Solution to the 1-D unsteady heat conduction equation with internal Joule heat generation for thermoelectric devices

Thermoelectric devices are semiconductor devices which are capable of either generating a voltage when placed in between a temperature gradient, exploiting the Seebeck effect, or producing a temperature gradient when powered by electricity, exploiting the Peltier effect. The devices are usually employed in environments with time-varying temperature differences and input/output powers. Therefore it becomes important to understand the behaviour of thermoelectric devices during thermal and electrical transients in order to properly simulate and design complex thermoelectric systems which also include power electronics and control systems. The purpose of this paper is to provide the transient solution to the one-dimensional heat conduction equation with internal heat generation that describes the transfer and generation of heat throughout a thermoelectric device. The solution proposed can be included in a model in which the Peltier effect, the thermal masses and the electrical behaviour of the system are considered too; this would be of great benefit because it would allow accurate simulations of thermoelectric systems. While the previous literature does not focus on the study of thermal transients in thermoelectric applications and usually considers constant the temperatures at the hot and cold sides, this paper proposes a dynamic exchange of heat through the hot and cold side, both in steady-state and transients. This paper also presents an analytical solution which is then computed by Matlab to simulate a physical experiment. Simulation results show excellent correlation with experimentally determined values, thus validating the solution