Separation Processes of Granular Materials by Sizes at the Sieve Classifiers

J. Chem. Chem. Eng. 7 (2013) 56-63

Separation Processes of Granular Materials by Sizes at the Sieve Classifiers

Fail Gabdulbarovich Akhmadiev* and R.F. Gizzjatov Kazan State Architectural Building University, Kazan 420043, Russia

Received: November 03, 2012 / Accepted: November 30, 2012 / Published: January 25, 2013.

Abstract: A mathematical model of the kinetics of thin-layer separation of granular materials by size stacked sieve classifiers based on the theory of Markov processes is constructed. The probability of screening in a cell in the shape and size of the holes and particle sieve material shared with the influence of the relative speed of their movement was determined. Residues on whole meal with sieves were identified by a kinetic model and numerical calculations, the efficacy of separation. Key words: Multilevel classifier, sieve separation process, probability sieving, the density distribution, Fokker-Planck equation.

1. Introduction

Separation of granular materials by size is an integral part of many processes, such as building materials, mineral processing, etc..

One of the most effective types of equipment used for the separation of granular materials into individual fractions by size, are sieve classifiers. Success in the development and implementation of the industrial practice of vibrating devices due to the fact that the vibration transforms the dry friction characteristic of the interaction between the particles of the granular mixture, such as viscous friction forces. As a result, the conditions for the manifestation of the differences in the parameters of the separation are appeared. These differences relatively small in static conditions and in the absence of vibration does not occur. As a result of the effects of vibration on the particles of the granular material come into vibrational forces that under appropriate conditions, lead to an increase in the separation process, i.e., opportunity to share with the little particles of different arguments separation.

Work flow in these devices consists of three *Corresponding author: Fail Gabdulbarovich Akhmadiev, Professor/Ph.D., research field: numerical simulation of technological processes. E-mail: [email protected].

processes of vibrational motion: (1) the movement of the material layer on the surface of the sieve; (2) the movement of smaller particles in the transverse direction to the surface of the sieve; (3) the penetration of fine particles through the sieve.

For the design of these devices is the speed of each of the three components of the separation process must be found. Best of all is the case with the calculation of the average rate of movement of the material along the vibratory sieve. Much more difficult studies the process of self-sorting. In Ref. [1], the process of redistribution of the particles in the layer by the Fokker-Planck equation is described. Separation as a diffusion process with an absorbing screen appears. Screen role does the surface of the sieve. Relatively well to the calculation process of sifting, it is probabilistic in nature.

Modeling separation processes dispersed materials of continuum mechanics, and one of the possible is justified, even though the practical implementation of this approach has to face a difficult problem circuit and solutions of the equations of mechanics of multiphase media [2].

Study of separation process of dispersed materials on a statistical basis is now promising and relevant.

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The process of separating materials for various characters is inherently stochastic. Therefore, the calculation of such processes without random factors inherent in it is incomplete and in many cases even impossible. Classification process on the screens is stochastic in nature, by virtue of the properties of the probability of shared materials, and the process of separation. Efficiency and quality of classification in general depend on many factors. This limits the possibility of generalisution of experimental study results for the particular process and dissemination to the general case. Therefore, to describe the kinetics of the separation of dispersed materials, statistical methods are proposed to use, in particular, the mathematical apparatus and the theory of Markov processes [3-5].

2. Mathematical Model of the Kinetics of the Separation of Granular Materials by Size

For the separation of granular materials by sizes single or multistage sieve classifiers are often used. Applicants to the classifier material is characterized by a distribution function of the sizes (diameter, length, width, etc.), described by the density of the probability of W(d) (Fig. 1), where d ( )nddd ,...,, 21 is a vector of the material sizes. It is required to divide the original material into m fractions of the sizes in range ( 1, +jj dd ).The proportion (number) of particles of the j-th fraction is defined by:

( )1

∫+

=j

j

jdWn

d

dd dd

In many cases of practical importance it is required to select a fraction of the size крdd≤ , where крd is determined by technological requirements, size and shape of mesh sieve. Then the proportion (concentration) of this fraction is equal to:

( ) .min

ddd

dd dWn

кр

∫=

Concentration jnd , or dn due to the influence of

random disturbing factors during the classification is a

Fig. 1 The density distribution of the particles of granular material by size.

