ISIJ International, Vol, 39 (1 999), No. 5, pp, 435-444

Ain

Fully Coupled Analysis of

Continuous RoundBillet

Fluid Flow,

Oasting

Heat Transfer and Stress

Jung-Eui LEE. HeungNamHAN.1) Kyu HwanOH2)and Jong-Kyu YOON2)

Formerly Graduate Student at School of Materials Science & Engineering, Secoul National University, Kwanak-ku, Seoul,151 -742, Korea, nowat Kwangyanglron &Steelmaking ResearchTeam.Technical Research Laboratories, POSCO,Cheonnam.545-090, Korea. 1) Sheet Products & Process Research Team, Technical Laboratories, POSCO,Pohang. 790-785,Korea. 2) School of Materials Science &Engineering, Seoul National University, Kwanak-ku, Seoul, 151 -742, Korea,

(Received on February 23. l998.• accepted in final form on August 17. 1998)

The thermal and vectorial fields in the strand and the temperature distribution in the mold were analyzedwith a finite difference method (FDM) considering the effects of turbulence and natural convection ofmolten steel. The thermo-elasto-plastic behaviors of the strand and the mold were analyzed with a finite

element method(FEM)taking into account the ferrostatic pressure dueto the gravity force and the mechanicalbehaviors of the strand in liquid phase, mushyzone and ~h, phase. The microsegregation of solute elementsin steel wasassessed to determine somecharacteristic temperatures and solid, 8-Fe and },-Fe fractions in

the mushyzone. The heat transfer coefficient betweenthe solidifying shell and the mold wall was iterativelydetermined with the coupled analysis of the fluid flow-heat transfer analysis by the FDMand thethermo-elasto-plastic stress analysis by the FEM.With the above procedure, the mathematical model hasbeen developed to predict the possibility of cracks in the strand, originated from the interdendritic liquidfilm in the mushyzone, through the fully coupled analysis of fluid flow, heat transfer and stress in thecontinuously cast round billet. The calculated mold temperature and heat flux at various casting speedsshowgood agreements with the reported experimental observations.

KEYWORDS:computer simulation; coupled analysis; fluid flow; cracks; heat transfer; microsegregation;thermo-elasto- plastic stress.

l. Introduction

Thecontinuous casting process, which has beenwidelyapplied in manysteel works because of its high pro-ductivity and other benefits, is still being actual]ystudied and developed. Nowadaysthe needs to improvethe design of casting machines with high castlng speed,high quality and defects free steel production have beenincreased. It has been reported that the primary causeof defects in the cast strand is the inhomogeneousheattransfer from the strand to the mold, which maybe dueto the entrapment of mold flux, the formation of air gapand so on.1 ~6) Therefore, the successful development ofthe process dependsmainly on the understandings andapplications of physical phenomenain the continuouscasting process. Manymathematical models7~18) havegiven the basic understandings about continuous casting

process and the effects of process variables.

To evaluate proper]y the heat transfer between thestrand and the mold with heat transfer coefncient or heatfiux Is important in the 'analysis of continuous casting

process. However, it is difficult to measureor evaluateit directly in plant caster. Somemodels7~12) have usedthe knownheat flux or data to estimate it. As mentionedby Kelly et al.,13) however, the heat flux determinedfrom a set of experimental conditions for a given caster

maynot be appropriate for other casters. Most of heattransfer from the strand to the mold is thermally resistedby air gap between the solidifying shell and the moldwall.13 - 17) Therefore, it is necessary to calculate direct-ly the formation of air gap in each casting condition.Kristlansson et a/,16) and Kim et a!.17) determined theheat transfer coefncient basedon the size of air gap fromthermal stress analysis, and then examined its effects onthe heat transfer and the occurrence of hot tears or cracksin the strand. Keliy et al.13) and O'Conneret a/.18) com-puted directiy the formatlon of air gap from the tem-perature-stress anaiysis consldering the analyzed fiuid

fiow of molten steel cl /'rio,"i. In their models, however,the changeof fluid flow due to the temperature distribu-tion and solidification of molten steel, and vice versa,

were not fully included. In this study, In order to properlyevaluate the heat transfer between the strand and themold under given casting conditions, the heat transfercoefncient wasiteratively obtained from the fully coupled

way of fiuid flow-heat transfer-defonllation behavioranalysls in the strand and the heat transfer-deformationbehavior analysis in the mold considering their mutualeffects on each other.

Another Important point In the mathematicai mod-eis of the continuous c'asting process is to predict thepossibility of cracks in the strand. A11 cracks observed

435 1999 ISIJ

ISIJ International, Vol. 39 (1999), No. 5

in continuously cast stee] except transverse cracks areoriginated from the mushyzone of low ductility.19~22)

This is associated with the microsegregation of solute

elements in steel. In order to determine the so]id, 8-Fe

and y-Fe fractions in the mushyzone as a function of

temperature and somecharacteristic temperatures such

as ZDT, LIT, ZSTand liquidus ternperature, the mi-crosegregation modei of solute elements in steel, which

wasdeve]oped in earlier works,21.23) wasadopted. Withthese results as well as the thermo-mechanical model, 17,23)

the thermophysical and mechanical properties and the

deformation behavior of steel in liquid region, mushyzone and two phases region, where both ~and yphasesexist, were assessed.

