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TYPE
Original Research
PAGE NO.
19-23
DOI
OPEN ACCESS
SUBMITED
09 February 2025
ACCEPTED
12 March 2025
PUBLISHED
08 April 2025
VOLUME
Vol.05 Issue04 2025
COPYRIGHT
© 2025 Original content from this work may be used under the terms
of the creative commons attributes 4.0 License.
Filtration of Suspensions
with The Formation of a
Nonlinearly Compressible
Sedimentary Layer
Parmonov J.T.
Samarkand State University of Architecture and Civil Engineering named
after Mirzo Ulugbek, Jizzakh State Pedagogical University, Uzbekistan
Nishanov.U.A.
Samarkand State University of Architecture and Civil Engineering named
after Mirzo Ulugbek, Jizzakh State Pedagogical University, Uzbekistan
Nurmatov K.J
Samarkand State University of Architecture and Civil Engineering named
after Mirzo Ulugbek, Jizzakh State Pedagogical University, Uzbekistan
Abstract:
The article considers the problem of filtration
of suspensions with the formation of a nonlinearly
compressible sedimentary layer. Based on numerical
calculations, the dependence of filtration characteristics
on nonlinear effects is established.
Keywords:
Suspension, filter, moving boundary, filter
layer, sediment, pressure.
Introduction:
Filtration of suspensions refers to
complex technological processes and depends on a
large number of micro- and macrofactors. Models of
filtration of suspensions are based on fundamental
equations of mechanics of multiphase, multicomponent
media and the theory of filtration consolidation [1,2,3].
The problem of pressure distribution in the sediment
layer is reduced to the solution of the classical parabolic
equation with moving boundary conditions. Real
physical conditions during filtration of suspensions are
such that the sediment is heterogeneous in its structure
and filtration-capacitive properties and consolidates
under the action of pressure and other forces. Thus, in
the general case, the consolidation coefficient is
variable, depending on the pressure distribution over
the sediment thickness. In this paper, the process of
sedimentation during filtration of suspensions is
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analyzed taking into account the variability of the
consolidation coefficient.
Let there be a flat filter element. The suspension is
placed in such a way that it contacts the initially
existing filter layer of some thickness. Under the action
of the pressure difference, which remains constant
over time, the process of separation of the suspension
begins, the thickness of the sediment continuously
increases due to the movement of the sediment -
suspension boundary. Thus, two regions can be
distinguished- the filter layer region, and - the sediment
region, where is the moving boundary. Due to the listed
conditions, the Stefan problem can be formulated for
the filtration process. Let be the pore pressure, - the
liquid pressure at the initial moment of time, at the
entrance to the sediment layer, at the exit from the filter
layer.
The mathematical model of filtration is written as
follows [1,2]
( )
2
2
z
p
p
b
t
p
=
,
T
t
0
,
2
z
,
(1)
( )
( )
z
p
z
p
0
0
,
=
,
1
z
,
(2)
( )
2
,
0
p
t
p
=
T
t
0
,
(3)
( )
1
,
p
t
t
h
p
=
,
T
t
0
,
(4)
dt
dh
l
z
p
=
,
T
t
0
,
(5)
Where
(
)
1
0
0
)
(
,
)
0
(
p
p
e
b
p
b
z
h
−
=
=
- consolidation ratio,
(
)
0
2
1
2
0
z
p
p
z
p
p
−
+
=
,
0
0
z
z
u
r
l
=
,
r
-
sediment resistivity,
- viscosity coefficient,
u
- external sedimentation coefficient,
- parameter.
Introducing a new variable
(
)
1
)
(
0
−
=
=
−
p
p
e
a
b
d
v
,
1
0
1
p
e
b
a
−
−
=
we will receive
+
−
=
1
ln
1
a
v
p
.
(6)
Then equation (1) is transformed to the form
=
z
p
v
v
k
z
t
v
)
(
)
(
,
(
)
a
v
v
k
+
−
=
1
)
(
.
