Modelingthe DirectMethanolFuelCell

Modeling theDirect Methanol Fuel Cell

Simulation of Methanol Crossover, Cathode Flooding and MixedPotential with an Analytical Butler-Volmer Based Concept

Dissertation rer. nat.

Faculty of Natural SciencesUniversity of Ulm

Germany

2011

Stefan N. Pofahl

Saarbrücken

Germany

Amtierender Dekan: Prof. Dr. Axel Groß

1. Gutachter: Prof. Dr. Werner Tillmetz2. Gutachter: Prof. Dr. Axel Groß

Tag der Promotion: 27. Februar 2012

ii

ABSTRACT

The Direct Methanol Fuel Cell (DMFC) is simulated in sequential mode in azero-dimensional model using Butler-Volmer equations to represent the electro-chemical reactions on both electrodes. It is shown that this approach is ableto predict the harmful methanol flow across the polymer electrolyte membrane(PEM) — the so called crossover — and the resulting loss of potential on the cath-ode, caused by the mixed potential that is built between the parasitic cathodicmethanol oxidation and the oxygen reduction. A linear model is presented, whichexplains the deviation between the real anode polarization and the Butler-Volmerequation with the difference in activity of methanol in liquid and vapor phase.The comprehensive model is validated qualitatively and quantitatively with datafrom literature as well as with measured data.

ZUSAMMENFASSUNG

Die Direktmethanolbrennstoffzelle (DMFC) wird sequentiell und nulldimen-sional simuliert, wobei die elektrochemischen Reaktionen an beiden Elektrodenmit der Butler-Volmer Gleichung abgebildet werden. Es wird gezeigt, daß mitdiesem Ansatz sowohl der schädliche Methanoltransport über die Polymerelek-trolytmembran (PEM) — der sogenannte Crossover — sowie der daraus resul-tierende Potentialverlust, der durch die Mischpotentialbildung zwischen der par-asitären kathodischen Methanoloxidation und der Sauerstoffreduktion entsteht,abgeschätzt werden können. Für die Anode wird ein lineares Model vorgestellt,daß die Abweichung der Anodenkennlinie vom Verlauf einer Butler-Volmer Glei-chung mit der unterschiedlichen Aktivität des Methanols in der Flüssig- undGasphase erklärt. Das Gesamtmodell wird sowohl an Hand von Literaturdaten, alsauch mit Hilfe von praktischen Meßdaten qualitativ und quantitativ validiert.

iii

Contents

Abstract — Zusammenfassung iii

Contents viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Preface xiii

1. Introduction and Motivation 11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3. Simulation Approaches for DMFC . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1. Constitutive Simulation Classes . . . . . . . . . . . . . . . . . . . . . . . 31.3.2. Single- or Multi-Domain Approach . . . . . . . . . . . . . . . . . . . . . 41.3.3. Dimensional Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.4. Models of the Electrochemical Reactions . . . . . . . . . . . . . . . . . . 41.3.5. Methanol Drag Across the Membrane . . . . . . . . . . . . . . . . . . . . 6

1.4. Classification of the Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5. Comparison of the Model to Other Works . . . . . . . . . . . . . . . . . . . . . 7

2. Model of the Direct Methanol Fuel Cell 92.1. Material and Energy Flows Between the Cell Parts . . . . . . . . . . . . . . . . 102.2. Anode Compartment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1. Anodic Bipolar Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.2. Anodic Diffusion Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3. Methanol Oxidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.4. Anodic Catalyst Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.5. Active Surface of the Electrode . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3. Model of the Polymer Electrolyte Membrane . . . . . . . . . . . . . . . . . . . 252.3.1. Water Transport Through the Membrane . . . . . . . . . . . . . . . . . . 262.3.2. Methanol Drag Through the Membrane . . . . . . . . . . . . . . . . . . 282.3.3. Carbon Dioxide Drag Through the Membrane . . . . . . . . . . . . . . 322.3.4. Proton Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4. Cathode Compartment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.1. Cathodic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.2. Cathodic Catalyst Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.3. Ascertainment of the OCP for the Oxygen Reduction . . . . . . . . . . 392.4.4. Cathodic Gas Diffusion Layer . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5. Electrical Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5.1. Electrical Losses in CL, DL and Stack . . . . . . . . . . . . . . . . . . . . 422.5.2. Inner Voltage Loss due to Membrane Ion Resistance . . . . . . . . . . 42

v

3. Implementation of the Model in AspenPlus 453.1. Guidelines of Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2. AspenPlus Process Flowsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3. Hierarchy of Iterative Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4. Strategies to Improve the Convergence Behavior . . . . . . . . . . . . . . . . . 50

3.4.1. Building Error Classes to Narrow Optimization Bounds . . . . . . . . . 513.4.2. Attenuate Oscillation of Values and Numerical Stability . . . . . . . . . 523.4.3. File Based Database to Improve Convergence . . . . . . . . . . . . . . . 53

3.5. Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5.1. Scaling Manipulated Values and Sampled Values . . . . . . . . . . . . . 543.5.2. Scaling Surface Fraction for Calculation of Mixed Potential . . . . . . . 56

3.6. Implementation of the Set of Butler-Volmer Equations . . . . . . . . . . . . . 58

4. Simulation Results 634.1. Transport Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2. Simulation of the Anode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3. Simulation of the Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.1. Cathode at Reference Temperature . . . . . . . . . . . . . . . . . . . . . 664.3.2. Oxygen Reduction Transfer Coefficient at Upper Temperature . . . . . 67

4.4. Variation of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.4.1. Electrode and Cell Polarization at Varied Temperatures . . . . . . . . . 674.4.2. Comparison of Methanol and Carbon Dioxide Crossover . . . . . . . . 69

4.5. Variation of Methanol Concentration . . . . . . . . . . . . . . . . . . . . . . . . 704.5.1. Methanol Concentration and Rest Potential . . . . . . . . . . . . . . . . 704.5.2. Crossover Growth with Increase of Methanol Concentration . . . . . . 71

4.6. Comparison of Two Membrane Types at Different Methanol Concentrations 724.7. Variation of Anode Stoichiometry to Optimum . . . . . . . . . . . . . . . . . . 734.8. Influence of Ion Resistivity on the Polarization . . . . . . . . . . . . . . . . . . 754.9. Global Parameter Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.10.Table of Parameters and Fit Results . . . . . . . . . . . . . . . . . . . . . . . . . 79

5. Discussion of the Model 835.1. Zero-dimensional Multi-domain Simulation . . . . . . . . . . . . . . . . . . . . 835.2. Limitations of a Single Cell Model . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3. Comparison of the Model to Other Modeling Approaches . . . . . . . . . . . . 84

5.3.1. Model of the Anodic Catalytic Layer . . . . . . . . . . . . . . . . . . . . . 855.3.2. Transition from Temkin to Langmuir Adsorption . . . . . . . . . . . . . 87

5.4. Anode Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.4.1. Potential Dependencies of Adsorption Processes . . . . . . . . . . . . . 895.4.2. Influence of the Vapor Phase on the Anode Polarization . . . . . . . . 905.4.3. Estimation of the Oxygen Drag from Cathode to Anode . . . . . . . . . 93

5.5. Electroosmotic Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.6. Comparison of the Modeling Results to Test Data . . . . . . . . . . . . . . . . . 95

vi

6. Summary and Conclusions 976.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2.1. Suitability of the Butler-Volmer Modeling Approach . . . . . . . . . . . 976.2.2. Key Elements of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2.3. Cell Voltage as Independent Variable . . . . . . . . . . . . . . . . . . . . 1016.2.4. Validation with own Cell Measurements . . . . . . . . . . . . . . . . . . 101

Acknowledgement 102

A. Nomenclature 105

Appendix 105

B. Parameter Values 113B.1. Composition of Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115B.2. Activities in Methanol-Water Mixtures . . . . . . . . . . . . . . . . . . . . . . . 116B.3. Diffusion Coefficient of Carbon Dioxide in Water . . . . . . . . . . . . . . . . . 116B.4. Water Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116B.5. Specific Electric Conductivity of Ionomer . . . . . . . . . . . . . . . . . . . . . 116B.6. Solubility of Oxygen in Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

C. Details on Implementation 119C.1. Set-Up of the File Based Database to Improve Convergence . . . . . . . . . . 119C.2. Influence of Scaling Factors on Number Representation . . . . . . . . . . . . . 120

D. Auxiliary Calculations 123D.1. Calculation of an Air Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . 123D.2. Algebraic Transformation of Butler-Volmer Equation . . . . . . . . . . . . . . . 123D.3. Butler-Volmer Equation for Fixed Concentration . . . . . . . . . . . . . . . . . 130

E. Measured Data 131E.1. Polarization of a DMFC with Acid Sulfur Doped PVC-Membrane . . . . . . . 131E.2. Polarization of a DMFC with Nafion™115-Membrane . . . . . . . . . . . . . . 132

E.2.1. Variation of Methanol Concentration . . . . . . . . . . . . . . . . . . . . 132E.2.2. Variation of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 133E.2.3. Cathode Polarization under Hydrogen Operational Mode . . . . . . . . 134

F. Software Tools 135

G. Glossary 137

Bibliography 155

Affidavit of originality — Eigenständigkeitserklärung 168

Curriculum Vitae 169

Index 171

vii

List of Figures

1.1. Schematic representation of a DMFC . . . . . . . . . . . . . . . . . . . . . . . . 1

2.1. Polarization scheme of a DMFC . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2. Functional layers of a single direct methanol fuel cell . . . . . . . . . . . . . . 102.3. Flow scheme of the anodic bipolar plate . . . . . . . . . . . . . . . . . . . . . . 102.4. Flow scheme of the anodic diffusion layer . . . . . . . . . . . . . . . . . . . . . 102.5. Flow scheme of the anodic catalyst layer . . . . . . . . . . . . . . . . . . . . . . 112.6. Flow scheme of the polymer electrolyte membrane . . . . . . . . . . . . . . . 112.7. Flow scheme of the cathode catalyst layer . . . . . . . . . . . . . . . . . . . . . 132.8. Flow scheme of the cathode diffusion layer and bipolar plate . . . . . . . . . 132.9. Scheme of a one channel meander flow field . . . . . . . . . . . . . . . . . . . 152.10.Side-face view across the functional layers ABP and ADL . . . . . . . . . . . . 152.11.Pressure profile across the layers ABP and ADL . . . . . . . . . . . . . . . . . . 162.12.Current density graph of the three causative electrochemical reactions . . . . 232.13.DMFC anode and cathode polarization curves . . . . . . . . . . . . . . . . . . 232.14.Polarization of the main DMFC reactions and the cathode mixed potential . 242.15.Electroosmotic water drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.16.Determination of the effective diffusion coefficient of the MEA . . . . . . . . 292.17.Diffusion coefficient of methanol in Nafion™11X . . . . . . . . . . . . . . . . . 302.18.Comparison of diffusion flows through two different membrane types . . . . 302.19.Ion conductivity of electrolyte membranes . . . . . . . . . . . . . . . . . . . . . 332.20.Transition from Temkin to Langmuir adsorption on the cathode . . . . . . . . 362.21.Comparison of the catalyst surface related current densities at the cathode . 392.22.Temperature dependency of the oxygen reduction OCP . . . . . . . . . . . . . 402.23.Schematic potential course across the fuel cell layers . . . . . . . . . . . . . . 42

3.1. Simulation process flowsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2. Stream types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3. AspenPlus models Mixer and FSplit . . . . . . . . . . . . . . . . . . . . . . . . . 473.4. AspenPlus separator models Flash2 and Sep . . . . . . . . . . . . . . . . . . . 483.5. AspenPlus RStoic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.6. Heater model illustrated with different icons . . . . . . . . . . . . . . . . . . . 493.7. j-slope over anode polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.8. Loop order of Design Specs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.9. Class definition for optimization loop boundaries . . . . . . . . . . . . . . . . 523.10.Best scaling factor of the surface fraction for the simulation . . . . . . . . . . 573.11.Structure of the Fortran77 calculator block C-BUTVOL . . . . . . . . . . . . . . 583.12.The nexus between the anode polarization and the different currents . . . . 603.13.Begin and end of iterative current calculation . . . . . . . . . . . . . . . . . . . 61

4.1. Temperature dependency of electroosmotic water drag coefficients . . . . . . 644.2. Anode polarization at reference temperature 298 K . . . . . . . . . . . . . . . . 65

ix

4.3. Anode polarization at 333 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4. Cathode polarization at reference temperature . . . . . . . . . . . . . . . . . . 674.5. Anode potential vs. SHE at varied temperatures . . . . . . . . . . . . . . . . . . 684.6. Cathode potential vs. SHE at varied temperatures . . . . . . . . . . . . . . . . 684.7. Fuel cell polarization at varied temperatures . . . . . . . . . . . . . . . . . . . 684.8. Carbon dioxide drag across the membrane at varied temperatures . . . . . . 694.9. Methanol crossover at varied temperatures . . . . . . . . . . . . . . . . . . . . 694.10.Rest potentials and OCV at 298 K for different methanol concentrations . . . 704.11.Rest potentials and OCV at 333 K for different methanol concentrations . . . 704.12.Crossover at different methanol inlet concentrations . . . . . . . . . . . . . . . 714.13.Polarization of anode and cathode at different methanol inlet concentrations 724.14.Cell polarization at different methanol inlet concentrations . . . . . . . . . . 724.15.Electroosmotic and diffusion transport at varied anode stoichiometries . . . 744.16.Varied methanol stoichiometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.17.Influence of the ion resistivity of the membrane on the polarization . . . . . 754.18.Global search for Butler-Volmer parameters . . . . . . . . . . . . . . . . . . . . 764.19.Global search for electroosmotic and flooding parameters . . . . . . . . . . . 764.20.Global search for Butler-Volmer parameters at given ξeo . . . . . . . . . . . . . 774.21.Global search for flooding parameters at given ξeo . . . . . . . . . . . . . . . . 774.22.Quality graphs of the Butler-Volmer parameters at steady ξeo , jfl and Θfl . . . 78

5.1. DOHLE model of anode polarization . . . . . . . . . . . . . . . . . . . . . . . . 855.2. MEYERS and NEWMAN model of anode polarization . . . . . . . . . . . . . . . 855.3. Simplified anodic Butler-Volmer equation at different concentrations . . . . . 875.4. Anodic Butler-Volmer equation at different concentrations . . . . . . . . . . . 875.5. Declining transfer coefficient in Langmuir adsorption region at 298 K . . . . 875.6. Declining transfer coefficient in Langmuir adsorption region at 333 K . . . . 875.7. Pure liquid phase in the porous anode electrode at OCV . . . . . . . . . . . . 905.8. Carbon dioxide evolution in the porous anode structure . . . . . . . . . . . . 905.9. Anode polarization depending on the aggregate state . . . . . . . . . . . . . . 915.10.Comparison of different anode transfer coefficients αAN at 298 K . . . . . . . 915.11.Comparison of optimum anode transfer coefficients αAN at 333 K . . . . . . . 935.12.Test results of the electrode polarization at varied concentration . . . . . . . 965.13.Test results of the cell polarization at varied concentration . . . . . . . . . . . 96

6.1. Anode polarization data compared to simulation results . . . . . . . . . . . . 986.2. Cathode polarization data compared to simulation results . . . . . . . . . . . 98

C.1. Spacing of the double precision number space . . . . . . . . . . . . . . . . . . 120C.2. Influence of scaling factor on the precision of the surface fraction . . . . . . 121

G.1. Galvani-, Volta- and surface potential . . . . . . . . . . . . . . . . . . . . . . . 144

x

List of Tables

3.1. Scaling factors for design specifications . . . . . . . . . . . . . . . . . . . . . . 55

4.1. OCP at varied methanol concentrations . . . . . . . . . . . . . . . . . . . . . . 704.2. Parameter range and step size for global optimization . . . . . . . . . . . . . . 764.3. Simulation parameters and fit results . . . . . . . . . . . . . . . . . . . . . . . . 79

5.1. Fit results for semi-empirical anode kinetics equations . . . . . . . . . . . . . 86

A.1. Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.2. Superscripts, subscripts, abbreviations . . . . . . . . . . . . . . . . . . . . . . . 108A.3. Operands and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

B.1. Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113B.2. Composition of dry air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115B.3. Simplified composition of dry air . . . . . . . . . . . . . . . . . . . . . . . . . . 115B.4. Activities in water-methanol mixtures . . . . . . . . . . . . . . . . . . . . . . . . 116B.5. Diffusivity of carbon dioxide in water . . . . . . . . . . . . . . . . . . . . . . . . 116B.6. Water drag coefficients of Nafion™ . . . . . . . . . . . . . . . . . . . . . . . . . 116B.7. Specific electric conductivity of Nafion™ . . . . . . . . . . . . . . . . . . . . . . 116B.8. Temperature dependence of ionic conductivity of Nafion™117 . . . . . . . . 117B.9. Water solubility of oxygen from air at ambient pressure . . . . . . . . . . . . . 117

E.1. WILLIAMS half cell polarization data at 298 K . . . . . . . . . . . . . . . . . . . 131E.2. WILLIAMS half cell polarization data at 333 K . . . . . . . . . . . . . . . . . . . 131E.3. Polarization of a Nafion™115-DMFC with varied methanol concentration . . 132E.4. Polarization of a Nafion™115-DMFC at 298 and 333 K . . . . . . . . . . . . . . 133E.5. Polarization of a DMFC-cathode under hydrogen operational mode . . . . . 134

xi

PREFACE

This work presents a new model of a liquid feed Direct Methanol Fuel Cell (DMFC).The functionality of the model was tested against the polarization data fromK. R. WILLIAMS et al. [148] from a scientific group at Shell Research Ltd. ThorntonResearch Centre, Great Britain, published in [148].

For further validation of the model, own measurements have been carried out atthe ZSW: Center for Solar Energy and Hydrogen Research1, Ulm, Germany (cf.chap. 5.6 on page 95).

The simulation was performed with the proprietary process engineering softwareAspenPlus by software supplier AspenTech.

For general aspects of the advantages and the discussed potentials of the DMFCand related topics see [2, 55]. Review articles about the scientific work on DMFCin general are [11, 78, 110, 126], review articles about modeling in particular are[24, 98].

Please note the regimentations concerning symbols and simplifications in thechapter A on page 105.

1 Zentrum für Sonnenenergie- und Wasserstoff-Forschung, ZSW

xiii

1. INTRODUCTION AND MOTIVATION

1.1. Introduction

Fuel cells are electrochemical reactors that realize the direct conversion of chem-ical energy of reactants to electrical energy. Among the different fuel cell types,the DMFC with a polymer electrolyte membrane (PEM) as electrolyte and liquidwater-methanol mixture as energy carrier is a promising power source for vehic-ular and various portable applications, like laptops, multimedia equipment andmobile power supplies. This type of fuel cell has several advantages, such as: lowemissions, a potentially renewable liquid fuel with a high power density, as well asfast and convenient refueling.

ADL ACL PEM CCL CDL

e-

e-

CHOH3

HO2

Air

CHOH3

HO2

2CO

Air

HO2

2CO

H+

CHOH3

HO2

- +

Fig. 1.1.: Schematic representation of adirect methanol fuel cell (DMFC).

A standard DMFC as shown schemat-ically in Fig. 1.1 comprises an anodeflow field incorporated in the anodebipolar plate (ABP), an anode diffu-sion layer (ADL), an anode catalystlayer (ACL), a polymer electrolytemembrane (PEM), a cathode cata-lyst layer (CCL), a cathode diffusionlayer (CDL) and a cathode flow fieldincorporated in the cathode bipolarplate (CBP). In the basic version of anideal DMFC the fuel flows through theABP, is transported by diffusion andconvection through the ADL, and is electrochemically oxidized at the ACL whereelectrons and protons are released. The released electrons are conducted to thecathode side by means of the external circuit and protons are transported throughthe PEM to the CCL. Simultaneously, oxygen or an oxygen-gas mixture is fed tothe CBP, is transported by diffusion and convection through the CDL into the CCL,where it reacts with protons and electrons, generating water [98].

The DMFC is driven by one of the most studied reactions in electrochemistry: theelectrooxidation of methanol. Nonetheless, the reaction path in the presence of acatalyst is still under discussion (cf. e.g. [43, 55, 88]). In contrast to the ideal DMFC,in the common DMFC there is a methanol flow across the membrane, caused

2 1. Introduction and Motivation

by the methanol permeability of the most commonly used polyperfluorosulfo-nic acid-membranes. Aside from the low reaction rates of the anodic methanoloxidation, this crossover is one of the main challenges of the DMFC. As a resultof the methanol drag into the cathode the electrode potential suffers from the socalled mixed potential (see chap. G on page 148) and the fuel conversion efficiencyηut is declining (cf. chap. G on page 142). However, the heat released by methanoloxidation at the cathode can be used in thermal management.

DMFC with alternative electrolytes such as: a) phosphoric acid membranes fortemperatures above the dew point of water and b) alkaline membranes are still ina pilot stage.

1.2. Motivation

There are several processes active in a DMFC that are contradictory, which bringsup the need for a detailed model of the different functional layers of the fuelcell. Up until now, there has been no study that bundles the different effectsof the whole liquid methanol fuel cell in a coherent simulation concept using aprocess engineering software on the basis of fundamental electrochemical kinet-ics via Butler-Volmer equations. In the DMFC a high fuel conversion efficiency isdemanded, and it is known that the diffusion coefficient of methanol in polyperflu-orosulfonic acid (Nafion™) is increasing with temperature and with the swelling ofa wet Nafion™-membrane. Hence, one would expect that the unwanted crossoveris increasing with temperature, but this is only true at low current densities. Ifthe anodic reaction rate is high enough, methanol crossover can be significantlyreduced (cf. chap. 2.3), this phenomenon is due to high methanol consumption inthe electrode, hence, low methanol concentration at the anodic membrane sur-face [105]. To cover such complex behaviors a modular zero-dimensional conceptwas designed.

1.3. Simulation Approaches for DMFC

The three publications [2, 77, 98] provide a comprehensive overview regardingthe simulation work on DMFC. The latest of these reviews is the article byOLIVEIRA [98], on which the following overview about the different definitionsof model classification is based.

1.3. Simulation Approaches for DMFC 3

1.3.1. Constitutive Simulation Classes

DMFC models can be devided into three constitutive simulation classes:analytical, semi-empirical and mechanistic models1.

Analytical DMFC Models

The analytical model is based on theoretically derived differential and algebraicwork. It operates with many simplifications, whereas empirical fitting is avoided.The numerical computational expense is moderate. Analytical models help tounderstand fundamental phenomena occurring within a fuel cell. They are notsuitable for quantitative predictions. Examples are: [75, 117, 120, 128]

Semi-empirical DMFC Models

Semi-empirical DMFC Models use data already available from existing fuel cellsand fuel cell systems. The data is taken to build simple empirical equationsthat are best suited to the data given. Modeling of systems with engineeringsoftware like Pro/II [135] or AspenPlus [142] may follow this route. Such worksassist engineers in constructing systems. Examples are: [1, 41, 69, 129]

Mechanistic DMFC Models

Mechanistic models focus on transport phenomena. The models are based onelectrochemistry and physics governing the conditions in the modeled object. Inmost cases this is only a part of the cell or the stack; e.g. some modeling works fo-cus on one of the electrodes [75], on the membrane [61] or on a diffusion layer [51].Three-dimensional studies — often named computational fluid dynamics calcu-lations (CFD) — go even more into detail and investigate particulars, such asparts of the media distribution structure. Mechanistic models are computationalresource-intensive. Most of the published works regarding DMFC are based onthis class. Examples are: [21, 73, 74, 95, 117, 143]

1 Alternative termes are: black, grey and white box models for empirical, semi-empirical and analyticmodels


1.3.2. Single- or Multi-Domain Approach

In addition to the constitutive methodology, there exists another basic criteria forthe comparison of different models: the type of the numerical solution strategy.Under the multi-domain or sequential modular mode the equations for differentparts of the simulated object are split into separate sets of equations and theseare solved in sequence. Under the single-domain or equation oriented mode thereis one unique set of equations for the whole model and all equations are solved inparallel. The advantages of the multi-domain mode over the single-domain modelie in the less complex structure of each module, and the convergence behaviorbeeing much more robust. The advantage of the single-domain mode is its lowerrequirement of machine time. Hence, this method is favorable for parameterstudies or computational fluid dynamics (CFD) calculations.

1.3.3. Dimensional Outline

There are four categories of the dimensional outline of a model, reaching from zeroto three-dimensional simulations. The zero-dimensional model uses one equationfor a specific calculation for a section of the modeled object, e.g. the diffusionacross a layer is described with one equation what implies the disregarding of theconcentration gradient inside the layer. One-, two- or three-dimensional modelswork with differential equations to describe the trend of concentrations, flows andother physical properties inside a section of the model along one, two or threecoordinate axes. Especially two- and three-dimensional models usually follow themechanistic simulation approach.

1.3.4. Models of the Electrochemical Reactions

There is no doubt that the reaction rate or the electrical current I in an elec-trochemical cell is dependent on the polarization U of the working electrode.Therefore, it makes sense to take the polarization of the electrode as the indepen-dent variable. In fact, the opposite is the case for the majority of models. Theadvantage of taking the electrical current as the independent variable is the simpli-fication of the model itself and of the system of equations. The reason is: via theFirst Faraday’s law the reaction rate is dependent on the electrical current, hence,the model becomes impassible to calculation errors of the open circuit potentials(OCP). Nevertheless, it is obvious that — from the physicochemical point of view —it is advantageous to use the polarization as the independent variable to describethe electrochemical process inside an electrode. The disadvantage of the simpli-

1.3. Simulation Approaches for DMFC 5

fied model grows with the complexity of the electrochemical mechanisms insidean electrode. As the electrochemistry of a DMFC is more complex than the one ofa polymer electrolyte membrane hydrogen fuel cell (PEM-HFC), the polarizationbased model reflects more accurately the electrochemistry of the DMFC.

Models of the DMFC-Anode

The electrooxidation of methanol inside a DMFC-anode is by far more complexthan the hydrogen oxidation in a PEM-HFC. The PEM-HFC-anode has nearlythe same characteristic as a dynamic hydrogen electrode (DHE) (cf. chap. G onpage 141), whereas the DMFC-anode shows a steep polarization behavior and itspolarization is influenced by the formation of a vapor phase inside the porouselectrode structure. Hence, a simple Tafel or Butler-Volmer equation (cf. Eq. G.15on page 154 resp. Eq. G.4 on page 140) provides poor accuracy in forecastingthe polarization characteristics at varied as well as at constant temperatures andmethanol concentrations.

Models of the DMFC-Cathode

The characteristics of the DMFC-cathode are also more complex, when comparedto a PEM-HFC-cathode. Though the main reaction is the same for both, theDMFC-cathode is penetrated by methanol, which results in two electrochemicalreactions on the same electrode. This phenomenon is called mixed potential (cf.chap. G on page 148).

The mixed potential leads to a lower electrode potential vs. the standard hydrogenelectrode (SHE). This parasitic potential loss can even be big enough at elivatedcurrent densities to provoke the cathode potential UCA falling down below thetransition potential of Utr = 0.8 V vs. SHE, where the transition from Temkin toLangmuir adsorption can be observed [121]. For the PEM-HFC simulation, thistransition of the adsorption mechanism is of no relevance because of the highercathode potential, however for the DMFC it is significant at low cell voltagescombined with an elevated methanol drag across the membrane (cf. Fig. 2.1 onpage 9). Most of the other models omit one or both peculiarities of a DMFC-cathode (e.g. [21, 135, 137]). Hence, as there are two main reactions on the DMFC-cathode, a second Butler-Volmer equation is defined for the crossover reaction.


1.3.5. Methanol Drag Across the Membrane

As mentioned above, the DMFC efficiency suffers from a parasitic methanol dragacross the membrane. This methanol flow consists of two components: diffusionand electroosmotic transport. Even though the crossover is crucial for the calcu-lation of the cathode potential, there are several models that neglect [21, 135] orsimplify [75] the crossover effect.

1.4. Classification of the Present Work

In this work, the approach is a zero-dimensional, analytical simulation of a poly-mer electrolyte membrane liquid direct methanol fuel cell (PEM-LDMFC) — ormore often called a Direct Methanol Fuel Cell (DMFC) — as a whole. The mainfocus lies a) in the simulation of the methanol drag from anode to cathode due toa water-dependent proton transport mechanism in the polyperfluorosulfonic acidmembrane, the so called crossover and b) the simulation of the cathode mixedpotential. In this work, both transport principles, electroosmosis and diffusion aremodeled in separate equations, which is in contrast to most of the other works(e.g. [43, 74, 135, 137]). For the simulation, the commercial software AspenPluswas used — a software designed particularly for dimensioning chemical plants.Therefore, the fuel cell is simulated in analogy to a chemical plant as an assemblyof different modules, such as pipes and reactors. One significant advantage ofthe use of an advanced engineering software is its capability to compute a widerange of the physical properties with a huge diversity of property models, like e.g.the composition and thermodynamic characteristics of the gas phase in a twophase mixed flow. This feature is especially useful in determining physical prop-erties such as the chemical activity of methanol and the carbon dioxide contentof water in the ACL. Furthermore, an analytical model of the DMFC inside anengineering software environment is easy to extend into a comprehensive modelof a more complex DMFC-system with heat exchangers, pumps, valves etc. Thiswould combine the advantages of semi-empirical simulation with the advantagesof analytical models.

1.5. Comparison of the Model to Other Works 7

1.5. Comparison of the Model to Other Works

In the review paper by OLIVEIRA [98] it is pointed out that the methanol trans-port across the membrane is one of the biggest challenges still to be overcome.Consistently, a DMFC model should predict this phenomenon and the resultingformation of the mixed potential on the cathode adequately. It is remarkablethat in some important papers on DMFC greatly simplified crossover and mixedpotential models are used. KULIKOVSKY computes the crossover based strictlyon a model for the anode. The electroosmotic coefficient of the membrane isignored. He determines the crossover rate by a dimensionless variable that is onlydependent on the thickness and the diffusion coefficients of the anodic diffusionand catalytic layer [75]. SCOTT takes the electroosmotic drag into account, butdisregards its temperature dependency [117]. There are only a few works that usean engineering software to model the DMFC, and there is no example of usingit for modeling the transport processes and complex interdependencies insidea DMFC. The common approach for this class of modeling work is to treat thewhole fuel cell more or less as a black box. One example is the simulation workof VON ANDRIAN [135]. She is simulating the entire system with the commercialengineering software Pro/II and is treating the fuel cell stack itself as a black boxneglecting crossover.

2. MODEL OF THE DIRECT METHANOLFUEL CELL

The DMFC is a galvanic cell, it donates electrical energy at variable ter-minal voltage dependent on the internal resistance of the connectedconsumer. For a better understanding of the electrical nature of theDMFC and the interdependencies of the polarizations with the fuelcell voltage and the open circuit half cell potentials see Fig. 2.1.

Fig. 2.1.: Polarization scheme of a DMFC. There are two contrary reactions with dif-ferent polarizations U on the cathode: the oxygen reduction (OR) and a parasiticmethanol oxidation, called the crossover reaction (CO). The open circuit potential(OCP) of the anodic and the cathodic methanol oxidation is in this example identi-cal Uoc, AN = Uoc, CO. Due to the crossover, there is a mixed potential at the cathodewhich leads to an OCP of the cathode vs. SHE Uoc, CA that is reduced compared withthe OCP of the pure oxygen reduction Uoc, OR. The upper line Uoc, OR relates to a mea-sured OCP of the real oxygen reduction reaction at or in a porous platinum basedelectrode, the value lies well below the theoretical value of 1.229 V vs. SHE (@298.15 K),cf. [99, 121, 148]. This deviation of the measured Uoc, OR is discussed on p. 38.

10 2. Model of the Direct Methanol Fuel Cell

CBPCDL

Fig. 2.2.: Sections or functional layers of asingle direct methanol fuel cell.

In the simulation model the DMFC isdivided into seven separate sections.The modeling is done in a sequentialmanner. The order is given in the figure2.2 on the right. It starts with the an-odic bipolar plate (ABP) and ends withthe cathodic bipolar plate (CBP). Thecalculation of the five inner layers —from the anodic diffusion layer (ADL) tothe cathodic diffusion layer (CDL) — isdone iteratively. This chapter describesthe physical and electrochemical prin-ciples of the simulation model. The nu-merical implementation is described inchapter 3.

2.1. Material and Energy Flows Between the Cell Parts

For a better understanding of the nexus between the different cell parts the mate-rial and energy streams are shown in the following flow diagrams.

shell

ABP

shell

heat

he

at

reactants

reactants

products

Fig. 2.3.: Flows: anodic bipolar plateand shell (ABP).

shell

ADL

reactants

reactants

products

shell

heat

Fig. 2.4.: Flows: anodic diffusionlayer (ADL)

The first layer in the model is the anodic bipolar plate (ABP), see Fig. 2.3. The mostimportant part of the ABP is the flow field, which is a system of small channelsin which the fuel is distributed to all parts of the geometric area of the anodicdiffusion layer (ADL). In order to have a small cell pitch, the hight of the flowfield distribution channels are below 1 mm, thus, dependent on the flow ratethere is a measurable pressure gradient along the ABP. Normally the anode feed

2.1. Material and Energy Flows Between the Cell Parts 11

is over-stoichiometric, that means, not all methanol is consumed. Hence, theABP has the function of a splitter, one part of the inflow is entering the ADLthe other part is heading for the anode outlet. The material of the ABP is eitheritself electroconductive or it contains an electroconductive coating in contactto the ADL. The ABP acts as a current collector and is electrically connected tothe anode terminal. Common materials for the ABP are stainless steel or carboncomposite.

The next layer is the ADL, see Fig. 2.4. It needs to be an electroconductive porousmedium. It distributes the fuel to the anodic catalyst layer (ACL) and transportsthe anodic exhaust (CO2) in the opposite direction to the anodic flow field. Twotransport effects are responsible for the methanol transport through the ADL:a) diffusion and b) convective transport due to the hydraulic pressure differencein the ABP. These transport effects are simulated: a) with the First Fick’s law andb) with the Darcy’s law. The diffusion layer consists of carbon cloth or of carbon-carbon composite paper, the latter one is often called toray paper after the mainmanufacturer Toray Industries, Inc. It also transports the electric current from theACL to the ABP.

shell

ACL

reactants

reactants

products

shell

reactants

protons electrons

heat heat

Fig. 2.5.: Flows: anodic catalyst layer(ACL)

Fig. 2.6.: Flows: polymer electrolytemembrane (PEM)

After the ADL follows the anodic catalyst layer (ACL), see Fig. 2.5. In this layer themethanol is electrochemically oxidized to carbon dioxide, electrons and protons(cf. Eq. 2.2 on page 17). The electrons establish the electric current; they areelectrically transfered via the ADL and ABP to the anode terminal. Aside fromthe electric conductivity, the openness for diffusion as well as convective transfer,the ADL needs to have likewise a proton conductivity. The electrode itself ismicro-porous it consists either of noble-metal powder or carbon powder / carbonnano-tubes with catalyst particles on the surface and intermediate spaces of thecarbon material, this variant is called supported catalyst (cf. p. 90). Due to thedistinctly lower reaction rate of the electrochemical methanol oxidation, the DMFC


anode needs a higher metal loading, in comparison to a hydrogen fuel cell anode;catalyst loadings above 0.05 kg/m2 are not unusual [49, 50]1, whereas the loadingfor PEM-HFC anodes lie well below 0.02 kg/m2. To keep the electrode thicknesslow, unsupported catalyst is often preferred. Methanol, water and carbon dioxidecan flow through the porous electrode structure; the electrons are transportedthrough the carbon support and / or the noble metal. For the proton conductivityand the mechanical stability the catalyst powder is agglutinated with an ionicpolymer e.g. polyperfluorosulfonic acid. The pressure gradient across the catalyticlayer is negligible, hence, only diffusion transport is assumed for the model of theACL.

