Modeling of aqueous suspension filtration when combining downward and upward flows
The physical domain is divided into two subareas of motion, and a nonlinear mathematical problem of water suspension filtration with linear kinetics of interphase detachment mass transfer is formulated with respect to each of them. The structure of gel-like deposit, the dependence of hydraulic condu...
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Poliakov, V.L. Kurganska, S.M. 2025-09-10T16:56:04Z 2025 Modeling of aqueous suspension filtration when combining downward and upward flows / V.L. Poliakov, S.M. Kurganska // Доповіді Національної академії наук України. — 2025. — № 1. — С. 31-40. — Бібліогр.: 14 назв. — англ. 1025-6415 https://nasplib.isofts.kiev.ua/handle/123456789/206391 628.16 https://doi.org/10.15407/dopovidi2025.01.031 The physical domain is divided into two subareas of motion, and a nonlinear mathematical problem of water suspension filtration with linear kinetics of interphase detachment mass transfer is formulated with respect to each of them. The structure of gel-like deposit, the dependence of hydraulic conductivity on its concentration, and the relationship of mass transfer coefficients (attachment and detachment) with the filtration rate are taken into account. The corresponding mathematical models contain interconnected clarification and filtration flow compartments. After the introduction of dimensionless variables and parameters, as well as the application of the operational method, rigorous solutions of both problems are obtained. As a result, the most important dependencies and equations were derived for engineering calculations of key characteristics of filtration — concentration of dispersed impurity in filtrate and head losses in distinguished subareas and common in the whole operating layer. The mentioned formalisms were used to determine the main technological times, which limited the time of continuous filter operation due to excessive deterioration of the filtrate quality and mechanical energy consumption for filtration through the clogged medium. As a consequence, the permissible time of its continuous operation (filter run) based on the criteria of effective filtration was established. The similar technological approach to the estimation of the filter performance was applied in parallel to the high-rate filter at the traditional single-fl ow suspension feeding and the two-fl ow feeding investigated above. Comparative analysis was performed on test examples with typical initial data for practice of clarification of aqueous suspensions. As a result, it was obtained that the division of the initial flow of suspension into two components coming through the upper and lower bed surfaces can contribute to a significant intensification of the technological process. At the same time, it is realistic to increase the duration of filter runs by 50 % and more, which leads to a tangible decrease in the cost of filtrate. Thus, application of well sorbing filtering materials becomes justified. Фізичну область розділено на дві підобласті руху, і стосовно кожної з них сформульовано нелінійну математичну задачу фільтрування водної суспензії за лінійної кінетики міжфазного відривного масообміну. При цьому враховано структуру гелеподібного осаду, залежність від його концентрації коефіцієнта фільтрації, зв’язок коефіцієнтів масообміну (прилипання та відриву) зі швидкістю фільтрування. Відповідні математичні моделі містять взаємопов’язані освітлювальний і фільтраційний блоки. Після введення безрозмірних змінних і параметрів, а також застосування операційного методу отримано суворі розв’язки обох задач. У підсумку виведено найважливіші залежності та рівняння, призначені для інженерних розрахунків ключових характеристик фільтрування — концентрації дисперсної домішки у фільтраті та втрат напору у виділених підобластях і загальних у всьому шарі завантаження. Зазначені формалізми використовували для визначення основних технологічних часів, що обмежувало час безперервної роботи фільтра внаслідок надмірного погіршення якості фільтрату та витрат механічної енергії на фільтрацію через засмічене середовище. Як наслідок, встановлювали допустимий виходячи з критеріїв ефективного фільтрування час його безперервної роботи. Подібний технологічний підхід до оцінки працездатності фільтра паралельно застосовували до швидкого фільтра за традиційної однопотокової подачі суспензії та дослідженої вище двопотокової. Порівняльний аналіз виконано на тестових прикладах з типовими для практики освітлення водних суспензій вихідними даними. У результаті отримано, що поділ вихідного потоку суспензії на дві складові, що надходять через верхню та нижню поверхні завантаження, може сприяти значній інтенсифікації технологічного процесу. При цьому реально збільшення тривалості фільтроциклів на 50 % і більше, що призводить до відчутного зниження вартості фільтрату. Таким чином, стає виправданим застосування фільтрувальних матеріалів, що добре сорбують. uk Видавничий дім "Академперіодика" НАН України Доповіді НАН України Механіка Modeling of aqueous suspension filtration when combining downward and upward flows Моделювання фільтрування водних суспензій у разі поєднання низхідного та висхідного потоків Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Modeling of aqueous suspension filtration when combining downward and upward flows |
