Advanced Multiphase Modeling of Solidification

Transcription

1 Advanced Multiphae Modeling o Solidiication with OpenFOAM A. Vakhruhev 1, A. Ludwig 2, M. Wu 1, Y. Tang 3, G. Hackl 3, G. Nitzl 4 1 Chritian Doppler Laboratory or Advanced roce Simulation o Solidiication and Melting, Univerity o Leoben, 8700 Leoben, Autria 2 Simulation and Modeling o Metallurgical rocee, Department o Metallurgy, Univerity o Leoben, 8700 Leoben, Autria 3 RHI AG, Technology Center, Magneittrae 2, 8700 Leoben, Autria 4 RHI AG, 1100 Vienna, Wienerbergtrae 9, Autria Abtract Accumulated reult o uing OpenFOAM a a tool to imulate variou phenomena accompanying cating procee are preented in thi paper. The deigned olidiication model incorporate ma and heat traner along with the decription o the multiphae olid / liquid region being ormed a a porou media [1]. Turbulent low interaction with the o called muhy zone i taken into account and it inluence on the olid hell ormation i examined extending reult o reerence [2,3]. Motion o the dierent non-metallic incluion i modeled within the Lagrangian rame o reerence [4] and veriied with the experimental data. A new approach o modeling elatic deormation o the olid hell during the withdraw proce in the unnel-haped mold type i propoed a a upplement to improve the algorithm with local numerical meh reinement preented in [5]. Keyword Solidiication, dicrete phae, turbulence, deormation. 1. INTRODUCTION Multiphae low phenomenon i o the great interet or the modern CFD ield invetigation, taking into account phae tranition proce a well a the momentum and energy exchange between the phae. Thee tak can be olved in the mot complex cae by incorporating the ull et o the Navier-Stoke equation or each o the ingle phae being under the conideration. Thi i o called Eulerian approach, where the decription o the velocity, denity, temperature, olid raction and other imulated ield ocue on the peciic location in the pace through which the multiphae low advance with the coure o time. reented tudie incorporate modeling reult or the around-cating procee, uch a dendritic olidiication, motion o the non-metallic incluion and ga bubble within the melt. The newet invetigation here concern tre-and-train analyi in the olidiied hell. An enthalpy-baed mixture olidiication model with conideration o turbulent low [6-9] wa introduced by the current author to model conventional continuou cating [4,10]. According to Voller et al., the treatment o the motion o the olid phae ha a dramatic inluence on the convection o the latent heat, hence on the hape o the evolving muhy zone [11-14]. New continuou cating technologie, e.g. the thin lab cating (TSC), or their advantage o integration o the cating-rolling production chain, energy aving, high productivity and near net hape, i likely to replace the conventional lab cating or 1

2 producing lat/trip product [15,16]. However, the requently reported problem like the enibility to breakout and edge/urace crack have challenged the metallurgit to conider modeling tool to optimize and control the TSC parameter [17-19]. The key iue or the modelling i to conider the evolution o the olid hell, which interact trongly with the turbulent low and in the meantime i ubject to continuou deormation due to the unnel hape (curvature) o the mould. Model ormulation and the analyi o the imulation reult are preented here. During modeling o the turbulent low and olidiication in the continuou cating proce to tackle particle and ga bubble motion, Lagrangian method give a huge advantage i it i exploited or the imulation o the additional phae with the relatively mall volume raction (<10%). Thereby, divere nonmetallic incluion and mall ga bubble can be decribed within Lagrangian rame o the reerence. In the preented work the reult o tudie, obtained uing commercial CFD otware FLUENT, are compared with the comprehenive experimental meaurement and with the imulation uing OpenFOAM olver, deigned by the author, incorporating the Dicrete hae Model (DM) and Dicrete Random Walk (DRW) model [4,20,21]. Momentum exchange between continuou and dicrete phae, which i rather important or the tranient imulation o the nonmetallic incluion and ga bubble motion during continuou cating, i taken into account by mean o two-way coupling technique. The advantage o the preented model conit o open-ource implementation uing OpenFOAM CFD otware package, which permit to combine dierent multiphae ubmodel and to apply or the multiphae imulation. 2. SOLIDIFICATION MODEL An enthalpy-baed mixture olidiication model [11-13] i applied. Thi mixture combine liquid -phae and olid -phae, which are quantiied by their volume raction, and. The morphology o the olid phae i uually dendritic, but here we conider the dendritic olid phae a a part o the mixture continuum. The mixture continuum change continuouly rom a pure liquid region, through the muhy zone (two phae region), to the complete olid region. The evolution o the olid phae i determined by the temperature according to a T relation (e.g. Gulliver-Scheil), 1 0 T T T T 1 liquidu 1 kp 1 T liquidu T T T Eutectic liquidu T T T. Eutectic (1) Only one et o Navier-Stoke equation, which i applied to the domain o the bulk melt and muhy zone, i olved in the Eulerian rame o reerence. u 0, (2) u u u p ( e u) Smon, (3) t 2

