Performance of a Nozzle to Control Bath Level Oscillations and Turbulence of the Metal-Flux Interface in Slab Molds

: The characterization of the turbulent ﬂow of liquid steel in a slab mold using a commercial nozzle was carried out through physical experiments and mathematical models. Six ultrasonic sensors were located at each side of the nozzle to obtain real-time plotting of the bath levels during the experimental time. An ultrasonic transducer located in the mold, 20 mm below the meniscus, determines the velocities and the turbulent variables along with the distance from the narrow face to the position of the nozzle’s outer wall. These data, together with the mathematical simulations, demonstrated a high correlation of bath level oscillations and the time-dependent behavior of the discharging jets. The ﬂow inside the mold shows low-frequency non-symmetric patterns without a severe turbulent in the meniscus. The source of this instability is the partial opening of the slide valve gate used to control the mass ﬂow of liquid.


Introduction
The fluid flow of liquid steel in continuous casting molds is critical to cast highquality slabs. The fluid flow pattern should include double roll flows, one upper and the other lower at both sides of the mold, to guarantee flow symmetry and uniform heat transfer conditions. Controlled thermal gradients in the narrow and broad faces of the mold permit the formation of a solidified shell with a strength large enough to support the liquid core [1,2]. Feeding in the mold-strand gap a lubricating flux decreases the friction forces between this shell and the mold walls. The casting process's three other essential variables are the bath level in the mold, the liquid metal speed, and flux melting rate at the metal-flux interface. The uncontrolled liquid flow near the metal-flux interface will usher powder entrainment to become trapped particles in the solidified shell. Non-melted powder induces sticking breakouts and bath level oscillations to promote transversal slab depressions and strand breakouts [3][4][5][6]. The literature in the field dealt with the effects of nozzle immersion depth, casting speed, steel grade, and thermo-solute convection on the fluid patterns of liquid steel in slab molds [7][8][9][10][11][12]. However, fluid flow in the nozzle is probably the most important tool to provide fluid flow patterns that fulfill the requirements mentioned here.
The importance of nozzle design on the fluid flow control of liquid steel has been recognized as a critical factor in slab quality [13][14][15], especially when slivers are frequent in the cast slabs. Specifically, a criterion establishes that the maximum melt velocity of 0.3 m/s at the melt surface, for any given casting speed, avoids flux entrapment and the derived generation of slivers [16,17]. In other words, the metal-slag interface is under the effects of shear stresses originated by the upper roll flow causing waves, leading to instabilities of Metals 2022, 12, 140 2 of 12 the top phase into the lower steel phase. Once reaching a critical shear stress and a corresponding interfacial fluid velocity these instabilities derive into traveling droplets flowing inside the liquid metal. These slags droplets will, eventually, form nonmetallic inclusions in the semifinished product, yielding a dirty final product. Hence, their elimination is a requirement for casting clean steel. However, this criterion for maximum melt velocity might be small, deriving into the transport of cold steel to the melt meniscus with a sensible heat that is not large enough to melt down the lubricator flux. Moreover, cold steel in the meniscus level will eventually solidify, creating strand breakouts conditions [18][19][20]. Other researchers reported that higher velocities of the melt in the meniscus than 0.3 m/s ensure suitable flux melting rates without entrainment of flux into the melt bulk [21,22]. Industrial experience indicates that a hot meniscus ensures a better slab surface, with smaller number of defects, such as slivers, cracks, slag entrapment, and transversal depressions [23,24]. Therefore, a suitable nozzle design working for some given specific conditions must deliver discharging jets yielding symmetric, double roll flows, hot steel at the meniscus level, and controlled turbulence leading to minimum velocity spikes along the casting time [22].
A commercial nozzle operating in a steel company located in the USA (see Figure 1a) yields slabs with high-quality surfaces and evidence of well-controlled friction forces between the mold and the shell, observing an overall good operation. Given these characteristics, an interest emerged to understand the reasons supporting the excellent performance of this nozzle and finding the design principles leading to successful casting processes. Therefore, the present research aims to characterize the turbulent flow of liquid steel in a slab mold using the nozzle in Figure 1a through physical experiments and mathematical models.
at the melt surface, for any given casting speed, avoids flux entrapment and the derived generation of slivers [16,17]. In other words, the metal-slag interface is under the effects of shear stresses originated by the upper roll flow causing waves, leading to instabilities of the top phase into the lower steel phase. Once reaching a critical shear stress and a corresponding interfacial fluid velocity these instabilities derive into traveling droplets flowing inside the liquid metal. These slags droplets will, eventually, form nonmetallic inclusions in the semifinished product, yielding a dirty final product. Hence, their elimination is a requirement for casting clean steel. However, this criterion for maximum melt velocity might be small, deriving into the transport of cold steel to the melt meniscus with a sensible heat that is not large enough to melt down the lubricator flux. Moreover, cold steel in the meniscus level will eventually solidify, creating strand breakouts conditions [18][19][20]. Other researchers reported that higher velocities of the melt in the meniscus than 0.3 m/s ensure suitable flux melting rates without entrainment of flux into the melt bulk [21,22]. Industrial experience indicates that a hot meniscus ensures a better slab surface, with smaller number of defects, such as slivers, cracks, slag entrapment, and transversal depressions [23,24]. Therefore, a suitable nozzle design working for some given specific conditions must deliver discharging jets yielding symmetric, double roll flows, hot steel at the meniscus level, and controlled turbulence leading to minimum velocity spikes along the casting time [22].
A commercial nozzle operating in a steel company located in the USA (see Figure 1a) yields slabs with high-quality surfaces and evidence of well-controlled friction forces between the mold and the shell, observing an overall good operation. Given these characteristics, an interest emerged to understand the reasons supporting the excellent performance of this nozzle and finding the design principles leading to successful casting processes. Therefore, the present research aims to characterize the turbulent flow of liquid steel in a slab mold using the nozzle in Figure 1a through physical experiments and mathematical models.

