*3.2. Coupled CFD–DEM Simulation*

The CFD–DEM simulation revealed a strong funneling effect in the settling droplet cloud. The effect was caused by the drag of the settling matte, pulling droplets towards the centerline of the cloud, and subsequently increasing the bulk density as the droplets became more closely packed. This caused an increased settling velocity which, in turn, increased the drag, and thus, created a self-sustaining funneling effect. However, the CFD–DEM simulation did not show the development of two light matte streams that can be seen at 15 s in Table 4. The development of the effect is shown in Figure 6.

**Figure 6.** Formation of the funneling effect in the slag at 5 s, 10 s, 20 s, and 40 s.

The funneling effect increased the settling efficiency of the matte by increasing the settling velocity, as presented in Figure 7. Also shown is a comparison to a calculated velocity. Stokes's law was used to calculate the settling velocity for an average droplet size for every 0.5 s, while the function calculator in Ansys CFD-Post was used to calculate the average downward slag flow velocity. These were summed to obtain the calculated velocity, which was found to match well with the results from the simulation. The average velocity decreased as the cloud reached the slag–matte interface. The decrease was caused by the slag scattering droplets on the outer sides of the bottom cloud. The scattering was caused by downward flowing slag being deflected by the matte. As the droplets were scattered, some of the smaller droplets were pushed sideways and slightly upwards, prolonging the residence time of the droplets. Some of these droplets were removed through the tapping hole. Also, some of the droplets

could have been trapped by the slag flows, preventing them from settling onto the matte layer and leading to copper losses.

**Figure 7.** Average velocity of droplets and calculated velocity.

In addition to the funneling effect, droplet size has a significant effect on settling. The droplet size increases as two droplets collide and coalesce. The droplet size can be seen to increase rapidly near the surface of the slag as drag decelerates the falling droplets. The average diameter of the droplets in the whole slag layer at 60 s was 627 μm. As the droplets started to settle and coalesce, the size increased rapidly, averaging around 1280 μm below a depth of 50 mm, as presented in Figure 8. However, the size distribution and the figure show that a significant number of the droplets coalesced very near the slag surface.

**Figure 8.** Droplet sizes in the slag layer at 60 s. The 50 mm settling distance is marked with a line.

The total feed rate was around 25,000 droplets per second. The cloud penetrated the slag layer in around 23 s, during which time over 500,000 droplets were injected in the slag. However, the droplet numbers in the simulation are significantly lower due to the high coalescence rate; 72% of all droplets in the simulation had coalesced. As can be seen in Figure 9, the droplet number peaks at around 135,000 droplets at the 12 second mark, but then begins to decrease. The decrease can be accounted for by the start of the funnel formation, which pulls the droplets closer together and thus enhances coalescence. However, the number of coalesced droplets does not show a similar peak during the funnel formation period, which can be explained by the increasing droplet diameter as the coalesced droplets further coalesce with each other. The average diameter stabilized soon after the funnel effect had fully formed. Also, the number of droplets in the simulation stabilized at around 124,000 after 30 s.

**Figure 9.** Droplet count and diameter development during the simulation.

#### **4. Discussion**

In the literature, [5,7] research regarding matte droplets settling through the slag phase was conducted at steady state using the Eulerian–Eulerian approach. In addition, the Eulerian–Lagrangian approach has been used to study the paths of individual droplets during settling and the effect of the slag tapping position and velocity on settling [6]. The conclusion drawn in these studies was that mostly droplets of 100 μm and below remained suspended in the slag phase, and that a higher tapping velocity and tap position close to the inlet disturbed the settling and entrapped the matte droplets. Similar conclusions were later drawn when using the Eulerian–Eulerian approach and transition state conditions while studying 100 μm, 300 μm, and 500 μm sized droplets individually [10]. The current investigation continued by adding the coalescence of droplets. Furthermore, instead of mono-sized droplets at the inlet, a mixture of different-sized droplets as reported in literature [33] were used for the inlet conditions, and the results from the full-scale settler were consistent with those reported earlier [6,7,10].

As expected, large droplets settled quickly and droplets of 500 μm and above mostly settled within 90 s. However, small droplets below 500 μm were still settling after 90 s, especially the 100 μm and 50 μm droplets, which were mostly dispersed inside the slag. Additionally, the formation of a vortex under the inlet and chaotic flows due to opposing currents of matte droplets trying to settle [7], and the lighter slag phase moving upwards were likewise observed in this work. Nonetheless, the coalescence rate of large droplets was relatively low, as indicated by the low volume fraction of 900 μm and 1300 μm sized droplets. In the future work, more droplets with a wider size distribution will be used as the inlet condition to get closer to the real situation.

The results in the down-scaled model and industrial scale geometry suggested a funneling settling of the matte phase. However, in the industrial scale simulation, there were several funnels whereas in the small model, only one central funnel formed. Intuitively, it is quite obvious that the large inlet area of the industrial settler would not produce only one single channel for the whole matte phase, but rather, several localized "nucleation" sites from where the funneled settling would start.

