Research Article A Mathematical Model of a Direct Propane Fuel …aix1.uottawa.ca/~ybourg/publications/JofChem_Khakdaman... · 2016-01-13 · Research Article A Mathematical Model

Research ArticleA Mathematical Model of a Direct Propane Fuel Cell

Hamidreza Khakdaman12 Yves Bourgault3 and Marten Ternan124

1Department of Chemical and Biological Engineering University of Ottawa 161 Louis-Pasteur Ottawa ON Canada K1N 6N52Center for Catalysis Research and Innovation University of Ottawa 30 Marie-Curie Street Ottawa ON Canada K1N 6N53Department of Mathematics and Statistics University of Ottawa 585 King Edward Avenue Ottawa ON Canada K1N 6N54EnPross Incorporated 147 Banning Road Ottawa ON Canada K2L 1C5

Correspondence should be addressed to Marten Ternan ternanbellnet

Received 13 October 2015 Revised 26 October 2015 Accepted 27 October 2015

Academic Editor Hanping Ding

Copyright copy 2015 Hamidreza Khakdaman et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

A rigorous mathematical model for direct propane fuel cells (DPFCs) was developed Compared to previous models it providesbetter values for the current density and the propane concentration at the exit from the anode This is the first DPFC model tocorrectly account for proton transport based on the combination of the chemical potential gradient and the electrical potentialgradient The force per unit charge from the chemical potential gradient (concentration gradient) that pushes protons from theanode to the cathode is greater than that from the electrical potential gradient that pushes them in the opposite direction Byincluding the chemical potential gradient we learn that the proton concentration gradient is really much different than thatpredicted using the previousmodels that neglected the chemical potential gradient Also inclusion of the chemical potential gradientmade this model the first one having an overpotential gradient (calculated from the electrical potential gradient) with the correctslopeThat is important because the overpotential is exponentially related to the reaction rate (current density)Themodel describedhere provides a relationship between the conditions inside the fuel cell (proton concentration overpotential) and its performanceas measured externally by current density and propane concentration

1 Introduction

The focus of this study is on the direct propane fuel cell(DPFC) which belongs to the polymermembrane electrolytefuel cell (PEMFC) family but consumes propane insteadof hydrogen as its feedstock The generation of electricalenergy in rural areas is our primary target application forDPFCs The cost of delivering electrical energy to rural areasis substantially greater than to urban areas because longertransmission lines are required to serve a comparativelysmall number of customers Therefore more costly fuel cellscan be justified for use in rural areas compared to urbanareas In addition the infrastructure for delivering liquefiedpetroleum gas (LPG) or propane to rural areas alreadyexists Two major advantages of DPFCs over hydrogen fuelcells are that the expense of hydrogen production plantsand of hydrogen transportstorage will be eliminated fromthe fuel cell energy production cycle However a drawback

associated with DPFCs is that the propane reaction rate ismuch slower than that of hydrogen Liebhafsky and Cairns[1] Bockris and Srinivasan [2] and Cairns [3] reviewed themajority of the DPFC experimental research that had beendone in the 1960s Only a little work has been performedsince then Polarization curves were reported both by Chenget al [4] and by Savadogo and Rodriguez Varela [5] forlow temperature PEMFCs Intermediate temperature (100ndash300∘C) proton conducting fuel cells were investigated by Heoet al [6] Solid oxide fuel cell (SOFC) studies were performedwith propane at 550ndash650∘C by Feng et al [7] and at 900∘C byYang et al [8]

The approaches to the modeling of fuel cells are sum-marized below Weber and Newman [9] have reviewed fourgroups of fuel cell models that consider transport of waterand protons in the electrolyte phase simple models diffusivemodels hydraulic models and combination models Thesimple models [10ndash13] describe proton transfer using Ohmrsquos

Hindawi Publishing CorporationJournal of ChemistryVolume 2015 Article ID 102313 13 pageshttpdxdoiorg1011552015102313

2 Journal of Chemistry

law with a constant ionic conductivity These models cannotpredict phenomena such asmembrane dehydration in whichwater content and thus ionic conductivity are variables Forwater movement a numerical value of the net water flux hasto be determined as the boundary condition at the interfacebetween the catalyst layers and the membrane

The diffusive models [14ndash19] predict the movement ofdissolved water and protons within the membrane as a resultof concentration and electrical potential gradients They areapplicable for the electrolyte systems with low water content(120582 lt 14 where 120582 is moles of water per mole of sulfonicacid sites in the Nafion membrane) where liquid water doesnot exist The diffusive models are referred to as single phasemodels of membranes and can predict proton distribution inthe electrolyte phase and membrane dehydration

At high water contents membrane pores are completelyfilled with liquid water and the water content is assumedto be uniform everywhere Therefore water diffusion doesnot occur and the convection mechanism causes protonand water transport The hydraulic models [20ndash23] weredeveloped for membranes with high water content Twophases liquid water and membrane are described by thehydraulic models Water velocity is calculated by Schloglrsquosequation [23] which is a function of electrical potentialgradient and pressure gradient Finally the hydraulic anddiffusive models are merged in the combination models [24ndash26] when calculations covering the whole range of watercontent are desirable This approach considers concentrationand pressure gradients as driving forces for water and protontransport

There are two possible approaches to dealing with thetransport properties in the diffusive models that is dilutesolution theory and concentrated solution theory [27] Masstransport in dilute electrolyte systems is usually describedby the Nernst-Planck equation [27] in which the flux of acharged species is a function of the concentration gradientof that species as well as the electrical potential gradi-ent For a noncharged species the potential gradient termin the Nernst-Planck equation disappears The membranetransport properties are not required to be constant in thisapproach

Employing concentrated solution theory leads to rigorousmodels that consider the interactions between all speciesKrishna [19] used Generalized Maxwell-Stefan (GMS) equa-tions to implement this approach for multicomponent elec-trolyte systems in generalWohr et al [17] also usedMaxwell-Stefan (MS) equations to model proton and water transportin PEM fuel cells in which the MS diffusion coefficients aremodified as a function of temperature and humidity Fullerand Newman [28] used the electrochemical potential of eachspecies as the driving force in the MS equations Fimrite etal [16] developed a transport model for water and protonsbased on the binary friction model The mole fraction andpotential gradients were considered in the electrochemicalpotential gradient expression Baschuk and Li [15] also usedMS equation but they calculated theMS diffusion coefficientsbased on experimental data available in the literature Thenthey validated those coefficients with experimental data forthe electroosmotic drag coefficient

A diffusive model has been developed in the presentstudy to investigate the movement of water and protonsin the electrolyte phase of a DPFC where the operationtemperature is above the boiling point of water One possiblestrategy for increasing the reaction rate in DPFC is to operateat temperatures of 150∘C or higher A membrane that canresist high temperature and show acceptable conductivity(50 Smminus1) has been developed in our research group [29]This membrane is composed of porous polytetrafluoroethy-lene (PTFE) that contains zirconium phosphate (Zr(HPO

