materials process design and control laboratory computational techniques for the analysis and...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


COMPUTATIONAL TECHNIQUES FOR THE ANALYSIS AND CONTROL OF ALLOY

SOLIDIFICATION PROCESSES

DEEP SAMANTAPresentation for Thesis Defense (B-exam)

Date: 21 December 2005

Sibley School of Mechanical and Aerospace Engineering

Cornell University




ACKNOWLEDGEMENTS

SPECIAL COMMITTEE: Prof. Nicholas Zabaras, M & A.E., Cornell University Prof. Ruediger Dieckmann, M.S & E., Cornell University Prof. Lance Collins, M & A.E., Cornell University

FUNDING SOURCES: National Aeronautics and Space Administration (NASA), Department

of Energy (DoE), Aluminum Corporation of America (ALCOA) Cornell Theory Center (CTC) Sibley School of Mechanical & Aerospace Engineering

Materials Process Design and Control Laboratory (MPDC)




OUTLINE OF THE PRESENTATION Introduction – alloy solidification processes. Objectives of the current research. Numerical model of alloy solidification with external magnetic

fields. FEM based computational techniques employed. Numerical Examples. Optimization problem for alloy solidification using magnetic

fields. Surface defect formation in aluminum alloys. Exploring the role of mold surface topography. Numerical Examples (parametric analysis). Important observations and conclusions. Suggestions for future study.




Introduction and objectives of the

current research




INTRODUCTION

Alloy Solidification Processes• Used in industry to obtain near net shaped objects – casting, welding,

directional and rapid solidification etc.• Highly coupled process that involves several underlying phenomena –

fluid flow, heat transfer, solute transfer, latent heat release and

microstructure formation.• Influences the underlying microstructure and properties of cast products.• Most cast or solidified alloys characterized by defects.

Phase change process involving more than one

chemical species

Appearance of solid and or crystalline phases

Alloy solidification




INTRODUCTION

COMPLEXITIES IN ALLOY SOLIDIFICATION PROCESSES

solidMushy zone liquid ~10-1 - 100 m

(b) Microscopic scale

~ 10-4 – 10-5m

solid

liquid(a) Macroscopic scale

qos g




Alloysolidification process

Fluid flow

Heat Transfer

Mass TransferPhase Change

Deformation

Shrinkage

Microstructure evolution

Non-equilibrium effects

INTRODUCTION




Close view of a freckle in a Nickel based super-alloy blade

(Ref: Beckermann C., 2000)

Freckles in a single crystalNickel based superalloy blade

Freckles in a cast ingot(Ref. Beckermann C.)

DEFECTS DURING ALLOY SOLIDIFICATION

Macrosegregation

• Oriented parallel to the direction of gravity in directionally (vertically) solidified cast alloys.• Concentration of solute element inside freckles varies a lot from the bulk.• Serve as sites for fatigue cracks and other types of failure

Freckles defects

Non – uniform solute concentration in bulk




(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 ALLOY SOLIDIFICATION

(a)

(d)(c)




Piping Micro shrinkage

Macroshrinkage

Different types of shrinkage porosity (ref. Calcom, EPFL, Switzerland)

Piping – occurs during early stages ofsolidification.

Macro shrinkage - leads to internaldefects.

Micro shrinkage – occurs late during solidification and between solidifyingdendrites.

DEFECTS DURING ALLOY SOLIDIFICATIONSurface defects in casting (Ref. ALCOA corp.)

(a) Sub-surface liquation and crack formation on top surface of a cast

(b) Non-uniform front and undesirable growth with non-uniform thickness (left) and non-uniform microstructure (right)




OBJECTIVES OF THIS RESEARCH

MacrosegregationLarge scale

distribution of

solute

Non – uniform

properties

on macro scaleThermosolutal

convection

Thermal and solutal

buoyancy in the liquid

and mushy zones

Control of macro-

segregation

MacrosegregationControl or suppression of

convection

Development of freckles

channels and other defects

Density variations in

terrestrial gravity

conditions




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

• Time varying fields

• Rotating magnetic fields

• Combination of magnetic field and field gradients





Engineered mold surface (Ref. ALCOA Corp.)

• In industry, the mold surface is pre-machined to control heat extraction in directional solidification

• This periodic groove surface topography allows multi-directional heat flow on the metal-mold interface

• However, the wavelengths should be with the appropriate value to obtain anticipated benefits.

