Physical and Numerical Simulation of Supersonic Gas Jets Blown onto a Liquid Sn40Wt Pct Bi Alloy Surface

The injection of gas into liquid metal using supersonic nozzles generally takes place in all top-blowing and partially in top-submerged lancing technology, where a reactive or non-reactive gas phase or a gas-solid/liquid fuel mixture are injected into a liquid medium through a top-lance system, such as the AOD, BOF,^[4,6‐9] the TBRC, and the TSL.^{[10,12,32,33]} These smelting and refining technologies use mainly cylindrical furnaces with a single central lance equipped with a multi-hole nozzle lance tip.

Figure 1 shows an example of a TBRC, representative of top-blowing technology, which is the focus of this paper. The influencing factors for gas injection and the typical reaction zones relevant to their metallurgical process are as follows:

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The interaction of the different gas and liquid phases and the resulting surface area are critical aspects for maximizing the reaction, mass, and heat transfer efficiency. Early experiments and initial industrial applications have shown that the physical processes and chemical reactions in the individual zones described above depend essentially on the type of gas injection in the liquid metal and the resulting momentum of the gas jet.^{[10,13,34‐36]} Therefore, in Figure 1, the highlighted cavity is the focus of this work.

The lance setup (number of nozzles, nozzle geometry, gas supply pressure, lance inclination angle, distance nozzle-liquid metal surface, and type of gas), which generates the gas jet, is of particular significance. The gas jet creates oxidizing, neutral, or reducing conditions in the furnace by regulating the air–fuel ratio in the gas control unit of the injection system to control the slag chemistry (via the oxygen partial pressure) in the bath.^[3] All of the above aspects influence the splashing and mixing phenomena; however, chemical reactions are excluded from this work.

A key requirement for the model experiment is that essential features of the industrial process are replicated as realistically as possible. In the present case, the focus is primarily on the direct impact that the gas jet has when it impinges on the liquid metal surface. For this reason, a supersonic gas jet is blown from a defined position above onto the free surface of a liquid metal at rest. The main difference between liquid metal and water is that the metal has a significantly higher density and surface tension. This suggests that the dynamics of the bath surface and the detachment behavior of drops and fluid elements are different than in water. The experimental setup used in this study is an integral part of the LIMMCAST facility at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR), which is designed for the physical modeling of flow processes in continuous steel casting.^[29] LIMMCAST is operated with the binary alloy Sn40wt pct Bi, and the liquidus temperature of which is approx. 170 °C. Relevant thermophysical properties of this model fluid are very similar to those of liquid steel.^[37]

The current experiments are carried out in a cylindrical vessel made of stainless steel with an inner diameter of D₁ = 600 mm and the height of the metal bath set at H₁ = 600 mm (cf. Figures 2(a), 4, and Table I). Electric heaters were installed to control the temperature of the liquid metal during the measurements, and the side wall is provided with thermal insulation. A lid is mounted on the container containing various adapters that are used for installing the gas supply or monitoring the liquid metal surface by means of video technology.

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Four Laval nozzles of different sizes (1.00 ≤ Ma ≤ 1.85) were designed and manufactured to generate the sonic and supersonic gas jets. A Laval nozzle is a tube which is pinched in the middle, first narrows (converging section), and then widens (diverging section), with the transition from one to the other being smooth, cf. Figure 4. The specific characteristic of a Laval nozzle is that the outlet velocity is predetermined by the ratios p^*/p₀, T^*/T₀, ρ^*/ρ₀, the volumetric gas flow rate \({\dot{V}}_{0}\) , the back-pressure p_∞, and ultimately by the nozzle geometry. In this study, however, it was decided to focus on just two nozzles that produce Mach numbers of Ma₃ = 1.00 and Ma₃ = 1.85 at their outlet (position (3), cf. Figure 4). The nozzles were designed according to the isentropic theory of flow,^[39] and the characteristic data apply to the nozzles used here, are:

For more details, cf. Table I. During the experiment, the respective nozzle is fitted to a gas lance with an outer diameter of 15 mm and vertically aligned from above with the liquid metal surface (cf. Figure 2(b)). The distance H between the nozzle outlet and the liquid metal surface can be varied over a wide range, including submerging the nozzle in the molten metal; this study focuses on distances H between 10 mm and 150 mm.