Experimental Work
The physical model employs a 1:1 water model scale of a slab caster made of transparent plastic sheets. The dimensions of the mold are 1450 × 235 mm 2 , the casting speed is 1.4 m/min, the equivalent flow rate is 7.95 × 10 −3 m 3 /s, and the nozzle immersion (measured from the upper edge of the port to the meniscus level) is 120 mm. The model fulfills the criteria of Reynolds and Froude numbers for scaling up liquid steel flow by using water; the surrounding pressure is 10,1325 Pa. The plastic mold has a height of 1700 mm, of which 200 mm remain immersed in a sealed tank filled with water. In the interior of the tank, an immersed pump elevates the liquid to an upper tundish to maintain a constant

Experimental Work
The physical model employs a 1:1 water model scale of a slab caster made of transparent plastic sheets. The dimensions of the mold are 1450 × 235 mm 2 , the casting speed is 1.4 m/min, the equivalent flow rate is 7.95 × 10 −3 m 3 /s, and the nozzle immersion (measured from the upper edge of the port to the meniscus level) is 120 mm. The model fulfills the criteria of Reynolds and Froude numbers for scaling up liquid steel flow by using water; the surrounding pressure is 10,1325 Pa. The plastic mold has a height of 1700 mm, of which 200 mm remain immersed in a sealed tank filled with water. In the interior of the tank, an immersed pump elevates the liquid to an upper tundish to maintain a constant bath level of 1 m. A valve and a flow meter inserted in the vertical pipe, transporting water from the tank to the tundish, control the water flow. Below the top of the upper tube nozzle (UTN), a replica of the actual sliding gate valve (SGV) of this caster is made of plastic to control water flow to the mold. Six ultrasonic sensors, three located at each side of the nozzle, placed 120 mm above the static level of the water bath, capture the analogic signals of the variations of the meniscus level during the experimental time. A data acquisition card converts these analogic signals into digital ones permitting an on-real time plotting of the bath levels during the measuring time. To capture the general feature of the flow, a red dye tracer, consisting of an aqueous solution of a food colorant, is injected by a syringe through a drill performed in the UTN, sealed with a rubber plug.
Another drill practiced in the midface of one of the narrow faces, located 20 mm below the water meniscus, holds an ultrasonic transducer of 10 million Hertz that receives the signals of the liquid's motion. The signals of the ultrasonic transducer are transformed into digital signals and processed through software to determine the velocities and the turbulent variables along with the distance from the narrow face to the position of the nozzle's outer wall. Figure 1b shows a simplified scheme of the experimental setup. Figure 1c shows a solid version of the interior walls of the nozzle drawn through a computational mesh.
The experimental procedure consists of fixing the water flow rate manipulating the valve and the flow meter readings installed in the vertical pipe, and maintaining the fixed bath level in the tundish. The slide gate opening corresponds to the desired casting speed reported equivalent to a flow area of 2.57 × 10 −3 m 2 . Once starting the closed-loop of the water flow, the system works for 15 min to attain steady-state conditions of the meniscus levels, corresponding to the turbulent flow conditions. At an arbitrary time, meniscus levels and turbulence measurements start storing all acquired data in a computer. Once all measurements conclude, the tracer is injected, and a conventional video camera records the images of its mixing kinetics. Although it is evident that a water model does not reproduce faithfully the current flows in slab molds, the overall flow patterns in casting systems will follow those reported here.

