Modeling of Trajectory and Residence Time of MetalDroplets in Slag-Metal-Gas Emulsions in OxygenSteelmaking

GEOFFREY BROOKS, YUHUA PAN, SUBAGYO, and KEN COLEY

In basic oxygen steelmaking, the major portion of the refining is realized through reactions betweenmetal droplets and slag. The residence time of metal droplets in the slag crucially influences the pro-ductivity. A model for the prediction of trajectory and residence time of metal droplets in slags hasbeen developed based on mechanics and chemical kinetics principles. When there is no decarburization,analysis of the ballistic motion of metal droplets in the slag predicts very short residence times (�1second). This result demonstrates that when decarburization is very weak, the metal droplets spenda very short time in the slag. This could explain in part the poor kinetic behavior in the end stage ofthe blow. During active decarburization metal droplets normally become bloated, resulting in adecreased apparent density. Accounting for this, the ballistic model predicts residence times rangingfrom 10 to 200 seconds, which are much more in keeping with practical experience and previouslaboratory studies. Excellent agreement between the model and laboratory measurements, combinedwith reasonable predictions of industrial residence times, shows that this model can be used to providea much improved understanding of theoretical aspects of oxygen steelmaking.

I. INTRODUCTION

IN basic oxygen steelmaking, large numbers of metaldroplets are generated and ejected into the slag through thehigh-speed injection of oxygen. The generation of dropletsaccelerates the overall kinetics of various steelmaking reac-tions through the creation of a large interfacial area. Sincethe invention of oxygen steelmaking, many studies[1–25] havebeen carried out to understand the kinetic phenomena asso-ciated with slag-metal-gas emulsions. In several stud-ies,[1,2,11–15] slag samples were recovered from industrialfurnaces and the size distribution of droplets in the slag wasmeasured. Other investigators have carried out more fun-damental studies on the interaction of metal droplets inoxidizing slags, including studies using X-ray fluoroscopyto observe reaction phenomena.[3–7] These experimentalobservations show that the decarburization of metal dropletsin oxidizing slags significantly affects the motion of metaldroplets through the rapid evolution of gas from the droplets.

In oxygen steelmaking, the refining rate is dependent notonly on the reaction kinetics but also on the residence time ofthe metal droplets in the slag.[10] The generation of dropletsthrough top blowing has been the subject of several studies,including recent papers by the authors.[16–25] Most of these stud-ies have concentrated on the mechanical aspects of droplet gen-eration and have largely ignored the issue of droplet residencetime. Subagyo et al.[24] developed a model to predict droplet

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 36B, AUGUST 2005—525

GEOFFREY BROOKS formerly Associate Professor, and YUHUA PAN,formerly Post Doctoral Fellow, with the Steel Research Centre, McMaster Uni-versity, Hamilton, ON, Canada L8S 4L7, are Principal Research Scientist andResearch Scientist, respectively, with CSIRO Minerals, Clayton South, VIC3169, Australia. Contact e-mail: yuhua.pan@csiro.au SUBAGYO, formerlyPost Doctoral Fellow, with the Steel Research Centre, McMaster University,is Senior Lecturer, with the Department of Mechanical and Industrial Engi-neering, Gadjah Mada University, Yogyakarta 55281, Indonesia. KEN COLEY,Associate Professor, is with the Steel Research Centre, McMaster University.

Manuscript submitted July 30, 2004.

generation rate and more recently proposed a method to pre-dict size distribution.[10] However, their attempts to predict dropletresidence time were not entirely successful. Therefore, it is thepurpose of the present work to analyze the ballistic behaviorof ejected metal droplets, in order to develop a model that willpredict residence times in slag under various conditions. In par-ticular, this study aims at combining the chemical kinetics andfluid mechanics of droplet behavior in the emulsion.

II. MODEL DEVELOPMENT

A. Model for Ballistic Motion of Metal Droplets in Slagwithout Decarburization

1. Governing equationsThe concept behind this study is to calculate the trajectory

of a single iron droplet moving in slag by using the ballisticmotion principle. This concept was originally applied by Subagyoet al.[10] to calculate the trajectory and residence time of metaldroplets vertically ejected into slag. In the present work, theangle of ejection is considered as schematically illustrated inFigure 1. If we assume the slag phase is quiescent, the follow-ing force balances can be established for the vertical (z-coordinate)and horizontal (r-coordinate) directions, respectively.z-direction:

[1]

r-direction:

[2]

where the subscript d stands for droplet; the subscripts z andr stand for coordinates in vertical and horizontal directions,respectively; FB, FG, FD, and FA are buoyancy, gravitation,drag, and “added mass” forces, respectively; �d is droplet

rdVd dur

dt� �FD,r �FA,r

rdVd duz

dt� FB �FG �FD,z �FA,z

526—VOLUME 36B, AUGUST 2005 METALLURGICAL AND MATERIALS TRANSACTIONS B

density; Vd is the droplet volume; and u is the velocity ofdroplet relative to slag velocity.

Furthermore, the relevant forces can be calculated asfollows:

[3]

[4]

[5]

[6]

[7]

[8]

where Adp is the project area of the droplet, m2.In addition, the drag coefficients, CD,z and CD,r, can be

approximately calculated by using the following equations:[9,10]

[9]

and

[10]

where

[11]

and

[12]

where Dd is the droplet diameter, �s is slag viscosity, andNRe is the Reynolds number of the droplet.

