Numerical Simulation of Macrosegregation Caused by Thermal-Solutal Convection and Solidification Shrinkage Using ALE Model

doi:10.1007/s40195-019-00897-0

Abstract

Abstract:

Solidification shrinkage has been recognized as an important factor for macrosegregation formation. An arbitrary Lagrangian-Eulerian (ALE) model is constructed to predict the macrosegregation caused by thermal-solutal convection and solidification shrinkage. A novel mesh update algorithm is developed to account for the domain change induced by solidification shrinkage. The velocity-pressure coupling between the non-homogenous mass conservation equation and momentum equation is addressed by a modified pressure correction method. The governing equations are solved by the streamline-upwind/Petrov-Galerkin-stabilized finite element algorithm. The application of the model to the Pb-19.2 wt%Sn alloy solidification problem is considered. The inverse segregation is successfully predicted, and reasonable agreement with the literature results is obtained. Thus, the ALE model is established to be a highly effective tool for predicting the macrosegregation caused by solidification shrinkage and thermal-solutal convection. Finally, the effect of solidification shrinkage is analyzed. The results demonstrate that solidification shrinkage delays the advance of the solidification front and intensifies the segregation.

Key words: Macrosegregation, Solidification shrinkage, Finite element method, Arbitrary Lagrangian-Eulerian (ALE)

Cite this article

Kang-Xin Chen, Hou-Fa Shen. Numerical Simulation of Macrosegregation Caused by Thermal-Solutal Convection and Solidification Shrinkage Using ALE Model[J]. Acta Metallurgica Sinica (English Letters), 2019, 32(11): 1396-1406.

share this article

0
/ / Recommend

Add to citation manager EndNote|Ris|BibTeX

URL: https://www.amse.org.cn/EN/10.1007/s40195-019-00897-0

https://www.amse.org.cn/EN/Y2019/V32/I11/1396

Figures/Tables 8

Fig. 1 a Schematic, b structured mesh for Pb-19.2 wt%Sn solidification problem

Table 1 Thermal-physical properties and computational parameters used in calculations

Properties	Units	Value
Liquid density, ρ_l	kg m^-3	10,000
Solid density, ρ_s	kg m^-3	10,800
Specific heat in liquid, c_l	J kg^-1 K^-1	154.7
Specific heat in solid, c_s	J kg^-1 K^-1	177.9
Thermal conductivity in liquid, k_l	W m^-1 K^-1	22.9
Thermal conductivity in solid, k_s	W m^-1 K^-1	39.7
Latent heat, L	J kg^-1	30,162
Liquid dynamic viscosity, μ_l	kg m^-1 s^-1	2.3?×?10^-3
Thermal expansion coefficient, β_T	K^-1	1.09?×?10^-4
Solutal expansion coefficient, β_w	(wt%)^-1	3.54?×?10^-3
Secondary dendrite arm spacing, λ₂	m	71?×?10^-6
Melting point of the pure metal, T_f	°C	327.5
Liquidus slope, m	K(wt%)^-1	-?2.334
Partition coefficient, k_P		0.31
Eutectic temperature, T_eut	°C	183.0
Eutectic composition, w_eut	wt%	61.9
Nominal concentration, w₀	wt%	19.2
Initial temperature, T₀	°C	287
Heat transfer coefficient	W m^-2 K^-1	1000
External temperature	°C	20
Diffusion coefficient in liquid, D_l	m² s^-1	1.05?×?10^-9

Table 1 Thermal-physical properties and computational parameters used in calculations

Properties	Units	Value
Liquid density, ρ_l	kg m^-3	10,000
Solid density, ρ_s	kg m^-3	10,800
Specific heat in liquid, c_l	J kg^-1 K^-1	154.7
Specific heat in solid, c_s	J kg^-1 K^-1	177.9
Thermal conductivity in liquid, k_l	W m^-1 K^-1	22.9
Thermal conductivity in solid, k_s	W m^-1 K^-1	39.7
Latent heat, L	J kg^-1	30,162
Liquid dynamic viscosity, μ_l	kg m^-1 s^-1	2.3?×?10^-3
Thermal expansion coefficient, β_T	K^-1	1.09?×?10^-4
Solutal expansion coefficient, β_w	(wt%)^-1	3.54?×?10^-3
Secondary dendrite arm spacing, λ₂	m	71?×?10^-6
Melting point of the pure metal, T_f	°C	327.5
Liquidus slope, m	K(wt%)^-1	-?2.334
Partition coefficient, k_P		0.31
Eutectic temperature, T_eut	°C	183.0
Eutectic composition, w_eut	wt%	61.9
Nominal concentration, w₀	wt%	19.2
Initial temperature, T₀	°C	287
Heat transfer coefficient	W m^-2 K^-1	1000
External temperature	°C	20
Diffusion coefficient in liquid, D_l	m² s^-1	1.05?×?10^-9

