A CFD analysis of the air entrainment rate due to a plunging steel jet combining mathematical models for dispersed and separated multiphase flows

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1 A CFD analysis of the air entrainment rate due to a plunging steel jet combining mathematical models for dispersed and separated multiphase flows arald Laux and tein Tore Johansen INTEF Materials Technology roup for Flow Technology Alfred etz vei 2B 734 Trondheim haraldl@matek.sintef.no Abstract The plunging jet in tapping of steel from converters or electric arc furnaces entrains gas into the steel bath contained in a ladle. as entrainment rate and the effect of the gas on the flow pattern in the ladle are studied by means of Computational Fluid Dynamics CFD. The work focuses on the mathematical modeling. Two numerical methods, one for free-surface flows and one for dispersed multiphase flow are combined. A transport equation for the average bubble diameter that incorporates the effect of coalescence and breakup is presented. Results from computations for one ladle and tapping configuration are shown. Our findings indicate that the gas entrainment rate is affected by the material properties of the liquid in the sense that the material properties determine the shape of the jet. Moreover, bubble sizes in the range of two-orders of magnitude are predicted in the ladle. The dispersed diameter model seems to work well. It showed the expected response to material parameters and the turbulent field. owever, the presented analysis is just a first step that was limited due to certain simplifications. In order to obtain better insight in the tapping process improvements of the analysis seem necessary.

2 Introduction In most of the steelplants worldwide the major part of alloying material is added during tapping of converters or electric arc furnaces. In a recent work it has been shown that the yield of the alloying process can be optimized if the alloy size, the alloy injection point, and the addition timing are chosen properly (Berg et al. []). As for plunging water jets it can be assumed that the plunging steel jet entrains air into the steel bath. Possible effects of such air entrainment to the flow pattern in the ladle, as bubbleinduced flow, have not been included into the analysis of Berg et al. []. In air-water models for the tapping process, however, e.g., Tanaka et al. [2] have shown that air entrainment can have a profound impact on the circulation pattern. Air entrainment is also important with respect to undesired nitrogen pick-up in steel (Choh et al. [3]). Of special interest are the air entrainment rate and the average size of the emerging bubbles. For air-water systems measurements of these quantities are easily accomplished and a large number of experimental works on this topic have been performed, Bin [4]. Corresponding experiments are difficult to perform in liquid metals under operation conditions and therefore, to our knowledge, almost no experimental correlations are found in the literature that allow calculating the air entrainment rate of a plunging metal jet. Therefore, such information must be obtained by either new experimental techniques or by other means. An alternative, that we apply here, is Computational Fluid Dynamics (CFD) which offers a detailed physical description of metallurgical flow phenomenons, Johansen [5]. We investigated a test problem where a continuous and smooth steel jet hits the melt surface. In this case a gas sheet forms around the jet and is drawn under the surface [4]. Due to the shear forces in the turbulent flow the sheet disintegrates in a certain depth and bubbles are formed that are dispersed by the turbulent flow and that rise eventually to the surface. The tapping of steel is a physically complex multiphase process, as it involves three physical phases: dispersed air bubbles, the slag layer on the steel bath surface and the liquid steel. The phases are separated at the free surface of the steel bath (steel-slag) and the slag layer (air-slag), but also mixed: air bubbles that are dispersed within the melt. Different numerical methods have been developed to treat separated (free-surface) and dispersed multiphase flows. New to our CFD analysis is that it combines mathematical models and numerical techniques for free-surface and for dispersed multiphase flows. It should be noted that the slag layer on top of the steel bath is assumed to be so thin that it can be neglected in the analysis. We present a two step analysis. In a first step the air entrainment rate is computed by using a volume-offluid model (VOF). The VOF model is designed for free-surface flows and provides the air entrainment rate. Because the typical length scales of the bubbles and the overall ladle geometry differ by orders of magnitude, the bubbles can not be resolved on a feasible computational mesh. In a second step, therefore, the computed air entrainment rate is used as inlet condition for a two-fluid model (designed for dispersed multiphase flows). The results of the two-fluid model computations show how the flow circulation pattern of the steel is affected by the entrained air and allows calculating the surface area of the bubbles in the steel bath. The latter quantity may be applied directly in the prediction of nitrogen