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Modeling of aqueous suspension filtration when combining downward and upward flows Poliakov, V.L. Kurganska, S.M. Механіка |
| title_short |
Modeling of aqueous suspension filtration when combining downward and upward flows |
| title_full |
Modeling of aqueous suspension filtration when combining downward and upward flows |
| title_fullStr |
Modeling of aqueous suspension filtration when combining downward and upward flows |
| title_full_unstemmed |
Modeling of aqueous suspension filtration when combining downward and upward flows |
| title_sort |
modeling of aqueous suspension filtration when combining downward and upward flows |
| author |
Poliakov, V.L. Kurganska, S.M. |
| author_facet |
Poliakov, V.L. Kurganska, S.M. |
| topic |
Механіка |
| topic_facet |
Механіка |
| publishDate |
2025 |
| language |
Ukrainian |
| container_title |
Доповіді НАН України |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| format |
Article |
| title_alt |
Моделювання фільтрування водних суспензій у разі поєднання низхідного та висхідного потоків |
| description |
The physical domain is divided into two subareas of motion, and a nonlinear mathematical problem of water suspension filtration with linear kinetics of interphase detachment mass transfer is formulated with respect to each of them. The structure of gel-like deposit, the dependence of hydraulic conductivity on its concentration, and the relationship of mass transfer coefficients (attachment and detachment) with the filtration rate are taken into account. The corresponding mathematical models contain interconnected clarification and filtration flow compartments. After the introduction of dimensionless variables and parameters, as well as the application of the operational method, rigorous solutions of both problems are obtained. As a result, the most important dependencies and equations were derived for engineering calculations of key characteristics of filtration — concentration of dispersed impurity in filtrate and head losses in distinguished subareas and common in the whole operating layer. The mentioned formalisms were used to determine the main technological times, which limited the time of continuous filter operation due to excessive deterioration of the filtrate quality and mechanical energy consumption for filtration through the clogged medium. As a consequence, the permissible time of its continuous operation (filter run) based on the criteria of effective filtration was established. The similar technological approach to the estimation of the filter performance was applied in parallel to the high-rate filter at the traditional single-fl ow suspension feeding and the two-fl ow feeding investigated above. Comparative analysis was performed on test examples with typical initial data for practice of clarification of aqueous suspensions. As a result, it was obtained that the division of the initial flow of suspension into two components coming through the upper and lower bed surfaces can contribute to a significant intensification of the technological process. At the same time, it is realistic to increase the duration of filter runs by 50 % and more, which leads to a tangible decrease in the cost of filtrate. Thus, application of well sorbing filtering materials becomes justified.
Фізичну область розділено на дві підобласті руху, і стосовно кожної з них сформульовано нелінійну математичну задачу фільтрування водної суспензії за лінійної кінетики міжфазного відривного масообміну. При цьому враховано структуру гелеподібного осаду, залежність від його концентрації коефіцієнта фільтрації, зв’язок коефіцієнтів масообміну (прилипання та відриву) зі швидкістю фільтрування. Відповідні математичні моделі містять взаємопов’язані освітлювальний і фільтраційний блоки. Після введення безрозмірних змінних і параметрів, а також застосування операційного методу отримано суворі розв’язки обох задач. У підсумку виведено найважливіші залежності та рівняння, призначені для інженерних розрахунків ключових характеристик фільтрування — концентрації дисперсної домішки у фільтраті та втрат напору у виділених підобластях і загальних у всьому шарі завантаження. Зазначені формалізми використовували для визначення основних технологічних часів, що обмежувало час безперервної роботи фільтра внаслідок надмірного погіршення якості фільтрату та витрат механічної енергії на фільтрацію через засмічене середовище. Як наслідок, встановлювали допустимий виходячи з критеріїв ефективного фільтрування час його безперервної роботи. Подібний технологічний підхід до оцінки працездатності фільтра паралельно застосовували до швидкого фільтра за традиційної однопотокової подачі суспензії та дослідженої вище двопотокової. Порівняльний аналіз виконано на тестових прикладах з типовими для практики освітлення водних суспензій вихідними даними. У результаті отримано, що поділ вихідного потоку суспензії на дві складові, що надходять через верхню та нижню поверхні завантаження, може сприяти значній інтенсифікації технологічного процесу. При цьому реально збільшення тривалості фільтроциклів на 50 % і більше, що призводить до відчутного зниження вартості фільтрату. Таким чином, стає виправданим застосування фільтрувальних матеріалів, що добре сорбують.
