Transient Thermo-fluid and Solidification Behaviors in Continuous Casting Mold: Evolution Phenomena

Abstract

A mathematical model coupling fluid flow with heat transfer as well as solidification in continuous casting mold is presented. The model features the formations of meniscus and slag films, including the growth of slag rim. Furthermore, the model describes the evolution of heat flux and thicknesses of shell and slag films from cast-start to steady-state in combination with actual operating conditions. The predictions in the developed model are in good agreement with plant measurements. The results show that a large amount of liquid slag infiltrates into the gap as the shell is withdrawn at a casting speed of 0.3 m/min, which creates the initial meniscus topography. The meniscus profile tends to bulge up at a higher casting speed, while the size of slag rim decreases. Large fluctuations of heat flux are found before forming a steady structure of slag films throughout the mold. Increasing in casting speed leads to thinner slag films and higher heat flux. This model provides a fundamental understanding on the influence of meniscus profile and slag films related to the casting speed on slab solidification, especially at the initial stage of casting process.

1. Introduction

In continuous casting of steel, surface defects arising in the mold severely affect the production efficiency and the steel quality. The complex and continuous behaviors in the mold, including fluid flow, heat transfer, chemical reactions and multiple phase transformations, which combined and interacted, are difficult to predict and control.^1,2,3) Figure 1 shows the transient phenomena in the mold during the continuous casting process. The liquid steel coming out from the submerged entry nozzle’s ports freezes against the water cooled mold walls to form a shell, which is withdrawn downward by a dummy bar at a given casting speed according to steel grade, slab size, pouring temperature, etc. Mold powder is added to the top of the molten steel, sinters, and melts to form a liquid slag pool that will infiltrate into a gap between the mold and the shell. This liquid slag close to the mold hot face cools and forms a solid film and that close to the high-temperature shell remains as a liquid film. The solid film sticking to the oscillating mold controls the horizontal heat transfer via its melting and crystallization characteristics. The liquid film provides a lubrication channel to prevent sticking of the shell to the mold. However, during the oscillation cycles, the periodic fluctuations in metal level, heat transfer and phsical force lead inevitably to irregular solidfication and microstructural evolution, which may cause the formation of defects such as deep oscillation marks and cracks.^4,5)

Fig. 1.

Schematic diagram of transient phenomena in the mold during continuous casting. (Online version in color.)

To understand the complex phenomena occuring in the continuous casting mold, many studies focusing on various aspects have been conducted in the past decades. Mathematical simulations and physical experiments have been developed and applied to study the flow patterns and the velocity fields in the mold region.^6,7,8) Banderas et al.⁹⁾ characterized the periodical behavior of turbulent flows using the large eddy simulation (LES) computational approach, which compared with the Digital Particle Image Velocimetry (DPIV) measurement. A similar model by Lopez et al.¹⁰⁾ included the effects of casting speed on fluid-flow structure, meniscus topography and kinetic energy distribution. Several computational models of heat transfer and solidification have focused on the formation of oscillation marks on the surface of continuously cast slabs.^{11,12,13,14,15)} From these studies a reasonable prediction of oscillation marks shape in the meniscus region emerged, which involved meniscus depression and overflow. Sengupta and Thomas¹⁴⁾ presented a detailed mechanism for the formation of oscillation mark and hook using a graphical animation that allowed the visualization of various events occurring near the meniscus. Lopez et al.¹⁵⁾ proposed a unified mechanism for the transient shell growth based on a multiphysics model that provided quantitative results regarding the influence of slag infiltration on shell solidification and OM morphology. Many analytical transient models of liquid slag flow and slag consumption have been formulated to investigate the interfacial slag behavior between the solidifying steel shell and the mold wall.^{16,17,18,19,20,21,22,23,24)} McDavid and Thomas²⁵⁾ applied a three-dimensional (3-D) FIDAP model to analyze the fluid flow and heat transfer of the top-surface flux layers. The predicted flux layer thicknesses were matched with experimental measurements, and a large recirculation zone in the liquid slag pool was observed by these model calculations. Shin et al.²⁶⁾ presented a semi-empirical model, which divided total powder consumption into two components: the lubrication consumption and the flux carried within the volume of oscillation marks, to predict mold lubrication phenomena based on the measurements from extensive plant trials on ultra-low carbon steels. This work was validated by the known trends of oscillation mark depth and slag consumption with various operation parameters, such as mold powder characteristics, casting speed and mold oscillation conditions.

