ww.sciencedirect.com

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 4

Available online at w

journal homepage: www.elsevier .com/locate/he

Numerical modeling and investigation of gascrossover effects in high temperature protonexchange membrane (PEM) fuel cells

Purushothama Chippar, Hyunchul Ju*

School of Mechanical Engineering, Inha University, 253 Yonghyun-Dong, Nam-Gu, Incheon 402-751,

Republic of Korea

a r t i c l e i n f o

Article history:

Received 14 April 2012

Received in revised form

24 July 2012

Accepted 29 July 2012

Available online 16 August 2012

Keywords:

High temperature proton exchange

membrane fuel cell (HT-PEMFC)

Polybenzimidazole (PBI)

Numerical modeling

Hydrogen crossover

Oxygen crossover

* Corresponding author. Tel.: þ82 32 860 731E-mail address: hcju@inha.ac.kr (H. Ju).

0360-3199/$ e see front matter Copyright ªhttp://dx.doi.org/10.1016/j.ijhydene.2012.07.1

a b s t r a c t

A gas crossover model is developed for a high temperature proton exchange membrane

fuel cell (HT-PEMFC) with a phosphoric acid-doped polybenzimidazole membrane. The

model considers dissolution of reactants into electrolyte phase in the catalyst layers and

subsequent crossover of reactant gases through the membrane. Furthermore, the model

accounts for a mixed potential on the cathode side resulting from hydrogen crossover and

hydrogen/oxygen catalytic combustion on the anode side due to oxygen crossover, which

were overlooked in the HT-PEMFC modeling works in the literature. Numerical simulations

are carried out to investigate the effects of gas crossover on HT-PEMFC performance by

varying three critical parameters, i.e. operating current density, operating temperature and

gas crossover diffusivity to approximate the membrane degradation. The numerical results

indicate that the effect of gas crossover on HT-PEMFC performance is insignificant in

a fresh membrane. However, as the membrane is degraded and hence gas crossover

diffusivities are raised, the model predicts non-uniform reactant and current density

distributions as well as lower cell performance. In addition, the thermal analysis demon-

strates that the amount of heat generated due to hydrogen/oxygen catalytic combustion is

not appreciable compared to total waste heat released during HT-PEMFC operations.

Copyright ª 2012, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights

reserved.

1. Introduction face another barrier, namely, the low tolerance of the anode

The application of perfluorosulfonic acid (PFSA) membranes,

such as DuPont’s Nafion� membranes typically used in proton

exchange membrane fuel cells (PEMFCs), is limited to low

temperatures (<90 �C) due to the stringent requirement for

membrane hydration to ensure good proton conductivity.

Therefore, PFSA membrane-based fuel cells suffers from

several issues raised by low temperature operation, such as

complicated water management requirements, high external

humidification, and cooling loads. Furthermore, PFSA

membrane fuel cells, particularly for residential applications,

2; fax: þ82 32 868 1716.

2012, Hydrogen Energy P23

platinum (Pt) catalyst to carbon monoxide (CO) which is

inevitably present in reformate fuel.

Recently, the operation of PEMFCs at elevated temperatures

(100 �Ce200 �C) has receivedmuchattention because of several

benefits, such as faster electrode kinetics, improved mass

transport, simple water management, and higher tolerance to

CO. Therefore, a high-temperature proton exchange fuel cell

(HT-PEMFC) is well suited for most distributed energy or

combined heat and power (CHP) applications in which

a hydrogen rich reformate gas is often used instead of pure

hydrogen. Themain focusofHT-PEMFC research resideson the

ublications, LLC. Published by Elsevier Ltd. All rights reserved.

Table 1 e HT-PEMFC model: governing equations.

Governing equations

Mass V$ðr u!Þ ¼ Sm (1)

Momentum Flow channels ðNavier� Stokes equationsÞ :�1= 3

2�V$ðru!u!Þ ¼ �Vpþ V$s (2)

Porous media ðDarcys equationsÞ :ru!¼ �ðK=nÞVp (3)

Species V$ð u!CiÞ ¼ V$�Deff

i VCi

�þ Si (4)

Charge Proton transport : V$�keffVFe

�þ SF ¼ 0 (5)

Electron transport : V$�seffVFs

�� SF ¼ 0 (6)

Energy V$�rCp u

!T� ¼ V$

�keffVT

�þ ST (7)

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 4 7705

developmentof alternativemembraneswhich canexhibit high

proton conductivity under low humidity conditions at the

elevated temperatures. One of the most promising candidates

is believed to be a phosphoric acid-doped polybenzimidazole

(PBI) membrane. Since Wainright et al. [1] first proposed the

use of phosphoric acid-doped PBImembranes for a HT-PEMFC,

considerable progress has been made in PBI membrane

development. Studies have reported good proton conductivity

[1], excellent thermal stability [2], and nearly zero electro-

osmotic drag [3]. In addition to membrane development,

several experimental efforts have been undertaken to investi-

gate the physiochemical properties of PBI membranes [4e8].

