### Abstract

A mathematical model is proposed for separating liquids which contain acrylic dispersions and returning expensive technological components dissolved within them to the production cycle. The main limiting factor is the formation of a sediment layer on the membrane surface. This process is unsteady, because the thickness of the sediment layer varies both in the time of the separation process and in the length of the membrane module. The influence of the sediment layer on the hydrodynamics and mass transfer efficiency, its productivity, and the quality of purification is determined. The calculation of the sediment layer and its influence on the separation process is performed using the microprocess method. This method is based on material and energy balances and takes into account viscosity and diffusion coefficients, which vary depending on the temperature and composition of the mixture. The dependences of the sediment thickness on time and the length of the membrane module and selectivity coefficients on the concentration of the colloidal solution are obtained to determine the effective parameters of the baromembrane installation.

## INTRODUCTION

Chemical, textile, mechanical engineering, motor transport, and other industries are sources of pollution of water bodies and soils by hardly oxidizable organic substances [1, 2].

Technological processes at industrial enterprises are very diverse; therefore, the concentration of impurities contained in the spent process fluids and their qualitative composition can vary widely. For example, in the textile industry, wastewater is generated during the processing of raw materials (wool, linen, and cotton), bleaching and dyeing of fibers, their strengthening with adhesives, chemical treatment and finishing of fabrics, etc.

The majority of the existing technological schemes for the treatment of industrial wastewater are aimed at the purification degree corresponding to the applicable sanitary norms and rules [1–5] and are established before their discharge into sewers or open water bodies. For this purpose, traditional technologies are used (gravity separation, flotation, electroflotation, reagent treatment, and other methods), which are economically justified when treating large volumes of wastewater [2, 4–6].

Reagent treatment of these liquids generates waste in which synthetic substances accumulate in high concentrations, are in a non-neutralized state [2], and must be disposed of. In addition, the above technological schemes of wastewater treatment are destructive and do not make it possible to recover very scarce and expensive drugs [5].

In this regard, there is a need to find promising methods for the regeneration of waste technological fluids of industrial enterprises which solve these problems.

Baromembrane processes (reverse osmosis and ultrafiltration) are among the most effective, reagent-free, and low-waste technologies for processing technological liquids [7–9].

The disadvantage of baromembrane processes is the cost of their operation, a significant part of which is the result of measures to prevent the formation of sediment on the surface of the membranes, which complicates the design of the apparatus and increases energy consumption [10, 11].

During the separation of liquids, the solvent mainly passes through the pores of the membrane, and the concentration of solute in the boundary layer increases. It will increase until the diffusion flux from the boundary layer into the shared flux is balanced by the flux of solute through the membrane, i.e. until dynamic equilibrium occurs [10]. Since substances with a high molecular weight (more than 500) are subjected to separation during ultrafiltration, the indicated diffusion flux is very small and the concentration of the solute can reach such a value that a gel layer forms on the side of the solution being separated [10, 12]. The formation of a sediment or gel layer on the surface of the membrane causes an increase in resistance and a decrease in flow through the pores of the membrane, and hence a decrease in the productivity of the process.

Quantitative modeling of the phenomenon of sedimentation is one of the key aspects in the development of such calculation methods.

There is a wide range of approaches to modeling precipitation based on various assumptions. These include modeling approaches at the micro and macro levels [13–18]. The approaches at the macro level ignore the fine structure and behavior of individual particles, while approaches at the micro level are based on modeling the expected behavior of microparticles using probabilistic and stochastic submodels. Differences in interpretation and mathematical approaches to the description of the radial migration of the dispersed phase [15, 19] from the surface layer are also observed. Below are the main approaches for the mathematical description of the process of sedimentation.

**Approaches****based on resistance models** are the most common when studying membrane separation processes. In general, the flow of solvent through the membrane is determined as

where Δ*Р* is the pressure drop, η is the dynamic viscosity, *R*_{m} is the membrane resistance, and Σ*R*_{i} is the total resistance of other factors.

