Modeling of different zones of as-cast structure of high carbon

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Modeling of different zones of as-cast structure ofhigh carbon steel ingotsTo cite this article: Zhiye Chen et al 2012 IOP Conf. Ser.: Mater. Sci. Eng. 33 012081

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https://doi.org/10.1088/1757-899X/33/1/012081

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Modeling of different zones of as-cast structure of high carbon steel ingots

Zhiye Chen, Sonja Arnsfeld, Dieter Senk Chair of Iron and Steel Metallurgy, Department of Ferrous Metallurgy, RWTH Aachen University, 52072 Aachen, Germany E-mail: [email protected]

Abstract. Ingot casting technology has been expanded to large parts during the last decades. As the ingot sizes increase, higher quality of the as-cast semi-products is demanded, with regard to the control of casting defects like macrosegregation and structure inhomogeneity. In order to investigate macrosegregation and to estimate the as-cast structure, theoretical study and simulation work on ingot solidification are carried out at the Department of Ferrous Metallurgy of RWTH Aachen University (IEHK). A solidification model has been developed, and based on that, modeling of the structure morphology has been performed. A proper coupling of the developed solidification model with experimental results from IEHK is under investigation. This solidification model is a two-phase FVM model applied for high carbon steel with 0.6 wt.% [C]. The temperature and concentration fields of the solid and liquid phases have been calculated and these results can provide information for further prediction of the solidification structure such as CET (Columnar to Equiaxed Transition) zone and casting defects in an ingot. The structure morphology model introduces a shape factor of grains as the quantitative criterion for identification of the structure morphology. It focuses on the interaction between nuclei density and resulting macroscopic structure, and can calculate the strictly columnar zone, CET zone with mainly columnar characteristics and enclosed equiaxed crystals, and the pure equiaxed zone. The results will be presented, and the correlation of CET zone with development of macrosegregation in the inner part of an ingot will be discussed.

1. Introduction Solidification is the most important process which influences the final quality of the as-cast semi-product. Today, the demand of heavy ingots and continuous cast products throughout the world for larger turbines and generator rotor shafts are increasing. As the size of as-cast semi-products increase, the problem of structure and segregation in large ingots and continuous cast products takes on new importance. In order to model the solidification structure, consideration of many solidification phenomena such as heat transfer, phase transformation, fluid flow, and the movement of the residual liquid and equiaxed grains is required, meanwhile the understanding of nucleation, growth kinetics, and species redistribution on interfacial scale is also demanded[1, 2].

Many modeling and simulation work, both analytically and numerically, has been done on the prediction of as-cast structure during solidification process since 1970’s[3-4]. Today, the multiphase volume-average model proposed by Beckermann has become the most promising model in the field of solidification simulation[2]. However, due to the lack of information, for instance the nuclei density in the melt, nucleation undercooling, solute redistribution mechanism at the solid/liquid interface, and the local heat transfer coefficient with consideration of gap formation during solidification and the sensitivity of this model on these parameters, prediction on solidification structure from this model

MCWASP XIII IOP PublishingIOP Conf. Series: Materials Science and Engineering 33 (2012) 012081 doi:10.1088/1757-899X/33/1/012081

Published under licence by IOP Publishing Ltd 1

remains uncertain, and the application of the model for steel ingots’ solidification is still very limited. Considerable works is needed to be done in order to improve this model and to validate the simulation especially for large ingot size. The present study uses the volume-averaged method to simulate the solidification process. The heat transfer, nucleation and grain growth are calculated from this solidification model and the results are used for the further calculation of the position of CET zone with mainly columnar characteristics and enclosed equiaxed crystals.

As an important feature of solidification structure, CET zone defines important properties of the as-cast products[5]. Different CET mechanisms have been proposed in the past decades[6] and the most famous two mechanism are the “mechanical blocking” mechanism proposed by Hunt[7] and the “solute interaction” mechanism proposed by Martorano and Beckermann[8]. Most of the mechanisms focus on the blocking effect of equaixed grains on columnar dendrites, either the geometrical impingement or the solutal interaction between columnar and equiaxed grains. However, the gradual transition of grains’ morphology from columnar to equiaxed grains are not quantified, and it seems in those models that the grains changes from columnar into equiaxed in a sudden, which appears to be in contradiction with that observed from experimental results. The present study tries to calculate the grain morphology transition from columnar to equiaxed during solidification quantitatively, where a geometry factor is used as the index of grain elongation degree based on theory of M. Rappaz[9].

