Modelling of Proton Exchange Membrane Fuel Cell Performance Based on Semi-empirical Equations

Technical Note

Modelling of proton exchange membrane fuel cell

performance based on semi-empirical equations

Maher A.R. Sadiq Al-Baghdadi1*

Department of Mechanical Engineering, College of Engineering, University of Babylon, Babylon, Iraq

Received 8 August 2004; accepted 29 November 2004

Available online 19 January 2005

Abstract

Using semi-empirical equations for modeling a proton exchange membrane fuel cell is proposed

for providing a tool for the design and analysis of fuel cell total systems. The focus of this study is to

derive an empirical model including process variations to estimate the performance of fuel cell

without extensive calculations. The model take into account not only the current density but also the

process variations, such as the gas pressure, temperature, humidity, and utilization to cover operating

processes, which are important factors in determining the real performance of fuel cell. The

modelling results are compared well with known experimental results. The comparison shows good

agreements between the modeling results and the experimental data. The model can be used to

investigate the influence of process variables for design optimization of fuel cells, stacks, and

complete fuel cell power system.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: PEM fuel cell; Electrochemistry; Modelling; Energy conversion

1. Introduction

The proton exchange membrane fuel cells (PEMFCs) are suitable for portable, mobile

and residential applications, due to their inherent advantages, such as high-power density,

Renewable Energy 30 (2005) 1587–1599

www.elsevier.com/locate/renene

0960-1481/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.renene.2004.11.015

* Fax: C218 41 632249.

E-mail address: [email protected] Present address: Mechanical and Energy Department, The Higher Institute for Engineering Comprehensive

Vocations, P.O. Box 65943, Yefren, Libya.

http://www.elsevier.com/locate/renene

Nomenclature

A active cell area (cm2)

ac chemical activity parameter for the cathode

C�HC proton concentration at the cathode membrane/gas interface (mol/cm3)

C�H2

liquid phase concentration of hydrogen at anode/gas interface (mol/cm3)

C�H2O water concentration at the cathode membrane/gas interface (mol/cm3)

C�O2

oxygen concentration at the cathode membrane/gas interface (mol/cm3)

E thermodynamic potential (V)

Efc thermodynamic efficiency

F Faraday’s constant (96,487 8C/mol)

i current density (A/cm2)

I current (A)

k0a ; k

0c intrinsic rate constant for the anode and cathode reactions, respectively

(cm/s)

l thickness of the polymer membrane (cm)

LHVH2lower heating value of hydrogen (J/kg)

m, n mass transfer coefficients

_mH2hydrogen mass flow rate (kg/s)

MWH2molecular mass of hydrogen (kg/mol)

Pa, Pc total pressure of anode and cathode, respectively (atm)

P�H2;P�

O2partial pressure of hydrogen and oxygen at the anode catalyst/gas interface

and cathode catalyst/gas interface, respectively (atm)

PsatH2O water saturation pressure (atm for Eqs. (12) and (14)), (bar for Eq. (15))

R gas constant (8.314 J/mol K)

Rinternal total internal resistance (U cm2)

rM membrane specific resistivity for the flow of hydrated protons (U cm)

T cell temperature (K)

Vcell cell voltage (V)

Wgross gross output power (W)

DGe standard state free energy of the cathode reaction (J/mol)

DGec standard state free energy of chemisorption from the gas state (J/mol)

Greek letters

hact activation over potential

hdiff diffusion over potential

hohmic ohmic over potential

x1,x2,x3,x4 semi-empirical coefficients for calculation of activation overpotential

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–15991588

simple and safe construction and quick startup even at low operating temperatures. Rapid

development recently has brought the PEMFC significantly closer to commercial reality.

Although prototypes of fuel cell vehicles and residential fuel cell systems have already

been introduced, it remains to reduce the cost and enhance their efficiencies. To improve

the system performance, design optimization and analysis of fuel cell systems are

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599 1589

important. Mathematical models and simulation are needed as tools for design

optimization of fuel cells, stacks, and fuel cell power systems. In order to understand

and improve the performance of PEMFC systems, several different mathematical models

have been proposed to estimate the behavior of voltage variation with discharge current of

a PEMFC. Recently, numerical modeling and computer simulation have been performed

for understanding better the fuel cell itself [1–19]. Numerical models are useful to simulate

the inner details of PEMFC, but the calculation required for these models is too extensive

to be used for system models. In system studies, it is important to have an adequate model

to estimate overall performance of a PEMFC in terms of operating conditions without

extensive calculations. But, few studies have focused on the simple models, which can be

used to investigate the impact of cell operating conditions on the cell performance and can

be used to design practical fuel cell total systems. In this paper, a lumped model is

presented. Empirical equations are useful for estimating the performance of PEMFC

stacks and optimization of fuel cell system integration and operation than numerical

models.