Also, several theoretical HT-PEMFC models have been

developed and introduced in the literature for understanding,

prediction, and optimization of key physical phenomena in

HT-PEMFCs [9e15]. Among these models, Cheddie and Mun-

roe [10] and Sousa et al. [15] account for the effects of gas

solubility into the phosphoric acid/PBI electrolyte. Cheddie

and Munroe [10] presented a two-dimensional (2D),

isothermal HT-PEMFC model wherein hydrogen and oxygen

dissolution into phosphoric acid of catalyst layers (CLs) were

taken into account. However, their model assumed CLs to

comprise only liquid phase electrolyte (phosphoric acid) and

solid-phase electron conducting regions, neglecting gas-phase

reactant transport through CLs. As a result, their numerical

predictions significantly overestimated mass transport loss in

the CLs. Sousa et al. [15] treated CLs as spherical agglomerate

porous structures and applied the CLmodel to a 2D isothermal

HT-PEMFC model. Their numerical predictions indicate that

an optimumphosphoric acid volume fraction in a CL is around

30%e55%. Most recently, Chippar and Ju [16] developed

a three-dimensional non-isothermal HT-PEMFC model and

investigated the impact of a coolant flow rate on multi-

dimensional distributions of species, temperature, and

current density as well as overall cell performance. However,

hydrogen and oxygen crossover through PBI membrane was

not considered in their model.

In this study, gas crossover phenomena in HT-PEMFCs are

newly modeled and implemented into the previous HT-PEMFC

model [16]. Previously, Nam et al. [17] developed the gas cross-

over model for low temperature- (LT-) PEMFCs and numerically

studied the influences of hydrogen and oxygen crossover on

two-phase transportandwateraccumulation insidecellsaswell

as overall cell performance. They also simulated decaying

polarization curves due to membrane degradation using gas

crossover evolution datameasured during long-term LT-PEMFC

operations. We adopted the gas crossover model of Nam et al.

[17] and modified it for HT-PEMFCs. The gas crossover model

rigorously accounts for hydrogen/oxygen dissolution into the

aqueous electrolyte phase and subsequent diffusion through

phosphoric acid-doped PBI membranes in HT-PEMFCs. Note

that, although Cheddie and Munroe [10] and Sousa et al. [15]

modeled the dissolution of the reactant gas into aqueous phos-

phoric acid, the effects of gas crossover through themembrane

on the thermal-electrochemical behavior of cells andoverall cell

performancewere not taken into consideration in theirmodels.

Due to hydrogen crossover from the anode side, a mixed

potential occurs on the cathode side,whereas oxygen crossover

results in hydrogen/oxygen catalytic combustion on the anode

side, which possibly redistributes species and charge profiles

inside HT-PEMFCs and consequently downgrades overall cell

performance. The gas crossover model presented in this paper

entails a detailed account of these gas crossover impacts.

2. Numerical model

The proposed three-dimensional, two-phase, non-isothermal,

electrochemical-transport coupled HT-PEMFC model is based

on our previous HT-PEMFC model [16], and it is further

improved by accounting for the effects of gas dissolution and

subsequent crossover through the PBI membrane. The model

considers all sub-components of an HT-PEMFC: membrane,

catalyst layers (CLs), gas diffusion layer (GDLs), gas channels

(GCs), and bipolar plates (BPs). The governing equations of the

HT-PEMFCmodel, relevant source terms, and electrochemical

properties at the anode and cathode CLs are summarized in

Tables 1e3, respectively. Readers are referred to our previous

publication [16] for a more detailed description of the model.

2.1. Model assumptions

The specific assumptions invoked in the present model are:

(1) Incompressible and laminar flow due to small pressure

gradient and flow velocities.

(2) Ideal gas mixture due to low pressure and high tempera-

ture HT-PEMFC operation.

(3) Isotropic and homogeneous porous layers (GDLs, CLs)

characterized by effective porosity and permeability.

2.2. Transport properties

The diffusivity of species i, in the gasmixture is defined as [18]

Di;M ¼ 1�xiPj¼n

jjsi

xj

Di;j

;where Di;j ¼1:013�10�7T1:75

p�c1=3i þc

1=3j

�2�

1Mi

þ 1Mj

�1=2

cH2¼7:07; cH2O ¼ 12:7; cN2

¼17:9; cO2¼ 16:6:

(22)

Table 2 e HT-PEMFC model: source/sink terms.

Sourceterms

Anode CL Cathode CL

H2

SH2¼ � ja

2F� nxover

H2

dCL� 2

nxoverO2

dCL(8)

O2

SO2¼ jc

4F� nxover

O2

dCL� 12

nxoverH2

dCL(9)

H2O

SH2O ¼ 2nxoverO2

dCL(10a) SH2O ¼ � jc

2Fþ nxover

H2

dCL(10b)

Mass

Sm ¼Xk

Sk ¼ �MH2

ja2F

þ�MO2

nxoverO2

�MH2nxoverH2

�dCL

(11a) Sm ¼Xk

Sk ¼ MO2

jc4F

�MH2Ojc2F

þ�MH2

nxoverH2

�MO2nxoverO2

�dCL

(11b)

Charge

SF ¼ ja (12a) SF ¼ jc (12b)

Heat

ST ¼ jahþ I2

keffþ DHH2

2nxoverO2

dCL(13a) ST ¼ jchþ I2

keffþ jc

dUo

dTT� IxoverH2

dCL

�hþ dUo

dTT

�(13b)

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 47706

Note that, in addition to molecular diffusion as defined in

the Eq. (22), species diffusion transport can also be controlled

by the Knudsen diffusion effect due to molecular-to-wall

collision. According to the kinetic theory, the Knudsen diffu-

sivity can be expressed as

Di;K ¼ dp

3

ffiffiffiffiffiffiffiffiffi8RTpMi

s: (23)

Table 3 e HT-PEMFC model: electrochemical properties.