The polymer electrolyte membrane (PEM), see Fig. 2.6, is dividing the fuel (metha-nol) from the oxygen (air). The PEM is not conductive for electrons, but for ions.Standard DMFC membranes are cation conductors, most frequently applied arepolyperfluorosulfonic acid membranes. The polyperfluorosulfonic acid containsprotons in a mobile form. They belong to a sulfonate group that is bound to anorganic tetrafluoroethylene based fluoropolymer-copolymer RSO−

3 . The polymer ishydrophilic and the water molecules inside the humidified polymer build a shellaround the protons. Due to the solubility of methanol and carbon dioxide in water,CH3OH & CO2 molecules become part of these hydrate shells, hence, the protontransport evolves an electroosmotic water, methanol and carbon dioxide drag.The methanol to water ratio at the anodic membrane surface is equal to the ratioelectroosmotic water / methanol drag, likewise the ratio H2O / CO2 drag. A sec-ond possible transport of water and methanol is diffusion. Convective hydraulictransport though the PEM is negligible [90]. Diffusion transport is considered forthe methanol transport. Because of a high water saturation on both sides of themembrane diffusion of water is not taken into account in the model. The anodicreaction is exothermic and the cathodic oxygen reduction endothermic, but thetemperature gradient across the membrane is not considered. The distance oflm < 2.0E-4 m between the electrodes is so tiny2 that the thermal energy transportfrom the anode to the cathode is sufficient to equal the two temperatures, even ata small temperature gradient.

In the cathodic catalyst layer (CCL), Fig. 2.7, two contrary reactions take place,the main oxygen reduction and the parasitic methanol oxidation, called crossoverreaction. The oxygen transport from the cathodic diffusion layer into the porouselectrode structure is modeled with the First Fick’s law, it is a dead end stream inthe model, no oxygen is crossing the membrane, nor does nitrogen (cf. chap. 5.4.3on page 93). A pressure gradient across the CCL is neglected and hence a con-vective transport. Just like the anode, the electrode structure needs to be openfor diffusion transport and conductive for electrons as well as for protons. The

1 0.05 kg/m2 = 5.0 mg/cm2

2 lm < 2.0E-4 m is the thickness of a swollen Nafion™117-membrane, in dry state the thickness isl⋆m < 1.78E-4 m

2.1. Material and Energy Flows Between the Cell Parts 13

CCL

shell

shell

reactants

crossover

protons electrons

reactants

products

products

heat

Fig. 2.7.: Flows: cathode catalystlayer (CCL)

CBPCDL

shell

shell

products

reactants

electrons

reactants

heat

Fig. 2.8.: Flows: cathode diffusionlayer (CDL) and bipolar plate (CBP)

electrons needed for the electrochemical reduction of oxygen are coming from theanode through the outside electric circuit. The transport of electrons and protonsinside the electrode structure occur analog to the ACL. The electrodes and protonsthat are produced in the crossover reaction recombine inside the electrode onneighboring reaction sites of the oxygen reduction, this phenomenon is calledlocal-element formation.

The cathodic diffusion layer (CDL), Fig. 2.8, has in principle the same conductiveand transport capabilites as the ADL. The difference lies in the hydrophobicqualities of the CDL. To avoid the formation of water films on the CCL, referedto as flooding, the CDL is often more hydrophobic in comparison to the ADL. Inaddition to diffision, there is a convective transport into the porous CDL. Thistransport is modeled with Darcy’s law. The cathodic bipolar plate (CBP), see Fig.2.8, is in the majority of cases made of the same material as the ABP. The flowfield may differ from the flow field in the ABP, due to the different viscosity of air.Again, the distribution channels have a low profile to keep the CBP thin and thisresults in a perceptible pressure drop between inlet and outlet of each CBP in astack. This pressure drop induces a convective transport of air into the CDL. Alikethe ABP has the CBP the function of current collector.

It now follows a more detailed description of the assumptions made and themathematic model of the different cell parts, also named functional layers.


2.2. Anode Compartment

The following conditions are assumed to be given in the anode compartment:

• The pressure in the anode compartment is equal to the pressure in thedistribution channels in the ABP and constant across the Y-plane of thediffusion layer.

• A convective transport is assumed only into the diffusion layer [74, 143].

• Until the water-methanol mixture in the electrode is saturated with carbondioxide, there is no gas phase, beyond this the gas in the pores consistssolely of carbon dioxide saturated with a water-methanol mixture.

• Diffusion and catalyst layer are thin and homogeneous, hence, Fick’s Firstlaw can be used to describe diffusion transports.

• The diffusion coefficients are specific to one medium with a characteris-tic tortuosity and hence, a tortuosity correction for the calculation of thediffusion transport is not necessary [18].

• The influence of oxygen reduction on the anode is not simulated (cf. chap.5.4.3).

2.2.1. Model of the Bipolar Plate

The pressure difference between the inlet and the outlet of every bipolar plate ∆pcan be approximated with a simple equation derived from the Hagen Poiseuilleequation [20] for laminar flow in a tube with the radius r and the length l :

m = π · r 4 ·ρ8 · ηV

· ∆p

l

⇔ ∆p = 8 · l

π · r 4 · m · ηVρ

; [p] = Pa ≡ kg

m·s2 = m

m4 ·kg·kg·m3

s·m·s·kgwith:dynamic viscosity, [ηV]= kg/(m·s); density, [ρ]= kg/m3; mass flow, [m]= kg/s

As a result, ∆pABP = (pi n, ABP −pout , ABP) is proportional to:

∆pABP ∼ m · ηVρ

⇒ ∆pABP = fg , ABP · m · ηVρ

2.2. Anode Compartment 15

At the inlet of the ABP pressure p, temperature T and methanol fraction xCH3OH

are given, hence ρ and ηV can be calculated using AspenPlus. If the pressuredifference ∆pABP at a given flow m of the water-methanol mixture through theABP is known or can be estimated a proper proportionality factor fg , ABP that isspecific for the given assembly of the bipolar plate with the bordering diffusionlayer can be calculated:

fg , ABP = ∆pABP ·ρm · ηV

; [ fg , ABP] = m-3 = kg·kg·s·m·s

s2·m·m3·kg·kg

Accordingly, the formula can be expressed as follows:

∆pABP = fg , ABP · m · ηVρ

; [p] ≡ kg

m·s2 = 1

m3 · kg·kg·m3

s·m·s·kg

This calculation of the pressure difference is needed for the calculation of convec-tive transport through the porous anodic diffusion layer in the next calculationstep. As mentioned above, there is only a single pressure value commonly usedfor the calculation of the physical properties in the entire anode compartment.Fig. 2.9 and Fig. 2.9 illustrate the two different main directions of flow in the neigh-bouring layers ABP and ADL. In the ABP is the flow direction dependent on theflow field and convective transport is dominant. Whereas in the ADL the ratiobetween convective and diffusive transport is dependent on the flow rate andtherefore the pressure gradient along the anode flow-flield.

InletFlowChannel

Outlet

Bipolar Plate

l

Fig. 2.9.

ABP PEMADL-ACLInlet(pin)

Outlet(pout) ACL-PEM

(pout-ABP)

xFig. 2.10.

Fig. 2.9, 2.10: Top and side-face view of the functional layers ABP and ADL. Fig. 2.9shows a one channel meander flow field, which is a typical design for test laboratorysingel-cells. Fig. 2.10 is the cross-section side-face of the four layers ADL, ABP, ACLand PEM.

The pressure gradients in Fig. 2.11 exist in reality, but as mentioned above, inthe model we assume a common pressure to calculate the physical properties.


ABP

𝑙 / m

𝑝/

PaADL

𝑥 / m

𝑝/

Pa

𝑝𝑖𝑛, adl

𝑝𝑜𝑢𝑡, abp

𝑝𝑖𝑛, adl = 𝑝𝑖𝑛,abp − 0.5 · Δ𝑝abp

𝑝𝑜𝑢𝑡, adl = 𝑝𝑜𝑢𝑡, abp

𝑝𝑖𝑛, abp

Fig. 2.11.: Sheme of pressure drop across the functional layers ABP and ADL. The pres-sure in the ABP decreases along the distribution chanels (coordinate l), whereas thepressure in the ADL changes meanly across the layer (coordinate x). At the interfaceADL-ACL the pressure is homogenous and equal to the output pressure of the ABP.The pressure change is for both sections in the range well below 1E+4 Pa.

Because of low pressure differences below 1E+4 Pa this can be considered as aproper approach to model the ABP.

Higher flow induces a higher pressure gradient between inlet and outlet of theABP, this leads to a higher mean pressure accross the ABP-ADL interface.

2.2.2. Model of the Anodic Diffusion Layer

Dependent on the pressure drop over the ABP the transport through the anodicdiffusion layer in a standard DMFC with a low concentrated water-methanolmixture may be dominated by the convective transport due to the pressure dropacross the layer. The pressure drop across the ADL is not constant, as is thepressure in the ABP-channels. At the inlet the pressure gradient has its maximumvalue, at the outlet we assume a zero pressure drop. The reason for that is thePEM with its characteristic as a flow barrier. To simplify matters in the model, themean pressure between inlet and outlet flow in the bipolar plate is taken as thepressure on the inlet plane and the bipolar outlet pressure is used as the pressureat the outlet plane of the diffusion layer, cf. Fig. 2.11. The mathematic nexus isshown in the following:

∆pADL = pi n, ABP −0.5 · (pi n, ABP −pout , ABP)−pout, ABP

If the following is defined:

pout , ADL ≡ pout , ABP

the result is: ∆pADL = 0.5 ·∆pABP


Now a simplified version of the Darcy’s law (see page 141) can be used to calculatethe convective flow through the diffusion layer:

∆p = ηV

khp· �� · l ⇔ ��= khp

ηV· ∆p

l⇒ Φ= �� · Afc/V (2.1)

with:Mixed component molar volume, [V ] = m3

mol ; geometric electrode area, [Afc] = m2;

velocity [𝑣] = ms ; hydraulic permeability, [khp ] = m2; total mole flow, [Φ] = mol

s

The hydraulic permeability constant khp needs to be determined for the diffusionlayer in use or needs to be fitted to the expected convective flow through thediffusion layer.

Additionally the methanol diffusion flow can be described by Fick’s First law :

ΦADL,CH3OH = DADL,CH3OH ·CCH3OH, ADL out −CCH3OH, ADL i n

lADL

· Afc

with:ADL-methanol diffusion coefficient, [DADL,CH3OH]= m2/s; methanol concentration,

[CCH3OH]= mol/m3; geometric electrode area, [Afc] = m2 ; ADL thickness, [lADL] = m

2.2.3. Reaction Path of the Methanol Oxidation

The gross exothermic reaction of the methanol oxidation is:

2CH3OH + 3O2 −→ 2CO2 + 4H2O

In the fuel cell the reaction is split into an anodic and a cathodic reaction:

anode: CH3OH + H2O −→ CO2 + 6H+ + 6e− (2.2)

cathode: 6H+ + 6e− + 1.5O2 −→ 3H2O

=⇒ CH3OH + H2O + 1.5O2 −→ CO2 + 3H2O (2.3)

As previously indicated, the methanol oxidation is the best known electrochemicaloxidation of an organic substance. The exact reaction path is contingent on the


catalyst in use. The standard DMFC-anode catalyst is a platinum-ruthenium-catalyst, for this binary Pt-Ru-catalyst HAMNETT [56] suggests the following reac-tion path [43, 67]:

CH3OH + 3Pt −→ PtRCOH + 3PtRH

PtRCOH + Pt −→ PtRCO + PtRH (2.4)

H2O + Ru −→ RuROH + H+ + e− (2.5)

RuROH + PtRCO −→ PtRCOOH + Ru (2.6)

PtRCOOH + RuROH −→ CO2 + H2O + Pt + Ru

PtRH −→ Pt + H+ + e−

RuROH −→ RuRO + H+ + e− (2.7)

PtRCO + RuRO −→ CO2 + Pt + Ru (2.8)

However, if a pure Pt-catalyst is used Ru has to be replaced by Pt in the equations.The crucial point for the electrooxidation of methanol is the presence of thecarbonyl group CO (2.4). Due to the fact that the Pt-adsorbed CO is blocking theactive surface of the Pt-catalyst, it is important to have an oxidant that is ableto crack the strong PtRCO-bond further to CO2. An adsorbed hydroxyl group OHon a neighboring catalyst site is such an oxidant (2.5). This can be achieved bythe use of mixed catalyst Pt-Ru. The advantage of using Ru is that this metalcatalyzes the H2O-splitting due to its good OH adsorption capabilities. In thenext step it supports the hydroxyl oxidation reaction (2.7), thus providing thenecessary adsorbed oxygen atom, which is needed for the unblocking of the Pt-catalyst surface (2.8). For further discussion of the electrooxidation of CO see[46,83]. This mechanism explains why it is probably favorable to have a ratio 50:50for Pt-Ru, which is in accordance to the work of GASTEIGER et al. [46]. However,WANG et al. [144] found in recent DEMS3 studies an optimum ratio of 80:20 Pt-Rufor an unsupported catalyst. For the discussion of this matter it is important tokeep in mind that it is generally difficult to compare different measurements ofcatalyst activities. E.g., when working with catalyst powders the desired adsorptionproperties may differ significantly, even if powders of the same product but fromdifferent lots are used. Measuring in a traditional full cell or on a DEMS cellwith a minuscule active area and a glassy carbon electrode holder can also resultin variations from the above. In the ZSW laboratories we observe differences inactivity of the same catalyst but from a different lot of up to 20 – 30%. Nevertheless,the metal ratio is only one of many properties that influence performance.

3 DEMS = Differential Electrochemical Mass Spectroscopy


HAMNETT [55] proposed the following simplified reaction scheme for the faradaicmethanol oxidation:

CH3OH + H2O + Pt −→ PtRCOOH + 5H+ + 5e−

∧ PtRCOOH −→ CO2 + Pt(s) + H+ + e−

=⇒ CH3OH + H2O −→ CO2 + 6H+ + 6e−

This simplified reaction path does not explicitly contain the catalyst poisoningstep of the carbonyl formation (2.4), but this reaction step is necessary for thecarboxyl formation (2.6) and hence, is considered.

It has become accepted knowledge that the rate-limiting step in the reaction pathis the oxidation of the Pt-adsorbed CO-molecule. HAMNETT [55] proposed thefollowing rate-determining steps:

PtRCO + RuROH −→ CO2 + H+ + e− (2.9)

∧ PtRCO + H2O −→ CO2 + 2H+ + 2e−

The reaction (2.9) correlates with the findings of TREMILIOSI-FILHO [133] andKAURANEN et al. [67]. Thus the reaction is of zero-order in respect to methanol,when all active Pt-atoms are occupied by carbonyl groups. This can be expected,as there is normally enough methanol in the anode feed if the concentration isabove 0.1 M [105]. Consequently, in order to increase the reaction rate, the scientisthas to enlarge the active catalyst surface and / or to raise the temperature [9]. Anincrease of the methanol feed might even be counterproductive, as this raises theunwanted crossover.

2.2.4. Model of the Anodic Catalyst Layer

For the model of the anodic catalyst layer it is presumed that there is only onerate determining step (RDS) for the electrochemical reaction at the anode [117]and that this reaction step can be described by a single kinetic equation, such asthe Butler-Volmer equation. It is further assumed that the activities of the reagentsand products of this RDS are proportional to the activities of the reagents andproducts of the gross reaction at the anode. On the basis of these assumptions


the Butler-Volmer equation can be used to simulate the electrochemical reactionfor the anode:

I AN = I 0AN(T,αAN, I✸0

AN ,∆G‡r ea, AN,aoc,red,aoc,ox )

· f BV(T,αAN,UAN, z‡AN,aox, AN,ar ed , AN)

= I✸0AN · aαAN

oc,red · a1−α

ANoc,ox · exp

−∆G‡rea, AN

R·

1

T− 1

T✸

·aox, AN

aoc,ox· exp

−αAN · z‡

AN ·F ·UAN

R ·T

− ar ed , AN

aoc,red· exp

(1−αAN) · z‡

AN ·F ·UAN

R ·T

(2.10)

A version of the Butler-Volmer equation is used, in which the temperature- andconcentration-dependent exchange current density I 0 is related to the exchangecurrent density of a referenced case. The result is a form with an exchange cur-rent density that is a product of a temperature and concentration independentexchange current density constant I✸0, a concentration term and a fraction oftwo Arrhenius equations. The I✸0 is related to the active catalyst surface SAN

available for the electrochemical RDS4. Hence, the value of I✸0 is very specific forthe examined electrode. The derivation of the Butler-Volmer equation, Eq. 2.10,can be found on page 123 et sqq. For more information about the Butler-Volmerequation cf. e.g. [12, 23, 54, 115, 139].

To calculate the current I the active surface related current density I has to bemultiplied by the active catalyst surface. In conjunction with the electrons trans-fered per methanol molecule z the reaction rate nCH3OH can be calculated in termsof amount of substance per time:

IAN = I (UAN) ·SAN

nCH3OH = IAN

F · zAN

, zAN = 6 , [nCH3OH] = mol/s

In some publications temperature dependent transfer coefficients α are discussedin theory as in [114] or postulated for the oxygen reduction, based on data [117].SCHMICKLER [114] concludes that electrochemical reactions, which significantlydepend on proton tunneling may have a temperature dependent α. The result of

4 To make the difference between the two current densities clear: the catalyst surface- and thegeometric electrode area-related current density, the two symbols: I and j are used. The differencebetween the two values is not always clear, e.g. SCOTT [120] used the same symbol for both andthe resulting erroneous model is discussed by KULIKOVSKY in [76].


the simulations support this theory; as a result a linear temperature relationshipas discussed by PARTHASARATHY [99] is used:

α(T ) = α✸+bα · (T −T✸) ; α✸ =αT ✸

(2.11)

⇒ bα = α(T )−α✸

T −T✸

The number of electrons transfered in the RDS is given by z‡AN, with its value

being one.5 The polarization of the anode UAN (see glossary on page 150) is thedifference between the active half cell potential UAN and the open circuit potentialUoc, AN vs. a common reference potential U✸, e.g. the potential of the standardhydrogen electrode (SHE) USHE:

UAN = (UAN −U✸)− (Uoc, AN −U✸) ∧ UAN >Uoc, AN

= UAN −Uoc, AN ⇒ UAN ≥ 0V

The open circuit half cell potential Uoc is calculated via the Nernst equation asfollows:

Uoc = Uid +UNernst

= Uid +R ·T

z ·F · νi · lnai (2.12)

More information about the Nernst equation and the second term of the equationis given on page 149. For the Nernst equation the activities or concentrations ofthe reagents and reaction products are needed. According to the rate-determiningstep reaction (Eq. 2.9 on page 19) the reduced species are H2O, RCO, whereasthe oxidized species is CO2. It is, however, not easy to estimate the activity of anadsorbed carboxyl group. To overcome this problem, it is assumed that the activityofRCO is proportional to the methanol activity. The activity of water is assumedto be constant at the value of one and the activities of methanol aox = aCH3OH andcarbon dioxide ared = aCO2

can be calculated by AspenPlus.

The Nernst equation is appropriate for one step reactions [12]. For the morecomplex reaction of methanol oxidation the following assumption is made:Equation 2.9 is the rate-determining step and this reaction step is a one stepreaction and so the Nernst equation is appropriate. The activity of the reagentscorresponds to the activities of H2O and CH3OH [43]. As the value for aCO2

theactivity of the saturated CO2-water-methanol-solution is taken. This results in:

Uoc = Uid +R ·T

z ·F · lnaCO2

aH2O · aCH3OH

5 Aside from the number of electrons transfered in the RDS, there is also the number of electronstransfered in the gross reaction, z‡

AN = zAN , the latter being 6.


Once the open cell potential and the activities used in the Nernst equation havebeen established, the ideal half cell or electrode potential vs. SHE Uid can becalculated.

Uid = R ·T

z ·F · lnaCO2

aH2O · aCH3OH−Uoc

In theory the ideal half cell potential Uid is only dependent on the change of theGibbs Free energy of the reaction ∆G , as ∆G is temperature dependent also Uid

is temperature dependent, cf. the glossary on page 147. In compliance with theconvention of electrical engineering electric energy that is produced by a galvaniccell is positive. As energy leaving the system is defined as negative in physics, aminus sign has to be positioned in front of Gibbs Free energy ∆G :

Uid(T ) = −∆G(T )

z ·FIn this study published OCP values are taken to calculate the ideal half cell poten-tial Uid [148]. The temperature dependency of Uid is following an Arrhenius behav-ior (see page 137); subsequently, a referenced open cell potential U✸

id =Uoc (T✸)is invented and a fraction of Boltzmann factors is formulated in the same way ascalculating the temperature dependent exchange current density for the Butler-Volmer equation (see Eq. D.15 on page 129). Furthermore, an activation energyat equilibrium conditions ∆G‡�

AN for the ideal half cell potential Uid is introduced.This activation energy can be calculated when the open circuit potentials (OCP)of at least two temperatures [99] are given:

Ui d ,AN(T ) = U✸i d , AN

· exp

−∆G‡�

AN

R·

1

T− 1

T✸

Now the formula to calculate the open half cell potential vs. SHE Uoc, AN at differenttemperatures and different methanol concentrations can be written as:

Uoc, AN = U✸i d , AN

· exp

−∆G‡�

AN

R·

1

T− 1

T✸

+ R ·T

z ·F · lnaCO2

aH2O · aCH3OH

The Butler-Volmer equation is also used for simulation of the electrochemical reac-tions on the cathode: the oxygen reduction and the contrary methanol oxidation,the so-called crossover reaction.

For a better understanding of the coherences between the three equations seeFig. 2.12. In this graph the polarization is shown on the x-coordinate, which isin accordance to the fact that the polarization is the driving force for the electro-chemical reaction. However, more common is the depiction in Fig. 2.13 with thecurrent I or the current density j on the abscissa and the electrode potential vs.SHE on the axis of ordinate. The nexus between the negative crossover currentdensity jΠCO and the positive cathode current density jΠCA can be expressed by theequation: jΠCA = jΠOR + jΠCO and is explained in Fig. 2.14.


Fig. 2.12.: Scheme of the polarization of the three causative electrochemical reactions ina DMFC with methanol drag across the membrane (crossover):a) anodic methanol oxidation (AN), b) cathodic methanol oxidation (CO, crossoverreaction), c) oxygen reduction (OR).While the polarization of oxidative reactions is positive, it is the opposite for reductivereactions. The reaction rates or currents are dependent on the polarization. The linejΠCA is the sum of the normalized current densities on the cathode (CA): jΠCA = jΠOR + jΠCO.The difference of current densities between oxygen reduction and crossover reaction atthe open circuit cathode potential is proportional to the methanol diffusion across themembrane jΠD ∼ ΦD . For the graph above a low membrane diffusivity was used, as itwas determined for the proprietary PVC-membrane utilized by WILLIAMS [148]; hence,the crossover is increasing with the absolute value of the cathode polarization, due tothe increasing of the electroosmotic drag, this behavior is in contrast to DMFC withpolyperfluorosulfonic acidmembranes. Note that the polarization of the crossoverreaction is more than twice the absolute value of the oxygen reduction reaction.

Fig. 2.13.: The polarization scheme in the more common form with the current densityon the x- and the half cell potential vs. standard hydrogen electrode (SHE) on the y-axis.In a fuel cell the reduction current density (jΠOR) is positive and the oxidation currentdensities (jΠAN, jΠCO) are negative [7].


Fig. 2.14.: Polarization curves of the three main reactions and the resulting mixedpotential of the cathode in the standard arrangement of current density as the x-axisand the cell voltage on the y-axis.

2.2.5. Active Surface of the Electrode

It is very difficult to determine the electrochemically active surface of a porouspolyperfluorosulfonic acid soaked electrode. The CO-stripping method is wellestablished for pure and solid Pt-electrodes [65], whereas for porous membranesthat carry organic impurities this method gives only qualitative results. In ourcase, even an estimation of the active surface of the electrodes is not possible, asWILLIAMS [148] did not mention the metal loading on the electrodes. As this isnot relevant in a general investigation, an active surface of unity for the referencecase can be utilized. This merely influences the reference exchange current densityI✸0. Nevertheless, a variability of the surface S and the thickness of the electrodel AC L, AN dependent on the metal load mcat is desired, therefore it is:

mcat , AN = mcat , AN · Afc

VAN = mcat , AN ·vcat , AN

l AC L, AN = VAN

Afc

S✸AN = VAN ·SAN

with:geometric electrode area, [Afc] = m2; catalyst loading, [mcat , AN] = kg/m2;

anode volume, [VAN] = m3; mass specific cat volume, [vcat , AN] = m3/kg;

volume specific surface, [SAN] = m2/m3 = m-1

2.3. Model of the Polymer Electrolyte Membrane 25

In addition, the original geometric area of the electrode Afc is also unknown.Therefore, a value of unity is taken for the geometric area. To be comparableto a real DMFC anode, the values for the mass specific catalyst volume vcat andthe volume specific surface S are singled out in a way to gain an electrode thick-ness of l AC L, AN = 2.5E-5 m = 25µm and a catalyst loading of mcat = 0.05 kg/m2 =5.0 mg/cm2.

2.3. Model of the Polymer Electrolyte Membrane

For the model of the polymer electrolyte membrane (PEM) the following assump-tions are made and the species transport through the membrane is restricted:

• Diffusion coefficients and the electroosmotic drag factor are temperaturedependent [134].


• Hydrated protons — the flow is proportional to the cell current I:ΦH+ = I /F.

• Water molecules — the transport is driven by electroosmotic drag, the waterflow is dependent on the cell current, the ratio "protons to water molecules"is given by the electroosmotic drag factor ξeo :ΦH2O = I ·ξeo/F.Water diffusion is negligible in a liquid feed DMFC with a low methanol con-centration6 in the feed stream, because it is assumed that the concentrationgradient across the membrane can be neglected.

• Methanol molecules — the flow consists of electroosmotic drag and diffu-sion ΦCH3OH =ΦCH3OH,eo +ΦCH3OH,D .

• Carbon dioxide — the transport is dependent on the mole fraction of dis-solved CO2 in the water-methanol mixture at the anode-membrane inter-face (AMI) and on the water flux: ΦCO2

= xCO2

AMI

·ΦH2O.

• The diffusion of oxygen from the cathode to the anode is intentionallyomitted (cf. chap. 5.4.3)

• The nitrogen diffusion is not considered, as it does not interfere with theanode polarization.

6 If the water transport through the ADL and ACL to the anode-membrane interface is hindered,there is a back diffusion of water from the cathode, this back-flow has to be considered especiallyfor methanol concentrations above 5 M


2.3.1. Water Transport Through the Membrane

There are three mechanisms that can promote water drag trough the membrane:a) diffusion; b) pressure induced transfer; c) electroosmotic drag.

The driving force for a diffusion transfer is a concentration gradient over themembrane. For the DMFC with low methanol concentration in the anode feed, itis obvious that there is nearly no concentration gradient of water, because of theoxygen reduction reaction

6H++1.5O2 +6e− −→ 3H2O (2.13)

and the water transfer through electroosmotic drag [91].

The concentration gradient grows with higher partial vapor pressure of water(higher temperatures), stronger airflow (Φai r ) and decreasing humidity of theincoming air. Higher total pressure results in a less effective drying of the CDL bythe outgoing air. In the liquid run DMFC with low methanol concentrations inthe feed the membrane on the anode side is always water saturated. Therefore,even if there is a methanol diffusion transfer through the membrane, there willbe no additional drag effect because of hydration of the methanol molecules. Noindication is given in the literature on the water diffusion being dependent on themethanol diffusion [123].In order to protect the membrane against mechanical stress big pressure gradientsover the membrane have to be avoided, which results in a negligible pressure in-duced water transport[61,88,123]. MEIER [88] observed that within practical limitsthe membrane thickness, the methanol concentration at the anode or the diffusionlayer have no significant influence on the water flux through the membrane.

Electroosmotic Water Transport

REN et al. [106, 107] found that at high enough current densities the electroosmoticdrag becomes the main transfer mechanism for water. At higher current densitiesthey found the water flux showing a linear relation to the current density. InFig. 2.15 on the next page the values of the water drag coefficient ξeo derived fromthis linear region are plotted.

They discovered that the beginning of this linear region depends mainly on temper-ature, the methanol concentration and on stoichiometry: for an oxygen run DMFCwith Nafion™117 membrane it starts at a temperature of 353 K and 𝒞CH3OH = 1M7

at 2000 A/m2. At lower temperatures, e.g. 333 K, and a pressure of 2.37·105Pa, the

7 To differentiate between the SI-unit for concentration mol/m3 and the non SI-unit M = mol/ltwo different symbols C and 𝒞 are applied.


Fig. 2.15.: Electroosmotic water drag coefficient ξeo for Nafion™ 117, 𝒞CH3OH = 1M

measured in the linear regime with liquid water-equilibrated membranes [107]

region of a linear relation between water flux across the membrane and currentdensity starts at 1000 A/m2 [107].

In this work the temperature dependency of the electroosmotic water drag wasalso found; hence, ξeo is defined as:

ξeo(T ) = ξeo(T✸) · exp

beo ·

1

T− 1

T✸

(2.14)

The value at reference temperature for the proprietary PVC-membraneused by WILLIAMS [148] was achieved by fitting (cf. chap. 4.1) and is set toξeo,PV C (T✸) = 1.8, the value for Nafion™117 was taken from literature [107] andset to: ξeo,Nafion™117(T✸) = 2.0. The electroosmotic water drag itself is linear to thefuel cell current Ifc:

Φeo,H2O =ξeo · Ifc

F; [Φ] ≡ mol

s= A

A ·smol

(2.15)


2.3.2. Methanol Drag Through the Membrane

The main methanol transport is caused by diffusion and by electroosmosis.

Methanol Diffusion Through PEM

Diffusion is influenced by three terms: a) length l , b) diffusion coefficient D andc) concentration difference ∆C . In a simple form, the diffusion flux density ΦD isdescribed with Fick’s First law :

ΦD = −D · ∆C

lm; [ΦD ] = mol/(m2·s) (2.16)

with: [D] = m2/s , [C ] = mol/m3

For standard polyperfluorosulfonic acid-membranes data for the methanol diffu-sion is available from existing literature or from the manufacturer, however theactual diffusion through the membrane electrode assembly (MEA) can differ sig-nificantly from that. If polarization data of the cathode with and without crossoveris at hand, the effective diffusion coefficient of the MEA can be calculated. AtOCV of the crossover-penetrated cathode the cathodic methanol reaction rate isproportional to the pure diffusion flow of methanol through the MEA (as there isno electroosmotic drag, jfc = 0 A/m2). The current density of the cathode withoutmethanol penetration jOR (pure oxygen reduction) at the open circuit mixed poten-tial Uoc,mp of the methanol penetrated cathode is proportional to the crossoverflow:

jORUoc,mp

∼ DCH3OH ; (DCH3OH: effective diffusion coeff. of the MEA)

ΦD,CH3OH = −DCH3OH ·CCH3OH,2 −CCH3OH,1

lm, [Φ] = mol

s · m2

with:

CCH3OH,2 = 0 mol/m3 ; (concentration in the cathode)

jCH3OH,D = ΦD,CH3OH ·6·F

=−DCH3OH · −CCH3OH,1 ·6·F

lm(2.17)

and: jCH3OH,D = jORU

OR= U

CA,oc, with crossover

DCH3OH =jCH3OH,D · lm

CCH3OH,1 ·6·F ; [D] ≡ m2

s= A·m·m3·mol

m2·mol·A·s(2.18)


In the publication of WILLIAMS [148] all necessary information is provided forcalculation of the effective diffusion coefficient at the reference temperature of298.15 K:

CCH3OH,1 = 1.0E+3 mol/m3

lm = 7.62E-4m

jORT ✸

Uoc,mp

= 144.0 A/m2 , graphically determinable, cf. Fig. 2.16

6.0 ·F = 5.78912E+5A·s/mol

D✸CH3OH = 144.0 A/m2 ·7.62E-04m

1.0E+3 mol/m3 ·5.78912E+5A·s/mol

= 1.895E-10m2/s (2.19)

Fig. 2.16.: Graphical determination of the effective diffusion coefficient of the electrodemembrane assembly:Plotted are the polarization data of the cathode with and without crossover [148]. Atthe open circuit half cell potential vs. SHE of the cathode with methanol penetra-tion (called mixed potential) Uoc,mp the methanol diffusion coefficient DCH3OH is

proportional to the current density of the cathode without mixed potential (withoutcrossover) jOR. In our case it is: jOR(@T = 298K, UOR =Uoc,mp = 0.89V) = 144.0 A/m2

The diffusion coefficient itself is temperature dependent. If a set of polarizationdata of the cathode with and without methanol crossover at different temperaturesis available, the temperature dependency can easily be fitted with an exponentialfunction equivalent to the form of equation 2.21 as in [117]. In the case of theWILLIAMS data [148], the effective diffusion can be ascertained only at referencetemperature, so it is recoursed to published temperature dependency information.WANG [143] used the formula published in the textbook of YAWS [152]:

D(T ) = 10dD− 999.778K

T

⇒ dD = lg(D✸)+ 999.778K

T✸(2.20)


Fig. 2.17. Fig. 2.18.

Fig. 2.17, 2.18: Temperature dependency of methanol diffusion coefficient and flow.Fig. 2.17 shows the difference between a simple exponent formula (cf. Eq. 2.20) for thediffusion coefficient and an ARRHENIUS like temperature influence (cf. Eq. 2.21) fora Nafion™11X membrane. Fig. 2.18 compares the current density equivalent (CDE)of the methanol diffusion flow through a Nafion™117 and a gold coated sulfuric aciddoped porous PVC-membrane [148] at a constant methanol concentration differenceacross the membrane of 𝒞CH3OH = 1M.

whereas SCOTT et al. [117] used an exponential form8:

D(T ) = D✸ · exp

−2436.0K ·

1

T− 1

T✸

(2.21)

For the simulation of the polarization data from WILLIAMS [148] the formula ofYAWS [152] was used. Fig. 2.17 shows that the difference to the formula used bySCOTT [117] is minor. A comparable temperature dependency of the diffusion atOCV conditions in a real DMFC cell was also found by THOMAS et al. [132]. It isalso possible to estimate the diffusion, by varying the methanol concentrationand comparing the OCV values, this method is described in [13].

