Transport Phenomena In Fuel Cells (Developments in Heat Transfer)

TRANSPORT PHENOMENA IN FUEL CELLS

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International Series on Developments in Heat Transfer Objectives The Developments in Heat Transfer book Series publishes state-of-the-art books and provides valuable contributions to the literature in the field of heat transfer. The overall aim of the Series is to bring to the attention of the international community recent advances in heat transfer by authors in academic research and the engineering industry. Research and development in heat transfer is of significant importance to many branches of technology, not least in energy technology. Developments include new, efficient heat exchangers, novel heat transfer equipment as well as the introduction of systems of heat exchangers in industrial processes. Application areas include heat recovery in the chemical and process industries, and buildings and dwelling houses where heat transfer plays a major role. Heat exchange combined with heat storage is also a methodology for improving the energy efficiency in industry, while cooling in gas turbine systems and combustion engines is another important area of heat transfer research. To progress developments within the field both basic and applied research is needed. Advances in numerical solution methods of partial differential equations, high-speed, efficient and cheap computers, advanced experimental methods using LDV (laser-doppler-velocimetry), PIV (particleimage-velocimetry) and image processing of thermal pictures of liquid crystals, have all led to dramatic advances during recent years in the solution and investigation of complex problems within the field. The aims of the Series are achieved by contributions to the volumes from invited authors only. This is backed by an internationally recognised Editorial Board for the Series who represent much of the active research worldwide. Volumes planned for the series include the following topics: Compact Heat Exchangers, Engineering Heat Transfer Phenomena, Fins and Fin Systems, Condensation, Materials Processing, Gas Turbine Cooling, Electronics Cooling, Combustion-Related Heat Transfer, Heat Transfer in Gas-Solid Flows, Thermal Radiation, the Boundary Element Method in Heat Transfer, Phase Change Problems, Heat Transfer in Micro-Devices, Plateand-Frame Heat Exchangers, Turbulent Convective Heat Transfer in Ducts, Enhancement of Heat Transfer and other selected topics.

Series Editor B. Sundén Lund Institute of Technology Box 118 22100 Lund Sweden

Associate Editors C.I. Adderley Rolls Royce and Associates Limited UK

S. del Guidice University of Udine Italy

E. Blums Latvian Academy of Sciences Latvia

M. Faghri The University of Rhode Island USA

C.A. Brebbia Wessex Institute of Technology UK

P.J. Heggs UMIST UK

G. Comini University of Udine Italy

C. Herman John Hopkins University USA

R.M. Cotta COPPE/UFRJ, Brazil

D.B. Ingham University of Leeds UK

L. De Biase University of Milan Italy

Y. Jaluria Rutgers University USA

G. De Mey University of Ghent Belgium

S. Kotake University of Tokyo Japan

G. de Vahl Davies University of New South Wales Australia

P.S. Larsen Technical University of Denmark Denmark

D.B. Murray Trinity College Dublin Ireland

A.C.M. Sousa University of New Brunswick Canada

A.J. Nowak Silesian University of Technology Poland

D.B. Spalding CHAM UK

K. Onishi Ibaraki University Japan

J. Szmyd University of Mining and Metallurgy Poland

P.H. Oosthuizen Queen’s University Kingston Canada

E. Van den Bulck Katholieke Universiteit Leuven Belgium

W. Roetzel Universtaet der Bundeswehr Germany

S. Yanniotis Agricultural University of Athens Greece

B. Sarler University of Ljubljana Slovenia

TRANSPORT PHENOMENA IN FUEL CELLS

Editors B. Sundén Lund Institute of Technology, Sweden.

M. Faghri University of Rhode Island, USA.

TRANSPORT PHENOMENA IN FUEL CELLS Series: Developments in Heat Transfer, Vol. 19 Editors: B. Sundén and M. Faghri

Published by

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Contents Preface Chapter 1: Multiple transport processes in solid oxide fuel cells P.-W. Li, L. Schaefer & M.K. Chyu 1 2

3

4

5

Introduction......................................................................................... Thermodynamic and electrochemical fundamentals for solid oxide fuel cells ........................................................................... 2.1 Operation with hydrogen fuel ..................................................... 2.2 Operation with methane through internal reforming and shift reactions.............................................................................. Electrical potential losses .................................................................... 3.1 Activation polarization................................................................ 3.2 Ohmic loss ................................................................................. 3.3 Mass transport and concentration polarization ........................... Computer modeling of a tubular SOFC............................................... 4.1 Outline of a computation domain................................................ 4.2 Governing equations and boundary conditions ........................... 4.3 Numerical computation............................................................... 4.4 Typical results from numerical computation for tubular SOFCs.. 4.4.1 The SOFC terminal voltage ............................................. 4.4.2 Cell temperature distribution ........................................... 4.4.3 Flow, temperature and concentration fields ..................... Concluding remarks ............................................................................

Chapter 2: Numerical models for planar solid oxide fuel cells S.B. Beale 1

Introduction......................................................................................... 1.1 History and types of solid oxide fuel cell ................................... 1.2 Survey of modeling techniques ................................................... 1.3 Thermodynamics of solid oxide fuel cells...................................

xv 1 1 3 4 8 11 12 14 18 19 21 22 26 27 27 30 31 34 43 44 44 45 46

2

3 4

1.4 Cell voltage and current .............................................................. 1.5 Activation losses ......................................................................... 1.6 Diffusion losses........................................................................... 1.7 Basic computational algorithm................................................... Computer schemes .............................................................................. 2.1 General scalar equation............................................................... 2.2 Continuity ................................................................................... 2.3 Momentum ................................................................................. 2.4 Heat transfer................................................................................ 2.5 Mass transfer............................................................................... 2.6 Numerical integration schemes ................................................... 2.7 Iterative procedure...................................................................... 2.8 Additional chemistry and electrochemistry................................. 2.9 Porous media flow ...................................................................... 2.10 Current and voltage distribution.................................................. 2.11 Advanced diffusion models ........................................................ 2.12 Thermal radiation........................................................................ Stack models ....................................................................................... Closure................................................................................................

Chapter 3: Electrochemical and thermo-fluid modeling of a tubular solid oxide fuel cell with accompanying indirect internal fuel reforming K. Suzuki, H. Iwai & T. Nishino 1 2

3

Introduction......................................................................................... General remarks on the mechanism of IIR-T-SOFC ........................... 2.1 Tubular cell................................................................................. 2.2 Internal reforming process .......................................................... 2.3 Electrochemical process.............................................................. 2.4 Purpose and key points of the analysis........................................ Numerical modeling............................................................................ 3.1 Computational domain and general assumptions for heat and mass transfer ................................................................ 3.2 Model for electrochemical reactions ........................................... 3.3 Model for internal fuel reforming ............................................... 3.4 Governing equations of velocity, temperature and concentration fields and boundary conditions ............................. 3.5 Discretization scheme................................................................. 3.6 Equations for electric potential and electric circuit .................... 3.7 Mass production or consumption rate of each chemical species through electrochemical and reforming reactions............. 3.8 Model for thermodynamic heat generation rates ......................... 3.9 Ohmic heat generation ................................................................

49 52 54 57 58 59 60 60 62 64 64 65 66 67 68 70 71 73 75

83 84 86 86 88 88 90 91 91 93 95 95 98 101 102 103 104

4

5

3.10 Radiation model.......................................................................... 3.11 Overall picture of the model ....................................................... Results and discussion......................................................................... 4.1 Results for a cathode-supported tubular SOFC without accompanying indirect internal reforming [16,43,44].................................................................................. 4.2 Results for the Base case with accompanying indirect internal reforming ....................................................................... 4.2.1 Thermal and concentration fields..................................... 4.2.2 Electric potential and current fields ................................. 4.2.3 Power generation characteristics...................................... 4.3 Strategies for the ideal thermal field ........................................... 4.3.1 Effect of gas inlet temperature ......................................... 4.3.2 Effect of air flow rate....................................................... 4.3.3 Effect of density distribution of catalyst ......................... Conclusions.........................................................................................

Chapter 4: On heat and mass transfer phenomena in PEMFC and SOFC and modeling approaches J. Yuan, M. Faghri & B. Sundén 1 2

3

4

5

Introduction......................................................................................... Fuel cell modeling development......................................................... 2.1 Basics of SOFCs and PEMFCs................................................... 2.2 Modeling development............................................................... 2.2.1 Modeling approaches....................................................... 2.2.2 Various existing models ................................................... Main processes in SOFCs and PEMFCs ............................................. 3.1 Gas transport............................................................................... 3.2 Electrochemical reactions ........................................................... 3.3 Heat transfer ............................................................................... 3.4 Various transport processes in the electrodes (porous layers) ............................................................................ 3.5 Other processes appearing in fuel cell components..................... Processes and issues in SOFC and PEMFC ........................................ 4.1 Water management in PEMFCs .................................................. 4.2 Fuel reforming issues in SOFC................................................... Modeling methodologies for transport processes in SOFC and PEMFC.............................................................................. 5.1 General considerations................................................................ 5.2 Assumptions................................................................................ 5.3 Governing equations ................................................................... 5.4 Boundary and interfacial conditions............................................ 5.5 Additional equations ...................................................................

104 107 108

108 113 115 116 119 121 121 123 123 125

133 133 135 135 137 137 138 141 142 142 143 143 144 144 144 145 146 146 147 147 149 150

6

7

5.6 Solution methodology................................................................. Results and discussions ....................................................................... 6.1 Mass transfer effects on the gas flow and heat transfer ............... 6.2 Porous layer effects on the transport processes ........................... 6.2.1 Transport processes in PEMFCs ...................................... 6.2.2 Transport processes in SOFCs ......................................... 6.3 Two-phase flow and its effects on the cell performance ............. Conclusions.........................................................................................

Chapter 5: Two-phase transport in porous gas diffusion electrodes S. Litster & N. Djilali 1

2

3

4

5

Introduction......................................................................................... 1.1 PEM fuel cells ............................................................................ 1.2 Porous media .............................................................................. 1.3 Porous media in PEMFC electrodes ........................................... Single-phase transport......................................................................... 2.1 Transport of a single-phase with a single component ................. 2.1.1 Permeability .................................................................... 2.2 Transport of a single-phase with two components ...................... 2.2.1 Effective diffusivity in porous media............................... 2.2.2 Determination of the binary diffusion coefficient .............. 2.2.3 Transport of a single-phase with more than two components...................................................................... 2.3 Knudsen diffusion....................................................................... 2.4 Determination of Knudsen diffusivity........................................ 2.5 Knudsen transition regime .......................................................... 2.5.1 Comparison of the diffusivities ........................................ Two-phase systems.............................................................................. 3.1 Two-phase regimes ..................................................................... 3.2 Hydrodynamics and capillarity in two-phase systems................. 3.2.1 Capillary pressure curves ................................................. 3.3 Relative permeability.................................................................. Multiphase flow models...................................................................... 4.1 Multi-fluid model ....................................................................... 4.1.1 Phase change.................................................................... 4.1.2 Application ...................................................................... 4.2 Mixture model ............................................................................ 4.3 Moisture diffusion model............................................................ 4.4 Porosity correction model ........................................................... 4.5 Evaluation of the multiphase models in the literature ................ Outstanding issues and conclusions ....................................................

151 152 152 155 155 160 163 168 175 175 175 177 178 181 181 181 182 183 185 185 186 187 187 187 188 189 191 195 197 200 200 201 203 205 207 208 208 208

Chapter 6: Numerical simulation of proton exchange membrane fuel cell T.C. Jen, T.Z. Yan & Q.H. Chen 1 2

Introduction......................................................................................... One-dimensional (1-D) model ............................................................ 2.1 General 1-D model ..................................................................... 2.1.1 Model description ........................................................... 2.1.2 Model assumptions .......................................................... 2.1.3 Governing equations........................................................ 2.1.4 Catalyst layers ................................................................. 2.1.5 Membrane........................................................................ 2.1.6 Boundary conditions........................................................ 2.1.7 Results and discussion ..................................................... 2.1.8 Summary.......................................................................... 2.2 General 2-D model ..................................................................... 2.2.1 Model description and assumptions................................. 2.2.2 Mathematical model ........................................................ 2.2.3 Boundary conditions........................................................ 2.2.4 Numerical procedures ..................................................... 2.2.5 Results and discussion ..................................................... 2.2.6 Concluding remarks......................................................... 2.3 Three-dimensional (3-D) model.................................................. 2.3.1 Model development ......................................................... 2.3.2 Mathematical model ........................................................ 2.3.3 Boundary conditions........................................................ 2.3.4 Discretization strategies................................................... 2.3.5 Solution algorithms ......................................................... 2.3.6 Results and discussion ..................................................... 2.4 Summary and conclusion ...........................................................

Chapter 7: Mathematical modeling of fuel cells: from analysis to numerics M. Vynnycky & E. Birgersson 1 2

Introduction ........................................................................................ PEFC................................................................................................... 2.1 Mathematical formulation for flow in the cathode ...................... 2.1.1 Channel............................................................................ 2.1.2 Porous backing ................................................................ 2.1.3 Boundary conditions........................................................ 2.2 Nondimensionalization ............................................................... 2.3 Parameters .................................................................................. 2.4 Narrow-gap approximation......................................................... 2.5 Further simplifications and observations ....................................

215 216 216 217 217 218 218 219 220 221 223 223 224 227 227 230 230 230 232 233 233 234 238 238 238 239 244

247 247 250 251 251 253 254 256 258 258 261

3

4

2.6 Numerics and results .................................................................. 2.6.1 Effect of A and Q ............................................................ 2.6.2 'Polarization surfaces' ...................................................... DMFC ................................................................................................. 3.1 Mathematical formulation for flow in the anode......................... 3.1.1 Channel............................................................................ 3.1.2 Porous backing ................................................................ 3.1.3 Boundary conditions........................................................ 3.2 Nondimensionalization ............................................................... 3.3 Parameters .................................................................................. 3.4 Narrow-gap approximation ......................................................... 3.5 Further simplifications and observations..................................... 3.6 Numerics and results................................................................... Conclusions.........................................................................................

Chapter 8: Modeling of PEM fuel cell stacks with hydraulic network approach J.J. Baschuk & X. Li 1 2

3

4 5

Introduction......................................................................................... Model formulation .............................................................................. 2.1 Stack flow model ........................................................................ 2.2 Manifold pressure loss ................................................................ 2.3 Cell pressure loss ........................................................................ 2.4 Mass consumed in the catalyst layers.......................................... 2.5 Boundary conditions ................................................................... Numerical procedure .......................................................................... 3.1 Outer iteration............................................................................. 3.2 Inner iteration ............................................................................. 3.3 Numerical procedure summary .................................................. Results and discussion......................................................................... Conclusions.........................................................................................

Chapter 9: Two-phase microfluidics, heat and mass transport in direct methanol fuel cells G. Lu & C.-Y. Wang 1 2

3

Introduction......................................................................................... Fundamentals of DMFC...................................................................... 2.1 Cell components and polarization curve .................................... 2.2 Thermodynamics......................................................................... 2.3 Methanol oxidation and oxygen reduction kinetics..................... Two-phase flow phenomena................................................................ 3.1 Bubble dynamics in anode .........................................................

264 264 268 270 271 271 271 272 272 273 273 273 277 278 283 284 285 288 290 292 293 294 294 295 297 298 298 309

317 317 320 320 322 324 325 325

4 5 6

7 8

3.1.1 Flow visualization................................................................ 3.1.2 Bubble diameter and drift velocity....................................... 3.1.3 Pressure drop ....................................................................... 3.2 Liquid water transport in cathode............................................... 3.2.1 Flooding in the cathode........................................................ 3.2.2 Flooding visualization.......................................................... Mass transport phenomena ................................................................. 4.1 Methanol crossover ........................................................................ 4.2 Water management in portable DMFC systems.............................. Heat transport.......................................................................................... Mathematical modeling and experimental diagnostics ............................ 6.1 Mathematical modeling .................................................................. 6.2 Experimental diagnostics ................................................................ 6.3 Model validation............................................................................. Application: micro DMFC ...................................................................... Summary and outlook .............................................................................

325 328 329 329 329 331 333 333 335 337 339 340 341 343 344 350

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Preface Fuel cells are expected to play a significant role in the next generation of energy systems and road vehicles for transportation. However, substantial progress is required in reducing manufacturing costs and improving the performance. Solid Oxide fuel cells (SOFC), Proton Exchange Membrane fuel cells (PEMFC) and Direct Methanol fuel cells (DMFC) are of current interest. Many of the associated heat and mass transport processes are not well understood and include multidimensional flow and heat transfer in multi phase flows, multicomponent transport of gaseous species in porous media and electrochemical reactions including heat generation. Depending on the fuel being used, modifications in the design of the next generation of fuel cells are needed. Therefore, additional transport processes even at micro scale level need to be investigated. Other important considerations are air management system for oxygen supply, water management and recovery or rejection of heat of exhaust products. Currently an extensive amount of research and development activities are carried out for fuel cells worldwide. The dissemination of results is through various generic and specialized journals and conference proceedings. There is no comprehensive book available to address the analysis of transport phenomena in fuel cells. This book aims to contribute to the understanding of the transport processes in SOFC, PEMFC and DMFC fuel cells. The nine chapters cover a wide range of topics and are invited contributions from some prominent scientists in the field. The first chapter presents the thermodynamic and electrochemical fundamentals of SOFC. Efficiency, energy distribution, chemical equilibrium, losses of electrical potential, ohmic losses and losses due to mass transfer resistance are discussed. Modeling and numerical simulations of the coupled transport processes which determine the local and overall electromotive force in a SOFC are provided. Chapter 2 discusses various numerical techniques to model single-cells and stacks of planar SOFC. In chapter 3 a numerical model for analysis of a tubular SOFC including indirect internal reforming is presented. Fundamental results and strategies to reduce the maximum temperature and temperature gradient of the cell are discussed. Chapter 4 provides numerical analysis of heat, mass transfer (species flow), two-phase transport and effects on the performance in SOFC and PEMFC. In chapter 5 information on transport

phenomena in the electrodes of PEMFC is provided. The physical characteristics of such electrodes are also discussed. The focus is on two-phase flow in porous media, with a discussion of driving forces and various flow regimes. Also, the mathematical models are summarized. Chapter 6 presents mathematical models and numerical simulations of PEMFC to evaluate effects of various designs and operating parameters on the fuel cell performance. The three-dimensional model can be used for optimisation of design and operation and serve as a building block for modeling and understanding of PEMFC stacks and systems. In chapter 7 scaling analysis, nondimensionalization and asymptotic techniques are used to identify the governing parameters in order to obtain a simplified model. Illustrations are provided for PEMFC and DMFC. In chapter 8 a mathematical model for a PEMFC stack is formulated. Distributions of pressure, fuel and oxidant mass flow rates in the stack are determined by a hydraulic network analysis. Finally chapter 9 provides an overview of the latest developments in the DMFC technology. Experimental and modeling works to elucidate critical transport phenomena, including two-phase mierofluidics, heat and mass transport are presented. All of the chapters follow a unified outline and presentation to aid accessibility and the book provides invaluable information for both graduate research and R & D engineers at industry and consultancy. We are grateful to the authors and reviewers for their excellent contributions. We also appreciate the cooperation and patience provided by the staff of WIT Press and for their encouragement and assistance in producing this volume. The editors would also like to thank the Wenner-Gren Center Foundation in Sweden for financial support. B. Sundén and M. Faghri 2005

CHAPTER 1 Multiple transport processes in solid oxide fuel cells P.-W. Li, L. Schaefer & M.K.. Chyu Department of Mechanical Engineering, University of Pittsburgh, USA.

Abstract In this topic, three important issues are discussed which concern the theoretical fundamentals and practical operation of a solid oxide fuel cell. The thermodynamic and electrochemical fundamentals of a fuel cell are reviewed in the Section 2. These fundamentals concern the ideal efficiency and energy distribution of a fuel cell’s conversion of chemical energy directly into electrical energy through the oxidation of a fuel. Issues of the chemical equilibrium for a solid oxide fuel cell with internal reforming and shift reactions (in case of methane or natural gas being used as the fuel), are also discussed in detail in this section. The losses of electrical potential in the practical operation of a fuel cell are elucidated in the third section, which includes a discussion about activation polarization, Ohmic loss, and the losses due to mass transport resistance. In the fourth section, the coupled processes of flow, heat/mass transfer, chemical reaction, and electrochemistry, which influence the performance of a fuel cell, are analyzed, and modeling and numerical computation for the fields of flow, temperature, and species concentration, which collectively determine the local and overall electromotive force in a solid oxide fuel cell, are examined in detail.

1 Introduction A fuel cell is a device that converts the chemical energy of a fuel oxidation reaction directly into electricity. It is substantially different from a conventional thermal power plant, where the fuel is oxidized in a combustion process and a thermalmechanical-electrical energy conversion process is employed. Therefore, unlike heat engines that are subjected to the Carnot cycle efficiency limitation, fuel cells can have energy conversion efficiencies generally higher than that of heat engines [1].

2 Transport Phenomena in Fuel Cells

Figure 1: Principle of operation of a SOFC. Ideally, the Gibbs free energy change of fuel oxidation is directly converted into electricity [1, 2] in a fuel cell. As is common in many kinds of fuel cells, the core component of a solid oxide fuel cell (SOFC) is a thin gas-tight ion conducting electrolyte layer sandwiched by a porous anode and cathode, as shown in Fig. 1. For a SOFC, this electrolyte is a solid oxide material that only allows the passage of charge-carrying oxide ions. To produce useful electrical work, free electrons released in the oxidation of a fuel at the anode must travel to the cathode through an external load/circuit. Therefore, the electrolyte must conduct ions while preventing electrons released at the anode from returning back to the cathode by the same route. The oxide ions are driven across the electrolyte by the chemical potential difference on the two sides of the electrolyte, which is due to the oxidation of fuel at the anode. This difference in the chemical potential is proportional to the electromotive force across the electrolyte, which, therefore, sets up a terminal voltage across the external load/circuit. The solid oxide electrolyte has sufficient ion conductivity only at high temperatures (from 600–1000 ◦ C). The high operating temperature of a SOFC also ensures rapid fuel-side reaction kinetics without requiring an expensive catalyst. In addition, the high temperature exhaust from a SOFC can be directed to a gas turbine (GT); thus, using a SOFC-GT hybrid system, one can achieve an efficiency of at least 66.3% based on the lower heating value (LHV, which means that the electrochemical product, water, is in a gaseous state) of the SOFC [3–6]. Since it operates via transport of oxide ions rather than that of fuel-derived ions, in principle, a SOFC can be used to oxidize a number of gaseous fuels. In particular, a SOFC can consume CO as well as hydrogen as its fuel, and therefore can be fueled with reformer gas containing a mix of CO and H2 [7, 8]. Recently, ammonia has also been reported as a fuel for SOFCs [9]. Since a SOFC operates under high temperatures, its energy conversion efficiency and component safety are both of concern to industry. In the following sections, the issues to be discussed will include: (1) the thermodynamic and electrochemical fundamentals of the energy conversion and species variation, (2) the potential losses in practical operation, (3) the influence of fluid flow and heat and mass transfer on operational efficiency and safety, and (4) the creation of a numerical model to simulate the performance and the fields of flow, temperature, and species concentration.

3

Multiple transport processes in solid oxide fuel cells

2 Thermodynamic and electrochemical fundamentals for solid oxide fuel cells To study the energy conversion efficiency and distribution of the conversion processes in a fuel cell, one must understand the basic principles. The chemical potential and, thereof, electromotive force across the electrolyte involve the interrelation of thermodynamics, electrochemistry, ion/electron conduction, and heat/mass transfer. In this section, the fundamentals of thermodynamics and the electrochemistry for a solid oxide fuel cell system are reviewed. The isothermal oxidation of a fuel A with oxidant B can be expressed by the following equation: aA + bB + · · · → xX + yY + · · ·.

(1)

The systematic changes of enthalpy, Gibbs free energy, and entropy production in the reaction are related by
H = T
S +
G.

(2)

In a solid oxide fuel cell, the operating temperature is from 600 ◦ C to 1000 ◦ C and the pressure of gases is relatively not high. Thus, the gas species of reactants and products can be treated as ideal gases, which allows the chemical enthalpy change to be expressed as:
H = (xhX + yhY + · · ·) − (ahA + bhB + · · ·),

(3)

where the h is the specific enthalpy. When a gas is pure, ideal, and at 1 atm, it is said to be in its standard state. The standard state is designated by writing a superscript 0 after the symbol of interest [10]. The Gibbs free energy which pertains to one mole of a chemical species is called the chemical potential. For an ideal gas at temperature of T and pressure of p, the chemical potential is expressed as: g = g 0 + RT ln

p , p0

(4)

where R is the gas constant and p0 is the standard pressure of 1 atm. One may omit the p0 in the denominator of the logarithm in eqn (4), but in such a case, the pressure p must be measured in atm. The systematic change of the Gibbs free energy in eqn (1) can be expressed in terms of the standard state Gibbs free energy and the partial pressures of the reactants and products:
G = (xgX + ygY + · · ·) − (agA + bgB + · · ·) = [xgX0 + ygY0 + · · ·] − [agA0 + bgB0 + · · ·] + RT ln =
G 0 + RT ln

(pX /p0 )x (pY /p0 )y · · · , (pA /p0 )a (pB /p0 )b · · ·

(pX /p0 )x (pY /p0 )y · · · (pA /p0 )a (pB /p0 )b · · · (5)

4 Transport Phenomena in Fuel Cells where
G 0 = (xgX0 + ygY0 + · · ·) − (agA0 + bgB0 + · · ·),

(6)

which is the Gibbs free energy change of the standard reaction at temperature T (i.e., with each reactant supplied and each product removed at the standard atmospheric pressure, p0 = 1 atm). The theoretical electromotive force (EMF) induced from the chemical potential (
G) is the Nernst potential: E=

−
G −
G0 RT (pA /p0 )a (pB /p0 )b · · · , = + ln ne F ne F ne F (pX /p0 )x (pY /p0 )y · · ·

(7)

where F(=96486.7 C/mol) is Faraday’s constant. The first part of the righthand side of the standard reaction is also called the ideal potential, which is denoted by: −
G0 E0 = , (8) ne F where ne is the number of electrons derived from a molecules of the fuel, when the fuel is oxidized in the reaction of eqn (1). While the Gibbs free energy change, −
G, converts to electrical power, the entropy production, −T
S, is the thermal energy that is released in the electrochemical oxidation of the fuel. Both the h and g 0 are solely functions of temperature for ideal gases, which are given in Tables 1(a) and 1(b) for the gas species involved in the reactions of a SOFC. While the electromotive force in a fuel cell is determinable from the chemical potentials as discussed above, the current to be withdrawn from a fuel cell, denoted by I , is directly related to the molar consumption rate of fuel and oxidant through the following expressions: mfuel =

I nfuel e F

;

mO2 =

I 2 nO e F

,

(9)

2 where nO e is for oxygen, and is the number of electrons per b molecules of oxygen is the number of obtained in the electrochemical reaction (in eqn (1)), and nfuel e electrons derived per a molecules of the fuel.

2.1 Operation with hydrogen fuel If a SOFC operates on hydrogen gas, the oxidation of hydrogen is the only electrochemical reaction in the fuel cell, which may be expressed by the following chemical equation: 1 H2 + O2 = H2 O (gas). (10) 2

Multiple transport processes in solid oxide fuel cells

5

Table 1(a): Enthalpy and standard state Gibbs free energy of species. Formula weight

CO 28.01

CO2 44.01

H2 2.016

T (K)

h (kJ/k mol)

g◦ (kJ/k mol)

h (kJ/k mol)

g◦ (kJ/k mol)

h (kJ/k mol)

g◦ (kJ/k mol)

298.15 300 320 340 360 380 400 420 440 460 480 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

−110530 −110476 −109892 −109309 −108725 −108141 −107555 −106967 −106377 −105887 −105195 −104599 −103102 −101588 −100053 −98507 −96938 −95353 −93749 −92129 −90499 −88840 −85495 −82100 −78662 −75187 −71680 −68145 −64585 −61004 −57404

−169474 −169816 −173796 −177819 −181877 −185927 −190035 −194201 −198337 −202671 −206763 −210999 −221737 −232568 −243573 −254677 −265838 −277193 −288569 −300119 −311659 −323340 −346965 −370940 −395082 −419587 −444280 −469265 −494515 −519824 −545324

−393510 −393441 −392687 −391916 −391128 −390326 −389507 −388675 −387827 −386966 −386094 −385205 −382938 −380603 −378207 −375756 −373250 −370704 −368112 −365480 −362821 −360113 −354626 −349037 −343362 −337614 −331805 −325941 −320030 −314079 −308091

−457254 −457641 −461967 −466308 −470688 −475142 −479627 −484141 −488719 −493318 −497982 −502655 −514498 −526583 −538822 −551316 −564800 −576704 −589622 −602720 −615996 −629413 −656576 −684317 −712432 −741094 −770105 −799541 −829350 −859479 −889871

0 53 630 1209 1791 2373 2959 3544 4131 4715 5298 5882 6760 8811 10278 11749 13223 14702 16186 17676 19175 20680 23719 26797 29918 33082 36290 39541 42835 46169 49541

−38968 −39217 −41866 −44521 −47241 −49953 −52721 −55508 −58305 −61157 −64062 −66968 −74970 −81849 −89432 −97171 −104977 −112898 −121004 −129114 −137290 −145520 −162291 −179363 −196672 −214158 −231910 −249899 −268095 −286471 −305189

The Nernst potential from this electrochemical reaction will be: E(H2 +1/2O2 =H2 O) =

RT
ln( pH2 /p0 )anode 2F 2F  0 + ln( pO2 /p0 )0.5 cathode − ln( pH2 O /p )anode . 0 −
G(H 2 +1/2O

2 =H2 O)

+

(11)

The ideal chemical potentials at the temperature T (K) can be calculated from the data given by handbooks [11]. As a convenient reference, Table 1(c) gives the

6 Transport Phenomena in Fuel Cells Table 1(b): Enthalpy and standard state Gibbs free energy of species. O2 31.999

Formula weight T (K) 298.15 300 320 340 360 380 400 420 440 460 480 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

H2 O (Gas) 18.015

CH4 16.043

h (kJ/k mol)

g◦ (kJ/k mol)

h (kJ/k mol)

g◦ (kJ/k mol)

h (kJ/k mol)

g◦ (kJ/k mol)

0 54 643 1234 1828 2425 3025 3629 4236 4847 5463 6084 7653 9244 10859 12499 14158 15835 17531 19241 20965 22703 26212 29761 33344 36957 40599 44266 47958 51673 55413

−61151 −61536 −65661 −69826 −74024 −78325 −82495 −86797 −91112 −95433 −99849 −104266 −115382 −126656 −138056 −149551 −161117 −172885 −184684 −196669 −208745 −220897 −245378 −270239 −295426 −320883 −346551 −372374 −398632 −424967 −451507

−241814 −241752 −241079 −240404 −239726 −239045 −238362 −237675 −236985 −236291 −235592 −234889 −233115 −231313 −229493 −227622 −225732 −223812 −221860 −219876 −217860 −215814 −211623 −207308 −202872 −198321 −193663 −188906 −184056 −179121 −174108

−298105 −298452 −302263 −306126 −309998 −313943 −317882 −321885 −325865 −329947 −334040 −338139 −348890 −359173 −369828 −380712 −391707 −402852 −414130 −425526 −436930 −448514 −471993 −495908 −520072 −544681 −569563 −594826 −620276 −646221 −672288

– −74448 −73718 −72974 −72213 −71432 −70631 −69808 −68962 −68094 −67202 −66287 −63892 −61356 −58671 −55853 −52897 −49818 −46613 −43296 −39866 −36336 −28981 −21274 −13254 −4956 3587 12347 21295 30406 39658

– −130398 −134166 −137948 −141801 −145684 −149591 −153598 −157578 −161658 −165698 −169837 −180327 −191016 −201899 −213073 −224347 −235898 −247638 −259566 −271666 −283936 −309041 −334834 −361264 −388416 −416113 −444293 −473235 −502574 −532432

ideal chemical potentials and enthalpies for the gas species that are typically utilized in a SOFC. Recognizing the electrochemical equilibrium in the anode gas mixture:
0 + RT ln( pH2 /p0 )anode + ln( pO2 /p0 )0.5
G = −
G(H anode 2 +1/2O2 =H2 O)  (12) − ln( pH2 O /p0 )anode = 0.

Multiple transport processes in solid oxide fuel cells

7

Table 1(c): Change of enthalpy and standard state Gibbs free energy of reactions. Reaction T (K) 298.15 300 320 340 360 380 400 420 440 460 480 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

H2 + 1/2O2 = H2 O (gas)

CH4 + H2 O = 3H2 + CO

CO + H2 O = H2 + CO2


H (kJ/mol)


G 0 (kJ/mol)

E0 (V)


H (kJ/mol)


G 0 (kJ/mol)


H (kJ/mol)


G 0 (kJ/mol)

−241.814 −241.832 −242.031 −242.230 −242.431 −242.631 −242.834 −243.034 −243.234 −243.430 −243.622 −243.813 −243.702 −244.746 −245.201 −245.621 −246.034 −246.432 −246.812 −247.173 −247.518 −247.846 −248.448 −248.986 −249.462 −249.882 −250.253 −250.580 −250.870 −251.127 −251.356

−228.561 −228.467 −227.567 −226.692 −225.745 −224.828 −223.914 −222.979 −222.004 −221.074 −220.054 −219.038 −216.229 −213.996 −211.368 −208.766 −206.172 −203.512 −200.784 −198.078 −195.268 −192.546 −187.013 −181.426 −175.687 −170.082 −164.378 −158.740 −152.865 −147.267 −141.346

1.184 1.184 1.179 1.175 1.170 1.165 1.160 1.155 1.150 1.146 1.140 1.135 1.121 1.109 1.095 1.082 1.068 1.055 1.040 1.026 1.012 0.998 0.969 0.940 0.910 0.881 0.852 0.823 0.792 0.763 0.732

– 205.883 206.795 207.696 208.587 209.455 210.315 211.148 211.963 212.643 213.493 214.223 214.185 217.514 218.945 220.215 221.360 222.383 223.282 224.071 224.752 225.350 226.266 226.873 227.218 227.336 227.266 227.037 226.681 226.218 225.669

– 141.383 137.035 132.692 128.199 123.841 119.275 114.758 110.191 105.463 100.789 96.073 82.570 72.074 59.858 47.595 35.285 22.863 10.187 −2.369 −14.934 −27.450 −52.804 −78.287 −103.762 −128.964 −154.334 −179.843 −205.289 −230.442 −256.171

– −41.160 −41.086 −40.994 −40.886 −40.767 −40.631 −40.489 −40.334 −40.073 −40.009 −39.835 −39.961 −38.891 −38.383 −37.878 −37.357 −36.837 −36.317 −35.799 −35.287 −34.779 −33.789 −32.832 −31.910 −31.024 −30.172 −29.349 −28.554 −27.785 −27.038

– −28.590 −27.774 −26.884 −26.054 −25.225 −24.431 −23.563 −22.822 −21.857 −21.241 −20.485 −18.841 −16.691 −14.853 −13.098 −12.232 −9.557 −7.927 −6.189 −4.697 −3.079 0.091 3.168 6.050 9.016 11.828 14.651 17.346 20.095 22.552

Substituting eqn (12) into eqn (11), the Nernst potential in another form for the electrochemical reaction of eqn (10) is obtained:  RT
0 0.5 E(H2 +1/2O2 =H2 O) = (13) ln( pO2 /p0 )0.5 cathode − ln( pO2 /p )anode . 2F Since the oxygen partial pressure at the anode is very low (on the order of 10−22 bar) due to the anode reaction [2], it does not cause an appreciable effect on the partial pressures of the other major species in the anode flow. Therefore, when calculating the partial pressures of hydrogen and water vapor for determining the Nernst potential of the electrochemical reaction of eqn (10), the oxygen partial pressure in anode flow stream is ignorable. Hereafter, for an electrochemical reaction

8 Transport Phenomena in Fuel Cells as in eqn (1), the common practice in determining the Nernst potential will be to use eqn (7), in which the partial pressure of oxygen on the cathode side and those of the fuel and product species on the anode side are used. The molar consumption rate of hydrogen and oxygen in the electrochemical reaction of eqn (10) can be easily derived from eqn (9) as: mH2 =

I ; 2F

mO2 =

I . 4F

(14)

2.2 Operation with methane through internal reforming and shift reactions As previously mentioned, it is necessary to have a high operating temperature in a solid oxide fuel cell in order to maintain sufficient ionic conductivity for the solid oxide electrolyte [2, 4]. This provides a favorable environment for the reforming of hydrocarbon fuels like methane. In fact, since a solid oxide fuel cell operates based on the transport of oxide ions through the electrolyte layer from the cathode side to the anode side, the reforming products of hydrogen and carbon monoxide in the fuel channel can both serve as fuels. Given this advantage, solid oxide fuel cells can directly utilize hydrocarbon fuels or, at least, methane as a pre-reformed or partly reformed gas with components of CH4 , CO, CO2 , H2 and H2 O. Therefore, the fuel reforming and shift reactions will occur in the fuel channel in a solid oxide fuel cell. The anode is, in fact, a good material to serve as the catalyst for such chemical reactions, since the high temperature in a SOFC means that no noble metals are needed for a catalyst [12]. If there are five gas species, CH4 , CO, CO2 , H2 , and H2 O, in the fuel channel, the solid oxide fuel cell will operate with internal reforming and shift reactions. Therefore, the electrochemical reaction and the coexisting chemical reactions of reforming and shift need to be considered for determining the species’mole fractions (which are crucial to the electromotive forces in the fuel cell). Reforming :

Shift :

CH 4 + H2 O ↔ CO + 3H2 . CO + H2 O ↔ CO2 + H2 .

(15)

(16)

Since the high operating temperature of a SOFC ensures rapid fuel reaction kinetics, it is a common practice to assume that the reforming and shift reactions are in chemical equilibrium [4] when determining the mole fractions of the species, which makes the computation significantly convenient. From the concept of chemical equilibrium, the reactants and products must satisfy the condition of
G = 0. Therefore, the mole fractions or partial pressures of the five gas species in the fuel stream are related through the following two simultaneous equations [13]:  p 3 

KPR =



pCO p0 p  p  CH4 H2 O p0 p0 H2 p0

 = exp −

0
Greforming

RT

 ,

(17)

Multiple transport processes in solid oxide fuel cells

p

KPS

CO2 p0

 p  H2



0
Gshift   p  = exp − = H2 O pCO RT p0

p0

9

 .

(18)

p0

The dominant electrochemical reaction has been reported to be the oxidation of H2 [12], which is primarily responsible for the electromotive force. However, at the same time, the electrochemical oxidation of the CO is also possible, and likely occurs to some extent in the solid oxide fuel cell. It has been reported that fuel cells operated by using mixtures of CO and CO2 have shown that the electrochemical oxidation of CO is an order of magnitude slower than that of hydrogen [14]. Nevertheless, there is no necessity to distinguish whether the electrochemical oxidation process involves H2 or CO in order to formulate the electromotive force. The following discussion will clarify this point. When the shift reaction of eqn (16) in the anodic gas is in chemical equilibrium, there is   

pCO2 pH 2 0 0
G = gCO2 + RT ln + gH2 + RT ln p0 p0

   pCO pH2 O 0 0 + g = 0. (19) + RT ln + RT ln − gCO H2 O p0 p0 Rearranging this equation gives: 

  pCO2 pCO 0 0 gCO − g + RT ln + RT ln CO 2 p0 p0 

  pH2 O pH2 0 0 = gH2 O + RT ln − gH2 + RT ln . p0 p0

(20)

1/2

Subtracting a term of [(1/2)gO0 2 +RT ln( pO2 /p0 )cathode ] from both sides of eqn (20), results in:    pCO pO2 1/2 1 0 pCO2 0 0 − RT ln − RT ln gCO2 − gCO − gO2 + RT ln 2 p0 p0 p0 cathode  1 pH 2 O = gH0 2 O − gH0 2 − gO0 2 + RT ln 2 p0   pO2 1/2 pH2 − RT ln − RT ln . (21) p0 p0 cathode It is easy to see that the left-hand side of eqn (17) is the Gibbs free energy change of the electrochemical oxidation of CO, and the right-hand side is that for H2 . Dividing by (2F) on both sides, eqn (17) is further reduced to: E(H2 +1/2O2 =H2 O) = E(CO+1/2O2 =CO2 ) ,

(22)

10 Transport Phenomena in Fuel Cells where E(H2 +1/2O2 =H2 O) is given in eqn (11), while the Nernst potential for the electrochemical oxidation of CO is E(CO+1/2O2 =CO2 ) =

RT
ln (pCO /p0 )anode 2F 2F  0 + ln (pO2 /p0 )0.5 cathode − ln (pCO2 /p )anode . 0 −
G(CO+1/2O 2 =CO2 )

+

(23)

It is preferable that the EMF of an internal reforming SOFC be calculated from the electrochemical oxidation of H2 ; however, the species’consumption and production are the results collectively determined from the reactions of eqns (10), (15) and (16). The above discussion clearly indicates that the electrochemical reaction can be assumed to be driven by the hydrogen, and the electrochemical fuel value of CO is readily exchanged for hydrogen by the shift reaction under chemical equilibrium. Therefore, only H2 is considered as the electrochemical fuel in the following analysis, and CO only takes part in the shift reaction. For convenience, the mole flow rates of CH4 , CO and H2 are denoted by their formulae. Assuming that, x¯ , y¯ , and z¯ are the mole flow rates, respectively, for the CH4 , CO, and H2 that are consumed in the three reactions given by eqns (15), (16) and (10) in the fuel channel, the coupled variations of the five species between the inlet and the outlet of an interested section of fuel channel are in the following forms [8, 15]: CH4 out = CH4 in − x¯ ,

(24)

COout = COin + x¯ − y¯ ,

(25)

= CO2 + y¯ , in

(26)

H2 out = H2 in + 3¯x + y¯ − z¯ ,

(27)

H2 Oout = H2 Oin − x¯ − y¯ + z¯ .

(28)

CO2

out

The overall mole flow rate of fuel, denoted by Mf , will vary from the inlet to the outlet of the section of interest in the fuel channel in the form of Mfout = Mfin + 2¯x.

(29)

Meanwhile, the partial pressures of the species, proportional to the mole fractions, must satisfy eqns (15) and (16) at the outlet of the section, which thus gives:  KPR =

COin +¯x−¯y Mfin +2¯x





CH4 in −¯x Mfin +2¯x

H2 in +3¯x+¯y−¯z Mfin +2¯x



3   2

H2 Oin −¯x−¯y+¯z Mfin +2¯x

p p0



,

(30)

Multiple transport processes in solid oxide fuel cells

 KPS = 

H2 in +3¯x+¯y−¯z Mfin +2¯x

COin +¯x−¯y Mfin +2¯x





CO2 in +¯y Mfin +2¯x

11



H2 Oin −¯x−¯y+¯z Mfin +2¯x

,

(31)

where the p is the overall pressure of the fuel flow in the section of interest. Since, as discussed in the preceding section, the oxidation of H2 is responsible for the electrochemical reaction, the consumption of hydrogen is directly related to the charge transfer rate, or current, I , across the electrolyte layer: z¯ = I /(2F).

(32)

From the electrochemical reaction, the molar consumption of oxygen on the cathode side can be calculated by using eqn (14). By finding a simultaneous solution for eqns (30)–(32), the species variations, x¯ , y¯ and z¯ , can be determined. Finally, with the reacted mole numbers of CH4 and CO determined, the heat absorbed in the reforming reaction and released from the shift reaction can be obtained: QReforming =
H Reforming · x¯ ,

(33)

Q Shift =
H Shift · y¯ .

(34)

Nevertheless, prior to finding a solution for eqns (30) and (31), the electric current of the fuel cell in eqn (32) must be known. This demonstrates that the processes in a SOFC feature a strong coupling of the species molar variation and the electromotive force, as well as interdependency of the ion conduction and current flow. The ion transfer rate or current conduction in a SOFC will be discussed in Section 3.

3 Electrical potential losses The ideal efficiency is never attained in practical operation for any fuel cell. In fact, there are three potential drops in a fuel cell that cause the actual output potential to be lower than the ideal electromotive forces of the electrochemical reaction. The nature of the fuel cell performance in response to loading condition can be realized by its polarization curve, typically shown as in Fig. 2. With an increase in current density, the cell potential experiences three kinds of potential losses due to different dominant resistances. The potential drop due to the activation resistance, which is the activation polarization, is associated with the electrochemical reactions in the system. Another potential drop comes from the ohmic resistance in the fuel cell components, when the ions and electrons are conducted in the electrolyte and electrodes, respectively. The third drop, which can be sharp at high current densities, is attributable to the mass transport resistance, or concentration polarization, in the flow of the fuel and oxidant. It is known from observing the Nernst equation that the electromotive force of a fuel cell is a function of the temperature and the gas species’ partial pressures at the electrolyte/electrode

12 Transport Phenomena in Fuel Cells

Figure 2: Over-potential in the operation of a fuel cell.

interfaces, which are directly proportional to their mole fractions. It is important to note that, in the fuel stream, fuel must be transported or diffused from the core region of the stream to the anode surface, and, also, the product of the electrochemical reaction must conversely be transported or diffused from the reaction site to the core region of the fuel flow. On the cathode side, oxygen must be transported and diffused from the core region of airflow to the cathode surface. Along with the fuel and air streams, the consumption of reactants or development of products will make the mole fractions of reactants decrease and those of the product increase. Due to these resistances in the mass transport process, the feeding of reactants and removing of products to/from the reaction site can only proceed under a large concentration gradient between the bulk flow and the electrode surface when the current density is high, which therefore induces a sharp drop in the fuel cell potential. As a consequence of all the above-mentioned potential drops, extra thermal energy will be released together with the heat (−T
S) from systematic entropy production. The heat transfer issues in a solid oxide fuel cell will be considered later in Section 4. 3.1 Activation polarization The activation polarization is the electronic barrier that must be overcome prior to current and ion flow in the fuel cell. Chemical reactions, including electrochemical reactions, also involve energy barriers, which must be overcome by the reacting species. The activation polarization may also be viewed as the extra potential necessary to overcome the energy barrier of the rate-determining step of the reaction to a value such that the electrode reaction proceeds at a desired rate [16–18].

Multiple transport processes in solid oxide fuel cells

13

The Butler-Volmer equation is a well-known expression for the activation polarization, ηAct : 

  ne FηAct ne FηAct − exp −(1 − β) , (35) i = i0 exp β RT RT where β, which is usually 0.5 for the fuel cell application [16], is the transfer coefficient; i is the actual current density in the fuel cell; and i0 is the exchange current density. The transfer coefficient is considered to be the fraction of the change in polarization that leads to a change in the reaction-rate constant. The exchange current density, i0 , is the forward and reverse electrode reaction rate at the equilibrium potential. A high exchange current density means that a high electrochemical reaction rate and good fuel cell performance can be expected. The ne in eqn (35) is the number of electrons transferred per reaction, which is 2 for the reaction of eqn (10). Substituting the value of β = 0.5 into eqn (35), one can obtain a new expression as follows:  ne FηAct (36) i = 2i0 sinh 2RT from which the activation polarization can be expressed as:      2  i i 2RT 2RT i or ηAct = + + 1. ηAct = sinh−1 ln  ne F 2i0 ne F 2i0 2i0 (37) For a high activation polarization, eqn (37) can be approximated as the simple and well-known Tafel equation [16]:  2RT i ηAct = . (38) ln ne F i0 On the other hand, if the activation polarization is small, eqn (37) can be approximated as the linear current-potential expression [16]: ηAct =

2RT i . ne F i0

(39)

Nevertheless, eqn (37) is recommended for its integrity and accuracy in calculating the activation polarization. The value of the exchange current density (i0 ) is different for the anode and cathode, and is also dependent on the electrochemical reaction temperature, the partial pressures of the gases [18, 19], and the electrode materials. The determination for i0 shows diversity in different literature [16–22]. There are formulations available in the literature [18–20], but some parameters used in the formulation are not well documented. On the other hand, empirical estimation of i0 is also

14 Transport Phenomena in Fuel Cells made in literature, like the work of Chan et al. [16]. An i0 of 5300 A/m2 for the anode and 2000 A/m2 for the cathode for a SOFC were used; however there was no comment on the selection methodology for setting these values in the paper [16]. Keegan et al. [17] also adjusted the i0 so as to obtain a simulation result to satisfy their experimental data; no report, however, is given about the adjusted i0 values in their paper. The present authors used a slightly higher value of 6300 A/m2 and 3000 A/m2 [23, 24], respectively, for the i0 of the anode and cathode, which resulted in very good agreement between the numerical simulated cell terminal voltage and experimental results from different researchers [25–28]. Nevertheless, i0 varies according to the temperature and pressures of the electrochemical reaction. For SOFCs working at temperature from 800 ◦ C–1000 ◦ C and pressures up to 15 atm, an i0 of 5300–6300 A/m2 for the anode and 2000–3000 A/m2 for the cathode are recommended from the study by the present authors [23]. 3.2 Ohmic loss The ohmic loss comes from the electric resistances of the electrodes and the current collecting components, as well as the ionic conduction resistance of the electrolyte layer. Therefore, the conductivity of the materials for the cell components and the current collecting pathway are the two factors most influential to the overall ohmic loss of a SOFC. In state-of-the-art SOFC technology, lanthanum manganite suitably doped with alkaline and rare earth elements is used for the cathode (air electrode) [20, 27], yttria stabilized zirconia (YSZ) has been most successfully employed for electrolyte, and nickel/YSZ is applied over the electrolyte to form the anode. Temperature could significantly affect the conductivity of SOFC materials. Especially for the electrolyte, for example, the resistivity could be two orders of magnitude smaller if its temperature increases from 600 ◦ C to 1000 ◦ C. The equations of resistivity for SOFC components suggested in literature are collected in Table 2.

Table 2: Data and equations for resistivity of SOFC components. Cathode ( · cm) Bessette et al. [29] Ahmed et al. [30] Nagata et al. [18] Ferguson et al. [31]

Electrolyte ( · cm)

Anode ( · cm)

Interconnect ( · cm)

0.008114e500/T

0.00294e10350/T

0.00298e−1392/T

–

∗ 0.0014

0.3685 + 0.002838e10300/T

∗ 0.0186

∗ 0.5

∗ 0.1

10.0e[10092(1/T −1/1273)]

∗ 0.013

∗ 0.5

T e1200/T 4.2×105

1 e10300/T 3.34×102

T e1150/T 9.5×105

T e1100/T 9.3×104

∗At temperature of 1000 ◦ C.

Multiple transport processes in solid oxide fuel cells

15

A careful check for the equations in Table 2 was conducted. The expressions by Bessette et al. [29] were found to be reliable, and to give nearly identical predictions as those by Ahmed et al. [30] and Nagata et al. [18]. The predicted data for anode resistivity by the equation of Ferguson et al. [31] shows significant discrepancies with the predictions by other equations. It is rational to assume that the passage of the charge-carrying species through the electrolyte, or the ion conduction through the electrolyte, is a charge transfer, like a current flow. In a planar type SOFC, as shown in Fig. 3, the current collects through the channel walls, also called ribs, after it moves perpendicularly across the electrolyte layer. The network circuit for current flow modeled by Iwata et al. [19] considers the channel walls as current collection pathways in a planar SOFC. However, the height and the width of the gas channel are both small (less than 3 mm), and the electric resistance through the channel wall might be negligible [30]. This simplification leads to the consideration that the current is almost exclusively perpendicularly collected, which means that the current flows normally to the trilayer of the cathode, electrolyte and anode. When calculating the local current density, the ohmic loss is thus simply accounted for in the following way [30]: I =
A ·

(E − ηaAct − ηcAct ) − Vcell , (δa ρea + δe ρee + δc ρec )

(40)

where
A is a unit area on the anode/electrolyte/cathode tri-layer, through which the current I passes; δ is the thickness of the individual layers; ρe is the resistivity of the electrodes and electrolyte; Vcell is the cell terminal voltage; and the denominator of the right-hand side term is the summation of the resistance of the tri-layer. The Joule heating due to current flow in the volume of
A×δ is expressed for the anode in the form of a QOhmic = I 2 · (δa ρea /
A). (41) This is also applicable to the electrolyte and cathode by replacing the thickness and resistivity accordingly.

Figure 3: Schematic of a planar type SOFC.

16 Transport Phenomena in Fuel Cells

Figure 4: Schematic of a tubular SOFC.

Figure 5: Ion/electron conduction network in a tubular SOFC. In case the current pathway is relatively long in a fuel cell, as, for example, in a tubular type SOFC (shown in Fig. 4), the current collects circumferentially, which leads to a much longer pathway [32] compared to that of a planar type SOFC. In order to account for the ohmic loss and the Joule heating of the current flow in the circumferential pathway, a network circuit [23, 25, 33, 34] for current flow may be adopted, as shown in Fig. 5. Because the current collection is symmetric in the peripheral direction in the cell components, only half of the tube shell is deployed with a mesh in the analysis. The local current routing from the anode to cathode through the electrolyte is determinable based on the local electromotive force, EMF, the local potentials in the anode and cathode, and the ionic resistance of the electrolyte layer, which yields the expression: I=

E − ηaAct − ηcAct − (V c − V a ) , Re

(42)

where V a and V c are the potentials in the anode and cathode, respectively. Re is the ionic resistance of the electrolyte layer given a thickness of δe and a unit

Multiple transport processes in solid oxide fuel cells

17

area of
A: Re = ρee · δe /
A,

(43)

ρee

where the is the ionic resistivity of the electrolyte. In order to obtain the local current across the electrolyte by using eqn (35), supplemental equations for V a and V c are necessary. Applying Kirchhoff’s law of current to any grid located in the anode, the equation associating the potential of the central grid point P with the potentials of its neighboring points (east, west, north, south) and the corresponding grid P in the cathode can be obtained: 

 a VSa − VPa VN − VPa VWa − VPa VEa − VPa + + + Rae Raw Ran Ras   VPc − VPa − (EP − ηPAct ) + = 0. ReP

(44)

In the same way for a grid point P in the cathode: 

 c VSc − VPc VEc − VPc VN − VPc VWc − VPc + + + Rce Rcw Rcn Rcs   VPa − VPc + (EP − ηPAct ) + = 0, ReP

(45)

where Ra and Rc are the discretized resistances in the anode and cathode respectively, which are determined according to the resistivity, the length of the current path, and the area upon which the current acts; ηPAct is the total activation polarization, including from both the anode side and the cathode side. With all of the equations for the discretized grids in both the cathode and anode given, a matrix representing the pair of eqns (37) and (38) can be created. When finding a solution for such a matrix equation for the potentials, the following approximations are useful: 1. At the two ends of the cell tube there is no longitudinal current flow, and, therefore, an insulation condition is applicable. 2. At the symmetric plane A–A, as shown in Figs 4 and 5, there is no peripheral current in the cathode and anode, unless the cathode or anode is in contact with nickel felt, through which the current flows in or out. 3. The potentials of the nickel felts are assumed to be uniform due to their high electric conductivities. 4. Since the potential difference between the two nickel felts is the cell terminal voltage, the potential at the nickel felt in contact with the anode layer can be assumed to be zero. Thus, the potential at the nickel felt in contact with the cathode will be the terminal voltage of the fuel cell. Once all the local electromotive forces are obtained, the only unknown condition for the equation matrix is either the total current flowing out from the cell or

18 Transport Phenomena in Fuel Cells the potential at the nickel felt in contact with the cathode. This highlights two approaches that can be taken when predicting the performance of a SOFC. If the total current taken out from the cell is prescribed as the initial condition, the terminal voltage can be predicted. On the other hand, one can prescribe the terminal voltage and predict the total current, i.e., the summation of the local current I across the entire electrolyte layer. Once the potentials are obtained in the electrode layer, the volumetric Joule heating in the electrode for a volume centered about P will be:  (VWa − VPa )2 (VNa − VPa )2 1 (VEa − VPa )2 a + + q˙ P = 2 Rae Raw Ran  (V a − V a )2 (
xP · r a ·
θP · δa ), + S a P (46) Rs  (VWc − VPc )2 (VNc − VPc )2 1 (VEc − VPc )2 c + + q˙ P = 2 Rce Rcw Rcn  (VSc − VPc )2 (
xP · r c ·
θP · δc ), + (47) Rcs   P − V c + V a )2  − η (E P P P Act q˙ Pe = (
xP · r e ·
θP · δe ), (48) ReP where the r and δ with the corresponding superscripts of a , c , and e are the average, radius and thickness, respectively, for the anode, cathode and electrolyte, and
xP and
θP are the P-controlled mesh size in the axial and peripheral directions, as shown in Fig. 5. The volumetric heat induced from the activation polarization in the anode and cathode is: P,a a a q˙ Act = IP · ηP,a Act /(
xP · r ·
θP · δ ),

(49)

P,c c c = IP · ηP,c q˙ Act Act /(
xP · r ·
θP · δ ).

(50)

The thermodynamic heat generation occurring at the anode/electrolyte interface in the area around P is: QPR = (
H −
G) · IP /(2F).

(51)

3.3 Mass transport and concentration polarization Due to their gradual consumption, the fractions of the reactants and oxidant will decrease, in the fuel and air streams, respectively, which will cause the electromotive force to decrease gradually along the flow stream. On the other hand, due to the mass transport resistance, the concentration of the gas species will encounter polarization in between the core flow region and the electrode surface, which will result in lower partial pressures for the reactants, but higher partial pressures for the products at the

Multiple transport processes in solid oxide fuel cells

19

Figure 6: The interrelation amongst concentration and other parameters. electrode surfaces. Therefore, the fuel cell terminal voltage will be lower than the ideal value that is indicated by the Nernst equation. At high cell current density, the increased requirements for the feeding of the reactants and removal of the products can make the concentration polarization higher, and, thus, the cell output potential will sharply decrease. In order to take the concentration polarization into account when calculating the electromotive force, the local partial pressures of the reactants and products at the electrode surface are used. However, this requires the solution of the concentration fields for the gas species in the fuel and oxidizer channels, which might be either simply based on a one-dimensional [35–37] or else based on a complicated twoor three-dimensional solution for the mass conservation governing equations [23, 38–40]. In fact, the concentration fields are strongly coupled with the gas flow, temperature, and the distribution of the electromotive force in the ways indicated in Fig. 6. First, the gas species mass fraction determines the gas properties in the flow field, while the flow fields affect the gas species concentration distribution and temperature. Second, the gas species concentration field and temperature distribution determines the electromotive forces, while the ion/electron conduction due to the electromotive force determines the mass variation and heat generations in the fuel cell. The inter-dependency of these parameters will be discussed in detail in the following section when modeling a SOFC in order to predict both the fuel cell performance and the detailed distributions of the temperature, gas species concentration, and flow fields.

4 Computer modeling of a tubular SOFC An operation curve for a SOFC that characterizes the average current density versus the terminal voltage is very important when designing a SOFC system or a hybrid

20 Transport Phenomena in Fuel Cells SOFC/GT system [41–44]. Other information, like the temperature and concentration fields in a SOFC, is also of high concern for the safe operation of both the SOFC itself and the downstream facilities if a hybrid system is under consideration. Although there have been some experimental data generated about the operational performance and temperature of SOFCs [25–28], rigorous experimental testing for a SOFC is still rather tough because of its high operating temperature. Therefore, numerical modeling of SOFCs is very necessary. The purpose of computer simulation for a SOFC is to predict the operational characteristics in terms of the average current density versus the terminal voltage (based on prescribed operating conditions). The operating conditions of a SOFC are solely determined by fixing the flow rates and the thermodynamic state of the fuel and oxidant, as well as a load condition such as terminal voltage, the current being withdrawn, and external load [45]. The flow rates and thermodynamic conditions of the fuel and oxidant may be called internal conditions, and the terminal voltage, current to be withdrawn, and external load may be designated external conditions. Like any kind of “battery,” the external load condition of a SOFC determines the consumption of the fuel/oxidant and the generation of products in the electrochemical reaction [46]; the only difference in a fuel cell is its continuous feeding of fuel/oxidant and removal of products and waste species. According to the different ways of prescribing the external parameters, the following three schemes might be designed in order to predict the other unknown parameters when constructing a numerical model for a SOFC: (1) Use the internal conditions and terminal voltage to predict the total current to be withdrawn. (2) Use the internal conditions and current to be withdrawn to predict the terminal voltage. (3) Use the internal conditions and external load to predict the terminal voltage and current density. The cost of iterative computation using the three schemes is quite different. In the first scheme, the cell terminal voltage is known, and thus the local current can be obtained, for example, by using eqn (42) and solving eqns (44) and (45), for a planar and tubular type SOFC, respectively, once the temperature and partial pressure fields of the gas species are available. The integrated value from the local current will be the total current to be withdrawn from the SOFC. In the second scheme, however, the terminal voltage needs to be assumed, and then checked by integrating the total current from the local current until the calculated total current agrees with the prescribed value. In this computation process, a proper method is needed to find the best-fit terminal voltage iteratively. The third scheme resembles the second scheme, in that one needs to assume a terminal voltage to find the total current. The computation will be stopped only when the voltage-current ratio equals the prescribed load. With an understanding of the principles of the energy conversion, chemical equilibrium, potential loss, and the operation of a SOFC, a computer model for a SOFC can now be constructed. Generally speaking, the modeling and computation for a tubular SOFC and a planar SOFC share rather common features except for the Ohmic losses and Joule heating, for which differences result from the different structures variation in the current pathway in the electrodes. Relatively speaking,

Multiple transport processes in solid oxide fuel cells

21

the tubular SOFC has a more complex current pathway in the electrodes [27] and will be discussed in the following analysis. Modeling works on planar type SOFCs are available in both the current author’s work and in the literature [17, 19, 30, 31, 39, 47–49]. In the following subsections, there are three issues that address the construction of a numerical model. 4.1 Outline of a computation domain In a practical tubular SOFC stack, multiple tubular cells are mounted in a container to form a cell bundle, as shown in Fig. 7. A pre-reformer might be put adjacent to the cell bundles [50, 51]. In order to conduct a modeling study with relatively less complexity, it is assumed that most of the single tubular SOFCs operate under the same environment of temperature and concentrations of gas species. This allows the definition of a controllable domain in the cross-section, which pertains to one single cell, as outlined by the dashed-line square in Fig. 7. It is then specified that there must be no flow velocity and fluxes of heat and mass across the outline. This will significantly simplify the analysis for a cell stack. Through analysis of the heat/mass transfer and the chemical/electrochemical performance for the single cell and its controllable area, one can obtain results very useful for evaluating the performance of an entire cell stack. Also considering the longitudinal direction, the heat and mass transfer in the above outlined square area enclosing the tubular SOFC are three-dimensional in nature. For a solution of the three-dimensional governing equations of momentum, energy, and species conservation, a large number of discretized mesh points are necessary, which results in an unacceptably heavy computational load. In order to reduce computational cost, the square area enclosing the tubular SOFC is approximated to be an equivalent circular area; therefore, the domain enclosing the single tubular SOFC is viewed as a 2-dimensional axi-symmetric one, as seen in Fig. 7.

Figure 7: Schematic of a tubular SOFC in a cell stack.

22 Transport Phenomena in Fuel Cells

Figure 8: Computation domain for a tubular SOFC.

It should be noted, though, that the zero-flux, or insulation of heat and mass transfer at the boundary remains unchanged, even given this geometric approximation. From the preceding discussions, an axi-symmetrical two-dimensional (x − r) computation domain is profiled as shown in Fig. 8, which includes two flow streams and a solid area of the cell tube and air-inducing tube. 4.2 Governing equations and boundary conditions Since the mass fractions of the species vary in the flow field, all of the thermal and transport properties of the fluids are functions of the local species concentration, temperature, and pressure; therefore, the governing equations for momentum, energy, and species conservation (based on mass fraction) have variable thermal and transport properties: ∂(ρu) 1 ∂(rρv) + = 0, ∂x r ∂r

  ∂(ρuu) 1 ∂(rρvu) ∂p ∂ ∂u 1 ∂ ∂u + =− + µ + rµ ∂x r ∂r ∂x ∂x ∂x r ∂r ∂r   ∂ ∂u 1 ∂ ∂v + µ + rµ , ∂x ∂x r ∂r ∂x   ∂(ρuv) 1 ∂(rρvv) ∂p ∂ ∂v 1 ∂ ∂v + =− + µ + rµ ∂x r ∂r ∂r ∂x ∂x r ∂r ∂r   ∂ ∂u 1 ∂ ∂v 2µv + µ + rµ − 2 , ∂x ∂r r ∂r ∂r r   ∂ ∂T 1 ∂ ∂T ∂(ρCpuT ) 1 ∂(rρCpvT ) + = λ + rλ + q˙ , ∂x r ∂r ∂x ∂x r ∂r ∂r

(52)

(53)

(54)

(55)

Multiple transport processes in solid oxide fuel cells



∂(ρuYJ ) 1 ∂(rρvYJ ) ∂YJ ∂ + = ρDJ ,m ∂x r ∂r ∂x ∂x

+

1 ∂ r ∂r

 rρDJ ,m

∂YJ ∂r

23

+ SJ . (56)

These equations are applied universally to the entire computation domain; however, zero velocities will be assigned to the solid area in the numerical computation. In the energy conservation equation, thermal energy from the chemical and electrochemical reactions (expressed by eqns (33), (34), (51)) and the Joule heating in electrodes and electrolyte (expressed by eqns (46)–(50)), represented by q˙ , are introduced as source terms in the proper locations in the fuel cell. Some terms due to energy diffusion driven by the concentration diffusion of the gas species are very small, and thus neglected [52, 53]. The boundary conditions for the momentum, heat and mass conservation equations are as follows: 1. On the symmetrical axis, or at r = 0: v = 0, and ∂φ/∂r = 0, where φ represents general variables except for v. 2. At the outmost boundary of r = rfo : there are thermally adiabatic conditions; impermeability for species and non-chemical reaction are also assumed, which gives v = 0, and ∂φ/∂r = 0, where φ represents general variables except for v. 3. At x = 0: the fuel inlet has a prescribed uniform velocity, temperature, and species mass fraction; the solid part has u = 0, v = 0, ∂T /∂x = 0, and ∂YJ /∂x = 0. 4. At x = L: the air inlet has a prescribed uniform velocity, temperature and species mass fraction; the gas exit has v = 0, ∂u/∂x = 0, ∂T /∂x = 0, and ∂YJ /∂x = 0; the tube-end solid part has u = 0, v = 0, ∂T /∂x = 0, and ∂YJ /∂x = 0. 5. At the interfaces of the air/solid, r = rair , and fuel/anode, r = rf : u = 0 is assumed. In the fuel flow passage, the mass flow rate increases along the x direction due to the transferring in of oxide ions. Similarly, a reduction of the air flow rate occurs in the air flow passage, due to the ionization of oxygen and the transferring of the oxide ions to the fuel side. Therefore, radial velocities at r = rair and r = rf are:  fuel,species  m ˙x  (57) vf = r=rf , fuel ρx  air,species  m ˙x r=r , vair = (58) air air ρx where m ˙ [kg/(m2 s)] is mass flux of the gas species at the interface of the electrodes and fluid, which arises from the electrochemical reaction in the fuel cell. The mass fractions of all participating chemical components at the boundaries of r = rair and r = rf are calculated with consideration of both diffusion and convection effects [54, 55]: ∂YJ (59) + ρxair YJ vair , m ˙ Jx ,air = −DJ ,air ρxair ∂r ∂YJ + ρxfuel YJ vf . m ˙ Jx , fuel = −DJ , fuel ρxfuel (60) ∂r

24 Transport Phenomena in Fuel Cells Table 3: Properties of SOFC materials. Thermal conductivity (W/(m K)) Cathode Electrolyte Anode Support tube Air-inducing tube Interconnector aAhmed

Cp (J/(kg K))

Density (kg/m3 )

d 11; c 2.0; b 2.0

b 623

a 4930

d 2.7; c 2.7; b 2.0

b 623

a 5710

c 11.0; d 6.0; b 2.0

b 623

a 4460

b 800

a 6320; b 7700

c 1.0 c 1.0 b 13; c 2.0; d 6.0

et al. [30]; b Recknagle et al. [39]; c Nagata et al. [18]; d Iwata et al. [19].

It is worth noting that the mass fluxes for the species in the above equations, eqns (57)–(60), strongly relate to the ion/electron conduction; the determination of mass variation and related mass flux that arise from the electrochemical reaction has been discussed (as expressed by eqns (30)–(32)) in Section 2. As a consequence, the mass/mole fraction at the solid/fluid interface, derived from eqns (59) and (60), will be used for the determination of the partial pressures and, thereof, the local electromotive forces by eqn (11). The properties of solid materials in a SOFC are given in Table 3, which show some variation based on the different literature sources. The single gas properties are available from references [11] and [56]. For gas mixtures, equations from references [11, 57] are available, and some selected equations from reference [11] for calculating the properties are listed in the following section. The mixing rule for the viscosity is:   1/2  1/4 2  n  Mj µi Xi µ i Mi −1/2 1 n µm = 1+ ; φij = 1/2 1 + , M µ Mi X φ 8 j ij j j j=1 i=1

(61) where µm (Pa · sec) is the viscosity for the mixture, and µi or µj are the viscosities of individual species (Pa · sec); Mi or Mj is the molecular weight of a species; Xi or Xj is the mole fraction; and when i = j, φij = 1. The mixing rule for the thermal conductivity of gases at atmospheric pressure or less is: n  Xi ki n km = ; j=1 Xj Aij i=1 (62)      1/2 2  3/4  T + Sij T + Si 1 µi Mj , Aij = 1+  T + Si 4 µj Mi T + Sj

Multiple transport processes in solid oxide fuel cells

25

where km [W/(m · K)] and ki [W/(m · K)] are the thermal conductivities of the mixture and species; Sij = C(Si Sj )1/2 , and C = 1.0, but when either or both components i and j are very polar, C = 0.73; for helium, hydrogen, and neon, Si or Sj is 79 K; otherwise, Si = 1.5Tbi and Sj = 1.5Tbj , where Tb is the boiling point temperature of species; and the unit of T is K. When the gas mixture is above atmospheric pressure, the following correction is applied to the km obtained above: k = k +

A × 10−4 (eBρr + C)  , 1/6 Tc M 1/2 5 Z 2/3 c

(63)

Pc

ρr < 0.5, 0.5 < ρr < 2.0, 2.0 < ρr < 2.8,

A = 2.702, A = 2.528, A = 0.574,

B = 0.535, B = 0.670, B = 1.155,

C = −1.000, C = −1.069, C = 2.016,

where k [W/(m · K)] is the gas thermal conductivity at the temperature T (K) and pressure P of interest in the mixture; k  [W/(m · K)] is the thermal conductivity at T and atmospheric pressure obtained by eqn (62); ρr = Vc /V is the reduced density; Vc (m3 /kmol) is the critical molar volume; V (m3 /kmol) is the molar volume at T and P; Tc (K) is the critical temperature; M is the molecular weight; Pc (MPa) is the critical pressure; Zc = Pc Vc /(RTc ) is the critical compressibility factor; and R is the gas constant, which is 0.008314 MPa · m3 /(kmol · K). The mixture critical properties are obtained via the following equations:   n  

 Tcm − Tpc Pcm = Ppc + Ppc 5.808 + 4.93 , (64) Xi ωi Tpc i=1

Tcm =

n   j=1

Vcm =

Xj Vcj n Tcj , i=1 Xi Vci

 i

(65)

φi φj vij (i = j),

(66)

j

where 2/3

Xj Vcj φj =  ; 2/3 n i=1 Xi Vci

vij =

Vij (Vci + Vcj ) ; 2.0

   Vci − Vcj   + C, Vij = −1.4684  V +V  ci

cj

and C is zero for hydrocarbon systems and is 0.1559 for systems containing a non-hydrocarbon gas. In all the above equations, Xi or Xj is the mole fraction of a species in the mixture; ωi is the acentric factor of a species; Pcm , Tcm and Vcm are the mixture critical properties; and Ppc and Tpc are the pseudocritical properties of the mixture, which are expressed as: Tpc =

n  i=1

Xi Tci ;

Ppc =

n  i=1

Xi Pci .

(67)

26 Transport Phenomena in Fuel Cells Table 4: Atomic diffusion volumes for use in eqn (68). Atomic and structural diffusion-volume increments v [11] C H O N Aromatic ring Heterocyclic ring

16.50 1.98 5.481 5.69 −20.2 −20.2

H2 N2 O2 CO CO2 H2 O

7.07 17.9 16.6 18.9 26.9 12.7

The gas diffusivity of one species against the remaining species of a mixture is expressed in the form of:

Dim

1 − Xi = , jj=i Xj /Dij

 0.5 0.01013T 1.75 M1i + M1j   Dij = , P[( vi )1/3 + ( vj )1/3 ]2

(68)

where units of T , P, and D are K, Pa, and m2 /sec, respectively; Mi or Mj is the molecular weight; and all vi or vj are group contribution values for the subscript component summed over atoms, groups and structural features, which are listed in Table 4. 4.3 Numerical computation In order to conduct a numerical computation for flow, temperature, and concentration fields in a SOFC, a mesh system with a sufficient grid number both in the r and x directions must be deployed at the computational domain. All the governing equations may be discretized by using the finite volume approach, and the SIMPLE algorithm can be adopted to treat the coupling of the velocity and pressure fields [58, 59]. The temperature difference between the cell tube and the air-inducing tube might be large enough to have radiation heat transfer; therefore, a numerical treatment based on the method introduced in the literature [60] can be used to consider the radiation heat exchange. As has been discussed at the beginning of Section 4, the computation may be based on the internal conditions and the current to be withdrawn; and, as a consequence of the simulation, the terminal voltage will be given as an output along with other operational details. The convenience of using this procedure in the simulation is discussed next. It is quite common in practice that the total current is prescribed in terms of the average current density of the fuel cell. Also, instead of the flow rates of fuel and air, the stoichiometric data are prescribed in terms of the utilization percentage of hydrogen and oxygen. This kind of designation of the operating conditions results in a convenient comparison of the fuel cell performance based on the same level of

Multiple transport processes in solid oxide fuel cells

27

average current density and the hydrogen and oxygen utilization percentage. The inlet velocities of fuel and air are, then, obtainable in the forms of:  RTf Acell icell ufuel = , (69) 2FUH2 XH2 Afuel Pf  Acell icell RTair , (70) uair = 4FUO2 XO2 Aair Pair where icell is the cell current density; Acell is the outside surface area of the fuel cell; Afuel and Aair are the cross-sectional inlet areas of the fuel and air; Pf , Pair and Tf , Tair are the inlet pressure and temperature of the fuel and air flows respectively; XH2 and XO2 are the mole fractions of hydrogen in the fuel and oxygen in the air, respectively; and UH2 and UO2 are the utilization percentage for hydrogen and oxygen. The computation process is highly iterative and coupled in nature. As the first step, the latest local temperature, pressure, and species’ mass fractions are used in the network circuit analysis to obtain the cell terminal voltage and local current across the electrolyte, and thus the local species’ transfer fluxes and local heat sources. In the second step, the local temperature, pressure and species’ mass fractions are, in turn, obtained through solution of the governing equations under the new boundary conditions determined by the latest-available species’fluxes and heat sources. The two steps iterate until convergence is obtained. 4.4 Typical results from numerical computation for tubular SOFCs The present authors have conducted numerical computations for three different single tubular SOFCs [23], which have been tested by Hagiwara et al. [26], Hirano et al. [25], Singhal [27], and Tomlins et al. [28]. The fuel tested by Hirano et al. [25] had components of H2 , H2 O, CO and CO2 ; therefore, there is a water-shift reaction of the carbon monoxide in the fuel cell to be considered together with the electrochemical reaction. The fuel used by the other researchers [26–28] had components of H2 and H2 O, where there is no chemical reaction except for the electrochemical reaction in the fuel channel. The dimensions of the three different solid oxide fuel cells tested in their studies are summarized in Table 5, in which the mesh size adopted in our numerical computation is also given. The operating conditions are listed in Table 6, including the species mole fractions and the temperature of the fuel and air in those tests, which are the prescribed conditions for the numerical computation. In the experimental work by Singhal [27], a test of the pressure effect was also conducted by varying the fuel and air pressure from 1 atm to 15 atm. It is expected that the experimental data for these SOFCs in different dimensions and operating conditions will facilitate a wide benchmark range for validation of the numerical modeling work. 4.4.1 The SOFC terminal voltage The computer calculated and the experimentally obtained cell terminal voltages under different cell current densities are shown in Fig. 9. The relative deviation of

28 Transport Phenomena in Fuel Cells Table 5: Example SOFCs with test data available. Data sequence: Outer diameter (mm)/Thickness (mm)/Length (mm) Singhal [27] Hagiwara et al. [26] Hirano et al. [25] Tomlins et al. [28] Air-inducing tube Support tube Cathode Electrolyte Anode Fuel boundary Grid number (r × x)

7.00/1.00/485 – 15.72/2.20/500 15.80/0.04/500 16.00/0.10/500 18.10/ – /500 66×602

6.00/1.00/290 13.00/1.50/300 14.40/0.70/300 14.48/0.04/300 14.68/0.10/300 16.61/ – /300 66×602

12.00/1.00/1450 – 21.72/2.20/1500 21.80/0.04/1500 22.00/0.10/1500 24.87/ – /1500 66×1602

Table 6: Species’ mole fractions, utilization percentages, and temperatures. Air O2 %−UO2 /N2 %/T (◦ C) I II III

fuel H2 %–UH2 /H2 O%/CH4 %/CO%/CO2 %/T (◦ C)

21.00–17.00/79.00/600.0

98.64–85.00/1.36 /0/ 0 /0 /900.0 55.70–80.00/27.70/0/10.80/5.80/800.0 ∗∗ 21.00–25.00/79.00/400.0 55.70–80.00/27.70/0/10.80/5.80/800.0 21.00–17.00/79.00/600.0 98.64–85.00/1.36 /0/ 0 /0 /800.0 ∗ 21.00–25.00/79.00/600.0

∗ Current

density = 185 mA/cm2 ; ∗∗ Current density = 370 mA/cm2 . I: Tested by Hagiwara et al. [26]. II: Tested by Hirano et al. [25]. III: Tested by Singhal [27] and Tomlins et al.[28]. the model-predicted data from the experimental data is no larger than 1.0% for the SOFC tested by Hirano et al. [25], 5.6% for that by Hagiwara et al. [26], and 6.0% for that by Tomlins et al. [28]. It is interesting to observe from Fig. 9 that, under the same cell current density, the cell voltage of the SOFC tested by Hagiwara et al. [26] is the highest and that by Hirano et al. [25] is the lowest. The mole fraction of hydrogen in the fuel for the SOFC tested by Hirano et al. [25] is low, which might be the major reason that this cell has the lowest cell voltage. Because the current must be collected circumferentially in a tubular type fuel cell, the large diameter of the cell tube investigated by Singhal [27] and Tomlins et al. [28] will lead to a longer current pathway. Thus, the cell voltages of these cells are lower than those found by Hagiwara et al. [26], even though the former investigators tested the SOFCs at a pressurized operation of 5 atm, which, in fact, helps to improve the cell voltage. Under a current density of 300 mA/cm2 , the cell voltage and power increase with the increasing operating pressure, as seen in Fig. 10. The agreement between our

Multiple transport processes in solid oxide fuel cells

29

Figure 9: Results of prediction and testing for cell voltage versus current density. (The operating pressure of the cell tested by Hagiwara et al. [26] and Hirano et al. [25] is 1.0 atm, and that by Tomlins et al. [28] is 5 atm.)

