An Improved Model of Cored Wire Injection in Steel Melts

1. Introduction

The cored wire injection method has been developed forthe addition of low-density alloys like Ca-alloys into thesteel bath. The Ca-recovery in this process is relativelyhigher than in the conventional and some other improvedaddition methods like shooting bullets, ‘pouring over’ etc.Factors such as minimum interaction with the slag, reducedliquid steel movement, possibility of suppressing the pre-mature evaporation and, thereby, improving utilisation, pos-sibility of preheating the filling material before release andsimple and inexpensive operations have given an edge tothis method over the others.

To realise these benefits, however, it is imperative thatthe filling material is released at such a depth in the ladlethat the resultant residence time is the highest.1) This paperpresents an understanding of the dissolution mechanism de-veloped through a numerical approach and thereby suggestsmodification in the operating and the design parameters toincrease the depth of penetration before the release of thepowder. The formation of a thermal contact resistance andthe freezing of the slag on the wire surface have been con-sidered for the first time in a study of cored wire dissolu-tion. The results of the model have been tested with theavailable study of zirconium, titanium and niobium dissolu-tion2–4) and also with the actual plant data.

The factors influencing melting and dissolution of ferro-alloys in steel have been studied by various workers.5–8) Insimilar studies on cored wire it has been established that,when a cored wire at room temperature is injected into theliquid steel solidifies on itself a layer of the steel from themelt.9–11) This layer, referred to as a shell, grows to a maxi-mum thickness and thereafter melts back leaving the origi-

nal wire surface uncovered. The metallic sheath or casingthen starts melting and finally releases the powder into themelt. A slag shell forms first and over that a steel shell so-lidifies, since the wire passes through the slag layer beforeentering the steel bath.

Figure 1 represents a typical cross-section of an injectedwire perpendicular to the axis of the cored wire. The for-mation of the slag and steel shells and their subsequent meltback, the melting of steel case and the point of release ofthe filling material have been studied through the followingmathematical formulation.

ISIJ International, Vol. 44 (2004), No. 7, pp. 1157–1166

1157 © 2004 ISIJ

An Improved Model of Cored Wire Injection in Steel Melts

Sarbendu SANYAL, Sanjay CHANDRA, Suresh KUMAR1) and G. G. ROY2)

R & D, Tata Steel, Jamshedpur – 831 007, India. E-mail: [email protected]) L.D. Shop 1, Tata Steel, Jamshedpur – 831 001, India. 2) Indian Institute of Technology, Kharagpur, 721 302, India.

(Received on October 31, 2003; accepted in final form on March 19, 2004 )

Mathematical models for tracking the melting of cored wire during its injection into the steel bath havebeen developed in the past though important aspects of the formulations have not been discussed in suffi-cient detail. As a result, it is difficult to use the results of these models to derive benefits for a specific steelmelting shop.

A general purpose mathematical model has been developed at R & D, Tata Steel, using the finite differ-ence approach with a fully implicit scheme to simulate the process of cored wire injection taking into ac-count the different operating practices encountered in the steel shop. Numerical simulation of this kind ofproblem, involving moving boundary, typically suffers from the limitation that the progressive solidificationof frozen layers that takes place is not made part of the thermal balance till it attains the size of a full nodeand thus the heat gained or lost by this “partial node” is not accounted for till such time. An alternative nu-merical formulation has been developed to rectify this.

Owing to the difficulty in making a direct validation, this model has been verified through a novel ap-proach. This work suggests that the use of different wire dimensions (13–18 mm diameter and 0.4–0.6 mmcasing), depending on the steel grades to be processed, is necessary in order to extract the maximum ben-efit.

KEY WORDS: mathematical model; steelmaking; calcium treatment; cored wire; injection metallurgy; de-oxidation; alloy addition.

Fig. 1. Schematic representation of the cross section of a cylin-drical cored wire addition.

2. Development of Mathematical Model

2.1. Formulation of ModelThe temperature distribution inside the cylinder shaped

cored wire for a fixed observer can be described by thesteady state heat conduction equation expressed in cylindri-cal coordinates (Fig. 2):

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

The temperature around the circumference of the coredwire is uniform and thus ∂T /∂q�0 and Eq. (1) can berewritten as

....(2)

Since the wire moves at a high speed, the heat transfer inthe z direction by bulk motion is much higher than the heatbrought in by conduction thereby permitting (k ∂T/∂z) termto be dropped. The Peclet number (Pe) has been calculatedfor this condition and found to be 2 000, which justifies theabove hypothesis. Thus the Eq. (2), takes the followingform.

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

The above formulation has been derived by the heat balancemethod for an observer fixed in space as shown in Fig. 2with the material passing with a velocity ‘v’. If the observerwere to move with the same speed of material in the z di-rection, he would notice a change in the temperature of thewire with time. Thus, if z is the distance travelled in time t

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

and the right hand side of Eq. (3) becomes

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

Thus, for a moving observer, the Eq. (3) takes the form ofan unsteady state heat transfer.

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

q̇ takes the values 0, �ve or �ve depending on whetherheat is generated or absorbed during internal freezing ormelting.

