State National Research Polytechnical University of Perm, Perm, Russia:

+A. I. Tsaplin
V. N. Nechaev, Researcher (Chair of General Physics), e-mail: nechvladimir@mail.ru

This paper shows the mathematical model, describing the dynamics of nonequilibrium heat and mass transfer in technological process of magnesiumthermic reduction of titanium from TiCl₄. The process is very power-consuming: more than 5 MW·hour of electric power is required for one titanium sponge ton obtaining. The rated operating conditions of reactor are shown in axially symmetric approximation. The boundary problem definition includes the equations with “temperature-vorticity-current function” variables for the area, initially placed by magnesium smelt. The model, realized by the finite-difference method on regular mesokurtic, takes into account the exothermal reaction heat, steam formation in the reactor and heat effects on its boundary. Investigation of influence of TiCl₄ mass and consumption and delay between its single portions introduction on thermal and hydrodynamic situation in the reactor volume carried out the multi-parametric calculations on the basis of the designed software complex of computer realization of mathematical model. These calculations provide the lowest consumption of the reactor-supplying energy. There are shown the results of numerical modeling of non-isothermal flow in the reactor volume with introduction of a single portion of TiCl₄. Formation of multi-vortical structure makes an influence on the transfer of reaction products into the lower part of reactor.
This work was carried out with the support of the Ministry of Education and Science of the Russian Federation within the basic part of the state task, project No. 1599 “Mathematical modeling of heat and mass transfer in melt and porous medium during the electromagnetic impacts”, as well as the grant of the Russian Foundation for Basic Research (project No. 13-08-96004-р_урал_а).

1. Zabelin I. V. 50 let po puti nauchnogo progressa (50 years on the way of scientific progress). Tsvetnye Metally = Non-ferrous metals. 2006. No. 8. pp. 90–93.
2. Parfenov O. G., Pashkov G. L. Problemy sovremennoy metallurgii titana (Problems of modern metallurgy of titanium). Novosibirsk : Publishing House of Siberian Branch of Russian Academy of Sciences, 2008. 279 p.
3. Nechaev V. N., Tsaplin A. I. Obzor sposobov polucheniya gubchatogo titana (An overview of methods for titanium sponge production). Titan = Journal “Titan”. 2015. No. 3 (49). pp. 4–13.
4. Eydenzon M. A. Magniy (Magnesium). Moscow : Metallurgiya, 1969. 352 p.
5. Tankeev A. B., Rymkevich D. A., Putina O. A., Putin A. A., Nechaev V. N. Intensifikatsiya tekhnologicheskogo rezhima protsessa vosstanovleniya v apparate proizvoditelnostyu 4.8–5 tonn gubchatogo titana za tsikl (Intensification of technological mode of reduction process in the unit with productivity of 4.8–5 t of titanium sponge for the cycle). Titan = Journal “Titan”. 2007. No. 1 (20). pp. 3–8.
6. Tsaplin A. I., Nechaev V. N. Chislennoe modelirovanie neravnovesnykh protsessov teplomassoperenosa v reaktore dlya polucheniya poristogo titana (Numerical modeling of non-equilibrium heat and mass transfer processes in a reactor for the production of porous titanium). Vychislitelnaya mekhanika sploshnykh sred = Computational Continuum Mechanics. 2013. Vol. 6, No. 4. pp. 483–490. DOI: 10.7242/1999-6691/2013.6.4.53

