eISSN 2658–5782

DOI 10.21662/mfs

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2026. Vol. 21. Issue 1, Pp. 4–10
URL: https://multiphasesystems.online/mfs2026.1.002,en
DOI: https://doi.org/10.21662/mfs2026.1.002
Criterion for Evaluating the Dominant Mechanism of Gas Separation Using a Selective Membrane
M.A. Anisimova 🖂, A.Ya. Gilmanov, A.P. Shevelev
University of Tyumen, Tyumen, Russia

Abstract

One of the most effective and technologically promising ways to separate components of a gas mixture is the use of selective polymer membranes. It is advisable to model and optimize operation of such installations based on fundamental approaches of mechanics of multiphase systems. The aim of this work is to develop a generalized imensionless criterion for evaluating the dominant mechanism of gas separation using selective fiber membrane. An important result is that the same membrane, depending on operating parameters, can function either predominantly in diffusion or filtration mode. The mathematical model underlying the analysis includes system of equations based on the laws of conservation of mass and Darcy’s law for porous medium. The closing relations connecting the component fluxes with their concentrations and pressure drop made it possible by numerical solution to obtain distributions of concentrations of key components (oxygen and nitrogen) along the length of membrane module. A dimensionless criterion is proposed that characterizes the dominant mechanism of mass transfer of gas mixture components through the body of selective membrane. A critical value of 1 is established for introduced criterion, which allows the selection of operating parameters for selective membrane system that guarantees the required mass transfer regime. Calculations have shown that when the value of proposed criterion is less than one, the filtration mechanism of mass transfer prevails, which leads to selective release of the lightest component of the mixture, nitrogen. If the criterion exceeds one, the diffusion mechanism becomes dominant, and the plant effectively separates oxygen.

Received: 20.01.2026
Accepted: 12.02.2026
Published: 31.03.2026
Citation

Anisimova MA, Gilmanov AYa, Shevelev AP. Criterion for evaluating the dominant mechanism of gas separation using a selective membrane. Multiphase Systems. 2026;21(1):4–10 (in Russian).

Keywords

selective membrane;
mass transfer;
dimensionless criterion;
concentration distribution;
filtration;
diffusion;
mechanics of multiphase systems

Article outline

One of the most effective and technologically promising ways to separate components of a gas mixture, such as air, is the use of selective polymer membranes. Depending on the type of material, structure, and shape of the membrane fibers, two dominant mass transfer mechanisms are distinguished: diffusion and filtration. Diffusion membranes work on the principle of dissolving gas in the material from which it is made, and then moving through it due to diffusion in the presence of a concentration gradient. Filtration membranes are porous structures, the size of the pore channels of which is comparable to the size of molecules. It is impossible to create a membrane with only one separation mechanism. A change in the separation mode may cause the gas taken away by the membrane to change. Installations for the separation of gas mixtures using selective membranes have 3 types of construction: flat, rolled and fiber. Fiber membranes are characterized by the widest ranges of operating pressures with a high effective working surface, achieved due to the microscopic size of these fibers and, as a result, a large number of them in a particular installation. Due to these circumstances, the latest design of the membrane installation was chosen for modeling. It is advisable to model and optimize the operation of such installations based on the fundamental approaches of the mechanics of a continuous multiphase medium, which have proven themselves well in solving problems of this type. The presence of various mechanisms of mass transfer of gases through the membrane, noted above, leads to the need to determine the operating modes of the installation, leading to the predominance of one process or another. The purpose of this work is to develop a generalized dimensionless criterion for evaluating the dominant mechanism of gas separation using a selective fiber membrane. It has been shown for the first time that the same membrane, depending on the operating parameters (pressure, mixture composition), can function either predominantly in diffusion or filtration mode. Three subtasks are identified: the movement of gases inside the membrane fiber with a length (the first subtask), their filtration and/or diffusion through the membrane body (the second subtask) and the movement of gases outside the membrane fiber due to the presence of a pipe for sampling products with a flow (the third subtask). The mathematical model underlying the analysis includes a system of equations based on the laws of conservation of mass and Darcy's law for a porous medium. The first and second subtasks are linked by specifying the source terms based on Fick's first law and the linear filtration law, which is similar in form to Henry's law. The closing relations connecting the component fluxes with their concentrations and pressure drop made it possible by numerical solution to obtain distributions of concentrations of key components (oxygen and nitrogen) along the length of the membrane module. A dimensionless complex equal to the ratio of the total diffusion rate of the components to the total filtration rate is chosen as the criterion characterizing the dominant mechanism of mass transfer of the components of a gas mixture. A critical value of 1 has been established for the introduced dimensionless criterion, which makes it possible to select the operating parameters of a selective membrane installation, at which the required mass transfer mode is guaranteed. Calculations have shown that when the value of the proposed criterion is less than one, the filtration mechanism of mass transfer prevails, which leads to the selective release of the lightest component of the mixture, nitrogen. If the criterion exceeds one, the diffusion mechanism becomes dominant, and the plant effectively separates oxygen. The distribution of nitrogen concentration along the length of the installation inside and outside the fiber is nonlinear, due to the different rate of pressure drop inside and outside the membrane fibers. With the specified parameters, the maximum oxygen concentration increases by 1.2% of the initial one. Thus, the diffusion mechanism of mass transfer makes it possible to purify gases more efficiently due to the high values of the characteristic process time. It is necessary to adjust the pressure at the ends of the installation in such a way that one of the operating modes of the membrane installation is guaranteed: filtration or diffusion, since in the transition mode nitrogen is primarily transferred due to the filtration mechanism and oxygen due to diffusion, the membrane ceases to be selective.

