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Model
Nicolas Leriche, Hervé Combeau, Charles-André Gandin, Miha Zaloznik
To cite this version:
Nicolas Leriche, Hervé Combeau, Charles-André Gandin, Miha Zaloznik. Modelling of Columnar-to-
Equiaxed and Equiaxed-to- Columnar Transitions in Ingots Using a Multiphase Model. MCWASP
XIV: International Conference on Modelling of Casting, Welding and Advanced Solidification Pro-
cesses, Jun 2015, Awaji island, Hyogo, Japan. pp.012087, �10.1088/1757-899X/84/1/012087�. �hal-
01254330�
Modelling of Columnar-to-Equiaxed and Equiaxed-to-
Columnar Transitions in Ingots Using a Multiphase Model
N Leriche
1, H Combeau
1, Ch-A Gandin
2and M Založnik
11
Université de Lorraine – Institut Jean Lamour, UMR CNRS 7198, 54011 Nancy, France
2
Mines ParisTech – CEMEF, UMR CNRS 7635, 06904 Sophia Antipolis, France
E-mail: herve.combeau@univ-lorraine.fr
Abstract. We present a new method to handle a representative elementary volume (REV) with a mixture of columnar and equiaxed grains in ingot castings in the framework of an Eulerian volume averaged model. The multiscale model is based on a previously established fully equiaxed model. It consists of a three-phase (extra-granular liquid, intra-granular liquid and solid) grain-growth stage coupled with a two-phase (solid and liquid) macroscopic transport stage accounting for grain and nuclei movement.
In this context, we take into account the formation of a columnar structure and its development using a simplified front-tracking method. Columnar solidification is coupled with the growth of equiaxed grains ahead of the columnar front. The particularity of the model is the treatment of concurrent growth of mixed columnar and equiaxed structures only in the volumes that contain the columnar front. Everywhere else, the structure is considered either fully columnar or fully equiaxed. This feature allows for reasonable computational times even in industrial size castings, while describing the solutal and mechanical blocking phenomena responsible for the Columnar-to-Equiaxed Transition.
After a validation of the model, we discuss the numerical results for a 6.2-ton industrial steel ingot by comparison with experimental measurements. Final maps for macrosegregation and grain structures size and morphology are analysed. Furthermore, we quantify the impact of nuclei formation through fragmentation along the columnar front on the result.
An attempt at predicting the occurrence of the Equiaxed-to-Columnar Transition in the later phases of the process is also made.
1. Introduction
The following work is based on a multiphase volume-averaged model developed by Combeau and Založnik, which makes use of a numerical splitting technique [1,2]. Several studies of this research group about industrial ingots have already been presented concerning macro-segregation and grain morphology [3-5]. However, the columnar and equiaxed structures were only distinguished by defining a zone close to the mold where the grains were set to be fixed.
Efforts have recently been made to predict, thanks to multiphase models, the competition in
industrial ingots between a columnar structure growing from the mold and transported equiaxed grains
[6,7]. This multiphase model has been able to reproduce qualitatively the centerline macro-segregation
in a 2.45-ton steel ingot. However, no comparison with CET observation was undertaken and the
description of the equiaxed grain morphology remained simplified.
In the columna Columna zone wh
The o detachm dendrites investiga 2. Descr The pre coworke granular grain str (ocathed In th structure previous morphol trunk is cubic lat columna
F g d The m structure the equia front nee the one behind t equiaxed ℓ of t consider gradient
e work prese ar structures ar-to-Equiax hich leads to a
origin of the ment of grain s. This is the ated in the pr ription of th esent model ers [1,2]. Thi r liquid [8] w ructure can b dral in 3D).
his framewor es and their s models in logical param
schematized ttice with ch ar front in the
Figure 1. Sch grain and its different pha main differe es are propag
axed grains t eds thus to b developed b the columna d grains. In the front is red that the d
in the liqui
ented thereaft from the m xed-Transitio
an Equiaxed equiaxed gr ns formed in e reason why resent model he model
is based o is model has within the gra be schematiz rk, the goal growth conc n the literat meters to mo d by a regula
haracteristic e volume.
hematic repr envelope tog ases.
nce between gated by a fr
take their ori be tracked du y Ludwig an ar front, (ii)
order to pro defined in direction of id ahead of
fter, a comple mold, their i on (CET). Th d-to-Columna
rains in a ste n contact wi y different m
l.
