Advanced approach to mathematical modeling of the impurities diffusion in the process of water softening with limited particles sorption

Abstract

This paper presents a compound mathematical model of convective diffusion to describe the changes in water hardness during its filtration in a porous medium. The skeleton of the porous medium is formed by ion-exchange resins, zeolites, or coal, and the accompanying sorption processes are modeled as nonlinear sources, which existing models often neglect. The constructed mathematical model consists of two sequential submodels. Submodel 1 describes filtration processes in the porous layer accompanied by nonlinear sorption until the concentration of the sorbed substance reaches saturation. Submodel 2 consists of the convection–diffusion equation of impurity particles in the aqueous solution and the equality of concentration of the sorbed substance to saturation concentration. To address the gaps in prior research, specifically the omission of nonlinear sorption dynamics and transition zone behavior in existing models, this study formulates appropriate problems of mathematical physics for the nonlinear model and conducts a numerical analysis using Neumann series expansion. Improving these gaps is critical because they affect the precision of impurity behavior predictions and the optimization of filtration processes. The results reveal that the transition zone from Submodel 1 to Submodel 2 occurs either from the upper boundary of the filter or in the lower half of the porous layer. Additionally, concentration peaks appear at solution boundaries, which act as transition points between submodels, offering a refined understanding of impurity behavior during water filtration. This approach provides new insights into optimizing filtration processes under limited sorption conditions.

Introduction

Water used by humans typically contains a significant amount of dissolved substances of various types (possibly excluding melt water)¹. The presence of these impurity components often determines the fundamental macroscopic properties of water, such as hardness, alkalinity, oxidation and so on.

Some of these substances are directly harmful to biological formations, and the use of such water for household purposes leads to damage to the technical systems of its distribution^2,3.

Therefore, the problem of water purification and improving its quality is urgent, both in order to reduce the negative impact on human health and the existence of the human biostructure, as well as reducing the indirect economic costs associated with the corresponding technical redistribution systems.

In most instances, different types of filters are employed for water purification, including mechanical, chemical, physicochemical, electrical, biological, and more^4,5. Each of these types of filters is effective for a specific type of pollution. For example, mechanical filters, whose operation is based on sorption processes, remove heavy metals and organochlorine substances from water. During the so-called biological filtration of water, due to the metabolic processes of microorganisms, organic compounds are extracted by decomposing them into nitrates⁴. Electric water purification systems make it possible to effectively purify water from oxidizable impurities, the most common of which are iron, manganese, and hydrogen sulfide⁴.

As a rule, the technological process of water purification involves the use of combinations of different types of filters at different stages. The reduction of water hardness, primarily caused by the presence of calcium and magnesium ions, plays an important role in this process. These ions enter the water due to the dissolution of rocks, chemical weathering, and the interaction of dissolved carbon dioxide with carbonate minerals^4,6. Reducing water hardness requires adding additional reagents into the solution, such as sodium carbonate (soda ash), calcium hydroxide, or sodium orthophosphate. Calculating the required quantity of the appropriate reagent becomes a separate and fairly complex technical problem in this case. Hence, to eliminate calcium and magnesium ions, filters containing ion-exchange resins, crushed zeolites, or coal, which form their porous skeleton are also used⁷. At the same time, such filters can be periodically regenerated, for example, by washing with a concentrated solution of sodium chloride.

It is advisable to determine the optimal parameters of the functioning of filters using appropriate mathematical models of filtration processes. This allows to save time and material, as well as financial resources, necessary for conducting natural experiments. In particular, the work⁸ describes the utilization of a mathematical model of the oxygen electrode and biological wastewater treatment to optimize the placement of the electrode in the bioreactor. To simulate the process of aerobic wastewater treatment, in⁹, an algorithm was constructed for solving the corresponding nonlinear perturbed problem of the convection–diffusion-mass transfer type. This algorithm allows for the automated control of impurity sedimentation in the biological filter.

The issue of mathematical modeling of the water softening process is the subject of research by many scientists. Thus, in the paper¹⁰ a mathematical model of the process of water softening using ion exchange pre-treatment of water to desalination is developed. While providing valuable insights into sodium-cationite filters, the study does not address nonlinear sorption processes in such systems. Obertas I. in the work¹¹ investigated changes in filtration coefficients, namely a formula for determining the filtration coefficient for a combined form of sediment formation is obtained. However, the study does not consider dynamic interactions or nonlinear effects. Similarly, Tokmachev M., Tikhonov N., et al.¹² investigated ion exchange pretreatment of saline waters for desalination, focusing on the removal of scale-forming components. While addressing the phenomenon of isothermal supersaturation in ion exchange beds, their model fails to account for saturation thresholds under dynamic conditions.

The purpose of this work is to construct a mathematical model of convective diffusion to describe the change in water hardness during its filtration in a porous body under conditions of limited sorption by the skeleton. Existing models have either failed to account for nonlinear sorption kinetics, the dynamic behavior at saturation thresholds, or the combined effects of convection and diffusion under these nonlinear conditions, applying them only in simplified or limited scenarios. By constructing a compound model with two sequential submodels, this work offers a more precise approach to predicting impurity concentration behavior. The findings contribute to both the theoretical understanding of convective diffusion under limited sorption and practical applications in water purification technologies.

Research methods

General approaches to the description of mass transfer processes in porous heterogeneous media are formulated in papers^13,14. Their application to specific problems also requires the use of certain continuum concepts and principles of thermodynamics of nonequilibrium processes^15–19.

The principles of the continuum thermodynamic approach and methods of mathematical physics were used to establish the initial relationships of this model. Nonlinear dynamic system approaches were applied to construct a linearized comppound model. Using the method of successive iterations the solutions are presented in the form of Neumann series in the vicinity of the linearized initial-boundary value problem formulations. Solutions of the homogeneous initial-boundary value problem and the Green’s function were obtained using integral transform methods.

