Lab Report Conduction With Free Convection

LAB REPORT
CONDUCTION WITH FREE CONVECTION
(FIN EXPERIMENT)

1
Contents
INTRODUCTION .......................................................................................................................... 2
FIGURE 1 ............................................................................................................................... 2
THEORY AND ANALYSIS.......................................................................................................... 4
TABLE :1.................................................................................................................................... 4
Equipment information........................................................................................................... 4
PROCEDURE................................................................................................................................. 5
RESULTS AND DISCUSSION ..................................................................................................... 6
TABLE :2a.................................................................................................................................. 6
FIGURE 2a ............................................................................................................................. 7
FIGURE 2b ............................................................................................................................. 8
TABLE :3.................................................................................................................................... 9
FIGURE :3a .......................................................................................................................... 11
FIGURE :3b .......................................................................................................................... 11
TABLE 5................................................................................................................................... 12
CONCLUSIONS........................................................................................................................... 13

2
INTRODUCTION
The object of this experiment is to measure the convection coefficient for a fin; in this case, the fin is a
cylindrical rod made of aluminum. Fins increase the surface area of a solid, and because the rate of heat
transfer is directly related to surface area, fins consequently increase the rate of heat transfer from that solid.
It is very important for the engineer to have accurate and precise convection coefficient values because these
values are used to calculate heat transfer rates, which can be very important in the design process.
Figure :1 provides a schematic of one of the fins of the temperature profile apparatus. Hot water entered the
base through the inlet, supplied heat to the fin, and was finally discharged at the outlet. The heat supplied to
the base was conducted in the z direction by the fin and thermocouples that were placed along the length of
the fin allowed temperature measurements to be taken at various distance intervals. Figure 7.1 also gives a
typical steady-state temperature profile for the fin pictured.
The fin employed in this experiment was aluminum with a thermal conductivity, k, of 401 W/m*K (Çengel, p.
868). The rod was 0.0261 m in diameter.
FIGURE 1: A schematic of one of the three fins of the temperature profile apparatus, and a corresponding
steady-state temperature profile (Janna, p. 33).
thermocouples
base

4
THEORYAND ANALYSIS
A differential equation can be written to model the one dimensional temperature profile of the fin in Figure 1:
𝑑2
𝑇
𝑑𝑧2
−
ℎ𝑃
𝑘𝐴
( 𝑇 − 𝑇∞
) = 0 (1)
where h is the convection coefficient in W/m²∙K, T is the temperature at a position along the z direction in °C,
𝑇∞ is the ambient temperature in °C, P is the perimeter around the fin in m, k is the thermal conductivity of
the fin material, and A is the cross sectional area of the fin in m².
The convection coefficient is taken to be a constant along the entire length of the fin and is an
average. The wall temperature, 𝑇 𝑊, takes the temperature value of the first thermocouple.
𝜃 =
𝑇−𝑇∞
𝑇 𝑊 − 𝑇∞
(2)
The variable m is a ratio of parameters often used to characterize fins, has units of 1/m, and is defined as:
𝑄𝑡 = √ℎ𝑃𝑘𝐴(𝑇𝑎 − 𝑇∞)tanh⁡( 𝑚𝑙) (3)
The fin efficiency, 𝜂 𝑒, is defined as a ratio of the actual heat transferred from the wall with the fin attached to
the heat transferred if the entire fin was the wall temperature. It can be shown that:
𝜂 𝑒 =
𝑄𝑡
𝑄𝑚𝑎𝑥
(4)
TABLE :1.
Equipment information.
Item Equipment Manufacturer Model Number
1 Pigcotel-thot-rod
fluke. 500 KIJ
2 Digital thrmomter Fluke 2175R
3 Power meter Vaiuc W8MT3VM
4 Voltmeter Type 2041
5 Stop watch Apple i-phone 6S
6 Fin apparatus vno ---------

5
PROCEDURE
1) The temperature profile apparatus was correctly wired to the series meter and hot water was
allowed to circulate through the base of the apparatus.
2) The temperatures of the thermocouples were periodically checked to ensure no fluctuation in
magnitudes so that steady-state was guaranteed to be achieved.
3) Once the system reached steady-state, temperatures were recorded at each of the thermocouples
along the length of the aluminum fin. There were a total of ten recorded temperatures and they are
included in Table 2.
4) The series meter and temperature profile apparatus were disconnected and the hot water,
circulating through the base, was turned off.
5) The distances between the thermocouples on the fin were measured and are shown in Table 2.

