Mathematical Simulation and Modeling of Steel Flow with

1. Introduction

During the last years control of steel flow in tundisheshas been performed through the employment of dams,weirs, impact pads and the like devices.1) More recently tur-bulence inhibitors have successfully replaced all those tra-ditional flow controllers because, in addition to all flow pa-rameters controlled by conventional devices, they are ableto kill melt turbulence in the pouring box decreasing airpickup, slag entrainment and bath surface instability. Dueto these advantages turbulence inhibitors are also very ef-fective to perform grade change operations at very smallsteel 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 inorder to enhance the tundish capability to float out inclu-sions. Yamanaka et al. investigated the tundish using argonbubbling through porous plugs. They claim a 50% improve-ment in the removal of inclusions in the 50 to 100 mmrange.6) Other industrial experiences report that argon bub-bling in the tundish is helpful to decrease the population ofinclusions in the final product.7,8) Marique et al.9) tested gasbubbling in a two-strand bloom tundish with six tones ca-pacity using a pipe distribution system embedded in themonolithic refractory lining. The holes in the pipe were2 mm diameter and were spaced every 100 mm. An argonflow rate of 3 Nm3/h was used during the casting time. Theyreported a decrease of inclusions of 25–50% though main-

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 bub-bling tundish as an inclusions floater, the performance of acombined arrangement of the two devices i.e., turbulenceinhibitor plus bubbling and the implications of bubblingflow rate on fluid velocity fields. In order to reach these objectives modern analytical tools like mathematical simu-lation, water modeling and velocity fields determinationsthrough Particle Image Velocimetry (PIV) technology areapplied in this research. The next lines describe the ex-perimental development, the analytical measurements, theanalysis and conclusions related with bubbling and its effects on fluid flow patterns of liquid steel in a tundish.

2. Experimental Procedure

A typical through type tundish of a Brazilian one-strandslab caster was chosen to perform this work and for thatpurpose a 2/5 scale model made of plastic with the geomet-ric dimensions shown in Figs. 1(a) and 1(b) was built.Residence Time Distribution (RTD) curves were deter-mined through the typical pulse input signal techniqueusing a red dye tracer.5,10) The output signals were recordedin a PC equipped with a data acquisition card. To model thegas curtain a strip 2.3 cm wide and 22 cm long of balsamwood was placed in the position indicated in Fig.1. Thisstrip has a pre-chamber below its lower surface, 0.4 cm

ISIJ International, Vol. 43 (2003), No. 5, pp. 653–662

653 © 2003 ISIJ

Mathematical Simulation and Modeling of Steel Flow with GasBubbling 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]) Instituto Politécnico Nacional, Department of Metallurgy and Materials Engineering, Apdo. Postal 75-874, México D.F. CP07300. 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 gasbubbling curtain was studied using mathematical simulations, PIV measurements and Residence TimeDistribution (RTD) experiments in a water model. The use of a bubbling curtain originates two recirculatingflows, upstream and downstream at each of its sides. The first one meets, at some point along the tundishlength and close to the bath surface, the downstream that is driven by the TI. After, free shear stresses pro-vided by the upstream make the downstream be directed toward the tundish bottom forming a bypassflow. At the other side, in the outlet box, there is strong recirculating flow which impacts the end wall andgoes directly toward the outlet. Two-phase flows simulated mathematically matched experimental flowfields measured with PIV measurements. Tundish performance for inclusions flotation is maximized whenonly the TI is used followed by using only the bubbling curtain. Increases of gas bubbling flow rate increasethe mixing processes in the tundish according to the RTD determinations.

KEY WORDS: tundish; PIV; two-phase flows; gas bubbling; mathematical simulations; steel.

thick made of painted steel sheet where air is introducedthrough a small tube with a diameter of 0.2 cm. Flow rate ofgas was metered by a flow meter with a capacity from 0 to1 200 cm3/min. Figure 2 shows the geometric dimensionsof the turbulence inhibitor.

In addition to the tracer dispersion measurements fluidflow was also monitored using a Particle Image VelocimetryTechnique (PIV). A green frequency double pulsed Nd :YAG laser with a wavelength of 532 nm was employed. Inorder to obtain short bursts of light energy, the lasing cavityis Q-switched so that the energy is emitted in 6–10 ns burstsopposed to pulses of 250 ms, which is the duration of theexciting lamp in the laser cavity. Output energy from thelaser is 20 mJ of Nd : YAG laser from the fibre bundle. Thisenergy is increased with light guides that can transmit 500mJ of pulsed radiation with an optical transmission that isgreater than 90% at 532 nm. Interrogation areas of 1�1mm in the flow were scanned with a resolution of 32�32 or64�64 pixels.

