Modelling the filtration processes of liquids from multicomponent ...

INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES

Volume 9, 2015

Modelling the filtration processes of liquids from multicomponent contamination in the conditions of authentication of mass transfer coefficient Andrii Safonyk

Nevertheless, through the complexity of the implementation and operation in practice of filtering are not widely known even filters with» continuously" non-uniform loads.

Abstract — A mathematical model of the process of purification of liquids from multicomponent pollution by n-layers magnetic filter is built, which takes into account the inverse effect of the determining factors (concentration of fluid contamination and sediment) on the medium characteristics (porosity factor), and includes the ability to identify an unknown mass transfer coefficient. We propose an algorithm for solving the corresponding nonlinear inverse singular perturbed problem of the "convection-mass transfer." Keywords — Model of the magnetic deposition, multicomponent, the inverse problem, authentication, condition of overdetermination, the asymptotic solution, perturbation. I. INTRODUCTION

T

HE process of cleaning liquid mediums from ferromagnetic impurities occurs most effectively in magnetized porous nozzles. Impurity particles of the mediums under effect of magnetic force factor F= H ⋅ gradH are deposited at the contact points of the c nozzle granules, where Fс the value can reach values of the order of 2·10 15 A² / m³ (H - magnetic field strength). At the initial time (t = 0) porous nozzle in ratio is "clean," that is not filled with impurity particles, its porosity - σ 0 . At the same time, with practical considerations, many-sphere filters are the most effective, as they provide a greater degree of purification in comparison with one sphere filters. Analysis of the results of studies [1, 14] indicates the presence of the complex structure of the interdependence of the different factors that determine the processes of filtration and filtering through porous mediums and were not considered in the traditional (classical, phenomenological) models of such systems. Filtering in the direction of decreasing the equivalent diameter of the loading granules is one of the generally recognized methods for increasing the efficiency of the filters [12]. In the complicated technological conditions, which are changing, the optimum granulometric composition would have to depend on time. Andrii Safonyk is with associate professor of the Department of automation, electrical engineering and computer integrated technologies of the Institute of Automation, Cybernetics and Computer Engineering, National University of Water Management and Nature Resources Use, Rivne, Ukraine (e-mail: [email protected])

ISSN: 1998-0140

Fig.1. Schematic representation of the n-layer filter

A mathematical model of a magnetic impurity deposition in the porous filter nozzle is proposed in the work [10] , which 189


σ ( ρ= ) σ 0 − εσ * ρ ( x, t ) , where σ 0 - initial porosity of backfill

takes into account the effect of the reverse influence of the characteristics of the process (sediment concentration) on the filtration parameters, with some coefficients of the considered process, were determined experimentally. In the present paper, a mathematical model of the process of filtering the liquids from multi-component contamination in the n-layers magnetic filter is built with the consideration an unknown mass transfer coefficient. We constructed an algorithm for solving the corresponding nonlinear inverse problem of the "convectionmass transfer." Solution of the corresponding inverse problem allows significantly to bring the numerical calculations nearer to the experimental data, to predict more exactly and to calculate the efficiency of the magnetic deposition of the impurities of different technological water-dispersed systems.

( x ∈ [ Ln −1 , Ln ] ); α 0 , α* , σ * , qi , ε - solid parameters, which characterize the corresponding coefficients; hi - coefficient characterizing the interaction between the different concentrations of impurities ε - a small parameter; [ Ln −1 , Ln ] -n-th sphere of the filter ( n = 1, 2, ..., l ); in equations (2) [] - an increase of the corresponding function at a given point x = Ln . III. ASYMPTOTIC BEHAVIOR OF THE SOLUTION Asymptotic approximation of the solution of the model problem ci ,1 ( x, t ) , L0 = 0 ≤ x < L1 ,  c ( x, t ) , L1 ≤ x < L2 , ci ( x, t ) =  i ,2 ...  c ( x, t ) , L ≤ x < L = L, n −1 n  i,n  ρ1 ( x, t ) , L0 = L ≤ x < L1 ,   ρ ( x, t ) , L1 ≤ x < L2 , ρ ( x, t ) =  2 ...  ρ ( x, t ) , L ≤ x < L = L, n −1 n  n

II. STATEMENT OF THE PROBLEM Let us consider spationally a one-dimensional filtering process of cleaning fluid in n-layers filter-layer with the thickness L (see Fig. 1), which is identified with the segment [0;L] of the axis 0x. Suggest [12] that the dirt particles can move from one state to another (processes of the capture, separation, sorption, desorption), at the same time the concentration of pollution affects the considered layer. Concentration of pollution is multi-component, = c c= = cm ) ( c1 ( x, t ) ,..., cm ( x, t ) ) where ci ( x, t ) ( x, t ) ( c1 ,...,

