pa E), Faculdade de Engenharia da Universidade do Porto,

Ciencia

5001-801 Vila-Real Codex, Portugal

a r t i c l e i n f o

systems (>120 C) due to the advantages they present, such asincreased tolerance to carbon monoxide and simplified water

management.

Mathematical models are widely used to simulate the

behavior of fuel cells and play an important role in the

comprehension of the phenomena occurring inside the fuel

PEMFC have been developed since the early 90s by Springer

et al. [1] and Bernardi and Verbrugge [2]. Springer et al.

[1] developed an isothermal model that provided insight

into the water transport mechanisms of the fuel cell and

their effect on the respective performance. Bernardi and

* Corresponding author. Tel.: 351 225081695; fax: 351 225081449.

Avai lab le at www.sc iencedi rect .com

w.

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 6 ( 2 0 1 1 ) 9 8 4 2e9 8 5 4E-mail address: mendes@fe.up.pt (A. Mendes).The interest in hydrogen fuel cells has increased because of

their ability to produce electricity with minimal environ-

mental pollution. Typical polymer electrolyte membrane fuel

cells (PEMFC) operate at temperatures below the water boiling

point; recently, attention has been drawn to high temperature

cell and the parameters that control the performance. Most of

experimental fuel cells results are obtained in steady-state

conditions and therefore mathematical models, considering

different spatial dimensions, have been developed for these

conditions. One-dimensional (1D)models for low temperaturephenomena not included in the proposed phenomenological model.

Copyright 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rightsreserved.

1. Introduction development of fuel cells devices. They allow a betterArticle history:

Received 5 March 2011

Received in revised form

27 April 2011

Accepted 29 April 2011

Available online 12 June 2011

Keywords:

PEM

Fuel cell

Dynamic model

Impedance spectra0360-3199/$ e see front matter Copyright doi:10.1016/j.ijhydene.2011.04.218a b s t r a c t

A dynamic one-dimensional isothermal phenomenological model was developed in order

to describe the steady-state and transient behavior of high temperature polymer electro-

lyte membrane fuel cells (PEMFC). The model accounts for transient species mass transport

at the bipolar plates and gas diffusion layers and the electric double layers charge/

discharge. To record the impedance spectra, a small sinusoidal voltage perturbation was

imposed to the simulator over a wide range of frequencies, and the resultant current

density amplitude and phase were recorded.

The steady-state behavior of the fuel cell, as well as the impedance spectra were

obtained and compared to experimental data of two different fuel cells equipped with

different MEAs based on phosphoric acid polybenzimidazole membrane. This approach is

new and allows a deeper analysis of the controlling phenomena. The model fitted quite

well the IeV curves for both systems, but fairly well the Nyquist plots. The differences

observed in the Nyquist plots were attributed to proton resistance in the catalyst layer and

the gas diffusion limitations to cross the phosphoric acid layer that coats the catalyst,Rua Roberto Frias, s/n 4200-465 PortobDepartamento de Qumica, Escola deal

s da Vida e do Ambiente, Universidade de Tras-os-Montes e Alto Douro, Apartado 1013,Laboratorio de Engenharia de Processos, Ambiente e Energia (LEPA

, PortugM. Boaventura a, J.M. Sousa a,b, A. Mendes a,*A dynamic model for high temmembrane fuel cells

journa l homepage : ww2011, Hydrogen Energy Perature polymer electrolyte

e lsev ie r . com/ loca te /heublications, LLC. Published by Elsevier Ltd. All rights reserved.

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 6 ( 2 0 1 1 ) 9 8 4 2e9 8 5 4 9843Verbrugge [2] developed an isothermal model for the cathode

electrode focusing on the cell polarization characteristics,

water transport and catalyst utilization. Both models

account only for diffusive mass transport and

electrochemical kinetics. Recently, more detailed 1D models

were developed. Ramousse et al. [3] included in the fuel cell

modeling the gas diffusion in the porous electrodes, water

balance in the membrane and heat transfer in the

membrane electrodes assembly (MEA) and bipolar plates.

Although a single domain models provide important

information concerning fluxes, species concentration,

temperatures, and electrical potentials, two-dimensional

(2D) models were developed in order to take into account

spatially affected phenomena, usually along a single

channel [4e8]. Three-dimensional (3D) models, with

improved transport models and based on computational

fluid dynamics were also developed [9e17]. Recently, Ko et

al. [18] obtained the local temperature, water content in the

MEA and gas velocity in the channel of the PEMFC, at

different operation temperature and under the cathode

starvation conditions using a commercial flow solver.

