Modeling PC Sui and N Djilali, University of Victoria, Victoria, BC, Canada & 2009 Elsevier B.V. All rights reserved. Introduction Two of the distinc...

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Modeling PC Sui and N Djilali, University of Victoria, Victoria, BC, Canada & 2009 Elsevier B.V. All rights reserved.

Introduction Two of the distinctive features of proton-exchange membrane fuel cells (PEMFCs), also referred to as polymer electrolyte membrane fuel cells, are the use of a thin ionomer membrane as the electrolyte and the central role of water transport in the operation of such fuel cells. The self-contained nature of the electrolyte, combined with low operating temperatures and scalability, has stimulated developments of PEMFCs for a broad range of applications ranging from transit buses to portable electronic devices. The power range as well as the operating conditions and design constraints vary considerably between these applications, making the potential role of modeling all the more critical to better understand and control the key transport processes and to facilitate design, optimization, and scaling. Road vehicle applications require, for instance, high-powerdensity stacks that are subjected to highly dynamic power cycles as well as extreme environmental conditions, including freezing. Conditioning of the reactants and active water and thermal management are typically required. Conversely, the design of micro-PEMFCs for portable applications is driven by the requirements of maximizing energy densities and system simplifications, and airbreathing designs with passive thermal and water management are preferred in such systems. In the most common plate-and-frame configuration illustrated in Figure 1, the membrane, which is typically 50–150 mm thick, is sandwiched between two thin anode and cathode catalyst layers (CLs) and a porous gas diffusion layer (GDL) to form the so-called membrane– electrode assembly (MEA). The MEA is then placed between two graphite or metallic bipolar plates with machined groves that provide flow channels for distributing fuel to the anode and oxidant to the cathode. In hydrogen-fueled PEMFCs, the focus of this article (see ‘DMFC – General’ Fuel Cells – Direct Alcohol Fuel Cells: Direct Methanol: Overview), electric current is produced by oxidizing hydrogen at the anode, releasing protons and electrons. The protons are transported to the cathode through the ionically conducting membrane, whereas the electrons are collected through the solid matrix of the porous anode (CL and GDL) and the bipolar plates and then flow to the cathode through an outer circuit producing useful electric current. The operation and the performance of a PEMFC rely on the effective coupling of reaction kinetics with the


distribution of reactants, and removal of product heat and water. This involves an intricate array of coupled thermofluid and charged species transport processes: ionic and water transport, including electroosmotic • drag (EOD), in the ionomer matrix; mass charged species transport coupled with re• actionand kinetics and heat generation in the porous CLs; multicomponent diffusion and two-phase transport • (vapor and liquid water) in the pores of the GDL, and electrical current through its solid matrix; and

multiphase, multicomponent flow in manifolds and • gas flow channels. The GDL, which facilitates transport of electrons to the collector plate and transport of reactants to the CL, is a porous medium that consists of electrically conducting carbon fibers (B10 mm diameter) that are randomly layered in a ‘paper’ structure or woven into a ‘cloth’ structure. To ensure effective reactant transport in the GDL, it is necessary to minimize flooding of the pores due to condensation of water, which is a reaction byproduct and is also introduced by the humidified gas stream. This is achieved in part by hydrophobic treatment of the GDL. In recent designs, a third microporous layer (MPL) is also sometimes used to facilitate water management. The CL is a composite consisting of ionomer and a carbon-supported catalyst (platinum), and provides the interface between the electron, reactant, and ion conducting phases required for the heterogeneous reactions to proceed. Polymer electrolyte membrane fuel

Collector plate Flow channel Hyd






Figure 1 Components of a plate-and-frame proton-exchange membrane fuel cell (PEMFC) illustrating the bipolar collector plates with the gas distribution flow channels, the gas diffusion layer (GDL), the catalyst layer (CL), and the polymer electrolyte membrane (PEM).

Fuel Cells – Proton-Exchange Membrane Fuel Cells | Modeling

cell electrodes are therefore multifunctional and facilitate a number of simultaneous processes: of the reactants to the reaction sites; • transport transport • membrane;of ions between the reaction sites and collection and conduction of electrons from the re• action sites to the bipolar plate; conjugate transfer through both the pores and • solid matrixheatof the CL, MPL and GDL; and transport of water (as vapor and liquid) with con• densation/evaporation. One of the major objectives of modeling is to predict overall performance given a set of design and operation parameters, and to quantify changes in various performance metrics when design or operation parameters are varied. Overall performance is usually quantified in terms of cell voltage (DEcell) and current density (i ):     DEcell ¼ Er  Zact;c  þ Zact;a   iRcell  Zconc


where Er is the maximum reversible potential, which is readily obtained from the Gibbs free energy of the reaction and products at their standard state (see ‘Thermodynamics’). The actual cell voltage achieved in practice is significantly lowered by the activation, ohmic, and mass transport losses represented by the last three terms on the right-hand side of eqn [1]. The task then is to predict these losses, determine the dominant factors that control them, and eventually identify methods for minimizing them. Modeling not only speeds up and lowers the cost of product development by reducing the amount of hardware prototyping, but it also allows identification and understanding of limiting processes and provides a platform for multiobjective optimization. The complexity and multitude of phenomena at play in a PEMFC, the very broad range of length and timescales and diversity of Table 1

