Nonlinear Inverse Problem for an Ion-Exchange Filter Model : Numerical Recovery of Parameters

This paper considers the problem of identifying unknown parameters for a mathematical model of an ion-exchange filter via measurement at the outlet of the filter. The proposed mathematical model consists of a material balance equation, an equation describing the kinetics of ion-exchange for the nonequilibrium case, and an equation for the ion-exchange isotherm. The material balance equation includes a nonlinear term that depends on the kinetics of ion-exchange and several parameters. First, a numerical solution of the direct problem, the calculation of the impurities concentration at the outlet of the filter, is provided. Then, the inverse problem, finding the parameters of the ion-exchange process in nonequilibrium conditions, is formulated. A method for determining the approximate values of these parameters from the impurities concentration measured at the outlet of the filter is proposed.


Introduction and Background
One of the key technologies for the preparation of water at thermal power plants and nuclear power plants is the use of ion-exchange materials.Water treatment using ionexchange methods is based on the passing of source water through a filter bed of ion-exchange material.That material is substantially insoluble in water but capable of ionexchange with ions contained in the treated water.Water is a good solvent of various salts that have a negative solubility coefficient; that is, when the water temperature rises, the amount of dissolved salts is reduced, and the excess amounts of these salts are precipitated in heated tubes, forming scale with a very low coefficient of thermal conductivity.Powergenerating units of thermal power plants (TPP) operate under severely corrosive conditions: high temperature (515-530 ∘ C) and pressure (15 MPa) hot steam [1][2][3].The main reason for failures of power equipment is failure of the most thermally loaded parts, namely, pipes in which the coolant water flows.Scale restricts flow of heat energy to the coolant (water) and does not allow for the cooling of pipe surfaces; this causes the surface metal temperature to increase, reaching the critical melting point.The result is an emergency stop of the boiler [3].Another problem is that the rate of corrosion of the metal under scale on the surface of the tubes sharply increases; that is, scale is a corrosion catalyst [4][5][6][7].Reducing metal corrosion requires not only using materials with high corrosion resistance but also taking all measures to reduce aggressiveness of the coolant medium, water [4,7,8].The primary source of harmful impurities is water from the water treatment plant, which is used to compensate for the loss of water coolant in the power plant cycle.The necessary water for the plant production process is usually taken from rivers and lakes.
This technology requires periodic regeneration [9,10] of the filters.Regeneration of ion-exchange filters results in highly mineralized, acidic, and alkaline waste water [10].After circulating through the energy plant, it is a highly mineralized aqueous solution.The high volume of water used produces an increase in the salinity of rivers and lakes when it is returned and causes harmful environmental impact due to the deterioration of the quality of the water.Approximately 25-30% of the water purifier output is for water treatment facility needs.For example, given a thermal power plant with 1200 MW of power, the water treatment plant capacity will be approximately 250 m 3 /hour; that is, an average of approximately 62.5 m 3 /hour or 547,500 m 3 /year of water is used for filter recovery.The switching from filtering (water treatment) to backwash, regeneration, and washing modes is carried out either by a time-scheduled mode or based on the volume of passed water and solution.Any of these procedures leads to the consumption of more water and reagents than needed.Thus, it is necessary to find alternative filter operation rules for the aim of reducing the water and reagents consumption.The problem of water treatment for thermal power plants using the ion-exchange method is a multiparametric task.The existing techniques for designing the operation of the water purification plant exclude the possibility of reaching an optimal mode of operation [9,11,12].
In this paper we propose a new method based on analysis of the water quality at the filter outlet.That is, we model the dynamics of the ion-exchange process at the filter, and then we solve an inverse problem of finding values for the process parameters.In other words, one can make decisions based on the real work conditions of the equipment.The operation of the water treatment plant under those operating rules can increase the operating cycle of the ion-exchange filter.The amount of waste water and water consumed for its own needs can be reduced.The method is easy to implement for water treatment equipment, either in an existing TPP or at the stage of designing a water treatment plant's equipment.That allows one to predict the stopping time of the filtration process and the starting time for a regeneration process.
Section 2 of the paper contains a description of the mathematical model of the ion-exchange filtration process.In Section 3, a description of the method to solve the direct problem is provided.Section 4 describes the statement and the main idea of solving the inverse problem.In Section 5, the method for the inverse problem with one unknown parameter is developed.In Section 6, results of the numerical solution of the inverse problem with two unknown parameters of the model are provided.The final section contains a brief summary of the research conducted.

