Analysis

Every image gathered during these tests corresponds to an array of pixel measurements, the first of which is the brightness of each pixel from 0 to 255. We created a program which would take this data and use it to locate the brightest point in the image and take 11 rows and columns from around this point, which were then averaged to construct a thermal profile.

Technically, the amplitude of a thermal profile given by this program, and therefore of all the graphs included in this paper, is in units of brightness from 0 to 255. As was tested previously, the brightness is directly proportional to the temperature, with some offset. So long as that is true, we were able to simply subtract the minimum temperature of the profile to ensure the amplitude is on an absolute scale. Then we only need to care about the width of the thermal profile.

Once the thermal profile of an image was obtained, it was fit to a standard gaussian model and the width recorded, using the 95% confidence values given by the fitting function as an estimation of uncertainty. This was done for each of seven images per test, with three tests per material. An example image from the thermal camera and associated thermal profile with gaussian fit is shown below.

Once the seven values for thermal profile width were obtained, they were plotted against time to create a linear graph. From equation 4 we know that the slope of this graph is 8 times the thermal diffusivity. An example of the linear fit is shown below.

This process was repeated three times for each substance, and the final value for the thermal diffusivity of each taken to be the average of the values derived from each of the three tests. As the variance between the values given by each test was much greater than the value given by MATLAB, our final uncertainty is derived from the difference between these values and the maximum. Table 1 shows our final values for the thermal diffusivity in each metal as well as a comparison with accepted values.

As seen in table 1, our data was by no means consistently accurate, but we do have explanations for why our measured values for the thermal diffusivity of brass and aluminum were so far off. In the case of aluminum, we have no guarantee that our sample was pure aluminum, though the accepted value we quote is for pure aluminum. Aluminum alloys with as little as 5% impurities can lower the thermal diffusivity by a substantial margin. Aluminum alloy 6061, for example, is extremely common and is rated to have between 95.8% and 98.6% aluminum. It has a thermal diffusivity of 69.03 mm2/s.

For our brass sample, the main issue we know we had is that the sample was too small. One of our primary assumptions when deriving equation 4 was that the edges of our samples would effectively held at a fixed temperature, but with the brass sample the thermal energy could diffuse to the edges before the end of our tests because of how small the sample was, which explains why our model would give the wrong values.

Furthermore, both the brass and aluminum samples were substantially thicker than our copper sample. Another of our main assumptions when deriving equation 4 was that the samples were two dimensional, so the added thickness of the samples further reduced our accuracy, seemingly by systematically reducing the measured thermal diffusivity.

Still, with something of a proof of concept given by our copper tests, we decided to test the two different weaves of carbon fiber. The only difference in the analysis process was that the program generated two thermal profiles by averaging either rows or columns, instead of averaging both the rows and columns together. In the case of the square woven carbon fiber, this resulted in two thermal profiles that were exceedingly similar to each other, as shown below. From these tests, we found that the thermal diffusivity along one axis of the square woven carbon fiber to be 2.100±.0.191 mm2/s and 1.750 ± 0.149 mm2/s. These values are within the bounds of what was expected when compared to the metals, but, because every batch of carbon fiber is unique, there is no accepted value to compare to.

The two values we measured for the thermal diffusivity of the woven carbon fiber along the different axes differ from each other by 0.350 ± 0.242 mm2/s, which was somewhat surprising, as we had expected them to match exactly. We assume that the main reason why the two different directions don’t agree perfectly is that the carbon fiber samples we were using are flexible, and thus the individual threads have the freedom to be off from each other by a not inconsequential margin.

For the unidirectional carbon fiber, we observed from the initial thermal images a strong anisotropy in the heat flow, as shown below. It is this anisotropy that separates our experiment from others of its kind, and is what we would have attempted to measure along the diagonals of the woven carbon fiber if we had more time.

Just from a quick glance at this figure we noticed that the thermal energy diffused somewhere between 2 and 3 times as quickly along the fibers as it did across them, so we hoped to see this reflected in the data. Applying the same process as before, we found that the thermal diffusivity along the fibers was 4.012 ± 0.057 mm2/s, and across the fibers it was 1.700 ± 0.136 mm2/s, so this seems to support our theory, but again there is no accepted value to compare these numbers against.

Unfortunately for this experiment, there is a major problem with applying equation 4 to the data taken across the fibers of the unidirectional carbon fiber sample. Across the fibers, the sample is no longer uniform, which breaks one of the assumptions that allowed us to derive equation 4. While our process did give a value within expected bounds the actual fit to the data is very inaccurate, as shown below. While the thermal profile across the fibers does not match a gaussian, its behavior is consistent, and perhaps future experiments could attempt to derive a model which does describe the heat flow across a nonuniform substance such as this carbon fiber in the future.