With the Eq. 2.18, the membrane thickness lm = 7.62E-4m [148]9 and the con-centration difference across the membrane of 1 M CH3OH, the effective dif-fusion coefficient of the PVC-membrane at T✸= 298 K can be calculated to:D✸

CH3OH,PVC = 1.895E-10 m2/s. For simulations with the Nafion™117-membrane

the reference diffusion coefficient was set to 80% of the value used by SCOTT [117]:D✸

CH3OH,Nafion™11X = 1.662E-10 m2/s. The methanol diffusion coefficient at 298 K

8 There is a minor difference between the value in the article cited by SCOTT [127] (2416.0 K) andthe value printed in the article from SCOTT [117] (2436.0 K).

9 In DMFC often thicker membranes are used in comparison to PEM-HFC, e.g. Nafion™115 insteadof Nafion™112. The last digit of the Nafion™ type specifies the thickness in "mil" (1 mil = 1E-3 in =2.54E-5 m) of the dry membrane. Thicker membranes have the advantage of lower methanolcrossover. In DMFC the membrane is fully humidified, hence, the values of the swollen Nafion™is relevant, which is about 12% higher in comparison to the dry membrane.


for the proprietary PVC-membrane is 14% higher than the one of Nafion™117,but the diffusion flow is lower because of the 3.2 times thicker PVC membrane, ascan be seen in Fig. 2.18.

Electroosmotic Methanol Drag Through Membrane

The electroosmosis term is caused by the superimposed electroosmotic water fluxthrough hydrophilic channels of the membrane and thus is also dependent onthe methanol concentration on the anode-membrane interface. The formula is asfollows:

ΦCH3OH,eo = ΦH2O,eo ·xH2O

xCH3OH

AM I

Influence of Temperature and Current Density on Crossover

At high current densities and high temperatures the situation can occur wherenearly all methanol is faradaically oxidized in the anodic catalytic layer before itcan reach the membrane surface [6, 28]. Under such conditions a low methanolcrossover is observed. REN et al. [105] observed a 50% reduction of methanolcrossover with a Nafion™117 membrane at 353 K above 4000 A/m2. However, ifthe methanol concentration on the membrane surface is higher, elevated tem-peratures and high current densities promote the methanol flux. The reasonsfor this are: the diffusion coefficient for methanol increases with temperatureand the convection rises along with the superimposed electroosmotic drag of thewater, which is proportional to the current density. Thus, in order to simulate themethanol crossover, it is crucial to implement a proper model for the methanolconcentration at the interface between the anode catalytic layer and the PEM.


2.3.3. Carbon Dioxide Drag Through the Membrane

The carbon dioxide flux is mainly dependent on the amount of dissolved CO2

in the water-methanol mixture and the flow rate of water through the mem-brane. The mathematical description is equivalent to the one of the metha-nol transport through the membrane (cf. chap. 2.3.2). The value for the diffu-sion coefficient at T✸= 298 K of CO2 through the proprietary PVC membrane isapproximated to 1.14 times the value in water: D✸

CO2,PVC = 2.298E-10 m2/s; the

value for water is published in [131]. This estimate is derived from the ratioD✸

CH3OH,PVC/D✸CH3OH,Nafion™117 = 1.14. The value for Nafion™117 is assumed to be

equal to the diffusivity in water: D✸CO2,Nafion™117 = 2.006E-10 m2/s (cf. Tab. B.5 on

page 116). The diffusion flux can reach high values in relation to the amount ofCO2 produced on the cathode side [94]. DOHLE [39] found that with low methanolconcentrations (𝒞CH3OH ≤ 1 M) and high current densities (jfc > 2000 A/m2) theamount of carbon dioxide passing from the anode to the cathode can even exceedthe amount of carbon dioxide formed at the cathode by methanol oxidation.

2.3.4. Proton Conductivity

The ionic conductivity σ of an ion exchange membrane is dependent on theeffective diffusion factor of protons DH+ , and with the diffusion factor itself beingdependent on temperature, BERNARDI and VERBRUGGE [18] gave the formula:

σ(T ) = F2

R ·T·DH+(T ) ·CH+ ; [σ] = S

m= A

V·m= A2·s3

kg·m3

There is a constant proton concentration CH+ in polymer electrolyte-membranes(PEM)10. Therefore, σ(T ) becomes proportional to DH+(T ), which can be alterna-tively derived:

DH+(T ) = D✸H+ · exp

bD ·

1

T− 1

T✸

∧ σ ∼ DH+

⇒σ(T ) = σ✸ · exp

bσ ·

1

T− 1

T✸

(2.22)

10 In Nafion™-membranes like Nafion™117 this concentration is given in an implicit form in thename. The first two digits multiplied by 100 indicate the so-called equivalent weight (EW). TheEW is defined as the weight of Nafion™ per mole of sulfonic acid group SO3H−. In the proprietarypolyvinyl chloride-membrane (PVC) used by WILLIAMS [148] the concentration is also constant,because the electrolyte in the porous PVC gold coated structure was constant at 6 M H2SO4.


WILLIAMS [148] mentioned as a typical internal voltage loss due to internal resis-tance a value of UR = 0.045 V at a current density of jfc = 1000 A/m2. The mem-brane thickness was lm = 7.62E-4m at a working temperature of Tfc = 333.15K. Asa result, the approximate electrical conductivity can be figured out as follows:

σ(T = 333.15K) = j · lm

UR= 1000A ·7.62E-4m

0.045V ·m2 = 16.93S

m

In order to compare this value to polyperfluorosulfonic acid-membranes, Fig. 2.19shows the conductivity data for fully humidified Nafion™117-membrane pub-lished by ZAWODZINSKY et al. [155]. The exponential factor bσ is established via aregression analysis and the function shifted so that it corresponds to the value ofthe electrical conductivity calculated for the gold coated PVC-membrane used byWILLIAMS. The result is shown as line c in Fig. 2.19 and is comparable with theexponential factor used by SCOTT [117](line d). For the temperature dependencyin the model the exponential factor fitted to the measured data of ZAWODZINSKY

was used.

Fig. 2.19.: a) Nafion™117 ion conductivity, published by ZAWODZINSKI [155];b) Calculated ion conductivity of WILLIAMS-PVC-membrane [148] at 333.15 K;c) Exponential conductivity function in the form of σ(T ) =σ✸ ·exp[bσ · (T −1 −T✸−1)],

bσ fitted to the data of ZAWODZINSKI [155] as bσ =−1359.3K;d) Exponential conductivity function with bσ =−1268.0K used by SCOTT for

Nafion™117 [117]


2.4. Cathode Compartment

For the cathode compartment the following assumptions are made:

• The pressure in the cathode compartment is equal to the pressure in thedistribution channels in the cathode bipolar plates (CBP) and constantacross the diffusion layer [19].

• The air in the pores is saturated with water [19].

• The layer is thin and homogeneous, so Fick’s First law can be used to de-scribe diffusion.

• The convective transport into the CDL due to pressure change along thecathode flow field can be modeled via the Darcy’s law.


2.4.1. Cathodic Reactions

On the cathode two reactions take place: the wanted oxygen reduction and theunwanted parasitic methanol oxidation.

Oxygen Reduction

The first step of the oxygen reduction is the adsorption of oxygen, for which atleast three models exist — two of them are named the GRIFFITHS and the PAULING

model [102]:

a) GRIFFITHS model: Pt

OO

b) PAULING model: Pt

O

O

For the oxygen reduction three reaction paths are in discussion — one direct andtwo indirect pathways. The direct is [54]:

O2 +2Pt −→ PtRO−ORPt

PtRO−ORPt+4H++4e− −→ 2H2O

ARICÒ [2] postulates a third adsorption model in which two active Pt-surface atomsin parallel are needed, which reduces the importance of this pathway. He suggests

2.4. Cathode Compartment 35

an alternative indirect reaction path, where two neighboring Pt-adsorbed species(a hydroxyl group and an oxygen atom) are involved in the rate-determiningstep (2.23):

Pt + O2 −→ PtRO2

PtRO2 + H+ + e− −→ PtRHO2

PtRHO2 + Pt −→ PtROH + PtRO ; (RDS) (2.23)

PtROH + PtRO + 3H+ + 3e− −→ 2Pt + 2H2O

In the more common second indirect model there is only one Pt-adsorbed species(RO2,RHO2,RH2O2) involved in each step, which leads to pathways over hydrogenperoxide as published in [43, 54, 102]:

Pt + O2 −→ PtRO2

PtRO2 + H+ + e− −→ PtRHO2 ; (RDS)

PtRHO2 + H+ + e− −→ PtRH2O2

PtRH2O2 + 2H+ + 2e− −→ Pt + 2H2O

Temkin and Langmuir Adsorption

In addition to the actual pathways of the cathodic reactions, the adsorption mech-anisms for the reacting species are important and dependent on the half cellpotential vs. SHE. At a specific transition potential Utr a change from the Temkinto the Langmuir adsorption mechanism is observed (cf. SEPA [121]). The transi-tion potential Utr for Pt on polyperfluorosulfonic acid soaked electrodes lies near0.8±0.2V vs. SHE [70]. Further PARTHASARATHY [99] showed that the transitionpotential is dependent on temperature. This dependency can be approximated aslinear to temperature decrease:

Utr (T ) = a +b ·T

Above Utr the Temkin adsorption is predominant, where oxygenated species (RO,ROH) are present on the surface of the electrode [100]. Below this potential, thuswithin the Langmuir adsorption region, the surface is nearly free of oxidizedspecies that could be reduced. This causes a decrease in the reaction rate (a lowertransfer coefficient α, cf. Fig. 2.20). The theory for both adsorption mechanisms isdescribed in the textbook of MASEL [85].

For more details see the review article regarding the oxygen reduction published byMARKOVIC[84] or the fundamental publications of DAMJANOVIC, PARTHASARATHY,SEPA [31, 99, 121].


Fig. 2.20.: Transition from Temkin to Langmuir adsorption on the cathode.There is a sharp decline of adsorbed oxygen species on the catalyst surface belowtransition potential Utr . A reduction of the transfer coefficient of about 50% can beobserved [101, 121]. The influence of the changing transfer coefficient at Utr = 0.8V vs.SHE is shown. A change of 47% from αtk = 0.67 to αlm = 0.37 is equivalent to a changeof Tafel slope from ϕTs =−0.092V/decade j to ϕTs =−0.171V/decade j and is equivalentto the change observed by PARTHASARATHY [101].Note that the x-axes has a logarithmic scale that leads to straight lines of the polariza-tion curves. Note: the parasitic methanol oxidation on the cathode leads to: jOR > jfc.

Cathodic Methanol Oxidation

In the liquid feed DMFC with polyperfluorosulfonic acidmembrane an undesirablecrossover of methanol is unavoidable. Therefore, the methanol oxidation on thecathode is also of interest. ARICÓ [2] published the following reaction route:

CH3OH + 3Pt −→ Pt3RCOH + 3H+ + 3e−

Pt3RCOH −→ PtRCO + 2Pt + H+ + e−

Pt + H2O −→ PtROH + H+ + e−

PtROH + PtRCO −→ 2Pt + CO2 + H+ + e−

2.4.2. Model of the Cathodic Catalyst Layer

Electrochemical Model

As stated above, two electrochemical reactions take place on the DMFC cathode— the oxygen reduction (OR) and the crossover reaction (CO). In analogy to theanode model, these reactions are simulated with two independent Butler-Volmerequations (cf. Eq. 2.10 on page 20). The Butler Volmer eq. for the crossover reactionhas the configuration of Eq. 2.10 on page 20, whereas the Butler Volmer eq. forthe oxygen reaction has the form of Eq. D.15 on page 130 because the oxygenfraction of air can be assumed as constant. As PARTHASARATHY [99] and others


gave strong evidence that the transfer coefficient is temperature dependent, thistemperature dependence for α is defined analogous to the anode (cf. Eq. 2.11on page 21). It is typical for the oxygen reduction that there are two adsorptionconducts, the Temkin (tk) and the Langmuir adsorption (lm) . At the transitionpoint Utr a sharp decline in the transfer coefficient is observable; PARTHASARATHY

measured a decrease of αlm < 0.5 ·αtk . In the literature this reduction in thereaction rate is often expressed as a change of Tafel slope ϕ

Tsor a change of

polarization per decade active surface related or geometric area related currentdensity ∆U/[1E+(i+1)−1E+i ] A/m2.

The relation between the three values ∆Udec , αor and ϕTs

pertains to the par-ticularity of 1 electron transfer into the electrode double layer during the rate-determining step (z‡ = 1) in the oxygen reduction:

ϕTs

= ∆U∆ ln I /I †

; [ϕTs

] = V

= [U(@10i+1 A/m2)−U(@10i A/m2)]

ln(10i+1)− ln(10i )= ∆Udec

ln(10)= ∆Udec

2.303∧ϕ

Ts OR = R ·T

−αOR ·F⇒αOR = − R ·T

ϕTs·F = − ln(10) · R ·T

∆Udec ·F; [α] =

J·Kmol·KJ·A·s

A·s·mol

= 1 (2.24)

(See also the definition of the Tafel slope on page 154)

Even if the slope of the polarization curve may have a buckling at the transitionpotential, the current density is continuous, resulting in the following relation be-tween the Butler-Volmer equation in the Langmuir region and the current densityat the transition potential Itr, tk = Itk

Utr

:

Itr, tk = Itr, l m

∧Itr, tk = I 0

tk · fBV (αtk ,UORUtr

, . . . )

∧Itr, lm = I 0

lm · fBV (αlm ,UORUtr

, . . . )

⇒I 0

l m = Itr, tk

fBV (αlm ,UORUtr

, . . . )


The following conclusion can be drawn from this connection: I 0lm is a dependent

variable and does not need to be expressed analog to the term for I 0tk ; hence,

the parameters I✸l m , ∆G‡r ea, lm are not necessary. The only fit parameter for the

Langmuir region becomes the transfer coefficient αlm . This results in four Butler-Volmer fit parameters for the oxygen reduction:I✸tk , ∆G‡

rea, tk , αtk , αlm and a transition potential Utr .

The change from the tk- to the lm-Butler-Volmer equation takes place in an epsilon-neighborhood ϵtr around the transition potential Utr . The blending of one withthe other adsorption mechanism is computed with a shifted Tangens hyperbolicusfunction11:

Υtk = 0.5 · tanhπ·{UCA −Utr }/ϵtr

+1.0

; Υtk = blending factor

I OR = Υtk · I tk + (1.0−Υtk ) · I lm

Both reactions need the catalyst surface: the higher the active surface relatedcurrent density I , the lower is the surface fraction Θ needed for the reaction.Depending on the operation conditions only a part of the total or reference surfaceS✸

CA may be utilizable for the reactions. The following regularities are given:

S✸CA = Sg

CA +SfCA ; (g= available, f = blocked)

SgCA = Sg

OR +SgCO

ΘOR = SgOR

SgCA

IOR = I OR ·ΘOR ·SgCA ∧ ICO = I CO · (1−ΘOR) ·Sg

CA (2.25)

ICA = −IAN ⇔ ICA + IAN = 0

ICA = IOR + ICO ⇔ ICA + IOR + ICO = 0

Note that the two reactions refer to different half cell open circuit potentials. Theeffective OCP for the oxygen reduction Uoc, OR lies above 0.9 V vs. SHE12, whereasUoc, CO lies in the region of the anodic methanol oxidation well below 0.2 V vs. SHE.Fig. 2.12 on page 23 and Fig. 2.21 explain the nexus.

11 To lead over from the Temkin to the Langmuir region, a function with a steady course from0 to 1 is needed, such as the shifted Tangens hyperbolicus function. In the argument of theTangens hyperbolicus function a factor is strived for that is high enough to bring the value of theexpression at the edges of the epsilon-neighborhood close to zero or unity. The constant π meetsthat demand.

12 The theoretical OCP for the oxygen reduction at 298.15 K lies at 1.229 V vs. SHE, but practicalvalues may lie below 1 V vs. SHE, cf. [99] and Fig. 2.22. As one reason for the low OCP theformation of a mixed potential between the oxygen reduction and the catalyst oxidation is indiscussion [121].


Fig. 2.21.: Comparison of the catalyst surface related current densities I at the cathode:The absolute values of the oxygen reduction and the cathodic methanol oxidation po-larization, UOR & UCO, are reverse and so are the catalyst surface related current den-sities, IOR & ICO. With increasing fuel cell currents Ifc the crossover reaction occupiesmore catalyst surface in relation to a given flow of methanol across the membrane.The open circuit potential of the oxygen reduction lies above 0.9 V vs. SHE, whereasthe polarization of the crossover reaction lies well below 0.2 V vs. SHE. At OCV thepolarization of the oxygen reduction is less then 20% of the crossover polarization.This leads to a difference in the order of magnitudes for the catalyst surface relatedcurrents or reaction rates of both reactions. Note that the right y-axes has a log-scale.

2.4.3. Ascertainment of the OCP for the Oxygen ReductionReaction

To be able to calculate the open half cell potential (OCP) at a given temperaturethe measured OCP values of at least two temperatures are needed. These valuescan not be fitted, because there is a flat response of the least squares13 to anychange in a varied OCP. This is due to the fact that in the Butler-Volmer equationthe activation energy ∆G‡

rea and the polarization U are both part of the superscriptto the base e and the polarization itself is calculated as the difference of OCP andthe working half cell potential vs. SHE (cf. chap. G on page 150). Therefore, themain influence of a varied OCP is a different value of the activation energy of thebest fit.

The polarization data of WILLIAMS [148] holds the OCP for the oxygen reductionreaction at the reference temperature, but no OCP value for a higher temperature,so comparable data from other sources is required. PARTHASARATHY et al. [99]published two series of OCP values for this reaction. One series was measuredafter an activation of the electrode, whereas the others are steady-state values; cf.Fig. 2.22, curves a) and b). Of course, these values differ from the values of the

13 Least squares is a term used for the sum of squared residuals and also the name for a popularoptimization criteria or method. The residual is defined as the difference between a measuredpoint and the predicted value.


Fig. 2.22.: Temperature dependency of the oxygen reduction open circuit half cell po-tential: two series of cathode OCP values Uoc, OR published by PARTHASARATHY [99],one after activation, one steady-state values[99] (a – c) and a fitted temperature depen-dency function for the OCP values for the cathode used by WILLIAMS [148] (d – f):a) PARTHASARATHY, after activation; b) PARTHASARATHY, steady-state values;c) PARTHASARATHY, fitted temperature dependency function with the equation

Uoc (T ) =U✸oc, OR ·exp[bOCP·(T −1 −T✸−1)] with a fit result of bOCP = 172.327 K;

d) WILLIAMS, temp. dependency function with the fitted bOCP used for graph c);e) WILLIAMS, measured cathode OCP Uoc, OR(298K) = 1.02V;f ) WILLIAMS, evaluated cathode OCP Uoc, OR(333K) = 1.08V

cathode used by WILLIAMS, but the temperature dependency should be equivalent.Correspondingly, the steady-state values of PARTHASARATHY are fitted with anequation that contains a fraction of Boltzmann factors analog to the Butler-Volmerexchange current density term (cf. Eq. D.15 on page 129):

UOR(T ) = U✸oc, OR · exp

bOCP ·

1

T− 1

T✸

(2.26)

The fitted superscript bOCP is used to ascertain a proper value for the OCP at333.15 K, cf. Fig. 2.22. With this evaluated Uoc, OR(@333.15K) together with theNernst voltage (Eq. 2.12 on page 21), which is calculated by ApenPlus, the idealhalf cell potential at 333.15 K can be obtained:

Uoc, OR = U✸i d , OR

· exp

−∆G‡�

OR

R·

1

T− 1

T✸

+ R ·T

z ·F · lna

2H2O

aO2

∧ z = 4


Cathode Flooding

The water flow across the membrane is probably the biggest challenge in thecathode of liquid-fed DMFC at ambient pressure. Due to the liquid-fed anode thewater drag in the DMFC is much higher in comparison to the hydrogen PEM-FC14.In addition to the water coming from the anode side, there is an intrinsic waterevolution in the cathode caused by the oxygen reduction reaction (cf. Eq. 2.13on page 26). Both water flows are proportional to the current density jfc. At lowcurrent densities the water flow is low enough to not disturb the oxygen reductionreaction. Under such conditions the air flow through the cathode flow field isable to carry the water out of the porous electrode structure to keep the surfaceaccessible for oxygen. However, at higher current densities jfc a so called cathode

flooding [53, 82, 87, 124, 146] is detectable. The current density jfln, at which theformation of a liquid water film on the catalyst surface starts, is dependent on theoperating conditions (e.g. water absorption power of the air, air flow, temperature)and the hydrophobic characteristics of the electrode as well as of the diffusionlayer. The emerging water film is hindering the oxygen transport to the catalystsurface, which leads to a decrease of the active reaction zone. As a result themodel predicts a linear flooding (growth of blocked or flooded catalyst surfaceSf

CA) with increasing current densities as soon as jfc > jfln:

SfCA ∼ jfc ⇔ Sg

CA ∼ 1− jfc ∧ jfc > jfln ∧ jfc = jCA

Θfl = max[min(aΘ+bΘ · jfc, Θfl UB), 0.0] ; SfCA = Θfl ·SCA (2.27)

with:

bΘ = 1−Θfl✸

1000 A/m2 − jfln

aΘ = −jfln ·bΘ

jfln = jfc at the begin of cathode flooding

Θfl✸ = flooded fraction of SCA at j✸fc = 1000 A/m2

Θfl UB = upper bound (maximum) flooded fraction of SCA, e.g. Θfl UB = 0.95

The parameters jfln and Θfl✸ may also be temperature dependent and so temper-ature laws analogous to Eq. 2.14 on page 27 are defined.

2.4.4. Model of the Cathodic Diffusion Layer and Bipolar Plate

The model is analogous to the anodic diffusion layer and bipolar-plate, thereforesee chapter 2.2.1 and 2.2.2.

14 PEM-FC = Polymer Electrolyte Membrane Fuel Cell


2.5. Electrical Effects

2.5.1. Electrical Losses Across the Catalyst Layers and ContactLosses

The ohmic losses caused by the catalyst and diffusion layers, as well as contactlosses between the bipolar- and end-plates are disregarded, below 10 000 A/m2

they are in the range of a few mV [18, 94]. Consequently these ohmic drops arenot part of the simulation. A schematic overview in Fig. 2.23 shows the voltageloss across the polymer electrolyte membrane that are caused by the diffusionresistance of hydrated hydronium ions 15. The ohmic losses across the catalyticlayer are below 1 mV [18], which results in a constant potential across the catalyticlayer in the Fig. 2.23.

Fig. 2.23.: Schematic potential course across the fuel cell layers. The potential lossesoutside the PEM are disregarded. Inside the anode a change of charge carriers takesplace from the negative electrons to positive hydrated hydronium ions (protons);hence, inside the PEM the direction of charge transfer of the cations is from the ‘posi-tive’ anode potential to the ‘negative’ cathode potential [10,23,150]. For the term surfacepotential χ cf. chap. G on page 144.

2.5.2. Inner Voltage Loss due to Membrane Ion Resistance

For calculation of the voltage loss due to ion resistance across the membraneUR,m and ohmic losses along the electron-conductive parts of the fuel cell Uec, fc

in the model, an iterative calculation is necessary, as the independent value is Ufc,

instead of jfc or Ifc. It starts with an initial value of U⊢R, fc and a first calculation of

Ifc. Now UR, fc = Rfc · Ifc is determined and this voltage loss is split into two parts.

15 WÖHR published a similar scheme in [150, fig.2.2]

2.5. Electrical Effects 43

In each electrode a voltage loss proportional to the related polarization fractionof total cell polarization ΞU is observed. This voltage loss of the half cell UR,hc issubtracted from the half cell polarization modulus without resistance loss |U| andis provoking a current decline. The new lowered polarization is called the innerpolarization UG:

Ifc = f 1(Ufc, UR, fc)

UR, fc = Rm · Ifc ∧ UR, fc :=UR,m (Uec, fc := 0)

Ufc = U0, CA −U0, AN −Ufc

UAN = f 2(Ufc) ∧ UCA = f 3(Ufc) (UCA ≤ 0)

Ufc = UAN −UCA = UAN +|UCA| (Ufc ≥ 0)

ΞUAN

= UAN

Ufc

UR, AN = ΞUAN·Ufc ∧ UR, CA = (1−ΞU

AN) ·Ufc (2.28)

UGAN = UAN −UR, AN ∧ UG

CA =UCA +UR, CA

3. IMPLEMENTATION OF THE MODEL INASPENPLUS

Fig. 3.1.: The AspenPlus process flowsheet (PFS). The first three letters of the unitnames are acronyms for the fuel cell parts: ABP = anode bipolar plate, ADL = anodediffusion layer, ACL = anode catalyst layer, PEM = polymer electrolyte membrane, CCL= cathode catalyst layer, CDL = cathode diffusion layer, CBP = cathode bipolar plate.The units themselves may consist of more than one AspenPlus-Model. The boxedlabels belong to streams, either material (solid) or energy (dashed). The length of unitlabels is restricted to six the length of stream labels to eight upper case chars, thisresults in names like: ACL1X-EF = Anode Catalyst Layer, block 1x — Exit Flow.

The model of the DMFC is computed in AspenPlus, a commercial software formodeling chemical plants. The software works in two modes: In a single- and amulti-domain approach, called equation or sequential oriented mode (cf. chap. Gon page 149). The calculations are steady-state. The strong point of AspenPlus is itsability to calculate the thermodynamic properties of pure and mixed componentswith a wide range of physical property methods. This is of great value if physicalproperties such as the activity of solved carbon dioxide in water or the fraction ofwater in the vapor and in the liquid phase are to be calculated. The simulationis of zero dimension, all parts are treated as boxes with the spatial expansion ofzero.

46 3. Implementation of the Model in AspenPlus

3.1. Guidelines of Implementation

The implementation of the DMFC into AspenPlus was achieved along the followingguidelines:

• The specialty of AspenPlus is the simulation of chemical plants, hence theimplementation of the fuel cell was done in the same manner.

• AspenPlus “knows” nothing about protons and electricity and only littleabout electrochemistry, so the reactions are simulated in continuouslystirred stoichiometric reactors and the proton flow across the membrane isreplaced by a hydrogen flow.

• As much as possible existing AspenPlus-Models are utilized, such as mixeror separator blocks.

• The cell was unitized in order to increase the flexibility of the comprehensivemodel and to be able to debug and work on the blocks in stages.

3.2. AspenPlus Process Flowsheet

Fig. 3.2.: Stream types.

The process flowsheet (PFS) is part of the graphical userinterface (GUI) of AspenPlus. The GUI is meant for usercomfort only and it is the tool to produce an Aspen in-put file (*.inp). This input file is examined by the coreapplication aspen.exe and the output is sent to the realAspenPlus engine apmain.exe. In principle, it is possibleto compute any simulation task in the AspenPlus inputfile language with a simple text editor. This was standardin the initial release of the software. However this modeof operation lacks vividness and only well experienced specialists can follow thisroute for more complex PFS. However, to make certain changes in a sequence ofsimulations, this approach makes sense. On AspenPlus PFS three visible types ofstreams can be present: heat, work, and material streams. On the flowsheet above,heat and material streams are depicted (Fig. 3.1). Depending on the AspenPlus-Model used there are various input and output ports for these stream types. Somemodels are usable for different stream types, e.g. multiplier blocks, or have portsfor different types, e.g. a compressor with inlet and outlet material port, an op-tional work stream input and a heat stream outlet. All streams are oneway only.In the sequential mode the process is flow driven, meaning that the precept of

3.2. AspenPlus Process Flowsheet 47

flow is e.g. the amount of substance per time ([Φ] = kmol/s)1. However, in theequation oriented mode it is differentiated between a pressure and a flow drivenprocess flowsheet. Simulations have been performed under both modes. Theselection was made for the sequential mode. The reasons for this decision are:a gain of robustness, a convenient working environment and primarily the fullsupport of user defined Fortan77 calculator blocks. Beside the visible streams, itcan also be worked with so-called transition blocks. They help keep the layout ofa flowsheet simple. When, e.g. the thermal energy outlet of an exothermic reactorin the upper right part of a PFS should be conected to the thermal energy input ofa heater block in the bottom left corner, a long energy stream across the PFS canbe avoided by using a transition block.

Mixer and Splitter

Fig. 3.3.: AspenPlus mod-els Mixer (left) and FSplit(right).

The AspenPlus-Models Mixer and FSplit are usable forall types of streams (Fig. 3.3). If the Mixer is connectedto material streams, the valid phases can be specified.If it is known that there is only vapor, the mode "va-por only" can be selected to save computation time.A second variable aside from the valid phases is theabsolute pressure or the pressure drop. Thus a Mixermay hold a passive part or it can be used as an adia-batic pressure changer.FSplit has the additional feature so that the exact splitfraction for multiple outlet flows can be specified. However, only a fraction of thetotal mixed flow can be forced; it is not possible to influence the composition ofthe outgoing flows. In the upper right corner in the flowsheet (Fig. 3.1) there isone FSplit block called CCL5X. Here the sum of all heat streams is divided intoan electric power and a heat outlet stream. The fraction is dependent on theButler-Volmer based electrochemical calculation: Pel =Ufc · Ifc.

1 In AspenPlus the SI derived unit for amount of substance (aos) is kmol rather than mol, but inthis work the unit mol is preferred. Note that kmol/s is only one of 27 aos related units for flows,in addition to another 38 mass and 76 volume based flow units, and customized units can bedefined as well.


Separator

Fig. 3.4.: Separator mod-els Flash2 (left) and Sep(right).

The two icons on the right (Fig. 3.4) stand for two dif-ferent types of separators that are used in the PFS.They are called Sep and Flash2. Sep is in principle anadvanced form of the splitter FSplit. The advantage isthe fact that the split fraction for each component ina mixed stream can be specified. The block ACL2X inFig. 3.1, e.g., separates the hydrogen from the mixedinput stream ACL1X-EF.Flash2 performs calculations of the physical proper-ties, especially the phase fractions and the compo-sition of gas and liquid phase. Specific calculationoptions, called property methods, are offered by As-penPlus and can be assigned in the determination ofphysical properties. Specific property methods can be determined, which haveto be used for these calculations. Marginal conditions, such as oversaturation orundersaturation of a substance in one of the phases, can also be arranged. In thepresent simulation the Flash2 block ACL3X is used to calculate the vapor fractionof the carbon dioxide in the two-phase flow coming from the first exit flow (E1) ofthe anode part ACL2X.

Continuously Stirred Stoichiometric Reactor

Fig. 3.5.: RStoic— Continuouslystirred stoichiomet-ric reactor.

If the reaction rate and the stoichiometry of a reaction ora set of reactions is known in advance, the RStoic model isappropriate (Fig. 3.5). Several reactions and reaction ratescan be defined, and the unit RStoic calculates the propertiesof the outlet flows for a specific temperature and pressureor it calculates the temperature if the reactor is adiabatic. Inthe case under consideration the temperature of the anodeand cathode catalytic layers are prescribed and the reactionrate is calculated with the Butler-Volmer equation in a calcu-lator block. By fixing the reaction rate, it is also possible tosimulate incomplete reactions.

3.3. Hierarchy of Iterative Solution 49

Heater

Fig. 3.6.: Heater model illustrated with differ-ent icons.

The use of the AspenPlus-ModelHeater as part of the membrane(see PEM2X in Fig. 3.1) gives agood example for the multifunc-tionality of some AspenPlus-Models. This is visualized bya variety of icons for one andthe same block. The zigzag linethrough the left icon in Fig. 3.6symbolizes a heating / cooling pipeand clarifies the functionality as a heat-exchanger, whereas the right symbolreminds one of a pressure changer or orifice. In the present simulation theHeater model is indeed used for block PEM2X in Fig. 3.1 as an isenthalpic pressurechanger, where it simulates a possible pressure gradient across the membrane.

3.3. Hierarchy of Iterative Solution

The concentration and the convective flows influence the electrochemical reac-tion and vice versa. Therefore, convergence loops are needed to consider thiscorrelation. As previously mentioned, the driving force of the electrochemicalreactions is the polarization. The polarization curve of Fig. 2.12 on page 23shows that above a distance of 0.2 V from the OCP, a small change of the po-larization |∆U| causes a significant change in the current density ∆j (UAN > 0.2Vor UOR < 0.2V). The fraction j-slope = ∆j/∆U changes in the form of the Butler-Volmer equation (see Eq. 2.10 on page 20) as an exponential function, see graphand caption of Fig. 3.7. It becomes clear that it is rather easy to simulate near theOCV, but the interesting part for the practical use of DMFC lies in the region ofhigher current densities.

Fig. 3.7.: j-slope or change of currentdensity at constant polarization stepsover anode polarization. The curve ofthe simulation result is nearly congru-ent with the exponential fit. The resultis expected, because the derivationof the simplified form of the Butler-Volmer equation is also in the form ofan exponential equation:fBF (x) = a · (b ·eb·x − c ·ec·x )⇒ f ′

BF (x) = a · (b·b·eb·x − c ·c·ec·x )


Fig. 3.8.: Loop order of Design Specs.

In theory it should be possible to solveall values in one optimization loopcalled design specification / DesignSpec, but that has the side effect ofbad convergence characteristics. Inthe first step two nested loops havebeen computed: in the inner loop theoptimization of water and methanoltransport through the membrane, andin the outer loop the residue of thetransport values. After intense exami-nation the calculation of the methanol diffusion and the electroosmotic transporthave been separated in two sequential Design Specs (see Fig. 3.8).

The electroosmotic transport is more complex in comparison to the diffusiontransport, as it depends on the methanol concentration on the membrane surfaceas well as on the current density. Another aspect is that the convective electroos-motic transport has precedence over the slower diffusion transport, so the formeris solved in the inner loop.

3.4. Strategies to Improve the Convergence Behavior

The separation of the optimization of the different transport mechanisms im-proves the robustness of convergence, but in critical regions (e.g. low methanolconcentrations and low cell voltages) the result is still not satisfactory. The follow-ing scope of influence is given to improve the results:

• Good initial estimate of the manipulated value

• Boundaries of the manipulated value as tight as possible

• Variation of the step size scope

• Good choice of the manipulated variable and the convergence criteria

• Oportune choice of the convergence algorithm

• Scaling of the manipulated and sampled variables

The selection of the convergence algorithm inside AspenPlus is constrained by thesimulation software, but the most important ones are available. The differentiationlies in three main types:

a) The direct or sequential search: The manipulated value is increased or de-creased by a fix step size from one edge to the other. This method is the mostrobust one, but is time consuming.