NRe,r �rsurDd

ms

NRe,z �rs uzDd

ms

CD,r � 0.44 (1000 � NRe,r � 10,000)

CD,r � 18.5NRe,r�0.6 (1.0 � NRe,r � 1000)

CD,r � 24NRe,r�1 (NRe,r � 1.0)

CD,z � 0.44 (1000 � NRe,z � 10,000)

CD,z � 18.5NRe,z�0.6 (1.0 � NRe,z � 1000)

CD,z � 24NRe,z�1 (NRe,z � 1.0)

FA,r �1

2Vdrs

dur

dt

FA,z �1

2Vdrs

duz

dt

FD,r �1

2AdpCD,rrsur

2

FD,z �1

2Adp CD,z rsuz

2

FG � Vd rd g

FB � Vd rs g

Introducing Eqs. [3] through [8] into Eqs. [1] and [2]yields

[13]

[14]

After rearrangement, we obtain

[15]

[16]

Considering

[17]

[18]

Eqs. [15] and [16] become

[19]

[20]

To simplify the presentation of Eqs. [19] and [20], the vari-ables K0, Kz, and Kr are introduced as follows:

[21]

[22]

[23]

[24]

Finally, we obtain

[25]

[26]

Equations [25] and [26] are the major differential equa-tions to be solved, and the model established based on solvingthese equations is called the “ballistic droplet motion model.”Since Eqs. [25] and [26] are highly nonlinear, in this workthey are solved numerically for uz and ur. Then, the coor-dinates of the droplet’s trajectory can be calculated as

[27]

[28]Lr � ∫0

turdt

Lz � ∫0

tuzdt

dur

dt� Krur

2

duz

dt� K0 � Kzuz

2

Kr � �3rsCD,r

2(rs � 2rd)Dd

Kz �3rsCD,z

2(rs � 2rd)Dd(when uz � 0)

Kz � �3rsCD, z

2(rs � 2rd)Dd (when uz � 0)

K0 �2(rs � rd)g

rs � 2rd

dur

dt� �

3rsCD,r

2(rs � 2rd)Ddur

2

duz

dt�

2(rs � rd)g

rs � 2rd�

3rsCD,z

2(rs � 2rd)Dd uz

2

Vd �1

6pDd

3

Adp �1

4pDd

2

dur

dt� �

rsCD,r

rs � 2rd

# Adp

Vd ur

2

duz

dt�

2(rs � rd)g

rs � 2rd�

rsCD,z

rs � 2rd

# Adp

Vduz

2

rdVd

dur

dt� �

1

2AdpCDrsur

2 �1

2Vdrs

dur

dt

rdVd

duz

dt�Vdrsg �Vdrdg �

1

2AdpCDrsuz

2 �1

2Vdrs

duz

dt

Fig. 1—Schematic illustration of ballistic motion of a metal droplet in slag.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 36B, AUGUST 2005—527

Furthermore, from Eq. [27], the residence time, , of thedroplet in slag can be evaluated from the following equation:

[29]

2. Numerical methodBy using a simple explicit forward differencing numerical

method,[26] the finite difference equations of Eqs. [25] and[26] are derived as

[30]

[31]

From Eqs. [30] and [31], the velocities at time-step i canbe calculated from those at previous time-step (i � 1), i.e.,

[32]

[33]

where uz(i) and ur(i) are the numerical solutions of Eqs. [25]and [26], from which the trajectory and residence time canbe further calculated.

3. Droplet trajectory and residence timeThe droplet trajectory can be calculated by using the

numerical values of uz and ur as

[34]

[35]

When Lz(i) is equal or close to zero, the corresponding totaltime lapse is defined as the residence time, i.e.,

[36]

B. Model for Motion of Bloated Metal Droplets in Slagwith Decarburization

Min and Fruehan[6] studied decarburization of Fe-C dropletsin oxidizing slags using X-ray fluoroscopy and the mea-surement of gas flow rates to determine the kinetics of decar-burization. By means of the X-ray fluoroscopy technique,they were able to observe the decarburization phenomena iniron droplets initially containing 2.3 to 4.2 wt pct carbonmoving in oxidizing slags containing 3 to 15 wt pct FeO.These authors reported that droplets became suspended inthe slag for 20 to 200 seconds before sinking to the bottomof the crucible. Molloseau and Fruehan[7] performed similarexperiments but with more oxidizing slags and by using aconstant volume-pressure technique. In their experiments, theslags contained 3 to 35 wt pct FeO and 1-g droplets with ini-tial carbon levels of 2.91 wt pct were used. They reportedthat the Fe-C droplets expanded during reaction due to therapid CO gas generation through decarburization at both thesurface and inside the metal droplet. Molloseau and Fruehan[7]

described these droplets as being “emulsified.” In this arti-cle, we describe these expanded droplets as being “bloated”

t �ai

�t

Lr(i) �ai

1

2 Cur(i) �ur(i � 1)D�t

Lz(i) �ai

1

2 Cuz (i) � uz (i � 1)D�t

ur (i) � ur (i � 1) � Kr(i � 1)ur (i � 1)2�t

uz(i) � uz(i � 1) � K0�t � Kz(i � 1)uz(i � 1)2�t

ur(i) � ur(i � 1)

�t� Kr(i � 1)ur (i � 1)2

uz(i) � uz(i � 1)

