Integrated Approach for Prediction of Hot Tearing

Transcription

1 Integrated Approach for Prediction of Hot Tearing SUYITNO, W.H. KOOL, and L. KATGERMAN Shrinkage, impoed train rate, and (lack of) feeding are conidered the main factor that determine cavity formation or the formation of hot tear. A hot-tearing model i propoed that will combine a macrocopic decription of the cating proce and a microcopic model. The micromodel predict whether poroity will form or a hot tear will develop. Reult for an Al-4.5 pct Cu alloy are preented a a function of the contant train rate and cooling rate. Alo, incorporation of the model in a finite element method (FEM) imulation of the direct-chill (DC) cating proce i reported. The model how feature well known from literature uch a increaing hot-tearing enitivity with increaing deformation rate, cooling rate, and grain ize. Similar trend are found for the poroity formation a well. The model alo predict a beneficial effect of applying a ramping procedure during the tart-up phae, which i an improvement in comparion with earlier finding obtained with alternative model. In principle, the model doe not contain adjutable parameter, but everal parameter are not well known. A full quantitative validation not only require detailed cating trial but alo independent determination of ome thermophyical parameter of the emiolid muh. DOI: / y Ó The Author() Thi article i publihed with open acce at Springerlink.com I. INTRODUCTION HOT tear are crack that initiate in the muhy zone. Thee crack are characterized by intergranular fracture and a mooth fracture urface due to the exitence of a liquid phae in the interdendritic region during cracking. [1] Hot tearing i one of the crucial problem encountered during the direct-chill (DC) cating proce. The occurrence of hot tear determine the productivity during the proce. Thee olidification defect have been undertood for a long time, but a quantitative prediction of their occurrence i till underdeveloped. In general, the olidification procee proceed in four tep that reflect the morphological development and interaction, namely: (1) the nucleated crytal float freely and the macrocopic behavior i cloe to the liquid behavior; (2) the nuclei are cloe and tend to attach to each other to form a porou network and the olidification hrinkage train i eaily counteracted by the liquid flow and olid arrangement; (3) the deformation of the olidified body caued by the olidification hrinkage and external train i not fully counteracted by the liquid flow and olid movement, o that olidification defect uch a hot tear and poroity are uually initiated; and (4) the grain are trongly interconnected, o that deformation of the olidified body will not reult in further defect. SUYITNO, Aociate Profeor, i with the Cating and Solidification Technology (CASTEC) Laboratory, Department of Mechanical and Indutrial Engineering, Gadjah Mada Univerity, Yogyakarta, Indoneia. Contact uyitno@ugm.ac.id W.H. KOOL, Aociate Profeor, and L. KATGERMAN, Profeor, are with Department of Material Science and Engineering, Faculty 3mE, Delft Univerity of Technology, 2628 CD Delft, The Netherland. Manucript ubmitted May 13, Article publihed online Augut 19, 2009 Quantitative prediction of hot tearing i not eay becaue of the complex interplay between macrocopic and microcopic phenomena. Prediction of hot tearing during DC cating i baed on two tep, namely modeling of the thermomechanical behavior during olidification [2 4] and the implementing of hot-tearing criteria into thi model. [5 10] The firt tep ue contitutive equation to decribe the thermomechanical modeling, to calculate tree and train in the billet. Computed tree or train indicate the hot-tearing tendency. In the econd tep, the reult of the firt tep are ued a input into a hot-tearing criterion. Several mechanim of hot tearing have recently been reviewed [1] and a recent article outline the requirement for a modern hot-tearing model and a criterion baed on thi model a well a the future development of hottearing reearch in term of mechanim of hot-crack nucleation and propagation. [11,12] Variou criteria that might enable the prediction of hot tear have been propoed. [13 20] Thee criteria can be claified into thoe baed on nonmechanical apect uch a feeding behavior, [13 15] thoe baed only on mechanical apect, [16 18] and thoe that combine thee feature. [19,20] Approache baed on nonmechanical criteria [13 15] put emphai on the feeding propertie in the muh and aume that the econd and third tage of olidification play the main role in the formation of a hot tear. In contrat, the mechanical criteria [16 18] emphaize the importance of the trength and train developed in the third and fourth tage of the olidification. Some approache [19,20] combine thee mechanical and nonmechanical method for the prediction of hot tearing. A comprehenive evaluation of exiting hot-tearing criteria for DC cating of aluminum billet i reported in Reference 8 through 10. In the aement of variou criteria for hot tearing, it i found that the RDG criterion, [19] which combine mechanical and feeding 2388 VOLUME 40A, OCTOBER 2009

