A Three-Dimensional Non-isothermal Model of High Temperature Proton Exchange Membrane Fuel Cells with Phosphoric Acid Doped Polybenzimidazole Membranes

AThree-Dimensional Non-isothermalModel of High Temperature ProtonExchange Membrane Fuel Cells withPhosphoric Acid DopedPolybenzimidazole MembranesK. Jiao1 and X. Li1*1 Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Received March 26, 2009; accepted December 17, 2009

1 Introduction

High temperature proton exchange membrane fuel cells(HT-PEMFCs) with operating temperatures higher than100 °C have attracted growing interests in the past decade.By comparing with conventional PEMFCs operating ataround 80 °C, HT-PEMFCs with elevated operating tempera-tures feature faster electrochemical kinetics, simpler watermanagement (presence of liquid water can be neglected),higher carbon monoxide (CO) tolerance (e.g. >1% CO at150 °C [1]) and easier cell cooling and waste heat recovery.

Although HT-PEMFCs have many attractive features, tech-nical challenges still remain and are mostly related to the pro-ton exchange membrane (PEM). Perfluorosulfonic acid(PFSA) polymer membranes (e.g. Nation membranes) widelyused in conventional PEMFCs suffer significant decrement inmechanical strength at the high operating temperature of HT-PEMFCs, and the much lower relative humidity (RH) in HT-PEMFCs than in conventional PEMFCs due to the signifi-

–[*] Corresponding author, [email protected]

AbstractHigh temperature proton exchange membrane fuel cells(HT-PEMFCs) with phosphoric acid doped polybenzimida-zole (PBI) membranes have gained tremendous attentionsdue to its attractive advantages over conventional PEMFCssuch as faster electrochemical kinetics, simpler water man-agement, higher carbon monoxide (CO) tolerance and easiercell cooling and waste heat recovery. In this study, a three-dimensional non-isothermal model is developed for HT-PEMFCs with phosphoric acid doped PBI membranes. Agood agreement is obtained by comparing the numericalresults with the published experimental data. Numericalsimulations have been carried out to investigate the effectsof operating temperature, phosphoric acid doping level ofthe PBI membrane, inlet relative humidity (RH), stoichiome-try ratios of the feed gases, operating pressure and air/oxy-gen on the cell performance. Numerical results indicate that

increasing both the operating temperature and phosphoricacid doping level are favourable for improving the cell per-formance. Humidifying the feed gases at room temperaturehas negligible improvement on the cell performance, andfurther humidification is needed for a meaningful perfor-mance enhancement. Pressurising the cell and using oxygeninstead of air all have significant improvements on the cellperformance, and increasing the stoichiometry ratios onlyhelps prevent the concentration loss at high current densi-ties.

Keywords: Cell Performance, High Temperature ProtonExchange Membrane Fuel Cell, Numerical Simulations,Phosphoric Acid Doped Polybenzimidazole Membrane,Three-Dimensional Non-isothermal Model

FUEL CELLS 10, 2010, No. 3, 351–362 © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 351

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DOI: 10.1002/fuce.200900059

Jiao, Li: A Three-Dimensional Non-isothermal Model

cantly increased vapour saturation pressure with tempera-ture also results in severe reduction of the proton conductiv-ity of the PFSA polymer membranes. Therefore, developingPEMs with high mechanical strength at the temperatureshigher than 100 °C and with high proton conductivity inanhydrous environments becomes the major challenge, andmost of the previous HT-PEMFC related researches focusedon this important issue [2]. Polybenzimidazole (PBI) mem-branes first proposed by Aharoni and Litt [3] have beeninvestigated in the previous studies and recognised as prom-ising PEMs when doped with a strong oxo-acid (e.g. phos-phoric acid or sulphuric acid) for HT-PEMFCs [4–6]. More-over, phosphoric acid doped PBI membrane first suggestedfor fuel cell applications by Wainright et al. [7] has attractedmost of the attentions due to its relatively higher proton con-ductivity and mechanical strength by comparing with theother types of acid doped PBI membranes.

The proton conductivity measurements of phosphoric aciddoped PBI membranes have been carried out [1, 7–10] and ithas been found that the temperature, phosphoric acid dopinglevel and surrounding RH all have significant effects on theproton conductivity. It was shown that the proton conductiv-ity of phosphoric acid doped PBI membranes increases withtemperature by following the Arrhenius law [7], and theexperimental measurements in [1, 8–10] also observed thatincreasing both the phosphoric acid doping level and sur-rounding RH all have significant improvements on the pro-ton conductivity. The experimental study in Ref. [11] reportedthat the thermal stability of a PBI membrane with a dopinglevel of 4.8 (4.8 phosphoric acid molecules per PBI repeatunit) is more than enough for use as a PEM in a HT-PEMFC.Weng et al. [12] concluded that the electro-osmotic drag(EOD) effect is negligible in PBI membranes, which couldfurther simplify the water management of HT-PEMFCs. Insitu tests of HT-PEMFCs were also conducted and promisingcell performances were obtained under various operatingconditions with good CO tolerance and acceptable durability[9, 13–15].

Numerical models for HT-PEMFCs with phosphoric aciddoped PBI membranes have also been developed in the pre-vious studies [15–21]. Cheddie and Munroe [16, 17] devel-oped a one-dimensional model, which were later extended totwo-dimensional [18] and three-dimensional [19] models. Athree-dimensional model similar to Ref. [19] was introducedin Ref. [15] as well. Both the steady and unsteady three-dimensional models were presented by Peng et al. [20, 21].However, the numerical models in [15–19] assumed constantproton conductivities of the membranes, and only the tem-perature dependence of the membrane proton conductivitywas considered in Ref. [20, 21]. As mentioned earlier, temper-ature, phosphoric acid doping level and surrounding RH allhave significant influences on the membrane proton conduc-tivity, therefore need to be fully accounted for in numericalmodels.

In this study, a three-dimensional non-isothermal modelof HT-PEMFCs with phosphoric acid doped PBI membranes

is developed. In this model, a semi-empirical correlation forcalculating the proton conductivity of phosphoric acid dopedPBI membrane is formulated based on the Arrhenius Lawand the previously reported experimental data to fullyaccount for the effects of temperature, phosphoric acid dop-ing level and surrounding RH, and therefore these effects onthe cell performance are all considered in this model. In thefollowing, the detailed description of the present model isgiven, followed by the results and discussion and the conclu-sion.

2 HT-PEMFC Model Formulation

2.1 Physical Problem

Figure 1 shows the computational domain and grid systemfor this study. All the major components of a HT-PEMFC areconsidered in this model, which are the bipolar plate (BP),flow channel, gas diffusion layer (GDL), catalyst layer (CL)and membrane. A single cell is considered with straight flowchannels. Hydrogen is supplied to the anode flow channel,air or oxygen is supplied to the cathode flow channel, andboth the anode and cathode inlet gases can be humidified.

