MAE300 Lab 5
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School
California State University, Long Beach *
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Course
300
Subject
Aerospace Engineering
Date
Dec 6, 2023
Type
Pages
22
Uploaded by ChancellorGorillaPerson971
MAE 300: Engineering Instrumentation and Measurement
Experiment No. 5
Experimental Investigation of Natural
Convection from a Sphere, Using Lumped
System Analysis
Submitted By:
Jack Martin
Cristina Garcia Franco
Allen Xavier
Date Performed: 4-18-2022
Date Submitted: 5-2-2022
Abstract
The objective of this experiment was to familiarize the students with the method
of determining the convective heat transfer coefficient of a heated stainless steel sphere
in an idle environment. The method used to find the convective heat transfer coefficient
was the lumped system approach. Students recorded the temperature of the stainless
steel sphere after it was heated to 124.8 °C and then students recorded the progressive
decrease in temperature every 30 seconds over the span of 1500 seconds. A small
torch was used to heat up the stainless steel sphere which was connected to a
thermocouple which displayed the stainless steel sphere’s temperature on a display
screen. The time was recorded by a phone stopwatch. After 1500 seconds, the
temperature of the stainless steel sphere was 42.0 °C. After the experiment was
conducted in the lab, students made all the calculations correlated to the lab. Students
plotted a graph of ln((T-T∞)/(Ti-T∞)) vs time from the recorded data. Using the graph, a
slope of -0.0011 was found. Using this slope, the overall effective convective heat
transfer coefficient, h
eff
, was calculated for and was found to be 13.0432. Following this
calculation, the total heat transfer out of the sphere, q, was calculated to be 0.9130. The
average temperature of the sphere, T
W
, was found to be 83.4
℃
. After these
calculations, the lumped system assumption was verified by calculating for the Biot #.
The biot number was calculated to be 0.0030 which is less than 0.01 therefore verifying
the lumped system assumption. After this, the average theoretical convection heat
transfer coefficient and its uncertainty, h_eff ± Δh, was calculated to be 13.04 ± 1.24.
Background Theory
The method of finding the convective heat transfer of a lumped object is the
lumped system approach. The lumped system approach for a sphere assumes that the
temperature is uniform throughout the object and that it is a function of time only. The
equation used to find the total heat transfer is:
Another equation is used for the total heat transfer out of the sphere which is
equal to the decrease of internal energy. The equation is as follows:
After equating the two equations for heat transfer, rearranging them and deriving
the with respect to temperature and time, you get the following equation:
In this equation, T is the temperature measured as a function of time, T
∞
is the
room temperature, T
i
is the initial temperature of the sphere at time t = 0, h
eff
is the heat
transfer coefficient, A is the area of the sphere,
⍴
is the density of the sphere, V is the
volume of the sphere, C
p
is the specific heat of the sphere, and t is the time respective
to temperature T.
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A graph can be plotted using ln((T-T∞)/(Ti-T∞)) vs time from which the slope of
the line can be found to aid in finding the heat transfer coefficient. The equation used to
find the heat transfer coefficient is:
The lumped system approach can be verified by the Biot number:
For a sphere, the equation for the Biot number is as follows:
In order for the lumped system approach to be vald, the Biot # must be < 0.01.
The constant K
S
for a brass sphere is 119 W/mK and 14 W/mK for a stainless steel
sphere.
The average theoretical heat transfer coefficient is found by calculating for the
Rayleigh number and Nusselt number. The Rayleigh number is found by using the
following equation:
Where g is the value of gravitational acceleration,
𝛽
= 1/T
f
(where as
T
f
= (T
W
+T
∞
)/2), D is the diameter of the sphere, T
W
is the sphere’s surface
temperature, T
∞
is the room temperature, v is the kinematic viscosity, adn Pr is the
Prandtl number (which can be found using a table for the physical property of air at
atmospheric pressure).
