Transcript Lecture Slides

ME 322: Instrumentation
Lecture 24
March 24, 2014
Professor Miles Greiner
Announcements/Reminders
• This week: Lab 8 Discretely Sampled Signals
– Next Week: Transient Temperature Measurements
• HW 9 is due Monday
• Midterm II, Wednesday, April 2, 2014
– Review Monday
• Extra Credit Opportunity, Friday, April 18, 2014
– Introduction to LabVIEW and Computer-Based
Measurements Hands-On Seminar
• NI field engineer Glenn Manlongat will walk through the
LabVIEW development environment
• 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 roomtemperature water
• 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
𝜌, 𝑐, 𝐷
TI
T
Environment Temperature
TF
TF
Initial Error
EI = TF – T I
T(t) ℎ
Error = E = TF – T ≠ 0
TI
t
t = t0
• 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: 𝜃 𝑡 =
–
𝜏=
𝜌𝑐𝐷
6ℎ
𝐸
𝐸𝐼
=
(spherical thermocouple)
𝑇𝐹 −𝑇
𝑇𝐹 −𝑇𝐼
=
𝑡−𝑡
− 𝜏𝐼
𝑒
Lab 9 Measured Thermocouple Temperature versus Time
100
90
Temperature, T [oC]
80
70
60
tR = 5.78 s
In Room
Temperature Water
tA = 3.36 s
In Air
50
40
30
20
tB = 0.78 s
In Boiling Water
10
0
0
• From this chart, find
1
2
3
4
5
6
7
8
Time, t [sec]
– 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: tT ~ D2rc/kTC
Type J Thermocouple Properties
Effective
Diameter D Density ρ
[in]
[kg/m3]
Value
3s Uncertainty
0.059
0.006
Thermal
Conductivity
kTC [W/mK]
Specific
Heat c
[J/kgK]
Initial
Transient
Time tT [sec]
45
24
421
26
0.18
0.10
8400
530
• 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.
𝑘𝐼𝑟𝑜𝑛 +𝑘𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛
2
𝑘𝐼𝑟𝑜𝑛 −𝑘𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛
2
– Best estimate: 𝑘𝐽 =
– Uncertainty: 𝑤𝑘𝐽 =
• tT ~ D2rc/kTC;
𝑊𝑡𝑇 2
𝑡𝑇
=?
TC Wire Properties (App. B)
Dimensionless Temperature Error
• 𝜃 𝑡 =
𝑇𝐹 −𝑇
𝑇𝐹 −𝑇𝐼
=
𝑇𝐵 −𝑇
𝑇𝐵 −𝑇𝑅
=𝑒
𝑡−𝑡𝐼
−
𝜏
– For boiling water environment, TF = TBoil, TI = TRoom
• During what time range t1<t<t2 does 𝜃 𝑡 decay
exponentially with time?
– Once we find that, how do we find t?
Data Transformation (trick)
• 𝜃 𝑡 =
𝑡−𝑡
− 𝜏𝐼
𝑒
𝑡𝐼
𝜏
=𝑒 𝑒
𝑡𝐼
𝜏
𝑡
−𝜏
= 𝐴𝑒 𝑏𝑡
– Where 𝐴 = 𝑒 , and b = -1/t are constants
• Take natural log of both sides
– ln 𝜃 = ln 𝐴𝑒 𝑏𝑡 = ln 𝐴 + 𝑏𝑡
• Instead of plotting 𝜃 versus t, plot ln(𝜃) versus t
– Or, use log-scale on y-axis
– During the time period when 𝜃 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 =
𝜌𝑐𝐷
6ℎ
– Calculate ℎ =
1
=− ,
𝑏
𝜌𝑐𝐷𝑏
−
6
(power product?),
𝑤ℎ 2
ℎ
=?
• Assume uncertainty in b is small compared to other components
• What does convection heat transfer coefficient depend on?
