Increase Hydraulic Pressure by Compressing the Roller
Shukhrat Khurramov
Tashkent University of Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan
https://doi.org/10.5281/zenodo.10471075
Keywords:
roller squeezing, moisture filtration, hydraulic pressure, squeezing area.
Abstract:
The results of the study of hydraulic pressure in the roll squeezing of wet materials are given.
Mathematical models of hydraulic pressure distribution in the squeezing zone are developed. It is
revealed that the hydraulic pressure in the compression zone increases from zero at the initial
contact point to a maximum at a point lying on the line of centers. The distribution patterns of
hydraulic pressure in the strain restoration zone depend on the length of its part, where the fluid
flows from the wet material into the roll coating.
Introduction
In many industries, roll technological machines are
widely used for the mechanical processing of various
materials. A special group is roller machines for
squeezing wet materials.
In the process of roller squeezing of wet materials, the
simultaneous occurrence of two phenomena is
observed - contact interaction and moisture filtration.
At the same time, a change in the indicators of contact
interaction affects the change in moisture filtration, and
vice versa. Therefore, the study of one of the
phenomena without taking into account the other does
not allow one to obtain reliable parameters of the
process of roller squeezing of wet materials.
Accordingly, to describe the roll squeezing of wet
materials, it is necessary to jointly solve two problems:
the first is contact interaction in two-roll modules
(contact problem); the second is moisture filtration in a
deformable inhomogeneous porous medium (hydraulic
problem).
The identity of the laws that take place during the
extraction of various materials causes a desire to reveal
the physical picture of the phenomenon of moisture
squeezing. Many researchers have tried to describe the
movement of a liquid during its roll squeezing [1–4]
References [5-10] are devoted to the study of the
phenomenon of contact interaction under the roll
squeezing of wet materials. Mathematical models of
the shape of roll contact curves, friction stresses, and
distribution of contact stresses in a generalized two-roll
module are developed in these studies.
One of the main hydraulic problems of the theory of
roller squeezing of fibrous materials is the modeling of
the hydraulic pressure distribution in the squeezing
area [2].
An analysis of the studies devoted to the hydraulic
problems of wet materials roll pressing [5-11] showed
that the existing models of hydraulic pressure
distribution in the pressing area were obtained with the
introduction of models of roll equipment and materials
that do not correspond to the real physical phenomena
of wet materials roll pressing. They do not consider the
presence of roll coating. However, in roll squeezing
machines at least one of the rollers has an elastic
coating. Therefore, the existing models of hydraulic
pressure distribution do not allow the disclosure of the
hydraulic phenomenon of wet materials roll pressing.
In [20], analytical dependencies were determined that
describe the distribution patterns of the hydraulic
pressure in the pressing area for a symmetrical two-roll
module. In order to further develop theoretical
concepts, [1-3], the object of study is a generalized
two-roll module, in which the rolls are located relative
to the vertical line with an inclination to the right at an
angle of
, have unequal diameters
)
(
2
1
D
D
with
elastic coatings, and a layer of wet (processed) material
has a uniform thickness and is fed tilted downward
with respect to the line of centers at an angle of
1
(Fig. 1).
Fig. 1 Scheme of a two-roll moduleof squeezing machine
Materials and methods
The lower roll contact curve (curve
A
1
A
2
) consists of
two zones
3
1
A
A
and
2
3
A
A
. In zone
3
1
A
A
, the
fibrous material and the roll coating are compressed,
and in
2
3
A
A
the strain is restored.
Let us first consider the process of fluid filtration in
zone
3
1
A
A
.
The equation of the contact curve of the lower roll in
the compression zone for the
considered two-roll module has the following form [4]:
,
0
)
(
,
)
cos(
)
cos(
1
1
11
1
11
11
1
11
11
11
11
11
1
11
+
+
−
+
+
+
+
=
k
k
R
r
(1)
,
)
sin(
)
sin(
1
21
1
*
1
21
11
1
11
11
−
+
=
m
H
m
k
;
))
1
(
)
1
(
(
)
(
))
1
(
)
1
(
(
)
(
1
*
11
*
11
11
11
11
11
11
0
11
*
11
*
11
11
11
11
*
11
*
11
11
H
m
A
m
A
l
m
A
h
m
A
m
A
l
m
A
ср
ср
−
−
−
+
−
−
−
−
=
)
sin(
)
sin(
21
11
1
21
1
0
11
+
−
=
h
;
,
)
(
2
)
(
2
sin
1
)
(
1
11
1
11
1
11
+
+
−
=
R
l
ср
here
−
11
m
is the coefficient of strengthening of the
points of the elastic coating of the lower roll under
compression,
−
*
1
m
is the coefficient of strengthening
of the points of the processed material under
compression.
