Transient motion of inclusion cluster in vertical-bending continuous casting caster considering heat transfer and solidification

Powder Technology 287 (2016) 315–329

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Transient motion of inclusion cluster in vertical-bending continuous casting caster considering heat transfer and solidification Zhongqiu Liu, Baokuan Li ⁎ School of Materials and Metallurgy, Northeastern University, Shenyang, 110819, China

a r t i c l e

i n f o

Article history: Received 20 July 2015 Received in revised form 10 October 2015 Accepted 16 October 2015 Available online 19 October 2015 Keywords: Inclusion cluster Large eddy simulation Heat transfer Solidification Continuous casting mold

a b s t r a c t A coupled three-dimensional finite-volume computational model has been developed to simulate the transient fluid flow, heat transfer and solidification processes in a vertical-bending continuous casting caster. The turbulence of molten steel inside the liquid pool is calculated using the large eddy simulation (LES). The enthalpy–porosity approach is used to simulate the heat transfer and solidification of steel in the caster. Based on the fractal theory and the conservation of mass, a kind of inclusion cluster model was developed. A new criterion was developed using the user-defined functions to model the motion and entrapment of inclusion cluster in the caster based on the Lagrangian approach. Firstly, the predicted growth of solidified shell was compared with the plant measurements, and the asymmetrical flow pattern was compared with the dye-injection observations of water model experiments. Secondly, the validated model was used to predict the instantaneous motion and entrapment distribution, statistical data, escape and entrapment positions of different inclusion clusters in the caster. Many known phenomena and other new predictions were reproduced in this part, and the center inclusion band defects in the steel plates found by the UT method can be interpreted using the current model. Finally, two methods were proposed to optimize the inclusion cluster motion and entrapment in the caster. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Continuous casting (CC) has been widely developed as the most important production process in the steel industry. In the casting process, molten steel flows from a ladle, through a tundish into a mold. Inside the mold, molten steel freezes against the water-cooled copper mold walls and forms a solid shell. Wide-Thick-Slab (WTS) CC technology [1] is presented for the production of slabs with wide ranging from 1500–3000 mm and thicknesses ranging from 200–400 mm. Combining the advantages of conventional CC technology, the WTS-CC can meet different production targets such as high casting capacities, highquality special steel production and highly economical plate production. And this casting process is highly flexible and well suited for casting slabs used for a wide range of applications, such as large marine engineering and shipbuilding, large bridge, large pressurized vessel, nuclear equipment, and so on. Through the LF, RH and tundish refining, most of large argon bubbles and non-metallic inclusions have been removed and the purity of molten steel has been raised [2,3]. However, many smaller bubbles and inclusions still stay inside the molten steel, would be carried deep into the mold and collide together to form larger clusters, and finally these large clusters would be entrapped by the solidified shell forming ⁎ Corresponding author at: No. 3-11, Wenhua Road, Heping District, Shenyang, 110819, China. E-mail address: [email protected] (B. Li).

http://dx.doi.org/10.1016/j.powtec.2015.10.025 0032-5910/© 2015 Elsevier B.V. All rights reserved.

defects, as shown in Fig. 1 which were the ultrasonic flaw detection maps (three cross sections: casting direction section, transverse section, and side section respectively) of two rolled steel plates, obtained from a vertical-bending WTS-CC caster. As can be observed from the visual analysis, the etch-pits, viewed as dots are the entrapped argon bubbles and non-metallic inclusions. There are many defects along the casting direction in the first plate, Fig. 1(a), most of them located at the thick center of the plate, be called “center inclusion band”. Compared with the result of Fig. 1(a), the number of inclusions decreases as shown in Fig. 1(b), however it can be seen that there is a macro cluster inclusion found inside this steel plate. Then this macro inclusion cluster is magnified into view, as shown in Fig. 1(c); it can be seen that the outside dimension of this macro inclusion is 5 × 300 × 2151 mm. And it is composed with various smaller inclusions, such as Al2O3 inclusions. However, according to many previous works [4–6], most of monomer bubbles and inclusions would locate at the inner-curved section of the vertical-bending continuous caster due to the buoyancy of bubbles and inclusions. Previous researches would not be able to explain the reason for the center inclusion band defect which locates on the center of slabs. Turbulent flow in the mold is important to the steel quality, because it influences the transport and entrapment of inclusions and bubbles, the fluctuation and shape of slag–metal interface, the transport and dissipation of superheat, the entrainment of mold flux at the top surface, and the growth of initial solidified shell. According to the recent plant observations and detections, the distribution of argon bubbles and

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Fig. 1. Ultrasonic testing of different steel plates.

non-metallic inclusions captured by solidified shell in the rolled steel plates was intermittent and asymmetric [7,8]; suggesting that the fluid flow inside the mold was unsteady and instable. Several physical modeling studies [7–10] have been carried out to reproduce the asymmetrical and oscillating fluid flow inside the casting mold. However, experimental measurements on an actual slab continuous casting machine are very difficult, dangerous and expensive. Numerical modeling provides an alternative tool to understand and solve this kind of problem. Most of the reported mathematical models have been performed using the Reynolds averaged Navier–Stokes (RANS) models [2,11,12],

such as the k-ε or Reynolds stresses. These models predict timeaveraged velocities with reasonable accuracy and at a reasonable computational. However, these models, limited by the RANS's nature, are not suited for modeling the evolution of transient flow pattern triggered by flow instabilities. In recent studies [7,8,13,14], the large eddy simulation (LES) has been successfully applied to obtain the transient asymmetrical flow inside the mold. Authors developed different LES models for the single phase (molten steel) flow [8] and the two-phase (molten steel and argon gas) flow [7] inside the slab mold, respectively. Both of simulation results agree acceptably well with the water model

