*2.1. Population Balance Model*

The high-volume fraction of the dispersed matte phase during the settling process in the FS settler was described by the population balance model (Eulerian–Eulerian approach). The dispersed phase model (DPM, Eulerian–Lagrange approach) method has limitations when the volume fraction is above 10%, since computational time becomes expensive. Several methods are available for droplet classifications using the population balance model. These include size class formation and the moment method. Size class formation divides the system into several size groups. Therefore, it is a more accurate way to estimate the system. Size classes are further classified into the homogeneous and inhomogeneous methods. The homogeneous method classifies the different size groups into a single velocity field, which means it treats all the size groups as one phase and, therefore, the results from this method are not as accurate. In contrast, the inhomogeneous method defines a velocity profile for each size group, so this method is more feasible for an accurate representation of a system that contains different size groups. The inhomogeneous method has been used for an oil, water and air system [25], and it accurately predicted the system. Therefore, this method was preferred for this study as well. However, defining the different size groups with the inhomogeneous method is not enough, since the model equations need to be parametrized to truly depict the scenario population balance, especially for coalescence efficiency and frequency, to capture the correct coalescence rate. Nonetheless, since it is not easy to obtain experimental data under the harsh conditions of the settler, these methods may give a good estimate of what is happening inside the settler.

The population balance model distributes different droplet sizes into various size groups. These groups represent a range of droplet sizes and the volume fraction of each droplet size is estimated through mathematical models. The equation representing the population balance model (PBE) in a control volume is shown in Equation (1) [26].

$$\frac{\partial f(\mathbf{x},\ \xi,\ t)}{\partial t} + \nabla \cdot (V(\mathbf{x},\ \xi,\ t) f(\mathbf{x},\ \xi,\ t)) = \mathcal{S}(\mathbf{x},\ \xi,\ t) \tag{1}$$

where *f* represents the density function with parameters: *x*, ξ, and *t*, where *x* represents the physical coordinates, and ξ represents the internal coordinates of the particle, for example, diameter. Internal coordinates could be either a scalar or a vector depending on how many properties of the droplet or particle you wish to include. Since in this work the focus is on the diameter of the droplet, ε is considered as a scalar. Finally, *t* represents the time, *V* represents velocity, and *S* an external source term.

If the death and birth of particles are not considered, then the number of particles in a control volume is constant, and

$$\frac{dN}{dx} = \frac{\partial f(\mathbf{x}, \boldsymbol{\xi}, t)}{\partial t} + \nabla \cdot (\mathcal{V}(\mathbf{x}, \boldsymbol{\xi}, t) f(\mathbf{x}, \boldsymbol{\xi}, t)) = \mathcal{S}(\mathbf{x}, \boldsymbol{\xi}, t) = 0 \tag{2}$$

Two different schemes, the class method and the moment method, are available for estimating the particle size distribution (PSD) in the domain. The class method divides the PSD into various discrete size groups and each size group is represented by the PBE. For some applications, the multiple size groups (MUSIG) model has been developed, which uses a homogeneous and inhomogeneous approach to solve the PSD field.

Inhomogeneous particle size distribution was used for this work since homogeneous PSD schemes use a single velocity field for all particle sizes groups and therefore, may not be an accurate representation of the velocity fields for different size groups. In contrast, inhomogeneous PSD uses different velocity fields for the size groups [26].

#### *2.2. Coalescence Model*

The Luo coalescence model [27] over-predicts the coalescence frequency and requires adjustment of coefficients in the equation using sensitivity analysis and validation with experimental data [28]. Therefore, the turbulence model was used for detailed analysis. However, a few results from the Luo model are presented as well for comparison. The turbulence model is governed by the following empirical equations.