To fill the vessel before starting the experiments, the liquid metal is pushed by pressurized argon from a storage tank into the vessel through a feed line in the bottom. The filling process is finished when a level of H₁ = 600 mm above the steel plate is reached. The diameter-to-height ratio of D₁/H₁ = 1 of the fluid volume then corresponds to the typical value in the industrial process.

The measurements presented in this study focus on a visual observation of the processes on the free surface due to the impact of the gas jet. The lid of the fluid container is equipped with three observation windows where two cameras and the illumination system are installed (cf. Figure 2(b)). One window is placed in the center of the lid, allowing a vertical line of sight to the central part of the bath surface. Two further windows are installed at a defined radial distance and an angle of 25  deg relative to the vertical position. Furthermore, the lid has a couple of feedthroughs through which, for example, individual sensors can be introduced into the measuring volume.

A full-frame mirrorless camera (Sony α7III) with a fixed focal length of 40 mm is used to observe the free metal surface and the space above. This camera allows frame rates of up to 100 frames/s in HD with an exposure time of 1/500 s used in most cases. All videos are recorded in full HD with a 16:9 aspect ratio and a resolution of 1920 x 1080 pixels.

Various lighting systems were tested. The Godox LC30B, Bi-Color LED lamp, which has a brightness capability of up to 6500 Lux, is used for all recordings presented in this study. However, one major challenge is that the liquid metal features a highly reflective surface, which carries the risk of severe overexposure.

A portable high-temperature water-cooled borescope (HTO-038 from CESYCO, OptoPrecision GmbH) is used to obtain a viewpoint from inside the vessel in addition to the access through the lid (cf. Figure 3). This provides a better observation of the impingement point of the supersonic jet and the vertical movement of the fluid filaments ejected from the bath by the gas jet. The borescope is mounted (aligned vertically) at one of the feedthroughs on the vessel lid (radial distance r = 240 mm). In principle, the tip of the borescope can be equipped with various types of lenses. The lens used in this series of experiments results in an optical axis that deviates 70  deg from the vertical axis. This optical lens system at the tip of the borescope opens a circular field of view with an aperture angle of 60  deg. To provide a certain degree of protection against hot liquid metal splashes and high temperatures, the lens system is sealed off from the outside by a sapphire glass window. In addition, the lens is shielded by a customized protective cover made of stainless steel. To record the metal surface during the experiments from this viewpoint inside the otherwise closed vessel, the borescope is connected to the Sony α7III camera.

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The simulation is based on the Unsteady Reynolds-averaged Navier–Stokes (URANS) equations:

In order to model the turbulence, the Transition SST model with standard settings is used. Since all equations are described in detail in,^[38] they are not presented here.

The time-dependent motion of the gas–metal interface is calculated using the volume of fluid (VoF) model.^[38] This surface tracking approach was developed for immiscible multiphase flows where the location of the interface between the phases is of interest. An additional transport equation for the volume fraction α_s of the secondary phase is solved as follows:

The volume fraction of the primary phase is α_p = 1 - α_s. The following applies to the secondary phase:

α_s = 0: cell is filled with primary phase, α_p = 1 - α_s,

α_s = 1: cell is filled with secondary phase,

0 ≤ α_s ≤ 1: cell shares both phases and the phase interface is located in the cell.

In a multiphase system with n phases, (n−1) equations are required. The coupling between the momentum conservation and the phases depends on the density ρ and the viscosity ν:

Thus, the liquid metal surface is treated as discontinuity of the material data. Since the problem involves a high momentum transfer from the supersonic argon jet into the liquid metal phase, the entire fluid domain is strained with domains of very high and low Re numbers. At the start of the simulation, in particular, this might lead to convergence issues that require manual user input.

