1

2-D MODELING OF A PROTON EXCHANGE MEMBRANE FUEL CELL

A THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OFMIDDLE EAST TECHNICAL UNIVERSITY

BY

ERTAN AGAR

IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR

THE DEGREE OF MASTER OF SCIENCEIN

MECHANICAL ENGINEERING

FEBRUARY 2010

Approval of the thesis:

2-D MODELING OF A PROTON EXCHANGE MEMBRANE FUEL CELL

submitted by ERTAN AGAR in partial fulfillment of the requirements for the degree ofMaster of Science in Mechanical Engineering Department, Middle East Technical Uni-versity by,

Prof. Dr. Canan OzgenDean, Graduate School of Natural and Applied Sciences

Prof. Dr. Suha OralHead of Department, Mechanical Engineering

Assoc. Prof. Dr. Derek K. BakerSupervisor, Mechanical Engineering Department, METU

Assist. Prof. Dr. Mehmet SankırCo-supervisor, Micro and Nanotechnology Graduate Program,TOBB ETU

Examining Committee Members:

Assist. Prof. Dr. Tuba Okutucu OzyurtMechanical Engineering, METU

Assist. Prof. Dr. Ilker TarıMechanical Engineering, METU

Assist. Prof. Dr. Ahmet YozgatlıgilMechanical Engineering, METU

Assoc. Prof. Dr. Derek K. BakerMechanical Engineering, METU

Assist. Prof. Dr. Mehmet SankırMicro and Nanotechnology Graduate Program, TOBB ETU

Date:

I hereby declare that all information in this document has been obtained and presentedin accordance with academic rules and ethical conduct. I also declare that, as requiredby these rules and conduct, I have fully cited and referenced all material and results thatare not original to this work.

Name, Last Name: ERTAN AGAR

Signature :

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ABSTRACT

2-D MODELING OF A PROTON EXCHANGE MEMBRANE FUEL CELL

Agar, Ertan

M.S., Department of Mechanical Engineering

Supervisor : Assoc. Prof. Dr. Derek K. Baker

Co-Supervisor : Assist. Prof. Dr. Mehmet Sankır

February 2010, 88 pages

In this thesis, a Proton Exchange Membrane Fuel Cell is modeled with COMSOL Multi-

physics software. A cross-section that is perpendicular to the flow direction is modeled in a

2-D, steady-state, one-phase and isothermal configuration. Anode, cathode and membrane

are used as subdomains and serpentine flow channels define the flow field . The flow velocity

is defined at the catalyst layers as boundary conditions with respect to the current density

that is obtained by using an agglomerate approach at the catalyst layer with the help of fun-

damental electrochemical equations. Darcy’s Law is used for modeling the porous media

flow. To investigate the effects of species depletion along the flow channels, a different type

of cross-section that is parallel to the flow direction is modeled by adding flow channels as

a subdomain to the anode and cathode. Differently, Brinkman Equations are used to define

flow in the porous electrodes and the free flow in the channels is modeled with Navier-Stokes

equations. By running parallel-to-flow model, mass fractions of species at three different

locations (the inlet, the center and the exit of the channel) are predicted for different cell po-

tentials. These mass fractions are used as inputs to the perpendicular-to-flow model to obtain

performance curves. Finally, by maintaining restricted amount of species by having a very

low pressure difference along the channel to represent a single mid-cell of a fuel cell stack, a

iv

species depletion problem is detected. If the cell potential is decreased beyond a critical value,

this phenomenon causes dead places at which the reaction does not take place. Therefore, at

these dead places the current density goes to zero unexpectedly.

Keywords: Proton Exchange Membrane Fuel Cell, Numerical Modeling, COMSOL Multi-

physics, Species Depletion Along the Channel, Catalyst Layer Agglomerate Model

v

OZ

PROTON GECIRGEN ZARLI YAKIT HUCRESININ IKI BOYUTLU OLARAKMODELLENMESI

Agar, Ertan

Yuksek Lisans, Makina Muhendisligi Bolumu

Tez Yoneticisi : Doc. Dr. Derek K. Baker

Ortak Tez Yoneticisi : Y. Doc. Dr. Mehmet Sankır

Subat 2010, 88 sayfa

Bu tez calısmasında, COMSOL ticari programı kullanılarak, proton gecirgen zarlı yakıt hucresinin

2 farklı kesit alanı iki-boyutlu olarak modellenmistir. Iki calısmada da sabit durum, sabit

sıcaklık ve tek fazlı akıs kosulları varsayılmıstır. Elektrokimyasal denklemler kullanılarak,

akıs hızı, akım yogunluguna baglı olarak kataliz yuzeyinde sınır kosulu olarak tanımlanmıstır.

Ayrıca, gozenekli ortamdaki akıs Darcy Kuralı ile modellenirken, girdilerin kanal boyunca

tukenmesi esnasında olusabilecek sonucları ongorebilmek icin akısa paralel olan kesit, Brinkman

denklemleri kullanılarak degistirilmis Navier-Stokes denklemleri ile tanımlanmıstır. Aslında

reaksiyona girecek turlerin tukenmesi, calısma kosulları ayarlanarak duzeltilebilmektedir. Fakat

yakıt hucresi yıgınlarının orta kısımlarında bulunan tek yakıt hucrelerinde bu ayarlamayı ya-

pabilmek oldukca zordur. Bu sebepten, basınc farkı cok azaltılarak girdi miktarı sınırlandırılıp,

yakıt hucreleri icin onemli bir problem olan girdi tukenmesi olayı, tek bir yakıt hucresi kul-

lanılarak modellenmistir. Model sonucunda, girdi tukenmesinin yakıt hucresi uzerinde olum-

suz etkileri oldugu saptanmıstır. Hatta hucre potansiyeli yeteri kadar azaltıldıgında, anot

kanalları uzerinde reaksiyon gerceklesmeyen bazı olu bolgeler olusmustur. Bu yuzden hucre

potansiyeli dusuruldugu halde, akım yogunlugunun gittikce azalarak yok oldugu saptanmıstır.

vi

Anahtar Kelimeler: Proton Gecirgen ZarlıYakıt Hucresi, Sayısal Modelleme, COMSOL,

Girdilerin Kanal Boyunca Tukenmesi, Kataliz TabakasıYıgılma Modeli

vii

To Mom and Dad

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ACKNOWLEDGMENTS

First and foremost, I would like to express my deepest gratitude to my supervisors, Dr. Derek

Baker and Dr. Mehmet Sankır for everything that they have done for me. I am very indebted

to them for the guidance and support they have provided not only in every stage of my thesis

but also in real life. I owe them lots of gratitude having me had academical engineering vision.

I would like to thank the Ministry of Industry and Trade (SAN-TEZ # 00277.STZ.2008 − 1)

for supporting this research.

I am forever indebted to my family for believing me and supporting my decisions everytime

in my life. Especially, I would like to thank my sister very much. Without her encouragement

and supports, I could not have chosen such a wonderful career.

Also I would like to thank my best friends, Volkan, Yigit, Anıl and Nazlı. Without them

nothing would be so meaningful.

Barıs helps me a lot wherever he is. I greatly appreciate his brilliant computer knowledge.

My acknowledgement would not be completed without expressing my appreciation to Oguzhan

Topcu for making my life easier. I feel privileged to have such a wonderful teacher and

brother.

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TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

OZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

CHAPTERS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 History of Fuel Cells . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Types of Fuel Cells . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Components of Proton Exchange Membrane Fuel Cells . . 9

1.1.3.1 Polymer Electrolyte Membrane . . . . . . . . 9

1.1.3.2 Catalyst Layer . . . . . . . . . . . . . . . . . 11

1.1.3.3 Gas Diffusion Layer . . . . . . . . . . . . . . 12

1.1.3.4 Bipolar plates . . . . . . . . . . . . . . . . . 13

1.2 Proton Exchange Membrane Fuel Cell Chemistry and Thermody-namics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Basic Reactions and Thermodynamics . . . . . . . . . . . 14

1.2.1.1 Effects of Temperature . . . . . . . . . . . . 16

1.2.1.2 Reversible Fuel Cell Efficiency . . . . . . . . 17

1.2.1.3 Effect of Pressure . . . . . . . . . . . . . . . 18

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1.3 PEM Fuel Cell Electrode Kinetics . . . . . . . . . . . . . . . . . . . 19

1.3.1 Voltage Losses . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.1.1 Activation Loss . . . . . . . . . . . . . . . . 20

1.3.1.2 Ohmic (Resistive) Losses . . . . . . . . . . . 22

1.3.1.3 Concentration Loss . . . . . . . . . . . . . . 23

1.3.1.4 Internal Currents and Crossover Losses . . . . 24

1.3.2 Actual Fuel Cell Potential . . . . . . . . . . . . . . . . . 24

1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1 Modeling of Proton Exchange Membrane Fuel Cells . . . . . . . . . 28

2.1.1 General Modeling Assumptions . . . . . . . . . . . . . . 28

2.1.2 1991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1.3 1992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.1.4 1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.1.5 1996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1.6 1997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.7 1998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1.8 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1.9 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1.10 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.11 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1.12 After 2004 . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1.13 Summary of Literature Review . . . . . . . . . . . . . . . 39

3 NUMERICAL MODELING OF A PEM FUEL CELL . . . . . . . . . . . . 40

3.1 Charge Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Porous Media Flow for Perpendicular-to-Flow Model . . . . . . . . 44

3.3 Porous Media Flow for Parallel-to-Flow Model . . . . . . . . . . . . 46

3.4 Maxwell-Stefan Mass Transport . . . . . . . . . . . . . . . . . . . . 47

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4 MODELING ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Perpendicular-to-Flow Model . . . . . . . . . . . . . . . . . . . . . 52

4.1.1 Conductive Media DC Application Mode . . . . . . . . . 53

4.1.1.1 Electrodes . . . . . . . . . . . . . . . . . . . 53

4.1.1.2 Membrane . . . . . . . . . . . . . . . . . . . 54

4.1.2 Darcy’s Law Application Mode . . . . . . . . . . . . . . 54

4.1.3 Maxwell-Stefan Diffusion and Convection Application Mode 55

4.1.3.1 Maxwell-Stefan for Anode Species . . . . . . 55

4.1.3.2 Maxwell-Stefan for Cathode Species . . . . . 56

4.2 Parallel-to-Flow Model . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.1 Incompressible Navier-Stokes Application Mode with BrinkmanEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.1 Perpendicular-to-Flow Model . . . . . . . . . . . . . . . 60

4.3.2 Parallel-to-Flow Model . . . . . . . . . . . . . . . . . . . 61

4.4 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . . . . . . . 62

6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7 FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

APPENDICES

A A Numerical, 3-D Investigation of Gas Flow Channels Modified with Distur-bances in PEMFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.1.1 Governing Equations . . . . . . . . . . . . . . . . . . . . 81

A.1.2 Computational Domains and Boundary Conditions . . . . 81

A.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 84

A.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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LIST OF TABLES

TABLES

Table 1.1 Enthalpies, entropies and Gibbs free energy for hydrogen oxidation reaction

at 25oC [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Table 1.2 ∆H, ∆G, ∆S values with different temperatures and resulting Erev values [2] 16

Table 4.1 Geometrical properties of Perpendicular-to-Flow Model . . . . . . . . . . . 53

Table 4.2 Boundary Conditions for the Conductive Media DC (Electrodes) Applica-

tion Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Table 4.3 Boundary Conditions for the Conductive Media DC (Membrane) Applica-

tion Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Table 4.4 Boundary Conditions for the Darcy’s Law Application Mode . . . . . . . . 55

Table 4.5 Boundary Conditions for the Maxwell-Stefan Diffusion and Convection Ap-

plication Mode for the Anode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Table 4.6 Boundary Conditions for the Maxwell-Stefan Diffusion and Convection Ap-

plication Mode for the Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Table 4.7 Geometrical properties of Parallel-to-Flow Model . . . . . . . . . . . . . . 58

Table 4.8 Boundary Conditions for the Incompressible Navier-Stokes Application Mode

with Brinkman Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Table A.1 Selected passes to be presented for 5, 10 and 15 passes . . . . . . . . . . . 84

Table A.2 Maximum y-velocity values for selected passes for 3 different flow fields . . 84

xiii

LIST OF FIGURES

FIGURES

Figure 1.1 The Schematic View of ”Gas Battery” invented by Sir William Grove in

1839 [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Figure 1.2 Commercial fuel cell powered cars . . . . . . . . . . . . . . . . . . . . . 4

Figure 1.3 Schematic of a single Proton Exchange Membrane Fuel Cell and funda-

mental reactions [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Figure 1.4 Schematic of MEA and its components . . . . . . . . . . . . . . . . . . . 11

Figure 1.5 Voltage losses in a Proton Exchange Membrane Fuel Cell and resulting

polarization curve, [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Figure 2.1 Gas diffusion electrode bonded to a solid polymer electrolyte [4] . . . . . . 29

Figure 2.2 Net flux of water across membrane. A value of the flux less than zero

indicates transport from the anode to the cathode; f is the air-to-fuel ratio [5] . . . 31

Figure 2.3 Scheme of a PEM half-cell with conventional and the new flow field . . . . 32

Figure 2.4 Freeze-fractured cross-section of MEA: (a) Nafion R© 119 membrane, (b)

impregnated active catalyst layer [6] . . . . . . . . . . . . . . . . . . . . . . . . 33

Figure 2.5 Geometry, coordinate axes, boundary types and characteristic dimensions

of the three domains [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 2.6 Flow diagram of the solution procedure used [8] . . . . . . . . . . . . . . 36

Figure 2.7 Agglomerate catalyst geometry [9] . . . . . . . . . . . . . . . . . . . . . 37

