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Department of Physics and Electronics, Ovidius University, Constanta, 900527, Romania,
E-mail: [email protected]
Received July 31, 2013
A proton exchange membrane fuel cell, also known as polymer electrolyte membrane
fuel cell (PEMFC), becomes in the last decade a prime candidate for applications in
power systems, submarines and aerospace. The single-cell PEMFC consists of a carbon
plate, a gas diffusion layer (GDL), a catalyst layer for each of the anode and the cathode
sides, as well as a PEM membrane at the center. This model of the fuel cell, created
from Electrochemistry Module of Comsol Multiphysics software package uses current
balances, mass transport equations (Maxwell-Stefan diffusion for reactants, water and
nitrogen gas) and momentum transport (Darcy’s law for the gas flows) to simulate a
PEM fuel cell’s behavior. Darcy’s velocity field at GDL channels, water diffusive flux,
current density distributions at anode catalyst electrode and membrane electrolyte
potential distribution across the length of the cell for three models of PEM cell models
(different from the point of view of membrane width) were computed in order to
investigate the cell model performance.
Key words: diffusive flux, electrolyte potential, current density.
Proton exchange membrane fuel cells (PEMFC) [1,2] are highly efficient
power generators, achieving up to 50–60% conversion efficiency, even at sizes of a
few kilowatts. PEMFCs have zero pollutant emissions when fueled directly with
hydrogen, and near zero emissions when coupled to reformers. These attributes
make them potentially attractive for a variety of applications, especially in
stationary and mobile power generators and electric vehicles [3]. There are several
compelling technological and commercial reasons for operating H2/air PEM fuel
cells at temperatures above 100°C. Rates of electrochemical kinetics are enhanced,
water management and cooling is simplified, useful waste heat can be recovered,
and lower quality reformed hydrogen may be used as the fuel.
The performance of the fuel cell system is characterized by current-voltage
curve (i.e. polarization curve). The difference between the open circuit potential of
the electrochemical reaction and cell voltage occurs from the losses associated with
the operation. The corresponding voltage drop is generally classified in three parts:
Rom. Journ. Phys., Vol. 59, Nos. 3–4, P. 285–294, Bucharest, 2014
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i) activation over-potential caused by the electrochemical reactions, ii) ohmic drop
across the polymer electrolyte, iii) mass transfer limitations of reactants.
These associated losses dominate over different current density ranges. For
low current densities, the activation over-potential is dominant. For high current
densities, which are of particular interest for vehicle applications because of higher
power density, the mass transfer limitations dominates the losses. For moderate
current densities, the ohmic drop across the polymer membrane dominates.
Moreover, for high current densities, water starts to exist in liquid form leading to a
two-phase transport of reactants to reaction site, which is an additional transport
phenomenon of PEM Fuel Cell operation.
In the last years, numerical modeling of PEFCs has received much attention.
Many two- and three-dimensional models have been developed in which the
computational fluid dynamics (CFD) method has been rigorously coupled with
electrochemical phenomena [4–7]. Electron transport and heat-transfer phenomena
have also been incorporated [8–10].
Chiang and Chu [11] investigated the effects of transport components on the
transport phenomena and performance of PEM fuel cells by using a threedimensional model. The impacts of channel aspect ratio (AR) and GDL thickness
were examined. It was found that a flat channel with a small AR or a thin GDL
generates more current at low cell voltage due to the merits of better reactant gas
transport and liquid water delivery.
The goal of the present work is to create a 2D isothermal, steady state PEM
fuel cell model in perpendicular to the gas flow direction to investigate the
performance of fuel cell from the point of view of activation over-potential caused
by the electrochemical reactions at cathode, ohmic drop across the electrolyte
membrane and water diffusive flux distribution at anode for three different
geometries of the PEM cell model.
The modeled section of the fuel cell consists of three domains: an anode (Ωa),
a proton exchange membrane (Ωm), and a cathode (Ωc) as indicated in Fig.1.a. Each
of the electrodes (gas diffusion layers) is in contact with an interdigitated gas
distributor, which has an inlet channel (∂Ωa,inlet), a current collector (∂Ωa,cc), and an
outlet channel (∂Ωa,outlet). The same notation is used for the cathode side.
The 2D modeling geometry is shown in Fig.1.b representing a cross-section
along the parallel channels of an along-the-channel test fuel cell. It is assumed that
the channels on the anode and cathode side appear in the same cross-section.
Finite element method modelling of a high temperature PEM fuel cell
Oxygen (O2) or air and hydrogen (H2) are supplied via gas channels in
counter-flow and diffuse through the gas diffusion layers to the reactive layers
where the following oxidation/reduction reactions take place.