Michaels and Blatt [15, 16, 20] suggested to considering the effect of the gel layer *R*_{g} as additional resistances. In their studies, they proceeded from assumptions regarding the existence of a gel-polarization regime independent of pressure. In accordance with this model, the gelling process begins under the condition that the surface concentration reaches the concentration of the beginning of gelling (Fig. 1).

Moreover, the gelling point depends on the size, shape, chemical structure, and degree of solvation of the macromolecules, but does not depend on the concentration of the bulk solution fed to the membrane.

The flow of solvent through the membrane, in this case, is described by Eq. (2) [21]:

In another version of the resistance method, the influence of the boundary layer *R*_{bl} is taken into account. Due to the increased concentration, the boundary layer will provide hydrodynamic resistance to the penetration of the solvent through the membrane [22]. The scheme of this model is shown in Fig. 2.

Resistances indicated in Fig. 2 work sequentially, therefore, the solvent flow through the membrane is calculated according to Eq. (3) [23]:

According to published data, the boundary layer and the gel layer have the most pronounced effect on the separation resistance when the size of the molecules is much larger than the pores of the membrane [23, 24].

In the case when the molecules of the solute are smaller than the pores of the membrane or close to their size, these molecules can penetrate into the pores, gradually reducing their effective radius, causing resistance due to the absorption of molecules on the inner surface of the pores until they completely block *R*_{a} [25, 26].

Models which take into account the combined effect of these resistances [27] most adequately describe the decrease in permeate flow during filtration.

**Osmotic pressure model.** Belford and Marks [19] presented an approach to modeling the resistance of a gel layer, which is a modification of the standard theory of filtration. It is noted in the present study that under conditions of achieving high flow rates, high retention levels, and low mass transfer coefficients, the concentration of macromolecules near the membrane can be very large and therefore osmotic pressure cannot be neglected. In particular, the authors propose replacing the transmembrane pressure drop Δ*Р*_{t} with a term (Δ*Р*_{t} – σΔπ), where σ is the Staverman coefficient. It was also proposed to take into account the influence of the micropore fouling process on the resistance growth by introducing a correction factor (1 + β*V*), where β is the mass transfer constant, and *V* is the transmembrane flow.

**Approach that takes into account the mechanisms of mass migration from the membrane surface.** In their early studies, Segré and Silberberg described experiments with diluted suspensions of spherical particles in pipelines. While the particle was moving through the pipeline, it moved away both from the pipe axis and from the pipe wall and reached equilibrium in the radial eccentric position with a radius slightly larger than the capillary radius *r** = 0.6*r* (Fig. 3). The discovered phenomenon was called the tubular pinch effect [28, 29].

The expression for determining the pinch effect is written as follows [28]:

where *w* is the average capillary flow rate and Re is the Reynolds number.

Thus, the volumetric flow rate of the liquid being separated leads to the separation of contaminant particles from the surface of the membrane and slows down the process of sediment formation on the ultrafilter. Therefore, increasing the volumetric flow rate is one of the most effective ways to control the concentration polarization, extend the life of the membrane, and maintain constant permeability.

**Approach based on modeling the trajectory of motion of an individual particle that takes into account the forces acting on the particle.** The process of selective deposition of particles in the sediment layer during flow microfiltration was considered by Brou [13]. In his work, the influence of physical properties, the conditions of the process, and the critical size of the particles forming the layer were investigated.

Ho and Zydney [14] described the study of the influence of the hydrodynamic profile and concentration of the dispersed phase on the layer formation rate.

In a number of early studies, a set of questions regarding the properties and structure of the formed layer remained outside the scope of consideration. Houi and Lenormand [30] presented a model describing the participation of spherical particles in the formation of a gel layer. It allows modeling the layer structure at different values of the Péclet criterion.

Tassopoulos [31] presents a stochastic discrete model that describes the process of layer formation using probabilistic characteristics to take Brownian motion into account.

