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Mathematical modeling process of liquid filtration taking into account reverse influence of process characteristics on medium characteristics |
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Andrii Safonyk 1*, Andrii Bomba 2 |
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1 Department of Automation, Electrical and Computer-Integrated Technologies, National University of Water Management and Nature Resources Use, Rivne, Ukraine 2 Department of Informatics and Applied Mathematics, Rivne State Humanitarian University, Rivne, Ukraine *Corresponding author E-mail: safonik@ukr.net |
Copyright © 2014 Andrii Safonyk, Andrii Bomba. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The article presents and solves the questions of accounting for reverse influence of process characteristics (the contamination concentration of liquid and sediment) on medium characteristics (the coefficients of porosity, filtration, diffusion, mass-transfer and others) by the example of liquid cleaning in magnetic and sorption filters. The algorithm of numerical-asymptotic approximation to the solution of the relevant model task which is described by the system of nonlinear singular perturbative differential equations of the type «convection-diffusion-mass-transfer». The proper correlations (formulas) are effective for conducting theoretical researches which are aimed at the «productivity» (in particular, optimization) of the parameters of filtration process (namely: time of protective action of load, sizes of filter, and others) in cases of predominance of convection and sorption components of the proper process above diffusive and desorption components, that takes place in large majority of filtration installations. The computer experiment was conducted on this basis. These ones results show the advantages of the offered model in comparing to classic.
Keywords: Filtration; Reverse Influence; Multicomponent Concentration; Magnetic Filter; Model of the Magnetic Sedimentation; Sorption Treatment; Asymptotic Upshots; Nonlinear Tasks.
1. Introduction
The analysis of researches results which was conducted in [1-17] testifies about the presence of difficult structure of the interrelations of different factors, which determined the processes of filtration and filtering through porous mediums, which was not taken into account in the “traditional” (classic, phenomenological) models of such systems. Taking into account the different interdependences, and also different additional factors which are inserting in a “initial” (base) model with the purpose of more deep study of process, often directs researchers to the necessity of construction of bulky and ineffective (in terms of numeral realization and practical using,) mathematical models. However in many practically important cases during researching of such processes it is possible to come in terms of modeling of different kind of perturbations of the known (idealizing, averaging, base) backgrounds.
In accordance with the researches, which were considered earlier, the article presents the questions of account of reverse influence of process characteristics (the contamination concentrations of liquid and sediment) on medium characteristics (the coefficients of porosity, filtration, diffusion, mass-transfer and others) on the example of liquid cleaning in magnetic and sorption filters.
2. Setting a task
Consider the one-dimensional process of cleaning liquid by filtration in the filter layer with thickness L, which is identified with the cut [0, L] axis 0x. This layer is placed that abscissa axis is perpendicular to its surface, and origin of coordinates is on its upper boundary. The particles of contamination of admixture substance can pass from one state in other (processes of capture-tearing away, sorption-desorption) at same time the contamination concentrations are influenced on the considered layer. A concentration of contamination is multicomponent. The proper process of filtration with the account of reverse influence of characteristics of process (concentrations of liquid and sediment contamination) on medium characteristics (coefficients of porosity, filtration, diffusion, mass-transfer and others) is described the following system of interconnected differential equations:
(1)
(2)
, (3)
where 
–
concentrations of admixtures in the liquid environment, which is filtered; 
– concentrations of admixtures, which are
sedimentationed in the filter attachment; 
–
coefficient, which characterizes the mass volumes of admixture particles
sedimentation for time unit; 
, 
– coefficient, which characterizes the
mass volumes of torn-off for that time from the granules of filing of admixture particles, 
,
– speed of filtration, 
– concentrations of admixture particles
at the input of the filter, 
– porosity of
filter attachment (
– the initial porosity
of attachment,
; 
, 
,
hard
parameters (they characterize the proper coefficients), 
– soft parameters and they founded an
experimental method), – small parameter, 
, 
–
pressure.
3. Algorithm (asymptotic) of the solution
Solution of system (1) in the terms (2) was founded in the kind of the asymptotic rows [9] – [17]:
,
, (4)
where 
– the
remaining members, 
, 
(
)
– the members of regular parts of asymptote, 
,
(
), 
,
(
)
– the functions of type of boundary layer
(accordingly corrections
at the input and at the output of
filtration flow), 
, 
, 
,
– the proper regulating transformations.