pick-up. Transient and axisymetric computations for both air-water and air-steel systems for one ladle configuration are conducted. Actual operation conditions in the steel plant are simulated. Mathematical modeling Volume-of-Fluid model VOF methods are designed for separated or free-surface flows. In a VOF simulation the time evolution of the interface between two immiscible fluids is followed in the computational domain. imulations can be performed by several variants of the VOF method. Important is that they are capable of tracking the sharp interfaces without excessive numerical diffusion. In this work we apply the VOF model of Johansen [6] which is based on straightforward conservation methods. The numerical model is implemented in an older version (3.4) of the commercial CFD code Fluent. Dispersed Diameter model The two-fluid model comprises governing equations for the liquid metal including equations for turbulence in the liquid phase (two-equation turbulence model) and the governing equations for the air bubbles. It was used to compute the effect of the entrained gas on the flow pattern in the ladle. everal features of the present modeling are new to two-fluid modeling:

3 ) New and essential to the model is that the auter mean diameter of the bubble size distribution function is described by a transport equation. This computed diameter is fully coupled to the friction coefficient in the drag term of the momentum equations and the dispersion terms in the equations for turbulent kinetic energy. 2) Coalescence and breakup are described by introducing corresponding time scales into the dispersed diameter transport equation. 3) We use an Eulerian-granular model [7,8] and apply it to the dispersed bubbles, i.e., we assume that the dispersed phase has effective properties, as a dynamic bubble phase viscosity, that are due to bubble-to-bubble interactions and not related to the material properties of the air. This approach has been successfully applied to flow of droplets dispersed in a gas [9]. The computations are performed with the commercial CFD code Fluent version The governing equations for Eulerian-granular multiphase flow and the two-equation model for particle-laden turbulent flows, the so-called dispersed turbulence model, are standard physical models in Fluent They can be looked up in the Fluent User s uide [7] and are therefore not given here. The transport equation for the average bubble/droplet diameter on the other hand is new and has been implemented into the source code of Fluent by the authors. It is therefore described in detail. Note that the dispersed turbulence model is a k-ε type model for fluid turbulence that comprises the effect of dispersion of particles by the fluid turbulence. It gives the best results for bubble volume fractions below %, but may, also be applied to cases with moderate volume fractions. Moreover, the well-known phenomenon of turbulence production by large bubbles [] is not included in the turbulence model. owever, the bubble volume fraction exceeds % percent only locally and is most often smaller than 5% and the plunging jet generates large values of turbulent kinetic energy such that we assume that the effect of the turbulence production due to bubbles is small. Transport equation for average dispersed diameter. The evolution of the mean of a local bubble diameter distribution is modeled as follows. Consider any volume within the system, e.g., typically the volume of a computational cell. All bubble sizes within the cell constitute a local diameter distribution. During any time interval, the diameter distribution may change due to: (i) the cell under consideration looses bubbles to downstream cells; (ii) it receives bubbles from upstream cells; (iii) some bubbles may break and form smaller ones; and (iv) others may coalesce and form larger bubbles. Thus, the local rate-ofchange of the mean of the local bubble diameter distribution, represented by the local time derivative, is balanced by convective and diffusive fluxes, and a source (equilibrium) term which accounts for breakup and coalescence. We can formulate this balance as ( ) ( ) ( ) W ρ + M ρ 8 M ' II M = ρ () τ UO ere, is the instantaneous mean bubble diameter, ρ = α ρ is the bubble bulk density which equals the product of the bubble volume fraction α d and material density ρ d, U d j is the mean bubble velocity in direction j, and D eff is the effective bubble dispersion coefficient. For a turbulent flow it is given by ρ ' II = α µ W (2) ρ ere, µ t is the turbulent viscosity of the fluid, which is computed by the k-ε model. The term on the right hand side of equation () is crucial to the model. It forces the mean bubble diameter towards its equilibrium diameter, which equals the local mean diameter of bubbles in a turbulent flow when completely adjusted to the local hydrodynamic conditions. Besides, the equilibrium term contains also the relaxation time τ rel which is characteristic for the time needed to reach the equilibrium. We solve the transport equation () coupled to, and simultaneously with, the governing equations of the two-fluid model. The numerical boundary conditions are the zero-gradient condition ( Q =, n coordinate normal to boundary) at walls, symmetry cells, and outlet cells. At inlet cells the value of must be known explicitly.

