In this article, the fractional-order differential equation of particle sedimentation was obtained. It considers the Basset force’s fractional origin and contains the Riemann–Liouville fractional integral rewritten as a Grunwald–Letnikov derivative. As a result, the general solution of the proposed fractional-order differential equation was found analytically. The belonging of this solution to the real range of values was strictly theoretically proven. The obtained solution was validated on a particular analytical case study. In addition, it was proven numerically with the approach based on the S-approximation method using the block-pulse operational matrix. The proposed mathematical model can be applied for modeling the processes of fine particles sedimentation in liquids, aerosol deposition in gas flows, and particle deposition in gas-dispersed systems.

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.

Granulated chelates are innovative fertilizers that are highly effective and versatile, and they ensure the best start-up effect for plants. The final properties of granules are influenced by the method of their preparation and the used substances. The diameters of the obtained granules, their size range, and the final costs of the produced fertilizer are of great importance. The paper describes granules that were produced using an agglomeration of ZnIDHA in a fluidized bed with the aid of an aqueous solution of this substance with a high dry matter content. The aim of the study was to evaluate the effect of surfactant addition to the solution on the evolution of granule size distribution during the process carried out in a batch mode and to access the possibility of describing the process dynamics using population balance approach. A sieve analysis was performed in order to determine the size of the granulate, and numerical calculations were performed to determine the value of the constant aggregation rate. Based on experimental studies, it can be seen that the increase in the diameters of granules is mainly caused by the agglomeration process, and to a lesser extent by the coating process. The addition of surfactant increased the median size of the granules in the initial granulation stage, and also lowered the surface tension. This in turn enables a lower spraying pressure to be used. A comparison of different aggregation kernels constituting an integral part of the population balance model proved that the physically motivated equipartition kinetic energy kernel performs best in this case. Moreover, the computational results show an increase in the aggregation rate when the surfactant additive is used and confirm that population balance allows the extraction of physical information about the granulation.

Fluidized-bed spray granulation (FBSG) enables manufacturing particles with desired characteristics, including particle size distribution (PSD), density, or dust content. This study investigated the effect of selected factors on the granules obtained in a continuous FBSG of chelated fertilizers for foliar applications. The effect of surfactant addition to the solution sprayed into the bed and perturbations of operating parameters on PSD and granules morphology was studied. The experiments were supplemented with calculations based on a population balance equation (PBE). It was shown that granules manufactured with the tenside addition are more regular in shape, and thus less prone to mechanical wear. It was demonstrated that increasing rotational mill speed does contribute to a slight increase in the amount of dust, but in the long term, it does not disturb the regular agglomeration process. The computational results confirm that, despite the complexity of the process, its description with PBE is feasible.