3 where u u u u u bulk melt region muhy zone olid region. (4) Here u, the olid velocity, i etimated by olving volume conervation and Laplace equation (ee below). The momentum ink due to the drag o the olid dendrite in the muhy zone i modeled by the Blake-Kozeny law: S mon K u u ) ( (5) The permeability, K, i modeled a unction o the primary dendrite arm pacing 1 [22]: K (6) 2 The energy equation applie to the entire domain, h t uh e T S e. (7) Here h i the enible enthalpy o the olid h h re cp dt. At a given temperature the T re liquid phae i aumed to have an enthalpy o h h L. Releae o latent heat by olidiication, L, i treated in the ource term o the energy equation, Se L t L u. (8) 3. TURBULENT FLOW MODEL FOR SOLIDIFICATION A low Reynold number k - model wa introduced by recott and Incropera [6-9] to handle the turbulence during olidiication. In current tudie a realizable k - model wa employed providing improved perormance or low involving boundary layer under trong preure gradient and trong treamline curvature. The governing equation or the turbulence are k t t uk t k G k r t,k t u C1 rt, The turbulent randtl number or k : 3 K T S C, (9) 2 r t, k =1.0, and or : 2, (10) k r t, =1.2; G i the hear production o turbulence kinetic energy; S 2S S ij ij Sij 0. 5 u j xi ui x j ; S k /. A imple approach i ued to modiy the turbulence kinetic energy in the muhy zone. It i aumed that within a coherent muhy zone turbulence i dampened by hear which i linearly,

4 correlated with the reduction o the muh permeability. The inluence o turbulence on the momentum and energy tranport are conidered by the eective vicoity, e t, and 2 the eective thermal conductivity, e t, where t Cμk, t tcp, rt, h, C μ i a unction o velocity gradient and enure poitivity o normal tree; r t, h i the turbulent randtl number or energy equation (0.85). 4. VELOCITIES IN THE DEFORMING SHELL A linear elaticity model [23] i urther impliied to etimate the olid velocity. I we aume that in the olid domain the elatotatic condition applie and the body orce i ignorable, the governing equation obtained i called Navier-Cauchy equation or elatotatic equation: 0 where i the diplacement vector. So called Lamé parameter E, E, (11), are, (12) where E repreent Young' modulu and i oion ratio. I the olid hell i incompreible and it deormation i at mall train ( 0. 5 ), then a volume conervation condition i ulilled: 0, (13) and the irt term o Eq. (11) i orced to zero a well: 0. (14) Tranorming Eq.(13) and (14) rom Lagragian rame into Eulerian rame by conidering u t, we obtain volume-conerved Laplace equation: u 0, u 0. In 2D cae, thee volume-conerved Laplace equation can be olved with a method (Method I) [24]. A tream unction and a curl o the olid velocity u are deined by: (15) u y y u, u y x x x, x u y, (16) with x y T u ( u, u ). Thereore, correponding ytem o Eq. (15) can be written in orm o: 4

