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REV. representative ½ arm space. Process. solid. g. ~ 50 m m. ~5 mm. sub-grid model. ~0.5 m. 1 of 19. Maco-Micro Modeling— Simple methods for incorporating small scale effects into large scale solidification models – Vaughan Voller, University of Minnesota.

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Computational grid size

REV

representative ½ arm space

Process

solid

g

~ 50 mm

~5 mm

sub-grid model

~0.5 m

1 of 19

Maco-Micro Modeling—

Simple methods for incorporating small scale effects into large scale solidification models–

Vaughan Voller, University of Minnesota

Scales in a “simple” solidification process model

Computational grid size

Enthalpy based

Dendrite growth model

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


Computational grid size

2 of 19

~ 0.1 m

chill

A Casting

The REV

casting

~10 mm

103

101

10-1

10-3

10-5

10-7

10-9

Nucleation

Sites

heat and mass tran.

equi-axed

columnar

grain

formation

The Grain

Envelope

growth

~ mm

Time Scale (s)

solute diffusion

The Secondary

Arm Space

nucleation

~100 mm

interface

kinetics

10-9 10 10-3 10-1

Length Scale (m)

The Tip

Radius

~10 mm

The Diffusive

Interface

f = 1

f = -1

~1 nm

Scales in Solidification Processes

(after Dantzig)

Can we build a direct-simulation of a Casting

Process that resolves to all scales?


Computational grid size

1 meter

6 decades

1 micron

3 of 19

Well As it happened not currently Possible

1000 20.6667 Year

“Moore’s Law”

2055 for tip

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

Plotted The three largest MacWasp Grids (number of nodes) in each volume


Computational grid size

~ 0.1 m

chill

A Casting

The REV

casting

~10 mm

103

101

10-1

10-3

10-5

10-7

10-9

Nucleation

Sites

heat and mass tran.

equi-axed

columnar

grain

formation

The Grain

Envelope

growth

~ mm

Time Scale (s)

solute diffusion

The Secondary

Arm Space

nucleation

~100 mm

interface

kinetics

10-9 10 10-3 10-1

Length Scale (m)

The Tip

Radius

~10 mm

The Diffusive

Interface

f = 1

f = -1

~1 nm

4 of 19

Scales in Solidification Processes

(after Dantzig)

To handle with current computational

Technology require a “Micro-Macro” Model

See Rappaz and co-workers

Example a heat and Mass Transfer model

Coupled with a Microsegregation Model


Computational grid size

5 of 19

Solidification Modeling

Process

REV

representative ½ arm space

solid

g

sub-grid model

~ 50 mm

Micro segregation—segregation and

solute diffusion in arm space

~5 mm

~0.5 m

Computational grid size

from computation

Of these values

need to extract

--

--

--


Computational grid size

A

C

6 of 19

Primary Solidification Solver

g

Transient mass balance

g

model of micro-segregation

Iterative loop

Cl

T

(will need under-relaxation)

Give Liquid Concentrations

equilibrium


Computational grid size

Micro-segregation Model

transient mass balance gives liquid concentration

Solute Fourier No.

Solute mass density

after solidification

Solute mass density

before solidification

Q -– back-diffusion

Solute mass density

of new solid (lever)

7 of 19

liquid concentration due to

macro-segregation alone

½ Arm space of

length l takes

tf seconds to solidify

In a small time step new solid forms

with lever rule on concentration

Need an easy to use approximation

For back-diffusion


Computational grid size

For special case Of Parabolic Solid Growth

In Most other cases

The Ohnaka approximation

and

And ad-hoc fit sets the factor

Works very well

8 of 19

The parameter Model --- Clyne and Kurz,

Ohnaka


Computational grid size

Need to lag

calculation one

time step and

ensure Q >0

m is sometimes take as a constant ~ 2 BUT

In the time step model a variable value can be use

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

9 of 19

The Profile Model

Wang and Beckermann


Computational grid size

A model by Voller and Beckermann suggests

If we assume that solid growth is close to parabolic

m =2.33 in

Parameter model

In profile model

10 of 19

Arm-space will increase in dimension with time

Coarsening

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

back diffusion 


Computational grid size

11 of 19

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

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

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

Calculating the transient value of g from

Remaining Liquid when

C =5 is Eutectic Fraction

Parameter or Profile


Computational grid size

Note

Wide variation

In Eutectic

12 of 19

Results are good across a range of conditions


Computational grid size

13 of 19

Predictions of Eutectic Fraction

With constant cooling

Co = 4.9

Ceut = 33.2

k = 0.16

Comparison with Experiments Sarreal Abbaschian Met Trans 1986


Computational grid size

14 of 19

Parabolic solid growth – No Second Phase – No Coarsening

Use 10,000 equal of Dg

C0 = 1,

k = 0.13, a = 0.4

Use

To calculate evolving segregation ratio


Computational grid size

15 of 19

Performance of Models under parabolic growth no second phase

in last liquid to solidify

Prediction of segregation ratio

(fit exponential through last two

time points)


Computational grid size

Solidification Solver

16 of 19

Calculate

Transient solute balance in arm space

predict T

Predict g

predict Cl

Two Models For Back Diffusion

Profile

Parameter

A

A little more difficult to use

Robust

Easy to Use

Poor Performance at

very low liquid fraction—

can be corrected

With this

Ad-hoc correction

Excellent performance

at all ranges

Account for coarsening

C

My Method of Choice


Computational grid size

Voller and Porte-Agel,

JCP 179, 698-703 (2002)

1000 20.6667 Year

“Moore’s Law”

Model Directly

2055 for tip

Tip-interface scale

current for REV of 5mm

(about 1018 nodes)

17 of 19

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

solid

~ 50 mm

~5mm

(about 106 nodes)

.5m


Computational grid size

18 of 19

riser

liquid

Parameter

Current estimate

mushy

empirical

y

solid

chill

Application – Inverse Segregation in a binary alloy

Shrinkage sucks solute rich

fluid toward chill – results

in a region of +ve segregation

at chill

100 mm

Fixed temp chill results

in a similarity solution


Computational grid size

19 of 19

Comparison with Experiments

Ferreira et al Met Trans 2004