4 Equilibrium diameter. The local mean diameter of bubbles that are dispersed in a turbulent flow field are given by Calderbank [].6 σ P Q & ρ µ = α + &.4 2 (3) ε µ ere, ε is the dissipation of turbulent kinetic energy, µ d is the material viscosity of the bubbles, and σ is the surface tension between bubbles and fluid. Calderbank [] studied experimentally gas-liquid and liquid-liquid systems for a wide range of parameters. For all systems studied n was close to.5. For airbubbles in water, the mean diameter was found to be independent of the viscosities and m=. We adopted m= and n=.5. Calderbank [] found for air-bubbles in water values of C =4.5 (dimensionless) and C 2 =9 µm. owever, his data were obtained for bubbles in an impeller stirred tank and the constants may not be valid for the plunging jet. Bin [4] reports air bubble sizes in water below C 2 are reported. Therefore, we used C =4 and C 2 =. Relaxation time. The time needed to relax to the mean diameter as given by equation (3) is controlled by the speed of the breakup or the coalescence process. We assume that if the instantaneous mean diameter is smaller than its equilibrium value, coalescence occurs more frequently than breakup and model then the relaxation timescale as a characteristic timescale for the coalescence process. On the other hand, if the mean diameter is larger than its equilibrium value then breakup occurs more frequently and the relaxation time becomes a characteristic timescale for the breakup process τ LI > % τ = (4) UO τ LI < & We restrict the relaxation time by the turbulent microscale that represents the smallest timescale in a turbulent flow: τ = τ, τ. The turbulent microscale is given by τ = 6 Y / ε, where ν is the UO UO N max N kinematic viscosity of the fluid. Breakup timescale. Breakup of a bubble occurs if the turbulent shear forces exceed the resistive surface tension forces [2,3]. The bubble is sheared by the turbulent eddies it is exposed to. Turbulent eddies that are large compared to the bubble size contribute not to breakup but move the bubble around. maller eddies contribute not either, as they are too small to shear the bubble. owever, eddies of comparable size may cause breakup. The lifetime of such eddies is thus the maximal time interval that is available for breakup and we can write τ = & τ (5) % % ere, C B is the proportionality factor and τ is the lifetime of eddies of size. Factor C B is in general a function of dimensionless numbers that characterize the different forces acting on the bubbles: Weber [2,3], bubble Reynolds [2], and Viscosity [3] number. We employed C B = and leave a more advanced modeling to future work. For droplets the dependence on these number has already been given [9,2,3]. If eddies which contribute most to breakup are located in the inertial subrange [4], then we can 3 calculate the eddy lifetime by using an estimate for the dissipation [4]: ε = X UPV O. ere l is an appropriate turbulent length scale and u rms =l/τ e is the turbulent velocity scale of eddies of size l. As discussed above, eddies with length scales of the order of the bubble diameter contribute most to breakup; thus, l= and τ = ε (6)

5 Timescale for coalescence. We assume that coalescing bubbles are brought into contact by turbulent velocity fluctuations, and model the characteristic coalescing time scale as V τ = & (7) & & X56 ere, C C is the proportionality factor, X is the turbulent rms velocity of the bubbles, and s is the 56 mean bubble spacing ( α α ) 3 π max V = 6 (8) α where, α max is the maximum bubble volume fraction. In isotropic turbulence X is per definition 56 proportional to the square root of the kinetic energy of the bubble phase turbulence 2 X 56 = N (9) 3 For the dilute flow considered in this work, k d is directly related to the turbulent kinetic energy of the fluid phase k [8] N N = + 6W () ere, 6W is the tokes number 6W = τ / τ, τ is the dynamic relaxation time of the bubbles, and τ is / / the characteristic time of the energetic turbulent eddies. Large bubbles with large slip velocities (t>>) cross turbulent eddies almost unaffected (k d << k). mall bubbles (t<<) follow turbulent