5 2 2, 2 2 x y x y (17) Thi method provide accurate olution, but it applie only to the 2D cae. An alternative and approximation method (Method II), applicable to both 2D and 3D, i to olve the one-phae Navier-Stoke equation with an ininite olid vicoity. In the current work the approximation method i to be jutiied by comparion with the method on the 2D bae. 5. MODELING DISCRETE HASE The incluion and bubble motion along with a highly turbulent low are under conideration in the preented numerical model. To peciy the continuou phae motion, a ixed inite volume meh i ued with a o-called collocated or non-taggered variable arrangement (Rhie and Chow [25], erić [26]), where all phyical value hare the ame control volume (CV), and all lux variable reide on the CV ace. The generalized orm o the divergence theorem i ued throughout the dicretization procedure to repreent ma and momentum conervation law in integral orm over the control volume. Nonmetallic incluion and ga bubble hereinater reerred to a Lagrangian particle and repreenting the dicrete phae are tracked within Lagrangian ramework. The Navier-Stoke equation with an aumption o the liquid incompreibility are ued to imulate the liquid melt motion. The equation ytem coniting o the continuity and momentum equation i u 0, (18) u u u p ( e u) S, (19) t The preence o the dicrete phae i deined by the ource term S p, which repreent the momentum exchange between Lagrangian particle and the liquid low. The etimation o it quantitative evaluation i decribed later. In the preented tudy, a RANS turbulence approach i ued baed on k model. To employ the DM theory, a deinition o the Lagrangian particle hould be introduced. Hereinater we conider a pherical particle with the diameter D and the denity. Thereby the ma o the particle with the volume V i etimated a m 1 3 V D. (20) 6 Next, it i required to track the antecedent particle trajectorie through the imulation domain. Thereto each Lagrangian object i provided with it own poition vector x in the Carteian ytem o coordinate. To determine the particle velocity u and the 5

6 correponding acceleration a, it i uicient to compute the time derivative o the trajectory vector x o the correponding order: a u x. (21) The deining equation o the motion in the Lagrangian rame i baed on the Newton Second Law. It bind the acceleration o the particle with the reulting orce, acting on it: m u F, (22) where a um o the external orce F originate in the inluence rom other Lagrangian particle a well a in the impact rom the urrounding continuou media motion. A number o the orce are taken into account in the preented work: particle drag, gravitational orce, lit orce, virtual ma orce a well a preure and tre gradient orce. For the detailed ormulation and decription o each orce pleae reer to publication [4]. It hould be noticed, that drag coeicient in the model depend on the low regime around the dicrete particle (ee Figure 1) or the proper decription o the incluion behavior in the turbulent low. Figure 1: Spherical particle drag law To repreent the particle / olid wall interaction the bouncing model i deined by the retitution actor wall and the wall riction coeicient wall. Splitting the particle velocity n vector into the normal u τ and the tangential u component, one can achieve a new particle * velocity u ater it interaction with the irm urace: u u u * * n * τ u * n ε u u wall 1 μ * τ n wall,, u τ. 6 (23)

7 Figure 2: Interaction o Lagrangian particle with continuou phae To couple the momentum in the both phae, the momentum exchange i determined in each inite volume o the numerical meh being ued or the continuou phae (ee Figure 2). For the meh element with the index k and o the volume V k the momentum exchange during time interval t i calculated a k 1 S V Δt k m k u u out k in. (24) The complexity o the dicrete phae interaction with the vicou low in the reality i deined by the tochatic nature o the turbulent low. The typical trajectory o the mall particle / ga bubble inide the turbulent eddy i repreented in Figure 3. Figure 3: article trajectory within the turbulent eddy To introduce uch behavior o the Lagrangian particle in the preented model a Dicrete Random Walk model can be employed [20,21]. It main aumption concern the introduction o the o-called eddy time and the croing time cale. Firt o them decribe the characteritic time, within which the eddy can exit until it i diipated. Second one deine the time interval uicient or the dicrete phae object to cro the eddy. Both parameter are baed on the local turbulence parameter. The intant velocity become a um o the mean and pulating component in the local point, and it luctuation amplitude depend on the kinetic energy (ee [4] or detail). Next we will proceed to the imulation reult and conider their application or the modelling o the real around-cating procee. Here ome new tudie being o the great importance or the imulation o the olidiication are dicued, and it wa an advantage o uing 7