4)2sdot

H2O or ZrP) in its pores ZrP is a known proton conductor

[30] Concentrated solution theory was used in which thebinary interactions between water protons and ZrP specieswere described

We are developing mathematical models of DPFCs inorder to understand this phenomenon and hopefully toenhance their performance The results reported here aremajor improvements over our previous model [10] Ourprevious model like the vast majority of fuel cell mod-els only used an electrical potential gradient to describemigration and neglected the proton concentration gradientin accounting for proton transport through the electrolytelayer As we noted previously [10] neglect of the protonconcentration gradient caused the overpotential gradient andthe electrical potential gradient in the electrolyte phase tobe incorrect The model being described here unlike themajority of fuel cell models includes both a valid electricalpotential gradient and a proton concentration gradient toaccount for proton transport by a combination of migrationand diffusion This model accounts for the influence of theproton concentration in the electrolyte phase and therebyovercomes the deficiencies mentioned above

2 Model Development

Thismodel solved the governing equations for theMembraneElectrode Assembly (MEA) consisting of the membranelayer anode layer and cathode layer A schematic of atypical DPFC is shown in Figure 1 The cell is composedof two bipolar plates two catalyst layers and a membranelayer Each bipolar plate has two sets of channels one forreactants and one for products The channels are connectedto each other through the catalyst layer Figure 1 shows thesechannels for the anode bipolar plate The interdigitated flowfields show a symmetric geometry with repetitive pieces Inorder to increase the computational speed only one of thesepieces was considered as the modeling domain Thereforethe modeling domain can be defined as the part of the MEAthat is located between the middle of a feed channel andthe middle of its adjacent product channel (cross sectionin Figure 1) That cross section is shown in Figure 2 as themodeling domain Its boundaries are shown as a dashed blackline

Previously it was shown that neglecting proton diffusionin the proton conservation equation (an assumption usedin many fuel cell models) led to incorrect results for theelectrolyte potential and overpotential profiles even thoughthe polarization curve was predicted correctly [10] Thepresent model includes both proton diffusion and migration

Journal of Chemistry 3

propane

Propane and water

Waterand air

Air

Anode bipolar plate

Anode catalyst layer

Membrane layer

Cathode catalyst layer

Cathode bipolar plate

Anode gas channels

Dead end

CO2 water andx

y

z

Figure 1 A direct propane fuel cell with interdigitated flow field

and propanePropaneand water

Air Water and air

Modeling domain

Anode land

Cathode land

CO2 water

y

x

Figure 2 Boundaries in the modeling domain

21 Governing Equations Three phases are present in theanode and cathode catalyst layers They are the ldquogas phaserdquocontaining reactants and products the ldquosolid catalyst phaserdquocontaining the carbon support and platinum and the ldquosolidelectrolyte phaserdquo The latter consists of a stationary ZrPmatrix = [Zr(HPO

4)2sdot H2O] containing mobile H

2O =

[Zr(HPO4)2sdot 2H2O] and mobile H+ = [Zr(HPO

4)2sdotH3O+]

species that can be transportedThemembrane layer containsthe ZrP electrolyte phase as well as PTFE

Conservation equations for momentum total mass andmass of noncharged species were solved for the gas phase ineach of the catalyst layers A list of equations that were usedfor the gas phase of both anode and cathode catalyst layers isshown as follows

Conservation of mass in gas phase

nabla sdot (120576G120588G) +119899

sum

119894

]119894MW119894119895

119911119865= 0 (1)

where 119894 = C3H8 H2O and CO

2for the anode and O

2

and H2O for the cathode

Conservation of momentum in gas phase

minusnabla119875 = 150 [120583G (1 minus 120576G)

2

1198632p1205763

G] (2)

Conservation of noncharged species in gas phase

nabla sdot (120576G119888G119910119894) minus nabla sdot (120576G119888G119863119894nabla119910119894) +]119894119895

119911119865= 0 (3)

where 119894 = C3H8and CO

2for the anode and O

2and

H2O for the cathode

Conservation of species in the electrolyte phase

4 Journal of Chemistry

for water

minus nabla sdot (119888ELY (1198611015840

H2OndashH2O minus 1198611015840

H2OndashH+) nabla119909H+) + nabla

sdot (119888ELY1198611015840

H2OndashH+

119865119909H+

119877119879nabla120601ELY) minus

119895

119911119865= 0

(4)

for proton

nabla sdot (119888ELY (1198611015840

H+ndashH+ minus 1198611015840

H+ndashH2O) nabla119909H+) + nabla

sdot (119888ELY1198611015840

H+ndashH+119865119909H+

119877119879nabla120601ELY) +

119895

119911119865= 0

(5)

Butler-Volmer equation in the anode

119895A = 1198950

A119860Pt [exp(120572A119865120578A119877119879

) minus exp(minus120572C119865120578A119877119879

)] (6)

where

1198950

A = 1198950refC3Ox(

119901C3

119901refC3

) exp[Δ119866Dagger

C3Ox

119877(1

119879ref minus1

119879)] (7)

120578A = Δ120601A minus Δ120601EQA = (120601PtA minus 120601ELYA

) minus (120601EQPtA minus 120601

EQELY) (8)

Butler-Volmer equation in the cathode

119895C = 1198950

C119860Pt [exp(120572A119865120578C119877119879

) minus exp(minus120572C119865120578C119877119879

)] (9)

where

1198950

C = 1198950refO2Rd(

119901O2

119901refO2

) exp[Δ119866Dagger

O2Rd

119877(1

119879ref minus1

119879)] (10)

120578C = Δ120601C minus Δ120601EQC = (120601PtC minus 120601ELYC

) minus (120601EQPtC minus 120601

EQELY) (11)

Equation (1) describes the total mass conservation in the gasphase of the catalyst layers The second term in this equationis the sink or source term describing the mass consumptionor production in the gas phase caused by electrochemicalreactions Equation (2) is the linear form of the Ergunequation It was used to calculate the pressure profiles inthe gas phase of the catalyst layers because they are packedbeds At the conditions used in this study the magnitudeof the quadratic velocity term in the Ergun equation wasmuch smaller than the linear term Hence only the linearterm in velocity was used in (2) Equations (1) and (2) weresolved together to calculate the velocity and pressure profilesin the gas phase of the catalyst layers Mass balances for eachof the individual gas phase species account for convectiondiffusion and reaction as shown in (3)

Equations (4) and (5) describe respectively water andproton conservation in the electrolyte phase of themembraneand catalyst layers Diffusion was described by concentratedsolution theory through the use of the GMS equations Thefollowing paragraphs illustrate the derivation of (4) and (5)

