**2.1 Gas flow**

The gas flow inside the cyclone separator is modeled using Large Eddy Simulations (LES), which has been successfully applied to flows involving cyclone separators [9–15]. LES has the potential to resolve most of the energy-carrying larger eddies directly, whereas smaller energy eddies are modeled using a sub-grid scale (SGS) model. The sub-grid scale refers to scales smaller than the grid size. As the grid size determines the scales of eddies that will be directly resolved, a smaller grid size results in higher accuracy at the cost of higher computational time. The Smagorinsky-Lilly model [16] with a Smagorinsky constant of 0.1 has been used for modeling the sub-grid scale stresses. The sub-grid scale stresses are calculated employing the Boussinesq hypothesis [17] similar to that in the Reynolds averaged Navier-Stokes (RANS), model.

As turbulence introduces erratic velocity fluctuations in the flow field, appropriate models are needed to capture the turbulent characteristics of the physical flow field into the numerical calculations. Many models have been developed over time to approximate the turbulent behavior of fluids, Spalart-Allmaras model, k-ε model and k-ω models are some that use the Boussinesq hypothesis. These models are relatively less expensive in a computational sense but possess shortcomings in many complex flows. Reynolds Stress equation model is another approach of modeling turbulence which uses three additional equations for determining the Reynolds stresses is considered in the present study. Accuracy of the Reynolds Stress model in the context of flows in cyclone separators has been well established in the literature. The Reynolds averaged continuity and Navier-Stokes equations are given as [2]:

$$\frac{\partial \overline{u\_i}}{\partial \mathbf{x}\_i} = \mathbf{0} \tag{1}$$

$$\frac{\partial \overline{u\_i}}{\partial t} + \overline{u\_i} \frac{\partial u\_j}{\partial \mathbf{x}\_j} = -\frac{\mathbf{1}}{\rho} \frac{\partial \overline{P}}{\partial \mathbf{x}\_i} + \nu \frac{\partial^2 u\_i}{\partial^2 \mathbf{x}\_j} + \frac{\partial}{\partial \mathbf{x}\_j} \mathbf{r}\_{ij} \tag{2}$$

where *u* is the mean velocity, *x* is the coordinate direction, *P* the average static pressure, *ρ* the gas density, *ν* the kinematic viscosity and *τ*ij the Reynolds stress tensor ( ' ' τ *ij i j* = −*u u* ) with *u'* as the fluctuating component of velocity, this term represents the transfer of momentum due to turbulence. Eq. (1) is the differential form for the conservation of mass for a small fluid element whereas Eq. (2) is the

differential form for the conservation of momentum. The terms on the left hand side of Eq. (2) collectively describe the acceleration of a fluid element, the first term denoting the temporal acceleration and the second denoting the convective acceleration. The terms on the right hand side represent the pressure force, viscous force and the Reynolds stress tensor.

#### **2.2 Particle motion**

The particle motion inside the cyclone separator is described using a Lagrangian approach, the equation for motion of the particle is given by Clift et al. [18]:

$$\frac{\partial \mathbf{u}\_{pl}}{\partial t} = F\_D \left( \mathbf{u}\_i - \mathbf{u}\_{pl} \right) + \frac{\left( \boldsymbol{\rho}\_p - \boldsymbol{\rho} \right) \mathbf{g}\_i}{\boldsymbol{\rho}\_p} + F\_i \tag{3}$$

Where *upi* is the particle velocity, first term on the right-hand side, denotes the drag force encountered by the particles due to fluid flow, the second term denotes forces due to gravitational accelerations and the third term includes all the additional forces that can be accounted for. The additional force in Eq. (3) comprises of virtual mass force, the pressure gradient force, the Brownian force and the Saffman's lift force some of which can be safely neglected for the case of particulate flow in cyclone separators [19]. The particle trajectories in the turbulent flow are solved by using the gas velocity derived from the gas flow field computed by the Large Eddy Simulations. The particles considered are small and interpenetrating in nature, which means the collisions between the particles are neglected in the present simulations. In cases where the particle concentration inside the cyclone separator is small, the exchange of momentum from the dispersed phase to the fluid phase can be neglected, this is known as a one way coupled simulation. Whereas when the exchange of momentum from the dispersed phase to the fluid phase becomes significant and is accounted for in the simulations, it is known as a two-way coupled simulation. In the present study, both one way and two-way coupled simulations have been employed. The effects of turbulence on the particles are incorporated using a discrete random walk model [20], this model simulates the interaction of the particles with discrete stylized fluid phase turbulent eddies. The turbulent eddies are characterized by Gaussian distributed random velocity fluctuations and the eddy time scale. Assuming anisotropy of the stresses, the instantaneous turbulent velocity fluctuations are given as [20]:

$$
\omega\_i' = \xi \sqrt{\overline{\mu\_i^2}} \tag{4}
$$

Each interaction of the particle with the turbulent eddy, is considered over a time scale which is shorter of the eddy time scale and the eddy crossing time, given by:

$$
\pi\_\epsilon = -T\_L \ln(r) \tag{5}
$$

Where *r* is a random number between 0 and 1 and *T*L is given by:

$$T\_L \approx 0.15 \frac{k}{\varepsilon} \tag{6}$$