α1 ( t ) , L0 = L ≤ x < L1 ,  α ( t ) , L1 ≤ x < L2 , α (t ) =  2 ... α ( t ) , L ≤ x < L = L, n −1 n  n is found in the form of asymptotic series [9-11]:

- concentration i- component of the impurity ( i = 1, m ) in liquid filter media. The corresponding mathematical model of the filtration process, taking into account the inverse effect of process characteristics (concentration of fluid contamination and sediment) on the characteristics of the medium (the coefficients of porosity, mass transfer, etc. [9-11]) will be introduced in the form of the following problem: m  ∂ (σ ( ρ ) ci ) ∂ ( vci ) + + β= ci εα ( t ) ρ − ∑ hi −1ci −1 ,  ∂t ∂x i=2   i m x t G x L x L = = ∈ < < < t < ∞} , 1, , , : , 0 ( ) n { n −1  n (1)  n 1, l − 1, =  m  ∂ρ β  q c  − εα ( t ) ρ , = ∑ i i    i =1   ∂t

ci x 0 = ci* (t= ), c t 0= 0,= = ρ t 0 0, [ ci ]x = L = 0 , [ ρ ]x = L = 0 =

Volume 9, 2015

k

ci , n ( x, t ) = ci , n ,0 ( x, t ) + ∑ ε j ci , n , j ( x, t ) + Rc ,i , n ( x, t , ε ) , j =1 k

ρ n ( x, t ) = ρ n ,0 ( x, t ) + ∑ ε j ρ n , j ( x, t ) + Rρ , n ( x, t , ε ) , (4) j =1 k

αn (t ) = α n ,0 ( t ) + ∑ ε jα n , j ( t ) + Rα , n (t , ε ) , j =1

where

-

Rc ,i , n , Rρ , n , Rα , n

the

remaining

terms,

( i 1,= ci , n , j ( x, t ) , ρ n , j ( x, t ) , α i , n (t ) = m; j 0,= k ; n 0, l )

-

(2)

terms of the regular parts of the asymptotic. Similarly, [10],in the result of the substitution (4) into (1) - (3) (3) α (t ) ∫ ρ ( x , t )dx = µ (t ) , and the implementation of the standard "equating procedure" 0 to determine the functions ci , ρi , α i ( i = 0, k ) , we arrive to where ρ ( x, t ) - the concentration of impurities, trapped by the such problems: filter backfill; β - coefficient characterizing the mass volumes ∂ci , n ,0  ∂ci , n ,0 ∂ρ0  m  v c qi ci , n ,0  , 0, σ β β + + = = ∑ i n 0 , ,0  of deposition of impurities per unit of time, α ( t ) - the required  ∂t ∂x ∂t  i =1   coefficient, which characterizes the mass volumes of isolated = ci , n (t ),= ci , n ,0 0,= ρ n ,0 t 0 0, ci , n ,0 x 0= = t =0 from granules the backfill particles µ n ( t ) - a function which k

n

Ln

Ln

characterizes the mass distribution of sediment over time (is found empirically [14]), the condition of overdetermination (3) - is intended to find α ( t ) ([11, 12]), v - filtration rate ci* ( t ) -

α n ,0 (t ) ∫ ρ n ,0 ( x , t )dx = µ (t ) , where

the impurities concentration at the inlet of the filter,

ISSN: 1998-0140

ci , n ( t ) = ci* (t ),

0

if

n = 0,

ρ n ( t ) = ρ n −1,0 ( Ln −1 , t ) , if n = 1, l ; 190

ci , n ( t ) = ci , n −1,0 ( Ln −1 , t ) ,


Volume 9, 2015

∂ci , n , j ∂ci , n , j  ∂ (σ ( x, t ) c1 ( x, t ) ) ∂ ( v ( x, t ) c1 ( x, t ) )  +v + β ci , j = g n , j ( x, t ) , + + β c1 = εα ( t ) ρ ( x, t ) ,  −σ * ρ n , j −1 ∂t ∂x ∂t ∂x    ∂ (σ ( x , t ) c ( x , t ) ) ∂ ( v ( x , t ) c ( x , t ) )  ∂ρ n , j  m  2 2  = β  ∑ qi ci , n , j  − g n , j ( x, t ) ,  + + β c2 =  ∂t (5)  i =1   ∂t ∂x   = εα ( t ) ρ ( x, t ) − h1c1 ( x, t ) , = c2, n , j c1, n , j 0,= 0,= 0, c1, n , j x 0 = = x 0= t 0   ∂ρ ( x, t ) i 1, m, = j 1, k , = n 1, l − 1; 0, c2, n , j = 0,=  ρ n , j t =  = β ( q1c1 ( x, t ) + q2 c2 ( x, t ) ) − εα ( t ) ρ ( x, t ) , = 0= t 0  ∂t Ln Ln Ln * * = c1 x 0 c= = c1 t 0= 0,= c2 x 0 c= c2 t 0= 0,= ρ t 0 0, (6) α n,0 (t ) ρ n, j ( x, t )dx + α n,1 (t ) ρ n, j −1 ( x, t )dx + ... + α n, j (t ) ρ n,0 ( x, t )dx = 0 .= 1 (t ), 2 (t ),=

