About speed approximate calculation of the grain mixture flow in a cylindrical vibrosieve

Abstract

Nonlinear first order differential equation describing the steady vertical movement of vibroliquefied layer of fine-grained mixture on the inner surface of the cylindrical sieve of vibrating centrifuge which rotating at a constant angular velocity around a vertical axis are deduced. Nonlinear quadratic-polygonal dependence of shear stress in a mixture from shear strain rate are laid in equations basis. It is assumed, that the rheological constant in this depending are parametrically associated with the physical and mechanical characteristics of the grain mixture and the characteristics of the mechanical vibrations of vibrating centrifuge. In addition to the equation a boundary conditions is formulated. It expresses the absent of slip of mixture on vibrosieve surface. Analytic solution of the boundary value problem in the form of squaring, that can not be expressed in terms of elementary or special tabulated functions are obtained. Therefore, to calculate of the integral (or flow velocity) two approximate methods based on well-known in the integral calculus Simpson’s formula are suggested. In the first method, called formula is applied directly to the integral calculation, and in the second addition is carried out a preliminary approximation of the integrand, so the basic part of integral is expressed a separate summand in analytical form. It is shown that the second method is more accurate than the first. Its error does not exceed tenths of a percentage set by comparing the approximate and exact results obtained by numerical computer integration. Effective using of Simpson's formula is possible due to the relatively small thickness of the moving layer grain mixture. In article provides a list of recent publications, where earlier in the continuum models of bulk material movement the quadratic rheological dependence were used. Formulated objective of the study, which is a derivation of a compact formula for approximate calculation of the velocity of a steady grain flows on the surface of the cylindrical sieve vibrating centrifuge. In a result of the calculations the effect of movable layer thickness and rheological constant to the grain mixture velocity are investigated.