Statistical analysis of kinetics of an adiabatic thermal explosion

Abstract

Stochastic phenomena in the kinetics of chemical reactions induced by a thermal explosion are studied. The statistical analysis is performed on the basis of nonlinear differential equations for gas temperature and concentration of reacting particles with random initial conditions, and also corresponding equations for the probability density function of the state variables of the system. The results of the numerical solution of these equations are presented in the graphical form as a statistical set of an ensemble of kinetic curves of concentration of reacting particles and gas temperature and as the time evolution of their probability density function. Small random fluctuations of gas temperature and particle concentration at the beginning of the reaction are demonstrated to induce significant fluctuations of the induction period.Key words: thermal explosion, chemical reactions, gas temperature, concentrations of reagents, fluctuations, induction period, statistical analysis, numerical methods, nonlinear differential equations, kinetic curves, evolution of the probability density function. 1.Studying the processes that accompany the thermal explosion is extremely important for both basic research and applications. These investigations were intensely pursued and were described in monographs and reviews (e.g., [1][2][3][4][5]). The formulation of the problem about the thermal explosion in the classical theory implies determining the basic characteristics of the phenomenon with specified initial and boundary conditions. These basic characteristics are the critical conditions and the induction period (time of self-sustained acceleration of the reaction before the combustible mixture becomes ignited).One of the issues of particular interest in analyzing the process of thermal self-ignition is the adiabatic explosion, where the heat released in chemical reactions is spent on heating the combustible mixture. This problem was solved in [4], as applied to combustion of hydrocarbons, on the basis of differential equations for gas temperature and concentrations of reagents. The state variables describing the system were assumed to take their mean values, and fluctuations were neglected. It was noted [6], however, that calculations on the basis of deterministic equations cannot ensure high accuracy in terms of the correct description of all features of the thermal explosion, because of internal thermodynamic fluctuations, which are always inherent in any physicochemical system and are responsible for the complicated random behavior of this system [6]. One possible reason for these fluctuations is the fact that ignition always occurs at a certain point, which can be randomly localized inside the vessel, and then the process propagates over the entire volume [3].As subsequent research showed [6], the allowance for concentration fluctuations results not only in quantitative but also in qualitative changes in the system behavior, in particular, they can initiate random transitions from one state to another, etc.From the mathemat...