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Coupling of CFD and PBE Calculations to Simulate the Behavior of an Inclusion Population in a Gas-Stirring Ladle JEAN-PIERRE BELLOT, VALERIO DE FELICE, BERNARD DUSSOUBS, ALAIN JARDY, and STE´PHANE HANS Gas-stirring ladle treatment of liquid metal has been pointed out for a long time as the processing stage is mainly responsible for the inclusion population of specialty steels. A steel ladle is a complex three-phase reactor, where strongly dispersed inclusions are transported by the turbulent liquid metal/bubbles flow. We have coupled a population balance model with CFD in order to simulate the mechanisms of transport, aggregation, flotation, and surface entrapment of inclusions. The simulation results, when applied to an industrial gas-stirring ladle operation, show the efficiency of this modeling approach and allow us to compare the respective roles of these mechanisms on the inclusion removal rate. The comparison with literature reporting data emphasizes the good prediction of deoxidating rate of the ladle. On parallel, a simplified zerodimensional model has been set-up incorporating the same kinetics law for the aggregation rate and all the removal mechanisms. A particular attention has been paid on the averaging method of the hydrodynamics parameters introduced in the flotation and kinetics kernels. DOI: 10.1007/s11663-013-9940-7 Ó The Minerals, Metals & Materials Society and ASM International 2013

I.

INTRODUCTION

THE reduction of the weight of high performance materials together with the improvement of mechanical properties and the increase of the recycling of used metal are new challenges which emphasize the importance of metal cleanliness. Ladle treatment of specialty steels has long been described as the secondary metallurgical process is mainly responsible for the non-metallic inclusion derived from the deoxidation process. As shown in Figure 1, the ladle is positioned in a sealed vessel and a primary pumping device allows the pressure to reach about 1 mbar. The main purpose of the degassing operation concerns the elimination or reduction of the dissolved gaseous components (nitrogen, oxygen, hydrogen) and the removal of sulfur from the liquid steel. Elements such as aluminum or calcium are introduced in the form of cored wires (Figure 1) with the aim of deoxidating the bath. Finally, an injection of argon through one or more porous plugs at the bottom of the ladle provides both mixing of the liquid metal to achieve thermal and chemical homogeneity and the entrapment of the inclusions by the bubbles well known as the flotation mechanism. The physical processes involved in gas-stirred ladles are numerous and complex owing to the 3-D and multiphase (metal-gas and inclusions) nature of the reactor. A precise JEAN-PIERRE BELLOT, Professor, VALERIO DE FELICE, BERNARD DUSSOUBS, and ALAIN JARDY, Scientists, are with the Institut Jean Lamour, UMR 7198, CNRS (‘LabEx DAMAS’)/ Universite´ de Lorraine, 54011 Nancy, France Contact e-mail: [email protected] STE´PHANE HANS, Scientist, is with the Aubert & Duval, Les Ancizes Steel Plant BP1, 63770 Les Ancizes, France. Manuscript submitted August 29, 2012. Article published online September 13, 2013. METALLURGICAL AND MATERIALS TRANSACTIONS B

description of the large number of interactions is necessary in order to obtain a good representation of the evolution of the inclusion population during the treatment. The mechanisms involved are: (a) the collisions between inclusions, which can lead to aggregation and even to agglomeration if reconstruction of the aggregate occurs, (b) the collisions with bubbles, which lead to the mechanism of floatation, (c) the entrapment at the interface between the liquid bath and slag coverage, (d) the separation induced by gravity, (e) the entrapment at the ladle walls. The modeling of the liquid steel ladle has already been the subject of many studies. Until the late ’80s, research on inclusion cleanness has been divided between thermodynamic studies aimed to determine the experimental equilibrium slag-metal-inclusion[1] and the first calculations on secondary metallurgical reactors.[2] Since the ’90s, the knowledge of thermo-chemical equilibrium has been capitalized on and has given birth to computing software, now widely used in the steel industry to predict the composition of stable phase inclusions.[3] Furthermore, the simulations of liquid metal processing were developed with an increasing level of sophistication. This type of work, performed under the leadership of KTH in Stockholm[4] and the University of UrbanaChampaign,[5] can be considered as a benchmark in this field. Numerical simulation tools have been set-up at KTH[6] coupling hydrodynamic, thermodynamic, and thermophysical modeling with the aim of evaluating the inclusion composition during ladle refining. Recently, a very innovative work was performed by Claudotte et al.[7] by coupling of a population balance equation VOLUME 45B, FEBRUARY 2014—13

zero-dimensional (0-D) geometry (based on the assumption of a perfectly stirred reactor). The main focus of this work is to investigate alternate approach for solving the PBE in the context of multiphase flow by using the class method (CM) and not the detailed modeling of the hydrodynamics in a gas-stirred ladle which has already presented in a recent publication.[13] The modeling integrates all the captures of particles at interfaces and the role of the different removal mechanism is compared.

