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In the current study, several topics related to numerical modeling of nonmetallic inclusions in continuous casting are reviewed: (1) the sources and the basic methods to study nonmetallic inclusions in continuous casting; (2) motion of inclusions in turbulent region of casting mold; and (3) interaction between particle and solidification front. According to the analysis, a multi-scale model was suggested to solve the complicate multi-phase transport phenomena in both turbulent melt pool and laminar flow mushy zone. The model can predict the final distribution of particles in steel.
The demand for clean steel increases every year. The requirement of clean steel involves the control of nonmetallic inclusion and amount elements, such as sulfur, phosphorus, nitrogen and etc. [1, 2] Among these requirements, control of nonmetallic inclusion is always a hot issue for both research work and industry manufacture since the remaining of inclusions inside steel matrix would damage the mechanical properties of steel product. For example, the inclusion results in the stress concentration inside the steel, as shown in Figure 1.
The formation of the nonmetallic inclusion can be classified as endogenous inclusion and exogenous inclusion, as shown in Figure 2. The endogenous inclusion is mainly from deoxidation and/or reoxidation. And the exogenous inclusion is from other sources, for example slag entrainment and/or refractory damage. Since inclusion problem is highly related to both refining and casting process , in the past couple of decades, there are many researches focusing on the methods to evaluate steel cleanliness [4, 5], sources of non-metallic inclusions [6, 7], and the efficiency of different methods to remove inclusions from melt, such as ladle stirring [8-12], slag refining [6, 13-18], tundish operation [19-23] etc.
In 1985, Mcpherson applied the word "Mold Metallurgy" to emphasize the importance of the mold to improve the steel cleanliness . Relevant phenomena in the mold region are shown in Figure 3 . In the casting mold the destinations of inclusions can be concluded as,
- Removed to top surface by fluid flow transport;
- Re-Entrained to the molten steel from the interface between the slag and the molten steel;
- Captured by the solidification shell;
- Recirculating in the molten steel;
- Attached to bubble surface, and bubble with inclusions either being removed to the top surface or captured by the shell
The mold flow pattern is very important to avoid defects because it affects particle transport: removal to the top surface or entrapment by the solidifying shell. The behavior of the top surface of the melt pool is particularity important since it controls the entrainment of top slag, surface defects, bubbles and particle removal, and many other problems. The research and operation about flow pattern control, including flow pattern stability [26-28], casting speed [29, 30], nozzle geometry [31, 32], argon gas injection [33-36], submergence depth [37, 38], electromagnetic force [39, 40] and caster curvature [41, 42], and top surface control, such as meniscus stagnation [43, 44], entrainment of mold slag [12, 45, 46], level fluctuations [24, 26], are concerned.
Except the flow pattern, the complexity phenomena in continuous casting mold involve heat transfer, shell solidification, bubble floating and mold lubrication. Recently, some research works focus on the engulfment of inclusions into solidification shell, since the inclusions with a certain size may cause serious problems to the steel quality. Solidification front was assumed to be planner and mushy zone (solid and liquid mixed zone) was treated like a porous media in some mathematic model [47-49] to analysis the movement of solidification front during solidification process. And a laboratory scale experiment based on polystyrene particle (around 100µm) and salt solution were established to investigate movement of inclusions near the solid wall and engulfment into the solidifying front by Hideyuki Y. and Itsuo O. et al . Also, engulfment and pushing of non metallic inclusions in steel melt by advancing melt/solid interface was observed by Shibata et al with the way of "In-situ" . However, the detailed and comprehensive research about interaction of inclusions with the solidification front in steel combined with the complicated phenomena during solidification in continuous casting mold is not enough. Understanding the particle-front interaction is particularly important for effecting better removal inclusions or control of the distribution of particles in steel. Simultaneity, improved mathematical model can be developed to enable better simulation of inclusion removal in casting processes.
The current paper aims to review the numerical modeling of nonmetallic inclusions in continuous casting mold. Several points, including (a) the basic method to study the inclusions in continuous casting; (b) motion of inclusions inside turbulent region; (c) interaction between inclusions and solidification front, were discussed. And a new model by considering the effect of turbulent flow and solidification front was suggested for future study.