Figure 2.8 Various polarization curves at Tcell = 80oC, and the anode/cathode stoi-

chiometry of 1.5 and 1.8 at 1.0A/cm2 [10] . . . . . . . . . . . . . . . . . . . . . 38

Figure 3.1 Two different types of modeling . . . . . . . . . . . . . . . . . . . . . . . 41

xiv

Figure 3.2 Modeled domain and boundary conditions for coupled free flow and porous

media flow [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Figure 4.1 Constants that are used for the modeling in COMSOL . . . . . . . . . . . 51

Figure 4.2 A Schematic View of the Perpendicular-to-Flow Model . . . . . . . . . . 52

Figure 4.3 Boundary Numeration for the perpendicular-to-flow model . . . . . . . . . 53

Figure 4.4 Schematical View of the Parallel-to-Flow Model . . . . . . . . . . . . . . 57

Figure 4.5 Current collector plate with serpentine flow field channel . . . . . . . . . . 57

Figure 4.6 Schematic of a fuel cell stack [12] . . . . . . . . . . . . . . . . . . . . . . 58

Figure 4.7 Critical points and boundaries for free mesh parameters . . . . . . . . . . 60

Figure 4.8 Mesh Statistics for Perpendicular-to-Flow Model . . . . . . . . . . . . . . 60

Figure 4.9 Mesh Statistics for Parallel-to-Flow Model . . . . . . . . . . . . . . . . . 61

Figure 5.1 Weight fractions of H2, O2 and H2Oc for V = 0.4V with respect to channel

position x [m] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 5.2 Weight fractions of H2, O2 and H2Oc for V = 0.7V with respect to channel

position x [m] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 5.3 Weight fractions of H2, O2 and H2Oc for V = 1V with respect to channel

position x [m] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 5.4 Current density variation with respect to channel position of x [m] for V =

0.7V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 5.5 Performance curve (V - i curve) of the parallel-to-flow model of a Proton

Exchange Membrane Fuel Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Figure 5.6 Selected positions to be investigated for the parallel-to-flow model. . . . . 67

Figure 5.7 Selected positions to be investigated for the perpendicular-to-flow model . 68

Figure 5.8 Cell Potential (vs) Local Current Density Graphs for each x = 0, x =

0.025m and x = 0.05m positions of the parallel-to-flow model . . . . . . . . . . . 69

Figure 5.9 The present model validation with an experimental data for an air-breathing

PEMFC yielding lower current density. The experimental data values were esti-

mated from a graph [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xv

Figure 5.10 Cell Potential versus Integrated Current Density Graphs for x = 0, x =

0.025m and x = 0.05m positions of the parallel-to-flow model . . . . . . . . . . . 71

Figure A.1 Three different models of the gas flow channel for 5, 10 and 15 number of

passes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Figure A.2 Disturbance types of the gas flow channel . . . . . . . . . . . . . . . . . . 83

Figure A.3 x and y components of the velocity distribution for straight flow channels . 85

Figure A.4 Improvement in y-velocity for 5 pass Geometry . . . . . . . . . . . . . . . 85

Figure A.5 Improvement in y-velocity for 10 pass Geometry . . . . . . . . . . . . . . 86

Figure A.6 Improvement in y-velocity for 15 pass Geometry . . . . . . . . . . . . . . 86

Figure A.7 Pressure drop relations for different disturbance types . . . . . . . . . . . 87

xvi

LIST OF ABBREVIATIONS

F: Faraday’s constant, 96485 [C/mol]n: Number of electrons per mole of H2i: Current density [A/m2]M: Molecular weight [kg/kmol]x: Mole fractionw: Weight fractionG: Gibbs free energy [kJ]H: Enthalpy of reaction [kJ]

Greek Symbols

ηconv: Energy conversion efficiencyi0: Exchange current density [A/m2]η: Gas viscosity [Pa.s]ρ: Density [kg/m3]ε: Porosityδ: Distance [m]λH2O: Drag coeff. of waterκ: Conductivity [S/m]κp: Permeability [m2]φ: Potential [V]

Subscripts

a: anodec: cathodeL: limitinge f f : effectiveeq: equilibriume: anode or cathodei: speciesre f : referencerev: reversiblein: internal

Abbreviations:

PFS A: Perfluorosulfonic AcidGDL: Gas Diffusion LayerHHV: Higher Heating ValueLHV: Lower Heating ValueS EM: Scanning Electron Microscope

xvii

CHAPTER 1

INTRODUCTION

1.1 General Background

The energy demand of the world is increasing day by day due to the rapid development of

technology. According to the International Energy Agency’s World Energy Outlook 2008,

world primary energy demand is expected to grow by 1.6% per year on average between 2006

and 2030 from 11730 Mtoe to just over 17010 Mtoe - an increase of 45% [14]. By consider-

ing that today 80% of total primary energy is provided by fossil fuels, our current dependency

on carbon based fuels is evident. This reliance on the combustion of fossil fuels has caused

severe problems such as global warming and an extensive decrease in the world’s fossil fuel

resources. Therefore, engineers and scientists are working to find clean and sustainable al-

ternative resources for energy and affordable and efficient power conversion devices for the

future.

The world needs an energy conversion device that has low pollutant emissions and is efficient.

Fuel cells are considered as one promising energy conversion technology. A fuel cell is an

energy conversion device that directly converts the chemical energy of a fuel into electrical

work through an electrochemical reaction. While a fuel cell is similar to battery, it is not

identical. It does not store energy in any manner, indeed it converts a continuous stream of

fuel into a continuous stream of electricity. Therefore, fuel and oxidant should be supplied

constantly. The fuel at the anode is oxidized to release electrons that are then transferred to

the cathode side. At the cathode, these electrons are used for reducing the oxidant. During

the red-ox reactions, the flow of electrons from anode to cathode is the current produced by

the fuel cell and an electrical potential difference exists between the two electrodes.

1

A heat engine also converts chemical energy into electricity, but with intermediate steps.

Through combustion, chemical energy is converted first into thermal energy, then a heat en-

gine is used for the conversion of this thermal energy into shaft work. Finally, at the last step,

an electrical generator converts shaft work into electrical work (electricity). This multi-step

process is based on a heat engine, so the overall conversion efficiency is limited by the Carnot

efficiency. To reach higher efficiencies, higher operating temperatures are needed. However,

the electrochemical energy conversion in fuel cells does not depend on a heat engine, and

therefore electrochemical reactions are not governed by Carnot’s law (although they are still

limited by the Second Law of Thermodynamics). The theoretical efficiency of low temper-

ature fuel cells is higher than the theoretical efficiencies of heat engines, which includes all

internal combustion engines. A fuel cell is silent and mechanically stable since there are no

moving parts, which yields the potential for higher reliability. Moreover, because fuel cells

can be designed over a very wide power production range, they have a very broad range of

applications [15, 16, 17].

1.1.1 History of Fuel Cells

The idea of fuel cells has been known to science for more than 150 years. In 1800, William

Nicholson and Anthony Carlisle described the electrolysis process in which electricity is used

to break water into hydrogen and oxygen [16]. After their work, a Welsh lawyer turned scien-

tist William Robert Grove (1811 - 1896) realized that he might combine several electrodes in

a series circuit, so he can recompose water and produce electricity. In 1839, he accomplished

this idea and made a device called a ”gas battery”, the first fuel cell. This device was filled

with a diluted sulfuric acid electrolyte solution in which separate platinum electrodes (con-

tacting oxygen and hydrogen separately) were submerged. In time, it was observed that as

well as the water level increasing, a current flow occurred as shown in Fig. 1.1.

After this discovery, fuel cells remained just a scientific curiosity for almost a century. Friedrich

Wilhelm Ostwald (1853 - 1932), the founder of Physical Chemistry, provided much of the the-

oretical understanding of how fuel cells operate. In 1893, he experimentally discovered the

role of many fuel cell components. Chemist Ludwig Mond (1839 - 1909) and his assistant

Carl Langer achieved 6 amps per square foot area of electrode at 0.73 V in 1889.

2

Figure 1.1: The Schematic View of ”Gas Battery” invented by Sir William Grove in 1839 [3]

At the same time, Charles R. Alder Wright (1844 - 1894) and C. Thompson developed a sim-

ilar cell, but due to leakage problem of gas streams, they could not reach high voltage values.

In 1896, a big advance in fuel cells occurred when William W. Jacques (1855 - 1913) sub-

merged a carbon electrode in an alkali electrolyte with air injection. It was called a carbon

battery [18]. Until the end of the 1950s, some scientific curiosity continued. Francis Tom Ba-

con constructed a fuel cell to repeat the experiment of Grove. He developed double-layered

electrodes and solved the problems of liquid flooding and gas bubbling. He also tried to find

a novel solution to the problem of corrosion by forming an oxide coating on the nickel elec-

trode. In 1959, he constructed a 6 kW fuel cell. After this time, the curiosity of industry was

raised. In the early 1960s, the first practical fuel cell was used in the U.S. space programs [2].

Although General Motors tried to make several applications in the 1960s, 1970s and 1980s,

fuel cells were used successfully only for space research. Fuel cells received little attention

for terrestrial applications until the 1990s [2]. Today most automobile manufacturers are try-

ing to develop fuel cell powered cars after realizing the great potential of proton exchange

membrane fuel cells for road vehicle applications (Fig. 1.2) [16].

3

(a) Mercedes F-Cell, fuel cell car by Daimler Chrysler (2005)

(b) Honda FCX Clarity, fuel cell car by Honda (2007)

Figure 1.2: Commercial fuel cell powered cars

1.1.2 Types of Fuel Cells

Fuel Cells are generally classified based on the electrolyte they use. The exception is DMFC

(Direct Methanol Fuel Cell) in which the fuel methanol is used for classification instead of

the electrolyte. The most common types of fuel cells and their abbreviations are as follows:

Polymer Electrolyte Membrane Fuel Cell (PEMFC); Molten Carbonate Fuel Cell (MCFC);

Solid Oxide Fuel Cell (SOFC); Direct Methanol Fuel Cell (DMFC); Alkaline Fuel Cell (AFC)

and Phosphoric Acid Fuel Cell (PAFC). A second classification can be done based on operat-

ing temperatures with Low temperature Fuel Cells operating below approximately 220oC and

High Temperature Fuel Cells generally operating between 600 - 1, 000oC. Low Temperature

Fuel Cells include Polymer Electrolyte Membrane Fuel Cells, Direct Methanol Fuel Cells,

Alkaline Fuel Cells, and Phosphoric Acid Fuel Cells. High Temperature Fuel Cells include

Molten Carbonate Fuel Cells and Solid Oxide Fuel Cells. Below, the types of fuel cells are

summarized in terms of the electrolyte used and the operating temperature range:

4

• Proton Exchange Membrane Fuel Cells: perforated sulphonated polymer electrolyte /

50 - 100oC

• Solid Oxide Fuel Cells: ceramic solid electrolyte / 600 - 1, 000oC

• Molten Carbonate Fuel Cells: molten carbonate solution / 650oC

• Alkaline Fuel Cells: alkaline electrolyte/ 50 - 200oC

• Phosphoric Acid Fuel Cells: stabilized phosphoric acid / 160 - 220oC

Proton Exchange Membrane Fuel Cells (PEMFC)

Proton Exchange Membrane Fuel Cells use perforated sulphonated polymer as the electrolyte.

Hence, it is additionally called a Polymer Electrolyte Membrane Fuel Cell. The electrolyte

for a Proton Exchange Membrane Fuel Cell is a solid polymer membrane (a thin plastic

film). This membrane is covered by highly dispersed metal alloy particles such as platinum

that is an active catalyst and allows protons to pass through to the electrolyte. The catalyst

helps the hydrogen atoms react to form electrons and protons. Those protons are transported

across the electrolyte to the cathode while the electrons flow through an external circuit and

produce electric power. Simultaneously, oxygen is supplied to the cathode and reacts with the

protons and electrons to produce water. Proton Exchange Membrane Fuel Cells operate in the

range of 50 - 100oC, which allows for rapid start-up. This ability to rapidly provide power

output makes them preferable for automotive power applications where quick start-ups are

required. Moreover if hydrogen is used as the fuel, Proton Exchange Membrane Fuel Cells

have zero emission of pollutants such as SOX , NOX and COX . When heated the N2 and O2

in air will form NOX regardless of whether the fuel is H2 or a hydro carbon. However, at

the low temperatures in a PEMFC, the formation of NOX is not favored thermodynamically

(equilibrium) or kinetically (rate). But as temperature increases, the equilibrium point for a

mixture of N2, O2, and NOX shifts toward NOX and the reaction rate increases, thus forming

NOX . Therefore the formation of NOX does not depend on fuel, but on the oxidant and

temperature [17, 19].

Alkaline Fuel Cells (AFC)

Alkaline Fuel Cells are one of the most developed fuel cell technologies. Alkaline Fuel Cells

have been used by NASA in the Apollo and Space Shuttle programs as they are among the

most efficient fuel cells and can generate electricity with up to 70% efficiency. Alkaline

5

Potassium Hydroxide is used as the electrolyte in these cells, which is a water based solu-

tion of potassium hydroxide (KOH). The Alkaline Fuel Cell’s operating temperature ranges

from 50oC to 100oC. Until recently Alkaline Fuel Cells for commercial applications were

very expensive but companies are trying to reduce fixed costs and improve the flexibility of

operations. Also, one other characteristic of Alkaline Fuel Cells is that they are very sensi-

tive to CO2 that may be in the fuel or air. The CO2 reacts with the electrolyte and poisons it

rapidly, and this reduces the fuel cell performance. Even the smallest amount of CO2 must

be removed. Therefore, Alkaline Fuel Cells require purified or cleansed hydrogen and pure

oxygen (not ambient air). However this purification process is costly. Another key issue that

the purification creates is although Alkaline Fuel Cells have the highest electrical efficiency

among all the fuel cells, the requirement of high purity fuel and oxidant places a great con-

straint on applications. Another key barrier is the operating time capacity. The operating hour

capacity must be upgraded from an average of 8, 000 hrs to 40, 000 hrs by solving current

material durability issues. On the other hand, Alkaline Fuel Cells are the cheapest fuel cells to

manufacture. This is because of the variety of low-cost catalysts that can be used [15, 19, 20].

