Magnetic Control

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FRECKLE SUPPRESSION IN DIRECTIONAL SOLIDIFICATION OF BINARY AND MULTICOMPONENT ALLOYS USING MAGNETIC FIELDS Deep Samanta and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/

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RESEARCH SPONSORS DEPARTMENT OF ENERGY (DOE) Industry partnerships for aluminum industry of the future - Office of Industrial Technologies CORNELL THEORY CENTER NATIONAL AERONAUTICS AND SPACE ADMINISTRATION (NASA) NASA Microgravity Materials Science program

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OUTLINE OF THE PRESENTATION Introduction and motivation for the current study Effect of magnetic fields and gradients on convection Numerical model of alloy solidification under the influence of magnetic fields and gradients Computational strategies for solving the coupled numerical system Numerical Examples : – damping convection and reducing macrosegregation during horizontal alloy solidification with magnetic fields and gradients – freckle suppression during directional solidification of alloys using magnetic fields Conclusions Current and Future Research

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Introduction and motivation for the current study

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Solidification is a commonly used method for obtaining near net shape objects in industry. Different casting processes place different restrictions on the solidification process. Homogenous material distribution is one of the key objectives. Solidification of alloys often accompanied by large scale solute variations. Macrosegregation results in non – uniform properties in the final cast alloy. Leads to significant material loss to remove these defects. Need to develop methods to remove these defects for better quality castings. INTRODUCTION Close view of a freckle in a Nickel based super-alloy blade (Ref: Beckermann C., 2000) Freckles in a single crystal Nickel based superalloy blade Freckles in a cast ingot (Ref. Beckermann C.)

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(b) (a) Macro-segregation patterns in a steel ingots (b) Centerline segregation in continuously cast steel (Ref: Beckermann C., 2000) (c) Freckle defects in directionally solidified blades (Ref: Tin and Pollock, 2004) (d) Freckle chain on the surface of a single crystal superalloy casting (Ref. Spowart and Mullens, 2003) DEFECTS IN CASTINGS (a) (d) (c)

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MACROSEGREGATION – CAUSES AND METHODS OF CONTROL Macrosegregation Large scale distribution of solute Non – uniform properties on macro scale Thermosolutal convection Thermal and solutal buoyancy in the liquid and mushy zones Macrosegregation Control of macro – segregation Macrosegregation Control or suppression of convection MEANS OF SUPPRESSING CONVECTION Control the boundary heat flux Multiple-zone controllable furnace design Rotation of the furnace Micro-gravity growth Electromagnetic fields Electromagnetic fields Constant magnetic fields Rotating magnetic fields Combination of magnetic field and field gradients

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Convection damping through magnetic fields and gradients

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1) Quasi – magnetostatic assumption 2) Non – relativistic assumption Maxwell’s equations for EM 0 Equations of Motion & Energy JxB – Lorentz force term – Kelvin force Joule heating Thermo-magnetic cross-effects = 0 + cross – effects BRIEF REVIEW OF MAGNETO-HYDRODYNAMICS

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Paramagnetic: Weakly attracted towards the field. Examples include N2, O2 gas. Diamagnetic: Weakly repelled by the field. Examples include Water, Germanium, Bismuth, Copper. Ferromagnetic: Strongly attracted towards the field. Examples include Iron, Nickel and Cobalt. Nature of the material corresponds to its magnetic susceptibility, . Diamagnetic  ~ -1e-8, Paramagnetic  ~ 1e-7, Ferromagnetic  ~ 10.  depends on temperature for paramagnetic materials. Curie’s law for paramagnetic materials   ~1/T. In the presence of a thermal gradient, a paramagnetic material experiences a body force in a magnetic field. An electrically conducting moving body experiences Lorentz damping force due to the magnetic field. Diamagnetic materials  = m ~ 1/T (m = mass magnetic susceptibility  constant). A magnetic field provides a means of control with or without a magnetic gradient. BRIEF REVIEW OF MAGNETO-HYDRODYNAMICS Behavior of a material in a magnetic field:

