Thermal Conductivity of Metals

UNDER CONSTRUCTION

Description of the Experiment

This experiment revolves around the conduction of heat. The user can study and compare how fast heat propagates through three bars of different materials.

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Experimental Apparatus

Two methods are used to study heat conduction through metals:

1. Single thermal pulse, applied to both ends, and subsequent evolution through the three monitored points;
2. Comparison of the output of each bar's first sensor.

Each bar has a 5mm diameter and 370mm length, and the three metals are iron, copper and brass.

Montagem: Sanduíche de lã de rocha cobrindo os cilindros dos metais.

The first 70mm of each bar have a heating resistor wrapped around them. 10mm away from the heat sink is the first batch of thermometers, the second being at 250mm and the third at 300. The heat sink keeps of of the ends at room temperature and is the reference point for the position axis. The bars are mounted in the middle of two 50mm thick layers of thermal insulation (rockwool) prevent heat convection and minimize thermal losses.

An electric current is applied to the heating resistor and this heats that end (through Joule's effect). The heating regime can be pulsed or periodic. The heat generated will run through the bar and be dissipated at the opposite end, with the sink.

By choosing the pulse width (it should be around one minute), heat propagation can be observed by measuring the three points described earlier.

Protocol

Run the experiment in pulse mode and observe how the pulse travels along the bars.

Note: The heat sink keeps the ends of the bars at room temperature, which means that the readings of one of the thermometers will not vary, having been chosen as the reference point for distance.

Advanced Protocol

In pulsed mode, the temperature reading has two components:

1. one comes from the oscillation of the heating itself, its period being the same as the heat source (on and off);
2. the other is the average heating of the bar as a whole, which is almost exponential.

Through a graphical fitting of a function to the average temperature, we can extract the oscillating values by subtracting the average. By analyzing the oscillating data, we can determine the heat propagation constant by Fourier analysis or a simple sinusoidal fitting.

For best results, please follow the procedure:

1. Execute two experiments in pulsed mode, selecting an interval of a few seconds between samples and a heating about a minute. In the first run, choose the maximum possible recording time and, after looking at these, do the second with a recording time depending on the results of the first;
2. Copy to the data to a spreadsheet or a science application, like Matlab or Origin;
3. Fit a math function to the thermal behavior of the bar;
4. Determine the period of oscillation for each experiment conducted and the phase shift between two consecutive thermometers, for each metal available;
5. Numerically adjust the function to the experimental data to get the highest precision possible.

The value for the thermal conductivity of each metal can be determined from the experimental data by using:

\[ k = \frac{C_p \rho n \omega (x_1 - x_2)^2}{2 (\phi_1 - \phi_2) ln(A_1/A_2)} \]

where \( C_p \) is the metal's heat capacity, \( \rho \) the density, \( \omega \) the angular frequency, \( (x_2 - x_1) \) the distance between thermometers, \( (f_1 - f_2) \) phase between signals (measured in radians) and \( A_1/A_2 \) the ratio between amplitudes.

We can also calculate the experimental uncertainty that k has, by using propagation formulas. Note: the diameter of the bars was measured up to a 0.01 mm and the sensor position up to 0.5 mm.

Afterwards, compare it to the textbook value.

There is a simple analysis that can be made to check the experimental data, assuming the initial stimulus is sinusoidal. This way we can take n=1 and the phase shift is determined by the distance between the extrema of the experimental data.

Experimental results for different metals

Metal	Specific heat \( C_p [J K^{-1} kg^{-1}] \)	Density \( \rho [kg m^{-3}] \)	Thermal conductivity \( k [W m^{-1} K^{-1}] \)
Brass	384	8500	113
Copper	394	8930	390
Iron	456	7860	81