Transcript Lecture Slides

ME 322: Instrumentation
Lecture 40
April 30, 2014
Professor Miles Greiner
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•
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•
Announcements/Reminders
This week: Lab 12 Feedback Control
HW 14 due now (Last HW assignment)
Review Labs 9, 10, 11, and 12; Today and Friday
Supervised Open-Lab Periods
• May 2-4, 2014, 11-2 Friday through Sunday (will those times work?)
• Extra Credit Lab 12.1 (due in class Monday, 5/5/2014)
• See Lab 12 instructions (study effect of DT, DTi, TSP, heater and TC locations)
• Check out Lab-in-a-Box for DeLaMare Library
• Only 0.5% of grade
• Lab Practicum Finals (May 6-14)
– Guidelines, New Schedule
• http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Tests/Index.htm
• How many of you will graduate this year (2014) or next year
(2015).
• If it will be later than 2015, was there something the ME Department did that
delayed your graduation?
Lab 9 Transient TC Response in Water and Air
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
1
2
3
4
5
6
7
8
• 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
• 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)
• Also calculate Biot number (dimensionless thermal size) and delay
time for center to respond
Time, t [sec]
LabVIEW VI
Dimensionless Temperature Error
𝜌, 𝑐, 𝐷
TI
T
Environment Temperature
TF
TF
T(t) ℎ
Initial Error
EI = TF – TI
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.
• Theory for a uniform-temperature TC predicts:
– Dimensionless Error: 𝜃 𝑡 =
𝐸
𝐸𝐼
=
𝑇𝐹 −𝑇
𝑇𝐹 −𝑇𝐼
=
𝑡−𝑡
− 𝜏0
𝑒
– Time Constant for a spherical thermocouple 𝜏 = 𝜌𝑐𝐷
6ℎ
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
• 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
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
1
2
3
4
5
6
7
8
Time, t [sec]
• 𝜃 𝑡 =
𝑇𝐹 −𝑇
𝑇𝐹 −𝑇𝐼
=
𝑇𝐵 −𝑇
𝑇𝐵 −𝑇𝑅
=𝑒
𝑡−𝑡0
𝜏
−
– For boiling water environment, TF = TBoil, TI = TRoom
– For room-temperature air and water, TF = TRoom, TI = TBoil
• How can we find the time range t1 < t < t2 when 𝜃 𝑡
decays exponentially with time?
Data Transformation (trick)
• 𝜃 𝑡 =
𝑡−𝑡
− 𝜏0
𝑒
= 𝐴𝑒 𝑏𝑡
𝑡0
𝜏
– 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
Find decay constant b using Excel
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]
• Use curser to find beginning and end times for straight-line period
– q exhibits random variation when it is less than q < 0.05
• Add a new data set using those data
• Use Excel to fit a y = Aebx to the selected data
– For this data b = -13.65 1/s
– Since b = -1/t, and t =
– Calculate ℎ =
𝜌𝑐𝐷𝑏
−
6
𝜌𝑐𝐷
6ℎ
1
𝑏
=− ,
(power product?),
𝑤ℎ 2
ℎ
=?
