For the qualitative application of ultrafiltration processes for the concentration and purification of food solutions, both experimental studies and a mathematical description of the processes of the membrane separation process of solutions from the standpoint of the development of computational mathematical models are required. In this work, by analytical solution of equations, that is, by the method of finite differences, mathematical equations are solved. To obtain the system, the flow continuity equations, convective diffusion equations, Navier-Stokes equations and flow equations with boundary conditions were solved in order to build a mathematical model of the process of ultrafiltration protein concentration in cheese whey in the production of rennet cheeses. As a result of the analytical solution of the equations, a system of mathematical equations was obtained that allows one to construct a profile of changes in the flow rates of the solution along the cross-section of the intermembrane channel and to determine the protein concentration in cheese whey along the length of the tubular ultrafiltration element BTU 05/2 of industrial type. The obtained mathematical model makes it possible to theoretically describe the process of ultrafiltration protein concentration in cheese whey along the entire length of the membrane channel of the tubular element under laminar and transient regimes of solution flow. The resulting system of mathematical equations makes it possible to find the numerical values of the mass flow rate of cheese whey, make it possible to calculate the specific output flow when the transmembrane pressure changes and to calculate the concentration of solutes in the secondary milk raw materials on the left and right ultrafiltration membrane of the intermembrane channel. The adequacy of the developed mathematical model was carried out by comparing the calculated and experimental data on the specific output flow when the transmembrane pressure in the intermembrane channel changes from 0.1 to 0.25 MPa with ultrafiltration concentration of cheese whey. The deviation of the calculated data found by the mathematical model from experimental studies obtained on a semi-industrial tubular ultrafiltration plant BTU 05/2 using semipermeable membranes, in which the active layer is made of fluoroplastic, hemisulphone and polyethersulfone, did not exceed 10%.

1. Worsztynowicz P., Bia?as W., Grajek W. Integrated approach for obtaining bioactive peptides from whey proteins hydrolysed using a new proteolytic lactic acid bacteria // Food chemistry. 2020. V. 312. P. 126035. doi: 10.1016/j.foodchem.2019.126035

2. Лазарев С.И. и др. Эмпирическая модель ультрафильтрационного концентрирования белка в подсырной сыворотке // Вестник Тамбовского университета. Серия: Естественные и технические науки. 2016. Т. 21. №. 2. С. 655–660. doi: 10.20310/1810-0198-2016-21-2-655-660

3. Лобасенко Б.А., Семенов А.Г. Математическая модель ультрафильтрации с учeтом гелеобразования в условиях периодической очистки мембраны // Техника и технология пищевых производств. 2010. № 3.

4. Лазарев К.С. и др. Математическая модель процесса электрогиперфильтрационного разделения промышленных растворов // Вестник тамбовского университета. Серия: естественные и технические науки. 2013. Т. 18. – №. 4–1.

5. B?r C., Mathis D., Neuhaus P., D?rr D. et al. Protein profile of dairy products: simultaneous quantification of twenty bovine milk proteins // International Dairy Journal. 2019. V. 97. P. 167-175. doi: 10.1016/j.idairyj.2019.01.001

6. Иванова С.А., Гарифулин Р.Ш., Чаплыгина Т.В. Моделирование процесса мембранного концентрирования белков молочного сырья // Техника и технология пищевых производств. 2011. № 1.

7. Mansor E.S., Ali E.A., Shaban A.M. Tight ultrafiltration polyethersulfone membrane for cheese whey wastewater treatment // Chemical Engineering Journal. 2021. V. 407. P. 127175. doi: 10.1016/j.cej.2020.127175

8. Вороненко Б.А., Пеленко В.В., Поляков С.В. Математическое описание мембранного разделения эмульсий // Научный журнал НИУ ИТМО. Серия «Процессы и аппараты пищевых производств». 2011. № 2.

9. Chamberland J. et al. Effect of membrane material chemistry and properties on biofouling susceptibility during milk and cheese whey ultrafiltration // Journal of Membrane Science. 2017. V. 542. P. 208-216. doi: 10.1016/j.memsci.2017.08.012

10. Рудобашта С.П., Махмуд С.Ю. Математическое моделирование процесса мембранной дистилляции // Известия высших учебных заведений. Химия и химическая технология. 2012. Т. 55. № 11.

11. Damar I., Cinar K., Gulec H. A. Concentration of whey proteins by ultrafiltration: Comparative evaluation of process effectiveness based on physicochemical properties of membranes // International Dairy Journal. 2020. V. 111. P. 104823. doi: 10.1016/j.idairyj.2020.104823

12. Лобасенко Б.А., Котляров Р.В., Сазонова Е.К. Математическое моделирование процесса мембранного концентрирования на основе кибернетического подхода // Фундаментальные исследования. 2016. № 2–1. С. 70–75.

13. Тимофеев А.Е., Лобасенко Б.А., Котляров Р.В. Разработка математической модели процесса мембранного концентрирования на основе передаточных функций // Техника и технология пищевых производств. 2013. Т. 28. № 1.

14. Лобасенко Б.А., Шушпанников А.С., Котляров Р.В. Разработка математической модели процесса мембранного концентрирования на основе методов информационного моделирования // Современные проблемы науки и образования. 2013. № 4. С. 89–89.

15. Corbat?n-B?guena M.J., ?lvarez-Blanco S., Vincent-Vela M.C. Evaluation of fouling resistances during the ultrafiltration of whey model solutions // Journal of Cleaner Production. 2018. V. 172. P. 358-367. doi: 10.1016/j.jclepro.2017.10.149

16. RedCorn R., Fatemi S., Engelberth A.S. Comparing end-use potential for industrial food-waste sources // Engineering. 2018. V. 4. № 3. P. 371-380. doi: 10.1016/j.eng.2018.05.010

17. Azad T., Ahmed S. Common milk adulteration and their detection techniques // Food Contamination. 2016. V.3. № 22. doi: 10.1186/s40550-016-0045-3

18. Vaziri M. et al. Hybrid of Adsorption and Nanofiltration Processes as a Capable Removal Method for HANs Removal // South African Journal of Chemical Engineering. 2021. V. 36. P. 1-7. doi: 10.1016/j.sajce.2020.12.002

19. Susanto H. et al. Preparation of low-fouling polyethersulfone ultrafiltration membranes by incorporating high-molecular-weight chitosan with the help of a surfactant // South African Journal of Chemical Engineering. 2020. V. 33. P. 133-140. doi: 10.1016/j.sajce.2020.07.003

20. Chollom M.N., Pikwa K., Rathilal S., Pillay V.L. Fouling mitigation on a woven fibre microfiltration membrane for the treatment of raw water // South African Journal of Chemical Engineering. 2017. V. 23. P. 1-9. doi: 10.1016/j.sajce.2016.12.003