Phosphoric Acid Fuel Cells (PAFC)

Phosphoric Acid Fuel Cells were the earliest fuel cells to be commercialized and they still

exist commercially at the present time. The development process started in 1960 and testing

lasted until the 1970s. PAFCs have improved significantly in terms of stability and perfor-

mance. These substantial improvements and characteristics have made the Phosphoric Acid

Fuel Cell a good choice for stationary applications such as implementations in buildings, ho-

tels, hospitals, airports, power plants, etc. A Phosphoric Acid Fuel Cell uses phosphoric acid

as an electrolyte with concentrations up to 100% which minimizes the water vapor pressure.

Just like Proton Exchange Membrane Fuel Cells, the positive charge carrier is the proton,

the protons move through the electrolyte and combines with the oxygen, usually from air, at

the cathode and electricity and by-product heat are produced. Phosphoric Acid Fuel Cells

produce electricity with efficiencies in the range from 37% to 40%. This efficiency is lower

than Alkaline Fuel Cells because of the slower oxygen reduction kinetics at the cathode. The

operating temperature of Phosphoric Acid Fuel Cells are between 160 and 220oC. Moreover,

at these operating temperatures, the waste heat is capable of heating hot water or generating

steam at atmospheric pressure. The ease of construction, electrolyte stability, thermal, chem-

ical and electrochemical stability, and long-term stability are other less important advantages.

6

However, these fuel cells are expensive in terms of catalysts as they use platinum. Also to

operate typical Phosphoric Acid Fuel Cells in the range of 5 to 200kW costs from 4, 000

to 4, 500 per kW, which is about five times greater than the cost targets for conventional

stationary applications [21]. To conclude, currently the primary goal for Alkaline Fuel Cells

are to improve and to gain experience with PAFC operation, maintenance, reliability and

performance [15, 19, 20].

Molten Carbonate Fuel Cells (MCFC)

Molten Carbonate Fuel Cells are in the class of high-temperature fuel cells with Solid Oxide

Fuel Cells. Molten Carbonate Fuel Cells started to be developed in the mid 1960s. Improve-

ments have been made in performance, fabrication methods, and endurance for natural gas

and coal-based power plants for military applications, industrial applications and electrical

utilities. At higher temperatures such as that in a Molten Carbonate Fuel Cell, fuel reforming

of natural gas can occur internal to a fuel cell, which allows Molten Carbonate Fuel Cells to

use natural gas directly without the need for a fuel processor. Molten Carbonate Fuel Cells use

an electrolyte composed of a molten mixture of carbonate salts (a mixture of lithium carbon-

ate and potassium carbonate, or lithium carbonate and sodium carbonate). The high operating

temperatures, however, has both advantages and disadvantages. To melt the carbonate salts

and make them conductive to carbonate ions, Molten Carbonate Fuel Cells must operate at a

high temperature of 650oC. The carbonate ions flow from the cathode to the anode where they

react with hydrogen to produce water, carbon dioxide, electrons (electricity) and by product

heat. Additionally, achieving high ion mobility through the electrolyte is another reason for

the high temperature. Molten Carbonate Fuel Cells can have efficiencies of 60%, and in addi-

tion to this, the efficiency can be increased by adding a reforming catalyst to reform the fuel

internally where CO and other Carbon-based fossil fuels can be used as a fuel. In terms of

durability, the high temperatures at which these cells operate decrease the cell life. Also, the

high temperature requires significant start-up time to reach operating conditions and Molten

Carbonate Fuel Cells respond slowly to changing power demands. These characteristics make

MCFCs more suitable for constant power applications. The carbonate electrolyte can also

cause electrode corrosion problems. Furthermore, since CO2 is consumed at the anode and

transferred to the cathode, introduction of CO2 and its control in the air stream becomes an

issue for achieving optimum performance that is not present in any other fuel cell.

7

Moreover, the sulphur tolerance of the Molten Carbonate Fuel Cells is low. One very impor-

tant disadvantage of MCFC is poor mechanical stability due to material problems arising from

high temperatures. These problems make MCFCs unsuitable for mobile and transportation

applications unlike Proton Exchange Membrane Fuel Cells and Alkaline Fuel Cells [15, 20].

Solid Oxide Fuel Cells

Solid Oxide Fuel Cells are by far the highest temperature fuel cells and operate in the range

of 600 - 1, 000oC. To operate at such high temperatures, the electrolyte is solid oxide; a thin,

solid ceramic material that is conductive to oxygen ions. Because the electrolyte is solid,

the cell can be cast into various shapes, such as tubular, planar, or monolithic. The solid

ceramic construction of the unit cell decreases corrosion problems in the cell. These fuel cells

are considered to be appropriate for industry and large scale central electricity generating

stations. They require significant start-up time to reach their operating temperature and these

cells respond slowly to changes in electricity demand. Also, the high temperatures require

more expensive materials for construction. The efficiency for generating electricity is about

60%, which is a very high relative to most other types of fuel cells. Moreover, the high

operating temperature allows cogeneration applications to create high-pressure steam which

can be used in other applications [15, 20]. Also, pairing a Solid Oxide Fuel Cell with a turbine

in a hybrid fuel cell can increase the efficiency of generating electricity up to 70% [22].

In conclusion, among all of the types of fuel cells, Proton Exchange Membrane Fuel Cells are

the most promising, most researched and demonstrated type of fuel cell [23]. For this reason,

in this work Proton Exchange Membrane Fuel Cells are investigated.

8

1.1.3 Components of Proton Exchange Membrane Fuel Cells

In this section, the key components of a basic Proton Exchange Membrane Fuel Cell are

explained. A schematic of a single Proton Exchange Membrane Fuel Cell is shown in Fig.

1.3.

Figure 1.3: Schematic of a single Proton Exchange Membrane Fuel Cell and fundamentalreactions [2]

1.1.3.1 Polymer Electrolyte Membrane

The most important component of a Proton Exchange Membrane Fuel Cell is a polymer elec-

trolyte membrane, which is made of a solid polymer membrane which should possess high

proton conductivity. Another important feature that a membrane should contain is the high

level of electrical insulation as well as low fuel crossover properties. The robustness of the

membrane in terms of thermal and chemical stability is critical in a fuel cell stack. The func-

tion of the membrane in a Proton Exchange Membrane Fuel Cell is to distribute and conduct

protons efficiently while preventing the passage of unreacted, reacted and inert gases, and

repelling the electrons by directing the electrons to travel through the external circuit.

9

If the most commonly used standard membrane is considered, which is DuPont’s Nafion R©,

the process is as follows [19, 20].

Perfluorosulfonic acid (PFSA) is the membrane material used in Nafion [12]. There are two

important advantages of using PFSA as a main material for membrane manufacturing. Since

the structure is based on a polytetrafluoroethylene backbone (PTFE, DuPont’s Teflon R©), they

are quite strong and stable on both the oxidation and reduction sides. When well-humidified

the proton conductivity of PFSA membranes reaches 0.1 S/cm, which means that the cell re-

sistance value are approximately 0.05 Ω/ cm2 for a 50 µ thick membrane corresponding to a

voltage loss of only 50 mV at 1 A/cm2. However, there are several disadvantages of PFSA

membranes. First, they are expensive at $ 25/kW. Because of low conductivity values for

low humidified PFSA membranes, an external humidification unit is often needed, which in-

creases the cost of the system. Moreover, elevated temperatures cause degradation of several

properties. For example, hydrogen permeability increases starting around 80oC due to mor-

phological changes in molecular shape. Therefore, if the temperature becomes higher than

80oC, the membrane performance life shortens significantly [12].

Although, in the literature a lot of research for finding an alternative to PFSA membranes is

presented, the present modeling work assumes Nafion R©.

Because this polymer electrolyte membrane is bonded to a catalyst layer with porous elec-

trodes as shown in Fig. 1.3, the membrane electrode assembly is commonly referred to as

the MEA. The components of the MEA of a single Proton Exchange Membrane Fuel Cell are

shown in Fig. 1.4.

10

Figure 1.4: Schematic of MEA and its components

1.1.3.2 Catalyst Layer

Slow kinetics at the electrodes limits the reaction speed. To speed up the kinetics, two methods

are commonly used:

• Raising the operating temperature

• Introducing a catalyst material (i.e., the membrane is coated with a catalyst layer)

The catalysts layer is located between the Gas Diffusion Layer (GDL) and membrane on each

side of the fuel cell. In most cases, platinum, which has a strong affinity for CO, is used as

the catalyst to overcome the slow kinetics associated with low temperature. However, using

platinum can lead to anode catalyst CO poisoning trouble when the source of hydrogen is

alcohol or hydrocarbon fuel due to residual CO in the fuel. If pure hydrogen is used, there

is no poisoning problem. For this reason, depending on the fuel source, the catalyst used in

Proton Exchange Membrane Fuel Cell can be modified to increase resistance to CO poisoning.

In the review of Mehta and Cooper [12], different catalyst layer designs are investigated for

the anode of the Proton Exchange Membrane Fuel Cell [20].

To optimize the effectiveness of the catalyst layer and efficiency in power production, the sur-

face area of the platinum should be increased by depositing it onto a layer of carbon powder.

This method is commonly used in commercially available catalyst layers [19].

11

For the cathode catalyst layer, platinum dispersed on carbon or other small particles have

superior performance for oxygen reduction. At lower temperatures, oxygen reduction rates

will be slow and to compensate for this, catalyst loading should be increased resulting in

higher fuel cell costs [24].

Historically catalysts had 2 - 4 mg Pt/cm2 Pt loadings, but today loadings are generally less

that 1 mg Pt/cm2 with the same performance. Also the thickness of the catalysts layer is ap-

proximately 5 - 30 µm. Today, most of the current research activities are focused on lowering

the Pt loadings further as well as using less expensive active metals [20].

1.1.3.3 Gas Diffusion Layer

Gas diffusion layers (GDL) are typically porous carbon cloth or carbon fiber paper. They

are located between the bipolar plate and catalysts layer on both the anode and cathode sides

of the fuel cell. The purpose of the Gas Diffusion Layers, which are also known as Porous

Transport Layer, are listed below [19]:

• To serve as a structural support bridging the catalyst layer and bipolar plate

• To distribute the reactant gases to the catalyst layer

• To provide an interface where ionization occurs

• To manage the membrane temperature by removing heat from the reaction site when

needed

• To keep the anode hydrated (maintaining water management)

• To transfer electrons from catalysts to bipolar plate and vice versa

The Gas Diffusion Layer is generally coated with polytetrafluoroethylene (PTFE). PTFE is a

hydrophobic material, which helps with water management by inhibiting liquid water from

saturating the GDL. Also this material provides high resistance to chemical attack.

12

1.1.3.4 Bipolar plates

Bipolar plates are a vital component of a fuel cell system. According to Tsuchiya and

Kobayashi [25] bipolar plates account for about 80% of the total weight and 45% of stack

cost for a 50 kW fuel cell stack in 2000. Even after producing 5 million units with mass

production, it is estimated that bipolar plates will account for around 37% of the total fuel

cell cost. Therefore to reduce the high cost and high weight to reach a significant market

penetration, bipolar plates should be designed carefully. There are five functions of bipolar

plates [26]:

1. Distribution of reactants through the flow field channels

2. Separation of reactants between adjacent cells

3. Providing an electrically conductive path between the anode and cathode of adjacent

cells

4. Transferring heat out of the cell (cell cooling)

5. Providing structural support for the MEA

In order to balance all these desired functions, Mehta and Cooper [12] have suggested the

following goals:

• Electrical conductivity: plate resistance < 0.01 Ω cm−2

• Thermal conductivity: as high as possible

• Hydrogen/gas permeability: < 10−4 cm3/scm2

• Corrosion resistance: corrosion rate < 0.016 mA/cm2

• Compressive strength: > 22 lb/in2

• Density: < 5 g/cm2

13

For these reasons, the material of the bipolar plate should be chosen carefully. Until now, a

lot of work has been conducted to improve these functions by investigating different bipolar

plate materials. In general, materials investigated so far can be classified as: Non-metal; Non-

porous graphite/electrographite metals; Non-coated and coated composites; Polymer - carbon

and polymer - metal [27].

Early attempts were made to manufacture bipolar plates with metallic materials. Tawfik et

al. [28], Wind et al. [29] and Davies et al. [30] reviewed metallic bipolar plates, and their

advantages and disadvantages are explained in these reviews. Busick et al. [31] described a

low-cost composite material. Moreover, several new materials have been tried by Hentall et

al. [32] to improve important properties, especially to decrease the cost of the bipolar plate

and fuel cell.

1.2 Proton Exchange Membrane Fuel Cell Chemistry and Thermodynamics

This section is prepared based on the chapter of Fuel Cell Basic Chemistry in Electrochemistry

and Thermodynamics in Mini-Micro Fuel Cells: Fundamentals and Applications [33]

1.2.1 Basic Reactions and Thermodynamics

Electrochemical reactions occurs both at the anode and cathode of the fuel cell and these

reactions are called half-cell reactions. Fundamental electrochemical reactions of PEM fuel

cells are as follows:

• Anode Oxidation Reaction:

H2 → 2H+ + 2e− (1.1)

• Cathode Reduction Reaction:

1/2O2 + 2H+ + 2e− → H2O (1.2)

• Overall Reaction:

H2 + 1/2O2 → H2O (1.3)

14

These reactions are for only proton exchange membrane fuel cells; for the other types of fuel

cells, different fundamental reactions can occur. Eqn. (1.3) is identical to the combustion

reaction of hydrogen. Since combustion is an exothermic process, heat is generated due to

this reaction.

