Numerical Modelling of Transport Phenomena and Macrosegregation during Ternary Alloy Solidification: Solutal Undercooling Effects

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1 Journa of Aied Fuid Mechanics Vo. No Avaiabe onine at ISSN EISSN DOI: /acadub.jafm Numerica Modeing of Transort henomena and Macrosegregation during Ternary Aoy Soidification: Souta Undercooing Effects S. Ganguy R &D Tata Stee Ltd. Jamshedur Jharkhand India Corresonding Author Emai: suva_@yahoo.co.in (Received Ari 7 08; acceted Setember 08) ABSTRACT In this aer a macroscoic mathematica mode is deveoed for simuation of transort henomena during ternary aoy soidification rocesses taking into account non-equiibrium effects due to souta undercooing. The mode is based on a fixed-grid enthay-based contro voume aroach. Microscoic features ertaining to non-equiibrium effects on account of souta undercooing are incororated through a nove formuation of a modified artition coefficient. The effective artition coefficient is numericay modeed by means of macroscoic arameters reated to the soidifying domain. Numerica simuations are erformed for ternary stee aoy by emoying the resent mode and the resuting convection and macrosegregation atterns are anayzed. It is observed that the consideration of non-equiibrium soidification in the resent mathematica aroach is abe to cature the thermo-souta convection and eads to rediction of accurate vaue of macrosegregation. The resuts from the resent mode matches we with the exerimenta observations ubished in the iterature. Keywords: Soidification; Modeing; Convection; Macrosegregation. NOMENCLATURE Nomencature shoud be in ahabetic order (A Z) and Greek etters shoud foow after Latin etters in ahabetic order (α β...). C c D f g h h c H secies concentration secific heat diffusion coefficient mass fraction voume fraction secific enthay convective heat transfer coefficient atent enthay K L T u v ρ ermeabiity atent heat temerature veocity comonent in x-dierction veocity comonent in y-direction dynamic viscosity density. INTRODUCTION Research efforts towards accurate mathematica modeing of aoy soidification rocess have been reorted in the ast (rescott and Incroera 996; Choudhary and Mazumdar 994; Kang et a. 005; Ganaoui et a. 00; Ganguy et a. 03). The roe of macroscoic transort henomena during soidification has been discussed in detai by many researchers (Brent et a. 988; Ganesan and oirier 990). The freezing of a soid invoves many comex hysica issues which determine the transort behavior governing the hase-transition rocess during soidification. The interactions of the soidifying met with the two-hase mushy region in the resence of both temerature gradient and soute concentration gradient roduces a resutant fow known as doube-diffusive convection. This thermo-souta convection drives the rejected soute eements away from the soid-iquid interface eading to a comosition variation in the domain thereby giving rise to macrosegregation (Femings 974). Severa studies have been undertaken to characterize the fuid fow and the deveoment of macrosegregation during soidification. Initia studies started with moving-grid mutie-domain formuations (Gadgi and Gobin 984) foowed by

2 S. Ganguy / JAFM Vo. No singe-domain continuum formuation (Bennon and Incroera 987; Voer et a. 989). These studies have highighted the imortant roe of thermosouta convection in the ong-range transort of soute and fina macrosegregation attern. An imortant asect in this context is roer accounting of microscoic issues in the macroscoic mode that dictate the macrosegregation behavior of the cast roduct. rediction of oca soute redistribution is generay given by Scheis equation (Femings 974) and is widey acceted. The mode roosed by Brody and Femings (966) takes into consideration the diffusion in the soid hase and rovides an imrovement of the Scheis mode. Cyne and Kurz (98) resented a mode for soute redistribution with raid soid state diffusion. Kobayashi (988) resented a modified microsegregation mode under the conditions of constant diffusion coefficient of soute in the soid constant equiibrium artition-ratio and araboic growth. It can be noted here that during soidification due to finite diffusivity