User:Oleg A. Mornev/Proposed/Zeldovich-Frank-Kamenetskii model

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Zeldovich – Frank-Kamenetskii Model in the Combustion Theory

The model mentioned (below it will be referred to as ZF model) has been suggested in 1937 for the mathematical description of combustion processes. ZF model relates to the class of nonlinear reaction diffusion models. Actually, it was one of the first models of the class mentioned, which gave non-trivial results of great importance: in particular, this model enabled to derive the general analytical expression for the velocity of stationary propagating plane flame front Before considering ZF model and describing the results of its authors, it is helpful to remind briefly some physical-chemical facts connected with its formulation. Generally, the combustion is realized as a non-equilibrium exothermal chemical reaction (i.e. the reaction followed by calorification), whose speed W nonlinearly depends on the temperature T. The qualitative view of this dependence is presented on the Fig. 1. The curve W(T) displays three specific branches: the branch 1, where the growth of temperature T leads only to minor monotonous increase of the speed W; and the branches 2 and 3, where small variations of T induce strong variations of W. The branch 1 corresponds to a slow spontaneous stage of the exothermal reaction; at this stage the quantity of heat evolved during reaction, is insignificant, and a flame is absent. On the contrary, the branch 2 (and partly the branch 3 near maximum point Tm on the Fig. 1) corresponds to the burning stage of the reaction, at which the reagents react violently, emitting a heat and a glow; the glow is perceived by an observer as a flame. For the space of the branches 2 and 3 the graph of the function W(T) has a bell-shaped form. The left part of this “bell”, the branch 2 that rises sharply over the range (Ti, Tm) (see Fig. 1), is responsible for the heat self-accelerating the exothermal reaction during the burning process. Indeed, if the temperature of some local site of the combustible medium takes on some value T1 belonged to the range (Ti, Tm), then violent calorification of this site raises its temperature and accelerates the reaction; as a result, the additional calorification occurs, leading to further increase in temperature. The speed of reaction culminates at the temperature Tm corresponding to the peak of the “bell” (see Fig. 1) and then decreases. This diminution of the value of W (it corresponds to the right part of the “bell”, the branch 3 on the Fig. 1) is caused by depletion of the reagents. At the temperature Ta on the Fig. 1 the reagents are entirely exhausted and the reaction comes to a stop. Aforesaid also elucidates the physical mechanism supporting the flame propagation: the heat evolved in the local burning site of combustible medium is transmitted to more cold sites located ahead of the flame front, raises their temperatures to the values corresponding to the branch 2 on the Fig. 1, and thereby sharply accelerates the exothermal chemical reactions in these sites, inducing a transference of the flame front along the medium. The processes considered above are influenced by different factors, among which one can mention the transport of reagents and combustion materials from the flame front to other regions of combustible medium. In itself this transport can be supported both by the passive diffusion and by the convection processes including turbulence, and the latter can also involve in the heat transfer. As it evidently follows from the latter notes, the theory of combustion phenomena and flame propagation must be very intricate from the mathematical point of view. It is a merit of Yakov Borisovich Zeldo-vich and David Albertovich Frank-Kamenetskii that they could distinguish a situation, in which a develop-ment of such theory proved to be possible; then they had constructed it. Our authors consider burning process and flame propagation in a gas phase in the situation, when all reagents involving in the exothermal reaction have close molecular weights. Besides, they neglect the convec-tion processes and the turbulence, i.e. they deal with the laminar combustion. In this situation the heat and mass transfer from the combustion zone is realized by the passive diffusion; hence, the full description of the phenomenon will be given by a set of reaction diffusion equations added with the heat conduction equation with a thermal source in its second part. This set of equations is still immense. The key step for reducing the set consists in the next consideration: if the molecular weights of all reagents are close to each other then their diffusivities are close too. (...)

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