cell layer level generalized dynamic modeling of a pemfc stack using vhdl-ams language
TRANSCRIPT
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 1
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Cell layer level generalized dynamic modelingof a PEMFC stack using VHDL-AMS language
Fei Gao, Benjamin Blunier*, Abdellatif Miraoui, Abdellah El-Moudni
Transport and Systems Laboratory (SeT) – EA 3317/UTBM, University of Technology of Belfort-Montbeliard,
Rue Thierry Mieg, 90000 Belfort, France
a r t i c l e i n f o
Article history:
Received 21 December 2008
Received in revised form
8 April 2009
Accepted 25 April 2009
Available online 4 June 2009
Keywords:
Fuel cells stack
PEM fuel cells
Dynamic Modeling
Energy conversion
Experimental tests
Hardware design languages
* Corresponding author. Tel.: þ33 (0)3 84 58 3E-mail addresses: [email protected] (F. G
[email protected] (A. El-Moudn0360-3199/$ – see front matter ª 2009 Interndoi:10.1016/j.ijhydene.2009.04.069
a b s t r a c t
A generalized, cell layer scale proton exchange membrane fuel cell (PEMFC) stack dynamic
model is presented using VHDL-AMS (IEEE standard Very High Speed Integrated Circuit
Hardware Description Language-Analog and Mixed-Signal Extensions) modeling language.
A PEMFC stack system is a complex energy conversion system that covers three main
energy domains: electrical, fluidic and thermal. The first part of this work shows the
performance and the advantages of VHDL-AMS language when modeling such a complex
system. Then, using the VHDL-AMS modeling standards, an electrical domain model,
a fluidic domain model and a thermal domain model of the PEMFC stack are coupled and
presented together. Thus, a complete coupled multi-domain fuel cell stack 1-D dynamic
model is given. The simulation results are then compared with a Ballard 1.2 kW NEXA fuel
cell system, and show a great agreement between the simulation and experimentation.
This complex multi-domain VHDL-AMS stack model can be used for a model based control
design or a Hardware-In-the-Loop application.
ª 2009 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights
reserved.
1. Introduction produced from fossil fuels or directly from renewable energy
Fuel cells are now a very active research field as they are
considered as one of the principal candidates for the future
clean and renewable energy solution in the world. Fuel cells
are energy conversion devices that convert chemical energy
stored in the fuel directly into electricity. Compared with the
classical electricity production system (heat engine plus
electrical alternator), electricity generation from fuel cells
systems has three major advantages [1]: firstly, the process of
electrical production is environmental friendly. Most of the
fuel cell systems use hydrogen or reformed natural gas as fuel
supply, the principal byproduct is water, and the CO2 emission
can be minimized, or reduced to zero whether hydrogen is
3 98.ao), benjamin.blunier@i).ational Association for H
using water electrolysis. Secondly, a high efficiency of energy
conversion can be obtained: around 50% for the fuel cell stack
itself and 40–45% for the fuel cell system including all
auxiliaries power consumption [1]. Thirdly, a fuel cell system
can be very compact. Unlike other electricity generators which
usually have some intermediate energy conversion steps
(chemical to thermal, thermal to mechanical, mechanical to
electrical), a fuel cell energy conversion is much less complex
(chemical to electrical), so that the number of components in
a fuel cell system can be reduced.
In the last ten years, researches for proton exchange
membrane fuel cell (PEMFC) have made a great improvement.
A PEMFC stack uses a solid proton exchange membrane as
utbm.fr (B. Blunier), [email protected] (A. Miraoui),
ydrogen Energy. Published by Elsevier Ltd. All rights reserved.
Nomenclature
Greek letters
d thickness, m
DS entropy change, J/mol K
3 gas diffusion layer porosity
3i empirical activation losses parameters i ˛ {1.4}
3m emissivity
l(z) membrane water content
l thermal conductivity, W/m K
m dynamic viscosity, Pa s
r density, kg/m3
rdry membrane dry density, kg/m3
s gas diffusion layer tortuosity
s Stefan–Boltzmann constant, W/m2 K4
Roman letters
A section of fluid channels flow direction, m2
a water activity
Cp thermal capacity, J/kg K
Dh hydraulic diameter, m
Dij binary gas diffusion coefficient, m2/s
Dl membrane mean water diffusion coefficient, m2/s
E electromotive force, V
F Faraday constant, C/mol
fdarcy Darcy fiction factor
h heat transfer coefficient, W/m2 K
istack fuel cell stack current, A
layer modeling layer
L channel length, m
Lplate bipolar plate perimeter, m
M molar mass, kg/mol
m mass, kg
Mn membrane equivalent mass, kg/mol
Nu Nusselt number
nsat electro-osmotic coefficient
P pressure, Pa
Q heat flow, J/s
q mass flow rate, kg/s
R perfect gas constant
r resistivity, U m
Re Reynolds number
Rmem membrane resistance, U
Slateral channels lateral surface (perpendicular to
modeling axis), m2
Splate-ext layer bipolar plate external surface, m2
SSection layer section (modeling axis z), m2
T temperature, K
V volume, m3
Vx x-layer voltage, V
Vs velocity, m/s
z distance on axis z, m
Subscripts
A anode
act activation
cata catalyst layer
C cathode
center midpoint of control volume
ch gas channels layer
crit critical point
diff back-diffusion
drag electro-osmotic effect
fluid fluid
GDL gas diffusion layer
in inlet
internal internal source
layer modeling layer
L cooling layer
left left side
mass mass transfer
mem membrane layer
out outlet
right right side
solid solid
S channels support plate layer
tot total
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electrolyte, it has all the advantages of fuel cells mentioned
above. Moreover, compared to the other fuel cell technologies,
PEMFCs have some other benefits: PEMFCs run at low
temperature conditions (mostly 60–80 �C), which give them
a relatively short start time at room temperature; there is no
electrolyte leakage risks as for the alkaline fuel cells thanks to
its 100% solid polymer membrane. All these advantages show
the special interest of using PEMFC in automotive
applications.
Despite all of this, there are still some issues need to be
considered for a PEMFC system. On the one hand, its lifetime
should be increased up to 5000 h for automotive application
and its cost should be reduced down to 45 US$ per kilowatt
[2,3]. On the other hand, the PEMFC stack is an open system; in
order to maintain its normal operation conditions, an appro-
priate control strategy should be applied to the PEMFC stack
and the system auxiliaries (i.e., air compressor, cooling
system, supervision, power converters). Moreover a PEMFC
stack is such a compact device that the measurements of
many system state parameters become quite difficult or too
expensive. For automotive applications, this problem is
particularly a key issue for control purposes as the fuel cell
system runs continuously in transient conditions, and the
system state observability is a key factor for an efficient
control. One good example for this kind of issue is the water
content of each membrane in the stack: the proton exchange
membrane should be maintained and fully hydrated during
operation to achieve good proton conductivity. A dry
membrane, or even a partial dry membrane, can lead to an
abnormal increase of membrane resistance, thus a perfor-
mance loss of PEMFC stack. Under some conditions (transient
state for example), the water contents of membrane should be
taken into account for an appropriate control strategy.
However the typical thickness of Nafion membrane is from
Fig. 1 – Thermocouple representation.
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50 mm to 200 mm, at this scale the measurement for control
purpose becomes very difficult, very expensive or even
impossible.
One possible solution for this PEMFC system state observ-
ability problem is using a model based control method. This
brings up the need of an accuracy of PEMFC stack model
including spatial (at least 1-D) and dynamic behavior for
transient state control. Since the model can predict detailed
system state parameters based on geometric parameters and
physical material properties (very few empirical parameters),
such a model can also be used to improve the design of fuel
cell systems.
To build such a complex physical system a rigorous
methodology has to be applied: the model has to be modular
and should be read and improved easily by someone who did
not build the model. The overall model should include multi-
domains continuous-time functionalities, like electrical,
electrochemical, fluidic (e.g., channels, compressor) and
thermal parts. If a control method is applied to the fuel cell
system, the fuel cell model has to able to be coupled with
discrete-time functionalities for the control purpose (DSP
controllers, FPGA, etc.). Such a model covers a range of
modeling in different energy domains and spans levels of
abstraction from low-level devices that make the components
to the top-level functional unit. If we encompass such range of
views of digital, analog and mixed-signal systems, the
complexity we are dealing with is really high and it is not
possible to comprehend such complex systems in their
entirety without a proper methodology.
Methods which deal with the complexity have to be
found in order to design, with some degree of confidence,
components and systems that meet the requirements.
Systematic methodology of design using a top-down design
approach has to be applied. This methodology decomposes
the system design in a collection of components that
interact which can be decomposed until a level where we
have sufficient details.
For this purpose Hardware Description Languages (HDLs)
have been developed first for the electronic (digital) domain.
Later on, the modeling demands for other energy domains
have resulted in the development and standardization of
VHDL-AMS (IEEE standard Very High Speed Integrated Circuit
Hardware Description Language-Analog and Mixed-Signal
Extensions) language [4].
The aim of this paper is to present a VHDL-AMS language
based, cell layer level dynamic PEMFC stack model which
includes electrical, fluid and thermal multi-domains modeling.
This model is based on the model of Blunier and Miraoui [4], but
includes many improvements like the thermal part and the
spatial effects taking into account each cell of the stack indi-
vidually (1-D model). A short introduction to VHDL-AMS
language and its modeling method are presented at first. After
a review of the existing model of PEMFC, a VHDL-AMS PEMFC
stack model is presented in three separate parts considered as
sub-models that can be described separately: electrical model,
fluidic model and thermal model. From these sub-models,
a complete PEMFC stack model is developed. Then, the model
simulation results are compared and discussed together with
the experimental results obtained from a Ballard NEXA 1.2 kW
PEMFC system.
2. VHDL-AMS language presentation
2.1. Overview (Blunier and Miraoui [4])
The detailed VHDL-AMS language description has been
introduced by Blunier and Miraoui [4], The same authors
presented a geometry-based VHDL-AMS air scroll compressor
model in [5].
The VHDL-AMS is designed to fill a number of needs in the
process design:
� it allows description of the structure of a system, that is,
how it is decomposed into subsystems from different
disciplines and how those subsystems are interconnected,
� it allows the specification of the function of a system using
familiar programming language and equation forms,
� it allows the design of a system to be simulated before being
manufactured: designers can compare alternatives and test
for correctness without the delay and expense of hardware
prototyping. For example, a fuel cell stack model containing
few empirical parameters as presented in this paper can be
simulated together with its auxiliaries and control strategies
model, before the entire fuel cell system manufacturing,
� it allows the detailed structure of a design to be synthesized
from a more abstract specification, allowing designers to
concentrate on more strategic design decisions and
reducing time to market.
