The presented article is an attempt to evaluate the progress in the development of the mathematical simulation of the pressure-driven membrane processes. It was considered more than 170 articles devoted to the simulation of reverse osmosis, nanofiltration, ultrafiltration, and microfiltration and the others published between 2000 and 2010 years. Besides the conventional approaches, which include the irreversible thermodynamics, diffusion and pore flow (and models which consider the membrane surface charge for nanofiltration process), the application of the methods the computational fluid dynamics, artificial neural networks, optimization, and economic analysis have been considered. The main trends in this field have been pointed out, and the areas of using approaches under consideration have been determined. The technological problems which have been solved using the mentioned approaches have also been considered. Although the question of the concentration polarization has not been considered separately, it was defined that, in many cases, the sufficiently accurate model cannot be designed without considering this phenomenon. The findings allow evaluating more thoroughly the development of the simulation of pressure-driven membrane processes. Moreover, the review allows choosing the strategy of the simulation of the considered processes. Keywords: membrane, simulation, model, reverse osmosis nanofiltration, ultrafiltration, microfiltration.

Introduction. The experimental examination of hypothesis about linear dependence of concentration polarization resistance from pressure was carried out and the influence of hydrodynamic condition on this resistance is determined. Materials and methods. The research was carried out with using of commercially available membrane modules TFC-75 type. The measurements of productivity were carried out with using of deionized water (total dissolved solids 5-15 mg/dm 3) and also NaCl solutions. The volumetric technique was used for flux measurements. The concentration was measured by conductometric technique. Results and discussion. The membrane resistance during reverse osmosis of deionized water did not change with applied pressure in experimental conditions and was equal Rm=7,549•10 13 m-1. The concentration polarization layer resistance (Rcp) increased from 0.65-1.29•10 13 m-1 to 1.46-1.83•10 13 m-1 with applied pressure increasing from 0.2 MPa to 0.6 MPa and from 0.65-1.46 m-1 to 1.29-1.83•10 13 m-1 with increasing of feed concentration from 100 mg/dm 3 to 600 mg/dm 3. This increasing of Rcp value with pressure was linear which is in agreement with previously reported data for the ultrafiltration process. Moreover, in considered range of applied pressure, the exponential dependence of index of concentration polarization from applied pressure could be approximated by a linear equation with correlation coefficient 0.93. Therefore, assumption about linear dependence of concentration polarization layer resistance from pressure is reasonable and could be extended to reverse osmosis process for mentioned above conditions. The increasing of concentration polarization layer resistance with increasing of applied pressure is determinated by higher values of transmembrane fluxes and lower values of mass transfer coefficient at higher values of applied pressure in the considered system. These results are in agreement with film theory of concentration polarization. Conclusions. The exanimated hypothesis is validated for reverse osmosis in considered range of applied pressure. The correlation between concentration polarization layer resistance and index of concentration polarization was defined.

The experimental determination of concentration polarization layer resistance during reverse osmosis of mineral salts solutions was carried out with the aim to estimate the influence of solution composition on the value of mentioned resistance. In experimental conditions, the membrane resistance remains constant (the mean value was 0.534•10 14 m-1) which means that the membrane compaction was not observed. Moreover, under experimental conditions, the hypothesis about linear dependence between the concentration polarization layer and applied pressure was confirmed for all solutions under investigations. It was defined that value of concentration polarization layer resistance different salt solutions was varied less than 10 % although under experimental conditions the diffusion coefficient values of magnesium sulfate were more than three times higher than corresponded values for other salts. The increasing of solutions concertation determines the increasing of concentration polarization layer resistance. At the same time, in previous study it was defined that changes in hydrodynamic regime in membrane module under similar conditions could determine the change in concentration polarization layer resistance in 3-5 times, while in both studies the trends of impact of hydrodynamic conditions still similar to the value of considered resistance decrease with Reynolds number increasing. Such results showed that in considered range of concentrations the hydrodynamic conditions have a lower influence on concentration polarization layer resistance than solution composition. The obtained results are in agreement with the film theory of concentration polarization.

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The cleaning or regeneration of fouled membrane modules is an essential procedure in the membrane equipment operation. Despite the development of some successful cleaning techniques, the predictions of the membrane separation process operation parameters after regeneration is still an unsolved problem. In our previous works, the attempt to develop the methodology of estimating the membrane productivity after the regeneration of the fouled spiral wound membrane modules by cleaning the subatmospheric pressure has been made. However, this methodology requires some improvement, including the correction of the dimensionless equation to calculate the mass transfer coefficient. In this work, a set of additional experiments was carried out, and the corrections of the mass transfer correlation were done using both new and previously obtained experimental data. As a result, the improved dimensionless equation was contained as Sh = 0.00045Re0.8Sc0.33(de/l). This equation is valid in the range of Reynolds number variation of 0.4–60.0 for the case of the regeneration of spiral wound modules and can be used for the prediction of the permeate flux after the regeneration procedure.