Lu and Hwang [32] considered the mechanism of sediment layer formation during the process in a dead-end mode at constant pressure. The concept of a critical friction angle between spherical particles was introduced to model the structure of the gel layer formed [32].

Given the compressibility of the layer, profiles of local properties, such as porosity and resistivity, were also obtained in this work. To study the mechanism of the formation of a layer, Lu and Hwang [32] analyzed the forces acting on a particle. This work is devoted to numerical modeling of local properties of the sediment, such as particle distribution, porosity, and resistivity under various conditions of the process. The model presented [32] is based on the following physical assumptions: the particle moves to the surface of the membrane under the influence of a liquid phase flow.

One of the modern approaches for calculating the membrane separation process, which takes into account the formation of sediment, is **the creation of mathematical models based on the equations of material balance and conservation of energy of moving flows.** This direction was developed by scientists such as Hvang and Kammermejer [12], Yu.I. Dytnerskii [7, 10], and others. Theoretical calculations for reverse osmosis, ultrafiltration, and microfiltration devices of hollow fiber and flat types were reported [6, 10, 12].

The analysis of the scientific works presented shows that there are a large number of approaches and methods for describing the processes of mass transfer in the membrane, which take the effect of sedimentation into account. Many models accurately describe the nature of mass transfer through a porous baffle, but their use requires a large amount of initial data, many of which are obtained during preliminary experiments.

It is of interest to develop such a technique for calculating ultrafiltration, which allows one to determine the main characteristics for both unsteady and steady state modes with minimal input data.

Thus, a theoretical and experimental study of ultrafiltration processes in tubular membrane modules with an internal filtering surface and the development of new engineering solutions based on them is a pressing task.

## THEORY

The proposed mathematical model is based on the theoretical calculations of Yu.I. Dytnerskii for reverse osmosis and ultrafiltration processes in hollow fiber and flat-bed apparatuses [10].

The mathematical model of the ultrafiltration of liquids containing macromolecular compounds includes the equation of material balance of the solution, permeate, and one of the components of the mixture, as well as the energy conservation equation of the initial mixture and permeate.

It is obvious that in the case of a relatively long pipe, as a result of friction losses against the walls and solvent suction through the pores of the membrane, the working pressure, rate of the volume flow, concentration, and temperature of the solution will change, and hence the sediment thickness will be distributed unevenly along the length of the tubular module. We consider the ultrafiltration process as unsteady; we take into account changes in all variables through the thickness of the sediment. The whole separation process of duration *t*_{s} is divided into *n* parts, depending on the properties and characteristics of the separation process

where *t*_{i} is the time of one interval, s; *t*_{s} is the time of the separation process, s; *n* is the number of intervals.

Within each interval, the process is considered as steady-state. The rate of sedimentation within each interval is assumed to be constant [10]:

where σ is the rate of sedimentation, m/s; *S* is the thickness of the sediment layer, m.

The quasistationary nature of the process implies a discreteness in the variation of parameters depending on the thickness of the sediment layer. To achieve the necessary calculation accuracy, a tubular channel of length *L* is divided into *m* equal sections of length *l*:

Then *m* equal sections are subject to calculation. All parameters and characteristics are taken as constant along the length and diameter of each section. For all *n* sections, for each *t*, the permeate concentrations and consumption as well as the rate of sediment formation and its thickness are calculated. For the first time interval, the membrane is considered to be clean *S* = 0. After time, the thickness of the sediment layer *S* = σ*t*. Thus, the height of the free section in the tube will be equal to *R* – *S* [10]. The flow direction diagram is shown in Fig. 4.

The equation of the material balance of the initial solution is:

where ρ_{1} is the density of the solution in the middle of the pressure channel of the elementary section, kg/m^{3}; ρ_{0} is the density of the solution near the pressure channel wall, kg/m^{3}.