n the fully been recent ain envelopes
ed as in Figu of this mo currently wit ture [10,11, del the colum ar assembly
length .
resentation o gether with t
n the two typ ont which in igin in the d uring solidif nd Wu [13].
contain the opagate the
each averag propagation the front ∇
ete model ha interaction w he columnar
ar Transition eel ingot can ith the mold mechanisms fo
coupled eq tly extended s following t ure 1 where odel is to ta th the develo
12] we cho mnar and eq
of sub-struc The length
of a
the Fig con (red
pes of struct nitially origin dendrite fragm
fication. To t Therefore, e growing fro front further ging volume of the colum
. Co
as been devel with the equ
grains can a n (ECT).
n be attribute d during fill or the format
quiaxed mod to take into the ideas of R
it was chose ake into acco
opment of m ose to use quiaxed struc ctures with a
ℓ gives a
gure 2. Sche ntaining both d, right) equi tures lies in nates at the m ments or nuc this end, we each represe ont or (iii) o
r to neighbo e, similarly t mnar trunks onsequently,
loped, includ uiaxed grain lso grow fro ed to various ing or fragm tion of equia
del develope account the Rappaz and T en to use a sq ount the app moving equiax the same e tures. In Fig a square env access to th
matic of a m h (blue, left) c iaxed grains.
their origins mold and is c clei. The pos
make use of ntative volum only contain oring volume to the proce is aligned w ℓ is simp
ding the grow ns and espec om a packed s causes inclu mentation of axed grains h
ed by Comb presence of Thevoz [9].
quare envelo parition of c axed grains. B
envelope sh gure 2, each c velope, arran he propagatio
mixed volum columnar an .
s. The colum connected to sition of the c
f a method s me can be lo n liquid plus
es, a maxim edure in ref with the loca ply the leng
wth of the cially the equiaxed uding the f existing have been
beau and f an inter-
Thus, the ope shape columnar Based on hape and columnar nged on a on of the
me nd
mnar sub- o it, while
columnar similar to ocated (i) possibly mal length f[13]. We al thermal gth of the
2
volume along the direction of ∇ . The velocity of the front at the columnar tips as well as the growth velocities of both the equiaxed envelopes and the columnar “sub-envelopes” are all calculated according to the LGK model [14].
In our model, a mixed volume containing both columnar and equiaxed structures as the one in Figure 2 is made of six phases. Three phases are associated with the equiaxed grains: the volume fraction of the solid phase as well as the inter- and extra-granular liquid fractions and respectively. Similarly, we can define the variables , and for the columnar structures such that: + + + + + = 1. In order to simplify the calculations, we suppose that the equiaxed grains present in a mixed volume containing the columnar front are fixed and only the liquid phase is allowed to move in these volumes. This approximation is only made for a layer of mixed volumes. The movement of equiaxed grains is accounted for ahead of the columnar front where columnar grains are not present. In the volumes containing only equiaxed structures, two flow regimes are considered. Where the local grains fraction is larger than the packing limit , the solid phase is fixed. The flow of the intragranular liquid is then described by a momentum equation for porous media including a Darcy term for the drag interactions. The permeability of the porous solid is modeled by the Kozeny-Carman law. For local grains fractions smaller than the packing limit
< , the solid phase is considered to be in the form of free-floating equiaxed grains.
In accordance to most volume-averaged models of the literature, two phenomena responsible for the Columnar-to-Equiaxed Transition (CET) are considered:
1) Solutal blocking of the front. Its description is intrinsic to our model. Indeed, when the equiaxed grains grow, they enrich the equiaxed extra-granular liquid and thus reduce the chemical undercooling used to calculate . This liquid medium becomes solutally well-mixed, in which case is negligible and the columnar tips are effectively blocked.
2) “Mechanical” blocking. It occurs when the equiaxed grain fraction in a mixed volume containing the front reaches a critical value . This parameter has been used in previous studies with values ranging from 0.2 [15] to 1 [16]. In the present work, we used = 0.5 according to the geometric criterion proposed by Hunt [17].
At the top of industrial steel ingots, it is common to observe an oriented dendritic structure [3]. It is believed that this structure originates from a packed equiaxed zone when the sources of nuclei and fragments have been depleted [4]. As a consequence, this phenomenon can be assimilated to the restart of a columnar structure during what we call here the Equiaxed-to-Columnar Transition (ECT).