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CONVECTION DAMPING USING MAGNETIC FIELDS Application of magnetic field on a moving fluid Conducting fluid Non - conducting fluid Produces both Lorentz + Kelvin forces Produces Kelvin force Kelvin force proportional to 1) mass magnetic susceptibility 2) Magnitude of Magnetic field superimposed with gradient Lorentz force proportional to 1) electrical conductivity 2) Magnitude of magnetic field Both damp convection

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Suppression/reversal of natural convection by exploiting the temperature/composition dependence of magnetic susceptibility – J.W. Evans et.al., J. Appl. Phys (2000) Applied to reversal of flow of aqueous salt solution System specifications Natural convection in a square cavity Cavity filled with water Left wall at 30 C. Right wall at 10 C Magnetic gradient corresponding to  = 2 MAGNETIC GRADIENTS IN FLOW CONTROL

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Suppression/reversal of natural convection by exploiting the temperature/composition dependence of magnetic susceptibility – J.W.Evans et.al., J. Appl. Phys (2000) Applied to reversal of flow of aqueous salt solution System specifications Natural convection in a square cavity Cavity filled with water Left wall at 30 C. Right wall at 10 C Magnetic gradient corresponding to  = 2 MAGNETIC GRADIENTS IN FLOW CONTROL

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Suppression/reversal of natural convection by exploiting the temperature/composition dependence of magnetic susceptibility – J.W.Evans et.al., J. Appl. Phys (2000) Applied to reversal of flow of aqueous salt solution System specifications Natural convection in a square cavity Cavity filled with water Left wall at 30 C. Right wall at 10 C Magnetic gradient corresponding to  = 2 MAGNETIC GRADIENTS IN FLOW CONTROL

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PREVIOUS WORK Effect of magnetic field on transport phenomena in Bridgeman crystal growth – Oreper et al. (1984) and Motakef (1990). Numerical study of convection in the horizontal Bridgeman configuration under the influence of constant magnetic fields – Ben Hadid et al. (1997). Simulation of freckles during directional solidification of binary and multicomponent alloys – Poirier, Fellicili and Heinrich (1997-04). Effects of low magnetic fields on the solidification of a Pb-Sn alloy in terrestrial gravity conditions – Prescott and Incropera (1993). Effect of magnetic gradient fields on Rayleigh Benard convection in water and oxygen –Tagawa et al.(2002-04). Suppression of thermosolutal convection by exploiting the temperature/composition dependence of magnetic susceptibility – Evans (2000). Solidification of metals and alloys with negligible mushy zone under the influence of magnetic fields and gradients; Control of solidification of conducting and non – conducting materials using tailored magnetic fields – B.Ganapathysubramanian and Zabaras (2004-05)

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EFFECTS OF CONVECTION ON SOLIDIFICATION Buoyancy driven thermal and solutal convection predominantly influence macrosegregation. Large scale freckling and distribution of solute severely degrades quality of the casting Control of thermosolutal convection is a key objective in controlling segregation in alloys Need to explore methods to damp convection during solidification Application of magnetic fields and magnetic field gradients during solidification of alloys. Production of both Lorentz and Kelvin damping forces Suitable for both non – metallic and metallic alloy solidification Successfully applied for the growth of metals and crystal growth

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Numerical model of alloy solidification under the influence of magnetic fields and gradients

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B(t) Mushy zone MELT SOLID g qs Application of magnetic field on an electrically conducting fluid produces additional body force – Lorentz force. This force is used for damping flow during solidification of electrically conducting metals and alloys. Application of a constant magnetic gradient also produces Kelvin force that acts directly on the thermosolutal buoyancy force. Combination of magnetic field and magnetic field gradients is suitable for all kinds of alloys. PROBLEM DEFINITION

