**2. Model Description**

The model assumes that the gas phase and the solid (particle) phase constitute an interpenetrating continuum. The different phases appear in the same calculated cell and are characterized by the volume fraction, α*i*, of each phase *i* (gas, solid). The gas-phase turbulence was described as a k–ε dispersed model and the gas-phase stress was considered in terms of effective viscosity. An advanced constitutive relation was adopted to describe solid stress.

#### *2.1. Conservation Equations*

In the process of gas–solid flow, both the gas and particle flows satisfy the conservation of mass and momentum. Given that there is no mass exchange between the solid particles and the gas phase, they are independent of each other. The mass conservation equation for phase *i* can be expressed as

$$\frac{\partial(\alpha\_i \rho\_i)}{\partial t} + \nabla \cdot (a\_i \rho\_i \mathcal{U}\_i) = 0,\tag{1}$$

$$
\Sigma a\_i = 1.\tag{2}
$$

The momentum conservation equation for phase *i* can be written as

$$\frac{\partial (a\_i \rho\_i l I\_i)}{\partial t} + \nabla \cdot (a\_i \rho\_i l I\_i l I\_i) = \nabla \cdot \tau\_i + a\_i \rho\_i \mathbf{g} + \mathbf{S}. \tag{3}$$

The source term, *S*, is generated by the momentum transfer between the gas and solid phases and is expressed as

$$S = \beta \{ \mathcal{U}\_{\bar{j}} - \mathcal{U}\_{\bar{i}} \}, \; j \neq i. \tag{4}$$

For α*<sup>g</sup>* > 0.8, coefficient β is based on the drag force of the fluid acting on a single particle, and for α*<sup>g</sup>* ≤ 0.8, β is described by Ergun's equation [37]. Thus, β can be expressed as

$$\beta = \begin{cases} \frac{3}{4} \mathbb{C}\_{D} \frac{\alpha\_{s} \alpha\_{\mathcal{S}} \rho\_{\mathcal{S}} \left[ \left| L\_{s} - \mathcal{U}\_{\mathcal{S}} \right| \right]}{d\_{s}} \alpha\_{\mathcal{S}}^{-2.65} & \alpha\_{\mathcal{S}} > 0.8\\ 150 \frac{\alpha\_{s}^{2} \mu\_{\mathcal{S}}}{\alpha\_{\mathcal{S}} d\_{s}^{2}} + 1.75 \frac{P\_{\mathcal{S}} \alpha\_{s} \left[ \left| L\_{s} - \mathcal{U}\_{\mathcal{S}} \right| \right]}{d\_{s}} & \alpha\_{\mathcal{S}} \le 0.8 \end{cases} \tag{5}$$

where *ds* is the solid (particle) diameter; the drag coefficient, *CD*, is given by

$$\mathcal{C}\_{D} = \begin{cases} \frac{24}{a\_{\circ} Re} \left[ 1 + 0.15 \left( a\_{\circ} Re \right)^{0.687} \right] & Re \le 1000\\ 0.44 & Re > 1000 \end{cases} \tag{6}$$

where *Re* is the particle Reynolds number and can be expressed as

$$Re = \frac{\rho\_{\mathcal{S}} d\_{\mathfrak{s}} \left| \mathcal{U}\_{\mathfrak{s}} - \mathcal{U}\_{\mathfrak{s}} \right|}{\mu\_{\mathfrak{s}}}.\tag{7}$$