2 condition, had the greatet potential with repect to the other criteria, but it did not predict cracking for the practical condition in which cracking will not occur. [10] Therefore, in thi tudy, a hot-tearing model i derived baed on cavity formation when there i inufficient feeding during olidification. The feeding during olidification i incorporated uing a tranient ma balance equation. The flow behavior of the emiolid tate ha been included in order to model the mechanical repone of the emiolid body. The cavity formed become a hot tear when a critical dimenion i achieved; otherwie, poroity will reult. The poibility of the formation of microporoity i not found for other model. The propoed model i applied in two type of imulation. Firt, the poroity growth, hot-cracking enitivity, and developed tre in the muh are calculated a a function of everal parameter, uing a contant parameter the train rate and cooling rate. Second, the propoed model i incorporated in a finite element method (FEM) imulation of DC cating an Al-4.5 pct Cu billet. The hot-cracking enitivitie are calculated a a function of everal parameter and are compared with thoe from the RDG criterion. [19] II. PHYSICAL MODEL Solidification i initiated by the formation of nuclei that grow to form a dendritic tructure. In the model, it i aumed that the microtructure i equiaxed. The floating grain grow, coaren, and reach the coherency point at which the dendrite touch each other. Cavitie are formed at triple junction between the grain. At and beyond the coherency point, tree can be tranmitted through the dendritic network. At the coherency point, the volume difference due to the thermal hrinkage and impoed deformation can be filled with liquid metal. During further olidification, the liquid network become interrupted and liquid pocket become iolated. Permeability become low and the dendritic network become trong. The liquid flow and afterfeeding of the volume difference become uppreed. During complete olidification, there are three poibilitie. The firt i that the liquid flow and afterfeeding (or even the olid diffuion after complete olidification) are ufficient to counteract thermal hrinkage and impoed deformation and, therefore, cavitie are not formed and a fully dene microtructure i found. The econd i that the liquid flow and afterfeeding are inufficient to counteract thermal hrinkage and impoed deformation and, therefore, cavitie are formed, leading to a microtructure containing poroity. The third i that the cavity dimenion reache a critical value, which lead to the formation of a hot crack. The critical cavity dimenion that lead to crack i determined by uing Griffith approach, [21,22] conidering that local condition are brittle. Thi approach i ued conidering the appearance of hot-tear urface that doe not how evidence of platic deformation. [11] Experimental obervation indicate that hot tear are generally found in the center of the billet and have a tarlike form (viible in the billet cro ection). [2,9,23] Thi mean that tree and train in the billet cro ection are dominant. To implify the complex threedimenional (3-D) condition of olidification during cating, it i therefore aumed that the tre, train, and train rate impoed by the muh are acting in the plane normal to the cating direction. Feeding take place in the cating direction. The tre, train, and train rate are calculated by a FEM. The feeding behavior i repreented by Feurer approach, which i derived from Darcy law. III. MATHEMATICAL MODEL The model that enable the prediction of the formation of microporoity or hot tear during DC cating conit of a micromodel that account for the local mechanical and feeding propertie of the muh, coupled with a macromodel of the DC cating proce (for example, by FEM imulation). The propoed model i illutrated in detail in Figure 1. The input data for the model provided by the macromodel are the temperature (T), cooling rate ( _T), and train rate (_e). The olidification model link the temperature with the olid/liquid fraction, which i needed in the contitutive model of the muh and in the feeding model. The contribution of feeding fe i compared with the local train rate fr, reulting from hrinkage and deformation. Above the coherency temperature, only a hrinkage term contribute to fr, and fr will be maller than fe. Below the coherency temperature, deformation force will contribute to fr. If at any moment during the olidification fr- fe become larger than a critical parameter f crit, a cavity i formed. The parameter f crit account for the nucleation Fig. 1 Schematic repreentation of the model: fr i the hrinkage + deformation rate and fe i the feeding rate. VOLUME 40A, OCTOBER

