Order reduction in models of spray ignition and combustion

In most papers focused on the system order reduction models, describing processes of heating, evaporation and ignition in fuel sprays, it is assumed that all functions in corresponding differential equations are sufficiently smooth and consequently Lipschitzian. In many cases, however, these functions are non-Lipschitzian. This means that the conventional approach to system order reduction, based on the theory of integral manifolds, cannot be applied. It is pointed out that the order reduction of systems with non-Lipschitzian non-linearities can be performed, using a concept of positively invariant manifolds. This concept is discussed and applied to the analysis of spray ignition based on five ODEs (for gas temperature, fuel vapour and oxygen concentrations, and droplet temperatures and radii). This system is reduced to single ordinary differential equations for the gas temperature or fuel concentrations. It is shown that the equation for gas temperature predicts an increase in gas temperature up to its limiting value during finite time. The reaching of this temperature is accompanied by the complete depletion of either fuel vapour or oxygen depending on their initial concentrations, as follows from the analysis of the equations for gas temperature and fuel concentration.