Numerical Optimization of Nozzle Ports to Improve the Fluidynamics by Controlling Backflow in a Continuous Casting Slab Mold

Abstract

In order to improve the fluid flow patterns inside the mold, a better understanding of the backflow phenomenon and its controlling parameters is necessary; then a mathematical simulation of the fluidynamics in the mold and the submerge entry nozzle (SEN) are carried out considering two typical nozzle designs and different modifications applied to the outlet ports. The numerical model considers isothermal three dimensional continuity and the Navier-Stokes equations in Cartesian co-ordinates, which are solved together with the k-ε standard turbulence and the Volume of Fluid (VOF) models through the volume finite method. The results show that the backflow phenomenon emerges from an inadequate SEN port design, when a boundary layer separation is generated before the steel is delivered to the mold. This separation occurs at the upper internal side of the port inducing a low pressure zone with high levels of kinetic energy dissipation, producing the backflow phenomenon. From the analysis, it is concluded that the implementation of a radius at the internal upper side of the port avoids the separation of the boundary layer, eliminating the backflow phenomenon which allows the use of the complete effective exit area of the port; this is reflected in a velocity decrease of the jets and consequently a velocity decrement of the bulk flow. Furthermore, the size of the radius controls the penetration angle of the jets, the impact point position and the meniscus deformation; which avoids the need of the inclination angle of the port.

1. Introduction

Design of submerged entry nozzles and casting powder selection for slab molds together with heat transfer and initial solidification are the most important issue to control surface quality of steel products. Specifically, various aspects of liquid steel flow control in molds are well related with design of the discharging ports. As reported for different authors, unsuitable fluid flow patterns induce various superficial defects on slabs like pencil pipes, longitudinal and subsurface cracks,^1,2,3,4,5,6) flux infiltration problems, etc. The severity of those problems can affect considerably the caster productivity due to downgraded steel, product rejection and customer claims.

Many of the mold researches have been done by experiments in water models^7,8,9,10) and by mathematical simulation^{11,12,13,14,15,16,17)} which has been used to analyze deeply the fluidynamics of the mold and to evaluate and improve nozzles designs to control the steel flow patterns. It is well known that nozzle ports have to be design to deliver liquid steel constantly and uniformly to maintain a constant liquid level in the mold and promote surface stability, and also to avoid meniscus freezing, clogging and crack formation.^{7,11,12,13,14)} It has been predicted that initial clogging may enhance the steel flow rate due to a potential streamlining effect before it becomes great enough to restrict the flow channel and this should be avoided or reduced by argon injection.¹²⁾ In order to understand the effect of the port design, different parameters such inclination angle, size and geometry have been studies together with the variation of the casting speed and the nozzle immersion. From these studies, it has been determined that an impropriated port design can encourage the formation of phenomena generating instabilities in the entry steel jet and consequently a non uniform flow pattern; for instance, the dynamic distortions inside the mold¹⁸⁾ as reported in previously published works.^18,19,20) An important phenomenon that has shown a significant effect on the jets stability is the backflow, which is present in almost all types of nozzles and it is intrinsically related to the port design. This phenomenon has been observed in many different research works,^{12,14,16,17,21,22,23,24,25,26)} where it was found that the gas injection reduces its size, the position of the slide gate orientation affects its intensity, the jets velocity increases when the backflow zone increases, the backflow is larger with shallower port angles and smaller with rectangular ports design,^{19,20,21,22,23,24)} and increasing the outlet port length the backflow is larger.²⁵⁾ Even all these results, the origin of the backflow phenomenon has not been determined and it is unclear the parameters to control or avoid the detrimental flow effects of this phenomenon. Having in mind this, the present numerical study has as a main objective to understand the origin and the effects of the backflow phenomenon in order to improve the fluidynamics of a conventional mold by controlling the delivered jets, through redesigning the SEN ports in order to inhibit its presence. In a further work the physical validation shall be made known.

