Internal SEN Design and Its Influence on Fluid Dynamics in Slab Molds: A Combined Numerical and Experimental Analysis

Abstract

The optimization of submerged entry nozzle (SEN) designs plays a pivotal role in achieving stable flow conditions and high-quality steel production during continuous casting. This study presents a comparative analysis of two SEN geometries under identical operational parameters using a combined approach of numerical simulation and physical modeling. A full-scale water model and a validated CFD framework based on the realizable k-$\epsilon$ and VOF models were employed to evaluate velocity distribution, turbulence intensity, free surface behavior, and flow symmetry. Results reveal that the SEN-2 design enhances flow stability near the meniscus region, promotes a consistent double-roll flow pattern (DRF), and reduces surface oscillations and sub-meniscus velocities, thereby minimizing the risk of mold flux entrapment. The proposed dimensionless KE number effectively quantifies the energy dissipation behavior of both designs, highlighting SEN-2’s superior hydraulic performance. This integrated methodology offers a robust evaluation framework for future nozzle development aimed at improving product quality without compromising productivity.

1. Introduction

The steel industry is facing a growing challenge due to the current demand for steel and the strict quality standards imposed by the market. An increase of 146 million metric tons in new steel production capacity is expected between 2025 and 2027 [1]. This scenario requires steel companies to maintain high level of production without sacrificing product quality, which is complicated by the inversely proportional relationship between productivity and quality in the melting shop. An interesting hypothesis put forward by experts in the field proposes that controlling the turbulence of liquid steel within the continuous casting mold may be key to increasing both productivity and steel quality [2,3]. Numerical simulation tools and physical modeling are used to address the technical challenges of the continuous casting process [2,3,4,5,6,7,8,9,10] and study the oscillation and asymmetry of the flow in the mold. They report that the symmetry of the liquid steel flow is strongly affected when the SEN is misaligned more than 4% in the direction of the slab mold width, when the discharge ports of the submerged entry nozzle (SEN) have an upward angle, and when there is an increase in flow rate. It can be a combination of these factors, or just one is enough to unbalance the flow symmetry within the mold. The effect of the steel feeding system from the tundish to the mold has also been studied [11,12,13], as well as the use of magnetic fields [14], the injection of argon gas [15,16,17,18], and other factors related to the clogging and design of the SEN [19,20,21,22,23,24,25] of an initial solidified layer with sufficient integrity to withstand the ferrostatic pressure of the liquid core [26,27]. In addition, a proper SEN design facilitates the feeding of steel into the meniscus area, promoting proper flux infiltration and reducing defects such as adhesions and ruptures [27,28,29]. However, the phenomenon of the SEN clogging represents a persistent challenge, addressed through strategies such as chemical treatments, argon gas insufflation, and SEN preheating [30,31,32,33,34,35]. The design of submerged entry nozzles plays a pivotal role in enhancing the quality and productivity of the continuous casting process. Recent efforts to improve heat extraction uniformity in large-scale, complex-shaped beam-blank molds have employed advanced numerical simulations, particularly using the NSGA-II genetic algorithm to optimize three-port submerged entry nozzle (SEN) designs. These configurations have demonstrated significant improvements in flow distribution, reduced thermal gradients along mold walls, and minimized thinning of the initial solidified shell [36]. In funnel-type molds used for thin slab casting, where casting speeds are higher than in conventional molds, a critical force balance has been identified at the internal bifurcation of the SEN, directly influenced by velocity fluctuations and ferrostatic pressure [37]. To maintain stable operating conditions, precise automation is required to regulate casting speed as the steel level in the tundish decreases. Alternatively, SEN designs that are less sensitive to immersion depth and velocity variations offer a more accessible solution. In this context, Debasish [38] proposed a four-port SEN design, evaluated through numerical simulation to assess the impact of turbulence on metal–slag interaction, level fluctuations, and breakout risk. The results revealed superior performance compared to the conventional two-port SEN design. Similarly, Sen et al. [39] emphasize that turbulence within the mold must remain within optimal limits: intense enough to promote flotation of inclusions and their capture by the mold powder, yet not excessive to avoid entrainment of powder particles into the molten steel. Achieving this balance depends directly on a SEN design tailored to the specific operating conditions of the continuous casting process.