�t� K0 �Kz(i � 1)uz (i � 1)2

Lz � ∫0

tuzdt � 0

to avoid confusion with the surrounding emulsified slag.When the decarburization rate was large enough, the dropletscould even float to the top surface of the slag layer. Oncedecarburization had slowed down, the droplets would sinkto the bottom of the vessel. Similar droplet bloating phe-nomena were also observed by Mulholland and Hazeldean,[3]

Gare and Hazeldean,[4] and Gaye and Riboud.[5] These find-ings demonstrate that during rapid decarburization, the dropletbecomes effectively less dense through the rapid generationof CO gas. The apparent lowering of density could be dueto small gas bubbles being temporarily trapped below thesurface of the metal droplet or to the formation of an inter-mediate gas “halo,” as proposed by Min and Fruehan.[6] What-ever the mechanism, it is clear that there must be a relationshipbetween the decarburization rate and the apparent densityof the droplet that governs the motion of the droplet in slagand hence the trajectory and residence time.

In the present analysis, it is assumed that the apparentdensity of the droplet is influenced by the decarburizationrate through the following equations:

[37]

[38]

where �d0 is initial droplet density, kg/m3; rc is decarbur-ization rate, wt pct/s; and rc* is threshold decarburizationrate for bloating, wt pct/s.

Equations [37] and [38] describe the fact that the dropletwill not bloat unless the decarburization rate is larger thana certain value, rc*, termed “threshold decarburization rate.”That is, when decarburization is very slow, the generatedCO gas inside the droplet can easily escape from the droplet,allowing the droplet to maintain its original size. However,if the decarburization rate is large enough, the generationrate of CO gas exceeds its escape rate, resulting in a certainamount of CO gas being trapped in the droplet causing thedroplet to bloat. Appendix A gives a qualitative descriptionof the bloating process of a metal droplet in slag, based onthe experimental results of Molloseau and Fruehan.[7]

Furthermore, it may be speculated that in Eq. [37] thethreshold decarburization rate, rc*, is a complex functionof interfacial properties of droplet and slag and will be subjectto further study by the authors. At present, however, thisparameter is determined from the experimental data available.In this work, the threshold decarburization rate, rc*, wasmainly evaluated from the experimental data of Molloseauand Fruehan.[7] Appendix A demonstrates that Eq. [37] givesa good qualitative approximation of the apparent density ofbloated droplets.

Figure 2 schematically depicts the motion of bloateddroplet observed by Molloseau and Fruehan.[7] In their work,at 1713 K, a 1-g Fe-C droplet was dropped into an oxidizingslag containing 20 wt pct FeO and X-ray fluoroscopy wasused to monitor the behavior of the droplet in slag. Theexperimental observation in this case was described in detailin Reference 7 and is briefly summarized as follows:

(1) After the droplet entered into the slag, decarburizationcommenced within 1 second.

(2) The droplet expanded (became bloated) and floated tothe surface of the slag.

rd � rd0 (if rc �� rc*)

rd � rd0 r *c

rc (if rc � r *c )

528—VOLUME 36B, AUGUST 2005 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig. 2—Schematic diagram of the behavior observed by X-ray fluoroscopy ofa Fe-C drop in a slag containing 20 wt pct FeO (Molloseau and Fruehan).[7]

Fig. 3—Measured and calculated carbon content in metal droplet and cal-culated decarburization rate in the experimental case shown in Fig. 2.

(3) The diameter of the droplet was increased more thantwice its original diameter or the volume over 10 timesof its original volume.

(4) The droplet floated on the slag surface for about 10 seconds.

(5) Finally, when the decarburization reaction became ratherweak, the droplet contracted and rapidly fell to the bot-tom of the slag bath. The time lapse to this momentmarked the residence time of the droplet in slag.

From this observation, as depicted in Figure 2, it can beestimated that the droplet residence time in slag is around11 seconds.

In the present work, the developed ballistic droplet motionmodel was also applied to the experimental case referred toearlier. In this application, the variation of the droplet densitywas assumed to obey Eqs. [37] and [38]. The carbon contentin the droplet was calculated from the amount of CO evolvedout of the droplet as a function of time, measured byMolloseau and Fruehan.[7] Then, through nonlinear regres-sion analysis, the relationship between carbon content andtime was cast into the following form:

[39]

where the values of the regression constants a and k are 1.47and 0.313, respectively.

Further, the decarburization rate, rc, was derived from Eq.[39] as

[40]

Equation [40] together with Eqs. [37] and [38] was finallyused in the ballistic droplet motion model to predict the res-

rc � �d[pct C]

dt� ake�kt

[pct C] � a(1 � e�kt)

idence time for the experimental case mentioned above.Figure 3 shows the carbon content in the droplet measuredby Molloseau and Fruehan[7] and calculated by Eq. [39] aswell as the decarburization rate calculated by Eq. [40].

According to the observed residence time (i.e., 11 seconds),by implementation of the model by using trial and errormethod, the threshold decarburization rate corresponding to20 wt pct FeO was found to be 5.72 10�3 wt pct/s.

In another experiment with a slag containing 5 wt pct FeO,Molloseau and Fruehan[7] described that, while the dropletremained in slag, the decarburization reaction could last about50 seconds and then the droplet slowly sank to the bottomof the bulk slag. Therefore, a rough estimate of the residencetime of Fe-C droplet in slag containing 5 wt pct FeO couldbe around 60 seconds. According to this value and againusing the trial and error method, the implementation of themodel determines the threshold decarburization rate corre-sponding to 5 wt pct FeO to be 1.43 10�3 wt pct/s.

