Mathematical modeling of mass transfer in electromembrane systems in galvanodynamic mode, taking into account electroconvection and the dissociation/recombination reaction of water molecules

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Abstract

Mass transfer in electrodialysis systems during intense current modes is accompanied by the emergence of additional transfer mechanisms that significantly affect their operational efficiency. According to modern concepts, for dilute electrolyte solutions, mechanisms such as electroconvection and the dissociation/recombination reactions of water molecules are particularly important. These processes have opposing effects on the effectiveness of electrodialysis technologies.
Mathematical models that take these mechanisms into account are actively used in membrane system research; however, they typically describe only the potentiostatic regime, in which a potential jump is established in the system. The interpretation of a vast database of experimental data for the galvanodynamic regime (at fixed current density) also requires theoretical analysis tools.
The aim of this work is to develop a mathematical model of mass transfer in the electrolyte solution layer at an ion-exchange membrane, considering electroconvection and water dissociation in the galvanodynamic regime. The model is based on a system of coupled Nernst–Planck–Poisson–Navier–Stokes equations, supplemented by a new galvanodynamic boundary condition for the potential.
Using the developed model, chronopotentiograms of the membrane system were calculated for the first time, taking into account the influence of both electroconvection and the dissociation/recombination reactions of water molecules. The results showed that the ratio of the concentration of water dissociation products to the concentration of salt ions determines the balance of the effects of electroconvection and dissociation.
The following options for balancing the effects of electroconvection and dissociation of water molecules are considered:

  1. electroconvection significantly influences mass transfer, while the influence of water dissociation is minimal;
  2. electroconvection and dissociation substantially affect transport processes:
    the formation of additional charge carriers from the dissociation of water molecules reduces the potential jump in the electrolyte layer, which decreases the intensity of electroconvection, while the development of electroconvection, in turn, slows down the dissociation process;
  3. the products of intense water dissociation slow down the development of electroconvection.

About the authors

Aminat M. Uzdenova

Umar Aliev Karachay–Cherkess State University

Author for correspondence.
Email: uzd_am@mail.ru
ORCID iD: 0000-0001-5951-9876
SPIN-code: 2810-2267
Scopus Author ID: 55821149800
http://www.mathnet.ru/person188273

Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Informatics and Computational Mathematics