Uniform front growth (left) and uniform microstructure (right) – obtained using grooved molds





Exploring methods to reduce defects during alloy solidification processes.

Developing a computational framework for modeling alloy solidification processes.

Studying the role of convection on macrosegregation. Employing constant or time varying magnetic fields to reduce

macrosegregation based defects. Designing appropriate mold surface topographies to reduce

surface defects in alloys.

Alloysolidification

process

Formation of various defects

Material, monetary and energy losses





Numerical model of alloy solidification underthe influence of magnetic fields




PREVIOUS WORK

• Effect of magnetic field on transport phenomena in Bridgman crystal growth – Oreper et al. (1984) and Motakef (1990).

• Numerical study of convection in the horizontal Bridgman 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, Felliceli 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)




B

Mushy zone

MELTSOLID gqs

• Solidification of a metallic alloy inside a cavity in terrestrial gravity conditions–

heat removal from left• Strong thermosolutal convection present – drives convection during solidification• Application of magnetic field on an electrically conducting moving fluid produces

additional body force – Lorentz force.• This force is used for damping flow during solidification of electrically

conducting metals and alloys.• The main aim of the current study is to investigate its effect on macro-

- segregation during alloy solidification.

PROBLEM DEFINITION

ql




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 transportequations

Volume-averagingprocess

Macroscopic governingequations

wk

dAk

1lg =

0lg =

(Ref: Gray et al., 1977)




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 in the momentum

equations.

• Interfacial resistance 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.

TRANSPORT EQUATIONS FOR SOLIDIFICATION




IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS

• Phenomenological cross effects – galvomagnetic, thermoelectric and

thermomagnetic – are neglected.

• The induced magnetic field is negligible, only field is the externally applied

field.

• Magnetic field assumed to be quasistatic.

• The current density is solenoidal.

• The external magnetic field is applied only in a single direction.

• The magnetic field is assumed to constant in space.

• Charge density is negligible

MAGNETO-HYDRODYNAMIC (MHD) EQUATIONS




GOVERNING EQUATIONS

Ref: Toshio and Tagawa (2002-04), Evans et al. (2000), Ganapathysubramanian B. and Zabaras (2004-05), Samanta and Zabaras, (2005)

Magnetic damping force

Thermosolutal buoyancy force term

Continuity equation

Momentum equationDarcy damping

term

Intertial and advective termsPressure terms Viscous or diffusive terms




GOVERNING EQUATIONS

where :

Ref: Toshio and Tagawa (2002-04), Evans et al. (2000), Ganapathysubramanian B. and Zabaras (2004-05), Samanta and Zabaras, (2005)

Electric Potential equation

Solute equation

Energy equation

convective term diffusive term Latent heat term

Convective term

diffusive term

Transient term

Transient term




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.

ε = Volume fraction of liquid phase.

PERMEABILITY EXPRESSIONS IN ALLOY SOLIDIFICATION

Kx = Ky = Kz = fn(ε,d)

Kx = Ky ≠ Kz

Kx = Ky = fn1(ε,d) Kz = fn2(ε,d)




CLOSURE RELATIONSHIPS INVOLVING BINARY PHASE DIAGRAM

Lever Rule : (Infinite back-diffusion)

Scheil Rule :(Zero back-diffusion)

ClC

T

(assumed constant for all problems)

• Phase diagram relationships depend on the state of the alloy – solid, liquid or mushy.

• These relationships are used for obtaining mass fractions and solute concentrations of liquid and solid phases.

• Lead to strong coupling of the thermal and solutal problems.




FEM based numerical techniques




COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES

• In the presence of strong convection, standard FEM techniques lead to oscillations in solution.

• Inappropriate choice of interpolation functions for pressure and velocity can lead to oscillations in pressure.

• Stabilized FEM techniques – 1) prevent oscillations 2) allow the use of same finite element spaces for interpolating pressure and velocity for the fluid flow problem.

• Stabilizing terms take into account the dominant underlying phenomena (convective, diffusive or Darcy flow regimes)

Convection stabilizing term

Darcy dragstabilizing term

Pressurestabilizing term

Convection stabilizing term

Fluid flowproblem

Thermal and speciestransport problems

StandardGalerkin FEformulation

Stabilizingterms

StabilizedFE

formulation+





• Multistep Predictor – Corrector method used for thermal and solute problems.

