A Simple Model on Dendrite Growth during Steel Continuous Casting Process

Wei Guo1,2, Lifeng Zhang1, Miaoyong Zhu2

1 Department of Materials Science & Engineering, Missouri University of Science and Technology (Missouri S&T), 223 McNutt Hall, Rolla, MO 65409-0330, USA, Email: zhanglife@mst.edu

2 School of Materials & Metallurgy, Northeastern University, Shenyang 110004, China,

Keywords: Steel Continuous Casting, Solidification, Microstructure, Dendrite Growth

Abstract Prediction and control of steel microstructure during continuous casting process is an essential

task for steel production. Studies on the micro- structural parameters, which are hardly measured directly during high temperature solidification process, have significant importance for that purpose. Dendrite growth is an important phenomenon during steel solidification process, which is the main existing style in the inner side of the solidifying steel. In the current paper, a simplified mathematical model for the dendrite growth was used to analyze and calculate the micro- structural parameters for both medium carbon steel and AISI304 type stainless steels under different casting conditions. The dendrite tip radius, dendrite growth velocity, dendrite arm spacing, temperature gradient, and dendrite tip temperature in the front of solid and liquid interface were investigated. Parametric studies for these two steel grades were also performed. The comparisons showed that the value of micro- structural parameters for stainless steels is all smaller than that of carbon steels due to its special composition and solidification mode characteristics.

Nomenclature C0 original liquid concentration, mass% CL

* equilibrium liquid concentration in front of solid/liquid interface, mass%CS

* equilibrium solid concentration in front of solid/liquid interface, mass%CP specific heat, J kg-1 K-1

CR cooling rate, oC s-1

Creq equivalent composition of Cr, mass%D i solid diffusion coefficient, = or phase, m2 s-1

DL liquid diffusion coefficient, m2 s-1

D0i pre-exponential diffusion coefficient, m2 s-1

Q i activation energy, J mol-1

Rg gas constant, 8.3145 J K-1 mol-1

Gc concentration gradient at the solid/liquid interface, mass% m-1

GT average temperature gradient in the interface, K m-1

GTS solid temperature gradient at the dendrite tip, K m-1

GTL liquid temperature gradient at the dendrite tip, K m-1

Iv(P) Invanstov function L latent heat, J kg-1

Nieq equivalent composition of Ni, mass% Pc solute Péclet number, Pc RV/(2DL)Pt heat Péclet number, Pt RV/(2 L)R dendrite tip radius, m T non-equilibrium liquid temperature, K

TL liquidus temperature, K TS solidus temperature, K Ttip dendrite tip temperature, K

T superheat, K V growth velocity of the dendrite tip, m/s Vc casting speed, m/min d1 primary dendrite arm spacing, md2 secondary dendrite arm spacing, mk0

/L equilibrium solute distribution coefficient, stands for ferritic phase or austenitic phase mL/ liquid slopes, stands for ferritic phase or austenitic phase tf local solidification time, s

L liquid thermal diffusion coefficient, m2 s-1

i Fourier number for solute element ii+ modified Fourier number

c a constant value, c = 0.1 i back-diffusion parameter S solid conduction coefficient, W m-1 K-1

L liquid conduction coefficient, W m-1 K-1

c minimum critical disturbance wavelength, m* stability parameter, assumed to be 1/(4 2) based on the marginal stability criterion c function of Pc, when Pc<1, c 1

dimensionless super-saturation c concentration dimensionless super-saturation Gibbs-Thomson coefficient, m K

ferrite phase austenite phase

Introduction The characteristic and capabilities of solidification microstructure were controlled by the

evolution and stability selection of interface shape of solidification. Solidification interface is a typical configuration of the non-equilibrium structure, which is related to heat transfer, mass transfer, interface dynamics and capillary functions. Research on the mechanism of solidification microstructure is to control the structure better. For steel continuous casting process, dendrite growth is a very common configuration during solidification. Its evolution is to form a stable structure under unstable conditions. In the current study, dendrite growth was investigated by a numerical method for both medium carbon steel and AISI304 type stainless steel. Some important micro- structural parameters were studied, and comparisons between these two steel grades were investigated, which have great importance on the analysis of solidification structure and the development for improving the steel qualities.