Russian Federation, 369202, Karachayevsk, Lenina st., 29

References

  1. Ran J., Wu L., He Y., et al. Ion exchange membranes: New developments and applications, J. Membr. Sci., 2017, vol. 522, pp. 267–291. DOI: https://doi.org/10.1016/j.memsci.2016.09.033.
  2. Slouka Z., Senapati S., Chang H. C. Microfluidic systems with ion-selective membranes, Annu. Rev. Anal. Chem., 2014, vol. 7, pp. 317–335. DOI: https://doi.org/10.1146/annurev-anchem-071213-020155.
  3. Gurreri L., Tamburini A., Cipollina A., Micale G. Electrodialysis applications in wastewater treatment for environmental protection and resources recovery: A systematic review on progress and perspectives, Membranes, 2020, vol. 10, no. 7, 146. DOI: https://doi.org/10.3390/membranes10070146.
  4. Rubinshtein I., Zaltzman B., Pretz J., Linder C. Experimental verification of the electroosmotic mechanism of overlimiting conductance through a cation exchange electrodialysis membrane, Russ. J. Electrochem., 2002, vol. 38, no. 8, pp. 853–863. EDN: VBRMLJ. DOI: https://doi.org/10.1023/A:1016861711744.
  5. Pismenskaya N. D., Nikonenko V. V., Belova E. I., et al. Coupled convection of solution near the surface of ion-exchange membranes in intensive current regimes, Russ. J. Electrochem., 2007, vol. 43, no. 3, pp. 307–327. EDN: LKOUSV. DOI: https://doi.org/10.1134/S102319350703010X.
  6. Nikonenko V. V., Mareev S. A., Pis’menskaya N. D., et al. Effect of electroconvection and its use in intensifying the mass transfer in electrodialysis (Review), Russ. J. Electrochem., 2017, vol. 53, no. 10, pp. 1122–1144. EDN: XNXZMK. DOI: https://doi.org/10.1134/S1023193517090099.
  7. Mani A., Wang K. M. Electroconvection near electrochemical interfaces: Experiments, modeling, and computation, Annu. Rev. Fluid Mech., 2020, vol. 52, pp. 509–529. DOI: https://doi.org/10.1146/annurev-fluid-010719-060358.
  8. Simons R. Strong electric field effects on proton transfer between membranebound amines and water, Nature, 1979, vol. 280, pp. 824–826. DOI: https://doi.org/10.1038/280824a0.
  9. Frilette V. J. Electrogravitational transport at synthetic ion exchange membrane surfaces, J. Phys. Chem., 1957, vol. 61, no. 2, pp. 168–174. DOI: https://doi.org/10.1021/J150548A010.
  10. Zabolotskii V. I., Nikonenko V. V., Korzhenko N. M., et al. Mass transfer of salt ions in an electromembrane system with violated electroneutrality in the diffusion layer: The effect of a heterolytic dissociation of water, Russ. J. Electrochem., 2002, vol. 38, no. 8, pp. 810–818. EDN: LHEZOD. DOI: https://doi.org/10.1023/A:1016849309018.
  11. Uzdenova A. M. Mathematical modeling of non-stationary ion transport in electromembrane systems given the dissociation (recombination) reaction of water molecules in galvanodynamic mode, Perspektivy Nauki, 2023, vol. 11, no. 170, pp. 104–112 (In Russian). EDN: USIXRO.
  12. Mishchuk N. A. Concentration polarization of interface and non-linear electrokinetic phenomena, Adv. Colloid Interface Sci., 2010, vol. 160, no. 1–2, pp. 16–39. EDN: MYAMRJ. DOI: https://doi.org/10.1016/j.cis.2010.07.001.
  13. Porozhnyy M. V., Shkirskaya S. A., Butylskii D. Y., et al. Physicochemical and electrochemical characterization of nafion-type membranes with embedded silica nanoparticles: effect of functionalization, Electrochim. Acta, 2021, vol. 370, 137689. EDN: INLSZS. DOI: https://doi.org/10.1016/j.electacta.2020.137689.
  14. Grossman G. Water dissociation effects in ion transport through composite membrane, J. Phys. Chem., 1976, vol. 80, no. 14, pp. 1616–1625. DOI: https://doi.org/10.1021/j100555a020.
  15. Rubinstein I. A diffusional model of “water splitting” in electrodialysis, J. Phys. Chem., 1977, vol. 81, no. 14, pp. 1431–1436. DOI: https://doi.org/10.1021/j100529a018.
  16. Rubinstein I., Shtilman L. Voltage against current curves of cation exchange membranes, J. Chem. Soc., Faraday Trans. 2, 1979, vol. 75, pp. 231–246. DOI: https://doi.org/10.1039/F29797500231.
  17. Kovalenko A. V., Urtenov M. Kh., Seidova N. M., Pismensky A. V. The influence of reaction of dissociation/recombination of molecules of water on transporting electrolyte 1:1 in the membrane systems in the diffusion layer. Part 1. Mathematical model, Nauchnyi Zhurnal KubGAU, 2016, no. 121, 122 (In Russian). EDN: WWSKMP. DOI: https://doi.org/10.21515/1990-4665-121-122.
  18. Kovalenko A. V., Urtenov M. Kh., Seidova N. M., Pismensky A. V. The influence of reaction of dissociation/recombination of molecules of water on transporting electrolyte 1:1 in the membrane systems in the diffusion layer. Part 2. Asymptotic analysis, Nauchnyi Zhurnal KubGAU, 2016, №122, 017 (In Russian). EDN: XBDYXZ. DOI: https://doi.org/10.21515/1990-4665-122-017.
  19. Urtenov M. K., Pismensky A. V., Nikonenko V. V., Kovalenko A. V. Mathematical modeling of ion transport and water dissociation at the ion-exchange membrane/solution interface in intense current regimes, Pet. Chem., 2018, vol. 58, no. 2, pp. 121–129. EDN: YSXUTS. DOI: https://doi.org/10.1134/S0965544118020056.
