Materials Process Design and Control Laboratory MODELING THE EFFECTS OF MOLD TOPOGRAPHY ON ALUMINUM...

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Materials Process Design and Control Laboratory

MODELING THE EFFECTS OF MOLD TOPOGRAPHY ON ALUMINUM CAST SURFACES

Lijian Tan and Nicholas Zabaras

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: http://www.mae.cornell.edu/zabaras/



Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

RESEARCH SPONSORS

DEPARTMENT OF ENERGY (DOE)

Industry partnerships for aluminum industry of the future

- Office of Industrial Technologies

ALUMINUM CORPORATION OF AMERICA (ALCOA)

Ingot and Solidification Platform

– Alcoa Technical Center

CORNELL THEORY CENTER



(b) Ripple formation (c) Non-uniform front growth(a) Surface porosity & surface crack


Surface defects in casting

• Surface defects removed by some post casting process ----- Scalping.

• These post processes will cause substantial energy and material lost.

• Surface defects were caused by non-uniform heat extraction, improper contact at metal/mold interface, inverse segregation and meniscus freezing etc.




Investigate the effects of mold topography

Mold topography

Gap nucleation

Microstructure

Growth pattern

Stress evolution

Heat transfer

Fluid flow

• For a single aspect, the effect of mold topography may be small. These effects interact each other and may lead to surface defects.

• Our aim is to establish a (thermo-mechanical) model to incorporate these effects.

Segregation

DefectsInteraction



Thermo-mechanical model


Shell formation Shell distortion &mold movement

Air gap formation

Reduction in heat transfer

Crack initiation

Assuming a sinusoidal mold topography, a study of the effects of wavelength or amplitudeon the solidification growth at early times is conducted



Definition of the thermal problem


• Heat conduction in the liquid melt, shell and mold

• At solid-liquid interface, the Stefan equation is satisfied

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

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

• At the mold-shell interface, the thermal resistance is related to the contact pressure or air-gap

( ) ( ) ( ) (liquid region)ll l l l Lc k x y t

t

( ) ( ) ( ) (shell region)ss s s s Sc k x y t

t

( ) ( ) (mold)mm m m m mc k x y

t

s ls l s l s sq q k k L v

n n

0gap

gap 0 0

( ) 01

m sm s s m

hq k k

n n h k

0

1( ) 0

( )m s

m s s m gapq k kn n R R P




Deforming FEM

• A deforming finite element analysis is used to account for phase change.

e eee ba b a aae e ee

cN N dV k N dV Nc N dV k dS

V n• An additional term would appear in the weak (heat or flow) form due to the mesh motion, e.g.

Mesh motion is compatible with the interface velocity

• The freezing interface velocity is computed from the Stefan equation

[ ]e eI I

a b b a s s l le e

LN N N k k d

n v n

0a v




• Equilibrium equation (neglect gravity and inertial force)

• Constitutive law (hypo-elastic)

• Evolution of plastic strain and state variable

• Decomposition of total strain

Definition of the deformation problem

0

e p

[ ]e eL 2

2 ( )I I3

eL I

( )p f s ( )s g s

1{ }

({ } { }

{

)

}{ }

e e eS S

eS

a b a ae e e

a

n

e

b

p

B DB t dV N dS B dV

B D t dV

u

• •

•

weak form

• Evolution of thermal strain (driven force)

I




Accounting for the initial stress at the time of solidification• The stress at a material point P can be easily computed by proper integration of the hypo- elastic law:

( )p f s ( )s g s t* should be also used for integration of the following derivatives:

*initialn t

where [ ],e eL n-1s*

n-1sv

if P

if P

tt and

n-11

n-1sl

s

-P I

if P

if P

ninitial

v t v t

Solid created during the time step t

Original solid shell

Current interface

Previous interface

P

n-1s




Mold/solid shell contact and friction algorithm

• Contact between mold and shell is the key of this thermo-mechanical model (a) Deformation affects heat transfer through the contact between mold and shell (b) Contact traction is unknown in the weak form of the deformation problem

• Use augmentation (Uzawa’s method) to calculate the contact traction (4 steps)

(a) Initialize the multipliers, λN, λT.

(b) Start a nested iteration to solve the displacement with contact forces governed by:

N N Nt g

1( )trial

T T T n nt m

trial trial

T Nt t

0

0

trial trialT

trialT T trial

N trialT

t if

t tt if

t

(c) Update the Lagrange multipliers (d) Return to the second phase, till converged

1{ } { } ({ } { }){ }e e e eS S S

pa b b a a n a

e e e e

B DB t dV u N dS B dV B D t dV • • •



Gap nucleation time: effects of wavelength

• At the very early stages of aluminum solidification, the contact pressure between mold and solid shell will drop at the trough due to thermal stress development. When this contact pressure drops to zero, gap nucleation is assumed to take place.