fluctuations in the fluid closely (k d k). The timescales are functions of the flow and the turbulent field variables, and explicit expressions are given in the Fluent User s uide [7]. The final expression for the coalescence timescale is ( α α ) 3 π max τ = & + 6W & & () 6 α 2 N 3 Note that C C = means that every encounter results in coalescence. A smaller coalescence probability can be modeled by assigning large values to C C. We employed C C =2. Definition of test problems Computations of gas entrainment into liquid steel and water caused by a plunging jet were performed. The material properties are summarized in Table. The plunging jet provides a volume flow of 4 & =4.96 m 3 /min. It issues from the tapping hole of an electric arc furnace with diameter of 8 mm and a velocity of 3.25 m/s (realistic operation conditions []) For all computations 2% of the ladle volume is filled with liquid. In the VOF simulations the liquid level rises as new fluid enters with the plunging jet, in the two-fluid simulations the flow field is computed only for the case of 2% filling. The modeled tapping system is shown in figure. Table I Material properties of steel, water, and air a. ystem ρ d (kg.m -3 ) µ d (mpa.s) ρ (kg.m -3 ) µ (mpa.s) σ (N/m) T (K) air-steel air-water a Note that index d for dispersed denotes the bubble phase (air). All computational runs used axisymmetric models of the cylindrical ladle. This was justified because the jet hits the liquid surface in the ladle center. Moreover, isothermal conditions. This is true for the airwater system. Also for the air-steel system this is a good assumption, as the filling time of the ladle is small (a few minutes) and the ladles are isolated and preheated []. The entrained gas is assumed to have the same temperature as the surrounding steel or water.

6 Volume-of-Fluid model Four simulations were performed using the VOF model. The computations differ in the material properties and the mesh resolution. The important numerical properties are summarized in table II. The computational domain covers the ladle up to 4% filling. The jet enters at the top and hits initially the liquid surface at 2% filling (after a fall height of.58 m inside the domain). The total fall height from the tapping hole to the inlet is 3.24 m, and the inlet velocity is 8.6m/s. Because of the gravitational acceleration it increases initially by.64 m/s before plunging onto the liquid surface at 2% filling. The jet diameter at the plunging point was.68 cm. As shown in figure the boundaries of the domain consist of walls for gas and liquid phase, the symmetry line through the center of the ladle, an inlet for the jet, and an outlet for the circulating gas above the liquid surface. The used boundary conditions are standard: no-slip condition at the walls; zero-gradient conditions at outlets and symmetry line; and a prescribed uniform velocity with zero gas volume fraction at the inlet. Figure Tapping configuration to the left, and computational domains for two-fluid and 92) simulations to the right. A time step control was implemented that adjusts the time step such that a maximal Courant number of approximately one was maintained. owever, the time step was mostly of the order of t = -4 s. The total computed time was six seconds. That was sufficient to establish a more or less stationary flow field in the ladle. Table II List over the performed computations. Run number Mesh resolution Model Materials VOF air-water VOF air-water 3 82 VOF air-water VOF air-steel two-fluid air-steel two-fluid air-water single-phase air-steel single-phase air-water Eulerian modeling Four runs were performed (table II). In runs 5 and 6 we applied the dispersed diameter model, and in runs 7 and 8 we solved the single-phase Navier-tokes equations to obtain the flow pattern without gas entrainment. It was assumed that the liquid surface is horizontal because this Eulerian type of modeling does not permit to compute the surface shape. Therefore, we computed the flow only in the part of the ladle that is filled with liquid. The computational domain included consequently the ladle geometry up to 2% filling (figure ).

