1. Introduction

There are several ways in the literature to estimate thedepth reached by a supersonic O2 flow in liquid steel. Theycan be summarized in three methods: energetic, dynamicand numerical methods. Energetic methods are based on anenergetic balance between the momentum of the jet and theenergy needed for the penetration.

The dynamic methods3) are based on the evaluation ofthe impact force of the jet upon the surface of the liquidsteel, which requires the inclusion of the features of thenozzle and of the jet:

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

where v0 is the jet velocity at the nozzle outlet. If the vol-ume of displaced liquid is constant and known, the depthreached by a vertical jet in a steel bath is given by:

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

where H is the height of the nozzle above the steel, g is thegravity constant, r is the density. The problem with the use

of this kind of method is the value of the proportionalityconstant b that must be determined experimentally throughhydro-dynamical trials, which not always offer reliable andsignificant results for the industrial practice. The numericalmethods are based essentially on a finite element or finitedifference calculation (i.e. CFD software).4) There are sev-eral problems bonded to the convergence of the solutionsfor high velocity jets. Problems are generated by the numer-ic definition of an interface surface and by the formation ofsteel drops in the jet itself. Moreover, as these techniquesare very time consuming, the use for an on-line control canfind a significant obstacle.

To overcome these problems, an update of the energeticapproach has been developed in this study. Energetic ap-proach has been integrated by the principle of the dynamicapproach for the approximation of the volume of the sur-rounded by the liquid phase.

The model elaborated for the injection of a supersonic jetcan lead to a computation about decarburisation phenome-na through the determination of the computed liquidsteel/injected O2 interface.

A thermodynamic and kinetic model of the bath duringthe refining period has been built. Its main aim is to providea fast tool to estimate the effect of oxygen and carbon injec-tion and to optimise the process in EAF.

DF

H

βπ ρsteelg

2

Fd

ρO2v0

2 02

4

ISIJ International, Vol. 44 (2004), No. 8, pp. 1342–1349

© 2004 ISIJ 1342

Simulation of Oxygen Penetration and Decarburisation in EAFUsing Supersonic Injection System

Francesco MEMOLI, Carlo MAPELLI,1) Pietro RAVANELLI2) and Marco CORBELLA

Techint S.p.A., Via Monte Rosa 93, 20149 Milano, Italy. 1) Dipartimento di Meccanica, Politecnico di Milano, Via la Masa34, 20156 Milano, Italy. 2) Tenaris Dalmine Meltshop, Piazza Caduti 6 Luglio 1944, 1, 4044 Dalmine (BG), Italy.

(Received on February 13, 2004; accepted in final form on May 19, 2004 )

The supersonic jet generated by a Laval nozzle in air or in steel has been modelled to vary degrees ofcomplexity and with different goals.1,2)

This paper describes the application of our model to an injection system based on fixed lances, placed inthe furnace walls without the help of CFD simulation, which is certainly very precise, but very time consum-ing and useless for an on line control.

The approach followed here is energy-based and it allows estimating the depth reached by the O2 duringits penetration within the slag and steel. It can be used for an on line control and for the whole descriptionof the injection process. The calculation of a theoretical interface surface allows investigating the chemicalprocess and the mass transport phenomena induced by the injection itself.

This simulation can be used to investigate the required injection parameters to ensure the maximummass exchange with the bath and to foresee new injection technologies performances.

Moreover, this model of the supersonic injection allows the construction of a decarburisation model thatcan evaluate the effects and the efficiency of the injection technology.

This article describes the jet-penetration model and the decarburisation model.The validation of the model has been performed by the comparison with the experimental data of the

Tenaris-Dalmine EAF and the KT Supersonic Injection System (the chemical package of TechintTechnologies for the multi-point injection of oxygen, carbon and other fines into the EAF).

KEY WORDS: oxygen injection; steel oxidation; electric arc furnace; penetration; steel refining.

The supersonic injection of O2 has been performed by animproved shape of the nozzle featuring the KT SupersonicInjection System developed by Techint.

The validation has been performed on an industrial EAFplant equipped by a burning injection system of three fixedlances endowed with special nozzle profile. The EAF isalso using a conventional door lance.