Figure 10: Effect of operating pressure on the terminal voltage and power. model-predicted results and the experimental ones by Singhal [27] is quite good, showing a maximum deviation of 7.4% at a low operating pressure. When the operating pressure increases from 1 atm to 5 atm, the cell output power shows a significant improvement of 9%. However, raising the operating pressure becomes less effective for improving the output power when the operating pressure is high. For example, the cell output power shows an increase of only 6% when the operating pressure increases from 5 atm to 15 atm. The reason for this is that the operating pressure contributes to the cell voltage in a logarithmic manner. Nevertheless,

30 Transport Phenomena in Fuel Cells pressurized operation of the fuel cell can improve the output power significantly. For example, when increasing the operating pressure from 1 atm to 15 atm, the cell output power can have an increment of 15.8%. There is no doubt from the above investigation that the investigators can satisfactorily predict the overall performance of a SOFC through numerical modeling and computation. On the basis of this good agreement with the overall fuel cell performance, the internal details of the flow, temperature, and concentration fields from numerical prediction can also be reliably presented. 4.4.2 Cell temperature distribution Because the measurement of temperature in a SOFC is very difficult, only three experimental data points, the temperature at the two ends and in the middle of the cell tube, were available from the work on Hirano et al. [25]. Figure 11 shows the simulated cell temperature distribution for the SOFC, for which Hirano et al. [25] provided the test data. The agreement of the simulated data and the experimental results is good in the middle, where the hotspot is located; relatively larger deviations between the predicted and experimental values appear at the two ends of the cell. Nevertheless, such a discrepancy is acceptable when designing a SOFC with respect to concerns about the prevention of excessive heat in the cell materials. The predicted temperature distributions for the fuel cells tested by Hagiwara et al. [26] and Tomlins et al. [28] are given in Fig. 12. Unfortunately, there was no experimental data on the cell temperature. Generally, the two ends of the cell tube have lower temperatures than the middle of the cell tube. However, at low current densities, the hotspot is located closer to the closed end of the cell. With an increase in current density, the hotspot shifts to the open-end side, and the hotspot temperature also decreases, which improves the uniformity of the temperature distribution along the fuel cell. It should be observed that the heat transfer between the cooling air and the cell tube at the closed-end region is dominated by laminar jet impingement, since the exit velocity from the air-inducing tube is quite low. However, the velocity of the exit air from the air-inducing tube affects the heat

Figure 11: Longitudinal temperature distribution in the fuel cell.

Multiple transport processes in solid oxide fuel cells

31

Figure 12: Predicted longitudinal temperature distribution for two SOFCs. transfer coefficient significantly. For the high current density case, the flow rate of air also becomes large accordingly. Thus, the heat transfer coefficient between the air and the fuel cell closed-end region is increased. This can suppress the temperature level of the closed-end region of the fuel cell significantly. Since the air receives a large amount of heat at the closed-end region, its cooling to the fuel cell in the downstream region becomes weak, and the uniformity of the cell temperature distribution becomes much better when the fuel cell operates at high current densities. 4.4.3 Flow, temperature and concentration fields Figure 13 shows the flow and temperature fields for the SOFC tested by Hirano et al. [25] at a current density of 185 mA/cm2 . The air speed in the air-inducing tube has a slight acceleration because the air absorbs heat and expands in this flow passage. After leaving the air-inducing tube, the air impinges on the closed end of the fuel cell, and then flows backwards to the outside. In this pathway, the air obtains heat from the heat-generating fuel cell tube and transfers the heat to the cold air in the air-feeding tube. It is easy to understand that the electrochemical reaction at the closed end of the fuel cell is strong because the concentrations of fuel and air are both high there. Therefore, the heat generation due to Joule heating and the entropy change of the electrochemical reaction is high at the upstream area of the fuel path. However, it is known from both experiments and computation that the closed-end region of the fuel cell does not demonstrate the highest temperature; therefore, it is believed that the cooling of the air in the closed-end area of the fuel cell is responsible for this. After being heated at the closed-end region, air exhibits a higher temperature, and its cooling ability to the cell tube is low when it is in the annulus between the air-inducing tube and the cell tube. At the cell open-end region, the air in the annulus can transfer heat to the incoming cold fresh air in the air-inducing tube, and this will help it to cool the fuel cell tube.

32 Transport Phenomena in Fuel Cells

Figure 13: Predicted flow and temperature fields for the SOFC reported by Hirano et al. [25] at a current density of 185 mA/cm2 .

From this airflow arrangement, the hotspot temperature of the cell tube may mostly occur in the center region in the longitudinal direction of the cell tube. The airflow has two passes, incoming in the air-inducing tube and outgoing in the annulus between the air-inducing tube and the cell tube. The heat exchange in between the two passes allows the air to mitigate its temperature fluctuation in the whole air path, and thus the temperature field in the fuel cell might be maintained as relatively uniform. Nevertheless, the heat generation, air and fuel temperature, and air-cooling to the fuel cell will collectively affect the temperature field in the fuel cell. Therefore, the hot spot position in a cell tube might shift more or less away from the center region depending on the operating condition of the fuel cell. Figure 14 shows the gas species’mole fraction contours for the same SOFC under the same operating conditions as discussed with respect to Fig. 13. In the air path, oxygen consumption at the closed-end region is relatively large, which leads to more densely distributed contour lines. The contour shape of oxygen also indicates a relatively larger difference of the mole fraction between the bulk flow and the wall of the cathode/air interface. This implies that the mass transport resistance on the air side might be dominant in lowering the cell performance if the stoichiometry of the oxygen is low. Feeding more air than is needed is already well applied in operational fuel cell technology.

Multiple transport processes in solid oxide fuel cells

33

Figure 14: Predicted fields of the mole fraction of the species for the SOFC reported by Hirano et al. [25] at a current density of 185 mA/cm2 . The hydrogen budget is collectively determined by the consumption by the electrochemical reaction and the generation from the water-shift reaction of CO. Since the consumption dominates, the hydrogen mole fraction decreases along the fuel stream. Corresponding to this hydrogen variation, consumption due to the water-shift reaction and production due to the electrochemical reaction cause the water vapor to increase gradually along the fuel stream. The water-shift of CO proceeds gradually in the fuel path, and thus the mole fraction of CO decreases but the CO2 increases. The shape of the contour lines of the species in the fuel path is

34 Transport Phenomena in Fuel Cells

Figure 15: Predicted streamwise molar flow rate variation for species in the fuel channel for the SOFC reported by Hirano et al. [25]. relatively flat from the cell wall to the bulk flow. This indicates that mass diffusion in the fuel channel is relatively stronger than that in the airflow. For a further illustration of the variation of the gas species, Fig. 15 shows the molar flow rate variation along the fuel path. In one third of the length from the fuel inlet, the hydrogen flow rate shows a faster decrease and the water flow rate shows a faster increase, indicating a strong reaction in the upstream region. The flow rate of CO and CO2 vary roughly in a linear style, and a small amount of CO still exists in the waste gas.

5 Concluding remarks Fuel cell technology is currently under rapid development. To improve SOFC performance, for high power density and efficiency, efforts have been made to reduce the three over-potentials: activation polarization, ohmic loss, and concentration polarization. Better understanding of these three over-potentials is also very important in developing accurate computer models for predicting the overall performance and internal details of a SOFC. The activation polarization relates to the porous structure of the electrode and electrocatalyst materials. The state-of-the-art in material and manufacturing processes for the electrodes and electrolyte has been reported by Singhal [27]. The reduction of ohmic losses also heavily relies on the reduction of electronic and ionic resistances in the electrodes and electrolyte. A shorter current collection pathway also helps to reduce ohmic loss. A new design, referred to as a high power density solid oxide fuel cell (HPD-SOFC), has been developed by Siemens Westinghouse Power Corporation [27, 32], and has a significantly shorter current pathway, and

Multiple transport processes in solid oxide fuel cells

35

thus improves the power density significantly. A planar structure also promises to have a shorter current pathway and thus a higher power density, and measures for reducing the ohmic loss in a planar type SOFC have been reported by Tanner and Virkar [61]. The reduction of mass transport resistance, or the concentration polarization, has not been given much attention. Mass transfer enhancement has been reported to be effective in polymer electrolyte membrane fuel cells (PEMFCs) for obtaining a higher cell current density [62] before a sharp drop in cell voltage (which is due to excessive concentration polarization). It might also be possible for SOFCs to obtain a higher current density by means of mass transfer enhancement. In a numerical model of a SOFC, the precise calculation of the over-potentials is very important in order to accurately predict the overall current-voltage performance. The heat generation from the over-potentials is also significant in computing the temperature, flow, and species concentration fields. With respect to the activation polarization, studies elucidating the data and equations for the exchange current density are still needed. For the prediction of ohmic losses, reliable property data for electrodes are required. Additionally, a method for analyzing a complex network circuit in a SOFC needs to be developed. The concentration polarization is considered in the numerical computation by using the local mole fractions of the species at the interface of the electrode and fluid when calculating the electromotive force by the Nernst equation. Because the porous electrodes also serve as the reaction site, there is no well-described model for the mass transport resistance in the electrodes. Adopting a lower exchange current density, which induces a larger over-potential of the activation polarization, may be a way to incorporate the mass transport resistance in the electrodes into the activation polarization. The method given by Hirano et al. [25] for the consideration of mass transport resistances in the electrodes is convenient, but may be too simple and needs more investigation. With the progress being made in computer modeling of SOFCs, it is expected that costs for research and development of SOFCs will be significantly reduced by using computer simulations in the future.

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Srinivasan, S., Mosdale, R., Stevens, P. & Yang, C., Fuel cells: reaching the era of clear and efficient power generation in the twenty-first century. Annual Review Energy Environment, 24, pp. 281–328, 1999. [2] Gardner, F.J., Thermodynamic processes in solid oxide and other fuel cells. Proc. Instn. Mech. Engrs., 211, Part A, pp. 367–380, 1997. [3] Suzuki, K., Iwai, H., Kim, J.H., Li, P.W. & Teshima, K., Solid oxide fuel cell and micro gas turbine hybrid cycle and related fluid flow and heat transfer. The 12th International Heat Transfer Conference, Grenoble, August 18–23, 1, pp. 403–414, 2002. [4] Harvey, S.P. & Richter, H.J., Gas turbine cycles with solid oxide fuel cells, Part I: Improved gas turbine power plant efficiency by use of recycled exhaust gases and fuel cell technology. ASME Journal of Energy Resources Technology, 116, pp. 305–311, 1994.

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Iwata, M., Hikosaka, T., Morita, M., Iwanari, T., Ito, K., Onda, K., Esaki, Y., Sakaki, Y. & Nagata, S., Performance analysis of planar-type unit SOFC considering current and temperature distributions. Solid State Ionics, 132, pp. 297–308, 2000. Minh, N.Q. & Takahashi, T., Science and Technology of Ceramic Fuel Cells, Elsevier: New York, 1995. Ota, T., Koyama, M., Wen, C.J., Yamada, K. & Takahashi, H., Object-based modeling of SOFC system: dynamic behavior of micro-tube SOFC. Journal of Power Sources, 118, pp. 430–439, 2003. Srikar, V.T., Turner, K.T., Andrew Ie, T.Y. & Spearing, S.M., Structural design consideration for micromachined solid-oxide fuel cells. Journal of Power Sources, 125, pp. 62–69, 2004. Li, P.W. & Chyu, M.K., Simulation of the chemical/electrochemical reaction and heat/mass transfer for a tubular SOFC working in a stack. Journal of Power Sources, 124, pp. 487–498, 2003. Li, P.W., Schaefer, L. & Chyu, M.K., Investigation of the energy budget in an internal reforming tubular type solid oxide fuel cell through numerical computation, Paper No. IJPGC2003-40126. Proc. of the International Joint Power Conference, June 16–19, Atlanta, Georgia, USA, 2003. Hirano, A., Suzuki, M. & Ippommatsu, M., Evaluation of a new solid oxide fuel cell system by non-isothermal modeling. J. Electrochem. Soc., 139(10), pp. 2744–2751, 1992. Hagiwara, A., Michibata, H., Kimura, A., Jaszcar, M.P., Tomlins, G.W. & Veyo, S.E., Tubular solid oxide fuel cell life tests. Proc. of the Third International Fuel Cell Conference, Nagoya, Japan, pp. 365–368, 1999. Singhal, S.C.,Advances in solid oxide fuel cell technology. Solid State Ionics, 135, pp. 305–313, 2000. Tomlins, G.W. & Jaszcar, M.P., Elevated pressure testing of the Simens Westinghouse tubular solid oxide fuel cell. Proc. of the Third International Fuel Cell Conference, Nagoya, Japan, pp. 369–372, 1999. Bessette N.F., Modeling and simulation for solid oxide fuel cell power system, Georgia Institute of Technology, Ph.D Thesis, 1994. Ahmed, S., McPheeters, C. & Kumar, R., Thermal-hydraulic model of a monolithic solid oxide fuel cell. J. Electrochem. Soc., 138, pp. 2712–2718, 1991. Ferguson, J.R., Fiard J.M. & Herbin, R., Three-dimensional numerical simulation for various geometries of solid oxide fuel cells. Journal of Power Sources, 58, pp. 109–122, 1996. Singhal, S.C., Progress in tubular solid oxide fuel cell technology. Electrochemical Society Proceedings, 99–19, pp. 40–50, 2001. Sverdrup, E.F., Warde, C.J. & Eback, R.L., Design of high temperature solidelectrolyte fuel cell batteries for maximum power output per unit volume. Energy Conversion, 13, pp. 129–136, 1973. Li, P.W., Suzuki, K., Komori, H. & Kim, J.H., Numerical simulation of heat and mass transfer in a tubular solid oxide fuel cell. Thermal Science & Engineering, 9(4), pp. 13–14, 2001.

38 Transport Phenomena in Fuel Cells [35] Aguiar, P., Chadwick, D. & Kershenbaum, L., Modeling of an indirect internal reforming solid oxide fuel cell. Chemical Engineering Science, 57, pp. 1665–1677, 2002. [36] Bessette, N.F. & Wepfer W.J., A mathematical model of a tubular solid oxide fuel cell. Journal of Energy Resources Technology, 117, pp. 43–49, 1995. [37] Campanari, S., Thermodynamic model and parametric analysis of a tubular SOFC module. Journal of Power Sources, 92, pp. 26–34, 2001. [38] Haynes, C. & Wepfer, W.J., Characterizing heat transfer within a commercialgrade tubular solid oxide fuel cell for enhanced thermal management. Int. J. of Hydrogen Energy, 26, pp. 369–379, 2001. [39] Recknagle, K.P., Williford, R.E., Chick, L.A., Rector, D.R. & Khaleel, M.A., Three-dimensional thermo-fluid electrochemical modeling of planar SOFC stacks. Journal of Power Sources, 113, pp. 109–114, 2003. [40] Li, P.W., Schaefer, L. & Chyu, M.K., Interdigitated heat/mass transfer and chemical/electrochemical reactions in a planar type solid oxide fuel cell. Proc. of ASME 2003 Summer Heat Transfer Conference, Las Vegas, Paper No. HT2003-47436, 2003. [41] Joon, K., Fuel cells – a 21st century power system. Journal of Power Sources, 71, pp. 12–18, 1998. [42] Watanabe, T., Fuel cell power system applications in Japan. Proc. Instn. Mech. Engrs., 211, Part A, pp. 113–119, 1997. [43] Veyo, S.E. & Lundberg, W.L., Solid oxide fuel cell power system cycles. Proc. of the International Gas Turbine & Aeroengine Congress & Exhibition, Indianapolis, Indiana, USA, Paper No. 99-GT-365, 1999. [44] White, D.J., Hybrid gas turbine and fuel cell systems in perspective review. Proc. of the International Gas Turbine & Aeroengine Congress & Exhibition, Indianapolis, Indiana, USA, Paper No. 99-GT-419, 1999. [45] Li, P.W., Schaefer, L., Wang, Q.M., Zhang, T. & Chyu, M.K., Multi-gas transportation and electrochemical performance of a polymer electrolyte fuel cell with complex flow channels. Journal of Power Sources, 115 pp. 90–100, 2003. [46] Cahoon, N.C. & Weise, G. (eds) The Primary Battery, Vol. II, John Wiley & Sons: New York, 1976. [47] Li, P.W., Schaefer, L. & Chyu, M.K., Three-dimensional model for the conjugate processes of heat and gas species transportation in a flat plate solid oxide fuel cell. 14th International Symposium of Transport Phenomenon, Bali, Indonesia, June 6–9, pp. 305–312, 2003. [48] Li, P.W., Schaefer, L. & Chyu, M.K., The energy budget in tubular and planar type solid oxide fuel cells studied through numerical simulation. International Mechanical Engineering Congress and Exposition 2003, (IMECE2003-42426), Washington, D.C., Nov. 16–21, 2003. [49] Burt, A.C., Celik, I.B., Gemmen, R.S. & Smirnov, A.V., A numerical study of cell-to-cell variations in a SOFC stack. Journal of Power Sources, 126, pp. 76–87, 2004.

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George, R.A. & Bessette, N.F., Reducing the manufacturing cost of tubular SOFC technology. Journal of Power Sources, 71, pp. 131–137, 1998. Krumdieck, S., Page, S. & Round S., Solid oxide fuel cell architecture and system design for secure power on an unstable grid. Journal of Power Sources, 125, pp. 189–198, 2004. Bird, R.B., Stewart, W.E. & Lightfoot, E.N., Transport Phenomena, John Wiley & Sons: New York, 1960. Williams, F.A., Combustion Theory, Benjamin/Cummings Publishing Co., 1985. Eckert, E.R.G. & Drake, R.M., Heat and Mass Transfer, 2nd Edition of Introduction to the Transfer of Heat and Mass, McGraw-Hill Book Company, Inc., 1966. Turns, S.R., Introduction to Combustion: Concept and Application, Second Edition, McGraw-Hill Higher Education: New York, 1999. Incropera, F.P. & DeWitt, D.P., Introduction to Heat Transfer, Third Edition, John Wiley & Sons, 1996. Todd, B. & Young, J.B., Thermodynamic and transport properties of gases for use in solid oxide fuel cell modeling. Journal of Power Sources, 110, pp. 186–200, 2002. Patankar, S.V., Numerical Heat Transfer and Fluid Flow, McGraw-Hill, 1980. Karki, K.C. & Patankar, S.H., Pressure based calculation procedure for viscous flows at all speed in arbitrary configurations. AIAA Journal, 27(9), pp. 1167–1174, 1989. Beckermann, C. & Smith, T.F., Incorporation of internal surface radiation exchange in the finite-volume method. Numerical Heat Transfer, Part B, 23, pp. 127–133, 1993. Tanner, C.W. & Virkar A.V., A simple model for interconnect design of planar solid oxide fuel cells. Journal of Power Sources, 113, pp. 44–56, 2003. Nguyen T.V., A gas distributor design for proton-exchange-membrane fuel cells. J. Electrochem. Soc., 413(5), pp. L103–L105, 1996.

Nomenclature a A Acell Aair , Afuel b B C Cp DJ ,m

Stoichiometric coefficient of chemical species. Chemical species. Area (m2 ). General variable. Outer surface area of fuel cell (m2 ). Inlet flow area of air and fuel, respectively (m2 ). Stoichiometric coefficient of chemical species. Chemical species. General variable. General variable. Specific heat capacity at constant pressure [J/(kg K)]. Diffusion coefficient of jth species into the left gases of a mixture (m2 /s).

40 Transport Phenomena in Fuel Cells E F g h H i i0 I k KPR , KPS L m m ˙ M Mf ne p, P q˙ Q r r a , rc , re R Ra , Rc , Re S T u U v V Vcell V a, V c W x X x¯ , y¯ , z¯ y Y z Z

Electromotive force or electric potential (V). Faraday’s constant [96486.7 (C/mol)]. Gibbs free energy (J/mol). Chemical enthalpy (kJ/kmol) or (J/mol). Height (m). Current density (A/m2 ). Exchange current density (A/m2 ). Current (A). Thermal conductivity (W/m K). Chemical equilibrium constant for reforming and shift reactions, respectively. Length (m). Mass transfer rate or mass consumption/production rate (mol/s). Mass flux [mol/(m2 s)]. Molecular weight (g/mol). Total mole rate of fuel flow (mol/s). Number of electrons involved in per fuel molecule in oxidation reaction. Pressure (Pa) or position. Volumetric heat source ( W/m3 ). Heat energy (W). Radial coordinate (m). Average radius of anode, cathode, and electrolyte layers (m). Universal gas constant [8.31434 J/(mol K)]. Discretized resistance in anode, cathode, and electrolyte (). Source term of gas species (kg/m3 ); General variable. Temperature (K). Velocity in axial direction (m/s). Utilization percentage (0–1). Velocity in radial direction (m/s); Diffusion volume in eqn (68). Specific volume (m3 /kmol). Cell terminal voltage (V). Potentials in anode and cathode, respectively (V). Width (m). Stoichiometric coefficient of chemical species; Axial coordinate (m). Chemical species. Mole fraction. Reacted mole rate of CH4 , CO and H2 , respectively in a section of interest in flow channel (mol/s). Stoichiometric coefficient of chemical species; Coordinate (m). Chemical species. Mass fraction. Coordinate (m). Compressibility factor.

Multiple transport processes in solid oxide fuel cells

41

Greek symbols θ δ
G
G 0
H
S
x λ µ ρ ρea , ρec ρee ρr ηAct

Circumferential position. Angle. Thickness of electrodes and electrolyte layers (m). Gibbs free energy change of a chemical reaction (J/mol). Standard state Gibbs free energy change of a chemical reaction (J/mol). Enthalpy change of a chemical reaction (J/mol). Entropy production [J/(mol K)]. One axial section of fuel cell centered at x position (m). Thermal conductivity [W/(m ◦ C)]. Dynamic viscosity (Pa s). Density (kg/m3 ). Electronic resistivity of anode and cathode respectively ( · cm). Ionic resistivity of electrolyte ( · cm). Reduced density. Activation polarization (V ).

Subscripts a c cell e, w, n, s E, W , N , S f i j m P R x X,Y
x

Anode. Cathode. Overall parameter of fuel cell. East, west, north, and south interfaces between grid P and it neighboring grids. East, west, north, and south neighboring grids of grid P. Fuel. Subscript variable. Gas species; Subscript variable. Mixture. Variables at grid P. Electrochemical reaction. Axial position. Chemical species. Variation in the channel section of
x.

Superscripts a b c e in out P R x

Anode. Sequence. Sequence. Cathode. Sequence. Electrolyte. Inlet of a channel section of interest. Outlet of a channel section of interest. Variables at grid P. Reaction. Axial position.

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CHAPTER 2 Numerical models for planar solid oxide fuel cells S.B. Beale National Research Council, Ottawa, Canada.

Abstract This article discusses various numerical techniques used to model single-cells and stacks of planar solid oxide fuel cells. A brief history of the solid oxide fuel cell (SOFC), and a survey of modeling efforts to-date are presented. The fundamental thermodynamics of the SOFC are introduced, together with the equations governing the ideal (Nernst) potential. Factors affecting operating cell voltages are then discussed. A simple iterative calculation procedure is described, whereby inlet mass factions, flow rates, and cell voltage are known, but outlet values, and current are required. This provides a paradigm for more complex algorithms set out in the remainder of the text. The next level of complexity is provided for by numerical integration schemes based on ‘presumed-flow’ methodologies, where the inlet flow rates of oxidant and fuel are assumed to be uniform. Local mass sources and sinks are computed from Faraday’s equation. These are then used to correct the continuity and mass transfer equations. Temperature variations are also computed; because heat and mass transfer effects affect the output of SOFCs, significantly. Some more advanced chemistry, heat and mass transfer issues are further detailed. The most detailed cell models are obtained using computational fluid dynamics codes, based on finite-volume and other techniques. Additional improvements to these (and other) codes involve the detailed modeling of the electric potential (in place of the Nernst equation formulation), and the analysis of the combined kinetics and mass transfer problem in the porous electrode media. Finally stack models are introduced. These may be full-scale computational fluid dynamics models or simpler models based on volume-averaging techniques.

44 Transport Phenomena in Fuel Cells

Figure 1: Schematic of typical planar SOFC.

1 Introduction 1.1 History and types of solid oxide fuel cell There are many types of fuel cell under development. Most popular among these are the proton exchange membrane fuel cell (PEMFC) and the solid oxide fuel cell (SOFC) [1–3]. The SOFC is a solid-state device that utilizes the oxygen ion conducting property of the ceramic zirconia. The operation of a SOFC differs from that of a PEMFC or a conventional battery, in that both electronic and ionic circuits are due to the flux of negatively charged particles; electrons and O2− ions. The positive ions or ‘holes’are immobilised in the solid matrix, at the time of fabrication. The book by Williams [4] details developments in SOFC technology prior to the 1960’s. A SOFC consists of an anode or fuel electrode, an electrolyte, and a cathode or air electrode. The anode is typically a porous cermet, for example the combination of NiO and Yttria-stabilised Zirconia (YSZ). A typical material for the cathode is Sr doped LaMnO3 , a perovskite. The electrolyte consists of a layer of YSZ, and is fabricated as thin as possible in order to reduce Ohmic losses. Fuel and air passages are required to supply the products and remove the reactants; the gas passages are typically rectangular micro-channels, though planar channels may also be utilized occasionally. In addition, porous media inserted between the fluid passages and electrodes may serve as gas diffusion layers. Both planar and cylindrical SOFC geometries are to be found; the latter avoid mechanical sealing problems, whereas the former allow for relatively large electrical power densities to be achieved. The discussion in this paper focuses on the planar design. In order to reach the necessary operating voltage, fuel cells are connected together in a stack by situating a metallic or ceramic interconnect between each cell. These are often made of stainless steel. Cells are connected electrically

Numerical models for planar solid oxide fuel cells

45

Figure 2: Planar SOFC stack. Courtesy Global Thermoelectric Inc. in series, but hydraulically in parallel. The interconnects also function as a housing for the flow channels for the air and fuel which are supplied via manifolds. The balance-of-plant makes sure that the working fluids are supplied under the desired operating conditions. Fuel cells may be operated in co-flow, counter-flow and cross-flow. The industrial SOFC designs encountered by the present author were for cross-flow. Hydrogen is normally provided by reforming methane or methanol in a reactor. The electrical efficiency of a system running on natural gas can exceed 50%. The remaining energy is released as heat. The basic operation of a SOFC is as follows: reduction takes place at the cathode. O2 + 4e− → 2O2− .

(1)

At the anode, if the fuel is H2 , the oxygen ion combines with hydrogen to form water, electricity and heat. H2 + O2− → H2 O + 2e− .

(2)

Thus the overall reaction is H2 + 12 O2 → H2 O. 1 kg + 8 kg

9 kg

(3)

1.2 Survey of modeling techniques Mathematical models of heat and mass transfer and electrochemistry provide tools for performance prediction of single-cells and stacks of fuel cells. Thermo-fluid simulations can predict the thermal and electrochemical performance, and structural analysis models can predict the mechanical behavior. Our concern here is

46 Transport Phenomena in Fuel Cells with the former. Among the first to perform calculations on SOFC performance were Vayenas and Hegedus [5]. Since then, the scope and detail of modeling have increased. Models have been developed at various scales: micro-scale or nano-scale studies aim at the development of better electrodes through mathematical analysis. Single-cell models are often the first consideration for fuel cell manufacturers embarking on a new product design, and are currently a target application for the vendors of commercial CFD codes. Stack modeling is necessary for the overall design and often involves parametric studies. Among the most salient issues that need to be addressed are the electric potential field, porous media transport phenomena, and chemical kinetics (shift reactions, internal reforming). The development of a model for conservation of heat, mass, and charge in SOFCs is detailed in [6–8]. Such models have been applied to SOFC stacks [9–12]. Shift and reforming reactions have also been considered [13, 14] as well as heat and mass transfer issues [15, 16]. The size and complexity of SOFC stacks requires the use of very large computers, if a conventional CFD code is employed. Because of this, simpler models based on engineering assumptions have sometimes been devised [10, 12, 17, 18] in order that personal computers may perform such calculations. The speed and memory requirements of a computer code will be limited by the application. For detailed SOFC design large-scale CFD codes will be required, whereas for real-time control, rapid response will preclude all but the simplest of schemes from being employed. 1.3 Thermodynamics of solid oxide fuel cells Newman [19] states that thermodynamics, kinetics, and transport phenomena are fundamental to understanding electrochemical systems. It is important to realise that transport phenomena are affected by thermodynamic properties and chemical kinetics. Physicochemical hydrodynamics [20, 21] is the study of such mutual interactions. It is timely to start with the thermodynamics of fuel cells. Figure 3 illustrates the notion of the fuel cell as a thermodynamic device. It is postulated

Figure 3: Fuel cell as a thermodynamic system.

Numerical models for planar solid oxide fuel cells

47

that the fundamental relation [22] for the internal energy, U , may be expressed as U = U (S, V , Nj , Q),

(4)

where S is entropy, V is volume, Nj is mole number of species j, and Q is charge number. Conservation of energy requires that, dU = dWHeat + dWMech + dWChem + dWElec ,  µj dNj + EdQ, dU = TdS − PdV +

(5) (6)

j

where T = (∂U /∂S)V ,Nj ,Q , P = − (∂U /∂V )S,Nj ,Q , µj = (∂U /∂Ni )S,V ,Q , and E = (∂U /∂Q)S,V ,Nj are temperature, pressure, chemical potential, and electric potential, respectively. Texts in physical chemistry invariably introduce Gibbs free energy, G, which for a chemical reaction which does not involve charge transfer may be written: dG = dU − TdS + PdV =



µj dNj .

(7)

j

The free energy is useful for analysis of situations where temperature and pressure are constant (e.g. at ambient). Note that for the situation under consideration dU − TdS + PdV  = j µj dNj . For an electrochemical process, where temperature and pressure are maintained constant and there is no change in internal energy, chemical energy is converted to electrical energy:  µj dNj = −EdQ, (8) j

where it is understood that Nj is positive for inflow and negative for outflow, i.e., the sum of chemical energy of the reactants less that of the products. The molar enthalpy of formation of the chemical reaction defined by eqn (3) is
H 0 = −247.3 kJ/mol at 1000 ◦ C, whereas the molar Gibbs energy of formation (i.e. the chemical energy of formation) is
G 0 = −177.4 kJ/mol, the negative sign indicating that heat will be released at a rate of 69.9 kJ/mol. The electrochemical reaction is thus exothermic. The difference between these terms
H 0 −
G 0 is frequently referred to in a somewhat cavalier fashion as the T
S 0 term. However there are other heat sources in a fuel cell, which can change the entropy (and internal energy) of the system. It is also worth noting that a fuel cell is not open to the atmosphere, so pressure and temperature can and do vary. It is therefore prudent to appreciate that eqn (8) is just a simplified form of eqn (6). Nonetheless it is eqn (8) that is invariably the point of departure for a fuel cell analysis, with irreversibilities being added to modify the basic postulate that ideally all chemical energy would be converted to electrical energy.

48 Transport Phenomena in Fuel Cells Since the air and fuel in a SOFC are high-temperature gases, we may treat them as multi-component perfect gases, such that PV = NRT ,

(9)

where R = 8.3144 × 103 J/mol.K is the gas constant, and N=



Nj

(10)

j

for the air-side and fuel-side gases. The chemical potentials are given by [22]:

where

µj = µ0j (T ) + RT ln P + RT ln xj = µ0j (T ) + RT ln Pj ,

(11)

 xj = Nj /N = Nj Nk

(12)

k

is mole fraction and Pj = xj P is the partial pressure of species j. Faraday’s law gives the number of moles of oxygen consumed by the electrochemical reaction as Q , (13) N˜ = nF where F = 96 485 Coulomb/mol is Faraday’s constant and n = 2 is the valence of oxygen. Thus the change in chemical energy is   
  µj
Nj = (14) νj µj Products − νj µj Reactants N˜ and νi are stoichiometric coefficients. For example, 

ν1 A1 + ν2 A2 → ν3 A3 , µi
Ni = (ν1 µ1 + ν2 µ2 − ν3 µ3 )N˜ = (
G 0 + RT ln Kp )N˜ ,

(15) (16)

 where
G 0 = [(νj µj )Products − (νj µj )Reactants ] is referred to as the molar Gibbs free energy for the chemical reaction. (NB: The author denotes
G = G/N˜ J/mol for consistency with the literature). The equilibrium constant, KP , is given by Kp =

P1ν1 P2ν2 . ν P3 3

(17)

This may be re-written in terms of mole fractions, xj , e.g., for a SOFC with hydrogen as fuel:    0.5  xH2 xO 1 0 2 µi
Ni =
G + RT ln Pa + RT ln N˜ , (18) 2 xH2 O

Numerical models for planar solid oxide fuel cells

49

where Pa is air pressure. Substituting molar flow rate for current using Faraday’s law we immediately obtain  µi
Ni . (19) E=− nF Since the valence of oxygen is 2, we have,   0.5 xH2 xO RT RT 0 2 + E=E + ln ln Pa 2F xH2 O 4F

(20)

which is the Nernst equation for a solid oxide fuel cell. The quantity E is the reversible or Nernst potential. In what follows below, we make the assumption that we may associate the thermodynamic pressure with the hydrodynamic pressure, and the thermodynamic potential with the electrostatic potential. For practical reasons, E 0 is redefined such that Pa is gauge (relative) pressure rather than absolute air pressure, in which case the latter term in eqn (20) is replaced by the form (RT /4F) ln (1+Pgauge /P0 ), and P0 is a reference operating pressure. The latter form is more useful when considering detailed transport models where values of pressure and temperature vary. Generally speaking Pgauge /P0 is very small so changes in E are largely due to variations in mole fraction and temperature. Values of E 0 are typically enumerated as polynomials, e.g., E 0 = a − bT .

(21)

Typical values would be a = 1.214 V and b = 3.2668 V/K. Higher order polynomials may also be found. The physical significance of the terms on the rightside of eqn (21) are a ≈
G 0 /nF and b ≈ T
S 0 /nF. Although these terms are themselves temperature dependent, this is a quite reasonable approximation in many situations. The process illustrated in Fig. 3 is an open thermodynamic cycle. Thus ˙ flows. Under these circumstances when a load is connected, a finite current i = Q there will be irreversible changes, and the actual cell voltage will be lower than the reversible Nernst potential. 1.4 Cell voltage and current The conduction mechanism in the ionic electrolyte differs fundamentally from that in the neighbouring electrodes and interconnects (electronic conduction). Therefore the electrolyte polarises at the edges and a double charge layer is established at the interfaces. The change in chemical potential across the interface,
µj , is balanced by the opposing change in the electric potential for electrochemical equilibrium, nF
φ. The sum of these terms is referred to as the electrochemical potential. Figure 4 illustrates the double charge layer in a fuel cell, while Fig. 5 shows the potential distribution. The Nernst potential is just the sum of the anodic and cathodic potentials, E=
φa +
φc . It is not possible to measure the potential,
φ, at a single electrode, though it is possible to measure changes from equilibrium, η = δ
φ,

50 Transport Phenomena in Fuel Cells

Figure 4: Double charge layer.

Figure 5: Electric potential at open circuit and with a current flowing. when a current is drawn. The double-layer is just a few nanometres thick. There is no net change in potential, in the absence of an electric current, within the bulk of the electrolyte. In other words, there are step changes in potential at the anode and the cathode, but zero change within the electrolyte, as shown in Fig. 5. Whenever a current, i, is drawn, the voltage drops and the actual cell voltage, V , is given by (22) V = E − ηe − ηa − ηc = iRl , where the terms ηe , ηa and ηc are electrolytic, anodic, and cathodic losses, respectively. These have often been referred to as overpotentials or polarisations in the electrochemistry and chemical engineering literature. Rl is the resistance of the external load. The potential difference across the electrolyte ηe = iRe = E − ηa − ηc − V

(23)

is sometimes referred to as the Ohmic overpotential. Here Re is the internal resistance of the electrolyte. The resistivity of the electrolyte is highly temperature dependent, and decreases with increasing temperature. There will also be a similar (usually smaller) Ohmic loss associated with metallic interconnect(s), and for

Numerical models for planar solid oxide fuel cells

51

convenience that latter will be ignored for now. Thus work is being done by the electric field on the ions in the electrolyte and interconnects, whereas work is being done against the field in crossing the electrodes. Although the anodic and cathodic overpotentials are themselves a function of the current, it is sometimes convenient for numerical purposes to lump all the losses into a single linearised internal resistance, R¯ ¯ (24) ηe + ηa + ηc = iR, ¯ where it is understood that R¯ = R(i), i.e. the slope of the V-i curve is not constant. The V-i curve or, more often the voltage-current-density, V-i , curve is the single most important characteristic to the electrochemical scientist. N.B. The reader will note that in this text, a ‘dot’ is used to denote a time derivative, and a ‘dash’ to denote a length derivative. Thus if q has units [J], then q˙  has units [W/m2 ]. While mole fractions are often found in electrochemical texts, in engineering problems it is much more common to utilize mass fractions, mj , which may be converted to/from molar fractions, xj , as follows: mj = xj

Mj , M

(25)

where the mixture molecular weight is just M =



x j Mj = 

j

1

mj j Mj

.

(26)

The reactions at the electrode surfaces lead to continuity and species sources/sinks in the working fluids. The mass fluxes, m ˙ j (in kg/m2 s), are related to the current  density, i = i/A, by Faraday’s law m ˙ j = ±

Mj i , 1000nj F

(27)

where Mj is the molecular weight of chemical species j (in g/mol), and n is the valence. It is trivial to re-write the Nernst equation in terms of mass fractions.     0.5 m m H 1 RT RT 2 O 2 + ln (Ma ) + 0.4643 + E = E0 + (28) ln ln Pa , 2F mH2 O 2 4F where Ma is the mixture molecular weight, eqn (26), for air, and   MH2 O ln ≈ ln (18) − ln (2) − 12 ln (32) = 0.4643. MH2 MO0.5 2

(29)

In electrochemistry, molar-based units are frequently encountered, but in transport phenomena, for a variety of reasons, mass-based units are preferred. This duality is a fact-of-life when dealing with fuel cells. Both have their uses and limitations.

52 Transport Phenomena in Fuel Cells 1.5 Activation losses Electrochemical reactions can only occur at finite rates. For a charged particle to pass across a double charge layer, it must possess more free energy than a certain minimum. From an engineering perspective the activation energy is similar to bulk-to-interphase energy, required for example in multi-phase systems involving phase-change. Figure 6 illustrates the basic principle for a single-step singleelectron electrochemical reaction A → B. The sharp sign, #, indicates the activated state. Eyring’s analysis [23] is useful for such rate-limiting situations. The analysis proceeds as follows: the probability density function for ions crossing the electrode boundary follows a Boltzmann distribution, and it can readily be shown that the forward rate constant, kf mol/sec, is given by kf = k0 exp (−
Gf /RT ).

(30)

A backward rate constant, kb , is similarly defined. For an open circuit (equilibrium), kf = kb = k0 . If a current flows, the potential difference deviates from the open circuit value, and the required free energy will also change as a result of the electric field:
G =
Gf − βF
φ, (31) where β is a symmetry coefficient, 0 ≤ β ≤ 1, which is a measure by which a change in electric potential affects the required activation energy, see Fig. 6. Under this condition, the forward rate will not equal the backward rate. The former is given by, if = FNA k0 exp (−
Gf /RT ) exp (−ηF
φ/RT ) = i0 exp (−βFη/RT ),

(32)

where xA is the mole fraction of compound A, and M is the mixture molecular weight. The quantity η = δ
φ =
φ −
φ0 is the so-called activation overpotential. Similarly, for the backwards reaction B → A the charge flux of charge is ib = i0 exp (−(1 − β)Fη/RT )

(33)

Figure 6: Schematic of concept of activation energy for a single-step single-electron transfer reaction.

Numerical models for planar solid oxide fuel cells

53

and the form of the Butler-Volmer equation, appropriate for a single-step singleelectron transfer process, is obtained, i = i0 [ exp (−βnFη/RT ) − exp ((1 − β)nFη/RT )].

(34)

The quantity i0 is the exchange current. In general i0 will be a function of molar concentration. That dependence can be theoretically determined by ensuring compatibility with the thermodynamic requirement E = E 0 + (RT /F) log (PB /PA ) for equilibrium, where PA and PB are partial pressures. Setting if = ib = i0 it follows 1−β β that the exchange current density is just i0 = Fk0 PA PB . The two terms in eqn (34) are essentially similar to Arrhenius-type expressions occurring in chemically reacting systems. The following points are noted: for small currents, eqn (34) reduces to a linear form, obtained by expanding it in a Taylor series about η = 0. At higher overpotentials (currents) an exponential form is obtained; for the latter situation one of the two terms in eqn (34) is negligibly small, depending on the direction of flux of charged particles. The exponential form is referred to as a Tafel equation. The ratio i/i0 determines if one is in the linear or Tafel regions. For a multi-step electrochemical reaction, the Butler-Volmer equation is generally written, i = i0 exp (nβf Fη/RT ) − exp (−nβb Fη/RT ) . (35) The quantities βf and βb are referred to as transfer coefficients. In deriving eqn (35) for a multi-step reaction [24, 25] it is generally assumed that at any electrode there is only one rate limiting step, with all other steps being considered at equilibrium. In view of these restrictions, it is probably wise to regard eqn (35) as being a semi-empirical formulation, for which βf and βb are determined experimentally. Equation (35) may be considered as an implicit form of the relationship of η = η(i, T ) which must be solved iteratively, for any given i0 and β. Typical values for planar SOFCs are i0 = 1 − 5 000 A/m2 . In general two sets of eqn (35) must be solved at both the anode and cathode to yield ηa and ηc for use in eqn (22), i.e. there are two values of i0 and four of β (N.B. β is often assumed to be ½). The overpotential at one electrode may be negligible compared to the other. Following the argument above for a single-step reaction for two electrodes, and imposing the requirement that for equilibrium, the sum of the potentials must equal the thermodynamic Nernst potential, eqn (20), and with the additional assumption that βf = βb = 1/2, it may be reasonably concluded that the concentration dependence at the cathode and i0 ∝ PH0.5 PH0.5 at the anode of the exchange current is i0 ∝ PO0.25 2 2 2O for a SOFC. Simplifying the Butler-Volmer equation has the advantage of rendering an explicit solution for η(i, T ); however the calculation procedure for the cell/stack is iterative by nature, and therefore there is little problem in using the implicit form, above. Charge transfer losses are particularly important at low current densities. The SOFC is a high-temperature device, and thus activation losses are generally less significant than in other fuel cells such as proton exchange membrane fuel cells, due to the presence of the exponent (RT )−1 . The subject of electrode kinetics, or electrodics, is an important one and the reader interested in more than the brief overview

54 Transport Phenomena in Fuel Cells provided here is referred to specialised texts such as Gileadi [24], Bockris et al. [25] and Bard and Falkner [26]. 1.6 Diffusion losses Mass transfer losses are due to diffusion becoming a rate limiting factor in providing O2 and H2 to the cathode and anode, and removing H2 O at the anode. Thus transport phenomena in the fuel and air channels can and do affect the electrical performance of the fuel cell. This becomes increasingly important at high current densities where concentration gradients are high. Mass transfer effects are sometimes handled in an analogous manner to charge transfer (activation) by introducing a concentration overpotential, also known as a concentration polarisation. The origin of these terms is historical. They may be written in terms of either molar or mass fractions. The latter is adopted here, namely  RT mb η=± (36) ln nF mw for inclusion in eqn (22). The symbols mb and mw denote bulk and wall values of mass fraction (in the interests of readability the indicial notation mj,b etc., is dropped) and the sign is positive for products and negative for reactants. There is no real advantage to defining overpotentials in the form of eqn (22). Rather, all that is necessary is the recognition that the Nernst equation be based on wall values, instead of those in the bulk of the fluid. Some authors [3, 27] suggest that diffusion effects in fuel cells be modelled using so-called equivalent film theory, introducing a Nernst length scale. That approach is not recommended here. The suggested approach for mass transfer calculations in SOFCs is outlined in Beale [28] and follows standard approaches [29–32] to the generalized engineering mass transfer problem. In fuel cells, heterogeneous chemical reactions on the electrode surfaces lead to sources and sinks in the continuity and species (mass fraction) equations. Let it be supposed that the mass flux, m ˙  , is given by m ˙  = gB,

(37)

where m ˙  = ±Mi  /1000nF is negative for O2 and H2 and positive for H2 O, and g is a mass transfer conductance. For historical reasons the symbol g is used to represent a generalised conductance, whereas the symbols h (Europe) and α (N. America) are reserved for heat transfer, with temperature as independent variable. There is clearly an element of redundancy in this convention. The driving force, B, is defined by B=

mb − mw . mw − m t

(38)

By convention, both m ˙  and B are positive for injection (H2 O at the anode) and negative for suction (O2 and H2 at the cathode and anode, respectively),

Numerical models for planar solid oxide fuel cells

55

while the conductance g is always positive. The subscript, t, refers to the so-called transferred-substance state (T-state) [29, 32, 33]. For multi-component mixtures, T-state values are given by, m ˙ j mj ,t =  , (39) m ˙  where m ˙  = j m ˙  j . Thus it can be seen from the stoichiometry of eqn (3) that for the air-side mt = 1 for O2 , whereas mt = −1/8 for H2 and mt = +9/8 for H2 O on the fuel-side. Equation (37) may be conveniently rewritten in dimensionless form B=

g∗ b, g

(40)

˙  → 0, and the blowing parameter, where g ∗ denotes the value of g in the limit m b, is given by m ˙  (41) b= ∗. g Equation (38) may be rearranged to obtain the required form 1 + mmbt B mw . = mb 1+B

(42)

Ideally actual mass transfer data for B(b) for the particular channel geometry under consideration are available; if not, the solution to the 1-D convectiondiffusion equation, B = exp (b) − 1 (43) may be used as an approximation. Figure 7 shows data from numerical simulations of fully-developed mass transfer in ducts of various geometries [34]. It can be

Figure 7: Mass transfer driving force as a function of blowing parameter, from [28].

56 Transport Phenomena in Fuel Cells seen that data normalized in the form of B vs. b (or equivalent) conforms well to the curve corresponding to eqn (43). The choice of non-dimensional number is important; for example use of a wall Reynolds number, Re, or Sherwood number, Sh, for the abscissa in place of the normalised convection flux, b, will not result in the data reducing to (roughly) a single curve. Equations (42) and (43) or equivalent, are sufficient to compute mass transfer losses in the form of eqn (22) in a manner which is reasonably rigorous while at the same time being reasonably simple to implement. Thus knowledge of g ∗ for air-side and fuel-side geometries together with experimental data or an analytical expression for B(b) is sufficient to obtain a reasonable estimate of diffusion losses in a fuel cell. Some deviations from Fig. 7 are to be anticipated for a variety of reasons (entry-length considerations, Schmidt number variations, etc.). If experimental data are available, these should always be used. It is to be noted that for many situations the mass flow rates are small, and under the circumstances g = g ∗ and B = b. Values of g ∗ may be obtained from the zero-mass flux Sherwood number, Sh∗ , g ∗ Dh . (44) Sh∗ =  Note that diffusion control is important in fuel cells at high current densities/mass fluxes, not in the limit Sh → Sh∗ . Figure 8 shows values of Sh∗ for fully-developed mass flow and scalar transport in rectangular ducts of various aspect ratios, as given in Table 43 of ref. [35]. It is to be noted that suction (e.g. at the cathode) reduces mass transfer, i.e. g/g ∗ < 1, whereas blowing (e.g. at the anode) has the opposite effect g/g ∗ > 1. Knowledge of the T-state value is also useful when performing detailed CFD simulations. These may be used to prescribe boundary conditions in the species equations (see Section 2.5). Although this introductory analysis is

Figure 8: Sh∗ (Nu∗ ) for rectangular ducts in the limit of zero blowing. From [34], data of Schmidt [35].

Numerical models for planar solid oxide fuel cells

57

Figure 9: Schematic representation of a SOFC with hydrogen as fuel. appropriate for mass transfer in the gas channels, Section 2.9 shows that it may be readily extended to consideration of more complex geometries involving combined mass transfer in channels and porous diffusion layers. Section 2.11 discusses some further considerations for mass transfer analysis in fuel cells. 1.7 Basic computational algorithm We are now in a position to construct a simple algorithm for computation of the performance for a fuel cell. Assume the following are known: cell voltage, V , ˙ a,in , m ˙ f ,in , and temperature, T , air pressure Pa , inlet flow rates for air and fuel, m mass fractions of the component species in the air, mO2 ,in , mN2 ,in , and the fuel, mH2 ,in , mH2 O,in , respectively. 1. Guess an initial value for the current i. 2. Calculate mass sources/sinks, s˙O2 , s˙H2 , s˙ H2 O , from Faraday’s law. Compute ˙ f ,out , and mass fractions, mO2 ,out , air and fuel outlet mass flow rates, m ˙ a,out , m mH2 ,out , mH2 O,out , mN2 ,out . 3. Calculate average of inlet and outlet bulk mass fractions and then compute wall mass fractions, using eqn (42). 4. Compute the anode and cathode overpotentials, ηa , ηc , from the Butler-Volmer equation eqn (35), based on wall values. 5. Compute the Nernst potential, E, from eqn (20), based on wall values. 6. Compute the current, i, from i=

E − ηa − ηc − V . Re

(45)

7. Repeat steps 2–6 until convergence is obtained. This simple procedure will form the basis for subsequent more complex procedures described further below. The continuity relationships in step (5) are easily ˙ a,in + s˙O2 and m ˙ f ,out = m ˙ f ,in + s˙H2 + s˙H2 O computed. For example; m ˙ a,out = m while the mass fraction equations are just mk,out = mk,in + s˙k /(˙sk + m ˙ k,in ), where k = O2 , H2 , H2 O, as appropriate.

58 Transport Phenomena in Fuel Cells One important note regards the requirement of prescribed current vs. prescribed voltage. Traditionally physicists work with prescribed voltage, and electrochemists with prescribed current, i (or current density, i ). If it is required that the galvanostatic condition, as opposed to the potentiostatic condition, be prescribed, then it is necessary to adjust the prescribed voltage until the required current is obtained. This is straightforward and two methods for achieving the same end are discussed in Section 2.7.

2 Computer schemes The numerical scheme described above is of limited practical utility, though it does set the stage for what follows. Numerical schemes vary from very simple schemes requiring a few seconds to run on a personal computer, to complex models which require large computers and long run times. The latter might be used for detailed cell design whereas the former could be employed in a system model including balance-of-plant or for real-time control. If a SOFC or stack is well-designed, the inlet flow streams will be uniform. However, there may be very substantial variations of mass fraction, temperature, and current density across the fuel cell. Also the flow velocities along the cell passages vary due to mass sources/sinks, and as a result, the pressure may also change in a non-linear fashion. Thus there are many circumstances where it is necessary to solve only the continuity and scalar transport equations, employing the previously-described algorithm, and not involve the pressure-corrected momentum equations in the solution. In other situations, for example when investigating a particular configuration of flow channels, solutions to the momentum equations are required. It is important to distinguish between situations where the domain is tessellated so finely that diffusive components (viscous effects, heat conduction, and gas diffusion) are captured, and situations where it is necessary to invoke a rate equation for the diffusion flux of some general field variable, . −

 ∂  = g
. ∂y w

(46)

The local gradient of  at the wall is thus replaced by the bulk-to-wall (or in some cases bulk-to-bulk) difference,
, thus invoking the concept of a conductance, g, discussed in detail above in Section 1.6. This notion is central to the classical subject of convective heat and mass transfer [36]. In general, when discretising a SOFC, a mesh will be constructed. It may be structured or unstructured, and it may or may not be boundary fitted (so that each cell corresponds to a given fluid/solid material). For detailed simulations where the diffusion terms are evaluated directly, a fine mesh, concentrated in the nearwall fluid boundaries, will be required as shown in Fig. 10(a). If a rate-equation assumption is invoked, a coarse body-fitted mesh, (b), may be employed, or (c) the mesh may be arbitrary corresponding to local volume averaging. In this case no

Numerical models for planar solid oxide fuel cells

59

Figure 10: Some possible grids used to discretise a planar SOFC (a) Detailed boundary-fitted (b) Boundary-fitted rate-based formulation (c) Volumeaveraged (non-boundary-fitted).

spatial distinction is made between the different materials in the fuel cell. All of these schemes have their advantages and disadvantages. For the latter case storage allocation may be required for more than one phase at each computational cell, whereas for case (b) it is necessary to keep track of all material properties and inter-phase coefficients (conductances), which may be numerous. For case (a), the grid may be so large as to render calculations intractable for large SOFC stacks. Thus there is a trade-off between detail and speed. The code designer is obliged to make some decisions at the outset regarding the nature of the mesh and the corresponding numerical scheme. Let it be supposed that the SOFC is composed of four distinct media: (1) air, (2) fuel, (3) electrolyte, and (4) interconnect. Let it further be assumed that the electrodes are sufficiently thin that they may be considered as forming the electrolyte surfaces. We shall relax these assumptions in due course. In the next section, we shall consider how to solve such a system using a finite-volume procedure [37]. 2.1 General scalar equation Let is be proposed that a prototype equation having the form   ∂(ρj εj j ) αjk (k − j ) + ∇ · (εj ∇j ) + εj s˙ + ∇ · ρεj u j j = εj ∂t j

(i)

(ii)

(iii)

(iv)

(v)

(47)

60 Transport Phenomena in Fuel Cells be adopted, where  is a generic scalar (continuity, mass fraction, enthalpy, etc.) α is a volumetric inter-phase transfer coefficient and  is an exchange coefficient. Let the terms in eqn (47) be referred to as, from left to right: (i) transient, (ii) convection, (iii) inter-phase transfer, (iv) diffusion or within phase transfer, and (v) source. If the volume fractions, εj , are constant they make no contribution to the overall balance, and can be eliminated prior to integrating the equations. Moreover, if a boundary-fitted mesh is employed, εj = 0 for all but one ‘phase’ for which εj = 1. For simplicity they will therefore be removed in the analysis. Generally-speaking, terms (iii) and (iv) are mutually exclusive. Also, although (iii) is listed here as a distinct term, it is almost always coded in the form of a linearised source term (v). For steady co-flow, counter-flow and cross-flow, the equations simplify further and in some situations, reduce to ordinary differential equations. Equation (47) may readily be converted to linear algebraic equations having the form, aW (W − P ) + aE (E − P ) + aS (S − P ) + aN (N − P )  (48) + CP (VP − P ) + SP = 0, where in the interests of brevity, the steady 2-D form has been adopted. Equation (48) is the form appropriate to the finite-volume method. The compass notation E = East, W = West, S = South, N = North, T = Previous time step, etc. has been adopted [37]. The terms SP and CP (VP − P ) are referred to as fixed source and linearised source terms, respectively. 2.2 Continuity It is usual to solve the phase continuity equations  ∂ρj

· ρj u j = m ˙  +∇ j . ∂t

(49)

The source terms in the air and fuel passages are due to the electrochemical reactions at the electrodes, namely:  m ˙ j = ± Mk i /1000nF, (50) k

where Mk is the molecular weight of chemical element k (O2 , H2 , H2 O) in fluid j, (air or fuel). Since the continuum equations are integrated, it is equivalent to prescribe source terms as being per unit volume or per unit area. 2.3 Momentum The decision to solve the momentum equations depends on the problem at hand. For single SOFCs where it is known, a priori, that the inlet flows are uniform

Numerical models for planar solid oxide fuel cells

61

and steady, there is little point to adding additional complexity to the code. For a 3-D SOFC stack problem, or for a single cell which has yet to be designed, the flow may vary substantially, and it is therefore necessary to solve a generic equation of the form: ∂(ρj u j )

j − Fj εj u j + ∇

· (µ∇

u j ),

· (ρj u j ; u j ) = −∇P +∇ ∂t

(51)

where the subscript j refers to either air or fuel as distinct phases. Since the phases do not intermingle, the inter-phase terms are of the form Fj εj (0 − u j ). The second term on the right-side of eqn (51) is normally absent for a detailed numerical analysis, Fig. 10(a). Conversely, for cases Figs 10(b) and (c), it is assumed that if transient and inertial effects were negligible, the overall pressure drop would be entirely due to fluid drag or resistance,

j = −Fj U

j = −εj Fj u j , ∇P

(52)

is a local bulk superficial velocity, u is a local bulk interstitial velocity, where U and the quantity Fj has been referred to as a ‘distributed resistance’[38], which for a homogeneous porous media of permeability, kDarcy , has the significance Fj = µ/kDarcy .

(53)

Generally-speaking the dependent variable for case Fig. 10(a) is the actual local velocity, u ; for (b) u is the local bulk interstitial velocity, and for (c) it may be

For fully-developed laminar duct flows with negligible chosen to be either u or U. mass transfer, τw C f∗= 1 = . (54) 2 Re 2 ρu Shah and London [35] suggest that for rectangular ducts, dimensions L × H : C = 24(1−1.3553α∗ +1.9467α∗2 −1.7012α∗3 +0.9564α∗4 −0.2537α∗5 ), (55) where α∗ = H /L or L/H , whichever is a minimum. The Reynolds number, Re = Dh ρu/µ, is based on a hydraulic diameter Dh = 4A/P, where A is the flow area and P is the wetted perimeter. It can readily be shown that Fj =

2C µ . ε Dh2

(56)

For a detailed numerical simulation, Fig. 10(a), knowledge of eqns (52) to (56) is irrelevant. The drag is directly obtained from the viscous term in eqn (51). Conversely, for a rate-based formulation, the viscous term is discarded in favour of eqns (52) to (56). In reality, F will also vary due to mass transfer effects, f/f ∗ = 1, as τ = µ|∂u/∂y| is a function of the shape of the velocity profile which depends on the rate of mass transfer. This profile is not necessarily similar to the scalar mass fraction profile [39]; eqn (43) or Fig. 7 cannot be used to correct F. Detailed experimental

62 Transport Phenomena in Fuel Cells or numerical data of f/f ∗ as a function of the momentum blowing parameter are required for the particular geometry under consideration. At present, this consideration is often simply ignored. Reference [18] contains a comparison of a distributed resistance formulation with a detailed CFD methodology for a SOFC stack including heat and mass transfer effects. 2.4 Heat transfer Equation (47) is considered appropriate for the analysis of heat transfer, with the solved-for variable generally being written in terms of temperature (not enthalpy or internal energy) as   ∂ ρj cj Tj

· (k ∇T

j) + ∇

· q ˙ + q˙  . (57) + ∇¯ · (ρcj u j Tj ) = αjk (Tk − Tj ) + ∇ ∂t j

A volumetric heat source occurs in the electrolyte due to irreversible Joule heating. This is given by i (E − V ) , (58) q˙  = He where He is the thickness of the electrolyte. For thin electrodes the energy of activation at the electrodes may be prescribed as per unit area with a magnitude q˙  = i η

(59)

normal to the electrode surface. An additional heat source is due to the entropy change as discussed in Section 1.3 and can be expressed as q˙  =

i (
G0 −
H 0 ). 2F

(60)

Technically this should be split into anodic and cathodic components. For SOFCs this source term is usually prescribed at the anode. Convective sources exist also by virtue of mass transfer at the walls and these must also be accounted for (see below). The reader will note that there is inter-phase heat transfer between solids and fluids. At the same time there is within-phase metallic conduction, for example in the metallic interconnects. This is one case where both terms (iii) or (iv) in eqn (47) are present. The volumetric heat transfer coefficients are computed as αV = U¯ A,

(61)

where A is the area for heat transfer, V is cell volume, and U¯ is an overall heat transfer coefficient, obtained using harmonic averaging, for example; H 1 1 , + = hA kAS cond U¯ A

(62)

Numerical models for planar solid oxide fuel cells

63

Figure 11: Shape factor for square ducts for the case, L/H = Lchannel /Hchannel = 2.

where Scond is a conduction shape factor. (N.B. the standard definition in heat transfer texts includes the term H /A, where A is the maximum surface area for heat transfer and H is the thickness of the solid region). Figure 11 shows an example of Scond for square geometry. These data were obtained by the present author by means of numerical simulation. The fuel cell designer should obtain Scond from a numerical simulation for the Laplacian system, ∇ 2 φ = 0, for the actual geometry under consideration. Fin theory may be employed, if there are significant temperature variations within the interconnects. Conduction shape factors may also be used when considering the potential distribution in the interconnects where the electric resistance is given by H r= . (63) σAScond It is to be noted that for both electric and thermal conduction, the shape function will vary as a function of the direction of the local current density or heat flux vector. It is thus an engineering approximation. Temperature variations in fuel cells are inevitable, even when the heat sources are entirely uniform (corresponding to uniform current density and resistance). Figure 12 shows typical simulated temperature variations for a cross-flow SOFC; Achenbach [10] compares the performance for co-flow and counter-flow as well. Temperature variations impact in a number of ways; in particular the electrolyte resistance, re , is extremely temperature dependent. This in turn influences the current density and hence the internal power dissipation. Temperature also affects the local Nernst potential, E0 . Generally speaking lower temperatures are required for mechanical stability, however if methane is used as fuel, higher temperatures are needed to assist the reforming reaction. The thermally conducting interconnect is beneficial; were it not present, the temperature gradients in Fig. 12 would be much greater.

64 Transport Phenomena in Fuel Cells

Figure 12: Temperature variation, deg. C, in a SOFC for cross-flow. 2.5 Mass transfer Treatment for mass transfer within the gas phases is particularly simple, namely: ∂(ρmk )

· (ρ uj mk ) = ∇ · (∇mk ) + ∇

· j  , +∇ k ∂t

(64)

where the diffusion term is absent for cases Fig. 10(b)(c). The reader will note that there are no inter-phase mass transfer terms. Mass sources/sinks at the wall are of the form ˙  mk ,t . (65) jk = m Thus mass transfer values are just the T-state values. The wall values, needed for the Nernst equation, are obtained directly by the calculation procedure, for case Fig. 10(a). For cases Fig. 10(b) and (c) these must be computed using eqns (42)– (43), or equivalent, from the local bulk mass fractions. This is particularly simple with the mass-based formulation, given above. 2.6 Numerical integration schemes The generalised coupled system of equations is quite complex, and we shall illustrate by considering the simplified case of the energy equation for the case of steady 2-D cross-flow, for which numerous terms may be eliminated. Specifically, in the fluids the inter-phase terms (Fig. 10(a)), and/or diffusion terms (Figs 10(b) and (c)) are absent or negligible; in the solids terms (i) and (ii) are absent. Thus we can write eqn (47) for the air and electrolyte in the following reduced forms, dTa = αae (Te − Ta ) + αai (Ti − Ta ), dx

· ke ∇T

e + αae (Ta − Te ) + αfe (Tf − Te ) + q˙  , 0=∇ (ρuc)a

(66) (67)

Numerical models for planar solid oxide fuel cells

65

where the subscripts a, f , e, i refer to air, fuel, electrolyte, and interconnect phases. Similar expressions may easily be written for the fuel and interconnect. These may readily be discretised as follows: ∗ − TaP ) = 0, c˙ a (TaW − TaP ) + αae V (TeP − TaP ) + αai V (TfP − TaP ) + aF (TaP (68)

dew (TeW − TeP ) + dee (TeE − TeP ) + des (TeS − TeP ) + den (TeN − TeP ) ∗ − TeP ) + q˙ P = 0, + αae V (TaP − TeP ) + αfe V (TfP − TeP ) + aF (TeP

(69)

where c˙ = mc ˙ = ρu
Ac and de = k
A/|P − E| etc., are convection and diffusion terms, respectively. This is a set of simplified 2-D finite-volume equations (eqn (48)) with  = T . For air: convection dominates and the linear coefficients are aW = c˙ a , aE = aS = aN = 0; the inter-phase terms may be linearised with coefficients Ce = αae V , Ci = αai V , and values Ve = TeP , Vi = TiP , and there are no internal sources of heat S = 0. For the electrolyte: the linkages are diffusive, aE = dew , etc., and the inter-phase terms are as above, namely Ca = αae V , Cf = αfe V , and ˙ P . Thus Va = TaP , Vf = tfP . This time, however, there is a heat source S = Q by careful consideration of the governing equations, it is possible to effect great savings in computer resources, when generating source code in-house. The symbol TP∗ denotes the value of TP at the previous iteration, and aF is an inertial or false time step coefficient. This mechanism is used to relax the scheme in order to procure convergence. 2.7 Iterative procedure The calculation proceeds precisely as in Section 1.7, except that calculations are performed on a per-unit-cell basis rather than at the mean of the inlet and outlet values. A local Nernst potential may be computed: E(x, y) = V + i r + ηa + ηc = V + ri i ,

(70)

¯ current where ri is a local internal resistance. Average values of Nernst potential, E,  ¯ density, i , electrolyte resistance, r¯e etc. are obtained by summation. The galvanostatic (constant current) condition is usually implemented for stack models to ensure overall charge conservation. When the overall current or mean current density, ¯i , is prescribed, some form of ‘voltage correction’, V , is required. For example a correction may be applied in the form V  = −ri ¯i ,

(71)

where V = V ∗ + V  is the desired cell voltage, and V ∗ is the value of V at the previous iteration; similarly ¯i = ¯i + ¯i∗ is the desired value and ¯i∗ is the previous computed value. Thus the mean current density, ¯i ∗ , is computed at the end of each iterative cycle compared to the desired value ¯i, and the voltage is corrected

66 Transport Phenomena in Fuel Cells accordingly. r¯i need not be exact; any reasonable value will procure conversion. An alternative approach taken in many codes is to directly set V = E¯ − r¯i ¯i .

(72)

If eqn (72) is employed, r¯i must be carefully computed using numerical integration over the surface of the SOFC. Because very small changes in V can effect large changes in i , relaxation is generally employed. We have used both eqns (71) and (72) with success in our numerical codes and schemes. The reader will note that if the full potential field is solved-for as a scalar variable, as discussed below in Section 2.10, there is no need to adjust the voltage. Another adjustment often made while performing fuel cell calculations is to change the inlet flow rates until particular values of the utilisation for fuel and air are obtained. This is straightforward. 2.8 Additional chemistry and electrochemistry One advantage of the SOFC is that CO and CH4 may be used as fuels in addition to pure H2 . This introduces additional kinetics and further complicates mass transfer calculations. If CO is employed as fuel, the anode surface reaction is CO + O2− → CO2 + 2e− .

(73)

If both CO and H2 are present, the two reactions at the anode are considered to be in parallel, but the net anode and cathode reactions are in series, so that    = iH + iCO . i  = iO 2 2

(74)

The Nernst potentials are not the same, ECO/O2 = EH2 /O2 , however charge transfer (activation) and mass transfer mechanisms will ensure that the two parallel reac and i  ) adjust until a single cell potential, V , is obtained, such that tion rates (iH CO 2 V = ECO/O2 − ηCO,a − ηc − i R = EH2 /O2 − ηH2 ,a − ηc − i R,

(75)

where ηCO,a and ηH2 ,a are evaluated iteratively so that ECO/O2 − ηCO,a = EH2 /O2 −  , and η  ηH2 ,a where ηCO,a is a function of iCO H2 ,a is a function of iH2 such that eqn (74) is identically satisfied.Although this adds complexity, nothing fundamental has changed in the basic algorithm. No further assumption need be made regarding the reaction kinetics, other than estimates for i0 and β for each anodic reaction, as discussed in Section 1.5. The CO and H2 may be produced by reforming methane CH4 + H2 O → CO + 3H2 .

(76)

The rate of reaction is considered to be kinetically controlled [10]. The water-gas shift-reaction is (77) H2 O + CO → CO2 + H2 . This process is generally considered to be thermodynamically controlled.

Numerical models for planar solid oxide fuel cells

67

Figure 13: Detail of gas diffusion layer.

2.9 Porous media flow Up to the present point, the electrodes have been assumed to be sufficiently thin as to be treated as 2-D objects. There are many situations where this is not true, and these should be treated as electro-chemically reacting porous media. Moreover, SOFCs are usually designed with gas diffusion layers, in the form of porous media, as illustrated in Fig. 13. Lchannel and Lrib are the length of the gas channel and the solid ‘rib’. Under the circumstances, Darcy’s law may be used to compute the flow field, as detailed in Section 2.3. Here is a situation where the volume fraction, ε, should be retained in eqn (47). Reynolds number based on a pore diameter is used to evaluate whether the Forcheimer-modified form [40] is required to account for inertial effects. Many codes compute within-phase mass transfer by assuming an effective diffusion coefficient  eff = , (78) τ where τ is a ‘tortuosity’ factor, which is a measure of the mean interstitial path length of the flow lanes per unit overall length. In general, it is to be noted that the rib design of SOFCs is one of competing interests [41]: a high ratio of Lchannel /Lrib , will increase Ohmic losses in the narrow ribs, whereas a low aspect ratio may result in non-uniform concentrations along the widely spaced gas diffusion layer. This is, of course, a function of the diffusivity (and permeability) of the gas diffusion layer as well as the current density. Recent work [34] has shown that the analysis developed in Section 1.6 may be extended to the combined or conjugate problem of simultaneous mass transfer in both the channels and diffusion layers, provided the driving force, b, is based on the overall or average conductance for the entire geometry, g¯ ∗ , obtained by harmonic averaging of the combined conductances of the gas channel and diffusion layer. The relative magnitude of mass transfer in the channels and diffusion layers is determined not only by the values Sh for the gas channel and diffusion layer, but also by the ratio eff /.

68 Transport Phenomena in Fuel Cells It is to be noted that within the electrodes the mean bulk mass fractions must differ from the mean wall values for a current to flow. Since it is not feasible to discretise the actual porous matrix for any practical geometry, the rate of mass transfer will need to be estimated, again using the methodology outlined in Section 1.6, in order that the Nernst equation be correctly prescribed. This will in general require knowledge of Sh∗ , for the particular porous media employed. Such data are rarely available. However bulk-to-wall mass transfer will be important at high current densities when the so-called ‘diffusion limit’ is reached, and should therefore be considered in future codes and methodologies. Heat transfer in porous media is often analyzed by assuming thermal equilibrium between the two phases, with algebraic averaging employed to compute an effective thermal conductivity keff = εkg + (1 − ε)ks , (79) where kg and ks are the gas-phase and solid-phase conductivities of the medium. When modeling fluid flow, heat and mass transfer in porous media, certain closure assumptions are made by virtue of the local volume-averaging procedure. These closures are quite reasonable and may be considered reliable. The question remains: is it possible to lump the analyses to develop a simpler, less computationally complex model? The answer would appear to be in the affirmative: certainly the techniques applied to heat and mass transfer predictions may readily be applied to the electric field, with volumetric sources and sinks of potential being applied across phase boundaries. Moreover, mass transfer in gas diffusion layers may be amenable to closed form analytic solutions. 2.10 Current and voltage distribution For planar geometries it is generally sufficient to compute the potential field using the Nernst equation (eqn (20)). This is a local 1-D model. Moreover it is often sufficient to neglect Ohmic losses in the metallic interconnects. We shall now relax these assumptions. The Nernst potential may be considered to be the sum of the potential differences across the anode and cathode, E =
φa +
φc , where RTf 2F RTa
φc =
φc0 − 4F


φa =
φa0 +



ln xH2 − ln xH2 O − ηa ,



ln Pa + ln xO2 − ηc ,

(80) (81)

where Tf and Ta are the fuel-side and air-side temperatures (not necessarily the same). As has been discussed, there is an element of arbitrariness in the choice of
φa0 and
φc0 . A number of codes now involve the solution of the Poisson equations for the electric field over the entire region (both ionic and electronic)

· (σ ∇φ)

= s˙  , ∇

(82)

Numerical models for planar solid oxide fuel cells

69

where σ is the ionic conductivity in the electrolyte or the electronic conductivity in the interconnects. Equation (82) is to be considered a statement of the principle

This is a simplified version of the scalar of conservation of charge, i = −σ ∇φ. equation, eqn (47), and presents no difficulty. If the electrodes are sufficiently thin, source terms prescribed according to eqn (80) and (81) account for step changes in potential across the anode and cathode. These may be applied as sources of equal and opposite magnitude on either side of the electrode boundary (a ‘planar’ dipole). Elsewhere in the interconnect and the electrolyte s˙  is zero, i.e., the Poisson system reduces to Laplace’s equation. Since this is a conjugate diffusion problem, harmonic averaging of the conductance, σ, should be employed. When the electrodes are of finite thickness, some care is required as electronic and ionic conduction zones overlap. Under the circumstances, storage is generally assigned for two different scalar potentials. The principle of superposition can be applied to the electric field potential, and other definitions of potential may be encountered. A variety of external boundary conditions may be applied to obtain the solution for the electric field potential. For example, (i) two voltages, V1 , and V2 may be applied at the boundaries of the interconnects (Dirichlet problem), alternatively (ii) a uniform current density may be prescribed at both boundaries (Von Neumann problem), with a single point within the assembly fixed to a reference potential, or (iii) the current density may be fixed at one interconnect and the voltage at the other (mixed boundary value problem). Figure 14 shows flux lines of current density computed from the electric potential. In this case the field is quite uniform, though in situations where mass fractions of the products/reactants vary substantially at high i , the iso-galvanic lines can be highly non-uniform. The additional effort required to obtain a solution for the electric potential should only be undertaken if significant non-uniformities in the current distribution are anticipated. This approach is generally combined with an analysis of flow and mass transfer in the porous gas diffusion layers on either side of the electrodes.

Figure 14: Particle traces along galvanostatic lines. From [34].

70 Transport Phenomena in Fuel Cells It is worthy of note that the Poisson system, eqn (82), can be considered to be a simplified version of the Nernst-Planck equation,

i = 2F (ρ u − ∇m)

− (σ ∇φ),

M

(83)

where for convenience we assume the presence of only a single charge carrier, O2− , in the electrolyte. Diffusion and convection are generally considered subordinate to migration in the SOFC electrolyte. 2.11 Advanced diffusion models The development derived in Section 1.6 is appropriate to so-called ‘ordinary’ diffusion, for which the diffusion flux is given by Fick’s law:

j  = −ρD∇m.