2.2. Multi energy domain modeling example
In order to provide a more demonstrative power of the VHDL-
AMS language, a simple example of a K-type (Chromel–Alumel
electrode) thermocouple model is presented in this section.
A thermocouple covers electrical and thermal domains: it
converts a temperature difference to an electrical potential
difference. A simple representation of a thermocouple is
illustrated in Fig. 1.
The corresponding VHDL-AMS model program is the
following:
1 library IEEE;
2 use IEEE.THERMAL_SYSTEMS.all;
3 use IEEE.ELECTRICAL_SYSTEMS.all;
4 entity Thermocouple is
5 –generic model constant parameter:
6 generic(
7 –Cold junction temperature, default value is 24 �C
8 Tc: temperature¼ 297.15);
9 –model physical nodes:
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10 port(
11 –electrical nodes:
12 terminal E1, E2: electrical;
13 –thermal nodes:
14 terminal T: thermal);
15 end entity Thermocouple;
16 architecture K_type of Thermocouple is
17 quantity d_V across i through E1 to E2;
18 quantity Th across T;
19 begin
20 physical relation:
21 Th�Tc¼¼ 0.2266þ 24152.109*(d_V)þ 67233.4248*
(d_V**2)þ 2210340.682*
(d_V**3)�860963914.9*(d_V**4)þ 4.83506*(10**10)*(d_V**5);
22 end architecture K_type;
The first three lines indicate the energy domain libraries
needed for the model. From line 4 to line 15 is the physical
entity declaration. A general type thermocouple is defined
here: it has one model constant parameter (cold junction
temperature) and three physical nodes. Tc is a temperature
type constant and can be set to any value in Kelvin when using
the model; E1 and E2 represent two wire-end nodes of elec-
trical nature, which has voltage and current properties;
T stands for the hot junction of thermal nature, which has
temperature and heat flow properties. The thermals nodes or
terminal which have the same physical nature can be con-
nected together in order to be coupled with other models.
From line 16 to line 22 is the architecture declaration named
K_type associated to the declared entity. The architecture
describes the physical behavior of an entity, each terminal
needs to be associated with across or/and through quantities,
which represent the physical properties of terminal (lines 17
and 18). At last, a set of differential and/or algebraic equations
is given to describe the relations between these quantities. For
the thermocouple example, the equation describes the rela-
tion between the voltage difference and the temperature
difference in a polynomial form for a K-type thermocouple.
Modeling with VHDL-AMS language has two particular
advantages. Firstly, an entity can contain different architec-
tures. In fact, the entity represents the physical nodes. In the
given example, the thermocouple entity is defined for an
ordinary thermocouple without special type. The architecture
is presented for a K-type thermocouple, but the other type of
thermocouple, like an E-type (Chromel–Constantan electrode)
or a T-type (Copper–Constantan electrode), can be also defined
in a different architecture for the same entity, because they
have the same kind of physical nodes. Furthermore, for
a given entity, the models can have different levels of
complexity, and can be defined in different architectures: one
can have a very detailed equation system description for
a very accurate simulation (e.g., physical-based model); and
others can have a simple level description (e.g., behavioral
description) in order to speed up the simulation, if it does not
need a high precision of the model.
Secondly, as shown in the thermocouple example, the
physical equation can be written in an implicit form. For the
thermocouple model, if the temperature is given, the model
will give the voltage difference as a calculated result. In that
case, the voltage difference does not need to be expressed by
an explicit function of the temperature. Because in VHDL-AMS
language, the terminal quantities do not need to be prefixed
whether it is an unknown system, the only requirement of
a system of equations is that the number of unknown
matches the number of independent equations, in order to
ensure that the system of equation is not under determined.
The possibility to write the physical equation in its nature
form makes the VHDL-AMS program easier to be understood.
3. Proton exchange membrane fuel cell stackmodel
3.1. Preliminary study for the needs of modeling
The mathematical fuel cell models can be divided into four
categories: zero-dimensional (0-D), one-dimensional (1-D),
two-dimensional (2-D) and three-dimensional (3-D). These
models are mainly physical-based models. Other models,
called empirical or behavioral models based on interpolated
maps or functions [6,7], sometimes used to simulate the
behavior of the fuel cell without a priori physical knowledge of
the fuel cell. However these models do not put forward
physical comprehension of the fuel cell systems. The further
presented models are physical-based models. The first fuel
cell models published in earlier time are simple zero-dimen-
sional or one-dimensional models [8–10]. In recent years,
thanks to the growing calculation power of computers, the
Computational Fluid Dynamic (CFD) methods have been intro-
duced in fuel cell modeling. These numerical methods (finite
differences, elements and volumes) enable a cell model to be
realized in two or three dimensions [11,12].
Zero-dimensional modeling is generally the simplest, but it
does not allow local phenomena to be studied. Multidimen-
sional modeling using CFD methods allows very accurate
results to be obtained. These models can be very useful in fuel
cell design. Indeed, these simulations enable the spatial
distributions for different quantities to be visualized
(temperatures, fluid velocities, liquid water, etc.) and help
improving bipolar plate geometry, for example, as well as
optimizing water management or studying the reliability and
durability of the fuel cell.
Even though a 2-D model or 3-D gives more detailed
information and precision, but for the fuel cell gas channel or
bipolar plate modeling, the particular flow field geometry
parameters are needed. These parameters are the property of
the fuel cell manufacturers, and are very difficult to be
obtained. Thus, it can be very difficult, without those param-
eters, to build precise two- or three-dimensional models. In
contrast, a 1-D model requires less parameter, so that it can be
used in a more general way.
To model the physical phenomena of the fuel cell, despite
the unparalleled accuracy of the 2-D and 3-D models, the 1-D
model still displays advantages over the other multidimen-
sional models:
� the equation number is reduced and the resolution of the
model is greatly speeded up as a result. This calculation
speed opens up the possibility of using the model in a real-
time application;
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� the accuracy of the simulation results of the model is highly
acceptable for most applications such as feed-forward
control, Hardware-In-the-Loop (HIL) applications, system
diagnosis, etc;
� there is a wide choice in terms of the implementation
method of the model: Matlab, C language, bond graphs,
VHDL-AMS, Fortran, etc.; in contrast, a 3-D model must be
implemented in dedicated business software for CFD
models like Comsol, Fluent, Ansys, etc.
In order to simulate the transient state of the fuel cell, such
as room temperature start or rapid load change, the model
should be able to predict the dynamic behavior of the stack.
Thus, a dynamic model is needed.
At last, the model is developed to be able to represent
different PEMFC stacks with different geometries and
different physical characteristics. The model possesses a set
of adjustable parameters which permit different fuel cell
stacks to be simulated. This is done by setting the adjustable
parameters into the ‘‘generic’’ fields of VHDL-AMS language,
as shown in the thermocouple example. In order to simulate
PEMFC stacks with the same VHDL-AMS program, only these
generic parameters need to be modified by using the appro-
priate values of the PEMFC stack (physical properties,
geometry values, etc.). We must notice that, these parame-
ters are not ‘‘empirical parameters’’ because they represent
geometry of the physical properties of the specific fuel cell
stack. These adjustable parameters, contrarily to empirical
parameters, do not need to be identified from stack experi-
mental tests.
In this paper, a one-dimensional, cell layer scale, dynamic
proton exchange membrane fuel cell stack model is
presented.
3.2. State of art of PEMFC modeling
Among the first PEMFC models, Springer et al. [8] have pre-
sented an isothermal, one-dimensional and steady state
model of a single PEMFC. Their model takes into account only
the membrane electrode assembly (MEA) and cannot there-
fore represent the stack as a whole.
Bernardi and Verbrugge [9] have put forward another 1-D
single PEMFC model enabling analysis of the factors that can
limit the performance of the cell. However, this model is
always in isothermal and steady state conditions.
Amphlett et al. [10] have developed a 1-D model of a single
cell by combining a mechanistic part and an empirical part.
But the fluid channels and cooling are not considered.
Alongside the steady state models, Amphlett et al. [13]
have introduced for the first time a 1-D dynamic model of the
fuel cell. The dynamic part is modeled for the complete stack.
Thus, the non-uniform distribution of the stack temperature
cannot be studied.
Wohr et al. [14] have put forward a more complete 1-D
thermal model. In their model there are no cooling channels
between the cells and a sharp temperature gradient is
therefore seen throughout the stack during the operation of
the cell.
Mann et al. [15] have introduced a generic 1-D model that
can be applied to different PEM cells with different
characteristics. However, this remains an isothermal and
steady state model. Moreover, it only includes the electrical
domain.
Baschuk and Li [16] have presented a 1-D model to study
the flooding phenomena in the fuel cell. Still the dynamic of
the cell is not introduced in this model.
Djilali and Lu [17] have presented yet another 1-D fuel cell
model, focusing on the modeling of non-isothermal and non-
isobaric effects. But this remains, nevertheless, a steady state
model; it cannot simulate the evolution of temperature
transients.
Wang et al. [18] have presented a complete 3-D fuel cell
model. Despite the complexity of their model, the experi-
mental validations are only applied to fuel cell polarization
curve of a single cell.
Yan et al. [19] have put forward a model of the fuel cell
membrane. Heat and water management in the membrane
are presented in detail. In their model the phenomena in the
fluid channels and diffusion layers are not studied.
Xue et al. [20] have put forward a dynamic, non-isothermal
1-D model of the fuel cell. The model is validated experi-
mentally for a cell. But the phenomenon of water transfer in
the membrane is highly simplified.
Shan and Choe [21] have presented a detailed 1-D model of
the fuel cell. The steady state and dynamic performances of
the stack are analyzed, but the simulation results are not
validated experimentally.
Bao et al. [22] have introduced a control-oriented fuel cell
system model. A fuel recirculation model and a compressor
model are coupled with a fuel cell model. However, their
model remains isothermal and in steady state.
Especially investigated in the intermediate temperature
PEMFC, Cheddie and Munroe [23] have presented a corre-
sponding two-phase 2-D model. The model predicts correctly
the polarization performance at 150 �C and 170 �C.
Lin et al. [24] have presented another 2-D PEMFC model
with consideration of axial convection in the gas channel.
However, their model is isothermal and in steady state, with
the fully hydrated membrane assumption.
Blunier and Miraoui [4] have introduced a method of fuel
cell modeling using VHDL-AMS language. Their cell model
comprises of the fluid and electrical domains. The perfor-
mance of the stack is obtained from the model of a single cell.
