Modeling Heat Transfer, Precipitate Formation, and Grain...
Transcript of Modeling Heat Transfer, Precipitate Formation, and Grain...
ANNUAL REPORT 2008UIUC, August 6, 2008
Kun Xu(PHD Student)
Department of Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
Modeling Heat Transfer, Precipitate
Formation, and Grain Growth during
Secondary Spray Cooling
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 2
Introduction: Transverse cracks
Crazing around a transverse crack at base of an oscillation mark on the as-cast top surface of a 0.2% C steel slab.
Note larger grain size atbase of oscillation mark.
Typical grain size ~1mm
Reference: Reference:E. S. Szekeres, 6th Internat. Conf. on Clean Steel, Balatonfüred, Hungary, June 2002.
Casting direction
1mm
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 3
Formation of surface cracks & precipitate embrittlement
STAGE II - Surface grains “blow” locally due to high temperature (>1350oC) and strain, especially at the base of deeper oscillation marks.
STAGE I - Normal solidification on mold wall. Surface grains are small but highly oriented.
STAGE III - Nitride precipitates begin to form along the blown grain boundaries. Microcracks initiate at weak boundaries.
STAGE IV - Ferrite transformation begins and new precipitates form at boundaries. Existing micro-cracks grow & new ones form.
STAGE V - At the straightener, microcracks propagate and become larger cracks, primarily on top surface of the strand.
γγ γγγγ
Reference:E. S. Szekeres, 6th Internat. Conf. on Clean Steel, Balatonfüred, Hungary, June 2002.
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 4
Larger grains beneath oscillation marks & surface depressions
Due to:* more heat flow resistance across gap* higher shell temperature* faster grain growth rate
Causes:* Tensile strain concentration area* Make hot grains actually align — “secondary
recrystallization”— “blown grains”* Embrittled with a large numbers of fine
precipitates at the weak grain boundaries —cracks open up along the boundaries —transverse cracks
mold
gap
Steel shell
Molten
steel pool
Effect of oscillation marks on grain size
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Project Overview
CON1D --- Temperature & Phase fraction
Equilibrium precipitation model --- Tendency for precipitation and equilibrium concentration of all elements
Kinetic model for precipitate growth --- Size distribution and volume fraction of possible precipitates
Grain growth model evolving with time, cooling history with the effect of pinning precipitates --- Grain size of steel casting
Predict ductility and transverse cracks --- Stress and strain analysis concerning coupled influence of precipitates and grain size
Final goal: Prevent cracks and assure quality
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 6
Size distribution and micrograph of AlN precipitate in CSP steel
Reference: J. A. Garrison, Aluminum nitride precipitation behavior in thin slab material, AIST 2005 proceeding, Volume II, June 2002.
Precipitates could nucleate and grow 1) in the liquid, 2) duringsolidification (segregation), 3) on grain boundaries or 4) inside the grains. Composition and size distribution evolve with temperature and time.
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 7
Example Problem: multiple peak distribution
Equilibrium precipitation model calculates AlN at different stages
Tliq=1528.1oC, Tsol=1501.5oC
No AlN forming in the liquid
Differential Scheil model for segregation
AlN begins to form at 1503oC; amount formed during solidification is 6.86×10-4wt%.
Equilibrium amount of AlN at 900oC is 0.01134; percent forming in solid is 94%.
Need kinetic model to correct error (94% vs 55%): Much of the AlN remaining in the solid should stay dissolved until 1060oC. At this low temperature, the nucleation and growth of new AlN particles in the solid is slow. During this time, the AlNparticles already present (from solidification) can grow quickly due to local high concentration from segregation. Finally, new nuclei form in the solid at lower temperature, grow slowly from the remaining fraction of Al and N.