8 OpenFOAM and the tranparency o it ource code to be able, or example, to couple luid low and tre-and-train analyi in the ingle CFD otware, which i rather hard or ome time impoible uing commercial otware. 6. LATENT HEAT AND FLOW REGIME INLUENCE In thi ection an importance o the latent heat advection due to the motion o the dendritic tructure along with the turbulent/laminar low regime inluence i tudied. A 2D benchmark to analyze mention phenomena i deined, a hown in Figure 4. The melt with nominal compoition o Fe-0.34wt.%C ill continuouly through the inlet into the domain with contant temperature (1850 K). The cating ection i gradually reduced to mimic the olidiication and hell deormation in TSC. Other material propertie being ued reer to [1]. Solid velocity i calculated with the coniguration o Figure 4(a). The whole domain i illed with the olid which i extruded downward with the contant peed u pull =0.07 m/, being et at the outlet. Free lip condition i applied at the wall and non-rotational condition ( u 0 ) i ued at the inlet. Right boundary repreent ymmetry plane. Flow-olidiication imulation i conigured in Figure 4(b). A ma balance between the inlet and the outlet i ulilled: uinain upull Aout, where A in and A out are the inlet and outlet urace area. At the wall ree-lip condition i aumed. (a) (b) Figure 4. Coniguration o a 2D benchmark (a) or olid velocity calculation and (b) or the olidiication low calculation. The geometry in vertical direction i caled by 1/8 (the ame hereater). In the preented tudie olid velocitie calculation i decoupled rom the imulation o the melt low. They are initially etimated by olving (15) with an aumption that the whole domain i illed with the olid and are ued later on or the olidiication modeling in the muh region only. Thereby liquid core in the center can accelerate or retard or balancing the total ma low rate. The calculated olid velocity, u, i hown in Figure 5 (a)-(b). Solid phae enter the domain in parallel to the traight wall. In the ection-reduction region the olid i extruded and it velocity i gradually increaed. The urace proile i orced to move along the curved wall. Comparion o the calculation reult by two dierent method (I and II) i made in Figure 5(b)-(d). The maximum error caued by method II, olving a impliied Navier-Stoke equation with an ininite olid vicoity, i 0.8%, alling in the engineering tolerance. Thi olid velocity will only be ued by the low-olidiication model, Eq.(5) and (8), in the region where olid phae exit. 8

9 2 (m /) u I,II (m/ ) u (%) a) b) c) d) Figure 5. Calculated olid velocity: (a) tream unction, (b) u obtained with Method I, (c) u obtained with ininite olid vicoity Method II and (d) velocity dierence I,II I II I between the two method ( u u u / u 100). Table 1. arameter tudy o the low-olidiication model Flow regime Treatment o latent heat (Eq.(8)) integral (vol.%) * Cae I laminar S 9.38 e Cae II laminar ignoring in S e Cae III turbulent S 8.81 e Cae IV turbulent ignoring in S e * integral : total olid phae (vol.%) in the whole calculation domain at the teady tate. = Cae I - Cae II ath I ath I ath I ath I I a) b) I c) Figure 6. redicted teady tate olidiication with a model conidering laminar low only: ditribution or (a) Cae I and (b) Cae II; (c) dierence in between Cae I and Cae II. In order to invetigate dierent model aumption, e.g. the inluence o olid velocity and turbulence, on the olid hell ormation by olidiication, 4 imulation cae are deined (Table 1). For the boundary condition reer to Figure 4(b). The predicted olid hell ormation or the Cae I and II (only laminar low i conidered) at the teady tate i hown 9

10 in Figure 6. Obviouly the treatment o the advection o latent in the energy equation i extremely important. Ignorance o the advection term, L u, in Cae II will to a great extent overetimate the olid hell thickne. More precie analye o the olid phae ditribution along ath I and II, marked in Figure 6(a) and (b), are made in Figure 7. a) b) Figure 7. Solid volume raction ditribution o dierent imulation cae along (a) ath I and (b) ath II are compared. Similar calculation were carried out or the turbulent low regime, but are not diplayed here becaue the global phae ditribution how imilar pattern to the Cae I and II. Intead the inluence o the turbulence on the olid hell ormation i analyzed (Figure 8). Comparion between Cae III and I how that the preence o turbulence hinder the olid hell ormation. Ignorance o the advection term, L u, will alo overetimate the olid hell thickne. More precie analye o the olid phae ditribution alone two ath I and II, marked in Figure 6(a) and (b), or the imulation Cae III and IV are alo made in Figure 7. Cae III = - = Cae I Cae III - Cae IV a) b) Figure 8. Inluence o turbulence on the olid hell ormation, i.e. the dierence in ditribution (a) between Cae III and I, (b) between Cae III and IV. Additionally a meh and time tep dependency o the numerical olution wa examined. A low latent heat relaxation actor (0.05) [1] along with a relatively large number o iteration (50 per time-tep) allowed uing a relatively large time tep without problem o divergence. It wa hown that the increae in time tep did not inluence the inal teady tate olution. To 10