However, the model presents limitations as it is considered
isothermal. Part of the thermal domain must be added into the
model to have a more complete model.
Park and Choe [25] have developed a dynamic model of
a 20-cell stack. However, mechanical losses in the supply
channels and water condensation in the form of liquid in the
channels are not taken into account.
Rismanchi and Akbari [11] have introduced a 3-D fuel cell
model using CFD method. But only a small area of electrode is
taken into account for the steady state simulations.
Jeon et al. [26] have also presented a CFD method based fuel
cell model. Their model is more concentrated in the gas flow
fields modeling, 4 different types of serpentine channels are
presented and discussed.
From the experimental identification and model adjust-
ment method, Kunusch et al. [27] have introduced a simple
linear state-space fuel cell model for a 7-cells stack. This kind
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of model can be used in control purpose, but the fuel cell non-
linearity cannot be investigated.
More recently, Sahraoui et al. [12] have presented another
2-D fuel cell model using CFD method. The optimum param-
eters for fuel cell high performance are investigated. However,
their model remains in steady state.
3.3. VHDL-AMS fuel cell stack model structure(Top-down approach)
3.3.1. Stack modelThe model of the complete stack represents the ultimate
objective of the work. In certain models of the literature the
model of the stack is based on the model of a single cell called
the mean equivalent cell, meant to represent the mean
performance of all the cells. The quantities of the stack are
simply obtained by multiplying the quantities of the mean
equivalent cell by the number of cells. For example, for a stack
of n cells, the total voltage will be equal to n times the voltage
of the mean equivalent cell. This hypothesis supposes that the
physical conditions for each cell should be identical, which is
not the case in physical reality.
In reality, each cell in the stack has its own physical
conditions and boundary conditions (different temperatures,
mass flows, resistance, etc.). The boundary conditions of a cell
will influence those of the adjacent cell, which will thus create
the different physical conditions in each cell. These differ-
ences are seen in every domain of the modeling, but especially
in the thermal domain due to the large thermal time constant
of the stack.
In this model the stacking method for building the model of
the stack has been used (see Fig. 2). The stack model is obtained
from the individual cell model (top-down), each cell model is
stacked one after another. The physical conditions of the cell n
are calculated from cell n� 1 and cell nþ 1. The physical
equations of each cell have the same formula (same cell model),
but they are interconnected and have different boundary
conditions. This represents the physical reality of the stack.
3.3.2. Individual cell modelWith the top-down approach of VHDL-AMS language, in the
individual cell model, a cell is then broken down into several
different ‘‘elementary layers’’ according to their position,
geometry and functionality (see Fig. 3).
In the model, each layer is associated with a system of
mathematical equations describing the relevant physical
1st cell 2nd cell 3rd cell
Fig. 2 – Elementary n cells ma
behaviors. The boundary conditions are given by the two
adjacent layers.
3.3.3. Elementary layer model structureEach elementary layer is divided again into three different
physical domains in VHDL-AMS modeling with the associated
physical terminals (nodes), as illustrated in Fig. 4:
Each domain is presented by a VHDL-AMS program bloc
containing the physical differential equations and algebraic
equations of the specific domain. The only link between them
is the value exchanges. For example, in order to calculate the
activation losses of a cell (located in the ‘‘electrical domain’’
bloc), the gas pressure is needed. These gas pressures are
calculated in the ‘‘fluidic domain’’ bloc and are sent as an
exchangeable value to the ‘‘electrical domain’’ bloc.
The advantage of this top-down structure is that each
domain has its own equation systems and can be described
separately regardless the others. If an improvement of equa-
tions is needed in a specific domain in the future, only the
concerned program bloc is need to be revised.
This modeling structure is held in the detailed modeling
sections hereafter.
3.4. Modeling hypotheses
Like any other mathematical models, some hypotheses are
used when modeling the PEMFC stack. These hypotheses are
given below.
H 1. The pressure drop in the channels is only due to the
mechanical losses of the gas crossing the straight channels
and to the flow of mass towards the gas diffusion layer. The
pressure drop due to local channel inflection and the pressure
drop due to water/vapor two-phase flow are neglected.
H 2. Water does not come out of the channels in the liquid
phase but only in the vapor phase. The water in liquid phase
is only considered for vapor saturation and pressure
computation.
H 3. Gas diffusion in the diffusion layers, catalyst layers and
membrane is considered in steady state.
H 4. There is no total pressure gradient in the diffusion layer.
The convective mass transport due to total pressure gradient
is neglected.
Nth cell
Last cooling layer
king up the entire stack.
Coo
nil gnil et
Ano dein
el tAnod e
outel t
Cathod e
ouelt t
Coo
il ngoutl et
Catho de
niel t
Fig. 3 – Structure of a basic cell of fuel cell stack.
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H 5. The voltage drop associated with activation losses is
negligible at the anode [28,29].
H 6. The ohm losses are considered only in the membrane
and the electrical resistance of the bipolar plate is negligible.
H 7. The gases in the cell layers are considered as perfect
gases.
H 8. The thermal capacity and thermal conductivity of each
layer remain unchanged during the stack operation.
H 9. The cell geometries remain unchanged during the stack
operation.
VHDL-AMS terminals to beconnected with others
Elementary layer
Electrical domainPhysical equations systems
Fluidic domainPhysical equations systems
Thermal domainPhysical equations systems
Current(s)
Voltage(s)
Pressure(s)
Mass flowrate(s)
Heat flow(s)
Temperature(s)
Values exchange
Values exchange
Fig. 4 – Representation of a multi-physical elementary
layer.
H 10. Gas pressure losses in the catalyst layer are negligible
(both anode and cathode sides).
3.5. Cell electrical domain modeling
3.5.1. Cooling channels, gas supply channel supports, gassupply channels and gas diffusion layersFollowing hypothesis H 6, the electrical resistance of these
layers (cathode side and anode side) is equal to 0.
3.5.2. Catalyst layersCathode side. From the temperature of the catalytic site TC,cata
(K) and the oxygen partial pressure PC;cata;O2 ðPaÞ the cathode
contribution to the electromotive force E can be calculated
[30]:
EC ¼ 1:229� 0:85� 10�3ðTC;cata � 298:15Þ
þ RTC;cata
2Fln
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPC;cata;O2
101325
r !(1)
where R¼ 8.314 is the perfect gas constant (J/mol K) and
F¼ 96,485 the Faraday constant (C/mol).
Indeed, the activation losses VC,act due to electrochemical
reactions [10] are:
VC;act ¼ 31 þ 32TC;cata þ 33T ln
PC;cata;O2101325
5:08� 106$e�ð 498
TC;cataÞ
!
þ 34TC;cata lnðistackÞ (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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 1 5505
where 31, 32, 33 and 34 are four empirical parameters, which
need to be identified from the stack static polarization curve,
istack the stack current (A).
The total voltage of the cathode catalyst layer is the sum of
cathode contribution to the electromotive force and activation
losses.
Anode side. From the temperature of the catalytic site TA,cata
(K) and the partial pressure of hydrogen PA;cata;H2 ðPaÞ, the
anode contribution to the electromotive force (V) can be
calculated [30]:
EA ¼RTA;cata
2Fln
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPA;cata;H2
101325
r !(3)
According to hypothesis H 5, the activation losses of the
anode side are negligible.
Membrane. From the water content of the membrane
being given in terms of the position on the Z-axis:
lðzÞ z˛½0ðanodeÞ; dmemðcathodeÞ� and the membrane
temperature Tmem (K), the resistivity of the membrane (U m)
can be calculated [8]:
rðTmem; lðzÞÞ ¼
8>>><>>>:
10:1933e
h1268ð 1
Tmem� 1
303Þi
if 0 < lðzÞ � 1
10:5193lðzÞ�0:326e
h1268ð 1
Tmem� 1
303Þi
if lðzÞ > 1
(4)
The total resistance of the membrane (U) is obtained by inte-
grating the resistivity over the thickness of the membrane dmem
(m), and then divided by its section surface Smem,Section (m2).
Rmem ¼R dmem
0 rðTmem; lðzÞÞdz
Smem;Section(5)
The expression of the water content l(z) varies according to
whether the stack current is different from zero (istack s 0) or
whether the current is zero (istack¼ 0). These two cases must
be considered separately in the calculation of the resistance of
the membrane.
If istack s 0: The expression of the resistivity of the
membrane is a piecewise function. This resistivity no longer
depends on the water content l(z), if l(z) is lower than 1. By
looking at the expression of l(z) (see fluidic domain modeling
equation (45)), the water content in the membrane is a z
monotonic function.
If one of the l(0) or l(dmem) is less than 1 and the other is
greater than 1, in order to calculate the integral with piecewise
function, the critical point of the distance zcrit (m) as l(zcrit)¼ 1
must be found first from equation (45):
zcrit ¼BA
ln
ð1� lð0ÞÞeA
Bdmem � ð1� lðdmemÞÞlðdmemÞ � lð0Þ
!(6)
with
A ¼ nsat$MH2O$istack22F
B ¼ rdry$Smem;Section$DClD
Mn
where nsat z 2.5 is the coefficient of electro-osmosis for
maximum hydration conditions [8], MH2O the molar mass of
the water (kg/mol), rdry the dry density of the membrane (kg/
m3), DClD
the mean water diffusion coefficient in the membrane
(m2/s), and Mn the equivalent mass of the membrane (kg/mol).
After integration of equation (5), the resistance of the
membrane (U) is obtained:
Rmem ¼1
Smem;Section
"Ba3
Aa1
"ABðBn2 � Bn1Þ � ln
a1 þ a2e
ABBn2
a1 þ a2eABBn1
!
þ a3Dzcrit
0:1933
##(7)
with
a1 ¼ 0:5193lð0ÞeA
Bdmem � lðdmemÞe
AB
dmem�1� 0:326
a2 ¼ 0:5193,lðdmemÞ � lð0Þ
eAB
dmem�1
a3 ¼ e
h1268
�1
Tmem� 1
303
�iand8>><>>:
Bn1 ¼0 Bn2 ¼ d Dzcrit¼0 if lð0Þ> 1 and lðdmemÞ>1Bn1 ¼0 Bn2 ¼ zcrit Dzcrit¼ dmem�zcrit if lð0Þ>1 and lðdmemÞ�1Bn1 ¼ zcrit Bn2¼ d Dzcrit¼ zcrit if lð0Þ� 1 and lðdmemÞ> 1Bn1 ¼0 Bn2 ¼0 Dzcrit ¼ dmem if lð0Þ�1 and lðdmemÞ�1
where l(0) is the membrane water content of the anode side,
and l(dmem) the membrane water content of the cathode side.