2. The Theoretical Background

2.1. About the Laval Nozzle

Here are reported Laval nozzle classical isentropic for-mulas (from Hugoniot)5):

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

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

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

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

These relations are submitted to: short conduct withsmall curvature and divergence, perfect O2, negligiblestresses due to viscosity and negligible energy loss and heatexchange (adiabatic flow). These equations show the outletvelocity of the jet given the characteristics of the nozzle andof the O2 inlet.

Once the geometry of the nozzle is fixed, if the environ-ment temperature is assumed as constant: outlet velocity,inlet pressure and O2 flow are mutually related by:

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

2.2. About Coherence Length

Supersonic O2 jet behaviour has been studied extensive-ly: O2 exiting the nozzle meets the outside air creating aturbulent mixing and transport region. This process in-volves the jet diameter increase and the decrease of jet ve-locity while rising distance from the nozzle. The coherenceregion is the region nearby the nozzle where the axial ve-locity and the Mach number are constants and equal to thevalue at the nozzle exit. Outside this region, axial velocitybecomes subsonic (developed lateral flux). According toThiring and Newby the coherence length is6):

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

This formula takes the density of the surrounding medium(r ambient) into account with the term d*:

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

Increasing the pressure of the jet the Mach number andthe geometry of the nozzle change and these lead to thechange of the coherent jet length.

In the same way, the increase of environment temperatureand of the flow rate of the jet, and the decrease of jet tem-perature cause the increase of the coherent length.

3. The Theoretical Model

3.1. Effect of the Liquid Steel Surface Tension

It is considered a spherical cavity generated by the jet inthe steel (radius R0); the energy balance is:

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

where D is the depth of the cavity in the molten bath (Fig.3).

Under the hypothesis of a developed flow of O2 the axialvelocity is given by the equation:

1

2

2

0

ρ ρσ

O ax2

steel2v ( )x D

R g

d d*

.

ρ

ρO

ambient

2

0 5

0

L

dM

T

Tjp

a*. . 4 2 1 1 12

Q AP P

P

P

P

throatin in throat

in

throat

in

2

11

1 1ρ γ

γ

γ γ γ

( ) / /

dM

dρρ

2 v

v

dA

A

M

M

dP

P

1 2

2γ

dA

AM

d ( )2 1

v

v

ISIJ International, Vol. 44 (2004), No. 8

1343 © 2004 ISIJ

Fig. 1. Improved shape of the Laval Nozzle used in the KT in-jection system.

Fig. 2. Layout of the EAF with O2 and C Injection points(Tenaris Dalmine EAF, Italy).

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

where x is the coordinate coaxial with the axis of the lanceand starts from the outlet of the nozzle, k is constant for agiven temperature and pressure of injected O2 and a givenconditions (i.e. pressure, temperature, chemical composi-tion) of the surrounding air. This is a good approximation,but for a more precise calculation one should consider the“jet pole” of the flow as suggested by the theory of devel-oped jets.

A standard value for the surface tension is assumed:

s1.5 J m2 ..............................(12)

The radius of the cavity R0 can be approximated as:

R0(He) sin f..............................(13)

Where the angle f has a standard value of 9° for a devel-oped jet of air in air. The balance equation is a third orderalgebraic equation whose real solution for He0.4 m is:

D11.91 cm

Ignoring the influence of the surface tension, with the sameinput data the solution is:

D11.95 cm

These results demonstrate that the surface tension has anegligible influence on the depth of penetration in liquidsteel. With a supersonic coherent jet this small differenceevanishes, thus this hypothesis is more confirmed.Moreover, the computation performed by the energetic ap-proach described in Eq. (10) suggests that the sphere shapeis not the most suitable one for the description of the O2 jetimpinging the steel. The surface of a paraboloid of revolu-tion can represent a more reliable description of the inter-face shape.

3.2. Axial Velocity Calculation for an Inclined Jet overa Steel Bath

The axial velocity for an inclined nozzle placed in thewalls of the furnace can be calculated through the use ofthe concepts exposed previously. A supersonic jet can bedivided in two main regions (Fig. 4): the coherent region,featured by a potential core and a thin turbulent mixinglayer, which is followed by the zone of transition toward thecompletely developed flow that characterizes the incoherentregion.