H2 + 1/2O2 → H2O + Heat (1.4)

The heat of a chemical reaction can be calculated by the difference between the heats of

formation of products and reactants. Therefore, the heat of reaction of Eqn. (1.4) is

∆H = (h f )H2O − (h f )H2 − 1/2(h f )O2 (1.5)

Due to the 2nd law of thermodynamics, not all of the heat of reaction can be converted into

useful work, which in our case is electricity.

The Gibbs Free Energy of a reaction is defined as

∆G = ∆H − T∆S (1.6)

where for a reversible reaction T∆S is the heat transfer to the reaction required by the 2nd

Law. For the reaction given by Eqn (1.3) ∆S < 0 and the heat transfer is out of the system. A

reversible reaction produces the maximum work given by

Wrev = −∆G = −∆H + T∆S (1.7)

For a reversible H2 Polymer Electrolyte Fuel Cell, Eqn. (1.7) indicates that of the total energy

released by the reaction −∆H, the portion −T∆S must leave the fuel cell by heat transfer to

satisfy the 2nd Law and thus is unavailable to do useful work.

∆H, ∆G and ∆S values of Eqn.(1.4) are given in Table 1.1. Also the enthalpy of reaction and

electricity generated can be calculated from this table.

15

Table 1.1: Enthalpies, entropies and Gibbs free energy for hydrogen oxidation reaction at25oC [1]

∆H(kJ/kmol) ∆S (kJ/kmol.K) ∆G(kJ/kmol)H2 + 1/2O2 → H2Oliq -286.02 -0.1633 -237.34H2 + 1/2O2 → H2Ogas -241.98 -0.0444 -228.74

The theoretical (reversible) potential (voltage difference) of fuel cell is:

Erev = −∆G/nF (1.8)

where,

• n: Number of electrons per molecule of H2 (2 kmol.e− / kmol H2)

• F: Faraday’s constant (96485 C/mol)

Therefore, at 25 oC, the reversible potential or voltage difference can be found as

Erev = 1.23V (1.9)

1.2.1.1 Effects of Temperature

The numerical values obtained above are calculated at a temperature of 25oC; however, if

the temperature is changed, every term will be affected by this change, because ∆H and ∆S

depend on temperature, so ∆G depends on temperature too. Consequently the reversible cell

voltage depends on temperature. The numerical values of ∆H, ∆G, ∆S and Erev of hydrogen

oxidation reaction for a proton exchange membrane fuel cell at different temperatures are

shown in Table 1.2.

Table 1.2: ∆H, ∆G, ∆S values with different temperatures and resulting Erev values [2]

T (K) ∆H(kJ/kmol) ∆S (kJ/kmol.K) ∆G(kJ/kmol) Erev(V)298.15 -286.02 -237.34 -0.1633 1.230333.15 -284.85 -231.63 -0.1598 1.200353.15 -284.18 -228.42 -0.1579 1.184373.15 -283.52 -225.24 -0.1562 1.167

16

As can be seen in Table 1.2, with temperature increasing from 25oC to 100oC, the theoretical

cell voltage decreases from 1.23 V to 1.167 V. Therefore voltage and temperature are inversely

related.

1.2.1.2 Reversible Fuel Cell Efficiency

The efficiency of any energy system can be found as the ratio of useful output energy to energy

input. This efficiency is also called the energy conversion efficiency. In a Proton Exchange

Membrane Fuel Cell, the useful energy output is the electricity produced and energy input is

the hydrogen oxidation reaction’s heat of reaction.

For theoretical (reversible) energy conversion via an electrochemical reaction, the electricity

produced is equal to the minus Gibbs free energy of the reaction. Therefore,

ηconv = ∆G/(−∆H) (1.10)

The theoretical efficiency depends on whether water is produced in the liquid or vapor phase.

The heating value is the amount of heat that may be generated by a complete combustion of

1 mole of hydrogen. The Higher Heating Value (HHV) corresponds to liquid water being

produced.

HHV = −∆H = 286.02 kJ/kmol (at 25oC) (1.11)

The Lower Heating Value (LHV) corresponds to vapor water being produced.

LHV = −∆H = 241.98 kJ/kmol (at 25oC) (1.12)

The difference between the HHV and LHV is about 45 kJ/kmol, which is the energy required

to evaporate liquid water.

Consequently, the maximum possible theoretical (reversible) efficiency of a fuel cell at 25oC

assuming water vapor is produced is:

ηrev = ∆G/∆H = 237.34/286.32 = 83% (1.13)

17

Internal combustion engines are the dominant technology currently. For internal combustion

engines, the energy conversion efficiency is defined by using the LHV of fuel. To compare

a fuel cell’s energy conversion efficiency to internal combustion engines, hydrogen’s lower

heating value is used to express the energy conversion efficiency.

ηrev = ∆G/∆H = 228.74/241.98 = 94.5% (1.14)

The energy conversion efficiency of fuel cells can be defined using either hydrogen’s HHV

or LHV based on whether liquid or vapor water is produced. In either case, the theoretical

(reversible) efficiency of fuel cells is much higher than that of internal combustion engines.

1.2.1.3 Effect of Pressure

A typical fuel cell can operate at pressures ranging from atmospheric pressure to 6 - 7 bar.

To understand more about reversible cell voltage, in addition to temperature, pressure effect

should be investigated.

For an isothermal process,

dG = vdP (1.15)

where,

• v: Molar Volume of species, m3/mol

For an ideal gas,

PV = RT (1.16)

Substituting Eqn. (1.16) into Eqn. (1.15),

dG = RTdPP

(1.17)

18

After integration,

G = G0 + RT lnPP0

(1.18)

Applying Eqn. (1.18) for hydrogen/oxygen reaction, assuming ideal gas behaviour;

∆G = ∆G0 + RT ln

PH2O

PH2 P0.5O2

(1.19)

Substituting Eqn. (1.19) into Eqn. (1.8) yields the Nerst Equation.

Erev = E0 +RTnF

lnPH2 P0.5

O2

PH2O(1.20)

where,

• P: Partial pressure of reactants and products

Note that, if liquid water is produced, then in Eqn. (1.20), PH2O = 1

Therefore, from Eqn. (1.20), higher reactant pressures result in higher cell potential; con-

versely if the reactants are diluted, the cell potential decreases.

1.3 PEM Fuel Cell Electrode Kinetics

This section is prepared based on the chapter Fuel Cell Basic Chemistry, Electrochemistry

and Thermodynamics in Mini-Micro Fuel Cells: Fundamentals and Applications [33]

The actual values of cell potential(V) and efficiency are always lower than the theoretical

values due to losses.

V = Erev −∑

∆Vlosses (1.21)

19

1.3.1 Voltage Losses

In a real fuel cell, the voltage losses are greater than zero for an open circuit and increase

monotonically with current density.

There are 4 different types of losses that affect the actual cell potential.

1.3.1.1 Activation Loss

A voltage gradient at the anode and cathode is required to drive each half reaction. The

relationship between current density and activation voltage loss is given by the Butler-Volmer

equation [34, 35].

i = i0

(exp

[−αredF(E − Erev)

RT

]− exp

[−αoxF(E − Erev)

RT

])(1.22)

Note that, by definition, the open circuit potential Erev at the anode is 0 and at the cathode is

1.23 V for 25 oC and atmospheric pressure [35].

Two important factors in the above equation that determine the current density and open

circuit potential are the transfer coefficient, α and exchange current density, i0.

Here, the transfer coefficient is an experimental parameter and in general

(αa + αc) =nν

(1.23)

where,

• n: number of electrons transferred in the overall reaction

• ν: is stoichiometric number defined as the number of times the rate-determining step

must occur for the overall reaction to occur once [36]

20

The exchange current density, i0, in electrochemical reactions is analogous to the rate constant

in chemical reactions. Several parameters determine the exchange current density including

electrode catalyst loading, catalyst, specie concentrations and temperature. The reference

exchange current density parameter, obtained at reference temperature and pressure, is [37]

i0 = ire f0 αcLc

Pr

Pre fr

γ exp(−

Ec

RT(1 −

TTre f

))

(1.24)

where,

• ire f0 : reference exchange current density (at reference temperature and pressure, typi-

cally 25 oC and 101.25 kPa) per unit catalyst surface area,

• αc: catalyst specific area,

• Lc: catalyst loading,

• Pr: reactant partial pressure, kPa,

• Pre fr : reference pressure, kPa,

• γ: pressure coefficient, 0.5 - 1,

• Ec: activation energy,

• R: gas constant, 8.314 J/(molK),

• T : temperature, K,

• Tre f : reference temperature, i.e. 298.15 K.

If the exchange current density is high, the surface of the electrode is more active. In a Proton

Exchange Membrane Fuel Cell, the exchange current density at the anode is much larger

than at the cathode. At a fixed voltage, increasing the exchange current density increases the

current generated. Therefore, for fixed voltage, increased exchange current density results in

decreased activation polarization losses. These losses happen at both the anode and cathode,

however oxygen reduction at the cathode requires much higher activation voltage loss than

hydrogen oxidation at anode due to slower reaction kinetics.

21

If the activation voltage loss is lower than the equilibrium potential value, such as those at the

fuel cell cathode, the first term in the Butler-Volmer Equation (Eqn. (1.22)) dominates;

∆Vact,c = Erev,c − Ec =RTαcF

ln(i

i0,c) (1.25)

Similarly, at the anode, where potentials higher that the equilibrium potential, the second term

in Eqn. (1.22) becomes predominant;

∆Vact,a = Erev,a − Ea =RTαaF

ln(i

i0,a) (1.26)

If the activation polarizations were the only losses in a fuel cell, and combining Eqns. (1.25)

and (1.26), the cell potential would be:

V = Erev −RTαcF

ln(i

i0,c) −

RTαaF

ln(i

i0,a) (1.27)

1.3.1.2 Ohmic (Resistive) Losses

Ohmic losses occur due to the electrical resistance of the electrode, ionic resistance to the

flow of ions in the electrolyte and contact resistance at the interface. Ohmic resistances are

expressed by Ohm’s Law:

∆Vohm = iRin (1.28)

where,

• Rin: Total cell internal resistance that includes ionic, electronic and contact resistances

Rin = Rin,ion + Rin,elec + Rin,contact (1.29)

Electrical resistances are usually negligible, even when graphite or graphite/polymer compos-

ites are used as bipolar plates. Ionic and contact resistances are approximately of the same

order of magnitude [37]

22

1.3.1.3 Concentration Loss

Hydrogen and oxygen are fed separately at the anode and cathode of a fuel cell, respectively.

Due to the decrease in their concentrations, there will be a slight pressure drop at the catalyst

layer as they are consumed as a result of a current being drawn.

The electrochemical reaction potential changes with concentration of the reactants, and this

relationship is given by the Nerst equation:

Vloss =RTnF

ln(CB

CS) (1.30)

where,

• CB: Bulk concentration of reactant,

• CS : Surface concentration of reactant at the catalyst

The flux of reactant (N) is proportional to the concentration gradient according to Fick’s Law;

N =D(CB −CS )

δA (1.31)

Moreover, with Faraday’s Law, at steady state the rate at which reactant species are consumed

is equal to the diffusion flux.

N =i

nF(1.32)

Combining Eqns. (1.31) and (1.32)

i =nFD(CB −CS )

δ(1.33)

While reactant is available at the catalyst surface, current can be obtained. The higher the

current density, the lower the surface concentration of reactant. But, when the surface con-

centration reaches zero, the diffusion rate reaches its maximum. The current density at this

limiting condition is the limiting current density:

23

iL =nFDCB

δ(1.34)

Combining Eqns. (1.30), (1.33) and (1.34) the voltage loss due to concentration polarization

can be obtained:

∆Vconc =RTnF

ln(iL

iL − i) (1.35)

Recently another approach has become more popular. This approach is not theoretical, instead

it is entirely empirical. The following empirical equation fits experimental results very well

[38].

∆Vconc = c exp(id

) (1.36)

where, c and d are empirical coefficients. Although they depend on cell design and operating

conditions, they have been suggested as 3 × 10−5 V and 0.125 A/cm2 respectively.

1.3.1.4 Internal Currents and Crossover Losses

In the ideal case, the polymer electrolyte membrane should not be electrically conductive

and should be impermeable to reactant gases. However, practically some small amount of

bulk hydrogen and electrons can pass through membrane. Each H2 molecule that passes from

anode to cathode reacts with oxygen at the cathode, resulting in two fewer electrons that travel

through the external circuit. Although, this loss can be negligible at higher current densities,

at open circuit, these losses can be significant.

1.3.2 Actual Fuel Cell Potential

Finally, if we consider the three different types of significant losses, the actual voltage is

V = Erev − (∆Vact + ∆Vconc)a − (∆Vact + ∆Vconc)c − ∆Vohm (1.37)

24

Substituting Eqns. (1.25), (1.26), (1.28) and (1.35) into (1.37),

V = Erev,T,P −RTαcF

ln(i

i0,c) −

RTαaF

ln(i

j0,a) −

RTnF

ln(iL,c

iL,c − i) −

RTnF

ln(iL,a

iL,a − i) − iRin (1.38)

The voltage losses in a Proton Exchange Membrane Fuel Cell can be shown graphically in

Fig. 1.5.

Figure 1.5: Voltage losses in a Proton Exchange Membrane Fuel Cell and resulting polariza-tion curve, [2]

1.4 Motivation

Before working on PEMFC’s in experimentally, it will be preferable to prepare several nu-

merical models. In this 2-D modeling of a PEM fuel cell study, the following are the main

motivation:

• Reducing cost of the project

• Flexibility in trying several different conditions

• Ease of convenience with foreseeing difficulties and other problems that would occur

in experimental work.