(84)

Such an analysis is appropriate [42] for non-dilute, multi-component mixtures only if  (85) D12 = D13 = · · · Djk = D = . ρ Kays et al. [31] suggest that this is indeed approximately the case for high temperature gases, as occur in SOFCs. If it is not the case, the simplest procedure is to calculate an effective diffusion component according to Wilkes [43] as 1 − xk Dk ,eff =  xj

(86)

j =k Djk

which is exact, only for dilute mixtures in a concentrated carrier fluid. A more complete analysis involves the solution of the Stefan-Maxwell equations [44] for the diffusion fluxes n 2 mk jj − mj jk

kM ) = M . ∇(m ρ Mj Djk

(87)

j=1

The set of eqns (87) may be regarded as implicit expressions for the diffusion fluxes, jk . These may then be substituted into eqn (64) in the usual fashion, to obtain the mass fractions, mk . Typically, the linear coefficients in the finite-volume equation (eqn (48)) are caste in the same form as for ‘ordinary’ diffusion and an additional source term added [45]. This has the effect of preserving the correct form of non-linearity in the transport equations, for the case of binary diffusion. If the pore size is sufficiently small that it approaches the mean free path of the colliding particles, Knudsen diffusion must also be considered. Finally surface diffusion may also occur. Multi-component diffusion in reacting porous media is a highly complex subject, considered beyond the scope of this brief introduction.

Numerical models for planar solid oxide fuel cells

71

2.12 Thermal radiation A SOFC is a high temperature device, and even though temperature differences within the passages may be small, they may nonetheless be such that radiative heat transfer is present. The relative importance of radiation in SOFCs is currently the subject of debate. Radiation may be important within the gas channels, within the electrolyte-electrode assembly, and also when thermal insulation layers are employed. Reference [46] reviews some of these issues. Surface-to-surface radiation in the micro-channels may be accounted-for by the network method [47],  q0j − q0k σTk4 − q0k = , (1 − εk )/εk Ak 1/Fk−j Ak N

q˙ k =

(88)

j=1

where εk and Ak are the emissivity and area of the kth element, q0k is the radiosity (the total of emitted and reflected radiation), Fk−j is a configuration factor [48], and σ = 5.67 × 10−8 W · m 2 K4 is the Stefan-Boltzmann constant. The terms (1 − εk )/εk Ak and 1/Fk−j Ak in the denominators are referred to as ‘surface’ and ‘space’ resistances respectively. By the construction a network of resistances, the unknown radiosities may be eliminated and the system of equations solved. Since the micro-channels are very narrow, only a few immediate neighbours (opposite and sides) in any given channel will be of any consequence, and the computation can then be much simplified. The additional work involved in computing surfaceto-surface radiation along the length of the channel is probably not justified. Yakabe et al. [49] describe a procedure for calculating surface-to-surface radiation in gas micro-channels, which differed from the standard analysis, eqn (88). VanderSteen et al. [50] suggest that surface-to-surface radiation could affect the temperature by as much as 25–50 K. Although the gas channels are amenable to a simplified analysis, the electrolyte, and possibly the electrodes of a SOFC constitute radiatively participating media. Murthy and Fedorov [51] suggest the exclusion of radiation effects in the participating electrolyte could result in the over-prediction of temperatures by as much as 100–200K for the case of an electrolyte-supported SOFC. By contrast, refs. [52, 53] suggest very modest differences in electrolyte temperature; less than 1 K. These disparities indicate that the need to include radiation effects is, at present, still a moot point. Reference [51] suggests that the electrolyte may be treated as being optically thin, while the electrodes may be considered as optically thick. A very limited amount of experimental data has been gathered in support of this, to date; however [52, 53] suggest that in fact, the electrodes are so optically thick as to be entirely opaque for all practical purposes. The general radiation heat transfer problem in a participating medium involves heat transfer ‘at a distance’; described mathematically by an integro-differential equation, referred to as the radiative transfer equation (RTE). For a grey

72 Transport Phenomena in Fuel Cells isotropically-scattering medium, this may be written in the simplified form, eb σs di = −(a + σs ) i + a + ds π 4π

!4π i d ω.

(89)

ω=0

(i)

(ii)

(iii)

(iv)

In this section only, the symbol i denotes radiant intensity (not electrical current), a and σs are absorption and scattering coefficients, s is displacement, ω is solid angle, and eb = σT 4 is black-body emissive power. The terms in eqn (89) represent (i) changes in intensity, i, due to: (ii) absorption and out-scattering, (iii) emission, and (iv) in-scattering, respectively. Detailed numerical solutions to the RTE are typically conducted using a discrete ordinate method or a Monte Carlo method. The former is more popular, at the present time, but suffers from ‘ray effects’ which can lead to numerical errors. The RTE solution results in a field of values for the radiation intensity from which a radiant flux vector, q ˙ , can be calculated for substitution in the energy equation. However, both the discrete ordinate, and Monte Carlo methods are very compute-intensive. Therefore some additional engineering assumptions are required if it is possible to invoke simpler radiation models within multi-channel fuel cells and stacks. For optically thin regions a multi-flux model may be considered, for which the 1-D two-flux model is the simplest. It is referred to, in the literature, as the SchusterSchwarzschild approximation. If it is assumed that the intensity is composed of two homogeneous components such that q˙ + = πi+ and q˙ − = πi− in the +x and −x directions, then it can readily be shown that eqn (89) may be written as,   1 d q¯ d (90) = 4a q¯ − eb , dx a + σs dx  where q¯ = q˙ + + q˙ − /2, and for which (the x-component of ) the radiant flux vector, q˙  = q˙ + − q˙ − , may be computed as q˙  = −

1 d q¯ . a + σs dx

(91)

Equation (90) is a diffusion-source equation, and is thus suitable for discretisation in the form eqn (48). It has been presented in the form developed by Spalding [54], from which it differs slightly mathematically, being consistent with the formulation in Siegel and Howell [48]. A six-flux model may similarly be constructed in terms of three field variables, q¯ x , q¯ y , q¯ z in 3-D, and has the advantage of being in the same form as the general scalar equation (eqn (47)). However for thin plane layers, a two-flux model may be sufficient. The discrete ordinate method may be regarded as a generalised multi-flux method [48]. For optically thick regions, a diffusion approximation is considered appropriate, with radiative conductivity defined by, kR =

16n2 σT 3 , 3a

(92)

Numerical models for planar solid oxide fuel cells

73

a is a (Rosseland mean) absorption coefficient. The term, kR , may simply be added to the normal conduction term, k, in eqn (57), i.e., keff = k +kR . There are however difficulties associated with use of the diffusion approximation near boundaries. This may be remedied by the use of a temperature modified slip boundary condition [48]. Thermal radiation is one mechanism by which it may be possible to actually control stack temperature and temperature gradients, by manipulating the optical properties of the component materials or employing radiation shields. Spinnler et al. [55, 56] propose the use of highly-reflective thermal insulation to control fuel cell temperatures by minimising losses to the environment. The above analysis adds considerably to the complexity of the calculation procedure, even so for situations where radiation is important; it is probably oversimplistic: In reality, the electrolyte assembly is a porous media with nonhomogeneous optical properties, including dependent scattering, while the interconnect is typically metallic and hence there may be both specular and diffuse components to the surface radiosity. The spectral nature of the radiative properties of the SOFC components also need to be determined. The subject has received little attention to-date and should be considered a topic for future research by scientists and engineers seeking to design high-temperature fuel cells. The four outstanding technical issues in numerical analysis of themo-fluids are; chemical kinetics multi-phase flow, radiative heat transfer and turbulence. For SOFCs, the chemical kinetics is a complex issue that needs to be addressed; multiphase flow is superficially not a problem, but the presence of porous media and the need for volume-averaging raises many of the same issues. Thermal radiation may be a matter for concern, as discussed above. Turbulence is not generally generated at the Reynolds numbers encountered within the passages of planar SOFCs, however turbulence may be encountered within the inlets and outlets and manifold passages of SOFC stacks.

3 Stack models SOFCs are stacked to increase the working voltage to a reasonable value. Fuel and air are typically introduced via manifolds (risers and downcomers). Various design configurations are possible. The main concerns of the stack designer are uniformity of flow and pressure gradients. If SOFCs were not operated in stacks, it probably would be feasible to conduct detailed numerical analysis using CFD codes without invoking a rate equation assumption, at least for simple geometries. The presence of numerous fuel cells, each with multiple channels, renders this situation unlikely for large stacks for the foreseeable future. A rough estimate of the relative magnitude of parasitic losses in the cells and manifolds can be made by assuming the flow is uniform, then analysing to see if this is a reasonable assumption. It may readily be shown from eqns (54) and (55), that for planar passages neglecting injection/suction that the pressure drop across the stack is just, ˙ 48LµQ 48Lµ¯u = , (93)
P¯ = 2 H nfc WH 3

74 Transport Phenomena in Fuel Cells

Figure 15: Pressure (Pa) in a 50-cell SOFC stack assembly. From [57]. where nfc is the number of fuel cells in the stack, 2H is the height of the (air or fuel) channels, W is the width of the cell, L is the length of the fuel cells in the ˙ = nfc BH u¯ is the volumetric discharge, assuming uniform flow flow direction, and Q conditions. A similar, though more complex, expression may be readily derived for rectangular passages. The basic principle of stack design is that the losses across the cells should be large in comparison to those in the inlet manifolds. Large changes in pressure drop are effected by changing the height of the passages, since
¯p ∝ 1/H 3 . Equation (93) is to be considered only approximate. Mass transfer will increase or decrease pressure for the cases of injection and suction, respectively. This purely inertial effect is in addition to the frictional f/f ∗ effects discussed earlier (recall Bernoulli’s law P + 12 ρu2 = constant). In the manifolds suction/blowing leads to all fluid being entirely evacuated over the stack height, H , and the latter must be considered. Berman [58] was the first to derive an analytical solution for the pressure distribution for a plane duct with mass transfer at both walls; analyses for suction/blowing at only one wall also exist [59, 60]. For rectangular geometry it is simple to utilize a numerical calculation (CFD) procedure. Figure 15 shows the pressure distribution in a 50-cell stack assembly; it can be seen that (1) in the inlet manifold the pressure gradient decreases due to suction, (2) there is relatively uniform pressure within the stack, and (3) an increasing pressure gradient is observed in the outlet manifold due to side-injection. The reader will note that different contour scales have been used in the stack and manifold regions and that the overall (parasitic) pressure losses are much larger in the former than the latter. As the stack height increases, the pressure difference between the manifolds at the top of the stack becomes much less than at the bottom and flow maldistributions may occur. Another interesting feature of fuel cell stacks is that secondary temperature distributions in the vertical direction arise, even when the flow and heat source term (due to Joule heating) are entirely uniform. Figure 16 shows the temperature distribution in a 10-cell stack obtained using a detailed numerical simulation. The oscillations in the temperature contours are due to the finite differences in temperature between

Numerical models for planar solid oxide fuel cells

75

Figure 16: Temperature distribution (deg. C) in a 10-cell SOFC stack with uniform heat sources. Adapted from [34].

the air, fuel, electrolyte and interconnects for each cell in the stack. The secondary temperature gradients are caused by these materials being ordered, creating a net heat flux in the vertical direction. The stack designer cannot necessarily presume that thermo-mechanical behavior is the same for a stack as for a single cell, as has been presumed by many researchers in the past.

4 Closure Transport phenomena in SOFCs influence every aspect of their design and operation. Fluid flow, heat conduction, convection, and radiation, mass transfer, electrochemistry, charge transfer kinetics and thermodynamics are all important mechanisms. In the last 5–10 years much progress has been made in the development of robust engineering computer codes to model transport phenomena and physico-chemical hydrodynamics in these devices. This is significant progress, however further developments are needed both in models and on input data of properties. Although it is currently fashionable to state that growth in this area is ‘rapid’, it remains to be seen whether recent events will continue to be sustainable. On the one hand, design methodologies may continue to evolve exponentially, as the product unfolds and becomes commercially viable in the years to come. On the other, it could be that analysis tools will plateau and languish, if the technology proves to be uncompetitive, and promises of economic viability prove to be a ‘bubble’. The author’s hope is that the former will prove to be the case, and that years from now, we will see improvements, not only in analysis methods, but also as a result, in the products. It is clear that if SOFCs, or indeed any other new technologies, are to supplant existing products and processes, they must be designed properly. Every available analysis tool and advanced scientific method needs to be brought to bear upon the problem, if good engineering is to replace good will and sway the public to buy new and untested products over established and functioning ones, in a competitive market place. In many ways the success or failure of fuel cells will ultimately depend on factors completely unrelated to the thermodynamic efficiency of the electrode-electrolyte assembly, so popular among research proposals. Probably reliability and safety

76 Transport Phenomena in Fuel Cells rank even higher than cost-per-unit energy, and while mathematical modeling alone cannot safeguard these attributes, careful engineering using established techniques such as those described in the chapters in this book (and already widely employed in design of conventional power-generation equipment) will likely prove a good investment, and aid in better understanding of equipment performance. To be competitive, fuel cells need to be at least as well or better designed than existing products. Finally, for a variety of reasons, numerical models of physico-chemical processes can now be developed in much shorter time-scales (months), and for much less cost than it takes to build complex experimental test rigs (years). While this situation cannot be blamed upon those who seek to construct models and codes, it is nonetheless an unhealthy situation; a balanced program of research should always involve the continuous feedback of experimental data to analysis tools, as well as the abstraction of that which is being modelled to the pragmatist. It is therefore imperative to establish reliable data bases of empirical data for code evaluation purposes.

Acknowledgements There are many people who have assisted the author in this endeavour, both directly and indirectly. Thanks are due to (in alphabetical order): Katherine Cook, Kyle Daun, Wei Dong, Eliezer Gileadi, Ron Jerome, Yongming Lin, Fengshan Liu, and Rod McMillan.

References [1] Appleby, A.J. & Foulkes, F.R., Fuel Cell Handbook, Van Nostrand Reinhold: New York, 1989. [2] Kordesh, K. & Simader, G., Fuel Cells and their Applications, VCH: New York, 1996. [3] Larminie, J. & Dicks, A., Fuel Cell Systems Explained, Wiley: Chichester, 2000. [4] Williams, K.R., An Introduction to Fuel Cells, Elsevier: Amsterdam, 1966. [5] Vayenas, C.G. & Hegedus, L.L., Cross-flow, solid-state electrochemical reactors: A steady-state analysis. Ind. & Eng. Chem. Fundamentals, 24, pp. 316–314, 1985. [6] Fiard, J.M. & Herbin, R., Comparison between finite volume and finite element methods for the numerical simulation of an elliptic problem arising in electrochemical engineering. Computational Methods in Applied Mechanical Engineering, 115, pp. 315–338, 1994. [7] Ferguson, J.R., Analysis of temperature and current distributions in planar SOFC designs. Proceedings Second International Symposium on Solid Oxide Fuel Cells, Athens, Greece, pp. 273–280, 1991. [8] Herbin, R., Fiard, J.M. & Ferguson, J.R., Three-dimensional numerical simulation of the temperature, potential and concentration distributions of a unit

Numerical models for planar solid oxide fuel cells

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77

cell for various geometries of SOFCs. Proceedings, European Solid Oxide Fuel Cell Forum l, Luzern, Switzerland, 1994. Karoliussen, H., Nisancioglu, K., Solheim, A., Odegard, R., Singhal, S.C. & Iwahara, H., Mathematical modeling of cross plane SOFC with internal reforming. Proceedings, Third International Symposium on Solid Oxide Fuel Cells, Honolulu, Hawaii, 1993. Achenbach, E., Three-dimensional modeling and time-dependent simulation of a planar solid oxide fuel cell stack. Journal of Power Sources, 73, pp. 333– 348, 1994. Bessette, N.F. & Wepfer, W.J., Electochemical and thermal simulation of a solid oxide fuel cell. Chemical Engineering Communications, 147, pp. 1–15, 1996. Bernier, M., Ferguson, J. & Herbin, R., A 3-dimensional planar SOFC stack model. Proceedings, Third European Solid Oxide Fuel Cell Forum, Nantes, France, pp. 483–495, 1998. Ahmed, S., McPheeters, C. & Kumar, R., Thermal-Hydraulic model of a monolithic solid oxide fuel cell. Journal of the Electrochemical Society, 138, pp. 2712–2718, 1991. Sira, T. & Ostenstad, M., Temperature and flow distributions in planar SOFC stacks. Third International Symposium on Solid Oxide Fuel Cells, Honolulu, Hawaii, pp. 851–860, 1993. Costamagna, P. & Honegger, K., J. Electrochem. Soc., 145(11), pp. 2712– 2718, 1998. Chan, S.H., Khor, K.A. & Xia, Z.T., Journal of Power Sources, 93, pp. 130–140, 2001. Beale, S.B., Lin, Y., Zhubrin, S.V. & Dong, W., Computer Methods for Performance Prediction in Fuel Cells. Journal of Power Sources, 11(1–2), pp. 79–85, 2003. Beale, S.B. & Zhubrin, S.V., A distributed resistance analogy for solid oxide fuel cells. Numerical Heat Transfer, Part B, in press. Newman, J.S., Electrochemical Systems, Prentice-Hall Inc.: Englewood Cliffs, N.J., 1973. Levich, V.G., Physicochemical Hydrodynamics, Prentice-Hall: Englewood Cliffs, N.J., 1962. Probstein, R.F., Physicochemical Hydrodynamics: An Introduction, Butterworths: Boston, 1989. Callen, Thermodynamics, John Wiley & Sons Inc.: New York, 1960. Glasstone, S., Laidler, K.J. & Eyring, H., The Theory of Rate Processes: The Kinetics of Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena. International Chemical Series, McGraw-Hill: New York, 1941. Gileadi, E., Electrode Kinetics for Chemists, Chemical Engineers, and Material Scientists, Wiley-VCH: New York, 1993. Bockris, J.O.M., Reddy, A.K.N. & Gamboa-Aldeco, M., Modern Electrochemistry. Vol. 2, Plenum: New York, 2000.

78 Transport Phenomena in Fuel Cells [26] [27] [28] [29] [30]

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Bard, A.J. & Faulkner, L.R., Electrochemical Methods, 2nd edn, John Wiley & Sons: New York, 2001. Fuel Cell Handbook, 5th edn, U.S. Department of Energy, National Energy Technology Laboratory: Morgantown/Pittsburgh, 2000. Beale, S.B., Calculation procedure for mass transfer in fuel cells. Journal of Power Sources, 128(2), pp. 185–192, 2004. Spalding, D.B., Convective Mass Transfer: An Introduction, Edward Arnold: London, 1963. Spalding, D.B., A standard formulation of the steady convective mass transfer problem. International Journal of Heat and Mass Transfer, 1, pp. 192– 207, 1960. Kays, W.M., Crawford, M.E. & Weigand, B., Convective Heat and Mass Transfer, 4th edn, McGraw-Hill: New York, 2005. Mills, A.F., Mass Transfer, Prentice Hall: Upper Saddle River, N.J., 2001. Kays, W.M. & Crawford, M.E., Convective Heat and Mass Transfer, 2nd edn, McGraw-Hill: New York, 1980. Beale, S.B., Conjugate mass transfer in gas channels and diffusion layers of fuel Cells. Proceedings 3rd International Conference on Fuel Cell Science, Engineering and Technology, eds R.K. Shah & S.G. Kandlikar, ASME: Ypsilanti, Michigan, 2005. Shah, R.K. & London, A.L., Laminar flow forced convection in ducts. Advances in Heat Transfer, eds T.F. Irvine & J.P. Hartnett, Academic Press: New York, 1978. Jacob, M., Heat Transfer, Wiley: New York, 1949. Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere: New York, 1980. Patankar, S.V. & Spalding, D.B., A Calculation Procedure for the Transient and Steady-state Behavior of Shell-and-tube Heat Exchangers. Heat Exchangers: Design and Therory Sourcebook, eds N. Afgan & E.U. Schlünder, Scripta Book Company: Washington, D.C., 1974. Beale, S.B., Mass transfer in plane and square ducts. International Journal of Heat and Mass Transfer, in press. Vafai, K. & Tien, C.L., Boundary and Inertia Effects on Flow and Heat Transfer in Porous Media. International Journal of Heat and Mass Transfer, 24(2), pp. 195–203, 1981. Ferguson, J.R., Fiard, J.M. & Herbin, R., Three-dimensional numerical simulation for various geometries of solid oxide fuel cells. Journal of Power Sources, 58, pp. 109–122, 1996. Knuth, E.L., Multicomponent diffusion and Fick’s law. Physics of Fluids, 2, pp. 339–340, 1959. Wilke, C.R., Diffusional properties of multicomponent gases. Chemical Engineering Progress, 46(2), pp. 95–104, 1950. Taylor, R. & Krishna, R., Multicomponent Mass Transfer, Wiley-Interscience: New York, 1993.

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Kleijn, C.R., van der Meer, T.H. & Hoogendoorn, C.J., A mathematical model for LPCVD in a single wafer reactor. Journal of the Electrochemical Society, 136(11), pp. 3423–3433, 1989. Damm, D.L. & Fedorov, A.G., Radiation heat transfer in SOFC materials and components. Journal of Power Sources, in press. Oppenheim, A.K., Radiation analysis by the network method. Transactions of the ASME, 78, pp. 725–735, 1956. Siegel, R. & Howell, J.R., Thermal Radiation Heat Transfer, 4th edn, Hemisphere: Washington, 2002. Yakabe, H., Ogiwara, T., Hishinuma, I. & Yasuda, I., 3-D model calculation for planar SOFC. Journal of Power Sources, 102, pp. 144–154, 2001. VanderSteen, J.D.J., Austin, M.E. & Pharaoh, J.G., The role of radiative heat transfer with participating gases on the temperature distribution in solid oxide fuel cells. Proceedings 2nd International Conference on Fuel Cell Science, Engineering and Technology, eds R.K. Shah & S.G. Kandlikar, Rochester, NY, 2004. Murthy, S. & Federov, A.G., Radiation heat transfer analysis of the monolith type solid oxide fuel cell. Journal of Power Sources, 124, pp. 453–458, 2003. Damm, D.L. & Fedorov, A.G., Spectral radiative heat transfer of the planar SOFC. Proceedings International Mechanical Engineering Congress and Exposition, ASME: Anaheim, California, 2004. Daun, K., J., Beale, S.B., Liu, G. & Smallwood, G.J., Radiation heat transfer in solid oxide fuel cells. Proceedings of 2005 Summer Heat Transfer Conference, ASME: San Francisco, California, 2005. Spalding, D.B., Mathematical Modelling of Fluid-mechanics, Heat-transfer and Chemical-reaction Processes: A Lecture Course. 1980, Computational Fluid Dynamics Unit, Imperial College, University of London: London. Spinnler, M., Winter, E.R.F. & Viskanta, R., Studies on high-temperature multilayer thermal insulations. International Journal of Heat and Mass Transfer, 47, pp. 1305–1312, 2004. Spinnler, M., Winter, E.R.F., Viskanta, R. & Sattelmayer, T., Theoretical studies of high-temperature multilayer thermal insulations using radiation scaling. Journal of Quantitative Spectroscopy and Radiative Transfer, 84, pp. 477–491, 2004. Beale, S.B., Ginolin, A., Jerome, R., Perry, M. & Ghosh, D., Towards a virtual reality prototype for fuel cells. PHOENICS Journal of Computational Fluid Dynamics and its Applications, 13(3), pp. 287–295, 2000. Berman, A.S., Laminar flow in channels with porous walls. Journal of Applied Physics, 24(9), pp. 1232–1235, 1953. Jorne, J., Mass transfer in laminar flow channel with porous wall. Journal of the Electrochemical Society, 129(8), pp. 1727–1733, 1982. Lessner, P. & Newman, J.S., Hydrodynamics and Mass-Transfer in a PorousWall Channel. Journal of The Electrochemical Society, 131(8), pp. 1828– 1831, 1984.

80 Transport Phenomena in Fuel Cells

Nomenclature A a B b c D Dh E eb F

G g H h i j KP k kDarcy k kr L M m N n nfc P Q ˙ Q q q0 R r S Scond s T U U¯

Area (m2 ) Absorption coefficient (m−1 ), coefficient in finite-volume equations, length (m) Width (m) Length (m) Specific heat (J/kgK) Diffusion coefficient (m2 /s) Hydraulic diameter (m) Nernst potential (V), electric potential (V) Black-body emissive power (W/m2 ) Distributed resistance (kg/m2 s), Faraday’s constant, 96.48 (Coulomb/mol), Radiation configuration factor () Gibbs free energy (J) Mass transfer conductance (kg/s) Height (m), enthalpy (J) Heat transfer coefficient (W/m2 K), Planck constant 6.626 × 10−34 (J.s) Current (A), radiant intensity (W/sr.m2 ) Diffusion source Equilibrium constant Thermal conductivity (W/mK), rate constant (mol/m2 ) Permeability (m2 ) Thermal conductivity (W/mK) Radiative conductance (W/mK) Length (m) Molecular weight (kg/mol) Mass fraction (kg/kg), mass (kg) Number of cells in stack (), mole number Charge number (), valence () Number of fuel cells in SOFC stack Pressure (Pa), partial pressure (Pa) Charge (Coulomb) Volumetric discharge (m3 /s) Heat source term (W) Radiosity (W) Gas constant, 8.314 × 103 (J/molK), resistance (Ohm) Resistance (Ohm/m2 ) Entropy (J/K) Conduction shape factor () Source term, specific heat (J/kgK) Temperature (◦ C) Internal energy (J), superficial velocity (m/s) Overall heat transfer coefficient (W/m2 K)

Numerical models for planar solid oxide fuel cells

u V W T xi

Interstitial velocity (m/s) Volume (m3 ), voltage (V) Work (J) Temperature (K) Mole fraction (mol/mol)

Greek symbols α α∗ β ε   φ η ν ρ µ σ σS τ

Volumetric transfer coefficient (W/m3 K) Aspect ratio () Transfer coefficient, symmetry coefficient (), Blockage factor () Volume fraction, void fraction (), emmissivity (m−1 ) Exchange coefficient (kg/ms) General scalar General scalar, electric field potential (V) Overpotential, polarisation (V) Stoichiometric coefficient Density (kg/m3 ) Chemical potential (J/mol), viscosity (W/mK) Electric conductivity (Ohm−1 m), Stefan-Boltzmann constant 5.67 × 10−8 (W/m2 K4 ) Scattering coefficient (m−1 ) Shear stress (N/m2 ), tortuosity ()

Non-dimensional numbers B b f Nu Sh Sc Re

Driving force Blowing parameter Friction coefficient Nusselt number Sherwood number Schmidt number Reynolds number

Subscripts 0 a b c e f i

Reference state Air, anode Bulk, backward Cathode Electrolyte Fuel, forward Interconnect, internal

81

82 Transport Phenomena in Fuel Cells l t w

Load total, transferred substance state Wall

Superscripts 0 ∗ . ’ + −

Reference state, equilibrium state Zero mass transfer Per unit time Per unit length Positive direction Negative direction

CHAPTER 3 Electrochemical and thermo-fluid modeling of a tubular solid oxide fuel cell with accompanying indirect internal fuel reforming K. Suzuki1 , H. Iwai2 & T. Nishino2 1 Department

of Machinery and Control Systems, Shibaura Institute of Technology, Japan. 2 Department of Mechanical Engineering, Kyoto University, Japan.

Abstract Development of fuel cells has been boosted by global and regional environmental issues. Among others, the solid oxide fuel cell (SOFC) has been drawing much attention as a unit for distributed energy generation. Numerical analysis is used as a powerful tool in research and development of fuel cells, especially of the SOFC for which important and necessary thermal management information has scarcely been supplied experimentally. A central issue for achieving maximum benefit from the numerical analysis is how to properly model the complex phenomena occurring in the fuel cells. This chapter presents a numerical model for a tubular SOFC including a case with indirect internal reforming. In this model, the velocity field in the air and fuel passages and heat and mass transfer in and around a tubular cell are calculated with a two-dimensional cylindrical coordinate system adopting the axisymmetric assumption. Internal reforming and electro-chemical reactions are both taken into account in the model. Electric potential field and electric current in the cell are also calculated simultaneously allowing their nonuniformity in the peripheral direction. A previously developed quasi-two-dimensional model was adopted to combine the assumed axisymmetry of velocity, heat and mass transfer fields and the peripheral nonuniformity of electric potential field and electric current. Details of those numerical procedures are described and examples of the calculated results are discussed. After presenting a few fundamental results, several strategies to reduce the maximum temperature and temperature gradient of the cell are examined for a case with indirect internal fuel reforming.

84 Transport Phenomena in Fuel Cells

1 Introduction One of the most important global issues in the present world is the suppression of global warming. Global warming is proceeding related to the increase of atmospheric concentration of carbon dioxide. Discharge of carbon dioxide into the atmosphere occurs in the eruption of volcanoes but the largest emission rate of carbon dioxide originates from power plants and from vehicles both of which support the modern world. Since drastic depression of energy demand and usage of automobiles cannot be accepted easily by people living in the modern society, development of energy conversion systems being free from the emission of carbon dioxide or, at least, of a system having a lower emission rate of carbon dioxide is important. Fuel cells are one of the prospective power generation systems for this purpose. There are five types of fuel cells characterized by the difference in materials for the electrolyte and in this relation by the difference in operation temperature. They are alkaline fuel cells, phosphoric acid fuel cells, polymer electrolyte fuel cells, molten carbonate fuel cells and solid oxide fuel cells (SOFCs). In this chapter, attention is paid to the last one, the SOFC. Solid oxide fuel cells (SOFCs) have an advantage in that not only hydrogen but also a variety of hydrocarbons can be used as fuels. This is because solid oxides used for the electrolyte, are oxygen-ion conducting materials. In the case of fuel cells using hydrogen-ion conducting or proton conducting materials as the electrolyte, the fuel should in principle be hydrogen only but, in the case of the SOFC, a variety of fuels including carbon monoxide in addition to hydrocarbons can react with oxygenions conducted across the electrolyte. High operation temperature provides another advantage of SOFCs. No catalyst is needed for the electrochemical reaction if they are operated at a reasonably high temperature except in the case of low temperature SOFCs [1]. In addition to this, high quality thermal energy of the effluent from the SOFCs can be recovered in various ways. Construction of a hybrid system that fuses the SOFC to a gas turbine to obtain high electricity generation efficiency is one of them [2–4]. In addition, steam reforming of hydrocarbon fuels is possible in or around the cells with the aid of high operation temperature. A variety of commercial materials can be used as a catalyst for the reforming reaction. Fuel reforming supplies hydrogen and carbon monoxide from methane as fuel, which makes the fuel cell operation simpler and additionally provides a means to cool and therefore to control the high temperature fuel cell. When the reforming reaction takes place on the anode with heat supplied directly from the electrochemical reaction, it is referred to as direct internal reforming (DIR). In this case, the cell structure is simple because no additional catalyst except the anode is needed. However, there are still difficulties in proper control both of reforming and electrochemical reactions within the anode itself [5–7]. On the other hand, the case in which the reforming reaction takes place on catalysts that are positioned apart from the anode is referred to as indirect internal reforming (IIR). This is the case treated in this chapter as will be discussed later. Another advantage of SOFCs is that the electrolyte is solid. This results in geometrical flexibility of the cell design. Several types of SOFCs having different geometry have been proposed, tested and developed so far. They are tubular SOFCs, planar

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85

Figure 1: Schematics of three types of SOFC having different geometry.

SOFCs and disk type SOFCs. Schematics of the three types of SOFC are illustrated in Fig. 1. Planar SOFCs and disk type SOFCs are prospective in a point that they can be stacked in a compact way and that they can have higher electric power density. Therefore, they are the subject of keen research and development activities [8–11]. On the other hand, tubular SOFCs are one of the most developed towards practical use. The two major advantages of the tubular cells are that they have relatively good thermal shock resistance and that there is no need for special sealing treatment. Owing to these advantages, a thousands-hour-long operation test of an atmospheric pressure 100-kW tubular SOFC system has successfully been performed [12]. In this system, cathode-supported SOFCs are used, in which air flows inside the tubular cell structure and fuel flows around the tubular cell along its axis. Natural gas is used as fuel and it is fully reformed in “in-stack reformers”, which are placed between the rows of cell bundles composed of many tubular cells. Reformers are heated indirectly by the cells. The cell structure is simpler in this case but the stack structure is rather complicated. On the other hand, in the case of anode-supported SOFCs, fuel flows inside the tubular cell and air flows outside the tubular cell. In this case, a fuel feed tube mounted inside the tubular cell, through which fuel is injected into the tubular cell, can be used as a reformer. The feed tube is filled with the catalyst material and indirect but internal fuel reforming (IIR) proceeds while fuel flows through it. This is the concept of indirect internal reforming type tubular SOFCs (IIR-T-SOFCs) to be discussed in this chapter. In this

86 Transport Phenomena in Fuel Cells case, although the cell structure is a little more complicated due to the catalyst to be embedded in it, the stack structure becomes fairly simple. There is no in-stack reformer so that the size and geometry of the stack can be designed more easily and more flexibly for various SOFC electricity generation systems having different power output capacity. In addition, there is a chance to control the local thermal field in the cell by adjusting the distribution density of the fuel reforming catalyst. This is one of the achievements of earlier numerical analyses [13, 14]. At present, development of this type of SOFC is just in a conceptual design stage. So it is meaningful to study its feasibility right now and numerical simulation is considered to be an effective tool for this purpose. Numerical simulation has become a useful means in research and development in various engineering fields. Fuel cells are no exception and various numerical simulations of fuel cells have been conducted by several research groups. Numerical modeling of fuel cells is complicated due to a variety of phenomena occurring in the cells such as multi-component gas flows with heat and mass transfer, electrochemical reactions between fuel and air, and electric potential field and ionic and electric current in the cell. Numerical simulation is especially important for SOFCs. They need good thermal management because the operation temperature is high. However, the detailed information necessary for management is not easily supplied from measurement. Numerical simulation is a sole means to supply such information. There are two types of numerical simulations applied to SOFCs. One is for a single cell [15–20] and another for a stacked-cell module [21–24]. In this chapter, the former type of numerical simulation is treated and in particular description is given to a numerical model developed for a tubular SOFC including a case of IIR-T-SOFC based on the model for the cathode-supported tubular SOFC reported in [15, 16]. Finally, some results are also presented as examples of the simulations made with the described model.

2 General remarks on the mechanism of IIR-T-SOFC Prior to describing the numerical modeling and simulated results, general remarks will be given to the phenomena in the indirect internal reforming tubular solid oxide fuel cells, the IIR-T-SOFCs, in this section. Except for the part related to internal fuel reforming, description to follow can be applied to a general case without internal fuel reforming including the case of a conventional cathode-supported tubular solid oxide fuel cell.

2.1 Tubular cell Figure 2 shows schematic views of two types of a tubular SOFC. One is a conventional Siemens-Westinghouse type cathode-supported tubular cell [25] and another is a single cell to be used in the IIR-T-SOFC stack. The latter type of fuel cell consists of a tubular cell and a feed tube inserted in it, basically similar to a design of

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87

Figure 2: Schematic view of a conventional tubular SOFC and IIR-T-SOFC.

Figure 3: Schematic view of a cell stack. conventional tubular SOFCs. The cell tube is composed of two electrodes sandwiching an electrolyte layer. Electrochemical reactions take place at both these electrodes, anode and cathode. The difference between the two cases illustrated in Fig. 2 is that the hydrocarbon to be used as fuel flows inside the feed tube and that the feed tube is filled with porous catalysts in the case of the IIR-T-SOFC. So fuel is reformed “indirectly” in each tubular cell. That is to say, fuel is reformed inside the feed tube, changes flow direction at the closed end of the cell tube, reacts electrochemically on the anode and then flows out of the cell. On the other hand, air flows outside the cell tube and reacts on the cathode. One drawback of this cell design is that the interconnects are exposed to the high-temperature air so that serious oxidization of the interconnects may occur. However, this problem is being resolved due to the development of new interconnect materials such as the doped-lanthanum chromite series in recent years [26, 27]. The tubular cell can be stacked either electrically in series or electrically in parallel as shown in Fig. 3. The number of the tubular cells in the stack is adjusted so

88 Transport Phenomena in Fuel Cells as to fit the required electricity capacity. When the number of the cells is sufficiently large, most of the cells in the core region of the stack should be in the same thermal and operating conditions. Therefore, attention is paid in the present study to a single representative tubular cell in the core region of the stack. 2.2 Internal reforming process One of the important phenomena occurring in the IIR-T-SOFCs is the internal reforming of fuel and it takes place, as mentioned above, inside the feed tube. Although a variety of hydrocarbons can be used as fuel, description is given here as an example for the case where methane, a major component of natural gas, is used as fuel. For methane, the steam reforming process is widely known as a conventional process for producing hydrogen [28, 29] and it proceeds on catalysts such as nickelalumina through the following two chemical reactions: Reforming reaction: CH4 + H2 O ←→ 3H2 + CO.

(1)

CO + H2 O ←→ H2 + CO2 .

(2)

Shift Reaction:

The equilibrium constant of the steam reforming reaction described by eqn (1) is reasonably large so that methane can almost totally be reformed if the system is kept at the operation temperature 700 ◦ C or higher. However, a supply of thermal energy is needed for the reforming reaction to proceed since it is strongly endothermic. Therefore, how methane is reformed inside the feed tube is highly dependent on the local conditions including not only temperature but also partial pressure of each chemical species and density of the catalysts. On the other hand, the water-gas shift reaction described by eqn (2) is a weak exothermic reaction. The important point is that the shift reaction is almost in equilibrium in the entire fuel passage because its reaction rate is much faster than that of the steam reforming reaction. 2.3 Electrochemical process The most important phenomenon occurring in the tubular cell is a series of events associated with the electrochemical reaction. In the case where methane is used as fuel and is totally reformed inside the feed tube, the electrochemical processes on the anode and cathode can be described by the following reactions: H2 + O2− → H2 O + 2e−

(anode),

(3)

CO + O2− → CO2 + 2e−

(anode),

(4)

(cathode).

(5)

1 2 O2

+ 2e− → O2−

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89

Figure 4: Basic concept of the electrochemical process.

Overall reactions are equivalent to the following: H2 + 12 O2 → H2 O,

(6)

CO + 12 O2 → CO2 .

(7)

Figure 4 illustrates the mechanism of the electrochemical processes in the tubular cell. On the cathode side, oxygen ions are produced by the combination of oxygen molecules in the air flow with electrons. The produced oxygen ions move through the electrolyte to the anode side. Electrons are released from the oxygen ions at the anode and the produced oxygen molecules react with hydrogen or carbon monoxide molecules there. The electrons released through the anode reactions move to the cathode of the neighbouring cell through the interconnect or return to the original cathode via an external circuit – i.e. electric current is generated through the tubular cells. The magnitude of the electric current directly relates to the rates of the electrochemical reactions and depends on a variety of factors. These factors can be classified into three groups: electromotive force, internal resistance, and external load of the cell. The electromotive force is an ideal cell terminal voltage to be achieved at zero electric current or when the electrochemical reaction proceeds very slowly. The internal resistance causes the loss of terminal voltage and includes three different types of overpotentials; namely ohmic overpotential, activation overpotential and concentration overpotential. Finally, the external load of the cell regulates the electric current to be realized with the fuel cell under operation and is actually equal to the cell terminal voltage divided by the magnitude of electric current.

90 Transport Phenomena in Fuel Cells The electromotive force is largely determined by the concentration of fuel. In general, it decreases as hydrogen and carbon monoxide are consumed and therefore as their concentrations become lower. As for the ohmic overpotential, ionic conductivity of the electrolyte is one of the important factors. It has a tendency to decrease as the temperature rises. This is the main reason why operation temperature must be high in the case of SOFCs. Electric conductivity of the electrodes also affects the magnitude of the ohmic loss of terminal voltage. Electric current flows through the electrodes basically in the circumferential direction as shown in Fig. 4. Activation overpotential occurs related to the sluggishness of electrochemical reaction and tends to decrease as the cell temperature rises. Concentration overpotential is also an important factor affecting the cell performance and is related to the radial concentration nonuniformity of participating chemical species, i.e. fuel and oxygen. At the reaction sites, their concentrations are lower so that the real electromotive force to be attained under such concentrations is lower than the counterpart that would be achieved using the concentration in the core flow regions. This difference is the concentration overpotential. These electromotive forces and internal resistances vary from place to place because temperature and gas composition are not uniform in the cell. Finally, as for the external load, electric current becomes larger and terminal voltage becomes lower as it decreases. Therefore, the output power of the cell, which is the product of the electric current and the terminal voltage, reaches a peak at a certain value of the electric current or of the external load. 2.4 Purpose and key points of the analysis In the development of good SOFCs, in particular, good IIR-T-SOFCs, there are three fundamental problems to be solved: how to improve the power generation performance, how to prevent thermal crack failure, and how to reduce the cost of production. Numerical analysis is expected to contribute to the solution of these issues. Among them, the thermal crack issue is an especially suitable one to be tackled with numerical analysis. Thermal crack failure is caused by a combination of several thermal conditions including the appearance of a hot spot or of excessively high temperature, the generation of excessively large spatial temperature gradients, the time changing rate of temperature in start-up of the fuel cells and the frequency of load change or of thermal conditions in practical operation. Cell temperature or electrochemical and other chemical reactions governing the cell temperature cannot be measured and such detailed information can only be supplied from numerical analysis. The thermal field in the cell is determined by the balance of several fundamental phenomena: heat generation accompanied with the electric current and the electrochemical processes, heat transfer among different parts of the cell having different temperature, heat absorption by the endothermic internal fuel reforming process, and heat removal by the air flow. Therefore, we must integrate all of the calculation for the gas flow fields, temperature fields both in gas flow passages and solid parts, electrochemical and fuel reforming processes and related mass transfer fields of participating chemical species, and ionic and electric current fields. This must be done with some assumptions to keep the computation load within the reach of engineering workstations but certainly must be done as accurately as possible in

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order to establish reasonable databases for the thermal field of the cell. As for the gas flow velocity, temperature and mass transfer fields, at least two-dimensional axisymmetric analysis would be required to study heat and mass transfer at a reasonably accurate level. Meanwhile, consideration of peripheral nonuniformity is requisite for the electric potential field. This is because, although ionic current occurs in radial direction across the electrolyte, electric current flows basically in the circumferential direction in the electrodes as shown in Fig. 4. Of course it is better if three-dimensionality is taken into account for all related fields, but it requires an extremely large CPU time. Hence what to assume and how to integrate those calculations are the key points of the numerical analysis on tubular SOFCs including the IIR-T-SOFCs. In this study, a previously developed quasi-two dimensional treatment [15, 16] is adopted as a base to develop the new model. This quasi-two dimensional model combines the assumed axisymmetry of the velocity, heat and mass transfer fields with the peripheral nonuniformity of the electric potential and electric current fields.

3 Numerical modeling This section presents the details of the numerical modeling of a tubular SOFC, particularly of an IIR-T-SOFC. This model is based on a previously developed model for a cathode-supported tubular SOFC to be operated with reformed fuel or especially with hydrogen [15, 16]. Therefore, the main parts of the model are common to both cases with and without accompanying internal fuel reforming. However, some general parts common to both cases will be outlined for simplicity referring to the references and other parts specific to the case with indirect internal reforming will be emphasized. As mentioned in the preceding section, various phenomena occur in the tubular cell such as gas flows, heat and mass transfer, internal reforming, electrochemical reaction, non-uniform electric potential field and ionic and electric currents and generation of heat and electricity. These phenomena are closely related to one another, thus they should be considered simultaneously in the numerical analysis. For the sake of the reader’s convenience, geometry of the cell and general assumptions adopted in the present model are described as a preparation to read the following parts. After this, modeling for the most basic phenomena in fuel cells, i.e. electrochemical reactions, will be described first. Next, reforming and shift reactions are described. Then, the modeling for the velocity, temperature and concentration fields will be discussed. Description will be made of the modeling for electric potential fields, electric current and ohmic heating. Finally, an overall picture of the model is given at the end of the section. 3.1 Computational domain and general assumptions for heat and mass transfer A longitudinal sectional view of the tubular cell is shown in Fig. 5 with the dimensions and the chemical reaction equations to be considered in this study. In the examples for IIR-T-SOFCs, of which results will be discussed later, the cell tube

92 Transport Phenomena in Fuel Cells

Figure 5: Longitudinal cross sectional view of a tubular fuel cell. geometry is assumed to be 500 mm long and to have a 6.9 mm inner radius and a 9.6 mm outer radius. For computational convenience, air is taken to flow through an annular space between a tubular fuel cell outer surface and a co-axial confining cylindrical wall of which the radius is 14.6 mm. A square drawn with a fine dotted line in Fig. 3 illustrates the average air flow space in a stack. The surface of this artificial confining wall is assumed to be thermally adiabatic and the area of the annular space is equated with the average air flow space for a single cell in a stack. This treatment is reasonable for a tubular cell located in the core of stack. The hemi-spherical tube end is replaced by a flat end in the model and the computational domain is indicated by the broken line in Fig. 5. The cell tube and fuel feed tube are solid and the feed tube is assumed to be filled with porous catalytic material, and the other areas are gas flow passages. In this study, except in a case of a hydrogen-fuelled cathode-supported tubular SOFC, fuel is assumed to be a mixture of fresh methane and recirculated effluent exhausted from the fuel cell, namely a mixture of hydrogen, steam, CO, CO2 and methane, and air is treated as a mixture of oxygen and nitrogen. All gas components are treated as ideal gases in the calculation of density and electromotive force. In the calculation of thermal fields, temperature is adopted as a variable to solve but the thermodynamic properties like specific heat at constant pressure, enthalpy and Gibb’s free energy and transport properties like viscosity and heat conductivity are treated variable with temperature. Their local values are evaluated at local temperature by consulting the tables of properties included in the program [30, 31]. Properties of mixture are evaluated with mixing laws by making use of the partial pressure of each chemical species [32]. Gaseous fluids are treated to be Newtonian and Reynolds number is lower than two hundred so that both air and fuel flows

Transport Phenomena in Fuel Cells

93

are assumed to be laminar, steady and axisymmetric. The effect of gravity is not considered. 3.2 Model for electrochemical reactions The most basic phenomena in the fuel cell are the electrochemical processes by which chemical energy of the fuel is directly converted into electricity. In this relation, consumption of fuel and oxygen occurs. In addition to electricity generation, heat is generated as an ineffective part of the used chemical energy of the fuel. Here it is first discussed how they are modelled. As mentioned in the preceding section and shown in Fig. 5, the electro-chemical reactions of hydrogen and CO, described by eqns (6) and (7), are considered in this study. The local electromotive forces to be generated by these reactions are calculated based on the following Nernst equation:   PH2 O R0 T 0 ln , (8) EH2 /O2 = EH2 /O2 − 0.5 2F PH2 PO 2   P T R 0 CO 2 0 ln ECO/O2 = ECO/O , (9) − 2 0.5 2F PCO2 PO 2

where  EH0 2 /O2

0 −
GH 2 /O2

=

2F

 0 = ECO/O 2

0 −
GCO/O 2

2F

,

(10)

.

(11)

The subscripts “H2 /O2 ” and “CO/O2 ” indicate the reactions of eqns (6) and (7), 0 0 respectively.
GH and
GCO/O are the changes of the standard Gibbs free 2 /O2 2 energy accompanied with the reactions, and F is the Faraday constant. It is noted here that the local partial pressures of chemical species at the surface of each electrode are assigned to the partial pressures in the above equations. This means that the concentration overpotential related to the concentration non-uniformity existing in the fuel and air flow passages is automatically taken into account in the electromotive forces described by eqns (8) and (9). The overall electrochemical reactions described by eqns (6) and (7) can be divided into the electrode reactions described by eqns (3), (4) and (5). It is necessary to consider the activation overpotential for each electrode reaction. In this study, the following equations suggested by Achenbach [18] are used to describe the activation overpotentials:  ηH2 = iH2 /O2

−1   2F PH2 0.25 Aa kH exp − , R0 T 2 Pfuel R0 T

(12)

94 Transport Phenomena in Fuel Cells  ηCO = iCO/O2

  −1 2F PCO 0.25 Aa exp − , kCO R0 T Pfuel R0 T

−1   PO2 0.25 4F Ac kO =i exp − , R0 T 2 Pair R0 T

(13)



ηO 2

(14)

where kH2 , kCO and kO2 are the coefficients of each equation: kH2 = 2.13 × 108 , kCO = 2.98 × 108 and kO2 = 1.49 × 1010 A/m2 . Aa and Ac are the activation energy: Aa = 1.1 × 105 and Ac = 1.6 × 105 J/mol. In addition, iH2 /O2 and iCO/O2 are the current densities in the electrolyte arising from the reactions of eqns (6) and (7), respectively, which are related to the consumption rates of hydrogen and carbon monoxide as will be discussed later. They are also related to the local current density i as follows: i = iH2 /O2 + iCO/O2 . (15) Here the following relations exist between the electromotive force, activation overpotential, ionic current density across the electrolyte, and the electric potential of the anode and cathode, Va and Vc : EH2 /O2 − (ηH2 + ηO2 ) − iRe he = Vc − Va ,

(16)

ECO/O2 − (ηCO + ηO2 ) − iRe he = Vc − Va ,

(17)

where Re is the ionic resistivity of the electrolyte, which can be estimated considering temperature dependency as suggested by Bessette et al. [19], and he is the thickness of the electrolyte. From the above relationships, the local current density i is obtained if the local potential difference Vc − Va is given, or vice versa. In the present model, interconnects are assumed to work ideally and potential difference Vc − Va is therefore assumed to be constant along the tube axis. It is therefore equal to the cell terminal voltage. As will be explained later, the cell terminal voltage is given to start the computation and both the current density and potential fields are calculated by making use of an equivalent electrical circuit model, details of which will be discussed later. In eqns (16) and (17), note that concentration overpotential to occur in relation with the nonuniform distributions of participating chemical species in air and fuel flow passages has been taken into account in eqns (8) and (9). However, the concentration overpotential to be generated by the diffusion resistance of participating chemical species inside thin anode and cathode layers has been ignored. An approximate method to treat this type of concentration overpotential has been proposed in [33]. However, microstructure of the porous electrodes, or more specifically tortuosity as well as porosity and permeability of the porous electrodes, is not normally available. The authors found that the magnitude of the ignored concentration overpotential is not large for a case without the indirect internal fuel reforming but in future this must be revisited more carefully when appropriate data for the structure of the cell supporting material and porous electrodes become available.

Transport Phenomena in Fuel Cells

95

3.3 Model for internal fuel reforming Steam reforming of methane proceeds in the fuel feed tube preceding the electrochemical reactions at the electrodes in the case of the IIR-T-SOFC. Therefore the modeling of internal fuel reforming is now discussed. As shown in Fig. 5, the steam reforming reaction described by eqn (1) and water-gas shift reaction described by eqn (2) are considered inside the feed tube, and the shift reaction is considered to further proceed outside the feed tube too. In this study, the reaction rates of the two reactions, Rst and Rsh , are locally calculated as follows:  −57840 1.2 W exp , (18) Rst = 1.75PCH cat 4 R0 T + − PCO PH2 O − ksh PH2 PCO2 . Rsh = ksh

(19)

Equation (18) is based on an empirical formula suggested by Odegard et al. which is referred to in reference [34]. Wcat is the filling mass density of the catalyst for the steam reforming and it is an important parameter in a sense that control of its distribution can be used as a means to change the distribution pattern of cell temperature. In some examples to be discussed later, it is actually demonstrated + − and ksh denote how this control is effective. R0 is the universal gas constant. ksh the rate constants of forward and backward water-gas shift reactions and the value of Rsh is determined following the method suggested by Lehnert et al. [35]. Their values vary from one place to another because of the nonuniform distribution of cell temperature. An important point is that these constants are so large that the shift reaction is always close to its equilibrium. Chemical equilibrium is represented by the equilibrium constant which is a function of temperature and is equal to the ratio between the reactant partial pressures and the product partial pressures as follows:   + 0 ksh −
Gsh pCO2 pH2 Ksh = − = = exp , (20) pCO PH2 O R0 T ksh 0 is the change of the standard Gibbs free energy accompanied with the where
Gsh shift reaction. This equilibrium constant is introduced into eqn (19) to calculate the rate of the shift reaction.

3.4 Governing equations of velocity, temperature and concentration fields and boundary conditions Air and fuel are continuously supplied to a fuel cell to keep its appropriate operation and they flow through their respective flow passages in and around the fuel cell. Electrochemical reactions are supplied by diffusion of the participating chemical species from the core flow region toward the reaction sites. Generated heat in the cell is partly used as a source to supply the energy to support the endothermic reforming reaction proceeding in the fuel feed tube but its main portion is transferred to and removed by fuel and air flows. Heat generation is an ineffective part of chemical

96 Transport Phenomena in Fuel Cells energy converted in the fuel cell and therefore occurs at a rate related to the rate of electricity generation. Electricity generation results from the electrochemical reaction of the fuel. The fuel consumption rate is related to the electric charge or ionic transfer rate across the electrolyte. Electric current occurs in the electrodes where the transferred electric charge is collected. Electric current occurs in a pattern consistent with the electric potential fields to be established in the electrodes. Ohmic heat generation occurs in accordance with the electric current in the electrodes and ionic current across the electrolyte. All these phenomena are interrelated to each other in a complicated manner. Therefore, all of the equations governing the phenomena must be integrated in a consistent manner and must be solved simultaneously in an iterative procedure. The most basic governing equations to be solved in the numerical model are the ones for the velocity field, temperature field and concentration field in the fuel cell. They are different among the three kinds of areas: gas area, solid area and porous area. For the gas area, the following two-dimensional continuity, momentum, energy and mass transfer equations for laminar flows apply: 1 ∂rρUr ∂ρUx + = 0, ∂x r ∂r   ∂Ux ∂Ux ∂P ∂ ∂Ux 1 ∂ ∂Ux ρUx + ρUr =− + µ + rµ , ∂x ∂r ∂x ∂x ∂x r ∂r ∂r   ∂Ur ∂Ur ∂P ∂ ∂Ur 1 ∂ ∂Ur µUr ρUx + ρUr =− + µ + rµ − 2 , ∂x ∂r ∂r ∂x ∂x r ∂r ∂r r   ∂ ∂T 1 ∂ ∂T ∂T ∂T + ρCp Ur = λ + rλ + Q, ρCp Ux ∂x ∂r ∂x ∂x r ∂r ∂r   ∂Yj ∂Yj ∂Yj ∂Yj ∂ 1 ∂ ρUx + ρUr = ρDjm + rρDjm + Sj . ∂x ∂r ∂x ∂x r ∂r ∂r

(21) (22) (23) (24) (25)

In the above equations, Ux and Ur are the x- and r-components of the velocity and T is the temperature. Yj is the mass fraction of chemical species j and Djm is the mass diffusivity of species j in the multicomponent mixture of gases. As described above, all of the thermodynamic and transport properties of each chemical species are treated as local variables varying with temperature and the properties of the mixture are evaluated based on mixing laws by making use of the mass fraction of each chemical species. The terms Q and Sj in eqns (24) and (25) denote the source terms to be described later. The effects of the electrochemical and reforming reactions on the thermal and concentration fields are included in these source terms. The effect of radiative heat transfer appears in the matching conditions at gassolid interfaces. However, viscous dissipation in the energy equation is neglected. Thermal diffusion in mass transfer and enthalpy transfer due to species mass transfer are not considered either in the present model.

Transport Phenomena in Fuel Cells

97

Table 1: Thermal conductivity of the solid area.

λs [W/mK]

Anode

Cathode

Electrolyte

Support tube

Feed tube

11.0

6.23

2.7

1.1

1.1

For the solid part, the governing equation to be solved is the following heat conduction equation:   ∂T ∂T 1 ∂ ∂ λs + rλs + Q, (26) 0= ∂x ∂x r ∂r ∂r where λs is the thermal conductivity of the solid. Values of λs of each material are treated to be constant and specific values used in the present study are shown in Table 1 [34]. For the porous area, the governing equations derived by the method of volumeaveraging are applied because the microstructure of porous media is generally too complicated to be considered directly in computations. In the adopted method, physical values are locally-averaged for a representative elementary volume, which is sufficiently larger than the scale of the fine structure of the porous medium and is sufficiently smaller than the scale of the porous body itself [36]. Consequently, the following transport equations of the averaged physical values are used in the present model: ∂ρUx 1 ∂rρUr + = 0, (27) ∂x r ∂r     ∂Ux ∂Ux ∂P 1 ∂ ∂Ux 1 ∂ ∂Ux 1 + ρUr =− + µ + rµ ρUx ε2 ∂x ∂r ∂x ε ∂x ∂x r ∂r ∂r " µ ρf − Ux − √ Ux Ux2 + Ur2 , (28) K K     ∂P 1 ∂ ∂Ur 1 ∂ ∂Ur µUr ∂Ur ∂Ur 1 + ρU = − + µ + rµ − ρU x r ∂x ∂r ∂r ε ∂x ∂x r ∂r ∂r ε2 r2 " ρf µ (29) − Ur − √ Ur Ux2 + Ur2 , K K   ∂ 1 ∂ ∂T ∂T ∂T ∂T ρCp Ux + ρCp Ur = λeff + rλeff + Q, (30) ∂x ∂r ∂x ∂x r ∂r ∂r   ∂Yj ∂Yj ∂Yj ∂Yj ∂ 1 ∂ ρUx + ρUr = ρDjm,eff + rρDjm,eff + Sj . ∂x ∂r ∂x ∂x r ∂r ∂r (31)

In principle, the physical values in the above equations represent the local “intrinsic” phase average of the gas (average over the gas volume, rather than the

98 Transport Phenomena in Fuel Cells Table 2: Parameters of the porous area. ε [–]

K [m2 ]

f [–]

λs [W/mK]

λeff [W/mK]

0.9

1.0 × 10−7

0.088

10.0

ελ + (1 − ε)λs

Djm,eff [m2 /s] √ (1 − 1 − ε)Djm

total volume). This is the case for Ux and Ur which denote the gas phase average of the local gas velocity components. However, this is not the case for temperature. A two-equation model for the thermal field [37] is not adopted in this study so that local thermal equilibrium between the two phases is assumed. Therefore, T denotes the local average both over the gas and solid phases. In addition, ε and K are the porosity and permeability of the porous medium, respectively. f is the inertia coefficient that depends on the Reynolds number and the microstructure of the porous medium [38], while λeff and Djm,eff are the effective thermal conductivity and the effective mass diffusivity of species j, respectively. Values of these parameters used in this study are shown in Table 2. The above three pairs of governing equations are properly conjugated to match at the boundaries between different areas. No-slip condition is applied for the velocity field at solid surfaces and gas-electrode interfaces. Impermeable condition is also applied to the artificial confining wall of air flow and the inner and outer surfaces of the fuel feed tube. At the gas-electrode interfaces, the normal velocity component is set equal to a value incurred by non-zero mass flow rate at the electrode due to the electrochemical reaction [16]. The artificial air-flow-confining wall is assumed to be thermally adiabatic and continuity of heat flux is applied at all of other solid surfaces and gas-electrode interfaces. At the gas-electrode interfaces, mass production or consumption rate of each chemical species is equated to a value regulated by the electrochemical reaction, which will be briefly discussed later. Other solid surfaces are treated as unreactive. In starting computation, inlet conditions must be given for air and fuel flows. In addition, the cell terminal voltage is given as an external condition to start the computation and electric current density to be achieved under the given terminal voltage is calculated. Through iteration, velocity, temperature and concentration fields are determined as their converged solutions. 3.5 Discretization scheme All the above governing equations can be solved only numerically and introduction of some sort of discretization is indispensable. So, a short description is given here about the discretization scheme adopted in the present model just as a hint for some readers who may want to develop their own home-made program. However, it may be worth noting that a number of commercial codes available for general use in a variety of thermo-fluid problems may be transformed into one applicable for fuel cell simulation. All the above governing equationsl listed in section 3.4, eqn (21) through eqn (31), and eqns (38) and (39) to appear in section 3.6 also can be expressed in

Transport Phenomena in Fuel Cells

99

Table 3: Variables and coefficients of each governing equation. Sˆ

Eqn

φ

ρˆ



(22)

Ux

ρ

µ

(23)

Ur

ρ

µ

(24)

T

ρCp

λ

∂P ∂x ∂P µUr − − 2 ∂r r Q

(25)

Yj

ρ

ρDjm

Sj

(26)

T

0

λs

Q

(28)

Ux

(29)

Ur

(30)

T

ρ ε2 ρ ε2 ρCp

µ ε µ ε λeff

" ρf ∂P µ − Ux − √ Ux Ux2 + Ur2 ∂x K K " µ ρf ∂P µUr − − 2 − Ur − √ Ur Ux2 + Ur2 ∂r εr K K Q

(31)

Yj

ρ

ρDjm,eff

Sj

−

−

the following general form:   ∂φ ∂φ ∂ ∂φ 1 ∂ ∂φ ˆ + ρU ˆ r =  + r + S, ρU ˆ x ∂x ∂r ∂x ∂x r ∂r ∂r

(32)

where φ represents the variable of interest. ρˆ and  are the coefficients of the convection term and the diffusion term, respectively, while Sˆ denotes the source term. Specific values of these factors are summarized in Table 3 for several equations. Although there are a variety of discretization schemes available, the finite volume method is adopted in the present model with the staggered grid system. In the staggered grid system, two types of grid arrangements are provided: the velocity grids to store the velocity components and the normal grids for the other scalars such as pressure, temperature and concentrations. The velocity grids are shifted in position from the normal grids as illustrated in Fig. 6 to stably couple the velocity and pressure fields [39]. Integrating over each control volume for the normal grids, painted grey in Fig. 6, eqn (32) can be discretized into the following form: {(ρU ˆ x φ)e − (ρU ˆ x φ)w }rm
r + {(r ρU ˆ r φ)n − (r ρU ˆ r φ)s }
x     ∂φ ∂φ ∂φ ∂φ rm
r +
x + Sˆ · rm
r
x −  r − r =  ∂x e ∂x w ∂r n ∂r s (33)

100 Transport Phenomena in Fuel Cells

Figure 6: Staggered grid for the Finite Volume Method.

in which the subscripts e, w, n and s indicate the positions of four faces of the control volume under consideration, and rm denotes the arithmetic mean of rn and rs . Note that the control volume for velocity components is different from the counterpart for a scalar because the staggered grids are employed. In order to interpolate the fluxes of convection and diffusion at the interfaces expressed by e, w, n and s, the Power-Law scheme suggested by Patankar [39] is adopted here. As a result, the fluxes on the interfaces can be represented by the values on the grid points as follows:  ∂φ rm
r = aE (φP − φE ), (34) ρU ˆ xφ −  ∂x e where aE is a coefficient determined by the Power-Law scheme. This is an example for the interface e, and similar expression can be adopted on the other interfaces. Consequently, eqn (33) can be transformed into the following algebraic equation: aP φP = aE φE + aW φW + aN φN + aS φS + b,

(35)

aP = aE + aW + aN + aS ,

(36)

where

b = Sˆ · rm
r
x.

(37)

The above discretized equations for velocity, temperature and concentration have to be solved simultaneously since gas properties are dependent on temperature and concentration of each gas component. To solve for velocity, the pressure field must be determined because the pressure gradient is included in the momentum equations. In order to solve both the velocity and pressure fields concurrently,

Transport Phenomena in Fuel Cells

101

the SIMPLE algorithm put forward by Patankar et al. [39] is adopted in this model. Other details of discretization may be found in [39]. 3.6 Equations for electric potential and electric circuit Here is described the method to calculate the electric potential fields to be generated in electrodes and accompanying flow of electric current. The mathematical description in this section is given directly in discretized form for the reader’s convenience. In the present model, it is assumed that ionic current flows only radially across the electrolyte but electric current through electrode layers only in the circumferential (θ) direction as shown in Fig. 4 based on the fact that the electric resistivities of the electrodes are much smaller than the ionic resistivity of the electrolyte. In addition, the interconnect is treated to act ideally so that the potential difference between its position attaching the anode, θ = 0◦ , and its position attaching the cathode, θ = 180◦ , is taken to be equal to the cell terminal voltage. Figure 7 illustrates the grid system adopted in the calculation of the electric potential field in the cell tube and accompanying electric current. According to Kirchhoff’s law, the following discretized equations can be derived for both of the two electrodes: (iu − id )ha − iPP · re
θ = 0,

(38)

(iu − id  )hc + iPP · re
θ = 0,

(39)

where iPP  is the current density given by eqn (15) along a particular line connecting the grid points P and P  located across the electrolyte as illustrated in Fig. 7. iu , id , iu and id  are the values of current density to occur respectively along the four lines connecting the two grid points separated in the circumferential direction, U and P and P and D in the anode layer and U  and P  and P  and D in the cathode layer. re is the radius of the mid surface of the cylindrical electrolyte shell layer.

Figure 7: Grid for the equivalent electrical circuit.

102 Transport Phenomena in Fuel Cells According to Ohm’s law, the electric current in the electrodes can be related to the difference in the electric potential between the related grid points: iu h a =

ha (VU − VP ), Ra ra δθu

iu hc =

hc (VU  − VP ), Rc rc δθu

id ha =

ha (VP − VD ), Ra ra δθd

id  hc =

hc (VP − VD ), Rc rc δθd 

(40) (41)

where Ra and Rc are the electric resistivities of the anode and cathode, respectively. They are estimated by the empirical formula suggested by Bessette et al. [19] in this model. Rewriting eqn (15) with E˜ defined by the following equation: E˜ = EH2 /O2 − (ηH2 + ηO2 ) = ECO/O2 − (ηCO + ηO2 ),

(42)

the electric current across the electrolyte iPP can be represented by the potential difference and E˜ as follows: iPP re
θ =

re
θ ˜ {EPP  − (VP − VP )}. Re he

(43)

Hence eqns (40) through (43) can be rewritten into the following algebraic equations: aP VP = aU VU + aD VD + aPP VP  − b,

(44)

aP VP = aU  VU  + aD VD + aPP  VP + b,

(45)

where aU =

ha , Ra ra δθu

aD =

ha , Ra ra δθd

aPP  =

aU  =

re
θ , Re h e

b=

hc , Rc rc δθu re
θ ˜ E, Re h e

aD  =

hc , Rc rc δθd

(46)

(47)

aP = aU + aD + aPP  ,

(48)

aP = aU  + aD + aPP  ,

(49)

eqns (44) and (45) can be solved numerically with a scheme similar to the one to solve momentum, heat and mass transfer equations described in the Section 3.5. 3.7 Mass production or consumption rate of each chemical species through electrochemical and reforming reactions The mass production rate of each chemical species by the electrochemical reactions is directly related to the electric current density caused by each of the electrochemical reactions, eqns (6) and (7), and its value, s˙j , is tabulated in Table 4. For example, the mass consumption rate of hydrogen, s˙ H2 , is equal to the product of the molecular

Transport Phenomena in Fuel Cells

103

Table 4: Mass production or consumption rate of each species by the reactions (6) and (7). s˙ H2

Eqn (6) (7)

−

iH2 /O2 MH2 2F 0

s˙H2 O

s˙CO

s˙CO2

iH2 /O2 MH2 O 2F

0

0

0

−

iCO/O2 MCO 2F

iCO/O2 MCO2 2F

s˙O2 iH2 /O2 MO2 4F iH /O − 2 2 MO2 4F

−

Table 5: Mass production or consumption rate of each species by the reactions (1) and (2). Eqn

SH2

SH2 O

SCO

SCO2

SCH4

(1) (2)

3Rst MH2 Rsh MH2

−Rst MH2 O −Rsh MH2 O

Rst MCO −Rsh MCO

0 Rsh MCO2

−Rst MCH4 0

iH

/O

2 2 weight MH2 and the molar consumption rate of hydrogen 2F , where the Faraday constant F designates the electric charge of one mole of electrons. It must be noted here that two moles of electrons participate into the electrochemical reaction of one mole of hydrogen as seen in Fig. 5. These species mass production rates have nonuniform distributions in the circumferential direction because the current density is nonuniform in that direction. In the present quasi-two-dimensional model, therefore, they are peripherally averaged before being introduced into axisymmetric two dimensional mass transfer equations for the concentration fields. The production or consumption rate of each chemical species by the reforming and shift reactions is also calculated as shown in Table 5. The value of mass production or consumption rate for each chemical species is introduced into the species mass transfer equation as a part of its source term.

3.8 Model for thermodynamic heat generation rates The thermodynamic heat generation rates related to the electrochemical reactions are equal to the generation rate of the ineffective part of the input energy, which cannot be converted into electricity, and are calculated as follows:  (−
HH2 /O2 ) (50) q˙ H2 /O2 = − E˜ iH2 /O2 , 2F  (−
HCO/O2 ) q˙ CO/O2 = (51) − E˜ iCO/O2 . 2F These heat generation rates are included in the calculations of the energy equation as a part of its source term. The thermodynamic heat generation rate by the reforming

104 Transport Phenomena in Fuel Cells reactions, are calculated based on the reaction rates discussed in the Sections 3.3 and 3.7, as follows: Qst = −
Hst Rst ,

(52)

Qsh = −
Hsh Rsh ,

(53)

where
Hst and
Hsh are the enthalpy change accompanied with each reaction. 3.9 Ohmic heat generation With the current density and electric potential fields obtained in Section 3.6, ohmic heat generation rates in the electrodes and electrolyte are calculated as follows: QP =

iu2 δθu + id2 δθd Ra , δθu + δθd

QP =

2 QPP = iPP  Re .

iu2 δθu + id2  δθd Rc , δθu + δθd

(54) (55)

These heat generation rates are nonuniform in the θ-direction. In the present quasitwo-dimensional method, they are peripherally averaged before they are introduced into the source term of the axisymmetric two-dimensional energy equation for the thermal field [15, 16]. Averaged results for the heat generation rates read as follows: π π qP
θ θ=0 qP
θ Qa = , Qc = θ=0 , (56) π π π qPP 
θ . (57) Qe = θ=0 π 3.10 Radiation model Now modeling of radiation heat transfer will be discussed. In the present model, the radiation heat transfer between the inner surface of the cell tube and the outer surface of the feed tube is considered. In the case of the IIR-T-SOFC, this plays an important role in transferring the heat generated by the electrochemical reactions at the tubular cell to the feed tube, and therefore, to the reaction site of endothermic fuel reforming. In the case of a tubular SOFC, the cell tube is slender in geometry. For example, in the case of a tubular IIR-T-SOFC to be studied in this chapter, the fuel flow passage is an annular space between two cylindrical surfaces facing to each other, i.e. anode surface and fuel feed tube outer surface. The width of the annular space is 2.4 mm. This is quite small compared to the cell tube length, 500 mm. This slenderness of the tubular cell has important implications for thermal radiation. The first of the implications to note here is that the optical length of the gas media flowing through the annular space is small. Emittance of the gas layer is given as the product of the optical length and absorption coefficient of gas which is an increasing function of the partial pressure of each chemical species participating

Transport Phenomena in Fuel Cells

105

in radiation. So absorption into or emission from gaseous media can be neglected or the annular space can be treated transparent to radiation so long as the operation pressure is not very high. Description of the modeling of radiation heat transfer will begin with an explanation about the computation method of radiation heat transfer for general use. Certainly, the first feature of thermal radiation in a tubular fuel cell just noted above is taken into account. For more details of this part of discussion, refer to the reference [40]. Now consideration is given to an enclosure covered by the inner surface of the anode and the outer surface of the fuel feed tube and this enclosure is considered to consist of N diffusely emitting and diffusely reflecting gray surfaces. The surfaces are each isothermal. Surface j has temperature Tj and emittance εj . The net rate of heat loss flux qi from a surface i is equal to the difference between the emitted radiation and absorbed portion of the incident radiation. Therefore, it is given as: qi = εi σTi4 − εi Ii , (58) where Ii is the radiation incident on surface i per unit time and unit area and absorptance of the surface i is replaced by the emittance εi taking into account Kirchhoff’s law of radiation. Here is introduced the radiosity Bj of a surface j. The radiosity is the sum of the reflected and emitted radiant fluxes. So the following relationship holds: Bj = εj σTj4 + (1 − εj )Ij .

(59)

A fraction of Bj is directed to the surface i and contributes to the incident radiation on the surface i. Such contribution from the surface j to the incident radiation on surface i per unit area is now calculated as: Iji = Fji Bj Aj /Ai = Fij Bj ,

(60)

where Fji and Fij are the shape factors and the reciprocity rule Aj Fji = Ai Fij has been used. Now similar contributions come from every surface so that Ii is finally expressed as follows: Ii =

N  j=1

Fij Bj =

N 

Fij (εj σTj4 + (1 − εj )Ij ).

(61)

j=1

Iterative calculation is needed to determine the value of Ii or the value of qi with the determined value of Ii . However, this iterative procedure needs a large computational load if a large number is assigned to N . Actually the computation is excessively large for the present purpose of using the result for optimum design or thermal management of a tubular SOFC. Therefore, introduction of a certain degree of approximation into the modeling is desirable. The procedure followed is to reduce the number of surfaces considered in the computation. The approximation adopted in the present model is again based on the slenderness of the tubular cell. For just an example, attention is paid to a geometrical situation

106 Transport Phenomena in Fuel Cells

Figure 8: Geometry of two facing surfaces. illustrated in Fig. 8. Two small elementary surfaces having 20 mm length, surfaces 1 and 2, are positioned in parallel to each other at a distance of 2.4 mm in this figure. An angle viewing one of them, say surface 1, from a point M on another surface 2 is about 153 degrees. The viewing angle of other parts, say surface 3, outside the surface 1 from the point M is only 27 degrees. So, the value of F21 is close to 1 and both of F23 and F32 are close to zero. Naming the surfaces situated on both sides of the surface 2 as surface 4, F24 = 0 and F22 = 0 hold. From the same reasoning as above both of F14 and F41 are close to zero too. So, based on the slender geometry of the tubular cell, radiation heat transfer between the two surfaces 1 and 2, directly facing each other, is considered. Then, it can be assumed that only two shape factors F21 and F12 take a non-zero value. Under this treatment, eqn (61) can be very much simplified. Certainly, by increasing the number of the surfaces, a better approximation is obtained but at the cost of computational time. Now assigning the suffix 1 to the anode inner surface and 2 to the fuel feed tube outer surface, F21 = 1, F12 = A2 /A1 and F11 = 1 − A2 /A1 hold under the adopted assumption. Thus eqn (61) can be approximated as: I1 =

2 

F1j Bj = F11 B1 + F12 B2 ,

(62)

F2j Bj = F21 B1

(63)

B1 = ε1 σT14 + (1 − ε1 )I1 ,

(64)

B2 = ε2 σT24 + (1 − ε2 )I2 .

(65)

j=1

I2 =

2  j=1

and eqn (60) as:

Similarly, eqn (58) can finally be approximated as: q1 = ε1 σT14 − ε1 I1 ,

(66)

q2 = ε2 σT24 − ε2 I2 .

(67)

In the present model, these equations are locally used at each axial position and these radiant fluxes are included in the calculation of heat transfer as a part of the heat flux both at the inner surface of the anode and at the outer surface of the fuel feed tube.

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107

Monochromatic radiation properties of ceramics are highly dependent on wavelength [41, 42]. Pore size of the electrodes and support materials is close to the wavelength of thermal radiation. Thinning the electrodes creates a similar situation that the electrode thickness comes closer to the wavelength too. This situation makes the problem complicated. Pore microstructure affects the wavelength dependency of the radiation properties of the fuel cell element materials and thin element layer can become semi-opaque. Therefore, sophisticated treatment of the radiation properties of the electrode and support materials is desirable. However, radiation properties of such materials have not been thoroughly studied yet. Therefore, for simplicity, the absorption coefficient of the solid materials of the fuel cell is assumed to be constant in the present model. This assumption however must be revisited when the micro structure and radiation properties of the fuel cell element materials have been thoroughly studied. Radiation heat transfer among the tubular cells in the module can be another factor affecting the performance of the fuel cell. This is very true in the case of the module having an “in-stack reformer”. However, in the case of the present IIRT-SOFC or a tubular SOFC using fuel reformed outside the module, every single tubular fuel cell located in the core of the module is surrounded by other tubular fuel cells similar in conditions including the temperature distribution on the outer surface of the cell tube. Again, the space between neighbouring tubular cells is small since tubular cells are stacked in a compact way in the module. So neglecting radiation heat transfer between the outer surfaces of neighbouring tubular cells is less serious in such cases of a tubular SOFC. However, it must be studied more seriously in the case of the module having an in-stack reformer and this heat transfer is a problem to be carefully studied in the modeling of cell stack modules. 3.11 Overall picture of the model In closing this section, an overall picture of the model is presented in Fig. 9. One of the main parts of this computation is the calculation of the temperature and concentration fields assuming their two-dimensionality. Based on the calculated temperature and species concentrations, the electromotive force and several resistances are calculated for the calculation of electric current and potential fields allowing their circumferential nonuniformity, which is another main part of this model. Based on the results of the electric fields, the peripheral average of the ohmic heat generation is calculated. Local current density is related to the mass production rate of participating chemical species. The heat generation rates and species mass production rates thus calculated are fed back to the calculation of the temperature and concentration. The gas properties and the reforming reaction rate are also interrelated with the temperature and concentration fields. Therefore, all of the above calculations are iterated simultaneously. As a result of iteration, all of the temperature, concentration and electric fields of the cell are obtained. Finally the cell performance, such as average current density, output power or energy conversion efficiency, is obtained. A more detailed description of this overall scheme can be found in the references [8, 15, 16].

