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The solidification process of alloy ingot is often accompanied by the phenomena of free dendrites growing and colliding with each other while moving, which has a non-negligible influence on the temperature field, flow field, solute field and microstructure of the ingot, and it is one of the key issues in the study of ingot solidification organization formation. The cellular automata-lattice Boltzmann (CA-LB) coupling model has been developed rapidly in recent years in dealing with the moving dendrites, which can not only maintain the morphology of the moving dendrites well, but also calculate the mutual collisions between the dendrites reasonably. In this work, the cell-automata-lattice Boltzmann model for simulating the growth of free dendrites is improved. Alternating direction implicit iteration method is used to solve the differential heat conduction equation, and the simulation parameters are not limited by stability conditions in this method. In this research, the accuracy of the flow-solid coupling of the model is verified by taking the flow around a circular cylinder for example, and the temperature field of the model is well coupled under the natural convection condition. Finally, the solidification process of Fe-0.34%C alloy ingots with or without equiaxed grain movement is simulated using this model. The simulation results show that the movement of equiaxed grains increases the contact probability with the neighboring dendrites, which leads to a more uniform grain size in the ingot; the movement of dendrites also changes the solute distribution in the center of the melt, especially increasing the size and range of the hot-top segregation; the movement of equiaxed grains is impeded by the columnar crystals, and therefore the CET region is not much affected by the movement of dendrites.
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Keywords:
- ingot /
- macroscopic segregation /
- numerical simulation /
- dendrite movement
[1] Qi X B, Chen Y, Kang X H, Li D Z, Gong T Z 2017 Sci Rep 7 45770
Google Scholar
[2] Rátkai L, Pusztai T, Gránásy L 2019 npj Comput. Mater. 5 113
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[28] 张照 2020 硕士学位论文 (天津: 河北工业大学)
Zhang Z 2020 M. S. Thesis (Tianjin: Hebei University of Technology
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 圆柱绕流物理模型
Figure 1. Model of the flow around a cylinder.
图 2 圆柱绕流的结果 (a) Fs = 0.02; (b) Fs = 0.038; (c) Fs = 0.138; (d) Fs = 0.363
Figure 2. Result of the flow around a cylinder: (a) Fs = 0.02; (b) Fs = 0.038; (c) Fs = 0.138; (d) Fs = 0.363.
图 3 圆柱绕流结果对比
Figure 3. Comparison of results.
图 4 Rayleigh–Bénard自然对流模型示意图
Figure 4. Schematic of the Rayleigh-Bénard natural convection model.
图 5 不同Ra数下系统达到稳态时的等温线分布 (a) Ra = 2500; (b) Ra = 5000; (c) Ra = 10000; (d) Ra = 50000
Figure 5. Isothermal distribution of the system reaching steady state for different Ra numbers: (a) Ra = 2500; (b) Ra = 5000; (c) Ra = 10000; (d) Ra = 50000.
图 6 稳态Nu数随Ra数的变化规律
Figure 6. Variation rule of steady state Nu number with Ra number.
图 7 铸锭物理模型
Figure 7. Physical model of ingot.
图 8 不考虑等轴晶运动时铸锭的凝固过程 (a) t = 2.0 s; (b) t = 4.5 s; (c) t = 4.6 s; (d) t = 4.8 s; (e) t = 5.2 s; (f) t = 5.6 s; (g) t = 6.0 s; (h) t = 7.0 s; (i) t = 10.0 s; (j) t = 18.0 s
Figure 8. Solidification process of ingot without considering the equiaxial crystal motion: (a) t = 2.0 s; (b) t = 4.5 s; (c) t = 4.6 s; (d) t = 4.8 s; (e) t = 5.2 s; (f) t = 5.6 s; (g) t = 6.0 s; (h) t = 7.0 s; (i) t = 10.0 s; (j) t = 18.0 s.
图 9 考虑等轴晶运动的铸锭凝固过程 (a) t = 2.0 s; (b) t = 4.5 s; (c) t = 4.6 s; (d) t = 4.8 s; (e) t = 5.2 s; (f) t = 5.6 s; (g) t = 6.0 s; (h) t = 7.0 s; (i) t = 10.0 s; (j) t = 18.0 s
Figure 9. Solidification process of ingot considering equiaxial crystal motion: (a) t = 2.0 s; (b) t = 4.5 s; (c) t = 4.6 s; (d) t = 4.8 s; (e) t = 5.2 s; (f) t = 5.6 s; (g) t = 6.0 s; (h) t = 7.0 s; (i) t = 10.0 s; (j) t = 18.0 s.