Hence
.
cos
sin
cos
1
11
2
11
11
11
11
1
11
11
11
k
R
k
r
+
=
(2)
In zone
3
1
A
A
, the fibrous material is compressed, so
the fluid flows from it into the roll coating along the
polar angle [2, 12].
The process of fluid flow is considered continuous and
steady.
The feed rate of the fibrous material in the contact area
is constant and equal to
.
m
v
The fluid rate in the contact area is variable and equal
to [12]:
),
)
(
)
((
3
11
3
1
11
11
11
+
+
+
−
=
b
u
0
)
(
11
1
11
+
+
−
, (3)
where
,
))
cos(
1
)(
1
(
3
)
cos(
1
11
11
11
11
11
0
11
1
11
1
11
+
+
+
+
=
k
k
h
R
v
b
m
.
)
sin(
)
sin(
21
11
1
21
1
0
11
+
−
=
h
In [2], assuming the working hypothesis of the
orthogonality of the maximum and minimum porosity,
the applicability of the generalized Darcy law for an
anisotropic medium was established
K
u
n
P
n
−
=
, (4)
with filtration factor
,
sin
cos
1
min
2
max
2
K
K
K
+
=
where
−
u
P
n
,
are the hydraulic pressure and
filtration rate in direction
n
;
−
is the fluid viscosity
coefficient;
−
max
K
is
the
maximum
filtration
coefficient in the direction across the surface of the
material (along the
Оу
axis);
−
min
K
is the minimum
filtration coefficient in the direction along the warp
threads of the material (along the
Оx
axis).
According to this dependence, the filtration direction
angle varies within
o
90
0
. On roller squeezing
machines, where the rollers have an elastic coating, at
each point of the roller contact curve, the resulting
filtration rate is directed relative to the direction of the
material feed at some angle of
−
o
90
, close to the
polar angle of
[2]. Therefore, for this case, we can
take
=
−
o
90
[12]. Then the expression
for the filtration coefficient takes the following form:
.
cos
sin
1
min
2
max
2
K
K
K
+
=
(5)
With formulas (3), (4), and (5), we obtain
)
)
(
)
((
)
(
cos
)
(
sin
3
11
3
1
11
min
11
11
2
max
11
11
2
11
11
11
+
+
+
+
+
+
=
K
K
b
n
P
n
assuming
2
11
11
2
)
(
1
)
(
cos
+
−
+
и
2
11
11
2
)
(
)
(
sin
+
+
)
(
)
)
(
)
((
)
(
1
)
(
11
11
3
11
3
1
11
2
11
max
11
min
11
max
11
min
11
11
11
11
+
+
+
+
+
−
−
=
+
d
dn
K
K
K
K
b
d
dP
n
. (6)
From Fig. 1, it follows that
)
cos(
11
11
11
+
=
r
n
.
Hence
)
sin(
)
cos(
)
(
11
11
11
11
11
11
+
−
+
=
+
r
r
d
dn
or considering equalities (1) and (2)
)
(
1
)
sin(
1
)
(
11
11
1
11
11
11
1
11
11
+
+
−
+
+
−
=
+
k
R
k
R
d
dn
.
With this in mind, from equality (6) we obtain
+
−
−
+
−
=
+
2
11
max
11
min
11
max
11
min
11
11
11
11
1
11
11
)
(
1
)
1
(
)
(
K
K
K
K
k
b
R
d
dP
n
)
)(
)
(
)
((
11
3
11
3
1
11
+
+
+
+
or being limited to the terms of the third power relative
to
)
(
11
+
).