The settling of individual matte droplets was simulated using CFD–DEM coupling. The funneling effect was also formed in the slag layer as the drag pulled the droplets towards the centerline of the settling cluster. The funneling effect increased the settling velocity of the droplets, and thus, increased the settling efficiency.

A user-defined model for coalescence was also included in the CFD–DEM simulation. The behavior of the droplets, and consequently, coalescence is practically impossible to observe in a real FS process due to the extreme environment in the furnace. However, the effects of the coalescence in the simulations seemed reasonable: the settling velocity of the droplets increased due to the increasing size of the droplets. Also, coalesced droplets created a relatively large size distribution in the slag layer. Compared to the Eulerian–Eulerian method, the coalescence rate was higher in the CFD–DEM simulation. This difference could have been caused by different methods for solving the droplet–droplet contact or coalescence criteria. Nonetheless, some of the smaller droplets did not coalesce and never settled, and consequently, they were entrained in the slag and would have caused copper losses.

Due to computational instabilities, the maximum droplet size had to be limited to 2 mm. This could have had some effect on settling, as some droplets could have grown more, leading to a smaller number of droplets with a larger average size. They would cause stronger drag, and thus strengthen the funneling effect. However, the number of such large droplets would most likely be relatively low, limiting their effect. Also, very large droplets could break into smaller ones due to inertial forces. Furthermore, a significant majority of the droplets would most likely be much smaller.

The funneling effect and coalescence together formed a kind of feedback loop. Coalescence increased the droplet size which increased drag, and subsequently strengthened the funneling effect. The effect caused the droplets to move closer to each other, which consequently increased the coalescence rate. Both phenomena increased the settling velocity. The results were in good agreement with the values calculated by combining the average slag flow velocity and Stokes's law velocity of an average-sized droplet.

The slag flow deflecting from the matte surface created a kind of dispersion layer above the slag–matte interface, as some of the smaller droplets were pushed sideways with the slag. This kind of layer has also been reported in the literature [20]. Some droplets from the dispersion layer were sucked by the tapping hole, leading to copper losses. Such a situation can be seen in Figure 6; however, the effect in the simulation is likely over-emphasized by the small geometry as the tapping hole is much closer to the matte–slag interface than it would be in a real settler.

The use of well-established CFD modeling with the commercial and widely used tool ANSYS Fluent, together with a coupled CFD–DEM modeling with EDEM software was the first computational attempt to understand the settling of matte droplets in the FS settler. The results of the scaled-down geometry obtained with both computational approaches suggest that droplet coalescence plays a crucial role in the resulting flow pattern. The matte droplets entering the slag surface layer coalesce rapidly causing accelerated settling which, in turn, drags the droplets towards a central path or channel down to the underlaying matte layer. In the full-scale geometry, in contrast, the flow of matte droplets, although quickly coalescing, seems to follow a more complicated settling pattern. This is believed to be due to the more complicated flow of the continuous slag phase induced by the input material flow.

A comparison of the two computational approaches for the matte settling phenomena can be seen in Figure 10, where the small-scale model results from the CFD and CFD–DEM models are shown. Based on the computational results of the small-scale model of the matte droplets settling through the slag layer, it seems to be fair to conclude that the two approaches used, CFD and coupled CFD–DEM, both predict the settling pattern to be channeled or funneled to a central "trail" of coalescing droplets. However, these results have to be validated by a physical model before this can be confirmed.

The relation of the funneling effect and size of the settling cloud requires further study. In a real FS furnace, the area corresponding to the inlet in the model is significantly larger. The diameter of the reaction shaft can be, for example, 4.5 m compared to the 15 cm of the scaled-down model in this study. Additionally, the inlet area would not be as well defined as in the computational study. Instead, there would be a gradient in the feed density. Further research is needed to evaluate the effect of these factors on the formation and strength of the funneling effect. Moreover, the small geometry of the model may affect the ease of forming the funneling effect as walls close to the cloud may create stronger turbulence in the slag. Thus, larger model geometry should also be studied and the funneling effect experimentally validated with a physical water–oil model. The results of the validation will be presented in a future article.

**Figure 10.** A comparison of the CFD and CFD–DEM results for matte settling in the small-scale settler model, revealing a similar funneling flow pattern at 15, 17, 18, and 20 s. Upper row: CFD, lower row: CFD–DEM.

Further development is needed to improve the accuracy of the coalescence model. An improved coalescence model could decrease memory issues (especially with the CFD–DEM computation), if the droplet size limit were increased. In addition, simulating the formation and effect of spinel particles should be considered. DEM is capable of simulating spinel particles too, but their attachment to droplets and formation require an additional model. The spinel model would also be supplemented by a reaction kinetics model. However, development of these two models would require further development in the software used as well. Additionally, a more powerful computer is required until full-scale settler simulations with CFD–DEM can be considered as even small-scale simulations take a long time. For example, the small cube simulation took over a month of computing and the larger model [9] took over two months with Intel Xeon E3-1230 v5 @ 3.4 GHz.