The computational domain is shown in Figure 4 and was meshed with ANSYS ICEM-CFD. It must be noted that the domain excludes the gas supply pipe and the nozzle. The mesh consists of 12.34 million hexahedrons and is locally refined in the regions of the free argon jet as well as directly above and below the liquid metal surface, cf. Figure 5. The smallest hexahedra have an edge length of 0.025 mm, the largest, especially near the vessel wall and in the upper argon domain, an edge length of 2.83 mm. For example, the nozzle outlet surface has been resolved by 100 mesh cells along the nozzle outlet diameter D₃. The mesh quality has been checked by the orthogonality criterion. The number of mesh cells represents a good compromise between computing time and spatial resolution of the metal drops formed. The initial phase interface (position (1), cf. Figure 4) between the liquid metal and the gaseous phase is defined as the interior surface.

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For example, the supersonic nozzle II is operated at its design conditions of \({\dot{V}}_{0}=\) 4.233 × 10^-3 Nm³/s, p₀ = 6.790 bar, p_∞ = 1.01325 bar, and β = 2.5 deg, i.e., over- or under-expansion phenomena such as shock and/or expansion waves do not occur. The gas flow inside the nozzle was not modeled because the outlet conditions (p₃ = p_∞, T₃, u₃, Ma₃) can be precisely predicted by the isentropic theory of flow,^[35] cf. Table I. Thus, the nozzle provides outlet conditions of u₃ = 404.00 m/s, T₃ = 138.56 K, and Ma₃ = 1.85. The distance H between the nozzle outlet A₃ and the liquid metal surface is in all cases, and for the inclined nozzle as well, cf. Figure 1, H = 50 mm.

For the numerical simulation, all further boundary conditions are chosen to meet the experimental setup. The inlet conditions for the mass flow rate \(\dot{m}=\) 7.546 × 10^-3 kg/s, the total (stagnation) temperature T₀ = T₃ = 293.15 K, and the turbulence intensity Tu₃ = 5 pct are set at the nozzle outlet (3). At the top of the vessel (2), a pressure boundary condition is used (p_∞ = 1.01325 bar, T_∞ = 493.15 K, Tu_∞ = 5 pct). All walls (4) are set as no-slip boundary conditions with standard settings.

The three-dimensional, transient, compressible, and multiphase flow of argon and Sn40wt pct Bi is computed with the ANSYS FLUENT 2024R2.01 pressure-based solver in double precision mode. For the VoF model, the implicit time-marching method with implicit body force formulation and volume fraction cutoff value of 1 × 10^-16 are used. The bounded second-order implicit linearization is used for temporal discretization. Mass, energy, and turbulence equations are spatially discretized by second-order upwind schemes. The momentum equation uses the bounded central differencing scheme, and the volume fraction equation uses the compressive approach for spatial discretization. The pressure–velocity correction is made with the SIMPLEC procedure, and the PRESTO scheme is used for pressure discretization. Default values for all under-relaxation factors were used. The calculations were performed on a HPC cluster with 64 parallel processes. For 1.84 s of real top-blowing process, the computing time was approx. four weeks.

Although the numerical simulation has been carried out for just a few seconds of real process, important conclusions can be drawn. During this period, the cavity and other fluid-dynamic characteristics have almost completely been formed so that the free surface A_c can be derived. In contrast to the experiment, the process parameters (\({\dot{V}}_{0}\), p₀, γ, H) can be easily changed, sometimes even without creating a new mesh, but only a numerical value has to be changed. It is expected that the calculation time can be further reduced by higher parallelization using 256 or more nodes and this will enable detailed case studies. However, it is also planned to transfer the validated ANSYS-FLUENT model to the OpenFOAM environment, whereby the computing power is no longer limited by the available licenses but only by the number of nodes. The OpenFOAM model can then be used to simulate industrial plants that are larger than the test facility by a factor of 10. Here, CFD is the only way to observe mixing and splashing phenomena inside the metallurgical reactor. Since the spatial scales of droplets and bubbles are also 10 times larger in the industrial plant than in the Sn40wt pct Bi vessel, the number of mesh cells required will hardly increase compared to the experiment and the calculation times will remain comparable. The consideration of the so far neglected slag phase does not pose a problem for the CFD code, as only one further equation has to be solved in the VoF model.

