Using a Numerical Model to Investigate the Analytical Limits of Thermal Diffusion

Our project explores the thermodynamics of heat transport through a thin metal rod, which is governed by the one-dimensional thermal diffusion equation. This equation has an analytical solution only when you assume a thin metal rod of infinite length where heat is applied instantaneously to an infinitesimal segment of the rod. We created a numerical model of the diffusion equation in order to investigate the situations where the analytical solution breaks down. Our model agrees with the analytical solution under the analytical conditions and can be extended to conditions that violate analytical assumptions, including: finite rod lengths, long heat pulses, and heat-sunk and free-floating rods. In addition, the numerical model can investigate situations that are difficult to replicate experimentally, such as testing various heater sizes. We compared our numerical model to experimental data in systems with both high and low heat loss (in air and in vacuum). Our results show that, when compared to the analytical solution, the numerical simulation more accurately models thermal diffusion in metal rods.