If istack¼ 0: The expression of the critical point of the
distance zcrit (m) in this case is calculated by (see fluidic
domain modeling equation (45)):
zcrit ¼ð1� lð0ÞÞ$dmem
lðdmemÞ � lð0Þ (8)
As above, the resistance of the membrane (U) after inte-
gration of equation (5) is given:
Rmem ¼1
Smem;Section$
�a3
a5
�ln
�a4 þ a5$Bn2
a4 þ a5$Bn1
��þ a3$Dzcrit
0:1933
�(9)
with
a4 ¼ 0:5193$lð0Þ � 0:326
a5 ¼ 0:5193$lðdmemÞ�lð0Þdmem
(10)
The values of Bn1, Bn2, Dzcrit are determined with the same
condition where istack s 0.
Finally, thevoltage Vmem canbegiven according toOhm’sLaw:
Vmem ¼ �Rmem$istack (11)
3.6. Cell fluidic domain modeling
3.6.1. Cooling channelsThe volume of the cooling channels is considered as a control
volume in the model.
First, the Reynolds number of the fluid in the channels can
be calculated in the following way [31]:
Re ¼ rVSDh
m(12)
where r is the fluid density (kg/m3), VS the mean fluid velocity
in the channels (m/s), Dh the hydraulic diameter of the
channels (m) and m the fluid dynamic viscosity (Pa s).
Based on hypothesis H 7 and the perfect gas law, if the fluid
in the cooling channels is of a gaseous type, the density of the
gas (kg/m3) is given by:
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 15506
r ¼ MPRT
(13)
If the coolant is in liquid form (e.g., water), the density of the
coolant is considered constant (no compressible fluid).
The fluid mean velocity (m/s) can be calculated from its
mass flow q (kg/s), its density r (kg/m3) and the total section of
the channels A (m2) (in the flow direction):
VS ¼q
rA(14)
The dynamic viscosity m (Pa s) can be calculated from
Sutherland empirical formula [32].
The pressure drops in the channels due to the global
mechanical losses are calculated using the Darcy–Weisbach
equation [33]:
DPk ¼ fdarcyrL2D
V2S;k k˛fin;outg (15)
withDPin ¼ PL;fluid;in � PL;fluid;center
DPout ¼ PL;fluid;out � PL;fluid;center
where fdarcy is the Darcy friction factor and L the length of the
channel (m).
The Darcy friction factor fdarcy can be obtained from the
empirical equations [33].
The dynamic behavior of the fluid is given by the mass
balance of the control volume:
V
�ddt
r
�¼ qL;fluid;in þ qL;fluid;out (16)
where V is the volume of the channels (m3) and r the fluid
density in the channels (kg/m3).
3.6.2. Gas supply channelsThe volume of the gas supply channels is considered as
a control volume in the model.
The total pressure of the midpoint of the channels (control
volume) (Pa), noted PC,fluid,center, is obtained by:
For cathode:
PC;fluid;center ¼ PC;O2 ;GDL þ PC;N2 ;GDL þ PC;H2O;GDL (17)
For anode:
PA;fluid;center ¼ PA;H2 ;GDL þ PA;H2O;GDL (18)
where PC;O2 ;GDL;PC;N2 ;GDL and PC;H2O;GDL are respectively, the
oxygen, nitrogen and vapor pressures at the cathode GDL
interface, PA;H2 ;GDL and PA;H2O;GDL are, respectively, the hydrogen
and vapor pressures at the anode GDL interface.
The pressure drops in the channels are calculated from (15)
Considering the hypothesis H 7, the dynamic of the fluid in
the gas supply channels is given by the mass balances of each
species in the control volume:
ddt
ml;gi;species;GDL ¼ qi;species;in þ qi;species;out þ qi;species;GDL (19)
mgi;species;GDL ¼
Pi;species;GDLVMspecies
RTi;ch(20)
with i ˛ {C (cathode), A (anode)}, species ˛ {O2, N2, H2O} at the
cathode side and species ˛ {H2, H2O} at the anode side.
qi,species,in, qi,species,out and qi,species,GDL are, respectively, the
mass flows of the species at the inlet, the outlet and the GDL
interface of the gas supply channels (kg/s); ml;gi;species;GDL the
mass of species in the channels (sum of liquid and gas form),
mgi;species;GDL the mass of species in gas form in the channels.
The gas mass flows toward gas diffusion layers are
imposed by the chemical reaction rate according to the stack
current of the fuel cell.
To take into account the water in liquid form in the chan-
nels, considering hypothesis H 2, the vapor pressure
Pi;H2O;GDLðPaÞ in the gas supply channels should be verified by
the following equations:
mgi;H2O;GDL ¼
(ml;g
i;H2O;GDL if ml;gi;H2O;GDL � msat
i;H2O
msati;H2O elsewhere
(21)
msati;H2O ¼
Psati;H2O$V$MH2O
RT(22)
with i ˛ {C (cathode), A (anode)}, Psati;H2O is the vapor saturation
pressure in the gas supply channels (Pa).
This saturation vapor pressure (Pa) is given from the
temperature T (K) in the channels [8]:
log10
Psat
H2O
105
!¼�2:1794þ0:02953ðT�273:15Þ�9:1837
�10�5ðT�273:15Þ2þ1:4454�10�7ðT�273:15Þ3 ð23Þ
It must be noted that the sum of the pressures Pi,species,GDL in
gas form is equal to the total pressure of the gas in the gas
supply channels (see (17) and (18)).
The mass flows of the species at the inlet or at the outlet of
the gas supply channels are obtained following the species
composition of the inlet or the outlet gas. An example of equa-
tions is given below only for the cathode inlet gas supply chan-
nels, similar equations can be applied to outlet and anode side.
Example for cathode inlet gas supply channels:
The inlet gas mass flows are obtained following the type of
oxidant (air, pure oxygen):
qC;O2 ;in ¼ qC;fluid;inkO2
PC;fluid;in � PC;H2O;in
�MO2
Cin(24)
qC;N2 ;in ¼ qC;fluid;in
kN2
PC;fluid;in � PC;H2O;in
�MN2
Cin(25)
qC;H2O;in ¼ qC;fluid;in$PC;H2O;in$MH2O
Cin(26)
with
Cin ¼ kO2$PC;fluid;in � PC;H2O;in
�$MO2
þ kN2$PC;fluid;in
� PC;H2O;in
�$MN2
þ PC;H2O;in$MH2O
andkO2¼ 0:22 kN2
¼ 0:78 if airkO2¼ 1 kN2
¼ 0 if pure oxygen
3.6.3. Gas diffusion layers (GDLs)According to hypothesis H 3, the dynamics in the gas diffusion
layers are not considered, that is, the inflow of each species is
equal to its outflow.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 1 5507
The phenomenon of gas diffusion of each species i in the
GDL is described by the Stefan–Maxwell equation [34]:
PiðdGDLÞ � Pið0Þ ¼dGDL$R$Ti;GDL
Ptot$SGDL;Section
Xjsi
Pi$qj
Mj� Pj$
qiMi
Dij(27)
For the cathode: i ˛ {O2, N2, H2O} and for the anode: i ˛ {H2,
H2O}; j stands for species other than species i and Dij the binary
diffusion coefficient between the species i and j (m2/s).
This set of N (N¼number of species) Stefan–Maxwell
equation represents N� 1 independent equations. Conse-
quently, a further condition must be added to ensure the
unique solution. Based on hypothesis H 4:Xi
Pið0Þ ¼X
i
PiðdGDLÞ (28)
For the cathode: i ˛ {O2, N2, H2O} and for the anode:
i ˛ {H2, H2O}.
The binary diffusion coefficient between species i and j
(m2/s) depends on the porosity 3 of the GDL and the tortuosity s
of the GDL, according to Slattery and Bird’s Gas Law and using
the Bruggemann correction [35]:
Dij ¼10:1325
Ptot
$a$
Tffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Tci$Tcj
p!b
$
�Pci$Pcj
1013252
�1=3
$Tci$Tcj
�5=12$
�10�3
Mi
þ 10�3
Mj
�$3s
(29)
The critical temperature Tc and pressure Pc of the gas are
given in Table 1 [35].
The coefficients a and b differ whether one of the species is
a polar gas or not [35].
If the pair of gases contains no polar gas:
a¼ 2.745� 10�4 and b¼ 1.823
If the pair of gases contains a polar gas (vapor in this case):
a¼ 3.640� 10�4 and b¼ 2.334
3.6.4. Catalyst layersAccording to hypothesis H 3, the dynamic mass conservation
in the catalyst layers is not considered.
For the cathode side. The following electrochemical reaction
takes place in this layer:
H2 þ12O2/H2Oþ electricityþ heat (30)
According to hypothesis H 3, the mass balance of water is:
qC;H2O;GDL þ qC;H2O;prod þ qC;H2O;mem ¼ 0 (31)
Table 1 – Critical properties of the gas.
Gas TC (K) PC (1.01325� 105 Pa)
Hydrogen 33.3 12.8
Air 132.4 37
Nitrogen 126.2 33.5
Oxygen 154.4 49.7
Water 647.3 217.5
The mass flows of the gases (kg/s) are imposed by the stack
current istack (A). The mass flow of oxygen (kg/s) at the cathode
side is
qC;O2 ;GDL ¼MO2
$istack
4F(32)
and the mass flow of water produced (kg/s) due to the elec-
trochemical reaction is
qC;H2O;prod ¼MH2O$istack
2F(33)
The mass flow of water through the membrane
qC;H2O;memðkg=sÞ is imposed by the membrane and, according to
hypothesis H 10,
PC;H2O;GDL ¼ PC;H2O;mem (34)
For the anode side. According to hypothesis H 3, the mass
balance of water is
qA;H2O;GDL þ qA;H2O;mem ¼ 0 (35)
and the mass flow of hydrogen (kg/s) is imposed by the stack
current istack (A) according to
qA;H2 ;GDL ¼MH2
$istack
2F(36)
The water mass flow through the membrane qA;H2Oðkg=sÞ is
imposed by the membrane and, according to hypothesis H 10,
PA;H2O;GDL ¼ PA;H2O;mem (37)
3.6.5. MembraneAccording to hypothesis H 3, the dynamic mass conservation
in the membrane is not considered. The water content in the
Nafion membrane is not uniform [28,29].
The water content coefficient l is defined as the relation-
ship of the number of water molecules per charged site
(sulphonate site). To calculate the water distribution in the
membrane, the boundary conditions of water content at the
cathode and the anode sides must be calculated.