  20. Urtenov M., Gudza V., Shkorkina I., Chubyr N. Theoretical analysis of the stationary transport of 1:1 salt ions in a cross-section of a desalination channel, taking into account the non-catalytic dissociation/recombination reaction of water molecules, Membranes, 2020, vol. 10, no. 11, 342. DOI: https://doi.org/10.3390/membranes10110342.
  21. Kovalenko A. V., Nikonenko V. V., Chubyr N. O., Urtenov M. Kh. Mathematical modeling of electrodialysis of a dilute solution with accounting for water dissociation-recombination reactions, Desalination, 2023, vol. 550, 116398. DOI: https://doi.org/10.1016/j.desal.2023.116398.
  22. Kovalenko A. V., Urtenov M. Kh., Chubyr N. O., et al. Mathematical modeling of the influence of the main temperature effects in stationary transport of ions of salt in the diffusion layer, Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2018, vol. 15, no. 3, pp. 78–86 (In Russian). EDN: YABSYX. DOI: https://doi.org/10.31429/vestnik-15-3-78-86.
  23. Kovalenko A. V., Urtenov M. Kh., Chubyr N. O., et al. Influence of temperature effects associated with the dissociation/recombination reaction of water molecules and joule heating of the solution on the stationary transport of salt ions in the diffusion layer, Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2018, vol. 15, no. 4, pp. 67–84 (In Russian). EDN: YRMLET. DOI: https://doi.org/10.31429/vestnik-15-4-67-84.
  24. Nikonenko V., Urtenov M., Mareev S., Pourcelly G. Mathematical modeling of the effect of water splitting on ion transfer in the depleted diffusion layer near an ionexchange membrane, Membranes, 2020, vol. 10, no. 2, 22. DOI: https://doi.org/10.3390/membranes10020022.
  25. Kovalenko A. V. Influence of the water dissociation to the electroconvection in membrane systems, Kondens. Sredy Mezhfaz. Gran., 2014, vol. 16, no. 3, pp. 288–293 (In Russian). EDN: SQBSBF.
  26. Kovalenko A., Urtenov M., Chekanov V. Kandaurova N. Theoretical analysis of the influence of spacers on salt ion transport in electromembrane systems considering the main coupled effects, Membranes, 2024, vol. 14, no. 1, 20. DOI: https://doi.org/10.3390/membranes14010020.
  27. Newman J., Thomas-Alyea K. E. Electrochemical Systems. NJ, USA, John Wiley and Sons, 2004, xx+647 pp.
  28. Uzdenova A. M. 2D mathematical modelling of overlimiting transfer enhanced by electroconvection in flow-through electrodialysis membrane cells in galvanodynamic mode, Membranes, 2019, vol. 9, no. 3, 39. DOI: https://doi.org/10.3390/membranes9030039.
  29. Uzdenova A. M. Time-dependent two-dimensional model of overlimiting mass transfer in electromembrane systems based on the Nernst–Planck, displacement current and Navier–Stokes equations, Computation, 2023, vol. 11, no. 10, 205. DOI: https://doi.org/10.3390/computation11100205.
  30. Cohen H., Cooley J. W. The numerical solution of the time-dependent Nernst–Planck equations, Biophys. J., 1965, vol. 5, no. 2, pp. 145–162. DOI: https://doi.org/10.1016/s0006-3495(65)86707-8.
  31. Brumleve T. R., Buck R. P. Numerical solution of the Nernst–Planck and Poisson equation system with applications to membrane electrochemistry and solid state physics, J. Electroanal. Chem., 1978, vol. 90, no. 1, pp. 1–31. DOI: https://doi.org/10.1016/s0022-0728(78)80137-5.
  32. COMSOL Multiphysics Reference Manual. https://doc.comsol.com/6.1/doc/com.comsol.help.comsol/COMSOL_ReferenceManual.pdf.
  33. Nikonenko V. V., Vasil’eva V. I., Akberova E. M., et al. Competition between diffusion and electroconvection at an ion-selective surface in intensive current regimes, Adv. Colloid Interface Sci., 2016, vol. 235, pp. 233–246. DOI: https://doi.org/10.1016/j.cis.2016.06.014.
  34. de Valenc˛a J. C., Wagterveld R. M., Lammertink R. G. H., Tsai P. A. Dynamics of microvortices induced by ion concentration polarization, Phys. Rev. E, 2015, vol. 92, no. 3, pp. 031003. DOI: https://doi.org/10.1103/physreve.92.031003.
  35. Filippov A. N., Akberova E. M., Vasil’eva V. I. Study of the thermochemical effect on the transport and structural characteristics of heterogeneous ion-exchange membranes by combining the cell model and the fine-porous membrane model, Polymers, 2023, vol. 15, no. 16, 3390. DOI: https://doi.org/10.3390/polym15163390.
  36. Urtenov M. A. K., Kirillova E. V., Seidova N. M., Nikonenko V. V. Decoupling of the Nernst–Planck and Poisson equations. Application to a membrane system at overlimiting currents, J. Phys. Chem. B, 2007, vol. 111, no. 51, pp. 14208–14222. DOI: https://doi.org/10.1021/jp073103d.
  37. Krol J. J., Wessling M., Strathmann H. Chronopotentiometry and overlimiting ion transport through monopolar ion exchange membranes, J. Membr. Sci., 1999, vol. 162, no. 1–2, pp. 155–164. DOI: https://doi.org/10.1016/S0376-7388(99)00134-9.