0.011 wt% Al, 0.009 wt% NkAl=0.6, kN=0.27
45%55%
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 8
Introduction: Particle Collision and diffusion
Particle collision
Loss of particle i by collision of other particle j, double loss
if both particle i
Generation of particle i by collision of two smaller particle j + particle i-j, only
count once for two reacting particles
ijΦ: Number density of size i particle (/m3)
: Collision coefficient for collision between size i and size j particle (m3/s)
1
, ,1 1
1(1 ) (1 )
2
M ii
i ij ij j j i j j i j j i jj j
dnn n n n
dtδ δ
−
− − −= =
⎡ ⎤= − + Φ + + Φ⎣ ⎦∑ ∑
in
Molecule Diffusion(i→i+1)
Molecule Diffusion
(i-1→i)
Particle Dissolution
(i+1→i)
Particle Dissolution
(i→i-1)
1 1 1 1 1 1 1D Di
i i i i i i i i i i
dnn n n n A n A n
dtβ β α α− − + + += − + + −Particle diffusion
: Dissociate rate per unit area of particle i (m-2s-1)
: Diffusion rate constant of particle i (m3/s)Diβ
iα
Collision happens only in liquid, and diffusion happens both in liquid and solid
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 9
Particle Size Group (PSG) Model
This model simulates nucleation (from individual molecule size 0.1nm) up to real particles (μm): particles contain from n=1 to 1010 molecules.
serious computation and memory storage issues arise with so many particle sizes.
solve problem with PSG model: use G groups of geometrically progressing size
:jr
, 1 :j jr +
Characteristic radius of group j particle
Threshold radius between group j and group j+1 particle
, 1
1,
( )rj j
j jj r
N n r dr+
−
= ∫ 1 ( 1) /31
j jV j V
j
VR r R r
V+ −= ⇒ =
, 1 1j j j jr r r+ +=
Total number density for each group
Average particle volume ratio
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 10
Effect of RV on PSG collision model
Range 2 (2.148>RV>1.755)
Range 3 (1.755>RV>1.588)
1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0
-1.0
-0.5
0.0
0.5
1.0
3 2
1 , 12 j j j
j
V V
V− +−
3 , 1j j j j
j
V V V
V− ++ −
2 , 1j j j j
j
V V V
V− ++ −
1 , 1j j j j
j
V V V
V− ++ −
, 12 j j j
j
V V
V+−
Fu
nct
ion
val
ue
RV
1
Range 1 (4.00>RV>2.148)
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 11
PSG model summary for collision
,
4.000 2.148
1 2.148 1.755
2 1.755 1.588
V
c j V
V
j R
k j R
j R
> >⎧⎪= − > >⎨⎪ − > >⎩
, 1
, 1
* 11* ( 1) /3 ( 1) /3 3 * ( 2) /3 ( 1) /3 3 * *
1*1
( ) ( )c j
c j
k jj j kj k j kk
j V V k V V j kk k kj j
dN V VVN R R N R R N N
dt V V
−
−
−−− − − −
−= =
+= + + +∑ ∑
, 1
, 1 ,
11
1, 11
(1 )c j
c j c j
k jj j kk
j jk k j k j k j jk jk kk k k k kj j
dN V VVN N N N N N
dt V Vδ
−
−
− ∞−
− −= = =
+⎡ ⎤= Φ + Φ − + Φ⎣ ⎦∑ ∑ ∑
Generally choose to cover the cases when RV<2.011max( , )j
j VV R j V−=,
* ( 1) /3 ( 1) /3 3 *(1 )( )c j
Gj k
j jk V V kk k
N R R Nδ − −
=
− + +∑
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 12
Validation of PSG model for collision
Decreasing the value of RV can increase the accuracy of PSG model
If largest particle for exact solution has Mmolecules, then the number of groups Gused in the PSG model must satisfy
So we choose
(Including boundary group NG≡0 at all t*)
(log ) 1 1VRG Ceil M= + +
1GVR M− >
Total number density of particles and M=1000, time step
Initial condition and for i>1 at
Boundary condition at all
1
M
T ii
n n=
=∑*1 1n = * 0in = * 0t =
* 0Mn = *t
* 0.001tΔ =
Turbulent collision
1/ 2 31.3( / ) ( )ij i jr rε νΦ = +
1E-3 0.01 0.10.0
0.2
0.4
0.6
0.8
1.0
Exact Solution R
V=3.0
RV=2.0
RV=1.6
nT/n
0
1.3r1
3(ε/ν)0.5n0tt* =
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 13
Validation of PSG model for collision (RV=3.0)
1E-3 0.01 0.11E-4
1E-3
0.01
0.1
1
1.3r1
3(ε/ν)0.5n0t
Exact Solution R
V=3.0
N6
N5
N4
N3
N2
N
j/n0
N1 Number of groups G=8
N1: size 1N2: size 2-5N3: size 6-15N4: size 16-46N5: size 47-140N6: size 141-420 N7: size 421-1262
Number range in each group
t* =
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 14
Validation of PSG model for collision (RV=2.0)
1E-3 0.01 0.11E-4
1E-3
0.01
0.1
1
1.3r1
3(ε/ν)0.5n0t
Nj/n
0
N8
N7
N6
N5
N4
N3
N2
N1 Exact Solution
RV=2.0
N1: size 1N2: size 2N3: size 3-5N4: size 6-11N5: size 12-22N6: size 23-45N7: size 46-90N8: size 91-181N9: size 182-362N10:size 363-724N11:size 725-1448
Number of groups G=12
Number range in each group
t* =
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 15
Develop PSG model for diffusion
leftjn
rightjn
1rightjn − 1
leftjn +
Generation to group j
Particles inside group except can diffuse still into group j; particles inside group except can dissolute still into group j. can diffuse into group j; can dissolute into group j.