random variable and at each i-th level of the classifier can be characterized by a probability density ( )tnW

ji ,,rd . To determine the density ( )tnWji ,,rd

Markov processes can be taken as an approximation and constructed the KFP (kolmogorov-fokker-planck) equation. In this case, the system of kinetic equations can be written as:

( ) ,1,,),,,( nitFtnFdt

dnii j

j =′+= zzrdd

(1)

,ij

ijdt

d VVr ′+= ffV

′+=∑k

k

ij

j dtd

m (2)

where kf are deterministic forces acting on the particle of fraction jd , f ′ are the random effects forces, iF is the kinetic function, which characterizes the rate of sifting of a granular material at the i-th level,

iF ′ is random kinetic function, z is the vector of the material properties,

ijV is velocity of the particles of

jd -th fraction at the i-th level. The KFP-equations in view of the kinetic Eqs. (1)

and (2) look like:

( )∑=

−∂∂

−=∂

∂ 3

1i

ix

k k

i WVxt

Wk

( ) ( )∑∑= = ∂∂

∂+

∂∂ 4

1

4

1

2

21

k s k

iiks

iisxx

WBWFn

jd

(3)

where ixk

V is the components of the average velocity of particles at the i-th level, i

ksB is the second infinitesimal moments (diffusion coefficients),

.4 jnx d= Eq. (3) describes the process of


58

classification at multilevel classifiers, and are solved with appropriate initial and boundary conditions.

To establish the kind of kinetic functions iF it is necessary to determine the probability of a particle sifting in a unit cell of the sifting surface as it moves over the holes of the sieve.

The probability of a particle sifting in a unit cell of the sieve is the main parameter by which is achieved the greatest effect of separation.

In the first approximation, the probability of sifting can be represented as a product of the probabilities of two independent events vг ppp = , where гp : geometric probability, depending on the size and shape of the particle and mesh vp : the probability that depends on the velocity of the particle.

Geometric probability is calculated by the formula ,4/2 abDSp мг = where ba 22 × is the mesh size,

D is the size of the holes of the cell, мS is the measure of favorable outcomes when at least half of the particle as it moves along the vibrating surface of the sieve may be at the hole area. Then, the particle can sift through the hole of the cell under its own weight.

The set of favorable outcomes is defined from following system of constraints:

( ) ( ) ( ){ },; 21 DrDrxxM ≤≤+= θθθ ( ) ( )( ) 2/cos2/sin1 θθπθθ dsignlr −+= ( ) ( )( ) 2/cos2/sin2 θθπθθ lsigndr −+=

where dl, are the linear dimensions of the rectangle circumscribing the greatest section of the particle, and dl > (Fig. 2). The coordinate x is the distance from the center of the circumscribed rectangle to the side face of the hole. Angle θ is the angle between the direction of the velocity of the particle and the axis parallel to the larger side of the circumscribed rectangle ( 2/0 Dx ≤≤ , 21 θθθ ≤≤ ).

Values 1θ and 2θ depend on the dimensions and shape of the particles and account the orientation of the particles on a plane, caused by vibration.

Sifting of a particle in a cell depending on the speed of movement of the vibrating surface is considered as a random effect with a normal distribution law. Then:

Fig. 2 Scheme of cell of the sieve.

( )( )duVup k

V

v

a22

0

2/exp211 σ

σπ−−−= ∫ (4)

where aV is the amplitude of the particle velocity relative to sieve, kV , σ are the parameters of the normal law, which are determined from experimental studies.

The kinetic Eq. (1) describing the process of thin-layer classification at i-th level for the selected fraction can be represented as:

( )=′+=∂∂

+∂

∂= tFtnF

xNV

tN

tdNd

iii

срiii

j,),,,( zzrd

( ) 0,11 ≡+−= −− oiiiiii tNN αηβαα

,2/ aPPV Vгсрi =α ni ,1= (5)

where ( )tiη is delta-correlated in time random function with mean equal to zero, characterizing the deviation of the number of particles of selected fraction from the average on the surface of the sieve.