The purpose of this study is to develop a mathemati-cal model for the prediction of cracks in the strand,

originated from interdendritic liquid film, through the

fu]ly coupled analysis of fluid fiow-heat transfer-de-

formation behavior in continuous casting process. Theformation of air gap, its effects on the fiuid flow-tem-perature-deformation behavior of the strand, the de-

velopment of solidifying shell, the possibility of cracksin the strand and the heat transfer-deformation behavi-

or of the mold were analyzed and the relationship be-

tween casting speed and heat transfer was also Investi-

gated.

2. Model Description

Thecalculation procedure of the mathematical modelfor the fully coupled analysis of the fluld flow-heattransfer-deformation behaviors of the strand and the

mold in continuous casting process is illustrated in Fig.

l and the order of its execution in detaii is as follows.24)

OMicrosegregation AnalysisCalculate somecharacteristic temperatures and frac-

tions in the mushzone as a function of temperature withmicrosegregation analysis and assumethe guessed heattransfer coefficlent netweenthe strand and the mold wall.

@Thermal and Vectorial Analysis in StrandAnalyze the thermal and vectorial fields of the strand

Including the effects of turbulence and natural con-vectlon. (by FDM)OThermal Analysis in Mold

Analyze the temperature distribution of the mold. (by

FDM)~) Iterative Calculation

Repeat step ~) - ~) until the converged solutions

of fluid flow and ternperature in the strand and the mo]dare obtained under the given heat transfer coefficient.

OThermo-Elasto-Plastic Stress Analysis In MoldAnalyze the thermo-elasto-plastic deformation be-

havior of the mold with a mold taper based on the

temperature of the moldcalculated at step O•(by FEM)

(~) Thermo-Elasto-Plastic Stress Analysis in StrandAnalyze the thermo-elasto-plastic deformation be-

havior of the strand wlth the slice model based on the

tenlperature of the moidcalculated at step (~).(by FEM)

~) Calculatlon of Air GapSize

Caiculate the size of air gap between the strand andthe mold due to solidification shrinkage and thermal

Input

Microsegregation AnalysisCalculate So id Fraction

Analysis of StrandFlu dFiow & l-lcat Transfcr

surface tem .ofmold surface tem .

ofstrand

Analysis of MoldHcat Transfcr

NoCheckConvergence

Ye5

heat transfer coefficient tem .fields in strand &mold

Analysis of Moldrhcrmo-Elasto- I'las lic Dcformation

dis lacement ofmold

Analysis of Strandrhcrmo-Elaslo*Plasl c Deformation

dis lacement ofstrand &mold

Analysis of Air GapCompulethc Fornlation ofAir Gap

Analysis of Heat Transfer CoefficientConlpute 1'1cErt Transfcr CoefrLCicnl

NoCheckConvergence

Yes

Prediction of CracksCalculalc Crack Susccptibilit} Coerncic]11

Output

End

Fig, l. Thc schcmatic calculation procedure for the fully

coupled analysis model.

contraction of the solidifying shell.

(~) Update of Heat Transfer Coefficient

Calculate the heat transfer coefficient between the

solidifying shell and the mold wall with the size of air

gap and the thermal resistance of mold flux.

R Iterative CalculationRepeat the whole procedure of step @- Runtil a

set of the converged solutions of heat transfer coefficient

is obtained.

R prediction of Cracks in the StrandCa]culate the yield strength of steel and the crack

susceptibility coefficient and analyze the possibility of

cracks in strand, which is originated from the inter-

dendritic liquid film.

3. Mathematical Formulations

3.1. Microsegregation Analysis

In order to determine solid, ~-Fe and y-Fe fractions

in mushyzone as a function of temperature, the mi-

crosegregation ana]ysls of solute elements of steel was

O1999 ISIJ 436

ISIJ International. Vol. 39 (1999), No. 5

Table 1. Chemical composition ofthe carbon steel used in

this study.1.0

Element C Si Mn P S0.8

WtO/Q Oll ol o54 O~)17 O013

Table 2. The calculated characteristic temperature of the

carbon steel

Temperature dcscription ZDT LIT ZST Liquidus temperature

ZOF 0.6OOCLL 0.4

~ 0.2

Solid fraction

Temperatu re( ~; )l O649]08

1457 1510l501

O1521

assessed.22~25) From the analysis, the characteristic

temperatures of ZDT(Zero Ductility Temperature, fully

solidified temperature), LIT (Liquid Impenetrable Tem-perature), ZST (Zero Strength Temperature) and liq-

uidus temperature (fully melted temperature) could beobtained. The composition of steel and the calculated

ZDT, LIT. ZSTand liquidus temperature are listed in

Tables I and 2, respective]y. TheLIT wasdefined by the

temperature at the solid fraction .~ of O.8.23) The ZDTof 1457'C is 38'C Iower than the calculated solidus

temperature based on FeCbinary equllibrium phasediagram. Figure 2shows the calculated solld, ~-Fe andy-Fe fractions as a function of the temperature duringsolidification of the steel. As shown in Fig. 2, the 8ly

phase transformation begins at the temperature of

l 489'C corresponded to,f*=0.9. Therefore, in the tem-perature region between 1457'C (ZDT) and 1489'C, thesolidification and the ~h, phase transformation occurtogether.