In the last 5 years, several steady-state models have been

developed for high temperature fuel cells based in phosphoric

acid doped polybenzimidazole; nonetheless, models for

Nafion based PEMFC outnumber models for high tempera-

ture PEMFC. Korsgaard et al. [19] and Scott et al. [20] developed

a zero-dimensional semi-empirical model to express the

relationship between voltage and current density; Scott et al.

considered the effect of mass transport through the

electrolyte film covering the catalyst layers. Cheddie and

Munroe presented two 1D non-isothermal models focused

on polarization performance of a fuel cell [21] and geometric

factors such as porous media characteristics, membrane and

catalyst properties [22]. Scott et al. [23] followed with an

isothermal 1D model which included the potential and

current distributions in the catalyst layers. Similarly to low

temperature PEMFC, multidimensional models were

developed. Hu et al. [24,25] constructed 2D models based on

electrochemical methods (cathode exchange current density

and cell internal resistance were supplied by linear sweep

voltammetry and electrochemical impedance spectroscopy

techniques) while Shamardina et al. [26] developed an

analytical pseudo 2D isothermal model accounting for the

crossover of reactant gases through the membrane.

Furthermore, Cheddie and Munroe [27] developed a 2D

model considering two phase media. These authors

considered that oxygen and hydrogen dissolve in the

electrolyte before undergoing electrochemical reaction,

assuming, this way, aqueous phase kinetics. Moreover,

steady-state 3D models were developed for phosphoric acid

doped polybenzimidazole fuel cells [28e30].

When integrated in a complex system that includes

humidifiers, compressors and reformers, among others, a fuel

cell has a response time associated to changes in the load or in

the gas feed concentrations. The dynamic response of PEMFC

is a central issue for mobile applications such as in automo-

biles. Therefore, transient models have been developed to

study the fuel cell response to changes in current density/voltage [31e37] and operation conditions [38e41], as well as its

own start-up [39,42,43]. Andreasen and Kaer [44] developeda control-oriented model for predicting the dynamic

temperature of a high temperature PEMFC stack. Moreover,

Wang et al. [45] developed a transient model to predict the

CO tolerance of high temperature PEMFC and observed the

transient degradations of fuel cell performance.

The dynamic models referred above take into account

three of the four main transient processes in a PEM fuel cell,

namely species mass transport, membrane hydration/dehy-

dration and heat transfer. The electric double layer charge/

discharge is often neglected, since its time constant is

considered very short [46,47]. The double layer is, neverthe-

less, essential for understanding the cell dynamics [48e50].

The double layer occurs in a thin layer adjacent to the reaction

interface: electrons accumulate on the surface of the electrode

and protons on the surface of the electrolyte and therefore the

interface stores electrical charge and acts as an electric

capacitor [35,47,48,51]. Upon a load change, it will take some

time for this charge to build up or dissipate [47,48,51,52]. High

temperature PEMFC dynamic models incorporating double

layer effect were developed by Zenith et al. [53] and Peng et al.

[52]. Zenith et al. developed a control-oriented model

considering only the transient behavior of the activation

overvoltage when changing the resistance of the external

load whereas Peng et al. developed a transient three-

dimensional model accounting for transient convective and

diffusive transport. These authors considered a macro-

homogeneous model to describe the catalyst layer.

Fuel cells are usually characterized by the steady-state

response, butmore detailed information can be obtained from

transient response measurements such electrochemical

impedance spectroscopy (EIS) experiments. This technique

determines the fuel cell impedance along the frequency

domain characterizing phenomena that occur at different

time scales [54]. The EIS spectra can be modeled using

electrical equivalent circuits (analogs composed mainly of

resistors and capacitors) and the corresponding parameters

can be obtained by fitting to the experimental results [55,56].

However, this analysis gives a simplified picture of the fuel

cell representation and it becomes difficult to assign

physical meaning to these parameters [57]. Alternatively, the

parameters of the EIS spectra can be obtained using

a dynamic phenomenological model derived from

conservation laws. Several impedance models were

developed for porous cathode electrodes. Springer et al. [58]

developed a model based on macro-homogeneous porous

electrode theory and considered the ternary diffusion on the

gas diffusion layer (GDL). Later, Jaouen et al. [54] considered

a spherical agglomerate model for the porous electrode.

They considered that gases are firstly transported by

diffusion and/or convection before dissolving in the

electrolyte phase and reaching the catalyst particles. This

work was further extended by Guo et al. [59]. The analysis of

EIS based on continuum mechanics approach was reviewed

by Gomadam and Weidner [57].

In this work, a dynamic one-dimensional isothermal

model was developed for high temperature PEMFC taking into

account two transient processes: the mass transport on the

bipolar plates and gas diffusion layers and the double layerscharge/discharge. The EIS experiments were mimicked by

imposing a small sinusoidal voltage perturbation to the

simulator, over a wide range of frequencies. The developed

model was used to simulate the steady-state and transient

behavior of two fuel cells systems operated at 160 C, based onphosphoric acid polybenzimidazole MEAs.