structures and materials, and the variety of end user requirements dictate a hierarchy of modeling strategies ranging from stack models that allow investigation of transient cell-to-cell coupling effects to component models of transport processes in CLs for instance, which can be as highly resolved as on the molecular scale. Models suitable for multidimensional analysis of basic transport processes, performance predictions, and design involve the solution of a set of classical continuum thermofluid equations that express conservation of mass, momentum, energy, and species in conjunction with model equations representing the heterogeneous electrochemical reactions and the transport processes specific to polymer electrolyte membrane (PEM) components. These include transport of charged species (ions and electrons) as well as transport of water in nonvapor form. The conservation of charged species appears as the electric potential in the electron conductor and ion conductor (electrolyte) phase. The conservation equations of water in nonvapor forms account for sorbed water in the electrolyte phase (l) and liquid water saturation (s) in the porous media. The conservation equations take the general form @rc þ r  ðr¯ucÞ þ r  ðJ¯c Þ ¼ Sc @t


with c ¼ P; u¯ ; Yi ; fþ ; f2 ; T ; l; s

The relevant variables and source terms in different components of a PEMFC are listed in Table 1 and discussed in the following text. The integration of these model equations with state-of-the-art computational fluid dynamics (CFD) methods has significantly expanded the capabilities and scope of PEMFC modeling in recent years. The CFD methods have advantages over other models, e.g., IV characteristics, impedance mode, dynamic models, HIL models, etc., in their capabilities of resolving actual geometry and physical time, which are

Variables and source terms in different regions of the proton-exchange membrane fuel cell (PEMFC) BPP

P Yi fþ f T l s SYi Sf þ Sf  ST Ss




















BPP, bipolar plate; CL, catalyst layer; GC, gas channel; GDL, gas diffusion layer; PEM, proton-exchange membrane/polymer electrolyte membrane. P, momentum; s, liquid water saturation in porous media; S, source term; Yi, gas species; T, temperature; l, water content in membrane; f þ , electrical field of ion conductor; f  , electrical field of electron conductor.


Fuel Cells – Proton-Exchange Membrane Fuel Cells | Modeling

T Coollant

Current flow

O2 Cathode

H2 Anode H2O H2

Figure 2 Three-dimensional (3D) simulation illustrating the distribution of reactants (H2 and O2), water, temperature, and electric current flow in the main components of a proton-exchange membrane fuel cell (PEMFC).

essential for engineering design of the fuel cell hardware. The CFD methods, on the contrary, are not an ideal tool for analysis in the frequency domain, nor are they suitable for analysis at system level due to high demand of computational time. Figure 2 illustrates the type of information that can be obtained from such computational simulations to unravel some of the complex interplays between the various transport phenomena. Such models are built upon an array of submodels that are specific to the dominant physicochemical processes in each component. The salient features of these submodels are reviewed below, and a discussion of their integration and application to cell and stack simulations is provided in the last part of this article.

Polymer Electrolyte Membrane The membranes of choice in low-temperature fuel cells are perfluorosulfonic acid (PFSA) membranes, such as Nafion, consisting of a polytetrafluoroethylene (PTFE) backbone with side chains ending in a sulfonic acid group (SO3H). These membranes combine good mechanical strength and chemical stability with high permselectivity for nonionized molecules, which limits the crossover of reactants. Most importantly, PFSA membranes exhibit good ionic conductivity, but this conductivity is highly dependent on hydration. Transport of ions and that of water are thus the two phenomena of prime interest. There is general consensus that a hydrated PFSA membrane forms a nanoporous two-phase system consisting of a water-ion phase distributed throughout a

partially crystallized perfluorinated matrix phase. Phenomenological models postulate the formation of approximately spherical clusters in regions with a high density of sulfonate heads. Under vapor-equilibrated conditions, the interfacial region between clusters consists of collapsed channels that allow for some ionic conductivity, because sorbed water molecules can dissociate from the sulfonate heads. When the membrane is equilibrated with liquid water, the collapsed channels fill with water to form continuous liquid pathways. Water sorption behavior of PEMs is commonly considered in terms of the water content l, defined as the number of sorbed water molecules per sulfonate head. Modeling of water content in PEMs is key to predicting ionic conductivity, and is central to overall performance calculations, because membrane resistance is usually the dominant contribution to the ohmic loss term in eqn [1]. Water transport in the membrane is determined by the hydration conditions at the anode and cathode interfaces and the balance between several transport mechanisms: diffusion, hydraulic permeation, and EOD, whereby a number of water molecules are ‘dragged’ with each proton migrating from anode to cathode. Ionic conductivity is typically characterized experimentally as a function of the activity of the water with which the membrane is equilibrated; these measurements are in turn related to the actual water content using experimentally determined sorption isotherms, which play a key role in membrane transport models. The high mobility/conductivity of protons under well-hydrated conditions is due to excess protons within the bulk water network that are found primarily in the