3.2. Thermal and Vectorial Analysis

The 2-dimensional axisymmetric governing equationswith cylindrical coordinate system are as follows.

(Continuity Equation):

laT~,("Pu.)+aa

(pu) O ..........(1)

(General Transport Equation):

0.0

YfSl \ 5fs - fs

l42O 1440 1460 1480 1500 1520 1540

TEMPERATURE,ocFig. 2. The ctllculated solid fr~rction /~. (5-Fe rl rctlon / and

,'-Fe f'raction :'/~ 2ts runction of temperature forO. I I wt(Vo carbon steel.

Table 3. Dependenlvariables, effective diffusion coemclentsand source ternls for governing equations on thecylindrical coordinate system.4 1)

c rc Sc

~; z =la

,-~,('"purip)+aa

(pu ip) I a rrac

r a' c ar

ca F

ac+~ c

,..,,.....(2)+Sez az

where p is density and u, and u= are the velocity

componentsin ,' and _• directlon, respective]y. ip can bethe dependent variables such as u,, u=, temperature T,

turbulent kinetic energy kand rate of energy dissipation

8. Fc and Sc rs the effective diffusion coefiicient and the

source term of ip, respectively. Fc and Sip for the de-

pendent variables are listed in Table 3. The gravity di-

rection is taken as .--direction.

The latent heat of solidificatlon could be taken into

account by the effective heat capacity method2s) andthe effect of natural convection due to the density changewith temperature was incorporated into the source termof momentumequation. The turbulence of molten steel

was considered by the well-known standard k8 tur-

ur~J~~i + I ~ au au ur( )r ( z_;!eff + ~L 2u)r'c!errar r ar ar !/etf ar etf2raz

uzJ~!i

+ i ~ au au( +~ +F) ) ir ( :'l eif r'd effaz r ar az Jeteff

azaz

T k + 'ItloT

k '/ + 'dt/ak G-CDpE

s p + l!tl(1E Ci eC'Ik-C2p e ~'1k

f au 2 au 2 au au 2) 2[(ar az az + ar) :( r(-) J( tr )Jurwhere G= 'Itl2 r + + +

'e'err = '1 + ul "/t = cl"pk21E

C"=0'09, CD=1~' Cl=1'44, C2*1~2, (1k=1 O' c E=1~3, cFT=0'9

bulence model. The schematic mold of round billet

casting and the important boundary conditions for theanalysis of fluid flow and heat transfer are lllustrated inFig. 3.13'23 ~25)

3.3. Stress Analysis

In order to describe the thermo-mechanical behaviorsof the strand in the mushyzone between the ZDTandthe ZSTand in the liquid region above the ZST, thefinite element model,17,23) which can take into accountthe ferrostatic pressure due to gravity force as well asthe volume reduction in the liquid region due to large

mold taper, solidification shrinkage and thermal con-traction of the strand, wasadopted.

The initial meshand the boundary conditions for theanalysis of deformation behaviors in the strand and themold are shownin Figs. 4(a) and 4(b), respectively. V.,

Vy and V. are the displacement components due todeformation in x, y, -' direction, respectively. As for theanalysis of deformation behavior in the mold, theaxisymmetric longitudinal section was adopted as thecalculation domain and the right top corner of themold was constrained.13) In the analysis of deforma-tion behavior in the strand, the normal componentofdisplacement is set to be zero at the symmetryboundaryof the center and the analyzed displacement of the moldis used as boundary conditions. The calculations werecarried out under the plane straln condition using the2-dimensional FEMbasedon the slice of the strand. The

437 C 1999 ISIJ

ISIJ International, Vol. 39 (1999), No. 5

JiI~L'I30mmradius

radius91 4mmH~1

IlIIllIlI

--'F

Fig. 3.

4~top of mold, reference position

(-= meniscus, 100 mm

1-- mold thickness

~-- mold length

12 mm

705 mm

cut off 820 mrn

(a)

(a) The schematic illustratiol

llow-he2lt transre]' analysis.

lIlIIlIlIl

h= 11W/m2.C,Ta = 30 oc

h= 11W/m2ac,Ta = 1500oc

u = O, adiabatic

see Text

symmetrycondition

(--- see Appendix A,

h= I ,i W/m2oc, Ta = 30 oc

h= 700Wlm2oc, Ta = 63 ac

of continuous ro Lllld bi let cast '

Dg 'llld (b)

flow rate balance

(b)

he boundary conditions fo]- fluid

v,c= o

•Lz x

91

(a)

Vr-~O

.4 mm

mold

/}~yx +HF,L-

IJ~v~t~~-- '- '~ "~l:1 1"""'~~

i

~L,,

(b)

Fig. 4.