rightjn
leftjn
1rightjn −
1leftjn +
Loss from group j
can diffuse into group j+1; can dissolute into group j-1.
leftjnright
jn
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 16
PSG model for diffusion
, 11 11 1
( )( ) ( )j j jD right left D right
j j j j j j j j jj j j
dN Floor VV VN N n A N n N n
dt V V Vβ α β+= − − − −
1, , 1 1,1 1 1 1 1 1
( ) ( ) ( )j j j j j jD right left leftj j j j j j j j
j j j
Ceil V Floor V Ceil VN n A n A n
V V Vβ α α− + −
− − + + ++ + −
The mass balance can be conserved by this equation. Of course, the exact distribution of particles inside each group is unknown.
1,j jV − , 1j jV +
, 1 1 1, 1max(1, ( / ) ( / ) 1)j j j j jDV Floor V V Ceil V V+ −= − +
4Di iDrβ π=
1,
2exp
Di m
i eqi i
Vn
A RTr
β σα⎛ ⎞
= ⎜ ⎟⎝ ⎠
Define possible molecules number range inside group j particle from to
1
1
11
j jV Vj
j
nq
n
−−
−
⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠
jj
j
Nn
DV=
1, 1( )1 1
j j jCeil V Vleftj jn n q − −−
−= , 1( )2
j j jFloor V Vrightj jn n q + −=
1
1
12
j jV Vj
j
nq
n
+ −+⎛ ⎞
= ⎜ ⎟⎜ ⎟⎝ ⎠
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 17
Estimation of and leftjn right
jn
1
1
12
j jV Vj
j
nq
n
+ −+⎛ ⎞
= ⎜ ⎟⎜ ⎟⎝ ⎠
Define average number density for each group
And we assume
1
1
11
j jV Vj
j
nq
n
−−
−
⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠
1, 1( )1 1
j j jCeil V Vleftj jn n q − −−
−=
, 1( )2
j j jFloor V Vrightj jn n q + −=
Geometric progression assumption
Maximum limit of and are Njleftjn right
jn
jj
j
Nn
DV=
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 18
0.01 0.1 1 10 1001E-4
1E-3
0.01
0.1
1
Lines: Exact solution Symbols: PSG model
N8N
6N
5
N4
N3
N2
Den
sity
Time
N1
Good match for just G=12 groups
Define total number density of particles and choose M=1000
Initial condition for any size i at
Boundary condition at all
* 0in = * 0t =1
M
T ii
n n=
=∑
* 0Mn = *t
*11 max
1,
4eq
nn
nΠ = = ⇒ Π ≈
0 20 40 60 80 1000
1
2
3
4
Su
per
satu
rati
on
t*
Choose time step * 0.01tΔ =
Validation of PSG model for diffusion (RV=2.0)
Supersaturation
Total number of molecules in all size particles* * * *
1
( ) 9[1 exp( 0.1 )]M
S ii
n t i n t=
= ⋅ = − −∑
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 19
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.41E-6
1E-5
1E-4
1E-3
0.01
0.1
Ni/N
1,eq
Particle radius (nm)
Exact solution PSG model
Further PSG model validation:Extend test problem to physical size distribution
max 5max
24 0.742 16
lnm
c c
Vr nm i N
RT
σΠ ≈ ⇒ = ≈ ⇒ ≈ ⇒Π
5 3 9 216.4 10 / , 0.294 , 300 , 10 /mV m mol r nm T K D m s− −= × = = =
2 23 3 1/3 * 61, 10.02 / , 6 10 , , 2.216 10eq iJ m N m r ri t tσ −= = × = = ×
N5: size 12-22 (0.673-0.824nm)N6: size 23-45 (0.836-1.045nm)N7: size 46-90 (1.053-1.317nm)N8: size 91-181 (1.322-1.662nm)N9: size 182-362 (1.665-2.094nm)N10:size 363-724 (2.096-2.639nm)
t=45.1μs
