Twodimensional modeling of electrochemical and transport phenomena in the porous structures of a PEMFC
Melik Sahraoui
a
, Chaﬁk Kharrat
b
, Kamel Halouani
b,
*
a
Institut Pre´ paratoire aux Etudes d’Inge´ nieurs de Tunis (IPEIT), Tunisia
b
UR: MicroElectroThermal Systems (METSENIS), Industrial Energy Systems Group, Institut Preparatoire aux Etudes d’Ingenieurs de Sfax(IPEIS), University of Sfax, B.P: 1172, 3018 Sfax, Tunisia
a r t i c l e i n f o
Article history:
Received 1 October 2007Received in revised form3 November 2008Accepted 4 November 2008Available online 18 February 2009
Keywords:
PEM fuel cellNumerical modeling Finite volumeHeatMass and charge transfer
a b s t r a c t
A twodimensional CFD model of PEM fuel cell is developed by taking into account theelectrochemical, mass and heat transfer phenomena occurring in all of its regions simultaneously. The catalyst layers and membrane are each considered as distinct regions withﬁnite thickness and calculated properties such as permeability, local protonic conductivity,and local dissolved water diffusion. This ﬁnite thickness model enables to model accurately the protonic current in these regions with higher accuracy than using an inﬁnitesimal interface. In addition, this model takes into account the effect of osmotic drag in themembrane and catalyst layers. General boundary conditions are implemented in a waytaking into consideration any given species concentration at the fuel cell inlet, such aswater vapor which is a very important parameter in determining the efﬁciency of fuel cells.Other operating parameters such as temperature, pressure and porosity of the porousstructure are also investigated to characterize their effect on the fuel cell efﬁciency.
ª
2008 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rightsreserved.
1. Introduction
As the human needs for energy increase and with them theprices of fossil fuels, it is becoming vitally essential to developalternative energy sources with minimum negative impact onthe environment. The fuel cell technology has been identiﬁedas one of the alternatives that offers better efﬁciency than thecombustion of hydrocarbons while emitting less pollution.The fuel cell is an electrochemical device that convertschemical energy directly into electricity with heat and wateras byproducts. The Proton Exchange Membrane Fuel Cell(PEMFC) is one of the fuel cell technologies that is widely usedbecause of its lower operating temperature with applicationsranging from powering vehicles to mini and micro devicessuch as laptops or cell phones.In recent years a lot of progress has been achieved in thedevelopment of cost effective fuel cell technologies, whichenabled more widespread use of this technology. However,moreresearch in design optimization and improved materialsneeds to be done in order to make the fuel cell technologyubiquitous. For the design optimization, modeling of thetransport phenomena and chemical reactions for two orthreedimensional representations of the fuel cells are verycritical in the development of efﬁcient fuel cells.Recently research on PEMFC technology has receiveda great deal of attention focusing on materials, size optimization, water and thermal management, and reliability. Instudying transport and electrokinetics, many researchershave focused on different aspects of the PEMFC and theapproaches in dealing with the problem are quite varied. Fuel
*
Corresponding author
. Tel.:
þ
216 74 241 223; fax:
þ
216 74246347.Email address: kamel.halouani@ipeis.rnu.tn (K. Halouani).