|
| issn |
1025-6415 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/206391 |
| citation_txt |
Modeling of aqueous suspension filtration when combining downward and upward flows / V.L. Poliakov, S.M. Kurganska // Доповіді Національної академії наук України. — 2025. — № 1. — С. 31-40. — Бібліогр.: 14 назв. — англ. |
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31ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2025. № 1: 31—40
C i t a t i o n: Poliakov V.L., Kurganska S.M. Modeling of aqueous suspension fi ltration when combining downward and
upward fl ows. Dopov. Nac. akad. nauk Ukr. 2025. № 1. P. 31—40. https://doi.org/10.15407/dopovidi2025.01.031
© Publisher PH «Akademperiodyka» of the NAS of Ukraine, 2025. Th is is an open access article under the CC BY-NC-
ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)
МЕХАНІКА
MECHANICS
https://doi.org/10.15407/dopovidi2025.01.031
UDC 628.16
V.L. Poliakov,
S.M. Kurganska
Institute of Hydromechanics of the NAS of Ukraine, Kyiv, Ukraine
E-mail: v.poliakov.ihm@gmail.com
Modeling of aqueous suspension fi ltration when combining down-
ward and upward fl ows
Presented by Corresponding Member of the NAS of Ukraine A.Ya. Oleynik
Th e physical domain is divided into two subareas of motion, and a nonlinear mathematical problem of water suspension
fi ltration with linear kinetics of interphase detachment mass transfer is formulated with respect to each of them. Th e
structure of gel-like deposit, the dependence of hydraulic conductivity on its concentration, and the relationship of mass
transfer coeffi cients (attachment and detachment) with the fi ltration rate are taken into account. Th e corresponding
mathematical models contain interconnected clarifi cation and fi ltration fl ow compartments. Aft er the introduction
of dimensionless variables and parameters, as well as the application of the operational method, rigorous solutions of
both problems are obtained. As a result, the most important dependencies and equations were derived for engineering
calculations of key characteristics of fi ltration — concentration of dispersed impurity in fi ltrate and head losses in
distinguished subareas and common in the whole operating layer. Th e mentioned formalisms were used to determine
the main technological times, which limited the time of continuous fi lter operation due to excessive deterioration of the
fi ltrate quality and mechanical energy consumption for fi ltration through the clogged medium. As a consequence, the
permissible time of its continuous operation (fi lter run) based on the criteria of eff ective fi ltration was established. Th e
similar technological approach to the estimation of the fi lter performance was applied in parallel to the high-rate fi lter
at the traditional single-fl ow suspension feeding and the two-fl ow feeding investigated above. Comparative analysis
was performed on test examples with typical initial data for practice of clarifi cation of aqueous suspensions. As a result,
it was obtained that the division of the initial fl ow of suspension into two components coming through the upper and
lower bed surfaces can contribute to a signifi cant intensifi cation of the technological process. At the same time, it is
realistic to increase the duration of fi lter runs by 50 % and more, which leads to a tangible decrease in the cost of fi ltrate.
Th us, application of well sorbing fi ltering materials becomes justifi ed.
Keywords: fi ltration, suspension, concentration, dual-fl ow, fi lter run, exact solution, head losses.