(a)

(b)Fig. 1 Computational domain and grid system (a: 3D view, 10× minifica-tion along the x-direction; b: 2D view on y–z plane).

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The cell properties and operating conditions are listed inTable 1.

2.2 Assumptions

In this model, steady state operating condition is assumed,and ideal gas behaviour and laminar flow (low flow veloci-ties) assumptions are used as well. Water is assumed to beproduced as vapour from the electrochemical reactions dueto the high operating temperatures. The membrane isassumed to be impermeable to all the gases. The water trans-port through the membrane is also neglected due to the dryenvironment in HT-PEMFCs and negligible EOD effect forPBI membranes [12]. Therefore, the hydration effect on thePBI membrane conductivity is approximated by using thevolume averaged RH of the CLs on both the anode and cath-ode sides (detailed in Section 2.3). Both the GDL and CL areassumed to be homogeneous and isotropic.

2.3 Conservation Equations

Ten conservation equations are developed including mass,momentum (x-, y- and z-velocities), different species trans-port (hydrogen, oxygen and vapour), electronic potential,protonic potential and energy, as shown in the followingequations:

Mass: ∇ � q�u� � � Sm (1)

Momentum:

∇ � q�u�ue2

� �� ∇p � l∇

� ∇�ue

� �� ∇

�uT

e

� ��

� 23

l∇ ∇ � �ue

� �� Su �2�

Species (i: hydrogen, oxygen and vapour):

∇ � q�uYi� � � ∇ � qDeffi ∇Yi

� �� Si (3)

Electronicpotential: 0 � ∇ � jeff

ele∇�ele

� �� Sele (4)

Protonicpotential: 0 � ∇ � jeff

pro∇�pro

� �� Spro (5)

Energy:

∇ � qCp�uT� �

� ∇ � keffg�s∇T

� �� ST (6)

Table 2 shows the transport parame-ters. The temperature dependence of thedynamic viscosities is considered, and

both the temperature and pressure dependences of the massdiffusivities are accounted for in this non-isothermal model.The thermal conductivities and specific heat capacities of thegas species, as well as the entropy change of reaction areassumed to be constant due to the small variations with tem-perature [22]. Since the product water is assumed to be invaporous phase, the entropy change of the reaction is also cal-culated based on vapour product. The corresponding sourceterms of the conservation equations are listed in Table 3.

Equations 1 and 2 represent the mass and momentum con-servations of the gas mixture, and Eq. (3) describes the spe-cies transport of hydrogen, oxygen and vapour. Superficialvelocity is used in this model to satisfy the mass conservationat the interfaces between GDLs (porous) and flow channels.As mentioned earlier, all the gas species are assumed to beideal gas, and the gas mixture density (q, kg m–3) is calculatedbased on the ideal gas law:

q � p RT�

i

Yi

Mi

� ��1

�7�

where p is pressure (Pa), T is temperature (K), R is the univer-sal gas constant (J K–1 kmol–1) and Yi and Mi (kg kmol–1) arethe mass fraction and molecular weight of species i. Thedynamic viscosity of the ideal gas mixture (l, kg m–1 s–1) iscalculated based on the kinetic theory [23] as:

Table 1 Cell properties and operating conditions.

Parameter Value

Channel length; width; depth; rib width 100; 1.0; 1.0; 1.0 mmThicknesses of membrane; CL; GDL 0.1; 0.01; 0.2 mmVolume fraction of electrolyte in CL x � 0�4Porosities of CL; GDL eCL�GDL � 0�3� 0�6Permeabilities of CL; GDL KCL�GDL � 6�2 × 10�13� 6�2 × 10�12 m2

Densities of membrane; CL; GDL; BP qmem�CL�GDL�BP � 1� 000 kg m�3

Specific heat capacities of membrane; CL;GDL; BP

Cp

� �mem�CL�GDL�BP

� 1� 650� 3� 300; 568; 1� 580 J kg�1 K�1

Thermal conductivities of membrane; CL;GDL; BP

kmem�CL�GDL�BP � 0�95� 1�0; 1�0; 20 W m�1 K�1

Electrical conductivities of CL; GDL; BP jCL�GDL�BP � 300� 300; 20� 000 S m�1

Stoichiometry ratios at 1.5 A cm–2 na�c � 2�0 or 4�0Relative humidities of inlet gases RHin

a�c � 0� 100% 25 o C or� 100% 80 oC�Absolute total pressures at outlets pout

a�c � 1 or 2 atmOperating temperatures Topt � 110� 150 or 190 �C

Table 2 Transport parameters.

Parameter Correlation/Value (T in K, p in Pa) Unit

Hydrogen dynamic viscosity lH2� 3�205 × 10�3�T�293�85�1�5�T � 72��1�0 kg m–1 s–1

Oxygen dynamic viscosity lO2� 8�46 × 10�3�T�292�25�1�5�T � 127��1�0 kg m–1 s–1

Water vapour dynamic viscosity lH2O � 7�512 × 10�3�T�291�15�1�5�T � 120��1�0 kg m–1 s–1

Hydrogen diffusivity in anode DaH2

� 1�055 × 10�4�T�333�15�1�5�101� 325�p� m2 s–1

Vapour diffusivity in anode DaH2 O � 1�055 × 10�4�T�333�15�1�5�101� 325�p� m2 s–1

Oxygen diffusivity in cathode DcO2

� 2�652 × 10�5 T�333�15� �1�5 101� 325�p� � m2 s–1

Vapour diffusivity in cathode DcH2 O � 2�982 × 10�5 T�333�15� �1�5 101� 325�p� � m2 s–1

Specific heat capacities of species Cp

� �H2

� 14283� Cp

� �O2

� 919�31� Cp

� �H2O

� 2014 J kg–1 K–1

Thermal conductivities of species kH2� 0�1672� kO2

� 0�0246; kH2O � 0�0261 W m–1 K–1

Entropy change of reaction DS � �44� 500 J kmol–1 K–1

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l ��

i

Xili�j Xiwij

�8�

wij �1 � li

lj

� �0�5Mi

Mj

� �0�25 �0�5

8 1 � Mi

Mj

� �� 0�5

�9�

and Xi is the mole fraction of species i,and i and j represent different species.

The effects of porous regions on gas-eous transport are considered by addingsource terms in the momentum equa-tions (based on the permeabilities of theporous materials, as shown in Table 3),and the effective mass diffusivities in thespecies transport equations are modifiedby considering the porosity and tortuos-ity based on the Bruggeman correlation:

Deffi � Die

1�5 �10�

where Di (m2 s–1) is the bulk mass diffu-sivity for species i calculated based onthe correlations in Table 2.