Meanwhile, several novel laboratory analogues have been developed to improve the understanding of chaotic events in the continuous casting mold.^27,28,29,30) Badri et al.^31,32) designed a mold simulator, which was used to obtain the solidified steel shells of the different grades of steel under conditions similar to those found in industrial casting operations, to investigate heat transfer phenomena during the initial phase of strand solidification. The use of this technique, coupled with the understanding developed from experimental and theoretical heat-transfer studies, suggested that the oscillations in heat flux corresponded directly with variations in the shell surface profile. Kajitani et al.^33,34) proposed a new mechanism to clarify the mold flux infiltration by using a cold model experiment in which silicone oil was infiltrated in the channel between the rotating belt and the acrylic plate. They confirmed the channel profile for different conditions, and also showed that the flux infiltrated from the final part of the positive strip time towards the latter half of the negative strip time. Similarly, a new experimental apparatus for simulating the mold oscillation of continuous casting was constructed in our previous work in which various oscillation factors affecting flux lubrication in the meniscus region were analysed and the stage of slag consumption was identified.³⁵⁾

The difficulties in studying the effects of casting parameters on slab solidification inside the mold have been recognised due to the interdependence among different variables. The development of transient multiphysics models plays a significant role in addressing these issues since it can provide a good description of natural phenomena occurring in the mold. Meng and Thomas^13,17) provided a detailed treatment of the interfacial gap between the shell and the mold by using a CON1D program with combined heat-transfer, liquid-flow, and solid-friction models. It simulated the solidification of steel shell and solid/liquid slag layers movement, including the effects of oscillation marks on heat transfer and powder consumption. This model has been widely used to predict the shell thickness, slag layer thickness, temperature distributions in the mold and shell, heat flux profiles, transient shear stress, and other related phenomena. Based on the output from the CON1D model, a computational model by Jonayat and Thomas,²⁴⁾ solved the Navier-Stokes equations, turbulence, and energy equations. This method was applied to describe the transient thermal-flow in the meniscus region during an oscillation cycle and to predict the slag consumption for arbitrary conditions, such as casting speed, frequency, stroke, and modification ratio, which was validated with both lab and plant measurements. Lopez et al.^21,36) developed a comprehensive model of heat transfer, fluid flow, and solidification, based on the VOF method for tracking the interfaces in the multiphase steel-slag-air system under transient conditions. A parametric study was carried out with this 2-D model, investigating the influence of various operational parameters on surface defect formation. Predictions of shell thicknesses and heat fluxes were in good agreement with prior, cold model experiments^33,34) and mold simulators.^31,32)

Previous works have provided various modelling approaches to gain insights into the correlation mechanisms of thermal-flow and solidification in the mold. However, these coupled and simultaneous problems are stepwise solved to reach steady-state, which will inevitably deviate from the events found in practices and limit their uses for extensive study. A mathematical model which addresses the limitations of concurrent phenomena in the continuous casting mold was presented in a series of two articles, predicting the evolution phenomena and the oscillation behaviors during the casting process from cast-start to steady-state under industrial operating conditions. In the present article the model development was described and the evolutions of meniscus and slag films for different casting speeds were predicted.

2. Model Development

2.1. Governing Equations

A transient model of fluid flow, heat transfer, and solidification has been developed by solving momentum, continuity, turbulence, and energy equations in a two dimensional domain. The Volume of Fluid (VOF) method is used to predict the steel/slag interface through tracking the volume fraction of each of the fluids throughout the domain. The continuity equation for the volume fraction is expressed as:   

$$\begin{array}{c}\frac{\partial }{\partial t}\left({\alpha }_{\text{steel}}{\rho }_{\text{steel}}\right)+\nabla \cdot \left({\alpha }_{\text{steel}}{\rho }_{\text{steel}}\overrightarrow{v}\right)\\ =S_{{\alpha }_{\text{steel}}}+\sum \left({\dot{m}}_{\text{slag-steel}}-{\dot{m}}_{\text{steel-slag}}\right),\end{array}$$(1)

where α is the phase fraction, ρ is the density, $\overrightarrow{v}$ is the velocity vector, $\dot{m}$ is the mass transfer between the phases and the source term and $S_{{\alpha }_{\text{steel}}}$ is zero.

