Numerical Simulation on Macrosegregation in Fe-C Alloy Under Solidification Shrinkage Through Interface Tracking-Dynamic Mesh Technique
DONG Shihu1,2, ZHANG Hongwei1,2(), LÜ Wenpeng1,2, LEI Hong1,2, WANG Qiang1,2
1Key Laboratory of Electromagnetic Processing of Materials, Ministry of Education, Northeastern University, Shenyang 110819, China 2School of Metallurgy, Northeastern University, Shenyang 110819, China
Cite this article:
DONG Shihu, ZHANG Hongwei, LÜ Wenpeng, LEI Hong, WANG Qiang. Numerical Simulation on Macrosegregation in Fe-C Alloy Under Solidification Shrinkage Through Interface Tracking-Dynamic Mesh Technique. Acta Metall Sin, 2024, 60(3): 388-404.
Macrosegregation is the mutual contribution of many factors, such as thermo-solutal buoyancy-induced flow, solidification shrinkage, and grain movements, during alloy solidification. Fe-C-based alloy ingot is apt to form carbon segregation due to C's relatively small partition coefficient and the large ingot cross-section size. Moreover, it is also easy to generate solidification shrinkage because of the density difference between the liquid and solid and among ferrite, austenite, and other solids in the alloy. The numerical studies on solidification shrinkage show that introducing the air (or slag) phase mainly fills up the shrinkage cavity, which appears in the top part of the ingot. Although it has minor effect on segregation, it creates severe difficulty in solving the continuum transport equations at air-alloy interface owing to the large difference in their physical properties, such as density and thermal conductivity. A method that tracks the boundary profile of the cavity due to solidification shrinkage was developed in the present work to study macrosegregation under solidification shrinkage while avoid solving the air-alloy interaction. Macrosegregation under solidification shrinkage in Fe-C alloy ingot was predicted through the traditional liquid-solid mixed continuum model. To this end, the melt-air interface position was determined through allocating the shrink in volume to the solidified, mushy and liquid zones by the dynamic mesh technique. The predicted shape of the shrinkage cavity was fitted with the experimental one in the literature. Comparing the impact of thermo-solutal buoyancy showed that the predicted maximum positive C segregation at the top part of the Fe-0.3%C alloy ingot decreased by 4.78% with the additional consideration of solidification shrinkage. However, the heat exchange between the surroundings' and ingot's top surface reduced the positive C segregation near the latter. The C concentration distribution at the upper part of the ingot was more consistent with the experimental results in the literature when a heat transfer coefficient of 2.0 W/(m2·K) was adopted at the ingot's top surface, besides the effects of the thermo-solutal buoyancy and solidification shrinkage are considered. Compared with the thermo-solutal buoyancy influence, the solidification shrinkage enhanced the solutal buoyancy impact in the mushy zone. This made a faster reverse circulation in the mainstream ahead of the solidification front and led to the maximum flow velocity all over the molten steel exceeding that of mere thermo-solutal buoyancy during the solidification. All of them accelerated the overall solidification rate of the ingot. However, the predicted negative segregation at the lower part of the ingot was lower than the experimental data in the literature because the present continuum model only consisted of a liquid-columnar mixture. The movement of equiaxial grains needs to be included in further consideration.
Fund: National Natural Science Foundation of China(51574074);National Natural Science Foundation of China(51425401);National Natural Science Foundation of China and Shanghai Baosteel(U1460108);National Natural Science Foundation of China and Shanghai Baosteel(U1560207);Natural Science Foundation of Liaoning Province(L20150183);Shenyang Municipal Natural Science Foundation(23-503-6-07)
Fig.1 2D configuration of 10.5 t ingot and corresponding boundary conditions (x, y—axes; h1, h2, h3, h4—heat transfer coefficients at bottom, side wall, neck part, and riser, respectively)
Fig.2 Layouts of volume evolution at ingot top during solidification shrinkage stage II (Δx—cell size in x-direction, xs—maximum of x-coordinate in solid zone at ingot top, xmush—maximum of x-coordinate in mushy zone at ingot top, A1—solidified zone, A2—mushy zone, A3—liquid zone, Vs—volume shrinkage at solidified zone, Vmush—volume shrinkage at mushy zone, VL—volume shrinkage at liquid zone, y1—maximum of y-coordinate for ingot at end of stage I, N—number of shrinkage mesh, fl—liquid volume fraction) (a) N = 1 (b) N = N(0 < fl < 0.9)
Parameter
Unit
Sn-5%Pb alloy*
Fe-0.3%C alloy
Ref.
