### 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
, ,
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:   =
–
=

6ℎ

=
(spherical thermocouple)
−
−
=
−
−

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.
+
2
−
2
– Best estimate:  =
– Uncertainty:  =
• tT ~ D2rc/kTC;
2

=?
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
– ln  = ln   = ln  +
• 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 =

6ℎ
– Calculate ℎ =
1
=− ,

−
6
(power product?),
ℎ 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

r
D
•  = ℎ  −  =
• ℎ≈

≈

≈
−

Conduction in Fluid
– h increases as k increase and object sized decreases
• ℎ=

–  =
ℎ

= 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 =
ℎ

ℎ

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
–
–

2
−  =  =

6

+Δ −(−Δ )
t =

2Δ
– Differentiation time step Δ = mΔ
• Sampling time step Δ
• 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)
• − =

=

= ℎ  −
– Divided by hA and
– Let the TC time constant be  =
•

ℎ
=

6ℎ
(for sphere)
+  =
– 1st order, linear differential equation, non-homogeneous
Solution

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

+  = 0
−
– 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:
–    +  + )(− +  −  +  −  = 0
=0
=0
• Find C, D and E in terms of A and M
– C=M
– E = −
–  2  +  = ;
• =
– E=

2 +1
−
2 +1
=0
Result
•  =  +   + ()
•  =  +

2
+1
•  =  +

[
2
+1
•  =  +

2 +1
+
−
()
2
+1
− ()  ]
−
– where tan() =
Compare to Environment Temperature
tD
T
•   =  + ()
•  =  +

2 +1
−  ; tan() =
• Same mean value
• If 1 >> =2 =
•  ≫ 2
2

• Then  =  +
– Minimal attenuation and phase lag

• Otherwise
•  =

=
<
2 +1
arctan()

2
=
arctan()
360
(since 0 =  − )
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
= ~(0.1  0.2),
2) VI
=  (volts) ×
40°c
Volt
3) Data Acquisition
f = 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.
~  ~
2  2
=
=

Pg 455 Properties for TC ,  ,
,  ,  →average for Iron and const
,  ,
Transform Trick
Straight Line
Fit data
1 <  < 2
1
= −13.67

1 −6ℎ
= =

ℎ
ℎ
2

=

2

+

2

+

Same for air & water
2

+

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?
= ℎ  −  =

≈
−

T

ℎ≈
≈

ℎ=

T
D
NuD ≡ Nusselt number
TF

r
```