The reaction rates (ja and jc, A m–3) forboth the anode and cathode CLs are cal-culated by using the Butler–Volmerequation:

ja � jref0�a

cH2

crefH2

� �0�5

exp2aaFRT

gact

� �� exp � 2acF

RTgact

� � ��11�

jc � jref0�c

cO2

crefO2

�exp4aaFRT

gact

� �� exp � 4acF

RTgact

� � ��12�

where F is the Faraday’s constant (C mol–1), and the correla-tions and values of the parameters related to the electroche-mical reactions are listed in Table 4. The reference exchangecurrent densities also vary with temperature due to thechange of electrochemical reaction kinetics [24], and the cor-relations are given in this table as well.

A semi-empirical correlation is formulated based on theArrhenius Law [7] and the previously reported experimentaldata [1, 9, 10] to fully account for the effects of temperature,phosphoric acid doping level and surrounding RH on themembrane proton conductivity (jpro, S m–1):

jpro �abT

exp�Ea

RT

� ��13�

where a and b are the two different pre-exponential factors,and Ea is the activation energy (J mol–1). Li et al. [9] showed

an almost linear relationship between Ea and the phospho-ric acid doping level, and therefore the following linearequation is obtained by fitting the experimental data inRef. [9]:

Ea � �619�6DL � 21� 750 �14�

where the unit of Ea is J mol–1; DL is the phosphoric acid dop-ing level of PBI membranes, which is defined as the numberof phosphoric acid molecules per PBI repeat unit. The pre-exponential factor a in Eq. (13) further accounts for the effectof phosphoric acid doping level on the membrane protonconductivity, and the following equation is formulated basedon the experimental measurements in Ref. [9]:

a � 168DL3 � 6� 324DL2 � 65� 760DL � 8� 460 �15�

The pre-exponential factor b in Eq. (13) considers theeffect of surrounding RH on the membrane proton conductiv-ity. The experimental measurements in Refs. [1, 10] allshowed almost linear relationships between the proton con-ductivity and the surrounding RH with different slopes atdifferent temperatures, therefore the following correlation isdeveloped to calculate b in the temperature range between100 and 200 °C (the operating temperature range of HT-PEMFCs):

Table 3 Source terms.

Source term Unit

Sm � SH2� SO2

� SH2 O kg m–3 s–1

Su � �lK�u in CL and GDL� �

0 in other zones� �

�kg m–2 s–2

SH2� � ja

2FMH2

in anode CL� �0 in other zones� �

�SO2

� � jc4F

MO2in cathode CL� �

0 in other zones� �

�kg m–3 s–1

SH2O �jc2F

MH2O in cathode CL� �0 in other zones� �

�kg m–3 s–1

Sele ��ja in anode CL� �jc in cathode CL� �0 in other zones� �

�� Sion �

ja in anode CL� ��jc in cathode CL� �0 in other zones� �

��

A m–3

ST �

ja gact� � � ∇�ele 2jeffele � ∇�pro

�� 2jeff

pro in anode CL� �� jcTDS

2F� jc gact� � � ∇�ele 2jeff

ele � ∇�pro

�� 2jeff

pro in cathode CL� �∇�ele 2jeff

ele in GDL and BP� �∇�pro

�� 2jeff

pro in membrane� �0 in other zones� �

��

W m–3

Table 4 Electrochemical parameters.

Parameter Correlation/Value (T in K) Unit

Overpotential (activation loss) gact � �ele � �pro VTransfer coefficient aa � ac � 0�5 DimensionlessVolumetric reference exchangecurrent density in anode

jref0�a � jref

0�a 353�15 K exp �1� 4001T� 1

353�15

� � ��jref0�a 353�15 K �� 109

A m–3

Volumetric reference exchangecurrent density in cathode

jref0�c � jref

0�c 353�15 K exp �7� 9001T� 1

353�15

� � ��jref0�c 353�15 K �� 104

A m–3

Reference hydrogen and oxygenmolar concentrations

crefH2

� crefO2

� 40�0 mol m–3

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b �1 � �0�01704T � 4�767�RHeff if 373�15 K ≤ T ≤ 413�15 K1 � �0�1432T � 56�89�RHeff if 413�15 K � T ≤ 453�15 K1 � �0�7T � 309�2�RHeff if 453�15 K � T ≤ 473�15 K

��

�16�

where the unit of T is K, and the effective RHeff is defined as:

RHeff � RH �in CL�RHavg �in membrane�

��17�

where in the CLs of both the anode and cathode, the local RHis used for the calculation of the proton conductivity of theelectrolyte; in the membrane, the water transport mechanismin PBI membranes are still unclear, however, since the PBImembrane conductivity changes almost linearly with the sur-rounding RH [1, 10], the hydration effect on the PBI mem-brane conductivity can be considered by using the volumeaveraged RH of the CLs on both the anode and cathode sides(RHavg). In the CLs of the anode and cathode, RH is calcu-lated as:

RH � XH2Oppsat

(18)

where XH2O is the mole fraction of vapour in the gas mixture,p (Pa) the pressure of the gas mixture and psat (Pa) the satura-tion pressure of water. The following correlation is used byfitting the experimental data in Ref. [22] to calculate psat (Pa)in the temperature range between 100 and 200 °C (the operat-ing temperature range of HT-PEMFCs):

psat � 0�68737T3 � 732�39T2 � 263� 390T � 31� 919� 000 �19�

where the units of psat and T are Pa and K, respectively.The effective electron conductivity in CL and GDL and

proton conductivity in CL are further modified with the por-osities (e) of the CL and GDL and the electrolyte volume frac-tion in CL (x). Based on the Bruggeman correlation, an expo-nent of 1.5 is used:

jeffele � 1 � e � x� �1�5jele �20�

jeffpro � x1�5jpro �21�

The specific heat capacity of the gas mixture Cp (J kg–1 K–1)in the energy conservation equation (Eq. 6) is determinedbased on the mass fraction of each species:

Cp ��

i

Yi Cp

� �i

�22�

For simplicity, the effective thermal conductivity(W m–1 K–1) in the energy conservation equation (Eq. 6) isassumed to be a volume averaged value:

keffg�s � ek � 1 � e � x� �ksld � xkmem �23�

where ksld (W m–1 K–1) represents the thermal conductivity ofthe electron conducting materials (platinum powders, carbonpowders etc.) in CL and all the solid materials in GDL and BP,kmem (W m–1 K–1) represents the thermal conductivity of themembrane, and the thermal conductivity of the gas mixture, k(W m–1 K–1), is calculated based on the kinetic theory [23] as:

k ��

i

Xiki�j

Xiwij�24�

wij �1 � ki

kj

� �0�5Mi

Mj

� �0�25 �0�5

8 1 � Mi

Mj

� �� 0�5 �25�

2.4 Boundary Conditions

At the inlets of the flow channels, the mass flow rates(kg s–1) are defined as:

