modelling of macrosegregation in steel ingots: benchmark

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Modelling of macrosegregation in steel ingotsbenchmark validation and industrial applicationTo cite this article Wensheng LI et al 2012 IOP Conf Ser Mater Sci Eng 33 012090

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Modelling of macrosegregation in steel ingots benchmark validation and industrial application

Wensheng LI1 Bingzhen SHEN2 Houfa SHEN1 and Baicheng LIU1 1 Key Laboratory for Advanced Materials Processing Technology MOE Department of Mechanical Engineering Tsinghua University Beijing 100084 China 2 CITIC Heavy Industries Co Ltd Luoyang 471039 China E-mail shentsinghuaeducn Abstract The paper presents the recent progress made by the authors on modelling of macrosegregation in steel ingots A two-phase macrosegregation model was developed that incorporates descriptions of heat transfer melt convection solute transport and solid movement on the process scale with microscopic relations for grain nucleation and growth The formation of pipe shrinkage at the ingot top is also taken into account in the model Firstly a recently proposed numerical benchmark test of macrosegregation was used to verify the model Then the model was applied to predict the macrosegregation in a benchmark industrial-scale steel ingot The predictions were validated against experimental data from the literature Furthermore macrosegregation experiment of an industrial 53-t steel ingot was performed The simulation results were compared with the measurements It is indicated that the typical macrosegregation patterns encountered in steel ingots including a positively segregated zone in the hot top and a negative segregation in the bottom part of the ingot are well reproduced with the model

1 Introduction The production of heavy steel ingots with improved structure and chemical homogeneities is of great concern in industry [1] Prediction of macrosegregation in industrial steel ingots is of great importance In the last few decades numerous multiphase models have been developed to depict the solidification of steel ingots and to predict the macrosegregation Beckermann and co-workers ([2] and references therein) proposed a multiphase model that accounts for melt convection and grain motion which bridges the length scales between global transport phenomena and microscopic grain growth kinetics Ludwig and co-workers [3-4] developed a series of multiphase solidification models The most sophisticated one is a five-phase model that accounts for columnar-to-equiaxed transition non-dendritic and dendritic crystal growth and columnar primary dendritic tip tracking as well as melt flow and grain motion An overview of published multiphase models is presented in reference [5]

The application of multiphase models to industrial steel ingots is a challenging problem mainly due to the large computational resources required to accurately resolve the variety of the phenomena over the process scale A pioneering application was performed by Combeau and co-workers [6-7] in which macrosegregation in an industrial 33-t steel ingot was measured and predicted Li et al [8] also presented simulations for macrosegregation in this benchmark steel ingot using a two-phase multi-scale solidification model

MCWASP XIII IOP PublishingIOP Conf Series Materials Science and Engineering 33 (2012) 012090 doi1010881757-899X331012090

Published under licence by IOP Publishing Ltd 1

In this paper recent progress made by the authors on modelling of macrosegregation in steel ingots is reported The proposed two-phase solidification model involves the simultaneous consideration of the coupled macroscopic phenomena of heat transfer fluid flow solute transport and solid movement and microscopic phenomena of grain nucleation and growth The model was verified with a numerical benchmark test of macrosegregation Then the predictions of macrosegregation in a benchmark 33-t steel ingot were validated against experimental data from the literature Finally application of the model to an industrial 53-t steel ingot is provided

2 Model description The volume-averaged two-phase solidification model was described previously [8] The governing equations and supplementary relations are summarized as follows

Mass conservation equations

s s s s s lsg g Mt

u (1)

l l l l l lsg g Mt

u (2)

Momentum conservation equations

bs s s s s s s s s s s s s ls l s( )g g g p g g K

t

u u u u g u u (3)

bl l l l l l l l l l l l l sl s l( )g g g p g g K

t

u u u u g u u (4)

Species conservation equations

V ss s s s s s s s s s ls

s

( )S D

g C g C C C C Mt

u (5)