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experimental measurements of instantaneous flow structures. The flow pattern in the lower recirculation zone is asymmetric and changeover; the direction of flow deviation is different from time to time. However, except authors, relatively little work has been reported on the transient asymmetrical flow inside the solidified shell of the mold using LES. The information is important for finding some effective methods to improve the quality of wide-thick-slab. Except turbulent flow, the motion and entrapment of inclusions are also affected by heat transfer, cooling rate, characteristics of the solidification front (dendrite arm spacing), and so on. Some model experiments were carried out to investigate the motion and entrapment of argon bubbles and inclusions on the solidified shell [15–20]. Mukai and Lin [15] proposed a velocity of small inclusion caused by interfacial tension gradient in the boundary layer of concentration formed in front of an advancing solid–liquid interface. Through a series of model experiments using molten steel, Ohno and Miki [16,17] found that the entrapment of inclusions at the solid–liquid interface was enhanced with increasing S concentration, and was reduced by the existence of a low velocity flow, e.g., 0.05 m/s. Experimental studies [18] have found that the morphology of the solidification front has a significant effect on the entrapment of inclusions, because it decides the micro-forces acting on inclusions and the inter-dendritic fluid flow. When the inclusions small enough to enter the gap between two dendrite arms, they would be entrapped. Due to the limitations of physical modeling, it is difficult to design an experimental study on the entrapment of inclusions and bubbles by the solidified shell. Many mathematical simulation studies [19–21] have focused on the entrapment of inclusions in the solidified shell of steel during continuous casting. Thomas et al. [19] considered normal and tangential force balances involving ten different forces acting on a particle in the boundary layer, and the primary dendrite arm spacing (PDAS) was considered for the particle entrapment. Zhang et al. [20] developed a three-dimensional numerical model to simulate the fluid flow, solidification, and motion of inclusions in runner steel, and the results were compared with the measurement. Lee et al. [21] studied the effect of thermal Marangoni force for the behavior of argon bubbles at the solidifying interface, revealing that the thermal Marangoni force could play an important role in the entrapment of argon bubbles. Unfortunately, there is not a unified and accurate capture criterion, because the capture mechanism of inclusions at the solid–liquid interface is not fully understood. The current work develops a LES model for investigating the transient turbulent flow, heat transfer, solidification, and particle transport and entrapment in the continuous casting mold. An inclusion cluster model is described based on the fractal theory and mass conservation. Then the models are applied to investigate the various important phenomena on inclusion cluster motion inside the mold. The predicted results of this model are compared with the plant measurements of the ultrasonic testing of the rolled steel plates, growth of solidified shell thickness, and the water model experiments.


 ∂ui ∂ ui u j 1 ∂p ∂ ¼− þ þ ρ ∂xi ∂x j ∂t ∂x j

ðυ þ υt Þ

∂ui ∂u j þ ∂x j ∂xi

!! þ Sm

ð2Þ

where u, t, P, ρ, and υ are the velocity, time, pressure, density, and kinematic viscosity respectively. The superscript “−” represents filtered. The subscripts i and j represent the three Cartesian directions and repeated subscripts imply summation. The terms on the right-hand side of Eq. (2) are respectively representing the pressure gradient, the stress, and the momentum source term. According to Smagorinsky SGS model [22], the turbulent viscosity term υt is described as follows: υt ¼ L2s jSj

ð3Þ

where Ls is the mixing length for sub-grid scales; Ls = min (kvdw, CsΔ), Δ = (ΔiΔjΔk)1/3 is the filter width; S is the characteristic filtered rate of pffiffiffiffiffiffiffiffiffiffiffiffi ∂u ∂ui strain, S ¼ 2Sij Sij, Sij ¼ 12 ð∂x þ ∂x j Þ. And kv is the von Kármán constant, j

i

kv = 0.42; dw is the distance to the closest wall; Cs is the Smagorinsky constant, Cs = 0.1, which has been successfully used to calculate the flow field in the mold [7,8]. The momentum source term (Sm) was given as follows [23]: Sm ¼ −

υ ðui −uc Þ Km

ð4Þ

where uc is the casting speed of slab; Km is the liquid phase permeability coefficients, which can be calculated using Carman–Kozeny equation [23]: 3

Km ¼

fl þ ξ 2

Dm ð1−f l Þ

:

ð5Þ

Here, fl is the volume fraction of liquid phase; ξ is a small constant (0.00001); Dm is a model constant, which depends on the morphology of porous media. 2.2. Enthalpy–porosity model Instead of tracking the liquid–solid front explicitly, the enthalpy– porosity formulation is used to simulate the heat transfer and solidification of steel in the continuous casting mold. The liquid–solid mushy zone is treated as a porous zone with porosity equals to the liquid fraction. The enthalpy (H) of the material is computed as the sum of the sensible enthalpy, h, and the latent heat, ΔH: Z H ¼ h þ ΔH; h ¼ href þ

2. Mathematical model formation

T

T ref

cp dT; ΔH ¼ f l
L:

