MATHEMATICAL MODELING SINGULARLY PERTURBED PROCESSES OF WATER SOFTENING ON SODIUM-CATIONITE FILTERS

Mathematical model of the process of water softening using ion exchange pre-treatment of waters to desalination, with a view to removal of scale forming components, such as calcium and magnesium, are formed in the paper. In this process, no additional chemicals, except for brines formed during desalination, are required for regeneration of ion-exchanger in operation cycles. An asymptotic approximation of a solution of a corresponding model problem is constructed. Theoretical description and modelling assumptions included the set of differential equations of mass balance, initial, boundary and operational conditions. The paper deals with the development of a computer model for description and prediction of the performance of ion exchange columns.


Introduction
Hight salinity levels in wastewater usually have negative consequences to equipment that comes in contact with such water, for example boiler room installations. Though many techniques have been developed to solve the problem, high capital costs or energy consumption have hindered their application in wider areas. One economical solution of reusing the wastewater of high salinity is applying natural or synthetic zeolites as ion exchanger and adsorbent.
The principle of work of softeners is based on the process of replacing calcium, magnesium and other metals with cations of sodium (or hydrogen). The process of softening the water is carried out by filtration from the top down through the cationite layer of insoluble nanoporous granules about 1 mm in diameter. After exhausting the cation exchanger, the quality of softening is significantly worse. To restore the ion-exchange resin's properties, an automatic transition to the regeneration mode occurs by passing through a layer of salt solution.
The mechanisms governing the salt removal process by zeolites are mainly ion exchange, adsorption, and salt storage. Factors such as zeolite's geochemical properties, pH, co-existing anions, concentration, valency, surface charge, and experimental conditions all influence the ion exchange process [6]. The adsorption kinetics of Ca on zeolites is mostly a pseudosecond-order type with an exothermic nature. Calcium removal by zeolites appears to be an effective water treatment technology for maximizing the beneficial use of poor-quality wastewater. However, challenges still remain and further work is required in areas of lowering operational cost and improving zeolite's regenerability. To overcome these challenges, researchers could make more efforts in technical improvements, including alterable surface properties and the incorporation of other approaches to achieve better salt removal outcome.

Literature review
The reaction of ion exchange, which is the basis of the cation exchange method of water softening, is a complex multi-stage process. On the basis of the known laws of sorption dynamics in [4] a one-dimensional mathematical model of the cation exchange filter operation under the conditions of the nonequilibrium dynamics of the mixed diffusion kinetics of exchange of Na  cations on 2 Ca  cations was proposed.
In this model, the mutual influence of diffusion of some cations on the diffusion of others was not taken into account. In addition, since different stages of the described process proceed at different speeds, for some members of the mathematical model equations, small parameters appear, that is, the model is perturbed, and the solution of the corresponding model problem permits the asymptotic development of a small parameter [5].
In this paper, the asymptotic approximation of the solution of a singularly perturbed problem of convection diffusionadsorption transfer of a solution of calcium salts through a cationite layer is explained.

Formulation of the problem
On the basis of existing ideas [1,6], when softening water in cationite, in the theory of ion exchange, it is customary to consider three stages of cation exchange: a) delivery of the cation from the cation-exchange filter solution to the surface of the cationite grain; b) the penetration of this cation into the inside of the cationite grain; c) chemical interaction of the cation with the functional group of the cation exchanger. All the listed stages usually proceed simultaneously, but their flow rates are different. In this regard, the equations describing the course of the most slowly proceeding ion exchange stage in the case under consideration will be described by equations of external diffusion transfer (kinetics) in the first stage and inside diffusion transfer (kinetics) in the second stage.
Thus, based on the foregoing, and taking into account the existing ideas, the process of water softening in the ionexchange filter consists of the following steps:  convection-diffusion-adsorption transfer of cations 2 Ca  and Na  in a filtration solution  diffusion transfer inside the grain of cation  The following assumptions are taken when compiling the model: the interaction of cations in the liquid and solid phases is absent, in particular, this substance is present in the liquid and solid phases mainly in the same form; the exchange of cations of sodium and calcium is independent of the presence of other cations; at each time, the cationic pairs that are in solution and adsorbed by the cationite grains are in equilibrium, which makes it possible to set jj cc  when deriving equation (6), introducing the appropriate correction to the constant 1,2 k ; a layer of a cation exchanger consists of grains (granules) of spherical shape with conditional radius R, to which, according to well-known recommendations, grains (particles) of any shape can be given. Note that the latter have a microporous structure, which ensures the outflow of cations into their pore space.

Materials and methods
We introduce dimensionless quantities as follows The case of the predominance of the convective component of the mass transfer on diffusion and mass transfer is considerer, that is when 1 00 ( , ) To assess the remaining members we have:   Distribution of the concentration of ion in the interparticle space is shown in Fig. 2 (1*-3*). As the experiment showed, the purification takes place at a certain point in time (contamination of microporous particles). Under given conditions it is 5 th  , after which the intensity of cleaning begins to decrease, although the timepieces are not yet completely polluted. The maximum deviation of the calculated concentration value at the filter outlet under given conditions, taking into account adsorption, is 14%.
In Fig. 3, and depicts the distribution of the concentration of pollutant in a microporous particle at moments of time

Conclusions
A mathematical model of a singularly perturbed process of convection diffusion and adsorption mass transfer of ions is formed 2 Ca  and Na  in a homogeneous environment particles of microporous structure, which unlike the existing, takes into account the mechanism of convective transfer, as well as specifies the calculation of the distribution of concentration in a microporous medium (filter). The asymptotic approximation of solutions of the corresponding boundary value problem is constructed. t is confirmed that despite the small velocity of the diffusion mass transfer phenomena in particle pores, over time this significantly affects the distribution of concentration in the part itself and throughout the filter.