Here, we can see that the FeO content in slag has a sig-nificant influence on the threshold decarburization rate rc*.From the values of rc* referred to earlier corresponding to5 and 20 wt pct FeO in slags and assuming a linear rela-tionship between rc* and (pct FeO), we obtain

[41]

From Eq. [40] one can further express the decarburizationrate as

[42]

which is equivalent to the first-order rate equation. Therefore,it may be reasonable to estimate the overall decarburizationrate using the first-order reaction rate principle, i.e.,

[43]

where [pct C] is the carbon content in the droplet, wt pct;[pct C]e is the equilibrium carbon content, wt pct; keff is theeffective rate constant, m/s; Aapp is the apparent surface areaof the droplet, m2; Vapp is the apparent volume of droplet,m3; and t is time, seconds.

rc ��d[pct C]

dt� keff

Aapp

Vapp ([pct C] � [pct C]e)

rc � �d[pct C]

dt� k([pct C] � a)

rc* � 2.86 10�4(pct FeO)

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 36B, AUGUST 2005—529

For modeling the experimental cases of Molloseau andFruehan,[7] the measured decarburization rates, representedin form of Eq. [40], were directly used in the bloated dropletmotion model. However, for modeling the real processes intop blown oxygen steelmaking, in which the decarburizationrates are normally unknown, Eq. [43] was applied in thepresent model to predict the decarburization rate of metaldroplets in slag. This necessitates estimations on the effectiverate constant keff. In the present work, keff was calculatedusing the following equation:

[44]

where Dc is the effective diffusivity of carbon in liquid iron,m2/s; ud is the overall velocity of the droplet, m/s; and Dd,app

is the apparent diameter of the droplet, m. Equation [44] isan application of Higbie’s penetration theory[27] to calculatethe mass transfer coefficient for carbon diffusion to thesurface of the droplet, as we assume that this process is thelimiting step of the overall decarburization reaction.

Strictly speaking, it is only possible to use Eq. [44] to pre-dict keff when the droplet velocity is known, e.g., duringdroplet ascending and descending stages (Appendix A). How-ever, when the droplet is floating on the top surface of theslag, it is difficult to evaluate the droplet velocity. Therefore,in order to use Eq. [44] to predict keff for the droplet floatingon the slag surface, the droplet velocity is calculated byassuming the droplet is moving in a hypothetical infinite slagmedium. That is, while the droplet is floating and movingon the top surface of the slag, it is assumed that the droplethas a velocity resulted from ballistic motion. In this way, thedecarburization kinetics for the whole process of the droplettraveling in slag (ascending floating descending) canbe simulated. As for the trajectory of the droplet, however,the maximum vertical position of the droplet (correspond-ing to slag surface) is set to a presumed slag height (2 m).

It has to be admitted here that the approach described herefor modeling the decarburization kinetics of Fe-C droplet float-ing on the top surface of slag is just an approximation. Nev-ertheless, it is certain that the droplet is not static but movingin some way (e.g., wandering or rotating) on the slag surface.Thus, considering the fact that there are no experimentaldata available to ascertain the moving velocity of the dropletfloating on the slag surface, using the ballistic motion velocity,as a first approximation, to approach the velocity of the dropletfloating on the slag surface seems acceptable.

It should also be noted here that, using Eq. [43], we areassuming that a steady-state first-order relationship can beused to describe the rate of decarburization. This assump-tion ignores the possibility of a change of mechanism dur-ing the reaction and higher order relationships that may beassociated with reaction control or mixed reaction and masstransfer control.

Integration of Eq. [43] leads to

[45]

from which the decarburization rate can be numerically cal-culated by

[46]rc �[pct C]i �[pct C]i�1

�t

C � Ce � (C0 � Ce)e�keff

Aapp

Vapp

t

→→

keff � 21Dcud /(pDd,app)

where [pct C]i is the carbon content in the droplet at the pre-sent time-step, weight percent; and [pct C]i�1 is the carboncontent in the droplet at the last time-step, weight percent.

Introducing Eqs. [37], [38], [41], [45], and [46] into Eqs.[25] and [26], the ballistic droplet motion model was furtherdeveloped to predict trajectory and residence time of themetal droplet in slag under the influence of decarburization.This model is termed in the present article as the “bloateddroplet motion model”, which accounts for the effects ofboth mechanics and chemical kinetics on the motion of metaldroplets in oxidizing slag.

C. Motion of Metal Droplets in Slag-Metal-GasEmulsions

It is expected that, in addition to the decarburization reac-tion, gas bubbles trapped in the slag phase also influence themotion of metal droplets in slag-metal-gas emulsions. Deoet al.[8] studied the terminal velocities of metal droplets andgas bubbles in slag-metal-gas emulsions for Reynolds numbersconfined to the Stokes flow regime. Subagyo and Brooks[9]

extended the approach of Deo and his coworkers for higherReynolds numbers up to 10,000. For a slag-metal-gas emul-sion, these authors assumed that the metal droplets could betreated as the dispersed phase in a slag-gas continuum so thatthe average density of this continuum can be defined as

[47]

where �sg is the average density of the slag-gas continuum,kg/m3; and �g is the volume fraction of the gas bubbles,which is defined as[9]

[48]

where Vg, Vm, and Vs are the volume of gas, metal, and slag,respectively, m3.