Mathematical Model Dendrite Growth Model

In the actual solidification process, the micro-structural parameters such as dendrite tip radius, dendrite growth velocity, temperature gradient and concentration gradient at the front of solid/liquid (S/L) interface, etc., couldn’t be measured directly during this high temperature continuous casting process, which can directly affect the dendrite growth stability, micro-segregation and the mechanical capability. A simplified numerical mathematical model was developed[1] according to the Invanstov theory[2], the interface stability theory[3] and the minimum critical disturbance wavelength criterion[4]

and the final expression for it can be obtained as follows:

0)1()exp()exp()1(12

1)(4

/00/

022* L

LcL

PccL

PLcL D

kmCdzz

zPPkC

LPD

Vc

(1)

If the dendrite growth velocity V is known in advance, with the solute Pelect number Pc from Eq.(1), the dendrite tip radius R and liquid concentration at the S/L interface CL

* can be obtained through Invanstov theory and the minimum critical disturbance wavelength criterion. The details about this model can be seen in the current author’s another paper[1].

Figure1 Relationship between dendrite growth velocity and casting speed in mushy zone

There exist great differences for calculating the dendrite growth rate between continuous casting process and ingot casting process essentially. In the steel continuous casting process, the dendrite growth rate was combined with the casting speed closely, seen Figure 1. For simplification[1], the dendrite growth rate can be expressed as follows:

2'

'

))((1)(xf

xfVVL

Lc

(2)

The following figure shows the variation of dendrite growth rate with the distance from meniscus when casting speed is 1.0m/min.

Figure 2 Calculated dendrite growth velocity according to casting speed

Microsegregation Modell With the progress of solidification, the concentration of each element in the remainder liquid

increases gradually. After the solid fraction reaches 0.9, the microsegregation of P and S increases remarkably, while the microsegregation of C, Si, and Mn changes gently. Since the liquid film, which melts the solidification front and induces the inner cracks, appears at the end of solidification, the microsegregation degree can be expressed by the ratio of the remainder liquid concentration to the initial concentration of the liquid steel:

0

*,

0

*,

kCC

CC isil

(3)

Solidification Path for Stainless Steell To predict the capabilities of stainless steels, it is necessary to determine the solidification mode

and solid transformation characteristics. Since the stainless steel has its own solidification characteristics that are different from other steels, the analysis of its microstructure evolution is closely related with the special solidification mode. For the solidification of austenitic stainless steels of 300 series, the following modes have been proposed[5]:

Mode A: Liq L+Mode B: Liq L+ L+ + +Mode C: Liq L+ L+ + +Mode D: Liq L+

These modes are illustrated in the Fe-Ni-Cr phase diagram, as shown in Figure 3. Complex compositions of austenitic stainless steels can be reduced to simple Fe-Ni-Cr ternary alloys by assuming Ni and Cr equivalent compositions, as suggested express below[6]:

Creq=%Cr+1.37(%Mo)+1.5(%Si)+2(%Nb)+3(%Ti) (4)

Nieq=%Ni+22(%C)+14.2(%N)+0.31(%Mn)+(%Cu) (5)

By using equivalent composition, it is possible to find the solidification sequence by analyzing Fe-Ni-Cr, as shown in Figure 4. The effect of the alloying elements on the solidification microstructure can be expressed by the ratio Creq/Nieq

[7]:

Creq/Nieq > 1.95 (mode A)

Creq/Nieq = 1.48-1.95 (mode B)

Creq/Nieq = 1.25-1.48 (mode C)

Creq/Nieq < 1.25 (mode D)

Figure 3 Fe-Cr-Ni Phase diagram at 70% Fe[8] Figure 4 Solidification mode diagram[5]

Methodology and Conditions The current study focused on the continuous casting of both medium carbon steel slabs with

dimension of 1040-1320mm in width and 220mm in thickness and AISI304 stainless steel slab of 750-1600mm in width and 220mm in thickness. The compositions of these two steel grades, the thermo-physical properties, the solute equilibrium distribution coefficients k0

/L used in calculation, the liquid-slope mL for each element are presented in Tables 1, 2, 3, 4 respectively. The equivalent compositions of Cr and Ni for stainless steel can be calculated from Eqs.(4) and Eq.(5) as Creq=18.7 and Nieq=10.13. So Creq/Nieq = 1.846, and then the steel solidifies as mode B: Liq L+ L+ + + . (Fig.3).

