 8 Discretely Sampled Signals

– Next Week: Transient Temperature Measurements• HW 9 is due Monday• Midterm II, Wednesday, April 1, 2015

– Review Monday• Trying to arrange for Extra Credit Opportunity

– Introduction to LabVIEW and Computer-Based Measurements Hands-On Seminar

• NI field engineer will walk through the LabVIEW development environment

• Potentially 1% of grade extra credit for actively attending• Time, place and sign-up “soon”

Transient Thermocouple Measurements

• Can a the temperature of a thermocouple (or other temperature measurement device) accurately follow the temperature of a rapidly changing environment?

Lab 9 Transient TC Response in Water and Air

• Start with TC in room-temperature air• Measure its time-dependent temperature when it is plunged into boiling

water, then room temperature air, then room-temperature water (all in ~8 seconds)

• Determine the heat transfer coefficients in the three environments, hBoiling, hAir, and hRTWater

• Compare each h to the thermal conductivity of those environments (kAir or kWater)

Dimensionless Temperature Error

• At time t = t0 a thermocouple at temperature TI is put into a fluid at temperature TF. – Error: E = TF – T

• Theory for a lumped (uniform temperature) TC predicts:– Dimensionless Error: – (spherical thermocouple)

T

tt = t0

TI

TF

Error = E = TF – T ≠ 0

T(t)

TI

TF

Environment Temperature

Initial Error EI = TF – TI

𝜌 ,𝑐 ,𝐷

h

t = t0

• From this chart, find– Times when TC is placed in Boiling Water, Air and RT Air (tB, tA, tR)– Temperatures of Boiling water (maximum) and Room (minimum) (TB, TR)

• Thermocouple temperature responds more quickly in water than in air• However, slope does not exhibit a step change in each environment

– Temperature of TC center does not response immediately • Transient time for TC center to start to respond: tT ~ D2rc/kTC (order of magnitude)

Lab 9 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]

Tem

pera

ture

, T [o C

]

tB = 0.78 sIn Boiling Water

tA = 3.36 sIn Air

tR = 5.78 sIn Room Temperature Water

• 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: – Uncertainty:

• tT ~ D2rc/kTC; = ? (58%, is that appropriately small?)

Type J Thermocouple Properties

Effective 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 0.18

3s Uncertainty 0.006 530 24 26 0.10

Not a sphere

TC Wire Properties (App. B)

Dimensionless Temperature Error

– For boiling water environment, TF = TBoil, TI = TRoom

• During what time range t1<t<t2 does decay exponentially with time?– Once we find that, how do we find t?

Data Transformation (trick)

– Where , and b = -1/t are constants • Take natural log of both sides

• Instead of plotting versus t, plot ln() versus t– Or, use log-scale on y-axis– During the time period when decays exponentially, this transformed

data will look like a straight line

To find decay constant b using Excel

• Use curser to find beginning and end times for straight-line period• Add a new data set using those data• Use Excel to fit a y = Aebx to the selected data

– This will give b = -1/t– Since t , – Calculate (power product?), ?

• Assume uncertainty in b is small compared to other components

• What conditions does the convection heat transfer coefficient depend on?

Thermal Boundary Layer for Warm Sphere in Cool Fluid

• ; • thickness ~ D

– h increases as k increases and object sized decreases

– = Dimensionless Nusselt Number

Conduction in Fluid

TD r

TF 𝛿

Thermal BoundaryLayer

Lab 9 Sample Data• http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%20

09%20TransientTCResponse/LabIndex.htm

• Plot T vs t– Find TB and TR

• Calculate q and plot vs time on log scale– In Boiling Water, TI = TR, TF = TB – In Room Temperature air and water, TI = TB, TF = TR

• Select time periods that exhibit exponential decay– Find decay constant for those regions– Calculate h and wh for each environment

• For each environment calculate – NuD – BiD

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 =

(TB-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 Roo

m

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 – NuD ; BiD

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

Air and Water Thermal ConductivitiesAppendix B

• kAir (TRoom)

• kwater (TRoom, TBoiling)

Lab 9 Extra Credit

• Measure time-dependent heat transfer rate Q(t) to/from the TC (when TC is placed into boiling water)

• 1st Law

– Differentiation time step • Sampling time step • Integer m

– What is the best value of m?

Measurement Results

• Choice of DtD is a compromise between eliminating noise and retaining 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 Tra

nsfe

r Q [W

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

What Do We Expect?Expected for Uniform Temperature TC

Measured

t0

TF = TB

Ti = TR

What do we measure?

ti

Expected for uniform temperature TC

Measured

Q

t

tT

Sinusoidally-Varying Environment Temperature

• For example, a TC in a car cylinder or exhaust line• 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, and there will be a phase lag.

TENV TTC

Heat Transfer from Fluid to TC

• Environment Temperature: TE = M + Asin(wt)

– Divided by hA (assume h is constant)– Let the TC time constant be (for sphere)

– 1st order, linear differential equation, non-homogeneous

Q =hA(T – T)

Fluid TempTF(t)

TD=2r

Solution

• Solution has two parts– Homogeneous and non-homogeneous (particular):– T = TH + TP

• Homogeneous solution

– Solution: – Decays with time, not important as t∞

• Particular Solution to whole equation– Assume )

• )• Plug into non-homogeneous differential equation to find

constants C, D and E

Particular Solution

• Plug in assumed solution form: )• Collect terms:

– +

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

=0 =0 =0

Result

• )

– where

Compare to Environment Temperature

• Same mean value• If 1 >> =

• Then – Minimal attenuation and phase lag

• Otherwise: TC signal amplitude is • Phase lag: (

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 ATC = 0.5 AENV?

• If the combustion gases may be assumed to have the properties of air at 600C, what is the required Nusselt number?

VI