∫

∫

0

∫

0

[c1 ]x = L

0

As a result of their decision, we have:  *  σ 0 x  qvi x σ x ⋅e , t ≥ 0 , ci , n  t −  v  v ci , n ,0 ( x, t ) =    σ0x , 0, t < v  t  m   ρ n ,0 ( x, t ) β e −α0t ∫  ∑ qi ci , n ,0 ( x, t )  eα0t dt + ρ n** ( x ) ,  0  i =1 µn (t ) , α n ,0 (t ) = Lk

∫ρ

n ,0

n

time dependence of the corresponding mass transfer coefficient α ( t ) = α n ( t ) (for different numbers of spheres n)

( x , t )dx

of sediment over time (see. Fig. 2, 3, 4). Growth of the mass transfer coefficient with time is explained that (in this case with experimental values µ ( t ) ) during the deposition of

0

particles of granules of porous filling the impurity particles are maximum saturated and by the hydraulic pressure probability of separation of particles from granules increases to time τ з the effective work of the filter.

t

− g n , j −1 ( x , t )dt t

g n , j −1 ( x , t )dt   )  e ∫0 q c x t dt , ( , i i ,n, j  ∫0  ∑ i =1 

∫ 0

2

Ln

j

α n , j (t ) = −

∑ α n, j − w (t ) ∫ ρn, j ( x, t )dx w =1

0

Ln

∫ρ

n ,0

,

( x , t )dx

Fig 2. Mass distribution of sediment µ n ( t ) with time (1 - for

0

= g n , j ( x, t )

where

(7)

n

We present the results of calculations using formulas (4) c (t ) = 2 mg/l, c2* (t ) = 1 mg/l, β = 60 s −1 , v = 250 m / h , L = 1 m. q= q= 1 ; σ 0 = 0.5 ; α* = 1 ; β* = 1 ; σ * = 1 ; 1 2 ε = 0.001 . As a result of the interpolation of the experimental data [14], we obtained the mass distribution µ ( t ) = µ n ( t ) and the

x

ρ n , j ( x, t ) = β e

n

* 1

 − λn , j ( x , f ( x ) + t − f ( x ) ) dx  σ e ∫0 − * × v  x  λn , j ( x , f ( x ) + t − f ( x ) ) dx ∫  ci , n , j ( x, t ) =  x gi , n , j ( x , f ( x ) + t − f ( x ) ) e 0 dx , ×∫ ρ n , j −1 ( x , f ( x ) + t − f ( x ) )  0 t ≥ f ( x ) ,  0, t < f ( x ) , t

= 0 , [ c2 ]x = L = 0 , [ ρ ]x = L = 0 ,

j

∑α

(t )ρ

n = 3 , 2- for n = 1 )

m

( x, t ) − ∑ h c

n, j − w n , j −1 i −1 i −1, n , j w 1 =i 2 =

( x, t ) ,

∂ρ n, j −1 ( x, t ) λn, j ( x, t ) = − q jσ * + β . Approximate values of the ∂t functions f j ( x ) are found by [9-11] array interpolation ∆x σ * ρ j −1 ( xi , ti ) . v Estimate of the remainder terms is similar to [9].

( xi , ti ) ,

i = 1, k , where xi =∆x ⋅ i , ti +1= ti +

IV. THE NUMERICAL CALCULATION To simplify the exposition, we assume that the concentration of contamination is two-component, then the problem (1) - (3) can be rewritten as: ISSN: 1998-0140

Fig. 3. Mass distribution of sediment the corresponding mass transfer

191


coefficient α n ( t ) with time (1 - for n = 3 , 2- for n = 1 )

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effective deposition of impurities in a magnetized filter nozzle depending on the output of water pollution. In the perspective, - modeling of filtration under incomplete data with respect to diffusion (see., For example. [10, 15, 16]). REFERENCES [1] [2]

[3]

[4]

[5] [6] [7] [8]

[9]

[10]

[11]

[12] [13]

Fig. 4. Graphs of distribution of the contamination concentration at the output filter at time t in the formulas (4) if n = 1 - (a) and n = 3 (b)

[14]

[15]

Figure 4 shows the contaminant concentration distribution on the filter output at time t in the formulas (4) under n = 1 -(a) and n = 3 - (b) respectively. As can be seen in the picture, the time of the protective effect of n-sphere filter (for n = 3 tp=80h and tp=84h) significantly more time for protective action for one-sphere filter (for n = 1 -tp=72h and tp=80h).

