The effect of liquid bound by microparticles (ions) on the membrane selectivity
Notion “critical concentration” was introduced for membrane processes, and a formula was proposed to calculate it. A mathematical problem of steady-state convective-diffusive mass transfer within the layer of concentration polarization was formulated and exactly solved, taking into account the hyd...
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irk-123456789-1929892023-07-28T15:27:55Z The effect of liquid bound by microparticles (ions) on the membrane selectivity Poliakov, V.L. Механіка Notion “critical concentration” was introduced for membrane processes, and a formula was proposed to calculate it. A mathematical problem of steady-state convective-diffusive mass transfer within the layer of concentration polarization was formulated and exactly solved, taking into account the hydration of microparticles (ions). For numerous examples, analyses were performed to estimate the effect of hydration and mass transfer parameters on the concentration of suspended microparticles at the membrane surface and its selectivity for uniform impurity. Стосовно мембранних процесів введено поняття критичної концентрації і запропоновано формулу для її визначення. Сформульовано і строго розв’язано математичну задачу усталеного конвективно-дифузійного масопереносу в шарі концентраційної поляризації з урахуванням гідратації мікрочастинок (іонів). На численних прикладах для однорідних домішок проаналізовано вплив параметрів гідратації і масопереносу на концентрацію завислих мікрочастинок на поверхні мембрани, її селективність. 2023 Article The effect of liquid bound by microparticles (ions) on the membrane selectivity / V.L. Poliakov // Доповіді Національної академії наук України. — 2023. — № 1. — С. 40-48. — Бібліогр.: 14 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2023.01.040 http://dspace.nbuv.gov.ua/handle/123456789/192989 532.546:628.16 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Механіка Механіка |
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Механіка Механіка Poliakov, V.L. The effect of liquid bound by microparticles (ions) on the membrane selectivity Доповіді НАН України |
| description |
Notion “critical concentration” was introduced for membrane processes, and a formula was proposed to calculate
it. A mathematical problem of steady-state convective-diffusive mass transfer within the layer of concentration
polarization was formulated and exactly solved, taking into account the hydration of microparticles (ions). For
numerous examples, analyses were performed to estimate the effect of hydration and mass transfer parameters on
the concentration of suspended microparticles at the membrane surface and its selectivity for uniform impurity. |
| format |
Article |
| author |
Poliakov, V.L. |
| author_facet |
Poliakov, V.L. |
| author_sort |
Poliakov, V.L. |
| title |
The effect of liquid bound by microparticles (ions) on the membrane selectivity |
| title_short |
The effect of liquid bound by microparticles (ions) on the membrane selectivity |
| title_full |
The effect of liquid bound by microparticles (ions) on the membrane selectivity |
| title_fullStr |
The effect of liquid bound by microparticles (ions) on the membrane selectivity |
| title_full_unstemmed |
The effect of liquid bound by microparticles (ions) on the membrane selectivity |
| title_sort |
effect of liquid bound by microparticles (ions) on the membrane selectivity |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2023 |
| topic_facet |
Механіка |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/192989 |
| citation_txt |
The effect of liquid bound by microparticles (ions) on the membrane selectivity / V.L. Poliakov // Доповіді Національної академії наук України. — 2023. — № 1. — С. 40-48. — Бібліогр.: 14 назв. — англ. |
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Доповіді НАН України |
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AT poliakovvl theeffectofliquidboundbymicroparticlesionsonthemembraneselectivity AT poliakovvl effectofliquidboundbymicroparticlesionsonthemembraneselectivity |
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2025-07-16T18:53:49Z |
| last_indexed |
2025-07-16T18:53:49Z |
| _version_ |
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| fulltext |
40 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 1: 40—48
C i t a t i o n: Poliakov V.L. The effect of liquid bound by microparticles (ions) on the membrane selectivity.
