# Analytical Modelling of Fuel Cells

**By**

- Andrei Kulikovsky, Research Centre Julich, Julich, Germany

In fuel cell research, the gap between fundamental electrochemical processes and the engineering of fuel cell systems is bridged by the physical modelling of fuel cells. This relatively new discipline aims to understand the basic transport and kinetic phenomena in a real cell and stack environment, paving the way for improved design and performance. The author brings his unique approach to the analytical modeling of fuel cells to this essential reference for energy technologists...

View full description### Audience

Professionals working in the energy market (transportation, residential, industrial, military,and aerospace segments), chemical engineers, mechanical engineers, and organic chemists /

### Book information

- Published: June 2010
- Imprint: ELSEVIER
- ISBN: 978-0-444-53560-3

### Reviews

"This is a well-written introduction to the modeling of fuel cells that the reader will no doubt want to supplement by referring to published research articles (a comprehensive bibliography is included). It is written by a physicist, but is clearly influenced by engineering rather than scienceā¦It uses āsimplifiedā models based on idealized systems, basic geometries and minimal assumptions, enabling the qualitative understanding of the causes and effects of the system phenomena. In this way it shows how analytical modeling can be applied to real fuel cell systems."--**Energy News** @

### Table of Contents

Contents

Preface

Introduction

1 Fuel cell basics

1.1 Fuel cell thermodynamics

1.1.1 The physics of the fuel cell effect

1.1.2 Open-circuit voltage

1.1.3 Nernst equation

1.1.4 Temperature dependence of OCV

1.2 Potentials in a fuel cell

1.3 Rate of electrochemical reactions

1.3.1 Butler-Volmer equation

1.3.2 Butler-Volmer and Nernst equations

1.3.3 Tafel equation

1.4 Mass transport in fuel cells

1.4.1 Overview of mass transport processes

1.4.2 Stoichiometry and utilization

1.4.3 Quasi-2D approximation

1.4.4 Mass conservation equation in the channel

1.4.5 Flow velocity in the channel

1.4.6 Mass transport in gas diffusion/backing layers

1.4.7 Mass transport in catalyst layers

1.4.8 Proton and water transport in membrane

1.5 Sources of heat in a fuel cell

1.6 Types of cells considered in this book

1.6.1 Polymer electrolyte fuel cells (PEFCs)

1.6.2 Direct methanol fuel cells (DMFCs)

1.6.3 Solid oxide fuel cells (SOFCs)

2 Catalyst layer performance

2.1 Basic equations

2.1.1 The general case

2.1.2 First integral

2.2 Ideal oxygen and proton transport

2.3 Ideal oxygen transport

2.3.1 Basic equations

2.3.2 Integral of motion

2.3.3 Equation for proton current

2.3.4 Low cell current

2.3.5 High cell current

2.3.6 The general polarization curve

2.3.7 Condition of negligible oxygen transport loss

2.4 Ideal proton transport

2.4.1 Basic equations

2.4.2 The x-shapes and polarization curve

2.4.3 Large zeta (zeta greater than 1)

2.4.4 Small zeta (zeta less than 1)

2.5 Optimal oxygen diffusion coeĀ±cient

2.5.1 Reduction of the full system

2.5.2 Optimal oxygen diffusivity

2.6 Gradient of catalyst loading

2.6.1 Model

2.6.2 Polarization curve

2.7 DMFC anode

2.7.1 The rate of methanol oxidation

2.7.2 Basic equations and the conservation law

2.7.3 The general form of the polarization curve

2.7.4 Small variation of overpotential in the active layer

2.7.5 Active layer of variable thickness

2.8 Heat balance in the catalyst layer

2.8.1 Heat transport equation in the CL

2.8.2 Reduction to boundary condition

2.8.3 Solution to the heat transport equation

2.9 Remarks on Chapter 2

3 One-dimensional model of a fuel cell

3.1 Voltage loss due to oxygen transport in the GDL

3.2 One-dimensional polarization curve of a cell

3.3 One-dimensional model of DMFC

3.3.1 Feed molecule concentration in the active layers

3.3.2 One-dimensional polarization curve of DMFC

3.4 Heat transport in the MEA of a PEFC

3.4.1 General assumptions

3.4.2 Equations