The development of software for numerical research includes the creation of the following computational modules:

• Module for calculating impurity concentrations in aqueous solution;

• Module for calculating concentrations of impurity adsorbed on the filter skeleton;

• Module for calculating the formation and spread of zones of saturation with sorbed impurity particles on the surface of the water filter skeleton.

In their creation, a free software development environment for the Free Pascal Compiler, namely the Lazarus integrated development environment, was used. All codes were run under work station Ms AMD Ryzen 5 3.6 GHz/Gigabyte GA-AB350M/DDR4 8 Gb/HDD 1000 Gb/GeForse GT730 1 GB/ATX 450 W/K + M (Manufactured by GIGA-BYTE Technology Co., Ltd, Nan-Ping, Taiwan).

Compound mathematical model

Consider a porous medium through which an aqueous porous solution flows, containing water molecules and ions migrating in the solution, which determine its hardness and can be adsorbed to the filter skeleton. We assume that an arbitrary region of the body consists of a skeleton and an aqueous solution that fills the pore space. The aqueous solution is a two-component solution consisting of water particles and an impurity. Impurity particles (ions that determine the hardness of the aqueous solution) exist in two states: in the convectively moving solution and on the surface of the filter skeleton.

As a rule, in real conditions, the sorption capacity of the surface of the skeleton is limited, and this value is experimentally measurable.

In the mathematical description of the nonlinear processes of convective diffusion of impurity particles, which are accompanied by the process of limited sorption onto the skeleton, we accept a number of assumptions that correspond to the main properties of the kaonite, zeolite, or coal material, the known kinetic characteristics of the processes under consideration, and the conditions of the experiment. Namely,

• In the process of filtration (convective diffusion), the solution, like the skeleton, can be considered incompressible. This means that the concentration of impurities is small and the change in flux density due to their transition to the skeleton-bound state can be neglected.

• Filtration and adsorption of impurity components of the solution (ions or ), as a rule, can be considered independently.

• The conditions of the experiment ensure the movement of the solution in the pores of the skeleton at a constant velocity () in each cross-section of the filtering plant.

• We will treat the exchange of the solution with the solid phase as the sink (source) of the impurity component when describing the filtration process. Let us assume that at each moment of time and at each point in space, a local thermodynamic equilibrium is maintained between the impurities in solution (adsorbent) and those bound to the skeleton (adsorbate).

• The sorption process at a certain point in space continues until the ion concentration reaches the experimentally measured value of the “saturation concentration”, and then the sorption process stops at that point.

• In most practical cases, the technological process is carried out under thermally homogeneous conditions, and the heat release during the considered processes is negligibly small. That is, the temperature of the solution and the solid phase are the same and constant.

The main processes under consideration are convective diffusion of impurities, i.e. ions of or , and their sorption onto the skeleton. These processes are described by using the approximation of continuum of mass centre^20,21 for the liquid phase and it is assumed that the velocity of convective movement of particles is approximately equal to the true velocity of the porous solution during filtration, which is ensured by the porosity of the medium and the distribution of pores according to their geometric characteristics, in particular their radius (the distribution of pores by radius is close to normal, which corresponds to real physical objects). Note that here the medium is considered isotropic and the surface and volume porosities are assumed to be equal.

As the basic relations of the model, we take the balance equations for the masses of each component of the system. If the change in the mass of the component occurs due to mass fluxes and internal sources²⁰, then the equations of the mass balance of the components take place.

Let us define a compound mathematical model of convective diffusion of an impurity substance under limited sorption by the skeleton, as two different submodels that are realized sequentially in the system (Fig. 1).

Fig. 1.

Scheme of compound mathematical model under limited sorption.

Here is the point, interval or zone where , is the point, interval or zone where , is the impurity concentration in the bound state, is its maximum value; is time, is the radius-vector of a running point.

According to the scheme of the compound mathematical model (Fig. 1), the concentration of the impurity in the aqueous solution and the concentration in the bound state are determined from Submodel 1²⁰

1

where is the coefficient of diffusion of impurity particles in the aqueous solution, is the Laplace operator, is the Hamilton’s nabla-operator, is the total density of the solution, is the velocity of convective movement of water, is the constant of the sorpsion process.

This system of equation describes the filtration processes in a porous layer accompanied by nonlinear sorption while the concentration of sorbed substance reaches saturation value .

If the condition is not fulfilled at a certain point , then the sorption process is stopped and the transition is made to Submodel 2, which consists of a set of the following relations

2

The first equation of (2) is the equation of convective diffusion of impurity²².

The set (2) describes the processes that occur in the body until the condition begins to be met again. Then we switch to Submodel 1, and so on (Fig. 1).

Note that both submodels can be realized simultaneously in the system. For example, at the point the concentration has not reached the value and Submodel 1 (1) is realized, and at the point and Submodel 2 (2) is realized.

Reduce Eqs. (1) and (2) to the dimensionless form, introducing new variables²⁰

3

where is the constant, and also , , , .

The natural dimensionless form (3) does not depend on the geometric parameters of the body under consideration. At the same time, this form for transfer processes in real systems compresses the time axis and stretches the spatial axis (since ).

Then we obtain

Submodel 1

4

where , ; .

Submodel 2

5

where . Here . We have formally left the diffusion coefficient in Eqs. (4) and (5) so that the solutions of the problems that will be obtained in this article can be used for the dimensional form, i.e., Eqs. (1) and (2).

We present the total flux of impurity particles in dimensionless variables (3) as follows.

Initial-boundary value problem of convective diffusion under limited sorption

Consider a porous layer of dimensionless thickness , in which the processes of mass transfer of impurities in the aqueous solution occur, accompanied by limited sorption of particles on the skeleton of the body. Taking into account the symmetry in two spatial coordinates, let us consider the one-dimensional case of the mentioned processes in the porous layer.

When the condition

6

is satisfied (for example, for small temporal intervals of process progression), Submodel 1 (4) consists of the nonlinear system of equations of convective diffusion allowing for the sorption of particles onto the skeleton, namely

7

This system of equations describes the filtration processes in the porous layer as long as the condition (6) is fulfilled. Let’s assume that the condition (6) is not satisfied at a certain point . Then sorption processes cease, and a transition to Submodel 2 occurs, which in the one-dimensional case reduces to the following set of relationships.

