Cooling and warming laws: an exact analytical solution

Abstract

This paper deals with temperature variations over time of objects placed in a constant-temperature environment in the presence of thermal radiation. After a historical introduction, the paper discusses cooling and warming laws, by taking into account first solely object–environment energy exchange by thermal radiation, and then adding object–environment heat exchange by convection. These processes are usually evaluated by approximating the law of exchange of thermal radiation by a linear relationship between power exchange and temperature difference. In contrast, in this paper an exact analytical solution considering Stefan's fourth power law is provided, under some general hypotheses, for both cases. A comparison with exponential approximations and with a historical law proposed by Dulong and Petit in 1817 is presented. Data of an experiment are used to test the analytical solution: the test has allowed evaluating the heat transfer coefficient h of the experiment and has shown that our solution provides a better fit with the measured values than any exponential function. The topic is developed in a way which can be suitable both for undergraduate students and for general physicists.