�ma � qanaIrefA2FcH2

; �mc � qcncIrefA4FcO2

�26�

cH2� pa � RHapsat� �

RTain

;

cO2�

0�21 pc�RHcpsat� �RTc

in

for supplying air� �

pc�RHcpsat� �RTc

in

for supplying oxygen� �

�� (27)

where na and nc are the stoichiometry ratios (listed in Table 1)for the anode and cathode, respectively. Iref is the referencecurrent density (1.5 A cm–2), and A is the flat active surfacearea of the CL (m2). In order to achieve the desired operatingtemperatures, constant temperatures are defined at the anodeand cathode inlets (Tin, K) as well as on the surrounding wallsto be equal to the operating temperatures (Topt, K). The heatfluxes at the surrounding walls are calculated correspond-ingly based on the temperatures defined. On the other hand,the temperatures at the surrounding walls can be calculatedwhen heat fluxes are defined. Due to the heat generationinside the cell, the real operating temperature in the cell willbe slightly higher than the demanded operating temperaturethat is specified at the boundaries. Constant pressures aredefined at the flow channel outlets, as listed in Table 1.

The electronic potentials on the end surfaces of the BPs forboth the anode and cathode are defined as:

�c�endele � 0� �a�end

ele � Vrev � Vcell � gtotal �28�

where Vrev and Vcell are the reversible and operating voltages,therefore gtotal represents the total voltage loss. For vapourproduct, the reversible cell voltage is calculated as:

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Vrev � 1�185 � 2�3 × 10�4 Topt � 298�15� �

� RTopt

2F

× ln pinH2

� 12

ln pinO2

� ��29�

In this equation, the inlet partial pressures of hydrogenand oxygen (atm), and the operating temperature (Topt, K)are used for the calculation. In this model, the open circuitvoltage is assumed to be equal to the reversible voltage. Thisassumption is usually valid because the gas and electroncrossover through the membrane can be neglected when ameaningful current is drawn from the cell, and this assump-tion has been used in most of the previous numerical models(such as in Refs. [16–21]).

3 Numerical Procedures

The conservation equations are discretised and solved inthe commercial computational fluid dynamics (CFD) soft-ware package Fluent 6.3 based on finite volume method.Equations and physical properties are written and implemen-ted by using its user defined function (UDF). The SIMPLEalgorithm is used which is based on semi-implicit method forpressure-linked equations for steady state problems. An alge-braic multigrid (AMG) method with a Gauss–Seidel typesmoother is used to accelerate the convergence. Strict conver-gence criteria with a residual of 10–8 are used for all variables.The total number of the computational grid shown inFigure 1 is 160,000, and structured grid is used for thewhole computational domain. Each grid in the flowchannel and BP has the same size with dimensions of1.0 mm × 0.1 mm × 0.1 mm, along the x-, y- and z-directions,respectively. The grid size along the y-direction is reduced to0.01, 0.001 and 0.02 mm for the membrane, CL and GDL,respectively, to ensure at least ten grids along the y-directionin each layer. Grid independent study was carefully carriedout. The difference in results (variations in the current densityat different cell voltages) by further increasing the number ofgrid (up to two times of the one used) is negligible. Such gridsystems were also considered sufficient and widely used inprevious three-dimensional simulations [e.g. Refs. 15, 24].

4 Results and Discussion

To examine the accuracy of the present model, the polari-sation curve obtained from this model is compared with theexperimental data in Ref. [9], as shown in Figure 2, whichindicates a good agreement. For the results shown in Fig-ure 2, the operating temperature is 170 °C, the cell operateswith hydrogen and oxygen without humidification at atmo-spheric pressure, the stoichiometry ratios are 1.87 and 3.74 forthe anode and cathode for a reference current density of1.5 A cm–2, and the phosphoric acid doping level for the PBImembrane is 6.2. Although there are other experimental dataavailable in literature, the information to specify the design

and operating parameters required by the model is usuallynot fully available for a meaningful comparison. The experi-mental data in Ref. [9] are provided with relatively morecomplete information and therefore are chosen for compari-son. All the cell design and operating parameters mentionedin Ref. [9] are incorporated in the simulation, and representa-tive values are used for the parameters not provided inRef. [9].

Based on the three-dimensional non-isothermal model ofHT-PEMFCs with phosphoric acid doped PBI membranesdeveloped in this study, numerical simulations are carriedout to study the effects of operating temperature, phosphoricacid doping level of the PBI membrane, inlet RH, stoichiome-try ratios of the feed gases, operating pressure and air/oxy-gen on the cell performance. The transport phenomena in thecell with different design and operating conditions are inves-tigated as well.

4.1 Effects of Operating Temperature on the Cell Performance

Figure 3 compares the cell performances at different oper-ating temperatures (190, 150 and 110 °C). For the comparisonin this figure, the cell operates with hydrogen and air withouthumidification at atmospheric pressure, the stoichiometryratio is 2 for both the anode and cathode for a reference cur-rent density of 1.5 A cm–2, and the phosphoric acid dopinglevel for the PBI membrane is 6. No apparent concentrationloss is observed for all the operating temperatures due to thehigh stoichiometry ratios and the avoided liquid water for-mation, similar observations have been shown in [15] as well.The peak power densities are obtained at a cell voltageof 0.4 V for all the operating temperatures. An increment ofthe peak power density of 0.065 W cm–2 (from 0.213 to0.278 W cm–2) is obtained by increasing the operatingtemperature from 110 to 150 °C, and the increment is0.062 W cm–2 (from 0.278 to 0.34 W cm–2) from 150 to 190 °C.

Current Density / A cm

CellVoltage/V

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2Model predictionsExperimental data [9]

-2

Fig. 2 Comparison between the present model predictions and experi-mental data [9] (the operating temperature is 170 °C, the cell operateswith hydrogen and oxygen without humidification at atmospheric pres-sure, the stoichiometry ratios are 1.87 and 3.74 for the anode and cath-ode for a reference current density of 1.5 A cm–2 and the phosphoric aciddoping level for the PBI membrane is 6.2).

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The almost linear and significant increment of the peakpower density with temperature indicates that operating thecell at high temperatures is furthersome, and the main rea-sons are the enhanced electrochemical kinetics and mem-brane proton conductivity at high operating temperatures.

The effect of operating temperature on the membrane pro-ton conductivity needs to be investigated carefully, becausethe surrounding RH (strongly affected by temperature) alsohas significant influence on the membrane conductivity, thusaffecting the cell performance. For the operating conditions inFigure 3, the RH in the anode remains zero due to the non-humidified inlet hydrogen gas and the neglected water trans-port through the membrane, and the RH in the cathodechanges dramatically with current density and operatingtemperature. Figure 4 shows the contours of RH in the mid-dle plane (y = 0.001705 m) of the cathode CL at differentoperating temperatures (190, 150 and 110 °C) and at a cellvoltage of 0.6 V. The other operating conditions in Figure 4are the same as in Figure 3. At 0.6 V, the cell current densitiesat 190, 150 and 110 °C are 0.433, 0.364 and 0.284 A cm–2,respectively, indicating that more water is produced at higheroperating temperatures. However, Figure 4 shows that theRH is much higher at a lower operating temperature, due tothe increased vapour saturation pressure with temperature.Similar distributions of the RH in the cathode CL for all thethree operating temperatures are shown in Figure 4: the RHincreases along the flow direction due to the accumulation ofproduct water, and it is higher under the land than under theflow channel because the water removal is easier under theflow channel than under the land. The volume averaged RHsof the CLs on both sides are 0.233, 0.527 and 1.39%, corre-sponding to the operating conditions in Figure 4a–c, respec-tively. By using these values together with the operating tem-peratures, the pre-exponential factor b (Eq. 16) is calculated tobe 1.035, 1.02 and 1.025 for Figure 4a–c, respectively. Sincethe pre-exponential factor b represents the effect of surround-

ing RH on the membrane proton conductivity, these similarvalues of b indicate that even the RH changes significantlywith operating temperature, the hydration effects on themembrane conductivity are similar when the other operatingconditions are similar. With the similar pre-exponential factorb at different operating temperatures, increasing the operat-ing temperature will increase the membrane proton conduc-tivity by following the Arrhenius Law, as shown in Eq. (13).Therefore, it can be concluded that increasing the operatingtemperature is favourable for better cell performance. Despitethat, the thermal stability of the phosphoric acid doped PBImembrane must be considered, and based on the experimen-tal data in Ref. [9], operating the cell at 190 °C probablyalready reaches the temperature tolerance for PBI membraneswith a phosphoric acid doping level of 6, and decreasing the

Current Density / A cm-2

CellVoltage/V

PowerDensity/W

cm-2

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

0

0.1

0.2

0.3

0.4

(a) Temperature = 190 C(b) Temperature = 150 C(c) Temperature = 110 Co

o

o

Fig. 3 Effects of operating temperature on the cell performance (the celloperates with hydrogen and air without humidification at atmosphericpressure, the stoichiometry ratio is 2 for both the anode and cathode for areference current density of 1.5 A cm–2 and the phosphoric acid dopinglevel for the PBI membrane is 6) (a: the operating temperature is 190 °C;b: the operating temperature is 150 °C; c: the operating temperature is110 °C).

0.0019 0.002

40.0029 0.0035

0.0040

0.0077

0.0045

0.0045

0.0051

0.0051

0.0056

0.0061

0.0061

0.0067 0.00720.007

7

0.00190.0024

0.0029 0.00350.0040

0.0045

0.0045

0.0051

0.0051

0.0056

0.0056

0.00610.0067

0.0072

0.0013

Z

X Flow direction

0.011 0.015

0.018 0.021

0.024

0.046

0.027

0.027

0.030

0.030

0.033

0.036

0.036

0.040 0.0430.046

0.011 0.015

0.018 0.0210.024

0.027

0.027

0.030

0.030

0.033

0.033

0.0360.040 0.043

0.008

Z

X Flow direction

0.004

0.005 0.007 0.008

0.009

0.017

0.010

0.010

0.011

0.011

0.013

0.014

0.014

0.015 0.0160.017

0.0050.007 0.008

0.009

0.010

0.010

0.011

0.011

0.013

0.013

0.0140.015 0.016

0.003

Z

X Flow direction

(a)

(b)

(c)Fig. 4 Contours of relative humidity in the middle plane(y = 0.001705 m) of the cathode catalyst layer (the cell operates at 0.6 Vwith hydrogen and air without humidification at atmospheric pressure, thestoichiometry ratio is 2 for both the anode and cathode for a referencecurrent density of 1.5 A cm–2 and the phosphoric acid doping level for thePBI membrane is 6) (a: the operating temperature is 190 °C; b: the operat-ing temperature is 150 °C; c: the operating temperature is 110 °C).

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phosphoric acid doping level is necessary for further heatingup of the cell. However, it is also known that decreasing thephosphoric acid doping level results in decrement in themembrane proton conductivity (as shown in Eqs. 13–15),therefore the effect of the phosphoric acid doping level of thePBI membrane on the cell performance needs to be deter-mined, which is investigated in the next subsection.

4.2 Effects of Phosphoric Acid Doping Level of the PBImembrane on the Cell Performance

The cell performances at different phosphoric acid dopinglevels (9, 6 and 3) of the PBI membrane are compared in Fig-ure 5. For the comparison in this figure, the cell operates withhydrogen and air without humidification at atmosphericpressure, the stoichiometry ratio is 2 for both the anode andcathode for a reference current density of 1.5 A cm–2 and theoperating temperature is 190 °C. Similar to Figure 3, the peakpower densities are also obtained at a cell voltage of 0.4 V inFigure 5 for all the phosphoric acid doping levels. The peakpower density increases by 0.137 W cm–2 (from 0.203 to0.34 W cm–2) when the phosphoric acid doping levelincreases from 3 to 6, and the increment reduces to0.095 W cm–2 (from 0.34 to 0.435 W cm–2) for the phosphoricacid doping levels from 6 to 9. By using Eq. (13) with zero RHand the operating temperature of 190 °C, the membrane pro-ton conductivities are calculated to be 1.89, 4.228 and6.815 S m–1, which correspond to the phosphoric acid dopinglevels of 3, 6 and 9, respectively. These values indicate thatthe increment in the membrane proton conductivity byincreasing the phosphoric acid doping level from 6 to 9 ishigher than from 3 to 6, which seems to contradict the pre-dicted cell performances. The reason is that with the incre-ment of the membrane proton conductivity, the current den-

sity increases as well, thus resulting in larger ohmic losses. Inspite of the reduced increment, the peak power density isincreased by about 28% for the phosphoric acid doping levelsfrom 6 to 9, which is still a significant improvement. Theseresults indicate that increasing the phosphoric acid dopinglevel of PBI membranes have significant improvements onthe cell performance. However, as mentioned earlier, thephosphoric acid doping level of 9 is not feasible with an oper-ating temperature of 190 °C [9], and therefore further studiesto increase the thermal stability while keeping the phosphoricacid doping level for PBI membranes are needed.

