ISIJ International, Vol. 43 (2003), No. 5, pp. 653–662
Mathematical Simulation and Modeling of Steel Flow with Gas Bubbling in Trough Type Tundishes A. RAMOS-BANDERAS, R. D. MORALES,1) L. GARCÍA-DEMEDICES and M. DÍAZ-CRUZ2) Graduate student, Instituto Politécnico Nacional, Department of Metallurgy and Materials Engineering, Apdo. Postal 75-874, México D.F. CP 07300. E-mail:
[email protected] 1) K&E Technologies S.A. de C.V. and Instituto Politécnico Nacional, Department of Metallurgy and Materials Engineering, Apdo. Postal 75-874, México D.F., CP 07300. E-mail:
[email protected] 2) Instituto Politécnico Nacional, Department of Metallurgy and Materials Engineering, Apdo. Postal 75-874, México D.F. CP 07300. E-mail:
[email protected] (Received on September 19, 2002; accepted in final form on December 11, 2002 )
Flow of steel in a one-strand slab tundish equipped with a turbulence inhibitor (TI) and a transversal gas bubbling curtain was studied using mathematical simulations, PIV measurements and Residence Time Distribution (RTD) experiments in a water model. The use of a bubbling curtain originates two recirculating flows, upstream and downstream at each of its sides. The first one meets, at some point along the tundish length and close to the bath surface, the downstream that is driven by the TI. After, free shear stresses provided by the upstream make the downstream be directed toward the tundish bottom forming a bypass flow. At the other side, in the outlet box, there is strong recirculating flow which impacts the end wall and goes directly toward the outlet. Two-phase flows simulated mathematically matched experimental flow fields measured with PIV measurements. Tundish performance for inclusions flotation is maximized when only the TI is used followed by using only the bubbling curtain. Increases of gas bubbling flow rate increase the mixing processes in the tundish according to the RTD determinations. KEY WORDS: tundish; PIV; two-phase flows; gas bubbling; mathematical simulations; steel.
1.
tenance of the porous life was difficult. On line with the state of the art summarized above three main research objectives emerge; an assessment of the bubbling tundish as an inclusions floater, the performance of a combined arrangement of the two devices i.e., turbulence inhibitor plus bubbling and the implications of bubbling flow rate on fluid velocity fields. In order to reach these objectives modern analytical tools like mathematical simulation, water modeling and velocity fields determinations through Particle Image Velocimetry (PIV) technology are applied in this research. The next lines describe the experimental development, the analytical measurements, the analysis and conclusions related with bubbling and its effects on fluid flow patterns of liquid steel in a tundish.
Introduction
During the last years control of steel flow in tundishes has been performed through the employment of dams, weirs, impact pads and the like devices.1) More recently turbulence inhibitors have successfully replaced all those traditional flow controllers because, in addition to all flow parameters controlled by conventional devices, they are able to kill melt turbulence in the pouring box decreasing air pickup, slag entrainment and bath surface instability. Due to these advantages turbulence inhibitors are also very effective to perform grade change operations at very small steel levels without the risk of breakouts at the caster.2–5) Another approach, different to those so far mentioned here, is the bubbling of argon gas from the tundish bottom in order to enhance the tundish capability to float out inclusions. Yamanaka et al. investigated the tundish using argon bubbling through porous plugs. They claim a 50 % improvement in the removal of inclusions in the 50 to 100 m m range.6) Other industrial experiences report that argon bubbling in the tundish is helpful to decrease the population of inclusions in the final product.7,8) Marique et al.9) tested gas bubbling in a two-strand bloom tundish with six tones capacity using a pipe distribution system embedded in the monolithic refractory lining. The holes in the pipe were 2 mm diameter and were spaced every 100 mm. An argon flow rate of 3 Nm3/h was used during the casting time. They reported a decrease of inclusions of 25–50 % though main-
2.
Experimental Procedure
A typical through type tundish of a Brazilian one-strand slab caster was chosen to perform this work and for that purpose a 2/5 scale model made of plastic with the geometric dimensions shown in Figs. 1(a) and 1(b) was built. Residence Time Distribution (RTD) curves were determined through the typical pulse input signal technique using a red dye tracer.5,10) The output signals were recorded in a PC equipped with a data acquisition card. To model the gas curtain a strip 2.3 cm wide and 22 cm long of balsam wood was placed in the position indicated in Fig.1. This strip has a pre-chamber below its lower surface, 0.4 cm 653
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Fig. 2.