A general procedure for the calculation of mass fluxesin multicomponent electrolyte systems was presented byKrishna [19] It has been proven that the Nernst-Planckequation is a limiting case of the GMS equations The GMSequations can be written as follows

119889119894=

119899

sum

119895=1

119895 =119894

119909119894

119869119895minus 119909119895

119869119894

119888ELYĐ119894119895119894 = 1 2 119899 minus 1 (12)

where 119889119894is a generalized driving force for mass transport

of species 119894 Because the summation of the 119899 driving forcesis equal to zero due to the Gibbs-Duhem limitation [31]only 119899 minus 1 driving forces are independent The equationto calculate the generalized driving force has been derivedbased on nonequilibrium thermodynamics [31] A simplifiedexpression for a solid stationary electrolyte (no convectionterm) [19] can be written as

119889119894= nabla119909119894+ 119909119894119911119894

119865

119877119879nabla120601ELY (13)

For a noncharged species such aswater 119911119894is equal to zero and

according to (13) the concentration gradient will be the onlydriving force

The migration term in (13) was obtained by representingion mobility by the Nernst-Einstein relation (D

119894= 119877119879u

119894)

This equation is applicable only at infinite dilution How-ever it can be used in concentrated solutions if additionalcomposition-dependent transport parameters such as the 1198611015840parameters in (19) are used to calculate the flux of ions [27] Itwill be shown in the following paragraphs that (18) representthe composition-dependent parameters

Equation (12) results in (119899minus1) independent equations thatcan be written in matrix form for convenience

119888ELY(

1198891

119889119899minus1

)

= minus(

11986111 sdot sdot sdot 119861

1119899minus1

d

119861119899minus11 sdot sdot sdot 119861

119899minus1119899minus1

)(

1198691

119869119899minus1

)

(14)

where the elements of the matrix of inverted diffusioncoefficients [119861] are given by

119861119894119894=

119899

sum

119895=1119895 =119894

119909119894

Đ119894119895

119894 = 1 2 119899 minus 1

119861119894119895=minus119909119894

Đ119894119895

119894 = 1 2 119899 minus 1 (119894 = 119895)

(15)

Journal of Chemistry 5

The fluxes of species 119869119894 can be calculated from (16) which is

the inversion of (14)

(

1198691

119869119899minus1

)

= minus119888ELY(

11986111

sdot sdot sdot 1198611119899minus1

d

119861119899minus11

sdot sdot sdot 119861119899minus1119899minus1

)

minus1

(

1198891

119889119899minus1

)

(16)

For the present electrolyte system containing three speciesmobile H

2O and H+ plus immobile solid ZrP (16) may be

written as

(

119869H2O

119869H+) = minus119888ELY(

1198611015840

H2OndashH2O 1198611015840

H2OndashH+

1198611015840

H+ndashH2O 1198611015840

H+ndashH+)(

119889H2O

119889H+) (17)

where [1198611015840] is the inverse of the matrix of inverted diffusioncoefficients Because ĐH

2OndashH+ = ĐH+ndashH

2O the elements of

[1198611015840] are calculated using (18) which are functions of the GMSdiffusivities and the species mole fractions in the electrolytephase

1198611015840

H2OndashH2O

=119909H2OĐH+ndashZrP + ĐH

2OndashH+

119909H+ + (ĐH+ndashZrPĐH2OndashZrP) 119909H

2O + ĐH

2OndashH+ĐH

2OndashZrP

1198611015840

H2OndashH+

=119909H2OĐH+ndashZrP

119909H+ + (ĐH+ndashZrPĐH2OndashZrP) 119909H

2O + ĐH

2OndashH+ĐH

2OndashZrP

1198611015840

H+ndashH2O

=119909H+ĐH

2OndashZrP

119909H2O + (ĐH

2OndashZrPĐH+ndashZrP) 119909H+ + ĐH

2OndashH+ĐH+ndashZrP

1198611015840

H+ndashH+

=119909H+ĐH

2OndashZrP + ĐH

2OndashH+

119909H2O + (ĐH

2OndashZrPĐH+ndashZrP) 119909H+ + ĐH

2OndashH+ĐH+ndashZrP

(18)

Combining sets of (17) and (13) results in two independentequations that can be used to calculate the fluxes of mobilespecies ( 119869H

2O and 119869rarr

darr

(Huarr+)uarr

) within the electrolyte phase

119869H2O = minus119888ELY119861

1015840

H2OndashH2O (nabla119909H2O)

minus 119888ELY1198611015840

H2OndashH+ (nabla119909H+ +

119865119909H+

119877119879nabla120601ELY)

(19)

119869H+ = minus119888ELY1198611015840

H+ndashH2O (nabla119909H2O)

minus 119888ELY1198611015840

H+ndashH+ (nabla119909H+ +119865119909H+

119877119879nabla120601ELY)

(20)

Equations (19) and (20) show that diffusion flux of eachspecies is a function of the concentration gradient of allspecies as well as of the potential gradient There are fiveunknowns in (19) and (20) 119869H

2O 119869H+ 119909H

2O 119909H+ and 120601ELY

Therefore three more equations are requiredZrP is immobile As a result the diffusion phenomenon

will effectively be the interchange of H+ and H2O species

Therefore for diffusion purposes we will only consider thedomain of the mobile species H+ and H

2O and will ignore

the immobile species ZrP On that basis (21) can be usedas a third equation Nevertheless the presence of ZrP isimportant because of its interaction with the mobile speciesSpecifically the values of the 1198611015840 coefficients for H+ and H

2O

were influenced by the presence of ZrP119909H2O + 119909H+ = 10 (21)

The differential equations for H2O andH+mass conservation

in the electrolyte phase can be expressed in molar units as

nabla sdot 119869H2O =

minus119895

119911119865

nabla sdot 119869H+ =119895

119911119865

(22)

where 119895 is the volumetric current production This quantitywhich appears in (1) (3) to (5) and (22) is the rate of produc-tion of protons in the anode Therefore it is positive in theanode 119895A and negative in the cathode 119895C It was calculatedusing the Butler-Volmer equation for the anode and cathode(6) and (9) respectively Exchange current densities at theanode and cathode are a function of the reactantsrsquo partialpressure and the operating temperature as shown in (7) and(10) The Butler-Volmer equation and its parameters for bothpropane oxidation and oxygen reduction were described inour previous communication [10] Complete conversion ofC3H8to CO

2was reported in experiments by Grubb and

Michalske [34] Equations (19) to (22) were combined and areshown as (4) and (5)

22 Numerical Procedure The numerical solution proce-dure is illustrated in Figure 3 Equations (1)ndash(11) define theproblem at steady state However a time derivative wasappended to each partial differential equation and a backwardEuler time stepping method was used to increase stabilitywhile converging to the steady-state solution The FiniteElementMethodwas used to discretize the partial differentialequations in space with all dependent variables discretized bya linear finite element except for the pressure that is taken asa quadratic

FreeFEM++ software has been successfully used to solvetwo-dimensional partial differential equations (1)ndash(11) It isopen-source software and is based on the Finite ElementMethod developed by Hecht et al [32] The calculated resultsfrom FreeFEM++ were exported to ParaView visualizationsoftware [35] for postprocessing ParaView is also open-source software

There is no proton loss through the exterior boundaries ofthe domain (Figure 2)Therefore the total rate of proton pro-duction in the anode intAnode 119895119889119881 has to be equal to the total

6 Journal of Chemistry

Momentummass at anodecathode

Gaseous species at anodecathode

Proton and water at anode

Proton and water at membrane

Proton and water at cathode

Define geometrygenerate mesh

Balance proton productionconsumption

Iteration

Iteration

Define problemdiscretize equations

Iteration

Iteration

Output result for postprocessing

Updating transfer

conditionsproperties

Figure 3 Modeling procedure

rate of proton consumption in the cathode intCathode(minus119895)119889119881In each case the electrical potential of the catalyst phase ofthe anode 120601PtA and that of the cathode 120601PtC had individualconstant values Then all the variables in the whole domainwere calculated However having fixed electrical potentialsof the anode and cathode catalyst phases does not guaranteethat the proton production at the anode will equal the protonconsumption at the cathode The difference between the rateof proton production and consumption can be minimized byshifting 120601ELY by a constant value because the production andconsumption rates are functions of the electrical potential inboth of their respective catalyst phases 120601PtA and 120601PtC andin the electrolyte phase 120601ELY Therefore the Newton methodwas used to force equal proton production and consumptionIn other words balancing intAnode 119895119889119881 and intCathode(minus119895)119889119881

acts as a constraint for the conservation of protons in theelectrolyte phase

The equations for the conservation of momentum totalmass and individual species in the gas phase of the anodeand cathode were solved by assuming there was no speciescrossover through the membrane Electrical potential pro-ton and water concentrations in the electrolyte phase of theanode cathode and membrane layers were coupled to eachother These variables were calculated by solving (4) (5)

and (21) iteratively in each layer Then the Robin method[10] was used to couple the solutions between layers In theRobin method both of the following transfer conditions areprogressively satisfied on the anode catalystmembrane inter-face and the membranecathode catalyst interface throughiterations of (a) the continuity of the variable (eg potential)and (b) the continuity of the flux (eg electrical current)

Figure 2 shows four types of boundary conditions for themodeling domain that is inlet outlet wall of the land andthe midchannel symmetry boundaries The flux of speciesin the gas phase is zero at the walls because there is notransfer through walls The zero flux condition is also true atthe midchannel symmetry boundaries The compositions ofthe gaseous species are known at the inlet of the anode andcathode catalyst layers It was assumed that no change in thecomposition of gasmixture occurred after leaving the catalystbed Therefore the composition gradients are zero in thedirection normal to the catalyst layer at the outlet boundariesThe zero flux condition is applied at all exterior boundariesfor the species in the electrolyte phase

23 Input Parameters The parameters used for the simula-tions are shown in Table 1 The GMS diffusivities Đ

119894119895 which

are used in (18) have to be calculated from the Fickiandiffusion coefficients 119863

119894119895 For ideal solutions the Fickian

diffusion 119863119894119895 can be used as Đ

119894119895in the Stefan-Maxwell

equations [26] because the concentration dependence ofFickian diffusion coefficients is ignored Experimental valuesfor 119863H+ndashZrP and 119863H

2OndashH+ are given in Table 1 Note that the

diffusivity of protons in ZrP is approximately two orders ofmagnitude smaller than the diffusivity of protons in waterThe movement of protons causes the electroosmotic flow ofwater [9] It was assumed that one water molecule is draggedby each proton H

3O+ that travels from anode to cathode

Therefore the diffusivity of water in ZrP was set equal tothe diffusivity of protons in ZrP [36] the smaller of the twoproton diffusivities in Table 1 Proton diffusivity and protonmobility are different quantities The three diffusivities inTable 1 were the ones used to calculate the 1198611015840 parameters in(18)

24 Model Validation The model predicts the performanceof a DPFC that (i) has interdigitated flow fields (ii) haszirconium phosphate as the electrolyte and (iii) operatesover a temperature range of 150ndash230∘C As there are noexperimental data for DPFCs having zirconium phosphateelectrolytes and interdigitated flow fields the model resultshave been compared to published results for DPFCs withother types of electrolytes and flow fields

Figure 4 compares the modeling results for zirconiumphosphate electrolyte with the experimental data for othertypes of electrolytes [34 37] The figure shows that thepolarization curve for ZrP-PTFE electrolyte is somewhatcomparable to that for the other electrolytes The differencebetween the polarization curves can be partially explained bythe difference between conductivities of the electrolytes Theproton conductivity of a nonmodified Nafion 117 approaches10 Smminus1 at 80∘C [38] The conductivity of the 95 H

3PO4

Journal of Chemistry 7

Table 1 Operational electrochemical and design parameters for simulations

Property ValueTemperature 119879 423ndash503KPressure 119875 1013 k PaProtonndashZrP diffusivity119863H+ndashZrP 31 times 10minus12m2 sminus1 [29]Protonndashwater diffusivity119863H2OndashH+ 29 times 10minus10m2 sminus1 [12]Ionic conductivity in membrane 120590ZrPPTFE 50 Smminus1 [24]Electrical resistivity in membrane 119877PTFE 10 times 1016ΩmCharge transfer coefficients 120572A and 120572C 10 [30]Equilibrium potential of catalyst phase at the anode 120601EQPtA 0136V [1]Equilibrium potential of catalyst phase at the cathode 120601EQPtC 1229VEquilibrium potential of electrolyte phase 120601EQELY 0136VApparent bulk density of carbon catalyst support 120588CAT 0259 gcatalyst mLminus1catalystSpecific surface area of carbon catalyst support in the anode and cathode 119860CAT 255m2catalyst g

minus1

catalyst

Gas phase volume fraction in anode and cathode 120576G 05Electrolyte phase volume fraction in anode and cathode 120576ELY 04Effective particle diameter in anode and cathode119863p 5 120583mLand width 119871

119882

2ndash8mmAnode and cathode thickness ThA ThC 200ndash400 120583mMembrane thickness ThM 100ndash200 120583mFluid channels width in bipolar plates 04mm

0

02

04

06

08

1

0 50 100

Cel

l pot

entia

l (V

)

(a) Savadogo and Rodriguez varela 2001 Nafion 117

(c) Model results ZrP

(a)

(b)(c)

minus2)

(b) Grubb 95 H3PO4

Current density (mA cm

Figure 4 Polarization curves of direct propaneoxygen fuel cellusing Pt anode and cathode (a) Experimental results [31] usingNafion 117 at 95∘C (b) Experimental results [32] using 95H

3

PO4

at200∘C (c)The present protonmigration and diffusionmodel resultsfor a solid ZrP-PTFE electrolyte at 150∘C

electrolyte is 35 Smminus1 at 200∘C [39] However the protonconductivity for the best ZrP-PTFE that has been developedin our laboratory is about 5 Smminus1 at 150∘C

3 Results and Discussion

Figure 5(a) shows the two-dimensional variation of theproton concentration in the electrolyte phase of the entiredomain that is the anode catalyst layer (AN) the membranelayer (ML) and the cathode catalyst layer (CA) The protonconcentration at the anode inlet close to the feed gas channelhas the highest value This would be expected because thepropanersquos partial pressure is higher at the anode inlet and thatcauses a higher propane oxidation reaction rate according toButler-Volmer equation (6) Because protons are produced inthe anode catalyst layer and consumed in the cathode catalystlayer the proton concentration is greater at the anode thanthe cathode The resulting proton concentration gradient isthe driving force for protons to diffuse from the anode to thecathode

The electrical potential variation in the electrolyte phaseof the catalyst layers and membrane is shown in Figure 5(b)As the reaction rate in the catalyst layers is not uniformcurrent density and electrical potential will be variableFigure 5(b) shows that the electrical potential is higher atthe cathode electrolyte phase than at the anode electrolytephase That electrical potential gradient is the driving forcefor protons to migrate from the cathode to the anode Thisprotonmigration (caused by the electrical potential gradient)is in the opposite direction to the proton diffusion (causedby the proton concentration gradient) that was discussedabove In reality protons are known to be transported fromthe anode to the cathode Therefore the dominant driving

8 Journal of Chemistry

AN

ML

CA

0416

0404

04080412

0400 0404 0408 0412 0416

Proton concentrationmole fraction

(a)

AN

ML

CA

00610065

0069

0073

0057 0061 0065 0069 0073

Electrolyte potential (V)

(b)

AN

ML

CA

(c)

Figure 5 (a) Proton concentration in the electrolyte phase of the anode membrane and cathode layers (b) Electrical potential profile for theelectrolyte phase of the anode membrane and cathode layers (c) Protonic flux from anode to cathode in the electrolyte phase The vectorslengths indicate the flux magnitude which varies from 0 to 17mA cmminus2 in this case

force is the proton concentration gradient Furthermore it canbe concluded that the electrical potential gradient is not thedominant driving force for proton transport

Figure 5(c) shows the magnitude and direction of pro-tonic flux in the electrolyte phase of the anode cathodeand membrane layers Protons are produced in the anodeand travel from the anode through the membrane layerand to cathode where they are consumed As discussedabove in Figure 5(a) the concentration driving force forproton flux was from anode to cathode and in Figure 5(b) theelectrical potential driving force for protons was in the oppo-site direction from cathode to anode Finally Figure 5(c)demonstrates that the net flux of protons is from the anodetoward the cathode As the net flux is the summation oftwo driving forces that are in opposite directions again onecan conclude that proton diffusion is dominant over protonmigration For the fuel cell to operate the net transport ofprotons must be from the anode to the cathode Thereforethe rate of proton diffusion must exceed the rate of protonmigration Figure 5(c) also shows that the arrowsrsquo lengths arebecoming longer (indicating that the proton flux increases) inthe 119910-direction from the anode landanode catalyst interfaceto the anode catalystmembrane interface asmore protons areproduced throughout the anode catalyst layer Similarly thearrowsrsquo length becomes shorter (as the proton flux decreases)in the 119910-direction frommembranecathode catalyst interfaceto the cathode catalystcathode land interface

There are two routes by which electrons can flow fromthe anode to the cathode The electron flux through theelectrolyte is shown in Figure 6 The electron flow ratethrough the electrolyte will be many orders of magnitude

AN

ML

CA

Figure 6 Electronic flux from anode to cathode in electrolyte phaseThe vectors lengths indicate the flux magnitude which varies from 0to 1119890 minus 11mAcmminus2 in the same case as in Figure 5(c)

smaller than the electron flow rate through the externalcircuit Although the vast majority of electrons flow throughthe external circuit the production and consumption ofthe miniscule number of electrons that flow through theelectrolyte have a distribution (Figure 6) that is similar to thedistribution of protons (Figure 5(c))

It is constructive to compare this model (migration plusdiffusion) with a migration-only model [10] A cross sectionof Figure 5(b) along the 119910-direction at the middle of thedomain (119909 = 119871

1198822) is shown in Figure 7(a) where the

electrical potential for the migration plus diffusion modelin the electrolyte phase (the left axis in Figure 7(a)mdashsolidline) is compared with that in the two solid catalyst phases(the right axis in Figure 7(a)mdashdashed lines) The electricalpotentials in each of the two solid catalyst phases (dashedline) are almost constant throughout their layers becausethese phases have high electrical conductivities The greater

Journal of Chemistry 9

0

02

04

06

08

1

12

006

007

008

009

00 01 02 03 04 05 06 07

Cathode AnodeMembrane

Cata

lyst

phas

e pot

entia

l (V

)

Elec

troly

te p

hase

pot

entia

l (V

)

Electrolyte phaseCatalyst phase

y-axis (mm)

(a)

00 01 02 03 04 05 06 07

Electrolyte phaseCatalyst phase

00

02

04

06

08

10

000

010

020

030

040

Cathode AnodeMembrane

Cata

lyst

phas

e pot

entia

l (V

)

Elec

troly

te p

hase

pot

entia

l (V

)

y-axis (mm)

(b)

Figure 7 Electrical potential profiles in the 119910-direction for the electrolyte and catalyst phases located at themiddle of the domain 119909-directionfor the cathode and anode catalyst layers and membrane layer The arrows point in the direction of the ordinate scale that applies to each ofthe three curves (a) Proton migration plus diffusion within the electrolyte phase (the present model) (b) Proton migration only within theelectrolyte phase [5]

electrical potential at the cathode than at the anode (bothin the catalyst phases and the electrolyte phase) provides adriving force that (a) pushes positively charged protons fromthe cathode to the anode via the electrolyte and (b) pushesnegatively charged electrons from the anode to the cathodevia both the external circuit (almost all the electrons) and theelectrolyte (a miniscule quantity of electrons) The flow rateof negatively charged electrons through the electrolyte phasefrom the anode to the cathode will be miniscule

The results of the migration plus diffusion model shownin Figure 7(a) correctly describe these phenomena In con-trast the results from the migration only model [10] are seenin Figure 7(b)Those calculations showed that themigration-only model produced incorrect results Specifically the elec-trical potential gradient in the electrolyte has the wrongslope The slope (gradient) predicted by the migration-onlymodel incorrectly drives the positively charged protons in theelectrolyte from cathode to anode In reality they move fromthe anode to the cathode in the electrolyte

Figure 8 compares the anodic and cathodic overpotentialfor two cases The solid lines in Figure 8 are the results fromthe migration plus diffusion model The dashed lines arethe results from a migration only model The dashed lines(migration-only) have a negative slope whereas the solidlines (migration plus diffusion) have a positive slope Sincethe overpotential is the electrochemical driving force forthe reaction (see (6) and (9)) it will always have its largestvalue adjacent to the anode land and decrease toward themembrane In summary the migration plus diffusion modelpredicted the correct behaviour while the migration-onlymodel predictions were incorrect

Figure 9 shows the propane mole fraction in the gasphase of the anode catalyst layer along the 119909-direction Forsimilar operating conditions the migration plus diffusion

y-axis (mm)00 01 02 03 04 05 06 07

minus02

minus015

025

03

Ove

rpot

entia

l (V

)

H+ migration and diffusionH+ migration only

Cath

ode

Mem

bran

e

Ano

de

Figure 8 Overpotential profile in the anode and cathode along 119910-axis at the middle of the modeling domain Solid lines (migrationplus diffusion) Dashed lines (migration only) [5]

model predicted different propane concentrations than themigration-only model This difference is caused by thedifferent overpotential profiles predicted by the two modelsThe difference in overpotentials for migration plus diffusioncompared to migration-only model is shown in Figure 8Those differences are small However those small differencesare in exponential terms as shown in (6) and (9) It isthe exponential terms that cause the large differences inconcentration shown in Figure 9 If proton diffusion in theelectrolyte phase is ignored the prediction of species distri-bution within the gas phase of the catalyst layers becomes

10 Journal of Chemistry

0

002

004

006

008

01

012

0 1 2 3 4 5

noitcarfelomenaporP

(a)

(b)

(a) H+ migration and diffusion(b) H+ migration only

x-axis (mm)

X = 11

X = 56

Figure 9 Propane mole fraction in the gas phase of the anodecatalyst layer along the 119909-direction at the middle of the anode cata-lyst layer (a) Proton migration plus diffusion within the electrolytephase (the present model) (b) Proton migration only within theelectrolyte phase [5]

0

02

04

06

08

1

0 20 40 60 80 100

(a)

(b)

Current density (mA cmminus2)

(a) H+ migration and diffusion(b) H+ migration only

Cel

l pot

entia

l (V

)

Figure 10 Modeling results for polarization curves of directpropaneoxygen fuel cells using a solid ZrP-PTFE electrolyte at150∘C (a) Proton migration and diffusion within the electrolytephase (the present model) (b) Proton migration only within theelectrolyte phase [5]

incorrect In other words the migration-only model can notcorrectly calculate either the proton concentration in theelectrolyte phase or the propane concentration in the gasphase

In Figure 10 the polarization curves for the migrationplus diffusion model are compared with the migration-onlymodel At a specific cell potential the cell current densitypredicted by the migration plus diffusion model is lowerthan that of the migration-only model That is because the

0

02

04

06

08

1

12

0 10 20 30 40 50 60 70

(a)

(b)

(c)

(d)

(e)

(d) H2 PEMFC(e) 95 H3PO4

(c) T = 230

(b) T = 190

(a) T = 150

Cel

l pot

entia

l (V

)

Current density (mA cmminus2)∘C∘C∘C

Figure 11 (a) (b) and (c) Predicted polarization curves for a directpropaneoxygen fuel cell at different operating temperatures (d)experimental data for a typical hydrogenoxygen PEMFC [33] and(e) experimental data for the best performed DPFC at 200∘C [32]

steady-state value for concentration occurs in the equationfor the exchange current density (7) and (9) This deviationmay appear to be small at some conditions In Figure 10 ata cell potential of 04V the migration plus diffusion modelpredicts a current density near 50mA cmminus2 In contrast themigration-only model predicts nearly 70mA cmminus2 That isone cannot conclude that a reasonable prediction of thefuel cell overall performance can be obtained using simplemodels that ignore the proton diffusion phenomenon in theelectrolyte In addition there are other phenomena for whichthemigration-onlymodel predicts results that are completelyerroneous

It would be desirable to expand the range of the polar-ization curve in Figure 10 to greater current densities andto smaller cell potentials Many attempts to obtain such awider range of values were made Unfortunately they were allunsuccessful As the current density increased convergenceto an acceptable numerical solution of the equations becameprogressively more difficult Convergence was not obtainedat values of current densities greater than those shownin Figure 10 The difficulty was caused by the exponentialnature of the Butler-Volmer equation in combination withthe complex Generalized Maxwell-Stefan equations Smallchanges in cell potential cause the current density calculatedfrom the Butler-Volmer equation to vary enormously Thesearch for superior convergence techniques is a topic that isbeing actively pursued in our laboratory

Activation overpotential and ohmic polarization are themajor sources of potential drop in a direct propane fuel cellAny change in the operating conditions or cell design thatresults in a decrease in activation overpotential and ohmicpolarization will improve the cell performance Figure 11shows the performance of a DPFC predicted by the model

Journal of Chemistry 11

at different operating temperatures It also shows the perfor-mance of a hydrogen PEM fuel cell at 80∘C [40] and that ofa DPFC at 200∘C having a phosphoric acid electrolyte [34]As temperature is increased from 150∘C to 230∘C the rateof reaction increases according to (7) and (10) This leadsto a decrease in the overpotential term in the Butler-Volmerequation and amajor improvement in the cell performance Itcan be concluded that the predicted performance of a DPFCoperating at 230∘C can approach that of a hydrogen PEMFCat 80∘C when both operate at current densities less than40mA cmminus2

4 Conclusions

The migration plus diffusion model described in this workwas shown to be superior to the migration-only model thatis used in many fuel cell modeling studies Specifically themigration-only model predicted values of electrical potentialin the electrolyte that are erroneousThe gradient of the elec-trolyte electrical potential predicted by the migration-onlymodel was in the wrong directionThe incorrect values of theelectrical potential in the electrolyte caused the values for theoverpotential to be incorrect Incorrect overpotential valuescaused the values calculated for the propane concentration tobe incorrect This work has shown that the predicted valuesfor steady-state current density and steady-state propaneconcentration become substantially different when the effectof proton diffusion in the electrolyte is included in themodelThe migration plus diffusion model described here has beenshown to be a major improvement over the migration-onlymodel that was used in earlier studies