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form of hydronium ions (H3O þ ). Proton diffusion can occur via two mechanisms, structural diffusion and vehicle diffusion, which combine to provide protonic defects that have exceptional mobility in bulk water. The other phenomenon linked to membrane conductivity is EOD. For a vapor-equilibrated membrane, the number of water molecules dragged per proton (the EOD coefficient) has been reported to be B1 over a wide range of water vapor activities, and B2.5 for a liquid-equilibrated membrane, which allows the formation and diffusion of larger ionic complexes (Eigen and Zundel ions). The most widely used model for determining membrane conductivity is due to Springer, Zawodzinski, and Gottesfeld. A generalized form of the water transport equation can be written as kp i Nl ¼ nd ðl; T Þ  Dw ðl; T Þrl  rp m F


where Nl is the molar water flux, i the ionic current, F the Faraday constant, l the water content defined earlier, nd the EOD coefficient, Dw the diffusion coefficient, k the permeability, m the viscosity, and p the pressure; the terms represent the EOD, diffusion, and hydraulic components. In Springer’s model, eqn [3] is solved in conjunction with Ohm’s law and an empirical relation that accounts linearly for the variation of conductivity, s, with water content:    1 1  s30 ; sSpringer ¼ exp 1268 303 273 þ Tcell s30 ¼ 0:005139l  0:00326 ðl > 1Þ

Np Nw



fDpp Dpw ¼ c fDwp Dww


rfm rl

This model provides a general constitutive relation in which the ionic flux Np and water flux Nw are explicitly coupled. And fm is the membrane potential, c is the molar concentration of water; f ¼ F/RT, where F is Faraday’s constant, and the Ds are generalized diffusion coefficients. This model results in significantly different water flux and conductivity predictions from earlier empirical models, particularly under medium to low relative humidity conditions. Such conditions prevail, for instance, in air-breathing fuel cells for portable devices that rely on passive water and thermal management to achieve significant system simplifications. Another aspect of the modeling framework of eqn [5] is that it is not specific to a particular membrane and it applies, with minor and physically based parameter adjustments, to the entire family of PFSA membranes. In closing this brief discussion on PEM modeling, it should be noted that conductivity is commonly measured as a function of the activity of the solvent with which the membrane is equilibrated. Consequently, another crucial element in the modeling of membrane transport is the sorption isotherms, which relate such measurements to the actual water content. Reliable isotherms and, in particular, proper representation of temperature sensitivity are critical to the model.

Catalyst Layer with ½4

where Tcell is the cell temperature (in 1C) and s30 is the conductivity (in S cm1) at 30 1C. The robustness of this model and the approximate accounting of the effect of water content have been particularly valuable in determining membrane resistance in computational fuel cell models under well-hydrated conditions that typically prevail in automotive fuel cell stacks. Several variants and improvements have been proposed, but the model has several inherent limitations because it is essentially a curve fit with restricted applicability. Physically based models with greater generality have recently been proposed, starting from the chemical potential as a general driving force. This approach has, for instance, been adopted in conjunction with multicomponent transport to derive the binary friction membrane model (BFM2), in which the generalized Stefan– Maxwell relations are applied to a free solution of ions (H3O þ ) and water in a membrane pore structure, yielding "


# ½5

Because of its composite structure and the convergence of virtually all transport processes within it, as illustrated in Figure 3, the CL is the most challenging domain from the view point of modeling. Within the confines of this B10-mm thick layer, transport of charged species (H þ , electrons), radicals, and noncharged species (gaseous water, oxygen, hydrogen, nitrogen, and liquid water) occurs in distinct pathways provided by the electrolyte, carbon, water, and gas pores. In large-scale models of multicell stacks, the disparity of scales makes even partial resolution of CL impractical, and by and large such stack models do not directly account for the layer composition and structure but rather model the CL as an interface. The effect of the CL on the overall cell performance is parameterized through a single reaction kinetics equation, which is an interface boundary condition between the GDL and the membrane. Such models do not account for spatial gradients in chemical species concentration and overpotential. Several variants of spatially resolved models have been proposed based on the volume-averaging method in conjunction with lumped parameters to account for macroscopic transport in the representative element volume (REV). These fall broadly into two categories: macrohomogeneous models and agglomerate models.


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Proton conducting media

Carbon supported catalyst

Agglomerate Model Electrically conductive fibers e−

H2O O2



Catalyst layer


Figure 3 Cathode side transport processes at the polymer electrolyte membrane (PEM)/catalyst/gas diffusion layer (GDL) interface.

In agglomerate models, the conductive carbon support and platinum particles are assumed to cluster in small agglomerates bonded and surrounded by electrolyte. The agglomerates are taken to be either cylindrical or spheres of ionomer (typically Nafion) filled with carbon and platinum particles, and the interagglomerate space or pore may be occupied by the electrolyte and/or a mixture of reactant/products. Oxygen is assumed to diffuse through the interagglomerate pores, dissolve into the electrolyte phase, and finally diffuse through the electrolyte inside the agglomerate to the reaction site, and the reaction is modeled according to well-established methods for porous catalysts. The agglomerate structure postulated some time ago has gained support from more recent experimental observations, and several studies have found that agglomerate models correlate better with experimental results. Transport from the bulk flow to the agglomerate is the limiting process. Transport of oxygen is controlled by two mass transfer ‘resistors’ in series, represented by the two terms in the denominator of the conservation equation (diffusive transport is discussed further in the section ‘Gas Diffusion Layer and Microporous Layer’): 0

