ISIJ International, Vol. 35 (1995), No. 6, pp. 715-722

Experimental Study and Modeling of the Precipitation of Non-metallic Inclusions during Solidification ot Steel

MarcWINTZ.Manuel BOBADILLA.Jean LEHMANNand Henri GAYEIRSID. Voie RomaineBP320, 57214 Maizi~res-l~s-Metz Cedex, France.

(Received on December26. 1994.• accepted in final form on April 24. 1995)

An original model for the calculation of the precipitation of non-metallic inclusions during solidification

has been developed at IRSID. The microsegregation equations and the equilibrium conditions betweenliquid steel and oxide, sulphide, nitride, carbide inclusions are combinedin a general multiphase equilibriumcode.

Sulphide precipitation in several steel grades: plate grades, mediumcarbon and bearing steels has beenanalyzed in laboratory samples quenched from a partially solidified state. For the various steel gradesinvestigated, the computed results of sulphides (Mn, Fe)S or (Mn, Fe,Cr)S compositions are in goodagreementwith the results of the experimental study.

Another example concerns the precipitation of oxide inclusions in semi-killed high-carbon steels: thecalculation represents very precisely the observed heterogeneous popuiation of oxide inclusions formed atdifferent stages of the industrial process.

KEYWORDS:microsegregation; sulphide; hot-cracking; slag treatment.

l .Introduction

Nonmetallic inclusions which mayappear in liquid

metal at various stages of its elaboration or during its

solidification have a strong influence of the mechanical,machinability or fatigue properties of the transformedproduct. For this reason, the control of non-metallicinclusions in steels at all stages of elaboration is one ofthe main challenges facing the steelmaker.

For someapplications, the presence of certain inclu-

sions with well defined composition may be sought-after. This is, for example, the case of steels with im-proved machinability where the presence of certain

types of sulphides is desired. Another future application

could be that of steels for plates or tubes where the

presence of certain types of oxides (TiO*, Zr02, complexsilicates) and sulphides (MnS) improves their toughnessafter welding.

However, for most applications, it is important to

obtain clean steels. This implies the control of the

cleanness of liquid metal at various stages of elaboration

as well as the one of precipitations occurring during steel

solidification.

In the past, efforts have been essentially oriented to-

wards inclusions formation in liquid steel. Morerecent-ly, studies have been initiated at IRSID to control pre-cipitations occurring during metal solidification. Their

aim is to propose an optimization of the chemicalcomposition to control microsegregations in order to

improve the internal quality of continuously castproducts. They consist in an experimental study of the

precipitation of inclusions during solidification and in

the development of a calculation model making it

possible to predict the composition and amount ofinclusions which form in the metal at various stages ofelaboration and during solidification.

After a description of the calculation model for pre-cipitations, the results of inclusion characterizations

madeon laboratory samples and on industrial products

are presented and discussed with respect to model pre-dictions. These studies concern:

- the effect of carbon and sulphur content on the so-lidification behavior of steels,

- the prediction of the inclusion population in a semi-killed steel.

2. Modeling of the Precipitation of Non-metallic In-

clusions during Solidification

Several models describing inclusions precipitationduring solidification have beendeveloped during the last

thirty years. The first ones were developed by Nilles,1)

Masui,2) and Pesch3,4) for the casting of rimmedsteels.

Considering the great state of agitation of the metallicbath, the solidification interface was considered plane.

Only the reaction of deoxidation by manganesewastakeninto account in the boundary diffusion layer in front ofthe interface.

In the case of solidification with a dendritic front,

Turkdogan5) and then Harkness6) treated the precipita-

tion of manganesesilicates in order to predict theconditions of blow-holes formation in the case of steels

71 5 C 1995 ISIJ

ISIJ International, Vol. 35 (1995), No. 6

deoxidized with silicon-manganese.

