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Advances in Mechanical Engineering
Volume 2010 (2010), Article ID 372795, 16 pages
http://dx.doi.org/10.1155/2010/372795
Research Article

A Continuum Model for Water Transport in the Ionomer-Phase of Catalyst Coated Membranes for PEMFCs

Chemical Engineering Department, Case Western Reserve University, Cleveland, OH 44106, USA

Received 19 August 2009; Revised 24 November 2009; Accepted 28 December 2009

Academic Editor: Chen Li

Copyright © 2010 Vladimir Gurau and J. Adin Mann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the problem of water transport in the ionomer-phase of catalyst coated membranes (CCMs) for proton exchange membrane fuel cells (PEMFCs), where microscopic-scale phenomena at the distributed interfaces between structural components control the water management. Existing models for water transport in CCMs describe the transport in systems which consist exclusively of an ionomer-phase. Interfacial water fluxes across distributed interfaces representing various mechanisms of water transfer between ionomer and catalyst layer pores are not captured properly in these models. Here we develop a continuum model for water transport in CCMs using the method of volume averaging. Water is exchanged between ionomer and the catalyst layer pores by electro-osmotic discharge (EOD) through the three-phase boundary (TPB) regions and by sorption and desorption across the ionomer-pore interfaces. While the former mechanism does not affect directly the water content in the ionomer-phase, it represents an effective mechanism for water transfer during fuel cell operation and controls directly the water saturation in the catalyst pores.

1. Introduction

Catalyst layers for proton exchange membrane fuel cells (PEMFCs) are heterogeneous porous structures consisting of interpenetrating phases for the transport of electrical charges, electroactive gaseous species, and water. During fuel cell operation, water is transferred between the catalyst layer pores and ionomer by parallel mechanisms. These phenomena control the water management and must be captured in models for water transport in the catalyst layer components. The macroscopic equation for water transport in catalyst coated membranes (CCMs) has been frequently derived for use in computational fluid dynamics (CFDs) simulations, by performing a water flux balance over a control volume representing 100% ionomer-phase. The resulting equation is thus valid solely for the water transport in proton exchange membranes (PEMs) and does not capture the interfacial transport phenomena at subgrid scale which are specific for catalyst layers. The consequence is that some of the mechanisms for water transfer between ionomer and catalyst layer pores have not been conceptualized and quantified until recently. Moreover, there seems to have been a lack of consensus in the interpretation of the macroscopic equation for water transport in CCMs. The reader is referred to [1, Section 7] of the critical overview of CFD multiphase models for PEMFCs [1]. In the previous modeling works, interfacial transport phenomena across the ionomer boundaries have been interpreted as mechanisms of water transport along the ionomer-phase, or vice versa. This had a strong impact on the prediction of the water content in the ionomer-phase of CCMs and on the liquid saturation in the diffusion media. The misinterpretation of the effect that various terms have on the prediction of water accumulation in ionomer may be attributed in part to the lack in the literature of a formal derivation of a continuum model for water transport in CCMs.

An objective of this paper is to present a continuum model for water transport in the ionomer-phase of catalyst layers for PEMFCs, starting from the point equations for conservation of charge and for water concentration in ionomer-phase. Another objective is to emphasize various mechanisms of water transfer between the ionomer-phase and its surroundings, some of which have not been conceptualized and quantified until recently [24]. Another objective is to identify those mechanisms that affect water accumulation in the ionomer-phase of the catalyst layer.

The study of water transport in the ionomer-phase of catalyst layers for PEMFCs requires the consideration of a hierarchy of length scales over which significant changes in the parameters related to structure, processes, and operating conditions occur. Figure 1 illustrates the hierarchical structure of the catalyst layer system representative of PEMFCs and some of its characteristic length scales are suggested.

372795.fig.001
Figure 1: Hierarchical structure for catalyst layer system.

Level I represents a CCM consisting of a PEM impermeable to gases and two catalyst layers bounded on each of its sides. The characteristic length scale associated with this level is the catalyst layer thickness (10𝜇m), which is the distance over which significant changes in current density, water content, and concentrations of electroactive species occur. This is the macroscale at which the fuel cell operation is modeled and the parameters associated with this level, such as current density, water concentration, or water flux represent macroscopic, averaged quantities.

Level II evokes that catalyst layers are porous structures with a random composite morphology, consisting of a mixture of carbon-supported Platinum (C/Pt) and solid ionomer. They provide continuous paths for the protonated complexes to reach the catalyst sites through the ionomer network, for the electroactive species through the porous network and for the electrons through the C/Pt matrix. Level II defines the microscale at which the principal microstructural features of the catalyst layer morphology are described. The C/Pt matrix consists of carbon grains (20–40 nm) which form agglomerates of 200–400 nm size. Their porous structure is characterized by a bi-modal pore size distribution, with primary pores of 20–40 nm inside the agglomerates and secondary pores of 40–400 nm between the agglomerates. The size of the catalyst particles dispersed on the carbon grains is 2–5 nm. Ionomer, due to its molecular size, cannot penetrate the intra-agglomerates, but exist only in the interagglomerate space [5]. Since the electrochemical reactions take place at the three-phase boundary between the gas pores, ionomer and catalyst sites, only the Pt particles in the interagglomerate pores are electrochemically active. During the catalyst fabrication process, the ionomer organizes itself into clusters with diameters of a few hundred nm [6, 7]. The length scale associated with this level is a representative configuration length such as diameter of the C/Pt agglomerate or diameter of the ionomer cluster.

Level III illustrates the ionomer details at nanometer scale. The most widely investigated solid ionomers, the perfluorinated sulfonic acid polymers (e.g., Nafion) are heterogeneous media. They consist of a hydrophobic polytetrafluoroethylene framework with pendant side chains ending in sulfonic acid groups SO3H. When exposed to water, the acid groups dissociate and release the protons H+ into the aqueous subphase while the remaining sulfonate groups SO3 hydrate. Perfluorinated sulfonic acid polymers separate in hydrophobic and hydrophilic domains. At nanometer-scale, their morphology consists of hydrophobic elongated domains that confer the polymer its morphological stability, and hydrated hydrophilic domains that consist of immobilized, negatively charged sulfonate groups SO3, water molecules, and mobile proton-bearing complexes. Even at low hydration levels, the hydrophilic domains define a well connected network of nanochannels for proton conduction and water transport. Their diameters range between 1 to 4 nm depending on water content 𝜆, defined as the number of water molecules per sulfonate group. One distinguishes two water environments in the hydrophilic region: (i) the first 6 water molecules per acid group (𝜆=6) representing the hydration water are strongly bound to the SO3 sites at the inner surface of the hydrophilic domains and can be only removed under vacuum conditions at high temperatures; (ii) additional water (𝜆>6) that fills the volume of the hydrophilic domains behaves like bulk water and is free to equilibrate with the water in the catalyst layer pores, outside the ionomer.