11 Mold wall Bulk liquid improve the accuracy a neceity to ue eparate reinement region or the temperature and olid raction ield wa approved previouly by author [5]. Thereby baed on the error analyi o the energy equation and the reolution criterion o the Gulliver-Scheil correlation (Eq.(1)), conecutive meh reinement were made (Figure 9). Eventually meh independent reult were obtained. temperature boundary layer muhy zone boundary layer Figure 9. Meh reinement to track the temperature boundary layer and olidiication ront u (m/) a) b) Figure 10. Quai teady tate imulation reult o an engineering TSC. a) 3D ditribution o the velocity vector ield and evolution o the olid hell (dark region in the 5 cro ection); b) zoomed velocity ield in the central plane near the narrow ace; c) detailed velocity (u y component) proile and olid volume raction along two path cro the muhy zone. Baed on the aorementioned model a imulation o the real engineering TSC (width 1726 mm and thickne 72 mm) wa perormed, and the calculation reult i hown in Figure 10. The calculation domain include the ubmerge entry nozzle and entire mold region and part o water cooled trand (till 2000 mm rom menicu). To enure the calculation accuracy numerical technique like parallel computing and meh adaptation are necearily applied. More than 1 million computational cell are ued to reolve the interdendritic low in the 11 c)

12 muhy zone (Figure 10(c)). In [2,3] Laplace equation wa olved or olid velocitie with a contant vertical component aumption retricting longitudinal deormation. The new approach doen t have uch a limitation and help to mimic the muh better. 7. MODELING DISCRETE HASE Baed on the implementation o the DM olver in the OpenFOAM CFD otware package, a number o imulation were made. Firt o all, comparion o the nonmetallic incluion and ga bubble behavior in the turbulent low wa made (Figure 11). article o two type (with denitie o 2700 and 5000 kg/m 3 accordingly) and argon bubble (denity 0.19 kg/m 3 ) were injected rom dierent point o the 2D continuou cating mold geometry with the teady tate luid low been etablihed. It i marked on the igure by the gray-colored tream line. Figure 11: Initial injection o the particle and ga bubble Bubble (blue circle) and particle (red dot), preented at the let draw, were ejected through SEN with the ame initial velocitie. On the right picture, black-dotted particle are injected at the region o the large top vortex at the lag region, other (red and green) at the SEN port. The only dierence in the imulation etup or green-marked particle i that turbulence / particle interaction i not taken into account (in other word, DRW model i witched o or that type o object). Hereby it i poible to etimate the igniicance o the eddy inluence on the particle trajectorie. Further one can ee (Figure 12) how the ditribution o the ga bubble and particle dramatically change ater they are involved by the low. Bubble, a it wa emphaized here beore, are trongly inluenced by the buoyancy orce counteracting the main low drag: even at the initial tage o the imulation ome o the ga bubble are already captured by the lag. Smaller particle motly ollow the melt motion. Additionally, they are pread out over the imulation domain due to the interaction with the turbulent low. 12

13 Figure 12: Ga bubble and olid particle ditribution ater injection i complete Figure 13: Developed ditribution o the ga bubble and olid particle during numerical imulation a) b) c) Figure 14: Ga bubble and olid particle ditribution in 3D mold geometry in (a) 1 econd (b) 2 econd (c) 3 econd ater injection 13