The initial meshand boundary conditions of (a) thestraLnd and (b) the mold for stress analysis. ElementA is ~l finile elemenl 2lt the surface of the strand.

reference temperatures for the thermal strain in the moldis 25'C. The deformation behaviors of both the strandand the mold were calculated based on the temperaturefields obtained by the FDM.

3.4. Criteria of CracksIt has been reported that since ~/y phase transfor-

mation of O.1 wto/o carbon steel begins at the solidus

temperature of equilibrium phase diagram and thehigh probability of cracking in 0.1 wto/o carbon steel

exists below this temperature.1'3'5,Is) Therefore, manystudiesl3'29~32) took Into account the ~/y phase trans-formation strain below this temperature. The steel dur-ing continuous casting, however, can not solidify alongthe equllibrium path due to rapid primary cooling ratein the mold and the secondary cooling zone.

In this study, all mechanical properties, which werebased on the non-equilibrium cooling path obtained bythe microsegregation analysis of the steel, were used.

The LIT was adopted as the reference temperature forthe thermal and ~h, phase transformation strains.23)

Therefore, all stresses except the ferrostatic pressure in

the liquid region occur below the LIT. Al] cracks ob-

served in the cast strand is associated wlth low ductility

of steel in the mushyzone during casting.1922,33~36)Thecracks due to hot tear in the mushyzone can occurwhen the strand is under tension and in the brittle

temperature range TI}, which was defined as the tem-perature range between the ZDTand the LIT.17-23)In order to predict the cracks in the strand, the cracksusceptibility coefficient, Sc, wasdefined as follows.24 ~ 37)

Sc=YM

for ~fs~,fsYc

=0 for O~,fs """-"-""'(3)

=0 for YMwhere Ifs Is the solid fraction which corresponds to theLIT, YMis the analyzed maximumprincipal stress andYc is the crltical yield strength for the cracking, whichwas analyzed based on Eq. (A-5).23) It is reported thatthe cracks, originated from interdendritic liquid film, is

mainly affected by the maximumprincipal stress, 17,37.38)

Therefore, the probability of cracking In cast strand canbe analyzed from Eq. (3).37) In this study, strain rate

was assumedto be I x 10~2/s.24.37) Whenthe steel is

@1999 ISIJ 438

ISIJ International, Vol. 39 (1999), No. 5

fully solidified and is below the ZDT, the possibility ofcracks due to the hot tear in the mushyzone disappears.

The predictlon of the cracks, originated and developed

from the solid steel, shou]d be based on new criterla

consldering the hardening by precipitates and the

ducti]ity of the steel.24)

3.5. Heat Transfer Coefficient betweenSolidifying Shell

and MoldHeat transfer coefficient between the solidifying shell

and the mold wascalculated with the series concept of

thermal resistance through the following equation. 17,37)

/1= I ..........(4)

Rl+R2+R3+R4+ll*

.....

where Rl, R2, R3 and R4 are the thermal resistance

between the mold and the mold flux, due to the air gap,due to the mold flux and between the mold flux and the

soiidifying shell, respectively. /1, is the heat transfer

coefficient by radiation through the air gap. Each ther-

mal resistance component was explained in previous

papers. 17.2 3)

3.6. Procedure Details in Numerical Calculation

The governing equations in Eqs. (1) and (2) werediscretized with the FDMbased on the control vo]umemethodproposed by Patankar.39) Thesolution of matrix

was obtained with the well knowntri-diagonal matrix

algorithm (TDMA). The staggered grid system for the

velocity fields was used and in order to consider effec-

tlvely the diffuslve/convective transport, the power lawscheme39)was adopted. In order to treat the pressuregradient terms in momentumequation, the SIMPLERalgorithm proposed by Karki40) was adopted. Thecalculatlng domains for the FDMare divided by 106 x88 grids for the strand and by 96 x 12 grids for the

mold. Thegrid system In the strand wasclustered near the

strand surface. The FEMmeshescontain 3000 nodes

and2842 isoparametric elements for the stand and I 140

nodes and I 034 isoparametric elements for the mold,respectively, as shownin Fig. 4. The position of radial

meshused in the FEMwas sameto that of the radial

grid used in the FDM.The temperature distribution of

the strand at a time t in the FEMwas obtained by the

interpolation of the thermal fields at the distance fromthe meniscus of molten steel, which was previously

calculated by the FDM.In this study, the convergenceof the solutions is determined by the following criteria.

