The motion of particles in coaxially-arranged cylinders

The purpose of the study outlined in the paper is to substantiate the technological and technical characteristics of the processes of moving a bulk material particle in the vertical direction. To determine the motion of the entire mass of material with a spiral, we used the simulation of steady-state particle motion using the d’Alembert equations in a cylindrical coordinate system.


Introduction
When calculating and designing vertically arranged spiral screw units, there is a need to use data that provide the relationship between the available parameters and the kinetics of motion of the entire flow of bulk material and its particles. A separate consideration deserves the case when the motion of the particle becomes constant (steady), and it moves along the inner surface of the cylindrical casing of the unit in the axial direction. Therefore, it is necessary to determine connections suitable for practical use in a particular spiral screw unit.

Materials and methods of research
We consider the case of a vertical arrangement of the unit, consisting of a spiral screw working body and two cylindrical coaxial housings (Fig. 1). The housings are fixedly mounted, and the spiral rotates between them around its axis with an unchanged angular velocity ω0.
For the fixed coordinates, we take the right-handed Cartesian coordinate system Oxyz, where Оz coincides with the vertical axis of the spiral. Suppose that the material point lies on the surface of the spiral, which is located between the coaxially mounted cylinders, while the point rotates around the axis with the angular velocity ω (Fig. 1).
During the movement of the material point by a spiral screw inside the coaxial cylinders, friction forces arise on the inner surface of the outer cylinder F2 and the surface of the spiral F1. Figure 1 shows the following notation: G -gravity, H; N2 is the force acting on the material point from the side of the inner surface of the outer cylinder, H; N1 is the force acting on the material point from the side of the surface of the spiral, while it makes an angle θ with the normal to the helical line of the spiral, and the normal in turn is the angle α with the axis Oz. With the vertical movement, the material point moves in the following directions: 1) absolute, along a helical path, along the inner surface of the outer cylinder; 2) carryingtogether with a spiral turn; 3) relative -along a spiral turn [1,2,3,4].
Let us consider in detail the forces acting on the material point (Fig. 2). Where υ0 is the velocity directed along the spiral; υn is the velocity vector of the carrying motion on a plane perpendicular to the axis of the spiral; υ is the absolute velocity vector. The axial velocity υ1 has a significant effect on the feed of a spiral screw unit and from Figure 2 one can see that it is defined as υ1 = υ sinβ = υ0 sin α. If we place the vector υ0 on the plane perpendicular to the axis of the spiral, then we get the vectors rω and υ1 which are directed at right angles to each other (Fig. 2), therefore, the expression for velocity is written: where r is the radius of the inner surface of the outer cylinder [5,6,7]. This explains that the values of υ1 are determined by the angular velocity ω. With the steady flow of bulk material, the equilibrium condition must be met Having projected the forces acting on the material point along the axes Y and X [8,9,10,11,12,13], we find: where f1-coefficient of friction of a particle on a spiral surface; f2 -particle friction coefficient on the inner surface of the cylinder. We multiply the equation (1) by f2, and add it to (2) multiplied by cosθ. We reduce the resulting equation by G and multiply by minus one, we get: [ ] From Fig. 2 we will find the trigonometric functions of the angle β: We transform in these equalities the expression under the root: Substituting in (4) and (5) By putting (7) and (8) We introduce in our consideration the criterion The sought root k of the equation (11)  If we take the friction coefficient of a material particle for steel 1 f = 2 f = 0.5 and plot the graphs of k in the function Consequently, for the motion of the mass up along the auger axis it is necessary to satisfy the condition We solve the problem of the axial velocity of the material in a vertical spiral screw unit.

Results of the research
The equation (12) describes the relative motion of a particle in a vertical spiral unit. Figure 3 shows the results of calculations of the axial velocity of the spiral and particle movement in a spiral screw unit with the following characteristics; r = 0.02 m is the inner radius of the outer cylinder; d = 0.004 m is the diameter of the spiral bar; r1 = 0.004 m is the average radius of the particle of the moved material; r2 = 0.018 m is the radius of the spiral.
The obtained equation and dependencies allow one to choose the optimal parameters in the calculation and design of spiral-helical conveyors.
The experimental studies were carried out on a unit for conveying soybeans with the use of a spiral screw working body located between vertically arranged coaxial cylinders. Figure 4 shows the results of the experimental studies on the movement of soybeans: spiral screw working bodies with an outer diameter of d = 33 mm, a helix pitch of S = 25 mm, a wire diameter of 4 mm, a cylinder length of L = 1 m, an inner diameter of the outer cylinder D1 = 36.5 mm, the outer diameter of the inner cylinder D2 = 21 mm, the angular velocity of rotation of the spiral in this case varied from ω = 40 to ω = 88 s -1 . The transferred material is soybeans with bulk density ρ = 750 kg / m 3 .

Conclusions
From this formula, one can determine the threshold critical value of the angular velocity of the spiral rotation, at which the particle begins to move in a vertical coaxial cylindrical casing. For experimental research conditions this is critical ωcrit= 25 s -1 .
The work was carried out in the framework of a grant of the President of the Russian Federation for state support of young Russian scientists -candidates of sciences МК-3511.2019.8