6
RESULTS ANDDISCUSSION
TABLE :2a
Experimental data obtained from the temperature profiles in solids apparatus and series meter for
copper rod.
Trial
Thermocouple position
beginning at base
Distance z
in in
Recorded
temperatures, T,
along fin in °F
1 1 0 485.2
2 2 436
3 4 371.6
4 6 362.6
5 8 350
6 10 336.4
7 12 324.2
8 14 317.2
9 16 302.6
10 18 298
 Ambient temperature: 74 F
 Power:2.2W
 Voltage:110V
 Shunt value:5

7
FIGURE 2a. Temperature represented as a function of distance from the base of the apparatus for the copper
rod.
Thermocouple position
beginning at base
Distance z
in in
Recorded
temperatures, T,
along fin in °F
1 0 484.6
2 2 425.6
3 4 387.4
4 6 382.6
5 8 350.8
6 10 332.4
7 12 324.6
8 14 316.8
9 16 305.6
10 18 300.2
0
100
200
300
400
500
600
0 2 4 6 8 10 12 14 16 18
temperature(F)
distance(in)
Run 1

8
 Ambient temperature: 73.8F
 Power:2W
 Voltage:100V
 Shunt value:5
FIGURE 2b. Temperature represented as a function of distance from the base of the apparatus for the copper
rod.
0
100
200
300
400
500
600
0 2 4 6 8 10 12 14 16 18
temperature(F)
distance(in)
Run 2

9
TABLE :3. Calculated convection coefficientfor the copper rod.
Trial
Temperature
(F)
Convection coefficient, h,
for the aluminum rod in
W/m2∙F
1
485.2
N/A
436 170.5117477
371.6 100.7993642
362.6 1.09642878
350 1.275162268
336.4 1.053853114
324.2 0.641164235
317.2 0.168616498
302.6 0.608705559
298 0.051006491

10
2
484.6
N/A
425.6 230.1711006
387.4 31.56539169
382.6 0.239512071
350.8 7.088024435
332.4 1.669303333
324.6 0.239512071
316.8 0.185478148
305.6 0.322287443
300.2 0.064764064

11
FIGURE :3a. Calculated convection coefficient, h, as a function of temperature, T, for the copper rod.
FIGURE :3b. Calculated convection coefficient, h, as a function of temperature, T, for the copper rod.
The mean value of the convection coefficient, ℎ̅, was determined to be 30.68956098 W/m2∙F in first run.
0
20
40
60
80
100
120
140
160
180
436 371.6 362.6 350 336.4 324.2 317.2 302.6 298
heattransfercoefficent(h)
temperature(F)
Run 1
0
50
100
150
200
250
484.6 425.6 387.4 382.6 350.8 332.4 324.6 316.8 305.6 300.2
heattransfercoefficent(h)
temperature(F)
Run 2

12
And 30.1717082 W/m2∙F in second run
.
TABLE 5. Theoretical fin efficiency and effectiveness for the copper rod.
Trial Fin efficiency, ηe Fin effectiveness, ηf
1 0.182274469 3552.323771
2 0.189347298 3757.545715
Table 6:
trial
1 Qmax 5165.97
Q no fin .265
2 Qmax 5221.37
Q no fin .262

13
CONCLUSIONS
This experiment successfully demonstrated a method that can be used to empirically determine the
convection coefficient from a fin. The temperature profile in solids apparatus was relatively simple to
operate and the experimental data was retrieved, without difficulty, from the series meter. Theory of
convection from fins was well modeled with this experiment. It provided a tangible application that is able to
support abstract theory, which enables the experimenter to more firmly grasp concepts.
One recommendation to improve the experiment would be to add a fixed length scale to the
apparatus against which the difference in length between thermocouples could be measured. The current
method of using a ruler held against the fin leaves significant room for user error. Also, the availability of
calipers by which to measure the diameter of the rod, or given dimensions of the rods would likely improve
the consistency of results in this experiment.

14
APPENDICES
 APPENDIX A - References
 APPENDIX B – Sample Calculations
 APPENDIX C – Original Data Sheet

15
APPENDIX A. References
References.
Janna, William S. (2008). Manual for Thermodynamics and Heat Transfer Laboratory. Department
of Mechanical Engineering. The University of Memphis, pp. 33 - 36.
Cengel, Yunus and Afshin Ghajar (2011). Heat and Mass Transfer. McGraw-Hill. New York, NY. p. 868.
APPENDIX B. Sample Calculations
The following sample calculations were performed for the thermocouple at the fifth position on the copper rod.
 𝜃 =
𝑇−𝑇∞
𝑇 𝑊 − 𝑇∞
=
336 −73.4
350 −73.4
= .9563
 𝑚 = −
ln( 𝜃)
𝑥
= .21
 𝜂 𝑒 =
1
𝑚𝐿
 𝜂 𝑓 = √ 𝑘𝑃
ℎ𝐴⁄ = √
102627∗2
30.68×,00053
= 3552.3

Lab Report Conduction With Free Convection

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Lab Report Conduction With Free Convection