7 As shown in figure the boundaries consist of the ladle side- and bottom walls, the symmetry line and the top surface. The latter is divided into inlets for the plunging jet and the gas sheet. The larger part of the top surface has a prescribed liquid mass flow rate out of the domain that balances that mass inflow by the jet. The inlet conditions for the gas sheet were taken from runs to 4: entrainment rate, 4 & / 4 & = OLT.2; and sheet thickness =.78 cm. The computations were performed with a time step of t = - 3 s until the circulation pattern in the ladle was established and the bubble dispersion close to the jet reached a quasi-stationary state. Results and discussion VOF simulations To obtain the exact gas entertainment rate from the VOF computations has proved to be a difficult task. Therefore, we computed the following integral at different heights below the liquid surface 4& 2π 5 α max, X UU (2) where R is a small radial distance from the centerline (R=), but larger than the jet radius (R jet =5.34 cm) at the plunging point. The integral gives the volume flow rate of air within and around the plunging jet. Only the air that moves downwards, u>, is included into 4 &. Variation of the integration distance R between and 2 cm gave approximately the same results. Thus, the air moves mainly downwards close to the jet. The integral varies in time, but after a strong initial transient it fluctuates about a more or less constant value. Therefore we averaged the calculated 4 & over the last three seconds of real time and used this averaged value in figure. Note that the resulting values are normalized with the flow rate of the liquid 4 & and are shown for different depths. The gas volume moving downwards around the jet OLT decreases with increasing depth. This is because some of the buoyant gas already rises towards the surface in form of bubbles. Thus, 4 & -values close to the ladle bottom do not reflect the correct entrainment rate. Also values close to the surface can not be used, as the plunging jet depresses the surface locally and thus close to the surface gas which is not actually entrained is included in the integral, equation (2). The variation of the 4 & -values between the different runs is smallest for h =.2 m below the surface (.8 < 4 & / 4 & <.25). Therefore, we used for the computation with the two fluid OLT model an entrainment rate of 4 & / 4 & =.2. OLT It is striking that the curves for the air-steel and the air-water system in figure 2 are almost equal. We would have expected a much larger entrainment rate for water because of the considerable lower density difference. We explain this as follows. A jet can be characterized by the Weber number based on the gas 2 density: : = ( ρ 8 ) σ. The subscript jet denotes the jet velocity and jet diameter, e.g., at the JDV MW MW plunging point. Large We indicate high velocity jets for which friction at the jet surface with the surrounding gas plays an important role, and which leads to total disintegration already shortly after it emerges from the nozzle [4]. On the other hand, We < 2 indicates more or less smooth jets that breakup only because of surface tension forces [5] and which are therefore much more stable than high velocity jets. Breakup is related to surface roughness of the jets. That we obtained almost the same results for water and steel jets strongly suggests that the surface roughness of the jet is the predominant feature for the gas entrainment. ere, the respective We numbers were.5 and 8 for steel and water jet. Thus, the steel jet is expected to be smooth and continuous (as modeled), whereas the water jet of a real system would have disintegrated after the given fall height and would have caused the larger gas entrainment rates that are predicted by empirical correlations [4]. Because the predicted entrainment rates are nearly equal for the two materials (runs and 4) we conclude that, for a given shape of the jet, the material properties play a minor role for the entrainment, but that they play a major role in the sense that they determine the shape of the jet. Thus, runs 2 and 3 (water, fine mesh) are also representative for steel. In the VOF simulations the general flow field is established approximately after 3 seconds real time. The results show that the jet does not spread at once after it enters the liquid bath, but is protected by a gas sheet against shearing with the surrounding fluid. In this manner the plug velocity profile of the jet almost reaches the bottom of the ladle. Moreover, the momentum carried by the jet is so large that the surface remains not horizontal but is disturbed and somewhat wavy. The waviness is due to a certain