This study focuses on a visual observation of the impact of the gas jet on the free bath surface of a liquid metal alloy. The images presented in the following sections impressively demonstrate the effect of the gas jet's momentum on the stability of the liquid metal surface. It is expected that this impact increases with the intensity of the gas jet, which in the experiment was varied by using different nozzles and by adjusting the distance between the nozzle outlet and the liquid metal surface.

The dynamics of various surface effects and the onset and development of certain instabilities can be observed very well in a short time span immediately after the gas jet is switched on. This enables the influences of different parameters to be characterized effectively. In cases where the exposure was particularly intense, the volume above the liquid metal bath is filled with a multitude of drops and fluid fragments, so that the individual configurations are very difficult to distinguish from one another on the basis of single images. Figures 6 and 7 show a series of images for nozzle I (Ma₃ = 1.00) and nozzle II (Ma₃ = 1.85). The liquid metal surface that the gas jet encounters is initially flat and at rest. The deformation of the surface starts at the point of impact of the jet. There, the fluid is displaced downward and radially to the side, creating surface waves that propagate from the center toward the side wall. A blowing cavity is formed directly below the nozzle, the depth of which is scaled with the intensity of the gas jet. As the intensity of the jet exceeds a certain threshold, the first drops detach from the surface. However, further evolution depends largely on the type of nozzle and the distance from the liquid metal.

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Figure 6 illustrates the start-up process for nozzle I at a distance of H = 50 mm (two top rows) and H = 10 mm (two bottom rows); the qualitative evaluation is started for the case H = 50 mm. The first images of each series [Figures 6(a), (g)] show the moment 0.15 s after the gas initially contacts the metal surface. Even after such a short time, a depression zone forms with a diameter approximately equal to that of the lance, consisting of a smaller cavity surrounded by an almost circular rim. As time proceeds, the area occupied by deformations initially expands [Figures 6(b), (c)]. The diameter of the directly disturbed zone reaches values of about 50 mm to 60 mm and then appears to remain almost constant and does not increase further [Figures 6(d) through (f)]. However, the degree of deflections has increased and the vertical indentation is greater. Smaller volumes of fluid rise and fall again. From the static images, it is difficult to discern the separation of small drops from the zone around the gas impact zone and their outward radial projection. To gain a better insight, refer to the supplementary material (video 1). A few larger drops are easier to follow because they quickly fall back into the bath at short distance from the point of impact of the gas jet, leaving circular patterns there [cf. Figure 6(e)].

The situation changes significantly when nozzle I is brought to within H = 10 mm of the bath surface, intensifying the effect of the jet. Already with the first gas contact, small amounts of fluid are released from the bath at the point of impact [Figure 6(g)]. A compact fluid film forms around the point of impact and moves upward. Individual fluid ligaments break away from this film and then break up into small drops, which move outward like a series of pearls on a string [Figure 6(h)]. A short time later, the upward movement comes to a standstill, the film ruptures and begins to fall back toward the fluid level [Figure 6(i)]. The moment of impact on the bath surface is seen in Figure 6(j). Since the falling film also moves outward, the deformation zone of the bath surface increases significantly [Figure 6(k)]. A corresponding video of this experiment can be found in the supplementary material (video 2).