Description Anode CL

Exchange current density � ratio of

the reaction surface to the

CL volume, airef0 (A/m3)

1.0 � 109

Reference H2/O2 molar

concentration, (mol/m3)

40.88

Transfer coefficients, a aa ¼ 0:5

Thermodynamic equilibrium

potential, U0 (V)

0

Surface overpotential,h (V)

fs � fe � U0 ðwith fs ¼ 0ÞTransfer current density, j (A/m3)

ja ¼ airef0;a

CH2

CH2;ref

!1=2�aa þ ac

RuTF

Electrochemical reactions :Xk

siMzi ¼ ne�;where

8<:

Mi ¼ chemicalsi ¼ stoichiomn ¼ number of

Hydrogen oxidation reactionðHORÞat the anode side : H2 � 2Hþ ¼

Oxygen reduction reactionðORRÞat the cathode side : 2H2O�O2 �

Therefore, the effective diffusivity of species in porous

media is obtained by combining both the molecular and

Knudsen diffusion effects with the effects of porosity and

tortuosity of the porous medium using the Bruggemann

correlation [19]:

Di ¼ 3n

�1

Di;Mþ 1Di;K

��1

: (24)

Cathode CL

1.0 � 104

40.88

ac ¼ 0:5

1:1669� 0:24� 10�3ðT� 373:15Þ (14)

(15) fs � fe � U0 ðwith fs ¼ VcellÞ (16)

h

�(17) jc ¼ �airef0;c

CO2

CO2;ref

!3=4

exp

�� ac

RuTFh

�þ IxoverH2

dCL(18)

formula of species ietry coefficientelectrons transferred

(19)

2e� (20)

4Hþ ¼ 4e� (21)

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 4 7707

The dissolution of species in the aqueous electrolyte phase

and their subsequent diffusion are described below. Accord-

ing to Cheddie and Munroe [10], the solubility and diffusivity

of oxygen in a concentrated phosphoric acid can be expressed

in terms of theweight percentage of phosphoric acid,mPA, and

temperature as

HPAO2

¼ 0:1exp

266664

�257:13ðmPAÞ2�431:08ðmPAÞ þ 178:45

�

þ�� 93500ðmPAÞ2þ156646ðmPAÞ � 64288

�T

377775; (25)

DPAO2

¼ 10�9exp

266664

�� 192:55ðmPAÞ2þ323:55ðmPAÞ � 125:61

�

þ�62010ðmPAÞ2�105503ðmPAÞ þ 40929

�T

377775: (26)

Henry’s constant of oxygen in the phosphoric acid-doped

PBI membrane can be obtained as a function of the volume

fraction of phosphoric acid in the membrane, 3PA as [10]

HPBIO2

¼ � 3PA�1:945h

HPAO2

þ 5:79�1� � 3

PA�1:8�i

: (27)

The 3PA in turn depends on the doping level of phosphoric

acid in the membrane, X as [10]

3PA ¼

4:81X� 2

þ 1

�1

: (28)

The X in the above equation can be computed based on the

phosphoric acid concentration, M as [10]

X ¼ 0:012M3 � 0:2111M2 þ 1:2363Mþ 0:7119: (29)

On the other hand, the oxygen diffusion coefficient in the

phosphoric acid-doped PBI membrane can be obtained by

Bruggemann’s relation as [10]

DPBIO2

¼ � 3PA�1:8

DPAO2: (30)

The dissolved concentration of oxygen at the gas/electrolyte

interface in the cathode CL is determined from the partial

pressure of oxygen pO2using Henry’s law as follows:

CPBIO2

¼ HPBIO2

pO2¼ HPBI

O2

�CgO2 ;memRT

�: (31)

Due to lack of studies regarding hydrogen dissolution

into concentrated phosphoric acid, the diffusivity and solu-

bility of hydrogen are assumed to be two times and four

times larger than those of oxygen, respectively, i.e. based on

the transport behavior of hydrogen and oxygen in water

systems [10]:

DPAH2

¼ 2DPAO2; (32)

HPAH2

¼ 4HPAO2: (33)

The proton conductivity of the phosphoric acid-doped PBI

membrane is correlated to the doping level and temperature

as follows [10]:

k ¼ 100T

exp

8:0219�

�2605:6� 70:1X

T

�: (34)

The effective proton conductivity in the CLs is obtained by

combining the effects of the volume fraction of themembrane

phase and tortuosity of the porous medium by using Brugge-

mann’s correlation:

keff ¼ 31:5mck: (35)

2.3. Gas crossover model and relevant source terms

The gas crossover model accounts for the influences of

hydrogen crossover (from the anode to cathode) and oxygen

crossover (from the cathode to anode) on electrochemical

processes and the resultant overall cell performance. The gas

crossover model has been described in detail in a previous

study [17] and hence only a brief summary is repeated here.