ð6Þ

The liquid fraction, fl, can be defined as

2.1. Large eddy simulation Turbulent flows are characterized by eddies with a wide range of length and time scales. LES is an effective approach which solves for large-scale eddies directly and uses a sub-grid scale (SGS) model for the small-scale eddies, which are smaller than the finite volume and therefore filtered during the simulation. In the current work, in order to separate out the large-scale eddies from the flow; the box type filtering function is used. Then the governing equations for transient turbulent flow are generally given as follows: ∇
ðρui Þ ¼ 0

317

ð1Þ

8 > > <

fl ¼ 1 T−T solidus fl ¼ > T liquidus −T solidus > : fl ¼ 0

; if TNT liquidus ; if T solidus bTbT liquidus

ð7Þ

; if TbT solidus

The latent heat content can vary between 0 (for a solid) and L (for a liquid). For solidification/melting problems, the energy equation is written as ∂ðρl H Þ þ ∇
ðρl ui HÞ ¼ ∇
ðk∇T Þ þ Se ∂t

ð8Þ

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where Se is source term of energy, Se ¼ ρl Luc ð1−f l Þ−ρl L

∂f l : ∂t

ð9Þ

Here, uc is the solid velocity due to the pulling of solidified material out of the domain (here referred to as the casting speed). 2.3. Particle transport and entrapment model Since the volume fraction of the inclusion clusters is expected to be small, the effect of the inclusions on the fluid flow can be neglected, and the particle transport equations can be calculated decoupled from the flow equations. Here, the Lagrangian approach [24] is used to calculate the transport of inclusion clusters. The motion of inclusion clusters can be simulated by integrating the following transport equation for each inclusion cluster: mp

duP ¼ F g þ F B þ F P þ F D þ F S þ F VM þ F M þ F Ma dt

ð10Þ Fig. 2. Schematic of solid–liquid coexisting zones during the solidification process.

where, the terms on the right hand side of Eq. (10) are gravitational force, buoyancy force, pressure gradient force, drag force, Saffman lift force, virtual mass force, Magnus force and Marangoni force. All forces in Eq. (10) are expressed as follows, and details can be seen in previous work. [25]  Fg þ FB ¼

 3 ρp −ρl πdp 6

ð11aÞ

g

3

FP ¼

ρl πdP Du 6 Dt

ð11bÞ

FD ¼

 
 1 2 πdp ρl C D u−up  u−up 8

ð11cÞ

6K s μ eff FS ¼ CL
ρp πdp F VM ¼ ρl C VM

FM ¼

ρl μ eff

!1=2





u−up



 π 3 du dup dp − 6 dt dt

 2 1 2 πd ρ C M ul −up  8 p l

 2 2 ∂σ dT ∂σ dC þ : F Ma ¼ − πdp
3 ∂T dx ∂C dx

ð11dÞ

ð11eÞ

ð11fÞ ð11gÞ

Fig. 2 shows a schematic illustration of solid–liquid coexisting zone of solidifying steel. The arm spacing has a great effect on the motion and entrapment of inclusion. The first dendrite arm spacing and secondary arm spacing decide the micro forces acting on inclusions and the fluid flow condition, thus affect the inclusion entrapment. In the current model, if the size of inclusion is larger than the first arm spacing, the dendrites would push the inclusion rather than entrap it. According to the work of Yamazaki et al. [26], the solid–liquid coexisting zone was classified into three zones of q1, q2, and q3 base on the fluidity of inter-dendritic liquid. In zone q1 (fl N 0.6), the primary crystals and molten steel can flow concurrently; and only inter-dendritic molten steel can flow among dendrites in the zone q2 (0.3 b fl ≤ 0.6); no flow in the zone q3 (fl ≤ 0.3). So the inclusions would be captured when they move into the zone q2. The motion and entrapment of inclusion cluster in the current model were controlled by an own-programmed user-defined function (UDF), the flow chart was shown in Fig. 3. When the temperature of molten steel is higher than liquidus temperature (1786 K), inclusion clusters

can transport in the liquid pool, and the motion of inclusion cluster is governed by the gravitational force, buoyancy force, pressure gradient force, drag force, lift force, virtual mass force and Magnus force. However, the inclusion cluster would be pushed to the mushy-zone when the direction of resultant force points to the mushy-zone, where the temperature is between liquidus temperature and solidus temperature (1730 K). In addition to the forces described above, one extra force is exerted on inclusion cluster, which is the Marangoni force. Finally, inclusion cluster would be captured if the liquid fraction at the location of particle is below 0.6. Then the entrapment locations of inclusion clusters would be exported to a separate file using another UDF. 2.4. Inclusion cluster model Due to the high interfacial energy, the shape of aluminum oxide inclusions was dendritic under the condition of high oxygen concentration [27]. Therefore, through the collision and aggregation of many separate individual inclusions, it is easily to form larger inclusion cluster, as shown in Fig. 4(a), the aluminum oxide inclusion cluster was found in a steel casting slab [28]. However, the size and density of inclusion are important to its motion and entrapment. Based on the fractal theory, Tozawa et al. [29] obtained the relationship of the measured diameter of an inclusion cluster (Dc) with the number (N) of small individual alumina inclusions (with diameter dp) in one cluster. 1:8

N ¼ D1:8 c =dp :

ð12Þ

According to the conservation of mass, making the equivalent inclusion clusters into aluminum oxide inclusions, its equivalent diameter (de) can be given as follows [30]: 3

3

de ¼ Ndp :

ð13Þ

Then the diameter of inclusion cluster can be calculated as: 5=3 −2=3

Dc ¼ de dp

:

ð14Þ

In order to estimate the density of inclusion cluster (ρc), the following equation is used [31]: ρc ¼ ð1−βÞρl þ βρAl2 O3

ð15Þ

where β is the volume fraction of alumina inclusions; ρAl2O3 is the density of alumina inclusions.