Previously, it was necessary to artificially “renucleate” new equiaxed grains to be able to solidify the top of the ingots even with a non-uniform nucleation law [1]. We propose here a more physical way to deal with these zones according to the ECT idea: when the remaining liquid in front of an equiaxed packed zone is emptied of grains and nuclei, a new columnar structure is initiated in the volumes directly adjacent to the packed zone. These columnar trunks will interact with further remaining equiaxed grains and this will possibly lead to a subsequent CET.
The sources of these equiaxed grains have already be mentioned. However, most of the models
applied to industrial ingots use either a three-parameter model [13] or a simple uniform model [1] to
simulate the volumic heterogeneous nucleation of [ . ] equiaxed grains at
undercooling Δ [ ] . Yet, it remains unclear if there are a significant number of nucleation sites in
non-inoculated alloys [18]. Another potentially important source of grains is through fragmentation of
the columnar dendrites, especially in industrial ingots [19]. That is why the goal of this work is also to
study the effect of a surface injection model at the columnar front. During the growth of the front, we
consider a constant surface flux of equiaxed grains [ . ] which will be added to the
adjacent liquid volumes that do not contain columnar structures. Because the initial sizes of the
fragments are unknown, they will all be fixed to 1 , i.e. the same value as for the grains formed by
heterogeneous nucleation. The only condition for the surface injection is that the quantity of solid in
of the nu Whe
3. Case The pres ingot ha consider concerni measure The i columna cast with material exotherm produces 5 minut remeltin the break Therefor hot-top originati a value Conc average columna only a v other pa study on fraction no pour equal to The n cm. The single In hours.
4. Resu Results f injection [10 ; 10 and to h moveme bordered The colo the exten
en modeling uclei and sup definition a sent study co as a square c r a simplified ing the therm
ment of the m ingot exhibi ar zone, betw
h a hot-top . Shortly aft mic powder s heat by a tes. This ex
g in the hot- kdown of the re, we could and proba ing from an E cerning the
of 1mm w measured ar zone. It is very limited i arameters of
n the same for equiaxed ring superhe
the liquidus numerical do e time step i
ntel Xeon X
lts and discu for volumic n at the colu 0 ] [ .
highlight the ent of the eq d by thick w our map indi nt of the mix
the volumic ppose that the and experim
oncerns a 6.
cross-section d 2D axisym mal boundary
morphologie its along the ween 7 and 1 consisting o ter the begin is released a chemical xothermic re
-top region a e columnar s expect a thi ably very f ECT scenario
new param was chosen w primary d noteworthy impact on th the model a ingot [5]. M d grains was
eat was con temperature omain was d is variable fr X5550 2.66 G
ussion injections o umnar front ] is activat e role of gra quiaxed grain white lines in
cates the equ xed columnar
heterogeneo ey move at th mental param 2-ton steel i n and a sligh mmetric geom y conditions es and macro e mold a rou
1 cm thick. T of a 3 cm th
nning of soli besides the reaction wh eaction caus and is believ structure by f nner column few column meter of th o.
which corres endritic spa that the valu he results in are reported Most notably
= 0.4 nsidered; i.e e = 1475 discretized in
rom 10 Ghz process
f equiaxed g ) is conside ted at the liq ains motion, ns. The black ndicate colum uiaxed grain r/equiaxed zo
ous nucleatio he same velo meters
ingot cast by htly conical metry of equ s, the thermo osegregation ughly unifor The ingot w hick insulatin
idification, a e hot-top an hich lasts f ses significa ved to promo fragmentatio nar zone in th nar structur he model,
sponds to th acing in th ue of ha our case. Th in a previou y the packin 4 , the alloy w
e. the initia 5 ° . The ini nto approxim to 0.1 . Th sor. The cloc
grains are sh ered and onl quidus temp , simulation k thick lines mnar zones w
fraction ones.
on of the grai ocity as the li
y Ascometal shape. In th ual mass whi
o-physical pr can be found rm as
ng an for nd ant ote on. he es he , he ad he us ng
was simplifie al temperatu itial mold tem mately 8600 c
he code is no ck time for
howed in Fig ly a single erature, i.e.
are carried s indicate the which have
in the wh Figure conditi
ins, we accou iquid.
