In this study, models for simulating transport phenomena occurring during solidification of a binary metal alloy are reviewed, with emphasis placed on the benefits and shortcomings of existing continuum and two-phase approaches. Linkages between the two approaches are discussed, and volume averaging procedures inherent in the two-phase model are used to develop an extension of the continuum model which retains its computational convenience, while eliminating its inability to treat important features such as solutal undercooling, nucleation, stereological characteristics of the solid/liquid interface, solid movement in the form of floating or settling crystals, and shrinkage. Two approaches to development of a mixture momentum equation are considered, one involving evolution from the liquid momentum conservation equation and the other involving summation of the liquid and solid momentum equations. Special features of both approaches are discussed. In a companion paper, additional models are developed to account for the transport (floating and settling) of solid crystals in the melt, solutal undercooling, and nucleation.