*2.1. Discrete Phase Model*

### 2.1.1. Liquid Flow

The governing equations of flow are continuity Equation (1) and the Navier–Stokes Equation (2)

$$\frac{\partial(\rho\_{\varepsilon})}{\partial t} + \nabla \cdot (\rho\_{\varepsilon} \mathfrak{u}) = 0,\tag{1}$$

$$\frac{\partial(\rho\_{\varepsilon}\mathfrak{u})}{\partial t} + \nabla \cdot (\rho\_{\varepsilon}\mathfrak{u}\mathfrak{u}) = -\nabla p + \mu\_{\varepsilon}\nabla^{2}\mathfrak{u} + \rho\_{\varepsilon}\mathfrak{g} + \mathfrak{S}\_{M\prime} \tag{2}$$

where *ρ<sup>c</sup>* is the continuous phase (liquid) density, *u* is the continuous phase (liquid) velocity, *μ<sup>c</sup>* is the continuous phase (liquid) dynamic viscosity and *p* is the pressure in the continuous phase (liquid). The term *ρcg* represents the gravitational force, where *g* is the gravitational acceleration.

The presence of particles in the flow can affect the flow field, which is accounted for by the additional momentum term *SM*. For turbulence, the RANS approach is adopted with the realizable *k* − *ε* turbulence model, where two additional transport equations for turbulence kinetic energy (k) and dissipation of turbulence kinetic energy (ε) are used.

$$\frac{\partial(\rho\_c k)}{\partial t} + \nabla \cdot (\rho\_c k u) = \nabla \cdot \left[ \left( \mu\_c + \frac{\mu\_l}{\sigma\_k} \right) \nabla k \right] + G\_k + G\_b - \rho\_c \varepsilon - Y\_M + S\_{k\prime} \tag{3}$$

$$\frac{\partial(\rho\_{\varepsilon}\varepsilon)}{\partial t} + \nabla \cdot (\rho\_{\varepsilon}\varepsilon \mathfrak{u}) = \nabla \cdot \left[ \left( \mu\_{\varepsilon} + \frac{\mu\_{t}}{\sigma\_{\varepsilon}} \right) \nabla \varepsilon \right] + \rho\_{\varepsilon} \mathbb{C}\_{1} \mathbb{S} \varepsilon - \rho\_{\varepsilon} \mathbb{C}\_{2} \frac{\varepsilon^{2}}{k + \sqrt{\nu \varepsilon}} + \mathbb{C}\_{1\varepsilon} \frac{\varepsilon}{\tilde{k}} \mathbb{C}\_{3\varepsilon} \mathbb{G}\_{\mathfrak{k}} + \mathbb{S}\_{\varepsilon \prime} \tag{4}$$

where the coefficients are

$$\mathcal{C}\_1 = \max\left[0.43; \frac{\eta}{\eta + 5}\right],\tag{5}$$

$$
\eta = \mathcal{S} \frac{k}{\varepsilon},
\tag{6}
$$

and the strain rate magnitude is

$$S = \sqrt{2S} \,\text{S}.\tag{7}$$

The model constants are *C*1*<sup>ε</sup>* = 1.44, *C*<sup>2</sup> = 1.9, *σ<sup>k</sup>* = 1.0 and *σε* = 1.2. The expression for eddy viscosity (*μt*) is

$$
\mu\_t = \rho\_\varepsilon \mathbb{C}\_\mu \frac{k^2}{\varepsilon} \tag{8}
$$

In contrast to the standard *k* − *ε* model, the coefficient (*Cμ*) is a function of mean strain, mean rotation rate, turbulence kinetic energy and dissipation of turbulence kinetic energy. In Equations (3) and (4), there are additional terms that describe the production of turbulence kinetic energy (*Gk*), buoyancy effects (*Gb*) and compressibility effects (*YM*). *σ<sup>k</sup>* and *σε* are the turbulent Prandtl numbers for the turbulent kinetic energy (*k*) and the dissipation of turbulent kinetic energy (ε), respectively. To account for the effects of particles on turbulence, source terms for the turbulence kinetic energy (*Sk*) and the dissipation of turbulence kinetic energy (*Sε*) are also included.