The expressions for the transient heat conduction in thepowder, casing and the shells can be written using Eq. (6)as follows:1. Powder:

0�r �rp, 0�t�tTOT

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

2. Casing:

rp�r�rc, 0�t�tTOT

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

The casing is mild steel and the respective property val-ues of mild steel has been considered for the calculation in-volving casing.3. Slag Shell:

rc�r�rsl, 0�t�tTOT

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

4. Steel Shell:

rsl�r�rs, 0�t�tTOT

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

The positions of rp, rc, rsl and rs have been indicated in Fig.1. The values of k, r and cp of the relevant liquid phasehave been considered in the Eqs. (8) and (10) when thepowder or the slag shell is molten. The tTOT is the total timeup to the point of release of the powder and consists of dif-ferent time intervals as

tTOT�tslag shell freezing�tsteel shell freezing�tsteel shell melting

�tslag shell melting�tcasing heating�tcasing melting ........(11)

In case of the absence of slag shell the Eq. (11) simplifiesto

tTOT�tsteel shell freezing�tsteel shell melting

�tcasting heating�tcasting melting..................(12)

where,

tslag shell freezing�tsteel shell freezing�tsteel shell melting

�tslag shell melting�tTSP.....................(13a)

and

tsteel shell freezing�tsteel shell melting�tSSP ............(13b)

1

r rk r

T

rc

T

t

∂∂

∂∂

ρ∂∂s s ps

�

1

r rk r

T

rc

T

t

∂∂

∂∂

ρ∂∂sl sl psl

�

1

r rk r

T

rc

T

t

∂∂

∂∂

ρ∂∂c c pc

�

1

r rk r

T

rc

T

t

∂∂

∂∂

ρ∂∂p p pp

�

1

r rkr

T

rq c

T

t

∂∂

∂∂

ρ∂∂

� �˙ p

ρ∂∂

ρ∂∂

∂∂

ρ∂∂

cT

zc

z

t

T

zc

T

tp p pv � �

z tz

t� �v v⋅ and

∂∂

1

r rkr

T

rq c

T

z

∂∂

∂∂

ρ∂∂

� �˙ pv

10

r rkr

T

r zk

T

zq c

∂∂

∂∂

∂∂

∂∂

ρ∂∂

� � � �˙ pv

T

z

� � �q̇ cT

zρ

∂∂pv 0

1 12r r

krT

r rk

T

zk

T

z

∂∂

∂∂

∂∂θ

∂∂θ

∂∂

∂∂

� �

ISIJ International, Vol. 44 (2004), No. 7

© 2004 ISIJ 1158

Fig. 2. Control Volume on which heat balance is done to get differential equation.

Equations (7)–(10) have been used for the present studyand solved with the relevant initial and boundary conditionsto determine 1. the temperature distribution inside the cored wire and

the solidified shell, 2. the total time taken for the melting of the casing and

the shell and for the release of the powder,3. the temperature of the powder at the time of release.

In indicating the initial conditions, boundary conditions(B.C.) and boundary equations, the listing has been done ina systematic way starting at the cored wire centre and startof computation and proceeding radially outwards towardsthe liquid steel bath.

2.1.1. Initial ConditionsPrior to immersion, the temperature of the cored wire can

be taken to be uniform at T0. Mathematically,

Initial Condition-1 T�T0 when t�0 and 0�r�rc

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

Similarly, the temperature of the slag layer and the steelbath is assumed to be constant during the period of compu-tation. Since the injection times are of the order of minutesand the weight of the total additions is less than 0.1% ofthe steel bath, this assumption is reasonable. Therefore;

Initial Condition-2 T�TSLAG when t�0 and r�rsl

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

Initial Condition-3 T�TBATH when t�0 and r�rs

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

2.1.2. Boundary ConditionsThough the problem has been expressed in terms of the

following boundary conditions (B.C.), all do not apply si-multaneously. Thus,a) Boundary conditions at the casing and the frozen layer

interface change once the slag shell starts melting in-ternally.

b) Boundary conditions at the powder region changesduring the internal melting of powder.

c) Boundary conditions at the surface changes when thecasing starts receiving heat directly from the melt.

2.1.2.1. B.C. at CentreB.C.1 represents the symmetric condition at the centre of

the cored wire:

B.C.1 0�t�tTOT, r�0 ..........................(16)

2.1.2.2. B.C. at Powder/Casing InterfaceAt the powder casing interface, the heat leaving the cas-

ing should be equal to the heat entering the powder, whichis depicted by B.C.2:

B.C.2 0�t�tTOT, r�rp .....(17)

2.1.2.3. B.C. at Casing/Shell InterfaceDuring the injection the wire first passes through the slag

layer present at the top of the steel bath in the ladle andfreezes a slag layer as the first shell on its surface. The im-perfect contact between the casing and this shell results in athermal contact resistance (RT) between the two. The for-mation of this thermal resistance between the casing andthe first shell and its effect on the heat flux to the wire havebeen considered for developing the boundary conditions atthe casing/shell interface as suggested by Argyropoulos2)

in his work on melting of static cylinders in metallic melts.This concept has hitherto not been applied to the study of

cored wire dissolution. This contact resistance to the heatflow, of course, ceases to exist under two conditions. Firstly,if the slag shell interface starts melting internally (i.e. afterthe time tm(sl), provided the tm(sl) is less than the time forcomplete melting of the steel shell). This has been repre-sented by B.C.3a., where the interface temperatures at theslag and casing are T*sl and T*c respectively (Fig.1).