108 Transport Phenomena in Fuel Cells

Figure 9: Overall picture of the IIR-T-SOFC model.

4 Results and discussion In this section, some calculation results obtained by making use of the numerical model described in the preceding sections are discussed. In section 4.1, discussion is first given to some results of the model applied to a conventional hydrogenfuelled tubular SOFC and its validation is presented by comparing some results with published experimental data. How this model is powerful as a tool for optimum design and thermal management of a tubular SOFC is also demonstrated in this section. In sections 4.2 and 4.3, discussion will be given to some results obtained by applying the model to an IIR-T-SOFC. In section 4.2, the results for a “Base case” will be presented so as to illustrate the essence of the phenomena in a tubular cell accompanying the indirect internal fuel reforming. Then, methods to lower and/or to make more uniform the temperature of the cell will be discussed along with other results in section 4.3. 4.1 Results for a cathode-supported tubular SOFC without accompanying indirect internal reforming [16, 43, 44] In this type of tubular SOFC, as shown in Fig. 2, air is supplied inside the tubular cell through the air feed tube and fuel flows around the tubular cell in the axial direction. So the innermost layer of the cell is the cathode and outermost layer is the anode. Electrolyte material is assumed to be Yttrium-stabilized Zirconia (YSZ) and the fuel is pure hydrogen. So the parts of the present model concerning fuel reforming and electrochemical reaction of carbon monoxide are inactivated. In this case, heat absorption by the fuel reforming does not exist either and heat transfer from the cathode inner surface to the outer surface of the air feed tube is less important. For this reason, radiation heat transfer between the two surfaces is ignored in this

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Table 6: (a) Inlet conditions. Inlet species and their weight concentrations Fuel side Air side

H2 H2 O O2 N2

Inlet temperature

Inlet pressures

870 ◦ C 870 ◦ C 600 ◦ C 600 ◦ C

1.013 × 105 Pa 1.013 × 105 Pa 1.013 × 105 Pa 1.013 × 105 Pa

89% 11% 21% 79%

Table 6: (b) Geometry of cathode supported cell. Components’

Thickness

Supporting tube Cathode Electrolyte Anode

1500 µm 1000 µm 50 µm 150 µm Diameter

Inner side of air-inducing tube Outer side of air-inducing tube Inner side of supporting tube Outer side of anode Outer diameter of fuel channel Length of cell unit

8.0 mm 9.0 mm 13.8 mm 19.2 mm 29.2 mmφ 500 mm

particular example. Inlet conditions of air and fuel flows are tabulated in Table 6(a). Geometry of a tubular cell is kept to be the one given in Table 6(b) unless otherwise stated. Fuel and air utilization factors are set to be 0.85 and 0.167 respectively. Illustration of local quantities shown in Figs 14–16 is for the case of an electric current density of 3500 A/m2 . Other details are found in reference [16]. Figure 10 illustrates the current-voltage (I-V) diagram of the studied tubular SOFC. Calculated results agree fairly well with the experiments performed by Hagiwara et al. [45]. For small current density conditions, simulated results are a little higher than the experiments. This over-prediction may be related to a simple treatment of activation overpotential. It is assumed in this particular computation that the activation polarization is proportional to current density and the value of the proportionality constant is adjusted at an optimum current density around 3000 A/m2 . However, at smaller current density, the cell temperature is lower, and a larger activation overpotential or a larger proportionality constant should have been adopted. In the examples to be discussed in the Section 4.2 and 4.3, more sophisticated treatment of activation overpotential is adopted as already discussed

110 Transport Phenomena in Fuel Cells

Figure 10: Cell voltage-current density diagram.

Figure 11: Conversion efficiency and output power.

in Section 3.2. Except for this point, obtained results look reasonable and the present model is basically validated. Figure 11 illustrates the cell electricity generation efficiency and the power to be achieved with a single tubular cell. With an increase of current density, power to be obtained increases up to a certain value but efficiency decreases. This is the normal tendency and optimum current density is determined by a trade-off between the two quantities having opposite tendency. Figure 12 shows the cell terminal voltage for four cases of tubular SOFCs having different tube length. All the calculated results almost overlap. In this computation, the fuel utilization factor is kept constant. This means that both air and fuel are supplied at a rate proportional to the cell tube length. Larger air flow rates enhance the cooling of the cell but this is countered by larger ohmic heating at lower temperatures. Under the same electric current density, therefore, both the concentrations of fuel and oxygen decrease at a rate proportional to the tube length almost keeping similarity

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Figure 12: Effect of cell length on output voltage at constant fuel utilization factor.

Figure 13: Effects of tube diameter on energy efficiency. in the temperature field among the three cases. This results in such a condition that both electrochemical reaction and heat generation proceed similarly or in a manner proportional to the cell tube length. Figure 13 shows the effects of the cell tube diameter. In the case of a cell tube having larger diameter, the length not only of electrolyte but also of electrodes is larger. Then larger ohmic loss occurs, reducing the terminal voltage, and ohmic heat generation also becomes larger because of the larger collection of electric current per unit length of interconnect. Therefore, a thinner tubular cell can have a higher performance. Figure 14 shows the axial distributions of the local EMF and current density. Both quantities show the same tendency to decrease in the downstream direction. Concentrations of hydrogen and oxygen in the first half of the cell are still high so that the EMF in that region is high too. Similarity between the current density distribution and the EMF distribution indicates that the axial electric current is minor compared to the peripheral one or that interconnect works well.

112 Transport Phenomena in Fuel Cells

Figure 14: Distribution of EMF and current density.

Figure 15: Temperature distribution in a cell.

Figure 15 shows the temperature distribution in the tubular SOFC unit. As for the fuel flow temperature, it rises downstream of the inlet and approaches 1000 ◦ C, then keeps an almost uniform value in the middle part of the cell and slightly decreases toward the exit end. In the air feed tube, air is heated after it flows into the tube. A noticeable transverse gradient of air temperature is found to exist in the annular space over the whole length of the air feed tube. This gradient drives heat transfer from the cell structure, where heat generation occurs, to the air flow in the annular space. This ensures effective cooling of the cell. Radiative heat transfer, ignored in the present computation, may flatten the streamwise temperature non-uniformity but is not expected to change the trend of the temperature distribution significantly. This is because the convective heat transfer considered in the present study is effective already to reduce the differences in surface temperature among various parts of the cell. So air acts as a coolant of the cell. One point related to the convective heat transfer should be noted here. In various numerical models proposed so far, a constant value for the Nusselt number is used in evaluating the local heat removal rate to the air flow. However, this is not a good assumption and actually the heat transfer coefficient is significantly nonuniform along the axial direction [46]. In relation to the discussion of the concentration polarization, distributions of hydrogen and oxygen concentrations are presented in Fig. 16. In the fuel passage, the concentration of hydrogen decreases downstream, especially steeply in the first part after the inlet. However, the radial non-uniformity of the hydrogen concentration is not significant

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Figure 16: Hydrogen and oxygen concentrations. so that the related concentration polarization is judged to be minor. In contrast to this, radial non-uniformity of oxygen concentration is noticeable. This difference is attributed to the difference in the diffusivity, i.e. the diffusivity of hydrogen is much higher than that of oxygen. Therefore, concentration polarization induced by the nonuniform distribution of species concentration in the flow space is not negligible at the cathode side but it can be ignored on the anode side. 4.2 Results for the Base case with accompanying indirect internal reforming In the examples for which results will be discussed here, cell tube geometry is assumed as is shown in Fig. 5 to be of 500 mm length, 6.9 mm inner radius and 9.6 mm outer radius. An adiabatic cylindrical wall artificially introduced to confine air flow for computational convenience has 14.6 mm in radius. Other details of cell geometry are tabulated in Table 7. The inlet and outlet conditions of the fuel and air are summarized in Table 8 and computational conditions for the Base case are shown in Table 9. In the present study, fuel mean velocity at the inlet is set equal to such a value that the limiting value of the average current density is about 5000 A/m2 (e.g. 0.923 m/s when the inlet temperature is 800 ◦ C). In contrast, the air mean velocity at the inlet is set to be 2.0 m/s except for some special cases to be specified later. The fuel and air temperatures at the inlets are changed from one case to another as will be described in a later section but unless otherwise stated they are the values given in the table. The temperature at the closed end of the cell tube is set equal to the air inlet temperature. Catalyst distribution “U-20” means that the catalyst mass density, Wcat , is set at 2.0 × 106 g/m3 uniformly inside the feed tube. This is such a value that the supplied methane fuel can be fully reformed even when the electrical circuit is open. In this case, fuel reforming is thermally supported just by the heat transferred from hot air and fuel flows. The case number for the Base case 2-800U20 indicates that the inlet velocity of the air is 2 m/s, the inlet temperature of the fuel and air is 800 ◦ C, and the catalyst distribution is the pattern U-20. “Neuman” appearing in the table means that zero axial gradient is assumed at the outlet for the variable under concern as its boundary condition. As mentioned before, the cell terminal voltage is given at the start of the computation as a condition to calculate the electric current and potential fields in

114 Transport Phenomena in Fuel Cells Table 7: Size of fuel cell elements. Components’

Thickness

Supporting tube Anode Electrolyte Cathode

1500 µm 150 µm 50 µm 1000 µm Diameter

Inner side of feed tube Outer side of feed tube Inner side of supporting tube Outer side of cathode Outer diameter of fuel channel Length of cell unit

8.0 mm 9.0 mm 13.8 mm 19.2 mm 29.2 mm 500 mm

Table 8: Gas inlet and outlet conditions. Velocity

Temperature

Fuel inlet

Poiseuille flow

Constant

Air inlet

Plug flow

Constant

Outlet

Neuman

Neuman

Molar fraction [−]

Total pressure [Pa]

H2 0.200 0.500 H2 O CO 0.020 CO2 0.030 0.250 CH4 O2 0.209 N2 0.791 Neuman

110000

110000 Neuman

Table 9: Computational conditions of the Base case.

Base case 2-800U20

Velocity (Fuel inlet) [m/s]

Velocity (Air inlet) [m/s]

Temperature (Fuel inlet) [◦ C]

Temperature (Air inlet) [◦ C]

Catalyst distribution

0.923

2.0

800

800

U-20

order to identify the given external load of the cell. Unless otherwise stated, the results of the Base case are shown with particular emphasis on the condition in which the cell terminal voltage is set at 0.55 V and accordingly the average current density becomes 3926 A/m2 .

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115

4.2.1 Thermal and concentration fields Figure 17 shows velocity vectors inside and outside the cell tube. Note that this figure is not to scale: the radial (r) direction is magnified ten times larger than the axial (x) direction. The fuel flow is accelerated inside the feed tube due to an increase in the total number of moles caused by the steam reforming. Fuel flows into the cell tube from the open end of feed tube and flows back to the tube outlet through annular fuel flow passage. Fuel flow inside the tube is almost in a stagnant condition at the bottom end. Temperature contours for the same case are shown in Fig. 18. Note that the darker tone corresponds to the higher local temperature. The temperature in and around the cell basically becomes higher downstream in relation to the air flow because of the heat generation accompanying the electrochemical process. However, the temperature is noticeably lower near the inlet of the fuel due to the endothermic effect of the steam reforming reaction. Here the x-direction distributions of the temperature inside the electrolyte and the feed tube are shown in Fig. 19. Conspicuous non-uniformity of temperature exists as seen in the figure. This is not the case of Siemens-Westinghouse type hydrogenfuelled tubular fuel cell [16] and must be avoided because it can lead to thermal crack failures of the cell. Furthermore, the maximum temperature of the electrolyte

Figure 17: Velocity vectors around the cell for an average current density of 3926 A/m2 in the Base case (2-800U20).

Figure 18: Temperature field in and around the cell for an average current density of 3926 A/m2 (2-800U20).

116 Transport Phenomena in Fuel Cells

Figure 19: Local temperature of the electrolyte and the feed tube for an average current density of 3926 A/m2 (2-800U20) and that in a similar case but not considering the effect of radiation. reaches about 1050 ◦ C, which is the hot spot temperature to be avoided from the material view point. By the way, the broken lines in Fig. 19 show the result in which the effect of radiation described in section 3.4 is excluded. From this result, it is confirmed that the radiation contributes noticeably to the heat transfer between the cell tube and the feed tube in this case with accompanying indirect internal fuel reforming. Figure 20 shows mole fraction contours of hydrogen, steam, CO, CO2 , methane and O2 , respectively. Note that the grey tone levels differ among each chemical species. Inside the feed tube, hydrogen and CO are produced while steam and methane are consumed by the steam reforming. Methane is rapidly reformed near the feed tube inlet because the amount of the catalyst is excessive for the operating condition of the cell under study. Meanwhile, outside the feed tube, hydrogen and CO are consumed while steam and CO2 are produced mainly by the electrochemical reactions. The reason for the considerable change of the mole fraction occurring at the bottom of the cell is that the fuel flow stagnates while the electrochemical reactions take place there. As for the air side, oxygen is consumed by the electrochemical reaction. However, variation of its mole fraction is not so large because the flow rate of air is adjusted so as to effectively remove the generated heat and is normally in excess of that required to complete the electrochemical reactions. In other words, the oxygen utilization factor is set normally as low as 0.2 or 0.3. The molar flow rates of each chemical species inside and outside the feed tube are shown in Figs 21(a) and (b), respectively. Characteristics of the electrochemical and reforming reaction described above can be confirmed in these figures. 4.2.2 Electric potential and current fields Distribution of electromotive force (EMF) is shown in Fig. 21. Although two different values of EMF, EH2 /O2 and ECO/O2 , are calculated in this study as described

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117

Figure 20: Mole fraction distributions of each chemical species for an average current density of 3926 A/m2 (2-800U20).

in section 3.6, these two differ very slightly from each other by the reason that the shift reaction is very close to equilibrium everywhere in the cell. Hence only EH2 /O2 is plotted here as the EMF. As seen in this figure, the EMF strongly depends on the mole fraction of hydrogen at the anode surface: the EMF has a peak near the outlet of the feed tube and becomes lower toward the cell tube outlet. However, it becomes a little higher near the cell tube outlet again due to the larger change of Gibb’s free energy,
G, at lower temperature.

118 Transport Phenomena in Fuel Cells

Figure 21: Molar flow rate of each chemical species of fuel for an average current density of 3926 A/m2 (2-800U20).

Figure 22: Electromotive force and current density distributions for an average current density of 3926 A/m2 (2-800U20). Distribution of current density is also shown in Fig. 22. Note that the distribution plotted in this figure is the peripheral average of local current density at each axial position. This figure indicates that the distribution of current density is not similar in shape to that of EMF. This is because the current density is also related to the activation overpotential and the ohmic loss which depend strongly on the temperature. As shown in Fig. 23, both the activation and ohmic overpotentials are significantly reduced around x = 0.4, where the cell temperature takes a peak value (see Figs 18 and 19). That is to say, the current density distribution depends on the thermal field as well as, or more than, on the concentration field. At the same time, it is also true that the temperature tends to become higher where the large current is generated, so that a kind of enhancement due to a mutually assisting interaction exists between the cell temperature and the electric current. Incidentally, the ohmic loss in the anode and cathode is closely related to the distribution of the current density in the θ-direction, which is essentially non-uniform. Details can be found in the references [13, 14].

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119

Figure 23: Activation overpotential and ohmic loss in the electrolyte for an average current density of 3926 A/m2 (2-800U20).

Figure 24: Temperature of the electrolyte for several different average current densities (2-800U20). 4.2.3 Power generation characteristics Before moving into the discussion on the power generation characteristics of the fuel cell, results of the electrolyte temperature for several cases of different average current densities are given here. Figure 24(a) shows the x-direction distribution of electrolyte temperature for the cases of average current density 2191, 3081 and 3926 A/m2 . All the three distributions have similar features as those described in section 4.1.1, but temperature becomes higher on the whole as the average current density becomes larger. The relation between the average current density and the maximum and average temperatures of the electrolyte is shown in Fig. 24(b). The power generation characteristics of the cell are significantly affected by the cell temperature. Figure 25 shows the cell-averaged EMF, activation overpotential and ohmic loss in the electrolyte for various values of average current density. The cell terminal voltage, which is practically given as the external condition to start the calculation in this study, is also plotted here. From this figure, the following

120 Transport Phenomena in Fuel Cells

Figure 25: Average EMF and overpotentials vs. average current density (2-800U20).

Figure 26: Output power and energy conversion efficiency vs. average current density (2-800U20). things are confirmed. First, the cell-averaged EMF goes down at a nearly constant rate as the average current density increases. This is largely because the local concentrations of hydrogen and CO decrease due to their larger consumption rates. The activation overpotential does not increase proportionally to the average current density countered by the rise in temperature as mentioned above. The ohmic loss in the electrolyte also does not increase for the same reason. However, the ohmic loss in the anode and cathode, which is basically larger than that in the electrolyte because the electric current flows long distances there in the θ-direction as mentioned in section 3.7, increases nearly in proportion to the average current density. From these results, the output power and energy conversion efficiency of the cell are also obtained as shown in Fig. 26. Here the efficiency is calculated based on the lower heating value (LHV) of the fuel consumed in the cell. Judging from

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121

Table 10: Computational conditions for the additional cases.

2-750U20 4-800U20 2-800U02 2-800L02

Velocity (Fuel inlet) [m/s]

Velocity (Air inlet) [m/s]

Temperature (Fuel inlet) [◦ C]

Temperature (Air inlet) [◦ C]

Catalyst distribution

0.880 0.923 0.923 0.923

2.0 4.0 2.0 2.0

750 800 800 800

750 800 800 800

U-20 U-20 U-02 L-02

these calculated results, fairly high performance seems to be achieved under the conditions for the Base case. 4.3 Strategies for the ideal thermal field As described in section 4.1.1, the thermal field in the cell obtained for the Base case is undesirable in a sense that the thermal field is seriously non-uniform and the maximum temperature is rather high. Hence the strategies to flatten and/or lower the distribution of cell temperature are discussed here. Decreasing the inlet temperatures of the gases is a primary method to lower the cell temperature. Increasing the air flow rate also seems to be effective when the cell temperature is higher than the inlet temperature of the air. To achieve a nearuniform thermal field, controlling the density distribution of catalyst inside the feed tube should be useful. In order to examine those effects, calculations for the four cases summarized in Table 10 have been conducted. In Case 2–750U20, the inlet temperatures of the gases are decreased down to 750 ◦ C and accordingly the fuel inlet velocity is reduced to 0.880 m/s to keep its molar flow rate at the same value as the Base case. In Case 4-800U20, the air flow rate is doubled from the Base case. In Case 2-800U02 and 2-800L02, the density distribution of catalyst is changed to “U-02” and “L-02”, which are described in Fig. 27. The amounts of the catalyst in U-02 and L-02 are optimised for the conditions of average current density about 4000 A/m2 . That is, fuel is completely reformed just near the end of the feed tube under such conditions. The calculation results for the above four cases are discussed below in sequence. 4.3.1 Effect of gas inlet temperature The temperature contours obtained at the average current density 4043 A/m2 in Case 2-750U20 are shown in Fig. 28, and the electrolyte temperature distribution in the x-direction is shown in Fig. 29. Owing to the lower gas inlet temperatures, the cell temperature decreases on the whole. However, the maximum temperature is not so reduced, and thus the non-uniformity of the temperature becomes more serious. This is because the activation overpotential becomes drastically larger where the cell temperature drops below 850 ◦ C and accordingly the current density distribution

122 Transport Phenomena in Fuel Cells

Figure 27: Mass density distribution of the catalyst inside the feed tube.

Figure 28: Temperature field in and around the cell for an average current density of 4043 A/m2 (2-750U20).

Figure 29: Effect of the gas inlet temperature on the electrolyte temperature distribution. becomes more noticeably non-uniform as shown in Fig. 30. Additionally, electricity conversion efficiency decreases by more than 5% compared to the Base case (see Table 11) due to the increase in activation overpotential caused by the decrease in temperature.

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123

Figure 30: Effect of the gas inlet temperature on the local current density distribution. Table 11: Calculation results for the Base case and the additional cases. Case & Average current density [A/m 2 ]

Terminal voltage [V]

Output power [W]

Efficiency (LHV) [%]

Maximum temperature (electrolyte) [◦ C]

Average temperature (electrolyte) [◦ C]

2-80-U20: 3926 2-750U20: 4043 4-800U20: 3920 2-800U02: 3911 2-800L02: 4051

0.55 0.48 0.50 0.54 0.52

58.2 52.3 52.8 56.9 56.8

41.6 36.3 37.8 41.0 39.5

1047.0 1023.3 968.0 995.4 974.2

957.6 913.2 909.5 942.6 939.2

4.3.2 Effect of air flow rate The temperature contours obtained for the average current density 3920 A/m2 in Case 4-800U20 are shown in Fig. 31, and the electrolyte temperature distribution in the x-direction is shown in Fig. 32. Owing to the increased flow rate of air, the coolant, the cell temperature, especially the maximum, is fairly reduced in comparison with the Base case, and accordingly the temperature gradient is also reduced. Therefore, to increase the air flow rate is quite effective for flattening the cell temperature distribution, though often it should be adjusted so as to fulfil the requirements to optimize the performance of a topping or bottoming device. 4.3.3 Effect of density distribution of catalyst The temperature contours obtained for the average current density 3911 A/m2 of the Case 2-800U02 are shown in Fig. 33. Since the amount of the catalyst inside the feed tube is optimised, the maximum temperature becomes lower and a sharp fall of the cell temperature near the fuel inlet observed in the Base case is mitigated.

124 Transport Phenomena in Fuel Cells

Figure 31: Temperature field in and around the cell for an average current density of 3920 A/m2 (4-800U20).

Figure 32: Effect of the air flow rate on the electrolyte temperature distribution.

Figure 33: Temperature field in and around the cell for an average current density of 3911 A/m2 (2-800U02).

The contours for the average current density 4051 A/m2 of the Case 2-800L02 are shown in Fig. 34. The thermal field becomes almost uniform in a large area of the cell by arranging the catalyst density in a linear distribution rather than in the uniform distribution. These effects of the density distribution of the catalyst on the thermal field are seen more clearly in Fig. 35, which shows the electrolyte temperature distribution in the x-direction.

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125

Figure 34: Temperature field in and around the cell for an average current density of 4051 A/m2 (2-800L02).

Figure 35: Effect of the catalyst distribution on the electrolyte temperature distribution. Figure 36 shows the mole flow rate of methane inside the feed tube for the above cases. As shown in this figure, the difference in the catalyst density distribution greatly affects how the internal reforming takes place inside the feed tube, which means how the heat generated by the electrochemical process is absorbed there. However, the total heat absorption rate inside the feed tube is not so changed accompanying the change in the catalyst density distribution because reforming is almost 100% accomplished in all of the above cases. Therefore, the average electrolyte temperature scarcely decreases and accordingly the power generation performance is little affected by the change of the catalyst density distribution (see Table 11).

5 Conclusions This chapter has presented a numerical model for a single tubular SOFC including a case with accompanying indirect internal fuel reforming or for an IIR-T-SOFC. The model developed is useful to analyze not only its electricity generation performance but also the details of the phenomena inside the cell. A variety of phenomena such as

126 Transport Phenomena in Fuel Cells

Figure 36: Effect of the catalyst distribution on the molar flow rate of methane inside the feed tube. heat and mass transfer, internal reforming, electrochemical reaction, heat generation and production of chemical species, and generation of electric field proceed inside the cell in a three dimensional manner. All of them are taken into consideration in the developed model but in a quasi-two dimensional way. All aspects of the numerical model and numerical procedures have been described in the first half of this chapter. Some numerical results obtained by using the developed numerical model have been presented, first for a conventional type tubular SOFC and then for an IIRT-SOFC. In the first part, it was demonstrated how the developed model works in analyzing not only the fuel cell performance but also the details of the phenomena in and around the tubular cell. In the second part, how to reduce the cell temperature gradient has been discussed. The thermal field of the cell is largely determined by the balance of three fundamental phenomena: heat generation with the electrochemical process, heat absorption with the internal reforming process, and heat removal by the air flow. In the electrochemical process, a kind of mutually enhancing interaction has been observed between the rise in cell temperature and increase in electric current. It has been confirmed that increasing the air flow rate or decreasing the air utilization factor is effective to flatten the temperature distribution. It has been also demonstrated that changing the density distribution of the catalyst inside the feed tube greatly affects how the reforming reaction takes place and therefore how the cell is cooled.

Acknowledgements This study has been carried out as one of the CREST projects titled “Micro Gas Turbine and Solid Oxide Fuel Cell Hybrid Cycle for Distributed Energy System” under the support of the Japan Science and Technology Agency (JST). The authors would like to extend their appreciation also to Dr. Peter Woodfield, Saga University, who gave us valuable discussions and critiques in finishing this chapter.

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130 Transport Phenomena in Fuel Cells

Nomenclature a A b Cp Djm e E E0 E˜ f F F11 , F12 , F21 , F22 h i J kH2 , kCO , kO2 + − ksh , ksh K Ksh M P q q˙ Q r R Rst , Rsh R0 s˙ Sj Sˆ T Ux , Ur V Wcat x Yj

coefficient in the discretized equations activation energy [J/mol] source term in the discretized equations specific heat at constant pressure [J/kg K] mass diffusivity of species j in multicomponent gas [m2 /s] emissivity [–] electromotive force [V] standard electromotive force [V] substantial electromotive force defined by eqn (42) [V] inertia coefficient [–] Faraday constant [C/mol] configuration factors [–] thickness [m] electric current density [A/m2 ] radiosity [W/m2 ] coefficients in eqns (12), (13) and (14) [A/m2 ] velocity constants of the water-gas shift reaction [mol/m3 Pa2 s] permeability [m2 ] equilibrium constant of the water-gas shift reaction [–] molecular weight [kg/mol] pressure [Pa] radiant flux [W/m2 ] heat generation per unit area of the electrolyte [W/m2 ] heat generation [W/m3 ] radial coordinate [m] resistivity [ m] reaction rates of the reforming reactions [mol/m3 s] universal gas constant [Pa m3 /mol K] mass generation per unit area of the electrolyte [kg/m2 s] mass generation of species j [kg/m3 s] source term in the general transport equation temperature [K] velocity components [m/s] electric potential [V] catalyst mass density [g/m3 ] axial coordinate [m] mass fraction of species j [–]

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Subscripts a c cat e eff s st sh 1 2 H2 /O2 CO/O2 e, w, n, s E, W , N , S u, u , d , d  U , U  , D, D P, P  PP 

anode cathode catalyst electrolyte effective solid steam reforming reaction: eqn (1) water-gas shift reaction: eqn (2) cell tube side in the radiation model feed tube side in the radiation model electrochemical reaction of hydrogen: eqn (6) electrochemical reaction of CO: eqn (7) interfaces on a control volume shown in Fig. 6 grid points in a control volumes shown in Fig. 6 interfaces on a control volume shown in Fig. 7 grid points in a control volumes shown in Fig. 7 grid points on the intended control volumes electrolyte between the grid points P and P 

Greek symbols 
G 0
H
r
x
θ δr δx δθ ε η θ λ µ ρ ρˆ σ φ

diffusivity in the general transport equation standard Gibbs free energy change enthalpy change control volume size for r direction control volume size for x direction control volume size for θ direction distance between grid points for r direction distance between grid points for x direction distance between grid points for θ direction porosity activation overpotential circumferential coordinate thermal conductivity viscosity density density in the general transport equation Stefan-Boltzmann constant variable in the general transport equation

[J/mol] [J/mol] [m] [m] [rad] [m] [m] [rad] [–] [V] [rad] [W/m K] [Pa s] [kg/m3 ] [W/m2 K4 ]

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CHAPTER 4 On heat and mass transfer phenomena in PEMFC and SOFC and modeling approaches J. Yuan1 , M. Faghri2 & B. Sundén1 1 Division

of Heat Transfer, Lund Institute of Technology, Sweden. of Mechanical Engineering and Applied Mechanics, University of Rhode Island, USA.

2 Department

Abstract There are similarities among various transport processes in solid oxide fuel cells (SOFCs) and proton exchange membrane fuel cells (PEMFCs), but also some differences. The present work concerns modeling and numerical analysis of heat, mass transfer/species flow, two-phase transport and effects on the cell performance in SOFCs and PEMFCs. Numerical calculation methods are further developed to enable predictions of convective heat transfer and pressure drop in flow ducts of the fuel and the oxidant. The unique boundary conditions (thermal, mass) for the flow ducts in fuel cells are identified and implemented. The composite duct consists of a porous layer, gas flow duct and/or solid current inter-connector (or -collector). The results from this study are applicable for other investigations considering overall fuel cell modeling and system studies, as well as the emerging field of micro-reactor engineering.

1 Introduction In a fuel cell, electrical energy is generated directly through the electrochemical reaction of oxidant (oxygen from air) and fuels (such as natural gas, methanol, or pure hydrogen) at two electrodes separated by an electrolyte. When pure hydrogen is used, the only products of this process are heat, electricity and water. Unlike a battery, fuel cells do not store energy. The energy conversion is achieved without making use of the materials that constitute an integral part of the fuel cell structure. It should be noted that fuel cells convert chemical energy directly into electricity without an intermediate combustion process.

134 Transport Phenomena in Fuel Cells One of the main factors that have influenced the development of fuel cells has been the increasing concern about the environmental consequences of fossil fuel utilization in the production of electricity and for the propulsion of vehicles. More importantly is the increasing global awareness of how industry activities influence the environment and how a sustainable energy development can be achieved with a tremendously increasing world population. Fuel cells may help to reduce our dependence on fossil fuels and diminish poisonous emissions into the atmosphere. The operation of a fuel cell requires a fuel electrode (Anode), oxidant electrode (Cathode), electrically-insulating ionic conductor (Electrolyte), and external electric circuit. In general, fuel cells can be classified according to the type of ionic conductor (Electrolyte) they use and the temperature range at which they operate. Several types of fuel cells are currently under development. Alkali fuel cells (AFCs) use alkaline potassium hydroxide as the electrolyte, and have been used for a long time by NASA on space missions. In proton exchange membrane fuel cells (PEMFCs) and phosphoric acid fuel cells (PAFCs), hydrogen fuel dissociates into free electrons and protons (positive hydrogen ions). The hydrogen protons migrate through the electrolyte to the cathode. The liquid-fed direct methanol fuel cell (DMFC) feeds a solution of methanol and water to the anode.At the cathode, oxygen from air, electrons from the external circuit and protons combine to form pure water and heat. All these three types are low temperature fuel cells. High temperature fuel cells, such as solid oxide fuel cells (SOFCs) and molten carbonate fuel cells (MCFCs) are of particular interest, because their high temperature operation allows natural gas to be used as a fuel, and the hybrid concept involving a combination of a fuel cell and a gas turbine becomes feasable. The overall system efficiency can be significantly increased in a hybrid system. Operation at a temperature of about 1000 ◦ C (conventional, electrolyte-supported planar design), 700 ◦ C (intermediate temperature, anode-supported design) and pressures greater than one atmosphere leads to solid oxide fuel cells (SOFCs) as one of the choices. It has been found that the proton exchange membrane fuel cell (PEMFC) system has some advantages, such as its relative simplicity of design and operation, low cost construction and self-starting at low temperatures. Both SOFC and PEMFC systems are expected to play a significant role in the next generation of primary or auxiliary power for stationary, portable, and automotive systems. During the last decades, a large amount of research activities have been carried out on fuel cells worldwide, with particular interest and focus on SOFC and PEMFC systems. High performance, low cost and high reliability have been considered as the primary aspects and concerns for fuel cells to compete with well-developed fossil fuel power technology, such as the internal combustion engine. Most of the work has focused on creating new materials and material processes for the manufacturing of specific systems so as to achieve good cell/stack performance while minimizing the final system cost and size. To help expand future market opportunities for fuel cells, additional fundamental understanding and research work are needed. More attention needs to be focused on detailed analysis of transport processes, even at

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micro scale level. Because of the particular sensitivities and complexities of fuel cells, analysis and optimization of fuel cell components/system can effectively be performed by numerical modeling and simulation. The present work concerns numerical analysis of heat and mass transport processes, and fluid flow in ducts of both planar type solid oxide fuel cells and proton exchange membrane fuel cells. Numerical models have been developed to enable predictions of convective heat transfer and pressure drop in flow ducts of the fuel and the oxidant. The composite duct consists of a porous layer, gas flow duct and/or solid current inter-connector (or collector). The results from this study are applicable for related investigations considering overall fuel cell modeling and system studies, e.g., by providing heat transfer coefficients for various conditions, as well as in the emerging field of micro-reactor engineering. In general, the work also contributes to the understanding of duct flow.

2 Fuel cell modeling development Despite the differences in terms cell structure/materials employed and operating conditions (such as temperature, pressure), supply of species to an active surface of cells in the stack is a common feature of fuel cells, such as SOFCs and PEMFCs. It is so because the performance degradation for the cells operating within the stack results from the unequal distribution of reactant mass flow among the cells. Understanding the various gas and heat transport processes is crucial for increasing the power density, reducing manufacturing costs and accelerating commercialisation of fuel cell systems. To this end, detailed modeling and improved simulation tools are required to fully characterize the complex multi-dimensional, multi-phase and multi-component transport processes on media that are both porous and electrochemically reactive. 2.1 Basics of SOFCs and PEMFCs There are various similarities in the transport processes occurring in SOFCs and PEMFCs. In the anode duct, the fuel (e.g., H2 ) is supplied and air (O2 + N2 ) is introduced in the cathode duct, and these ducts are separated by the electrolyte/electrode assembly. Reactants are transported by diffusion and/or convection to the electrode/electrolyte (SOFC) or catalyst/electrolyte (PEMFC) interfaces, where electrochemical reactions take place.An electrochemical oxidation reaction at the anode produces electrons that flow through the inter-collector (bipolar plate, for PEMFC) or -connector (for SOFC) to the external circuit, while the oxide ions (in SOFCs) or protons (in PEMFCs) pass through the electrolyte to the opposing electrode. The electrons return from the external circuit to participate in the electrochemical reaction at the cathode. In the electrochemical reaction process, part of the oxygen is consumed in the cathode duct, while the hydrogen is consumed in the anode duct. Heat and water (H2 O) are the only by-products during the process. The water generated is injected into the anode duct further along the duct in SOFCs, while in

136 Transport Phenomena in Fuel Cells PEMFCs, it enters into the cathode duct. The electrochemical reactions in SOFCs can be written as: Cathode reaction: Anode reaction:

1 2 O2

+ 2e− → O2− .

H2 + O2− → H2 O + 2e− .

(1a) (1b)

and for PEMFCs: Cathode reaction:

1 2 O2

+ 2e− + 2H+ → H2 O.

H2 → 2H+ + 2e− .

Anode reaction:

(2a) (2b)

The overall reaction is as follows: 1 2 O2

+ H2 → H2 O.

(3)

Due to the flow resistance in the fuel cells, the pressure drop (
P) along the ducts and in the manifolds can cause non-uniform flow distribution. Furthermore, the output of electrical energy will differ in terms of voltage potential and in some cases even gas re-circulation occurs. At some severe conditions, the lack of gas in some channels can cause the irreversible damage to the fuel cell components. The pressure drop depends on the channel and manifold structures, flow streams etc. However, the temperature is always non-uniform even when there is a constant mass flow rate in the ducts. This is caused by the heat transfer and phase change (in PEMFCs), which in turn causes fluctuation in the available
T. Heat transfer occurs in the following manner: •

• •

Between the cell component layers and the flowing air and fuel streams. This can be described in terms of heat transfer coefficients ha (for air channel), hf (for fuel channel) due to forced convective heat transfer with or without natural convection; Between the fuel and air streams across the interconnect layer in terms of the overall heat transfer coefficient, U ; In solid structures in terms of heat conduction with different thermal conductivities, ki (i = electrolyte, electrodes and current interconnect layers).

For the electrolyte and porous layers, often referred to as the membrane electrode assembly (MEA) in PEMFCs, the overall principal energy balance can be written as: Qc + hf Af (Te − Tf,av ) + ha Aa (Te − Ta,av ) = Qs ,

(4)

where Qc is the heat conduction in the solid structure, Qs is the heat source to account for the electrochemical heat generation, ohmic heating caused by the electrical resistance due to the current flow; h is the convective heat transfer coefficient; T is the temperature. Equation (4) shows that the heat transfer coefficients in the fuel and oxidant ducts are important. There are certainly some specific aspects and phenomena which need to be carefully investigated for different applications.As an example, Nafion® membranes are

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often employed in the electrolytes in PEMFCs. These membranes possess high ion conductivity by selecting perfluorosulphonic acid copolymers with a short pendant group. However, the performance of the membranes, in terms of electrical conductivity, strongly depends on the water content. Water management in the membrane is one of the major issues in PEMFCs. As discussed later in this chapter, there are several factors affecting the water content in the membrane, such as the water drag through the electrolyte (electro-osmotic), back diffusion of product water from the cathode to the anode. For the case of excessive accumulation of water vapor in the cathode, condensation may occur. Consequently, different methods of water management have been proposed and investigated. Another issue is the temperature range in which the membranes are stable. Therefore, both water and thermal management are coupled and need to be carefully balanced, as will be discussed later. SOFCs employ solid oxide material as electrolyte and are, therefore, more stable. There are no problems with water management, liquid water flooding in the cathodes or slow oxygen reduction kinetics in SOFCs. On the other hand it is difficult to find suitable materials for operation at high temperatures. There are other processes which only occur in SOFCs, such as internal reforming of fuels as pure hydrogen is not used and co-generation of heat/electricity with other power systems (such as gas turbines). For PEMFCs, external reforming is needed to handle hydrocarbon fuels. It should be mentioned that the reforming issues are not treated in this chapter but instead focus is on the composite duct flows in SOFCs and PEMFCs. 2.2 Modeling development Modeling has already played an important role in fuel cell development since it facilitates a better understanding of parameters affecting the performance of fuel cells and fuel cell systems. Moreover, water management within the cathode is a key consideration in the design of the PEMFCs. However, knowledge of the behavior of liquid water in electrodes is limited by the inability to make in situ measurements. Better understanding of the transport of water in the PEMFC electrode can be obtained from models that capture the important physical processes. This is particularly evident in works published in the open literature during recent years. There has been a range of models developed, from simple lumped models of individual cell channels to more complex three-dimensional detailed ones of complete stacks. 2.2.1 Modeling approaches There are several issues which affect the choice of modeling strategies, and should be considered before selecting a fuel cell modeling approach. The most important factors are the objectives and features of the model, which should be clearly defined and specified. These include the following issues, such as steady-state/transient, theoretical/semi-empirical, components/system study, lumped/multidimensional, accuracy/time/flexibility, validation/documentations and so on. The development of modeling and simulation tools is a cost and time consuming process. The level of

138 Transport Phenomena in Fuel Cells user knowledge and available resources (such as personnel and computer facilities etc.) are also constraints to include in the decision process. Based on detailed electrochemical, fluid dynamics, species/current transport and heat transfer relationships, a theoretical fuel cell modeling approach usually employs the basic equations, such as the Stefan-Maxwell equation for gas-phase transport, and the Butler-Volmer equation for cell voltage [1]. The electrochemical and transport processes are tightly coupled. For proper water and thermal management, this approach includes not only the electrochemical reaction but also thermaland fluid-dynamic equations. Multi-component species transport and heat transfer are important for providing a detailed picture of all processes in the fuel cell and the system. The output of the study can provide details of the processes, such as fuel cell species distribution/flow pattern, current density/temperature distribution, voltage and pressure drop, etc. On the other hand, based on experimental data specific to the applications and operating conditions, semi-empirical fuel cell models have also been developed during the recent years. Both approaches have advantages and disadvantages. The theoretical modeling approach is flexible to applications and operating conditions, and may be appreciated when detailed studies are desired. However, development and implementation of this approach takes a longer time, and it is difficult to validate due to lack of detailed data in the open literature. At present, the most readily available data are simply the overall I -V characteristics for a cell or stack. While the semi-empirical approach is already validated to some extent, it does not provide sufficient details. It should also be noted that it must be modified for each new application or operating conditions, and may not be suitable in some cases. During recent years, reliable commercial fuel cell codes (or in some cases, a fuel cell module added to an already existing software) have been developed. For instance, detailed SOFC and PEMFC models have been included in the Chemical Engineering Module of FEMLAB, based on the platform of the MathWorks simulation code MATLAB [2]. More detailed modules associated with other commercial codes are being further developed and will be incorporated into standard cell and stack modeling when they have been validated. 2.2.2 Various existing models In the last few years, attempts to simulate the velocities, pressures, temperatures, mass fractions, electric currents and potentials in fuel cells with given boundary conditions have been presented. This section only covers literature dealing with the relevant problems of heat transfer and gas flow modeling in SOFCs and PEMFCs. 2.2.2.1 SOFC modeling For conventional electrolyte-supported planar structure, Vayenas [3] created a two-dimensional mixing cell model for cross-flow to simulate the distribution of gas species, temperature, and current density. For a unit cell with square and rectangular ducts, constant temperature was assumed for the solid walls, and the gas-phase composition was uniform. Another model was developed by Ahmed et al. [4]. This could be used to simulate the electrochemistry and thermal hydraulics in a cross-flow monolithic SOFC with alternating layers

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139

of anode, electrolyte, cathode, and interconnect. Based on the average thermal and compositional conditions, a Nusselt number (= hd /k) of 3.0 based on the equivalent diameter of the rectangular flow ducts was used for the convective heat transfer between the gases and the solid surfaces. In this model, the generated heat was released in the electrolyte, and the pressure drop in the channels was modeled by assuming fully developed laminar flow. Bernier et al. [5] proposed a three-dimensional mathematical model to compute the local distribution of the electrical potential, temperature, and concentration of the chemical species. In the gas ducts, the thermal flux is mainly convective and conductive from the duct walls to the other solid parts. A computer code was developed by Melhus and Ratkje [6], which was used to find simultaneous solutions of all conservation equations for mass, energy and momentum in a quasi three dimensional simulation for single flat SOFCs. The reduction of the mass of O2 in the cathode chamber was assumed to be regained in the anode as H2 O for the mass balance. Regarding the distribution of the gas flow, Boersma and Sammes [7] developed a model to simulate the non-uniform gas flow distribution along the height of a fuel cell system. For the anode-supported design, the porous anode usually has a thickness of 1.5–2 mm [8–11], which is the thickest component and supporting structure, while a thin layer of electrolyte (∼10 µm) is deposited on its surface. The transport path length for the fuel gases from the flow duct to the anode/electrolyte interface where the electrochemical reaction happens, is at least equal to the porous anode thickness. The transport rate of fuel gases is controlled by the porous layer microstructure (e.g., pore size, permeability, and porosity), pressure gradient between the flow duct and porous layer, gas composition, etc. There are various polarizations or losses (such as ohmic, activation and concentration) affecting the overall performance of SOFCs. It has been revealed that one of the principal losses in this design is attributed to concentration polarization caused by limitation of gaseous species transport through the porous anode [8], because the size of the porous anode might be bigger than that of the flow duct. 2.2.2.2 PEMFC modeling It has been found that the proton exchange membrane fuel cell system has some advantages, such as its relative simplicity of design and operation, low cost construction materials and self-starting at low temperatures. However, water management is a critical issue for PEMFC, i.e., a high water-content must be maintained in the membrane in order to obtain acceptable ion conductivity. Water molecules are transferred from the anode duct to the cathode duct of the membrane by electro-osmosis during PEMFC operations. If this transport rate of water is higher than that of back diffusion, the anode gases will dry out, and the membrane becomes dehydrated and too resistive to conduct current. On the other hand, cathode flooding occurs when the water removal rate fails to reach its minimum transport rate, which is caused by both the transport from the anode duct mentioned above and the generation of water by the electrochemical reaction H+ /O2 at the catalyst surface [12]. Consequently, a sufficient amount of water must be supplied to the anode duct to make up for the loss due to net water transfer from

140 Transport Phenomena in Fuel Cells the anode, and water should be removed at a sufficient rate from the cathode duct to keep an active catalyst surface for reaction. Another challenging issue in PEMFCs is thermal management. In order to prevent excessive operating temperature and drying out of the membrane, the heat generated by the electrochemical reaction should be removed properly. The thermal management has a strong impact on the fuel cell performance, by affecting the transport of water and gaseous species as well as the electrochemical reactions in the cells. Both thermal and water management of PEMFCs are unique compared to other types of fuel cells. Various water and/or thermal management systems have been claimed to be efficient by different approaches of humidification designs (injection of liquid/vapor water) and operating conditions [13–17] and anode water removal [18]. It should be noted that most of the models are one- or two-dimensional, and are limited by the isothermal assumptions and also neglect the potentially significant gas pressure drop within the electrodes. However, these studies provided a fundamental framework to build multi-dimensional and multi-component models, such as a fully threedimensional [19–20], some of which include the heat transfer effects on the overall performance of fuel cells [20]. More recently, researchers have started to investigate two-phase water flow, by taking into account effects of heat generation/transfer and/or pressure drop [21–25]. 2.2.2.3 Other processes relevant to fuel cell modeling In fuel cells, the gaseous reactant flows at both the cathode and anode are subject to fluid injection and suction along the porous interface to the electrolyte. However, such processes are often simulated as flow in porous wall ducts at constant heat flux boundary condition [26]. Because both heat transfer and pressure distributions are significantly affected, fluid flow and heat transfer in ducts with mass transfer in porous walls have received a great deal of attention in the past decades [26–28]. Hwang et al. [26] simulated flow and heat transfer in a square duct in the range of −20.0 < Rem < 20.0, with boundary conditions of one porous wall subjected to a constant heat flux, while the other three walls were assumed to be adiabatic and impermeable. A better understanding of thermal engineering applications is required when porous materials are present in the duct flow. Because of its simplicity and reasonable accuracy within a certain range of applications, the Darcy model has been used for the majority of the existing studies on gas flow and heat transfer in porous media. It has been found that the Darcy model has some limitations, and inertial forces should be taken into account when the interstitial flow velocity (i.e., the flow through pores of a porous medium) is not small [29]. It was reported in [30] that heat transfer can be significantly affected by Darcy number and the thermal conductivity ratio (between thermal conductivity of the porous media and that of the fluid). Various types of interfacial conditions between a porous medium and a gas flow duct were analyzed in details for both gas flow and heat transfer in [31]. It was found that as some of the gas flow penetrates sideways into the porous layer, the remaining gas flows downstream at decreasing flow rates. The static pressure in such a duct changes along the main flow stream due to the following reasons:

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the friction between the gas flow and the internal surfaces of the duct creates pressure losses, and the mass permeation across the interface between the flow duct and the porous layer implies that mass and momentum are transferred to/from the porous layer [32]. The permeation process in the porous media is usually considered as an overall mass transport with a constant permeability. 2.2.2.4 Further challenges As discussed above, various models have been developed and improved for both SOFCs and PEMFCs. However, the modeling improvements are still limited due to, among other things, the limited material data available in the open literature. At a minimum, temperature and/or concentration dependent electrical resistivity and electrochemical activation energies, as well as the diffusion characteristics of the porous materials are pre-requisites to model development. Another critical factor limiting modeling development is the lack of test data that can be used to validate models.

3 Main processes in SOFCs and PEMFCs The major processes significant to fuel cell characteristics are similar in SOFCs and PEMFCs. These processes are the species transport, electrochemical reactions, electronic and ionic transport, and heat transfer and distribution. Figure 1 shows

Figure 1: Schematic sketch of a unit cell for: (a) PEMFC and (b) SOFC.

142 Transport Phenomena in Fuel Cells a unit cell structure of fuel cells (representing PEMFCs in this case). It includes various components, such as fuel and oxidant gas ducts, electrolyte (polymer electrolyte membrane for PEMFCs), anode and cathode diffusion layers, catalyst layers in between them, and current inter-conductors. 3.1 Gas transport In a fuel cell stack, the gas transport processes consist of: • • •

•

The fuel and oxidant gases flow separately through the gas manifolds where no electrochemical reactions occur; The fuel and oxidant gases flow along cell ducts where there is absorption of the reactants and the injection of reactive products from/to active surface; In the porous layers (electrodes), transport of the reactant gases occur towards the catalyst layer (in PEMFCs), or active surfaces (in SOFCs), and the exhaust gases are rejected to the cell ducts through the open pores; The exhaust gases from each cell are discharged through the gas output manifolds.

Fuel cell ducts and manifolds should be designed/configured to have appropriate gas flow rate and flow uniformity to the reactive surface. An important concern is the net pressure loss, which should be as low as possible to reduce parasitic power needed to operate pumps or compressors. Consequently, a laminar flow regime is effected in most of the fuel cells by employing small velocity and cross-sections in the manifolds and ducts [33]. The appropriate mass flow rate of reactants (fuels and oxidants) is determined by various factors, such as the requirement for the electrochemical reaction, proper thermal and water management, and fuel reforming reaction (SOFCs) etc. In PEMFCs, water management is critical to avoid the membrane dry out and cathode flooding. To deal with this concern, the oxidant flow rate may be increased to reduce the excess water generation. 3.2 Electrochemical reactions At the active surface, the electrochemical reactions take place as described in eqns (1) and (2) for SOFCs and PEMFCs, respectively. The overall cell reaction is shown as eqn (3). The impacts of the electrochemical reactions on the gas mass balance are represented by the absorption of reactants and generation of products at the active surfaces, in terms of mass flux rate J (kg/m2 s). The mass flux rate is related to local current density I (A/m2 ) and reads as follows when pure hydrogen is used as fuel. •

SOFC anode I MH2 , 2F

(5)

I MH2 O . 2F

(6)

JH2 = − JH 2 O =

Transport Phenomena in Fuel Cells •

PEMFC anode I MH2 , 2F α = − I · MH2 O . F

JH2 = − JH 2 O •

SOFC cathode JO2 = −

•

143

I MO2 . 4F

(7) (8)

(9)

PEMFC cathode I MO2 , 4F 1 + 2α = I · MH2 O . 2F

JO2 = − JH 2 O

(10) (11)

In the equations above, α is the net water transport coefficient, for the case of PEMFCs, which represents the net water transport through the membrane by electroosmotic drag and back diffusion due to the water concentration difference, and hydraulic permeation due to the pressure difference between the two sides. Other symbols can be found in the nomenclature. It should be noted that negative sign (−) in the above equations represents gas consumption, while plus (+) means gas generation. 3.3 Heat transfer As mentioned before, the heat transfer processes include various aspects, such as the convective heat transfer between the solid surface and the gas streams, conductive heat transfer in the solid and/or porous structures. Furthermore, heat generation occurs at the active surface in association with the electrochemical reactions and cell losses. The heat generation at the active surface, qb , can be expressed as follows [20]: I (12) qb = −
HH2 O MH2 O − I · Vcell , 2F where
H is the enthalpy change of formation of water. The first term on the RHS in the above equation accounts for the amount of heat energy released by the water formation, while the second one accounts for the current density generated by the electrochemical reaction. When the inhomogeneous current density is taken into consideration, the total local heat generation must be defined due to local joule heating. 3.4 Various transport processes in the electrodes (porous layers) Electrodes for fuel cells are generally porous to ensure maxium active surfaces, and to allow the injection of the generated products to the ducts. The mass transfer is dominated by the gas diffusion and/or convection, as discussed later in this chapter. This is ensured by the open pores of the electrodes, in terms of permeability

144 Transport Phenomena in Fuel Cells and/or porosity. Another requirement of a porous electrode is to also have a good ionic conductivity, because the ionic particles are transported via the solid matrix of the porous layer. In general, the electrodes should have a balanced performance and long-time stability. In PEMFCs, a catalyst material is frequently employed, such as platinum or platinum/ruthenium, whereas SOFCs utilize much cheaper catalyst materials such as nickel due to reduced activation polarization at higher temperature. 3.5 Other processes appearing in fuel cell components The electrolyte of fuel cells transport ions created by the electrochemical reaction at one electrode to the other. In PEMFCs the proton is transported through the electrolyte, while in SOFCs the oxygen ion is transferred. Reducing the electrolyte thickness and internal ohmic losses is a major requirement. On the other hand, the electrolyte should be impermeable to gases (fuel and oxidant) for the purpose of minimizing reactant crossover. The cell inter-connectors or -collectors involve heat transfer by thermal conduction and current collection. Consequently, high electrical conductivity and thermal conductivity are the basic requirements. The materials having the following features are often employed, i.e., impermeability to reactants and chemically stable in oxidizing and reducing environments. In general, PEMFCs need more expensive materials, individually machined graphite or even gold-plated stainless steel materials. It is well-known that the polarization curve, which represents the cell voltage behaviour against operating current density (V -I curve), is the standard measure of the performance for fuel cells, and depends on both the operating conditions and the component design. The operating conditions include the working temperature, partial pressures of fuel and oxidant and their utilization rate, and/or the water concentration in the components. On the other hand, the design parameters could be the porosity, tortuosity, thickness of the electrodes (concentration loss), thickness of the electrolyte (ohmic loss), and the electrode/electrolyte interface (activation loss).

4 Processes and issues in SOFC and PEMFC Despite various similar processes discussed above, there are certainly processes and/or phenomena which are connected to a specific fuel cell type and should be carefully considered. Among these, water management is a challenging issue for PEMFC, while internal fuel reforming at the anode side and high operating temperature have unique features relating to mass transport and heat transfer if fuels other than pure hydrogen are used in SOFCs. Moreover, thermal radiation may be important in SOFCs while it is not in PEMFCs. 4.1 Water management in PEMFCs Polymer electrolyte membranes in PEMFCs are basically water filled to have high proton conductivity, as mentioned earlier. Factors influencing the water content in

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145

the electrolyte are generally two transport processes, i.e., water drag through the electrolyte membrane (a shell of H2 O is transported via the electrolyte for every proton transported), and back diffusion of generated water from the cathode into the anode through the electrolyte. The first one is often referred to as electro-osmotic transport in the literature, and the latter one is due to the gradient of water content in the electrolyte. The effective electro-osmotic coefficient α is an important parameter to represent water transport between the anode and the cathode. It includes the effects of both electro-osmosis and water back diffusion. Water management in the electrolyte is one of the major issues in PEMFCs. This is because during PEMFC operation anode gases can be dried out if the electroosmosis transport rate is higher than that of back diffusion, which consequently causes the electrolyte membrane to become dehydrated and too resistive to conduct current. On the other hand, water is generated at the cathode active surface and transported to the cathode duct. Cathode flooding may occur when the water removal rate fails to reach its threshold transport/generation rate. Both dry-out and water flooding should be avoided, and various water management schemes have been proposed. More detailed discussions on these issues can be found in [13–18]. It should also be noted that condensation can occur in the cathode duct when local vapor saturation condition occurs in the duct. As discussed later in this work, this case mainly happens at high current densities and low operating temperatures of fuel cells. When such condensation occurs the transport process becomes two-phase, which besides flooding the cathode layer can considerably complicate the modeling procedure as no experimental results are available for two-phase flow in PEMFCs. Instead, much attention has been paid on numerical investigations to reveal the relationships between the water saturation, proton conductivity (ohmic loss), the level of catalyst flooding (activation loss), and the effective diffusivity of the porous layer (concentration loss); see [21–26]. 4.2 Fuel reforming issues in SOFC Because of the high operating temperature, an SOFC can convert not only hydrogen into electricity, but can also reform hydrocarbon compounds into reactant fuels. For instance, methane can be converted to H2 and CO2 in a steam reforming process within the anode of SOFCs. This reforming process takes place at the surface and in a very thin layer of the anode porous nickel cermets (ceramic metal) [10]. It is often referred to as internal reforming in the literature. The methane reforming reaction in this case can be written as follows: CH4 + H2 O → CO + 3H2 , CO + H2 O → CO2 + H2 ,


H = 206 kJ/mol,

(13a)


H = −41 kJ/mol.

(13b)

Equation (13b) is usually referred to as water gas shift reaction. The overall reforming reaction is: CH4 + 2H2 O → CO2 + 4H2 .

(13c)

146 Transport Phenomena in Fuel Cells It should be mentioned that the above processes in eqn (13c) are net endothermic and the overall balance of the reaction requires external heat input. This heat can be supplied by the exothermic electrochemical reaction, as given in eqns (1) and (2). Due to the fast reforming reaction compared to the electrochemical reaction, the endothermic steam reforming process may lead to local sub-cooling, and/or mechanical failure due to thermally induced stresses [10].

5 Modeling methodologies for transport processes in SOFC and PEMFC 5.1 General considerations A typical configuration of a simulated fuel cell duct is shown in Fig. 2. This section focuses on the establishment of equations for the analysis of the fuel cell ducts appearing in SOFCs and PEMFCs. The duct under study includes the gas flow duct, porous layer (anode/cathode) and solid current inter-connector or collector. The variables to be solved for are the gas transport velocities in the x, y and z directions, the gas mass concentration/distribution, pressure drop, temperature distribution and convective heat transfer coefficient (in term of Nusselt number, Nu). The liquid water saturation and its effects on the current density distribution are modeled for PEMFCs. The governing equations consisting of species (H2 , H2 O, O2 , etc.), momentum and energy equations, are solved for various sub-domains. Auxiliary equations are employed to calculate the source terms in the governing equations. A unified framework is developed for both SOFC and PEMFC, by implementing specific source terms to account for different transport processes and boundary conditions.

Figure 2: Schematic drawing of a composite fuel cell duct (Anode-supported planar SOFC) under consideration.

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147

5.2 Assumptions Steady laminar flow of incompressible fluid is considered. The appropriate mass flow rate of reactants is determined by several factors related to various requirements, e.g., maintaining proper water balance and thermal management. Water management concerns in PEMFCs may need an increased flow rate, which can be described by a parameter named the stoichiometric ratio for an electrode reaction. In general, the gas flow duct should be designed to minimize pressure drop, while providing adequate and evenly distributed mass transfer through the porous layer for the electrochemical reaction. Thus, typically laminar flow with Re numbers of the order of 100–1000 have been employed for fuel cells [33]. For simplicity, the following additional assumptions are applied: • •

•

• •

The inlet-velocity and axial-temperature distributions of the species are assumed uniform, and the inlet Reynolds number is assumed to be Rein (=Uin Dh /ν)  1; An electrochemical reaction is assumed to occur at the interface between electrodes and electrolyte. The released heat is simulated as a wall heat flux qb at one duct wall (the bottom wall in Fig. 2); When the porous layer is thick and needs to be considered (as for the ducts in PEMFCs and anode-supported SOFCs), it is assumed to be homogeneous and characterized by effective parameters, such as porosity, permeability and thermal conductivity, and the fluid in the porous layer is in thermal equilibrium with the solid matrix; Mass consumption and generation are simulated by a mass transfer flux (see following sections for more details); In PEMFCs, liquid water appears in the form of small droplets in the gas species. A multi-phase mixture model is employed to describe two-phase flow and heat transfer in the composite duct. This model enables the calculation of the individual phase velocities from the mixture flow field when the capillary flow due to the pressure gradient and gravity-induced phase migration are considered. For simplicity, the interfacial shear force and surface tension force between the liquid water and the gas phase are neglected; liquid water has the same pressure as the gas species.

5.3 Governing equations The governing equations to be solved are the continuity, momentum, energy and species equations. The mass continuity equation is written as ∇ · (ρeff V) = Sm .

(14)

The source term Sm in the above equation accounts for the mass balance caused by the reaction from/to the active surface Aactive (bottom surface in Fig. 2). For PEMFCs, it corresponds to the hydrogen and water consumption on anode side, oxygen consumption and water generation on the cathode side, respectively.

148 Transport Phenomena in Fuel Cells These are given by [19, 20],  Aactive I α Sm = SH2 + Sa,H2 O = − MH2 − I · MH2 O , 2F 2F V  Aactive I (1 + 2α)I Sm = SO2 + Sc,H2 O = − MO2 + MH2 O , 4F 2F V

(15) (16)

where V refers to control volume at the active site. The momentum equation reads ∇ · (ρeff VV) = −∇P + ∇ · (µeff ∇V) + Sdi .

(17)

The inclusion of the source term Sdi allows eqn (17) to be valid for both the porous layer and the flow duct Sdi = −(µeff V/β) − ρeff BVi |V|.

(18)

The first term on the right hand side of the above equation accounts for the linear relationship between the pressure gradient and flow rate according to Darcy’s law. The second term is the Forchheimer term which takes into account the inertial force effects, i.e., the non-linear relationship between pressure drop and flow rate. In eqn (18), β is the porous layer permeability, and V represents the volume-averaged velocity vector of the species mixture. For example, the volume-averaged velocity component U in the x direction is equal to εUp , where ε is the porosity, Up the average pore velocity (or interstitial velocity). This source term accounts for the linear relationship between the pressure gradient and flow rate by the Darcy law. It should be noted that eqn (17) is formulated to be generally valid for both the flow duct and the porous layer. The source term is zero in the flow duct, because the permeability β is infinite. Equation (17) then reduces to the regular NavierStokes equation. For the porous layer, the source term (eqn (18)) is not zero, and the momentum eqn (17) with the non-zero source term in eqn (18) can be regarded as a generalized Darcy model. The energy equation can be expressed as  keff ∇T + Swp , (19) ∇ · (ρeff VT ) = ∇ · cpeff where Swp is the heat source associated with the water phase change (condensation/ vaporization) for the case of PEMFC (see [20]). Swp = Jwl × hwl ,

(20)

where Jwl is the mass flux of liquid water by phase change, and hwl is the water latent heat. The species conservation equations are formulated in the general equation, ∇ · (ρeff Vφ) = ∇ · (ρeff Dφ,eff ∇φ) + Sφ ,

(21)

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149

where φ is the mass fraction. The above equation is solved for the mass fraction of H2 in the case of an SOFC anode, and O2 , H2 O(v) and H2 O(l) in the case of a PEMFC cathode, respectively. The concentration of the inert species, nitrogen, is determined from a summation of the mass fractions of the other species. For PEMFC, the source term in eqn (21) includes water vapor and liquid water caused by the phase change. It is written as follows [20]   Pw,sat − Pwv  massi . (22) Swv = −Swl = MH2 O P − Pw,sat Mi i

In eqn (22), massi refers to mass of species i; Pwv refers to partial pressure of water vapor, Pw,sat is the saturation pressure at the local temperature, while P is local pressure. If the partial pressure of water vapor is greater than the saturation pressure, water vapor will condense, and a corresponding amount of liquid water is formed. The water vapor partial pressure Pwv in the above equation is calculated based on its concentration and local pressure of the gas mixture, while the saturated pressure Pw,sat at the local temperature reads [21], log10 Pw,sat = −2.179 + 0.029T − 9.183 × 10−5 T 2 + 1.445 × 10−7 T 3 . (23) 5.4 Boundary and interfacial conditions A constant flow rate U = Uin is specified at the inlet of the gas flow duct, while U = 0 is specified at the inlet and the outlet of the inter-connector (or collector) and porous layer. The other boundary conditions employed can be written as: U = V − Vm = W = 0, −ρeff Dφ,eff

∂φ = Jφ ∂y

U = V = W = 0,

−keff

∂T = qb , ∂y

at bottom wall (y = 0),

q = 0 (orT = Tw ),

(24)

Jφ = 0

at top and side walls,

(25)

∂U ∂V ∂T ∂φ = =W = = =0 ∂z ∂z ∂z ∂z at mid-plane (z = a/2).

(26)

It should be noted that all the walls for the above boundary conditions are on the external surfaces of the solid layer and porous layer. In eqn (24), Vm is the wall velocity of mass transfer caused by the electrochemical reaction (see eqns (1) and (2)). The detailed procedure to obtain Vm was presented in [35], and the final form of the source term in eqn (14) is given by Sm = ρeff Rem

ν a , Dh A

(27)

150 Transport Phenomena in Fuel Cells where Rem = Vm Dh /ν is the wall Reynolds number caused by the electrochemical reaction. The other variables can be found in the nomenclature list. qb in eqn (24) is the heat source caused by the reaction and was given in eqn (12) (see [20]). Among various interfacial conditions between the porous layer and gas flow region, the continuity of velocity, shear stress, temperature, heat flux, mass fraction and flux of species (for the oxygen, water vapor and liquid water, respectively) are adopted, i.e., U− = U+ , (µeff ∂U /∂y)− = (µf ∂U /∂y)+ ,

(28)

T− = T+ , (keff ∂T /∂y)− = (kf ∂T /∂y)+ ,

(29)

φ− = φ+ , (ρeff Dφ,eff ∂φ/∂y)− = (ρeff Dφ,eff ∂φ/∂y)+ .

(30)

Here the subscript + (plus) is for the fluid side, while − (minus) is for the porous layer side. Moreover, the thermal interfacial condition eqn (29) is also applied at the interface between the porous layer and solid layer with ks instead of keff . 5.5 Additional equations It should be noted that the properties in the above equations with subscript ‘eff’ are effective ones. For the flow duct, the effective properties are reduced to regular values of the species mixture based on the species composition, or regarded as constant values in some cases. In the porous layer, there are many factors affecting the effective properties such as the microstructure or nanostructure of the porous layer, species composition and local temperature etc. It is not easy to obtain more accurate values at this moment because the available data of the porous layer structure are still limited. It has been found that setting µeff = µf and ρeff = ρf provide good agreement with experimental data [43]. For the sake of simplicity, this approach is adopted here as well. To reveal the porous layer effects, parameter studies can be carried out for the conductivity keff and species diffusion coefficients Dφ,eff by employing the ratios, θ, θk = keff /kf ,

(31)

θD = Dφ,eff /Dφ .

(32)

In eqn (31), kf is the species mixture conductivity in the porous layer, and is estimated by [39],   −1  1  xi xi kfi + kf = · , (33) 2 kfi i

where xi is the mole fraction, and kfi conductivity of the species component. The diffusion coefficients Dφ in eqn (32) are the values of the species components in the species mixture, i.e., DO2 , DH2 O (v) and DH2 O (l) for oxygen, water vapor and liquid

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151

water, respectively. However, the binary diffusion coefficients of the components in pure air are used as estimations of Dφ in the calculations [23]. For the case of PEMFC, the effective diffusivity ratios are corrected by applying the so-called Bruggemann correction [23, 24] to account for the effects of porosity in the porous layer, (34) θD = ε1.5 . It should be noted that the thermal physical properties of the species mixture, such as the density ρf , viscosity νf are estimated as functions of the local concentration. According to Dalton’s law, the relative humidity of the species mixture is defined as Pwv P = xwv , (35) η= Pw,sat Pw,sat where P is the pressure, Pwv the water vapor partial pressure, Pw,sat the saturation pressure identified in eqn (23), xwv water vapor molar fraction. The liquid phase saturation s is employed to describe the liquid water volume fraction in the species mixture. It reads [21] ρφw − ρg φwv , (36) s= ρwl − ρg φwv where φ is mass fraction, ρg gas phase density, ρwl liquid phase density. The density of the two-phase species mixture is ρ = ρg (1 − s) + ρwl s.

(37)

5.6 Solution methodology Due to the similarity of the conservation equations for SOFC and PEMFC, eqns (14, 17, 19, 21) can be written in the general form as  (38) ∇ · (ρeff V) = ∇ · ,eff ∇ + S , where  denotes any of the dependent variables,  is the diffusivity and S is a source term or sink term. Because there is no analytical solution to eqn (38), computational fluid dynamics (CFD) methods have to be employed to obtain the discrete solutions. Once in this form, the equations are integrated over control volumes. The boundary conditions are introduced as source terms in control volumes neighboring boundaries whenever appropriate. Details of various numerical schemes can be found in [34]. The numerical analysis, performed in this chapter, has been conducted by developing various source terms/boundary conditions related to the fuel cell transport processes, and implementation into general purpose and in-house developed CFD codes. The finite-volume method is used in the codes. More detailed discussion on the codes can be found in [35]. It should be noted that the source term in eqn (14) (accounting for mass transfer effects) is zero in most of the regions, and non-zero only in the regions neighboring boundaries, where mass transfer caused by the electrochemical reaction occurs

152 Transport Phenomena in Fuel Cells (bottom wall in Fig. 2). The CFD procedures are modified accordingly and the source term Sm is implemented in the pressure correction equation to adjust the mass balance due to mass transfer. For the composite duct, it is clear that no gas flow is present in the solid inter-connector (or -collector). Equations (17) and (21) are solved by applying high viscosity values and only the heat conduction equation, derived from the energy eqn (19), is solved for this domain. In order to evaluate the performance of the numerical method and codes, test calculations considering grid sensitivity, code performance and validation have been carried out. These results can be found in [35] and the literature cited there. Due to the lack of experimental data for fuel cells, it should be mentioned that the computational codes have been validated by comparisons with fully developed/developing conditions in pure duct flow with mass transfer and ducts for various thicknesses of the porous layer.

6 Results and discussions In the first section, dimensionless pressure differences and convective heat transfer coefficients, represented by friction factors and Nusselt numbers, respectively, are presented for conventional SOFC design, i.e., the porous electrodes (anode/cathode) are very thin and negligible, compared to the flow ducts. For the PEMFC and anodesupported SOFC design, the porous layer is thick and should be included in the analysis. The importance of gas flow and heat transfer in the porous layer and the effects on the transport processes in the flow duct are presented and discussed. For simplicity, the thermal boundary condition TBC-I refers to that of the constant heat flux at one wall, and constant temperature at the other three walls. TBC-II refers to the combination of constant heat flux at one wall, and thermal insulation at the remaining walls. 6.1 Mass transfer effects on the gas flow and heat transfer For electrolyte-supported SOFC ducts, an overall mass balance is considered for the flow duct when the species consumption and generation happen in a fully developed flow. ˙m = m ˙ out , (39) m ˙ in + m where m ˙ m is the mass flow rate from the active wall caused by the electrochemical reaction. Because Rem ( = Vm Dh /v) is assumed to be very small compared to the main flow Reynolds number, the mass transfer rate m ˙ m is given by: dm ˙ m = ρaVm dx

(40)

in which, dx is the increment in the main flow direction. The change of mass flow rate due to the mass transfer (suction/injection) reads:  ∂Ubulk ∂Ubulk dx − ρAUbulk = ρA dx. (41) dm ˙ = ρA Ubulk + ∂x ∂x

Transport Phenomena in Fuel Cells

153

By combining eqns (40) and (41), the mass flow rate change can be written [35]. Sm = ρ

v a ∂Ubulk = ρRem , ∂x Dh A

(42)

where ∂Ubulk /∂x is the velocity gradient in the main flow direction induced by the mass transfer. To characterize the overall pressure difference between inlet and outlet, either a pressure coefficient Cp or an apparent friction factor fapp of the gas flow in a duct can be employed as Cp = 4fapp =

(Pin − P) , 2 /2) (ρUbulk

(43a)

dP Dh , 2 (ρUbulk /2) dx

(43b)

where Ubulk is the mean velocity of the main flow, Dh is the hydraulic diameter defined in the conventional manner, dP/dx the pressure gradient along the main flow direction. The bulk velocity is calculated as: # UdA (44) Ubulk = # dA and the hydraulic diameter is defined as: Dh =

4A , P∗

(45)

A is the cross-sectional area and P ∗ is the wetted perimeter. It is clear that the mass transfer contributes to a change of the main flow velocity, thus the local Reynolds number varies as a function of x and mass flow rate even when the flow is fullydeveloped. It should be noted that the fluid pressure is affected by inertia forces and wall friction. As an example, the pressure difference increases for the case of injection, and it is larger than for the case without mass transfer. The apparent friction factor fapp is employed in this study because it incorporates the combined effect of wall shear and the change in momentum flow rate due to the effects of mass generation and consumption by the electrochemical reaction. For fully developed flow with mass transfer, the Nusselt number can be derived from an energy balance in the duct. As mentioned earlier, the mass flow through the porous wall is much smaller than the main flow, and the total heat flow rate qb (per unit length of the duct) can then be calculated by (Fig. 2): qb = ρUbulk cp A

dT − ρcp Vm a (Tbulk − Tw ) . dx

(46)

This equation can be rearranged for a rectangular duct as: ρUbulk cp A dT qb Dh Dh dx Dh = − ρcp Vm a ∗ . ∗ ∗ (Tbulk − Tw ) kP Tbulk − Tw kP kP

(47)

154 Transport Phenomena in Fuel Cells The left hand side is the definition for the Nu and the first part on the right hand side is the definition of the Nusselt number without mass transfer (Nuf ). Nu =

qb Dh , (Tbulk − Tw )kP ∗

(48)

Nuf =

ρUbulk cp A dT dx Dh , (Tbulk − Tw ) kP ∗

(49)

where Tw , the wall temperature, is the same on three walls but varies on the fourth wall for TBC-I. Tbulk is the bulk flow mean temperature in the cross section, and is calculated as # T |U | dA . (50) Tbulk = # |U | dA The second part of eqn (47) is the contribution to the Nusselt number by mass transfer. This term is rewritten as Num = −PrRem

b . P∗

(51)

Equation (47) is then rewritten as: Nu/Nuf = 1 + Num /Nuf .

(52)

The Nusselt number Nuw can be defined as: Nuw =

hw Dh qw Dh = , k k(Tw − Tbulk )

(53)

Nuw =

hwDh q w Dh , = k k(T w − Tbulk )

(54)

where Nuw and Nuw are spanwise variable and average Nusselt numbers of the heated wall at location x, respectively. qw is the wall heat flux; Tw and T¯ w are spanwise variable and average temperature of the heated wall, respectively. The dimensionless axial distance x∗ in the flow direction for the hydrodynamic entrance region is defined as: (55) x∗ = x/(Dh Re). Mass transfer effects on the friction factor and Nusselt number are shown in Fig. 3 for a rectangular duct with TBC-I. For the case of mass injection from the porous wall, additional mass is induced to the duct and thus the axial velocity increases. As clarified earlier, the fapp Re is related to the pressure gradient as well as changes in the momentum flux in the main flow direction. As can be seen from Fig. 3(a), fapp Re always increases for mass injection (Rem > 0), while it decreases for mass suction (Rem < 0). On the other hand for heat transfer, the temperature of the fluid will increase due to the heat induced by mass injection into the fluid, while a decrease

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Figure 3: Mass transfer (Rem ) effects on the fully developed convective flow: (a) the apparent friction factor and (b) Nusselt number in a rectangular duct at TBC-I, SOFC.

appears for the case of mass suction. The Nu/Nuf is thus reduced by mass injection, which can be seen in Fig. 3. A large aspect ratio has a significant effect on both fapp Re and Nu/Nuf , while a small aspect ratio gives less effect. Both fapp Re and Nu/Nuf approach the values for the case without mass transfer (Rem = 0), if the aspect ratio becomes about 0.1. The figures show also that the fapp Re and Nu/Nuf has a minimum when the aspect ratio is unity, i.e., a square duct. More discussion can be found in [35]. Comparisons of fapp Re and Nu for a developing flow with and without mass transfer in a rectangular duct are shown in Fig. 4 with TBC-II. It is clear that the apparent friction factor fapp Re is decreased while the Nusselt number Nu is increased. More discussion can be found in [35, 45]. 6.2 Porous layer effects on the transport processes 6.2.1 Transport processes in PEMFCs In this section, the gas flow and heat transport in a porous layer are included for PEMFCs. The geometry of the duct considered is similar to that in [19, 20], i.e., overall channel dimension is 10 cm ×0.20 cm ×0.16 cm (x ×y ×z), gas flow duct is 10 cm×0.12 cm×0.08 cm (x×y×z), while the diffusion layer is 10 cm×0.04 cm× 0.16 cm (x × y × z). The thickness ratio hr (thickness of porous layer hdiff over total height h of the channel) is 20%. The CFD procedure used is modified accordingly and the source term Sm is implemented in the pressure correction equation to adjust the mass balance due to mass transfer. The conditions and parameters considered as the base case are shown in Table 1, along with the references for the various parameters used in the model.

