The paper describes the modeling of the solidification process over a planar substrate which is being pulled through a melt bath at a constant speed. The complexities generated by the presence of the two moving/unknown boundaries are dealt with by solving the transient governing transport equations numerically, and by using an efficient finite volume scheme employing a multizone adaptive grid generating technique and a curvilinear finite volume approach. The temperature distributions in the solid and melt, the melt flow field in the bath, the thickness of the solidified layer, and the shape of the free surface are calculated and reported.