A single momentum equation that depends on the volume fractions of all phases through the effective density ρ_mix and the effective viscosity μ_mix is solved.   

$$\begin{array}{l}\frac{\partial }{\partial t}\left({\rho }_{\text{mix}}\overrightarrow{v}\right)+\nabla \cdot \left({\rho }_{\text{mix}}\overrightarrow{v}\overrightarrow{v}\right)=-\nabla p+\nabla \cdot \\ \left[{\mu }_{\text{mix}}\left(\nabla \overrightarrow{v}+\nabla {\overrightarrow{v}}^T\right)\right]+{\rho }_{\text{mix}}\overrightarrow{g}+{\overrightarrow{F}}_{\sigma }-S_{\text{m}},\end{array}$$(2)

  

$${\rho }_{\text{mix}}={\alpha }_{\text{steel}}{\rho }_{\text{steel}}+\left(1-{\alpha }_{\text{slag}}\right){\rho }_{\text{slag}},$$(3)

  

$${\mu }_{\text{mix}}={\alpha }_{\text{steel}}{\mu }_{\text{steel}}+\left(1-{\alpha }_{\text{slag}}\right){\mu }_{\text{slag}},$$(4)

where p is the pressure, $\overrightarrow{g}$ is the gravitational force vector and the last two terms are the momentum sinks due to surface tension and solidification. ${\overrightarrow{F}}_{\sigma }$ is calculated through the CSF model of Brackbill et al.³⁷⁾   

$${\overrightarrow{F}}_{\sigma }={\sigma }_{\text{slag-steel}}\frac{{\rho }_{\text{mix}}\kappa \nabla {\alpha }_{\text{slag}}}{\frac{1}{2}\left({\rho }_{\text{slag}}+{\rho }_{\text{steel}}\right)},$$(5)

where ${\sigma }_{\text{slag-steel}}$ is the interfacial tension between steel and slag, and κ is the local surface curvature as defined by Brackbill et al.³⁷⁾

The energy equation, also shared among the phases, is shown below.   

$$\frac{\partial }{\partial t}\left({\rho }_{\text{mix}}{E}_{\text{mix}}\right)+\nabla \cdot \left(\overrightarrow{v}\left({\rho }_{\text{mix}}{E}_{\text{mix}}+p\right)\right)=\nabla \cdot \left(K_{\text{eff}}\nabla T\right),$$(6)

where E_mix is the enthalpy of mixture, and K_eff is the effective thermal conductivity for the phase mixture.

For turbulence modeling, the realizable k–ε model is used to solve two additional transport equations for the turbulence kinetic energy, k, and its rate of dissipation, ε, as given in Appendix. Since the enthalpy-porosity technique treats the mushy region as a porous medium, the sinks are added to the momentum and turbulence equations in the mushy and solidified zones to account for the presence of solid matter, respectively.   

$$S_{\text{m}}=\frac{{\left(1-\beta \right)}^2}{\left(0.001+{\beta }^3\right)}{A}_{\text{mush}}\left(\overrightarrow{v}-{\overrightarrow{v}}_{\text{pull}}\right),$$(7)

  

$$S_{\text{t}}=\frac{{\left(1-\beta \right)}^2}{\left(0.001+{\beta }^3\right)}{A}_{\text{mush}}\varphi ,$$(8)

where β is the liquid volume fraction, A_mush is the mushy zone constant, ${\overrightarrow{v}}_{\text{pull}}$ is the pulling velocity, and ϕ represents the turbulence quantity being solved (k and ε). The mushy zone constant measures the amplitude of the damping; the higher this value, the steeper the transition of the velocity of the material to zero as it solidifies. Here, A_mush is set to 1×10⁸.

