mechanism analysis of free-surface vortex formation during

© 2014 ISIJ 1592

ISIJ International, Vol. 54 (2014), No. 7, pp. 1592–1600

Mechanism Analysis of Free-Surface Vortex Formation during Steel Teeming

Hong-Xia LI,1) Qiang WANG,1,2)* Hong LEI,1) Jia-Wei JIANG,1) Zhan-Cheng GUO2) and Ji-Cheng HE1)

1) Key Laboratory of Electromagnetic Processing of Materials (Ministry of Education), Northeastern University, Shenyang,Liaoning Province, 110819 China.2) State Key Laboratory of Advanced Metallurgy, University of Science and Technology Beijing, Beijing, 100083 China.

(Received on August 28, 2013; accepted on March 17, 2014)

To develop an effective technology to prevent vortex forming in ladles, the mechanism underlying vor-tex formation during steel teeming must be studied. For simulations using the numerical simulation soft-ware Fluent, a geometric model of the ladle was generated, and initially water teeming in the ladle wassimulated for different initial tangential velocities. The results obtained help to verify the validity of thenumerical computations. Similar simulation conducted for steel showed that the tangential velocityincreases from the liquid level to the bottom of the ladle, and the flow at the bottom is related to vortexformation. The inference is that vortex formation begins from the ladle bottom. Causes for vortex forma-tion during teeming are discussed to help lay a theoretical basis for ladle design in preventing vortex for-mation during steel teeming.

KEY WORDS: free-surface vortex; slag entrapment in the ladle; clean steel; critical height; volume-of-fluidmodel (VOF).

1. Introduction

For the steel industry, reducing inclusion content in meltis important in producing clean steel. During continuouscasting, vortex always forms above the nozzle when moltensteel flows from the ladle to the tundish. The vortex quicklyleads to slag entrapment. In particular, when the steel liquidlevel is below a critical height at the end of teeming, slagand even air are strongly entrapped. It results in inhibitinginclusion floating, causing nozzle corrosion and clogging,increasing molten steel reoxidation, and seriously affectingthe quality of steel. Therefore, if the principle underlying theformation of the free-surface vortex can be more clearlyunderstood, controlling the vortex might be possible andhelpful in designing against vortex formation.

Over recent years, many researchers in related fields havedone much in studying factors leading to formation and crit-ical submergence depth of the free-surface vortex; alsocalled critical height for short, the critical submergencedepth is the vortex depth when the vortex extends to nozzle.In regard to theories and experiments, Lewellen et al.1) stud-ied the formation of the inspiration vortex when a small ori-fice is opened at the bottom in viscous and incompressibleflows. Piva et al.2) found that vortex formation dependsmainly on the initial maximum tangential velocity and thenozzle location relative to the cylinder axis. In addition, theyalso analyzed why the height of vortex formation is lowerin the ladle of an off-centered nozzle than that of a centered

nozzle. Lin et al.3) analyzed several factors in detail (includ-ing initial liquid level, nozzle diameter, nozzle eccentricity,and slag layer) which influenced the critical height of vortex.Odgaard,4) Gordon,5) and Reddy and Piekford6) also proposeda formula to estimate critical height. Sankaranarayanan etal.7) pointed out that vortex were formed from two distinctfeatures, namely, vortexing funnels and non-vortexing fun-nels, which are determined by two entirely different sets ofvariables. Their research is important for continuous castingproduction. Kojola et al.8) exactly predicted the criticalheight of non-vortexing funnels using spherical control vol-ume and cylindrical control volume models. In regard tonumerical simulation, Zhao et al.9) solved the free-surfaceproblem using the multiphase volume of fluid (VOF) model.They obtained through simulations the structure and move-ment of free-surface vortex, and considered that this vortexwas caused by locally concentrated rotational energy andwas similar to the Rankine vortex. Kuwana et al.10) usedparticle image velocimeter to measure velocities for water-teeming in a ladle, and simulated this model using compu-tational fluid dynamics (CFD) software. They also foundthat vortex formation is related to the initial tangentialvelocity, and obtained a relation among critical height,Froude number, and Reynolds number. However, they didnot specifically analyze the vortex forming process. Davilaet al.11) used a VOF model to simulate steel teeming underthermal insulated and thermal non-insulated conditions.They pointed out that the vortex formation height increaseddue to buoyancy caused by temperature gradients. Althoughresearchers have done much work on free-surface vortexand have made some advance, there are still no explicit