2. Mathematical Model

2.1. Fundamental Equations

A mathematical model was developed considering the isothermal three dimensional continuity and the Navier-Stokes equations shown in Eqs. (1) and (2) in Cartesian co-ordinates, which were solved together with the k-ε standard turbulence model and the Volume of Fluid model, through the volume finite method embedded in FLUENT code.²⁷⁾   

$$(\nabla \cdot u)=0$$(1)

  

$$\rho \left[\nabla \cdot uu\right]=-\nabla P+{\mu }_{eff}{\nabla }^{2}u+\rho g$$(2)

Where, ${\mu }_{eff}=\mu +{\mu }_t$ and ${\mu }_t=\rho C_{\mu }\frac{k^2}{\mathit{\epsilon }}$

2.2. The k-ε Standard Turbulence Model

The turbulence model is defined under isothermal conditions as follows,   

$$\rho \frac{\partial k{u}_i}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+G_k-\rho \mathit{\epsilon }$$(3)

  

$$\rho \frac{\partial \mathit{\epsilon }{u}_i}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\mathit{\epsilon }}}\right)\frac{\partial \mathit{\epsilon }}{\partial x_j}\right]+C_{1\mathit{\epsilon }}\frac{\mathit{\epsilon }}{k}{G}_k-C_{2\mathit{\epsilon }}\rho \frac{{\mathit{\epsilon }}^2}{k}$$(4)

Model constants are: C₁ = 1.44, C₂ = 1.92, σ_ε = 1.3, σ_k = 1.0, C_μ = 0.09.

2.3. Multiphase Model

The Volume of Fluid (VOF) model was employed to solve the multiphase system steel-air. This scheme performs the calculation of the interface between the phases (p and q) present at each cell, based on their fraction as shown.^27,28)   

$${\rho }_{mix}={\alpha }_{\rho }{\rho }_{\rho }+(1-{\alpha }_q){\rho }_{\rho }$$(5)

  

$${\mu }_{mix}={\alpha }_{\rho }{\mu }_{\rho }+(1-{\alpha }_q){\mu }_{\rho }$$(6)

A unique continuity equation for the transient system is derived depending on the number of phases; therefore, the Eq. (7) is divided by the amount of phase q in the cell. Mass exchange between phases can be modelled by introducing an additional source term ( $S_{{\alpha }_q}$ ).   

$$\frac{\partial }{\partial t}({\alpha }_q{\rho }_q)+\nabla \cdot ({\alpha }_q{\rho }_q\overrightarrow{v})=S_{{\alpha }_q}+\sum _{p=1}^n({\dot{m}}_{pq}-{\dot{m}}_{qp})$$(7)

The VOF model solves a single set of momentum transfer equations when two or more phases coexist in the cell.   

$$\begin{array}{c}\frac{\partial }{\partial t}\left({\rho }_{mix}\stackrel{\rightharpoonup }{v}\right)+\nabla \cdot \left({\rho }_{mix}\stackrel{\rightharpoonup }{v}\stackrel{\rightharpoonup }{v}\right)\hfill \\ =-\nabla p+\nabla \left[{\mu }_{mix}\left(\nabla \stackrel{\rightharpoonup }{v}+\nabla \overrightarrow{u}\right)\right]-S_s+S_{\sigma }\end{array}$$(8)

The tracking of the interface is accomplished by an implicit method, which solves the face fluxes ( $\dot{m}$ ) in each grid cell through Eq. (9).   

$$\begin{array}{c}\frac{{\alpha }_q^{n+1}{\rho }_q^{n+1}-{\alpha }_q^n{\rho }_q^n}{\partial t}V+\sum _f\left({\rho }_q^{n+1}{U}_f^{n+1}{\alpha }_{q.f}^{n+1}\right)\\ \hfill =\left[S_{\alpha q}+\sum _{p=1}^n\left({\dot{m}}_{pq}-{\dot{m}}_{qp}\right)\right]V\end{array}$$(9)

2.4. Model Description and Considerations

Figure 1 shows the model dimensions including the geometries of the submerged entry nozzle (SEN) and the mold, where the position and opening of the slide gate remains constant. The inlet velocity is calculated to maintain the desired casting speed at the outlet. A pressure inlet condition is applied at the mold top (P = 101325 Pa, T = 300 K) to simulate the effects of a system open to the atmosphere. The effects of mold oscillating motion were not considered. The casting conditions employed on the simulations are presented in the Table 1. The governing equations are discretized and solved by the computational segregated-iterative method. The non-linear governing equations were linearized using the implicit approach, in combination with the VOF method to define sharp interfaces, and the discretization was performed using the first-order upwinding scheme. The body weighted scheme was used for pressure interpolation; this scheme computes the cell-face pressure by assuming that the normal gradient of the difference between pressure and body forces is constant. The algorithm used for pressure-velocity coupling was that known as SIMPLEC. Convergence criterion was obtained when the residuals of the output variables reached values equal or smaller than 1 × 10^–4. The computational grids, Fig. 2, are around 90% structured with 1100000 cells and these were constructed using GAMBIT^® v2.4.6. All CFD simulation cases are indicated in Table 2.