14 Finally, ater the motion o the particle and bubble i developing in the coure o time (Figure 13), ollowing concluion can be made: ga bubble motly tend to rie to the lag urace; however ome o them are till brought with the melt low downward and leave the imulation domain through the outlet; thereby they can be entrapped into the inal product and acilitate the poroity ormation; particle incluion are trongly inluenced by the melt low; turbulence dramatically enhance their ditribution (e.g. green colored particle trictly ollow the tream line a oppoed to other type with the DRW model taken into account). Diperion eect i even more ampliied in 3D real geometrie (Figure 14). Obtained reult emphaize the importance o the low pattern in the continuou cating mold, which can either enhance or reduce the poroity ormation and nonmetallic incluion entrapment into inal product. reented analyi o the reult, obtained with the developed olver or the imulation o the nonmetallic incluion and the ga bubble in the continuou cating mold, utain the eiciency o the numerical experiment concept or the continuou cating epecially when it i rather diicult to receive meaurement and experimental data. Open-ource CFD package OpenFOAM proved to be an excellent deign tool or the implementation and urther development o the DM approach or the multiphae low imulation. 8. VERIFICATION OF THE DM MODEL A an extenion o the preented work a veriication o the DM olver wa carried out baed on the obtained reult o the water modeling experiment regarding the particle low. article low experiment included tundih water low with the particle been injected through the nozzle above the trial device. article being lighter than water tended to reach the water / air urace due to the buoyancy orce, which competed with the trong low, dragging the particle toward the reervoir outlow. Tranparent wall o the tundih permitted to capture the particle motion during the trial with the peed camera, giving an impreion regarding the turbulent low behavior. Fig. 15. Wooden rame at the tundih and the particle, being captured at the urace (voetalpine Stahl laboratory, Linz, Autria) Figure 15 how the inal tage o the water modeling experiment with the particle been captured within emerged wooden rame on the top o the tundih. Figure 16(a), howing 14

15 intantaneou picture o the turbulent low tream line (OpenFOAM imulation) tree the complexity o the convectional mechanim inluencing particle motion. a) b) Fig. 16. article imulation in tundih: (a) turbulent low tream-line and (b) patching the top urace to count the captured particle To compare experimentally meaured data with the modeling reult the top urace o the imulation domain (repreenting the lag layer) wa divided to the correponding number o patche exactly conorming to the location o the cell o the capturing rame in the experiment (ee Figure 16(b)). 20,00% 18,00% 16,00% 14,00% 12,00% 10,00% 8,00% 6,00% 4,00% 2,00% 0,00% 20,00% 20,00% 18,00% 18,00% 16,00% 16,00% 14,00% 14,00% 12,00% 12,00% 10,00% 10,00% 8,00% 8,00% 6,00% 6,00% 4,00% 4,00% W5 W5 2,00% W4 W4 2,00% W4 W3 W3 W3 0,00% W2 W2 0,00% W2 A B C W1 A B C W1 A B C W1 Experiment FLUENT OpenFOAM a) W5 18,00% 18,00% 18,00% 16,00% 16,00% 16,00% 14,00% 14,00% 14,00% 12,00% 12,00% 12,00% 10,00% 10,00% 10,00% 8,00% 8,00% 8,00% 6,00% 6,00% 6,00% 4,00% 2,00% 0,00% 4,00% 2,00% W5 W4 0,00% W3 W2 A B C W1 FLUENT v. OpenFOAM W5 W4 W3 W2 A B C W1 Experiment v. FLUENT b) 4,00% W5 2,00% W4 W3 0,00% W2 A B C W1 Experiment v. OpenFOAM Fig. 17. Comparion o the experimental data with the imulated one: (a) total amount o the particle being captured; (b) dierence between acquired data et 15

16 Figure 17 repreent quantitative comparion between meaured and imulated reult. One can ee that FLUENT imulation a well a one done with OpenFOAM howed good agreement with the experiment along the tundih wall. FLUENT olver underetimated the amount o the captured particle in the central part o the tundih, wherea in OpenFOAM imulation we got a perect agreement. On the contrary, the mean part between wall and center plain wa much better treated by the FLUENT DM, than OpenFOAM one. The undamental o the imulation reult deviation rom the experimental obervation conit o the particle orce treatment in the dierent model ormulation and hould be tudied in the uture. 9. SUMMARY A long term experience with CFD otware development ha hown that the OpenFOAM package can be extenively ued or the combination o low-olidiication calculation o the conventional cating a well a or the thin lab cating application where the deorming olid hell i oberved. Modelling reult o a 3D thin lab cating are preented to demontrate the unctionality o the numerical model. For particle and bubble motion and extended DM model wa developed and the irt veriication with the water modelling are carried out. Further model development or the around-cating procee imulation with OpenFOAM (e.g. ree urace imulation along with the olidiication) and veriication againt experiment are deired. 10. ACKNOWLEDGMENT The inancial upport by RHI AG, the Autrian Federal Minitry o Economy, Family and Youth and the National Foundation or Reearch, Technology and Development i grateully acknowledged. 16