7. Kholpanov L. P., Prokudina L. A. Mathematical Modeling of Unstable Mass Transfer Complicated by Chemical Reactions. Theoretical Foundations of Chemical Engineering. 2005. Vol. 39, No. 1. pp. 36–46.
8. Kholpanov L. P., Polyakov Yu. S. Mathematical Modeling of Turbulent Heat and Mass Transfer with Chemical Conversions. Theoretical Foundations of Chemical Engineering. 2006. Vol. 40, No. 5. pp. 454–464. DOI: 10.1134/S0040579506050022.
9. Permikin D. V., Zverev V. S. Mathematical model on surface reaction diffusion in the presence of front chemical reaction. International journal of heat and Mass Transfer. 2013. Vol. 57, No. 1. pp. 215–221.
10. Deka R. K., Paul A. Magnetic Field Effects on Unsteady Convectively Driven Flow Past an Infinite Vertical Cylinder in the Presence of Chemical Reaction. Magnetohydrodynamics. 2014. Vol. 50, No. 1. pp. 45–58.
11. Nekrasov A. K., Nekrasova E. I., Kholpanov L. P. Mathematical Simulation of Dispersed Phase Dynamics for the Nonisothermal Free Convection of a Heterogeneous Medium in a Vertical Cylindrical Reactor. Theoretical Foundations of Chemical Engineering. 2008. Vol. 42, No. 2. pp. 142–149.
12. Svetlov S. A., Spiridonov F. F., Alekseenko S. A. Modeling of the liquid flow in the upstream portion of a cylindrical channel. Theoretical Foundations of Chemical Engineering. 2005. Vol. 39, No. 1. pp. 57–61.
13. Belkacem Ould Said, Noureddine Retiel, El Hadi Bouguerra. Numerical Simulation of Natural Convection in a Vertical Conical Cylinder Partially Annular Space. American Journal of Energy Research. 2014. Vol. 2, No. 2. pp. 24–29.
14. N. Ameer Ahamad, Hasan Ahmed Mohamed Hassan El Arabawy, Syed Iqbal Ahmed. Visualization of Natural Convection in a Vertical Annular Cylinder with a Partially Heat Source and Varying wall Temperature. International Journal of Engineering Research and Applications. 2014. Vol. 4, No 9. pp. 185–197.
15. Sheremet M. A. Matematicheskoe modelirovanie nestatsionarnoy sopryazhennoy termogravitatsionnoy konvektsii v zamknutom naklonnom tsilindre (Mathematical simulation of unsteady natural convection in an inclined cylinder). Vestnik Nizhegorodskogo universiteta imeni N. I. Lobachevskogo = Vestnik of Lobachevsky University of Nizhni Novgorod. 2011. No. 4 (3). pp. 1272–1274.
16. Sheremet M. A. Prostranstvennye rezhimy sopryazhennoy estestvennoy konvektsii v vertikalnom tsilindre v usloviyakh teploobmena s vneshney sredoy (3D conjugate natural convection in a vertical cylinder on the assumption of heat transfer to the environment). Vychislitelnaya mekhanika sploshnykh sred = Computational Continuum Mechanics. 2010. Vol. 3, No. 4. pp. 112–123.
17. Khoroshev A. S., Shakhov V. G. Modelirovanie laminarnogo svobodnokonvektivnogo techeniya v dlinnom vertikalnom tsilindre (Simulation of laminar buoyancy induced flow in long vertical cylinder). Izvestiya Samarskogo nauchnogo tsentra Rossiyskoy akademii nauk = Proceedings of the Samara Scientific Center of the Russian Academy of Sciences. 2011. Vol. 13, No. 4. pp. 72–76.
18. Kirpo M., Jakovis A., Baake E., Nacke B. Analysis of experimental and simulation data for the liquid metal flow in a cylindrical vessel. Magnetohydrodynamics. 2007. Vol. 43, No. 2. pp. 161–172.
19. Abricka M., Gelfgat Yu., Krumiš J. The influence of combined electromagnetic fields on the heat and mass transfer in a cylindrical vessel with the melt. Magnetohydrodynamics. 2007. Vol. 43, No. 2. pp. 173–182.
20. Denisov S., Dolgikh V., Kolesnichenko I., Khalilov R., Khripchenko S., Verhille G., Plihon N., Pinton J.-F. Flow of liquid metal in a cylindrical crystallizer generating two-directional mhd-stirring. Magnetohydrodynamics. 2010. Vol. 46, No. 1. pp. 69–78.
21. Nekrasova E. I., Kholpanov L. P., Nekrasov A. K. Matematicheskoe modelirovanie mekhaniki mnogofaznykh sred pri mezhfaznom vzaimodeystvii (Mathematical modeling of multiphase medium mechanics during the interphase connections). Sovremennaya nauka : sbornik nauchnykh statey (Modern science : collection of scientific proceedings). 2010. No. 2 (4). pp. 159–164.
22. Tarasov A. V. Metallurgiya titana (Titanium metallurgy). Moscow : Akademkniga, 2003. 328 p.
23. Babichev A. P., Babushkina N. A., Bratkovskiy A. M. et al. Fizicheskie velichiny : spravochnik (Physical magnitudes: reference book). Under the editorship of I. S. Grigorev, E. Z. Meylikhov. Moscow : Energoatomizdat, 1991. 1232 p.
24. Yatsenko A. P., Petrunko A. N., Shcherban R. A. Osobennosti razdeleniya magniya i khlorida magniya v protsesse magnietermicheskogo vosstanovleniya tetrakhlorida titana (Peculiarities of decomposition of magnesium and magnesium chloride in the process of magnesium thermal reduction of titanium tetrachloride). Sbornik trudov Mezhdunarodnoy konferentsii “Ti–2008 v SNG”. Sankt-Peterburg, 18–21 maya 2008 (Collection of proceedings of International conference “Ti–2008 in the CIS countries”. Saint Petersburg, May 18–21, 2008). Kiev : G. V. Kurdyu mov Institute for Metal Physics of the N. A. S. of Ukraine, 2008. pp. 104–112.