Macrohomogeneous Model In macrohomogeneous models, which stem from early models for electrochemical systems in which a liquid electrolyte occupies most of the porous electrode, the salient macroscopic phenomena as well as the composition of the CL are taken into account by relating the porous CL properties to the volume fraction of each material – catalyst, electrically conductive material, and electrolyte – and by modeling electronic and ionic transport and diffusion of reactants through the void spaces. Such models are primarily used to resolve the processes in the cathode where the limiting oxygen reduction reaction (ORR) takes place. Oxygen is assumed to diffuse within the pores of the wetproofed CL. An alternative approach assumes the pores to be flooded with water and oxygen to diffuse in the dissolved state. Assuming that the CL is a homogeneous medium and electronic resistance is negligible, a simple one-dimensional (1D) model for conservation of oxygen can be written as DOeff2 r2 CO2 ¼ 

1 ri 4F


where current production follows the Butler–Volmer equation (see ‘Kinetics’): ri ¼ ai0

! CO2  aa f Z e  eac f Z CO0 2

where aa, as are transfer coefficients.


B B ri ¼ nF B @

1 1 d aCO 2 Deff

C C C 1A þ kE


where d is the electrolyte thickness on the agglomerate, k¼

 ai0 2:3Z exp  b nF


tanhðmLÞ mL


with the Thiele modulus mL ¼ L

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k CO 2 Deff


There is some controversy about the electrolyte-filled agglomerate structure, and some insights from molecular dynamic simulations suggest that the agglomerates are ionomer free. Additionally, extensions of the agglomerate models have been proposed that account for liquid water and for species adsorption and desorption on the catalyst surface. A related model commonly implemented in CFD software assumes that mass transfer in the pores is the limiting process. For such models, the so-called reaction– diffusion balance equation is solved to relate the concentration in the pore to that on the catalyst surface, which is

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used in the concentration term of the Butler–Volmer equation. Microscopic Models Macroscopic CL models discussed so far are derived using the volume-averaging method. These macroscopic models rely on assumptions regarding the microstructure of the CL to prescribe the macroscopic transport parameters. Progress in achieving comprehensive macroscopic models is hindered by the lack of detailed morphological and compositional information from available experimental techniques. Atomistic simulations, such as coarse-grain molecular dynamics, are starting to be employed to model the formation of microstructures in the CL, and the socalled direct numerical simulation (DNS) models have also been recently proposed whereby representative geometric structures are reconstructed and transport processes are resolved at the individual pore level. DNS models resolve individual pores and capture flow physics at continuum scales. The size of the computational domain for a DNS is approximately the size of the REV, and could pave the way for multiscale strategies to link atomistic simulations to macroscopic models.


electrochemical reactions, models such as adsorption models, multistep and competing reactions and degradation mechanisms can be included to describe the heterogeneous reactions in the CL. Accurate modeling of these reactions in conjunction with the resolution of salient transport phenomena can greatly enhance understanding of the limiting processes in fuel cells under normal operating conditions and also provide insights into durability issues.

Gas Diffusion Layer and Microporous Layer As noted earlier, the GDL has several functions (see Figure 1). In general, the GDL is modeled as a homogeneous porous medium, and there is a rich literature that pertains to multiphase flow and multicomponent gas diffusion in such media. The challenges in modeling transport in the GDL lie first in the complexity and coupling of the conjugated transport of gas, water, electricity, and heat through the solid and void phases, and second in the fibrous anisotropic structure of the material and its wetting (hydrophobic) characteristics. Two-Phase Flow

Electrochemical Reactions Electrochemistry and reaction kinetics play a central role in all coupled transport phenomena within the CL. In a multidimensional CFD framework, the electrochemical reactions are typically treated as source terms in the corresponding conservation equations. In PEMFCs, the primary electrochemical reactions include the hydrogen oxidation reaction (HOR) HOR

H2 ! 2Hþ þ 2e


and the ORR ORR

O2 þ 4Hþ þ 4e ! 2H2 O


There are additional reactions of interest for specific scenarios and operating conditions, such as carbon corrosion, COR

C þ 2H2 O ! CO2 þ 4Hþ þ 4e


reactions related to platinum dissolution and the formation of a platinum band in the membrane þ

Pt þ H2 O"PtO þ 2H þ 2e

At higher current densities, or when local water and heat management is inadequate, water condensation takes place leading to potential ‘flooding’ of the electrode. To partially remedy this, GDLs are treated with PTFE to impart hydrophobicity and promote capillary transport of water. Enhanced understanding of liquid water transport and improvement of passive water management characteristics are central to GDL modeling and design. The capillary number (ratio of viscous forces to interfacial tension forces) is generally small in PEMFC electrodes, and liquid transport is dominated by capillary diffusion. Generally, liquid water in the GDL is characterized in terms of saturation, i.e., the volume fraction of liquid water in the pore. The governing equations for water transport can be obtained by extending the classical relationship for the hydrodynamics of a single phase in porous media (Darcy’s law) to each phase, which yields momentum conservation equations for both gas and liquid phases: kg ug ¼  rpg mg


kl ul ¼  rpl ml



PtO þ 2Hþ "Pt2þ þ H2 O


Pt2þ þ H2 -Pt þ 2Hþ


and adsorption of contaminants such as carbon monoxide and sulfur on the catalyst surface. In addition to these

where u is the Darcy flux, k the permeability, rp the pressure gradient, and m the viscosity of the fluid in question. Defining capillary pressure as pc ðsÞ  pl  pg and noting that this is a function of saturation, the mass flux of liquid