in the iquid hase there may be accumuation of soute within a diffusion boundary ayer adjacent to the interface. This gives rise to a concentration gradient at the interface which in turn causes a oca change in the iquidus temerature. This is commony known as souta undercooing and is rimariy governed by iquid secies diffusivity dendrite growth rate thickness of the souta boundary ayer and the interface geometry (Raaz 989; Chakraborty et a. 00). In a the revious studies souta undercooing effect is essentiay negected whie modeing soidification. However for accurate rediction of the fina macrosegregation attern in the soidified roduct it is imerative that macroscoic mode for muticomonent systems shoud reresent ertinent microscoic issues within the macroscoic framework in a metaurgicay consistent manner. Athough there are attemts to incororate the effect of souta undercooing through micro-macro modeing (Raaz 989) simiar considerations in the framework of macroscoic modeing of muticomonent soidification is very rare. Aso mode for ternary aoy systems which can characterize reevant microscoic issues in the macroscoic framework in the resence of convective fow fied is not yet avaiabe in the iterature. Towards this the resent research endeavour is a nove attemt to cature the intricate characteristics of non-equiibrium soidification fow and macrosegregation. In the resent work a macroscoic mathematica mode for ternary aoy soidification is deveoed with articuar emhasis on non-equiibrium soidification considerations. The numerica aroach is essentiay based on fixed-grid enthay-based continuum formuation (Bennon and Incroera 987; Voer et a. 989). Microscoic features arising out of non-equiibrium effects due to souta undercooing are incororated through a nove formuation of modified artition-coefficient. The numerica imementation of the roosed agorithm correates the microscoic features with the overa convection fied thereby resenting a method for accurate rediction of fina macrosegregation behavior.. MATHEMATICAL MODELLING The foowing section outines the mathematica modeing rocedure and the soution techniques.. Governing Equations A two-dimensiona rectanguar moud containing iquid ternary meta aoy is considered where the boundary was are subjected to different heat transfer conditions as deicted in Fig.. Unidirectiona soidification takes ace from the eft boundary with the initia and boundary conditions shown in Fig.. Fig.. Comutationa domain with boundary secifications. Foowing a fixed-grid continuum formuation with a singe-domain aroach (Bennon and Incroera 987) the governing equations for mass momentum energy and secies transort can be written as foows u 0 () t ( u) u ( u) u Su Sb t () Equations () and () are continuity equation and momentum equation resectivey. Here u are the density veocity viscosity and ressure resectivey and the subscrit denotes the iquid hase. In Eq. () S denotes source term reresenting the fow u resistance in the mutihase region and is a function of the morhoogy of the soid-iquid mushy region. These source terms are evauated from Darcy s mode (Morvan et a. 999) of viscous fow through a orous medium (assuming zero veocity of soid hase and isotroic ermeabiity) as S u i i (3) K where S denotes the source term corresonding to i 30

3 s S. Ganguy / JAFM Vo. No veocity u i in the aroriate direction and ermeabiity. For the urose of modeing the above term K is rescribed as a function of iquid voume fraction ( ). Towards this the Carman- g Kozeny reation (Brent et a. 988) is used within a range of vaidity of 0 g 0. 5 i.e. 0 g 3 g K K (4) Here K o is orosity constant. It may be mentioned here that the above-mentioned Carman-Kozeny reation is found to be reasonaby vaid ti the iquid fraction ( ) attains a vaue of 0.5 which is g the critica vaue of iquid fraction ( cr g K is ) for aication of the Carman-Kozeny mode. For g 0.5 a hybrid mode is used (Morvan et a. 999) A o (5) A Fg s 3 g K GK where 0 g 0.4 A (6) Here F and G are described according to the rheoogy of susensions (Brenner 970) as F 0.5 arctan 00 G 0.5 arctan 00 cr where 0. 