Mathematical foundations of fractal heat and mass transfer in the two-phase zone of the metal melt

Authors

  • T. Selivyorstova
  • V. Selivyorstov
  • L. Yvanova

Keywords:

heat, mass, diffusion, fractal, model, melt, solidification, two-phase zone

Abstract

The problem of improving progressive and creating new technologies in metallurgy and foundry production is relevant for obtaining high-quality cast metal and castings. The microscopic and macroscopic properties of casting templates significantly depend on the thermophysical parameters of the casting system and the casting mold, namely, the width of the two-phase zone of melt solidification, the initial temperature of the melt, the cooling rate of the casting, the cooling gradient of the melt, and the temperature on the surface of the casting mold. In order to obtain a fine-grained metal structure. The article presents the results of experimental studies, indicating the fractal nature of structure formation in a two-phase zone of the solidifying metal melt. The thermodynamic statement of the non-stationary problem solidifying of binary systems is considered. Transfer equations are described that are adequate for media with fractal geometry. The mathematical apparatus for describing the curing process from the standpoint of heat and mass transfer in a two-phase zone and diffusion in fractal media is presented. It is shown that the mathematical apparatus of fractional calculation makes it possible to effectively describe the fractal nature of diffuse processes. The analysis of the thermal and mass transfer processes in the melt of the metal, which is in the rare state, and their description using the mathematical apparatus of fractional calculation, have been carried out.

References

Sokolovskaya, L. (2013). On the choice of rational thermal regimes for casting steel ingots with shot. Metal and casting of Ukraine, 9(244), 32–35.

Bohacek, J., Kharicha, A., Ludwig, A., Wu, M., & Karimi-Sibaki, E. (2018). Heat transfer coefficient at cast-mold interface during centrifugal casting: Calculation of Air Gap. Metallurgical and Materials Transactions B, 49(3), 1421–1433. https://doi.org/10.1007/s11663-018-1220-0

Dotsenko, Y., Dotsenko, N., Tkachyna, Y., Fedorenko, V., & Tsybulskyi, Y. (2018). Operation optimization of holding furnaces in special casting shops. Technology Audit and Production Reserves, 6(1(44), 18–22. https://doi.org/10.15587/2312-8372.2018.150585

Selivorstov, V.Iu., Dotsenko, Yu.V., Khrychykov, V.Ie., Nosko, O.A., & Kutsova, V.Z. (2010). Osoblyvosti hazodynamichnoho vplyvu na strukturoutvorennia lytoi instrumentalnoi shtampovoi stali. Visnyk Natsionalnoho tekhnichnoho universytetu «KhPI». Seriia: Novi rishennia u suchasnykh tekhnolohiiakh, 1(57), 59–67. Vylucheno iz http://vestnik2079-5459.khpi.edu.ua/article/view/46636

Ivanova, V. S., Balankin, A. S., Bunin, I. Zh., & Oksogoev, A. A. (1994). Synergetics and fractals in materials science. M.: Science.

Hills, R. N., Loper, D. E., & Roberts, P. H. (1983). A thermodynamically consistent model of a mushy zone. The Quarterly Journal of Mechanics and Applied Mathematics, 36(4), 505–540. https://doi.org/10.1093/qjmam/36.4.505

Furmański, P. (2000). Modeling of transport phenomena during solidification of binary systems. Computer Assisted Mechanics and Engineering Sciences, 7(3), 391-402.

Stefanescu, D. M. (n.d.). Numerical macro-modeling of solidification. Science and Engineering of Casting Solidification, Second Edition, 1–30. https://doi.org/10.1007/978-0-387-74612-8_6

Worster, M. G. (1986). Solidification of an alloy from a cooled boundary. Journal of Fluid Mechanics, 167(-1), 481. https://doi.org/10.1017/s0022112086002938

Alexandrov, D. V., Nizovtseva, I. G., Malygin, A. P., Huang, H.-N., & Lee, D. (2008). Unidirectional solidification of binary melts from a cooled boundary: Analytical solutions of a nonlinear diffusion-limited problem. Journal of Physics: Condensed Matter, 20(11), 114105. https://doi.org/10.1088/0953-8984/20/11/114105

Alexandrov, D. V., Alexandrova, I. V., & Ivanov, A. A. (2017). The role of nonlinear mass transport on directional crystallization with a mushy layer. AIP Conference Proceedings. https://doi.org/10.1063/1.5012481

Stefanescu, D. M., & Ruxanda, R. (2004). Fundamentals of Solidification. Metallography and Microstructures, 71–92. https://doi.org/10.31399/asm.hb.v09.a0003724

Stefanescu, D. M., & Ruxanda, R. (2004). Solidification structures of steels and cast irons. Metallography and Microstructures, 97–106. https://doi.org/10.31399/asm.hb.v09.a0003725

Nakhushev, A.M. (2003). Fractional calculus and its application. Moscow: Fizmatlit.

Beybalayev, V. D. (2010). Mathematical model of heat transfer in fractal structure mediums. Mathematical Models and Computer Simulations, 2(1), 91–97. https://doi.org/10.1134/s2070048210010096

Kovalenko, A. N., Zarichnyak, Y. P., Ivanov, V. A., & Bolshev, K. N. (2021). Fractal characterization of heat and mass transfer processes in nanostructured materials. Paper presented at the Journal of Physics: Conference Series, , 2096(1) doi:10.1088/1742-6596/2096/1/012168

Published

2022-04-08