## 1 Introduction
OpenFOAM has even more flexibility due to its open-source nature, but the requirement to know the C++ programming language and the theoretical foundations of computational fluid dynamics can become a significant obstacle. In the case of FlowVision, the software package does not have the ability to describe particles using the Lagrange approach (i.e., there is no discrete element model). CFD-DEM is based on the Euler-Lagrange approach (the medium is described using the Euler approach, and the particles are described using the Lagrange approach), but a Two-Fluid Model (TFM) is also widespread, based on the Euler-Euler approach (both phases, the medium and particles, are described using the Euler approach) [16]. [PERSON]'s approach considers phases continuous and interpenetrating [5; 16]. The Lagrange approach considers particles discrete objects, i.e., the trajectory of each particle or their group is tracked separately [5; 16]. In the case of CFD-DEM, the medium is continuous and the particles are discrete objects [16]. This method is mainly used for relatively small systems and it is difficult to apply them to industrial apparatuses, since they can contain hundreds of thousands of particles, which would require huge computing powers [16]. In such cases, methods based on the Euler approach, such as CFD-TFM, are used, but they require closure equations necessary to calculate the values of additional parameters of the method, the choice of which greatly affects the accuracy of the calculation, which restricts their use [5; 47]. The calculation of the change of particle position requires the determination of the resultant force at each moment in time [4; 5; 47]. In addition to the effect of the gas flow, other particles can also affect the particle, which are either in direct contact or collide with it [4; 5; 47]. There are two models for determining particle collisions that are used in DEM: the hard-sphere model [51] and the soft-sphere model [40; 42; 52]. The hard-sphere model is event-oriented: the trajectories of particles are determined by binary instantaneous collisions that preserve momentum [42; 52]. In the intervals, the particles fly freely at a constant velocity, so the evolution of the system occurs from collision to collision [42; 52]. Collisions are processed one after the other in the order they have occurred [5; 37; 42; 46; 47; 48]. At high particle densities, such event-oriented methods are numerically impracticable; since collisions occur too often, the time interval, as well as relative particle velocities, tend to zero [5; 37; 42; 46; 47; 48]. In turn, the soft-sphere model is time-oriented and works both for dilute systems (albeit with less efficiency) and for dense systems, since the system evolves with a certain time step [5; 37; 40; 42; 47; 48]. The choice of this value should be taken especially seriously. The collision of particles occurs with a small interpenetration where the repulsive forces are determined based on one of the selected rheological schemes [5; 37; 40; 42; 47; 48]. Smooth and non-smooth DEM are used within the soft-sphere model [47; 53]. The main difference between DEM types lies in the way collisions are calculated: smooth DEM calculates collisions explicitly with a small time step locally for each pair of particles [47; 53]. In non-smooth DEM, the mutual penetration of particles is prevented by changing their velocity, and collisions are calculated implicitly and on a global scale [47; 53]. Since the soft-sphere model is time-oriented, it becomes possible to simulate the fluidized bed in any regime of its operation: from stationary to turbulent or pneumatic transport [47; 53]. A detailed calculation of particle collisions makes it possible to simulate reactors with various internals used to organize the fluidized bed, and determine the extent of their effect on the behavior of the bed. The internals of the reactor used to arrange the particle bed are also called distribution devices. These are various perforated plates, baffles, louver baffles, etc. [54; 55; 56; 57; 58; 59; 60].