5 Here g cr g cr g g cr g 4 g (7) (8) is the critica iquid fraction for aication of Carman-Kozeny equation. It is worthwhie to mention here that in accordance with the hysics of dendritic morhoogy occurring due to the evoution of soid dendrites in iquid matrix during hase change henomena in muticomonent soidifying system (Morvan et a. 999) severa modes are used in contemorary iterature for accurate rediction of mushy zone fow hysics (Odenburg et a. 99) and associated rheoogica characteristics (Brenner 970). The rheoogy of susensions (Brenner 970) of soid in a iquid is a comex function of its hysica roerties and of rocesses that occur on the scae of the susended soid artice morhoogy. Some imortant factors are voume fraction of different constituent hases hase/artice interaction nature of buk fow fied. In the resent study deending on the voume fraction of the iquid in the hase-changing domain different functiona forms describing the hysics of soid-iquid interaction are identified for aroriate modeing of mushy zone fuid fow. Accordingy deending on the oca iquid voume fraction ermeabiity factor is determined and Eqs. (3) - (7) is used to evauate the soid-iquid interaction force. The buoyancy source term S in Eq. () b constitutes of both therma and souta effects and is given by the differences between oca and initia vaues of temerature and iquid soute concentrations resectivey. In the resent context density and concentrations are imicity reated through Boussinesq aroximation foowing the conventiona assumtion that the imortant density changes are those associated with buoyancy force in the iquid (souta buoyancy and therma buoyancy) (Ganaoui et a. 00; rescott and Incroera 996). In the case of a ternary aoy the buoyancy term in Eq. () can be written as: Sb= g[ T ( T T0 ) s( C C ) ( C C0)] 0 s (9) Here T and s are the therma voumetric coefficient of exansion and souta voumetric coefficient of exansion resectivey is the initia temerature and C 0 T 0 is the initia soute concentration. Subscrit and refer to eement and eement resectivey. The therma energy conservation equation is given as foows: ( T) ( ut). g g T s t g H u H c t (0) where k here g k Tk c k and H is the atent enthay. denotes voume fraction and subscrits and s corresonds to iquid and soid hases resectivey. T is the temerature c is the secific heat and k is the therma conductivity. The secies conservation equation is given as foows: t () C. uc. DC Sc Here D is the mass diffusion coefficient and the source term. Equation () reresents the genera form of secies conservation equation and is determined by the microstructure under consideration. In case of non-equiibrium soidification reresented by a coumnar dendritic microstructure we have D S C gsdsk g D () t g C k C g s t s S c is (3) In the resent case of a ternary aoy system two indeendent secies transort equations need to be considered; one equation for each constituent secies. The reevant boundary conditions are as foows: T x Left wa: u =0 v =0 k h T T c 3

4 S. Ganguy / JAFM Vo. No ( C ) x =0 ( C ) x Right wa: u =0 v =0 ( C ) x =0 (4) T 0 x ( C ) x =0 =0 (5) To wa: u =0 v =0 T 0 y ( C ) y =0 ( C ) y =0 (6) Bottom wa: u=0 v=0 ( C ) y T 0 y =0 ( ) y =0 (7) C Here u and v are veocity comonents aong x-axis and y-axis resectivey is the convective heat h c transfer coefficient. The initia conditions are written as: At t=0 u=0 T =Ti ( C )= ( Ci ) ( C ) = ( Ci ). Modeing of Souta Undercooing (8) For the urose of modeing the microscoic advection effects artition coefficient in Eq. k (3) has to be aroriatey modified. The basis of such modification ies on the hysics of macroscoic soidification modes using the ever rue or Schei s equation describing the microscoic soute conservation with the assumtion of wemixed soute in the iquid state. In ractice the soute gets accumuated on the iquid hase within diffusion boundary ayer next to the interface. This gives rise to souta undercooing which can be quantified as the difference between the interfacia and voume-averaged iquid secies concentration. This rimariy refers to a change in oca iquidus temerature due to the change in secies concentration and deend on the thermodynamic and hysica roerties such as diffusivity boundary ayer thickness morhoogica arameters voume fraction etc. Souta boundary ayer characteristics at the interface affects