* Corresponding author: E-mail: [email protected]: http://dx.doi.org/10.2355/isijinternational.54.1592

ISIJ International, Vol. 54 (2014), No. 7

1593 © 2014 ISIJ

explanations for the formation of the free-surface vortex.Shapiro12) and Binnie13) confirmed experimentally that vortexin the northern hemisphere tended to rotate counterclock-wise due to the Coriolis force caused by the Earth’s rotation.Trefethen et al.14) also confirmed experimentally that vortexin the southern hemisphere tended to rotate clockwise. How-ever, Haugen et al.15) considered this effect was not signifi-cant for small-scale vortex. Moreover, Pedlosky et al.16)

reported that counterclockwise and clockwise directionswere both possible when vortex occurred. Suh et al.17)

pointed out that when compared with initial tangentialvelocity, the Coriolis effect can be neglected under normalflow conditions except when fluids had completely settledfor a long period of time.

In summary, some researchers studied the factors of vor-tex formation and proposed different formulas about thecritical height of vortex through experimental, dimensional,and mathematical analysis. These experimental studies werehowever unable to give a detailed analysis of changes in theentire flow field in the vessel at different times, especiallywhen the measurement of parameters of the molten steel isdifficult in practice. Therefore, it is difficult to use experi-mental methods to analyze the mechanism of vortex forma-tion in the ladle during steel teeming. Moreover, in respectto numerical simulations, researchers have already studiedtemperature effects on vortex formation, and arbitrary initialtangential velocities in simulations. Nonetheless, the simu-lation results were not compared with experimental results,thus the validity of the simulation was not confirmed. Forthis study, a geometric model of a ladle similar in manneras described in the research literature7) was generated forcomputer simulations. Teeming of water in the ladle wassimulated using the Fluent simulation software, and thenumerical simulation method verified by comparing simu-lation results with experimental data. Steel teeming in thesame ladle configuration was simulated using the samemethods. The vortex formation at the end of steel teemingwas analyzed. In this paper, a description of results is givenalong with a discussion of the mechanism and rule of vortexformation, to provide a reference and basis for preventingvortex formation in ladles during steel teeming.

2. Numerical Solution

2.1. Mathematical ModelThe mathematical model for fluid flow in a ladle is based

on the following assumptions: the fluid is viscous andincompressible; the effect of temperature is negligible; tosimplify modeling, the ladle is assumed cylindrical; wallthickness of the ladle is neglected; the nozzle is centered inthe ladle; the no-slip condition (zero velocity) for fluid flownear the ladle wall is assumed; and the slag layer on top ofthe molten steel is also neglected. The effect of Coriolisforce on vortex intensity can be represented by EquationR0 = V/fL, where R0 is Rossby number, V is the rationalvelocity of vortex, f is Coriolis parameter, L is the horizontallength of vortex. For small-scale flows in the bath, R0 isabove 103.18) This means that f is very small. Therefore, theCoriolis force has little effect on the vortex in the ladle likethat in the bath. The Coriolis effect is also neglected.