2. Investigation Methods
The basic way to study the behavior of inclusions during continuous casting includes industrial trial, physical simulation and numerical simulation.
2.1 Industrial trial
With systematic sampling in each process, the cleanliness of steel product can be studied by metallographic microscope observation, slime (electrolysis) and other methods . The information from samples, for example inclusions amount, size, distribution, composition, is helpful to optimize process parameters and evaluate the quality of steel product. Figure 4 show the inclusion morphologies inside steel. Also, the source or formation of these inclusions inside steel can be found by tracking the trace elements, such as Na, K and Mg.
2.2 Physical simulation
Physical simulation is a common method to use water or low melting point metal to simulate the continuous casting process. The basic principle for physical simulation is geometry similarity. Besides the geometry similarity, the design of the model has to follow some similarity numbers, such as, Reynolds number (Re) and Froude number (Fr):
The advantage of using water to simulate molten steel is that it is easy to observe and measure the fluid flow inside metallurgical vessels or equipment . So water model is widely accepted for continuous casting simulation, as shown in Figure 5. However, in order to simulate some special phenomena in continuous casting process, for example the effect of magnetic field on molten steel, low melting point metal was employed. In the past decade, mercury [55-60], tin , Gallium , and Bi-Pb-Sn  were applied to simulate molten steel inside casting mold. And to study the motion of inclusion inside the steel, both plastic particle and glass bead are employed. Even some researcher has applied a special oil to simulate the liquid inclusions in steel .
2.3 Numerical simulation
Due to the advantages of accurate, fast and low cost, numerical simulation is widely used to study the fluid flow related phenomena in continuous casting. Figure 6 shows the comparison of water model measurement and numerical simulation result. The excellent performance of numerical simulation makes it become a shape tool to design vessel and conduct the manufacture process. The detailed discussion about numerical modeling of nonmetallic inclusions in casting mold will be given in the next section.
3. Numerical Modeling of Nonmetallic Inclusions
3.1 Motion of inclusions in turbulent region
3.1.1 Forces acting on inclusion
Before particles touch the solidification front, they are transported by fluid flow in turbulent field. The movement of particle is governed by the following forces, which are exerting on particle in turbulent flow field, including drag force, gravitational force, virtual mass force, magnus force, saffman force etc. The expressions of these forces are given in Table 1.
3.1.2 Mathematical formulas
In order to simulate the motion of inclusion inside turbulent field, it is necessary to calculate the turbulent fluid flow at first. Normally, three-dimensional (3-D) single-phase steady turbulent fluid flow in the SEN and continuous casting mold was modeled by calculating the continuity equation and Navier-Stokes equations, using standard k-e two-equation turbulence model as expressed below,
The first term in Eq. is the drag force per unit particle mass, the second term is the gravitational force, the third term is the "virtual mass" force accelerating the fluid surrounding the particle , and the fourth term is the force stemming from the pressure gradient in the fluid.
To incorporate the "stochastic" effect of turbulent fluctuations on the particle motion, the "random walk" model  is used. In this model, particle velocity fluctuations are based on a Gaussian-distributed random number chosen according to the local turbulence kinetic energy. The random number is changed, thus producing a new instantaneous velocity fluctuation, at a frequency equal to the characteristic lifetime of the eddy. The instantaneous fluid velocity is then given by:
3.1.3 Application case
Figure 6 shows the motion of inclusion inside casting mold under the different condition. 20,000 inclusions, each with a diameter of 50 µm, were injected into the submerged entry nozzle (SEN) from the top over a 2 second time period. The result indicates that with SEN one-sided clogging, inclusions travel longer before escaping from the top or entering the bottom than they do in the nonclogging case.
3.2 Motion of inclusions near solidification front
3.2.1 Forces acting on inclusion
By the directly observation [71, 72], when a liquid containing insoluble foreign particles is solidified, these particles can interact in different ways with an approaching solidification front : (a) the solidification front engulfs a particle instantaneously on contact; (b) the solidification front engulfs the particle after pushing it over some distance; (c) the solidification front pushes the particle ahead and segregate it in the last freezing liquid.
