Effect of bottom argon blowing flow rate on evolution behavior of steel-slag interface
Keywords:
Numerical Simulation, Discrete Phase, Multiphase Flow, Slag Entrapment, Slag eyes
Abstract
In this research, the computational fluid dynamics (CFD) software FLUENT is used, which employs the finite volume method, to integrate discrete phase models and multiphase flow models in numerical simulations based on a prototype steel ladle from a particular facility. The simulations aim to investigate the slag entrapment phenomenon in bottom argon blowing. The slag layer is filled with DPM (Discrete Phase Model) particles whose densities are consistent with slag. These particles are used to simulate actual non-metallic inclusions in the slag. If the height of a particle is less than the minimum height of the slag layer, it is thought to have been entrained into the molten steel. By using the User Defined Function (UDF), the tracking of this particle is stopped. The simulation results reveal that the slag eyes have a tendency to increase in size as the argon flow rate increases. The slag eyes area is generally small when the argon flow rate is below 500 L/min. However, there is a noticeable increase in the slag eyes area when the argon flow rate exceeds 1000 L/min. The number of particles entrained into the molten steel increases as the argon flow rate increases; the number of particles entering the steel increases gradually below 1000 L/min and dramatically over 1000 L/min.
References
[1] C. Liu, X. Gao, S. Ueda, M. Guo, S-Y. Kitamura. Composition Changes of Inclusions by Reaction with Slag and Refractory: ISIJ Int., 60 (2020), 1835-1848. https://doi.org/10.2355/isijinternational.ISIJINT-2019-695.
[2] X. Liu, Q. Jia, C. Liu, A. Xiao, G. Li, Z. He, Q. Wang. Metallurgical Mechanism Guided Machine Learning to Predict Slag Entrapment Behavior during Ladle Refining with Bottom Blowing. Metall. Mater. Trans. B. 55 (2024), 1869-1880. https://doi.org/10.1007/s11663-024-03072-8.
[3] M-A. Van Ende, I-H. Jung. A Kinetic Ladle Furnace Process Simulation Model: Effective Equilibrium Reaction Zone Model Using FactSage Macro Processing. Metall. Mater. Trans. B. 48 (2016), 28-36. https://doi.org/10.1007/s11663-016-0698-6
[4] Y. Zhang, Y. Ren, L. Zhang. Kinetic study on compositional variations of inclusions, steel and slag during refining process. Metallurgical Research & Technology,115 (2018). https://doi.org/10.1051/metal/2018059.
[5] H. Hu, L. Yang, Y. Guo, F. Chen, S. Wang, F. Zheng, B. Li. Numerical Simulation of Bottom-Blowing Stirring in Different Smelting Stages of Electric Arc Furnace Steelmaking.Metals,11 (2021). https://doi.org/10.3390/met11050799.
[6] J. Li, LF Refining Technology (Metallurgical Industry Press, Beijing, 2009).ISBN: 978-7-5024-4520-1. (in Chinese)
[7] Z. Liu, B. Li, A. Vakhrushev, M. Wu, A. Ludwig. Physical and Numerical Modeling of Exposed Slag Eye in Continuous Casting Mold using Euler–Euler Approach. steel research international,90 (2018). https://doi.org/10.1002/srin.201800117.
[8] K. Krishnapisharody, G. A. Irons. A Model for Slag Eyes in Steel Refining Ladles Covered with Thick Slag. Metall. Mater. Trans. B, 46 (2014), 191-198. https://doi.org/10.1007/s11663-014-0184-y.
[9] S. Chatterjee, K. Chattopadhyay. Physical Modeling of Slag ‘Eye’ in an Inert Gas-Shrouded Tundish Using Dimensional Analysis. Metall. Mater. Trans. B.47 (2015), 508-521. https://doi.org/10.1007/s11663-015-0512-x.
