A Library for Numerically Simulating Indirect Heating of Granular Materials by the Induction Method
Keywords:
induction heating, granular materials, discrete element method, finite volume method, interdisciplinary modeling
Abstract
The article addresses the development of the TempTransferFoam library intended for modeling and studying induction heating of granular materials. The library offers a unique set of tools for conducting an accurate and comprehensive analysis of heating processes, which opens up new prospects for research and optimization of production processes in the industry. The relevance of the study is stemming from the need to overcome the complexity of designing and numerical modeling of induction heating systems, including those involving integration of the analysis of electromagnetic, thermal, and dynamic processes in granular media. Owing to a combined use of discrete element and finite volume methods, the library acquires the ability to perform adaptive grid refinement in modeling an electromagnetic field, solve heat transfer problems in multibody systems, and simulate the dynamics of granular materials with a high degree of detail. To solve complex interdisciplinary problems of induction heating, a technique is proposed that makes it possible to implement a qualitative transfer of thermal fields in the form of boundary conditions into calculation of the dynamics of a medium consisting of granular materials. An example of using the developed library is given, which confirms the validity of its application for effective modeling of electromagnetic and thermal processes in granular materials.
References
1. Boateng A.A. Rotary Kilns: Transport Phenomena and Transport Processes. Elsevier, Butterworth-Heinemann, 2016, 360 p.
2. Лисиенко В.Г. и др. Вращающиеся печи: теплотехника, управление и экология. М.: Теплотехник, 2004, 687 с.
3. Волков Д. А. и др. Технологии производства литой дроби из Fe-C-cплавов. – Литейное производство, 2012, № 1, с. 14–17.
4. Хацаюк М.Ю. Теория и моделирование магнитогидродинамических процессов в электротехнологических комплексах металлургического назначения: дис. … доктора техн. наук. СПб., 2020, 338 с.
5. Слухоцкий А.Е. и др. Установки индукционного нагрева. Л.: Энергоиздат, 1981, 328 с.
6. Лупи С., Форзан М., Алиферов А.И. Индукционная высокочастотная импульсная закалка стальных изделий. – Электричество, 2023, № 10, с. 48–56.
7. Демидович В.Б. Цифровое моделирование электромагнитных процессов в технологических индукционных устройствах. – Электричество, 2021, № 7, с. 26–32.
8. Baake E., Nacke B. Energy and Environmental Aspects of Induction Melting Processes. – In book: Induction Heating and Heat Treatment, 2014, pp. 548–554, DOI:10.31399/asm.hb.v04c.a0005908.
9. Demidovich V.B. Computer Simulation and Optimal Designing of Energy-Saving Technologies of the Induction Heating of Metals. – Thermal Engineering, 2012, 59 (14), pp. 1023–1034, DOI:10.1134/S0040601512140030.
10. Mickey H.S., Fairbanks D.F. Mechanics of Heat Transfer to Fluidized Beds. – AIChE Journal, 1955, DOI: 10.1002/aic.690010317.
11. Basakov A.P. The Mechanism of Heat Transfer between a Fluidized Bed and Surface. – International Chemical Engineering, 1964, 4 (2), р. 320.
12. Cook C.A., Cundy V.A. Heat Transfer between a Rotating Cylinder and a Moist Granular Bed. – International Journal of Heat and Mass Transfer, 1995, 38 (3), рр. 419–432.
13. Armstrong L.M., Gu S., Luo K.H. Study of Wall-to-Bed Heat Transfer in a Bubbling Fluidised Bed Using the Kinetic Theory of Granular Flow. – International Journal of Heat and Mass Transfer, 2010, 53 (21-22), pp. 4949–4959, DOI:10.1016/j.ijheatmasstransfer.2010.05.047.
14. ANSYS Rocky DEM [Электрон. ресурс], URL: https://www.esss.co/en/ansys-simulation-software/ansys-rocky (дата обращения 12.04.2024).
15. Kloss C., Goniva C. LIGGGHTS: A New Open Source Dem Simulation Software. – 5th International Conference on Discrete Element Methods (DEM5), 2010.