the soute distribution in the buk domain which again affects the convective fow fied resuting in overa macrosegregation attern. Incororation of such considerations in the macroscoic modeing framework cas for devising a method to cature the effect of souta undercooing. Towards this an effective artition coefficient ( ) is defined and correated with the equiibrium artition coefficient k ) as ( k k k R D k e k (9) Here k is given by the ratio of soid comosition to buk iquid comosition R is the rate of interface movement is the diffusion boundary ayer thickness D is diffusion coefficient in the iquid hase. Equation (9) assumes immense significance as it reates the comosition of soid to the aoy comosition and growth conditions. The arameter is often defined as the oca souta ecet R D number and takes into account the effect of souta undercooing. Above consideration is necessary to determine the overa fow behavior and the fina macro-segregation attern. 3. NUMERICAL MODELLING The coued governing differentia equations described in the receding section are soved using a finite voume methodoogy (atankar 980). The transort equations are soved according to the SIMLER agorithm (atankar 980). The above agorithm is aroriatey modified to account for hase change considerations during nonequiibrium soidification due to souta undercooing. This is achieved by modification of artition coefficient for convective effects in the agorithm. For the urose of imementation it is imortant to characterize the ratio of advection to diffusion strength defining the souta ecet number e R D. The convective strength can be exressed as where is hase V A change rate er unit voume is the voume of the contro voume interfacia area. The diffusion strength can be defined as where is the diffusion D L ref V A is the soid-iquid ength scae in the iquid. The iquid-hase mass conservation equation is discretized to obtain the hase change rate as: g t g u ref (0) An imicit forward differencing scheme is adoted to discretize the transient term in Eq. (0). The convective source term is discretized using uwind formuation. The strength of diffusive transort can be roery evauated with rescrition of macroscoic estimates of diffusion ength scae made as DL xt xt t t () ref where x is the ocation of the mushy-iquid interface and t is the time ste. For accurate rediction of the iquid fraction in the resent enthay-based aroach the atent heat content of each comutationa ce is udated according to the temerature and/or secies concentration vaues redicted by macroscoic conservation equations during each iteration within a time ste. In the resent context an iterative udate scheme (Brent et a. 988) is adoted as H = [ H ] + [ ] n a a n [{ h } c F { H } ] () 0 n n where H is the atent heat content of the 3

5 S. Ganguy / JAFM Vo. No Tabe Thermodynamic arameters of eements for ternary aoy Eement Carbon Manganese Nicke (C ) (Mn) (Ni) Cin (Wt %) (/Wt %) D (m/s) s T 0-9 Ds (m/s) (K/Wt%) k C comutationa ce around grid oint h is the sensibe enthay c is the secific heat reaxation factor n is the iteration eve a 0 is a a and are the coefficients of the finite voume discretization equation (atankar 980) and is the inverse of the atent heat function. It is to be noted here that in the above equation needs to be devised consistenty with the microscoic hysics foowed in the resent mathematica formuation so that accurate resuts ertaining to secific soidification mode is obtained. For the non-equiibrium soidification situation with souta undercooing the function can be exressed as k H F H c Tm c T m TL (3) where c the atent heat sovent T L F is the secific heat of the mixture L is T m L F F is the meting oint of the is the iquidus temerature and the effective artition coefficient. 