Table 1. Compositions of S20 steel and stainless steel type304

Composition (mass%) C Si Mn P S Cr Ni N Mo Cu Sn B Pb

S20 Steel 0.18 0.20 1.0 0.024 0.01 0.08 0.08 0.0048

Stainless Steel Type 304 0.050 0.40 1.10 0.000 0.000 18.10 8.05 0.045 0.00 0.00 0.00 0.002 0.000

Table 2. Thermo-physical properties of carbon steel and stainless steel

S20 AISI304 UnitLiquidus temperature, TL 1513.2 1461.0 oCSolidus temperature, TS 1446.1 1400.0 oCSuperheat, T 20 30 oCSpecific heat in mushy zone, CP 700 750 J kg-1 K-1

Latent heat, L 274950 26800 J kg-1

Thermal conductivity of liquid steels, L 78.111 12-15(293K) W m-1 K-1

Thermal conductivity in mushy zone, m 39.955 W m-1 K-1

Density of liquid multi-component steels, L 7200 7880 kg m-3

Table 3. Equilibrium partition coefficients of various solute elements[9]

Element C Si Mn P S Cr Ni Cu Mo Nk0

/L

k0/L

0.19 0.77 0.76 0.23 0.05 0.95 0.83 0.53 0.80 0.250.34 0.52 0.78 0.13 0.035 0.86 0.95 0.88 0.585 0.48

Table 4. Liquid slopes for carbon steels and stainless steel[10]

Solute S20 AISI304

mL (oC/%) mL/ mL/

C -78 -96 -55 Si -7.6 -11 -11

Mn -4.9 -4.0 -5.1 P -34.4 -37 -28 S -38.0 -44 -32 Cr -1.04 -3.755+0.188CCr+0.020CNi -3.328-0.047CCr+0.097CNi

Ni -4.69 -4.792+0.020CCr-0.060CNi -1.440+0.097CCr-0.112CNi

Cu — -8.9 -3.6 Mo — -2.2 -5.3 N -60.0 — —

The diffusion coefficients for typical solutes in liquid, ferritic, and austenitic iron-based alloys are presented in Table 5 [10]. Most of the data are based on the measurements of binary Fe-X alloys [10].The data are expressed in the conventional form of the Arrhenius equation:

D i=D0i exp(-Q i/ RgT) (6)

Table 5. Diffusion constants D0 i and activation energies Q i for solutes in liquid, ferritic, and austenitic iron-based alloys [10]

Solute Liquid Ferritic Austenitic

D0Li(cm2/s) QL

i (J/mol) D0Fi(cm2/s) QF

i(J/mol) D0Ai(cm2/s) QA

i(J/mol)

C 7.67 10-2 106000 0.013 81400 0.076 134600

Si 5.10 10-4 38300 8.000 248900 0.300 251400

Mn 4.60 10-3 70300 0.760 224000 0.486 276100

P 1.34 10-2 99200 2.900 230100 0.010 182800

S 4.33 10-4 35600 4.560 214600 2.400 223400

Cr 2.51 10-3 66900 2.400 239800 0.0012 219000

Ni 1.35 10-2 89200 1.600 240000 0.340 282400

Cu 5.30 10-5 2.600 240000 0.700 286000

Mo 5.82 10-4 41800 3.470 241400 0.068 246900

N 9.76 10-3 72800 0.008 79100 0.910 168500

For carbon steel, the Gibbs-Thomson coefficient is 1.9 10-7 K m[11]; for AISI 300-series stainless steels, the Gibbs-Thomson coefficient for ferritic F/L and austenitic phase A/L have values of around 2.8×10-7 and 3.4×10-7 m K[10] respectively. Some assumptions should be adopted for the current calculation, which are the same as the former paper[12].

Figure 5 shows the method for the calculation of dendrite growth in the current model.

Figure 5 Flow chart for calculation

Results and Discussions Comparisons of Microstructure Parameters for Medium Carbon Steel and Stainless Steel

Figures 6 and 7 show the variations among the dendrite tip radius, dendrite growth rate, secondary dendrite arm spacing, temperature gradient, cooling rate and solid shell growth for both medium carbon steel and stainless steel. Dendrite tip radius and secondary dendrite arm spacing increase with the increasing of the distance from surface to center of the shell, while the growth rate, temperature gradient and cooling rate decrease with the increasing of the distance from surface to center of the shell. The fluctuation of these parameters contributed to the variation of the casting speed. The values of the stainless steel were all smaller than those of medium carbon steel. That’s because type304 stainless steel contains less C and more Cr and Ni, and there exist the phase transition during solidification. The diffusion of austenitic phase was very slow, postponing the dendrite growth rate and increasing the dendrite tip radius. The variations of the temperature gradient and the cooling rate, seen from Fig.7, were larger in the mold than in the secondary cooling zone due to the greater heat transfer intension in the mold.