For comparison, the same constellation with nozzle II is shown in Figure 7. At the beginning, the formation of the cavity where the distance is H = 50 mm [Figure 7(a)] is even more pronounced than for the nozzle I. The further chronological sequence reveals significant differences compared to nozzle I. Figures 7(b) and (c) clearly show that a multitude of individual drops are released from the bath and catapulted away. Moreover, the formation and detachment of splashes and liquid metal fragments can be seen, which are considerably larger than the individual drops but do not form a continuous film either. In Figures 7(a) through (c), bubbles can also be seen that remain below the liquid metal surface. However, these were not created at that time but are a remnant from the previous experiment. These bubbles have not yet burst because they are stabilized by the oxide film on the surface over a longer period of time. From Figure 7(c) onwards, it can be clearly seen how larger filaments of liquid detach from the bath. When these large number of fragments fall back, a significantly larger area of the bath surface is disturbed [Figure 7(d)]. During the further course, the formation of further fluid fragments occurs quasi-continuously although seemingly randomly. This is accompanied by the appearance of a large number of splashes and drops, which are probably released directly from the surface or result from the further disintegration of the larger fluid filaments [Figures 7(e), (f)], cf. supplementary material (video 3).

As distance H decreases, the effect of the gas jet is further intensified. This can already be seen in Figure 7(g), where, in addition to the appearance of the cavity, the emerging formation of the liquid films and the emission of separated drops are observed. The gas jet introduces considerably more momentum into the surface deformation. The volume of fluid films that are ejected increases significantly. The films quickly disintegrate into individual fragments that reach a comparatively high ejection height before falling back [Figures 7(h) through (k)]. The impact of these large fragments on the liquid metal causes enormous disturbances, deflections of the bath surface, and surface waves [Figure 7(l)]. The attached video provided in the supplementary material (video 4) impressively shows that within a very short time the space above the liquid metal bath is almost completely filled with drops, splashes, and projecting liquid metal films, so that it is almost impossible to capture the individual fluid particles in their entirety, as well as to follow their trajectory and development of the shape. The fact that all these image series were recorded in an extremely short time interval of only 0.15 s underscores the high dynamics of the processes.

In addition to visual access from above through the container lid, the experimental setup offers the option of observing the process from the side using a borescope. Figure 8 illustrates the disintegration of the metal surface when exposed to nozzle I. The four individual images compare different distances between the nozzle and the bath level. The images were taken at least 30 s after the gas jet was switched on, i.e., when the blowing process was already in a quasi-stationary state and no longer within the start-up phase as in Figures 6 and 7. At a distance of H = 150 mm, it can be seen that only a few small drops form. The cavity and concentric wave fronts can be distinguished; however, the disturbed zone remains limited to a specific area [Figure 8(a)]. If nozzle I is brought 50 mm closer to the bath surface (H = 100 mm), the liquid metal surface appears much more agitated, and significantly, more drops are ejected from the surface. On the surface, there are indications of smaller fluid films, but these do not detach from the bath. Here, too, only a limited area of the free surface is affected by the perturbations [Figure 8(b)]. As the distance decreases further [Figure 8(c) H = 50 mm, Figure 8(d) H = 10 mm], the number of drops and splashes, the size of the liquid metal films, and the height that the diverse fluid fragments can reach above the bath surface continue to increase. The videos for all the experiments shown in Figure 8 can be found in the supplementary material (videos 5–8).