Fig. 1 – The 2D modeling geometry: a) sub-domain and boundary labels and b) model geometry.
On the anode, hydrogen oxidation occurs according to:
H 2 ⇒ 2 H + + 2e −
(no water molecules are assumed to be involved in the proton transport).
At the cathode, oxygen reacts together with the protons to form water according to:
O2 + 4 H + + 4e − ⇒ 2 H 2O
Protons reach the cathode directly through the membrane. Beside water at the
cathode side, heat is produced due to strong exothermic reaction.
Mass conservation or continuity equation tells that the change of mass in a
unit volume must be equal to the sum of all species entering or exiting the volume
in a given time period.
This law applies to the flow field plates, GDL and the catalyst layer.
Momentum conservation relates net rate of change of momentum per unit volume
due to convection, pressure, viscous friction and pore structure. This law applies to
the flow field plates, GDL and the catalyst layer. Species conservation relates the
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net rate of species mass change due to convection, diffusion and electrochemical
reaction. The most commonly used one is the Stefan-Maxwell diffusion equation
[12]. Charge conservation corresponds to the continuity of current in a conducting
material. This is applied to the GDL, catalyst layer and the membrane.
In this model set-up the following assumption are used: i) steady-state
operating conditions, ii) as the cell is operated at higher temperatures single-phase
water-flow is assumed, iii) since the velocity of the gases is low, laminar flow is
assumed, iv) all gases are treated as ideal, v) all electrochemical reactions are
gaseous phase reactions and there is no crossover of gases through the membrane,
vi) all material parameters are isotropic and homogenous.
COMSOL Multiphysics, which is a commercial solver based on the finite
element technique, is used to solve the governing equations. The stationary
nonlinear solver is used since the source terms of the current conservation equation
make the problem non-linear. Furthermore, the convergence behavior of this nonlinear solver is highly sensitive to the initial estimation of the solution. To
accelerate the convergence, the following procedures are adopted. The Conductive
Media DC module is firstly solved based on the initial setting. Secondly, Darcy’s
Law and Incompressible Navier-Stokes modules are solved together using the
results from the previous calculation as initial conditions. After the previous two
modules converge, all the coupled equations including Maxwell-Stefan Diffusion
and Convection module are solved simultaneously until the convergence is
The boundary conditions for the model in this study are as follows: i)
continuity at all internal boundaries, ii) no slip boundary condition for all channel
walls, iii) all initial values set to zero, iv) velocity and temperature defined at
channel inlet, step function used for these two parameters in time dependent study,
v) no backpressure at channel outlet, convective flux boundary conditions applied,
vi) constrain outer edges set to zero for both inlet and outlet, vii) bipolar plates on
the most side of the cell set to electric ground and cell operation potential, viii)
HTPEM fuel cell is insulated from environment.
Finite element method modelling of a high temperature PEM fuel cell
Using the aforementioned numerical procedures, the 2D model geometry as
described in Fig. 1.b was simulated at a single voltage of 0.7V, for different width
dimensions of PEM membrane: 50µm, 100µm and 200µm. From this point of
discussions, we will refer at three PEM cell models for witch the simulations are
computed, named CM1, CM2 and CM3 for 50µm, 100µm and 200µm PEM
membrane width cell geometries.
The cell operated at a constant temperature of 120°C, and the reference
pressure pref was 1 atm.
Nafion type membrane was modeled in this case, with the chosen value of 3
for electro-osmotic drag coefficient H2O/H+, determined previously at different
temperatures [13, 14].
The cell models dimensions are listed in Table 1. The physicochemical
parameters values that were used in this study are presented in Table 2.
Table 1
Cell geometry
Parameter name
Cell length
Current collectors height
Inlet and Outlet channels width
Inlet and Outlet channels height
GDL layers width
PEM membrane width
Table 2
Physicochemical parameters
Parameter name
GDL conductivity
GDL permeability
Nafion membrane conductivity
Fluid viscosity
Anode inlet pressure
Cathode inlet pressure
Exchange current density, anode
Exchange current density, cathode
Specific surface area
Effective binary diffusivity, H2_H2O
Effective binary diffusivity, O2_N2
Effective binary diffusivity, H2O_N2
Effective binary diffusivity, O2_H2O
Inlet weight fraction H2
1.1·pref [Pa]
1.1·pref [Pa]
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Table 2 (continued)
Inlet weight fraction O2
Cathode inlet weight fraction, H2O
Molar mass, H2
Molar mass, O2
Molar mass, H2O
Molar mass, N2
Inlet molar fraction, H2
Reference concentration, H2
Reference concentration, O2
Inlet molar fraction, O2
It is very important to know how reactants are transported into the membrane.