156 Transport Phenomena in Fuel Cells

Figure 4: Mass transfer effects on fapp Re and Nu for developing flow and heat transfer in a rectangular duct (aspect ratio 2:1) at Re = 250, Rem = −1, TBC-II, SOFC. Table 1: Parameters implemented as the base case in the PEMFC study. Cases

Parameter

Anode inlet

Mole fraction of H2 /Water vapor H2 O/Tin ,◦ C Air with Tin ,◦ C Permeability of diffusion layer β, m2 Porosity ε Cell voltage Vcell , V Net water transport coefficient α Current collector thermal conductivity k, W/mK

Cathode inlet Operating condition

Value

Reference

0.53/0.47/80

[20]

80 2 × 10−10 0.7 0.53 0.3 5.7

[20] [20] [19] [20] [19] [20]

The local current density I of the cell is essential for source term calculations. Because the gas flow and heat transfer in the porous layer and flow ducts are of major interest initially, the approach used in [21] is adopted by prescribing a local current density at the porous layer surface close to the catalyst layer (bottom surface in the study). It should be mentioned that this limitation is abandoned in the following section, where the local current density will be calculated. For a similar geometry, the width-average local current density profiles along the duct length in [19] are adapted to account for various operating conditions (co-/counter-flow, humid/dry air, membrane thickness and cell voltage). The current density of the base case in this study refers to the case by Yi and Nguyen [19], while the current densities of cases 1, 2 and 3 correspond to cases 1.0, 1.1 and 1.2 from [19], respectively. The current

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Figure 5: Effects of: (a) thickness and (b) permeability of porous layer on axial velocity profiles in the cathode duct at θk (= keff /kf ) = 1, PEMFC. density of the base case is the largest, and case 3 represents the smallest one while the remaining cases have current densities in between the others. A parameter study with various constant current densities has also been performed. Figure 5(a) presents porous layer thickness effects on the main flow velocity profile (close to the exit) for a fixed value of the permeability (βi = 2.0 × 10−10 ). The heights of the gas flow duct and solid current collector are kept constant, and the thickness ratio hr is approached by varying the porous layer thickness and the total height of the duct. It is obvious that the fluid velocity in the porous layer decreases significantly by increasing the thickness of the porous layer. Figure 5(b) corresponds to the case where the thickness of the porous region equals 20 percent of the duct height and the effect of permeability is shown. Clearly, in the porous layer the fluid flow rate is low, and it becomes significantly reduced when the permeability is low (e.g., βi = 2.0 × 10−11 ). However, the corresponding velocity profiles in the gas flow duct look very similar (see Fig. 5(b)). In Fig. 5(a), it is noticed that the velocity gradient at the interface region between the porous layer and gas flow duct becomes sharper as the thickness ratio of the diffusion layer increases. As a result, a larger fapp Re can be found in Fig. 6(a) with a larger thickness ratio of the porous layer (i.e., 40%). Compared to Fig. 6(a), it is found that the permeability at constant thickness ratio has less effect on fapp Re, which is consistent with the small effect of permeability on the velocity gradient at the interface region, shown in Fig. 6(b). The effects of various current densities on the cross-sectional averaged Nu are shown in Fig. 7(a). As seen here, Nu of the base case is smaller than those of other cases. A large current density (i.e., base case) can contribute to a large net mass injection to the duct, and to a bigger temperature difference between the heated surface and fluid because more mass and thermal energy are inserted into the duct by the mass injection. A parameter study for various overall current densities with

158 Transport Phenomena in Fuel Cells

Figure 6: Effects of: (a) thickness and (b) permeability of the porous layer on fapp Re of cathode duct at θk = 1, PEMFC.

Figure 7: Effects of: (a) current density and (b) permeability on Nu along the main flow direction of the cathode duct at Re = 300, θk = 1.0, hr = 20%, PEMFC. constant values has been performed, and the results are also shown in Fig. 7(a). The conclusion is that a large current density (1.5 A/cm2 in Fig. 7(a) affects the decrease of Nub . Figure 7(b) shows Nu with various permeabilities at a fixed current density (base case) and other porous layer parameters for the cathode duct. It is found that by increasing the permeability (βI = 2.0 × 10−9 ) Nu will increase, otherwise Nu will decrease [45]. As mentioned earlier, there exist some studies in the available literature modifying the Darcy model limitations. For example, the term accounting for friction due to macroscopic shear was included in the model. This model is usually referred

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Table 2: The inertial coefficient B in eqn (18). Model

The inertial coefficient

References

BFD-1 BFD-2

B = εF/(K)0.5 B = 1.75(1 − ε)/(ε3 d )

[29, 40, 41, 42] [43]

Table 3: The Forchheimer coefficient F in BFD-1 model. Model

The Forchheimer coefficient

BFD-1a BFD-1b

F = 1.8/(180ε5 )0.5 F = 0.143ε−1.5

References [29] [40]

Figure 8: (a) fapp Re and (b) Nu of a cathode duct by generalized BD and BFD models. to as the Brinkman-extended Darcy (BD) model. It was also suggested that a term representing the inertial energy of the fluid should be included in the Darcy’s model and this is often referred to as the Forchheimer term [29, 40]. The inertial coefficient B in eqn (18) depends very much on the microstructure or nanostructure of the porous medium, and theoretical determination of it is not easy. As an example, two models from the literature concerning B are given in Table 2. It is clear that model 2 in the table needs more detailed information about the porous medium microstructure or nanostructure in a PEMFC, e.g., particle diameter d , which is not available at the present moment. Therefore, only model 1 is used here. Table 3 shows methods how to determine the Forchheimer coefficient F in model 1. Both methods have been adopted in this study, together with a parametric study. Figures 8(a) and (b) show the variations of the apparent friction factor ratio fapp Re/fd Re (where fd is the friction factor for fully developed flow) and the Nusselt number Nu along the axial direction with various models. It is clear that both

160 Transport Phenomena in Fuel Cells

Figure 9: Effects of the inertial energy on: (a) fapp Re and (b) Nu of a cathode duct predicted by the generalized BFD models, PEMFC. fapp Re/f d Re and Nu rapidly decay due to both the hydrodynamic and thermodynamic boundary layer development. Compared to the BD model, the generalized BFD models predict a larger apparent friction coefficient and a smaller Nu for the PEMFC cathode duct, as seen in Figs 8(a) and (b). This is because the additional inertial force due to the Forchheimer term (B) retards the side-flow through the porous layer thereby allowing the axial velocity and its gradient to increase. A similar effect occurs due to mass injection into the duct. On the other hand, the retarded flow from the porous layer induces more heat into the duct. It is worth noting that the BFD model has significant effects on the decrease of Nu, but has small effects on the increase of fapp Re. It is also clear that both BFD-1a and 1b models provide similar results for Nu, which are close to that at F = 104 in the figure. From Table 3, it can be verified that the Forchheimer coefficient F from both models is in the order of 104 for this specific PEMFC case (ε = 0.7) [46, 47]. Parametric studies of the inertial effects, in terms of the Forchheimer coefficient F, have been conducted for a wide range, covering PEMFC cases. Results are shown in Figs 9(a) and (b). With reference to fapp Re in Fig. 9(a), it is noticed that an increase of F (BFD Models vs BD Model) can increase fapp Re, but all the cases provide similar fapp Re, i.e., the Forchheimer coefficient F has a limited effect on fapp Re. On the other hand for heat transfer, significant changes of Nu are expected by increasing the Forchheimer coefficient F, i.e., the inertial force with a larger Forchheimer coefficient forces more gas from the porous layer to the flow duct. Consequently, more heat induced by the retarded flow can be transferred into the duct from the heated wall, and the convective heat transfer coefficient, Nu, decreases, see Fig. 9(b). 6.2.2 Transport processes in SOFCs For anode-supported SOFCs, the following duct geometries are employed [9]: length of the duct L = 20 mm; width of the porous layer a = 2 mm, and its thickness

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Figure 10: (a) Permeation Reynolds number Rep , (b) fapp Re and Nu along the main flow direction at the base case condition, SOFC. hp = 2 mm; while the width of the flowing duct b = 1 mm, and its height hd = 1 mm. Fuel gas is 0.80 in mass fraction of H2 , and 0.20 of water vapor with an inlet temperature Tin = 750 ◦ C; thermal conductivity ratio kr (= keff /kf ) = 1, effective dynamic viscosity µeff = µf ; porosity ε = 0.5 and permeability βI = 1.7 × 10−10 m2 ; Rein = 100, Rem = 1.0, DH2 , f = 3 × 10−4 m2 /s. Figure 10 shows the permeation Reynolds number Rep , the friction factor fapp Re and the Nusselt number Nu along the main flow direction of an anode-supported SOFC duct. Similar to the wall Reynolds number Rem (for mass transfer across the bottom wall), the permeation Reynolds number is defined as Rep = Vp Dh /ν for gas permeation across the interface. Here, Vp is the velocity caused by the gas permeation across the interface. It is found that Rep has a large negative value (i.e., permeation into the porous anode layer) at the inlet region, see Fig. 10(a). Due to the decreasing pressure gradient along the duct, permeation into the porous layer becomes smaller. On the other hand, H2 O generation caused by the electrochemical reaction at the bottom wall, together with back permeation described above, contributes to a mass injection into the flow duct. This is confirmed by a small but positive (i.e., back flow into the flow duct) Rep shortly downstream the inlet in Fig. 10(a). For a pure flow duct with impermeable walls, fapp Re decays rapidly from the inlet, and levels off to a constant value as the convective gas flow becomes fully developed (see Fig. 10(b)). For the anode duct with a porous layer, the friction factor, fapp Re, is small at the inlet region but increases rapidly at the entrance region and also levels off to a near-constant value shortly downstream along the main flow direction. Similar to a suction flow from a duct, there is certainly a decrease in the friction factor from that of the pure forced convection, as mentioned before. For increasing x∗ , the Rep becomes smaller and its contribution to the decrease of fapp Re is less significant. This contribution will be zero when Rep = 0. Along the flow direction beyond this position, the gas flow is possibly affected by the following

162 Transport Phenomena in Fuel Cells

Figure 11: (a) H2 and (b) H2 O mass concentration distribution along the main flow direction of an SOFC anode duct. mechanisms: secondary flow and back permeation to increase fapp Re, convective flow to decrease fapp Re. It can be clearly observed that fapp Re in Fig. 10(b) is nearly constant downstream from the entrance region. Further downstream in the flow duct, the secondary flow and back permeation balance each other and the effects on the main flow disappear. The Nu for the composite duct has a similar behaviour as that of the pure flow duct. However, a slightly bigger Nu can be observed for the composite duct (see Fig. 10(b)), due to the mass permeation into the porous layer. From the discussion above, it is clear that mass permeation across the interface has more significant effects on the gas flow than on the heat transfer, both in the entrance region and further downstream. H2 and H2 O concentration profiles along the main flow direction are shown in Figs 11(a) and 11(b), respectively. It is found that the H2 concentration decreases, while H2 O increases along the main flow in the porous layer and the flow duct. This is due to the consumption of H2 and generation of H2 O during the electrochemical reaction. Moreover, the gradients of the H2 and H2 O concentrations in the direction normal to the active surface (the bottom surface in Fig. 11) are larger close to the reaction sites compared to the interface areas of both porous layer and flow duct [48]. The performance of the anode-supported SOFC duct is also analysed using the vertical component of the total hydrogen mass flux vector at the active site (bottom surface), which can be expressed as:   ∂mH2 JH2 , y b = ρeff mH2 V + ρeff DH2 ,eff . (56) ∂y b It should be noted that the first term on the right hand side represents the convection transport, while the second term is the contribution due to the diffusion. Figure 12 shows a comparison of the hydrogen mass fluxes by convection, diffusion and the total value for the base condition. It can be seen that convection mass flux has a large negative value (i.e., fuel species transport is to the reaction sites) at the

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Figure 12: Various modes of hydrogen flux at the active surface for the base case condition, SOFC. inlet region. Due to the decreasing pressure difference along the duct, convection becomes weaker downstream. On the other hand, water generation caused by the reaction at the active sites, together with the back permeation mentioned above, contributes to species flowing back to the flow duct. This is confirmed by a small positive value of the convection flux. It is also clear that the hydrogen diffusion flux is small in the entrance region due to a small hydrogen concentration gradient as discussed above, but it increases and reaches a peak value at a certain position of the duct. This is caused by the reaction. The diffusion flux maintains almost a constant value further downstream. By comparing the absolute values of convection and diffusion fluxes, it is found that the convection is stronger in the entrance region; however, the diffusion dominates the species transport downstream. The position for this occurrence is about 1/6 length from the inlet for this specific case. Consequently the total flux from eqn (56) is controlled by convection in the entrance region, and by diffusion for the rest of the duct, as seen in Fig. 12. 6.3 Two-phase flow and its effects on the cell performance As mentioned earlier, the important feature of the two-phase flow model implemented in the work described in this chapter is based on the two-phase mixture approach to account for the phase change and its effects on the gas flow and heat transfer. The approach is to model the liquid water transported by the multicomponent gas mixture in terms of the convection and the diffusion. The amount of water undergoing phase-change is calculated based on the partial pressure of water vapor and the saturation pressure. It is worthwhile to note that two-phase flow and heat transfer are concerned to get local pressure, temperature and species component composition. The model is therefore considered as a non-isothermal

164 Transport Phenomena in Fuel Cells

Figure 13: (a) Velocity vectors and (b) contours of temperature T along the main flow direction of a composite duct, PEMFC. and -isobaric model. As mentioned above, the source term in eqn (21) is for the water phase change. When the partial pressure of water vapor is greater than the saturation pressure, water vapor will condense to liquid water. Consequently, the mass fraction will be reduced in the main gas flow, together with a release of water latent heat, which continues to occur until the partial pressure equals the local saturation pressure. On the other hand, if the partial pressure is lower than the saturation pressure, the saturated water, if any, will evaporate. It should be mentioned that the source term in eqn (22) concerning the water phase-change and the associated heat source term in eqn (20) correspond to the control volumes where two-phase water appears, and these are not treated as the boundary conditions. In this section, the main results of numerical simulations are reported and discussed for a cathode duct of PEMFCs. Figure 13(a) shows the velocity profile along the main flow direction, in which the scale of the vector plots (i.e., 2 m/s) is a reference value of the maximum velocity. As shown in the figure, a parabolic profile is clearly observed in the flow duct. On the other hand, because the species have difficulties penetrating into the porous layer, the velocity in the porous layer is very small except in the region close to the flow duct. Figure 13(b) shows that the temperature increases along the main flow direction. The variation in temperature distribution can also be observed in the vertical direction with a slightly larger value close to the bottom surface. These effects are created by the heat generation due to both the reaction close to the active surface, and the latent heat release by water condensation in the two-phase region, caused by the increase in the water vapor concentration in this area. It is worthwhile to note that the temperature is non-uniformly distributed. By considering the local temperature distribution, the effects on the saturation pressure can be found, which is not the case for the isothermal assumption employed by different authors in literature [49]. Water activities in the duct are shown in Fig. 14. The mass concentration of the water vapor at the entrance is 23%, which corresponds to the saturated case at the base condition (Tin = 70 ◦ C). It can be observed that water vapor is generated at the bottom surface, and is transported back to the flow duct through the porous layer.

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Figure 14: Mass concentration profiles for: (a) water vapor and (b) liquid water along main flow direction of PEMFC cathode duct at the base case.

Figure 15: Liquid saturation profile along the main flow direction in a composite duct at the base case, PEMFC. For this reason, larger mass fractions of water vapor can be found in the porous layer close to the bottom surface. Therefore, the partial pressure of water vapor is greater in the regions mentioned above, and smaller at the interfacial region and the flow duct. Based on the calculated partial pressure and the saturation condition, the liquid water content was predicted and shown in Fig. 14(b). It is found that the liquid water appears in both the flow duct and the porous layer, with the largest mass fraction (around 10%) in the porous layer close to the exit. Because the saturation pressure is proportional to the local temperature, smaller saturated pressures can be expected for the flow duct compared to the porous layer. This is the reason why the liquid water can appear in the flow duct as well, but with smaller mass fractions (less than 5%). A proper gradient of the liquid water concentration should be kept for the liquid water to be driven out of the porous layer. Figure 15 shows the corresponding liquid water saturation level along the main flow direction of the composite duct. As shown in eqn (36), liquid saturation is represented by the volume occupied by the liquid phase divided by the total flow

166 Transport Phenomena in Fuel Cells

Figure 16: Liquid water flux at the active surface for: (a) the main flow direction and (b) the cross-sections in a composite duct for the base case, PEMFC. volume of the duct. It is clear that the liquid saturation s is zero in the single-phase gas flow region. It is so because the species density reduces to the gas phase density, and the water mass fraction equals the mass fraction of the water vapor as well. The liquid saturation s is 1.0 for the case of the single-phase liquid flow. As shown in Fig. 15, the liquid saturation s is zero in the inlet region and increases along the flow direction. It is also true that the liquid saturation s decreases from the active site to the flow duct at the same x due to the liquid water transport, as discussed later in this paper. The liquid water occupies more volume in the porous layer with the largest value of s appearing in the corner close to the active surface at the exit of the duct, while the single-phase gas species occupy most of the flow duct except that small values of s can be observed at the interface region after a certain distance downstream the inlet. As discussed above, the liquid water mass composition and saturation level have highest values at the active surface, consequently larger contribution to the species flow and heat transfer can be expected. In this section, the liquid water flow in the porous layer is evaluated by the liquid water mass flux at the active site, as shown in eqn (56). It should be noted that a positive value represents liquid water flowing from the active site to the porous layer and then the flow duct. Figure 16(a) shows the comparison of the liquid water mass fluxes by convection, diffusion and the total value for the base case. It is found that both convection and diffusion mass fluxes have very small values at the inlet region. Due to the phase change, when the water vapor is generated and the saturation condition is reached, the liquid water can be accumulated in this region. Its mass concentration and gradient along the duct become larger. Bigger values can therefore be observed for both the convection and diffusion downstream a certain distance from the inlet. As shown in Fig. 16(a), the saturated water transport is dominated by diffusion which takes place mainly in the porous layer. In fact, the convective contribution to the total water flux is estimated to be about 15%

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Figure 17: Local current density distribution for: (a) the active surface and (b) the cross-sections in a cathode duct of a PEMFC at the base case. or less. Consequently the total flux from eqn (56) can be said to be controlled by the diffusion for this specific case. Figure 16(b) shows the predicted total values of the cross-sectional liquid water mass flux at various locations. As shown, the mass flux is zero at the inlet region. It is found that the total mass flux has almost uniform values in the cross sections, except a weak liquid water flow for the site below the solid layer (z/h = 0.2−0.4), which has a long transport distance from/to the flow duct [50]. As mentioned previously, the local current density (I ) is an important parameter of the cell, and essential for the source term calculations related to mass injection/ suction and heat generation by the electrochemical reaction. In this section, the local current density is calculated considering the effects of the liquid water saturation, and can be expressed as follows, 

 (1 − s)φO2 O2,trans F (57) exp − Vover , I = Io φO2 ,ref RT b where I is the local current density based on the Tafel equation along the active surface, Io the exchange current density per real catalyst area, φO2 the oxygen species mass concentration, O2,trans the transfer coefficient for the oxygen reaction, Vover the cathode over-potential, φO2,ref the oxygen reference concentration (e.g., P = 1 atm) for the oxygen reaction. Other symbols are given in the nomenclature. The (1 − s) factor in eqn (57) accounts for the effect of liquid water saturation on the surface availability of the reaction site. Figure 17(a) shows a local current density distribution on the active surface (in the x-z plane). It can be seen that the current density is high near the entrance, and then decreases along the main flow direction. This is because the oxygen transfer to the reaction site is larger near the entrance region, which is dominated by the oxygen convection. The reduced current density downstream is due to the small oxygen transport rate controlled by the diffusion. It is also true that the current density is non-uniformly distributed in the cross-section with a smaller

168 Transport Phenomena in Fuel Cells

Figure 18: Comparison of the calculated results with the experimental data for polarization curves (V -I curve) of PEMFC, from [52]. value in the corner region beneath the solid current collector. This is due to a longer transport distance from the flow duct to the active site. With reference to Fig. 17(b), a similar conclusion can be drawn, i.e., the current density is non-uniform along the main flow direction and in the cross-section as well. It should be noted that the liquid water saturation affects this non-uniform distribution as indicated by eqn (57). The simulation results from a 3-D PEMFC model developed in [52] are compared with experimental data available for various parameters. Figure 18 shows a typical V -I polarization curve at operating and humidification temperatures of 70 ◦ C on both the anode and cathode sides. From the figure it is found that the comparison is favorable at low and medium current densities, but this is not true for the case for the high current densities. It is obvious that the simulation produces a higher current density. As claimed in [52], the low current density of the experimental results may be caused by the presence of liquid water content in the catalyst layers and the gas diffusion layers, which the model employed in [52] neglected.

7 Conclusions Based on the similarities of major transport processes in SOFCs and PEMFCs, a unified framework of computational fluid dynamics (CFD) methodology has been developed by implementing specific source terms to account for transport processes and boundary conditions. Numerical investigations and analyses for species flow and heat transfer have been presented for ducts of the type appearing in fuel cells. Models were developed for ducts with the thermal boundary conditions appearing in SOFCs and PEMFCs. Also, different duct configurations were considered, such as rectangular and trapezoidal cross sections, and composite geometries including porous layer, flow duct and solid current inter-collector. It was found that the electrochemical reaction related mass consumption/generation in the electrodes,

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duct configuration (such as the characteristics of the porous layer) have major effects on the gas flow and heat transfer. Furthermore, water management involving two-phase flow in PEMFCs has been studied, and its importance for the cell performance has been characterized.

Acknowledgements Financial support was provided by the Swedish National Programme for Stationary Fuel Cells. The Swedish Energy Agency (STEM) is the financial supporter and Elforsk is handling the programme. The Wenner-Gren Foundation supported the collaboration between Professors Faghri and Sundén.

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Nguyen, T.V. & White, R.E., A water and heat management model for proton exchange membrane fuel cells. J. Electrochem. Soc., 140, pp. 2178–2186, 1993. Singh, D., Lu, D.M. & Djilali, N., A two-dimensional analysis of mass transport in proton exchange membrane fuel cells. Int. J. of Eng. Sci., 37, pp. 431–42, 1999. Okada, T., Xie, G. & Meeg, M., Simulation for water management in membranes for polymer electrolyte fuel cells. Electrochem. Acta, 43, pp. 2141–2155, 1998. Kazim, A., Liu, H.T. & Forges, P., Modelling of performance of PEM fuel cells with conventional and interdigitated flow fields. J. Applied Electrochem., 29, pp. 1409–1416, 1999. Um, S., Wang, C.Y. & Chen, K.S., Computational fluid dynamics modeling of proton exchange membrane fuel cells. J. Electrochem. Soc., 147, pp. 4485– 4493, 2000. Voss, H.H., Wilkinson, D.P., Pickup, P.G., Johnson, M.C. & Basura, V., Anode water removal: a water management and diagnostic technique for solid polymer fuel cells. Electrochimica Acta, 40, pp. 321–328, 1995. Dutta, S., Shimpalee, S. & Zee, J.W.V., Three-dimensional numerical simulation of straight channel PEM fuel cells. J. Appl. Electrochemistry, 30, pp. 135–146, 2000. Shimpalee, S. & Dutta, S., Numerical prediction of temperature distribution in PEM fuel cells. Num. Heat Transfer (Part A), 38, pp. 111–128, 2000. You, L. & Liu, H., A Two-phase and multi-component model for the cathode of PEM fuel cells. ASME IMECE2001/HTD-24273, pp. 1–10, 2001. He, W., Yi, J.S. & Nguyen, T.V., Two-phase flow model of the cathode of PEM fuel cells using interdigitated flow fields. AIChE Journal, 46, pp. 2053– 2064, 2000. Nguyen, T.V., Modeling two-phase flow in the porous electrodes of proton exchange membrane fuel cells using the interdigitated flow fields. Electrochem. Soc. Proc., 99–14, pp. 222–241, 2000. Wang, Z.H., Wang, C.Y. & Chen, K.S., Two-phase flow and transport in the air cathode of proton exchange membrane fuel cells. J. Power Sources, 94, pp. 40–50, 2001. Djilali, N. & Lu, D., Influence of heat transfer on gas and water transport in fuel cells. Int. J. Therm. Sci., 41, pp. 29–40, 2002. Hwang, G.J., Cheng, Y.C. & Ng, M.L., Developing laminar flow and heat transfer in a square duct with one-walled injection and suction. Int. J. Heat Mass Transfer, 36, pp. 2429–2440, 1993. Kinney, R.B., Fully developed frictional and heat-transfer characteristics of laminar flow in porous tubes. Int. J. Heat Mass Transfer, 11, pp. 1393–1401, 1968. Quaile, J.P. & Levy, E.K., Laminar fow in a porous tube with suction. ASME J. Heat Transfer, 97, pp. 66–71, 1975.

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[29] Teng, H. & Zhao, T.S., An extension of darcy’s law to non-stokes flow in porous media. Chem. Eng. Sci., 55, pp. 2727–2735, 2000. [30] Alkam, M.K., Al-Nimr, M.A. & Hamdan, M.O., Enhancing heat transfer in parallel-plate channels by using porous inserts. Int. J. Heat Mass Transfer, 44, pp. 931–938, 2001. [31] Alazmi, B. & Vafai, K., Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer. Int. J. Heat Mass Transfer, 44, pp. 1735–1749, 2001. [32] Wang, J., Gao, Z., Gan, G. & Wu, D., Analytical solution of flow coefficients for a uniformly distributed porous channel. Chem. Eng. Journal, 84, pp. 1–6, 2001. [33] Kee, R.J., Korada, P., Walters, K. & Pavol, M., A generalized model of the flow distribution in channel networks of planar fuel cells. J. Power Sources, 109, pp. 148–159, 2002. [34] Versteeg, H.K. & Malalasekera, W., An Introduction to Computational Fluid Dynamics, the Finite Volume Method, Longman Scientific & Technical: England, 1995. [35] Yuan, J., Computational Analysis of Gas Flow and Heat Transport Phenomena in Ducts Relevant for Fuel Cells, Ph.D. Thesis, ISBN 91-628-55409/ISSN 0282-1990, Lund Institute of Technology, Sweden, 2003. [36] Shah, R.K. & London, A.L., Laminar flow forced convection in ducts (Chapters VII and X). Advances in Heat Transfer, eds T.F. Irvine & J.P. Hartnett, Academic Press: New York, 1978. [37] Comiti, J., Sabiri, N.E. & Montillet, A., Experimental characterization of flow regimes in various porous media-III: limit of darcy’s or creeping flow regime for Newtonian and purely viscos non-Newtonian fluids. Chem. Eng. Sci., 55, pp. 3057–3061, 2000. [38] Natarajan, D. & Nguyen, T.V., A Two-dimensional, two-phase, multicomponent, transient model for the cathode of a proton exchange membrane fuel cell using conventional gas distributors. J. Electrochem. Soc., 148, pp. A1324– 1335, 2001. [39] Warnatz, J., Maas, U. & Dibble, R.W., Combustion-Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation, Springer: Berlin, Germany, 1996. [40] Chikh, S., Bounedien, A. & K. Bouhadef., Non-darcian forced convection analysis in an annulus partially filled with a porous material. Num. Heat Transfer, 28, pp. 707–722, 1995. [41] Marafie, A., & Vafai, K., Analysis of non-Darcian effects on temperature differentials in porous media. Int. J. Heat Mass Transfer, 44, pp. 4401–4411, 2001. [42] Vafai, K. & Kim, S.J., Fluid mechanics of the interface region between a porous medium and a fluid layer – an exact solution. Int. J. Heat Fluid Flow, 11, pp. 254–256, 1990. [43] Poulikakos, D. & Renken, K., Forced convection in a channel filled with porous medium, including the effects of flow inertial, variable porosity, and Brinkman friction. ASME J. Heat Transfer, 109, pp. 880–888, 1987.

172 Transport Phenomena in Fuel Cells [44] Yuan, J., Rokni, M. & Sundén, B., Simulation of fully developed laminar heat and mass transfer in fuel cell ducts with different cross sections. Int. J. Heat Mass Transfer, 44, pp. 4047–4058, 2001. [45] Yuan, J., Rokni, M. & Sundén, B., A numerical investigation of gas flow and heat transfer in proton exchange membrane fuel cells. Num. Heat Transfer, 44, pp. 255–280, 2003. [46] Yuan, J., Rokni, M. & Sundén, B., Three-dimensional computational analysis of gas and heat transport phenomena in ducts relevant for anode-supported solid oxide fuel cells. Int. J. Heat Mass Transfer, 46, pp. 809–821, 2003. [47] Yuan, J. & Sundén, B., A numerical investigation of heat transfer and gas flow in proton exchange membrane fuel cell ducts by a generalized extended darcy model. Int. J. Green Energy, 1, pp. 47–63, 2004. [48] Yuan, J., Rokni, M. & Sundén, B., Gas flow and heat transfer analysis for an anode duct in reduced temperature SOFCs. Fuel Cell Science, Engineering and Technology, eds R.K. Shah & S.G. Kandlikar, pp. 209–216, FUELCELL2003-1721, ASME, 2003. [49] Yuan, J. & Sundén, B., Two-phase flow analysis in a cathode duct of PEFCs. Electrochemica Acta, 50, pp. 677–683, 2004. [50] Yuan, J. & Sundén, B., Numerical simulation of two-phase flow and heat transfer in a composite duct. Proc. of ASME International Mech. Eng. Cong. and R&D Expo, IMECE 2003-42228, Washington, DC, CD-ROM, 2003. [51] Beale, S.B., Some aspects of mass transfer within the passages of fuel cells. Fuel Cell Science, Engineering and Technology, eds R.K. Shah & S.G. Kandlikar, pp. 293–300, FUELCELL2003-1721, ASME, 2003. [52] Wang L., Husar, A., Zhou, T. & Liu, H., A parametric study of PEM fuel cell performance. Int. J. Hydrogen Energy, 28, pp. 1263–1272, 2003.

Nomenclature A a b B BD BFD d cp D Dh Dhr fapp F H hb hd

area, m2 width of porous layer, m width of flow duct, m microscopic inertial coefficient, 1/m Brinkman-extended Darcy model Brinkman-Forchheimer-Darcy model sphere diameter of porous layer, m specific heat, J/(kg K) diffusion coefficient, m2 /s hydraulic diameter, m diameter ratio apparent friction factor Faraday constant (96487 C/mol) or the Forchheimer coefficient enthalpy, J/kg heat transfer coefficient, W/(m2 K) height of the duct, m

Transport Phenomena in Fuel Cells

hp hr hwl I J k kr M MEA Nub O2,trans P P∗ q q Re Rem Rep S s Sdi T TBC-I TBC-II Ui V v Vcell Vm Vp x, y, z x∗ x∗∗

thickness of porous layer, m thickness ratio (hp /hd ) water latent heat, J/kg current density, A/m2 mass flux of species, kg/(m2 s) or kg/(m3 s) thermal conductivity, W/(m K) thermal conductivity ratio (keff/kf ) molecular weight, kg/kmol membrane electrolyte assembly spanwise average Nusselt number transfer coefficient for the oxygen reaction pressure, Pa wetted perimeter, m heat flux, W/(m2 ) heat flow rate per unit length of duct, (W/m) Reynolds number (UDh /ν) wall Reynolds number (Vm Dh /ν) permeation Reynolds number (Vp Dh /ν) source term liquid water saturation source term in momentum equations temperature, ◦ C thermal boundary condition I thermal boundary condition II velocity components in x, y and z directions, respectively, m/s volume of control volume at active site, m3 velocity vector, m/s cell voltage, V mass transfer velocity at bottom wall, m/s permeation velocity across interface, m/s Cartesian coordinates hydrodynamic dimensionless axial distance (x/(Dh Re)) thermal dimensionless axial distance (x∗ /Pr)

Greek symbols α βi ε φ  η µ ν ρ

173

net water transport coefficient permeability of diffusion layer, m2 porosity mass fraction arbitrary variable relative humidity, % dynamic viscosity, kg/(m s) kinematic viscosity, m2 /s density, kg/m3

174 Transport Phenomena in Fuel Cells Subscripts a active av b bulk c e eff f H2 H2 O in m O2 out p s w sat wl wp wv

anode or air at active site average bottom wall bulk fluid condition cathode electrolyte effective parameter fluid or fuel hydrogen water vapor inlet mass transfer oxygen outlet porous layer or permeation solid layer wall saturation water liquid water phase change water vapor

CHAPTER 5 Two-phase transport in porous gas diffusion electrodes S. Litster & N. Djilali Institute for Integrated Energy Systems and Department of Mechanical Engineering, University of Victoria, Canada.

Abstract The accumulation of liquid water in electrodes can severely hinder the performance of PEMFCs. The accumulated water reduces the ability of reactant gas to reach the reaction zone. Current understanding of the phenomena involved is limited by the inaccessibility of PEMFC electrodes to in situ experimental measurements, and numerical models continue to gain acceptance as an essential tool to overcome this limitation. This chapter provides a review of the transport phenomena in the electrodes of PEM fuel cells and of the physical characteristics of such electrodes. The review draws from the polymer electrolyte membrane fuel cell literature as well as relevant literature in a variety of fields. The focus is placed on two-phase flow regimes in porous media, with a discussion of the driving forces and the various flow regimes. Mathematical models ranging in complexity from multi-fluid, to mixture formulation, to porosity correction are summarized. The key parameters of each model are identified and, where possible, quantified, and an assessment of the capabilities, applicability to fuel cell simulations and limitations is provided for each approach. The needs for experimental characterization of porous electrode materials employed in PEMFCs are also highlighted.

1 Introduction 1.1 PEM fuel cells A polymer electrolyte membrane fuel cell (PEMFC) is an electrochemical cell that is fed hydrogen, which is oxidized at the anode, and oxygen that is reduced at the cathode. The protons released during the oxidation of hydrogen are conducted

176 Transport Phenomena in Fuel Cells

Figure 1: Schematic of a proton exchange membrane fuel cell.

through the proton exchange membrane (PEM) to the cathode. Since the membrane is not electronically conductive, the electrons released from the hydrogen gas travel along the electrical detour provided and electrical current is generated. These reactions and pathways are shown schematically in Fig. 1. At the heart of the PEMFC, is the membrane electrode assembly (MEA). The MEA is typically sandwiched by two flow field plates (referred herein as the current collectors) that are often mirrored to make a bipolar plate when cells are stacked in series for greater voltages. The MEA consists of a proton exchange membrane, catalyst layers, and gas diffusion layers (GDL). As shown in Fig. 1, the electrode is considered herein to comprise the components spanning from the surface of the membrane to the gas channel and current collector. A more detailed schematic of an electrode (the cathode) is illustrated in Fig. 2. The electrode provides the framework for the following transport processes: 1. The transport of the reactants and products to and from the catalyst layer, respectively. 2. The conduction of protons between the membrane and catalyst layer. 3. The conduction of electrons between the current collectors and the catalyst layer via the gas diffusion layer. An effective electrode is one that correctly balances each of the transport processes. The transport process of primary interest is the efficient delivery of oxygen to the catalyst layer and adequate expulsion of water from the electrode. The word

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177

Figure 2: Transport of gases, protons, and electrons in a PEM fuel cell cathode.

adequate is used because if water is removed too rapidly from the electrode, fuel cell performance diminishes due to electrolyte dry out. To efficiently conduct protons, the membrane and catalyst layers must maintain high levels of humidification. However, if liquid water is allowed to accumulate in the electrodes, the reactant delivery is reduced and performance is lost. An effective electrode is one that correctly balances each of the transport processes. The transport process of primary interest is the efficient delivery of oxygen to the catalyst layer and adequate expulsion of water from the electrode. The word adequate is used because if water is removed too rapidly from the electrode, fuel cell performance diminishes due to electrolyte dry out. To efficiently conduct protons, the membrane and catalyst layers must maintain high levels of humidification. However, if liquid water is allowed to accumulate in the electrodes, the reactant delivery is reduced and performance is lost. 1.2 Porous media Porous media is, by definition, a multiphase system. The solid portion of porous media is one of the phases. In general, the solid phase of the porous media either is dispersed within a fluid medium or has a fluid network within a continuous solid phase. In the case of a continuous solid phase, the fluid medium occupies pores and the characteristic length is the diameter of the pores. Whereas, if the solid phase is dispersed, as with a bed of sand, the particle size is the characteristic length. Many forms of porous media readily deform with the application of internal and external forces. However, these effects are commonly disregarded because of their complexity. In addition to pore diameter, or particle size, there are two more characteristics of flow paths in porous media: porosity and tortuosity. The porosity is the fraction

178 Transport Phenomena in Fuel Cells of the bulk volume that is accessible by an external fluid. The determination of the porosity either can neglect inaccessible inclusions in the solid or corrected for the inclusions. The tortuosity is the characterizing parameter that arises when fluid in a porous medium cannot travel in a straight path. Instead, the fluid follows through a tortuous path, which is longer than the point-to-point distance. The indirect fluid path reduces diffusive transport because of the increased path length and reduced concentration gradient. Typically, the solid phase is considered inert (with the exception of heat transfer). As well, deformation is typically neglected. This simplifies the modeling by neglecting momentum or mass transfer within the solid phase. This simplification explains why the flow of a single-phase in porous media is generally considered a single-phase system. Multiphase flow in porous media typically refers to a porous solid with more than one phase occupying the open volume. 1.3 Porous media in PEMFC electrodes PEMFC electrodes feature two regions of porous media; the gas diffusion layer and the comparatively more dense catalyst layer. The gas diffusion layer is much thicker and open than the catalyst layer. The catalyst layer features significantly lower void space because of the impregnation of a proton conducting ionomer (typically Nafion). These contrasts can be seen in Fig. 3, which shows a crosssection of an entire MEA. Presently, most catalyst layers are fabricated by applying ink containing Nafion and carbon-supported catalyst to either the membrane or the gas diffusion layer. Because the catalyst layer is so thin (∼15 µm) and applied as a Nafion solution, it can be considered spatially homogeneous. However, the properties of catalyst can be expected to change with further penetration into the GDL. The magnitude of this variation depends on whether the catalyst layer is applied to the GDL or

Figure 3: Scanning electron micrography (SEM) image depicting a carbon cloth gas diffusion layer that features a woven fiber structure. Reprinted from [1], Copyright (2004), with permission from Elsevier.

Two-phase transport in porous gas diffusion electrodes

179

the membrane. Catalyst layers typically feature a porosity (or void fraction) of approximately 5–15%, and pore diameters of roughly 1 µm. Two materials are typically employed as gas diffusion layers in PEM fuel cells; carbon cloth and carbon paper. Both materials are fabricated from carbon fibers. Carbon cloth, visible in Fig. 3, is constructed of woven tows of carbon fibers. Alternatively, carbon paper (see Fig. 4) is formed from randomly laced carbon fibers. Both carbon cloth and carbon paper have approximate pore diameters of 10 µm and porosities ranging between 40–90%. However, carbon cloth is generally available in thicknesses between 350–500 µm, whereas carbon paper is available in thicknesses as low as 90 µm. In addition, the two gas diffusion layer structures vary by spatial uniformity and degree of anisotropy. Carbon cloth, because of its woven structure, is spatially heterogeneous on a macroscopic scale, while carbon paper is spatially homogeneous because of its random lacing. Moreover, the woven nature of carbon cloth results in three degrees of macroscopic anisotropy. This is in contrast to the two degrees in carbon paper. All three forms of porous media in PEM electrodes are summarized in Table 1. Generally, gas diffusion layers are treated with a PTFE (Teflon) solution to increase the hydrophobicity of the medium. This is done to aid water management in the electrode. The hydrophobicity causes water droplets to agglomerate at the free surface of the gas diffusion layer. However, the Nafion in catalyst layers is hydrophilic and will absorb and retain liquid water. Thus, the liquid water produced travels from a saturated catalyst layer to the free surface of the gas diffusion layer. It has been theorized that, within the gas diffusion layer, the condensation will only take place in cracks in the carbon fibers, which are hydrophilic [2]. Thus, it is likely that the PTFE treatment of gas diffusion layers lessens the condensation rate. Condensation typically occurs since the fuel and oxidant gases are generally saturated with water vapour and thus the product water forms as liquid or forms as

Figure 4: Microscope image depicting the random fiber structure of a GDL formed of Toray carbon paper.

180 Transport Phenomena in Fuel Cells Table 1: Summary of porous media in PEM electrodes. Porous media

Spatial uniformity

Catalyst Homogeneous Layer Carbon Heterogeneous Cloth Carbon Homogeneous Paper

Dimensions Pore Thickness of anisotropy Porosity diameter [µm] [µm] Isotropic

5–15%

∼1

∼15

3-D

40–60%

∼10

350–500

2-D

40–90%

∼50

100–300

Figure 5: Environmental scanning electron micrography (ESEM) image depicting condensatin on Toray carbon paper. (a) At time = t. (b) At time = t +
t. Reprinted from [2], Copyright (2003), with permission from Elsevier.

vapour that rapidly condenses. Moreover, the depletion of the hydrogen and oxygen results in the condensation of excess water in the gas streams. All the traits of liquid in a hydrophobic medium are visible in Fig. 5, which shows two images of condensation in PTFE-treated carbon paper. First, the liquid water has formed as droplets instead of a film. As well, it can be seen that over time, with greater levels of liquid water present, the droplets have connected and travelled toward areas of greater liquid accumulation. In addition, the droplets are disperse indicating condensation occurs in randomly oriented cracks in the surface of the carbon fibers. The objective of this review is to present methods of modeling multiphase and multicomponent transport within a porous medium that are applicable to gas diffusion electrodes. Air is a multicomponent mixture that consists of nitrogen, oxygen, and water vapour. To ensure these modeling approaches are as transparent as possible, the review will start with single-phase, single-component transport in porous media. The description will then extend to include multicomponent flow

Two-phase transport in porous gas diffusion electrodes

181

and heat transfer. Subsequently, the review will proceed to illuminate the modeling of multiphase systems.

2 Single-phase transport 2.1 Transport of a single-phase with a single component In the case of a single-phase with a single component in a porous medium under isothermal conditions, there are two equations required to describe the bulk hydrodynamic behavior. These equations are the conservation of mass and the conservation of momentum. The conservation of mass in porous media is expressed as: ∂(ερ) + ∇ · (ρu) = 0. (1) ∂t In this case, u is the superficial velocity that takes into account the facial porosity and is related to the average interstitial velocity (the average velocity in the pores) by: u = ui ε. (2) In an isothermal system, the transport of a single fluid/species in porous media is driven by the pressure gradient∇P. It is universally accepted when modeling porous media as a continuum to use the generalized Darcy’s equation form of the momentum conservation equation: k (3) u = − ∇P, µ where k is the permeability and k/µ is the viscous resistance. 2.1.1 Permeability The permeability of a porous medium represents its ability to conduct fluid flow through its open volume. Permeability can be obtained by applying the conservation of mass and momentum to a pore-scale model. Since most porous media feature complex geometry and are anisotropic, solutions of the permeability have only been obtained for idealized conditions. The fibrous media found in the gas diffusion layer of PEMFC electrodes is very complex and three-dimensional. Two forms of pore-scale modeling are capillary models and drag models. In capillary models, the Navier-Stokes equation is applied for ducts in serial, parallel, and networks. Drag models for approximating permeability are an application of the Navier-Stokes equation to flow over objects. CFD is now being employed at a pore level to determine permeability [3, 4]. However, if pore sizes are small enough that molecular effects require consideration (the flow inside the pore can no longer be considered a continuum), the CFD approach is inaccurate. Instead, computationally intensive methods, such as lattice-Boltzmann models, can be employed. Another approach for evaluating permeability is the semi-heuristic hydraulic radius method, commonly referred to as the Carman-Kozeny theory. The relationship for the permeability is obtained by applying the equation for Hagen-Poiseuille

182 Transport Phenomena in Fuel Cells flow to a pore with an approximated hydraulic diameter. The derivation results in the following equation: k=

2 ε3 dpor

36kk (1 − ε)2

,

(4)

where dpor is the characteristic pore diameter and kk is Kozeny constant. The Kozeny constant is evaluated from a shape factor (2 for circular capillaries) multiplied by the tortuosity factor (roughly 2.5 for a packed bed). Thus, the permeability of a packed bed is often obtained by using a Kozeny constant of 5. An alternative to analytical/semi-analytical methods is to rely on empirical data to determine for the permeability value. An outline for determining permeability experimentally is given by Biloé and Mauran [5]. 2.2 Transport of a single-phase with two components The presentation of single-phase multicomponent transport will begin with the equation for the conservation of a single species in a binary mixture. If there are more than two species that can be considered dilute, their diffusion can be approximated as binary diffusion with the species of the greatest concentration (the background species). With the addition of new species and variation in concentration, the viscosity and density of the mixture change. These changes can be accounted for in a variety of manners. The simplest method is volume averaging. However, Bird et al. [6] offers more theoretical approximations for the mixture viscosity. These values should then be used in the momentum conservation equation (see eqn (3)). With the exception of the dependence of the viscosity and density on the concentrations of the species, the hydrodynamic equations are unchanged. Herein, mass fluxes (nA ), mass concentration (ρA ), and mass fractions (yA ) will be used to describe velocities and distributions in multicomponent systems. These variables can be transformed to their molar counterparts by each species’ molar mass. The mass fraction of A is related to the concentration by yA = ρA /ρ, and the mass flux of A is related to the velocity (uA ) of A by nA = ρA uA . The species conservation equation in porous media for binary or dilute multicomponent mixtures is presented in vector form as: ∂(ερA ) + (∇ · nA ) = SA , ∂t

(5)

where SA is the volumetric source/sink of A. The mass flux of A(nA ) is evaluated as the mass-average flux of A(ρA u) plus the relative mass flux of A( jA ). It should be remembered that u is the superficial velocity expressed in eqn (2). The relative mass flux of A( jA ) is the Fickian diffusion term and is a function of the effective eff ) and the gradient in the concentration diffusivity of A in the second species B(DAB of A(∇ρA ). eff ∇ρA . jA = −DAB

(6)

Two-phase transport in porous gas diffusion electrodes

183

Figure 6: Schematic of driving forces of momentum, energy, and mass transfer. The transport coefficient associated with each phenomenon is listed. Reproduced from [6]. Subsequently, the mass flux of A (nA ) can be evaluated with the expression: eff nA = ρA u − DAB ∇ρA .

(7)

By substituting the mass flux of A in eqns (7) and (5), the mass transport equation for species A is revealed.   ∂(ερA ) eff ∇ρA + SA . + ∇ · (ρA u) = ∇ · DAB ∂t

(8)

However, this is a simplification if the system under consideration is not isothermal. In addition to a mass flux, the gradient in the concentration of A also drives a flux of energy. This is the Dufour effect. Mass flux of A can also be attributed to a temperature gradient. This is the Soret effect. The present review will not investigate these two additional effects, as they are generally negligible in fuel cell applications. A matrix of fluxes and their driving forces are shown in Fig. 6. 2.2.1 Effective diffusivity in porous media In order to account for the geometric constraints of porous media, the open space diffusivity is often corrected with geometric factors. The Bruggemann correction used by Berning and Djilali [7], and many others modeling gas diffusion layers in PEMFCs, modifies the diffusivity for porous regions with a function of the porosity: Deff = ε1.5 D.

(9)

However, in many pieces of literature and fundamental studies [5, 8, 9], a function of both the porosity and the tortuosity factor is adopted. The tortuosity factor is the square of the tortuosity. The tortuosity is the actual path length over the pointto-point path length as shown in Fig. 7. The tortuosity factor often varies between

184 Transport Phenomena in Fuel Cells

Figure 7: Schematic of tortuosity.

2 and 6, and values as high as 10 have been reported [10]. The effective diffusivity is obtained with the following relationship: Deff = τ=

ε D, τ 

Actual path length Point to point path length

(10)

2 .

The ε/τ term is sometimes referred to as the formation factor. The porosity ε is a result of the area available for mass transport. The square of the tortuosity is present because of the extended path length and reduced concentration gradient that are both represented by the tortuosity. The derivation of this result is presented by Epstein [9]. The Bruggemann correction and the second expression are ±5% equivalent in the region of 0.4 < ε < 0.5 with tortuosity equal to a low value of 1.5. Under all other ranges there is a significant difference between the correlations. It should be noted that the Bruggemann correction, which is widely employed and quoted, was obtained from a study on the electrical conductivity of dispersions [10]. The exponent of 1.5 was an empirically determined factor found for a specific case in the De La Rue and Tobias paper [10]. Electrical conductivity measurements are presently one of the only methods of determining the tortuosity. Measurements of the electrical conductivity are taken when a non-conductive porous media is saturated with a conductive fluid. However, this method cannot be applied to gas diffusion layers, which are electrically conductive. Nevertheless, the formation factor is determined by the ratio of the effective conductivity ke for a porous medium saturated with a fluid of known conductivity kf . ε ke = . τ kf

(11)

In order to use the Bruggemann correction, it is more appropriate to replace the exponent of 1.5 with the Bruggemann factor α. The Bruggemann factor can then

Two-phase transport in porous gas diffusion electrodes

185

be presented as a function of the porosity and the tortuosity factor. ε , τ log10 (ε/τ) α= . log10 (ε)

εα =

(12)

In a typical case where the porosity is equal to 0.5 and the tortuosity factor is 3, the value of the Bruggemann factor α is 2.585. This indicates that commonly employed exponent of 1.5 may not be applicable to gas diffusion layers. Finally, a good rule of thumb for the effective diffusivity in typical porous media is a reduction of an order of magnitude. 2.2.2 Determination of the binary diffusion coefficient Binary diffusivities DAB are typically calculated based on an empirically developed formulation presented by Cussler [8]. This is an effective method for determining the diffusivity in a numerical model, and agrees well with published empirical data. The empirical method was implemented by Berning and Djilali [7] and is expressed in Cussler [8] as: √ T 1.75 1/MA + 1/MB   , DAB = (13) 1/3 1/3 P φ +φ B

A

where φ is the diffusion volume and M is the molar mass. Other expressions, such as the Chapman-Enskog theory [8], require the use of tabulated temperature dependant values that complicate the procedure for determining the binary diffusivity. With the knowledge of the diffusivity at given pressure Po and temperature To , the above expression can be further simplified to:  Po T 1.75 . (14) DAB = DAB (To , Po ) P To 2.2.3 Transport of a single-phase with more than two components If considering a system with n components (n > 2) that are not dilute, the evaluation of the diffusion becomes much more complex. With n components, the diffusive flux of each species depends on the concentration gradient of the other n−1 species. This dependence is evident in the Maxwell-Stefan equations for multicomponent diffusion, which are expressed as: ∇ci =

n  xj Ni − xi Nj j=1, j =i

Dijeff

.

(15)

In the above equation, integer subscripts i and j have replaced the letter subscripts A and B as we are no longer considering a binary system. Above, ci is the molar concentration, xi, j is the mole fraction, and Ni, j is the molar flux. However, this is a difficult expression to include in a finite volume CFD code. Berning [11] suggests the use of an equivalent approach that is termed the generalized Fick’s law.

186 Transport Phenomena in Fuel Cells 2.3 Knudsen diffusion With diffusion in porous media, it is acknowledged that the diffusion mechanism varies with the length scale of the porous media. The Knudsen number is the nondimensional parameter commonly employed to characterize the flow and diffusion regimes in micro-channels. The Knudsen number is the ratio of mean free path to pore diameter. When the mean free path is large in comparison to the pore diameter, the probability of molecule-molecule interaction is small and moleculewall collisions dominate. The expression for the Knudsen number is [8]: Kn = =

λ dpor kB T , √ dpor 2πσii2 P

(16)

where λ is the mean free path, dpor is the pore diameter, kB is the Boltzmann constant, σii is the collision diameter, and P is the pressure. Flow in porous media can be categorized into three regimes by the Knudsen number [12]: 1. Continuum Regime, Kn < 0.01; 2. Knudsen Regime, Kn > 1; 3. Knudsen Transition Regime, 0.01 < Kn < 1. In most of the literature [12, 13], the Knudsen regime is defined by Kn > 1. However, Karniadakis [14] states the transition region corresponds to 0.1 < Kn < 10, and the Knudsen regime to Kn > 10. In addition, Karniadakis notes that at Kn > 1 the concept of macroscopic property distribution breaks down. It is also evident in various plots in Chapter 5 of in Karniadakis [14] that there is a significant difference in the flow behavior for 0.1 < Kn < 1, and much less variation in the flow characteristics in the region 1 < Kn < 10. Thus, if the flow is in the upper region of the transition regime, the flow is still dominated by Knudsen diffusion according to Karniadakis. It is therefore reasonable to assume that a strictly Knudsen regime is present when Kn > 1. In the Knudsen Regime, molecule-wall collisions dominate over moleculemolecule collisions. Similar to molecular diffusion, flux in the Knudsen regime is influenced by the gradient of the concentration of a species. The gradient of the partial pressure is the driving force. The partial pressure gradient is equal to that of the concentration for constant pressure conditions. However, a new diffusivity is defined (Knudsen Diffusivity DKn ). Knudsen diffusivity is corrected in the same manner as the molecular diffusivity in porous media. Since viscous and ordinary diffusion is negligible in the Knudsen Regime [12, 13], eqn (7) reduces to: nA = jA eff = −DKn ∇ρA

(17)

Two-phase transport in porous gas diffusion electrodes

187

and the mass transport equation (see eqn (8)) is:   ∂ερA eff ∇ρA + SA . = ∇ · DKn ∂t

(18)

2.4 Determination of Knudsen diffusivity The Knudsen diffusion coefficient is determined from the kinetic theory of gases and is expressed as [5, 8, 15, 16]:  8RT 1 DKn, A = dp . (19) 3 πMA It can be inferred from the previous equation that the Knudsen diffusivity is independent of the other species present in a system. This is because of negligible collisions between molecules. One species cannot “learn” about the presence of other species [13]. It follows that no additional considerations are necessary for systems with more than two components. Quoting from Cunningham [13], “In the Knudsen regime, there are as many individual fluxes present as there are species (as in molecular diffusion), and these fluxes are independent of each other (in contrast to molecular diffusion).” 2.5 Knudsen transition regime The transition regime is present when 0.01 < Kn < 1. In this regime, both molecular diffusion and Knudsen diffusion (slip flow) are present. A common way to evaluate this regime is the Dusty Gas Model (DGM) [12, 13, 15, 17, 18]. The DGM is derived by considering the solid matrix as large stationary spheres suspended in the gas mixture as one of the species present. The formulation is rigorously explained by Cunningham [13] and employed for modeling solid oxide fuel cells by Suwanwarangkul [18]. Often, the effective DGM diffusivity (DDG ) is approximated in the case of equal molar masses [13, 19]. In these cases, the effective DGM diffusivity is calculated by: DDG =

DKn, A DAB . DKn, A + DAB

(20)

This diffusivity can subsequently be corrected for porous media with eqn (10) and then replace the binary diffusion coefficient in the mass transport equation (eqn (8)). 2.5.1 Comparison of the diffusivities The three diffusivities that have been presented (binary molecular diffusivity DAB , Knudsen diffusivity DKn, A , and an effective diffusivity retrieved from the dusty-gas model with equimolar diffusion DDG ) are now compared to a range of Knudsen numbers. This is done to depict the applicability of each diffusivity to the three

188 Transport Phenomena in Fuel Cells

Figure 8: Comparison of the binary molecular diffusivity DAB , Knudsen diffusivity DKn, A , and an effective diffusivity retrieved from the dusty-gas model with equimolar diffusion DDG over a range of Knudsen numbers. Oxygen in nitrogen for a temperature of 350 K and a pressure of 1 atm. diffusion regimes. The Knudsen number has been varied from 0.01 (continuum regime) to 10 (Knudsen regime). The diffusivity presented is that of oxygen in nitrogen at a temperature of 350 K and a pressure of one atmosphere. Figure 8 depicts the three diffusivities. Figure 8 illustrates the significant difference in approximations of the diffusivities depending on the regime. Images of Toray carbon paper show voids with widths between roughly 1 and 100 µm, corresponding to Knudsen numbers between 0.1 and 0.001. A Knudsen number of 0.1 in Fig. 8 is on the boundary of the region where Knudsen diffusion is shown to dominate. This indicates that Knudsen diffusion could be of concern, but is not significant for the carbon paper shown in Fig. 2. Nevertheless, the smaller pore diameters (∼1 µm) in the catalyst layer require the consideration of Knudsen diffusion.

3 Two-phase systems A large variety of applications exist for models encompassing multiphase flow, heat transfer, and multicomponent mass transfer in porous media. These include thermally enhanced oil recovery, subsurface contamination and remedy, capillaryassisted thermal technologies, drying processes, thermal insulation materials, trickle

Two-phase transport in porous gas diffusion electrodes

189

Figure 9: Schematic of the volume fractions.

bed reactors, nuclear reactor safety analysis, high-level radioactive waste repositories, and geothermal energy exploitation [20]. As well, this combination of transport phenomena is present when modeling the flooding of PEM fuel cell electrodes. The phase distribution is potentially the result of viscous, capillary, and gravitational forces. The additional phase can be formed by phase change or is introduced externally into the system. Each phase can also be a multicomponent mixture and the components of each phase can in some cases be transported across phase boundaries. When modeling the diffusion layer of a PEMFC it is generally accepted that the second phase, liquid water, is comprised of single component and there is only transfer of water across the phase boundary. For porous media in which the void space is occupied by two-phases, the bulk porosity ε is divided between the liquid εl and gas εg volume fractions. The liquid saturation sl is the volume occupied by the liquid εl divided by the open pore volume ε. This relationship is depicted in Fig. 9 and eqn (21). εs + εl + εg = 1, εl + εg = ε, εl Sl = . ε

(21)

3.1 Two-phase regimes Two-phase flow exists in three possible regimes; pendular, funicular, and saturated. The regime present at any time and location depends on the saturation. To some degree it also depends on the wettability. The aforementioned regimes are illustrated in Fig. 10. The pendular regime is predominant for low saturations where the liquid phase is discontinuous. The term “pendular” stems from the pendular rings that form around sand grains in this regime. The funicular regime occurs when the liquid is continuous and travels through the pores in a funicular (corkscrew) manner. When the saturation approaches unity, the liquid saturated regime emerges and the pores are fully occupied by the liquid.

190 Transport Phenomena in Fuel Cells

Figure 10: Schematics of two-phase regimes in porous media. (a) Pendular, (b) funicular, (c) saturated. The saturation level at the transition between the funicular and pendular regimes corresponds roughly to what is termed the immobile saturation (also referred to as the irreducible saturation). This saturation level is found when no more water can be removed from a two-phase test sample in a permeation test, often featuring a centrifuge. The final weight of the sample is compared to the dry weight and the immobile saturation sim is determined. This immobility is the result of surface tension. From herein, the saturation s is the reduced saturation, which is the actual liquid saturation sl scaled as follows [21]: s=

sl − sim . 1 − sim

(22)

The immobile saturation can be expected to be quite high for PEM fuel cell electrodes. This stems from the results presented by App and Mohanty [22], who showed the dependence of sim on the capillary number Ca: Ca =

uµ , σ

(23)

where u and µ are the velocity and viscosity of the invading phase and σ is the interfacial tension. The capillary number is the ratio of viscous forces to interfacial tension forces. High capillary numbers arise from a viscous force much greater than the surface tension forces. This could be the result of a high-velocity. One condition in which a small capillary number applies is the case of negligible velocity in either phase. The capillary number is small in PEMFC electrodes because mass transport is dominated by diffusion and the velocity term u is quite small. App and Mohanty stated that in porous cores the immobile saturation for their low capillary number cases was sim = 0.18, whereas for large capillary numbers the immobile saturation approached zero. In addition, Kaviany [23] stated that the immobile saturation increases as the pore size is reduced. The dependence of the immobile saturation on the capillary number can be explained by the deformation of droplets under the viscous stress of a high velocity invading phase. The viscous forces elongate the droplets that eventually bridge and form a continuous phase. At this moment the phase regime of the displaced phase transforms from a pendular regime to a funicular regime and capillary flow

Two-phase transport in porous gas diffusion electrodes

191

Figure 11: Schematic of two processes responsible for surpassing the immobile saturation: Increasing the capillary number through higher velocities and increasing the saturation of the liquid. is initiated. Figure 11 illustrates how an increased capillary number, due to greater velocities, allows the saturation to surpass the immobile saturation. In addition, the same figure depicts how increasing liquid saturation causes the transformation from a pendular regime to a funicular regime (where the liquid is capable of motion) when the saturation surpasses the immobile saturation. 3.2 Hydrodynamics and capillarity in two-phase systems The relationships developed for the hydrodynamics of a single-phase in porous media will now be applied to each phase in the two-phase system (gas and liquid). The conservation equations for the gas and liquid phases are:  ∂(1 − sl )ερg + ∇ · ρg ug = S˙ g , ∂t ∂sl ερl + ∇ · (ρl ul ) = S˙ l , ∂t

(24)

where (1 − sl )ε and sl ε in the first terms in the equations represents the volume of each phase. Respectively, S˙ g and S˙ l are the volumetric sources of gas and liquid. These sources, or sinks, can arise from phase change, in which case S˙ g = −S˙ l , or from an external source. The single-phase momentum equation (eqn (3)) is adapted to the two-phase system in a similar fashion. The only notable difference between the single and two-phase cases is that the permeability is phase specific for the gas (kg ) and liquid (kl ). These permeabilities are a correction of the bulk permeability (k) for the effect of the reduced area open to each phase due to the presence of the other phase. kg ∇Pg , µg kl ul = − ∇Pl . µl

ug = −

(25)

192 Transport Phenomena in Fuel Cells

Figure 12: Hydrostatic representation of capillary pressure when the liquid is the non-wetting phase. It is evident that the system is not fully defined with the general conservation of mass and momentum equations. The four equations above cannot evaluate the five variables (sl , ug , ul , Pg , and Pl ) that must be solved. This is because a prominent phenomenon in multiphase flow in porous media has not been introduced into the equation set. Expressions for the capillary pressure are employed as the constitutive relationship that completes the system of equations. Capillarity and capillary pressure are the result of interfacial tension, which is the surface free energy between two immiscible phases. The microscopic capillary pressure is directly proportional to the interfacial tension and inversely proportional to the radius curvature of the interface. Thus, the lesser the radius of curvature, the more dominant the effects of capillary pressure. This is the microscopic definition of the capillary pressure, which is typically formulated as: Pc ∝

σ , r

(26)

where r is the characteristic radius of the liquid/gas interface. The macroscopic definition of the capillary pressure Pc , the pressure difference between the wetting gas and non-wetting liquid pressures, is included in the two-phase momentum equations (eqn (25)). This is shown in hydrostatic form in Fig. 12. Figure 13 is an attempt to use the microscopic and macroscopic definitions of the capillary pressure to explain capillary motion in a pore. At the end of the pore where the liquid radius is smaller (lower local saturation), the capillary pressure is greater than at the end with the larger liquid radius (greater local saturation). Because the liquid pressure is the sum of the capillary pressure and the gas pressure, the hydrodynamic pressure of the liquid is greater at the end of the pore with the smaller radius. Therefore, the bulk motion of the liquid is toward the end with the greater radius (and local saturation). Pc = Pl − Pg .

(27)

At this point in the discussion, the transport of the liquid water in the electrodes of PEMFCs should be revisited. The transport of liquid water from low to high saturation, as shown in Fig. 13, is counter-intuitive and could lead to

Two-phase transport in porous gas diffusion electrodes

193

Figure 13: Schematic of capillary diffusion, in which the liquid is the non-wetting phase. incorrect conclusions. In a broad sense, the transport depicted in Fig. 13 illustrates the penchant for water, in hydrophobic media, to move to ever-increasing pore diameters according to the capillary pressure’s inverse proportionality to the liquid radius. Ultimately, the largest radius can be attained when the liquid reaches the gas channel. Conversely, the water invades smaller pores in the catalyst layer due to the hydrophilic nature of the electrolyte. However, the electrolyte phase in the catalyst layer offers a second mode of water transport. Water can be transported through the catalyst layer’s electrolyte in a similar fashion to that of the electrolyte membrane. Though, a review of these transport issues is beyond the scope of the present chapter and shall be reserved for a separate discussion of transport phenomena in polymer electrolyte membranes. Subsequent to the definition of the macroscopic capillary pressure, the momentum equations take the form: ug = −

kg ∇Pg , µg

kl kl ul = − ∇Pg − ∇Pc . µl µl

(28)

When interfacial tension is observed as driving mass transport due to a gradient in the capillary pressure, the phenomena is referred to as capillary diffusion. The mass flux of the liquid phase due to capillary diffusion can be obtained from the second term of the liquid momentum equation (eqn 28): m ˙ l, s = −

ρkl ∇Pc . µl

(29)

194 Transport Phenomena in Fuel Cells If the gradient of the capillary pressure is assumed to rely only on the saturation gradient, the liquid transport due to capillary diffusion emerges as:  ρkl dPc ∇sl m ˙ l, s = − (30) µl dsl and thus the capillary diffusivity is often defined [20]:  kl dPc . D(sl ) = µl dsl

(31)

Now the momentum equations for the two-phase system (eqn (28)) can be reformulated to eliminate the liquid pressure field from the equation set. Inserting the definition of capillary diffusivity into the liquid momentum equation yields a function of the gas pressure and the liquid saturation. ul = −

kl ∇Pg − D(sl )∇sl . µl

(32)

It is important to note that the interfacial tension, which capillary diffusion is a result of, is not constant in a non-isothermal and multicomponent system [20]: σ = σ(T , cA ).

(33)

Therefore, diffusion due to interfacial forces can be driven by temperature and concentration gradients in addition to saturation. These two transport mechanisms are termed thermal- and solutal-capillary diffusion:  ρkl ∂Pc ∂σ ∇T , m ˙ l, T = µl ∂σ ∂T (34)  ρkl ∂Pc ∂σ ∇cA . m ˙ l, cA = µl ∂σ ∂cA As with capillary diffusion, the thermal- and solutal-capillary diffusivities can be derived. However, these terms are often not included because of their negligible contributions to the total mass flux. Another characteristic of two-phase flow in porous media, which needs to be introduced before proceeding, is the surface tension between the liquid phase and solid matrix. The surface tension is dependant on the wettability, or the hydrophobicity, of the liquid/solid interface. Figure 14 depicts the effect of hydrophobicity on the contact angle θ. Hydrophobic interfaces feature a contact angle greater than 90◦ . Teflon (PTFE) features a contact angle of 108◦ . For hydrophobic solids, the gas is the wetting phase. Hydrophilic interfaces feature a contact angle less than 90◦ . In this case, the liquid is the wetting phase. The contact angle θ can be calculated from the gas-liquid σ, gas-solid σgs , and liquid-solid σls interfacial tensions: cos (θ) =

σgs − σls . σ

(35)

Two-phase transport in porous gas diffusion electrodes

195

Figure 14: Contact angle for hydrophobic and hydrophilic fluid/solid interfaces.

Figure 15: General form of the capillary pressure curve. There have been corrections developed for the contact angle in porous media as an alternatively to using the contact angle on a flat plate. It has been found that the effective contact angle porous media is less than the actual [24]. This indicates that the porous form of a material is less hydrophobic than the bulk form. An important note on surface tension is that it is known that transport of liquid water increases with the contact angle (more hydrophobic). This trend could be due to the reduced contact area between the porous media and the liquid (see Fig. 14), which increases the influence of the viscous forces exerted by the invading phase. This increased liquid transport is a reason for the impregnation of electrode diffusion layer with PTFE. 3.2.1 Capillary pressure curves In order to use the capillary pressure as a constitutive relationship, an expression for the capillary pressure is derived. It would be too difficult to determine the capillary pressure microscopically as in Fig. 13. Thus, a volume averaging approximation is utilized. The starting point for derivation this constitutive relationship is the form of capillary pressure curves obtained in experiment. In these experiments, the capillary pressure is measured in a sample and compared to the estimated level of saturation [24]. The form of curves generated is shown in Fig. 15.

196 Transport Phenomena in Fuel Cells It was postulated by Leverett [25] that the capillary pressure versus saturation relationship could be presented in the non-dimensional form: Pc J (s) = σ

 1/2 k . ε

(36)

This function is typically referred to as the Leverett J-function. Occasionally, a cosine of the contact angle θ is included in the Leverett J-function [2]: Pc J (s) = σ cos (θ)

 1/2 k . ε

(37)

However, Anderson [24] stated that this is not valid for modeling the effects of wettability on capillary pressure. Udell [21] later used data presented by Leverett [25] to determine the J-function for the porous media in Leverett’s experiments. Udell then compared experimental results with a one-dimensional steady-state model for packed sand. The porosity was varied from 0.33–0.39 and permeabilities from 1.39–10.3×10−12 m2 . Good agreement between the model and the experiment was presented. The J-function Udell obtained is often referred to as the Udell function, which is expressed as: J (s) = 1.417(1 − s) − 2.120(1 − s)2 + 1.263(1 − s)3 .

(38)

There are also many other J-functions [23], including Scheidegger’s [26]: J (s) = 0.364(1 − e−40(1−sl ) ) + 0.221(1 − sl ) +

0.005 . sl − 0.08

(39)

The above J-function (eqn (39)) is then employed to calculate the capillary pressure with the formula: σ cos (θ) Pc = J (s). (40) (k/ε)1/2 As well, van Genuchten [27] presented a relation for the capillary pressure, which is a function of saturation and requires empirical constants: Pc =

1m s −1 α

m = 1 − 1/n.

1/n

, (41)

Kaviany [23] stated that the capillary pressure curves must be bounded by the relationship: dPc lim = −∞. (42) sl →sim ds Observing the dPc /ds derivative in eqn (30), it could be inferred that Kaviany’s boundary (eqn (42)) would predict a very large mass flux at sl ≈ sim . However, this is also the point at which the liquid is in transition between the pendular and

Two-phase transport in porous gas diffusion electrodes

197

Figure 16: Capillary pressure curves from various relations and experimental results. For Udell and Scheidegger’s: k ≈ 10−11 m2 , ε = 0.35, and σ = 0.0644 N/m (air/water). For Anderson’s plot: air and water in an interfacial Teflon core. For van Genuchten’s: n = 2 and σ ≈ 7 × 10−21 . funicular regimes. Thus, the high capillary pressure is due to the immobility of the discontinuous phase. Therefore, the mass flux is limited. Figure 16 presents these various capillary pressure relationships as a function of saturation. Experimental results presented by Anderson [24] are also plotted. It is evident that the experimental results abide by Kaviany’s boundary, whereas the capillary pressure expressions do not. Another flaw of these expressions is that the result is the same whether imbibition (increasing sl ) or drainage (decreasing sl ) is being considered (when it has been clearly shown that there is a significant difference in reality [23, 24]. Many other expressions for the capillary pressure exist. However, they are not presented herein. As with Leverett and Udell’s work, these relations are for packed sand and other representations of soil. There is a significant lack of material evaluating capillary pressure in fibrous porous media. 3.3 Relative permeability When two or more phases occupy the same pores, the amount of pore space available for each phase is reduced. Referring to eqn (4) it is acknowledged, at least theoretically, that the permeability is a function of the porosity. Therefore, the permeability must be adjusted for the volume fractions occupied by different phases. The permeabilities of the gas and the liquid are now expressed as: kg = krg k, kl = krl k,

(43)

198 Transport Phenomena in Fuel Cells

Figure 17: General form of the relative permeability functions. where k is the single-phase permeability. Figure 17 depicts the general form of relative permeability functions used for the gas and liquid phases. In addition, the immobile saturation for the phases is mathematically accounted for in the relative permeability. The relative permeabilities are commonly expressed as: krg = (1 − s)3 ,

(44)

krl = s ,

(45)

3

where s is the reduced saturation (see eqn (22)). Some explanation for the cubic form of these equations can be found in the Carman-Kozeny equation (see eqn (4)), where the single-phase permeability is roughly proportional to ε3/(1 − ε)2 . Recalling that the more dominant the interfacial tension the higher immobile saturation, it is evident that interfacial tension and the wettability/hydrophobicity has an effect on the relative permeability. The effect of wettability on the relative permeabilities was surveyed by Anderson [28]. Figure 18 presents the relative permeability of gases for a large range of contact angles. The case of θ = 108◦ is that of nitrogen displacing water in a Teflonized core. This is a good approximation of the PEMFC electrode. Equation (44) is plotted to evaluate its validity. Firstly, it is evident that the gas permeability presented by Anderson is apparently bimodal for all degrees of wettability. However, the often prescribed cubic function is monotonic. It is also evident in Fig. 18 that eqn (44) would underestimate the relative permeability at low saturations and would predict values significantly higher than the experimental results show in the high saturation regions. Figure 19 depicts the effect of wettability on the displacing phase (water) permeability as presented by Anderson [28]. The plots indicate that the immobile saturation resides between 0.16 and 0.20. The plot also depicts a uniform effect of wettability on the relative permeability. It is clear in the plot that a hydrophobic porous structure aids the transport of water by increasing the permeability of the structure. Again, the cubic relative permeability function (eqn (45)) is plotted for

Two-phase transport in porous gas diffusion electrodes

199

Figure 18: Relative permeability of displaced phase (gas) for various degrees of hydrophobicity. θ = 108◦ corresponds to nitrogen displaced by liquid water in an artificial Teflon core (Anderson, 1987b).

Figure 19: Relative permeability of displaced phase (liquid) for various degrees of hydrophobicity. θ = 108◦ corresponds to nitrogen displaced by liquid water in an artificial Teflon core (Anderson, 1987b).

200 Transport Phenomena in Fuel Cells comparison. It can be seen that the cubic function follows the liquid curves to a higher degree than the gas curves. It is evident that eqn (45) approximately predicts the relative permeability for interfaces featuring a contact angle of 100◦ , which is slightly hydrophobic. It is noted that the relative permeability is seen to increase rapidly at higher saturations, allowing effective water transport. However, the gas phase is shown in Fig. 18 to reach its immobile saturation at a liquid saturation of 0.6 for the hydrophobic cases. At this point, the permeability of the gas reduces to zero.

4 Multiphase flow models With a clear understanding of the transport phenomena in porous media with nonisothermal multiphase flows, the models prescribed in literature can be evaluated for use in PEM fuel cell electrodes. Figure 20 classifies the models by their features. The distinguishing features include the accounting of liquid water, convection of the liquid water by the gas, transport of liquid water due to surface tension effects (capillary diffusion), or whether the liquid is considered to be stationary. The characterization of the various multiphase models that can be applied to the gas diffusion layer in a PEM will start with most generalized case (multi-fluid) and work toward the most specific version (porosity correction). 4.1 Multi-fluid model The multi-fluid model presented herein is the application of the equations developed previously to a porous medium occupied by air and liquid water. The air is treated as

Figure 20: Classification of PEMFC electrode models.

Two-phase transport in porous gas diffusion electrodes

201

a multicomponent mixture and the liquid phase is considered as immiscible water. In the multi-fluid model, as employed for fuel cells by Berning and Djilali [7], each phase is modelled with its own set of field equations. The two-phases are coupled by the relative permeabilities, which are sensitive to saturation, and phase change terms. The steady state mass and momentum conservation equations governing this model are: ∇ · (ρg ug ) = m ˙ p c, ˙ pc ∇ · (ρl ul ) = m

(46)

and, ug = −

kg ∇Pg , µg

kl ul = − ∇Pg − D(sl )∇s. µl

(47)

The species conservation equation in Berning and Djilali [7] was applied only to the gas phase and is the single-phase species conservation equation (eqn (8)) with modification of the liquid saturation to account for the reduced volume fraction open to the gas. The steady state conservation of species A in the gas phase can be written as:   (48) ∇ · (ug ρA ) = ∇ · DAeff ∇ρA + S˙ A , where DAeff is the diffusivity of species A and S˙ A is the phase change source term, which is zero except in the water vapour conservation equation. 4.1.1 Phase change To complete the multi-fluid model, an expression for the rate of phase change between the gas and liquid phases must be introduced. The phase change between the gas and liquid can be either evaporation or condensation, depending on the local properties. However, the knowledge of molecular dynamics during phase change is still considered limited [29]. A starting point for the exploration of phase change is the kinetic theory of gases. The main principle when modeling phase change with kinetic theory is that there is a maximum amount of vapour that can be accommodated at vapour/liquid interface. This maximum accommodation is the mass transfer limiting characteristic. Thus, through the application of the kinetic theory of gases, the maximum rate of evaporation for a liquid can be determined. The goal of recent research in kinetic phase change is the approximation of evaporation and condensation coefficients. The coefficients are ratios of actual mass transfer to the theoretical maximum rate. Eames et al. [29] and Marek and Straub [30] offer reviews of previously obtained values for a variety of circumstances. The kinetic theory approximation allows for the consideration of thermal equilibrium, or differences in the temperature between the gas and liquid phases.