References

  1. Ahmad AL, Chong MF, Bhatia S. Mathematical modeling of multiple solutes system for reverse osmosis process in palm oil mill effluent (POME) treatment. Chemical Engineering Journal. 2007;132(1–3):183–193. https://doi.org/10.1016/j.cej.2006.12.022
  2. Hanspal NS, Waghode AN, Nassehi V, Wakeman RJ. Development of a predictive mathematical model for coupled stokes/Darcy flows in cross-flow membrane filtration. Chemical Engineering Journal. 2009;149(1–3):132–142. https://doi.org/10.1016/j.cej.2008.10.012
  3. Лазарев СИ, Головин ЮМ, Хорохорина ИВ, Лазарев ДС, Родионов ДА. Влияние структуры поверхностного ацетатцеллюлозного слоя на транспортные характеристики ультрафильтрационных композиционных мембран. Инженерно-физический журнал. 2021;94(2):400–407. https://elibrary.ru/obzjpp
    Lazarev SI, Golovin YM, Khorokhorina IV, Lazarev DS, Rodionov DA. Influence of the Structure of the Surface Cellulose Acetate Layer on the Transport Characteristics of Ultrafiltration Composite Membranes. Journal of Engineering Physics and Thermophysics. 2021;94(2):384–391. https://doi.org/10.1007/s10891-021-02308-7
  4. Alkandari SH, Castro-Domingue B. Advanced and sustainable manufacturing methods of polymer-based membranes for gas separation a review. Frontiers in Membrane Science and Technology. 2024;3:1390599. https://doi.org/10.3389/frmst.2024.1390599
  5. Parulekar PJ, Bhagat PG, Mehta T, Nichani R, Nair A. Chemical Plant Utility –- Nitrogen System Design. International Journal for Research. 2021;9(11):1560-1567 https://doi.org/10.22214/ijraset.2021.39047
  6. Singh R, Prasad B, Ahn Y-H. Recent developments in gas separation membranes enhancing the performance of oxygen and nitrogen separation: A comprehensive review. Gas Science and Engineering. 2024;123:205256. https://doi.org/10.1016/j.jgsce.2024.205256
  7. Ricci E, Minelli M, De Angelis MG. Modelling Sorption and Transport of Gases in Polymeric Membranes across Different Scales: A Review. Membranes. 2022;12(9):85. https://doi.org/10.3390/membranes12090857
  8. Astorino C, De Nardo E, Lettieri S, Ferraro G, Pirri CF, Bocchini S. Advancements in Gas Separation for Energy Applications: Exploring the Potential of Polymer Membranes with Intrinsic Microporosity (PIM). Membranes. 2023;13(12):903. https://doi.org/10.3390/membranes13120903
  9. Purkait MK, Singh R, Mondal P, Haldar D. Membrane materials and modification for thermal induced membrane separation processes. In: Thermal Induced Membrane Separation Processes. 2020:41–53. https://doi.org/10.1016/B978-0-12-818801-9.00003-7
  10. Hegde VH, Doherty MF, Squires TM. A two-phase model that unifies and extends the classical models of membrane transport. Science. 2022;377(6602):186–191. https://doi.org/10.1126/science.abm7192
  11. Akbari B. Preparation of hydrophobic flat sheet membranes from PVDF-HFP popolymer for enhancing the oxygen permeance in nitrogen/oxygen gas mixture / B. Akbari, A. Lashanizadegan, P. Darvishi, A. Pouranfard // Chinese Journal of Chemical Engineering. 2020. С. 1–16.
  12. Chong KC, Lai SO, Thiam HS, Lee SS. Oxygen/Nitrogen Gas Separation by Polyetherimide Hollow Fiber Membrane: Effects of Bore Fluid Rate on Permeance and Selectivity. Journal of Applied Membrane Science \& Technology. 2020;24(2):73–83. https://doi.org/10.11113/amst.v24n2.184