3.4. Strategies to Improve the Convergence Behavior 51

b) A linear technique, for which a linear relation between manipulated variableand the convergence criterion or quality is assumed. For simple optimizationtasks this is the preferential algorithm, as it is robust and fast.

c) The Newton algorithm: This method is good for non linear problems e.g. apronounced exponential relation between manipulated variable and conver-gence criteria. It is based on the computation of the derivatives of the qualityfunction. The calculation can be difficult and is also time consuming.

In addition to these algorithms there is the popular Broydon method [103], whichcombines the advantages of the linear and the Newton ansatz. As long as the linearapproach is successful it remains linear and switches to the Newton algorithm onlyif it becomes necessary. This method is well implemented in AspenPlus and turnedout to be the best choice for all three loops of the simulation (see Fig. 3.8). Butapart from the method used it is good to have narrow investigation bounds and agood initial value for the manipulated value. Together with the result of the lastcycle and/or last cycles, it is likely to make an accurate assumption regarding thebounds and the initial value for the two nested loops in the simulation. However,this is not possible for the outer loop.

Dependent on the sensitivity of the quality function y for changes of the manipu-lated variable, also called x, it may be necessary to scale both separately. This wasalso true for the DMFC model used in this work.

The minimum absolute step size for x inside a AspenPlus Design Spec is ∆x =1E-10. If the step size reaches this threshold the convergence process comesto a hold. It also stops when the quality function is too flat, particularly if ∆yis too small in relation to ∆x. Both problems have been solved by scaling themanipulated variable and the quality function with two different scale factors (cf.Tab. 3.1 on page 55).

3.4.1. Building Error Classes to Narrow Optimization Bounds

In the AspenPlus optimization loops, called design specifications, it is not possibleto adjust the investigation boundaries during a run, as they are fixed over allcycles. It is useful to have as much information about the optimization task aspossible, before the entrance in the loop, in order to set proper boundaries. Narrowboundaries fasten the simulation and increase the probability for convergence.Such information are e.g.: ∆x and ∆y over all cycles of the last run, importantflows or concentrations, the current cycle number of the paling design specification.

Classes dependent on such informations are defined. At the initial call of thenested optimization loop, the run is started in class 1. That means the boundariesare set loose and the initial point of the x-value is just put in the middle. With


Fig. 3.9.: Principle of classes definition ahead of the two inner AspenPlus Design Spec-ifications for the diffusion and electroosmotic transport across the membrane. Theboundaries of every class are set according to the results of the previous run(s). Thevariables are: I_run the cycle index of the outer loop, c_meth methanol concentrationat the membrane entrance, c_min lower threshold for the methanol concentration,Q quality function, tol error tolerance, x manipulated variable, ∆x difference of xbetween last two runs of the inner Design Spec, y sampled value

increasing cycle numbers and better qualities of the last run the boundaries areset closer around the last result of the manipulated variable. Additionally, specialclasses are defined for extreme situations, like concentration values near zero, seeFig. 3.9. For the innermost loop, the optimization of the electroosmotic transport,60 classes have been defined, whereas for the middle loop, the design specificationfor the diffusion across the membrane, 45 classes are been selected.

3.4.2. Attenuate Oscillation of Values and Numerical Stability

Because the simulation is not equation oriented but sequential the system isvulnerable against oscillation. To reduce this tendency three methods are used:a) arithmetic averaging b) extrapolation by linear regression fit of the last valuesand c) good positioning of calculations in the sequence. Methods a) & b) arecombined with the monitoring of the value itself and of values that are related toit.

One example for method c) is the calculation of cathode flooding: the floodinghas an important influence on the polarization and the polarization is calculatedin the inmost loop in the calculator block of the Butler-Volmer equation. Thus,one could argue it is best to determine the flooding firsthand before or after thatcalculation, but this would in turn disturb the convergence of the electroosmotictransport calculation. Therefore, the evaluation is accomplished in the middleloop before the entry to the innermost optimization loop. To lower the oscillationeven further a weighted mean is built.

For the method b), the linear extrapolation, a routine based on the algorithm inthe textbook “Numerical Recipes of Fortran 77” [103] was developed. It does not

3.4. Strategies to Improve the Convergence Behavior 53

only build a linear regressed extrapolated point for the next iteration, but it alsofilters out irregularities, e.g. values of zero or oscillation. The extrapolated value isonly used if certain conditions of irregularity are met, else the pristine sampledvalue persists unchanged. This routine is used to trowel the sampled values ofthe concentration at the membrane entrance and the calculated values of themethanol diffusion and electroosmotic transport flow across the membrane. Thissmoothing procedure accelerates convergence in critical situations, such as smallvalues of those three crucial variables.

Numerical Stability and Thresholds in AspenPlus

The obstacle of the so called numerical stability is a general issue of discrete binarynumber representation. If two numbers of a calculation step differ considerablyin the order of magnitude, the rounding or truncation error is significant. To avoidthis situation in the computation of physical properties, minimum thresholdsfor the flows are introduced. In AspenPlus this marginal limit has the defaultvalue of RGLOB_RMIN= 1E-12 mol/s; that limit has been decreased to the allowedminimum of 1E-27 mol/s (2). In addition to the restriction of minimum flow, othercalculations need a minimum fraction of substance, e.g. the calculation of theconcentration or the activity of methanol in water that fail below the fraction ofx LB

CH3OH = 7.76E-17. This results in a minimal detectable concentration in a water-

methanol mixture at ambient conditions of C LBCH3OH = 4.28E-12 mol/m3. With this

in mind, the threshold for concentrations is set to C LBCH3OH = 1E-11 mol/m3. Conse-

quently, if the point of zero methanol concentration at the electrode-membraneinterface is to be modeled, the descent from C LB

CH3OH to zero has to be smoothed

down, which is accomplished using the routine mentioned above.

3.4.3. File Based Database to Improve Convergence of the OuterDesign Specification

When a convergence loop is nested in an outer loop there is a opportunity toset the optimization boundaries close and decide on good initial values for themanipulated value. If there is no outer loop there are three possibilities: a) suchliberal bounderies that the optimal result will surely fall within them; b) build anartificial loop outside, so that the original outer loop becomes nested; c) build adatabase.In the simulation of this work the outer Design Spec of Fig. 3.8 is named DS-ADL02.For this optimization loop the extremes of the boundaries are well defined, themanipulated variable (up-scaled set point) x SP∝

CH3OHADL2XADL2X-E1

is changing the split

2 In AspenPlus the value is 1E-30, because a scaled unit of kmol/s is used instead.


fraction of block ADL2X for the exit flow ADL2X-E1, hence, the unscaled valuexCH3OH

ADL2XADL2X-E1

lies between zero and one:

xCH3OHADL2XADL2X-E1

=x SP∝

CH3OHADL2XADL2X-E1

F ∝x,DS-ADL02

∧ xCH3OHADL2XADL2X-E1

∈ [0..1]

with:F ∝

x,DS-ADL02 : Scaling factor for manipulated variable of Design Spec DS-ADL02

That implies variant a) as a possible strategy because the outer limits for x areknown exactly. This method is inefficient and fails in difficult situations dueto loose bounderies. Variant b) is also time consuming and is increasing thecomplexity of the simulation by adding a fourth loop level. Therefore, a file-baseddatabase has been developed and it has been proven to be an effective approachto increase the convergence. Details regarding this strategy and the set-up of thedata base are documented in chap. C.1 on page 119.

3.5. Scaling

3.5.1. Scaling Manipulated Values and Sampled Values ofAspenPlus Design Specs

The Design Specs or convergence blocks in AspenPlus have a value that definesthe minimum increment between two manipulated or x-values. By default thisincrement ∆xmi n is 1E-04, whereas the permitted minimum is 1E-10, but thisis still not sufficient for the needed precision of the optimization loops. Themanipulated value is scaled in order to achieve improved specificity. Equallyimportant for good convergence is that the order of magnitude between x and thesampled value y must not differ significantly. It was figured out that scaling thequality function Q also helps to avoid a flat response of Q in relation to ∆xmi n .Through experiments suitable scaling factors have been determined, see Tab. 3.1on the facing page.

3.5. Scaling 55

Design Spec ManipulatedVariable x

ScalingFactor x

SampledValue y

ScalingFactor y

Qualityfunction Q

DS-ADL02 xSP∝CH3OH

ADL2XADL2X-E1

F ∝x,DS-ADL02

ΦSVCH3OH

ADL2X-E1

F ∝y,DS-ADL02

F ∝y,DS-ADL02· (ΦPD−ΦSV)

Calc. of dif-fusion andconvectiveflow throughthe diffusionlayer ADL

Set point forproportionalfactor of splitterADL2X

1.0E-09 Sum of diffusionand convectiveflow.

1.0E+10 Scaling factortimes thedifferencebetweenpredicted andsampled flows

DS-PEM1X ΦSP∝CH3OH,D

PEM1XPEM1X-E1

F ∝x,DS-PEM1X

· Afc

6·F

ΦSVCH3OH,D

PEM1X-E1

F ∝y,DS-PEM1X

· 6·FAfc

F ∝y,DS-PEM1X· (ΦPD

D −ΦSVD )

· 6·FAfc

Calc. of dif-fusion flowthrough themembranePEM

Up-scaled setpoint for me-thanol diffusionflow

1.17E-10 ·Afc

Calculated byFick’s First law

1.0E+09/Afc

DS-EODRG ΦSP∝CH3OH,eo

PEM1XPEM1X-E1

F ∝x,DS-EODRG

· Afc

6·F

ΦSVCH3OH,eo

PEM1X-E1

F ∝y,DS-EODRG

· 6·FAfc

F ∝y,DS-EODRG· (ΦPD

eo −ΦSVeo )

· 6·FAfc

Calc. of elec-troosmotictransportthrough PEM

Up-scaled setpoint for elec-troosmoticmethanol trans-port

1.17E-10 ·Afc

Water dragthrough PEMtimes methanol-water propor-tion

1.0E+09/Afc

Table 3.1.: Scaling factors for the three design specifications in use and the definitionof the quality functions. Tiny scaling factors for the manipulated variable increase thevalue of x. The combination of the x-scaling factors with Afc/(6·F) leads to currentdensity equivalents of the methanol flows, F is the Faraday constant and Afc is thegeometric area of the electrode. For the Design Specs DS-PEM1X and DS-EODRG thesame scaling factors are appropriate, because the sampled values are of the sameorder of magnitude.


3.5.2. Scaling Surface Fraction for Calculation of Mixed Potential

The analytical equation for the relation between the two currents on the cathodeis given with the Eq. 2.25 on page 38, but this this approach induces numericaldifficulties, when the surface fraction for the oxygen reduction is determinedas:

I CECO = ΦCH3OH, CO ·6 ·F ∧ I BV

CO = I BVCO · (1−ΘOR) ·Sg

CA

Equating of I CECO with I BV

CO and rearranging yields:

I CECO = I BV

CO · (1−ΘOR) ·SgCA ⇔ ΘOR = I BV

CO ·SgCA − I CE

CO

I BVCO ·Sg

CA

(3.1)

The reason for these problems is that I BVCO ·Sg

CA and the current equivalent ofthe crossover flow3 I CE

CO can be of different order of magnitude (cf. Fig. 2.21 onpage 39). Similar rounding errors as described above in chap. 3.4.2 on page 53occur. To solve these, one idea is to make use of the different spacing of thebinary represented double precision numbers according to the so called IEEE 754standard4. The larger the scaling factor and hence, the larger the scaled variablevalue, the wider the gap to the next floating point number. By introducing ascaling factor F ∝

ΘOR

≫ 1.0 and defining a scaled surface fraction Θ∝OR, a surface

fraction in the range within the number 1.0 is prevented; as a result the differentspacing of large numbers can be exploited:

Θ∝OR = F ∝

ΘOR

·ΘOR (3.2)

An investigation on this matter shows: the scaling factor has a helpful effect, butthe effect is slight. Chap. C.2 on page 120 illustrate in detail the influence ofscaling on spacing and number representation. Nevertheless, a systematic test ofthe influence of the scaling factor on the polarization curve in its critical region atthe end of the polarization curve (cf. chap. 3.4 on page 50) has been accomplished.The result is shown in Fig. 3.10 on the facing page where it proves the positiveeffect of the scaling factor on the rounding errors.

But even with the use of a scaling factor, the phenomenon of zero surface for thecrossover reaction Sg

CO = (F ∝Θ

OR−Θ∝

OR) ·SgCA = 0, as a result of a numerical artefact,

remains. In order to cure that remaining obstacle the value of the scaled surface

3 To differentiate between the crossover current calculated by the Butler-Volmer equation and thecurrent equivalent of the (sampled) methanol crossover flow the symbol I for the latter is markedwith the superscript CE .

4 IEEE = Institute of Electrical and Electronics Engineers, IEEE 754 = IEEE Standard for floating-point arithmetic.

3.5. Scaling 57

Fig. 3.10.: Influence of the scaling factor (SF) on the simulation result in the criticalzone of high current densities and high flooding. Without scaling (SF = 1E00) thedisturbance of the simulation is evident, even a scaling factor of 10 improves thesituation significantly. The best result in the range of SF = [1E00 .. 1E25] was found ata value of SF = 1E14.

fraction Θ∝OR should be monitored and the understanding of the situation can

be utilized: The case of zero surface for the crossover reaction can only occur ifI BV

CO ·SgCA ≫ I CE

CO. This implies that the active surface related current density I CO

is so immense, that the needed surface area is very tiny. When I CECO is the current

equivalent of the methanol crossover flow, the following can be written:

jCO =

a)

I BVCO·(F ∝Θ

OR

− Θ∝OR

)·SgCA

Afc: Θ∝

OR = F ∝Θ

OR

b)I CE

CO

Afc: Θ∝

OR = F ∝Θ

OR

∧ UCO > 0V

(3.3)

with:Geometric area related current density, [jCO] = A/m2; catalyst surface related currentdensity, [ICO] = A/m2; accessible catalyst surface, [Sg

CA] = m2

The situation of case b) (Θ∝OR = F ∝

ΘOR

∧ UCO > 0 V) can only occur as a consequenceof a rounding error, hence it can be assumed that all methanol, crossing themembrane, will be completely electrochemically oxidized.


3.6. Implementation of the Set of Butler-VolmerEquations

Fig. 3.11.: Structure of theFortran77 calculator blockC-BUTVOL.

The core of the simulation is a Fortran 77 calcu-lator block named C-BUTVOL, which is dividedinto three main parts:a) initial calculations, among other values thecalculation of the open circuit cell voltageUOCVFC = Uoc, fc,b) calculation of values for the conditionUSPFC ≥ UOCVFC (U SP

fc ≥ Uoc, fc),c) calculation of the currents and the electrodepolarization for U SP

fc < Uoc, fc.These calculations are done regardless of theflows in the fuel cell inlets. The only flow that isconsidered is the total methanol crossover flowfrom the last cycle ΦCH3OH,co of the innermostloop DS-EODRG. For the OCV calculation only the inlet methanol concentrationand the current methanol diffusion coefficient for the membrane are relevant. Tomark the currents that are calculated based only on the Butler-Volmer equations,the superscript of the symbols is BV.

The main outcome of the calculator block C-BUTVOL are the three reaction ratesξr for the two reactor blocks ACL1X, CCL1X (cf. process flowsheet 3.1 on page 45).ξr

BV

ACL1X, ANfor the reactor block ACL1X, the anodic methanol oxidation (AN, cf. Eq. 2.2

on page 17) and two for the reactor block CCL1X: a) ξrBV

CCL1X, ORfor the oxygen reduc-

tion (OR, cf. Eq. 2.23 on page 34) and b) ξrBV

CCL1X, COfor the crossover reaction (CO, cf.

Eq. 2.24 on page 36).

These values are absolute and not related to the geometric area of the electrodes.But in the Butler-Volmer equation with the sign convention of ATKINS [7] for fuelcells, the currents or current densities for oxidation reactions are negative, hence, itis necessary to multiply the currents of the two methanol oxidations I BV

AN, I BVCO with

minus one. In accordance with the First Faraday law the current is proportional tothe reaction rate and together with the equations 2.3 on page 17 the result is:

ξrBV

ACL1X, AN= − I BV

AN

6·F ∧ I BVAN ≤ 0 ∧ |νCH3OH| = 1

ξrBV

CCL1X, OR= I BV

OR

6·F ∧ I BVOR ≥ 0

ξrBV

CCL1X, CO= − I BV

CO

6·F ∧ I BVCO ≤ 0 ; [ξr ] ≡ mol

s= A ·mol

C= C ·mol

s ·C

3.6. Implementation of the Set of Butler-Volmer Equations 59

The reaction rates are calculated as maximal values and are checked in othercalculator blocks before the evaluation of the reactor blocks. If these tests revealthat there is a shortage of reactants, the values for the reaction rates will be cut.The OCV is calculated with the following fundamental equations:

I BVfc = −I BV

AN = 0 (@OCV) ∧ I BVOR + I BV

CO + I BVAN = 0

⇒ 0 = I BVOR + I BV

CO

If I BVfc is zero, the electroosmotic transport is zero. Therefore, the current equivalent

of the diffusion flow I CED

is equal to the crossover current I BVCO (cf. Eq. 2.17 on

page 28):

I BVCO = −I CE

D ∧ I CED =

DCH3OH,m ·CCH3OHADL2X-E1

·6 ·F · Afc

lmwith:DCH3OH,m : diffusion constant for methanol through the membrane;

Afc: geometric area of the electrodes; lm : membrane thickness;CCH3OH

ADL2X-E1

: methanol concentration at anode inlet

Now the surface fraction for the oxygen reduction can be calculated with theEq. 3.1 on page 56 and a bracketing search for the cathode potential vs. SHE canbe performed, where the sum of the two cathode currents is zero, I BV

OR + I BVCO = 0.

The bracketing algorithm is taken from [103]. It is fast and easy to implement,but it has no border check. Therefore, before the entrance in the bracketingsearch it is necessary to calculate in advance the maximal polarization at OCVconditions. This calculation is done with the golden section search [103]. The valueof the OCV is changing only slightly at constant temperature and methanol inletconcentration, so this calculation is done only from time to time.

Along with the calculation of the OCV Uoc, fc, the up-scaled lower bound of thesurface fraction for the oxygen reduction reaction ΘLB∝

OR is determined here (cf.Eq. 2.25 on page 38 and Eq. 3.2 on page 56):

I CEOR

OCV

= I BVOR

OCV

·ΘLBOR ·F ∝

ΘOR

·SgCA ∧ I CE

OR

OCV

= −I CECO

OCV

⇔

ΘLB∝OR =

I CEOR

OCV

I BVOR

OCV

·SgCA

∧ SgCA = SCA (no flooding)

This value is needed, else the situation of zero oxygen reduction current causedby a surface fraction Θ∝

OR of zero can occur. This happens if |I CECO| ≥ |I BV

CO ·SgCA| in

Eq. 3.1 on page 56, one example for that are high crossover diffusion rates due tohigh methanol inlet concentration and/or a large membrane diffusion coefficient


Fig. 3.12.: The nexus between the anode polarization UAN and the different currents ata fuel cell voltage of Ufc = 0.3 V. The (negativ) oxygen reduction polarization amountsto UOR =Uoc, AN −Uoc, OR +UAN +Ufc (cf. Fig. 2.1). At the simulation optimum the mod-uli of the positive cathode and the negative anode current should be equal |ICA|

!= |IAN|⇒ |ICA,opt| = |IAN,opt|. The crossover current I CE

CO is limited by the methanol flow at themembrane surface.

for methanol. With the knowledge of ΘLB∝OR the up-scaled surface fraction can be

defined as:

Θ∝OR = max

min

F ∝Θ

OR

· (SgCA · I BV

CO − I CECO)

SgCA · I BV

CO

, F ∝Θ

OR

, ΘLB∝OR

The nexus between the anode polarization UAN and the different currents is ex-plained by Fig. 3.12; Fig. 3.13 depicts the trend of the unscaled surface fractionΘOR against the anode polarization at the beginning and the end of the iterativecalculation in the inner Design Spec DS-EODRG of the reaction rates or currents ata fuel cell voltage of 0.3 V.

If the set point for the fuel cell voltage U SPfc is below the OCV Uoc, fc, the calculation

of the mixed potential and the currents will be performed. For the establishingof the mixed potential, at first, the maximal crossover flow is calculated that thecathode can handle at a given fuel cell voltage Ufc. And subsequently it is searchedfor the optimum through the bracketing algorithm. As the cell voltage is belowOCV, the fuel cell current I BV

fc is greater zero. Following the important equationsthat define the relations between the different voltages, polarizations and currents

3.6. Implementation of the Set of Butler-Volmer Equations 61

Fig. 3.13.: The trends of the current sum ΣI = I AN + I CA and the surface fraction forthe oxygen reduction reaction ΘOR over the anode polarization at the same fuel cellvoltage at the beginning (left) and the end (right) of the iterative current calculation.During the cycles the current values may come to unrealistic values as seen on theleft. At the optimum — marked by a circle — the absolute value of the negative anodecurrent and the cathode or fuel cell current are equal: I CA,opt = |I AN,opt | = Ifc,opt . Theoptimum for the fuel cell current Ifc,opt starts at very high 4500 A and ends at 1500 A.The surface fraction is bounded below by the surface fraction at OCV conditions ΘLB

OR.

are presented; to simplify matters the model case without voltage loss due to ionicresistance is used here (cf. 2.1 on page 9):

U SPfc < Uoc, fc ⇒ I BV

fc > 0

I BVCO = −I BV

OR − I BVAN ∧ I BV

CO

!= I CECO ≡ ΦCH3OH, CO ·6 ·F

UCO = UAN +Ufc

UOR = Uoc, AN +UAN +Ufc −Uoc, OR ∧ UOR < 0

UUBAN = Uoc, OR −Uoc, AN −Ufc

To find the mixed potential of the cathode UCA5 the anode polarization UAN is

varied in the range of [0 ..UUBAN ] until (−I BV

OR − I BVAN) is equal to the current equivalent

of the methanol flow through the membrane (ΦCH3OH, CO ·6 ·F).

5 In a non ideal DMFC with a crossover of methanol, the mixed potential UCA,mp is identical withcathode potential: UCA,mp ≡UCA

4. SIMULATION RESULTS

The model was tested with the polarization data published by WILLIAMS et al. [148].He published half cell polarization data at two temperatures: 298.15 and 333.15 K,with the lower temperature being used as the reference temperature T✸.

To simulate the DMFC, some parameters needed to be fitted, which was done inseven stages:

• Some transport parameters

• Anode parameters at reference temperature T✸= 298.15 K

• Anode parameters at the higher temperature

• Cathode parameters under Temkin condition at T✸

• Transfer coefficient under Langmuir condition at T✸

• Cathode parameters under Temkin condition at the higher temperature

• Transfer coefficient under Langmuir condition at the higher temperature

The table 4.3 at the end of this chapter documents the fit results. To comparethe PVC-membrane with a Nafion™117-Membrane a second set of parametershas been assembled based on transport parameters for the polyperfluorosulfonicacid-membrane found in literature and the remainder of the parameters used forthe WILLIAMS-DMFC.

4.1. Transport Parameters

The most important transport parameters for the modeling of the crossover phe-nomenon are the diffusion coefficient for methanol and the electroosmotic dragcoefficient for water through the membrane. The effective diffusion coefficientfor methanol at the reference temperature, T✸= 298.15 K is easy to calculate, cf.Eq. 2.19 on page 29. The temperature dependence can be estimated to be equalto the temperature dependence of the methanol diffusion through water[143],cf. Eq. 2.20. The electroosmotic drag coefficient and other properties of the solidelectrolyte used in [148] are not available in literature, because WILLIAMS [148]used his own proprietary membrane made out of porous PVC. Subsequently,

64 4. Simulation Results

there is no data in literature about the properties of this type of membrane. Thenumber of water molecules that establish the hydration structure of protons in theelectrolyte is identical with the electroosmotic drag coefficient ξeo and is mainlydependent on the molecule structure and dipolar character of the solvent andthe temperature. The electrolyte character is slightly different inside Nafion™ incomparison to H2SO4 doped porous PVC, but in both cases the solvent is domi-nated by water molecules. Accordingly, different values in the same domain forthe electroosmotic coefficient ξeo are to be expected. Because of the similarityof the solvent it is probable that the temperature behavior is analog to the tem-perature dependence of ξeo in Nafion™, see Fig. 2.15 on page 27. The range of

Fig. 4.1.: Temperature dependent electroosmotic water drag coefficients ξeo for twodifferent membrane types. Nafion™117 data was published in [107] (indicated by⃝), data for the sulfuric acid containing PVC-membrane was derived by means offitting from the polarization data of WILLIAMS [148] at the temperatures 298 and 333 K(indicated by ✸); the results for the latter are: ξeo(@298 K) = 1.8 and ξeo(@333 K) = 2.8.The two curves are plotted with ARRHENIUS like equations: ξ✸·exp[beo ·(T −1 −T✸−1)](cf.chap. 2.3.1) and ascertaining of beo by the means of least square search against themeasured or simulated data.

ξeo for Nafion™117 at temperatures below 335.0 K is [1.5 .. 3.0] [107]. Now it isinteresting, if the results from the parameter fit are in a comparable scope andshow the same tendency with rising temperature. Fig. 4.1 shows the result of thefitting (cf. chap. 4.9 on page 75).

The other transport coefficients: like diffusion of oxygen and methanol throughthe diffusion layers and the convective transport coefficients are fitted in a waythat the resulting flows are meaningful.

4.2. Simulation of the Anode 65

4.2. Simulation of the Anode

WILLIAMS et al. published polarization curves for the anode at the two tempera-tures 298 K and 333 K. Fig. 4.2 shows the polarization data for the reference temper-ature T✸= 298K and the simulation results for two transfer coefficients αAN. Thebest fit for the entire range of measured current densities is a value of α✸

AN = 0.56,but this value is not suited to simulate the current densities near the open circuitvoltage. Furthermore, this value is quite high in relation to α✸

AN = 0.24 [143], whichcorresponds to the experimental results of REN [108]. On the other hand, a valueof α✸ = 0.2 is only correct for low polarization (U< 0.4V).

Fig. 4.2.: Anode polarization at reference temperature T✸= 298 K, pressure p= 1E+5Pa,methanol concentration CCH3OH= 1M, membrane: porous H2SO4-doped PVC [148]

The polarization data and the simulation results for T = 333K are plotted in Fig. 4.3.Here the situation is simular: a singular value for the transfer coefficient can notsimulate the polarization over the whole range of anode polarization. In addition

Fig. 4.3.: Anode polarization at 333 K, pressure p = 1E+5Pa, methanol concentrationCCH3OH= 1M, membrane: porous H2SO4-doped PVC [148].


to this phenomenon, a second peculiarity can be seen in the graph: a tempera-ture independent transfer coefficient α✸

AN =αAN(@333K) = 0.56 results in a distinctdivergence from the measured data. However, with a temperature dependenttransfer coefficient α(@333K) = 0.45, the simulation is in good conformity withthe measured data [148]. This finding induced the implementation of a temper-ature dependent transfer coefficient for the anode (cf. chap. 2.2.4, Eq. 2.11 onpage 21). Further discussion, which other processes effect the anode polarizationcan be found in chap. 5.4 on page 89.

4.3. Simulation of the Cathode

It is vital to have a proper model of the crossover to simulate the mixed potentialat the cathode. The effective diffusion flow can be calculated with the currentdensity of the oxygen reduction without crossover at the OCP value of the mixedpotential of the real DMFC cathode, see the Eq. 2.18 on page 28. As mentionedbefore the electroosmotic drag coefficient ξeo has to be fitted, because there is nodata available for the proprietary membrane used by WILLIAMS in [148]. Other fitparameters are the exchange current density constant I✸0, the activation energy∆G‡

r ea, OR and the transfer coefficient for the oxygen reduction αOR.

4.3.1. Cathode at Reference Temperature

The polarization of the cathode at reference temperature T✸= 298 K is shown onFig. 4.4. It is obvious that it is not possible to come close to the measured datawith a pure Butler-Volmer based model (first curve "No Flooding - No LM-Region").The resulting polarization curves are necessarily concave due to the mathematicalnature of the exponential function in use (the Butler-Volmer equation). Nonethe-less, being aware that there is a well known flooding problem on the cathode,cf. [53, 124], the introduction of a flooding model as described in chap. 2.4.3 onpage 41 makes sense (second curve "Flooding - No LM-Region"). In the next step,it is taken into account that there is a sharp change of the kinetic at the transitionfrom the Temkin to the Langmuir adsorption region at the transition potentialUtr near 0.8 V vs. SHE (third curve "Flooding - with LM-Region", cf. chap. 2.4.1 onpage 35). With the flooding model and the Langmuir adsorption region there arethree additional fit parameters:Start of flooding j fln, flooding fraction Θfl✸ and the transfer coefficient in theLangmuir adsorption region αOR,LM . The fit at T✸ was done in two stages, firstξeo , I✸0, αOR, j fln, Θfl✸ in the Temkin region and in the second stage αOR,LM inthe Langmuir region. The decline of α from the Temkin to the Langmuir potentialdomain is expected, but the simulation result of 10% reduction is smaller than theexpected 45±5% [101].

4.3. Simulation of the Cathode 67

Fig. 4.4.: Cathode polarization at reference temperature 298 K. Comparison of pub-lished values [148] with simulation results obtained from three varied models. Thedifference between the graph with and without Langmuir region is small, the declineof the transfer coefficient from the Temkin to the Langmuir adsorption is rd. 10%, thevalues are: αtk = 0.74, αlm = 0.66.

4.3.2. Oxygen Reduction Transfer Coefficient at UpperTemperature

The value of the fitted αlm = 0.6 at 333 K1 corresponds to data in literature; it isonly 5 % under the value published by PARTHASARATHY [101], the decline of α fromTemkin to Langmuir adsorption is 17%.

4.4. Variation of Temperature

4.4.1. Electrode and Cell Polarization at Varied Temperatures

Figure 4.5 demonstrates, that the methanol oxidation on the anode raises withtemperature and increasing polarization. This causes a decline of methanol con-centration at the membrane surface, hence, the crossover flow thins out. The OCPof the cathode in Fig. 4.6 is decreasing with temperature because of increasingmethanol diffusion through the membrane, due to the rise of the temperaturedependent diffusion transport of methanol across the membrane. This trend,however, changes at higher current densities. The diamond symbol ✸ in Fig. 4.6lies on the crossing point of the 298 and 333 K curves.

1 αlm = 0.6 = -0.110 V/(decade ∆jOR), cf. Eq. 2.24 on page 37


Fig. 4.5.: Anode potential vs. SHE at variedtemperatures. The transfer coefficientis temperature dependent α = α✸ +bα ·(T −T✸), it decreases from 0.56 (T = 298 K)down to 0.38 (T = 353 K)

Fig. 4.6.: Cathode potential vs. SHE atvaried temperatures. It is interesting thatthe polarization curves for 298 and 333 Kcross at current density of jfl = 448.0 A/m2,which is marked with a diamond ✸.

The polarization curves of the full cell in Fig. 4.7 show the expected increase of thecurrent density at a given fuel cell voltage with rising temperature. Nonetheless,the increase of current density slows down with rising temperature, this is causedby the opposite trend of the cathode polarization, as is visible in Fig. 4.6.

Fig. 4.7.: Fuel cell polarization at varied temperatures. As in Fig. 4.6 and 4.5 crossoverand cathode flooding is considered.

4.3. Simulation of the Cathode 69

4.4.2. Comparison of Methanol and Carbon Dioxide Crossover atVaried Temperatures

In addition to the methanol crossover, there is also a carbon dioxide crossover fromthe anode across the membrane. The transport mechanisms for both substancesare comparable: The transport relies on the electroosmotic drag and the diffusion.These transport modes depend mainly on the species concentration at the anode-membrane interface. In the model assumptions, the state of aggregation for waterat the membrane surface is defined as liquid, thus the concentration is limited bythe species solubility in water or in the water-methanol mixture. The solubility ofmethanol in water is ideal, but the solubility of carbon dioxide is declining withtemperature.

Despite the superimposition of these two flows, one method to determine themethanol crossover is the measurement of carbon dioxide in the cathode off-gas. This method is described by VALDEZ and NARAYANAN[134]. As mentionedin chap. 2.3.3 DOHLE [39] reported that the CO2-crossover at low methanol feedconcentrations (𝒞CH3OH ≤ 1 M) and high current densities (jfc > 2000 A/m2) canexceed the CO2 evolution on the cathode caused by the crossover reaction. Hence,the model differentiates between three carbon dioxide flows in the cathode off-gas: a) atmospheric, b) CO2-drag and c) product gas from the cathodic methanoloxidation. In Fig. 4.8 and Fig. 4.9 the flows b) and c) are depicted for steady anodeand cathode supply. The results are qualitatively comparable to the measurementsof VALDEZ and NARAYANAN[134] and the simulation of WANG & WANG [143] andGARCIA et al. [44]. The simulation results for the CO2-drag (Fig. 4.8) furthers thepractice of gauging the methanol crossover behavior of a DMFC by measuringthe CO2-off-gas concentration, (cf. e.g. [58]): the CO2-drag is below 5 % of theproduct gas from the cathodic methanol oxidation. The decline of CO2-drag withtemperature is caused by the lower solubility of CO2 at higher temperatures.

Fig. 4.8. Fig. 4.9.

Fig. 4.8, 4.9: Comparison of the carbon dioxide and methanol drag across the mem-brane at varied temperatures; membrane type: Nafion™117, stoichiometric fac-tors: λn

AN(@1000 A/m2) = 1.7, λnCA(@1000 A/m2) = 5.8. Fig. 4.8: current density equiv-

alents (CDE) of the CO2 drag from anode to cathode side. Fig. 4.9: CDE of methanolcrossover.


4.5. Variation of Methanol Concentration

4.5.1. Methanol Concentration and Rest Potential

The influence of the methanol inlet concentration on the anode open circuit po-tential (OCP) or rest potential Uoc, AN is shown on Fig. 4.10 and Fig. 4.11. Themethanol diffusion coefficient DCH3OH, pm is increasing with temperature from thereference temperature 298 K on the left graph, to 333 K on the right graph. Addi-tionally, the crossover of methanol decreases the OCP of the cathode Uoc, CA. Theanode OCP Uoc, AN profits by increasing temperature and methanol concentration.The modest methanol diffusivity2 of the gold plated porous PVC-membrane usedby WILLIAMS [148] results in a gain of Uoc, fc =Uoc, CA −Uoc, AN for low as well as forhigh methanol inlet concentrations. Simular measured and empirical simulatedOCV curves for varied methanol inlet concentration are published in [128]. Theexact OCP values are given in Tab. 4.1.