• Backward – Euler fully implicit method for time discretization and Newton-Raphson method for solving fluid flow problem.

• 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.

• 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 (Tezduyar et al.) 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.

• Combination of direct and iterative solvers used to realize the transient solution.




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, 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.)


n = n +1

Solve for velocityand 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)




Numerical Examples

1) Solidification of an aqueous binary alloy – effect of

convection (no magnetic fields).

2) Convection damping during horizontal solidification

of a Pb-Sn alloy.

3) Convection damping during directional solidification

of a Pb-Sn alloy.




SOLIDIFICATION OF AN AQUEOUS BINARY ALLOY

RaT = 1.938x107 RaC = -2.514x107

Temperature of hot wall, Thot = 311 KTemperature of cold wall, Tcold = 223KInitial temperature, T0 = 311KInitial concentration, C0 = 0.7Solutal flux on all boundaries = 0 (adiabatic flux condition)vx = vy = 0.0 on all boundaries

Initial and boundary conditions(a) (b) (c) (d)

(a) Velocity and mass fraction (b) isotherms (c) solute concentration (d) liquid solute concentrationThermal solutal convection is very strong and large scale solute distribution occursEffect of thermosolutal convection seen in all other fields




DAMPING CONVECTION IN HORIZONTAL ALLOY SOLIDIFICATION

Mushy zone

MELTSOLID g

qs = h(T – Tamb)

• Solidification of Pb – 10% Sn alloy studied under the influence of magnetic fields (initial temperature = 600 K) (RaT = -2.678x107 RaC = 4.941x108).

• This alloy is characterized by a large mushy zone and strong convection.

• Macrosegregation is severe and extent of segregated zone is large.

• A magnetic field of 5 T applied in the z direction.

• Lorentz force responsible for convection damping.

• Effect of Lorentz force on macrosegregation to be studied.

L = 0.08 m

H = 0.02 m




(a) No magnetic field (b) A magnetic field of 5 T in the z direction

(i) Isotherms (ii) velocity vectors and liquid mass fractions (iii) isochors of Sn

(iv) liquid solute concentration

(iv)

(i)

(ii)

(iii)

HORIZONTAL SOLIDIFICATION OF A METAL ALLOY (Pb – Sn)




FRECKLE SUPPRESSION IN 2D DIRECTIONAL SOLIDIFICATION

ux = uz = 0

ux =

uz =

0ux

= u

z =

0ux = uz = 0

T/t = r

T/

x =

0 T/x =

0

T/z = G

C/x =

0C/

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• Lorentz force occurs once magnetic field is applied.

Important parameters

L x B = 0.04m x 0.007mC0 = 10% by weight Tin (Sn)

Insulated boundaries on the rest of faces

g

Direction of

solidification

Constant magnetic field of 3.5 T applied in x direction

B0

(RaC = 6.177x107)




(b)

0.11050.10870.10700.10520.10340.10160.10040.09990.09810.09630.09460.09380.09280.09100.08920.08750.0857

0.90.80.70.60.50.40.30.20.1

(i) CSn (ii) fl(i) CSn (ii) fl

(a)

0.90.80.70.60.50.40.30.20.1

0.09870.09700.09550.09400.09250.09090.08940.0879

(a) No magnetic field (b) Magnetic field (3.5 T)• Significant damping of convection throughout the cavity• Freckle formation is totally suppressed homogeneous solute distribution• (a) ΔC = Cmax – Cmin = 2.63 wt %Sn (t = 800 s) (b) ΔC = Cmax – Cmin = 1.3 wt % Sn (t = 800 s)





OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD

Mushy zone

MELTSOLID g

qs

L

H

Micro-gravity based growth is purely diffusion based

Objective is to achieve some sort of reduced gravity growth

ql

Growth under diffusion dominated conditions leads to :

• A uniform solute concentration profile due to reduced convection.

• Reduction of defects and sites of fatigue cracking.

• Uniform properties in the final cast alloy.

• Reduction in rejection rate of cast alloy components




Time varying

magnetic fields

Temporal variations

in thermosolutal

convection

Non-linear finite dimensional optimal control problem to determine time variation

Design parameter

set {b} = {b1 b2,…,bn}

Measure of convection in the entire domain and time interval considered

Cost Functional:

Minimization of this cost functional yields design parameter set that leads to a growth regime where convection is minimized.





