Numerical simulation of steel ingot solidification process

Journal of Materials Processing Technology 160 (2005) 156–159

Numerical simulation of steel ingot solidification process Z. Radovic∗ , M. Lalovic Department of Metallurgy, Faculty of Metallurgy and Technology, University of Montenegro, Podgorica, Serbia and Montenegro Received 4 November 2003; received in revised form 10 May 2004; accepted 5 July 2004

Abstract In this paper, two-dimensional numerical model of ingot solidification is presented. Numerical model is based on the Fourier’s differential equation, including equation of latent heat, which is released during crystallisation, as well as energy of lattice defects. For solving of the numerical model a finite difference method was used. On the basis of numerical model results, it is possible to calculate temperature distribution, temperature gradient, distribution solid and liquid phase and increment of solid fraction. After steel casting in mould, the measurements of temperature changes during steel ingot solidification, under laboratory conditions, were carried out. Cooling curves obtained by experiments were compared with the ones obtained on the basis of numerical model solidification, and good agreement of results was obtained. © 2004 Elsevier B.V. All rights reserved. Keywords: Ingot solidification; Numerical model; Fourier’s differential equation

1. Introduction

2. Model description

Macrostructure forming is a process, which depends on the conditions and type of solidification. Optimization of this parameter decreases level of structure heterogeneity. However, theoretical analysis of crystallization, often cannot give the answer to real relationship between casting parameters and structure morphology. This correlation was defined by modeling of the solidification process. The results of this simulation are compared with experimental results [1–3]. Although good agreement with experiments achieved, many models are relied on many approximations, and theirs application to multicomponent steel ingot solidification has been very limited. More recently, solidification models have been formulated that relay on fully coupled numerical solutions of mass, momentum, energy, and species conservation equation for a solid/liquid mixture [4,5]. Vannier and Combnau [6] attempted to extend recent binary alloy solidification models that couple mass, momentum, and energy conservation in all regions (solid, mush, and bulk liquid) to model steel solidification considering only buoyancy driven flow.

Two-dimensional numerical model is based on the system of differential equation, which are define of heat and mass transfer, with boundary conditions, which characterise steel ingot solidification process. As base, Fourier’s differential equation of heat conduction is used:
2  ∂T ∂ T ∂2 T ρcp =λ +q (1) + ∂t ∂x2 ∂y2

∗

Corresponding author. Tel.: +381 265 471. E-mail addresses: [email protected] (Z. Radovic), [email protected] (M. Lalovic). 0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.094

After some mathematical operation and introducing a thermal diffusivity: a=

λ cp ρ

Eq. (1) becomes:
2  ∂T ∂ T ∂2 T =a + q + ∂t ∂x2 ∂y2

(2)

(3)

where ρ is the density (kg/m3 ), cp the heat capacity (J/kg K), t the time (s), λ the thermal conductivity (Wm K), T the temperature (K), q the latent heat, (kJ/kg). Parameter q is a ratio of latent heat and heat capacity, i.e.  q = q/cp . The values of parameter q can vary during steel

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157

solidification: 

q = 0,

za T > T1

H dfs ∂T q = , cp dT ∂t 

Hv q = Xv cp





(4) za Ts ≤ T ≤

2D λ21



 exp

π2 Dt − 4λ21

(5)  ,

za t < Ts

(6)

Table 1 Chemical composition of examined steel (%) C Si Mn Cr Ni V W Mo P S

1.64 0.35 0.32 12.46 0.29 0.21 0.45 0.64 0.032 0.029

where H is the latent heat of crystallization (J/kg), fs the solid fraction, Hv the energy of lattice defects (J/kg), Xv the relativity volume contraction, λ1 the primary dendrite arm spacing (m). The above equations define heat, which is released during liquid cooling (4), solidification (5) and solid steel cooling (6). In classical models, heat evolved after solidification, assumed to be equal a zero, i.e. for T < Ts , q = 0. However, experimental investigations [7] show that lattice defects energy, during solidification increase solid free energy, proportionally to defects type. Lattice defects and vacancy are condensed in the already solidified part of crystal and increase enthalpy of the solid and thus the latent heat will decrease. The changes of solid fraction during solidification can be written as [8]. 
 dfs 1 π(T − Tl ) =− 1 − sin (7) dT (Tl − Ts )(1 − 2/π) 2(Ts − Tl ) Fig. 1. Schematic diagram of the area of model variables.

and (Tl − T ) + fs =


  2 π(T − Tl ) (Ts − Tl ) 1 − cos π 2(Ts − Tl ) (Tl − Ts )(1 − 2/π)

(8)

The heat flux in the mould is assumed to be uniform. It is calculated using the measured mould temperature at the inlet and outlet.

