Development of mathematical model of extraction from lupine with cheese serum by applying low-frequency mechanical vibrations

Abstract

It is noted that the model is designed to create the largest possible pressure change in the cheese whey in the extractor, since the rate of transfer of the target components is proportional to the pressure difference at the ends of the capillaries. The mathematical description of impregnation as the main or important auxiliary operation is given in detail. The equations for the impregnated part of the capillary, the ratio of impregnation rates at different times are given. From the above dependencies, the equation Washburne regarding the time of impregnation. The formulas for calculating the volume of extractant passed through the capillary, serum and forced out of the capillary air taking into account the viscous resistance of the latter. After integration of the equation of the speed of capillary impregnation of the obtained expression allows to estimate the final value of the impregnation in the initial stage. For different cases of capillary impregnation expressions are written at atmospheric pressure, vacuuming and overpressure. The introduction of dimensionless values allowed to simplify the solution and to obtain an expression for calculating the time of pore impregnation. The analysis of the equation of dimensionless impregnation time taking into account the application of low-frequency mechanical vibrations is made. It is noted that the processes of impregnation and extraction occur simultaneously, so the impregnation time is often neglected, which impoverishes the understanding of the physics of the process, reduces the accuracy of the calculation. Taking into account the diffusion unsteadiness of the process of substance transfer due to hydrodynamic unsteadiness, the equation containing the effective diffusion coefficient is written. The equation of unsteady diffusion for a spherical lupine particle in a batch extractor is supplemented with initial and boundary conditions. Taking into account the balance equation, the kinetic equation of the process is obtained. We studied the distribution of pores in the particle lupine along the radii and squares, the calculated value of the porosity of the particle. The values of De and Bi are determined by the method of graphical solution of the balance equation, the equation of kinetics and the parameters included in these equations. Conclusions on the work.