Fig. 2 Distributions of tin concentration at time t?=?400 s predicted using different meshes: a Mesh I, b Mesh II, c Mesh III

Table 2 Comparison of maximum and minimum concentrations in midplane of casting for the three meshes

Solidification time (s)	w_max (wt%)				w_min (wt%)
Solidification time (s)	Mesh I	Mesh II	Mesh III	Ref [10]	Mesh I	Mesh II	Mesh III	Ref [10]
50	20.27	20.26	20.27	20.0	18.86	18.61	17.83	18.8
100	20.27	20.26	20.27	20.1	18.62	18.48	17.66	18.6
200	20.27	20.26	20.50	20.2	18.36	18.26	17.74	18.3
400	20.27	20.26	20.73	20.2	17.90	17.88	17.48	17.4

Table 2 Comparison of maximum and minimum concentrations in midplane of casting for the three meshes

Solidification time (s)	w_max (wt%)				w_min (wt%)
Solidification time (s)	Mesh I	Mesh II	Mesh III	Ref [10]	Mesh I	Mesh II	Mesh III	Ref [10]
50	20.27	20.26	20.27	20.0	18.86	18.61	17.83	18.8
100	20.27	20.26	20.27	20.1	18.62	18.48	17.66	18.6
200	20.27	20.26	20.50	20.2	18.36	18.26	17.74	18.3
400	20.27	20.26	20.73	20.2	17.90	17.88	17.48	17.4

Fig. 3 Solidification sequence and fluid flow of Pb-19.2 wt%Sn alloy at different times a-d correspond to the ALE shrinkage model, e-h correspond to the simplified model without shrinkage: a, e 50 s, b, f 100 s, c, g 200 s, d, h 1000 s

Fig. 4 Evolution of temperature field of Pb-19.2%Sn alloy at different times a-d corresponds to the ALE shrinkage model, e-h corresponds to the simplified model without shrinkage: a, e 200 s, b, f 400 s, c, g 600 s, d, h 800 s

Fig. 5 Evolution of Sn concentration distribution of Pb-19.2%Sn alloy at different times a-d corresponds to the ALE shrinkage model, e-h corresponds to the simplified model without shrinkage: a, e 200 s, b, f 400 s, c, g 600 s, d, h 800 s

Fig. 6 Concentration of tin in mid-height of casting at different times: a 50 s, b 100 s, c 200 s, d 400 s, e 600 s, f 800 s

References

[1]	M.C. Flemings, ISIJ Int. 40, 833-841 (2000)
[2]	C. Beckermann, Int. Mater. Rev. 47, 243-261 (2002)
[3]	W.D. Bennon, F.P. Incropera, 2161- 2170 (1987)
[4]	C. Beckermann, R. Viskanta, PhysicoChem. Hydrodyn. 10, 195-213 (1988)
[5]	X. Ma, D. Li, ISIJ Int. 54, 356-358 (2014)
[6]	K.C. Chiang, H.L. Tsai, Int. J. Heat Mass Transf. 35, 1763-1770 (1992)
[7]	Q.Z. Diao, H.L. Tsai, Metall. Mater. Trans. A 25, 1051-1062 (1994)
[8]	M.J.M. Krane, F.P. Incropera,Metall. Mater. Trans. A 26, 2329-2339 (1995)
[9]	J.C. Heinrich, D.R. Poirier, Model. Simul. Mater. Sci. 12, 881-899 (2004)
[10]	D. Samanta, N. Zabaras, Int. J. Numer. Methods Eng. 64, 1769-1799 (2005)
[11]	S. Zhang, J. Yanke, D.R. Johnson, M.J.M. Krane, Int. J. Numer. Methods Heat 24, 468-482 (2014)
[12]	T.M. Wang, S. Yao, X.G. Zhang, J.Z. Jin, M. Wu, A. Ludwig, Acta Metall. Sin. 42, 584-590 (2006)
[13]	T.M. Wang, T.J. Li, Z.Q. Cao, J.Z. Jin, T. Grimmig, A. Buhrig- Polaczek, Acta Metall. Sin. 42, 591-598 (2006)
[14]	M. Wu, A. Kharicha, A. Ludwig, IOP Conference Series: Materials Science and Engineering, vol 84. 12006(2015)
[15]	M. Wu, A. Ludwig, A. Kharicha, Appl. Math. Model. 41, 102-120 (2017)
[16]	M. Bellet, V.D. Fachinotti, Comput. Methods Appl. Mech 193, 4355-4381 (2004)
[17]	M. Bellet, O. Jaouen, I. Poitrault, Int. J. Numer. Methods Heat Fluid Flow 15, 120-142 (2005)
[18]	J. Ni, F.P. Incropera, Int. J. Heat Mass Transf. 38, 1271-1284 (1995)
[19]	T. Belytschko, W.K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, 2nd edn. (Wiley,Hoboken, 2014), pp. 417-475
[20]	Y.F. Cao, Y. Chen, D.Z. Li, H.W. Liu, P.X. Fu, Metall. Mater. Trans. A 47, 2927-2939 (2016)