Fig. 1.

Geometry of mold and SEN in mm, a) Mold upper view, b) Mold lateral view, c) SEN, d) Slide gate, and e) SEN ports.

Table 1. Simulation parameters.

Parameter	Value
Casting speed, (m/min)	1.0
Air zone, (m)	0.1
Nozzle immersion*, (m)	0.14
Viscosity of the liquid steel, (Pa s)	0.0064
Density of the liquid steel, (kg/m3)	7100
Viscosity of the air, (Pa s)	1.7894 × 10–5
Density of the air, (kg/m3)	1.225
Interfacial tension between air and steel, (N/m)	1.6

*  Distance from the free surface to the upper exit port position

Fig. 2.

Computational grid, a) Mold and b) SEN.

Table 2. Simulated cases.

Case	Port type	Inclination angle	Radius (mm)	Code
1	Circular	0°	0	C-0-R0
2	Circular	15°	0	C-15-R0
3	Rectangular	0°	0	Re-0-R0
4	Rectangular	10°	0	Re-10-R0
5	Rectangular	0°	70	Re-0-R70
6	Rectangular	0°	35	Re-0-R35
7	Rectangular	0°	15	Re-0-R15
8	Rectangular	0°	10	Re-0-R10

3. Results and Discussion

3.1. Analysis Inside the SEN

First of all, the steel flow inside the SEN was studied, predicting the velocity profiles at the central symmetrical plane parallel to the narrow mold face, as shown in Fig. 3. In this figure, it is observed a fluid acceleration due to the area reduction produced by the opening of the slide gate; this velocity increment does not accomplish a homogenous profile when it arrives at the port entry level, producing a recirculation pattern in the nozzle pool. It was found that this flow pattern is independent of the port design, but this phenomenon already changes the trajectory of the outlet flow. Then, in order to study the recirculation effect, velocity profiles and velocity contours at the SEN ports exit were predicted for C-0-R0 and C-15-R0 designs. The results are shown in Figs. 4 and 5, respectively; where can be seen that the flow in the right port shows a slide tendency towards the right hand side and the left port have the same behavior but towards the left hand side. This implies that the jets will have a bent towards the outer wall of the mold radius. The same pattern remains even when the port does not have inclination angle, but it is less appreciated because the fluid leaves the port straight forwards. Also, this figure shows that the output flow does not use the total effective exit area of the port since in the upper side edge there is a fluid flowing backwards, this is what is known as backflow phenomenon. Considering the phenomena discussed above, it is expected to have a degree of dynamic distortion of the jets, affecting the whole fluidynamics of the mold as been previously demonstrated.¹²⁾ Therefore, firstly it is necessary to eliminate or reduce as much as possible the described phenomena to achieve a better jets stability. Nevertheless, the slide gate effect can only be eliminated by changing it for another flow control device; this option it may not be feasible, consequently as the objective of the present work indicates, the research is focus on the improvement of the nozzle delivering steel flow by the elimination of the backflow phenomenon.

Fig. 3.

Velocity profiles inside the SEN, a) At the inlet level, b) At the slide gate position and c) At the SEN bottom.

Fig. 4.

Velocity profiles at the SEN ports, a) and b) Original port C-0-R0; c) and d) Original C-15-R0.

Fig. 5.

Velocity contours at the SEN ports, a) and b) Original port C-0-R0; c) and d) Original port C-15-R0.

3.2. Analysis of the Cylindrical and the Rectangular Port Designs

In previous works, the SEN ports were redesign changing the original geometry and considering different inclination angles searching for the optimal design parameters. It was concluded that rectangular port geometry works better than the typical cylindrical port.^13,14) Furthermore, a curve wall at the upper part of the nozzle was also considered to improve the entry flow.²⁶⁾ Nevertheless, in these studies the factors that originate the backflow are not analyzed. Therefore, the first change in the proposed new SEN design was to change the geometry of the port from cylindrical to rectangular, trying to keep the same exit port area.