Fig. 1.
The geometric dimensions of the tundish (m).
thick made of painted steel sheet where air is introduced through a small tube with a diameter of 0.2 cm. Flow rate of gas was metered by a flow meter with a capacity from 0 to 1 200 cm3/min. Figure 2 shows the geometric dimensions of the turbulence inhibitor. In addition to the tracer dispersion measurements fluid flow was also monitored using a Particle Image Velocimetry Technique (PIV). A green frequency double pulsed Nd : YAG laser with a wavelength of 532 nm was employed. In order to obtain short bursts of light energy, the lasing cavity is Q-switched so that the energy is emitted in 6–10 ns bursts opposed to pulses of 250 m s, which is the duration of the exciting lamp in the laser cavity. Output energy from the laser is 20 mJ of Nd : YAG laser from the fibre bundle. This energy is increased with light guides that can transmit 500 mJ of pulsed radiation with an optical transmission that is greater than 90 % at 532 nm. Interrogation areas of 11 mm in the flow were scanned with a resolution of 3232 or 6464 pixels. The laser light was placed in a desired plane by means of a computer-controlled positioner with three-dimensional (3D) movements to send a longitudinal laser-sheet located in the axial-symmetrical plane of the tundish. In order to follow the fluid flow the fluid was previously seeded with polyamide particles with a density of 1 030 kg/m3 and 20 m m of diameter. A cross-correlation procedure using, Fast Fourier Transforms (FFT), allowed to process the recorded signals and a Gaussian distribution function was used to determine the location of the maximum of the peak displacement with sub-pixel accuracy. The signals were detected by a Sony coupled charged device (CCD) and the recordings were processed through a commercial Flow Map software in order to obtain the vector velocity fields and other derived parameters. Vorticity fields of the flow patterns were derived from the velocity fields, as determined by the PIV measurements, using a finite center difference scheme,11)
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Geometric dimensions of the inhibitor (m).
Fig. 3.
Scheme of the Particle Image Velocimetry equipment employed in the experiments of physical modeling.
Table 1. Tundish arrangements simulated in the mathematical model.
ωk
∂ui ∂u j .............................(1) ∂x j ∂xi
Path lines were determined also using the same method of finite differences according to the definition given by, dy v y ...................................(2) dx v x 654
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Figure 3 shows a scheme of the complete experimental setup and Table 1 shows the experimental program. 3.
The second terms in the right hand side members of Eqs. (7) and (8) are the momentum transfer interaction amounts between both phases expressed through their relative velocity fields and Kgl is usually known as the interphase momentum-exchange coefficient.14) This term can be interpreted as the drag force between phases due to their relative movement. This coefficient is given by;
Theory of Multiphase Flows
3.1. Eulerian–Eulerian Model The bubbly flow was simulated using an Eulerian– Eulerian model12) where water was considered as the primary phase (l) and air as the secondary one (g). In this model the continuity and momentum equations for each phase have to be solved together with a suitable turbulence model for each phase. In the present work the k–e model13) was employed for the primary phase, while turbulence of the secondary phase was modeled through the k–e equations of the liquid phase. The continuity equations for both phases are (see the list of symbols):
α g ρl | ug ul | 3 ..........(9) K gl 3π vl d b f b / V CD 4 db where db is the average diameter of the bubble, f b the friction coefficient between the bubble and the continuous phase, V is bubble’s volume and CD is the drag coefficient which is given by; C D
∂ ∂ (α l ρl ) (α l ρl uli )0 ....................(3) ∂t ∂xi
for Reynolds numbers 1 000 and has a value of 0.44 for Re1 000. The Reynolds definition in this two-phase problem is
∂ ∂ (α g ρg ) (α g ρg ugi )0 ..................(4) ∂t ∂xi
Re
The following constraint should be obeyed:
The effective density of any phase q is,
rˆ qa q r q ...................................(6)
∂u ∂ul j ..............(12) τ ij ρquiu jµ e li ∂xi ∂x j
Momentum balance equation for the liquid phase is, ∂ ∂ ∂p (α l ρl uli ) (α l ρl ulj uli )α l l ∂t ∂x j ∂xi
where m e is the sum of the molecular and turbulent viscosities, m em lm t and the turbulent viscosity is calculated using the turbulent kinetic energy and the dissipation rate of the kinetic energy of the continuous phase l,
∂uli ∂ulj ∂ α l ( µ l µ t ) ∂x j ∂xi ∂x j p1 a l r l gi .....................................................................(7) n
∑K
gl ( ugi uli )
µ t C µ ρl
the corresponding momentum balance for the gaseous phase is;
k l2 .............................(13) εl