Introduction. Th e cost of the purifi cation of contaminated water signifi cantly depends on the costs
of its fi ltration. Th erefore, it is natural that throughout the history of water treatment the intensifi ca-
tion of the technological process of clarifi cation on rapid fi lters remained relevant. Since the total
cost of the process consists of capital and operating costs, two ways were realized to reduce the cost
32 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2025. No 1
V.L. Poliakov, S.M. Kurganska
of separating suspensions of diff erent origin, namely, the improvement of fi lter design and clarifi ca-
tion technology. In the fi rst case, mainly the design was improved, in particular, layered structures
were used, they were given a special curvilinear shape, the size of fi ltering material grains was se-
lected, etc. By supplying contaminated water prepared in a special way, the eff ective operation of the
fi lters was prolonged, thus saving on operating costs. Obviously, the second way deserves special
attention with regard to operating fi lters. Various techniques and methods ensuring its implemen-
tation in practice have been the subject of theoretical and experimental studies, the results of which
are set forth in extensive literature, and we can mention, for example, the following works [1—5].
One of indicative examples of the technological improvement of direct-fl ow fi ltration on rapid
fi lters can be non-traditional methods of suspension feeding on them. Th ey assume splitting of
the initial suspension fl ow (hydraulic load) into two or more components with their subsequent
localization at separate sections of the packed bed [6]. Usually, the suspension enters the fi lter bed
only through its upper (downward fi ltration) or lower (upward) surfaces [7]. Th e fi ltrate is taken
from the opposite side. However, it is not technically diffi cult to feed the suspension simultane-
ously from both sides. At the same time, it is structurally feasible to implement the suspension
infl ow into the operating layer from the inside. In the fi rst case it is possible to increase the effi -
ciency of the technological process by varying the ratio between 1V and 2V , as well as the position
of the drainage device ( dL ). In the second case, the increase of the operating period is achieved
by selecting the position of the internal feeder ( sL ) and also the above-mentioned ratio. In both
cases, minimal design changes are required. It is important that it is possible to achieve a signifi -
cant clarifying eff ect by selecting 1V , 2V , dL , sL , which is refl ected in a noticeable extension of
the continuous operation of the fi lter. From a physical point of view, this result can be explained by
a more uniform distribution of deposit in the bed. A natural consequence of the sharp reduction
of the maximum clogging of a fi lter media is a signifi cant reduction of the total head losses in it.
Just the purpose of the article was to quantify the clarifying eff ect due to the separate feeding
of aqueous suspension through the upper and lower surfaces of the packed bed of a water treat-
ment fi lter.
It is easier to evaluate the consequences of splitting suspension feeding formally using math-
ematical modeling methods. For its initial evaluation it is enough to limit ourselves to the theoreti-
cal study of dual-fl ow fi ltration. Just below we analyze the eff ectiveness of the above-mentioned
technique of the intensifi cation of rapid fi ltration, namely, if the suspension is fed into the fi lter
bed simultaneously through its upper and lower boundaries )0 a( ndz L with constant rates,
respectively, 1V and 2 1V V V , where V is the hydraulic load on the fi lter. Filtrate is taken away
by drainage device inside the fi lter media at the depth dL ( )dz L . Washing of the clogged bed is
carried out by the fl ows of the treated water in reverse directions. Th us, a single domain of motion
in the conventional fi ltration (single-fl ow) is divided into two unrelated sub-domains.
It is preliminary assumed that the detachment fi ltration, linear kinetics of mass transfer be-
tween liquid and solid phases take place; the hydraulic conductivity of the clogged medium is
generalized as a nonlinear function of the volumetric deposit concentration SS [8—10]
1 2
0 0 0( ) ( ) [1 ( ) ]k km m
S k S Sk S k f S k S n , (1)
where 0k , 0n are the hydraulic conductivity and porosity of the clean bed, 1, 2k km are the
empirical coeffi cients.
33ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2025. № 1
Modeling of aqueous suspension fi ltration when combining downward and upward fl ows
Th en the formulation of the corresponding mathematical problem includes for the upper bed
section (0 ≤ z ≤ Ld) the system of equations
1 1
1 0C SV
z t
, (2)
1
1 1 1 1
l q
V V
S V C V S
t
, (3)
1
1 0 1( )k
hV k f S
z
, (4)
1 2( ) {1 [ ( ) ] }k km m
k i i if S S S ( 1, 2)i (5)
and the boundary conditions operator
1 1 0 10, 0; 0, ; d dt S z C C z L h H . (6)
H ereinaft er it was assumed: iC is the volumetric concentration of the suspended solids within
i th section ( 1, 2)i ; iS and ih are the volumetric concentration of the deposited particles and
piezometric head there; dH is the constant piezometric head at the fi lter outlet ( )dz L . Th e fol-
lowing relationship between the concentrations of the deposit and deposited particles was used in
the expression for kf (5) [11]
( )S i iS S S , (7)
where is the functional bulk factor. In addition, rate coeffi cients of the suspension particles at-
tachment and detachment are related to the fi ltration rate V as follows [12-13]
,l q
V VV V , (8)
where ,V V are the reduced rate coeffi cients of the suspension particles attachment and de-
tachment, which do not depend on the fl ow characteristics. Similarly for the bottom section
(Ld ≤ z ≤ L) will be
2 2
2 0,C SV
z t
(9)
2
2 2 2 2 ,l q
V V
S V C V S
t
(10)
2
2 0 2( ) ,k
hV k f S
z
(11)
2 20, 0; , .d dt S z L h H (12)
34 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2025. No 1
V.L. Poliakov, S.M. Kurganska
and also (5) at 2i . Aft er introducing dimensionless variables and parameters in the standard way:
0 0 0
, ,i i
i i
C SC S
C n C
0
, ,z Vtz t
L n L
0 ( )i i d
kh h H
VL
, ,i
i
VV
V
–1, ,ld
d V V
LL V
L
–1q
V VV , 1 2V V V , 0C problem (2)—(6), (9)—(12) is transformed to the following
form, specifi cally,
for the upper section (0 ≤ z ≤ Ld)
1 1
1 0,C SV
z t
(13)
1
1 1 1 1,l q
V V
S V C V S
t
(14)
1
1 1( ) ,k
hV f S
z
(15)
1 2( ) {1 [ ( ) ] } ;k km m
k i i if S S S (16)
1 1 10, 0; 0, 1; , 0dt S z C z L h ; (17)
for the lower section (Ld ≤ z ≤ 1)
2 2 0,C S
z t
(18)
2
2 2 ,V V
S C S
t
(19)
2
2 2( ) ;k
hV f S
z
(20)
2 2 20, 0; 1, 1; , 0.dt S z C z L h
(21)
Th e rigorous solution of the problem (13) — (21) was obtained by the operational method.
Th e course of the solution of a similar problem is given, for example, in [10]. Th erefore, the fol-
lowing are only the main calculation formulae that are necessary to gain a full understanding of
the results and consequences of the suspension clarifi cation in a dual-fl ow fi lter with diff erent fl ow
directions. So, they will be for the upper section
1
1 1 1
1
1
1 0 1
1
1 0 1
0
( ) (2 )
(2 ) ,
ql
VV
q
V
V tV L l q
e V V d
t
Vq l q
V V V d
C t e e I V L t
V e I V L d
(22)
35ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2025. № 1
Modeling of aqueous suspension fi ltration when combining downward and upward fl ows
1
1 1 1
1 1 0 1
0
( , ) (2 ) ,
ql
VV
t
VV zl l q
V V VS z t V e e I V z d
(23)
Index " "e means that this characteristic refers to the outlet from the fi lter.
Besides, the head losses 1h are accurately calculated by the formula
1 1
10
( ) .
( ( , ))
dL
k
dzh t V
f S z t
(24)
Th e following dependencies are recommended for similar calculations for the lower section
–1
1 1
1
(1 ) (1 ) (1 ) 1
2 0 1
(1 ) 1
1 0 1
0
( ) (2 (1 ) (1 ) )
(1 ) (2 (1 ) (1 ) ) ,
l q
V d V
q
V
V L V t l q
e V V d
t
Vq l q
V V V d
C t e e I V L t
V e I V L d
(25)
2( , )S z t
–1
1 1(1 ) (1 ) (1 ) 1
1 0 1
0
(1 ) (2 (1 ) (1 ) ),
l q
V V
t
V z Vl l q
V V VV e e I V z (26)
1
2 1
2
( ) (1 ) .