The nanometer-size regions at the interface between ionomer, C/Pt and catalyst layer pores filled with electroactive gaseous species are referred to as three-phase boundary (TPB) regions. These are electrochemically active regions where the hydrogen oxidation reaction (HOR) or the oxygen reduction reaction (ORR) takes place. These regions are dynamic and their behavior depends on the hydration state of the ionomer. At the interface, the SO3 -ending side chains of the ionomer structure unfold towards the Pt particles as the water content in the ionomer increases and the interfacial region becomes hydrophilic [8]. At the cathode, the TPB regions contain (i) protonated complexes emerging from the hydrophilic domains of the ionomer and engaging in the ORR at the Pt surface, (ii) water molecules introduced in the interfacial region by the protonated complexes via electro-osmotic drag, (iii) water molecules produced in the ORR and desorbed from the Pt surface, and (iv) oxygen molecules diffusing from the catalyst layer pores and engaging in ORR at the Pt surface.

At the nanometer-scale level, proton transfer is interpreted as a combination of proton transport in bulk-like water and interfacial interactions with the SO3 groups at the polymer-water interface [9]. Membranes at a high degree of hydration exhibit proton transport resembling that in bulk water [10]. In water, excess protons H+ hydrate to form hydronium ions H3O+, which in turn are hydrogen bonded to the three water molecules in its first solvation shell and form Eigen clusters H9O+4. Fluctuations between Eigen and Zundel (H5O+2) complexes generate a transport mechanism for protons along the hydrogen bond network called the Grotthus mechanism or structure diffusion [10]. When an electrical field is applied, the structure diffusion relays the protons in the direction of the field along the hydrogen bond network without inducing a concerted transport of the water molecules. In addition to this mechanism, the electrical field exerts electrokinetic body forces on the protonated complexes, superposing a hydrodynamic motion of the protons over the structure diffusion. This transport is called the vehicle mechanism. The proton flux caused by structure diffusion and vehicle mechanism 𝑁H+=𝑁structdiH++𝑁vehmechH+ generates a current density along the ionomer from anode to cathode 𝑖=𝑁H+𝐹.

The distribution of excess protons relative to the immobile SO3 groups is the result of electrostatic attractive interactions between the counterions and structural inhomogeneities in the vicinity of the electrified interface. The decreased dielectric constant in the interfacial region determines a stabilization of excess protons away from the fixed sulfonate groups, towards the center of the channels [11]. Since water within the hydrophilic regions of the ionomer and liquid water outside the ionomer forms a continuous medium, it has been postulated [12] that excess protons could leave the ionomer and migrate throughout the liquid water in the catalyst layer pores, towards the Pt sites in the C/Pt intra-agglomerates. These are regions where ionomer, due its molecular size cannot penetrate. This is an intriguing hypothesis, since it leads to the conclusion that all the Pt sites, including those in the TPB regions and those in the intra-agglomerates are electrochemically active. This hypothesis however, breaches the condition of electroneutrality. In liquid water, one expects a total depletion of protons within 1-2 nm away from the most peripheral array of sulfonate groups [11]. We will consider that protons may exit the hydrophilic domains of the ionomer into the adjacent liquid water, but will continue to migrate within a few nanometers away from the most peripheral array of SO3 groups towards the TPB regions. This thin layer of liquid water (1-2 nm thick) may be assimilated to the hydrophilic domains and may be considered to be integral part of the ionomer.

In their motion generated by the vehicle mechanism, excess protons carry the water molecules in their solvation shells as well as other water molecules due to viscous interactions, generating a water transport mechanism called electro-osmotic drag. The electro-osmotic drag coefficient 𝑛drag is defined as the ratio of the water flux 𝑁dragH2O through the ionomer to the flux of protons 𝑁H+ when current is passed under conditions of no water concentration gradient. The flux of water induced by electro-osmotic drag increases the water concentration along the ionomer from anode to cathode. This effect is balanced by the chemical diffusion of water, 𝑁diH2O=𝐷H2O𝑐H2O, which acts in the direction from regions with higher water concentration towards regions with lower water concentration.

Whereas at nanometer scale (Level III) the ionomer represents a heterogeneous system with the characteristic length of only a few nanometers, the average size of the ionomer clusters (100 nm) is large enough so that at the microscale (Level II) the ionomer-phase may be considered homogeneous. The consequence of this assumption is that one may prescribe microscopic, point equations for the transport of protons and water in the ionomer-phase. This is considered a trivial practice when modeling transport phenomena in PEM systems consisting of 100% ionomer-phase, in which case the microscale (Level II) and macro scale (Level I) are indistinguishable. In the case of catalyst layers, the microscopic point equations governing the transport in the ionomer-phase at micro scale (Level II) must be solved along with the boundary conditions describing various mechanisms of water and proton transport across ionomer-TPB and ionomer-catalyst layer pore interfaces. However, this direct analysis is impractical due to the complex morphology of the catalyst. The practical solution is to apply the method of volume averaging [1318] for the derivation of continuum transport equations that are valid anywhere in the catalyst layer. These transport equations, called volume-averaged equations, may be solved at the macro scale (Level I) and allow the consideration of the catalyst layer as a macro-homogeneous domain. The volume-averaged equations retain the information regarding interfacial transport across ionomer-phase boundaries.

In the following sections we will present the local, microscopic equations governing proton and water transport in ionomer-phase (Level II) along with their boundary conditions and we will derive the volume-averaged equations for conservation of charge and water transport in catalyst layers (Level I). The scope of this analysis is to emphasize various mechanisms of water transfer between the ionomer-phase and its surroundings, some of which have not been conceptualized and quantified until very recently. For this analysis we consider the following assumptions.

(1)At the micro scale (Level II) the ionomer-phase is considered a homogeneous medium for the transport of water and proton-bearing complexes.(2)The TPB regions are envisioned as surfaces of discontinuity between the ionomer-phase and electrically conductive solid phase (C/Pt), rather than lines of discontinuity between ionomer, C/Pt and catalyst layer pores filled with electroactive species. Indeed, the diffusion coefficient of oxygen is sufficiently large, so that oxygen can diffuse efficiently through the ionomer towards the Pt sites up to 200 nm [19] (compared to 1 nm interface thickness) and the reaction rate is not controlled by mass transport limitations in the Nafion coating. Furthermore, this interface is considered immaterial and therefore unable of accumulating mass and charge, namely, 𝜕𝑐H+,𝑖𝑠/𝜕𝑡=0,𝜕𝑐H2O,𝑖𝑠/𝜕𝑡=0.(3)Protons may enter and exit the ionomer-phase only when they cross the TPB regions (ionomer/electrically-conductive solid interfaces, labeled is) as they participate in electrochemical reactions. In order to satisfy ionomer-phase electroneutrality, protonated complexes do not leave the ionomer across the interfaces with the liquid water in the catalyst layer pores and thus are unable to migrate towards the Pt sites in the intra-agglomerate space. The liquid water layer 1-2 nm thick adjacent to the ionomer-phase and containing protonated complexes is assimilated to the hydrophilic domains and is considered to be integral part of the ionomer. Therefore, the proton flux across the ionomer-fluid interfaces, labeled if, is zero, 𝑐H+,𝑖𝑣H+,𝑖𝑛𝑖𝑓=0.(4)The C/Pt agglomerates are fixed solid structures. Their interface with the ionomer is fixed once the ionomer has been hydrated and the velocity of this interface is zero, 𝑤𝑖𝑠=0. As a consequence, the ionomer-phase may swell or contract as a function of hydration only across the interfaces with the fluids in the catalyst layer pores (if-interfaces). Even if we relax this constraint, the velocity of the ionomer-solid interface 𝑤is is negligibly small compared to the proton mobility 𝑣H+,𝑖 or velocity of water molecules 𝑣H2O,𝑖.

