Study of Grain Flow in a Cylindrical Vibrating Sieve

Abstract

The paper proposes a modified hydrodynamic model of steady grain flow of a heterogeneous mixture on the surface of a vertical cylindrical vibrating sieve. The main hypothesis of the model is the dependence of the mixture's porosity in the moving annular layer on the particle velocity. A linear dependence is assumed, where an increase in velocity leads to an increase in the mixture's porosity. To determine the grain flow velocity, the problem is reduced to solving a non-homogeneous Bessel-type differential equation. To simplify the solution, the "freezing" method of the variable coefficient is applied, which is permissible due to the relatively small thickness of the moving grain mixture layer compared to the sieve radius. As a result of this approximation, the velocity dependence on the radial coordinate is expressed using elementary functions. A compact formula for calculating the maximum grain flow velocity and the average velocity is derived by integrating the respective expression in elementary functions. Additionally, an approximate formula for calculating the sieve's throughput based on the mass of the overflow fraction is proposed. Simpson's integration method is used to avoid complex calculations of special functions for large arguments via asymptotic formulas. The study shows that the sieve's throughput is significantly affected by the porosity of the grain mixture. To assess the accuracy of the approximate formulas, numerical integration of the original Bessel-type differential equation was performed on a computer. Comparative analysis of the results confirmed that the proposed simplifications introduce minimal errors and ensure the adequacy of the theoretical results. Thus, by transitioning to the simplified differential equation, approximate formulas for calculating the key characteristics of grain flow in a vertical cylindrical vibrating sieve were developed and tested, taking into account the porosity variation of the mixture depending on the particle velocity. The generalization of theoretical results obtained using hydrodynamic models of grain mixture flow in a pseudo-fluidized state under vibrations only slightly complicated the model but resulted in compact and practically convenient formulas for use.