25

1.5 Overview of Thesis

The present work mainly includes two-dimensional mathematical modeling of a Proton Ex-

change Membrane Fuel Cell. In Chapter 1, background information about fuel cells and

fundamental relations for a Proton Exchange Membrane Fuel Cell is introduced. Important

models in the reviewed papers are discussed in Chapter 2. The fundamental equations of the

numerical modeling is explained in Chapter 3. How the present model was developed are

given in Chapter 4 in detail. Chapter 5 discusses the results of the model with figures and

graphics. Conclusions about the results and some future recommendations are given in Chap-

ter 6. In addition to the 2-D modeling work using Comsol Multiphysics presented in the main

part of this thesis, a separate 3-D CFD study was completed using Fluent to investigate the

effect of modifying the gas flow channel with disturbances. This CFD work is presented as

Appendix A.

26

CHAPTER 2

LITERATURE REVIEW

Since the early 1990s, there has been increasing interest in the modeling of Proton Exchange

Membrane Fuel Cells to understand the factors that affect fuel cell performance. The mod-

els have evolved from simple one-dimensional models to complex three-dimensional models.

Although a fuel cell has a simple design, indeed it is very complicated to model. Fuel cells is

a very interdisciplinary subject that includes material science, electrochemistry, heat transfer,

fluid mechanics, thermodynamics, etc. Also, to have satisfactory performance, optimization

of the operational condition is needed. Experiments can be used for finding optimum con-

ditions and obtaining the best performance. However, dealing with the various disciplines

makes isolation of one factor that is investigated experimentally very difficult.

Moreover, to develop a complicated experimental setup to test a fuel cell is an expensive and

time-consuming process. To overcome these problems, modeling can be used. Modeling

gives us flexibility in trying several different conditions while spending less money and time.

On the other hand, dealing with several phenomena that occur within various disciplines and

coupling fundamental equations for each discipline is a difficult task.

In the literature, there are different approaches for modeling. Some researchers concentrated

on general principles of a fuel cell to gain an understanding of the interactions and reactions,

while others focused more on a single component to improve this specific part. Also, these

approaches can be classified according to dimensionality and complexity of the modeling.

In the present work, the literature review is organized chronologically from relatively simple

models presented in papers from the early 1990s to more complex models presented more

recently.

27

Moreover, over 100 papers were initially reviewed, and of these 15 were considered directly

applicable to the present work. Only these 15 papers are discussed here with the help of two

reference dissertations and a review paper [39, 40, 41]. Additionally, the reviewed papers that

were published after 2004 tend to be too narrow for the present work. Therefore these most

recent papers are not discussed herein.

2.1 Modeling of Proton Exchange Membrane Fuel Cells

2.1.1 General Modeling Assumptions

The quality of a model is defined based on its assumptions. Scientists and engineers use

assumptions to understand the real world with modeling while neglecting issues that do not

significantly affect the actual results. Therefore to understand a model and to make comments

about its results and conclusions, it is important to understand all assumptions completely.

To avoid repetition in the following parts, general assumptions common to all of the papers

that are investigated are given as follows:

• The gas species and mixtures are considered as ideal and compressible.

• Electrodes, membrane and channels are assumed to be isotropic materials.

• The local heat transfer resistances of each phase (gaseous, liquid and solid) are assumed

to be zero, i.e. the temperature of solid/liquid/gaseous interface are always the same.

• Produced water from the reduction reaction is assumed to be a liquid.

• There is an equilibrium between liquid water and water vapor.

2.1.2 1991

The first important Proton Exchange Membrane Fuel Cell models were developed by both

Bernardi and Verbrugge [4] and Springer et al. [42] at the beginning of the 1990s with the

help of phosphoric acid fuel cell investigations and empirical studies previously completed.

28

In 1991, Bernardi and Verbrugge [4] published a one-dimensional, isothermal and steady-

state model for a single half cell. The system is considered to be composed of four regions as

seen in Fig. 2.1.

Figure 2.1: Gas diffusion electrode bonded to a solid polymer electrolyte [4]

The model consists of three main parts: 1) a fully hydrated membrane; 2) an active catalyst

layer which is formed by the overlap of gas diffusion electrode and membrane; and, 3) a

gas diffusion layer referred to as a gas diffuser. In addition to assuming membrane as fully

hydrated, other important assumptions are as follows:

1. Gas pressure within gas diffuser is constant.

2. Gas phase is in equilibrium with liquid-water phase both at gas diffuser and gas cham-

ber.

3. The gas pores in the active catalyst layer are hydrophobic (do not contain liquid water).

4. Volume fraction of gas pores in the active catalyst layer is uniform.

29

With these assumptions, it is concluded that, at low current densities, the reaction rate distribu-

tion is uniform throughout the catalyst layer volume. However, at higher current densities, the

reaction rate distribution is highly non-linear and most of the reactions occur in the portion of

the active catalyst layer near the oxygen-rich catalyst layer/gas diffuser interface. Therefore,

concentrating the catalyst on this interface may help to reduce the cost efficiently.

Almost at the same time, Springer et al. published another steady-state, one-dimensional and

isothermal Proton Exchange Membrane Model, which was the first model to model Nafion R©

117 membranes [42]. The model consists of a cathode gas channel, cathode gas diffuser,

membrane, anode diffuser and anode gas channel. The effects of different humidification

values were considered with an empirical correlation between the membrane water content

and the activity of Nafion R©, which is a function of relative humidity. Since anodic voltage and

concentration losses were negligible, this model only accounted for cathodic activation loss

and ohmic loss. Moreover, for the water transportation at the membrane, electro-osmotic drag

and diffusion were considered, which means that the pressure driven flow in the membrane

was ignored. Springer et al. used this model to understand the effects of stoichiometry and

humidifier temperature while investigating water transport in membrane.

2.1.3 1992

In this year Bernardi and Verbrugge [43] extended their original model published one year

before [4]. The biggest difference in this extended model is that it consists of complete cell

components for cathode and anode. Ohmic and activation polarization losses, membrane

dehydration and transport limitations are investigated as performance limiting factors. Like in

their first work, the model validation was made with a scale factor ”a” of the exchange current

density, expressed as aio,e f f so that experimental data matched with the results predicted by

their model. The effect of different membrane properties on the performance of the Proton

Exchange Membrane Fuel Cell was investigated.

30

2.1.4 1993

Springer et al. [44] published another paper which was an extension of their work in 1991.

This model considered gas transportation limitations in the cathode, which was not taken

into account previously. They observed that the anode losses for a well humidified Proton

Exchange Membrane Fuel Cell with pure hydrogen feed can be neglected. The model results

were validated with a correction factor.

After all of these one-dimensional approaches, again in this year Fuller and Newman [5] pub-

lished a steady-state, two-dimensional and non-isothermal model of the Membrane Electrode

Assembly (MEA). The model was solved in one-dimensional space and solutions were then

integrated along the channel length. Therefore the model may be described as a quasi-two-

dimensional model. Fuller and Newman assumed that the gas outside the gas diffusion layer

is uniform. They constructed the model with three parts. First, a cross-section of the unit

cell with constant temperature was considered. Second, the co-current flow of the air and

fuel streams was analyzed. In this case, as the air and fuel flow down, they are consumed and

water is produced. Third, temperature variation across the assembly is assumed to be zero and

no heat transfer along the channel direction is assumed. With constant cell potential input,

local current density and flux of water change, as can be in Fig. 2.2.

Figure 2.2: Net flux of water across membrane. A value of the flux less than zero indicatestransport from the anode to the cathode; f is the air-to-fuel ratio [5]

31

In the same year, Nguyen and White [45] developed a steady-state, two-dimensional and

non-isothermal model by considering convective water transport across the membrane by a

pressure gradient and heat removal by natural convection, co-current flow and counter flow

heat exchangers. The influence of the liquid water content on the ionic conductivity and the

enthalpy change due to phase change was taken into account. Heat transfer by conduction in

the gas phase was neglected. As a result, water concentration, temperature, partial pressure

and current density profiles were presented along the channel with different humidification

designs. Also voltage losses due to the oxygen reaction were discussed.

2.1.5 1996

Nguyen [46] investigated the effectiveness of an interdigitated flow field by experimentally

comparing the polarization curves of an interdigitated flow field with conventional parallel

design (Fig. 2.3).

Figure 2.3: Scheme of a PEM half-cell with conventional and the new flow field

32

With this new design, inlet and outlet of the channels become dead-ended so that reactant

gases are forced to pass through the gas diffusion layer. According to the results, the mass-

transport limited region in the V − i curve has been significantly extended, and the maximum

power density value was increased significantly.

2.1.6 1997

Broka and Ekdunge [6] compared two different approaches for catalyst layer modeling: a

pseudo-homogeneous film model and an agglomerate model. In the first approach, the cat-

alyst layer is modeled as a macropseudohomogeneous film consisting of four superimposed

media: a diffusion medium, an ionic conduction medium, an electronic conduction medium

and a catalytic medium where the electrochemical reactions take place. However, in the ag-

glomerate model the active layer is assumed to consist of a mixture of small agglomerates

of carbon, platinum and Nafion R©, that are separated by gas pores. According to simulation

results and Scanning Electron Microscopy observations, seen in Fig. 2.4, the active layer of

the electrodes shows an agglomerate structure. Also the agglomerate model better fits the

experimental data. Therefore an agglomeration is suggested.

Figure 2.4: Freeze-fractured cross-section of MEA: (a) Nafion R© 119 membrane, (b) impreg-nated active catalyst layer [6]

33

2.1.7 1998

A Proton Exchange Membrane Fuel Cell model based on computational fluid dynamics tech-

niques was published this year by Gurau, Liu and Kakac [7]. This was again a two-dimensional,

steady state and non-isothermal model. They defined the domain, gas channel, gas diffuser

and catalyst layer as coupled, so there was no need to define boundary conditions at the in-

terfaces. According to this property, this model was very important since it accounts for the

interaction between gas channels and the fuel cell itself. The important assumptions made in

here are:

• Laminar flow within the flow field

• Negligibly thin catalyst layer such that they are considered as interfaces

• No interaction between liquid water and water vapor

• All materials are considered as homogeneous

• Viscous dissipation is neglected

Moreover, as can be seen from Fig. 2.5, the domain was separated into three regions: humid-

ified hydrogen, humidified air and liquid water.

They found a non-linear oxygen mole fraction along the channel and tried to understand the

effects of several parameters such as temperature, porosity and fluid velocity on fuel cell

performance. To validate their model, as done previously, they used a scaling factor of aio,e f f

in the Butler-Volmer equation for matching their polarization curve to experimental data.

34

Figure 2.5: Geometry, coordinate axes, boundary types and characteristic dimensions of thethree domains [7]

2.1.8 2000

A transient Proton Exchange Membrane Fuel Cell model was published by Um et al. [8]. This

was a similar CFD model as in Gurau et al. [7]. The importance of this model was that the

transient behavior of a Proton Exchange Membrane Fuel Cell was investigated by changing

the cell potential, and the reacting time of the current density was shown to be on the order of

several seconds. Also by using Henry Law, the difference in oxygen concentration between

the liquid and the gas phases was accounted for. Additionally, Um et al. realized that cell

voltage decreases when reformate hydrogen is used instead of pure hydrogen.

35

Three dimensional models started to be published in 2000. A three dimensional, steady state

and isothermal model was published by Dutta et al. [47]. The commercial software Fluent

(4.48) was used with a modification of source addition in the electrochemical reactions. To

overcome the much more computational time intensive problem in three dimensional analysis,

they used a solver procedure as shown in Fig. 2.6.

Figure 2.6: Flow diagram of the solution procedure used [8]

They conclude that the diffusion layers help create a larger reaction area and thus results in

higher current density although they are resistances to convective flow.

2.1.9 2002

A comprehensive, three-dimensional and non-isothermal Proton Exchange Membrane Fuel

Cell model has been developed by Berning et al. [48]. Like that used in Dutta et al. [47], this

model considered the anode, cathode and membrane, and the catalyst layers were assumed as

interfaces. Although, it is a single-phase model, all major transport phenomena were taken

into account. As a result, three-dimensional reactant gas, temperature distribution and net flux

of water through the membrane are given. Also, the authors suggested a non-homogeneous

catalyst layer to solve the non-uniformity problem of current density.

36

2.1.10 2003

Siegel et al. [9] published a two-dimensional steady-state model, which includes liquid water

transport within the pores of the electrode. They used the agglomerate catalyst geometry

(Fig. 2.7) to model the catalyst layers where the electrochemical reactions occur. They used

a 50cm2 ElectroChem cell operating at various conditions to validate their model.

Figure 2.7: Agglomerate catalyst geometry [9]

They concluded that the void fraction of the catalyst layer has a large effect on the fuel cell

performance. Moreover, increasing the characteristic length results in a decrease in perfor-

mance due to the diffusive resistance to reactant flow.

In the same year, Natarajan and Nguyen [49] expanded their previous two-dimensional model

to a three-dimensional model to better understand the fundamental principles along the length

of the channel. At the gas channels, only convection was considered, however in the diffusion

layer, diffusion was assumed to be the only mechanism for transport. After validation of this

model, they concluded that oxygen concentration variation in gas channels due to reaction

and dilution caused by water evaporation strongly affected the variation of current density

along the channel. Therefore, controlling the operating conditions like higher temperature

and lower inlet humidity to get better water removal increases the cell performance.