2. Microscopic Equations for Conservation of Charge and Water Transport in the Ionomer-Phase

The catalyst layer system under consideration is a porous mixture with a solid matrix consisting of an electrically conductive phase-s (carbon-supported Pt) and a proton-conductive phase-i (ionomer) available for protons and water transport. The primary and secondary pores of the C/Pt are filled with gaseous electroactive species and liquid water. For the derivation of the governing equations for the concentration of protons and water in ionomer we will be focusing on the transport phenomena in the ionomer and will not make distinction between the fluid phases-f in the primary and secondary pores of the carbon-supported Pt.

2.1. Microscopic Equation for Conservation of Charge in Ionomer-Phase

The microscopic equation which governs the proton transport in ionomer-phase (equation for conservation of charge) can be expressed as

𝜕𝑐H+,𝑖𝜕𝑡+𝑐H+,𝑖𝑣H+,𝑖=0(1)

in which the second subscript i indicates that the variables are related to the ionomer phase. At the TPB regions (is-interface), the boundary condition associated with (1) is

𝜕𝑐H+,𝑖𝑠𝜕𝑡+𝑠𝑐H+,𝑖𝑠𝑣H+,𝑖𝑠=𝑐H+,𝑖𝑣H+,𝑖𝑤𝑖𝑠𝑛𝑖𝑠+̇𝑐H+,𝑖𝑠(B.C.1)

in which 𝑠 represents the surface gradient operator along the is-interface and 𝑛𝑖𝑠 is the unit vector normal to the is-interface and pointing outwards from the ionomer-phase. The terms in the left-hand side (LHS) represent proton accumulation and proton transport in the TPB region. The terms in the right-hand side (RHS) represent the interfacial molar flux of protons between the bulk ionomer and the TPB region and the molar rate of proton production in HOR at anode or proton consumption in ORR at cathode. Considering the second and the fourth assumptions and considering the proton flux within the TPB region constant (𝑐H+,𝑖𝑠𝑣H+,𝑖𝑠=const), the resulting boundary condition at the TPB regions reduces to the following expression:

𝑐H+,𝑖𝑣H+,𝑖𝑛𝑖𝑠+̇𝑐H+,𝑖𝑠=0,(B.C.1a)

which states that the proton flux between the ionomer and the TPB region is balanced by the rate of proton production at anode during HOR or the rate proton consumption at cathode during ORR.

2.2. Microscopic Equation for Water Transport in Ionomer-Phase

The microscopic equation that governs the water transport in ionomer is

𝜕𝑐H2O,𝑖𝜕𝑡+𝑐H2O,𝑖𝑣H2O,𝑖=0.(2) The local velocity of water may be decomposed into an average hydrodynamic fluid velocity and a diffusion velocity

𝑣H2O,𝑖=𝑣(H2O/H+),𝑖+𝑢H2O,𝑖,(3) such that the total water flux 𝑁H2O,𝑖=𝑐H2O,𝑖𝑣H2O,𝑖 is the sum of the electro-osmotic drag flux 𝑁dragH2O,𝑖=𝑐H2O,𝑖𝑣(H2O/H+),𝑖 and the diffusion flux 𝑁diH2O,𝑖=𝑐H2O,𝑖𝑢H2O,𝑖.

Considering that TPB regions do not accumulate mass and the velocity of the TPB regions is zero (second and fourth assumptions) and considering the water flux within the TPB region to be constant, the boundary condition associated to (2) at the TPB regions reduces to the following form:

𝑐H2O,𝑖𝑠𝑣H2O,𝑖𝑠𝑛𝑖𝑠𝑓=𝑐H2O,𝑖𝑣H2O,𝑖𝑛𝑖𝑠+̇𝑐H2O,𝑖𝑠,(B.C.2)

which states that the water entering the TPB region from the ionomer-phase and the water produced in the ORR exits the TPB region into the catalyst layer pores (Figure 2).

372795.fig.002
Figure 2: Water fluxes between the TPB regions (is-Interface), ionomer (i), and catalyst layer pores (f).

Unlike protons, water may be transferred by sorption or desorption across the if-interfaces between the ionomer and the catalyst layer pores. At these interfaces, the boundary condition associated to (2) is

𝑐H2O,𝑖𝑣H2O,𝑖𝑤𝑖𝑓𝑛𝑖𝑓=𝑐H2O,𝑓𝑣H2O,𝑓𝑤𝑖𝑓𝑛𝑖𝑓.(B.C.3)

3. Volume-Averaged Equations forConservation of Charge and Water Transport in CCMs

The volume-averaged transport equations are obtained by application of the averaging theorems to the local, microscopic transport equations. An averaging control volume (Figure 3) has volume Vol fixed in space and time surrounded by an enveloping surface of area 𝐴. Its size 3Vol must be larger than the length scale associated with Level II, which may be the diameter of the C/Pt agglomerates or diameter of the ionomer clusters, but smaller compared to the computational subdomain (length scale associated to Level I). The averaging control volume must include all catalyst layer ingredients, including ionomer-phase, electrically conductive C-Pt agglomerates with fluids in its primary and secondary pores and the TPB regions defined by their contiguity.

372795.fig.003
Figure 3: Averaging volume.

The ionomer-phase within the averaging control volume has volume Vol𝑖 surrounded by the interfacial area 𝐴𝑖 with the following constituents: ionomer-C/Pt interfacial area 𝐴𝑖𝑠, ionomer-fluid interfacial area 𝐴𝑖𝑓 and the control volume entrances and exits crossing the ionomer 𝐴𝑖𝑒 (Figure 3). The volume fraction of ionomer within an averaging control volume is defined as 𝜀𝑖=Vol𝑖/Vol. The volume Vol𝑖 occupied by ionomer in an averaging control volume may change in time when the hydrated hydrophilic domains expand or contract as a function of hydration level. The ratio between area 𝐴𝑖𝑒 available for protons and water to enter or exit the control volume and the total enveloping surface area 𝐴, 𝜀𝐴=𝐴𝑖𝑒/𝐴, is called directional surface permeability tensor. For a property 𝜓𝑖 which can be a scalar or a tensor associated with the ionomer-phase are defined the local volume average 3𝜓𝑖=1/VolVol𝑖𝜓𝑖𝑑𝑣, the intrinsic volume average 3𝑖𝜓𝑖=1/Vol𝑖Vol𝑖𝜓𝑖𝑑𝑣, and the intrinsic surface average 2𝑖𝜓𝑖=1/𝐴𝑖𝑒𝐴𝑖𝑒𝜓𝑖𝑛𝑖𝑑𝐴. The local volume and the intrinsic volume averages satisfy the relation

3𝜓𝑖=𝜀𝑖3𝑖𝜓𝑖.(4) The following averaging theorems that relate the averages of the space and time derivatives of property 𝜓𝑖 to the space and time derivatives of the average of 𝜓𝑖 have been proposed [13, 14]