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 :

n sensitivity problems

to be solved

Gradient

information

Obtained from

sensitivity field

Direct ProblemContinuum

sensitivity equationsDesign differentiate

with respect to

Non – linear conjugate gradient method





DESIGN OBJECTIVESFind the optimal magnetic field B(t) in [0,tmax]determined by the set {b} 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

Direct ProblemSingle domain volume averaged equations foralloy solidification

Differentiate with

respect to design

parameters

Discretize in space and time

Continuum sensitivity

equations

Optimization

problemSensitivity of each variable with respect to the design parameters

Gradient information,

step size in

nonlinear CG

algorithm





(t)

CONVECTION DAMPING IN 2D HORIZONTAL SOLIDIFICATION

T = Ti = 580 K

C = C0 = 10 % by wt. Sn

Isotropic permeability (Kozeny Carman relationship)

[0,120]t Î

L = 0.08 m

H = 0.02 m





Time varying optimal magnetic field Cost functional

5 design variables (5 CSM problems solved)

CG Iterations

Co

stfu

nct

ion

al

0 2 4 6 8 10 12

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Op

timal

mag

net

icfie

ld(T

)

0 20 40 60 80 100 1202

3

4

5

6

7

8





(a)

(b)

(c)

(i) No magnetic field

Comparison of results at time t = tmax = 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 - Use of a time varying optimal magnetic field results in a near diffusion based growth - Near homogeneous solute concentration profile obtained.

(ii) Optimal magnetic field

0.90.80.70.60.50.40.30.20.1

0.2110.1930.1750.1570.1390.1210.1030.0850.067

579.1578.1575.1563.2549.6532.5514.3488.2449.1

0.90.80.70.60.50.40.30.20.1

0.1080.1060.1040.1030.1010.0990.0970.0950.093

577.6572.6563.3546.2524.7495.5470.3430.5





Time (in sec)

Maximum velocity magnitude (mm/s) – No magnetic field

Maximum velocity magnitude (mm/s) –

Optimal magnetic field

40.0

120.0

74.9

96.9

0.310

0.170

Time (in sec)

ΔC = Cmax – Cmin (wt % Sn) - No magnetic field

ΔC = Cmax - Cmin (wt % Sn) - Optimal magnetic field

40.0

120.0

10.04

17.57

0.85

1.52

Time (in sec)

Elemental Peclet number (Pe) – No magnetic field

Elemental Peclet number (Pe) – Optimal magnetic

field

40.0

120.0

6.810

8.811

0.028

0.012

Elemental length,

h = 1 x 10-3 m

Convection damping

Macro--segregation suppression

Diffusion dominated

growth regime




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• Lorentz force – primary damping force once magnetic field is applied.

Important parametersL x B x H = 0.01m x 0.01m x 0.02m

C0 = 10% by weight Tin

vx = vy = vz = 0 on all surfaces

A magnetic field applied in x

Insulated boundaries on the rest of faces


(RaC = 7.721x106)




- Formation of freckles in the absence of magnetic field (t = 800 s). - Thermosolutal convection is strong and leads to large scale solute distribution(at t = 400 s, ΔC = Cmax – Cmin = 5.97 wt. % Sn at t = 800 s, ΔC = Cmax – Cmin = 7.4 wt. % Sn)

X Y

Z

0.1180.1090.1030.0950.0870.0800.0720.0640.056

X Y

Z

0.980.900.800.700.600.500.410.310.210.11

(a) Concentration of Sn (t = 800 s) (b) Liquid Mass fraction (t = 800 s)





FRECKLE SUPPRESSION WITH OPTIMAL MAGNETIC FIELD

Time varying optimal magnetic field Cost functional

4 design variables (4 CSM problems solved)

Time (s)

Op

timal

mag

net

icfie

ld(T

)

0 200 400 600 8003

3.5

4

4.5

5

5.5

6

6.5

CG Iterations

Co

stfu

nct

ion

al

0 1 2 3 4 5 6 7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1




FRECKLE SUPPRESSION WITH OPTIMAL MAGNETIC FIELD

- Complete suppression of freckles in the presence of optimal time varying magnetic field. - Thermosolutal convection causing convection is totally suppressed. - Homogeneous solute distribution in the solidifying alloy (at t = 400 s, ΔC = Cmax – Cmin = 0.4 wt. % Sn at t = 800 s, ΔC = Cmax – Cmin = 1.65 wt. % Sn)

(a) Concentration of Sn (t = 800 s) (b) Liquid mass fraction (t = 800 s)

X Y

Z

0.1130.1110.1090.1070.1050.1030.102

X Y

Z

0.950.900.850.800.750.700.660.61




Surface defect formation in

Aluminum alloys




• Aluminum industry relies on direct chill casting for aluminum ingots.