2. Fuel cell model

A simple dynamic phenomenological model, accounting for

the transient mass transport (on the bipolar plates and GDLs)

and the double layers charge/discharge, was developed in

order to describe the steady-state behavior and the imped-

ance spectra of high temperature PEMFC. This approach is

new and, to the best knowledge of the authors, was never

performed before despite its relevance, as it will be shown

below. The impedance spectra were obtained imposing

a small voltage perturbation to the simulator and recording

water drag coefficient through the membrane wasneglected;

membrane was assumed impermeable to gases; anode and cathode double layer capacitanceswere assumedto be homogeneous;

catalyst layers were assumed very thin and having homo-geneous properties (planar catalyst layer model).

2.1. Mass balances

The unsteady state partial and total mass balance equations

for the flow fields (anode or cathode) are described in the

following by Equations (1) and (2):

VFF

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 gt 0; pGDLi cz pFFi;in for both anode and cathode (6)

z 0; pGDL;ai ct pFF;ai and z da; NGDL;ai ct Scl;ai (7)

z 0; NGDL;ci ct Scl;ci and z dc; pGDL;ci ct pFF;ci (8)where the superscripts a and c refers to anode and cathode

electrodes, respectively, and cl means catalytic layer. d is

the gas diffusion layer thickness and Si is a source term,

defined as:

SH2 jar2F

; SO2 jcr4F

; SH2O jcr2F

(9)

For species that do not take part in the reaction, Si 0. F isthe Faraday constant and jr is the reaction current density,

calculated by the ButlereVolmer equation [61]:

jr j0

pclipi; in

!g0B@eanFhact

DN2 P dx PGDL;c

x0 b pN2 dx x0 0 (24-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 6 ( 2 0 1 1 ) 9 8 4 2e9 8 5 49846vpGDL;a

i

vq"v

vx

DGDL;a

i P

GDL;a v

vx

pGDL;a

i

PGDL;a

!bpGDL;ai

vPGDL;a

vx

!#sref

sGDL;a

(22-a)

vpGDL;c

i

vq"v

vx

DGDL;c

i P

GDL;c v

vx

pGDL;c

i

PGDL;c

!bpGDL;ci

vPGDL;c

vx

!#sref

sGDL;c

(22-b)

DGDL;aH2 PGDL;a ddx

pGDL;a

H2

PGDL;a

!

x1 bpGDL;a

H2

dPGDL;a

dx

x1

dref

powder was pressed to the gas diffusion layer with heated

plates at 160 C and 5.5 bar for 2 min. The in-house MEA wasthen placed in an Electrochem single cell and closed with

the eight screws, applying a torque of 3.5 N m to each screw.

The fuel cell was operated at 160 C, 2 bar and 1.0% of

MEAs were operated at atmospheric pressure with stoichi-

ometry of 2 for air and 1.2 for hydrogen and using non-

straightforward method for EIS spectra analysis, but the

transposition of the capacitance and resistance values into

physical significant parameters is not always evident or easy.

An alternative way to relate directly the EIS spectras

parameters with the electrochemical system is to use

dynamic phenomenological models. The model presented in

this study considers two transient processes, namely the

mass transport at the bipolar plate and gas diffusion layer and

the double layer charge/discharge, either for the anode or for

the cathode. When a sinusoidal perturbation is applied to the

model, three semi-circles can appear in the Nyquist plot

showing the impedance related with the mass transport (at

low frequencies) and the combination of charge transfer

resistancewith catalyst double layer capacitance of anode and

cathode (at medium to high frequencies). The semi-circles are

Fig. 2 e Simple electrical analogs of fuel cell systems.

Fig. 3 e Nyquist (a) and Bode (b) plots obtained using the

Equations (27)e(30), representing the electric circuit of

Fig. 2 (a), and the model, using data from Table 1 and

ja0[10 A$cmL2.

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 6 ( 2 0 1 1 ) 9 8 4 2e9 8 5 4 9847humidified inlet gases, at 160 C. The MEAs were activated atconstant load of 0.20 A$cm2 and 160 C for at least 100 h [68].

For both MEAs, the polarization curve was obtained start-

ing at OCV and decreasing the potential until 300 mV. AC

impedance spectroscopy was obtained in the frequency range

from 100 kHz to 100 mHz with a perturbation amplitude of

5 mV, using a Zahner IM6e electrochemical workstation

coupled with a potentiostat PP-241.

4. Results and discussion

The most common way to analyze EIS spectra is using an

equivalent electrical circuit. Fitting this equivalent circuit

model to the EIS spectra, it is possible to obtain the parameters

that characterize the electrochemical system. The use of

equivalent electric circuits can be a very easy and

Table 1 e Parameters used in phenomenological modelfor simulator validation.

Symbol Value Unitsrelative humidity (RH) in both streams, after potential

cycling activation [67]. The cell was fed with 1.7 cm3$s1 ofhydrogen and with 5.0 cm3$s1 of air (STP).