8

The set (8) describes the processes taking place within the body until condition (6) is fulfiled again. Then a transition to Submodel 1 occurs, and so on (Fig. 1).

Linearize the compound model (6)-(8). Then Submodel 1 is reduced to the form.

9

and Submodel 2, the set (8), will remain unchanged.

We impose the first-type initial and boundary conditions, which correspond to the case of a clean water filter at a certain initial moment, and for there is the constant mass source acting on the upper surface of the body.

10

For the linearized problems, the initial and boundary conditions (10) remain unchanged.

Analytical solutions for the nonlinear initial-boundary value problem (6)-(8) cannot be found exactly. Therefore, we will employ the method of successive approximations to construct approximate solutions for this system of equations in the form of integral Neumann series. These are obtained by expanding the corresponding functions in the vicinity of solutions of the linearized initial-boundary value problems.

Construction of the solution for initial-boundary value problem of convective diffusion under limited sorption

Let condition (6) be satisfied. Then we reduce the nonlinear initial-boundary value problem (7) and (10) to the system of integral equations. Treating the nonlinear terms in (7) as “sources”, we express the solution of the initial-boundary value problem as a sum of the solution of the homogeneous initial-boundary value problem and the convolution of Green’s functions with the source^23–26. In this case, the homogeneous initial-boundary value problem coincides with the linearized problem (9) and (10)²⁷. Then we obtain the following system of integral equations

11

where and are Green’s functions of the problem (7) and (10).

We construct the solution of the system of integral Eqs. (11) using the method of successive iterations²⁸ by expanding the functions and into series around the solutions of the homogeneous linear initial-boundary value problem (9) and (10). We take and as the zero approximation.

To obtain the first iteration²⁸, we write the values of the concentration functions and at the point since the system of integral Eqs. (11) holds for all points in the domain , including when and . So we have

12

Substituting the expressions (12) into the right-hand side of the integral Eqs. (11), we obtain

13

Write the values of concentrations and in the point and substitute them into the right-hand side of (13), then we obtain the second iteration. By continuing this operation infinitely many times, we will get infinite integral Neumann series²⁸, namely:

14

Note that the Neumann series for the function is positively increasing, while for the function , it is alternately changing.

Furthermore, it should be noted that the formulas (14) hold for arbitrary initial and boundary conditions imposed in the initial-boundary value problems of filtration, accounting for the sorption process in the porous medium.

For approximate computations of the concentration functions and , we limit ourselves to the first two terms of the Neumann series (14). In other words, calculations are performed using the following formulas:

15

16

Formulas (15) for the concentration of impurities in the solution and (16) for the adsorbed substance on the filter skeleton contain the “homogeneous part”, i.e. the solution of the convection–diffusion problem in a medium with traps for migrating impurity particles²⁹, and the “inhomogeneous part” that describes the influence of the nonlinear component of the sorption processes within the body. The “inhomogeneous” term is proportional to the coefficient , which could be one of the normalizing coefficients in the transition to dimensionless variables, particularly in the naturally dimensionless form²⁰, then 1.

Finding the solution for homogeneous initial-boundary value problem of convective diffusion and Green functions

Find the solutions of the homogeneous initial-boundary value problem (9) and (10). The second equation of system (9) can be integrated

17

For constructing the solution of the first partial differential Eq. (9) under corresponding initial and boundary conditions (10) we reduce this problem to the problem with zero boundary conditions, we apply Laplace integral transforms in the time³⁰ and such the finite integral transform in the spatial coordinate³¹

18

Performing the appropriate calculations for the function , we find

19

where , , , .

To find the concentration of particles in the bound state, we substitute the obtained formula (19) into the relation (17) and integrate it. As a result, considering the initial conditions (12), we have

20

It should be noted that the expression (20) for the concentration of bound particles contains two terms directly proportional to the time of the convective diffusion process, accompanied by the sorption process.

Now we find the Green’s functions and . By definition, they are solutions of the corresponding system of equations with a point source and zero initial and boundary conditions³², that is,

21

Intergate the second equation of the system (8)

22

To find the function , we apply Laplace transforms in time and (18) in spatial coordinate to the first equation of the system (9), and Eq. (18). Then we get the equation in the images

23

where is the parametr of Laplace transformation. Here we take into account³³ that , .

The solution of (23) has the form

24

Then, applying the inverse Laplace transform to expression (24), using the shift theorem³², we find

where is Heaviside step function^32,33.

Let us apply the inverse transformation (18) to the obtained expression and take into account the shift theorem. So we have

25

Then, by substitution, we find the Green’s function

26

To find the concentration functions described by Submodel 1, we substitute the obtained solutions of the homogeneous initial-boundary value problem (19) and (20), as well as the expressions for the Green’s functions (25) and (26) into relations (15) and (16).

Submodel 2. The solution of the first equation of the set (9) under the initial and boundary conditions (10) is as follows²⁰

27

where .

The solutions of the linearized initial-boundary value problem (9) and (10) coincide with the solutions of the homogeneous problem for the concentration of particles migrating by diffusion and convection mechanisms and accompanied by sorption of particles onto the body skeleton (19) and (20) for , .

Analysis of the distribution of zones of Submodels 1 and 2

Let us first investigate the formation and spread of zones of saturation with sorbed impurity particles on the surface of the water filter skeleton based on the calculation formulas (15) and (16), taking into account the relations (19), (20), (25), (26), and (27). Numerical integration of double integrals was carried out by the cubature method according to the formula of rectangles, where the division of the integration region is 50 × 50 elementary subareas, the area of which is equal to ^34,35. The following values of the coefficients are taken as basic parameters: 0.3; 1;  = 3; 1; 1; 0.5. In this case the accuracy of the calculation of series by in formulas (19), (20), (25), (26) and (27) is .

