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
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