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The corresponding endoscopic images for nozzle II are presented in Figure 9. It is immediately obvious that this case differs significantly from the situation with nozzle I. Even at a large distance of H = 150 mm, the space above the bath level consists of a large number of drops and splashes [Figure 9(a)]. Furthermore, the liquid metal surface is severely disturbed over a larger area; in particular, drop impacts can be detected at greater distances from the point of impact of the gas jet. Within the disturbance zone, intensive billowing can be observed with the formation of smaller fluid films, which, however, do not rise too far above the bath level. With H = 100 mm, the energy of the gas jet is sufficient to eject larger liquid metal films that move away from the point of impact in a funnel shape [Figure 9(b)]. The acceleration is obviously so strong that these films do not remain stable and coherent for a long time, but rather quickly disintegrate into small splashes and drops that are propelled quite a distance. This also clearly shows the area affected by the disturbance, which now takes up the majority of the entire bath surface. This tendency continues if the distance is reduced to H = 50 mm [Figure 9(c)]. The liquid metal films increase in size, their shape is complex and heavily frayed, and splashes and drops at the edges are generated constantly. When nozzle II is situated directly on the bath surface at a distance of H = 10 mm, large liquid metal films are catapulted out of the metal bath [Figure 9(d)]. The energy is sufficient for these bulky and heavy fragments to reach a considerable height, so that the region where the metal films split into splashes and drops is outside the field of view of the borescope. As for nozzle I in Figure 8, the corresponding videos can be found in the supplementary material (videos 9 - 12). The videos also reveal the occurrence of large-scale metal bath movements in the form of dynamic surface waves or sloshing of the liquid metal. In the case of small nozzle distances of 10 mm < H < 50 mm, in particular, it can be seen that the re-entry of the large liquid metal fragments into the bath not only leads to the production of further splashes but also triggers and amplifies sloshing motions due to their considerable mass.

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The images presented so far confirm the impression that the bath level and ejected drops and larger fluid fragments above are highly dynamic and extremely volatile, especially when hitting the high-intensity gas jet. Figure 10 contains a time series of six images acquired in succession with a time step of ∆t = 0.02 s for nozzle II located at a distance of H = 50 mm, cf. supplementary material (video 3). When focusing on the central area around the point of impact of the gas jet, similar structures can be seen across the series of images, which nevertheless show a considerable rate of change in this short period of time. The structures appearing on a larger radius, on the other hand, seem to be somewhat more stable, since they are not directly within the zone dominated by the gas jet. In Figure 10, an example of a structure is marked by a red circle, which apparently represents a smaller fluid film that protrudes slightly from the bath; this flow pattern is shown in all the images. In the last image [Figure 10(f)] this fluid film mergers with another similar structure. It is interesting to note that this fluid ligament apparently moves along an azimuthal path, which suggests a rotating bath motion. This phenomenon could be deceptive, however, as it is not evident from the images how closely this structure is connected to the surface of the bath. A closer inspection of more frames of this sequence over a longer timespan reveals that this structure originally forms in a neighboring zone closer to the lance, at a moment when the gas jet is ejecting a larger fluid ligament from the bath. From the two-dimensional images, it is not possible to clarify whether this is a detached part of the fluid ligament or a surface wave created by falling splashes. Only the latter would allow a conclusion to be drawn about the bath motion. Since we did not perform parallel flow measurements in the present study, this cannot be conclusively verified here.

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A supersonic gas jet affected by splashing drops behaves differently from a classic supersonic jet that enters an undisturbed and gaseous environment under regular conditions. The classic jet shows a core region with a length of about 5 times the nozzle outlet diameter D₃, where pressure p, velocity u, temperature T, and density ρ are nearly constant and then decrease further downstream; the overall length of the supersonic region is approx. 10 – 20 × D₃. Such a jet has a small cross-sectional area and remains undisturbed over a relatively long distance. In the present case, the situation is somewhat different, as the splashing metal drops significantly disturb the expansion of the jet and vice versa. Drops of different size, form, velocity, and momentum are ejected out of the cavity and upward into the gaseous region and clearly change the spread of the jet. As a result, the jet oscillates significantly in all directions, i.e., back and forth. Both the jet’s core region and the supersonic region are significantly shorter, i.e., the jet quickly loses velocity and momentum.

As an example, Figure 11 shows instantaneous results for the sonic nozzle I (Ma₃ = 1.00, γ = 90 deg) and supersonic nozzle II (Ma₃ = 1.85, γ = 90 and 45 deg) at different angles and how this arrangement generally affects the splashing of metal drops. The gas–metal interface, i.e., the surface of the liquid metal bath and all drops, is again represented by the volume fraction α = 0.5.

