Control volume capacitance method for solidification modelling

The apparent and e!ective heat capacitance methods are popular "nite element formulations used for solidi"cation problems where conduction predominates over other heat transfer mechanisms. These methods involve the speci"cation of element or nodal capacitances to accommodate for the release of latent heat. Unfortunately, they su!er from a major drawback in that energy is not correctly transported through elements so providing a source of inaccuracy. In this paper a capacitance method that considers the elements as control volumes is introduced. Elemental capacitances are prescribed so as to ensure that the Unsteady Flow Energy Equation (USFEE) for each element is satis"ed. The new method allows for the transport of mass arising from volumetric shrinkage and ensures that energy is correctly transported. A comparison is made between the method and the well-known temporal and spatial approximations of apparent heat capacitance, and e!ective capacitance. It is shown that these approaches can be highly inaccurate when energy transport is used as a criteria for judging them. Copyright ( 1999 John Wiley&Sons, Ltd