Fig. 4.10. Fig. 4.11.

Fig. 4.10, 4.11: Cathode rest potentials Uoc, CA and open circuit fuel cell voltages Uoc, fcat different methanol concentrations are displayed together on the left y-scales ofeach plot. The values of the anode OCP Uoc, AN are given on the opposite y-scales.Figure 4.10 depicts the curves for the reference temperature 298 K, and Fig. 4.11 showsthe values for T = 333 K.

Cathode OCP Anode OCP𝒞CH3OH T = 298 K T = 333 K T = 298 K T = 333 K

mol/l V vs. SHE

0.5 0.911 0.899 0.112 0.0621.0 0.888 0.872 0.111 0.0592.0 0.863 0.843 0.107 0.0563.0 0.846 0.824 0.106 0.0555.0 0.825 0.807 0.104 0.0538.0 0.806 0.774 0.103 0.052

Table 4.1.: Simulated open circuit half cell potentials for the fuel cell data byWILLIAMS [148] at two temperatures and varied concentrations.

2 The diffusion coefficient is in fact greater in relation to a Nafion™117-Membrane, but the thick-ness compensates for the higher diffusion coefficient (cf. Tab. 4.3 on page 79).

4.5. Variation of Methanol Concentration 71

4.5.2. Crossover Growth with Increase of Methanol Concentration

The anode polarization benefits from higher methanol concentrations, hence itwould be fine to work with methanol concentrations above 10 M CH3OH. At thesame time the crossover is increasing, thereby overcompensating for the posi-tive effect on the anode polarization. Fig. 4.12 shows the two components of the

Fig. 4.12.: Crossover for the WILLIAMS [148] proprietary gold coated porous PVC mem-brane at methanol inlet concentrations of a) 0.5, b) 1, c) 2, d) 3, e) 6 M CH3OH, temper-ature Tfc = 298 K, nominal methanol stoichiometric factor λn

AN(@1000 A/m2) = 7. Theelectroosmotic drag ΦCH3OH,eo over the fuel cell current density jfc is shown on theleft diagram. On the right diagram, the lower curves indicate the diffusion transportΦCH3OH,D . The sum of diffusion and electroosmotic transport, the crossover flowsΦCH3OH, CO, are given as the upper curves. The methanol flow rates are converted tocurrent density equivalents (CDE): j =ΦCH3OH ·6 ·F/Afc.

crossover flow: on the left graph the electroosmotic and on the right the diffu-sion current density equivalents are depicted; the sum of both, the crossover, isoutlined as the upper curves on the right. The nominal methanol stoichiometricfactor at 1000 A/m2, λn

CH3OH(@1000 A/m2), is equal for the six concentrations to en-sure comparability. This common value needs to be above 6.8 to avoid methanolshortage in the anode.

On the right graph of Fig. 4.12 it is interesting to see that the diffusion curve of6 M CH3OH (the lower curve) comes closest to the point of methanol shortageat 1100 A/m2. The reasons are: a) the total water-methanol flow through theanode at const. λn

CH3OH is decreasing with increasing methanol concentration,which reduces the convective flow into the anode; b) the anode activity and thusthe methanol consumption is increasing with higher methanol concentration.The graphs of Fig. 4.12 confirm two things: 1.) the fuel efficiency ηut is sufferingfrom higher methanol concentrations; 2.) the positive effect of higher methanolconcentrations on the activity of the anode can only be exploited in a small rangeof optimized operating points for λn

CH3OH and jfc.


4.6. Comparison of Two Membrane Types at DifferentMethanol Concentrations

The proprietary PVC-membrane invented by WILLIAMS [148] behaves strangelyin comparison to the more common Nafion™-membrane, the methanol dragis advantageously less pronounced, which leads to the crossing of the cathodepolarization curves in Fig. 4.6 on page 68. To verify the model the electroosmotic

Fig. 4.13.: Polarization of anode and cathode for two different electrolyte membranesat varied anode methanol inlet concentration of 0.5 – 6 M CH3OH. The electrolyte onthe left diagram is a proprietary PVC-membrane used by WILLIAMS [148] and on theright it is Nafion™117. Temperature: 333 K, nominal methanol stoichiometric factor:λ

nAN@1000 A/m2 = 7.

Fig. 4.14.: Cell polarization for two different electrolyte membranes at varied anodemethanol inlet concentration. The parameters are identical to Fig. 4.13.

4.7. Variation of Anode Stoichiometry to Optimum 73

drag factor, the methanol diffusion and the membrane thickness were exchangedwith the parameters of Nafion™117, the literature value of the diffusion coefficientwas generously decreased by 20% to consider the diffusion contribution of the ACLto the effective diffusion coefficient (the values are documented in the Tab. 4.3 onpage 79). The diagrams on the left of Fig. 4.13 and Fig. 4.14 depict the polarizationof the DMFC with the WILLIAMS PVC-membrane and the diagrams on the rightillustrate the situation with Nafion™117-membrane: the anode polarizations arenearly identical (Fig. 4.13). The small differences are caused by the differingmethanol flow through the membrane and thus through the ACL. The cathodeOCPs in Fig. 4.13 are higher for the PVC-membrane, due to the lower methanolpermeability. The methanol drag in the DMFC with PVC-membrane is low enough,to allow for a crossing of the cathode curves 3 and 6 M CH3OH on the left diagramof Fig. 4.13. Hence, the good protection of the cathode by the WILLIAMS PVC-membrane leads to the good result for the simulation of 6 M CH3OH solution. Thepolarization curves on the right of Fig. 4.13 and Fig. 4.14 validate the practicalexperience with Nafion™-membrane equipped DMFC, that concentrations below1 M CH3OH lead to a weak performance above 600 – 800 A/m2 and concentrationsabove 1 M CH3OH show bad results on the whole current density range.

4.7. Variation of Anode Stoichiometry to Optimum

Increasing the anode stoichiometry at a given methanol inlet concentration raisesthe anode inlet flow, which also effects the convective transport through thediffusion layer (cf. chap. 2.2.2 on page 16) and, of course, the diffusion transportthrough the membrane. Due to the transport barriers from the flow field tothe reaction zone and because of methanol crossover, the nominal methanolstoichiometric factor3, also refered to as nominal anode stoichiometric factor λn

AN,needs to be well above 1 to avoid reactant shortage.

Most effected by the variation of the anode stoichiometric factor λnAN is the cath-

ode potential vs. SHE as can be seen on the right of Fig. 4.16. The electroosmotictransport is small at 333 K, in comparison to the diffusion transport, as can beseen on Fig. 4.15. The simulation drastically demonstrates that at normal op-eration temperatures, e.g. 333 K, at high methanol concentration of 6 M CH3OHand Nafion™117 as membrane, it is important to have good control of the anodestoichiometric factor, else the current density equivalent of the crossover flow mayreach extreme values of more than 2000 A/m2 (cf. Fig. 4.15 on the right). This is inagreement with practical experience. At this operation conditions a lot of extraheat is produced on the cathode and the OCV is low.

3 The difference between λAN and λn

AN is described in the glossary on page 152.


Fig. 4.15.: Electroosmotic and diffusion transport at varied anode stoichiometries.Methanol inlet concentration 6 M CH3OH, temperature 333 K, membrane Nafion™117.

Fig. 4.16.: Varied nominal methanol stoichiometry λnAN. Depicted on the left graph

is the sum of electroosmotic and diffusion transport. Parameters are the same as inFig. 4.15.

4.8. Influence of Ion Resistivity on the Polarization 75

4.8. Influence of Ion Resistivity on the Polarization

The ion resistivity of the membrane is not to neglect, in the case of our DMFC data,the loss is in the range of 12% at a temperature of 333.15 K and a cell polarizationof 0.44 V (Ufc= 0.38 V, jfc = 1000 A/m2), see Fig. 4.17. The voltage loss is split in ananodic and a cathodic part, therefore, a higher polarization in both electrodes isneeded to compensate for the electric current loss, see Eq. 2.28 on page 43.

Fig. 4.17.: Influence of the ion resistivity of the membrane on the polarization. Theresistivity is calculated according to the information given by WILLIAMS [148] that thevoltage loss at 1000 A/m2 and 333.15 K is typically 0.045 V.

4.9. Global Parameter Optimization

As mentioned in chap. 2, it was necessary to identify some parameters by meansof fitting. The polarization curves published by WILLIAMS [148], documented onpage 131, have been used as target polarization curves. The curves printed inlit. [148] have been digitized with the software WinDIG (cf. chap. F on page 135)and smooth graphs through the digitized points have been implemented in aFortran block inside the AspenPlus code via a cubic spline interpolation [103]. Asa result it was possible to calculate the difference between target graph and anypoint on the simulated polarization curve of the electrodes and hence, it waspossible to calculate the quality criteria of least squares. It is of general interestto know if there is a global optimum of the five parameters that influence thepolarization curve of the oxygen reduction in the Temkin adsorption region or ifthere are different optima. To evaluate the topography of the quality over the five


αTK I 0TK �A/m2 ξeo jfln�A/m2 Θfl✸

min 0.68 3.20 1.0 120 0.6max 0.75 5.62 2.0 180 0.8

step size 0.0025 0.02 0.1 10 0.025iterations 29 122 11 7 9

Table 4.2.: Parameter range and step size for global optimization.

Fig. 4.18. Fig. 4.19.

Fig. 4.18, 4.19: Global search result for fit-parameters in the Temkin region at 298 K.Fig. 4.18: Normed quality of least squares over normed Butler-Volmer parametersαTK ∈ [0.68 .. 0.75] and I 0

TK ∈ [3.68 .. 4.94]. The graph of αTK is fitted with a polyno-mial of degree 5, and the single values are marked with a circle ◦. Fig. 4.19: Normal-ized quality of Least Squares over normalized electroosmotic drag ξeo ∈ [1.0 .. 2.0],flooded fraction of cathode at 1000 A/m2 Θfl✸ ∈ [0.6 .. 0.8] and begin of floodingjfln ∈ [120 .. 180 A/m2].

parameters αTK, I 0TK, ξeo , jfln, Θfl✸, a global optimization was performed. Each

parameter set was tested on 5 cell voltages and each of the parameters weremanipulated in a wide range, to ensure that the optimum lies within the ranges.Tab. 4.2 shows the variation step size and the range of the 5 parameters. In total,more than 1E+7 points have been investigated4 in this search and the simulationslasted over 4 weeks on 3 PCs. To plot the quality graph for a specific parameter, allresults were sorted in ascending order, in the first run after the quality and in thesecond after one of the parameters. The sorting was undertaken with Octave (cf.chap. F). Quality and the parameter values were normalized to maximum value inorder to be able to compare the different parameters to each other:

XΠ = X /Xmax

4 Total number of investigated points: 5 · 29 · 122 · 11 · 7 · 9 = 12 259 170

4.9. Global Parameter Optimization 77

Fig. 4.20. Fig. 4.21.

Fig. 4.20, 4.21: Global search result for fit-parameters at constant electroosmotic dragfactor ξeo= 1.8. Search ranges are equivalent to Fig. 4.18 and 4.19.Fig. 4.20: Normalized quality of Least Squares over normalized Butler-Volmer param-eters αTK and I 0

TK. The graph of I 0TK is fitted with a polynomial of degree 11, the single

values are marked with small triangles. Fig. 4.21: Normalized quality of Least Squaresover normalized flooded fraction of cathode at 1000 A/m2 Θfl✸ and begin of floodingjfln.

Fig. 4.18 shows the result for the two Butler-Volmer parameters αTK and I 0TK in

the Temkin region for the oxygen reduction. The values of best fit for the transfercoefficient αTK are heavily distorted, thus to see a possible trend, a polynomialapproximation of degree 5 is shown as the line for αTK. But even this compensatingcurve shows no clear optimum. The second line of Fig. 4.18 depicts the referenceexchange current density. I 0

TK follows a distinct trend, but the optimum is notclearly pronounced, as it falls within a broad range.

The line for the electroosmotic factor of Fig. 4.19 also shows no clear minimum,but it is reasonable that this factor for the gold coated, sulfuric acid soaked porousPVC-membrane lies in the near range to the values for Nafion™117 (cf. Fig. 2.15),because in both membranes water molecules dominate the solvent. Hence, thevalue of ξeo= 1.8 (ξΠeo= 0.8) is much more realistic compared to ξeo= 1.0 (ξΠeo=0.0). The quality graph of the flooding fraction Θfl✸ show a clear trend, but theoptimum is located in a flat vale. Best pronounced is the optimum of the begin offlooding jfln.

In the next step of data analysis the data sets were confined to packs with anelectroosmotic drag factor of ξeo = 1.8. The results are shown in Fig. 4.20 and 4.21.The graph of αTK in Fig. 4.20 is smooth without polynomial approximation andthe minimum is identical to that in Fig. 4.18. The values of I 0

TK are bit noisy, butthe trend is clear; nevertheless the minimum is still difficult to detect on the plot.The flooding parameters jfln, Θfl✸ are now easy to ascertain from the Fig. 4.21.


Fig. 4.22.: Quality graphs of the Butler-Volmer parameters αTK, I 0TK in the Temkin region

for steady variables: ξeo = 1.8, jfln= 150 A/m2, Θfl✸= 0.725 at 298 K as the result of aglobal parameter optimization.

The result of the data analysis is for the electroosmotic factor and the floodingparameters:ξeo = 1.8, jfln= 150 A/m2, Θfl✸= 0.725.

If these values are used to filter the data, the result is the final plot Fig. 4.22 for theButler-Volmer coefficients, the minimum valleys are still platy but the optima aredistinct.

It is noticeable that the graphs for the reference exchange current density are roughin three plots Fig. 4.18, 4.20, 4.22: the reason is the discontinuity for the I 0

TK values.To flatten the curves it would be necessary to downsize the increment for theparameter I 0

TK, but this would increase the computing time even further. The stepsize of ∆I 0

TK = 0.02 A/m2 is a compromise to narrow the computing time. If desired,smaller increments can be used in a second parameter search in special parts ofthe parameter space.

4.10. Table of Parameters and Fit Results 79

4.10. Table of Parameters and Fit Results

Symbol Value Unit

Anode Bipolar PlateFlow field channel cross-section area A ABP 1.0E-03 m2

Pressure drop proportionality factor fg , ABP 5.077417E+05 m-3

Anode Diffusion LayerDiffusion layer thickness l ADL 4.0E-04 m

Diffusion coefficient methanol DCH3OH, ADL 1.38E-09 m2/s

Diffusion coefficient water DH2O, ADL 3.71E-09 m2/s

Hydraulic permeability constant khp, ADL 7.1411E-15 m2

Anode Catalyst LayerGeometric electrode area Afc 1.0 m2

Catalyst metal loading per area mC at , AN 5.0E-02 kg/m2

Mass specific volume vC at , AN 5.0E-04 m3/kgVolume-specific catalyst surface SAN 4.0E+04 m-1

Anode Parameters, Methanol Oxidation

Activation energy at ideal open circuit condition ∆G‡�AN 9.845429E+03 J/mol

Reference exchange current density I✸0AN 1.346 A/m2

Activation energy ∆G‡r ea, AN −6.802607E+03 J/mol

Transfer coefficient @298 K α✸AN 0.560

Transfer coefficient @333 K αAN ,333K 0.445Transfer coefficient slope ∆α

∆T AN −3.285714E-03

Cathode Bipolar PlateFlow field channel cross-section area AC BP 1.0E-03 m2

Pressure drop proportionality factor fg ,C BP 1.127794E+02 m-3

Cathode Diffusion LayerDiffusion layer thickness lC DL 4.0E-04 m

Diffusion coefficient oxygen DO2,C DL 1.70E-06 m2/s

Hydraulic permeability constant khp,C DL 2.3504E-14 m2

Cathode Catalyst LayerCatalyst metal loading per area mC at ,C A 5.0E-02 kg/m2

Mass specific volume vC at ,C A 5.0E-04 m3/kg

Table 4.3.: Simulation parameters and fit results.


Continuation of tab. Parameters

Symbol Value Unit

Volume-specific catalyst surface SC A 4.0E+04 m-1

Cathode, Oxygen Reduction, tk-Region

Activation energy at ideal open circuit condition ∆G‡�OR −1.074589E+03 J/mol

Reference exchange current density I✸0tk 3.8 A/m2

Activation energy, temkin adsorption ∆G‡r ea, tk −2.174968E+04 J/mol

Reference transfer coefficient @298 K α✸tk 0.735

Transfer coefficient @333 K αtk,333K 0.720Transfer coefficient slope ∆α

∆T tk −4.28571E-04

Cathode, Oxygen Reduction, lm-RegionReference transfer coefficient @298 K α✸

lm 0.660

Transfer coefficient @333 K αl m,333K 0.600Transfer coefficient slope ∆α

∆T lm −1.714286E-03

Cathode, Crossover ReactionReference exchange current density I✸0

CO 1.346 A/m2

Activation energy ∆G‡r ea,CO −6.802607E+03 J/mol

Transfer coefficient @298 K α✸CO 0.560

Transfer coefficient @333 K αCO,333K 0.445Transfer coefficient slope ∆α

∆T CO −3.285714E-03

Cathode, Flooding ParameterReference start of flood. as current dens. @298 K j fl✸ 1.5E+02 A/m2

Flooding begin as current density @333 K jfl,333K 1.7E+02 A/m2

Exponential factor for start of flooding bfl −3.55208E+02 K

Reference flooded fraction at 1000 A/m2 Θfl✸ 0.725Flooded fraction at 1000 A/m2, @333 K Θfl,333K 0.340

Exponential factor flooded fraction cfl 1.504836E+03 K

PVC Membrane ParametersMembrane thickness lm 7.62E-04 m

Effective ref. diffusion coefficient, 298 K DCH3OH 1.895E-10 m2/s

Diffusion exponent factor, Methanol dD,CH3OH −6.36912

Electroosmotic reference coefficient ξ✸eo 1.8Electroosmotic coefficient @333 K ξeo,333K 2.8

Exponential factor electroosmotic coeff. beo −1.264022E+03 KIonic Conductivity @298 K σ298K 1.0489E+01 S/m

Reference Ionic Conductivity @333 K σ✸333K 1.6933E+01 S/m


4.10. Table of Parameters and Fit Results 81

Continuation of tab. Parameters

Symbol Value Unit

Exponential factor electroosmotic coeff. bσ −1.3593E+03 KDiffusion coefficient, CO2 @298 K DCO2

2.2983E-09 m2/s

Molar diffusion activation energy, CO2 ∆GD

CO2−1.850E+04 J/mol

Nafion™117 Membrane ParametersMembrane thickness (dry) l⋆m 1.78E-04 m

Membrane thickness (swollen) lm 2.00E-04 mEffective ref. diffusion coefficient, 298 K DCH3OH 1.66151E-10 m2/s

Diffusion exponent factor, Methanol dD,CH3OH −6.31537

Electroosmotic reference coefficient ξ✸eo 1.99Electroosmotic coefficient @333 K ξeo,333K 2.84

Exponential factor electroosmotic coeff. beo −1.081704E+03 KIonic Conductivity @298 K σ298K 1.693E+01 S/m

Reference Ionic Conductivity @333 K σ✸333K 2.71E+01 S/m

Exponential factor electroosmotic coeff. bσ −1.3593E+03 KDiffusion coefficient, CO2 @298 K DCO2

2.0133E-09 m2/s

Molar diffusion activation energy, CO2 ∆GD

CO2−1.850E+04 J/mol


5. DISCUSSION OF THE MODEL

This thesis presents an analytical, zero-dimensional model of the DMFC. Thecalculation of the kinetics for the anode as well as for the cathode is based onthe Butler-Volmer equation. The simulation is implemented in a chemical engi-neering software. The model has been tested on the DMFC polarization data ofWILLIAMS [148]. The advantage of this data is the existence of a polarization curveof the cathode without crossover and a pair of polarization data for the half cellpotentials vs. SHE at two temperatures. The disadvantage of this data is that themembrane of the investigated DMFC is a proprietary membrane hardly compara-ble to the more common polyperfluorosulfonic acid membranes like Nafion™117.However, the data is sufficient to prove the proper qualitative functionality of themodel. In addition to the validation based on literature data, own DMFC measure-ments with a Nafion™115 membrane have been performed and are discussed inthis chapter.

5.1. Zero-dimensional Multi-domain Simulation

The model is based on a zero-dimensional or one equation mode (cf. chap. 1.3) andneeds an engineering software, e.g. AspenPlus, to calculate the physical properties.Yet, as long as no optimization of the flow field is wanted, the loss of informa-tion due to the zero-dimensional mode is small, inasmuch as the core functionallayers ADL, ACL, PEM, CCL and CDL are relatively thin (l < 3E-4 m). If the flowfield design should be optimized, the model can be extended to a pseudo three-dimensional model by subdividing the cell into small segments. Each segmentis a micro cell; all cells are connected to the flow fields via the diffusion layersand can exchange flows at the cell borders to the neighboring micro cells. Themain drawback to this approach within engineering software, such as AspenPlus,ProII, HYSYS, etc., is that these programs are not prepared to work in a way CFDsoftware does. On the other hand, CFD software is not well equipped to performdifficult property calculations. To overcome this limitation, the software supplierANSYS provides an interface between his CFD-software FLUENT and AspenPlus.Another aspect of the model that is unfavorable for CFD calculations is the multi-domain structure (cf. chap. 1.3.2). However, this multi-domain structure is notindispensable for the model; the structure can be transformed inside AspenPlusinto a single-domain structure.

84 5. Discussion of the Model

5.2. Limitations of a Single Cell Model

Simulating fuel cells with an engineering software, such as AspenPlus, has theadvantage that it is easy to implement system models rather than isolated single-cell models. It is standard to work with cell-stacks in a system, instead of single-cells. The main differences between a single-cell model and a stack model are:

• The inlet flow conditions of the single-cells in the stack are different fromthe stack entrance conditions and may differ for each cell.

• In addition to a model for the flow field of the cells, a model for the stackdistribution system is also needed.

• In a liquid feed DMFC leakage currents, due to increased ion conductivityof the water-methanol mixture, need to be addressed by the stack model.

• As there could be a temperature gradient inside the stack, a model for theheat exchange is required.

This list clarifies that it is a substantial modification to transform the single-cellmodel into a stack model. Nevertheless, a cell model, already implemented inan engineering software, is a good starting point for a stack model inside such asoftware.

5.3. Comparison of the Model to Other ModelingApproaches

This thesis is likely the first attempt to model the electrochemistry of a DMFCusing three Butler-Volmer equations1. There are no works published that model theelectrochemical (parasitic) methanol-oxidation on the cathode via a Butler-Volmerequation. A common approach is a combination of a kinetic model on the anodecombined with one common Tafel equation2 on the cathode [67, 70, 92, 98].

1 For Butler-Volmer equation see on page 1382 For Tafel equation see on page 154

5.3. Comparison of the Model to Other Modeling Approaches 85

5.3.1. Model of the Anodic Catalytic Layer

The model works with a zero-order kinetic model, but for concentrations belowthe so called transition concentration of 𝒞CH3OH, tr = 0.1 M a reaction order of oneis predominant [93, 105]. For a static zero-order model of the anode kinetics thischange of reaction order is no restriction above feed concentrations 𝒞CH3OH, i n >0.2 M3, and the influence on the cathode mixed potential is negligible becauseof the high polarization of the cathodic methanol oxidation — thus high surfacerelated current densities I CO (cf. Fig. 2.21 on page 39). Even one-dimensionalsimulations work with a constant methanol concentration across the electrode(e.g. [75]). In the literature there is no example of a Butler-Volmer equation variantthat reproduces the change of reaction order from zero to one below the transitionconcentration 𝒞CH3OH, tr . Only semi-empirical equations are published, whichregard the change of reaction order across the electrode. In the review article byOLIVEIRA [98] regarding DMFC-modeling, two different semi-empirical equationsfor the anode kinetics are discussed that consider the change of reaction orderfor low concentrations. One is invented by DOHLE [41] and the other is publishedby MEYERS & NEWMAN [92]; both are suited to simulate the anode at constanttemperatures, as the fit parameters differ with changing temperatures. Fig. 5.1 andFig. 5.2 demonstrate that both show a perfect compliance with the measured dataof WILLIAMS [148]. The eq. published by DOHLE [41] is a bit more adaptable to themeasured data due to the logarithmic term. The eq. of MEYERS & NEWMAN [92]has the advantage that the polarization UAN is the free variable of the simulation.Tab. 5.1 shows the fit-parameter for the WILLIAMS data at 298 K.

Fig. 5.1. Fig. 5.2.

Fig. 5.1, 5.2: Plots of the semi-empirical models of MEYERS and NEWMANN [92] andDOHLE [41], the fit parameters are given in Tab. 5.1. Both curves show a sharp increaseof polarization below methanol concentrations of 𝒞CH3OH < 0.2 M

3 The model uses the arithmetic mean of methanol inlet and membrane interface activity. Hence,at an inlet concentration of 𝒞CH3OH, i n > 0.2 M, the corresponding mean concentration lies above0.1 M


DOHLE: p1 p2 p3 p4

UAN = p1 +p2· ln j +p3·exp

p4·j𝒞

3.947E-3 3.079E-2 1.121E-1 5.271E-4

MEYERS: j⋆ a b K

jAN = j⋆·𝒞·a·exp(b·UAN

)𝒞+K ·a·exp(b·U

AN) 5.84E-2 3.151 2.436E+1 4.1E-5

Table 5.1.: Fit results against the anode polarization data of WILLIAMS [148] at T =298.15 K and 𝒞 = 1 M for semi-empirical anode kinetics equations, the first is suggestedby DOHLE [41] and the second by MEYERS and NEWMANN [92].

To compare the two semi-empirical equations with the analytical Butler-Volmerequation the temperature is set equal to the reference temperature T = T✸ anda huge stoichiometric factor for methanol λn

CH3OH(@1000 A/m2) ≫ 20 is assumed,which leads to activity relations equal to one: aox, AN/aoc,ox = ar ed , AN/aoc,red ≈ 1and thus to a simplified version of the anodic Butler-Volmer Eq. 2.10 on page 204;this variant is marked with the star symbol ⋆:

I⋆AN = I 0⋆AN · f BV⋆

with:

I 0⋆AN = I✸0

AN · aαAN

oc,red · a1−α

AN

oc,ox

f BV⋆ = exp

αAN ·bBV ·UAN

−exp(1−αAN) ·bBV ·UAN

bBV = z‡

AN ·FR ·T

(5.1)

The behavior of this simplified form of the Butler-Volmer equation is plotted on Fig.5.3, the polarization is growing with decreasing methanol concentrations, evenwithout a transformation of the reaction order below the transition concentration.However, the influence is less pronounced than in the other two models. Fig.5.4 shows a more realistic picture. The activity relations are not set to unity,which leads to a comparatively sharp change of the polarization at low methanolconcentrations / activities. Table B.4 on page 116 shows the difference of theactivity and the concentration values of water-methanol solutions with differentmolar concentrations, as computed by AspenPlus.

4 A huge stoichiometric factor raises a huge anode inlet flow and hence, increases the convectiveflow through the anode diffusion layer. This leads to a mean methanol concentration inside theanode catalyst layer near the inlet concentration. With the assumptions of chap. 2.2.4 on page 19the chemical activity of the oxidized species is aox, AN = aCH3OH and the one of the reduced speciesis ared, AN = aCO2

.

5.3. Comparison of the Model to Other Modeling Approaches 87

Fig. 5.3. Fig. 5.4.

Fig. 5.3, 5.4: Figure 5.3 depicts the influence of varied methanol concentrations onthe simplified Butler-Volmer equation in the form of Eq. 5.1. Figure 5.4 shows thesame for the plain Butler-Volmer equation (cf. Eq. 2.10 on page 20). The curve forthe concentration of 2 M CH3OH is left out in this plot, because the average anodicmethanol activity ared, AN has to be below the anodic methanol activity at open cellconditions aoc,red. For both figures the temperatures are set equal to the referencetemperature T = T✸.

5.3.2. Transition from Temkin to Langmuir Adsorption

One of the important improvements of the presented model in comparison toother DMFC cathode models is the implementation of the Langmuir adsorptioninto the Butler-Volmer based model (cf. chap. 2.4.2). Due to the mixed poten-tial and hence, the higher cathode polarization, the transition from Temkin toLangmuir adsorption has more significance in a DMFC in comparison to a PEM-HFC.

Fig. 5.5. Fig. 5.6.

Fig. 5.5, 5.6: Influence of a declining transfer coefficient below a cathode potentialof Utr =0.8 V vs. SHE. The cathode potentials UCA relate to the left and the fuel cellvoltages to the right y-scales. The measured data [148] at 298 K (Fig. 5.5) and 333 K(Fig. 5.6) are charted with a circle ⃝. The upper curves of the cathode and cell polar-ization are simulated with constant transfer coefficients αtk = αl m . The transitionpoint from tk- to lm-region is marked with ✸.


The published cathode data of WILLIAMS [148] show at 298 K in Fig. 5.5 and at 333 Kin Fig. 5.6 an unusually steady transfer coefficient around the transition potentialUtr from the Temkin to the Langmuir region. According to the work of SEPA [121]and others, a decline of αOR of 50±5% is expected (cf. chap. 2.4.2). However, ascan bee seen in Tab. 4.3 on page 79 the decline of the transfer coefficient for thebest fit is at 298 K only 10% and at 333 K only 17%. To demonstrate the declineof the polarization slope at the transition potential Utr polarization curves areadded with transition coefficients for the Langmuir region αlm , which fit into theratio αl m/αtk = 0.5±0.05 found by PARTHASARATHY [99]. The fitted αl m values arein good agreement with the data of PARTHASARATHY, while the fitted αtk valuesare relatively low compared to that. This discrepancy may be caused by the specialtype of membrane electrode assembly used by WILLIAMS.

Transport Across the Diffusion Layers

Aside from the diffusion, the presented model includes a convective transportdue to pressure difference into the anode and cathode diffusion layers (ADL +CDL). Semi-empirical works, such as [1, 135], but also the DMFC anode modelfrom KULIKOWSKY [75] of the year 2003 neglect convective transport into the ADL.Whereas BERNARDI & VERBRUGGE [19] as well as others [14, 43, 51, 143] considerconvective transport into the diffusion layers via the Darcy’s law. The convectivetransport due to the velocity of the media in the flow field channels and hence,a hydraulic pressure drop across the diffusion layers, is strongly influenced bythe dynamic viscosity ηV (cf. Eq. 2.1 on page 17). As the dynamic viscosity is lowerfor the gas phase, the convective transport is more important for the cathode.But for modern diffusion layers with a good penetrability on the anode side, theconvective transport becomes also important for the transport of the liquid water-methanol mixture to the ADL-ACL interface. Therefore the conclusion is thatthe consideration of convective transport into the diffusion layer is necessary, tomodel the influence of varied inlet flows on the cell perfomance.

Crossover Flow

As mentioned in the chap. Introduction on page 1 ff., the flow of methanol acrossthe membrane, the so called cross over flow is the most critical transport in aDMFC, it consists of two elements: a) diffusion and b) convection — both tem-perature dependent. In some works the crossover flow is neglected, especiallyin older works or system simulations, e.g. [1, 21, 70, 135]. Other publications thatconsider the diffusion transport via the First Fick’s law use an isothermal diffusioncoefficient, e.g. [63, 124]. Works that include an electroosmotic transport, utilizemostly a constant drag coefficient ξeo , e.g. [38,44,63,117,143]. In the present thesisboth transport modes are implemented with temperature dependent parameters;

5.4. Anode Polarization 89

this increases the correctness of the cathode polarization predictions at variedtemperatures considerably.

5.4. Anode Polarization

5.4.1. Potential Dependencies of Adsorption Processes

A potential change of the adsorption behavior of reactant species may influencethe kinetics on the anode, as can be observed on the cathode (cf. chap. 2.4.1on page 35) and hence, a similar effect my be visible on the DMFC-anode. Onpure platinum anodes a sharp increase of the reaction rate of the electrochem-ical oxidation of methanol can be observed at room temperature above 0.5 Vvs. SHE [45, 46, 79]. Below this potential the adsorption of OH-species that areneeded in the RDS-reaction step is hindered (cf. Eq. 2.9 on page 19). To lowerthis transition voltage of the OH-adsorption, binary catalysts, such as Pt-Ru, areadvantageous (cf. chap. 2.2.3 on page 17). GASTEIGER et al. investigated the elec-trooxidation of methanol on well defined unitary and binary Pt-anodes. The cyclicvoltammogram of a Pt-Ru catalyst with an atomic ratio of 54:46 did not showthe sharp increase of the OH-adsorption rate at 0.5 V vs. SHE, which is typicalfor unitary Pt-anodes. Additionally, the graph from 0 to 0.5 V vs. SHE is planer incomparison to the voltammogram of pure Pt-anodes [45].

Another aspect that confirms the need for a more complex theory of the methanoloxidation are the actual polarization curves of DMFC-anodes: the anode potentialstays well below the transition potential of 0.5 V vs. SHE, as can be seen in thefigures 4.2, 4.3, 5.12 on the pages 65, 65, 96.

These findings justify an anode theory with one set of Butler-Volmer parametersfor the DMFC. If cyclic voltammetry analysis show a different behavior at anodepolarizations below 0.3 V for the investigated catalyst a second set of Butler-Volmerparameters can be established, as it is described for the transition from Temkin toLangmuir adsorption on the cathode (cf. chap. 2.4.2 on page 36).


5.4.2. Influence of the Vapor Phase on the Anode Polarization

As shown in chap. 4.2, it is not possible to have a perfect match of the data givenin [148] with a pure Butler-Volmer approach; either the low current range or thehigh current range is fit well (cf. Fig. 4.2 on page 65). WANG & WANG ascertainedthe anode transfer coefficient to a value of αAN = 0.24 [143] by means of fittingthe data published in [108]. The discrepancy to the value of αAN = 0.56 used inthis work requires some explanations. One explanation is the increasing carbondioxide evolution in the anode with growing current densities. This may reducethe active catalyst surface in contact with the liquid water-methanol solution, ascan be seen in Fig. 5.8.

Fig. 5.7. Fig. 5.8.

Fig. 5.7, 5.8: Pictures of the porous anode structure, based on a so called carbon sup-ported catalyst. Catalytic particles have to be in a three phase contact with: a) theelectrolyte, b) an electron conductive phase (the support or another catalyst particle)and c) the fuel. Fig. 5.7 depicts the situation at OCV. Fig. 5.8 shows carbon dioxideevolution at raising current densities.