For thin-layer separation granular material moves along the sieve and in the case of the independent operation of each level of the classifier KFP-Eq. (2) can be written as:

( ) ( ) ( )2

2

21

jjn

WBn

WFxWV

tW i

iiiiсрii

dd ∂∂

+∂

∂−

∂∂

−=∂

∂ (6)

The initial distribution ( )0,, xnWji d is determined

by the composition of the separated material and organization of its supply to the classifier.

The distribution at the input source is given by the


59

fractional composition of product ( ) ( )

jjnWtnW ii dd =,0, (7)

The concentration of the considered fraction can vary from zero to one. Then the boundary conditions can be written as:

( ) 0,,0 =txWi (8)

( ) 0,,1 =txWi (9)

In practice it is possible to neglect by time of transition of the device on the established mode of operation. It is therefore sufficient to determine the steady-state solution of Eq. (6). It can be found by separation of variables. Then, solutions of Eqs. (6)-(9) have the form:

( ) ( ) ( )exp,0

jjjnY

VxAnWxnW kср

k

kk

стii ddd ⎟

⎟⎠

⎞⎜⎜⎝

⎛−+= ∑

∞

=

λ

where ( )j

nYk d are eigenfunctions corresponding to eigenvalues of the kλ of differential equation:

( ) 022=+

′−′′ k

kkik Y

BBYFY

ii

λ provided constB i =

For cases where the source material should be divided into two groups: large-sized (unsifted) and small-sized (sifted) or you want to select a target fraction, the authors consider the solution of the kinetic Eq. (5) for a single sieve with the following initial and boundary conditions:

( ) ,0,0 =xN j ( ) =0,tN j ( ),tN jо (10)

where ( ),tN jо determines the number of particles of

j-th fraction ( 1=j —unsifted, 2=j —sifted fractions) per unit length of the sieve in its initial section.

Using the change срVxtxz /, −== τ , the general solution can be written as:

( ) ( )ττ zNzN jj += ,,

( ) ξξαξτηβ dzVVV ср

j

ср

z

ср⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛+∫ exp

01

1 (11)

Where, ( ) ( )exp, ⎟

⎟⎠

⎞⎜⎜⎝

⎛−= z

VNzN

ср

jjо

j αττ, ,2/ apV j

срj =α

jp —probability of sifting of j-th fraction in a sieve

cell. Let’s define the numerical characteristics of the

process ),( zN j τ mathematical expectation and variance:

( ) ( ) ( ),,, zNzNzM jjN j τττ >==<

( ) ( ) ( )2 2, [ ,. , ]j jj

N Nz N z M zσ τ τ τ≤ − ≥

2exp14

21

1

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

Δ zVср

j

jαβ

α

where 1Δ : spectral density of white noise. In calculating the dispersion

2jNσ is used filtering

property of the delta function:

( ) ( ) ( ) 0,00 >=−∫+

−

εδε

ε

zfdzzzzfo

o

z

z For separation into several fractions multilevel

classifiers where sieves are arranged one above the other are often used. In the scheme of separation from large to small, large (first) fraction is removed from the upper level. Products for the separation at the next top level are the −i th particles, sifted from the upper

1i − level. For the chosen scheme of separation the general

solution of differential Eq. (5), taking into account the initial and boundary conditions (9), for the first three levels of the classifier (i=1, 2, 3) can be written as:

( ) ( ) ( )exp,,1

1

11

01

1

11111 ξξαξτηβττ dz

VVVzNzN

jzjj

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛++= ∫

( ) ( ) ( ) +⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛++= ∫ ξξαξτηβττ dz

VVVzNzN

jzjj

2

2

22

02

2

22222 exp,,

( ) ( )expexp0 1

1

11

01̀

2

2

21

11 ξγξγαγτηξααβ ξ

ddVV

zVVV

z jjj

∫ ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

( ) ( ) ( ) +⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛++= ∫ ξξαξτη

βττ dz

VVVzNzNjz

jj

3

3

33

033333 exp

33,,

( ) ( ) ξγξγαγτηξαβα ξ

ddVV

zVVV

z

o

jjj

∫ ∫ ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

0 2

2

222

3

3

32

22 expexp

321

112

VVV

jj αβα+ × × ( ) ( )expexp

0 0 2

2

3

3 ξγαξα ξ

Vz

V

z jj

∫ ∫⎢⎢⎣

⎡⎢⎣

⎡×⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

( ) ,exp0 1

111

1 ξγθθτηγθαγ

dddVV

j

∫⎥⎥⎦

⎤⎥⎦

⎤⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

where, ij

iij

i apV 2/=α , jip : probability of sifting

of j-th particles probability at the i-th sieve, ia2 : step of i-th sieve, −iV the average velocity of particles at


60

the i-th sieve, ( )zN ij

i ,τ : the average value of solutions of the Eq. (5).

