1. Materials Process Design and Control Laboratory CONTROL OF CONVECTION IN THE SOLIDIFICATION OF ALLOYS USING TAILORED MAGNETIC FIELDS B. Ganapathysubramanian, D. Samanta and N. 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/

2. Materials Process Design and Control Laboratory 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

3. Materials Process Design and Control Laboratory OUTLINE OF THE PRESENTATION  Introduction and motivation for the current study  Numerical models of alloy solidification under the influence of magnetic fields (Direct Problem)  Numerical examples ( 2D and 3D direct problems with constant magnetic fields)  Optimization problem in alloy solidification using time varying magnetic fields  Numerical Examples : – 2 D optimal control problem with time varying magnetic fields (alloy with a mushy zone) – 2 D optimal control problem with time varying magnetic fields and constant magnetic gradients (cyrstal growth problem – no mushy zone)  Conclusions  Current and Future Research

4. Materials Process Design and Control Laboratory Introduction and motivation for the current study

5. Materials Process Design and Control Laboratory Solidification is a commonly used method for obtaining near net shape objects in industry. Different 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.)

6. Materials Process Design and Control Laboratory (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 DURING SOLIDIFICATION (a) (d) (c)

7. Materials Process Design and Control Laboratory 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 Constant magnetic fields Rotating magnetic fields Combination of magnetic field and field gradients Development of freckles channels and other defects

8. Materials Process Design and Control Laboratory 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)

9. Materials Process Design and Control Laboratory Numerical models of alloy solidification under the influence of magnetic fields and gradients

10. Materials Process Design and Control Laboratory B(t) Mushy zone MELT SOLID g qsqs 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

11. Materials Process Design and Control Laboratory GOVERNING EQUATIONS (Ref: Toshio and Tagawa (2002-04), Evans et al. (2000), Zabaras and Ganapathysubramanian B., 2004-05) INTERFACE SOLID

12. Materials Process Design and Control Laboratory NUMERICAL MODEL: 1 Free adiabatic surface 20 mm solid liquid insulated The solid part and the melt part modeled seperately Moving/deforming FEM to explicitly track the advancing solid- liquid interface Transport equations for momentum, energy and species transport in the solid and melt Individual phase boundaries are explicitly tracked. Interfacial dynamics modeled using the Stefan condition and solute rejection Different grids used for solid and melt part SALIENT FEATURES :

13. Materials Process Design and Control Laboratory NUMERICAL MODEL: 2 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 wkwk dA k (Ref: Gray et al., 1977)

14. Materials Process Design and Control Laboratory 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. The solid is assumed to be stress free and pore formation is neglected. Constant thermo-physical and transport properties, including thermal and solute diffusivities viscosity, density, thermal conductivity and phase change latent heat. The melt flow is assumed to be laminar The solute diffusion in the solid is negligible compared to that of the liquid A macroscopically stable sharp interface exists between the solid and liquid regions TRANSPORT EQUATIONS FOR SOLIDIFICATION (MODEL 1)

15. Materials Process Design and Control Laboratory 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 dV k but can globally vary Darcy drag force is assumed to be linear in velocity and quadratic drag term is neglected TRANSPORT EQUATIONS FOR SOLIDIFICATION (MODEL 2)

16. Materials Process Design and Control Laboratory 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 is applied only in a single direction Spatial variations in the magnetic field negligible due to small size of problem domains Charge density is negligible, MAGNETO-HYDRODYNAMIC (MHD) EQUATIONS Volume averaged current density equation : Volume- averaging process Assumption of interfacial fluxes : Electromagnetic force per unit volume on fluid : Current density :

17. Materials Process Design and Control Laboratory GOVERNING EQUATIONS: MODEL 2 where : (Ref: Toshio and Tagawa (2002-04), Evans et al. (2000), Zabaras and Ganapathysubramanian B., 2004-05)

18. Materials Process Design and Control Laboratory 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

19. Materials Process Design and Control Laboratory 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)

20. Materials Process Design and Control Laboratory (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

21. Materials Process Design and Control Laboratory COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES 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 methods used for discretizing governing volume averaged equations. For the thermal and solute sub-problems, SUPG technique used for discretization. The fluid flow sub-problem is discretized using a modified form of the SUPG-PSPG technique incorporating the effects of Darcy drag force in the mushy zone. (Ref:Zabaras and Samanta: 04,05). Both velocity and pressure and solved simultaneously and convergence rate is improved.