V ll l l l l l l l l l ls

l

( )S D

g C g C C C C Mt

u (6)

Energy conservation equations

s s s l l l s s s s l l l l s s l l ls[( ) ] [( ) ] [( ) ]g c g c T g c g c T g k g k T M Lt

u u (7)

Species balance at the solid-liquid interface

V s V ls s s ls l l l ls

s l

( ) ( ) 0S D S D

C C C M C C C M

(8)

Grain population balance

s

nn N

t

u (9)

In the above equations g is the volume fraction ρ is the density Mls is the interfacial phase change rate p is the pressure μ is the viscosity Kls (= Ksl) is the interphase friction coefficient C is the concentration Sv is the interfacial area concentration δ is the solute diffusion length D is the mass diffusivity c is the specific heat k is the thermal conductivity L is the latent heat n is the grain density and N is the grain nucleation rate The subscripts ldquosrdquo and ldquolrdquo refer to solid and liquid respectively and the superscript ldquordquo represents equilibrium at the solid-liquid interface

The volume fractions of solid and liquid sum to unity gs + gl = 1 The densities of both phases are assumed to be equal and constant ρs = ρl = ρ except in the buoyancy terms according to the Boussinesq approximation b

s sl(1 ) (10)

bl T ref C l ref[1 ( ) ( )]T T C C (11)


2

where βsl is the solidification volume shrinkage βT and βC are the thermal and solutal expansion coefficients respectively Tref and Cref are reference values for the temperature and concentration

The interfacial friction coefficient is calculated by the Gidaspow correlation [9]

265s l lD s l l s

p

ls 2s l l s

s l s2l p p

3 02

4

150 175 02

g gC g g

dK

g gg

g d d

u u

u u

(12)

where DC is the drag coefficient and pd is the characterized particle diameter

The diffusion lengths are modeled as

s ss l

10 2

d d (13)

where ds is the grain diameter The viscosity of the solid phase is given by

sc25

s sc ls l

s

(1 ) gg g g

g

(14)

where gsc is the critical solid volume fraction (ie the packing limit)

3 Numerical procedure The model equations presented above are similar in structure and take the general form as

k k k k k k k k k k kg g g St

u (15)

where Ψk is a typical representative variable associated with phase ldquokrdquo the expression for the diffusion coefficient Γk and the source terms Sk can be deduced from the parent equations The general governing conservation equation can be discretized using a finite volume method [10] A staggered grid is employed for the discretization of the momentum equations A power law scheme is adopted to estimate the convection-diffusion flux at the control volume faces The transient term is treated fully implicitly

The resolution of the velocity-pressure coupling for a multiphase flow system can be performed by the IPSA (inter-phase slip algorithm) A detailed description of this algorithm is provided in reference [11] A variant of IPSA is proposed for the numerical implementation of the present model The present approach could also be viewed as an extension of the well-known single-phase solution algorithm SIMPLEC (semi-implicit method for pressure-linked equations-consistent) to multiphase flows In the derivations the philosophy of a multiphase pressure-based solution procedure called the mass conservation-based algorithm (MCBA) [12] is followed as well

31 Pressure correction equation To derive the pressure correction equation the mass conservation equations of the various phases are added to yield the global mass conservation equation given by

old

k k k kk k k

k

( ) ( )( ) 0

g gV g

t

u S (16)

where the operator represents the operation f

f=NB

( ) (17)

and S is the surface vector at the cell face The algorithm consists of two stages the predictor stage and the corrector stage In the predictor

stage the momentum equations can be solved based on a guessed pressure field p The resulting velocity fields denoted by

ku which satisfy the momentum equations will not in general satisfy the


3

mass conservation equations Thus a corrector stage is needed in order to obtain velocity and pressure fields that satisfy both equations The corrections for pressure and velocity denoted by p and ku respectively can be introduced in the following manner

k k kp p p u u u (18) Hence the momentum equations solved in the predictor stage are

k k kNB kNB k kNBA A g V p u u B (19)