Mathematical Modeling of the Ion-Exchange Filtration Process
Several mathematical models have been proposed to describe the dynamics of ion-exchange processes [13][14][15][16].In general, these models consist of three equations: mass balance equation, kinetic equation, and isotherm equation.The ionexchange kinetics equation determines the speed of delivery of ions to the surface of the ion-exchanger.Since the system comprises a kinetic equation, this system of equations models an ion-exchange filter operating in nonequilibrium conditions.In the following, we describe each of them as presented in [11].

Mass Balance Equation.
Sorption of an ions mixture by an ion-exchange material is subject to the law of mass conservation.Thus, the mathematical model for the ionexchange in the filter also includes a mass balance equation: where (i)  ap is the flow velocity in the free section of the filter in m/sec; (ii)  is the porosity of the ion-exchange material in m 3 /m 3 ; (iii) / is the rate of change of the ion concentration in the solution, (, ), along the filter axis () in (mEq/kg)/m; (iv) / is the rate of change of the ion concentration in the solution, (, ), with respect to time () in (mEq/kg)/sec; (v) / is the rate of change of the ion concentration in the ion-exchange material, (, ), at the point  with respect to time () in (mEq/kg)/sec; (vi)  is the thickness of the filter material in m.

Kinetic Equation.
Processes of the anion exchange and cation exchange of fresh natural waters, as well as processes of ion-exchange of condensed water, run in accordance with the mechanism of the external diffusion kinetics.One of the equations that is customarily used to describe the rate of ion-exchange in the external diffusion kinetics is the Nernst kinetics equation [9].However, the Nernst theory is not always convenient for describing the diffusion kinetics because of the large number of experimental variables in the equation.To overcome this difficulty, a simpler alternative is proposed in [9].An equation from the general theory of mass transfer is used, which expresses the hypothesis that the rate of ion-exchange is proportional to the deviation of the ion concentration in the sorbent from the corresponding equilibrium value.The equation is mathematically written in the following form: where (i)  is the mass transfer coefficient (proportionality) in m 3 /m 3 ; (ii)  is the concentration of exchanging ions in the sorbent in mEq/kg; (iii)  is the concentration of the exchanging ions in the solution in mEq/kg; (iv)  is the equilibrium ion concentration with respect to the value  in the solution in mEq/kg, which borders with the surface of the ion-exchange material, in the filtration case  > .

