Transient and Steady-State Temperature Distribution in Foil-Wound Solenoids and Other Electric Apparatus of Rectangular Cross Section

The formal solution is obtained for the transient thermal response of a rectangular region with orthotropic thermal conductivity, Joulean heating, and general boundary conditions (where the temperature in the surrounding medium is a function of both space and time). In the most practical case, where the surrounding medium is at a constant ambient temperature, the transient solution takes the form of a rapidly converging doubly infinite sum whose leading term is a very simple and accurate approximation for the thermal response. Graphical aids are presented which allow the designer to quickly obtain estimates of the temperature distribution, the maximum temperature, the average temperature, the heat flow from the boundaries, and the thermal time constant. A new and interesting result is that the solution indicates that no steady-state condition exists if the heat transfer coefficients on the boundaries are below certain critical values, or if the current density is above a critical value.