22. Materials Process Design and Control Laboratory All fields known at time t n Advance the time to t n+1 Solve for the concentration field (solute equation) Solve for the temperature field (energy equation) Solve for liquid concentration, liquid volume fraction (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 (3D only)

23. Materials Process Design and Control Laboratory Numerical Examples (constant magnetic fields) 1) 2D Alloy solidification 2) 3D directional solidification

24. Materials Process Design and Control Laboratory DAMPING CONVECTION IN HORIZONTAL ALLOY SOLIDIFICATION Mushy zone MELT SOLID g q s = h(T – T amb ) Solidification of Pb – Sn alloy studied under the influence of magnetic fields A magnetic field of 5 T applied in the y direction. Lorentz force responsible for convection damping. Effect of Lorentz force on macrosegregation to be studied L = 0.08 m H = 0.02 m

25. Materials Process Design and Control Laboratory PropertySymbolValueUnits Thermal conductivity of solid k s 1.85x 10 -2 kW m -1 K -1 Thermal conductivity of liquid k l 1.85x 10 -2 kW m -1 K -1 Heat capacity of solidcscs 0.167 kJ kg -1 K -1 Heat capacity of liquid c l 0.167 kJ kg -1 K -1 Latent heat h f 37.6kJ kg -1 Equilibrium partition coefficient κ0.31 Coefficient of thermal expansion βTβT 1.2x10 -4 K -1 Coefficient of solutal expansion βsβs 0.515 K -1 Density of solid ρ s 1.01 x 10 4 kg m -3 Density of liquid ρ l 1.01 x 10 4 kg m -3 PropertySymbol ValueUnits Viscosity of liquid μ2.495 x 10 -3 kg m -1 s -1 Eutectic temperature T e 456.0 K Melting temperatureTmTm 600.0 K Initial temperature T i 600.0 K Ambient temperature T amb 298.0 K Initial liquid solute concentration C l,0 0.1 Gravity g 9.81ms -2 Slope of the liquidus line m liq -233.0 K Convective heat transfer coefficient h conv 1.01 kW m -2 o C -1 Important properties and initial conditions for this example HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Pb – Sn) (Ref: C. Beckermann, 2002 )

26. Materials Process Design and Control Laboratory (a) No magnetic field or gradients (b) A magnetic field of 5 T in the y direction (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)

27. Materials Process Design and Control Laboratory 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 V max = 1.32 mm/s at t = 40 sec V max = 4.98 mm/s at t = 160 sec 2. absence of magnetic field and gradients V max = 74.4 mm/s at t = 40 sec V max = 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; HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Pb – Sn)

28. Materials Process Design and Control Laboratory 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 C 0 = 10% by weight Tin v x = v y = v z = 0 on all surfaces A magnetic field 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

29. Materials Process Design and Control Laboratory Liquid mass fraction at t = 1800 s (a) NO magnetic field (b) magnetic field of 5T applied 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

30. Materials Process Design and Control Laboratory (a) (b) Solute concentration at t = 1800 s (a) (NO magnetic field) (b) magnetic field of 5T applied in the 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

31. Materials Process Design and Control Laboratory OPTIMIZATION PROBLEM USING TAILORED MAGNETIC FIELDS 1: Continuum sensitivity and adjoint equations 2: Continuum sensitivity equations