While the axial velocity in the first region can be consid-ered as constant, in the second one follows the law given bythe Eq. (11). For these reasons the velocity of O2 along theaxis of symmetry of the jet during the passage through theair is:

........(14)

This situation verifies when the nozzle is positioned abovethe slag.

The profile of the velocity which features the severaltransverse sections which compose the O2 jet impinging thesteel can be computed on the basis of a Gaussian distribu-tion of the velocity whose maximum is centered along thesymmetry axis of the jet (Appendix A). The jet can meetthe slag in two different ways: as a coherent flow or as a de-veloped flow. The theory developed for the surrounding aircan be applied also in the case of the jet flowing throughthe slag. As a matter of fact the Thiring formula (8) can be

v v

vv

axial, air p

axial, airair

p airp

( )

( )( )

x x L

xk r

x L r kx L

0

0 0

02

vv

ax ( )xk r

x 0 0

ISIJ International, Vol. 44 (2004), No. 8

© 2004 ISIJ 1344

Fig. 3. Shape of a vertical jet and spherical cavity.

Fig. 4. Regions of a supersonic jet. The coherent region is featured by the potential flow core and a thin turbulent mixinglayer. The fully developed flow characterizes the non-coherent region.

used to estimate the length of the coherent flow in slag. Thecalculation is simple if the nozzle area is surrounded by theslag (this is the case encountered during the refining periodwith a foamy slag). In case the jet loses its coherence in theimpact on the steel bath (this hypothesis is justified by cal-culating the Eq. (8) for an oxygen jet in liquid steel) theaxial velocity can be estimated through the energy balancein a generic point of the axis inside the bath:

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

The maximum vertical depth (from the surface) reachedby the flow is calculated at the point where the axial veloci-ty goes to zero (xmax):

D(xmaxxsteel) sin a ........................(16)

Through this model the velocity profile of the O2 jet is com-pleted and plotted (Fig. 5).

3.3. The Calculation of the Interface Surface

In order to avoid the use of a CFD software or a statisticanalysis the interface surface between the O2 jet and thesteel bath has been assumed schematically not as a sphere,but as a revolution paraboloid whose axis is the same of thejet one. This representation seems to be more realistic for acoherent supersonic jet, because it can permit to simulate ajet in which the penetration depth is larger than the radiusof its transversal section. In this scheme the free surface ofthe bath will be represented as a plane surface. This surfaceis simple to be handled mathematically and it can be deter-mined as a function of the main parameters involved in theinjection practice. If the jet is not perpendicular to the bath,then the paraboloid will have a certain angle a measuredfrom the plane to its axis as shown below (Fig. 6).

In order to evaluate the equation of these surfaces, theyare represented in a three dimensional Cartesian space

(x, y, z) (Fig. 6). The general equation for a paraboloid is:

x2y22pz ...............................(17)

Where the constant p can be determined imposing twophysical conditions: the depth reached by the jet into thesteel bath (previously calculated) and the volume of theparabolic section inside the bath, equal to the fluid dis-placed by the flow (provided the impulsive force generatedby the O2 jet ):

............................(18)

Provided the depth of the jet penetration into the bath, theshape of the paraboloid is determined by imposing that thegeometrical volume of the paraboloid is equal to the oneoccupied by the impinging jet. Since the paraboloid is in-clined, the surface has been determined by an iterative pro-cedure (Appendix B). Once known the value of the parame-ter p, the surface of interface can be determined by integrat-ing the Eq. (17).

S∫G dS..................................(19)

Where G is the integration dominium. Moreover, the inter-section between paraboloid and plane of the bath can givean approximate measure of the impinging region.

3.4. Oxidation and Decarburization

Assumed that the refining period—of an EAF—is thetime starting from the complete melting of the metalliccharge until tapping, during this period the steel inside thefurnace may be considered a liquid phase.