B.C.3a 0�t�tm(sl), r�rc

......(18)

Secondly, the contact resistance also ceases to exist aftercomplete melting of both the shells (tTSP) as depicted inB.C.3b.

B.C.3b 0�t�tTSP�tm(sl), r�rc

......(19)

Thus, as soon as the slag shell becomes molten the thermalcontact resistance becomes zero (B.C.3c).

B.C.3c tm(sl)�t�tTSP, r�rc RT�0 and T*sl�T*c........................................(20)

2.1.2.4. B.C. at Slag Shell/Steel Shell InterfaceThe wire enters the steel bath after passing through the

slag layer and the steel shell forms over the slag shell. Theamount of heat leaving the steel shell should enter the slagshell as shown in B.C.4 below. The freezing of the last layerof the slag shell and the first layer of the steel shell occuralmost simultaneously from the respective liquid phasesand so the interface between these two layers has been as-sumed to be in perfect thermal contact and no contact resis-tance has been considered.

B.C.4 tsl�t�tSSP, r�rsl ....(21)

2.1.2.5. B.C. at Wire SurfaceThe temperature at the surface of the wire is at the melt-

ing temperature (TMP) of the respective liquid. B.C.5a repre-sents the temperature at the surface of the slag shell duringthe travel through the slag layer.

B.C.5a 0�t�tsl, r�rsl T�TMP(sl) ........................(22)

B.C.5b represents the temperature at the surface of the steelshell during its freezing and melting.

B.C.5b tsl�t�tSSP, r�rs T�TMP(bath)...................(23)

After melting of steel shell (tSSP), the slag shell starts melt-ing and so the temperature at the surface of the slag shell isagain at the melting point of slag as shown in B.C.5c.

B.C.5c tSSP�t�tTSP, r�rsl T�TMP(sl) ..................(24)

When the shell melts completely (time�tTSP) and the casingstarts melting, the temperature at the surface of the casingthen becomes the melting point of the casing:

B.C.5d tTSP�t�tTOT, r�rc T�TMP(c)...................(25)

2.1.2.6. B.C. at Solidification FrontBoundary conditions (6a–6d) represent the heat balance

for the moving solidification (or melting) front at the

kT

rk

T

rqsl s

∂∂

∂∂

� � �

kT

rk

T

rq q

T T

Rc slsl c

T

and∂∂

∂∂

� � � ��* *

kT

rk

T

rq q

T T

Rc slsl c

T

and∂∂

∂∂

� � � ��* *

kT

rk

T

rqp c

∂∂

∂∂

� � �

∂∂T

r�0


1159 © 2004 ISIJ

shell/liquid steel or casing/liquid steel interface where l isthe latent heat of fusion of the respective melt. The B.C.6ais the heat balance for solidification of slag shell during thetravel of the wire through the slag layer.

B.C.6a 0�t�tsl, r�rsl

...(26)

The B.C.6b represents the heat balance for the solidificationand melting of steel shell when the wire travels through thesteel bath.

B.C.6b tsl�t�tSSP, r�rs

...(27)

When the steel shell melts completely, the melting of slagshell is governed by the B.C.6c.

B.C.6c tSSP�t�tTSP, r�rsl

...(28)

The melting of casing is represented by B.C.6d. The tMS(c) isthe time when the casing starts melting.

B.C.6d tMS(c)�t�tTOT, r�rc

...(29)

2.1.2.7. B.C. at Bulk LiquidThe B.C.7a and 7b demonstrate that the temperature of

the slag layer and steel bath respectively far from the coredwire can be regarded as being constant.

B.C.7a 0�t�tsl, r→∞ T�TSLAG ......................(30a)

B.C.7b 0�t�tTOT, r→∞ T�TBATH...................(30b)

2.1.2.8. Additional B.Cs. for Unique SituationsThere may be a situation when the shell is completely

molten, but the surface of the casing is yet to reach its melt-ing point. The casing would then start receiving heat direct-ly from the bath before commencement of its melting. TheB.C.8 represents the heat balance for such condition. ThetMS(c) is the time when the casing starts melting.

B.C.8 tTSP�t�tMS(c) r�rc

..............(31)

If the slag shell reaches its melting point before the com-plete melting of steel shell, the temperature of the node rep-resenting the outer layer of the slag shell is held constant atits melting temperature until it gains heat equal to the latentheat of fusion to initiate melting. B.C.9 represents the heatbalance for such a situation which is relevant for the periodwhen the slag shell is melting (tm(sl)).

B.C.9 t�tm(sl)�tSSP rc�r�rsl and Tsl�TMP(sl),

..........(32)

Similarly, the internal melting of the powder is demonstrat-ed by the B.C.10 which is relevant for the period when thepowder is melting (tm(p)).

