In the Eulerian approach, both the fluids and solid phases are treated as interpenetrating continua in a computational cell that is much larger than the individual particles, but still small compared to the size of the process scale. Therefore, continuum equations are solved for both phases with an appropriate interaction term to model them. This, in turn, means that constitutive equations for inter- and intra-phase interaction are needed. Since the volume of a phase cannot be occupied by the other phases, the concept of phasic volume fraction is introduced. Location-based mapping techniques are applied and local mean variables are used in order to obtain conservation equations for each phase. The advantage of this approach is its reasonable computational cost for practical application problems, making it the most used granular-fluid modeling technique in use today [38].

The problem relies on the fact that finding general equations for granular systems is difficult due to the changing nature of how solids flow. However, the capacity of the continuous approach to produce accurate results is directly dependent on the constitutive relations adopted for modeling interactions between the phases and the rheology of the particulate material, which are quite difficult to obtain [54].

Moreover, due to the continuum interpenetrating approach, no individual particle information is available, and this might be the data sought. Not to mention that prescribing a particle size distribution can increase your computational cost, since in general several phases are created to model several particle sizes.

In the Lagrangian approach, the fluid is still treated as continuum by solving the Navier-Stokes equations, while the dispersed phase is solved by tracking a large number of particles through the flow field. Each particle (or group of particles) is individually tracked along the fluid phase by the result of forces acting on them by numerically integrating Newton's equations that govern the translation and rotation of the particles [55].

This approach is made considerably simpler when particle-particle interactions can be ignored. This requires that the dispersed second phase occupies a low volume fraction, which is not the reality in the majority of the industrial applications. Due to the fact that no particle interaction is resolved, the model is inappropriate for modeling applications where the volume fraction of the second phase cannot be ignored, such as fluidized beds. For applications such as these, particle-particle interactions need to be taken into account when solving the dispersed phase.