[10] L. Han, X. Li, Y. Liu. Water Model Experiment of 70t Bottom-Blown Argon Ladle. Laboratory Research and Exploration,30 (2011), 29-34. https://kns.cnki.net/kcms2/article/abstract?v=W694F5cljyBRjrq3JEwoPR5aHVSxP-UU1zmFY-T7ChPJyty6SXfM10QprrnyybgENz7KnhWgNVG3X4bBmvIRFB24irrV0WSvS5wJoL62Y_kwzgLCbtElCgpf-acDk_fqa4qdy8fLF0fcArIBC6hscZPYGVAWcz7yYqY3W2MNxYyoDIeVLTaBVQTBiuyO8c-Z&uniplatform=NZKPT&language=CHS (in Chinese).
[11] Amaro-Villeda, A. M., Ramirez-Argaez, M. A., & Conejo, A. N. (2014). Effect of Slag Properties on Mixing Phenomena in Gas-stirred Ladles by Physical Modeling. ISIJ International, 54(1), 1-8. doi:10.2355/isijinternational.54.1
[12] Linmin, L., Zhongqiu, L., Baokuan, L., Hiroyuki, M., & Fumitaka, T. (2015). Water Model and CFD-PBM Coupled Model of Gas-Liquid-Slag Three-Phase Flow in Ladle Metallurgy. ISIJ International, 55(7), 1337-1346. doi:10.2355/isijinternational.55.1337.
[13] Jardón-Pérez, L. E., González-Morales, D. R., Trápaga, G., González-Rivera, C., & Ramírez-Argáez, M. A. (2019). Effect of Differentiated Injection Ratio, Gas Flow Rate, and Slag Thickness on Mixing Time and Open Eye Area in Gas-Stirred Ladle Assisted by Physical Modeling. Metals, 9(5). doi:10.3390/met9050555.
[14] Dipak, M., Palani, D., & Rajagopal, S. (2017). Modeling and Optimisation of Gas Stirred Ladle Systems. ISIJ International, 57(2), 286-295. doi:10.2355/isijinternational.ISIJINT-2015-701.
[15] Jardón Pérez, L. E., Amaro-Villeda, A., Conejo, A. N., González-Rivera, C., & Ramírez-Argáez, M. A. (2017). Optimizing gas stirred ladles by physical modeling and PIV measurements. Materials and Manufacturing Processes, 33(8), 882-890. doi:10.1080/10426914.2017.1401722.
[16] Z. Li, W. Ouyang, Z. Wang, R. Zheng, Y. Bao, C. Gu. Physical Simulation Study on Flow Field Characteristics of Molten Steel in 70t Ladle Bottom Argon Blowing Process. Metals,13 (2023). https://doi.org/10.3390/met13040639.
[17] K. Krishnapisharody, G. A. Irons. Modeling of slag eye formation over a metal bath due to gas bubbling. Metall. Mater. Trans. B, 37 (2006), 763-772. https://doi.org/10.1007/s11663-006-0058-z.
[18] M. Eranandhanthan, D. Mazumdar. Modeling of Slag Eye Area in Argon Stirred Ladles.ISIJ Int.,50 (2010), 1622–1631. https://doi.org/10.2355/isijinternational.50.1622.
[19] B. G. Thomas, Q. Yuan, S. Mahmood, R. Liu, R. Chaudhary. Transport and Entrapment of Particles in Steel Continuous Casting. Metall. Mater. Trans. B,45 (2013), 22-35. https://doi.org/10.1007/s11663-013-9916-7.
[20] Amaro-Villeda, A. M., Ramirez-Argaez, M. A., & Conejo, A. N. (2014). Effect of Slag Properties on Mixing Phenomena in Gas-stirred Ladles by Physical Modeling. ISIJ International, 54(1), 1-8. doi:10.2355/isijinternational.54.1.
[21] X. Guo, J. Godinez, N. J. Walla, A. K. Silaen, H. Oltmann, V. Thapliyal, A. Bhansali, E. Pretorius, C. Q. Zhou. Computational Investigation of Inclusion Removal in the Steel-Refining Ladle Process. Processes, 9 (2021). https://doi.org/10.3390/pr9061048.
[22] Z. Tan, Y. Yu, X. Deng, J. Li. Simulation Analysis of Influence of Argon Gas Injection Hole Position on Flow Field in Steel Ladle. Jom, (2024). https://doi.org/10.1007/s11837-024-06415-7.
