The article shows the possibility of adapting a one-parameter diffusion model to the membrane separation process by taking into account the permeability of one of the walls of the channel of rectangular cross section under consideration. The structure of the hydrodynamic flow is studied, which allows determining the behavior of the solute concentration field on the membrane surface and evaluating the effectiveness of the measures used to reduce the concentration polarization using hydrodynamic methods due to velocity variations. Investigations of the process of microfiltration of unfiltered unpasteurized beer were carried out on an experimental installation of flow microfiltration. The microfiltration of beer under the flow-through mode of the organization of the process was carried out at the following technological parameters: temperature 2–60 °C, working pressure 0.08–0.25 MPa, speed of the divided flow above the membrane surface 2–3 m/s. The study of hydrodynamics of membrane processes, in view of their highest complexity and specificity, makes it possible today to create a theoretical description in general form and only for one phase or component. It is most convenient to use the diffusion mathematical model for the theoretical description. In mathematical modeling of hydrodynamic processes involving membranes, it is impossible to objectively quantitatively take into account most of the factors due to their large diversity and variability. It should be noted the absence of a unified and generally accepted theory of mass transfer in the study of membrane processes, which is a significant deterrent. The particular complexity of transmembrane transport arises in the case of the imposition of hydrodynamic instabilities of variable intensity, since Any (even insignificant) change in the regime parameters of the microfiltration process leads to different conditions for the formation (or destruction) of the surface layer, which inevitably affects the boundary conditions.

mathematical modeling, microfiltration, membrane module, concentration polarization, specific permeability, equations of motion of a viscous incompressible fluid, diffusion model, hydrodynamic structure of the flow

1. Akhmadiev F.G., Farakhov M.I., Bekbulatov I.G., Isyanov Ch.Kh. Mathematical modeling of the process of filtering two-phase suspensions in tubular filters in non-isothermal conditions. Teoreticheskiye osnovy khimicheskoy tekhnologii [Theoretical foundations of chemical technology]. 2016. vol. 50. no. 1. pp. 44. (in Russian).

2. Gan Q., Howell J.A., Field R.W., England R. et al. Beer clarification by microfiltration – product quality control and fractionation of particles and macromolecules. Journal of Membrane Science. 2001. vol. 194. рp. 185.

3. Hunt J.W., Brouchaert C.J., Raal J.D., Treffry-Goatley K. et al. The unsteady-state modeling of cross-flow microfiltration. Desalination. 1987. vol. 64. рp. 431.

4. Babenyshev S.P., Chernov P.S., Mamai D.S. Modeling the process of membrane filtration of liquid systems. Nauchnyy zhurnal KubGAU [Scientific Journal of KubSAU]. 2012. no. 76 (02). pp. 1–11. (in Russian).

5. Bazhenov V.I., Ustyuzhanin A.V. Mathematical model of biological wastewater treatment taking into account hydrodynamic and non-stationary conditions. Vestnik IrGTU [Proceedings of ISTU]. 2014. no. 11 (94). pp. 128–133. (in Russian).

6. Gorbunova Yu.A., Timkin V.A. Hydrodynamics of the processes of micro – and ultrafiltration separation of milk and cottage cheese. Agrarnyy vestnik Urala [Agrarian Bulletin of the Urals]. 2016. no. 06 (148). pp. 70–75. (in Russian).

7. Lobasenko B.A., Pavsky V.A. Determination of the concentration of solutes in the boundary layer on the membrane surface. Izvestiya vuzov. Pishchevaya tekhnologiya [News of universities. Food technology]. 2001. no. 2–3. pp. 68–70. (in Russian).

8. Semenov A.G. Development of gel contamination of the membrane during tangential ultrafiltration of a solution of high-molecular compound. Tekhnika i tekhnologiya pishchevykh proizvodstv [Technique and technology of food production]. 2011. no. 1 (20). (in Russian).

9. Becker V.F. Modelirovaniye khimiko-tekhnologicheskikh ob"yektov upravleniya [Modeling of chemical-technological objects of management]. Moscow, RIOR : INFRA-M, 2014. 142 p. (in Russian).

10. Timashev S.F. Fizikokhimiya membrannykh protsessov [Physico-chemistry of membrane processes]. Moscow, Khimiya, 1998. (in Russian).

11. Bryk M.T. at al. Ul'trafil'tratsiya [Ultrafiltration; ed. Pilipenko A.T.]. Kiev, Nauk. dumka, 1989. (in Russian).

12. Schmitz P., Houi D., Wandelt B. Hydrodynamic aspects of crossflow microfiltration. Analysis of particle deposition at the membrane surface. Journal of Membrane Science. 1992. vol. 71. рp. 29.

13. Antipov S.T., Kretov I.T., Shakhov S.V., Klyuchnikov A.I. Concentration polarization in the process of clarifying beer. Pivo i napitki [Beer and drinks]. 2001. no. 3. pp. 18. (in Russian).

14. Antipov S.T., Shakhov S.V., Ryazanov A.N., Klyuchnikov A.I. et al. Membrannyy apparat s izmenyayushcheysya vysotoy kanalov [Membrane apparatus with changing the height of the channels]. Patent RF, no. 2147459, 2000.

15. Laptev A.G., Lapteva E.A. Determination of turbulent mixing coefficients in one – and two-phase media using the Taylor model. Fundamental'nyye issledovaniya [Fundamental research]. 2015. no. 2. pp. 2810–2814. (in Russian).