Available at www.sciencedirect.comjournal homepage: www.elsevier.com/locate/he
03603199/$ – see front matter
ª
2008 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.ijhydene.2008.11.012
international journal of hydrogen energy 34 (2009) 3091–3103
cellmodelscanbecategorizedasanalytical,semiempiricalornumerical as summarized by Chedie and Munroe [1].Simple analytical models limited to onedimensionalrepresentations, such as the ones developed by Standaertet al. [2,3] and by Charradi and Halouani [4], are able to predict
the cell polarization curve for a given operating currentdensity using ohm’s law and Butler–Volmer equation.However, analytical models are unable to predict accuratelytransfer processes inside the cell but remain useful for quickcalculations of simple systems. More detailed analysis of themultiple transport and electrochemical mechanisms requiresthe use of numerical models (CFD).Recently numerical modeling from one to threedimensional representations of fuel cells has received the mostattention.BernardiandVerbrugge[5]presentedasteadystate,one dimensional, isothermal model for a fully hydratedmembrane. This model studied the species transport in thegas, liquid and solid phases, the inﬂuence of the porosity of the electrodes and the effects of membrane properties. Theyidentiﬁed inefﬁciencies due to unused species and low catalyst utilization at practical operating current densities. YouandLiu[6]presentedanisothermalonedimensionalmodelof the cathode catalyst layer. Their model predicts the currentlimit condition at high current densities. Their model showsthattheporosityandpolymercontentinthecatalystlayerhasto be optimized in order to produce the highest currentdensity. Fuller and Newman [7] presented a transient, twodimensional, nonisothermal membrane electrode assemblymodel. The model showed that the performance of these cellsimproves when gas streams are saturated with water. Gurauet al. [8] used a steady state, two dimensional, nonisothermalmodel describing regions of the gas channels, gas diffusers,catalyst layers and membrane. Their modelshows that a nonuniform reactant distribution has an important impact on thecurrent density. But, this model is unable to predict theconcentration overpotential due to mass transport limitationin the catalyst layer at high current densities.Eaton[9]usedatransient,onedimensionalmodelforheat,mass and charge transfer in a PEM (Proton ExchangeMembrane). Through this model, which accounts for a pressure gradient across the fuel cell, the authors concluded thatthe water ﬂux from anode to cathode increases with current.But the water ﬂux may be offset by the application of a pressure gradient between the anode and cathode. Genevey [10]presented a transient, onedimensional comprehensivemodel of the cathode catalyst layer. The model is based on anagglomerate catalyst layer structure and includes the transport of thermal energy, gas species, liquid water, and chargetransfer. Nguyen et al. [11] developed a threedimensionalmodel including the bipolar plates with serpentine gas ﬂowchannels. Their model does not take into account the liquidwater phase and the local overpotential on the cathode side iscalculated locally from the potential equation. The overpotential on the anode side is neglected. Their model showedthe current nonuniformity due to the presence of the solidplates and it is found that local maximal values for currentproduction change with cell usage conditions.Ramousse et al. [12] developed a steady state and onedimensional model for heat and mass transfer in a PEMFC.Their results show that temperature gradient in themembrane can be affected by temperature difference of the inlet gases and this effect can be much bigger than the
Nomenclature
A
area
c
concentration
c
p
speciﬁc heat
D
mass diffusivitye Thickness
F
Faraday’s constant
H
fuel cell overall height
I
current density
j
volumetric current density
h
a
anode height
h
c
cathode height
K
permeability
k
thermal conductivity
L
fuel cell lengthM molar massp pressure
R
universal gas constant
S
source term
T
temperature
u x
direction velocity
u
average inlet velocity
v y
direction velocity
V
th
theoretical voltage
V
cell
working cell voltage
y
i
mass fraction
Greek
l
water content
3
porosity
4
potential
s
protonic conductivity
m
dynamic viscosity
r
density
Subscripts
0 reference valueAmb ambienta anodec cathodeCL catalyst layerdrag frictional drag DW dissolved watere electronic
i,j
species (H
2
, O
2
, H
2
O, N
2
)hs heatsink source termm membraneohm ohmiceff effectiverev reversibleth theoreticalWV water vapor
international journal of hydrogen energy 34 (2009) 3091–3103
3092
gradient due to resistive heating. Their results also show thatthe temperature is greatly affected by the thermal conductivity of the membrane. Hmidi and Halouani [13] developeda steady state, onedimensional numerical model of a PEMFCstudying the effect of temperature, pressure and currentdensity. This model showed that the water content increaseswith current density but decreases with an increase of temperature or pressure whereas the membrane potentialincreases with both temperature and pressure.Siegel et al. [14] developed a twodimensional model forPEMFC with a modiﬁed Butler–Volmer model in order to takeinto account the agglomerate geometry and take into accountthe pore level mass transfer limitation on the performance of the fuel cell. Their model uses the Thiele’s modulus to incorporate the pore level mass transfer limitation. Siegel et al. [15]extended the previous work to include the effect of liquidwater and have improved further the electrochemical model.Their results show that the agglomerate geometry and liquidwater inﬂuence signiﬁcantly the efﬁciency of the fuel cell.Sivertsen and Djilali [16] developed threedimensionalmodel for PEMFC using a commercial CD tool (Fluent) withuser deﬁned functions (UDF) to incorporate the electrochemistry in the membraneelectrode assembly. In theirmodel they assumed that the membrane is fully humidiﬁed,therefore, limiting the validity of the model to high inletrelative humidity. Phase change for water vapor is not takeninto account. The results produced by their model were ingood agreement with the measurements. However, the modeldoes not predict, the accurate variation of the slope of thepolarization curve when the fuel cell is operating at highcurrent density and near the current limit.Um and Wang [17] examined water transport in PEMFCusinga 3D model. Their modelassumed that only watervaporis present and they used it to solve for the gaseous phase of water throughout the entire MEA. In the polymer they useda ﬁctive gaseous phase assumed to be in equilibrium with thewater content. With their model they examined the effect of humidity ratio of inlet mixtures for different values of membrane thickness, and for coﬂow and counter ﬂowconditions. Their results show with the counter ﬂow designand at low humidity ratio, the current density produced iscomparable to that of the coﬂow condition with fullyhumidiﬁed inlet mixtures.Wu et al. [18] presented an optimization methodologyusing design of experiments and a few results from a CFDthreedimensionaltoolinordertoﬁndthebestsolutionwhichmaximizes power density. Their results show that manysolutions can be found. This approach avoids running a largenumber of threedimensional solutions, which may takea long time to get results. Other studies have examined theoptimization of PEMFC by proposing different ﬂow conﬁgurations in the channel by using ﬂow bafﬂes (Liu et al. [19]),variablescross section(Liu et al. [20] and Yan et al. [21]). These
studies have shown that improved performance can beobtained with the above modiﬁcations since they help inspreading reacting species from the channel to the catalystlayer. Harvey et al. [22] using a single phase model for a threedimensional representation of a PEMFC examined threemodeling approaches for the catalyst layers which are theagglomerate model as discussed by Siegel et al., the discretevolume where the pores are assumed to be ﬁlled with water,and the third is the thin ﬁlm model where the catalyst layer isused as a boundary condition. Their results indicate that thebest model that predicts well the experiments is the agglomerate model, which can predict the sharp drop off in voltagewhen the current limit is reached. Kamarajugadda andMazumder [23] used a twodimensional single phase model toinvestigate different models for membrane conductivity andthey found that the choice of the membrane model can affectthe current density results signiﬁcantly.In this work, a mathematical model of PEMFC is developedin order to build a numerical tool enabling the design optimizationofPEMFCforawiderangeofparameters.Thismodelintroduces a different approach in dealing the pore level masstransfer than previously presented as in Siegel et al. [14,15].ThismodelusesintheButler–Volmerequationsgoverningtheelectrochemical source terms a ﬁctitious pore level concentration instead of the volume averaged concentration. Theﬁctitious concentration on the surface of the catalyst particleallows to take into account the pore level mass transferaround the catalyst particles and predict the current limitcondition. The pore level model is based on the local solutionof the mass transfer equation in the polymer and arounda spherical catalyst particle. Using the presented model, theperformance of the fuel cell is presented for many parameterssuchasspeciesinletpressure,relativehumidity,temperature,membrane thickness, and catalyst layer porosity.
2. Overview of fuel cells
A proton exchange membrane fuel cell (PEMFC) is an electrochemical device that converts the chemical energy of a hydrogen oxidation reaction directly into electrical energywith production of heat and water. A schematic representation of a PEMFC is shown in Fig. 1 with all its components,namely:
The gas channel (GC), which is machined into the collectorplate, serves as channels for the reacting gas species.
The gas diffusion layer (GDL) is made of a porous materialsuch as carbon cloth or carbon paper. The porous nature of the GDL facilitates reactant distribution across the catalystlayer while being a good electrical conductor, providing a low electrical resistance connection between the catalyst
Fig. 1 – Computational domain.
international journal of hydrogen energy 34 (2009) 3091–3103
3093
layer and the collector. Also the GDL facilitates the removalof liquid water produced at the cathode and heat producedin the membrane and catalyst layers.
The catalyst layer (CL) main role is to accelerate the electrochemical reaction. It is a mixture of the three differentcomponents,namelymembranepolymer,GDL,andcatalystparticles supported by the structure. Platinum is the typicalcatalyst used in PEMFC. The catalyst enables the chemicalreaction to occur. The overall reaction in the anode CL isgiven by:H
2
/
2H
þ
þ
2e
and in the cathode CL by:2H
þ
þ
2e
þ
1/2O
2
/
H
2
O. The overall cell reaction is givenby: H
2
þ
1/2O
2
/
H
2
O
þ
Heat.
The proton exchange membrane (PEM), sandwichedbetween the two fuel cell electrodes, is a polymer networkcontainingcovalentlybondednegativelychargedfunctionalgroups capable of exchanging ions. The polymer matrixconsists of polymer backbone, which can be hydrocarbonbased polymers such as polystyrene and polyethylene ortheir ﬂuorinated polymer analogs. For fuel cell applications,sulfonic acid functional group
ð
SO
13
Þ
is the most widelyused.
3. Numerical model
The twodimensional representation of a PEMFC used in thisstudy is shown in Fig. 1. In this study, the single domainmodeling approach is used. The continuity, momentum,energy, and species transfer equations for all regions of thecomputational domain of the fuel cells can be written in thesame manner. However, the diffusion,convection, and sourcetermswillvarydepending onthelocal properties oftheregion(e.g., GC, GDL, CL, and PEM). As an example, the source termfor hydrogen consumption is activated in the anode CL only,whereas consumption of oxygen and production of H
2
O isactivated only in the cathode CL.The PEMFC model is reduced to the following transportequations for the entire domain with variable diffusion andsource terms which will be described below for each region.The continuity equation is given by:
v
r
u
v
x
þ
v
r
v
v
y
¼
S
H
2
þ
S
O
2
þ
S
H
2
O
þ
S
d
(1)The momentum equations in the
x
 and
y
directions withthe Darcy term included are given by:
v
r
uu
v
x
þ
v
r
vu
v
y
¼
v
p
v
x
þ
vv
x
m
v
u
v
x
þ
vv
y
m
v
u
v
y
3m
Ku
(2)
v
r
uv
v
x
þ
v
r
vv
v
y
¼
v
p
v
y
þ
vv
x
m
v
v
v
x
þ
vv
y
m
v
v
v
y
3m
Kv
(3)The Darcy term is included in order to account for the porousstructure of the GDL and CL.The energy equation is given by:
vv
x
3
r
c
p
f
uT
þ
vv
y
3
r
c
p
f
vT
¼
vv
x
k
eff
v
T
v
x
þ
vv
y
k
eff
v
T
v
y
þ
S
rev
þ
S
ohm
þ
S
hs
(4)The species equation is given by:
v
uc
i
v
x
þ
v
vc
i
v
y
¼
vv
x
D
i
;
eff
v
c
i
v
x
þ
vv
y
D
i
;
eff
v
c
i
v
y
þ
S
i
(5)The diffused water equation in the polymer phase is givenby:
vv
x
D
DW
v
c
DW
v
x
þ
vv
y
D
DW
v
c
DW
v
y
þ
S
drag
þ
S
d
¼
0 (6)The protonic potential equation is given by:
vv
x
s
m
v
f
m
v
x
þ
vv
y
s
m
v
f
m
v
y
þ
S
p
¼
0 (7)The model assumes:
Steady state;
Compressible and laminar ﬂow;
Newtonian ﬂuid;
Ohmic losses due to electrons transfer are neglected;
Liquid water is not taken into account;
Dispersion in the porous media is neglected;
Each region is isotropic and homogeneous;
Local thermal equilibrium between the porous solid structure and the ﬂuid (GDL, CL, PEM);
H
2
is present in the anode only, O
2
and N
2
are present in thecathode only, and H
2
O vapor is present in the entirecomputational domain except in the membrane.
3.1. The gas channel
In the gas channels, the continuity Eq. (1) has no source termsin both of these regions since there are no chemical reactionsoccurring. The momentumEqs. (2) and (3) in both channelsdonot have the darcean terms since they are not porous. Thesourcetermsfortheenergy, mass,and potentialarealsonil inthese regions. The properties of the gaseous mixture in allregions of the computational domain are calculated by taking intoaccountthe different speciespresentlocally.The mixturedensity, viscosity and mass diffusion are respectively givenby:
r
¼
X
i
M
i
c
i
(8)
m
¼
X
i
r
i
rm
i
¼
X
i
M
i
c
i
r m
i
(9)and
D
i
;
mix
¼
1
y
i
=
X
j
s
i
y
j
D
ij
(10)where
y
i
is the molar fraction of species
i
is given by:
y
i
¼
c
i
P
j
c
j
(11)The diffusion coefﬁcient given in Eq. (10) is based ona simpliﬁcation of the Stefan–Maxwell equations [8]. Thebinary diffusion coefﬁcient
D
ij
for two species
i
and
j
is scaledusing the local temperature and pressure [14]:
D
ij
ð
T
;
p
Þ¼
D
0
ij
T
0
;
p
0
p
0
p
TT
0
1
:
5
(12)
international journal of hydrogen energy 34 (2009) 3091–3103
3094
The speciﬁc heat and thermal conductivity of the ﬂuidmixture are given by:
c
p
¼
X
i
r
i
r
c
p
;
i
¼
X
i
M
i
c
i
r
c
p
;
i
(13)
k
f
¼
X
i
y
i
k
i
¼
X
i
c
i
P
j
c
j
k
i
(14)wherethespeciesproperties
k
i
,
c
p,i
,and
m
i
aregivenasafunctionof temperature [24].