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319

Fig. 3. Flow chart of the motion and entrapment criterion.

Based on the analysis above, the inclusion cluster model can be simplified as a globe, as shown in Fig. 4(b), which was used in the current calculation. From Eq. (15), it can be seen that the density of inclusion cluster is almost the same as that of molten steel when the β is set to a small constant, as reported by Asano and Nakano (β = 0.03) [31], which was measured from some agglomerated inclusion clusters. But Miki and Thomas [2] suggest that the average cluster density is 5000 kg/m3, when the clusters embed about equal volumes of steel and Al2O3 (β = 0.5). In order to study the effect of β on the transient motion and entrapment of inclusion cluster, various β values (0.03 to 1.0) were analyzed based on the flow fields of LES inside the liquid pool. 2.5. Numerical details The considered continuous casting caster have a vertical, straight mold (800 mm length), a straight section of secondary cooling zone (1900 mm length), and a curved section of secondary cooling zone (with a radius of 10.25 m). In the current work, two calculation models were adopted: Case-1 is 23.5 m length (Fig. 5) and Case-2 is 6 m length which was given in author previous works [25]. Details of geometry and operating conditions for Case-1 and Case-2 have been shown in Table 1. The boundary conditions for heat transfer and solidification solution are as follows: (1) a constant liquidus temperature 1786 K is set at the top surface; (2) heat flux on wide and narrow wall of the actual mold is a function of distance down the mold, as shown in Eq. (16), which is similar to the form proposed by Savage and Pritchard [32], ψ is constant for wide and narrow wall which is 0.275 and 0.295 respectively; (3) convection heat transfer occurs in the walls of the secondary cooling zone of the continuous caster, and the average heat transfer coefficient

for wide and narrow face is 350 and 300 W/(m·K), respectively. (4) Both the initial and inlet temperature of molten steel are 1801 K. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ 2:68−ψ 60hm =V C :

ð16Þ

However, LES approach needs to run for a sufficiently long time to obtain a stable statistic flow field. The computational cost involved with LES is normally orders of magnitudes higher than that for steady RANS calculations in terms of memory and CPU time. In order to save the calculation time, this calculation was first carried out using the standard k-ε turbulence model to obtain a steady flow field. Running until the flow field is reasonably converged and then run LES until the flow becomes statistically steady. The best way to see the fully developed flow and statistically steady is to monitor solution variables (e.g., velocity components) at selected locations in the flow. Additionally, it will help in reducing the time needed for the LES simulation to reach a statistically stable mode. The time step size for LES is 2.5 × 10−4 s, which is determined by the criterion that the maximum Courant–Friedrichs–Levy (CFL) number must be less that one (Δt ≤ Δz/ |u|). Various dispersed inclusion clusters with sizes of 5 to 500 μm are injected randomly into the SEN in 1 s. The number for each size group is 3000. The initial locations are uniformly distributed at the inlet surface. It is assumed that particles are reflected once touching the walls inside and outside the SEN, and escape once reaching the top surface, and leave the system once entering the bottom of the domain. In the current work, these particles which flow out of the calculation region bottom with the outflow were considered to be captured at a deeper position.

Fig. 4. Plant inclusion cluster (a) [28] and schematic of inclusion cluster model (b).

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Fig. 5. Calculation model of case-1 and predict the solidification end location.

3. Model validation 3.1. Growth of solidified shell In order to validate the heat transfer and solidification model used in the current work, a calculation model contains solidification end location was considered, the total calculation length of Case-1 is 23.5 m, as shown in Fig. 5. And in this figure, the shape of solidified shell (0.33 liquid fraction isosurface) inside the calculation domain with casting speed of 1.2 m/min was also shown. The predicted results of the current model were compared with the plant measurement results, as shown in Table 2, it is found that the predictions of the model agree well with the measured solidification end location, and the error between the predicted result and the measurements is within 3%. So the current model can Table 1 Geometry and thermo-physical properties for the simulation conditions. Parameters

Values (Case-1 and Case-2)

SEN submergence depth, mm SEN outport angle, (°) SEN outport sizes, mm Width of mold, mm Thickness of mold, mm Length of actual mold, mm Radius of curvature, mm Length of vertical section of secondary cooling zone, mm Length of bending section of secondary cooling zone, mm Casting speed, m·min−1 Molten steel density, kg·m−3 Single inclusion density, kg·m−3 Molten steel viscosity, kg·m−1·s−1 Specific heat at constant pressure, J·kg−1·K−1 Enthalpy of fusion, kJ·kg−1 Solidus temperature, K Liquidus temperature, K

300 15 downward 80 × 70 1728 230 800 10,250 1900 3400 for Case-1; 20, 800 m for Case-2 1.2 7020 3500 0.0056 710 270 1730 1786

accurately predict the solidification end location in the vertical bending caster. However, it is very time consuming with LES model to predict the transient turbulent flow, heat transfer, solidification in the entire continuous casting caster (Case-1). In order to save the calculation time, Case-2 is used to calculate the multi-physics fields and particle transport inside the caster. According to the previous validation, the model can reasonably predicts the trend of the thickness of solidified shell over the entire Case-2 domain. The thickness of solidified shell at the three referenced locations, yield an average value of 17 mm at the actual mold end(0.8 m below top surface); 32 mm at the curved part of the secondary cooling zone(2.7 m below top surface); 48 mm at the outlet of the calculated domain. The thickness of solidified shell is very important, because it controls the entrapment positions of bubbles and inclusions. 3.2. Asymmetrical flow inside the liquid pool Plant observations have found that the defects in the steel plates are intermittent, suggesting that they are related to fluid flow transient and asymmetry. In order to visualize the flow pattern and validate the mathematical model, a one-third-scale water model was established, including tundish, SEN and mold (contain the secondary cooling zone), and the detail parameters were shown in author's previous work [25]. Black tracer was injected to the bottom of a tundish to observe the flow pattern in mold. The transient mixing phenomena were recorded by a video camera. Fig. 6(a) and (b) shows the flow pattern obtained Table 2 Comparison between the predicted results and the plant measurements. Casting speed