The laser light was placed in a desired plane by means ofa computer-controlled positioner with three-dimensional (3-D) movements to send a longitudinal laser-sheet located inthe axial-symmetrical plane of the tundish. In order to fol-low the fluid flow the fluid was previously seeded withpolyamide particles with a density of 1 030 kg/m3 and 20mm of diameter. A cross-correlation procedure using, FastFourier Transforms (FFT), allowed to process the recordedsignals and a Gaussian distribution function was used to de-termine the location of the maximum of the peak displace-ment with sub-pixel accuracy. The signals were detected bya Sony coupled charged device (CCD) and the recordingswere processed through a commercial Flow Map softwarein order to obtain the vector velocity fields and other de-rived parameters.

Vorticity fields of the flow patterns were derived from thevelocity fields, as determined by the PIV measurements,using a finite center difference scheme,11)

.............................(1)

Path lines were determined also using the same method offinite differences according to the definition given by,

...................................(2)dy

dxy

x

�v

v

ωki

j

j

i

u

x

u

x� �

∂∂

∂∂

ISIJ International, Vol. 43 (2003), No. 5

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Table 1. Tundish arrangements simulated in the mathematicalmodel.

Fig. 1. The geometric dimensions of the tundish (m).

Fig. 2. Geometric dimensions of the inhibitor (m).

Fig. 3. Scheme of the Particle Image Velocimetry equipmentemployed in the experiments of physical modeling.

Figure 3 shows a scheme of the complete experimentalsetup and Table 1 shows the experimental program.

3. 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 prima-ry phase (l) and air as the secondary one (g). In this modelthe continuity and momentum equations for each phasehave to be solved together with a suitable turbulence modelfor each phase. In the present work the k–e model13) wasemployed for the primary phase, while turbulence of thesecondary phase was modeled through the k–e equations ofthe liquid phase. The continuity equations for both phasesare (see the list of symbols):

....................(3)

..................(4)

The following constraint should be obeyed:

a l�ag�1...................................(5)

The effective density of any phase q is,

rq�aqrq ...................................(6)

Momentum balance equation for the liquid phase is,

�a lr lgi .....................................................................(7)

the corresponding momentum balance for the gaseous phaseis;

�agrggi.............................................................(8)

where the indices i and j�1,2 and 3 represent x, y and z di-rections, respectively; ui�(u, v and w) are the velocity com-ponents in these three directions; the subscripts l and g de-note liquid an gas phases, respectively; a is the volumefraction; r is the fluid’s density, m l and m t are the molecularand turbulent viscosities. Repeated indices imply summa-tion. Since the liquid density is two or three orders of mag-nitude higher than the gas density the turbulence modelingof the gas phase was simplified by considering only theequations of the turbulent kinetic energy and its dissipationrate of the continuous phase. This procedure simplifies con-siderably the computing effort.

The second terms in the right hand side members of Eqs.(7) and (8) are the momentum transfer interaction amountsbetween both phases expressed through their relative veloc-ity fields and Kgl is usually known as the interphase mo-mentum-exchange coefficient.14) This term can be interpret-ed as the drag force between phases due to their relativemovement. This coefficient is given by;

..........(9)

where db is the average diameter of the bubble, f b the fric-tion coefficient between the bubble and the continuousphase, V is bubble’s volume and CD is the drag coefficientwhich is given by;

....................(10)

for Reynolds numbers �1 000 and has a value of 0.44 forRe�1 000. The Reynolds definition in this two-phase prob-lem is

........................(11)

The third terms on the right hand sides of Eqs. (7) and (8)are the stress strain tensors of the q th phase, which wasconsidered using the Boussinesq approximation15) using thefluctuating Reynolds stresses:

..............(12)

where m e is the sum of the molecular and turbulent viscosi-ties, m e�m l�m t and the turbulent viscosity is calculatedusing the turbulent kinetic energy and the dissipation rateof the kinetic energy of the continuous phase l,

.............................(13)

