Modeling and nonlinear dynamics of oscillation processing systems

Advanced oscillation processing systems are involved in the primary mode of completely new methods of application of vibration technology for efficient transport-related separation of chemicals, minerals, pharmaceuticals, foods, metallic powders (lead, copper, zinc, steel) and all types of powder products common in many industries: abrasive, powder metallurgy, paint and varnish, diamond, construction, mining and chemical. These new screen-less methods can be effectively used in the abrasive industry to produce materials in which more than 90% of the grains are isometric. Grinding wheels made from such grains are twice as effective as those made from regular grains which are unclassified by shape. Powder products generally are separated by particle size or shape without forming dust. In the diamond tool industry the vibrating equipment is used for selecting isometric, plane and needle-shaped diamond grains. In the agriculture and food industry this advanced machinery can utilized for removing harmful inclusions in grains. In the mining and chemical industries high efficiency processing machinery with accurate separations in sizes from 15 mm down to 20 μm can be used for many types of the ore. Separation is based on the velocity differences of the particles due to the existence of oscillating driving force. If components are to be separated, their relative friction coefficients on a vibrating deck may be pertinent. The friction coefficient is a function of the particle shape and the particle size. A common correlation of the friction coefficient f and of the adhesion force Fa to the particle size D in μm is that of the author [1]:

f 1 = 0,9∙10^-0,001D                             (1)

F a /mg = 7,46∙D^-0,24 – 1, 68.                  (2)

The above functionality is useful to understand the phenomenon of effective process.

There are two types of operating equipment: 1) harmonic motion of a deck, where direction of oscillation does not have a lateral tilt, and 2) vibration characterized by a lateral tilt with respect to the vertical direction. The direction of vibration is also inclined at the angle of vibration β with respect to the surface of the vibrating deck. A vibratory effect appears in the process so that particles with different sizes, shapes, or coefficients of friction, therefore with different angles of vibration separation α0 move with different velocities over a separating surface. A subset of the two types of operation conditions are the 1st and 2nd basic separation mechanisms common to the 1st and 2nd types of machinery. The acceleration W0 = Aω2sinβ / g cosα is the operating condition factor for these two types of equipment, where A is the displacement amplitude of the deck, ω is the angular frequency of the simple harmonic motion, β is the angle of vibration, g is the acceleration of gravity, and α is the longitudinal tilt of the deck. The force driving the separation could be several orders of magnitude greater than of gravity. Particles will be intensively tossed upwards after a brief contact with the vibrating surface at phase angles cosδ0* = πp1 ++R and cos δ* = πp1 + R cos ε for machinery of the 1st and 2nd types, respectively, W01+R where ε is the lateral tilt of the deck with respect to horizon, λ (see below) and R are the coefficients of momentary friction and elasticity in accordance with Newton’s theory of impact, the coefficient p defines regime of continuous rate of particle flight over vibrating deck. The particle velocities along the x and z axes are represented by for the equipment of the first type

πpg 1 - R2

Vx =     (     ctgβcosa-      sina)(3)

• x ω 1+R

π pg 2 - λ 1 - R

Vz =(     -)cosasin ε(4)

ωλ1+R for the equipment of the second type

πpg 1 - R2

V =     (     ctgβcosacosε -      sina)(5)

• x ω 1+R

  ВЕСТНИК БУРЯТСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА 1/2013

• π pg 2 - λ

Vz =            cos a sin ε .                                    (6)

ωλ

To maximize the operation that must be dealt with developing an optimal process the phase angle δ 0 and parameter p can be determined as follows:

cos δ 0 = ± 1, p = W 0 (1+R) / π (1-R).

for the equipment of the first type

Vx =Aω(cosβ-sinβtga)(7)

q

Vz = Aωsinβsinε( 1 - 1)(8)

q for the machinery of the second type

Vx = Aω(cosβ - sinβ tga)(9)

qcos ε sinβ

Vz =Aω .(10)

q cos ε

Dividing equations (8), (10) by Eqs. (7), (9), respectively, the expressions for the trajectory of the particle movement for both types can be written dz Vz (1 - q) sinε dx Vx qctgβ - tga

dz dx

Vz

sin ε

Vx  qctg β cos ε - tg a

The XZ plane of the vibrating deck with its longitudinal and lateral axis in the X and Z directions, respectively, has the length l and the width b of the deck,

respectively. Solving Eqs (11) - (12), separability is defined analyt- ically as follows for the machinery of the first type dx dx = z=b=

I dq

b  ctgβ- tga sinε  (1-q)2

dz d z =

I dq

ll x = = sinε

ctg β - tg a (qctg β -tg a ) ²

for the machinery of the second type

D_xII = dctg ε ctg β

D =l    sin 2εctgβ zII   4 (qcosεctgβ-tga)2 .

In the design and operation of separation processes, each process will give a maximum removal of a single component from the mixture, if qctgβ / tgα = 1 and cosε ctgβ / tgα = 1. The particle velocity in its movement in the longitu- dinal direction can be reduced to zero. It can be used to write the angles of vibration separation for the 1st and 2nd types of equipment, as α0λ = arctg (qctgβ), α0λ = arctg (qcosε ctgβ).

Conclusions

The analytical and inventive aspects of design include: estimate the velocities of each components; evaluation its angles of separation; calculation the separability of the process; designing an efficient equipment and its control strategy.