Download - 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.
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.
M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–15991590
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)
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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
K0.9514x2
0.00312x3
7.4!10K5x4
K0.000187The 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
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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)
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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$LHVH2
(22)
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Once the output voltage of the stack is determined for a given output current, the gross
output power is found as:
Wgross Z IVcell (23)
The output current is correlated with the hydrogen mass flow rate by the equation [17];
_mH2Z
IMWH2
2F(24)
Thus, the thermodynamic efficiency of the fuel cell can simplifies as follows;
Efc Z2VcellF
MWH2$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.
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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].
Fig. 4. Predicted polarization curve, efficiency and power density at 72 8C cell temperature and 1 atm reactant
pressures.
M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–15991596
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.
Fig. 5. Predicted polarization curve, efficiency and power density at 72 8C cell temperature and 5 atm reactant
pressures.
M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599 1597
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.
M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–15991598
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.
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