II. HYDRODYNAMIC MODELING OF THE LIQUID-BUBBLES MIXTURE A. Description of the Model Fig. 1—Schematics of a ladle refining facility with degassing.

(PBE) with the computational fluid dynamics (CFD) to model the convective transport of particles and the probability of collisions due to the turbulent molten steel flow and the changes of particle properties due to breakup and aggregation. A variant of the quadrature method of moments (QMOM) has been adapted by the authors for the prediction of inclusion population in terms of chemical composition and size taken into consideration nucleation, diffusion growth, and aggregation. The combination of CFD and PBE is more and more adopted for modeling dispersed phases in complex metallurgical reactor. Hence, a fairly comprehensive and recent presentation of the modeling approaches is given by Zhang[8] and an application to the aluminum industry is proposed by Mirgaux et al.[9,10] However, most models used by these authors should be regarded as general models of processes that do not accurately describe the behavior and the capture of particles at interfaces (refractory wall, surface, and bubbles). Among the few experimental studies in this field, the contribution of Tohoku University[11] represents a laudable effort in order to make original experiments in hot or cold models. The innovative experiment of Xie et al.,[12] often referred to as the ‘‘Oeters’ ladle,’’ to measure the hydrodynamics of a metal bath stirred by argon bubbles is an important contribution to this research. In this paper we present the development, using the Fluent CFD code as a basis, of a 3-D simulation model taking into account the industrial geometry and operating conditions. This modeling is divided into two parts. First, the two-phase flow turbulent bubble-liquid steel is simulated for the 3-D geometry of the ladle and a strong coupling is achieved between the liquid metal and the gas. This hydrodynamic model has been applied to a 60-t steel ladle and the comparison has been set-up with the experimental measurement of the mixing time. In the second step, the behavior of oxide inclusions is simulated through the resolution of the PBE either in a 3-D geometry (taking into account the macroscopic transport of the particles by the molten steel flow) or in a 14—VOLUME 45B, FEBRUARY 2014

As the overall flow is 3-D in nature, we have developed a hydrodynamic model within the frame of the CFD code FLUENT. In order to simulate the twophase gas-liquid flow, an Eulerian-Eulerian scheme was chosen. This hydrodynamic simulation has already been the issue of a former publication and more detailed information is available in Reference 13. In this representation, the behavior of the dispersed phase (i.e., Ar bubbles) is governed by a set of partial differential equations that express continuity (1) and momentum transfer (2), similar to the continuous phase (i.e., liquid steel). This two-fluid model is expressed as follows:   @ ðak qk Þ þ r  ak qk uk ¼ 0 @t

½1

   @ ak qk uk þ r  ak qk uk uk @t ¼ ak rp  r  ðak sk Þ þ ak qk g þ Fk

½2

In our model, the local pressure p is assumed to be identical for the two phases. Fk represents all the interaction forces between the phases (Fl = Fg) per unit volume, coupling two sets of equations (for liquid and gas). Four interaction forces are accounted for: drag force, lift force, added mass effect, and turbulent dispersion. A particular effort has been paid on the expressions of these forces taken into account deformation of the bubble into a spherical cap.[13] The turbulence of the continuous phase flow (liquid steel motion) is calculated with the standard k-e model, and then results are used to simulate the turbulence of the dispersed phase flow (bubbles). An additional contribution to the liquid effective viscosity is introduced for modeling the turbulence induced by the bubble motion (Bubble-Induced Turbulence, BIT). At the inlet (porous plugs) and outlet (free surface of the bath) of the bubble flow, the model incorporates additional source terms in Eqs. [1] and [2] with the aim of modeling gas injection and degassing. A large number of user defined functions (UDF) has been incorporated in order to take into account the specific features of the hydrodynamic model.[13] METALLURGICAL AND MATERIALS TRANSACTIONS B

