Computational grid size

l

TLe

~0.5 m

Process

~5 mm

REV

Maco-Micro Modeling—Simple methods for incorporating small scale effects into large scale solidification models– Vaughan Voller, University of Minnesota

1 of 19

Can we build a direct-simulation of a Casting Process that resolves to all scales?

Scales in a “simple” solidification process model

~ 50 m

solid

representative ½ arm space

sub-grid model

g

Enthalpy basedDendrite growth model

chill

A Casting

The REV

NucleationSites

columnar equi-axed

The GrainEnvelope

The SecondaryArm Space

The TipRadius

The DiffusiveInterface

~ 0.1 m

~10 mm

~ mm

~100 m

~10 m

~1 nm

103

101

10-1

10-3

10-5

10-7

10-9

10-9 10 10-3 10-1

Length Scale (m)

interfacekinetics

nucleation

solute diffusion

growth

grainformation

casting

heat and mass tran.

Time Scale (s)

Scales in Solidification Processes

2 of 19

(after Dantzig)

Can we build a direct-simulation of a Casting Process that resolves to all scales?

1.0E+02

1.0E+04

1.0E+06

1.0E+08

1.0E+10

1.0E+12

1.0E+14

1.0E+16

1.0E+18

1.0E+20

1.0E+22

1.0E+24

1.0E+26

0 20 40 60 80 100 120

Year-1980

No

de

s

Well As it happened not currently Possible

1000 20.6667 Year

“Moore’s Law”

Voller and Porte-Agel, JCP 179, 698-703 (2002) Plotted The three largest MacWasp Grids (number of nodes) in each volume

2055 for tip6 decades

1 meter

1 micron

3 of 19

chill

A Casting

The REV

NucleationSites

columnar equi-axed

The GrainEnvelope

The SecondaryArm Space

The TipRadius

The DiffusiveInterface

~ 0.1 m

~10 mm

~ mm

~100 m

~10 m

~1 nm

103

101

10-1

10-3

10-5

10-7

10-9

10-9 10 10-3 10-1

Length Scale (m)

interfacekinetics

nucleation

solute diffusion

growth

grainformation

casting

heat and mass tran.

Time Scale (s)

Scales in Solidification Processes

To handle with current computational Technology require a “Micro-Macro” Model

See Rappaz and co-workers

Example a heat and Mass Transfer modelCoupled with a Microsegregation Model

4 of 19

(after Dantzig)

l

TLe

~0.5 m

~ 50 m

solid

~5 mm

Computational grid size

Process REVrepresentative ½ arm space

sub-grid model

]C[ll

0

ss C)g1(dCg

g

]H[ HTc)g1(Tcg llss

from computationOf these values need to extract

lCg g

0

s

s

s dCg

1C T

Solidification Modeling

-- -- --

5 of 19

Micro segregation—segregation and solute diffusion in arm space

C

A

C

)C,....,C,C(GTrk

l

r2l

r1l

r

Primary Solidification Solver

H

Tc]H[)g1(

l

1rpr

Transient mass balance

equilibrium

g

ClTIterative loop

g

model of micro-segregation

(will need under-relaxation)

6 of 19

Give Liquid Concentrations

oldl

oldl

oldoldl C)g1(CC)g1(C

liquid concentration due to macro-segregation alone

Micro-segregation Model

oldg

g

In a small time step new solid forms with lever rule on concentration C

C

Q -– back-diffusion

Need an easy to use approximationFor back-diffusion

transient mass balance gives liquid concentration

QC)g1(C)gg(C)g1( slr

lloldr

sold

l

Solute mass densitybefore solidification

Solute mass densityof new solid (lever)

Solute mass densityafter solidification

7 of 19

g

sCQ

x,

t

t

f

2fDt

Solute Fourier No.

½ Arm space of length takestf seconds to solidify

)CC(gdt

dCgtQ old

lll

The parameter Model --- Clyne and Kurz,

)Scheil(0

ddg

g)lever(1

Ohnaka 8 of 19

2

1

d

dgg

21

g

For special case Of Parabolic Solid Growth

And ad-hoc fit sets the factor

2

2and

14

4

In Most other casesThe Ohnaka approximation

21

2

Works very well

QC)g1(C)gg(C)g1( slr

lloldr

sold

l

)CC)(1m(g

Qs

sl

The Profile Model Wang and Beckermann

baC ms

Need to lagcalculation onetime step andensure Q >0

9 of 19

g

0

m

s

s

s b1m

gadC

g

1C

QC)g1(C)gg(C)g1( slr

lloldr

sold

l

1

s

s

l

C

C2~m

m is sometimes take as a constant ~ 2 BUTIn the time step model a variable value can be use

Due to steeper profile at low liquid fraction ----- Propose

bagC ml

Coarsening Arm-space will increase in dimension with time

3/1n

This will dilute the concentration in the liquid fraction—can model be enhancing the back diffusion

g

sCQ

A model by Voller and Beckermann suggests 1m

n2

2

1mg 3

13

4

If we assume that solid growth is close to parabolic

1

s

s

l

C

C2~m

In profile model

m =2.33 inParameter model

1.0g 3

4

10 of 19

Remaining Liquid when C =5 is Eutectic Fraction

11 of 19

No Coarsening

0.10.110.120.130.140.150.160.17

0.001 0.01 0.1 1 10

Fourier No.