• Aluminum ingots are often characterized by defects in surface due to non-uniform heat extraction, improper contact at metal/mold interface, inverse segregation, air-gap formation and meniscus freezing etc.

• These surface defects are often removed by post casting process: such as scalping/milling.

• Post-processing leads to substantial increase of cost, waste of material and energy.

• The purpose of this work is to reduce scalp-depth in castings.

• Detailed understanding of the highly coupled phenomenon in the early stages of solidification is required.

INTRODUCTION




INTRODUCTION

Surface defects in casting (Ref. ALCOA corp.)

(a)

(c)

(b)

(a) Sub-surface liquation and crack formation on top surface of a cast(b) Non-uniform front and undesirable growth with non-uniform thickness (left) and non-uniform microstructure (right)(c) Ripple formation




Engineered mold surface (Ref. ALCOA Corp.)

• In industry, the mold surface is pre-machined to control heat extraction in directional solidification.

• This periodic groove surface topography allows multi-directional heat flow on the metal-mold interface.

• However, the wavelengths should be with the appropriate value to obtain anticipated benefits.

INTRODUCTION

Uniform front growth (left) and uniform microstructure (right) – obtained using grooved molds




Numerical model for deformation of

solidifying alloys




PREVIOUS WORK

• Zabaras and Richmond (1990,91) – hypoelastic rate-dependent small deformation model to study the deformation of solidifying body.

• Rappaz (1999), Mo (2004) – deformation in mushy zone with a volume averaging model: Continuum model for deformation of mushy zone in a solidifying alloy and development of a hot tearing criterion.

• Mo et al. (1995-98) – Surface segregation and air gap formation in DC cast Aluminum alloys.

• Hector and Yigit (2000) – semi – analytical studies of air gap nucleation during solidification of pure metals using a hypoelastic perturbation theory.

• Hector and Barber (1994,95) – Effect of strain rate relaxation on the stability of solid front growth morphology during solidification of pure metals.

• Chen et al. (1991 – 93), Heinrich et al. (1993,97) – Inverse segregation caused by shrinkage driven flows or combined shrinkage and buoyancy driven flows during alloy solidification.

• A thermo-mechanical study of the effects of mold topography on the solidification of Al alloys- Tan and Zabaras (2005)




PROBLEM DEFINITION

• Solidification of Aluminum-copper alloys on sinusoidal mold surfaces.

• With growth of solid shell, air – gaps form between the solid shell and mold due to imperfect contact – which leads to variation in thermal boundary conditions.

• The solid shell undergoes plastic deformation and development of thermal and plastic strain occurs in the mushy zone also.

• Inverse segregation caused by shrinkage driven flow affects variation in air – gap sizes, front unevenness and stresses developing in the casting.




Fluid flow

Heat transfer

Casting domain

Heat transfer

Mold

Contact pressure/ air gap criterion

Solute transport

Inelastic deformation

Phase changeand mushy zone

evolution

Deformable or non-deformable mold

SCHEMATIC OF THE HIGHLY COUPLED SYSTEM

• There is heat transfer and deformation in both mold and casting region interacting with the contact pressure or air gap size between mold and casting. • The solidification, solute transport, fluid flow will also play important roles.




GOVERNING TRANSPORT EQUATIONS FOR SOLIDIFICATION

Initial conditions :

Isotropic permeability :

Continuity equation

Momentum equation

Energy equation

Solute equation




MODEL FOR DEFORMATION OF SOLIDIFYING ALLOY

• For deformation, we assume the total strain to be decomposed into three parts:

elastic strain, thermal strain and plastic strain.• Elastic strain rate is related with stress rate through an hypo-elastic constitutive law• Plastic strain evolution satisfy this creep law with its parameters determined from

experiments (Strangeland et al. (2004)).• The thermal strain evolution is determined from temperature decrease and shrinkage.