The Mathematical Model
The mathematical simulations consist of an Unsteady Reynolds-Averaged Navier Stokes (URANS) computational scheme based on the realizable k-ε model [25] to complement the experimental work described above. The simulations ensure that the fluid of water is in line with the experimental measurements described above. The realizable k-ε turbulence model is simple, contains an alternative formulation for the turbulent viscosity, and predicts the boundary layer characteristics under large pressure gradients. Additionally, the realizable model provides the best performance of all the k-ε model versions for several conditions of separated flows and flows with complex secondary flow features. The equations governing the motion of an incompressible fluid include one equation for continuity and three equations for momentum transfer expressed by averaged on time according to the Reynolds decomposition (u i = u i + u i ): The Reynolds stresses, τ R = −u i u j is approximated through the hypothesis of the eddy viscosity, µ t , according to the Boussinesq expression: where S ij is the mean deformation rate tensor. The relation between the eddy or turbulent viscosity and the kinetic energy (k = 1 2 u i u j ) and its dissipation rate ε is: where C µ is not a constant; it is a function of the mean strain and rotation rates, the angular velocity of the system rotation, and the turbulence field simulates the fields of the kinetic energy and its dissipation rate to calculate the eddy viscosity, by solving the k and ε balances: The no-slipping boundary condition applies to all walls in the system, and the link between the wall and the outer flow, from the boundary layer, is through the log-wall function [26]. The boundary condition of the velocity inlet is calculated by an energy balance using Bernoulli's Equation, applied from the bath surface in the tundish to the UTN. The shear boundary condition applies on the bath surface, to take into account the presence of the upper layer phase. The present model is focused on the simulation of the water model experiments. In the outlet, a pressure boundary condition governs the exit of the fluid outside the mold.
In the discretization of the governing Equations, a Second-Order Upwind method was employed [27]. The Green-Gauss-based model approximates the derivatives in the cell centroid, the body force weighted approach works to compute the pressure field. The computing unstructured-tetrahedral mesh consists of 2,591,343 cells and 486,328 nodes. The averaged skewness of the computing mesh is 0.20588 with a standard deviation of 0.01652. The averaged orthogonal quality of the mesh is 0.88292 with a standard deviation of 8.713 × 10 −2 . The PISO algorithm helps to make the pressure-velocity coupling. The sum of all residuals of the flow variables smaller than 10 −4 establishes the convergence criterion of the numerical computations.
The solution of the motion equations provides the velocity fields using a previously calculated, or assumed, pressure field. These velocity fields employed in Equations (5) and (6) permit the calculation of the k and ε fields which, once substituted, in Equation (4), yields the field of the eddy viscosity. The procedure requires an iterative process, provided by the PISO algorithm, using as the convergence criterion the fulfillment of the mass balance provided by the Equation of continuity, expressed through the pressure field.

