Simple analysis and working equations for the solidification of cylinders and spheres

Abstract

The analysis of solidification processes is complicated by a nonlinear boundary condition at the moving solid-liquid interface, and exact solutions are rare. Various attempts to predict the rate of solidification are available in the literature but most of the results seem to be of limited use for operation and design studies on metallurgical processes. In this article we present a physical model which can be solved analytically for the most commonly encountered boundary conditions; that is constant temperature at the cooling wall or finite heat transfer to the cooling fluid. The model is based on the assumption of a linear temperature profile in the solidified shell and a corresponding differential removal of internal energy. As a result one obtains a very simple expression for the soli-dification time as a function of the space variable and the pertinent system parameters. By comparison with numerical results the prediction error is shown to be less than 10 pet over a wide range of parameter combinations. In extreme situations, where a larger error may occur, equally accurate working equations can be generated by slightly modifying the basic results.