Following the C-0-R0 and C-15-R0 analysis, the velocity profiles and the velocity contours at the exit rectangular port designs (Re-0-R0 and Re-10-R0) were calculated, these are shown in Figs. 6 and 7, respectively. The results show similar exit flow pattern with a tendency towards the outer wall radius of the mold and the backflow phenomenon at the upper edge of the port. However, it is important to notice that the delivered liquid presents less dispersion comparing to the cylindrical design. In order to compare the flow pattern inside the SEN between the cylindrical and the rectangular cases, velocity profiles at the central symmetrical plane were determined; these are shown in Fig. 8. This figure shows that there is not a significant flow pattern improvement between designs, for this reason a quantitative comparison was necessary. To achieve this, a line at the centre of the exit port, parallel to the vertical SEN axis, was drawn; along the line the flow velocity values were calculated and the results are shown in Fig. 9, where the fraction of the port occupied by the backflow for the four cases is practically the same; subsequently, these results eliminate the port angle as a control variable to reduce this phenomenon. Nevertheless, the velocity values for the rectangular design shows that the backflow fraction is reduced about 5% by considering this port change. It should be noticed that the exit velocity magnitude does not show a considerable change, which does not encourage a significant flow pattern variation in the bulk flow.

Fig. 6.

Velocity profiles at the SEN ports, a) and b) Rectangular port Re-0-R0; c) and d) Rectangular port Re-10-R0.

Fig. 7.

Velocity contours at the SEN ports, a) and b) Rectangular port Re-0-R0; c) and d) Rectangular port Re-10-R0.

Fig. 8.

Velocity profiles at the central-symmetrical plane of the SEN, a) Cylindrical design C-0-R0, b) Cylindrical design C-15-R0, c) Rectangular design Re-0-R0, and d) Rectangular design Re-10-R0.

Fig. 9.

Velocity magnitude along a line located in the outer right port edge, from its upper side to its lower side.

3.3. Application of the Internal Upper Port Radius

From fundamental knowledge of the boundary layer theory,²⁹⁾ previous works^25,26) and the present results, it can be established that there is boundary layer separation at the instant the steel flow reaches the upper internal side of the port, generating a low pressure zone with strong changes in the fluctuant velocity which are reflected in high kinetic energy dissipation, as can be seen in Figs. 10 and 11, respectively. As a consequence of the low pressure, the flow is pulled back into that zone. Consequently, it is evident that in order to reduce or eliminate the backflow phenomenon is necessary to redesign the internal upper side of the port to prevent the boundary layer separation. In this work the modification consisted in the consideration of a radius in the mentioned position, which will allow the flow to continue its trajectory closer to the internal upper wall of the port. In order to study carefully the effect of this modification, the inclination angle of the ports was considered constant, at zero degrees. The biggest applied radius was 70 mm and decreased until 10 mm, (Table 2). The analysis of these cases started comparing the velocity profile calculated at the nozzle central symmetrical plane, shown in Fig. 12. The results show that in a radius range from 70 to 15 mm, the backflow phenomenon is eliminated and this began happening again with a 10 mm radius. To show these results quantitatively the velocity magnitude was calculated considering a line located at the exit port and the results are shown in Fig. 13, where it can be confirmed that using radius bigger than 10 mm the backflow can be avoided. The elimination of this phenomenon allows the total use of the effective exit area of the port by the delivered metal, having as a consequence a decrease of the exit flow velocity. It is also necessary to determine the variation of the low pressure and the high kinetic energy dissipation at the port zone where the backflow take place for the four cases with internal upper port radius. Figures 14 and 15 show respectively these variables, where it can be seen that the low pressure zone disappears, eliminating the suction effect and consequently the energy dissipation strongly decreases; once a small backflow reappears the low pressure gradients emerge again. Observing the consequences of the boundary layer separation the backflow existence was detected; however, due to the grid size and the employed k-e turbulence model, it is not possible to observe the typical vortexes that take place in the zone where there is a boundary layer separation. However, attempting to observe this phenomenon, velocity fixed vectors and the path lines at the close up view on the backflow phenomenon zone were calculated and shown in Fig. 16. The velocity vectors fields show how the flow is not able to follow the SEN wall geometry because of the strong step between the SEN body and the ports (Figs. 16(a) and 16(b)). However, for the reason mentioned above, it is not possible to observe the typical vortexes expected in the position of the boundary layer separation. For that, it is needed to give more attention to the path lines; it is know that if the path lines are closer, the flow velocities are more variable in that zone, and if these are more separate the flow velocities are more structured with fewer changes. According with the results, it can be observed in the upper side of the port that the flow has more variations when the backflow phenomenon is present and those variations decrease when the backflow is eliminated (Figs. 16(c) to 16(f)). From this analysis, it can be established that changing the port design from cylindrical to rectangular with a radius bigger than 10 mm, in the internal upper side of the port, can be sufficient to control and/or eliminate the backflow phenomenon.