Many important phenomena that occur in fuel cells arenot described by polarization curves Meaningful values forvariables internal to the fuel cell for example overpotentialand reactant concentration are essential for the understand-ing of fuel cell performance At some operating conditionsvariables external to the fuel cell for example current densityand the exit concentration of propane are substantiallydifferent when proton diffusion in the electrolyte is includedin the model The insight obtained using the migration plusdiffusion model is far more useful than that obtained fromthe migration-only model

Nomenclature

119860Pt Platinum surface area per catalyst volume(m2Pt m

minus3

catalyst)119860CAT Specific surface area of catalyst support

(Vulcan carbon) in the anode and cathode(m2catalyst kg

minus1

catalyst)[119861] Matrix of inverted binary diffusion

coefficients (smminus2)[1198611015840

] = [119861]minus1 Inverse of the matrix of inverted binary

diffusion coefficients (m2 sminus1)119888 Molar concentration of mixture (kmolmminus3)119888119894 Molar concentration of species 119894 (kmolmminus3)119889119894 Generalized driving force for mass diffusion

(mminus1)

119863119894 Diffusion coefficient of species 119894 in the gas mixture

(m2 sminus1)D119894 Diffusion coefficient of ion 119894 in a solution (m2 sminus1)

119863119894119895 Fickian binary diffusion coefficient (m2 sminus1)

Đ119894119895 Generalized Maxwell-Stefan diffusivities for the

pair 119894-119895 in a multicomponent mixture (m2 sminus1)119863p Effective particle diameter (120583m)119865 Faradayrsquos constant 96485 (C kmolminus1charge)Δ119866Dagger Activation energy for the exchange current

density (kJ kmolminus1)119895 Volumetric current density rate of production of

proton in electrodes (Amminus3catalyst)1198950 Exchange current density at operating conditions

(Amminus2Pt )1198950ref Reference exchange current density at the

reference conditions (Amminus2Pt )119869 Current density (mA cmminus2)119869119894 Molar diffusion flux of species 119894 with respect to

119899th component (solvent) velocity (kmolmminus2 sminus1)119871119882 Land width in the flow field (mm)

MW119894 Molecular weight of species 119894 (kgmolminus1)

119899 Number of species119901119894 Partial pressure of species 119894 (kPa)

119875 Total pressure (kPa)PTFE Polytetrafluoroethylene119877 Universal gas constant 8314 (kJ kmolminus1 Kminus1)119877PTFE Electrical resistivity in membrane (Ωm)119879 Temperature (K)Th Thickness of catalyst layers and membrane (120583m) Superficial velocity of gas mixture (ms)u119894 Mobility of ion 119894 in a solution (cm2sdotmolJsdots)

119883 Propane conversion ()119909 Cartesian coordinate119910 Cartesian coordinate119910119894 Mole fraction of species 119894 in the gas phase

119909119894 Mole fraction of species 119894 in the electrolyte phase

119911 Moles of transferred electrons in anode andcathode reactions (kmolelectrons kmolminus1propane)

119911119894 Charge number of species 119894 (kmolcharge kmolminus1species)

ZrP Zirconium phosphate

Greek Letters

120572A and 120572C Anodic and cathodic charge transfercoefficients

120576 Volume fraction120578 Overpotential (V)120582 Moles of water per mole of sulfonic acid sites120583 Dynamic viscosity (kgmminus1 sminus1)]119894 Stoichiometric coefficient of species 119894 positive

for reactants and negative for products120588 Mass density (kgmminus3)120588CAT Apparent bulk density of catalyst support

(kgcatalyst mminus3

catalyst)120590ZrPPTFE Ionic conductivity in membrane (Smminus1)120601 Electrical potential (V)

12 Journal of Chemistry

120601EQPt Equilibrium potential of catalyst phase (V)120601EQELY Equilibrium potential of electrolyte phase

(V)

Subscripts and Superscripts

A AnodeC CathodeC3 Propane

C3Ox Propane oxidation reaction on Pt catalyst

ELY Electrolyte phase in the membrane anodeand cathode catalyst layers containingsolid ZrP and mobile H

2O and H+

EQ Equilibrium stateG Gas mixture119894 Species in gas or solid phase propane

water CO2 O2 H+ and ZrP

ML Membrane layerO2Rd Oxygen reduction reaction on platinum

catalystPt Platinum catalystref Reference conditionsZrP Zirconium phosphate in the electrolyte

phase

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Financial assistance is gratefully acknowledged DiscoveryGrant was awarded from the Canadian federal governmentrsquosNatural Sciences and Engineering Research Council Aproject within theOntario Fuel Cell Research and InnovationNetwork (OFCRIN) was funded both by the Ontario provin-cial governmentrsquos Ontario Research Fund and by the Net-workrsquos industrial sponsors One of the authors (HamidrezaKhakdaman) is grateful to the Ontario government for theaward of an Ontario Graduate Scholarship

References

[1] H A Liebhafsky and E J Cairns ldquoThe direct hydrocarbon fuelcell with aqueous electrolytesrdquo in Fuel Cells and Fuel Batteriespp 458ndash523 Wiley New York NY USA 1968

[2] J O Bockris and S Srinivasan ldquoFuel cells their electrochem-istryrdquo in Electrochemical Combustion of Organic Substances pp357ndash411 McGraw-Hill New York NY USA 1969

[3] E J Cairns ldquoAnodic oxidation of hydrocarbons and thehydrocarbon fuel cellrdquoAdvances in Electrochemical Sciences andEngineering vol 8 pp 337ndash391 1972

[4] C K Cheng J L Luo K T Chuang and A R SangerldquoPropane fuel cells using phosphoric-acid-doped polybenzim-idazole membranesrdquo Journal of Physical Chemistry B vol 109no 26 pp 13036ndash13042 2005

[5] O Savadogo and F J Rodriguez Varela ldquoLow temperaturedirect propane electrolyte membrane fuel cellsrdquo Journal of NewMaterials for Electrochemical Systems vol 4 pp 93ndash97 2001

[6] P Heo K Ito A Tomita and T Hibino ldquoA proton-conductingfuel cell operating with hydrocarbon fuelsrdquoAngewandte ChemieInternational Edition vol 47 no 41 pp 7841ndash7844 2008

[7] Y Feng J Luo and K T Chuang ldquoConversion of propane topropylene in a proton-conducting solid oxide fuel cellrdquoFuel vol86 no 1-2 pp 123ndash128 2007

[8] C Yang J Li Y Lin J Liu F Chen and M Liu ldquoInsitu fabrication of CoFe alloy nanoparticles structured(Pr04

Sr06

)3

(Fe085

Nb015

)2

O7

ceramic anode for directhydrocarbon solid oxide fuel cellsrdquo Nano Energy vol 11pp 704ndash710 2015

[9] A Z Weber and J Newman ldquoModeling transport in polymer-electrolyte fuel cellsrdquo Chemical Reviews vol 104 no 10 pp4679ndash4726 2004