Industries [ he framewor ich is shown roperties as d elsewhere [
ed as a binary ure of the mperature is cells for an a ot parallelize a typical sim
gure 4. No fr class of nuc with ∆ ≈
out (wo) w e CET where formed follo hole ingot, c e 3. Geom ions for the 6
unt for the m
[3]. The expe rk of our m n in Figure 3
well as expe [3-5].
ry Fe-1.01 wt liquid stee set to 25 ° average cell ed and was mulation is
ragmentation clei in t 0 . For co without and reas the dash
owing an EC characterizing metry and b 6.2-ton steel
movement
erimental model, we 3. Details
erimental
t.%C and l, , is . size of 1 run on a about 60
n (surface the range mparison (w) with hed zones CT event.
g notably boundary
ingot.
4
F b h t l g Without zone is n 4 with the top o liquid at solidifica explains leading t then blo complex advancin trapped between the relati ingot. C junction the wedg without be empti predicted assuming An exam composi composi central without
Figure 4. Pr black lines) t have assume the ECT are left map corr grains.
accounting negligible an 0,5 of the ingot. F
t the bottom ation favour why the f to a thinner ocked upward x thermal co
ng front set in a clockw the hot-top ively poorer onsequently, and favour ge-like shap solid movem ied of equiax d in these c g a fixed so mple is sho ition, ̅ , is ition . Th segregation solid movem
redicted map the CET for ed no nuclea shown as ha responds to
for grain mo nd the ingot i . For lower g For = 10 m of the ing rs the growth front is bloc columnar zo ds in the ing onditions and up an enric wise convect and the mo r and hotter l
, higher ther the columna pe of the fro ment a com xed grains o cases. The m
lid are simil own in Figu
normalize he dashed cu
profile fo ment. Except
ps of the fina increasing n ation underc atched zones fixed solid p
otion and for is predicted t grain densitie got near the .
h of equiaxe cked at the one. The colu
got. In the h d the interac ched, cold liq
tion cell. At old, this first liquid from t rmal gradient ar structure w nt there. Co mputational v or nuclei, no macrosegreg lar to previo ure 5. Here ed with t
urve represe or = 10
for a negativ
al equiaxed g nuclei densit cooling. The s bordered by phases and th
r values of to have a full
es, we notice , sweepin beginning ed grains. Th
bottom firs umnar front hot-top regio ction with th
quid which t the junctio t liquid mee the core of th
ts arise at th which explain onsidering th volume cann ECT event gation profil ous results [5
e the averag the nomin ents the axi ve segregatio .
grain fraction ties from 10
columnar z y white lines he (w) right
greater than ly equiaxed e a roughly u
g of colder y his of
st, is on, he on is ets he his ns not hat es is 5]. ge nal ial
on
Figure central where
n as w
to 10 [ zones predict
s. For each c map to mov
n 10 structure. Th uniform colu yet not signif
e 5. Compa l segregation
= 100
well as (thic . ] . W ted followin case, the (wo ving equiaxe
. , the c his is shown umnar zone, ficantly enric
arison of t n with measu
̅
[%]
ck We ng ed o)
columnar in Figure except at ched
the axial urements,
].
at the bottom of the ingot and an inversion at the top, a strong positive segregation is predicted which is not in agreement with the experimental results. For other values of , the results without solid movement are qualitatively similar. Especially, the position and height of the main positive segregation are the same, even though the intensity of this segregation can differ.
When taking into account the movement of the equiaxed grains, we notice a very dissimilar repartition of the structures in Figure 4. Starting from = 10 . , we see a clear distinction between a lower zone where the columnar thickness is very thin and an upper part which is entirely columnar. As increases, the equiaxed zone becomes larger and its boundary with the columnar zone moves upward. The top of the ingot always remains columnar. This can be explained by the fact that the motion of the equiaxed grains is controlled primarily by sedimentation. The grains first appear in the vicinity of the columnar front where the undercooling is maximal. Then, the grains grow rapidly and slide down along this region, thus participating to the formation a packed layer at the bottom of the ingot. Because of liquid convection, the whole ingot quickly becomes undercooled (except the top part) and the majority of the nuclei are activated. The newly formed grains directly fall at the bottom of the ingot and favour the growth of the packed layer. The local packing times (the times it takes for the grain in the equiaxed zone to become locally packed) are short and the sedimentation ends at ≈ 300 in all cases. This value is to be compared with the total solidification time for the ingot, ≈ 9000 .After that, virtually no equiaxed grains remain in the liquid as evidenced by the negligible values of in the upper part of the ingot. Once the equiaxed grains have packed and nearly all the nuclei are activated, the columnar structure is free to develop in the remaining liquid. Remarkably, an ECT takes place at the top of the packed layer. The columnar structures originating from this ECT subsequently meet those growing from the mold. This is seen in Figure 4 with = 10 [ . ] where a dashed blue region and a plain blue region are connected. The segregation profiles along the axis are shown in Figure 5 for three different densities of nuclei. The results are similar to previous calculations [5]. The best fit of the experimental results is found for = 10 . but fails predicting the CET and is still not satisfying when considering the measured segregation profile.