Dopov. Nac. akad. nauk Ukr. 2023. No 1. P. 40—48. https://doi.org/10.15407/dopovidi2023.01.040
© Publisher PH «Akademperiodyka» of the NAS of Ukraine, 2023. This is an open acsess article under the
CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)
МЕХАНІКА
MECHANICS
https://doi.org/10.15407/dopovidi2023.01.040
UDC 532.546:628.16
V.L. Poliakov
Institute of Hydromechanics of the NAS of Ukraine, Kyiv
E-mail: v.poliakov.ihm@gmail.com
The effect of liquid bound by microparticles (ions)
on the membrane selectivity
Presented by Corresponding Member of the NAS of Ukraine O.Ya. Oliinyk
Notion “critical concentration” was introduced for membrane processes, and a formula was proposed to calculate
it. A mathematical problem of steady-state convective-diffusive mass transfer within the layer of concentration
polarization was formulated and exactly solved, taking into account the hydration of microparticles (ions). For
numerous examples, analyses were performed to estimate the effect of hydration and mass transfer parameters on
the concentration of suspended microparticles at the membrane surface and its selectivity for uniform impurity.
Keywords: membrane, microparticle, impurity, hydration, concentration, selectivity, diffusion, mass transfer,
concentration polarization.
The most important indicator of the ability of porous membranes to separate a finely dispersed or
dissolved impurity from a liquid containing it and thus obtain a permeate or concentrate with de-
sired properties is selectivity [1-3]. It generally and in relative units (percentage) characterizes the
distribution of the initial impurity between the products of membrane separation. The selectivity
limit is 1 (100 %) and corresponds to the ideal separation of a two-component physical system. It is
impossible to achieve such a separation because of the heterogeneity of the impurity and the struc-
ture of the pore space of the membrane. Nevertheless, it is possible to achieve a high selectivity of the
membrane action, as a rule, due to the reasonable choice of its type and characteristics. Under cer-
tain conditions, liquid films can be one of the reasons for the decrease in membrane selectivity. They
usually form on the surface of microparticles (small colloids, macromolecules, ions) and actively
interact with them at the molecular level. The thickness of such a liquid film is often comparable to
the size of a microparticle — collector, a pore. The liquid enclosed in a film has anomalous properties,
for example, an increased viscosity [4]. This fact is indirectly confirmed by the data of experimental
studies of a gel-like deposit that accumulates when suspensions are filtered through granular media
of a rapid filter [5]. The indicated thickness of the aforementioned film can vary within very large
limits under the influence of various external factors. Accordingly, the set of microparticles can be
41ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 1
The effect of liquid bound by microparticles (ions) on the membrane selectivity
both strongly and weakly hydrated. Here and below, their hydration, as well as any hydrophilic solid
surfaces, is understood as the process of formation of a liquid film physically associated with them.
Below, the behaviour of only those microparticles that are commensurate with the membrane
pores will be considered in detail. Then two characteristic situations can occur. In the first case,
the microparticle, previously retained at the entrance to the membrane pore, now freely passes
into it due to the thinning of its surface film. In the second situation, the microparticle loses its
ability to penetrate inside the membrane due to the binding of an additional amount of liquid.
If there are many such particles, then in both situations described, a significant change in their
selectivity should be expected.
During the operation of membranes of various types, an impurity accumulates in the immedi-
ate vicinity of their working surface. It is here, in the diffusion boundary layer, that the degree
of microparticle hydration can decrease much due to the intense interaction between them. As
a result, a very sharp deterioration in the quality of the permeate occurs. The purpose of this re-
search is a theoretical assessment of the specified effect of dehydration of a homogeneous impurity
on the efficiency of separation of a fine suspension (solution) by micro- and ultrafiltration (when
using reagents), reverse osmosis. At the same time, it is still assumed that the concentration of mi-
croparticles near the membrane surface is insufficient for their structuring. Therefore, a deposited
layer is not formed. The following condition for the penetration of a microparticle into a mem-
brane pore is of fundamental importance for the analysis performed below by analytical methods
h p pr r< χ (1)
where ,h pr r are the radii of a hydrated microparticle and pore, the empirical coefficient pχ is less
than 1 and, according to [6] is equal to 1/3. Condition (1) applies to the entire membrane if its
pore space has a regular structure and is formed by many identical cylindrical pores. In the case of
an irregular porous space, equivalent dimensions should be used. The second basic position here
is the law of hydration, which establishes the relationship between the degree of hydration of mi-
croparticles and the content of impurities in the liquid. It will be generalized for
( )w
h
i
C
f C
C
= , (2)
where ,w iC C are the volumetric concentrations of liquid bound by microparticles and non-hy-
drated microparticles; C is the volumetric concentration of hydrated microparticles, so that
= +i wC C C . (3)
It follows from (2), (3) that
( )
( ) ( )
1 ( )
h
w h
h
Cf C
C C F C
f C
= =
+
. (4)
Thus, the volumetric concentration of the liquid, bound at the membrane surface, 0wC will be
= =0 0 0( ) ( )w w h h hC C C F C , (5)
where Ch0 is the volumetric concentration of hydrated particles at the membrane surface.