Strain measure :

Elastic strain

13 T sh sw T I

qe

Thermal strain

( )P f s e1

00

exp( ) exp( )m

s

mQf w

RT

e

Plastic strain




MODELING DEFORMATION IN MUSHY ZONE

low solid fraction high solid fraction

• Liquid or low solid fraction mush - zero thermal and plastic strains. (Without any strength) • Solid or high solid fraction mush - thermal and plastic strains start developing gradually.

0 for <

1 for

ths s

ths s

gw

g

The parameter w is defined as:

• Low solid fractions usually accompanied

by melt feeding and no deformation due to

weak or non – existent dendrites

leads to zero thermal strain.

• With increase in solid fraction, there is an increase in strength and bonding ability of

dendrites to non – zero thermal strain.

• The presence of a critical solid volume fraction is observed in experiment and varies for different alloys.

13 T sh sw T I

qe




IMPORTANT PARAMETERS FOR DEFORMATION IN MUSHY ZONE

Cu (wt%)

Cri

tica

lso

lidfr

act

ion

2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Al - 7 pct Si - 0.3 pct Mg

Volumetric thermal expansion coefficient5 17.65 10T K

Volumetric shrinkage coefficient( ) 0.10417sh s l l

Strain-rate scaling factor5 1

0 9.00 10 s

Stress scaling factor3

0 5.5 10 MPa

Activation energy1154Q J mol

Creep law exponent0.4m

Mushy zone softening parameter6.3

1

00

exp( ) exp( )m

ps

mQg

RT

Creep law for plastic deformationRef. Strangeland et al. (2004)

Critical solid fraction for different copper concentrations in aluminum-copper alloy Ref: Mo et al.(2004)

Al – Cu – 0.3 pct Mg




THERMAL RESISTANCE AT THE METAL-MOLD INTERFACE

Contact resistance:

At very early stages, the solid shell is in contact with the mold and the thermal resistance between the shell and the mold is determined by contact conditions

Example: Aluminum-Ceramic Contact

• Before gap nucleation, the thermal resistance is determined by pressure

• After gap nucleation, the thermal resistance is determined by the size of the gap

Heat transfer retarded due to gap formation

Uneven contact condition generates an uneven thermal stress development and accelerates distortion or warping of the casting shell.




MOLD – METAL BOUNDARY CONDITIONS

• The actual air – gap sizes or contact pressure are determined from the contact sub problem.• This modeling of heat transfer mechanism due to imperfect contact is very crucial for studying the non-uniform growth at early stages of solidification.

Consequently, heat flux at the mold – metal interface is a function of air gap size or contact pressure:

= Air-gap size at the interface

= Contact pressure at the interface




• The thermal problem is solved in a region consisting of both mold and casting to account for non-linear (contact pressure/air gap dependent) boundary conditions at the mold – metal interface.

• Deformation problem is solved in both casting and mold (if mold deformable) or only the casting (if mold rigid, for most of our numerical studies).

• Solute and momentum transport equations is only solved in casting with multistep predictor – Corrector method for solute problems, and Newton-Raphson method for solving heat transfer, fluid flow and deformation problems.

• Backward – Euler fully implicit method is utilized for time discretization to make the numerical scheme unconditionally stable.

• The contact sub-problem is solved using augmentations (using the scheme introduced by Laursen in 2002).

• All the matrix computations for individual problems are performed using the parallel iterative Krylov solvers based on the PETSc library.





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)

Is the error in liquid concentration and liquid mass fraction less than tolerance

No

Solve for velocity and pressure fields (momentum equation)

Yes(Ref: Heinrich, et al.)

Decoupled

momentum

solver

SOLUTION ALGORITHM AT EACH TIME STEP

n = n +1

Solve for displacementand stresses in the casting(Deformation problem)

Contact pressure or

air gap obtained from

Contact sub-problem

Check if convergence satisfied

Convergence criteria basedon gap sizes or contactpressure in iterations




Numerical examples




SOLIDIFICATION OF Al-Cu ALLOY ON UNEVEN SURFACES

Deformation

problem

Heat Transfer(Mold is rigid and non-

deformable)

Solidification

problem

We carried out a parametric analysis by changing these four parameters

1) Wavelength of surfaces (λ)

2) Solute concentration (CCu)

3) Melt superheat (ΔTmelt)

4) Mold material (Cu, Fe and Pb)Both the domain sizes are

on the mm scale

• Combined thermal, solutal and

momentum transport in casting.• Assume the mold is rigid.• Imperfect contact and air gap

formation at metal – mold interface




SOLIDIFICATION COUPLED WITH DEFORMATION AND AIR-GAP FORMATION

1mm 5mm

• Because of plastic deformation, the gap formed initially will gradually decrease.• As shown in the movies, a 1mm wavelength mold would lead to more uniform growth and less fluid flow.