B. Hydrodynamics Prediction for a 60-t Ladle The numerical simulation is applied to a full scale 60 t steel ladle. Argon is injected through two porous plugs, located at the base of the ladle, into a bath of molten steel contained in a slightly tapered cylindrical vessel. Two hydrodynamic conditions have been simulated, first with a relatively small argon flow rate (less than 30 Nl/min for each plug), and second with a higher stirring intensity corresponding to 70 Nl/min for each plug. The model assumes a uniform liquid temperature equal to 1873 K (1600 °C). In Figure 2, the liquid steel velocity and argon plume region on a plane passing through the porous plugs of the ladle are shown in the case of relatively weak aeration. One can observe the shapes of the two bubble plumes rising from the two porous plugs (the isosurface of gas volume fraction equal to 1 pct is drawn). These regions are characterized by weak bath aeration as the distance from the porous plug increases. The gas volume fraction was found to be below 5 pct in each plume. The liquid metal flow is associated with two recirculation zones in each half of the plane of symmetry. The liquid steel velocities are consistent with the magnitude found in the literature for equivalent industrial configurations.[14,15] C. Mixing Time: Measurement and Calculation A shrewd way to characterize the turbulent hydrodynamic behavior of the bath consists of introducing a chemical tracer and measuring the time necessary for a full homogenization of the liquid phase. Such a measurement of a characteristic mixing time directly provides some valuable information on the efficiency of the bubble-induced stirring.[16] Furthermore, this experimental measurement can be easily compared to the model predictions, thus validating the hydrodynamic simulations.[17]

It was decided to use copper as a tracer. The computation of solute transfer requires the solution of an additional transport equation:     ll;t @al ql xCu þ divðal ql ul xCu Þ ¼ div al ql DCu þ grad xCu ; @t Sct

½3 where DCu is the molecular diffusivity of diluted copper in liquid steel (taken here equal to 1.6 9 1010 m2 s1[18]), ll,t is the turbulent viscosity of the liquid, and Sct is the turbulence Schmidt number, assumed to be equal to 0.7. In the right hand side of Eq. [3], the molecular diffusion can be neglected in comparison with the turbulent diffusion. During the industrial trials, copper platelets were added into the metal bath, and samples were taken at time intervals of about 50 seconds. Since the location of platelet immersion is not accurate and their behavior in the bath is not well known (for example the melting rate is influenced by the formation of a solid steel shell and the tracking of the sample in the bath is unknown), several simulations were conducted with varying points of addition of copper platelets (1 to 3) and a fixed sampling location (P). Figure 3 shows a relatively strong dependence of mixing time with initial tracer location, especially when the addition is simulated on the symmetry axis of the ladle; consequently, resulting in low values for mixing time compared to other addition locations. The numerical simulation seems to underestimate the bath mixing time, but the order of magnitude (2 minutes) is in agreement.

III.

MODELING THE INCLUSION BEHAVIOR

The behavior of the inclusion population, defined by a distribution function of particle size (Ni is the number of inclusions of class ‘‘i’’—volume of the particle is between vi and vi+1—per m3 of liquid steel), is described by the PBE. Equation [4] represents the macroscopic transport phenomena of inclusions (left member), and mesoscopic phenomena such as bubble-inclusion (flotation Zbi) and inclusion-inclusion (aggregation Bi  Di) interactions: @al Ni þ divðal ul Ni Þ ¼ al ðBi  Di Þ  al Zbi  Si @t

Fig. 2—Predicted velocity of the liquid steel along with the argon plumes (isosurface of the 1 pct gas volume fraction) in a vertical plane passing through the porous plugs for a weak aeration. METALLURGICAL AND MATERIALS TRANSACTIONS B

½4

In Eq. [4], Si is the inclusion gravity separation term. The numerical solution of the transport equation [4] has been obtained following two approaches. The first method uses the spatial 3-D mesh developed in the hydrodynamic modeling and integrates this equation in each cell of the grid. This is a 3-D modeling combining CFD and solution of the local PBE. The second method integrates the Eq. [4] on the volume of the reactor VR assuming a perfect spatial homogeneity of the inclusion distribution Ni at each time step. This is a 0-D modeling based on the solution of the PBE applied to the whole volume of the reactor. VOLUME 45B, FEBRUARY 2014—15

P

1 2

3

3 2 P

1

0,200

Cu concentration (%)

0,175

0,150

0,125 Addition location 1 0,100

Addition location 2 Addition location 3

0,075

Measurements 0,050 20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

Time (s)

Cu addition

Fig. 3—Predicted Cu concentration profiles at the P sampling point compared with plant measurements for three platelet addition locations in the gas-stirred ladle treatment.