The aim of this research is to develop a model for investigating the performance of a

PEM fuel cell at different operation variables using semi-empirical equations. Model

validation against the experimental data of Chahine et al. [11] is presented.

2. Background

The fundamental structure of a PEM fuel cell can be described as two electrodes (anode

and cathode) separated by a solid membrane acting as an electrolyte (Fig. 1). Hydrogen

fuel flows through a network of channels to the anode, where it dissociates into protons

that, in turn, flow through the membrane to the cathode and electrons that are collected as

electrical current by an external circuit linking the two electrodes. The oxidant (air in this

study) flows through a similar network of channels to the cathode where oxygen combines

with the electrons in the external circuit and the protons flowing through the membrane,

Fig. 1. Schematic of a single typical proton exchange membrane fuel cells.


thus producing water. The chemical reactions occurring at the anode and cathode electrode

of a PEM fuel cell are as follows:

Anode reaction: 2H2/4HCC4eK

Cathode reaction: O2C4HCC4eK/2H2O

Total cell reaction: 2H2CO2 /2H2OCelectricityCheat

The products of this process are water, DC electricity and heat.

3. Mathematical model

Useful work (electrical energy) is obtained from a fuel cell only when a current is

drawn, but the actual cell potential (Vcell) is decreased from its equilibrium thermodynamic

potential (E) because of irreversible losses. When current flows, a deviation from the

thermodynamic potential occurs corresponding to the electrical work performed by the

cell. The deviation from the equilibrium value is called the over potential and has been

given the symbol (h). The over potentials originate primary from activation over potential

(hact), ohmic over potential (hohmic) and diffusion over potential (hdiff).

Therefore, the expression of the voltage of a single cell is:

Vcell Z E Chact Chohmic Chdiff (1)

The reversible thermodynamic potential of the H2CO2 reaction previously described is

given by the Nernst equation:

E Z E0 CRT

zFln½P�

H2ðP�

O2Þ0:5� (2)

where E0 is a reference potential and the partial pressure terms are related to the hydrogen

and oxygen concentrations at the anode and cathode. Further expansion of this equation

return [11,12]:

E Z 1:229 K0:85!10K3ðT K298:15ÞC4:3085!10K5 T lnðP�H2ÞC

1

2lnðP�

O2Þ

� �

(3)

Activation overpotential arises from the kinetics of charge transfer reaction across the

electrode–electrolyte interface. In other words, a portion of the electrode potential is lost in

driving the electron transfer reaction. Activation overpotential is directly related to the

nature of the electrochemical reactions and represents the magnitude of activation energy,

when the reaction propagates at the rate demanded by the current.

The activation overpotential can be divided into the anode and cathode overpotentials.

The equation for the anode overpotential is [11–13]:

hact;a ZKDGec

2FC

RT

2Flnð4FAk0

aC�H2ÞK

RT

2FlnðiÞ (4)


The respective equation used for calculating the cathode overpotential is:

hact;c ZRT

aczFln zFAk0

c expKDGe

RT

� �ðC�

O2Þð1KacÞðC�

HCÞð1KacÞðC�

H2OÞac

� �K lnðiÞ

� �

(5)

where zZ1 is the number of equivalents involved in the cathode reaction.

In order to have a single expression of the activation overpotential, Eqs. (4) and (5) can

be combined and written in a parametric form as follows:

hact Z x1 Cx2T Cx3T½lnðC�O2Þ�Cx4T½lnðiÞ� (6)

where the terms xi are semi-empirical coefficients, defined by the following equations:

x1 ZKDGe

aczF

� �C

KDGec

2F

� �(7)

x2 ZR

aczFln zFAk0

c ðC�HCÞ

ð1KacÞðC�H2OÞ

ac� �

CR

2F½lnð4FAk0

a C�H2Þ� (8)

x3 ZR

aczFð1 KacÞ (9)

x4 ZKR

aczFC

R

2F

� �(10)