1) Mold material - Cu

2) CCu = 8 wt.%

3) ΔTmelt = 0 oC

Air gap is magnified

200 times.

• Preferential formation of solid occurs at the crests and air gap formation occurs at the trough, which in turn causes re-melting.




TRANSIENT EVOLUTION OF IMPORTANT FIELDS (λ = 3 mm)

(a) Temperature

(b) Solute concentration

(c) Equivalent stress

(d) Liquid mass fraction


1) Mold material - Cu

2) CCu = 5 wt.%

3) ΔTmelt = 0 oC

• We take into account solute transport and the densities of solid and liquid phases are assumed to be different.

• Inverse segregation, caused by shrinkage driven flow, occurs at the casting bottom.This is observed in (b).

(d)(c)

(b)(a)




TRANSIENT EVOLUTION OF IMPORTANT FIELDS (λ = 5 mm)

• For other wavelengths, similar result is observed: (1) preferential formation of solid occurs at the crests (2) remelting at the trough due to the formation of air gap.

• For λ = 3mm, the solid shell unevenness decreases faster than for λ = 5mm.

(d)(c)

(b)(a)

0.0600.0580.0570.0550.0540.0520.0510.0500.049

901.5886.9872.3857.7843.1828.5813.9799.3784.7770.1

18.316.114.412.610.8

9.07.25.43.61.8

0.90.80.70.60.50.40.30.20.1

(a) Temperature

(b) Solute concentration

(c) Equivalent stress

(d) Liquid mass fraction




Max. equivalent stress σeq variation with λ

• σeq first increases and then decreases

• Initially, σeq is higher for greater λ

• Later (t=100 ms), stress is lowest for

5 mm wavelength.

Time (s)

Max

imu

meq

uiv

alen

tstr

ess

(MP

a)

0.025 0.05 0.075 0.15

10

15

20

25

30

35

40

45 = 3 mm= 5 mm= 7 mm= 9 mm

Time (s)

Air

gap

size

(m)

0.025 0.05 0.075 0.1

2E-06

4E-06

6E-06

8E-06

1E-05

1.2E-05

1.4E-05

1.6E-05= 3 mm= 5 mm= 7 mm= 9 mm

Air-gap size variation with wavelength λ• Initially, air-gap sizes nearly same for

different λ• At later times, air-gap sizes increase

with increasing λ

ΔTmelt = 0 oC, CCu = 5 wt.%, mold material = Cu

VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT STRESS




VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT STRESS

• σeq first increases and then decreases

• Variation of σeq with Cu concentration

is negligible after initial times

Time (s)

Max

imu

meq

uiv

alen

tstr

ess

(MP

a)

0.025 0.05 0.075 0.10

5

10

15

20

25

30

35

40

451% Cu3% Cu5% Cu7% Cu9% Cu

Time (s)

Gap

size

(m)

0.025 0.05 0.075 0.10

2E-06

4E-06

6E-06

8E-06

1E-05

1.2E-05

1.4E-05

1.6E-05 1% Cu3% Cu5% Cu7% Cu9% Cu

• Air-gap sizes increase with time• Increasing Cu concentration leads to

increase in air-gap sizes

ΔTmelt = 0 oC, λ = 5 mm, mold material = Cu

Increase of solute concentration leads to increase in air-gap sizes, but its effect on stresses are small.




EFFECT OF INVERSE SEGREGATION – AIR GAP SIZES

• Differences in air-gap sizes for different solute concentrations are more pronounced in the presence of inverse segregation.

Time (s)

Gap

size

(m)

0.025 0.05 0.075 0.10

2E-06

4E-06

6E-06

8E-06

1E-05

1.2E-05

1.4E-05

1.6E-05 1% Cu3% Cu5% Cu7% Cu9% Cu

(a) With inverse segregation (b) Without inverse segregation

inverse segregation actually plays an important role in air-gap evolution.