37

2.1.11 2004

Investigations in the relation between mass transport and fuel cell performance using straight

and iterdigitated flow fields were carried out by Um and Wang [10]. This model was based

on extending their previous work done in 2000 [8] to three-dimensions. Model validation

was done like in the previous work, but also an extra experimental polarization curve taken

from the Penn State Electrochemical Engine Center was added. Finally, they indicated that

interdigitated flow field improves mass transport of oxygen and thus increases fuel cell per-

formance by increasing mass transport limiting current density as seen in Fig. 2.8.

Figure 2.8: Various polarization curves at Tcell = 80oC, and the anode/cathode stoichiometryof 1.5 and 1.8 at 1.0A/cm2 [10]

2.1.12 After 2004

After 2004, the reviewed papers tend to be too specific for the present work [50, 51, 52]. The

models presented in these papers used the fundamental approaches and equations stated in

the reviewed papers in the previous sections and extended their models to meet more specific

needs. Therefore to understand the main issues about Proton Exchange Membrane Fuel Cell

modeling, the papers after 2004 are not necessary.

38

2.1.13 Summary of Literature Review

According to the models presented in the papers reviewed, an agglomerate approach for the

modeling of the catalyst layer is more realistic than other approaches. The biggest difference

of three-dimensional modeling of a Proton Exchange Membrane Fuel Cell is not only having

more detailed results but also understanding the effects of species dissipation along the chan-

nels. However, three-dimensional modeling is a very time consuming process due to having

more complex geometry and higher number of degrees of freedom.

Therefore for the present study, the agglomerate approach is used for catalyst layer modeling.

To eliminate the longer convergence times for a three-dimensional model, a two-dimensional

model for all of the main components of a Proton Exchange Membrane Fuel Cell are used with

two different types of models: First, the species depletion is investigated in a cross-sectional

model which is parallel to the flow. Second, a cross-sectional model, perpendicular to the

flow direction is developed by using the outputs from the first model to understand the effect

of species depletion. So, by eliminating the longer convergence time of a three-dimensional

model, species depletion can be investigated with the help of two two-dimensional models.

Both of these models are developed with COMSOL 3.5a commercial software.

39

CHAPTER 3

NUMERICAL MODELING OF A PEM FUEL CELL

This study is based on numerical modeling of a proton exchange membrane fuel cell with a

serpentine flow field design. In particular, it consists of two different types of cross-sectional

models. The first model is for a cross-section perpendicular to the flow in the channel and

is termed the perpendicular-to-flow model herein, The second model is for a cross-section

parallel to the flow in the flow channel along the channel center line and is termed the parallel-

to-flow model (Fig. 3.1). Although they do not have the same subdomains, they use the same

approach and the same fundamental mathematical equations.

Humidified hydrogen and humidified air are sent from the channel inlets. At the anode, hydro-

gen reacts with the catalyst layer and is consumed to produce protons that carry ionic current

to the cathode. In the present study, each proton is assumed to drag one molecule of water

from anode to cathode. At the cathode, these protons react with oxygen to form water at the

catalyst layer.

This is a single-phase model, so there is only the gaseous phase of species (e.g., water in

the liquid phase is neglected). Both humidified hydrogen and humidified air are considered

as ideal gases and transported via convection and diffusion. The electrode and channel have

homogeneous properties such as porosity, permeability and conductivity.

For the catalyst layer reactions, an agglomerate model is used. The catalyst layer is assumed

to contain small agglomerates consisting of carbon, catalyst metal and electrolyte that are

separated by gas pores. The reactants from the diffusion layer diffuse first through the gas

pores, then through the electrolyte to reach the catalyst sites. This approach originated from

an analytical solution of a diffusion-reaction problem in a spherical porous particle [53].

40

(a) Cross-sectional modeling

(b) Along-the-channel modeling

Figure 3.1: Two different types of modeling

3.1 Charge Balances

To describe the electrical potential distribution in a PEM fuel cell, conservation of charge

equations are used.

∇ · (−κs∇φs) = 0 (3.1)

∇ · (−κm∇φm) = 0 (3.2)

41

Here, κs is the solid-phase electronic conductivity (S/m) and κm is the membrane ionic con-

ductivity (S/m). The potential (V) across the membrane is denoted by φm and that between

the electrodes by φs. This means that Eqn. (3.1) is defined at C and E in Fig. 3.1, and Eqn.

(3.2) is used for region D.

Like most of the work investigated in the literature review, this work uses the Butler-Volmer

Equation (Eqn. (1.22)) as a boundary condition at the catalyst layer to obtain the relation

between voltage and current density.

For the membrane, the inward normal ionic current densities at the anode and cathode bound-

aries, ia and ic, are defined as follows [54]:

ie = Lact(1 − εmac) jagg,e (3.3)

where;

• index e stands for ’a’ (anode) and ’c’ (cathode)

• Lact is the active layer thickness (m)

• εmac is the macroscopic porosity

• iagg,e is the current densities given by agglomerate model

In the agglomerate model, the current density of the active catalyst layer can be described

locally, because there are agglomerates of ionic conductor materials and electrically conduct-

ing particles covered partially with catalyst. This local current density can be expressed with

an analytical solution of the diffusion equation and the Butler-Volmer Equation (Eqn. (1.22))

[54]:

iagg,e = −6neDagg

R2agg

(1 − λe coth λe)βe (3.4)

In Eqn. (3.4), λ and β values for the anode and cathode should be known. In this model, they

can be found separately with the formulation given below.

42

λa =

√i0aS R2

agg

2FcH2,re f Dagg(3.5)

λc =

√i0cS R2

agg

2FcO2,re f Daggexp(−

F2RT

ηc) (3.6)

βa = [cH2,re f − cH2,re f exp(−2FRT

ηa)] (3.7)

βc = cO2,agg (3.8)

where,

• Dagg is the agglomerate gas diffusivity (m2/s)

• Ragg is the agglomerate radius (m)

• ne is the charge transfer number (1 for H+ and −2 for O−22 )

• S is the specific area of the catalyst inside the agglomerate (1/m)

• F is the Faraday’s constant (C/mol)

• ci,re f are the reference concentrations of the species (mol/m3)

• ci,agg are the corresponding concentrations in the agglomerate surface (mol/m3)

• i0a and i0c are the exchange current densities (A/m2)

• R is the gas constant (J/(molK))

• T is the temperature (K)

Moreover, the overvoltages at the anode and cathode (ηa and ηc) are given by

ηa = φs − φm − Eeq,a ηc = φs − φm − Eeq,c (3.9)

where Eeq,a denotes the equilibrium voltage.

43

While oxygen and hydrogen are being consumed by the reaction, the adsorbed oxygen and

hydrogen concentrations at the surface of the agglomerate in the catalyst layer are related to

the molar fractions of the respective species in the gas phase which is calculated by Henry’s

law [54].

cagg,H2 =pH xH

KHcagg,O2 =

pOxO

KO(3.10)

where, K is the Henry’s constant (Pa.m3/mol).

For the electric potential of the electrodes, all of the boundary conditions are modeled as an

electric insulation boundary condition except the boundaries that are having contact with the

current collector plates as shown in the Fig. 3.1-a (B and G). The potential difference between

the anode and cathode current collector plates gives the total cell potential. The voltage at the

anode is set as to a reference value of 0 while the voltage at the cathode is inputted as Vcell. The

cell voltage is the difference between these voltages and therefore is an independent parameter

in this model and the current density is a dependent parameter. The difference becomes the

cell potential.

φs = 0 at B/C Fig. 3.1-a φs = Vcell at G/E Fig. 3.1-a (3.11)

3.2 Porous Media Flow for Perpendicular-to-Flow Model

In this model, Darcy’s law is used for the porous media flow. In a porous medium the transport

of momentum caused by shear stress in the fluid is neglected because pore walls prevent

momentum transport to the fluid. The continuity equation and equation of state for the fluid

flowing in the pores provide a complete mathematical model for porous media flows for which

the pressure gradient is the major driving force.

The gas velocity in A and F (Fig. 3.1-a) is given by the continuity equation.

∇.(ρu) = 0 (3.12)

where ρ is the density of the gas mixture (kg/m3) and u denotes the gas velocity (m/s). Ac-

cording the Darcy’s law, the velocity depends on the gradient of pressure, the viscosity of the

44

fluid and the structure of the porous media [55];

u = −κp

η∇p (3.13)

where,

• κp is the electrode’s permeability (m2)

• η is the gas viscosity (Pa.s)

• p is the pressure (Pa)

Moreover, the density of the gas mixture can be found by

ρmix =p

RT

∑i

Mixi (3.14)

where x is the mole fraction.

At the electrode boundary, between C/D and E/D, for the anode and cathode respectively (Fig.

3.1-a), the gas velocity is calculated from the total mass flow given by the electrochemical

reaction rate [54].

−n.u |anode=ianode

ρF

(MH2

2+ λH2OMH2O

)(3.15)

−n.u |cathode=icathode

ρF

(MO2

4+ (

12

+ λH2O)MH2O

)(3.16)

where λH2O is the drag coefficient of the water. In this model each proton is assumed to drag

one molecule of water from anode to cathode. So λH2O is constant and equal to 1.

45

3.3 Porous Media Flow for Parallel-to-Flow Model

For the parallel-to-flow model, The Brinkman Equations are used to define the porous media

flow. However, as discussed above, for the perpendicular-to-flow model, Darcy’s Law was

used. To understand the reason for this difference in modeling approaches one must under-

stand the Brinkman Equations in depth and the available libraries in COMSOL.

The Brinkman Equations are used for porous media flows, for which the momentum transfer

within the fluid due to shear stresses is of importance. This means that this equations is

an extended version of Darcy’s Law to account for the viscous transport in the momentum

balance. More importantly, The Brinkman Equations application mode in COMSOL makes it

very easy to model systems consisting of porous media and free flow domains. If the channels

would like to be investigated in a model like in Fig. 3.1-b, this application mode should be

used. Also, by using this application mode, one eliminates the need for describing boundary

condition for the interface between A/C and E/F (interface of free flow and porous media flow,

Fig. 3.2). This application mode allows the user to model both domains as a single integrated

domain. So the interface is not considered as a boundary condition.

Figure 3.2: Modeled domain and boundary conditions for coupled free flow and porous mediaflow [11]

46

Flow in the free channel is described by the Navier-Stokes equations [55].

∇.(ρmixu) = 0 (3.17)

∇.[−η

(∇u + (∇u)T

)+ pI

]= −ρmix(u.∇)u (3.18)

where,

• η is the dynamic viscosity of the fluid (Pa.s).

• u is the fluid velocity (m/s).

• ρmix is the density of the fluid mixture (kg/m3).

• p is the pressure (Pa).

In the porous domain, The Brinkman equations describe the flow [55];

∇.(ρmixu) = 0 (3.19)

∇.

[−η

εp

(∇u + (∇u)T

)+ pI

]= −

η

κpu (3.20)

Here,

• κ denotes permeability of the porous medium (m2).

• εp denotes porosity which is dimensionless.

3.4 Maxwell-Stefan Mass Transport

Fick’s Law, which is used more commonly for mass transportation, is based on the assumption

that species dissolved in a solution interact only with the solvent. While defining diffusion

coefficients, such interactions do not depend on concentrations. However, in concentrated

solutions or gas mixtures, where relative concentrations are of the same order of magnitude,

47

all species interact with each other and therefore their diffusion coefficients depend on con-

centrations in addition to temperature and pressure.

In this model, there are two species, H2 and H2O, at the anode and three species, O2, N2 and

H2O, at the cathode. According to the Maxwell-Stefan mass transport theory, Maxwell-Stefan

multi-component diffusion is governed by the equation given below [55].

∂

∂tρωi + ∇.

−ρωi

N∑j=1

Di j

MM j

(∇ω j + ω j

∇MM

)+ (x j − w j)

∇pp

+ ωiρu + DT

i∇TT

= Ri

(3.21)

The computational modeling is seeking mass fraction, ωi. For this model, the Ri source term

is assumed to be zero, since temperature driven diffusion can be neglected. Also, since this is

a steady-state model, the very first term in Eqn. 3.21 goes to zero. For example, at the cathode

with three species (oxygen=1, water=2 and nitrogen=3), the mass transport is described by

the following three equations [54]:

∇.

−ρω1

∑j

[D1 j(∇x j + (x j − ω j)(∇p/p))]

= −(ρu.∇ω1) (3.22)

∇.

−ρω2

∑j

[D2 j(∇x j + (x j − ω j)(∇p/p))]

= −(ρu.∇ω2) (3.23)

ω3 = 1 − ω1 − ω2 (3.24)

Here, Di j is the Maxwell-Stefan diffusivity matrix (m2/s), which is calculated from the binary

diffusivities specified in the mathematical model. Also, at the inlets, feed-gas mass fractions

are given as inputs and at the outlets, the boundary condition is set to a convective boundary

condition, which means that the flux at the outlet is dominated by the convection. At the

anode, the mass flux of hydrogen and at the cathode the mass fluxes of oxygen and water

are determined by the electrochemical reactions as follows and they are used as a boundary

condition at the electrode-membrane boundary for modeling the mass transportation [54].

48

−n.NH2 |anode=ianode

2FMH2 (3.25)

−n.NO2 |cathode=icathode

4FMO2 (3.26)

−n.NH2O |cathode=icathode

F

(12

+ λH2O

)MH2O (3.27)

49

CHAPTER 4

MODELING ANALYSIS

In Chapter 3 the fundamental equations for mathematical modeling are introduced. In this

chapter, how this model was developed in COMSOL commercial software is explained. Val-

ues for each input, boundary conditions and subdomain settings for each application are pre-

sented.

Constants that are used for both perpendicular-to-flow and parallel-to-flow models are pre-

sented in Fig. 4.1.

As can be seen in Fig. 4.1, it is important to emphasize that cell potential is given as an input,

and current density is obtained as an output. Inlet weight fractions of species are given. At

the cathode, it is assumed that oxygen consists of 21% of the dry air by volume.