The hydrogen crossover through the membrane causes

a mixed potential at the cathode CL due to facile hydrogen

oxidation kinetic and large surface overpotential at the

cathode. Therefore, the final form of the ORR kinetic expres-

sion can be determined as Eq. (18) in Table 3 where the second

term in the right-hand side of Eq. (18) represents the effect of

the hydrogen crossover. Under the assumption that crossed

hydrogen is uniformly and completely oxidized in the cathode

CL, the hydrogen crossover current density, can be defined as

below:

IxoverH2¼ 2Fnxover

H2¼ 2FDPBI

H2

CPBIH2

���aCL

dmem: (36)

On the other hand, the oxygen crossover through the

membrane leads to catalytic hydrogen/oxygen combustion in

the anode CL due to small potential difference between the

solid and electrolyte phases at the anode. The influences of

the hydrogen and oxygen crossover on mass, species

(hydrogen, oxygen, andwater), and energy balance are seen in

their source/sink terms in Table 2.

2.4. Numerical implementation, computational domain,and boundary conditions

The HT-PEMFC model described in Section 2 is numerically

implemented in a commercial computational fluid dynamics

(CFD) program, FLUENT, basing on its user defined functions

(UDF). The convergence criteria for all species and energy

calculation residuals are set to 10�8. Fig. 1 shows the mesh

configuration of the simple single-straight channel geometry.

The physical properties and, cell dimensions and operating

conditions are given in Tables 4 and 5, respectively. The

isothermal boundary condition is applied to the anode and

cathode wall of the computational cell for temperature

calculations. In addition, the no-flux condition is applied to

the outer faces for flow and species transport equations

except for the channel inlets and outlets. The inlet velocities

in the anode and cathode GCs can be expressed as functions of

Fig. 1 e Mesh configuration of three-dimensional, single-

channel HT-PEMFC geometry.

Table 5 e Cell dimensions and operating conditions.

Description Value

Cell length 0.1 m

Anode/cathode channel/rib width 1 � 10�3 m

Anode/cathode channel height 0.7 � 10�3 m

Thickness of the anode/cathode GDLs 250 � 10�6 m

Thickness of the anode/cathode CLs 10 � 10�6 m

Thickness of the membrane 70 � 10�6 m

Anode/cathode inlet pressure 1.0 atm

Anode stoichiometry 2.0 (Pure H2)

Cathode stoichiometry 2.0 (Air)

Anode/cathode inlet temperature 373 K, 453 K

RH of the anode/cathode inlet 0.0%

Phosphoric acid doping level 6.2

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 47708

the anode and cathode stoichiometric ratios, the operating

current density, the cross-sectional areas of the anode and

cathode GCs, and the concentrations of hydrogen and oxygen,

which are functions of the anode/cathode inlet pressure and

temperature:

uin;a ¼ xaðI=2FÞAmem

CH2Aa;chan

and uin;c ¼ xcðI=4FÞAmem

CO2Ac;chan

: (37)

3. Results and discussion

To examine the effects of gas crossover on HT-PEMFCs, we

assessed the effects of three critical parameters, namely,

operating current density, operating temperature, and gas

crossover diffusivity itself. Regarding the effect of operating

current density, it is evident that the impact of gas crossover is

Table 4 e Physical properties.

Description Value

Porosity of GDL, CL 0.6, 0.3

Volume fraction

of ionomers in CL

0.4

Permeability of GDL, CL 1 � 10�12, 1.0 � 10�13 m2

Electronic conductivity

in the GDL, CL, BP

1250, 300, 14000 S m�1

Specific heat capacities

of GDL, CL, membrane, BP

568, 3300, 1650, 2930 J kg�1 K�1

Specific heat capacities

of species e H2, O2, N2, H2O

14430, 929, 1042, 1968 J kg�1 K�1

Thermal conductivities

of GDL, CL, membrane, BP

1.2, 1.5, 0.95, 20 W m�1 K�1

Thermal conductivities

of species e H2, O2, N2, H2O

0.2040, 0.0296, 0.0293,

0.02378 W m�1 K�1

more significant at lower current density operation due to the

lower reactant consumption rate and resultant higher

concentration in the CL that leads to a higher dissolution rate

of the reactant gas into the aqueous electrolyte phase. The

hydrogen and oxygen crossover diffusivities given by Cheddie

and Munroe [10] imply that the degree of gas crossover

through a phosphoric acid-doped PBI membrane is consider-

ably altered by operating temperature. Therefore, the oper-

ating temperature is another critical factor to control the gas

crossover behavior inside an HT-PEMFC. Finally, several

degradation mechanisms of the PBI membrane have been

reported in the literature, such as chemical degradation [20],

thermal degradation [21] and phosphoric acid evaporation

[22]. In particular, it should be noted that the phosphoric acid

loss due to evaporation not only decreases the membrane

conductivity but also increases the gas crossover diffusivities

for hydrogen and oxygen [10]. Although all of thesemembrane

degradationmechanisms appear to be highly localized, due to

the lack of experimental data on degradations, we assume the

membrane to be degraded uniformly; thus, the gas crossover

diffusivity is also uniformly raised as a function of the degree

of membrane degradation.