The use of such semi-empirical coefficients gives a significant degree of flexibility when

the model is applied to simulate a specific fuel cell stack, as the terms xi can be obtained by

a fitting procedure based on the measured polarization curve of the stack. At the same

time, these coefficients have a significant mechanistic background. The values used here

for the coefficients xi are the ones proposed in Ref. [11] and also with the works of

Maxoulis et al. [12] and Fowler et al. [13] and are shown as

x1
K0.9514
x2
0.00312
x3
7.4!10K5
x4
K0.000187
The water concentration levels at the anode and cathode may play an important role in

the activation losses of a fuel cell, as the reactants H2 at the anode and O2 at the cathode

side must diffuse through a water film to reach the catalyst active sites. The anode side of

the membrane is expected to present a quite low water concentration, while the water film

at the cathode side is expected to be significantly thicker due to the production of water

there.

The effective hydrogen concentration at the anode catalyst sites, which can be

approximated, by the hydrogen concentration at the anode water–gas interface, is


expressed as [12];

C�H2

ZP�

H2

1:09!106exp 77T

� � (11)

The partial pressure of hydrogen, in turn, depends on the water content in the anode

channel (which is assumed constant, due to the zero-dimension approach) based on the

expression;

P�H2

Z1

2Psat

H2O

� �1

exp 1:653!iT1:334

xH2O

K1

24

35 (12)

The effective oxygen concentration at the cathode catalyst sites, which can be

approximated, by the oxygen concentration at the cathode water–gas interface, is

expressed as [12];

C�O2

ZP�

O2

5:08!106exp K498T

� � (13)

The partial pressure of oxygen at the water–gas interface is related to the water

concentration at the cathode channel with the expression;

P�O2

Z ðPsatH2OÞ

1

exp 4:192!iT1:334

xH2O

K1

24

35 (14)

The saturation pressure of water vapor can be computed from Berning et al. [14];

log10 PsatH2O ZK2:1794 C0:02953ðT K273:15ÞK9:1837!10K5

!ðT K273:15Þ2 C1:4454!10K7ðT K273:15Þ3(15)

Ohmic overpotential result from electrical resistance losses in the cell. These resistances

can be found in practically all fuel cell components: ionic resistance in the membrane,

ionic and electronic resistance in the electrodes, and electronic resistance in the gas

diffusion backings, bipolar plates and terminal connections. This could be expressed using

Ohm’s Law equations such as:

hohmic ZKiRinternal (16)

The total internal resistance is a complex function of temperature and current. A general

expression for resistance is defined to include all the important membrane parameters [13];

Rinternal ZrM :l

A(17)


The following empirical expression for Nafion membrane resistivity is proposed [13];

rM Z181:6 1 C0:03 i

A

� �C0:062 T

303

� �2 iA

� �2:5h i

14 K0:634 K3 iA

� �� exp 4:18 TK303

T

� �� (18)

Diffusion overpotential is caused by mass transfer limitations on the availability of the

reactants near the electrodes. The electrode reactions require a constant supply of reactants

in order to sustain the current flow. When the diffusion limitations reduce the availability

of a reactant, part of the available reaction energy is used to drive the mass transfer, thus

creating a corresponding loss in output voltage. Similar problems can develop if a reaction

product accumulates near the electrode surface and obstructs the diffusion paths or dilutes

the reactants.

As proposed by Berning et al. [14], Chahine et al. [15], and Hamelin et al. [16], the total

diffusion overpotential can be represented by the following expression:

hdiff Z m expðniÞ (19)

The diffusion overpotential is directly related to the concentration drop of reactant

gases, and thus inversely to the growth rate n of byproducts of the electrochemical

reaction in the catalyst layers, flow fields, and across the electrode. A physical

interpretation for the parameters m and n was not given, but Berning et al. [14] found

in their study that m correlates to the electrolyte conductivity and n to the porosity of

the gas diffusion layer. Both m and n relate to water management issues. A partially

dehydrated electrolyte membrane leads to a decrease in conductivity, which can be

represented by m, whereas an excess in liquid water leads to a reduction in porosity and

hence to an early onset of mass transport limitations, which can be captured by the

parameter n.