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NUMERICAL MODEL Single domain model based on volume averaging is used. Single set of transport equations for mass, momentum, energy and species transport. Individual phase boundaries are not explicitly tracked. Complex geometrical modeling of interfaces avoided. Single grid used with a single set of boundary conditions. Solidification microstructures are not modeled here and empirical relationships used for drag force due to permeability. SALIENT FEATURES : Microscopic transport equations Volume- averaging process Macroscopic governing equations wk dAk (Ref: Gray et al., 1977)

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IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS Only two phases present – solid and liquid with the solid phase assumed to be stationary. The densities of both phases are assumed to be equal and constant except in the Boussinesq approximation term for thermosolutal buoyancy. Interfacial drag in the mushy zone modeled using Darcy’s law. The mushy zone permeability is assumed to vary only with the liquid volume fraction and is either isotropic or anisotropic. The solid is assumed to be stress free and pore formation is neglected. Material properties uniform (μ, k etc.) in an averaging volume dVk but can globally vary Darcy drag force is assumed to be linear in velocity and quadratic drag term is neglected TRANSPORT EQUATIONS FOR SOLIDIFICATION

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IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS Phenomenological cross effects – galvomagnetic, thermoelectric and thermomagnetic – are neglected The induced magnetic field is negligible, the applied field Magnetic field assumed to be quasistatic The current density is solenoidal, The external magnetic field and gradient are applied only in a single direction Spatial variations in the magnetic field negligible due to small size of problem domains Charge density is negligible, Electromagnetic force per unit volume on fluid : Current density : MAGNETO-HYDRODYNAMIC (MHD) EQUATIONS

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IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS Volume averaged current density equation : Volume- averaging process Assumption of interfacial fluxes : Boussinesq approximation for the body force terms : Diamagnetic materials Paramagnetic materials (Ref: Toshio and Tagawa 2000, Evans C.G., 2000)

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GOVERNING EQUATIONS where : (Ref: Toshio and Tagawa (2002-04), Evans et al. (2000), Zabaras and Ganapathysubramanian B., 2004-05)

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Anisotropic permeability (obtained experimentally and from regression analysis for directional solidification of binary alloys, Heinrich et al., 1997) Isotropic permeability (empirical relation based on Kozeny – Carman relationship) d = dendrite arm spacing – important microstructural parameter. PERMEABILITY EXPRESSIONS IN ALLOY SOLIDIFICATION

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CLOSURE RELATIONSHIPS Lever rule : Scheil rule : Lever and Scheil rule form the lower and upper limits of liquid mass fractions Other models take into account back diffusion or solidification histories (paths) History based solidification eliminates equilibrium assumptions (No back diffusion) (Infinite back diffusion) Segregation models needed for closure of the numerical model Relationships between auxiliary field variables derived from thermodynamic relations Linear phase diagram with constant slopes of solidus and liquidus lines used Finite Back Diffusion : Back diffusion parameter : (Ref: Kurz and Fisher, 1989) (Ref: Flemings, 1970)

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(solidification) (re – melting) (solidification) (re – melting) Solidification histories explicitly taken into account Equilibrium assumption is not invoked Microsegregation in solid phase modeled H(f) is a function that gives I for old values of f HISTORY BASED SEGREGATION MODEL (Ref:Heinrich et al.1997 - 99, 2004) CLOSURE RELATIONSHIPS

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Computational strategies to solve the coupled numerical system

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COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES SUPG based finite element discretization technique used for thermal and solutal problems to stabilize advective effects. A modified form of SUPG-PSPG technique used for fluid flow incorporating effects of the Darcy drag force in the stabilizing parameters. Stabilizing parameters take into account the underlying regime  advective / diffusive /Darcy dominated. Fluid flow problem

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Stabilizing techniques needed to accommodate equal-order velocity-pressure interpolations Absolutely necessary for convection dominated problems Stabilizing terms derived by minimizing the momentum equation residual or a subgrid scale approach Stabilized FE formulation for the momentum equation (Ref:Zabaras and Samanta, 2004) Advection stabilizing term Darcy drag stabilizing term Pressure stabilizing term COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES

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Convection stabilizing parameter Pressure stabilizing parameter Darcy stabilizing parameter Stabilizing parameters are time constants representing the dominant underlying phenomenon Smooth transition between advective, diffusive or Darcy dominated flow regimes The Darcy stabilizing term is chosen to provide a linear additional term – (1 – ε)wh The combined shape function is (Ref:Zabaras and Samanta, 2004) (Ref: Tezduyar T.E., 1992) COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES

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Stabilized finite element formulation for energy equation : Implicit factor (Scheil rule / History) (Ref: Heinrich, Poirier and Felicilli et al.) COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES Modified energy equation :

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Multistep Predictor – Corrector method used for thermal and solute problems. Backward – Euler fully implicit method for time discretization and Newton-Raphson method for solving heat transfer, fluid flow and deformation problems. Thermal and solutal transport problems along with the thermodynamic update scheme solved repeatedly in a inner loop in each time step. Fluid flow and electric potential problems decoupled from this iterative loop and solved only once in each time step. Stabilized finite element formulation of the solute equation: (Ref: Incropera, 1987 Heinrich, 1991) COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES

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All fields known at time tn Advance the time to tn+1 Solve for the concentration field (solute equation) Solve for the temperature field (energy equation) Solve for liquid concentration, mass fraction and density (Thermodynamic relations) Inner iteration loop Segregation model (Scheil rule) SOLUTION ALGORITHM AT EACH TIME STEP Is the error in liquid concentration and liquid mass fraction less than tolerance No Yes (Ref: Heinrich, et al.) COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES n = n +1 Solve for velocity and pressure fields (momentum equation) Decoupled momentum solution only once in each time step Check if convergence satisfied Solve for the induced electric potential

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Numerical Examples

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DAMPING CONVECTION IN HORIZONTAL ALLOY SOLIDIFICATION Mushy zone MELT SOLID g qs = h(T – Tamb) Solidification of Pb – Sn alloy studied under the influence of magnetic fields and field gradients Lorentz force dominates and Kelvin force is negligible A magnetic field of 5 T combined with a gradient of 20 T/m Effect of Lorentz force on macrosegregation to be studied Diamagnetic susceptibility, χ = -1.36 x 10-9 m3/kg L = 0.08 m H = 0.02 m

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Important properties and initial conditions for this example HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Pb – Sn) (Ref: C. Beckermann, 2002 )

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(a) No magnetic field or gradients (b) A magnetic field of 5 T combined with a gradient of 20 T/m (i) Isotherms (ii) velocity vectors and liquid mass fractions (iii) isochors (iv) liquid solute concentration (iv) (i) (ii) (iii) HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Pb – Sn)

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Maximum solute concentration differences : 1. in the presence of magnetic field and gradients ΔC = 0.43 % wt. Sn at t = 40 sec ΔC = 1.79 % wt. Sn at t = 160 sec 2. absence of magnetic field and gradients ΔC = 9.81 % wt. Sn at t = 40 sec ΔC = 14.86 % wt. Sn at t = 160 sec Maximum velocity magnitudes 1. in the presence of magnetic field and gradients Vmax = 1.32 mm/s at t = 40 sec Vmax = 4.98 mm/s at t = 160 sec 2. absence of magnetic field and gradients Vmax = 74.4 mm/s at t = 40 sec Vmax = 149.2 mm/s at t = 160 sec Significant damping in thermosolutal convection in the whole cavity Freckle formation is largely inhibited Substantial reduction in macrosegregation and solute concentration variations Horizontal velocity damped out; small vertical velocities are induced causing small cyclic perturbations in concentration HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Pb – Sn)

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HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Al – Cu) Important properties and initial conditions for this example (Ref: C. Beckermann 1995, Tsai 1993 )