The parametric study was carried out at two operating

temperatures (100 �C and 180 �C) under various operating

current densities and hydrogen/oxygen crossover diffusiv-

ities. Four different cases of gas crossover diffusivity are

defined here. Case 1 assumes that the membrane is perfectly

impermeable to hydrogen and oxygen; hence, their crossover

diffusivities are set to zero. Case 2 represents the case of

a fresh phosphoric acid-doped PBI membrane, and the cross-

over processes for hydrogen and oxygen are approximated

using the crossover diffusivities given by Cheddie andMunroe

[10]. To consider a degradedmembrane, the hydrogen/oxygen

crossover diffusivities were further raised by one and two

orders of magnitude for cases 3 and 4, respectively.

Fig. 2 shows the hydrogen and oxygen concentration

profiles in the anode and cathode CLs for cases 1 to 4 at the

operating current density of 0.2 A cm�2 based on the operating

temperatures of 100 �C and 180 �C. As shown in Fig. 2(a) and

(b), the hydrogen concentration under the land region is lower

than that under the channel region due to the longer transport

path from the anode flow channel. In a comparison of cases 1

to 4 for each operation temperature, the hydrogen distribu-

tions for cases 1, 2, and 3 are almost identical, indicating that

Fig. 2 e Hydrogen concentration contours (in mol mL3) in the anode CL and oxygen concentration contours (in mol mL3) in

the cathode CLs at an operating current density of 0.2 A cmL2: (a) Hydrogen; T[ 100 �C, (b) Hydrogen; T[ 180 �C, (c) Oxygen;

T [ 100 �C and (d) Oxygen; T [ 180 �C.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 4 7709

the effect of crossover though the fresh membrane is negli-

gible (case 2), and even a ten-fold increase in the hydrogen-

permeation coefficient due to membrane degradation is still

acceptable for HT-PEMFC operations (case 3). However, more

severe hydrogen depletion is observed in case 4, which can be

attributed to a combined result of the higher degree of

hydrogen crossover from the anode through the membrane

and the higher rate of catalytic hydrogen/oxygen combustion

at the anode CL driven by stronger oxygen crossover from the

cathode in case 4. In addition, higher hydrogen depletion was

predicted at the higher operating temperature because the

amount of hydrogen crossover increases with temperature.

The same trend is observed in the oxygen concentration

contours in the cathode CL in Fig. 2(c) and (d) where oxygen

depletion near the cathode outlet region is more severe with

a higher degree of membrane degradation (case 4) and/or

a higher operating temperature (180 �C). Note that the severe

oxygen depletion in case 4 is attributed to both its higher

oxygen crossover rate from the cathode to anode aswell as the

higher hydrogen crossover from the anode to cathode that

leads to additional ORR andmixed potential at the cathode CL.

Fig. 3(a) shows the local current density distributions in the

membrane for cases 1 to 4 at the operating current density of

0.2 A cm�2 and the operating temperature of 100 �C. In all

cases, the local current density near the land region is lower

than near the channel region along the in-plane direction (Z ).

Along the cathode flow direction (Y ), the local current density

continuously decreases toward the cathode outlet. These

trends indicate that oxygen depletion is the sole factor in

determining the current density distribution for all the cases.

In a comparison of cases 1 to 4, the current density distribu-

tions for cases 1, 2, and 3 are almost identical, indicating that

the degree of gas crossover up to a ten-fold increase in the gas-

permeation coefficient due to membrane degradation has

a negligible influence on HT-PEMFC performance. However,

spatial non-uniformity in the current density profile is clearly

increased in case 4 due to the higher degree of gas crossover

through the membrane. Fig. 3(b) displays the local current

density contours at 180 �C. As compared with Fig. 3(a), the

local current densities near the cathode outlet are reduced

due to the higher degree of hydrogen and oxygen crossover at

the elevated temperature.

Fig. 4 e Crossover current density distribution (in A mL2) in the membrane at an operating current density of 0.2 A cmL2:

(a) T [ 100 �C and (b) T [ 180 �C.

Fig. 3 e Local current density distribution (in A mL2) in the membrane at an operating current density of 0.2 A cmL2:

(a) T [ 100 �C and (b) T [ 180 �C.

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 47710

Fig. 5 e Cathode overpotential distribution (in volts) in the CL at an operating current density of 0.2 A cmL2: (a) T [ 100 �Cand (b) T [ 180 �C.