We divide the right and left sides of Eq. (4) by ρ_{1}*u*_{a} and introduce quantity θ, which is a dimensionless coefficient of proportionality of the decrease in the volume of the solution during the filtering process

Then Eq. (7) after the transformations is written in the form

The equation of the material balance of permeate

We introduce notation φ for membrane selectivity

Then Eq. (10) after the transformations is written in the form

The characteristics of the ultrafiltration process, and therefore the overall performance of the apparatus as a whole, are greatly influenced by the hydraulic resistance to flow in the pressure and drainage channels. To determine it, we will use the energy balance that takes into account the change in hydrostatic pressure along the length of these channels with the following assumptions: we assume that the pressure along the height of the pressure channel and the longitudinal velocity *w* in the drainage channel are constant; changes in kinetic energy is not taken into account considering low longitudinal velocities [10].

where Δ*Р*_{g} is the loss of friction pressure and local resistance (dissipation work).

The frictional pressure loss Δ*P*_{1g} can be calculated using the Darcy–Weisbach formula

where λ_{1} is the coefficient of friction losses along the length of the pressure channel and *d*_{TP} is the diameter of the pressure channel.

The coefficient of friction losses along the length for a hollow channel is determined from the ratio proposed by Berman [10]

where ν is the kinematic viscosity coefficient, m^{2}/s.

The determination of friction pressure loss in the drainage channel is difficult because of the heterogeneity of the diameters, pore sizes, and their curvature, as well as their deformation under the action of the working pressure. In this case, using the filtration coefficient *K*_{f}, we have

where *b* is the dimensionless approximation coefficient (see Eq. (24)) and *K*_{f} is the filtration coefficient, m^{3}/(m^{2} s Pa).

After integration and transformation of Eqs. (13) and (14), we have

During ultrafiltration, it is necessary to take into account the diffusion flux *j* of the dissolved substance directed from the membrane to the channel axis (Fig. 5) [10].

The movement of matter near the membrane surface is carried out both by the diffusion flux *j* and convective flux *C*_{0}*V*_{0}, and the release is carried out only by the convective flux *C*_{2}*V*_{2}.

Then the equation of material balance of the dissolved substance, which approaches the surface of the membrane in the pressure channel and passes into the permeate from its opposite side, can be written as

The diffusion flux *j* at the interface is determined by the Fick law [6, 10, 21] and the concentration profile [10]

where *k* is a dimensionless coefficient taking into account the ratio of the thickness of the viscous sediment layer δ and the diffusion layer Δ along the length of the membrane module, *k* = δ/Δ [10]; *A* and *b* are the coefficients of approximation of the equations of the velocity profile *u*(η) and concentration *C*(η) [10].

The Reynolds number Re in Eqs. (23) and (24) is determined by the expression [10]

where ν_{0} is kinematic viscosity of the solution at the surface of the membrane.

The speed of permeate and its concentration are determined by the formulas [10]:

where φ is the selectivity and *g* is the specific productivity, m^{3}/m^{2} h MPa.

To close the mathematical model, an expression is determined for calculating the sedimentation rate σ. For this, it is necessary to write the equation of the material balance of the solution and the dissolved substance, taking into account the diffusion flux and the motion of the sediment along the membrane in the boundary layer (Fig. 6).

For the control volume *a*_{0}*a*_{s}*e*_{s}*e*_{0} (Fig. 6), the material balance of the solute has the form

The material balance of the solution will take the form

For the control volume *a*_{1}*е*_{1}*a*_{s}*e*_{s} (Fig. 2), the material balance of the dissolved substance is written in the form

where \({{\tilde {C}}_{{\text{e}}}}\) is the concentration of solute at the end of the section.

The unknown quantities in Eqs. (28)–(30) are *j*, *V*_{0}, and σ. Solving them together, we have:

After determining *j* from Eq. (21), we find the value of parameter *k*

Then we find the reduced concentration [10]

The following general formula was proposed in [10] for calculating the velocity profile in a dimensionless form

where *u*(η) = 1 when η ≥ 1; η is the dimensionless ordinate of the sediment, the beginning of which is on the surface of the sediment layer; it varies within η = (0, 1).