B.C.10 t�tm(p)�tTOT r�rp and Tp�TMP(p),

...........(33)

2.1.2.9. Evaluation of the Heat Transfer CoefficientThe heat transfer coefficient, i.e., h, from the bath to the

shell/casing surface has been deduced from the followingdimensionless correlations.9)

NuL�0.4 Re0.5 Pr0.453 for laminar flow i.e. Re�104

........................................(34)

NuL�0.0296 Re0.8 Pr0.33 for laminar flow i.e. Re�105

........................................(35)

NuL for a transition flow has been obtained by proportionat-ing the values at Re of 104 and 105. The x and v in the Nuand Re numbers, defined as follows,

NuL�hx / k ...............................(36)

Re�xvr / m ...............................(37)

refer to the bath height and the wire injection speed respec-tively. The relevant physico-chemical properties of liquidslag12,13) and liquid steel2) are given in Table 1.

2.2. Solution MethodologyAs the set of partial differential Eqs. (7)–(10) is too com-

plex for analytical solution a numerical approach has beentaken to predict the melting time and the distance travelledas a function of time. These equations have been convertedinto finite difference equations using the fully implicitscheme. The calculation domain in radial direction(powder�casing�shell) has been discretised into nodes (0,1,..., M�2, M�1, M as shown in Fig. 3), each node beingthe centre of a control volume. The change in the outershell thickness, dr, of the moving solidification front, hasbeen calculated as per the Eqs. (26)–(29). A positive valueof dr indicates an increase in the thickness of the shell.When the shell thickness just exceeds the size of a node, anew node is added and the temperature of this new nodecalculated by a linear interpolation of the adjoining nodaltemperatures. A negative value of dr indicates melting ofthe shell (or casing after complete melting of shell as perEq. (29)) and a node is reduced when an appropriateamount of the shell melts. The reduction of a surface nodemakes the node underneath, the surface node where the

kT

rk

T

r

r

tp

in

p

out

p pp∂

∂∂∂

ρ λ∂∂

� �

kT

rk

T

r

r

tsl

in

sl

out

sl slsl∂

∂∂∂

ρ λ∂∂

� �

kT

rh T Tc

Bath,Casing

BATH c

∂∂

� �( )

kT

r

r

th T Tc

Bath,Casing

c c BATH MP(c)

∂∂

ρ λ∂∂

� � �( )

kT

r

r

th T Tsl

Bath,Shell

sl sl BATH MP(sl)

∂∂

ρ λ∂∂

� � �( )

kT

r

r

th T Ts

Bath,Shell

s s BATH MP(bath)

∂∂

ρ λ∂∂

� � �( )

kT

r

r

th T Tsl

Slag,Shell

sl sl SLAG MP(sl)

∂∂

ρ λ∂∂

� � �( )


© 2004 ISIJ 1160

Table 1. Physico-chemical properties.

temperature is known. Thus, no separate effort is needed tocompute the new surface temperature. During the calcula-tion of the surface node temperatures, the actual shell vol-ume at that point of time has been considered instead ofconsidering only the half-node volume as discussed inAppendix-A. This has enhanced the accuracy of the modeland in the opinion of the authors represents an improve-ment over the existing numerical modelling techniques forcored wire modelling.

A computer program for the solution of the above equa-tions has been developed in FORTRAN. To check the accu-racy and consistency of the basic numerical solutions, themodel has been adjusted to run under the limiting conditionof an infinitely long cylinder subjected to a sudden changeof surface temperature for which analytical solutions areavailable. The results were within the acceptable limits ofdeviations (0.5%).

3. Model Validation

3.1. Validation with Published WorkThough few published work on cored wire injection are

available,9–11) their simplified assumptions and limited vali-dation have forced the authors to compare the model resultswith the published work on the dissolution of static metalliccylinders in quiescent/inductively stirred steel melt.2–4) Thevalidation results with the work on dissolution of pureTitanium, Niobium and Tantalum cylinders in a quiescentsteel melt have been presented in Fig. 4 to Fig. 6. The TataSteels’ R & D model has been modified to suit the experi-mental conditions of the published works.

Titanium—Figure 4 shows the variation of the centretemperature during dissolution of a 2.54 cm diameter Ticylinder in molten steel at 1 590°C. Ti shows a change ofphase from a-Ti to b-Ti at 882°C. The critical temperaturefor initiation of the exothermic dissolution reaction at theoriginal cylinder surface/shell interface is 1 090°C.2,4) Theshell/original cylinder interface achieves this temperature atthe eighth second after immersion in the bath.

Niobium—In contrast to Ti, Nb does not undergo phase

change. However, it exhibits an exothermic dissolution inliquid steel,2,3) which starts at 1 370°C. Figure 5 shows thegrowth and decay of radius of a 3.81 cm diameter Nb cylin-der when immersed in liquid steel at 1 600°C.

Tantalum—The predictions for the variation of radiusand the centre temperature of a 2.54 cm diameter Ta cylin-der during its dissolution in molten steel at 1 600°C is pre-sented in Fig. 6 along with the published data.2)

3.2. Validation with Plant ResultsAn innovative approach for indirect validation with the

plant results has been attempted and discussed here. Themodel predicts the zone or level of the steel bath where thepowder is released. The direct verification of this parameterin a steel ladle is nearly impossible. No simple method ex-ists for collecting sample from different depth in a ladle ofmolten steel.

To assess the variation in Ca and Si content samples werecollected from the tundish at definite intervals during cast-ing. The steel in the tundish was assumed to represent aparticular zone of the ladle depending on the flow rate andthe time of sample collection. The samples were subse-quently analysed for Ca and Si and the results havematched well with the predictions. In the opinion of the au-thors, these evidences are supportive but cannot be taken asconclusive.