17 REFERENCES [1] M. Wu, A. Vakhruhev, G. Nummer, C. eiler, A. Kharicha and A. Ludwig, Importance o Melt Flow in Solidiying Muhy Zone. Open Tranport henomena J. Bentham Open. 2, pp (2010) [2] A. Vakhruhev, A. Ludwig, M. Wu, Y. Tang, G. Nitzl and G. Hackl, Modeling o Turbulent Melt Flow and Solidiication rocee in Steel Continuou Cater with the Open Source Sotware ackage OpenFOAM. roc. OSCIC 10, Munich, Nov. 4-5, pp (2010) [3] A. Vakhruhev, A. Ludwig, M. Wu, Y. Tang, G. Nitzl and G. Hackl, Modeling o Heat Traner and Solidiication rocee in Thin Slab Cater. roc. ECCC 2011, Düeldor, June 27 July 01, S18, pp (2011) [4] A. Vakhruhev, A. Ludwig, M. Wu, Y. Tang, G. Nitzl and G. Hackl, Modeling o nonmetallic incluion and ga bubble motion in continuou cater. roc. STEELSIM 2011, Düeldor, June 27 July 01, S17, pp. 1-8 (2011) [5] A. Vakhruhev, A. Ludwig, M. Wu, Y. Tang, G. Nitzl and G. Hackl, Coupling the turbulent low with the olidiication procee in OpenFOAM. roc. OSCIC 11, ari Chantilly, Nov. 3-4, pp (2011) [6] recott J and Incropera F 1994 Tranp. hen. in Mater. roc. & Manu.- ASME HTD [7] recott J and Incropera F 1994 J. Heat Traner [8] recott J, Incropera F and Gakell DR 1994 Tran. ASME [9] recott J and Incropera F 1995 Tran. ASME [10] eiler C, Thoma BG, Wu M, Ludwig A and Kharicha A 2008 Steel Re. Int [11] Voller VR and rakah C 1987 Int. J. Heat Ma Traner [12] Voller VR, Brent AD and rakah C 1989 Int. J. Heat Ma Traner [13] Voller VR, Brent AD and rakah C 1990 Appl. Math. Modeling [14] Chakraborty R and Dutta 2011 Metall. Mater. Tran. 42B in pre (DOI: / ). [15] Yin RY 2009 J. Iron & Steel Re. 16 (Supplement 1) 1. [16] Birat J and Bobadilla M 2006 roc. McWAS XI (Ed: Gandin CA and Bellet M, TMS ublication) 33. [17] Tang Y, Krobath M, Nitzl G, Eglaeer C and Morale R 2009 J. Iron & Steel Re. 16 (Supplement 1)

18 [18] Thoma BG 2001 Brimacombe Lecture, 59 th Electric Furnace Con. (Iron & Steel Soc.) 3. [19] Tian X, Zou F, Li B and He J 2010 Metall. Mater. Tran. 41B 112 [20] Graham, D. I.; Jame,. W.: Turbulent diperion o particle uing eddy interaction model; Int. J. Multiphae Flow, 1996, vol. 22, Iue 1, [21] Graham, D. I.: On the inertia eect in eddy interaction model; Int. J. Multiphae Flow, 1996, vol. 22, Iue 1, [22] Gu J and Beckermann C 1999 Metall. Mater. Tran. 33A [23] Slaughter WS 2002 The Linearized Theory o Elaticity (Boton: Birkhäuer). [24]Roache J 1998 Hermoa ubliher) Fundamental o Computational Fluid Dynamic (New Mexico: [25] Rhie, C. M.; Chow, W. L.: A numerical tudy o the turbulent low pat an iolated airoil with trailing edge eparation; 3rd Joint Thermophyic, Fluid, lama and Heat Traner Conerence, St. Loui, Miouri, 1982 [26] erić, M.: A Finite Volume method or the prediction o three-dimenional luid low in complex duct; hd thei, Imperial College, Univerity o London,