НАУ

Ma

Abst

create a ma

regard to th

Key w

Intro

In th

grain, and

pebbles. T

drying, pr

large adm

which hav

impurities

Feat

through a

these par

considere

and impro

cylindrical

Skal

axis with

vibration.

by the rota

right end Σ

Math

processes

«fast» mo

movemen

2

ωaj =

environme

grains. Oc

atoms or

these con

which, tha

УЧНИ ТРУД

athemati

B

tract: To des

athematical m

he influence o

words: scalp

oduction

he pile of g

d there ar

This negat

rimary and

mixtures of

ve a size

s. Most wid

ture of the

apertures o

rticles to t

ed as a de

ove the qu

l screens i

perator co

a constant

The mixtur

ation and v

Σ2, forming

hematical

s. We sha

otion of gr

nts device.

(a - the a

ent arise

ccurs chao

molecules

nditions, su

at same law

ДОВЕ НА

cal mode

Bogdanovic

scribe the pro

model of the

of separation

perator, cylin

rain that is

re large im

tively affec

d seconda

grain hea

much larg

despread a

e process

of a sieve.

the surfac

eterminant.

uality of th

mpose vib

nsists of a

t angular v

re of grain

vibration im

g a layer of

Fig

modeling

ll mention

ranular me

At a certa

amplitude

instant ten

otic movem

of the gas

ubject to t

w of the N

РУСЕНСК

el of the

ch Sergey

ocess of sepa

process on t

n.

ndrical sieve,

s served at

mpurities: p

cts the resu

ry treatme

ap use mac

ger than th

re the drum

of skalper

The probl

ce of a sie

For a con

he process

ration.

cylindrical

velocity Ω

is served

mpacts grai

f variable c

g.1. Settleme

g: Inside

the main o

edia, whic

ain intensity

of vibrati

nsile force

ment collid

s. This phe

he laws of

avier-Stok

КИЯ УНИВ

- 175 -

process

, Vladimír

aration of the

the basis of t

grain mass,

t the proce

pieces of

ults of the

ent, calibra

chines her

he size of

m scalpera

ration is re

lem with th

eve. This

nsiderable

s of skalp

l sieve dru

(Fig.1). Th

inside, in t

n waking u

ross-sectio

ent scheme о

the drum

of them. In

ch is provi

y of these

on oscilla

es that lea

ding betwe

enomenon

f dynamics

kes and the

ВЕРСИТЕТ

s of skalp

Kročko, Hr

e grain mixtu

the laws of d

vibration.

essing afte

straw, ear

further pro

ation. It is

reinafter ca

f the grain

ators [1].

elatively ea

his process

step in th

e increase

peration pr

m S0, whic

he rotary m

he initial re

up and coa

on with a fr

оf a skalpera

skalperto

n the futur

ded by th

impact, de

tions, ω -

ad to a br

en the par

leads to th

s of viscou

e phenome

- 2014, то

peration

risto Beloe

ure in the dru

dynamics of v

r harvestin

rs, weeds,

ocesses of

recommen

alled skalp

ns, but sm

asy passag

s is in the

he process

in product

oposed to

ch revolves

movement

egion of the

rse impurit

ee surface

ator

r have the

e we will c

e influenc

etermined

- their fre

reach of c

rticles, sim

he fact tha

us medium

enological

м 53, сери

grain he

ev

um scalperat

viscous liquid

ng , togethe

, lumps of

f processin

nded for r

perators, s

maller than

ge of grain

rapid pen

s of skalp

tivity of sk

o rotation m

s around a

of the dru

e drum (ne

ties are mo

e S1.

e complex

consider a

ce of smal

by the cha

equency),

contact be

milar to the

at the envir

m, rheolog

coefficient

ия 1.1

eap

tor asked to

d with

er with the

f clod and

g of grain:

removal of

ieve holes

the large

n particles

etration of

peration is

kalperators

movement

horizontal

um overlap

ear Σ1) and

oved to the

x dynamic

regime of

ll vibration

aracteristic

within the

tween the

motion of

ronment in

ical law of

ts of which

e

d

f

s

e

s

f

s

s

t

l

p

d

e

c

f

n

c

e

e

f

n

f

h

НАУЧНИ ТРУДОВЕ НА РУСЕНСКИЯ УНИВЕРСИТЕТ - 2014, том 53, серия 1.1

- 176 -

depends on the intensity of the vibration. So, the stress tensor in the Cartesian system of

allelic worldview is:

2ˆ ˆ

ˆ v 2

3

p div Vσ δ λ μ μ⎛ ⎞

= − + − +⎜ ⎟

⎝ ⎠

, (1)

where ,λ μ - bulk and shear viscosity coefficients, in this case reveal-related functions of

the intensity of vibration , p - pressure, ˆδ - a single tenzor,

( ) ( ) ( )( )1 1 2 3 2 1 2 3 3 1 2 3

v v , , ,v , , ,v , ,x x x x x x x x x=

- the velocity field of a continuous medium, ˆ

V -

tensor of deformation velocities equal to:

1 2 1 3 1

1 1 2 1 3

2 1 2 3 2

1 2 2 2 3

3 1 3 2 3

1 3 2 3 3

v 1 v v 1 v v

2 2

1 v v v 1 v v

2 2

1 v v 1 v v v

2 2

ik

x x x x x

V

x x x x x

x x x x x

⎛ ⎞⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂

+ +⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂

⎝ ⎠ ⎝ ⎠⎜ ⎟

⎜ ⎟⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂

⎜ ⎟= + +⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂

⎝ ⎠ ⎝ ⎠

⎜ ⎟

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟

∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

(2)

At low velocities of motion of continuous media can be considered that this

movement is made with preservation of volume, as in the case of an incompressible fluid.