Though the diffusivity of methanol in the gas phase is higher in comparisonto the liquid phase, the methanol activity in the carbon dioxide gas phase ismuch smaller than the activity in the liquid phase. To make an estimation of theeffect, the gas phase is assumed to be fully saturated with water-methanol vapor.With this assumption an activity calculation in the anodic vapor phase has beenperformed with AspenPlus. The obtained activity values of methanol vapor in fullyhumidified carbon dioxide can be utilized to calculate the extreme situation of anactive catalyst surface covered completely with vapor phase. AspenPlus calculatesthe methanol activity for a 1 M methanol solution in saturated carbon dioxideat 298 K to a

VPCH3OH = 4.5E-03, which is 14.8% of the activity of a liquid 1 M CH3OH-

solution (cf. Tab. B.4 on page 116). In the simulation model with a nominallambda of λn

AN(@1000 A/m2) = 5.7 this activity shrinks at current densities above1000 A/m2 to values below a

VPCH3OH = 3.5E-03 at the membrane interface. The course


of this activity value along the current density was ascertained by an AspenPlussimulation and taken to calculate the mean activity inside the catalytic layeragainst the current density. Further calculations were performed with Octave5

outside AspenPlus using the data of the AspenPlus simulations. The calculationresult for the polarization of the catalyst surface in sole contact with the vaporphase is given in Fig. 5.9. For the determination of the mixed case of liquid andgas phase in contact with the active layer (cf. Fig. 5.10), a linear influence of thecurrent density is assumed: the higher the current, the higher the surface fractioncovered with the gas phase. The linear correlation considers the fact that thecarbon dioxide evolution is growing linearly with the current or current densityaccording to the First Faraday’s law (cf. Eq. G.10 on page 146).

Fig. 5.9. Fig. 5.10.

Fig. 5.9, 5.10: Figure 5.9 shows as the upper and the lower curve the two extrema ofpolarization curves for the anode: a) The whole active catalyst surface is covered withfuel in gaseous state (carbon dioxide saturated with water-methanol vapor); b) Thecatalyst surface is covered only by a liquid water-methanol mixture. The middle curvedepicts the realistic situation of a mixed case. The ratio of polarization cases a) and b)is modeled with a linear equation, comparable to the flooding Eq. 2.27 on page 41.Figure 5.10 is a plot of the best fit of mixed cases with different transfer coefficients forthe reference temperature of 298 K, the polarization data points of [148] are markedwith circles.

5 Cf. chap. F on page 135


The linear regularity is analogous to Eq. 2.27 on page 41:

SVPAN ∼ jfc ⇔ SLP

AN ∼ 1− jfc ∧ jfc > jVPn ∧ jfc = |jAN|SVP

AN = ΘVP ·SAN = max[min(a +b · jfc, 0.95), 0.0] ·SAN

with:

b = 1−ΘVP✸

1000 A/m2 − jVPn

a = −jVPn ·b

jVPn = jfc at the begin of vapor phase coverage of active surface SAN

ΘVP✸ = vapor covered fraction of SAN at j✸fc = 1000 A/m2

The Butler-Volmer equation for the vapor phase is in principle identical to theEq. 2.10 on page 20; the only differences are the lower values of the activities:

ar ed , AN ≡ aVPCH3OH ≪ a

LPCH3OH

aox, AN ≡ aVPH2O ≪ a

LPH2O

In order to simplify matters, the activity fraction of the oxidized species (water) isset to unity. With the abbreviation bBV of Eq. 5.1 on page 86 the complete formulais:

I VPAN = I✸0

AN · aαAN

oc,red · a1−α

AN

oc,ox · exp

−∆G‡rea, AN

R·

1

T− 1

T✸

·

expαAN ·bBV ·UAN

−a

VPCH3OH

aLPoc,CH3OH

· exp(1−αAN) ·bBV ·UAN

Now the half cell current or current density can be calculated:

IAN = (1−ΘVP) ·SAN · I LPAN + ΘVP ·SAN · I VP

AN ∧ jAN = IAN

Afc

With the data of three AspenPlus simulations with different anode transfer co-efficients, a optimum search for the best parameters jVPn and ΘVP✸ has beenperformed6. The transfer coefficients are comprised of three values: the valuefrom WANG & WANG αAN = 0.24 [143] complemented by one 0.02 smaller andone 0.02 larger value. The optimum for the two vapor phase formation pa-rameters jVPn and ΘVP✸ is the same for all three sets of simulation data:

6 The current density and the vapor covered surface fraction ΘVP are interdependent, hence, aiterative solution is necessary.


Fig. 5.11.: Comparison of three anode trans-fer coefficients at 333 K for the case of mixedphases. The circles show the data of [148].The optimum lies near αAN= 0.25.

the start of vapor phase coverage jVPn

is below 5 A/m2, and at current den-sities above 1000 A/m2 nearly all ac-tive surface is covered with vaporphase, ΘVP✸ > 0.95. This is in agree-ment with the visual impression ofexperiments performed with specialtest cells equipped with transparentanodic bipolar plates. Such experi-ments have been carried out by W.TAAMA in collaboration with P. ARGY-ROPOULOS [128] and by VOLKER HAR-BUSCH7.

These optimum vapor phase forma-tion parameters were used to calcu-late the resulting mixed phase polar-ization curves. Figures 5.10 and 5.11show the results: with transfer coeffi-cients αAN, similar to the value found in literature [143], the outcome is a perfectmatch with the measured data from [148].

5.4.3. Estimation of the Oxygen Drag from Cathode to Anode

Oxygen transport is a serious issue for PEM fuel cells, as the oxygen forms highlyoxidative hydrogen peroxide and hence degrade the membrane and the anode. Yet,for the anode polarization the oxygen drag is of minor relevance, as WEBER & NEW-MANN argue in [146]. Nevertheless, there are several works that model the oxygentransport through the membrane for PEM-HFC, e.g. [18, 86]; a comprehensive listof other related works can be found in [146]. To verify the statement of WEBER

& NEWMANN, the maximum amount of oxygen that may be transported to theanode through fully hydrated Nafion™1158 with a thickness of 1.43E-4 m is calcu-lated here. The following assumptions are made: the oxygen mole fraction at thecathodic membrane surface is equal to the dry air value, xO2

= 20.83 (cf. Tab. B.2on page 115), the temperature is T = 298 K. All oxygen that reaches the anodewill be entirely reduced near the membrane surface, hence, it is C2 = 0.0 mol/m3.ZHANG et al. published the following diffusivity properties of O2 in Nafion™117

7 VOLKER HARBUSCH — firm siqens, Munich, Germany — refered that his impression is: at elevatedcurrent densities there is a closed vapor film on the anode and he expects that there is only vaporpresent in the porous anode catalyst layer.

8 For the calculations the thinner membrane Nafion™115 is selected. The thickness of the drymembrane is 5 mil (0.127E-3 m), but in the DMFC the membrane is swollen.


at T = 303 K and p = 304.0 kPa: activation energy ∆GD = −27550.0 J/mol, diffu-sion coefficient D = 5.51E-10 m2/s. Whereas BERNARDI [18] published for ambientpressure and T = 353 K a diffusion coefficient of D = 1.2E-10 m2/s. For these pre-sumptions the current density equivalents (CDE) can be calculated as follows:

D(T ) = D✸ · exp

∆GD

R·

1

T− 1

T✸

(5.1)

ΦO2,m = −D ·CO2,2 −CO2,1

lm; [Φ] = mol

s · m2

jCDEO2

= ΦO2,m · zO2·F (zO2

= 4, cf. Eq. 2.13 on page 26)

The value for the oxygen concentration in the liquid phase was calculated withAspenPlus as CO2,2 = 0.255 mol/m3. Next, the CDE can be calculated according tothe two literature values of the diffusion coefficient for oxygen in Nafion™11X:

BERNARDI:

D(298.15K) = 1.2E-10m2

s· exp

−27550.0

8.3145

J·mol·K

J·mol·

1

298.15− 1

353.15

1

K

jCDEO2

= 6.77E-10m2

s· 0.255

1.43E-04

mol

m3·m·4 ·96485.338

A·s

mol

= 1.47E-2A

m2

ZHANG:

D(298.15K) = 5.51E-10m2

s· exp

−3313.5 ·

1

298.15− 1

303.15

jCDEO2

= 6.62E-10m2

s·6.89E+8

A·s

m4

= 3.16E-1A

m2

At elevated temperatures the solubility of oxygen in water is lower (cf. Tab. B.9 onpage 117), but the increase of the diffusivity with temperature is strong enough toovercompensate this, for the higher diffusivity published by ZHANG the CDE at353 K is: jCDE

O2= 0.6 A/m2. These calculations assume the worst-case scenario, with

no water flow from anode to cathode and no oxygen consumption in the cathode,hence, the real oxygen flows are expected to be below these estimations.

5.5. Electroosmotic Drag Coefficient

REN et al. published measured data of the temperature dependent water drag co-efficient for the electroosmotic drag coefficient for water ξeo through Nafion™117[107]. He found that the electroosmotic transport is increasing with temperature

5.6. Comparison of the Modeling Results to Test Data 95

(cf. Fig. 2.15 on page 27). For the proprietary membrane used by WILLIAMS [148]no data was given for ξeo . However, it is a known fact that both electrolytes aredominated by water molecules9. Hence, it is to be expected that the drag coeffi-cients do not vary to much from each other. It was interesting to determine if it ispossible to ascertain a meaningful value by means of fitting for the electroosmoticwater drag coefficient inside the H2SO4-doped porous gold plated PVC-membrane.The results are encouraging; the values lie near to the values for Nafion™ and theyshow a comparable temperature dependency, cf. Fig. 4.1 on page 64.

5.6. Comparison of the Modeling Results to Test Data

During the final phase of the work on this thesis own electrode polarization mea-surements on a full cell against a reversible hydrogen electrode (RHE)10 have beenaccomplished. The polarization data were validated by additional measurementsagainst a mercury sulfate electrode. The RHE was coupled with the PEM viaa Nafion™115-strip dipping in 300 K warm 0.5 M H2SO4 solution. The electricalimpedance was measured with a potentiostat11 with variable excitation frequency;hence, it was possible to measure at a zero phase angle. Of course, only a quali-tative comparison with the simulation results of Fig. 4.13 and 4.14 on page 72 ispossible; the simulation was done with the parameters for the WILLIAMS DMFCand the membrane coefficients of a Nafion™117-membrane.

Taking this into account leads to the following conclusion: the simulation predictsa steeper decline of the cell polarization for higher concentrations, which is causedby the higher crossover. At higher current densities the bad effect of the elevatedcrossover lessens. Due to the lower methanol diffusivity of the WILLIAMS-PVC-membrane in comparison to a Nafion™115-membrane, this leads to a crossingof the cell polarization curves (Fig. 4.14 on page 72, left). The simulation for theNafion™117 shows the same tendency (Fig. 4.14, right). However, this changes forthe Nafion™115-membrane with an increase of above 20% methanol diffusivity inrelation to a Nafion™117-membrane12. The crossover stays high enough at tallercurrent densities to avoid a positive stagnation of the cathode polarization for the3 M curve in Fig. 5.12.

9 In a liquid fed DMFC the electrolyte Nafion™ is fully humidified.10 RHE: HydroFlex from manufacturer Gaskatel, Germany. The dynamic hydrogen electrode (DHE)

HydroFlex has been transformed into a RHE by extension with an external hydrogen supply.11 Potentiostat: IM6 from the manufacturer ZAHNER-Elektrik, Germany12 Nafion™117 is 28.6% thicker than Nafion™115. Hence, in this work a 28±5% increase of diffusiv-

ity is expected. This is in accordance e.g. to [151], but in contrast to[112], in which a decrease of6.3% is published.


Fig. 5.12. Fig. 5.13.

Fig. 5.12, 5.13: Galvanostatically measured polarization curves of a DMFC single cell:geometric area, Afc = 0.0025m2; catalyst loading (unsupported catalyst from JohnsonMatthey), mcat , AN = 0.059kg/m2 = 5.9mg/cm2 Pt-Ru; mcat , CA = 0.067kg/m2 Pt; diffusionlayer for anode and cathode, SGL 10BB; Nafion™115-membrane; mean electricalcell resistance at zero phase angle, Rfc = 6.5 mΩ. Operating conditions: temperature,Tfc = 333K ; atmospheric pressure on anode and cathode; methanol concentrationvaried from 0.5 up to 3.0 M CH3OH; air at ambient humidity; nominal stoichiometriccoefficients at 1000 A/m2, λn

AN = 7, λnCA = 5.

6. SUMMARY AND CONCLUSIONS

6.1. Summary

The present thesis describes a comprehensive, physicochemical model of a DMFC.The model is based on individual Butler-Volmer equations for the three key ki-netic reactions on both electrodes: a) methanol oxidation, b) oxygen reductionand c) crossover reaction. Mass transport effects are simulated for the five innerfunctional layers (ADL – CDL, cf. Fig. 2.2 on page 10) of a DMFC. Beside diffusiontransport, the convective transport is computed a) through the diffusion layers,ADL & CDL, by Darcy’s law, cf. Eq. 2.1 on page 17, and b) through the PEM bythe electroosmotic drag, cf. Eq. 2.15 on page 27. The variation of temperatureand methanol inlet concentration is enabled by the use of special variants of theButler-Volmer equation (cf. chap. D.2 on page 123) and temperature laws for theimportant kinetic and transport parameters, such as transfer coefficients or theelectroosmotic drag coefficient. The phenomenons of two phase flow in the anode,cathode flooding and cathode mixed potential are considered. The ionic resistanceof the membrane is calculated iteratively and responds upon the polarization ofeach electrode, as shown in Fig. 4.17 on page 75. The electronic resistance is ofminor influence on the single-cell performance [18] and hence, not simulated. Theprocesses are defined in a one-equation or zero-dimensional mode (cf. chap. 1.3.3on page 4). The model is implemented inside a chemical engineering software; thenumerical realization is documented with the example of the proprietary softwareAspenPlus. The good agreement of the simulation with the polarization data fromliterature [148] is shown in Figs. 6.1 & 6.2. The conformity of the model with ownmeasurements is shown in chap. 5.6.

6.2. Conclusions

6.2.1. Suitability of the Butler-Volmer Modeling Approach

It is known that the widespread modeling approach, based on a Tafel or Butler-Volmer equation for the cathode coupled with a kinetic model for the anode (cf.chap. 5.3 on page 84), cannot predict the polarization of a DMFC over the wholerange of meaningful cell voltages. However, this is not a problem with the Butler-Volmer approach itself. The work shows that the analytical Butler-Volmer equationis not only appropriate to simulate the kinetics, but that, if combined with model

98 6. Summary and Conclusions

Fig. 6.1. Fig. 6.2.

Fig. 6.1, 6.2: Comparison of the polarization data from WILLIAMS [148] for the tempera-tures 298 and 333 K (symbols) with the simulation results (lines).Fig. 6.1: The model for the anode contains a special version of the Butler-Volmer equa-tion with concentration and temperature sensitive exchange current density termsand a linear law for vapor phase evolution inside the porous electrode structure.Fig. 6.2: The model for the cathode accounts for the mixed potential and the cathodeflooding.

statements for the other key processes inside a DMFC beside the actual kinetics,it is suitable for a comprehensive model of the cell polarization. This is true overthe entire range of realistic fuel cell voltages at varied temperatures and methanolconcentrations (cf. Fig. 6.1 and Fig. 6.2).

To set up a proper model to describe the characteristic curve of a DMFC, thefollowing phenomena have to be accounted for:

• Anode and cathode kinetics need to be described by separate Butler-Volmerequations.

• A fraction of the electrochemically active anode surface is covered by a gasphase.

• Due to methanol crossover, the cathode polarization is formed by a mixedpotential.

• The transfer coefficients α for both electrodes may be temperature depen-dent [114].

• The exchange current density I is temperature and concentration dependent.

• The adsorption mechanism on the cathode is changing at the transitionvoltage from Temkin to Langmuir adsorption (Utr = 0.8 V vs. SHE).

• Transport coefficients for diffusion and electroosmotic drag — D, ξeo — aretemperature dependent.

• The ohmic loss induced by the ionic resistivity of the PEM is split into ananodic and a cathodic fraction.

6.2. Conclusions 99

6.2.2. Key Elements of the Model

Introduction of a Temperature and Concentration Invariant ExchangeCurrent Density Constant

A new variant of the Butler-Volmer equation was developed that incorporates atrinominal definition for the active surface related exchange current density I . Thethree parts are: a) an exchange current density constant I 0, b) a concentrationterm and c) a temperature term. The derivation of the new invented variant of theButler-Volmer equation is documented in chap. D.2 on page 123.

Influence of the Anodic Carbon Dioxide Evolution

The anode reaction, Eq. 2.2 on page 17, generates carbon dioxide and the reactionextent is coupled via the First Faraday’s law with the electrical current, hence,grows linearly with the current or current density. This CO2 evolution causes theformation of a carbon dioxide vapor phase inside the porous anode layer and leadsto a rising fraction of electrochemically active surface covered by a fully humidifiedwater-methanol-carbon-dioxide mixture. As the chemical activity of methanolin the vapor phase is less than 20 % of the chemical activity in the liquid phasethe kinetics at both surface fractions need to be described by separate Butler-Volmer equations. Calculations show that the hypothesis of a linear influence ofcarbon dioxide evolution on the fraction of vapor phase covered active catalystsurface results in a perfect match between simulation results and the measuredpolarization data found in [148] (cf. chap. 5.4.2 on page 90).

Mixed Potential of the Cathode

Two electrochemical reactions take place on the cathode, the oxygen reductionand the parasitic methanol oxidation. Accordingly, one Butler-Volmer equation isnot sufficient to simulate the mixed potential. A second Butler-Volmer equationfor the crossover reaction is added to the cathode model to calculate the surfacefraction of the cathode needed to oxidize the crossover flow of methanol. Thisenables the model to predict the loss of the cathode-OCP due to crossover andhence can model the effect of changing methanol transport properties of themembrane on polarization.


Temperature Dependent Transfer Coefficients

The simulation results confirm the observation of the temperature dependencyof the transfer coefficients. The fit results for the temperature dependency of thecathode transfer coefficient αCA are in agreement with the literature [99, 114]. Thetemperature influence on the αAN is less pronounced.

Implementation of Langmuir Adsorption for the Oxygen Reduction

Own measurements show that the cathode mixed potential can reach values belowthe adsorption transition potential Utr = 0.8 V vs. SHE, even for DMFC with rela-tively thick polyperfluorosulfonic acid membranes like Nafion™115 and cathodeswith a high catalyst loading above 0.05 kg/m2. The Langmuir adsorption startsfor methanol concentrations of 2 M and a temperature of 333 K at cell voltagesbelow 0.61 V (cf. Tab. E.3 on page 132) and for a concentration of 1 M CH3OHbelow 0.39 V (cf. Tab. E.4 on page 133). Therefore, to simulate DMFC cathodes athigher crossover rates the transition from Temkin to Langmuir adsorption needsto be implemented.

Temperature Dependency of Transport Coefficients

The necessity of temperature dependent diffusion and electroosmotic drag co-efficients increases with temperature and methanol concentration. A correcttemperature law for the methanol diffusion coefficient is more relevant than it isfor the electroosmotic drag coefficient (cf. Fig. 4.12 on page 71).

Split of the Ionic Resistivity Losses

The polarization gradients of the two electrodes differ considerably, as can be seenin Fig. 2.1 on page 9. Due to the ionic resistivity of the MEA, this leads at a given(common) current loss to different values of resistivity induced polarization losses,as can be seen in Fig. 4.17 on page 75. Hence, it is important to split the ionicloss in the ratio: outer electrode polarization to cell polarization (cf. Eq. 2.28 onpage 43).

6.2. Conclusions 101

6.2.3. Cell Voltage as Independent Variable

Most of the other fuel cell simulation works use the current Ifc or current densityjfc as the independent variable. Hence, these models can not work with the

Butler-Volmer equation1; they use the Tafel equation2 or other empirical equationsinstead. The use of the Tafel equation has only an effect near the open (half-)cell potential, therefore, in most of the cases — especially for PEM-HFC — theloss of information seems to be marginal. The main difference lies in the logicof these models: if the current is the free variable, it is not crucial to set up anaccurate model for the rest potentials of the electrodes to determine the transportrelevant values (e.g. concentration gradients). The consumption and productionof substances is already determined by the cell current via the First Faraday’slaw (cf. Eq. G.10 on page 146) and need not to be predicted on the basis ofsimulations. This may simplify matters, but the downside is that inaccuraciesof the kinetic models may remain undetected. Another aspect is the numericalrobustness or the ease of convergence: the potential based Butler-Volmer approachis much more vulnerable to oscillation or other numerical obstacles (cf. chap. 3.4on page 50). The work shows that the analytical potential based Butler-Volmeransatz is appropriate to simulate the electrochemistry of a DMFC and that thenumerical obstacles are manageable.

6.2.4. Validation with own Cell Measurements

To prove the model the polarization data of WILLIAMS [148] have been successfullysimulated (cf. chap. 4.2 to 4.4.1 on pages 65–67). Additionally, own measurementswith a Nafion™115-based DMFC full cell have been carried out at the Centerfor Solar Energy and Hydrogen Research, ZSW. This validation shows, that thequalitative model predictions for the influence of increasing crossover due tohigher methanol concentration is in accordance with a real DMFC (cf. chap. 5.6on page 95).

1 It is not possible to transpose the Butler-Volmer eq. to make it equal to the polarization U.2 TAFEL invented his equation in 1905 as an empirical description of his experiments.


ACKNOWLEDGEMENTS

I thank Dr. LUDWIG JÖRISSEN for the opportunity to accomplish the scientific workon this interesting topic and the supervision of the doctoral thesis.

The experimental work at the ZSW had been accomplished by the student ap-prentice UTE RIEK under the supervision of Dr. VIKTOR GOGEL.

MATTHIAS MESSERSCHMIDT, Dr. LUDWIG JÖRISSEN and Dr. JOACHIM SCHOLTA, allZSW, provided some valuable literature.

The proofreading was done by KARIN HINSON, North Las Vegas, USA,Dr. SUSAN FISCHER and Dr. LUDWIG JÖRISSEN, both Ulm, Germany.

— Thank you very much for your contribution!

Appendix

A. NOMENCLATURE

The notation of physical values is equal to the definition in [34]:

X = {X } · [X ]

variable = numerical value · unit

The symbols are adopted mainly from DIN e.V.[34, 36, 37], VDI e.V.[136] and IU-PAC [60], but some special values are taken from electrochemical literature suchas [115].

Unless otherwise indicated, free Gibbs enthalpy refers to free reaction energyreferenced in the amount of substance at constant standard pressure p✷ andreference temperature T✸:

∆G T=T ✸

p=p✷

=∆G ; [∆G] = J/mol

In electrochemical books the reference temperature T✸ frequently varies from thestandard temperature, which is T✷= 273.15 K (0.0°C)[35].

Table of Symbols

Variable Description Unit1

Roman symbols

a chemical activity 1

cp (specific) mass related heat capacity J/(kg·K)

cp cp related to the amount of substance (AOS) J/(mol·K)

kB Boltzmann constant J/K

Table A.1.: Symbols.

1 Most of the units are base units of the “International System of Units” as documented in [33],but some are derived from that to be confirm with the common usage in other electrochemicalpublications.

106 A. Nomenclature

Continuation of tabular Symbols

Variable Description Unitk pre-exponential factor m / smκek electrokinetic permeability of the membrane m2

mκhp hydraulic permeability of the membrane m2

l length/distance m

m mass kg

ni amount of substance i per time mol/s

n conversion ratio 1

p pressure N/m2= Pa= 10-5bar

�� velocity m/s

v mass specific volume m3/kg

wi mass fraction 1

xi mole fraction of substance i , oriented on themixture 1

z, z+,z−

valency, positive or negative elementarycharge of an ion [37] 1

z elementary charge exchanged in the RDS 1

A geometric area m2 = 104cm2

C concentration in SI base units mol/m3

𝒞 concentration in M = mol/dm3= kmol/m3

D diffusion coefficient m2/s

E energy J = kg·m2/s2

F ∝ scaling factor 1

∆G change of Gibbs Free energy related to AOS J/mol

∆H change of enthalpy related to AOS J/mol

I amperage A = C/s

J area-related amperage[36] A/m2 = 0.1· mA/cm2

JCO methanol crossover expressed as a parasiticcurrent density[37]

A/m2

M molar weight kg/mol

Q electric charge C

R ohmic resistance Ω = V/A = J/(A2·s)

∆S change of entropy related to AOS J/(mol·K)

S catalyst surface m2

S Volume specific surface m2/m3 = m-1

U✷ standard equilibrium voltage between SHEand another half cell (EMF) V


107

Continuation of tabular Symbols

Variable Description UnitUtheo ideal theoretical cell voltage, see Eq. G.16 on

page 154 V = J/(A·s)

U Galvani or inner potential V = J/C

U polarization V = J/(A·s)

P power W = J/s = kg·m2/s3

Greek symbols

α charge-transfer coefficient 1

γ activity coefficient 1

ϵpor e effective porosity [97] 1

ηut Faraday or utilization efficiency 1

ηid ideal or thermodynamic efficiency 1

ηU voltage efficiency 1

ηV dynamic viscosity [60] N·s/m2 = kg/(m·s)

µ chemical potential J/mol

µ∗ electrochemical potential J/mol

ν stoichiometry coefficient, ν> 0 for products 1

ξeo electroosmotic drag coefficient 1

ξr extent of reaction mol

ϑ celsius temperature[37] °C

Θ fraction 1

λn nominal stoichiometric factor 1

ρ molar density mol/m3 = 10-3mol/dm3

σ conductivity S/m = A/(V·m) = A2·s3/(kg·m3)

ϕTs

Tafel slope V = kg·m2/(A·s3)

χ surface potential V = J/(A·s)

Φ flow of substance mol/s

Φ flow density mol/(m2·s)

ψα VOLTA or outer potential V = J/C


108 A. Nomenclature

Table of Super- and Subscripts

Abbreviation Description

Superscript symbols

Xn X is indicating the begin of a specific region

X † norm of 1.0 of the value X

X ‡ activated complex of the RDS[7]

X✷ standard value[35, 37]

X� value at equilibrium

XΠ normalized to unity

X✸ value at reference condition

X ∝ scaled variable

X⋆ special value or form of X

X G X of inner part of something

Subscripts and other abbreviations (some also used as superscripts)

0 or ✷ standard [35, 37]

@ at

ads adsorbed

aos amount of substance

ABP anodic bipolar plate

ACL anodic catalytic layer

ADL anodic diffusion layer

AMI anode-membrane interface

AN anode

BP bipolar plate

BV Butler-Volmer equation

CA cathode

cat catalyst

ch chemical

cmi cathode-membrane interface

CE current equivalent

CO crossover flow, crossover reaction /

cathodic methanol oxidation

conc concentrated, pure

const constant

cp cell pitch

Table A.2.: Superscripts, subscripts, abbreviations.

109

Continuation of tabular Superscripts, subscripts, abbreviations

Abbreviation Descriptioncr common reference

CBP cathodic bipolar plate

CCL cathodic catalytic layer

CDL cathodic diffusion layer

CDE current density equivalent

CFD computational fluid dynamics (software)

de disk electrode

diss dissolved

DL diffusion layer

D diffusion

eff effective

ec electron conductivity

el electrical

elst electrostatic

eo electroosmotic

et electron transfer

EMV electromotive force

fc fuel cell

fl flooding

Fo Faradaically oxidized

GALV GALVANI

gro gross

GDL gas-diffusion-layer

GUI graphical user interface

hp hydraulic permeability

HHV higher heating value

id ideal

in incoming

IUPAC International Union of Pure and AppliedChemistry

LB lower bound (minimum)

lim limiting

LP liquid phase

ls least squares

m membrane

mp mixed potential

MEA membrane electrode assembly


110 A. Nomenclature


Abbreviation Descriptionnom nominal

oc open circuit

op over-potential

out outgoing

ox oxidation

OCV open circuit voltage

OCP open circuit potential

OR oxygen reduction

phys physical

pl polarization

por pore

PFS process flowsheet

PSA perfluorosulfonic acid (Nafion™)

PV predicted / calculated value

Q source, sink

r rest

rea reaction

red reduction

rev reversible

R ion or electrical resistivity

RDS rate-determining step

s surface

sf superficial

S storage

SHE standard hydrogen electrode

SV sampled value

te thermodynamic equilibrium

theo theoretical

th thermal

thn thermo neutral

tot total

TK Temkin region

ut (fuel) utilization

UB upper bound (maximum)

VP vapor phase

VOLT VOLTA

wv working value


111


Abbreviation Descriptionφ transport


Table of Operands and Notation

Operands Description

X := X is set / defined to

exp x ex

x, y, z space coordinates x, y, z

X flow of X

x small letter: X related to mass

X related to area

X related to volume

X related to the amount of substance[136]˙X area related flow of X

XXX matrix

X constant X

∥X ∥ absolute valueφX transport term XQ X source, sink term XS X storage term X

Table A.3.: Operands and notation.

B. PARAMETER VALUES

Parameter Description Value Unit

CH+, N 11X proton concentration / fixedcharge site concentration ofNafion™ 11X [18]

1200.0 mol/m3

DCH3OH diffusion coefficient of CH3OH inH2O [52]

2.8·10−9 m2/s

DH+,N117353.15K

proton diffusion coefficient inN117 (Nafion™)[18]

4.5·10−9 m2/s

e0 elementary charge is the absolutevalue of the electric charge of anelectron[54, p.7]

16021.773·10−15 C = A · s

F Faraday constant[52] 96498.0 C/mol = A·s/mol

F Faraday constant, CODATA-value[29] (selected)

96485.3383 C/mol

∆GcrCH3OH Gibbs Free energy for the metha-

nol oxidation at common refer-ence conditions crT = 298.15 Kand p0 = 101 325.0 Pa (see Eq. 2.2on page 17)[140, p. 27]

−702.5 ·103 J/mol

∆H crR,CH3OH reaction enthalpy for the metha-

nol oxidation at T cr = 298.15 Kand p0 = 101 325.0 Pa (see Eq. 2.2on page 17)[140, p. 27]

−726.6 ·103 J/mol

kB Boltzmann constant [29],kB =R/NA

13806.50·10−27 J/K

p✷ol d standard pressure[35] (used be-

fore 1980, 101 325.0 Pa = 1 atm)101325.0 Pa = 10−5 bar

Table B.1.: Parameter values.

114 B. Parameter Values

Continuation of tabular Parameter values

Parameter Description Value Unit

p✷ standard pressure after IUPAC[12, 60]

1.0 ·105 Pa = 10−5 bar

NA Avogadro’s number[29] 602214.15·1018 1/mol

R universal gas constant[29] 8.314472 J/(mol·K)

T✷ standard temperature[35] 273.15 K

crUCH3OH theoretical open circuit cellvoltage of the galvanic cellfor the methanol oxidation atcommon reference conditionsT cr = 298.15 K and pcr = p0 =101 325.0 Pa (see Eq. 2.2 onpage 17)[140, p. 27]

1.214 V

crηid thermodynamic or ideal effi-ciency of energy conversion forthe methanol oxidation in a gal-vanic cell at T cr = 298.15 K andp0 = 101 325.0 Pa (see Eq. 2.2 onpage 17)[140, p. 27]

96.7% 1

σ✷N 117 Electrical conductivity of

Nafion 117, wet membrane,ambient temperature[17],

7.0 S/m

ξeo total (electroosmotic) waterdrag coefficient for liquid equi-librated perfluorosulfonic acidmembranes Nafion™ N117 atT ≈ 303.15 K [104, 154]

2.5 mol H2O/mol H+

Table B.1.: Parameter values.

B.1. Composition of Air 115

B.1. Composition of Air

percent byvolume1

percent byweight

mole weight percent byamount ofsubstance2

Nitrogen 78.1 75.51 28.0134 78.06156

Oxygen 20.93 23.01 31.998 20.82535

Argon 0.9325 1.286 39.948 0.93228

Carbon dioxide 0.03 0.04 44.01 0.02632

Neon 0.0018 0.0012 20.183 0.00172

Helium 0.0005 0.00007 4.0026 0.00051

Krypton 0.0001 0.0003 83.8 0.00010

Hydrogen 0.00005 0.000004 2.016 0.00006

Remain 0.00505 0.152426 29.021 0.15211

Sum 100.0% 100.0% 28.96 100.0%

Table B.2.: Actual average chemical composition of air.

percent byweight

mole weight percent byamount ofsubstance

Nitrogen 75.88046 28.0134 78.44454

Oxygen 23.01000 31.998 20.82535

Carbon dioxide 1.10954 44.01 0.73011

Sum 100.0% 28.96 100.0%

Table B.3.: Simplified chemical composition of air.

1 Conditions under the terms of DIN ISO 2533 (Dez. 1979)[109]:

101 325.0 Pa; 288.15 K (15°C)2 Calculated from molar weight of air, its constituents and the percent by weight; the data

from [8, 135] differ slightly from that.

116 B. Parameter Values

B.2. Activities in Methanol-Water Mixtures

[𝒞CH3OH] = mol/dm3 : 0.05 0.1 0.5 1.0 2.0 3.0 6.0

[aCH3OH] = 1 : 0.0015 0.0031 0.0152 0.0306 0.0614 0.0924 0.1867

[aH2O] = 1 : 0.9991 0.9981 0.9909 0.9816 0.9628 0.9433 0.8814

Table B.4.: Activities in water-methanol mixtures calculated with AspenPlus.

B.3. Diffusion Coefficient of Carbon Dioxide in Water

[T] = K : 293 298 303 313 333 353 368

[D] = m2/s (measured) : 1.76 1.94 2.20 2.93 4.38 6.58 8.20

[D] = m2/s (fitted) : 1.77 2.01 2.27 2.87 4.40 6.42 8.30

Table B.5.: Diffusivity of carbon dioxide in water. Measured data is digitized from [131].Fitting of the data with an Arrhenius like equation (cf. Eq. 5.1 on page 94) results in an

activation energy of ∆GD

CO2=−18500.5 J/mol

B.4. Water Drag Coefficient

Ionomer Water content Temp. Drag

H2O/SO3H− K Coefficient

T.A. ZAWODZINSKI[155] Nafion 117 22 303.15 2.0 - 2.9

XIAOMING REN[104, 154] Nafion 117 22 303.15 2.5

Table B.6.: Water drag coefficients of Nafion™.

B.5. Specific Electric Conductivity of Ionomer

Ionomer Water content Temp. ConductivityH2O/SO3H− K S/m

T. A. ZAWODZINSKI[155] Nafion 117 22 303.15 10.0P. D. BEATTIE[17] Nafion 117 22 303.15 7.0

Table B.7.: Specific electric conductivity of Nafion™.

B.6. Solubility of Oxygen in Water 117

Temperature Dependence of Ionic Conductivity of Nafion™117

[T] = K 302.93 313.28 323.15 333.51 343.28 353.67 363.57

[σ] = S/m 10.01 10.22 10.93 13.12 14.60 17.63 18.92

Table B.8.: Temperature dependence of ionic conductivity of Nafion™117, data is takenfrom digitized plot in [155].

