TO CALCULATE THE TRAJECTORY OF A DISPERSED PARTICLE
Authors:
1.
Kazan State Agrarian University
(Physics and Mathematics, Professor)
2.
Kazan state agrarian University
(Department of Physics and Mathematics, Associate Professor of the Department)
Russian Federation
Russian Federation
3.
Kazan State Agrarian University
Type:
Article
Pages:
from 91
to 95
Status:
Published
Received:
24.12.2025
Accepted:
25.12.2025
Published:
25.12.2025
Language:
Russian
Keywords:
DISPERSED PARTICLE, TRAJECTORY CALCULATION, RELATIVE VELOCITY, CENTRIFUGAL FIELD
Abstract (English):
The description of the motion of dispersed media is the basis for mathematical modeling of many technological processes. In this article, the motion of dispersed particles is described in the Lagrangian coordinate system, taking into account the velocities and differential characteristics of the motion of a continuous medium recorded in the Eulerian coordinate system. The velocity of a dispersed particle is represented as the sum of the velocities of the continuous phase and the relative velocity. Relative velocity is expressed in terms of the relative velocity modulus, which is a scalar quantity. The direction of movement is set by the angle between the velocity vectors of the main stream and the relative velocity. The description of the trajectory of the dispersed inclusions is reduced to the numerical calculation of changes in two scalar quantities over time - the modulus of relative velocity and the angle of rotation. The corresponding trajectory calculation equations are written for an arbitrary orthogonal coordinate system using metric coefficients. The components of the vector of mass forces, as well as the initial conditions for solving a system of scalar equations, are determined taking into account the process in a particular apparatus. Numerical calculations are carried out using the example of a description of the operation of a pneumomechanical grain husker consisting of a bladed disk rotating inside a cylindrical surface. Grain material is fed to the disk, which is accelerated and ejected by centrifugal force. When hitting a cylindrical surface, peeling occurs, the quality of which is determined by the direction and speed of grain movement at the time of contact. The value of the contact angle upon impact with the wall can be influenced by rotating the cylindrical surface in the opposite direction. At the same time, two zones with opposite directions of air flow appear in the workspace, which will affect the flight path of the grain. The sizes of these zones and the flow rates in them depend on the angular velocities of the disk and the outer surfaces. Consequently, it becomes possible to control the direction of flight of the grain at the moment of its impact on the wall by changing the values of the rotational speeds of the units of the unit.
The description of the motion of dispersed media is the basis for mathematical modeling of many technological processes. In this article, the motion of dispersed particles is described in the Lagrangian coordinate system, taking into account the velocities and differential characteristics of the motion of a continuous medium recorded in the Eulerian coordinate system. The velocity of a dispersed particle is represented as the sum of the velocities of the continuous phase and the relative velocity. Relative velocity is expressed in terms of the relative velocity modulus, which is a scalar quantity. The direction of movement is set by the angle between the velocity vectors of the main stream and the relative velocity. The description of the trajectory of the dispersed inclusions is reduced to the numerical calculation of changes in two scalar quantities over time - the modulus of relative velocity and the angle of rotation. The corresponding trajectory calculation equations are written for an arbitrary orthogonal coordinate system using metric coefficients. The components of the vector of mass forces, as well as the initial conditions for solving a system of scalar equations, are determined taking into account the process in a particular apparatus. Numerical calculations are carried out using the example of a description of the operation of a pneumomechanical grain husker consisting of a bladed disk rotating inside a cylindrical surface. Grain material is fed to the disk, which is accelerated and ejected by centrifugal force. When hitting a cylindrical surface, peeling occurs, the quality of which is determined by the direction and speed of grain movement at the time of contact. The value of the contact angle upon impact with the wall can be influenced by rotating the cylindrical surface in the opposite direction. At the same time, two zones with opposite directions of air flow appear in the workspace, which will affect the flight path of the grain. The sizes of these zones and the flow rates in them depend on the angular velocities of the disk and the outer surfaces. Consequently, it becomes possible to control the direction of flight of the grain at the moment of its impact on the wall by changing the values of the rotational speeds of the units of the unit.
Keywords:
DISPERSED PARTICLE, TRAJECTORY CALCULATION, RELATIVE VELOCITY, CENTRIFUGAL FIELD
DISPERSED PARTICLE, TRAJECTORY CALCULATION, RELATIVE VELOCITY, CENTRIFUGAL FIELD