3𝜓𝑖=3𝜓𝑖+1Vol𝐴𝑖𝑓+𝐴𝑖𝑠𝜓𝑖𝑛𝑖𝑑𝐴,(5a)3𝜓𝑖=3𝜓𝑖+1Vol𝐴𝑖𝑓+𝐴𝑖𝑠𝜓𝑖𝑛𝑖𝑑𝐴,(5b)3𝜕𝜓𝑖𝜕𝑡=𝜕3𝜓𝑖𝜕𝑡1Vol𝐴𝑖𝑓+𝐴𝑖𝑠𝜓𝑖𝑤𝑖𝑛𝑖𝑑𝐴(5c)

in which 𝑛𝑖 is the unit outward vector normal to the interfaces surrounding the ionomer and 𝑤𝑖 is the velocity vector of the ionomer-phase interfaces within the averaging volume. It can be shown (see, e.g., [17]) that

3𝜓𝑖=𝜀𝐴2𝑖𝜓𝑖.(6)

3.1. Volume-Averaged Equation for Conservation of Charge in CCMs

Application of the averaging theorems (5b) and (5c) to the microscopic transport equation (1) yields the volume-averaged equation for conservation of charge

𝜕3𝑐H+,𝑖𝜕𝑡+3𝑐H+,𝑖𝑣H+,𝑖+1Vol𝐴𝑖𝑓+𝐴𝑖𝑠𝑐H+,𝑖𝑣H+,𝑖𝑤𝑖𝑛𝑖𝑑𝐴=0.(7) We note first that 𝜕/𝜕𝑡3𝑐H+,𝑖=0 since due to electroneutrality, the volume-averaged proton concentration is constant and equal to the concentration of the immobile negatively charged sulfonate groups SO3. The last term in (7), which emerges as a consequence of employing the averaging theorems, represents the interfacial flux of protons across the ionomer boundaries laying inside the averaging volume. According to the third assumption, the flux of protons across the if-interface is zero and the area of integration reduces to 𝐴𝑖𝑠. Since the protons crossing the is-interface converge towards the Pt sites (Figure 4), the area of integration 𝐴𝑖𝑠 reduces further to the total catalyst area 𝐴Pt in the averaging volume. According to the fourth assumption 𝑤𝑖𝑠=0 and the term reduces to 1/Vol𝐴Pt𝑐𝐻+,𝑖𝑣𝐻+,𝑖𝑛𝑖𝑠𝑑𝐴. Considering relation (6), (7) becomes

372795.fig.004
Figure 4: The proton flux across the ionomer/solid electrically conductive phase (C/Pt) interface 𝐴𝑖𝑠.

𝜀𝐴2𝑖𝑐H+,𝑖𝑣H+,𝑖+1Vol𝐴Pt𝑐H+,𝑖𝑣𝐻+,𝑖𝑛𝑖𝑠𝑑𝐴=0(8) in which 2𝑖𝑐H+,𝑖𝑣H+,𝑖=2𝑖𝑁H+,𝑖 represents the macroscopic flux of protons along the CCM (macro scale) and 𝑐H+,𝑖𝑣H+,𝑖𝑛𝑖𝑠=𝑁H+,𝑖𝑛𝑖𝑠 represents the local interfacial flux of protons crossing the TPB regions (nanometer scale) when protons engage in electrochemical reactions.

The volume-averaged equation for conservation of charge (8) states that:

“The change in proton flux across a control volume (averaging volume) is equal to the total interfacial flux of protons across the TPBs distributed in the control volume, when protons engage in electrochemical reactions”.

The total interfacial flux of protons across the TPBs within the averaging volume 1/Vol𝐴Pt𝑐H+,𝑖𝑣H+,𝑖𝑛is𝑑𝐴 is a subgrid-scale phenomenon. It represents a distributed source of protons at anode and a distributed sink of protons at cathode. In a membrane this term is zero since 𝐴Pt=0.

Considering the boundary condition (B.C.1a) we can write

1Vol𝐴Pt𝑐H+,𝑖𝑣H+,𝑖𝑛𝑖𝑠𝑑𝐴=1Vol𝐴Pṫ𝑐H+,𝑖𝑠𝑑𝐴=𝐴PtVolPṫ𝑐H+,𝑖𝑠,(9) where the operator Pt defines a surface average over the effective Pt surface area Pt𝜓𝑖𝑠=1/𝐴Pt𝐴Pt𝜓𝑖𝑠𝑑𝐴 and 𝐴Pt/Vol  [m2/m3] represents the effective catalyst layer surface area (within the TPB regions) per unit geometric volume. Pṫ𝑐H+,𝑖𝑠 represents the averaged rate of proton production during HOR or proton consumption during ORR.

The volume-averaged equation for conservation of charge can now be written as

𝜀𝐴2𝑖𝑁H+,𝑖=𝐴PtVolPṫ𝑐H+,𝑖𝑠,(10) which states that:

“The change in proton flux across a control volume (averaging volume) is equal to the volumetric rate of proton production during HOR or proton consumption during ORR”.

Multiplying(8)by the Faraday constant we obtain the equation for current density

𝜀𝐴2𝑖𝑖+1Vol𝐴Pt𝑖𝑛is𝑑𝐴=0(11) in which 2𝑖𝑖=2𝑖𝑁H+,𝑖𝑖𝑘𝐹 is the macroscopic, longitudinal current density representing the macroscopic flux of protons along the ionomer, throughout the anode catalyst layer, membrane and cathode catalyst layer and 𝑖𝑘=(𝑖𝑥,𝑖𝑦,𝑖𝑧) are the versors of the macroscopic coordinate system (Figure 1). The 𝑖=𝑁H+,𝑖𝑛𝑖𝑠𝐹 is the local, transversal current density generated by the protons that engage in electrochemical reactions. The local transversal component of the current density defines the charge transfer current 𝑗=(1/Vol)𝐴Pt𝑁H+,𝑖𝑛𝑖𝑠𝐹𝑑𝐴  [A/cm3] [1] and the equation for current density may be written as

𝜀𝐴2𝑖𝑖+𝑗=0.(12)

3.2. Volume-Averaged Equation for Water Transport in the Ionomer-Phase of CCMs

Application of the averaging theorems (5b) and (5c) to the microscopic equation for water transport in ionomer (2) and using relations (4) and (6) yield the volume-averaged equation for water transport in CCM

𝜕𝜀𝑖3𝑖𝑐H2O,𝑖𝜕𝑡+𝜀𝐴2𝑖𝑁dragH2O,𝑖+𝜀𝐴2𝑖𝑁diH2O,𝑖+1Vol𝐴𝑖𝑓+𝐴𝑖𝑠𝑐H2O,𝑖𝑣H2O,𝑖𝑤𝑖𝑛𝑖𝑑𝐴=0(13) in which the total water flux was written as the sum between an electro-osmotic drag flux and a diffusion flux (Section 2.2).

