Comparison of methods for solving the system of ordinary differential equations of chemical kinetics equations

Аuthors

Galayko N. M.^1*, Severina N. S.^2**

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. ,

*e-mail: ninagalayko@gmail.com
**e-mail: severina@mai.ru

Abstract

This paper considers questions, related to comparison of different methods for solving rigid systems of differential equations, which describe distribution of a detonation wave in a flat channel of constant cross section. This assignment compares the results obtained by several methods, e.g. Euler, Gear, Pirumov.

Thermodynamic method for studying chemical reactions suggests the theoretical possibility of the testing processes under different conditions. If pressure and temperature are constant, then spontaneous process flow is possible only in the direction of reducing the Gibbs energy. However, this condition does not allow to estimate the speed of the transition from the initial to the final.

Chemical reactions can take place at different speeds — from slow to very explosive. In some cases it is necessary to increase the reaction rate, in other contrary, reduce. Problems of chemical kinetics is the study of the rate of chemical processes. It is necessary to determine velocity dependence on various parameters for the rational chemical reactions.

Modern computer mathematics systems allow to give a quick and clear forecast of the chemical behavior of the system over time. Systems of equations of chemical kinetics contain differential equations, the main feature of which is low sensitivity of the solution in the presence of rapidly decaying perturbation. (Rigid system of differential equations). There are various ways of implementing the software solution of a mathematical model chemical kinetics processes.

This assignment compares three methods for solving differential equations, which describe the chemical interaction of the components of the gas mixture in the problem of the distribution of the stationary detonation waves in a flat channel of constant cross section, filled with stationary stoichiometric hydrogen-air mixture.

This assignment solves system of equations describing the one-dimensional inviscid steady flow of multicomponent reactive gas mixture behind the shock wave propagating with constant velocity D. To close the system uses thermal and caloric equation of state of the combustible mixture, which is a mixture of perfect gases.

According to comparison, Pirumov’s method requires the smallest number of steps for all velocities behind the shock wave. All calculation methods showed fairly similar results at speeds approximately equal speed Chapman Jouget, but Pirumov’s method gave the solution with the least amount of steps. Comparison also showed that Gere’s method gives a solution quickly than Euler’s method at all speeds.

Keywords:

rigid system of differential equations, numerical methods, equations of chemical kinetics, detonation waves

References

1. Dmitry M. Davidenko, Iskender Gökalp, Emmanuel Dufour, Philippe Magre. Systematic Numerical Study of the Supersonic Combustion in an Experimental Combustion Chamber // 14th AIAA/AHI Int. Space Planes and Hypersonic Systems and Technologies Conference, AIAA Paper 2006-7913.

2. Maas U., Warnatz J. Ignition processes in hydrogen-oxygen mixtures // Combust. and Flame. 1988. V.74. № 1. P.53-69.

3. Kamzolov V. N., Pirumov U. G. Izvestiya AN SSSR. Mekhanika zhidkosti i gaza, 1966, № 6, pp. 25-33.

4. Mudrov A. E. Chislennye metody dlya PEVM na yazykakh Beisik, Fortran i Paskal’ (Numerical methods for PC in the languages BASIC, FORTRAN and Pascal), Tomsk, Rasko, 1991, 271p.

5. Gurvich L.V., Vejc I.V., Medvedev V.A. Termodinamicheskie svoistva individual’nykh veshchestv (Thermodynamic properties of individual substances), Moscow, Nauka, 1979, 344 p.

6. Gidaspov V.Yu.,Severina N.S. Elementarnye modeli i vychislitel’nye algoritmy fizicheskoi gazovoi dinamiki. Termodinamika i khimicheskaya kinetika (Basic models and computational algorithms of physical gas dynamics. Thermodynamics and chemical kinetics), Moscow, Faktorial Press, 2014, 84 p.

7. Ibragimova L.B., Smekhov G.D., Shatalov O.P. Fiziko-khimicheskaya kinetika v gazovoi dinamike, 2009, Vol.8, available at: http://chemphys.edu.ru/issues/2009-8/articles/204/

8. Pirumov U.G. Vychislitel’nye aspekty resheniya zadach okhrany okruzhayushchei sredy (Computational aspects of solving environmental), Moscow, MAI, 1988, 68 p.

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