16. Kozicki J., Donze F.V. YADE‐OPEN DEM: An Open‐Source Software Using a Discrete Element Method to Simulate Granular Material. – Engineering Computations, 2009, 26 (7), pp. 786–805, DOI:10.1108/02644400910985170.
17. Fonte C.B., Oliveira J.A., Almeida L.C. DEM-CFD Coupling: Mathematical Modelling and Case Studies Using ROCKY-DEM® and ANSYS Fluent®. –The 11th International Conference on CFD in the Minerals and Process Industries, CSIRO, 2015, рр. 7–9.
18. Kloss C. et al. Models, Algorithms and Validation for Opensource DEM and CFD–DEM. – Progress in Computational Fluid Dynamics, an International Journal, 2012, 12 (2-3), pp. 140–152, DOI:10.1504/PCFD.2012.047457.
19. Caulk R.A., Bruno C. An open Framework for the Simulation of Coupled Thermo-Hydro-Mechanical Processes in Discrete Element Systems. – 8th International Conference on Discrete Element Methods, 2019.
20. Weber N. et al. Numerical Simulation of the Tayler Instability in Liquid Metals. – New Journal of Physics, 2013, 15 (4), DOI:10.1088/1367-2630/15/4/043034.
21. Beckstein P., Galindo V., Vukčević V. Efficient Solution of 3D Electromagnetic Eddy-Current Problems within the Finite Volume Framework of OpenFOAM. – Journal of Computational Physics, 2017, 344 (11), рр. 623–646, DOI:10.1016/j.jcp.2017.05.005.
22. Busse C. et al. Numerical Modeling of an Inductively Coupled Plasma Torch Using OpenFOAM. – Computers & Fluids, 2021, vol. 216, p. 104807.
23. Khatsayuk M.Yu. et al. Numerical Simulation of Process of Electromagnetic Casting and Technology Features. – Metallurgical and Materials. Transaction B, 2023, 54 (4), pp. 1768–1783, DOI:10.1007/s11663-023-02791-8.
24. The Foam-Extend Project [Электрон. ресурс], URL: http://www.foam-extend.org (дата обращения 08.11.2023)
25. Jasak H. Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows, 1996.
26. Weller H.G. et al. A Tensorial Approach to Computational Continuum Mechanics Using Object-Oriented Techniques. – Computers in Physics, 1998, 12 (6), pp. 620–631, DOI:10.1063/1.168744.
27. OpenFOAM – Free Open Source CFD [Электрон. ресурс], URL: http://www.openfoam.org (дата обращения 12.04.2024).
28. Ai J. et al. Assessment of Rolling Resistance Models in Discrete Element Simulations. – Powder Technology, 2011, 206 (3), pp. 269–282, DOI:10.1016/j.powtec.2010.09.030.
29. Aruliah D.A. et al. A Method for the Forward Modelling of 3-D Electromagnetic Quasi-Static Problems. – Mathematical Models and Methods in Applied Sciences, 2001, 11 (1), DOI:10.1142/S0218202501000702.
30. Guermond J.L. et al. An Interior Penalty Galerkin Method for the MHD Equations in Heterogeneous Domains. – Journal of Computational Physics, 2007, 221 (1), pp. 349–369, DOI:10.1016/j.jcp.2006.06.045.
31. Malik M., Fan E.S.-C., Bussmann M. Adaptive VOF with Curvature‐Based Refinement. – International Journal for Numerical Methods in Fluids, 2007, 55(7), pp. 693–712, DOI:10.1002/fld.1490.
---
Исследование выполнено за счет гранта ФГБУ «Фонд содействия развитию малых форм предприятий в научно-технической сфере» (Фонд содействия инновациям), договор № 4125ГС1/68579.
#
1. Boateng A.A. Rotary Kilns: Transport Phenomena and Transport Processes. Elsevier, Butterworth-Heinemann, 2016, 360 p.
2. Lisienko V.G. et al. Vrashchayushchiesya pechi: teplotekhnika, upravlenie i ekologiya (Rotary Kilns: Heat Engineering, Management and Ecology). M.: Teplotekhnik, 2004, 687 p.