Considering previous advancements, this study focuses on the internal design of the submerged entry nozzle (SEN), aiming to properly manage the high kinetic energy generated by the elevated flow velocities that this device must withstand. More importantly, it seeks to control the dissipation of turbulent kinetic energy, based on the core premise of this research group: that turbulence intensity can only be effectively regulated by controlling energy dissipation through optimized internal SEN geometries. To this end, a comprehensive analysis of molten steel flow dynamics within the slab mold was conducted, comparing two distinct SEN configurations under identical operating conditions. Quantitative measurements and qualitative observations were employed to evaluate the hydrodynamic behavior induced by each design, enabling a solid assessment of their relative performance in terms of flow stability and potential impact on final slab quality. The methodological strategy integrates numerical simulations, performance evaluation using the dimensionless KE number, and physical experimentation, forming a robust approach to address current challenges in the steel industry. This multidisciplinary framework facilitates a deeper understanding of the complex flow phenomena within the mold and supports the development of optimized SEN designs. Furthermore, the results aim to establish a standardized evaluation methodology capable of predicting the efficiency of any SEN configuration prior to its fabrication, thereby enhancing process reliability and product quality.

Case Study

This study complements two earlier investigations conducted by the same research group, both centered on two specific SEN designs (see Figure 1). The first study [32] focuses on decreasing the nozzle clogging through flow dynamics control, while the second [23] examines the chemical interaction between inclusions and the inner walls of the SEN. Together, these contributions provide valuable insights into the mechanisms underlying clogging phenomena. In the present work, the two SEN designs are evaluated against specific operating conditions intended to optimize their metallurgical performance in continuous casting: (a) to promote rapid melting of the lubricating powder, flow velocities in the meniscus region must be maintained within the range of 0.2 to 0.35 m/s; (b) a double- roll flow (DRF) pattern is pursued to ensure uniform heat transfer across the slab mold; (c) to regulate the initial formation of the solidified shell, it is essential to avoid zero-flow zones in the meniscus region and to prevent excessive velocities exceeding 0.5 m/s; (d) standing waves at the mold’s free surface must remain below 15 mm in heigh and 0.5 m/s in velocity to minimize the entrapment of slag particles (such entrapment is typically induced by shear layer instabilities and the formation of vortices with lengths greater than 20 mm); and (e) finally, establishing the performance threshold of each SEN configuration, three casting speeds (0.9, 1.2, and 1.4 m/min) were tested and arranged into six operating scenarios, as presented in

Table 1. Physical and numerical study cases.

No. Case	Design	Casting Speed	No. Case	Design	Casting Speed
Case 1	SEN-1	0.9 m/min	Case 4	SEN-2	0.9 m/min
Case 2	SEN-1	1.2 m/min	Case 5	SEN-2	1.2 m/min
Case 3	SEN-1	1.4 m/min	Case 6	SEN-2	1.4 m/min

In all study cases, the same immersion depth (120 mm) of the SEN was used.

.

2. Materials and Methods

In the present study, both physical modeling and mathematical simulation were conducted under isothermal conditions. This methodological decision was made to isolate and more clearly analyze the hydrodynamic phenomena of liquid steel flow within the continuous casting mold, without the influence of thermal gradients. Although the actual continuous casting process involves significant heat transfer, during the initial stages of mold filling, flow behavior is primarily governed by inertial, geometric, and turbulence effects rather than thermal effects. Therefore, the use of isothermal conditions enables an accurate representation of flow distribution, the formation of recirculation zones, and jet interaction, without introducing additional complexities associated with thermal coupling.

2.1. Description of the Physical Model

A full-scale physical model of the slab mold and corresponding SEN design was constructed to replicate the hydrodynamic conditions of the continuous casting process [36,37,38,39]. The model was assembled using translucent plastic panels (15 mm thick), with overall dimensions of 1.450 m × 0.235 m × 1.7 m. To model the behavior of liquid steel flow within the mold, the setup was partially submerged (0.7 m) in a water-filled reservoir designed to mimic the continuity of the slab strand. The reservoir was equipped with a 10 HP submersible pump that recirculated water through a vertical pipeline incorporating a flow meter and precision control valve. This configuration enables regulation of flow through one-third of a full-scale tundish replica, which included a stopper rod to control the casting speed. The SEN was centrally positioned at an immersion depth of 0.12m, measured from the top of the discharge port to the mold’s free surface. To model ferrostatic pressure effects observed in industrial casting, the tundish was filled to a height of 1m and held constant throughout experimentation. Casting speed was finely tuned using the integrated flow meter, control valve, and stopper rod, thus ensuring a consistent mold level and uninterrupted flow into the cavity. Experimental conditions were stabilized for 15 min prior to initiating data collection, which encompassed video recording and quantitative measurements. A schematic of the experimental setup is presented in Figure 2.