The Nusselt number is found using the following equation:
And finally the
average theoretical convection heat transfer coefficient can be
found using the following equation which is related to the Nusselt number, thermal
conductivity of the air, and the diameter of the sphere:
The uncertainty for the theoretical heat transfer coefficient can be found using the
following equation:
Where U
h
can be found using the following equation:
Aft6er finding both the average theoretical convection heat transfer coefficient
and its uncertainty, they can written in the following form:
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Experimental Procedure
The experiment was started by the instructor heating the stainless steel sphere to
124.8 °C. The stainless steel sphere was connected to a thermocouple. The students
then recorded the temperature of the sphere in 30 second increments for a total of 1500
seconds. The temperature readings were shown on a display panel and the time was
recorded by a phone stopwatch. After the students recorded the temperature vs. time,
the students began their calculations. First the students calculated ln((T-T∞)/(Ti-T∞)) for
each time/temperature and then plotted a graph of ln((T-T∞)/(Ti-T∞)) vs time. Then the
students calculated for the total heat transfer coefficient using the plotted graph.
Following that, the students calculated for the corrected heat transfer coefficient
between 90 seconds and 1500 seconds. After that, the students verified the lumped
system by use of the Biot number. Next, the students calculated the average theoretical
heat transfer coefficient. Finally, the students derived and calculated the uncertainty for
the experimental heat transfer coefficient.
Raw Data (Steel Sphere)
Time (s)
Sphere T (°C)
0
124.8
30
121.9
60
118.7
90
115.2
120
111.6
150
107.8
180
104.5
210
102.0
240
99.5
270
96.0
300
92.8
330
89.7
360
88.1
390
86.1
420
84.3
450
81.9
480
80.0
510
78.0
540
76.2
570
74.4
600
72.6
Time (s)
Sphere T (°C)
630
70.8
660
69.5
690
67.8
720
66.5
750
64.7
780
63.6
810
61.9
840
60.8
870
59.8
900
58.6
930
57.5
960
56.1
990
55.0
1020
54.1
1050
53.1
1080
52.2
1110
51.2
1140
50.5
1170
49.7
1200
48.6
1230
47.8
1260
47.1
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Time (s)
Sphere T (°C)
1290
46.3
1320
45.6
1350
45.2
1380
44.5
1410
43.8
1440
43.1
1470
42.3
1500
42.0
Calculations/Results
Time (s)
Sphere T (°C)
T (K)
ln((T-T∞)/(Ti-T∞))
0
124.8
398.0
0.00
30
121.9
395.1
-0.03
60
118.7
391.9
-0.06
90
115.2
388.4
-0.10
120
111.6
384.8
-0.14
150
107.8
381.0
-0.18
180
104.5
377.7
-0.22
210
102.0
375.2
-0.25
240
99.5
372.7
-0.28
270
96.0
369.2
-0.33
300
92.8
366.0
-0.37
330
89.7
362.9
-0.42
360
88.1
361.3
-0.44
390
86.1
359.3
-0.47
420
84.3
357.5
-0.50
450
81.9
355.1
-0.54
480
80.0
353.2
-0.57
510
78.0
351.2
-0.61
540
76.2
349.4
-0.64
570
74.4
347.6
-0.67
600
72.6
345.8
-0.71
630
70.8
344.0
-0.75
660
69.5
342.7
-0.77
690
67.8
341.0
-0.81
720
66.5
339.7
-0.84
750
64.7
337.9
-0.88
780
63.6
336.8
-0.90
810
61.9
335.1
-0.95
840
60.8
334.0
-0.97
870
59.8
333.0
-1.00
900
58.6
331.8
-1.03
930
57.5
330.7
-1.06
960
56.1
329.3
-1.10
990
55.0
328.2
-1.14
1020
54.1
327.3
-1.16
1050
53.1
326.3
-1.20
1080
52.2
325.4
-1.22
1110
51.2
324.4
-1.26
1140
50.5
323.7
-1.28
1170
49.7
322.9
-1.31
1200
48.6
321.8
-1.35
1230
47.8
321.0
-1.38
1260
47.1
320.3
-1.41
1290
46.3
319.5
-1.44
1320
45.6
318.8
-1.47
1350
45.2
318.4
-1.49
1380
44.5
317.7
-1.52
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1410
43.8
317.0
-1.55
1440
43.1
316.3
-1.58
1470
42.3
315.5
-1.62
1500
42.0
315.2
-1.64
Slope = -0.0011
Tw = (124.8+42.0)/2 = 83.4
Biot # = 0.0030
Biot # < 0.01, therefore lumped system assumptions are valid
Raleigh Number Calculations
For T = 0s
Interpolation Calculations
Thermal Conductivity (K)
K = 0.0263 + (346.58-300)*(0.03-0.0263)/(350-300)
K = 0.02975
Kinematic Viscosity (v)
v = 0.0000159 + (346.58-300)*(0.0000209-0.0000159)/(350-300)
v = 2.056 E-5
Prandtl Number (Pr)
Pr = 0.707 + (346.58-300)*(0.7-0.707)/(350-300)
Pr = 0.7004
Ra_1 = 33384.5
Nu_1 = 8.13
therefore h_theo1 = 8.13*0.2975/0.01905 = 12.697
For T = 1500s
Tf_2 = 305.18K
Beta_2 = 1/Tf = 0.00328