2.2. Geometric Model and Boundary Conditions

The model domain and its boundaries are shown in Fig. 2. A 2-D vertical section parallel to the wide face through the center of the caster is chosen, which consists of half SEN, slab mold, and 1 m of slab legth after the mold exit. Steel is poured from an inlet surface to the mold domain through a SEN whose nozzle port is submerged 0.152 m below the meniscus and 0.252 m below the top of the 0.9 m long mold. The angle of the nozzle jet is 15 deg downward. The inlet velocity of steel is calculated by a mass balance:   

$$v_{\text{inlet}}=\frac{v_{\text{c}}{A}_{\text{outlet}}}{A_{\text{inlet}}}.$$(9)

where v_c is the casting speed, A_inlet and A_outlet are the areas of SEN inlet and mold outlet, respectively. An initial powder slag layer of 50 mm is added on the top of the steel free surface. The powder inlet boundary is given a constant flux as the supply of liquid slag to lubricate the shell. The mold cold face is defined by a convection boundary that removes the heat to the cooling water:   

$$q_{\text{w}}=h_{\text{w}}(T_{\text{w}}-T_{\text{s}}),$$(10)

where q_w is the heat flux of mold cold face, h_w is the effective convection heat transfer coefficient, T_s is the local surface temperature, and T_w is the water temperature that is assumed to increase linearly from the inlet temperature, T_wi to the outlet temperature, T_wo. The shell face is modeled as a combined convection/external radiation boundary condition. The oscillation of mold domain is presented as a sinusoidal curve:   

$$v_{\text{m}}=2\pi sfcos(2\pi ft),$$(11)

where frequency, f = 170−20v_c, and amplitude, s = (2+4v_c)/2. The interface between the molten steel and the mold domains is coupled in both velocity and heat flux, and moves with the mold oscillation. The values used in the boundary conditions are listed in Table 1. In order to predict the phenomena in the thin gap between the mold and the shell, high degree of mesh refinement was applied near the interface. The total mesh size consists of 364181 elements ranging in size from 50 μm to 4 mm.

Fig. 2.

Schematic of model domain and boundaries. (Online version in color.)

Table 1. Variables used in boundary conditions.

Superheat temperature, Tsh	25	K
Water flow rate, vw	10	m/s
Inlet water temperature, Twi	303	K
Outlet water temperature, Two	310	K
External water temperature, Twe	298	K
External radiation temperature, Ter	318	K

2.3. Steel and Slag Properties

2.3.1. Steel Properties

The current work adopts different choices for steel properties, including several constants calculated as functions of composition, as given in Table 2. Both thermal conductivity and specific heat are calculated as functions of carbon content, temperature, and phase fraction, which are fitted from the measured data compiled by Harste,³⁸⁾ as shown in Fig. 3. The thermal conductivity of liquid is artificially increased by a factor of 6.5 to account for the influence of convection due to the turbulent flow in the liquid steel pool.³⁹⁾

Table 2. Steel composition and properties.

Steel composition (wt pct): C=0.05, Si=0.03, Mn=0.25, P=0.024, S=0.01, Al=0.04.
Liquidus temperature, Tliq	1806.1	K
Solidus temperature, Tsol	1780.7	K
Steel density, ρ steel	7400	kg/m3
Steel latent heat, Lsteel	272	kJ/kg
Steel emissivity, Esteel	0.8	–
Steel viscosity, μ steel	0.0065	Pa·s
Fraction solid for shell thickness location, fs	0.3	–

Fig. 3.

(a) Thermal conductivity and (b) specific heat of steel.

2.3.2. Slag Properties

During operation, mold slags absorb some inclusions from the molten steel, which changes their compositions and properties. In this work, the constant molten slag composition is used to calculate temperature-dependent slag properties, as given in Table 3. The viscosity of slag is modeled differently for melting and solidifying, as shown in Fig. 4. The mold powder on the top domain is characterized as it sinters and melts to form the liquid, according to the rheological data used in the particle fluidization field⁴⁰⁾ and the McDavid’s model.²⁵⁾ The exponential function by Meng and Thomas¹³⁾ is applied to model the liquid viscosity:   

$$\mu ={\mu }_{\text{o}}{\left(\frac{T_{\text{o}}-T_{\text{fsol}}}{T-T_{\text{fsol}}}\right)}^n,$$(12)

where the parameters, T_fsol and n are chosen empirically to fit the measured data and μ_o is the viscosity measured at 1573 K. To avoid numerical problems, the maximum viscosity of resolidified slag is truncated at 10⁴ Pa s.

Table 3. Molten slag composition and properties.