• Assume uncertainty in b is small compared to other components
Dimensionless Temperature Error versus Time 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]
• Decays exponentially during two time periods:
– In air:
• t = 3.83 to 5.74 sec, b = -0.3697 1/s
– In water:
• t = 5.86 to 6.00 sec, 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
• Water environments have orders of magnitude
higher h (and b) than air
– Similar to kFluid
• Nusselt numbers 𝑁𝑢𝐷 =
ℎ𝐷
𝑘𝐹
(power product) are
more dependent on flow conditions (steady versus
moving) than environment composition
• Biot number 𝐵𝑖𝐷 =
ℎ𝐷
𝑘𝑇𝐶
(dimensionless size)
Air and Water Properties (bookmark)
Lab 9 Sample Data
•
http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab
%2009%20TransientTCResponse/LabIndex.htm
• Plot T vs t
– Find TB, TR, tB, tA, 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 =
𝜌𝑐𝐷𝑏
−
6
and wh for each environment
• For each environment calculate
– NuD =
– BiD =
ℎ𝐷
𝑘𝐹𝑙𝑢𝑖𝑑
ℎ𝐷
𝑘𝑇𝐶
Lab 10 Vibration of a Weighted Cantilever
Beam
L
Clamp
LB
E
MW
W
T
E (given)
LT MT
• Accelerometer Calibration Data
– i.e. C = 616.7 mV/g
– Use calibration constant for the
issued accelerometer
– Inverted Transfer function: a = V/C
• During Final you will be given values
and uncertainties of E, W, T
• Measure (3s uncertainty)
• MT, MW: Analytical balance, 𝑤𝑀 = 0.1 g
– LB, LE, LT: Tape measure, 𝑤𝐿 = 1/16 in
Accelerometer
Table 1 Measured and Calculated Beam Properties
LE
LB
Clamp
MW
W
T
E (given)
Accelerometer
LT MT
Units
Elastic Modulus, E [Pa] [GPa]
Beam Width, W
[inch]
Beam Thickness, T
[inch]
Beam Total Length, LT [inch]
End Length, LE
[inch]
Beam Length, LB
[inch]
Beam Mass, MT
[g]
Intermediate Mass, MI
[g]
Combined Mass, Mw
[g]
• Intermediate mass (later)
Value
63
0.99
0.1832
24.00
0.38
10.00
196.8
21.9
741.2
3s
Uncertainty
3
0.01
0.0008
0.06
0.06
0.06
0.1
1.5
0.1
Figure 2 VI Block Diagram
Convert to Dynamic Data
Convert to Dynamic Data
Converts numeric, Boolean, waveform and array data types to the
dynamic data type for use with Express VIs.
Statistics
This Express VI produces the following measurements:
Time of Maximum
Figure 1 VI Front Panel
Disturb Beam and Measure a(t)
Aluminum
Steel
• Use a sufficiently high sampling rate to capture the peaks
– fS > 2fM
– When plotting a versus t, use time increment Dt = 1/fS
• Looks like 𝑎 𝑡 = 𝐴𝑒 𝑏𝑡 sin(2𝜋𝑓𝑡 + 𝜙)
– Is b constant?
• Measure f from spectral analysis ( fM )
• Find b from exponential fit to acceleration peaks
Figure 4 Acceleration Oscillatory Amplitude Versus Frequency
• The sampling period and frequency were T1 = 10 sec and fS =
200 Hz.
– As a result the system is capable of detecting frequencies between 0.1
and 100 Hz, with a resolution of 0.1 Hz.
– To plot aRMS vs t, use frequency increment Df = 1/T1
• The frequency with the peak oscillatory amplitude is fM = 8.70 ±
0.05 Hz. This frequency is easily detected from this plot.
Fig. 5 Peak Acceleration versus Time
Aluminum
Steel
• For aluminum, exponential decay changed at t = 2.46 s
• During the first and second periods the decay rates are
– b1 = -0.292 1/s
– b2 = -0.196 1/s
• Decay “constant” b was not constant
Equivalent Endpoint Mass
LE
LB
Clamp
MW
LT MT
ME
Beam Mass MB
• Beam is not massless,
– Its mass affects its motion and natural frequency
𝑀𝑇
𝐿𝑇
• 𝑀𝐸𝑄 = 𝑀𝑊 + 𝐿𝐸 + 0.23𝐿𝐵
= 𝑀𝑊 + 𝑀𝐼 (linear sum)
– 𝑀𝑊 = mass of weight, accelerometer, pin, nut
– 𝑀𝐼 = 𝐿𝐸 + 0.23𝐿𝐵
•
𝑤𝑀𝐼 2
𝑀𝐼
=
𝑤𝑀𝑇 2
𝑀𝑇
+
𝑀𝑇
𝐿𝑇
𝑤 𝐿𝑇 2
𝐿𝑇
(contribution form beam mass)
+
𝑤𝐿𝐸
2
+ 0.23𝑤𝐿𝐵
𝐿𝐸 +0.23𝐿𝐵 2
2
Beam Equivalent Spring Constant, KEQ
F
LB
d
• 𝐾𝐸𝑄 =
𝐹
𝛿
=
𝐹
𝐿3
𝐵𝐹
3𝐸𝐼
– Power product?