3 of the cavity. In the imulation, f crit i taken equal to zero, i.e., it i aumed that a nucleation effect of the cavity can be neglected. In the cae in which the diameter of the cavity exceed a critical diameter determined by Griffith model, the cavity will reult in a hot tear. In the Griffith model, the critical diameter depend on the tre in the muh. The tre r follow from the contitutive model. Baed on thi, three poibilitie arie: a dene microtructure without microporoity, the formation of microporoity, or the formation of hot tear. A. Solidification Model In the olidification model, the liquid fraction f l a a function of temperature i determined uing the following equation: [24] T m T 1 2a k k 1 f l ¼ 1 2a k 4 5 and T m T l ½1Š a ¼ a 1 exp exp a 2 2a where T m i the melting temperature of the pure metal, T l i the liquidu temperature, T i the temperature, k i the partition coefficient, a i the back-diffuion coefficient, and a * i the modified dimenionle olid-tate back-diffuion parameter. B. Contitutive Model of the Muh The tre developed in the muh i computed uing a contitutive model that will depend on the olid fraction, train rate, and temperature. Here, we will ue the following expreion: [20] r ¼ r o expðbf Þexp mq ðþ _e m ½2Š RT where Q i the activation energy given by the olid phae deformation behavior, m i the train rate enitivity coefficient, R i the ga contant, r o and b are material contant, and _e i the train rate. C. Shrinkage, Deformation, and Feeding Term A tranient ma conervation equation i applied to a 3-D element in the olidifying billet of which a complete derivation i hown in the Appendix. It read: ¼ þ q _e þ fe ½3Š where _e and fe are the train rate and feeding rate, repectively, f, f l, and f v are the olid, liquid, and cavity fraction, repectively, and q and are the denitie of the olid and liquid, repectively. The contribution of hrinkage and deformation fr read: fr ¼ q þ q S _e ½4Š The feeding term fe i baed on the interdendritic flow in a muh that ue the Carman Kozeny approximation [25,26] and read a follow: fe ¼ K P gl 2 K ¼ k2 ð1 f Þ f 2 P ¼ P o þ P m P c P c ¼ 4c ½8Š k where K i the permeability, k the i econdary dendritic arm pacing, g i the vicoity of the liquid phae, L i the length of the porou network, c i the olid-liquid interfacial energy, P i the effective feeding preure, and P o, P m, and P c are the atmopheric, metallotatic, and capillary preure, repectively. In the model, k and P c are independent of the temperature. The L i taken a the length of muh from the coherency until the end of olidification. For the feeding term fe, only the z direction (cating direction) i taken into account. Under compreive condition, P i negative, reulting in a negative value of fe. D. Cavity Nucleation If fr < fe (Eq. [4] and [5]), feeding will be ufficient and the actual volumetric flow rate per unit volume will be equal to the hrinkage and deformation rate. If fr> fe, feeding will be inufficient and a cavity will form and grow if fr jfej f crit, where f crit i a term decribing the nucleation of the cavity. The f crit can be expreed in a critical preure (P crit ) for cavity nucleation, a follow: f crit ¼ K P crit gl 2 The value of the critical depreion preure ha to be determined from the experimental data. A value 2 kpa, uch a i ued in Reference 5, will reult in a mall value of f crit. Therefore, in thi calculation, f crit i taken a equal to zero and it i aumed that fr jfej i alway equal to the cavity volume. If the condition for the formation of a cavity i fulfilled, v =@t i a meaure for cavity growth. The fraction f v and the diameter d of the cavity are determined by f v ¼ Z t t dt ½5Š ½6Š ½7Š ½9Š ½10Š d ¼ 3 1=3 2p f vv char ½11Š 2390 VOLUME 40A, OCTOBER 2009

4 V char ¼ Cd 3 g ½12Š where t liq i the time that correpond with the tart of olidification, V char i the characteritic volume of the local geometry (cavity and grain), and d g i the diameter of the grain. The term C i a packing parameter accounting p for the packing of the grain. It i equal to 2 ffiffi 2 for fcc and p8 ffiffi for bcc packing. 3 3 In Griffith approach, [21,22] the relation between the critical cavity length a crit and the tre r in the muh for the cavity to propagate a crack i expreed in Eq. [13]: E a crit ¼ 4c pr 2 ½13Š where c e i the urface tenion of the liquid metal and E i the Young modulu of the muh. The tre in the muh i given by Eq. [2]. To account for the irregularity of the cavity hape, a contant C 1 i introduced: a ¼ C 1 d ½14Š which relate the longet cavity length with the diameter of a pherical cavity. The C 1 i larger than or equal to 1. From Eq. [11], [12], and [14], a i calculated. If a a crit, a crack will develop. E. Hot-Cracking Senitivity The hot-cracking uceptibility (HCS) i defined by HCS ¼ a crit ½15Š a Ued qualitatively, thi definition mean an increaing uceptibility for hot cracking with increaing value. Ued quantitatively, it mean that if the hot-cracking enitivity i higher than 1, a hot tear will develop. IV. SIMULATION A. Simulation with Contant Parameter A part of the calculation in the model wa performed for contant _e, _T, andl, where the train rate _e were varied from 10 5 to , the cooling rate _T were varied from 0.1 through 10 K/, and L wa taken a 0.1 m. Simulation wa done for an Al-4.5 pct Cu alloy. The parameter ued in the calculation are given in Table I. Further, it i aumed