The injection of oxygen in the bath produces two maineffects: the oxidation of the elements in solution with thesteel, and the generation of heat that contributes to conse-quent electrical energy savings. Both these phenomena haveto be described by a single model, since they are closely re-lated.

The energy developed by the oxidation is valuable by thereaction enthalpy, once the number of reacting moles areknown.

Vr

ρ π

ρO

steel

2v0

202

g1

2

1

22 0

2

ρ ρ α ρO Osteel slag steel

steel steel2 2sinv

vg( )

( )( )x

k x r

xh x

ISIJ International, Vol. 44 (2004), No. 8

1345 © 2004 ISIJ

Fig. 5. Radial velocity in a nozzle covered by slag: the jet main-tains its coherence until the impact with steel. In this sim-ulation the nozzle is positioned 400 mm above the steelbath with an angle of 40°.

Fig. 6. The paraboloid surface describing the O2-steel interface.

Then the problem is to choose the chemical species oxi-dized in the bath and the calculation of the number ofmoles of a certain chemical species in solution that are oxi-dized by oxygen during a fixed time range.

The first point can be solved through the use of oxygenpotentials.

Considering an oxidation reaction:

p[metal]O2→← f(oxide) .....................(20)

According to Wagner theory the oxygen potential for thisreaction is:

DmO2Dm*O2

pRT ln(a[metal])fRT ln(a(oxide))......(21)

Since thermodynamic states that the element with the low-est oxygen potential is the first to be oxidized, then theproblem of the choice of the chemical species to be oxi-dized is solved for a certain instant. The values of Dm* foroxidation reactions can be evaluated as:

Dm*O2AB ·T log TC ·T ...................(22)

See Table 1 for the values of A, B and C used. Phosphorushas not been considered, because the used scrap has a lowphosphorus content.

To quantify the moles reacting, kinetics of the bath mustbe taken into account.

The diffusion laws give the number of the moles for acertain species that can move to the interface surface withthe jet in D t. In terms of concentrations (z):

z i(D t)z i(t)b SD t...........................(23)

where the transport constant b i for each species can be cal-culated in turbulent conditions as7):

b ic Di1/2ut

3/2r1/2steels1/2 ......................(24)

Since the difference between the diffusion constants for theelements dissolved in the steel bath are not significant forthe description of the treated industrial case, only a singlevalue for Dj has been used for all the species to simplify thecomputation:

Di0 ·5 ·108 m2 s1 ........................(25)

The velocity ut is the average value of the steel velocity incorrespondence of the cavity generated by the O2 flow in-side the bath. The velocity of steel can be calculated apply-ing the momentum transfer between jet and bath:

....................(26)

and ut is the average value over the range covered by the jetinside the bath:

......(27)

3.5. The Carbon Injection and Its Integration in theModel

In the common melt shop practise and in the observedheats during the refining period of the melting operation,the carbon injection in the slag takes place too.

This is needed for some main reasons: first of all to avoidan excessive oxidation of the slag, to increase scrap yield,to promote the formation of foamy slag and to protect thefurnace walls from the direct exposition to the electric arc.

The role played by the carbon injection in the decarburi-sation phenomena has been studied in the same way operat-ed for the oxygen injection through the use of kinetic fac-tors x (each one different for each component of the slag):

Dz ix iz iD t ...............................(28)

and with the introduction of carbon potentials (analogous tothe oxygen ones) for the reduction reactions in the slag:

............(29)

The reactions of SiO2 and Al2O3 were neglected becausetheir reduction potential is too high in the EAF process.

The moles of each species in the bath is completed byadding the contribute of the reduction from the slag andsubtracting the oxidised moles of each species. In this way,combining in the same time interval the effect of the twophenomena of the oxygen and carbon injection, the modelcan estimate the composition of slag and steel bath (and theequilibrium between them) dynamically during the wholerefining period. The total chemical energy developed is cal-culated by adding the enthalpies of oxidation for each oxi-dation reaction in an interval, and subtracting the reductionones in the slag. The oxygen activity in the bath is constant-ly updated considering the equilibrium with the FeO in theslag8):

..................(30)

4. Implementation of the Model

If the refining time is divided in several intervals, in eachof them a thermodynamic description of the interface foreach oxidation reaction can be performed through the useof the potential of oxygen.

