There are numerous systems in which heat and mass transfer are accompanied by a phase transformation and in which the motion of the melt cannot be neglected. The Stefan numbers for the solid and the melt, i.e. the ratios of sensible and latent heats based on boundary and melt temperatures, govern important parameters such as the melting rate, the heat transfer rate, and the characteristic melt film or boundary layer thicknesses. A method has been developed for studying these systems whereby the energy balance at the phase interface is combined with appropriate integral forms of the governing equations. The importance of the Stefan numbers is demonstrated by studying the contact melting of crystalline polymers in an extruder and the convective melting of glacial ice in seawater.