The last terms in the right hand sides of Eqs. (7) and (8) provide the buoyancy driven momentum transfer by the loss of density in the two-phase flow. The turbulent equations for the kinetic energy and its dissipation rate for the liquid phase are:
∂ ∂ ∂p (α g ρg ug j ) (α g ρg ug j ugi )α g ∂t ∂x j ∂xi ∂ugi ∂ug j α g µ g ∂xi ∂x j a gr ggi .............................................................(8) K gl (uli ugi )
ρl | ul ug | d b ........................(11) µl
The third terms on the right hand sides of Eqs. (7) and (8) are the stress strain tensors of the qth phase, which was considered using the Boussinesq approximation15) using the fluctuating Reynolds stresses:
a la g1...................................(5)
24 (10.15 Re0.687 ) ....................(10) Re
∂ ∂x j
µt ∂ ∂ ∂ (α l ρl k l ) (α l ρl uli k l ) α l ρl l ∇k l σk ∂t ∂xi ∂xi a lr l(Pe l)a lr lP k l ......................(14)
where the indices i and j1,2 and 3 represent x, y and z directions, respectively; ui(u, v and w) are the velocity components in these three directions; the subscripts l and g denote liquid an gas phases, respectively; a is the volume fraction; r is the fluid’s density, m l and m t are the molecular and turbulent viscosities. Repeated indices imply summation. Since the liquid density is two or three orders of magnitude higher than the gas density the turbulence modeling of the gas phase was simplified by considering only the equations of the turbulent kinetic energy and its dissipation rate of the continuous phase. This procedure simplifies considerably the computing effort.
and µt ∂ ∂ ∂ (α l ρl ε l ) (α l ρl uli ε l ) α l ρl l ∇ε l σε ∂t ∂xi ∂xi α l ρl
εl (Clε PC2ε ε l )α l ρl Π ε l .............(15) kl
The signs P k l and P e l represent the influence of the dispersed phase (gas) on the continuous phase l. The previous equations contain five empirical constants that produce rea655
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sonable results in a wide field of applications, their standard values are as follows13): C11.44, C21.92, Cm 0.09, s k1.00
3.3. Initial and Boundary Conditions Continuity and momentum transfer equations were simultaneously solved together with the kl and e l equations. Non-slipping conditions were applied at all solid surfaces. Wall functions19) were used at nodes close to any wall. Gradients of velocity, turbulent kinetic energy and its dissipation rate were assumed zero on the bath surface and at all solid surfaces. Initial conditions for velocity, turbulent kinetic energy and its dissipation rate in the ladle shroud are given by,
and s e 1.30.
3.2. Eulerian–Lagrangian Model Inclusion trajectories were calculated using a Lagrangian particle tracking approach,16) which solves a transport equation for each inclusion as it travels through the previously calculated flow field of water and air using the Eulerian– Eulerian approach. The mean local-inclusion velocity com¯ i) needed to obtain the particle path are calculatponents (u ed from the following balance which includes the drag and buoyancy forces relative to water;
UinQ/Anozzle ..............................(18) 3 k in U in2 ∗ (0.0073) .......................(19) 2
ρi ρq dui FD (ui ui ) g j .................(16) dt ρi
e in2kin3/2/Dnozzle ............................(20) Equations (19) and (20) express boundary conditions for a turbulent flow in pipes according to the theory of the k –e model. The initial size of the bubbles, as a function of the gas flow rate and the size of the orifice, was calculated using the equation of Sano20)
To simulate the chaotic effect of the turbulence eddies on the inclusion trajectories, a discrete random-walk model was applied.17) In this model, a fluctuant random-velocity vector (uI) is added to the calculated time-averaged vector ¯ i) in order to obtain the inclusion velocity (ui) at each (u time step as “i” travels through the fluid. Each random component of the inclusion velocity is proportional to the local turbulent kinetic energy level, according to the following equation: uiζ i ui2 ζ i
2k p 3
db6.18doQ2/3 .............................(21) where do is the interior orifice diameter. The gas strip in the bottom of the tundish was divided in 1 mm squared mesh and the total gas flow rate was divided into half the number of squares leaving one square without gas and the other working like a tuyere. In every live cell with a gas flow rate, the size of the bubble was calculated using Eq. (21) and an equivalent gas velocity employed as the initial condition to couple the gas phase flow with water flow in the two-phase domain. The computing domain was divided into 250 000 hybrid cells using the Volume Finite Method21) so that there was not the need for changing spatial coordinates. The governing equations were solved using the SIMPLEC algorithm.21) This model was run in two Silicon Graphics Work-Stations and the results were stored in CD’s for further analysis and presentations. Mathematical simulations included cases A, B, C, D, E, F and G such as is described in Table 1.
