Transient heat conduction in a torus: theory and application

Abstract

In production processes of the packaging and plastic industry cooling-problems of ring-like blanks can arise. In this paper, a simple mathematical model for such bodies consisting of foam materials with small heat conductivities is presented using toroidal co-ordinates. The corresponding Biot numbers are very high because the initial heated blanks are exposed to an efficient cooling stage based on forced convection. Thus, from a mathematically point of view the transient heat conduction in a torus with an isothermal surface is present. The fundamental eigenvalues of the problem, i.e. the thermal time constants, are obtained using the Rayleigh–Ritz-method with series of toroidal functions as trial functions for a wide range of the ratio between the inner and the outer radius of the torus. The analytical results are successfully compared with direct numerical simulations based on the method of finite differences. It is found that even for a comparable thick torus a simple correlation arising from the infinite cylinder idealisation leads to a satisfying expression for the thermal toroidal behaviour. This enables a suitable estimation formula for the cooling times required in the considered production process.