Results and Discussion
Figure 2a-c shows the tracer mixing characteristics at 0.5, 1.5, and 4.5 s, respectively, after its injection in the UTN. The right jet leaves the nozzle with apparently larger velocity than the left one as the tracer has advanced a longer distance, as seen in Figure 2a. Figure 2b confirms the observation mentioned above since, as seen, the right jet has impacted the narrow face, and the upper roll flow is about to be revealed. Figure 2c shows the upper and lower flows well developed at both sides of the nozzle. Figure 2d shows a selected image, derived from the files of the mathematical simulations, resembling, as close as possible, the trajectory of the tracer. The results in Figure 2a-d shows good qualitative matching between the flow in the mold and the mathematical model results.
The maximum velocities at each time identified in the velocity profiles of the right port are plotted in Figure 4c and compared with the critical velocities calculated with Equation (7), assuming a standard flux layer thickness of 10 mm. The three horizontal lines labeled as A 1 , A 2 , and A 3 correspond to the critical velocities using the fluxes presented in the precedent lines.
From the velocities results in Figure 4c, it is evident that powder A 3 is the most suitable since most of the velocity peaks of the liquid are below this critical velocity corresponding to this powder. The instantaneous variations of the upper of upper roll flows, going from the narrow walls toward the nozzle, must provide a quantitative idea about the flow dynamics. Indeed, Figure 4a,b shows the time-varying velocity profiles, calculated through the mathematical model, along a line traced from the narrow wall to the nozzles' wall (20 mm below the meniscus surface, i.e., the same position of the ultrasonic transducer). The numbers in the plots indicate the times corresponding to the smallest and largest velocity profiles. The numbers in the color-scale of Figure 4a The maximum velocities at each time identified in the velocity profiles of the right port are plotted in Figure 4c and compared with the critical velocities calculated with Equation (7), assuming a standard flux layer thickness of 10 mm. The three horizontal lines labeled as A1, A2, and A3 correspond to the critical velocities using the fluxes presented in the precedent lines.
From the velocities results in Figure 4c, it is evident that powder A3 is the most suitable since most of the velocity peaks of the liquid are below this critical velocity corresponding to this powder. To complement the information about the unsteady nature of the fluid flow in the mold provided in Figures 4a,b and 5a-f shows the numerical velocity fields in the mold's central plane of symmetry at different times. Twenty seconds after starting the time count, the flow looks very symmetric; sixty seconds later, both jets flow down the mold, and the To complement the information about the unsteady nature of the fluid flow in the mold provided in Figure 4a,b and Figure 5a-f shows the numerical velocity fields in the mold's central plane of symmetry at different times. Twenty seconds after starting the time count, the flow looks very symmetric; sixty seconds later, both jets flow down the mold, and the loss of symmetry begins. After another sixty seconds, in Figure 5c, the flow is asymmetric, and the upper roll flow on the right side is more developed than the upper roll flow on the left. In Figure 5d, the flow comes back to a symmetric flow pattern during the next sixty seconds. Figure 5e shows a loss of flow symmetry sixty seconds after the generation of the last symmetric flow, but it is rapidly recovered after a few seconds (following periods of 14 s). The upper roll flow on the left side advances unevenly than the upper jet on the right side. Forty seconds later, in Figure 5f, the flow symmetry is recovered. Therefore, for 280 s, almost five minutes, the flow loses its symmetry twice, though the overall double roll flow pattern is always present. For the flow control standards generally observed in slab molds, the flow provided by this nozzle is considered stable. The loss of symmetry is due to requirements of maintaining the turbulent kinetic energy balance: the dissipation rate of kinetic energy takes place just on the wall and the production of the kinetic energy is located just some mm from the wall. Other terms, such as pressure and viscous transportation of kinetic energy, complement the total balance. When the system starts kinetic energy part of this energy is released to keep the energy balance resulting in the loss of symmetry. It is noteworthy to underline that those higher velocities in the meniscus ensure a rapid melting of the mold flux and that using the flux A 3 will avoid excessive powder entrainment. loss of symmetry begins. After another sixty seconds, in Figure 5c, the flow is asymmetric, and the upper roll flow on the right side is more developed than the upper roll flow on the left. In Figure 5d, the flow comes back to a symmetric flow pattern during the next sixty seconds. Figure 5e shows a loss of flow symmetry sixty seconds after the generation of the last symmetric flow, but it is rapidly recovered after a few seconds (following periods of 14 s). The upper roll flow on the left side advances unevenly than the upper jet on the right side. Forty seconds later, in Figure 5f, the flow symmetry is recovered. Therefore, for 280 s, almost five minutes, the flow loses its symmetry twice, though the overall double roll flow pattern is always present. For the flow control standards generally observed in slab molds, the flow provided by this nozzle is considered stable. The loss of symmetry is due to requirements of maintaining the turbulent kinetic energy balance: the dissipation rate of kinetic energy takes place just on the wall and the production of the kinetic energy is located just some mm from the wall. Other terms, such as pressure and viscous transportation of kinetic energy, complement the total balance. When the system starts kinetic energy part of this energy is released to keep the energy balance resulting in the loss of symmetry. It is noteworthy to underline that those higher velocities in the meniscus ensure a rapid melting of the mold flux and that using the flux A3 will avoid excessive powder entrainment. The design of this nozzle develops a thin sublaminar boundary layer, δl, of variable thickness in its the interior wall [29]: where * = is the velocity friction and is the wall's strain-stress. This nozzle develops