Like to [17], after a substitution (4) in (1) and
application of standard “procedure of equation”, for finding of functions 
and 
(
) we come to such tasks:
(5)
(6)
As a result of their solving we have:
,
,
Where
,
. The approximate values of functions 
are founded by way of interpolation of
array,
, 
,
where 
, 
.
The
functions 
, 
,
(
), 
, 
() which were assigned for the removal of
inconsistencies, which were brought by the built
regular parts, 
, 
in areas around the points with some
accuracy 
(input and output of filtration
flow), that is providing implementation of
terms: 
, 
,
, 
.
These functions are founded like to [17]. We are have proper task analogical to [9] for the estimation of
remaining members.
Fig. 1: The Efficiency of Treatment Process
4. Numerical calculations
4.1. Magnetic filter
Let us look at the
process of cleaning of liquid mediums from ferromagnetic admixtures in
magnetized porous nozzles that is one of main tasks of exception of corrosion
products admixtures as a result of continuous corrosion of technological
equipment. The admixture particles of mediums at working of magnetic power
factor 
settling in points of the
contact of nozzles granules, where value 
can
arrive the size at the value in order 2·10
А²/m³
(Н – magnetic field intensity). In initial moment of time (t=0) porous nozzle
is relatively “clean”, that is unsaturated admixture particles, its porosity –. In the process of settling of admixtures
the size of porosity 
is gradually
diminishing, the coefficient of hydraulic resistance is increasing and
accordingly in the case of reserve of the system, the size of overfall of
pressure 
in the porous nozzle. The
Efficiency of cleaning process of medium remains at enough high level during
definite time (time
of filtercycle, time of protective action of
filter). At the accumulation of critical mass of admixtures in the volume of
porous nozzle which is characterized by the size of working capacity of
absorption, efficiency of cleaning process which equals the relation of
difference of concentrations of admixtures at input and output of filter to the
concentration at input, is diminishing and the treatment regime passes to the
non-stationary stage (Fig. 1). As known from [17], at
, certain amount of admixtures settled in
the pores layers of nozzle yet. Greater their part “breaks away” and darts out
with medium which is cleaning. Gradually, barns on length of porous nozzle are
maximally saturated admixtures and are self-switching-off at achievement of sometime
efficiency of cleaning is diminishing to
the zero.
The process of magnetic settling of admixtures, which
is realized in magnetic filter (
) with
homogeneous granular filter nozzle, is realized by operation of laws, the
prototype of which is a classic model of filtration [15], taking into account
reverse influence of the besieged particles on porosity and coefficient 
, and on the coefficient of filtration also
[17].
(7)
(8)
, (9)
Where –coefficient
which characterizes the mass volumes which were torn-off during that time from
the granules of nozzle of admixture particles;
(10)
v – Speed of filtration (
,
which characterizes locking of technological process), 
– the porosity of filter nozzle ( – the initial porosity of nozzle),
(11)
– Coefficient
of filtration, 
– limit of filling by
sediment,
(12)
,
– hard parameters (they are characterized
the proper coefficients), 
– soft parameters and they founded
an experimental method), – pressure.
Such character of change of porosity and coefficient
of the torned-off particles is explained that at the increase of admixture
particles in nozzle, the proper parameters of filtration change. As a system is
reserved, the change of coefficient of filtration causes the change of size of
overfall of pressure 
in porous nozzle.
The solution of system (7) in the terms (8) is founded similar to (1)-(2) in the form of asymptotic series (4) (see [9], [17]):
Fig. 2: The Distribution of Concentrations of Admixtures in the Liquid and Sediment
Along the Filter at the Time Moment 
Hours, 
Hours.
|
|
|
|
Fig. 3: The Variation |
Fig.
4: The
Distribution Of Filter
Efficiency |
According to [4] coefficients of captured admixture
particles and the torned-off sediment
particles are calculated using the formula: 
[13], where 
- free
parameter, H - magnetic field
intensity, - speed of filtration, 
- diameter
of granular filter nozzle.
The results of calculations
by the formulas look like (3) at 
mg/l, 
,
m, 
s-1,
s-1, 
kА/m,
mm, 
.