8 overshoot along the sidewall. owever, this waviness reduces with increasing liquid level in the ladle. We observed also that close to the plunging point liquid droplets were lifted from the surface and carried into the gas above, and that gas was entrained by the surface waves. Whereas this gives a good picture for he air-water system, a slag layer on top of the steel bath would certainly damp the surface waves. The slag layer could not be modeled in the present analysis. Dispersed diameter model The two-fluid model computations were conducted assuming a horizontal liquid surface. The region close to the jet is not affected by this assumption. Figures 3 to 5 show results for the jet region (R = to 2 cm) at three different depths: close to the surface, at /3 of the total depth and close to the bottom. Figure 3 compares the velocity profile of steel as predicted by the two-fluid model (run 5) and as predicted by the corresponding single-phase flow computation (run 7). The constant-velocity core of the profile is larger for all depths when the gas entrainment is modeled (two-fluid). The protection effect of the gas sheet on the jet that was observed in the VOF simulations is also reproduced in the two-fluid simulation although less pronounced. For water no results are shown since they are very similar. The jet and the bubbles also reach the ladle bottom as predicted in the VOF simulations (figure 7). Figure 5 shows that the volume fraction profiles close to the jet are approximately the same for water and steel. A peaky profile, as for the two upper depths in Fig. 6, is maintained almost to the bottom of the ladle. This confirms again the results from the VOF simulations. Moreover, the spreading of the air sheet seems to be larger, and the air is farther away from the jet center for the water air-system (figure 5). That is due to generally lower liquid density and smaller bubbles in water (figure 4). maller bubbles in a less dense fluid are better dispersed by the fluid turbulence due to smaller tokes numbers and thus k d is closer to k, equation (). The larger bubble diameters in steel could be expected because of the higher ratio of surface tension to liquid density, table I and equation (3). The bubble are small in the jet core because at the inlet the jet core is bubble free. In radial direction towards the sidewall, first a zone of large bubble diameters is passed (figures 4 and 6). The large bubbles in this zone originate in the disintegrating gas sheet. The next zone is one of small bubbles in which turbulent shear forces cause the breakup. Then again a zone with large bubbles follows. In this zone the turbulent shear forces are comparatively small and the smaller bubbles dispersed into this zone start to coalesce to larger diameters and rise. In the eye of the flow circulation (located where > mm is shown in figure 6) the residence time of the bubbles becomes large and causes the large bubble diameters there. Close to the bottom wall the velocity gradients are high and the bubbles breakup and form smaller average diameters (figure 6). The predicted effect of the bubbles on the liquid flow pattern was, for the investigated ladle configuration, small. Only close to the gas sheet down to some centimeters below the liquid surface negative fluid velocities, i.e., bubble-induced upflow is predicted (figure 3). This upflow, approximately m/s for steel, may reduce the possibility of particles to be carried away by the jet when added very close to the jet. In the work of Berg et al. [] the particles were injected 2 to 3 cm away from the centerline, outside of the bubble-induced flow zone, which thus would not have affected their results for the alloying efficiency. The waviness of the liquid surface, on the other hand, would impact the predicted alloying efficiency. Again, since the slag layer on the steel melt was not modeled no certain conclusion with respect to effect of the waviness can be made. Finally, because the liquid velocities for 2% filling (large fall height) are still high compared to the terminal velocities of the largest bubbles, bubble-induced liquid motion is expected to gain importance at higher filling levels of the ladle. Conclusions Based on the results of the computations and the analysis we draw the following main conclusions with respect to the studied tapping process in general:. The gas entrainment rate is mainly caused by the shape of the jet surface at the plunging point. 2. The gas entrainment rate is only affected by the material properties of the liquid in the sense that the material properties determine the shape of the jet. The main conclusions that apply for the studied ladle configuration are as follows: 3. The entrained bubbles induce liquid flow reversal only in a small region next to the gas sheet.