The Celtec e P1000 MEA cathode side was loaded with

0.8 mg$cm2 of platinum and the anode with 1 mg$cm2 ofplatinum; the total MEA thickness was 905 mm and the active

area was 20.25 cm2. The membrane thickness was approxi-

mately 120 mm. Accordingly to suppliers specifications, theFuel cell current density jcell 0.23 A$cm2

Anode pressure Pa 1 bar

Cathode pressure Pc 1 bar

Membrane proton conductivity k 0.06 S$cm1

Membrane thickness dm 100 mm

GDL electrical conductivity s 2.22 S$cm1

GDL thickness anode da 375 mm

GDL thickness cathode dc 375 mm

Cathode exchange current density jc0 8 106 A$cm2Anode transfer coefficient aa 0.5 e

Cathode transfer coefficient ac 0.2 e

Anode capacitance Cadl 5 mF$cm2

Cathode capacitance Ccdl 50 mF$cm20.0 0.1 0 .2 0 .3 0 .4 0.50 .0

0 .1

0 .2

-Z

''/

c

m2

Z '/ cm 2

A nalytica l so l. (a ) P hen. M odel

1 10 100 1000 10000

0.20

0 .25

0 .30

0 .35

0 .40

0 .45

|Ph

as

e|/

Frequency/ H z

Z/

c

m2

A na lytica l so l. (a ) P hen . M ode l

0

10

20

30

a

bassociated to a time constant (sEIS), which depends on the

values of the capacitance (C ) and resistance (R) (sEIS R$C ).

4.1. Simulator validation

The phenomenological model was first used to predict the

response of two very simple fuel cell systems, represented by

the electrical analogs in Fig. 2.

The electric equivalent circuit of a fuel cell is composed by

cathode and anode analogs and ohmic losses, connected in

series. The electric analog of the anode and cathode can be

well represented by a resistance (representing the charge

transfer resistance (it was attributed a value of

RcTrf 227 U$cm2), element 2 represents cathode double layercapacitance Ccdl 50 mF$cm2 and element 3 is the ohmicresistance Rohm 200:5 U$cm2. Additionally, element 4represents the anode charge transfer resistance

Rohm 118 U$cm2 and element 5 represents anode doublelayer capacitance Cadl 5 mF$cm2. The Nyquist and Bodeplots of these electric circuits were also obtained by using the

following equations:

Z Z0 jZ00 (27)With Z

Z02 Z002

q. The real and imaginary parts of the

impedance can be obtained as [69]:

Z0 Rohm RaTrf

1 u2Cadl2RaTrf2 RcTrf

1 u2Ccdl2RcTrf2 (28)

Z00 u"

Cadl

RaTrf

21 u2Cadl2RaTrf2

Ccdl

RcTrf

21 u2Ccdl2RcTrf2

#(29)

For the analog of Fig. 2 (a), RaTrf 0. The phase shift wasobtained by

Z00

0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .60 .0

0 .1

0 .2

0 .3

Z '/ c m 2

-Z

''/

c

m2

A n a ly t ic a l s o l. ( b ) P h e n . M o d e l

0 .1 1 10 100 1000 100000.15

0 .20

0 .25

0 .30

0 .35

0 .40

0 .45

0 .50

0 .55

|Ph

as

e|/

Frequency/ H z

Z/

c

m2

A na ly tica l so l. (b ) P hen . M ode l

0

10

20

b

a

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 6 ( 2 0 1 1 ) 9 8 4 2e9 8 5 49848Fig. 4 e Nyquist (a) and Bode (b) plots obtained the using

Equations (27)e(30), representing the electric circuit oftransfer resistance) in parallel with a capacitance (represent-

ing the double layer capacitance), called RC analog. In both

electric analogs, element 1 represents the cathode charge

Fig. 2 (b) and the model, using data from Table 1

and ja0[0:1 A$cmL2.

Table 2 e Parameters used to simulate the fuel cells behavior.

In-house ME

Fuel cell temperature T 160

Anode pressure Pa 2

Cathode pressure Pc 2

Anode gas supply rate Qa 1.7

Cathode gas supply rate Qc 5.0

MEA active area A 4.4

Membrane proton conductivity k 0.025

Membrane thickness dm 100

GDL electrical conductivity s 2.22

GDL porosity anode ea 0.76

GDL porosity cathode ec 0.76

GDL gas permeability b 5.8 108GDL thickness anode da 375

GDL thickness cathode dc 375

H2 effective diffusivity DGDLH2 0.519

O2 effective diffusivity DGDLO2 0.130

H2O effective diffusivity anode DGDL;aH2O

0.519

H2O effective diffusivity cathode DGDL;cH2O

0.147

Anode exchange current density ja0 0.5

Cathode exchange current density jc0 5 106Anode transfer coefficient aa 0.5

Cathode transfer coefficient ac 0.2

Reaction order anode g 0.5

Reaction order cathode g 14 arctanZ0

(30)

A set of operating conditions and properties, listed in Table

1, were chosen for run the developed phenomenological

model. To predict the impedance plots of the electrical

analogs in Fig. 2 (a) and (b) it was assumed an exchange

current density for the anode (ja0) of 10 A$cm2 and

0.1 A$cm2, respectively, and no mass transfer resistance atthe GDLs. A high anode exchange current density will make

anode charge transfer resistance negligible.