3. Volkov D.А. et al. Liteynoe proizvodstvo – in Russ. (Foundry Production), 2012, No. 1, pp. 14–17.
4. Khatsayuk M.Yu. Teoriya i modelirovanie magnitogidrodina-micheskih protsessov v elektrotekhnologicheskih kompleksah metallur-gicheskogo naznacheniya: dis. … doktora tekhn. nauk (Theory and Modeling of Magnetohydrodynamic Processes in Electrotechnological Complexes of Metallurgical Purpose: dis. ... Dr. Sci. (Eng.)). SPb., 2020, 338 p.
5. Sluhotskiy A.E. et al. Ustanovki induktsionnogo nagreva (Induction Heating Installations). L.: Energoizdat, 1981, 328 p.
6. Lupi S., Forzan М., Aliferov А.I. Elektrichestvo – in Russ. (Electricity), 2023, No. 10, pp. 48–56.
7. Demidovich V.B. Elektrichestvo – in Russ. (Electricity), 2021, No. 7, pp. 26–32.
8. Baake E., Nacke B. Energy and Environmental Aspects of Induction Melting Processes. – In book: Induction Heating and Heat Treatment, 2014, pp. 548–554, DOI:10.31399/asm.hb.v04c.a0005908.
9. Demidovich V.B. Computer Simulation and Optimal Designing of Energy-Saving Technologies of the Induction Heating of Metals. – Thermal Engineering, 2012, 59 (14), pp. 1023–1034, DOI:10.1134/S0040601512140030.
10. Mickey H.S., Fairbanks D.F. Mechanics of Heat Transfer to Fluidized Beds. – AIChE Journal, 1955, DOI: 10.1002/aic.690010317.
11. Basakov A.P. The Mechanism of Heat Transfer between a Fluidized Bed and Surface. – International Chemical Engineering, 1964, 4 (2), р. 320.
12. Cook C.A., Cundy V.A. Heat Transfer between a Rotating Cylinder and a Moist Granular Bed. – International Journal of Heat and Mass Transfer, 1995, 38 (3), рр. 419–432.
13. Armstrong L.M., Gu S., Luo K.H. Study of Wall-to-Bed Heat Transfer in a Bubbling Fluidised Bed Using the Kinetic Theory of Granular Flow. – International Journal of Heat and Mass Transfer, 2010, 53 (21-22), pp. 4949–4959, DOI:10.1016/j.ijheatmasstransfer.2010.05.047.
14. ANSYS Rocky DEM [Electron. resource], URL: https://www.esss.co/en/ansys-simulation-software/ansys-rocky (Date of appeal 12.04.2024).
15. Kloss C., Goniva C. LIGGGHTS: A New Open Source Dem Simulation Software. – 5th International Conference on Discrete Element Methods (DEM5), 2010.
16. Kozicki J., Donze F.V. YADE‐OPEN DEM: An Open‐Source Software Using a Discrete Element Method to Simulate Granular Material. – Engineering Computations, 2009, 26 (7), pp. 786–805, DOI:10.1108/02644400910985170.
17. Fonte C.B., Oliveira J.A., Almeida L.C. DEM-CFD Coupling: Mathematical Modelling and Case Studies Using ROCKY-DEM® and ANSYS Fluent®. –The 11th International Conference on CFD in the Minerals and Process Industries, CSIRO, 2015, рр. 7–9.
18. Kloss C. et al. Models, Algorithms and Validation for Opensource DEM and CFD–DEM. – Progress in Computational Fluid Dynamics, an International Journal, 2012, 12 (2-3), pp. 140–152, DOI:10.1504/PCFD.2012.047457.
19. Caulk R.A., Bruno C. An open Framework for the Simulation of Coupled Thermo-Hydro-Mechanical Processes in Discrete Element Systems. – 8th International Conference on Discrete Element Methods, 2019.
