To assist industrial engine design processes, 3-D computational fluid dy- namics simulations are widely used, bringing a comprehension of the underlying physics unattainable from experiments. However, the multiphase flow description involving the liquid jet fuel injected into the chamber is still in its early stages of development. There is a pressing need for a spray model that is time efficient and accurately describes the cloud of fuel and droplet dynamics downstream of the injector. Eulerian descriptions of the spray are well adapted to this highly unsteady configuration. The challenge is then to capture accurately the evaporating spray polydispersity in this framework. The Eulerian Multi-Size Moment model, a high-order (in size) moment model has proved to be well adapted for injection simulations with moving geometries. Moreover, it requires less computational effort as compared to existing methods, with a single section for the size phase space. Academic test cases have demonstrated its great potential for industrial applications using one-way coupling. In order to draw comparisons with experimental data, a two-way coupling framework accounting for the droplet-gas turbulence interactions is developed and validated through homogeneous test-cases under both an academic framework and realistic internal combustion engine conditions.