Modeling the columnar-to-equiaxed transition with a three-phase Eulerian approach

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Materias Science and Engineering A 413 414 (2005) 109 114 Modeing the coumnar-to-equiaxed transition with a three-phase Euerian approach Andreas Ludwig, Menghuai Wu Christian-Dopper-Lab for Mutiphase

18, A-8700 Leoben, Austria Received in revised form 1 Juy 2005 Abstract A three-phase Euerian approach is deveoped to mode the coumnar-to-equiaxed transition (CET) during soidification.

1 Materias Science and Engineering A (2005) Modeing the coumnar-to-equiaxed transition with a three-phase Euerian approach Andreas Ludwig, Menghuai Wu Christian-Dopper-Lab for Mutiphase Modeing of Metaurgica Processes, Department of Metaurgy, University of Leoben, Franz-Josef-Str. 18, A-8700 Leoben, Austria Received in revised form 1 Juy 2005 Abstract A three-phase Euerian approach is deveoped to mode the coumnar-to-equiaxed transition (CET) during soidification. The three phases are the parent met, the soidifying coumnar dendrites and the soidifying equiaxed grains. They are considered as spatiay interpenetrating and interacting continua. We sove the conservation equations of mass, momentum, species and enthapy for a three phases. Additionay we define and sove an additiona transport equation for the number density of equiaxed grains which aso accounts for grain nuceation. Diffusion controed growth for both coumnar and equiaxed phases, drag forces, species partitioning at the soid/iquid interface, heat of fusion, etc. are taken into account with the corresponding cosure aws. A binary stee (Fe 0.34 wt.% C) ingot casting as benchmark was simuated to demonstrate the mode potentias. Preiminary resuts of the mixed coumnar and equiaxed soidification incuding the motion of the coumnar tip front, the occurrence of the CET, the formation of macrosegregations, and the resuting met convection and grain sedimentation and their infuence on the fina macroscopic phase distribution are presented Esevier B.V. A rights reserved. Keywords: Soidification; Mutiphase; Coumnar-equiaxed transition; Macrosegregation; Sedimentation; Convection 1. Introduction In recent years, the authors deveoped two two-phase modes, one for equiaxed soidification [1 4] and another for hypermonotectic soidification [5 7]. Both modes are based on an Euerian Euerian approach. This paper reports about the extension of the two-phase equiaxed soidification mode to a threephase mode for a mixed coumnar and equiaxed soidification with consideration of the so-caed coumnar-to-equiaxed transition (CET). With an additiona (coumnar) phase, one more set of conservation equations for mass, enthapy and species must be soved. No additiona momentum equation is soved, as the coumnar phase is considered to be stationary. The most chaenging point is the appropriate definition of the cosure aws for the phase exchanges and interactions: e.g. the competitive growth of both soid phases, and the mechanica interaction between them. The equiaxed phase which is aowed to nuceate Corresponding author. Te.: ; fax: E-mai address: menghuai.wu@notes.unieoben.ac.at (M. Wu). everywhere in buk met, grows and moves freey. The stationary coumnar phase is regarded to start from the mod wa and grows thus preferentiay aong the heat fow direction. During soidification, both phases grow competitivey. A socaed coumnar-to-equiaxed transition (CET) occurs at the end of soidification when the growing coumnar dendrite tips are bocked by equiaxed grains. Eary studies reported two bocking mechanisms: one is by mechanica bocking when the oca voume fraction of equiaxed grain enveopes exceeds a certain imit [8], the other is by soft-bocking when the oca constitutiona undercooing disappears [9,10]. The presented three-phase mode tracks the coumnar tip front expicity, and incudes both of the above-mentioned CET mechanisms. However, in our work the impact of the met convection and equiaxed grain transport on the occurrence of CET is aso considered. 2. Mode 2.1. Genera assumptions (1) Three phases are defined: primary iquid phase (), equiaxed phase (e), and coumnar phase (c). The corresponding phase /$ see front matter 2005 Esevier B.V. A rights reserved. doi: /j.msea