......................(17)
where z is a random number, normally distributed between 1 and 1, which changes at each integration step. For Lagrange modeling initial conditions for velocities of inclusions were the input velocity of water thorough the shroud, for those inclusions that take contact with the bath surface a trap boundary condition was used, it implies that the calculations for determining the trajectories are stopped, through this procedure the absorption of inclusions by slag can be simulated and finally those inclusions that impact the walls are assumed to obey elastic reflection. No attachment mechanisms of inclusions on the bubbles surfaces were assumed, see Ref. 18) for further details. In this way the effects of liquid flow patterns affected by gas bubbling on the flotation phenomena of inclusions were isolated from other factors like particle’s coalescence, wall adhesion of particles, etc. Ten simulations for each case of inclusions trajectories were performed including 500 inclusions for each one with a lineal size distribution from 1 to 100 m m. Then for each case shown in Table 1 the trajectories of 5 000 particles were calculated and the inclusions absorbed by the upper slag were considered as fractions of the total number of injected inclusions. Since the fluid simulated here is water, virtual inclusions with a density of 500 kg/m3 were considered just to emulate the relationship of densities between alumina inclusions and liquid steel, which keep a density ratio of about 0.5. Density of water and its viscosity at room temperature are 1 000 k/m3 and 0.001 Pa · s. These values were employed in the model for the continuous phase. Air density and viscosity at room standard conditions are 1.225 kg/m3 and 1 · 79105 Pa · s.
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4.
Results and Discussion
Figure 4(a) shows the mathematical simulations of water flow in a 3D (three dimensional) view in the tundish model using only the turbulence inhibitor (TI), without the injection of air (Case A). Is clear that the TI decreases the exiting fluid velocities toward the outlets providing a plug flow throughout the vessel’s volume. With this flow pattern gentle turbulence is produced, which may be suitable for floating inclusions. However, when air is bubbled with a flow rate of 596 ml/min the fluid flow pattern suffers radical changes by the generation of two recirculating flows at each side of the bubbling curtain as is seen in the 3D view of Fig. 4(b). The origins of these recirculating flows are the shear stresses between both phases at the liquid-bubbles interfaces calculated through Eqs. (9), (14) and (15). One of the recirculating flows, at the left side of the bubbling curtain, makes the fluid flow upstream with an opposing direction to the velocity vectors exiting from the TI, which go 656
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Fig. 5.
The Mathematical Simulation of path lines in the tundish model. (a) With turbulence inhibitor, (b) with TI plus a gas flow rate of 596 ml/min.
Fig. 6. Fig. 4.
Three dimensional views of water velocity vectors in the tundish model. (a) With turbulence inhibitor (TI), (b) TI and a gas flow rate of 596 ml/min.
downstream. At some point close to the bath surface both flows, upstream and downstream, meet and after this point the recirculating flow drives, by free shear stresses, the down streaming fluid below the bath surface, along the tundish bottom, forming a bypass flow. In the outlet box, on right side of the bubbling curtain, the recirculating flow is intensified due to the momentum transfer applied by the down streaming flows of the liquid onto the bubbling curtain. Recirculating flows with lower and higher intensities can be expected with gas flow rates of 240 and 913 ml/min, respectively. Simulation of path lines for both cases are presented in Figs. 5(a) and 5(b) where the difference of fluid flow patterns of both cases is clearly seen. When gas is bubbled, Fig. 5(b), the path lines of the flow are directed toward the outlet following the recirculating flow. To test the validity of these mathematical simulations flow fields at both sides of the bubbling curtain were calculated for gas flow rates of 240, 596 and 913 ml/min and the results are shown in Figs. 6(a), 6(b) and 6(c), respectively.
