A model for grain growth based on the novel description of dendrite shape

Transcription

1 ARCHIVES of FOUNDRY ENGINEERING Publihed quarterly a the organ of the Foundry Commiion of the Polih Academy of Science ISSN ( ) Volume 7 Iue 4/ /4 A model for grain growth baed on the novel decription of dendrite hape O. Wodo*, N. Sczygiol Faculty of Mechanical Engineering and Computer Science, Czetochowa Univerity of Technology, Armii Krajowej 21, Czetochowa, Poland *Correponding author. addre: olga.wodo@ici.pcz.pl Received ; accepted in revied form Abtract We ue novel decription of dendritic hape in the micro olid phae growth model. The model decribe evolution of both primary olid olution dendrite and eutectic that form between arm and grain in the lat tage of olidification. Obtained reult how that our approach can be ued in grain growth model to determine more reliable eutectic ditribution. In the paper no kinetic connected with the eutectic tranformation i taken into account. However, thi doe not affect the eutectic ditribution becaue at the beginning of eutectic reaction all liquid phae wa aumed to fully tranform into eutectic. Reult for olid phae growth model baed on thi decription are preented. The obtained reult of eutectic ditribution are epecially important in the hypoeutectic alloy olidification cae, where the eutectic grain grow between formed olid olution grain. Thu, the ditribution of olid olution grain become crucial due to it influence on the delay in olid fraction increae of eutectic grain. Keyword: Solidification Proce, Application of Information Technology to the Foundry Indutry, Dendritic Crytallization, Simple Geometry Grain Growth Model. 1. Introduction Solidification i one of the manufacturing procee for many material, epecially for alloy. The final propertie of cating depend on the microtructure formed during thi proce. Thu, control of ome quantitie on grain cale i eential in modeling of cating. The olidification of hypoeutectic alloy involve following phenomena: nucleation and growth of primary dendrite and eutectic. Dendrite are probably the mot often oberved type of tructure in olidification of cat alloy, thu many imulation method have been propoed, both determinitic [1,2] and tochatic [3,4]. Alo eutectic olidification ha received coniderable attention from reearcher. However, the two-tage olidification, a can be oberved in hypoeutectic alloy, till need both experimental and theoretical tudy. In thi paper we preent how a novel decription of the hape of equaixed dendrite can be integrated with the micro olid phae growth model [5]. Thi method belong to the imple geometry grain growth model [6]. Model form thi group ue phere, cube or octahedron to decribe the grain hape [7,8]. Our decription i intermediate between the pherical and the real hape of dendrite with different-order arm. The paper i organized a follow. We tart from giving ome detail of the propoed hape decription. Next, we provide a brief recall of the olid phae growth model. Reult of imulation are preented and dicued in Section 4. In the lat ection concluion and poible extenion are preented. ARCHIVES of FOUNDRY ENGINEERING Volume 7, Iue 4/2007,