The VOF multiphase and renormalization-group (RNG)

k – ε models are used in the simulation. The VOFformulation19) relies on the fact that two or more fluids (orphases) are not interpenetrating. In each control volume, thesum of volume fractions of all phases is 1. If the qth fluid’svolume fraction in the cell is denoted αq, then the followingthree conditions are possible: αq = 0, the qth fluid is emptyin the cell; αq = 1, the qth fluid is full in the cell; 0 < αq <1, the interface between the qth fluid and one or more otherfluids is contained in the cell. The RNG k – ε model19) isderived using a rigorous statistical technique (based on theRNG theory). It is similar in form to the standard k – ε mod-el, but it includes the following refinements: the RNG modelhas an additional term in its ε equation that significantlyimproves the accuracy for rapidly strained flows; the effectof swirl on turbulence is included in the RNG model, thusenhancing the accuracy for swirling flows; the RNG theoryprovides an analytical formula for turbulent Prandtl num-bers, whereas the standard k – ε model uses user-specified,constant values; whereas the standard k – ε model is a high-Reynolds-number model, the RNG theory provides an ana-lytically derived differential formula for effective viscositythat accounts for low-Reynolds-number effects. These fea-tures make the RNG k – ε model more accurate and reliablefor a wider class of flows than the standard k – ε model. Thegoverning equations used in the simulations are as follows:Continuity equation:

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

Momentum conservation equation:

........ (2)

Turbulent kinetic energy equation (k):19)

...... (3)

Turbulent dissipation rate equation (ε):19)

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

In these equations, ρ is the density; ui velocity in the idirection; xi length in the i direction; P the pressure; τij thestress tensor; gj gravitational acceleration in the j direction;μeff the effective viscosity, in the high-Reynolds-number limit,

; Gk the generation of turbulence kinetic energy

due to the mean velocity gradients; Gb the generation of tur-bulence kinetic energy due to buoyancy; YM the contributionof the fluctuating dilatation in compressible turbulence tothe overall dissipation rate, neglected because the fluid isincompressible; Sk user-defined source terms; Rε the maindifference between the RNG k – ε and standard k – ε modelslies in the additional term in the ε equation; and Sε user-defined source terms. The model constants: C1ε = 1.42;

∂∂

+∂∂

=t x

ui

i( ) ( )ρ ρ 0

∂∂

+∂∂

= −∂∂

+∂

∂+

tu

xu u

P

x xgi

ji j

i

ij

ji( ) ( )ρ ρ

τρ

∂∂

+∂∂

=∂∂

∂∂

⎛

⎝⎜⎜

⎞

⎠⎟⎟ + + − − +

tk

xku

x

k

xG G Y S

ii

jk eff

jk b M k

( ) ( )ρ ρ

α μ ρε

∂∂

+∂∂

=∂∂

∂∂

⎛

⎝⎜⎜

⎞

⎠⎟⎟

+ +

t xu

x x

CkG C G

ii

jeff

j

k b

( ) ( )

( )

ρε ρε α με

ε

ε

ε ε1 3 −− − +Ck

R S2

2

ε ε ερε

μ ρεμeff Ck

=2

© 2014 ISIJ 1594


C2ε = 1.68; C3ε = 1 when buoyant shear layers in the mainflow direction is parallel to the direction of gravity, C3ε = 0for buoyant shear layers perpendicular to the gravitationalvector; Cμ = 0.0845; αk is the inverse effective Prandtl num-ber for k, αk ≈ 1.393; αε is the inverse effective Prandtl num-ber for ε, αε ≈ 1.393.

2.2. Geometric ModelThe ladle dimensions are as encountered in the research

literature,7) see Table 1.The ladle model is meshed using a combination of struc-

tured and non-structured grids.

2.3. Boundary Conditions, Initial Conditions and Solu-tion Method

The inlet boundary is a pressure inlet and the exit bound-ary is a pressure outlet. The gauge pressures of inlet and out-let are 0, and the operation pressures of inlet and outlet are101 325 Pa (1 atm). The flow direction of inlet is perpendic-ular to the boundary. The flow direction of outlet is obtainedby iterative operation when the pressure conditions are set.The velocity values of inlet and outlet are also obtained byiterative operation. The velocities for the ladle wall andnozzle wall are 0 because of no-slip wall. The turbulentintensity I is calculated by I = 0.16Re

