A GBC-based Approach to Solving Heat Conduction Problems: a Case of Bodies with Thin Multilayer Coatings

Usually, the calculation for bodies with laminated coatings is connected with formulating and solving appropriate problems of mathematical physics for multilayer systems, which are cumbersome and ineffective for practical purposes and are typically used as standards in elaborating approximate methods. This paper suggests an approach essentially simplifying solving the problems of the determination of thermal state of constructions with thin multilayer coatings. It is based on the application of generalized boundary conditions (GBC) that gives a possibility to simplify the analytical solution of such problems, since the simulation of the influence of coatings by these conditions reduces the solution of the problem for a non-homogeneous body to the solution of a one-media problem. The derivation of GBC for the case of linear heat conduction for an orthotropic coating has been based on the operator method application (Podstryhach & Shvets, 1978) that does not require any a priori assumptions as to the distribution of temperature over the thickness of a coating. In contrast to classical ones, these boundary conditions can contain time derivatives and derivatives along a coating, that allows us, in particular, to take into account the divergence of thermal fluxes along a coating in the relationships of a balance type. The GBC on thermal parameters make it possible to formulate and solve nonclassical boundary value problems of heat conduction for the determination of thermal state of bodies with thin piecewise homogeneous coatings. Simultaneously, the restoration formulae for the distribution of temperature over the thickness of every layer of a coating in terms of boundary values of temperature in a body and the environment are constructed. The effectiveness of the suggested approach has been shown by the comparison of the results, obtained according to this approximated approach, with the exact solution of a test problem (Vendin, 1994).