In addition, according to Deo et al.[9] the average viscosityof the slag-gas continuum can be defined as

[49]

When considering the motion of metal droplets in a slag-gascontinuum, the developed model here for bloated dropletmotion can be applied to slag-metal-gas emulsions by simplyreplacing slag phase density (�s) and viscosity (�s) with theslag-gas continuum density (�sg) and viscosity (�sg), respec-tively. In the present modeling work, the following valuesof physical properties were adopted:[10] �s � 2991.4 kg/m3,�s � 0.0709 Pa � s, �m � 7000 kg/m3, �m � 0.00568 Pa � s,and �g � 1.25 kg/m3.

III. RESULTS AND DISCUSSION

A. Trajectory and Residence Time of Metal Droplets inSlag without Decarburization

1. Influence of metal droplet size on its trajectory in slagFigure 4(a) shows the trajectories predicted by the ballistic

droplet motion model for metal droplets ejected in a 30-degangle and at different ejection velocities. It can be seen fromthis figure that the droplets do have parabolic-like trajectories

msg �2

3# ms

(1 � fg1/3)

(rsg � rg)

(rs � rg)

fg �Vg

Vg � Vm � Vs

rsg � rgfg � rs(1 � fg)

530—VOLUME 36B, AUGUST 2005 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig. 5—Influence of ejection angle on the trajectory of the metal dropletin slag without decarburization.

Fig. 6—Influence of droplet diameter and ejection velocity on the resi-dence time of a metal droplet in slag without decarburization.

in slag. The calculations were performed based on 2-m-thickslag layer with a density of 2991.4 kg/m3 and a viscosity of0.0709 Pa � s. The droplet ejection velocity for top blownoxygen steelmaking was determined by nonlinear regressionanalysis on data calculated by Subagyo et al.,[10] from theexperimental data of Koria and Lange.[22] This relationshipis shown in Figure 4(b). A general feature in Figure 4(b) isthat in top blown oxygen steelmaking, a larger droplet hasa lower ejection velocity. Figure 4(a) depicts that the motionof metal droplets in slag is influenced by both the size (diam-eter) of the droplets and the ejection velocity.

2. Influence of ejection angle on droplet trajectoryin slag

Figure 5 shows the effect of the ejection angle on droplettrajectory. We can see that when the ejection angle increases,the residence time decreases along with decreased maximumheight. For a 60-deg angle of ejection, the droplet residencetime is only about half of that for vertical ejection. Therefore,when there is no decarburization, droplet ejection angle hasa marked influence on the residence time of the droplets.

Figure 6 illustrates the residence time for different dropletdiameters ejected in a 30-deg angle and at different ejectionvelocities. For convenience in comparison and analysis, theejection velocity shown in Figure 4(b) is redrawn in this fig-ure. It can be seen from Figure 6 that the residence time firstincreases with the droplet size but decreases if the dropletis too large because its ejection velocity becomes lower.

It can be concluded from Figure 6 that for fully densedroplets the residence times of all droplets are very short,less than one-third of a second. This result prompts an impor-tant finding for top blown oxygen steelmaking; the resi-dence time of dense droplets is insufficient to support thereaction rates observed in practice. Subagyo et al.[10] sug-gested that residence times of around 60 seconds arerequired. These workers assumed that the bubble-inducedstirring in the slag leads to extended residence times. How-ever, this effect could only be inferred but not directly cal-culated. Therefore, the bloated droplet would seem to be abetter representation of reality for the part of the blow whereactive decarburization occurs because it combines rigorousfluid mechanics with experimental data, resulting in morereliable predictions.

Fig. 4—Influences of droplet diameter on ejection velocity and trajectoryof metal droplet in slag without decarburization.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 36B, AUGUST 2005—531

Fig. 7—Results predicted by the bloated droplet motion model for an exper-imental case of Molloseau and Fruehan[7] (cf., Fig. 2).

B. Trajectory and Residence Time of Metal Droplets inSlag with Decarburization

1. Results of application of bloated droplet motionmodel to laboratory experiments

Figure 7 shows the results of application of the bloateddroplet motion model to the experimental case of Molloseauand Fruehan[7] (cf., Figure 2). Figure 7(a) gives the variationof carbon content in a Fe-C droplet with time, in comparisonwith the observed values, while Figure 7(b) shows the dropletvertical position as a function of time. From Figure 7(b) wecan see that the droplet residence time is approximately11.5 seconds, which is very close to the value of about 11 sec-onds observed from Molloseau and Fruehan’s work.[7]

After this model validation, the bloated droplet motionmodel was further used to predict droplet trajectories andresidence times for other experimental cases of Min andFruehan[6] and of Molloseau and Fruehan.[7] Figure 8 illus-trates the model’s predicted results. For the same droplet

size, the droplet residence time generally decreases withincreasing FeO content in slag. This is because higher FeOcontent in slag results in a larger rate of decarburization. Ifthe droplet size is the same, its carbon will be removed ina shorter time. On the other hand, larger droplets containmore carbon, which needs more time to be removed. There-fore, this model predicts that larger droplets have longer res-idence times than smaller droplets in slag and this isconsistent with the experimental work of Fruehan andcoworkers.[6,7] Significantly, the model predicted that resi-dence times for some experimental cases of Min and Frue-han[6] are in the range of 80 to 190 seconds, which falls wellwithin the range of their observations (20 to 200 seconds).

