2.1 MC Model for FL Formation

Following the modeling framework for FL formation described by Rodrigues et al.^[17] a computationally efficient MC model was used to account for slag solidification. It was originally proposed for metal alloy solidification by Voller and Prakash^[31] and was applied here to slag solidification. Only the key features of the model are outlined here, as the details of the approach were presented in an earlier publication.^[17]

Industrial slag^[11] with the composition listed in Table 1 was considered. Figure 2 shows the solidification path (f_s–T curve) obtained from FactSage 8.2 using the FToxid database, assuming thermodynamic equilibrium solidification conditions. To improve accuracy, additional experimental data are necessary to establish a more reliable f_s–T curve, particularly for nonequilibrium solidification conditions.

2.2 Multiphase VOF Model

Depending on the local α_slag, the material properties were either purely representative of one phase (when α_slag = 0 or 1) or representative of a mixture of phases (when 0 < α_slag < 1). The conservation equations are listed in Table 2.

Sensible enthalpy was defined by $h=h_{\text{ref}}+{\int }_{T_{\text{ref}}}^T{c}_{\mathrm{p}}\mathrm{d}T$, where h_ref = 0 J kg⁻¹ and T_ref = 298.15 K. The source terms ${\stackrel{\rightharpoonup }{\mathrm{S}}}_{\mathrm{U}}$ and S_H are described in Section 2.1. The term S_S in Equation (9) is the energy sink associated with the fuming rate. During fuming, endothermic chemical reactions occur in the slag. Although the model does not explicitly solve these reactions, the sink term accounts for their effect on the system's enthalpy. Consequently, the sink term is only applied to a predefined region of the slag bath where the fuming reaction occurs. For commercial reasons, the details of this fuming reaction are not presented here. Note that a laminar model was employed due to the absence of an accurate turbulence model for multiphase solidification processes.

To fully couple the MC solidification model with the VOF model, the source terms used in the conservation equations require the pre-factor α_slag. This ensured that the source term acted in the correct proportion for each corresponding phase. For instance, even if the local temperature fulfills the solidification condition, S_H will still be zero if no slag phase (α_slag = 0) is present.

2.3 Modeling the Two Types of FL Formations

From a modeling standpoint, two types of FL can be identified in an SF furnace, as shown in Figure 1. 1) FL formation in the slag bath region (inset A of Figure 1): the slag solidifies from a liquid state in the molten slag (f_s = 0.0) to a solid state in the FL (f_s = 1.0). This transition occurs along a spatial gradient of the solid slag known as the mushy zone (0.0 < f_s < 1.0). As the slag bath region is primarily occupied by the slag phase (α_slag = 1.0), the model described in Section 2.1 and 2.2 can be applied directly. This is referred to as the slag-in-bath approach. 2) FL formation in the freeboard region (inset B of Figure 1): splash-induced liquid slag drops collide with the reactor wall and solidify. Unlike the slag bath, in this region both gas and solidifying slag coexist locally (α_slag < 1.0), and they form a slurry zone (which includes gas, liquid slag, and solid slag). As a result, the model described in Section 2.1 and 2.2 must be extended as described later. It is referred to as the slag-in-freeboard approach.

Figure 3 is introduced to understand the computational challenges associated with the modeling of FL formation in the freeboard region. Figure 3ai–iii shows a schematic sequence of events leading to the collision and solidification of a liquid slag drop on the reactor wall in the freeboard region. The liquid slag drops originate from splashing. As shown in Figure 3aiii, after colliding with the cold wall, the drop adhered to the wall and solidified, forming an FL layer. Figure 3bi–iii shows the modeling concept for the scenario described earlier. Computational cells (black lines) are shown with an arbitrary refinement. The gray scale reflects α_slag: lighter cells indicate lower α_slag, whereas darker cells indicate higher α_slag. Drop adhesion to the wall is ensured by the combination of no-slip boundary conditions (BCs) at the wall and a substantial increase in slag viscosity during solidification. This prevents the drop from being deformed by external forces, such as incoming flow or gravity, allowing it to remain stationary and adhere firmly to the wall. Figure 3ci–iii shows rendered images of the preliminary simulation results obtained using the slag-in-freeboard approach. The shape of the drop is an interpolation of the cell center values of α_slag (with CFD-Post from ANSYS Fluent). The FL layer comprises two layers of computational cells adjacent to the wall. Note that the mesh was more finely resolved (≈4 mm) near the wall to capture the FL with the necessary detail. As expected, the drop solidified and remained stationary after colliding with the wall.