For the turbulent velocity fields,4)

Residual sources of ip I x l0~3

where ip is dependent variable such as y., u., kand 8For the temperature fields,

"'~ T']d~TT"'~

For the fully coupled analysis of velocity, temperatureand stress fields,13)

f~7;1~~rl~T~~~r~dold

_ cinew 12~/ IJLuairgap~uairgrap) I x l0~2

/~TT~~~~dnew\2~/ L Luair gap'

Table 4. Thecasting conditions used in this stud y. 13'26 ~ 23)

Parameters

Casting speed (m!min)

Mold taper ('/,)

Superheat( t )

l 5 2 2.S 3

1218

30

..(5)

1.o,L

oQ:- 0.8

LLoQ)z 0.6oF~ 0.4

>~J>h 0.2

Jwoe 0.00

HF•STARTFROMINITIALIZED FIELDS

HI* STARTFROMCONVERGEDFIELDSATOTHERCONDITION

439

5 10 15

NUMBEROFITERATION

25

Fig. 5. The relative variation of alr gap with the numberof

iteration during the fully coupled analysis.

where cl*~' g"p

is the size of air gap at a node in the surface

between the strand and the mold and the analyzed air

gap size was underreiaxed to take the stable analysis offluid flow and heat transfer and the underrelaxationfactor is taken as 0.7.24)

4. Results and Discussion

The casting conditlons used in this study are sum-marized in Table 4. The thermo-physical and mechani-cal properties of the carbon steel and the mold are list-

ed elsewhere.9' 17,24,42)

4.1. Fluid Flow, Heat Transfer and Deformation Be-havior in Strand and Mold

The relative variation of air gap size in Eq. (5) duringthe iteration of the fully coupled analysis of the fluid

flow-heat transfer-deformation behaviors in the strand

and the mold is shownin Fig. 5whencasting speed is

3m/mln. At the first calculation step, the guessed air

gap size and heat transfer coefficient between the strand

and the mold wall were assumed to be 0.0mmand

l 500W/m2.C, respectively. As shownin this figure, the

relative variation of air gap size was converged within

25 iterations whenthe analysis started from the aboveinitialized condditions. Using the results converged at

the other casting conditions, the relative variation of air

gap slze wasconverged within 20 iterations and the sameresults of air gap size, temperature distribution, velocity

fields could be obtained.

The calculated air gap size through the fully coupledanalysis is shownin Fig. 6. As shownin this figure, the

air gap starts to form just below the meniscus. Themaximumair gapsize of O.OI Immforms at about 40mmbelow the meniscus. At the lower part, as the moldtaper compensates for the gap, its size decreases andthe strand becomesclose to the mold again. The maxi-

mumair gap size is smaller than that of bloom andslab of about 0.2mmat the corner region. 16.43) The

C 1999 ISIJ

ISIJ International, Vol. 39 (1999), No. 5

Fig. 6.

EEC,JO:~

LLOaOHUJ!:H

EOQ~LLUJ

OZhCO

O

O

1OO

200

300

400

500

600

700

0,0

CASTINGSPEED=3.0m/minMOLDTAPER=1.21 8%

oo 0.003 0.006 O.009

AIRGAP,mmair

0,0 12

The calculated air gap along the distance rrom the

mold top at the casting speed of 3m/mln.

600

450Eo~

~(* 300:DJLL

h 150LU:E

oo

Computed. Experimental

~~l

Fig. 8

300

c~:

:)H200o(LJo_~;LU

C; 100~:~

"'t' Experimental atCompufed

IIIll ExperimentQl at

---- Computed

r=2.75mm

r=8.75mm

100 200 300 400 500 600 700

DISTANCEFROMTHETOPOFMOLD,mmThe comparison of the calculated and experimentalheat flux betweenthe slrand and lhe moldal the casling

speed of 3m!min.

o

meniscus mod exit

O

Fig. 7.

1OO 200 3ao 400 50a 5aoDISTANCEFROlvf THE TOP Or MOLD,mm

The calculated and experimentali3'-'(') temperatureprofile of the mold at 2.75 mmand 8.75 mmfrom the

hot face inlo the moldat the casting speedof 3m/min.

700

~~Cf~lO~:

F~Ofl,

O~:~

Op~F~

F~L)Z;

~(/2;::~

o

100

200

300

400

500

600

700

c

c

B

c

c

8

B

A= 50 'c

B= Ioa 'c

c= 150 'c

D=2ao 'c

formation and size of air gap is determined by thecombinations of the thermal contraction and solidifica-tion shrinkage of steel, the mold taper and the ferrostatic

pressure of molten steel. In case of bloomand slab wherethe shapes of cross section are rectangular, the thermalcontraction of the strand is localized near the cornerreglons and the air gap size becomeslarger.1'3'24.43) In

case of the round billet where the shape of cross sectionis, clrcular, the radial thermal contraction of the strandis uniformly distributed along its circumferential region.3)