32. Materials Process Design and Control Laboratory MATERIAL SOLIDIFICATION PROCESS DESIGN UNDER DIFFUSION CONDITIONS MATERIAL SOLIDIFICATION PROCESS DESIGN UNDER DIFFUSION CONDITIONS DESIGN OBJECTIVES Find the optimal magnetic field such that, in the presence of coupled thermocapillary, buoyancy, and electromagnetic convection in the melt, a flat solid- liquid interface with diffusion dominated growth is achieved Growth under diffusion dominated conditions ensures: Flat solid-liquid interface. This is crucial in crystal growth Uniform temperature gradients along the interface. This results in reduced stress in the cooling crystal. Found to be directly related to the life time of the component Uniform solute distribution. This leads to a homogeneous crystal. Further, this also results in reduced dislocations Suppression in temperature and solute fluctuations leading to reduced defects in the crystal Micro-gravity based growth is purely diffusion based Objective is to achieve some sort of reduced gravity growth MELT SOLID G,V q os q ol g B

33. Materials Process Design and Control Laboratory INVERSE-DESIGN PROBLEM INVERSE-DESIGN PROBLEM INVERSE PROBLEM STATEMENT Find the magnetic field b(t) in [0, t max ] such that melt convection is suppressed With a guessed magnetic field, solve the following direct problem for: Melt region: Temperature field: T(x, t; b) Temperature field: T(x, t; b) Concentration field: c(x, t; b) Concentration field: c(x, t; b) Velocity field: v(x, t; b) Velocity field: v(x, t; b) Electric potential:  (x, t; b) Electric potential:  (x, t; b) Solid region: Temperature field: T s (x, t; b) Temperature field: T s (x, t; b) Measure of deviation from diffusion based growth MELT SOLID G,V q ol g B(t) q os

34. Materials Process Design and Control Laboratory NONLINEAR OPTIMIZATION APPROACH TO THE INVERSE SOLIDIFICATION PROBLEM Continuum sensitivity problem Define the inverse solidification problem as a unconstrained spatio- temporal optimization problem Solve the above unconstrained minimization problem using the nonlinear Conjugate Gradient Method (CGM) Needs design gradient information Needs descent step size Continuum adjoint problem Find a quasi- solution: o  L 2 ([0, t max ]) such that Find a quasi- solution: b o  L 2 ([0, t max ]) such that S ( o )  S ( )  o  L 2 ( [0, t max ]) S ( b o )  S ( b )  b o  L 2 ( [0, t max ])

35. Materials Process Design and Control Laboratory DIRECT CONTINUUM PROBLEM AND THE CONTINUUM SENSITIVITY PROBLEM MELT CRYSTAL INTERFACE DIRECT EQUATIONS SENSITIVITY EQUATIONS

36. Materials Process Design and Control Laboratory DIRECT CONTINUUM PROBLEM AND THE CONTINUUM SENSITIVITY PROBLEM MELT CRYSTAL INTERFACE DIRECT EQUATIONS SENSITIVITY EQUATIONS

37. Materials Process Design and Control Laboratory THE CONTINUUM ADJOINT PROBLEM Find an analytical expression for the gradient of the cost functional Find operators (similar to the corresponding sensitivity operators) that satisfy the following relationships: Using integration by parts; Green’s theorem; Reynolds transport theorem and some vector algebra

38. Materials Process Design and Control Laboratory THE CONTINUUM ADJOINT PROBLEM Adjoint equations Gradient of the cost functional given in terms of the direct and the adjoint fields Using integration by parts; Green’s theorem; Reynolds transport theorem and some vector algebra 12341234

39. Materials Process Design and Control Laboratory OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD DETAILS OF THE CONJUGATE GRADIENT ALGORITHM Make an initial guess of b(t) and set k = 0 Solve the direct and adjoint problems for all required fields Set p k = -S’ ( b 0 ) if (k = 0) else p k = -S’ ( b k ) + γ p k-1 Set γ = 0, if k = 0; Otherwise Calculate S(b) and S’(b) = S({b} k ) Check if S(b k ) ≤ ε tol γ=γ= Calculate the optimal step size α k αk =αk = Set b opt = b k and stop Update {b} k+1 = {b} k + α p k Yes No b(t) – optimal magnetic field Solve the sensitivity problem for all the required fields