Figures 2, 3, 4 and 5 show the zones where the concentration of sorbed particles has reached its maximum possible value, i.e., condition (6) is not satisfied (marked in green), and the zones where the concentration of sorbed impurity has not reached the value , i.e., condition (6) is fulfilled (marked in white). Here, the dimensionless spatial variable is plotted along the abscissa axis, and a dimensionless temporal variable is plotted along the ordinate axis. Tables 1, 2, 3 and 4 show the points from which the formation of the zone, where condition (6) is not fulfilled, begins and the transition to Submodel 2 occurs. The table of values of the point from which the formation of the green zone begins, corresponds to the figures of the zones of fulfillment/non-fulfillment of condition (6) shown above.

Fig. 2.

Zones of saturation/nonsaturation of the skeleton surface by impurity particles that are formed under different values of the velocity of convective transfer .

Fig. 3.

Zones of saturation/nonsaturation of the skeleton surface by impurity particles that are formed under different values of .

Fig. 4.

Zones of saturation/nonsaturation of the skeleton surface by impurity particles that are formed for different values of the filter thickness at 3.

Fig. 5.

Zones of saturation/nonsaturation of the skeleton surface by impurity particles that are formed for different values of the filter thickness at 4.5

Table 1.

The initial point of forming the saturation zone for different values of the convective transfer velocity at 1.

Value ,	1, 0.1	3, 0.5	3.3, 0.5	3.8, 0.5	4.5, 0.5	6, 0.5
Point	(0.501, 0)	(0.501, 0)	(0.5001, 0)	(0.4609, 0.42)	(0.3699, 0.56)	(0.196, 0.68)

Table 2.

The initial point of forming the saturation zone for different values of the maximum value of the concentration at 3 and 1.

Value	0.1	0.5	0.7
Point	(0.101, 0)	(0.501, 0)	(0.701, 0)

Table 3.

The initial point of forming the saturation zone for different values of the filter thickness at 3.

Value , 3	0.5	1	1.5
Point	(0.501, 0)	(0.501, 0)	(0.366, 0.84)

Table 4.

The initial point of forming the saturation zone for different values of the filter thickness at 4.5

Value , 4.5	0.5	1	1.5
Point	(0.501, 0)	(0.3699, 0.56)	(0.1382, 1.08)

Note that the formation of the zone, where condition (6) is not fulfilled and the transition from Submodel 1 to Submodel 2 occurs, begins either from the upper boundary of the filter or in the lower half of the porous layer. The location of the initial point depends on the magnitude of the convective transfer velocity. Thus, the formation of the saturation zone from the surface is characteristic for low velocities 3.3 (Fig. 2a–c, Table 1). For 3.3 the values , and the larger the value of , the closer the point shifts to the lower surface of the filter (Fig. 2d–f, Table 1). It should also be noted that the higher the convective transfer velocity, the earlier the formation of the saturation zone begins (Table 1). For example, if increases twice (from to ), then the value of the initial time of saturation zone formation is approximately twice as small 0.4 (Table. 1). Also, the larger the value of , the larger the green zone where condition (6) is not satisfied (Fig. 2a–f).

Figure 2 shows the zones where the concentration reached or did not reach the value , for different values of the velocity of convective transfer 1, 0.1 (Figure a), 3, 0.5 (Figure b), 3.3, 0.5 (Figure c), 3.8, 0.5 (Figure d), 4.5, 0.5 (Figure e), 6, 0.5 (Figure f). In Fig. 3 it is demonstrated the influence of the saturation concentration value on the formation of zones of fulfillment/non-fulfillment of condition (6). Here Figure a is built for the value 0.1, Figure b – for 0.5, Figure c – for 0.7 at 3. In the Table 2 it is presented the corresponding values of the initial point of the formation of the zone where condition (6) is not fulfilled.

Figure 4 (for 3) and Fig. 5 (for 4.5) show the zones where the concentration of sorbed particles has reached its maximum possible value for different thickness of the filter 0.5 (Figure a), 1 (Figure b), 1.5 (Figure c). Table 3 (for 3) and Table 4 (for 4.5) demonstrate the points , from which saturation zones begin to form for the same dimensionless thicknesses of the porous layer.

Moreover, for small values of , the transition from one submodel to another either does not occur (for small times of the convective diffusion process) or occurs once for (Figs. 2a-2c). For large values of the convective transport velocity (Figs. 2d–f), there is no transition to another submodel on the time interval . Furthermore, there exists a time such that there are two transitions between submodels in the interval , and there is only one such transition for . For example, 0.501 i 0.55.

The influence of the maximum concentration of impurity particles that can participate in sorption processes on the formation of saturation/non-saturation zones is also significant. With that the higher the value of , the later the saturation zone begins to form, but the location of the initial point does not change (Fig. 3, Table 2). Also, the smaller the value of , the larger the zone where condition (6) is not fulfilled. The value of does not affect the number of transitions between submodels (Fig. 3).

For small filter thicknesses, the point of origin of the zone formation is the same, namely (0.501, 0), regardless of the value of the convective transfer velocity (Fig. 4a and b). Such a situation is true for , and for example for there is already the existing interval of this zone in the interval . As the thickness of the filter increases, the location of the initial point shifts to the lower boundary of the filter and the formation of the saturation zone begins earlier (Table 4). The larger the value of , the larger the zone is (Figs. 4 and 5).

We also note that for small filter thicknesses for there is one transition between submodels (Figs. 4a,b and 5a). For larger thicknesses of the porous layer (for example, for , , Fig. 4, or , , Fig. 5), there exists such a time that two transitions between submodels occur in the interval , and there is one such transition for . For the given data, 0.501 and further increase in the value of does not affect the value of .

Numerical analysis of concentrations of impurity particles in aqueous solution and in bounded state

The distributions of impurity particle concentrations that migrate under conditions of limited sorption in the porous layer, calculated using formulas (15) and (16) taking into account the relations (19), (20), (25), (26), and (27), are given in Fig. 6–13. Figures a illustrate the distributions of the concentration of particles in the aqueous solution , while in Figure b, graphs of the concentration function of the sorbed substance are provided .