2 2. Model decription The aim of the paper i to how the application of novel decription for the dendrite hape in the grain growth model propoed in [5]. Thi model belong to the group of imple geometry grain growth model. One important feature of the choen grain growth model i that it can be eaily coupled with the macrocopic heat flow computation [9]. The grain in original model i aumed to be a phere, in which the dendrite with different-order arm i growing. The olid phae i approximated by a phere a well. The volume of liquid between different-order arm i approximated by volume of pace between the olid phae phere and the grain phere. In Fig. 2(a) region within conidered volume element are preented. Becaue in the model phere have been utilized to decribe hape of both grain and olid phae, we decided to overcome thi implification partly. To achieve thi, we have applied the following decription of dendrite: R( θ, φ) = 0KL (1 A F in(2θ ) ) (1 A in(2φ ) ) θ = 0K2π, φ = 0Kπ 2 in (2φ ) φ (0, π 4) (3π 4, π ) F = 1 φ ( π 4,3π 4) where L i the main arm length and A i hape parameter. In the Fig. 1 example hape i preented. The length of main arm i equal to 1 and the A-hape parameter equal 0.4. For the ake of clarity we alo preented one eighth part of dendrite. Three region within the conidered volume element can be conidered (ee Fig. 2): olid phae, no back-diffuion i conidered (labeled (1)), interdendritic liquid phae, aociated with the liquid between the higher-order dendrite arm, where complete mixing i aumed and the olute concentration i equal to the olid/liquid interface olute concentration (labeled (2)), liquid phae outide the grain, where diffuion i conidered and diffuion equation i olved (labeled (3)). a) b) c) Fig. 1. The hape of dendrite for ued decription (a) whole dendrite and one-eighth part In the model we conider the volume element in which one equiaxed, dendritic grain grow. The ize of the volume element i computed uing the intanteou nucleation 1 3 model( R tot = ( 4 3π n) ). The nucleation begin when the given undercooling i reached and all grain nucleate. The number of nucleu i computed according to the grain denity, n. Within the volume element the homogenou temperature i aumed, which i valid for element with mall Biot number. Fig. 2. Region within conidered volume element 184 ARCHIVES of FOUNDRY ENGINEERING Volume 7, Iue 4/2007,

3 In Fig. 2 another three region can be ditinguihed: the olid phae, grain and liquid phae outide the grain. If we aume that any fraction i computed with repect to the total volume of conidered region, we can define the following fraction: grain fraction f g = Vg Vtot, olid phae fraction f = V Vtot, internal fraction f i = f f g. Here, Vg i the volume of the grain, V i the volume of the olid phae and V tot i the volume of the conidered volume element. In Fig. 2 comparion of variou hape i given. A can be een, when improved hape function i applied the main arm of the dendrite are in contact with the grain urface. Thi agree with aumption about modeled phenomena. Our new hape function require an additional parameter, that i hape parameter, A. However, it can be eaily computed on the bai of the olute and heat balance. Another modification of the model can be made (ee Fig. 2(c)). The hape of the grain can differ from the pherical one and can be decribed by propoed hape function. A a reult value of two hape parameter are required. The conequence of uch modification i hrinkage of the grain region and extenion of the liquid region outide the grain. Thi, however doe not permit to conider diffuion proce in radial coordinate. Moreover, another variable Ag, which i a hape parameter of grain, i introduced into the model. Such modification can be intereting when more accurate hape of the grain i deired, but it make computation more complex. Therefore, in the reminder of the paper the hape of region hown in Fig.2(b) i utilized. Within the volume element three region are ditinguihed according to the olute tranport. We reformulated the olute balance and write it uing the volume intead of radiu. In thi way decription of the olid phae can be ued without ubtantial model reformulation In the liquid outide the growing grain olute diffuion i oberved and non-tationary olute diffuion equation i olved for thi region. The olute content within thi region i calculated on the bai of olute concentration: Rtot 2 Cl = 4π c ( r) r dr Lg The econd region i liquid phae inide the grain where complete mixing i potulated. Thu, the term in olute balance i eaily computed having the volume of grain and olid phae c * V. The third region i connected with the olid phae, g V where no olute diffuion i aumed. In thi way, no equation ha to be olved. The olute content within the olid phae region in every time tep can be computed uing the value of the olute content from the previou time tep, which i given by: C = C * + c k V V In the conidered volume element the olute balance ha to be conerved, thu, taking into account above aumption, we can write the following olute balance: * C + c k V V V V g + C + l = C0 * where c i the olute concentration at olid-liquid interface and in interdendritic liquid phae in the current time tep. All ymbol with aterik are aociated with the quantity for olid-liquid interface, wherea ymbol with apotrophe indicate the quantity in the current time tep. Here, C 0 = c 0 V tot i the total olute content within the conidered volume. We aumed that within the conidered volume element the temperature gradient i neglected, and the temperature i aumed to be uniform. Moreover, we may potulate that the temperature i related to the olute concentration at the olid-liquid interface by the value of the liquidu lope, m. Conequently, uing the relation between the heat that flow from the volume element and the variation in enthalpy, we can write the heat balance a follow: Q ext dc df S tot = ( ρ cp m L ) V dt dt * where L i the volumic latent heat, ρ cp i the volumic pecific heat, and S tot i the urface area of conidered volume element. In order to determine the volume of the growing grain we utilize the model for the dendrite tip. Becaue the hape of the grain i a phere having calculated length of the main arm, R g, we can compute the volume of the grain. Here, we aume that dendrite tip i controlled only by olute diffuion and therefore the growth model preented in [10] can be applied. It capture the relation between the olutal undercooling and the velocity of the dendrite tip. 3. Reult and dicuion Reult preented in thi Section were obtained for the Al-Si alloy. The propertie of the material ued in computation are lited in table 1. The following input parameter for the preented model were applied: nucleation undercooling, ΔT N, volume element ize Rtot and cooling rate, dt dt. The latter can be related to external heat flow, Q ext, by: dt dt = 3Qext ( ρ cprtot )). The influence of the above parameter on the olidification path and the validation of the model can be found in [5]. Becaue the aumption for the model are the ame, we decided not to preent them again, but only remind the parameter which can be adjuted. The firt tage of olidification wa imulated olving the heat and olute balance equation imultaneouly. In every time tep the fraction of grain wa computed utilizing the growth law for tot ARCHIVES of FOUNDRY ENGINEERING Volume 7, Iue 4/2007,

