Sultan Qaboos University

College of Engineering

Petroleum and Chemical Engineering Department

Chemical 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 :

24th October 2012

I

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.

II

Nomenclature:

Symbols description unit

V Voltage V

I Current A

P Power W

Tcold Temperature of the cold face °C

Thot 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

III

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) 12

IV

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

V

List of Tables:

Table 1 : Experimental Data and Calculated Thermal Conductivity For Aluminium Rod 4 Table 2 : 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

1

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 heat 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 thermal 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.

푷(푾) = 푰(푨풎풑) ∗ 푽(푽)(ퟏ)

푸(푾)= 푲(푾/풎. °푪) ∗ 푨(풎ퟐ)∗ ∆푻(°푪)∆풙(풎) (ퟐ)

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.

2

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.

Apparatus consist: 1. Linear heat transfer unit H111A as shown in figure 2.

Figure 2 : Linear Heat Transfer Unit H111A

Figure 1: Apparatus Used In This Experiment

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2. Thermal conductivity taster as shown in figure 3.

3. Water source. 4. Pipes. 5. Thermocouples. 6. Dewar vessel. 7. Sample of Aluminium and sample of Stainless Steel were used in this

experiment.

Procedure:

Figure 3 : Thermal Conductivity Taster

Figure 4 : Thermal Conductivity Tester Transparent views: Exploded View (a) and Side View (b)

4

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:

Table 1 : Experimental Data and Calculated Thermal Conductivity For Aluminium Rod

K(W/m.°C) Aint(m2) ΔXint(m) Thot(°C) Tcold (°C) Power (W)

Current (A)

Voltage (V)

26.68897 0.00049 0.03 40.95714 33.17143 3.4 0.068 50 45.5654 0.00049 0.03 63.14286 46.24286 12.6 0.126 100

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.

5

By repeating the same procedure for a Stainless Steel at three different voltages, the results are shown in the following table:

K(W/m.°C) Aint(m2) ΔXint(m) Thot(°C) Tcold (°C) Power (W)

Current (A)

Voltage (V)

3.385432 0.00049 0.03 56.77143 31.02857 1.426 0.046 31 8.115337 0.00049 0.03 69.02857 32.42857 4.86 0.081 60 9.310402 0.00049 0.03 90.94286 36.32857 8.32 0.104 80

50 V ـــــــــــــــــ

100 V ـــــــــــــــــ

Figure 5: Experimental Data and Extrapolation for Aluminum Rod

Table 2 : Experimental Data and Calculated Thermal Conductivity for Stainless Steel Rod

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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:

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

80 V ـــــــــــــــــ

60 V ـــــــــــــــــ

31 V ـــــــــــــــــ

Figure 6 : Experimental Data and Extrapolation for Stainless Steel Rod

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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.

8

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

9

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 =πD4 =

π(2.5 × 10 )4 = 0.000490874m

10

Temperature Extrapolation

Table 6 : Calculation of Hot tempreture by Extrapolation

T (x) r T (0.03) Trial 1 46.07143-170.476x 0.804161 40.95714

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

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

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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