Fig. 6.

Distributions of impurity concentrations (a) and (b) at different moments of dimensionless time for  = 3.

Fig. 13.

Distributions of impurity concentrations (a) and (b) depending on the filter thickness at the moment of dimensionless time for .

Figure 6 (for  = 3) and Fig. 7 (for  = 6) show the distributions of impurity concentration in the body at different moments of dimensionless time 0.05, 0.3, 0.6, 0.9, 1.2 (curves 1–5, respectively).

Fig. 7.

Distributions of impurity concentrations (a) and (b) at different moments of dimensionless time for  = 6.

Figure 8 (for ) and Fig. 9 (for ) show the distributions of impurity concentrations (Figure a) and (Figure b) for different values of the convective transfer velocity 1, 3, 3.3, 3.8, 4.5, 6 (curves 1–6).

Fig. 8.

Distributions of impurity concentrations (a) and (b) for different values at the moment of dimensionless time

Fig. 9.

Distributions of impurity concentrations (a) and (b) for different values at the moment of dimensionless time

Figure 10 (for , ) and Fig. 11 (for , ) illustrate the distributions of impurity concentrations (Figure a) and (Figure b) depending on the maximum value of the concentration of impurity capable of being sorbed on the body’s skeleton, 0.1, 0.3, 0.5, 0.6, 0.8 (curves 1–5).

Fig. 10.

Distributions of impurity concentrations (a) and (b) for different values at the moment of dimensionless time for .

Fig. 11.

Distributions of impurity concentrations (a) and (b) for different values at the moment of dimensionless time for .

Figure 12 (for , ) and Fig. 13 (for , ) show comparative distributions of functions (Figure a) and (Figure b) for different filter thicknesses 0.4, 0.6, 0.8, 1, 1.2 (curves 1–5).

Fig. 12.

Distributions of impurity concentrations (a) and (b) depending on the filter thickness at the moment of dimensionless time for .

Let us analyze the concentration distributions constructed according to formulae (15) and (16). In the aqueous solution the concentration of particles is a smooth function as long as the condition (6) for the concentration is fulfilled (curves 1 and 2, Fig. 6a). In the presence of intervals where (curves 3–5, Fig. 6b and curves 2–5, Fig. 7b), at the ends of such intervals in the distributions for appear peaks of growth of the values of this function (curves 3–5, Fig. 6a and curves 2–5, Fig. 7a). Note that the ends of such intervals are points and/or transitions from one Submodel to another (Fig. 1).

As the filtration process continues, the concentration of impurity particles in the aqueous solution decreases after a sharp increase in the initial interval of the process (Figs. 6a and 7a), while the concentration of sorbed particles increases (Figs. 6b and 7b).

Moreover, the further increase in the filtration process duration practically does not affect the concentration of particles in the solution; in other words, curves 5a and 5b in Fig. 3a characterize a certain steady-state distribution. For relatively large values of the convective transfer velocity ( = 6), there is a significant increase in the function from the upper surface of the filter , reaching the maximum at the point 0.68 (and with the change of time, the point of the maximum is always around 0.66), and then decreasing to zero at the lower boundary of the body , ensuring the satisfaction of the boundary condition (2) (Fig. 7a). Here, the difference between the values of at moments 0.01 and 0.3 reaches 19%. As the duration of the filtration process increases, the decrease in concentration slows down until it reaches a steady-state distribution for 4. For relatively small values of the convective transfer velocity ( = 3), as the nondimensional time increases from 0.01 to 0.3, the concentration values decrease to 16% (Fig. 6a).

The value of the convective transfer velocity significantly affects the behavior and values of the functions and (Figs. 8 and 9). The smaller the values of the coefficient , the closer to the linear distribution are the concentration functions in the aqueous solution and in the bounded state (curves 1, Figs. 8 and 9). With an increase in the value of , both the concentrations and increase across the entire range (Fig. 8). This can be explained by the fact that the sorption process does not have sufficient time to operate at full capacity at higher convective transfer velocity. For significant durations of the convective diffusion process at large values of , the effect of a sharp increase in the concentration function in the vicinity of the ends of the interval where disappears (curve 6, Fig. 9a). As for the function , the lower the velocity of convective transfer, the narrower the interval in which condition (6) is not fulfilled (Fig. 9). We also note that with the increase of the value significantly increases and shifts to the lower boundary of the layer (Fig. 8a).

The value of the maximum concentration of impurity particle s that can be sorbed on the filter skeleton significantly affects the concentration of sorbed particles and has a minor effect on their concentration in the aqueous solution (Fig. 10 and 11). An increase in the sorption capacity of the water filter leads to an increase in the function over the entire interval, regardless of whether the zone starts from the upper boundary (Fig. 10b) or from the middle of the layer (curves 3 and 4, Fig. 11b). At the same time, increasing from 0.1 to 0.8 leads to an increase in the values of the function up to 11%. At the same time, the characteristic sharp growth of the concentration function in the solution at the point of transition from Submodel 2 to Submodel 1 occurs closer to the lower boundary of the body for smaller values of (curves 1–3, Fig. 10a), and the magnitude of this local growth is significantly smaller. Thus, for , and . With an increase in , the change in value of has even less influence on the function (Fig. 11a), but the effect on the function remains the same (Fig. 11b).

Note that an increase in the thickness of the filter leads to an increase in concentrations of both and within the body (Fig. 12 and 13). Additionally, the smaller the value of , the closer the concentration functions approach the linear distribution (curves 1, Fig. 12 and 13). If the zone starts at the point (Fig. 12), then the larger , the deeper the point of transition from Submodel 2 to Submodel 1 is located (Fig. 12b). For this case, when the filter thickness is small, there is no noticeable effect of transitioning to another model in the distributions of the function (curves 1 and 2, Fig. 12a). With an increase in , the local sharp increase in concentration in the aqueous solution around the point significantly increases, for example, and (Fig. 12b). If the zone where starts at the point (curves 4 and 5, Fig. 13), then the thicker the filter, the wider the interval and the smaller the value of where the transition from Submodel 1 to Submodel 2 occurs (Fig. 13b). Conversely, the larger the point where the transition from Submodel 2 to Submodel 1 occurs. For this case, the jumps in the function at the transition points between submodels are small compared to the values of the concentration of impurity particles in the aqueous solution (Fig. 13a), for example, and .

