*2.3. Turbulence Equations*

Turbulence predictions were obtained from a k–ε dispersed model. The transport equations were expressed as follows:

$$\frac{\partial}{\partial t} \left( a\_{\mathcal{S}} \rho\_{\mathcal{S}} k\_{\mathcal{S}} \right) + \nabla \cdot \left( a\_{\mathcal{S}} \rho\_{\mathcal{S}} \mathcal{U}\_{\mathcal{S}} k\_{\mathcal{S}} \right) = \nabla \cdot \left( a\_{\mathcal{S}} \frac{\mu\_{t,\mathcal{S}}}{\sigma\_k} \nabla k\_{\mathcal{S}} \right) + a\_{\mathcal{S}} \mathcal{G}\_{\mathcal{S},\mathcal{S}} - a\_{\mathcal{S}} \rho\_{\mathcal{S}} \varepsilon\_{\mathcal{S}} + a\_{\mathcal{S}} \rho\_{\mathcal{S}} \Pi\_{k\_{\mathcal{S}}} \tag{8}$$

$$\frac{\partial}{\partial t} \Big( a\_{\mathcal{S}} \rho\_{\mathcal{S}} \varepsilon\_{\mathcal{S}} \Big) + \nabla \cdot \Big( a\_{\mathcal{S}} \rho\_{\mathcal{S}} \mathcal{U}\_{\mathcal{S}} \varepsilon\_{\mathcal{S}} \Big) = \nabla \cdot \Big( a\_{\mathcal{S}} \frac{\mu\_{t,\mathfrak{g}}}{\sigma\_{\mathcal{E}}} \nabla \varepsilon\_{\mathcal{S}} \Big) + a\_{\mathcal{S}} \frac{\varepsilon\_{\mathcal{S}}}{k\_{\mathcal{S}}} \Big( \mathcal{C}\_{1\varepsilon} \mathcal{G}\_{k,\mathfrak{g}} - \mathcal{C}\_{2\varepsilon} \rho\_{\mathcal{S}} \varepsilon\_{\mathcal{S}} \Big) + \alpha\_{\mathcal{S}} \rho\_{\mathcal{S}} \Pi\_{\varepsilon\_{\mathcal{S}}} \tag{9}$$

where Π*kg* and Πε*<sup>g</sup>* are source terms that can be included to model the influence of the dispersed phases on the continuous phase. The constants for the k–ε model were taken as σ*<sup>k</sup>* = 1.00, σε = 1.30, *C*1<sup>ε</sup> = 1.44, and *C*2<sup>ε</sup> = 1.92 [26].
