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

Shlomo Hareli,Vladimir Gol’Dshtein,Ophir Nave

doi:10.3390/app11209739

Shlomo Hareli, Vladimir Gol’Dshtein + Show 1 more

Open Access

https://doi.org/10.3390/app11209739

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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.

Highlights

A polydispersed fuel spray consists of tens of thousands of droplets with completely different radii scales [1,2,3]
Three probability density functions (PDFs) that are widely used for theoretical interpretations of experimentally measured droplet radii distributions are the Rosin–Rammler distribution [28], the Nukiyama–Tanasawa distribution [29], and the well-known gamma distribution
In the region close to rm = 1, we observed an almost linear behaviour of the maximal radius rm as a function of time. This corresponds to initial fast motion. Close to this almost “linear” region, our model is valid; we can study the various types of dynamics that describe the particle size distribution (PSD)

Summary

Introduction

A polydispersed fuel spray consists of tens of thousands of droplets with completely different radii scales [1,2,3]. Analytical solutions are not available; numerical methods are required Because solving such a high number of equations with computers cannot be completed in any reasonable time, a ‘parcel’ approach was adopted [4,5,6,7,8,9,10,11]. Polydispersity is modelled using smooth PDFs that correspond to the initial distribution of the size of the fuel droplets. This approximation of a polydisperse spray is more accurate than the traditional ‘parcel’. PDFs correspond to the initial distribution of the size of the fuel droplets) This allowed us to obtain reasonable estimates of thermal explosion limits, but the detailed properties of a spray’s self-ignition are beyond the scope of this method. We plan to check the main results for more complicated models in the future

Model Description

Results and Discussion

Conclusions