Exchange Isotherm Equation.
The adsorption isotherm equation is used to determine the capacity of each sorbent for various sorbates.In the ion-exchange theory, isotherms of Langmuir [17], Freundlich [18], and Redlich-Peterson [19] are commonly used.These studies have shown that, for a mathematical description of the ion-exchange process in nonequilibrium conditions, the Langmuir isotherm equation described in [9] may be used.The corresponding mathematical expression can be written as follows: Here, we get that (i)  = / 0 is the relative concentration of exchanging ions in the solution phase in mEq/kg, where  0 is the maximal possible concentration of exchanging ions in the original solution and  is the constant of the exchange isotherm ( > 1 for a filtration process and  < 1 in a regeneration case); (ii)   is the relative concentration of additional ions in the ion-exchange material.These ions form a weakly dissociated compound.  =   / 0 where   is the concentration of ions in the original solution in mEq/kg that form a weakly dissociated compound; (iii)  is the relative concentration of exchanging ions in the ion-exchange phase in mEq/kg. = / 0 , where  0 is the total exchange capacity of the ion-exchange material.
Thus, taking into account (1)-( 3), the mathematical model of the ion-exchange filtration process for nonequilibrium conditions can be written as follows: Because exchange isotherm equation ( 3) is expressed in terms of relative concentrations of exchanging ions in the solution phase and the ion-exchange phase, we can write the equation of balance and kinetics in terms of relative concentrations.By dividing the equation of ( 1) by the value of  0 and the equation of ( 2) by the value of  0 , we rewrite (4) in terms of the functions (, ) = (, )/ 0 and  = (, )/ 0 , given that  =  0 : Let us introduce the following notations for the dimensionless constants appearing in the system of equations: 0 <  < 1 is a constant dealing with possible chemical reactions with impurities of other ions in solution.
Given the notations in (6), the system of equations in (5) can be rewritten as Here, we use the function () to denote the relationship between the relative concentration of exchanging ions in the solution phase  and the relative concentration of exchanging ions in the ion-exchanger phase : The result is the system of equations in ( 7) that describes the model of the ion-exchange filter operating under nonequilibrium conditions.The following are boundary conditions for system (7): The physical meaning of these terms is as follows.The condition (0, ) =  0 () indicates that, over the ionexchanger layer ( = 0), at any time , there is a solute with constant concentration  0 ().(, 0) = 0 shows that initially ( = 0) the absorbed ion is absent at any level of layer  of the ion-exchanger.The equation (, 0) = 0 indicates that, at the initial time ( = 0), the concentration of ions absorbed from the solution in the filter material is negligibly small.Calculations showed that a change in porosity of the ionexchange material  virtually had no impact on the period of time before reaching the permissible concentration of the ion to be removed at the output of the filter, so the average value of 0,325 is used for calculation.

The Numerical Solution of the Direct Problem
Direct problem (7), (9) with known constants  and  is nonlinear; however, the nonlinear function () satisfies the Lipschitz condition, so it can be shown that the solution of problem ( 7), (9) exists and is unique.The proof of the uniqueness and existence of the solution to the direct problem is out of the scope of this study.For the numerical solution of problem ( 7), ( 9) we apply the finite-difference method.According to this method, the partial derivatives in the system of equations of ( 7), ( 9) are replaced with their difference analogues: Here,  = /  is the time step, and ℎ = / is the step in space on a uniform grid.
The first equation of the direct problem is approximated by the implicit difference scheme of point-to-point computation, since when using the calculation parameters characteristic of the technological process the explicit scheme appeared to require a very small step equal to, in terms of time,  = 0,000014 for  = 128.Due to the nonlinear nature of the problem, in order to ensure consistency of the calculation, the time step to be used for the point-topoint computation must be equal to  = 0,000125 for  = 128.The deviation in the results of calculations using the explicit scheme and the point-to-point computation scheme was calculated using the following formula: where  im 1 (  ) is the relative concentration of Na + cation to be removed at the outlet of the filter when calculating using the implicit finite-difference scheme;  ex 1 (  ) is the relative concentration of Na + cation to be removed at the outlet of the filter when calculating using the explicit finite-difference scheme.
The deviation was equal to 0,075%.To investigate the influence of parameters  and  on the output data the grids of parameters  and  are given at their characteristic ranges of variation.Then, for each pair of discrete values of  and , the concentration function  1 () is calculated at the output of the filter on a grid of time values.
We compared the results with data from a real case.The data were collected at the 1200 MW thermal power plant located in the Pavlodar Region of the Republic of Kazakhstan.At the thermal power plant, a 3-stage scheme of deep water desalination is provided, and the capacity of the main water treatment plant is 300 t/h.
In Figure 1, we show the characteristic graph of  depending on  at filter layer  = 2,5 m.
Figure 1 shows changes in the relative concentration of Na + cations in time during operation of the filter in the water filtration mode at the output of the filter according to experimental and calculation data.According to the calculations, the relative concentration of Na + cations during the first 40 hours of the filter operation is extremely low, ranging from 0.000178 to 0.000477; then breakthrough of Na + cations is observed which leads to an increase in the relative concentration of Na + cations at the output from the filter.The filtering process ends when the maximum of the relative concentration of Na + cations at the output of the filter is achieved equal to 0.297, with the filtration duration being 55 hours.According to the experiment data, the maximum relative concentration of cation removed is achieved at the moment in time being 55 hours.Comparison of the graphs shows that the mathematical model approximation (7), (9) using implicit finite-difference scheme (10) provides satisfactory results.
In Figure 2, we show the characteristic graph of  depending on .
Figure 2 shows changes in the relative concentration of Na + cations depending on the height of the layer of the ionexchange material in the filter based on calculation data 20 hours after the beginning of the filtration process.When filtering water through the ion-exchanger layer, the ionexchange process flows layer by layer.The total height of the ion-exchange layer is 2.5 m.The graph shows three zones as follows: 0 to 0.3 m high layers of depleted ion-exchanger, which cease to be part of the ion-exchange processes (I); 0.3 to 1.3 m high working ion-exchanger layers in which the ion-exchange processes take place (II); and 1.3 to 2.5 m high layers of fresh ion-exchanger which are not yet involved in the ion-exchange processes (III).As the depletion progresses the upper layers of the ion-exchanger cease to be part of the ion-exchange processes.Instead of them fresh ion-exchanger layers are involved in operation which lie under the working layer, and the ion-exchange zone progressively moves as the water treatment process continues.