The equation giving the local water content is given by
[28,29]:
l ¼
0:0043þ 17:81aH2O � 39:85a2H2O þ 36a3
H2O if 0 < aH2O � 114þ 1:4
aH2O � 1
�if 1 < aH2O � 3
(38)
where aH2O is the water activity, computed from the water
local vapor partial pressure PH2OðPaÞ and the local vapor
saturation pressure Psat (Pa) [28,29]:
aH2O ¼PH2O
Psat(39)
where the saturation pressure is calculated from (23).
Using (38) and (39), the water content at the cathode side
l(dmem) and the anode side l(0) can be calculated based on the
vapor pressure of each side.
Then, the mean water content is given:
ClD ¼ lð0Þ þ lðdmemÞ2
(40)
In the membrane two antagonistic phenomena called the
electro-osmotic drag and the back-diffusion are distinguished
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 15508
The electro-osmotic drag occurs when the protons are sub-
jected to an electrical field. Under an electric field, the protons
migrate through the pores of the membrane (from the anode
to the cathode) and drag them with one or several water
molecules. This mass flow (kg/s) can be expressed by [28,29]:
qH2O;drag ¼nsat$lðzÞ
11istack
2F$MH2O (41)
The back-diffusion phenomenon occurs when the concen-
tration of water at the cathode side is higher than the
concentration of water at the anode side, the water diffuses
from the cathode to the anode. This phenomenon counter-
balances the effect of the electro-osmotic drag. This mass flow
(kg/s) can be expressed by [28,29]:
qH2O;diff ¼ �rdry
Mn$D
ClD$dlðzÞ
dz$Smem;Section$MH2O (42)
The mean water diffusion (m2/s) coefficient is given from
the membrane temperature Tmem (K) and the mean water
content ClD (0� l� 22) [36]:
DClD¼ 10�4$e
�2416ð 1
303� 1Tmem�
CdðClDÞ (43)
with
CdðClDÞ ¼
8>><>>:
10�6 if ClD < 210�6ð1þ 2ðClD� 2ÞÞ if 2 � ClD � 310�6ð3� 1:67ðClD� 3ÞÞ if 3 < ClD < 4:51:25� 10�6 if ClD � 4:5
By determining the mass balance of water with hypothesis
H 3 the total mass flow of water (kg/s) in the membrane can be
computed:
qH2O;net ¼ qH2O;drag þ qH2O;diff (44)
This equation is a differential equation of l(z) derivated by z.
The solution of this equation depends on istack and the
analytical result depends on whether the current is zero or
not.
lðzÞ ¼(
1AqH2O;net þ k1e
ABz if istacks0
�1BqH2O;netzþ k2 if istack ¼ 0
(45)
with
A ¼ nsat$MH2O$istack
22F
B ¼ rdry$Smem;Section$DClD
Mn
The constants k1 and k2 are determined from the boundary
conditions of l: l(0) at the anode side and l(dmem) at the
cathode side.
Thus, the expression of the total mass flow of water (kg/s)
can be expressed:
� If istack s 0:8<: qH2O;net ¼
A$�
lð0Þ$eABdmem � lðdmemÞ
�e
AB
dmem�1
k1 ¼ lðdmemÞ � lð0Þe
AB
dmem�1
(46)
� If istack¼ 0:
qH2O;net ¼ Bðlð0Þ�lðdmemÞÞ
dmem
k2 ¼ lð0Þ(47)
3.7. Cell thermal domain modeling
3.7.1. Cooling channelsIn this layer, the bipolar plate channel solid part and the fluid
channels volume are considered as two control volumes.
The fluid channels gas volume (1st control volume). The heat
flows from the anode support plate and cathode support plate
(J/s), noted QL,A to chan and QL,C to chan, respectively, are due to
heat transfers by forced convection between the coolant and
the support plates. According to Newton’s cooling law [37]
these flows can be written:
QL;i to chan ¼ hforced$SSection;ch$TL;i � TL;fluid
�i˛fC;Ag (48)
where hforced is the forced convection heat transfer coefficient
(W/m2 K), SSection,ch the side surface of the fluid channel
volume in contact with the anode (or cathode) support plates
(m2), TL,i the temperatures (K) at the interface between the
anode or cathode support plate and the cooling layer,
respectively, and TL,fluid the temperature (K) of the channels
fluid volume (control volume).
The forced convection heat transfer coefficient is calcu-
lated by the following relationship [37]:
hforced ¼Nu$lfluid
Dh(49)
where Nu is the Nusselt number of the fluid, lfluid the thermal
conductivity of the fluid (W/m K) and Dh the hydraulic diam-
eter of the cooling channels (m).
The Nusselt number can be given by the empirical equa-
tions [38]:
Nu ¼ 3:657þ0:0677$
Re$
Cp;fluidm
lfluid
DhL
�1:33
1þ 0:1$Cp;fluidm
lfluid$Re$Dh
L
�0:3 (50)
where Cp, fluid is the coolant thermal capacity (J/kg K).
In addition, the heat flow between the lateral surfaces of
the channels fluid volume and the plate solid part (J/s) in the
same cooling layer must also be taken into account. This flow
is calculated by:
QL;solid to fluid ¼ hforced$Slateral$TL;solid � TL;fluid
�(51)
where Slateral is the total lateral surface of the channels (m2),
and TL,solid the temperature (K) of the plate solid part.
Aside from the heat flows due to the forced convection
mentioned above, the convective heat flow (J/s) due to mass
flows (kg/s) entering or leaving the control volume can be
written:
QL;fluid;k ¼ qL;fluid;k$Cp;fluid$TL;fluid;k � TL;fluid
�k˛fin;outg (52)
The outlet temperature of the fluid, TL,fluid, out (K) is
considered equal to the temperature of the control volume
TL,fluid (K):
TL;fluid;out ¼ TL;fluid (53)
Thus, the temperature dynamic can be obtained from the
energy balance in the cooling channels gas volume:
�rfluid;Vchan;Cp;fluid
�dTL;fluid
dt¼ QL;A;chan þ QL;C;chan þ QL;solid to fluid
þ QL;fluid;in þ QL;fluid;out ð54Þ
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 1 5509
where rfluid is the coolant density (kg/m3), and Vchannels the
volume of the cooling channels (m3).
Bipolar plate channels solid part (2nd control volume). Unlike
fluids, the heat flows between solid materials are transferred by
the phenomenon of conduction according to Fourier’s Law [37]:
QL;i;solid ¼2$lplate$SSection;solid
dL
TL;i � TL;solid
�i˛fC;Ag (55)
where lplate is the bipolar plate thermal conductivity (W/m K),
SSection,solid the side surface of the channel solid part in contact
with the anode (or cathode) support plates (m2), and dL the
cooling layer thickness (m).
Indeed, regarding the scale of the bipolar plates, the heat
flows due to natural convection and radiation (J/s) must also
be considered according to Newton’s cooling law:
QL;convþnatþrad ¼ hconvþnatþrad$Splate-ext$Tamb � TL;solid
�(56)
where hconvþnatþrad is the combined natural convection and
radiation heat transfer coefficients (W/m2 K), Splate-ext the
external lateral surface of the cooling plate (m2), and Tamb the
ambient temperature (K).
The combined heat transfer coefficient by natural
convection and radiation (W/m2 K) can be calculated from the
following equation [37]:
hconvþnatþrad ¼ Nu$lamb
Lplate|fflfflfflfflfflffl{zfflfflfflfflfflffl}natural convection
þ3m$s$�
T2L;solidþT2
amb
�TL;solidþTamb
�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
radiation
(57)
where lamb is the thermal conductivity of the ambient air (W/
m K), L the bipolar plate perimeter (m), 3 the bipolar plate
emissivity, and s¼ 5.6704� 10�8 the Stefan–Boltzmann
constant (W/m2 K4).
The Nusselt number of the natural convection can be
calculated from the empirical equations [38].
The temperature dynamic can therefore be obtained from
the energy balance in the cooling channel solid part:
�rplate;Vsolid;Cp;plate
�dTL;solid
dt¼ QL;A;solid þ QL;C;solid � QL;solid to fluid
þ QL;convþnatþrad
(58)
where Vsolid is the channel solid part volume (m3).
3.7.2. Channels support platesThe channels solid plate volume is considered as a control
volume in the model.
The conduction heat flow (J/s) from cooling channels side
and gas channels side can be written as:
Qi;j¼2$lplate$SSection;S
dS
Ti;j�Ti;S
�i˛fC;Ag; j˛fcooling;chang (59)
where SSection,S is the side section of the bipolar channels support
plates (m2), dS the support plates thickness (m), Ti,chan and
Ti,cooling the temperatures (K) at the interface between the gas
supply channels layer and the cooling layer, respectively, and
Ti,S the channels support plate temperature (K) (control volume).
Indeed, the heat flow due to natural convection and radi-
ation Qi,convþnatþrad (J/s) can be obtained from (56) and (57) by
giving the channels support plate temperature Ti,S (K).
The temperature dynamic can thus be obtained from the
energy balance in the anode channels support:
�rplate;Vsolid;Cp;plate
�dTi;Supportplate
dt¼Qi;coolingþQi;chanþQi;convþnatþrad
(60)
where i ˛ {C, A}, Vsolid is the channels support plate layer
volume (m3).
3.7.3. Gas supply channelsIn this layer, the bipolar plate channel solid part and the gas
channels volume are considered as two control volumes.
The gas channels void volume (1st control volume). The heat
flow due to forced convection (J/s) can be written as in the
cooling channel section, from equations (48)–(51).
The heat flow due to the inlet/outlet mass transfer (J/s) can
be obtained from the expression in (52).
In the gas supply layers, the gases diffuse through the gas
diffusion layers (GDLs). The heat flow due to the mass transfer
through the GDL during the fuel cell stack operation can be
calculated by:
Qi;mass;GDL ¼"X
i
�qi;l;GDL$Cp;l
�#$Ti;GDL � Ti;fluid
�(61)
with i ˛ {C, A} and l ˛ {O2, N2, H2O} for cathode or l ˛ {H2, H2O}
for anode.
The outlet temperature of the mixed gas, Ti,fluid,out (K) is
considered equal to the temperature of the control volume
Ti,fluid (K):
Ti;fluid out ¼ Ti;fluid (62)
The temperature dynamic can thus be obtained from the
energy balance in the gas supply channels:
�rfluid;Vchan;Cp;fluid
�dTi;fluid
dt¼Qi;Supportplate;chanþQi;GDL;chanþQi;fluid in
þQi;fluidoutþQi;mass;GDLþQi;solidtofluid
(63)
where Vchan the gas supply channels volume (m3).