( ( , ))
d kL
dzh t V
f S z t
(27)
Th e relative impurity concentration in the fi ltrate and the total head losses in the bed are cal-
culated as the following sums
1 1 1 2( ) ( ) (1 ) ( ),e e eC t V C t V C t (28)
1 2( ) ( ) ( )h t h t h t . (29)
Finally, the clogging of the lower bed surface is proposed to be calculated based on the formula
1(1 )
2 2 1( ) (1, ) (1 ) (1 ).
q
V Vl qV
e
V
S t S t V e
(30)
Th e clarifying effi ciency of the variety of the dual-fl ow fi ltration under consideration was eval-
uated on a number of test cases with the following fi xed data: 0.01,V 1,q * *0.1, 8,C h
–3 –6( ) 2.5 10 2 10 ,S S 3( ) [1 ( ) ]kf S S S . In addition, two values of (1 3, 1)l , character-
istic for Brownian (dp ≤ 10–6) and non-Brownian (dp ≥ 10–6) suspension particles, respectively,
were chosen. Finally, either their typical values or wide ranges of possible values are adopted for
1, ,d VV L . Th e main subject of the calculations was the relative duration of the fi lter run ft as a
key technological parameter. For its establishment, the relative technological times pt and ht were
previously found with the involvement of the criterion equations [14]
1 2( ) ( ) ( )e p e p e pC t C t C t C ., (31)
36 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2025. No 1
V.L. Poliakov, S.M. Kurganska
1 2( ) ( )h hh t h t h ., (32)
where ht is the time of reaching maximum allowable head losses in fi lter bed, pt is the time of
protective eff ect of fi lter bed, C., h . are the relative maximum permissible values of impurity
concentration in the fi ltrate and head losses in the fi lter bed.
At already known relative values of the fi lter run duration ft , the relative excesses (or reduc-
tions under unfavorable conditions) ft corresponding to them were determined due to the
splitting of a hydraulic load. Th us, *f f ft t t , where *ft is the duration of the fi lter run at only
downward fi ltration under similar conditions.
Th e dependences for the relative concentrations ieC , S and head losses ih were involved:
(24)—(26) for the fi rst section (0 ≤ z ≤ Ld) and (27)—(29) for the second section (Ld < z ≤ 1).
In order to establish the conditions that contribute to the productive operation of the dual-fl ow
fi lter and to evaluate the gain therein due to the distribution of the hydraulic load between the
upper and lower bed surfaces, the relative duration ft was calculated depending on the position
of the fi ltrate runoff ( dL ), as well as on the ratio between the upper 1( )V and lower 1( )V V fl ow
rates and, fi nally, on the absorption capability of the fi lter bed ( )V . Th e mass exchange coeffi -
cients were assumed to be independent of the fi ltration fl ow direction. First of all, the sensitivity
of the relative duration ft to the position of the drainage device was analyzed at equal upper and
lower fl ow rates 1 2( 0.5)V V . It was found out that the technological process is limited in time
only by the protective capability of a bed for coarse impurities, and it is reasonable to place the
drainage device in the middle of the fi lter media ( 0.5)dL . Th us, it is possible to increase ft by
32.7 % in comparison with the traditional single-fl ow fi ltration. In case of fi nely dispersed impu-
rity, fi rstly, the eff ect of splitting hydraulic load is even greater and the maximum value of ft
(1.643) is reached at * 0.256dL L , and the fi lter operation has to be stopped at *dL L , because
of the excessive deterioration of the fi ltrate quality, and due to the excessive head losses at *dL L .