The mass transfer coefficient m decreases linearly with cell temperature but it has two

dramatically different slopes as shown by the following expressions [15].

m Z 1:1!10K4 K1:2!10K6ðT K273:15Þ for T R312:15 K ð39 8CÞ (20)

m Z 3:3!10K3 K8:2!10K5ðT K273:15Þ for T !312:15 K ð39 8CÞ (21)

The thermodynamic efficiency of the fuel cell Efc can be determined as the ratio of output

work rate Wgross to the product of the hydrogen consumption rate _mH2and the lower heating

value of hydrogen LHVH2[17].

Efc ZWgross

_mH2$LHVH2

(22)


Once the output voltage of the stack is determined for a given output current, the gross

output power is found as:

Wgross Z IVcell (23)

The output current is correlated with the hydrogen mass flow rate by the equation [17];

_mH2Z

IMWH2

2F(24)

Thus, the thermodynamic efficiency of the fuel cell can simplifies as follows;

Efc Z2VcellF

MWH2$LHVH2

(25)

4. Results and discussion

Model validation involves the comparison of model results with experimental data,

primarily for the purpose of establishing confidence in the model. To validation the

mathematical model presented in the preceding section, comparisons were made to

Fig. 2. Comparison between the model predictions and experimental results of Chahine et al. [15].

Fig. 3. Predicted polarization curve, efficiency and power density at 31 8C cell temperature and 1 atm reactant

pressures.


the experimental data of Chahine et al. [15] for a single cell operated at different

temperatures. Fig. 2 compares the computed polarization curves with the measured ones.

The calculated curves show good agreement with the experimental data for all

temperatures.

Polarization curves with different cell temperature and different operating pressures

are shown in Figs. 3–5. The pressures of anode and cathode sides were kept the same.

The performance of the fuel cell increases with the increase of the cell temperature,

as will be shown in Figs. 3 and 4. The exchange current density increases with the

increase of fuel cell temperature, which reduces activation losses. Another reason for

the improved performances is that higher temperatures improve mass transfer within

the fuel cells and results in a net decrease in cell resistance (as the temperature

increases the electronic conduction in metals decreases but the ionic conduction in the

electrolyte increases). This may explain the improvement of the performance [18,19].

The shifting of the polarization curves towards higher voltage at higher current

densities when increasing the cell temperature is due to the increase of conductivity

of the membrane. Also, Figs. 3 and 4 show that the maximum power density shifts

towards higher current density with an increasing temperature as a result of reduced

ohmic loss [14].


pressures.


Figs. 4 and 5 show that the performance of the fuel cell improves with the increase of

pressure. The higher open circuit voltage at the higher pressures can be explained by the

Nernst equation. The overall polarization curves shift positively as the pressure increases.

Another reason for the improved performances is the partial pressure increase of the

reactant gases with increasing operating pressure.

Changes in operating pressure have a large impact on the inlet composition and, hence,

on the power density, as will be shown in Figs. 4 and 5. The maximum power density shifts

positively with an increasing pressure because of the rate of the chemical reaction is

proportional to the partial pressures of the hydrogen and the oxygen. Thus, the effect of

increased pressure is most prominent when using air. In essence, higher pressures help to

force the hydrogen and oxygen into contact with the electrolyte. This sensitivity to

pressure is greater at high currents.

Return to Figs. 3–5, the maximum power corresponds to relatively high current. At the

peak point, the internal resistance of the cell is equal to the electrical resistance of the

external circuit. However, since efficiency drops with increasing current, there is a tradeoff

between high power and high efficiency. Fuel cell system designers must select the desired

operating range according to whether efficiency or power is paramount for the given

application.


pressures.


The influence of cathode/anode pressure on the performance of PEMFC at 72 8C is

shown in Fig. 6. It is clearly shown that cathode pressure is more effective to the fuel cell

performance than anode pressure. This result demonstrates that an increase in the cathode

pressure results in a significant reduction in polarization at the cathode (Eqs. (3) and (6)).

Performance improvements due to increased pressure must be balanced against the energy

required to pressurize the reactant gases. The overall system must be optimized according

to power, efficiency, cost, and size.

5. Conclusion

An empirical model of a PEM fuel cell has been developed and the effect of operation

conditions on the cell performance has been investigated. The objective was to develop an

empirical model that would simulate the performance of fuel cells without extensive

calculations. The results of the present study indicate the operating temperature and

pressure can be optimized, based on cell performance, for given design and other operating

conditions. For most applications, and particularly for steady operation, a fuel cell does not

Fig. 6. Effect of cathode/anode pressure on the fuel cell performance.


have to be operated at its maximum power, where the efficiency is the lowest. When higher

nominal cell potential is selected, savings on fuel cost offset the cost of additional cells.