=
3𝐸 𝑊𝑇 3
𝐿3𝐵 12
=
𝐸𝑊 𝑇 3
4 𝐿𝐵
Predicted Frequencies
• Undamped
– 𝑓0𝑃 =
𝜔0𝑃
2𝜋
=
𝐾𝐸𝑄
1
2𝜋
𝑀𝐸𝑄
– Power Product?
• Damped
– 𝑓𝑃 =
𝜔𝑃
2𝜋
=
1
2𝜋
𝐾𝐸𝑄
𝑀𝐸𝑄
−
𝜆
2𝑀𝐸𝑄
2
=
1
2𝜋
𝐾𝐸𝑄
𝑀𝐸𝑄
− 𝑏2
– Power product?
– If
𝑏2
≪
𝐾𝐸𝑄
𝑀𝐸𝑄
, then 𝑓𝑃 ≈ 𝑓0𝑃 , and 𝑤𝑓𝑃 ≈ 𝑤𝑓0𝑃
• Measured Damping Coefficient
– 𝜆𝑀 = −2𝑀𝐸𝑄 𝑏
Table 2 Calculated Values and Uncertainties
Equivalent Mass, MEQ
Units
Value
[kg]
0.7631
3s
Uncertainty
0.0005
2445
124
Equivalent Beam Spring
[N/m]
Constant, kEQ
Predicted Undamped
Frequency, foP
[Hz]
9.0
0.2
Measured Damped
Frequency, fM
[Hz]
8.70
0.05
Decay Constant, b1
[1/sec]
-0.292
-
0.45
0.00
9.0
0.2
3.5%
-
-0.196
-
0.30
0.00
9.0
0.2
3.5%
-
Damping Coefficient, lM [Ns/m]
Damped Frequency, fp
[Hz]
Percent Difference
(fP/fM-1)*100%
Decay Constant, b2
[1/sec]
Damping Coefficient, lM [Ns/m]
Damped Frequency, fp
Percent Difference
(fP/fM-1)*100%
[Hz]
•
The equivalent mass is not strongly affected by the intermediate mass
•
The predicted undamped and damped frequencies, fOP and fP, are essentially the
same (frequency is unaffected by damping).
•
The confidence interval for the predicted damped frequency fP = 9.0 ± 0.2 Hz does
not include the measure value fM = 8.70 ± 0.05 Hz.
Time and Frequency Dependent Data
•
http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%201
0%20Vibrating%20Beam/Lab%20Index.htm
• Plot a versus t
– Time increment Dt = 1/fS
• Plot aRMS versus f
– Frequency increment Df = 1/T1
• Measured Damped (natural) Frequency, fM
– Frequency with peak aRMS
1
2
– Uncertainty 𝑤𝑎𝑅𝑀𝑆 = Δ𝑓 =
1
2𝑇1
• Exponential Decay Constant b (Is it constant?)
– Show how to find acceleration peaks versus time
• Use AND statements to find accelerations that are larger than the ones before
and after it
• Use If statements to select those accelerations and times
• Sort the results by time
• Plot and create new data sets before and after 2.46 sec
– Fit data to y = Aebx to find b
Thermal Boundary Layer for Warm Sphere in Cool Fluid
Thermal Boundary
Layer
T
TF
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 (power product?)
Lab 9
Transient TC Response in Air & Water
Wire yourself
TC → Conditioner
(+)TC → White Wire
(-)TC → Red Stripe
Conditioner to → MyDAQ
Com → (-)
Vout → (+)
Write VI Easy Fig 1 & 2 will not be given
Acquire Data
Fs = 1000 Hz
Ti = 8 sec
At least 2 seconds in each environment.
•Room temp water
•Boiling water
•Room temp air
•Room temp water
Fig 3 Plot T Vs. t
ID time tB , tA , tR
ID Temp
TRoom = Tmin
TBoil = Tmax
Fig 4 For boiling water
vs. t
Identify: Start & end times of exponential decay period (looks linear)
•Select exp decay data y
•Add data to plot
to that data
•Fit
Show results on the plot
•Find b
Units s-1
Fig 5 Room Temp Air & Water
vs. t
Find
Find
Table 2
Lab 10: Vibrating Beam
You will be given beam and its E and WE
VI fig 1 &2
Table 2
Undamped Predicted Frequency if b = 0, λ = 0
Measured Damping Coeff
If
Then Wfp ≈ Wfop
Is