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water can be expressed as m_ l ¼ 

rkl rkl dpc rs ¼ rDcap rs rpc ¼  ml ml ds


kg and k1 are two important properties representing the phase-specific permeabilities, which are obtained by correcting the bulk permeability (k) for the effect of the reduced area open to each phase due to the presence of the other phase. There are not yet any definitive relations applicable to PEM electrodes for specifying the relative permeabilities and the capillary pressure, which are both functions of saturation; this is an area of active ongoing research using a number of emerging experimental and numerical approaches. Multicomponent Gas Diffusion The transport of gaseous species in a fuel cell electrode is primarily dominated by diffusion. The general mathematical representation of the multicomponent diffusion–convection processes is provided by the Stefan–Maxwell equation. Because this equation is relatively intractable, in practice, a generalized Fick’s law is often used, with the flux Ji of species i given as X rYi Dj rYj Di rM  rYi M j rrM X  Dj Yj M j

Ji ¼ rDi rYi þ


where Yi is the mass fraction, r the density, and M the molecular weight of the mixture. In GDLs, the effective diffusivity, which accounts for the geometric constraints imposed by the structure of the porous medium (porosity and tortuosity), is commonly evaluated using the Bruggeman ‘correction’: Deff ¼ e1:5 D


When liquid water is also present in the medium, additional changes should be considered accordingly. In many fundamental studies, a correction in terms of both porosity (e) and tortuosity factor (t), defined as the square of the actual path length to the direct length, is adopted: e Deff ¼ D t


Recently, computational models, such as Lattice–Boltzmann, volume-of-fluid methods, and capillary pore networks, have started to emerge to calculate the transport properties of the GDL and to establish the constitutive relationships for two-phase flows. Microporous Layer An MPL made of fine carbon particles with binder is often applied on the CL before the GDL is assembled.

The MPL is believed to act as a moisture barrier to prevent membrane dehydration, while simultaneously repelling excess liquid water out of the CL, thanks to its high degree of hydrophobicity. The transport of water through the MPL is conceptually similar to that described above for the GDL except for some differences in transport properties. In particular, because the characteristic length scale of the pores is much smaller than in the GDL, Knudsen diffusion needs to be taken into account. The Knudsen regime occurs when the pore length scale is an order of magnitude or less than the mean free path of the diffusing molecules, and molecule–wall collisions start to dominate over molecule–molecule collisions. As in classical molecular diffusion, flux in Knudsen diffusion depends on the concentration gradient of a species, and Knudsen diffusivity is corrected in the same manner as the molecular diffusivity in porous media. Another issue specific to the MPL and CL is that with porosity- and tortuosity-based corrections (eqns [16] and [17]), diffusive transport will be allowed to occur even in the case of very small porosity, when continuous pore-topore connectivity may no longer exist. It has therefore been suggested that the percolation theory is more appropriate for the determination of the transport parameters, as this approach allows for a minimum volume fraction of the transporting phase to form a percolation network.

Gas Channel, Bipolar Plate, and Stack The design of the flow field plate has a critical impact on practical fuel cell engineering in many aspects. The flow distribution in the unit cell affects both the local electrochemical reactions and global stack-level flow sharing. An ill-designed flow field plate may result in the maldistribution of flow streams in the unit cell and the entire stack, causing excessive power loss due to local understoichiometry. Motion of Water Droplets in Gas Channel With recent experimental observations of liquid water transport inside the flow field plate, modeling of water transport in the gas channel has gained more attention. Simulations employing numerical techniques, such as the volume-of-fluid method, show that the movement, in particular the detachment of water from the GDL surface, is sensitive to factors such as wall geometry and flow conditions in the gas channel, as shown in Figure 4. Flow Sharing in the Unit Cell and Stack Numerical calculations using the flow network model and CFD computations have demonstrated that the dimensions of the headers, which are the ducts where fresh

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Y (m)

0.0002 −0.001 0.0001 −0.0006



0 0 01 0.00 00 2 X (m 0.0 )

−0.0002 0


Z (m

t = 1.0 ms




Y (m)

0.0002 −0.001 −0.0008

0.0001 −0.0006 −0.0004

0 0

01 0.00 0.000 2 X (m )



Z (m


t = 1.3 ms Y



Y (m)

0.0002 −0.001 0.0001 −0.0006 −0.0004

0 −0.0002


00 1 0.0 02 X (m 0.00 )




Z (m

t = 1.4 ms

Figure 4 Dynamics of a water droplet emerging from a pore into a gas flow channel with a hydrophobic gas diffusion layer (GDL) surface. Initial deformation and shedding phase due to interaction with gas flow in the channel are tracked using volume-of-fluid computational fluid dynamic (CFD) simulations.

