Measurement of Thermal Conductivity in a Copper Cylinder using the One Dimensional

Heat Equationthegreedyturtle

The Ohio State University, Department of Physics, Columbus, OH 43210

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Fourier’s Law of Thermal

Conductivity (3-D)

Thermal Conductivity

Fourier, Joseph, The Analytical Theory of Heat. 1822. Article 68.

Continuity Equation (1-D)

F is total Heat Flux in x, y, and z,

is temperature difference per x unit length,

K is proportionality constant called Thermal Conductivity

u is measured as temperature

And when K does not vary with

distance…

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One Dimensional Heat Equation

Thermal Conductivity

𝜕𝑢𝜕𝑡

=−𝐾𝜕𝑢𝜕𝑥

• Temperature uniform over cross-sections• Heat transfer is by conduction• Heat transfer only along x-axis• No heat escapes from sides (perfect insulation)

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Experimental Setup

Thermal Conductivity

Cylindrical Copper Rod-1.9 cm diameter-76.2 cm long

11 Copper Constantin Thermocouples- about 8 cm apart

NI 9213 Thermocouple input module

Fig: Shaw, C. H. (January 01, 1955). Intermediate Laboratory Experiment in Heat Conduction. American Journal of Physics, 23, 2, 89.

5° C Tap Water runs constantly through the cold end.

Water is trapped the section surrounding the hot end.

The water is heated in the hot end to about 60° constant temperature and held until there is a linear thermal gradient across the entire rod.

5° C Tap Water runs through both ends and cooling measured.

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Thermal Conductivity

Wtshymanski, http://en.wikipedia.org/wiki/File:Thermocouple_circuit.svg

Measuring TemperatureThermocouples•Seebeck Effect

Thermal Gradient produces an emf

Reference Temperature required•NI 9213 Thermocouple Input

Autocalculates reference temp

Automatically converts Voltage to temp

Copper-Constantin (Type T) have sensitivity of ~1.5 °C [9213 Manual]

Can only take measurements every 1 second to get this sensitivity

Initial room temp measurements show a 0.3° range from separate thermocouples, requiring daily calibrations

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Thermal Conductivity

Complete 2.7.2104Data Run

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Thermal Conductivity

2.7.2104 Cooling Curve

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Thermal Conductivity

2.7.2104 Hot End Exponential Decay

𝑢(76.2𝑐𝑚 , 𝑡)=14.18773 𝑒(− 𝑡445.27641 )+42.56943 𝑒(− 𝑡

30.0113 )+7.00428

Setting boundary conditions for Hot and Cold ends

Not actually at exactly 5° during measurements

Repeat for Cold End

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Thermal Conductivity

2.7.2104 Cold End Exponential Decay Fit

Setting boundary conditions for Cold End

Not actually at exactly 5° during measurements

𝑢 (0𝑐𝑚 , 𝑡 )=6.72732+3.89729𝑒− 𝑡1049.39903

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Thermal Conductivity

Modelling the Heat EquationUsing Thermal Conductivity of Copper (K )as 1.1026667 cm2/sBoundaries:

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Thermal Conductivity

Grey Wireframe is Mathematica Numerically Calculated. The largest difference is at the lower edge, where the final bar is warmer in the center.

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Thermal Conductivity

Correlation of 0.99542

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Thermal Conductivity

Adding InsulationEthofoam Insulation R-Value is based on the insulation’s

thickness:

Conduction Equation:

Again,,

New equation:

(20° is Room Temp)

Correlation up by 0.00351 to 0.99893 End Temps

Match Better

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Thermal Conductivity

Adding RadiationBlackbody Radiation is based on the temperature of the surface vs. surroundings, and the total surface area.

Stefan-Boltzmann’s Law:

Stefan–Boltzmann constant:

And so when ignoring convection…

Correlation up another 0.00032 to 0.99925

Final Equation:

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Final Thoughts

Thermal Conductivity

Verify the computer saves data!Make sure the hoses are secure in the

sink!Omega CN 800 doesn’t always work,

verify that it is heating with LabView.Make sure the computer saves your data!Use a quality data analysis program,

or analyze one thermocouple at a time.Wait a long time to get the initial gradient correct, the end steady state does not take long to reachAdditional analysis of convection could possibly be added, but a different method would be better, with fewer variables, and with more careful measures of the room and temperatures surrounding the box.Speaking of which, measure the room temperature too!