Formulation of the Inverse Problem
During the operation process, the measurement of the impurities concentration at the outlet of the filter is available: Here,  is the horizon observation time.
Due to the complexity of the sorption process and heterogeneity of the mixture to the filter, the parameters  in model (7) and  and  in (8) cannot be determined accurately.In practice, we can operate with approximate values of , , and .In some cases, when the main ions of the filtered mixture are known, we can assume that the value of  is given.Then, we still have two uncertain model parameters, the exchange isotherm parameter  and parameter .Having the measurements of ( 13), the problem is actually reduced to an inverse problem, which can be stated as follows: Find the model parameters  and  of problem ( 7), (9) via boundary measurements (13).
Although there are examples of formulations and solutions of inverse problems for filtration processes in the literature [20], this statement of the inverse problem is new.The standard method to solve inverse problems is quasisolution [21], according to which the unknown parameters of the model are determined by minimizing the residual function.
Earlier, in [22], we investigated the behavior of the residual function of the following form: with the fixed value of the mass transfer coefficient .On the basis of the direct numerical simulation described in [23], we have shown that function ( 14) is not convex.Additionally, the low sensitivity of function (14) with respect to the model parameters  and  is established.This means that the routine use of gradient methods is inefficient.As a result, we propose an alternative approach to the solution of the formulated inverse problem.The main idea of the method is the following; as we have two unknown parameters of the model,  and , it is necessary to find at least two integral characteristics of measured data (13), which clearly define these parameters.

Solution of the Inverse Problem with One Unknown Parameter, 𝑚, of the Model
For solving the inverse problem of finding two unknown parameters  and  first consider the solution for a simple problem.We solve the problem with one unknown parameter  of the model.Assume first that the parameters  and  in model ( 7), ( 9) are known and state the problem of determining the value of .The natural measured value is the total amount of impurity  passed through during the ion observation period: Due to the extremely large scatter of (, , ), it is more convenient to use its logarithm,  1 = log.It is established numerically that the function  1 () and the value  1 depend on the ratio  = /.This fact indicates that, using measured data (13), it is possible to recover the ratio  = /, and it might be difficult to recover the variables  and  simultaneously.It follows that the models with the same value of the function (, ) = / are equivalent with respect to measured data (13) and observed value of  1 = log.In Figures 3 and 4, we show the characteristic graphs of log depending on  and .Thus, the following equivalence principle is obtained for the problem of filtration ( 7)-( 9): mathematical models with different  and  and the same  = / are equivalent with respect to the observed value  1 and measured data (13).
This result seems to be natural if we expand the function () with respect to  as follows: One can see that the function () can be approximated as () = / and depends on the ratio  = / for small values of .Note that work parameters of the filter use small values of , which explains why we obtain the same output data with the same value of the ratio  for different  and .Calculations show monotonic dependence  1 () of  in the practically acceptable range of  and .This implies that if we know the value of , then, on the basis of the observed value of  1 , the value of  = / is uniquely determined, and, through it, in turn, the value of  = / can be calculated.Thus, we have two uncertain parameters of the model, the parameters  and  = /, which is associated with exchange isotherm (3).For practical applications of this result, we have generated a table of values of  1 = log for a range of values of  in the interval [1.0, 5.0].It is enough to interpolate the data of the generated table to recover the desired value of  0 using the measured value of log 0 .Table 1 shows the values of the exact ( exact ) and restored ( rest ) parameter  values for hydrogencation exchange filters used in the first stage of make-up water treatment.
The test results showed satisfactory accuracy of  variable restoration.