that the deformation wa only in the cro ection of the billet. B. Simulation with FEM Modeling of DC Cating Billet Another part of the calculation in the model wa performed after incorporating the model in an FEM code. The DC cating of an Al-4.5 pct Cu alloy billet with a 100-mm radiu and 1000-mm length i imulated. The computation procedure i imilar to that performed in Reference 4. An axi-ymmetric model i ued in thi work. Due to the ymmetry, only a half ection of the billet and bottom block need to be modeled. For the Table I. Parameter Ued in Calculation and the Appropriate Reference Parameter Value Unit Reference a k T m 933 K 35 r o 4.5 Pa 20 m a Q 160 kj/mol 20 E 40 MPa 20, 33 c 0.84 J/m 2 35 q 2790 kg/m kg/m 3 35 k 8Æ10 5 m g PaÆ 35 d g 5Æ10 4 m C 1 1 p C 2 ffiffi 2 imulation, a coupled computation of the tre and the temperature field i applied uing four-node rectangular element with four Gauian integration point. In the imulation, the ingot remain in a tationary poition, while the mold and the impingement point of the water flow move upward with a velocity equal to the cating peed. The continuou feeding of liquid metal i implemented by activating horizontal layer of element incrementally. The computational domain i hown in Figure 2. After every time tep, the HCS i computed at every node. Computation i performed for two phae in the billet. The firt phae i ditant from the beginning of the billet (tart-up phae) by 0 to 400 mm. Four cating condition, denoted 1 through 4, are applied in the computation to calculate the hot-tearing tendency a a function of the axial poition. The cating mode are hown in Figure 3. The econd region i at a ditance of 750 mm from the beginning of the billet (teady-tate phae). The cating peed elected were contant and were equal to 120, 150, and 180 mm/min. Here, the hottearing tendency i calculated a a function of the radial poition in the billet. Mot of the parameter ued in the incorporation with FEM modeling of DC cating a billet are alo given in Table I. However, three parameter were adjuted and are given in Table II. Thee adjutment parameter were taken from the experimental data. [9] V. RESULTS A. Simulation with Contant Parameter In the imulation with a contant train rate, cooling rate, and length of porou network, the main emphai wa on the calculation of the feeding rate or mechanical parameter a a function of the olid fraction. 1. Cavity nucleation In Section III, the quantitie fe and fr, which are the volume fraction per unit time fed by the liquid flow and VOLUME 40A, OCTOBER

5 Table II. Modified Parameter Ued in the Calculation of DC Cating Parameter Value Unit Reference E 10 GPa k 1Æ10 5 m 9 d g 3Æ10 4 m 9 Fig. 2 Computational domain of the DC-cat billet. The C 1 through C 7 correpond with boundary condition defined in Ref. 4. Fig. 3 Cating mode applied for imulation of the tart-up phae of DC cating. Cating condition: (e) 1,(h) 2,(D) 3, and (x) 4. the volume fraction per unit time related to hrinkage and impoed deformation, repectively, were introduced. Figure 4 how the value of fe and fr a a function of the olid fraction for variou train rate and cooling rate. The value of fr become lower at high Fig. 4 Parameter fr (1 through 6) and fe v olid fraction. (a) Effect of train rate at cooling rate of 1 K/; train rate: (1) , (2) , (3) 5Æ10 4 1, (4) , (5) 5Æ10 3 1, and (6) (b) Effect of cooling rate for train rate of 5Æ ; cooling rate: (1) 0.1 K/, (2) 1 K/, (3) 5 K/, and (4) 10 K/. The fe i independent of train rate or cooling rate. olid fraction, becaue the hrinkage term i dominant l =@T decreae with increaing olid fraction. The value of fr increae with an increaing train rate or 2392 VOLUME 40A, OCTOBER 2009

6 cooling rate, according to Eq. [4]. The value of fe decreae a a function of the olid fraction, becaue feeding become more difficult at higher olid fraction. For the effect of the train rate or cooling rate on the fe, microtructural parameter are the mot influential factor for the value of fe. Becaue in thi calculation thee parameter are kept contant, there i no effect of the train rate or cooling rate on fe. The interection point of the fe and fr curve, indicated in the figure, give the olid fraction at which the growth of cavitie will begin. It i een that growth occur earlier (i.e., at lower fraction of olid) for higher train rate and cooling rate. Thi i evident, becaue the amount of volume per unit time, which hould be fed, increae with higher train rate and cooling rate. 2. Poroity and Hot Tearing Figure 5 and 6 preent both the tre developed in the muh a a function of olid fraction (calculated from Eq. [2]) and the critical tre in the Griffith approach, i.e., the tre calculated from Eq. [13] for a cavity, with length a calculated from Eq. [14]. In the figure, the HCS i alo given (Eq. [15]). Developed tre and HCS increae for an increaing olid fraction and train rate. Fig. 5 (a) Developed tre and critical tre and (b) HCS v olid fraction for variou train rate; cooling rate: 1 K/. Developed tre (1 through 5 ); critical tre and HCS (1 through 5). Strain rate: (1) , (2) , (3) 5Æ10 4 1, (4) , and (5) 5Æ Fig. 6 (a) Developed tre and critical tre and (b) HCS v olid fraction for variou cooling rate; train rate: 5Æ Developed tre (1 through 4 ); critical tre and HCS (1 through 4). Cooling rate: (1) 0.1 K/, (2) 1 K/, (3) 5 K/, and (4) 10 K/. VOLUME 40A, OCTOBER