2.2. Description of Physical Experiments

To assess flow velocities within the sub-meniscus region (defined as 20 mm below the free surface), a high-frequency ultrasonic transducer operating at 1 MHz was deployed. The sensor was mounted at the center of the narrow face of the plastic mold, precisely 20 mm beneath the mold surface to ensure accurate spatial targeting. The transducer emits an ultrasonic beam that interacts with the outer wall of the SEN, generating echo signals whose frequency shifts are recorded over time. These Doppler shifts are then processed to produce instantaneous velocity profiles along the beam axis, revealing dynamic flow behavior in the vicinity of the SEN. The specific location and orientation of the transducer within the experimental setup are depicted in Figure 3a,c. The free surface level within the mold cavity was continuously monitored for the entire duration of the experiment (300 s) using six ultrasonic sensors strategically distributed around the SEN. Three sensors were placed on the left side of the SEN and three on the right, at equidistant points to ensure symmetrical coverage of surface fluctuations. The specific positions were as follows: Sensor 1 was located 50 mm from the narrow face, Sensor 2 at 300 mm, and Sensor 3 at 550 mm. Sensors 4, 5, and 6 were installed at identical distances on the opposite side, providing a mirrored sensor layout. Figure 3b,d illustrate this arrangement.

To reveal the flow pattern inside the mold, a red tracer (aqueous solution) is pulse-injected through a perforation in the outer wall of the SEN [40]. The mixing kinetics of the tracer within the plastic mold is recorded on video. Figure 4 displays the first seconds captured of the flow behavior corresponding to the SEN 1 design.

2.3. Description of Numerical Model

2.3.1. The Realizable k-$\epsilon$ Model

To complement the experimental investigations described previously, six computational case studies were configured and resolved, as summarized in Table 1. Each simulation was conducted by solving the unsteady Reynolds-averaged Navier–Stokes (URANS) equations for incompressible viscous flow, utilizing the volume of fluid (VOF) multiphase model [41,42,43] in conjunction with the realizable k-$\epsilon$ turbulent model. The realizable k-$\epsilon$ model introduces notable improvements over the standard formulation by altering the transport equation for the turbulent kinetic energy dissipation rate ($\epsilon )$ and by redefining the turbulent viscosity (${\mu }_t$) as a dynamic function dependent on the local flow field rather than as a constant parameter. This refinement enhances the model’s predictive capabilities in complex flow environments, such as the SEN and mold cavity. The choice of the realizable k-$\epsilon$ turbulence model is primarily based on its greater stability in simulations involving complex meshes, such as those generated inside the submerged entry nozzles. The governing transport equations for turbulent kinetic energy ($k$) and its dissipation rate ($\epsilon$) in the realizable k-$\epsilon$ framework are expressed as follows:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho k〈U_j〉\right)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right) {\displaystyle \frac{\partial k}{\partial x_j}}\right]+G_k-\rho \epsilon$$

(1)

$${\displaystyle \frac{\partial \left(\rho \epsilon \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho \epsilon 〈U_j〉\right)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[{\displaystyle \frac{\partial }{\partial {\epsilon }_j}}\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right)\right]+\rho C_1{S}_{\epsilon }-\rho C_2{\displaystyle \frac{{\epsilon }^2}{k+\sqrt{\nu \epsilon }}}$$

(2)

where $G_k$ represents the generation of turbulence kinetic energy due to the mean velocity gradients, and ${\sigma }_{\epsilon }$ is the turbulent Prandtl number for $\epsilon$. The pseudo-constant, $C_1$, is defined as follows:

$$C_1=\mathit{max}\left[0.43, {\displaystyle \frac{\eta }{\eta +5}}\right], \eta =S{\displaystyle \frac{k}{\epsilon }}, S=\sqrt{2S_{ij}{S}_{ij}}$$

(3)

with the turbulent viscosity determined by:

$${\mu }_t=\rho C_{\mu }{\displaystyle \frac{k^2}{\epsilon }}$$

(4)