The last terms in the right hand sides of Eqs. (7) and (8)provide the buoyancy driven momentum transfer by the lossof density in the two-phase flow. The turbulent equationsfor the kinetic energy and its dissipation rate for the liquidphase are:

�a lr l(P�e l)�a lr lP k l ......................(14)

and

.............(15)

The signs P k l and Pe l represent the influence of the dis-persed phase (gas) on the continuous phase l. The previousequations contain five empirical constants that produce rea-

� � �α ρε

ε α ρε ε εl ll

ll l l l lk

C P C( )2 Π

∂∂

∂∂

∂∂

t x

uxi

ii

t

( ) ( )α ρ ε α ρ ε α ρµσ

εε

l l l l l l l l ll

l� � ∇

∂∂

∂∂

∂∂

t

kx

u kx

ki

ii

t

k

( ) ( )α ρ α ρ α ρµσl l l l l l l l l

ll� � ∇

µ ρεµt l

l2

l

�Ck

τ ρ µij i ji

j

j

i

u uu

x

u

x�� q e

l l∂∂

∂∂

Re l l g b

l

��ρµ

| |u u d

CD ReRe� �

241 0 15 0 687( . ).

K d f V Cu u

dgl l b b Dg l g l

b

� ��

33

4π

α ρv /

| |

� � � �K u ux

u

x

u

xi ij

i

j

j

igl l g g g

g g( )∂

∂∂∂

∂∂

α µ

∂∂

∂∂

∂∂t

ux

u up

xjj

j ii

( ) ( )α ρ α ρ αg g g g g g g g� ��

� � � � ��

K u ux

u

x

u

xi ij

i

j

j

ip

n

gl g l l l tl l( ) ( )

∂∂

∂∂

∂∂

∑ α µ µ1

∂∂

∂∂

∂∂t

ux

u up

xij

j ii

( ) ( )α ρ α ρ αl l l l l l l ll� ��

∂∂

∂∂t x

ui

i( ) ( )α ρ α ρg g g g g� �0

∂∂

∂∂t x

ui

l l i( ) ( )α ρ α ρl l l� �0


655 © 2003 ISIJ

sonable results in a wide field of applications, their standardvalues are as follows13):

C1�1.44, C2�1.92, Cm�0.09, s k�1.00 and se�1.30.

3.2. Eulerian–Lagrangian Model

Inclusion trajectories were calculated using a Lagrangianparticle tracking approach,16) which solves a transport equa-tion for each inclusion as it travels through the previouslycalculated flow field of water and air using the Eulerian–Eulerian approach. The mean local-inclusion velocity com-ponents (ui) needed to obtain the particle path are calculat-ed from the following balance which includes the drag andbuoyancy forces relative to water;

.................(16)

To simulate the chaotic effect of the turbulence eddies onthe inclusion trajectories, a discrete random-walk modelwas applied.17) In this model, a fluctuant random-velocityvector (u�I) is added to the calculated time-averaged vector(ui) in order to obtain the inclusion velocity (ui) at eachtime step as “i” travels through the fluid. Each randomcomponent of the inclusion velocity is proportional to thelocal turbulent kinetic energy level, according to the follow-ing equation:

......................(17)

where z is a random number, normally distributed between�1 and 1, which changes at each integration step. ForLagrange modeling initial conditions for velocities of inclu-sions were the input velocity of water thorough the shroud,for those inclusions that take contact with the bath surface atrap boundary condition was used, it implies that the calcu-lations for determining the trajectories are stopped, throughthis procedure the absorption of inclusions by slag can be simulated and finally those inclusions that impact thewalls are assumed to obey elastic reflection. No attachmentmechanisms of inclusions on the bubbles surfaces were as-sumed, see Ref. 18) for further details. In this way the ef-fects of liquid flow patterns affected by gas bubbling on theflotation phenomena of inclusions were isolated from otherfactors like particle’s coalescence, wall adhesion of parti-cles, etc. Ten simulations for each case of inclusions trajec-tories were performed including 500 inclusions for eachone with a lineal size distribution from 1 to 100 mm. Thenfor each case shown in Table 1 the trajectories of 5 000 par-ticles were calculated and the inclusions absorbed by theupper slag were considered as fractions of the total numberof injected inclusions. Since the fluid simulated here iswater, virtual inclusions with a density of 500 kg/m3 wereconsidered just to emulate the relationship of densities be-tween alumina inclusions and liquid steel, which keep adensity ratio of about 0.5. Density of water and its viscosityat room temperature are 1 000 k/m3 and 0.001 Pa · s. Thesevalues were employed in the model for the continuousphase. Air density and viscosity at room standard condi-tions are 1.225 kg/m3 and 1 ·79�10�5 Pa · s.