Bipolar plate channels solid part (2nd control volume). In this
section the heat flow due to the conduction can be written as:
Qi;j;solid ¼2$lplate$SSection;solid
dch
Ti;j � Ti;solid
�(64)
with i ˛ {C, A} and j ˛ {Supportplate, GDL}, SSection,solid the plate
solid part side surface in contact with the channels support
layer (m2), and dch the gas supply channels layer thickness (m).
In addition, the heat flow due to the natural convection and
radiation Qi,convþnatþrad (J/s) can be obtained from (56) and (57)
giving Ti,solid the channels solid part temperature (K).
Thus, the temperature dynamic can be obtained from the
energy balance in the solid part of the channel layers:
�rplate;Vsolid;Cp;plate
�dTi;solid
dt¼ Qi;Supportplate;solid þ Qi;GDL;solid
þ Qi;convþnatþrad � Qi;solid to fluid (65)
with i ˛ {C, A}.
Where Vsolid is the channels solid part volume (m3).
Fig. 5 – Experimental fuel cell stack.
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 15510
3.7.4. Gas diffusion layers, catalyst layers and membranelayerThese layers have a similar thermal model. In general, the
control volume of these porous layers can be presented as
follows:
The volume of these layers is considered as a control
volume.
The heat flow due to the conduction can be written using
Fourier’s Law:
Qcond;i;adjlayer ¼2$llayer$SSection;layer
dlayer
Ti;adjlayer � Tlayer
�(66)
where i ˛ {left, right}, Ti,adjlayer is the temperature (K) at the
interface between the layer and the adjacent layers and Tlayer
the layer temperature (K) (control volume).
Because the gases flow in and out of these layers, the heat
flows due to mass transfer from the neighboring layers during
the operation of the fuel cell can be written as:
Qmass;i;adjlayer ¼"X
l
�ql;i;adjlayerCp;l
�#$Ti;adjlayer � Tlayer
�(67)
where i ˛ {left, right}, l ˛ {O2, N2, H2O} for the cathode gas
diffusion layer and the cathode catalyst layer, l ˛ {H2, H2O} for
the anode gas diffusion layer and the anode catalyst layer,
l ˛ {H2O} for the membrane layer.
The total heat flow from the adjacent layers can be given:
Qi;adjlayer ¼ Qcond;i;adjlayer þ Qmass;i;adjlayer (68)
where i ˛ {left, right}.
In addition, the catalyst and membrane layers have the
internal heat sources:
For cathode catalyst layers, an internal heat source due to
the variation in entropy during the electrochemical reaction
and to the activation losses can be obtained [28]:
Qinternal ¼ �istack$TC;cata$DS
2F|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Entropie changes part
þ istack$VC;act|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}Activation losses part
(69)
where DS¼�163.185 is the entropy change (J/mol K) during
the electrochemical reaction, and VC,act the activation losses
(V) in the cathode catalyst layer.
According to hypothesis H 5, for the anode catalyst layers,
the internal heat sources due to the anode activation losses
are negligible.
For the membrane layer, a source of heat due to the Joule
effect of the membrane resistance can be obtained according
to Joule’s Law:
Qinternal ¼ i2stack$Rmem (70)
The temperature dynamic can therefore be obtained from
the energy balance in these layers:
dTlayer
dt¼ Qleft;adjlayer þ Qright;adjlayer þ Qinternal
rgas$Vgas$Cp;gas þ rlayer$Vlayer$Cp;layer
(71)
where rgas is the mean gas density in the porous layer (kg/
m3), Vlayer the layer volume (m3), Cp;gas the gases mean
thermal capacity (J/kg K), and Cp,layer the layer thermal
capacity (J/kg K).
4. Experimental validation
4.1. Test platform
The experimental test has been performed with a 1.2 kW
Ballard Nexa 47 cells stack (Fig. 5). This stack is supplied by
compressed air and hydrogen. It is also air-cooled by means of
a fan (forced convection).
During the experimental tests, most of the experimental
data measurements were done by the stack Ballard system
controller, such as: inlet air mass flow, inlet air temperature,
stack current, stack output voltage etc. However, the
controller does not measure the individual cell’s voltages and
individual cell’s temperatures. In order to measure these
physical quantities, additional instrumentation was added.
The individual cells voltages were measured with a differen-
tial voltage acquisition module of National Instrument, and
the individual cell temperature profiles were captured by an
FLIR infrared camera (see Figs. 6 and 7).
The bipolar plate emissivity is needed for the infrared
temperature measurement. In order to minimize the
measurement errors, a non-reflective black band was painted
on the top of the bipolar plates (see Fig. 5), to ensure that the
emissivity of this band is equal to 1 (like a blackbody). A
thermocouple was added to a point of this black band, and the
measured values were then compared to those of the infrared
camera, on purpose of the camera calibration.
Fig. 7 shows an infrared camera image of the stack. The
black painted band can be visualized clearly in the image.
The data acquisition frequency was fixed to 1 Hz. All the
measured values from the different sensors were centralized
and treated in a LabView environment and then stored in a file
for future data processing use.
Fig. 6 – Experimental test platform.
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 1 5511
4.2. Model parameters for Ballard Nexa fuel cell stacksimulation
To simulate Ballard Nexa fuel cell stack, the model must be
adjusted to the real geometry and physical property of the
Nexa stack.
As mentioned in the Modeling section above, this model is
a generalized PEMFC model that can simulate several PEMFC
stacks, because it has a set of adjustable parameters. These
adjustable parameters are listed in the tables at the end of
paper, the geometrical values and physical properties for the
Ballard Nexa fuel cell stack given in the table are obtained
Fig. 7 – Infrared camera image during the stack operation
(stack top view).
from the manufacturer datasheet and from our laboratory
database.
For the electrical domain modeling, the four empirical
parameters values of the cathode activation losses formula
are listed.
31¼�0.948
32¼ 3.86� 10�3
33¼ 7.6� 10�5
34¼�1.93� 10�4
These four coefficients are identified from the single cell
static polarization curve, by using a linear regression tech-
nique. The detailed identification method can be found in
Amphlett et al. [10].
4.3. Results and discussions
To validate the Nexa cell model, a simulation is run under the
same conditions as the experimental tests. The current profile
applied to the Nexa fuel cell system is delivered by an elec-
tronic load linked to the stack. This profile is then applied to
the model in the simulation. Thus, the same physical exper-
imental conditions and boundary conditions are used during
simulation.
4.3.1. The first experimental validation: long current stepThe first current profile of the fuel cell stack is presented in
Fig. 8(a). A long current step of 28.6 A is applied at 73 s, and
kept at this value until 825 s. This step current profile allows
the stack to reach its steady state operating point for constant
current.
By applying the same current profile, the model predicts the
total voltage of the 47 cells Nexa stack with a good accuracy (see
Fig. 8(b)). As shown in the figure, the stack dynamic behavior is
well predicted by the model, but there exists little errors
between the simulation results and those of experimentation.
A possible reason for that is because of the integrated humid-
ifier of Nexa system as suggested by Blunier and Miraoui [4]: the
performance of the humidifier is unknown and the hypothesis
adopted in simulation is that the cathode air input is humidi-
fied at 70% regardless of the performance conditions. In reality,
the relative humidity of the cell’s air input depends on the cell’s
previous performance conditions: the higher the current of the
cell, the more water is produced and, as a result, the hygrom-
etry conditions of the air input will be high (hysteresis
phenomenon). Another possible reason is that the exact
membrane thickness of the Nexa cell is not known a priori. It
has been assumed that is a Nafion 115 type membrane [39], but
the real membrane thickness cannot be measured precisely,
which affects the membrane resistances prediction.
A voltage error analysis is shown in Fig. 8(c). It has to be
noticed that in the model, each cell voltage is calculated
individually, and the overall absolute voltage prediction error
is less than 2 V for a 47 cells stack model which gives a 6%
maximum relative error between the model and the experi-
mental results.
The above model enables the mean temperature of the
fuel cell to be predicted. It also permits to predict the individual
cells temperatures. Fig. 8(d)–(f) show the transient temperatures
0 200 400 600 800 100025
30
35
40
45
50
55
60
Time (Second)
Stack vo
ltag
e (V
)
SimulationExperimentation
0 5 10 15 20 25 30−4
−3
−2
−1
0
1
2
3
4
Stack current (A)
stack vo
ltag
e sim
ula
tio
n
erro
rs (V
)
0 200 400 600 800 1000
25
30
35
40
45
50
55
Time (Second)
3th
cell b
ip
olar p
late
tem
peratu
re (°C
)
SimulationExperimentation
0 200 400 600 800 1000
25
30
35
40
45
50
55
60
65
Time (Second)
24th
cell b
ip
olar p
late
tem
peratu
re (°C
)
SimulationExperimentation
0 200 400 600 800 1000
25
30
35
40
45
50
55
60
Time (Second)
45th
cell b
ip
olar p
late
tem
peratu
re (°C
)
SimulationExperimentation
0 200 400 600 800 10000
5
10
15
20
25
30a b
c d
e f
Time (Second)
Stack cu
rren
t (A
)
Stack voltageStack current
Voltage simulation errors 3d cell bipolar plate temperature
24th
cell bipolar plate temperature 45th
cell bipolar plate temperature
Fig. 8 – First experimental validation: long current step (1/3).
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 15512
prediction of the 3rd, 24th and 45th cells, which are located at
the two sides and the center of the stack. The results given here
display a good consistency between the experimental and
simulated results. Indeed, the model accurately predicts the rise
and fall of the temperature according to the variation in current.
In experiments the thermal time constant of the system is
slightly lower than the one predicted by the model. Two reasons
can be advanced for this difference:
1. The model simulates only the stack; the heat transfer
between the cell and its auxiliaries is not taken into
account. In reality, these auxiliaries are in contact with the
stack and also have an influence on the dynamic of the cell
temperatures.
2. The thermal characteristics of the bipolar plate, the diffu-
sion layer and the membrane are not exactly known a priori.
The values used in the simulation are only an estimation
and this can generate uncertainties over the simulated
dynamic temperature. A more detailed knowledge of the
physical parameters of the materials would normally give
more accurate results.