4. Routes for Cored Wire Melting

With reasonable validation obtained, the model was runfor several conditions to develop an understanding of themelting behaviour of cored wire. The results were analysedand the different routes of melting have been grouped intofour broad categories as discussed here. Mainly the pres-ence or absence of slag at the ladle top and the bathTemperature Above the Bath Liquidus (TABL), commonlyknown as ‘superheat’, dictate these melting behaviours.

Whenever the first liquid (slag/steel) freezes on the cas-ing, as discussed before, an interfacial resistance is devel-


1161 © 2004 ISIJ

Fig. 4. Centre temperature of a 2.54 cm diameter Ti cylinder dur-ing dissolution in liquid steel at 1 590°C.

Fig. 5. Radius of a 3.81 cm diameter Nb cylinder during dissolu-tion in liquid steel at 1 600°C.

Fig. 6. Centre temperature and radius of a 2.54 cm diameter Tacylinder during dissolution in liquid steel at 1 600°C.

Fig. 3. Schematic of mesh distribution.

oped between the casing and the frozen layer (shell) due toimproper thermal contact between these two. The formationof this resistance at the surface of an immersed metal andits magnitude has been studied by many workers.2,14–16) Theexact value of the resistance depends on factors like theratio of thermal expansion coefficient and the ratio of ther-mal diffusivity of the immersed metal and the shellmetal,14,15) the thickness of air and/or the oxide film at theinterface.16) The resistances for different metal combina-tions have been derived experimentally,2,14–16) which varyfrom 1.9 to 9.1 cm2 s °C/J and particularly for steel/othermetal combination2) the range is 1.9 to 2.1 cm2 s °C/J. As nowork is available for determining the resistance in case ofsteel immersed in liquid steel, the authors has assumed thevalue as 1.9 cm2 s °C/J for the present study which seems tobe a reasonable from the above discussion. However, exper-iments have been planned at R & D, Tata Steel to verify thisassumption. The resistance subsequently vanishes in eitherof the two conditions: complete melting of the shell or in-ternal melting of either casing or the first shell.

The shell growth occurs due to the large gradient of tem-perature between the surface and the interior. The first shellgrows to a certain maximum thickness until the heat sup-plied to the surface through convection becomes equal tothe heat conducted inside.

Route 1—No Slag Cover at the Ladle Top along with HighTABL (�60°C)

In this case, the shell thickness (stage C1 of Fig. 7(a)) isrelatively low owing to the larger amount of heat suppliedthrough convection from the melt of high TABL. On com-plete melting of the shell the melting of the casing does notstart immediately. This is because the interfacial resistanceprevents the casing from being heated within such a shortperiod of shell melt-back. The casing then starts receivingheat directly by convection from the melt (stage E1).Subsequently, it melts and releases a solid but hot powderto the bath (stage G1).

Route 2—No Slag Cover at the Ladle Top along with LowTABL (�60°C)

The convective heat input to the wire is less in this casedue to the low TABL. So, this route sees a thicker shell andlonger melt-back time (stage C2 and D2 of Fig. 7(a)). Thislong duration enables the powder to start melting internallymuch before the casing melts. On the complete melting ofthe casing partially or completely molten powder is re-leased in the bath.

Route 3—Presence of Slag at the Ladle Top along withHigh TABL (�60°C)

The presence of the slag, as the first shell, changes thefreezing and melting behaviour. This route is similar to‘Route 1’ with the exception of a larger shell thickness anda longer melting period (Fig. 7(b)).

Route 4—Presence of Slag at the Ladle Top along withLow TABL (�60°C)

In this case, the steel shell period (i.e., the duration of existence of a steel shell) is significantly higher due to thelower heat input from the bath. The slag shell starts meltinginternally during the initial steel shell period (stage F4 ofFig. 7(b)) and the interfacial resistance between the casingand the slag shell disappears. This leads to an increasedheat conduction from the steel shell to the casing throughthe molten slag. The steel shell then resumes its growth andreaches a maximum thickness which is less than the maxi-mum thickness achieved during the initial growth. The sub-

sequent melting of steel shell exposes the casing to the meltand melting of the casing starts. This period (stage G4) issubstantially lower than other routes (F1, F2, G3). Thepowder also starts melting by this time. Finally, a partiallyor completely molten powder is released into the bath.

Figure 8 presents a typical set of graphs showing thevariation of casing and shell thickness during the abovementioned freezing and melting periods for both the “no-slag” and the “with-slag” conditions. As already mentioned,the shell initially grows with time before melting back. Thethickness remains constant during the direct heating of cas-ing in case of the curve of “no-slag” condition. In the caseof curve of the “with-slag” condition, the secondary growthof steel shell starts and reaches a second peak. It can beclearly seen that after the complete melting of the steelshell, there is no holding for direct heating of the casing inthe later cases and the melting of casing is also relatively


© 2004 ISIJ 1162

Fig. 7a. Melting scheme of cored wire (no slag shell).

Fig. 7b. Melting scheme of cored wire (with slag shell).

faster.

5. Impact of Wire Parameters on Melting Behaviour

The four parameters of the wire which effect the distancetravelled are discussed below. The distance travelled is thedistance travelled by the wire before the powder is set freeinto the melt and is an indicator of the point of release ofthe powder in the ladle.