Then, since the O2 flow is known by imposing the tech-nologic parameters of the process, the number of moles ofoxidized elements can be evaluated starting from the onewith the lowest potential and proceeding towards the higherpotential ones. Only the free moles of oxygen left by eachoxidation can react with the elements with higher poten-tials.

As a matter of fact from Eq. (24) the amount of eachspecies coming to the surface can be estimated. If there isoxygen left from the oxidation of the first element featured

[ ]

( ).

a

a Tn

O

FeO

6 320

2 765

1

n

q

nq n( ) [Me O C Me] CO ↑→←

ux x

x dxx

x

τρρ

O

steel max steelax

2

max

steel1 ∫

v ( )

v vsteelO

steelO

2

2( ) ( )x x

ρρ

ISIJ International, Vol. 44 (2004), No. 8

© 2004 ISIJ 1346

Table 1. coefficients for DG° calculation for the most impor-tant oxidation reactions (from Kubaschewski).

by the lowest oxygen potential it becomes available for theoxidation of the next lower element in the oxygen potentialscales and so on.

On every time interval the activities of the elements atthe surface can be evaluated considering the moles con-sumed in the former time step by the oxidation reactionsand the diffusion laws. The temperature is evaluated atevery time step by adding the heat developed by each reac-tion in terms of temperature difference on steel. This is trueunder the adiabatic hypothesis for the reaction surface:

DEj (J)Hj (J mol1) [m(t)m(tD t)] (mol) .....(31)

The enthalpy can be evaluated by:

DHjH°j cp, j[T298.15] ....................(32)

where H°j is the value of enthalpy at standard conditions forthe j-th reaction and cp is the specific heat at constant pres-sure. They can be easily evaluated according to Barin–Knacke9) data.

The sequence of this computation is developed for eachtime step taken into account to follow the thermo-chemicalevolution of the considered system.

The decarburisation process can be limited by the flow ofoxygen and by the carbon diffusion. When the oxygen con-sumes all the carbon available on the reaction surface andcan oxidize also elements with higher potential, decarburi-sation is limited by the carbon diffusion to the interface. Onthe other hand, when the carbon consumes all the oxygen atthe interface, the decarburization is controlled by the oxy-gen flow.

The application of the model in the condition of the ob-served industrial process shows that this transition takesplace in the range of 0.2–0.4% (weight) of carbon in thebath. This matches the data presented in the bibliographyabout this topic.10)

The same can be observed for the other elements consid-ered in the model.

5. Experimental Procedure and Results

The model exposed in this article has been implementedin dedicated software. In order to use it, it must be validated(the kinetic constants must be tuned) on a real EAF. Thecalibration has been done on the TenarisDalmine furnaceand the original software has been modified in order to sim-ulate the different types of lances present in the plant. Thesoftware has been calibrated over 100 heats and it has beenchecked on other hundreds heats.

Temperature, oxygen activity in the steel and the compo-sition of slag and steel have been monitored through a se-ries of sampling trials performed during the refining periodof each heat.

Through these experiments the estimation of the kineticconstants has been possible and the results are shown inTable 2.

Once those constants have been obtained, the model hasbeen tested over 100 heats (Figs. 8 and 9).

The experimental curve has been obtained through thesampling of liquid steel in different steps of the refining pe-riod and their analysis through spectroscopic analysis orSEM spectrometer.

ISIJ International, Vol. 44 (2004), No. 8

1347 © 2004 ISIJ

Fig. 7. The scheme of the implemented decarburisation modelscheme.

Table 2. Kinetic constants obtained by the software calibra-tion.

Fig. 8. Values predicted by the software vs. values measured forcarbon.

Fig. 9. Values predicted by the software vs. values measured forFeO in slag.

Since the model describes the trend of the concentrationsin the bath and in the slag also the output curves of theother elements can be compared to the experimental ones,here are shown only the most important ones, and they offera very reliable results.

6. Conclusions

The decarburisation model described in this article canevaluate dynamic wise the composition of the bath and slagduring the refining period of the melting operation withinthe EAF. This calculation mainly depends on the character-istics and the technology of the injection system.

