Finite Volume Algorithms for Heat Conduction

Many modern computational fluid dynamics computer programs are developed by using the finite volume discretization method. It has an excellent numerical capability for capturing changes in conserved quantities such as mass, momentum and energy. In many cases, thermal energy is transferred from fluids to some adjacent solid mass. Head accumulation in this solid matter is an important engineering issue. To capture this energy transfer, it is important to have heat conduction algorithms that function well with fluid dynamics codes. The present work tackles this problem by presenting an algorithm for solving the heat equation in finite volume form. Although this derivation is cast in two dimensions, it may be readily generalized to three dimensions. Example problems are solved involving heat conduction within a section of an annular ring. Along the boundaries we enforce both Dirichlet and Neumann boundary conditions. For code validation, our numerical solutions, based upon the Douglas-Rachford ADI time integration scheme, are compared with exact mathematical solutions.