4. RESULTS & DISCUSSIONS k Numerica simuations are erformed for the case of a (Fe-0.08wt%C-0.wt%Mn) stee aoy with iron carbon and manganese as the constituent eements. The iquidus temerature of the ternary stee aoy system for the resent case is 805 K whie the met initiay is at a uniform temerature of 830K inside the domain. The heat transfer coefficient (hc) at the eft wa is taken as 5 W/m.K. Tabe summarizes the secies-deendent thermodynamic characteristics as adoted in the resent work. The reevant thermohysica roerties for stee are isted in Tabe. Different grid sizes ( and 00 00) are used for the resoution of the comutationa domain of size (0. m 0. m) to check for grid indeendence. For this urose magnitude of variation of soute concentration in the iquid is examined for otimum soution. It has been observed that the grid size of roduces comutationay economica resuts with a variation of 0.5% among a the different cases studied in the resent investigation. It is aso found that further refinement in grid sacing does not ater the numerica redictions areciaby. is Tabe Thermohysica roerties of stee arameter Vaue Secific heat (c ) 787 J/Kg. K Therma conductivity of soid (ks) 30 W/m.K Therma conductivity of iquid (k) 7 W/m.K Therma exansion coefficient ( ) K - Density ( ) 7300 kg/m 3 Viscosity ( ) kg/m.s Latent heat of fusion (L) J/kg T Figure deicts the convection attern for the seected aoy system at time t=0 s after beginning of soidification. Once cooing is started at the eft vertica wa of the moud the cod and dense fuid descends aong the interface and turns near the bottom of the cavity as demonstrated by the streamines in Fig.. The streamines show that the fow in the buk met is rimariy countercockwise with a net downward fow occurring near the interface. With rogress in soidification rejected soute tends to reduce the mushy zone fuid density and guides the fuid in the uward direction. The streamines aso show the deveoment of a minor vortex due to souta buoyancy effects near the bottom corner of the cavity. The deveoment of this minor vortex is due to souta gradient buid-u caused by the transortation of soute by the therma buoyancy driven major vortex. Fig.. Streamfunction ots at time =0 s for Fe-C-Mn stee aoy. 33

6 S. Ganguy / JAFM Vo. No Figure 3 iustrates the comosition variation of carbon aong the ongitudina (x-axis) direction at different time intervas during the soidification of the resent aoy system. For the urose of comarison resent resuts are otted aong with those without the convection-correction of the artition-coefficient. Figures 3(a)-3(b) shows the comosition variation at t=00 s and t=00 s resectivey and at a vertica ocation y=0.05m. Rejection of soute during soidification eads to a shar increase in soute concentration near the interface. The thermay driven fow being dominant carries the rejected soute downward aong the interface giving rise to a comosition variation. When the souta undercooing effects are not considered the goba doube-diffusive convection effects rimariy contros the buk fuid fow in the domain. However with the consideration of souta undercooing (resent mode) there is an additiona infuence of souta convection due to soute buid-u at the ti of the soidified dendrites. Owing to enhanced strength of resutant convective fow the rejected soute secies are transorted more effectivey resuting in comosition variation in the domain. Negecting such non-equiibrium effects due to undercooing may ead to the rediction of amost homogeneous comosition distribution in the buk domain with a fatter curve as deicted in Fig. 3. Simiar behavior is aso observed for the comosition variation of the other eements. Fig. 3. Variation of iquid concentration in terms of mass fraction of soute (carbon) (a) at t = 00 sec (b) at t = 00 sec. Figure 4 iustrates the deveoment of macrosegregation of the constituent eements during soidification. Macrosegregation effects are cacuated as foows: C mix ( ) i M i dv V V Co i (4) where the subscrit i denotes soute eement under concern. Figure 4(a) shows the macrosegregation of carbon and manganese for Fe-0.08wt% C- 0.wt%Mn aoy In order to better quantify the nove simuation mode deveoed in the resent study a searate case study is undertaken by changing the soute comosition of the aoy system. Accordingy Fig. 4(b) deicts the macrosegregation deveoment for Fe-0.08wt% C-0.wt%Ni aoy for which the thermodynamic arameters are given in Tabe. Figures 4(a)- 4(b) aso comares the resent mode resuts with the case when no souta undercooing effects are considered. For the articuar case when the souta undercooing effects are not incororated in the mode (i.e. no correction of k ) the eve of