The empirical model of electrochemical reactions and current distribution as presented

herein is shown to be provide a computer-aided tool for design and optimization of future

fuel cell engines with much higher power density and lower cost.

References

[1] Hu M, Gu A, Wang M, Zhu X, Yu L. Three dimensional, two phase flow mathematical model for PEM fuel

cell: part I. model development. Energy Conversion Manage 2004;45(11–12):1861–82.

[2] Hu M, Gu A, Wang M, Zhu X, Yu L. Three dimensional, two phase flow mathematical model for PEM fuel

cell: part II. Analysis and discussion of the internal transport mechanisms. Energy Conversion Manage

2004;45(11–12):1883–916.

[3] Jung H-M, Lee W-Y, Park J-S, Kim C-S. Numerical analysis of a polymer electrolyte fuel cell. Int

J Hydrogen Energy 2004;29(9):945–54.

[4] Nguyen PT, Berning T, Djilali N. Computational model of a PEM fuel cell with serpentine gas flow

channels. J Power Sources 2004;130(1–2):149–57.

[5] Siegel NP, Ellis MW, Nelson DJ, von Spakovsky MR. A two-dimensional computational model of a

PEMFC with liquid water transport. J Power Sources 2004;128(2):173–84.


[6] Amphlett JC, Mann RF, Peppley BA, Roberge PR, Rodrigues A. A model predicting transient responses of

proton exchange membrane fuel cells. J Power Sources 1996;61(1–2):183–8.

[7] Lee JH, Lalk TR, Appleby AJ. Modeling electrochemical performance in large scale proton exchange

membrane fuel cell stacks. J Power Sources 1998;70(2):258–68.

[8] Ceraolo M, Miulli C, Pozio A. Modelling static and dynamic behaviour of proton exchange membrane fuel

cells on the basis of electro-chemical description. J Power Sources 2003;113(1):131–44.

[9] von Spakovsky MR, Olsommer B. Fuel cell systems and system modeling and analysis perspectives for fuel

cell development. Energy Conversion Manage 2002;43(9–12):1249–57.

[10] Siegel NP, Ellis MW, Nelson DJ, von Spakovsky MR. Single domain PEMFC model based on agglomerate

catalyst geometry. J Power Sources 2003;115(1):81–9.

[11] Mann RF, Amphlett JC, Hooper MAI, Jensen HM, Peppley BA, Roberge PR. Development and application

of a generalised steady-state electrochemical model for a PEM fuel cell. J Power Sources 2000;86(1–2):

173–80.

[12] Maxoulis CN, Tsinoglou DN, Koltsakis GC. Modeling of automotive fuel cell operation in driving cycles.

Energy Conversion Manage 2004;45(4):559–73.

[13] Fowler MW, Mann RF, Amphlett JC, Peppley BA, Roberge PR. Incorporation of voltage degradation into a

generalised steady state electrochemical model for a PEM fuel cell. J Power Sources 2002;106(1–2):

274–83.

[14] Berning T, Djilali N. Three-dimensional computational analysis of transport phenomena in a PEM fuel

cell—a parametric study. J Power Sources 2003;124(2):440–52.

[15] Chahine R, Laurencelle F, Hamelin J, Agbossou K, Fournier M, Bose TK, et al. Characterization of a

Ballard MK5-E proton exchange membrane fuel cell stack. Fuel cells 2001;1(1):66–71.

[16] Hamelin J, Agbossou K, Laperriere A, Bose TK. Dynamic behavior of a PEM fuel cell stack for stationary

applications. Int J Hydrogen Energy 2001;26(6):625–9.

[17] Cownden R, Nahon M, Rosen MA. Modelling and analysis of a solid polymer fuel cell system for

transportation applications. Int J Hydrogen Energy 2001;26(6):615–23.

[18] Rowea A, Li X. Mathematical modeling of proton exchange membrane fuel cells. J Power Sources 2001;

102(1–2):82–96.

[19] Yerramalla S, Davari A, Feliachi A, Biswas T. Modeling and simulation of the dynamic behavior of a

polymer electrolyte membrane fuel cell. J Power Sources 2003;124(1):104–13.

Modelling of Proton Exchange Membrane Fuel Cell Performance Based on Semi-empirical Equations

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