Thermal diffusion in the advanced lab: Evaluating numerical simulation against the analytical model

Traditionally, physicists have used analytical models of physical systems to understand and predict the system's behavior, with experiments often revealing the limitations of the analytical models. We can overcome the limits of the analytical model by creating a numerical model. The numerical model often builds on the same foundations as the analytical model but can describe situations that not only mimic the analytical model but also the situations that violate the explicit assumptions of the analytical model.

In this talk, we demonstrate the versatility of a numerical model of thermal diffusion in metal rods. In our experiment, we applied heat pulses to a copper rod and measured the temperature at various locations along the rod. We can solve for the temperature as a function of time analytically if we assume an infinitesimally short heat input, an infinitely long rod, and infinite heat sinks at the ends. Our experiment used a short (10 cm) copper rod, heat pulses extending up to 20 seconds, and aluminum heat sinks of a known, non-infinite mass. Under these conditions, the analytical model fails to predict the rod's temperature accurately. Using a numerical model, we can accurately reproduce the experimental results for all the conditions that violate the analytical model.

This experiment can be easily replicated in an advanced laboratory class to demonstrate to students how a numerical model builds on the analytical model and can more accurately predict the behavior of a complex system.