2. Mathematical Model Description

2.1 The solidification model The solidification model in the present study is a two-phase model which describes the globular

solidification for a binary Fe-C alloy. The solid and liquid phases in a representative elementary volume (REV) are treated separately and the field properties are averaged. The melt is considered to be stagnant before solidification. The melt convection and the grain movement are neglected.

2.1.1. Governing equations The governing equations are based directly on the model of Wu and Ludwig[10]. The conservation

equations which govern the mass balance in the two-phase system are

(1)

(2)

where ρl and ρs are density of liquid and solid steel, fl and fs are the volume-average volume fraction of liquid and solid phase. ul and us are velocity of liquid and solid, and here ul=us=0. Mls and Msl are the source terms which indicate the interfacial mass transfer due to phase transformation. The heat conservation equations are

(3)

(4)

where Tl and Ts are the volume-average temperature and enthalpy of the liquid and solid phase, and λl and λs are the heat conductivity coefficients. The species conservation equations are

(5)

(6)

where cl and cs are the volume-average composition of liquid and solid steel, and Dl and Ds are the diffusion coefficient of carbon in the liquid and solid steel. The source terms Cls and Csl indicate the interfacial solute redistribution due to solidification and remelting.

slllll Mufft l =⋅∇+∂∂ →

)()( ρρ

lssssssssssss CcDfcufcft

+⋅∇⋅∇=⋅∇+∂∂ →

)()()( ρρρ

lssssss Mufft

=⋅∇+∂∂ →

)()( ρρ

lsssssssssss QTfhufhft

+⋅∇⋅∇=⋅∇+∂∂ →

)()()( λρρ

sllllllllllll CcDfcufcft

+⋅∇⋅∇=⋅∇+∂∂ →

)()()( ρρρ

slllllllllll QTfhufhft

+⋅∇⋅∇=⋅∇+∂∂ →

)()()( λρρ


2

2.1.2. Nucleation and growth The nucleation is considered as heterogeneous nucleation based on M. Rappaz’s theory.

(7)

where N is the nucleation rate, ΔT is the constitutional undercooling, nmax is the maximum nuclei distribution, ΔTN is the mean nucleation undercooling, and ΔTσ is the standard deviation of the Gaussian distribution of nucleation rate. The conservation equation for nucleation is:

(8)

where n is the volume-averaged nuclei distribution. In order to define the growth of the globular/equiaxed dendrites, grains are considered to be sphere shaped and the grain growth velocity is in the radius direction, which is according to the theory of Kurz and Fisher[11]. At the liquid/solid interface, thermodynamic equilibrium is assumed. The grain growth velocity v is given by:

(9)

where de is the volume-averaged diameter of equiaxed grains, cs* and cl* are the interfacial concentrations of the solid and liquid phases. The volume-averaged mass transfer between liquid and solid phase Mls and Msl are defined as

(10) (11)

2.1.3. Boundary conditions and parameters The solidification model is a 1-D axis symmetrical model applied on a 100 kg square-shaped steel

ingot with cast iron mould (see figure 1). It is considered that the ingot is isolated at the top and the bottom, which means that this model could also be applied for a CC-billet with the width of 140 mm. The temperature of the mould inner surface is measured from a real ingot cast at IEHK RWTH Aachen University with the same geometry and the same steel grade, and then fitted as a function of time during and imported to the model as the boundary condition. Other parameter and properties used for simulation are summarized in table 1 and illustrated in Figure 1.