42 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 1
V.L. Poliakov
If microparticles both in the absence and in the presence of surface films are justified to be
considered spherical, then the radius and concentration of hydrated particles are related by the
following relation
3( ) 1 ( )h i hr C r f C= + , (6)
where ri is the radius of non-hydrated particles.
It should be emphasized that the impurity in suspensions is often substantially heteroge-
neous in practice. If it contains, among other things, a fraction of microparticles with sizes close
to the (equivalent) pore diameter and their total volume is only a small part of the volume of
the entire suspension, then the dehydration process will not noticeably affect the quality of the
permeate. However, the selectivity of the membrane under certain conditions will seriously de-
teriorate with an increase in the number of such particles. For a reliable prediction of its change
in the general case of a polydisperse impurity, first of all, it is necessary to choose such a hydra-
tion law that would correctly reflect the effect of the impurity concentration as a whole on the
dehydration of the separated fraction. Then the form of law (2) does not change but C should
be interpreted as the total volumetric concentration of the impurity. Therefore, it is necessary to
perform mathematical modelling of the behavior of all impurity fractions in the layer of concen-
tration polarization.
The critical concentration crC is of key importance for assessing the possible consequences of
the initial hydration of an impurity (in the initial suspension or solution) and its subsequent dehy-
dration in the indicated layer. With its development, a formal transition of the current concentra-
tion at the membrane surface 0hC through the value corresponding to the conditions under consi-
deration crC is possible. This event will be accompanied by an abrupt change in the membrane selec-
tivity in the corresponding direction. To calculate the marked value, a general formula is proposed
–1 3 3( 1)cr h p pC f r= χ − . (7)
If the law of hydration is linear so that,
max( )h hf C C C= −σ , (8)
then is received for crC
3 3
max
1
( 1 )cr p p
h
C C r= + −χ
σ
. (9)
In the case of the exponential form of the law, namely,
–
max( ) hC
hf C C e α= , (10)
will be true
χ −
= −
α
3 3
max
11
ln p p
cr
h
r
C
C
. (11)
43ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 1
The effect of liquid bound by microparticles (ions) on the membrane selectivity
Here p p ir r r= ; max max 0C C C= is the limiting relative volumetric content of bound liq-
uid (in the absence of interaction between hydrated particles), 0C is the initial volumetric con-
centration of hydrated microparticles; hσ , hα are empirical coefficients. The exponential and
linear forms are fundamentally different. In the first case, complete dehydration of microparti-
cles is achieved only asymptotically (at C →∞ ), and in the second case — at a finite value of
max( )hC C σ .
To reliably assess the effectiveness of the membrane, first of all, it is necessary to know the
concentration of suspended (unstructured) microparticles on its surface 0hC . Just here the con-
centration C reaches not just the maximum, but often a very large value (reverse osmosis) [7].
