Explosive systems with reactant consumption. I. Critical conditions

It has been shown in a general way that the problem of stability in an explosive system with reactant consumption can be solved by well known mathematical techniques, hitherto unapplied in this area. General criteria which must be satisfied by empirical solutions to this problem, particularly with respect to the Semenov limit ε{lunate} → 0 are examined. The requirement of continuity of the critical parameters is shown to rule out the (π{variant} + λ) inflection criterion. The methods presented above are not restricted to the simple system described by equations (1) and (2), but in principle can be used to calculate the region of stability, and hence criticality, in more complex systems such as autocatalytic reactions, including intermediates, and chain thermal explosions. The requirement of continuity of the solutions as ε{lunate} → 0 (which is absolutely essential if all the work with ε{lunate} = 0 is not to be discarded) implies that the solution is a discontinuous function of the parameter a for sufficiently small ε{lunate}. The discontinuity will occur at the critical value of a for a given ε{lunate} and we will only be able to prove stability in regions not including this discontinuity, which may well disappear for sufficiently large ε{lunate} (see Appendix).