7 Fig. 7 Cavity fraction v fraction olid. Dahed curve indicate tart of tear development. (a) Effect of train rate at cooling rate of 1 K/; train rate: (1) , (2) , (3) , (4) , and (5) 5Æ (b) Effect of cooling rate for train rate of 5Æ ; cooling rate: (1) 0.1 K/, (2) 1 K/, (3) 5 K/, and (4) 10 K/. The HCS increae for an increaing cooling rate. The interection point of the developed tre and the critical tre for cavity growth are aumed to be the tranition from development of microporoity to the initiation of hot tearing. Figure 7 how the cavity fraction fv a a function of olid fraction for the variou train rate and cooling rate. Cavitie are initiated at the olid fraction at which fv tart to deviate from zero. Thee point correpond with the interection point found in Figure 4 and indicate the beginning of the formation of poroity. In addition, the interection point found in Figure 5 and 6 are indicated in Figure 7. They are connected by a dahed line, which indicate the fraction olid value at Fig. 8 Three region (A = microporoity and hot tear are abent, B = microporoity develop but hot tear do not form, and C = hot tear are formed) that give the condition of train rate and olid fraction. Cooling rate: (a) 1 K/ and (b) 5 K/. which hot crack are formed. Lowering the train rate or cooling rate will increae the olid fraction at which hot tear tart to develop. At certain train rate, the hot tear will not develop. In Figure 8, the boundarie of three region are given a a function of the train rate and olid fraction for the cooling rate 1 or 5 K/. In region v =@t equal zero or negative and, conequently, no microporoity or hot tear are developed. In region v =@t i poitive but a i maller than a crit. Only microporoity will develop. Region C, in which a i larger than a crit, mark the condition for which hot tearing will occur. The beginning of the formation of poroity or hot crack increae with the train rate and cooling rate VOLUME 40A, OCTOBER 2009

8 Fig. 9 Effect of grain ize on the developed tre in the muh (1 through 4 ) and the critical tre (1 through 4). Strain rate: Grain diameter: (1,1 ) 600 lm, (2,2 ) 300 lm, (3,3 ) 100 lm, and (4,4 ) 30 lm. Fig. 11 HCS a a function of ditance from the bottom block for the four cating condition. B. Incorporation in FEM Simulation of DC Cating Billet In the imulation tudie of the DC cating of a billet, the emphai wa on the calculation of the HCS a a function of variou parameter. Here, the approach taken wa imilar to that ued in an earlier tudy, in aeing everal hot-tearing criteria in literature. [10] 1. Start-up phae Figure 11 how the hot-tearing uceptibility for the variou cating condition during tartup. The uceptibility i maximum approximately 70 mm from the bottom of the billet. Uing a ramping procedure reduce the uceptibility, although it doe not influence it teady-tate value. The hot-cracking enitivity i higher for the higher cating rate. Becaue the HCS value i maller than 1, a hot tear will not form. In Figure 12, the development during olidification of the HCS 50 mm from the bottom of the billet i hown. The uceptibility increae with the olid fraction. A hot crack will not form. Fig. 10 Effect of packing parameter on the developed tre in the muh (1 through 3 ) and the critical tre (1 through 3). Strain rate: Packing parameter: (1,1 ) 2.83, (2,2 ) 2.34, and (3,3 ) Effect of grain ize and packing parameter The effect of the grain ize on the developed tre in the muh and the critical tre i hown in Figure 9.Iti found that a maller ize reduce the developed tre (and hot-tearing tendency) and that, below a certain grain ize, a hot tear will not develop. In Figure 10, the effect of the packing parameter on the developed tre in the muh and the critical tre i hown. The packing parameter hardly influence the tre or the location of the interection point. 2. Steady-tate phae Suceptibilitie are conidered teady tate 750 mm from the bottom of the billet. The effect of the cating peed on the HCS in the teady tate i hown in Figure 13 a a function of the ditance from the center of the billet. The uceptibilitie increae with increaing cating peed and are maximum in the center of the billet. At a ditance of approximately 70 mm and higher from the center, the uceptibility i cloe to zero. In addition, for a cating peed of 180 mm/, the uceptibility will be lower than 1 a hot crack will not form. The development of the uceptibility during olidification i hown in Figure 14. The value of the uceptibility tart to deviate from zero level at a certain fraction olid that i dependent on the cating peed. The higher the cating peed, the lower the olid fraction for which the value deviate from zero. VOLUME 40A, OCTOBER

