session iii: computational modelling of solidification processing
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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?. - PowerPoint PPT PresentationTRANSCRIPT
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