Fig. 6 e Overall polarization curves for cases 1e4 at the

operating temperatures of 100 �C and 180 �C.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 4 7711

To analyze the degree of hydrogen crossover, the hydrogen

crossover current density was calculated by Eq. (36) and

plotted in Fig. 4 for the same simulation cases (at 0.2 A cm�2

under two operating temperatures of 100 �C and 180 �C). For allcases, the hydrogen crossover current density decreases

toward the anode downstream because the hydrogen

concentration in the anode CL is high near the anode inlet and

continuously depleted along the flow direction by not only

HOR but also hydrogen permeation through the membrane

and the hydrogen/oxygen catalytic combustion due to the

oxygen crossover. In a comparison of cases 2 to 4, the

hydrogen crossover current densities of case 4 are roughly two

orders of magnitude larger than those of case 2, which clearly

indicates that the amount of hydrogen crossover flux is

directly proportional to the considered hydrogen crossover

diffusivity. In addition, much higher hydrogen crossover

current density was predicted with the higher operating

temperature (180 �C) since the hydrogen crossover diffusivity

is a function of increasing temperature.

Fig. 5 shows the cathode overpotential distributions in the

cathode CL for cases 1 to 4 at the operating temperatures of

100 �C and 180 �C. The cathode overpotential increases toward

the land region and cathode downstream due to lower local

oxygen concentrations there as seen in Fig. 2(c) and (d). More

importantly, a comparison of the 100 �C and 180 �C cases

indicates that operating the cell at the higher temperature

significantly reduces the cathode overpotential, although the

available oxygen concentration for ORR in the cathode CL is

lower at 180 �C (see Fig. 2(d)). This is mainly due to the

enhanced electrochemical kinetics of ORR at the elevated

operating temperature.

Fig. 6 shows the effects of hydrogen and oxygen crossover

through the membrane on cell polarization curves at two

different operating temperatures (100 �C and 180 �C). First,

superior cell performance is achieved at the higher operating

temperature due to improved ORR kinetics, better proton

conductivity, and more efficient mass transport with

increasing temperature. Further, the polarization curves

clearly demonstrate that the impact of gas crossover is more

significant at lower current densities, because the hydrogen

and oxygen concentrations remaining in the CLs are higher

under lower current density operations. In addition, the

polarization curves for cases 1 to 3 are similar to each other at

both temperatures (100 �C and 180 �C), which means that

a ten-fold increase in the gas-permeation coefficient (case 3) is

acceptable for HT-PEMFC operations under wide ranges of

operating current density and temperature. However, a more

pronounced effect of gas crossover is seen in case 4, particu-

larly at the higher operating temperature (180 �C). These

results imply that careful attention to suppress gas crossover

is required for low current density and/or high temperature

operations.

Tables 6 and 7 summarize the overall heat balance and the

individual heat sources for cases 1 to 4 under the operating

current density of 0.2 A cm�2 at 100 �C and 180 �C, respectively.The simulation results show that the largest part of the total

waste heat release is due to irreversible ORR reaction at the

cathode, that is, roughly 70% of the total heat generation. In

Table 6 e Summary of energy balance results under operating current density of 0.2 A cmL2 at 100 �C.

Case 1 Case 2 Case 3 Case 4

(1) Anode CL Irreversible reaction heat, [W] ST;irrev;a ¼ RV

j:hdV 0.00344 (0.985%) 0.00345 (0.985%) 0.00345 (0.977%) 0.00348 (0.903%)

Ohmic joule heating, [W] ST;joule;a ¼ RV

I2

keffdV

0.00399 (1.141%) 0.00399 (1.139%) 0.00399 (1.129%) 0.00400 (1.038%)

H2/O2 catalytic combustion heat [W] SxoverT;a ¼ DH SxoverO20.0 0.0000376 (0.011%) 0.000374 (0.106%) 0.00352 (0.911%)

(2) MembraneOhmic joule heating, [W] ST;joule;mem ¼ R

V

I2

keffdV

0.02678 (7.659%) 0.02675 (7.643%) 0.02675 (7.572%) 0.02675 (6.935%)

(3) Cathode CL Irreversible reaction heat, [W] ST;irrev;c ¼RV

j:hdV 0.26724 (76.43%) 0.26740 (76.41%) 0.26883 (76.10%) 0.28304 (73.37%)

Ohmic joule heating, [W] ST;joule;c ¼RV

I2

keffdV

0.01232 (3.524%) 0.01231 (3.516%) 0.01232 (3.487%) 0.01245 (3.227%)

Mixed potential and entropic heat due to

hydrogen crossover, [W] SxoverT;c ¼ �IxoverH2

dCL

�hþ T

dU0

dT

� 0.0 0.000152 (0.043%) 0.00152 (0.428%) 0.01489 (3.859%)

Entropy heat, [W] ST;rev;c ¼RVj$

�TvU0

vT

�dV

0.03588 (10.26%) 0.03589 (10.26%) 0.03605 (10.21%) 0.03763 (9.755%)

Sum (1)þ(2)þ(3), [W] 0.3497 0.3499 0.3533 0.3858

Table 7 e Summary of energy balance results under operating current density of 0.2 A.cmL2 at 180 �C.