and spent reactant gases flow in and out of the stack, respectively, play an important role in the flow sharing of a stack. For the inlet header, pressure tends to increase downstream due to mass leaving the header (to enter the unit cells) along the flow path, whereas for the outlet header, pressure decreases due to the acceleration of flow downstream. The optimization of flow distribution inside a stack can therefore be achieved by the proper header design. The flow distribution inside a unit is equally important as that in the stack. The flow sharing in a unit cell

is affected by the design of the entrance region where flow enters from the header into the unit cell. The total pressure drop, cell performance, capability of water removal, and durability of a unit cell are closely related to flow distribution. Effects of Bipolar Plate Ribs In a typical PEMFC plate-and-frame design, the ribs of the flow field plate come into contact with the GDL. The ribs have significant effects on the coupled transport


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between the GDL and the flow field plate. For instance, the dimensions of the rib versus gas channel affect the electron conduction and heat transfer between the GDL, as well as the gas/liquid water transport between the CL and the gas channel. The current density distribution under the rib and gas channel is therefore greatly influenced by the limiting process of the coupled transport among mass, electron, and proton. Furthermore, the analysis is complicated by the compression force applied to the bipolar plate, which may alter the transport properties of the GDL (porosity, permeability, thermal and electrical conductivity, etc.). The uneven distribution of transport properties in the GDL, due to the compression from the rib, is expected to have a significant impact on the coupled transport phenomena near the rib area as well as on cross-channel flows induced by pressure differences between gas channels.

Dynamic Response There are many circumstances in which a fuel cell and its balance of plant (BoP) have to operate under dynamic conditions. For example, a cell experiences dynamic conditions during start-up or shut down, or responds to dynamic loads, etc. Some transport processes in a fuel cell system are dynamic in nature, e.g., freeze-start, contamination, and degradation. Analysis of the dynamic response of a fuel cell requires knowledge of the characteristic timescales, which vary over a wide range in a fuel cell system. So far, system dynamics models have focused on the transient transport of water because of its central impact on virtually all coupled transport phenomena. The dynamic response of water in a fuel cell is a combined result 1 Forward sweep Backward sweep

0.9 0.8

Cell voltage (V)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0





Curent density (Acm2)

Figure 5 Predicted polarization curves illustrating hysteresis between forward and backward sweeps.

of hydration/dehydration and liquid water movement in the MEA and the channel, and often exhibits hysteresis, as shown in Figure 5. This behavior is a result of the combination of the evolution of hydration/saturation in the membrane and electrode and of the different timescales associated with evaporation and condensation (Figure 5). Electrochemical impedance spectroscopy (EIS) is a powerful diagnostic tool based on the analysis of the fuel cell response in the frequency domain. Interpretation of the EIS signal follows a prescribed equivalent circuit. The low-frequency regime of common EIS diagnostics falls within a similar range to that of typical CFD transient models.

Conclusions The combination of improved physical understanding of transport phenomena in PEMFCs and advanced CFD methodologies has allowed significant progress in modeling and simulation of PEMFC systems, and functional computational tools are already used for analysis and design of novel systems. However, truly predictive capabilities have yet to be achieved over a broad range of operating conditions and design parameters. Because the usefulness of simulation tools depends to a large extent on their usability and turnaround time, a central issue is the resolution over a wide spectrum of length and timescales of the array of transport phenomena and associated components. Some of the key challenges to achieve the next level of functionality are devising improved macroscopic modeling of the key transport processes, such as membrane transport, liquid water transport in porous electrodes, and the dynamics of the transient processes, such as a phase change under freezestart conditions and degradation. A hierarchy of models is, in the short- to mid-term, the most pragmatic route. Comprehensive three-dimensional (3D) CFD methods are often prohibitively time-consuming and, consequently, are not yet practical as screening tools to systematically explore a wide range of design options and operating conditions. Mathematical models and solution procedures using simplified models with reduced dimensions, e.g., the so-called along-the-channel models, have been proposed to address the issue of CFD time expense. Such approaches are computationally efficient, but no systematic study has yet been conducted to quantitatively assess the effects of the neglected dimensionality and thus establish clearly the bounds of validity. Key transport processes in a PEMFC take place over length scales of the order of nanometers, which are unresolvable in practical large-scale models of single cells or stacks. These phenomena are instead described using volume-averaging methods; however, in order for such

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macroscopic models to be representative, improved linkage and bridging with detailed microscopic models, such as pore-scale resolved numerical simulations and molecular dynamics, are required through multiscale modeling and homogenization procedures. With the continuing reliability on low-temperature PEMs for most ongoing developments, heat and water management will remain a critical issue, and improved understanding of the phase change processes (liquid– vapor, ice–liquid, etc.) and characterization of the ‘state of water’ over a wide range of length scales, temperatures, and contact boundary conditions is required to improve the rather rudimentary treatment of phase change in current CFD-based fuel cell models. The development of comprehensive 3D CFD models has been largely motivated by the need for simulationbased design, analysis, and optimization tools. To fulfill these functions reliably, systematic and rigorous validation of these simulation tools is essential, and will have to largely rely on the development of new ex situ and in situ characterization and measurement techniques. Measurement tools such as current mapping, water content mapping, droplet density distribution etc., are being developed and employed for validation of computational models. Such validation data will allow a much greater degree of confidence on (1) the prescription of parameters such as effective conductivity, relative permeability, and electrochemical properties of the CL and (2) the reliability of the submodels for liquid water transport, membrane resistance, degradation, etc.