202 Transport Phenomena in Fuel Cells The expression for the rate of mass transfer per unit area of gas/liquid interface (m ˙ H2 O ), when the gas and liquid are in thermal equilibrium, was presented by Eames et al. [29] as:  m ˙ H2 O = γk

MH2 O 2πR

1/2 (Ps (T ) − Pv ),

(49)

where Ps (T ) is the water vapour saturation pressure and Pv is the partial pressure of the water vapour in bulk gas stream. γk is the kinetic evaporation coefficient, which is equivalent to the condensation coefficient under thermal equilibrium conditions. The magnitude of evaporation coefficients measured in experiments can range from 0.001 to 1 [30]. A second approach assumes the liquid phase exists in the form of a spherical droplet. In such a case, the mass transfer is determined from the diffusion rate between the bulk gas and the surface of the droplet. In addition, a mass transfer Nusselt number is employed as a dimensionless measure of a droplet’s ability to exchange mass. In its present form, this method applies only to systems featuring local thermal equilibrium. The Nusselt number for mass transfer (Num ) from a liquid droplet to the surrounding gas is [6]: Num = 2.0 + 0.6Re1/2 Sc1/3 ,

(50)

where Sc is the Schmidt number (µg /(ρg DH2 Og )) and Re is the Reynolds number (ρg Dd urel /µg ). Dd is the droplet diameter and urel is the relative velocity between the gas and the droplet. The droplet’s mass transfer coefficient (γd ) can be obtained by multiplying the Nusselt number by the diffusivity of water vapour in the gas (Dvg ) and the inverse of the droplet diameter (1/Dd ). γd = Dvg Num /Dd .

(51)

The mass transfer coefficient is subsequently multiplied by the difference between the density of water vapour in water-saturated air (ρs ) and the density of water vapor in the bulk gas (ρ v ) to approximate the mass transfer per unit area of gas/liquid interface. See Fig. 21 for clarification of these two densities. m ˙ H2 O = γd (ρs − ρv ).

(52)

This method of determining mass transfer rate was implemented in the modeling of a PEM fuel cell by Berning [7, 11]. A simplification required to use this model was to use a mean droplet diameter rather than the actual. Another appropriate assumption is that since the velocities in the gas diffusion layer are small enough, the Nusselt number reduces to 2.0 [6]. Therefore, eqn (52) can be expressed as: m ˙ H2 O =

2Dvg (ρs − ρv ). Dd

(53)

Two-phase transport in porous gas diffusion electrodes

203

Figure 21: Schematic of droplet evaporation for thermal equilibrium. Berning and Djilali [7] also included a correction factor (ω) for reduced phase change rates in porous media: m ˙ H2 O =

2ωDvg (ρs − ρv ). Dd

(54)

When implementing either of the two aforementioned methods of calculating the rate of phase change, the total interfacial area between the gas and liquid (Agl ) phase ˙ H2 O ). This is typically must be determined to calculate the total mass transfer (M achieved with the area of a spherical droplet of a mean diameter (πDd2 ). The number of droplets (nd ) in a representative volume (V ) is the volume of liquid (εsl V ) divided by the volume of a single droplet ( 16 πDd3 ). nd =

6εsl V

.

(55)

Agl =

6εsl V Dd

(56)

πDd3

Thus, the interfacial area is:

and the total mass transfer in volume V is: ˙ H2 O = Agl m ˙ H2 O . M

(57)

4.1.2 Application The multi-fluid model is the most general and, conceptually, the most flexible as it relies on less restrictive assumptions. An example of the application of this model to fuel cells is shown in Figs 22–24 [11]. Figure 22 shows iso-contours of oxygen and water vapour concentrations. In these plots, as in subsequent figures, the top and bottom of the vertical axis correspond to the catalyst layer/ GDL and channel/GDL interfaces, respectively, and the planes correspond to successive locations from the inlet to the outlet of the fuel

204 Transport Phenomena in Fuel Cells

Figure 22: Molar oxygen concentration (left) and water vapour distribution (right) inside the cathodic gas diffusion layer at a current density of 0.8A/cm2 [11].

Figure 23: Rate of phase change [kg/(m3 s)] (left) and liquid water saturation [-] (right) inside the cathodic gas diffusion layer at a current density of 0.8A/cm2 [11].

Figure 24: Velocity vectors of the gas phase (left) and the liquid phase (right) inside the cathodic gas diffusion layer at a current density of 0.8A/cm2 [11].

cell section. The molar oxygen concentration contours are similar to those found with a single-phase version of the model, with more pronounced oxygen depletion under the land areas of the collector plate. However, because of phase change, the concentration of water is relatively uniform. The 0.8% difference can be attributed to the temperature field’s influence on the saturation pressure of water.

Two-phase transport in porous gas diffusion electrodes

205

The left-hand side of Fig. 23 presents the phase change in the cathode’s GDL. Positive values correspond to evaporation, which can be found under the land areas due to the increased pressure drop. The pressure drop, an artifact of the increased resistance to gas transport below the land area, reduces the vapour pressure below the saturation point. Negative values, indicating condensation, are most prevalent at the catalyst layer/GDL interface because of the oxygen consumption and the production of water vapour. On the right-hand side of Fig. 23 the distribution of liquid water saturation in the cathode’s GDL is shown. For this current density (0.8 A/cm2 ), a maximum saturation of 10% is obtained under the land area at the end of the channel. The gradient of the saturation is from high levels at the catalyst layer to low levels at the channel interface. This reflects the implementation of the well-posed hydrophilic formulation of the capillary transport, in which water travels from high to low saturation. The velocity vectors of the gas and liquid phases are presented in Fig. 24. The gas phase, shown on the left, indicates the bulk transport of gas to the catalyst layer. In a single-phase model, the bulk motion of the gas is in the opposite direction due to the removal of product water vapour. However, when phase change and capillary transport are accounted for, the removal of product water is, in the most part, by the liquid phase. This is evident in the plots of the liquid phase velocity vectors on the right-hand side of Fig. 24. The broader applicability of the multi-fluid model comes at the cost of solving for an additional set of field equations and the required coupling of the phases. This makes the numerical solution much more challenging and computationally intensive. In particular, the convergence rates and the numerical stability of the model can be problematic under some operating conditions. Alternative models based on various levels of simplifications are presented below. Though they are less general, these models can be more practical and can be effective in simulating transport in the GDL, provided they are used for appropriate regimes. 4.2 Mixture model The mixture model has been used to model two-phase flow in PEM electrodes by several researchers, including Wang et al. [31] and You and Liu [32]. The main theme of the mixture model is the description water transport, as vapour and liquid, with traditional mixture theory practices. The resulting equation set is mathematically equivalent to the multi-fluid model [20]. The reformulation is obtained by utilizing phase quantities that are relative to that of the mixture. The set equations employed in the mixture model for steady state conditions are as follows: The mixture conservation equation is: ∇ · (ρu) = 0.

(58)

The mixture momentum equation is: u=−

K ∇P, ρν

(59)

206 Transport Phenomena in Fuel Cells where ν is kinetic viscosity of the mixture. The mixture species conservation equation is expressed as:    
∇ · (γA ρuyA ) = −∇ · (ερD∇yA ) + · · · ∇ · ε ρk sk DAk (∇yAk − ∇yA )   −∇ · 





k=g, l

yAk j k  ,

(60)

k=g, l

where the advection correction factors γA in eqn (60) account for the specific velocity fields encountered by each species. The advection correction factor is formulated as:  g ρ λl yAl + λg yA (61) γA = g, ρl sl yAl + ρg sg yA where the λk terms are the relative mobilities for the gas and liquid phases. They are expressed as: λg =

krg /νg , krg /νg + krl /νl

krl /νl λl = . krg /νg + krl /νl

(62)

The mixture quantities are evaluated as: ρ = ρl sl + ρg sg , ρu = ρl ul + ρg ug , g

ρyA = ρl sl yAl + ρg sg yA , g

ρDA = ρl sl DAl + ρg sg DA ,

(63)

ρh = ρl sl hl + ρg sg hg , ν=

1 . (krl /νl ) + (krg /νg )

The individual phase velocities are extracted from the solution in the post processing stage. These velocities are found with the addition of relative velocities to the mixture velocity (ρg, l ug, l = λg, l ρu + jg, l ). Accounting only for capillary diffusion, the relative mass flux term j k in eqn (60) is expressed as:  λl λg dPc ∇sl , jl = k ν dsl (64) jg = −j l .

Two-phase transport in porous gas diffusion electrodes

207

The liquid saturation is calculated and updated with each iteration. It is determined by comparing the total water concentration with the concentration of water vapour required to saturate the gas phase. If the concentration of water is greater than saturation concentration, then liquid water must be present. Using the expression for the total density of water (ρH2 O ) as a function of liquid water saturation (sl ), and the saturated gas and liquid concentration of water (ρs and ρl respectively), ρH2 O = sl ρl + (1 − sl )ρv ,

(65)

the saturation can be determined. It is important to note that ρs is the mass of water vapour per unit volume of water vapour saturated air (the mass fraction of the water vapour in the air multiplied by the bulk density of water vapour). ρs can be extracted from the temperature dependent saturation pressure of air (Ps (T )), the absolute gas pressure (P), and the bulk density of water vapour (ρv ): ρs =

Ps (T ) ρv . P

(66)

Considering a pure liquid phase, eqn (65) can be rearranged as: sl =

ρH2 O − ρs . ρl − ρs

(67)

Thus, the saturation can be calculated from the total water concentration, temperature, and pressure. 4.3 Moisture diffusion model The moisture diffusion model, also referred to as unsaturated flow theory [20], was developed to determine the transport of liquid water when the only driving force is capillarity. Luikov [33] and Whitaker [34] are considered pioneers of this formulation. This method was applied to PEM fuel cells by Natarajan and Nguyen [35]. The steady state transport equation for liquid water in the moisture diffusion model can be expressed as (Wang and Cheng, 1997):  (68) ∇ · D(sl )∇sl + S˙ l = 0, where ρl kl ∂pc , µl ∂sl S˙ l = −S˙ g, H2 O

D(sl ) =

(69) (70)

and S˙ is the mass source due to phase change. The moisture diffusion model could be incorporated into a CFD code by treating the liquid phase as a scalar species with no convection terms. This would be a moderately easy method of incorporating two-phase flow into a single-phase fuel cell model. The mass sources need to be calculated in the form of a rate as in the multi-fluid model. In addition, the porosity would require a correction based on the liquid saturation.

208 Transport Phenomena in Fuel Cells 4.4 Porosity correction model The porosity correction model simplifies the present two-phase problem by neglecting the transport of liquid water. The saturation level is computed with each iteration. In the Kermani et al. [36] model, the temperature and the level of saturation are calculated iteratively by the internal energy and density of the water in the system. Subsequently, the volume fraction open to the gas phase is reformulated as: εg = ε(1 − s).

(71)

This model is particularly efficient when saturation levels are low (below the immobile saturation limit for the liquid water). This model would be the simplest to append to an existing single-phase fuel cell model. 4.5 Evaluation of the multiphase models in the literature Table 2 is provided to help determine which model should be employed given the conditions of the system, the porous media considered, and the computational resources available. At one extreme is the multi-fluid model. The multi-fluid model is a strong candidate when an abundance of computational resources is available and stable phase coupling can be achieved. At the other extreme is the porosity correction model. This model is an ideal candidate, due to its computational efficiency when considering saturation levels below the immobile value.

5 Outstanding issues and conclusions A number of fundamental issues need to be addressed in order to devise reliable predictive tools for two-phase transport in gas diffusion electrodes. These include: •

•

The hydrodynamic and diffusive properties of the porous media in the electrodes need to be characterized. The structure of PEM fuel cell gas diffusion layers is typically fibrous. In the case of carbon cloth layers, the porous matrix is constructed from woven tows of fibers producing macro- and micro-pores. Alternatively, carbon paper layers are a formation of randomly laced fibers. It is obvious that the architecture of gas diffusion layers is significantly different to that of packed beds, or cylindrical pores in a monolithic structure that are often the object of porous media studies. It is also clear that these fibrous layers are anisotropic. Some parameters to be resolved are the three-dimensional tensors for the area porosity, permeability, and tortuosity. At present, the properties can only be implemented in an isotropic form in the available commercial CFD codes. The capillarity in the electrode’s porous media requires significant research. Current expressions used to determine capillary pressure in electrodes are based on studies of packed beds and rarely include the influence of wettability. Issues to be resolved include the presentation of a capillary pressure versus saturation

Two-phase transport in porous gas diffusion electrodes

209

Table 2: Advantages, disadvantages, and areas of application for each of the multiphase flow models. Multiphase flow model

Advantages

Disadvantages

Multi-fluid model

– Generalized form. – Interphase transfer models can be used. – Can resolve complex liquid motion. – Models convection of liquid by the gas. – Can model species diffusion in liquid.

Mixture model

– Reduced number of variables. – Models the influence of the gas pressure on the liquid.

Moisture diffusion model

– One additional equation over onephase model. – Can employ phase change models.

– Highest number of variables. – Needs the most computational resources. – Coupling of the phases can lead to unstable models. – Requires a multiphase CFD code. – May have trouble converging at higher saturations (liquid and gas have significantly different velocity fields). – Cannot employ interphase transfer models. – Large number of mixture quantities to calculate. – Does not account for the influence of the gas pressure on the liquid. – Cannot model interphase transfer of heat and species.

Porosity correction

– No additional transport equations over the one-phase model.

– Does not account for liquid motion.

Areas of application – Best employed for high saturation conditions because of the need for greater liquid resolution. – When the influence of the gas on the liquid is equivalent to that of the surface tension. – Best used when the gas pressure is the dominant force on the liquid or when capillary forces drive the liquid in the same direction. – High capillary number (i.e. large pores and high permeability).

– When surface tension is the dominant force on the liquid. – Low capillary numbers (i.e. small pores and low permeability). – Conditions where the liquid saturation does not exceed the immobile saturation (i.e. low relative humidities, very small pores, and low current densities).

210 Transport Phenomena in Fuel Cells curve for a gas diffusion layer, the immobile saturation levels for gas and liquid, and the influence of wettability on those properties. In closing this discussion, we note that mass transport limitations continue to be a significant hindrance to achieving higher current densities in PEM fuel cells. Water management within these fuel cells is a key consideration in their design. Knowledge of the behavior of liquid water in electrodes is limited by the inability to make in situ measurements. Better understanding of the transport of water in the PEMFC electrode can be obtained from models that capture the important physical processes. Several specific models of two-phase mass transport have been outlined and discussed in this chapter. These models, once implemented in a CFD code, will be able to help fuel cell designers improve their understanding of the transport of liquid water, as well as the transport of reactant and product gases, in the porous electrodes. It is clear from this review that a critical area to advance modeling of two-phase transport in gas diffusion electrode is further experimental characterization of the porous materials employed in PEM fuel cells.

References [1]

[2]

[3] [4] [5] [6] [7]

[8] [9] [10] [11]

Schulze, M., Schneider, A. & Gülzow, E., Alteration of the distribution of the platinum catalyst in membrane-electrode assemblies during PEFC operation. J. Power Sources, 127, pp. 213–221 Nam, J.H. & Kaviany, M., Effective diffusivity and water-saturation distribution in single- and two layer PEMFC diffusion medium. Int. J. Heat Mass Transfer, 46, pp. 4595–4611, 2003. Ngo, N.D. & Tamma, K.K., Microscale permeability predictions of porous fibrous media. Int. J. Heat Mass Transfer, 44, pp. 3135–3145, 2001. Papathanasiou, T.D., Flow across structured fiber bundles: a dimensionless correlation. J. Multiphase Flow, 27, pp. 1451–1461, 2001. Biloé, S. & Mauran, S., Gas flow through highly porous graphite matrices. Carbon, 41, pp. 525–537, 2003. Bird, R.B., Stewart, W.E. & Lightfoot, E.N., Transport Phenomena, John Wiley and Sons: New York, 1960. Berning, T. & Djilali, N., A three-dimensional, multi-phase, multicomponent model of the cathode and anode of a PEM fuel cell. J. Electrochem. Soc. 150, pp. A1589–A1598, 2003. Cussler, E.L., Diffusion–Mass Transfer in Fluid Systems, Cambridge University Press: New York, 1997. Epstein, N., On tortuosity and the tortuosity factor in flow and diffusion through porous media. Chem. Eng. Sci., 44(3), pp. 777–779, 1989. De La Rue, R.E. & Tobias, C.W., On the conductivity of dispersions. J. Electrochem. Soc., 106(9), 1959. Berning, T., Three-Dimensional Computational Analysis of Transport Phenomena in a PEM Fuel Cell. PhD thesis, University of Victoria, 2002.

Two-phase transport in porous gas diffusion electrodes

[12] [13] [14] [15] [16]

[17]

[18]

[19]

[20] [21]

[22]

[23] [24] [25] [26] [27]

[28] [29]

211

Kast, W. & Hohenthanner, C.R., Mass transfer within the gas-phase of porous media. Int. J. Heat Mass Transfer, 43(5), pp. 807–823, 2000. Cunningham, R.R. & Williams, R.J.J., Diffusion in Gases and Porous Media, Plenum Press: New York, 1980. Karniadakis, G.E. & Beskok, A., Micro Flows-Fundamentals and Simulation, Springer-Verlag: New York, 2002. Feng, C. & Stewart, W.E., Practical models for isothermal diffusion and flow of gas in porous solids. Ind. Eng. Chem. Fundam., 12(2), 1973. Taylor, R., Calculation of steady-state multicompnent mass transfer rates in porous media in the transition region. Ind. Eng. Chem. Fundam., 21, pp. 63– 67, 1982. Staia, M.H., Cambell, F.R. & Hills, A.W.D., Measurement of gaseous diffusion coefficents in porus reaction products. Ind. Eng. Chem. Res., 26, pp. 438– 446, 1987. Suwanwarangkul, R., Croiset, E., Fowler, M.W., Douglas, P.L., Entchev, E. & Douglas, M.A., Performance comparison of Fick’s, dusty-gas and StefanMaxwell models to predict the concentration overpotential of a SOFC anode. J. Power Sources, 122, pp. 9–18, 2003. Mezedur, M.M., Kaviany, M. & Moore, W., Effect of pore structure, randomness and size on effective mass diffusivity. AIChE Journal, 48(1), pp. 15–24, 2002. Wang, C.Y. & Cheng, P., Multiphase flow and heat transfer in porous media. Advances in Heat Transfer, 30, pp. 30–196, 1997. Udell, K.S., Heat transfer in porous media considering phase change and capillarity-the heat pipe effect. Int. J. Heat Mass Transfer, 28(2), pp. 485– 495, 1985. App, J.F. & Mohanty, K.K., Gas and condensate relative permeability at nearcritical conditions: capillary and Reynolds number dependence. J. Petrol. Sci. Eng., 36, pp. 111–126, 2002. Kaviany, M., Principles of Heat Transfer in Porous Media, Springer-Verlag: New York, 1991. Anderson, W.G., Wettability literature survey – Part 4: Effects of wettability on capillary pressure, J. Petrol. Technol., pp. 1283–1299, 1987. Leverett, M.C., Capillary behaviour in porous solids. AIME Trans., 142, pp. 152–169, 1941. Scheidegger, A.E., The Physics of Flow through Porous Media, University of Toronto Press: Toronto, 1974. van Genuchten, M. Th., A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Amer. J., 44, pp. 892–898, 1980. Anderson, W.G., Wettability literature survey – Part 5: Effects of wettability on relative permeability. J. Petrol. Technol., pp. 1453–1468, 1987. Eames, I.W., Marr, N.J. & Sabir, H., The evaporation coefficient of water: a review. Int. J. Heat Mass Transfer, 40(1), pp. 2963–2973, 1997.

212 Transport Phenomena in Fuel Cells [30]

[31]

[32] [33]

[34]

[35]

[36]

Marek, R. & Straub, J., Analysis of the evaporation coefficient and the condensation coefficent of water. Int. J. Heat Mass Transfer, 44, pp. 39–53, 2001. Wang, Z.H., Wang, C.Y. & Chen, K.S., Two-phase flow and transport in the air cathode of proton exchange membrane fuel cells. J. Power Sources, 94, pp. 40–50, 2001. You, L. & Liu, H., A two-phase and transport model for the cathode of PEM fuel cells. Int. J. Heat Mass Transfer, 45, pp. 2277–2287, 2002. Luikov, A.V., Systems of differential equations of heat and mass transfer in capillary-porous bodies (Review). Int. J. Heat Mass Transfer, 18, pp. 1–14, 1975. Whitaker, S., Simultaneous heat, mass and momentum transfer in porous media: A theory of drying. Advances in Heat Transfer, 13, pp. 119–203, 1977. Natarajan, D. & Nguyen, T.V., Three-dimensional effects of liquid water flooding in the cathode of a PEM fuel cell. J. Power Sources, 115, pp. 66– 80, 2003. Kermani, M.J., Stockie, J.M. & Gerber, A.G., Condensation in the cathode of a PEM fuel cell. Proc. of the 11th Annual Conference of the CFD Society of Canada, 2003.

Nomenclature AD Agl c Ca dpor D DAB Dij D(sl ) Dd j k kg, l kB kk Kn m ˙ H2 O M n N Num P

Droplet area Area of gas-liquid interface Molar concentration Capillary number Pore diameter Diffusion coefficient Diffusion coefficient in a binary system Diffusion coefficient in a multicomponent system Capillary diffusion coefficient Droplet diameter Relative mass flux Permeability Relative permeability Boltzmann constant Kozeny constant Knudsen number Phase change mass transfer Molecular weight Mass flux Molar flux Mass transfer Nusselt number Pressure

Two-phase transport in porous gas diffusion electrodes

Pc Ps R s sim sg, l S Sc t T x y u ui V VD

Capillary pressure Saturation pressure Universal gas constant Reduced saturation Immobile saturation Phase saturation Mass source term Schmidt number Time Temperature Mole fraction Mass fraction Superficial velocity Average interstitial velocity Volume Volume of droplet

Greek symbols γ γA ε λ λg, l µ ν ρA ρ σ σii τ φ ω

Mass transfer coefficient Advection correction factor Porosity Mean free path Relative mobility Viscosity Kinematic viscosity Density of species A Density Interfacial tension Collision diameter Tortuosity factor Diffusion volume Phase change correction factor

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CHAPTER 6 Numerical simulation of proton exchange membrane fuel cell T.C. Jen, T.Z. Yan & Q.H. Chen Department of Mechanical Engineering, University of Wisconsin-Milwaukee, USA.

Abstract This chapter presents general mathematical models and numerical simulation for proton exchange membrane fuel cell (PEMFC) to evaluate the effects of various designs and operating parameters on the PEMFC performance. Both one and two-dimensional models are presented; the advantages and weaknesses of using one-dimensional and two-dimensional models are discussed in detail. Subsequently, a general three-dimensional model is developed and described in detail. This threedimensional general model accounts for electrochemical kinetics, current density distribution, hydrodynamics, and multi-component transport. It starts from basic transport equations including mass conservation, momentum equations, energy balance, and species concentration in different elements of the fuel cell sandwich, as well as the equations for the phase potential in the membrane and the catalyst layers. These governing equations are coupled with chemical reaction kinetics by introducing various source terms. It is found that all these equations are in a very similar form except the source terms. Based on this observation, all the governing equations can be solved using the same numerical formulation in a single domain without prescribing the boundary conditions at interfaces between different elements of the fuel cell. Detailed numerical formulations are presented in this chapter. Various parameters, such as velocity field, local current density distribution, species concentration variation along the flow channel, under various operation conditions are computed by solving these governing equations in a single domain consisting of gas channel, catalyst layer and membrane. The performance of the PEMFC affected by various parameters such as temperature, pressure, and the thickness of the membrane is investigated. The numerical results are further validated with experimental data, which are available in the literature. This general three-dimensional model can be used for the optimization of PEMFC design and operation. It can also serve as a building block for the modeling and understanding of PEMFC stacks and systems.

216 Transport Phenomena in Fuel Cells

1 Introduction Although the fuel cell has received much attention in recent years, the underlying concept is still under heavy investigations [1]. The general concept of fuel cell operations can be characterized as gas-mixtures transport and transformation of species by electrochemical reactions. Therefore, one of the most challenging problems in fuel cell research is to predict the performance of the fuel cell. Using a mathematical model with numerical procedures to simulate the transport phenomena is a powerful tool to understand the fundamental physical and chemical processes of a fuel cell. Thus, reliable mathematical models are essential for fuel cell design, operation and optimization.

2 One-dimensional (1-D) model The earliest attempts to simulate the phenomena in PEM fuel cells were mainly using 1-D models. The pioneering work in 1-D modeling includes Savinell and Fritts [2], Ridge [3], Fritts and Savinell [4], Yang [5], Verbrugge and Hill [6], Bernerdi and Verbrugge [7], and Springer [8]. These earliest works mostly considered water and thermal management. During PEM fuel cell operations, water molecules are carried from the anode side to cathode side of the membrane by electro-osmosis. If this transport rate of water is higher than that by back diffusion of water, the membrane will become dehydrated and too resistive to conduct current. On the other side, if there is too much water, i.e. cathode side flooding may occur in the pores of the gas diffuser and hinder the transportation of reactants to the catalyst side. Consequently, proper water management is required to maintain high membrane conductivity and prevent flooding. Moreover, the effect of reacting gas dilution by high vapor pressure must be taken into consideration [1]. Thus, the water management is critical for efficient performance. Verbrugge and Hill [6] have carried out extensive modeling of transport properties in perfluorosulfonate ionomers based on dilute solution theory. False [9] reported an isothermal water map based on hydraulic permeability and electro-osmotic drag data. Fuller and Newman [10] applied concentrated solution theory and employed literature data on transport properties to produce a general description of water transport in fuel cell membranes. Bernardi and Verbrugge [7] took a different approach, in which transport through the gas diffusion electrodes was considered. They assumed the membrane to be uniformly hydrated, corresponding to an “ultra-thin membrane” case. Springer [8] presents an isothermal model, which includes transport through the porous electrodes. Their model required inputs from calculated diffusivities, which are needed to correct for porosity, and from experimentally determined transport parameters for the transport through the membrane. Most of these models treated flow channel as being perfectly well fixed, with no pressure and concentration difference along the gas channel. These research works were particularly useful in classifying the different models for porous gas diffusion electrodes and providing the key properties of the membrane required for numerical simulation. These 1-D models provided the sub-model bases for many following 2-D or 3-D research works.

Numerical simulation of proton exchange membrane fuel cell

217

Here, we introduce a general 1-D PEM fuel cell model based on the research work by Bernardi and Verbrugge [7]. 2.1 General 1-D model 2.1.1 Model description The fuel cell consists of a membrane sandwiched between two gas-diffusion electrodes as shown in Fig. 1. The gas-diffusion electrodes are porous composites made of electronically conductive material mixed with hydrophobic polytetrafluorethylene and supported on carbon cloth. The gaseous reactants can transport through the electrodes during operation. The electrochemical reactions occur in the catalyst layer. Humidified hydrogen enters anode gas chamber, transports through the anode gas diffuser, and dissolves into catalyst layer. Hydrogen molecule is oxidized and generates protons and electrons. Protons go into membrane and electrons are received by carbon conductor. The overall electrochemical reaction can be expressed as: 2H ↔ 4H+ + 4e-. (1) 2

Gaseous reactant O2 from air, usually mixed with H2 O, enters into cathode gas channel, transports through porous gas diffuser, and dissolves into cathode catalyst layer. Hydrogen protons diffuse into the cathode catalyst layer from anode catalyst layer through the membrane. On the surface of the catalyst particles, the oxygen is consumed along with the protons and electrons, and the product, H2 O, is produced along with the waste heat. The overall electrochemical reactions occurring at the reaction site may be represented as: 4H+ + 4e- + O2 → 2H2 O + heat.

(2)

Therefore, the overall electrochemical reaction of the PEM fuel cell is: 2H2 + O2 → 2H2 O + heat + electric energy.

Figure 1: Schematic diagram of PEMFC.

(3)

218 Transport Phenomena in Fuel Cells The proton, dissolved oxygen, and dissolved hydrogen concentrations depend on the kinetic expressions of four-electron-transfer reaction for oxygen reduction and the proton-transfer reaction for hydrogen oxidation. 2.1.2 Model assumptions The cell is assumed to operate at steady state conditions. Since the cell thickness is very small compared with the other dimensions of the cell, a one-dimensional approximation is used in the model formulation. The entire system is taken to be at constant temperature, and the gases are assumed to be ideal and well mixed in the chambers. The steady state and isothermal assumptions are valid because these conditions are often achieved in a small single-cell experimental investigation. The model requires some properties inputs such as water-diffusion coefficients, electroosmotic drag coefficients, water sorption isotherms, and membrane conductivities, which are all measured in single-cell experimental studies. The temperature of fuel cell is assumed to be well controlled. The heat transfer is supposed to be very efficient so that the heat generation, which is due to the irreversibility of the electrochemical reaction, ohmic resistance, as well as mass transport overpotentials, can be taken out quickly. The inlet gaseous temperature is assumed to be preheated to the cell temperature. Fully hydrated membrane, wet gas diffuser, and saturated chamber gases are considered. The total gas pressure within the gas channel is assumed to be constant and same as the pressure in gas diffuser since the gas velocity is very slow, the pressure variation can be neglected along the gas channel. However, the pressure can vary between anode and cathode. With these assumptions the cell model is formulated with transport equations for the electrodes, catalyst layers and membrane as described below. 2.1.3 Governing equations The governing equations constituting the mathematical model of the PEM fuel cell are derived by applying the conservation equations for an ideal gas in porous medium, the Butler-Volmer equations, and the Stefan-Maxwell equation for gasphase transport. Electrodes: Continuity: d (ρv) = 0. dx

(4)

d (ρi v + ρi Vi ) = mi . dx

(5)

Species:

Potential:

I d s (6) = − eff , dx σ where ρi is the partial density of species i, ρ is average density, v is the massaveraged velocity in x direction, Vi is the species i diffusion velocity, mi is the mass

Numerical simulation of proton exchange membrane fuel cell

219

of species i, s is the electrical potential in the solid matrix of the electrode, I is cell current density, and σ eff is the effective electrical conductivity. The diffusion velocity Vi can be determined from the Stefan-Maxwell equation for multi-component gas diffusion,    xi xj  (7) ∇xi = Vj − Vi , Dijeff i where Dieff is the effective binary diffusion coefficient for species i in j, and xi is the mole fraction for species i. Species i for cathode electrode can be taken as O2 , N2 and H2 O; species i for anode electrode can be taken as H2 and H2 O. 2.1.4 Catalyst layers The catalyst layers are very thin compared to the other components in PEMFC, but they are the heart of the fuel cell. Electrochemical reaction occurs in this region to produce electrical energy and products. In this region, the transfer of mass and energy is coupled with reaction kinetics when the cell is loaded and results in a potential difference between electrodes. How this potential difference varies as functions of mass transfer, electrode kinetics, and energy flux determines the fuel cell performance. The mathematical description of the active-catalyst-layer region is based on Butler-Volmer relations [13, 14] for electrochemical reaction and species diffusion. The macro-homogeneous approach is used in developing the governing equations. Current in the catalyst layer can transfer to electronically-conductive solid portion of the catalyst layer (carbon and catalyst particles). The rate of electrochemical reaction is given by the Butler-Volmer expression [13, 14] . The mass transfer is modeled by using species conservation and Fick’s Law of diffusion. The governing equations can be expressed as follows:   CO2 αc Fηc di ref exp − = −aj0 , (8) dx CO2 ref RT where i is the current density, a is the catalyst reactive surface area per unit volume, j0ref is the reference exchange current density at the reference concentration C ref , CO2 is the oxygen concentration, αc is the transfer coefficients in Butler-Volmer relation, F is the Faraday’s number, R is the universal gas constant, ηc is the overpotential at cathode side, and T is the reaction temperature. From the material balance based on standard porous-electrode theory, we have: si di dNi =− , dx nF dx

(9)

where Ni is the superficial flux of species i, and si is the stoichiometric coefficient for species i in the cathode reaction and anode reaction. The right hand side of eqn (9) is the source term for the species conservation equation as shown below. For hydrogen: eff DH 2

di d 2 CH2 dCH2 + =v dx dx dx2



sH 2 nF

.

(10)

220 Transport Phenomena in Fuel Cells For oxygen:

d 2 CO2 dCO2 di  sO2  =v + . 2 dx dx dx nF

(11)

di  sw  d 2 CH2 O dCH2 O + − . = v dx dx nF dx2

(12)

eff DO 2

For H2 O: eff DH 2O

Potential:

d , (13) dx where i is the current density in the electron-conducting solid, and σ eff is the conductivity of the electronically conductive catalyst layer. i = −σ eff

2.1.5 Membrane The membrane of PEM fuel cell acts as the hydrogen proton conductor. The transport processes in the membrane can be described by the conservation of species. The flux of species in the membrane is determined by the net effect of electroosmotic drag, diffusion due to concentration gradient, and the convection due to a pressure gradient. A form of the Nernst-Planck equation is used to describe the flux of species in the membrane [15, 16]. Ni = −zi

F d m dCi Di Ci − Di + vCi , RT dx dx

(14)

where Ni is the superficial flux of species i, zi is charge number of species i, Ci is the concentration of species i, and Di is the diffusion coefficient of species i, and m is electrical potential in membrane. v is the velocity of H2 O, which is generated by electric potential and pressure gradient, and can be described by a form of Schögl’s equation: kp dp d m k zf cf F − , (15) v= µ dx µ dx where k is electro-kinetic permeability, µ is pore-fluid viscosity, zf is fixed-site charge, cf is fixed-charge concentration, and kp is hydraulic permeability. Current conservation:

di = 0. dx

(16)

dv = 0. dx

(17)

Mass conservation for liquid flow:

Electric potential: i F d = − + cH + v, dx σ σ F2 DH + cH + . σ= RT

(18) (19)

Numerical simulation of proton exchange membrane fuel cell

221

Furthermore, the potential and pressure profiles throughout the membrane region are assumed to be linear with constant velocity. This is basically true for fullydeveloped porous media case. 2.1.6 Boundary conditions Before solving the governing equations, appropriate boundary conditions must be specified. The temperature, pressure, relative humidity, flow rate, compositions of the reactant gases both in anode and cathode channels are specified according to the cell operation condition. In the following model description, c− is the anode catalyst layer, c+ is the cathode catalyst layer, d− is the anode gas diffuser; d+ is the cathode gas diffuser, and m is membrane. lc− is the thickness of anode catalyst layer, lc+ is the thickness of cathode catalyst layer, lm is the thickness of membrane, ld− is the thickness of anode gas diffuser, ld+ is the thickness of cathode gas diffuser. At the interface between the anode gas diffuser and catalyst layer, the solid phase potential gradient is related to the cell operating current density by Ohm’s law:  d m  c i= σ −, (20) dx c− eff c

where σeff− is electronic conductivity of anode catalyst layer. The membrane concentration of oxygen set to be zero as, cO2 = 0.

(21)

Since the hydraulic pressure distribution throughout the anode gas diffuser is linear, we can impose the boundary condition of pressure at the interface between anode gas diffuser and catalyst layer as: po = p− −

µ

d l v −, d− d− s kp

(22)

d

where p− is the pressure in anode gas chamber, kp − is anode gas diffuser permed ability, and vs − is water velocity in anode gas diffuser. Because the total water flux is continuous, we have  d d c  (23) Cw vs − + Nw− = Cw εm− εm wv c , −

d

where Nw− is water vapor flux from Stefan-Maxwell equation [17], which is sat

d

Nw− = −

I xw − , 2F (1 − xwsat− )

where

(24)

sat

sat−

xw

=

pw − . p−

(25)

222 Transport Phenomena in Fuel Cells The concentration of hydrogen is given by sat cH = (1 − xwsat ) 2

p− . KH2

(26)

At the interface between the membrane and anode catalyst-layer, the current continuity should be satisfied:   d   d   = σ . (27) σm eff dz m dz c− The superficial flux of liquid water is continuous: c

v|m = εm− v|c−

(28)

and the flux of dissolved hydrogen is continuous through the internal boundary:    dcH2  eff dcH2  DH2 = DH2 . (29)  dx m dx c− At the membrane /cathode-catalyst-layer interface, the current, superficial flux of liquid water, and the flux of dissolved oxygen are continuous, and the dissolved hydrogen concentration is zero.   d   c+ d   σm = σ (30) eff dz  , dz m c+ c

DO2

v|m = εm+ v|c+ ,    dcO2  eff dcO2  = DO2 ,  dx m dx c+ cH2 = 0.

(31) (32) (33)

At the cathode-catalyst-layer/gas-diffuser interface, the current in the solid phase is continuous.   c d solid  d+ d solid  = σ (34) σeff+ eff dx c+ dx d+ and the superficial flux of water is given by  d d c  ρvs + + Nw+ = ρεm+ εm wv c , +

where

(35)

sat

d Nw+

xw + I = sat 4F 1 − xw + − xN2 +

xN2 rw

(36)

and rw is the diffusivity ratio given by rw =

Dw−N2 . Dw−O2

(37)

Numerical simulation of proton exchange membrane fuel cell

223

The concentration of dissolved-oxygen in the catalyst layer is given by sat

sat = (1 − xN2 − xw + ) cO 2

p+ , K O2

(38)

where KO2 is Henry’s constant, and sat

sat+

xw

=

pw + . p+

(39)

2.1.7 Results and discussion Ticianelli et al. [11] and Srinivasan et al. [18] published their experimental results respectively. Their experimental data have been used extensively as standard validation by many numerical studies such as Bernardi and Verbrugge [7]. Since most of the numerical simulation parameters and properties are based on their experimental results, these parameters and properties are adopted in this chapter. Table 1 shows the detailed physical parameters and properties. Figure 2 shows the comparison of Bernardi and Verbrugge’s [7] calculated results with Ticianelli’s experimental data [11]. Note that, in their model the exchangecurrent density a+ ioref was adjusted to yield model results that are suitable mimic the experimental results. With this one adjustable parameter, the agreements between model and experimental results are quite good. Figure 3 shows the components of the overall cell polarization for the base case. At the low current densities (less than 100 mA/cm2 ), the activation overpotential of the oxygen reduction reaction is almost entirely responsible for the potential losses of the cell. For current densities greater than 200 mA/cm2 , the ohmic potential loss due to the membrane and electrodes become more significant, and the cathode activation overpotential reaches a relatively constant value. Figure 4 shows the effect of membrane thickness on the potential. Generally, the electronic conductivity is proportional to the thickness of the membrane. The results show the fuel cell potential increases as the membrane thickness decreases from 7 mil (1 mil = 1/1000 inch) to 2 mil under dry membrane state. The modeling value for the wet membrane thickness was obtained by assuming that the ratio of the wet to dry thickness was a constant. The comparison between experimental results (not shown) and model predictions is generally satisfactory. The effect of cathode pressurization on the fuel cell potential is depicted in Fig. 5 When the cathode pressure decreases from 5 to 3 atm, the open-circuit potential is only slightly affected by the change in cathode pressure. The cell potential decrease is primarily due to oxygen mole fraction decreased because of lower pressure; this leads to the increase in cathode activation overpotential loss. 2.1.8 Summary 1-D model provides a good preliminary foundation for PEM fuel cell modeling. These 1-D models base on the fundamental transport properties where the potential losses incurred by the activation overpotential of the anode and cathode reactions,

224 Transport Phenomena in Fuel Cells Table 1: The values of the parameters. Wet membrane thickness, lm Gas diffuser thickness, ld+ = ld− Catalyst layer thickness, lc+ = lc− Relative humidity of inlet air Relative humidity of inlet hydrogen Inlet fuel and air temperature, T0 Cell temperature, T Inlet nitrogen-oxygen mole ratio Air-side pressure, p+ Fuel-side pressure, p− Thermodynamic open-circuit potential, U Estimated proton diffusion coefficient, DH+ Estimated ionic conductivity, σm Fixed charge site concentration, cf Charge of fixed site, zf Dissolved oxygen diffusivity, DO2 Electrokinetic permeability, k Hydraulic permeability, kp Pore-fluid (water) viscosity, µ Pore-fluid (water) density, ρ sat Saturated water vapor pressure, pw Henry’s law constant for oxygen in membrane, KO2 Henry’s law constant for hydrogen in membrane, KH2 Volume fraction membrane in active layer, εm,c Volume fraction water in membrane, εw,m d = σc Electronic conductivity, σeff eff Reference kinetic parameter of anode, ajoref Reference kinetic parameter of cathode, ajoref Cathodic transfer coefficient, αc Anodic transfer coefficient, αa ref H+ reference concentration, cH + ref O2 reference concentration, cO 2

ref H2 reference concentration, cH 2

0.023 cm 0.026 cm 0.001 cm 100% 100% 105 ◦ C 80 ◦ C 0.79/0.21 5 atm 3 atm 1.194 V 4.5 × 10−5 cm2 /s 0.17 ohm/cm 1.2 × 10−3 mol/cm3 −1 1.2 × 10−6 cm2 /s 1.13 × 10−15 cm2 1.58 × 10−14 cm2 3.565 × 10−4 kg/m · s 0.054 mol/cm3 0.467 atm 2 × 105 atm · cm3 /mol 4.5 × 104 atm · cm3 /mol 0.5 0.28 0.53 ohm/cm 1.4 × 105 A/cm3 1 × 10−5 A/cm3 2 1/2 1.2 × 10−3 mol/cm3 3.39 × 10−6 mol/cm3 5.64 × 10−5 mol/cm3

and the ohmic losses incurred by the membrane. The membrane is assumed fully hydrated and the cell is assumed isothermal. However, it cannot model the depletion of the reactants and the accumulation of products in the flow direction. 2.2 General 2-D model Two-dimensional (2-D) models are developed to improve the earlier 1-D models. Fuller and Newman [19] modeled and solved the transport across the fuel cell

Numerical simulation of proton exchange membrane fuel cell

225

Figure 2: Model and experimental data for fuel cell at 80 ◦ C, p+ = 5 atm, p− = 3 atm.

Figure 3: The contribution of fuel cell. sandwich at certain location along the gas channel and then integrated in the second direction. In this 2-D model, the gas outside the gas diffusers was assumed to have uniform composition in the direction across the cell. Nguyen and White [20], Amphlett et al. [21], and Yi and Nguyen [22] developed pseudo 2-D models accounting for composition changes along the flow path. These models are useful for small single cells, however, when it is applied to large-scale fuel cells, particularly under high fuel utilization conditions, the applicability is limited.

226 Transport Phenomena in Fuel Cells

Figure 4: The effects of membrane thickness on fuel cell potential.

Figure 5: Effect of cathode gas-channel pressure.

Later, Gurau et al. [23] presented a 2-D model of transport phenomena in PEM fuel cells. They developed a model to understand the transport phenomena such as the oxygen and water distributions. They considered the interaction between the gas channels and the rest of the fuel cell sandwich. Yi and Nguyen [24] also formulated a 2-D model to explore hydrodynamics and multi-component transport in the cathode

Numerical simulation of proton exchange membrane fuel cell

227

Figure 6: PEM fuel cell schematic diagrams. of PEMFCs with an interdigitated flow field. More recently, Um et al. [25] presented a transient 2-D model based on finite-volume CFD approach. They also explored hydrogen dilution effects on PEMFC running on reformate gas. The following section describes a general two-dimensional model for electrochemical and transport processes occurring inside a PEMFC. 2.2.1 Model description and assumptions Figure 6 shows a typical 2-D PEM fuel cell. It includes the collector plates, fuel and gas channels, gas-diffusers, catalyst layers and membrane. Hydrogen and water vapor flow through anode fuel channel and humidified air is fed into the cathode channel. Assume that hydrogen oxidation and oxygen reduction reactions occur only within active catalyst layers. Additional assumptions of this 2-D model are described below: • • • • • • •

The gas mixtures including anode channel and cathode channel are ideal gases. The density is treated as constant. The gas flows are laminar and incompressible. The electrodes, gas diffusers, catalyst layers, and membrane are isotropic and homogeneous. The heat generated under reversible condition is neglected. Only steady state condition is considered. Cell temperature is held constant. The contact electrical potential drop of different cell components is negligible.

2.2.2 Mathematical model A single-domain approach developed by Um et al. [24] is used in this section to describe fuel cell transport processes. Therefore, no boundary conditions are required at the interface of different fuel cell components.

228 Transport Phenomena in Fuel Cells The governing equations can be written as: Mass conservation: ∂u ∂v + = 0. ∂x ∂y

(40)

Momentum conservation:   2 1 ∂u ∂u ∂p ∂ u ∂2 u + Sx , + ρ u +v = −ε + µ ε ∂x ∂y ∂x ∂x2 ∂y2   2 1 ∂v ∂v ∂p ∂ v ∂2 v + Sy . + ρ u +v = −ε + µ ε ∂x ∂y ∂y ∂x2 ∂y2

(41) (42)

Species conservation:   2 ∂ 2 Xk ∂Xk ∂Xk 1 ∂ Xk + u +v = εDk + Sk . ε ∂x ∂y ∂x2 ∂y2

(43)

Charge conservation:   ∂ ∂ ∂ ∂ σm + σm + S = 0, ∂x ∂x ∂y ∂y

(44)

where ε is porosity (for gas channel ε equals 1), S is the source term. Table 2 shows the values for different region of fuel cell. Xk is the mole fraction of species k, Dk is diffusivity of species k. Table 2: Source terms for the above governing equations.

Gas channel Gas diffuser

Catalyst layer

Membrane

Sx

Sy

Sk

S

0 µ 2 − ε u K

0 µ 2 − ε v K

0

0

µ ∂ − εc u + E Kp ∂x

µ ∂ − ε c v + E Kp ∂y

∂ µ − ε m u + E Kp ∂x

∂ µ − εm v + E Kp ∂y

0

0

For H2 −

ja 2Fctotal

For anode ja

For H2 O

jc 2Fctotal

For cathode jc

For O2 −

jc 4Fctotal

0

0

where E = KKp zf cf F, K is electro-kinetic permeability, Kp is hydraulic permeability of membrane, zf is fixed site charge, cf fixed charge concentration, and F is Faraday constant. ctotal is the total species concentration. ja and jc are transfer

Numerical simulation of proton exchange membrane fuel cell

current densities defined by the Bulter-Volmer equations:    a  c

XO2 αa F αa F ref exp ja = aj0 ηc − exp − ηC , RT RT XOref2  1/2   a  c

XH2 αa F α F ref jc = aj0 exp ηa − exp − a ηa , XHref RT RT

229

(45)

(46)

2

where σm is the proton conductivity in the membrane, which was correlated by Springer et al. [8] as:   1 1 − (0.005139λ − 0.00326). (47) σm = exp 1268 303 T The water content λ, in eqn (47) can be expressed as follows [25]: λ = 0.043 + 17.81A − 39.85A2 + 36A3 λ = 14 + 1.4(A − 1)

for 0 < A < 1, 1 ≤ A ≤ 3.

(48)

Note that A is defined as the vapor activity at the cathode gas diffuser catalyst layers interface assuming thermodynamic equilibrium, which is given by A=

Xw p . psat

(49)

Here Xw is the water vapor molar fraction. The saturated water partial pressure, psat is expressed by the following empirical equation. log10 psat = −2.1794 + 0.2953T − 9.1837 × 10−5 T 2 + 1.4454 × 10−7 T 3 . (50) Once the electrical potential is determined in the membrane, the local current density can be calculated by: i(y) = −σm

∂e |interface . ∂x

Then the averaged current density is determined as follows: ! 1 i(y)dy. iavg = L mem

(51)

(52)

The cell voltage can be calculated as: E = E0 − |ηa | − |ηc | − iavg /σm ,

(53)

where E0 is the reference open-circuit potential of the fuel cell, which can be expressed as a function of temperature [25]: E0 = 0.0025T + 0.2329.

(54)

230 Transport Phenomena in Fuel Cells 2.2.3 Boundary conditions The boundary conditions for u, v, p, XH2 , XH2 O , XO2 , e at inlet of the channel are fixed. uin,anode = uo− ,

vin,anode = 0,

pin,anode = po− ,

uin,cathode = uo+ ,

vin,cathode = 0, −

pin,cathode = po+ ,

XH2 O,anode = XHo2 O , +

XH2 O,cathode = XHo2 O , where superscript or subscript o− and o+ represent inlet condition at anode and cathode, respectively. All velocities at solid walls are set to be zero due to no-slip conditions. The boundary condition for the electric potential is no-flux everywhere along the boundaries of computational domain. At the outlet, both channels are assumed sufficiently long that the velocity and species concentration are fully developed. 2.2.4 Numerical procedures Gurau et al. [23] solved these governing equations in three different domains: the cathode gas channel-gas diffuser-catalyst layer for air mixture, the cathode gas diffuser-catalyst layer-membrane-anode catalyst layer-gas diffuser for liquid water, and the anode gas channel-gas diffuser-catalyst layer for hydrogen. The continuity equations and momentum equations for the gas mixture were solved first in the coupled gas channel and gas diffuser domains. Then the species concentration equations together with the Butler-Volmer equations were solved iteratively. After convergence is achieved, the transport equations related to water vapor flow as well as the equations for cell potential and current density can be solved. Since the source term and boundary conditions for the two domains have to be matched at the interface, new level iterations have to be introduced. When they solved the electrochemical cell efficiency, they assumed the total overpotential first and then calculated the transfer current density together with other unknowns. Once the correct current density is found, the ohmic losses can be calculated, and the cell potential is set to be the open potential minus the sum of the total over-potential, and the total ohmic losses. Um et al. [25] solved the governing equations in a single domain. Although some species are practically non-existing in certain regions of a fuel cell, the species transport equations can still be applied throughout the entire computational domain by using the large source term technique, which is often assigned a sufficiently large source term in this sub-region, that effectively freezes the non-exsiting species mole fraction to zero. 2.2.5 Results and discussion Figure 7 shows the basic profile of fuel cell potential-current density characteristic for a 2-D model solution [26]. The potential loss generally has three sources: (1) activation polarization (act), (2) ohmic polarization (ohm), and (3) concentration polarization (conc). The activation polarization loss is dominant at low current

Numerical simulation of proton exchange membrane fuel cell

231

Figure 7: Fuel cell voltage-current density.

Figure 8: Effect of the gas diffuser porosity on the voltage-current density characteristic. density. At this point, electronic barriers have to be overcome prior to current and ion flow. Activation losses increase slightly as current increases. Ohmic polarization (loss) varies directly with current, increasing over the whole range of current because cell resistance remains essentially constant when temperature does not change too much. Gas transport losses occur over the entire range of current density, but these losses become prominent at high limiting currents where it becomes difficult to provide enough reactant flow to the cell reaction sites. Figure 8 demonstrated the effect of gas diffuser porosity on fuel cell performance presented by Gurau et al. [23]. In their work, an isothermal condition at T = 353 K and an inlet air velocity U0 = 0.35 m/s with 100% humidity were assumed. For lower values of porosity, lower values of the limiting current density were found. The cell performance in the region where the concentration over-potentials are predominant was also predicted.

232 Transport Phenomena in Fuel Cells

Figure 9: Effect of the air inlet velocity on the performance. Figure 9 presents the current density for different air inlet velocity at the cathode gas channel, assuming the gas diffuser porosity ε = 0.4 and a constant temperature T = 353 K with 100% humidity. For higher inlet air velocity, more oxygen is fed; therefore, more oxygen is likely to arrive at the catalyst layer, with the result of a higher limiting current density. This is explained by the fact that for the same pressure field, the axial momentum transfer across the gas channel-gas diffuser interface becomes more important for higher velocities. This is a consequence when more “fresh” air is able to arrive at the catalyst layer. For the inlet air velocity higher than approximately 2 m/s, the limiting current becomes constant, which shows that there is a limiting effect of the momentum transfer across the gas channel-gas diffuser interface. Figure 10 shows the oxygen mole fraction field in the cathode gas channelgas diffuser coupled domain for a case of current density Iavg = 2.89 × 103 A/m2 , Ui = 0.35 m/s, and T = 353 K (iso-thermal case) with humidity = 100%. The oxygen is consumed in the catalyst layer. The mole fraction is decreased along the flow direction. The oxygen consumption depends on the operating current density. The higher current density, the more oxygen is consumed, thus, the faster mole fraction decreases along the flow direction. A limiting current density may occur when the oxygen or hydrogen is completely depleted at the reaction surface. The hydrogen mole fraction field in the anode gas channel-gas diffuser coupled domain has a similar distribution as the oxygen distribution in cathode side. In the present model, it is assumed that water only exists in the vapor state. However, if the electrochemical reaction rate is sufficiently high, the amount of water produced is condensed into liquid phase. Under this situation, two-phase flow model has to be considered. 2.2.6 Concluding remarks The 2-D model is a significant improvement over the 1-D models and could provide simulations that are more realistic. It can predict the transport phenomena in the

Numerical simulation of proton exchange membrane fuel cell

233

Figure 10: Oxygen distribution in the cathode gas channel and gas diffuser. entire fuel cell sandwich, including the gas channels. No assumptions are necessary for the distribution of the species concentrations or current density. The input data are only those parameters that can be controlled in real fuel cell applications. 2.3 Three-dimensional (3-D) model Recently, a few research groups extended 2-D model to 3-D model such as Shimpalee and Dutta [27], Zhou and Liu [28], Um and Wang [29], and Jen et al. [30]. One of the major improvements of the 3-D model over the 2-D model is its ability to study the blocking effect of the collector plates and the effectiveness of the interdigitated flow field [31]. Here we introduce a generalized 3-D model as well as some numerical approaches and results. To solve the 3-D model, many different numerical method have been used such as CD-Star [24], FLUENT [27], Semi-Implicit Method [28], and Vorticity-Velocity Method [30]. In this section, 3-D formulations are developed in such a way that they allow using the same code for solving the Navier-Stokes equations of the gas channel, gas diffuser and catalyst layers in a coupled domain. The potential equation and species concentration equations are solved in a single domain of the whole fuel cell. The solutions of the hydrodynamics of the flow and polarization curves are analyzed and presented in details. The results of this study may be beneficial for further and more complete analyses of the performance of fuel cells. 2.3.1 Model development A typical PEM fuel cell configuration is shown in Fig. 11. The physical fuel cell model consists of anode gas channel, anode gas diffuser that formed by porous

234 Transport Phenomena in Fuel Cells

Figure 11: The schematic of PEMFC.

media, anode catalyst layer, membrane, cathode catalyst layer, cathode gas diffuser, and cathode gas channel. In reality, when the fuel cell works, fuel and oxidant can be viewed as a steady, laminar, developing forced convection flow in an isothermal rectangular channel, and penetrating through the gas diffuser to catalyst layers. In the model presented here, the flow is assumed to be steady, constant properties, and incompressible. The viscous dissipation, compression work and buoyancy are assumed negligible. The gas mixtures are considered as perfect gases, and the species concentrations are considered as constant at the inlet of the channel. The concentrations along the gas channel and the gas-diffuser will vary due to diffusionconvection transport and electron kinetics in catalyst layers, and the distributions will depend on the gas properties and reaction rate. Water transport in and out of the electrodes is assumed to be in the form of vapor only. This assumption may be questionable in this model, in particular when the reactants flow into the gas channel under saturated conditions. The water generation rate is very likely to exceed its removal rate and thus condensation formed in the cathode. As a result, twophase flow forms in the cathode channel. This complex case is, however, neglected in this chapter. The gas-diffuser, catalyst layers, and the membrane materials are considered as isotropic porous media. Contraction of the porous media is also neglected. 2.3.2 Mathematical model The governing equations base on conservation of mass, momentum, energy and species. The model governing equations can be written as: Mass conservation: ∂u ∂v ∂w + + = 0. ∂x ∂y ∂z

(55)

Numerical simulation of proton exchange membrane fuel cell

Momentum conservation:   2 ρ ∂u ∂u ∂u ∂p ∂ u ∂2 u ∂2 u + 2 + 2 + Sx , u +v +w = −ε + µ ε ∂x ∂y ∂z ∂x ∂x2 ∂y ∂z   2 ∂v ∂v ∂v ∂p ∂ v ρ ∂2 v ∂2 v u +v +w = −ε + µ + Sy , + + ε ∂x ∂y ∂z ∂y ∂x2 ∂y2 ∂z 2   2 ∂w ∂w ∂w ∂p ∂ w ∂2 w ∂2 w ρ + Sz . + + u +v +w = −ε + µ ε ∂x ∂y ∂z ∂z ∂x2 ∂y 2 ∂z2

235

(56) (57) (58)

Species conservation:  2  1 ∂ Xk ∂ 2 Xk ∂2 Xk ∂Xk ∂Xk ∂Xk + Sk . + + u +v +w = εDk ε ∂x ∂y ∂z ∂x2 ∂y2 ∂z2

(59)

Charge conservation:    ∂ ∂ ∂ ∂ ∂ ∂ σm + σm + σm + S = 0, ∂x ∂x ∂y ∂y ∂z ∂z

(60)

where ε is porosity, for gas channel ε equals 1, for gas diffuser ε = εd , for catalyst layer ε = εc , and for membrane ε = εm , and S is the source term. Table 3 shows the source terms for different region of fuel cell. The parameters of above governing equations are the same as the 2-D model. The boundary conditions and numerical procedures are similar to the 2-D case and will not repeat here. Here we introduce a new method, which was used by Jen et al. [30], to simplify the 3-D model by making the usual parabolic assumption. With the parabolic 2 2 2 assumption, the diffusion term in axial direction such as ∂∂zu2 , ∂∂z v2 , ∂∂zw2 , as well as ∂ 2 Xk ∂z 2

can be neglected. Furthermore, the modified pressure P may be defined as P(x, y, z) = p(z) + p∗ (x, y),

(61)

where p(z) is the pressure over the cross section at each axial location, and p∗ (x, y) is the pressure variation in the x, y direction, which drives the secondary flow. Pressure gradient for axial direction: ∂p ∂p∗ ∂P = + , ∂z ∂z ∂z

(62)

∗

∂p where ∂p ∂z  ∂z is due to the useful parabolic assumption. So the axial pressure gradient can be written as:

∂p ∂P = = f (z). ∂z ∂z

(63)

Gas channel −

Gas diffuser

Catalyst layer

Membrane

−

−

Sx

Sy

Sz

Sk

Sφ

0

0

0

0

0

µ − ε2 w K

0

0

µ 2 ε u K

∂e µ εc u + E Kp ∂x

∂m µ εm u + E Kp ∂x

−

−

−

µ 2 ε v K

∂e µ εc v + E Kp ∂y

∂m µ εm v + E Kp ∂y

−

−

∂e µ εc w + E Kp ∂z

∂m µ εm w + E Kp ∂z

For H2 −

ja 2Fctotal

For anode ja

For H2 O

jc 2Fctotal

For cathode jc

For O2 −

jc 4Fctotal

0

0

236 Transport Phenomena in Fuel Cells

Table 3: Source terms for the above governing equations.

Numerical simulation of proton exchange membrane fuel cell

237

And we also have: ∂p∗ ∂P = , ∂x ∂x ∂P ∂p∗ = . ∂y ∂y

(64) (65)

With the above parabolic flow assumptions, it is now possible to simply march through the computational domains in the main flow direction without worrying about the downstream conditions as those in elliptic flow cases. A novel vorticityvelocity method was used here to solve this problem. The axial vorticity function can be defined as: ∂u ∂v ζ= − . (66) ∂y ∂x Applying it to continuity eqn (55) for u and v, respectively. ∇ 2u =

∂ζ ∂2 w − , ∂y ∂x∂z

∇ 2v = −

∂ζ ∂2 w − . ∂x ∂y∂z

(67) (68)

Cross differentiation of x and y momentum equations of gas channel to eliminate pressure terms yield:  ∂ζ ∂ζ ∂u ∂v ∂w ∂u ∂w ∂v ∂ζ + + − u +v +w +ζ ∂x ∂y ∂z ∂x ∂y ∂y ∂z ∂x ∂z  2 ∂2 ζ µ ∂ ζ (69) + 2 . = 2 ρ ∂x ∂y Using the same method we can get similar equation for gas diffuser    1 ∂ζ ∂ζ ∂ζ ∂u ∂v ∂w ∂u ∂w ∂v ε2 µ u +v +w +ζ + + − + ζ ε ∂x ∂y ∂z ∂x ∂y ∂y ∂z ∂x ∂z κ  ∂2 ζ µ ∂2 ζ (70) + 2 . = ρ ∂x2 ∂y An additional constraint, which will be used to determine f (z), is that global mass conservation must be satisfied as follows: !! (1 + r)2 wdxdy = Uin De2 , (71) 4r where r = ab is the aspect ratio of the gas channel, Uin is inlet velocity, De is hydraulic diameter (4A/S).

238 Transport Phenomena in Fuel Cells 2.3.3 Boundary conditions Boundary conditions at the gas channel entries, such as gas mixture velocities, pressure, and component concentrations, are specified. No slip boundary conditions are specified at the gas channel walls. At the interface between the gas channel and electron collectors, the boundary conditions of components concentration are assumed as: ∂Xi = 0. (72) ∂n For the membrane potential equations, the boundary conditions are: ∂ = 0. ∂y

(73)

This means that no proton current leaves top and bottom boundary. In x-direction,      ∂   = 0. (74)  anode = 0,  catalyst ∂x cathode surface catalyst surface No boundary conditions are needed at the interface between gas channel and gas diffuser because they are coupled in a single domain. 2.3.4 Discretization strategies In order to solve the above equations, some terms need to be discretized furthermore. The strategies of discretization is as follows: ∂w ∂u ∂v The values of ∂w ∂x , ∂y , ∂x , and ∂y are discretized using central differencing at ∂v ∂ ∂z , ∂z are computed by two points backward ∂ζ ∂ζ ∂x∂z , ∂y∂z , ∂x and ∂y are calculated by using backward

each grid point. The values of

∂u ∂z

∂2 w

and ∂2 w

differencing. The values of differences axially and central differences in the transverse directions. The values of vorticity on the boundary walls can be evaluated by following equations. ζ1 1 ,j = 2

v2, j 1 1 (ζ1, j + ζ2, j ) = (u1 1 , j+1 − u1 1 , j−1 ) − , 2 2 2
y 2
x

ζ1, j = −2v2, j /
x + (u2, j+1 − u2, j−1 )/(2
y) − ζ2, j ,

(75) (76)

where u1 1 , j+1 = 12 u2, j+1 ; u1 1 , j−1 = 12 u2, j−1 2

2

2.3.5 Solution algorithms 1. Solve velocity and pressure field for anode domain first. The initial values of the unknowns u, v, and ζ are assigned to be zero at the entrance, z = 0. Uniform inlet axial velocity (i.e., w = 1) is used. Note that ζ = 0 at z = 0 results from the vorticity definition. 2. Discretizing eqns (69), (70) with power-law scheme [33] and solving them in the coupled domain. Vorticity ζ can be calculated for gas channel, gas diffuser and catalyst layer.

Numerical simulation of proton exchange membrane fuel cell

239

3. The elliptic-type eqns (67) and (68) are solved for u and v iteratively. During the iteration process, the values of vorticity on the boundaries are evaluated using eqn (76). 4. Using control volume method with power-law scheme to discretize momentum equation in z direction, plug in new values u and v solved in step 2, one can solve axial velocity w in coupled domain with constraint (71) to meet the requirement of the constraint flow rate. 5. Steps 3 – 4 are repeated at a cross section until the following convergence criterion is satisfied for the velocity components u and v:    n  max ui,n+1 j − ui, j    < 10−5 , (77)  n+1  max ui, j  where n is the nth iteration of steps 3 – 4. 6. Repeat steps 1– 5 for cathode domain. One can calculate velocity distributions for gas channels, gas diffusers and catalyst layers in cathode domain. 7. With obtained solutions u, v, and w for both anode and cathode channels, species concentration equations and potential equations for a single domain from anode electrolyte to cathode electrolyte can be solved iteratively. These steps are repeated until the following convergence criterion is satisfied:        max Xkm+1 − Xkm  m+1 − m    ,    < 10−6 , max  (78)  m+1  m+1  max Xk  where m is the mth iteration of step 6, and k is the kth species. 8. Steps 3 –7 are repeated at the next axial location until the final z location is reached. 9. Once the electrolyte potential is obtained, the local current density can be calculated along the axial direction using following equation: ∂ . ∂x The average current density is then determined by I ( y, z) = −σm

Iavg =

11 bL

!

!

b

dy 0

(79)

L

I ( y, z)dz.

(80)

0

2.3.6 Results and discussion 3-D models are generally more accurate and more detailed in transport phenomena than 2-D models. Jen et al. [30] investigated the secondary flow patterns in fuel cell channels. Zhou and Liu [28] analyzed detailed temperature distribution in fuel cell. Um and Wang [29] studied the effectiveness of the interdigitated flow field on fuel cell performance. Figure 12 shows the comparison of Zhou and Liu’s [28] 3-D model numerical results with experimental data [32]. The operating conditions are: cathode flow rate is 1200 cm3 /s; anode flow rate is 1200 cm3 /s; cathode temperature

240 Transport Phenomena in Fuel Cells

Figure 12: 3-D Comparison of computational result with experimental data. is 60 ◦ C; anode temperature is 60 ◦ C; cathode pressure is 3 atm; anode pressure is 1 atm. It can be found that the 3-D model numerical results for the polarization curve agree well with the experimental data except in the region of the concentration polarization loss dominate. Even if some discrepancies can be seen in this region, the basic trend is still good. Note that the polarization of concentration region was not given in Ticianelli et al. [11, 12] experimental investigations. Most early 1-D or 2-D models did not investigate this region because no experimental data are available for this region. 2.3.6.1 Secondary flow pattern In Jen et al’s 3-D model [30], a general secondary flow pattern in fuel cell channel were also investigated as shown in Fig. 13. At the location z¯ = 0.0005, as shown in Fig. 13(a), the secondary flow moves out from porous media in the core region. This is simply due to the acceleration of the core gas channel flow and the strong porous media resistance to push the fluid out of the gas diffuser in the early entrance region. Figure 13(b) shows the vector secondary flow pattern at z¯ = 0.008, which has two pairs of vortices, with one small pair counter rotating cells near the corner of the gas channel. It is observed that, near the core region, a fairly uniform secondary flow running from the left side to right side into the gas diffuser, and there are outflows from the gas diffuser to gas channel near the top and bottom wall. It is also interesting to see two counter rotating cells at the top and bottom left corner, which are generated due to the corner effect of the gas channel. In nearly fully developed region at z¯ = 0.2 (Fig. 13(c)), the secondary flow strength has decreased significantly, and the effect of cross-sectional convection is negligible. 2.3.6.2 Temperature distributions Zhu and Liu [28] analyzed the temperature distribution in PEMFC. The temperature distributions across the whole fuel cell

Numerical simulation of proton exchange membrane fuel cell

241

Figure 13: Secondary flow vector plots. sandwich depend on the heat generation of the chemical reaction, Joule heating of the current, and cooling from channel walls. Due to very effective cooling, constant cell wall temperature is assumed; the typical temperature distribution is shown in Fig. 14 as an illustrative case. The inlet temperature of the air and fuel are both assumed to be 82 ◦ C, inlet pressure at anode/cathode is 1 atm/3 atm. The major heat source is the chemical reaction and Joule heating in the cathode side catalyst layer. The maximum temperatures are located within the cathode side catalyst layer near air entrance region, as shown in Fig. 14. This is due to the high chemical reaction rate. The outer walls of the gas channels are kept at the constant temperature of 82 ◦ C, heat is transferred out of the fuel cell effectively by the cooling along the outer walls of the gas channel so the inside of the fuel cell is prevented from overheating. The temperature distribution profiles can be used to improve fuel cell design to avoid membrane burning and drying out. 2.3.6.3 Water concentration variation Water management is very important for PEM fuel cell performance. During fuel cell operation, water within membrane is driven from the anode side to the cathode side by electro-osmosis, and at the same time it is driven in the opposite direction by diffusion [28]. If the water generation is more than water transport from the cathode, it causes cathode side flooding. On the other hand, if the water loss from anode is more than water supplies, it causes membrane dehydration which leads to high ohmic losses. The conductivity of the polymer electrolyte is a strong function of the degree of hydration, and flooding of

242 Transport Phenomena in Fuel Cells

Figure 14: Temperature distributions across the fuel. the cathode. It is common practice to saturate inlet fuel and air streams. However, due to the water generation along the cathode and electrical-osmosis drag from the anode to cathode, the water vapor concentration increases along the cathode channel. As the inlet cathode stream is already fully saturated, the added water from the chemical reaction and osmosis drag causes the cathode side to become over-saturated. In other words, liquid water exists along the cathode side and this could lead to flooding. To avoid this phenomenon, it is then suggested that the water content in the cathode inlet stream should be reduced. Figures 15 and 16 show the water vapor distributions along the cathode side and anode side when the hydrogen stream at the anode is fully saturated and the inlet air stream at the cathode is completely dry. Water vapor concentration along the cathode is increased due to the combined effect of water generation from the chemical reaction and electrical-osmosis drag from the anode to cathode as shown in Fig. 15. Under such operation conditions, the water vapor concentration along the anode is reduced due to the electrical-osmosis drag as shown in Fig. 16. Under this operation condition, membrane flooding can be avoided. However, the water vapor concentration near the inlet of the cathode is very low, the membrane dehydration may occur. Therefore, a proper water vapor added to the cathode stream and fully saturated anode stream are beneficial for fuel cell performance. 2.3.6.4 The effect of the flow fields Um and Wang [29] investigated the effects of conventional flow fields and interdigitated flow fields on fuel cell performance. Conventional flow is usually straight streamtraces due to them all having open inlet and outlet. In contrast to conventional flow fields, interdigitated flow channels indicate that air comes in along the lower channel, penetrates through the porous backing layer and

Numerical simulation of proton exchange membrane fuel cell

243

Figure 15: Water vapor mole fraction varies along the flow direction in cathode channel.

Figure 16: Water vapor mole fraction varies along the flow direction in the anode channel. exits through the upper channel. During the process, more oxygen is brought to the cathode catalyst reaction site by the forced convection, which results in better cell performance. Figure 17 shows the effect of the interdigitated flow field on the cell performance. There is a little difference in the current densities until cell potential reaching 0.55 V because the cell potential loss for current densities below 1 A/cm2 is primarily dominated by the ohmic resistance. However, for Iavg > 1 A/cm2 , the cell polarization curve begins to be limited by mass transport. In this region the positive

244 Transport Phenomena in Fuel Cells

Figure 17: The polarization curve for conventional and interdigtitated flow fields at 353 K. role of the interdigitated flow field becomes apparent. It is seen that the mass transport limiting current density is much improved by the use of interdigitated flow field. 2.4 Summary and conclusion The numerical simulation presented here enables prediction of phenomena in the entire fuel cell sandwich, including the two gas flow channels, two gas diffusers, two catalyst layers and membrane. It is able to predict detailed distributions of velocity fields, species concentration, current density, temperature, and polarization curves. It can be used to understand the interacting, complex electrochemical and transport phenomena that cannot be visualized experimentally. It also provides the ways to increase fuel cell power output for future fuel cell design, such as increase cell temperature, pressure, decrease the membrane thickness, choosing proper water vapor contents, and flow fields etc.

References [1] [2] [3]

[4]

Crowe, B.J. Fuel Cells: A Survey, NASA Rep. (SP-5115), Washington, DC, 1973. Savinell, R.F. & Fritts, S.D., Theoretical performance of a hydrogen-bromine rechargeable SPE fuel cell. J. Power Sources, 22, p. 423, 1988. Ridge, S.J., White, R.E., Tsau, Y., Beaver, R.N. & Eisman, G.A., Oxygen reduction in a proton exchange membrane test cell. J. Electrochem. Soc., p. 136, 1902, 1989. Fritts, S.D. & Savinell, R.F., Simulation studies on the performance of the hydrogen electrode bonded to proton exchange membranes in the H2 -Br2 fuel cell. J. Power Source, 28, pp. 301–315, 1990.

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[5] Yang, S.C., Cutlip, M. B. & Stonehart, P., Further development of an approximate model for mass transfer with reaction in porous gas-diffusion electrodes to include substrate effects. Electrochem. Acta, 34, p. 703, 1989. [6] Verbrugge, M.W. & Hill, R. F., Ion and solvent transport in ion-exchange membranes. J. Electrochem. Soc., 137, pp. 886, 1990a. [7] Bernardi, D.M. & Verbrugge, W.M., Mathematical model of a gas diffusion electrode bonded to a polymer electrolyte. AIChE, 37, pp. 1151–1163, 1991. [8] Springer, T.E., Zawodinski, T.A. & Gottesfeld, S., Polymer electrolyte fuel cell model. Electrochem. Soc., 136, pp. 2334–2341, 1991. [9] False, J.L., Vanderborgh, N.E. & Stroeve, P., The influence of channel geometry on ionic transport, Diagrams. Separators, and Ion-Exchange Membranes, 86, pp. 13, 1986. [10] Fuller, T.F. & Newman, J., Water and heat management in solid-polymerelectrolyte fuel cell. J. Electrochem. Soc., 140, pp. 1218–1225, 1993. [11] Ticianelli, E.A., Derouin, C. R., Redondo, A. & Srinivasan, S., Methods to advance technology of proton exchange membrane fuel cells. J. Electrochem. Soc., 135, pp. 2209–2214, 1988a. [12] Ticianelli, E.A., Derouin, C. R. & Srinivasan, S., Localization of platinum in low catalyst loading electrodes to attain high power densities in SRE fuel cells. J. Electroanal. Chem., 251, pp. 275–295, 1988b. [13] Newman, J., Electrochemical Systems, Prentice-Hall: Englewood Cliffs, NJ, 1973. [14] Bard, A.J. & Faulkner, L.R., Electrochemical Methods, Wiley: New York, 1980. [15] Nernst, W., Die elektromotorische wirksamkeit der Jonen: I. Theorie der diffusion. Z. Physic Chem., 4, pp. 129, 1889. [16] Planck, M., Ueber die erregung vol electricitat und warme in electrolyten. Ann, d. Phys. U. Chem., 39, p. 161, 1890. [17] Bird, R.B., Stewart, W.E. & Lightfoot, E.N., Transport Phenomena, Wiley: New York, 1960. [18] Srinivasan, S., Manko, D.J., Koch, H., Enayetullab, M.A. & Appleby, J.A., J. Power Sources, 29, p. 367, 1990. [19] Fuller, T.F. & Newman, J., Water and thermal management in solid-polymerelectrolyte fuel cells. J. Electrochem. Soc., 5, pp. 1218, 1993. [20] Nguyen, T.V. & White, R.E., A water and heat management model for proton-exchange membrane fuel cells. J. Electrochem. Soc., 140, pp. 2178–2186, 1993. [21] Amphlett, J.C., Baumert, R.M., Mann, R.F., Peppley, B.A. & Roberge, P.R., Performance modeling of ballard mark IV solid polymer electrolyte fuel cell. J. Electrochemical Society, 142, pp. 1, 1992. [22] Yi, J.S. & Nguyen, T.V., An along-the-channel model of PEM fuel cells, J. Electrochem. Soc., 145, pp. 1149–1159, 1998. [23] Gurau, V., Liu, H.T. & Kakac, S., Two dimensional model for proton exchange membrane fuel cells. AIChE Journal, 44, pp. 2410–2422, 1998.

246 Transport Phenomena in Fuel Cells [24] Yi, J.S. & Nguyen, T.V., Multi-component transport in porous electrodes of proton exchange membrane fuel cells using inerdigitated gas distributors. J. Electrochem. Soc., 146, pp. 38–45, 1999. [25] Um, S. & Wang, C.Y., Computational fluid dynamics modeling of proton exchange membrane fuel cells. Journal of The Electrochem. Soc., 147, pp. 4485–4493, 2000. [26] Fuel Cell Hand Book, Fifth edition, EG&G Services Parsons, Inc. Science Applications International Corporation, 2000. [27] Shimpalee, S. & Dutta, S., Effect of humidity on PEM fuel cell performance. Part-II Numerical Simulation, HTD, 364,1. Heat Transfer Division. ASME 1999. [28] Zhou, T. & Liu, H., A general three-dimensional model for proton exchange membrane fuel cells. I.J. Trans. Phenomena, 3(3), pp. 177–198, 2001. [29] Um, S. & Wang, C.Y., Three dimensional analysis of transport and reaction in proton exchange membrane fuel cells. Proceedings of the ASME Fuel Cell Division – 2000: The 2000 ASME 5, 10, Walt Disney World Dolphin, Orlando, FL. [30] Jen, T. C., Yan, T.Z. & Chan, S.H., Chemical reaction transport phenomena in a PEM fuel cell. International Journal of Heat and Mass Transfer, 46, pp. 4157–4168, 2003. [31] Wood, D.L., Yi, J.S. & Nguyen, T.V., Effect of direct liquid water injection and interdigitated flow field on the performance of proton exchange membrane fuel cells. Electrochem. Acta, 43, pp. 3795–3809, 1998. [32] Huang, Z., Experimental and mathematical studies for PEM fuel cell performances, M.S. thesis, University of Miami: Coral Gables, Florida. [33] Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere: Washington, DC, 1980.

CHAPTER 7 Mathematical modeling of fuel cells: from analysis to numerics M. Vynnycky & E. Birgersson FaxénLaboratoriet, Royal Institute of Technology, Stockholm, Sweden.