The interaction of insoluble particles with an advancing solid/liquid interface has been the subject of numerous theoretical studies since the early work of Ulmann et al. in 1964 [72-76]. Determining whether or not a particle is pushed or engulfed by the solidifying front can be obtained by analyzing forces acting on the particle [77-83]. Usually, it is considered that there is a repulsive force, FI (thermo molecular interaction force) which acts to push the particle away from the front, and there is an opposing drag force, FD that pushes the particle towards the front. In order to obtain the particle velocity achieved during interactions with a solidification front, a force balance is performed on particle ,
Typically, Repulsive force is derived from repulsive van der Waals interaction between the surfaces. Conversely, the origin of drag force is from a low-pressure region under the particle which is caused by the viscous losses experienced by the flow of melt into the thin gap from the bulk melt far away from the particle. When a solidification front approaches the particle to minor distance repulsive force in the gap become large and push the particle forwards. The movement of particle causes melt refill the gap resulting in a fluid dynamic drag force acting on the particle and prevents its motion , as illustrated in Figure 7 .
Due to the van der Waals force between a spherical particle and a plane wall, the expression of repulsive forces can be presented as follow ,
According to authors' view [87, 88], for the high interfacial energy systems of metal of particles, the behavior of the particle in front of solidification interface is mainly determined by the relationship of interfacial energies between the particle/solid/liquid phases. Kennedy and Clyne  suggested that for many metal/particle systems a force due to interfacial energy differences between the particle and solid with the liquid phase acts at the solid liquid interface which either engulf or push particles. It was stated that the particle movement was influenced by the variation of free energy between the particle and the solidification interface. If the interfacial energy difference ??0 is smaller than zero, and the contact angle between solid and particle is less than 90°, the particle could be engulfed into the solid even at very low solidification rate. The particle would be pushed along by the advancing solid liquid interface, if ??0 was greater than zero [73, 90, 91].
In the equation (10), the drag force is hydrodynamic in origin. But based on the different cases, the different expression can be used, as listed in Table 2.
Also, the existence of flow velocity in front of solidifying shell is known to be important, because of the detaching force by lift force caused by the velocity gradient, turbulence dissipation etc . According to Han's experiments , under certain fluid flow conditions, almost all insoluble particles in the melt can be rejected by the growing solid, even for solid growing at very high rates. This indicates that particle pushing is likely to be a result of fluid flow rather than of direct surface energy interactions. In addition, a model experiment using water was also performed to analyze behavior of non-metallic inclusions near the solidification shell under the fluid flow . It indicates that the main factor avoiding the nonmetallic inclusion entrapment is the force given by the horizontal flow. The Saffman force was recognized when the particles removed from the solidifying shell. Moreover, effects of the flow direction with respect to the rising/sedimentation velocity on the inclusion entrapment were also reported [96-98]. However, it is not still understood how the particle reach the solidifying front under the turbulent flow and are engulfed into the solidifying front.
3.2.2 Morphology and movement of solidification front
The interaction between particle and solidification front is very sensitive to changes in the geometrical arrangement [72, 99, 100]. G. Wilde  conclude that the morphology of the solid/liquid interface and its dependence on under cooling are crucial parameters for the different interaction modes. According to Juretzko , the term "engulfment" is used to describe incorporation of a particle by a planar interface, and the term "entrapment" is used for particle incorporation by a cellular or dendritic interface, because the mechanisms are rather different, as show in Figure 8.
The growth of crystal is a complicate process, overall solidification front morphology is governed by both scales as its shape changes  due to the thermal conduction phenomena, strongly anisotropic surface tension and kinetics, material properties, possibility even impurities, the detailed theory review was given by B. Caroli, et al. .
In the continuous casting process, the solidifying interface of molten steel is usually dendrite-like . For the case of dendritic solidification, the effect drives the conditions away from marginal stability thus leading to tip-splitting. Such morphology change increases the effective radius of the interface and changes the local curvature thus increasing the probability for engulfment . The interaction of particles and solidification front, particle in front of the dendrite tip and between dendritic primary or secondary arms, is illustrated in Figure 9. Particles that are entrapped by the first few side branches near to the dendrite trunk (a), can eventually become incorporated (b) and some can reside near the interior of the grains after complete solidification (c) . Thus, the velocity of the dendrite tip and the lateral growth velocity of the dendrite trunk should be considered in the analysis of critical velocities for particle incorporation during dendritic solidification.
