Sultan Qaboos University College of Engineering Petroleum and Chemical Engineering Department
Chemical Engineering Engineering Lab.1 ( CHPE4312 ) - Section # ( 20 )
Exp. 4 ( Thermal Conductivity ) By : Hilal Mohammed Ali Al Ghefeili
88549
Maher Mohammed Hamed Al Busaid
88654
Ahmed Hamed Mohamed Al Qasmi
88805
Mahamood Nasser Hamed Al Rawahi
88812
To : Dr. Mohammed Al abri
Due Date : th
24 October 2012
Abstract: Thermal conductivity coefficient was calculated for different specimens; Aluminium and Stainless Steel separately. By applying different voltages hence, different power, and noting the temperatures at different points of the specimens. It was found that there is a linear relationship between the length and the measured temperature at different voltages. The thermal conductivity coefficient values were calculated at every voltage applied and found to be different from each other due to heat losses at each trial.
I
Nomenclature:
Symbols
description
unit
V
Voltage
V
I
Current
A
P
Power
W
T cold
Temperature of the cold face
°C
T hot
Temperature of the hot face
°C
∆Xint
Thickness of the specimen
m
Aint
Surface area of the specimen
m2
K
Thermal Conductivity coefficient
W/m.°C
II
Table of Contents: Abstract:
I
Nomenclature:
II
Table of Contents:
III
List of Figures:
IV
List of Tables:
V
Introduction:
1
Experimental Setup:
2
Equipment:
2
Apparatus consist:
2
Procedure:
3
Results:
4
Discussion of Results:
6
Conclusion:
7
References:
8
Appendices:
9
Appendix (A) : Raw Data
9
Appendix (B) : Calculations
9
Appendix (C) : Matlab Code to Extrapolate Data and Calculate (k)
III
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List of Figures: Figure 2 : Linear Heat Transfer Unit H111A 2 Figure 1: Apparatus Used In This Experiment 2 Figure 3 : Thermal Conductivity Taster 3 Figure 4 : Thermal Conductivity Tester Transparent views: Exploded View (a) and Side View (b) 3 Figure 5: Experimental Data and Extrapolation for Aluminum Rod 5 Figure 6 : Experimental Data and Extrapolation for Stainless Steel Rod 6
IV
List of Tables: Table1 : Experimental Data and Calculated Thermal Conductivity For Aluminium Rod 4 Table2 : Experimental Data and Calculated Thermal Conductivity for Stainless Steel Rod 5 Table 3 : Actual Thermal Conductivity for Aluminium and Stainless Steel 6 Table 4 : Experimental Data for Aluminium Rod 9 Table 5 : Experimental Data for Stainless Steel Rod 9 Table 6 : Calculation of Hot tempreture by Extrapolation 10 Table 7 : Calculation of Cold Temperature by Extrapolation 10
V
Introduction: Heat transfer is one of the most complex topics in engineering and when the heat transfer throughout a material is studied a term called "Thermal Conductivity coefficient" arises. This term is a measure of how well the material is as a he at conductor. This term is being studied and calculated throughout this experiment. This term is of great importance if the engineer is designing a heat exchanger for example or in a simple way if he/she is designing an air conditioning system for a building, because knowing the the rmal conductivity coefficient values of several material at different temperatures will help him/her so much in the process of choosing which material he/she will be using as an insulator for the building walls his/her air conditioning design will be used. This study and experiment can be very helpful for students with small knowledge in the field of heat transfer in which they can know a great amount of information from this report especially if they are interested how to measure the thermal conductivity coefficient for any material, and later on uses that data on designing some cooling or heating systems. To measure the thermal conductivity coefficient for a material we need two important equations. One, is the power equation from which we can make an assumption and equal it to rate of heat conducted. Second, is the Fourier’s law of heat conduction which relates the rate of heat conducted to the thermal conductivity coefficient and the surface area of the conduction and the temperature difference throughout the thickness of the material that the conduction is occurring.
() = ( ) ∗ ( )
() = (/. °) ∗ () ∗
()
∆(°) ∆()
()
where (P) is the power which in the experiment equals to (Q) the rate of heat conducted. In this experiment, thermal conductivity coefficient of Aluminium and Stainless Steel will be calculated at different voltages which means different power using the two equations stated above.
1
Experimental Setup:
Equipment: In this experiment we use a certain apparatus in order to measure the thermal conductivity (k) of material which relates its ability to conduct heat. The apparatus used in this experiment is shown in figure 1.
