Microsegregation Models and their RoleIn Macroscale Calculations

Vaughan R. Voller University of Minnesota

After Flemings (Solidification Processing) and Beckermann (Ency. Mat)

1m

l

TLe

What is Macrosegregation

Csolid

Cliquid

Partitioned solute atsolid-liquid interface

Redistributed by Fluid and solid motion

convection

grain motion

shrinkage

Macro (Process) Scale Equations

Equations of Motion (Flows)

mmREV

Heat:

)TK()HcT(Ut

]H[l

0UCt

]C[ll

Solute Concentrations:

Assumptions for shown Eq.s: -- No solid motion --U is inter-dendritic volume flow To advance to the next time step we need find REV values for

•T temperature•Cl liquid concentration

•gs solid fraction•Cs distribution of solid concentration

Need four relationships which can be obtained from a micro-scale model

Under the assumptions of:

1. Equilibrium at solid-liquid interface2. Perfect solute mixing in the liquid3. Identification of a solid-liquid interface length scale (e.g., a ½ secondary arm space)

Possible Relationships are

Xs(t)

Xl(t)~1/3 ~ 10’s m

coarsening

llsl

X

0ssl C)XX(dC]C[X

s

HTc)XX(TcX]H[X llslsssl

Definitions of mixture terms in arm space

T=G(Cl1,Cl

2…….)

4. Account of local scaleredistribution of soluteduring solidification ofthe arm space

3.

1

1.

2.

Thermodynamics

Primary

+Secondary

Clk <--> Cs

k (interface)

Clk=F(Cl

1,Cl2…….)

gs= Xs/Xl

solidLiquid

The Micro-Scale Model

~ 1 m

ll

slX

0s

lC

X

XXdC

X

1]C[

s H

X

)XX(cT]H[

l

sl

Macro Inputs

Xs(t)

Xl(t)~1/3

sCq

back

Define:

CXXCCCXXqXX

1C old

lloldl

oldolds

oldl

oldbold

sl

*

macro coarseningback

Average Solute Concentration in Xl-Xsold During time step Treat like initial sate for a new problem

Iteration: Guess of T in [H] Xs –Xsold Assume Lever on C* Cl--, T

new solid thermo

ff

f

f

t

t,

xX

,Dt4

s2s

2s 0,

CC

3***. At each time stepapprox. solid solute profile as

lpows

pow kCXXa

Choose to satisfyMass Balance

1. Numerical Solution in solid

Requires a Micro-Segregation Model that to estimate “back diffusion” of solute into the solid at the solid-liquid interface

solidLiquid

~ 1 mXs(t)

Xl(t)~1/3

sCq

back

( Three Approaches)

d

dCCls

2. Approximate with “average” parameter

Function of can be correctedfor coarsening

solid

Testing: Binary-Eutectic Alloy. Cooling at a constant rate

Predictions of Eutectic Fraction at end of solidification

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.001 0.01 0.1 1 10

Fourier Number

Fra

cti

on

Eu

tec

tic

coarseningNumerical back diff model

Approx profile model

4.4

4.5

4.6

4.7

4.8

4.9

5

5.1

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

zZ

eut

Zliq

lever

Gulliver -Scheil

Effect of MicrosegregationOn Macrosegregation

= 0.2

Coarsening

g

A uni-directional solidification of a Of binary alloy cooled from a fixed chill.

Microsegregation (back diffusion into solid) modeled in terms off rate of change of solute in liquid

ddCC

ls

No Coarsening

, = 1

=0, = 0

Solute concentration in mushy region

l

TLe

~ 1 mm

ll

slX

0s

lC

X

XXdC

X

1]C[

s

HX

)XX(cT]H[

l

sl

1m

Summary

T ,g, Cs and Cl

~ 1 m

solid

Microsegregation

And Themodynamics

From macro variables

Find REVvariables

Accounting for

at solid-liquid interface

Key features

-- Simple Equilibrium Thermodynamics

-- External variablesconsistent with macro-scaleconservation statements

-- Accurate approximate accounting of BD and coarsening at each stepbased on current conditions

l

TLe

1m

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

~ 1 m

solid

~mm(about 109)

Model directly (about 1018)

Tip-interface scale

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

2010 for REV of 1mm

2055 for tip

xxllME

xU

)g1(xx

p)g1()U()U(

t

U

yyravlllMEg)g1(

yV

)g1(yy

p)g1()V()V(

t

U

Modeling the fluid flow requires a Two Phase modelThat may need to account for:

Both Solid and Liquid Velocities at low solid fractionsA switch-off of the solid velocity in a columnar regionA switch-off of velocity as solid fraction g o.

An EXAMPLE 2-D form of the momentum equations in terms of the interdentrtic fluid flow U, are:

turbulence

Buoyancy

)(K)g1(suu

l

Friction between solid and liquidAccounts for mushy region morphologyRequires a solid-momentum equation

Extra Terms