Cell layer level generalized dynamic modeling of a PEMFC stack using VHDL-AMS language

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

Avai lab le at www.sc iencedi rect .com

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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.

mailto:[email protected]




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

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

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 5499

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.


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:


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;


� 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


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.


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)


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:


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.


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


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� 1TmemÞ�

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


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:


�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:


�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.


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.


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

)


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

)


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

)


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).


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


Individual cells voltage at 350 s

0.05 0.1 0.15 0.2 0.25 0.3

40

45

50

55

60


in

divid

ual cells b

ip

olar p

lates

tem

peratu

res (°C

)

temps = 350 s


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).


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


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


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


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

)


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


0.05 0.1 0.15 0.2 0.25 0.3

0.65

0.7

0.75

0.8

0.85

0.9


in

divid

ual cells vo

ltag

es (V

)

temps = 250 s


0.05 0.1 0.15 0.2 0.25 0.3

30

32

34

36

38

40

42

44

46


in

divid

ual cells b

ip

olar p

lates

tem

peratu

res (°C

)

temps = 250 s


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).


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


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


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


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


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


an

od

e ch

an

nels H

2O

p

artial

pressu

res (P

a)


0.05 0.1 0.15 0.2 0.25 0.38

9

10

11

12

13


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


in

divid

ual cell m

em

bran

e

resistan

ces (O

hm

)


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).


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

)


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


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


in

divid

ual cells vo

ltag

es (V

) temps = 120 s


0.05 0.1 0.15 0.2 0.25 0.3

40

45

50

55

60


in

divid

ual cells b

ip

olar p

lates

tem

peratu

res (°C

)

temps = 120 s


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).


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


in

divid

ual cells vo

ltag

es (V

) temps = 310 s


0.05 0.1 0.15 0.2 0.25 0.3

40

45

50

55

60


in

divid

ual cells b

ip

olar p

lates

tem

peratu

res (°C

)

temps = 310 s


0.05 0.1 0.15 0.2 0.25 0.3

40

45

50

55

60


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


in

divid

ual cell m

em

bran

e

resistan

ces (O

hm

)


0.05 0.1 0.15 0.2 0.25 0.330

32

34

36

38

40

42

44


co

olin

g ch

an

nels fo

rced

co

nvectio

n h

(W

/(m

2.K

))


0.05 0.1 0.15 0.2 0.25 0.38

8.5

9

9.5

10

10.5

11

11.5

12


bip

olar p

lates n

atu

ral co

nvectio

n

an

d rad

iatio

n h

(W

/(m

2.K

))


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).


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)


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 length 1.256� 10�1 (m)

Number of channels 18 –

Table 5 – Cathode channel support plates layer.


Layer volume 8.32646� 10�6 (m3)

Table 6 – Anode channel support plates layer.


Layer volume 1.20750� 10�5 (m3)

Anode gas thermal conductivity 1.6705� 10�1 (W/m K)


Layer volume 6.37250� 10�6 (m3)



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 conductivity 2� 10�1 (W/m K)



Table 11 – Membrane layer.

Material dry density 1.97� 103 (kg/m3)

Material equivalent mass 1.0 (kg/mol)


Material thermal conductivity 2.1� 10�1 (W/m K)




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Cell layer level generalized dynamic modeling of a PEMFC stack using VHDL-AMS language

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