Mathematical modeling process of liquid filtration taking into account reverse influence of process characteristics on medium characteristics

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.


Introduction
The analysis of researches results which was conducted in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][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.

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: where  

Algorithm (asymptotic) of the solution
Solution of system (1) in the terms (2) was founded in the kind of the asymptotic rows [9] - [17]: the proper regulating transformations. Like to [17], after a substitution (4) in (1) and application of standard "procedure of equation", for finding of functions , c ij and j  ( 0, jn  ) we come to such tasks: As a result of their solving we have: jn  ) which were assigned for the removal of inconsistencies, which were brought by the built regular parts, in areas around the points with some accuracy 0, x x L  (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.

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 F H gradH c  settling in points of the contact of nozzles granules, where value F c can arrive the size at the value in order 2·10 15 А²/m³ (Нmagnetic field intensity). In initial moment of time (t=0) porous nozzle is relatively "clean", that is unsaturated admixture particles, its porosity -0  . 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 P  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 nonstationary stage (Fig. 1). As known from [17], at t з   , 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 n  efficiency of cleaning is diminishing to the zero. The process of magnetic settling of admixtures, which is realized in magnetic filter ( 0 xL ) 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]. 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]):   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 tookv const  , 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, 0 xL ) pressure gradient, especially using the formula   v grad P   we can find the time of passage more than critical value gradient and to solve proper "decisions of automatization". The change grad P is shown in Figure 3 with time.
As we can see (Fig. 4), if the case   ** c t c const  the filter efficiency is unchanged practically until the time moment з  , which confirms the known fact of filter efficiency distribution with time.

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: The solution of system (13) at the terms (14) we are founding similar to the general problem in the form of asymptotic series (see [9], [17]). The results of calculations by formulas (4)   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).

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 masstransfer. The proposed algorithm for solving the proper problem, in particular, includes: determineting the time з  protective action of filtering nozzle, determineting of the limit overfall of pressure P  and value grad P at change    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).