Comparative values of the first terms of the Neumann series

To investigate the contribution of the solution of the linearized initial-boundary value problem (6), (8)-(10) to the solution of the nonlinear problem (6)-(8), (10), that is, initial-boundary value problems of convective diffusion under conditions of limited sorption, a numerical analysis of the functions and will be conducted compared to the next two terms of the Neumann series (14):

It should be noted that the double integrals and are calculated using the cubature method³⁶ with the division of the integration domain into elementary rectangular subdomains. Quadruple integrals and are calculated by the method of double cubatures (Fig. 14) with successive division of partial areas of integration into rectangular elements, and as a result, the total four-dimensional area of integration is divided into elements.

Fig. 14.

Scheme for calculating the quadruple integral (double quadrature method).

Tables 5, 6, 7 and 8 show the calculated data of the functions , , , and , on the interval at , , . Table 5 shows the corresponding data for the convective transfer velocity 3 at time 0.3, Table 6 illustrates data for 3, 0.6, Table 7 is calculated for 6, 0.05, and Table 8 – for 6, 0.3 accordingly.

Table 5.

Data for 3, 0.3, , , .

0	1	1	0	0	0.3	0.3	0	0
0.055556	0.998961	0.994208	0.004448	3.05	0.295115	0.294875	2.38	1.54
0.111111	0.996196	0.986365	0.009182	6.49	0.291375	0.290867	5.05	3.30
0.166667	0.990838	0.97602	0.013801	1.02	0.288326	0.28753	0.000791	5.23
0.222222	0.981965	0.962572	0.017996	1.40	0.285512	0.284419	0.001086	7.30
0.277778	0.968827	0.945466	0.02159	0.001772	0.282517	0.28113	0.001377	9.44
0.333333	0.950623	0.924012	0.024486	0.002125	0.278903	0.277237	0.001655	1.16
0.388889	0.926586	0.897452	0.026689	0.002445	0.274237	0.272317	0.001906	1.36
0.444444	0.895855	0.864994	0.028147	0.002713	0.268019	0.265884	0.002119	1.55
0.5	0.857526	0.825748	0.028861	0.002917	0.259774	0.257473	0.002285	1.70
0.555556	0.810641	0.778778	0.02882	0.003044	0.248959	0.24655	0.002391	1.82
0.611111	0.754189	0.723088	0.028019	0.003081	0.235021	0.232574	0.002428	1.88
0.666667	0.687088	0.657637	0.026434	0.003017	0.217352	0.214948	0.002385	1.88
0.722222	0.608305	0.581338	0.024122	0.002845	0.195353	0.19308	0.002254	1.81
0.777778	0.516669	0.493102	0.021015	0.002552	0.168398	0.166355	0.002027	1.64
0.833333	0.411113	0.391871	0.017114	0.002128	0.135872	0.134165	0.001694	1.39
0.888889	0.290474	0.276535	0.012372	0.001567	0.097221	0.095962	0.001249	1.03
0.944444	0.153768	0.146198	0.00671	0.000859	0.052016	0.051325	0.000685	5.69
1	2.41	2.79	3.46	3.37	7.27	7.53	2.64	1.96

Table 6.

Data for 3, 0.6, , , .

0	1	1	–	–	0.5	0.5	–	–
0.055556	0.980393	0.980393	–	–	0.5	0.5	–	–
0.111111	0.959771	0.959771	–	–	0.5	0.5	–	–
0.166667	0.937431	0.937431	–	–	0.5	0.5	–	–
0.222222	0.913016	0.913016	–	–	0.5	0.5	–	–
0.277778	0.886057	0.886057	–	–	0.5	0.5	–	–
0.333333	0.856008	0.856008	–	–	0.5	0.5	–	–
0.388889	0.822277	0.822277	–	–	0.5	0.5	–	–
0.444444	0.784227	0.784227	–	–	0.5	0.5	–	–
0.5	0.741167	0.741167	–	–	0.5	0.5	–	–
0.555556	0.843329	0.773394	0.062536	0.007399	0.483836	0.478728	0.005065	4.36
0.611111	0.786582	0.717505	0.061508	0.007569	0.453197	0.447987	0.005165	4.52
0.666667	0.718225	0.652044	0.058696	0.007485	0.415862	0.410722	0.005094	4.5
0.722222	0.637147	0.57596	0.054071	0.007115	0.3709	0.366026	0.004831	4.3
0.777778	0.542169	0.488198	0.047545	0.006426	0.317352	0.312957	0.004355	3.95
0.833333	0.432147	0.387725	0.039032	0.005391	0.254284	0.250603	0.003648	3.34
0.888889	0.305836	0.273452	0.028399	0.003985	0.180807	0.178088	0.002694	2.4
0.944444	0.162154	0.144497	0.015466	0.002191	0.096217	0.094723	0.001481	1.3
1	2.0	2.85	7.28	8.06	1.55	1.61	5.55	4.64

Table 7.

Data for 6, 0.05, , , .