From a modeling standpoint (Figure 3b), the parameters and material properties used in the conservation equations (Equation (7)–(9)) vary with the local α_slag, as determined by the mixture laws (Equation (4-5)–(6)). When T > T_liquidus, f_s = 0 and no adaptation is required for the VOF model. However, when T < T_liquidus, solidification occurs, which means that f_s varies as shown in Figure 2, the latent heat is released into the domain according to Equation (1), the drag force is updated according to Equation (2), and the slag viscosity varies according to Equation (13) (Section 2.4). Notably, the slag viscosity can increase by several orders of magnitude with decreasing temperature. This can cause the slag solid/liquid mixture properties to dominate the corresponding mixture equation in the VOF conservation equations, even in computational cells with low α_slag. This could incorrectly cause the computational cells with minimal solidifying slag to behave as a solid structure and obstruct the flow from reentering (or flowing out of) the computational cell.

A solution to this problem is to refine the mesh near the reactor wall. This increases the likelihood of α_slag = 1.0; therefore, the model operates similarly to the slag-in-bath approach (Case I in Figure 4a). However, this mesh refinement cannot eliminate the scenario of α_slag < 1.0 (Case II in Figure 4a). Therefore, a new algorithm is proposed for these cases.

Note that when any liquid slag drops splash on the front of an already formed FL layer, it adds its corresponding mass to that layer and increases its thickness. Also, solidification is only activated when α_slag ≠ 0; that is, no solidification of the gas phase is allowed.

2.4 Simulation Settings

The simulation settings are shown in Figure 5. It includes the calculation domain, momentum and thermal BCs, energy-source term, and initial conditions. Because of symmetry, only one-third of the furnace was simulated. A mesh with 3.6 million cells was used, with local refinement near the wall (with 4 mm cells) to guarantee an adequate resolution of the FL. Variable time-stepping was used based on the maximum Courant number (set to 0.9). However, the fine mesh near the furnace walls and the relatively high inlet gas velocity resulted in a time-step ranging between 10⁻⁵ and 10⁻⁴ s, leading to an extremely high calculation cost. For example, simulating only 200 s of real-time processing would require ≈1 year for a high-performance cluster (using 56 CPU cores). This computational cost makes simulating the entire FL formation process on the freeboard infeasible. Therefore, the FL formation in the freeboard region was simplified. A thin (0.0235 m) FL layer was patched on the surface of the furnace in the freeboard region following the recommendations of the industry. This ensured a reasonable local heat transfer (although not fully accurate) within a reasonable timeframe, even if no slag splashing reached the upper portion of the furnace. A mesh independence study could not be performed for this 3D domain due to the prohibitive computational cost of running simulations with a mesh that is 4 or 16 times larger than the current configuration. Instead, such a study was conducted in an analogous 2D domain, demonstrating consistent results across different mesh sizes. A PISO velocity–pressure coupling method was used, along with a second-order spatial discretization scheme for most equations. For the volume fraction, the Compressive Interface Capturing with Adaptive Mesh Refinement (CICSAM) discretization method was used because it offered good interfacial sharpness while still ensuring model stability.

The material properties were primarily sourced from industrial data reported by Nagraj^[22] (see Table 3). The Boussinesq approximation was used to account for the natural convection of the slag. As for the gas, due to the limitations of the VOF model, a constant gas density was necessary for calculation convergence. The surface tension at the interface between the gas and liquid slag drop was neglected due to the high Weber number (We ≫ 1) during splashing events.