This may create preconditions for the formation of a layer of deposit, which already has the prop-
erties of a solid porous medium. Thus, the value 0hC can determine not only the quality of separa-
tion of the liquid-impurity system but also the performance of the membrane apparatus. It should
be emphasized that the diffusion boundary layer stabilizes already at the beginning of the operat-
ing period [8]. Therefore, it is sufficient to use the steady-state model of convective-diffusion mass
transfer for approximate calculations of the impurity concentration profile in it. Within this layer,
both transport mechanisms play an equally important role [9]. The presence of a semipermeable
flat or cylindrical surface in cross-flow technologies for the separation of liquid systems enhances
the diffusion transfer of microparticles due to a sharp increase in shear stresses in the near-wall
zone. It is important to note that the change in the flow of hydrated particles is balanced with the
decrease in the liquid associated with them. Therefore, a special term appears in the basic equation
of one-dimensional steady-state mass transfer that characterizes the intensity of the release of the
bound liquid. The equation in relation toCδ takes on the following form
[ ( ) ] 0w
z
dC dCd
D C V C V
dz dz dz
δ
δ δ+ − = . (12)
Here, the dependence of the diffusion coefficient on their concentration, which is character-
istic of especially small particles (macromolecules), is additionally taken into account [10]. At the
boundaries of the considered boundary layer, traditional conditions are set, namely, [11]
,h hz C Cδ= −δ = ; (13)
0z = , δ
δ δ+ =( )z p
dC
D C VC VC
dz
; (14)
where hδ is the thickness of the separated layer, V is the constant rate of the liquid flow in the
layer and the membrane, and pC is the impurity concentration in the permeate. It is possible to
get rid of the unknown quantity wC in equation (12) by using expression (4). If an interlayer with
completely dehydrated microparticles does not form in the vicinity of the membrane surface, then
the concentration of bound water on it should be calculated using the formula (5). Therefore,
integrating the equation (12) with the use of condition (14) gives
δ
δ δ δ+ − − =0( ) [ ( ) ( )]z h h h p
dC
D C VC V F C F C VC
dz
. (15)
44 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 1
V.L. Poliakov
The solution of equation (15) under condition (14) is represented as the following inverse
integral function
δ ξ ξ
= δ −
ξ − ξ − +∫
0
( )1
( ) ( )
h
C
z
h
h p h hC
D d
z
V F C F C
. (16)
Then the concentration of the hydrated impurity on the membrane surface can be easily
found by selection from the equation
ξ ξ
= δ
ξ − ξ − +∫
0
0
( )
.
( ) ( )
h
h
C
z
h
h p h hC
D d
V
F C F C
(17)
In the subsequent quantitative analysis, however, the same equation will be used, but already
in a dimensionless form, namely,
δ
ξ ξ
=
ξ − ξ − +∫
0
0
( )
( ) ( )
h
h
C
z
h p h hC
D d
Pe
F C F C
, (18)
where the Peclet number Peδ is 0h zV Dδ ; 0zD is the value of the coefficient zD for the non-
hydrated impurity; =0, 0, 00h p h pC C C , 00C is the scale for impurity concentrations selected de-
pending on the specific filtration conditions; 0z z zD D D= .
Obviously, at 0h crC C> and, therefore, h h pr r> χ the membrane in the case under consid-
eration will completely retain a homogeneous impurity, even though i p pr r< χ . Therefore, the
selectivity of the membrane concerning the impurity will be 100 %. As the concentration 0hC
decreases to reach a critical value crC , the membrane loses its ability to separate the suspension
(solution). The impurity entering the diffusion boundary layer will pass through it in transit and
thus the selectivity of the membrane will drop to 0. The impurity concentration profile within the
specified layer will quickly stabilize and 0hC equalize with crC and then remain invariant until a
new change in technological conditions occurs. To prevent a mass breakthrough of microparticles
through the membrane, it is necessary to correct in advance the parameters of the baromembrane
process (pressure, flow rate, concentration) so that, as a result, the value 0hC determined from
equation (18) will be less than crC .
Of interest is the time crt to reach the critical level of fluid contamination on the membrane
surface h0(C )crC= . It can be easily estimated if the effect of impurity dehydration is neglected.