[23] G. Wang, G. Cheng, Y. Zhang, L. Chen, L. Hui, Q. Wang, H. Cai. Numerical and Physical Simulation of Mixing Process in Argon-Stirred Ladles with Single and Dual Bottom Injection. Jom, (2024). https://doi.org/10.1007/s11837-024-06618-y.
[24] F. Su, L. Fang, Z. Kang, H. Zhu. Numerical Simulation Om Heat Transfer of Multi-Layer Ladle in Empty and Heavy Condition. Front. Heat Mass Transf., 20 (2023).doi:10.5098/hmt.20.14.https://cdn.techscience.cn/uploads/attached/file/20230425/20230425161938_89003.pdf.
[25] Y. Liu, H. Bai, H. Liu, M. Ersson, P. G. Jönsson, Y. Gan. Physical and Numerical Modelling on the Mixing Condition in a 50 t Ladle.Metals,9 (2019). https://doi.org/10.3390/met9111136.
[26] X. Li, B. Li, Z. Liu, R. Niu, Y. Liu, C. Zhao, C. Huang, H. Qiao, T. Yuan. Large Eddy Simulation of Multi-Phase Flow and Slag Entrapment in a Continuous Casting Mold.Metals,9 (2018). https://doi.org/10.3390/met9010007.
[27] W. Liu, H. Tang, S. Yang, M. Wang, J. Li, Q. Liu, J. Liu. Numerical Simulation of Slag Eye Formation and Slag Entrapment in a Bottom-Blown Argon-Stirred Ladle. Metall. Mater. Trans. B, 49 (2018), 2681-2691. https://doi.org/10.1007/s11663-018-1308-6.
[28] L. Wang, J. Yang, Y. Liu. Numerical Investigation for Effects of Polydisperse Argon Bubbles on Molten Steel Flow and Liquid Slag Entrapment in a Slab Continuous Casting Mold.Metall. Mater. Trans. B, 53 (2022), 3707-3721. https://doi.org/10.1007/s11663-022-02634-y.
[29] X. Li, B. Li, Z. Liu, D. Wang, T. Qu, S. Hu, C. Wang, Gao R. Evaluation of Slag Entrapment in Continuous Casting Mold Based on the LES-VOF-DPM Coupled Model. Metall. Mater. Trans. B, 52 (2021), 3246-3264. https://doi.org/10.1007/s11663-021-02253-z.
[30] G. Chen, Q. Wang, S. He. Assessment of an Eulerian multi-fluid VOF model for simulation of multiphase flow in an industrial Ruhrstahl–Heraeus degasser. Metallurgical Research & Technology, 116 (2019).doi: 10.1051/metal/2019049.
[31] A. Gupta, R. Kumar, R. K. Singh. Assessment of Critical Vortexing Height to Prevent Slag Entrapment During Tundish Teeming. Metals and Materials International, 28 (2021), 1246-1256. https://doi.org/10.1007/s12540-021-01014-6.
[32] X. Zhao, J. Zhang, F. Gao, X. Wang, L. Wang. Transient simulation of slag entrapment in a tundish. Journal of Physics: Conference Series, 2390 (2022).doi: 10.1088/1742-6596/2390/1/012080.
[33] Aydogdu, M. (2023). Analysis of the effect of rigid vegetation patches on the hydraulics of an open channel flow with Realizable k-ε and Reynolds stress turbulence models. Flow Measurement and Instrumentation, 94. doi:10.1016/j.flowmeasinst.2023.102477.
[34] R. D. Morales, S. Garcia-Hernandez, J. D. J. Barreto, A. Ceballos-Huerta, I. Calderon-Ramos, E. Gutierrez. Multiphase Flow Modeling of Slag Entrapment During Ladle Change-Over Operation. Metall. Mater. Trans. B, 47 (2016), 2595-2606.
https://doi.org/10.1007/s11663-016-0663-4.