Case 1 Case 2 Case 3 Case 4

(1) Anode CL Irreversible reaction heat, [W] ST;irrev;a ¼ RV

j:hdV 0.00235 (0.846%) 0.00236 (0.844%) 0.00238 (0.801%) 0.00253 (0.568%)

Ohmic joule heating, [W] ST;joule;a ¼ RV

I2

keffdV

0.00202 (0.727%) 0.00202 (0.722%) 0.00202 (0.683%) 0.00208 (0.465%)

H2/O2 catalytic combustion heat [W] SxoverT;a ¼ DH SxoverO20.0 0.00023 (0.084%) 0.00226 (0.763%) 0.01562 (3.493%)

(2) MembraneOhmic joule heating, [W] ST;joule;mem ¼ R

V

I2

keffdV

0.01165 (4.193%) 0.01164 (4.160%) 0.01165 (3.929%) 0.01178 (2.639%)

(3) Cathode CL Irreversible reaction heat, [W] ST;irrev;c ¼RV

j:hdV 0.21214 (76.33%) 0.21292 (76.10%) 0.21996 (74.18%) 0.28598 (64.07%)

Ohmic joule heating, [W] ST;joule;c ¼RV

I2

keffdV

0.00623 (2.241%) 0.00623 (2.225%) 0.00629 (2.122%) 0.00725 (1.625%)

Mixed potential and entropic heat due

to hydrogen crossover, [W] SxoverT;c ¼ �IxoverH2

dCL

�hþ T

dU0

dT

� 0.0 0.00072 (0.258%) 0.00717 (2.420%) 0.06686 (14.98%)

Entropy heat, [W] ST;rev;c ¼RVj$

�TvU0

vT

�dV

0.04354 (15.669%) 0.04367 (15.607%) 0.04476 (15.10%) 0.05426 (12.156%)

Sum (1)þ(2)þ(3), [W] 0.2779 0.2798 0.2964 0.4463

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 47712

addition, it should be noted that the heat generated by

hydrogen/oxygen catalytic combustion on the anode side is

about 3.5% in the most severe gas crossover case in this study

(case 4 at 180 �C and 0.2 A cm�2). Therefore, the contribution of

the hydrogen/oxygen chemical reaction does not seem to be

significant to the total heat release during HT-PEMFC opera-

tions, even when two orders of magnitude greater gas cross-

over diffusivities are considered.

4. Conclusions

In this study, a gas crossover model that considers the

dissolution of hydrogen/oxygen into the electrolyte phase and

subsequent diffusion through a phosphoric acid-doped PBI

membrane was developed and incorporated into a HT-PEMFC

model developed in an earlier study [16]. The main interest of

this study is to numerically assess the impact of gas crossover

on HT-PEMFC performance. The gas crossover model rigor-

ously accounts for the major outcomes of hydrogen and

oxygen crossover, i.e. a mixed potential at the cathode CL and

the hydrogen/oxygen catalytic combustion at the anode CL.

The numerical results show that the gas crossover has

a negligible influence on overall cell performance in a fresh PBI

membrane (case 2) and a moderately degraded membrane

(case 3), which was assumed to have one order of magnitude

higher crossover diffusivities than those of the fresh

membrane. However, the effect of gas crossover begins to

appear in more a severely degraded membrane with two

orders of magnitude larger gas crossover diffusivities (case 4).

A comparison of case 4 with cases 1 to 3 clearly shows that the

increased effect of gas crossover increases the degree of non-

uniformity in the hydrogen, oxygen, and current density

distributions. In addition, the simulation results for case 4

indicate that gas crossover is more detrimental to cell opera-

tion at a higher operating temperature and/or lower current

density due to more facile crossover of hydrogen and oxygen

with elevated temperature and due to higher reactant

concentration in the CL with lower current density. Finally,

the thermal analysis carried out in this study demonstrated

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 4 7713

that the heat generated via hydrogen/oxygen catalytic

combustion at the anode CL is not significant, occupying only

3.5% of the total waste heat release in the worst gas crossover

case in this study (case 4 at 180 �C and 0.2 A cm�2). This paper

contributes to enhancing the fundamental understanding of

the gas crossover phenomena occurring during HT-PEMFC

operation. As an extension of this work, our efforts are

underway to numerically study the effects of local heteroge-

neous gas crossover due to membrane pinhole formations.

Acknowledgment

This work was supported by the New & Renewable Energy

R&D program (grant no. 2010T100200501) of the Ministry of

Knowledge Economy of the government of the Republic of

Korea. The authors gratefully acknowledge this support.

Nomenclature

A area, m2

c specific heat, J kg�1 K�1

C molar concentration, mol m�3

Di mass diffusivity of species i, m2 s�1

F Faraday constant, 96487 C mol�1

H Henry’s constant, mol m�3 atm�1

i0 exchange current density, A m�2

I operating current density, A m�2

j transfer current density, A m�3

K hydraulic permeability, m2

M molecular weight, kg mol�1

p partial pressure, Pa

Q heat, watt

R universal gas constant, 8.314 J mol�1 K�1

S source term in the conservation equation

T temperature, K

u! fluid velocity and superficial velocity in a porous

medium, m s�1

U0 thermodynamic equilibrium potential, V

Vcell cell potential, V

X doping level

Greek symbols

a transfer coefficient

3 porosity

3mc volume fraction of the ionomer phase in the CL

f phase potential, V

h overpotential, V

m viscosity, kg m�1 s�1

r density, kg m�3

s viscous shear stress, N m�2

k ionic conductivity, S m�1

x stoichiometry flow ratio

Superscripts

e electrolyte

eff effective value in the porous region

g gas

ref reference value

Subscripts

a anode

c cathode

CL catalyst layer

GC gas channel

GDL gas diffusion layer

H2 hydrogen

i species index

in channel inlet

m mass equation

mem membrane

O2 oxygen

u momentum equation

w water

F potential equation

0 standard condition, viz., 298.15 K and 101.3 kPa

(1 atm)

r e f e r e n c e s

[1] Wainright JS, Wang JT, Weng D, Savinell RF, Litt MH. Acid-doped polybenzimidazoles: a new polymer electrolyte. JElectrochem Soc 1995;142(7):L121e3.

