Method for determining particle growth dynamics in a two-component alloy
https://doi.org/10.17073/0368-0797-2020-2-135-139
Abstract
The paper deals with issue of particle growth in a two-component alloy. The particle is formed from the products of chemical reactions that occur at the phase boundary. Generalized mathematical model of particle growth includes diffusion equations, mass transfer equations in boundary layer, and equation characterizing change in radius of the growing particle. The paper proposes an approach that allows reduction of the initial issue to system of PDEs and ODE that describes the state of growing particle. This approach provides basis for developing numerical method for calculating radius of growing particle as a function of time, based on the obtained equations. The computational scheme involves the finite-difference analogues of equations with an additional regularizing functional that ensure stability of the method with respect to accumulated computational error. In order to verify reliability of the proposed computational scheme and to obtain experimental error estimates of numerical solutions, computational experiments were carried out. In the experiments, radius of growing particle is determined with respect to the time via the proposed method. Also, comparative analysis of the calculated radius with test values was carried out and experimental estimates of deviations of the calculated radius from the test functions were obtained. The results of the experiment presented in the work indicate sufficient accuracy of the developed numerical method.
Keywords
multicomponent alloy,
particle growth,
new phase formation,
diffusion equation,
mass transfer,
numerical method,
regularization,
error estimate
Supporting agencies: The work was financially supported by the Ministry of Science and Higher Education of the Russian Federation as part of the basic part of the state task “Development, research and implementation of algorithms for processing data of dynamic measurements of spatially distributed objects”, TDA 8.9692.2017 / 8.9 of 02.17.2017.
About the Author
N. M. Yaparova
South Ural State University (NRU)
Russian Federation
Russian Federation
Cand. Sci. (Phys.–Math.), Assist. Professor, Head of the Chair of Computational Mathematics and High-Performance Computing
Chelyabinsk
References
1. Kolmogorov A.N. To statistical theory of metals crystallization. Izvestiya AN SSSR. Seriya Matematicheskaya. 1937, vol. 1, no. 3, pp. 355–359. (In Russ.).
2. Avrami M. Granulation, phase change, and microstructure kinetics of phase change III. Journal of Chemical Physics. 1941, vol. 9, no. 2, pp. 177–184.
3. Volmer Max. Kinetics of Phase Formation: (Kinetik Der Phasenbildung). Central Air Documents, 1966, 219 p. (Russ. ed.: Volmer M. Kinetika obrazovaniya novoi fazy. Мoscow: Nauka, 1986, 208 p.).
4. Shneidman V.A., Goldstein E.V. Nucleation time lag at nanosizes. Journal of Non-Crystalline Solids. 2005, vol. 351, no. 8, pp. 1512–1521.
5. Wang D., Liu Y., Han Y., Zhang Y., Gao Z. Kinetic consideration for the incubation of the phase transformation and its application to the crystallization of amorphous alloy. Applied Physics A. 2008, vol. 92, no. 3, pp. 703–707.
6. Drozin A.D., Gamov P.A., Dudorov M.V., Roshchin V.E. Model for nanocrystal growth in an amorphous alloy. Russian Metallurgy (Metally). 2012, vol. 2012, no. 11, pp. 1002–1005.
7. Drozin A.D., Yaparova N.M. Probabilistic-statistical testing method for the techniques of metallographic determination of the amount of non-metallic inclusions in metal. Chernye Metally. 2018, no. 8, pp. 19–22.
8. Lipiński T., Wach A. Size of non-metallic inclusions in high-grade medium carbon steel. Archives of Foundry Engineering. 2012, vol. 14, no. 4, pp. 55–60.
9. Belyaev I.V., Grigorovich K.V., Kol’chugina N.B., Shibaev S.S. Effect of the purity of starting materials on the structure and properties of permanent magnets. Inorganic Materials. 2010, vol. 46, no. 3, pp. 291–294.
10. Sadovskii V.M. Equations of the dynamics of a liquid crystal under the influence of weak mechanical and thermal perturbations. AIP Conference Proceedings. 2014, vol. 1629, pp. 311–318.
11. 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.
12. Izmail-Zadeh A.T., Korotkii A.I., Naimark B.M., Tsepelev I.A. Three-dimensional numerical simulation of the inverse problem of thermal convection. Computational Mathematics and Math. Physics. 2003, vol. 43, no. 4, pp. 587–599.
13. Prokudina L.A. Nonlinear development of the marangoni instability in liquid films. Journal of Engineering Physics and Thermophysics. 2016, vol. 89, no. 4, pp. 921–928.
14. Tikhonov A.N., Goncharskii A.V., Stepanov V.V., Yagola A.G. Chislennye metody resheniya nekorrektnykh zadach [Numerical methods for solving ill-posed problems]. Мoscow: Nauka, 1990, 232 p. (In Russ.).
15. Vabishchevich P.N., Samarskii A.A. Finite-difference schemes for convection-diffusion problems on irregular meshes. Computational Mathematics and Mathematical Physics. 2000, vol. 40, no. 5, pp. 692–704.
16. Fra̧ckowiak A., Botkin N.D., Ciałkowski M., Hoffmann K.-H. A fitting algorithm for solving inverse problems of heat conduction. International Journal of Heat and Mass Transfer. 2010, vol. 53, no. 9-10, pp. 2123–2127.
17. Yaparova N.M., Shestakov A.L. Method for temperature measuring inside a cylindrical body based on surface measurements. In: 14 th IMEKO TC10 Workshop on Technical Diagnostics 2016: New Perspectives in Measurements, Tools and Techniques for Systems Reliability, Maintainability and Safety. 2016, pp. 8–12.
18. Vasil’ev V.V., Vasilyeva M.V., Kardashevsky A.M. The numerical solution of the boundary inverse problem for a parabolic equation. AIP Conference Proceeding. 2016, vol. 1773, no. 1, article 100010.
19. Lukyanenko D.V., Shishlenin M.A., Volkov V.T. Solving of the coefficient inverse problems for a nonlinear singularly perturbed reaction-diffusion-advection equation with the final time data. Communications in Nonlinear Science and Numerical Simulation. 2018, vol. 54, pp. 1339–1351.
20. Samarskii A.A. Vvedenie v teoriyu raznostnykh skhem [Introduction to difference theory]. Мoscow: Nauka, 1971, 553 p. (In Russ.).
Review
For citations:
Yaparova N.M. Method for determining particle growth dynamics in a two-component alloy. Izvestiya. Ferrous Metallurgy. 2020;63(2):135-139. (In Russ.) https://doi.org/10.17073/0368-0797-2020-2-135-139
Views:
584
Email this article 