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Article information
2016 , Volume 21, ¹ 2, p.88-97
Shanin S.A., Knyazeva A.G.
On the numerical solution of non-isothermal multicomponent diffusion with variable coefficients
Purpose. The numerical implementation of the coupling models of multicomponent diffusion requires the development of special numerical algorithms which take into account the interrelation between several physical phenomena with different temporal and spatial scales. Traditional algorithms for numerical implementation for similar models are unstable and lead to the negative concentrations. The main object of this work is to develop an algorithm for solving the equation system of multicomponent diffusion with coefficients that could change sign thus leading to the ill-posed problem. Methodology. The basic idea of the suggested algorithm consists in the determination of critical concentration coinciding which the solubility limit. Mathematically, the concentration equaling to critical one leads to the uncertainty of the type “zero to infinity”. Using the L’Hospital rule we obtain the diffusion equation, which correctly describes the diffusion in the vicinity of solubility limit and eliminate the appearance of negative values for concentrations. Findings. As a result, the developed algorithm for the numerical investigation of the problem was implemented numerically. It was shown that the separation of the feature leads to the replacement of the parabolic diffusion equation with the equation of the third order that is correct in a small vicinity of a critical concentration. The examples of numerical solution are presented for the problem of the evolution of coating composition during deposition.
[full text]
Keywords: multicomponent diffusion , cross effects, associated model
Author(s):
Shanin Sergey Alexandrovish
PhD.
Position: Senior Fellow
Office: Tomsk Polytechnic University
Address: 634050, Russia, Tomsk, Lenin Avenue, 30
Phone Office: (3822)421480
E-mail: shanin_s@mail.ru
Knyazeva Anna Georgievna
Dr. , Professor
Position: Head of Laboratory
Office: Tomsk Polytechnic University
Address: 634050, Russia, Tomsk, Lenin Avenue, 30
Phone Office: (3822)701777
E-mail: anna@ispms.tsc.ru
References:
[1] Wangard, W., Dandy, D.S., Muller, J. A numerically stable method for integration of the multicomponent species Diffusion equations. Journal of Computational physics. 2001; (174):460-472.
[2] Voroshnin, L.G., Vityaz', P.A., Nasybulin, A.Kh., Khusid, B.M. Mnogokomponentnaya diffuziya v geterogennykh splavakh [Multicomponent diffusion in heterogeneous alloys ]. Minsk: Vysheyshaya shkola; 1984: 144. (In Russ.)
[3] Errst Kozeschnik Multicomponent Diffusion simulation bused on finite elements . Metallurgical and materials transactions A. 1999; (30A):2576-2582.
[4] Ansgar Jungel and Viktoria Stelzer Existence analysis of Maxwell-Stefan systems for multicomponent mixtures. Available at: http://arxiv.org/abs/1211.2394V1.
[5] Vincent Giovangigli Mass conservation and singular multicomponent Diffusion algorithms. Impact of computing in science and engineering. 1990; (2): 73-97.
[6] Hywel Rhys Thomas, Majid Jedighi, Philip Janus Vardon Diffusive reactive transport of multicomponent chemicals under coupled thermal, hydraulic, chemical and mechanical conditions. Geotechnical & Geological Engineering. 2012; (30):841-857.
[7] Simon Emmanuel, Andrea Cortis, Brian Berkowitz Diffusion in multicomponent systems: a free energy aproach . Chemical physics. 2004; (302): 21-30.
[8] Tikhonov, A.N., Arsenin, V.Ya. Metody resheniya nekorrektnykh zadach [Methods of solving incorrect problems]. Moscow:Nauka; 1979: 288. (In Russ.)
[9] Volpert, A.I., Posvyanskiy, V.S. On the positive solution of multi-component diffusion and chemical kinetics. Chemical physics. 1984; 3(8):1200-1205.
[10] Stark, J.P. Solid state diffusion. New York - London: Willey - Interscience Publ., 1976.
[11] Haase, R. Thermodynamics of irreversible processes. Massachussets: Addison-Wesley ; 1969.
[12] Eremeev, V.S. Diffuziya i napryazheniya [Diffusion and strain]. Ìoscow: Energoatomizdat; 1984: 240. (In Russ.)
[13] Geguzin, Ya.E. Diffuzionnaya zona [The diffusion zone]. Ìoscow: Nauka; 1979:343. (In Russ.)
[14] Knyazeva, A.G. Modes of development from the initial nucleus of solid-phase reaction limited diffusion. Combustion, Explosion and Shock Waves. 1996; 32(4):72–76. (In Russ.)
[15] Samarskiy, A.A., Nikolayev, E.S. Metody resheniya setochnykh uravneniy [Methods for solving grid equations]. Ìoscow:Nauka; 1978: 591. (In Russ.)
[16] Knyazeva, A.G., Shanin, S.A. Model of the coating growth under the conditions of magnetron sputtering deposition. Russian physics journal. 2010; (1):76-81. (In Russ.)
[17] Shanin, S.A., Knyazeva, A.G., Pobol, I.L., Denizhenko, A.G. Numerical and an experimental research of influence of technological parametres on phase and a chemical compound êàðáèäíîãî a coat growing in pulsing electroarc plasma. Chemical Physics and mezoskopiya. 2012; 14(4):525–535. (In Russ.)
Bibliography link:
Shanin S.A., Knyazeva A.G. On the numerical solution of non-isothermal multicomponent diffusion with variable coefficients // Computational technologies. 2016. V. 21. ¹ 2. P. 88-97
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