Mathematical investigation of granular material movement at the impact mill

D. K. Zhirov, M. R. Koroleva

Institute of Mechanics, Ural Branch of the Russian Academy of Sciences, Izhevsk, Russia

Abstract: The mathematical model of particles movement on the centrifugal impact mill blade is presented. The equations for particle displacement (3) and particle velocity (4) on the blade surface were obtained. Besides was received expression for supporting force (2). Similar equations were obtained without friction force — (5), (6).
Numerical calculations of the time and velocity of particle escape from mill straight blade into air space with friction force and without it were made. Was demonstrated, that ignoring friction force leads to incorrect results and conclusions. Obtained equations allow determine velocity and angle of departure of particles, value of the supporting force towards blades wear, depending on the rotor speed, blade angle inclination, friction coefficient and particle mass. The development model allows determine optimal geometry of the impact deck.
The blade angles $r_n / R$ that are considered in this work ranged from 0 to 0.55. Increasing the inclination angle increases the time departure, and decreases the angle, velocity and supporting force of the particle when it escape from blade surface. Change in supporting force from design parameters determines the blades operating life. The optimum operating parameters of the system «damage efficiency / working surfaces wear» can be obtained. For example, when the disk diameter is 0.250 m, rotor angular velocity is 628 rad/s, using the radial blades leads to escape velocity is 94 m/s, supporting force - 13 N, impact impulse - 0.0135 N. When $r_n / R = 0.55$, velocity departure is increased to 85 m/s, supporting force to 5.8 N, and impact impulse to 0.0105 N. This means that angle blade change affects the working surfaces wear and herewith impact impulse reduced slightly.
For effective grinding of materials, particle's impact angle on deck should be close to 90$^{\circ}$. This fact reached in case, when is used toothed deck with deck angle equal to 45$^{\circ}$.

Keywords: centrifugal impact mill, mathematical modeling, particle movement, departure velocity, supporting force, impact impulse.

UDC: 622.73:519.6