In order to investigate the cells space distribution charac-
teristics (voltage and temperatures), Fig. 9(a) and (b) show
0.05 0.1 0.15 0.2 0.25 0.3
0.55
0.6
0.65
0.7
0.75
0.8
a b
d
Stack horizontal axis (m)
in
divid
ual cells vo
ltag
es (V
)
temps = 350 s
SimulationExperimentation
Individual cells voltage at 350 s
0.05 0.1 0.15 0.2 0.25 0.3
40
45
50
55
60
Stack horizontal axis (m)
in
divid
ual cells b
ip
olar p
lates
tem
peratu
res (°C
)
temps = 350 s
SimulationExperimentation
Individual cells temperature at 350 s
Individual cells voltages (measurements) Individual cells voltages (simulation)
Individual cells temperature (measurements) Individual cells temperature (simulation)
c
fe
Fig. 9 – First experimental validation: long current step (2/3).
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 1 5513
individual cell voltages and temperatures of the stack at 350 s.
Again, the model results show a good agreement with the
experimentation. A visible ‘‘boundary effect’’ of the stack is
well seen: the cells at the end sides of the stack do not have
the same operating conditions as the other cells; moreover,
both of the end side conditions are not even symmetrical. It
has to be noted that the measured voltages of the last cells,
from 0.25 m to the end of stack, are quite ‘‘unstable’’, an
important voltage difference can be distinguished. This is
because the Nexa fuel cell stack is operated in ‘‘dead-end’’
mode at the anode side. The outlet of the anode is normally
closed, during the stack operation, the impurities species from
the hydrogen supply or from the membrane gas permeation
can be accumulated at the anode channels, especially at the
last end cells. This dead-end ‘‘impurities’’ effect is not taken
into account in the model.
Fig. 9(c)–(f) give a more direct way to understand the time
and space variation of the cell voltages and temperatures. The
model predicted values are very close to the experimental
values. The dynamic of the individual cell voltages and
temperatures are correctly reproduced by the model. As
explained in the previous paragraph, the ‘‘unstable’’ voltage at
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 15514
the beginning of the experimental (Fig. 9(a)) is due to the dead-
end mode of the Nexa fuel cell stack.
The stack is cooled by the ambient air. The mean air
temperature at the outlet of the cooling channels is given in
Fig. 10(a). The temperature variation of the cooling air outlet is
very close to the stack temperature variation.
Fig. 10(b) shows the membrane water content evolution in
the central cell (24th). The anode side, cathode side and
average water content are drawn. In this stack, the anode
hydrogen inlet is not humidified. For a high current step, the
anode water content decreases sharply because of the water
electro-osmotic drag through the membrane. However, at the
cathode side, the water content increases because water is
produced in the cathode catalyst layers. This asymmetrical
water content effect can lead the membrane to be partially
dehydrated in transients. This means that the membrane
local resistance should be accounted in order to have an
accuracy of fuel cell model.
In the model, the forced convection heat transfer coeffi-
cients of each cell cooling channels are calculated at each
simulation time step as shown in Fig. 10(c). This figure shows
a slight coefficients difference between those in the central
cells channels and in the end side cells channels. It has to be
noted that, the forced convection heat transfer coefficient is
depending on the air cooling fan speed. In this experimenta-
tion, the air cooling fan is almost turning under a constant low
0 200 400 600 800 1000
25
30
35
40
45
50a
c
Time (Second)
Co
olin
g air o
utlet
tem
peratu
re (°C
)
0.05 0.1 0.15 0.2 0.25 0.330
32
34
36
38
40
42
44
46
Stack horizontal axis (m)
co
olin
g ch
an
nels fo
rced
co
nvectio
n h
(W
/(m
2.K
))
time = 250 s time = 350 s time = 550 s time = 800 s
Cooling air outlet temperature
Forced cooling coeffcient (simulation)
Fig. 10 – First experimental valid
speed, but if the fan speed changes, the forced convection
heat transfer coefficient changes too.
The total heat transfer coefficients of the bipolar plates of
each cell, taking into account natural convection and radia-
tion, are calculated also at every time-instant, as shown in
Fig. 10(d). The model predicted values are between 9 and 11
(W/m2 K), which is very close to the published data from [13].
We should notice that, under our test conditions, the value of
the radiation part of this composed heat transfer coefficient
varies from 4.67 to 5.79 (W/m2 K). That means the radiation
part contributes about 50% for this composed coefficient
value. Thus, we can conclude that the natural convection heat
transfers and radiation heat transfers have almost the same
heat exchange level.
4.3.2. The second experimental validation: short current stepchangesThe second experiment shows a dynamic current steps
change. The current profile is given in Fig. 11(a). Under this
current profile, the Nexa stack is always operated under
temperature transient conditions.
With this current profile, the stack voltages obtained in
simulation and in experimentation are compared in Fig. 11(b).
The predicted values show a great dynamic agreement with
the experimentation. An error analysis is given in Fig. 11(c),
the maximum absolute stack voltage difference between the
b
d
0 200 400 600 800 10002
4
6
8
10
12
14
16
18
Time (Second)
24th
cell m
em
bran
e w
ater
co
nten
ts
cathode sideanode sideaverage
0.05 0.1 0.15 0.2 0.25 0.3
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
Stack horizontal axis (m)
bip
olar p
lates n
atu
ral co
nvectio
n
an
d rad
iatio
n h
(W
/(m
2.K
))
time = 250 s time = 350 s time = 550 s time = 800 s
24th
cell membrane water content
Natural and radiation coeffcient (simulation)
ation: long current step (3/3).
0 200 400 600 800 1000 1200 14000
5
10
15
20
25
30
35
40a b
c d
e f
Time (Second)
Stack cu
rren
t (A
)
0 200 400 600 800 1000 1200 140025
30
35
40
45
50
55
60
Time (Second)
Stack vo
ltag
e (V
)
SimulationExperimentation
0 5 10 15 20 25 30 35 40−3
−2
−1
0
1
2
3
Stack current (A)
stack vo
ltag
e sim
ulatio
n erro
rs (V
)
0 200 400 600 800 1000 1200 14002
4
6
8
10
12
14
16
18
Time (Second)
24th
cell m
em
bran
e
water co
nten
ts
cathode sideanode sideaverage
0.05 0.1 0.15 0.2 0.25 0.3
0.65
0.7
0.75
0.8
0.85
0.9
Stack horizontal axis (m)
in
divid
ual cells vo
ltag
es (V
)
temps = 250 s
SimulationExperimentation
0.05 0.1 0.15 0.2 0.25 0.3
30
32
34
36
38
40
42
44
46
Stack horizontal axis (m)
in
divid
ual cells b
ip
olar p
lates
tem
peratu
res (°C
)
temps = 250 s
SimulationExperimentation
Stack current Stack voltage
Voltage simulation errors 24th cell membrane water content
Individual cells voltage at 250 s Individual cells temperature at 250 s
Fig. 11 – Second experimental validation: short current step changes (1/3).
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 1 5515
experimental and the model results is no more than 2 V for
a 47 cells stack dynamic simulation which means that the
relative stack voltage error is always less than 6%.
Fig. 11(d) shows the central cell (24th) membrane water
content at the cathode side, the anode side and its average
value. As in the previous test, the anode water content is
always much lower than the one at the cathode side. Espe-
cially at high stack currents, the membrane anode side water
content is not far from 4th, which is very close to membrane
dehydration conditions. However, the cathode side water
content is always kept around 14th, not only because the
water is produced at cathode side, but also because the
cathode air supply is humidified by an integrated humidifier.
The voltages and temperatures space distribution of indi-
vidual cells in the stack at 250 s are also given in Fig. 11(e) and
(f). The ‘‘boundary effect’’ is shown clearly on the temperature
space profiles, the temperature difference between the central
cells and the end side cells can reach 5 �C. For a large stack like
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 15516
the Nexa stack, the cells temperatures cannot be perfectly
regulated because of the high heat power generation. In this
case, this kind of ‘‘boundary effect’’ must be considered in an
accuracy of stack model to ensure the correct prediction
results.
Fig. 12(a)–(d) show the cells voltages and temperatures time
and space evolutions, from the model prediction and the
experimentation. The stack dynamic behavior is very well
predicted by the model over the time and space.
The vapor’s partial pressures at the cathode and anode
channels are shown in Figs. 12(f) and 13(b). The water
production of the cell is proportional to the current delivered.
0.05 0.1 0.15 0.2 0.25 0.31.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
x 104
Stack horizontal axis (m)
cath
od
e ch
an
nels O
2 p
artial
pressu
res (P
a)
time = 250 s time = 950 s time = 1400 s
Individual cells voltages (measurements)
Individual cells temperature (measurements)
Cathode channels O2 pressures (simulation)
a
c
e f
Fig. 12 – Second experimental validatio
The vapor pressure dynamic variation in the cathode/anode
channels is due to the water diffusion through the membrane,
the water produced at cathode catalyst layers and the gas
pressures dynamics in the cathode and anode channels.
Oxygen pressures in each cathode channels layers and
hydrogen pressures in each anode channels layers at different
time-instants are given in Figs. 12(e) and 13(a), respectively.
An interesting phenomenon is seen: oxygen and hydrogen
pressures in the end cell channels are slightly higher than the
ones of the central cells. This partial pressure difference
depends on the temperature of the individual cells (see
Fig. 11(f)). Indeed, the total pressures in the channels are
0.05 0.1 0.15 0.2 0.25 0.34000
5000
6000
7000
8000
9000
10000
11000
12000
Stack horizontal axis (m)
cath
od
e ch
an
nels H
2O
p
artial
pressu
res (P
a)
time = 250 s time = 950 s time = 1400 s
Individual cells voltage (simulation)
Individual cells temperature (simulation)
Cathode channels vapor pressures (simulation)
b
d
n: short current step changes (2/3).
0.05 0.1 0.15 0.2 0.25 0.31.3
1.4
1.5
1.6
1.7
1.8
a b
c d
x 105
Stack horizontal axis (m)
an
od
e ch
an
nels H
2 p
artial
pressu
res (P
a)
time = 250 stime = 950 stime = 1400 s
0.05 0.1 0.15 0.2 0.25 0.32000
3000
4000
5000
6000
7000
8000
9000
10000
Stack horizontal axis (m)
an
od
e ch
an
nels H
2O
p
artial
pressu
res (P
a)
time = 250 stime = 950 stime = 1400 s
0.05 0.1 0.15 0.2 0.25 0.38
9
10
11
12
13
Stack horizontal axis (m)
cell m
em
bran
es averag
e
water co
nten
ts
time = 250 s time = 950 s
time = 1400 s
0.05 0.1 0.15 0.2 0.25 0.3
1.2
1.4
1.6
1.8
2
2.2x 10−3
Stack horizontal axis (m)
in
divid
ual cell m
em
bran
e
resistan
ces (O
hm
)
time = 250 stime = 950 stime = 1400 s
Anode channels H2 pressures (simulation) Anode channels vapor pressures (simulation)
Cells membrane average water content (simulation) Cells membrane resistance (simulation)
Fig. 13 – Second experimental validation: short current step changes (3/3).