The results presented in Figure 5 remain qualitatively valid when considering different nucleation undercoolings ∆ . A three parameter heterogeneous nucleation model [20], based on a Gaussian distribution, has also been tried. Comparable results were obtained, albeit with a more cone-shaped equiaxed zone.
It can be concluded that a volumic injection of nuclei fails to predict the observations. The volumic source of nuclei can be connected with heterogeneous nucleation or grains formed at the mold surface during the filling stage and then detached. It is already known that the fragmentation of dendritic columnar structures is another important source of equiaxed grains in industrial castings [19]. Thus, we investigate the influence of fragmentation on the results. The surface injection model mentioned earlier is now used for different flux values [ . ] at the columnar front. The results are summarized in Figure 6 where no volume densities of nuclei were considered for these calculations. All calculations include grain movement as simulations with fixed solid do not produce realistic results. This stems from the fact that the fragments appear only in the liquid besides the front and need to be carried by the liquid to fill the centre of the domain. It is first noticeable that for = 10 . . , the equiaxed zone has a much more pronounced cone shape than in previous volumic injection scenario.
The fraction of equiaxed grains entrapped in the columnar structures is also not negligible and the predicted structure can become considerably mixed especially at the top of the ingot, where remelting takes place. When the fragmentation flux is increased, a thin columnar zone is predicted in the lower part of the ingot whereas there is still a thick columnar zone in the upper part of the ingot.
Nevertheless, the columnar zone at the bottom is not negligible with a CET at about 5 cm from the mold providing that ≥ 4.10 . . . The upper part is not entirely columnar in contrast with the case with the volumic density of nuclei scenario and the front is blocked at mid-radius. The
6
columna that the present i
Figu fract for colu
The C of the eq place, pr These fir the quick columna fragmen the ingot in about top). As contrary the core The m As we c volumic positive thick.
5. Summ Modellin resulting different Simu model or
ar zone direc columnar st in the hot-top
ure 6. Pred tion
increasing umnar front.
CET predicte quiaxed grai roducing new rst grains wi k build-up o ar region. Th
ts as well as t with a norm t 1300 s expl s evidenced y to the previ
of the ingot.
macrosegreg can see, the nucleation m segregation
mary and co ng of the co g CET and E ulations were t origins of r the injectio
ctly besides t tructures can p [3].
dicted final as well as th fragmentatio
ed which fits ins. As soon w equiaxed g
ll not only c of a vertical p
his build-up s the new one
mal successio laining why
in Figure 6, ious nuclei d . ation profile results at th model. In th
is found. Th
onclusions olumnar stru
CT have bee e carried out the equiaxe on of fragme
the hot-top i n re-melt du
equiaxed he CET and on fluxes a
s better the ex n as the colu
grains which ontribute to t packed zone p takes place
es coming fr on of horizon
the columna , there are f density mode es at the centr
he bottom o he top half of his correspon
ucture in ing en implement
for a 6.2-ton ed grains we ents in the liq
is thinner be ue to the re-
grain ECT at the
F s
f xperiment ca umnar structu h will at first
the growth o e next to the e in about rom the fron
ntal layers. T ar zone is th few columna el. Only disp
reline are rep of the ingot
f the ingot t nds roughly t
got casting, nted in a form n industrial s ere studied:
quid ahead o
ecause of the -heating indu
Figure 7. Co segregation w
= 100 fluxes.
an be explain ures appear t instantly se of a flat pack front whose 200 s and c t in the uppe The whole in hicker in the
ar structures persed spots o
ported in Fig are compara the results de to the positio
its interacti merly purely
steel ingot ca either a cla of the colum
e thermal co uced by the
omparison of with the mea
̅
