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

Abstract

The dynamics of the particle size distribution (PSD) of polydispersed fuel spray is important in the evaluation of the combustion process. A better understanding of the dynamics can provide a tool for selecting a PSD that will more effectively meet the needs of the system. In this paper, we present an efficient and elegant method for evaluating the dynamics of the PSD. New insights into the behaviour of polydispersed fuel spray were obtained. A simplified theoretical model was applied to the experimental data and a known approximation of the polydispersed fuel spray. This model can be applied to any distribution, not necessarily an experimental distribution or approximation, and involves a time-dependent function of the PSD. Such simplified models are particularly helpful in qualitatively understanding the effects of various sub-processes. Our main results show that during the self-ignition process, the radii of the droplets decreased as expected, and the number of smaller droplets increased in inverse proportion to the radius. An important novel result (visualised by graphs) demonstrates that the mean radius of the droplets initially increases for a relatively short period of time, which is followed by the expected decrease. Our modified algorithm is superior to the well-known `parcel’ approach because it is much more compact; it permits analytical study because the right-hand sides of the mathematical model are smooth, and thus eliminates the need for a numerical algorithm to transition from one parcel to another. Moreover, the method can provide droplet radii resolution dynamics because it can use step functions that accurately describe the evolution of the radii of the droplets. The method explained herein can be applied to any approximation of the PSD, and involves a comparatively negligible computation time.

The Impact of the Polysidpersed Fuel Distribution on the Ignition Characteristic

“…For lean fuels and reach fuels, such parameters may be different. We hope that this paper showing the importance of droplets radius distribution, combined with the tool for assessing the fuel dynamics [10] will encourage experimenting with more advance PSDs.…”

Simplified Interfacial Area Modeling in Polydisperse Two-Phase Flows under Explosion Situations

“…Analyzing Equation (19) for the total number of particles N 2 , Equation (25) for the volume fraction α 2 , and Equation (28) for the specific interfacial area A I , it appears that all relations involve an integral term, related to the PDF, that can be written under the generic form,…”

Probabilistic Analysis of an RL Circuit Transient Response under Inductor Failure Conditions

“…Similarly, the expression for the interval corresponding to the interval l > b for F L (l) is obtained using (31) in the following.…”

“…In these instances, probability theory alongside conventional circuit analysis techniques may be put to use for the investigation of different circuit parameters. Probability theory has been effectively used for the analysis and provision of solutions to various problems in multiple disciplines in science and engineering [26][27][28][29][30][31][32]. For example, it is extensively used in communication systems and signal processing applications [33,34].…”