Mathematical septic condition expression of incompressibility is written in the form:

v 0div =

. (3)

Movement of fluid through a porous medium can be roughly described as a

movement it through the curved cylindrical channels. Mathematical description of such a

movement gives filtration theory. The basic law of filtration - Darcy's law - indicates the

proportional dependence of the velocity of motion of the medium pressure gradient with

the proportionality coefficient d

K [2]:

vd

K p= ∇

These considerations suggest that the rate of passage of a grain flow through the

holes of the drum is proportional to the pressure drop in grain flow inside and outside of

the sieve cloth cylinder :

( )vr d a

K p p= − . (4)

You must consider the impact of surface sieve flow in the form of friction, submitting

to the law of the Coulomb friction:

k

T f p= (5)

and strength of hydraulic resistance:

c

T wτ

λ= , (6)

where ,k c

T T - forces per unit area of the wall; f - dry sliding friction coefficient; p -

pressure environment; λ - coefficient of hydraulic resistance, defined empirically; wτ

-

relative tangent component of the speed of «liquid» particles of the internal surface of a

sieve.

On the free surface are dynamic boundary conditions. As for over grain flow is

absent, the pressure outside the scope of the grain mixture is equal to zero. Then the

dynamic conditions meet the condition of absence of tension. Denote by n

p

tension,

according to the formulas of Cauchy, we obtain the following expression:

НАУЧНИ ТРУДОВЕ НА РУСЕНСКИЯ УНИВЕРСИТЕТ - 2014, том 53, серия 1.1

- 177 -

1

3

, 1

0n k ki i

i kS

p n eσ

=

≡ =∑

, (7)

where k k

n n e=

- the unit normal to the surface 1

S , external to the volum of grain (on

repeated indices according to the rules of tensor calculus here is the summation from 1 to

3), ( 1,2,3)i

e i =

- vector basis in some Cartesian coordinate system.

The kinematic condition:

v v vz x y

F F F

t x y

∂ ∂ ∂

= + +

∂ ∂ ∂

. (8)

Assume condition: consumption of grain flow1

Q and depth of filling 0z = in are

assigned values. Taken condition of the incompressibility environment gives the first

equation model , having in the chosen coordinate system type:

vv v

0

yx z

x y z

∂∂ ∂

+ + =

∂ ∂ ∂

,. (9)

The following equation, called the vector nature, the equation of motion expressing

the second law of mechanics of a continuous medium:

v

ˆ

d

div g

dt

ρ σ ρ= +

, (10)

where ρ - the density of the medium (in this case the constant value); g

- the intensity of

the external forces acting on the environment (in this case equal to acceleration free fall),

ˆσ - stress tensor, which has components, determined by the ratio of (1).

Expressions (9) and (10) are a mathematical model of the process scalperation.

Conclusions

1. The mixture of grain in rotating drumskalperator superimposed vibrations obeys

the laws of dynamics of viscous medium, rheological law which is similar to the law of the

Navier-Stokes equations, and the phenomenological coefficients of which depends on the

intensity of the vibration.

2. The process of separation of granular mixture in skalperator can be considered as

a viscous fluid through a porous medium.

3. Identified correlation between the factors affecting the process of skalperator,

allow you to create a mathematical model of this process.

References

1. Zik O. Rotationally induced Segregation of Granular Materials /O. Zik, D. Levine,

S.G. Lipson, S. Shtrikman, J. Stavans //Phjsical. rew. let. V.-73 5 1994. Pp. 644-647.

2. Darcy Henry. Les fontaines publiques de la ville de Dijon: exposition et application

des principes à suivre et des formules à employer dans les questions de distribution

d'eau..../ Henry Darcy — Paris: V. Dalmont, 1856. — 647 p.

About the author

Assistant Bogdanovich Sergey, Department of theoretical mechanics and machine

parts Kharkiv national technical University of agriculture named Petro Vasilenko, Kharkiv,

Artioma st. 44, 61002, E-mail: bogdanovichserg@gmail.com

dr.h.c. prof. Ing.Vladimír Kročko, CSc., Department of Quality and Engineering

Technologies, Faculty of Engineering, Slovak University of Agriculture in Nitra, Tr. A.

Hlinku 2, 949 76 Nitra, Slovak Republic, e-mail: vladimir.krocko@uniag.sk

Prof. Hristo Beloev, DTSC, University of Ruse, 7017, 8 Studentska Str. Bulgaria

This paper has been reviewed.