Molten slag composition (wt pct): SiO2=38.52, CaO=24.53, Al2O3=6.41, MgO=1.84, Na2O=9.41, Fe2O3=0.63, CaF2=11.93, Li2O=0.4.
Slag solidification temperature, Tfsol	1346	K
Slag viscosity at 1573 K, μo	0.416	Pa·s
Exponent for temperature dependent viscosity, n	1.4	–
Slag-steel interfacial tension, σslag-steel	1.35	N/m
Slag density, ρ slag	2500	kg/m3

Fig. 4.

Viscosity-temperature model for melting and solidifying slag. (Online version in color.)

The effective thermal conductivity of slag is also simulated using two different models during heating and cooling, as shown in Fig. 5. Below the sintering temperature, the thermal conductivity of powder is fixed at 0.3 W/m K.⁴¹⁾ As the powder sinters and melts, the conductivity gradually increases. For both melting and solidifying slag, a constant effective thermal conductivity of 3 W/m K above the melting temperature is adopted, including the conduction and the radiation through the slag.^42,43) In the totally solid state, a constant value of 0.5 W/m K is used, which is similar to the previous models.^24,25)

Fig. 5.

Thermal conductivity-temperature model for melting and solidifying slag. (Online version in color.)

The temperature denpendent specific heat of slag, as shown in Fig. 6, is based on reasonable estimates from the partial molar heat capacity values for individual components.^44,45) It includes a sharp increase at the glass transition temperature which corresponds to the transition between glass slag and liquid. The contact resistance of interfacial gap generally increases as the increasing slag surface roughness in the casting direction. Thus, the thermal contact resistance in this study, expressed as a linear function, increases from 8×10⁻⁵ m² K/W at the meniscus to 2.5×10⁻⁴ m² K/W at the mold exit, which corresponds to the model of Meng and Thomas⁴⁶⁾ and the measurements of Yamauchi.⁴⁷⁾ The interfacial tension between the steel and the slag is calculated to be 1.35 N/m, using a function of slag and steel compositions.⁴⁸⁾

Fig. 6.

Temperature dependent specific heat of slag.

2.4. Solution Procedure

The transient model developed in this study couples fluid flow with heat transfer and solidification during mold oscillation. These coupled behaviors are solved simultaneously on the basis of industrial operating procedure using ANSYS FLUENT 14.0. The solution starts from the calculation of powder slag on the steel free surface for 60 s, which produces a liquid slag as the start mold powder. This step is to establish a fully developed metal flow and solid shell growth conditions before pulling the steel shell. Then, the shell is withdrawn according to the casting speed curve from cast-start to steady-state, where a stepped increase in casting speed was adopted to ensure the process stability, as shown in Fig. 7. The casting speed increases 0.3 m/min per 60 s since the starting speed is set to 0.3 m/min. As the casting speed reaches a plateau (1.2 m/min), the simulation runs for another 60 s to obtain the correlation data. Meanwhile, the mold domain moves in accordance with the oscillation equation, which depends on the casting speed as well. The calculation is run with a fixed time step of 0.001 s on an Intel (R) Xeon (R) CPU with 12 × 3.47 GHz cores PC.

Fig. 7.

Casting speed curve from cast-start to steady-state during the casting process. (Online version in color.)

3. Results and Discussion

3.1. Formations of Meniscus and Slag Films

The model used in this study makes no assumptions regarding the interface between liquid slag and molten steel, where the fluid flow couples with heat transfer and solidification are solved simultaneously, which successfully predicts the formations of meniscus, slag rim and films between mold and shell, as shown in Fig. 8. This approach produces a thick solidified shell as well as an initial liquid pool and sintered layer corresponding to the slag properties after 60 s’ standing, as shown in Fig. 8(a). The liquid slag close to the mold freezes to form the original slag rim, while the shell tip grows along the steel/slag interface. Figure 8(b) shows a sudden fall in the liquid slag near the mold when the solidified shell starts to be pulled down at the casting speed. The shell tip is immersed in the molten steel and melts away. Meanwhile, the liquid steel starts to bulge out and overflows, which creates the curved shape due to the interfacial tension between liquid slag and molten metal, accompanying with strong instibility in metal level. The overflowing steel freezes anew along the steel/slag interface creating a thin shell, which welds onto the edge of the previous shell. The liquid slag close to the mold forms a solid film connected to the slag rim and the rest continues to flow downward. As the solidified shell is withdrawn downward, the newly formed shell of uneven thickness is pushed to the mold wall by the static pressure of steel, which results in an obvious shrinkage of the slag channel (Fig. 8(c)). After 9 s from the shell withdrawal, the meniscus gains its equilibrium shape, where a regular shell formation and a uniform slag film are developed (Fig. 8(d)). It indicates there is a short transition period when the meniscus shape changes continually as the combined results of shell withdrawal, static pressure of steel and surface tension differences between molten steel and liquid slag. Then, the liquid slag continues to flow into the gap along this dynamically balanced meniscus, forming a steady structure of slag films throughout the mold length. Further, the curvature of meniscus and the thickness of slag films vary gradually as the casting speed increases. These results are analyzed in detail in the next section.