Parameters

fs = 0.15

fs = 0.7

fs = 1.0

1.2 m/min

Plant measurements (mm) Model predictions (mm) Error (%)

17741 18200 2.59

21471 21950 2.23

22259 22700 1.98

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321

Fig. 6. Fluid flow pattern obtain from the water model experiment (a), (b) and the LES model (c), (d) [25].

from the water model experiment (water flow rate was 40.08 L/min). The left recirculation domain in the lower recirculation zone was much bigger than the right one in one of the repeated experiments, as shown in Fig. 6(a). After some time, in Fig. 6(b), it changed and became a mirror image of the previous one. The transient flow pattern inside the liquid pool obtained from this mathematical model (steel flow rate was 636 L/min; the casting speed was 1.2 m/min) is shown in Fig. 6(c) and (d). The LES results further reveal the significant flow asymmetry in the lower recirculation zone of the mold, the shape of the jets and the upper and lower recirculation zones agree well with the dye-injection observation.

4.2. Instantaneous motion and entrapment of inclusion clusters In the current work, the obtained transient fluid flow field of 100 s was used as the initial field to investigate the inclusion cluster behavior. Fig. 9 shows the distribution of the inclusion clusters (β = 0.5, Dc = 50 μm) inside the liquid pool at different times. The purple isosurface represents the 0.6 liquid fraction of steel. The blue dot represents the

4. Results and discussion 4.1. Typical instantaneous flow field Fig. 7 shows a typical instantaneous velocity field in the center-plane parallel to the wide face inside the liquid pool at an arbitrary time of 100 s. The growth of solidified shell thickness is also shown in this figure. It can be seen that the molten steel flow is very chaotic in the upper recirculation zone of the mold. Molten steel emerges from the SEN port as a jet, diffuses as it traverses across the liquid pool, and splits into two recirculation zones after impinging on the narrow face of the mold. Fig. 8(a) to (d) reveals the velocity profiles in various transverse sections taken at four locations down the calculation domain. The solidified shell thickness increases with increasing the strand length. The velocity profiles are inhomogeneous in these sections of the strand. Many small multiple vortices are found in these flow zones, and their positions and intensity are ever-changing. In addition, those vortexes make the flow field in the mold more complex. And these vortices are important to the transport of particles, because the rotary motion of molten steel can spin the particles. The predicted flow field results also reveal that the growth of solidified shell has an important impact on the fluid flow and should not be neglected.

Fig. 7. Transient velocity fields inside the liquid pool (Y–Z center-section).

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Fig. 8. Transient velocity fields inside the liquid pool (X–Z cross-sections) (a) −0.005 m, (b) −0.08 m, (c) −2.7 m, and (d) −5 m below the top surface.

inclusion clusters. It can be seen that most inclusion clusters have flowed into the mold with the strong downward steel jet at the SEN ports, in Fig. 9(a), at 1.5 s after injection. Then 1.5 s later, some of them can rise to the meniscus and are removed, while most of them would continue following the jet flow and disperse in the vortices, as shown in Fig. 9(b). At 10 s after injection, in Fig. 9(c), many inclusion clusters are well-dispersed throughout the upper recirculation zone, and other inclusion clusters are branched forming two downward streams. In Fig. 9(d), by 30 s after injection, some inclusion clusters have moved to the curved part of the secondary cooling zone from the right side of the caster, which is caused by the asymmetrical flow. At 100 s after injection, in Fig. 9(e), some inclusion clusters have flowed out of the calculated domain, and it is difficult for these inclusions to float to the top surface again, so they would be continue moving inside the deeper liquid pool and captured at some deeper locations of this caster.

In this model, inclusion clusters touch 0.6 liquid fraction isosurface are always assumed to be entrapped, and the number of entrapped inclusion clusters and the positions of every inclusion clusters are recorded using the UDF. Fig. 10 shows the three-dimensional entrapment positions of inclusion clusters (β = 0.5, Dc = 50 μm) on the solidified shell at various times corresponding to Fig. 9. The blue isosurface represents the 0.6 liquid fraction of steel. The entrapped inclusions are represented with red dots. Corresponding to Fig. 9(a), at 1.5 s, some inclusions are found in the regions near the SEN ports, as shown in Fig. 10(a). 3 s after injection, in Fig. 10(b), more inclusions are entrapped along the jet directions. With the development of time, 10 s later (Fig. 10c), many inclusions had been entrapped at the upper region of the mold, and some of them are located adjoining the narrow face of the mold from the thickness direction, because the shell thickness at wide face of the mold is thin. At 30 s, in Fig. 10(d), several inclusions had been found at the curved part of the secondary cooling zone. Most

Fig. 9. Transient inclusion cluster distribution inside the liquid pool (a) 1.5 s, (b) 3.0 s, (c) 10 s, (d) 30 s, and (e) 100 s.