During studies on the possibilities of continuouslycasting weakly deoxidized grades at the beginning of the

80's, taking into account the phenomenaof competi-tion between gas evolution and inclusion precipitation

proved to be important. Several solidification modelswith precipitation of oxide inclusions were developed at

that time, at IRSID7) and elsewhere.8 ~ Io)

Towardsthe end of the 80's, NSCresearchers, duringthe development of the so-called "Metallurgy of oxidesin steels",11) began also to be interested in the pre-cipitation of other types of inclusions. Thus, TiO*, Zr02based oxides formed during solidification of steels for

tubes or plates, and on which manganesesulphides

precipitate, can serve as nucleation sites for acicular

ferrite after welding or heat treatment. This newresearch

topic required the development of more- sophisticated

models, capable notabiy of integrating thermodynamicmodels of complex oxides. Thus the modeldeveloped at

IRSID combines, in a samecalculation algorithm, the

equations for microsegregation and a program of multi-

phaseequilibriuml 2) to describe inclusions precipitation.

The calculation tools developed in parallel by NSC13)correspond to the juxtaposition of two calculation

models, one solving the microsegregation equations,

the other using a standard multiphase equilibrium

software (such as SOLGASMIXor THERMOCALC)to calculate the distribution of elements between liquid

metal and inclusions.

The description of the kinetics of solidification in the

IRSIDmodel is based on the classical hypotheses of the

ClyneKurz microsegregation modell4) which takes into

account a partial diffusion of alloying elements in solid

phase:redistribution of the solutes inside a closed spaceelement in which the geometry of the solid/liquid

interface is plane,

- the size of the space element corresponds to the

half secondary dendrltic spacing,

- the temperature is homogeneousin the space element,

- the densities of the liquid and the solid are equal,

the composition of the liquid is homogeneous(Xj

represents the molar concentration of i in the liquid),

and the thermodynamicequilibrium is realized at the

interface,

- a parabolic growth rate is assumed,

- as in the approximation proposed by Brody andFlemings,15) the concentration gradient in the solid at

the solid/liquid interface corresponds to the one of nodiffusion in solid:

(aX~(1, t)/al)1=~=dX~'/d~ ...................(1)

~: coordinate of the solid/liquid interface,

X~' : molar concentration of i in the solid at the

solid/liquid interface.

Figure I showsthe redistribution of the solute i in the

solid and liquid phases.

With these hypotheses, Clyne and Kurz propose, for

each solute, a single differential equation making it

possible to describe the enrichment of the residual liquid

in alloying elements during the advanceof solidification:

C 1995 ISIJ 71 6

X!l

homogeneouss*Xi

liquid

and

solid inclusions.diffusion precipitation

X

half-dend*ite*+idth

Fig. l. Redistribution of the solute i in the liquid and solid

phases in the case of partial diffusion in solid phase.

(1 - ki) ' XjdX! = '(dA/L)

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

' I- (1 -20(; • ki) ' (A/L)

with ki : Partition coefficient of i between liquid andsolid

L : half secondary dendrltic spacingc(j = o(i( I - exp( -

1lcci)) 0.5 exp( - I12c(i)

oci =D: • flL2

D; : diffusion coefficient of i in the solid

t* : Iocal solidification time.

The classical treatment of inclusions precipitation

during solidification is the juxtaposition of these micro-segregation equations with a liquid metalinclusionsequilibrium model. In the IRSID model, the calcula-

tion is based on the materials balance expressed at

each instant for each solute by the following relation-

shi p:

~ =-fo1 l(t)

..........(3)n (t)

LXis(1, t)dl

n:(t) : numberof moles of iin the solid at time tX~(1, t) : molar concentration of i in the solid at

distance I at time t~(t) : coordinate of the solid/1iquid interface at

time t.