20. Weber N. et al. Numerical Simulation of the Tayler Instability in Liquid Metals. – New Journal of Physics, 2013, 15 (4), DOI:10.1088/1367-2630/15/4/043034.
21. Beckstein P., Galindo V., Vukčević V. Efficient Solution of 3D Electromagnetic Eddy-Current Problems within the Finite Volume Framework of OpenFOAM. – Journal of Computational Physics, 2017, 344 (11), рр. 623–646, DOI:10.1016/j.jcp.2017.05.005.
22. Busse C. et al. Numerical Modeling of an Inductively Coupled Plasma Torch Using OpenFOAM. – Computers & Fluids, 2021, vol. 216, p. 104807.
23. Khatsayuk M.Yu. et al. Numerical Simulation of Process of Electromagnetic Casting and Technology Features. – Metallurgical and Materials. Transaction B, 2023, 54 (4), pp. 1768–1783, DOI:10.1007/s11663-023-02791-8.
24. The Foam-Extend Project [Electron. resource], URL: http://www.foam-extend.org (Date of appeal 08.11.2023)
25. Jasak H. Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows, 1996.
26. Weller H.G. et al. A Tensorial Approach to Computational Continuum Mechanics Using Object-Oriented Techniques. – Computers in Physics, 1998, 12 (6), pp. 620–631, DOI:10.1063/1.168744.
27. OpenFOAM – Free Open Source CFD [Electron. resource], URL: http://www.openfoam.org (Date of appeal 12.04.2024).
28. Ai J. et al. Assessment of Rolling Resistance Models in Discrete Element Simulations. – Powder Technology, 2011, 206 (3), pp. 269–282, DOI:10.1016/j.powtec.2010.09.030.
29. Aruliah D.A. et al. A Method for the Forward Modelling of 3-D Electromagnetic Quasi-Static Problems. – Mathematical Models and Methods in Applied Sciences, 2001, 11 (1), DOI:10.1142/S0218202501000702.
30. Guermond J.L. et al. An Interior Penalty Galerkin Method for the MHD Equations in Heterogeneous Domains. – Journal of Computational Physics, 2007, 221 (1), pp. 349–369, DOI:10.1016/j.jcp.2006.06.045.
31. Malik M., Fan E.S.-C., Bussmann M. Adaptive VOF with Curvature‐Based Refinement. – International Journal for Numerical Methods in Fluids, 2007, 55(7), pp. 693–712, DOI:10.1002/fld.1490
---
The study was financially supported by the Grant of the Federal State Budgetary Institution Foundation for Assistance to Small Innovative Enterprises in Science and Technology (Foundation for Innovation Assistance), agreement No. 4125GS1/68579
2. Лисиенко В.Г. и др. Вращающиеся печи: теплотехника, управление и экология. М.: Теплотехник, 2004, 687 с.
3. Волков Д. А. и др. Технологии производства литой дроби из Fe-C-cплавов. – Литейное производство, 2012, № 1, с. 14–17.
4. Хацаюк М.Ю. Теория и моделирование магнитогидродинамических процессов в электротехнологических комплексах металлургического назначения: дис. … доктора техн. наук. СПб., 2020, 338 с.
5. Слухоцкий А.Е. и др. Установки индукционного нагрева. Л.: Энергоиздат, 1981, 328 с.
6. Лупи С., Форзан М., Алиферов А.И. Индукционная высокочастотная импульсная закалка стальных изделий. – Электричество, 2023, № 10, с. 48–56.
7. Демидович В.Б. Цифровое моделирование электромагнитных процессов в технологических индукционных устройствах. – Электричество, 2021, № 7, с. 26–32.
8. Baake E., Nacke B. Energy and Environmental Aspects of Induction Melting Processes. – In book: Induction Heating and Heat Treatment, 2014, pp. 548–554, DOI:10.31399/asm.hb.v04c.a0005908.
9. Demidovich V.B. Computer Simulation and Optimal Designing of Energy-Saving Technologies of the Induction Heating of Metals. – Thermal Engineering, 2012, 59 (14), pp. 1023–1034, DOI:10.1134/S0040601512140030.
