| Popis: |
A modified hydrodynamic model of a stable grain flow of an inhomogeneous mixture over the surface of a vertical cylindrical vibrating sieve is proposed under the assumption that the porosity of the mixture in the moving annular layer depends on the velocity of movement. A linear dependence of the porosity of the mixture on the velocity of movement are accepted, where higher speed corresponds to higher porosity.. The calculation of the velocity is reduced to solving an inhomogeneous differential equation of the Bessel type. Further, by "freezing" the variable coefficient of this equation, the problem has been simplified. This simplification is permissible due to the fact that the thickness of the moving layer of the mixture is much less than the radius of the vibrating sieve. As a result, the dependence of the velocity on the radial coordinate is expressed through the elementary functions. A compact formula for determining the maximum grain flow velocity is obtained. By integrating, in elementary functions, the formula for the average velocity in the layer is obtained. An approximate formula for the performance of the vibrating sieve by the mass of the exit fraction is derived. For this, it is proposed to calculate the corresponding integral approximately by the Simpson formula, so as not to calculate the values of special functions of large argument using the asymptotic formulas. It is shown that the named productivity significantly depends on the porosity of the grain mixture. In order to obtain information about the actual errors of the approximate formulas, we additionally carried out the numerical integration of the original non-simplified Bessel-type equation on a computer. A comparative analysis of the calculation results confirmed the small errors of the simplifications introduced into the equation of motion, as well as the adequacy of the theoretical results obtained. By passing to a simplified differential equation, approximate formulas were derived and tested for calculating the main characteristics of the grain flow along a vertical cylindrical vibrating sieve, taking into account the change in porosity in the grain mixture layer from the velocity of movement. The work summarizes the known theoretical results obtained using hydrodynamic models of the motion of grain mixtures fluidized by vibrations. The generalization carried out slightly complicated the theory, because the final calculation formulas are quite compact and convenient in practical implementation. |