(
)
(
)
(
)
1
(
)
(
11
11
3
11
max
11
min
11
max
11
min
11
11
11
3
1
11
11
1
11
+
+
−
+
−
+
+
=
d
K
K
K
K
k
b
R
dP
n
(7)
After integration, we get
,
)
(
2
)
(
11
2
11
4
11
max
11
min
11
max
11
11
11
C
K
K
K
c
P
n
+
+
−
+
−
=
(8)
where
))
cos(
1
)(
1
(
12
)
)(
cos(
1
11
11
11
11
11
0
11
min
11
3
1
11
1
11
2
1
11
+
+
+
+
+
=
k
k
h
K
R
v
с
m
Constant
11
C
is determined by initial condition
0
))
(
(
1
11
11
=
+
−
n
P
:
.
)
(
)
(
2
4
1
11
max
11
min
11
max
11
2
1
11
11
+
−
−
+
=
K
K
K
C
Then we obtain
,
)
)
(
)
((
2
)
)
(
)
((
2
11
2
1
11
max
11
min
11
max
11
2
11
2
1
11
11
11
+
+
+
−
−
+
−
+
=
K
K
K
c
P
n
(9)
where
.
0
)
(
11
1
11
+
+
−
This formula determines the patterns of distribution of
hydraulic pressure along the contact curve of the lower
roll in the compression zone.
The patterns of distribution of hydraulic pressure along
the contact curve of the
lower roll in the strain restoration zone are determined
similarly:
,
)
)
(
)
((
2
)
)
(
)
((
2
12
2
4
14
max
12
min
12
max
12
2
12
2
4
14
12
12
+
+
+
−
−
+
−
+
=
K
K
K
c
P
n
(10)
where
;
0
2
12
12
+
+
;
1
0
),
(
1
2
12
1
4
14
+
=
+
))
cos(
1
)(
1
(
12
)
)(
cos(
2
12
12
12
12
12
0
12
min
12
3
2
12
2
12
2
1
12
+
+
+
+
+
=
k
k
h
K
R
v
с
m
.
The patterns of distribution of hydraulic pressure along
the contact curve of the upper roll are determined
likewise.
They have the following form:
,
2
)
)
(
)
((
)
)
(
)
((
2
21
2
1
21
max
21
min
21
max
21
2
21
2
1
21
21
21
−
+
+
−
−
−
−
−
=
K
K
K
c
P
n
(11)
where
,
0
)
(
21
1
21
−
−
−
;
))
cos(
1
)(
1
(
12
)
)(
cos(
1
21
21
21
21
21
0
21
min
21
3
1
11
1
21
2
2
21
−
+
+
−
−
=
k
k
h
K
R
v
с
m
,
2
)
)
(
)
((
)
)
(
)
((
2
22
2
4
24
max
22
min
22
max
22
2
22
2
4
24
22
22
−
−
+
−
−
−
−
−
=
K
K
K
c
P
n
(12)
where
;
0
2
22
22
−
−
;
1
0
),
(
2
2
22
2
4
−
=
−
))
cos(
1
)(
1
(
12
)
)(
cos(
2
22
22
22
22
22
0
22
min
22
3
2
22
2
22
2
2
22
−
+
+
−
−
=
k
k
h
K
R
v
с
m
.
Thus, analytical dependencies (9)-(12) are determined,
which describe the patterns of distribution of hydraulic
pressure in the pressing zone for the generalized two-
roll module shown in Fig. 1.
Graphs of changes in hydraulic pressure along the roll
contact curve are shown in Fig.2.
Fig. 2. Graphs of changes in hydraulic
pressure along the roll contact curve:
;
2
1
2
;
4
1
1
12
14
12
14
=
−
=
−
.
4
,
4
3
3
12
14
12
14
=
−
=
−
Results
Mathematical models of hydraulic pressure distribution
in the squeezing zone were developed.
Conclusions
From the analysis of the calculated data and
graphs, it follows that the hydraulic pressure in the
compression zone increases from zero at the initial
point of contact, to a maximum at a point lying on the
line of centers. The distribution patterns of hydraulic
pressure in the strain restoration zone depend on the
length of its part, where the fluid flows from the wet
material into the roll coating.
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