The numerical simulation of heat conduction in irregularly‐shaped materials of thermally‐dependent properties

The numerical simulation of conduction heat transfer in arbitrarily‐shaped regions is relevant to many engineering applications including the casting of plastics and metals, cold region problems and latent heat storage. This class of problem offers its particular challenges. The accurate computational simulation of conduction with thermally‐dependent properties requires a solution technique with good conservation properties. The boundary‐fitted co‐ordinates of an arbitrary solution domain typically yield grids which are non‐orthogonal and which, consequently, make energy conservation difficult to model. In the current article, a technique is described to simulate accurately the conduction heat transfer in materials of thermally‐dependent properties in irregular domains. The method combines boundary‐fitted co‐ordinates with the finite volume method, FVM, to produce a numerical technique which will accurately solve this non‐linear conduction problem using a grid which may be highly skewed. This capability is achieved through a unique treatment of the cross‐derivative terms that arise when the heat conduction equation is transformed to a non‐orthogonal grid. The cross‐derivative terms represent the non‐normal components of the heat fluxes into the skewed control volume. The tangent components of the heat fluxes are interpreted in a special way to produce finite difference expressions which accurately model the cross‐derivative partial. The numerical procedure is validated by comparing it against a purely analytical mathematical method. Although the numerical results have been obtained using a highly skewed grid, they exhibit close agreement with the analytically‐derived solution.