Calculation of optimal flow of ionite during water treatment in the column of continuous action

. In the article an algorithm for calculating the continuous ion exchange column is proposed. By analogy with the calculation of the optimal phlegm number of the rectification column this algorithm takes into consideration depreciation charges for capital costs associated with the size of the column and working capital associated with the consumption of ionite and its regeneration. This algorithm can be used to calculate depreciation charges for capital costs related to column sizes and working capital regarded to ionite consumption and regeneration. Examples of ionite consumption calculation at typical and proposed algorithms of calculation and technological and geometric parameters used in column of water purification from sodium cations are given. Thus, in a typical calculation, when the water being purified from sodium cations moves in the ideal displacement mode and the excess of ionite is 1% of its minimum consumption. The optimal ionite consumption corresponding to the minimum of capital costs and working capital, which requires an increase in the ionite consumption by 5.6% of its minimum consumption. In this case the cost of ion exchange increases by 4.5%. It confirms the recommendations for industrial operation and design of ion exchange columns and increases the minimum ionite consumption by 1÷10%. The continuous ion exchange column was calculated including the longitudinal diffusion. Necessity of the further increase of the ionite consumption with regard to the longitudinal diffusion is shown. In the article an example of the ion exchange process calculation in comparison with the typical one, when the displacement mode for the treated water is ideal, is given. Similar equations for the dependences of the concentration of extracted ions in the purified solution and ionite granules are proposed if the equilibrium dependence is described by a linear equation. The working line is dependent on the concentration of the extracted ions in an ionite and on their concentration in the purified water, first, because of a jump in concentration at the inlet and a concave shape of the working line (the latter is usually a straight one in case of the mass-transfer processes) reduces the local and average driving forces of mass-transfer process significantly. It leads to increasing of the moving layer height of the ion exchanger in the column 2 times, and can lead to crossing the line of equilibrium by reducing the concentration jump at the entrance. Consequently, it is necessary to increase ionite consumption by more than 18% compared to its minimum consumption.


Introduction
Engineering calculations of ion exchange column geometric and technological parameters must be carried out taking into consideration energy and resource savings. The main cost of working capital is the consumption of ionite while the capital costs usually depend on the size of ion exchange column.
Standard algorithms for calculating continuous ion exchange columns are known to be based on calculations of basic mass exchange processes (rectification, absorption, adsorption and extraction), on determination from material balance with regard to equation of minimum flow rate of reflux, sorbent or extractant and their increase from several percent (ion exchange consumption) up to several tens of percent (in sorption and extraction processes) and even several times, that is, hundreds of percent (in the process of rectification) [1][2][3][4][5][6][7][8].
Attempts are made to take into account the flow pattern (e.g., longitudinal diffusion) and its effect on column sizes [9][10][11][12][13][14][15], but the economic costs associated with working capital (ionite consumption and regeneration) and with the depreciation charges for capital costs and their minimization are neither considered nor calculated at typical algorithms.
The purpose of the work is to determine the function of costs minimization for the shock absorption depending on the size of the column and on the working equipment depending on the flow rate of the ionite and its regeneration, as well as the effect of longitudinal diffusion on technological parameters and geometric dimensions of the ion-exchange column.

The calculation of the ionite columns
According to the articles [1,16,17] described minimization function "C" can be represented as a function where V is the column volume, Gx is the ionite consumption, and a is the weight coefficient of the relationship between depreciation charges for capital expenditures proportional to the column volume and working capital proportional to the ionite consumption. Tables 1 and 2 show the initial, reference data and calculated parameters of a continuous ion exchange column for removing sodium cations from water, calculated using the standard algorithm and the formula (1). A typical algorithm for calculating an ion exchange column with the specified initial data is presented in the article [18].  20 The equilibrium coefficients of the Langmuir equation a,b

Discussion of calculation results and conclusions
As it is evident from the comparison of the results of calculations of technological and geometric parameters of a continuous ion exchange column using the standard algorithm and the proposed algorithm for optimizing ionite consumption with regard to the minimum capital costs and working capital, the latter requires an increase in ionite consumption from 1.372 kg/hour to 1.434 kg/hour, that is 4.3%. In a year, this increase will be 500 kg/year. However, it is obvious from the graph shown in figure 2, the "C" minimization function is reduced from a value of 5.3014 to 5.3. It leads to a reduction in capital expenditures by 0.026%. Thus, the proposed algorithm for calculating the optimal ionite consumption with regard to depreciation charges for capital expenditures and working capital confirms the correctness of increasing ionite working consumption compared to its minimum consumption by 1-10%.
Proposed algorithm makes it possible to determine exact value of optimum flow rate of ion exchange column operating in mode of ideal displacement by cleaning solution.
The increase in longitudinal diffusion in the liquid phase often requires a greater increase in ionite consumption compared to the ideal displacement mode. It can be explained by a jump in the concentration at the entrance of the liquid phase flow in the column. A decrease in CH to C0, in this case, can lead to a value of the concentration of C0< CK* and the intersection of the working line and the equilibrium one (Fig.1). This is what happened in this case. Only an increase in the ionite flow rate from GX=1.434 kg/hour to 1,61 kg/hour, i.e. an excess of its flow rate from 5,6% to 18,6% in comparison with the minimum flow rate, allows the ion exchange process to be carried out in case the entire working line is under equilibrium line (Fig.1).
A response curve was taken on a laboratory column with ionite granules. Its dispersion was σ 2 =0.048 [19,20]. If the Pecle number of longitudinal diffusion is calculated by the formula it is equal to Pe=41.7.
In Fig.1 a graph of the working line (line 5) is shown. It has a jump in the input concentration from cH= 0.1 kg/m3 to c0 = 0.0928 kg/m 3 and the working line obtained by solving a non-uniform differential equation of the second order [10][11][12][13][14] *) ( when the equilibrium line can be approximated by the equation In this case, the deviation from the equation does not exceed 6%. The analytical solution of the differential equation (2) with regard to the formula (3) has the form The coefficients c1, c2 and Q are found from the boundary conditions and the equation of the working line takes the form where g=dc/dh -is the concentration gradient of the extracted component from the purified solution and h is the dimensionless height. According to the results of calculations and the graph of the working line (Fig.1, line 5) the working line not only does not reduce the concentration of extracted ions at the input c from cH to c0 due to a jump in concentrations due to longitudinal diffusion, but it also has a concave appearance (that is, the working line is no longer straight), which further reduces the local and average driving force of mass transfer. It leads to an increase in the number of transfer units, the average residence time, and the required height of the moving ionite layer.
Thus, the calculation presented in the article takes into account that the flow mode of the liquid phase does not correspond to an ideal displacement, but to a longitudinal diffusion. It changes the position of the working line significantly, firstly, shifting it towards the equilibrium line at the entrance and reducing the doubleacting force of the mass transfer process. Secondly, it requires an increase in the ionite consumption compared to the ideal displacement. At the same time, the proposed https://doi.org/10.1051/ conf/20 2016101084 ICEPP-2020 e3s E3S Web of Conferences 161, 010 (2020) 84 calculation of the optimal ionite consumption, considering capital costs and working capital, allows us to calculate the ionite consumption accurately and reduce the cost of the ion exchange process. Thus, it is advisable to replace the standard algorithm for calculating the ion exchange column with its exact calculation considering the formula (1) and longitudinal diffusion.