Figure 2 shows the distribution
of admixtures concentration in liquid and
sediment in the certain time moment. Thence,
giving at the output filter (when)
the permissible concentration 
mg /
l, we find time its
protective action: 
hours, that on 4 hours is differed from data which conducted by the test method [13]. At this
filter will accumulate sediment by
weight 240 g.
We emphasize
that in the process of calculation we took,
though the coefficient of filtration (and porosity also) decreases by sticking
to the walls (filling) solid particles. This enables to find in each
cross-section filter (each point x,
) pressure gradient, especially using the formula 
we can find the time of passage more than
critical value gradient and to solve proper "decisions of
automatization". The change 
is shown in
Figure 3 with time.
As we can see (Fig. 4), if the case 
the filter efficiency is unchanged practically
until the time moment
, which confirms the known fact of filter efficiency distribution with time.
4.2. Sorption filters
The process of filtering in sorption filters does not require closed system. So speed of filtering is not a constant and the speed is changing along the filter over time usually. For simplification calculations, we assume that the concentration of pollution is one-component. Also we must consider the reverse effect on the porosity and coefficients which is characterizing the settling of particles of dirt and sediment particles tearing-off [17] and longitudinal diffusion. Coming from the above facts system (1) - (2) can be rewritten as:
(13)
(14)
The results of calculations by formulas
(4) when 
mg/l, 
m, 
s-1, 
s-1,
, are ,
, 
,
, 
.
Fig. 5: A). The Distribution of Admixtures Concentrations at the Output Filter During the Time Of Protective Action: 1 -According To Model of Minz; 2 - According To Formulas (4), at D = 0:78 Mm, V = 10 M/Hour.
B). The Distribution of Admixtures Concentrations at the Output Filter During the Time of Protective Action: 1 -According To Model of Minz; 2 - According To Formulas (4), at Mm, M/Hour.
Fig. 6: A) The Distribution of Admixtures Concentrations Along the Filter During at the Time Moment T = 26 Hours: 1 -According To Model of Minz; 2 - Founded by Formulas (4), At D = 0:78 Mm, V = 10 M/Hour.
B) The Distribution of Sediment Concentrations along the Filter during At the Time Moment T = 26 Hours: 1 -According To Model of Minz; 2 - Founded By Formulas (4), At D = 0:78
In the figures 5-6 were illustrated the comparative characteristics of the test data obtained and calculated by the classical model of Minz [14] and calculated by formulas (4). So the results of calculations by formulas (7) are providing greater accuracy in comparison with the classical model calculation formulas of Minz. Also the obtained results allow calculating the dynamics of promoting concentration of contamination and sediment along the filter (Fig. 7- 8).
5. Conclusion
In the work, the mathematical model was built ,which taking into account reverse influence of process characteristics (the contamination concentration of liquid and sediment) on medium characteristics (the coefficients of porosity, filtration, diffusion, mass-transfer and others) on the example of liquid cleaning in magnetic and sorption filters, namely:
Mathematical
model is the built, transferenced on the process that describes the
regularities of magnetic settling of admixtures in porous filtering nozzle, the regularities of
accumulation (“skidding”) admixtures in the nozzle, and also takes into account the reverse influence of
sediment concentration on coefficients of porosity, filtration and
mass-transfer. The proposed algorithm for solving the proper problem, in
particular, includes: determineting the time protective
action of filtering nozzle, determineting of the limit overfall of pressure and value at
change 
and
.
The results of calculations of concentration distribution and mass amount of admixtures in height filtering porous nozzle
for different time moments
are given, values of coefficient filtering for different values of lengt L of nozzle, which corresponds to time of
protective action (filtercycle) of nozzle. In this model provides the possibility of automatic
control of process of efficient settling of admixtures in magnetic filtering nozzle depending on the initial data of
the water medium which is cleaned;
The offered mathematical model is transferenced on the process of sewage treatment in sorption filters with taking into account reverse influence of sediment concentration on the medium characteristics and variable speed of filtering. The results of calculations of concentration distribution and mass amount of admixtures in height filtering porous nozzle for different time moments, values of filtering coefficient for different values of filtering speed, and characteristics of filling of filter are given. There were conducted comparative characteristics of the data which were obtained through research and calculated on base of classic model of Minza and formulas obtained by us (including, according to data presented in fig. 5-6, we see that the accuracy of calculations by formulas proposed by us is more higher in compared to estimates obtained by the classical Minz model).
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