In the realizable k-$\epsilon$ model, $C_{\mu },$ it stops being constant and is calculated from the following expression:

$$C_{\mu }={\displaystyle \frac{1}{A_0+A_s\frac{k{U}^{\ast }}{\epsilon }}}$$

(5)

$$U^{\ast }=\sqrt{S_{ij}{S}_{ij}+{\overline{\Omega }}_{ij}{\overline{\Omega }}_{ij}}$$

(6)

$${\overline{\Omega }}_{ij}={\Omega }_{ij}-2{ϵ}_{ij}{\omega }_k$$

(7)

where ${\Omega }_{ij}$ is the mean rotation velocity tensor and ${\omega }_k$ is the angular velocity, ${ϵ}_{ij}$ is the Civi-Levita symbol. The constants $A_0$ and $A_s$ are expressed as follows:

$$A_0=4.04, A_S=\sqrt{6} \cos \varphi$$

(8)

The parameters used in the model are expressed as follows:

$$\varphi ={\displaystyle \frac{1}{3}}{\cos }^{-1}\left(\sqrt{6}W\right),W={\displaystyle \frac{S_{ij}{S}_{jk}{S}_{ki}}{{\tilde{S}}^3}},\tilde{S}=\sqrt{S_{ij}{S}_{ij}},S_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u_j}{\partial x_i}}+{\displaystyle \frac{\partial u_i}{\partial x_j}}\right)$$

(9)

2.3.2. The Volume of Fluid (VOF)

The multi-fluid volume of fluid (VOF) model [41,42,43,44,45,46] offers an integrated framework that couples the classical VOF approach with the Eulerian multiphase model, enabling advanced simulation of interfacial dynamics in multiphase systems. This hybrid model is particularly suited for capturing the transient behavior of the interface between the steel’s free surface and the surrounding air during continuous casting processes. Designed for incompressible, immiscible fluids with no interphase mass transfer (i.e., phase change), the multi-fluid OF formulation allows the local material properties (such as density and viscosity) to be computed as weighted averages, based on the volume fractions of each fluid phase present within computational cells. This treatment enables precise tracking of the interface and accurate prediction of surface instabilities, wave motion, and mold level fluctuations. The effective fluid properties within each cell can be expressed as:

$${\rho }_{mix}={\alpha }_{\rho }{\rho }_{\rho }+\left(1-{\alpha }_q\right){\rho }_{\rho }$$

(10)

$${\mu }_{mix}={\alpha }_{\rho }{\mu }_{\rho }+\left(1-{\alpha }_q\right){\mu }_{\rho }$$

(11)

A continuity equation applicable to transient systems has been derived, with its structure contingent upon the number of phases present within the system. To adequately represent interphase mass transfer dynamics, a source term, denoted as $S_{\alpha q}$, is introduced. This term accounts for the mass exchange between phases and is explicitly defined in Equation (3):

$${\displaystyle \frac{1}{{\rho }_q}}\left[{\displaystyle \frac{\partial }{\partial t}} \left({\alpha }_q{\rho }_q\right)+\nabla ·\left({\alpha }_q{\rho }_q\overrightarrow{v}\right)=S_{\alpha q}+\sum _{p=1}^n({\dot{m}}_{pq}-{\dot{m}}_{qp})\right]$$

(12)

The volume fraction of the secondary phase (air) is determined by solving Equation (12), which governs its temporal and spatial evolution within the system. In contrast, the volume fraction of the primary phase is evaluated by enforcing a constraint condition, ensuring that the sum of volume fractions for all phases remains conserved, typically satisfying:

$$\sum _{q=1}^n{\alpha }_q=1$$

(13)

A unified set of momentum equations is solved across the entire computational domain, ensuring consistency in the treatment of fluid motion. The resulting velocity fields are commonly shared among all phases, allowing interphase interaction through the coupling of momentum. The momentum equation, detailed below, incorporates phase (dependent properties), namely, the density $\rho$ and dynamic viscosity $\mu$, which are weighted by the corresponding volume fractions of each phase. This formulation enables accurate representation of multiphase dynamics and interfacial momentum exchange.

