Об одном методе решения задачи кристаллизации многокомпонентного раствора в цилиндрической ампуле

19 / 21

О.В. Щерица, А.О. Гусев, О.С. Мажорова
136 ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 5
Abstract Keywords
The paper presents a self-consistent model of multicompo-
nent alloy crystallization in a cylindrical ampule. The
mathematical model accounts for the heat and matter
transfer in both solid and liquid phases. We described the
system by the interface position and radially average tem-
perature and concentrations. Special efforts are required to
solve a corresponding one dimensional phase transition
problem in multicomponent alloy. To handle evolution of
solid/liquid interface, the moving boundary problem is
mapped to a new coordinate system. We obtained a con-
servative and implicit finite difference scheme in a new
coordinate system with control volume technique and
constructed a fully implicit coupled approach. Further-
more, we solved a corresponding set of nonlinear equations
by Newton method for the unknown vector, whose com-
ponents are concentrations of all species, interface rate and
temperature. The proposed method was used for numerical
simulation of the crystallization process of А–В–С solution
Stefan problem, phase transition,
mathematical simulation
Received 24.10.2016
© BMSTU, 2017
REFERENCES
[1] Samarskii A.A., Vabishchevich P.N. Vychislitel'naya teploperedacha [Computational heat
transfer]. Moscow, URSS Publ., 2003. 784 p.
[2] Samarskii A.A., Vabishchevich P.N., Iliev O.P., Churbanov A.G. Numerical simulation of
convection/diffusion phase change problems — a review. Journal of Heat Mass Transfer, 1993,
vol. 36, no. 17, pp. 4095–4106.
[3] Muray W.D., Landis F. Numerical and machine solutions of the transient heat conduction
problems involving melting or freezing. Journal of Heat Transfer, 1959, vol. 81, pp. 106–112.
[4] Vermolen F.J., Vuik C. A mathematical model for the dissolution of particles in multicom-
ponent alloys. J. of Computational and Applied Math., 2000, vol. 126, no. 1-2, pp. 233–254.
DOI: 10.1016/S0377-0427(99)00355-6
Available at:
[5] Mazhorova O.S., Popov Yu.P., Shcheritsa O.V. An algorithm for solving a phase transition
problem in a multicomponent system. Differential Equations, 2004, vol. 40, no. 7, pp. 1051–
1059. DOI: 10.1023/B
Available at:
[6] Mazhorova O.S., Popov Yu.P., Shcheritsa O.V. Conservative scheme for the thermodiffu-
sion Stefan problem. Differential Equations, 2013, vol. 49, no. 7, pp. 869–882.
DOI: 10.1134/S0012266113070094
Available at:
[7] Illingworth T.C., Golosnoy I.O. Numerical solutions of diffusion-controled moving
boundary problems which conserve solute. Journal of Computation Physics, 2005, vol. 209,
no. 1, pp. 207–225. DOI: 10.1016/j.jcp.2005.02.031
Available at: