Journal of The Electrochemical Society, 150 ~12! A1589-A1598~2003!0013-4651/2003/150~12!/A1589/10/$7.00 © The Electrochemical Society, Inc.

A1589

A 3D, Multiphase, Multicomponent Model of the Cathodeand Anode of a PEM Fuel CellT. Berning and N. Djilali z

Institute for Integrated Energy Systems, University of Victoria, Victoria, British Columbia V8W 3P6, Canada

A computational fluid dynamics multiphase model of a proton-exchange membrane~PEM! fuel cell is presented. The modelaccounts for three-dimensional transport processes including phase change and heat transfer, and includes the gas-diffusion layers~GDL! and gas flow channels for both anode and cathode, as well as a cooling channel. Transport of liquid water inside thegas-diffusion layers is modeled using viscous forces and capillary pressure terms. The physics of phase change is accounted for byprescribing local evaporation as a function of the undersaturation and liquid water concentration. Simulations have been performedfor fully humidified gases entering the cell. The results show that different competing mechanisms lead to phase change at bothanode and cathode sides of the fuel cell. The predicted amount of liquid water depends strongly on the prescribed materialproperties, particularly the hydraulic permeability of the GDL. Analysis of the simulations at a current density of 1.2 A/cm2 showthat both condensation and evaporation take place within the cathode GDL, whereas condensation prevails throughout the anode,except near the inlet. The three-dimensional distribution of the reactants and products is evident, particularly under the land areas.For the conditions investigated in this paper, the liquid water saturation does not exceed 10% at either anode or cathode side, andincreases nonlinearly with current density.© 2003 The Electrochemical Society.@DOI: 10.1149/1.1621412# All rights reserved.

Manuscript submitted January 13, 2003; revised manuscript received June 9, 2003. Available electronically November 12, 2003.

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The operation of proton-exchange membrane~PEM! fuel cellsdepends not only on the effective distribution of air and hydrogbut also on the maintenance of an adequate cell operating tempture and fully humidified conditions in the membrane. The fuhumidified state of the membrane is crucial to ensuring good ioconductivity and is achieved by judicious water management. Wcontent is determined by the balance between various water trport mechanisms and water production. The water transport menisms are electro-osmotic drag of water~i.e., motion of water mol-ecules attaching to protons migrating through the membrane fanode to cathode!; back diffusion from the cathode~due to nonuni-form concentration!; and diffusion and convection to/from the aand hydrogen gas streams. Water production depends on the elcurrent density and phase change. Without control, an imbalabetween production and removal rates of water can occur. Thisresult in either dehydration of the membrane, or flooding ofelectrodes, which are both detrimental to performance.

A common water management technique relies on the humidcation of the air and hydrogen gas streams. At higher current deties, the excess product water is removed by convection via thestream, and the rate of removal is controlled by adjusting moiscontent in concert with pressure drop and temperature in the flchannels. Thermal management is also required to remove theproduced by the electrochemical reaction in order to prevent dryout of the membrane, which in turn can result not only in reducperformance but also in eventual rupture of the membrane. Themanagement, which is performed via forced convection coolinglarger stacks, is also essential for the control of the water evaption or condensation rates.

The operation of a fuel cell and the resulting water and hdistributions depend on numerous transport phenomena inclucharge-transport and multicomponent, multiphase flow, and htransfer in porous media. The complexity and interaction of thprocesses and the difficulty in making detailedin situ measurementshave prompted the development of a number of numerical modThe theoretical framework was laid out in early one-dimensionumerical models of the membrane-electrode.1-3 A quasi-two-dimensional model based on concentrated solution theory wasproposed by Newman and Fuller,4 and a full two-dimensional modeincluding flow channels but no electrodes was also presentedNguyen and White.5 This model was refined in a number of subsquent studies to account for the porous electrodes and interdigitgas distribution.6,7

z E-mail: ndjilali@uvic.ca

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One of the first models accounting fully for transport in the gdistribution channels and applying the methods of computatiofluid dynamics for PEM fuel cell was published by Gurauet al.8

This 2D steady-state model included both the MEA and the gas flchannels and considered the gas-liquid phases in separate comtional domains,i.e., the interaction between both phases was nconsidered. A similar model was also presented by Umet al.9 Withthe exception of Ref. 10, the three-dimensional models reportedthe literature were mostly developed using commercial computional fluid dynamic ~CFD! codes.11-14 These models providecomprehensive details on the distribution of reactants and allow ssitivity analysis and prediction of performance under various contions, but they do not account for phase change and liquid waconcentration in the electrodes.

Models providing information on liquid-water saturationflooding include the one-dimensional steady-state model of Bascand Li15 in which the degree of water flooding was determineby matching the predictions to the experimental polarization cuusing a trial-and-error method. Shimpaleeet al.16 extended theirsingle-phase 3D model to account for liquid water as a componof the fluid mixture. The impact of liquid water on transport in thgas-diffusion electrode was, however, not accounted for. Nguyegroup presented a 2D multiphase transport model of the cathgas-diffusion layer.17 The other important contributions are the 2multiphase cathode models18,19 based on Wang’s two-phase flowmixture theory.20 In this approach, each species/phase in the porelectrode is modeled using an individual mass conservation eqtion, but a single momentum equation is solved to obtain the velity field of the entire mixture. This means that each componenconvected according to the global mixture velocity and diffusedcording to a unique diffusivity tensor. The liquid water and vapconcentrations are determineda posterioribased on the temperaturand pressure field. In addition to being restricted to 2D andcathode side of a cell, the simulations presented so far usingmixture model are restricted to low humidity inlet feed streams.

Overall, the coupling of computational fluid dynamics methoologies with electrochemistry has allowed the development ofcreasingly more representative fuel cell models. In the present wseveral new steps have been taken: (i ) the model accounts fully forthree-dimensional transport and two-phase flow; (i i ) both cathodeand anode are included; (i i i ) the two-phase model accounts fomomentum transport in the liquid and gas phase through sepatransport equations and includes exchange terms between phThis model was implemented into the commercial code CFX 4and numerical simulations corresponding to operation with high

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Journal of The Electrochemical Society, 150 ~12! A1589-A1598~2003!A1590

midity inlet are presented and analyzed. The simulations provinsight into the phase-change mechanisms at the cathode sidesreveal some interesting multiphase effects at the anode.

Model Description

Model assumptions.—A schematic of the domain modeled ishown in Fig. 1. The model includes both anode and cathode,reactant flow channels as well as a water cooling channel. Transand gradients in all three-directions are accounted for. The framwork of the single-phase PEM fuel cell model described in Ref.was used. In this paper we focus on modeling of two-phase transand phase change in the electrodes and the prediction of the onspore plugging or flooding under various operating conditions.

The following assumptions are made in implementing the twphase model:~i! no liquid water enters the cells at the inlets;~ii ! thegases entering the cell are fully humidified;~iii ! the product water isin the liquid phase;~iv! two-phase flow inside the porous media cabe described by the unsaturated flow theory~UFT!;20 ~v! the liquidphase and the gas phase share the same pressure field inside thchannels; and (v i ) both phases occupy a certain local volume fration inside the porous media and their interaction is accountedthrough a multifluid approach.

In addition to three-dimensionality and the inclusion of both aode and cathode, the present model differs substantially from eastudies in that momentum transport in the liquid and gas phasaccounted for through separate transport equations that includechange terms between phases. It should be noted that the anodecathode transport are decoupled. The coupling of the two siwould significantly add to the computational cost and complexityan already very involved problem and would also require a detamembrane model accounting fully for the variation of transpoproperties as a function of water content and temperature.

Modeling domain.—Symmetry conditions are applied at thboundaries in they direction and thez direction; thus only half of agas-flow channel needs to be modeled and a quarter of the wcooling channel. This reduces computational requirements forcomplex problem. Two separate computational domains have toset up for the gas-diffusion layers. Domain I consists of the gas-flchannels, separated by the impermeable membrane and the graplates, and domain II consists of the gas-diffusion layers tocommodate the heat transfer through the solid matrix of the pormedium.

Modeling equations.—The model is implemented in a commercial Computational Fluid Dynamics codesCFX 4 using a set ofcustom written user subroutines. The code uses the finite volu

Figure 1. Computational domain employed for the multiphase model. Tmembrane is currently considered a heat-conducting solid.

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approach to solve the discretized Navier-Stokes equations andneric transport equation. The mathematical model for the gas phis essentially identical to the single-phase model presented inearlier study14 with the addition of exchange terms associated wiphase change and modified permeability as discussed below.therefore focus on describing the modeling and implementationthe liquid water transport and phase change.

Gas flow channels.—The liquid water flow inside the gas-flow chan-nels is governed by the Navier-Stokes equations for an incompreible fluid. The continuity equation for the liquid water inside thchannel is given by

¹ • ~r lr lul! 5 0 @1#

where r l denotes the volume fraction of the liquid water,r l is thedensity, andul is the velocity vector. The momentum equation is thstandard momentum equation for an incompressible Newtonian fl

¹ • $@rul 3 ul 2 m~¹ul 1 ~¹ul!T!#% 5 2r l¹p @2#

Gas-diffusion layers.—In practice it is expected that most of thephase change will take place in the porous gas diffusion electrodThe physics of multiple phases through a porous medium, whichchallenging in itself, is further complicated here with phase chanand the sources and sinks associated with the electrochemical rtion. The equations used to describe transport in the gas-diffuslayers are given below.

Mass transfer in the form of evaporation (mphase. 0) and con-densation (mphase, 0) is assumed, so that the mass balance equtions for both phases read

¹ • @~1 2 s!«rgug# 5 mphase @3#

and

¹ • ~s«r lul! 5 mphase @4#

Note that the saturations is identical to the liquid water volumefraction r l and is introduced here simply to keep with the acceptenomenclature. Since the sum of all volume fractions hasequal unity, the volume fraction of the gas-phaser g is equivalent to(12 s).

The momentum equation for the gas phase reduces to Darclaw, which in this case is based on the relative permeability for tgas phasekp

g . The relative permeability accounts for the reduction ipore space available for one phase due to the existence of the sephase. Different approaches can be used to mathematically descthe relative permeability, the simplest of which is21

kpg 5 ~1 2 s!kp

0 @5#

and

kpl 5 skp

0 @6#

wherekp0 is the permeability of the dry electrode ands is again the

saturation of liquid water inside the GDL. With this, the momentumequation for the gas phase reduces to

ug 5 2kp

g

mg¹pg 5 2~1 2 s!

kp0

mg¹pg @7#

Two liquid water transport mechanisms are considered; shear, whdrags the liquid phase along with the gas phase in the directionthe pressure gradient, and capillary forces, which drive liquid wafrom high to low saturation regions.21 Starting from Darcy’s law wecan write

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Journal of The Electrochemical Society, 150 ~12! A1589-A1598~2003! A1591

ul 5 2kp

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where the liquid water pressure stems from the gas-phase preand the capillary pressure according to20

¹pl 5 ¹pg 2 ¹pc 5 ¹pg 2]pc

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Introducing this expression into Eq. 8 yields a liquid water velocfield equation

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where the diffusivityD(s) is defined as20

D~s! 5 2skp

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]pc

]s@11#

The functional variation of capillary pressure with saturationpc(s)is prescribed following Leverett22 who has shown that

pc 5 sS «

kp0D 1/2

f ~s! @12#

wheres is the interfacial liquid/gas tension,« is the porosity, andf (s) is the empirical function proposed by Udell23

f ~s! 5 1.417~1 2 s! 2 2.12~1 2 s!2 1 1.263~1 2 s!3

@13#

Note that the above expression merely describes the overall shathe capillary pressure function as a function of saturation, whichof cubic nature, whereas Eq. 12 provides the order of magnituwhich is predominantly determined by the permeabilitykp

0 . Thisparameter varies over several orders of magnitude in the litera~e.g., Ref. 2 and 18!.

An alternative expression for the liquid water velocity was sugested by Natarajan and Nguyen,17 who, however, neglected theconvective part of the liquid water velocity

ul 5 2skp

0r l

m lgS 2

]c

]s D¹s @14#

wherer l is the liquid water density andg is gravity. The term]c/]sis the functional that describes the dependence of permeabilitycapillary head on saturation, according to

]c

]s5 2A 3 D$eA(s 2 C) 1 e2A(s2C)% @15#

whereA, C, andD are constants equal to 3.7, 0.494, and 0.01cm, respectively. Again a liquid water diffusivity can be defined

D~s! 5 skp

0r l

m lgS 2

]c

]s D @16#

Figure 2 shows the liquid water diffusivities prescribed by Waet al. based on Leverett’s relation and using a dry electrode perability of kp

0 5 10210 cm2. The corresponding liquid water diffu-sivities obtained by Natarajan and Nguyen17 are four orders of mag-nitude smaller, which means that extremely high saturation gradiwould be necessary in order to induce the flux of liquid water. Texplains why the results reported in Ref. 17 show liquid water sarations close to unity under the shoulders of the collector platenear the catalyst layer. It should be noted that the function propoby Leverett is based on experimental data in soil-like porous me

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whereas Natarajan and Nguyen adjusted their data to fit their pdicted polarization curve to measurements. It is nonetheless difficto explain a four-order of magnitude differences between the liquwater diffusivity in soils and a porous gas diffusion layers, anclearly experimental data is required to resolve this issue. Inpresent study we use Leverett’s function to determine the liquwater diffusivity.

The liquid phase consists of pure water, while the gas phasemulticomponents. The transport of each species in the gas phasgoverned by a general convection-diffusion equation in conjunctiwhich the Stefan-Maxwell equations to account for multispecies dfusion, as described in Ref. 14, 24, with the addition of a sourterm accounting for phase change

¹ • @r g~«grgugygw 2 «grgDgi¹ygw!# 5 «g~r gSgw 1 mphase!

@17#

whereygw is the mass fraction of water vapor.The energy equation reads as follows

¹ • @r g~«grgugHg 2 «glg¹Tg!# 5 2«gmphaseDHevap @18#

where DHevap denotes the heat of evaporation or condensation@J/kg# at 80°C. The gas phase and the liquid phase are assumed tin thermodynamic equilibrium,i.e., the liquid water and the gasphase are at the same temperature.

Implementation of phase change.—In order to account for the mag-nitude of phase change inside the GDL, expressions are requirerelate the level of over- and undersaturation as well as the amounliquid water present to the amount of water undergoing phachange. In the case of evaporation, such relations must be depenon (i ) the level of undersaturation of the gas phase in each convolume and on (i i ) the surface area of the liquid water in the controvolume. The surface area can be assumed proportional to the volfraction of the liquid water in each cell. A plausible choice for thshape of the liquid water is droplets, especially since the cataarea is Teflonated.

The evaporation rate of a droplet in a convective stream depeon the rate of undersaturation, the surface area of the liquid dropand a~diffusivity dependent! mass-transfer coefficient. The flux owater due to phase change can be represented by25

Nw 5 kxmpD2xw0 2 xw`

1 2 xw0@19#

Figure 2. Variation of the liquid water diffusivity as a function of saturationusing the Leverett’s relation.

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Journal of The Electrochemical Society, 150 ~12! A1589-A1598~2003!A1592

whereD is the diameter of the droplet,xw0 is the molar concentra-tion of water at the gas/liquid interface,xw` is the bulk concentra-tion of water vapor,kxm is the transfer rate of water in@mol/(m2s)#

andNw is the flux of water from the liquid phase into the gas phain @mol/s#. The bulk concentrationxw` is known by solving thecontinuity equation of water vapor. To obtain the concentrationthe surface, it is reasonable to assume thermodynamic equilibbetween the liquid phase and the gas phase at the interface,i.e., therelative humidity of the gas in the immediate vicinity of the liquid100%. Under that condition, the surface concentration can be calated based on the saturation pressure and is only a functiotemperature.

The heat-transfer coefficient for convection around a spherwell established, and by invoking the analogy between convecheat and mass transfer, the following mass-transfer coefficientkxm isobtained25

kxm 5cgDwg

D F2.0 1 0.60S Dv`rg

mgD 1/2S mg

rgDwgD 1/3G @20#

wherecg is the concentration of air in@mol/m3#, Dwg is the diffu-sion coefficient of water-vapor in air in@m2/s#, v` is the free-streamvelocity in @m/s#, and rg is the air density in@kg/m3#. All theseproperties can be readily calculated. It should be noted that bothheat-mass transfer analogy and the above correlation were defor boundary-layer-type of flow. Fundamental work is requiredestablish alternative relations for droplet evaporation in a pormedium. In any case since only an order of magnitude estimatthe evaporation rate is required for the numerical procedure to aequilibrium, this approach is adequate in the context of this stud

It is further assumed that all droplets have a specified diamD, and the number of droplets in each control volume is founddividing the total volume of the liquid phase in each control volumby the volume of one droplet

ND,CV 5r lVCV

1

6pD3

@21#

The above expression can be used to obtain an order of magnfor the rate of evaporation of the liquid water in each control vume. Because the choice of nominal droplet size is subject to a luncertainty, along with the assumption that the droplets either eor not ~they do not shrink due to evaporation! the overall expressionis scaled by a factorc, so that the overall molar flux for a givencontrol volume is given by

Nw 5 cND,CVkxmpD2xw0 2 xw`

1 2 xw0@22#

The sensitivity of the solution toc was assessed by varying its valuby orders of magnitude. For evaporation-driven cases, the relahumidity of the gas phase was found to remain 100% for valuesc several orders of magnitude smaller than 1.0, confirming thatrate of evaporation is indeed fast enough to justify the commoused assumption of a fully humidified gas phase.

In the case when the calculated relative humidity in a contvolume exceeds 100% condensation occurs. The case of condtion is more complex, because it can occur on every solid surfarea, but the rate of condensation can be different when it taplace on a wetted surface. In addition, the overall surface areeach control volume available for condensation shrinks with ancreasing amount of liquid water present. It the current implemention, we assume the rate of condensation depends only on theof oversaturation of the gas phase multiplied by a constant. Thibelieved to be a good approximation for low saturation valuescase the saturation is high, the shrinkage of the available conde

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tion surface area with an increasing saturation has to be accountfor. We see in the Results section, however, that the predicted locsaturation values do not exceed 10%. For operating conditions odesigns resulting in high levels of liquid saturation, a more generaapproach would need to be implemented. Thus

mcond 5 2cCxw0 2 xw`

1 2 xw0@23#

Catalyst layers.—The catalyst layers are treated as thin interfaceswhere the oxygen and hydrogen are depleted and liquid water anheat are produced. The depletion and production rates depend onlocal current densityi , which is described by the Tafel equation. Incommon with most other multidimensional modeling studies, a simplification is introduced by assuming a constant local activationoverpotentialh. The local current density at the cathode side be-comes thus a function of the local oxygen distribution only. Thisapproach allows the use of the average current density as an inpvariable when performing systematic simulations for various operating parameters such as the stoichiometric flow ratio. For detailsplease refer to Ref. 14 and 24.

Cooling water channel.—In the water channel, the Navier-Stokesequations and the energy equation for an incompressible fluid asolved,i.e.,the continuity equation

¹ • ~r lul! 5 0 @24#

the momentum equation

¹ • ~r lul 3 ul 2 m l¹ul! 5 2¹S p 12

3m l¹ulD

1 ¹ • @m l~¹ul!T# @25#

and the energy equation

¹ • ~r lulH l 2 l l¹Tl! 5 0 @26#

where the total enthalpyH is calculated using the static~thermody-namic! enthalpyh

H l 5 hl 11

2ul

2 @27#

Table I. Geometrical, operational, and transport parameters atbase case conditions.

Parameter Symbol Value Unit

Channel length l 0.03 mElectrode thickness te 0.203 1023 mInlet fuel and air temperature T 60 °CAir side pressure pc 1 atmFuel side pressure pa 1 atmRelative humidity of inlet gases j 100 %Electrode porosity « 0.5 -Dry hydraulic permeability kp

0 10214 m2

Condensation constant C 1025

Surface tension s 6.25 N/cmEnthalpy of evaporation DHevap 2.363 106 J/kgLiquid water diffusivity in channels D l 1025 m2/sWater-vapor diffusivity Dwg 2.923 1025 m2/sBinary diffusivity (Tref 5 307.1 K) DH2-H2O 0.915 cm2/sBinary diffusivity (Tref 5 298.0 K) DH2-CO2

0.646 cm2/sBinary diffusivity (Tref 5 307.5 K) DH2O-CO2

0.202 cm2/sBinary diffusivity (Tref 5 308.1 K) DO2-H2O 0.282 cm2/sBinary diffusivity (Tref 5 293.2 K) DO2-N2

0.220 cm2/sBinary diffusivity (Tref 5 307.5 K) DH2O-N2

0.256 cm2/s

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Journal of The Electrochemical Society, 150 ~12! A1589-A1598~2003! A1593

The fluid in the cooling channels is assumed to be pure liquid wathence, no additional species equation is required.

Graphite plates.—The collector plates consist of graphite and servto transfer electrons towards the gas-diffusion layers and to theaction sites~current collectors!. Currently, only heat conduction isconsidered in the solid plates

¹ • ~lgr¹T! 5 0 @28#

Boundary conditions.—Symmetry boundary conditions are ap-plied in thez direction and they directions, thereby reducing thesize of the computational domain and computational costs. In thexdirection, zero flux conditions are applied at all interfaces except fthe flow channels. At the inlets of the gas-flow channels, the incoming velocity is calculated as a function of the desired current densand stoichiometric flow ratio, as described in Ref. 14. At the outletthe pressure is prescribed for the momentum equation and a zegradient conditions is imposed for all scalar equations. For the wacooling channel, the inlet velocity and the temperature are specifi

Modeling parameters.—With the exception of the parameterslisted in Table I, the physicochemical and geometric parametersidentical to those used in the single-phase simulations of Berniet al.14 The reference binary diffusivities for the Stefan-Maxwelequation are listed and were scaled for temparture and pressusing

Figure 3. Average molar oxygen concentration at the catalyst layer as funtion of the current density.

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reg

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D ij 5 D ij~T0 ,p0!p

p0S T

T0D 1.5

@29#

In order to reduce the computational requirements, which are muhigher for the two-phase flow model, the channel length has beshortened from 5 to 3 cm. The thickness of the gas diffusion layer0.2 mm with a porosity of 0.5. Operation is assumed at atmosphepressure and at a nominal temperature of 60°C, and the incomgases are fully humidified. It was mentioned before that in the mutiphase case the permeability of the dry electrode becomes onethe central parameters. A value of 10210 cm2 is used, which is con-sistent with the value used by Natarajan and Nguyen.17 As the val-ues for the permeability cited in the literature range over ordersmagnitude, this will be an important parameter to investigate in thfuture.

The last parameters listed in Table I are specific to the multiphaflow and phase change. The condensation constant has been adjuso that the relative humidity inside the gas diffusion layers remainbelow 100.1%, assuming fast condensation, which is not limited bmass transport to and from the condensation surfaces. A standvalue was used for the enthalpy of evaporation~e.g., Ref. 26!.

The droplet size and the scaling parameter for evaporation weadjusted so that the relative humidity inside the gas diffusion layremained above 99.9% in the presence of liquid water, again assuing a fast rate of phase change.

Numerical procedure.—The model presented here was implemented into a commercial computational fluid dynamics code~CFX4.3 from AEA Technology!. In order to accommodate the varioustransport mechanisms, an extensive suite of user subroutines habe implemented along with customized iteration procedures.single data point required approximately 10,000 iterations, and moof the simulations were performed on a single node of an IBM SPcomputer. Due to the complexity of this model with a large spatiavariation in competing transport and phase-change mechanismconvergence was assumed when the overall mass imbalancebelow 0.1%.

Results and Discussion

Prior to proceeding with a detailed analysis of the results, ituseful to discuss some of the phase-change mechanisms. The ceparameter for determining the direction of phase change is the retive humidity of the gas phase

f 5pH2O

psat(T)@30#

c-

Figure 4. Molar oxygen~left! and water vapor~right! fraction inside the cathodic gas-diffusion layer at a current density of 1.2 A/cm2. The cathode-catalystlayer correspond to the upper side and the cathode-channel interface to the lower.

Journal of The Electrochemical Society, 150 ~12! A1589-A1598~2003!A1594

Figure 5. Pressure drop@Pa# and temperature distribution@K# inside the cathodic gas-diffusion layer at a current density of 1.2 A/cm2. ~Cathode-catalyst layercorresponds to the upper side and the cathode-channel interface to the lower.!

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i.e., the ratio of partial pressure of the water vapor in the gas phto the saturation pressurepsat, which is a function of temperatureonly. According to Dalton’s law the partial pressure of a speciesequal to its molar fractionx multiplied with the total pressure of thegas phasepg , i.e.,

f 5 xH2O

pg

psat(T)@31#

When the relative humidity is below 1.0~or 100%! in the presenceof liquid water, this give rise to evaporation. Condensation, onother hand, occurs when the relative humidity exceeds 100% inpresence of condensation surfaces, which are in abundance inthe gas-diffusion layer. The gas-diffusion layer of a PEM fuel cellparticularly interesting for phase-change considerations, becausthree parameters on the right side of Eq. 31 vary, resulting in copeting directions of phase change as follows.

1. The molar water fractionxH2O increases inside the GDL, simply as a result of reactant consumption. Provided the relative humity of the incoming air is at 100%, this process alone would leadcondensation.

2. The thermodynamic pressurepg of the gas-phase changes inside the GDL. This is a very interesting effect and, depending onincoming gas condition, it can lead to either evaporation or condsation. In the first place, there is a pressure drop inside the GDLagain to reactants consumption. This pressure drop depends stro

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on the permeability of the gas-diffusion layer,i.e., for the sameamount of consumed reactants, the pressure drop will be highera lower permeability. The bulk velocity of the gas phase is directeinto the GDL, and is governed by Darcy’s law. Thus, the pressudrop inside the GDL depends strongly on the permeability. The patial pressure of the water vapor decreases with the gas-phase psure pg . Hence, this effect alone would lead to undersaturatiocausing evaporation. A special case arises when the incoming airelatively dry, in which case most of the product water will evaporate. The cathodic half-cell reaction produces two water moleculfor each oxygen molecule consumed. This means that for hievaporation rates, every oxygen molecule inside the gas phasebe replaced by two water vapor molecules, thereby causing ancrease in the gas-phase pressure. As a result, the bulk flow of thephase would then be directed from the catalyst layer towards tchannel, as has been observed by Wanget al.18 This means that theoxygen would have to diffuse towards the catalyst interface againthe bulk flow of the gas phase, which will possibly cause a decreain the limiting-current density.

3. The saturation pressurepsat(T) increases with an increase intemperature caused by the heat production term due to the elecchemical reaction. The order of magnitude of the temperaturecrease depends primarily on the thermal conductivity of the gadiffusion layer. It was found with an earlier version of this modethat the temperature can rise by a few degrees Kelvin,14 and thiseffect alone would lead to evaporation of liquid water.

Figure 6. Rate of phase change~left! in @kg/(m3 s)# and liquid water saturation~right! inside the cathodic gas-diffusion layer at a current density of 1.2 A/m2.~Cathode-catalyst layer correspond to the upper side and the cathode-channel interface to the lower.!

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Journal of The Electrochemical Society, 150 ~12! A1589-A1598~2003! A1595

Figure 7. Gas-phase velocity vectors~left! and liquid water velocity vectors~right! inside the cathodic gas-diffusion layer at a current density of 1.2 A/cm2.The vector scaling is 5@~cm/s!/cm# for the gas phase and 200@cm/~cm/s!# for the liquid phase.~Cathode-catalyst layer correspond to the upper side and t

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The net phase change is a result of the balance between tcompeting, coupled, and spatially varying mechanisms. It shouldnoted that the first two effects are also of importance inside theflow channels; the oxygen depletion from inlet to outlet resultsoversaturation and condensation at the walls and channel/GDLterface, whereas the overall pressure drop along the channel walone cause evaporation. For the straight channel section considhere, the total pressure drop is relatively small and hence the oxydepletion effect dominates, causing condensation. The dominmechanisms highlighted in this discussion are relevant to cawhere the incoming air is at a high humidification level, as is thcase in practical fuel cell operation.

Cathode side transport.—Investigations of phase-change behaior are usually focused on the cathode side, since this is where wis produced. Also, mass-transport limitations are almost alwayssociated with the cathode. An estimate of the limiting current desity for the case considered in this paper can be obtained by exining the average molar oxygen fraction at the catalyst layerfunction of current density, shown in Fig. 3. The maximum curredensity predicted is about 2.4 A/cm2. This relatively high current isdue to the high oxygen molar fraction brought about by the lowater vapor molar fraction dictated by the relatively low operatintemperature of 60°C, as well as to the high stoichiometric flow raof 3.0 and a relatively high gas diffusion layer of 0.5. A detailenumerical parametric study27 shows that these parameters greatinfluence the limiting-current density, and a recent experimen

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parametric study reports comparable limiting current densities fsimilar operating and humidification temperatures.28

The discussion and detailed analysis focuses on simulations cresponding to a current density of 1.2 A/cm2. Figure 4 shows theoxidant and water vapor distributions inside the cathode. Note thas pointed out earlier, the simulations take advantage of the symmtry of the system and only include half of channel and half of thland area. For clarity, the solutions are mirrored and a full channand the adjacent land areas are displayed in the results. The oxyconcentration at the catalyst layer is qualitatively similar to thapredicted in single-phase simulations. It is highest under the chanel, and exhibits a three-dimensional behavior with a fairly significant drop under the land areas, and a more gradual depletionwards the outlet. The molar water vapor fraction, however, remaialmost constant throughout the gas-diffusion layer. In the absencephase change, this would not be the case, as the nitrogen and wvapor fraction would increase as the oxygen fraction decreases.

Figure 5 shows the pressure distribution and temperature distbution inside the cathodic gas-diffusion layer at the same curredensity. Clearly, the pressure drop inside the gas-diffusion layer vaies strongly with the hydraulic permeability of the carbon fiber paper. The permeability is set at 10210 cm2 in this study, but as pointedout earlier, values ranging over several orders of magnitude afound in the literature. The magnitude of the pressure drop signicantly impacts the magnitude of the evaporation term accordingEq. 31; this and the effect of varying permeability was beyond th

Figure 8. Molar hydrogen fraction~left! and pressure drop@Pa# inside the anodic gas-diffusion layer at a current density of 1.2 A/cm2. ~Cathode-channelinterface correspond to the upper side and the cathode-catalyst interface to the lower.!

Journal of The Electrochemical Society, 150 ~12! A1589-A1598~2003!A1596

Figure 9. Rate of phase change~left! in @kg/(sm3)# and liquid water saturation~right! inside the anodic gas-diffusion layer at a current density of 1.2 A/cm2.~Cathode-channel interface correspond to the upper side and the cathode-catalyst interface to the lower.!

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scope of the present study. It should be noted that in the caselyzed here a pressure drop occurs partly because of the fully humfied incoming gas at the cathode side. With a dry inlet cathodestream, most of the product water will evaporate, but, sinceevery reacting oxygen molecule two water molecules are produin the gas phase, the gas phase pressure will therefore increase.can in fact be observed in the low inlet humidity simulations oWanget al.18 The effect of phase change on the temperature disbution is small overall in this particular case, mostly because offully humidified inlet gases that result in a low amount of net phachange. The small temperature increase at the catalyst layer ismarily due to the heat production term implemented here.14

The rate of phase change and the resulting liquid water saturaare shown in Fig. 6. Slight condensation occurs almost throughthe GDL ~values being between22 and 0!. The condensationcaused by the oxygen depletion effect appears to be the domineffect in this particular case. A strong evaporation term exists atinterface between the GDL and the channel, particularly at the inIn this area the oxygen depletion effect is not yet significant andoutweighed by the combined effect of pressure drop and temperaincrease. The resulting liquid water saturation in the cathodic Gis below 10% throughout the entire GDL, with maximum saturatiofound under the land area. The high spatial variation of the satution demonstrates again the three-dimensional nature of transprocesses in PEMFCs. The range of liquid water saturations pdicted here is similar to that reported by Wanget al.18 and You

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and Liu.19 Natarajan and Nguyen17 reported liquid water saturationlevels close to 100% near the catalyst and under the shouldersspite of the completely dry incoming gas stream. As discussed elier this is a consequence of the extremely low liquid water diffusivity chosen in that study. Clearly, liquid water saturation depenstrongly on the specified capillary pressure function~Eq. 12 and 13!and again, the permeability of the gas-diffusion layer becomes tcentral parameter.

The velocity field inside the cathodic gas-diffusion layer ishown in Fig. 7. The pressure gradient induces bulk gas flow frothe channels into the GDL. This is in contrast with the low humiditinlet stream case~high evaporation! results of Wanget al.where theflux is directed from the GDL to the channel. The maximum velocitin they direction is around 7.83 1023 m/s, and is thus of the sameorder of magnitude as that reported in Ref. 18. The liquid water fluis directed from the GDL into the channel,i.e., in the opposite di-rection of the gas-phase velocity. The highest value of the liquiphase velocity is around 2.53 1024 m/s, occurring under the el-bow of the collector plate land. Inside the GDL the liquid watevelocity is in the range of 1025 m/s.

Anode side transport.—One of the unique features of the multiphasmodel presented here is that the anode side is included as well. Tunderlying mechanisms of phase change are the same for cathand anode sides, but their magnitudes differ and in addition tphase-change process takes place in a binary rather than a ter

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Figure 10. Gas-phase velocity vectors~left! and liquid water velocity vectors~right! inside the anodic gas-diffusion layer at a current density of 1.2 A/cm2. Thevector scaling is 5@~cm/s!/cm# for the gas phase and 200@~cm/s!/cm# for the liquid phase.~Cathode-channel interface correspond to the upper side andcathode-catalyst interface to the lower.!

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Journal of The Electrochemical Society, 150 ~12! A1589-A1598~2003! A1597

Figure 11. Average liquid water saturation inside the gas-diffusion layersa function of current density.