a thin sublaminar boundary thickness of about 10 μm along most of its interior wall. However, close to the port, the thickness increases well above 100 μm due to the The design of this nozzle develops a thin sublaminar boundary layer, δ l , of variable thickness in its the interior wall [29]: where v * = τ w ρ is the velocity friction and τ W is the wall's strain-stress. This nozzle develops a thin sublaminar boundary thickness of about 10 µm along most of its interior wall. However, close to the port, the thickness increases well above 100 µm due to the boundary layer separation phenomenon, originated by the sudden change of the flow direction through the ports. The uniformity of the boundary layer thickness along the body of the nozzle leads to a plug-flow of the liquid, see Figure 6a. The expansions and contractions of the cross-section areas of the nozzle give origin to plug-like flows of high and low velocities. The rounded corners of the upper side of the port avoid the severe flow separations in the boundary layer in these regions. A source of flow instability is the asymmetric flow formed by the closed area of the SGV as seen in the longitudinaltransversal view perpendicular to the plane of the ports, Figure 6b. The flow is biased inside of the nozzle because of the higher velocities through the open area of the SGV, leaving a stagnant region in the upper right zone, just beneath the closed area. The design of this nozzle consisting of expansions and contractions of the cross-section flow area helps to decrease the flow-biased characteristics. These flow control features help to straighten the stream coming from the tundish making a contribution to stabilize the meniscus in the mold. This design is different from other nozzle designs consisting of simple straight tubes. However, there is a recirculating flow in the bottom due to the presence of a low pressure-region generating and unfilled volume. Figure 6c,d shows the velocity fields in the right port in two planes; the first is in the central longitudinal axis of the pipe (PT1), and the second is just the outer frame of the port (PT3). Backflows occupy a proportion of the flow area of the port in the axial plane. The liquid flows through the upper left corner in Figure 6c; this area is larger in the outer plane of the port, see Figure 6d. The fluid leaves the nozzle through the outer frame of the port by a biased flow oriented toward its lower-left corner. The upper left corner is close to characteristics of stagnant conditions, leaving a reduced area utilization for liquid flowing into the mold. Figure 6e,f show the corresponding velocity fields in the axial and outer frame of the left port, respectively. In the left port, the backflows are a mirror image of those found in the right one. The fluid leaves the nozzle through the outer frame by a biased flow oriented in the lower right corner. Therefore, the source of flow instability is the biased flow inside the nozzle prevailing in both ports. It explains the low frequency of periodic-asymmetric flows despite the uniform boundary layer thickness throughout the interior wall of the nozzle.   Figure 7a shows the probability density function (PDF) of the measured velocities in the region of maximum velocity (located at 0.33 m from the mold's narrow face, labeled as L33) during a period of flow instability. The overall PDF is divided into two PDFs of positive and negative velocities, indicating the generation of backflows toward the narrow  Figure 7a shows the probability density function (PDF) of the measured velocities in the region of maximum velocity (located at 0.33 m from the mold's narrow face, labeled as L33) during a period of flow instability. The overall PDF is divided into two PDFs of positive and negative velocities, indicating the generation of backflows toward the narrow wall. In contrast, Figure 7b shows the PDF of positive velocities close to the nozzle (located at 0.60 m from the nozzle's wall, labeled as L60). The meaning of this later velocity PDF in the region adjacent to the nozzle is the permanent existence of a flow-oriented toward the nozzle. Figure 7c shows the velocity profile along with the distance between the narrow face and the nozzle's wall. The gray region in Figure 7c shows the standard deviation of velocities, and the green curve is the average velocity profile. The regions close to the narrow face and the nozzle remain permanently, with small and positive velocities. The interpretation of the flow structure is feasible through the autocorrelation function, defined as: Normalized by R xx (0) = 1. From the Schwartz inequality, R xx < 1 for all τ = 0; physically, the autocorrelation function expresses the correlation between the velocities at a time t and at other times (t + ∆t). Figure 8b shows the autocorrelations at the distance of 0.33 m (L33), and 0.60 m (L60) from the narrow mold's face, corresponding to the high turbulence regions and closer to the nozzle, respectively. The position L33 yields a fast-decreasing autocorrelation to 0.2 after seven seconds and to 0.6 at position L60 after 65 s. These differences explain the different velocity PDFs for both positions, as seen in Figure 7a,b.
wall. In contrast, Figure 7b shows the PDF of positive velocities close to the nozzle (located at 0.60 m from the nozzle's wall, labeled as L60). The meaning of this later velocity PDF in the region adjacent to the nozzle is the permanent existence of a flow-oriented toward the nozzle. Figure 7c shows the velocity profile along with the distance between the narrow face and the nozzle's wall. The gray region in Figure 7c shows the standard deviation of velocities, and the green curve is the average velocity profile. The regions close to the narrow face and the nozzle remain permanently, with small and positive velocities. The interpretation of the flow structure is feasible through the autocorrelation function, defined as: Normalized by 0) = 1. From the Schwartz inequality, < 1 for all ≠ 0; physically, the autocorrelation function expresses the correlation between the velocities at a time t and at other times (t + Δt). Figure 8b shows the autocorrelations at the distance of 0.33 m (L33), and 0.60 m (L60) from the narrow mold's face, corresponding to the high turbulence regions and closer to the nozzle, respectively. The position L33 yields a fastdecreasing autocorrelation to 0.2 after seven seconds and to 0.6 at position L60 after 65 s. These differences explain the different velocity PDFs for both positions, as seen in Figure 7a,b.

