The Evolutions in Time of Probability Density Functions of Polydispersed Fuel Spray-The Continuous Mathematical

The dynamics of the particle-size distribution of the polydispersed fuel spray are important for the evaluation of the combustion process. In this paper, we presented the particle-size distribution change in time which gives a new insight into the behavior of the droplets during the self-ignition process. Semenov was the first to shows that self-ignition in the homogeneous case can be qualitatively and even quantitatively described by simplified models \cite{first_Math_Semenov_1928}. A simplified model of the polydisperse spray is used for a study of combustion processes near the initial region. This model involves a time-dependent function of the particle-size distribution. Such simplified models are particularly helpful in understanding qualitatively the effect of various sub-processes. Our main results show that during the self-ignition process, the droplets' radii decrease as expected, and the number of smaller droplets increases in inverse proportion to the radius. An important novel result (visualized by graphs) demonstrates that the mean radius of the droplets, at first increases for a relatively short period of time, and that is then followed by the expected decrease. It means that the maximum of the mean radius is not located at the beginning of the process as expected. We only have a heuristic explanation of this phenomenon, but an analytic study is planned for the future. Our modified algorithm is superior to the well known `parcel' approach because it is much more compact, it permits an analytical study since the right-hand sides are smooth, and thus eliminates the need for a numerical algorithm transitioning from one parcel to another, The method explain herein can be applied to any approximation of the particle-size distribution, and it involves comparatively negligible computation time.