(7)
After transforming conditions (2) - (5) we arrive at
( )
( )
(
)
1
|
0
0
0
−
=
=
−
=
z
p
t
e
a
z
v
v
,
(8)
(
)
1
|
2
2
0
−
=
=
−
=
p
z
e
a
v
v
,
(9)
(
)
1
|
1
1
)
(
−
=
=
−
=
p
t
h
z
e
a
v
v
,
(10)
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dt
dh
l
z
p
v
v
k
=
)
(
)
(
.
(11)
To solve problem (7)-(11) we use the finite difference method [4].
Let's introduce a uniform grid
z
with a step
f
j
m
i
if
z
z
i
i
+
=
=
=
+
=
,...,
1
,
0
,
|
2
1
, where in time
we will use a non-uniform grid [4]
T
t
N
m
m
j
t
t
t
t
t
N
m
j
j
j
=
+
+
=
=
+
=
=
=
−
,
,...,
2
,
1
,
0
,
|
1
with variable pitch
0
j
. The time interval step should be selected
1
,
+
j
j
t
t
so that the moving boundary moves
exactly one step along the spatial grid. This approach is known as the grid node front catching method [4]. For the
boundary node of the dynamic stack we use the notation
( )
f
i
t
f
z
j
j
=
=
.
We approximate equation (7) with a purely implicit scheme
−
−
−
=
−
+
−
+
+
+
+
+
+
f
v
v
f
v
v
f
v
v
j
i
j
i
i
j
i
j
i
i
j
i
j
i
1
1
1
1
1
1
1
1
1
,
1
,...
2
,
1
−
=
N
i
,
(12)
где
( ) ( )
j
i
j
i
i
v
k
v
k
1
5
,
0
−
+
=
.
Approximation of the initial and boundary conditions (8)-(10) gives
(
)
1
exp
0
0
−
=
i
i
p
a
v
,
(
)
mf
p
p
if
p
p
i
2
1
2
0
−
+
=
,
m
i
,...
1
,
0
=
,
(13)
2
1
0
v
v
j
=
+
,
(14)
1
1
v
v
j
i
=
+
,
1
1
h
i
z
j
+
=
.
(15)
Condition (11) taking into account
1
+
j
f
dt
dh
we discretize it like this
(
)
0
1
1
1
1
1
1
=
+
−
+
+
+
−
+
+
j
j
i
j
i
j
i
f
l
f
v
v
a
v
,
f
i
z
j
1
+
=
.
(16)
Equation (12) is reduced to the form
i
j
i
i
j
i
i
j
i
i
F
v
B
v
C
v
A
−
=
+
−
+
+
+
+
−
1
1
1
1
1
,
j
i
,
1
=
,
(17)
where
2
f
A
i
i
=
,
2
1
f
B
i
i
+
=
,
2
1
2
1
f
f
C
i
i
i
+
+
+
=
,
j
i
i
v
F
=
.
To solve the nonlinear difference problem (12) - (16) at each new time layer
1
+
=
j
t
t
iterative processes can be
used. The simplest of these is associated with iterative refinement of the time step
1
+
j
[4]. Let the initial
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approximation be given
0
1
+
j
. Given
s
j
1
+
the corresponding approximation for
1
+
j
i
v
we find their solutions to the
following nonlinear difference problem
i
j
s
i
i
j
s
i
i
j
s
i
i
F
v
B
v
C
v
A
−
=
+
−
+
+
+
+
−
1
,
1
1
,
1
,
1
,
j
i
,
1
=
,
(18)
2
1
,
0
v
v
j
s
=
+
,
(19)
1
1
,
v
v
j
s
i
=
+
,
в
узел
f
i
z
j
i
1
+
=
.
(20)
To solve this problem we use a three-point sweep. Relation (16) is used to determine the time step and in the
simplest case we have [4]
(
)
1
1
1
1
1
1
1
1
−
+
+
−
+
+
+
−
+
=
f
v
v
a
v
lf
j
i
j
i
j
i
s
j
,
в узле
f
i
z
j
i
1
+
=
.
