Analytical Solution of Unsteady State Conduction Problem with Heat Source

Abstract

Separation-of-variables method is one of the analytical solution methods to solve unsteady state heat conduction problems. But unsteady state conduction with heat source is an inhomogeneous problem. It can not be solved by separation-of-variables method directly. The combination of variables division method and separation-of-variables method was applied in this paper to deal with heat conduction with heat source in an infinite plate wall. The problem was divided to a steady state and inhomogeneous heat conduction problem, and an unsteady state and homogeneous heat conduction problem by variables division method. The steady state and inhomogeneous problem can be integrated directly. The unsteady state and homogeneous problem can be transformed to the problem that can be solved by separation-of-variables method directly through variable substitution. The unsteady state temperature field in the infinite plate wall was then obtained.