$${\displaystyle \frac{{\alpha }_q^{n+1} {\rho }_q^{n+1}-{\alpha }_q^n{\rho }_q^n}{∆t}} V+\sum _f\left({\rho }_q{U}_f^n{\alpha }_{q·f}^n\right)=\left[S_{\alpha q}+\sum _{p=1}^n({\dot{m}}_{pq}-{\dot{m}}_{qp})\right]V$$

(14)

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_{mix}\overrightarrow{v}\right)+\nabla ·\left({\rho }_{mix}\overrightarrow{v}\overrightarrow{v}\right)=-{\nabla }_p+\nabla \left[{\mu }_{mix}\left(\nabla \overrightarrow{v}+\nabla \overrightarrow{u}\right)\right]+\rho g+S_{\sigma }$$

(15)

2.3.3. Numerical Procedure

The governing equations were discretized using the finite volume method and solved through a segregated–iterative computational approach (Figure 5). The nonlinear momentum equations were linearized using an implicit scheme. Spatial discretization was performed using the second-order upwind scheme, while pressure interpolation was carried out using the body force weighted algorithm [43]. The PISO algorithm [43] was employed to couple the pressure and velocity fields. To model the interaction between two immiscible phases, the volume of fluid (VOF) model was selected as the most appropriate approach. The computational mesh consisted of 2,227,697 structured cells and 486,328 nodes, with an average skewness of 0.10852 and a standard deviation of 9.982 × 10⁻². The average orthogonal quality was 0.92652, with a standard deviation of 6.097 × 10⁻². The total simulation time was 300 s, using a fixed time step of 0.01 s. For the initial turbulence calculations, and based on the Reynolds numbers obtained for each of the three simulated casting speeds (Re₁ = 96,261.990; Re₂ = 128,359.161; and Re₃ = 147,589.339), the turbulence intensity and turbulent length scale were set to 15% and 0.00525 m, respectively. These values were uniformly applied as boundary conditions at both the inlet and outlet of the numerical domain. Finally, a fixed pressure outlet condition (P = 1 atm) was applied at the mold exit to avoid constraining the flow direction, thereby preserving a more realistic representation of the system. Additionally, no-slip boundary conditions were imposed on all solid surfaces of the SEN/mold configuration to accurately model the fluid–wall interaction.

3. Results and Discussion

3.1. General Flow Pattern

To begin the comparison between designs and simultaneously validate the numerical simulations, cases 3 and 6, shown in Table 1, were selected. Case 3 corresponds to the SEN-1 design operating at a casting speed of 1.4 m/min, while Case 6 represents the SEN-2 design under the same casting speed. These cases were selected as they represent extreme operating conditions, with the aim of highlighting more contrasting differences and evaluating the agreement between the experimental approach and the numerical simulations. Figure 6 illustrates three key moments during the tracer injection in the physical experiment: (1) when the jets reach the narrow face of the mold (4.5 s); (2) when the upper recirculating flows reach the free surface (6 s); and (3) when these upper recirculation’s reach the SEN wall (7 s), fully forming the double-roll flow (DRF) pattern.

Analyzing the key moments shown in Figure 6 reveals good agreement between the experiments and the simulation. The first moment, when the jets reach the narrow face of the mold, was also used to measure their impact depth. The difference in impact depth between the two jets of SEN 1 in the physical experiment was 145 mm, while its counterpart in the numerical simulation was 43 mm. For the SEN 2 design, the difference in impact depth between its discharging jets was 106 mm in the physical experiment and 14 mm in the numerical simulation. The second moment corresponds to the point when the upward flow reaches the meniscus region, marked by points (1, 2) for SEN-1 and (5, 6) for SEN-2. These points indicate that the upward stream generated by SEN-2 (points 5 and 6) rises more rapidly compared to Nozzle-1 (points 1 and 2). The final moment shows that SEN-1 (points 3 and 4), and the corresponding numerical results confirm this observation.

