COMPUTER PREDICTION OF TECHNOLOGICAL REGIMES OF RAPID CONE-SHAPED ADSORPTION FILTERS WITH CHEMICAL REGENERATION OF HOMOGENEOUS POROUS LOADS

Abstract. Mathematical models for predicting technological regimes of filtration (water purification from the present impurities), backwashing, chemical regeneration and direct washing of rapid cone-shaped adsorption filters, taking into account the influence of temperature effects on the internal mass transfer kinetics at constant rates of the appropriate regimes, are formulated. Algorithms for numerical-asymptotic approximations of solutions of the corresponding nonlinear singularly perturbed boundary value problems for a model cone-shaped domain bounded by two equipotential surfaces and a flow surface are obtained. The proposed models in the complex allow computer experiments to be conducted to investigate the change of impurity concentrations in the filtration flow and on the surface of the load adsorbent, temperature of the filtration flow, filtration coefficient and active porosity along the filter height due to adsorption and desorption processes, and on their basis, to predict a good use of adsorbents and increase the protective time of rapid cone-shaped adsorption filters with chemical regeneration of homogeneous porous loads.


Introduction
Any water needs to be purified before it can be used for domestic and drinking water supply. The main methods of water purification are clarification, decolorization and disinfection. The final stage is its purification from various impurities, in particular, calcium and magnesium salts, the total content of which determines the hardness of the water, as well as iron removal, in rapid adsorption filters with chemical regeneration of porous loads [4,6]. They use natural (bentonite, montmorillonite, peat), artificial (activated carbon, artificial zeolites, polysorbs) and synthetic (nanostructured carbon sorbents) materials as adsorbents [17]. The rate of the adsorption process depends on the concentration, nature and structure of the impurities, filtration rate and temperature seepage, and type and properties of the adsorbent [5]. Maintaining a constant set filtration rate is achieved by automatically adjusting the increase in the opening of the valve on the filtrate pipeline as the resistance of the filter load increases due to the accumulation of impurity particles in it. The impulse to increase the opening of the valve on the filtrate pipeline is a change in the water level on the filter (controlled by a float device) or water flow in the filtrate pipeline (controlled by a throttle device and a differential pressure gauge) [11]. When the latch is fully open, the filter is switched off to regenerate the porous load. First, the backwash regime with a high water supply rate (2-3 times higher than the filtration rate), which lasts for 5-20 minutes and allows the filter material of the porous load to loosen and large particles of impurities to be removed. Next, a regime of chemical regeneration is carried out with a high feed rate of a solution of a certain reagent (potassium permanganate KMnO 4 is usually used), which starts the process of chemical restoration of the adsorption capacity of the porous load, and lasts for 10-30 minutes. Impurity particles from the filter material pass into the reagent solution. Finally, a regime of direct rinsing at a high water supply rate, lasting up to 10 minutes, seals the filter material of the porous load and removes residues of impurities and the chemical solution of the reagent.
The increasing needs for purified water in industrial enterprises and the growing cost of filter materials require research, on the one hand, into more optimal use of adsorbents and increasing the duration of filters by choosing their shape, in particular, taking into account the influence of changes in the temperature of the filtration flow along the filter height on the process of adsorption water purification, and on the other hand, into restoration of the filtration properties of porous loads by chemical regeneration for their reuse [4,6].

