A numerical procedure is demonstrated for the solution of moving convective boundary problems using a general finite element program and a stationary mesh. The proposed technique will allow for the modeling of various time- and temperature-dependent heat transfer coefficient profiles and removes the restriction of a constant convective boundary velocity from the solution. Quenching of a hot surface that occurs during rewetting is effectively modeled for the case of a step function heat transfer coefficient profile. The accuracy of the numerical scheme is examined by comparing the results with an analytical rewetting solution. A final example demonstrates the general capability of this finite element technique in modeling the effects of the nonlinear temperature dependency of the material thermal conductivity and heat capacity on the rewetting solution.