A generalized Stefan-Maxwell (concentrated-solution) model of coupled hydronium and water transport in the polymer electrolyte membrane of a PEM fuel cell, formulated by Baschuk and Li (2010) with three effective binary diffusion coefficients mapped from experimental data, is restricted to vapour-equilibrated, single-phase operation: the pressure-gradient term of the Stefan-Maxwell driving force is set to zero, and liquid water, phase change, and two-phase transient behaviour in the membrane and catalyst layers are excluded. The open problem is to extend this formulation to the regimes it excludes - liquid-equilibrated membranes under hydraulic pressure differences, and condensation/evaporation and liquid-vapour coexistence in the catalyst layers and membrane - as one unified, thermodynamically consistent model.

Background

A polymer electrolyte membrane (PEM) fuel cell converts hydrogen and oxygen into electricity across a perfluorosulfonic-acid (PFSA) ionomer membrane, of which Nafion is the archetype: a polymer bearing acidic side groups whose hydrated domains conduct protons. Cell performance, durability, and water management depend on how water and protons (present in the membrane as hydronium ions, H3O+\mathrm{H_3O^+}) move through this membrane, because proton conductivity requires hydration, while excess liquid water floods the porous electrodes. The membrane water content λ\lambda (moles of water per mole of sulfonic acid groups) is fixed by the equilibrating environment. Membranes in contact with water vapour take up less water than membranes in contact with liquid water of the same thermodynamic activity; models therefore distinguish vapour-equilibrated (VE) and liquid-equilibrated (LE) regimes, and this uptake anomaly is the long-standing puzzle known as Schroeder's paradox. Water flux through the membrane combines diffusion down the water-activity gradient, electro-osmotic drag (water convected by the proton current), and - when connected liquid pathways exist, as in the LE regime - hydraulic, Darcy-type flow driven by a pressure gradient.

Several model families describe this coupled transport. The semi-empirical Springer model prescribes conductivity, water diffusivity, and electro-osmotic drag as fitted functions of λ\lambda (Springer, Zawodzinski, and Gottesfeld, 1991). The Bernardi-Verbrugge model describes an effectively liquid-filled membrane hydraulically through Schlögl's electrokinetic equation - an extension of the Nernst-Planck flux equations that adds pressure-driven convection of the electrolyte through a charged porous medium - (Bernardi and Verbrugge, 1992). The Weber-Newman model treats VE and LE as distinct transport modes, the LE water flux being a hydraulic flow driven by the liquid pressure gradient with a constant LE conductivity (Weber and Newman, 2004). Concentrated-solution or generalized Stefan-Maxwell (SM) theories take a more fundamental route, originating with Mehta, Morse, Mason, and Daneshpajooh (1976): the mobile species - here water and hydronium in an effectively immobile polymer matrix - interact through binary friction (binary diffusion) coefficients, and the driving force on each species is written in terms of gradients of activity, pressure, and electrostatic potential. The binary friction model of Fimrite, Carnes, Struchtrup, and Djilali (2005) is a member of this family.

Within this last family, Baschuk and Li (2010) derived a two-species generalized SM model of the membrane in closed form. Inverting the SM relations gives the water and hydronium fluxes in terms of the mole-fraction gradient and the electrolyte-potential gradient, with overall diffusion coefficients written as explicit algebraic functions of three effective binary diffusion coefficients - water-hydronium, water-polymer, and hydronium-polymer. These coefficients are obtained from experimental measurements by explicit mapping rules: a Nernst-Einstein relation (which links a species' diffusivity to its electrical conductance) from the equivalent conductance for water-hydronium, a Zawodzinski self-diffusion fit corrected by the activity derivative for water-polymer, and a conductivity fit for hydronium-polymer. With the mapped coefficients the model reproduces the Motupally-Becker-Weidner water-permeation measurements and literature electro-osmotic drag data, and it is coupled to gas-phase water transport in the porous layers as a single computational domain by assuming local equilibrium between the gas phase and the membrane water activity, avoiding internal boundary matching between subdomains.

Two restrictions are built into this formulation. The general SM driving force for species α\alpha in the membrane liquid phase,

dℓ,eα=−xℓ,eα∇ln⁡aℓ,eα−xℓ,eαV^ℓ,eαRTℓ,e∇Pℓ,e−zαxℓ,eαFRTℓ,e∇Φe,d_{\ell,e}^{\alpha}=-x_{\ell,e}^{\alpha}\nabla\ln a_{\ell,e}^{\alpha}-\frac{x_{\ell,e}^{\alpha}\widehat{V}_{\ell,e}^{\alpha}}{\mathcal{R}T_{\ell,e}}\nabla P_{\ell,e}-\frac{z_{\alpha}x_{\ell,e}^{\alpha}\mathcal{F}}{\mathcal{R}T_{\ell,e}}\nabla\Phi_{e},

where xx is mole fraction, aa activity, V^\widehat{V} partial molar volume, Pℓ,eP_{\ell,e} the electrolyte pressure, Φe\Phi_e the electrolyte potential, zz valence, R\mathcal{R} the gas constant, TT temperature, and F\mathcal{F} Faraday's constant, is applied with its pressure-gradient term set to zero - an approximation valid when no hydraulic pressure difference acts across a vapour-equilibrated membrane. And only vapour-equilibrated, single-phase operation is described: there is no free liquid water in the membrane or in the catalyst layers (the porous, ionomer-containing electrodes adjacent to the membrane), no condensation or evaporation, and no transient two-phase behaviour. The operating regimes that dominate flooding and transient water management in fuel cells are therefore outside the model's validity range: the formulation's own scope - vapour-equilibrated operation with the pressure-gradient term inactive, coupled to gas-phase water transport in a single domain - marks the gap to be closed, and the field-level openness of liquid-equilibrated and two-phase membrane transport modelling is documented in the open literature (see Dickinson and Smith, 2020).

