## 3 Numerical Model and Method  
### Numerical Model  
Most previous studies of the behavior of a fluid-solid fluidized bed used a two-dimensional coupling model, most of the time between the DEM method and the lattice Boltzmann method (LBM) [4; 19; 20; 21]. Two-dimensional models did not give a very clear account of the mechanical and hydraulic properties involved in the study of the behavior of a granular bed subjected to a localized upward flow. To realistically give account for these granular behaviors and for a quantitative approach to this problem, we developed a three-dimensional model in this work, based on the coupling of two particulate methods, EDEM and ANSYS-FLUENT. EDEM, used for describing solid particle dynamics, is a commercial software of discrete element method (DEM) expanded for taking into account the bulk material simulation. The continuous model, based on the Eulerian multiphase models code in CFD-FLUENT, is used for solving the fluid dynamics equation. Figure 2 provides an overview of EDEM-FLUENT coupling process flow. Figure 1: Schematic representation of the experimental device. The present study, which concerns the case of dense granular materials immersed, takes into account the approach of deformable bodies in contact. This approach allows for multiple particle interactions (including possible local deformation at the point of contact that is frequent in dense fluidized beds). Particle movement is tracked individually according to [PERSON]'s motion laws:  
\[m_{p}\frac{du_{p}}{dt}=F_{P}+F_{D}+F_{C} \tag{1}\]  
\[I_{p}\frac{d\omega_{p}}{dt}=T_{p} \tag{2}\]  
where \(m_{p}\) is the particle mass, \(u_{p}\) is the particle velocity, \(F_{P}\) is the particle gravity force, \(F_{D}\) is the fluid-particle drag force, \(F_{C}\) is the contact force, \(\omega_{p}\) is the particle angular velocity, \(I_{p}\) is the particle inertia momentum, \(T_{p}\) is the applied resulting torque on the particle. The drag force represents the energy dissipation resulting from the effects of the fluid on the particles during the movement. It is expressed by:  
\[F_{D}=\frac{\beta V_{p}}{(1-\varepsilon)}\,\left(u_{f}-u_{p}\right) \tag{3}\]  
where \(V_{p}\) is the volume of particle, \(\beta\) is the interphase transfer term, \(\varepsilon\) is the porosity. The particle gravity force is expressed by:  
\[F_{P}=m_{p}\,\,\xi=V_{p}\left(\rho_{p}-\rho_{f}\right)\xi \tag{4}\]  
The soft sphere model, proposed by [PERSON] and [PERSON] [22] and used for the first time by [PERSON] et al. [23], was adopted in this study to solve all the problems of particle deformation and collision of each particle. Forces are studied as the three models (spring, dashpots and a slider).These particle-particle and particle-wall contact forces are calculated as a function of spring constants and damping coefficients. The following enumerated  
Figure 2: EDEM-FLUENT coupling process flow. equations are used to account for particle overlaps in the model of soft spheres during particle collisions:  
\[F_{Ci}=\sum_{j}\bigl{(}f_{n,ij}+f_{t,ij}\bigr{)} \tag{5}\]  
\[T_{pi}=\sum_{j}\bigl{(}r_{i}n_{ij}\times f_{t,ij}\bigr{)} \tag{6}\]  
In which \(F_{Ci}\) is the net contact force acting on a particle \(i\) because of its neighbors, \(T_{pi}\) is the couple acting on a particle \(i\) because of its neighbors, \(f_{n,ij}\) and \(f_{t,ij}\) are the normal and tangential forces on particle \(i\) by particle \(j\).