Figure 1: Apparatus Used In This Experiment
Apparatus consist: 1. Linear heat transfer unit H111A as shown in figure 2.
Figure 2 : Linear Heat Transfer Unit H111A
2
2. Thermal conductivity taster as shown in figure 3.
Figure 3 : The rmal Conductivity Taster
3. 4. 5. 6. 7.
Water source. Pipes. Thermocouples. Dewar vessel. Sample of Aluminium and sample of Stainless Steel were used in this experiment.
Procedure:
Figure 4 : Thermal Conductivity Tester Transparent views: Exploded View (a) and Side View (b)
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First of all, we connect the water pipe to the thermal conductivity taster. In this experiment it already done by the technician. Next, the thermocouples were connected from the thermal conductivity taster to the H111A device. Next, the tightening knob shown in figure (2.b) was sealed off and clamp was opened and a small quantity of grease as shown in figure (2.a) was spread all over a sample of Aluminium and it was inserted in. After that, the clamp was closed and the tightening knob was sealed on. next, the water was opened and insured to be flowing in with a rate of 1.5Litres/minute and the H111A device was turned on. then, the voltage was set to 50 V. Moreover, the temperature was left until it stabilized for approximately 40 minutes. After that, the reading values of T1, T2, T3, T6, T7, T8 and I were recorded. Next, the voltage value was increased to 100 V and the temperature was left until it stabilized again for another 40 minutes. Then, the reading values of T1, T2, T3, T6, T7, T8 and I were recorded again. Finally, the previous steps were repeated again for a sample of Stainless Steel starting with volt values of 31, 60 and 80 V.
Results: Table1 : Experimental Data and Calculated Thermal Conductivity For Aluminium Rod
Voltage (V) 50 100
Current (A) 0.068 0.126
Power (W) 3.4 12.6
T cold (°C)
Thot(°C)
33.17143 40.95714 46.24286 63.14286
2
ΔXint(m)
Aint(m )
K(W/m.°C)
0.03 0.03
0.00049 0.00049
26.68897 45.5654
The table above shows the variables of Fourier Equation that was either measured or calculated for Aluminium rod. According to the figure below, there is a linear relationship between the distance from/to the rod and the measured temperature . The two temperatures for the two sides of the rod were calculated by extrapolating the three measured temperatures of each phase. The standard deviation for the linear relationship between the temperature and length is greater than 0.73 for the hot phase and not less than 0.98 for the cold phase. The two sides Aluminium rod temperatures were generated from those linear relationships.
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50 V 100 V
Figure 5: Experimental Data and Extrapolation for Aluminum Rod
By repeating the same procedure for a Stainless Steel at three different voltages, the results are shown in the following table: Table2 : Experimental Data and Calculated Thermal Conductivity for Stainless Steel
Voltage (V) 31 60 80
Current (A) 0.046 0.081 0.104
Power (W) 1.426 4.86 8.32
Tcold (°C)
Thot(°C)
31.02857 56.77143 32.42857 69.02857 36.32857 90.94286
5
ΔXint(m) Aint(m2) 0.03 0.03 0.03
0.00049 0.00049 0.00049
K(W/m.°C) 3.385432 8.115337 9.310402
As in the Aluminium rod, there is a linear relationship in the Stainless Steel rod between length and the measured temperature at different voltages. This relationship will applied again to extrapolate the two phase temperatures as show in the figure below:
80 V 60 V 31 V
Figure 6 : Experimental Data and Extrapolation for Stainless Steel
Discussion of Results: Table 3 : Actual Thermal Conductivity for Aluminium and Stainless Steel
Specimen
Thermal Conductivity Range (W/m.°C)
Aluminium
204.3 - 250
Stainless Steel
16.3 - 24
It is clear from the previous tables, that there is a huge difference between the exact value and the experimental conductivity of Aluminium and Stainless Steel. In this experiment the minimum errors percentage of the two materials are 72.5% and 10.3% in 6
order. This big difference is due to, firstly the heat loss that was ignored in the thermal conductivity calculations. The strength of this effect decrease as the voltage increase, because there is a proportional electrical relationship between the power loss and the voltage. Secondly, the recorded temperatures were not taken at the stability moment. Finally, the effect of the extrapolation method helps to decrease the accuracy of the experimental results.