A Celtec MEA Units Reference

160 C [exp. condition]1 bar [exp. condition]

1 bar [exp. condition]

3.5 cm3$s1 [exp. condition]13.8 cm3$s1 [exp. condition]20.25 cm2 [exp. condition]

0.090 S$cm1 [determined]120 mm BASF

2.22 S$cm1 [30]0.6 e [determined]

0.6 e [determined]

5.8 108 cm2 [52]375 mm [determined]

375 mm [determined]

0.727 cm2$s1 [60]0.176 cm2$1 [60]e cm2$s1 [60]0.206 cm2$s1 [60]0.4 A$cm2 [determined]5 106 A$cm2 [determined]0.5 e [determined]

0.2 e [determined]

0.5 e [22]1 e [22]

0.2

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 6 ( 2 0 1 1 ) 9 8 4 2e9 8 5 4 98490.0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0 1 .10 .0

0 .1

C urren t dens ity/ A cm -2

0 .3

0 .4

0 .5

0 .6

0 .7

olta

ge lo

ss

/ V

A c tiva tion anode

olta

ge lo

ss

/ V

A c tiva tion ca thode

0 .02

0 .03

0 .04

0 .05b0.3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

1 .1

Vo

ltage

/ V

E xpe rim en ta l P hen . m ode l

aFig. 3 shows the Nyquist and Bode plots obtained by

Equations (27)e(30), representing the electric analog of Fig. 2

(a) (analytical sol. (a)), and predicted by the model using data

from Table 1 and ja0 10 A$cm2. Fig. 4 shows the Nyquist andBode plots predicted by Equations (27)e(30), representing the

electric analog of Fig. 2 (b) (analytical sol. (b)), and obtained by

the model using data from Table 1 and ja0 0:1 A$cm2.Nyquist and Bode plots obtained from the phenomenological

model and by the analytical equations show that the model

describes correctly electrochemical systems that can be well

described by elemental electric analogs.

4.2. Fuel cell modeling

The purpose of a phenomenologicalmodel is to provide a good

fitting of the experimental values and to assist understanding

the physical problem and to optimize the operating and

design conditions. In the present case, two fuel cells operating

at 160 C and equipped with two different MEAs, based onphosphoric acid doped PBI membranes, were considered: an

in-house MEA and a Celtec e P1000 MEA. The experimental

(exp.) conditions and parameters used in the simulation of

both fuel cells are shown in Table 2.

0.0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0 1 .10 .0

0 .1

0 .2

VV

C u rren t dens ity/ A cm -2

O hm ic

0 .00

0 .01

Fig. 5 e Experimental and simulated IeV curves for the

Celtec e P1000 MEA operated at 160 C, 1 bar and withunhumidified air and hydrogen gas flow (a), and sources of

voltage loss obtained from simulation: anode (right Y-axis)

and cathode activation losses and ohmic losses (b). The

model parameters are presented in Table 2.0.0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .70 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

1 .1

Vo

ltage

/ V

C u rren t dens ity/ A cm -2

E xpe rim en ta l P hen . M ode l

0 .3

0 .4

0 .5

0 .6

olta

ge lo

ss

/ V

olta

ge lo

ss

/ V A c tiva tion anode

A ctiva tion ca thode

0.10

0 .15

0 .20

a

bThe values of transfer coefficient (ac) and exchange current

density (jc0) for the cathode were estimated by fitting the Tafel

equation to experimental IeV curves for Celtec e P1000 MEA

and for very low current densities and are ac 0.2 (or nac 0.8)and jc0 5 106 A$cm2, respectively. These values are inagreement with the literature for similar operating conditions

[22,26]. The same cathode exchange current density was also

used for the in-house MEA, since the experimental values of

the IeV curves in the relevant region are very few. Moreover,

Lui et al. [70] showed that the exchange current density of

oxygen reduction does not markedly change with relative

humidity (up to 10%) or the phosphoric acid loading in

a platinum interface with phosphoric acid doped PBI.

The membrane conductivity, the transfer coefficient (aa)

and the exchange current density of the anode (ja0), the GDL

porosity (e) and the anode and cathode capacitances were

obtained by fitting the model to the IeV curve for each MEA

and to the impedance spectra at fuel cell current density of

0.18 A$cm2 for the in-house MEA and at 0.20 A$cm2 for theCeltec e P1000 MEA.

4.2.1. IeV modelingThe experimental and the modeled IeV curves are shown in

Fig. 5 (a) for the Celtece P1000MEA and in Fig. 6 (a) for the in-

0.0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .70 .0

0 .1

0 .2 VV

C u rren t dens ity/ A cm -2

O hm ic

0 .00

0 .05

Fig. 6 e Experimental and simulated IeV curves for the in-

house MEA operated at 160 C, 2 bar and 1.0% RH (a) andsources of voltage loss obtained from simulation: anode

(right Y-axis) and cathode activation losses and ohmic

losses (b). The model parameters are presented in Table 2.

observed at cathode activation overpotential for Celtec e

P1000 MEA at current densities higher than 0.90 A$cm2.