Fig. 10.

Pressure contours at the central symmetrical plane of the SEN, a) Cylindrical design C-0-R0, b) Cylindrical design C-15-R0, c) Rectangular design Re-0-R0, and d) Rectangular design Re-10-R0.

Fig. 11.

Kinetic energy dissipation contours at the central symmetrical plane of the SEN, a) Cylindrical design C-0-R0, b) Cylindrical design C-15-R0, c) Rectangular design Re-0-R0, and d) Rectangular design Re-10-R0.

Fig. 12.

Velocity profiles at the central symmetrical plane of the SEN, without inclination angle, a) Re-0-R70, b) Re-0-R35, c) Re-0-R15, and d) Re-0-R10.

Fig. 13.

Velocity magnitude along a line located in the outer right port edge, from its upper side to its lower side.

Fig. 14.

Pressure contours at the central symmetrical plane of the SEN, a) Re-0-R70, b) Re-0-R35, c) Re-0-R15, and d) Re-0-R10.

Fig. 15.

Kinetic energy dissipation contours at the central-symmetrical plane of the SEN, a) Re-0-R70, b) Re-0-R35, c) Re-0-R15, and d) Re-0-R10.

Fig. 16.

Fixed velocity vector and path lines in a close up view of the backflow phenomenon, a) C-0-R0, b) C-15-R0, c) Re-0-R70, d) Re-0-R35, e) Re-0-R15, and f) Re-0-R10.

3.4. Effects of the Backflow Elimination on the Bulk Flow

Now that it is known the groundwork of the backflow phenomenon and how can be eliminated, it is necessary to establish the benefits of its elimination on the bulk flow. The analysis starts with one of the greatest benefits achieved, being able to use the total of the effective exit area of the port. This promotes a jets velocity decrement which will affect the patterns of the whole mold. Having in mind this, a further analysis of the most transcendent variables, such as global velocity profiles in the mold, the meniscus oscillation and the impinging jets point was carried out. Figure 17 shows the velocity profiles at the central symmetrical plane of the mold, Figs. 17(a) and 17(b) for the cylindrical SEN cases and Figs. 17(c) to 17(f) for the rectangular cases with a radius at the internal upper side of the port; it can be observed that all cases present the typical two roll flow characteristics for a slab mold. For the C-0-R0 and C-15-R0 cases the jets are compact and its trajectory is strongly affected by the inclination angle port, and the meniscus fluctuation is higher when the port does not have inclination angle. For the rectangular cases the results illustrate more dispersed jets with a decrement on its velocity; in addition, the jets penetration angle shows a direct dependency of the internal upper port radius. This has as consequence that the impinging jet point and the velocity at the meniscus are also radius dependent. In Figs. 17(c)–17(f) can be observed that the jets start to compact and its angles increase as the internal upper port radius decreases until the backflow reappears (Fig. 17(f)). In order to observe in more detail the effect on the impinging point, it was determined for all cases and the results are shown in Fig. 18. Figure 18(a) shows that changing the port geometry from cylindrical to rectangular do not show any particular change in the impinging point position. However, Fig. 18(b) shows that for rectangular port cases with internal port radius, as the radius increases the impinging point position gets deeper. Therefore, the flow velocity at the meniscus is expected to be smaller inducing a better stability. In order to prove this, the maximum positive and negative meniscus deformation were predicted and are shown in Fig. 19. Following the same idea of the previous figure, Fig. 19(a) shows a comparison between the cylindrical and the rectangular port designs, where the values for the meniscus deformation are similar as could be expected since they show almost the same impinging points. On the other hand, when the internal upper port radius is applied and the backflow is eliminated, the meniscus deformation values decrease as a function of the increment in the radius, as shown in Fig. 19(b).

