Mathematical modeling of a pneumatic separator operation

Abstract

A mathematical model of the seed separation process for vegetable and melon crops in a novel pneumatic separator with a cylindrical working surface is discussed. The research goal is to develop a model describing seed motion under the combined action of gravity, friction, centrifugal force, air resistance, aerodynamic force, and vacuum suction force. Based on theoretical mechanics and aerodynamics, a second-order nonlinear differential equation was derived to describe the angular motion of a seed along the inner surface of a rotating drum. The equation was solved numerically using the fourth-order Runge-Kutta method implemented with Python (Matplotlib) and Mathcad software. The initial conditions were set to zero values for both angular position and angular velocity. Under the specified parameters (seed mass of 0.0003 kg, vacuum pressure of 280 Pa, drum rotation speed of 0.16 m s, and friction coefficient of 0.61), the model predicts a stable seed separation angle of 23° from the drum top point. The geometric analysis showed that the maximum allowable separation angle for seeds to enter the charging chute was 57°9’ that ensured a sufficient technological margin. It was found that the key adjustable parameter determining separation accuracy was the vacuum pressure in the air chambers. The developed model enables not only to predict seed behavior during separation but also optimize both the separator structural design and operational settings to improve grading quality and minimize seed loss. The results obtained are of practical value for the development of high-efficiency equipment intended for use in the agro-industrial sector.