2 110 A. Ludwig, M. Wu / Materias Science and Engineering A (2005) voume fractions f, f e and f c are subject to f + f e + f c =1. Both, the primary iquid phase and the equiaxed phase are aowed to move. The coumnar phase is assumed to stick to the wa, and soidify from the wa towards the buk met aong the heat fow direction. No momentum equation for the coumnar phase is soved. (2) Idea morphoogies for both soid phases are assumed: spherica for the equiaxed (gobuar) grains and cyindrica for the coumnar (ceuar) primary dendrites. (3) The grain size of the equiaxed grains is expicity cacuated, whie a constant vaue for the primary dendrite arm spacing is assumed for the coumnar phase. (4) The Boussinesq approach is used to mode thermo-souta convection, grain sedimentation and sedimentation-induced met convection. The voume shrinkage is ignored. (5) Grains created or brought into the mod during fiing, and fragmentation (or segmentation) of the dendrites are not modeed Nuceation of equiaxed grains and grain transport The number density of equiaxed grains (m 3 ), n, is cacuated with t n + ( u en) = N e (1) Here, u e is the (voume averaged) veocity of the equiaxed phase. The nuceation rate (m 3 s 1 ), N e, is modeed with a threeparameter heterogeneous nuceation aw [1,2] Mass conservation and grain growth kinetics Mass conservation equations are t (f ρ ) + (f ρ u ) = M e + M c, (2) t (f eρ e ) + (f e ρ e u e ) = M e + M ce, (3) t (f cρ c ) + (f c ρ c u c ) = M c + M ec, (4) where ρ, ρ e, ρ c, are the densities and u, u e, u c the veocities of the different phases. The source terms M c (= M c ) is the mass transfer rate (kg/m 3 /s) from iquid to coumnar phase, and M e (= M ce ) from iquid to equiaxed phase by soidification (positive) or meting (negative), and M ce (= M ec ) from coumnar to equiaxed phase by the mechanism of fragmentation (positive) or by attaching (negative). The fragmentation is ignored, so we chose M ce =0. Diffusion controed grain growth kinetics at the micro scae is considered. For equiaxed soidification an idea spherica morphoogy is considered so that the grain growth veocity in the radius direction can be soved anayticay [11] v Re = dr e dt = D R e c c c c s = D R e (1 k) ( 1 c c ) (5) Here, c and cs (c s = kc ) are the equiibrium concentrations adjacent to the soid/iquid interface, and from the phase diagram c = (T T f )/m yieds. D is the diffusion coefficient in the iquid and R e = d e /2 the radius of the grain. With Eq. (5), we can define the voume averaged mass transfer rate for gobuar equiaxed soidification by considering the tota surface area of the spherica grains and the infuence of pingement by an Avrami-factor f M e = v Re (nπd 2 e )ρ ef (6) For coumnar soidification two different cases are distinguished: (i) tip regions and (ii) growing coumnar trunks behind the tip region. We trace the tip front of the coumnar grains by a method described in Section 2.8. For the voume eements which have aready been pasted by the tip front a diffusion controed growth mode around cyindrica dendrite trunks is used. With the simiar anaytica method of [11], the growth veocity in radia direction of such a cyindrica trunk is approximated by v Rc = dr c dt = D c ( c R c c cs n 1 Rf R c ), (7) where now R c = d c /2 is the average radius of a cyindrica dendrite trunk and R f = λ 1 /2 is haf of the primary dendrite spacing λ 1. So we can define the voume averaged mass transfer rate for those voume eements by considering the tota surface area of coumnar dendrite trunks per voume S A = πd c /λ 2 and an Avrami-factor f ( ) πd c M c = v Rc ρ c f (8) λ 2 For the eements containing the growing coumnar tips, the mass transfer rate M c for coumnar soidification is written by considering both the tip growth veocity, v tip, and radia growth veocity, v Rc M c = v Rc n c (πd c )ρ f + v tip n c (πr 2 tip )ρ f (9) The first term on the eft hand side of Eq. (9) denotes the mass transfer rate due to the growth in radia direction and the second term that of the growth in tip direction. n c = 4f c /(πdc 2 ) is the number density of the coumnar trunks. The dendrite tip veocity v tip and the tip radius R tip are cacuated according to [11,12] Momentum conservation The momentum equations for the parent met and the moving equiaxed phase are t (f ρ u ) + (f ρ u u ) = f p + τ + F B + U c + U e, (10) t (f eρ e u e ) + (f e ρ e u e u e ) = f e p + τ e + F Be + U e + U ce, (11)