Fig. 17.

Velocity profiles at the central symmetrical plane of the mold, a) C-0-R0, b) C-15-R0, c) Re-0-R70, d) Re-0-R35, e) Re-0-R15, and f) Re-0-R10.

Fig. 18.

Impinging point of the jets at the narrow mold walls for all cases, a) Without radius and b) With different radius values.

Fig. 19.

Maximum meniscus deformation for all cases, a) Without radius and b) With different radius values.

According to the complete analysis exposed above, it can be concluded that eliminating the backflow phenomenon allows the use of the complete effective exit area of the port, which is reflected in a velocity decrease of the jets and consequently a velocity decrement of the bulk flow. Furthermore, the size of the radius controls the penetration angle of the jets, the impinging point position and the meniscus deformation; which may avoids the need of the inclination angle of the port. Finally, it is determined that the better cases to replace the cylindrical original designs are as follow: Re-0-R10 for C-0-R0 and Re-0-R15 for C-15-R0.

The present study confirms that mathematical simulation is a powerful tool for the evaluation of new SEN designs at low cost and time; this always should to be considered during the process design for optimal results. Nevertheless, the final proposed designs require a physical validation, which is considered a further work by the authors.

4. Conclusions

In order to understand the backflow phenomenon and its controlling parameters, a mathematical simulation of the flow patterns in the mold and inside the SEN was carried out considering a typical cylindrical nozzle design and the different variations applied to the exit port. The following conclusion can be drawn from the simulation results and its analysis:

(1) The backflow phenomenon emerges from an inadequate SEN port design where a separation of the boundary layer is generated before the steel is delivered.

(2) The boundary layer separation at the upper internal zone of the port induces a low pressure zone with high levels of kinetic energy dissipation, producing flow suction backwards to the upper edge of the port which generates the known backflow phenomenon.

(3) Modifying the port design from cylindrical to rectangular allows the compaction of the delivered jets flow, but it does not diminish the backflow intensity.

(4) The implementation of a radius at the internal upper side of the port avoids the boundary layer separation, eliminating the low pressure zone and the high dissipation; this allows the total use of the effective exit area of the port reducing the jets velocity and consequently the bulk flow velocities are reduced with a more stable flow pattern.

(5) Increasing the internal upper radius of the port, the jets impinging point is deeper and the meniscus deformation smaller. Therefore, with the implementation of this radius, it is possible to control the jets penetration angle, which may eliminate the need of the port inclination angle.

Acknowledgements

The authors gratefully acknowledge the National Council of Science and Technology (Mexico) and the institutions IPN, DGEST, ITM, SNI, and PROMEP for all support given to carry out this work.

Nomenclature

∇: Gradient, velocity change on the coordinate axes (x, y, z)

u, v, w: Velocities on the coordinate axes (x, y, z), m/s

P: Pressure, Pa

μ, μ_t, μ_eff: Dynamic viscosity, turbulent and effective, Pa s

ρ: Fluid density, kg/m³

C_μ: Viscosity constant

k: Turbulence kinetic energy, m²/s²

ε: Dissipation of turbulent kinetic energy, m²/s³

G_k: K due to generation of the mean velocity gradients

σ_k σ_ε C_μ C₁C₂: Constants model k-ε

ρ_mix: Mixed Phase density, kg/m³

μ_mix: Mixed Phase viscosity, Pa s

α: Volume fraction, %

α_p α_q: Volume fraction of the p_th and q_th phases, %

μ_p α_q: Viscosity fraction of the p_th and q_th phases, %

S_αq: Source term or constant of VOF model

ρ_q ρ_p: Density fraction of the q_th and p_th phases, kg/m³

m_pq m_qp: Mass exchange between p and q phases, kg/s

S_sS_σ: Exchange rate of the stress tensor

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