Nomenclature Symbols and Units a b c CO2 D Dcap D eff Dij Dw E Er f F i J k kg kl mL

specific surface area (m  1) Tafel slope molar concentration (mol m  3) molar concentration of oxygen (mol m  3) gas diffusivity; diffusion coefficient capillary diffusivity effective diffusivity binary diffusivity (m2 s  1) diffusion coefficient efficiency factor maximum reversible potential variable defined as F/RT Faraday constant (96 487 C mol  1) current density (A m  2) molar flux (mol m  2 s  1) reaction rate permeability of gas (kg mN  1 s  1) permeability of liquid (kg mN  1 s  1) Thiele modulus

˙ m M n nd Np Nw Nk p pc pg pl P R Rcell s S t T Tcell u ug ul Y aa,ac d DEcell e g gact,a gact,c gconc K k l q qm r r30 / /m s w

mass flow rate (kg s  1) molecular weight of the mixture number of electrons drag coefficient ionic flux water flux molar water flux pressure capillary pressure (N m  2) Pressure of gas (N m  2) Pressure of liquid (N m  2) momentum universal gas constant resistance of cell (O) saturation source term time (s) temperature (K) cell temperature velocity (m s  1) velocity of gas (m s  1) velocity of lquid (m s  1) mass fraction transfer coefficients electrolyte thickness on agglomerate change in cell voltage porosity overpotential (V) activation overpotential (anodic) activation overpotential (cathodic) overpotential due to mass transport permeability water content viscosity (Nm  2 s  1) density (kg m  3) density of membrane (kg m  3) surface tension (N m  1); conductivity conductivity at 30 1C electrical potential (V) membrane potential tortuosity factor variable in the conservation equation

Abbreviations and Acronyms 3D BFM2 BoP CFD CL DNS EIS EOD GDL HIL


three-dimensional binary friction membrane model balance of plant computational fluid dynamics catalyst layer direct numerical simulation electrochemical impedance spectroscopy electroosmotic drag gas diffusion layer hardware in the loop


Fuel Cells – Proton-Exchange Membrane Fuel Cells | Modeling


hydrogen oxidation reaction membrane–electrode assembly microporous layer oxygen reduction reaction polymer electrolyte membrane/protonexchange membrane proton-exchange membrane fuel cell perfluorosulfonic acid polytetrafluoroethylene representative element volume

See also: Electrochemical Theory: Kinetics; Thermodynamics; Fuel Cells – Direct Alcohol Fuel Cells: Direct Methanol: Overview.

Further Reading Bernardi DM and Verbrugge MW (1992) A mathematical-model of the solid-polymer-electrolyte fuel-cell. Journal of the Electrochemical Society 139(9): 2477--2491. Berning T and Djilali N (2003) A 3D, multiphase, multicomponent model of the cathode and anode of a PEM fuel cell. Journal of the Electrochemical Society 150(12): A1589--A1598. Bi W, Gray GE, and Fuller TF (2007) PEM fuel cell Pt/C dissolution and deposition in Nafion electrolyte. Electrochemical and Solid State Letters 10(5): B101--B104. Bird RB, Stewart WE, and Lightfoot EN (2006) Transport Phenomena 2nd edn. New York: Wiley. Broka K and Ekdunge P (1997) Modelling the PEM fuel cell cathode. Journal of Applied Electrochemistry 27(3): 281--289. Djilali N and Sui PC (2008) Transport phenomena in fuel cells: From microscale to macroscale. International Journal of Computational Fluid Dynamics 22(1–2): 115--133. Fimrite J, Struchtrup H, and Djilali N (2005) Transport phenomena in polymer electrolyte membranes – I. Modeling framework. Journal of the Electrochemical Society 152(9): A1804--A1814. Goddard W, Merinov B, Van Duin A, et al. (2006) Multi-paradigm multiscale simulations for fuel cell catalysts and membranes. Molecular Simulation 32(3–4): 251--268. Helfferich F (1962) Ion Exchange. New York: McGraw Hill. Iczkowski RP and Cutlip MB (1980) Voltage losses in fuel-cell cathodes. Journal of the Electrochemical Society 127(7): 1433--1440. Kamarajugadda S and Mazumder S (2008) On the implementation of membrane models in computational fluid dynamics calculations of polymer electrolyte membrane fuel cells. Computers & Chemical Engineering 32(7): 1650–1660. Kaviany M (1995) Principles of Heat Transfer in Porous Media, 2nd edn. New York: Springer. Kreuer KD, Paddison SJ, Spohr E, and Schuster M (2004) Transport in proton conductors for fuel-cell applications: Simulations, elementary reactions, and phenomenology. Chemical Reviews 104(10): 4637--4678. Litster S and Djilali N (2007) Mathematical modelling of ambient airbreathing fuel cells for portable devices. Electrochimica Acta 52: 3849--3862. Lum KW and McGuirk JJ (2005) Three-dimensional model of a complete polymer electrolyte membrane fuel cell – model formulation, validation and parametric studies. Journal of Power Sources 143(1–2): 103--124. Markicevic B, Bazylak A, and Djilali N (2007) Determination of transport parameters for multiphase flow in porous gas diffusion electrodes using a capillary network model. Journal of Power Sources 171(2): 706--717. Mazumder S and Cole JV (2003) Rigorous 3-D mathematical modeling of PEM fuel cells – I. Model predictions without liquid water transport. Journal of the Electrochemical Society 150(11): A1503--A1509.