Value of front unevenness and maximum equivalent stress for various wavelengths • one cannot simultaneously reduce both stress and front unevenness• when wavelength greater than 5mm, both unevenness and stress increase wavelength less than 5 mm is optimum

Maximum equivalent stress (MPa)

Fro

ntu

nev

enn

ess

(m)

16 20 24 28 320

0.0004

0.0008

0.0012

0.0016

With inverse segregationWithout inverse segregation

= 1 mm

= 9 mm= 7 mm

= 3 mm= 5 mm

Copper concentration (wt %)

Eq

uiv

alen

tstr

ess

atd

end

rite

roo

ts(M

Pa)

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

Equivalent stress at dendrite roots• The highest stress observed for 1.8% copper alloy suggests that aluminum copper alloy with 1.8% copper is most susceptible to hot tearing• Phenomenon is also observed experi-mentally Rappaz(99), Strangehold(04)

VARIATION OF EQUIVALENT STRESSES AND FRONT UNEVENNESS

Time t = 100 ms




CONCLUSIONS AND OBSERVATIONS

• Magnetic fields are successfully used to damp convection during solidification of metallic alloys in terrestrial gravity conditions.

• Near homogeneous solute element distributions obtained.

• Suppression of freckle defects during directional solidification of alloys achieved.

• An optimization problem solved to determine time varying magnetic fields that damp convection and minimize macrosegregation in solidifying alloys.

• Optimal time varying magnetic fields take into account variations in thermosolutal convection – superior to constant magnetic fields.

• Reduction in current and power requirements possible.




• Early stage solidification of Al-Cu alloys significantly affected by non – uniform boundary conditions at the metal mold interface.

• Variation in surface topography leads to variation in transport phenomena, air-gap sizes and equivalent stresses in the solidifying alloy.

• Air-gap nucleation and growth significantly affects heat transfer between metal and mold.

• Distribution of solute primarily caused by shrinkage driven flows and leads to inverse segregation at the casting bottom.

• Presence of inverse segregation leads to an increase in gap sizes and front unevenness.

• Effects of surface topography more pronounced for a mold with higher thermal conductivity

• Computation results suggest that an Al-Cu alloy with 1.8% Cu is the most susceptible to hot tearing defects. An optimum mold wavelength should be less than 5mm.

• Overall aim is to develop techniques to reduce surface defects in Al alloys by modifying mold surface topography.

CONCLUSIONS AND OBSERVATIONS




Suggestions for future research




MULTILENGTH SCALE SOLIDIFICATION MODELING

• Ability to resolve morphology of microstructural entities like dendrites.• Effect of various instabilities on the actual growth front morphology can be studied.• Key to understanding effects of micro – scale phenomenon on macro scale and

vice – versa.• Can avoid very fine grids for macro – scale simulations.• Lay the foundations of multi – length scale robust design.

Importance of multi – length scale modeling

Large scale casting

metres Small scale phenomena

• Dendritic growth• Macro defects

• Degrade quality of casting

Macroscopic

transport phenomenon

Evolution of

microstructure

Multi – length scale

solidification model




compute average of physical quantities on fine scale

Macro scalegrid

compute equivalent parameters

(permeability)


Macroscopic governing equations based on volume averaging to compute fields like velocity, temperature and solute concentration

Boundary conditionsfor micro problem

Microstructure evolution model

On each elementmicro scale grid

1) Interface temperature condition

2) Thermal Flux jump at the interface

3) Concentration flux jump at the interface

4) Thermal, solutal and momentum

problems in the liquid phase

5) Thermal problem in the solid phase.

6) Curvature effects

7) Tracking the interface position

Averaging techniques

Upscaling methods basedon homogenization

Allows variation of macro variables on both scales

Multi-length scale direct

problem




Multi-length scale design

problem

Design macro variables (magnetic field or boundary heat flux to get a

desired microstructure


Multi-length scale direct

problem

Volume – averaged continuum macro model

Microstructure evolution model

Probabilistic nucleation model

Averaging/upscaling

techniques

CSM based optimization method

Cast components with desired properties and microstructure




MICROSTRUCTURE EVOLUTION DURING EARLY STAGE SOLIDIFICATION

Microstructure evolution

Surface parametersand mold topographyin transport processes

Interfacial heat transfer

Varying stresses in solid

Inverse segregation

Air gap formation(non uniform contact and shell remelting)

Metal/mold interaction

Shell growth kinetics• uneven growth• distortion

Combined effect of several phenomena on microstructure evolution (during early stages of solidification)




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