–1/8.19) Re is Reynolds

number, , where is hydraulic diameter, is

the average velocity. The hydraulic diameter of pressureinlet is the ladle diameter and that of pressure outlet is thenozzle diameter. The average velocity of outlet is calculatedby , where, g is gravitational acceleration, H issurface height. And the average velocity of inlet is calculat-ed by . So turbulent intensity of pressureinlet is obtained, Iin = 0.06. The velocity of inlet isdecreased because of viscous losses and decreasing liquidhead. So it is set as Iin ≈ 0.03, that means turbulent intensityof inlet is set as 3%. It is considered air can’t flow back intoladle through the outlet. Turbulent intensity of pressure out-let is backflow turbulent intensity, so it is set as 0. The geo-metric model is a whole volume, the volume fractions oftwo phases and the initial velocity are set using a user-defined function (UDF). The numerical algorithms used tosolve the governing equations are known as the “finite vol-ume implicit algorithm” and “pressure-implicit with split-ting of operators” algorithms. Fluid teeming is simulatedusing the unsteady state, and a time step of 0.01 s. Waterteeming was simulated initially to validity the model andsimulation methods by comparing the results with existingexperimental data.7) Next, molten steel teeming was simu-lated using the same simulation methods. The physical

parameters of both fluids are listed in Table 2.

3. Verification of Simulation Method

3.1. Initial Tangential Velocity DistributionIn an ideal, incompressible, and inviscid liquid, the veloc-

ity distribution of a free-surface vortex and forced vortex aredescribed as in Eqs. (5) and (6),20) respectively. With rdenoting the distance from nozzle center, the tangentialvelocity, uθ is:

.............................. (5)

............................. (6)

The vortex, formed at the end of steel teeming, is a com-bination-type vortex like the Rankine vortex.20) The tangen-tial velocity distribution of Rankine vortex is described as:

....................... (7)

Where ω is the angular velocity and a the vortex core radius.In this study, the two initial tangential velocity distribu-

tions are introduced by way of a UDF program expressed inthe form:

..................... (8)

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

The conversion between the velocity in theCartesian coordinate system and the velocity inthe cylindrical coordinate system is:When x ≠ 0 and y ≠ 0

....... (10)

When

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

That means

Table 1. Model dimensions of the ladle.

Parameter Value

Diameter of ladle, D (m) 1.16

Height of ladle, Hladle (m) 1

Initial liquid level, H0 (m) 0.5

Diameter of nozzle, d (m) 0.0765

Length of nozzle, l (m) 0.1235

R d ve = ρ μ d v

v gHout ≈ 2

v v A Ain out out in≈vin

Table 2. Physical parameters of fluids.

Fluid Parameter Value

Water

Density, ρ (kg/m3) 998.2

Dynamic viscosity, μ (Pa⋅s) 0.001003

Surface tension, σ (N/m) 0.0728

Steel

Density, ρ (kg/m3) 7 000

Dynamic viscosity, μ (Pa⋅s) 0.0053

Surface tension, σ (N/m) 1.6

u rθ = constant

u rθ = constant

u r r a

u a r r a

θ

θ

ω

ω

= ≤ ≤

= ≥

⎧⎨⎪

⎩⎪

02

u rθ ω= = 0.2069 rad/s

u m sθ = 0 06. /

( , , )u u ux y z

( , , )u u ur zθ

u uy

uy

u uy

uy

u

x

y

=+

++

=+

++

θ

θ

y

x

x

x

x

x

y

x

2r

2

2r

2

2 2

2 2

zz zu=

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x yu u

u u

x y

z z

= == =

=

⎧⎨⎪

⎩⎪0

0


1595 © 2014 ISIJ

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

Where, (x,y,z) is the coordinate value of any point in theflow field.