[2] Samms SR, Wasmus S, Savinell RF. Thermal stability ofproton conducting acid doped polybenzimidazole insimulated fuel cell environments. J Electrochem Soc 1996;143(4):1225e32.

[3] Weng D, Wainright JS, Landau U, Savinell RF. Electro-osmoticdrag coefficient of water and methanol in polymerelectrolytes at elevated temperatures. J Electrochem Soc1996;143(4):1260e3.

[4] Qingfeng Li, Hjuler HA, Bjerrum NJ. Oxygen reduction oncarbon supported platinum catalysts in high temperaturepolymer electrolytes. Electrochim Acta 2000;45(25e26):4219e26.

[5] Qingfeng Li, Hjuler HA, Bjerrum NJ. Phosphoric acid dopedpolybenzimidazole membranes: physiologicalcharacterization and fuel cell application. J Appl Electrochem2001;31(7):773e9.

[6] Ma YL, Wainright JS, Litt MH, Savinell RF. Conductivity of PBImembranes for high-temperature polymer electrolyte fuelcells. J Electrochem Soc 2004;151(1):A8e16.

[7] He RH, Qingfeng Li, Bach A, Jensen JO, Bjerrum NJ.Physiochemical properties of phosphoric acid dopedpolybenzimidazole membranes for fuel cells. J Membr Sci2006;277:38e45.

[8] Liu Z, Wainright JS, Litt MH, Savinell RF. Study of the oxygenreduction reaction (ORR) at Pt interfaced with phosphoricacid doped polybenzimidazole at elevated temperature andlow relative humidity. Electrochim Acta 2006;51:3914e23.

[9] Cheddie D, Munroe N. Mathematical model of a PEMFC usinga PBI membrane. Energy Convers Manage 2006;47:1490e504.

[10] Cheddie D, Munroe N. A two phase model of an intermediatetemperature PEM fuel cell. Intl J Hydrogen Energy 2007;32:832e41.

[11] Cheddie D, Munroe N. Three dimensional modeling of hightemperature PEM fuel cells. J Power Sources 2006;160:215e23.

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 8 ( 2 0 1 3 ) 7 7 0 4e7 7 1 47714

[12] Scott K, Pilditch S, Mamlouk M. Modeling and experimentalvalidation of a high temperature polymer electrolyte fuelcell. J Appl Electrochem 2007;37:1245e59.

[13] Jiao K, Li X. A three-dimensional non-isothermal model ofhigh temperature proton exchange membrane fuel cells withphosphoric acid doped polybenzimidazole membranes. FuelCells 2010;10(3):351e62.

[14] Siegel C, Bandlamudi G, Heinzel A. Systematiccharacterization of a PBI/H3PO4 sol-gel membrane-modeling and simulation. J Power Sources 2011;196:2735e49.

[15] Sousa T, Mamlouk M, Scott K. An isothermal model ofa laboratory intermediate temperature fuel cell using PBIdoped phosphoric acid membranes. Chem Eng Sci 2010;65:2513e30.

[16] Chippar P, Ju H. Three-dimensional non-isothermalmodeling of a phosphoric acid-doped polybenzimidazole(PBI) membrane fuel cell. Solid State Ionics, in press.

[17] Nam J, Chippar P, Kim W, Ju H. Numerical analysis of gas-crossover effects in polymer electrolyte fuel cells (PEFCs).Appl Energy 2010;87:3699e709.

[18] Bird RB, Stewart WE, Lightfoot EN. Transport phenomena.New York: John Wiley & Sons; 1960.

[19] Meredith RE, Tobias CW. In: Tobia CW, editor. Advances inelectrochemistry and electrochemical engineering, vol. 2.New York: Interscience Publishers; 1962.

[20] Liao JH, Li QF, Rudbeck HC, Jensen JO, Chromik A, Bjerrum NJ,et al. Oxidative degradation of polybenzimidazolemembranes as electrolytes for high temperature protonexchange membrane fuel cells. Fuel Cells 2011;10(3):351e62.

[21] Zhang J, Song C, Zhang J. Accelerated lifetime testing forproton exchange membrane fuel cells using extremely hightemperature and unusually high load. J Fuel Cells Sci Tech2011;8(051006):1e5.

[22] Yu S, Xiao L, Benicewicz BC. Durability studies of PBI-basedhigh temperature PEMFCs. Fuel Cells 2008;8:165e74.