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With nozzles I, II acting in the vertical direction [Figures 11(a), (b)], a wide range of drops is pulled out from the annular edge of the blowing cavity and splashes upward toward the lance tip, which can increase the risk of a nozzle blockage or damage. Thus, the high transient jet is considerably disturbed and deflected, hits another region of the turbulent cavity domain, and pulls out more drops there, which again move upward toward the lance tip. At every time unit, a large number of drops is always present in the jet region so that the propagation of the jet is disturbed.

If the supersonic jet hits the metal surface at an angle of γ = 45 deg and with the same axial distance H = 50 mm between nozzle outlet and metal bath surface [cf. Figure 11(c)], the cavity becomes elliptical, the diameter increases, the depth decreases, and the drops splash against the sidewall of the vessel. However, in this case, the jet spreads more freely and shows an increased supersonic jet length. The depth H_c of the blowing cavity [cf. Figure 11(b)] can be specified if the vertical distance between the deepest location of the cavity and the mean metal level is averaged over the simulated process time; the same applies for the diameter D_c of the cavity, and the results are shown in Table III.

Figure 12 shows exemplary results for nozzle II, which is vertically aligned at a distance of H = 50 mm above the metal surface at rest. The jet is characterized by a red colored iso-volume containing a velocity range of 100 m/s < u < 400 m/s. It should be noted again that nozzle II was designed for p₀ = 6.790 bar, \({\dot{V}}_{0}\) = 4.233 × 10^-3 Nm³/s, p_∞ = 1.01325 bar, and β = 2.5 deg; these input data result in theoretical nozzle outlet values of u₃ = 404.00 m/s, Ma₃ = 1.85, and T₃ = 138.56 K. However, the CFD results are somewhat lower. For the blowing period of 0 s < t < 1.84 s, time-averaged values of u₃ = 389.6 m/s, Ma₃ = 1.620, and T₃ = 149.7 K result at the outlet. Directly below the nozzle, the jet expands slightly, so that local gas velocities higher than 400 m/s occur in the jet core region.

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If two monitoring points are considered on the vertical y-axis, the first point at the level of the free metal surface (y = 0), and the second point 5 mm below the nozzle outlet (y = 45 mm), the time-averaged velocity, and the fluctuation velocity are \(\overline{u }\)(y = 0) = 102.8 m/s and \({u}_{rms}^{\prime}\) (y = 0) = 55.9 m/s or \(\overline{u }\) (y = 45 mm) = 391.5 m/s and \({u}_{rms}^{\prime}\) (y = 45 mm) = 7.2 m/s. Figuratively speaking, the jet contains flow regimes with extreme fluctuations that hit the metal surface and create a very unstable, oscillating cavity.

For the numerical simulation, nozzle II can be subjected to the maximum flow rate of \(\dot{m}\) = 7.546 × 10^-3 kg/s right at the start (t = 0) of the calculation, if an initial step of ∆t = 1 × 10^-6 s is chosen. During the initial blowing period (0 s < t < 0.5 s) of gas onto the liquid metal, the deepest cavity is formed with a mean penetration depth of H_c ≈ 100 mm. In the further course of blowing (0.5 s < t < 1.84 s), a mean penetration depth of H_c ≈ 80 mm and a mean diameter of D_c ≈ 70 mm are achieved (cf. Table III), i.e., the quasi-steady-state cavity is approximately as deep as it is wide. Drops and ligaments are torn from the edge of the cavity via shear forces and splash toward the lance tip. The mean velocity of the liquid metal surface at the edge of the cavity is about 4 m/s. Due to the highly stochastic behavior of the surface, new drops and cavities are constantly forming, i.e., the entire region below the nozzle is extremely chaotic. However, this is advantageous for the TBRC process because it allows the metal bath to be mixed very well and to react chemically in all zones.