In the calculation. the volume element is supposedto

contain only one mole of chemical elements. Integration

which takes into account Brody-Flemings approxima-tion makesit possible to obtain an equation expressing

the numberof moles in the solid (N~(~)) with respect to

successive molar concentrations at the interface (X~(A)):

l rA

~ = 20c;•A'X~'(~)+(1-2c(j)J, X~'(1)dl....N(~)

L(4)

Thevarious steps for the calculation of the numberof

moles of alloying elements in solid metal are presented

in detail in the appendix. Insertion of this equation into

a multiphase equilibrium program describing the solid

metal/liquid metal/inclusions equilibria makesit possi-

ble to calculate the equilibrium concentrations at the

interfaces at each decrement of liquid fraction. The ad-

vantages of the methodused are the following:

simple integration in a complex chemical equilibrium

program, with better convergence than the iterative

methodsused in the other reported approaches,

- independence of the equilibrium conditions between

ISIJ International, Vol. 35 (1995), No. 6

the phases. It is thus possible to choose, for the The solidification range (difference between liquidus

description of the chemical properties of these phases, temperature and temperature of end of solidification) de-

the best adapted thermodynamicformalism. pends on steel chemical composition and on cooling rateduring solidification. For the experiments summarizedin Table l, the cooling rates were identical (0.25'C/s),

3. Effect of Carbonand Sulphur Contents on the Solid-so that the measureddifferences between the five steel

ification Behavior of Steelsgrades are solely governed by the differences in chemical

The presence of segregated internal cracks in con- composition. During solidification ofbinary Fe-Calloys

tinuously cast products depends both on the thermo- at cooling rates considered in this study and thosemechanical stresses in the continuous caster and on generally reached in industrial conditions, carbon dif-

the physico-chemical and metallurgical properties of fusion in the solid metal is fast enoughso that its com-the steel. According to many studies,16~21) the me- position is uniform at each momentof solidification.

chanical properties of the steels depend mainly on The end of solidification then occurs at the solidustheir chemical composition and it has been shownthat temperature of the alloy, that is 30'C below liquidus forthe presence of carbon and sulphur, in particular, ag- Fe-0.160/0C and 100'C for Fe-O.98010C.Thesevalues aregravates problems of hot-cracking.22~24) It is generally muchsmaller than the measured solidification rangesaccepted that the hot-cracking susceptibility of steels is (80'C for steel I at 0.160/0 Cand 165'C for steel 5athigher when the rate of disappearance of liquid metal 0.980/0 C). Thesedifferences are due to the other solutes

towards the end of solidification is slow.25 ~28) This rate for which the microsegregation, amplified by their slowis strongly affected by the microsegregation of sulphur diffusion in the solid, contributes to a substantial increase

and by precipitation of manganesesulphides, effects in solidification range. Precipitations also play a role onwhich for a given sulphur content are usually analysed the value of the solidificati.on range as they modify thein terms of limiting values for the ratio Mn/S. The goal composition of the final liquid metal.

of this study is to get a clearer insight, from laboratory In the case of steels I to 3, the differences in the

experiments and theoretical calculations, on the effect measured solidification ranges are essentially due toof carbon, sulphur and manganesecontents on the different contents in Sand P. In particular, these ex-solidification range, on the nature and conditions of perimental results show that an increase in sulphursulphides precipitation, andon the rate ofdisappearance content leads to an increase of the solidification rangeof liquid metal towards the end of solidification. of about 8'C per 100ppmS. This large increase in

The laboratory solidification experiments consist in solidification range with nominal sulphur content results

quenching directionally solidified steel rods in a con- from the very large microsegregation of this element,ventional vertic~l apparatus with a static induction mainly because of its very low solubility in the solid.

coil under controlled thermal conditions. A detailed Indeed, the values of sulphur partition coefficients

description of the methodand of the exploitation of the between ferrite and liquid and austenite and liquid aremetallographic examination of the samples (determina- 0.05 and 0.004, respectively.30)

tion of solidification range, . .

