About motion of inhomogeneous grain mixture on vertical vibrosieve

Abstract

A mathematical hydrodynamic model has been developed for the steady motion of an inhomogeneous fine-grained grain mixture on the surface of a vertical cylindrical vibrating sieve, under the assumption that the specific gravity of the mixture depends on the velocity of its movement. A linear dependence of the porosity of the mixture from the velocity is accepted when the porosity is higher where the velocity of movement is higher.
Determining the velocity as a function of the radial coordinate is reduced to solving an inhomogeneous differential equation of the Bessel type. For the given boundary conditions, the analytical solution is expressed in terms of the modification Bessel function and the Macdonald function of zero indices. By integrating this equation, a formula for the average velocity of the throw out fraction on the sieve surface is obtained in cylindrical functions. Using asymptotic representations of special functions for large values of the arguments, approximate formulas for the velocity of a grain of a flow, as an annular layer, are derived in elementary functions. It is shown that the derived theoretical dependences, as a special case, follow the well-known formulas in the hydrodynamic model for the velocity of a homogeneous layer of a grain mixture. Examples of calculations are given, in which the influence of changes in porosity on the grain flow rate is shown. A comparison is carried out and full correspondence is established between the numerical results, to which the obtained analytical solutions and numerical computer integration of the output differential equation of motion. The developed mathematical model makes it possible to take into account the dependence of the porosity of a grain mixture on its velocity of movement along a vertical cylindrical vibrating sieve. The resulting dependence generalizes the previously obtained theoretical results. To calculate the grain flow velocity in real conditions, one can use the derived approximate asymptotic formulas, without calculating the cylindrical functions of large arguments.