5.1. Wire Speed The effect of speed on the melting time and thus on the

distance travelled has been shown in Figs. 9 and 10 for twodifferent thermal conductivities of the filling materials. Asexpected, these two figures indicate the decrease of themelting time with the increase in speed. However, the de-crease in the melting time on account of this factor is notnecessarily accompanied by a decrease in the distance trav-elled. On the contrary, as evident from the Figs. 9 and 10,the distance travelled, initially increases with speed (seg-ment a–b) and reaches a maximum at a certain speed(speed b) and then decreases (segment b–c). The position ofthis point ‘b’ changes with the TABL.

The change in the distance travelled by the wire with in-crease in the speed of injection is dependent on the relativedominance of the two competing factors. The increase inspeed clearly implies that if the melting time were to re-main unchanged, the distance travelled would be more.However, since the heat transfer coefficient also increaseswith the speed, as quantified in Eqs. (34) and (35), the melt-ing time decreases. Clearly, whether the injected wire willmove deeper or not would be dictated by whether the de-crease in melting time is significantly higher or not. In theregion of speed lower than the point ‘b’, the first factordominates and thus, the distance travelled increases withthe speed. After point ‘b’, the dominance of the second fac-tor prevails and so as the speed increases the distance trav-elled decreases in this region. This is a significant result. Itsuggests that depending on the prevailing conditions in asteel shop, an increase in speed may not necessarily helpthe wire travel nearer to the bottom of the ladle before re-lease of the powder.

5.2. Wire DiameterIf the wire diameter is decreased, the total heat require-

ment for melting of the wire decreases as there is less wiremass to be melted and the release of the powder occurs ear-lier. This is clear in the curves of Fig. 11 (for 10 mm dia.Wire) and Fig. 10 (for 13 mm dia. Wire).

5.3. Packing Density of PowdersThe discussions thus far have been presented considering

a very high thermal conductivity11 of the powder, whichmay be true for a densely packed powder (or solid) CaSi.The melting behaviour changes significantly, however, if theconductivity changes to a lower value as suggested byRibiere.3) The distance travelled for a wire with lower ther-mal conductivity of powder (Fig. 10) is lesser than the dis-tance travelled for the case of wire with higher thermal con-ductivity (Fig. 9). The lower conductivity of the powder re-duces the amount of heat transferred from the casing to thecentre of the wire. This reduction of heat conduction, inturn, reduces the thickness of the shell and causes fastermelting of the shell as well as the casing. As is evidentfrom Fig. 10, when the bath temperature is 1 600°C or high-er and the speed is higher than 2 m/s, the powder is be re-leased much before it reaches the ladle bottom (which isusually 3 m for 140 MT capacity ladle).

5.4. Casing ThicknessThe preceding discussion highlights the problem of early

release of powder in case of high bath temperatures. Thismay result in higher evaporation loss as suggested byPellicani et al.1) as well as loss of unreacted powder by re-action with the top slag.


1163 © 2004 ISIJ

Fig. 8. Variation of radius of cored wire during injection.

Fig. 9. Travelled distance and time before melting (high thermalconductivity) 13 mm wire–0.4 mm casing.

Fig. 10. Travelled distance and time before melting (low thermalconductivity) 13 mm wire–0.4 mm casing.

Fig. 11. Travelled distance and time before melting (low thermalconductivity) 10 mm wire–0.4 mm casing.

To find out the suitable dimensions of the wire for suchapplications, the model was run with different wire diame-ters and casing thicknesses. Figure 12 suggests that the13 mm wire with 0.8 mm casing17) is more suitable than0.4 mm casing wire in case of high superheat melts as theformer reaches the closer to the ladle bottom before releas-ing the powder.

6. Impact Of Steel Bath Parameters on Melting Behav-iour

The parameters of the steel bath which effect the meltingof CaSi cored wire can be divided in two broad categories:

6.1. Temperature above Bath Liquidus (TABL)As the TABL is increased, the maximum thickness of

shell decreases due to the increased amount of heat sup-plied from the bath to the wire. Additionally, the total timefor melting decreases as well as shown in Figure 13. As thetotal shell period (TSP) decreases, the time available for thecasing to be heated up by the shell is reduced. Thus, thoughthe total time for melting is lower with increasing TABL,the horizontal portion of the graph is longer indicating thelarger time interval for which the casing is in direct thermalinfluence of the bath.

As discussed previously, the lower TABL brings in theunique feature of secondary shell growth in case of with-slag conditions (curves 1a and 2a of Fig. 14). However, anincrease in the TABL in this case reduces its possibilities.Another very prominent feature of melting in all with-slagcondition is the extremely fast melting of slag shell aftercomplete melting of the steel shell; the difference betweenthe melting temperature of the slag and the temperature ofthe bath is responsible for that. The model was run underthree thicknesses of slag layers viz., 1 cm, 3 cm and 5 cm.No appreciable difference was noticed in the melting be-haviour of melting in these three cases.

The total melting time of the shell and casing is of acade-mic interest as the plant operator is more interested to knowthe point of release of the powder in the ladle and its varia-tion with different operating parameters. Figure 9 shows thevariation of melting time and the corresponding distancetravelled for a typical wire specification. As it is evident,the wire melts much beyond the 3 m bath depth (assumingwire travels vertically inside the bath) and thus, there wouldbe unmelted wire at the ladle bottom in case of lowerTABLs. These results are different if a wire with differentthermal properties is used as has been discussed in a pre-ceding section.

