An asymptotic solution for short-time transient heat conduction between two similar contacting bodies

Abstract

A short-time asymptotic solution is developed for the problem of a half-space, part of the surface of which is raised to a prescribed temperature. A Green's function formulation is used to demonstrate that the heat flux at the surface can be determined from a one-dimensional analysis of the local heat conduction problem, except in the immediate vicinity of the edge of the heated area, where there is a boundary layer, the thickness of which grows with time. This boundary layer is then analyzed in more detail, using Williams' asymptotic technique. In particular, the additional total heat flux to the half-space due to the boundary layer is determined and hence a two-term asymptotic expression for the transient thermal resistance is obtained, which depends only on the area and the perimeter of the heated region. The results are compared with existing solutions for the case of a circular heated area and show good agreement up to Fourier numbers of the order of 0.3. Errors at large times—including the steady state—are still moderate, not exceeding 30% except for heated areas of large aspect ratio.