  13. Shakil R, Tarek YA, Rumon MH. Polymeric Membranes for O2/N2 Separation. Materials Research Foundations. 2021;113:171–202. https://doi.org/10.21741/9781644901632-6
  14. Patil M, Hunasikai SG, Mathad SN, et al. Enhanced O2/N2 separation by QuaternizedMatrimid/Multiwalled carbon nanotube mixed-matrix membrane. Heliyon. 2023;9(11):21992. https://doi.org/10.1016/j.heliyon.2023.e21992
  15. Adhikari B, Orme CJ, Klaehn JR, Stewart FF. Technoeconomic analysis of oxygen-nitrogen separation for oxygen enrichment using membranes. Separation and Purification Technology. 2020;268:118703. https://doi.org/10.1016/j.seppur.2021.118703
  16. González-Revuelta D, Fallanza M, Ortiz A, Gorri D. Thin-Film Composite Matrimid-Based Hollow Fiber Membranes for Oxygen/Nitrogen Separation by Gas Permeation. Membranes. 2023;13(2):218. https://doi.org/10.3390/membranes13020218
  17. Сигунова АА, Мищенко ЕС. Мембранные технологии в газопереработке: опыт и преемственность. Деловой журнал Neftegaz.RU. 2023;(10(142)):108–112. https://elibrary.ru/shpsxc
    Sigunova AA, Mishchenko ES. Membrane technologies in gas processing: experience and continuity. Business magazine Neftegaz.RU. 2023;(10(142)):108–112. (in Russian)
  18. Гильманов АЯ, Деменчук МА, Шевелев АП. Определение параметров установки с селективными мембранами. Вестник Тюменского государственного университета. Физико-математическое моделирование. Нефть, газ, энергетика. 2021;7(3(27)):71–88. https://doi.org/10.21684/2411-7978-2021-7-3-71-88
    Gilmanov AYa, Demenchuk MA, Shevelev AP. Determination of Unit Parameters with Selective Membranes. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy. 2021;7(3(27)):71–88.
  19. Koutsonikolas DE, Pantoleontos G, Karagiannakis G, Konstandopoulos AG. Development of H2 selective silica membranes: Performance evaluation through single gas permeation and gas separation tests. Separation and Putification Technology. 2021;264:118432. https://doi.org/10.1016/j.seppur.2021.118432
  20. Neyertz S, Brown D, Salimi S, Radmanesh F, Benes NE. Molecular Characterization of Membrane Gas Separation under Very High Temperatures and Pressure: Single-and Mixed-Gas CO2 /CH4 and CO2/N2 Permselectivities in Hybrid Networks. Membranes. 2022;12(5):526. https://doi.org/10.3390/membranes12050526
  21. Frikha S, Frikha N, Gabsi S. Modeling of the flow inside a pore in vacuum membrane distillation. Euro-Mediterranean Journal for Environmental Integration. 2021;6:66. https://doi.org/10.1007/s41207-021-00275-2
  22. Tas-Koehler S, Lerch A. Application of computational fluid dynamics technique in reverse osmosis/nanofiltration processes. In: Basile A, Ghasemzadeh K (ed.). Current Trends and Future Developments on (Bio-) Membranes. 2021:63–79. https://doi.org/10.1016/B978-0-12-822294-2.00004-7
  23. Malakhov AO, Volkov VV. Mixed-Gas Selectivity Based on Pure Gas Permeation Measurements: An Approximate Model. Membranes. 2021;11(11):833. https://doi.org/10.3390/membranes11110833
  24. Jasim DJ, Mohammed TJ, Harharah HN, Harharah RH, Amari A, Abid MF. Modeling and Optimal Operating Conditions of Hollow Fiber Membrane for CO2/CH4 Separation. Membranes. 2023:13(6):557. https://doi.org/10.3390/membranes13060557
  25. Darabi M, Pahlavanzadeh H. Mathematical modeling of CO2 membrane absorption system using ionic liquid solutions. Chemical Engineering and Processing: Process Intensification. 2020;147:107743. https://doi.org/10.1016/j.cep.2019.107743