Solutions ( )zN ij

i ,τ for the first three levels (i=1, 2, 3) taking into account the fact that separated material is supplied only at the beginning of the upper (first) level, take the form:

( ) ( )exp,1

111011 ⎟⎟

⎠

⎞⎜⎜⎝

⎛−= z

VNzN

jjj αττ

( ) ( )2

2

2

121022 exp, z

VVNzN

jjjj ααττ ⎜⎜

⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

( )22111

1 ///exp VVzV

jjj

ααα−⎟⎟

⎠

⎞⎟⎟⎠

⎞⎜⎜⎝

⎛−−

( ) ( ) ⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎢

⎣

⎡= eexp,

2

2

32

2131033 z

VVVNzN

jjjjj αααττ

( )( )−−−⎟⎟⎠

⎞⎟⎟⎠

⎞⎜⎜⎝

⎛− )////(/exp 33112211

1

1 VVVVzV

jjjjj

ααααα

−−−− /))exp()(exp(2

2

3

3 zV

zV

jj αα

( ) ⎥⎦

⎤−− )//)///(( 33222211 VVVV jjjj αααα

For the remaining levels of the classifier ( ni ,4= ) solutions of differential Eq. (5) are determined similarly.

To determine the average velocity cpV and amplitude of the particle velocity aV in the calculation of the probability of sifting in the cell depending on Eq. (4) is necessary to investigate the vibrational motion of the separated material at the surface of the sieve.

The study of vibrational motion of granular material for the case of thin-layer classification can be carried out within the framework of the description of motion of single particles. For harmonic vibrations the rectilinear inclined sieve differential equation of relative motion of the particles of the vibrating surface can be represented as:

( ) ( ) xfgAx ср∓

−±

−=ρ

ραωρ

ρβωcos

sinsincos

cos2

where A and ω are the amplitude and frequency

of oscillations, and α and β are angles of slope and

vibration, ρ is the angle of friction, срf is the coefficient of resistance to the motion of a particle in the medium. The upper and bottom signs corresponds to the forward and backward movement.

The regular mode of bi-directional movement with instant stops without flipping has a practical interest. This mode is often used in separating machines and allows you to make full use of the separated material residence time on the sieve surface. In this mode, the relative velocity in the positive and negative directions has the form:

( ) −−⎢⎣

⎡+

= ±± )cossin((

11

02 ττω

τ ff

axo

−−−− ±±± )))(exp()cossin( 00 ττττ ff

)))(exp(1( 00

⎥⎦

⎤−−−− ±

± ττffz

where ,

cos)(cos2

ρρβω ∓Aa =±

( )( ) ,

cossin

2 ρβρα

ω ∓±

=± Agz ,/ωсрo ff = ,tωτ = ±τ :

moments of the beginning of sliding in the positive (“+” sign) and negative (“-” sign) directions, defined by the system of nonlinear equations:

( )( )⎩

⎨⎧

=+=

+−

−+

02,0

πττ

xx

When the particle moves along the inclined surface of a vibrating sieve with respect to its velocity changes its direction twice every ωπ /2 second, therefore, the amplitude of the relative velocity аV of the normal law kV used in Eq. (4), accept two different values. For small angles in the range (0-5), the amplitude of the velocity can be defined as follows:

( ).max ττττ +≤≤ −+

= xVа

Required rate of displacement can be obtained with different combinations of the basic parameters of operation of device: the amplitude A and frequency of oscillations ω , the slope angle α and the angle of vibration β . The problem lies in the choice of a such combination that would provide optimal operation conditions. The average rate of vibrational motion can


61

be determined by the formula:

( ) ( ) πωττττπτ

τ

τ

τ

2/)(2

dxdxVср ∫∫+

−+

+

−

−

+

+=

3. Identification of Kinetic Models and Evaluation of Separation

Identification of mathematical models of the object is an important step in constructing of an adequate model. During identification of the constructed mathematical models it is necessary to define parameters kV and σ . They are defined from sieves residues by comparing the experimental and calculated values of the coefficient of extraction or separation efficiency.