40. Materials Process Design and Control Laboratory Design definition: Find the time history of the imposed magnetic field/gradient, such that diffusion based growth is achieved in the presence of thermocapillary, buoyancy and electromagnetic forces Material characteristics: Binary alloy/pure material, Conducting INVERSE DESIGN PROBLEM Antimony-doped Germanium growth Prandtl number: 0.007 Thermal Rayleigh number: 200000 Solutal Rayleigh number: 10000 Lewis number: 1000 Marangoni number: 0 Stefan number: 0.034 Setup Specifications Solidification in a rectangular cavity Dimensions 2cm x 2cm Fluid initially at 40 C superheat Left wall kept at 40 C below melting Driven by thermal and solutal buoyancy along with electromagnetic effects Minimize the cost functional

41. Materials Process Design and Control Laboratory INVERSE DESIGN PROBLEM -RESULTS Optimal magnetic field (x60 Ha) to be applied along with a magnetic gradient of 2 mT/cm Stopping tolerance ~ 5e-4. Super linear convergence using the Fletcher Reeves CG method

42. Materials Process Design and Control Laboratory INVERSE DESIGN PROBLEM - RESULTS Comparison of isotherms and total velocity contours In the case of a uniform magnetic field, the Lorentz force inhibits flow only in the horizontal direction, so due to the buoyant forces and continuity, there is some motion in the horizontal direction on the top. The uniform magnetic field damps out convection to some extent. But there is still some amount of convection resulting in skewed isotherms. Compare with the optimal magnetic field along with the superimposed magnetic gradient

43. Materials Process Design and Control Laboratory INVERSE DESIGN PROBLEM - RESULTS Comparison of the evolution of the velocity and temperature fields for ref case and the optimal magnetic field (Right). Velocity is damped out to a large extent. The maximum velocity for the optimal case is 0.80 compared to 36.0 for the other case. Temperature evolution is primarily conduction based as can be seen by the motion of the isotherms.

44. Materials Process Design and Control Laboratory Time varying magnetic fields Temporal variations in thermosolutal convection Non-linear optimal control problem to determine time variation Choosing a polynomial basis Design parameter set DESIGN OBJECTIVES Find the optimal magnetic field B(t) in [0,t max ]determined by the set such that, in the presence of coupled thermosolutal buoyancy, and electromagnetic forces in the melt, diffusion dominated growth is obtained leading to minimum macrosegregation in the cast alloy OPTIMIZATION PROBLEM USING TAILORED MAGNETIC FIELDS Measure of deviation from diffusion based growth Cost Functional:

45. Materials Process Design and Control Laboratory OPTIMIZATION PROBLEM USING TAILORED MAGNETIC FIELDS Define the inverse solidification problem as an unconstrained spatio – temporal optimization problem Find a quasi – solution : B ({b} k ) such that J(B{b} k )  J(B{b})  {b}; an optimum design variable set {b} k sought Gradient of the cost functional: Sensitivity of velocity field : m sensitivity problems to be solved Gradient information Obtained from sensitivity field Direct Problem Continuum sensitivity equations Design differentiate with respect to Non – linear conjugate gradient method

46. Materials Process Design and Control Laboratory CONTINUUM SENSITIVITY EQUATIONS FOR ALLOY SOLIDIFICATION

47. Materials Process Design and Control Laboratory Unknown variables in the sensitivity problem : Auxiliary variables : CONTINUUM SENSITIVITY EQUATIONS FOR ALLOY SOLIDIFICATION Initial conditions : = 0 Continuum sensitivity problems solved are linear in nature. Each optimization iteration requires solution of the direct problem and m linear CSM problems. In each CSM problem : Thermal and solutal sub-problems solved in an iterative loop with error in and being the bounds The flow and potential sub - problem are decoupled from the main iterative loop and solved only once.