3.1 Dynamics of SF Process

This section discusses the key features of the flow dynamics observed during the SF process. Figure 7 shows three sequential snapshots of molten slag splashing across the freeboard, where the blue iso-surface represents the slag/gas interface at α_slag = 0.1. Splashing events originate from the interaction between the hot gas plumes (created upon injection through the SPT) and the molten slag bath. After rising into the freeboard, the collapsing slag splash diverts the trajectory of the subsequent gas plumes, resulting in the observed variations in splashing directions across sequential snapshots (Figure 7a–c). This is crucial for achieving a wider and more uniform coating on the freeboard walls. In addition, the splashing motion directly influences the flow dynamics of the gas in the freeboard, creating a constantly changing flow field. Notably, this dynamic splashing behavior persists throughout the entire SF process, even after establishing a global energy balance. In the molten slag bath (semitransparent red region shown in Figure 7), the four streamlines illustrate the complex slag flow pattern, which is driven by hot gas injection and plume motion and their interactions with splashing events. The arrows on each streamline indicate the flow direction. Note that the streamlines reach most regions of the 3D domain across the snapshots, confirming the chaotic flow behavior in the slag bath.

Figure 8a shows the iso-surface of the slag/gas interface at α_slag = 0.5, overlaid with a contour of T. The results were obtained at t = 206 s. At this moment, a significant volume of molten slag is observed ascending into the freeboard region, indicating an imminent splashing event. Furthermore, the green and yellow regions in the predominantly blue temperature contour indicate that a recent splashing event on the freeboard surface caused a localized temperature increase. This suggests that frequent, successive splashing events can hinder local FL formation. In addition, a hot gas plume created by the SPT is visible in the slag bath, as shown in Figure 8a. This plume is expected to disrupt the slag interface, leading to further splashing.

Figure 8b shows the contour of T, overlaid with temperature isotherms, in a vertical 2D plane intersecting the center of the SPT. The high-temperature zones near the inlet correspond to hot gas plumes, as confirmed in Figure 8c. The combined effect of the plume motion, splashing dynamics, and, to a lesser extent, natural slag buoyancy promotes a relatively homogeneous temperature distribution in the slag bath (uniform color in the region). The average temperature, excluding the hot plume areas, is 1435.45 K. This is only 0.3 K above T_liquidus. In contrast, the freeboard region exhibits significantly lower temperatures, although it is influenced by splashing events. The hot zones in this region indicate the recent or ongoing presence of hot molten slag, which increases the local gas temperature and influences the FL formation kinetics.

Figure 8c shows the contour of α_slag, overlaid with black velocity vectors, in a vertical 2D plane intersecting the center of the SPT. The vectors indicate the direction and magnitude of the flow (the arrows range from 0 to 10 m s⁻¹). The largest vectors arise from the gas plume motion in the slag bath and the motion of the splashing slag in the freeboard region. Additional white arrows are overlaid in Figure 8c to emphasize the most significant flow patterns. The inset in Figure 8c shows the FL layer formed on the wall. The first layer in contact with the wall has f_s = 1, indicating that T < T_solidus. The subsequent layers have 0 < f_s < 1, indicating the development of a slurry zone where T_solidus < T < T_liquidus. Moreover, the velocity vectors confirm that flow occurs mostly in the molten phase. As f_s increases, the vectors gradually decrease until they completely disappear within the fully solidified FL.

3.2 Slag Splashing and FL Formation in Freeboard

Figure 9 shows the FL evolution as the molten slag approaches and collides with the reactor wall in the freeboard region. This demonstrates the effectiveness of the slag-in-freeboard approach introduced in Section 2.3. Although the actual simulation results are in 3D, a simplified 2D representation is shown here for clarity. The irregular mesh in Figure 9 stems from the fact that it represents a cross section of the original 3D mesh, which consists of regular hexahedral elements near the wall and tetrahedral elements elsewhere.