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mixture. Figure 8 shows the gas-phase pressure and the hydrdistribution inside the anodic gas-diffusion layer. The molar hydgen fraction is almost constant inside the GDL. This may sesurprising initially, but is explained by the fact that consumptionhydrogen leads to a direct increase of the molar water vapor fracsince the anode gas steam is a binary mixture, and hence a highof oversaturation~see Eq. 31! results in a strong condensation potential inside the gas-diffusion layer. This in turn affects the gphase pressure. The pressure drop inside the anodic gas diffulayer is higher than the one at the cathode, because of thecondensation terms here. The virtually constant hydrogen mfraction is a particularly interesting feature when considering ttransport of the reactants towards the catalyst is predominantlydiffusion.

Figure 9 shows the rate of evaporation and the liquid water sration in the anodic gas-diffusion layer. As pointed out above, cdensation occurs throughout the GDL due to hydrogen depletLiquid water is thus produced inside the anodic GDL, which shohelp counteract the tendency for membrane dehumidificationserved on the anode side in previous studies.3 Transport or drivingout of ~condensed! liquid water in the GDL towards the channerequires the build up of a sufficient capillary pressure. The resulliquid water saturation is presented in the right side of Fig. 9. Toverall saturation for the conditions considered is lower than atcathode side. Since the predicted saturation levels depend primon the capillary pressure function, it is clear that the permeabilitythe carbon fiber paper plays a determining role.

The velocity of both gas and liquid phases at the anodeshown in Fig. 10. The gas-phase velocity direction is alwaysrected from the channel to the GDL, as there is no product watercan evaporate. The maximum velocity at the anode side is relatihigh, around 1.43 1022 m/s. The liquid water velocity field in theanodes exhibits a similar pattern to that in the cathodes with a mmum velocity of about 1.13 1024 m/s.

Bulk mass-transport balance.—Figure 11 shows the liquid watesaturation, averaged over the entire cell, as a function of curdensity. The average liquid water saturation inside the gas diffuslayers increases in a nonlinear monotonic fashion for both anodecathode sides. The volume fraction of liquid water remains be10% at the cathode side and 6% at the anode side for all curdensities. This suggests that it should be possible to keep the mbrane fully humidified without an external humidification schemparticularly on the anode side, and it would be interesting to invtigate and verify this experimentally. A liquid water saturation valof 10% is far below the pore-plugging regime, and it will be inteesting to explore in the future under what conditions this levelsaturation rises and to what extent.

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The overall mass flow balances for cathode and anode sideshown in Fig. 12 and 13, respectively. In both cases, the amounliquid water leaving the cell increases almost linearly with the curent density, and the mass flow rate is roughly an order of magnitsmaller than the mass flow of the gas phase. The small differebetween outlet and inlet mass flow rates at the cathode is anoindication of the relatively small evaporation rate of product watThis is also demonstrated by examining the net phase change ploin Fig. 14 for various current densities. The net phase changnegative on both cathode and anode sides, implying that on aveand in addition to product water, incoming water vapor condenwhen transiting through the cell. The zero line corresponds tosituation where all product water would leave the cell in the liquphase.

Conclusions

A three-dimensional computational model of a PEM fuel cell hbeen developed. This model includes multiphase multicompontransport and heat transfer for both cathode and anode. Simulatperformed with this model for operation with fully humidified reactant streams show that the liquid water volume fraction~saturation!increases nonlinearly with increasing current density. Phase chais driven by the balance of three separate processes, temperchange, reactant gas depletion, and pressure drop inside thewhich combined contribute to evaporation or condensation in a nintuitive manner. The predicted saturation levels depend heavilythe permeability of the diffusion material. A systematic parametstudy of the effect of permeability was beyond the scope ofpresent study, but results indicate that for large values of the Gpermeability, liquid water saturation in excess of 20% can be

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Journal of The Electrochemical Society, 150 ~12! A1589-A1598~2003!A1598

tained at the anode side. These saturation levels decrease withmeability because all of the liquid water at the anode is producedcondensation at the channel/GDL interface. At the cathode sideopposite is true, the liquid water saturation increases with decreapermeability, because most of the water is produced via the elecchemical reaction and can become trapped inside the GDL.level of saturation decreases with increasing permeability.

The simulations presented in this paper clearly illustratethree-dimensional nature of the transport mechanisms, fluxes,resulting distributions of reactants, temperature, pressure, and wThe model provides several improvements and allows simulatiunder representative operating conditions, including high inletmidity, but the added complexity comes with a significantly highcomputational overhead. In addition to investigating the effectpermeability and wetting~Teflonation! characteristics of the gas diffusion electrode material, future work will focus on incorporatin

Figure 14. Rate of phase change inside the gas diffusion layers. Negavalues indicate overall condensation. Average rate of production of wthrough the electrochemical reaction is shown for reference.

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membrane transport and fully coupling the anode and cathode tranport, as well as on implementing more efficient numerical strategieincluding parallel computing to allow practical use of the model fosystematic parametric studies. It is clear that further progress othese fronts will require not only validation data under well-documented operating conditions but also experimental data fmany of the transport parameters of the gas-diffusion electrode.

The University of Victoria assisted in meeting the publication costs of tharticle.

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