3 A. Ludwig, M. Wu / Materias Science and Engineering A (2005) where τ and τ e are the stress strain tensors. With the Boussenisq approximation, the therma-souta buoyancy force F B acting on the iquid and the buoyancy force F Be acting on the free moving equiaxed grains are appied [4]. Each of the momentum exchange terms U e, U c and U ce incudes two parts: the part due to phase transformation and the part due to drag force, for exampe, U e = U p e + U e d and U c = U p c + U c d. Detais to treat these momentum exchange term between the iquid phase and the equiaxed phases are described in [1,2]. The same idea is used for the momentum exchange between the iquid and the coumnar phase. However, we cacuated the iquid-coumnar drag force by considering a mushy zone permeabiity as proposed in [13]. For the momentum exchange between the coumnar and the equiaxed phase a simpe approach is used. We assumed that when the oca voume fraction of the coumnar phase is more than a critica vaue of fc free = 0.2, an infinite drag force coefficient between both soid phases appies and thus the equiaxed grains are captured. When the voume fraction of the coumnar phase is smaer than this critica vaue, no drag force between both soid phases exists and thus the motion of the equiaxed grains is not affected by the coumnar front. The choice of the critica vaue for fc free = 0.2 is here arbitrary. Therefore, further parameter study on the infuence of this vaue on the fina resut is necessary Species conservations The voume averaged concentration c in the iquid phase, c e in the equiaxed phase and c c in the coumnar phase are obtained by soving the species concentration equations: t (f ρ c ) + (f ρ u c ) = (f ρ D c ) + C c + C e, (12) t (f eρ e c e ) + (f e ρ e u e c e ) = (f e ρ e D e c e ) + C e + C ce, (13) t (f cρ c c c ) + (f c ρ c u c c c ) = (f c ρ c D c c c ) + C c + C ec, (14) where D, D e, D c are the diffusion coefficients. The diffusive species exchange at the phase interface is ignored, but the species partitioning at the phase interface during soidification (and meting) is taken into account [1,2]. As we ignore any phase exchange between equiaxed and coumnar phases, we set C ce Enthapy conservation We sove the enthapy conservation equation for each phase t (f ρ h ) + (f ρ u h ) = (f k T ) + Q c + Q e, (15) t (f eρ e h e ) + (f e ρ e u e h e ) = (f e k e T e ) + Q e + Q ce, (16) t (f cρ c h c ) + (f c ρ c u c h c ) = (f c k c T c ) + Q c + Q ec, (17) where the enthapies are defined via h = T T ref c p() dt + h ref and h e = h c = T e T ref c p(s) dt + h ref e with specific heat of the iquid c p() and the soid phase c p(s). T ref and h ref e are defined so that the enthapy difference between iquid and any soid, h h e and h h c, is equa to the atent heat of fusion. Further detais to treat the atent heat can be found in previous pubications [1,2]. By soving Eqs. (15) (17), three different temperatures are obtained, T, T e and T c. The therma equiibrium condition (T T e T c ) is satisfied by appying a quite arge voumetric heat transfer coefficient of 10 8 (W m 3 K 1 ) between the phases Auxiiary quantities A mixture concentration for the description of macrosegregations, c mix, is defined by c mix = c ρ f + c e ρ e f e + c c ρ c f c. (18) ρ f + ρ e f e + ρ c f c The average diameter of the equiaxed grains, d e, is cacuated from the foowing reation f e = n 4π ( ) 3 de. (19) 3 2 The average diameter of the coumnar dendrite trunks, d c,is cacuated by reating the cross section area of a singe coumnar trunk, π(d c /2) 2, to the maxima avaiabe area of each trunk when the dendrites are ranked in hexagon. Thus, we get f c = 3 dc 2 4 λ 2. (20) In the case the dendrites are ranked in square, the prefactor of Eq. (20) woud be π/4 instead of 3/4. In this paper, the average dendrite arms spacing, λ, is assumed to be constant and given. For the resuts presented in this paper we took λ = 1 mm Coumnar tip front tracking and tip front bocking mechanism The coumnar tip front tracking is based on the assumption that coumnar dendrite trunks grow from the wa into the buk met. However, no growth-preferred crystaine orientation is considered. (1) Each contro voume is indexed with a coumnar status marker, i c, which indicates whether a contro voume contains the coumnar tip front (i c = 1); coumnar dendrite