10. Mickey H.S., Fairbanks D.F. Mechanics of Heat Transfer to Fluidized Beds. – AIChE Journal, 1955, DOI: 10.1002/aic.690010317.
11. Basakov A.P. The Mechanism of Heat Transfer between a Fluidized Bed and Surface. – International Chemical Engineering, 1964, 4 (2), р. 320.
12. Cook C.A., Cundy V.A. Heat Transfer between a Rotating Cylinder and a Moist Granular Bed. – International Journal of Heat and Mass Transfer, 1995, 38 (3), рр. 419–432.
13. Armstrong L.M., Gu S., Luo K.H. Study of Wall-to-Bed Heat Transfer in a Bubbling Fluidised Bed Using the Kinetic Theory of Granular Flow. – International Journal of Heat and Mass Transfer, 2010, 53 (21-22), pp. 4949–4959, DOI:10.1016/j.ijheatmasstransfer.2010.05.047.
14. ANSYS Rocky DEM [Электрон. ресурс], URL: https://www.esss.co/en/ansys-simulation-software/ansys-rocky (дата обращения 12.04.2024).
15. Kloss C., Goniva C. LIGGGHTS: A New Open Source Dem Simulation Software. – 5th International Conference on Discrete Element Methods (DEM5), 2010.
16. Kozicki J., Donze F.V. YADE‐OPEN DEM: An Open‐Source Software Using a Discrete Element Method to Simulate Granular Material. – Engineering Computations, 2009, 26 (7), pp. 786–805, DOI:10.1108/02644400910985170.
17. Fonte C.B., Oliveira J.A., Almeida L.C. DEM-CFD Coupling: Mathematical Modelling and Case Studies Using ROCKY-DEM® and ANSYS Fluent®. –The 11th International Conference on CFD in the Minerals and Process Industries, CSIRO, 2015, рр. 7–9.
18. Kloss C. et al. Models, Algorithms and Validation for Opensource DEM and CFD–DEM. – Progress in Computational Fluid Dynamics, an International Journal, 2012, 12 (2-3), pp. 140–152, DOI:10.1504/PCFD.2012.047457.
19. Caulk R.A., Bruno C. An open Framework for the Simulation of Coupled Thermo-Hydro-Mechanical Processes in Discrete Element Systems. – 8th International Conference on Discrete Element Methods, 2019.
20. Weber N. et al. Numerical Simulation of the Tayler Instability in Liquid Metals. – New Journal of Physics, 2013, 15 (4), DOI:10.1088/1367-2630/15/4/043034.
21. Beckstein P., Galindo V., Vukčević V. Efficient Solution of 3D Electromagnetic Eddy-Current Problems within the Finite Volume Framework of OpenFOAM. – Journal of Computational Physics, 2017, 344 (11), рр. 623–646, DOI:10.1016/j.jcp.2017.05.005.
22. Busse C. et al. Numerical Modeling of an Inductively Coupled Plasma Torch Using OpenFOAM. – Computers & Fluids, 2021, vol. 216, p. 104807.
23. Khatsayuk M.Yu. et al. Numerical Simulation of Process of Electromagnetic Casting and Technology Features. – Metallurgical and Materials. Transaction B, 2023, 54 (4), pp. 1768–1783, DOI:10.1007/s11663-023-02791-8.
24. The Foam-Extend Project [Электрон. ресурс], URL: http://www.foam-extend.org (дата обращения 08.11.2023)
25. Jasak H. Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows, 1996.
26. Weller H.G. et al. A Tensorial Approach to Computational Continuum Mechanics Using Object-Oriented Techniques. – Computers in Physics, 1998, 12 (6), pp. 620–631, DOI:10.1063/1.168744.
27. OpenFOAM – Free Open Source CFD [Электрон. ресурс], URL: http://www.openfoam.org (дата обращения 12.04.2024).
28. Ai J. et al. Assessment of Rolling Resistance Models in Discrete Element Simulations. – Powder Technology, 2011, 206 (3), pp. 269–282, DOI:10.1016/j.powtec.2010.09.030.