9 Fig. 12 HCS in the lat tage of olidification 50 mm from the bottom. Fig. 14 HCS development during the lat tage of olidification at cating peed: (1) 120 mm/min, (2) 150 mm/min, and (3) 180 mm/min. Fig. 13 HCS a a function of ditance from the center of the billet. Cating peed: (1) 120 mm/min, (2) 150 mm/min, and (3) 180 mm/min. Becaue the Young modulu at emiolid temperature i not well known, the effect of the Young modulu on the HCS i hown in Figure 15. A higher Young modulu will reduce the HCS. The effect of the grain ize on the HCS i hown in Figure 16. An increaing grain ize will reult in an increaing HCS, which i relevant for the center of the billet. VI. DISCUSSION A. Aumption in the Preented Model In an aement tudy [10] of variou criteria for hot cracking that were integrated into a FEM imulation Fig. 15 HCS development during the lat tage of olidification Young modulu: (1) 10 GPa, (2) 1 GPa, and (3) 0.1 GPa. Center of billet. Cating peed: 120 mm/min. and applied to DC cating, it i found that the RDG criterion, [19] which combine apect of mechanical a well a feeding condition, had the greatet potential but did predict cracking for the practical condition in which cracking will not occur. In the preent model, everal approache and aumption differ from thoe in the RDG model. In both model, the baic equation i that of ma conervation; in the RDG approach, however, it i taken a teady tate, wherea in the preent approach it i tranient, allowing for the formation of a cavity volume. The RDG conider the ma balance equation, wherea in the preent model, the equation i train baed. In the RDG, feeding follow the hrinkage and give rie to a 2396 VOLUME 40A, OCTOBER 2009

10 The hot-tearing enitivitie preented in Figure 5(b) and 6(b) alo indicate increaing enitivity for an increaing train rate or increaing cooling rate. For all the train rate and cooling rate applied here, the HCS value become higher than 1, which mean that hot cracking will take place. However, ome parameter ued in thee calculation (for example, the Young modulu) are rather uncertain and have a ignificant influence on the HCS value. For the Young modulu, we did ue in thee calculation the only value known to u, [20,33] but we believe that a higher value might be more realitic, one that predict that crack are not formed for ome of the train rate and cooling rate parameter, which i more in agreement with indutrial practice. Fig. 16 HCS a a function of the ditance from the center. Grain ize: (1) 100 lm, (2) 300 lm, and (3) 500 lm. Cating peed: 120 mm/min. preure drop that may become critical. In the RDG approach, a crack i formed if the preure drop become o high that the preure in the liquid of the muh become lower than a critical preure for cavity formation. For uch a cae, a cavity will form that alway give rie to the formation of a hot tear. In the preent approach, a cavity will grow if the volume that can be fed per unit time become maller than the volume change due to hrinkage and deformation. A cavity will lead to poroity, and only when it ize will urpa a critical ize will a crack be formed. Conidering the muh a brittle, the critical ize i determined from the Griffith approach [21] and depend on the tre in the muh. It become maller for higher tree. Other difference are that the RDG conider a columnar olidification tructure, wherea the preent model aume an equiaxed tructure, uch a i commonly found in DC-cat billet, and that the RDG i eentially a two-dimenional approach, wherea the preent model i, in principle, 3-D. B. Obervation with Contant Parameter Simulation A Figure 8 demontrate, the formation of poroity or hot crack i promoted by higher train rate and higher cooling rate. Thee obervation are in line with thoe found by other author. [19,33] The obervation that maller grain ize reduce the tendency for poroity or hot tearing (Figure 9) i confirmed by Reference 27 through 29 and i upported by the beneficial effect of the addition of grain refiner. Figure 8 indicate that, during the lat tage of olidification, poroity i alway formed. The model not only indicate whether poroity will be found but alo predict the amount of poroity. Stree in the muh at which hot tear are formed (Figure 5, 6, 9, and 10) are in the range of 0.1 to 2 MPa. Thee value are in agreement with the fracture tree found in the tenile tet of emiolid aluminum alloy. [30 32] C. Obervation with DC-Cat Simulation Incorporation of the model into a FEM imulation lead to reult comparable to thoe obtained with the imulation with contant parameter. The HCS increae with the cating peed and i maximum in the center of the billet. It i higher for a larger grain ize. The FEM imulation alo how (Figure 11) that the uceptibility at approximately 80 mm from the bottom i highet and that it reduce when the cating peed i lowered. It mean that the application of a ramping procedure during the tart-up phae ha ignificance. In Figure 17 and 18, the hot-cracking uceptibilitie obtained in thi tudy are compared with the depreion preure (which are a meaure for HCS), obtained in an earlier tudy [10] with the RDG model. [19] The condition ued in the imulation were