soute redistribution shows an initia rising trend foowed by asymtotic saturation of the concentration distribution. On the other hand with the incusion of non-equiibrium effects due to souta undercooing the additiona strength of souta convection intensifies the overa advective transort of the soute secies in the fow domain thereby giving rise to an enhanced vaue of macrosegregation. The soute concentration rofie shows a rising trend for both the case studies as comared to the case when no souta undercooing are considered. Overa the macrosegregation distributions for the eements (carbon manganese and nicke) are anaogous with the segregation intensity deending on the artition coefficient of the eement. In the aoy under consideration in the resent study the Mc vaue (i.e carbon macrosegregation) is higher than (i.e macrosegregation for manganese) and M Mn M Ni (i.e macrosegregation for nicke) as seen from the Figs. 4(a)-4(b). A comarative study between existing exerimenta data (Ferreira et a.004) and resent numerica simuation is erformed. The reorted exerimenta data in the iterature refers to soidification of a ternary aoy system in a vertica water-cooed mod (Ferreira et a.004). The casting assemby is a directionay soidified water-cooed moud and exeriments were erformed with A8.wt%-Cu3wt%-Si aoy under controed soidification conditions. The thermohysica roerties of this aoy are given in Tabe 3. Figure 5 comares the simuated macrosegregation eve of coer and the corresonding exerimenta resuts (Ferreira et a.004). It is observed that the redictions from the resent mode agrees we with 34

7 S. Ganguy / JAFM Vo. No the corresonding exerimenta resuts; the average error being ess than 0.5% and the maximum difference is no more than 3.8% from the iterature. It can be deduced from the receding observation that the consideration of nonequiibrium henomena arising out of souta undercooing renders better redictive abiity to the resent mode which ead to more accurate determination of the macrosegregation characteristics. Tabe 3 Thermohysica roerties of the A-Cu- Si aoy arameter Vaue Secific heat (c ) 089 J/Kg. K Therma conductivity of 9 W/m.K soid (ks) Therma conductivity of 88 W/m.K iquid (k) Soute Diffusivity (D) m /s Density ( ) 698 kg/m 3 artition coefficient (Cu/Si) Liquidus soe (Cu/Si) Latent heat of fusion (L) 0.05/ / K/Wt% 380 J/kg Fig. 5. Comarison between exerimenta resuts and numerica redictions. 5. CONCLUSION In the resent work a macroscoic mathematica mode is deveoed for studying transort henomena and macrosegregation during ternary aoy soidification rocess. An attemt has been made to cature the non-equiibrium effects due to souta undercooing by adoting a nove methodoogy based on fixed-grid enthay-contro voume aroach. Numerica simuation for muticomonent stee aoy soidification is undertaken and the overa transort behaviour and macrosegregation characteristics are anaysed. Evoution of segregation atterns indicates the nature of comosition distribution in the soidifying domain and rovides quantitative estimation of the macrosegregation. Simuation case-studies oints towards the fact that the non-equiibrium effects on account of souta undercooing strengthen the advective effect which in turn can ead to enhanced vaue of macrosegregation. The mode redicted resuts are aso comared with the avaiabe exerimenta data and good match can be observed. ACKNOWLEDGEMENTS Fig. 4(a). redicted carbon and manganese macrosegregation eves during soidification. Fig. 4(b). redicted carbon and nicke macrosegregation eves during soidification. The author woud ike to thank the management of Tata Stee India for giving ermission to ubish this work. REFERENCES Bennon W. D. and F.. Incroera (987). A continuum mode for momentum heat and secies transort in binary soid-iquid hasechange systems-i. Mode formuation. Internationa Journa of Heat Mass Transfer Brenner H. (970) Rheoogy of two-hase systems. Annu. Rev. Fuid Mech Brent A. D. V. R. Voer and K. J. Reid (988). The enthay orosity technique for modeing convection-diffusion hase change: aication to the meting of a ure meta. Numerica Heat Transfer Brody H. D. and M. C. Femings (966) Soute 35

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