[35] A. Srivastava, K. Chattopadhyay. Macroscopic Mechanistic Modeling for the Prediction of Mold Slag Exposure in a Continuous Casting Mold. Metall. Mater. Trans. B, 53 (2022), 1018-1035. https://doi.org/10.1007/s11663-021-02396-z.
[36] Xufeng, Q. (2021). Study on gas-liquid two phase flow behavior under effects of annular gas curtain and swirling flow at tundish nozzle. (Doctoral dissertation, Wuhan University of Science and Technology, Wuhan, Hubei, P.R. China). Retrieved from https://link.cnki.net/doi/10.27380/d.cnki.gwkju.2021.000533. (in Chinese)
[2] X. Liu, Q. Jia, C. Liu, A. Xiao, G. Li, Z. He, Q. Wang. Metallurgical Mechanism Guided Machine Learning to Predict Slag Entrapment Behavior during Ladle Refining with Bottom Blowing. Metall. Mater. Trans. B. 55 (2024), 1869-1880. https://doi.org/10.1007/s11663-024-03072-8.
[3] M-A. Van Ende, I-H. Jung. A Kinetic Ladle Furnace Process Simulation Model: Effective Equilibrium Reaction Zone Model Using FactSage Macro Processing. Metall. Mater. Trans. B. 48 (2016), 28-36. https://doi.org/10.1007/s11663-016-0698-6
[4] Y. Zhang, Y. Ren, L. Zhang. Kinetic study on compositional variations of inclusions, steel and slag during refining process. Metallurgical Research & Technology,115 (2018). https://doi.org/10.1051/metal/2018059.
[5] H. Hu, L. Yang, Y. Guo, F. Chen, S. Wang, F. Zheng, B. Li. Numerical Simulation of Bottom-Blowing Stirring in Different Smelting Stages of Electric Arc Furnace Steelmaking.Metals,11 (2021). https://doi.org/10.3390/met11050799.
[6] J. Li, LF Refining Technology (Metallurgical Industry Press, Beijing, 2009).ISBN: 978-7-5024-4520-1. (in Chinese)
[7] Z. Liu, B. Li, A. Vakhrushev, M. Wu, A. Ludwig. Physical and Numerical Modeling of Exposed Slag Eye in Continuous Casting Mold using Euler–Euler Approach. steel research international,90 (2018). https://doi.org/10.1002/srin.201800117.
[8] K. Krishnapisharody, G. A. Irons. A Model for Slag Eyes in Steel Refining Ladles Covered with Thick Slag. Metall. Mater. Trans. B, 46 (2014), 191-198. https://doi.org/10.1007/s11663-014-0184-y.
[9] S. Chatterjee, K. Chattopadhyay. Physical Modeling of Slag ‘Eye’ in an Inert Gas-Shrouded Tundish Using Dimensional Analysis. Metall. Mater. Trans. B.47 (2015), 508-521. https://doi.org/10.1007/s11663-015-0512-x.
[10] L. Han, X. Li, Y. Liu. Water Model Experiment of 70t Bottom-Blown Argon Ladle. Laboratory Research and Exploration,30 (2011), 29-34. https://kns.cnki.net/kcms2/article/abstract?v=W694F5cljyBRjrq3JEwoPR5aHVSxP-UU1zmFY-T7ChPJyty6SXfM10QprrnyybgENz7KnhWgNVG3X4bBmvIRFB24irrV0WSvS5wJoL62Y_kwzgLCbtElCgpf-acDk_fqa4qdy8fLF0fcArIBC6hscZPYGVAWcz7yYqY3W2MNxYyoDIeVLTaBVQTBiuyO8c-Z&uniplatform=NZKPT&language=CHS (in Chinese).
[11] Amaro-Villeda, A. M., Ramirez-Argaez, M. A., & Conejo, A. N. (2014). Effect of Slag Properties on Mixing Phenomena in Gas-stirred Ladles by Physical Modeling. ISIJ International, 54(1), 1-8. doi:10.2355/isijinternational.54.1
[12] Linmin, L., Zhongqiu, L., Baokuan, L., Hiroyuki, M., & Fumitaka, T. (2015). Water Model and CFD-PBM Coupled Model of Gas-Liquid-Slag Three-Phase Flow in Ladle Metallurgy. ISIJ International, 55(7), 1337-1346. doi:10.2355/isijinternational.55.1337.