  26. Ghobadi J, Ramirez D, Khoramfar S, Kabir MM, Jerman R, Saeed M. Mathematical modeling of CO2 separation using different diameter hollow fiber membranes. International Journal of Greenhouse Gas Control. 2021;104:103204. https://doi.org/10.1016/j.ijggc.2020.103204
  27. Ortiz-Albo P, Takaba H, Kumakiri I, Crespo JG, Neves LA. An overview of Molecular Simulations Studies in Mixed Matrix Membranes for Gas Separation Processes. Journal of Membrane Science. 2022;8(4):549436. https://doi.org/10.22079/jmsr.2022.549436.1536
  28. Pacheco MJ, Vences LJ, Moreno H, Pacheco JO, Valdivia R, Hernández C. Review: Mixed-Matrix Membranes with CNT for CO2 Separation Processes. Membranes. 2021:11(6):457. https://doi.org/10.3390/membranes11060457
  29. Cao Z, Kruczek B, Thibault J. Monte Carlo Simulations for the Estimation of the Effective Permeability of Mixed-Matrix Membranes. Membranes. 2022;12(11):1053. https://doi.org/10.3390/membranes12111053
  30. Atlaskin AA, Trubyanov MM, Kirillov SY, et al. Transient dynamics in a membrane module with a pulsed change of retentate: Modeling and experimental study of an unsteady-state mode of membrane gas separation process. Separation and Purification Technology. 2021;259:118201. https://doi.org/10.1016/j.seppur.2020.118201
  31. Majid OA, Kuznetsova M, Castel C, Favre E, Hreiz R. Impact of Concentration Polarization Phenomena on Gas Separation Processes with High-Performance Zeolite Membranes: Experiments vs. Simulations. Membranes. 2024;14(2):41. https://doi.org/10.3390/membranes14020041
  32. Lundin S-TB, Ikeda A, Hasegawa Y. On the Maximum Obtainable Purity and Resultant Maximum Useful Membrane Selectivity of a Membrane Separator. Membranes. 2024;14(6):143. 10.3390/membranes14060143
  33. Lin JYS, Ovalle-Encinia O. Dual-Phase Ionic-Conducting Membranes: Pressure Dependence of Gas Permeation Flux. Journal of Membrane Science Letters. 2023;3(1):100041. https://doi.org/10.1016/j.memlet.2023.100041
  34. Rivero JR, Nemetz LR, Da Conceicao MM, Lipscomb G, Hornbostel K. Modeling gas separation in flat sheet membrane modules: Impact of flow channel size variation. Carbon Capture Science \& Technology. 2023;6:100093. https://doi.org/10.1016/j.ccst.2022.100093
  35. Da Conceicao M, Nemetz L, Rivero J, Hornbostel K, Lipscomb G. Gas Separation Membrane Module Modeling: A Comprehensive Review. Membranes. 2023;13(7):639. https://doi.org/10.3390/membranes13070639
  36. Бекман ИН, Романовский ИП. Феноменологическая теория диффузии в гетерогенных средах и ее применение для описания процессов мембранного разделения. Успехи химии. 1988;57(6):944–958. https://www.uspkhim.ru/RCR3369
    Bekman IN, Romanovskii IP. A Phenomenological Theory of Diffusion in Heterogeneous Media and Its Use in Describing Membrane Separation Processes. Russian Chemical Reviews. 1988;57(6):530–538. https://doi.org/10.1070/RC1988v057n06ABEH003369
  37. Freger V, Ramon GZ. The solution-diffusion model “Rumors of my death have been exaggerated”. Journal of Membrane Science Letters. 2024;4(2):100084. https://doi.org/10.1016/j.memlet.2024.100084
  38. Campo MC, Magalhães FD, Mendes A. Separation of nitrogen from air by carbon molecular sieve membranes. Journal of Membrane Science. 2010;350(1–2):139–147. https://doi.org/10.1016/j.memsci.2009.12.021