2 Casting Plant & Technology 4/2012

Authors: Olivier Jaouen, Frederic Costes and Patrice Lasne, Transvalor, Mougins

Ingot casting simulation with ThercastOptimization of ingot casting processes need a good knowledge of the thermo-mechanical state in order to analyze important issues like gap formation, stress and deformation of the solidified shell. Casting parameters (casting speed, cooling strategy, flow rate …) have a heavy influence on the internal quality of the final product. In that context, a numerical simulation package has been developed within the objective of supplying a helping tool for an accurate analysis of those phe-nomena. This paper presents a numerical method that simulates the creation and evolution of in-ternal defects from solidification to the first forming operations (cogging or hot-rolling)

IntroductionIn the process of ingot casting, it ap-pears that the first solidified zones oc-cur mush before the end of the pour-ing and the remaining liquid areas are still present even well after the end of the filling. For sure, behavior of the different metal phases is fully coupled during the process. It appears that de-fects like porosities, cracks or hot tears take their roots from the strains, stress-es and distortions occurring at the first

instants of solidification in the brittle temperature range (BTR) of the alloy. Depending on the tonnage, solidified areas at the end of the pouring of in-gots can represent up to 30 % to 40 % (Figure 1) of the total mass. Hence, it is easy to imagine that in such amount of transformed alloy, defects have al-ready occurred in the shell. Within this framework, thermo-mechanical modeling is of interest for steel mak-ers. It can be helpful in the adjustment

of the different process parameters in order to improve casting productivity while maintaining a satisfying product quality. Here, parameters are flow rate driving, initial pouring temperature or superheating, exothermic powder effi-ciency, ingots shape, etc. for the ingot caster concerned. However, optimiza-tion of the parameters requires a quite complex model that delivers very pre-cise responses. From this point of view, the use of a CFD model sequenced with

Casting powder burns on just filled casting ingots (Photo: Ascometal)

Casting Plant & Technology 4/2012 3

a structure model to simulate respec-tively the liquid and the solid phas-es, and to forecast the defects, is not well suited. Indeed, it is necessary to take into account at the same time li-quid, mushy and solid areas in a cou-pled model. In addition, at each in-stant and locally, the air gap should be taken into account for its influence on the heat transfers between metal shell and molds that dramatically change throughout the solidification.

In this paper, Thercast by Transvalor S. A., Mougins, France, software ded-icated to the simulation of metal so-lidification is presented. The thermo-mechanical models developed in this software are presented. The way of tak-ing into account the coupling between metal and molds during solidification is shown. A model of determination of the liquid and mushy zones’ constitut-ed equation parameters is developed. Industrial applications in ingot casting are proposed.

An original mixed thermo-me-chanical modelThercast is a commercial numerical package for the simulation of solidifi-cation processes: shape casting (found-ry), ingot casting, and direct-chill or continuous casting. A 3-D finite ele-ment thermo-mechanical solver based on an Arbitrary Lagrangian Eulerian (ALE) formulation is used.

Thermal model The thermal problem treatment is based on the resolution of the heat transfer equation, which is the gener-al energy conservation equation:

dH(T)dt

==.(l(T)=T)

(1)

where T0 is the temperature, λ (W/m/°C) denotes the thermal con-ductivity and H (J) the specific enthal-py which can be defined as:

H(T)=er(t)Cp(t)dt+g

1(T)Lr(T

s)

T

To (2)

T0 (°C) is an arbitrary reference temper-ature, ρ (kg/m3) the density, Ts (°C) the

solidus temperature, Cp (J/kg/°C) the specific heat, gl the volume fraction of liquid, and L (J/kg) the specific latent heat of fusion. In the one-phase mod-elling, gs is previously calculated using the micro-segregation model PTIMEC_CEQCSI [13]. The boundary conditions applied on free surface of the mesh of the metal could be of classical different types:» average convection: -λ∇T.n = h(T - Text)

where h (W/m²/°C) is the heat trans-fer coefficient, and Text is the external temperature,

» radiation: -λ∇T.n = εσstef (T4 - Text4,

where ε is the steel emissivity, σstef is the Stephan–Boltzmann constant,

» external imposed temperature: T = Timp,

» external imposed heat flux: -λ∇T.n = Φimp denotes the outward normal unit vector.