Conclusions
A commercial nozzle delivers steel flows into a slab mold yielding reasonable turbulence control and stability of the flux-metal interface. Physical and mathematical models help unveil the nozzle design effects on fluid flow in the confined mold's space. From the results obtained using these modeling tools, the following conclusions can be drawn: • The variations of bath levels during the casting are highly correlated, especially at both sides of the nozzle, and between the mold's narrow face and the nozzle's wall. • The maximum velocity at the meniscus, recorded by the physical and mathematical models, is 0.55 m/s, which is well above the maximum velocity recommended to avoid flux entrainment into the liquid metal.

Conflicts of Interest:
On behalf of all authors, the corresponding author states that there are no conflicts of interest.

Conclusions
A commercial nozzle delivers steel flows into a slab mold yielding reasonable turbulence control and stability of the flux-metal interface. Physical and mathematical models help unveil the nozzle design effects on fluid flow in the confined mold's space. From the results obtained using these modeling tools, the following conclusions can be drawn:

•
The variations of bath levels during the casting are highly correlated, especially at both sides of the nozzle, and between the mold's narrow face and the nozzle's wall.

•
The maximum velocity at the meniscus, recorded by the physical and mathematical models, is 0.55 m/s, which is well above the maximum velocity recommended to avoid flux entrainment into the liquid metal.