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We investigate a thermodynamically consistent isothermal solutal phase-field mode1 describing the solidification of a binary alloy. The system is characterized by two variables: phase-field and concentration. The phase-field locally describes the phase state of the alloy (liquid, solid or intermediate). The evolution of these variables is described by a parabolic system with Neumann boundary conditions. After presenting the construction of the model, we investigate formal asymptotic limits when the liquid-solid interface width becomes small. Limit models are found in the form of generalized Stefan problems, which account for effects of the local interface curvature and velocity. We then introduce a finite element in space, semi-implicit Euler in time numerical scheme. The convergence of this scheme is proved thanks to the introduction of a generalized elliptic projector. We use this scheme to simulate dendritic growth in alloys, and investigate the stability of physical simulations with respect to various numerical parameters.