The electro-osmotic drag coefficient is obtained experimentally [20] by measuring the current density (total proton flux) and the water flux throughout a PEM (𝜀𝐴=1) and is defined as

𝑛drag=𝑁dragH2O,𝑖𝑁H+,𝑖.(14) Using (8), (11), and (14), the second term in (13) may be further written as

𝜀𝐴𝑛drag2𝑖𝑁H+,𝑖=𝑛drag𝜀𝐴2𝑖𝑁H+,𝑖+𝜀𝐴2𝑖𝑁H+,𝑖𝑛drag=𝑛drag𝜀𝐴2𝑖𝑖𝐹+𝜀𝐴2𝑖𝑖𝐹𝑛drag.(15) The term 𝑛drag(𝜀𝐴2𝑖𝑖)/𝐹 represents the volumetric rate at which water accumulates in the ionomer-phase of a control volume as a result of the difference between the incoming and outgoing proton fluxes. In cathode catalyst layers, the proton flux 2𝑖𝑁H+,𝑖 decreases in the direction of the current as protons are consumed in the ORR at a rate of (𝜀𝐴2𝑖𝑖)/𝐹(8).In this case, the water flux generated by electro-osmotic drag will decrease as well, the difference between the outgoing and incoming fluxes representing the water accumulated in the ionomer-phase of a control volume. In anode catalyst layers the proton flux increases in the direction of the current as protons are produced in the HOR and the water flux generated by electro-osmotic drag will increase as well. In this case, water in ionomer will be depleted.

The second term, 𝑛drag𝜀𝐴2𝑖𝑖/𝐹, represents the volumetric rate at which water accumulates locally in the ionomer-phase of a control volume when the hydrodynamic motion of the protons occurs along a water concentration gradient, or equivalently, along a gradient of hydrophilic domain sizes. This term is the macroscopic representation of the phenomenon called “stripping off” the water molecules in the peripheral solvation shells of the protonated complexes when the latter are transported from larger hydrophilic domains (higher water content) to smaller hydrophilic domains (lower water content) [1]. When the hydrodynamic motion of the protons is in the opposite direction, from smaller to larger hydrophilic domains (lower to higher water contents), it describes a local depletion of water. In this case, the protonated complexes engage more water molecules in the hydrodynamic motion as the number of molecules in their solvation shells increases.

The third term in (13) represents water transport by diffusion and is equal to (𝜀𝐴𝐷H2O,𝑖2𝑖𝑐H2O,𝑖).

The last term in (13) representing the fluxes of water across the interface with the fluids in the catalyst layer pores and across the TPB regions may be deconvolved as

1Vol𝐴𝑖𝑓+𝐴𝑖𝑠𝑐H2O,𝑖𝑣H2O,𝑖𝑤𝑖𝑛𝑖𝑑𝐴=1Vol𝐴𝑖𝑓𝑐H2O,𝑖𝑢H2O,𝑖𝑤𝑖𝑓𝑛𝑖𝑓𝑑𝐴+1Vol𝐴𝑖𝑠𝑐H2O,𝑖𝑢H2O,𝑖𝑛𝑖𝑠𝑑𝐴+1Vol𝐴Pt𝑐H2O,𝑖𝑣(H2O/H+),𝑖𝑛𝑖𝑠𝑑𝐴.(16) The first term in the RHS represents the water exchanged between ionomer and catalyst layer pores by sorption and desorption. The second term in the RHS represents the water exchanged between ionomer and TPB regions by diffusion at nonequilibrium and is equal to

1Vol𝐴𝑖𝑠𝐷H2O𝜇H2O𝑛𝑖𝑠𝑑𝐴.(17) The last term represents the volumetric rate at which water molecules in the solvation shells of the protonated complexes are dragged in or out of the ionomer when they cross the TPB as they participate in the HOR or ORR. This transfer mechanism was called “electro-osmotic discharge” (EOD) [1]. Using (14) and (11), this term is equal to

1Vol𝐴Pt𝑐H2O,𝑖𝑣(H2O/H+),𝑖𝑛𝑖𝑠𝑑𝐴=𝑛drag1Vol𝐴Pt𝑐H+,𝑖𝑣H+,𝑖𝑛𝑖𝑠𝑑𝐴=𝑛drag𝜀𝐴2𝑖𝑖𝐹,(18) and therefore it cancels with the first term in the RHS of (15). Indeed, the water that would accumulate in the ionomer-phase of a control volume at cathode as a result of the difference between the incoming and outgoing electro-osmotic drag fluxes is removed from the ionomer by electro-osmotic discharge.

Inserting expressions (15)–(18) in (13) and considering the relation between molar concentration of water in ionomer and water content 𝑐H2O,𝑖=𝜆𝜌dry/EW, one obtains the conservation equation for water content in CCMs

𝜌dryEW𝜕𝜀𝑖3𝑖𝜆𝜕𝑡+𝜀𝐴2𝑖𝑖𝐹𝑛drag𝜌dryEW𝜀𝐴𝐷H2O,𝑖2𝑖𝜆+1Vol𝐴𝑖𝑓𝑐H2O,𝑓𝑣H2O,𝑓𝑤𝑖𝑓𝑛𝑖𝑓𝑑𝐴1Vol𝐴𝑖𝑠𝐷H2O𝜇H2O𝑛𝑖𝑠𝑑𝐴=0.(19) The integral terms in (19) which are subgrid-scale phenomena represent distributed sources or sinks for water in ionomer that are zero in a PEM. If water does not accumulate in the TPB regions, it will be at equilibrium with the adjacent water in ionomer and the last term in (19) is zero. The volume-averaged equation for water transport in the ionomer-phase of CCMs may be finally written as

𝜌dryEW𝜕(𝜀𝑖𝜆)𝜕𝑡bulkaccumulation+𝜀𝐴𝑛drag𝑖𝐹``strippingo''waterfromperipheralsolvationshells𝜌dryEW𝜀𝐴𝐷H2O,𝑖𝜆diusion+𝑆𝜆sorption/desorptionwaterexchangedbysorption/desorption=0,(20) where for the sake of simple notations the averaging symbols were dropped. 𝑆𝜆sorption/desorption represents the volumetric rate of water exchanged between ionomer and catalyst layer pores by sorption and desorption and must be determined experimentally. Note that even though the EOD term 𝑛drag(𝜀𝐴𝑖)/𝐹 does not appear explicitly in the transport equation (20), it does represent an effective mechanism of water transfer between ionomer and catalyst layer pores during fuel cell operation. Indeed, boundary condition (B.C.2) indicates that EOD water and ORR water exit the TPB regions into the catalyst layer pores (see Figure 2). EOD and ORR water represent distributed sources for the water in catalyst layer pores and must appear explicitly in its transport equation. This transfer mechanism may be accessed experimentally using an electrochemical cell with Pd-H electrodes [21]. Zawodzinski et al. [21] performed electro-osmotic drag measurements across membranes exposed on both sides to deionized water and compressed between two Pd-H electrodes. Current was passed between the electrodes, thus passing protons through the membrane from anode to cathode. The amount of water dragged across the membrane with the protons was determined by measuring the change in height of water capillary columns. Since the membrane was equilibrated at both sides with liquid water, it results that the mechanism responsible for passing water between the ionomer and the chambers filled with liquid water was not sorption/desorption, but EOD.

In PEMs, 𝜀𝑖=𝜀𝐴=1, 𝑆𝜆sorption/desorption=0 and the volume-averaged equation for water transport becomes

𝜌dryEW𝜕𝜆𝜕𝑡+𝑛drag𝑖𝐹𝜌dryEW𝐷H2O,𝑖𝜆=0.(21) In this case, the term standing for EOD is zero, since in PEMs 𝑖=0.