Then it is justified to use the non-stationary equation of convective-diffusion mass transfer in the
following form
[ ( ) ]z
C C
D C V C
z z t
δ δ
δ δ
∂ ∂∂ + =
∂ ∂ ∂
, (19)
Its approximate solution was obtained by replacing the local derivative in equation (19) with
the expression
0h h
h
z dC
dt
δ −
δ
,
45ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 1
The effect of liquid bound by microparticles (ions) on the membrane selectivity
which accurately reflects its behaviour at the
boundaries of the concentration polarization
layer. Then, taking into account the values of
the impurity flow at its boundaries, as well as
the initial condition
0t = , 0
0hC C= ;
the following simple formula is derived for crt
0
0
0
( )
2 ( )
h cr
cr
C C
t
V C C
δ −
=
−
. (20)
The above calculation dependencies and
equations are illustrated by several examples
with initial data, which make it possible to
evaluate the possible consequences of (de)hy-
dration of a dispersed impurity for the sepa-
rating effect of membranes of various types.
The relative characteristic concentrations crC , 0hC were the subject of numerous calculations.
The ratio between them determines the efficiency of the separation of a liquid system. The selectiv-
ity of the membrane, due to the homogeneity of the impurity under consideration and the regular-
ity of the pore space, can take only extreme values — 0 at 0h crC C> and 1 at 0h crC C . The law of
hydration is chosen in the form (10). First of all, using formula (11), the value crC in combination
with с hα is determined as a function of the equivalent pore size of the membrane. In this case, the
hydration potential varied (parameter maxC ). The critical concentration, in principle, can take
any non-negative values. The spherical microparticles must be so small in the first limiting case
( 0)crC = , that even at the maximum degree of hydration they freely penetrate the membrane. In
particular, it will not be able to retain the impurity at max 3C = if p prχ exceeds 1.587. In another
limiting case ( )crC →∞ , mobile microparticles will not get inside the membrane due to their large
size, even if there is no bound water on their surface at all. The calculation results of crC are shown
in Fig. 1 in the form of a series of graphs of the generalized dependence of hα crC on p prχ .
However, the main attention during the quantitative analysis of the membrane process is paid
to the influence on the concentration polarization of the ability of mobile microparticles to hydra-
tion, as well as the diffusion and convective mechanisms of mass transfer. In this regard, the choice
of the effective diffusion coefficient is of great importance. According to the recommendations
[12], the following two-term expression is accepted for the relative value zD
1 2
3 3( ) ( ) ( )z h hh hD C a r C b r C
−
= + . (21)
Here, the first term describes molecular diffusion, and the second describes diffusion induced
by shear stresses [13, 14]. The scale for zD is chosen in such a way that
1h ha b+ = .
– –
–
ahCcr
2.0 prp1.51.0
0
1
1
2
2
3
3
4
Fig. 1. Dependence p p( r )h crCα χ : 1 — max 10C = ; 2 —
max 7C = ; 3 — max 5C = ; 4 — max 3C =
46 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 1
V.L. Poliakov
Thus, the ratio between the coefficients
ha and hb determines the relative contribu-
tions of each of the noted components to the
diffusion mass transfer. Incidentally, molecu-
lar diffusion prevails with smaller micropar-
ticles, and the second component becomes
essential for larger ones. Outside the layer of
concentration polarization 1hC = and hence
3
max(1) (1) 1 h
h h ir r r C e α= = + . (22)
The corresponding value zD , taking into
account (22), will be
1
3
max(1) (1 )h
z hD a C e
−α= + +
2
3
max(1 )h
hb C e α+ + . (23)
To isolate the effect of impurity dehydra-
tion, it is advisable to supplement equation
(18) with a formula. The indicated formula
makes it possible to calculate 0hC , provided
that the amount of liquid initially bound by
it does not change later. Then it follows from
equation (15) that
0 exp
(1)h
s
Pe
C
D
δ= . (24)
The measure of concentration polariza-
tion can be the concentration 0hC , which is
the highest in the entire region of the flow
of a liquid system. The model parameters
max( , , , )h hC a Peδα varied continuously or
with a small increment over a broad range
when calculating 0hC . At the same time,
–Ch0
–Cmax5.0 7.52.50
100
1
50
2
150
3
Fig. 2. Dependence 0 max( )hC C : 1 — 0.05hα = ; 2 —
0.1hα = ; 3 — 0.2hα =
–Ch0
ah0.50 0.750.250
100
1
50
2
150
3 4
Fig. 3. Dependence 0 ( )h hC a : 1 — max 3C = ; 2 —
max 5C = ; 3 — max 7C = ; 4 — max 10C =
47ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 1
The effect of liquid bound by microparticles (ions) on the membrane selectivity
their basic values were: 0.2hα = , 0.5ha = ,
10Peδ = . Based on equation (18), the de-
pendence of 0hC on the parameter maxC was
specified in the first series of examples for
three values hα (Fig. 2). The graphs corre-
sponding to them are shown by solid lines. At
the same time, 0hC was determined without
taking into account the dehydration of the
impurity using formula (24). If its initial hy-
dration is nevertheless taken into account,
then the relationship between 0hC and maxC ,
hα is preserved. The curve corresponding
to 0.2hα = and wC const= is given in Fig.