Figure 9a1 shows a molten slag drop approaching the reactor wall with a preexisting FL layer. The inset shows a magnified view of the corresponding f_s distribution near the wall. Most of the FL layer consists of dark red computational cells (f_s = 1), which indicates that T < T_solidus. However, one green computational cell remains partially molten (i.e., T_solidus < T < T_liquidus). Figure 9a2 confirms the elevated temperature of the molten slag drop, particularly within its core, where T > T_liquidus (enclosed by the black isoline). The velocity vectors indicate the movement of the molten slag drop toward the wall, which displaces the surrounding gas.

Figure 9b1 shows the moment after the molten slag drop collides with the reactor wall. Upon impact, the molten slag spreads across the preexisting FL layer, creating a localized hot zone at the FL front (black isoline shown in Figure 9b2). Despite the elevated temperature of the molten slag, the preexisting FL layer remains unaffected because of its lower temperature. This is evident from the identical appearance of the FL layers in the insets of both Figure 9a1,b1. The dark blue gap near the top of the spreading zone shown in Figure 9b1 reveals that the molten slag is temporarily unable to cover a small number of computational cells near the preexisting FL layer. However, this gap will be filled shortly afterward.

Figure 9c1 shows a later stage when the molten slag drop flows downward and leaves behind a thin coating that completely covers the preexisting FL layer. Owing to the continuous heat exchange between the preexisting FL layer and the molten slag, the temperature in this thin coating falls below T_liquidus. This is evident from the temperature drop and retraction of the black isoline shown in Figure 9c2. Consequently, the slag solidifies, forming a fresh FL layer on top of the preexisting layer. This fresh FL is identified by the new cyan cells in the inset of Figure 9c1 and only occurs in computational cells where α_slag ≥ 0.65 (as defined in the slag-in-freeboard approach described in Section 2.3).

Figure 10 shows a broader 3D perspective of a slag splashing event and the corresponding FL formation in the freeboard region. This perspective is intended to enhance our understanding of the process dynamics. The iso-surface shows the slag/gas interface at α_slag = 0.5, overlaid with a contour of f_s. Special focus is directed toward the splashing event occurring on the upper-right side of the freeboard; the insets provide a magnified view of the impact region. The semitransparent red color on the symmetry planes represents the molten slag bath. For clarity, the existing FL layer on the reactor freeboard wall is not shown (the background is transparent).

Figure 10a shows a splash event when the molten slag collides with the upper-right side of the freeboard. As the temperature of the slag exceeds T_liquidus, it remains in a completely molten state (f_s = 0.0). Upon impact, the molten slag spreads and begins to slide down the wall due to gravity. Consequently, the inset in Figure 10b shows a larger downward-extending slag coating. Throughout this sequence, heat is transferred from the molten slag to the cooler FL layer, resulting in a gradual decrease in the slag temperature. As shown in Figure 10c, the molten slag layer starts to solidify, as confirmed by the change in the f_s color contour from dark blue (molten) to cyan (partially solidified). In addition, as the slag solidifies, its viscosity increases, causing it to become more rigid and adhere more tightly to the existing FL layer. Figure 10d shows the later stage of the process with continued solidification of the slag coating on the wall. f_s increases further, as demonstrated by the change in f_s color contour from cyan to dark green.

Parallel to the aforementioned events, a molten-slag wave coats the lower left side of the freeboard (Figure 10a). This wave retracts in subsequent moments, leaving behind a thinner coating that gradually solidifies to form a localized FL layer, as indicated by the cyan or green colors shown in Figure 10b,c. However, this new FL layer is interrupted by a new splashing event in the central region shown in Figure 10d. The impact of this molten slag travels toward the left side, potentially remelting the recently formed FL layer in that area.

3.3 GNEB

At ≈t = 190 s, GNEB is established. This implies that the system reaches an equilibrium state where the overall energy input balances the energy output, even with ongoing flow dynamics. Table 4 lists the simulated heat fluxes on specific reactor surfaces after achieving GNEB. These results were compared with industrial furnace data from Aurubis–Beerse for validation. For commercial reasons, the heat sources related to the fuming reactions are not presented here. The heat flux from the inlet is not shown because it is a predefined BC (see Figure 5).