29. Aruliah D.A. et al. A Method for the Forward Modelling of 3-D Electromagnetic Quasi-Static Problems. – Mathematical Models and Methods in Applied Sciences, 2001, 11 (1), DOI:10.1142/S0218202501000702.
30. Guermond J.L. et al. An Interior Penalty Galerkin Method for the MHD Equations in Heterogeneous Domains. – Journal of Computational Physics, 2007, 221 (1), pp. 349–369, DOI:10.1016/j.jcp.2006.06.045.
31. Malik M., Fan E.S.-C., Bussmann M. Adaptive VOF with Curvature‐Based Refinement. – International Journal for Numerical Methods in Fluids, 2007, 55(7), pp. 693–712, DOI:10.1002/fld.1490.
---
Исследование выполнено за счет гранта ФГБУ «Фонд содействия развитию малых форм предприятий в научно-технической сфере» (Фонд содействия инновациям), договор № 4125ГС1/68579.
#
1. Boateng A.A. Rotary Kilns: Transport Phenomena and Transport Processes. Elsevier, Butterworth-Heinemann, 2016, 360 p.
2. Lisienko V.G. et al. Vrashchayushchiesya pechi: teplotekhnika, upravlenie i ekologiya (Rotary Kilns: Heat Engineering, Management and Ecology). M.: Teplotekhnik, 2004, 687 p.
3. Volkov D.А. et al. Liteynoe proizvodstvo – in Russ. (Foundry Production), 2012, No. 1, pp. 14–17.
4. Khatsayuk M.Yu. Teoriya i modelirovanie magnitogidrodina-micheskih protsessov v elektrotekhnologicheskih kompleksah metallur-gicheskogo naznacheniya: dis. … doktora tekhn. nauk (Theory and Modeling of Magnetohydrodynamic Processes in Electrotechnological Complexes of Metallurgical Purpose: dis. ... Dr. Sci. (Eng.)). SPb., 2020, 338 p.
5. Sluhotskiy A.E. et al. Ustanovki induktsionnogo nagreva (Induction Heating Installations). L.: Energoizdat, 1981, 328 p.
6. Lupi S., Forzan М., Aliferov А.I. Elektrichestvo – in Russ. (Electricity), 2023, No. 10, pp. 48–56.
7. Demidovich V.B. Elektrichestvo – in Russ. (Electricity), 2021, No. 7, pp. 26–32.
8. Baake E., Nacke B. Energy and Environmental Aspects of Induction Melting Processes. – In book: Induction Heating and Heat Treatment, 2014, pp. 548–554, DOI:10.31399/asm.hb.v04c.a0005908.
9. Demidovich V.B. Computer Simulation and Optimal Designing of Energy-Saving Technologies of the Induction Heating of Metals. – Thermal Engineering, 2012, 59 (14), pp. 1023–1034, DOI:10.1134/S0040601512140030.
10. Mickey H.S., Fairbanks D.F. Mechanics of Heat Transfer to Fluidized Beds. – AIChE Journal, 1955, DOI: 10.1002/aic.690010317.
11. Basakov A.P. The Mechanism of Heat Transfer between a Fluidized Bed and Surface. – International Chemical Engineering, 1964, 4 (2), р. 320.
12. Cook C.A., Cundy V.A. Heat Transfer between a Rotating Cylinder and a Moist Granular Bed. – International Journal of Heat and Mass Transfer, 1995, 38 (3), рр. 419–432.
13. Armstrong L.M., Gu S., Luo K.H. Study of Wall-to-Bed Heat Transfer in a Bubbling Fluidised Bed Using the Kinetic Theory of Granular Flow. – International Journal of Heat and Mass Transfer, 2010, 53 (21-22), pp. 4949–4959, DOI:10.1016/j.ijheatmasstransfer.2010.05.047.
14. ANSYS Rocky DEM [Electron. resource], URL: https://www.esss.co/en/ansys-simulation-software/ansys-rocky (Date of appeal 12.04.2024).