Figure 2 shows the velocity fields of hydrogen and oxygen at the anode and
cathode sides of flow channel at the operation condition of 0.7 V for the cell model
with 50µm PEM membrane width (CM1). These fields indicated the mass transport
at each section of the fuel cell very well. Figure 2 indicated that the flow-velocity
magnitude attains its highest values at the current collector corners; an almost
constant distribution of the field (at high values) are located close to the membrane
sides in the Y direction, which meant that hydrogen and oxygen were supplied to
the membrane by the diffusion mechanism.
Fig. 2 – Typically component of Darcy’s velocity field (m/s) at anode (left side) and cathode
(right side) GDL channels for CM1 cell model.
To ensure a fully hydrated membrane, fuel and oxidant (air) streams are fully
or partially humidified before entering the fuel cell. The ionic conductivity of the
proton-conducting membrane is strongly dependent on its degree of humidification,
or water content, with high ionic conductivities at maximum humidification.
Finite element method modelling of a high temperature PEM fuel cell
Fig. 3 – Water diffusive flux distribution (Kg/m s) at the anode catalyst layer
for the following cell models: a) CM1, b) CM2 and c) CM3.
In Figure 3 it was presented water diffusive flux distribution at the anode
electrode to show the degree of humidification of the PEM membrane in the
current collector area. From Fig. 3.a we can see that CM1 cell model had the
largest and most compact area for the maximum diffusive flux distribution in the
current collector zone, presenting in this way the highest level of humidification at
the PEM membrane interface. Spikes with the highest flux distribution were
present at the corners of the anode current collector area for all the cell models.
Figure 4 shows the plots of the current density distribution at the anodic
catalyst layer as a function of cell length for all the cell models.
Fig. 4 – Current density distribution plots at the anode electrode
for the cell models CM1, CM2 and CM3.
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The current density was uneven with the highest density in the cell’s upper
region, the cell model CM1 presenting the maximum variation domain of the
current density along the cell length, of about 1085 A/m2. This means that the
oxygen-reduction reaction rate in the cathode determines the current-density
distribution. From Figure 4 we can see that the maximum current density arises
close to the air inlet, probably because of the high concentration of hydrogen and
oxygen and high electro-osmotic mass flux at the inlet region.
Figure 5 shows the distribution of electrolyte membrane potentials across the
membrane length (y–coordinate). An obvious potential drop should be found
within the membrane due to the theoretical low proton conductivity of membrane.
From Figure 5 we observed that ohmic (resistance) losses are very small on both
anode and cathode electrodes in the case of cell model CM1, conclusion validated
by the smallest voltage drop (of 6.9 mV) across the electrolyte membrane for this
The activation potential is the potential difference above the equilibrium
value required to produce a current which depends on the activation energy of the
redox event. Reaction overpotential is an activation overpotential that specifically
relates to chemical reactions that precede electron transfer. This overpotential can
be reduced or eliminated with the use of homogeneous or heterogeneous
The electrochemical reaction rate and related current density is dictated by
the kinetics of the electrocatalyst and substrate concentration. A high overpotential
should be found on the cathode side of the PEM fuel cell due to the slow kinetic
reaction of oxygen reduction.
Fig. 5 – Membrane electrolyte potential distribution across the length of the cell (y direction)
for the cell models CM1, CM2 and CM3.
Finite element method modelling of a high temperature PEM fuel cell
As we can see from Fig.6, the cell model CM1 presented the smallest
overpotential drop of 7 mV along the cell length, suggesting a fastest
electrochemical reaction rate of the oxygen in this particular case.
Fig. 6 – Reaction overpotential distribution at the catalyst layer in cathode side
of the cell for the cell models CM1, CM2 and CM3.
A 2D numerical model simulating the performance of a PEM fuel cell
operating at 120oC has been successfully applied in this study, for three different
values of membrane width.
From Darcy’s velocity field distribution at the anode and cathode GDL
channels it was showed that hydrogen and oxygen were supplied to the PEM
membrane by the diffusion mechanism.
The most proper water diffusive flux distribution at the anode catalyst
electrode for a highest level of humidification at the PEM membrane interface was
established for CM1 cell model, with 50 µm membrane width.
The current density distribution at the anode electrode was determined by the
oxygen-reduction reaction rate, CM1 model presenting the maximum variation
domain of the current density along the cell length.
CM1 cell model showed the smallest ohmic loses by registering smallest
voltage drop (of 6.9mV) across the electrolyte membrane and also the smallest
overpotential drop along the cell length.
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