The density of the gas mixture in anode and cathode are defined as follows, respectively;

ρmix =p

RT(xH2 MH2 + xH2Oa MH2O

)at the anode (4.1)

ρmix =p

RT(xO2 MO2 + xH2Oc MH2O + xN2 MN2

)at the anode (4.2)

50

(a) Constants Part 1

(b) Constants Part 2

Figure 4.1: Constants that are used for the modeling in COMSOL

51

4.1 Perpendicular-to-Flow Model

In this model, there are three different application modes used for defining the fundamental

phenomena occurring in a simple standard fuel cell. The first mode, Conductive Media DC,

is used for charge balances and modeling the cell potential (vs) current density relation in the

cell. The second mode is Darcy’s Law, which is used for porous media flow within electrodes.

The last mode is Maxwell-Stefan Diffusion and Convection Mode.

A schematic view of the Perpendicular-to-Flow model is shown in Fig. 4.2. As understood

from the figure, serpentine or parallel flow field is used for current collector plates. Adjacent

channels feed the electrode and membrane with input species. Also the geometrical properties

of this figure can be seen in Table 4.1.

Figure 4.2: A Schematic View of the Perpendicular-to-Flow Model

The driving force of Darcy’s Law for porous media flow can be a pressure difference along the

flow direction or a concentration gradient (partial pressure gradient). Here, a concentration

gradients between the channel and the catalyst layer are developed Fig. 4.2. At the inlets,

mass fractions are defined as boundary conditions. At this point, it is important to emphasize

that these mass fractions are taken from the parallel-to-flow model and the position along the

channel determines the inlet weight fractions of the species.

52

Therefore, conclusions about species depletion or 3-D aspects of the Proton Exchange Mem-

brane Fuel Cell modeling can be obtained.

Table 4.1: Geometrical properties of Perpendicular-to-Flow Model

Object Dimensions Electrode Membrane Current Collector at the interfaceWidth 0.25mm 0.1mm Not ApplicableHeigth 2mm 2mm 1mm

4.1.1 Conductive Media DC Application Mode

This mode is defined for the two subdomains: Electrodes and membrane.

4.1.1.1 Electrodes

Only the subdomain settings for the electrodes are defined, and the membrane is deactivated

as shown in Fig. 4.3. Also in this figure, the boundaries for the perpendicular-to-flow model

are numerated to clearly communicate the boundary conditions for each application mode.

Figure 4.3: Boundary Numeration for the perpendicular-to-flow model

53

In this mode, the solid phase electrical conductivity (κs), which is given as a constant, is

defined for the conductivity relation. The boundary conditions are given in Table 4.2 below.

Table 4.2: Boundary Conditions for the Conductive Media DC (Electrodes) Application Mode

Settings Boundaries 1 − 6,14 − 15, 18 − 19,21 − 22

Boundary8

Boundary10

Boundary13

Boundary17

Type Electric Insulation ElectricPotential

InwardCurrentFlow

InwardCurrentFlow

ElectricPotential

Value 0 −ia −ic Vcell

4.1.1.2 Membrane

In this application mode, κm, ionic conductivity, is used for defining the conductivity relation.

Only the membrane subdomain is selected and the other two electrodes are deactivated. The

boundary conditions are given in Table 4.3 below.

Table 4.3: Boundary Conditions for the Conductive Media DC (Membrane) ApplicationMode

Settings Boundary 10 Boundaries11-12

Boundary 13

Type Inward CurrentFlow

ElectricInsulation

Inward CurrentFlow

Value ia ic

4.1.2 Darcy’s Law Application Mode

The electrodes are selected to set the subdomains and the membrane is deactivated as shown

in Fig. 4.3. For the inputs, κp, solid phase conductivity, η, permeability of the electrodes, and

ρmix density of the gas mixture (Eqn. 4.1 and Eqn. 4.2) are given. The boundary conditions

for this model are in Table 4.4.

54

Table 4.4: Boundary Conditions for the Darcy’s Law Application Mode

Settings Boundary4

Boundaries1, 22

Boundary21

Boundary10

Boundary13

Type PressureCondition

PressureCondition

PressureCondition

Inflow/

OutflowInflow/

Outflowp0 pre f pre f pre f

u0 −ua uc

In Table 4.4, ua and uc are boundary expressions defined at boundaries 10 and 13 respectively

(Fig. 4.3);

ua = ia( MH2

2+ λH2OMH2O

)/(Fρmix) (4.3)

uc = ic( MO2

4− (0.5 + λH2O)MH2O

)/(Fρmix) (4.4)

All of the remaining boundaries within defined subdomains are set as insulation/symmetry

boundary condition by default.

4.1.3 Maxwell-Stefan Diffusion and Convection Application Mode

This application mode is used for anode and cathode electrodes separately. For the inputs, the

density of the gas mixture for anode and cathode are defined (Eqn. 4.1 and Eqn. 4.2). The

velocity values are taken from the Darcy’s Law Application Mode.

4.1.3.1 Maxwell-Stefan for Anode Species

Boundary conditions are given in Table 4.5.

Table 4.5: Boundary Conditions for the Maxwell-Stefan Diffusion and Convection Applica-tion Mode for the Anode

Settings Boundary 1 Boundary 4 Boundary 10Type Mass Fraction Mass Fraction FluxωH2 ωH2,in ωH2,in

η0 −iaMH2/(2F)

55

All of the remaining boundaries within defined subdomains are set as insulation/symmetry

boundary condition by default.

4.1.3.2 Maxwell-Stefan for Cathode Species

Table 4.6 shows the boundary conditions for Maxwell-Stefan application mode at the cathode

electrode.

Table 4.6: Boundary Conditions for the Maxwell-Stefan Diffusion and Convection Applica-tion Mode for the Cathode

Settings Boundary 21 Boundary 22 Boundary 13Type Mass Fraction Mass Fraction FluxωO2 ωO2,in ωO2,in

ωH2Oc ωH2Oc,in ωH2Oc,in

η0(ωO2) −icMO2/(4F)η0(ωH2Oc) −icMH2O(0.5 +λ)/F

All of the remaining boundaries within defined subdomains are set as insulation/symmetry

boundary condition by default.

4.2 Parallel-to-Flow Model

In this model everything is the same with the cross-sectional model except instead of using

the Darcy’s Law application mode for porous media flow, the Brinkman Equation application

mode is used due to the fact that in this case there is a combination of free flow in the channel

and porous media flow (Fig. 3.2). The Brinkman Equation application mode in COMSOL

software is used to couple these two flows.

56

Figure 4.4: Schematical View of the Parallel-to-Flow Model

As seen from Fig. 4.4, this model contains the straight open channel model of the serpentine

flow field. Instead of having several turns (Fig. 4.5), one straight open channel is investigated

to understand the effect of species depletion on the performance of the fuel cell.

Figure 4.5: Current collector plate with serpentine flow field channel

The geometrical properties of Fig. 4.4 are given in Table 4.7.

57

Table 4.7: Geometrical properties of Parallel-to-Flow Model

Object Dimensions Electrode Membrane Flow ChannelsWidth 50mm 50mm 50mmHeight 0.25mm 0.1mm 0.75mm

4.2.1 Incompressible Navier-Stokes Application Mode with Brinkman Equations

Both flow channels and electrodes are selected for subdomain settings at the anode and cath-

ode. Since there is a combination of free flow and porous media flow, a pressure difference

should be created between the inlet and the exit of the flow channels. In this model;

pa,in = pc,in = 1.00001pre f (4.5)

As noted the pressure difference is very low. In the present study, the aim is to investigate

the effects of species depletion. In a single Proton Exchange Membrane Fuel Cell, species

depletion can be avoided by controlling operating conditions. However this phenomenon is

very important in fuel cell stacks in which single fuel cells are connected serially (Fig. 4.6).

Especially at the mid-cells, the pressure difference can be difficult to control and may be small,

which results in having a restricted amount of species in the flow channels. For this reason,

in this study, a very low pressure difference is given as an input to assure a restricted amount

of species to investigate species depletion. Although a single Proton Exchange Membrane

Fuel Cell is modeled, the overall goal is to demonstrate the species depletion phenomenon

that generally occurs in fuel cell stacks.

Figure 4.6: Schematic of a fuel cell stack [12]

58

Also, as an input, ε, porosity and κp, permeability are given for the electrode subdomains at

the anode and cathode.

Boundary conditions are given in Table 4.8.

Table 4.8: Boundary Conditions for the Incompressible Navier-Stokes Application Mode withBrinkman Equations

Settings Boundary 1 Boundaries10, 16

Boundary 12

Type Inlet Pressure Exit Pressure Inlet pressureValue pa,in pre f pc,in

All of the boundary conditions other than those given in Table 4.8 are no slip condition for

the wall type boundary condition.

The only difference between this model and the perpendicular-to-flow model is the use of the

Brinkman Equation application mode. The details of the other application modes are identical

and are not repeated for brevity.

59

4.3 Meshing

4.3.1 Perpendicular-to-Flow Model

For this model, free mesh parameters are defined for critical boundaries and points as shown

in Fig. 4.7.

Figure 4.7: Critical points and boundaries for free mesh parameters

• Maximum element size for critical points shown in Fig. 4.7 is 1e−5m.

• Maximum element size for critical boundaries of 10 and 13 shown in Fig. 4.7 is 5e−5m.

The general mesh statistics for the perpendicular-to-flow model are given in Fig. 4.8.

Figure 4.8: Mesh Statistics for Perpendicular-to-Flow Model

60

4.3.2 Parallel-to-Flow Model

For this model,

• Maximum element size for boundaries 5, 6, 7 and 13 shown in Fig. 4.4 is 5e−4m

• Maximum element size for boundaries 4 and 14 shown in Fig. 4.4 is 2e−4m

The general mesh statistics for the parallel-to-flow model are given in Fig. 4.9.

Figure 4.9: Mesh Statistics for Parallel-to-Flow Model

4.4 Solution Procedure

COMSOL uses a linear system solver with a maximum of 25 iterations and 10−6 relative tol-

erance. To obtain quick convergence a sequence arrangement is operated. First, Conductive

Media DC application modes are solved. Second, all of the application modes are solved with

the converged initial values obtained from the previous solution. Such a sequence develop-

ment are very useful to speed up the convergence time.

61

CHAPTER 5

RESULTS AND DISCUSSIONS

As explained in Chapter 4, the parallel-to-flow model is run to estimate the weight fractions of

each species (H2 and H2Oa for the anode and O2, H2Oc and N2 for the cathode) along the flow

channel. As the species pass along the channel the weight fractions change due to reactions.

Additionally, for both of the present models, the cell potential is given as an input and models

are solved for current density. Therefore to investigate the effect of species depletion along

the channel, the parallel-to-flow model is run for voltage inputs ranging between 0.3 - 1.2V in

0.1V increments. According to the parallel-to-flow model, the weight fractions of H2 at the

anode side and O2 and H2Oc at the cathode side as a function of channel position are given

in Figs. 5.1, 5.2 and 5.3. To represent the general trend, voltage values of 0.4, 0.7 and 1V are

selected to be shown in these figures.

According to Figs. 5.1, 5.2 and 5.3;

• As potential decreases, H2 and O2 depletion increases as expected. If reactants are

present, lowering the potential causes an increase in current density and current density

is proportional to reaction rate.

• If the potential is decreased enough, for example to V = 0.4V , O2 is completely con-

sumed before the cathode flow reaches the end of the channel. Specifically, O2 is almost

completely consumed by the midpoint of the channel (Fig. 5.1-b). This means that fuel

is not being used. Since O2 is depleted as it flows, the counter flowing H2 cannot react

until the last half of the channel (Fig. 5.1-a).

62

(a) Weight fraction of H2 at the anode

(b) Weight fraction of O2 at the cathode

(c) Weight fraction of H2O at the cathode

Figure 5.1: Weight fractions of H2, O2 and H2Oc for V = 0.4V with respect to channelposition x [m]

63

(a) Weight fraction of H2 at the anode

(b) Weight fraction of O2 at the cathode

(c) Weight fraction of H2O at the cathode

Figure 5.2: Weight fractions of H2, O2 and H2Oc for V = 0.7V with respect to channelposition x [m]

64

(a) Weight fraction of H2 at the anode

(b) Weight fraction of O2 at the cathode

(c) Weight fraction of H2O at the cathode

Figure 5.3: Weight fractions of H2, O2 and H2Oc for V = 1V with respect to channel positionx [m]

65

Moreover, with the parallel-to-flow model, one can obtain the current density variation over

the catalyst layer for each voltage value. For example, the current density variation along the

anode catalyst layer for V = 0.7V is shown in Fig. 5.4.

Figure 5.4: Current density variation with respect to channel position of x [m] for V = 0.7V

For each potential value ranging between 0.3 - 1.2V in 0.1V increments, spatial current density

variations can be obtained, similar to that in Fig. 5.4. By integrating current densities over

the channel length of 5cm, an integrated current density for each potential value is obtained.

The potential versus current density graph is known as the performance curve (or polarization

curve) of a fuel cell. The performance curve of the parallel-to-flow model for the integrated

current density of a Proton Exchange Membrane Fuel Cell is given in Fig. 5.5.

Comparing Fig. 5.5 with Fig. 1.4, in the present model activation (∼ 0-250 A/m2) and ohmic

(∼ 250-1000 A/m2) and concentration loss (∼ 1000-1150 A/m2) regions can be identified.

66

Figure 5.5: Performance curve (V - i curve) of the parallel-to-flow model of a Proton Ex-change Membrane Fuel Cell

Additionally, to further investigate the effect of species depletion on the local current densities,

the three positions x = 0, x = 0.025m and x = 0.05m are selected in the parallel-to-flow model

as shown in Fig. 5.6.