The simulation results generally agree well with the industrial data for most reactor surfaces. However, the heat fluxes at the walls exhibit discrepancies exceeding 15%. This can be explained by the dynamic behavior of slag splashing and hot gas plumes, which can cause temporary variations in the heat transfer across the slag bath and freeboard walls. For instance, a few seconds later, the simulation results show heat fluxes of −1.53 and −2.09 MW on these surfaces during a large splashing event. However, in both cases, when the aggregate heat flux through both the slag bath and freeboard walls is considered, the simulation results strongly agree with the industrial data, with error margins falling below 1%.

Table 5 lists a comparison between the simulation results and industrial data for the average surface temperatures at specific reactor surfaces. Note that the “Walls” listed in Table 5 correspond to the inner surface of the reactor. The overall agreement is good, with all surface temperatures matching industrial data within ±5%.

3.4 FL Thickness

Figure 11 shows the state of the FL layer at t = 225 s (i.e., after GNEB was reached). Figure 11a shows the FL thickness distribution on the reactor wall, whereas Figure 11b shows the variation of the FL thickness along the reactor height for three azimuthal angles (φ): 0°, 22°, and 45°. In both figures, the FL front is defined by the solid slag volume fraction at f_s = 0.1.

In the slag bath region (height range between 0 and 2 m in Figure 11b), the FL thickness is smallest along φ = 0° due to the influence of the hot gas plumes, which raises the local temperature and hinders solidification. Notably, the FL thickness vanishes at a height of 0.5 m because it coincides with the location of the SPT. As the azimuthal angles deviate from φ = 0°, the impact of the hot gas plumes weakens, and the FL thickness increases. For instance, at both φ = 22° and φ = 45°, the slag bath exhibits a consistent FL thickness of 3 cm, except for a slight increase to 3.65 cm near the interface at φ = 45°. This localized increase in FL thickness at φ = 45° can be attributed to the dynamic slag/gas interface, which results in periods where the area is momentarily devoid of hot slag from the bath. These intervals allow the existing thin layer of molten slag to cool and solidify, leading to an intermittently thicker FL layer at this specific location (before the hot slag returns and remelts this excess FL). As shown in Figure 11a, the FL thickness throughout the slag bath demonstrates symmetry around φ = 0°.

In the freeboard region (height range between 2 and 5.5 m, as shown in Figure 11b), the FL thickness exhibits a dependence on φ. At φ = 0°, a uniform FL thickness of 3.65 cm persists up to a height of 3.5 m before gradually decreasing to 2.35 cm (the FL thickness assumed as the initial condition in the simulation) at a height of 4.5 m. As φ deviates from 0°, the height at which the 3.65 cm FL thickness is maintained progressively decreases. For example, at φ = 22°, it reaches 3.0 m, whereas at φ = 45°, it reaches only 2.5 m. Unlike in the slag bath, the FL thickness in the freeboard region lacks symmetry around φ = 0° because of the varying frequency, direction and location of splashing and the resulting local heat transfer.

The average measured FL thickness in the industrial furnace operated by Aurubis–Beerse is 3 cm. In the slag bath region, the simulation results exhibit good agreement with the measured values, particularly at a distance from the SPT inlet that minimizes the influence of hot-gas plumes. In the freeboard region, the agreement is good in areas that experience at least one splashing event near the GNEB condition (i.e., once the slag is only slightly superheated). Clearly, at t = 225 s, no splashing events reached the top corner areas, and thus they retained their pre-defined FL thickness.

4.1 Analysis of Thermal Balance during SF Operation

Achieving GNEB in the SF process means that thermal equilibrium is preserved between different heat sources and sinks. These include 1) heat input from the superheated slag bath; 2) heat input from the injected hot gas from the SPT; 3) heat removal by the cooling system, the off-gas outlet and the bottom surface; 4) energy sinks due to the fuming reaction; and 5) latent heat is caused by the melting or solidification of the FL.