Starting from the cavity, ring-shaped surface waves run radially outward from the center of the vessel to the wall, are reflected, and hit the front of the next incoming wave. As a result, the metal surface moves evenly upward and downward with a frequency of f ≈ 0.4 Hz; this frequency can also be seen in the experimental data in a first approximation.

An important metallurgical variable is the overall size of the gas–metal interface (the sum of the free surface and the surface of all drops formed) since most chemical reactions take place here; in the following, this surface is referred to as A_c. If the volume fraction α = 0.5 is used as a basis, the surface A_c of the initially smooth metal level increases from A_c = 0.126 m² to a time-averaged (0.5 s < t < 1.84 s) surface of A_c = 0.201 m² which corresponds to a 60 pct increase, cf. Table III. However, the real surface is somewhat larger due to the unresolved drops and limited resolution of the fractal surface. To gain an impression of the transient phenomena simulated for nozzle II (p₀ = 6.790 bar, \({\dot{V}}_{0}\) = 0.254 Nm³/min, Ma₃ = 1.85, H = 50 mm, γ = 90 deg), cf. the supplementary material (videos 13,14).

As an example, Fig. 13 shows the vertical nozzle I (Ma₃ = 1.00, γ = 90 deg) and inclined nozzle II (Ma₃ = 1.85, γ = 45 deg) both located at a distance of 50 mm above the liquid metal surface. In the case of sonic blowing [Figures 13(a), (b)], there is much less splashing, and the effective free metal surface A_c is far smaller compared to the supersonic situation, cf. Table III. The qualitative agreement with the HZDR experiment [cf. Figure 8(c)] is good, although the metal drops in the experiment tend to splash somewhat higher into the surrounding area.

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In the case of angled blowing with supersonic nozzle II [Figures 13(c), (d)], the metal drops are ejected toward the tank wall and increase the risk of metal-slag building-up there. In general, far fewer drops enter the actual gas jet, which therefore oscillates significantly less while showing high velocities at the same time. The blowing cavity becomes oval and is less deep than with vertical blowing, cf. Table III. From a qualitative point of view, the number of drops formed is lower than with the vertically aligned nozzle II, and the drops are larger and form individual ligaments. The effective surface induced by the inclined nozzle II is A_c = 0.191 m² and is therefore comparable to that of vertical blowing; compared to the static metal bath, this corresponds to an increase of 51 pct., cf. Table III and supplementary material (videos 15,16).

Figure 14 shows a comparison between the numerical (a) through (e) and the experimental (f) through (j) results during vertical blowing with supersonic nozzle II. The boundary conditions of the experiment and the numerical simulation are largely identical, so that a direct, albeit qualitative comparison is possible. However, it must be borne in mind that the computational domain (D₁ = 400, H₁ = 200 mm) for the numerical case for limiting the number of mesh cells is smaller than the one in the experiment (D₁ = 600, H₁ = 600 mm). It should also be noted that the image section of the numerical case [cf. Figures 14(a) through (e)] is slightly larger than in the experiment [Figures 14(f) through (j)]. Nevertheless, in terms of surface movement and drop formation, both methods show good qualitative agreement, which is particularly clear when evaluating the individual video sequences in the supplementary material, cf. videos 13, 14.

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The results presented in Table III and the graphical representations in Figures 11, 12, 13, and 14, in relation to the physical observations from the experiments, suggest the following:

CFD simulations have proven to be a reliable tool for predicting the surface area at the gas–metal interface. This is particularly important for enhancing the diffusion of volatile compounds from liquid phases—especially slag—into the gas phase. As a result, the flue dust formed can be collected for further processing and material recovery.

Additionally, liquid splashing increases the momentum and energy transfer, thereby improving energy control. It also intensifies the interaction between the liquid metal and the refractory lining, which can lead to increased refractory wear.

The improved understanding of mixing behavior provides valuable insights for furnace design, particular with regard to interfacial reactions and related metallurgical phenomena.