3.2. Velocities at Sub-Meniscus Region

The flow velocity in the sub-meniscus region (20 mm below the free surface) is of critical importance, as this zone involves interaction between the molten steel and the mold flux layer [47,48]. This flux layer consists of three distinct phases: (1) molten flux in liquid state, which is in direct contact with the steel and acts as a lubricant between the copper mold and the solidifying steel strand; (2) flux in pasty state; and (3) solid flux. For this reason, it is essential to avoid excessively high velocities in this region (above 0.5 m/s) [49], as they can lead to the entrainment of molten flux particles into the molten steel [50]. Additionally, the flow pattern (particularly unidirectional velocities) should be controlled to prevent the vortex formation [51,52,53] that may draw liquid flux particles into the steel bath. Figure 7a,c show the probability density function (PDF) of the velocities measured by the ultrasonic transducer during the water model experiments. The PDF corresponding to SEN-1 (Figure 7a) indicates the presence of a bidirectional flow, from the narrow face toward the SEN wall and in the reverse direction, with notable velocity magnitudes exceeding 0.4 m/s in both directions. In contrast, the PDF for SEN-2 (Figure 7c) illustrates a unidirectional flow from the narrow face of the mold toward the SEN wall, with significantly lower velocities around 0.2 m/s.

Figure 7b,d, corresponding to SEN-1 and SEN-2, respectively, present velocity profiles obtained from numerical simulations at different time steps (colored lines), while the gray shading zone represents the experimental velocity measurements over time captured by the ultrasonic transducer. As previously noted, SEN-1 exhibits greater temporal velocity variance, particularly at the midpoint between the mold’s narrow face and the SEN wall. Conversely, SEN-2 demonstrates a more stable velocity distribution. These findings suggest that the SEN-1 design has a higher probability of inducing mold flux entrapment, both due to shear layer instability and vortex formation [49,54]. To provide a clear comparison of the performance of each SEN design, Figure 8 presents the average velocity profiles at the sub-meniscus level, obtained from both the experimental and numerical approaches. Additionally, velocity vectors corresponding to 300 s of simulation (representing the steady state condition) are included to visualize the internal flow symmetry within the mold.

Figure 8 clearly illustrates the difference between the SEN-1 and SEN-2 designs. SEN-1 exhibits velocities ranging from 0.4 to 0.45 m/s in the central region between the narrow mold wall and the SEN wall. In contrast, the SEN-2 design reduces these velocities in the same area, maintaining them between 0.3 and 0.35 m/s. It is important to note that the average velocity profiles obtained experimentally and those calculated through numerical simulation do not match perfectly. This discrepancy arises because the ultrasonic velocity profiler (UVP) captures both the magnitude and direction of the velocity vectors, which directly influences the final averaged values. Nevertheless, the overall trend observed in both approaches is consistent, indicating that the general flow behavior is adequately represented by both the experimental and numerical methods.

Analyzing the flow field represented by velocity vectors—corresponding to the instantaneous state at second 300, when steady-state conditions are reached—it is evident that the SEN-1 design exhibits greater asymmetry between the jets emerging from its discharge ports. This asymmetry leads to the presence of high velocity counterflow streams, which interfere with the proper formation of the desired double recirculation flow (DRF) pattern. In contrast, the SEN-2 design maintains flow symmetry at its discharge ports, promoting a more stable and well-defined formation of the DRF pattern. This suggests that geometric modifications in the nozzle design can significantly influence the flow behavior within the mold, with direct implications for process stability and product quality.

3.3. Free Surface Behavior

The behavior of the molten steel flow emerging from the discharge ports governs the integrity and shape of the jets, which, in turn, define the overall flow pattern within the mold. Consequently, the free surface exhibits a distinct response that can be interpreted as a butterfly effect [55,56,57] (where minor variations in initial conditions lead to substantial changes in surface flow dynamics).

To analyze the behavior of the free surface, six level sensors were installed at the top of the physical model (Figure 3b), allowing wave height measurements to be recorded over a period of 120 s. Figure 9 displays the results obtained from testing the SEN-1 design.

Figure 9a displays the free surface oscillations at the mold’s extremities, where wave magnitudes of 15 and −15 mm are recorded, driven by upward flow along the narrow mold walls. In the central mold region, monitored by sensors 2 and 5 (Figure 9b), the graphs overlap, indicating dynamic agreement on both sides of the SEN. However, near the SEN (sensors 3 and 4 in Figure 9c), the oscillations intensify, reaching values of 14 and −30 mm, which implies a high likelihood of mold power entrapment under real casting conditions. Finally, based on the statistical covariance identity $Cov\left(X_i{X}_j\right)=Cov\left(X_j{X}_i\right)$, covariances were calculated between sensors (Figure 9d). The resulting values, ranging from 20 to 27, reflect a highly unstable free surface. The same methodology was applied to analyze the free surface behavior using the SEN-2 design. The corresponding results are presented in Figure 10.