6.2. Steel GradeThe melting temperature of the liquid steel bath has an

impact on the melting time of the wire. The melting behav-iour of the cored wire for three steel grades, viz. low car-bon, medium carbon and high carbon, with three differentmelting temperatures have been furnished schematically inTable 2. The difference between the bath temperature andthe melting temperature of the bath is constant at 100°C forthese three grades of steel, the expectation that melting be-haviour in these cases would be similar is however not met.The distance travelled by the wire is different for differentgrades and the reason for the same are discussed. The actu-al bath temperatures are 1 625°C, 1 600°C and 1 560°C forlow carbon, medium carbon and high carbon grades respec-tively. The melting time (tTOT) in Eqs. (11) and (12) is influ-enced by tcasing melting. The increase in tcasing melting (keeping allothers terms of the RHS of Eqs. (11) and (12) constant), in-creases the time for complete melting of wire (i.e. tTOT).This is the case for the steel baths with low melting temper-atures (like high carbon grades). As the temperature differ-ence between the steel bath and the melting temperature of


© 2004 ISIJ 1164

Fig. 13. Variation of shell and casing thickness of CaSi coredwire (no slag shell), 13 mm dia. and 0.4 mm thick cas-ing.

Fig. 14. Variation of shell and casing thickness of CaSi coredwire (with slag shell), 13 mm dia. and 0.4 mm thick cas-ing.

Fig. 12. Variation of travelled distance at differentwire dimensions.

the casing is low, the time, tcasing melting, increases significant-ly which ultimately increases the tTOT although the tTSP is al-most same. Thus, when processing a high carbon heat, thepowder may be released close to the bottom, as shown inFig. 15, though the TABL is of the same order as in othersteel grades. The examination of plant data supports this asseen that the efficiency of wire injection in terms of calci-um yield is higher in the high carbon heats. According tothe authors, the release of the powder closer to the ladlebottom is the main cause. This is a significant observationfor Tata Steel, which casts low to high carbon heats. Thus,the optimum wire dimension as well as the injection speedneeds to be different for different grades of steel. A nomo-gram (Table 3) has been developed to help the plant opera-tor to choose the injection speed for a particular wire di-mension according to the grade of the steel beingprocessed.

7. Conclusion

a. A model has been developed to study the behaviour ofcored wire additions in the molten steel bath. This modelincorporates improvement over the existing models fordissolution studies as the concept of thermal resistanceat the interfaces has been used. The presence of molten

slag at the ladle top alters the melting behaviour of thewire to a great extent, though the exact amount of theslag, within the limits encountered in the plant, does notimpact the melting behaviour significantly. The modelsvalidation with plant results has been done in a uniquemanner.

b. Four different schemes of wire melting have been pro-posed based on the model predictions. It has been shownthat the powder can be in a molten condition at the timeof release.

c. Estimation of the effects of different bath parametersand wire parameters on the distance travelled has beendone. The effects of TABL and the speed of injection aresignificant. The thermal conductivity of the powder alsoplays a major role.

d. The dependency of the efficiency of the injectionprocess on the grade of steel to be processed has beenassessed and a modification in wire dimension and oper-ating parameters has been suggested. The increase in thewire diameter and the casing thickness shows favourableimpact on the distance travelled for certain grades ofsteel, whereas, for some other grades the existing prac-tice is in order.

Acknowledgement

The authors appreciate the cooperation and help extend-ed by the personnel of the L.D. Shop 1 and RAC 1 of TataSteel during the trials.

Nomenclaturer : Variable radius (cm)k : Thermal conductivity (J/s · cm · °C)T : Temperature (°C)r : Density (gm/cm3)cp: Specific heat (J/gm· °C))l : Latent heat of fusion (J/gm)t : Time step (s)

tsl: Residence time inside the slag layer of the ladletop (s)

tm(sl): Time for internal melting of slag shell (s)tm(p): Time for internal melting of powder (s)tTSP: Time for complete melting of shell (i.e. Total Shell

Period) (s)tSSP: Time for melting of steel shell (i.e. Steel Shell

Period) (s)tTOT: Time for complete melting of shell and casing and

release of powder (s)tMS(c): Starting point for melting of casing (s)

RT: Thermal resistance at the casing/shell interface,(cm2· s. °C/J)

T*sl: Temperature of the slag shell and casing respec-tively at casing/slag shell

T*c: interface in imperfect thermal contact (°C)q�: Heat flux (J/s · cm2)

TMP: Melting temperature (°C))TBATH: Bath temperature (°C))

NuL: Nusselt numberRe: Reynolds numberPr: Prandtl numberh : Heat transfer co-efficient (J/s · cm2· °C))x : Distance travelled (cm)v : Velocity (cm/s)m : Dynamic viscosity (g/cm· s)

Subscriptsp: Powderc: Casing


1165 © 2004 ISIJ

Fig. 15. Variation of travelled distance in different grades ofsteel.

Table 2. Melting behaviour of 13 mm diameter 0.4 mm casingcored wire at constant bath superheat (schematic).

