We have derived a new set of closure equations for the rheologic properties of a dense gas–solid fluidised bed consisting of a multi-component mixture of slightly inelastic spheres, using the Chapman–Enskog solution procedure of successive approximations, where the particle velocity distribution of all particle species is assumed to be nearly Maxwellian around the particle mixture velocity with the particle mixture granular temperature. In this theory, differences in the mean velocities (i.e. particle segregation) and granular temperatures of the particle species result from higher order perturbation functions. Special attention is paid to assure thermodynamic consistency between the radial distribution function and the chemical potential of a hard-sphere particle specie appearing in the diffusion driving force when applying the revised Enskog theory, which is often overlooked. In the resulting closure equations, the rheologic properties of the particle mixture are explicitly described in terms of the particle mixture velocity and granular temperature, and the diffusion velocity and granular temperature of the individual particle phases can be computed from the mixture properties, which is a major advantage with respect to the numerical implementation. A new numerical solution strategy has been devised, which is an extension of the well-known SIMPLE algorithm and takes the compressibility of the solids phase directly into account, which allows for much larger time steps (about a factor of 10 larger). In Part 2 the simulation results obtained with the new model are compared with experimental data and discrete particle model (DPM) simulations.