4. Calculations

Equations (12) and (20) for current density and water content in CCM were solved along with the volume averaged-equations for the conservation of mass, species, and momentum in the PEMFC components using ANSYS-CFX 4.4 software. The 3-dimensional computational domain consisting of an anode channel, anode gas diffusion layer (GDL), CCM, cathode GDL, and cathode channel is illustrated in Figure 5. It extends from the channel inlets to channel outlets and from the symmetry plane along the channels to the symmetry plane between two adjacent channels.

372795.fig.005
Figure 5: Computational domain and mesh.

The transport equations of the isothermal, single-phase model are presented in Appendix A, where for the sake of simple notations the averaging symbols were dropped. The reader may turn to [1] for a detailed presentation of the transport equations and their constitutive relations. The model considers that hydrogen and air enter the channels fully saturated. In the anode flow field, hydrogen is consumed in the HOR (A.17) and (C.27) and water is exchanged between the catalyst layer pores and ionomer by EOD (water leaves the pores) and by sorption/desorption at nonequilibrium equation (A.18), (C.28), and (C.30). In the cathode flow field, oxygen is consumed in the ORR (A.15) and (C.26) and water is produced in the ORR and is exchanged between the catalyst layer pores and ionomer by EOD (water enters the pores) and by sorption/desorption at nonequilibrium equations (A.16), (C.25), (C.28), and (C.29).

Figure 6 illustrates the current density distribution in the membrane for the case of an operating (average) current density of 0.46 A/cm2 and a cell voltage of 0.7 V. As hydrogen and oxygen are consumed, their concentrations decrease along the channels. Since the rates of HOR and ORR depend on their concentrations (C.33) and (C.34), the current density distribution follows the distributions of hydrogen and oxygen, being higher above the channel inlets and lower above the land in the vicinity of the outlets.

372795.fig.006
Figure 6: Current density distribution in the membrane for an operating current density of 0.46 A/cm2 and cell voltage of 0.7 V.

In Figure 7 are shown the current density profiles along two lines crossing the CCM situated in the regions marked 1 and 2 in Figure 6. Current density increases at anode (protons are produced in the HOR), is constant in the membrane, and decreases at cathode (protons are consumed in the ORR). The current density increases in region 1, and decreases in region 2 when the operating current increases. This is due to a decreasing oxygen and hydrogen concentrations in region 2 when the operating current increases. The slight increase in current density at the cathode-membrane interface indicates the increase in proton flux as the ionomer fraction reduces from 100% in membrane to 30% in catalyst layer.

372795.fig.007
Figure 7: Current density profiles along two lines crossing the CCM and situated in the regions marked 1 and 2 in Figure 6; operating conditions: (i) 0.46 A/cm2 and 0.7 V, (ii) 0.81 A/cm2 and 0.6 V, (iii) 1.1 A/cm2 and 0.5 V.

Figures 8 and 9 show the water content profiles across the CCM, along the lines marked 1 and 2 in Figure 6. In catalyst layers, the water in the ionomer-phase equilibrates with the water vapor at its local activity 𝑎 and/or the liquid water present in the catalyst layer pores according to isopiestic curves such as the one for Nafion 117 equation (C.44) [21]. Supersaturated conditions are encountered at cathode and the ionomer-phase is surrounded by a liquid water film. At cathode, the water content in the ionomer-phase is therefore constant at any operating regime (𝜆=22). At anode, the vapor concentration (activity) decreases with increasing operating current. This determines the water in the ionomer-phase of the anode catalyst layer to equilibrate at lower values when the operating current increases. Inside the membrane, water is depleted when the current density flows along a positive gradient of water content (from smaller to larger hydrophilic domains) according to 𝑖𝑛drag/𝐹. In this case, the protonated complexes engage more water molecules in hydrodynamic motion as the number of water molecules in their solvation shells increases. The electro-osmotic drag coefficient equation (C.48) was measured by the Los Alamos National Laboratories group [22, 23] for membranes immersed in liquid water (𝑛drag=2.5 ) and for membranes at equilibrium with water vapor over a wide range of activities (𝑛drag=1 ). Water inside the ionomer-phase will be therefore depleted only when the water content is higher than 14. Above this value, the water content will depart from a linear profile (see the membrane area towards the cathode catalyst layer in Figure 8). This feature is less evident when the local current density is lower (Figure 9). Note that in our computations, we allowed the electro-osmotic drag coefficient to increase linearly when 𝜆 increases from 14 to 22.

372795.fig.008
Figure 8: Water content profiles across the CCM (along line 1) for three operating regimes.
372795.fig.009
Figure 9: Water content profiles across the CCM (along line 2) for three operating regimes.

Figures 10 and 11 depict the water flux profiles in the ionomer-phase across the CCM along lines 1 and 2. The electro-osmotic drag flux 𝑖𝑛drag/𝐹 increases at anode following the increase in current density, is constant in membrane where the water content is below 14 (areas towards the anode catalyst layer), increases in membrane areas towards the cathode catalyst layer (electro-osmotic drag coefficient increases), and decreases with the current at cathode. The back-diffusion flux 𝐷H2O,𝑖𝜆 is highest in the membrane areas towards the cathode catalyst layer, where it acts to bring back the water carried away by the protonated complexes in electro-osmotic drag. Note that the latter mechanism displaces water from areas with lower water content towards areas with higher water content. The back-diffusion flux is constant in the membrane areas close to the anode catalyst layer where the water content gradient is constant. In region 2 (Figure 10), the electro-osmotic flux is lower than in region 1 (Figure 9) as the current density is lower (see Figures 6 and 7), but the back-diffusion flux is similar in both regions. The resulting total water flux is positive in region 1 and negative in region 2. As a consequence, water is transported from anode to cathode across the CCM in region 1 and is transported in the opposite direction in region 2.

372795.fig.0010
Figure 10: Water flux profiles across the CCM (along line 1) for an operating current of 0.64 A/cm2.
372795.fig.0011
Figure 11: Water flux profiles across the CCM (along line 2) for an operating current of 0.64 A/cm2.

Figure 12 illustrates the water vapor distribution in the anode catalyst layer at the interface with membrane. Water mass fraction decreases downstream as water is transported from anode to cathode, but increases towards the channel outlets (region 2), where water is transported across the CCM in the opposite direction.

372795.fig.0012
Figure 12: Water vapor distribution in the anode catalyst layer at the interface with membrane an operating current of 0.64 A/cm2.

5. Conclusions

We derive a continuum model for water transport in the ionomer-phase of CCMs for PEMFCs using the method of volume averaging. Water accumulates in the ionomer-phase of CCMs as a result of

(i)“stripping off” the water molecules in the peripheral solvation shells of the protonated complexes when the hydrodynamic motion of the latter occurs along a gradient of water content (along a gradient of hydrophilic domain sizes), and expressed by 𝜀𝐴𝑛drag𝑖/𝐹,(ii)the difference between the incoming and outgoing water diffusive fluxes, expressed by (𝜌dry/EW)(𝜀𝐴𝐷H2O,𝑖𝜆),(iii)exchanges by sorption and desorption between the ionomer-phase distributed in catalyst layers and the catalyst layer pores, 𝑆𝜆sorption/desorption.

A second mechanism of water transfer between the ionomer-phase and the TPB regions distributed in the catalyst layers, referred to as electro-osmotic discharge and expressed by 𝑛drag(𝜀𝐴𝑖)/𝐹, does not affect the water accumulation in the ionomer-phase but affects the water accumulation in the catalyst pores. This term does not appear explicitly in the equation for water transport in the ionomer-phase, (20) and (21), but appears as source/sink terms in the transport equations for water in the catalyst layer pores.