图 10 不同情况下铸锭完全凝固时的形貌 (a) 等轴晶运动; (b) 等轴晶不运动
Figure 10. Morphology of a fully solidified ingot in different cases: (a) Equiaxed grain motion; (b) equiaxed grain immobility.
图 11 不同情况下铸锭完全凝固的溶质分布 (a) 等轴晶运动; (b) 等轴晶不运动
Figure 11. Solute distribution in the fully solidified ingot for different cases: (a) Equiaxed grain motion; (b) equiaxed grain immobility.
图 12 等轴晶运动与否和实验结果的偏析程度对比
Figure 12. Comparison between the presence or absence of equiaxed crystal movement and the degree of segregation observed in experimental results.
表 1 不同位置的${C_{\text{e}}}$与${D_{\text{e}}}$的取值方法
Table 1. Methods of taking values of ${C_{\text{e}}}$ and ${D_{\text{e}}}$ at different locations.
液相 固相 界面 界面与固相之间 界面与液相之间 ${C_{\text{e}}}$ ${C_{\text{l}}}$ ${C_{\text{s}}}/k$ ${C_{\text{l}}}$ ${C_{\text{l}}}$ ${C_{\text{l}}}$ ${D_{\text{e}}}$ ${D_{\text{l}}}$ ${D_{\text{s}}}$ ${D_{\text{l}}}$ ${D_{\text{s}}}$ ${D_{\text{l}}}$ 
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表 2 模拟用到的参数
Table 2. Parameters used for simulation.
Physical parameter Value Solidus temperature, ${T_{\text{m}}}/K$ 1811.15 Liquidus temperature, ${T_0}/K$ 1723.15 Liquidus slope, ${m_{\text{l}}}/({{\mathrm{K}} {\cdot} {{\text{%}} ^{-1}}})$ –80.45 Interface kinetics coefficient, ${\mu _{\text{k}}}/({1{0^{ - 3}}\;{\mathrm{m}} {\cdot} {{\mathrm{s}}^{ - 1}} {\cdot} {\mathrm{K}}})$ 2 Kinetic anisotropy strength, ${\delta _{\text{k}}}$ 0.3 Thermodynamic anisotropy, ${\delta _{\text{t}}}$ 0.3 Liquid Viscosity, $\nu /({1{0^{ - 3}}\;{\mathrm{kg}} {\cdot} {{\mathrm{m}}^{ - 1}} {\cdot} {{\mathrm{s}}^{ - 1}}})$ 3.6 Diffusivity in liquid, ${D_{\text{l}}}/( {{{10}^{ - 8}}\;{{\text{m}}^2} {\cdot} {{\text{s}}^{ - 1}}} )$ 2 Diffusivity in solid, ${D_{\text{s}}}/( {{{10}^{ - 9}}\;{{\text{m}}^2} {\cdot} {{\text{s}}^{ - 1}}} )$ 1 Partition coefficient, $k$ 0.36 Average Gibbs-Tomson
coefficient, $ {\bar \varGamma /( {{{10}^{ - 7}}\;{\text{m}} {\cdot} {\text{K}}} )}$1.9 Specific heat capacity, ${C_{\text{p}}}/( {{\text{J}} {\cdot} {\text{k}}{{\text{g}}^{ - 1}} {\cdot} {{\text{K}}^{ - 1}}} )$ 455 Convective heat transfer
coefficient, $h/({\mathrm{W}} {\cdot} {{\mathrm{m}}^{ - 2}} {\cdot} {{\mathrm{K}}^{ - 1}})$600 Thermal conductivity, $\lambda /({\text{W}} {\cdot} {{\text{m}}^{ - 1}} {\cdot} {{\text{K}}^{ - 1}})$ 30 Thermal expansion coefficient, $ {\beta _{\text{T}}}/( {{{10}^{ - 4}}\;{{\mathrm{K}}^{ - 1}}} ) $ 2 Solutal expansion coefficient, $ {\beta _{\text{c}}}/( {{{10}^{ - 2}} \;{{\text{%}} ^{ - 1}}} ) $ 1.1 Latent heat, $L/( {{{10}^3}\;{\text{J}} {\cdot} {\text{k}}{{\text{g}}^{ - 1}}} )$ 269.55 Density, $\rho /( {{\text{kg}} {\cdot} {{\text{m}}^{ - 3}}} )$ 7001
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表 3 不同位置的平均浓度以及偏析程度
Table 3. Average concentration and segregation of index (SI) at different locations.
