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Optimal Shape Design for Polymer Electrolyte Membrane

Fuel Cell Cathode Air Channel: Modelling, Computational

and Mathematical Analysis

Jamal Hussain Alsmail

Thesis submitted to the Faculty of Graduate and Postdoctoral Studies

in partial fulfillment of the requirements for the degree of Doctor of Philosophy in

Mathematics 1

Department of Mathematics and Statistics

Faculty of Science

University of Ottawa

c© Jamal Hussain Alsmail, Ottawa, Canada, 2012

1The Ph.D. program is a joint program with Carleton University, administered by the Ottawa- Carleton Institute of Mathematics and Statistics

Abstract

Hydrogen fuel cells are devices used to generate electricity from the electrochemical

reaction between air and hydrogen gas. An attractive advantage of these devices is

that their byproduct is water, which is very safe to the environment. However, hy-

drogen fuel cells still lack some improvements in terms of increasing their life time

and electricity production, decreasing power losses, and optimizing their operating

conditions. In this thesis, the cathode part of the hydrogen fuel cell will be consid-

ered. This part mainly consists of an air gas channel and a gas diffusion layer. To

simulate the fluid dynamics taking place in the cathode, we present two models, a

general model and a simple model both based on a set of conservation laws governing

the fluid dynamics and chemical reactions. A numerical method to solve these mod-

els is presented and verified in terms of accuracy. We also show that both models

give similar results and validate the simple model by recovering a polarization curve

obtained experimentally. Next, a shape optimization problem is introduced to find

an optimal design of the air gas channel. This problem is defined from the simple

model and a cost functional, E, that measures efficiency factors. The objective of

this functional is to maximize the electricity production, uniformize the reaction rate

in the catalytic layer and minimize the pressure drop in the gas channel. The impact

of the gas channel shape optimization is investigated with a series of test cases in

long and short fuel cell geometries. In most instances, the optimal design improves

efficiency in on- and off-design operating conditions by shifting the polarization curve

ii

Abstract iii

vertically and to the right.

The second primary goal of the thesis is to analyze mathematical issues related to the

introduced shape optimization problem. This involves existence and uniqueness of

the solution for the presented model and differentiability of the state variables with

respect to the domain of the air channel. The optimization problem is solved using

the gradient method, and hence the gradient of E must be found. The gradient of E

is obtained by introducing an adjoint system of equations, which is coupled with the

state problem, namely the simple model of the fuel cell. The existence and uniqueness

of the solution for the adjoint system is shown, and the shape differentiability of the

cost functional E is proved.

Acknowledgements

I would like first to express my deep appreciation to my beloved and great mother

Zahra, who strove for me to be successful. I also greatly thank my grandfather Hus-

san, for his care and support to us since our childhood. I thank every member of my

family for their good wishes to me.

I would like to express my love and gratitude to the person I admire my wife, Nada–

for her care and support in my life with her.

It is my deep pleasure to express my gratitude to my supervisor Prof. Yves Bourgault–

who made completing my thesis possible. I thank him for his good wishes, strong

support and continuous encouragement to me. I would also like to express my grat-

itude to my first supervisor Prof. Arian Novruzi; I really appreciate his advices and

previous support to me.

I would like to thank my teachers and everyone who taught me and cared about

me, in particular my uncle Mohamed and my aunt Afifa.

Thanks to everyone who makes my success possible.

iv

Dedication

It is my honor to dedicate my PhD thesis to my parents–Zahra and Hussain, my

grandparents Hussan and Khaireyya, my wife Nada and my children Mohamed and

Kendeel.

I also like to dedicate my thesis to my family and my city–Awamia.

v

Contents

List of Figures ix

List of Tables xi

0.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Introduction 4

1.1 Fuel Cells and main objective of the thesis . . . . . . . . . . . . . 4

1.2 Plan of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Statement of Problems 8

2.1 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 The General Model . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 The Simplified Model . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Mathematical Analysis 24

3.1 Some Background in Shape Calculus . . . . . . . . . . . . . . . . 25

3.1.1 Differentiation of ζ → f ◦ (I + ζ) and ζ → f (ζ) ◦ (I + ζ) . . 27

3.1.2 Local differentiation . . . . . . . . . . . . . . . . . . . . . . 28

3.1.3 Differentiation of cost functionals defined through an inte-

gral form over Ωζ or ∂Ωζ . . . . . . . . . . . . . . . . . . . . 29

3.2 Derivative of the Cost Functional, E . . . . . . . . . . . . . . . . 31

vi

CONTENTS vii

3.3 Existence of the shape derivative of the state variables . . . . . . 45

3.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.2 Variational Formulation of the Steady State Problem . . . . 64

3.3.3 Shape differentiability of the state variables . . . . . . . . . 66

3.4 Existence and Uniqueness for the Adjoint Problem . . . . . . . . 78

3.4.1 The Adjoint Problem in Strong From . . . . . . . . . . . . . 78

3.4.2 Variational Formulation of the adjoint problem . . . . . . . 79

3.4.3 Fixed Point Formulation of the Adjoint Problem . . . . . . . 88

4 Numerical Methods 92

4.1 The steady state problem . . . . . . . . . . . . . . . . . . . . . . 92

4.2 The adjoint problem . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3 Finite element formulations of the steady state and the adjoint

problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Numerical Results 102

5.1 Description of the test cases . . . . . . . . . . . . . . . . . . . . . 103

5.2 Verifying the numerical solution of the simplified model . . . . . 103

5.3 Comparing the numerical solutions of the simplified and general

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4 Validating the simplified model . . . . . . . . . . . . . . . . . . . 106

5.5 Optimal Shape Design of the Air Channel . . . . . . . . . . . . . 108

5.5.1 Long air channel: l = 0.4m . . . . . . . . . . . . . . . . . . . 110

5.5.2 Short air channel with l = 0.02m and ε = 0.4 . . . . . . . . 112

5.6 Long versus short channel design . . . . . . . . . . . . . . . . . . 114

6 Conclusion 123

6.1 The contribution of the thesis . . . . . . . . . . . . . . . . . . . . 123

CONTENTS viii

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Bibliography 129

List of Figures

1.1 2d cross-section of hydrogen fuel cells. . . . . . . . . . . . . . . . . . 5

2.1 2d cross-section of the cathode part of HFC . . . . . . . . . . . . . . 8

5.1 Domains with the air channel and GDL, and the cross-section x =

0.2m at which two numerical solutions are compared. . . . . . . . . 106

5.2 The two meshes used to verify relative mesh independence of the

solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3 Comparison of the numerical solutions, obtained along a vertical

cross-section at x = 0.2m, obtained using Mesh 1 (solid line) and

Mesh 2 (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4 Surface plot of the numerical solutions for the simplified (at right)

and general (at left) models for the long geometry. . . . . . . . . . . 116

5.5 Solution of the simplified model (solid line) and general (dashed line)

models on a vertical cross-section at x = 0.2m. . . . . . . . . . . . . 117

5.6 The numerical solution of the simplified model for the short air

channel geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.7 Validating the simplified model . . . . . . . . . . . . . . . . . . . . . 118

5.8 The numerical solution while minimizing only the total variance of

the oxygen mass flux on M , for the long channel geometry (a =

1, b = e = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

ix

LIST OF FIGURES x

5.9 The numerical s