The final solutions are k k kNB kNB k kNB

A A g V p u u B (20)

Subtracting the above two equations from each other and rearranging yields the following equation

k kNB k kNB kNB k kNB NBA A A g V p u u u (21)

The velocity-correction formula can be obtained by omitting the first term on the right-hand side (RHS) of equation (21) k k kg D p u (22)

where kk kNBNB

VD

A A

By combining equations (16) (18) and (22) the final form of the pressure correction equation is obtained as

old

k k k kk k k k k k k

k k

( ) ( )( ) ( )

g gg g D p V g

t

S u S (23)

The corrections are then applied to the velocity and pressure fields by using the following equations

k k k kp p p g D p u u (24)

32 Solution procedure The discretization equations are solved iteratively in the following sequence (1) Guess the interfacial phase change rate Mls by adopting the value of the previous time-step (zero

for the first time-step) (2) Solve the velocities us and ul using the IPSA algorithm (3) Solve the temperature and concentration fields (T Cs and Cl) from the conservation equations (4) Check the status of each cell If the cell is within the mushy zone go on to the next step otherwise

set Mls = 0 and return to step (2) (5) Solve the grain density n from the conservation equation (6) Calculate the phase change rate Mls from the interfacial species balance and return to step (2) (7) Repeat until convergence before advancing to the next time-step

4 Benchmark validation The model was verified using a numerical benchmark recently proposed by Bellet et al [13] The test is concerned with solidification of a binary Sn-10wtPb alloy in a two-dimensional rectangular mould cavity of 60 mm height and 100 mm width The mould cavity initially filled with the alloy at the liquidus temperature was cooled symmetrically on both vertical walls Full details of the problem are available in reference [13] Due to the symmetry a half of the full mould domain was simulated It is noteworthy that a fixed and rigid solid phase is assumed The mesh size used in the present simulation was 05 mm and the time-step was 002 s Figure 1 shows the predicted final segregation maps in the cavity The y-axis refers to the vertical centreline of the full mould cavity The principal characteristics of macrosegregation formed can be recognized a positive segregation zone along the vertical axis and at the bottom a negative segregation pocket in the upper right part of the ingot and channel segregates in the bottom central part It can be concluded that the present model retrieves well the macrosegregation patterns predicted in the literature [14]


4

Figure 1 Final segregation pattern in Sn-10wtPb alloy (a) Literature simulation (b) Present prediction

Further the model was used to predict the macrosegregation in a benchmark 33-t steel ingot about

2 m in height and 06 m in average diameter The movement of equiaxed crystals as well as the pipe formation at the top of the ingot was taken into account in the simulation Equiaxed nucleation was described with a heterogeneous three-parameter Gaussian nucleation law (TN = 5 K T = 2 K and nmax = 5 1012 m3) The mesh size was 10 15 mm The time-step was variable in the order of 001 s The predicted final macrosegregation pattern for carbon is shown in figure 2a Two features can be observed first a strong positive segregation is shown at the top of the ingot secondly a conically shaped negative segregation zone is present at the bottom of the ingot Measured and predicted carbon macrosegregation profiles at the vertical ingot centreline are compared in figure 2b Although some discrepancies could be found between the results plotted the present simulation reproduces generally well the measured segregation tendency

Figure 2 (a) Predicted final macrosegregation in benchmark steel ingot (b) Measured and predicted segregation along ingot centreline

5 Industrial application The model was applied to a 53-t steel ingot cast and analyzed by the steel plant of CITIC Heavy Industries Co Ltd The ladle composition and the dimensions of the ingot are reported in table 1 and


5

figure 3 respectively The liquid metal was top poured at the temperature of 1550 oC in a cast iron mould with a refractory hot top A layer of exothermic powder was overlaid after teeming The solidification time for such an ingot was about 15 hours During solidification the time evolution of the temperature was recorded by thermocouples located at different positions in the mould and in the solidifying ingot Finally carbon concentration was measured in a longitudinal section of the hot top