However, the calculated value of ft is only slightly less than the maximum value (1.632) even at
0.5dL . Th erefore, the value 0.5dL is fi xed in subsequent calculations. At the same time, it is
inadmissible to set dL too low, which may cause complete non-functionality of the fi lter. Indeed,
it is not diffi cult to specify such a critical value crL , that the fi rst portion of a suspension will be
insuffi ciently clarifi ed at Ld ≤ Lcr. Proceeding from dependences (22), (25) the following equation
concerning crL is derived
–1 –1 –1 –1
1 1 1 1[ (1 ) ] (1 ) (1 )
1 * 1(1 ) 0
l l l l
V cr V cr VV V L V L VV e C e V e . (33)
Equation (33) is easily solved in the general case by the fi tting procedure, and the simple for-
mula follows from it in the special case 1 2 0.5V V
–10.52
* *
–1
ln
0.5
l
V
cr l
V
C C e
L
. (34)
Both roots of equation (34) have physical meaning and correspond to two critical positions
of the drainage device, which are equidistant from the center of the bed and its nearby inlet. Spe-
cifi cally, in the second series of examples, crL equals to 0.1267 and 0.8733, respectively, at 1 3l
37ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2025. № 1
Modeling of aqueous suspension fi ltration when combining downward and upward fl ows
and already 0.2022crL and 0.7978 at 1l . And further attention was emphasized on the rela-
tionship between ft and 1V or V . Curves describing the excess of ft over the basic value for
comparison *ft (single fl ow fi ltration with parameters 1 1)dV L at 0.5dL and changing 1V
from 0 (upward fi ltration through a bed with half the height) to 1 (downward fi ltration through a
similar bed) are shown in Fig. 1. Naturally, the reduction of the bed height in both limiting situ-
ations 1 0V and 1 twice causes a serious (one and a half times) reduction of ft at the same
total hydraulic load V . If the suspension is fed through the both surfaces with equal fl ow rates,
the increment of ft is maximum and will be 32.7 % at 1 3l , and it doubles (63.2 %) at 1l .
Also, the technological times pt , ht , ft were calculated as functions of V at the fi xed values of
1V (0.7), dL . Th e two series of plots obtained for the fi ne and coarse impurities based on (22) —
(27) are shown in Fig. 2. Here the solid lines highlight the curves of the dependence ( )f Vt the
most important for practice, which are continuous and have one fracture each. Comparison of
the optimum values of ft in three considered cases of dual- and single-fl ow fi ltration is indica-
tive. A comparison of the peak values of ft shows that the corresponding eff ect can be estimated
at 30 %. Also, an increase in the times ht , ft is observed here with an increase in V from the
Δtf
—
Vl
—
1.6
1
2
1.2
0.8
0.4
0.80.60.40.20
tp , th , tf
—— —
aV
—
400
1
2
3
4 5
6
300
200
100
8 106420
Fig. 1. Dependence 1ft V : 1 — 1 3l ,
2 — 1l
Fig. 2. Dependencies ( )p Vt , ( )h Vt ,
( ) :f Vt 1—3 — ht , 4—6 — pt , ft — solid
lines; 1, 6 — dual-flow ( 1)l ; 2, 4 —
dual-flow ( 1 3)l ; 3, 5 — single-flow
38 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2025. No 1
V.L. Poliakov, S.M. Kurganska
value 7.9. However, the observed even more signifi cant eff ect should not be overestimated due to
the possible inadequacy of linear kinetics of mass exchange during prolonged fi ltration through a
well-absorbing media.
Th us, due to simultaneous feeding of a suspension into the bed in two places (not only through
the upper and lower surfaces), it is possible to signifi cantly intensify the clarifi cation process also
in the section of the bed, which practically did not take part in it in the traditional (downward
or upward) fi ltration. Th us, fi rstly, the absorption resource of the entire fi lter media is more fully
implemented, and secondly, the concentration profi les of the deposit and deposited particles are
smoothed out and the mechanical energy expenditure is reduced accordingly. However, the qual-
ity of the fi ltrate deteriorates faster due to the reduction of the fi ltration rate and the weakening of
the absorption ( 0)l . Th erefore, it is necessary to preselect the technique of feeding a suspension
into the fi lter bed, based on the results of the joint technological analysis of dual-fl ow and single-
fl ow fi ltration.
Conclusions. It is established on the basis of strictly solving the mathematical problem of
detachment fi ltration of water suspension at linear kinetics of interphase mass transfer that its si-
multaneous feeding through the upper and lower surfaces contribute to a signifi cant reduction of
mechanical energy consumption for the technological process, which can be expressed in several
tens of percent.
Separation of the initial suspension fl ow, despite the minimal deterioration of water treat-
ment quality, allows to increase the duration of the fi lter run to the same extent, and in practical
terms — to use well sorbing fi ltering materials.