Thus the size in the round billet is smaller than that in

the bloom and slab under the samemold taper.Figure 7 shows the calculated mold temperatures

comparaedwith the experimental data,13'26) which werereported In the samecaster wlth this study. The mo]dtemperatures were measuredat 2.75 and 8.75 mmfromthe hot face of mold along the casting direction. Thecalculated results show in good agreements with the

measuredvalues. Themagnltudeand location of the peaktemperature as well as the mild upward trend of the

temperatures toward the mold exit are well predicted.Figure 8 shows the calculated heat flux between

the strand and the mold comparedwith experimental

80 90 Ioo 110DISTACNEFROMCENTER,mm

Fig. 9. The calculated temperature profile in the mold at thecasting sr)eed of 3m/mln.

data,13'26) Near the meniscus, as the liquld steel orpartially solidifying shell is in good contact with moldwal] through the mold flux,13) the heat flow to the moldsharply increased. The heat transfer to the mold mainlydependson the contact heat transfer coefficient betweenthe strand and the moldwall, which is strongly dependenton the surface temperature of the strand as shown in

Appendix D. In order to consider the higher heat flowto the mold due to the good contact, whenthe strandsurface is not fully solidified at the initial stage of casting,the contact heat transfer coefficient of 6OOOW/m2.C wasused.13) Therefore, the heat flow to the mold becomeslarger near the meniscus and the temperature distribu-tion shows a sharp peak near the meniscus. As thesolidification of the strand proceeds further, the surfacegrid of the strand is fully solidified and the strandshrinks, and then the solidifying shell is detached fromthe mold wall due to the formation of air gap and theheat flow to the mold decreases. However, as the moldtaper compensatesthe gap, the strand becomesclose tothe mold again, as shownin the calculated air gap profilein Fig. 6. Therefore, the heat flow to the mold increases

C 1999 ISIJ 440

ISIJ International. Vol. 39 (1999). No. 5

~~~~~~OO*

OH~:

OC~:

F~

Fi}

C-~;

~Cl'

;::~

o

lOO

200

300

400

500

600

700

original dcforrncd

(a) (b)

Fig, lO. The mold geometries which are (a) original and (b)

defonTled at the casting speedof 3m/min, respectively

Thedistortion of the mold is magnified by 200 times.

o

A=0.26B=0.23C=0.20D=0.17E=0.14F::O. 11G=0.08H=0.05l:::0.02

I

I

B

A

B

C

D

C

D

A

D

A c

B

C

D

F

EI

EE

100C'JO 200=LLO~o 300

FUJ:rF 400EOaeLL 500UJOZ~c

600F,OO 700

o.ooo o.005 ool o ool 5 o020

DISPLACEMENT,~~Fig. Il. The displacement of the hot face of the mold at the

casting speed of 3m/min.

again. The temperature increase in the middle region ofthe mold, as is shown in Fig. 9, can be well explainedwith this decrease of alr gapsize. Thehighest temperatureof the mold appears in the hot face near the meniscus.The original and deformed geometries of the mold areshown in Figs. 10(a) and 10(b), respectively. The dis-

tortion of the mold in Fig. 10(b) is magnified by 200times. The original geometry of Fig, lO(a) is distortedinto the deformed geometry of Fig, lO(b) because thehigh ternperature gives large thermal expansion, andconsequently makesthe large displacement of the mold.Figure 11 shows the radial displacement of the hot faceof the moldalong the casting direction. Thedisplacementof the mold has the maximumvalue below the meniscus.Thedeformation behavior and displacement of the moldcorrespond to its temperature profile in Fig. 9.

The calculated stream line about velocity fields andthe temperature distribution in the strand are shownin

Figs. 12(a) and 12(b), respectively. As shown in thisfigure, a main melt circulation denoted as 'A' in the

441

A=1549'C

B=1545'C

C=1541 'C

D=1537'C

E=1533~F=1529 'C

G=1525'C

H=1521 Cl=1501 C

(a) (b)

Fig. 12. Thecalculated distribution of(a) stream line and (b)

temperature in the strand at the casting speed of

3m/min.

figure develops over the entire strand and anotherrecirculating flow denoted as 'C' appears near the upperregion of the strand, which can cause the thin solidify-ing shell. The distortion of isotherma] contour near thesubmergednozzle is due to the fiuid flow. As going tothe mold exit, the stream lines are dlstributed uniformlyalong the radial direction. The entire flow pattern de-veloped in the strand is analogous to the results re-ported by Aboutalebi et a/.11) and Choudharyet al.44)

In this study, however, the recirculatlng flow pattern nearthe nozzle appears because the submerged nozzle is

included.

4.2. Cracks in tbe Solidifying Shell

The radial distributions of maximumprincipal stressand temperature at various distances from the meniscusare shownin Figs. 13(a) and 13(b), respectively. At theinitial stage of casting, the maximumprincipal stress in

high at the surface of the strand and its value cecreasesdownto 200mmbelow the meniscus and then increasesagain downto 400mmbelow the meniscus. The regionof the strand having TB is apt to crack under tensile

stress. Fig. 13(c) shows the variation of crack suscep-tibility coefficient Sc along the distance from the centerof the strand at various distance below the meniscus. Atthe initial stage of casting such as 50mmand 100mmdownbelow the meniscus, the Sc is high at the surfaceregion. As the solidification proceeds downto 200-500

mmdownbelow the meniscus, the Sc at the surfaceregion becomesalmost zero, however, at the 85 mmfromthe center of strand showshigh Sc Values. These results

showthat the surface cracks can be formed at the initial

stage of casting and the internal cracks can be found at

C 1999 ISIJ

ISIJ International, Vol. 39 (1999), No. 5

s=,i,owo(h,,,J~5z~:

a:~:D:~

~E

8

6

4

2

o7

DISTANCEFROMMENISCUS-1- 50 mm-1- IOOmm ~-A- 200 mm-v- 300mm-x- 40Omm-+- 500 mm jvY~:/:

l~,F+/jl l'/~~/ l'l'

+/~//~/l'

,1 l'

~-~::r+-$; .1' -A-A-A-L-A-

75 80 9085

RADIALDISTANCEFROMCENTER,mm(a)