Paddison SJ (2003) Proton conduction mechanisms at low degrees of hydration in sulfonic acid-based polymer electrolyte membranes. Annual Review of Materials Research 33: 289--319. Park J and Li X (2008) Multi-phase micro-scale flow simulation in the electrodes of a PEM fuel cell by lattice Boltzmann method. Journal of Power Sources 178(1): 248--250. Pathapati PR, Xue X, and Tang J (2004) A new dynamic model for predicting transient phenomena in a PEM fuel cell system. Renewable Energy 30(1): 1–22. Secanell M, Karan K, Suleman A, and Djilali N (2007) Multi-variable optimization of PEMFC cathodes using an agglomerate model. Electrochimica Acta 52(22): 6318--6337. Shah AA, Sui PC, Kim GS, and Ye S (2007) A transient PEMFC model with CO poisoning and mitigation by O-2 bleeding and Rucontaining catalyst. Journal of Power Sources 166(1): 1--21. Shimpalee S, Greenway S, Spuckler D, and Van Zee JW (2004) Predicting water and current distributions in a commercial-size PEMFC. Journal of Power Sources 135(1–2): 79--87. Sivertsen BR and Djilali N (2005) CFD based-modelling of proton exchange membrane fuel cells. Journal of Power Sources 141(1): 65--78. Springer TE, Zawodzinski TA, and Gottesfeld S (1991) Polymer electrolyte fuel-cell model. Journal of the Electrochemical Society 138(8): 2334--2342. Springer TE, Zawodzinski TA, Wilson MS, and Gottesfeld S (1996) Characterization of polymer electrolyte fuel cells using AC impedance spectroscopy. Journal of the Electrochemical Society 143(2): 587–599. Sui PC and Djilali N (2006) Analysis of coupled electron and mass transport in the gas diffusion layer of a PEM fuel cell. Journal of Power Sources 161(1): 294--300. Sui PC, Kumar S, and Djilali N (2008a) Advanced computational tools for PEM fuel cell design – Part 1: Development and base case simulations. Journal of Power Sources 180(1): 410--422. Sui PC, Kumar S, and Djilali N (2008b) Advanced computational tools for PEM fuel cell design – Part 2: Detailed experimental validation and parametric study. Journal of Power Sources 180(1): 423--432. Sun W, Peppley BA, and Karan K (2005) An improved two-dimensional agglomerate cathode model to study the influence of catalyst layer structural parameters. Electrochimica Acta 50(16–17): 3359--3374. Sunde´n B and Faghri M (eds.) (2005) Transport Phenomena in Fuel Cells. Southampton, UK: WIT Press. Taylor R and Krishna R (1993) Multicomponent Mass Transfer (Wiley Series in Chemical Engineering). Wiley-Interscience. Thampan T, Malhotra S, Tang H, and Datta R (2000) Modeling of conductive transport in proton-exchange membranes for fuel cells. Journal of the Electrochemical Society 147(9): 3242--3250. Theodorakakos A, Ous T, Gavaises A, Nouri JM, Nikolopoulos N, and Yanagihara H (2006) Dynamics of water droplets detached from porous surfaces of relevance to PEM fuel cells. Journal of Colloid and Interface Science 300(2): 673--687. Um S, Wang CY, and Chen KS (2000) Computational fluid dynamics modeling of proton exchange membrane fuel cells. Journal of the Electrochemical Society 147(12): 4485--4493. Vishnyakov A and Neimark AV (2000) Molecular simulation study of Nafion membrane solvation in water and methanol. Journal of Physical Chemistry B 104(18): 4471--4478. Wang GQ, Mukherjee PP, and Wang CY (2006) Direct numerical simulation (DNS) modeling of PEFC electrodes – Part II. Random microstructure. Electrochimica Acta 51(15): 3151--3160. Wang Y and Wang CY (2005) Transient analysis of polymer electrolyte fuel cells. Electrochimica Acta 50(6): 1307--1315. Weber AZ and Newman J (2004) Modeling transport in polymerelectrolyte fuel cells. Chemical Reviews 104(10): 4679--4726. Yi JS and Nguyen TV (1998) An along-the-channel model for proton exchange membrane fuel cells. Journal of the Electrochemical Society 145(4): 1149--1159. Zhu X, Sui PC, and Djilali N (2008) Three dimensional numerical simulations of water droplet dynamics in a PEMFC gas channel. Journal of Power Sources 181(1): 101–115.