Table 1 Ladle composition of 53-t steel ingot

Chemical species C S P Si Mn Cr Mo

Mass (wt) 041 0003 0007 023 059 10 016

Figure 3 Schematic of 53-t steel ingot Figure 4 Predicted final macrosegregation in fully solidified ingot

Symmetry along the centreline axis was assumed A grid system consisting of 15376 cells was

used with mesh size of 20 mm At the beginning of solidification the time-step was in the order of 001 s As solidification proceeds the time-step was increased finally up to 05 s The steel alloy was simplified as a binary Fe-C alloy with a nominal composition of C0 = 041 wtC neglecting the other chemical species The thermophysical properties used are reported in the literature [8 15] The aforementioned nucleation parameters were adopted in the simulation

Figure 4 shows the predicted final macrosegregation pattern for carbon in the fully solidified ingot A negative segregation zone is observed at the bottom of the ingot This zone has a characteristic cone-shape A strong positive segregation is present in the hot top part Further a striking feature can be observed the A-segregates appeared as strong banded channel segregations are predicted in the top part of the ingot

The predicted evolution of the ingot solidification is shown in figure 5 The equiaxed grains nucleated descend along columnar layer at the surface and entrain the surrounding liquid with them inducing a downward melt flow at the mould side The solute-lean grains are blocked at the bottom part of the ingot and lead to a negative segregation The motion of grains has been progressively weakened as solidification proceeds (figure 5b) At a later stage of solidification (figure 5c) the thermosolutal convection dominates and establishes a counter-clockwise flow descending at the centre and ascending at the surface of the ingot The A-segregates develop in the hot top as a result of instabilities in the mushy zone growth that perturb the fluid flow at the scale of a few centimeters [6] As can be seen the thermosolutal convection is also responsible for the negative segregation at the


6

bottom of the ingot and a positive one in the hot top part In addition the maximum values of liquid velocity shown in figures 5a through c are 778 102 636 104 and 890 105 m s1 respectively

Figure 5 Predicted solidification sequence Left solid fraction Right macrosegregation and liquid velocity (a) 05 h (b) 25 h and (c) 67 h

Figure 6 (a) Schematic showing different transverse sections in ingot hot-top Measured and predicted segregation (b) Section A (c) Section B (d) Section C

Figure 6 presents the comparison of the predicted and measured macrosegregation variations along

transverse sections of the hot top From the centre to the ingot surface the measured segregation experiences a transition from positive to negative The predictions are generally in good agreement


7

with the measured tendencies However both the positive segregation near the centre and the negative segregation near the ingot surface are underestimated in the simulations Possible sources for the discrepancies could include uncertainties in the thermophysical data two-dimensional simplification of geometry or measurements errors It should also be noted that the source of free-floating grains and the globulardendritic morphological transition need to be determined and quantified in modelling Furthermore the predicted A-segregates are indicated with oscillations in the simulated segregation profiles (figures 6b though d) Nonetheless there are not sufficient measurement points to observe the possible A-segregates Future experiments should be designed and performed considering more details to provide a sound basis for assessing the model predictions

6 Conclusions A two-phase solidification model tackling melt flow grain motion and pipe formation has been presented The performance of the model in the prediction of macrosegregation was evaluated on literature benchmarks of a laboratory-scale Sn-10wtPb ingot and an industrial 33-t steel ingot The model was applied to simulate the solidification of an industrial 53-t steel ingot Positive segregation was predicted in the hot top and conically shaped negative segregation zone was predicted in the bottom part of the ingot The prediction of the formation of A-segregates in the top part of the ingot was made possible as well The results show that the model is able to predict important macrosegregation patterns Nonetheless it is shown from comparisons with the measurements that the positive segregation in the hot top is under-predicted by the model The simulations need to be extended to three dimensions using more accurate thermophysical properties Further numerical work and experiments remain to be done to properly account for the microstructure and grain morphological transitions in the macroscopic solidification model

