Chemical kinetics, critical case equations

Belles prediction of the limits of detonability takes the following course. He deals with the hydrogen-oxygen case. Initially, the chemical kinetic conditions for branched-chain explosion in this system are defined in terms of the temperature, pressure, and mixture composition. The standard shock wave equations are used to express, for a given mixture, the temperature and pressure of the shocked gas before reaction is established (condition 1 ). The shock Mach number (M) is determined from the detonation velocity. These results are then combined with the explosion condition in terms of M and the mixture composition in order to specify the critical shock strengths for explosion. The mixtures are then examined to determine whether they can support the shock strength necessary for explosion. Some cannot, and these define the limit. 

One of the main conditions in the theorem on the passage to the limit is the condition for the existence of an isolated root z =

singularly perturbed equations, and particularly in most problems of chemical kinetics, this condition is not satisfied because the reduced equation does not have an isolated root. Instead, it has a family of solutions depending on one or several parameters. This case will be called the critical case. 

The kinetics of the thermal decomposition of solids are reviewed, with emphasis on topological considerations. The general model of nucleation in the bulk of the reactant is explored in detail and the kinetic equations appropriate to this model are derived. It is pointed out that a multistage nucleation process leads to a power law whenever the characteristic time for nucleus formation is long compared with the observation time, and that the assumption of equal rate constants for successive steps is unnecessarily restrictive. The problem of the induction period is examined and two possible reasons for the critical time to, namely the use of an incorrect model, and time-dependent growth rates (including, as a special case, aggregation without chemical decomposition) are advanced. Finally, the consequences of nucleation only on the surface of the reactant are mentioned briefly. 

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