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A thermodynamically consistent method for the investigation of the coarsening behavior and in particular for the prediction of the secondary dendrite arm spacing (SDAS) in multi-component alloys is proposed which is based on the numerical simulation by means of a phase-field model. Existing variants of the phase-field model equations for multi-component systems were considered and their advantages and disadvantages were discussed. For the investigation of the coarsening behavior the variant described by the mixture composition and the entropy change was chosen. The method is applied to a high-alloy tool steel where it was found that elements such as C, Si, Mn decrease the SDAS whereas Cr increases. The resulting dependencies of the SDAS on alloy composition were compared to the analytical prediction of the coarsening model. For this aim the analytical model of the coarsening behavior in multi-component alloys [Rappaz and Boettinger, Acta Mater. 47 (1990)] was extended by taking into account the cross dependencies between the components in multi-component diffusion and the case of slow diffusion in the solid phase. The equilibrium parameters used in the phase-field model and in the analytical model were obtained from Thermo-Calc through global equilibrium calculations using the database TCFE7. The difference between both methods was found to be smaller than 2 pct in the investigated composition region.