Acknowledgments This work was financially supported by the National Major Science and Technology Project of China (No 2011ZX04014-052) and the National Basic Research Program of China (No 2011CB012901)

References [1] Lesoult G 2005 Mater Sci Eng A 413ndash414 19ndash29 [2] Beckermann C 2002 Int Mater Rev 47 243ndash261 [3] Wu M and Ludwig A 2009 Acta Mater 57 5621ndash31 [4] Wu M Fjeld A and Ludwig A 2010 Comput Mater Sci 50 32ndash42 [5] Pardeshi R Dutta P and Singh A K 2009 Ind Eng Chem Res 48 8789ndash804 [6] Combeau H Zaloznik M Hans S and Richy P E 2009 Metall Mater Trans B 40 289ndash304 [7] Zaloznik M and Combeau H 2009 Modeling of Casting Welding and Advanced Solidification

Processes ndash XII ed Cockroft S L and Maijer D M (Warrendale PA TMS) pp 165ndash72 [8] Li W S Shen H F and Liu B C 2011 Int J Miner Metall Mater Accepted for publication [9] Gidaspow D 1994 Multiphase Flow and Fluidization Continuum and Kinetic Theory

Description (New York Academic Press) [10] Patankar S V 1980 Numerical Heat Transfer and Fluid Flow (New York Hemisphere) [11] Karema H and Lo S 1999 Comput Fluids 28 323ndash60 [12] Moukalled F Darwish M and Sekar B 2003 J Comput Phys 190 550ndash71 [13] Bellet M Combeau H Fautrelle Y Gobin D Rady M Arquis E Budenkova O Dussoubs B

Duterrail Y Kumar A Gandin C A Goyeau B Mosbah A and Zaloznik M 2009 Int J Therm Sci 48 2013ndash16

[14] Combeau H Bellet M Fautrelle Y Gobin D Arquis E Budenkova O Dussoubs B Duterrail Y Kumar A Mosbah S Quatravaux T Rady M Gandin C A Goyeau B and Zaloznik M 2011 TMS 2011 Annual Meeting Supplemental Proc vol 2 (Hoboken John Wiley amp Sons Inc) pp 755ndash62

[15] Li W S Shen H F and Liu B C 2010 Steel Research Int 81 994ndash1000


8

Modelling of macrosegregation in steel ingots benchmark validation and industrial application

Wensheng LI1 Bingzhen SHEN2 Houfa SHEN1 and Baicheng LIU1 1 Key Laboratory for Advanced Materials Processing Technology MOE Department of Mechanical Engineering Tsinghua University Beijing 100084 China 2 CITIC Heavy Industries Co Ltd Luoyang 471039 China E-mail shentsinghuaeducn Abstract The paper presents the recent progress made by the authors on modelling of macrosegregation in steel ingots A two-phase macrosegregation model was developed that incorporates descriptions of heat transfer melt convection solute transport and solid movement on the process scale with microscopic relations for grain nucleation and growth The formation of pipe shrinkage at the ingot top is also taken into account in the model Firstly a recently proposed numerical benchmark test of macrosegregation was used to verify the model Then the model was applied to predict the macrosegregation in a benchmark industrial-scale steel ingot The predictions were validated against experimental data from the literature Furthermore macrosegregation experiment of an industrial 53-t steel ingot was performed The simulation results were compared with the measurements It is indicated that the typical macrosegregation patterns encountered in steel ingots including a positively segregated zone in the hot top and a negative segregation in the bottom part of the ingot are well reproduced with the model

1 Introduction The production of heavy steel ingots with improved structure and chemical homogeneities is of great concern in industry [1] Prediction of macrosegregation in industrial steel ingots is of great importance In the last few decades numerous multiphase models have been developed to depict the solidification of steel ingots and to predict the macrosegregation Beckermann and co-workers ([2] and references therein) proposed a multiphase model that accounts for melt convection and grain motion which bridges the length scales between global transport phenomena and microscopic grain growth kinetics Ludwig and co-workers [3-4] developed a series of multiphase solidification models The most sophisticated one is a five-phase model that accounts for columnar-to-equiaxed transition non-dendritic and dendritic crystal growth and columnar primary dendritic tip tracking as well as melt flow and grain motion An overview of published multiphase models is presented in reference [5]