The lumped thermal parameter, GR, when G is the thermal gradient and R is the solidification rate, has the dimension of cooling rate and the dendrite arms spacing (DAS) is observed to vary with cooling rate . The higher cooling rate is, the more uniform particles are distributed. Besides, with the increase of the cooling rate, the matrix grain size is smaller. These trends can be understood due to the fact that the growth of secondary dendrite arm will interrupt particle movement and accumulation. The higher cooling rate is, the denser the secondary dendrite arms are, and the smaller the secondary arm space is .
3.2.3 Effect of surfactants in front of phase interface
Surfactants always spread in front of phase interface and cause concentration gradient. The presence of a surfactant reduces the interfacial tension of a liquid near the interface and results in a interfacial tension gradient inside a boundary layer [109, 110]. The interfacial tension gradient along the interface between liquid and a foreign particle causes a driving force, interfacial tension driving force FIT, which drives particles towards the direction of the lower interfacial tension side and promotes the entrapment of them by the solidifying interface [111, 112], as shown in Figure 10.
Surfactant elements, O, S, N, Ti, etc. in molten steel are rich in front of solidifying interface and result in a concentration gradient layer, where the maximum concentration value is on the solidifying interface and minimum concentration value is on the layer with the distance dC away from the solidifying interface. So outside this boundary layer, the solute is assumed mixed very well and there is no concentration gradient in the rest molten steel pool . According to solidifying theory , the distribution of solute in the concentration boundary layer dC follows the equation,
Once foreign particles such as bubbles and inclusions enter the concentration gradient boundary layer in front of solidifying interface of liquid steel, the boundary layer produces an interfacial tension gradient around the particles [113, 115]. The force FIT, caused by the interfacial gradient drives the bubbles towards solidifying interface with higher surfactant concentration . Kaptay et al.  derived force from the aspect of energy. They gave the surface area of sphere particle, A, and the total interfacial energy, G, and the driving force, FIT, as:
However, Mukai and Lin  indicated that when a particle moves in the liquid with interfacial tension gradient, not only the total interfacial energy between the particle and liquid, but other kinds of energy such as the enthalpy of liquid and particle also change, which result in the change of total energy of system. Therefore, Kaptay et al.  overestimates the interfacial force and the interfacial force acting on the particle along x direction can not be derived from the derivative of the total interfacial energy. Mukai et al. [115, 118] derived the driving force that acts on sphere particle located in liquid with an interfacial tension gradient, as shown in equation ,
The relationship between interfacial tension driving force and other parameters, such as particle distance from solidifying interface, solidifying velocity, thickness of boundary layer, particle radius and surfactant concentration, can be obtained as shown in Figure 11,
3.3 Prediction of particle position on solidification front
Because the mushy zone in continuous casting only occupy a small part, and the calculation of particle engulfment or pushing in front of solidification requires mesh, which is much smaller than the mesh that has been used for solving the macroscopic transport and solidification. In order to save the calculation time and computer source, the separate models and computation domains will be applied in the study. The fluid flow, heat transfer, and the movement of particles in the turbulent field of continuous casting mold will be calculated in macroscopic zone. The information about the number and distribution of particle on the front of mushy zone, and the fluid flow and heat transfer condition near the mushy zone obtained from the previous model can be imported to mushy zone model and the further calculation work can be done. Finally, some models, such as dendrite growth model, solution segregation model and interaction between particles and solidification front model will be coupled in mushy zone. The quantitative result and final particle position in solidification front will be obtained. The detailed schematic of solution method is illustrated in Figure 12.
In this paper, several topics related to numerical modeling of nonmetallic inclusions in continuous casting are reviewed: (1) the sources and the basic methods to study nonmetallic inclusions in continuous casting; (2) motion of inclusions in turbulent region of casting mold; (3) interaction between particle and solidification front. Based on the analysis, a multi-scale model was suggested to solve the complicate multi-phase transport phenomena in both turbulent melt pool and laminar flow mushy zone. By this method the final distribution of particles on solidification front can be obtained.
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