Abstract The mathematical modeling of transport phenomena in fuel cells is rendered difficult by the vast array of effects that have to be considered: flows are often highly three-dimensional, non-isothermal, multiphase, multicomponent and timedependent, and occur over several media: flow channels, porous electrode, catalytic layer and electrolyte. Contrary to the most recent trends in fuel cell modeling, which typically involve extensive 3D computational fluid dynamics, the approach we advocate here involves the use of scaling arguments, nondimensionalization and asymptotic techniques to identify the main governing parameters in a model and, subsequently, to derive a reduced model. The benefit of this is a model that is considerably cheaper to compute, but which does not sacrifice any significant physical features. These ideas are illustrated initially for the case of 2D momentum and multicomponent mass transfer in the performance-limiting electrode of a polymer electrolyte fuel cell (PEFC) and a direct methanol fuel cell (DMFC). For the second case, quantitative comparison is provided of the computed results from, and the computation times for, the full and reduced models; all in all, good agreement is achieved, but with two orders of magnitude less computing time being required for the reduced formulation.

1 Introduction The recent explosive increase in interest in fuel cells has been accompanied by an increase in the use of mathematical modeling as a tool, both to interpret experimental results and to provide insight on how to improve fuel cell design and performance. The task is exacerbated by the vast array of physical phenomena that can occur simultaneously: for example, in the operation of a typical seven-layer polymer electrolyte fuel cell (PEFC), consisting of two sets of flow channels, an anode and

248 Transport Phenomena in Fuel Cells

Figure 1: Heart of a PEFC. cathode porous electrode (often referred to as porous backings or gas diffusion layers), anode and cathode catalytic layers (also referred to as active layers) and a polymer electrolyte as membrane shown in Fig. 1, some, or all, of the following phenomena are thought to be of importance: • • • • • • • • • • • • • •

water transport; phase change in the porous backings and flow channels; conjugate heat transfer between bipolar plates/cooling channels and bipolar plates/flow channels; two-phase (gas and liquid) flow in the cathode porous backing and flow channels; electron transport in the cathode porous backing; ohmic heating in the membrane and the catalytic layers; water production at the cathode catalytic layer; convective heat/mass transfer in the flow channels; electro-osmotic drag in the membrane; electrochemical reactions at the catalytic layers; diffusion of H2 /O2 through the membrane; proton transport in the membrane; heat generation in the catalytic layers; Knudsen diffusion in the porous backing.

In addition, flows can often be highly three-dimensional, as is evident from the figure, and fuel cell operation for a variety of applications can be time dependent.

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249

The state of the art in the modeling of transport phenomena in fuel cells is widely recognized to involve the use of 3D computational fluid dynamics (CFD) to describe flow in the channels and porous backings, coupled to models for the electrochemical reactions in the anode and cathode catalytic layers, as well as water transport through the membrane; most recent examples of this for PEFCs are [1–6] with, in several cases, the implementation of computational models in commercially available software [7–10]. There can be little doubt that 3D CFD, provided it is carried out correctly, is able to provide, geometrically at least, the most detailed model description of transport phenomena in a fuel cell. It does, however, have its drawbacks, mostly associated with lengthy computing times: 1. To solve the model equations numerically for one combination of operating parameters can be prohibitively expensive; furthermore, there are often many possible such combinations (humidification, temperature, anode/cathode pressure, stoichiometry), and there may well be numerous model constants also, particularly for multi-phase flows where empirical relations have to be used. 2. Often, it is stacks of cells, rather than individual cells, that are of interest; this increases the number of mesh points required for a numerical solution. 3. It is difficult to incorporate this approach into system studies, the model equations for which can be solved several orders of magnitude more quickly. 4. The simulation of cells operating in transient mode is also often of most interest. 5. There is no real possibility to use the models further, e.g. as the basis for fuel cell control strategies or for optimisation studies, again because of the lengthy computing times that this would entail. 6. It would be difficult, if not impossible, to identify model simplifications that could be exploited to reduce computing times. There is, therefore, a need for modeling that describes the physics that 3D CFD describes, but without the prohibitively large computational requirements. Prior to the use of CFD, this was first done for PEFCs through one-dimensional models [11–14], and later through pseudo-two-dimensional ‘along-the-channel’ models [15–18]. The respective disadvantages of these approaches are: 1. One-dimensional approaches, whilst they are able to address some aspects of the three issues related to fuel cell performance mentioned above, are not able to address these questions at a local level: that is to say, where oxygen depletion occurs, where there is flooding or inadequate heat removal. 2. ‘Along the channel’ models result in ordinary differential equations with the coordinate along the fuel cell as the independent variable and do not satisfy mass, momentum and energy balance locally, although they may do so globally; consequently, they are formally valid only in a global sense. In both cases, almost all studies have been carried out in dimensional variables; it is therefore difficult to establish whether model simplifications are possible, or what the relative importance of competing physical effects might be.

250 Transport Phenomena in Fuel Cells In this article, we fill the niche that exists between the types of modeling indicated above by highlighting the power of a reduced model approach. How this is carried out will be shown in later Sections 2 and 3, but the general idea will be to take the full model equations, e.g. as would be implemented in a CFD model, nondimensionalize, identify key dimensionless parameters and with luck (if there exist parameters which are much larger or smaller than one) reduce the model’s complexity, but without reducing its practical usefulness. It should be emphasized that, in the present context, the reduced model approach is not the same as a simple reversion to the non-CFD fuel cell models mentioned earlier [11–18]; rather, it starts with an analysis of the full equations, and reduces these to a formulation whose solution, if not obtainable analytically, is then almost invariably less computationally-demanding than the original one. The emphasis then is on making the behavior of a complex model more transparent by means of mathematical analysis; through this approach, time spent doing ‘pen-and-paper’ analysis at an early stage of model development is recouped later through a much clearer understanding of the model’s structure, as well as a much faster solution time. The development presented here reflects our own efforts in establishing reduced nondimensional models for steady state multicomponent flow in slender fuel cell geometries, with a particular focus on the hydrodynamics in the performancelimiting electrode of a particular type of cell: cathode for the PEFC and anode for the DMFC. The essence of the mathematical approach advocated can, however, be used for more general problems: the whole cell, multiphase flow, transient behavior, etc. We begin with a formulation for gaseous flow in the cathode of a PEFC, from which we derive a reduced model. Subsequently, we show that that model itself can be reduced further to describe flow in the anode of a DMFC. In both cases, we find algebraic relations between certain variables that would not have been apparent had a dimensional formulation been used. Furthermore, the reduced model equations are parabolic and are solved numerically using a marching scheme; this is found to be considerably faster than a solution to the full elliptic formulation of the original equations, and not significantly less accurate. The work is based on already published work [19, 20], and on work still in progress [21].

2 PEFC A schematic diagram of the operation of a PEFC is given in Fig. 2. Essentially, this entails a polymer electrolyte membrane sandwiched between two gas-diffusion electrodes, which are each adjacent to flow channels contained within bipolar plates. The oxidant, usually oxygen from air which is either dry or humidified to some extent, is fed in at the inlet of the channel on the cathode side, and is transported to the electrolyte/cathode interface; the fuel on the other hand, normally hydrogen, is fed at the anode channel inlet and is transported to the electrolyte/anode interface.

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Figure 2: Schematic of a PEFC. Both interfaces contain catalyst, often platinum, to accelerate the reactions 2H2 → 4H+ + 4e− +

−

O2 + 4H + 4e → 2H2 O

at the anode,

at the cathode,

(1) (2)

in the course of which an electric current is produced to drive a given load. In particular, the reaction at the cathode also produces both heat and water as byproducts, the latter of which may be present throughout the system as either vapor or liquid, or both; the production of the former can lead to temperatures at the catalytic layer in the order of 80–90 ◦ C. Optimal fuel cell performance is achieved at typical voltages of around 0.5 V at current densities of about 1 Acm−2 . 2.1 Mathematical formulation for flow in the cathode In what follows, we begin by considering a steady 2D flow of a gaseous air/water mixture in the slender channel and porous backing of a fuel cell. 2.1.1 Channel Consider flow in a channel of height hf , adjacent to a porous medium of length L and height hp (see Fig. 3). The equations of continuity of mass and momentum for the mixture are taken as ∇ · (ρv) = 0,  2µ ∇ · (ρv ⊗ v) = −∇ p + ∇ · v + µ∇ 2 v − ρgj, 3

(3) (4)

where ρ, µ, v are the density, viscosity, and mass-averaged velocity of the mixture, respectively, g is the acceleration due to gravity and j is the unit vector in the positive y-direction; for later use, it is also convenient to define p , the modified

252 Transport Phenomena in Fuel Cells

Figure 3: Cathode of a PEFC. pressure, given in terms of the pressure p by 2 p = p + µ∇ · v. 3 The continuity equation for each of the species is given in terms of the mass flux, n, by (5) ∇ · ni = 0, i = O2 , H2 O, N2 , with

n ρMi xi ρ  Mi Mj Dij ∇xj , v+ 2 ni = M M

i = O2 , H2 O, N2 ,

j=1

where xi is the mole fraction of species i, and the second term on the right-hand side is the mass diffusive flux for an ideal gas mixture [22], due to concentration diffusion alone, relative to the mass-averaged velocity v. Also, (Mi )i=1,..,O2 ,H2 O,N2 are the molecular weights, M = MO2 xO2 + MH2 O xH2 O + MN2 xN2 , and (Dij )i, j=1,..,O2 ,H2 O,N2 are the multicomponent diffusion coefficients, given by ( ) *+ xk Mk /Mj Dik − Dij Dij = Dij 1 + , i, j, k = O2 , H2 O, N2 ; xi Djk + xj Dik + xk Dij  here, the Stefan-Maxwell diffusion coefficients, Dij i, j=1,..,n , are independent of composition, and can in principle be measured experimentally. Equation (5) can then be recast in the form of two transport equations,  



 ρv xO2 ρ ∇xO2 ∇· =∇· , (6) M ∇xH2 O M xH2 O M2 where M = M N2



DO2 ,N2 DH2 O,N2

 DO2 ,N2 0 − DH2 O,N2 MO2 DH2 O,O2

MH2 O DO2 ,H2 O . 0

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253

Here, use has been made of the relation xO2 + xH2 O + xN2 = 1 to eliminate xN2 . For the mixture density, we use the constitutive relation for an ideal gas, ρ=

pM , RT

(7)

where T is the temperature and R is the universal gas constant (8.314 kg m2 s−2 mol−1 K−1 ). We note, in addition, the possibility that the mixture viscosity, µ, will not necessarily be constant either, although we have treated it as so here. 2.1.2 Porous backing For the porous backing, volume-averaging techniques are required, as is care in distinguishing between intrinsic and superficial quantities. First, let B be a quantity (either scalar, vector, or tensor) associated with the gas phase, and let the quantity B be the local volume (or superficial) average of B, defined by ! 1 B ≡ Bd V, (8) V V (g) and let B(g) be the intrinsic volume average of B in the gas phase, where ! 1 g) ( B ≡ Bd V. V (g) V (g)

(9)

Also, let γ be the porosity, given by γ = V (g) /V. A comparison of eqns (8) and (9) shows that the local and intrinsic volume average for the gas phase is given by B = γ B(g) . (10) With these definitions, the governing equations in the porous backing become   ∇ · ρ(g) v = 0, (11)  κ κ  ,  -(g) (12) ∇ p + ρ(g) gj + ∇ 2 v , µ γ    , -    , -  3 g) g ( ( ) g g ( ) ) ( ρ v xO2  γ 2 ρ  (g) ∇, xO2 ∇· = ∇ ·  -(g) , 2 M ( g)  , g ( ) M  g xH 2 O ∇ x ( ) H2 O M  v = −

(13) where

, M(g) = MN2 ,  −

DO2 ,N2

DH2 O,N2 ,

,

-(g) -(g)

0

MO2 DH2 O,O2

-(g)

,

DO2 ,N2

-(g)  -(g) 

DH2 O,N2 , -(g)  MH2 O DO2 ,H2 O 0

,

254 Transport Phenomena in Fuel Cells with ,

Dij

-(g)

 

 *  Mk /Mj Dik − Dij , = Dij 1 + , -g  xi (g) Djk + xj ( ) Dik + xk (g) Dij  xk (g)

)

i, j, k = O2 , H2 O, N2 . Here, (12) is Darcy’s law with Brinkman extension, and the Darcy law permeability tensor is assumed to be isotropic and constant, leading simply to a constant value of the permeability, κ. Formal details of how (11)–(13) can be arrived at are given in [19]. 2.1.3 Boundary conditions 2.1.3.1 Inlet, outlet, upper wall, vertical walls For boundary conditions in the channel, we prescribe inlet velocity and composition at x = 0, 0 ≤ y ≤ hf , so that u = U in ,

v = 0,

in xO2 = xO , 2

in xH2 O = xH , 2O

(14)

where v = (u, v). At the upper channel wall (0 ≤ x ≤ L, y = hf ), there is no slip, no normal flow and no componental flux, so that u=v=

∂xO2 ∂xH2 O = = 0. ∂y ∂y

(15)

At the outlet at x = L, 0 ≤ y ≤ hf , we have constant pressure and no diffusive componental flux, so that p = pout ,

∂v ∂xO2 ∂xH2 O = = = 0. ∂x ∂x ∂x

(16)

At the vertical walls of the porous electrode (x = 0, L, −hp ≤ y ≤ 0), we prescribe no normal flow, no tangential shear and no mass flux for the gas components, so that -(g) , -(g) , ∂ xO2 ∂ xH2 O ∂ v u = = = = 0, (17) ∂x ∂x ∂x where v = (u, v). 2.1.3.2 Channel/porous backing interface In addition, matching conditions are required for the fluid-porous interface at y = 0, 0 ≤ x ≤ L. The conditions for continuity of normal velocity and normal stress are given respectively as v = v , p−µ

∂ v ∂v = p(g) − µeff , ∂y ∂y

(18) (19)

where µeff (=µ/γ) is termed the effective viscosity of the porous medium. The remaining two conditions that are required have been the subject of longstanding

Transport Phenomena in Fuel Cells

255

debate ever since the work of Beavers and Joseph [23], and a summary of possible options for the momentum equation is given by Alazmi and Vafai [24]; of these, the most relevant for this application is one due to Ochoa-Tapia and Whitaker [25] when inertial effects are important: u = u ,

(20)

µ ∂ u ∂u β1 µ (21) −µ = 1 u + β2 ρu2 , γ ∂y ∂y κ2 respectively. Here, β1 and β2 are O (1) constants which would need to be determined experimentally, although it turns out here that the leading order problem is dictated more by (20) than by (21). Finally, analogous volume-averaging techniques at the interface to those used for heat transfer by [26] are required for the mole fraction transport equations. We do not pursue the details, but simply assume the point values for the mole fractions of O2 and H2 O in the channel to be equal to their intrinsic values in the porous backing, so that , -(g) , -(g) (22) xO2 = xO2 , xH2 O = xH2 O , at y = 0, and in addition that the point values for the mole fraction fluxes of O2 and H2 O are equal to their superficial values in the porous medium, so that , , nO2 · N = nO2 · N , nH2 O · N = nH2 O · N , where N is the outward unit normal; using (18) and (22), we arrive at , - 

 ( g) 3 ∂ ∂ xO2 xO2 , = γ2 , ∂y xH2 O (g) ∂y xH2 O

(23)

respectively. 2.1.3.3 Catalyst/porous backing interface At y = −hp , we would expect u, v, xO2 (g) and xH2 O (g) to match to their counterparts in the catalytic layer, although naturally this approach would require us to model this layer, and then by extension the membrane and the corresponding regions on the other electrode. An alternative approach, often adopted when the flow field in the porous backing and gas channels rather than the electrochemistry in the catalyst and the membrane is of interest, is to prescribe a functional form for the current density, I , at this interface. Using Faraday’s Law, the superficial mass flux of the reactant is given as a function of current density, so that , MO2 I nO2 · N = − , (24) 4F where F is the Faraday constant. The corresponding expression for water is then taken to be - MH2 O (1 + 2α)I , , (25) nH2 O · N = 2F

256 Transport Phenomena in Fuel Cells where α is a parameter accounting for the total water transport by all mechanisms, e.g. electro-osmosis, in the membrane; typical values encountered in the literature are α = 0.3 [28] and 0.5 ≤ α ≤ 1.7 [18, 29]. Furthermore, since nitrogen does not participate in the reaction at the catalytic layer, , nN2 · N = 0. (26) -(g) , -(g) , This leads to the following boundary conditions for v, xO2 and xH2 O : ρ(g) v =

I  (2 + 4α)MH2 O − MO2 , 4F

(27)

and

, -  , -  

3 (g) ρ(g) v xO2 (g) ∂ γ 2 ρ(g) I xO2 −1 M . − = , ,   (g) 2 ∂y xH2 O (g) 4F 2(1 + 2α) M (g) xH2 O M (g) (28)

2.2 Nondimensionalization Writing x˜ = p˜ =

x , L

, v v˜ = in , U

y v , v˜ = in , L U

y˜ =

ρ˜ =

ρ , [ρ]

ρ ˜ ( g) =

ρ(g) , [ρ]

,  -(g) out  out (g) − pout p − pout (g) = p − p , p˜  = p − p , ,p˜  -(g) = p ˜  , p ,     2 2 2 2 [ρ] U in [ρ] U in [ρ] U in [ρ] U in I , I˜ = [I ] Mi =

M=

Mi , [M ]

Re =

M , [M ]

˜ ij = D

[ρ] U in L , µ ˜ = M

M(g) =

Dij , [D] Sc =

˜ ijeff = D µ , [ρ] [D]

M , [M ] [D]

M (g) , [M ] Dijeff [D]

,

Da =

c˜ =

c , [ρ] / [M ]

i, j = O2 , H2 O, N2 κ , L2

Fr =

U2 , gL

, -(g) M(g) ˜ M , = [M ] [D]

where [ρ] is a density scale, [D] is a diffusion scale, [I ] is a current density scale and [M ] is a molecular weight scale (all to be either determined or specified shortly), and Re, Sc, Da and Fr are the Reynolds, Schmidt, Darcy and Froude numbers, respectively, we drop the tildes and arrive at the following nondimensionalized forms. For the channel (0 ≤ x ≤ 1, 0 ≤ y ≤ hf /L), ∇ · (ρv) = 0,

(29)

Transport Phenomena in Fuel Cells

 2δ2 ∇ · (ρv ⊗ v) = −∇ p + ∇ · v + δ2 ∇ 2 v−Fr −1 ρj, 3    



ρ ρv xO2 δ2 ∇xO2 M ∇· = ∇· , ∇xH2 O M xH2 O Sc M2

257 (30) (31)

where δ2 = Re−1 , and for the porous medium (0 ≤ x ≤ 1, −hp /L ≤ y ≤ 0),   ∇ · ρ(g) v = 0,

(32)

 ,  -(g) δ2 2 2 v v = −∇ p − Fr −1 ρ(g) j, +δ ∇ (33) γ ε2    , -   , -   g) g) ( ( g g 2 ( ) ) ( ρ v xO2 δ  3 ρ  (g) ∇, xO2 = ∇ · γ 2  ∇· -(g) , 2 M g)  , ( g) ( Sc M xH 2 O ∇ xH 2 O M(g) (34) where ε2 = Da. The boundary conditions are now u = 1,

in in , xH2 O = xH , at x = 0, 0 ≤ y ≤ hf /L; (35) xO2 = xO 2 2O ∂xH2 O ∂xO2 (36) = = 0, at 0 ≤ x ≤ 1, y = hf /L; u=v= ∂y ∂y

v = 0,

∂v ∂xO2 ∂xH2 O = = = 0, at x = 1, 0 ≤ y ≤ hf /L; (37) ∂x ∂x ∂x -(g) , -(g) , ∂ xO2 ∂ xH2 O ∂ v u = = = = 0, at x = 0, 1, −hp /L ≤ y ≤ 0. ∂x ∂x ∂x (38) p = 0,

The boundary conditions for 0 ≤ x ≤ 1, y = −hp /L are now:  u = 0, ρ(g) v = 

I  2(1 + 2α)MH2 O − MO2 4

,

(39)

, -  , -  3 (g) ρ(g) v xO2 (g) ∂ δ2 γ 2 ρ(g) xO2 (g) M − , ,   (g) 2 ∂y xH2 O (g) M(g) xH2 O Sc M(g) =

I 4



−1 , 2(1 + 2α)

(40)

258 Transport Phenomena in Fuel Cells where  = [I ] [M ] /FU in [ρ]. Finally, the boundary conditions along the fluidporous interface on y = 0 reduce to v = v , p − δ2

(41)

∂v ∂ v = p(g) − δ2 , ∂y ∂y

(42)

u = u ,   1 ∂ u ∂u β1 β2 ρu2 , − = u+ γ ∂y ∂y ε δ2

(43) (44)

and ,

xO2

-(g)

= xO2 ,

,

xH2 O

-(g)

= xH2 O ,

(45)

, - 

 (g) ∂ ∂ xO2 xO2 . γ = , ∂y xH2 O (g) ∂y xH2 O 3 2

(46)

2.3 Parameters Typically, U in ∼ 1 ms−1 , hf ∼ 10−3 m, hp ∼ 3 × 10−4 m, L ≥ 10−2 m, [I ] ∼ 104 Am−2 , pout ∼ 1 atm ∼ 105 kg m−1 s−2 , T ∼ 300–350 K, 0.1 ≤ γ ≤ 0.5, 0.3 ≤ α ≤ 1.7, µ ∼ O(10−5 ) kg m−1 s−1 . In addition, MO2 = 0.032 kg mol−1 , MH2 O = 0.018 kg mol−1 , MN2 = 0.028 kg mol−1 , F = 96487 Asmol−1 , from which we note that Mmin ≤ M ≤ Mmax , where Mmin = M |xH2 O =1, xO2 =0 = 0.018 kg mol−1 , Mmax = M |xH2 O =0, xO2 =1 = 0.032 kg mol−1 . Further, we use the constitutive relation for an ideal gas in order to obtain the density scale [ρ]; with p ∼ pout , we have ρ ∼ 1 kg m−3 , so that [ρ] ∼ 1 kg m−3 seems appropriate. For [D], we take O(10−5 ) m2 s−1 from available literature, e.g. [11, 12]. Thence, for the nondimensional parameters Re, Sc, Da, Fr, , we arrive at Re ∼ 104 ,

Sc ∼ 1,

Da ≤ 10−6 ,

Fr ∼ 1,

 ≤ 10−2 ,

σ ∼ 10−2 ,

so that δ ∼ 10−2 and ε ≤ 10−3 . 2.4 Narrow-gap approximation Typically, hf /L, hp /L  1, which leads us to further rescaling as follows. Writing X = x, P = p,

v y , U = u, V = , U  = u, σ σ , - , P  = p , P = p , P  = p ,

Y =

V  =

v , σ

Transport Phenomena in Fuel Cells

259

where σ = hf /L, we simplify further by neglecting terms in O (σ) or lower, although we retain for the time being terms which contain multiples of σ and the other dimensionless parameters. We introduce the dimensionless parameters
,  and , given by
= δ2 /σ 2 ,

 = σ 2 /ε,

 = /σ,

 −1 and note that an alternative expression for
is
= Reσ 2 , i.e. the reciprocal of the reduced Reynolds number. We have now, for the channel, ∂ ∂ (ρU ) + (ρV ) = 0, ∂X ∂Y  ∂U ∂P  ∂2 U ∂U +V =− +
2, ρ U ∂X ∂Y ∂X ∂Y 0=−

∂P  , ∂Y

(47) (48) (49)

    



∂ xO2 ∂ ∂ 1 xO2 ρ
∂ M ρ U +V = , ∂X ∂Y M xH2 O Sc ∂Y M2 ∂Y xH2 O

(50)

and for the porous medium,   ∂  (g) ∂  ( g) ρ U  + ρ V  , ∂X ∂Y , -(g) ε ∂2 U  ε ∂ P U  = − , +
 ∂X γ ∂Y 2 , -(g) 1 ∂ P ε ∂2 V  V  = − , +
2 ∂Y γ ∂Y 2 0=

 (g )

ρ

∂ ∂ U  + V  ∂X ∂Y



1

,

M(g)

 3 2


∂  γ ρ(g) (g) ∂ =  2 M Sc ∂Y ∂Y M(g) Note also that

  P  = P + O δ2 ,

,

xO2

(51)

(52)

(53)

-(g) 

xH2 O

-(g)

 -(g)  xO  , 2 -(g)  . xH2 O

,

(54)

  ,  -(g) = P(g) + O δ2 , P

and since δ2  1, henceforth, we use the actual pressure rather than the modified pressure. In addition, the gravitational terms in (49) and (53) are O(Fr −1 σ) and

260 Transport Phenomena in Fuel Cells have therefore been dropped. The boundary conditions are: for 0 ≤ X ≤ 1, Y = 1, ∂xO2 ∂xH2 O = = 0; ∂Y ∂Y

(55)

V = V  ,

(56)

P = P(g) ,

(57)

U = U  ,   β1 σ β2 σ 1 ∂ U  ∂U − = U+ ρU 2 , γ ∂Y ∂Y ε δ2

(58)

U =V = for 0 ≤ X ≤ 1, Y = 0,

,

xO 2

-(g)

∂ γ ∂Y 3 2

= xO2 ,

,

-(g)

(59)

= xH2 O ,

(60)

 ∂ xO2 = ; -(g) , ∂Y xH2 O xH 2 O

(61)

,

xO2

xH2 O

-(g) 

 for 0 ≤ X ≤ 1, Y = −H =hp /hf ,  U  = 0,

ρ(g) V  = 

I  2(1 + 2α)MH2 O − MO2 4

,

(62)

, -  , -  3 (g) ρ(g) V  xO2 (g)
γ 2 ρ(g) xO2 g) ∂ ( M − , , 2  ( g) ∂Y xH2 O (g) M(g) xH 2 O Sc M(g) =



I −1 . 4 2(1 + 2α)

(63)

The neglect of streamwise diffusion terms will of course imply that not all of the original boundary conditions at X = 0 and 1 in this reduced formulation can be satisfied and those terms would need to be reinstated for X ∼ O(σ) and 1−X ∼ O(σ). This is beyond the scope of interest here, and for a consistent formulation we simply retain in in , xH2 O = xH , at X = 0, 0 ≤ Y ≤ 1; xO2 = xO 2 2O , -(g) -(g) , ∂ xO2 ∂ xH 2 O U  = = = 0, at X = 0, −H ≤ Y ≤ 0. ∂X ∂X

U = 1,

(64) (65)

For the initial discussion, we proceed under the assumption that ,
,  ∼ O(1); later, we will consider   1 also. Further simplification is now possible by noting from (52) that U  = 0 to leading order, which reduces (51) and (54) still further. In addition, Vynnycky and Birgersson [19] identify a porous boundary

Transport Phenomena in Fuel Cells

261

1

layer of thickness ε 2 which has to be accounted for, although this proves to be of no consequence for (54). Furthermore, the leading order equations prove to be independent of β1 and β2 ; these, and a variety of other details can be found in [19]. 2.5 Further simplifications and observations Invoking the constitutive relation for an ideal gas in dimensionless variables, with     in 2  2 [ρ] U in [ρ] U P(g) (66) P, ρ(g) = M(g) + ρ = M+ pout pout for the channel and porous medium respectively, which can be reduced to just ρ = M, ρ(g) = M(g) , respectively, for the pressures and velocities being considered here. The reduced system of equations is now, for 0 ≤ X ≤ 1, 0 ≤ Y ≤ 1, ∂ ∂ (ρU ) + (ρV ) = 0, ∂X ∂Y  ∂U dP ∂2 U ∂U +V =− +
2, ρ U ∂X ∂Y dX ∂Y





      ∂ ∂
∂ M ∂ xO2 xO2 xO2 + = , U V xH2 O xH2 O ∂X ∂Y Sc ∂Y M ∂Y xH2 O and for 0 ≤ X ≤ 1, −H ≤ Y ≤ 0,  I  ρ(g) V  =  2(1 + 2α)MH2 O − MO2 , 4

(67) (68) (69)

(70)

1 ∂ P(g) , (71)
2 ∂Y , -  , -  

3 (g) (g)
γ 2 I xO2 −1 xO2 g) ∂ ( V  , M . = -(g) − , ∂Y xH2 O (g) 4 2(1 + 2α) Sc M(g) xH 2 O V  = −

(72) The boundary conditions are: for 0 ≤ X ≤ 1, Y = 1, U =V =

∂xH2 O ∂xO2 = = 0; ∂Y ∂Y

(73)

and for X = 0, 0 ≤ Y ≤ 1, U = 1,

in xO2 = xO , 2

in xH2 O = xH , at X = 0, 0 ≤ Y ≤ 1; 2O

(74)

no boundary conditions as such prove to be necessary for X = 0, −H ≤ Y ≤ 0, since only ordinary differential equations are solved for −H ≤ Y ≤ 0. At Y = 0

262 Transport Phenomena in Fuel Cells for 0 ≤ X ≤ 1, porous and fluid quantities are matched through U = 0, V = V  , P = P(g) , ,

x O2

-(g)

∂ γ ∂Y 3 2

= xO2 ,

, ,

xO2

,

xH 2 O

= xH2 O ,

(76)



∂ xO2 . ∂Y xH2 O

(77)

xH2 O

-(g) 

-(g) =

-(g)

(75)

In general, I will not be constant; even more generally, it cannot be described a priori, but is determined by considering the transport of species in the catalyst, membrane and the anode side also. However, a common practice in studies which emphasize the investigation of flow in the porous backing and the gas channel is simply to prescribe a current density as a function of mole fraction. For example. if we use the dimensional form of the Tafel law given by He, Yi and Nguyen [27],  αc Fη aρ exp , I= M RT where αc (= 2) is the transfer coefficient of the oxygen reduction reaction, eqn (2), η is the overpotential for the oxygen reaction and a ( = 10−6 Am mol−1 ) is a constant related to the exchange current density and oxygen reference concentration for the oxygen reaction, we obtain the appropriate scale for [I ] as [I ] =

 αc Fη a [ρ] exp ; [M ] RT

(78)

consequently, in dimensionless form, , -g , -(g) -(g) , -(g)  ρ(g) xO2 ( ) = xO2 . I ρ , xO2 , xH2 O = g) ( M 

(g)

,

(79)

A dimensional quantity of importance for the determination of polarization curves is the average current density, Iav , which is then given by ! Iav = [I ]

1

I dX . 0

This completes the formulation and necessary definitions. The data given in Section 2.3 for the base case physical parameters indicates that
∼ O (1). Obviously, taking channels with smaller aspect ratio, or operating the fuel cell at lower inlet gas velocity would increase
, motivating consideration of the lubrication theory limit (
 1), since it provides qualitatively useful analytical solutions, as well as a quantitative comparison with our numerical method; this is done in [19].

Transport Phenomena in Fuel Cells

263

An interesting and instructive simplification occurs when   1 [21]. We introduce the asymptotic series   χ = χ0 + −1 χ1 + O −2 ,

where χ = (U , V , P, ρ) ,

  χ = χ( 0) + −1 χ( 1) + O −2 ,

 where χ = xO2 , xH2 O , M, M

into eqns (67)–(77). These, taken at leading order in −1 , are for the most part unchanged, except that eqn (70) and the two equations contained within (72) appear to collapse onto just one equation: (0) xO = 0 at 0 ≤ X ≤ 1, Y = −H. 2

(80)

Examining (70) and (72) at the next order in  gives, respectively, ρ0 V0 =

1 4



(1)

2(1 + 2α)MH2 O − MO2 xO2

at 0 ≤ X ≤ 1, Y = −H,

(81)

and  V0

(0)

xO2 (0)

xH2 O



3

∂
γ2 M(0) − (0) Sc M ∂Y



(0) xO 2 (0)

xH2 O



  −MO2 1 (1) = x . 4 2(1 + 2α)MH2 O O2

(82)

At first sight, this suggests difficulties since the leading order problem appears to (1) require information from higher orders; however, we note that xO can be eliminated 2 via (81) and the necessary boundary conditions are (80) and  V0

0 (0)

xH2 O



3


γ2 ∂ − M(0) (0) Sc M ∂Y

ρ0 V0 2(1 + 2α)MH2 O − MO2

= 



(0)

xO2



(0)

xH2 O 

−1 . 2(1 + 2α)

(83)

This reveals that the limiting current for a particular cell can be found, regardless of how the electrochemistry in the active layer has been modelled: once ρ0 and V0 have been computed, the (dimensionless) limiting current density profile would be 4ρ0 (X ) V0 (X ) Ilim (X ) = . 2(1 + 2α)MH2 O − MO2 Further, these observations suggest that the use of (80) and (83) as boundary conditions should assist numerical convergence for higher current densities.

264 Transport Phenomena in Fuel Cells 2.6 Numerics and results The simplified parabolized equations were solved numerically using the KellerBox discretization scheme and Newton iteration (see, for example, Cebeci and Bradshaw [30]). The system of partial differential equations to be solved in the channel, (67)–(69), is of 8th order, and this is coupled to a 6th order system of ordinary differential eqns (70)–(72), in the porous region. As is well-known, the scheme is second-order accurate in both time-like and space-like variables, and we omit any further details here. As an indication of the speed of the computations, we note that a typical run with 500 points across, and 200 points along, the channel required around 100 CPU seconds on a 500 MHz Compaq Alphaserver with 3 GB RAM. Results are presented for the Tafel law given in dimensionless form by (79), and used previously for PEFC studies by [18, 27, 29]. Throughout, we keep γ = 0.3, T = 353 K, and concentrate more on the effect of changes in channel height and length, porous backing thickness and permeability, pressure, inlet speed and composition. Physically realistic and implementable changes in any of these will result in, at most, an order of magnitude change in the relevant dimensionless parameter. The most sensitive parameter is , which varies over several orders of magnitude as the cell voltage Ecell decreases; note here that we revert to using the cell voltage rather than the overpotential, η, with the two being related by Ecell = E0 − η, where E0 (=1.1 V) is termed the open circuit voltage of the fuel cell. Analytical solutions are given in [19] for when
 1. Here, however, we concentrate on solutions for physically more realistic operating parameters when
∼ O (1). 2.6.1 Effect of
and  We show first results for Ecell = 0.75 V , corresponding to  = 10.2, ranging over several orders of magnitude in
, and compare these with the analytical results in the lubrication theory limit. Figures 4 and 5 are for intrinsic oxygen and water mole fraction at Y = −H, respectively,  and demonstrate that the lubrication solution works well for
as low as O 102 ; this is, however, considerably higher than the base case physical values given in Section 2.3, which correspond to
= 1.89. Figure 6 shows the streamwise velocity U at Y = 12 , and illustrates the extent of deviation from the classical value 32 . An interesting limit occurs as Ecell is decreased. In this case,  increases although the quantity xO2 (g) at Y = −H remains O(1); this corresponds to the attainment of the limiting current and the corresponding plots are given in Figs 7–9; observe that in Figs 7 and 8 the limiting values for intrinsic oxygen and water mole fraction for the analytical solution are reached very rapidly, so that in Fig. 7 the curve for xO2 (g) effectively lies on the X -axis. As regards the numerics, it was found that considerably more outer loop iterations for the density were required as  was increased. For instance, whereas 4 iterations

Transport Phenomena in Fuel Cells

265

, -(g) Figure 4: Comparison of analytical solution for xO2 at Y = −H with numerical solutions for
= 1.89, 1.89 × 101 , 1.89 × 102 (Ecell = 0.75 V).

, -(g) Figure 5: Comparison of analytical solution for xH2 O at Y = −H with numerical solutions for
= 1.89, 1.89 × 101 , 1.89 × 102 (Ecell = 0.75 V).

266 Transport Phenomena in Fuel Cells

Figure 6: Comparison of analytical solution for U at Y = 12 with numerical solutions for
= 1.89, 1.89 × 101 , 1.89 × 102 (Ecell = 0.75 V).

, -(g) Figure 7: Comparison of analytical solution for xO2 at Y = −H with numerical solutions for
= 1.89, 1.89 × 101 , 1.89 × 102 (Ecell = 0.65 V).

Transport Phenomena in Fuel Cells

267

, -(g) Figure 8: Comparison of analytical solution for xH2 O at Y = −H with numerical solutions for
= 1.89, 1.89 × 101 , 1.89 × 102 (Ecell = 0.65 V).

Figure 9: Comparison of analytical solution for U at Y = 12 with numerical solutions for
= 1.89, 1.89 × 101 , 1.89 × 102 (Ecell = 0.65 V).

268 Transport Phenomena in Fuel Cells sufficed for Ecell = 0.75 V, it was common for 20–30 to be necessary for Ecell = 0.65 V. In addition, there were difficulties in initiating the marching scheme at X = 0 for higher values of ; we surmise this to be due to the phenomena illustrated at the end of the previous section. Whilst setting the channel inlet values as an initial guess for the first step along the channel was adequate for lower values of , this was found not be sufficient for Ecell lower than 0.71 V; for those cases, the firststep solution for Ecell = 0.71 V had to be used instead, then enabling numerical solutions to be obtained for higher and higher values of  until the limiting current was reached. 2.6.2 ‘Polarization surfaces’ It is customary for fuel cell performance to be given in terms of a polarization curve where the cell potential, Ecell , is given as function of the average current density, Iav . Generally speaking, if the analysis is done dimensionally, this leads to a vast number of graphs for each alteration made in one of the physical parameters. However, a major benefit of the nondimensional analysis carried out here is that the results can be expressed considerably more compactly by plotting polarization ‘surfaces’; individual polarization curves will therefore be curves lying on those surfaces. We explain this as follows. From the nondimensionalization given above, the emergent non-dimensional parameters were
, ,  and Sc. In addition, there is γ, which we in , x in and H, whose effect on fuel cell performance hold fixed in this study, and xO H2 O 2 one would like to explore. First, we observe that, in the parameter range of interest,  has no effect on Iav , since the dimensionless density is independent of pressure and the pressure in the channel serves as a boundary condition for the pressure in the porous medium. In addition, a change in Sc can only be effected by changes in [ρ], which only occurs if the cathode is run at a different pressure. Consequently, a tidy in , xin and representation of Iav is to plot it as a function of
and , for fixed Sc, xO O2 2 H, the benefit of this being that the effect of four parameters, hf , L, U in and Ecell , are displayed on one graph; since  can vary over several orders of magnitude, it proves more convenient to use log () as a variable. Examples of this are given below. Figure 10 gives polarization surfaces for H = 0.3, with the pressure at 1 and 3 atmospheres. The limiting current phenomenon is observed as  increases, and its value is observed to increase moderately with increasing
, but strongly with increasing pressure. Figure 11 shows a similar plot, except with computations now for pout = 1 atm, for H = 0.15 and H = 0.6. Average current densities are found to be higher for the thinner porous backing, and in both cases a limiting value is evident as  is increased. Figure 12 compares the base case for pout = in = 0.21 with two other cases at 1 atm which have differing inlet 1 atm and xO 2 in = 1) and partially humidified air, for which x in = compositions: dry oxygen (xO O2 2 in = 0.36 (corresponding to 76% relative humidity) [1, 2, 16]. As is 0.13 and xH 2O evident, increased oxygen content at the inlet raises the average current density; for in = 1, convergence difficulties were experienced for quite low values of , which xO 2 explains the rather narrow range of values presented for this case, but nonetheless the average current density is much higher than that for the other two cases.

Transport Phenomena in Fuel Cells

Figure 10: Polarization surfaces for pout = 1, 3 atm (H = 0.3).

Figure 11: Polarization surfaces for H = 0.3, 0.6 ( pout = 1 atm).

269

270 Transport Phenomena in Fuel Cells

Figure 12: Comparison of analytical solution for U at Y = 12 with numerical solutions for  = 1.89, 1.89 × 101 , 1.89 × 102 (Ecell = 0.75 V).

Figure 13: Schematic of a DMFC.

3 DMFC A schematic diagram of a DMFC is given in Fig. 13. Essentially, this entails a polymer membrane sandwiched between two gas-diffusion electrodes, which are each adjacent to flow channels contained within bipolar plates. The oxidant, usually humidified oxygen from air, is fed in at the inlet of the channel on the cathode side, and is transported to the electrolyte/cathode interface; the fuel on the other hand,

Transport Phenomena in Fuel Cells

271

normally dilute liquid methanol, is fed at the anode channel inlet and is transported to the electrolyte/anode interface. Both interfaces contain catalyst to accelerate the reactions CH3 OH + H2 O → CO2 + 6H+ + 6e− at the anode, O2 + 4H+ + 4e− → 2H2 O at the cathode,

(84) (85)

in the course of which an electric current is produced to drive a given load. Optimal fuel cell performance is achieved at typical voltages of around 0.4 V at current densities of about 0.1 Acm−2 . 3.1 Mathematical formulation for flow in the anode In what follows, we begin by considering a steady 2D flow of a liquid water/ methanol/carbon dioxide mixture in the slender channel and porous backing of a DMFC. For brevity, we highlight here the similarities and differences with the PEFC cathode model already derived in Section 2.3. Throughout, we make the following associations from the three species in the cathode of a PEFC to those in the anode of a DMFC: O2 ⇔ CH3 OH,

H2 O ⇔ CO2 ,

N2 ⇔ H2 O.

3.1.1 Channel Equations (3)–(6) are unchanged. However, the diffusion tensor in the channel and the porous backing can be simplified to a diagonal tensor, since the system can be treated as a dilute solution, rendering the cross terms redundant, i.e 

0 DCH3 OH,H2 O , 0 DCO2 ,H2 O ,  -(l) DCH3 OH,H2 O 0 = MH2 O -(l) . , 0 DCO2 ,H2 O

M = MH2 O M(l)

For the mixture density, the constitutive relation for an ideal gas is now of course entirely inappropriate. By extrapolating literature data to 70 ◦ C, the influence of methanol on the density is estimated to lower the density by 0.4% at 2 wt.% methanol in the mixture [31]; this contribution can safely be neglected, whence we take the density of the mixture, ρ, as that of pure water, and hence constant. 3.1.2 Porous backing Superscripts (g) are replaced by (l) to denote liquid phase.

272 Transport Phenomena in Fuel Cells 3.1.3 Boundary conditions 3.1.3.1 Inlet, outlet, upper wall vertical walls These remain, for the most part, unchanged, although we assume a developed flow profile at the inlet, so that   2  y y in u = 4U , v = 0. (86) − hf hf 3.1.3.2 Channel/porous backing interface These remain unchanged. 3.1.3.3 Catalyst/porous backing interface The only changes are the replacement of (24)–(26) by I (1 + 6αCH3 OH )MCH3 OH nCH3 OH · N = − , 6F - IMCO2 , , nCO2 · N = 6F , I (1 + 6αH2 O )MH2 O nH2 O · N = − , 6F

,

(87) (88) (89)

where αH2 O and αCH3 OH are parameters accounting for the water and methanol transport across the membrane. In the following, we have neglected the contribution from the methanol crossover, given by αCH3 OH . This is a reasonable assumption for any practical DMFC application, since a high crossover leads to an unfavourable mixed potential at the cathode as well as loss of fuel at the anode. If the crossover cannot be neglected, terms accounting for the transport of methanol in the membrane can easily be added; see for example Scott et al.[32]. Equations (27) and (28) give , , -(l) -(l) then instead the following boundary conditions for v, xCH3 OH and xCO2 : ρ(l) v = −

I  (1 + 6αH2 O )MH2 O + MCH3 OH − MCO2 , 6F

(90)

and ρ(l) v M (l)

, , -(l)  -(l)  

3 xCH3 OH xCH3 OH γ 2 ρ(l) I −1 ∂ ¯ = M . , -(l) −  , (l) 2 ∂y 6F 1 xCO2 xCO2 M (l)

(91)

3.2 Nondimensionalization This is unchanged except that (39) and (40) are now 

u = 0,

I  ρ v = − (1 + 6αH2 O )MH2 O + MCH3 OH − MCO2 6 (l)

, (92)

Transport Phenomena in Fuel Cells

, , -(l)  -(l)  3 xCH3 OH xCH3 OH δ2 γ 2 ρ(l) (l) ∂ , -(l) , -(l) −  2 M ∂y M(l) xCO2 xCO2 Sc M(l)  I −1 . = 1 6

273

ρ(l) v

(93)

3.3 Parameters Typically, U in ∼ 10−2 ms−1 , xCH3 OH ∼ O(10−2 ), hf ∼ 10−3 m, hp ∼ 1.8 × 10−4 m, L ≥ 10−2 m, κ ∼ 10−12 m2 , [I ] ∼ 4 × 103 Am−2 , pout ∼ 1 atm ∼ 105 kg m −1 s−2 , T ∼ 300–350 K, 0.1 ≤ γ ≤ 0.9, αH2 O = 2.5 (see [33]), αCH3 OH = xCH3 OH αH2 O (see [32]), µ ∼ O(10−4 ) kg m−1 s−1 , ρ ∼ 978 kg m−3 . In addition, MCH3 OH = 0.032 kg mol−1 , MCO2 = 0.044 kg mol−1 , MH2 O = 0.018 kg mol−1 , F = 96487 Asmol−1 , R = 8.314 Jmol−1 K−1 , A = 157, B = 0.61, E0 = 0.504 V, EA = 0.5 V, αA = 7.9. For [D], we take O(10−9 ) m2 s−1 , which is the typical scale for liquid diffusion. Thence, for the nondimensional parameters Re, Sc, Da, , we arrive at Re ∼ 103 ,

Da ≤ 10−6 ,

Sc ∼ 50,

Fr ∼ 10−2 ,

 ≤ 4×10−4 ,

σ ∼ 10−2 ,

so that δ ∼ 3 × 10−2 and  ≤ 10−3 . 3.4 Narrow-gap approximation The details for this are unchanged except for 

U  = 0,

(94)

I  (1 + 6αH2 O )MH2 O + MCH3 OH − MCO2 , 6 , , -(l)  -(l)  3 x ρ(l) V  xCH3 OH
γ 2 ρ(l) ∂ CH OH 3 (l) 2 M , -(l) −  , -(l) ∂Y M(l) (l) xCO2 xCO2 Sc M ρ(l) V  = 

I = 6

 −1

(95)



1

.

(96)

Significantly,  ∼ 10−2 , which will enable us to make simplifications that were not possible for the PEFC cathode. 3.5 Further simplifications and observations Since the system is dilute, xH2 O  xCH3 OH , xCO2 , implying that we will be able to write in ; xCH3 OH + xCO2 = 1 − xH 2O

274 Transport Phenomena in Fuel Cells thus, we only need to solve the mass transfer equation for methanol and this relationship will provide the mass fraction of carbon dioxide. The mean molecular mass M(l) reduces to MH2 O , since the contribution of carbon dioxide and methanol is negligible. The fact that the density is constant enables us to write simply ρ(l) = 1. This also simplifies the diffusion coefficient DCH3 OH,H2 O , which we can treat as constant for the dilute system; thus, an appropriate choice for the characteristic diffusion coefficient scale, [D], is [D] = DCH3 OH,H2 O . With the simplifications outlined above, the dimensionless form of the equations for the channel is ∂U ∂V + = 0, ∂X ∂Y U

∂U ∂U ∂P ∂2 U +V =− +
2, ∂X ∂Y ∂X ∂Y ∂P = 0, ∂Y

U

∂xCH3 OH ∂xCH3 OH
∂2 xCH3 OH ; +V = ∂X ∂Y Sc ∂Y 2

(97) (98) (99) (100)

these are the same as eqns (67)–(69) for constant density ρ, and with P  = P. For the porous backing, we have ∂ V  = 0, ∂Y

(101)

U  = 0,

(102)

∂ P(l) = −
2 V  , ∂Y , -(l) , -l 3 ∂ xCH3 OH
γ 2 ∂2 xCH3 OH V  . = ∂Y Sc ∂Y 2

(103)

(104)

However, V  is still not correctly scaled, since it is of order (1); thus, we set V  Vˆ  = , (105)  so that (101), (103) and (104) become, respectively, ∂Vˆ  = 0, ∂Y ∂P(l) = −
2 Vˆ , ∂Y -(l) , -(l) , ∂ xCH3 OH
3 ∂2 xCH3 OH ˆ . = γ2 V  ∂Y Sc ∂Y 2

(106) (107)

(108)

Transport Phenomena in Fuel Cells

275

The interface condition for V at Y = 0, the second equality in eqn (75), implies that V will also be of order . Setting Vˆ = V /, the governing equations in the channel, (97)–(100), are then given by ∂U ∂Vˆ + = 0, ∂X ∂Y U

∂U ∂P ∂2 U ∂U + Vˆ =− +
2, ∂X ∂Y ∂X ∂Y ∂P = 0, ∂Y

U

∂xCH3 OH ∂xCH3 OH
∂2 xCH3 OH . + Vˆ = ∂X ∂Y Sc ∂Y 2

(109) (110) (111) (112)

Now, since  ∼ O(10−2 )  1, we can safely neglect terms of O(), which corresponds to V = 0; furthermore, eqns (102) and (58) imply that U = 0 at Y = 0. Consequently, we find that the inlet condition for a fully developed laminar velocity profile satisfies the momentum equation downstream, and thence that, at leading order, the velocity decouples from the mass transfer in the channel; all that remains to be solved there is 4(Y − Y 2 )

∂xCH3 OH
∂2 xCH3 OH . = ∂X Sc ∂Y 2

(113)

Now, eqns (95) and (106) combine to give, for the entire porous backing, Vˆ  = −

I ; 6

(114)

where  = (1+6αH2 O )MH2 O +MCH3 OH −MCO2 . Then, since the current density is a function of X alone, we have Vˆ  = Vˆ (X ). We can integrate (108) and, using (114) and (96), we arrive at -(l) , -(l)
3 ∂ xCH3 OH I  , I 2 − xCH3 OH − γ = − MH2 O . 6 Sc ∂Y 6

(115)

Furthermore, choosing [M ] = MH2 O , whence MH2 O = 1, eqn (115) becomes -(l) , -(l) , 6
3 ∂ xCH3 OH γ2 = 1.  xCH3 OH + ScI ∂Y

(116)

Integrating eqn (116) gives ,

xCH3 OH

-(l)

  ScI 1 + C exp − (X , Y ) = Y , 3  6
γ 2

(117)

276 Transport Phenomena in Fuel Cells where C is an integration constant to be determined shortly. At Y = −H, we have   -(l) , ScI 1 + C exp xCH3 OH (X , −H) = H , (118) 3  6
γ 2 and at Y = 0,

,

-(l)

1 + C, (119)  whence the integration constant C is given by the methanol mass fraction at the interface between the porous backing and the channel. Recalling the boundary conditions for the plain/porous interface, eqns (60) and (61), for the methanol transport equation, we combine these two to obtain just one boundary condition for the channel at Y = 0 :  1 ∂xCH3 OH (X , 0) ScI + xCH3 OH (X , 0) − = 0. (120) ∂Y 6
 xCH3 OH

(X , 0) =

This is valid for any profile I . As for the PEFC, I will not in general be constant. In order to take the porous effects of the actual active layer into account, a previous model [34] is used to generate an expression for the local current density that can be used for the boundary conditions at the active layer/porous backing. This current density implicitly accounts for the same effects as were considered in that paper, i.e. pore diffusion in the active layer, finite ionic conductivity and the complex methanol oxidation kinetics: in dimensional form,    , -(l) B αA F (E ) exp − E A 0 ρ xCH3 OH RT ,  (121) I =A M (l) 1 + exp αA F (EA − E0 ) RT

which gives the appropriate scale for [I ] as  [I ] = A

in ρxCH 3 OH

M (l)



B

exp

αA F RT

1 + exp





(EA − E0 )

αA F RT

,

(EA − E0 )

and consequently, in dimensionless form, , -(l)  = I xCH3 OH

,

xCH3 OH

-(l) B

in xCH 3 OH

.

(122)

, -(l) This is valid for any profile I , which is a function of the value of xCH3 OH at Y = −H; for the profile considered in this paper, I can be determined in terms of xCH3 OH (X , 0) through the transcendental equation   1 1 1 I in B = I (X , 0) − + x exp , (123) xCH CH OH 3 3 OH   6 3

where  = ScH/(
γ 2 ).

Transport Phenomena in Fuel Cells

277

In summary, the adaption of the reduced model for the cathode of a PEFC to the anode of a DMFC is based on the fact that the anode operates at a large water fraction and that the magnitude of the dimensionless parameter  is much smaller than 1. 3.6 Numerics and results As for the PEFC, we resort to a numerical scheme to solve the transport equation for methanol in the channel, since no further analytical simplifications are possible. This entails solving eqn (113), subject to the boundary conditions (55) and (120) and the inlet condition for methanol (64). The governing equations are once again parabolic, for which a Modified Box discretization scheme is suitable [35]. The scheme leads to a block tridiagonal matrix, allowing fast computations. The resulting system of non-linear equations is solved with a Newton-Raphson-based algorithm in MATLAB 6. To confirm the validity of the reduced model, its predictions were compared with numerical results obtained using two other softwares [7, 8], wherein the full elliptic governing equations and boundary conditions are implemented. Comparison, shown in Fig. 14, was carried out in terms of the local superficial current density obtained along the anode for a variety of values for the nondimensional parameters
and . These were obtained by varying the inlet velocity, U in , and keeping

Figure 14: Verification of the reduced model. ( · · · ) corresponds to the CFX-4.4 solution with 104 number of nodes,(–) is the Femlab solution for ∼103 adapted nodes and markers are for the reduced model, with 104 cells. (✩):
= 0.932, (
):
= 2.79, (×):
= 9.32, (+):
= 27.9, (
):
= 93.2, ():
= 279.

278 Transport Phenomena in Fuel Cells all other physical parameters constant; in particular, the base case corresponds to
= 9.32 and  = 3.76 × 10−2 . Several features are apparent. First, for the higher (
, )-combinations, the reduced model starts to deviate from the full solution, since the normal velocity due to the electrochemical reaction now is of the same order of magnitude as the streamwise inlet velocity, i.e.  is no longer much smaller than unity and the normal velocity in the channel is no longer negligible. Nevertheless, the local current densities from the reduced model remain close to the current densities from the full elliptic equations, even for these combinations. Second, we note discrepancies between the values obtained at the inlet and outlet by the two commercial solvers. In particular, convergence difficulties were encountered with FEMLAB 2.2 as regards the resolution of the corners, and the solutions presented are actually for a channel that is extended at the inlet and outlet; such difficulties were not encountered with CFX-4.4. Since our principal interest was only to verify the reduced model, these differences were not investigated further. A final comment here which illustrates the benefit of the reduced model approach concerns a comparison of the computing times for the three methods. On a 1 Ghz AMD PC, with 512 MB SDRAM the reduced model with 104 cells took ∼5 CPU seconds to converge, whereas Femlab 2.2 required 1 − 2 CPU minutes. The CFX-4.4 code required 1 − 2 CPU minutes on a 500 MHz Compaq Alphaserver with 3 GB RAM. Mesh independent solutions were found for the reduced model for 103 cells, allowing for computational times of less than 1 second.

4 Conclusions In this chapter, we have considered the use of reduced models for describing transport phenomena in fuel cells as an alternative to computational fluid dynamics. Although the approach appears, at first sight, to be mathematically intensive, requiring nondimensionalization, rescaling and analysis, as well as a certain level of intuition and experience to ensure that the reduced model is indeed consistent, it does have several advantages: 1. reduced models are, computationally speaking, orders of magnitude less intensive than, yet still as practically useful as, CFD models; 2. physical trends become much more apparent, e.g. it is not necessary to perform computations in order to carry out a sensitivity analysis, since the dependence of the model on a particular parameter will become apparent as the model is being analyzed; 3. a nondimensional analysis opens the possibilities for model modification, so that a model used in one context (here, the cathode of a PEFC) can be used in another (the anode of a DMFC). Although the problems treated here address only the interaction of kinetics and transport in the performance-limiting electrodes of the PEFC and DMFC, there is little doubt that the approach can be used successfully when dealing with extensions of these models for multiphase flow, transient behavior, whole cell models, etc.,

Transport Phenomena in Fuel Cells

279

even for other types of fuel cells. Furthermore, the reduced model approach is the one most likely, in practice, to integrate successfully the modeling of transport phenomena at the heart of a fuel cell with system studies, stack modeling and fuel cell optimization.

References [1]

[2] [3]

[4]

[5] [6]

[7] [8] [9] [10] [11]

[12]

[13] [14] [15] [16] [17]

Dutta, S., Shimpalee, S. & van Zee, J.W., Three-dimensional numerical simulation of straight channel PEM cells. Journal of Applied Electrochemistry, 30, pp. 135–146, 2000. Shimpalee, S. & Dutta, S., Numerical prediction of temperature distribution in PEM fuel cells. Numerical Heat Transfer, Part A, 38, pp. 111–128, 2000. Dutta, S., Shimpalee, S. & van Zee, J.W., Numerical prediction of mass exchange between cathode and anode channels in a PEM fuel cell. Int. J. Heat and Mass Transfer, 44, pp. 2029–2042, 2001. Berning, T., Lu, D. & Djilali, N., Three-dimensional computational analysis of transport processes in a PEM fuel cell. J. Power Sources, 106(1), pp. 284–294, 2002. Jen, T.-C., Tan, T. & Chan, S.-H., Chemical reacting transport phenomena in a PEM fuel cell. Int. J. Heat and Mass Transfer, 46, pp. 4157–4168, 2003. Lee, W.-K., Shimpalee, S. & van Zee, J.W., Verifying predictions of water and current distributions in a serpentine flow field polymer electrolyte membrane fuel cell. J. Electrochem. Soc., 150(3), pp. A341–A348, 2003. CFX, www.waterloo.ansys.com/cfx/. Femlab, www.comsol.se/femlab/version23.php. Fluent, www.fluent.com. Star-CD, www.cd-adapco.com/apps/STAR-CDfcell.htm. Bernardi, D.M. & Verbrugge, M.W., Mathematical model of a gas diffusion electrode bonded to a polymer electrolyte. AIChE Journal, 37, pp. 1151– 1163, 1991. Bernardi, D.M. & Verbrugge, M.W., A mathematical model of the solidpolymer-electrolyte fuel cell. J. Electrochem. Soc., 139, pp. 2477–2491, 1992. Springer, T.E., Zawodzinski, T.A. & Gottesfeld, S., Polymer electrolyte fuel cell model. J. Electrochem. Soc., 138, pp. 2334–2342, 1991. Gurau, V., Barbir, F. & Liu, H., An analytical solution of a half-cell model for PEM fuel cells. J. Electrochem. Soc., 147, pp. 2468–2477, 2000. Dannenberg, K., Ekdunge, P. & Lindbergh, G., Mathematical model of the PEMFC. Journal of Applied Electrochemistry, 30, pp. 1377–1387, 2000. Fuller, T.F. & Newman, J., Water and thermal management in solid-polymerelectrolyte fuel cells. J. Electrochem. Soc., 140, pp. 1218–1225, 1993. Nguyen, T.V. & White, R.E., A water and heat management model for protonexchange-membrane fuel cells. J. Electrochem. Soc., 140, pp. 2178–2186, 1993.

280 Transport Phenomena in Fuel Cells [18] Yi, J.S. & Nguyen, T.V., An along-the-channel model for proton exchange membrane fuel cells. J. Electrochem. Soc., 145, pp. 1149–1159, 1998. [19] Vynnycky, M. & Birgersson, K.E., Analysis of a model for multicomponent mass transfer in the cathode of a polymer electrolyte fuel cell. SIAM Journal on Applied Mathematics, 63(4), pp. 1392–1423, 2003. [20] Birgersson, K.E., Nordlund, J., Ekström, H., Vynnycky, M. & Lindbergh, G., A reduced two-dimensional one-phase model for analysis of the anode of a DMFC. J. Electrochem. Soc., 150(10), pp. A1368–A1376, 2003. [21] Vynnycky, M. & Birgersson, K.E., On the role of heat transfer in the cathode of a polymer electrolyte fuel cell, in preparation. [22] Bird, R.B., Stewart, W.E. & Lightfoot, E.N., Transport Phenomena, Wiley and Sons, Inc.: USA, pp. 767–770, 2002. [23] Beavers, G.S. & Joseph, D.D., Boundary conditions at a naturally permeable wall. J. Fluid Mech., 30, pp. 197–207, 1967. [24] Alazmi, B. & Vafai, K., Analysis of fluid flow and heat transfer interfacial conditions between a porous layer and a fluid layer. Int. J. Heat and Mass Transfer, 44, pp. 1735–1749, 2001. [25] Ochoa-Tapia, J.A. & Whitaker, S., Momentum jump condition at the boundary between a porous medium and a homogeneous fluid: inertial effects. J. Porous Media, 1, pp. 201–217, 1998. [26] Ochoa-Tapia, J.A. & Whitaker, S., Heat transfer at the boundary between a porous medium and a homogeneous fluid: the one-equation model. J. Porous Media, 1, pp. 31–46, 1998. [27] He, W., Yi, J.S. & Nguyen, T.V., Two-phase flow model of the cathode of PEM fuel cells using interdigitated flow fields. AIChE Journal, 46, pp. 2053– 2064, 2000. [28] Wang, Z.H., Wang, C.Y. & Chen, K.S., Two-phase flow and transport in the air cathode of proton exchange membrane fuel cells. J. Power Sources, 94, pp. 40–50, 2001. [29] Nguyen, T.V., Modeling two-phase flow in the porous electrodes of proton exchange membrane fuel cells using the interdigitated flow fields. Electrochemical Society Proceedings, 99–14 pp. 222–241, 2000. [30] Cebeci, T. & Bradshaw, P., Momentum Transfer in Boundary Layers, Hemisphere Publishing: Washington, 1997. [31] Landolt-Börnstein, New Series IV/1, Springer: Germany, p. 117, 1977. [32] Scott, K., Argyropoulos, P. & Sundmacher, K., A model for the liquid feed direct methanol fuel cell. J. Electroanal. Chem., 477, pp. 97–110, 1999. [33] Baxter, S.F., Battaglia, V.S. & White, R.E., Methanol fuel cell model: anode. J. Electrochem. Soc., 146(2), pp. 437–447, 1999. [34] Nordlund, J. & Lindbergh, G., A model for the porous direct methanol fuel cells anode. J. Electrochem. Soc., 149(9), pp. A1107–A1113, 2002. [35] Tannehill, J.C., Anderson, D.A. & Pletcher, R.H., Computational Fluid Mechanics and Heat Transfer, Taylor & Francis: USA, 1997.

Transport Phenomena in Fuel Cells

Nomenclature a A B B C Dij Di, j Da E F Fr g h H = hp /hf I L M Mi M(k) , M(k) M, Mi ni N p, p P, P  R Re = [ρ]U in L/µ Sc = µ/([ρ][D]) T u, v, v, U in U , V , Vˆ V xi x, y x˜ , y˜ , X , Y

constant for current density (PEFC), Am mol−1 experimentally fitted parameter experimentally fitted parameter (DMFC) scalar, vector or tensor quantity (PEFC) integration constant diffusion coefficients, m2 s−1 binary diffusion coefficients for a pair (i, j), m2 s−1 Darcy number potential, V Faraday’s constant, A s mol−1 Froude number gravity, ms−2 height, m dimensionless height of porous backing current density, Am−2 length, m mean molecular mass, kg mol−1 molar mass of species i, kg mol−1 diffusion tensor of phase k, kg mol−1 m2 s−1 dimensionless molar mass total mass flux of species i, mol m−2 s−1 outward unit vector pressure, Pa dimensionless pressure gas constant, J mol−1 K−1 Reynolds number Schmidt number temperature, K velocities, m s−1 dimensionless velocities volume, m3 mole fraction of species i coordinates, m dimensionless coordinates

Greek symbols α αA αc β1 , β2 γ 3  = ScH/(
γ 2 )

coefficient for water transport in membrane Tafel slope (DMFC) transfer coefficient for oxygen reduction (PEFC) constants for channel/porous backing interface porosity dimensionless number

281

282 Transport Phenomena in Fuel Cells δ2 = Re−1
= 1/(Reσ 2 ) ε = Da2 η κ  = [i][M ]/(ρU in F), µ ρ σ = hf /L  = σ 2 /ε  = ((1 + 6αH2 O )MH2 O +MMeOH − MCO2 ) χ  = /σ

dimensionless number dimensionless number parameters for relative permeabilities overpotential (PEFC), V permeability, m2 dimensionless number dynamic viscosity, kg m−1 s−1 density, kg m−3 dimensionless number dimensionless number dimensionless number dummy variable for asymptotic series dimensionless number

Subscripts 0 (E0 ) 0 (E0 ) 0,1 A av cell CH3 OH CO2 eff f H2 O i, j, k lim min, max N2 O2 p

open circuit voltage (PEFC), V experimentally fitted parameter (DMFC), V index for series expansion anode average cell methanol carbon dioxide effective flow channel water species i, j, k limiting min, max values nitrogen oxygen porous backing

Superscripts (0), (1) (g) in (l) out

index in asymptotic series gas phase inlet liquid phase outlet

Miscellaneous [ ]

characteristic value of a variable

CHAPTER 8 Modeling of PEM fuel cell stacks with hydraulic network approach J.J. Baschuk & X. Li Department of Mechanical Engineering, University of Waterloo, Canada.

Abstract Polymer electrolyte membrane (PEM) fuel cells convert the chemical energy of hydrogen and oxygen directly into electrical energy. Waste heat and water are the reaction by-products, making PEM fuel cells a promising zero-emission power source for transportation and stationary co-generation applications. In this study, a mathematical model of a PEM fuel cell stack is formulated. The distributions of the pressure and mass flow rate for the fuel and oxidant streams in the stack are determined with a hydraulic network analysis. Using these distributions as operating conditions, the performance of each cell in the stack is determined with a mathematical, single cell model that has been developed previously. The stack model has been applied to PEM fuel cell stacks with two common stack configurations: the U and Z stack design. The former is designed such that the reactant streams enter and exit the stack on the same end, while the latter has reactant streams entering and exiting on opposite ends. The stack analyzed consists of 50 individual active cells with fully humidified H2 or reformate as fuel and humidified O2 or air as the oxidant. It is found that the average voltage of the cells in the stack is lower than the voltage of the cell operating individually, and this difference in the cell performance is significantly larger for reformate/air reactants when compared to the H2 /O2 reactants. It is observed that the performance degradation for cells operating within a stack results from the unequal distribution of reactant mass flow among the cells in the stack. It is shown that strategies for performance improvement rely on obtaining a uniform reactant distribution within the stack, and include increasing stack manifold size, decreasing the number of gas flow channels per bipolar plate, and judicially varying the resistance to mass flow in the gas flow channels from cell to cell.

284 Transport Phenomena in Fuel Cells

1 Introduction Polymer electrolyte membrane (PEM) fuel cells convert the chemical energy of hydrogen and oxygen directly and efficiently into electrical energy with by-products of heat and liquid water. PEM fuel cells also have a high power density, quick start-up and load following characteristics, making them attractive zero emission power sources [1]. Before PEM fuel cells can be successfully commercialized, the production cost must be reduced from the current estimate of approximately $200/kW to $30/kW [2]. Increasing the energy conversion efficiency and power output of the PEM fuel cells could decrease the cost per kW, and thus several empirical and mathematical modeling studies have been undertaken for the purpose of understanding and predicting PEM fuel cell performance. In order to satisfy the power demand of most applications, several PEM fuel cells must be connected in series to form a PEM fuel cell stack. However, heat and water management strategies, which are successful for single PEM fuel cells, are difficult to implement in a stack environment; the efficiency and power output of a PEM fuel cell operating within a stack are lower than the performance of a PEM fuel cell operating independently [3]. Thus, single cell PEM fuel cell models cannot be directly applied for PEM fuel cell stack optimization. This is because the operating conditions for each cell in a stack are typically not the same as the conditions at the stack inlet, and are different among the cells themselves due to the non-uniform reactant flow distribution among the cells, influenced by the pressure loss associated with each flow passage. Several modeling studies of PEM fuel cell stacks exist in the published literature. Empirical models, originally developed for a single PEM fuel cell, have been extended to model PEM fuel cell stacks. The empirical, single cell model of Kim et al. [4] was applied to a stack by Chu et al. [5]. The stack voltage was characterized as a function of current density using terms that represented activation overpotential, ohmic overpotential, and mass transport limitations. The generalized steady state electro-chemical model (GSSEM) [6] has been applied to both single PEM fuel cells and stacks. Stack voltage has a functional dependence on the partial pressure of the reactants, current density and temperature through terms accounting for activation, concentration and ohmic overpotential, CO poisoning, and performance degradation due to aging [7]. Mathematical PEM fuel cell models have also been extended to simulate stack performance. The single cell model of Nguyen and White [8] was isothermal, two-dimensional, steady state, and incorporated mass transport in the electrode backing, the electro-chemical reaction of the cathode, and proton migration in the polymer electrolyte. By modeling the reactant flow in the gas flow channels and stack manifold as a pipe network, Thirumulai and White [9] extended the single cell model to simulate stack performance. Due to the exothermic nature of the electro-chemical reactions occurring within a PEM fuel cell, thermal management within a stack is a significant consideration for stack design. Maggio et al. [10] investigated the temperature and current density distribution in a PEM fuel cell stack using a three-dimensional model. The temperature distribution in the cooling plate, cooling water and membrane

Modeling of PEM fuel cell stacks with hydraulic network approach

285

electrode assembly was found through application of conservation of energy, while the electro-chemical performance of the PEM fuel cell was determined with the empirical relationship of Patel et al. [11]. The model of Maggio et al. [10] was developed for a stack operating in steady state, but the model of Lee and Lalk [12] allowed for a non-steady state simulation. As with the model of Maggio et al. [10], the model of Lee and Lalk [12] determined the temperature distribution within the stack using conservation of energy and the voltage of each cell in the stack was found with the empirical model of Kim et al. [4]. When used as a power source for stationary or transportation applications, a PEM fuel cell stack requires auxiliary equipment for providing fuel, oxidant and heat removal; the stack operates as a component in a larger energy conversion system. Barbir et al. [13] developed a model of a PEM fuel cell stack system consisting of a stack, air compressor subsystem for the stack oxidant supply, gasoline reformer subsystem for the stack fuel supply, and cooling subsystem. The electro-chemical performance of the stack was modeled using an empirical, linear current-voltage relationship and the system water balance and efficiency was investigated at various operating pressures and temperatures. A PEM fuel cell stack system consisting of a stack, air compression subsystem, compressed hydrogen supply subsystem, and cooling subsystem was modeled by Cownden et al. [14]. The stack voltage and power output were determined with the GSSEM of [6] and the efficiencies of both the stack and system were examined. This study formulates a PEM fuel cell stack model. The reactant distribution within the stack is modeled by treating the stack manifold and gas flow channels as a pipe network, and the voltages of the cells in the stack are determined with the single cell, steady state, isothermal model developed previously by the present authors [15]. U and Z configuration stacks operating with humidified hydrogen or reformate as the fuel and humidified oxygen or air as the oxidant are simulated, strategies for reducing the unequal distribution of reactants within the stack are examined, and methods for the improvement of stack performance are described based on the results of the present study.

2 Model formulation In general, a PEM fuel cell stack consists of several PEM fuel cells connected in series, as illustrated in Fig. 1. The cathode side of the PEM fuel cell is exposed to the oxidant, while fuel is introduced to the anode side of the cell. Each cell in the stack consists of several components. The bipolar plates conduct electrons and have grooves, referred to as gas flow channels, that supply fuel or oxidant to the PEM fuel cell. The anode and cathode electrode backings also conduct electrons and allow reactants to access the catalyst layer. In the catalyst layer, electro-chemical reactions convert the chemical energy of the fuel and oxidant to electrical energy. Reduction of oxygen occurs in the cathode catalyst layer and the oxidation of hydrogen occurs in the anode catalyst layer. The polymer electrolyte membrane conducts the proton produced by hydrogen oxidation to the cathode for participation in the reduction of oxygen.

286 Transport Phenomena in Fuel Cells

Figure 1: Schematic of a polymer electrolyte membrane fuel cell stack. The fuel and oxidant for each PEM fuel cell are supplied by the stack manifold, with the anode manifold supplying fuel and the cathode manifold supplying oxidant. The major component of the fuel for a PEM fuel cell is hydrogen, with carbon dioxide and carbon monoxide being present if reformate fuel is utilized. The presence of carbon monoxide severely degrades the performance of a PEM fuel cell through the mechanism of CO poisoning [16]. Mitigation of CO poisoning is possible with oxygen or air bleeding, whereby 1 to 4% oxygen is added to the fuel; thus oxygen and nitrogen can also be present in the fuel stream. The oxidant used in a PEM fuel cell is oxygen, with nitrogen being present if air is used as the oxygen supply. The gas flow channels remove the water produced by the electro-chemical reactions within the MEA and supply the humidity required to avoid polymer electrolyte membrane dehydration; thus liquid and vapor phase water are present in both the oxidant and fuel streams. In addition to fuel and oxidant, water is circulated through cooling plates in order to remove the heat produced by the PEM fuel cells and maintain a constant stack temperature. The stack performance, often measured in terms of the stack voltage, can be determined by: Estack =

N cell  1

Ecell −

N cell 

ηcp ,

(1)

1

where Ncell is the total number of fuel cells in the stack, Ecell is the voltage of each cell (from bipolar plate to bipolar plate), and ηcp is the ohmic loss due to a cooling plate. In this study, the voltage of each cell is found with the single cell model of

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Baschuck and Li [15]. The single cell model is one-dimensional and assumes that the cell is isothermal and operating in steady-state with fully humidified reactants. Cell voltage is calculated with: Ecell = Erev − ηa − |ηc | − 2ηbp − 2ηe − ηm ,

(2)

where Erev is the reversible cell voltage, ηa and ηc are the overpotentials attributed to the anode and cathode catalyst layers, respectively. The voltage losses caused by the bipolar plate, electrode backing and polymer electrolyte membrane are denoted by ηbp , ηe , and ηm , respectively. The reversible cell voltage is the cell potential obtained at thermodynamic equilibrium. It is a function of temperature and reactant concentration through a modified version of the Nernst equation. The cell voltage is reduced from the reversible cell voltage by the overpotentials associated with the various components of the PEM fuel cell. The voltage losses attributed to the bipolar plate and electrode backing are the result of electron migration; the overpotential is calculated by considering the electrode backing and bipolar plate as electrical resistances. Proton migration is responsible for the voltage loss in the polymer electrolyte membrane and thus the voltage loss is determined by the Nernst-Planck equation. The conductivity of the polymer electrolyte is a function of hydration, but the single cell model [15] assumes that the reactants and the polymer electrolyte are fully humidified; thus the conductivity is constant. Therefore, the polymer electrolyte membrane overpotential is a function of the membrane properties, such as conductivity and thickness, and current density. For a PEM fuel cell operating with CO-free fuel, the cathode catalyst layer overpotential is the major voltage loss. The anode and cathode catalyst layer overpotentials are found by considering species conservation, proton and electron migration within the catalyst layers. Proton and electron migration within the catalyst layers are related to the protonic and ionic current through Ohm’s law. Species conservation requires modeling of reaction kinetics and mass transport. Oxygen reduction is modeled with the Butler-Volmer equation in the cathode catalyst layer, while in the anode catalyst layer the adsorption and desorption of H2 , CO and O2 , the electro-oxidation of the adsorbed hydrogen and carbon monoxide, and the heterogeneous oxidation of H2 and CO by O2 are included in the reaction kinetics. The reaction rates in the catalyst layers are functions of overpotential and reactant concentrations; the functional dependency on concentration necessitates consideration of mass transport. The concentrations within the catalyst layers are influenced by resistance to mass transport from the gas flow channels to the electrode backing, within the electrode backing, and within the catalyst layer. The mass transfer from the gas flow channels to the gas flow channel/electrode backing interface is calculated using a logarithmic mean concentration relationship. Mass transport within the electrode backing and catalyst layers is assumed to be through diffusion only and the diffusion coefficients are formulated such that a variable amount of liquid water can exist within the pore space of the electrode backing and catalyst layers; thus the PEM fuel cell can be simulated with a variable degree of water flooding.