The droplet size influences the decarburization rate viaEqs. [43] and [44]. As indicated by these equations, a largerdroplet has a smaller Aapp /Vapp ratio and keff value, resultingin a lower decarburization rate, which would keep the dropletbloated for a period as long as the decarburization rate ishigher than the threshold decarburization rate. The dropletsize may also influence the threshold decarburization rate.The details of these mechanisms have yet to be studied.

2. Influence of droplet size and ejection angleFigure 9 illustrates the model predicted trajectories of

bloated droplets in top blown oxygen steelmaking for dif-ferent droplet sizes and ejection angles, assuming a slagdepth of 2 m. The modeling conditions are the same as thesimulation cases without decarburization (cf., Figures 3 and4). It can be seen from Figure 9(a) that, under the influenceof the decarburization reaction, the droplet trajectories areno longer parabolic, and the motion of the droplet is dom-inated by buoyancy.

Figure 10 gives the variation of bloated droplet positionwith time. It shows that larger droplets will spend longertimes on the surface of the slag. We can predict from theseresults that very small droplets (�1 mm) do not reach thetop of the slag because of fast decarburization; while thelarge droplets (�2.5 mm) are able to float to the top of theslag layer because they contain more carbon and requiremore time for decarburization.

In addition, if we examine Figure 9(b), we find that theejection angle does not have any significant influence on

Fig. 8—Residence times predicted by the bloated droplet motion modelfor experimental cases of Min and Fruehan[6] (for droplet diameter �11.3 mm) and Molloseau and Fruehan[7] (for droplet diameter � 6.5 mm).

532—VOLUME 36B, AUGUST 2005 METALLURGICAL AND MATERIALS TRANSACTIONS B

the maximum height because droplets ejected at any angleare able to reach the top surface of the slag. This demon-strates the dominant influence of decarburization on the res-idence time of bloated droplets in slag in top blown oxygensteelmaking.

The results obtained from the bloated droplet motionmodel suggest that the carbon content in the droplet and thedecarburization rate have a dominant influence on the metaldroplet residence time in top blown oxygen steelmaking.The more carbon the droplet has, the longer the residencetime. When there is decarburization inside the droplet, thedroplet will become bloated. Accordingly, the model sug-gests that the buoyancy force dominates the motion of thedroplet in slag.

It should be pointed out here that another plausible expla-nation for the residence times measured by Fruehan and cowork-ers in the laboratory and by others in industrial trials[11–14] isthat the rapid generation of CO bubbles in the emulsion resultsin significant slag motion. The present bloated droplet motionmodel has ignored slag motion induced by bubble generation,though the observations of Fruehan and coworkers are consistentwith the notion that the trajectory of the droplets is dominatedby buoyancy, and that, at least in the laboratory, slag motiondoes not play a significant role. Certainly, the effect of themotion of slag on droplet trajectory in basic oxygen furnace(BOF) converters is worthy of further study.

C. Residence Time of Metal Droplets in Metal-Slag-GasEmulsions

Figure 11 shows the residence times of bloated dropletsin slag-metal-gas emulsions with different volume frac-tions of gas hold-up. It can be seen from this figure that themore the slag holds gas bubbles the shorter the residencetime of the droplets. In addition, the volume fraction of gasalso influences the distribution of residence time for differ-ent sizes of the droplets. If the gas volume fraction in slagis below about 0.7, the residence time first decreases withthe initial diameter of the droplet and then increases fordroplets initially larger than 2.5 mm. There exists a mini-mum residence time associated with a droplet size of about2.5 mm. However, if the gas volume fraction is higher than0.7, the relationship between residence time and droplet size

Fig. 9—Metal droplet trajectories predicted by the bloated droplet motionmodel for different droplet size and ejection angle.

Fig. 10—Variation of metal droplet vertical position in slag in top blownoxygen steelmaking.

Fig. 11—Influences of initial diameter of metal droplets and volume frac-tion of gas bubble on the residence time of metal droplets in slag-metal-gas emulsions.

ings of Fruehan and coworkers[6,7] and with industrialobservations.[11–14]

4. The excellent agreement between the bloated dropletmodel and laboratory experiments, combined with thevery reasonable predictions of industrial residence times,suggests the bloated droplet model will be an excellenttool for predicting the kinetic behavior in top blown oxy-gen steelmaking.

ACKNOWLEDGMENTS

The authors acknowledge the McMaster Steel ResearchCenter for financial support of this research project. Thanksare also given to Dr. Diancai Guo, McMaster Steel ResearchCenter, for valuable discussions on writing this article.

APPENDIX A

Qualitative description of the bloating process of metaldroplet in slag due to decarburization

When a Fe-C droplet enters an oxidizing slag, due todecarburization, the size change of the droplet can be char-acterized by five distinguishable stages, as illustrated inFigure A1. The size of the droplet may be influenced bysuch factors as decarburization rate (i.e., CO generation rate),rc, CO escape rate, rc

esc, threshold decarburization rate, rc*,and steady-stage decarburization rate, rc

ss. According to thedifferences among these decarburization rates, as shown in

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 36B, AUGUST 2005—533

Table I. Comparisons of Bloated Droplet Motion ModelPredictions with Laboratory Experiments and Plant Mea-

surements/Estimations on the Residence Time of MetalDroplets in Slag in Top Blown Oxygen Steelmaking

Residence Investigators Methods Time, s

Schoop et al.[11] Indirect plant measurement �60from which the residence time was calculated based on chemical analysis and kinetic model.