And the counterclockwise direction of velocity is posi-tive, clockwise is negative. The direction of initial velocityis assumed counterclockwise by UDF. The two initial veloc-ity distributions are the constant angular velocity of theforced vortex and the constant tangential linear velocitywith value taken from the literature.7) Thus, whether thefree-surface vortex like a Rankine vortex can be formed canfinally be revealed. The result in the literature 7) is initialtangential velocity value but not the initial tangential veloc-ity distribution, so the initial velocity distribution is assumedas the two types. One distribution is constant angular veloc-ity 0.2069 rad/s, this is equivalent to the mean tangentiallinear velocity 0.06 m/s. The other distribution is constanttangential linear velocity 0.06 m/s. As long as the initial tan-gential velocity distribution is close to the actual interactioneffect, the simulation results obtained through iterative com-putation will be consistent with the experiment result.Hence, the appropriate distribution of the initial tangentialvelocity which produces a dimple height for the vortex closeto the experimental result7) can be determined.

Figure 1 shows the simulation results for two differentinitial tangential distributions. The initial liquid levels H0

are 0.5 m in two cases. Because the velocity field is sym-metric with respect to the center axis, half of the curves forthe two initial tangential velocity distributions can be com-pared. After the nozzle of the ladle is opened, the entire flowfield is seen to tend gradually towards the distribution for aRankine vortex. This behavior always occurs regardless ofthe initial tangential velocity distribution. And vortex coreradius decreases with the distance to the ladle bottomdecreasing in both left and right figures. The values of thevortex core radius are greater than the nozzle radius in leftfigure and the values are less than the nozzle in right figure.But these values of vortex core radius are approximatelyclose to the nozzle radius. Figure 2 shows the heights of theliquid surface started to dimple at 0.34 m and 0.38 m for the

two different initial tangential velocity distributions. Com-paring with the experimental value of 0.39 m reported in 7),the latter is much closer. Therefore, the results show that theinitial tangential velocity should be set to a constant tangen-tial linear velocity value. That is closer to actual interactioneffect.

3.2. Verification of ModelWith the same simulation method, the draining of water

is simulated with initial tangential velocities of 0.06 m/s,0.03 m/s, and 0.01 m/s all in a counterclockwise rotation.Figure 3 shows a comparison of dimple height between oursimulations and experimental results from literature 7); Hdimple

is the height of the liquid when the fluid surface starts toform a dimple. In the same order as the initial tangential

Fig. 1. Radial distribution of tangential velocity at different heightswhen liquid surface begins to dimple.

u u

u u

r

z z

θ = =

=

⎧⎨⎪

⎩⎪

0

Fig. 2. Height for dimple formation start, (a) initial condition: uθ =0.2069 rad/s, H0 = 0.5 m and (b) initial condition: uθ =0.06 m/s, H0 = 0.5 m.

© 2014 ISIJ 1596


velocity, the dimple heights were 0.38 m, 0.31 m, and0.16 m, respectively, which compare favorably with theexperimental results from 7) of 0.39 m, 0.35 m, and 0.23 m.Their trends are consistent, and therefore the simulationmethod used in the study is correct. Moreover, the dimpleheight of the vortex increases as the initial tangential veloc-ity increases; in other words, the larger the initial velocity,the sooner the vortex occurs.

4. Simulation Results and Discussion

Steel teeming is simulated with the same method. Thevalues of the initial tangential velocities are set the same,0.06 m/s, 0.03 m/s, and 0.01 m/s respectively, with a coun-terclockwise rotation. The simulation results are as follows.

4.1. Critical Height of VortexThe critical height Hcritical is defined as the height of vor-

tex when the vortex extends all the way to the nozzle. Crit-

ical height and its formation time are plotted against the ini-tial tangential velocity in Fig. 4. When the initial tangentialvelocity was 0.06 m/s, the time it took for the air column toreach the nozzle was 19 s, and the critical height was about0.29 m. When the initial tangential velocity was 0.03 m/s,the time extended to 29 s, and critical height lowered to0.2 m. For an initial tangential velocity of 0.01 m/s, the timewas 46 s, and critical height was about 0.07 m. Based on theabove data, the critical height of the vortex increases as theinitial disturbance increases.

