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## Analytical Method for Determining Porosity in the Layer of Vibro Separated Grain Mixture

### Abstract

The formulas have been entered and approved for the calculation of porosity distribution on the thickness of layer of fine-grained mixture during its separation by the inclined flat or vertical cylinder vibro sieves. It has been attained as a result of approximation of tabular information of the obtained numerical computer integration of the specially worked out nonlinear differential equations of the second order in a dimensionless form. For approximation, the function of degree coefficients and index is used for the degrees which are certain by the Aitken's method. Coefficients of the entered analytical dependence are the vibro sieves related to the parameters obtained by mechanical descriptions of the separated material. Coefficients of the entered analytical dependence are related to the parameters of vibro sieves and mechanical descriptions of the separated material. In the case of cylinder vertical vibro sieve the action of centrifugal force is also taken into account. The method of mixture porosity calculation does not need a computer numerical integration of nonlinear differential equations conducted by other authors for solving this problem. Comparison of numerical results of the proposed analytical method of calculation with the ones described in literature, have confirmed its high accuracy results, for the differences do not exceed one percent. The expounded method is universal enough and simple in use, besides it opens the possibilities of subsequent analytical integration of differential equalizations of motion at the calculation of kinematics descriptions of grain flow. The developed method gives the opportunity to also solve the inverse task when, according to experimental measurements of porosity values of grain mixtures on the thickness of movable separated layer, it is needed to find the value of phenomenological permanent that is included in the expressions of coefficients of initial differential equalization. In this way, the adequacy of the mathematical model is improved. The use of approximation of degree considerably simplifies the method of authentication of differential equalization coefficients. In the article, the examples of grain mixture porosity calculation as well as the examples of phenomenological permanent authentication have been resulted after experimental calculations for both the variants of vibro sieves.
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## About motion of grain mixture of variable porosity in the cylindrical sieve of vibrocentrifuge

### Abstract

The analytical method of calculation of a withstand motion of fine-grained mixture is worked out in the vertical cylindrical sieve of vibrocentrifuge. Integration of differential equalization of motion is show out the reserved formulas for the calculation of kinematics descriptions of grain flow. The two-parameter continual model of the state of separation mixture is used in researches, as a heterogeneous continuous environment with variable specific mass (by porosity) on the thickness of movable layer of friable material. Change of specific mass on a radial coordinate in the cylindrical layer of mixture approximated by the function of degree, the coefficients of that are certain by the Aitken's method. Due to such approximation, the analytical decision of differential equalization of the grain flow, shown out with the use of two-parameter rheological dependence, in that the constituent of linear viscid resistance is complemented by the constituent of remaining internal dry friction, proportional overpressure in mixture, is built. An analytical decision is expressed as squaring that is not expressed through elementary functions in closed form, the close method of calculation of integral offers that is why, with the use of partial sum of row of degree. The results got close formulas result in that well comport with the results of numerical computer integration of squaring. Such method the continual models of grain flow on vibrosieves, it is assumed in that friable material is fully confined internal dry friction, are generalized known for, as a result of vibrodilution. The examples of calculations are considered, where influence of different factors is investigational, in particular values of rheological coefficients and change of porosity, on kinematics descriptions. It is set that calculation kinematics descriptions of grain flow substantially depend not only on the thickness of movable layer and rheological constants, and also from the concentration of grains near free surface of the mixture. Thus, worked out here a method of research of vertical grain flow in the cylindrical sieve of vibrocentrifuge can be an alternative to other methods in that for the calculation of motion of grain mixture of variable porosity conducted numerical computer integration of nonlinear differential equalizations.
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