Velocity fields of water under isothermal conditions obtained by mathematical simulation, the left side is before the air curtain and the right side is after the curtain, (a) gas flow rate of 240 ml/min (Case B), (b) gas flow rate of 596 ml/min (Case C) and (c) gas flow rate of 913 ml/min (Case D).
These calculations are compared with the flow fields determined through PIV measurements shown in Figs. 7(a), 7(b) and 7(c) for the mentioned flow rates of gas. The comparison indicates us that the mathematical simulations predict with reliability flow fluid of water in the tundish model. Vorticity measurements through the PIV technology for a gas flow rate of 596 ml/min are shown in Fig. 8(a) while Fig. 8(b) shows the corresponding vorticity map for the tundish without gas bubbling. Figures 9(a) and 9(b) show the corresponding path lines with and without gas bubbling, respectively. When gas is bubbled high positive vorticity, indicating counterclockwise rotational motion, are observed close to the bath surface at the left side of the gas curtain. On the other side, right one, the vorticity are high and negative indicating clockwise rotating motion close to the bath surface and decrease closer to the tundish bottom. No gas flow conditions clearly specify the non-existence of strong rotational fluid motions as is seen in the path lines of Fig. 9(b). 657
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Fig. 9.
Fig. 7.
Fig. 8.
Velocity fields of water under isothermal conditions obtained by physical modeling using Particle Image Velocimetry (PIV) measurements, the left side is before the air curtain and the right side is after the curtain, (a) gas flow rate of 240 ml/min (Case B), (b) gas flow rate of 596 ml/min (Case C) and (c) gas flow rate of 913 ml/min (Case D).
Simulation of velocity fields at the axial-longitudinal plane of the tundish are shown in Figs. 10(a), 10(b), 10(c) and 10(d) for no gas flow, and flow rates of 240, 596 and 913 ml/min (Cases A, B, C and D), respectively. As was mentioned above, first gas bubbling changes the fluid flow patterns due to the shearing of liquid by gas at the gas–liquid interfaces. Second when gas flow rate is increased the upstream flow of liquid at the left side of the bubbling curtain is intensified annihilating to some degree, depending on flow rate of gas, the down streaming momentum provided to the liquid by the TI. Distribution patterns of volume fractions of the gas phase at the same plane are presented in Figs. 11(a), 11(b) and 11(c) for the flow rates of gas of 240, 596 and 913 l/min, respectively. As can be seen the bubbling curtain is bent by the downstream liquid toward the outlet box and gas bubbles are entrained into the liquid until the end wall. However, gas phase is also entrained by the liquid flowing upstream and the volume gas fractions upstream are able to reach the ladle shroud when the flow rates of gas are high. Velocity fields in a plant view close to the bath surface for a tundish without gas flow and with gas flow rates of 596 and 913 ml/min are shown in Figs. 12(a), 12(b) and 12(c), respectively. In Fig. 12(a) fluid flows downstream following a symmetric pattern, close to the outlet, at the level of the half dams, the velocity is increased due to the narrowing central space left by those dams. In Figs. 12(b) and 12(c) is evident that the liquid flows downstream following recirculating flows also in the horizontal planes so actually gas bubbling provides a complex flow of axial and vertical mixings decreasing the fraction of plug flow observed for the tundish with only the TI as is reported in Fig. 4(a). These results demonstrate also that the liquid flow bends in a non-symmetrical way the bubble curtain. Consequently, the flow is non-symmetrical in the horizontal
Average vorticity contours in the tundish model through physical simulation, the left side is before the air curtain and the right side is after the curtain. (a) Gas flow rate of 596 ml/min (Case C) and (b) no gas flow (Case A).
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Stream lines in the tundish model through physical simulation, the left side is before the air curtain and the right side is after the curtain. (a) Gas flow rate of 596 ml/min (Case C) and (b) no gas flow (Case A).