4.2. Distribution of Tangential VelocityThe simulation results show that the radial distributions

obtained for the tangential velocities are similar for each ofthe initial disturbances. The results for initial tangentialvelocities 0.01 m/s (left figure) and 0.06 m/s (right figure)are compared in Fig. 5(a) which shows the tangential veloc-ity distribution at different heights at 4 s after nozzle open.These results are similar and in accord with the tangential

Fig. 3. Comparison of dimple height between simulation results and water model experiment results,7) (a) height for dim-ple formation start vs initial tangential velocity and (b) schematic diagram of dimple height formation start in theladle.

Fig. 4. Critical height and time of vortex extended to nozzle, (a) critical height and time vs initial tangential velocity and(b) schematic diagram of critical height of vortex extended to nozzle in the ladle.


1597 © 2014 ISIJ

velocity distribution of a Rankine vortex. Although the vor-tex does not occur at 4 s, the tangential velocity increaseswith decreasing distance from the bottom of the ladle. Notethat even before the surface begins to dimple, a vortexmotion has already developed at the bottom of the ladle.Moreover, tangential velocities increase as the initial distur-bance increases. And the values of the vortex core radius arealso close to the nozzle radius. Figure 5(b) shows the radialdistribution of the tangential velocity at a height of 0.05 m(for the initial tangential velocities 0.01 m/s (left figure) and0.06 m/s (right figure)). Trends are similar in both simula-tion results. The simulation results show that the formationprocess progresses first with the vortex appearing, thenforms an air column, and finally extends down to the nozzle,the tangential velocities gradually increasing at each sameheight to the bottom. And vortex core radius decreases asthe time increasing in both left and right figures. Accordingto the conservation of angular momentum, when the fluidflows towards the nozzle, the angular velocity of fluidincreases quadratically as distance decreases from the centerof nozzle. Therefore the vortex gradually forms and extendsto the nozzle as time evolves.

From the analysis given above, we know that the tangen-

tial velocity increases as depth increases. More importantly,the vortex formation is closely related to the flow at the bot-tom. Therefore, we infer that vortex formation starts at thebottom of the ladle.

4.3. Analysis of the Mechanism of Vortex Formation atthe End of the Steel Teeming Process

The energy in a free-surface vortex derives mainly fromgravity acting on the fluid during the steel teeming, but vis-cous forces also play a role. The situation is similar to theformation of a tornado where the temperature differenceleads to the rise of air and the shear force makes the airrotate. For the free-surface vortex, when the nozzle isopened at the bottom of the ladle, then with the effect ofgravity, the static pressure in the molten steel near the nozzleis no longer balanced, thus forcing the molten steel to flowout rapidly. Meanwhile, the tangential disturbance to themolten steel leads to its rotation. The static pressure con-tours (Fig. 6(a)) and the velocity field (Fig. 6(b)) are givenfor the fluid in the ladle when the nozzle is open. The semi-circle arc demarks the area near the nozzle. Region I is thevortex core region, with the vortex diameter being approxi-mately equivalent to the nozzle diameter. Because of thestatic pressure difference, the molten steel near the nozzlein region I flows towards the nozzle. However, this volumeof molten steel is also subjected to extrusion forces and vis-cous shearing stresses of the fluid in region II. Thereforeaxial velocity increases with decreasing distance to the noz-zle and hence is a maximum in the center of nozzle. Owingto the molten steel outflow, a horizontal static pressure dif-ference exists at the interface of regions I and II. Moreover,its direction points towards the nozzle; this causes radialflow and increases the tangential velocity. The horizontalstatic pressure difference gradually decreases in areas faraway from the center of nozzle in region I. Thus the flowin the axial direction plays a dominant role near the nozzlein region I.