112 A. Ludwig, M. Wu / Materias Science and Engineering A 413 414 (2005) 109 114 trunks (i c = 2); or no trunks or tips (i c = 0).

4 112 A. Ludwig, M. Wu / Materias Science and Engineering A (2005) trunks (i c = 2); or no trunks or tips (i c = 0). A contro voumes are initiaized with i c = 0, except the boundary (wa) eements where i c =1. (2) For each contro voume an equivaence cyinder is considered with a radius of ref and a height of ref. The voume of the cyinder is chosen to be equa to that of the corresponding contro voume: πref 3 = V. As no preferred crysta growth orientation is predefined, the equivaence cyinder is thought to be orientated parae to the oca heat fow direction. (3) The coumnar front grows parae to the equivaence cyinder with a growth veocity, v tip, which is determined from the LGT mode [11,12]. The actua position of the front is tracked by evauating the integra = t v tip dt, starting as soon as the front first enters the contro voume. (4) For > ref the coumnar tip front has grown out of the equivaence cyinder. In this case a neighboring contro voumes which are sti empty (i c = 0) wi be reached by the front. Thus, the coumnar status markers of these voumes are set to i c = 1, whereas the marker of the considered voume is set to i c =2. (5) A mass transfer from the iquid to the coumnar phase is ony considered for those contro voumes i c 0. (6) In order to mode the mechanica bocking mechanism by Hunt [8] the tip growth veocity is set to zero, v tip 0, as soon as the oca voume fraction of equiaxed, f e, increases over the critica threshod of f e,cet = This event defines what is known in the iterature as CET. The above described procedure for the CET automaticay eads to the so-caed soft-bocking mechanism proposed by Martorano et a. [10], asv tip vanishes when the oca constitutiona undercooing disappears Numerica impementation The conservation equations are numericay soved with the contro-voume based finite difference CFD software FLUENT 6.1. The cosure aws are impemented as user defined functions. The FLUENT formuation is fuy impicit, hence there is no stabiity criterion for the time step, t. However, the time steps used in practice impact on the accuracy of the cacuation, thus the reiabiity of the numerica resut. The t must be determined empiricay by test simuations. For the benchmark simuation in Section 3 a time step of t =10 4 swas used to start the simuation, and ater change to 10 3 s. For each time step, up to 60 iterations were necessariy to decrease the normaized residua of c, c e, c c, f e, f c, u, u e, p and n beow the convergence imit of 10 4, and h, h e, and h c beow that of Benchmark The soidification of a binary stee (Fe 0.34 wt.% C) ingot with a reativey sma size (diameter: 66 mm, height: 170 mm) was simuated (see Fig. 1). An axis-symmetrica simuation is made. The grid used consists of 690 eements with a mean size of 6 mm 2. The ingot is fied instantaneousy with temperature Fig. 1. Schematic of the simuated benchmark (average grid size 2.5 mm 2.5 mm). of 1785 K. A properties and parameters used for the simuation are isted in Tabe 1 and Fig. 1. Fig. 2 shows the soidification sequence of the ingot casting. Soidification starts as soon as the temperature drops down beow iquidus ( K). At 5 s, the equiaxed grains start to sink, and induce met convection. The met is dragged by the sinking grains downwards aong the wa and then rises again in the midde of the casting. Two symmetrica vortices form. In addition to the grain-sedimentation-induced met convection, therma and soute buoyancy aso drives met convection. The met near the Tabe 1 Properties and parameters used in simuation Therma physica Therma dynamics Process parameters c p() = c p(s) = J kg 1 K 1 k = n max = m 3 D = m 2 s 1 m = K T N =5K D e = D c = m 2 s 1 T f = 1811 K T =2K h ref e = J kg 1 Γ = mk h f = h ref k = k e = k c = W m 1 K 1 β T = K 1 β c = wt.% 1 ρ = ρ e = ρ c = 7027 kg m 3 ρ = 294 kg m 3

