Building on previous treatments of transport phenomena occurring during the solidification of binary alloys, a model is constructed to simulate the casting of ternary alloys. Specifically, the continuum mixture equations for the transport of mass, momentum, energy and species are modified to account for a third component. To close the model, supplemental thermodynamic relations, which are taken from the ternary equilibrium phase diagram and which treat the several primary solid phases, binary eutectic troughs, ternary eutectic and peritectic reactions and the formation of an intermetallic compound, are presented. Finally, a calculation procedure is developed for the thermodynamic model. Numerical results based on the application of this model are discussed in a companion paper [Krane and Incropera, Intenational Journal of Heat and Mass Transfer, 1997, 40 3837-3847].