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323

Fig. 10. Transient entrapment positions of inclusion clusters on the solidified shell (a) 1.5 s, (b) 3.0 s, (c) 10 s, (d) 30 s, and (e) 100 s.

of entrapped inclusions are located within the scope of 1 m below the top surface, less entrapped inclusions are found at the lower region of the domain. By 100 s after injection (Fig. 10e), more inclusions are entrapped at the curved part of the secondary cooling zone of the mold.

4.3. Statistics of inclusion clusters motion Fig. 11 shows the transient removal ratio of different inclusion clusters (β = 0.03and β = 1.0, respectively) from the top surface of the mold. Inclusion clusters can reach the top surface about 2.8 s after the first injection. The change of removal times for β = 0.03 inclusion clusters is the same as β =1.0 inclusion clusters. Two concentrated periods for inclusion removal from top surface are 2.8–5 s and 5–20 s, which indicates that most inclusions escape from the top surface with the upper recirculation jet during 2.8 to 5 s after injection; then some inclusions that are well dispersed in the upper roll gradually float to the top surface during 5 to 20 s. However, few inclusions can be removed from the top surface after 20 s injection. Compared the removal ratio between β = 0.03 and β = 1.0 inclusion clusters, some clear differences can be found. For β = 0.03 inclusion clusters, as shown in Fig. 11(a), the removal ratio decreases with increasing inclusion diameters when the inclusion is smaller than 100 μm; and then increases with increasing inclusion diameters when the inclusion is larger than 100 μm. Due to the small density difference between molten steel and inclusion clusters, the buoyancy of β = 0.03 inclusion clusters is weak, only 10 to 17% of the β = 0.03 inclusion clusters in the range from 5 to 500 μm float to the top surface of the mold. For β = 1.0 inclusion clusters, in Fig. 11(b), the removal ratio decreases with increasing inclusion diameters when the inclusion is smaller than 50 μm, and only 10 to 12% of these small inclusion clusters can be removed from the top surface. Due to the larger buoyancy of greater inclusion clusters (larger than 100 μm), an increasing amount of the inclusions is removed from the top surface. The removal ratio of inclusions from the top surface increases from 20 to 50% when the diameter of inclusion cluster increases from 100 to 500 μm. The total removal ratio of larger inclusions for β = 1.0 inclusion clusters is about 3 times as that of β = 0.03 inclusion clusters. The entrapment histories of different inclusion clusters (β=0.03 and β =1.0, respectively) by the solidified shell are shown in Fig. 12(a) and

Fig. 11. Removal ratio from the top surface for different inclusion clusters (a) β=0.03 and (b) β=1.0.

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Fig. 12. Entrapment ratio by solidified shell for different inclusion clusters (a) β=0.03 and (b) β=1.0.

Fig. 13. Escape ratio from the outlet for different inclusion clusters (a) β=0.03 and (b) β=1.0.

4.4. Escape and entrapment positions of inclusion clusters (b). Most inclusion clusters have been entrapped by the solidified shell approximately 50 s after the first injection, where the local solidified shell thickness is smaller than 17 mm. For β=0.03 inclusion clusters, in Fig. 12(a), the final entrapment ratio by solidified shell increases with increasing the inclusion diameter from 5 to 100 μm, and decreases slowly when the inclusion diameter is larger than 100 μm, because larger inclusion clusters would escape from the top surface. For β = 1.0 inclusion clusters, in Fig. 12(b), the final entrapment ratio by solidified shell increases with increasing the inclusion diameter from 5 to 50 μm, and decreases with increasing inclusion diameters when the inclusion is larger than 50 μm. Especially for the 500 μm inclusion diameters, only 48% of them is entrapped by the solidified shell. Fig. 13 shows the transient escape ratio of different inclusion clusters (β = 0.03 and β = 1.0, respectively) from the outlet of the calculation domain. It can be seen that some inclusion clusters could escape from the bottom of the domain until 62 s after injection. The change of escape times for β = 0.03 inclusion clusters is the same as β = 1.0 inclusion clusters. Due to the larger buoyancy of large inclusion clusters, the final escape ratio of inclusion cluster decreases with increasing the inclusion diameters from 5 to 500 μm. The difference of final escape ratio between the β =0.03 inclusion clusters and β =1.0 inclusion clusters is small when the inclusion diameter is smaller than 100 μm. But there is obvious difference for larger inclusion cluster (≥ 200 μm), the final escape ratio of the 500 μm β = 0.03 inclusion clusters is 11%, the value of the 200 μm β = 1.0 inclusion clusters is 6%. However, no β = 0.03 inclusion cluster can escape from the outlet of the domain when the diameter is larger than 200 μm.