Table. 1 Assumed parameters and properties for the simulation

Initial melt temperature [°C] 1500 °C

Initial mould temperature [°C] TM = 80 °C

Density of liquid and solid [kg/m3] ρl = 7000, ρs = 7027

Thermal conductivity [W/mK] λl = 33.94, λs = 33.94 Carbon diffusion coefficient

[m2/s]Dl =2×10-8, Ds = 5.6×10-10

Specific heat capacity [J/kgK] cpl = cps = 808.25

Segregation coefficient of carbon k = 0.2+5.63×10-4T-3.98×10-7T2 [12]

Figure 1. Schematic diagram of assembly of half ingot and mould

2)(21

max

2)(

σ

σπT

TT N

eT

ndt

TdN ΔΔ−Δ

−

⋅Δ⋅

⋅Δ

=

)1()1(

22***

*

l

l

e

l

sl

ll

e

l

cc

kdD

cccc

dDv −⋅

−⋅

=−−

⋅⋅

=

lsels fdnvM ⋅⋅⋅= ρπ )( 2

slls MM −=

Nnutn

s =⋅⋅∇+∂∂ →

)(


3

2.2 The CET model The CET model uses the analytical model for eutectic growth developed by M. Rappaz[9]. The

following equations describe a nucleated grain growing under the same conditions at the impingement point of the former grains.

(12)

(13)

(14)

The equation (14) describes the relation between the undercooling of the nucleus to the undercooling of the columnar solidification front. A nucleus, growing ahead of the solidification front can never get a higher undercooling than the solidification front itself. At the grain center position, the undercooling is

(15) At the impingement point of two grains (t = test) the undercooling of the two grains must be equal:

(16)

ΔT0(test) needs to be numerically solved in order to calculate the shape factor S:

(17)

Based on the temperature field model and concentration field calculated from the solidification model, parameters such as isotherm velocity vT, temperature gradient G, undercooling of the nuclei ΔTn, undercooling of the columnar front ΔTcol are defined in (18)-(21).

(18)

(19)

(20)

(21)

2.3 Numerical solution In the present study, conservation equations are

discretized based on fully implicit volume finite difference scheme, and a Matlab program has been developed to solve these conservation equations. The REV size is 3.5 mm, and the time step is 0.01 s. The error value for residual difference is 10-7, and the iteration times in each time step varies from 16 to185.

The CET model simulation was carried out in several steps (Figure 2): Figure 2. Program structure of the CET model

))11(1

11(0

Pek

cmv

DGTT

lcol

⋅−−−⋅⋅+⋅

⋅=Δ

))/(tanh(1

))/(tanh()/(

'''

'''

nTestgn

nnTestgcolTest TvLtTT

TTvLtTTvLtT

Δ−−Δ⋅Δ+

Δ+Δ−−ΔΔ=−Δ π

))(tanh(1

))(tanh()(

'''

''0

nestgn

nestgcolest TtTT

TtTTtT

Δ−Δ⋅Δ+

Δ−ΔΔ=Δ

col

ncoln T

TTTΔΔ

Δ=Δ '

col

Tg T

tvGtTΔ

⋅⋅=Δ )('

)/()(0Testest vLtTtT −Δ=Δ π

1)/(

)(0

0

>−

=Testg

estg

vLtRtR

S

)1,1(),1(),(),1(++−+

−+⋅=

jifjifjifjif

dtdxv

ss

ssT

),()1,( jiTjiTT ssn −+=Δ

dxjiTjiT

G ll )1,(),( −−=


4

a) Calculation of the estimated time test for two particles to collide. Due to the periodic characteristics, test is numerically solved from (12) and (13).

b) Calculation of the shape factor S according to (17). c) Comparison between test and the time t for solidification front of columnar dendrites to overgrow

the growing particles. If test < t, CET or equiaxed grains can form, if test >> t, columnar structure will develop, and if test > t, grown particles may be enclosed by the columnar dendrites. This is defined as the static “time criterion”. The shape factor S was calculated firstly without progressing of the columnar front (without time criterion). Since in reality CET and equiaxed zone can only occur when the particles have enough time to grow, S was then calculated with consideration of the growth of columnar dendrites (with time criterion).

3 Results and Discussion

3.1 Solidification sequence The present solidification model is applied to a Fe-0.6wt.% [C] alloy. The simulated solidification

time is about 600 seconds. The evolution of temperature, mass transfer rate between the liquid and solid phase, and the liquid concentration in the ingot during solidification are shown in Figure 3.