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.1

0.2

0.3

Experim enta l P hen. m odel

Z '/ cm 2

-Z

''/

c

m2

0 .5

0 .6

0 .7

P hen. m ode l

|2 40

50

60

E xperim en ta l

a

b

Fig. 9 e Fuel cell electrical equivalent circuit. Elements 1

and 4 represent anode and cathode charge transfer

resistance, elements 2 and 5 represent anode and cathode

double layer capacitance, elements 6 and 7 represent the

resistance and capacitance associated with gas phase

mass transfer; element 3 is the ohmic resistance.

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 6 ( 2 0 1 1 ) 9 8 4 2e9 8 5 498500.1 1 10 100 10000.0

0 .1

0 .2

0 .3

0 .4

Ph

as

e|/

Z/

c

m

0

10

20

30house MEA. The predicted IeV curves are quite close to the

experimental values. Fig. 5 (b) and Fig. 6 (b) also show the

simulated anode and cathode activation losses and ohmic

losses as a function of the current density for the Celtec e

P1000 MEA and the in-house MEA, respectively. As expected,

the major contribution for the voltage loss was due to the

cathode activation. Since the mass transport overpotential is

more pronounced for higher current densities, an increase is

Frequency/ H z

Fig. 7 e Experimental and simulated Nyquist plot (a) and

Bode plot (b) for the Celtec e P1000 MEA operated at

160 C, 1 bar and with unhumidified air and hydrogen gasflow, at 0.20 A$cmL2 current density. The model

parameters are presented in Table 2.

0.0 0.1 0 .2 0.3 0.4 0.50.0

0.1

0.2

100 H z M odel

100 H z Exp. 10 H z M odel

E xp. Phen. m odel 0.3 A cm E xp. Phen. m odel 0.4 A cm E xp. Phen. m odel 0.5 A cm

-Z

''/

c

m2

10 H z Exp.

Fig. 8 e Experimental and simulated Nyquist plots for the

Celtec e P1000 MEA operated at 160 C, 1 bar, and withunhumidified air and hydrogen gas flow, at 0.30 A$cmL2,

0.40 A$cmL2 and 0.50 A$cmL2 current density. The model

parameters are presented in Table 2.4.2.2. EIS modeling e Celtec e P1000 MEAThe experimental and predicted Nyquist and Bode plots of the

Celtece P1000MEA at fuel cell current density of 0.20 A$cm2

are shown in Fig. 7. The best fit to the impedance spectra was

obtained for Cadl 20 mF$cm2 and Ccdl 74 mF$cm2.Since the relative importance of the several phenomena

occurring in the fuel cell varies with the current density, the

phenomenologicalmodelwascomparedwith theexperimental

results for the Celtec e P1000 MEA at higher current densities

(0.3 A$cm2, 0.4 A$cm2 and 0.5 A$cm2), using the same

parameters used for current density 0.2 A$cm2 eTable 2. Theonly parameters fitted for each current density were the

anode and the cathode capacitances. According to the

experimental data, the membrane conductivity at current

densities 0.4 A$cm2 and 0.5 A$cm2 increased from0.09 S$cm1 to 0.1 S$cm1. Fig. 8 shows the experimental andbest fitting Nyquist plots. A capacitance value of 40 mF$cm2

was used for the anode, for all current densities, and

capacitances of 100 mF$cm2, 120 mF$cm2 and140 mF $ cm2 were used for the cathode side, for currentdensities of 0.30 A$cm2, 0.40 A$cm2 and 0.50 A$cm2,

respectively.

0.3 Experim enta l0.0 0 .1 0.2 0.3 0 .4 0.5 0.60 .0

0 .1

0 .2

107 H z EC

10 H z EC

1 H z EC

107 H z E xp .

10 H z E xp.

E lectrica l c ircu it (EC ) fitting

Z '/ cm 2

-Z

''/

c

m2

1 H z Exp .

Fig. 10 e Experimental Nyquist plot and predicted by the

Thales software using the electric circuit of Fig. 9, for the

Celtec e P1000 MEA operated at 160 C, 1 bar, and withunhumidified air and hydrogen gas flow, at 0.20 A$cmL2

current density. The model parameters are presented in

Table 2.

The phenomenological model captures the qualitative

response of the fuel cell impedance at 0.2 A$cm2 (Fig. 7).Nonetheless, at this point it is interesting to compare the

model results with the parameters obtained by fitting the

equivalent electric circuit presented in Fig. 9 to the

experimental spectrum (Fig. 7). Resistances 1 and 4

represent the charge transfer resistance of anode and

cathode whereas capacitances 2 and 4 represent the double

layer capacitance of anode (Cadl) and cathode (Ccdl). An extra

RC was included to describe the mass transfer resistance of

the reactant species to the active sites [56,71]. Elements 6

and 7 represent the resistance and capacitance associated

with mass transfer. The ohmic resistance represented by

element 3 includes membrane proton resistance and the

electronic resistance of the electrodes.

