## Newton's Law of Cooling Heats Up.

##### Abstract

Newton's Law of Cooling (NLC) is a differential equation that attempts to predict the rate at which a body will cool when placed in an environment with an essentially constant ambient temperature. If the ambient temperature does not remain constant, there will be a discrepancy between the body temperature predicted by NLC and the temperature observed experimentally. The focus of our research has been to investigate this discrepancy (the body actually cools slower than predicted by the model) and to propose physically-motivated models which better fit the data. Because the data plotted as a fairly smooth curve, we first tried to interpolate a small number of strategically selected data points using a type of exponential decay function that met certain requirements for continuity and concavity. To determine the function parameters we used Newton's Method for systems, an iterative root-finding technique, but found that we were unsuccessful due to computational limitations and the sensitivity of the function. We then returned to NLC and modified it to allow the ambient temperature to vary with time, taking the form of a cubic spline. This model, as well as the original NLC, were solved analytically and parameters were estimated using least-squares non-linear regression and Newton's Method. Plans for future research involve developing a PDE model to describe the heat transfer between the three compartments of the experimental setup and experimentally measuring temperature changes in the compartments as well.