A. Description of the 3-D Model The transient solution to this equation can be obtained by separating the transport and collision operators.[19] In the first part of the time step, the transport equation of the quantity Ni is solved using the Finite Volume Method: @al Ni þ divðal ul Ni Þ ¼ 0 @t

½5

In the second part of the time step, the population balance (Eq. [6]) is solved in each control volume applying the cell average technique[20] which is a variant of the fixed pivot method of Kumar and Ramkrishna[21] leading to a significant reduction of numerical diffusion: @al Ni ¼ al ðBi  Di Þ  al Zbi  Si @t

½6

The aggregation and flotation kernels are an issue of great development and the reader will find details of the physical phenomena and of the applied models in Reference 22. Concerning the separation induced by gravity, following the decomposition of particle velocity into local fluid velocity and Stokes velocity, the source term Si for the transport equation is: Si ¼ divðal us Ni Þ 16—VOLUME 45B, FEBRUARY 2014

where us is the vertical Stokes velocity in the case of small inclusions whose particle Reynolds number is lower than 1. The inclusion entrapment at the liquid metal/slag interface is modeled following the approach based on a deposition law developed initially by Wood[23] and adapted to a free liquid surface condition by Xayasenh.[24]

½7

B. Description of the 0-D Model In order to model the ladle, a simplified approach is possible: the homogeneous ladle system approach (0-D model). The ladle is assumed to be perfectly agitated, so that all magnitudes of interest are considered constant within the bath. Thus, macroscopic transport of inclusions is not considered and the evolution of the system is calculated by solving the PBE (Eq. [8]) where the quantities are volume averaged over the entire ladle. @Ni ¼ ðBi  Di Þ  Zbi  Si  Ci @t

½8

These quantities are: gas hold up, the bubble size and velocity, the sliding velocity, the turbulent kinetic energy, and its dissipation rate. From a practical perspective, the homogeneous ladle model code has been directly derived from the numerical METALLURGICAL AND MATERIALS TRANSACTIONS B

heterogeneous (3-D) ladle code. Thus, agglomeration and flotation models are identical. On the contrary, separation induced by gravity and the surface capture contribution have to be computed on the size of the entire reactor of height H: Si ¼

us N i H

½9

Ci ¼

ud N i ; H

½10

where ud comes from Reference 24. C. Average of Hydrodynamic Quantities A crucial point in the 0-D approach is represented by the average of hydrodynamic properties inside the reactor. The underlying assumption is that the liquid metal in the reactor is sufficiently uniform (high mixing effects) in order to ensure that all the variables of interest present small deviation respect to their average value. In particular, quantities that derive from the 3-D hydrodynamic simulation, such as the turbulent dissipation rate, are capital in the calculation of agglomeration and flotation kernels and thus directly determine the results and the accuracy of the 0-D model. Thus, we focus on the right estimation of the agglomeration kernel. We consider for the sake of simplicity the Saffman and Turner collision kernel model. We recall that good estimation of the average agglomeration kernel calculated from local values of the 3-D simulation is crucial to obtain good agreement between 3-D and 0-D results. The collision kernel bij, for two given classes i and j, only depends on the square root of the turbulent dissipation rate: þ bTurbulent bij ¼ bStokes ij ij

Stokes

bij

Turbulent

bij

½11

!    2pgql  qp  dp;i þ dp;j 2 d2p;i d2p;j  ¼ ½12 2 2 2 9ll rffiffiffiffiffiffi   8p dp;i þ dp;j 3 e 0:5 ¼ ! bTurbulent  di;j e0:5 ij 5 m 2 ½13

Taylor series expansion of Eq. [13] and integration over the ladle volume VR is thus performed:   @bi;j  @ 2 bi;j  ðe  eÞ2   þ  ðe  eÞ þ bi;j ðeÞ ¼ bi;j ðeÞ þ @e e @e2 e 2 ½14 Z

Z @ 2 bi;j  ðe eÞ2  dv þ Oðe eÞ4  : bi;j ðeÞdv  bi;j ðeÞVR ¼ @e2 e 2 ½15

METALLURGICAL AND MATERIALS TRANSACTIONS B

With e ¼

1 VR

Z edv

½16

VR

Equation [15] shows that the difference between the average value of the collision kernel over the ladle and its estimation obtained via the mean value of turbulent dissipation rate is more important when the variance of e in the ladle increases. In order to avoid that error, after some mathematical manipulations, Eq. [15] suggests the proper estimation of the average turbulent dissipation rate which corresponds to the best estimation of the average collision kernel:  2 R e1=2 dv2 e ¼ e1=2 ¼ VR 

½17

Equation [17] provides, new estimation of e which guarantees the required accurate prediction of the average collision kernel for the 0-D model.