Usually, when electrical equivalent circuits are employed

to study fuel cell EIS spectras, the capacitance of anode and

cathode is replaced by a constant phase element, mainly

because of the electrodes porous structure (the capacitance

caused by the double layer charging is distributed along the

pore depths [57,72]). To compare the electrical equivalent

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 6 ( 2 0 1 1 ) 9 8 4 2e9 8 5 4 9851circuit with the phenomenological model, pure capacitive

elements were considered, because the model assumes flat

electrodes. The Thales software (Zahner-Elektrik GmbH) was

used to fit this electrical model to the experimental data,

Fig. 10. From the electrical circuit fitting to the experimental

data, it was obtained a capacitance of 10 mF cm2 and74 mF cm2 for the anode and cathode, respectively. Thesevalues are close to the ones obtained by the proposed model.

For higher current densities, the model also predicts

qualitatively the Nyquist spectra. However, the predicted

cathode semi-circles are associated to lower frequencies than

the experimental ones (Fig. 8). The model results were

compared with the parameters obtained from fitting the

electrical circuit of Fig. 9 to the experimental spectra

(Fig. 11). The capacitances predicted by the model are also

higher than the capacitances obtained by fitting the

0.0 0.1 0 .2 0 .3 0.40 .00

0.05

0.10

0.15

0.20

500 H z

107 H z

Experim enta l E lectrica l c ircu it fitting Phen. m ode l

Z '/ cm 2

-Z

''/

c

m2

10 H z

Fig. 11 e Experimental Nyquist plot and predicted Nyquist

plots by Thales software using the electric circuit analog of

Fig. 9 (blue line) and by the phenomenological model

(green line), for the Celtec e P1000MEA operated at 160 C,1 bar, and with unhumidified air and hydrogen gas flow, at

0.50 A$cmL2 current density. The model parameters are

presented in Table 2. (For interpretation of the references tocolour in this figure legend, the reader is referred to the

web version of this article.)electrical circuit to experimental points. For instance, at

current density 0.50 A$cm2, the electric circuit best fittingwas for Cadl 10 mF$cm2 and Ccdl 62 mF$cm2. These twoparameters were then introduced in the phenomenological

model and the resultant Nyquist plot is shown in Fig. 11

together with the original electric circuit fitting. As it can be

seen, the results from the equivalent electric circuit fits

quite well the experimental results, contrarily to the model

ones.

The proposed dynamic model is very simple and considers

a planarmodel for the catalyst layer at the anode and cathode.

According to this approach, the characteristic semi-circles

associated to the anode and cathode depend only on the

interfacial kinetics of the electrochemical reactions, with the

diameter of the EIS loops determined by the charge transfer

resistances. Two other processes, however, can affect the

kinetic loop, namely the diffusion coefficient of gases towards

the catalyst layer and the resistance associated with the

proton leaving the catalyst layer [58]. When proton transport

resistance within catalyst layer becomes noticeable, a 45

branch should appear at high frequencies in the Nyquist

plot [54,57,58]. Furthermore, Springer et al. [58] showed that,

with air cathodes (at low temperature operation) the charge

transfer resistance of the cathode decreases for higher

current densities until reaching a minimum value;

afterward, the charge transfer resistance increases due to

a depletion of oxygen within the catalyst layer. The gas

diffusion on the phosphoric acid before undergoing

electrochemical reaction as well as proton conductivity

within the catalyst layer were not considered in the present

model, and, therefore, qualitative differences in the kinetic

loop would be expected, especially at high current densities.

For increasingly high current densities, the charge transfer

resistance at the cathode decreases as a result of the

increasing driving force for the oxygen reduction reaction

(Fig. 8); at the same time, the oxygen concentration within

the catalyst layer decreases. So, considering similar

capacitances, the Nyquist plot predicted by the model has

lower charge transfer resistances than the experimental

plot. The mass transfer resistance associated with the

transport in the gas diffusion layer (represented by the semi-

circle at low frequencies) is the only mass transport

resistance inserted in the model.

4.2.3. EIS modeling e in-house MEAThe experimental and simulated Nyquist and Bode plots of

the in-house MEA at 0.18 A$cm2 are shown in Fig. 12 (a). Thebest fit to the impedance spectra was obtained for

Cadl 40 mF$cm2 and Ccdl 80 mF$cm2. The model Nyquistplot is in agreement with the experimental values, Fig. 12

(a). However, the semi-circles are associated to much lower

frequencies.

The phenomenological model predictions were again

compared with the parameters obtained from fitting the elec-

trical circuit of Fig. 9 to the experimental spectra (Fig. 12 (b)).

