Approximate formulae for the thermalization of two blocks in an insulated system

Abstract

When two blocks are put in thermal contact, heat flows from the hotter block to the cooler until thermal equilibrium is reached. On thinking about the evolution of this system towards equilibrium, some interesting questions arise. Could we establish a good criterion for identifying the end of the heat transfer? If so, typically how long does it take for thermal equilibrium to be reached? What is the typical outline of the temperature distribution during the heat transfer? Although heat conduction is an important topic in science and engineering courses, only the last question is addressed in textbooks for undergraduates and it is answered based upon the Fourier series solution of the heat equation. In this work, I begin by showing that the first term of the exact Fourier series solution is itself an accurate approximation for the temperature profile for , where is the diffusive time constant for heat conduction in this system. I then introduce a ‘practical’ thermalization time which depends upon the difference of initial temperatures and the instrumental error of the thermometer, which satisfactorily answers the above questions.