3.3. Initial and Boundary Conditions

Continuity and momentum transfer equations were si-multaneously 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 dissi-pation rate were assumed zero on the bath surface and at allsolid surfaces. Initial conditions for velocity, turbulent ki-netic energy and its dissipation rate in the ladle shroud aregiven by,

Uin�Q/Anozzle ..............................(18)

.......................(19)

e in�2kin3/2/Dnozzle ............................(20)

Equations (19) and (20) express boundary conditions for aturbulent flow in pipes according to the theory of the k–emodel. The initial size of the bubbles, as a function of thegas flow rate and the size of the orifice, was calculatedusing the equation of Sano20)

db�6.18doQ2/3 .............................(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 dividedinto half the number of squares leaving one square withoutgas and the other working like a tuyere. In every live cellwith a gas flow rate, the size of the bubble was calculatedusing Eq. (21) and an equivalent gas velocity employed asthe initial condition to couple the gas phase flow with waterflow in the two-phase domain. The computing domain wasdivided into 250 000 hybrid cells using the Volume FiniteMethod21) so that there was not the need for changing spatial coordinates. The governing equations were solvedusing the SIMPLEC algorithm.21) This model was run intwo Silicon Graphics Work-Stations and the results werestored in CD’s for further analysis and presentations.Mathematical simulations included cases A, B, C, D, E, Fand G such as is described in Table 1.

4. Results and Discussion

Figure 4(a) shows the mathematical simulations of waterflow in a 3D (three dimensional) view in the tundish modelusing only the turbulence inhibitor (TI), without the injec-tion of air (Case A). Is clear that the TI decreases the exit-ing fluid velocities toward the outlets providing a plug flowthroughout the vessel’s volume. With this flow pattern gen-tle turbulence is produced, which may be suitable for float-ing inclusions. However, when air is bubbled with a flowrate of 596 ml/min the fluid flow pattern suffers radicalchanges by the generation of two recirculating flows at eachside of the bubbling curtain as is seen in the 3D view ofFig. 4(b). The origins of these recirculating flows are theshear stresses between both phases at the liquid-bubbles in-terfaces calculated through Eqs. (9), (14) and (15). One ofthe recirculating flows, at the left side of the bubbling cur-tain, makes the fluid flow upstream with an opposing direc-tion to the velocity vectors exiting from the TI, which go

k Uin in2�

3

20 0073∗ ( . )

u uk

i i i i�� ζ ζ22

3p

du

dtF u ui

i ii

ij� � �

�D

q g( )ρ ρ

ρ


© 2003 ISIJ 656

downstream. At some point close to the bath surface bothflows, upstream and downstream, meet and after this pointthe recirculating flow drives, by free shear stresses, thedown streaming fluid below the bath surface, along thetundish bottom, forming a bypass flow. In the outlet box, onright side of the bubbling curtain, the recirculating flow isintensified due to the momentum transfer applied by thedown streaming flows of the liquid onto the bubbling cur-tain. Recirculating flows with lower and higher intensitiescan be expected with gas flow rates of 240 and 913 ml/min,respectively. Simulation of path lines for both cases are pre-sented in Figs. 5(a) and 5(b) where the difference of fluidflow patterns of both cases is clearly seen. When gas isbubbled, Fig. 5(b), the path lines of the flow are directed to-ward the outlet following the recirculating flow.

To test the validity of these mathematical simulationsflow fields at both sides of the bubbling curtain were calcu-lated for gas flow rates of 240, 596 and 913 ml/min and theresults are shown in Figs. 6(a), 6(b) and 6(c), respectively.

These calculations are compared with the flow fields deter-mined through PIV measurements shown in Figs. 7(a), 7(b)and 7(c) for the mentioned flow rates of gas. The compari-son indicates us that the mathematical simulations predictwith reliability flow fluid of water in the tundish model.Vorticity measurements through the PIV technology for agas flow rate of 596 ml/min are shown in Fig. 8(a) whileFig. 8(b) shows the corresponding vorticity map for thetundish without gas bubbling. Figures 9(a) and 9(b) showthe corresponding path lines with and without gas bubbling,respectively. When gas is bubbled high positive vorticity,indicating counterclockwise rotational motion, are observedclose to the bath surface at the left side of the gas curtain.On the other side, right one, the vorticity are high and nega-tive indicating clockwise rotating motion close to the bathsurface and decrease closer to the tundish bottom. No gasflow conditions clearly specify the non-existence of strongrotational fluid motions as is seen in the path lines of Fig.9(b).