Figure 6 shows the temperature distributions on the y–zplane at x = 0.05 m (middle along the flow direction) at dif-ferent phosphoric acid doping levels (9, 6 and 3) of the PBImembrane and at a cell voltage of 0.3 V (the maximum cur-


CellVoltage/V

PowerDensity/W

cm-2

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

0

0.1

0.2

0.3

0.4

0.5

(a) Doping level = 9(b) Doping level = 6(c) Doping level = 3

Fig. 5 Effects of phosphoric acid doping level of the PBI membrane on thecell performance (the cell operates with hydrogen and air without humidi-fication at atmospheric pressure, the stoichiometry ratio is 2 for both theanode and cathode for a reference current density of 1.5 A cm–2 and theoperating temperature is 190 °C) (a: the phosphoric acid doping level is9; b: the phosphoric acid doping level is 6; c: the phosphoric acid dopinglevel is 3).

191.4

190.7

192.0

192.7

193.4

196.1

194.7

Anode

Cathode

Y

Z

190.8

190.4

191.2

191.6

192.0

193.6

192.8

Anode

Cathode

Y

Z

191.2

190.6

191.8

192.3

192.9

195.3

194.1

Anode

Cathode

Y

ZFig. 6 Temperature distributions (unit: °C) on the y–z plane at x = 0.05 m (middle section along the flow direction) (the cell operates at 0.3 V with hydro-gen and air without humidification at atmospheric pressure, the stoichiometry ratio is 2 for both the anode and cathode for a reference current density of1.5 A cm–2 and the operating temperature is 190 °C) (a: the phosphoric acid doping level is 9; b: the phosphoric acid doping level is 6; c: the phosphoricacid doping level is 3).

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rent densities in Figure 5). The other operating conditions inFigure 6 are the same as in Figure 5. As mentioned earlier,the demanded operating temperature of 190 °C is controlledby fixing the temperatures on the surrounding walls as theboundary condition and at the inlets of the anode and cath-ode to be equal to the demanded operating temperature, anddue to the heat generation inside the cell, the temperatureinside the cell becomes slightly higher than the demandedoperating temperature. Similar temperature distributions areobtained at different phosphoric acid doping levels: the high-est temperature is observed in the membrane electrodeassembly (MEA) away from the surrounding walls, slightlyshifted to the cathode CL due to the reversible and activa-tional heat generations in that layer. However, the magni-tudes of the highest temperatures are different, and they areabout 196.1, 195.3 and 193.6 °C corresponding to phosphoricacid doping levels of 9, 6 and 3, respectively. It can be noticedthat the maximum temperature difference (between the cath-ode CL and the surrounding walls) in the cell is about 6.1 °Cat a high current density of 1.39 A cm–2 (phosphoric aciddoping level of 9). This is caused by the current density dif-ferences–higher current densities result in higher heat genera-tion rates.

4.3 Effects of Inlet RH on the Cell Performance

Figure 7 presents the comparison of the cell performanceswith different inlet RHs (0, 0.25 and 3.8% at 190 °C). For thecomparison in this figure, the cell operates with hydrogenand air at atmospheric pressure, the stoichiometry ratio is 2for both the anode and cathode for a reference current densityof 1.5 A cm–2, the operating temperature is 190 °C, and thephosphoric acid doping level of the PBI membrane is 6. TheRHs of 0.25 and 3.8% at 190 °C are equivalent to 100% RHs at

25 and 80 °C, respectively, meaning that the feed gases arefully humidified at room temperature and at 80 °C. It can beseen from Figure 7 that humidifying the feed gases at roomtemperature has almost negligible improvement on the peakpower density (2%, from 0.34 to 0.346 W cm–2), and the peakpower density is increased by about 14% (from 0.34 to0.0.386 W cm–2) by increasing the RH from 0 to 100% at80 °C. By using Eq. (13) with the phosphoric acid doping lev-el of 6 and the operating temperature of 190 °C, the mem-brane proton conductivities are calculated to be 4.228, 4.386and 6.638 S m–1, which correspond to 0 RH, 100% RH at25 °C and 100% RH at 80 °C, respectively. The calculated val-ues indicate that humidifying the feed gases at room temper-ature has almost negligible enhancement on the cell perfor-mance, and further humidification is needed to obtain ameaningful improvement. However, obtaining a RH of 100%at room temperature is much easier than achieving a 100%RH at 80 °C. To obtain a 100% RH at 80 °C, liquid waterinjection is needed if humidified at room temperature,otherwise the temperature of the humidifier needs to beincreased to at least 80 °C. Both of the humidification meth-ods require more complex system design as well as extrapower consumption. Therefore, humidifying the feeds gasesmay not be a very manoeuvrable way to improve the cell per-formance. Despite that, promising cell performance can beobtained without humidification, as seen from the numericalresults in this paper as well as the experimental data inRefs. [9, 15].

Not only increasing the membrane proton conductivity,humidifying the feed gases also decreases the concentrationsof the reactants (the reactants are supplied at higher flowrates but lower concentrations). Figure 8 shows the hydrogenand oxygen molar concentrations in the anode and cathode,respectively, on the y–z plane at x = 0.05 m (middle along theflow direction) at different inlet RHs and at a cell voltage of0.4 V (peak power densities). The inlet RHs for Figure 8a–care the same as for the curves a, b and c in Figure 7. The otheroperating conditions in Figure 8 are the same as in Figure 7.Figures 8a–c show a similar profile of both the hydrogen andoxygen molar concentrations. The molar concentrations ofboth the hydrogen and oxygen are the highest in the flowchannels, and are reduced in the CLs due to the consump-tions by the electrochemical reactions. However, the magni-tudes of the hydrogen and oxygen molar concentrationsunder different inlet RHs are different. The hydrogen andoxygen molar concentrations are slightly decreased byincreasing the inlet RHs from 0 to 100% at 25 °C (Figure 8aand b), and they are reduced by about 50% when humidify-ing the feed gases from 0 RH to 100% RH at 80 °C (Figure 8aand c). According to the Butler–Volmer equation (Eqs. 11 and12), as well as the modified Nernst equation (Eq. 29), deceas-ing the molar concentrations of the reactants may result insignificant decrement in the cell performance. Therefore, theeffects of the concentration loss also need to be consideredwhen humidifying the feed gases.


CellV

oltage

/V

PowerDensity/W

cm-2

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

0

0.1

0.2

0.3

0.4

(a) Inlet RH = 0%(b) Inlet RH = 0.25%(c) Inlet RH = 3.8%

Fig. 7 Effects of inlet RH on the cell performance (the cell operates withhydrogen and air at atmospheric pressure, the stoichiometry ratio is 2 forboth the anode and cathode for a reference current density of 1.5 A cm–2,the operating temperature is 190 °C and the phosphoric acid doping levelof the PBI membrane is 6) (a: the inlet RH is 0% for both the anode andcathode; b: the inlet RH is 0.25% at 190 °C (100% at 25 °C) for both theanode and cathode; c: the inlet RH is 3.8% at 190 °C (100% at 80 °C) forboth the anode and cathode).