K_2 = 0.0263 + (305.18-300)*(0.03-0.0263)/(350-300)
K_2 = 0.0267
v_2 = 0.0000159 + (305.18-300)*(0.0000209-0.0000159)/(350-300)
v_2 = 0.0000164
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Pr_2 = 0.707 + (305.18-300)*(0.7-0.707)/(350-300)
Pr_2 = 0.706
Ra_2 = 11678.2
Nu_2 = 6.72
h_theo2 = 9.419
H_ave = (12.697+9.419)/2 = 11.058
Tw = (124.8+42)/2 = 83.4
q = 11.058*0.001140092*83.4
q = 1.05
Derivation of h_eff Uncertainty
Uh = (Uh1+Uh2)/2 = (0.1749+0.0153)/2 = 0.0951
Δh = Uh*h_eff = 0.0951*13.0432 = 1.24
h_eff = h_eff ± Δh
h_eff = 13.04 ± 1.24
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Discussion of Results
The lumped system approach for a sphere assumes that the temperature is
uniform throughout the object and that it is a function of time only. This was found to be
true as the Biot # was less than 0.01. Using this approach students were able to find the
experimental heat transfer coefficient h. This was done using a graph of ln((T-T∞) /
(Ti-T∞)) vs time, displaying a trendline slope of -0.0011 which was then used to
calculate the experimental h, using the equation h_eff =-(slope*p*V*Cp)/A which gave
13.0432 as the result. Next students calculated the average temperature of the sphere
as 83.4 degrees to then be used in the calculation of the total experimental heat transfer
from the sphere. This was found to be 0.9130 using the equation q=(h_eff)*A*(Tw-T∞).
Finally, the theoretical h would have to be found using the equations seen in the results
above. To calculate the Raleigh number and therefore Nusselt number, values of
thermal conductivity, kinematic viscosity, and the prandtl number are all interpolated
from a table of air properties at the temperature Tfilm, calculated once using the T=0s
and then again at T=1500s. These 2 sets of results were used to find h1 and h2, which
were averaged to find the theoretical h and therefore theoretical q, 11.058 and 1.05
respectively. Finally the uncertainty of the experimental h is derived from the the graph
equation, yielding the true range of h as 13.04 ± 1.24
Conclusion
Students recorded the temperature change of the steel sphere after it was
heated to 124.8 degrees Celsius, then in thirty second intervals students recorded the
temperature change until 1500 seconds. Using these values students then created a
graph which gave the slope of -0.0011, they would then calculate h_eff, using values
from the slope and density. Students were able to calculate the heat transfer and the
average temperature of the sphere using Raleigh number and Nusselt values. After
calculating the heat transfer and average temperature students were able to calculate
the Biot number to verify the lumped system assumption. Finally, the theoretical heat
transfer coefficient was calculated by finding the Nusselt number which required finding
the Rayleigh number. Using the h1 and h2 values students were able to find the
theoretical h 11.058. Following this calculation, the uncertainty for the theoretical heat
transfer coefficient was calculated. The theoretical h was 11.058 while the experimental
h was 13.04 ± 1.24. While there seems to be a large difference in the values, the
uncertainty of the experimental h yields the minimum value 11.8, which is much closer
to the theoretical value. This large uncertainty can be attributed to the large uncertainty
of the thermocouple used as well as the extended period of time the experiment was
performed at.
References
Hoang, Huy. Heat Transfer and Experiment 5. California State University, Long Beach
Mechanical and Aerospace Engineering Department.
Lab Report Format and Presentation. California State University, Long Beach
Mechanical and Aerospace Engineering Department.
Rahai, Hamid R. Laboratory Experiments. California State University, Long Beach
Mechanical and Aerospace Engineering Department.
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Group Contribution
Allen
●
Abstract
●
Background Theory
●
Experimental Procedure
●
References
Cristina
●
Discussion of Results
●
Conclusion
Jack
●
Raw Data
●
Calculations and Results
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