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A uniformly distributed mathematical model (based on Semenov's theory of thermal explosions) is formulated to model the thermal behaviour of cellulosic materials in compost piles. The model consists of a mass balance equation for oxygen, a heat balance equation and incorporates the heat release due to biological activity within the pile. Singularity theory and degenerate Hopf bifurcation theory are used to investigate the generic properties of the model as well as to determine the loci of different singularities: the isola, cusp, double-limit points, boundary limit set, double-Hopf bifurcation, generalised Hopf (Bautin) bifurcation and Bogdanov-Takens bifurcation. These loci divide the secondary parameter plane into different regions of solution behaviours. Conditions under which biological activity can result in the initiation of an elevated-temperature branch within the compost pile which does not pose the risk of spontaneous ignition are identified. These are the ideal conditions for composting. The regions of parameter space when spontaneous ignition is likely are also determined. © 2013.

We consider the self heating process in a two dimensional spatially dependent model of a compost pile which incorporates terms that account for self heating due to both biological and oxidation mechanisms. As moisture is a crucial factor in both the degradation process and spontaneous ignition within a compost pile, this model consists of four mass-balance equations, namely, energy, oxygen, vapour and liquid water concentrations. Analyses are undertaken for different initial water contents within the compost pile. We show that when the water content is too low, the reaction is almost negligible; whereas when it is too high, the reaction commences only when the water content evaporates and the water ratio drops to within an appropriate range. However, for an intermediate water content range, the biological reaction is at its optimum and there is a possibility of spontaneous ignition within the compost pile.

Oxygen index methods have been widely used to measure the flammability of polymeric materials and to investigate the effectiveness of fire-retardants. Using a dynamical systems framework we show how a limiting oxygen index can be identified with an appropriate bifurcation. The effectiveness of fire-retardants in changing the limiting oxygen index is calculated by unfolding the bifurcation point with a suitable non-dimensionalised variable, which depends upon the mode of action of the additive. In order to use this procedure it is essential the model is non-dimensionalised so as to retain the variables of interest as distinct continuation parameters.

The attenuation of strong plane shocks produced by hypervelocity impact is studied. Analytical equations are developed which describe the path of the shock front. Since the entropy change across the decaying shock is included in the derivation, these equations are applicable to both weak and strong shocks. The comparison of the graphical and analytical results shows that the simplifying assumptions made for the analytical solution are valid. Calculations using our analytical equations show that late-stage equivalence exists in one-dimensional like-material impacts. Two impacts are equivalent if the quantity du to the alpha power is the same for both cases, Here, alpha is between 1 and 2, and, therefore, the basis of equivalence is between equal energy and equal momentum. It is also shown that the spall velocity can be considerably higher than the impact velocity in hypervelocity like-material impacts., To be presented at the Hypervelocity Impact Symposium (7th) held in Tampa, FL on 17 Nov 1964.