Eut

etic Numerical

Profile

Parameter

Coarsening

0.10.110.120.130.140.150.160.17

0.001 0.01 0.1 1 10

Fourier No.

Eut

ecti

c

QC)g1(C)gg(C)g1( slllold

soldl

oldl

Constant Cooling of Binary-Eutectic Alloy With Initial Concentration C0 = 1

and Eutectic Concentration Ceut = 5, No Macro segregation , = 0.1

Use 200 time steps and equally increment 1 < Cl < 5

Calculating the transient value of g from

Parameter or Profile

oldlCC

21

2

Coarsening

0.10.110.120.130.140.150.160.17

0.001 0.01 0.1 1 10

Fourier No.

Eut

ecti

c

No Coarsening

0.10.110.120.130.140.150.160.17

0.001 0.01 0.1 1 10

Fourier No.

Eut

etic Numerical

Profile

Parameter

Results are good across a range of conditions

0

0.020.04

0.060.08

0.10.12

0.14

0.001 0.01 0.1 1 10Fourier No.

Eut

etic

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.001 0.01 0.1 1 10Fourier No.

Eut

etic

Note

Wide variation

In Eutectic

12 of 19

0

2

4

6

8

10

0.01 0.1 1 10

Fourier Number

wt%

Co

nc

en

tra

tio

n

Approximation

Experiment

Numerical

No-Coarsening

Predictions of Eutectic FractionWith constant cooling

Co = 4.9

Ceut = 33.2

k = 0.16

Comparison with Experiments Sarreal Abbaschian Met Trans 1986

13 of 19

oldlCC

QC)g1(C)gg(C)g1( slllold

soldl

oldl

Parabolic solid growth – No Second Phase – No Coarsening Use 10,000 equal of g

C0 = 1, = 0.13, = 0.4

0

l

C

kC

Use

To calculate evolving segregation ratio

14 of 19

00.5

1

1.52

2.53

3.5

44.5

5

0.00010.0010.010.11

Liquid Fraction

Seg

rega

tion

Rat

io

Full NumericalSolution

Profile

Profile Fixed m=2

Parameter

Parameter

1

s

s

l

C

C2~m

2

2

14

4

21

2

)CC)(1m(g

Qs

sl

1

s

s

l

C

C2~m

QC)g1(C)gg(C)g1( slllold

soldl

oldl

Performance of Models under parabolic growth no second phase

0

l

C

kCPrediction of segregation ratio in last liquid to solidify(fit exponential through last twotime points)

15 of 19

2

2

14

4

)CC(gdt

dCgtQ old

lll

0102030405060708090

100

0.01 0.1 1 10Fourier No.

Seg

rgat

ion

Rat

io a

t g

=1 analytical

profile

paramter

0

2

4

6

8

10

0.01 0.1 1 10

Fourier No.

Seg

rega

tion

Rat

io a

t g

=1

)CC(g12

2tq old

ll

Parameter

Robust Easy to UsePoor Performance at very low liquid fraction— can be corrected

)CC)(1m(g

tqs

sl

1

s

s

l

C

C2~m

Profile

A little more difficult to use

With this Ad-hoc correctionExcellent performanceat all ranges

Two Models For Back Diffusion

A

C

]C[

]H[

C

Predict g

QC)g1(C)gg(C)g1( srl

rl

rl

oldrs

oldl

predict Cl

predict T

Calculate C

Transient solute balance in arm space

Solidification Solver 16 of 19

1mg 3

13

4

Account for coarsening

My Method of Choice

1.0E+02

1.0E+04

1.0E+06

1.0E+08

1.0E+10

1.0E+12

1.0E+14

1.0E+16

1.0E+18

1.0E+20

1.0E+22

1.0E+24

1.0E+26

0 20 40 60 80 100 120

Year-1980

No

de

s

1000 20.6667 Year

“Moore’s Law”

current for REV of 5mm

Voller and Porte-Agel, JCP 179, 698-703 (2002)

l

TLe

.5m

I Have a BIG Computer Why DO I need an REV and a sub grid model

~ 50 m

solid

~5mm(about 106 nodes)

17 of 19

2055 for tip

Model Directly

(about 1018 nodes)

Tip-interface scale

chill

solid

mushy

liquid

riser

y

Application – Inverse Segregation in a binary alloy

100 mm

Shrinkage sucks solute rich fluid toward chill – results in a region of +ve segregation at chill

Fixed temp chill results in a similarity solution

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

4.4 4.6 4.8 5 5.2 5.4

Concentration

Sim

ilari

ty V

aria

ble

back diff.

Scheil

lever

similarity(lever)

similarity(Scheil)

)CC(g12

2tq old

ll

Parameter

1mg 3

13

4

2fDt

Current estimate

empirical

18 of 19

Ferreira et al Met Trans 2004

Comparison with Experiments

19 of 19