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4.4 Effects of Stoichiometry Ratios of the Feed Gases, OperatingPressure and Air/Oxygen on the Cell Performance

Figure 9 compares the effects of stoichiometry ratios of thefeed gases, operating pressure and air/oxygen on the cell per-formance. For the comparison in this figure, the cell operatesat 190 °C, and the phosphoric acid doping level of the PBImembrane is 6. The cell operates with hydrogen and air oroxygen at 1 or 2 atm, and the stoichiometry ratio is 2 or 4 forboth the anode and cathode for a reference current density of1.5 A cm–2. It can be noticed that increasing the stoichiometry

ratio from 2 to 4 for both the anode and cathode has almostnegligible improvement on the cell performance, and pres-surising the cell from 1 to 2 atm results in an increment in thepeak power density of 11%, and the increment is 15% byreplacing the supplied air with oxygen at 1 atm. Since theother design and operating parameters are kept the same, thechanges of the cell performance are only attributed to thechanges of the concentrations of the reactants.

The oxygen molar concentrations in the middle plane(y = 0.001705 m) of the cathode CL at a cell voltage of 0.4 V(peak power densities) are shown in Figure 10. The operatingconditions for Figure 10a–d are the same as for the curves a,b, c and d in Figure 9 at a cell voltage of 0.4 V, respectively.The general oxygen molar concentration distributions aresimilar in Figure 10a–d. The highest molar concentration is atthe inlet under the flow channel, decreasing along the flowdirection due to the reactant consumption, the oxygen molarconcentration under the land is also lower than under thechannel due to the slower mass transport. It can be noticedthat increasing the stoichiometry ratios of the feed gases hasnegligible effects on increasing the oxygen molar concentra-tion at the inlet, and it is increased by about 17% at the exit.The overall improvement on the oxygen molar concentrationis limited, indicating that further increasing the stoichiometryratio from a high value has insignificant improvements onthe cell performance (a stoichiometry ratio of 2 is already veryhigh). Figure 10 also shows that pressurising the cell from 1to 2 atm doubles the oxygen molar concentration, and repla-cing the supplied air with oxygen increases the oxygen molarconcentration about five times, which all obey the ideal gaslaw. The hydrogen molar concentration is also doubled whenpressurising the cell from 1 to 2 atm. The oxygen molar con-centrations shown in Figure 10 confirms the changes ob-

26.316

26.215

26.115

26.282

26.182

Anode

4.870

3.234 3.561

4.542

4.215

Cathode

Y

Z

12.907

12.73212.644

12.819

12.468

Anode

2.624

0.883 1.231

2.276

1.927

Cathode

Y

Z

25.384

25.280

25.176

25.349

25.245

Anode

4.715

3.057 3.389

4.384

4.052

Cathode

Y

ZFig. 8 Hydrogen and oxygen molar concentrations (unit: mol m–3) in the anode and cathode, respectively, on the y–z plane at x = 0.05 m (middle sec-tion along the flow direction) (the cell operates at 0.4 V with hydrogen and air at atmospheric pressure, the stoichiometry ratio is 2 for both the anodeand cathode for a reference current density of 1.5 A cm–2, the operating temperature is 190 °C, and the phosphoric acid doping level of the PBI mem-brane is 6) (a: the inlet RH is 0% for both the anode and cathode; b: the inlet RH is 0.25% at 190 °C (100% at 25 °C) for both the anode and cathode; c:the inlet RH is 3.8% at 190 °C (100% at 80 °C) for both the anode and cathode).


CellVoltage/V

PowerDensity/W

cm-2

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

0

0.1

0.2

0.3

0.4

(a) Air, St = 2, 1 atm(b) Air, St = 4, 1 atm(c) Air, St = 2, 2 atm(d) Oxygen, St = 2, 1 atm

Fig. 9 Effects of stoichiometry ratios of the feed gases, operating pressureand air/oxygen on the cell performance (the cell operates with hydrogenand air or oxygen at 1 or 2 atm, the stoichiometry ratio is 2 or 4 for boththe anode and cathode for a reference current density of 1.5 A cm–2, theoperating temperature is 190 °C and the phosphoric acid doping level ofthe PBI membrane is 6) (a: hydrogen and air at 1 atm, and the stoichiome-try ratio is 2 for both the anode and cathode; b: hydrogen and air at1 atm, and the stoichiometry ratio is 4 for both the anode and cathode; c:hydrogen and air at 2 atm, and the stoichiometry ratio is 2 for both theanode and cathode; d: hydrogen and oxygen at 1 atm, and the stoichiom-etry ratio is 2 for both the anode and cathode).

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served in Figure 9, indicating that pressurising the cell oroperating with pure oxygen all have significant improve-ments on the cell performance.

5 Conclusion

In this study, a three-dimensional non-isothermal model hasbeen developed for HT-PEMFCs with phosphoric acid dopedPBI membranes. By formulating a semi-empirical correlationfor calculating the proton conductivity of phosphoric aciddoped PBI membrane based on the Arrhenius Law and the pre-viously reported experimental data, and by incorporating thiscorrelation into the present model, the effects of temperature,phosphoric acid doping level and surrounding RH on the cellperformance can be fully accounted for. The numerical resultsof this model are compared with the published experimentaldata, and a good agreement is obtained. Numerical simulationshave been carried out to investigate the effects of operating tem-perature, phosphoric acid doping level of the PBI membrane,inlet RH, stoichiometry ratios of the feed gases, operating pres-sure and air/oxygen on the cell performance. It is found thatincreasing both the operating temperature and phosphoric aciddoping level are favourable for achieving better cell perfor-mance. However, since the thermal stability of the phosphoricacid doped PBI membrane decreases with the increments inboth the temperature and phosphoric acid doping level, themaximum allowed operating temperature and phosphoric aciddoping level should not be exceeded. The numerical resultsalso show that humidifying the feed gases at room temperaturehas almost negligible enhancement on the cell performance,and further humidification is needed to obtain a meaningfulperformance improvement, which requires more complex sys-tem design and extra power consumption. Therefore, humidi-fying the feeds gases may not be a very manoeuvrable way toimprove the cell performance. Despite that, the simulationresults show that promising cell performance can be obtainedwithout humidification. Pressurising the cell and using oxygeninstead of air as the feed gas all have significant improvementson the cell performance, and increasing the stoichiometry ratiosfor the anode and cathode only help prevent the concentrationloss at high current densities.

Acknowledgements

The financial support by the Natural Sciences and Engi-neering Research Council of Canada (NSERC) via a strategicProject Grant (grant no. 350662-07) and by Auto21 is greatlyappreciated.

List of Symbols

Latin Letters

a, b Pre-exponential factors to calculate membrane pro-ton conductivity

4.54 4.33 4.123.91

3.69

3.69

3.48

3.48

3.27

3.06

3.06

2.85 2.64

4.754.54

4.33 4.123.91

2.43

3.69

3.69

3.48

3.48

3.27

3.27

3.062.85 2.64

2.43

4.96

Z

X Flow direction

4.73 4.56 4.404.07

3.91

4.23

3.74

4.07

3.583.74

3.42 3.25

4.894.73

4.56 4.404.07

3.09

3.91

4.23

3.74

4.07

3.58

3.91

3.42 3.253.09

5.05

Z

X Flow direction

8.85 8.38 7.917.44

6.98

6.98

6.51

6.51

6.04

5.57

5.57

5.10 4.63

9.328.85

8.38 7.917.44

4.16

6.98

6.98

6.51

6.51

6.04

6.04

5.575.10 4.63

4.16

9.79

7.44

Z

X Flow direction

21.53

20.58

18.69 17.74

15.8423.42

22.48

19.63

13.00

16.79 14.90

13.95

13.00

24.37

Z

X Flow direction

(a)

(b)

(c)

(d)Fig. 10 Oxygen molar concentrations (unit: mol m–3) in the middle plane(y = 0.001705 m) of the cathode catalyst layer (the cell operates at 0.4 Vwith hydrogen and air or oxygen at 1 or 2 atm, the stoichiometry ratio is2 or 4 for both the anode and cathode for a reference current density of1.5 A cm–2, the operating temperature is 190 °C and the phosphoric aciddoping level of the PBI membrane is 6) (a: hydrogen and air at 1 atm, andthe stoichiometry ratio is 2 for both the anode and cathode; b: hydrogenand air at 1 atm, and the stoichiometry ratio is 4 for both the anode andcathode; c: hydrogen and air at 2 atm, and the stoichiometry ratio is 2 forboth the anode and cathode; d: hydrogen and oxygen at 1 atm, and thestoichiometry ratio is 2 for both the anode and cathode).

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A Cell geometric area (m2)c Molar concentration (mol m–3)Cp Specific heat capacity (J kg–1 K–1)D Mass diffusivity (m2 s–1)DL Phosphoric acid doping level for PBI membraneEa Activation energy (J mol–1)F Faraday’s constant (96487 C mol–1)I Current density (A cm–2)j Reaction rate (A m–3)j0 Volumetric exchange current density (A m–3)k Thermal conductivity (W m–1 K–1)K Permeability (m2)�m Mass flow rate (kg s–1)

M Molecular weight (kg kmol–1)p Pressure (Pa)R Universal gas constant (8.314 J mol–1 K–1)RH Relative humidityS Source terms, entropy (J kmol–1 K–1)T Temperature (K)Topt Cell operating temperature (K)�u Velocity (m s–1)V Electrical potential (V)X Mole fractionY Mass fraction

Greek Letters

a Transfer coefficiente Porosityg Over potential (V)j Electrical conductivity (S m–1)l Dynamic viscosity (kg m–1 s–1)n Stoichiometry ratioq Density (kg m–3)� Electrical potential (V)x Volume fraction of ionomer in catalyst layer

Subscripts and Superscripts

a Anodeact ActivationBP Bipolar platec Cathodecell Cell characteristicCL Catalyst layereff Effecitiveele Electronicg,s Gas and solid phasesGDL Gas diffusion layerH2 HydrogenH2O Water Vapouri, j The i-th and j-th componentsin Inletm Mass (for source term)mem MembraneO2 Oxygenopt Operating condition

out Outletpro protonicref Reference staterev Reversiblesat Saturationsld Solid phase excluding the membrane electrolyteT Energy (for source term)u Momentum (for source term)

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[2] J. Zhang, Z. Xie, J. Zhang, Y. Tang, C. Song, T. Navessin,Z. Shi, D. Song, H. Wang, D. P. Wilkinson, Z. Liu,S. Holdcroft, J. Power Sources 2006, 160, 872.

[3] S. M. Aharoni, M. H. Litt, J. Polym. Sci. Part A: Polym.Chem. 1974, 12, 639.

[4] S. Wasmus, A. Valeriu, G. D. Mateescu, D. A. Tryk, R. F.Savinell, Solid State Ionics 1995, 80, 87.

[5] R. F. Savinell, M. H. Litt, US Patent, US 5,525,436, 1996.[6] O. Savadogo, J. Power Sources 2004, 127, 135.[7] J. S. Wainright, J. T. Wang, R. F. Savinell, M. H. Litt,

J. Electrochem. Soc. 1995, 142, L121.[8] R. Bouchet, E. Siebert, Solid State Ionics 1999, 118, 287.[9] Q. Li, H. A. Hjuler, N. J. Bjerrum, J. Appl. Electrochem.

2001, 31, 773.[10] R. He, Q. Li, G. Xiao, N. J. Bjerrum, J. Membr. Sci. 2003,

226, 169.[11] S. R. Samms, S. Wasmus, R. F. Savinell, J. Electrochem.

Soc. 1996, 143, 1225.[12] D. Weng, J. S. Wainright, U. Landau, R. F. Savinell,

J. Electrochem. Soc. 1996, 143, 1260.[13] Q. Li, R. He, J. O. Jensen, N. J. Bjerrum, Fuel Cells 2004,

4, 147.[14] T. J. Schmidt, J. Baurmeister, J. Power Sources 2008, 176,

428.[15] E. U. Ubong, Z. Shi, X. Wang, ECS Trans. 2008, 16, 79.[16] D. Cheddie, N. Munroe, Energy Convers. Manage. 2006,

47, 1490.[17] D. Cheddie, N. Munroe, J. Power Sources 2006, 156, 414.[18] D. Cheddie, N. Munroe, Int. J. Transp. Phenom. 2006, 8,

51.[19] D. Cheddie, N. Munroe, J. Power Sources 2006, 160, 215.[20] J. Peng, S. J. Lee, J. Power Sources 2006, 162, 1182.[21] J. Peng, J. Y. Shin, T. W. Song, J. Power Sources 2007, 179,

220.[22] D. R. Lide (Ed.), Handbook of Chemistry and Physics, 88th

Ed., CRC Press, 2007.[23] R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport Phe-

nomena, John Wiley & Sons, New York, USA, 1960.[24] Q. Ye, T. V. Nguyen, J. Electrochem. Soc. 2007, 154,

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Documents

A Three-Dimensional Non-isothermal Model of High Temperature Proton Exchange Membrane Fuel Cells with Phosphoric Acid Doped Polybenzimidazole Membranes