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HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Al – Cu) (iv) (i) (ii) (iii) (a) No magnetic field or gradients (b) A magnetic field of 5 T combined with a gradient of 20 T/m (i) Isotherms (ii) velocity vectors and liquid mass fractions (iii) isochors (iv) liquid solute concentration (t = 60 sec) (a) ΔC = 0.855 wt % Cu (b) ΔC = 0.006 wt % Cu

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HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Al – Cu) (iv) (i) (ii) (iii) (i) Isotherms (ii) velocity vectors and liquid mass fractions (iii) isochors (iv) liquid solute concentration (t = 144 sec) (a) ΔC = 1.16 wt % Cu (b) ΔC = 0.006 wt % Cu (a) No magnetic field or gradients (b) A magnetic field of 5 T combined with a gradient of 20 T/m

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FRECKLE SUPPRESSION IN 2D DIRECTIONAL SOLIDIFICATION (BINARY ALLOY) ux = uz = 0 ux = uz = 0 ux = uz = 0 ux = uz = 0 T/t = r T/x = 0 T/x = 0 T/z = G C/x = 0 C/x = 0 T(x,z,0) = T0 + Gz C(x,z,0) = C0 Mushy zone permeability assumed to be anisotropic Formation of freckles and channels due to thermosolutal convection Only Lorentz force present and Kelvin force negligible Important parameters L x B = 0.04m x 0.007m C0 = 10% by weight Tin (Sn) Insulated boundaries on the rest of faces g Direction of solidification magnetic field of 2T combined with a gradient of 10 T/m in x dir

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FRECKLE SUPPRESSION IN 2D DIRECTIONAL SOLIDIFICATION (BINARY ALLOY) (b) (i) CSn (ii) fl (i) CSn (ii) fl (a) (a) No magnetic field/gradient (b) Combined magnetic field/gradient (2T with 10 T/m) Significant damping of convection throughout the cavity Freckle formation is totally suppressed  homogeneous solute distribution (a) ΔC = Cmax – Cmin = 2.63 wt %Sn (t = 790 s) (b) ΔC = Cmax – Cmin = 1.3 wt % Sn (t = 790 s)

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z x y T/z = G T/t = r g Direction of solidification Mushy zone permeability assumed to be anisotropic Formation of freckles and channels due to thermal – solutal convection Only Lorentz force present and no Kelvin force Important parameters L x B x H = 0.01m x 0.01m x 0.02m C0 = 10% by weight Tin vx = vy = 0 on all surfaces A combined magnetic field and gradient applied in x Insulated boundaries on the rest of faces No: of unknowns in fluid flow solver = 110864 No: of unknowns in thermal solver = 27716 No: of unknowns in solutal solver = 27716 FRECKLE SUPPRESSION IN 3D DIRECTIONAL SOLIDIFICATION (BINARY ALLOY)

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Liquid mass fraction at t = 1800 s (a) (NO magnetic field) (b) magnetic field of 5T combined with a gradient of 20 T/m in x dir Freckles present in (a) but absent in (b) Suppression of thermosolutal convection by magnetic field. (b) (a) FRECKLE SUPPRESSION IN 3D DIRECTIONAL SOLIDIFICATION (BINARY ALLOY)

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(a) (b) Solute concentration at t = 1800 s (a) (NO magnetic field) (b) magnetic field of 5T combined with a gradient of 20 T/m in x dir Freckles present in (a) but absent in (b) (a) ΔC = 10.5 wt % Sn (b) ΔC = 1.97 wt % Sn  drastic reduction in concentration variations. FRECKLE SUPPRESSION IN 3D DIRECTIONAL SOLIDIFICATION (BINARY ALLOY)