identical. It i een that hape of the correponding curve are rather imilar; therefore, in the qualitative ene, there i not much difference between the prediction for the RDG model and for the preent model. In the RDG model, depreion preure exceeding a critical value will lead to the formation of a hot crack. A a critical value, a value of 2 kpa i given. [5] The depreion preure in Figure 17 and 18 are coniderably higher and will therefore alway lead to the formation of a hot crack. In the preent model, HCS value lower than 1 will not lead to the formation of a hot crack and it i een that, for the condition imulated here, a hot crack will not be formed. In Reference 10, we aeed everal hot-cracking criteria on their predictive capability of four major obervation in indutrial experience. In Table III, a imilar aement i done with the preent model. It i een that the predictive capability of the preent model i excellent on thee four iue. A full quantitative aement hould require detailed cating trial and an independent determination of the Young modulu in the muhy tate. D. Preent Limitation Validation of the model hould take place in direct comparion with actual cating trial. In thi validation, ome model parameter are more critical or le known VOLUME 40A, OCTOBER

11 Fig. 17 (a) HCS and (b) depreion preure a a function of ditance from the bottom block for the four cating condition. Fig. 18 (a) HCS and (b) depreion preure a a function of ditance from the center of the billet. Cating peed: (1) 120 mm/min, (2) 150 mm/min, and (3) 180 mm/min. than other. Let u aume, a ha been done in thi tudy, a contant train rate and a contant cooling rate. For the calculation of the fraction olid in which poroity tart to appear, we et fe equal to fr and, uing Eq. [4] through [8], we derive the following equation: Kk 2 ð1 f Þ 2 gl 2 f 2 l ¼ 1 q P o þ P m þ q _e k ½16Š The expreion at the left ide i related to the amount of poroity found after cating (in cae hot tear are abent). At the right ide, the parameter L l =@T are well known or can eaily be determined. Parameter Table III. Predictive Capability of the Model on Four Major Obervation Obervation Practice Prediction A increaing enitivity ye ye for higher cating peed B highet enitivity in billet ye ye center C ramping might have poitive ye ye effect D crack will be formed no no 4c /k i relatively mall compared with P 0 + P m, o it influence on the poroity i relatively low. At the left ide, the permeability of the muh K i a critical parameter in determining when poroity tart to form 2398 VOLUME 40A, OCTOBER 2009

12 and the final amount obtained. However, the value of K i relatively uncertain and i alo difficult to determine. Therefore, regarding poroity, the permeability i the main parameter under conideration for poible adjutment in a validation experiment. For the determination of when a hot crack tart to appear, we derive the following equation from Eq. [11] through [14]: " f v ¼ 4c # E 1 2p 1=3 3 pr 2 ½17Š C 1 d g 3C If f v i maller than the right-ide term, a hot tear will not be formed, wherea if f v i higher, a hot tear will form. The variou parameter at the right ide are d g, which can be experimentally determined, and C, C 1, c, E, and tre r in muh. The parameter combination (c ÆE/r 2 ÆC 1 ÆC 1/3 ) i dominant in the formation of hot crack and i therefore the main expreion under conideration for poible adjutment in a validation experiment. The mot unknown parameter i aumed to be the Young modulu E, which i difficult to determine becaue of the low trength and high brittlene of the muh. The data in the literature on the Young modulu of a muh are carce. The value in the tudy with the contant parameter i taken from Reference 20 and 33. The value taken in the imulation with DC cating i higher, in order to be more in agreement with the reult in actual cating experiment. VII. CONCLUSIONS A model i propoed for prediction of the formation of microporoity and hot tear during DC cating. It aume that volume change during olidification reulting from the mechanical condition hould be fed by liquid. If feeding i inufficient, a cavity volume i formed. A new element in the model i that the formation of a cavity lead to microporoity and that if it ize exceed a certain critical ize, a hot crack will form. The muh i conidered brittle and therefore the critical ize i derived from fracture mechanic. The model how feature well known from the literature uch a increaing uceptibility for microporoity formation and hot tearing with increaing deformation rate, increaing cooling rate (i.e., increaing cating peed), and increaing grain ize. The model alo indicate a higher enitivity in the billet center. After incorporating thi model in an FEM imulation for a DC cating billet, thee obervation are not only confirmed, but it i alo found that the application of a ramping procedure during the tart-up phae might give a beneficial effect. A uch, the preent model improve the reult found in an earlier aement tudy baed on the criteria known from the literature. Further, in thi tudy, key parameter