(21)
Now, to solve the difference problem (18) - (20), we use the counter sweep method [3]. Here
)
(
0
,
t
h
m
m
i
=
- internal node. In the area
m
i
0
the solution is calculated using the right-hand sweep formulas
1
1
,
1
1
1
,
+
+
+
+
+
+
=
i
j
s
i
i
j
s
i
v
v
0
,
1
,...,
2
,
1
−
−
=
m
m
i
,
(21)
где
i
i
i
i
i
A
C
B
−
=
+
1
,
1
,...,
2
,
1
−
=
m
i
,
0
1
=
,
(22)
i
i
i
i
i
i
i
A
C
B
A
F
−
+
=
+
1
,
1
,...,
2
,
1
−
=
m
i
,
2
1
v
=
,
(23)
and in the region
)
(
t
h
i
m
- according to the left-hand sweep formulas
1
1
,
1
1
,
1
+
+
+
+
+
+
=
i
j
s
i
i
j
s
i
v
v
,
1
,...,
1
,
−
+
=
N
m
m
i
,
(24)
where
1
+
−
=
i
i
i
i
i
B
C
A
,
m
N
N
i
,...,
2
,
1
−
−
=
,
0
=
N
,
(25)
1
1
+
+
−
+
=
i
i
i
i
i
i
i
B
C
B
F
,
m
N
N
i
,...,
2
,
1
−
−
=
,
0
v
N
=
.
(26)
Considering (21), (24) in the node
m
i
=
we find
m
m
m
m
m
j
s
m
v
−
+
=
+
1
1
,
.
(27)
Based on the obtained numerical results, graphs of the
moving boundary were constructed.
( )
t
h
, pressure
distribution in the filter and sediment layer (Fig. 1-4).
From the presented graphs, one can estimate the
growth of the sediment layer and the pressure
distribution in it.
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European International Journal of Multidisciplinary Research and Management Studies
Growing importance
contributes to a corresponding
change in the consolidation coefficient, which in turn
affects the growth of the sediment layer and the
pressure distribution in it. With an increase in the value
lagging growth dynamics can be observed
( )
t
h
. The
graphs also show the dynamics of pressure at fixed
points of the sediment layer. As the sediment layer
grows, the pressure at fixed points decreases.
REFERENCES
Федоткин И.М., Воробьев Е.И., Вьюн В.И.
Гидродинамическая
теория
фильтрования
суспензией. Киев:Вища. шк.,Головное изд
-
во.1986.
-
166с
Федоткин И.М. Математическое моделирование
технологических процессов. Киев: Вища шк.,
Головные изд
-
во. 1988.
-
415с.
Нигматулин Р.И. Динамика многофазных сред. Т. II.
М., Наука, 1987.
-
389с.
Самарский А.А., Вабищевич П.Н. Вычислительная
теплопередача. –М.: Едиториал УРСС, 2003
-
784 с.
Самарский А.А. Теория разностных схем. –
М.:
Наука, 1977.
-
656с.
0,0011
0,0031
0,0051
0,0071
0,0091
0
50
100
150
200
250
300
350
0,25
0,28
0,31
0,34
0,37
0,4
0,43
0,0011
0,0031
0,0051
0,0071
0,0091
0,25
0,28
0,31
0,34
0,37
0,4
0,43
0,0011
0,0031
0,0051
0,0071
0,0091
0,0111
0,35
0,37
0,39
0,41
0,43
0
50
100
150
200
250
300
350
400
Fig. 1. Change in the thickness of the filter layer at
different
6
10
−
=
Pа
-1
;
7
10
−
=
Pа
-1
;
8
10
−
=
Pа
-1
.
350
=
t
s.
t
,с
)
(
t
h
,
м
p
, М Pа
,
м
Fig. 2. Distribution of pressure across the sediment thickness
at different
6
10
−
=
Pа
-1
;
7
10
−
=
Pа
-1
;
8
10
−
=
Pа
-1
350
=
t
s.
Рис 3. Distribution of pressure across the sediment
thickness.
6
10
−
=
Pа.
150
=
t
s
;
300
=
t
s
;
400
=
t
s
Fig. 4. Pressure distribution at fixed points of the sediment layer
6
10
−
=
Pа.
3
10
03
,
4
−
=
z
м );
3
10
75
,
4
−
=
z
м
3
10
65
,
5
−
=
z
м
p
, М Pа
p
, М Pа
)
(
t
h
,
м
t
,с