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Fig. 10. Mathematical simulation of velocity fields at the axial–longitudinal plane of the tundish. (a) No gas flow (Case A), (b) gas flow rate of 240 ml/min (Case B), (c) gas flow rate of 596 ml/min (Case C) and (d) gas flow rate of 913 ml/min (Case D).
planes. Just to see the fluid flow in the bare tundish Fig. 13(a) shows a 3D view of the fluid flow pattern inside the tundish and clearly this flow is disordered with a s-shaped main stream forming a string bypass. Figure 13(b) shows the velocity field of the axial- longitudinal plane which should be compared with Figs. 4 and 6. In this case a chaotic-bypass flow is observed, which may not be suitable for floating inclusions. Fluid flow parameters derived from the experimental RTD curves are shown in Table 2 where is seen that increasing the flow rate of the bubbling gas decreases the plug volume fraction and increases the mixing fraction of the liquid phase and this is in total agreement with the flow patterns mathematically simulated and physically determined through PIV measurements. Increasing the flow rate of gas increases the length and magnitude of the recirculating flows in the volumes located at both sides of the bubbling curtain and the vertical and horizontal mixing processes. Visual representations for trajectories of ten particles with a lineal size distribution from 5 to 50 microns for the cases of the tundish using the TI, the TI plus a gas bubbling flow rate of 596 ml/min and the bare tundish are shown in Figs. 14(a), 14(b) and 14(c), respectively. Apparently the first case is more efficient for floating inclusions followed
Fig. 11. Volume fractions104 of the gas phase at the axiallongitudinal plane. (a) Gas flow rate of 240 ml/min (Case B), (b) gas flow rate of 596 ml/min (Case C) and (c) gas flow rate of 913 ml/min (Case D).
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Fig. 12. Mathematical simulation of velocity fields in a plant view close to the bath surface of the tundish model. (a) No gas flow (Case A), (b) gas flow rate of 596 ml/min (Case C) and (c) gas flow rate of 913 ml/min (Case D).
Fig. 13. Mathematical simulation of water velocity vectors in the tundish model for the case of the tundish without flow modifiers (Case G). (a) A three dimensional view, (b) at the axial-longitudinal plane.
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ISIJ International, Vol. 43 (2003), No. 5 Table 2.
Experimental results of RTD curves.
Fig. 14. Trajectories of non-metallic inclusions throughout the tundish model by mathematical simulation. (a) With only turbulence inhibitor (TI) (Case A), (b) gas flow rate of 596 ml/min (Case C) and (c) without flow modifiers (Case G).
by the second and third cases. Massive simulations of particles trajectories lead to the results shown in Fig. 15(a) for ten simulations, each one with 500 particles, as was described above in the Lagrange simulation model. In that Figure is seen that the tundish with only the TI renders the highest performance to float and absorb inclusions and the bare tundish yields the worst one. Is also interesting to see that a gas flow rate of 596 l/min with the half dams and without the TI yields the second best system to flout out inclusions. The combination of TI plus gas bubbling worsens with increases of gas bubbling flow rate due to the intensification of the recirculating flows already discussed. Figure 15(b) summarizes the results presented in Fig. 15(b) allowing a clearer view about the performances of the seven cases for inclusions flotation processes. 5.
Conclusions
A one-strand slab tundish was mathematically simulated and water modeled to study the influence of gas bubbling curtains on the fluid flow patterns produced by turbulence inhibitors. The study included the bare tundish, the employment of half dams and the combination of half dams with gas bubbling without a TI. The main conclusions drawn from this study are as follows: (1) The mathematical simulations describe well the actual two-phase flows in the tundish since its results match
Fig. 15. Behavior of non-metallic inclusions through mathematical simulation for the different tundish model arrangements.
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with experimental flow fields determined through PIV technology. (2) A combination of a TI with gas bubbling derives in complex recirculating flows of liquid with vertical and horizontal mixing processes that originate bypassing streams toward the outlet. Higher flow rates of gas lead to decreases of plug flow fraction of the fluid inside the vessel. (3) The best results, for the purpose of inclusions flotation, are obtained for a tundish with the TI followed by gas bubbling, with 596 ml/min and with the half dams and without the TI. The worst case, for the same objective, is the bare tundish. (4) Thus gas bubbling itself yields very good results for floating inclusions but is not superior to a well designed TI for the same purpose. (5) A tundish without gas bubbling and without a TI but equipped with half dams provides acceptably high performances for inclusions flotation. If optimum flotation conditions are not required the simple arrangement of half dams is acceptable.