• For rigid mold (with an topographyamplitude=1 µm, wavelength=1-4 mm), under liquid pressure 8000 Pa, the gap nucleation time is in the order of seconds

Physical Conditions:Liquid pressure P=8000 PaThermal resistance at mold-shell interface R=10-5 m2 oC sec J-1




Gap nucleation time: effects of mold conductivity

• Mold conductivity affects gap nucleation time

• The higher the conductivity, the quicker the gaps nucleate from the mold surface

In this calculations, the deformation of the mold is neglected to illustrate the effects of mold conductivity.

Physical conditions:Liquid pressure P=10000 PaMold thickness h=0.5 mmThermal resistance at mold-shell interface R=10-5 m2 oC sec J-1

Wavelength=2 mm




• When the wavelength is relatively small, the evolution of the contact pressure at the trough is mainly affected by the conductivity of the mold, i.e. the deformation of the

mold does not play a crucial role

Gap nucleation time: effects of mold material (deformable mold)

Physical Conditions:Liquid pressure P=10000 PaMold thickness h=0.5 mmThermal resistance at mold-shell interface R=10-5 m2 oC sec J-1

Wavelength=10 mm, (20 mm, 30 mm in the next two slides)


• As the wavelength increases, the Ptr-t line is about to show a “turn-around” pattern (i.e. the pressure not reaching zero values). This is defined as the `critical wavelength’

in the analytical studies of L. Hector.

From this figure, we note that the critical wavelength is slightly above 20 mm.

In Hector’s analytical study, the critical wavelength is 16.60 mm, for iron mold and 14.03 mm for lead mold under the same conditions.





• When the wavelength is greater than the critical value, the Ptr-t curve shows a turn- around pattern before the contact pressure reaches zero





•The pressure would not decrease to 0 for an iron or lead mold, so a large wavelength is preferred.

In practice, we can never obtain a such a smooth mold topography with amplitude 1 µm and wavelength 30 mm.



Shell thickness at gap nucleation time

• The shell thickness at gap nucleation time plays an important role in deformation. The thicker the shell, the more its ability to prevent distortion or warping.

• From our calculations, high melt pressure is the preferred option to achieve larger shell thickness at gap nucleation time.




Shell growth after gap nucleation• After gap nucleation, the heat flux between mold and shell is determined by the contact condition• Gap formation substantially affects heat transfer from the mold to the shell

ConclusionA smaller wavelength is preferred as the growth pattern will become stable faster


1mm 5mm Growth front

Alcoa chill cast experimentalresults Vcast = 25mm/s



Fluid flow


A = 1 mm λ = 10 mm

6.64E-10

6.64E-10

-6.40E-08 6.40E-08

(e)

-1.24E-07 1.24E-07

(f)

A = 0.232 mm λ = 1 mm

• Two cells will be formed within each wavelength• As solidification proceeds, these cells would be merged into a global circulation



Effects of fluid flow


Potential effects:• Improve heat transfer rate• Changes in solid-liquid front morphology • Segregation

At the start of solidification, the fluid flow is weak, maximum stream function is only about 2.0.10-6 m2/s.

So its effect to heat transfer and shell growthpattern is small.But its effects on segregation could be substantial.

Segregation is one of our current interests.

Flow flow at the start of solidification(7ms)



Stress development

If the mold surface is perfectly smooth, no gap nucleation would occur and stress would develop uniformly.

For a sinusoidal mold topography, the development of stress is determined mainly by the temperature gradient. Stress is large near the crest and small near the trough.

The gap caused by thermal stress will substantially affect heat transfer.





Microstructure

• Chill surface: λ=1~20mm A=0.232mm

• Amplitude of growth front ~ mm Gap between mold-shell ~ 10 μm

In a macro scale

• Roughness ( A typical surface after grind ) λ=40R =2~200μm R=0.05~5μm

• Critical radius of homogeneous nucleation r*(hom) = 2γ/ΔG = 1nmCritical wavelength of heterogeneous nucleation is much larger than critical wavelength r*(het) = r*(hom) /S(θ) ~ μm

• Resulted surface grain size (H. Biloni) δ=50~200μm (R=5 μm)

In a micro scale



References

•“A stabilized finite element method for flow in porous media and solidification systems”, Proceedings of the Seventh U.S. National Congress on Computational Mechanics, presented at the Symposium on ‘Stabilized and Multi-length scale methods’, Seventh U.S. National Congress on Computational Mechanics, Albuquerque, New Mexico, July 27-31, 2003

•“A stabilized volume-averaging finite element method for flow in porous media and binary alloy solidification processes”, Nicholas Zabaras and Deep Samanta, International journal of Numerical Methods in Engineering, in press.

•“A thermomechanical study of the effects of mold topography on the solidification of Aluminum”, Lijian Tan and Nicholas Zabaras, Metallurgical and Materials Transactions B, submitted.


•“Solidification and macrosegregation of aluminum alloys on uneven surfaces”, Deep Samanta and Nicholas Zabaras, Int. Hournal of Heat and Mass Transfer, in preparation.

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Materials Process Design and Control Laboratory MODELING THE EFFECTS OF MOLD TOPOGRAPHY ON ALUMINUM...

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