[16]

Elimelech M. Predicting collision efficiencies of colloidal particles in porous media, Water Research., 26(1), 1-8, 1992. Elimelech M. Particle deposition on ideal collectors from dilute flowing suspensions: Mathematical formulation, numerical solution and simulations. Separations Technology, 4, 186-212, 1994. Jegatheesan V. Effect of surface chemistry in the transient stages of deep bed filtration, Ph Dissertation, University of Technology Sydney, pp. 300, 1999. Johnson P.R. and Elimelech, M., Dynamics of colloid deposition in porous media: Blocking based on random sequential adsorption, Langmuir, 11(3), 801-812, 1995. Ison C.R. and Ives, K.J., Removal mechanisms in deep bed filtration, Che. Engng. Sci., 24, 717-729, 1969. Ives K.J. Theory of filtration, special subject No.7, Int. Water Supply congress, Vienna, 1969. Ives K.J. Rapid filtration, Water Research, 4(3), 201-223, 1970. Petosa A.R., Jaisi, D.P., Quevedo, I.R., Elimelech, M., and Tufenkji, N. "Aggregation and Deposition of Engineered Nanomaterials in Aquatic Environments: Role of Physicochemical Interactions", Environmental Science & Technology, Volume 44, September 2010, pages 6532-6549. Bomba А. I. Neliniyni syngulyarno-zbureni zadachi typu "konvekciya – dyphuziya" / Bomba А. I., Baranovskiy S. V., Prisyazhnyk І. M. – Rivne: NUWGP, 2008. – 252 p. Bomba А. I. Neliniyni zadachi typu philtraciya- konvekciya – dyphuziya -masoobmin zа umov nepovnyh danyh / Bomba А. I., Gavriluk V.І., Safonyk А.P., Fursachyk О.А. // Monografiya. – Rivne : NUVGP, 2011. – 276 p. Ivanchov N.I. Оb opredelenii zavisyaschego оt vremeni starshego koefficienta v parabolichescom uravnenii / I.N. Ivanchov// Sib. mat. jurnal. - 1998.—Т. 39, N 3.—PP. 539-550. Kabanihin S.I. Obrannye i nekorekynye zadachi /S.I. Kabanihin. Novosibsrsk: Sibsrskoe nauchnoe izdatelstvo, 2009. – 458 p. Mints D.М. Teoreticheskie osnovy tehnologii ochistki vody / Mints D. М. // М. : Stroyizdat, 1964. - 156 p. Sanduliak А.V. Оchistka jidkostey v magnitnom pole / Sanduliak А. V. – Lvov : Vyshaya shkola, izdatelstvo pry Lvov. un-ta, 1984. - 166 p. Sergienko I.V., Deineka V.S. Resheniya kombinirovannyh obratnyh zadach dlya parabolicheskih mnogokomponentnyh raspredelennyh sistem // Kibernetika i systemnyi analiz. – 2007. – №5. – PP. 48-71. Sergienko I.V., Deineka V.S. Identyfikaciya gradientnymy metodamy parametrov zadach dyffuzii veshchestva v nanoporistoy srede // Problemy upravleniya i informatiki. – 2010. – №6. PP. – 5-18.

Associate prof., PhD, Andrii Safonyk e-mail: [email protected]

V. CONCLUSIONS

Associate professor of the Department of A mathematical model of the process of purification of automation, electrical engineering and computer liquids from multicomponent contamination by the n-layers integrated technologies of the Institute of magnetic filter is built, which takes into account the inverse Automation, Cybernetics and Computer effect of the determining factors (concentration of fluid Engineering, National University of Water Management and Nature Resources Use, Rivne, contamination and sediment) on the medium characteristics Ukraine. (porosity coefficient), and includes the ability to identify an Engaged in scientific mathematical modeling of unknown mass transfer coefficient. We propose an algorithm natural and technological processes, computer simulation of technological for solving the corresponding perturbed problem, which, in processes, computer techniques and computer technologies, programming. particular, suggests the possibility of determining the time t3 of the protective action of the filter. The results of numerical calculations are given. In the limits of this model, the possibility is provided of automated process control of the

ISSN: 1998-0140

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