The coefficients of kinetic Eq. (5) jiα determine

the probability of passage of particles of the j-th fraction through the i-th sieve openings per unit time and thus the speed of the process is characterized. According to Eq. (5), the coefficients are explicitly dependent on the mean velocity and the probability of sifting, therefore, also depend on the process control parameters: the amplitude A and frequency of oscillation ω of the working body, angles of slope and vibration α and β . Parameters of normal distribution of velocity as a function of Eq. (4) kV and σ are determined experimentally for some well-defined high-speed modes of operation of the apparatus and take into account the influence of factors that are not explicitly included in the relationship.

Therefore, estimates of coefficients of kinetic equations used to reconcile a mathematical model of separation with the real process. The average value of the j-th fraction output from i-th sieve based on solutions (11) per the time

itΔ is determined by the formula:

exp00 0

=⎟⎟⎠

⎞⎜⎜⎝

⎛−∫ ∫

Δ

i

jij

i

t Lj

i VdtdzzV

Ni i αα

,exp10 ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−Δ= i

i

jij

iii LV

NtV α (12)

where jiN 0

is the number of particles of the j-th fraction per unit length of i-th sieve at its initial section.

Expression j

iii NtV 0Δ in Eq. (12) determines the total number of particles of j-th fraction, coming on the i-th sieve during time

itΔ . Then the coefficient of extraction of j-th th fraction from i-th sieve, using Eq. (12) is determined by the formula:

,exp ⎟⎟⎠

⎞⎜⎜⎝

⎛−= i

i

jij

i LVαη ni ,1= ; 1, += iij

where j i= —the large-sized (unsifted) fraction, 1j i= + —the next sized (sifted) fraction.

The effectiveness of separation of −i th sieve can be calculated from equation ( ) ,%1001 1 ×−= +i

iiiiE ηη

ni ,1= , where 1+iiη —unsifted proportion of

small-sized fraction in a large-sized one (the proportion of impurities in the target product).

On the basis of experimental researches for each sieve of the qualifier statistical assessments of probability of sifting are defined as:

,ln2 ,, sji

i

isji L

aP η−= ni ,1= ; 1, += iij , sks ,1= (13)

where sk is the amount of experimental points included in the sample ( sk ≥ 4), carried out at different speed regimes.

Experimental values of extraction coefficient sj

i,η are determined by the unsifted residuals at sieves.

The choice of high-speed mode (parameters βαω ,,,A ) and characteristics of the sieves ( ii DL , ) is necessary to conduct in view of requirements for final products of the separation of the first and second particle size fractions. The first fraction, not being sifted, should go out from the sieve, and the second being sifted, leaves a slight unsifted residue:

,1, ∗> isi

i ηη ,0 2,1 ∗+ << isi

i ηη ni ,1=

Values ∗∗ 21 , ii ηη are selected in accordance with the requirements of the final products of separation.

Based on statistical estimates of the probability of sifting Eq. (13) for selected speed regimes the observed values p-quintile of the velocity (dimensionless rate at


62

which the velocity probability is high as p) are determined

),/5,0( ,1, jiг

sji

sji ppU −Φ= −

ni ,1= ; 1, += iij ; =s sk,1

where [ ]∗Φ−1 is the operation of Laplace function argument computing (the calculation of roots of the equation ))( pх =Φ ;

( ) ( )dttхx

∫ −=Φ0

2 2/exp21π

Estimated value of −p quantile of the velocity can be found by the formula:

( )/, ji

jik

sia

sji VVU σ−=

ni ,1= ; 1, += iij ; ,1 sks = (14)

where, siaV is the amplitude of the relative velocity of

the particles on the i-sieve. The unknown parameters of Eq. (14)

ji

jkiV σ, are

determined by comparing the observed and calculated values of p- quantile of the velocity:

min)(,

2,

1

,j

ij

ki

s

V

sji

k

s

sji UU

σ→−∑

=

(15)

For fractions that do not have unsifted residues (small-sized), the parameters are determined from the distribution of the approximating function obtained on the basis of the optimal values of Eq. (15) for the fractions with residues.