Table 3. Nomogram for wire injection (schematic).

s: Steel shellsl: Slag shellin: Interface between the zone of the powder at melt-

ing point and the hotter zoneout: Interface between the zone of the powder at melt-

ing point and the colder zoneBath: Steel bath

REFERENCES

1) F. Pellicani, B. Durand and A. Gueussier: Proc. of First Int. CalciumTreatment Symp., The Institute of Metals, London, (1988), 15.

2) P. G. Sismanis and S. A. Argyropoulos: Can. Metall. Q., 27 (1988),123.

3) P. G. Sismanis and S. A. Argyropoulos: Proc. of 5th ProcessTechnology Conf., ISS, AIME, Warrendale, PA, (1985), 167.

4) S. A. Argyropoulos and R. I. L. Guthrie: Metall. Trans. B, 15B(1984), 47.

5) S. A. Argyropoulos and R. I. L. Guthrie: Proc. of Steelmaking Conf.,AIME, Pittsburg, (1982), 156.

6) J. Schade, S. A. Argyropoulos and A. McLean: Can. Metall. Q., 30(1991), 213.

7) Q. Jiao and N. J. Themelis: Can. Metall. Q., 33 (1993), 75.8) R. Kumar, Sanjay Chandra and Amit Chatterjee: Tata Search,

(1997), 78.9) Y. E. Lee: Proc. of Conf. on ‘Development of Ladle Steelmaking and

Continuous Casting’, The Metallurgical Society of the CanadianInstitute of Mining and Metallurgy, Canada, (1990), 154.

10) M. Rebiere, Y. Fautrelle and Y. D. Terrail: Proc. of 6th Int. Conf. onRefining Processes- Scaninject VI, MEFOS, Lulea, (1992), 267.

11) N. Bannenberg, K. Harste and O Bode: Proc. of 6th Int. Conf. onRefining Processes- Scaninject VI, MEFOS, Lulea, (1992), 247.

12) J. H. Ludley and J. Szekely: J. Iron Steel Inst., 204 (1966), 12.13) K. C. Mills and B. J. Keene: Int. Mater. Rev., 32 (1987), 1.14) S. A. Argyropoulos, N. J. Goudie and Michael Trovant: Proc. of

‘Fluid Flow Phenomena in Metal Processing’, TMS, Warrendale,PA, (1999), 535.

15) N. J. Goudie and S. A Argyropoulos: Can. Metall. Q., 34 (1995), 73.16) F. A. Mucciardi: M. Eng. Thesis, McGill University, (1977).17) S. Sanyal and S. Chandra: Indian Patent (Applied on 11th February,

2004).

Appendix. Heat Balance around Surface Node

The heat balance for a surface node (aacc) can be writ-ten as follows (Fig. A-1):

Heat-in through Heat-in through Heat gained bysurface bb � surface cc � surface bbcc

(Term-I) (Term-II) (Term-III)......................................(A-1)

According to the available technique of discretisation,the calculation of surface node temperature based on Eq.(A-1) is done by considering only the half node (bbcc).Thus, the Eq. (A-1) becomes

Heat-in through Heat-in through Heat gained bysurface bb � surface cc � surface bbcc

(Term-I) (Term-II) (Term-III)......................................(A-2)

The volume of the ‘shell’ (aabb as shown in Fig. A-1)is ignored as it is negligible as compared to the volume of afull node. The time when this ‘shell’ matures to a thicknessof a full node, then only it is made part of the calculation byforming a new node (say M�1). The new surface nodeagain remains a half node till its maturity to a full node andthe calculation goes on like this. Thus, the thermal effect ofthe ‘shell’ is getting ignored until the formation of the newnode even if sometimes its size becomes very close to a fullnode. This leads to an error in estimation of the total ther-mal effect on the wire which leads to faulty prediction ofmelting time as well as the inner temperatures. Figure A-2presents the predicted variation of the radius of a 3.81 pureNiobium cylinder during its dissolution in liquid steel usingthis method along with the published data.2,3) As the nodesize is reduced from 0.1 cm to 0.005 cm, the position of ra-dius curve changes and the time for complete melting re-duces. It is evident that the error increases as the node sizeis increased. Thus, it becomes essential to check the depen-dency of the model on the node size. This is the drawbackof the above modelling technique.

A new method has been developed by the authors to takecare of this problem. During the calculation of the surfacenode temperatures, the actual shell volume (i.e., total vol-ume of Dr/2 and shell in Fig. A-1) at that point of time hasbeen considered instead of considering only the half-nodevolume (i.e., volume of Dr/2). Figure A-3 presents the pre-dicted variation of the radius of a 3.81 pure Niobium cylin-der during its dissolution in liquid steel, using the modifiedmethod, along with the published data.2,3) It is clear that thedependency of the model on the node size has significantlyreduced and in the opinion of the authors this represents animprovement in the existing numerical modelling tech-niques in such conditions.


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Fig. A-1. Heat balance around surface node.

Fig. A-2. Variation of radius of 3.81 cm Nb cylinder- results withdifferent node sizes using un-modified heat balance.

Fig. A-3. Variation of radius of 3.81 cm Nb cylinder- results withdifferent node sizes using modified heat balance.

Documents

An Improved Model of Cored Wire Injection in Steel Melts