3.2. Gas diffusion layer (GDL)
As mentioned earlier, the gas diffusion layer uses an electrically conductive porousstructure and beingporous allows theconvective and diffusive migration of the various species. Thevarious transport quantities are volume averaged overa representative elementary volume, which contains manypores (Kaviany [25]). For the sake of model simplicity, we usedthe same notation as in the plain medium, however anychanges due to the presence of the solid matrix is taken intoaccount in the properties as will be discussed below.AllthesourcetermsinboththeanodeandcathodeGDLarenil with the exception of the darcean terms of the momentumEqs. (2) and (3) and the energy Eq. (4) with the heat source
ð
S
hs
Þ
which represents the heat transfer from GDL to bipolarplates. The Darcy source term accounts for the pressure dropcaused by frictional drag in the pore structure and is directlyproportional to the averaged local velocity. The species andenergy conservation equations keep the same form as thosegiven in the gas channel domain but with averaged variables(velocities, temperature and concentrations) and correctedﬂuid property (diffusion coefﬁcient) to account for the porousstructure.Thespecieseffectivediffusioncoefﬁcientsaregivenby [5]:
D
i
;
mix
;
eff
¼
D
i
;
mix
3
1
:
5
(15)where
3
is the porosity and
D
i
,mix
is given by Eq. (10).The effective thermal conductivity of the porous mediumis given by:
k
eff
¼
3
k
f
þð
1
3
Þ
k
s
(16)where
k
f
is the conductivity of the ﬂuid mixture given by Eq.(14) and
k
s
is the solid conductivity. The source term
S
hs
in theenergy equation represents the heat removal from the systemand is activated on the channels/GDL interfaces at
y
¼
h
c
and
H
h
a
. This source term is given by (per unit depth)
S
hs
¼
k
al
=
h
ð
T
T
amb
Þ
;
h
¼
h
a
or
h
c
where
k
al
is the thermalconductivity of aluminum.
3.3. Catalyst layer (CL)
The catalyst layer can be described as a union of threedifferent parts: the porous matrix from the GDL, the polymerand the catalyst particles. Therefore, the catalyst layer ismodeled as a volume region and not simply as an interfacebetween GDL and membrane. This description gives thepossibility of coexistence of gas species, dissolved water andprotons in the catalyst layer region. As mentioned earlier thismodelassumes thatnophase changeforwateroccurs (monophase model). The governing equations in the CL layers aresimilar to those applied in the GDL except for activated sourceterms to account for chemical reactions.The source terms for the continuity Eq. (1) are only activated in the catalyst layers and are given by
S
H
2
¼
j
a
=
2
F
;
S
O
2
¼
j
c
=
4
F
;
S
H
2
O
¼
j
c
=
2
F
.These expressions model the consumption of hydrogenand oxygen respectively in the anode and cathode catalystlayers and production of water vapor in the cathode catalystlayer. The anode and cathode local current densities per unitvolume,
j
a
and
j
c
, are expressed in terms of the Butler–Volmerequation for the anode is as follows:
j
a
¼
2
A
cv
i
0a
;
ref
c
eff H
2
c
H
2
;
ref
!
g
H2
sinh
a
a
h
a
ð
2
F
Þ
RT
(17)which is used for modeling the reaction 2H
2
/
4H
þ
þ
4e
andfor the cathode:
j
c
¼
2
A
cv
i
0c
;
ref
c
eff O
2
c
O
2
;
ref
!
g
O2
sinh
a
c
h
c
ð
4
F
Þ
RT
(18)which is used to model the reaction 4H
þ
þ
4e
þ
O
2
/
2H
2
O.
A
cv
(m
1
) is the speciﬁc surface reaction area of the catalystlayer.
c
eff O
2
and
c
eff H
2
are respectively the effective oxygen andhydrogen concentrations at the surface of the agglomeratewhile
c
H
2
;
ref
and
c
O
2
;
ref
are H
2
and O
2
reference concentrations.In order to model, the concentration of the reactants at thesurface of the catalyst particle and incorporate the microscopic mass transfer, the local mass diffusion equation issolved for half a sphere embedded in the polymer as shown inFig. 2. This allows the calculation of a ﬁctitious surfaceconcentration on the catalyst surface. A representation of the domain around the platinum particle is depicted inFig. 2. The mass transfer equation in spherical coordinates isgiven bydd
r
r
2
d
C
d
r
¼
0 (19)
GAS (Pore) Pt H
+
O O O Polymer
Carbon
e

e

R
0
R
P
H H
Fig. 2 – Microscopic model of mass diffusion aroundplatinum particles.
international journal of hydrogen energy 34 (2009) 3091–3103
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