Thermal Boundary Layer for Warm Sphere in Cool Fluid
Thermal Boundary
Layer
TF
T
𝛿
r
D
• 𝑄 = 𝐴ℎ 𝑇𝐹 − 𝑇 =
• ℎ≈
𝑘𝐹
𝛿
≈
𝑘𝐹
𝐷
𝑑𝑇
𝐴𝑘𝐹
𝑑𝑟 𝐹𝑙𝑢𝑖𝑑
≈
𝑇𝐹 −𝑇
𝐴𝑘𝐹
𝛿
Conduction in Fluid
– h increases as k increase and object sized decreases
• ℎ=
𝑘𝐹
𝑁𝑢𝐷
𝐷
– 𝑁𝑢𝐷 =
ℎ𝐷
𝑘𝐹
= Dimensionless Nusselt Number
Lab 9 Sample Data
•
http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab
%2009%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 regions that exhibit exponential decay
– Find decay constant for those regions
– Calculate h and wh for each environment
• For each environment calculate
– NuD =
– BiD =
ℎ𝐷
𝑘𝐹𝑙𝑢𝑖𝑑
ℎ𝐷
𝑘𝑇𝐶
Fig. 4 Dimensionless Temperature Error versus Time in Boiling Water
qBOIL = (TB-T(t))/(TB-TR)
1
0.1
For t = 1.14 to 1.27 s
q = 1.867E+06e-1.365E+01t
0.01
0.8
0.9
1
1.1
1.2
1.3
1.4
Time, t [sec]
•
•
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
1
qRoom
In Air
For t = 3.83 to 5.74 sec
q = 2.8268e-0.3697t
In Room Temp Water
For t = 5.86 to 6.00 sec
q = 2E+19e-7.856t
0.1
0.01
3
3.5
4
4.5
5
5.5
6
6.5
7
Time t [sec]
•
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.
Lab 9 Results
Environment
Boiling Water
Air
Room Temperature
Water
h
b [1/s] [W/m2C]
Wh
[W/m C]
kFluid
[W/mC]
2
NuD
Lumped (Bi
Bi
< 0.1?)
hD/kFluid hD/kTC
-13.7
-0.37
12016
325
1603
43
0.680
0.026
26
19
0.403
0.011
no
yes
-7.86
6915
923
0.600
17
0.232
no
• 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
Air and Water Thermal Conductivities
Appendix 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
–
–
𝑑𝑇
𝜋
𝑑𝑇
2
𝑄 − 𝑊 = 𝜌𝑐𝑉 = 𝜌𝑐𝐷
𝑑𝑡
6
𝑑𝑡
𝑑𝑇
𝑇 𝑡+Δ𝑡𝑑 −𝑇(𝑡−Δ𝑡𝑑 )
t =
𝑑𝑡
2Δ𝑡𝑑
– Differentiation time step Δ𝑡𝑑 = mΔ𝑡𝑠
• Sampling time step Δ𝑡𝑠
• Integer m
– What is the best value of m?
Measurement Results
3.5
Dt = 0.001 sec
Dt = 0.01 sec
Dt = 0.1 sec
Dt = 0.05 sec
3
Heat Transfer Q [W]
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
0.6
0.8
1
1.2
1.4
1.6
Time t [sec]
• Choice of DtD is a compromise between eliminating
noise and responsiveness
What Do We Expect?
Expected for Uniform Temperature TC
T F = TB
Ti = TR
Measured
t0
What do we measure?
Expected for uniform temperature TC
Q
Measured
tT
t
ti
Sinusoidally-Varying Environment
Temperature
TENV
TTC
• For example, a TC in a car 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.