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 1 5517
practically identical but the vapor pressures depending on the
temperature are different. The oxygen partial pressure at the
cathode and the hydrogen partial pressure at the anode
increase when the temperature falls, because the vapor
saturation pressure decreases.
The cell membrane resistances at different time-instants
are given in Fig. 13(d). The membrane resistance depends on
the temperature and its water content (Fig. 13(c)). Tempera-
ture has a great influence on the membrane resistivity
(exponential dependence): when the temperature falls, resis-
tivity rises. The membrane water content is also depending on
the temperature, as it is linked to the saturation vapor pres-
sure. The influence of the temperature and the water content
is well illustrated by these simulation results: the cell’s
membrane resistances almost have the same shape as the
cell’s temperature (see Fig. 11(f) at 250 s), and rise when their
water content decreases.
4.3.3. The third experimental validation: very rapid currentchangesThe aim of the third experimentation is to validate the model
with a very high dynamic current profile covering the overall
stack current range, from 0 A to 45 A as shown in Fig. 14(a).
Fig. 14(b) shows a comparison between the experimental
and model voltages. With this high dynamic current change,
the model results still show a very good accuracy compared
with the experimental measurements. A detailed model
voltage prediction error analysis is given in Fig. 14(c). Over
the wide stack current values, the stack voltage errors are
mainly between �1 V and 1.5 V giving a relative error no
more than 5%.
The central cell (24th) membrane water content is given
in Fig. 14(d). The membrane water content changes are
directly related to the stack current variations. Generally,
for the Nexa stack, the anode side membrane water content
variation is more significant than the one at the cathode
side.
To show the model performances, the cell voltages and
temperatures space distribution at 120 s and 310 s are shown
in Figs. 14(e)–15(b). Again, the model demonstrates a good
agreement with the experimental profiles.
Thanks to the structure of the model, each individual lay-
er’s temperatures can be predicted (1 cell model has 10
modeling layers). These layer scale temperature space distri-
butions at different time-instants are given in Fig. 15(c). In this
figure, the temperature distribution into a single cell can be
0 100 200 300 400 5000
5
10
15
20
25
30
35
40
45
50a b
c d
e f
Time (Second)
Stack cu
rren
t (A
)
0 100 200 300 400 50025
30
35
40
45
50
55
60
Time (Second)
Stack vo
ltag
e (V
)
SimulationExperimentation
0 10 20 30 40−3
−2
−1
0
1
2
3
Stack current (A)
stack vo
ltag
e sim
ulatio
n
erro
rs (V
)
0 100 200 300 400 5002
4
6
8
10
12
14
16
18
Time (Second)
24th
cell m
em
bran
e
water co
nten
ts
cathode sideanode sideaverage
0.05 0.1 0.15 0.2 0.25 0.3
0.5
0.55
0.6
0.65
0.7
0.75
0.8
Stack horizontal axis (m)
in
divid
ual cells vo
ltag
es (V
) temps = 120 s
SimulationExperimentation
0.05 0.1 0.15 0.2 0.25 0.3
40
45
50
55
60
Stack horizontal axis (m)
in
divid
ual cells b
ip
olar p
lates
tem
peratu
res (°C
)
temps = 120 s
SimulationExperimentation
Stack current Stack voltage
Voltage simulation errors 24th
cell membrane water content
Individual cell voltage at 120 s Individual temperature at 120 s
Fig. 14 – Third experimental validation: very rapid current changes (1/2).
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 15518
clearly seen. Normally, the cathode catalyst layers have the
highest temperature compared to other layers. It is where the
electrochemical reaction takes place and most of the heat is
generated in this layer.
The membrane resistances of each cell at 120 s and 310 s
are shown in Fig. 15(d). The membrane resistances between
the central cells and the end side cells are different. Several
factors can explain these disparities but the temperature is
the most influential parameter as it acts exponentially on the
membrane resistance.
At last, the forced convection heat transfer coefficients in
the cooling channels and the total heat transfer coefficients of
bipolar plates are shown in Fig. 15(e) and (f).
5. Conclusion
A complete cell layer level generalized dynamic PEMFC stack
model has been developed. This model covers three major
physical domains: electrical, fluidic and thermal.
0.05 0.1 0.15 0.2 0.25 0.3
0.55
0.6
0.65
0.7
0.75
0.8a b
c d
e f
Stack horizontal axis (m)
in
divid
ual cells vo
ltag
es (V
) temps = 310 s
SimulationExperimentation
0.05 0.1 0.15 0.2 0.25 0.3
40
45
50
55
60
Stack horizontal axis (m)
in
divid
ual cells b
ip
olar p
lates
tem
peratu
res (°C
)
temps = 310 s
SimulationExperimentation
0.05 0.1 0.15 0.2 0.25 0.3
40
45
50
55
60
Stack horizontal axis (m)
tem
peratu
re (°C
)
time = 120 stime = 310 s
0.05 0.1 0.15 0.2 0.25 0.3
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3x 10−3
Stack horizontal axis (m)
in
divid
ual cell m
em
bran
e
resistan
ces (O
hm
)
time = 120 stime = 310 s
0.05 0.1 0.15 0.2 0.25 0.330
32
34
36
38
40
42
44
Stack horizontal axis (m)
co
olin
g ch
an
nels fo
rced
co
nvectio
n h
(W
/(m
2.K
))
time = 120 stime = 310 s
0.05 0.1 0.15 0.2 0.25 0.38
8.5
9
9.5
10
10.5
11
11.5
12
Stack horizontal axis (m)
bip
olar p
lates n
atu
ral co
nvectio
n
an
d rad
iatio
n h
(W
/(m
2.K
))
time = 120 stime = 310 s
Forced cooling convection (simulation) Natural and radiation convection (simulation)
Stack temperature profiles (simulation) Cells membranes resistances (simulation)
Individual temperature at 310 sIndividual cell voltage at 310 s
Fig. 15 – Third experimental validation: very rapid current changes (2/2).
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 1 5519
The advantages of VHDL-AMS for the complex system
modeling purpose are shown and discussed. The top-down
approach method allows the user to decompose a complex
system into sub levels until a level which gives sufficient
details for modeling. With this approach, an efficient PEMFC
model structure is given. This structure makes the model to be
very comprehensive, and if future improvements are needed,
the changes to be applied to the model can be reduced to
minimum.
The model is compared with a real Nexa 1.2 kW fuel cell
system (47 cells), and it is able to predict the stack behavior
with a very good accuracy. This model is capable to
predict every single cell performance with the given
stack boundary conditions. Three experimental tests are
Table 7 – Cathode gas supply channels layer.
Cathode gas type air
Cathode gas molar mass 2.89634� 10�4 (kg/mol)
Cathode gas thermal capacity 1.012� 103 (J/kg K)
Cathode gas thermal conductivity 2.63� 10�2 (W/m K)
Layer thickness 6.858� 10�4 (m)
Layer volume 1.01210� 10�5 (m3)
Channels volume portion 59.07%
Solid volume portion 40.93%
Channels length 8.807� 10�1 (m)
Number of channels 6
Table 8 – Anode gas supply channels layer.
Anode gas type hydrogen
Anode gas molar mass 2.0� 10�3 (kg/mol)
Anode gas thermal capacity 1.43� 104 (J/kg K)
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 15520
performed with different stack current profiles. The error
analysis shows that, the relative stack voltage prediction
error is less than 6%, and the stack temperature difference
between the simulation and experimentation can reach
about 5 �C (relative error less than 10% in Celsius unit). The
fuel cell stack time and space states evolutions are shown
and discussed.
This model has a set of adjustable parameters which are
considered as model inputs depending on the stack to be
simulated. In this article only a Nexa fuel cell model is dis-
cussed, but this model can be easily adjusted to any other
PEMFC stack for simulation purpose, as long as the correct
parameters are given to the model.
Appendix.A. Numerical parameter and Nexa stackproperties
Table 2 – Nexa stack property.
Number of cells 47
Table 3 – Bipolar plates properties (For cooling andcathode/anode supply channels layers).
Plate density 1.8336� 103 (kg/m3)
Plate thermal capacity 8.79� 102 (J/kg K)
Plate thermal conductivity 5.2 (W/m K)
Plate height 1.256� 10�1 (m)
Table 4 – Cooling channels layer.
Coolant type air –
Coolant molar mass 2.89634� 10�4 (kg/mol)
Coolant thermal capacity 1.012� 103 (J/kg K)
Coolant thermal conductivity 2.63� 10�2 (W/m K)
Layer thickness 3.1� 103 (m)
Layer volume 4.57498� 10�5 (m3)
Channels volume portion 64.19%
Solid volume portion 35.81%
Channels length 1.256� 10�1 (m)
Number of channels 18 –
Table 5 – Cathode channel support plates layer.
Layer thickness 5.642� 10�4 (m)
Layer volume 8.32646� 10�6 (m3)
Table 6 – Anode channel support plates layer.
Layer thickness 8.182� 10�4 (m)
Layer volume 1.20750� 10�5 (m3)
Anode gas thermal conductivity 1.6705� 10�1 (W/m K)
Layer thickness 4.318� 10�4 (m)
Layer volume 6.37250� 10�6 (m3)
Channels volume portion 64.93%
Solid volume portion 35.07%
Channels length 2.264 (m)
Number of channels 2
Table 9 – Cathode/anode gas diffusion layers (GDLs).
Material density 2� 103 (kg/m3)
Material thermal capacity 8.4� 102 (J/kg K)
Material thermal conductivity 6.5 (W/m K)
Layer thickness 4� 10�4 (m)
Layer volume 5.9032� 10�6 (m3)
Porosity 0.4
Tortuosity 1.5
Table 10 – Cathode/anode catalyst layers.
Material density 3.87� 102 (kg/m3)
Material thermal capacity 7.7� 102 (J/kg K)
Material thermal conductivity 2� 10�1 (W/m K)
Layer thickness 6.5� 10�5 (m)
Layer volume 9.5927� 10�7 (m3)
Table 11 – Membrane layer.
Material dry density 1.97� 103 (kg/m3)
Material equivalent mass 1.0 (kg/mol)
Material thermal capacity 1.1� 103 (J/kg K)
Material thermal conductivity 2.1� 10�1 (W/m K)
Layer thickness 1.27� 10�4 (m)
Layer volume 1.8743� 10�6 (m3)
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 e n e r g y 3 4 ( 2 0 0 9 ) 5 4 9 8 – 5 5 2 1 5521
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