A numerical analysis has been performed on the two-dimensional rectangular gallium melting problem using the enthalpy method. The major advantage of this method is that the physical domain is discretized with fixed grids without transforming variables and the interface conditions of phase change are accounted for the definition of suitable source terms in the governing equations. But in the fixed method, there is some ambiguity in defining the porosity constant which has no physical interpretation. If the velocity correction is included in the momentum equation, for the appropriate range of porosity constant, the realistic predictions are obtained. The object of the present work is to predict the phase change within the molten steel with thin riser slab using the modified enthalpy-porosity method. The computational procedures for predicting velocity and temperature are based on the finite volume method and the non-staggered grid system. The influence of natural convection on the melting process is considered. A comparison with the experimental results shows that the modified method is better than the previous one.