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ux = uz = 0 ux = uz = 0 ux = uz = 0 ux = uz = 0 T/t = r T/x = 0 T/x = 0 T/z = G C/x = 0 C/x = 0 T(x,z,0) = T0 + Gz C(x,z,0) = C0 g Direction of solidification FRECKLE SUPPRESSION IN A MULTICOMPONENT ALLOY (Ni-Al-Ta ALLOY) L x B = 0.02m x 0.007m C10 = 5.8 % by wt. Al C20 = 15.2% by wt. Ta Insulated boundaries on the rest of faces Boussinesq approximation for a multicomponent alloy : (Ref. Heinrich et al. (1997-98) Segregation rules defined for each species :

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(solidification) (re – melting) FRECKLE SUPPRESSION IN A MULTICOMPONENT ALLOY (Ni-Al-Ta ALLOY) (solidification) (re – melting) H(f) is a function that gives Ij for old values of f Some important properties Thermal expansion coefficient, βT = 1.15 x 10-4 K-1 Solute partition coefficients = 0.54 (Al), 0.48 (Ta) Solutal expansion coefficients, βc1 βc2 = 2.26 (Al) , -0.382 (Ta) Eutectic temperature = 1560 K Liquidus surface slopes, mliq1, mliq2 = -517.0 (Al), -255.0 (Ta) Thermal conductivity = 0.08 kWm-1K-1 (both s and l) Heat capacity = 0.66 kWm-1K-1 (both s and l) Electrical conductivity of the alloy = 2.1217 x106 ohm.m Latent heat = 290.0 kJ Curie temperature = 633.0 K (Ref: Heinrich et al, 1997-98)

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FRECKLE SUPPRESSION IN A MULTICOMPONENT ALLOY (Ni-Al-Ta ALLOY) where :

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FRECKLE SUPPRESSION IN A MULTICOMPONENT ALLOY (Ni-Al-Ta ALLOY) (a) No magnetic field/gradient (b) Combined magnetic field/gradient (2T with 10 T/m) Significant damping of convection throughout the cavity Freckle formation is suppressed  homogeneous solute distribution for both species (a) ΔC = Cmax – Cmin = 0.4 wt %Al (t = 35 s) (b) ΔC = Cmax – Cmin = 0.005 % Al (t = 35 s) (b) (i) CAl (ii) CTa (a) (iii) fl (i) CAl (ii) CTa (iii) fl

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CONCLUSIONS Magnetic fields successfully used to damp convection during solidification of alloys. Near homogenous solute element distributions obtained. Suppression of freckle defects during directional solidification of alloys possible. Minimization of solute concentration variations leads to uniform properties in the final cast product. Demostration of successful elimination of some casting defects using magnetic fields in terrestrial gravity conditions

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CURRENT AND FUTURE RESEARCH Computational design of crystal growth processes Optimize crystal growth with improved growing speeds Coupling of models to predict stresses in the cooling crystal with growth simulator Design for improved quality and defect control Computational design of binary alloy solidification processes Melt flow control Control of thermal, flow and segregation conditions within the mushy zone Control of segregation patterns and defects in the product Multi-length scale design of solidification processes Effect of magnetic fields and gradients on underlying microstructure Controlling magnetic fields to obtain a desired microstructure that yields uniform properties Cast components with desired properties and microstructure

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RELEVANT PUBLICATIONS D. Samanta and N. Zabaras, “Modeling melt convection during solidification of alloys using stabilized techniques”, in press in International Journal for Numerical Methods in Engineering. B. Ganapathysubramanian and N. Zabaras, “Using magnetic field gradients to control the directional solidification of alloys and the growth of single crystals”, Journal of Crystal growth, Vol. 270/1-2, 255-272, 2004. B. Ganapathysubramanian and N. Zabaras, “Control of solidification of non-conducting materials using tailored magnetic fields”, Journal of Crystal growth, Vol. 276/1-2, 299-316, 2005. B. Ganapathysubramanian and N. Zabaras, “On the control of solidification of conducting materials using magnetic fields and magnetic field gradients”, International Journal of Heat and Mass Transfer, in press. CONTACT INFORMATION http://mpdc.mae.cornell.edu/