At part/molds interface, heat transfers are taken into account with a Fourier type equation:

1R

eq

-l=T.n= (T -Tmold

)

(3)

where Tmold is the interface tempera-ture of the mold and Req (W/m²/°C)-1, the heat transfer resistance that can depend on the air gap and/or the lo-cal normal stress, as presented below:

1Req

=

Req

=

+Rsiƒ e

air>0

+Rsiƒ e

air=0

+

min( , +5 R0

1R

air

11

1R

s

1R

0

1

Rrad

(4)

liquid

viscoplastic behavior Elastisc – viscoplastic behavior

semi-liquid semi-solifraction solid

Solid fraction 0 1

temperature

liqui

dus

cohe

renc

y

solid

us

Figure 1: State of solidification of a small ingot (~300 kg) just after the end of pouring – high percentage of already solidified material

Figure 2: Schematic representation of the rheological behavior of the different phases of the metal in solidification conditions

K SIMULATION

4 Casting Plant & Technology 4/2012

where

eair

lair

Rair

= and

es

ls

Rs=

, with eair and es respectively the air gap and an even-tual other body (typically slag) thick-ness and λair and λs the air and the even-tual other body thermal conductivity. R0 is a nominal heat resistance depend-ing on the surface roughness,

1

sstef

(T 2+T 2mold

)(T+Tmold

)«

Rrad

=+ -1

1«

mold

, with εmold the emissivity of the mold, Rσ = 1/Aσm

n a heat resistance taking into account the normal stress σn , A and m being the pa-rameters of the law.

Mechanical modelAt any time, the mechanical equilib-rium is governed by the momentum equation:

∇.σ+ρg - ργ = 0, where σ, is the Cauchy stress tensor, g is the gravity vector, and γ is the acceleration vector.

Taking into account the very dif-ferent behaviors of liquid and solid metal is realized by a clear distinction between constitutive equations asso-ciated to the liquid, the mushy and the solid states. In order to fit the com-plex behavior of solidifying alloys, a hybrid constitutive model is consid-ered. In the one-phase modelling, the liquid (respectively, mushy) metal is considered as a thermo-Newtonian (respectively thermo-viscoplastic, VP) fluid. In the solid state, the metal is as-sumed to be thermo-elastic-viscoplas-tic (EVP) (Figure 2). Solid regions are treated in a Lagrangian formulation, while liquid regions are treated using ALE [17].

More precisely, a so called, transient temperature or coherency temperature is used to distinguish the two different behaviors. It is typically defined be-tween liquidus and solidus, and usual-ly set close to solidus temperature. For more information, the interested read-er can refer to [1] and [3].

In such a model, physical data, hence numerical data, take values among a set of a huge range, from some Pa to hundreds of GPa. If getting data at low temperatures is quite usual, it is not the case for the high temperatures closed to solidus and above. The acquisition of rheological data at high tempera-ture, small strain and low strain rate is a non easy problem. From this find-ing, Bellet [5] has proposed an extrapo-lation model of the solid data to liquid data for fields like viscosity and strain rate sensitivity in case of viscoplastic behavior. The viscoplastic behavior is formulated with the well known pow-er law:

s=K(T)Îã3m+1« m (5)

where s=K(T)Îã3m+1« mi is the von Mises flow stress,s=K(T)Îã3m+1« m

the equivalent plastic strain rate, T the temperature, K the viscoplastic consis-tency and m the strain rate sensitivity. It is to be noted that the Newtonian be-havior is obtained in case of m = 1 and K = ηl where ηl is the dynamic viscosity of the liquid. The proposed model is summarized as the definition of K and m in three domains defined using four parameters:

» gl, cohe the liquid fraction at coheren-cy temperature

» gl, susp the liquid fraction beyond which a suspension model is used

»

−K−g

1 the sensitivity of K to gl between solidus and coherency point

»

−m−g

1 the sensitivity of m to gl between solidus and coherency point

The two first parameters are arbitrary defined; the two last ones can be exper-imentally deduced.

Figure 3: illustration of the temperature distribution at the end of the pour-ing (left) and the corresponding solidification zones (right). Note the discon-tinuous values at ingot/mold interface due to the HTC depending on air gap.

Figure 4 : zoom on the ingot/mold interface. Solid skin on the left (blue is sol-id, red is liquid). air gap modifying the continuity of temperature diffusion (on the right).

K SIMULATION

6 Casting Plant & Technology 4/2012

If the liquid fraction is:In the interval [0,gl,cohe]

K(g1)=K(0)+ g

1

m(g1)=m(0)+5

−K−g

1

g1

−m−g

1

(6)

In the interval [gl,cohe, gl,susp]

K(g1)=K(g

1, cohe)a K(g

1, susp)1-a

m(g1)=a(m(g

1, cohe)-1)+15

(7)

where

g1, susp

-g1

g1, susp

-g1, cohe

a =

In the interval [gl,susp,1],

K(g1)=h(g

1)=h(g

1=1)(1+ )g

1+ 1 g

12

m(g1)=1

552

52

2

(8),

where the last expression of K comes from the Batchelor suspension model [17]

The values of K and m are continuous along the three intervals, so that, K(gl = 0) and m(gl = 0) are deduced from the solid state constitutive model and are taken at solid temperature or just below. K( gl,cohe) and m(gl,cohe) are deduced from (6) to ini-tialize (7). K(gl, susp), used in (7), results from the suspension model. The value of

η(gl = 1) = ηl (9)

is taken a priori.