w : Flow vorticity Subscripts b : Bubble e : Effective i : Direction coordinate in : Inlet g : Gas l : Liquid nozzle : Properties of the ladle shroud or nozzle t : Turbulent o : Orifice REFERENCES 1) L. J. Heaslip, A. McLean and I. D. Sommerville: Continuous Casting, Vol. 1, ISS, Warrendale, PA, (1983), 67. 2) R. D. Morales, S. López-Ramírez, J. Palafox-Ramos and D. Zacharias: Ironmaking Steelmaking, 28 (2001), 23. 3) J. Palafox-Ramos, J. de J. Barreto-Sandoval, S. López-Ramírez and R. D. Morales: Ironmaking Steelmaking, 28 (2001), 101. 4) R. D. Morales, M. Díaz-Cruz, J. Palafox-Ramos, S. López-Ramírez and J. De J. Barreto-Sandoval: Steel Res., 72 (2001), 11. 5) R. D. Morales, S. López-Ramírez, J. Palafox-Ramos and D. Zacharias: ISIJ Int., 39 (1999), 455. 6) H. Yamanaka: Tetsu-to-Hagané, 69 (1983), S213. 7) K. Nakajima and M. Kawasaki: Tetsu-to-Hagané, 73 (1987), 110. 8) Y. Nuri, K. Umezawa, T. Ohashi, R. Itchi, K. Mizuguchi and S. Yokoi: Trans. Iron Steel Inst. Jpn., 24 (1984), B16. 9) C. Marique, A. Dony and P. Nyssen: Tundish metallurgy, Vol. II, ISS, Warrendale, PA, (1990), 49. 10) H. Scott Fogler: Elements of Chemical Reaction Engineering 2nd ed., Prentice Hall Int., London, Sydney and Mexico, (1992), 708. 11) M. Raffel, C. Willert and J. Kompenhans: Particle Image Velocimetry, Springer Verlag, Berlín, New York, N.Y., (1998), 160. 12) M. W. Reeks: J. Fluid Mech., 83 (1977), 529. 13) B. E. Launder and D. B. Spalding: Comp. Meth. Appl. Mech., 3, (1974), 269. 14) R. Clift, J. R. Grace and M. E. Weber: Bubbles, Drops and Particles, Academic Press, New York N.Y., London, San Francisco, (1978), 125. 15) D. C. Wilcox: Turbulence Modeling for CFD 2nd ed., DCW Industries, Anaheim, CA, (2000), 178. 16) C. T. Crowe, M. D. Sharma and D. E Stock: Fluid Eng., 99 (1977), 325. 17) C. Crowe, M. Sommerfield and Y. Tsuji: Multiphase Flows with Droplets and Particles, CRC Press, New York, N.Y., London, (1998), 191. 18) S. López-ramírez, R.D. Morales, J. De J. Barreto-Sandoval and R. D. Morales: Metall. Mater. Trans., 32B (2001), 615. 19) D.A. Anderson: Computational Fluid Mechanics and Heat Transfer, McGraw Hill Books Co., New York N.Y., Mexico City, (1984), 223. 20) M. Sano and K. Mori: Savard/Lee Int. Symposium on Bath Smelting, TMS, Warrendale, PA, (1992), 465. 21) T Chung: Computational Fluid Dynamics, Cambridge University Press, London, (2002), 696.
Acknowledgments
The authors are very indebted to National Council of Science and Technology of Mexico for the financial support to this project through a scholarship provided to ARB. Thanks are given also to SNI and IPN, both institutions have provided to the Group of Mathematical Simulation of Materials Processing and Fluid Dynamics a decided support through all these years. Nomenclature CD : Drag coefficient db : Bubble diameter do : Interior diameter of an orifice. f b : Friction coefficient between bubbles and liquid FD : Drag force Kpq : Interphase momentum-exchange coefficient kl : Turbulent kinetic energy of liquid phase kgl : Covariance of the continuous and dispersed phases p : Pressure uij : Velocity vector of phase “j” in direction “i” Vgl : Relative velocity between phases Greek symbols a j : Volume fraction of phase “j” e : Dissipation rate of turbulent energy P gl : As defined in Eq. (16) in the text r j : Density of phase “j” m : Fluid viscosity
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