The dimensionless ordinate is determined from the relation [10]

To determine the function of changing the concentration along the ordinate axis (Fig. 6), we can use the analogy of momentum and mass transfer [10]. If the kinematic viscosity and diffusion coefficient are equal and in the case of a gradientless flow in the longitudinal direction, the profile of the dimensionless concentration of the dissolved substance along the channel height should coincide with the profile of the dimensionless velocity. Then the concentration profile can be approximated by a function similar to the velocity distribution function [10]

In Eq. (38), *C*(η) = 1 for η ≥ 1/*k* and the dimensionless ordinate is in the range η = (0, 1/*k*).

Then, from Eqs. (32) and (33), the values of the transverse component of the flow velocity in the pressure channel *V*_{0} and the sedimentation rate σ are determined.

## EXPERIMENTAL

The selectivity φ and the specific productivity *g* of the membrane in Eqs. (26) and (27) are determined experimentally.

The studies were conducted in a laboratory setup with tubular ultrafilters. To determine the significance of factors affecting the permeability and selectivity of the membranes, the method of mathematical planning of the experiment was used, namely a full factorial experiment according to the plan presented in Tables 1 and 2.

The following parameters were chosen as the main parameters: for specific permeability―the pressure drop *P*, the solution flow rate over the membrane ϑ, and the viscosity of the solution to be separated ν; for selectivity―the pressure drop *P*, the concentration of impurities in the initial solution *C*_{1}, and the viscosity of the separated solution ν [32].

Ultrafiltration was carried out for wastewater containing acrylic dispersions based on ethyl acrylate, butyl acrylate, methyl methacrylate (appretan 9211, 9212, Hoechst, Germany).

The separation was carried out using polymer anisotropic tubular membranes (NPO Vladipor) based on fluoroplastic (UFFK) and polysulfonamide (PSA) manufactured according to TU (State Standard) 6559-88, 605-221-734-83, and 655-4-88.

The determination of the technological parameters characterizing the separation process was carried out as follows: (i) the pressure at the inlet and outlet of the membrane element in the pipeline was measured by spring manometers; (ii) the fluid flow rate supplied by the pump to the membrane element was determined both according to the characteristics of the pump and by measuring the volume of fluid with a calibrated flask using a stopwatch; (iii) the experimental specific productivity was determined by the amount of ultrafiltrate obtained per unit time from a unit of the working surface (m^{3}/m^{2} s) and was calculated by the formula

where *V* is the permeate volume; (iv) the viscosity of the solutions was measured by a capillary glass VPZhT-2 viscometer; (v) the temperature of the initial solution and ultrafiltrate was measured using a mercury technical TL-4 thermometer with a division value of 0.1°C; (vi) the amount of permeate was determined by a volumetric flask with a volume of 1000 mL and a stopwatch; (vii) the concentration of polyacrylates was determined using the developed accelerated method based on light scattering of acrylate particles in an aqueous solution using reagents (glacial acetic acid, sodium hydroxide) and a KFK-2 photoelectrocolorimeter [32]; (viii) the selectivity (the retention of membranes) was determined by the formula

To determine the significance of the parameters, the full factorial experiment method was used.

The experimental conditions are given in Tables 1 and 2. The planning matrix and the results of the full factorial experiment are given in Tables 3 and 4.

After processing the data by regression analysis and checking the mathematical model for adequacy, dependencies were obtained to determine the specific permeability of the PSA and UFFK membranes, which in the named quantities have the form:

The regression equations Eqs. (41)–(44) were obtained as a result of experimental studies presented in previously published works of the authors [32]. In practice, to determine the selectivity and specific permeability, it is enough to conduct an experiment on the separation cell without mixing, changing the following parameters: pressure *P*, temperature *T*, and the size of the sediment layer *S*.