~~~rDJLL

hUJ:::

UJOO(UJ>

250

225

20o

175

150

125

1001

1600

1500

1400

13OO

12OO

11OO

1000

900

o

-e- Computed,0.11C, D.54Mn

-~* Experimental, 0.11C, 0.54Mn

-D- Experimental, O.2C. 1.4Mn

~/p-'~//~-

Fig. 14.

OhUJJJ,OLLOUJO(:DF~UJCL=UJh

7

,q, , , ,.I~,~,~,~N:: I~+\+\ \

I:;1 -I-

\\+\

\\\\\ \.\\+TB +\ \\1 \\'\.\DISTANCEFROMMENISCUS+\ ~~ :\

\ \\ v\I\-1- 50 mm

-.- IOOmm \\~ \v~-A- 200mm

300 mm-x- 400 mm \+

-+- 500 mm \\o 75 80 85 90

RADIALDISTANCEFROMCENTER,mm(b)

EO

U)U)LUZ~(

OIHJJLUIU)

O 3.51.5 2.0 2.5 3.0

CASTINGSPEED,m/min

The comparison of caiculated heat flux with ex-perimental valuesl3'2e.4s) at various casting speed.

3

2

1

1.O

i=1x' ro-2/s

/xr'x\ ~~:~~(:DISTANCEFROMMENISCUS

-1- 50mm-'- Ioo mm-JL- 200 mm //;1 :'_-v- 300 mm-x- 400 mm /1 +"I+/-+- 500 mm /;J

+\'r// I ++/f/

+

\+

A

\,A JL-A-JL-Jb,~ l

o

CASTINGSPEED

15m,mhl

--- 2Om!mhl

---- 25m!mn3Om/min

Fig.

mold exit

Fig, 15.

U:LLUJ

OO~J~i

FO.UiOQ):DQ))(O~O

0.8

0.6

0.4

o.2

o,o7o 75 80 85 90

RADIALDISTANCEFROMCENTER,mm(c)

13. Thecalculated distributions of(a)maximumprincipal

stress, (b) temperature and (c) crack susceplibility

coefficient of the strand at various distances from the

meniscus at the casting speed of 3m/min,

the middle stage of casting.

4.3. Effects of Casting SpeedCasting speed is one of important parameters

influencing the productivity of caster, the formation ofinternal and external cracks and the segregation of sol-

ute elements,1'3) Figure 14 shows the computed aver-

age heat fluxes at various casting speed comparedwiththe reported experimental values.13'26,4s) All the calcu-

lations were carried out In the coupled waydiscussed in

Sec. 2. The measuredand computedaverage heat flux

increase with increasing casting speed. The calculated

and measuredvalues for O. I IC, O.54Mnsteel showgoodagreements.

O IOO 200 300 400 500 600 700 800

DISTANCEFROMMENISCUS,mmThe calculated shell thickness in the strand at var-ious casting speed.

The stream lines about velocity fields are investigated

at various casting speed. The stream lines at variouscasting speedshowsimilar distributions. With increasing

casting speed, the main recirculating flow is developedlargely in the strand and the another recirculating flow

near the meniscusappears well. Increasing casting speedresults in higher flow rate and heat flux and the decreaseof time that the strand spends in the moid, which cancause the thin solidifying shell. The sufficient shell

thickness at the mold exit is necessary to prevent the

possible breakouts. Therefore, it is Important to deter-

mine mold taper and cooling conditions in the mold toadjust the casting speed. The calculated shell thickness

at different casting speed is shown in Fig. 15. The cal-

culated sheli thickness decreases with increasing cast-

ing speed. However, the development of solidifying

shell becomessmaller with increasing casting speed. Atthe high casting speed, the thin solidifying shell Is dueto the recirculating flow of high temperature liquid nearthe submergednozzle and the decrease of dwell time of

molten steel.

Figure 16 showsthe temperature variation of elementA, which iocates at the surface of the strand in Fig. 4,

along the distance from the meniscus at various casting

speed. The temperature of e]ement A decreases during

the casting and with increasing casting speed, the tem-

perature is higher. Below the mold exit, the tempera-ture increase of element A is due to the recalescence.