.) can be found in the In the case of steels 4and 5, the microsegregation ofreference noted.29) Acooling rate of 0.25'C/s wasapplied all solutes (Cr, Mn,Si, SandP) contributes to the increase

to all samples. After solidification, Iongitudinal and ofsolidification range by 65'Cwith respect to Fe~).40/0Ctransverse cross sections containing the quenchedmushy and Fe0.980/.C alloys, respectively. Betweenthese twozone were analyzed metallographically. Sulphides were samples, the effect of an increase in Cr content from 0.01

observed in the transverse cross sections without etching. to I .37 "/o is practically compensatedby the effect of the

Chemical analyses were obtained with an EDS(Energy decrease in Mncontent from I .5 to 0.35 "/*. For steel 4,

Dispersive Spectrometry) attachment on the scanning the combinedcontribution ofsolutes other than Cto theelectron microscope. increase in solidification range is larger than for steel l

Five steel grades were studied: three plate grades with which has almost identical contents in these solutes. Thisdifferent Mn/S ratios, a mediumcarbon steel and a result isdue to the fact that the microsegregation ofmostbearing steel. Thechemical compositions of the five steels alloying elements is larger whenthe steel carbon contentstudied, and the measuredand calculated solidification increases.31'32) Indeed, a higher carbon content leads to

ranges are reported in Table l. the formation of larger amountsof austenite during so-

Table l. composition and sohdification range of the sted grades studied.

weight ('/*) soudification range ("c)

Steel

C CI' Mn Si S PMn/S

Measured Calculated

l2345

O. 160.150.13

0.40

0.98

0.01

0.01

0.01

0.01

l.37

l .53

I.2 l1.40

l .50

0.35

0.32

0.320.32

0.30

0.26

0.0 170.023

0.077O.O150.023

0.0140.0030.0020.0 170.019

905318

lOO15

80~10

64~ lO

l05 + 10

l l0~ lO

l65 :i: 20

9288

107l08180*

* Calculation stopped at I o/o liquid.

71 7 C 1995 ISIJ

ISIJ International, Vol. 35 (1995). No. 6

Table 2. Analyzed and calculated average composition of sulphides.

Sulphides composition ('/.)

Steel Mn/SAnalysis Calculation

MnS FeS CrS MnS FeS CrS

l2345

905318

lOO15

94.7 + I.4

95.2 + I.4

92.9 + I.3

94. I+ I.3

75.7 +6.3

5.3 + 1.4

4.8 + I.4

7.1 + I.3

5.9 + I.3

11.2 + 1.6 13.1 +4.9

95.8

93*9

92.495.8

72.0

4.2

6. l7.6

4.2

18.l 9.9

70

60

50

40

30

20

10

O

FeScontent of the sulphides (%)

I OdlD

Li~i:~~

D Cl

40

35

30

25

20

15

10

5

FeScontent of the sulphides (%)

t

I Mn/S=12

c] Mn/S=15

' Mn/S=20

Mn/S= 30

A Mn/S=40

~ Mn/S= 50

60 1OO80O 20 40

Mn/Sratio

Fig. 2. FeS content of the sulphides vs. Mn/S ratio (experi-

mental results).

lidification by peritectic reaction (steel 4) or becauseaus-tenite is the primary phase (steel 5), and the micro-segregation is thus increased as the solubility of mostelements in austenite is smaller than in ferrite33) andthe diffusion coefficients are smaller.34)

Table I showsa good consistency betweencomputedand measuredsolidification ranges. In the calculation,

the value of MnSsolubility product in liquid iron,

log(aM.s/aM. ' as)) =2955/T- I .80, has been deducedfrom the experimental study of Ito et al.35) For steel 5of high carbon content, the calculation predicts, for

residual liquid fractions of a few percent, a large in-

crease of sulphur content and very slow disappearanceof this final liquid: I "/o liquid would remain 180'Cbelow liquidus temperature, whereas the experimentalsolidification range is 165~20'C. It is very likely that

this discrepancy is due to an improper thermodynamicrepresentation of this liquid metal with high contentsin manganese,carbon and sulphur. Workis in progressto improve this description.