Current heat-transfer models rely on Equation (14) to estimate steady-state FL thickness and temperature profiles across the layers of various systems.^{[25, 34, 35]} HTC_bath is a critical factor in these estimations. However, because it depends on complex factors, such as flow dynamics, multiphase transport, microstructure, and material properties, assigning an arbitrary value is unrealistic. Only comprehensive 3D simulations can determine the heat flux from the slag bath into the FL, from which the HTC_bath can be derived. The current model estimated the steady-state HTC_bath and q_bath for the SF process to be ≈390 W (m⁻² K⁻¹) and 50 kW m⁻², respectively.

Figure 12 shows the steady-state temperature profile across the reactor wall, the FL, the sub-liquidus layer, and the molten slag bath. The first and second dots correspond to the temperatures of the outer and inner reactor wall surfaces, respectively. The remaining dots correspond to the temperature at the corresponding computational cell center in the FL (nonpermeable zone), the sub-liquidus layer (permeable zone), and the molten slag bath. This curve was obtained at one location in the slag bath region; however, it is representative of the temperature profiles obtained at any location in the slag bath region if a sufficient distance from the SPT is considered to eliminate the influence of hot gas plumes.

The temperature at the inner reactor wall is T_w = 398 K. The temperature profile across the FL exhibits a nearly constant gradient until it approaches T_f = 1309 K. This critical temperature corresponds to the threshold at which the slag transforms from a solid-like state to a liquid-like state, as indicated by the viscosity curve shown in Figure 6. This implies that in this model, T_f is not an arbitrarily chosen temperature, but rather depends directly on the rheological behavior of the slag.^[11] When T ≤ T_f, the slag behaves as a static, non-permeable FL structure, which results in a large temperature gradient due to the small thermal conductivity of the FL. Conversely, when T > T_f, the slag transitions to a semisolid state with increased permeability. The viscosity in this permeable zone is higher than that in the molten slag but significantly lower than that in the non-permeable zone (FL). The corresponding temperature profile exhibits a gradual decrease in slope as it approaches the molten slag bath (Figure 12). The constant temperature profile across the different layers of the system overtime confirms the steady-state condition (Equation (14)) of the process.

4.2 Analysis of FL Layer and Splashing

In the slag bath, the simulation results show good agreement with industrial data in regions farther from the SPT but underestimate or even miss the FL formation closer to the inlet (Figure 9). This discrepancy arises from the strong influence of the hot-gas plumes, which is a consequence of assuming an SPT inlet at the reactor wall. In industrial furnaces, this issue has been rectified with a special nozzle design, where the SPT extends slightly inside the furnace, strategically positioning the gas plume farther away from the wall and reducing its direct impact on FL formation. However, this particular geometric feature was not considered in the current model because it would introduce excessive complexity. The small details of the extended SPT would require a very fine mesh in the region and an even smaller time-step size in the calculations than those currently used. This would significantly increase the simulation time. As a result, the findings presented in Section 3.4 must be analyzed with this limitation in mind, especially when analyzing Figure 9. It is highly probable that extending the SPT inlet location further into the slag bath would reduce the influence of the hot-gas plume on the FL thickness in this region.

In the freeboard region, the simulation results demonstrate that the splashing frequency and height vary with the azimuthal angle, leading to nonuniform FL formation across the freeboard. The lack of symmetry in the freeboard region is a consequence of the dynamic nature of splashing and its significant impact on FL formation patterns. At φ = 0°, the splashing height is largest, resulting in a thicker FL layer at higher levels. As the azimuthal angle deviates from φ = 0°, the splashing height reduces, causing the thicker FL layer to fade down. In addition, the splashing frequency influences the temperature distribution, which, in turn, affects the FL distribution. Frequent splashing hinders FL formation because it does not give enough time for the FL front to cool down enough for the subsequent splashing slag to solidify.