Figure 17 shows the displacement of the mold alongthe distance from the mold top at various casting speed.

C 1999 ISIJ 442

ISIJ International, Vol. 39 (1999), No. 5

EEU'DO,,)ZUJ~:~

OO~

UJOZq:hU,

O

o

2oo

400

600

6

CASTINGSPEED

- 15m!rr~n

- --- 2Om/mjn

25m/min

3Om/rT~n

mold exit

o 8 O Iooo 1200 1400 16Oo

TEMPERATUREOFELEMENTOFA, ocFig. 16. ThehistoryoftemperatureofelementAinFig. 4(a).

analysis of the crack susceptibi]ity coefficient, the sur-face crack cou]d form at the initial stage of casting andthe Internal crack could be found at the midd]e stageof casting. With increasing casting speed, the averagedheat flux was increased and the deve]opment of solid-

ifying shell was s]ow and the intensity of main recir-

culating flow was increased and the another recircu-

lating flow near the submergednozzle were developed,which could cause the slow development of solidifyingshell.

oEE IOOC'JO~tL

20O

OO.OF 300UJ::HE 40OOOeU-UJ

500OZ

60O

S7oo

o.o

CASTINGSPEED1.5 m,min

- -- - 2.0 m/min2.5 mlmin3.0 m/min

Oo o005 oOlo ool 5 0.0 2o

DISPLACEMENTOFHOTFACEMOLD,mmFig. 17. Thedisplacementat thehotfaceofthemold atvarious

casting speed.

The displacement showsthe similar behaviors as shownin Flg. lO. With increasing casting speed, the displace-

mentof the hot face of mold around 100mmdownfromthe mold top increases due to the higher mold tem-perature. The peak of the displacement around 300-500mmdown from the mold top movesdownalongcasting directlon with increasing casting speed.

5. Concluslons

A mathematical model was developed for the fully

coupled analysls of the fluid fiow-heat transfer-de-

formation behavior in continuous casting process ofthe round billet. The thermal and vectorial fields of thestrand and the temperature distribution of the moldwereanalyzed with the FDM.The thermo-elasto-plastic de-formation behavior of the strand were analyzed withthe 2-dimensional s]ice method by the FEM.The de-

formation behavior of the mold, which is at steadystate during the casting, was analyzed in the longi-tudinally sectioned region. In order to evaluate the heatflow and the formatlon of air gap between the strandandmold, the iterative wayof the fluid flow-heat transferanalysis and the thermal stress analysis wascarried out.

The calculated mold temperature and heat flux were in

good agreements with the reported values. From the

Acknowledgments

This work has been supported by Research Institute

of AdvancedMaterials in Seoul Natlonal Unlversity,Seoul, Korea through KangWonIndustries, LTD.,Pohang, Korea, for which authors are grateful andauthors also wish to thank Dr. K. Kim, TechnologyDevelopment Group, M/U FAB. Department, Semi-conductor R&D,SamsungElectronics, Yongin, Kyungi-Do, Korea for helpful discussions.

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

12)

13)

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l6)

17)

l8)

19)

20)

21)

22)

23)

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Appendix

A. Heat Transfer Coefficient betweenMold and CoolingWater13)

Mold wall Heat transfer coefficient

B. Flow Curve in the Fully Solidified Steell7,37)

Q I IPn~;p=Aexp - RT[sinh(fiK)] .(A- I)

temperature (w/m2'C)

Y0=K8tp'...

..........(A-2)

where A, fi and ,11 are constants, Ris gas constant, Qis

activation energy for deformation. K is strength co-efficient, Yo is yield stress, ep Is effectlve plastic strain

andn is strain hardening exponent. (A = I .047 x 1010s~ 1f3=0.01308MPa~1. Q=326.3kJ/mol, m=0.2008, n=0.4298).

C. Critical Yield Strength for Cracking in the MushyZone17.3 7)

Brittle temperature region TB

T13=LIT-ZDT ..............(A-3)

Critical strain for cracking 8c

Ifr

8cikTB "~

"~~"~(A-4)

where ~and k are constant and ~=I x 10 Is

With Eqs. (A-1), (A-2),

=81Yc

q) n exp(Q/RT). . . . .

(A-5)m

T'B'p Awhere l=m-k•n and the values of kand (p are 0.2739and O.07104, respectively.

In mushyzone,

Yc=~fsY6c+~fsr~•••••••

••••••••••(A-6)

where Y~c and Y~c are the crltical yieid stresses of ~and

yphases, respectively.

D. Heat Transfer Coefficient between Mold Flux andStrand Surface43)

Temperature description

Ol20125

l30135

l40145150

Heat transferTemperature, coemcient,

'C whn2*C

30 OOO30 OOO

31OOO34 OOO41 OOO54 OOO70 OOO90 OOO

Mold flux crystalline temperatureMold flux softening temperatureMetal solidus temperatureMetal liquidus temperature

l 030

l 150

l 450

l 523

1ooo

2ooo

6ooo20ooo

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