Similar to region I, when molten steel flows out of thenozzle, flows in the axial and radial directions are formedand the tangential velocity is increased by the horizontal andvertical static pressure differences in region II. Molten steelin region II is not directly above the nozzle, thus the verticalpressure difference is lower than that in region I. Becauseof the shear stress from the down-flow in region I, the axialvelocity is larger at points closer to the interface betweenregions I and II. Molten steel in region II flows towardsregion I because of the horizontal static pressure differenceat the interface of these regions. In region II, the static pres-sure difference decreases as distance to the interfacedecreases. With the effect of viscous shear stress, the radialvelocity increases as distance decreases to the interface.Tangential velocity, which is also affected by the horizontalpressure difference that acts as centripetal force, increaseswith decreasing distance to the interface. Thus radial andtangential velocities reach maximums near the interface. Formolten steel in region II far away from the nozzle, all threevelocity components decrease rapidly with increasing dis-tance to the nozzle. This is because vortex motion begins toform at the nozzle, thus the static pressure remains almostin balance at locations away from the nozzle. Similar toregion II, because region III is further away from the nozzle,

Fig. 5. Radial distribution of tangential velocities: (a) at differentheights at 4 s after nozzle open and (b) at different time at0.05 m.

© 2014 ISIJ 1598


the static pressure balance has not been disturbed and there-fore fluid rotates slowly in this region.

With the time increasing, the static pressure balance isdestroyed gradually in the area far away from the nozzle inregions I and II; the law of the molten steel flow in this areais consistent with that near the nozzle in regions I and II.Flows in region III are relatively calm because of the dis-tance from the nozzle. In this region, the static pressureremains almost in balance, so all three velocity componentsare very small; molten steel spirals slowly into region II.Figure 7(a) shows radial profiles of the velocity compo-nents at heights 0.05 m and 0.35 m away from the bottom.As these profiles are axial-symmetric, only half of thecurves are shown in comparison. Velocity components at thehigher height of 0.35 m are smaller than those at the lowerheight of 0.05 m. At the higher height, the tangential veloc-ity is larger than either the axial or radial velocities. In

contrast, at the lower height, in the area near the nozzle, theaxial velocity is larger than the other two components. In thearea distant from the nozzle, the tangential velocity is thelargest of the three velocities. Figure 7(b) shows that theflow of molten steel is spiraling outward from the nozzle.

5. Vortex Prevention

5.1. Height for Dimple FormationComparing vortex and non-vortex funnels, the tangential

velocities of vortex funnels accelerate the surface dimple.Kojola et al.8) derived a hydraulic model formula accordingto the Bernoulli equation for a cylindrical control volume inthe absence of viscous forces:

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

............... (14)

Where QTH and QTO are the fluxes flowing into and outof the cylindrical control volume, d is the diameter of nozzleand l is the nozzle length (defined in Table 1), t is air layerthickness, H is surface height. Note that H and t are vari-ables in Eqs. (13) and (14), whereas the other parameters arealready known. Also, when QTH is greater than QTO, thevelocity of the outflow will continually increase driven bythe static pressure. Moreover, with decreasing surfaceheight, gravity cannot provide enough energy. Thus, QTH is

Fig. 6. Pressure and velocity for molten steel in the ladle when thenozzle is open: (a) static pressure contours and (b) velocityvector field.

Fig. 7. Velocity distribution and streamline diagram when liquidsurface begins to dimple: (a) velocity distributions ofthree directions at different heights and (b) a streamlinediagram.

Qd

g H l tTH = + +⎛

⎝⎜

⎞

⎠⎟

π ρρ

22

142

Q d g HTO = −⎛

⎝⎜

⎞

⎠⎟

2

32 1 2

1

3 2πρρ

/


1599 © 2014 ISIJ

not sufficient, and finally a surface dimple appears. Com-pared with non-vortex funnels, in addition to a deficiency invortex energy at the end of steel-teeming process, the tan-gential motion and viscous force also lead to a decrease inQTH. Thus the amount of fluid in the control volume is notenough, and the surface rapidly forms a dimple. With anincrease in the tangential disturbance, the vortex is formedearlier and air is also trapped in the vortex column. There-fore, the control and dispersion of the initial disturbance andthe tangential velocity at the bottom of the ladle are veryimportant. Figure 3 shows the comparison between theresults and the value calculated by Kojola’s equation. Thedot line in this figure indicates that the dimple height of non-vortex funnel is 0.069 m, which is calculated by Eqs. (13)and (14). The dimple heights for vortex in both theexperiments7) and simulation are higher than this value. Andthe dimple height for vortex decreases with the initial tan-gential velocity decreasing. So, if the initial tangentialvelocity is controlled to be low enough, the dimple heightfor vortex can be closed to the dimple height of non-vortexfunnel, and the slag entrapment will be reduced greatly.