Conclusion: The thermal conductivity of the Aluminium and Stainless Steel rods was measured. The measured thermal conductivity of the Aluminium rod at 50 V and 100 V were 26.68897 and 45.5654 W/m.°C respectively. The measured value of the thermal conductivity at 31 V of Stainless Steel was 3.385432 W/m.°C. And values of 8.115337 W/m.°C and 9.310402 W/m.°C at 60 V and 80 V in order. The actual thermal conductivity of the Aluminium is 204.3-250 W/m. °C which shows a minimum error of 72.5%. For the Stainless Steel the exact thermal conductivity is 16.3-24 W/m.°C with a minimum error of 10.3%. There is a big difference between the actual and measured values of the thermal conductivity. A minimum error was found to be 72.5% and 10.3% for Aluminium and Stainless Steel respectively. That appears due to the heat loss, the resistivity of the device. Also errors may happen because the temperature probably was recorded before the stability moment. On the other hand, extrapolation method is not an accurate method.
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References: 1. http://en.wikipedia.org/wiki/List_of_thermal_conductivities , 30-10-2012 2. Çengel, Yunus A., Heat Transfer, A Practical Approach, 2nd Edition, McGraw Hill, (2003) 3. Lab Manual (Experiment 4) 4. http://hyperphysics.phy-astr.gsu.edu/hbase/tables/thrcn.html , 31-10-2012
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Appendices: Appendix (A) : Raw Data Aluminium specimen Table 4 : Experimental Data for Aluminium Rod
I(A)
V(V) T1(°C)
T2(°C)
T3(°C) T6(°C)
T7(°C)
T8(°C)
Trial 1
0.068
50
46.8
43.7
42.6
31.7
29.1
27.4
Trial 2
0.126
100
77.8
70.6
67.2
42.6
36.3
32
Stainless Steel specimen Table 5 : Experimental Data for Stainless Steel Rod
I(A)
V(V) T1(°C)
T2(°C) T3(°C) T6(°C)
T7(°C) T8(°C)
Trial 3
0.046
31
61.1
58.7
58
30
27.9
26.9
Trial 4
0.081
60
75.2
71.6
70.8
31.1
28.7
27.2
Trial 5
0.104
80
100.3
95.1
93.6
34.4
31.4
28.9
Appendix (B) : Calculations The Cross sectional Area
A =
πD 4
=
π(2.5 × 10 ) 4
9
= 0.000490874 m
Temperature Extrapolation Table 6 : Calculation of Hot tempreture by Extrapolation
Trial 1
46.07143-170.476x
r 0.804161
Trial 2
76.22857-436.19x
0.852477
63.14286
Trial 3
60.51429-124.762x
0.772879
56.77143
Trial 4
74.28571-175.238x
0.733703
69.02857
Trial 5
99.02857-269.524x
0.771184
90.94286
T (x)
T (0.03) 40.95714
Table 7 : Calculation of Cold Temperature by Extrapolation
T (x)
r
T (0.06)
Trial 1
44.48571-188.571x
0.995126
33.17143
Trial 2
74.07143-463.81x
0.993353
46.24286
Trial 3
39.31429-138.095x
0.999857
31.02857
Trial 4
42.71429-171.429x
0.996678
32.42857
Trial 5
50.61429-238.095x
0.981162
36.32857
Appendix (C) : Matlab Code to Extrapolate Data and Calculate (k)
clc clear all a=[46.8 43.7 42.6 77.8 70.6 67.2 61.1 58.7 58 75.2 71.6 70.8 100.3 95.1 93.6]; b=[31.7 29.1 27.4 42.6 36.3 32 30 27.9 26.9 10
31.1 28.7 27.2 34.4 31.4 28.9]; g=0.0075; o=[]; for i=1:5 x=[0 g 3*g]'; y=[a(i,:)]'; n=length(x); z=[ones(n,1) x ]; h=(z'*z)\(z'*y); m=length(h); sr=sum((y-z*h).^2); st=sum((y-mean(y)).^2); r2=1-sr/st; T=h(1)+0.03*h(2); o=[o;[r2 h' T]]; end for i=1:5 x=[9*g 11*g 12*g]'; y=[b(i,:)]'; n=length(x); z=[ones(n,1) x ]; h=(z'*z)\(z'*y); m=length(h); sr=sum((y-z*h).^2); st=sum((y-mean(y)).^2); r2=1-sr/st; T=h(1)+0.06*h(2); o=[o;[r2 h' T]]; end P=[3.4 12.6 1.426 4.86 8.32]; A=pi*(2.5e-2)^2/4; dx=0.03; k=[]; for i=1:5 k=[k;P(i)*dx/A/(o(i,4)-o(i+5,4))]; end k
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