Fig. 8.

Formations of meniscus and slag films. (Online version in color.)

3.2. Meniscus Profile and Slag Rim for Different Casting Speeds

Figure 8 has described the process of forming meniscus and slag films at a start casting speed. It is well known that the casting speed is a critical factor in process stability and surface quality. The model can make predictions of chaotic behavior in the mold at different stages from cast-start to steady-state, which provides new insights into the evolution of turbulence, heat transfer and solidification. Figure 9 shows the temperature contours in the meniscus for different casting speeds. The effect of changes in casting speed is consistent with previous investigations, showing increases in velocity magnitude in the steel jet and decreases in residence time of solidified shell. The solidified shell with high surface temperature is closer to the mold at a high casting speed. Apparently the maximum temperature in the copper mold increases, but its location is at approximately 50 mm below the meniscus for either low or high casting speeds. On the other hand, although the meniscus is formed completely within 9 s, the casting speed can considerably affect the meniscus profile. In order to stabilize the casting process, the model is solved incipiently at a low casting speed of 0.3 m/min, which allows small fluctuations in heat flux and flow field at the initial stage of casting. The meniscus stability is identified whose profile looks flat (Fig. 9(a)). With the increasing of casting speed, the recirculating flow coming from the mold wall is intensified, causing a small hump in the meniscus topography (Fig. 9(b)). To better distinguish the changes in the meniscus curvature, the meniscus profile for different casing speeds are summarized in Fig. 10. As is seen, the meniscus shape tends to bulge up during the process of increasing casting speed. In other words, the high casting speed produces a small radius of meniscus curvature. This result is consistent with the observations by Matsushita.⁴⁹⁾

Fig. 9.

Temperature contours in the meniscus for different casting speeds. (Online version in color.)

Fig. 10.

Meniscus profile for different casting speeds. (Online version in color.)

As we know, the liquid slag close to the mold hot face forms a solid layer attached to the mold wall and grows up into a slag rim near the meniscus, which would affect considerably slag infiltration and oscillation mark formation during the oscillation cycles of mold. In this study, the shape of slag rim is defined according to a sudden inrease in viscosity occured at the break temperature of slag based on the function of temperature and composition. Therefore, the slag rim profile represents the two dimensional heat transfer near the meniscus, which is closely related to the casting speed, as shown in Fig. 11. Since the slag rim has accumulated for a few seconds before shell withdrawal, there is a large rim whose thickness is more than 10 mm at the starting casting speed of 0.3 m/min, which means a closer distance from the meniscus. As the casting speed increases, the edge of slag rim begins to melt and its profile shrinks due to the higher heat flux. This leads to the increase in the distance between slag rim and meniscus, though the meniscus level rises as well, as shown in Fig. 10. At 1.2 m/min, the predicted thickness of slag rim maintains at 3.5–3.8 mm, which is closer to that measured under the same operating conditions in plant.

Fig. 11.

Slag rim profile for different casting speeds. (Online version in color.)

3.3. Evolution Phenomena

Previous works^24,26,36) have shed light on the effect of casting speed on fluid flow, heat transfer, and solidification in the mold, but the continuous and associated phenomena are not described, especially at the initial stage of casting process. Figure 12 shows the variations of meniscus level and liquid pool depth measured at a location 40 mm from the mold during the casting process. The liquid pool is considered as a liquid slag container, whose depth reflects the vertical heat transfer through the slag layers. There is a certain thickness of liquid slag formed on the steel surface before shell withdrawal. Here the initial steel free surface is regarded as a reference line for the meniscus level that depends on many factors, such as steel composition, flow field in the mold, slag property, argon blowing, cooling condition, and casting speed. Since the shell is pulled at a low casting speed of 0.3 m/min, a large amount of liquid slag flows downward, causing a slight decline in liquid pool depth. The withdrawal of solidified shell leads to the meniscus bulging, rising the liquid level abruptly in the first few seconds, as shown in Fig. 8(b). Then, the meniscus level continues to increase steadily with increasing casting speed, but the upward trend slows down. On the other hand, the intensive jet flow enhances the vertical heat transfer from the molten steel to the mold slag, deepening the liquid pool.

