Mineral flotation is a process whereby particulates containing mineral bearing constituents preferentially adhere to gas bubbles in a liquid medium. This is a way to separate and upgrade ores or other mineral matter. In order to simulate this unit operation hydrodynamically, a three-phase system for the liquid phase and dispersed solids and bubbles must be employed. Additional simulation features are the cell geometry, the boundary conditions, drag terms, variable viscous effects, and the treatment of the adhesion of particles to bubbles. This paper, the first of a series of two, details the formulation of the governing equations, the numerical implementation and shows some results for a two-dimensional, incompressible case. The formulation and implementation are sufficiently general and this simulation was applied to the flotation of coal-oil agglomerates as detailed in Part II of this series.