For woven sieves dependence of the parameters

kV and σ from cells and particles sizes can be represented as:

( ) ( )2

, lBDAr

glDV VVk −=

( ) ( ),, σσσ BlDVAlD k += (16)

where r is the radius of the lateral face of the woven sieve, l-the characteristic size of the j-fraction, D-the size of the holes of the i-th sieve cells.

For penetrative sieves parameters of speed probability can be found from the following approximating function:

( ) ( )lBDAlglDV Vvk −=, (17)

Dependency ratio Eqs. (16) and (17) σσ BABA vv ,,, determined by comparing the

calculated values kV and σ and their optimal values found previously from Eq. (15):

vv BA

jik

n

i

i

ijjik VlDV

,

2

1

1

min)),(( →−∑∑=

+

=

,

σσσσ

BA

ji

n

i

i

ijji lD

,

2

1

1

min)),(( →−∑∑=

+

=

Type of dependency Eqs. (16) and (17) agrees with the theoretical formulas, known in the scientific literature.

The adequacy of the mathematical model is established by comparing the calculated and experimental values of the coefficient of extraction or separation efficiency.

4. Calculating Experiment and Discussion

According to the obtained solutions numerical calculations (numerical experiment) were carried out for the specific conditions of the classifiers. For the separation granular material with polymer-based particles which have a cylindrical shape sized in the range of 0.2 to 1.2 mm and 0.4 mm in diameter was used. Granular material density сρ was 1,150 kg/m3, porosity -23%, content of fraction of 0.4-0.8 mm size in the separated material was 70%-85%. The angle of sieves α changed in the range 0-5o, the oscillation amplitude A: 3-5 mm, the oscillation frequency of the working body πω 2 : 6-8 s-1, the angle of vibration: 5-10o, the height of the exit slit of the hopper HO: 1.5-3 mm, sieves length L: 1.5 m. The average speed was varied in the range 0.05-0.07 m/s, the velocity amplitude Va: 0.15-0.20 m/sec.

The most important parameters that determine the speed mode of vibration-displacement regime, the probability of sifting in a cell and the separation efficiency are the amplitude A and oscillation frequency ω , and the angles of slope α and vibration β , the length and size of sieve cell L and D . Calculations show that for a given separation and


63

Fig. 3 The distribution of the linear density of selected fractions along the first and second screens.

purity of the chosen values ohDL ,, of design parameters are the optimal values of speed regime, therefore, the parameters βαω ,,,A for which the efficiency takes its maximum value. Given the demands on the purity of separation (the content of non-targeted is at most 5%), separation efficiency was

90%-93%, while the relative error of the calculated values of the separation efficiency and experimental does not exceed 15%.

Fig. 3 shows the results of calculations of linear densities 1N and 2N for different fractions along the first and second sieves. These solutions allow to set and solve the optimization problem of the classifiers.

=11α 5.78 × 10-3; =2

1α 8.42 × 10-2; =31α 2.57 ×

10-1; =41α 9.08 × 10-1; =1

2α 3.66 × 10-4; 22α = 2.77 ×

10-3; =32α 7.18 × 10-2; =4

2α 6.93 × 10-1, (с-1); =срV 0.05 m/c; the sizes of fractions, 1: (0.8-0.9); 2:

(0.7-0.8); 3: (0.6-0.7); 4: (0.5-0.6), (mm).

References [1] Nepomnyashy, Y. A. Kinetics of the Some Processes of

Processing of Disperse Materials. Theoretical Bases of Chemical Technologies 1973, 7, 754-763.

[2] Nigmatullin, R. I. Dynamics of Multiphase Environments. Science 1987, 464.

[3] Tihonov, V. I.; Mirinov, M. A. Markoff Process. Sov. Radio 1977, 206.

[4] Kazakov, V. A. Introduction in the Theory of the Markov-Processes and Some Radio Engineering Problems. Sov. Radio 1973, 231.

[5] Klyatskin, V. I. Stochastic Equations: V.2. Coherent Phenomena in Stochastic Dynamical Systems. Fizmatlit 2008.

Separation Processes of Granular Materials by Sizes at the Sieve Classifiers

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