*2.3. Theoretical Analysis of CFD-DEM Coupling*

Fluent is one of the most functional and applicable CFD software in recent years, and it has good applications mainly in industries related to fluids, heat transfer, and chemical reactions. The discrete element simulation software Rocky is a general CAE software based on the discrete element method, which can be used to simulate and analyze the mechanical behavior of granular materials and their effects on material handling equipment. It has been widely used in agricultural machinery, mining equipment, chemical and food grade pharmaceuticals, and other fields Many processes in various industries involve the simultaneous flow of fluids and particles. In these cases, it is important to consider the fluid flow in order to first obtain the correct particle behavior. This is then determined by the particle-level interactions between the fluid, the particles, and the boundaries. Therefore, it is obvious that a modeling approach is needed to deal with particle fluid systems, and there are two methods commonly used to solve them: Eulerian and Lagrangian methods.

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 intraphase interactions 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.

In the Lagrangian approach, the fluid is still treated as a 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. 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. Now, numerous authors have published their work using the Euler–Lagrange type of model to study granular flow [28–31].

The coupled CFD-DEM approach is an effective alternative for modeling particulate fluid systems because it captures the discrete nature of the particle phase while maintaining computational tractability. This is achieved by solving the fluid flow at the cell level rather than at the detailed particle level. By reducing the required fluid calculations, this technique expands the range of devices and processes that can be studied with numerical simulations. In the coupled CFD-DEM approach, the fluid flow is obtained by the traditional continuous medium approach, providing information to calculate the fluid forces acting on individual particles, while the particle motion is obtained by using the discrete particle approach. The gas phase is numerically simulated by CFD, the particles are solved by the DEM

method, and the exchange of energy is coupled through the gas–solid phase interaction. Currently, Rocky has two methods to perform the coupling between particles and fluid: the multiphase coupling approach relies on the Eulerian Multiphase Model of Fluent, where the material particles are represented by another dedicated phase, and the multiphase approach supports an arbitrary number of fluid phases; the single coupling approach is achieved by setting the fluid domain in Fluent as a porous medium, which enables the material particles to influence.

When the Multiphase Model is set to Eulerian in the Fluent case, the averaged mass conservation equation is given by

$$\frac{\partial}{\partial t} \left( \mathfrak{a}\_f \rho\_f \right) + \nabla \cdot \left( \mathfrak{a}\_f \rho\_f \mathfrak{u} \right) = 0 \tag{14}$$

whereas the averaged momentum conservation equation is written as

$$\frac{\partial}{\partial t} \left( \mathfrak{a}\_f \mathfrak{p}\_f u \right) + \left( \mathfrak{a}\_f \mathfrak{p}\_f u u \right) = -\mathfrak{a}\_f \nabla p + \nabla \cdot \left( \mathfrak{a}\_f \Pi\_f \right) + \mathfrak{a}\_f \mathfrak{p}\_f \mathfrak{g} + F\_{\mathfrak{p} \to f} \tag{15}$$

where *α<sup>f</sup>* stands for the fluid volume fraction, *p* is the shared pressure, *ρ<sup>f</sup>* is the fluid density, *u* is the fluid phase velocity vector, and *Tf* is the stress tensor of the fluid phase, defined as

$$
\Pi\_f = \mu\_f \left(\nabla u + \nabla u^T\right) + \left(\lambda\_f - \frac{2}{3}\mu\_f\right) \nabla \cdot u \Pi \tag{16}
$$

In Equation (15), *Fp*→*<sup>f</sup>* represents the source term of momentum from an interaction with the particulate phase, calculated according to the expression

$$F\_{p \to f} = -\frac{\sum\_{p=1}^{N} F\_{f \to p}}{V\_c} \tag{17}$$

where *Vc* is the computational cell volume, *N* is the number of particles inside the computational cell volume, and *Ff*→*<sup>p</sup>* accounts for the forces generated by the fluid on the particles.

When the Multiphase Model is turned off in the Fluent case, Rocky adapts the Fluent setup to treat the DEM particles as a porous media and to assign to the fluid phase momentum and energy source terms (that account for fluid–particle interactions) calculated by Rocky during coupled simulations. The porosity distribution of the domain is a function of the concentration of the solid phase as the simulation progresses.

Considering a single-phase flow through a porous medium and assuming that there is no mass transfer between phases, the averaged mass conservation equation of the fluid phase is given by

$$\frac{\partial}{\partial t} \left( \gamma \rho\_f \right) + \nabla \cdot \left( \gamma \rho\_f u \right) = 0 \tag{18}$$

where *γ* is the porosity of the medium. Likewise, the averaged momentum conservation equation is

$$\frac{\partial}{\partial t} \left( \gamma \rho\_f \mu \right) + \nabla \cdot \left( \gamma \rho\_f \mu \mu \right) = -\gamma \nabla p + \nabla \cdot \left( \gamma \Pi\_f \right) + \gamma \rho\_f \gleftarrow{} + F\_{p \to f} \tag{19}$$

and the averaged energy conservation equation is

$$\frac{\partial}{\partial t} \left( \gamma \rho\_f h\_f \right) + \nabla \cdot \left( \gamma \rho\_f u h\_f \right) = \gamma \frac{\partial p}{\partial t} + \gamma \Pi\_f : \nabla u - \nabla \cdot \gamma q\_f + \mathcal{Q}\_{p \to f} \tag{20}$$

The porosity *γ* is defined as the relative volume occupied by the void spaces of the porous region. As a single-phase coupled simulation runs, Rocky estimates the porosity of each cell as

$$
\gamma = 1 - \alpha\_s \tag{21}
$$

where *α<sup>s</sup>* is the local volume fraction of the solid phase at the current time step.
