(414i) A Comprehensive Analysis of Transient Heat Conduction in Solid Slabs Using Tailor-Made Integral Transforms

A comprehensive analysis of transient heat conduction in rectangular slabs of isotropic and anisotropic solids is presented. The most general situation is considered with arbitrary space- and time-dependent internal heat sources and arbitrary space- and time-dependent environment surrounding the solid. The solution to the energy balance for the solid is facilitated using tailor-made integral transforms associated with linear conduction operators. The configurations considered are a thin slab (C1), a long rod with rectangular cross-section (C2), and a slab with three comparable dimensions (C3). The general solution is applicable for the following total combinations of boundary conditions (BCs): 9 for C1, 81 for C2, and 729 for C3. The linear conduction operators are self-adjoint, with real valued, non-positive eigenvalues and independent eigenfunctions. The closed form solution to the energy balance in solid, subject to appropriate BCs and arbitrary initial temperature distribution in the slab, is very convenient to determine spatiotemporal variations in solid temperature. Four specific examples are analyzed in detail and pertinent numerical illustrations are provided.