5.2. Critical Height of VortexFigure 8 shows the relation between dimensionless criti-

cal height and Reynolds number Red, Weber number Wed,Froude number Frd. Conducting scaling analysis, see Eqs.(15) to (17):10)

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

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

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

Where d is the diameter of nozzle (defined in Table 1).In Fig. 8, it is found that the gravity is more important on

vortex formation process than viscosity force and surfacetension. When the Froude number Frd is same, the dimen-sionless critical heights of water and steel are equal in thesame conditions. So without regard to temperature, water isa good fluid for simulating steel teeming in the ladle withthe same structure. Because the critical height of vortex dur-ing steel teeming process is difficult to measure in practice,this simulation is a good method. The water model experi-ments and steel simulation are helpful to propose new vor-tex prevention approach.

In summary, as time evolves, the static pressure differ-ence gradually increases. Furthermore, because of viscousforce, the flow of fluid in all three directions near the nozzlespreads to the places away from the nozzle. Therefore, thecontrol and dispersion of the initial disturbance and the tan-gential velocity at the bottom of the ladle are very important.The vortex core radius is primarily close to the nozzle radiusand the vortex funnel formation starts from the bottom.Therefore, the electromagnetic force can be considered forvortex prevention near the nozzle in ladle. The effectivesteady electromagnetic field and rotational electromagnetic

field will be studied to prevent vortex during steel teemingin future. By this method, the critical height of vortex funnelcan be reduced to the critical height of non-vortex funnel orreduce lower than non-vortex funnel.

Ru d

ed =ρ

μθ2 2( / )

Wu d

ed =ρ

σθ2 2( / )

Fu

g drd =θ2

2( / )

Fig. 8. Dimensionless critical height as a function of each dimen-sionless parameter: (a) Reynolds number, (b) Weber num-ber, (c) Froude number.

© 2014 ISIJ 1600


6. Conclusions

In this paper, the free-surface vortex during steel teemingwas simulated using a numerical simulation method. Thecritical height of the vortex under different initial distur-bance conditions and the radial profiles of the tangentialvelocity during the whole process were investigated. More-over, formation and underlying mechanisms of free-surfacevortex were discussed. The conclusions are as follows:

(1) The vortex formed during steel teeming is similar toa Rankine vortex; its tangential velocity distribution has thesame characteristics as that for the Rankine vortex. And thevortex core radius is approximately the nozzle radius.

(2) The critical height of the vortex is an importantparameter for the free-surface vortex. The more intense theinitial disturbance is, the higher is the surface dimple heightas well as the critical height. In other words, vortex is easilyformed when the initial disturbance is greater.

(3) The simulation results show that formation beginswith a vortex, then an air column develops, and finally thevortex extends down to the nozzle. The tangential velocityof the vortex always increases with decreasing distance tothe bottom of the ladle.

(4) Gravity provides the energy for vortex formation.Aside from the static pressure imbalance, the initial tangen-tial disturbance also leads to vortex formation. The forma-tion is closely related to the pattern of fluid flow at thebottom of the ladle. Therefore, to prevent vortex forming,design considerations must be given to controlling the flowat the bottom. More importantly, flow mechanisms at thebottom of the ladle need to be further studied.

AcknowledgementsThis work was supported by Open Foundation of State

Key Laboratory of Advanced Metallurgy of University ofScience and Technology Beijing (KF12-07), the Science andTechnology Program of Liaoning Province (No. 201122109)and the 111 Project (No. B07015).

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