The application of multiphase models to industrial steel ingots is a challenging problem mainly due to the large computational resources required to accurately resolve the variety of the phenomena over the process scale A pioneering application was performed by Combeau and co-workers [6-7] in which macrosegregation in an industrial 33-t steel ingot was measured and predicted Li et al [8] also presented simulations for macrosegregation in this benchmark steel ingot using a two-phase multi-scale solidification model


Published under licence by IOP Publishing Ltd 1




s s s s s lsg g Mt

u (1)

l l l l l lsg g Mt

u (2)



t

u u u u g u u (3)


t

u u u u g u u (4)



s

( )S D

g C g C C C C Mt

u (5)


l

( )S D

g C g C C C C Mt

u (6)



u u (7)



s l

( ) ( ) 0S D S D

C C C M C C C M

(8)


s

nn N

t

u (9)



s sl(1 ) (10)



2



265s l lD s l l s

p

ls 2s l l s

s l s2l p p

3 02

4

150 175 02

g gC g g

dK

g gg

g d d

u u

u u

(12)



s ss l

10 2

d d (13)


sc25

s sc ls l

s

(1 ) gg g g

g

(14)




u (15)




old

k k k kk k k

k

( ) ( )( ) 0

g gV g

t

u S (16)


f=NB

( ) (17)





3









where kk kNBNB

VD

A A


old


k k

( ) ( )( ) ( )

g gg g D p V g

t

S u S (23)








4







5




Mass (wt) 041 0003 0007 023 059 10 016







6







7











8




s s s s s lsg g Mt

u (1)

l l l l l lsg g Mt

u (2)



t

u u u u g u u (3)


t

u u u u g u u (4)



s

( )S D

g C g C C C C Mt

u (5)


l

( )S D

g C g C C C C Mt

u (6)



u u (7)



s l

( ) ( ) 0S D S D

C C C M C C C M

(8)


s

nn N

t

u (9)



s sl(1 ) (10)



2



265s l lD s l l s

p

ls 2s l l s

s l s2l p p

3 02

4

150 175 02

g gC g g

dK

g gg

g d d

u u

u u

(12)



s ss l

10 2

d d (13)


sc25

s sc ls l

s

(1 ) gg g g

g

(14)




u (15)




old

k k k kk k k

k

( ) ( )( ) 0

g gV g

t

u S (16)


f=NB

( ) (17)





3









where kk kNBNB

VD

A A


old


k k

( ) ( )( ) ( )

g gg g D p V g

t

S u S (23)








4







5




Mass (wt) 041 0003 0007 023 059 10 016







6







7











8



265s l lD s l l s

p

ls 2s l l s

s l s2l p p

3 02

4

150 175 02

g gC g g

dK

g gg

g d d

u u

u u

(12)



s ss l

10 2

d d (13)


sc25

s sc ls l

s

(1 ) gg g g

g

(14)




u (15)




old

k k k kk k k

k

( ) ( )( ) 0

g gV g

t

u S (16)


f=NB

( ) (17)





3









where kk kNBNB

VD

A A


old


k k

( ) ( )( ) ( )

g gg g D p V g

t

S u S (23)








4







5




Mass (wt) 041 0003 0007 023 059 10 016







6







7











8









where kk kNBNB

VD

A A


old


k k

( ) ( )( ) ( )

g gg g D p V g

t

S u S (23)








4







5




Mass (wt) 041 0003 0007 023 059 10 016







6







7











8







5




Mass (wt) 041 0003 0007 023 059 10 016







6







7











8




Mass (wt) 041 0003 0007 023 059 10 016







6







7











8







7











8











8

modelling of macrosegregation in steel ingots: benchmark

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