are identified for which the value are rather uncertain and which may therefore act a fitting parameter in a validation tudy. OPEN ACCESS Thi article i ditributed under the term of the Creative Common Attribution Noncommercial Licene which permit any noncommercial ue, ditribution, and reproduction in any medium, provided the original author() and ource are credited. APPENDIX The derivation of Eq. [3] ue the conervation of ma equation baed upon the general framework for the volume averaged conervation equation a preented in Reference ðf ðf l þrðf q v Þ ¼ C ½A1Š þrðf l v l Þ ¼ C ½A2Š where f n i the volume fraction of phae n (n =, l, and v for the olid, liquid, and cavity, repectively), q k denote the ma denity, v k i the velocity, and C i the interfacial ma tranfer due to phae change. Adding Eq. [A1] and [A2] give the ma conervation equation for the two-phae (olid-liquid) ðf q Þ ð f þrðf q v Þþrðf l v l Þ ¼ 0 ðf q @ ðf l ¼ f þ l ½A4aŠ ½A4bŠ r: ðf q v Þ f q v x; f q v y; f q v z; f q v x; ¼ v x; þ f v x; q Þ f q v y; ¼ v y; þ f v y; q Þ f q v z; ¼ v z; þ f v z; q Þ ½A5dŠ r: ðf l v l Þ f l v x;l f l v y;l f l v z;l f l v x;l ¼ lv x;l þ fl v x;l Þ ½A5fŠ VOLUME 40A, OCTOBER

13 @y f l v y;l ¼ lv y;l þ fl v y;l Þ f l v z;l ¼ lv z;l þ fl v z;l Þ ½A5hŠ Subtitution of Eq. [A4a], [A4b], and [A5a] through [A5h] to Eq. [A3] þ þ l þ v x; þ f v x; q Þ þ v y; þ f v q ð v z; þ f v z; q Þ lv x;l þ fl v q ð lv y;l þ fl v y;l Þ lv z;l þ fl v z;l Þ¼0 [A6] 1 ¼ f l þ f þ f v ½A7Š f ¼ 1 f l f l v Subtitution of Eq. [A9] into Eq. [A6] and diviion by þ þ q þ f v q Þþ þ f f v z; v z; þ þ f lv ð f f v y; v y; f v q q f lv x;l f f l v y;l lv y;l Þ f f l v lv z;l þ Þ ¼ 0 [A10] Auming that and q are contant give þ q þ f v f q v x; f v y; f lv x;l f lv y;l f lv z;l ¼ 0 ½A11Š Rearrangement of term in Eq. [A11] give ¼ þ q _e þ fe ½A12Š where _e i the train rate and fe i the feeding rate. REFERENCES 1. D.G. Ekin, Suyitno, and L. Katgerman: Prog. Mater. Sci., 2004, vol. 49, pp I. Farup and A. Mo: Metall. Mater. Tran. A, 2000, vol. 31A, pp M. M Hamdi, A. Mo, and C.L. Martin: Metall. Mater. Tran. A, 2002, vol. 33A, pp Suyitno, W.H. Kool, and L. Katgerman: Metall. Mater. Tran. A, 2004, vol. 35A, pp J.M. Drezet and M. Rappaz: in Light Metal 2001, J.L. Anjier, ed., TMS, Warrendale, PA, 2001, pp J.M. Drezet, M. M Hamdi, S. Benum, D. Mortenen, and H. Fjaer: Mater. Sci. Forum, 2002, vol , pp M. M Hamdi, S. Benum, D. Mortenen, H. Fjaer, and J.M. Drezet: Metall. Mater. Tran. A, 2003, vol. 34A, pp Suyitno, W.H. Kool, and L. Katgerman: in Light Metal 2003, P.N. Crepeau, ed., TMS, Warrendale, PA, 2003, pp Suyitno, D.G. Ekin, V.I. Savran, and L. Katgerman: Metall. Mater. Tran. A, 2004, vol. 35A, pp Suyitno, W.H. Kool, and L. Katgerman: Metall. Mater. Tran. A, 2005, vol. 36A, pp Suyitno, D.G. Ekin, and L. Katgerman: Mater. Sci. Eng., A, 2006, vol. 420, pp D.G. Ekin and L. Katgerman: Metall. Mater. Tran. A, 2008, vol. 38A, pp U. Feurer: Quality Control of Engineering Alloy and the Role of Metal Science, Delft Univerity of Technology, Delft, The Netherland, 1977, pp T.W. Clyne and G.J. Davie: Br. Foundryman, 1981, vol. 74, pp L. Katgerman: J. Met., 1982, vol. 34 (2), pp N.N. Prokhorov: Ru. Cat. Prod., 1962, vol. 2, pp I.I. Novikov: Goryachelomkot Tvetnykh Metallov i Splavov (Hot Shortne of Non-Ferrou Metal and Alloy), Nauka, Mocow, 1966 (in Ruian). 18. B. Magnin, L. Katgerman, and B. Hannart: in Modeling of Cating Welding and Advanced Solidification Procee VII, M. Cro and J. Campbell, ed., TMS, Warrendale, PA, 1995, pp M. Rappaz, J.-M. Drezet, and M. Gremaud: Metall. Mater. Tran. A, 1999, vol. 30A, pp M. Braccini, C.L. Martin, and M. Suery: in Modelling of Cating Welding and Advanced Solidification Procee IX, P.R. Sahm, P.N. Hanen, and J.G. Conley, ed., Shaker Verlag, Aachen, Germany, 2000, pp A.A. Griffith: Philo. Tran. R. Soc. London, 1920, vol. A221, pp A.A. Griffith: Proc. 1t Int. Cong. for Applied Mechanic, C.B. Biezeno and J.M. Burger, ed., Waltman, Delft, The Netherland, 1924, pp B. Commet, P. Delaire, J. Rabenberg, and J. Storm: in Light Metal 2003, P.N. Crepeau, ed., TMS, Warrendale, PA, 2003, pp W. Kurz and D.J. Fiher: Fundamental of Solidification, Tran Tech Publication, Aedermanndorf, Switzerland, T.S. Piwonka and M.C. Fleming: Tran. AIME, 1966, vol. 236, pp K. Kubo and R.D. Pehlke: Metall. Tran. B, 1985, vol. 16B, pp J. Campbell: Cating, Butterworth, Oxford, United Kingdom, W.S. Pellini: Foundry, 1952, vol. 80, pp T.W. Clyne and G.J. Davie: Solidification and Cating of Metal, TMS, Warrendale, PA, 1979, pp P. Ackermann and W. Kurz: Mater. Sci. Eng., 1985, vol. 75, pp W.M. Van Haaften, W.H. Kool, and L. Katgerman: Mater. Sci. Forum, 2000, vol , pp P. Winiewki: Doctoral Thei, Univerity of Pittburgh, Pittburgh, PA, M. Sue ry, C.L. Martin, M. Braccini, and Y. Bréchet: Adv. Eng. Mater., 2001, vol. 3 (8), pp J. Ni and C. Beckermann: Metall. Tran. B, 1991, vol. 22B, pp Y.S. Touloukian and E.H. Buyco: Thermophyical Propertie of Matter, Vol. 4: Specific Heat, Metallic Element and Alloy, IFI/ Plenum, New York, NY, VOLUME 40A, OCTOBER 2009