Heat Transfer from Fluid to TC
Fluid Temp
TF(t)
Q =hA(T – T)
T
D=2r
• Environment Temperature: TE = M + Asin(wt)
• 𝑄−𝑊 =
𝑑𝑈
𝑑𝑡
=
𝑑𝑇
𝜌𝑐𝑉
𝑑𝑡
= ℎ𝐴 𝑇𝐸 − 𝑇
– Divided by hA and
– Let the TC time constant be 𝜏 =
•
𝑑𝑇
𝜏
𝑑𝑡
𝜌𝑐𝑉
ℎ𝐴
=
𝜌𝑐𝐷
6ℎ
(for sphere)
+ 𝑇 = 𝑇𝐸 𝑡
– 1st order, linear differential equation, non-homogeneous
Solution
𝑑𝑇
𝜏
𝑑𝑡
•
+ 𝑇 = 𝑇𝐸 𝑡 = 𝑀 + 𝐴𝑠𝑖𝑛(𝜔𝑡)
• Solution has two parts
– Homogeneous and non-homogeneous (particular):
– T = TH+TP
• Homogeneous solution
–
𝑑𝑇𝐻
𝜏
𝑑𝑡
+ 𝑇𝐻 = 0
−𝑡
– Solution: 𝑇𝐻 = 𝐴𝑒 𝜏
– Decays with time, not important as t∞
• Particular Solution to whole equation
– Assume 𝑇𝑃 = 𝐶 + 𝐷𝑠𝑖𝑛 𝜔𝑡 + 𝐸𝑐𝑜𝑠(𝜔𝑡)
𝑑𝑇𝑃
𝑑𝑡
•
= 𝜔𝐷𝑐𝑜𝑠 𝜔𝑡 − 𝜔𝐸𝑠𝑖𝑛(𝜔𝑡)
• Plug into non-homogeneous differential equation to find
constants C, D and E
Particular Solution
𝑑𝑇𝑃
𝜏
𝑑𝑡
•
+ 𝑇𝑃 = 𝑀 + 𝐴𝑠𝑖𝑛(𝜔𝑡)
• Plug in assumed solution form:
– 𝜏(𝜔𝐷𝑐𝑜𝑠 𝜔𝑡 − 𝜔𝐸𝑠𝑖𝑛(𝜔𝑡)) + 𝐶 + 𝐷𝑠𝑖𝑛 𝜔𝑡 + 𝐸𝑐𝑜𝑠(𝜔𝑡) =
𝑀 + 𝐴𝑠𝑖𝑛(𝜔𝑡)
• Collect terms:
– 𝑐𝑜𝑠 𝜔𝑡 𝜏𝜔𝐷 + 𝐸 +𝑠𝑖𝑛 𝜔𝑡)(−𝜏𝜔𝐸 + 𝐷 − 𝐴 + 𝐶 − 𝑀 = 0
=0
=0
• Find C, D and E in terms of A and M
– C=M
– E = −𝜏𝜔𝐷
– 𝜏𝜔 2 𝐷 + 𝐷 = 𝐴;
• 𝐷=
– E=
𝐴
𝜏𝜔 2 +1
−𝜏𝜔𝐴
𝜏𝜔 2 +1
=0
Result
• 𝑇𝑃 = 𝐶 + 𝐷𝑠𝑖𝑛 𝜔𝑡 + 𝐸𝑐𝑜𝑠(𝜔𝑡)
• 𝑇𝑃 = 𝑀 +
𝐴
𝑠𝑖𝑛
2
𝜏𝜔 +1
• 𝑇𝑃 = 𝑀 +
𝐴
[𝑠𝑖𝑛
2
𝜏𝜔 +1
• 𝑇𝑃 = 𝑀 +
𝐴
𝜏𝜔
2 +1
𝜔𝑡 +
−𝜏𝜔𝐴
𝑐𝑜𝑠(𝜔𝑡)
2
𝜏𝜔 +1
𝜔𝑡 − (𝜏𝜔)𝑐𝑜𝑠 𝜔𝑡 ]
𝑠𝑖𝑛 𝜔𝑡 − 𝜙
– where tan(𝜙) = 𝜏𝜔
Compare to Environment Temperature
tD
T
• 𝑇𝐸 𝑡 = 𝑀 + 𝐴𝑠𝑖𝑛(𝜔𝑡)
• 𝑇𝑃 = 𝑀 +
𝐴
𝑠𝑖𝑛
𝜏𝜔 2 +1
𝜔𝑡 − 𝜙 ; tan(𝜙) = 𝜏𝜔
• Same mean value
• If 1 >> 𝜏𝜔=𝜏2𝜋𝑓 =
• 𝑇 ≫ 2𝜋𝜏
2𝜋𝜏
𝑇
• Then 𝑇𝑃 = 𝑀 + 𝐴𝑠𝑖𝑛 𝜔𝑡
– Minimal attenuation and phase lag
𝐴
• Otherwise
• 𝑡𝐷 = 𝜙
𝜔
=
<𝐴
𝜏𝜔 2 +1
arctan(𝜏𝜔)
𝑇
2𝜋
=𝑇
arctan(𝜏𝜔)
360
(since 0 = 𝜔𝑡𝐷 − 𝜙)
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?