Fig. 12.

Meniscus level and liquid pool depth measured at a location 40 mm from the mold. (Online version in color.)

Figure 13 presents a typical output from the simulation at 300 s, where the heat flux, shell surface temperature, thicknesses of shell and solid/liquid slag films are measured at the casting speed of 1.2 m/min. These results are obtained based on the steady casting parameters when the casting speed reaches a plateau, which will be verified through comparing with the data measured in plant. The peak heat flux is located at 35 mm below the meniscus, where the thickness of solid slag film is smallest, as shown in Fig. 13(d). Then, heat flux decreases with the distance below the meniscus due to increasing gap resistance. It is noted that the peak heat flux position is inconsistent with the location of peak mold temperature, which is about 15 mm below the heat flux peak (50 mm below the meniscus) because of the vertical heat transfer experienced in the meniscus region. The predicted shell thickness at mold exit is about 25 mm, which agrees with measurements on a breakout shell that occurred under similar casting conditions.¹³⁾ The liquid slag film runs out at 600 mm below the meniscus, where the shell surface temperature drops below the slag solidification temperature of 1346 K, as shown in Fig. 13(b). The solid slag film continues to accumulate with the slag carried within the oscillation marks.

Fig. 13.

Predicted heat flux, shell surface temperature, and thicknesses of shell and slag films for casting speed of 1.2 m/min. (Online version in color.)

Based on the results obtained above, the evolutions of heat flux, shell surface temperature, and thicknesses of shell and slag films for a position 35 mm below the meniscus during the casting process are investigated in Fig. 14. At the beginning of casting procedure, a large amount of liquid slag infiltrates into the gap between the solidified shell and the mold wall as the shell is withdrawn at a casting speed of 0.3 m/min, which results in the excessive but uneven thickness of slag films near the meniscus. It should be noted that the heat transfer between the shell and the mold is mainly decided by the thicknesses of slag films at this stage since the casting speed is constant. Both liquid and solid slag films gradually thin out when the solidified shell moves to the mold wall under the static pressure of steel, increasing heat flux slightly. Then, the structure of slag films reaches a steady. As the increasing of casting speed proceeds, the thinner steel shell produced with less solidification time increases surface temperature, which leads to thinner slag films and higher heat flux. There is an inverse correlation between the solid slag film and the heat flux, where decrease in solid slag thickness is associated with the increasing in heat flux due to the lower thermal resistance. It can be seen that there is a period of adjustment to gain stable fluid flow and heat transfer in the mold at the low casting speed because of the formations of meniscus and slag films. Then all variables show stepped change as increasing in casting speed. Extreme fluctuations always occur at every practical case of increasing casting speed, and tend to weaken in further process. This will inevitably make differences in slag infiltration and initial shell solidification during the mold oscillation. A detailed analysis of oscillation behaviors for different casting speeds amd a mechanism to predict oscillation marks will be presented in the next article.

Fig. 14.

Evolution of heat flux, shell surface temperature, and thicknesses of shell and slag films at a location 35 mm below the meniscus during the casting process. (Online version in color)

4. Conclusions

The current work presents a numerical model to predict the transient thermal-flow and solidification behaviors in the continuous casting mold. Both the formations of meniscus and slag films and their continuous changes at different casting speeds are described for understanding the evolution phenomena during the process from cast-start to steady-state, which matches reasonably with plant measurements. The following conclusions are derived from this model:

(1) A balanced meniscus is gained in 9 s since the shell starts to be pulled down, accompanying with a large amount of slag infiltration, which develops non-uniform slag films and an irregular shell formation.

(2) A large slag rim is formed at a casting speed of 0.3 m/min, resulting in a closer distance from a flat meniscus profile. With the increasing of casting speed, the meniscus cuvre begins to bulge up, while the shape of slag rim shrinks.

