# Thermal Conductivity of Metals

## Contents

# 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.

**Links**

- Video: rtsp://elabmc.ist.utl.pt:554/radiare.sdp
- Laboratory: Advanced in e-lab.ist.eu[1]
- Control room: [Unavaliable]
- Level: ****

<swf height="200" width="500">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/ConducaoCalorFinal.swf</swf>

# Experimental Apparatus

Two methods are used to study heat conduction through metals:

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

Each bar has a 5mm diameter and is 370mm long and each one is made of a different metal: iron, copper and brass.

The first 70mm of each bar are wrapped in a heating resistor. 10mm away from the heat sink is the first batch of thermometers, the second being at 250mm and the third at 300mm. The heat sink keeps one 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 material (mineral wool) that prevent heat convection and minimize thermal losses.

In a pulsed or periodic regime, an electric current is applied to the heating resistor that heats that end through Joule's effect. The heat generated will run through the bar and be dissipated at the opposite end in 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 the periodic mode, the temperature reading has two components:

- one comes from the oscillation of the heating itself, its period being the same as the heat source (on and off);
- 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:

- Execute two experiments in periodic mode, selecting a time interval of a few seconds between samples and a heating time of the order of one minute. For the first run, choose the maximum acquisition time possible, and for the second choose a reasonable value based on the data from the first.
- Copy the data to a spreadsheet or preferably to a science application, like Matlab or Origin;
- Fit a math function to the thermal behavior of the bar;
- Determine the period of oscillation for each experiment conducted and the phase shift between two consecutive thermometers, for each metal available;
- Numerically adjust the function to the experimental data to get the highest precision possible.

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.

A simple analysis can be done 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 maximum of the two experimental curves.

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 |