0	1	1	0	0	0.05	0.05	0	0
0.055556	1.079914	1.079086	0.00081	1.79	0.053666	0.053636	2.93	3.69
0.111111	1.172491	1.170722	0.001728	4.06	0.059808	0.05974	6.73	8.57
0.166667	1.277056	1.274266	0.002722	6.79	0.067953	0.067838	0.000114	1.48
0.222222	1.39297	1.389058	0.003813	9.94	0.077845	0.077674	0.000171	2.24
0.277778	1.518917	1.513749	0.005033	0.000135	0.089152	0.088916	0.000236	3.14
0.333333	1.652249	1.645707	0.006367	0.000174	0.101561	0.101251	0.000309	4.17
0.388889	1.789284	1.781215	0.007851	0.000217	0.114642	0.114251	0.00039	5.32
0.444444	1.924843	1.915183	0.009399	0.000261	0.12789	0.127415	0.000475	6.56
0.5	2.05177	2.0405	0.010965	0.000306	0.140692	0.14013	0.000561	7.83
0.555556	2.160601	2.147827	0.012427	0.000347	0.152258	0.151614	0.000643	9.06
0.611111	2.239107	2.225083	0.013641	0.000382	0.161554	0.160837	0.000715	1.02
0.666667	2.271512	2.256729	0.014378	0.000405	0.167329	0.16656	0.000768	1.10
0.722222	2.238094	2.223102	0.014577	0.000415	0.167907	0.167113	0.000793	1.15
0.777778	2.114209	2.099914	0.013893	0.000402	0.161266	0.160488	0.000777	1.13
0.833333	1.869286	1.856689	0.012236	0.000362	0.144739	0.144031	0.000706	1.0
0.888889	1.466538	1.456782	0.009469	0.000287	0.115064	0.114497	0.000566	8.38
0.944444	0.860899	0.855245	0.005485	0.00017	0.068241	0.067903	0.000338	5.02
1	5.5	5.7	2.72	7.51	-6.00	-5.87	1.37	1.90

Table 8.

Data for 6, 0.3, , , .

0	1	1	0	0	0.3	0.3	0	0
0.055556	1.076149	1.069811	0.005984	3.54	0.322584	0.322336	2.47	8.25
0.111111	1.163642	1.149241	0.013582	8.19	0.350353	0.349779	5.72	1.92
0.166667	1.261457	1.237453	0.022601	1.40	0.382913	0.381933	0.000977	3.31
0.222222	1.368661	1.33373	0.032824	2.11	0.419792	0.418322	0.001464	5.02
0.277778	1.483839	1.436848	0.044065	0.002926	0.460337	0.4583	0.00203	7.05
0.333333	1.49582	1.49582	–	–	0.5	0.5	–	–
0.388889	1.598039	1.598039	–	–	0.5	0.5	–	–
0.444444	1.698105	1.698105	–	–	0.5	0.5	–	–
0.5	1.790281	1.790281	–	–	0.5	0.5	–	–
0.555556	1.867042	1.867042	–	–	0.5	0.5	–	–
0.611111	1.918641	1.918641	–	–	0.5	0.5	–	–
0.666667	1.932443	1.932443	–	–	0.5	0.5	–	–
0.722222	1.892236	1.892236	–	–	0.5	0.5	–	–
0.777778	1.778591	1.778591	–	–	0.5	0.5	–	–
0.833333	1.566117	1.566117	–	–	0.5	0.5	–	–
0.888889	1.366227	1.273892	0.085253	0.007083	0.466906	0.462006	0.004881	1.92
0.944444	0.800824	0.746035	0.050558	0.004232	0.274922	0.271994	0.002916	1.15
1	7.74	1.02	2.34	1.70	1.22	1.34	1.18	4.31

As follows from the given data in Tables 5, 6, 7 and 8, the absolute values of the second terms of the Neumann series and are two orders of magnitude smaller than the first terms of the series and , accordingly, for small and medium time intervals of the considered process. With an increase in , the difference between the first and second terms of the Neumann series decreases, and, for example, for 0.3 at (Table 6) reaches one order of magnitude. The third terms of the Neumann series and are an order of magnitude or less smaller than the second terms and . At the same time, the smaller the value of the coefficient and the longer the time of the convective diffusion process, the larger the absolute values of the second and third terms of the Neumann series, at the same time, the longer the interval, when the concentration of sorbed particles reached its maximum for Submodel 2. Note that the change in the coefficients and weakly affects the values of the second and third terms of the Neumann series.

According to the data given in the Tables 5, 6, 7 and 8, the difference between the values of the function of the concentration of impurity particles in the solution calculated according to the nonlinear model and the corresponding values calculated according to the linearized model is up to 7%, as it presented in the Table 9.

Table 9.

Estimation of the difference between the concentrations calculated by the nonlinear and linearized models.

Table No
	In absolute values	In % to	In absolute values	In % to
Table 5	< 0.032	Up tp 3.2%	< 0.0024	Up tp 0.82%
Table 6	< 0.07	Up tp 7%	< 0.0053	Up tp 1.05%
Table 7	< 0.015	Up tp 0.66%	< 0.0008	Up tp 0.48%
Table 8	< 0.093	Up tp 4.78%	< 0.005	Up tp 0.99%

The difference between the impurity concentration values calculated using linear and nonlinear models in the solution is one order of magnitude larger than the corresponding difference in concentrations in the bound state. This applies to both absolute values and percentages (Table 9).

In our opinion, the linearized compound model of convective diffusion under conditions of limited sorption describes the corresponding physical process quite well. Only for low values of convective transfer velocity or when highly precise calculations are required, it may be necessary to solve the initial-boundary value problems of convective diffusion in the nonlinear formulation.

Discussion

This work presents a nonlinear mathematical model of convective diffusion that describes changes in water hardness during filtration through a porous medium, comprising ion-exchange resins (kaolinites), finely ground zeolites, or coal. The model accounts for accompanying sorption–desorption processes, treated as nonlinear sources with limited skeleton absorption. Based on mass balance equations and kinetic relationships, we constructed a compound mathematical model of convective diffusion for impurities with limited sorption. This model consists of two sequential submodels: Submodel 1 captures filtration processes in the porous layer accompanied by nonlinear sorption until the sorbed substance concentration reaches its saturation value. When this saturation is achieved, the model transitions to Submodel 2, which includes two equations: the equation of convective diffusion of impurity particles in an aqueous solution and the equality of the concentration of the sorbed substance and the saturation concentration. At the same time, both submodels can be simultaneously implemented in the system.

