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


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.


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


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


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.


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%


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


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


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)


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 − −

=

− + +∑


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* =


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* =


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* =


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


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

+ −+⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠


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=


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=

= ⋅ = − −∑


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


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


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


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


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⇒ ≈


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)


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


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.


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.


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


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


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


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)


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!

modeling heat transfer, precipitate formation, and grain...

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