*2.2. Dense Discrete Phase Model*

#### 2.2.1. Liquid Flow

The governing equations of flow presented in Section 2.1.1 are modified to account for the presence of the particles via their volume fraction

$$\frac{\partial(a\_c \rho\_c)}{\partial t} + \nabla \cdot (a\_c \rho\_c \mathbf{u}) = 0,\tag{19}$$

$$\frac{\partial(\mathbf{a}\_{t}\rho\_{t}\mathbf{u})}{\partial t} + \nabla \cdot (\mathbf{a}\_{t}\rho\_{t}\mathbf{u}\mathbf{u}) = -\mathbf{a}\_{t}\nabla p + \nabla \cdot \left[\mathbf{a}\_{t}\mu\_{t}\left(\nabla \mathbf{u} + \mathbf{u}^{T}\right)\right] + \mathbf{a}\_{t}\rho\_{t}\mathbf{g} + K\_{M}(\mathbf{v} - \mathbf{u}) + \mathbf{S}\_{M,\text{explici}}\tag{20}$$

where *α<sup>c</sup>* is the continuous-phase volume fraction, *KM* is the particle-averaged interphase momentum exchange coefficient (implicit part of the momentum exchange with discrete phase) and *SM*,*explicit* is the momentum source term resulting from the displacement of fluid in the presence of a discrete phase.