For this reason it must be calibrated (for what concernsonly the chemical part since the penetration model is not in-fluenced by the kinetics of the chemical species in the bath)in order to define the kinetic constants of the process.

This means that the physical part about the impact of theoxygen jet inside the steel bath can be used for any plantwithout being modified, while for its chemical effects onemust know the values of the kinetic constants. For a rapidcalculation, with less precision, values obtained for analo-gous cases can be applied.

Once calibrated, the model gives the dynamical evolutionof the composition of the bath during the refining period forevery element implemented with this method (except phos-phorus for the reasons mentioned above). Through thestudy of the reaction equilibria also the slag compositionevolution can be evaluated.

This is very important for the refractory life and for thecreation of a plant procedure that can prevent its wear.

The calibrated model can estimate the effects of anychange of the operative parameters: the positioning of theburning-injection lances in the furnace, the pressure andtemperature of the O2 flow, a different nozzle design.

At present time it has been used and validated on thebasis of the decarburisation improvements brought by thenew injection technology equipped by an improved nozzledesign developed by Techint Technologies and named KTInjection System. This system is often applied in a multi-point configuration which can assure a better homogenisa-tions of the steel during the decarburisation process.

Nomenclatureg : Specific heat ratio of a gass : Surface tension (J m2)a : Angle of the jet (deg)b : Transport coefficient (m/s)c : Proportional kinetic constant for bathz i: Concentration of moles of the specie i

(mol m3)( · ): Concentration of an oxide in the slag (mol m3)

Dm*O2: Oxygen standard potential (kJ/mol)

DmO2: Oxygen potential (kJ/mol)

rambient: Density of the medium where the jet flows(kg/m3)

rO2: Jet density (kg/m3)

x i: Proportional kinetic constant for oxide i slag(s1)

r in: Density of the injected gas at the entry transver-sal section of the nozzle (kg/m3)

r steel: Density of the liquid steel (kg/m3)

D t: Time interval (s)[ · ]: Concentration of an element in the steel

(mol/m3)A: Transversal section of the nozzle (m2)

Athroat: The smallest transversal section of the nozzle(m2)

a: ActivityCp, steel: Calorific power of the steel bath (J kg1 K1)

D: Depth of the cavity in the steel (m)Di: Diffusion constant in liquid steel for a given

species i (m2/s)E : Energy (J)g: Gravity acceleration (m/s2)

hsteel: Vertical depth in the bath calculated at a certainabscissa (m)

H: Enthalpy (J mol1)He: Height of the exit transversal section of the noz-

zle above the plane level of the bath (m)DH°: Enthalpy at standard conditions (J mol1)

k : Constant of developed flow (for a given gas anda given medium)

ksteel (air): Constant for a developed flow of oxygen insteel (air)

Lp: Coherence length (m)Mj: Mach number during the iso-entropic expansion

of the injected gas m: Number of moles (mol)

mFe: Moles of Fe in the bath (mol)P: Pressure (Pa)

Pin: Pressure at the entry section of the nozzle (Pa)Pthroat: Pressure at the smallest section of the nozzle

(Pa)Q: Gas flow rate (m3/s)R: Perfect gas universal constant (J mol1K1)

R0: Radius of the cavity formed in the liquid steel(supposed spherical) (m)

r: Radius of the cylinder-Region 2 (Appendix B)(m)

r0: Radius of the outlet section of the nozzle (m)S: Interface between the gas flow and the steel

bath (m2)T: Absolute temperature (K)

Tj: Absolute temperature of the injected gas duringthe iso-entropic expansion within the nozzle

(K)ut: Average velocity of steel in correspondence of

the jet surface (m/s)v: Velocity (m/s)V: Volume of the cavity formed by the gas flow in

the steel (m3)V0: Velocity of the flow at nozzle outlet (m/s)vax: Axial velocity (component of the velocity along

the axis of symmetry of the jet) (m/s)vaxial,air: Velocity of the injected gas along the axis of

symmetry of the jet during the passage throughthe surrounding air (m/s)

vslag: Axial velocity of the gas after the passagethrough the slag layer (m/s)

