A one-dimensional solidification model has been developed to study the directional solidification of dendritic alloys. It is based on the resolution of the heat flow equation using a two-interface front tracking technique. The two interfaces are defined by imaginary limits, assumed to be macroscopically flat, which correspond to the positions of the growing dendritic and eutectic interfaces. These delimit the three regions that are considered: liquid, mushy zone and solid. Growth kinetics laws are applied to the interfaces by velocity vs temperature relationships. It was found that, if complete solidification was carried out directionally up to the top of the ingot (i.e. formation of a fully columnar structure), then the velocity of the dendrite tips first increased during the stage of the superheat loss, then decreased when no substantial thermal gradient remained in the liquid ahead of the growing dendritic interface. Applied to directional solidification experiments carried out with aluminium-silicon alloys, the model shows that this maximum velocity was reached when the top position of the mushy zone (i.e. the dendritic interface) reached about two-thirds the length of the ingots. This position being in the vicinity of the columnar-to-equiaxed transition (CET) observed in the longitudinal section of the ingots, a CET scenario is proposed based on a constrained-to-unconstrained growth transition, leading to breakdown of the columnar dendritic front.