20 s 60 s 100 s

a) T (1387~1500 °C) T (1216~1495 °C) T (1171~1481 °C) b) Mls (0~322 kg•m-3•s-1) Mls (0~1017 kg•m-3•s-1) Mls (0~1017 kg•m-3•s-1)

c) cl (0.60~1.76 wt.% [C]) cl (0.60~1.98 wt.% [C]) cl (0.65~1.91 wt.% [C]) Figure 3. Temperature (a), mass-transfer rate (b), liquid concentration (c) filed during solidification Due to the heat extraction from the side wall, solidification starts at the outer surface of the ingot. At

t = 20 s, the melt in contact with the mould is cooled down to 1387 °C (a), which is far below the liquidus temperature (1475 °C) and is approaching to the solidus temperature (1360 °C), and the mass transfer rate Mls of 0~322 kg/(m³s) (b) indicates the nucleation and growth of equiaxed grains. Meanwhile, the melt is enriched during grains growth from the initial concentration of 0.6 wt.% [C] up to 1.76 wt.% [C]. As the time increase to 60 s and 100 s, the melt is further cooled down and the mass

Outer surface

Ingot center


5

transfer rated is increased to 1017 kg•m-3•s-1, the melt concentration is also enriched to ~1.9 wt.% [C]. This will lead to the magnitude of the thermal buoyancy effect of the melt flow and therefore influence the macrosegregation pattern.

The distribution of nuclei and the nucleation rate at different time of the ingot is illustrated in Figure 4. The initial nuclei distribution is set to be 1×109 m-3, and at t = 20 s the nuclei distribution is increased from 1×109 m-3 up to 5×109 m-3 in the mushy zone. It can be observed that the nucleation rate obeys the Gaussian distribution. The maximum nucleation rate at t = 20 s is ~ 8.5×109 m-3s-1, which is larger than that of ~ 5×109 m-3s-1 at t = 100 s. At the outer surface of the ingot the temperature decreases faster than in the center, and the undercooling rate d(ΔT)/dt by both heat removal and solute enrichment is larger at the boundary, therefore the nucleation rate is larger at the boundary of an ingot.

a) t = 20s b) t = 100s

Figure 4. Particle distribution and nucleation rate along the x-axis

3.2 Comparison of calculated T-curves to experiments The measured and predicted cooling curves are compared in Figure 5 at the ingot center, outer

surface and the mould inner wall. The temperature was measured by thermocouples during pouring and solidification, and the pouring stage is not illustrated here. The predicted temperature curve at the outer surface fits to the measured curved quite well in the first half stage of solidification, and the curves at the ingot center show certain difference. At the early solidification, the predicted temperature stays at the liquidus temperature for ca. 100 s, and then it starts to decrease slowly, and the measured temperature stays at the liquidus temperature for relative long time of ca. 500 s and after that decrease dramatically. The most obvious difference stays at the very late stage of solidification (after 500 s), which might be due to the assumption that the grain growth at the solid/liquid interface is in thermodynamic equilibrium. Since at this time the liquid in the bulk melt is also enriched with solute, the interfacial solute redistribution must not stay in thermodynamic equilibrium any longer.

3.3 The CET map As it was mentioned, solidification started from the cooled outer surface to the ingot center. It is considered that for fs > 0.65, the morphology of the solidification structure would not be able to change, therefore the region with fs > 0.65 is not calculated in the present study. The calculated CET map (one column represents 5 mm of the ingot width) at different time is shown in Figure 6, and different colors are chosen to mark different solidification structures (Time progress is 1 s in each diagram):

• Globulitic structure for S < 1 → green • Equiaxed structure for 1 < S < 5 → red • CET for 5 < S < 10 → orange • Columnar structure for S > 10 → blue


6

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400 500 600solidification time / s

Tem

pera

ture

/ °C

Figure 5. Comparison of measured and predicted cooling curves in the ingot and in the mould

Without the time criterion With the time criterion

a) t1: 3 - 4 s

b) t2: 47 - 48 s

c) t3: 87 - 88 s Figure 6. Calculated CET map of the ingot at different solidification time At the beginning of solidification, globulitic chill zone with the width of 5 mm can form (a).