657 © 2003 ISIJ

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. Velocity fields of water under isothermal conditions obtained bymathematical simulation, the left side is before the air curtain andthe 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 flowrate of 913 ml/min (Case D).

Fig. 4. Three dimensional views of water velocity vectors inthe tundish model. (a) With turbulence inhibitor (TI),(b) TI and a gas flow rate of 596 ml/min.

Simulation of velocity fields at the axial-longitudinalplane 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 and913 ml/min (Cases A, B, C and D), respectively. As wasmentioned above, first gas bubbling changes the fluid flowpatterns due to the shearing of liquid by gas at the gas–liq-uid interfaces. Second when gas flow rate is increased theupstream flow of liquid at the left side of the bubbling cur-tain is intensified annihilating to some degree, dependingon flow rate of gas, the down streaming momentum provid-ed to the liquid by the TI. Distribution patterns of volumefractions of the gas phase at the same plane are presented inFigs. 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 bub-bling curtain is bent by the downstream liquid toward theoutlet box and gas bubbles are entrained into the liquiduntil the end wall. However, gas phase is also entrained bythe liquid flowing upstream and the volume gas fractionsupstream are able to reach the ladle shroud when the flowrates of gas are high. Velocity fields in a plant view close tothe bath surface for a tundish without gas flow and with gasflow rates of 596 and 913 ml/min are shown in Figs. 12(a),12(b) and 12(c), respectively. In Fig. 12(a) fluid flowsdownstream following a symmetric pattern, close to theoutlet, at the level of the half dams, the velocity is increaseddue to the narrowing central space left by those dams. InFigs. 12(b) and 12(c) is evident that the liquid flows down-stream following recirculating flows also in the horizontalplanes so actually gas bubbling provides a complex flow ofaxial and vertical mixings decreasing the fraction of plugflow observed for the tundish with only the TI as is reportedin Fig. 4(a). These results demonstrate also that the liquidflow bends in a non-symmetrical way the bubble curtain.Consequently, the flow is non-symmetrical in the horizontal


© 2003 ISIJ 658

Fig. 7. Velocity fields of water under isothermal conditions ob-tained by physical modeling using Particle ImageVelocimetry (PIV) measurements, the left side is beforethe air curtain and the right side is after the curtain, (a)gas flow rate of 240 ml/min (Case B), (b) gas flow rate of596 ml/min (Case C) and (c) gas flow rate of 913 ml/min(Case D).

Fig. 8. Average vorticity contours in the tundish model throughphysical simulation, the left side is before the air curtainand the right side is after the curtain. (a) Gas flow rate of596 ml/min (Case C) and (b) no gas flow (Case A).

Fig. 9. Stream lines in the tundish model through physical simu-lation, the left side is before the air curtain and the rightside is after the curtain. (a) Gas flow rate of 596 ml/min(Case C) and (b) no gas flow (Case A).

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 tundishand clearly this flow is disordered with a s-shaped mainstream forming a string bypass. Figure 13(b) shows the ve-locity field of the axial- longitudinal plane which should becompared with Figs. 4 and 6. In this case a chaotic-bypassflow is observed, which may not be suitable for floating in-clusions.

Fluid flow parameters derived from the experimentalRTD curves are shown in Table 2 where is seen that in-creasing the flow rate of the bubbling gas decreases theplug volume fraction and increases the mixing fraction ofthe liquid phase and this is in total agreement with the flowpatterns mathematically simulated and physically deter-mined through PIV measurements. Increasing the flow rateof gas increases the length and magnitude of the recirculat-ing flows in the volumes located at both sides of the bubbling curtain and the vertical and horizontal mixingprocesses.

