Unsteady heat conduction in two-dimensional two slab-shaped regions. Exact closed-form solution and results

The unsteady heat conduction analysis for multi-directional piecewise-homogeneous bodies is generally held to be complex and demanding, possibly explaining why practical guidelines for thermal field calculation are few and far between. The proposed solution method represents an extension of the new, ‘natural’ analytic approach derived in companion papers for solving one-dimensional multi-layer problems of time-dependent heat conduction. As the approach is new, it is presented in full, together with the complete temperature double-series solution prepared for computer implementation. By setting thermal diffusivity ratio unitary and assuming a uniform distribution of initial temperature, it emerges that, all other things being equal, the transient thermal response can be expressed as the product of two, separated, one-directional solutions, one across the layers and the other along the composite slab. The formulation deals properly with thermal conductivity ratios of all magnitudes. An efficient and accurate procedure of computing eigenvalues is given. Graphical and numerical output is presented and discussed.