Figure 6 Variation of dendrite growth rate, dendrite tip radius and secondary dendrite arm spacing with shell thickness for both medium carbon steel and stainless steel

Figure 7 Variation of temperature gradient and cooling rate with shell thickness for both medium carbon steel and stainless steel

Comparisons of Micro-segregation for Medium Carbon Steel and Stainless Steel Figures 8 and 9 show the comparisons of C, P, S microsegregation with the variation of

dendrite growth rate for medium carbon steel and stainless steel. The numerical values of stainless steel were larger than those of carbon steel. The microsegregation for stainless steel was more serious.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.75.4

5.5

5.6

5.7

5.8

5.9

6.0

Growth rate, mm/s

CL* /C

0

C

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.75.7

5.8

5.9

6.0

6.1

6.2

6.3

6.4

6.5

6.6

Growth rate, mm/s

CL* /C

0

C

(a) Medium carbon steel (b) Stainless steel

Figure 8 Variation of C micro-segregation with the dendrite growth rate for both medium carbon steel and stainless steel

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.74

5

6

7

22

24

26

28

30

Growth rate, mm/s

CL* /C

0

P S

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7789

1011

242628303234

Growth rate, mm/s

CL* /C

0

P S

(a) Medium carbon steel (b) Stainless steel

Figure 9 Variation of P and S micro-segregation with the dendrite growth rate for both medium carbon steel and stainless steel

Parametric Studies Effect of * on Dendrite Tip Radius and Secondary Dendrite Arm Spacings

The establishment of dendrite growth model was based on the solute transfer and dendrite tip stability. The parameter * was first introduced by Langer and Müller-krumbhaar[4, 13]. They proposed that the dendrite tip selected the largest stable radius value and referred to this dendrite tip selection process as the marginal stability criterion. This marginal stability criterion shows that * is a numerical constant whose value was 1/(4 2). For different systems, * took different values [14]. There was no evidence to show the accurate value of * for steels continuous casting process. For current calculation, the value of 1/(4 2)=0.025 was used, but different stability coefficient * were used in order to validate and study its effects on some micro-structural parameters. Figure 10 and 11 show the effects of * on dendrite tip radius and secondary arm spacing with shell growth for both medium carbon steel and stainless steel. The larger this value is, the smaller the dendrite tip radius and secondary dendrite arm spacing are for both these two kinds of steel grades, but the values of stainless steel are smaller than those of carbon steel. The high value of stability coefficient * is attributed to the higher anisotropy in the system. For a continuous casting system, the anisotropy of crystals in micro-scale exists everywhere, and the higher value of * shows more reasonable. There should be further studies for the analysis of *.

0 20 40 60 80 1000

5

10

15

20

Den

drite

radi

us,

m

Distance from the surface, mm

Stainless Steel, *= 0.025 Stainless Steel, *= 0.05 Medium Carbon Steel, *= 0.025 Medium Carbon Steel, *= 0.05

Figure 10 Effect of * on the dependency of dendrite radius on shell thickness

0 20 40 60 80 1000

200

400

600

800

1000

Sec

onda

ry a

rm s

paci

ng,

m *= 0.025*= 0.01*= 0.05

Distance from the surface, mm0 20 40 60 80 100

0

5

10

15

20

25

30

35

40

45*= 0.025*= 0.01*= 0.05

Sec

onda

ry a

rm s

paci

ng,

m

Distance from the surface, mm

(a) Medium carbon steel (b) Stainless steel

Figure 11 Effect of * on the dependency of secondary dendrite arm spacing on shell thickness

Effect of Casting Speed on Dendrite Tip Radius and Secondary Dendrite Arm Spacings Studies on micro-structural parameters should be based on the actual solidification conditions

during steel continuous casting. Effects of different casting speed Vc 0.8m/min and 1.0m/min on these parameters were investigated under 20 oC superheat. The dependence of dendrite tip radius and the secondary dendrite arm spacing on the casting speed for both medium carbon steel and stainless steel are shown in Figure 12 and 13. Both dendrite tip radius and secondary dendrite arm spacing increase with increasing casting speed from the surface to center of the solid shell. An increase of the casting speed promotes larger spacing due to its lower residence time of the strand in the spray cooling zone. This behavior of interdendritic spacing is more evident in the center of the slab. Increase in casting speed leads to a coarser interdendritic spacing.