Can we measure time-dependent heat
transfer rate, Q vs. t, to/from the TC?
1st Law
Differential time step
Measurement Results
3.5
Dt = 0.001 sec
Dt = 0.01 sec
Dt = 0.1 sec
Dt = 0.05 sec
3
Heat Transfer Q [W]
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
0.6
0.8
1
1.2
1.4
1.6
Time t [sec]
• Choice of dtD is a compromise between eliminating
noise and responsiveness
Lab 9 Transient Thermocouple Response
T
TI
Environment Temperature
TF
TF
T(t)
Faster
Initial Error
EI = TF – T I
Slower TC
Error = E = TF – T ≠ 0
TI
t = t0
t
• 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?
VI
Lab 9
1) Setup
TC-Signal Conditioner
MyDAQ
Boiling water, Room temp water, Air,
Measure TC “diameter” D
𝑊𝐷 = ~(0.1 𝑜𝑟 0.2)𝐷, 𝑠𝑡𝑎𝑡𝑒
2) VI
𝑇𝑇𝐶 = 𝑉𝑠𝑐 (volts) ×
40°c
Volt
3) Data Acquisition
f𝑠 = 1000Hz T1 = 8 s
2 sec in each
a) Boiling water
b) Air at room temp
c) Water at room temp
Save data
Initial Transient Time
Note: Slope does not exhibit a step change when TC enters
new environment.
Predict order of magnitude of initial transient time tT for TC
center to begin to respond.
𝛿~ 𝛼𝑡𝑇 ~𝐷
𝐷 2 𝐷 2 𝜌𝑐
𝑡𝑇 =
=
𝛼
𝐾
Pg 455 Properties for TC 𝜌, 𝑐 , 𝐾
𝜌, 𝑐 , 𝐾 →average for Iron and const
𝜌, 𝑐 , 𝐾
Transform Trick
Straight Line
Fit data
𝑡1 < 𝑡 < 𝑡2
1
𝑏 = −13.67
𝑠
1 −6ℎ
𝑏= =
𝜏 𝜌𝑐𝐷
𝑊ℎ 𝐵
ℎ𝐵
2
𝑊𝜌
=
𝜌
2
𝑊𝑐
+
𝑐
2
𝑊𝐷
+
𝐷
Same for air & water
2
𝑊𝑏
+
𝑏
2
Small
Lab 9
Find h in:
Boiling Water
Room Temper Air and water
Why does h vary so much in different environments? Water, Air
What does h depend on?
𝑄 = 𝐴ℎ 𝑇𝐹 − 𝑇 = 𝐴𝑘𝐹
𝑑𝑡
𝑑𝑟 𝐹𝑙𝑢𝑖𝑑
≈ 𝐴𝑘𝐹
𝑇𝐹 −𝑇
𝛿
T
𝑘𝐹 𝑘𝐹
ℎ≈
≈
𝛿
𝐷
ℎ=
𝑘𝐹
𝑁𝑢𝐷
𝐷
T
D
NuD ≡ Nusselt number
TF
𝛿
r