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Ⅶ Ⅷ Ⅸ 运动 0.227 0.257 0.292 0.309 0.323 0.441 0.282 0.289 0.351 偏析/% –33.2 –24.4 –14.1 –9.1 –5.0 29.7 –17.1 –15.0 3.2% 不运动 0.228 0.278 0.298 0.3 0.341 0.384 0.278 0.289 0.345 偏析/% –32.9 –18.2 –12.4 –11.8 0.3 12.9 –18.2 –15.0 1.5%
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[1] Qi X B, Chen Y, Kang X H, Li D Z, Gong T Z 2017 Sci Rep 7 45770
Google Scholar
[2] Rátkai L, Pusztai T, Gránásy L 2019 npj Comput. Mater. 5 113
Google Scholar
[3] 张士杰, 王颖明, 王琦, 李晨宇, 李日 2021 物理学报 70 238101
Google Scholar
Zhang S J, Wang Y M, Wang Q, Li C Y, Li R 2021 Acta Phys. Sin. 70 238101
Google Scholar
[4] Sakane S, Takaki T, Ohno M, Shibuta Y, Aoki T 2020 Comput. Mater. Sci. 178 109639
Google Scholar
[5] Ren J K, Chen Y, Cao Y F, Sun M Y, Xu B, Li D Z 2020 J. Mater. Sci. Tech. 58 171
Google Scholar
[6] Yamanaka N, Sakane S, Takaki T 2021 Comput. Mater. Sci. 197 110658
Google Scholar
[7] Zhang S J, Zhu B F, Li Y B, Zhang Y, Li R 2024 Comput. Mater. Sci. 245 113308
Google Scholar
[8] Liu L, Pian S, Zhang Z, Bao Y C, Li R, Chen H J 2018 Comput. Mater. Sci. 146 9
Google Scholar
[9] Wang Q, Wang Y M, Zhang S J, Guo B X, Li C Y, Li R 2021 Crystals 11 1056
Google Scholar
[10] Sakane S, Aoki T, Takaki T 2022 Comput. Mater. Sci. 211 111542
Google Scholar
[11] Meng S X, Zhang A, Guo Z P, Wang Q G 2020 Comput. Mater. Sci. 184 109784
Google Scholar
[12] Takaki T 2023 IOP Conf. Ser. Mater. Sci. Eng. 1274 012009
Google Scholar
[13] Flemings M C, Mehrabian R, Nereo G E 1968 T. Metall. Soc. AIME 239 1449
[14] Bennon W D, Incropera F P 1987 Int. J. Heat Mass Tran. 30 2161
Google Scholar
[15] Bennon W D, Incropera F P 1987 Int. J. Heat Mass Tran. 30 2171
Google Scholar
[16] Beckermann C, Viskanta R 1988 Physicochemical Hydrodynamics 10 195
[17] Gu J P, Beckermann C 1999 Metall. Mater. Trans. A 30 1357
Google Scholar
[18] Wu M, Ludwig A, Kharicha A 2016 Appl. Math. Model. 41 102
Google Scholar
[19] Zhang Z, Bao Y, Liu L, Pian S, Li R 2018 Metall. Mater. Trans. A 49 2750
Google Scholar
[20] Zhang S, Li Y, Zhang S, Zhu B, Li R 2025 Int. J. Therm. Sci. 211 109737
Google Scholar
[21] Rappaz M, Thévoz P H 1987 Acta Metall. 35 2929
Google Scholar
[22] Zhu M F, Lee S Y, Hong C P 2004 Phys. Rev. E 69 061610
Google Scholar
[23] Sun D K, Zhu M F, Pan S Y, Yang C R, Raabe D 2011 Comput. Math. Appl. 61 3585
Google Scholar
[24] Zhu M, Stefanescu D 2007 Acta Mater. 55 1741
Google Scholar
[25] Wen B, Zhang C, Tu Y, Wang C, Fang H 2014 J. Comput. Phys. 266 161
Google Scholar
[26] Mei R, Yu D, Shyy W, Luo L S 2002 Phys. Rev. E 65 041203
Google Scholar
[27] Drummond J E, Tahir M I 1984 Int. J. Multiphase Flow 10 515
Google Scholar
[28] 张照 2020 硕士学位论文 (天津: 河北工业大学)
Zhang Z 2020 M. S. Thesis (Tianjin: Hebei University of Technology
[29] Shan X 1997 Phys. Rev. E 55 2780
Google Scholar
[30] Clever R M, Busse F H 1974 J. Fluid Mech. 65 625
Google Scholar
[31] Wu M, Könözsy L, Ludwig A, Schützenhöfer W, Tanzer R 2008 Steel Res. Int. 79 637
Google Scholar
[32] Luo S, Wang W, Zhu M 2018 Int. J. Heat Mass Tran. 116 940
Google Scholar
[33] Ge H H, Li J, Guo Q T, Ren F L, Xia M X, Yao J H, Li J G 2021 Metall. Mater. Trans. B 52 2992
Google Scholar
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