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 20
AlN Calculation in Compact Strip Production
5 3 211.33 10 / , 0.174 , 0.75 /mV m mol r nm J mσ−= × = =
Chemical composition: 0.06wt% C, 0.462% Mn, 0.004% P, 0.004% S, 0.017% Si, 0.011% Al, 0.009% N
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 21
Solubility of AlN in austenite
6770log 1.03AlNK
T= − +
850 900 950 1000 1050 11000.000
0.005
0.010
0.015
We
igh
t o
f d
iss
olv
ed e
lem
en
ts a
nd
pre
cip
ita
tes
Temperature(oC)
[Al] [N] AlN
1071.8oC
Solubility product:
0 0[ ] [ ]
Al N
Al Al N N
M M
− −=
[ ][ ] AlNAl N K=
Focus calculation from 1071.8oC to 900oC
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 22
Diffusion of Al and N in austenite
Diffusion coefficient: 3 2345003 10 expAlD
RT− ⎛ ⎞= × −⎜ ⎟
⎝ ⎠
Choose diffusion of Al (for and ) to calculate the formation of AlN (since Al diffuses much slower than N and mole fraction of N is larger than that of Al)
900 950 1000 1050 11001E-13
1E-12
1E-11
1E-10
Al
D (
m2/s
)
Temperature(oC)
N
5 1685009.1 10 expND
RT− ⎛ ⎞= × −⎜ ⎟
⎝ ⎠
Djβ
jα
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 23
0.1 1 10 1001E-8
1E-4
1
10000
1E8
1E12
1E16
1E20
1E24
Nu
mn
er D
ensi
ty (
m-3)
Radius (nm)
ic=8
0.38nm
4.9nm
Stable precipitates
30 1.9144e+025/msteelM A
Al
AlN N
M
ρ×= ≈3 37.8 10 /steel kg mρ = ×
[ ]11.0 exp( 0.001*4 * * )s M MN N Dr N tπ= − −
Calculation of AlN size distribution
/ 20 /oTypical cooling rate T t C sΔ Δ ≈
Current time step ∆t=4*10-8s for stability, implicit format needs to be considered in the future work if the processing time is larger
G=40, packing factor 0.74Radius 0.174nm~1.58μm1~5.5×1011 molecules
8.6Total time t s⇒ ≈
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 24
Compare AlN size distribution with measurement
Good match with small-size precipitate distribution (governed by solid-state diffusion).
0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Rel
ativ
ve F
req
uen
cy
Radius (nm)
Simulation Experiment
Need to include liquid collisions, solidification segregation and subsequent growth to model larger AlN particles.
Radius (nm)
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 25
The influence of pinning precipitates on the grain growth
Grain growth model in the presence of growing precipitates
Update the value for the growing precipitates for each step
Some parameters to fit grain size for mainly Ti-microalloyed or Nb-microalloyed steels are given, but the influence of combined effect of many alloying elements still needs more work
(1/ 1)*0
1 1exp
nappQdD f
Mdt RT D k r
−⎛ ⎞ ⎡ ⎤= − −⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠
/f r
2*1 0 1
expt app
t
QI M dt
RT
⎛ ⎞= −⎜ ⎟
⎝ ⎠∫2
2 1 1
1exp( )
ts
t
QI c dt
T RT= −∫ 2
0 1
Ikm
f I
⎛ ⎞= ⎜ ⎟⎝ ⎠
0 0
0Lim
rD k
f=
Reference: I. Anderson, O. Grong, Acta Metall. mater. Vol. 43, No. 7, pp2673-2688, 1995.