B.6. Solubility of Oxygen in Water

[T] = K 298.15 303.15 313.15 323.15 333.15 353.15

[CO2] = mol/m3 0.255 0.233 0.196 0.167 0.141 0.086

[xO2] = 1 4.63E-6 4.63E-6 4.63E-6 4.63E-6 4.63E-6 4.63E-6

Table B.9.: Water solubility of oxygen from air in water at ambient pressure as calcu-lated by AspenPlus.

C. DETAILS ON IMPLEMENTATION

C.1. Set-Up of the File Based Database to ImproveConvergence of Outer DesignSpec

As can be seen in Fig. 3.8 on page 50 the iterative calculation of the cell polariza-tion is performed in three interlaced Design Specs, to increase the convergenceof the outer loop a data base has been constructed. Dependent on the mostimportant values that influence the reaction rate on the anode and the passagecharacteristics of the diffusion layer and the membrane, a directory and a filename convention have been defined: all 55 input parameters that define a com-plete assembly of a DMFC are stored in a text file and each assembly has its ownID-number; subsequently, the assembly ID-number results in the correspondingupper directory name of 001\ .. 999\. The syntax of database file names is:1.) ID-# (3 chars), 2.) Geometric electrode area (A, 7 chars), 3.) Cell tempera-ture (T, 5 chars), 4.) Exchange current density of the anode reaction (I0, 13chars), 5.) Transition coefficient αAN (TC, 7 chars), 6.) Inlet methanol concentra-tion (CVME, 10 chars), 7.) Total anode inlet flow (FL, 13 chars), 8.) Electroosmoticdrag coefficient (EO, 7 chars), 9.) Diffusion coefficient for methanol (DF, 13 chars),10.) Convective flow coefficient of the anodic diffusion layer (CF, 13 chars),11.) Cathode flooding parameters (FD, 11 chars).Together with the file extention “.csv” and 10 delimiter characters “_” the resultsare file names with the length of 122 characters1. The structure of these data basefiles is: 80 columns and three rows.Each column is reserved for one specific cell voltage in the range of [0.20, 0.21,. . . , 0.99]. The second row contains the best unscaled x-Value xCH3OH

ADL2XADL2X-E1

of the

last run and the third line holds the unscaled y or quality QDS-ADL02. Prior to theentrance in the outer Design Spec DS-ADL02 the file name pattern is built andthe existence of the file is checked. If the file does not exist, an initial file will beset up with unmistakable initial values for the unscaled x and y values. If the filealready exists and realistic values of a previous run are available, the boundariesand the start value for the manipulated value will be set contingent upon thequality. The better (lower) the modulus of the quality, the closer the boundaries. Ifno data of a previous run is available the boundaries are set as wide as necessary

1 A file name example is given here:001_A=01.00_T=335_I0=0.1650E+01_TC=0.44_CVME=01.00_FL=0.5200E-03_EO=2.30_DF=0.7125E-10_CF=0.1950E-05_

FD=050#0.50.CSV

120 C. Details on Implementation

to ensure that the optimum lies in between, this variant is refered to as variant a)in chap. 3.4.3 on page 53.

C.2. Influence of Scaling Factors on NumberRepresentation

As discussed in chap. 3.5.2 on page 56 the scaling of the surface fraction ΘOR

increases the convergence of the iterative calculation of the cathode polarization.The reason is the different spacing dependent on the order of magnitute of a binaryrepresented. In the simulation, the standard number format is double precision,these numbers are 8-byte binary represented. Fig. C.1 shows the influence ofthe order of magnitude on the spacing double precision numbers: the larger thenumber, the wider the gap to the next floating point number.

(1D12-SPACING(1D12)): 999999999999.99987792968750(1D13-SPACING(1D13)): 9999999999999.9980468750000(1D14-SPACING(1D14)): 99999999999999.984375000000(1D15-SPACING(1D15)): 999999999999999.87500000000(1D16-SPACING(1D16)): 9999999999999998.0000000000(1D17-SPACING(1D17)): 99999999999999984.000000000(1D18-SPACING(1D18)): 999999999999999872.00000000(1D19-SPACING(1D19)): 9999999999999997952.0000000(1D20-SPACING(1D20)): 99999999999999983616.000000

Fig. C.1.: Spacing of double precision values (8 byte) in ascending order of magnitude.SPACING(x) is a function of Fortran and it computes the space to the next neighbor ofa given x.

In Fig. C.2 on the next page the numerical result of the computation with a variedscaling factor is shown. Two conclusions can be drawn from this: a) the scalingfactor has a helpful effect and b), the effect is slight.

C.2. Influence of Scaling Factors on Number Representation 121

Input values: SA = 1.0D0 ; Calculation in two steps:JSACO = 0.82685D+23; FRSAOR = (SACA * JSACO - ICO)/(SACA * JSACO)ICO = 532.59D0 ; FRSAOR = SF * FRSAOR

Output of variation of scale factor: SF = [1D00 .. 1D25]:SF=10^00 -> FRSAOR= 1.0000000000000; SF=10^13 -> FRSAOR= 10000000000000.0000000000000SF=10^01 -> FRSAOR= 10.000000000000; SF=10^14 -> FRSAOR= 100000000000000.000000000000SF=10^02 -> FRSAOR= 100.00000000000; SF=10^15 -> FRSAOR= 1000000000000000.00000000000SF=10^03 -> FRSAOR= 1000.0000000000; SF=10^16 -> FRSAOR= 10000000000000000.0000000000SF=10^04 -> FRSAOR= 10000.000000000; SF=10^17 -> FRSAOR= 100000000000000000.000000000SF=10^05 -> FRSAOR= 100000.00000000; SF=10^18 -> FRSAOR= 1000000000000000000.00000000SF=10^06 -> FRSAOR= 1000000.0000000; SF=10^19 -> FRSAOR= 10000000000000000000.0000000SF=10^07 -> FRSAOR= 10000000.000000; SF=10^20 -> FRSAOR= 100000000000000000000.000000SF=10^08 -> FRSAOR= 100000000.00000; SF=10^21 -> FRSAOR= 1000000000000000000000.00000SF=10^09 -> FRSAOR= 1000000000.0000; SF=10^22 -> FRSAOR= 10000000000000000000000.0000SF=10^10 -> FRSAOR= 10000000000.000; SF=10^23 -> FRSAOR= 99999999999999991611392.000SF=10^11 -> FRSAOR= 100000000000.00; SF=10^24 -> FRSAOR= 999999999999999983222784.00SF=10^12 -> FRSAOR= 1000000000000.0; SF=10^25 -> FRSAOR= 10000000000000000905969664.0

Fig. C.2.: Influence of scaling factor F ∝ΘOR

= SF on the precision of the up-scaled surfacefraction Θ∝

OR = FRSAOR. The variables are: total cathode catalyst surface SA = S✸CA,

electrochemical active cathode catalyst surface SACA = SgCA, surface related current

density of the crossover reaction JSACO = I BVCO, crossover current equivalent ICO = I CE

CO.Note that the Fortran code:FRSAOR = SF*(SACA * JSACO - ICO)/(SACA * JSACO)is less precise, than performing the calculation in two steps, multiplying by SF in thesecond step.

D. AUXILIARY CALCULATIONS

D.1. Calculation of a Simplified Chemical Compositionof Air

The goal is to have the same amount of oxygen in the mixture and to have simularphysical properties of the simplified chemical composition of dry air. To simplifymatters the components are reduced to three:

xN2+ xO2

+ xCO2= 1 ; (xO2

= 20.82535)

≡ xO2= 1− xN2

− xCO2xN2

MN2+ xO2

MO2+ xCO2

MCO2= MAir ; (MAir = 28.96g)

≡ xN2MN2

+ (1− xN2− xCO2

)MO2

+xCO2MCO2

= MAir

≡ xN2=

xCO2(xO2

MO2− xCO2

MCO2)+ MAir − MO2

MN2− MO2

numeric solution: xCO2= 0.73011%⇒ xN2

= 78.44454%

D.2. Algebraic Transformation of Butler-VolmerEquation

In this chapter the derivation of the following specific form of the Butler-Volmerequation for reduction reactions is documented:

I red = I 0red ·a(1−α)

oc,ox · aαoc,red · exp

−∆G‡

rea

R·

1

T− 1

T✸

·

aox

aoc,ox· exp

−α · z‡ ·F ·UR ·T

− ared

aoc,red· exp

(1−α) · z‡ ·F ·U

R ·T

(D.1)

124 D. Auxiliary Calculations

The equation for oxidation reactions is equivalent and is given in Eq. 2.10 onpage 20.

WÖHR [150] used in his work a simular variant that can be used for fixed concen-trations and mutable temperatures; see Eq. D.15 on page 130. Nevertheless, hetook the sign convention for electrolysis instead of fuel cells (cf. ATKINS [7]) anddoes not state how this variation is derived from the conventional form of theButler-Volmer equation Eq. G.4 on page 140. SCOTT used in his work [117] thesame term for the temperature dependency of the exchange current density I , buthe made no comment about the derivation, either.

The following deductions are mainly taken from BARD and FAULKNER [12]:

In contrast to the Tafel equation the Butler-Volmer ansatz includes the forwardand the backward reaction in each electrode:

oxidised reactant Areduction−−−−−−−−−−→←−−−−−−−

oxidationreduced product B ⇔ Aox

red−−−−−→←−−−ox

Bred

Hence, the outer electrode current density is the sum of two inner current densitiesfor back and forth reaction:

I = I red + I ox

= zred ·F ·kred ·Cox + zox ·F ·kox ·Cred (D.2)

With the activation Gibbs function [7, 12, 80] the rate constants kox and kred can bedefined as:

kox = kox · exp

−∆G‡

ox

R ·T

kred = kred · exp

−∆G‡red

R ·T

In the theory of the activation Gibbs function the e-function expresses the prob-ability of surmounting the reaction barrier and k is called the frequency [12]or pre-exponential factor [80] and is related to the frequency of attempts of thespecies to react with each other.

The gross activation energy ∆G‡gro depends on two constituent parts.

A) the energy to move the reacting species1 against the electrostatic field into thereaction zone:

Eox,elst = z‡ox ·F · (1−α) ·∆φ

Er ed ,elst = z‡red ·F ·α ·∆φ

B) the free Gibbs activation energy for the reaction itself:∆G‡

ox,rea and ∆G‡r ed ,rea

1 To differentiate between the electrodes transfered in the gross reaction z and the elementarycharge transfered in the RDS we introduce for the latter the modified symbol z‡.

D.2. Algebraic Transformation of Butler-Volmer Equation 125

That results in the gross activation energy ∆G‡gro for the oxidation reaction:

∆G‡ox,gro = ∆G‡

ox,rea + Eox,elst

= ∆G‡ox,rea + z‡

ox ·F · (1−α) ·∆φ

and for the reduction reaction:

∆G‡r ed ,gro = ∆G‡

r ed ,rea + Er ed ,elst

= ∆G‡r ed ,rea + z‡

red ·F ·α ·∆φ

Next, the rate constants are defined as:

kox = kox ·exp

−(∆G‡

ox,rea + z‡ox ·F · (1−α) ·∆φ)

R ·T

= kox ·exp

−∆G‡

ox,rea

R ·T

· exp

−z‡

ox ·F · (1−α) ·∆φR ·T

kred = kred ·exp

−(∆G‡r ed ,rea + z‡

red ·F ·α ·∆φ)

R ·T

= kred ·exp

−∆G‡r ed ,rea

R ·T

· exp

−z‡red ·F ·α ·∆φ

R ·T

z‡ox and z‡

red are substituted with z‡:

z‡ox + z‡

red = 0 ⇒ z‡ox =−z‡ and z‡

red = z‡

Note: The variables zox and zred are equal to the electrons that are transfered inthe gross reaction, whereas z‡ is equal to the electrons exchanged in the rate-determining electrochemical step.Substitution of the rate constants kox, kred in D.2 on the preceding page yields:

I = zred ·F ·Cox

· kred · exp

−∆G‡r ed ,rea − z‡ ·F ·α ·∆φ

R ·T

+zox ·F ·Cred

· kox · exp

−∆G‡

ox,rea + z‡ ·F · (1−α) ·∆φR ·T

(D.3)


To reduce this equation, the equilibrium situation with an equilibrium electrodepotential φ� equal to the formal potential φ0 is investigated. This potential isknown from the Nernst equation [20]; — the result is:

I� = IEquilibrium

= 0A

m2

C�ox = C�

red (formal potential assumption)

∆φ� = φ�−φ0 = 0V

⇒ E�ox,elst = E�

r ed ,elst = 0J

mol

After two substitutions: zox =−z and zred = z this leads to:

I� = I 0,�red + I 0,�

ox

⇔ 0 = z ·F ·C�ox · kred · exp

−∆G‡r ed ,rea

R ·T

(D.4)

−z ·F ·C�red · kox · exp

−∆G‡

ox,rea

R ·T

Rearranging and cancelling of z ·F leads to:

C�red · kox · exp

−∆G‡

ox,rea

R ·T

= C�

ox · kred · exp

−∆G‡r ed ,rea

R ·T

The removal of the activation Gibbs function results in:

C�red ·k�

ox = C�ox ·k�

red

With an electrode potential equal to the formal potential an equity of concentra-tions is defined: C�

red = C�ox [12], therefore, the two rate constants k�

ox and k�red are

equal; a new standard or intrinsic rate constant with the symbol k0 is invented:

k0(T ) = kox · exp

−∆G‡

ox,rea

R ·T

= kred · exp

−∆G‡r ed ,rea

R ·T

(D.5)

= k · exp

−∆G‡

rea

R ·T

The new definitions for the rate constants are now:

kred = k0(T ) · exp

−z‡ ·F ·α ·∆φR ·T

kox = k0(T ) · exp

z‡ ·F · (1−α) ·∆φ

R ·T


At that point an equilibrium condition with ∆φ� = (φ� −φ0) different fromzero is observed. At equilibrium the current density I� is zero, and the con-centration at the reaction zone is equal to the bulk or open circuit concentration:C = Cbulk = Coc . Nevertheless, a balanced faradaic activity is still present, whichcan be expressed in terms of exchange current density I 0[12]:

I� = I 0red + I 0

ox = 0

= z ·F ·k0(T ) ·Coc,ox · exp

−z‡ ·F ·α ·∆φ�

R ·T

− z ·F ·k0(T ) ·Coc,red · exp

z‡ ·F · (1−α) ·∆φ�

R ·T

With the substitution bBV = z‡·FT ·R ,

after rearranging and canceling of z ·F·k0(T✸) the formula is now expressed asfollows:

Coc,ox · exp(−α·bBV ·∆φ�) = Coc,red · exp((1−α)·bBV ·∆φ�)

⇔ exp((1−α)·bBV ·∆φ�)

exp(−α·bBV ·∆φ�)= Coc,ox

Coc,red

with:

exp((1−α)·bBV ·∆φ�− (−α·bBV ·∆φ�)) = exp(bBV ·∆φ�·(1−α+α))

⇔ exp(bBV ·∆φ�) = Coc,ox

Coc,red(D.6)

raise both sides to the −α power, comes to:

⇔ exp(−α ·bBV ·∆φ�) = Coc,ox

Coc,red

−α(D.7)

The exchange current density of the reduction reaction is2:

I 0red(T,C ) = z ·F ·k0(T ) ·Coc,ox · exp(−α·bBV ·∆φ�) (D.8)

Substitution of D.7 into D.8 equals:

I 0red(T,C ) = z ·F ·k0(T ) ·C(1−α)

oc,ox ·Cαoc,red (D.9)

Using the definition of the standard rate constant D.5 in equation D.3 the Butler-Volmer equation can be formulated:

I = z ·F ·k0(T ) ·Cox ·e−α·b

BV ·∆φ − Cred ·e(1−α)·bBV ·∆φ

(D.10)

with: ∆φ=φ−φ0 (D.11)

2 For the following rearrangements also I 0ox can be taken.


Dividing D.10 by D.9 yields in:

I

I 0red

= z ·F ·k0(T )

z ·F ·k0(T )·

Cox ·e−α·bBV ·∆φ

C(1−α)oc,ox ·Cα

oc,red

− Cred ·e(1−α)·bBV ·∆φ


oc,red

(D.12)

The following is used:

C(1−α)oc,ox = Coc,ox

Cαoc,ox

and Cαoc,red = Coc,red ·Cα−1

oc,red

After cancel of z ·F·k0 and substitution the result is:

I

I 0red(T,C )

=

Cox

Coc,ox·e−α·b

BV ·∆φ · Coc,ox

Coc,red

α

− Cred

Coc,red·e(1−α)·bBV ·∆φ ·

Coc,ox

Coc,red

−(1−α)

The ratios (Coc,ox/Coc,red)α and (Coc,ox/Coc,red)−(1−α) can be evaluated from theequations D.6 and D.7: Coc,ox

Coc,red

α= e(α·bBV ·(φ�−φ0)) (raise D.7 to the −1 power) Coc,ox

Coc,red

−(1−α)

= e(−(1−α)·bBV ·(φ�−φ0)) (raise D.6 to the −(1−α) power)

By substitution it is incidental:

I

I 0red(T,C )

=

Cox

Coc,ox·e−α·b

BV ·(φ−φ0) ·eα·bBV ·(φ�−φ0)

− Cred

Coc,red·e(1−α)·bBV ·(φ−φ0) ·e−(1−α)·bBV ·(φ�−φ0))

The eq. is multiplied by I 0, and U= (φ−φ�)

as well as I 0red = I 0

ox(T ) = I 0 are substituted:

I = I 0(T,C ) ·

Cox

Coc,ox·e−α·b

BV ·U − Cred

Coc,red·e(1−α)·bBV ·U

(D.13)

The advantage of this revision is the removal of the more theoretical formal po-tential φ0 (known based on the theory of the Nernst equation) with the easy tomeasure voltage between open circuit φ� and working potential φ. The disadvan-tage is the temperature and concentration dependency of the exchange currentdensity I 0(T,C ).


The definition of the exchange current density from Eq. D.4 at reference conditionsin combination with the Eq. D.5 is taken to form:

I 0✸red

T ✸

C✸ox

= z ·F ·k0(T✸) ·C✸ox

∧I 0✸

oxT ✸

C✸red

= z ·F ·k0(T✸) ·C✸red (D.14)

For a reduction reaction I 0(T ) is substituted in Eq. D.13 with the definition of Eq.D.9 and divided D.13 by I 0

red(T✸) in the form of D.14:

I red

I 0✸red

=z ·F ·k0(T ) ·C(1−α)

oc,ox ·Cαoc,red

z ·F ·k0(T✸) ·C✸ox

·

Cox

Coc,ox·e−α·b

BV ·U − Cred


Canceling of z ·F and rearranging equals:

I red

I 0✸red

=C(1−α)

oc,ox ·Cαoc,red

C✸ox

· k0(T )

k0(T✸)

·

Cox

Coc,ox·e−α·b

BV ·U − Cred


Eq. D.5 is utilized and multiplied by I 0✸red :

I red = I 0✸red ·


oc,red

C✸ox

·k · exp

−∆G‡

rea

R ·T

k · exp

−∆G‡

rea

R ·T✸

·

Cox

Coc,ox·e−α·b

BV ·U − Cred


After canceling the frequency factor k in the third term in the equation above theresult can be rearranged to obtain a more compact form:

exp

−∆G‡

rea

R ·T

exp

−∆G‡

rea

R ·T✸

= exp

−∆G‡

rea

R·

1

T− 1

T✸


With the substitution I 0red = I 0✸

red/C✸ox

the result is a Butler-Volmer equation for reduction reactions applicable for differ-ent concentrations and temperatures:

I red = I 0red · C(1−α)

oc,ox ·Cαoc,red · exp

−∆G‡

rea

R·

1

T− 1

T✸

·

Cox

Coc,ox·e−α·b

BV ·U− Cred


The influence of C(1−α)

oc,ox ·Cαoc,red on the exchange current density is in accordance

with Butler-Volmer derivation in [113]. Finally, activities instead of concentrationsare taken and the result is the Butler-Volmer equation 2.10 on page 20.

D.3. Butler-Volmer Equation for Fixed Concentrationand Fluctuating Temperatures

In the case of fixed concentration at OCP and unsteady temperatures a specialform for both, reduction and oxidation reaction, can be shaped. In this case, theconcentration term can be simplified, because both concentrations of the oxidizedor reduced species are identical for the OCP and the reference condition:

Coc,ox ≡ C✸ox ∧ Coc,red ≡ C✸

red

And thus the concentration fractions become:


oc,red

C✸ox

=Coc,red

Coc,ox

α∧

Cαoc,red ·C(1−α)

oc,ox

C✸red

= Coc,ox

Coc,red

(1−α)

Now a new standard or reference exchange current density I✸0 is to be invented:

I✸0(C ) = I 0ox ·

Coc,red

Coc,ox

α∨ I✸0(C ) = I 0

red · Coc,ox

Coc,red

(1−α)

This results in:

I = I✸0(C ) ·exp

−∆G‡

rea

R·

1

T− 1

T✸

·

Cox

Coc,ox· exp

−α · z‡ ·F ·UR ·T

− Cred

Coc,red· exp

(1−α) · z‡ ·F ·U

R ·T

(D.15)

This variety is used with a different sign convention by WÖHR in [150] and foundto be in accordance with empirical data [99], cf. SCOTT [117]. It is also applied tothe oxygen reduction in this work.

E. MEASURED DATA

E.1. Polarization of a DMFC with Acid Sulfur DopedPVC-Membrane

Anode Cathodepure OR-reaction mixed potential

V vs. SHE A/m2 V vs. SHE A/m2 V vs. SHE A/m2

0.11 0.0 1.02 0.0 0.89 0.0

0.32 18.6 0.93 35.0 0.88 28.4

0.34 38.5 0.90 101.0 0.87 98.1

0.36 58.5 0.88 172.0 0.85 173.7

0.37 78.7 0.87 243.0 0.84 283.7

0.38 101.6 0.85 374.0 0.82 432.4

0.40 199.6 0.84 628.0 0.79 718.1

0.42 300.5 0.82 998.0 0.78 834.3

0.44 401.8 0.77 940.0

0.45 499.9 0.76 1015.1

0.49 800.9 0.75 1096.8

Table E.1.: Half cell polarization data at 298 K [148].

Anode Cathodemixed potential

V vs. SHE A/m2 V vs. SHE A/m2

0.06 0.0 0.87 0.0

0.23 17.2 0.85 158.5

0.26 37.1 0.83 297.6

0.27 58.7 0.77 996.8

0.28 79.2

0.29 100.6

0.31 198.5

0.32 300.6

0.34 401.3

0.35 502.4

0.38 799.9

0.40 998.7

Table E.2.: Half cell polarization data at 333 K [148].

132 E. Measured Data

E.2. Polarization of a DMFC withNafion™115-Membrane

E.2.1. Variation of Methanol Concentration

Anode Cathode0.5 M 1.0 M 2.0 M 3.0 M 0.5 M 1.0 M 2.0 M 3.0 M

[jfc] = A/m2 [UAN vs. SHE] = V [UCA vs. SHE] = V

0.0 0.05 0.03 0.03 0.06 0.89 0.85 0.83 0.77

0.1 0.09 0.04 0.03 0.07 0.88 0.85 0.82 0.76

0.5 0.10 0.05 0.04 0.09 0.88 0.85 0.81 0.76

1.0 0.13 0.06 0.05 0.13 0.88 0.85 0.80 0.76

2.0 0.18 0.10 0.08 0.15 0.88 0.84 0.80 0.77

3.0 0.21 0.16 0.13 0.16 0.88 0.84 0.80 0.76

5.0 0.22 0.20 0.17 0.17 0.88 0.84 0.80 0.76

10.0 0.23 0.21 0.19 0.18 0.88 0.84 0.80 0.76

20.0 0.24 0.23 0.20 0.19 0.88 0.84 0.79 0.75

50.0 0.27 0.24 0.21 0.20 0.88 0.84 0.79 0.75

100.0 0.28 0.26 0.22 0.19 0.88 0.84 0.78 0.73

200.0 0.31 0.28 0.24 0.21 0.87 0.83 0.77 0.73

350.0 0.33 0.30 0.26 0.25 0.87 0.83 0.77 0.74

500.0 0.34 0.32 0.29 0.27 0.87 0.83 0.77 0.74

750.0 0.36 0.34 0.31 0.29 0.86 0.82 0.77 0.72

1000.0 0.38 0.36 0.33 0.31 0.85 0.82 0.76 0.71

1200.0 0.39 0.37 0.34 0.32 0.85 0.81 0.75 0.70

1500.0 0.41 0.39 0.36 0.33 0.85 0.81 0.75 0.69

1750.0 0.43 0.39 0.37 0.32 0.84 0.80 0.75 0.66

2000.0 0.44 0.41 0.38 0.32 0.84 0.80 0.74 0.63

2400.0 0.47 0.42 0.39 0.30 0.83 0.79 0.72 0.58

2800.0 0.57 0.45 0.40 — 0.80 0.78 0.70 —

Table E.3.: Galvanostatically measured polarization of a DMFC with Nafion™115-membrane with different mathanol inlet concentrations.Operating conditions: temperature, Tfc = 333K; atmospheric pressure on anode andcathode; methanol concentration varied from 0.5 up to 3.0 M CH3OH; air at ambienthumidity; nominal stoichiometric coefficients at 1000 A/m2, λn

AN = 7, λnCA = 5.

Cell set-up: graphite bipolar plates with single meander flow field, geometric elec-trode area, Afc = 0.0025m2; catalyst loading (unsupported catalyst from JohnsonMatthey), mcat , AN = 0.059kg/m2 = 5.9mg/cm2 Pt-Ru; mcat , CA = 0.067kg/m2 Pt; diffusionlayer for anode and cathode, SGL 10BB; Nafion™115-membrane; mean electrical cellresistance (zero phase angle) Rfc = 6.5 mΩ.

E.2. Polarization of a DMFC with Nafion™115-Membrane 133

E.2.2. Variation of Temperature

298 K 333 K

jfc UAN UCA U R⋆fc jfc UAN UCA U R⋆

fc

A/m2 V A/m2 V

0.0 0.07 0.87 0.80 0.0 0.03 0.85 0.82

0.1 0.06 0.87 0.80 0.1 0.04 0.85 0.81

0.5 0.06 0.86 0.80 0.5 0.05 0.85 0.80

1.0 0.07 0.86 0.80 1.0 0.06 0.85 0.79

2.0 0.08 0.86 0.78 2.0 0.10 0.84 0.74

3.0 0.10 0.86 0.76 3.0 0.16 0.84 0.69

5.0 0.14 0.85 0.72 5.0 0.20 0.84 0.65

10.0 0.29 0.86 0.57 10.0 0.21 0.84 0.63

20.0 0.31 0.86 0.56 20.0 0.23 0.84 0.61

50.0 0.33 0.86 0.53 50.0 0.24 0.84 0.60

100.0 0.35 0.86 0.51 100.0 0.26 0.84 0.58

200.0 0.39 0.86 0.47 200.0 0.28 0.83 0.56

350.0 0.42 0.85 0.44 350.0 0.30 0.83 0.54

500.0 0.44 0.85 0.42 500.0 0.32 0.83 0.52

600.0 0.46 0.85 0.40 750.0 0.34 0.82 0.50

750.0 0.48 0.84 0.38 1000.0 0.36 0.82 0.48

800.0 0.49 0.84 0.37 1200.0 0.37 0.81 0.47

850.0 0.50 0.84 0.36 1500.0 0.39 0.81 0.45

900.0 0.51 0.84 0.35 1750.0 0.39 0.80 0.44

1000.0 0.52 0.84 0.34 2000.0 0.41 0.80 0.42

1100.0 0.54 0.83 0.32 2400.0 0.42 0.79 0.40

1200.0 0.55 0.83 0.31 2800.0 0.45 0.78 0.38

Table E.4.: Galvanostatically measured polarization of a DMFC with Nafion™115-membrane at temperatures of 298 and 333 K. Methanol concentration is 1.0 M CH3OH,the other operating conditions and the cell set-up are analog to E.3. Electrode poten-tials are relative to SHE and the fuel cell voltage is corrected by the ionic resistancevoltage U R⋆

fc =Ufc − Ifc ·Rm .

134 E. Measured Data

E.2.3. Cathode Polarization under Hydrogen Operational Mode

298 K 303 K 333 K

jfc UCA U R⋆fc Rm UCA U R⋆

fc Rm UCA U R⋆fc Rm

A/m2 V V mΩ V V mΩ V V mΩ

0.0 1.00 1.00 17.7 1.02 1.02 14.7 1.01 0.99 7.7

10.0 0.92 0.91 18.5 — — — — — —

20.0 0.90 0.89 17.8 0.94 0.94 15.6 0.97 0.95 8.5

50.0 0.88 0.87 17.6 0.92 0.91 14.8 0.95 0.93 8.5

100.0 0.86 0.85 16.9 0.90 0.89 14.0 0.93 0.91 7.7

200.0 0.84 0.83 14.8 0.87 0.86 13.6 0.92 0.89 8.2

300.0 0.83 0.82 14.6 — — — — — —

400.0 0.83 0.82 13.7 — — — — — —

500.0 0.82 0.81 12.7 0.84 0.82 12.3 0.89 0.85 8.0

600.0 0.81 0.80 11.6 — — — — — —

800.0 0.80 0.79 11.3 0.82 0.80 10.7 0.87 0.84 8.6

1000.0 0.79 0.78 10.7 0.80 0.79 10.8 0.86 0.82 7.6

1500.0 0.76 0.76 10.8 0.78 0.77 10.2 0.84 0.80 8.1

2000.0 0.75 0.74 10.7 0.76 0.75 10.2 0.82 0.79 8.3

2500.0 0.73 0.72 10.1 — — — — — —

3000.0 0.72 0.71 10.2 0.73 0.71 9.7 0.79 0.76 8.0

4000.0 0.69 0.69 11.2 0.70 0.69 10.0 0.77 0.74 8.4

5000.0 0.66 0.65 10.7 0.67 0.67 10.3 0.75 0.71 7.4

6000.0 0.64 0.63 11.3 0.65 0.64 10.6 0.72 0.69 7.4

7000.0 0.60 0.59 12.3 0.62 0.62 11.2 0.69 0.67 7.8

8000.0 0.57 0.57 14.2 0.59 0.61 13.0 0.67 0.64 7.6

Table E.5.: Galvanostatically measured cathode polarization of a DMFC withNafion™115-membrane under hydrogen operational mode at three temperatures. Cellset-up and operating conditions are analog to E.3. In deviation of this, the stoichiome-trie was: λn

H2(@1000 A/m2) = 12 and λ

nAi r (@1000 A/m2) = 20. For U R⋆

fc cf. Tab. E.4. Ref-

erence electrode was a mercury sulfate electrode with a potential of UHg/Hg2SO4=

0.671 V vs. SHE.

F. SOFTWARE TOOLS

In addition to the commercial thermodynamic simulation software AspenPlusfrom AspenTech, version 12.1.14 Build 273, several other software have been used,the most relevant being:

Compiler

To add user defined routines inside of the AspenPlus model Fortran77 code wasused; the compiler was Compaq Visual Fortran, version 6.6-1684.

Coding Environment and File Management

Fortran-coding and the file management was done in the free editor and projectmanagement tool PSpad.

Mathematical Software

For numerical optimization and calculation work outside of AspenPlus two soft-ware have been used:

• Linear algebra: octave (a free Matlab clone)

• Tabular calculations: MS Excel

Graphic Digitizer

In order to isolate numerical data from graphs in scientific papers, the free digitiz-ing software WinDIG vers. 2.5 by D. LOVY was used.

Editors

Beside PSpad as mentioned above, some editors were used for specific pur-poses: XEmacs (LATEX, gnuplot, code-debugging), Notepad++ (DOS-batch-files),SciTE(sticky notice board, Octave), emerald (Fortran samples)

136 F. Software Tools

Plotting

Plotting was done with GnuPlot as stand-alone software and in combination withOctave.

Nassi-Shneiderman Editor

Nassi-Shneiderman diagrams are created using the program Structorizer.

Drawing

Free hand drawing tool was Inkscape.

G. GLOSSARY

The literature at times makes it difficult to determine which definition or whichunit is used. At other times authors use different definitions for the same value, i.e.,the theoretical cell voltage Utheo, where three definitions are used. It is also unclearif the transfer coefficient or symmetry factor α of the Butler-Volmer equation isrelated to the oxidation (older German literature such as [139]) or to the reduction(international literature). For other symbols the units can differ: [µ] = J·mol-1

[37, 54] or [µ] = J[115]. In order to clarify some definitions are given here:

Term Explanation

Analytical model There are three categories of fuel cell models [24]:a) Analytical models work with many simplifying assump-tions and no empirical fitting.b) Semi-empirical models combine theoretically derived dif-ferential and algebraic work with empirically fitted relations.c) Mechanistic models avoid empirically fitted relations1.

Anode The anode absorbs electrons, either from an anion(2Cl− → Cl2 +2e−) or from the oxidation reaction(CH3OH+H2O → CO2 + 6H+ + 6e−) [54]. According to theinternationally accepted Convention of Stockholm (IUPAC1953 [25, 81]) the anode is depicted on the left side of fuelcell illustration.

Arrhenius behavior When the reaction rate can be described by the Arrheniusequation the reaction shows an Arrhenius behavior [52]:

k = k · exp

−∆G‡

R ·T

with:Reaction rate constant, [k]= mol·s-1; frequency factor [12] orpre-exponential factor, [k]=[k]

Bruggemannrelationship

Calculation rule for determining the diffusion coefficient inthe gas diffusion layer (referred to in [97]).

1 A pioneering work in the field of mechanistic PEM-modeling was published in 1992 by BERNARDI

and VERBRUGGE [18, 19]. Most of the recent works belong to the group of mechanistic modeling.