Figure 10a–c show an overlap of data across the three analyzed positions, indicating the presence of dynamic symmetry on both sides of the SEN. Additionally, the magnitudes of the surface waves are smaller compared to the results obtained from SEN-1 (Figure 9). It is important to highlight that the oscillations detected in the water model can be directly correlated with those observed in the actual steel reactor by applying a correction factor of 0.59 ($h_{steel}=0.59h_{water})$ [58,59]. This factor accounts for the significant difference in density between the two fluids. Finally, as the most notable difference, the covariance values calculated for the level sensors are significantly lower than those recorded in SEN-1, ranging from 1 to 3.5. It is worth noting that low covariance values indicate a more stable free surface.

3.4. Comparative Analysis of SEN Performance at Different Casting Speeds

In the previous sections, a comparison was carried out between Case 3 and Case 6, both corresponding to the highest casting speed (1.4 m/min), with the aim of identifying significant differences between the two SEN designs under evaluation. In this section, following the validation of the numerical model, an integrated analysis of all six simulated cases (Table 1) is presented. The goal of the optimized design (SEN-2) is to enhance turbulent kinetic energy dissipation within the nozzle, mitigate excessive velocities at the discharge ports, and, consequently, improve flow symmetry in the mold, thereby reducing disturbances at the free surface. Figure 11 shows the contours of kinetic energy (k), contours of kinetic energy dissipation ($\epsilon$), and velocity vectors generated by each SEN design at the three casting speeds (0.9, 1.2, and 1.4 m/min).

Figure 11 clearly highlights the improvement of the SEN-2 design over the original SEN-1, particularly in terms of turbulent kinetic energy generation, k, and the reduction in velocities at the discharge ports. However, it is necessary to quantitatively assess the improvement achieved. To this end, the maximum values of turbulent kinetic energy and its dissipation were analyzed and extracted for each simulated case; these values are presented in

Table 2. Values of mass flow rate, $\dot{m}$, turbulent kinetic energy, k, and turbulent kinetic energy dissipation, ε, extracted from Figure 11 and used for the calculation of ${\epsilon }^{1/3}$.

No. CASE	Speed Casting $\left(\raisebox{1ex}{$\mathbf{m}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{m}\mathbf{i}\mathbf{n}$}\right.\right)$	Mass Flow Rate $\dot{\mathit{m}}\left(\raisebox{1ex}{$\mathbf{k}\mathbf{g}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.\right)$	Turbulent Kinetic Energy, $\mathbf{k}, \left(\raisebox{1ex}{${\mathit{m}}^2$}\!\left/ \!\raisebox{-1ex}{${\mathit{s}}^2$}\right.\right)$	Turbulent Dissipation Rate, $\mathit{\epsilon },\left(\raisebox{1ex}{${\mathit{m}}^2$}\!\left/ \!\raisebox{-1ex}{${\mathit{s}}^3$}\right.\right)$	$${\mathit{\epsilon }}^{1/3}$$
CASE 1	0.9	36.2892	0.12	7	1.91293118
CASE 2	1.2	43.3865	0.22	14.8	2.45520205
CASE 3	1.4	55.6403	0.27	22	2.80203933
CASE 4	0.9	36.2892	0.08	3.5	1.51829449
CASE 5	1.2	43.3865	0.15	8.2	2.01652968
CASE 6	1.4	55.6403	0.20	12.3	2.30835024

. Subsequently, the dimensionless KE number (17), developed by this research group, was employed to build the graph of Figure 12. This parameter correlates the operating conditions (casting speed) with flow characteristics (turbulent kinetic energy dissipation) to evaluate the performance of the SEN designs. Further details regarding the definition of this number can be found in reference [2].

$$KE={\displaystyle \frac{Q}{\sqrt[3]{\epsilon }}}$$

(16)

Figure 12 illustrates the two slopes that characterize the performance of each SEN design across varying mass flow rates corresponding to the evaluated casting speeds. Notably, the slope associated with SEN-2 is steeper (+4.27) than that of SEN-1, indicating a more favorable energy dissipation behavior. The dimensionless KE number serves as an indicator of the SEN capacity to stabilize turbulent structures and promote symmetrical flow within the mold. A higher slope in this context reflects enhanced hydraulic performance and improved control over flow asymmetries and surface disturbances. To emphasize how even subtle differences in slope can influence flow behavior, Figure 13 presents the velocity contours on a central symmetric plane, the narrow mold walls, and the free surface for all six simulated cases (at 300 s of simulation). These visualizations confirm the capacity of SEN-2 to achieve a more homogeneous and stable flow distribution under similar operating conditions.

