Transient three-dimensional heat conduction in heterogeneous media: Integral transforms and single domain formulation

摘要

A hybrid numerical-analytical methodology based on integral transforms is employed for solving transient three-dimensional heat conduction problems in heterogeneous media comprising arbitrarily space variable thermophysical properties, including configurations with multiple subdomains of different materials and geometries. The Generalized Integral Transform Technique (GITT) is used to solve for both the temperature distribution and the corresponding eigenvalue problem accounting for the spatially variable coefficients. A previously introduced single domain reformulation strategy is adopted when multiple subdomains are involved, so as to rewrite the heat conduction equation in terms of coefficients with abrupt variations in the spatial coordinates. The hybrid numerical-analytical approach is demonstrated for three classes of problems in which the thermophysical properties undergo significant variations throughout the domain, such as: (ⅰ) FGMs (Functionally Graded Materials) with three-dimensional space variable thermophysical properties; (ⅱ) composite medium with inclusions of different geometries and thermophysical properties; (ⅲ) a multi-scale/multi-material heat conduction problem in a IGBT (Insulated Gate Bipolar Transistor) module. In each considered application, both the temperature eigenfunction expansions and the associated eigenvalue problem solutions are critically analyzed in terms of convergence rates and co-verified with purely numerical solutions.