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Session III: Computational Modelling of Solidification Processing Analytical Models of Solidification Phenomena V. Voller QUESTION: As models of solidification process and phenomena become more complex do analytical solutions of limit cases become less useful? Answer: Yes, limit cases that admit analytical solutions are often physical r removed from the process/phenomena of interest to be useful. wer: With a bit of searching and a little innovation it is possible to build ted limit solutions of process/phenomena that admit analytical solutions.
Transcript

Session III: Computational Modelling of Solidification Processing

Analytical Models of Solidification Phenomena

V. Voller

QUESTION: As models of solidification process and phenomena become

more complex do analytical solutions of limit cases become less useful?

Short Answer: Yes, limit cases that admit analytical solutions are often physically too far removed from the process/phenomena of interest to be useful.

Counter Answer: With a bit of searching and a little innovation it is possible to build physically sophisticated limit solutions of process/phenomena that admit analytical solutions.

A Key Moment In History of the Computational Modelling of Solidification Processing

65 years ago

Heating of a steel Ingot

Followed the standard modeling paradigm of Validation---comparing computations to measurement

observed

calculated

A very early (first ?) paper using numerical modeling of heat transfer in metals processing.

Pre-digital

The Differential Analyser: An analog machineBuild for ~ $30 in 1934.

Cra

nk,

19

47

A Key Moment In History of the Computational Modelling of Solidification Processing

At that time these were using state of the art computations

First use of Enthalpy Method for Solidification Model

Early application of Crank-Nicolson

And they recognized the need to Verify their calculations via comparison with appropriate analytical solutions

But in today's world with solutions obtained with sate of the art digital technologies

Differential Analyzer

Allow us to solve much more complex systems

soli

d mus

h liqui

d

cool

mold

One-D solidification of an alloy controlled by heat conduction

vs

Crystal growth in an under cooled alloy

vs. Distributed Graphics Processing Units (GPUS)

Is there still a place/role/opportunity for the meaningful use of analytical solutions?

growth of an initially spherical seed in an under cooled alloy

In fact for the problem shown there is a rich source of available analytical solutions

Carslaw and Jaeger, Conduction of Heat in Solids, (1959). (CJ)

Rubinstein, The Stefan Problem (1971) (R)

Alexiades and Solomon, Mathematical Modeling of Melting and Freezing Processes, (1984). (AS)

Dantzig and Rappaz, Solidification (2009) (DR)

Two Examples

One –D Solidification of a supper heated Binary Alloy-with a planar front (R, AS, DR)

solid liquid alloy

T<Teq

u

Front movement

Conc. History(kappa = 0.1)

Temp. History Symbols Numerical

Lines analytical

Constitutional undercoolingCalls into question planar assumption

conc. profile

Tm

Solidification of a spherical seed in an under cooled PURE melt

Analytical Solutions in

Carslaw and Jaeger –(also solutions for planar and cylindrical case)

Dantzig and Rappaz -(considers Surface Tension in limit of zero Stefan number)

Here we will demonstrate how these solutions can be coupled to model the solidification of A spherical seed in in an UNDER COOLED BINARY ALLOY With assumption of no-surface under cooling and no growth anisotropy.

solid liquid alloy

T<Teq

u

Tm

fixTT fix

s kCC

Temperature

Conc.

Solid Liquid

Dimensionless Governing Equations

Rrr

Tr

rrt

T

,1 22

t

R

r

TCStTRT

R

fixfix

),1( 0TrT

Rrr

Cr

rrLet

C

,11 22

Heat

1rC dt

dRCk

r

CCRC fix

R

fix )1(,

Concen.

sensible/latentStefan No.

thermal/massLewis No.

fixed values in solid

Temperature

Conc.

Solid Liquid

Similarity Solution

tR 2Assume

Can then show that value of Lambda follows from solving the following set of equations

0)1( fixfix CStT

0)()( 0212 22

TTerfcee fix

0)1()()1(212 22

fixLeLefix CLeerfcLeeeLeCk

Liquid temp and con . Then given by

Rr

t

rerfce

r

t

erfce

TTT

RrT

T trfix

fix

,22)(

)(2

0,

4/00

2

2

Rr

t

Lererfc

Lee

r

t

LeerfcLee

C

RrkC

C tLer

Le

fix

fix

,22)(

)1(21

0,

4/2

2

Solution can tell us something about the nature of the Lewis Le number and Verify Numerical Algorithms for coupling of solute and thermal fields in crystal growth codes

0

0.5

1

1.5

2

2.5

0 20 40 60

position

conc

entr

atio

n

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 20 40 60

position

tem

pera

ture

1.0,1.0,5.0,2 0 StkTLe

conc. profile Numerical (enthalpy) symbols

Lines analytical

Le ~1 similar thickness ofSolute and thermal layers

0

1

2

3

4

5

3 4 5 6

position

conc

entr

atio

n

0

5

0 20

-0.5

-0.4

-0.3

-0.2

-0.1

0

3 4 5 6

position

tem

pera

ture -0.5

-0.3

0 20

1.0,1.0,5.0,50 0 StkTLe

Le >>1 much thinnersolute layer

Outline of Enthalpy Solution in Cylindrical Coordinates

Assume an arbitrary thin diffuse interface whereliquid fraction

01 f

fTH Define

Throughout Domain a single governing Eq

0,1

rr

Tr

rrt

H

0lim0

r

Tr 0TrT

For a PURE material Numerical Solution Very Straight-forward

)(1

112

ii

noutii

ninn

ii

newi TTrTTr

rr

tHH

Initially

99.,0

15.0,1,5.0

11

0

fT

HfTT iii

seed

Set

Transition: When

1 and 0 1 newi

newi ff 99.0 1

newif

An explicit solution

10 ifIf ]1],0,(max[min newnewi Hf

Update Liquid fraction

newi

newi

newi fHT

Update Temperature

R(t)

0

2

4

6

8

10

12

14

0 20 40 60

time

soli

d-fr

ont R

(t)

Enthalpy

Analytical

Excellent agreement with analytical when predicting growth R(t)

)1( kCfkCC f

0,11

r

r

Cr

rrLet

C f

Can extend to the case of a binary alloy by defining a mixture solute as

Explicitly solving

0lim0

r

Cr

1rCWith

newp

newp

newp fTH

newPnew

p

newfpnew

pnewp f

kkf

CStCStT

)1(1)1(

Liquidus line

10 pfIf Update Liquid fraction

Quad eq. in newpf

newp

newp

newp fHT

Update Temperature

2-D enthalpy solutionsOf cy. seed growth inan undercooled pure melt

SimilaritySolution

Each one based on a different seed geometry and front update

ALL are wrong—Since there is no imposed anisotropy

Conclusions

Analytical models of solidification phenomena are important tools in advancingour understanding of solidification processes.

There is a rich source of available analytical solutions that can be adapted to provide meaningful solutions for a variety of solidification process and phenomena of current interest

e.g. Coupling of thermal and solute fields in crystal growth

---Beyond this they can be used to bench-marking the predictive performance of large multi-scale, general numerical solidification process models.

These solutions are useful

-- In the first instance they allow for a clear and direct understanding of the behavior and interaction of key elements in a solidification system

e.g. role of Lewis number

And Grid Anisotropy


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