The velocity contours shown in Figure 13 reveal significant differences in the performance of the designs analyzed. These images correspond to the 300-s mark of the simulation, at which point the system is considered to have reached a steady state. At this stage, the velocities along the narrow walls and the free surface exhibit stable magnitudes. For the SEN-1 design, velocities in these regions exceed 0.3 m/s, except in Case 1, where the casting speed is the lowest (0.9 m/min). In contrast, the SEN-2 design maintains velocities below 0.3 m/s in the same areas. Furthermore, analysis of the discharge jets indicates that the SEN-2 design promotes a more symmetric flow pattern compared to SEN-1, regardless of the casting speed applied. To further clarify the flow symmetry, velocity vectors were captured in the central plane of the mold in Figure 14, allowing for the identification of the recirculation centers associated with the desired DRF pattern.

Although both SEN designs successfully generate the desired double-roll flow (DRF) pattern, only the SEN-2 design maintains a noticeable symmetry between the recirculation centers. The asymmetry observed in the SEN-1 design could be a contributing factor to quality issues, as flow symmetry and moderate velocities in critical regions (such as the narrow walls and the free surface) are essential for efficient heat transfer from the strand to the mold. Finally, to further analyze the flow behavior within the mold, turbulence intensity contours were calculated and compared across the six simulated cases, as shown in Figure 15.

Turbulence intensity is a parameter that quantifies the magnitude of velocity fluctuations in a turbulent flow. It is calculated using the following expression [43]:

$$I_t={\displaystyle \frac{\sqrt{u^2}}{〈U_e〉}}$$

(17)

where $u$, is the turbulent velocity fluctuations, and, $U_e$, is the mean flow velocity.

Figure 15 illustrates the turbulence intensity contours within the SEN, the slab mold, and a close-up of the discharge ports. Although the intensity values are similar between both SEN designs, the contour morphology at the outlet ports is more pronounced in the SEN-1 design compared to SEN-2. This indicates that SEN-1 generates higher velocity fluctuations, which translates into greater turbulent kinetic energy and, consequently, higher turbulence intensity. However, when analyzing turbulence intensity contours, it can be challenging to clearly distinguish which design performs better based solely on visual or qualitative data. For this reason, the use of the dimensionless KE number becomes particularly important, as it provides a more objective, quantitative metric. This allows for faster and more reliable decision-making under real operating conditions.

4. Conclusions

This study has demonstrated the significant influence of SEN internal geometry on flow dynamics within the slab mold during continuous casting. The comparative analysis of SEN-1 and SEN-2 under identical operating conditions leads to the following conclusions:

Enhanced hydraulic performance of SEN-2: The SEN-2 design promotes a more symmetrical and stable velocity distribution, particularly in the sub-meniscus region. This reduces the likelihood of mold flux entrapment and improves the surface quality of the cast slab.

Reduction in turbulence intensity and free surface oscillations: SEN-2 exhibits lower fluctuations at the free surface and more efficient dissipation of turbulent kinetic energy, fostering more uniform heat transfer and minimizing defects associated with meniscus instability.

Robust validation through physical and numerical approaches: The combination of CFD simulations using the realizable k-$\epsilon$ turbulent model and full-scale physical experimentation confirm the consistency of results and establish a reliable framework for future SEN evaluations.

Operational relevance: SEN-2 consistently achieves the desired double-roll flow (DRF) pattern across all tested casting speeds (0.9, 1.2, and 1.4 m/min), emerging as a viable solution to enhance process efficiency without compromising throughput.

KE dimensionless number: The dissipation patterns of turbulent kinetic energy, as quantified by the dimensionless KE number, corroborate the superior energy dissipation capacity of the SEN-2 design (particularly under elevated casting speeds). This enhanced performance reflects improved momentum regulation and flow stabilization within the mold cavity.

The results obtained can serve as a foundation for future studies that incorporate coupled thermal models, including the interaction between flow, heat transfer, and solidification. In this way, the aim is to build a progressive understanding of the continuous casting process, starting with the fundamental fluid dynamics aspects.