Theory

Thermal diffusivity is defined by its place in the heat equation, which relates the time derivative of the temperature, T, to the Laplacian of the same. More succinctly,

The constant α which relates the two is the thermal diffusivity. The heat equation can be solved in any number of ways, depending on the conditions of the system being modeled. In our case, we assume that the samples we are using are uniform, effectively 2D, and sufficiently large that the edges of the samples will remain at fixed temperature. Each of these assumptions are simply approximations which simplify the math, and throughout the course of the experiment we encountered samples for which each of these assumptions became a worse approximation, the effects of which will be shown later. We also assume that radiative and convective contributions to heat loss in the system are negligible compared to the diffusive term, which is reasonable as the thermal conductivity of the substances we used is more than 100 times that of air. Thermal conductivity is measured in Watts per Meter*Kelvin which makes it more useful when comparing how easily energy flows through a material. With these assumptions, we can apply Fourier’s Theorem to find a general solution to the heat equation,

where T0 is the fixed temperature of the edges, L is the length of the sides of the sheet in question, and Cn are the Fourier coefficients, given by

From equations 2 and 3 we find that, for our assumptions, the equation for the temperature at any point on the samples can change wildly depending on the initial conditions, but will always return to the fixed temperature T0 given enough time. In this experiment, the initial conditions are set by introducing a heat source to the samples, creating a thermal profile like a delta function. In the case of the initial thermal profile appearing exactly as a delta function, the solution to the heat equation is given as

where A is the amplitude r is the radius from the center of the thermal profile, where temperature is maximized. Naturally, equation 4 is for the idealized initial condition. In reality, the initial thermal profile must have some width, which can be accounted for by adding an extra term into the denominator of the gaussian in equation 4. This is taken into account for our analysis, but not recorded as that value isn’t required to determine the thermal diffusivity. For our method, we need only know that the width of the gaussian grows with time, at a rate proportional to 8 times the thermal diffusivity. Figure 1 provides a visualization.

Figure 1: Normalized visualization of equation 4, for α = .04, and t = 0, .5, 1, and 2 seconds

For this experiment, we began by measuring the thermal diffusivities of copper, aluminum, and a brass alloy consisting of 70% copper and 30% zinc, as we could compare the values obtained from this method with accepted values from literature as a sort of proof of concept. This done, we measured the thermal diffusivity of two different thread patterns of carbon fiber, unidirectional and square weave, to see how the non-uniformity affected the thermal profile and, by extension, thermal diffusivity. To see the effects of the non-uniformity, we had to measure the thermal profile separately along the two axes of the samples. For the unidirectional carbon fiber, this corresponds to measuring parallel and perpendicular to the direction of the threads, while for the square weave the threads are woven perpendicular to start, so we measured parallel to both directions.