Restoring Parameters 𝑚 and 𝛾
Suppose now that we know parameter  and do not know two parameters,  and .To calculate the range of change of , we can use definition (6) of : In practice, parameter  is calculated according to [9,24] using the empirical formula where  ap is the rate of flow of fluid in the unit in m/s,  is the ion-exchange grain size in , and  is an empirical coefficient.
For the calculation, we use the parameter values that are typical for the hydrogen-cation exchange unit of the first stage of water treatment,  ap = 0,0056 m/s, and the value of  for the ion-exchange material varies from 0.315 to 1.250 mm.The coefficient  for the exchange of H + ions for Na + is 1.Based on this data, the values of  in the processes of interest are in the range of 1.523 × 10 3 ≤  ≤ 3.165 × 10 3 .Suppose that another observable quantity that is sensitive to the values of the unknown parameters of the model is the logarithm of the impurities concentration at the outlet of the filter at time  = : The partial derivatives  1 (  ,   )/,  1 (  ,   )/,  2 (  ,   )/, and  2 (  ,   )/ are approximately calculated using the tables via formulas of finite differences.The relations in (21) represent a linear system of equations with indeterminate  0 ,  0 .This system is solved numerically and defines approximate values of the unknown parameters.Table 2 shows the values of the exact ( exact ,  exact ) and restored ( rest ,  rest ) parameters  and  for the hydrogen-cation exchange filters used in the first stage of make-up water treatment.The test results show satisfactory accuracy of  and  variable recovery.
The developed method of recovering the parameters was verified with errors ±1%, ±3%, ±5%, and ±7% entered in the exact values ,  1 ().The mean relative errors of  and  parameters recovery are shown in Table 3.

Conclusion and Discussions
In the paper a mathematical model of the ion-exchange filter operation mode is considered (7), (9).The direct problem for the ion-exchange filter model was numerically solved.For calculation it is necessary to know the values of  and , which in practice are not available for measurement.In the paper a method for the determination of these coefficients based on the output values of ion absorbed at the outlet of the filter was proposed; the inverse problem was formulated.The method is based on the definition of the integral characteristics of the input data that accurately describe these parameters.The integral characteristics were determined.The integral characteristics are the logarithm of the number of impurity ions passing through during the observation period and the increase of the logarithm of the concentration of impurities at the outlet of the filter.The inverse problem was numerically solved given the unknown  and  parameters.Resetting the parameters using the proposed method yielded satisfactory results.
The developed method can be used in simulation of the ion-exchange unit operation mode.This will allow determining the optimal mode of operation facilitating reduction of the amount of regeneration and washout water.

Figure 1 :
Figure 1: The dependence of  on .

Figure 2 :
Figure 2: The dependence of  on .

Figure 3 :Figure 4 :
Figure 3: The dependence of log on the parameter .

Table 1 :
The exact ( exact ) and restored ( rest ) parameter  values.
By solving the direct problem in the range of variation of parameters  and , we have generated tables of values of data for  1 (, ) and  2 (, ).Based on these two tables, we can calculate the appropriate values of  0 and  0 from the measured values of  1,0 and  2,0 .Below is described a numerical method for calculating  0 and  0 .Thus, suppose that we have a table of values for the two measured variables at the filter outlet,  1 (  ,   ) and  2 (  ,   ).Suppose that  1,0 and  2,0 are measured values.Using the tables, we can determine a pair of values,   , , for which the minimum deviation is reached:     1 (  ,   ) − 1,0     2 +      2 (  ,   ) −  2,0     2 = min ,       1 (  ,   ) −  1,0      2 +       2 (  ,   ) −  2,0      2 .(20)

Table 2 :
The values of the exact and recovered parameters  and .