A Method for Analyzing Heat Conduction With High-Frequency Periodic Boundary Conditions

A special method is developed for calculating the steady periodic temperature solution in solid bodies with high-frequency boundary conditions. The numerical difficulty associated with steep gradients and rapid temperature variation near the boundary is addressed by confining all transient temperatures to a narrow boundary layer of constant depth. The depth of the layer is specified in advance and depends only on the period of the boundary disturbance and the thermal diffusivity of the material. The transient solution in the surface layer is represented by a polynomial in its transverse coordinate, with time-varying coefficients determined by a Galerkin method. This solution is coupled with the steady interior solution by imposing continuity of temperature and time-averaged heat flux at the interface. Although the method is sufficiently general to handle nonlinear boundary conditions, it turns out to be particularly useful in the important case of a time-varying heat transfer coefficient. In the latter case, it is possible to decouple the solution process and determine the solution in the transient surface layer separately from the solution in the steady interior. This reduces the effort of determining the complete steady periodic solution to little more than a routine steady analysis. Comparison with an exact solution shows that the polynomial representation for the transient solution in the surface layer converges very rapidly with increasing order. Moreover, the solution at the surface turns out to be relatively insensitive to the choice of the layer depth as long as it is greater than a certain minimum value. An application to permanent mold casting is given, illustrating both the utility and accuracy of the method in a practical context.