Visual representations for trajectories of ten particleswith a lineal size distribution from 5 to 50 microns for thecases of the tundish using the TI, the TI plus a gas bubblingflow rate of 596 ml/min and the bare tundish are shown inFigs. 14(a), 14(b) and 14(c), respectively. Apparently thefirst case is more efficient for floating inclusions followed


659 © 2003 ISIJ

Fig. 10. Mathematical simulation of velocity fields at the axial–longitudinal plane of the tundish. (a) No gas flow (CaseA), (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 of913 ml/min (Case D).

Fig. 11. Volume fractions�10�4 of the gas phase at the axial-longitudinal 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).


© 2003 ISIJ 660

Fig. 12. Mathematical simulation of velocity fields in a plant view close to the bath surface of the tundish model. (a) Nogas 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 flowmodifiers (Case G). (a) A three dimensional view, (b) at the axial-longitudinal plane.

by the second and third cases. Massive simulations of parti-cles trajectories lead to the results shown in Fig. 15(a) forten simulations, each one with 500 particles, as was de-scribed above in the Lagrange simulation model. In thatFigure is seen that the tundish with only the TI renders thehighest performance to float and absorb inclusions and thebare tundish yields the worst one. Is also interesting to seethat a gas flow rate of 596 l/min with the half dams andwithout the TI yields the second best system to flout out in-clusions. The combination of TI plus gas bubbling worsenswith increases of gas bubbling flow rate due to the intensifi-cation of the recirculating flows already discussed. Figure15(b) summarizes the results presented in Fig. 15(b) allow-ing a clearer view about the performances of the sevencases for inclusions flotation processes.

5. Conclusions

A one-strand slab tundish was mathematically simulatedand water modeled to study the influence of gas bubblingcurtains on the fluid flow patterns produced by turbulenceinhibitors. The study included the bare tundish, the employ-ment of half dams and the combination of half dams withgas bubbling without a TI. The main conclusions drawnfrom this study are as follows:

(1) The mathematical simulations describe well the ac-tual two-phase flows in the tundish since its results match


661 © 2003 ISIJ

Table 2. Experimental results of RTD curves.

Fig. 14. Trajectories of non-metallic inclusions throughout the tundish model by mathematical simulation. (a) With onlyturbulence inhibitor (TI) (Case A), (b) gas flow rate of 596 ml/min (Case C) and (c) without flow modifiers(Case G).

Fig. 15. Behavior of non-metallic inclusions through mathemati-cal simulation for the different tundish model arrange-ments.

with experimental flow fields determined through PIV tech-nology.

(2) A combination of a TI with gas bubbling derives incomplex recirculating flows of liquid with vertical and hori-zontal mixing processes that originate bypassing streamstoward the outlet. Higher flow rates of gas lead to decreasesof plug flow fraction of the fluid inside the vessel.

(3) The best results, for the purpose of inclusions flota-tion, are obtained for a tundish with the TI followed by gasbubbling, with 596 ml/min and with the half dams andwithout the TI. The worst case, for the same objective, isthe bare tundish.

(4) Thus gas bubbling itself yields very good results forfloating inclusions but is not superior to a well designed TIfor the same purpose.

(5) A tundish without gas bubbling and without a TIbut equipped with half dams provides acceptably high per-formances for inclusions flotation. If optimum flotationconditions are not required the simple arrangement of halfdams is acceptable.

Acknowledgments

The authors are very indebted to National Council ofScience and Technology of Mexico for the financial supportto this project through a scholarship provided to ARB.Thanks are given also to SNI and IPN, both institutionshave provided to the Group of Mathematical Simulation ofMaterials Processing and Fluid Dynamics a decided sup-port through all these years.

Nomenclature

CD : Drag coefficientdb : Bubble diameterdo : Interior diameter of an orifice.f b : Friction coefficient between bubbles and liquid

FD : Drag forceKpq : Interphase momentum-exchange coefficient

kl : Turbulent kinetic energy of liquid phasekgl : Covariance of the continuous and dispersed phases p : Pressure

uij : Velocity vector of phase “j” in direction “i”Vgl : Relative velocity between phases

Greek symbolsa j : Volume fraction of phase “j”e : Dissipation rate of turbulent energy

Pgl : As defined in Eq. (16) in the textr j : Density of phase “j”m : Fluid viscosity

w : Flow vorticity

Subscriptsb : Bubblee : Effectivei : Direction coordinate

in : Inletg : Gasl : Liquid

nozzle : Properties of the ladle shroud or nozzlet : Turbulento : Orifice

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Mathematical Simulation and Modeling of Steel Flow with