The sulphides observed in the plate grades and in the

mediumcarbon steel (steels I to 4) are solid solutions

(Fe, Mn)S. The sulphides observed in the bearing steel

(steel 5) are solid solutions (Fe, Mn,Cr)S. In eachsample,

a quantitative analysis of a large numberof sulphide

precipitates has beenmade,and the average compositionhas beendetermined. Figure 2indicates, as a function ofthe steel Mn/S ratio, the average FeS contents of the

sulphides obtained in our samples, as well as those

analyzed in industrial samples by Bandi et al.36) Thereis a satisfactory agreement between the two sets of ex-perimental data. For Scontents around 0.015-0.025 "/o,

the FeScontent of the sulphides has avalue around 5.5 ~/*

O'II~:, H

O 200l501OO50

Tl - T ('C)

Frg. 3. FeScontent ofthe sulphides vs. temperature difference

with respect to liquidus temperature.

if the Mn/Sratio is higher than about 20, and it increases

sharply if the Mn/Sratio decreases below about 15.

Thecomposition of sulphides formed during solidifica-

tion wasalso calculated with the model. The average of

the measuredandcalculated MnS,FeSandCrScontents

of the sulphides are reported in Table 2. The calculated

values of FeScontent are in satisfactory agreementwith

the experimental values. This is also true for the CrScontent of the sulphides for steel 5. The results of the

calculation agree with the fact that the FeScontent in

the sulphides increases when the Mn/S ratio becomessmaller than about 20 to 15.

The model has been used to predict the effect of iron

rich sulphide precipitations on the solidification behavior

of steels. To that effect, the solidification behavior of

plate grades containing identical carbon (O. 15o/*), silicon

(0.30~/.), phosphorus (0.0150/0) and sulphur (0.030"/.)

contents, and, various manganesecontents from 0.36 ol.

(Mn/S=12) to 1.5"/* (Mn/S=50) was simulated bycalculation. Figure 3 shows the variations of the

computed FeS content of sulphides as a function of

temperature difference with respect to liquidus tempera-ture. Whenthe Mn/Sratio is equal to 50, the FeScontentof the sulphides is very low and remains practically

constant throughout solidification (around 5"/.). Whenthe Mn/S ratio is equal to 12, the Fe~ content of the

sulphides is higher and increases from about 19 ~/o at the

beginning of sulphide precipitation to about 37 o/o at the

end of solidification. This large variation of FeScontentsin the sulphides at low Mn/S ratios, as solidification

progresses, is certainly an explanation for the scatter

shownby the results reported in Fig. 2. This difference

in precipitation pattern of sulphides, as the Mn/Sratio

C 1995 ISIJ 71 8

ISIJ International, Vol. 35 (1995), No. 6

0,9

0,8

0,7

0,6

0,5

0,4

0,3

0,2

O, lO

Liquid fraction

----- Mn/S:: 50

Mn/S= 15

Table 3. Composition ofthe semi-killed steel grade in whichoxyde inclusions have been analyzed.

C('/.) Mn('/.) Si("/.) Al(ppm) Ca(ppm)Mg(ppm)O(ppm)

0.712 l.030 0.357 8 3 o.4 16

(Mn,Fe)S (Mn,Fe)S

O 50 100 150 200

Tl - T('C)

Fig. 4. Liquid traction •s. temperature difference with respect

to hquidus temperature.

varies at constant Scontents, has an impact on the

solidification range. Indeed, as the Mn/S ratio is de-

creased, the Scontent in liquid metal towards the endof solidification is increased, owing to the delayed sul-

phides precipitation; as a consequence, the solidification

range increases and the rate of disappearance of liquid

metal is slowed down. This is illustrated in Fig. 4whichshows the evolution of residual liquid fraction as afunction of temperature, for two Mn/S ratios. ForMn/S= 15, the precipitation of sulphides starts 107'Cbelow liquidus temperature and the solidification rangeis equal to 169'C, these two values being 69 and 85'C,respectively, for Mn/S= 50.

A similar analysis has recently been proposed byAlvarez de Toledo et al.28) to define a criterion for the

influence of Sand Mn/S ratio on the hot ductility ofsteels. It uses a cruder evaluation of microsegregations

than the present model and considers that pure MnSis

precipitated. This moreprecise evaluation showsthat the

FeScontent of sulphide phases cannot be neglected asit plays an important role on the rate of disappearanceof the last liquid films at the end of solidification, in

particular in the conditions prone to hot-cracking sus-

ceptibility.