3. Results and discussion Presented model is applied to the solidification of steel ingot (150 mm × 150 mm × 400 mm). For investigations, high-alloy tool steel 165XCrMoW (DIN), produced in vacuum inductional furnaces Table 1. In experimental procedure, temperature changes measurements during ingot solidification were carried out. Apparatus for temperature measurements consists of thermocouples (Pt–Rh 18Pt), which are fixed on the level 2/3 ingot height, at different position, software adapter of signal and computer display. Since, a square cross-section is considered, in Eq. (1) can be used that x = y. Process of heat and mass transfer is assumed to be symmetrical in the direction of ingot centerline, so a quarter of ingot cross-section was used as a representative

Table 2 Values of model initial data [10,11,12] Data ρ a0 Ti Ts H Hv Xv D λ1 Tm T∝ b b1 Tpov ta λ0 Tg τ Xg h Cpl Cps

Values 7700–3.12 × 10.3 × 10− 6 1663 1371 270 × 103 32 × 103 0.04 2 × 10−9 350 × 10−6 473 300 4.91 × 10−8 4.726 × 10−8 1778, 1693 30 20 1200 1 0.07 0.0035 970 460

SI units 10−4 T

kg/m3 m2 /s K K J/kg J/kg – m2 /s m K K W/m2 K W/m2 K K s W/m K s s m m J/kg K J/kg K

158

Z. Radovic, M. Lalovic / Journal of Materials Processing Technology 160 (2005) 156–159

Fig. 2. Solid and liquid phase distribution, (Tcasting : 1778 K).

Fig. 5. 3-D temperature field during ingot solidification, (Tcasting : 1778 K).

from t = 0 to t = Tg , where Tg is a complete time of process. The cooling of outer ingot surface is defined by G1 boundary condition, but ingot center cooling by G3 boundary condition. The heat flux at the outer ingot surface is given by sum of convection and radiation to the moment of air gap formation [9] 4 − T4 ) 1.24(Tpov − T∞ )1.33 + σε(Tpov ∂T ∞ = ∂x λ

(10)

and Fig. 3. Increment of solid phase fraction for different distance from surface (x = 0.01–0.07 m; Tcasting : 1778 K).

area. On the basis of the given approximation Eq. (1) can be applied to the case of a one-dimensional solidification:
2  ∂T ∂ T + q (9) =a ∂t ∂x2 Domain of solidification numerical model, i.e. temperature variation in x-direction, for t time is shown in Fig. 1. Parameter x changes from x = 0, at the outer ingot surface, to x = Xg in the center of ingot. This variation is defined by the G2 boundary condition. Values of parameter t change

Fig. 4. Temperature distribution as function of time, (Tcasting : 1778 K).

j T1

j − T0

j−1

=h

1.24(T0

1.33

− T∞ )

j−1 4

λ0 + m4 (T0

j−1 4

+ b((T0

4) ) − T∞

j−1 3

) + m3 (T0

j−1 2 + m2 (T0 )

j−1 + m 1 T0

)

(11)

After gap formation, the heat transfer is primarily by radiation and can be expressed as: 4 − T4) σ(Tpov ∂T m = ∂x λ((1/εp ) + (1/εi ) − 1)

(12)

Fig. 6. Temperature distribution during ingot solidification for different distance from surface.

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159

Temperature of liquid steel during solidification is measured by three thermocouples, on the different position (surface of ingot, center and 1/2 distance s–c). Fig. 7 show a comparison of the measured and predicted values of temperature. In general, the agreement can be considered good, especially on the ingot surface.

4. Conclusions

Fig. 7. Comparison of measured and predicted temperature.

and j T1

j − T0

j−1 4

=h

b1 ((T0

) − Tm4 )

j−1 4

λ0 + m4 (T0

j−1 2

+ m2 (T0

j−1 3

) + m3 (T0

(13)

)

j−1

) + m 1 T0

where Ts is the surface temperature (K), Tm the temperature of mould (K), σ the Stephen–Boltzmann constant, ε the emissivity, εi , εm are the emissivity of ingot and mould, respectively. After definition of boundary conditions, using polynomial interpolation method, the following expressions are obtained: a = a0 + k4 T 4 + k3 T 3 + k2 T 2 + k1 T

(14)

λ = λ0 + m4 T 4 + m3 T 3 + m2 T 2 + m1 T

(15)

cp = cp0 + n4 T 4 + n3 T 3 + n2 T 2 + n1 T

(16)

The finite difference method was used for solution of numerical model equations, and temperature at grid point, i.e. j Ti = T (xi , tj ) is obtained. Some parameters called initial on given data for model equations solution are given in Table 2. Using the Eqs. (7) and (8), distribution of liquid and solid phase across ingot section during solidification are obtained (Fig. 2). The increment of solid phase fraction as a function of solidification time, for different casting temperatures is presented in Fig. 3. Temperature distributions across the ingot section and temperature field can be see in Figs. 4–6.

Using a two-dimensional numerical model, developed for the process solidification of steel in mould, the explicit values of the temperature at any position in the ingot section were calculated. On the basis of finite difference method it is possible to describe temperature distribution, temperature gradient as well as solid and liquid fraction during ingot solidification. In contrast to classical solidification model, the presented model includes the lattice defects energy, which influences to the decreasing in latent heat in the final balance. Cooling curves obtained by experiments were compared with the ones obtained on the basis of the numerical model solidification, and good agreement of the results was obtained.

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