3.4.3 Exact solutions

3.4.4 Temperature profiles

3.4.5 How to measure thermal conductivities of MEA layers

3.4.6 One-sided fluxes from the MEA

3.4.7 Heat crossover through the membrane

3.5 Heat transport in the MEA of a DMFC

3.5.1 Assumptions

3.5.2 Equations

3.5.3 Heat transport under open-circuit conditions

3.5.4 How to measure thermal conductivities of MEA layers

3.5.5 Temperature profiles and discussion

3.5.6 Exact solutions

4 Quasi-2D model of a fuel cell

4.1 Gas dynamics of channel flow

4.1.1 Momentum balance in the cathode flow

4.1.2 The limit of low flow velocity

4.2 A model of PEFC

4.2.1 Oxygen concentration and local current along the channel

4.2.2 Cell polarization curve

4.2.3 Water crossover and the polarization curve

4.2.4 Local polarization curves

4.3 A model of PEFC with water management

4.3.1 Model and governing equations

4.3.2 Solution at constant flow velocity

4.3.3 Close to the limiting current density

4.3.4 The general case

4.3.5 Model validation

4.3.6 Limiting current, optimal feed composition

4.3.7 Constant oxygen stoichiometry

4.4 Degradation wave

4.4.1 Model

4.4.2 Wave propagation

4.4.3 Cell potential

4.4.4 Two scenarios of cell performance degradation

4.4.5 Accelerated testing of aging phenomena

4.5 Gradient of catalyst loading along the oxygen channel

4.5.1 Low cell current

4.5.2 High cell current

4.5.3 The effect of transport loss in the GDL

4.6 A model of SOFC anode

4.6.1 Basic equations and the local polarization curve

4.6.2 Hydrogen concentration in the channel

4.6.3 Cell voltage

4.6.4 Low current: z-shapes

4.6.5 Low current: Polarization curve

4.6.6 High current: z-shapes and polarization curve

4.6.7 Remarks

4.7 A model of DMFC

4.7.1 Continuity equations in the feed channels

4.7.2 Solution for the case of lambda^{a} = lambda^{c}

4.7.3 Cell depolarization at zero current: mixed potential

4.7.4 Cross-linked feeding

4.7.5 Oxygen and methanol utilization, mean crossover current density

4.7.6 Remarks

4.8 DMFC: The general case of arbitrary lambda a and lambda c

4.8.1 Equation for local current

4.8.2 Numerical solution

4.9 DMFC: Large methanol stoichiometry, small current

4.9.1 The shape of the jumper

4.9.2 Plateau

4.9.3 Critical air flow rate

4.9.4 Experimental verification

5 Modelling of fuel cell stacks

5.1 Temperature field in planar SOFC stack

5.1.1 General assumptions

5.1.2 The general equation for bipolar plate temperature

5.1.3 Heat balance in the air channel

5.1.4 Heat balance in the BP

5.1.5 Cell polarization curve

5.1.6 Boundary conditions

5.1.7 Method of asymptotic expansion

5.1.8 Asymptotic solution

5.1.9 Local current

5.1.10 Example: Oxide-dominated stack resistivity

5.1.11 Remarks

5.2 Temperature gradient in SOFC stack

5.2.1 Stack and air temperatures

5.2.2 Temperature gradient

5.3 Thermal waves in SOFC stack

5.3.1 Basic equations

5.3.2 Stability analysis

5.3.3 Flow temperature is constant

5.3.4 Solution: The general case

5.3.5 Role of boundary conditions

5.3.6 Remarks

5.4 Heat effects in DMFC stack

5.4.1 General assumptions

5.4.2 Equations for stack and flow temperature

5.4.3 Asymptotic solution: The general case

5.4.4 Optimal stack temperature

5.5 Mirroring of current-free spots in a stack

5.5.1 Equation for bipolar plate potential

5.5.2 Cell polarization curve

5.5.3 Spot shape and numerical details

5.5.4 Numerical results

5.5.5 Analysis of equations: The length of mirroring

5.5.6 Remarks

5.6 Hybrid 3D model of SOFC stack

5.6.1 Thermal model

5.6.2 Electric problem

5.6.3 Numerical details

5.6.4 Numerical results

5.6.5 Analysis of governing equations

5.6.6 Temperature stratification

5.6.7 The mechanism of anomalous heat transport

5.7 Power generated and lost in a stack

5.7.1 The nature of voltage loss in bipolar plates

5.7.2 Power dissipated in a bipolar plate

5.7.3 Power dissipated in a thin bipolar plate

5.7.4 Useful power generated by the individual cell

5.7.5 Illustration: A 1D case

Bibliography

Abbreviations

Nomenclature *