4 the dendrite tip. Then, the diffuion equation wa olved in the region outide the grain and olute content within thi region wa determined. Table 1. Material propertie utilized in computation Property Value Unit ρ c p J 3 ( m deg) L D 9 J 3 m 3 10 Γ m deg T 660 deg m 2 m m 7. 7 deg % k c % e The olute and the heat balance provide information about the olute concentration at the olid-liquid interface and the volume of olid fraction in current time tep. In every time tep the lat tage of the imulation wa to compute the olute content within the olid phae region. Thi value wa then utilized in olute balance in the next time tep. Thi procedure wa repeated until the eutectic concentration wa etablihed within liquid phae. Then, the econd tage of olidification, aociated with eutectic tranformation, began. From that time tep computation were done in order to keep the temperature contant and equal to eutectic temperature. Thu, no nucleation and kinetic aociated with the eutectic tranformation wa taken into account. Becaue at the beginning of the eutectic reaction the liquid wa aumed a fully tranform into eutectic, and the olid olution fraction and grain fraction were known, two parameter were computed: the fraction of eutectic that form between arm, fea = fg f, and the fraction of eutectic that form between grain, feg = 1 f g. Computed fraction are preented in the table 2. reached, at maller value. Conequently, more eutectic phae i formed between grain. Increaing ize of the volume element, the fraction of olid increae lower and le latent heat i generated. Thi lead to increae in undercooling and accelerate the grain growth. Thi in turn increae the grain fraction. a) b) c) Table 2. Fraction of olid phae and eutectic between grain and arm c f f R tot 0 [ μ m] % f Eg Ea In Fig 3(a) and 4(a) the temperature, grain and olid fraction profile are preented. In the temperature profile the characteritic recalecence, aociated with the undercooling neceary to drive equiaxed olidification, can be oberved. The fraction profile differ for variou total volume and initial concentration. Increaing the initial concentration, the fraction of grain increae initially more rapidly to top, when the eutectic temperature i Fig. 3. Reult for volume element of different radiu, ( c 0 = 7%, dt/dt = 45[deg/], ΔTN = 0) (a) temperature, grain fraction and olid fraction profile (b) region for the time tep when the eutectic temperature i achieved, it profile for z=0 (c) correponding olid olution dendrite 186 ARCHIVES of FOUNDRY ENGINEERING Volume 7, Iue 4/2007,

