Mathematical modelling and improvement of ion exchange water treatment energy-saving technologies in power engineering

A mathematical model of the ion-exchange filter regeneration process of a wastewater treatment plant is presented in this paper. In the model the ionite filter is considered as a chemical through-passage reactor with bulk material, final length, periodical action in a stationary mode. The one-parameter diffusion model used for the description with the Dankwerts boundary conditions makes it possible to calculate with sufficient accuracy the output concentration curves of the reagent and products.


Introduction
In thermal and electric energy production systems (primarily at heat and power plants), wastewater treatment plants (WTP) are the main source of wastewater, which are highly mineralized, energy-and resource-consuming (reagents). Various technologies are proposed and implemented at many heat and power plants, which allow to reduce reagents unit cost and wastewater volumes. First of all, this refers to the most widespread ionite methods. Such technologies as block arrangement of multistage wastewater treatment plants filters, block-counter-current regeneration, counter-current regeneration, treated regeneration solution reuse are introduced. At the same time, regime characteristics and constructive features were chosen empirically, which doesn't guarantee optimal solutions.
The purpose of the work is mathematical modeling of the ionite filters regeneration process based on the of fundamental laws of physics, chemistry, hydrodynamics.

Theory
The ionite filter is similar to a cylinder loaded with a fixed solid ionite, in which the flow of water and reagent solution transfers along one axis (z) and the exchange chemical reaction of the reagent for chemo-sorbed ions (product) proceeds. Therefore, it is logical to consider it as a flow chemical reactor, and use the one-parameter diffusion model to describe it [1,2].
The equation of the component mass conservation according to convective and diffusion flows and chemical reactions in the flow reactor ( Figure 1) has the form: where L D -longitudinal diffusion coefficient or longitudinal mixing coefficient, r w -the rate of heterogeneous ion exchange chemical reaction,  To solve the diffusion model equation (1), it is necessary to make reasonable assumptions and simplifications. Direct reagent exchange chemical reaction to the product should formally be recorded as: where P c -concentration (content, fraction) of the product in the solid phase of ionite. According to the laws of chemical thermodynamics, in the case when a solid is involved in the reaction, its concentration is taken equal to one. Therefore, it is possible to record: where the constant K has a more complex meaning than the homogeneous reaction speed constant. In practice [3], we should return to the dimensionless form of the basic equation record (1) As a result, we have: It is accepted that the multiplier The equation (4) will be written as: For the product ( P ) there must be a similar equation. To simplify the problem, we accept that the chemical reaction is the equivalent product substitution for the reagent in the ionite solid phase and there is no specific reagent accumulation (sorption). Then the sum of reagent substance (the number of Using the number of equivalents moles as the substance quantity and the concentration of equivalents moles as the substance concentration -it is possible to get rid of the reagent ions charges and product and to simplify writing of equation To solve the equation (4) we should formulate restrictions. We consider that ion exchange filter in the regeneration stage is: 1) Liquid phase filled with solid grains 2) In stationary mode is flow-through ( ). In the dimensionless form of the record we have: At the final length filter outlet, it is assumed that the reaction flow is interrupted with a reaction stop (loading the ionite only inside the filter): To solve the obtaining problem of the response function (output concentration curves ) (θ c ) at the outlet of the filter in the original equation (5), the coordinate is fixed at 1 = L and the only variable remains time ( ) θ . In addition, since the case of the periodic action reactor with an obvious sharp change in conditions at the borders, it is advisable to zoning or, in terms of time, splitting the backwashing process into stages.
Two stages are visible for the regeneration process: 1 -regeneration is the income of a permanent concentration -normalized indicator of heterogeneous reaction rate, In Figure 2 the calculated concentration curves of the reagent and product at different criteria Pe are presented, allowing to visualize the characteristics of the model.

Comparison of theoretical (calculated) values with experimental data
Comparison of theoretical (calculated) output concentrations values with experimental data obtained during regeneration of industrial ion filter shows good accordance. In Figure 3 the output concentration curves of alkali, silicon dioxide (product 1), organic substances (product 2) at regeneration of the second stage anionite filter of chemical water purification-2 on the Nizhnekamsk HPP-1 by 2.6% alkaline regeneration solution with flow rate SiO ) 8.5 min. Therefore, the reaction proceeds at an average speed, but sufficient to separate and backwash most (but not 100%) of the filter. For organic substances K = 0.22, halfreaction time 3.5 hours. That is, during the regeneration, only a part of the organic substances is removed, which explains the ionite "poisoning". SiO , product 2 (organic substances) at regeneration of the second stage anionite filter of chemical water purification-2 on the Nizhnekamsk HPP-1. Solid lines -the diffusion model calculation.

Conclusions
The obtained data indicates that the implemented regeneration mode and the filter design do not achieve the ideal displacement mode, at which the theoretically possible minimum reagent flow, minimum wastewater volume and maximum product output are possible. Consequently, it is possible to carry out scientific research to improve both the design and the technology of regeneration.