Related Articles 15

[1]	Shu-Ming Wang, Jiang-Shan Li, Yan-Xin Wang, Xiao-Fang Zhang, Qing Ye. Thermal Shock Behavior Analysis of Tungsten-Armored Plasma-Facing Components for Future Fusion Reactor [J]. Acta Metallurgica Sinica (English Letters), 2018, 31(5): 515-522.
[2]	Zhao-Yang Jin, Nan-Nan Li, Kai Yan, Jing-Xin Chen, Dong-Lai Wei, Zhen-Shan Cui. Controlling Flow Instability in Straight Spur Gear Forging Using Numerical Simulation and Response Surface Method [J]. Acta Metallurgica Sinica (English Letters), 2018, 31(1): 82-96.
[3]	Yun-Long Wang, Kai-Kun Wang, Yu-Wei Wang, Guang-Chen Li. Experiment and Simulation for Rolling of Diamond-Cu Composites [J]. Acta Metallurgica Sinica (English Letters), 2017, 30(8): 791-800.
[4]	Zhen-Hu Duan, Hou-Fa Shen, Bai-Cheng Liu. A Numerical Study of the Effect of Multiple Pouring on Macrosegregation in a 438-Ton Steel Ingot [J]. Acta Metallurgica Sinica (English Letters), 2015, 28(9): 1123-1133.
[5]	Galina Lasko, Ulrich Weber, Siegfried Schmauder. Finite Element Simulations of Crack Propagation in Al₂O₃/6061Al Composites [J]. Acta Metallurgica Sinica (English Letters), 2014, 27(5): 853-861.
[6]	Huang Zhenyi, Shi Qi, Chen Fuqiang, Shi Yunfeng. FEM Simulation of the Hydrogen Diffusion in X80 Pipeline Steel During Stacking for Slow Cooling [J]. Acta Metallurgica Sinica (English Letters), 2014, 27(3): 416-421.
[7]	Zhuanzhao YANG, Daoxin LIU,Xiaohua ZHANG. Finite Element Method Analysis of the Stress for Line Pipe with Corrode Groove During Outdoor Storage [J]. Acta Metallurgica Sinica (English Letters), 2013, 26(2): 188-198.
[8]	Yixuan TAN,Saiyi LI. Finite element simulations of deformation behavior in equal channel angular pressing using a rotated die [J]. Acta Metallurgica Sinica (English Letters), 2012, 25(5): 357-364.
[9]	Masaaki TABUCHI, Hiromichi HONGO. Evaluation of microstructure and creep damage in high-Cr ferritic steel welds [J]. Acta Metallurgica Sinica (English Letters), 2011, 24(3): 225-234.
[10]	Jianping TAN, Guozhen WANG, Fuzhen XUAN and Shan-Tung TU. Creep crack growth in a Cr-Mo-V type steel: experimental observation and prediction [J]. Acta Metallurgica Sinica (English Letters), 2011, 24(2): 81-91.
[11]	Dongrong LIU, Xiuhong KANG, Baoguang SANG, Dianzhong LI. Numerical study of macrosegregation formation of ingot cast in normal sand mold and water-cooled sand mold [J]. Acta Metallurgica Sinica (English Letters), 2011, 24(1): 54-64.
[12]	Yongsheng WANG,Chenxi JI,Jiongming ZHANG,Xinhua WANG,Wanjun WANG. Investigation of the solute transportation coupled with heat transfer and fluid flow during twin-roll strip casting process [J]. Acta Metallurgica Sinica (English Letters), 2009, 22(5): 345-352.
[13]	Fengping YANG,Qin SUN,Wei HU. Yield criterions of metal plasticity in different stress states [J]. Acta Metallurgica Sinica (English Letters), 2009, 22(2): 123-130.
[14]	Fengli SUI,Liqing CHEN,Xianghua LIU,Lintao WANG,Wei LI. Temperature field analysis and its application in hot continuous rolling of Inconel 718 superalloy [J]. Acta Metallurgica Sinica (English Letters), 2009, 22(2): 81-90.
[15]	Y.X. Hu, Z.Q. Yao. FEM simulation of residual stresses induced by laser shock with overlapping laser spots [J]. Acta Metallurgica Sinica (English Letters), 2008, 21(2): 125-132 .

Metrics

Viewed

Full text

Abstract

Comments

Abstract
Figures/Tables
References
Related Articles
Recommended Articles
Metrics
Comments

TOP