48. Materials Process Design and Control Laboratory VERIFICATION OF THE CONTINUUM SENSITIVITY PROBLEM Comparison of sensitivities from CSM and FDM with a perturbation of 0.05 T (t = t max = 120 s) (Positive perturbation and forward finite difference) (a) Solute concentration senstivity, (b) Sensitivity of v x, x (c) Sensitivity of v y, y Reference magnetic field = 2 T Continuum sensitivity method (a) (b) (c) Finite difference method

49. Materials Process Design and Control Laboratory VERIFICATION OF THE CONTINUUM SENSITIVITY PROBLEM (a) (b) (c) Comparison of sensitivities from CSM and FDM with a perturbation of -0.05 T (t = t max = 120 s) (Negative perturbation and backward finite difference) (a) Solute concentration senstivity, (b) Sensitivity of v x, x (c) Sensitivity of v y, y Reference magnetic field = 2 T Continuum sensitivity method Finite difference method

50. Materials Process Design and Control Laboratory OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD Mushy zone MELT SOLID g q s = h(T – T amb ) L = 0.08 m H = 0.02 m  An optimization problem to determine the coefficients for a 2D horizontal Pb-Sn alloy solidification problem  Initial concentration = 10% by wt. Sn (C 0 )  Initial temperature = 580 K (T 0 )  Total time interval : 0 – 120 seconds  Isotropic permeability of the mushy zone  Thermodynamic relations in mushy zone determined by Scheil rule. Initial guess: GES = max i=1 N = Quantities of interest :

51. Materials Process Design and Control Laboratory OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD DETAILS OF THE CONJUGATE GRADIENT ALGORITHM Make an initial guess of {b} and set k = 0 Solve the direct and sensitivity problems for all required fields Set p k = -J’ ( {b} 0 ) if (k = 0) else p k = -J’ ( {b} k ) + γ p k-1 Set γ = 0, if k = 0; Otherwise Calculate J({b} k ) and J’({b} k ) = J({b} k ) Check if (J({b} k ) - J({b} k-1 ))/J({b} k-1 ) ≤ ε tol γ Calculate the optimal step size α k αk =αk = Set {b} opt = {b} k and stop Update {b} k+1 = {b} k + α p k Yes No {b} opt – final set of design parameters Minimizes J({b} k ) in the search direction p k Sensitivity matrix M given by

52. Materials Process Design and Control Laboratory OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD Stopping criterion Relative error ε tol = 2e-2 (a) |v| max with no and optimal magnetic fields Optimal Vs initial field with time (b) GES with no and optimal magnetic fields (a) (b)

53. Materials Process Design and Control Laboratory DIRECT PROBLEM WITH THE OPTIMAL MAGNETIC FIELD Comparison of results at time t = t max = 120 s (a) Isotherms (b) Solute concentration (c) Liquid mass fractions and velocity vectors - Convection is almost fully damped throughout the solidification process. - Significant reduction in macrosegregation (more than that of a constant magnetic field) - Use of a time varying optimal magnetic field results in a near diffusion based growth - Near homogeneous solute concentration profile obtained. (a) (b) (c) (i) No magnetic field (ii) Optimal magnetic field

54. Materials Process Design and Control Laboratory 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. Demonstration of successful elimination of some casting defects using magnetic fields in terrestrial gravity conditions An optimal control problem solved using time varying magnetic fields to suppress convection and eliminate macrosegregation. Use of tailored magnetic fields more effective in reducing convection. Time varying magnetic fields used for optimizing growth of crystals.

55. Materials Process Design and Control Laboratory 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

56. Materials Process Design and Control Laboratory 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, 2005.  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/