Problem Statement

Extend the generalized Stefan-Maxwell model of Baschuk and Li (2010) for coupled hydronium and water transport in the polymer electrolyte membrane - with its three experimentally mapped effective binary diffusion coefficients and its single-domain coupling to gas-phase water transport - to the regimes its formulation excludes, as one unified objective: a thermodynamically consistent formulation valid across vapour-equilibrated and liquid-equilibrated operation for transient water-management simulation. Specifically:

(1) Retain and quantify the pressure-gradient term in the Stefan-Maxwell driving force dℓ,eαd_{\ell,e}^{\alpha}, so that the formulation describes membranes equilibrated with liquid water and membranes subject to hydraulic pressure differences across their thickness, including mixed vapour/liquid-equilibrated operation, with the reduction to the published vapour-equilibrated model explicit when the pressure gradient vanishes.

(2) Incorporate phase change (condensation and evaporation) and the coexistence, permeation, and transport of liquid and vapour water within the catalyst layers and the membrane into the same formulation, in a way that is consistent with non-equilibrium thermodynamics (no double-counting of flux contributions between Stefan-Maxwell friction and added hydraulic terms).

The deliverable is the extended formulation with its parameterization (including re-mapping of the effective binary diffusion coefficients to liquid-equilibrated conditions as needed); the extended formulation and its parameterization are what resolve the task, and a quantitative assessment (analytical or numerical) against available experimental water-transport data exercising the liquid-equilibrated regime, where such data exist, is desirable but not required for resolution.

References

  1. J. J. Baschuk and Xianguo Li, 'Modeling of ion and water transport in the polymer electrolyte membrane of PEM fuel cells', International Journal of Hydrogen Energy 35 (2010) 5095-5103. DOI: 10.1016/j.ijhydene.2009.10.032. https://doi.org/10.1016/j.ijhydene.2009.10.032 (source paper of the model to be extended)
  2. Edmund J. F. Dickinson and Graham Smith, 'Modelling the Proton-Conductive Membrane in Practical Polymer Electrolyte Membrane Fuel Cell (PEMFC) Simulation: A Review', Membranes 10 (2020) 310. DOI: 10.3390/membranes10110310. https://doi.org/10.3390/membranes10110310
  3. A. Z. Weber and John Newman, 'Transport in Polymer-Electrolyte Membranes - II. Mathematical Model', Journal of The Electrochemical Society 151 (2004) A311-A325. DOI: 10.1149/1.1639157. https://doi.org/10.1149/1.1639157
  4. T. E. Springer, T. A. Zawodzinski and S. Gottesfeld, 'Polymer Electrolyte Fuel Cell Model', Journal of The Electrochemical Society 138 (1991) 2334-2342. DOI: 10.1149/1.2085971. https://doi.org/10.1149/1.2085971
  5. Dawn M. Bernardi and Mark W. Verbrugge, 'A Mathematical Model of the Solid-Polymer-Electrolyte Fuel Cell', Journal of The Electrochemical Society 139 (1992) 2477-2491. DOI: 10.1149/1.2221251. https://doi.org/10.1149/1.2221251
  6. G. D. Mehta, T. F. Morse, E. A. Mason and M. H. Daneshpajooh, 'Generalized Nernst-Planck and Stefan-Maxwell equations for membrane transport', The Journal of Chemical Physics 64 (1976) 3917-3923. DOI: 10.1063/1.432021. https://doi.org/10.1063/1.432021
  7. J. Fimrite, B. Carnes, H. Struchtrup and N. Djilali, 'Transport phenomena in polymer electrolyte membranes - II. Binary friction membrane model', Journal of The Electrochemical Society 152 (2005) A1815-A1823. DOI: 10.1149/1.1952647. https://doi.org/10.1149/1.1952647
  8. Sathya Motupally, Aaron J. Becker and John W. Weidner, 'Diffusion of Water in Nafion 115 Membranes', Journal of The Electrochemical Society 147 (2000) 3171. DOI: 10.1149/1.1393879. https://doi.org/10.1149/1.1393879
  9. L. Karpenko-Jereb, P. Innerwinkler, A.-M. Kelterer, C. Sternig, C. Fink, P. Prenninger and R. Tatschl, 'A novel membrane transport model for polymer electrolyte fuel cell simulations', International Journal of Hydrogen Energy 39 (2014) 7077-7088. DOI: 10.1016/j.ijhydene.2014.02.083. https://doi.org/10.1016/j.ijhydene.2014.02.083
  10. Hamidreza Khakdaman, Yves Bourgault and Marten Ternan, 'A Mathematical Model of a Direct Propane Fuel Cell', Journal of Chemistry (2015), article 102313. DOI: 10.1155/2015/102313. https://doi.org/10.1155/2015/102313
  11. Jian Zhao, Xianguo Li, Chris Shum and John McPhee, 'A Review of physics-based and data-driven models for real-time control of polymer electrolyte membrane fuel cells', Energy and AI 6 (2021) 100114. DOI: 10.1016/j.egyai.2021.100114. https://doi.org/10.1016/j.egyai.2021.100114