IV.

RESULTS AND DISCUSSION

With the aim of highlighting the capability of such numerical models to predict the time evolution of an inclusion population, a typical initial distribution of inclusions is considered in the 60 t stirred ladle for the hydrodynamics conditions corresponding to a high stirring intensity. According to the literature and in particular the review proposed by Zhang and Thomas[25], we have used a log-normal distribution as an initial particle size distribution (psd) with a total mass content equal to 0.176 kg/m3 corresponding to 7.9 ppm of total oxygen (TO) content—considering a population of calcium aluminate inclusions. This distribution matches relatively well with those analyzed in industrial ladle since one kilogram of metal contains 5 9 106 inclusions smaller than 20 lm but only 5.7 9 104 larger than 20 lm. Unfortunately the log-normal law cut off the distribution for the inclusions larger than 30 lm so that the density number Ni for inclusion size larger than 50 lm is equal to zero. It does not fit with accurate cleanliness measurements detecting rare (but existing) large inclusions (>50 lm). The initial distribution is plotted in Figure 4 where the inclusion population is rendered discretely into twenty different sized classes. In order to compare the two approaches, we investigate the evolution of the numerical density of inclusions remaining in the ladle over time, the evolution of the total mass of inclusions, and the Sauter’s diameter. The global rate of inclusion removal corresponding to a deoxidation rate is also presented and discussed (comparison with experimental data from literature is provided). VOLUME 45B, FEBRUARY 2014—17

Fig. 4—Initial and 300 s psd for the 0-D and 3-D models.

Fig. 5—Flotation and separation induced by gravity elimination rates vs time: 0-D and 3-D comparison.

A. Size Distribution In Figure 4 we compare the initial psd (white) with the one calculated after 300 seconds of treatment, respectively, for the 3-D model (gray) and 0-D model (black). In both cases larger size particles appear during treatment due to the agglomeration of smaller size particles (their number strongly decreases with time). For small classes, where the number of inclusions is very important, the slight overestimation of the agglomeration and flotation frequencies in the 0-D homogeneous system are almost not noticeable. In contrast, for large classes, where the number of inclusions is small and in which flotation and separation phenomena become important, there is a slight difference between the two models. This slight difference for the large classes can be readily explained by the comparison between flotation 18—VOLUME 45B, FEBRUARY 2014

and settling rates reported in Figure 5. The weak underestimation of the settling rate by the 0-D model is not balanced by the overestimation of the floatation rate. Otherwise we notice that the inclusion removal rate gradually increases with time because the psd moves toward larger classes and the removal efficiencies increases with the inclusion size. The surface entrapment represents at this stage of the treatment a second order effect [the rate has O(107 kg/s) magnitude] respect to flotation and settling. B. Mass and Sauter’s Diameter Evolution In order to further compare the two approaches, we reported on the graph in Figure 6 the evolution of the total mass of inclusions remaining in the ladle over time METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig. 6—Total mass of inclusions vs time.

Fig. 7—Evolution of Sauter’s diameter over time.

and of the Sauter’s diameter, for both 0-D and 3-D models. The results calculated by the homogeneous model are provided for both e and e in order to point out the importance of the averaging method already discussed in Section III–C. Figures 6 and 7 show really good agreement between the models when e is used for the collision kernel calculation. On the contrary the choice of e leads to an important overestimation of the agglomeration and floatation phenomena. The small difference observed between the two models, allow us to conclude that the 0-D model is a very efficient approach to predict the evolution of the total number of inclusions within the ladle. C. Deoxidation Kinetics A quite common tool for analyzing the inclusion removal is given by the deoxidation rate assuming a constant nature of inclusions (calcium aluminate in the METALLURGICAL AND MATERIALS TRANSACTIONS B

present case studied) during the process. The deoxidation kinetics are usually represented by a first order linear ODE, reported in Eq. [18]: dC0 ¼ K0 C0 ; dt

½18

where C0 represents the TO and K0 is a kinetic constant (min1). Zhang and Thomas[25] have plotted industrial efficiencies in a diagram for different stirring practices (Figure 8). Although the deoxidation mechanism does not fit with a first order law (it can be easily explained by the nonlinear mechanism played by the aggregation, see Reference 13 for detailed explanations) we observe in Figure 8 that the range of variation of the calculated deoxidation constant K0 (1.9 9 102 min1 initially and 3.3 9 102 min1 at t = 5 minutes) matches relatively well with reported industrial data in the diagram. VOLUME 45B, FEBRUARY 2014—19

Fig. 8—Effect of stirring power on deoxidation rate according to Zhang and Thomas.[25]

Fig. 9—Map of the total number density Ntot (#/m3) at t = 300 s in the symmetry plane.