## RESULTS AND DISCUSSION

The driving force of ultrafiltration is the pressure drop in front of and behind the membrane. At the same time, the creation of certain hydrodynamic regimes in the apparatus is a necessary condition for preventing the formation of sediment. To reduce its negative effect, it is necessary to turbulize the liquid layer adjacent to the membrane surface to accelerate the transfer of solute to the core of the shared stream (Fig. 7).

From the dependences shown in Fig. 7 it follows that with an increase in pressure drop by two times the productivity increased by 1.5–2 times, and with an increase in the flow velocity above the membrane from 2.3 m/s (Re = 13 000) to 4.8 m/s (Re = 37 000), the productivity increased by three to four times. This means that the rate of flow of the solution has a stronger effect on the permeability and the amount of sediment formation.

The results of solving Eqs. (31)–(33) for aqueous solutions of acrylate dispersion separated by Vladipor polymer-type tubular membrane modules of type BTU-0.5/2 according to (State Standard) TU 605-221-704-83 are shown in Figs. 8–10.

Analyzing Fig. 8, it can be concluded that the initial concentration of the solution weakly affects the duration of the unsteady ultrafiltration period. The latter substantially depends on the output flow velocity *u*, increasing with increasing channel diameter and with decreasing velocity.

Figure 9 shows the topography of the gel layer along the channel for different time points for the PSA membrane. In the initial section of the channel with a length of less than 20% of its length, a sediment layer is not formed. Active sedimentation occurs at a concentration distance from the initial value of *C*_{H} to the gelation concentration *C*_{s}.

Figure 10 shows the effect of the velocity of the solution at the input to the channel on the thickness of the sediment layer *S* for PSA and UFFK membranes.

For the given conditions of the separation process, there is a value of velocity *u*_{c} at which no precipitate forms.

This value depends on the height of the intermembrane channel, the concentration of the initial solution and its temperature, and the pressure drop across the membrane. With a change in these parameters, the value of *u*_{c} can be shifted in one direction or another. The velocity *u*_{c} can be found by extrapolating the dependence *S*(*w*) to a point corresponding to the value *S* = 0.

To determine the adequacy of the obtained model, a comparative analysis of the experimental and calculated concentration values for giving time intervals for the PSA membrane, which are given in Table 5, was performed.

At various durations of the wastewater separation process with an initial acrylate dispersion content of 3 kg/m^{3}, the concentration value in the reservoir for collecting the concentrate was measured and compared with those calculated previously. As can be seen from the data given in Table 5, the difference between the actual and calculated concentration values is less than 10%, which confirms the validity of the proposed ratios used in calculations.

## CONCLUSIONS

The proposed mathematical model can be used to calculate ultrafiltration processes complicated by the formation of a precipitate on the separation surface, having a minimum of information about the properties of the solution.

Using this model, we can solve direct tasks to determine the filtration rate with known installation parameters and process modes as well as inverse tasks, when for given performance and characteristics of a shared medium, it is necessary to calculate the geometric parameters of the apparatus.

To calculate the filtration rate *V*_{2}, the extraction coefficients *K*_{e} and solution purification coefficients *K*_{p}, the total volume, and average permeate concentration, the following initial data are needed: the working pressure *P*_{1}, initial concentration *C*_{1} of solution and its temperature, physicochemical properties of the dissolved substance and solution―*D*, η, and ρ; the geometric dimensions of the membrane apparatus―*R*, *L*; the velocities of the solution in the pressure channel; in addition, the change in specific permeability *g* and selectivity φ over time in the separator cell without mixing are required to be determined experimentally, which greatly simplifies the calculation procedure.

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Fedosov, S.V., Osadchy, Y.P. & Markelov, A.V. Modeling of Ultrafiltration Process Taking Into Account the Formation of Sediment on Membrane Surface.
*Membr. Membr. Technol.* **2, **169–180 (2020). https://doi.org/10.1134/S251775162003004X

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### Keywords:

- wastewater treatment
- ultrafiltration
- sediment layer
- material balance
- energy balance
- diffusion coefficient