applied sciences  
The Evolutions in Time of Probability Density Functions of Polydispersed Fuel Spray--The Continuous Mathematical Model  
[PERSON]; [PERSON]; [PERSON] The Evolutions in Time of Probability Density Functions of Polydispersed Fuel Spray--The Continuous Mathematical Model. 2021  
The commonly used approximations of the initial experimental polydisperse size distributions are the Rosin-Rammler distribution, Nukiyama-Tanasawa distribution, and gamma distribution [17; 18]. These approximations provide comparatively accurate representations of experimental data, which allow extrapolation of values beyond the measurement range. These approximations permit comparatively easy calculations of parameters of interest, because they have already been built into computer software (e.g., MATLAB, MATHEMATICA). However, to the best of our knowledge, there is no single mathematical theory that accurately and precisely fits experimental data. Therefore, in the absence of a theoretical basis, the Rosin-Rammler distribution is commonly used and allows us to describe combustion dynamics using raw experimental data. Most attempts to theoretically describe combustion processes that accompany polydispersed sprays have resulted in complex systems of highly nonlinear partial differential equations [19; 20]. The main difficulty is describing the dynamics of individual droplets or groups of droplets that compose the fuels. [PERSON] pioneered the description of thermal explosion processes for gaseous fuels using simple ordinary differential equations [21]. History has shown that simplified Semenov-type models can often describe the main characteristics of self-ignition in the homogeneous case [22]. However, the models built on [PERSON]'s work that focus on sprays quickly became very complex when accounting for various droplet radii [23]. The well-known 'parcel' method divides droplets into separate sections, but due to computational complexity, this approach fails to estimate a realistic distribution. In the literature, we did not find any study that used more than 5-6 'parcels', and we attributed this to the increased computational complexity with each additional parcel. In [24], the authors proposed using smooth probability density functions (PDFs) that approximate discrete droplet radii distributions instead of a separate analysis of individual droplets, the 'parcel' approximation. In this study, 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' approximation, and permits analytical treatment of the corresponding simplified model. We note that any analytical treatment of the 'parcel' models is almost impossible because the right-hand sides of the mathematical model are discontinuous. This is the main reason that they are infrequently used. In [24], the PDF approximations were compared with the 'parcel' approximations for 300 'parcels' and found to be more accurate. Details about this comparison can be found in [24]. The corresponding new models are low-dimensional and compact. In the case of ignition processes, such models use a PDF. The density function is a smooth function of only one variable, which is the droplet's maximal or mean radius. These models allow us to monitor different PDFs. The main disadvantage of this approach is the rigid shape of PDFs in time (the smooth 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. In this study, we succeeded in resolving this problem partially. Here, we propose an equation for the evolution of PDFs over time. We begin with the initial PSD and study its dynamics. This modified method was applied to the well-known Rosin-Rammler, Nukiyama-Tanasawa, and gamma distribution approximations of experimental distributions.