The Mathematical Model for the Secondary Breakup of Dropping Liquid

Ivan Pavlenko,Szymon Woziwodzki,Bernard Michałek,Izabela Kruszelnicka,Vsevolod Sklabinskyi,Radosław Olszewski,Sylwia Włodarczak,Marek Ochowiak,Vitalii Ivanov,Oleksandr Liaposhchenko,Michał Doligalski,Maksym Skydanenko,Marcin Mrugalski,Dobrochna Ginter-Kramarczyk

doi:10.3390/en13226078

Abstract

Investigating characteristics for the secondary breakup of dropping liquid is a fundamental scientific and practical problem in multiphase flow. For its solving, it is necessary to consider the features of both the main hydrodynamic and secondary processes during spray granulation and vibration separation of heterogeneous systems. A significant difficulty in modeling the secondary breakup process is that in most technological processes, the breakup of droplets and bubbles occurs through the simultaneous action of several dispersion mechanisms. In this case, the existing mathematical models based on criterion equations do not allow establishing the change over time of the process’s main characteristics. Therefore, the present article aims to solve an urgent scientific and practical problem of studying the nonstationary process of the secondary breakup of liquid droplets under the condition of the vibrational impact of oscillatory elements. Methods of mathematical modeling were used to achieve this goal. This modeling allows obtaining analytical expressions to describe the breakup characteristics. As a result of modeling, the droplet size’s critical value was evaluated depending on the oscillation frequency. Additionally, the analytical expression for the critical frequency was obtained. The proposed methodology was derived for a range of droplet diameters of 1.6–2.6 mm. The critical value of the diameter for unstable droplets was also determined, and the dependence for breakup time was established. Notably, for the critical diameter in a range of 1.90–2.05 mm, the breakup time was about 0.017 s. The reliability of the proposed methodology was confirmed experimentally by the dependencies between the Ohnesorge and Reynolds numbers for different prilling process modes.

Highlights

Scientific research of the processes of vibrational prilling [1,2], granulation [3,4] and separation of gas-dispersed systems [5,6] is an essential problem that has not been wholly studied is the mechanism of the breakup of dropping liquid
Where We = ρp v0 2 R/σ is the Weber number as the ratio of the specific inertia and the surface tension forces
It should be noted that the critical value of the Weber number Wecr = 3π/41/3 ≈ 5.94 obtained from this formula allows considering individual cases of solving the Equation (17)

Summary

Introduction

Scientific research of the processes of vibrational prilling [1,2], granulation [3,4] and separation of gas-dispersed systems [5,6] is an essential problem that has not been wholly studied is the mechanism of the breakup of dropping liquid. In this regard, a crucial hydrodynamic criterion that determines the behavior of droplets in a heterogeneous environment [7] is the Weber number (We) [8].

Methods

Results

Discussion

Conclusion