(3) Thickness of solid slag film and heat flux maintains an inverse relationship throughout the casting process. The fluctuation in any variable (depth of liquid pool, thicknesses of shell and slag films, heat flux, etc.) always occurs at every practical case of increasing casting speed, and tends to weaken in further process.

Acknowledgements

The authors are especially grateful to the National High Technology Research and Development Program of China (2015AA03A501) and the National Natural Science Foundation of China (U1660204).

Appendix

The realizable k–ε model contains an alternative formulation for the turbulent viscosity and a modified transport equation for the dissipation rate, which satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of turbulent flows.^A1)   

$$\begin{array}{l}\frac{\partial }{\partial t}\left({\rho }_{\text{mix}}k\right)+\nabla \cdot \left({\rho }_{\text{mix}}k\overrightarrow{v}\right)\\ =\nabla \cdot \left[\left({\mu }_{\text{mix}}+\frac{{\mu }_{\text{t}}}{{\sigma }_{\text{k}}}\right)\nabla k\right]+G_{\text{k}}-{\rho }_{\text{mix}}\epsilon ,\end{array}$$(A1)

  

$$\begin{array}{l}\frac{\partial }{\partial t}\left({\rho }_{\text{mix}}\epsilon \right)+\nabla \cdot \left({\rho }_{\text{mix}}\epsilon \overrightarrow{v}\right)\\ =\nabla \cdot \left[\left({\mu }_{\text{mix}}+\frac{{\mu }_{\text{t}}}{{\sigma }_{\mathrm{\epsilon }}}\right)\nabla \epsilon \right]+{\rho }_{\text{mix}}{C}_{1}S\epsilon -{\rho }_{\text{mix}}{C}_2\frac{{\epsilon }^2}{k+\sqrt{v\epsilon }},\end{array}$$(A2)

where $C_1=max\left[0.43,\frac{\eta }{\eta +5}\right]$, $\eta =S\frac{k}{\epsilon }$, $S=\sqrt{2S_{\text{ij}}{S}_{\text{ij}}}$. The generation of turbulence kinetic energy, G_k, is   

$$G_{\text{k}}={\mu }_{\text{t}}{S}^2.$$(A3)

The eddy viscosity, μ_t, is computed by combining k and ε as follows:   

$${\mu }_{\text{t}}={\rho }_{\text{mix}}{C}_{\mu }\frac{k^2}{\epsilon },$$(A4)

where,   

$$C_{\mu }=\frac{1}{A_0+A_{\text{s}}\frac{k{U}^{*}}{\epsilon }},$$(A5)

  

$$U^{*}=\sqrt{S_{\text{ij}}{S}_{\text{ij}}+{\tilde{\mathrm{\Omega }}}_{\text{ij}}{\tilde{\mathrm{\Omega }}}_{\text{ij}}}$$(A6)

and   

$${\tilde{\mathrm{\Omega }}}_{\text{ij}}={\mathrm{\Omega }}_{\text{ij}}-2{\epsilon }_{\text{ijk}}{\omega }_{\text{k}},$$(A7)

  

$${\mathrm{\Omega }}_{\text{ij}}={\overline{\mathrm{\Omega }}}_{\text{ij}}-{\epsilon }_{\text{ijk}}{\omega }_{\text{k}},$$(A8)

where $\mathrm{\hspace{0.17em} }{\overline{\mathrm{\Omega }}}_{\text{ij}}$ is the mean rate-of-rotation tensor viewed in a moving reference frame with the angular velocity, ω_k. The model constants A₀ and A_s are given by   

$$A_0=4.04, \mathrm{\hspace{0.17em} }{A}_{\text{s}}=\sqrt{6}cos\varphi ,$$(A9)

where $\varphi =\frac{1}{3}{cos}^{-1}\left(\sqrt{6}W\right)$, $W=\frac{S_{\text{ij}}{S}_{\text{jk}}{S}_{\text{ki}}}{{\tilde{S}}^3}$, $\tilde{S}=\sqrt{S_{\text{ij}}{S}_{\text{ij}}}$, $S_{\text{ij}}=\frac{1}{2}\left(\frac{\partial u_{\text{j}}}{\partial x_{\text{i}}}+\frac{\partial u_{\text{i}}}{\partial x_{\text{j}}}\right)$.The other constants are C₂=1.9, σ_k=1.0, and σ_ε=1.2.

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