Constant volume fraction f0 for precipitate
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 26
Kinetic model for austenite grain growth
Grain growth model(1/ 1)
*0
1 1exp
nappQdD f
Mdt RT D k r
−⎛ ⎞ ⎡ ⎤= − −⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠
Reference: C. Bernhard, J. Reiter, Simulation of austenite grain growth in continuous casting, AISTech 2007.
CW: secondary cooling intensity
When temperature drops into mixed-phase region (transformation at ~800oC), then the
ferrite will form and brand new grains cause the cracks much more difficult to form. And the already existing grains in
austenite almost will not change because of low growth rate at low temperature
The change of depth of oscillation marks influences the temperature history at these locations. Larger depth causes higher temperature and thus larger grain size beneath oscillation marks.
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 27
Validation of austenite grain size model
* 3 20 4 10 / , 0.5M m s n−= × =
% % 0.14 % 0.04 %Pwt C wt C wt Si wt Mn= − +
167686 40562( % )app PQ wt C= +
Calculation from starting temperature given to 900oC
The average cooling rate 5oC/s
initial grain size is zero
Reference: J. Reiter, C. Bernhard, H Presslinger, Determination and prediction of austenite grain size in relation to product quality of the continuous casting process, MS&T 06, Cincinnati, USA.
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 28
Grain growth in real continuous casting
0 3 6 9 12 15800
900
1000
1100
1200
1300
1400
1500
1600
Tem
per
atu
re (
C)
Distance from meniscus (m)
Temperature
0 3 6 9 12 150.0
0.2
0.4
0.6
0.8
Gra
in s
ize
(mm
)
Distance from meniscus (m)
Grain size
0.010.0060.0040.0150.0150.210.061006
%N%B%V%Nb%Ti%Si%Mn%CName
Mold length: 0.95mStart of first spray zone: 0.85mEnd of last spray zone: 11.25m
Tliq=1524.0oC, Tsol=1498.5oCCalculation is from liquidus temperature and initial grain size is zero
Casting velocity 3.7m/min Grain size at solidus temperature ~0.159mm
By cooling rate from Tliq to Tsol
CR=105.4oC/s
λ1=K(CR)m(C0)n
~0.184mm
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 29
Conclusions for precipitates
1) The precipitates can form in the different stages. This causes different locations (removed from liquid, between dendrites, on the grain boundary or inside the grain) and different number and size distributions of precipitates. Our equilibrium precipitation model can predict tendency of precipitation for different stages and explain multiple peak size distribution qualitatively
2) The PGS model gives a good match for particle collision within a wide range of RV and accuracy increases with decreasing values of RV
3) A new PSG method is developed for particle diffusion. The results match the exact Kampmann case with reasonable error. This model makes it possible to simulate precipitates distribution for realistic processes with tolerable computation resources
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 30
Conclusions for grain growth
4) The results from austenite grain growth model can match the experimental grain size for a wide range of chemical composition
5) Grain growth model is applied to simulate a practical continuous casting process. Starting from a reasonable small initial size, the grains approach 60% of their final grain size by mold exit. Without precipitates, they are large enough to cause ductility problems.
6) The higher temperature and the corresponding larger grain growth rate and lack of fine precipitates are likely the controlling factors to cause coarse grains and susceptibility to cracks especially beneath oscillation mark roots
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 31
Future work
1) Consider local different concentrations of element due to segregation into the kinetic PSG model to analyze the volume fraction and size distribution of the precipitates forming at different stages
2) The kinetic PSG model needs to extend to complex system with possible multi precipitates formation and simulate precipitates evolution for the practical steel grades
3) With the knowledge of the steel chemical compositions and cooling history, coupled precipitate and grain growth model can be used to predict and track precipitate formation and the grain size in a more physical and accurate way, especially larger grains beneath oscillation marks from different temperature history
4) Consider the relation between the tensile stress (or strain) and grain size at the presence of precipitates on the intergranular fracture (finally prevent cracks and assure quality)
University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Kun Xu 32
Acknowledgements
• Continuous Casting Consortium Members (Baosteel, Corus, Goodrich, Labein, LWB Refractories, Mittal, Nucor, Postech, Steel Dynamics, Ansys Inc.)
• Prof B.G.Thomas
• Other Graduate students of CCC group
Thank you for attendance!