The paper presents a numerical study of phase-change problems, using finite-volume procedures of the 'enthalpy' formulation. A mathematical model is developed, and is applied to predict the transient behaviour of internal solidification in a cooled pipe, for flows at laminar Reynolds numbers. Work is directed towards predicting the production of solid crust when both diffusion and convection effects are important. Results are presented, and they are compared with available experimental measurements. It is concluded that the procedure is robust and flexible and that the results are promising.