To our best knowledge, the previous models for water transport in CCMs have been exclusively derived for use in CFD analysis intuitively, or by performing a balance of water fluxes across a control volume consisting of 100% ionomer-phase. These equations are therefore valid solely in PEMs and cannot capture correctly the mechanisms of water transfer between the ionomer-phase and the catalyst pores or the mechanisms that affect water accumulation in the ionomer-phase.

Appendix Group

Appendices

A. Transport Equations in the Flow Field

A.1. Mass

In both channels, 𝜕𝜌𝜕𝑡+𝜌𝑈=0.(A.1) In GDLs, 𝜕𝜕𝑡𝜀𝑉𝜌+𝜀𝜌𝑈=0.(A.2) In cathode catalyst layer, 𝜕𝜕𝑡𝜀𝑉𝜌+𝜀𝜌𝑈=𝜀𝑉H𝑌satH2O𝑌H2Ȯ𝑚H2O+̇𝑚O2+𝜀𝑉̇𝑚sorption+H𝑌satH2O𝑌H2Ȯ𝑚desorption+̇𝑚EOD,𝑐.(A.3) In anode catalyst layer, 𝜕𝜕𝑡𝜀𝑉𝜌+𝜀𝜌𝑈=𝜀𝑉̇𝑚H2̇𝑚sorptioṅ𝑚EOD,𝑎+H𝑌satH2O𝑌H2Ȯ𝑚desorption.(A.4)

A.2. Momentum

In both channels, 𝜕𝜕𝑡𝜌𝑈+𝜌𝑈𝑈𝜇𝑈+𝑈𝑇=𝑝.(A.5) In both GDLs, 𝜕𝜕𝑡𝜀𝑉𝜌𝑈=𝜀𝑉𝑝+𝜀𝑉𝜇𝐾𝑉𝑈.(A.6) In cathode catalyst layer, 𝜕𝜕𝑡𝜀𝑉𝜌𝑈=𝜀𝑉𝑝+𝜀𝑉𝜇𝐾𝑉𝑈+𝜀𝑉̇𝑚O2̇𝑚sorption𝑈.(A.7) In anode catalyst layer, 𝜕𝜀𝑉𝜌𝑈𝜕𝑡=𝜀𝑉𝑝+𝜀𝑉𝜇𝐾𝑉𝑈+𝜀𝑉̇𝑚H2̇𝑚sorptioṅ𝑚EOD,𝑎𝑈.(A.8)

A.3. Species

In cathode channel, 𝜕𝜌𝑌O2𝜕𝑡+𝜌𝑈𝑌O2𝜌𝐷O2𝑌O2=0.(A.9) In both channels, 𝜕𝜌𝑌H2O𝜕𝑡+𝜌𝑈𝑌H2O𝜌𝐷H2O𝑌H2O=0.(A.10) In anode channel, 𝜕𝜌𝑌H2𝜕𝑡+𝜌𝑈𝑌H2𝜌𝐷H2𝑌H2=0.(A.11) In cathode GDL, 𝜕𝜀𝑉𝜌𝑌O2𝜕𝑡+𝜀𝜌𝑈𝑌O2𝜀𝜌𝐷O2𝑌O2=0.(A.12) In both GDLs, 𝜕𝜀𝑉𝜌𝑌H2O𝜕𝑡+𝜀𝜌𝑈𝑌H2O𝜀𝜌𝐷H2O𝑌H2O=0.(A.13) In anode GDL, 𝜕𝜀𝑉𝜌𝑌H2𝜕𝑡+𝜀𝜌𝑈𝑌H2𝜀𝜌𝐷H2𝑌H2=0.(A.14) In cathode catalyst layer, 𝜕𝜀𝑉𝜌𝑌O2𝜕𝑡+𝜀𝜌𝑈𝑌O2𝜀𝜌𝐷O2𝑌O2=𝜀𝑉̇𝑚O2.(A.15) In cathode catalyst layer, 𝜕𝜀𝑉𝜌𝑌H2O𝜕𝑡+𝜀𝜌𝑈𝑌H2O𝜀𝜌𝐷H2O𝑌H2O=𝜀𝑉H𝑌satH2O𝑌H2Ȯ𝑚H2O+𝜀𝑉̇𝑚sorption+H𝑌satH2O𝑌H2O×̇𝑚desorption+̇𝑚EOD,𝑐.(A.16) In anode catalyst layer, 𝜕𝜀𝑉𝜌𝑌H2𝜕𝑡+𝜀𝜌𝑈𝑌H2𝜀𝜌𝐷H2𝑌H2=𝜀𝑉̇𝑚H2.(A.17) In anode catalyst layer, 𝜕𝜀𝑉𝜌𝑌H2O𝜕𝑡+𝜀𝜌𝑈𝑌H2O𝜀𝜌𝐷H2O𝑌H2O=𝜀𝑉̇𝑚sorption+̇𝑚EOD,𝑎+H𝑌satH2O𝑌H2Ȯ𝑚desorption.(A.18) where the Heaviside step function is defined as, 𝐻(𝑥)=1if𝑥0,0if𝑥<0.(A.19)

B. Transport Equations in the CCM

B.1. Charge

In cathode catalyst layer, 𝜀𝐴𝜎Φ𝑖+𝑗𝑐=0,(B.20) see also  (12)

In membrane, 𝜎Φ𝑖=0.(B.21) In anode catalyst layer, 𝜀𝐴𝜎Φ𝑖+𝑗𝑎=0.(B.22)

B.2. Water Content

In both catalyst layers, 𝜌dryEW𝜕𝜀𝑖𝜆𝜕𝑡+𝜀𝐴𝑖𝐹𝑛drag𝜌dryEW𝜀𝐴𝐷H2O,𝑖𝜆+𝑆𝜆sorption/desorption=0(B.23) see also  (20)

In membrane, 𝜌dryEW𝜕𝜆𝜕𝑡+𝑖𝐹𝑛drag𝜌dryEW𝐷H2O,𝑖𝜆=0,(B.24) see also  (21)

C. Constitutive Relations

Expression

̇𝑚H2O=𝜔𝑐𝜈H2O𝑊H2O,(C.25)̇𝑚O2=𝜔𝑐𝜈O2𝑊O2,(C.26)̇𝑚H2=𝜔𝑎𝜈H2𝑊H2,(C.27)̇𝑚sorption/desorption=𝐶𝜆𝜆𝜆equilibrium𝜌dryEW𝑊H2O(C.28) (see [4]),

̇𝑚EOD,𝑐=𝑛drag𝑗𝑐𝐹𝑊H2O(C.29) (see [4])

̇𝑚EOD,𝑎=𝑛drag𝑗𝑎𝐹𝑊H2O(C.30) (see [4]),

𝐷𝑖=1𝑌𝑖𝑖𝑗𝑊𝑔/𝑊𝑗𝑌𝑗/𝐷𝑖𝑗𝑖=O2,H2O,N2atcathode,𝑖=H2,H2Oatanode(C.31) (see [24]),

𝑝𝐷𝑖𝑗=𝑎1𝑇𝑇cr,𝑖𝑇cr,𝑗𝑏1𝑝cr,𝑖𝑝cr,𝑗1/3𝑇cr,𝑖𝑇cr,𝑗5/12×1𝑊𝑖+1𝑊𝑗104,(C.32) (see [25])