4. Control of the Composition of Inclusions in Semi-killed Steels

In semi-killed steels generally used for bar and wire

products, the objective is to obtain oxide inclusions that

will remain glassy and plastic during metal shapingoperations, and to avoid the formation of hard phasessuch as alumina or spinels that can precipitate directly

in the metal or result from the crystallization of original-

ly liquid inclusions. A wide variety of endogeneousin-

clusions compositions are usually found in these steel

grades which can a priori belong to three categories:

deoxidation products, inclusions precipitated duringmetal solidification, and inclusions arising from late

reoxidations. Oneof the applications of the precipitation

model is to compute, from the global analysis of the castmetal, the theoretical composition of inclusions formedat equilibriurn, in liquid metal and during solidification,

assumingthat once formed these inclusions do not react

again with the metal. The composition of reoxidation

71 9

inclusions can also be simulated by arbitrarily increasing

the oxygen content to levels that will consumethe trace

oxidizable elements (Al. Ca, Mg), and start reacting withthe weakerdeoxidizors (Si andMn). Obviously, the exact

contents of oxides of the trace elements (e.g. Al203) in

these reoxidation inclusions will dependon the severity

of the reoxidation and on the residence time available

to them for equilibration with the liquid metal.

Basedon these calculations, the analysis of the variousinclusions is used, if there is coherency, to assess their

origin.

The following example concerns an industrial con-tinuously cast high-carbon steel, of global compositionshownin Table 3. Note the very low contents of the steel

in trace elements, A1. Caand Mg. Thecalculated inclu-

sions compositions depend very muchon these con-tents, so that a very accurate analysis is necessary for

a meaningful diagnosis. The analysis wasmadeon the

as-cast sample in which about 50 inclusions have beenobserved andquantitative microprobe analysis made.A11

the observed inclusions belong to one of the three

categories listed in Table 4 in which the calculated

compositions are compared to the range of analyzedcompositions. Theagreement is quite satisfactory.

Unfioated deoxidation products amount, according to

the calculation, to about 5ppmof oxygenout of the total

oxygencontent of 16ppm.Theyare composedessential-

ly of Si02. Al203 and CaOand contain small amountsof MgOand MnO.They were liquid whenprecipitated

and, consequently, have spherical shapes in the as-cast

product. During cooling, a (Mg, Mn)O-Al203 spinel

phase has crystallized in someof them, the matrix re-

maining homogeneousand glassy with a compositiononly slightly modified by this crystallization.

Oncethese inclusions have precipitated, the Caand

Mgcontents remaining in solution in the liquid metal

are practically zero. The ones that will form later will

consequently have a composition in the Si02-MnO-Al203 system. Theyrepresent nearly 70 o/o of the residual

inclusions (11ppmof oxygenout of the 16ppmanalyzed)

and are globally distributed amongtwo large classes:

- inclusions formed during metal solidification. Theones precipitated at the beginning of solidification arelocated in the alumina primary phase field, and alu-

mina crystals appear in someof themduring cooling.

During the later stages of solidification, as a result ofsegregations, their Si02 content increases, and inclu-

sions close to silica saturation maybe formed.

- inclusions resulting from late reoxidations by the at-

mospherethat, according to the calculation, have anaverage composition of 49010Si02~7010MnOandtraces of A1203. Theseinclusions are liquid whenthey

are formed. A Iarge amountof silica crystals will then

precipitate during cooling, the composition of the

C 1995 ISIJ

ISIJ International. Vol. 35 (1995), No. 6

Table 4. Composition of inclusions precipitated at various

micro-analyses.stages of elaboration. Comparison between calculation and

Deoxidation Solidification Reoxidation

Computed Analyzed Computed Analyzed Computed Analyzed

Matrixo/oSi02

o/oAl203

"/*CaOoloMnO

'/oMgO

31

25

33

6

5

3236

26=33

l8-303-8

25

40(65)

35(18)

25

(19)

35-39(5255)2026(5-8)

30-37(2535)

39

61

37~0

5259

Precipitates Spinel Spinel

(Mg-Mn)O-Al203 (Mg-Mn)O-Al203Alumina Alumina Silica Silica

(-): composition of inclusions formed at the end of solidification.