[13] Jardón-Pérez, L. E., González-Morales, D. R., Trápaga, G., González-Rivera, C., & Ramírez-Argáez, M. A. (2019). Effect of Differentiated Injection Ratio, Gas Flow Rate, and Slag Thickness on Mixing Time and Open Eye Area in Gas-Stirred Ladle Assisted by Physical Modeling. Metals, 9(5). doi:10.3390/met9050555.
[14] Dipak, M., Palani, D., & Rajagopal, S. (2017). Modeling and Optimisation of Gas Stirred Ladle Systems. ISIJ International, 57(2), 286-295. doi:10.2355/isijinternational.ISIJINT-2015-701.
[15] Jardón Pérez, L. E., Amaro-Villeda, A., Conejo, A. N., González-Rivera, C., & Ramírez-Argáez, M. A. (2017). Optimizing gas stirred ladles by physical modeling and PIV measurements. Materials and Manufacturing Processes, 33(8), 882-890. doi:10.1080/10426914.2017.1401722.
[16] Z. Li, W. Ouyang, Z. Wang, R. Zheng, Y. Bao, C. Gu. Physical Simulation Study on Flow Field Characteristics of Molten Steel in 70t Ladle Bottom Argon Blowing Process. Metals,13 (2023). https://doi.org/10.3390/met13040639.
[17] K. Krishnapisharody, G. A. Irons. Modeling of slag eye formation over a metal bath due to gas bubbling. Metall. Mater. Trans. B, 37 (2006), 763-772. https://doi.org/10.1007/s11663-006-0058-z.
[18] M. Eranandhanthan, D. Mazumdar. Modeling of Slag Eye Area in Argon Stirred Ladles.ISIJ Int.,50 (2010), 1622–1631. https://doi.org/10.2355/isijinternational.50.1622.
[19] B. G. Thomas, Q. Yuan, S. Mahmood, R. Liu, R. Chaudhary. Transport and Entrapment of Particles in Steel Continuous Casting. Metall. Mater. Trans. B,45 (2013), 22-35. https://doi.org/10.1007/s11663-013-9916-7.
[20] Amaro-Villeda, A. M., Ramirez-Argaez, M. A., & Conejo, A. N. (2014). Effect of Slag Properties on Mixing Phenomena in Gas-stirred Ladles by Physical Modeling. ISIJ International, 54(1), 1-8. doi:10.2355/isijinternational.54.1.
[21] X. Guo, J. Godinez, N. J. Walla, A. K. Silaen, H. Oltmann, V. Thapliyal, A. Bhansali, E. Pretorius, C. Q. Zhou. Computational Investigation of Inclusion Removal in the Steel-Refining Ladle Process. Processes, 9 (2021). https://doi.org/10.3390/pr9061048.
[22] Z. Tan, Y. Yu, X. Deng, J. Li. Simulation Analysis of Influence of Argon Gas Injection Hole Position on Flow Field in Steel Ladle. Jom, (2024). https://doi.org/10.1007/s11837-024-06415-7.
[23] G. Wang, G. Cheng, Y. Zhang, L. Chen, L. Hui, Q. Wang, H. Cai. Numerical and Physical Simulation of Mixing Process in Argon-Stirred Ladles with Single and Dual Bottom Injection. Jom, (2024). https://doi.org/10.1007/s11837-024-06618-y.
[24] F. Su, L. Fang, Z. Kang, H. Zhu. Numerical Simulation Om Heat Transfer of Multi-Layer Ladle in Empty and Heavy Condition. Front. Heat Mass Transf., 20 (2023).doi:10.5098/hmt.20.14.https://cdn.techscience.cn/uploads/attached/file/20230425/20230425161938_89003.pdf.
[25] Y. Liu, H. Bai, H. Liu, M. Ersson, P. G. Jönsson, Y. Gan. Physical and Numerical Modelling on the Mixing Condition in a 50 t Ladle.Metals,9 (2019). https://doi.org/10.3390/met9111136.