[10] H Khakdaman Y Bourgault and M Ternan ldquoComputationalmodeling of a direct propane fuel cellrdquo Journal of Power Sourcesvol 196 no 6 pp 3186ndash3194 2011

[11] E Carcadea H Ene D B Ingham et al ldquoNumerical simulationof mass and charge transfer for a PEM fuel cellrdquo InternationalCommunications in Heat and Mass Transfer vol 32 no 10 pp1273ndash1280 2005

[12] D Cheddie and N Munroe ldquoParametric model of an interme-diate temperature PEMFCrdquo Journal of Power Sources vol 156no 2 pp 414ndash423 2006

[13] SUmandC YWang ldquoThree-dimensional analysis of transportand electrochemical reactions in polymer electrolyte fuel cellsrdquoJournal of Power Sources vol 125 no 1 pp 40ndash51 2004

[14] J C Amphlett R M Baumert R F Mann B A Peppley P RRoberge and T J Harris ldquoPerformancemodeling of the BallardMark IV solid polymer electrolyte fuel cell IMechanisticmodeldevelopmentrdquo Journal of the Electrochemical Society vol 142 no1 pp 1ndash8 1995

[15] J J Baschuk and X Li ldquoModeling of ion and water transportin the polymer electrolyte membrane of PEM fuel cellsrdquoInternational Journal of Hydrogen Energy vol 35 no 10 pp5095ndash5103 2010

[16] J Fimrite B Carnes H Struchtrup and N Djilali ldquoTransportphenomena in polymer electrolyte membranes II Binary fric-tion membrane modelrdquo Journal of the Electrochemical Societyvol 152 no 9 pp A1815ndashA1823 2005

[17] M Wohr K Bolwin W Schnurnberger M Fischer WNeubrand and G Eigenberger ldquoDynamic modelling and simu-lation of a polymermembrane fuel cell includingmass transportlimitationrdquo International Journal of Hydrogen Energy vol 23no 3 pp 213ndash218 1998

[18] J J Baschuk and X Li ldquoA comprehensive consistent andsystematic mathematical model of PEM fuel cellsrdquo AppliedEnergy vol 86 no 2 pp 181ndash193 2009

[19] R Krishna ldquoDiffusion in multicomponent electrolyte systemsrdquoThe Chemical Engineering Journal vol 35 no 1 pp 19ndash24 1987

[20] T Berning D M Lu and N Djilali ldquoThree-dimensionalcomputational analysis of transport phenomena in a PEM fuelcellrdquo Journal of Power Sources vol 106 no 1-2 pp 284ndash2942002

[21] J J Baschuk and X Li ldquoModelling of polymer electrolytemembrane fuel cells with variable degrees of water floodingrdquoJournal of Power Sources vol 86 no 1 pp 181ndash196 2000

Journal of Chemistry 13

[22] S Um C-Y Wang and K S Chen ldquoComputational fluiddynamics modeling of proton exchange membrane fuel cellsrdquoJournal of the Electrochemical Society vol 147 no 12 pp 4485ndash4493 2000

[23] M W Verbrugge and R F Hill ldquoIon and solvent transport inion-exchangemembranes I Amacrohomogeneousmathemat-ical modelrdquo Journal of the Electrochemical Society vol 137 no 3pp 886ndash893 1990

[24] A Z Weber and J Newman ldquoTransport in polymer-electrolytemembranes II Mathematical modelrdquo Journal of the Electro-chemical Society vol 151 no 2 pp A311ndashA325 2004

[25] C Ziegler H M Yu and J O Schumacher ldquoTwo-phasedynamic modeling of PEMFCs and simulation of cyclo-voltammogramsrdquo Journal of the Electrochemical Society vol 152no 8 pp A1555ndashA1567 2005

[26] A Z Weber and J Newman ldquoEffects of microporous layersin polymer electrolyte fuel cellsrdquo Journal of the ElectrochemicalSociety vol 152 no 4 pp A677ndashA688 2005

[27] J Newman and K E Thomas-Alyea Electrochemical SystemsWiley-Interscience Hoboken NJ USA 3rd edition 2004

[28] T F Fuller and J Newman ldquoWater and thermal management insolid-polymer-electrolyte fuel cellsrdquo Journal of the Electrochem-ical Society vol 140 no 5 pp 1218ndash1225 1993

[29] A Al-Othman A Y Tremblay W Pell et al ldquoZirconium phos-phate as the proton conducting material in direct hydrocarbonpolymer electrolyte membrane fuel cells operating above theboiling point of waterrdquo Journal of Power Sources vol 195 no9 pp 2520ndash2525 2010

[30] Y-I Park J-D Kim and M Nagai ldquoHigh proton conductivityin ZrP-PTFE compositesrdquo Journal of Materials Science Lettersvol 19 no 19 pp 1735ndash1738 2000

[31] R Taylor and R KrishnaMulticomponentMass TransferWileyNew York NY USA 1993

[32] F Hecht O Pironneau A LeHyaric and K OhtsukaldquoFreeFEM++ Version 312rdquo 2011 httpwwwfreefemorgff++indexhtm

[33] G Psofogiannakis Y Bourgault B E Conway and M TernanldquoMathematical model for a direct propane phosphoric acid fuelcellrdquo Journal of Applied Electrochemistry vol 36 no 1 pp 115ndash130 2006

[34] W T Grubb and C J Michalske ldquoA high performance propanefuel cell operating in the temperature range of 150∘ndash200∘CrdquoJournal of The Electrochemical Society vol 111 no 9 pp 1015ndash1019 1964

[35] ParaView 39-Open Source Scientific Visualization 2011httpwwwparavieworg

[36] S Sang QWu and K Huang ldquoPreparation of zirconium phos-phate (ZrP)Nafion1135 composite membrane and H+VO2+transfer property investigationrdquo Journal of Membrane Sciencevol 305 no 1-2 pp 118ndash124 2007

[37] O Savadogo and F J Rodriguez Varela ldquoLow-temperaturedirect propane polymer electrolyte membranes fuel cell(DPFC)rdquo Journal of New Materials for Electrochemical Systemsvol 4 no 2 pp 93ndash97 2001

[38] W Vielstich A Lamm and H A Gasteiger Eds Handbook ofFuel Cells Wiley Chichester UK 2003

[39] DDobosElectrochemicalData AHandbook for Electrochemistsin Industry and Universities Elsevier New York NY USA 1975

[40] C ToriM Baleztena C Peralta et al ldquoAdvances in the develop-ment of a hydrogenoxygen PEM fuel cell stackrdquo InternationalJournal of Hydrogen Energy vol 33 no 13 pp 3588ndash3591 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Analytical ChemistryInternational Journal of

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CatalystsJournal of