Literature review
As an analysis of the literature sources shows, in particular [2,3,5,7,8,12,13,15,16,18,19], a significant contribution to the development of the theoretical foundations of filtering liquids through porous loads has been made by many scientists, both domestic and foreign. Note that mathematical models for predicting the technological processes of filtration and regeneration of porous loads by domestic researchers often use the model of D. M. Mintz [15] with constant rates of the respective processes and temperature, or some modification (improved model). In [10], its spatial generalization is proposed to predict the process of water purification from impurities in rapid cone-shaped filters while maintaining a constant filtration rate. The model proposed in this work is more efficient for theoretical studies aimed at optimizing the filtering process parameters (duration, shape, filter size, layer height, etc.) by introducing additional equations to determine the change in active porosity and filtration coefficient of filter load along its height, taking into account diffusion in the filtration flow and on the surface of the load grains. An urgent task is to generalize the appropriate model for computer prediction of technological regimes of filtration, backwashing, chemical regeneration and direct washing of rapid cone-shaped adsorption filters, taking into account the influence IAPGOŚ 4/2020 p-ISSN 2083-0157, e-ISSN 2391-6761 of temperature effects on the internal mass transfer kinetics at constant rates of the appropriate regimes.
These models in the complex will allow providing computer experiments to predict a better use of adsorbents and increasing the protective time of rapid cone-shaped adsorption filters with chemical regeneration of homogeneous porous loads by taking into account not only the change in the filtration flow rate along the filter height, but also the effect of temperature on the coefficients that characterize the rates of mass transfer during adsorption and desorption, as well as on filtration coefficient.

Formulation of the problem
Let's develop a model of technological regimes of filtration, backwash, chemical regeneration and direct washing of rapid cone-shaped adsorption filters with chemical regeneration of a homogeneous porous load. We assume that in the filtration regime, the convective components of mass transfer and adsorption outweigh the contribution of diffusion and desorption, and in the backwash, chemical regeneration and direct washing regimes, the convective components of mass transfer and desorption outweigh the contribution of diffusion and adsorption. In addition, due to changes in the temperature of the filtration flow due to adsorption and desorption processes, the influence of temperature effects on the internal kinetics of mass transfer is taken into account. We assume that the convective components of mass transfer and adsorption outweigh the contribution of diffusion and desorption. In addition, the impact of temperature effects on the internal kinetics of mass transfer is taken into account due to changes in the temperature of the filtration flow due to adsorption and desorption processes. So, for the domain (0, ) ) bounded by smooth, orthogonal interconnecting lines, by two equipotential surfaces S  , S  and by the flow surface S  (Fig. 1), the corresponding spatial model problems for predicting technological regimes of rapid cone-shaped adsorption filters, taking into account the reverse influence of process characteristics (impurity concentration, respectively, in the filtration flow and on the surface of the adsorbent) on the load characteristics (filtration coefficients, porosity, adsorption, desorption) will consist of equations describing the motion of the filtration flow and the equation of continuity: Next are equations for determining the change in impurity concentrations in the filtration flow and on the surface of the load adsorbent, temperature of the filtration flow, filtration coefficient and active porosity along the filter height, respectively, for the filtration regime: backwashing, chemical regeneration and direct washing regimes: which are supplemented by the following boundary conditions, respectively, for filtration and direct washing regimes: backwash and chemical regeneration regimes: , and initial conditions: ,, is respectively the potential and the velocity vector of the filtration,

Materials and methods
The problem is solved in the same way as in [10]

ABB A n CDD C n ADD A BCC B ABCD A B C D ABB A n CDD C n ADD A BCC B ABCD A B C D ABB A n CDD C n ADD A BCC B ABCD
backwash, chemical regeneration and direct wash regimes: , 0,

CDD C n ABB A n ADD A BCC B ABCD A B C D CDD C n ABB A n ADD A BCC B ABCD A B C D CDD C n ABB
the initial conditions (8) Similar to [9], problems (1), (9), (13) and (1), (11), (13) are replaced by the more general direct problem of finding a spatial analogue of the conformal mapping of the one-connected domain \ G  z to the corresponding domain of complex potential which is rectangular parallelepiped  is the full filtration flow, with subsequent finding of conditions of "gluing" on the banks of conditional section  . The algorithm for solving these problems is obtained in [9], in particular, the velocity field v , parameters Q  , * Q , Q and a number of other variables are found. By replacing variables backwash, chemical regeneration and direct wash regimes: ,, , …, for the model problem of predicting the filtration regimes:   ) and (16)