Fig. 14(a) and (b) shows the β = 0.03 inclusion clusters final escape positions on the top surface after 200 s injection respectively for 10 μm and 200 μm. The results indicate that most inclusion clusters tend to escape from the top surface closer to the narrow wall of the mold (−0.86 to −0.4 m and 0.4 to 0.86 m). The 10 μm and 200 μm inclusion cluster (β = 1.0) final escape positions on the top surface have been shown in Fig. 14(c) and (d). It can be seen that larger buoyancy of 200 μm inclusion clusters makes these inclusions disperse well along the width direction of the mold, as shown in Fig. 14 (d). Compared with the result of β = 0.03 inclusion clusters (Fig. 14a), the final escape positions of 10 μm inclusion clusters are similar (Fig. 14c). All the results show that inclusion escape positions on the top surface are asymmetric, corresponding to the asymmetrical molten steel flow in the liquid pool. Fig. 15 shows the entrapment position distributions of different inclusion clusters on the solidified shell after 200 s injection, which are obtained from the width direction and thickness direction, respectively. The entrapped inclusions are represented with small dots. Most of entrapped inclusions are located within the scope of 1 m below the top surface, less entrapped inclusions are found at the lower region of the domain. Small amounts of inclusions are entrapped at the curved part of the secondary cooling zone of the mold, and due to the buoyancy of inclusion, more of them are located at the inner-curved section of the mold. For β = 0.03 inclusion clusters, in Fig. 15(a) and (b), the entrapment position distribution of smaller inclusions is very similar with that of larger inclusions. The reason may be the density difference between the inclusion cluster and molten steel is small, the buoyancy of

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Fig. 14. Predicted final escape positions on the top surface for different inclusion clusters (a) Dc = 10 μm , β = 0.03, (b) Dc = 200 μm , β = 0.03, (c) Dc = 10μm , β = 1.0, and (d) Dc = 200μm,β=1.0.

Fig. 15. Predicted entrapment positions on the solidified shell for different inclusion clusters (a) Dc = 10μm,β= 0.03, (b) Dc =200μm, β= 0.03, (c) Dc =10μm,β=1.0, and (d) Dc = 200μm,β=1.0.

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Fig. 16. Predicted escape positions on the outlet for different inclusion clusters (a) Dc =10μm,β=0.03, (b) Dc =200μm,β=0.03, (c) Dc =10μm,β=1.0, and (d) Dc =200μm,β=1.0.

The influence of inclusion cluster model on the motion and entrapment of the inclusion clusters was investigated by changing the β value using the current mathematical model. The predicted removal ratios from the top surface of different β values are shown in Fig. 17. For

different inclusion cluster models with β from 0.03 to 1.0, the removal ratio is similar when the diameter is smaller than 100 μm, and increases with β and inclusion diameter when the diameter is larger than 100 μm. The removal ratios of different inclusion sizes varying from 10% to 18% for β = 0.03 inclusion clusters, and increases a lot for β = 1.0 inclusion clusters, which is varying from 12% to 57%. Finally, a few inclusion clusters can flow out of the calculation region with the steel outflow. Fig. 18 shows the escape ratios of different inclusion clusters from the outlet of the calculation domain. When the inclusion diameter is smaller than 100 μm, the same size of inclusions cluster has approximately the same escape ratio for various β values. However, the escape ratio decreases from 29% to 11% with increasing the inclusion cluster diameter from 5 to 100 μm. The escape ratio decreases with increasing the β value when the inclusion diameter is larger than 100 μm. For example, the escape ratio of 200 μm inclusion cluster decreases from 13% to 8% with increasing β value from 0.03 to 1.0; and the escape ratio of 400 μm inclusion cluster decreases from 12% to 0 with increasing β value from 0.03 to 0.5. In the current particle motion and entrapment model, the total entrapment of inclusion clusters contain two parts: one part was entrapped by the solidified shell and another part was flowing out of the calculation region. The final entrapment ratios of different inclusion clusters are shown in Fig. 19. For smaller inclusion clusters with diameter from 5 to 100 μm, the variation of total entrapment ratio is small for different β values, the largest entrapment ratio is close to 90%. The total entrapment ratio decreases with increasing the β value when the inclusion diameter is larger than 100 μm. For example, the entrapment ratio

Fig. 17. Removal ratio from the top surface for different β values.

Fig. 18. Escape ratio from the outlet for different β values.

inclusion is weak, so the motion of inclusion cluster is mainly controlled by the molten steel flow. For β = 1.0 inclusion clusters, in Fig. 15(c) and (d), the density difference between the inclusion cluster and molten steel is larger, the buoyancy of inclusion becomes stronger, and then more inclusions are entrapped at the inner-curved section of the mold, especially for the larger inclusion clusters, as shown in Fig. 15(d). Many smaller inclusion clusters can be carried deeper into the liquid pool of the strand due to the effect of strong steel jet flow. And 62 s after first injection, some bubbles would flow out of the calculating domain(case-2). Fig. 16 shows the distributions of different inclusion clusters at the outlet of the calculation domain. For β = 0.03 inclusion clusters, in Fig. 16(a) and (b), the inclusions are almost uniform distribution in the center of the liquid pool. There is no obvious movement trend to the inner-section of the caster. The predicted inclusion cluster distribution agrees well with measured data of UT at the rolled steel plates (Fig. 1). But the distribution of these β = 1.0 inclusion clusters on the outlet is asymmetric with more along the inner curved region, 55.44% for 10 μm inclusion clusters and 63.64% for 200 μm inclusion clusters. Therefore, the density difference between inclusion and steel is important to the inclusion motion at the curved part of the caster. 4.5. Effect of inclusion cluster model

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Fig. 19. Total entrapment ratio for different β values.

of 300 μm inclusion cluster decreases from 85% to 50% with increasing β value from 0.03 to 1.0; and the escape ratio of 500 μm inclusion cluster decreases from 85% to 49% with increasing β value from 0.03 to 0.7. However, the total entrapment ratio is approximately the same for 400 μm and 500 μm large inclusion cluster when the β value is larger than 0.6. 4.6. Optimizing inclusion cluster motion In the vertical-bending caster, since the liquid domain extends far into the bending region of the caster and the inclusions float upwards as long as the steel is not solidified, the inclusions accumulate in the inner side of the cater cross section area and the well-known formation of the quarter depth inclusion band arises, as shown in Fig. 20, similar with the motion of single inclusion. However, through the collision and aggregation of many separate individual inclusions, it is easy to

Fig. 21. Schematic of inclusion clusters floating to the top surface of the mold.