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2. Figure 1. Two-dimensional cross-section of a layer of $\mathrm{NaCl}$ electrolyte at the surface of a cation-exchange membrane (CEM). The flows of salt ions $\mathrm{Na}^+$, $\mathrm{Cl}^-$, as well as hydrogen $\mathrm{H}^+$ and hydroxyl $\mathrm{OH}^-$, generated in the extended space charge region (SCR) in the overlimiting mode

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3. [Figure 2. Chronopotentiograms $d_{\varphi}'(t)$ and average values of the thickness of the electroconvective vortex layer $d_{ec}(t)$, calculated for concentration $c_0=10$ mol/m$^3$ at current density $i=10$ A/m$^2$ (a), (b); $c_0=1$ mol/m$^3$, $i=1$ A/m$^2$ (c), (d); $c_0=0.1$ mol/m$^3$, $i=0.1$ A/m$^2$ (e), (f). Calculations with (solid lines) and without (dashed lines) considering EC, and with (blue lines) and without (red lines) considering DRRW. Fig. (a) also shows an experimental chronopotentiogram for the Neosepta CMX cation exchange membrane in the 10 mol/m$^3$ $\mathrm{CuSO}_4$ (black line) at $i=10$ A/m$^2$ from reference [34]

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4. Figure 3. The relationship between the concentrations of hydrogen ions and the cation of the salt $c_{3av}/c_{1av}$ (a) and the concentrations of hydroxyl ions and the anion of the salt $c_{4av}/c_{2av}$ (b). Fig. (c) shows an enlarged fragment of Fig. (b). The calculation results are presented with (solid lines) and without (dashed lines) consideration of EC, and with consideration of RDRW DRRW for $c_0=10$ mol/m$^3$, $i=10$ A/m$^2$, $c_0=1$ mol/m$^3$, $i=1$ A/m$^2$, $c_0=0.1$ mol/m$^3$, $i=0.1$ A/m$^2$

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5. Figure 4. The distribution of $\mathrm{Na}^+$ ion concentrations (showed in color) and the flow of the solution (white lines) at the time $t = 140$ s, calculated without considering (a) and with consideration of (b) for $c_0=1$ mol/m$^3$, $i=1$ A/m$^2$

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