15. Kloss C., Goniva C. LIGGGHTS: A New Open Source Dem Simulation Software. – 5th International Conference on Discrete Element Methods (DEM5), 2010.
16. Kozicki J., Donze F.V. YADE‐OPEN DEM: An Open‐Source Software Using a Discrete Element Method to Simulate Granular Material. – Engineering Computations, 2009, 26 (7), pp. 786–805, DOI:10.1108/02644400910985170.
17. Fonte C.B., Oliveira J.A., Almeida L.C. DEM-CFD Coupling: Mathematical Modelling and Case Studies Using ROCKY-DEM® and ANSYS Fluent®. –The 11th International Conference on CFD in the Minerals and Process Industries, CSIRO, 2015, рр. 7–9.
18. Kloss C. et al. Models, Algorithms and Validation for Opensource DEM and CFD–DEM. – Progress in Computational Fluid Dynamics, an International Journal, 2012, 12 (2-3), pp. 140–152, DOI:10.1504/PCFD.2012.047457.
19. Caulk R.A., Bruno C. An open Framework for the Simulation of Coupled Thermo-Hydro-Mechanical Processes in Discrete Element Systems. – 8th International Conference on Discrete Element Methods, 2019.
20. Weber N. et al. Numerical Simulation of the Tayler Instability in Liquid Metals. – New Journal of Physics, 2013, 15 (4), DOI:10.1088/1367-2630/15/4/043034.
21. Beckstein P., Galindo V., Vukčević V. Efficient Solution of 3D Electromagnetic Eddy-Current Problems within the Finite Volume Framework of OpenFOAM. – Journal of Computational Physics, 2017, 344 (11), рр. 623–646, DOI:10.1016/j.jcp.2017.05.005.
22. Busse C. et al. Numerical Modeling of an Inductively Coupled Plasma Torch Using OpenFOAM. – Computers & Fluids, 2021, vol. 216, p. 104807.
23. Khatsayuk M.Yu. et al. Numerical Simulation of Process of Electromagnetic Casting and Technology Features. – Metallurgical and Materials. Transaction B, 2023, 54 (4), pp. 1768–1783, DOI:10.1007/s11663-023-02791-8.
24. The Foam-Extend Project [Electron. resource], URL: http://www.foam-extend.org (Date of appeal 08.11.2023)
25. Jasak H. Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows, 1996.
26. Weller H.G. et al. A Tensorial Approach to Computational Continuum Mechanics Using Object-Oriented Techniques. – Computers in Physics, 1998, 12 (6), pp. 620–631, DOI:10.1063/1.168744.
27. OpenFOAM – Free Open Source CFD [Electron. resource], URL: http://www.openfoam.org (Date of appeal 12.04.2024).
28. Ai J. et al. Assessment of Rolling Resistance Models in Discrete Element Simulations. – Powder Technology, 2011, 206 (3), pp. 269–282, DOI:10.1016/j.powtec.2010.09.030.
29. Aruliah D.A. et al. A Method for the Forward Modelling of 3-D Electromagnetic Quasi-Static Problems. – Mathematical Models and Methods in Applied Sciences, 2001, 11 (1), DOI:10.1142/S0218202501000702.
30. Guermond J.L. et al. An Interior Penalty Galerkin Method for the MHD Equations in Heterogeneous Domains. – Journal of Computational Physics, 2007, 221 (1), pp. 349–369, DOI:10.1016/j.jcp.2006.06.045.
31. Malik M., Fan E.S.-C., Bussmann M. Adaptive VOF with Curvature‐Based Refinement. – International Journal for Numerical Methods in Fluids, 2007, 55(7), pp. 693–712, DOI:10.1002/fld.1490
---
The study was financially supported by the Grant of the Federal State Budgetary Institution Foundation for Assistance to Small Innovative Enterprises in Science and Technology (Foundation for Innovation Assistance), agreement No. 4125GS1/68579
Published
2024-05-30
Issue
Section
Article