Price[12] Plant measurement with 120 30radioactive gold isotope tracer technique.

Kozakevitch[14] Prediction based on the carbon and phosphorus 60 to 120contents in metal droplet from plant measurement.

Present Predictions using bloated 10 to 200work droplet motion model on

experimental cases of Molloseau and Fruehan[7] and Min and Fruehan[6] (cf., Figure 8)

Present Predictions using bloated droplet 20 to 80work motion model on metal-slag-gas

emulsions with 15 pct FeO and gas volume fraction less than 85 pct (cf., Figure 11)

changes dramatically. There exists a maximum residencetime which can be related to the droplet size and volumefraction of gas in the emulsion. For instance, the maximumresidence time is around 40 seconds for 5-mm droplets andfg � 0.85; it decreases to about 30 seconds for 2.5-mmdroplets and fg � 0.9. Therefore, the maximum residencetime shifts from larger droplets to smaller ones as gas hold-up increases in slag. Comparing the residence time predic-tions with the practical estimates shown in Table I, it canbe seen that the bloated droplet motion model provides veryreasonable predictions for residence times in top blownoxygen steelmaking.

IV. CONCLUSIONS

A mathematical model for simulating the motion of denseand bloated droplets in slag has been developed. From themodel predictions on trajectory and residence time for metaldroplets in slag with and without decarburization, the fol-lowing conclusions can be drawn.

1. A model based on ballistic motion principles can be usedto predict the residence time of dense metal droplets inslag in the end stage of blow in top blown oxygen steel-making.

2. For dense droplets, it is predicted that the residence timeof the droplets in slag is less than 1 second in top blownoxygen steelmaking. This implies that the poor kineticcondition found in the end stage of blow in top blownoxygen steelmaking may be in part due to the short res-idence time of metal droplets traveling in the slag.

3. The same model, modified to include the reductionin density of droplets due to decarburization, predictsresidence times consistent with the experimental find-

Fig. A1—A schematic illustration of the bloating process of Fe-C dropletin oxidizing slag: (a) variation of decarburization rate with time and (b)variation of droplet diameter with time.

u velocity of droplet (m/s)ud overall velocity of droplet (m/s)Aapp apparent surface area of bloated droplet (m2)Adp project area of droplet (m2)[pct C] average carbon content in iron droplet (wt pct)[pct C]e equilibrium carbon content in iron droplet (wt pct)CD drag coefficient (—)DC diffusivity of carbon in liquid iron (m2/s)Dd diameter of droplet (m)Dd,app apparent diameter of bloated droplet (m)(pct FeO) average content of FeO in slag (wt pct)FA added mass force (N)FB buoyancy force (N)FD drag force (N)FG gravitational force (N)K0 variable defined by Eq. [21]Kr variable defined by Eq. [24]Kz variable defined by Eq. [22] or [23]Lr horizontal coordinate of droplet’s trajectory (m)Lz vertical coordinate of droplet’s trajectory (m)NRe Reynolds number (—)Vapp apparent volume of bloated droplet (m3)Vd droplet volume (m3)Vg volume of gas in emulsion (m3)Vm volume of metal in emulsion (m3)Vs volume of slag in emulsion (m3)

Greek symbols�g volume fraction of gas bubble in emulsion (—)�d density of droplet (kg/m3)�d0 initial density of droplet (kg/m3)�g density of gas (kg/m3)�s density of slag (kg/m3)�sg average density of slag-gas emulsion (kg/m3)�s viscosity of slag (Pa � s)�sg average viscosity of slag-gas emulsion (Pa � s) residence time of droplet in slag (s)

REFERENCES1. H.W. Meyer, W.F. Porter, G.C. Smith, and J. Szekely: J. Met., 1968,

July, pp. 35-42.2. C. Cicutti, M. Valdez, T. Pérez, J. Petroni, A. Gómez, R. Donayo, and

L. Ferro: Study of Slag-Metal Reactions in an LD-LBE Converter, Proc.Slag Conf., Stockholm, 2000.

3. E.W. Mulholland, G.S.F. Hazeldean, and M.W. Davies: J. Iron SteelInst., 1973, vol. 211, pp. 632-39.

4. T. Gare and G.S.F. Hazeldean: Ironmaking and Steelmaking, 1981,No. 4, pp. 169-181.

5. H. Gaye and P.V. Riboud: Metall. Trans. B, 1977, vol. 8B, pp. 409-15.6. D.J. Min and R.J. Fruehan: Metall. Trans. B, 1992, vol. 23B, pp. 29-37.7. C.L. Molloseau and R.J. Fruehan: Metall. Trans. B, 2002, vol. 33B,

pp. 335-44.8. Brahma Deo, Arun Karamcheti, Amitava Paul, Pankaj Singh, and R.P.