𝜔𝑐=𝑎𝑖ref0𝑐𝜈H+𝐹𝑌O2𝑌refO2exp𝜂𝑐𝑏𝑐exp𝜂𝑐𝑏𝑐,(C.33)𝜔𝑎=𝑎𝑖ref0𝑎𝜈H+𝐹𝑌H2𝑌refH2exp𝜂𝑎𝑏𝑎exp𝜂𝑎𝑏𝑎,(C.34)𝜂𝑐=Φ𝑠Φ𝑖𝑈0,(C.35)𝜂𝑎=Φ𝑠Φ𝑖,(C.36)𝑗𝑐=𝜔𝑐𝐹𝜈H+,(C.37)𝑗𝑎=𝜔𝑎𝐹𝜈H+,(C.38)𝑖=𝜀𝐴𝜎Φ𝑖(C.39)𝜎=(0.5139𝜆0.326)exp126813031𝑇(C.40) (see [22]),

𝐷H2O,𝑖=𝜕ln𝑎𝜕ln𝑐𝐷self-diusion𝜆,(C.41)𝐷self-diusionH2O,𝑖=1010exp241613031𝑇×2.5630.33𝜆+0.0264𝜆20.000671𝜆3(C.42) (see [26]),

𝑆𝜆sorption/desorption=𝐶𝜆𝜆𝜆equilibrium𝜌dryEW(C.43) (see [4]),

𝜆equilibrium=0.043+17.18𝑎39.85𝑎2+36.0𝑎3,𝑎[0,1)(vapor),22,𝑎=1(liquid),(C.44) (see [21]),

𝑎=𝑊𝑝𝑌H2O𝑊H2O𝑝satH2O,(C.45)𝑊=𝑖𝑌𝑖𝑊𝑖𝑖=O2,H2O,N2atcathode,𝑖=H2,H2Oatanode,(C.46)ln𝑝satH2O=5800.2206𝑇+1.39149930.048640239𝑇+0.41764768×104𝑇20.14452093×107𝑇3+6.5459673ln𝑇(C.47)𝑛drag=1if2𝜆14,2.5if𝜆=22.(C.48) (see [22, 23]),

Nomenclature

3:Volume average operator
3𝑖:Intrinsic volume average operator
2𝑖:Intrinsic surface average operator
𝐴:Interfacial area [cm2]
𝑎:Effective catalyst surface area per unit volume, [cm2/cm3] (C.33), (C.34)
𝑎:Water activity (C.44), (C.45)
𝑎1:Constant in Slattery-Bird correlation (C.32)
𝑏1:Exponent in Slattery-Bird correlation (C.32)
𝑐H2O:Molar concentration of water in ionomer [mols H2O/cm3]
𝑐H+:Molar concentration of protons [mols H+/cm3]
̇𝑐H+:Molar rate of proton production or consumption [mols H+/cm2s]
̇𝑐H2O:Molar rate of water production in ORR [mols H2O/cm2s]
𝐶𝜆:Rate of water sorption in ionomer, [s-1] (C.43)
𝐷H2O:Diffusion coefficient of water in ionomer [cm2/s]
EW:Equivalent weight [mols SO3/g]
𝐹:Faraday constant 96487 [C/mols H+]
𝐻(𝑥):Heaviside step function
𝑖ref0:Reference exchange current density, [a/m3]
𝑖:Current density [A/cm2]
𝑖𝑥,𝑖𝑦,𝑖𝑧:Versors of the macroscopic coordinate system
𝑗:Charge transfer current [A/cm3]
𝐾𝑉:Absolute viscous permeability, [cm2]
̇𝑚:Mass source, [g/cm3s]
𝑛drag:Electro-osmotic drag coefficient [mols H2O/mols H+]
𝑛𝑖:Unit vector normal to the 𝑖-phase interfaces and pointing outwards from 𝑖
𝑛𝑖𝑠:Unit vector normal to the 𝑖𝑠-interface and pointing from 𝑖 to 𝑠
𝑛𝑖𝑠𝑓:Unit vector in the 𝑖𝑠 surface normal to the 𝑖𝑠𝑓-interface with 𝑓-phase and pointing outwards from 𝑖𝑠
𝑛𝑒:Number of electrons participating in electrochemical reaction
𝑁H+:Total proton flux [mols H+/cm2 s]
𝑁structdiH+:Proton flux caused by structure diffusion [mols H+/cm2 s]
𝑁vehmechH+:Proton flux caused by the vehicle mechanism [mols H+/cm2 s]
𝑁H2O:Total water flux in ionomer [mols H2O/cm2 s]
𝑁dragH2O:Water flux in ionomer generated by electro-osmotic drag [mols H2O/cm2 s]
𝑁diH2O:Diffusion water flux in ionomer [mols H2O/cm2 s]
𝑝:Pressure, [Pa]
𝑆𝜆:Distributed source for water content in ionomer [mols H2O/cm3 s]
𝑡:Time [s]
𝑇:Temperature, [K]
𝑢:Diffusion velocity [cm/s]
𝑈0Thermodynamic equilibrium potential, [V]
𝑈:Velocity vector, [cm/s]
𝑣:Velocity [cm/s]
𝑣(H2O/H+):Proton and water velocity in hydrodynamic motion [cm/s]
Vol:Averaging volume
𝑤:Interface velocity [cm/s]
𝑊:Molecular weight, [g/mols]
𝑌:Mass fraction.

Greek Symbols

𝛼𝑐𝑐,𝛼𝑐𝑎:Cathodic and anodic transfer coefficients
𝜀:Surface porosity
𝜀𝑖Volume fraction of ionomer in the averaging volume
𝜀𝐴:Directional surface permeability of ionomer
𝜀𝑉:Volumetric porosity
𝜂:Activation polarization, [V]
𝜆:Water content [mols H2O/mols SO3 ]
𝜇:Chemical potential [J]
𝜔𝑐:Electrochemical reaction rate at cathode, [mols/cm3 s]
𝜌:Density, [g/cm3]
𝜌dry:Density of dry ionomer [g/cm3]
𝜇:Viscosity, [Ns/cm2]
𝜈:Stoichiometric coefficient
𝜎:Ionomer conductivity, [Ω1cm-1]
Φ:Potential, [V].

Subscripts

𝑎:Anode
𝑐:Cathode
cr:Critic
EOD:Electro-osmotic discharge
,𝑖:Parameter defined in ionomer-phase
,𝑠:Parameter defined in the solid electrically conductive phase
,𝑓:Parameter defined in any of the fluid phases in catalyst layer pores
,𝑖𝑠:Parameter defined in the at 𝑖𝑠-interface (three-phase boundary region)
,𝑖𝑓:Parameter defined in the at 𝑖𝑓-interface
,𝑖𝑠𝑓:Parameter defined at the line between 𝑖, 𝑠 an 𝑓 phases.

Superscripts

𝑎:Anodic
𝑐:Cathodic
sat:Saturation.

Acknowledgment

This material was prepared with the financial support of the U.S. Department of Energy under Award no. DE-PS36-07G097012.

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