The droplet collision rate is determined by the local shear within the eddy in the suspension mixture and is represented by the following equation [29]

$$a\left(L\_i, L\_j\right) = \zeta T \sqrt{\frac{8\pi}{15}} \dot{\nu} \frac{\left(L\_i + L\_j\right)^3}{8} \tag{3}$$

where . Υ is the shear rate of the droplet and ς*T* is the capture efficiency coefficient of turbulent collision. Finally, the aggregation or coalescence rate is determined by the equation presented below [30]. The Brownian effect is negligible in turbulent flows and on droplet scale, so it was excluded.

$$a\left(L\_i, L\_j\right) = \zeta T 2^{\frac{3}{2}} \sqrt{\pi} \frac{\left(L\_i + L\_j\right)^2}{4} \sqrt{\left(\mathcal{U}\_i^2 + \mathcal{U}\_j^2\right)}\tag{4}$$

### *2.3. Coupled CFD–DEM*

Matte settling through the slag was also investigated with CFD coupled to a discrete element method (DEM). With the CFD–DEM method, individual droplets can be simulated by approximating them as soft spheres that are affected by slag flow. In the coupling, CFD is used to calculate the slag flow and drag forces for each droplet, while DEM is used to solve the contact forces and movement of the droplets. Additional models can be used to take more complex phenomena, such as coalescence, into account. CFD and DEM are not computed simultaneously; instead, first CFD solves one time step and transfers the drag forces to DEM, which then calculates until the CFD time step is reached and returns the droplet locations to CFD. Generally, DEM calculates several time steps between each CFD time step as the CFD time steps are longer. The basic principle of CFD–DEM simulation is illustrated in Figure 2.

**Figure 2.** Illustration of the computational fluid dynamics–discrete element method (CFD–DEM) calculation process.

In CFD–DEM simulation, coalescence was calculated using the bubble coalescence probability by Wang et al. [31], which was calculated for every droplet–droplet contact. If the probability was at least 50%, the droplets were considered to have coalesced.

### *2.4. Geometry Dimensions and Materials*

As CFD–DEM is a computationally demanding method, the FS settler model had to be scaled down for studying the settling phenomenon. The high computational demand is caused by using two methods in parallel and solving interactions of a very large number of individual elements. The geometry for the slag was a cube with 20 cm sides, as presented in Figure 3. As both slag and matte descend from the reaction shaft, the droplet and slag inlet was set at the center of the top face of the cube, while the tapping hole was depicted with a small tube on one side. The inlet and tapping hole had diameters of 15 cm and 1.5 cm, respectively. The slag flow rate (70 t/h) was taken from the literature [6] and scaled down with the reaction shaft diameter (4.5 m) reported in the literature [32]. The matte droplet feed rate was set as 40% of the slag feed rate and varied in size according to a normal distribution. The values of the slag and matte parameters used in the simulation are listed in Table 1.

**Figure 3.** Geometry and mesh of the CFD model for CFD–DEM simulation. Inlet (blue) on the top and tapping hole (red) on the right side.


**Table 1.** Values of the slag and matte parameters.

An additional user-defined model was used to simulate coalescence of the matte droplets. Coalescence was approximated by deleting the coalescing droplets and creating a new one in their center of mass with a mass equal to the sum of the deleted ones. Due to the limitations of the model, coalescence was instantaneous. The maximum size for coalesced droplets was limited to 2 mm, as larger droplets caused the software to use all the available memory due to the coalescence affected too many droplets at once.

This scaled-down settler geometry with similar slag and matte physical properties was modeled with CFD software as well. However, in addition to the previous research in this regard (single particle size of 500 μm) [10] and the CFD–DEM modeling part, in this CFD modeling the particle size distribution (PSD) scheme was also applied. The final selection and distribution of different droplets present in the initial mixture (PSD) were taken from the work of Jun 2018 [33], which is presented in Table 2.



Parallel with the development of these computational models, experimental investigations were ongoing on laboratory scale. The kinetics of slag–matte reactions [34,35] at temperatures prevailing in the FSF was studied in a vertical tube furnace in air and argon (oxygen-free) atmospheres simulating the settler area reactions between matte droplets and partly and nonreacted feed particles. Additionally, the main impurity elements (Sb, As, Bi) and pieces of waste electric and electronic equipment (WEEE) [36] were used in the experiments.