W : Weight of a mole of Fe (kg/mol)x : Axial coordinate taken from the nozzle outlet

(m)

ISIJ International, Vol. 44 (2004), No. 8

© 2004 ISIJ 1348

x: xxsteelksteel r0 sin(a) (m)xmax: Abscissa where the axial velocity is zero (m)xsteel: Abscissa of the intersection of the jet axis with

the plane of the bath surface (m)X : Radial distance transversal to the jet axis (m)j : Gaussian distribution constant (1/m)

V1,V2,V3: Volumes of the regions in which the cavityformed by the gas jet into the steel is subdivid-ed. (m3)

W2: Integration dominium of V2

W3: Integration dominium of V3

p, q, n, f : Stoichiometric coefficients

REFERENCES

1) C. Mapelli and G. Sandonà: 7th Europ. Electric Steelmaking Conf.,AIM, Milano, (2002).

2) U. della Foglia and C. Mapelli: 7th Europ. Electric SteelmakingConf., AIM, Milano, (2002).

3) S. R. Story and R. J. Fruehan: CISR Report, (1997).4) L. Gu and G. A. Irons: Electric Furnace Conf. Proc., ISS,

Warrendale, PA, (1999), 269.5) B. Allemand, P. Bruchet, C. Champinot, S. Melen and F. Porzucek:

2000 Process Technology Conf. Proc., ISS, Warrendale, PA, (2000),927.

6) M. W. Thiring and M. P. Newby: Proc. 4th. Int. Symp. Combustion,Williams & Wilkins, Baltimore, (1953), 789.

7) F. Oeters: Stahl Eisen, (1989), 155.8) E. T. Turkdogan: Physical Chemistry of High Temperature

Technology, Academic Press, New York, (1980).9) I. Barin and O. Knacke: Thermochemical properties of inorganic

substances, Springer-Verlag Berlin, Heidelberg, New York, VerlagStahleisen mbH., Düsseldorf, (1973).

10) H. G. Holtmann and E. B. Pretorius: AISE Steel Technology, (2003).11) J. Szekely and N. J. Themelis: Rate Phenomena in Process

Metallurgy, Wiley-Interscience, John Wiley & Sons, Inc., New York,(1971), 710.

Appendix A. Transversal Velocity Approximation

Once the velocity axial profile is known, under the hy-potesis of a Gaussian trend11) along the transversal direction(X is the radial coordinate which takes as zero the axis ofthe jet) of the jet, a very simple 3d-model of the trend of thevelocities is given by:

v(x,X)vax(x) · exp(jX)....................(A-1)

where the constant j shell be determined case by case infunction of the medium and the condition of the atmo-sphere inside the furnace.

Appendix B. Geometrical Volume Calculation

Under the hypothesis of a paraboloid shape, the cavitycan be divided into three regions (Fig. A-1).

The region 1 volume is:

.............................(B-1)

where the meaning of r is clearly expressed in the drawing(Fig. A-1).

For region 2:

.........................................(B-2)where W2, the integration dominium, is the projection ofthe outlined surface in Fig. A-1 on the x–y plane.

For region 3:

V3∫W3rdV .............................(B-3)

where W3, the integration dominium, is the projection ofthe region 3 (Fig. A-1) on the x– y plane. This value can beobtained through numerical calculations.

The sum V1V2V3 gives the total geometrical volumeof the cavity and it is a function of p and of r. Their valuecan be obtained imposing the depth reached by the jet pre-viously calculated (16) and the physical volume of movedliquid (18).

Once known the constant p, the dominium G (19) can beeasily determined intersecting the paraboloid (17) with thesteel bath plane:

....................(B-4)˜)

˜ )zD

x sin(

cot(α

α

π αr3 cot

V rdV dzd drr

2002

cot( cos( ) cot(α ρ α α

π

π

ρ ρ θ) )

∫∫∫∫Ω

Vp

r1

4

2

π

ISIJ International, Vol. 44 (2004), No. 8

1349 © 2004 ISIJ

Fig. A-1. Subdivision of the jet cavity into regions.