Fig. 20. Schematic of inclusion motion in the bending section of the mold.

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form larger inclusion cluster, and the density of inclusion cluster would increases according to Eq. (15). And due to the decreases of temperature in the solidification end location, the viscosity of the molten steel would increase quickly. So the floatation movement of inclusion clusters would be suppressed, they will transport with the weak steel flow inside the liquid pool, and finally be entrapped by the solidified shell, as shown in Fig. 20. Based on the previous analysis, the fate of the inclusion clusters entering the mold through the SEN can be divided into three categories: ① One part reaches the top surface of the mold and is absorbed by the covering mold flux, and this part is safe for the slab quality; ② One part is entrapped by the solidified shell in the vertical section of the caster, forming surface quality defects; and ③ the remaining inclusion clusters freeze with the molten steel in the curved section of the caster, forming the quarter inclusion band and center inclusion band. In order to reduce the quality defects, it is very important to optimize the inclusion cluster motion inside the liquid pool of the caster. A main method is to promote the floatation of inclusion clusters to the top surface of the mold. According to the inclusion clusters motion and entrapment characteristics, the way of the inclusion clusters removing from the top surface can be divided into three categories, as shown in Fig. 21, as follows: ① One part directly floats to the top surface adjacent the SEN wall; ② One part flows with the upper recirculation steel jet near the narrow wall of the mold and is removed adjacent narrow wall; and ③ another part floats to the top surface with the lower recirculation flow. Based on the results of Fig. 10(b) and Fig. 15, the part-② is the main way for removing the inclusion clusters, less inclusion clusters would be removed through the other two ways. So through strengthening the upper recirculation flow intensity is important to promote the floatation of inclusion clusters. And some methods can be considered to implement this process, such as adjusting the structure of the SEN and mold, appropriately increasing the casting speed, taking argon gas injection at the SEN. Another method is to promote the collision and aggregation of the small inclusion clusters, and generate large size of inclusions clusters whose diameter should be larger than 100 μm. From the results of Figs. 17 to 19, it can be seen that the β value is also important to the motion and entrapment of the inclusion clusters. The total entrapment ratio decreases so much as the β value is larger than 0.5. However, it is difficult to measure and control the β value in the complex continuous casting process. Further work should be considered to reduce the β value of the inclusion clusters. 5. Conclusion A mathematical model has been developed to simulate the transient fluid flow, heat transfer and solidification processes in a verticalbending continuous casting caster. Based on the fractal theory and the conservation of mass, a kind of inclusion cluster model was developed. The motion of different inclusion clusters inside the liquid pool is calculated using the Lagrangian approach based on the transient flow field. The following conclusion can be drawn: 1) The center inclusion band defects located in the center of plant steel plates were found by the UT method, and the distribution of inclusion clusters is intermittent and asymmetric. 2) LES predictions of the instantaneous velocity field are composed of various small recirculation zones and multiple vortices, and their positions and intensity are ever-changing. 3) The transient distribution of inclusion clusters inside the liquid pool and the entrapment positions on the solidified shell are asymmetric in the caster. 4) Two concentrated periods for inclusion clusters removal from top surface are found: 2.8 to 5 s and 5 to 20 s. Most inclusion clusters are entrapped at approximately 50 s after the first injection. Some inclusion clusters would escape from the bottom until 62 s after injection.

5) Most inclusion clusters tend to escape from the top surface closer to the narrow wall. Most entrapped inclusion clusters are located within the scope of 1 m below the top surface. Small amounts of inclusions are entrapped at the curved part of the secondary cooling zone. 6) The β value of the current inclusion cluster model is important to the motion and entrapment of the inclusion clusters. The total entrapment ratio decreases so much as the β value is larger than 0.5. Further work should be considered to reduce the β value of the inclusion clusters.

Nomenclature cp specific heat at constant pressure C solute concentration CD drag coefficient CL Saffman force model constant CVM virtual mass force model constant CM Magnus force coefficient Cs sub-grid scale model constant de equivalent diameter dp particle diameter dw distance to the closest wall Dc inclusion cluster diameter Dm model constant fl volume fraction of liquid phase FB buoyancy force FD drag force Fg gravitational force FM Magnus force FMa Marangoni force FP pressure gradient force FS Saffman lift force FVM virtual mass force g acceleration of gravity h sensible enthalpy href reference enthalpy hm distance below the meniscus H enthalpy ΔH latent heat k thermal conductivity kv Von Kármán constant Km liquid phase permeability coefficients Ks constant, Ks =2.594 L latent heat of the material Ls mixing length for sub-grid scales N number of small individual alumina inclusions P pressure q heat flux S characteristic filtered rate of strain Se energy source term Sm momentum source term t time T temperature Tliquidus liquidus temperature Tsolidus solidus temperature u velocity of molten steel up velocity of particle uc casting speed β volume fraction of alumina inclusion ρ density ρc density of inclusion cluster υ molecular kinematic viscosity υt turbulent kinematic viscosity Δ filtering width

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Δi Δj Δk σ ξ μeff ψ −

grid spacing in X direction grid spacing in Y direction grid spacing in Z direction surface tension small constant effective viscosity constant filtered

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