Finite element implementation of the eigenfunction series solution for transient heat conduction problems with low Biot number

Analytical solutions to transient heat conduction problems are often obtained by superposition of a particular solution (often the steady-state solution) and an eigenfunction series, representing the terms that decay exponentially with time. Here, a finite element realization of this method is presented in which conventional finite element discretization is used for the spatial distribution of temperature and analytical methods for the time dependence. This leads to a linear eigenvalue problem whose solution then enables a general numerical model of the transient system to be created. The method is an attractive alternative to conventional time-marching schemes, particularly in cases where it is desired to explore the effect of a wide range of operating parameters. The method can be applied to any transient heat conduction problem, but particular attention is paid to the case where the Biot number is small compared with unity and where the evolution of the system is very close to that with zero heat loss from the exposed surfaces. This situation arises commonly in machines such as brakes and clutches which experience occasional short periods of intense heating. Numerical examples show that with typical parameter values, the simpler zero heat loss solution provides very good accuracy. One also shows that good approximations can be achieved using a relatively small subset of the eigenvectors of the problem.