Solidification processes are crucial for many industrial applications, and understanding these processes is important if one wants to improve various material production techniques. The dynamics of some solidification processes can be modelled using Stefan problems, a type of free boundary problem. The aim of this dissertation is to study different aspects of the extended Stefan problem for the solidification of binary alloys with a special interest in the manufacturing of metallurgical grade silicon. First, we derive the extended Stefan problem in an arbitrary n-dimensional domain from basic physical principles, and from this we derive more specific models for different geometries of interest. Then, we perform a linear stability analysis for the exact self-similar solutions that can be found in the semi-infinite and infinite one-dimensional planar geometries. Introducing self-similar perturbations, we find that the solutions to both problems are unstable regardless of the parameter choice, and that the instability that arises is driven by the moving boundary. Having understood the relevant dynamics in unbounded domains, we next direct our attention to the extended Stefan problem in finite domains, which are of greater relevance to realistic applications. We first consider a one-dimensional finite planar geometry. In order to obtain analytical solutions we use the method of matched asymptotic analysis in the large Lewis number limit. The analysis requires us to consider ten different layers, spread over four different time regimes. We find good agreement with numerical simulations calculated using a fixed boundary method, and with experimental data from the cast of metallurgical grade silicon performed by our industrial sponsor, Elkem. Finally, we study the three-dimensional spherically symmetric geometry, in order to model the inward solidification of a binary alloy taking the form of a sphere. To determine the asymptotic solutions, we consider the large Lewis number limit and the small Stefan number limit. We distinguish several layers, and we notice qualitative differences between the spherical and planar geometries. To validate the asymptotic results, we compare them with numerical simulations, finding good agreement.