0 20 40 60 80 1000

5

10

15

20

Distance from the surface, mm

Den

drite

radi

us,

m

Stainless Steel, Vc=1.0 m/min Stainless Steel, Vc=0.8 m/min Medium Carbon Steel, Vc=1.0 m/min Medium Carbon Steel, Vc=0.8 m/min

0 20 40 60 80 1000

5

10

15

200

400

600

800

Distance from the surface, mmSec

onda

ry d

endr

ite a

rm s

paci

ng,

m Stainless Steel, Vc=1.0 m/min Stainless Steel, Vc=0.8 m/min Medium Carbon Steel, Vc=1.0 m/min Medium Carbon Steel, Vc=0.8 m/min

Figure 12 Effect of casting speed on the dependency of dendrite radius on shell thickness

Figure 13 Effect of casting speed on the dependency of secondary dendrite arm spacing on shell thickness

Conclusions - A simplified numerical mathematical method was developed to analyze the important micro-

structural parameters during continuous casting solidification for both medium carbon steel and stainless steel. For these two kinds of steel grades, the dendrite tip radius and secondary dendrite arm spacing increase with the distance from surface to center, while dendrite growth velocity, temperature gradient and cooling rate were decreasing.

- The larger the stability coefficient value is, the smaller the dendrite tip radius and secondary dendrite arm spacing are for both these two kinds of steel grades. The higher value of * shows more reasonable in continuous casting system due to the anisotropy of crystals in micro-scale everywhere.

- Both dendrite tip radius and secondary dendrite arm spacing increase with increasing casting speed from the surface to center of the solid shell. An increase of the casting speed promotes a coarser interdendritic spacing due to its lower residence time of the strand in the spray cooling zone.

- The values in the calculation results for stainless steel were smaller than those for carbon steel except the microsegregation due to the special solidification characteristic of the stainless steel. The microsegregation for stainless steel was more serious.

References [1] W. Guo, L. Zhang, M. Zhu, Modeling on Dendrite Growth of Medium Carbon Steel during Continuous Casting, submitted to ISIJ Int, 2009. [2] G.P.Invanstov, Temperature Field around Spherical, Cylindrical, and Needle-shaped Crystals Which Grow in Supercooled Melts, Dokl Akad Nauk SSSR, vol 58, 1947, p 567-570. [3] W.W.Mullins, R.F.Sekerka, Stability of a Planar Interface during Solidification of a Dilute Binary Alloy, Journal of Applied Physics, 1964, p 444-451. [4] J.S.Langer, H.Muller-Krumbnaar, Stability Effects in Dendritic Crystal Growth, Journal of Crystal Growth, vol 42, 1977, p 11-14. [5] G.K.Allan, Solidification of Austenitic Stainless Steels, Ironmaking and Steelmaking, vol 22(6), 1995, p 465-477. [6] Jernkontoret, A Guide to Solidification of Steels, 1977, Stockholm. [7] K.Rajasekhar, C.S.Harendranath, R.Raman and S.D.Kulkarni, Microstructural Evolution during Solidification of Austenitic Stainless Steel Weld Metals: A Color Metallographic and Electron Microprobe Analysis Study , Mater. Charact., vol 38, 1997, p 53-65. [8] I. D.Marinescu, Heat Convection by Coolant and Chips, Metal Handbook of Machining with Grinding Wheels 8, 1973, p 424-425.

[9] S.K.Choudhary, S. Ganguly, Morphology and Segregation in Continuous Cast High Carbon Steel Billets, ISIJ Int. vol 47(12), 2007, p 1759-1766. [10] J. MIETTINEN, Thermodynamic-Kinetic Simulation of Constrained Dendrite Growth in Steels, Metal Mater Trans B, vol 31B, 2000, p 365-379. [11] M.M.Hamdi, H. Combeau, G. Lesoult, Modeling of Heat Trandfer Coupled with Columnar Dendritic Growth in Continuous Casting of Steel, International Journal of Numerical methods for Heat & Fluid Flow, vol 9, 1999, p 296-317. [12] W. Guo, L. Zhang and M. Zhu, Modeling on Dendrite Growth for Carbon Steel of Continuous Casting Process, Iron and Steel (in Chinese), 2009, submitted and acepted. [13] J.S.Langer and H.Müller-krumbhaar, Acta. Metall. 26, 1681;1689;1697 (1978). [14] R.Trivedi and W.Kurz, Dendrite Growth, International Material Review, vol 39, 1994, p 49-74.