5

Si02

~l =2ppm

30

22.5

O_=15ppm

selection. Fig. 5showsthe calculated values, using IRSIDslag model, of dissolved aluminumand oxygencontentsresulting from equilibrium of the steel grade consideredwith CaO-Si02Al203 Iadle slags. To obtain an A1content qf 5ppmwith Ocontent as low as possible, it is

necessary to aim for a slag composition such thatSi02 ICaO~~0.9 andAl203 IO*/o

.Thesarne calculation

cap be done for industrial slags containing, in addition,

MgO.MnO,CaF2'37)

8

CaO Al203

Fig. 5. Aluminumandoxygencontentsatequilibriumbetweensteel containing 0.35 o/o Si and CaO-Al203-Si02slags

at 1550'C.

residual eutectic liquid having a base of Si02 andMnOin the ratio 39 o/0/61 "/* and containing small amountsof A1203.It appears that the most harmful amongthese in-

clusions, that is the ones leading to the formation of hardcrystallized phases (spinels andalumina), are the residual

deoxidation inclusions and the inclusions formed at the

beginning of solidification.

Oneof the waysto avoid these hard crystallized phasesis to limit the Al content of liquid metal to even lowervalues than in the present situation. For instance, for the

steel grade considered, a decrease of total Al contentfrom 8to 5ppmwould result in a large decrease of the

alumina content of inclusions (from 25 to 130/, for

deoxidation inclusions andfrom 35 to 28 '/, for inclusions

precipitating at the beginning of solidification), and aswitch from spinel and alumina primary phase fields

to, respectively, melilite and spessartite crystallization

domains. The only practical methodfor decreasing the

liquid metal contents in trace elements (Al, Ca, Mg) be-

low the levels that would result in the precipitation ofdeleterious inclusions consists in establishing the

equilibrium of the metal with a ladle slag of appropriatecomposition. As an illustration of the method for slag

C 1995 ISIJ 720

5. Conclusions

Analysis of sulphide precipitation in three plate gradeswith different Mn/Sratio, in a mediumcarbon steel andin a bearing steel containing 0.02~0.0050/0 Sshowedthat, if the Mn/Sratio is higher than about 20, the FeScontent of the sulphides is low (about 5.5 o/o). Whenthe

Mn/Sratio is smaller then about 15, the iron content ofthe sulphides increases considerably, the solidification

range increases and the rate of disappearance of liquid

at the end of solidification decreases with, as a con-sequence, aggravation of hot-cracking. Themodel for in-

clusions precipitation during solidification can be used

as a tool to define optimal steel composition in order toreduce the hot-cracking sensitivity.

In semi-killed steels, endogeneousinclusions belong tothree families: residual deoxidation inclusions (silico-

aluminates of Ca and Mg), inclusions precipitating

during solidification (silico-alurninates of Mn), andpossibly reoxidation inclusions (Mn silicate). If highlyoxidizable elements (A1, Ca, Mg) are not maintained at

low enoughlevels, harmful phases (spinels, alumina,. .

.)

can appear at the time of inclusions precipitation duringsolidification or crystallize during cooling of initially

liquid inclusions. Silica crystals can also form at coolingof reoxidation inclusions. This is quite accuratelypredicted using the model, which can also be used todefine optimal slag treatments for the control of Al, Caand Mg.

Work is in progress to improve further the modelcapabilities by improving the thermodynamic descrip-

tions of concentrated metal phases and of someof the

ISIJ International. Vol. 35 (1995), No. 6

precipitates. Taking into account other types of in-

clusions (nitrides,. .

.) is also being considered.