A. Ludwig, M. Wu / Materias Science and Engineering A 413 414 (2005) 109 114 113 Fig. 2. Soidification sequences. Both f c and f e are shown with 60 gray eves.

5 A. Ludwig, M. Wu / Materias Science and Engineering A (2005) Fig. 2. Soidification sequences. Both f c and f e are shown with 60 gray eves. The maximum and minimum vaues of them are given. The arrows of both veocities are ineary scaed starting from zero to the maximum vaue. The coumnar tip front with soid ine overaps the quantities f c and f e. wa reveaed a ower temperature and is thus heavier than the buk met (β T > 0) and enhance the fow. On the contrary, the effect of soute buoyancy is reverse compared with therma buoyancy. The met near the mod is enriched with soute, thus is ess dense (β c > 0) and so tends to rise. From the fow pattern of Fig. 2, it becomes obvious that the joint effect of therma buoyancy and grain-sedimentation-induced fow dominates the overa convection pattern. Sedimentation, of course, infuences the distribution of equiaxed grains. The equiaxed grains sink down, and sette at the bottom region, where the voume fraction of the equiaxed phase reaches a quite high eve, i.e. 39% at t =5s. In the course of further cooing, the voume fraction of the coumnar phase at the mod wa increases. In addition, the equiaxed grains continue to nuceate, sink and grow. They sette and pie up in the ower region of the ingot. At t =20s, the voume fraction of coumnar phase near the wa reaches 97% and the voume fraction of equiaxed phase in the ower part of the ingot reaches 78%. In the mean time the coumnar tip front moves inwards. As described in Section 2.4, it is assumed that the equiaxed grains can freey move unti the voume fraction of coumnar dendrite trunks reaches 20%, when they are bocked and incorporated in the coumnar front. At t = 60 s, the coumnar tip front in the midde of the casting meets. Therefore, two cosed coumnar tip front ines are seen: one in the upper region where the soid fraction is sti ow, and a second in the ower part of the casting, where the soidification is neary competed by equiaxed grains (f e 0.99). Here the coumnar tip front has aready been bocked by the presence of many equiaxed grains. The coumnar tip front ine in the upper part is sti abe to move. At t = 90 s the soidification of the whoe casting is amost finished. The coumnar tip front in the upper part of the casting has disappeared. However, in the ower part of the casting the coumnar tip front sti remains. This remaining coumnar tip front ine indicates the position of the CET. Within the CET ine ony equiaxed grains exist, whie out of the ine both coumnar phase and equiaxed phases coexist. The macrosegregation is shown in Fig. 3. A cone shaped negative segregation is predicted in the ower part of the ingot, where high sedimentation rate occurs. It is numericay evidenced that the mechanism of the sedimentation of equiaxed

114 A. Ludwig, M. Wu / Materias Science and Engineering A 413 414 (2005) 109 114 current unti the whoe casting had soidified and the met fow disappeared.