Melting point of pure solvent Tf
K
505.15
1805.15
[30]
Liquidus slope m
K·%-1
-1.286
-80.45
[30]
Equilibrium partition coefficient k
-
0.0656
0.36
[30]
External temperature
K
298.15
300
[30]
Melt density ρl
kg·m-3
7000
7027
[18]
Solid density ρs
kg·m-3
7000
7324
[18]
Specific heat cp
J·kg-1·K-1
260
500
[30]
Thermal conductivity ks
W·m-1·K-1
55
34
[30]
Latent heat L
J·kg-1
6.1 × 103
2.71 × 105
[30]
Viscosity μ
kg·m-1·s-1
1 × 10-3
4.2 × 10-3
[18]
Thermal expansion coefficient βT
K-1
6 × 10-5
1.07 × 10-4
[30]
Solutal expansion coefficient βc
%-1
-5.3 × 10-4
1.4 × 10-2
[30]
Secondary dendrite arm spacing λ2
m
6.5 × 10-5
5 × 10-4
[30]
Diffusion coefficient of C in liquid Dl
m2·s-1
1 × 10-8
2 × 10-8
[18]
Reference temperature Tref
K
499.15
1782
Reference concentration (mass fraction) cref
%
5
0.3
Reference temperature in enthalpy definition T0
K
273
273
Time step Δt
s
0.1
0.05
Table 1 Thermodynamic and physical parameters of modelling alloys[18,30,31]
Fig.3 Relative concentration ((c - c0) / c0) distribution of Pb for Sn-5%Pb alloy at 400 s (c—concentration, c0—initial concentration)
Fig.4 Relative concentration distributions of Pb for Sn-5%Pb alloy at end of solidification, positioned at height of 0.005 m (a), 0.025 m (b), 0.035 m (c), and 0.055 m (d) (The experimental[33] and other simulated[31,32] results are shown in Fig.4)
Fig.5 Relative concentration distributions of C in ingot at end of solidification (a) N = 1 (b) N = N(0 < fl< 0.9)
Fig.6 Relative concentration distribution of C along ingot centerline (The experimental[36] and simulated[19] results are shown in Fig.6)
Fig.7 Formation of shrinkage cavity by consideration of shrinkage + buoyancy (left—liquid volume fraction, right—relative concentration distribution of C (t—time) (a) t = 50 s (b) t = 1050 s (c) locally enlarged part of Fig.7b (Δy1—shrinkage height at stage I, Δy2—shrinkage height at stage II) (d) t = 3700 s (e) t = 14200 s (f) t = 22000 s
Fig.8 Formation of shrinkage cavity by consideration of only shrinkage (left—liquid volume fraction, right—relative concentration distribution of C (a) t = 50 s (b) t = 1050 s (c) t = 3700 s (d) t = 14200 s (e) t = 20000 s (f) t = 25000 s
Fig.9 Liquid fractions and velocity distributions (a-f, i, k) and locally enlarged parts (g, h, j, l) during solidification considering buoyancy (a-d) and shrinkage + buoyancy (e-l) at different time (ul, max—the maximum velocity in melt) (a, e) t = 300 s (b, f-h) t = 600 s (c, i, j) t = 5000 s (d, k, l) t = 20000 s (g, h) locally enlarged part of Fig.9f (j) locally enlarged part of Fig.9i (l) locally enlarged part of Fig.9k
Fig.10 Solutal and thermal buoyancy curves (a-c) and locally enlarged parts (d-f) at solidification front in ingot with only consideration of buoyancy at different time in the model (βc(cl - cref)—solutal buoyancy per gravity, |βT(T - Tref)|—absolute value of thermal buoyancy per gravity) (a, d) y = 0.5 m (b, e) y = 1.0 m (c, f) y = 1.5 m
Fig.11 Solutal and thermal buoyancy curves (a-c) and locally enlarged parts (d-f) at solidification front in ingot with consideration of shrinkage + buoyancy at different time in the model (a, d) y = 0.5 m (b, e) y = 1.0 m (c, f) y = 1.5 m
Fig.12 Relative concentration distributions of C in ingot at end of solidification (a) consideration of shrinkage + buoyancy (solidification finish at 22000 s) (b) consideration of only shrinkage (solidification finish at 25000 s) (c) consideration of only buoyancy (solidification finish at 31000 s)
Fig.13 Relative concentration distributions of C along ingot (a) x = 0 (b) x = -0.32 m
Fig.14 Relative concentration distributions of C in ingot with different heat transfer conditions at ingot top (a) adiabatic at ingot top (b) heat transfer coefficient at ingot top htop = 1.0 W/(m2·K) (c) htop=2.0 W/(m2·K)
Fig.15 Relative concentration distributions of C along ingot centerline (Top-0: adiabatic at ingot top; Top-1.0: htop = 1.0 W/(m2·K); Top-2.0: htop = 2.0 W/(m2·K))
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