Nomenc]ature

t : solidification time

l : coordinate in the space elementt* : Iocal solidification timeL: half secondary dendritic spacing

~(t) : coordinate ofthe solid/liquid interface at time tki : Partition coefficient of i between liquid and

solid

D; : diffusion coefficient of i in the solid

n~(t) : numberof moles of i in the solid at time tN~(~) : numberof moles of i in the solid

Xi~(1, t) : molar concentration ofiin the solid at distance

l and at time tX?'(~) : molar concentration of i in the solid at the

solid/1iquid interface

l)

2)

3)

4)

5)

6)

7)

8)

9)

lO)

l l)

l2)

13)

l4)

l5)

l6)

l7)

18)

l9)

20)

21)

22)

23)

24)

25)

26)

27)

28)

29)

REFERENCESP. Nilles: J, lron Stee! Inst., 7(1964), 601

.

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Appendix Calculation of the Numbersof Molesof Com-ponents in Solid Metal

In the case of a plane interface, the amountof solute

i contained in the solid metallic phase is given at anytime t by the sum:

n~(t)-ro

Xi'(1, t)dli _

1JA(t).

L .

.(A- I)Thefirst derivative of this quantity with respect to time

t is calculated by differentiation:

dn;(t) I d~(t)

+J[h(t) aXi'(1, t)

o=- X~(~(t), t)

' dldt L dt at

.(A-2)

Fick's law enables us to replace the first derivative ofthe concentration with respect to time by a derivative

with respect to the space coordinate l:

dn:(t) I x (A(t), t)d~(t)

+f:(t) ~ D

aX'(1' t)

dl=-( ~ ;'

dt L dt al al

.(A-3)

Theresulting expression can be integrated as follows:

' -(D~' ' +X~,(A(t),t)'dt_

aX~(~(t), t)dn~(t) I d~(t)

dt L ' al

.

(A-4)

The derivative eX~/al is null at the point (O, t) for

obvious symmetryreasons.In the last expression, the derivative of Xi~, function

of I and t, with respect to I can be replaced by thederivative of Xi*', considered as a function of A(X~,*(~) =Xi(~(t), t)), by using the approximation proposed byBrody and Flemings:

-~(_

dXi~'(~)

..

dn~(t) l d~(t)

+X~'(~)'D~' .(A-5)

dt L ' d~ dtTo transform the derivative of the amountof solute

i, n:, function of time t, in the derivative of the samequantity considered as a function of ~, and denoted N~,the following rule is used:

d.f(t) df*(A) dA..........(A-6)

dt ~ d~ dt ""'

Thus, the derivative of the amount N~ of solute i,

721 C 1995 ISIJ

ISIJ International, Vol.

function of ~, is obtained by dividing the expression ofdn:/dt by d~/dt:

dN~(A)_

I dX~'(~) /i d~.

~l

\D: •

/+X~ (A)/ ""'(A-7)

d~ L d~ dt

The growth rate expression enables us to write thederivative of ~as a function of ~. In the case of aparaboliclaw, A=K/, the derivative of ~is d~/dt =O.5 K2/~ andthe derivative of the mole number N~ can then beexpressed as a function of ~only:

-~( +;t?'(~))

....(A-8)_

dXi*'(~)dN~(A) 1 (2D;lK2) . ~'

d~ L d~

The integration by parts of this equation gives the

amountof solute i in the solid metallic phase:

~Johl

= 20(i ' ~'X~'(~) + (1

-20(i) X~'(O) dON(A)

L.(A-9)

35 (1995). No. 6

Setting c(i =D; •ts/L2 =2D;lK2, or if the coefficient oei is

replaced by c(; as proposed by Clyne and Kurz, this

equation can be rewritten:

~ ,•1= 2ccf

' A'X~'(~) + (1 -

2cc;)N(A)

LX' (O) dO

.(A- IO)

The case of an infinite diffusion is given by cej=0.5:

~N~(A)= L' X:'(~) ..........(A-1 l)

and the case of no diffusion is given by oc;=0:

~ =1N(~)

LX:'(O) dO

. . . . .(A- 12)

C 1995 ISIJ 722