6 114 A. Ludwig, M. Wu / Materias Science and Engineering A (2005) current unti the whoe casting had soidified and the met fow disappeared. The modeing resuts described above reproduce the sense, which was described by Campbe [14] based on the understanding to cassica experiments. The positive segregation at the top region of the ingot can be expained by the convection of the segregated met in buk region. This kind of positive segregation coincides with the eary experimenta resuts of Campbe [14] and Nakagawa and Momose [15]. Finay, it must be mentioned that channe segregations, which are frequenty found in stee ingots, are not predicted with the recent mode. The reason for that is that meting was not taken into account in this simuation and that the used numerica grid is too coarse. 4. Summary A three-phase Euerian soidification mode is deveoped, and preiminariy used to simuate a reduced stee ingot. The resut has shown the potentia of the mixed coumnar and equiaxed soidification incuding CET. The simuated soidification sequence, the sedimentation of the equiaxed grains, the movement of the coumnar tip front and the fina macroscopic phase distribution fit to the widey accepted expanations of experimenta findings, as summarized by Campbe [14]. However, no quantitative evauation was made yet. We rather suggest that further comprehensive parameter studies are necessary in order to anayze the importance of the different modeing assumptions. Fig. 3. Predicted macrosegregation c mix. The quantity of c mix is shown with both isoines and gray scae: ight for negative segregation and dark for positive segregation. The vaues accompanying c mix isoines are in unit of percentage. The CET (back ine) is aso drawn together with c mix. grains is responsibe for the negative segregation. The soute poor equiaxed grains pie up at the bottom of the ingot, and the soute rich residua met rises. The c mix distribution profies are approximatey simiar to the CET profie. The positive segregation zone which forms at the top part of the ingot is mainy due to met convection. The soute rich met rises as the equiaxed grains sink. The soute redistribution in the met is strongy dependent on the met convection pattern. As shown in Fig. 2, two symmetrica met vortices occur in the ingot. In the casting center the fow current transport soute rich met from the bottom region towards the top. As the met hits the casting top, it diverges into two side streams, resuting in a eft- and a right-hand side region enriched with soute eements. Obviousy, the positive segregated zones are not stationary during soidification, they move with the fow References [1] A. Ludwig, M. Wu, Meta. Mater. Trans. A 33 (2002) [2] M. Wu, A. Ludwig, A. Bührig-Poaczek, P.R. Sahm, Inter. J. Heat Mass Transfer 46 (2003) [3] M. Wu, A. Ludwig, Adv. Eng. Mater. 5 (2003) [4] M. Wu, A. Ludwig, J. Luo, Mater. Sci. Forum (2005) [5] M. Wu, A. Ludwig, L. Ratke, Mode. Simu. Mater. Sci. Eng. 11 (2003) [6] M. Wu, A. Ludwig, L. Ratke, Meta. Mater. Trans. A 34 (2003) [7] A. Ludwig, M. Wu, M. Abodano, L. Ratke, Mater. Sci. Forum (2006) [8] J.D. Hunt, Mater. Sci. Eng. 65 (1984) [9] C.Y. Wang, C. Beckermann, Meta. Mater. Trans. A 25 (1994) [10] M. Martorano, C. Beckermann, Ch.-A. Gandin, Meta. Mater. Trans. A 34 (2003) [11] W. Kurz, D.J. Fisher, Fundamentas of Soidification, Trans Tech Pubications, Aedemannsdorf, Switzerand, [12] J. Lipton, M.E. Gicksman, W. Kurz, Mater. Sci. Eng. 65 (1984) [13] J.P. Gu, C. Beckermann, Meta. Trans. A 30 (1999) [14] J. Campbe, Castings, Butterworth Heinemann Ltd., Oxford, [15] Y. Nakagawa, A. Momose, Tetsu-to-Hagane 53 (1967)

Identification of macro and micro parameters in solidification model

BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vo. 55, No. 1, 27 Identification of macro and micro parameters in soidification mode B. MOCHNACKI 1 and E. MAJCHRZAK 2,1 1 Czestochowa University