2 by a dotted line. Finally, in the absence of
primary hydration, sD no longer depends on
maxC , hα and is equal to 1. Then the calcu-
lated value 0hC reaches 42.2 10⋅ and, therefore, a dense layer of the deposit is necessarily formed.
The desired concentration also demonstrates high sensitivity concerning the factors that
determine mass transfer in the diffusion boundary layer. In the second series of examples, the
coefficient ah was used. As can be seen from Fig. 3, the molecular component of the diffusion
mechanism turns out to be more significant under given operating conditions. The observed
sharp rise in the calculated curves is explained by the fact that the hydration of microparticles
weakens molecular diffusion. This, in turn, contributes to a more intense accumulation of impu-
rities in the aforementioned layer and, as a consequence, the dehydration process. According to
(22), the effect caused by dehydration, first of all, affects the indicated diffusion and therefore
curve 1, calculated like all the others based on equation (18) and at a smaller maxC , is located
above the others.
Similarly, the value 0hC is related to the number Peδ , and, in essence, to the permeate veloc-
ity. Their increase means an adequate increase in convective mass transfer. In the final series of
calculations, a change of Peδ from 1 to 20 was allowed. The curves presented in Fig. 4 indicate
the danger of uncontrolled accumulation of suspended matter in the layer of concentration po-
larization with an excessive acceleration of the permeate flow with its subsequent deposition on
the membrane surface. In addition, this figure also shows a dotted line curve that does not take
into account the dehydration of the initially hydrated particles. Its significant discrepancy with
other curves convincingly confirms the generally positive role of the impurity dehydration effect
in membranous processes.
Thus, the processes of hydration and dehydration of dispersed and dissolved impurities, which
provide changes in the size of microparticles up to ions, and their number in the diffusion bound-
–Ch0
Pe10 1550
100
1
50
2
150
3
4
Fig. 4. Dependence 0 ( )hC Peδ : 1 — max 3C = ; 2 —
max 5C = ; 3 — max 7C = ; 4 — max 10C =
48 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 1
V.L. Poliakov
ary layer, can have a noticeable effect on the separation of liquid systems by membrane tech-
nologies. Their reliable forecast makes it possible to adjust the operating conditions and thereby
increase the efficiency of membrane devices.
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Received 09.09.2022
В.Л. Поляков
Інститут гідромеханіки НАН України, Київ
E-mail: v.poliakov.ihm@gmail.com
ВПЛИВ ЗВ’ЯЗАНОЇ МІКРОЧАСТИНКАМИ (ІОНАМИ) РІДИНИ
НА СЕЛЕКТИВНІСТЬ МЕМБРАНИ
Стосовно мембранних процесів введено поняття критичної концентрації і запропоновано формулу для її
визначення. Сформульовано і строго розв’язано математичну задачу усталеного конвективно-дифузійно-
го масопереносу в шарі концентраційної поляризації з урахуванням гідратації мікрочастинок (іонів). На
численних прикладах для однорідних домішок проаналізовано вплив параметрів гідратації і масопереносу
на концентрацію завислих мікрочастинок на поверхні мембрани, її селективність.
Ключові слова: мембрана, мікрочастинка, домішки, гідратація, концентрація, селективність, дифузія, ма-
соперенос, концентраційна поляризація.
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