Chhabra: Iron Steel Inst. Jpn. Int., 1996, vol. 36 (6), pp. 658-66.9. Subagyo and G. Brooks: Iron Steel Inst. Jpn. Int., 2002, vol. 42 (10),

pp. 1182-84.10. Subagyo, G.A. Brooks, and K. Coley: Can. Metall. Q., 2005, vol. 44 (1),

pp. 119-29.11. J. Schoop, W. Resch, and G. Mahn: Ironmaking and Steelmaking,

1978, vol. 2, pp. 72-79.12. D.J. Price: in Process Engineering of Pyrometallurgy, M.J. Jones, ed.,

The Institution of Mining and Metallurgy, London, 1974, pp. 8-15.13. B. Trentini: Trans. TMS-AIME, 1968, vol. 242, pp. 2377-88.14. P. Kozakevitch: JOM, 1969, vol. 22 (7), pp. 57-68.15. A. Chatterjee, N.O. Lindfors, and J.A. Wester: Ironmaking and Steel-

making, 1976, vol. 3 (1), pp. 21-32.

Figure A1(a), the five stages of the change in the size ofFe-C droplets in slag may be interpreted as follows.

(1) Incubation stage (A B): This stage is very short intime (less than 1 second). In this stage the CO generationrate increases dramatically while the CO gas is able toescape the droplet at the same rate, i.e., rc

esc � rc. Thehigher the CO generation rate, the higher the CO escaperate. As a result, the droplet retains its original size (diam-eter), i.e., Dd � D0, as indicated by the line a b inFigure A1(b).

(2) Bloating stage (B C): In this stage, the CO genera-tion rate is higher than the CO escape rate (rc � rc

esc),resulting in a certain number of CO bubbles beingtrapped in the droplet. As the volume of CO trapped inthe droplet increases, the droplet gets bloated withincreasing diameter (Dd � D0), as shown by the line b

c in Figure A1(b).(3) Steady stage (C D): In this stage, the decarburization

rate first reaches its maximum value and then decreasessharply. The droplet expands to such a state that the COescape rate keeps equal to CO generation rate (rc

esc �rc). That is, the higher the CO generation rate, the higherthe CO escape rate. In this dynamic steady state thedroplet reaches its maximum diameter, i.e., Dd � Dmax,as indicated by the line (c d) in Figure A1(b).

(4) Contracting stage (D E): When the decarburizationrate decreases to below the steady-state decarburizationrate (rc � rc

ss), the CO generation rate becomes smallerthan the CO escape rate, i.e., rc � rc

esc. This leads to thecontraction of the droplet (Dd � Dmax), as shown bythe line (d e) in Figure A1(b).

(5) Settling stage (E F): After the decarburization ratebecomes lower than the threshold decarburization rate,i.e., rc � rc*, the CO escape rate is eventually equal tothe CO generation rate (rc

esc � rc) so that the dropletkeeps its original diameter (Dd � D0) again, as shownby the line (e f) in Figure A1(b).

As a summary, the bloating process of carbon-containingmetal droplet in an oxidizing slag can be generally interpretedin terms of the five stages referred to. The expansion of thedroplet closely follows the magnitude of the decarburizationrate. This indicates that the motion of carbon-containingmetal droplet in an oxidizing slag is crucially influencedby the decarburization rate due to the changing apparentdensity of the droplet. Consequently, it is not surprising thatdecarburization rate dominates the motion (and hence theresidence time) of the droplet in slag.

NOMENCLATURE

a regression constant in Eq. [39] (wt pct)g gravitational acceleration (m/s2)i time-step index (—)k regression constant in Eq. [39] (1/s)keff effective decarburization rate constant (m/s)rc decarburization rate (wt pct/s)rc* threshold decarburization rate for droplet

bloating (wt pct/s)t time (s)�t time-step (s)

→

→→

→→

→→

→

→

→

534—VOLUME 36B, AUGUST 2005 METALLURGICAL AND MATERIALS TRANSACTIONS B

22. S.C. Koria and K.W. Lange: Metall. Trans. B, 1984, vol. 15B, pp. 109-16.23. S.C. Koria and K.W. Lange: Ironmaking and Steelmaking, 1986, vol. 13

(5), pp. 236-40.24. Subagyo, G.A. Brooks, K.S. Coley, and G.A. Irons: Iron Steel Inst.

Jpn. Int., 2003, vol. 43 (7), pp. 983-89.25. Subagyo, G.A. Brooks, and K. Coley: Interfacial Area in Top Blown

Oxygen Steelmaking, Steelmaking Conf. Proc., ISS, Warrendale, PA,2002, vol. 85, pp. 749-62.

26. Gary J. Lastman and Naresh K. Sinha: Microcomputer-Based NumericalMethods for Science and Engineering, Saunders College Publishing, a Divi-sion of Holt, Rinehart and Winston, Inc., New York, NY, 1989, pp. 211-15.

27. R. Higbie: Trans. Am. Inst. Chem. Eng., 1935, vol. 35, pp. 365-89.

16. R.C. Urquhart and W.G. Davenport: Can. Metall. Q., 1973, vol. 12 (4),pp. 507-16.

17. N. Standish and Q.L. He: Iron Steel Inst. Jpn. Int., 1989, vol. 29, No. 6,pp. 455-461.

18. Q.L. He and N. Standish: Iron Steel Inst. Jpn. Int., 1990, vol. 30, No. 4,pp. 305-309.

19. Q.L. He and N. Standish: Iron Steel Inst. Jpn. Int., 1990, vol. 30, No. 5,pp. 356-361.

20. G. Turner and S. Jahanshahi: Trans. Iron Steel Inst. Jpn., 1987, vol. 27,pp. 734-39.

21. S.C. Koria and K.W. Lange: Ironmaking and Steelmaking, 1983, vol. 10(4), pp. 160-68.

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