The electric circuit best fitting was for Cadl 1:3 mF$cm2 andCcdl 5:5 mF$cm2. These two parameters were thenintroduced in the model and the resultant Nyquist plot isshown in Fig. 12 (b), together with the original electric circuit

fitting.

0 .2 1 0 H z M o d e l 1 0 H z E x p .

1 0 0 H z E x p .

c

m

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 6 ( 2 0 1 1 ) 9 8 4 2e9 8 5 498520 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .90 .0

0 .1 1 0 0 H z M o d e l

0 .8 9 H z M o d e l

Z '/ c m 2

-Z

''/ 1 0 0 0 H z E x p .

0 .0

0 .1

0 .2

0 .3

0 .4

1 0 H z

1 0 7 H z

E xp e r im e n ta l E le c tr ic a l c irc u it f itt in g P h e n . m o d e l

-Z

''/

c

m2

1 0 0 0 H z

b0 .3

0 .4 E x p e r im e n ta l (E xp .) P h e n . M o d e l

2

aThe use of lower values of capacitance in the simulator

originated a different Nyquist plot compared to the experi-

mental values. The model predicts a semi-circle at lower

frequencies that is related to the gas diffusion layer

resistances.

At low current densities, it is expected that the charge

transfer resistance be dominated by the electrochemical reac-

tions. Nonetheless, the proton resistance and the gas diffusion

limitations in the catalyst layer can also contribute to the

effective charge transfer resistance making it noticeable in the

Nyquist plot. Indeed, this MEA has no ionomer in the catalyst

layer and proton-bridge between catalyst and electrolyte is

assured by migrating phosphoric acid from the electrolyte

membrane. As a consequence, for a similar value of capaci-

tance, a lower value of charge transfer resistance is obtained

using the model when compared to the experimental values.

5. Conclusions

A dynamic one-dimensional isothermal phenomenological

model was developed for simulating high temperature poly-

mer electrolyte membrane fuel cells. The model was used to

0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9

Z '/ c m 2

Fig. 12 e Experimental and simulated Nyquist plot (a);

experimental Nyquist plot, predicted Nyquist plot by

Thales software using the electric circuit analog of Fig. 9

(blue line) and by the phenomenological model (green line)

(b), for the in-house MEA operated at 160 C, 2 bar, and 1.0%relative humidity, at 0.18 A$cmL2 current density. The

model parameters are presented in Table 2. (For

interpretation of the references to colour in this figure

legend, the reader is referred to the web version of this

article.)simulate the steady-state and the impedance spectra of two

fuel cells operated at 160 C, one equipped with an in-houseassembled MEA and the other with Celtec e P1000 MEA. To

obtain the impedance spectra a small voltage perturbation

over a wide range of frequencies was imposed to the

simulator.

The steady-state behavior was captured and the model

proved to predict qualitatively well the Nyquist plots of both

fuel cell systems. The observed differences were assigned to

proton resistance and the gas diffusion limitations within the

catalyst layer, which were not considered in the phenome-

nological model.

A new phenomenological model is under development

that takes into account the porous structure of the catalyst

layer.

Acknowledgments

The work of M. Boaventura was supported by FCT (Grant

SFRH/BD/28187/2006). The present work was also partially

supported by FCT projects PTDC/EQU-EQU/70574/2006 and

PTDC/EQU-EQU/104217/2008.

Nomenclature

A active area, cm2

Cdl differential capacitance, F$m2

Dr diffusivity, cm2$s1

D effective diffusivity, cm2$s1

Etherm thermodynamic voltage, V

E0 standard state reversible fuel cell, V

F Faraday constant, C$mol1

f frequency, Hz

J0 amplitude of current response, A$cm2

j0 exchange current density, A$cm2

jcell fuel cell current density, A$cm2

jr reaction current density, A$cm2

k membrane proton conductivity, S$cm1

N flux, mol cm2$s1

n electrons transferred, mol

P total pressure, bar

p partial pressure, bar

Q volumetric flow rate, cm3$s1

R resistance, U$cm2

< gas constant, bar $ cm3$mol1$K1DS^ entropy change, J$K1

S source term, mol $ cm2$s1

T temperature, K

t time, s

V volume, cm3

Vcell operating voltage, V

V0 amplitude of the perturbation, V

z spatial coordinate, cm

Z impedance, U$cm2

Z0 impedance magnitude, U$cm2

0

Z real component of impedance, U$cm2

Z00 imaginary component of impedance, U$cm2

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A dynamic model for high temperature polymer electrolyte membrane fuel cells1 Introduction2 Fuel cell model2.1 Mass balances2.2 Fuel cell voltage2.3 Dimensionless equations2.4 Solution of the model equations

3 Experimental4 Results and discussion4.1 Simulator validation4.2 Fuel cell modeling4.2.1 IV modeling4.2.2 EIS modeling Celtec P1000 MEA4.2.3 EIS modeling in-house MEA

5 Conclusions Acknowledgments Nomenclature References