From Equations (8) and (9) results a continuous viscosity which values do not vary a lot from (9) in the interval [gl,susp,1]. Hence, the velocity vectors within the mushy zone are not so dif-ferent from those within the liquid zone. It results an evolution of mushy zone and solidifying front quite dif-fused and convected within the liquid zone. Compared to what it is supposed to be reality and other models like Dar-cy model [14], it does not seem to be compatible. Indeed, in case of Darcy model, the solidifying front is mov-ing like a thermal evolution. This re-sults from the fact that, in the Darcy model, the permeability factor yields velocity values within the mushy zone much below than the velocity values into the liquid zone. It results a qua-si nil convection of the mushy zone. In order to prevent this convection of the mushy zone into the liquid areas, a non continuous value of the viscosi-ty at liquid temperature has been intro-duced into the extrapolation model. It results a stabilized solidifying front evolution closed to a Darcy type model would yield. So that (9) becomes

η(g1 =1-) = ηl- (10)

where η1 >> η1.

Defects criteriaPrecise prediction of defects like mac-ro-porosities and/or hot tears is quite appreciated by steel makers. Several hot tear criteria are present through-out literature. Some are based on ther-mal considerations, others are fed with stresses, strain and/or strain rate. In [9] the conclusion of the authors tends to prove that the criterion of Yamanaka et al [10] is pertinent to forecast loca-tion of hot tears in solidification con-ditions. The expression of this criteri-on is the following:

εc =es=K(T)Îã3m+1« m^ dt (11)

BTR

where is the brittle temperature r ange defined when gl t 0, typically 0<gl <0.1,

introduced by Won et al [12] and s=K(T)Îã3m+1« m^ rep-

resents a norm associated to the dam-

Figure 5: global shape of the ingot after 11400 s of cooling. Note the differ-ence of the air gap thickness and the free surface shape. Note the secondary shrinkage due the lack of the powder (right)

Figure 6: distribution of the air gap after 11400 s of cooling for the two cases. Without internal shrinkage, the air gap thickness is bigger (on the left).

K SIMULATION

8 Casting Plant & Technology 4/2012

aging components of the strain rate tensor, expressed in tensile stress axis orthogonally of the crystal growth di-rection [9]. The critical value εc de-pends on steel composition. However, Yamanaka introduced, by experimental observations, a threshold value 2 % of the criterion above which, the odds of hot tears creation are high. Modelling experience tends to show that the same criterion applied with a lower thresh-old, 0.5%, gives distribution that fits quite well the macro-porosities evolu-tion in solidification conditions.

ApplicationsThe model presented above can be applied for ingot casting application or continuous casting applications. The differences are mainly set in the boundary conditions and in the treat-ment on the feeding metal. Here ingot casting application is focused.

Ingot casting applicationsIn case of ingot casting application, the pouring is piloted in flow rate, constant or not. The level of the metal into the mold is defined with a plane, which lo-cation is defined following the flow rate vs time. Both air and metal are taken into account into the ingot. As present-ed before theses phases are mainly treat-ed with an ALE model, whereas the sol-

id phase is actualized with a Lagrangian scheme. Such a scheme allows taking into account the solid shell of the ingot throughout the solidification. It means that even though the filling stage is not achieved, in case of solidification of the ingot skin, the air gap can be caught as soon as it occurs. Hence, the heat trans-fers are so modified between cooling metal and mold following (4), giving a strong thermo-mechanical coupling of all the domains in the cooling system. Moreover, strain and stress being cal-culated in the solid zones while pour-ing, it is possible to forecast defects cre-ation and evolution within the mushy and solid shell of cooling metal. This is true from stress and strain birth till the end of complete solidification of the ingot using (11). Other kind of results is the possibility to predict macro sec-ondary piping or shrinkage in case of local lack of exothermic powder for ex-ample. Actually, from the coupling be-tween VP and EVP models, a relevant state of stresses within the metal is pre-dicted. This state yields, after a specific analysis of the localization of the liq-uid areas compared to the solid ones, a criterion providing the opening of the mushy zone of the metal. The sec-ondary shrinkage results from the mass conservation throughout the solidifi-cation of the steel.

A specific study has been launched on small ingot (1,600 kg) casting case in two different situations. The aim of the study was to calibrate exothermic

powder used on the top of the riser. The first case takes an exothermic powder sufficiently calibrated and the second one an exothermic powder too small.