5 a) 3(b) and 4(b)). A can be oberved, with increae of the initial concentration of an alloy, application of our hape function i more deired and accurate. Moreover, thi tendency become more evident when the volume element ize increae (ee Fig. 4(c)). The ditribution of olid olution grain become crucial due to it influence on the delay in olid fraction increae of eutectic grain. It i epecially important when the kinetic of eutectic tranformation i taken into account. 4. Concluion b) c) In the paper a novel, analytical decription for equiaxed dendrite hape ha been integrated with the grain growth model ha been hown. The detail of the reformulated grain growth model have been preented. In our approach any decription of the olid phae can be ued without ubtantial model reformulation. Obtained reult how that uing uch approach the eutectic ditribution can be computed more reliable. Epecially, when the internal fraction of olid phae, in the time tep when the eutectic temperature i reached, wa relatively high. In the paper the eutectic tranformation i aumed to be iothermal. Thu, amount of olid phae that olidifie during particular time tep, i computed o that the temperature i contant and equal to eutectic temperature. In thi way, no undercooling aociated with eutectic tranformation i taken into account. We believe that taking into account the kinetic of eutectic tranformation would be intereting extenion of thi approach, motly becaue the ditribution of liquid of eutectic concentration i computed more reliable, influencing the delay in eutectic tranformation. Furthermore, the interaction between grain could be alo included into invetigation. Acknowledgement The reearch project wa financed from the government upport of cientific reearch for year , under grant No. N /3389. Fig. 4. Reult for volume element of different radiu, ( c 0 = 9%, dt/dt = 45[deg/], ΔTN = 0) (a) temperature, grain fraction and olid fraction profile (b) region for the time tep when the eutectic temperature i achieved, it profile for z=0 (c) correponding olid olution dendrite Moreover, increaing the growth rate the grain diffuion layer decreae. Thi delay the moment when the layer reache the limit of conidered region. A reult additional grain growth i timulated. All thi make that le liquid remain between grain, which influence the eutectic ditribution. In Fig 3(c) and 4(c) the hape of olid olution grain, when the eutectic temperature wa reached, are preented. In order to how more preciely the pread of all three region (conidered volume element, grain, and olid olution phae) projection for z = 0 are alo preented (Fig Reference [1] W. Kurz, B. Giovanola, and R. Trivedi. Theory of microtructural development during rapid olidification. Acta Metall., 34(5): , 1986 [2] J.C. Ramirez and C. Beckermann. Examination of binary alloy free dendritic growth theorie with a phae-field model. Acta Mater., 53: , [3] Ch.-A. Gandin, J.-L. Debiolle, M. Rappaz, and Ph. Thevoz. A three dimenional cellular automaton-finite element model for the prediction of olidification grain tructure. Metall. Mater. Tran A, 30A: , [4] M. Plapp and A. Karma. Multicale finite-differencediffuion monte-carlo method for imulating dendritic ARCHIVES of FOUNDRY ENGINEERING Volume 7, Iue 4/2007,

6 olidification. Journal of Computational Phyic, 165: , [5] Ph. Thevoz, J.-L. Debiolle, and M. Rappaz. Modeling of equiaxed microtructure formation in cating. Metall. Tran. A, 20A: , [6] D.M. Stefanecu. Science and Engineering of Cating Solidification. Kluwer Academic Publiher, 2002 [7] L. Natac and D.M. Stefanecu. Macrotranportolidification kinetic modeling of equiaxed dendritic growth: Part i. model development and dicuion. Metall. Mater. Tran. A, 27A: , [8] O. Nielen, B. Appolaire, H. Combeau, and A. Mo. Meaurement and modeling during equiaxed olidification of al-cu alloy. Metall. Mater. Tran. A, 32A: , [9] M. Rappaz and Ph. Thevoz. Solute diffuion model for equiaxed dendritic growth. Acta Metall., 34: , [10] H. Eaka and W. Kurz. Columnar dendrite growth: [11] A comparion of theory.j Cryt. Growth, 69: , ARCHIVES of FOUNDRY ENGINEERING Volume 7, Iue 4/2007,