Figure 5.6: Selected positions to be investigated for the parallel-to-flow model.

67

For these 3 positions and for each potential value, weight fractions of H2, O2 and H2Oc from

the parallel-to-flow model are used as inputs to the perpendicular-to-flow model. Therefore,

for each position along the channel for the parallel-to-flow model, current density variations

over the catalyst layer for the perpendicular-to-flow model can be obtained. In this case, 3

more positions are selected to investigate local current densities along y-direction as shown in

Fig. 5.7. Along the y-direction, lines of symmetry occur at y = 0, 1, 2, . . .mm. Therefore only

one half of the perpendicular-to-flow model along y-direction is necessary. For this reason,

the local points of y = 0, y = 0.5mm and y = 1mm are selected for more detailed investigation.

Another significance of these points is that y = 0 represents the catalyst beneath the center of

the flow channel, y = 1mm represents the catalyst beneath the center of the land (bipolar plate)

and y = 0.5mm represents the catalyst beneath the interface of the land and flow channel.

Figure 5.7: Selected positions to be investigated for the perpendicular-to-flow model

For each x = 0, x = 0.025m and x = 0.05m positions of the channel, cell potential (V) versus

local current density (i) graphs are given in Fig. 5.8.

68

(a) Cell Potential (vs) Local Current Density Graph for x = 0

(b) Cell Potential (vs) Local Current Density Graph for x = 0.025m

(c) Cell Potential (vs) Local Current Density Graph for x = 0.05m

Figure 5.8: Cell Potential (vs) Local Current Density Graphs for each x = 0, x = 0.025m andx = 0.05m positions of the parallel-to-flow model

69

As can be understood from Fig. 5.8, along the y-direction, y = 0, which is the beneath the

center of the flow channel, has the best performance when compared to the positions beneath

the center of the land and interface of the land and channel. Although at the higher voltages

the three y-positions have almost the same current density trends, at lower voltages the local

current densities beneath the channel are significantly higher than those beneath the land.

Moreover, for a counter flow Proton Exchange Membrane Fuel Cell, relatively good perfor-

mance results are obtained at the exit of the H2 channel, which is the inlet of the O2 (Fig.

5.8-c). The same result can be understood from Fig. 5.10, the graph of cell potential versus

integrated current density over the y-direction for x = 0, x = 0.025m and x = 0.05m positions.

The present model can be validated with the experimental data of an air-breathing Proton Ex-

change Membrane Fuel Cell yielding lower current density [13]. Recall that in this model,

there occurs restricted amount of species due to very low pressure difference along the chan-

nel, which results in decrease in the performance of the fuel cell. For this reason to be able to

compare the performance curve of the present model, the experimental data should be selected

for a fuel cell yielding lower current density. According to Fig. 5.9, the performance curve

of the model fits very well to the experimental data for an air-breathing Proton Exchange

Membrane Fuel Cell. The only important difference is that in the experimental data, the open

circuit cell potential value does not start from maximum theoretical value of 1.2 V as in the

present model, instead it starts from 0.9 V .

Figure 5.9: The present model validation with an experimental data for an air-breathingPEMFC yielding lower current density. The experimental data values were estimated from agraph [13].

70

Figure 5.10: Cell Potential versus Integrated Current Density Graphs for x = 0, x = 0.025mand x = 0.05m positions of the parallel-to-flow model

For x = 0 and x = 0.025m, the local current density reaches its maximum value at V = 0.8V

and V = 0.7V , respectively. For voltages less than V = 0.7V , the high current densities at

the O2 inlet cause O2 to be depleted before reaching the cathode exit resulting in zero local

current densities (Fig. 5.10).

Therefore, by having a very low pressure difference along the channel of a single Proton

Exchange Membrane Fuel Cell to restrict the amount of species to represent a mid-cell in a

fuel cell stack (Fig. 4.6), species depletion is detected as shown in Fig. 5.10.

71

CHAPTER 6

CONCLUSIONS

In the present study, two different types of Proton Exchange Membrane Fuel Cell models are

developed. In the parallel-to-flow model, a very low pressure difference is assumed along the

flow channel to restrict the amount of species that is given to the model. After running parallel-

to-flow model for cell potentials ranging between 0.3 - 1.2V in 0.1V increments, local weight

fractions of species are obtained with respect to channel position. These weight fractions

and their corresponding cell potentials are used as inputs to the perpendicular-to-flow model

to investigate the species depletion. Also, in the perpendicular-to-flow model the following

three positions of the catalyst are selected to investigate the current density variations along y-

direction: beneath the center of the flow channel; beneath the center of the land; and beneath

the center of the channel and land.

According to the results:

• While decreasing the cell potential while restricting the amount of species by using a

very low pressure difference, H2 and O2 depletion increases due to increasing current

densities at the O2 inlet. If cell potential is decreased enough, depletion of reactants

cause dead places in the fuel cell at which point the reaction does not take place.

• Along the y-direction, the catalyst beneath the center of the flow channel has the highest

current density values for a given cell potential as expected. Because bulk flow of the

species takes place in the flow channel, locations closer to the flow channel along the

y-direction have higher local current densities.

72

• If reactants are restricted by having a very low pressure difference to represent the

behaviour of a mid-cell within a fuel cell stack, depletion of species is detected. At the

inlet of the anode channels local current density values go to zero even as cell potential

decreases.

• For a single Proton Exchange Membrane Fuel Cell, species depletion can be avoided

by arranging appropriate operating conditions. However for the mid-cells in fuel cell

stacks, the properties of species cannot be managed easily. To overcome species deple-

tion problems, alternative solutions should be found.

73

CHAPTER 7

FUTURE WORK

This thesis represents a first-step in developing reliable numeric models for fuel cells. The

overall goal is to use these models to support the design and development of fuel cells. The

more narrow goal for the present work is to combine models for different fuel cell components

to develop a simple model of an overall fuel cell. As appropriate for the first-step in numeric

modeling of a fuel cell, many simplifying assumptions are made. The subsequent students

who continue this line of research can improve the model as follows:

• This model accounts for only the vapor phase of the water. For this reason, condensa-

tion, which may be a significant problem for mass transportation of the reactants in real

life, should be accounted for by extending this model to two-phase.

• Water diffusion in the membrane is another important phenomenon. In the present

model, the drag coefficient of water from anode to cathode is taken as constant. How-

ever, this parameter depends strongly on the hydration level and the design of the mem-

brane. Therefore a more sophisticated model for mass transport through the membrane

should be developed that not only accounts for a variable drag coefficient but also for-

ward and back diffusion of water.

• The proton conductivity of the membrane is taken as constant. However, proton con-

ductivity is a material property that changes with humidification level.

• The exchange current density values are taken as constant in this model. However,

exchange current densities vary with species concentration. Modeling the decrease in

exchange current density with decreasing reactant concentrations should reduce the rate

at which reactants are depleted.

74

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APPENDIX A

A Numerical, 3-D Investigation of Gas Flow Channels Modified

with Disturbances in PEMFC

A.1 Introduction

With the rapid development of technologies, the energy demand of the world increases day

by day. Because of the difficulties of compelling people to restrict their energy consumption

for their daily life, engineers and scientists are working to find an alternative solution to this

problem. Due to the fact that hydrogen has several important properties as an energy carrier,

one of the most popular ideas for powering the next generation of devices is H2 Polymer

Electrolyte Fuel Cells, which is widely known as Proton Exchange Membrane Fuel Cell.

In a typical Proton Exchange Membrane Fuel Cell, longitudinal channels are used to distribute

reactants across the cell. However to reach the catalyst layer to initiate the reactions, the re-

actant flow direction should shift from the longitudinal to the transverse direction, which is

perpendicular to the flow direction. For this reason, Kuo et al. (Improvement of performance

of gas flow channel in pem fuel cells. Energy Conversion and Management 49, 2776-2787.

2008) proposed a novel design that has several disturbances in the flow field to increase the

momentum transfer to the transverse direction. In this work, with wave-like, trapezoid-like

and ladder-like disturbances, catalyst reaction performance improved significantly due to hav-

ing higher input velocity to the diffusion media and also better heat transfer performance was

obtained. Kuo et al. made a 2-D analysis and looked at one channel, in fact a 0.1 cm portion

of any pass, but they did not investigate the 3-D flow characteristics of the hydrogen with

several passes over the bipolar plate.

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Therefore, the present work directly builds on the work by Kuo et al. and contributes to the

scientific literature by investigating the 3-D flow analysis of the total gas flow channel with

several passes in a bipolar plate to understand the effect of disturbances on the performance

of a PEM fuel cell.

A.1.1 Governing Equations

With the assumptions of negligible viscous dissipation and work performed by pressure forces

and constant thermophysical properties, continuity, Navier-Stokes and energy equations for

steady-state condition can be written as;

ρ(∇.V) = 0 (A.1)

∇P = µ∇2V (A.2)

α∇2T = 0 (A.3)

A.1.2 Computational Domains and Boundary Conditions

In the present work, for the same bipolar plate base area (40 mm x 40 mm), models for 5, 10

and 15 number of passes of the gas flow channel are developed, as shown in Fig A.1. The

cross-sectional area of each channel is the same (1.5 mm x 1.5 mm). Moreover to investigate

the effect of disturbances on the performance of a fuel cell, one control unit that has a straight

flow channel and two cases of rectangular-disturbed and wave-disturbed flow channels are

modeled. A 2-D view of these disturbed channels can be seen in Fig A.2.

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(a) 5 passes (b) 10 passes

(c) 15 passes

Figure A.1: Three different models of the gas flow channel for 5, 10 and 15 number of passes

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(a) Rectangular-Disturbed Channel

(b) Wave-Disturbed Channel

Figure A.2: Disturbance types of the gas flow channel

In real applications, the exit of the gas flow channels is open to the atmosphere. Therefore,

the exit boundary condition is set as open to the atmosphere. Also the catalyst layer (upper

boundary) and bipolar plate (bottom boundary) are assumed to have constant temperatures.

According to these considerations, boundary conditions can be written as;

• Vinput = 1.5 m/s

• Pexit = Pgage = 0

• Tupper = 353 K

• Tbottom = 333 K

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A.2 Results and Discussions

For each pass, velocity profiles at the mid-planes, which is perpendicular to the channels, are

constructed and a velocity analysis is presented using them. Since there are several passes for

each channel type, three representative passes are selected as shown in Table A.1. Moreover,

to eliminate complex flow at the inlet and exit of each pass, the first and last 1.5 mm is

neglected.

Table A.1: Selected passes to be presented for 5, 10 and 15 passes

No:1 No:2 No:35 passes 1st 3rd 5th

10 passes 1st 5th 10th

15 passes 1st 8th 15th

After running the model, maximum y-velocities are given in Table A.2. As can be understood

from Table A.2, maximum y-velocity values for the straight channel is very low, even in

the first pass. However, this value increases significantly for the rectangular-disturbed and

wave-disturbed flow field designs. This means that with a pure velocity analysis of this novel

flow field of the gas flow channels in PEM fuel cells, improved momentum transfer from the

longitudinal to transverse direction is obtained.

Table A.2: Maximum y-velocity values for selected passes for 3 different flow fields

5 Passes1st 3rd 5th

Straight 0.2162 m/s 0.1313 m/s 0.0935 m/sWave 0.9076 m/s 0.9266 m/s 0.8843 m/sRectangular 1.1106 m/s 1.1134 m/s 1.0290 m/s

10 Passes1st 5th 10th

Straight 0.2257 m/s 0.1405 m/s 0.0845 m/sWave 0.8525 m/s 0.8846 m/s 0.9105 m/sRectangular 1.1730 m/s 1.0744 m/s 1.1618 m/s

15 Passes1st 8th 15th

Straight 0.2408 m/s 0.1185 m/s 0.0970 m/sWave 0.8486 m/s 0.8939 m/s 0.8971 m/sRectangular 1.0990 m/s 1.167 m/s 1.2233 m/s

84

Fig. A.3 represents the velocity distribution in the straight channel and it is clear that the

y-velocity is almost zero. However, improvement in the momentum transfer of hydrogen gas

to the perpendicular direction at which catalysis reaction occurs is shown in Figs. A.3, A.4,

A.5 and A.6.

(a) Velocity in x-direction (b) Velocity in y-direction

Figure A.3: x and y components of the velocity distribution for straight flow channels

(a) y-velocity distribution in the rectangular-disturbed channel

(b) y-velocity distribution in the wave-disturbed channel

Figure A.4: Improvement in y-velocity for 5 pass Geometry

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(a) y-velocity distribution in the rectangular-disturbed channel

(b) y-velocity distribution in the wave-disturbedchannel

Figure A.5: Improvement in y-velocity for 10 pass Geometry

(a) y-velocity distribution in the rectangular-disturbed channel

(b) y-velocity distribution in the wave-disturbedchannel

Figure A.6: Improvement in y-velocity for 15 pass Geometry

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Moreover, increasing the flow length results in a pressure drop increase. For the 10 pass flow

field, the pressure drop relation with channel disturbance type is given in Fig. A.7. Although

the rectangular-disturbed channel geometry has more maximum y-velocity than the wave-

disturbed channel geometry, pressure drop in the former one is greater than the later one.

(a) Straight Channel Geometry (b) Rectangular-Disturbed Channel Geometry

(c) Wave-Disturbed Channel Geometry

Figure A.7: Pressure drop relations for different disturbance types

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A.3 Conclusions

• With disturbances, there is an important improvement in the momentum transfer toward

the catalyst layer without losing surface area.

• Rectangular-disturbed channels have more maximum y-velocity value than wave-disturbed

ones.

• To obtain this improvement, the pressure drop will increase more than 10 % to maintain

the same boundary conditions in rectangular-disturbed channels.

88