Figure 3 illustrates the distribution of the temperature (on the left) and the so-lidified skin (on the right) of the ingot at the end of the filling. Even though the cases are not the same, this result is in good agreement with Figure 1. That illustrates the fact that solidification be-gins a long time before the end of pour-ing and the amount of solidified mass is significant once the filling is achieved. In addition the influence of the air gap on the temperature evolution during the cooling process is relevant. Indeed, it appears that, in such small ingot, much before the end of filling, air gap is created due to the shrink of the so-lidified skin of the ingot involving non continuous temperature distribution at ingot/mold interface (Figure 4).

Figure 5 shows the global shape of the ingot after 11,400 s of cooling for the two cases (efficient powder on the left, lack of powder on the right). In the first case, it appears that the ingot has a priori no defect. The free surface shape is quite standard. On the right, the picture shows the results of the bad calibration of the exothermic powder, with the internal open shrinkage. The presence of air gap on all the height of the ingot can be noticed for the two cases.The distribution of air gap after 11,400 s of cooling for the two cases is illustrated on Figure 6. It appears that

Figure 7: illustration of the 1st prin-cipal component of the stress ten-sor (tensile state) during cooling. The white lines represent the mushy zone and the black lines, the velocity vec-tors (in the second case).

Figure 8: response of the hot tearing criterion in its porosities application. Standard results showing a low density zone on the central axis of the ingot.

K SIMULATION

10 Casting Plant & Technology 4/2012

the air gap thickness is lower in the sec-ond case. This is issued from the fact that a certain amount of energy and de-formation has been absorbed by the in-ternal shrinkage. In the first case, with no internal shrinkage, the global en-ergy is dissipated through the mold, yielding a bigger air gap thickness.

Figure 7 illustrates the fact that Ther-cast is able to take into account the sol-

id behavior as well as the liquid one at the same time. This result concerns the second case. It shows both tensile stress during solidification and veloci-ty vectors within the liquid area. Loop flows are resulting from standard nat-ural convection movement, following the Boussinesq effect.

The distribution of the mushy zone, indicated by the white lines, results

from the modification of the model (10). It also illustrates the lack of the exothermic powder as the upper free surface is solidifying before the core of the ingot. The defect criterion in its ap-plication of prediction of macro porosi-ties is illustrated in Figure 8. The area of low density in the lower part of the in-got is indicated by the lowest values of the criterion while the macro porosities, present just below the internal shrink-age, are indicated by the highest val-ues. The criterion indicates that odds of getting hot tears are quite low as the maximum values in those cases do not reach the critical threshold. Ingot skin getting solidified rapidly, the cooling metal does not remain in the BTR long enough under tensile stresses to create strain yielding hot tears.

With a chained operation, it is possi-ble to follow how hot tears and porosi-ties evolute while the ingot is forging. Indeed, in open die forging process, dis-tribution of porosities and/or hot tear-ing can be initialized with the results of the casting operation. In such chained processes, the complete history of the defects is reproduced. Yet, to improve the final quality of the product the pa-rameters from the steel making till the forging operations have to be adjust-ed. The accuracy of the model has been validated on an industrial 64 tons in-got. The mold has been equipped by thermo couples. Figure 9 illustrates the

Figure 10: Comparison of the final shape of the ingot calculated and in reality. Note the shape of the free surface that is perfectly matching the real one, re-sulting from the exothermic powder effect of the top of the ingot.

0

Zeit in s

T 15 T 16 T 18 T 19

40000 8000020000 60000

1000

800

600

400

200

0

Tem

pera

tur i

n °C

Figure 9: Comparison of calculated and real temperature evolution on thermo couples in the mold of a 64 tons ingot. Lines are simulation results, plots are measures. On the left, location of the thermo couples in the mold. One the right, temperature evolution throughout the cooling time for the thermo couples 15, 16, 18, 19.

K SIMULATION

12 Casting Plant & Technology 4/2012

comparison between measurement and simulation temperatures. The very good matching between simulation and real-ity issues from the fact that all the com-plex phenomena during the casting are taken into account. The air gap, heat transfer evolution and shrinkage are perfectly estimated by Thercast.

Figure 10 shows the comparison of the final shape of the ingot between simulation and reality. Here the effect and the shape of the exothermic pow-der are perfectly carried out.

ConclusionThercast is industrially used. It allows determining the thermo-mechanical

behavior of the cooling metal in con-tinuous casting processes. Its origi-nal model of treating the solidifying metal, associated to specific boundary conditions leads to forecast accurate-ly the defects of slabs or billets. It per-mits to better understand the impact of process parameters. With such a tool, steel makers are able to control and optimize their process. These differ-ent examples illustrate how nowadays numerical models could be used in the steel industry to improve the quality of production and the productivity.

We would like to acknowledge Ascometal and ArcelorMittal Le Creuzot.

www.transvalor.com

References:www.giesserei-verlag.de/cpt/references

Product brochure on ThercastPicture codes can be scanned by users of smart phones, e.g. with the “Bar-coo” app

http://bit.ly/SI7aWj