138 G. Glossary

Butler-Volmerequation

The reaction in an electrochemical cell takes place in the(electric) double layer of the electrode [96]. The Butler-Volmer equation is a fundamental equation that describesthe dependency between the reaction rate of electrochemi-cal reactions and the polarization of such reactions. For thefollowing specific assumption this equation can be derivedanalytically. The simplest class of electrochemical reactions,the electron-transfer reactions and here the plainest form ofthis class of reactions, the so called outer-sphere electron-transfer reactions are taken by SCHMICKLER [115] to derivethe Butler-Volmer equation. He provides the following ex-ample for such a reaction:

[Ru(NH3)6]3+ + e−reduction−−−−−−−−−→←−−−−−−−−−oxidation

[Ru(NH3)6]2+ (G.1)

In the course of this class of reactions no bonds are brokenor formed, the reactants are not specially adsorbed, andcatalysts do not interact with reactants. Typically such reac-tions involve metal ions surrounded by inert ligands, whichprevent adsorption [115]. Because the reaction takes placein the double layer of an electrode, phenomenological treat-ment assumes that the Gibbs energy of activation ∆G‡ is alsodependent on the electrode potential φ2; a contribution ofan electrostatic effect on the overall reaction rate can beobserved. This contribution is dependent on the electricalcharge that is exchanged in the reaction z‡ and the potentialof activation φ‡. The magnitude of the electrostatic activa-tion energy can reach a maximum of |E max

elst | = |z‡·F·(φ−φ00)|,where φ00 is the standard equilibrium potential3. With theintroduction of a dimensionless transfer coefficient α alsotermed symmetry factor in the range of α ∈ [0..1] the electro-static contribution to the reduction reaction4 can be writtenin the form of:

E ‡elst,red = z‡ ·F ·α · (φ−φ00)

For one electron reactions, as in the example reaction above,z‡ is equal to unity and can be removed in the equation. Ifa Taylor series of first order is utilized to approximate the

2 φ is also termed Galvani potential, inner potential or coulomb potential [12].3 Another common symbol for the standard equilibrium potential is E◦.4 The transfer coefficient α can also be assigned to the oxidation reaction as in [115, 139, 140] and

the polarization of the cathodic current is also customary. In this work the definitons of ATKINS [7]are used.

139

Gibbs energy of activation ∆G‡(φ) around φ00 that leadsto:

∆G‡red(φ) = ∆G‡

red(φ00)+∂∆G‡

red

∂φ

φ00· (φ−φ00)

In the next step the partial derivative is substituted:

∂∆G‡

∂φ

φ00= α ·F

An analogues mathematical treatment of the oxygen reac-tion (back reaction) with the symbol β for the anodic trans-fer coefficient and a minus sign to take into account theopposite impact of the higher electrode potential equatesto:

∆G‡ox(φ) = ∆G‡

ox(φ00)−β · (φ−φ00)

For the special case of equal surface concentration of bothredox ions5 from G.1 Cs

ox = Csred it can be shown that β =

1−α (cf. [115]), which results in:

∆G‡red(φ) = ∆G‡

red(φ00) + F · α · (φ−φ00)

∆G‡ox(φ) = ∆G‡

ox(φ00) − F ·(1−α)· (φ−φ00) (G.2)

Using concepts familiar from chemical kinetics, the overallsurface related reaction rate v can be written as:

v = kred ·Csox −kox ·Cs

red ; [v] = mol

s · m2

with:Cs

ox /red, surface concentration, [Cs] = molm3 ; kred /ox, reaction

rate constant, [k] = ms

Using absolute rate theory, the rate constants can be ex-pressed as:

kred = kred ·exp

−∆G‡red

R ·T

kox = kox ·exp

−∆G‡

ox

R ·T

(G.3)

Further reorganization yields the following form of the equa-tion6:

I = F ·k0 ·Csox ·exp

−α ·F · (φ−φ00)

R ·T

−F ·k0 ·Cs

red ·exp

(1−α) ·F · (φ−φ00)

R ·T

5 The equity of concentrations is called in [12] the formal potential assumption.6 The native form of the Butler-Volmer equation was published by J.A. Butler (1924) and M. Volmer

(1930), but with the opposite sign convention for the redox currents [113].

140 G. Glossary

Analogous to eq. G.3 k0 can be defined as follows:

k0 = k · exp

−∆G‡(φ00)

R ·T

If the equilibrium potential φ0 is calculated using the Nernstequation one has:

φ0 = φ00 +R ·T

F· ln

Csred

Csox

⇔ φ00 = φ0 −R ·T

F· ln

Csred

Csox

Reorganization and aggregation of the constants into anexchange current density I 0 amounts to:

I = I 0 ·

exp

−α ·F · (φ−φ0)

R ·T

− exp

(1−α) ·F · (φ−φ0)

R ·T

To simplify the equation further we introduce two substitu-tions: a) the polarization7 U= φ−φ0 and b) a tempera-ture dependent Butler-Volmer parameter bBV = z‡ ·F·(R·T )−1.With these substitutions a short form can be written as fol-lows:

I = I 0 ·exp(−α ·bBV ·U)−exp({1−α} ·bBV ·U)

(G.4)

The exchange current density can be interpreted as an indi-cator for the rate to reach the equilibrium of the reaction;it is proportional to the reaction rate per active catalyticsurface area Φ:

Φ� = I 0

F · z; [Φ] = mol

s·m2

Cathode The cathode delivers electrons to a neutral species (Cl2 +2e− → 2Cl−) or adsorbs cations (2H++2e−+0.5O2 → H2O)from the electrolyte [54]. According to the internationallyaccepted Convention of Stockholm (IUPAC 1953, [25,81]) thecathode is depicted on the right side of fuel cell illustration.

Chemical Potential The chemical potential ∆Grea, if positive, is the specific worknecessary to overcome the pure chemical interactions. Itdoes not include electrostatic forces to move substance iagainst an electrostatic field into a phase A . See also electro-chemical potential.

7 More common is the symbol η for the polarization, the polarization is also called overpoten-tial [113]. The symbol η is also quite common as efficiency symbol, hence, in this work a differentsymbol is selected.

141

Darcy’s law Darcy’s law dates back to the year 1856. For sufficiently lowfluid velocities, it describes the single-phase flow through aporous medium [14, 15, 20, 43, 51]

vsf =κhp

ηV·∆p

l+ρ · g

If hydrostatic pressure is negligible, it can be transformed to:

∆p = ηVκhp

· vsf · l

with:Superficial velocity, [vsf ] = m/s ; cinematic viscosity, [ηV] = N·

s/m2; hydraulic permeability, [κhp ] = m2

A variant form of the Darcy’s law is taken in [18] to calculatethe superficial velocity of liquid water in the cathodic diffu-sion layer, caused by the hydraulic pressure gradient acrossthis layer:

C DLv LPH2O,sf = −

C DLκhp,sf

ηVLPH2O

· ∂p

∂z

“The superficial velocity describes the volume rate of flowthrough a unit cross-sectional area of the solid plus fluid,averaged over a small region of space — small in respect tothe macroscopic dimensions in the flow system, but largewith respect to the pore size” [20].

Double layer The first scientist who introduced a model of the interfacebetween the electrode and the electrolyte was HELMHOLTZ.His model consists of an electric double layer. The layer nextto the electrode is called the Inner Helmholtz plane and theadjoining layer is called the Outer Helmholtz plane. Thismodel is neglecting the stirring effect of thermal motion onmolecules and so the Gouy-Chapman model postulates adiffuse double layer. As both are extreme perceptions STERN

formulated a mixed model, the Stern model. In the Sternmodel the ions next to the electrode are constrained into arigid Helmholtz plane while outside that plane the ions aredispersed as in the Gouy-Chapman model [7].

Dynamic hydrogenelectrode (DHE)

A pseudo-reference electrode assembly, simulating a re-versible hydrogen electrode (RHE) with an approximately 20to 40 mV more negative potential. While its potential is lessdefined, it has the advantage of not requiring a hydrogengas supply. Typically, it is a glass tube containing two in-ternal electrodes, of which at least one is a platinized plat-inum electrode immersed in the same electrolyte solution

142 G. Glossary

as found in the working cell. The two electrolytes are inionic contact through a separator. A small, constant currentis enforced between the two electrodes with the platinizedplatinum being the cathode, typically carrying a currentdensity of 1 mA/cm2, which results in a small amount of hy-drogen evolution. Here the cathode is used as the referenceelectrode. Voltages that are measured against a commonreference (cr) like the DHE are refered to as potentials vs. crUcr / UD HE .

Evans diagram Potentiostatic curves of the oxidation and reduction reactionat one electrode in a common diagram to determine themixed potential (see below) [48].

Efficiency There are several efficiencies defined and some of themhave different names:

a) Faraday [2, 94] or Coulombic efficiency [77]:The ratio between the effectively exchanged electrons per re-acted molecule (zeff ) and the theoretically possible numberof exchanged electrons (z).

ηFA =zeff

z

b) Fuel cell conversion or(fuel) utilization efficiency [93, 105]:Indicate fuel losses due to leak flows and / or fuel dragacross the membrane (cross over), for the DMFC it can bewritten as:

ηut =ΦCH3OH, Fo, AN

ΦCH3OH, Fo, AN +ΦCH3OH, leak +ΦCH3OH, CO

=jfc

jfc + j leak + jCO

(G.5)

(Index Fo means faradaically or electrochemically oxidized.)

If the total combustion of methanol to CO2 is given, theproduct (ηut ·ηFA) becomes 1.

c) Thermo neutral efficiency [140]:What often is called thermo neutral or ideal efficiency isstrictly speaking not an efficiency. It can have values above

143

1.0, so it is rather an (ideal) thermo-neutral conversion rationthn [72, 124]:

ηthn = nthn = ∆G

∆H

= ∆H −T ·∆S

∆H= 1− T ·∆S

∆H

d) Voltage efficiency[94]:

ηU =Ufc

Utheo(G.7)

(for theoretical cell voltage Utheo cf. G.16 on page 154)

Electromotive force Electromotive force (EMF) is sometimes also referred to aselectro-motoric force [72], which is the old terminology forthe ideal cell voltage Uid (see Eq. G.12 on page 148).

Electrochemicalpotential

The electrochemical potential µ∗ is a thermodynamic value,which is the potential of a system to provide work resultingfrom changing the amount of substance of the componenti in the system. The definition is:

µ∗i =

∂G

∂ni

p=constT=const

; [µ∗] = J·mol-1

The electrochemical potential is the sum of the chemicaland the specific electrostatic work:

E = Ech + Eelst , [E ] = J·mol-1

µ∗i = ∂Gch

∂ni+ z ·F ·φ

= µi + z ·F ·φ , [φ] =V = J·C-1

φ is termed the Galvani potential and is described bolow.Electrochemical and chemical potential are identical foruncharged particles (z = 0).

Electroosmoticdrag

There are two definitions:a) methanol transport evolved from the proton transportthrough the membrane [44];b) water transport caused by the proton transport [104].The electroosmotic drag coefficient ξeo is the number of wa-ter molecules per proton carried across a proton-conductivemembrane as current is passing under conditions of no con-centration gradient [155]. In the case b) the electroosmotic

144 G. Glossary

methanol drag can be calculated by multiplying the waterdrag with the methanol fraction at the anode-membraneinterface xAM I . The electroosmotic drag coefficient is de-pendent on the membrane type, membrane thickness andtemperature.

Φeo,H2O = ξeo

Ifc

F; [ξeo] = molH2O/molH+

Φeo,CH3OH = Φeo,H2O · xAM I

ElectrostaticPotential

In contrast to electrochemical potentials (see above), elec-trostatic potentials describe solely physical effects of chargetransfer, chemical reactions are not considered. Two dif-ferent electrostatic potentials are defined, the outer- orVolta potential ψ and the inner or Galvani potential φ; thecommon unit is V = J/C.

�100 nm

Fig. G.1.: Galvani-, Volta- and surface potential [7, 12].

a) Volta potentialThe work E that is necessary to move an electrically chargedparticle from the infinity near the surface of the electro-chemical phase A is the product of the outer or Volta poten-tial ψA and the electrical load of the particle. SCHOTTKY8

established in 1914 a distance to the electrode-surface ofd = 10−6m, to ensure the particle is not interacting with thedouble layer of the electrode [139]. The Volta potential can

8 Lit.: W. SCHOTTKY ◦ Über den Einfluß von Strukturwirkungen, besonders der Thomsonschen Bildkraft,auf die Elektronenemission der Metalle ◦ Physikalische Zeitschrift, vol.15, pp. 872–878, Germany,1914

145

be measured directly[115, 139]. For z transfered electronsper reacting molecule the equation is:

EVOLT = z ·F ·ψA ⇔ ψA = EVOLT

z ·F (G.8)

[E ] = J/mol, [F] = C/mol, [ψ] = J/C =V

b) Galvani PotentialThe work E needed to bring the same charged particle intothe electrochemical phase A is the product of the inner orGalvani potential φA

9 and the electrical load of this particle.The equation is analogous to Eq. G.8.

EGALV = z ·F ·φA ⇔ φA = EGALV

z ·FIn conjunction with the electrostatic force, there can alsobe other barriers on the surface, such as adsorbed ions ordipoles, that interact with the loaded particle on its way intothe phase [42,115]. This results in the so called surface poten-tial χ. It is not measurable and it should rather be termed asurface voltage rather then a potential because its referencepoint, the outer potential ψ, is a floating value in relation tothe infinity. According to Fig. G.1 on the preceding page ittakes effect:

φ=ψ+χ

Because of the unknown surface potential χ the Galvanipotential φ is also not measurable for one phase. Only thedifference of the inner potentials of two phases, e.g. the vol-tage between two phases or electrodes, is measurable. Thewidely used Stockholm convention [25, 81] for galvanic cellsdefines the phase with the lower potential (the anode) as leftside in symbolic depictions of a galvanic cell. If the letterA is assigned to the left and B to the right, the followingnotations are equivalent:

Uoc = φB − φA= φright − φleft

9 The Galvani potential is also termed the coulomb potential [12].

146 G. Glossary

In equilibrium (I = 0) the difference between two electrodesor half cells is called open circuit cell voltage (OCV):

Uoc = (φB −φSHE ) − (φA −φSHE )

= ∆φB − ∆φA

= φB − φA (G.9)

Under standard conditions the open circuit cell voltage Uoc

is equivalent to the standard voltage or standard equilibriumvoltage U✷

oc:

U✷oc = φ✷

B − φ✷A

The Galvani potential is also called the absolute electrodepotential. One example is the Galvani potential of a SHE.By definition it can not be measured, but it can be math-ematically approximated. The IUPAC[60] defines the ab-solute electrode potential of the SHE at T = 293.15 K andp = 101 325.0 Pa to φSHE = 4.4±0.02V, whereas the half cellpotential U✷

SHE is set to 0.0 V.

Faradaic oxidation Electrochemical oxidation without direct contact to the oxi-dant (anodic reaction).

Faraday’s laws a) First Faraday’s law :The mass of electrochemically converted species at one elec-trode is proportional to the ion flux:

M ∼ Q ∧ Q = Iconst · t or Iconst =Q

t(G.10)

b) Second Faraday’s law :The fraction of the converted masses of two substances isequal to the fraction of molar ion equivalents [54, 92]:

M1

M2=

M1

z1

M2

z2

Fick’s law The Fick’s First law was derived by ADOLF FICK in the year1855. It describes the steady-state diffusion through a ho-mogeneous layer [52].

Φi = −D · ∂Ci

∂z(G.11)

or:

Φi = −D · ∂Ci

∂z· A

147

Gibbs Free energy The total energy that a system holds at a given tempera-ture and pressure is defined by the Gibbs Free energy or theactivation Gibbs function [7].

GTp = HTp −T ·STp , [G] = J

The absolute value of Gibbs Free energy is fictive and cannot be measured [7]. What is measurable is the change ofGibbs Free energy ∆G between two situations:

∆G = G T2

p2

− G T1

p1

When ∆G is negative, the change of the Gibbs Free energyis the maximum energy that a system can deliver to theproximity, when ∆G is positive the environment donatesenergy to the system. If the following reaction is assumed:

reactant A + reactant B → product C + product D

νA · r A + νB · rB → νC ·pC + νD ·pD

the change of Gibbs Free energy will be calculated as:

∆G = ξr ·

|νC | ·GCT2

p2

+ |νD | ·G DT2

p2

− |νA| ·G AT1

p1

− |νB | ·G BT1

p1

[G ] = J/mol , [ξr ] = mol , [ν] = 1

If the Gibbs Free energy is negative a reaction can occurwithout net energy supply to the system (cf. definition ofstoichiometric factor on page 152 and [7, 72]).

Galvani potential See Electrostatic potential on page 144.

Helmholtz plane See double layer on page 141.

Ideal cell voltage If the change of Gibbs Free energy ∆G is taken as the nu-merator for the calculation of the theoretical open circuitcell voltage, it is called the ideal cell voltage [140], zero-current cell potential [7], rest potential or electromotive forceEMF [54]. Uid refers to the reversal (open) cell voltage at

148 G. Glossary

standard temperature T✷ and standard pressure p✷ (cf. p.113). The electrochemical convention defines the battery orfuel cell voltage Ufc as positive [25], whereas the thermody-namic convention defines energy that is leaving the systemas negative. Therefore the right side of the equation has tobe multiplied by −1.

Uid =U✷rev = E MF = −∆G

z ·F (G.12)

with: lnaox

ared= 0

For the influence of the activities ai on the cell voltage seeEq. G.13 on the next page. If the voltage is measured againstthe SHE Uid becomes the reference half cell potential U✷.

Leakage current The leakage current or crossover current ICO is the flux ofmethanol from anode to cathode multiplied by a factor of6F, as in the oxidation reaction six protons and electronsare released from each molecule of methanol (see Eq. 2.3on page 17). Every methanol molecule that diffuses acrossthe electrolyte (membrane) instead of being consumed atthe anode corresponds to a loss of current. This leakage ofmethanol leads to a loss of fuel efficiency. Furthermore, italso adversely affects cathode performance as the methanolthat reaches the cathode depolarizes it (see mixed potentialbelow) [93].

Limiting current The limiting current Il i m is the upper limitation of the cur-rent and is representing the maximum of mass transport.

Mechanisticmodels

See Analytical models on page 137.

Mixed potential The phenomenon of mixed potential was first used by WAG-NER and TRAUD in 1938 to describe the formation of rust10.The total current/potential curve is build from an oxidationand a reduction partial current, each with its own symmetryfactor and exchange current density. If two rivaling reac-tions take place, as on the DMFC cathode, the potentialwhere both currents are equal and opposite is termed mixedpotential [139, 141].

10 Lit.: C. WAGNER, W. TRAUD ◦ Über die Deutung von Korrosionsvorgängen durch Überlagerung vonelektrochemischen Teillvorgängen und über die Potentialbildung an Mischelektroden ◦ Zeitschriftfür Elektrochemie, pp. 391–454, vol. 44, 1938

149

Multi-domainapproach

If the equation system is solved in a sequence, it is termed amulti-domain solution strategy; in AspenPlus it is called theSM-mode (sequential modular mode). If all equations areput into one single system of equations it is called single-domain approach (in AspenPlus it is called equation orientedmode/eo). The advantage of the sequential modular methodis that there are fewer conversion problems and that differ-ent parts can be debugged more easily. The single-domainapproach has the feature that the numerical resolution ismuch more straight forward, what is advantageous espe-cially for parameter studies on the system.

Nernst equation The activities ai of the species in a electrochemical cell canhave an influence on the open cell voltage Uoc

11. This influ-ence can be described by the Nernst equation [7, 12, 54]:

Uoc = Uid +R ·T

z ·F · νi · lnai

= Uid +R ·T

z ·F · ln

a

νpipi

a

|νri|

ri

(G.13)

ai = γi ·Ci

C✷i

For the model reaction:

oxidized species(ox)+ z ·e reduction−−−−−−−−→←−−−−−−−−oxidation

reduced species(red)

it looks like:

Uoc = Uid +R ·T

z ·F · lnared

aox

Uid: ideal cell voltage, see Eq. G.12; pi = index of product(right side); api

: activity of the product species i , [a] = 1;ν: stoichiometry coefficient, ν > 0 for products, ν < 0 forreactants, [ν] = 1; z: elementary charge transfered permolecule, [z] = 1; γ: activity coefficient, [γ] = 1; Ci : molarconcentration, [C ] = mol·m-3

Nominal methanolstoichiometry

The nominal methanol stoichiometric factor λn is not cor-rected by the parasitic methanol oxidation and losses on thecathode [94] (see also stoichiometric factor below).

11 In the context of the Nernst equation it is more common to use the symbol E◦ for the cell voltageat equilibrium. But the symbol E is outside the electrochemical community reserved for energy,with [E] = J; therefore, the voltage symbol U is prefered in this work.

150 G. Glossary

Over-potential The over-potential is indicated with η in most books, whichcollides with the symbol for the efficiency; the unit is [η] =V = J/(A·s). To clarify that it is different from the cell voltageUfc, but has the same unit, U is selected as a symbol for theover-potential in this work. It is defined by VETTER [139] asthe difference of the terminal voltage or inner potential andideal voltage of the species i :

Ui = ∆φwv, i − ∆φoc, i

= ∆φwv, i − Uoc, i (see [7])

= Uwv, i − Uid (G.14)

Uwv is the working value cell voltage and Uoc is called opencircuit cell voltage (see Eq. G.9 on page 146). Note that U canbe the sum of different interferences. If no crystallizationovervoltage exist, the definition is as follows [140]:

U(I ) =Uet (I ) + UD(I ) + Urea(I )

=Uelectron transfer + Udiffusion + Ureaction

In this work the sign convention of ATKINS [7] for fuel cellsis used, which defines the anodic current IAN as negative,while the polarization UAN is positive:

IAN < 0V ⇔ IAN =−Ifc ⇔ UAN > 0V

The difference between the terms over-potential and polar-ization is that the over-potential is related only to the reac-tion of on species i [139], whereas polarization is related tothe entire reaction (see also the definition of polarizationon the current page).

Parasitic current For parasitic current or crossover current ICO see leakagecurrent on page 148.

Polarization VETTER [139] defined the difference in the electrode poten-tial between open circuit and operation condition polariza-tion.

Upolarization(I ) = Uworking value(I )−Uideal voltage

= Ui (I )

Another more descriptive word for the voltage losses be-tween open circuit voltage and the terminal voltage, closedcircuit voltage or working value voltage is over-potential or

151

overvoltage U [72, 139], see also Eq. G.14. In the event thatthere is only one electrode process i , polarization and over-potential are identical [139]:polarization(Upol ) = over-potential(Ui ), (

i = 1).

Potential There is a basic difference between electrostatic potential —an electrical value with the unit V — and the thermodynamicor electrochemical potential — a specific energy with theunit J ·mol-1. The difference of electrostatic potentials iscalled voltage: U =∆ψ.

Reaction extend The reaction extend ξ, also refered to as reaction rate orextent of reaction per time, describes the speed of a reaction;the unit is [ξ]= mol/s. The stoichiometry coefficient ν ofeither a reagent or a product, multiplied by the reactionextend, results in the amount of substance consumed of thisreagent or produced of this product in the reaction per time(see also the term stoichiometry).

Reversible hydro-gen electrode

A reversible hydrogen electrode (RHE) is a reference electrode,more specific a subtype of the standard hydrogen electrodes(SHE) for electrochemical processes and differs from it bythe fact that the measured potential does not change withthe pH so that they can be directly used in the electrolyte12.The name refers to the fact that the electrode is in the actualelectrolyte solution and not separated, e.g. by a salt bridge.The hydrogen ion concentration is therefore not 1, but cor-responds to that of the electrolyte solution; in this way wecan achieve a stable potential URHE with a changing pHvalue. The potential of the RHE correlates to the pH value:

URHE = (0.000−0.059 ·pH)V

Schlögl’s equation SCHLÖGL introduced his equation in 195513, which de-scribes the velocity of water transport in the membranepores through hydraulic pressure and potential gradients[19, 24]:

porv LPH2O =

mkek

ηVLPH2O

· − z ·Cconst ·F ·∂U

∂z−

mkhp

ηVLPH2O

· ∂p

∂z

with:

12 Lit.: YU CAI, ALFRED B. ANDERSON ◦ The Reversible Hydrogen Electrode: Potential-Dependent Ac-tivation Energies over Platinum from Quantum Theory ◦ The Journal of Physical Chemistry B,American Chemical Society, vol. 108, pp. 9829–9833, 2004

13 Lit.: R. SCHLÖGL ◦ Zur Theorie der anomalen Osmose ◦ Zeitschrift für physikalische Chemie,München, vol. 3, pp. 73–102, 1955

152 G. Glossary

Permeability of the PEM (hp = hydraulic pressure, ek = elec-trokinetic), [mk] = m2; dynamic viscosity of pore-water,[ηV] = N·s/m2; transfered elementary charge, z = 1; chargeconcentration, [Cconst] = mol/m3.

Semi-empiricalmodels

See Analytical model on page 137.

Standard hydrogenelectrode

SHE is the acronym for standard hydrogen electrode. TheSHE is an ideal device and can not be constructed. At stan-dard conditions real hydrogen electrodes can approximateit, and its properties can be defined by extrapolation [12].The potential of the SHE at 293.15 K and 101 325.0 Pa is setto U✷

SHE = 0.0 V; its Galvani or absolute electrode potentialis set by the IUPAC [60] to φSHE = 4.4±0.02V. See also dy-namic hydrogen electrode (DHE) on page 141 and standardpotential on the current page.

Single-domainapproach

See Multi-domain approach on page 149.

Standardconditions

A definition of standard pressure p✷ or p0 and temperatureT✷ or T 0, see page 113 for values.

Standard potential To be able to compare different half cell potentials with eachother a common reference potential is needed. There aredifferent reference systems in use. One of them is the stan-dard hydrogen electrode (SHE)[12,54]. For this electrode thestandard or reference potential is defined to as U✷

SHE = 0V.The calculation of U✷ is the same as for the ideal cell voltageon page 147, with the exception that U✷ is always based onstandard conditions:

U✷ =Uidp✷

T ✷

= −∆G✷

z ·F , with: lnaox

ared= 0

For the values of p✷ and T✷ see on page 113.

Stoichiometricfactor

The stoichiometry defines the quantitative relation betweenreactants and products.It differentiates stoichiometry coefficients ν and stoichiomet-ric factor λ. The first factor is used to define the relationbetween the extent of reaction ξr and each component ofa chemical equation, and the latter is used to describe theratio between reactant i ri to the maximum possible extentof reaction ξr UB, hence the amount of product j p j .In symbolic notation a reaction is written as follows:

νr1·r1 + νr2

·r2 −→ νp1·p1 + νp2

·p2

153

The stoichiometry coefficients ν for reactants are negativeand for products positive, which leads to:

−1CH3OH + −1H2O −→ 3H2 + 1CO2

Predominantly, in reaction equations the absolute values forν are used:

1CH3OH + 1H2O −→ 3H2 + 1CO2

With the introduction of ξr one gains:

νr1·ξr ·r1 + νr2

·ξr ·r2 −→ νp1·ξr ·p1 + νp2

·ξr ·p2

The product of extent of reaction and stoichiometry coeffi-cient is the amount of reactant consumed or product gener-ated by a reaction. The extent of reaction per time is calledthe reaction rate, or sometimes reaction extent, as per As-penPlus ξr :

ξr ·νr1= nr1

; [ξr ] = mol, [ν] = 1

ξr ·νr1= nr1

; [ξr ] = mol/s

To describe the amount of a reactant supplied n INr1

in rela-tion to the quantity of reactant needed nr1

= ξr ·νr1to react

completely with the other reactants {nr2, .. , nri

}, the termstoichiometric factor λ is used. If the value is above 1, notall reactant will be consumed by the reaction:

ξr UB = min

nr2

ν2, .. ,

nri

νi

⇒ λ1 =

n INr1

ξr UB ·νp1

=n IN

r1

nr1

To describe the reactant stoichiometry used to operate a fuelcell a slightly different definition is more common. Here anideal fuel cell is assumed with a fuel utilization efficiencyηut of 1. This special stoichiometric factor is called nominalstoichiometric factor λn:

ξrn ·νr1

!=Ifc

z1 ·F⇒ λ

n =n IN

r1

ξrn ·νr1

with:z1, electrons transfered per reactant molecule r1; F, Faradayconstant, [F] = C/mol = A·s/mol

In real fuel cells, however, ηut is smaller, indicating a λn wellabove 1 is needed to convert the other reactants completely.If a polarization curve is going to be characterized and con-stant flows of reactants are used, the value λn is mostly re-lated to a specific value of the fuel cell current or currentdensity.

154 G. Glossary

Tafel slope The original Tafel equation from 1905[48] is:

U = a − b · lnj

j†, [U] = V (G.15)

[b] orϕ

Ts

= V (Tafel slope)

In the current literature the minus sign is replaced by a plussign: U= a+b·ln(j/j†). To cancel the unit of j it is divided byj† =1·[j] (the unit of j is often [j]= mA/cm2). U is called thepolarization or over-potential and is described on page 150.In a variant form of the equation it becomes clear that thetransfer coefficient α and the exchange current density I 0 canbe calculated with the Tafel equation, see[7] and Eq. G.4. Forsimplification the exchange current I 0 is often not related tothe unknown electrochemically active catalyst surface, butto the geometric electrode area. To differentiate betweenboth, a different symbol j0 for the latter case is taken:

U = a + b · ln j

j†

Uox = −R ·T

{1−α} ·F · lnj0

j†

+ −R ·T

{1−α} ·F · ln jox

j†

Ured =R ·T

−α ·F · lnj0

j†

+ R ·T

−α ·F · ln jred

j†

Note the difference between the Butler-Volmer equation andthe Tafel equation: the TAFEL approach is not consideringan active equilibrium. Therefore there is only one term, foreither the oxidation (ox) or the reduction (red). More oftenthe Tafel equation in written in the exponential form:

j

j†= exp

U−a

b

jred = exp

b

a

·j† · exp

UR·T−α·F

jred = j0 · exp

−α·FR·T ·U

Theoretical cellvoltage

There are three possible definitions of the theoretical cellvoltage Utheo; to calculate Utheo following energies are used:a) the Gibbs Free energy ∆G , b) the gross enthalpy ∆H gro andc) the net calorific enthalpy ∆H net . Often the net calorificvalue is used [94]; in this case the definition is:

U nettheo =

−∆H net

z ·F (G.16)

(see also ideal cell voltage)

Bibliography

[1] P. ARGYROPOULOS, KEITH SCOTT, A.K. SHUKLA, C. JACKSON ◦ A semi-empirical model of the direct methanol fuel cell performance Part I. Modeldevelopment and verification ◦ Journal of Power Sources, vol. 123, pp. 190–199, 2003

[2] ANTONINO S. ARICÒ, S. SRINIVASAN, V. ANTONUCCI ◦ DMFCs: From funda-mental aspects to technology development ◦ Fuel Cells, no. 2, pp. 133–161,2001

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Index

abbreviations, 108ABP, 108ACL, 108activation Gibbs function, 124, 147activity, 149ADL, 108AMI, 25, 108analytical model, 137anode, 137Arrhenius

behavior, 22, 30, 137equation, 64, 116, 137

Arrhenius equation, 20

bracketing algorithm, 59bracketing search, 59Broydon method, 51Bruggemann relationship, 137Butler-Volmer equation, 20, 123–130, 138

carbon support, 90cathode, 140cathode flooding, 41, 52, 66CBP, 109CCL, 109CDE, see current density equivalentCDL, 109cell voltage

ideal, 147open circuit, 146theoretical, 154

cell potentialzero-current, 147

CFD, 3, 4, 109charge-transfer coefficient, see transfer

coefficientchemical potential, 140circuit voltage

closed, 150CO, 108coefficient

electroosmotic water drag, 27conversion ratio, 143conversion ratio

thermo neutral, 143coulomb potential, 145crossover, 6

current, 148flow, 88reaction, 12

cubic spline interpolation, 75current

parasitic, 150current density

equivalent, 71exchange, 154

Darcy’s law, 11, 13, 17, 88, 97, 141Design Spec, 50design specification, 50DHE, 141double precision number, 120double layer, 141DS-EODRG, 55, 58, 60dynamic hydrogen electrode, 5, 141dynamic viscosity, 88

efficiency, 142conversion, 142coulombic, 142Faraday, 142ideal, 143thermo neutral, 142thermodynamic, 143utilization, 142voltage, 143

electrochemical potential, 143electrode potential, 2electromotive force, 143, 147electroosmotic drag, 26, 143electrostatic potential

Galvani, 144

172 Index

inner, 144outer, 144Volta, 144

EMF, see electromotive forceequation oriented mode, 4Evans diagram, 142exchange current density, 20, 127, 154extent of reaction, see reaction extend

Faradaic oxidation, 146Faraday’s law

First, 4, 91, 99, 101, 146Faraday’s laws, 146Fick’s First law, 17, 28, 146Fick, Adolf, 146flooding, see cathode floodingformal potential, 126frequency factor, 124, 137fuel efficiency, 2functional layer, 10, 13

Galvani potential, 145Gibbs Free energy, 147golden section search, 59Gouy-Chapman model, 141Griffiths model, 34GUI, 46

Hagen Poiseuille equation, 14Helmholtz plane, 147

Inner, 141Outer, 141

hydraulic permeability, 17

inner polarization, 43inner potential, 145interpolation

cubic spline, 75intrinsic rate constant, see standard rate

constantIUPAC, 109

Langmuir adsorption, 5, 35, 66, 87Langmuir region, 67LCCN, 155leakage current, 148least squares, 76limiting current, 148local-element, 13

MEA, 109

methanol stoichiometry, 149mixed potential, 2, 5, 148multi-domain

approach, 149mode, 4

Nernst equation, 21, 149Newton algorithm, 51nominal methanol stoichiometry, 149normalized value, 76numerical stability, 53

OCPoxygen reduction, 9, 38

OCV, 146open circuit

cell voltage, 146potential, 4, 9, 70

open half cell potential, 39over-potential, 150overvoltage, 150

crystallization, 150oxygen reduction, 34

Pauling model, 34PEM-FC, 41PEM-HFC, 5PEM-LDMFC, 6PFS, 46, 110Poiseuille, see Hagen Poiseuille equationpolarization, 150

anode, 89potential, 151

absolute electrode, 146coulomb, 138electrochemical, 143electrostatic, 144Galvani, 138, 145inner, 138, 145standard equilibrium, 138Volta, 144

pre-exponential factor, see frequency fac-tor

process flowsheet, 46

rate determining step, 19, 21reaction extend, 151reference electrode, 152reference potential, 21rest potential, 147

Index 173

reversible hydrogen electrode, 95, 141,151

RHE, see reversible hydrogen electroderounding error, 53

sequential modular mode, 4, 149SHE, see standard hydrogen electrodesingle-domain

approach, 149mode, 4

spline interpolationcubic, 75

standardconditions, 152equilibrium voltage, 146hydrogen electrode, 21, 152potential, 152rate constant, 126voltage, 146

Stern model, 141Stockholm convention, 137, 140, 145stoichiometric factor, 152

nominal, 73, 153stoichiometry coefficient, 152subscripts, 108superscripts, 108supported catalyst, 11, 90surface potential, 42, 145surface voltage, see surface potentialsymbols, 105symmetry factor, see transfer coefficient

Tafel equation, 154Tafel slope, 37, 154Temkin adsorption, 5, 35, 66, 87three phase contact, 90transfer coefficient, 20, 138transition concentration, 85transition potential, 5, 35, 100truncation error, 53

velocitysuperficial, 141

Volta potential, 144voltage

efficiency, 107standard, 146standard equilibrium, 146

zero-current cell potential, 147

zero-order, 85ZSW, xiii, 18, 101, 102

Modelingthe DirectMethanolFuelCell

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