Due to the large scale of application of rapid fi ltration in the water industry and in particular
in water treatment technologies, it is possible to signifi cantly reduce the cost of fi ltrate by separate
feeding of contaminated water to fi lters.
39ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2025. № 1
Modeling of aqueous suspension fi ltration when combining downward and upward fl ows
REFERENCES
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3. Jegatheesan, V., & Vigneswaran, S. (2005). Deep bed filtration: mathematical models and observations. Crit.
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4. Shevchuk, E.A., Mamchenko, A.V., & Goncharuk, V.V. (2005). Technology of direct flow filtration of natural
and waste water through granular loads. Water Chemistry and Technology, 27, 4, pp. 369-384 (in Russian).
5. Zhurba, M.G. (1980). Water purification on granular filters. Lviv: Vyshchaya shkola, Publishing House at Lviv
State University (in Russian).
6. Adelman, M. J., Monroe, L. Weber-Shirk, M. L., Cordero, A. N., Coffey, S. L., Maher, W. J., Dylan Guelig, D.,
Jeffrey, C. Will, J. C., Stodter, S. C., Hurst, M. W., & Lion, L. W. (2012). Stacked filters: novel approach to rapid
sand filtration. J. Environ. Eng., 138, pp. 999-1008.
7. Orlov, V. O. (2005). Water purification filters with granular filter media. Rivne: NUWEE (in Russian).
8. Bai, R., & Tien, C. (1997). Particle detachment in deep bed filtration. J. Colloid Interface Sci., 186 (2),
pp. 307-317.
9. Mints, D. M., & Meltser, V. Z. (1970). Hydraulic resistance of granular porous medium in the process of
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Received 28.12.2024
40 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2025. No 1
V.L. Poliakov, S.M. Kurganska
В.Л.Поляков,
С.М. Курганська
Інститут гідромеханіки НАН України, Київ, Україна
E-mail: v.poliakov.ihm@gmail.com
МОДЕЛЮВАННЯ ФІЛЬТРУВАННЯ ВОДНИХ СУСПЕНЗІЙ
У РАЗІ ПОЄДНАННЯ НИЗХІДНОГО ТА ВИСХІДНОГО ПОТОКІВ
Фізичну область розділено на дві підобласті руху, і стосовно кожної з них сформульовано нелінійну мате-
матичну задачу фільтрування водної суспензії за лінійної кінетики міжфазного відривного масообміну.
При цьому враховано структуру гелеподібного осаду, залежність від його концентрації коефіцієнта
фільтрації, зв’язок коефіцієнтів масообміну (прилипання та відриву) зі швидкістю фільтрування.
Відповідні математичні моделі містять взаємопов’язані освітлювальний і фільтраційний блоки. Після вве-
дення безрозмірних змінних і параметрів, а також застосування операційного методу отримано суворі
розв’язки обох задач. У підсумку виведено найважливіші залежності та рівняння, призначені для
інженерних розрахунків ключових характеристик фільтрування — концентрації дисперсної домішки у
фільтраті та втрат напору у виділених підобластях і загальних у всьому шарі завантаження. Зазначені
формалізми використовували для визначення основних технологічних часів, що обмежувало час
безперервної роботи фільтра внаслідок надмірного погіршення якості фільтрату та витрат механічної
енергії на фільтрацію через засмічене середовище. Як наслідок, встановлювали допустимий виходячи з
критеріїв ефективного фільтрування час його безперервної роботи. Подібний технологічний підхід до
оцінки працездатності фільтра паралельно застосовували до швидкого фільтра за традиційної
однопотокової подачі суспензії та дослідженої вище двопотокової. Порівняльний аналіз виконано на те-
стових прикладах з типовими для практики освітлення водних суспензій вихідними даними. У результаті
отримано, що поділ вихідного потоку суспензії на дві складові, що надходять через верхню та нижню
поверхні завантаження, може сприяти значній інтенсифікації технологічного процесу. При цьому реально
збільшення тривалості фільтроциклів на 50 % і більше, що призводить до відчутного зниження вартості
фільтрату. Таким чином, стає виправданим застосування фільтрувальних матеріалів, що добре сорбують.
Ключові слова: фільтрування, суспензія, концентрація, двопотоковий, фільтроцикл, точний розв’язок,
втрати напору.
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