Nernst-Planck modeling of multicomponent ion transport in a Nafion membrane at high current density Citation for published version (APA): Moshtari Khah, S., Oppers, N. A. W., de Groot, M. T., Keurentjes, J. T. F., Schouten, J. C., & van der Schaaf, J. (2017). Nernst-Planck modeling of multicomponent ion transport in a Nafion membrane at high current density. Journal of Applied Electrochemistry, 47(1), 51-62. https://doi.org/10.1007/s10800-016-1017-2 DOI: 10.1007/s10800-016-1017-2 Document status and date: Published: 01/01/2017 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 26. Sep. 2021 J Appl Electrochem (2017) 47:51–62 DOI 10.1007/s10800-016-1017-2 RESEARCH ARTICLE Nernst–Planck modeling of multicomponent ion transport in a Nafion membrane at high current density 1 1 2 1 S. Moshtarikhah • N. A. W. Oppers • M. T. de Groot • J. T. F. Keurentjes • 1 1 J. C. Schouten • J. van der Schaaf Received: 13 June 2016 / Accepted: 18 October 2016 / Published online: 1 November 2016 Ó The Author(s) 2016. This article is published with open access at Springerlink.com Abstract A mathematical model of multicomponent ion experimental data. The effect of current density on the transport through a cation-exchange membrane is devel- observed increase in voltage drop and the decrease in oped based on the Nernst–Planck equation. A correlation permselectivity has been analyzed using the available for the non-linear potential gradient is derived from current qualitative membrane swelling theories. The observed non- density relation with fluxes. The boundary conditions are linear behavior of the membrane voltage drop versus cur- determined with the Donnan equilibrium at the membrane– rent density can be explained by an increase in membrane solution interface, taking into account the convective flow. pore diameter and an increase in the number of active Effective diffusivities are used in the model based on the pores. We show how the membrane pore diameter correlation of tortuosity and ionic diffusivities in free increases and dead-end pores open up when the current water. The model predicts the effect of an increase in density is increased. current density on the ion concentrations inside the mem- brane. The model is fitted to the previously published & J. van der Schaaf [email protected] 1 Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands 2 Akzo Nobel industrial Chemicals B.V., P.O. Box 247, 3800 AE Amersfoort, The Netherlands 123 52 J Appl Electrochem (2017) 47:51–62 Graphical Abstract Keywords Multicomponent ion transport Á Nernst– t Time (s) Planck Á Membrane resistivity Á Membrane selectivity Á ti Ion transport number (–) High current density T Temperature (K) W Weight percentage (wt%) List of symbols x Length (m) x0 Dimensionless length (–) Latin symbols V Volume (m3) a Activity coefficient V Partial molar volume (m3 mol-1) 2 A Membrane cross-sectional area (m ) zi Valence (–) C Concentration (mol m-3) z Dimensionless length (–) -3 dh Hydrodynamic permeability (kg.s.m ) dp Pore diameter (m) Greek symbols D Diffusion coefficient (m2 s-1) d Membrane thickness (m) 2 -1 Dij Effective diffusion coefficient (m s ) u Electrical potential (V) f Fraction in cluster (–) D Difference (–) F Faraday constant (C mol-1) r Gradient (–) -1 -1 hf Hydration factor (–) j Conductivity (ohm m ) I Current density (A m-2) q Density (g cm-3) J Flux (mol m-2 s-1) l Chemical potential (J mol-1) K Donnan equilibrium constant (–) g Dynamic viscosity (Pa s) -1 3 -2 -1 Kanolyte Mass transfer coefficient in anolyte (m s ) m Convective volume flux (m m s ) N Number of pore channels (–) e Porosity (–) P Pressure (Pa) s Tortuosity (–) r Radius (m) k Actual pore length (m) R Gas constant (J mol-1 K-1) 123 J Appl Electrochem (2017) 47:51–62 53 Superscript and subscript The morphological structure of an ion-exchange mem- act Active brane is also important in modeling the transport process in A Anode the pore volume of the membrane. This is because any Am Anolyte-membrane change in the morphology, i.e., the number and size of c Centigrade liquid pores, can alter the effective diffusivities inside the Cm Catholyte-membrane pores and as a result the transport process. There have been Don Donnan several studies on the morphological structure of ion-ex- e Electrolyte change membranes, particularly the Nafion membrane, i.e., i Species core–shell model proposed by Fujimura et al. [4], a sand- int Interface wich-like model proposed by Haubold et al. [5], and a rod- L Left like model proposed by Rubatat et al. [5]. The cluster- m Membrane network model presented by Mauritz et al. [6] is one of the p Pore earliest models widely used for understanding the proper- R Right ties of Nafion membranes. This model is based on small- s Solution phase angle X-ray scattering (SAXS) measurements and esti- tot Total mates the cluster network of Nafion to consist clusters with w Water a diameter of *4 nm with 1-nm channels connecting the pore clusters. Also, among the early models, Yeager et al. proposed a three-phase model without any strict geometry of the clusters with an interphase between the hydrophobic 1 Introduction and hydrophilic regions [6]. Schmidt et al. simulated par- allel water channel models for the structure of the Nafion Ion-exchange membranes have several industrial applica- membrane with water channel diameters of 1.8 and 3.5 nm tions, including fuel cells, the Chlor–Alkali process, and with an average of 2.4 nm at 20 vol% water [7]. Gebel water electrolysis. In order to explain the mass transfer in et al. have studied the structural evolution of perfluoro- the membrane at high current densities, a suitable mathe- sulfonated ionomer membranes from dry to a highly matical model is required. There are different approaches swollen state with small-angle X-ray scattering (SAXS) to describe the transport of ions inside the membrane. measurement [8]. They characterized the structural evolu- Rohman and Aziz have reviewed mathematical models of tion of Nafion 115 and 117 of 1100 EW. They concluded ion transport in electrodialysis. They have proposed three that as the membrane gets hydrated and swells, the cluster types of phenomenological equations in their irreversible sizes increase. As a result of opening up of the pores, the thermodynamic approach: (1) the Maxwell–Stefan (MS) pore clusters connect. At even higher hydration level, the equation which takes the interaction between each pair of authors observed that the swelling process for water con- components into account; (2) the Kedem–Katchalsky (KK) tent larger than 50 % causes an inversion of the structure equation that considers the membrane as a geometric from a reverse micellar structure to a connected network of transition region between two homogenous compartments; polymer rod-like particles [8]. Based on their assumption and (3) the Nernst–Planck (NP) equation which describes for water swollen state, the cluster radii is 2 nm and it diffusion and electro-migration in the ionic transport increases to 2.09, 2.18, and 2.35 nm for the case of N- without taking into account the interaction between ions. methylformamide, ethanol, and formamide, respectively The latter is widely used because of its simplicity [1]. [8]. Psaltis et al. have compared the Nernst–Planck and Max- The transport of ions in the membrane with the Nernst– well–Stefan approaches to transport predictions of ternary Planck approach has been studied by other authors as well. electrolytes. They have concluded that using binary diffu- Verbrugge et al. [9] have developed ion and solvent sivities (neglecting interaction between different solute transport within a sulfuric acid/perfluorosulfonic acid species) and the full Maxwell–Stefan model does not affect membrane. Bouzek et al. [10, 11] also modeled the ion the final steady-state concentrations profiles in the elec- transport inside the membrane with and without consider- trolyte solution of a multicomponent system.