D. Three-Dimensional Effects in Inclusion Transport In Figure 9 the total number density map (Ntot in particles/m3) is shown. Non-homogeneity in the ladle implies that the characteristic times of aggregation mechanisms, flotation, and entrapment cannot be regarded as infinitely large in comparison with the transport characteristic time. In the area of the plume and near the free surface, high turbulence and retention rates activate these mechanisms and lead to a local reduction of Ntot respect to the average value in the ladle. It is interesting to notice that despite the steel bath is not perfectly mixed the average technique makes accurate the homogeneous 0-D model.

V.

CONCLUSIONS

Nowadays the numerical modeling affords the possibility to simulate complex three-phase metallurgical 20—VOLUME 45B, FEBRUARY 2014

reactor by combining CFD and Population Balance resolution. Hence, a mathematical model of inclusion behavior in a gas-stirred ladle has been built up step by step. First, the turbulent flow is simulated using the EulerEuler method, taking into account all the interaction forces between the two phases (gas bubbles and liquid steel), i.e., drag, lift, added mass, and turbulent dispersion. A satisfactory validation is obtained with the prediction of the bath mixing time. Second, the numerical model handles both agglomeration and removal mechanisms (flotation, gravity separation, free surface entrapment) together with the convective transport of inclusions into the melt. The coupling of convective transport equation and the PBE is achieved using a splitting technique. The discrete CM with the cell average feature was adopted to solve the PBE and was implemented into the Fluent CFD code. As far as we know, the combination of CM technique (including all the removal mechanisms) and CFD calculation represents a new approach for the simulation of the gas-stirred ladle process. As an illustration the model is applied to an industrial type ladle with a population of inclusions where the size initially ranges between 2 and 40 lm. The results emphasize that the aggregation mechanism plays a major role by translating the PSD toward the large size and leading to an increase of the Sauter’s diameter. The removal mechanisms are mainly the flotation and gravity separation, whose efficiency increases with the inclusion size. Comparison between the calculated deoxidation rate and some literature data reported by Zhang and Thomas[25] shows a relatively good agreement, but it has been demonstrated that the deoxidation mechanism does not fit with a first order law, which is often used in the literature. It was also shown that, in this range of stirring intensity and for the confined system considered, a 0-D simplified approach provides an excellent prediction of the variations of psd with process time. It appears that the 0-D simulation is an elegant way for simulating the behavior of an inclusion population because of the strong reduction of the computational effort. However, the turbulent dissipation rate must be carefully averaged in order to guarantee a proper estimation of the aggregation and floatation mechanisms. We have highlighted that the square of the mean square root provides an accurate calculation of the average collision kernel.

NOMENCLATURE

VARIABLES Bi d Di Ci C0 DCu F

Birth rate (m3 s1) Diameter (m) Death rate (m3 s1) Capture rate (m3 s1) Total oxygen content (ppm m3) Diffusion coefficient of copper in liquid steel (m2 s1) Interaction force density (kg m2 s2) METALLURGICAL AND MATERIALS TRANSACTIONS B

g H K K0 Ni P t u Si V Zbi

Gravitational constant (m s2) Reactor height (m) Kinetic energy of turbulence (m2 s2) Kinetics constant (min1) Number density of inclusion of class i (m3) Pressure (Pa) Time (s) Fluid velocity (m s1) Settling rate (m3 s1) Volume (m3) Flotation rate (m3 s1)

GREEK SYMBOLS a b e l q s x

Volume fraction Collision kernel (m3 s1) Dissipation rate of the kinetic energy of turbulence (m2 s3) Dynamic viscosity (kg m1 s1) Density (kg m3) Tensor of viscous sheer stress (kg m1 s2) Mass fraction

SUBSCRIPTS d g i k l p s

Deposition Gas Bin Fluid phase index Liquid Particle Settling

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METALLURGICAL AND MATERIALS TRANSACTIONS B

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