[26] X. Li, B. Li, Z. Liu, R. Niu, Y. Liu, C. Zhao, C. Huang, H. Qiao, T. Yuan. Large Eddy Simulation of Multi-Phase Flow and Slag Entrapment in a Continuous Casting Mold.Metals,9 (2018). https://doi.org/10.3390/met9010007.
[27] W. Liu, H. Tang, S. Yang, M. Wang, J. Li, Q. Liu, J. Liu. Numerical Simulation of Slag Eye Formation and Slag Entrapment in a Bottom-Blown Argon-Stirred Ladle. Metall. Mater. Trans. B, 49 (2018), 2681-2691. https://doi.org/10.1007/s11663-018-1308-6.
[28] L. Wang, J. Yang, Y. Liu. Numerical Investigation for Effects of Polydisperse Argon Bubbles on Molten Steel Flow and Liquid Slag Entrapment in a Slab Continuous Casting Mold.Metall. Mater. Trans. B, 53 (2022), 3707-3721. https://doi.org/10.1007/s11663-022-02634-y.
[29] X. Li, B. Li, Z. Liu, D. Wang, T. Qu, S. Hu, C. Wang, Gao R. Evaluation of Slag Entrapment in Continuous Casting Mold Based on the LES-VOF-DPM Coupled Model. Metall. Mater. Trans. B, 52 (2021), 3246-3264. https://doi.org/10.1007/s11663-021-02253-z.
[30] G. Chen, Q. Wang, S. He. Assessment of an Eulerian multi-fluid VOF model for simulation of multiphase flow in an industrial Ruhrstahl–Heraeus degasser. Metallurgical Research & Technology, 116 (2019).doi: 10.1051/metal/2019049.
[31] A. Gupta, R. Kumar, R. K. Singh. Assessment of Critical Vortexing Height to Prevent Slag Entrapment During Tundish Teeming. Metals and Materials International, 28 (2021), 1246-1256. https://doi.org/10.1007/s12540-021-01014-6.
[32] X. Zhao, J. Zhang, F. Gao, X. Wang, L. Wang. Transient simulation of slag entrapment in a tundish. Journal of Physics: Conference Series, 2390 (2022).doi: 10.1088/1742-6596/2390/1/012080.
[33] Aydogdu, M. (2023). Analysis of the effect of rigid vegetation patches on the hydraulics of an open channel flow with Realizable k-ε and Reynolds stress turbulence models. Flow Measurement and Instrumentation, 94. doi:10.1016/j.flowmeasinst.2023.102477.
[34] R. D. Morales, S. Garcia-Hernandez, J. D. J. Barreto, A. Ceballos-Huerta, I. Calderon-Ramos, E. Gutierrez. Multiphase Flow Modeling of Slag Entrapment During Ladle Change-Over Operation. Metall. Mater. Trans. B, 47 (2016), 2595-2606.
https://doi.org/10.1007/s11663-016-0663-4.
[35] A. Srivastava, K. Chattopadhyay. Macroscopic Mechanistic Modeling for the Prediction of Mold Slag Exposure in a Continuous Casting Mold. Metall. Mater. Trans. B, 53 (2022), 1018-1035. https://doi.org/10.1007/s11663-021-02396-z.
[36] Xufeng, Q. (2021). Study on gas-liquid two phase flow behavior under effects of annular gas curtain and swirling flow at tundish nozzle. (Doctoral dissertation, Wuhan University of Science and Technology, Wuhan, Hubei, P.R. China). Retrieved from https://link.cnki.net/doi/10.27380/d.cnki.gwkju.2021.000533. (in Chinese)
Published
2025/07/31
How to Cite
Li, J., xia, wenwu, mou, qiqiang, & yu, yue. (2025). Effect of bottom argon blowing flow rate on evolution behavior of steel-slag interface. Journal of Mining and Metallurgy, Section B: Metallurgy, 61(1), 85-98. Retrieved from https://aseestant.ceon.rs/index.php/jmm/article/view/55212
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