Without the time criterion (a, left), equiaxed zone can form, and with the time criterion (a, right), the formation of the equiaxed zone is suppressed by columnar dendrites growth. In the middle of solidification (47-48 s), for the case without the time criterion, a mixed structure of CET and columnar structure is dominating right before the equiaxed zone. For the case with the time criterion, the formation of equiaxed structure is suppressed by columnar growth, and a narrow CET zone indicated by

exp.T, ingot center

exp. T, near the mould wall

exp. T, mould

simu.T, ingot center

simu.T, near the mould wall

simu. T, mould

x / mm

Tim

e / s

Outer surface Ingot center

Ingot center

Tim

e / s

Ti

me

/ s

Outer surface x / mm Ingot center

Outer surface x / mm Ingot center x / mm Outer surface

Outer surface x / mm Ingot center Outer surface x / mm Ingot center

fs < 0.001 fs < 0.001

fs > 0.65

fs > 0.65

fs > 0.65

fs > 0.65

fs < 0.001 fs < 0.001


7

the orange stripe is formed in front of the columnar zone. The progressing of CET zone can be observed from b) and c) and the CET zone covers around 25-30 mm. The blue area on the right side of the CET zone indicates the potential columnar structure that grains are able to develop, since here test>t, the columnar dendrites can overgrow the growing equiaxed grains / CET structure.

Figure 7 shows the sulfur print of the as-cast ingot along the longitude direction. The chill zone and columnar zone is ca. 3 mm and 25 mm in width, respectively. The CET zone which includes the mixture of columnar and equiaxed dendrites covers a range of ca. 18 mm. Compared with figure 6, the predicted chill zone and columnar zone show nearly the same width, and the predicted CET zone is ca. 50% wider than the observed, and the center equiaxed zone can be predicted only for the case without the static time criterion. Thus a suitable time criterion which balances the development of the equiaxed grains and the growth of the columnar solidification front need to be further investigated.

Figure 7. Sulfur print of the as-cast ingot

4 Conclusions The multiphase/multiscale model of Ludwig has been applied to simulate the solidification of a 100

kg square shaped ingot for a Fe-0.6 wt.% [C] steel. The comparison of cooling curves between simulation and experiments shows that at the first half stage of solidification, the predicted cooling curves fit the measured curves quite well, and at the late stage of solidification (after 500 s) the curves start to depart from each other. The melt concentration during solidification can be enriched up to ca. 1.9 wt.% [C], which provides a strong thermal buoyancy force to melt convection in the mushy zone.

Based on the results from the solidification model, a CET model was developed to predict the solidification structure after the theory of M. Rappaz. The globulitic chill zone and columnar zone were predicted to be similar to the experimental results. The enclosure of CET/equiaxed structure in columnar structure was also predicted for the case with consideration of columnar dendrites growth.

Acknowledgements Authors would like to thank the support of Promotionsstipendium from RWTH Aachen University

and to acknowledge the technical assistance provided by M.Sc. Syed Bilal Hussain and B.Sc. Henning Bernhardt during numerical simulation.

References [1] Gu J P and Beckermann C 1999 Metall. and Mater. Trans. A 30 1357 [2] Ludwig A and Wu M 2002 Metall. and Mater. Trans. A 33 3673 [3] Flemings M C 1966 Trans. Metall. Soc. AIME 236 625 [4] Ni J and Beckermann C 1991 Metall. Trans. B 22 349 [5] Wang J, Wang F, Li C and Zhang J 2010 Steel Research Inter. 81 150 [6] Flood S.C and Hunt J D 1998 ASM Handbook vol 15 (Materials Park) p 130 [7] Hunt J D 1984 Mater. Sci. Eng. 65 75 [8] Martorano M A and Beckermann C 2003 Metall. and Mater. Trans. A 34 1657 [9] Rappaz M 1994 Acta Metal. Mater. 42 2365 [10] Wu M and Ludwig A 2006 Metall. and Mater. Trans. A 37 1613 [11]Kurz W and Fisher D J 1989 Fundamentals of Solidification (Trans. Tech. Publications, Switzerland) p 65 [12] Chuang Y and Schwerdtferger K 1973 Arch. Eisenhüttenkunde 44 341

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Modeling of different zones of as-cast structure of high carbon

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