About nonlinear model of a grain flow of an inhomogeneous mixture along an inclined flat vibrosieve

Abstract

Using the quadratic rheological dependence of the Savage’s type, the steady motion of a layer of a finegrained mixture of variable porosity is described, along a flat vibrosieve inclined to the horizon. In this case, the distribution of the specific mass over the height of the moving layer is approximated by a square trinomial, the coefficients of which depend on the amplitude and frequency of oscillations of the vibrosieve, and also on the mechanical properties of the grain mixture. In the expressions for the coefficients, there is a factor that takes into account presence on sieve surface of the segregation process intensifiers (ribs, riffles, etc.). Due to shown approximation, we succeeded in solving of analytically compiled nonlinear differential equation of motion of the first order. The solution is represented in the form of an integral, which is not expressed in a closed form through known functions. Therefore, two versions of the approximate calculation of the integral are proposed. The first variant is based on the expansion of the integrand function in the power series and replacement of the series by its partial sum. In the second variant, the principal part of the integral is expressed in terms of elementary functions, and the additional part (residual) is determined approximately by the Simpson’s formula. Thus, approximate calculation formulas are obtained for calculating the velocity of the grain flow, the performance of the vibrating sieve and its specific loading. The influence of various factors, in particular rheological constants, on the calculated kinematic characteristics of the grain flow was studied. A comparison of the numerical results, to which the proposed calculation method leads with published in the literature, is compared. A good correspondence of the numerical results obtained by different methods is established, which confirms the consistency of the derived calculation formulas. In contradistinction to wellknown studies, the developed model does not require the numerical integration of differential equations in calculating porosity of the mixture and kinematic characteristics of the flow on a flat vibrosieve.