The simulation did not achieve full freeboard coverage through splashing within the simulated timeframe (Figure 9). In real-world operations, the FL is maintained on the reactor wall across multiple processing cycles. When the reactor undergoes complete cleaning to remove the existing FL layer, a specific procedure is employed to promote splashing and accelerate FL formation on the freeboard wall. This procedure was not included in the simulation. Therefore, the underestimated FL thickness in the top corner areas of the freeboard (where the FL thickness retained the predefined value) can be attributed to the limited simulation duration and absence of a dedicated splashing procedure. In the remaining areas of the freeboard, the simulation results for the FL thickness agree well with industrial measurements.

We recognize that the study of drop impact on solid surfaces is an extensively researched topic. The proposed slag-in-freeboard model is not intended to replace these sophisticated approaches, as it lacks their precision. Instead, the slag-in-freeboard model provides a practical numerical solution within an Eulerian framework for scenarios involving liquid splashing and solidification, capturing their impact on local heat transfer.

4.3 Other Considerations

All of the results discussed in Section 3 occur after t = 170 s (particularly if FL formation is analyzed) due to excessive slag superheat prior to this point. For the formation of an FL, molten slag must splash onto the freeboard and cool rapidly enough to solidify below T_liquidus. FL formation in the freeboard only occurs after t = 170 s when the slag's superheat is reduced sufficiently.

The large calculation domain for the SF process, the refined mesh, and high flow velocities in certain regions result in significant computational time requirements. Achieving 200 s of real-time operation required ≈6 months. This period was dedicated to establishing GNEB, which is a crucial criterion for validating the accuracy of a model. Note that including the FL patch in the freeboard region reduced the simulation time by half compared with a setup without the FL patch (discussed in Section 2.4). With the GNEB established, the model is well suited for conducting parametric studies. Any subsequent simulation using new parameters can achieve a new equilibrium much sooner. Although the model is not suitable for real-time assessment, it can be effectively employed to produce long-term strategic changes in industrial furnaces.

In Figure 9, the FL front was assumed to be at f_s = 0.1, which corresponds to a higher temperature (1425 K) than the previously mentioned T_f of 1309 K. This difference reflects a practical consideration of real operation. After the molten slag is tapped from the furnace, a thin layer likely remains on the reactor's inner wall and solidifies before measurements are obtained in the reactor. This contributes to a thicker overall FL. Therefore, a higher temperature of 1425 K was used for the FL front shown in Figure 9 instead of the original T_f value of 1309 K.

In Figure 10a1, the α_slag distribution in the FL layer transitioned from deep red near the wall to orange in subsequent cells. This shift arises from the modeling choices discussed in Section 2.4, where the first three cells adjacent to the wall were initialized with α_slag = 1 and f_s = 1. Conversely, the subsequent cells reflect the FL formation calculated using the numerical model, which ranges from 0.65 to 1.0. The numerical threshold α_slag,cr = 0.65 for FL formation in the freeboard region (referred to as the slag-in-freeboard modeling approach) is introduced in this study. Laboratory experiments on slag solidification under bath conditions have reported α_slag for FL samples between 0.6 and 1.^[11] However, no data is available in the literature for α_slag in scenarios involving slag splashing in the freeboard region.

According to Bellemans et al.^[36] reducing the average operational temperature of slag (even below T_liquidus) can enhance the energy efficiency of the SF process. The reduced slag temperature lowers the heat flux, minimizes the load on the cooling system, and reduces energy consumption. However, to achieve a positive net throughput, these energy savings must offset the potential reduction in production output caused by the lower operating temperature. This delicate balance underscores the crucial role of precise control in optimizing overall system performance. The proposed model can serve as a valuable tool for examining and understanding the implications of such modifications.

Another important topic for discussion is the chemistry incorporated in the model. Currently, the model does not explicitly solve for the chemical reactions occurring in the slag. Instead, it accounts for the effects of endothermic chemical reactions on the system's enthalpy. For simplicity, a constant sink term was assumed throughout the model, implying that the fuming rate remains constant during the process. However, this approach oversimplifies the situation. In reality, as the temperature of the slag and the ZnO content decrease, the fuming rate is expected to diminish overtime. This decrease is also expected to affect the solidification path illustrated in Figure 2. This variation is not considered in the current model, and no analysis was conducted to assess its impact on the final results.