9 4. The surface is more or less wavy, however for the air-steel system, these waves are expected to be damped by a slag layer (which, however, was not part of this analysis). 5. Very buoyant alloy particles that are added close to the jet may have a reduced residence time due to the bubble-induced motion. owever, alloy particles added elsewhere seem to be unaffected. 6. The jet plunges to the bottom and it is protected by the air sheet from the shearing with the surrounding fluid. Thus, the constant velocity profile in the jet core is maintained to larger depths. The conclusions with respect to the new modeling presented in this work (transport equation for an average bubble diameter in the frame of the two-fluid model) are: 7. The dispersed diameter model seems to work well. It confirmed the results of the VOF simulations qualitatively, and it showed the expected response to material parameters and the turbulent field. 8. The predicted bubble sizes varied over two orders of magnitude in the ladle. Thus, a constant bubble diameter approach seems to be a crude simplification. ere, we have to stress that the presented analysis is just a first step that was limited due to certain simplifications. To obtain better insight in the tapping process following improvements seem necessary: 9. The VOF method has to be extended such that three phases (air, slag and steel) can be simulated in one domain. Before this is possible, computations of slag-steel systems should be carried out.. The dispersed diameter model needs to be verified. This could be done for air-water systems for which experimental correlations and measurements exist.. Configurations with larger filling heights should be studied, as addition of alloy particles is mainly recommended for filling heights between 4 and 8 % depending on the type of alloy []. Also the ladle geometry and mass flow rate should be varied in future simulations. 2. Certain problems minor with the existing modeling are connected to the definition of inlet and outlet boundary conditions, and the determination of the entrainment rate. They should be overcome in the direct continuation of this work. References.. Berg,. Laux,.T. Johansen and O.. Klevan, Flow patterns and alloy dissolution during tapping of steel furnaces, accepted by Ironmaking and teelmaking (998). 2. M. Tanaka, D. Mazumdar and R.I.L. uthrie, Motions of alloying additions during furnace tapping in steelmaking processing operations, Met. Trans. B, 24B (993), T. Choh, K. Iwata and M. Inouye, Estimation of oxygen and nitrogen absorption of liquid steel during tapping from converter, Transactions IIJ, 23 (983), A.K. Bin, as entrainment by plunging liquid jets, Chem. Eng. ci., 48(2) (993), T. Johansen, Applications of computational fluid dynamics in optimisation and design of metallurgical processes, Modeling, Identification and Control, 8(2) (997),. 6..T. Johansen, Large-scale simulation of separated multiphase flows, 3 rd Int. Conference on Multiphase Flows, Lyon June 8-2, (998). 7. Fluent User s uide: Release 4.4, Fluent Inc., Lebanon, N 3766 (996). 8.. Laux, Modeling of dilute and dense dispersed fluid-particle two-phase flow (Ph.D. thesis 998:7, Norwegian University of cience and Technology, 998). 9.. Laux, (INTEF report TF24 F97586, 997)...T. Johansen, Robertson, Woje and T.A. Engh, Fluid dynamics in bubble stirred ladles: Part Experiments, Met. Trans. B, 9B (998),745.. Calderbank, Physical rate processes in industrial fermentation Part I: The interfacial area in gasliquid contacting with mechanical agitation, Trans. Instn. Chem. Engrs., 36 (958), Konno, Arai and aito, The effects of viscous and inertial forces on drop breakup in an agitated tank, J. Chem. Eng. Japan, (6) (977), Konno, Matsunaga, Arai and aito, imulation model for breakup in an agitated tank, J. Chem. Eng. Japan, 3() (98), Tennekes and J.L. Lumley, A first course in turbulence, (972) MIT Press, Boston. 5. Y. hijun, X. Maozhao and C. Baixin, The breakup and atomization of a viscous liquid jet, Acta Mechanica inica (English eries), 2(2) (996), 24.

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