For the nonlinear equations of the model, we formulated appropriate mathematical physics problems and derived solutions in the form of Neumann series around the linearized initial-boundary value problem. Numerical analysis of the solutions for the nonlinear initial-boundary value problem corresponding to various physical conditions was carried out. Additionally, a comparative analysis was performed by comparing these solutions with the solutions of linearized initial-boundary value problems.

We investigated the formation and distribution of saturation zones with sorbed impurity particles on the water filter’s skeleton surface. It was observed that the transition from Submodel 1 to Submodel 2 begins either from the upper boundary of the filter or from the lower half of the porous layer. The starting point for saturation zone formation is influenced by the magnitude of the convective transfer velocity. And the higher the velocity of convective transfer, the closer the starting point of the formation of the saturation zone moves to the lower surface of the filter. It is also shown that for small values of the convective transfer velocity, the transition from one submodel to another either does not occur (for small times of the convective diffusion process) or occurs only once. For large values of this coefficient within the short time interval, there is no transition to another submodel. As the time of the process increases after the starting point of the formation of the saturation zone, two transitions between submodels occur. Then, there is a moment in time beyond which only one such transition occurs.

The concentration distributions of impurities in the aqueous solution and those sorbed onto the filter skeleton were also analyzed. In the aqueous solution, the concentration of particles is a smooth function until the concentration of sorbed particles reaches its maximum value. In the presence of saturation zones, the concentration distribution of particles in the aqueous solution has sharp peaks at the boundaries of these zones, which correspond to the points of transition from one submodel to another. With an increase in the filtration process duration, the concentration of impurity particles in the water solution decreases after a sharp initial increase, while the concentration of sorbed particles increases. Moreover, the further increase in the duration of the process has little effect on the concentration of particles in the solution, indicating that the function reaches a certain steady-state distribution.

The convective transfer velocity significantly influences the behavior of concentration functions for impurity particles in both the solution and on the filter skeleton. The smaller the value of the convective transfer velocity, the closer to the linear distribution the concentration functions in both the aqueous solution and the bounded state. As this coefficient increases, both concentrations increase over the entire interval, which can be explained by the fact that the sorption process does not have enough time to reach its full capacity. For significant durations of the convective diffusion process and high values of the convective transfer velocity, the effect of a sharp increase in the concentration function in the solution near the ends of the interval, where the concentration of sorbed particles has reached a maximum, disappears. It was established that the lower the convective transfer velocity, the narrower the interval in which the concentration of sorbed particles reached the saturation concentration. It is shown that with an increase in the convective transfer velocity, the maximum concentration of the impurity in the aqueous solution increases significantly and shifts to the lower boundary of the layer.

The influence of the maximum concentration of impurities capable of being sorbed on the skeleton of the body is significant for the concentration of impurities bound to the skeleton and insignificant for particles in the solution. At the same time, an increase in the sorption capacity of the water filter leads to an increase in the function of the concentration of particles in a bound state over the entire range, regardless of where the saturation zone begins from the upper boundary or in the middle of the layer.

The thickness of the filter does not qualitatively change the distributions of the investigated concentrations. It is shown that an increase in the filter thickness leads to an increase in impurity concentrations in both states. Moreover, the thinner the filter, the closer the concentration functions approach a linear distribution. It was established that if the saturation zone starts from the upper boundary of the layer, then the thicker the filter, the deeper the point of transition from Submodel 2 to Submodel 1 is located.

Finally, the contribution of nonlinear terms to the solutions is generally minimal. In most cases, for small to medium time intervals, the absolute values of the second terms in the Neumann series are two orders of magnitude smaller than the first terms. The third terms are at least one order of magnitude smaller than the second terms. The coefficient of convective transfer velocity has the most significant impact on the difference between solutions of linear and nonlinear initial-boundary value problems. Specifically, lower values and extended durations of the convective diffusion process lead to larger absolute values of the second and third terms of the Neumann series.

Conclusion

In conclusion, this study develops a nonlinear mathematical model of convective diffusion to describe the dynamics of impurity concentration during water filtration under conditions of limited sorption. The research demonstrates the interplay between nonlinear sorption kinetics, saturation zone formation, and convective transfer velocity in influencing impurity concentration distributions. While linear models provide adequate approximations for many scenarios, the study highlights the necessity of addressing nonlinear effects under specific conditions, such as low convective transfer velocities or when precise evaluations of filtration efficiency are required. The analysis reveals that impurity concentration distributions in both the aqueous solution and the filter skeleton are significantly affected by sorption capacity, filter thickness, and process duration. It is shown that lower velocities result in near-linear concentration distributions, while higher velocities lead to increased concentrations due to insufficient sorption time. Prolonged diffusion and high velocities diminish sharp concentration peaks near boundaries. Lower velocities confine the saturation zone, whereas higher velocities shift the maximum impurity concentration to the lower boundary. Sorption capacity primarily influences skeleton-bound impurities, with minimal impact on solution concentrations. Increased sorption capacity raises bound impurity concentrations across the filter. Filter thickness increases concentrations in both states, with thinner filters producing near-linear distributions. Thicker filters deepen the transition point between Submodel 2 and Submodel 1 when the saturation zone starts near the upper boundary.

The transition between Submodel 1 and Submodel 2, governed by saturation dynamics, emerges as a critical aspect of filtration modeling. These findings underscore the importance of considering nonlinear effects, particularly in applications requiring high precision, such as industrial water treatment and resource optimization. Future research could extend the model to account for multi-component impurities and dynamic filter regeneration processes, enhancing its applicability to diverse water purification scenarios.

Author contributions

Y.C. and O.C. conceptualization, Y.B. data curation, O.C. and P.P. formal analysis, Y.B. and M.G. investigation, Y.C. and O.C. methodology, ; P.P. and M.G. project administration, A.C. and M.G. resources, Y.B. software, Y.C. and P.P. supervision, O.C. and Y.B. validation, Y.B. visualization, Y.B., and O.C. roles/writing—original draft, A.C. and O.C. writing—review & editing, all authors reviewed the manuscript.

Data availability

All data generated or analysed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.