288 Transport Phenomena in Fuel Cells Through consideration of species conservation, proton and electron migration, ordinary differential equations are generated which can be solved for the catalyst layer overpotential. Therefore, the catalyst layer overpotential depends on reactant concentration, current density, and the properties of the gas flow channels, electrode backing and catalyst layer, such as porosity, thickness, and conductivity. Determination of the reversible cell potential and overpotentials requires several input parameters, which can be classified as operating or design parameters. Design parameters depend on the manufacture of the PEM fuel cell and include properties, such as conductivity and porosity, and geometric dimensions. Design parameters can be further classified according to the components of a PEM fuel cell; thus there are bipolar plate, electrode backing, catalyst layer, and polymer electrolyte membrane design parameters. The operating parameters include current density, temperature, pressure, reactant composition and stoichiometry. The PEM fuel cells in a stack will have the same design parameters. Due to the series connection, the current density in each cell will be equal. As well, the circulation of cooling water allows each cell to have the same temperature. However, the pressure, reactant composition and stoichiometry can vary from cell to cell if the mass flow rate and pressure distributions within the stack are unequal. Therefore, the stack model presented here consists of two parts: the single cell model and the stack flow model. The single cell model, as described above, determines the voltage of each cell in the stack based on the cell inlet pressure, temperature, stoichiometry, and reactant composition in the gas flow channels, as well as the current density and design parameters. In order to find the cell inlet pressure, temperature, stoichiometry and reactant composition, the mass flow rate and pressure distributions among each cell within the stack must be determined; this constitutes the stack flow model. The voltage loss attributed to the cooling plate in eqn (1) is determined by assuming that the cooling plate has the same overpotential as the bipolar plate in the single cell model. The single cell model is described in detail elsewhere [15] and only the stack flow model will be presented here. 2.1 Stack flow model The mass flow rate and pressure distributions within the stack are coupled; thus they must be solved simultaneously. The fuel or oxidant flow within the stack is modeled as a pipe network, and is illustrated in Fig. 2. Two stack configurations are considered in this study: the U and Z configurations. Other stack configurations can be treated similarly. The U configuration is illustrated in Fig. 2(a) and is characterized by the stack inlet and outlet being on the same end of the stack. The Z configuration, which is shown in Fig. 2(b), has the flow inlet and outlet on opposite ends of the stack. For both the U and Z configuration, the section of stack manifold that supplies the reactants to the gas flow channels of the PEM fuel cells is referred to as the top section, while the gas flow channels exit into the bottom section of stack manifold. If two PEM fuel cells and the connecting manifold sections are considered, as illustrated in Fig. 2, then the pressure losses in the gas flow channels and manifold

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Figure 2: Schematic of (a) U and (b) Z stack manifold configurations. sections are related:  i  i  i i+1  i+1 i i i m ˙ cell,in −
Ptop m ˙ top −
Pcell m ˙ cell,in + α
Pbot m ˙ bot = 0,
Pcell (3) where α equals −1 for the U configuration, 1 for the Z configuration, and i = 1 at the stack inlet. Each pressure loss is a function of the mass flow rate in the manifold section or gas flow channel; thus the pressure and mass flow rate distributions must be solved simultaneously. The mass flow rate in the top sections of the manifold can be related to the mass flow rate entering the gas flow channels of each PEM fuel cell: i  j ˙ stack − m ˙ cell,in , (4) m ˙ itop = m in j=1

290 Transport Phenomena in Fuel Cells where m ˙ stack is the mass flow rate at the inlet of the stack. The mass flow rate in the in bottom sections of the stack manifold depends on the manifold configuration:

m ˙ ibot =

 i  j    m ˙ cell,out         

j=1 N cell 

Z configuration (5)

j m ˙ cell,out

U configuration,

j=i+1

where m ˙ icell,out is the mass flow rate exiting the gas flow channels. Because of the reactions occurring within the anode and cathode catalyst layers, the mass flow rate at the inlet of the gas flow channels is not equal to the value at the outlet. The inlet and outlet mass flow rates are related: m ˙ icell,out = m ˙ icell,in −
m˙r ,

(6)

where
m ˙ r is the mass consumed in the catalyst layers. Hence, the mass generated will be written with a minus sign. The anode and cathode values of
m ˙ r differ and are functions of current density. Equation (3), in conjunction with eqns (4) and (5), can be applied to generate Ncell −1 equations with Ncell unknown m ˙ icell,in values. One final equation is required i to solve for the unknown m ˙ cell,in ’s, and this equation is the mass conservation for the stack as a whole: N cell  = m ˙ icell,in . (7) m ˙ stack in i=1

With the addition of eqn (7), determination of the mass flow rate and pressure distributions within the stack is possible. However, the relationship between the pressure loss and mass flow rate must be determined first, as well as the value of
m ˙ r for the anode and cathode gas flow channels. These are presented in the next sections. 2.2 Manifold pressure loss The pressure loss in either the top or bottom section of stack manifold is found with: (8)
P =
Pm +
Pf , where
Pm is the pressure loss due to the change in momentum of the fluid and
Pf is the pressure loss due to the wall friction. The losses due to the branching of the flow, or minor losses, are not included in this formulation because appropriate coefficients are not available for the flow conditions encountered. However, a separate experimental investigation is under way to develop empirical correlation for the minor loss coefficient associated with the branching/confluence flow, and

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291

therefore, minor loss will be integrated in future studies. The pressure loss due to a change in momentum is given by:
Pm =

m ˙ (Vout − Vin ), Am

(9)

where m ˙ is the mass flow rate in the manifold section, Am is the manifold crosssectional area and V is the velocity. The velocity within the manifold section can be found with: m ˙ V = , (10) ρAm where ρ is the density of the fluid. The fluid in both the manifold and gas flow channels consists of a mixture of a multi-component gaseous phase and liquid water. The density of this mixture depends on pressure, temperature and the mass fraction of the gaseous phase; Appendix A describes the calculation of density. The pressure loss due to friction in the manifold sections is given by:
Pf =

2Cf Ls ρave (Vave )2 dhstack

,

(11)

where Cf is the friction coefficient, Ls is the length of the manifold section between the gas flow channels of the PEM fuel cells, and dhstack is the hydraulic diameter of the stack manifold. The subscript “ave” refers to the arithmetic average of the inlet and outlet values; because density is a function of pressure and mass fraction of the gas phase, the inlet and outlet values will differ. The distance Ls depends on the thickness of the PEM fuel cell and is equal to the thickness of two bipolar plates, one MEA and one cooling plate. In this study, the cooling plate is assumed to have the same dimensions as the bipolar plate. The friction coefficient is a function of the Reynolds number [17]: ( Redh ≤ 2000 16(Redh )−1 Cf = (12) −1/4 0.079(Redh ) Redh ≥ 4000, where Redh is the Reynolds number of the flow based on the hydraulic diameter: Redh =

ρave Vave dhstack . µave

(13)

The viscosity of the fluid is denoted by µ and is calculated with the equations described in Appendix A. As in eqn (11), the subscript “ave” denotes the arithmetic average of the inlet and outlet values. For Reynolds numbers between 2000 and 4000, a linear relationship is used for the friction coefficient:   Redh − 2000  Cf = Cf |Redh = 2000 + Cf |Redh = 4000 − Cf |Redh = 2000 , (14) 2000 where Cf |Redh = 2000 and Cf |Redh = 4000 are the friction coefficient values at Reynolds numbers of 2000 and 4000, respectively.

292 Transport Phenomena in Fuel Cells 2.3 Cell pressure loss The pressure loss calculation for the PEM fuel cell gas flow channels is similar to the manifold sections, with the total pressure loss being calculated with eqn (8). However, due to the change in mass flow rate between the inlet and outlet of the gas flow channels, the formulation of
Pm and
Pf differ from the manifold section formulation. The pressure loss due to momentum change is:
Pm =

1 (m ˙ out Vout − m ˙ in Vin ) , Afc

(15)

where Afc is the cross-sectional area of a gas flow channel. The bipolar plate, with the manifold and gas flow channels, is illustrated in Fig. 3. Two flow channel configurations are shown: serpentine and parallel. Both configurations allow for several gas flow channels to exist on a single bipolar plate. In this study, the gas flow channels on a bipolar plate are assumed to have the same resistance to mass flow; hence all of the gas flow channels on a single bipolar plate have the same mass flow rate. Therefore, the pressure losses of all the gas flow channels on the bipolar plate are found by considering only the flow in one gas flow channel. The mass entering and exiting one gas flow channel can be found with: m ˙ cell,in , nc m ˙ cell,out = , nc

m ˙ in = m ˙ out

(16) (17)

where nc is the number of gas flow channels per bipolar plate.

Figure 3: Illustration of the bipolar plate with serpentine and parallel gas flow channel configurations.

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The pressure loss due to friction is determined with:
Pf =

2Cf Lfc ρave (Vave )2 dhfc

,

(18)

where Lfc is the length of a gas flow channel and dhfc is the gas flow channel hydraulic diameter. The determination of the friction coefficient, average density and average velocity in eqn (18) is the same as for the stack manifold sections. This implies that the effect on the friction coefficient of wall suction due to the reactants flowing into the catalyst layers for electrochemical reactions and the wall blowing due to the reaction products coming out of the cathode catalyst layer are not accounted for. The wall suction/blowing might in reality have significant impact on the friction coefficient, their impact on the transport of momentum, heat and mass is being investigated numerically and will be incorporated later once the relevant information is available. Further, in the above approach for the pressure loss calculation associated with reactant stream in the flow channels built on bipolar plates, the minor loss associated with the bend or turn of the flow direction, as mandated by the serpentine flow channel design, has not been included due to the lack of relevant information. An experimental study is currently under way to measure the associated pressure loss in the serpentine flow channels, and correlations for the minor loss coefficient will be developed and included to improve the model formulated here. However, it might be pointed out that the minor loss for the branching/confluence flow associated with the flow in the manifolds and in the serpentine flow channels will affect the total pressure loss within the stack, but is relatively small in its impact on the reactant mass distribution among the cells in the stack. Therefore, their effect on the final results shown in this study is small as well, and can be neglected. 2.4 Mass consumed in the catalyst layers The amount of mass consumed in the anode and cathode catalyst layers differ, but in general can be written as: 
m ˙r =
m ˙ ir , (19) i=species

where the summation is over all of the species present in the catalyst layers. For the anode catalyst layer, the species present are H2 , CO, CO2 , O2 , N2 and H2 O. The amount of H2 , CO and O2 consumed in the anode catalyst layer can be calculated using Faraday’s law: O 2 m ˙H
m ˙ CO
m ˙r2 Iδ Acell r r , + −2 = ˆ ˆ ˆ 2F M H2 M CO M O2

(20)

where Iδ is the current density, Acell is the active area of a PEM fuel cell, F is the Faraday constant, and Mˆ is the molecular weight. The concentration of CO is

294 Transport Phenomena in Fuel Cells typically at the ppm level, while the concentration of O2 is 1 to 4 percent; as a result the amount of O2 and CO removed from an anode gas flow channel will be much less than the amount of H2 removed. Thus, in order to simplify the solution 2 ˙O procedure, the values of
m ˙ CO r are neglected. Since the amount of CO r and
m consumed is negligible, the mass flow rate of CO2 will not change between the inlet and outlet of a gas flow channel. Nitrogen, if present, does not react in the anode catalyst layer. The water produced in the cathode catalyst layer is removed by the cathode gas stream in practical PEM fuel cell stack operation, and thus this study assumes that no water is added or removed from the anode gas flow channels. Therefore, the only species that is consumed in the anode catalyst layer is H2 , and the mass consumed for the anode is: 2 ˙H
m ˙ r =
m r .

(21)

In the cathode catalyst layer, the species present are O2 , N2 , and H2 O. Nitrogen does not react in the cathode catalyst layer and the amount of oxygen consumed can be calculated with Faraday’s law: 2
m ˙O r =

Iδ Acell ˆ M O2 . 4F

(22)

All of the water produced by the PEM fuel cell is assumed to enter the cathode gas flow channels; thus the amount of water consumed (actually generated, hence the negative sign in the equation below) in the cathode catalyst layer becomes: 2O = −
m ˙H r

Iδ Acell ˆ M H2 O . 2F

(23)

The negative sign denotes that the water produced by the cathode catalyst layer is added, not removed from, the cathode gas flow channels. Therefore, the amount of mass consumed by the cathode catalyst layer becomes: 2 ˙O ˙ rH2 O .
m ˙ r =
m r +
m

(24)

2.5 Boundary conditions The boundary conditions for the above model formulation are specified at the stack inlet, that include the temperature, pressure, reactant composition. The reactant flow rate is determined based on the stack current density specified and the stoichiometry of the reactant desired. Then the stack performance and the reactant at the stack outlet are determined by the model presented earlier. The following section on numerical procedure provides the details on how the boundary conditions are implemented for the model formulated.

3 Numerical procedure As with the single cell model, the input parameters for the stack flow model are classified as operating and design parameters. The design parameters are the stack

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295

manifold dimensions and stack configuration. Operating parameters include the stack current density, temperature, and the pressure, stoichiometry and reactant composition at the stack inlet. The stack stoichiometry is defined as: Sastack = Scstack =

2FNHstack 2 Ncell Iδ Acell 4FNHstack 2 Ncell Iδ Acell

,

(25)

,

(26)

where the inlet molar flow rates of hydrogen and oxygen to the anode and cathode and NOstack . Using stoichiomesides of the stack, respectively, are denoted by NHstack 2 2 try and current density, the inlet molar flow rates of hydrogen in the anode manifold and oxygen in the cathode manifold can be calculated. The mass flow rate at the stack inlet can be determined with the molar flow rates and reactant composition. The relationship between pressure and mass flow rate in the manifold sections and gas flow channels is non-linear; therefore the mass flow rate and pressure distributions must be calculated using an iterative procedure. The numerical solution begins with assumed values of m ˙ icell,in and two levels of iteration are then required. The outer level of iteration solves for the pressure and mass flow rate distributions within the stack. For a given mass flow rate, the procedure for determining the pressure loss in the gas flow channels or stack manifold is iterative; thus an inner i ,
P i and
P i for the level of iteration is required to find the values of
Pcell top bot estimate of mass flow rate. 3.1 Outer iteration Using estimated values of m ˙ icell,in , values for m ˙ itop and m ˙ ibot can be calculated using eqns (4) and (5). Applying these values of mass flow rate to eqn (3) results in a pressure residual:  i  i  i i+1  i+1 i i i rpi =
Pcell m ˙ cell,in −
Ptop m ˙ top −
Pcell m ˙ cell,in + α
Pbot m ˙ bot . (27) The residual can be set to zero by changing the assumed mass flow rates:  i  i i i ˙ icell,in −
Ptop ˙ itop 0 =
Pcell m ˙ cell,in +
m m ˙ top +
m  i i+1  i+1 i ˙ i+1 ˙ ibot . m ˙ cell,in +
m ˙ bot +
m −
Pcell cell,in + α
Pbot m

(28)

Combining eqns (27), (28) and using a Taylor series expansion yields a relationship ˙ itop and
m ˙ ibot : between the
m ˙ icell,in ,
m −rpi =

i d
Pcell

dm ˙ icell,in −


m ˙ icell,in −

i+1 d
Pcell

dm ˙ i+1 cell,in

i d
Ptop

dm ˙ itop


m ˙ i+1 cell,in + α


m ˙ itop

i d
Pbot

dm ˙ ibot


m ˙ ibot ,

(29)

296 Transport Phenomena in Fuel Cells where the derivatives are calculated numerically [18]. From eqns (4) and (5),
m ˙ itop i i and
m ˙ bot can be rewritten in terms of
m ˙ cell,in :
m ˙ itop = −

i 


m ˙ icell,in ,

(30)

j=1


m ˙ ibot

=

 i    j 
m ˙ cell,in         

j=1 N cell 

Z configuration (31)

j
m ˙ cell,in

U configuration.

j=i+1

Thus, eqn (29) can be used to make Ncell − 1 equations for the Ncell values of
m ˙ icell,in ; it becomes for the Z configuration stack: −rpi =

i d
Pcell

dm ˙ icell,in −


m ˙ icell,in +

i+1 d
Pcell

dm ˙ i+1 cell,in

i i  d
Ptop

dm ˙ itop


m ˙ i+1 cell,in +

j


m ˙ cell,in

j=1

i i  d
Pbot

dm ˙ ibot

j


m ˙ cell,in .

(32)

j=1

and for the U configuration stack: −rpi =

i d
Pcell

dm ˙ icell,in −


m ˙ icell,in +

i+1 d
Pcell
m ˙ i+1 cell,in i+1 dm ˙ cell,in

i i  d
Ptop

dm ˙ itop −

j=1

i i  d
Pbot

dm ˙ ibot


m ˙ icell,in

The final equation needed to solve for the conservation eqn (7): N cell 
m ˙ icell,in = 0.

j


m ˙ cell,in j


m ˙ cell,in .

(33)

j=1

comes from the overall mass

(34)

i=1

Equations (34) and either (32) or (33) can be solved for the corrections to the assumed mass flow rate distribution with a linear equation solver, such as LU decomposition [18]. The mass flow rate distribution estimation is then updated: m ˙ icell,in |new = m ˙ icell,in |old +
m ˙ icell,in .

(35)

˙ icell,in are found until conUsing the updated values of m ˙ icell,in , new values of
m vergence, which is achieved when: N cell  

  i rp  ≤ 1 × 10−3 Pa.

i=1

(36)

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3.2 Inner iteration The pressure loss in the gas flow channels and stack manifold are functions of the inlet and outlet mass flow rate, density, and viscosity. Density and viscosity are functions of the mass fraction of the gas phase (χ), the species mole fractions in the gas phase (xi ), and the pressure. However, the outlet pressure is not known until the pressure loss is calculated; hence an iterative procedure is required to determine the pressure loss in the gas flow channels or stack manifold. From the estimated mass flow rates in the outer iteration, the pressure, mass fraction of gas phase and species mole fractions in the gas phase are known at the inlet of the gas flow channel or stack manifold. In order to obtain values of χ and xi at the outlet, the mass flow rate of each species at the outlet must be known. For the species other than water, calculation of the mass flow rate is straightforward. The mass flow rates of the species other than water at the inlet and outlet of the stack manifold sections are equal; the mass flow rates of H2 in the anode and O2 in H O the cathode gas flow channels are reduced by
m ˙ r 2 and
m ˙ r 2 , respectively, and all other non-water mass flow rates are the same at the inlet and outlet of the gas flow channels. The amount of water exiting in liquid or vapor phase depends on the outlet pressure of the gas flow channels or stack manifold section. The maximum mole fraction of water in the gaseous phase at the outlet is: max = xH 2 O,out

H2 O Psat k Pout

,

(37)

H2 O k is the estimated value of is the saturation pressure of water and Pout where Psat outlet pressure. With this mole fraction, the maximum mass flow rate of water vapor can be found:

m ˙ max H2 O(g) ,out =

max Mˆ H2 O(g) xH 2 O(g) ,out



max 1 − xH 2 O(g) ,out

i =H2 O(g) ;H2 O()

m ˙ i,out . Mˆ i

(38)

The total mass flow rate of water (liquid and vapor) is: ˙ H2 O(g) ,in + m ˙ H2 O() ,in −
m ˙ rH2 O , m ˙ total H2 O,out = m

(39)

where
m ˙ rH2 O is only non-zero for the cathode gas flow channels. Using eqns (38) and (39), the mass flow rates of the liquid and vapor water can be found:  m ˙ total m ˙ max ˙ total H2 O,out H2 O(g) ,out ≥ m H2 O,out (40) m ˙ H2 O(g) ,out = max max m ˙ H2 O(g) ,out m ˙ H2 O(g) ,out < m ˙ total H2 O,out ,  0 ˙ total m ˙ max H2 O(g) ,out ≥ m H2 O,out (41) m ˙ H2 O() ,out = total max max m ˙ H2 O,out − m ˙ H2 O(g) ,out m ˙ H2 O(g) ,out < m ˙ total H2 O,out .

298 Transport Phenomena in Fuel Cells Knowledge of the outlet mass flow rates allows for the calculation of the outlet mole fractions in the gas phase and the mass fraction of the gas phase: xi,out = 

m ˙ i,out /Mˆ i

j =H2 O()

m ˙ j,out /Mˆ j

m ˙ H2 O() ,out χout = 1 −  . m ˙ i,out

,

(42)

(43)

With the values of pressure, mole fraction, mass fraction and mass flow rate at the inlet and outlet, the pressure loss in either the gas flow channel or stack manifold can be determined with eqn (8). This value of
P can then be used to generate a new estimate for the outlet pressure: k+1 Pout = Pin −
P.

Iteration continues until:

   P k+1 − P k   out out   ≤ 1 × 10−6 .  k   Pout

(44)

(45)

3.3 Numerical procedure summary Using the inner and outer iterations, the mass flow rate and pressure distributions in the stack are solved with the following procedure, for a given current density output from the stack: 1. An initial estimate for m ˙ icell,in is made. i ,
P i , and 2. Using the inner iteration and the estimated value of m ˙ icell,in ,
Pcell top i are calculated.
Pbot 3. The values of
m ˙ icell,in are calculated and used to generate new values for i m ˙ cell,in . 4. Steps 2 to 3 (the outer iterations) are repeated until convergence. 5. The current density output can be varied, and the above procedures repeated in order to determine the stack performance for a range of loading conditions. The solution of the stack flow model provides the mass flow rate, composition and pressure at the inlet of each PEM fuel cell in the stack. These values are used by the single cell model of [15] to calculate values of Ecell for each cell. The cell voltages, along with the value of ηcp also calculated by the model of [15], are used in eqn (1) to determine the stack voltage.

4 Results and discussion Figure 4 compares the performance of a single PEM fuel cell operating independently with U and Z configuration stacks consisting of 50 cells. This comparison

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299

Figure 4: Polarization curves comparing the performance of a single PEM fuel cell with 50 cell, U and Z configuration stacks operating with (a) H2 /air, (b) reformate/air and (c) H2 /O2 reactants. The stack polarization curves are plotted by dividing the stack voltage by the number of cells in the stack.

is accomplished by comparing the single cell voltage to the stack voltage divided by the number of cells in the stack (average cell voltage). Both the single cell and the cells of the stacks have an active area of 240 cm2 and a serpentine gas flow channel configuration with three gas flow channels per bipolar plate. The relevant design parameters for various cell and stack components, such as the bipolar plate, electrode backing, catalyst layer and polymer electrolyte membrane are given in Table 1. The anode and cathode sides of the cells and stacks are assumed to be the same. The cross-sectional areas of both the anode and cathode stack manifolds are equal to 1.54 cm2 , and the design parameters for the anode and cathode sides of the cell or stack are considered to be equal. The single cell and stacks operate

300 Transport Phenomena in Fuel Cells Table 1: Design parameters for various cell and stack components. Parameter

Value

ρbp W L hp hc ws wc nc ng

6 × 10−5  · m 0.155 m 0.155 m 0.002 m 0.002 m 0.00262 m 0.002 m 3 11

Electrode Backing

ρebulk δe φe

6 × 10−5  · m 2.5 × 10−4 m 0.4

Catalyst Layer

δc mPt fPt m κs

2.0465 × 10−5 m 0.004 kg/m2 0.2 0.9 72700 S/m

Polymer Electrolyte Membrane

δm KE Kp CH+ DH +

1.64 × 10−4 m 7.18 × 10−20 m2 1.8 × 10−18 m2 1200 mole/m3 4.5 × 10−9 m2/s

Stack

wm hm Ncell

0.0124 m 0.0124 m 50

Bipolar Plate

with anode and cathode inlet pressures of 250 kPa, stoichiometries of 1.1 and 2, respectively, and a temperature of 358 K. The performance of a single cell and stack operating with fully humidified hydrogen as the fuel and fully humidified air, consisting of 21% O2 and 79% N2 , as the oxidant (H2 /air reactants) is compared in Fig. 4(a). Figure 4(b) compares single cell and stack performance with fully humidified reformate, consisting of 75% H2 and 25% CO2 , as the fuel and fully humidified air as the oxidant (reformate/air reactants), while single cell and stack performance with fully humidified hydrogen as the fuel and fully humidified oxygen as the oxidant (H2 /O2 reactants) is compared in Fig. 4(c). For operation with H2 /air and reformate/air reactants, the average cell voltage in the stack is less than the voltage of a single cell operating independently.

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301

Figure 5: Voltage of each cell in 50 cell U and Z configuration stacks operating with (a) H2 /air reactants and at the current density of 0.41 A/cm2 and (b) reformate/air reactants and at the current density of 0.32 A/cm 2 .

The difference between cell and stack performance is greater when reformate/air reactants are utilized. Operation with H2 /O2 reactants results in the average cell voltage in the stack being almost the same as the single cell voltage; this agrees with the experience that a stack designed for H2 /O2 reactants, in general, will not operate properly for H2 /air or reformate/air reactants. From Fig. 4, it is also apparent that the Z configuration stacks have a better performance than the U configuration stacks. The performance differences between the stacks and single cells are caused by voltage variations among the cells in the stack. This cell-to-cell voltage variation is illustrated for U and Z configuration stacks in Fig. 5, with Figs 5(a) and 5(b) showing cell-to-cell variation for H2 /air and reformate/air reactants, respectively. The average cell voltage in the stack of Fig. 5 is approximately 0.6, making the current density used for the H2 /air simulation 0.41 A/cm2 and 0.32 A/cm2 for the reformate/air simulation. The voltage of each cell in the stack is compared to the voltage of a single cell operating independently, with the cells of the stack being numbered starting at the stack inlet. Near the stack inlet and outlet, the cells of the Z configuration stacks have a higher voltage than the single cell operating independently, while only the cells near the stack inlet have a higher performance than the single cell for the U configuration stacks. However, the majority of the cells in the stacks have voltages less than the single cell, which results in the average cell voltage of the stack being less than the single cell voltage.

302 Transport Phenomena in Fuel Cells The cell-to-cell voltage variation can be quantified by the voltage spread of the stack: E max − E min SE = cellN cell × 100%, (46) 1 cell 1 Ecell Ncell max and E min are the maximum and minimum cell voltages, respectively, where Ecell cell within the stack. For the H2 /air reactants of Fig. 5(a), the voltage spread for the U configuration stack is 9.8% and 5.0% for the Z configuration stack. Operation with reformate/air reactants yields voltage spreads of 15% and 6.2% for the U and Z configuration stacks, respectively. Although not illustrated in Fig. 5, operation with H2 /O2 reactants and at the current density of 0.88 A/cm2 , which corresponds to the average cell voltage of 0.6 V, results in voltage spreads of only 0.0021% and 0.0018% for the U and Z configuration stacks, respectively. These voltage spread values show that a high voltage spread corresponds to a poor stack performance when compared to a single cell. Z configuration stacks have a better performance and lower voltage spread than U configuration stacks, while operation with reformate/air reactants results in the lowest performance and the highest voltage spread. Therefore, in practice, a low voltage spread and uniform cell-to-cell performance is desirable in order to maximize cell performance. The cell-to-cell voltage variations shown in Fig. 5 are the result of the pressure loss distribution, leading to non-uniform distribution of mass flow rate among the cells within the stack. This unequal distribution of mass flow rate among the cells in the stack creates cell-to-cell variations in the anode and cathode stoichiometry. To illustrate how the mass flow rate distribution affects the cell voltage distribution in a stack, the anode and cathode cell stoichiometry in a Z configuration stack is presented in Fig. 6. Operation with H2 /air reactants and at the current density of 0.41 A/cm2 results in little anode stoichiometry variation among the cells in the stack, as illustrated in Fig. 6(a). However, the cathode stoichiometry variation is similar to the cell voltage distribution shown in Fig. 5(a). The cathode stoichiometry varies more than the anode stoichiometry due to the larger mass flow rate in the cathode gas flow channels. The average Reynolds number in the anode gas flow channels is around 32, but due to the presence of inert N2 gas, the Reynolds number is approximately 640 in the cathode gas flow channels. The use of reformate/air reactants and a current density of 0.32 A/cm2 results in variation in both the anode and cathode stoichiometry, as illustrated in Fig. 6(b). The anode stoichiometry shows more variation for reformate fuel than for hydrogen fuel due to the higher flow rate caused by the presence of the inert CO2 ; the average Reynolds number in the corresponding gas flow channels is about 112 for the reformate fuel. In a manner similar to the cell voltage variation, the cell-to-cell variation in stoichiometry can be quantified by the stoichiometry spread of the stack:

SS =

max min − Scell Scell × 100%,  Ncell 1 1 Scell Ncell

(47)

Modeling of PEM fuel cell stacks with hydraulic network approach

303

Figure 6: Anode and cathode stoichiometry of each cell in a 50 cell, Z configuration stack operating with (a) H2 /air reactants and at the current density of 0.41 A/cm2 and (b) reformate/air reactants and at the current density of 0.31 A/cm 2 .

max min where Scell and Scell are the maximum and minimum values, respectively, of cell stoichiometry within the stack. For the H2 /air reactants of Fig. 6(a), the anode stoichiometry spread is only about 1%, while the cathode stoichiometry spread is 50%. The use of reformate/air reactants, as illustrated in Fig. 6(b), results in an anode stoichioimetry spread of 10% and a cathode stoichiometry spread of 42%. The cathode stoichiometry spread is greater for the case with H2 /air reactants than for reformate/air reactants because the H2 /air case uses a larger current density. With a larger current density, the mass flow rate increases and generates a larger stoichiometry spread; the average Reynolds number in the cathode gas flow channels, when reformate/air reactants and a current density of 0.32 A/cm2 are used, is approximately 501 while, as mentioned previously, the average Reynolds number in the cathode gas flow channels is about 640 for the H2 /air case. However, the larger anode stoichiometry spread results in a larger cell voltage spread when reformate/air reactants are used. Obtaining a uniform mass flow rate distribution within a PEM fuel cell stack reduces the voltage spread and improves stack performance. The degree of flow uniformity through the side branches of a manifold system depends on the ratio of cross-sectional area between the side branches and manifold, and the resistance to mass flow in the side branches and manifold [19]. The ratio of cross-sectional area between the side branches and manifold is defined as:

nb Ab , f¯ = Am

(48)

304 Transport Phenomena in Fuel Cells

Figure 7: Effect of stack manifold cross-sectional area on voltage spread for (a) Z and (b) U configuration stack with 3 gas flow channels per bipolar plate. The current density for H2 /air reactants is 0.41 A/cm2 , reformate/air reactants is 0.32 A/cm2 , and H2 /O2 reactants is 0.88 A/cm2 . where nb is the number of side branches, Ab is the cross-sectional area of each side branch, and Am is the cross-sectional area of the manifold. For the PEM fuel cell stacks considered in this study, eqn (48) becomes: nc Ncell Afc . f¯ = Am

(49)

In principle, a uniform mass flow rate distribution is achieved when f¯ approaches zero. However, nearly uniform flow can be obtained in reality at a finite value of f¯. From eqn (49), f¯ can be reduced by increasing the cross-sectional area of the stack manifold. The effect of stack manifold cross-sectional area, or manifold area, on voltage spread for stacks operating with H2 /air reactants and a current density of 0.41 A/cm2 , reformate/air reactants and a current density of 0.32 A/cm2 , and H2 /O2 reactants and a current density of 0.88 A/cm2 is illustrated in Fig. 7.Although the different mass flow rates in the anode and cathode manifold could warrant different manifold areas, the results shown in Fig. 7 are for anode and cathode manifold areas that are equal. It is seen that the voltage spread decreases as the manifold area is increased if H2 /air or reformate/air reactants are used. Within the range of manifold areas considered in Fig. 7, the voltage spread is almost zero and independent of

Modeling of PEM fuel cell stacks with hydraulic network approach

305

Table 2: Critical stack manifold areas and f¯ values, corresponding to a voltage spread of 1%, for stacks using three gas flow channels per bipolar plate. Reactants H2 /air reformate/air H2 /air reformate/air

Configuration

Current Density

Manifold Area

f¯

Z Z U U

0.41 A/cm2 0.32 A/cm2 0.41 A/cm2 0.32 A/cm2

3.3 cm2 3.8 cm2 4.9 cm2 5.6 cm2

1.8 1.6 1.2 1.1

manifold area if H2 /O2 reactants are utilized. In reality, the critical manifold area, below which significant cell-to-cell voltage variations occur, is much smaller for the H2 /O2 reactants. The U configuration stacks exhibit greater cell-to-cell variation than the Z configuration stacks, while operation with reformate/air reactants results in a larger voltage spread than operation with H2 /air reactants. This is evident in Table 2, which lists the critical manifold areas and f¯ values corresponding to a voltage spread of 1%. The U configuration stacks require a larger manifold area than the Z configuration stacks in order to obtain a voltage spread of 1%. For a given stack configuration, reformate/air reactants require a larger manifold area than H2 /air reactants; thus stacks designed for H2 /air reactants may not operate effectively if switched to reformate/air reactants. Decreasing the number of gas flow channels per bipolar plate (nc ) can also reduce the value of f¯ in eqn (49), resulting in less cell-to-cell voltage variation. To illustrate this, Fig. 8 shows the effect of manifold area on voltage spread for U and Z configuration stacks employing a bipolar plate with one serpentine gas flow channel. Table 3 tabulates the bipolar plate design parameters used for the simulation results of Fig. 8. Other than the value of nc used (3 in Fig. 7 and 1 in Fig. 8), all other parameters used in the simulations of Fig. 8 are the same as in Fig. 7, such as stack configuration, reactant composition and current density. The trends illustrated in Figs 7 and 8 are also similar. Increasing the manifold area decreases the voltage spread if H2 /air or reformate/air reactants are used, while the manifold area does not affect the voltage spread when H2 /O2 reactants are used. For a given manifold area, the Z configuration stack has a smaller voltage spread than the U configuration stack. Although reducing the number of gas flow channels does not affect the general relationship between voltage spread and manifold area, a reduction in nc significantly decreases the magnitude of the voltage spread. Therefore, the manifold area corresponding to a voltage spread of 1% would be expected to be smaller if one, rather than three, gas flow channel grooves exist on each bipolar plate. Table 4 lists the critical manifold areas and f¯ values corresponding to a voltage spread of 1% for stacks utilizing one gas flow channel per bipolar plate. Comparing the entries of Tables 2 and 4, it is evident that the critical manifold areas for the nc = 1 stacks are approximately one-third of those when nc = 3. However, less variation occurs

306 Transport Phenomena in Fuel Cells

Figure 8: Effect of stack manifold cross-sectional area on the voltage spread for (a) Z and (b) U configuration stack with 1 gas flow channel per bipolar plate. The current density for H2 /air reactants is 0.41 A/cm2 , reformate/air reactants is 0.32 A/cm2 , and H2 /O2 reactants is 0.88 A/cm2 . Table 3: Bipolar plate design parameters utilizing one gas flow channel per bipolar plate. Parameter

Value

ρbp W L hp hc ws wc nc ng

6 × 10−5  · m 0.155 m 0.155 m 0.002 m 0.002 m 0.00213 m 0.002 m 1 37

between the f¯ values, with the f¯ values for stacks with nc = 1 being approximately 20% larger than the values with nc = 3. Both the manifold cross-sectional area and the number of gas flow channels can influence the pressure loss of the stack. The pressure loss for the cathode stream in a Z configuration stack is illustrated in Fig. 9. Air is used as the cathode reactant and

Modeling of PEM fuel cell stacks with hydraulic network approach

307

Table 4: Critical stack manifold areas and f¯ values, corresponding to a voltage spread of 1%, for stacks using one gas flow channels per bipolar plate. Reactants H2 /air reformate/air H2 /air reformate/air

Configuration

Current Density

Manifold Area

f¯

Z Z U U

0.41 A/cm2 0.32 A/cm2 0.41 A/cm2 0.32 A/cm2

1.1 cm2 1.4 cm2 1.6 cm2 2.0 cm2

2.2 1.8 1.6 1.3

Figure 9: Cathode stream stack pressure loss for a Z configuration stack operating with H2 /air reactants, a current density of 0.41 A/cm2 , and 1 or 3 gas flow channels per bipolar plate (nc ).

the current density is 0.41 A/cm2 . The pressure loss increases by approximately 9 times if the number of gas flow channels per bipolar plate is reduced from three to one, caused by the combined effects of a lengthened flow path and higher flow velocity in the gas flow channel. However, the general effect of manifold area on the stack pressure loss is the same regardless of the number of gas flow channels. As the manifold area is increased, the pressure loss decreases initially, but then becomes independent of manifold area. This plateau in the pressure loss/manifold area plot indicates that the manifold area is sufficiently large such that the friction loss attributed to the manifold does not contribute to the overall stack pressure loss; the stack pressure loss depends almost solely on the gas flow channels. In practice, nc = 1 is the minimum that can be used. However, depending on the

308 Transport Phenomena in Fuel Cells cell size, the flow path and the momentum loss in the gas flow channel may be too large, resulting in an excessive pressure loss; this large pressure loss increases the parasitic power required for gas compression and decreases the overall stack effeciency. Thus, nc = 3 is often used for large stacks consisting of large cells. In a pipe network, the distribution of mass flow rate is influenced by the flow resistance in each pipe; pipes with a high resistance to mass flow have smaller mass flow rates than pipes with a low resistance to mass flow. Therefore, one method of reducing the cell-to-cell variation of mass flow rate and cell voltage in a PEM fuel cell stack is to alter the resistance to mass flow in the gas flow channels. In this study, the resistance to mass flow in the gas flow channels is altered through the addition of a term in the pressure loss due to friction, eqn (18):
Pf = ζ

2 2Cf Lfc ρave Vave

dhfc

,

(50)

where ζ is the flow resistance parameter that, if greater than one, increases the resistance to mass flow in the gas flow channel. Practically, the resistance to mass flow can be increased by either decreasing the gas flow channel hydraulic diameter, increasing the length of the gas flow channel, or installing a flow obstruction in the gas flow channel. Reducing voltage spread by varying the flow resistance parameter among the cells in the stack can be illustrated by considering a Z configuration stack operating with H2 /air reactants and a current density of 0.41 A/cm2 . In order to reduce cell-tocell cathode stoichiometry variation, the flow resistance parameter for the cathode gas flow channels is set according to: ζi =

Sci Scmin

,

(51)

where Sci is the cathode stoichiometry of cell i in Fig. 6(a) and Scmin is the minimum cathode stoichiometry in Fig. 6(a). Using these values of ζi , the cellto-cell variation of cathode stoichiometry is greatly reduced when compared to the variation resulting from using a constant ζ = 1, as illustrated in Fig. 10. The variable values of ζi are shown in the inset of Fig. 10. The reduction in cathode stoichiometry variation results in a 0.5% voltage spread, which is much smaller than the 5% voltage spread achieved by using ζ = 1 for all cells in the stack. Significantly, the use of the variable flow resistance parameter only increases the cathode stack pressure loss from 1.8 kPa to 2.0 kPa; this increase of 11% is much smaller than the pressure loss increase of over 800% incurred if the voltage spread reduction is achieved by reducing the number of gas flow channels per bipolar plate from three to one. Thus, the energy required to overcome the stack pressure loss would be less if a variable ζi is used, allowing for greater system efficiency. However, varying the flow resistance from cell to cell may be difficult to implement since it requires the customization of each bipolar plate. Finally, it should be pointed out that the above modeling results are based on a 5 kW 50 cell PEM fuel cell stack. However, we don’t have access to the test results

Modeling of PEM fuel cell stacks with hydraulic network approach

309

Figure 10: Cathode stoichiometry for a Z configuration stack using a constant flow resistance parameter of ζ = 1 and the variable flow resistance parameter (dashed curve), which is given in the inset. for the validation of the present model. Thus the above results should be treated as qualitative for the moment. However, the single cell model used in this study has been validated against the single cell test results [15], and the stack flow model has been validated against experimental results for a different application [23, 24]. In this sense, the present model can be used as a useful tool for the design and optimization of PEM fuel cell stacks.

5 Conclusions The performance of 50 active cell, U and Z configuration stacks were simulated using a mathematical model that consisted of two parts. The first part of the model determined the distribution of mass flow rate and pressure in the stack manifold and the gas flow channels of the fuel cells through a hydraulic network analysis. The results of the hydraulic network analysis were used as an input parameter for the second part of the model, which calculated the voltage of each cell in the stack with a previously developed, mathematical model. Therefore, the distribution of mass flow rate within the stack influenced the voltage of each cell in the stack; a small mass flow rate in a cell resulted in a low cell voltage. This relationship between mass flow rate distribution and individual cell voltage lead to the performance of fuel cells operating within a stack being lower when compared to a fuel cell operating independently. The magnitude of the performance difference between

310 Transport Phenomena in Fuel Cells single cells and cells within stacks was larger for U than for Z configuration stacks, and greater when the anode/cathode reactant compositions were fully humidified reformate/air, rather than H2 /air. Eliminating the performance differential could be achieved by ensuring that each cell in the stack had the same mass flow rate, and three methods of attaining a uniform mass flow rate distribution were examined. The first method was increasing the cross-sectional area of the stack manifold, while the second involved decreasing the number of gas flow channels per bipolar plate. Finally, a uniform distribution of mass flow rate within the stack could be achieved by varying, from cell to cell, the resistance to mass flow in the gas flow channels.

Acknowledgments This work is a part of a large research project on PEM fuel cells and related technologies supported by the Auto 21 NCE, NRC Institute for Fuel Cell Innovation, Hydrogenics Corporation and PalCan Fuel Cells Ltd. Partial funding is also provided by the Natural Sciences and Engineering Research Council of Canada.

Appendix A: Property determination The fluid in the gas flow channels and stack manifold is assumed to be composed of a multi-component gas phase with liquid water droplets. The density and viscosity of the fluid are required to calculate the pressure loss in the gas flow channels and stack manifold. The density of the fluid is given by [20]: χ 1−χ 1 = + , ρ ρg ρH2 O()

(52)

where ρg is the density of the gas mixture and ρH2 O() is the density of liquid water. The density of the gas phase is calculated with the ideal gas relation: ρg =

 i=species

xi P Mˆ i , RT

(53)

where the summation includes all gaseous species present in the gas flow channel or stack manifold. The liquid water density can be found in [21] and is equal to 968 kg/m 3 at a temperature of 85◦ C. The viscosity of the fluid in the gas flow channels and stack manifold can be found with [20]: 1 1−χ χ + , = µ µg µH2 O()

(54)

Modeling of PEM fuel cell stacks with hydraulic network approach

311

where µg is the viscosity of the gas mixture and µH2 O() is the viscosity of liquid water. The Wilke correlation is used to find the viscosity of the gas mixture [22]: N 

xi µi N j=1 xj φij i=1   −1/2   1/2  ˆ 1/4 2 ˆ Mj Mi 1 1 + µi  , φij = √ 1 + ˆ µ 8 j Mj Mˆ i

µg =

where the individual gas viscosities are found using a power law [17].  n T µi = . µi,293 K 293 K

(55)

(56)

The viscosity of liquid water can be found with the relationship [17]:  µH2 O() = −1.704 − 5.306  + 7.003  2 , ln µo =

273 K , T

(57)

µo = 1.788 × 10−3 kg/(m · s).

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Hamelin, J., Agbossou, K., Laperriere, A., Laurencelle, F. & Bose, T., Dynamic behavior of a PEM fuel cell stack for stationary applications. International Journal of Hydrogen Energy, 26(6), pp. 625–629, 2001. Bar-On, I., Kirchain, R. & Roth, R., Technical cost analysis for PEM fuel cells. Journal of Power Sources, 109(1), pp. 71–75, 2002. Costamagna, P. & Srinivasan, S., Quantum jumps in the PEMFC science and technology from the 1960s to the year 2000 Part II. Engineering, technology development and application aspects. Journal of Power Sources, 102(1–2), pp. 253–269, 2001. Kim, J., Lee, S.M., Srinivasan, S. & Chamberlin, C., Modeling of proton exchange membrane fuel cell performance with an empirical equation. Journal of the Electrochemical Society, 142(8), pp. 2670–2674, 1995. Chu, D., Jiang, R. & Walker, C., Analysis of PEM fuel cell stacks using an empirical current-voltage equation. Journal of Applied Electrochemistry, 30(3), pp. 365–370, 2000. Amphlett, J., Baumert, R., Mann, R., Peppley, B., Roberge, P. & Harris, T., Performance modeling of the Ballard Mark IV solid polymer electrolyte fuel cell II. Empirical model development. Journal of the Electrochemical Society, 142(1), pp. 9–15, 1995.

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Fowler, M., Mann, R., Amphlett, J., Peppley, B. & Roberge, P., Incorporation of voltage degradation into a generalised steady state electrochemical model for a PEM fuel cell. Journal of Power Sources, 106(1–2), pp. 274–283, 2002. Nguyen, T. & White, R., A water and heat management model for protonexchange-membrane fuel cells. Journal of the Electrochemical Society, 140(8), pp. 2178–2186, 1993. Thirumulai, D. & White, R., Mathematical modeling of proton-exchangemembrane-fuel-cell stacks. Journal of the Electrochemical Society, 144(5), pp. 1717–1723, 1997. Maggio, G., Recupero, V. & Mantegazza, C., Modelling of temperature distribution in a solid polymer electrolyte fuel cell stack. Journal of Power Sources, 62(2), pp. 167–174, 1996. Patel, D., Maru, H., Farooque, M. & Ware, C., Methodology for predictive testing of fuel cells. Journal of the Electrochemical Society, 131(12), pp. 2750–2756, 1992. Lee, J. & Lalk, T., Modeling fuel cell stack systems. Journal of Power Sources, 73(2), pp. 229–241, 1998. Barbir, F., Balasubrumanian, B. & Neutzler, J., Trade-off design analysis of operating pressure and temperature in PEM fuel cell systems. Proceedings of the ASME Advanced Energy Systems Division, pp. 305–315, New York, 1999. ASME Advanced Energy Systems Division, The American Society of Mechanical Engineers. Cownden, R., Nahon, M. & Rosen, M., Modelling and analysis of a solid polymer fuel cell system for transporation applications. International Journal of Hydrogen Energy, 26(6), pp. 615–623, 2001. Baschuk, J. & Li, X., Mathematical model of a PEM fuel cell incorporating CO poisoning and O2 (air) bleeding. International Journal of Global Energy Issues, 20(3), pp. 245–276, 2003. Oetjen, H.F., Schmidt, V., Stimming, U. & Trila, F., Performance data of a proton exchange membrane fuel cell using H2 /CO as fuel gas. Journal of the Electrochemical Society, 143(12), pp. 3838–3842, 1996. White, F., Fluid Mechanics, Third Edition, McGraw-Hill: New York, 1994. Press, W., Teukolsky, S., Vetterling, W. & Flannery, B., Numerical Recipes in C: The Art of Scientific Computing, Second Edition, Cambridge University Press: Cambridge, 1992. Idelchik, I., Handbook of Hydraulic Resistance, Third Edition, CRC Press: Boca Raton, 1994. Wallis, G., One-dimensional Two-phase Flow, McGraw-Hill: New York, 1969. Reynolds, W. & Perkins, H., Engineering Thermodynamics, Second Edition, McGraw-Hill: New York, 1977. Reid, R., Prausnitz, J. & Sherwood, T., The Properties of Gases and Liquids, McGraw-Hill: New York, 1977. Zhang, J. & Li, X., Coolant flow distribution and pressure loss in ONAN transformer windings - Part I: Theory and model development. IEEE Transactions on Power Delivery, 19(1), pp. 186–193, 2004.

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Zhang, J. & Li, X., Coolant flow distribution and pressure loss in ONAN transformer windings, Part II: Optimization of design parameters. IEEE Transactions on Power Delivery, 19(1), pp. 194–199, 2004.

Nomenclature Acell Am Afc Cf CH+ DH+ ,ref dh E fpt fw F hc hm hp Iδ KE Kp m H2 O() L Lfc Ls m ˙ mPt
m ˙r Mˆ i Ncell nc ng Nistack
P
Pf
Pm P R Redh rpi

Active area of fuel cell (m2 ) Cross-sectional area of stack manifold (m2 ) Cross-sectional area of gas flow channel (m2 ) Wall friction coefficient Fixed charge concentration in the polymer electrolyte (mole/m3 ) Diffusion coefficient of H+ in the polymer electrolyte (m2 /s) Hydraulic diameter (m) Cell or stack voltage (V) Mass ratio of platinum to carbon support Volume fraction of electrode backing void flooded by liquid water Faraday constant (96495 C/mole) Depth of gas flow channel (m) Height of stack manifold cross-section (m) Thickness of solid portion of bipolar plate (m) Cell current density (A/m2 ) Electrokinetic permeability of polymer electrolyte membrane (m2 ) Hydraulic permeability of polymer electrolyte membrane (m2 ) Fraction of catalyst layer void space occupied by polymer electrolyte Fraction of catalyst layer void space occupied by liquid water Length of fuel cell active area (m) Length of gas flow channel (m) Length of stack manifold between cells (m) Mass flow rate (kg/s) Platinum mass loading per unit electrode area (kg/m2 ) Amount of mass consumed in the catalyst layer (kg/s) Molecular weight of species i (kg/mole) Number of cells in a stack Number of bipolar plate gas flow channels Number of times a serpentine gas flow channel traverses the bipolar plate Molar flow rate of species i at the stack inlet (mole/s) Pressure loss (Pa) Pressure loss due to friction (Pa) Pressure loss due to momentum change (Pa) Pressure (Pa) Universal gas constant (8.314 J/mole · K) Reynolds number Pressure residual (Pa)

314 Transport Phenomena in Fuel Cells S S T V W wc wm ws xi

Spread (%) Stoichiometry Fuel cell or stack temperature (K) Velocity (m/s) Width of fuel cell active area (m) Width of gas flow channel (m) Width of stack manifold cross-section (m) Width of gas flow channel support (m) Mole fraction of species i

Greek symbols α δc δe δm ζ η ks µ ρ ρebulk ρbp φe φc χ

Equals −1 for U and 1 for Z configuration stacks Thickness of catalyst layer (m) Thickness of electrode backing (m) Thickness of polymer electrolyte membrane (m) Flow resistance parameter Overpotential (V) Conductivity of catalyst layer solid phase (S/m) Viscosity (N · s/m2 ) Density (kg/m3 ) Resistivity of electrode backing ( · m) Resistivity of bipolar plate ( · m) Porosity of electrode backing Porosity of catalyst layer Mass fraction of the gaseous phase

Subscripts a ave bot bp c cell e fc g in  m out rev top

Anode Average value Stack manifold bottom section Bipolar plate Catalyst layer; cathode Fuel cell Electrode backing Gas flow channel Gas Inlet value Liquid Polymer electrolyte membrane Outlet value Reversible Stack manifold top section

Modeling of PEM fuel cell stacks with hydraulic network approach

Superscripts cell k max min new old stack

Fuel cell Iteration number Maximum value Minimum value Current value of an iterative parameter Previous value of an iterative parameter Stack

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CHAPTER 9 Two-phase microfluidics, heat and mass transport in direct methanol fuel cells G. Lu & C.-Y. Wang Department of Mechanical Engineering and Electrochemical Engine Center (ECEC), The Pennsylvania State University, USA.

Abstract This chapter provides an overview of the latest developments in direct methanol fuel cell (DMFC) technology. We begin by describing major technological challenges that DMFCs presently face for portable power, and demonstrate that the fundamental transport processes of methanol, water and heat, along with methanol oxidation kinetics, hold the key to successfully address these challenges. We then describe complementary experimental and modeling work to elucidate the critical transport phenomena, including two-phase microfluidics, heat and mass transport. We explain how the better understanding of these basic transport phenomena leads to a paradigm shift in the design of portable DMFCs, and show experimental evidence of surprisingly low methanol and water crossover through a very thin membrane, Nafion® 112. Fuel efficiency resulting from low methanol crossover reaches 78% and net water transport coefficient through the membrane is found to be less than unity. These salient characteristics will enable highly concentrated methanol to be used directly and hence lead to much higher energy density for next-generation portable DMFCs. Finally, the latest research on micro-DMFCs is reviewed.

1 Introduction A direct methanol fuel cell (DMFC) is an electrochemical cell that generates electricity based on the oxidation of methanol and reduction of oxygen. Figure 1 illustrates the cell construction and operating principles of a DMFC. An aqueous methanol solution of low molarity acts as the reducing agent that traverses the anode flow field. Once inside the flow channel, the aqueous solution diffuses through the backing layer, comprised of carbon cloth or carbon paper. The backing

318 Transport Phenomena in Fuel Cells

Figure 1: Operating schematic of a DMFC [1].

layer collects the current generated by the oxidation of aqueous methanol and transports it laterally to ribs in the current collector plate. The global oxidation reaction occurring at the platinum-ruthenium catalyst of the anode is given by: CH3 OH + H2 O → CO2 + 6H+ + 6e− .

(1)

The carbon dioxide generated from the oxidation reaction emerges from the anode backing layer as bubbles and is removed via the flowing aqueous methanol solution. Air is fed to the flow field on the cathode side. The oxygen in the air combines with the electrons and protons at the platinum catalyst sites to form water. The reduction reaction taking place on the cathode is given by: 3 2 O2

+ 6H+ + 6e− → 3H2 O.

(2)

These two electrochemical reactions are combined to form an overall cell reaction as: CH3 OH + 32 O2 → CO2 + 2H2 O. (3)

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Extensive work on DMFCs has been conducted by many groups, notably Halpert et al. [2] of Jet Propulsion Laboratory (JPL) and Giner, Inc, Baldauf and Preidel [3] of Siemens, Ren et al. [4] of Los Alamos National Laboratory (LANL), Scott and co-workers [5–7] of University of Newcastle upon Tyne, and Wang and co-workers [8–11] of the Pennsylvania State University. A comparative study of DMFC with H2 /air polymer electrolyte fuel cells (PEFC) was presented by the LANL group [4–12], demonstrating that a DMFC requires platinum-ruthenium and platinum loadings roughly five times higher to achieve power densities of 0.05 to 0.30 W/cm2 . A number of extensive reviews have been published in recent years as worldwide DMFC research activities grew exponentially. Gottesfeld and Zawodzinski [13] briefly summarized research at Los Alamos intended for transportation application, and pointed out areas for improvement if DMFC technology is to become a serious power plant candidate for transportation. Among these challenges, reducing catalyst loading to compete with reformed/air fuel cells is perhaps the greatest, and presents a difficult task for the foreseeable future. In a later book chapter, Gottesfeld and Wilson [12] discussed perspectives on DMFC for portable applications. Lamy et al. [14] provided an in-depth review of DMFC fundamentals, including the reaction mechanisms of methanol oxidation, use of various binary and ternary electrocatalysts, effects of electrode structure and composition on the activity of methanol oxidation, and development of proton conducting membranes with low methanol crossover. It was projected that DMFCs will be commercialized as portable power sources before the year 2010 and that a quantum jump in technology will occur, making it possible to drive DMFC-powered vehicles ten years thereafter. Arico et al. [15] reviewed recent advances in DMFC from both fundamental and technological aspects. The fundamental aspects concerned electrocatalysis of methanol oxidation and oxygen reduction in the presence of methanol crossover, and the technological aspects focused upon the proton conducting membranes, as well as MEA fabrication techniques. Neergat et al. [16] provided an excellent review of new materials for DMFC, including novel proton conducting membranes and electrocatalysts. Narayanan et al. [17] and Muller et al. [18] discussed, in detail, the paramount importance of water balance to the portable DMFC system. As expected, a DMFC exhibits lower power densities than that of a H2 /air PEFC, which at present requires anode and cathode platinum loadings of less than 1 mg/cm2 to achieve power densities of 0.6 to 0.7 W/cm2 . However, the DMFC has the advantages of easier fuel storage, no need for humidification, and simpler design. Thus, DMFC is presently considered a leading contender for portable power application. To compete with lithium-ion batteries, the first and foremost property of a portable DMFC system must be higher energy density in Wh/L. This requirement entails overcoming four key technical challenges: (1) low rate of methanol oxidation kinetics on the anode, (2) methanol crossover through the polymer membrane, (3) water management, and (4) heat management. The present chapter deals with the fundamental transport processes of methanol, water and heat underlying DMFCs. The basic transport phenomena, along with electrochemical kinetics, are critical to addressing the four technical challenges outlined above. Section 2 summarizes the thermodynamics and electrochemical

320 Transport Phenomena in Fuel Cells kinetics of DMFCs. Section 3 discusses the two-phase micofluidic phenomena in the DMFC anode and cathode, respectively, based on experimental observations. Section 4 describes mass transport phenomena in DMFC with focus on methanol crossover and water management issues. Section 5 treats heat transfer in DMFC and its coupling with water transport. Section 6 presents a review of mathematical modeling and experimental diagnostic techniques presently under active research. Finally, in Section 7 we present an exciting application of DMFC technology to power microsystems.

2 Fundamentals of DMFC 2.1 Cell components and polarization curve The heart of a DMFC is a membrane-electrode assembly (MEA) formed by sandwiching a perfluorosulfonic acid (PFSA) membrane between an anode and a cathode. Upon hydration, the polymer electrolyte exhibits good proton conductivity. On either side of this membrane are anode and cathode, also called catalyst layers, typically containing Pt-Ru on the anode side and Pt supported on carbon on the cathode side. Here the half-cell reactions described in eqns (1) and (2) are catalyzed. On the outside of the MEA, backing layers made of non-woven carbon paper or woven carbon cloth, shown in Fig. 2, are placed to fulfill several functions. The primary purpose of a backing layer is to provide lateral current collection from the catalyst layer to the ribs as well as optimized gas distribution to the catalyst layer through diffusion. It must also facilitate the transport of water out of the catalyst layer. This latter function is usually accomplished by adding a coating of hydrophobic polymer, polytetrafluoroethylene (PTFE), to the backing layer. The hydrophobic character of the polymer allows the excess water in the cathode catalyst layer to be expelled from the cell by the gas flowing inside the channels, thereby alleviating flooding. The microstructure of the catalyst layer is of paramount importance for the kinetics of an electrochemical reaction and species diffusion. Figure 3 shows scanning

Figure 2: SEM micrographs of (a) carbon paper and (b) carbon cloth.

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electron microscopy (SEM) images of such microstructures of the DMFC anode and cathode, respectively, where high surface areas for electrochemical reactions are clearly visible. A cross-sectional SEM of a MEA segment consisting of a backing layer, a microporous layer (MPL) and a catalyst layer, is displayed in Fig. 4 [9]. The MPL, with an average thickness of 30 µm, overlays a carbon paper backing layer. The anode catalyst layer of about 20 µm in thickness covers the MPL. In the anode, this MPL provides much resistance to methanol transport from the feed to the catalyst sites, thus reducing the amount of methanol crossover. In the cathode, the MPL helps alleviate cathode flooding by liquid water [19]. Figure 5 displays a voltage vs. current density polarization curve of a typical DMFC. The thermodynamic equilibrium cell potential for a DMFC, as calculated in Section 2.2, is approximately equal to 1.21 V. However, the actual open circuit voltage in DMFCs is much lower than this thermodynamic value, largely due to

Figure 3: SEM images of electrodes.

Figure 4: Cross-sectional SEM micrograph of backing layer, microporous layer, and catalyst layer.

322 Transport Phenomena in Fuel Cells Thermodynamic reversible cell Potential 1.21

Voltage (V)

Voltage drop due to fuel crossover

Activation overpotential Mass transport loss

Ohmic loss 0 Current density (mA/cm 2)

Figure 5: Schematic of DMFC polarization curve.

fuel crossover. Methanol crossover is an important topic in DMFCs and thus will be fully elaborated in Section 4.1. On closed circuit, the polarization curve can be categorized into three distinctive regions: kinetic control, ohmic control, and mass transport control. The kinetic control region of DMFC is dictated by slow methanol oxidation kinetics at the anode as well as oxygen reduction kinetics at the cathode. In this region a DMFC suffers the voltage loss second only to the low open circuit voltage caused by methanol crossover. More detailed discussion of this aspect is provided in Section 2.3. The area where cell voltage decreases nearly linearly in the polarization curve is recognized as the ohmic control region. For a DMFC where the polymer electrolyte is usually well hydrated, the voltage loss in this section is minimal. The last portion is referred to as the mass transport control region, whereby either methanol transport on the anode side results in a mass transport limiting current, or the oxygen supply due to depletion and/or cathode flooding becomes a limiting step. The cell voltage drops drastically in the mass transport control region. 2.2 Thermodynamics The thermodynamic equilibrium potential of a fuel cell can be calculated from:
E = −


h − T
s
g =− . nF nF

(4)

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Table 1: Thermodynamic data of fuel cell reactions (per mole of fuel) [20]. Reaction PEFC DMFC

T (K)


g (kJ/kg)


h (kJ/kg)


s (kJ/kg K)

n


E (V)

ηrev

298 298

−237 −704

−285 −727

−162 −77

2 6

1.23 1.21

0.83 0.97

PEFC: H2 + 12 O2 → H2 O; DMFC: CH3 OH + 32 O2 → CO2 + 2H2 O. Table 1 lists thermodynamic data of common fuel cell reactions at 25 ◦ C and 1 atm. For the liquid-feed DMFC, n = 6 and the thermodynamic cell potential is 1.21 V, similar to that of the H2 /air PEFC. The thermodynamic efficiency of a fuel cell is defined as the ratio of maximum possible electrical work to the total chemical energy, i.e.: ηrev =


g nF
E = .
h −
h

(5)

As shown in Table 1, the theoretical thermodynamic efficiency of DMFC reaches 97% at 25 ◦ C. The practical energy efficiency, however, is much lower after accounting for voltage and fuel losses. The voltaic efficiency is defined as the ratio of the actual electric work to the maximum possible work, with the former given by: Wact = −nFVcell ,

(6)

where Vcell is the cell voltage at a current of I . Hence the voltaic efficiency can be written as: ηvoltaic =

Wact −nFVcell −nFVcell Vcell = = = . Wmax
g −nF
E
E

(7)

For example, if the cell is running at 0.4 V, then the voltaic efficiency is only 33%. This low efficiency is caused by substantial overpotentials existed in both the anode and cathode of a DMFC. In a DMFC, there is also fuel efficiency due to methanol crossover defined as: ηfuel =

I , I + Ix over

(8)

where Ix over is an equivalent current density caused by methanol crossover under the operating current density of I . The total energy efficiency of DMFC is therefore given by: η = ηrev ηvoltaic ηfuel .

(9)

Suppose that the fuel efficiency, ηfuel , in a DMFC is 80%, the total energy efficiency becomes η = 97% × 33% × 80% = 25.6% with cell voltage of 0.4 V.

324 Transport Phenomena in Fuel Cells In comparison, for a PEFC η = 83% × 0.7/1.23 = 40.5% with the cell voltage of 0.7 V. The energy efficiency of the PEFC is relatively higher owing largely to its negligibly small fuel crossover and overpotential for hydrogen oxidation on the anode. It is evident from eqn (6) that in order to achieve higher energy-conversion efficiency, one must control methanol crossover so as to maintain high fuel efficiency (e.g. >80%). In addition, it is desirable to operate DMFCs at higher voltages. Thus, high-voltage performance is a high priority for portable DMFC development. Waste heat produced in the DMFC can thus be expressed as: Q=

IVcell − IVcell = IVcell (1/η − 1). η

(10)

By substituting the definition of the total energy efficiency, another expression of heat generation results: Q = (−
h)

I + Ix over − IVcell , nF

(11)

where the first term on the right hand side represents the chemical energy of methanol consumed for power generation and by crossover, while the second term stands for the electric energy generated. 2.3 Methanol oxidation and oxygen reduction kinetics Combined with methanol crossover, slow anode kinetics lead to power density in a DMFC that is three to four times lower than that of a hydrogen fuel cell. Much work has been focused on the anodic oxidation of methanol [21]. A multi-step mechanism of electrocatalytic oxidation of methanol at the anode was postulated [22, 23]. Different anode catalyst structures of Pt-Ru were developed [24] and several anode catalysts other than Pt-Ru were explored [25–27]. Additionally, the effects of the anode electrochemical reaction on cell performance were experimentally studied [28–30]. Lamy et al. [14] and Arico et al. [15] provided extensive reviews of the most recent work on electro-catalysis. More active catalysts for methanol oxidation would enable a certain power density to be realized at higher cell voltage, and hence directly impact the energy efficiency of the cell, which translates to the energy density if the amount of fuel carried by a DMFC system is fixed. The activation overpotential of methanol oxidation reaction (MOR) can be described by Tafel kinetics of the following form:  I , (12) ηa = ba log I0,a where ba and Io,a are the Tafel slope and exchange current density of MOR, respectively. A convenient method to characterize anode activation polarization uses a MeOH anode vs. H2 cathode cell as proposed by Ren et al. [28], to be discussed in more detail in Section 7.

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The oxygen reduction reaction (ORR) on the DMFC cathode is similarly slow, causing a high cathode overpotential. Thus, a Tafel expression is usually used to describe ORR kinetics as follows:  I ηc = bc log , (13) Io,c where the Tafel slope for ORR is around 70 mV/decade in the absence of methanol oxidation. However, in DMFCs, ORR takes place simultaneously with oxidation of crossover methanol, and consequently bc for a DMFC becomes greater than that for a H2 /air PEFC.

3 Two-phase flow phenomena 3.1 Bubble dynamics in anode On the anode side, carbon dioxide is produced as a result of MOR. If CO2 bubbles cannot be removed efficiently from the surface of the backing layer, they remain, covering the backing surface and hence decreasing the effective mass transfer area. In addition, flow blockage results, particularly in channels of small dimensions as required in micro or compact portable fuel cells with maximum volumetric power and energy densities. Therefore, gas management on the anode side is an important issue in DMFC design. Argyropoulos et al. [5, 31] was perhaps among the first to observe the two-phase flow pattern in the anode channel under various operating conditions. This flow visualization on the anode side yields valuable understanding of bubble dynamics in a DMFC. This study was, however, undertaken under low cell performance. Most recently, Lu and Wang [32] developed an improved transparent DMFC to visualize bubble dynamics on the anode side and liquid droplet (and flooding) dynamics on the cathode. This latest study is described in detail in the next two subsections. 3.1.1 Flow visualization Figure 6 displays a diagram of the experimental setup, which consists of an electronic load system to characterize polarization behaviors of the fuel cell, a peristaltic pump to deliver the liquid fuel, an electric heater with temperature controller, pressure relief valves, flow meters and pressure gauges. A Sony digital video camera recorder was used in experiments for flow visualization, and still pictures were captured according to the time sequence when the movie was edited offline. Also, a Nikon N70 camera with a micro-Nikkor lens (60 mm f/2.8D) was utilized to obtain clear pictures of small-size objects. Figure 7 shows a picture of the transparent fuel cell. The cell was constructed of a pair of stainless steel plates mated with a polycarbonate plate.Atotal of eight parallel flow channels (1.92 mm width, 1.5 mm depth, 1 mm rib width) were machined through the stainless steel plate to form an active area of approximately 5 cm2 . The surface of the stainless steel plate contacting the MEA was coated with 30 nm Cr and 300 nm Au to minimize contact resistance. A transparent polycarbonate plate

326 Transport Phenomena in Fuel Cells

Figure 6: Experimental setup for flow visualization.

Figure 7: Photo of the transparent fuel cell.

covered the stainless steel plate, forming a window to allow direct observation of flow behaviors. The polycarbonate plate was concave in design, while the stainless steel plate had a matching convex pattern. This unique design avoided flow leakage between neighboring parallel channels. Cell inlet and outlet manifolds were also machined in the polycarbonate plates.

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Figure 8: Images of bubble dynamics in the DMFC anode using a MEA with carbon paper as backing layer for 2 M MeOH feed and non-humidified air at 100 mA/cm2 and 85◦ C. Figure 8 shows a sequence of images at various times for an MEA with hydrophobic carbon paper as the backing layer at the feed temperature of 85 ◦ C and the current density of 100 mA/cm2 . The images, one second apart, were captured from the movie, with time resolution 1/30 second. In addition, the time of the first image was chosen arbitrarily due to the fact that two-phase flow is a regularly periodic event and the cell was operated at steady-state. As shown in Fig. 8, the CO2 bubbles nucleate at certain locations and form large and discrete gas slugs in the channel. The CO2 bubbles are large in size (∼2 mm) and confined by the channel dimensions, elongated in shape, and distributed discretely on the backing layer along the anode channel. This bubble flow is commonly categorized as Taylor bubbles. The bubble motion is governed by the momentum of liquid flow, the force of buoyancy on the bubble, and the surface tension between bubbles and substrate. It can be seen from Fig. 8 that the bubbles are held on the carbon paper by strong surface tension until they grow into larger slugs for detachment, clearly indicative of the dominant

328 Transport Phenomena in Fuel Cells

Figure 9: Bubble behavior on the anode side using hydrophilic carbon cloth for 1 M MeOH feed and non-humidified air at 100 mA/cm2 and 85 ◦ C. effect of surface tension in bubble dynamics in DMFC. Once the bubbles grow to a sufficient size, they detach and sweep along the backing surface in the channel. This sweeping process clears all small bubbles pre-existing on the backing surface, making new bubbles grow from the smallest size to full detachment diameter. As a result, the two-phase flow becomes regularly intermittent. The flow pattern on the MEA with carbon paper is characterized as bubble flow or slug flow, depending on the accumulation of the bubbles. Figure 9 shows the bubble patterns on the MEA with hydrophilic carbon cloth also at the feed temperature of 85 ◦ C. Since the carbon cloth has a much rougher surface, it is challenging to capture sharp still pictures due to light deflection, although the two-phase flow could be observed clearly in the experiments and movies. Alternatively, a Nikon N70 camera with a micro-Nikkor lens was employed for still photos. It is seen that the CO2 bubbles are produced more uniformly and with smaller size (∼0.5 mm) from the hydrophilic carbon cloth. Therefore, the flow on the MEA with carbon cloth is characterized as a bubble flow. The differences in bubble behaviors between hydrophobic carbon paper and hydrophilic carbon cloth can be explained by considering the fundamental process of bubble growth. It is insightful to compare the differences in the pore structure of carbon paper and carbon cloth. Figure 2 shows SEM micrographs of these two substrates. Clearly, carbon cloth has more regularly distributed pores, whereas carbon paper is more of a random porous medium. This difference in the pore size distribution gives rise to the fact that CO2 bubbles emerge more uniformly from the carbon cloth than carbon paper. 3.1.2 Bubble diameter and drift velocity The bubble detachment diameter from the backing layer is strongly correlated with surface wettability. Consider a bubble growing and detaching from a single pore

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329

of known diameter, dp , and surface contact angle of θ. Assuming, as was indicated in experimental observations, that the bubble detachment process is dominated by buoyancy and surface tension effects, the force balance predicts that the diameter of the bubble at detachment, db , is [33]:  db =

4dp σ sin θ g(ρl − ρg )

1 3

.

(14)

With the typical pore size of 10 µm for both carbon paper and carbon cloth [34], eqn (14) calculates the bubble detachment diameter of 0.68 mm for the hydrophobic carbon paper (e.g. θ = 100◦ ) and 0.38 mm for the hydrophilic carbon cloth (e.g. θ = 10◦ ). These theoretical estimates are consistent with experimental observations. Once detached, bubbles stay spherical in shape due to strong surface tension. If these bubbles are smaller than the channel dimension, the bubble drift velocity through the liquid can be estimated from the correlation of the bubble rising velocity through an infinite, stagnant liquid as obtained from the balance between inertia and gravitational forces [35], i.e.: ub =

db2 g(ρl − ρg ) . 12µl

(15)

3.1.3 Pressure drop For the purpose of gross estimation, the two-phase frictional pressure drop through the anode channel of a DMFC may be approximated by assuming a homogenous flow. Usually, the two-phase flow in the DMFC anode is laminar as the flow rates of both phases are quite small. For a 50 cm2 DMFC operated under typical conditions, the anode pressure drop is of the order of a few kPa. 3.2 Liquid water transport in cathode The importance of flooding on the cathode side in H2 /air PEFCs has been emphasized in the literature [36–39]. Similarly, water management on the cathode in a DMFC was identified as a key issue [40]. A proper level of liquid water existing in the DMFC cathode helps to hydrate the polymer membrane, thus increasing the proton conductivity. However, severe flooding should be avoided so as to maintain the cathode performance. 3.2.1 Flooding in the cathode In the cathode, water is produced by the oxygen reduction reaction as well as transported from the aqueous anode due to diffusion and electro-osmotic drag. Parameters governing liquid water formation and distribution in the cathode include the stoichiometry (or volumetric flow rate) of the inlet air, current density, cell temperature, and membrane water transport properties such as the diffusion coefficient and electro-osmotic drag coefficient.

330 Transport Phenomena in Fuel Cells Formally, the water flux arrived at the cathode by diffusion, electro-osmosis, and hydraulic permeation across the membrane can be expressed as [40] jm = −D


Cc−a I Km I ρl + nd −
Pc−a =α . δm F µl MH 2 O F

(16)

Clearly, the three terms on the right hand side in eqn (16) represent three modes of water transport through the membrane, respectively. The molecular diffusion is driven by the concentration gradient. The electro-osmotic drag is proportional to the current density, and the permeation is driven by the hydraulic pressure difference. The net water flux through the membrane can be conveniently quantified by a net water transport coefficient, α, as defined in eqn (16). This important parameter dictates water management strategies in DMFC systems. It is a combined result of electro-osmotic drag, diffusion and convection through the membrane. For thick membranes like Nafion® 117, α approaches the electro-osmotic drag coefficient as the other two modes of water transport are weakened with increasing membrane thickness. The electro-osmotic drag coefficient nd for Nafion electrolyte in contact with liquid water depends on the temperature [41], as shown in Fig. 10. The relation can also be fitted as: nd = 1.6767 + 0.0155T + 8.9074 × 10−5 T2 ,

(17)

where T is the cell temperature in ◦ C. The total rate of water transported to and produced at the cathode is given by:  1 I . (18) jH2 O = α + 2 F

Figure 10: Water drag coefficient as a function of temperature [41].

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331

At steady state, this must be balanced by the removal rate through cathode air flow. Suppose that completely dry air is fed into the cathode channel, its molar flow rate can be expressed as a function of air stoichiometry such that: jair =

1 I ξ , 0.21 4F

(19)

where ξ is the stoichiometry defined at the current density of I . The depletion rate of oxygen due to the ORR is simply given by: j O2 =

I . 4F

(20)

A simple water balance thus yields the relative humidity of air at the cathode exit as follows:   I 1 + α F 2 pH2 O ptotal  × RHexit = = 1 I I 1 I psat (T ) psat (T ) ξ − + +α 0.21 4F

=

4F

1 + 2α ξ 0.42

F

2

ptotal × . p sat (T ) + 0.5 + 2α

(21)

A critical air stoichiometry is obtained when the relative humidity RHexit = 100%, i.e.: 

ptotal ξcri = 0.21 (2 + 4α) − (1 + 4α) . (22) psat (T ) This threshold stoichiometry represents the formation of liquid water and thus characterizes the state of cathode flooding. If the actual stoichiometry is smaller than that given in eqn (22), the cathode exhaust air will carry liquid water and hence cathode flooding likely occurs. On the other hand, if the actual air stoichiometry is higher, the cathode exhaust air is under-saturated. In this case, cathode flooding is avoided; however, there is too much water loss through evaporation in the case of large air flowrate. Recovery of water vapor in an external condenser proves to be a difficult task for a compact portable system. Hence, for portable DMFC systems, air stoichiometry ought to be designed to be smaller than the critical value given in eqn (22), implying at the same time that a small amount of cathode flooding is inevitable in portable systems. 3.2.2 Flooding visualization Visualization on the cathode side is a useful diagnostic tool to understand the nature of flooding. Figure 11 displays an image of water drop formation on carbon paper treated with PTFE with non-humidified air preheated to 85 ◦ C and fed at a volumetric flow rate of 68 mL/min using the transparent cell as shown in Fig. 7. The cell current density was 100 mA/cm2 . It is shown in Fig. 11 that water droplets are attached on the surface of the carbon paper due to its decreased hydrophilicity

332 Transport Phenomena in Fuel Cells

Figure 11: Water droplet formation at cathode using Toray carbon paper for 2 M MeOH feed and non-humidified air (68 mL/min and 1 psig) at 100 mA/cm2 and 85 ◦ C.

Figure 12: Cathode flooding on single-sided ELAT carbon cloth for 2 M MeOH feed and non-humidified air (161 mL/min and 1 psig) at 60 mA/cm2 and 85 ◦ C. at elevated temperatures. It was observed that while the droplets grow slowly, the cell voltage drops gradually when the current density is fixed. Figure 12 shows an image of flooding on the single-sided ELAT carbon cloth with non-humidified air preheated to 85 ◦ C. It is seen from Fig. 12 that the surface of carbon cloth is nearly free of liquid droplets due to its higher hydrophobicity, but liquid droplets or ‘sweating’ can be found in contact corners between the stainless steel rib and carbon cloth. This is because the rib surface is rather hydrophilic. Interestingly, it appears that sweating inside corners between the ribs and carbon cloth gas diffusion layer occurs at a rather low current density of 60 mA/cm2 and a high air flow rate of 161 mL/min. In comparison, no such sweating is seen

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between the stainless steel ribs and carbon paper GDL at a higher current density of 100 mA/cm2 and a lower airflow rate of 68 mL/min (see Fig. 11). Much remains to be learned about the fundamental process of flooding occurrence and its relation with the backing layer material.

4 Mass transport phenomena 4.1 Methanol crossover Methanol crossover occurs due to the inability of the commonly-used perfluorosulfonic acid (PFSA) membrane to prevent methanol from permeating its polymer structure. Diffusion and electro-osmotic drag are the prime driving forces for methanol transport through the polymer membrane and eventual reaction with platinum catalyst sites on the cathode, leading to a mixed potential on the cathode. This mixed potential on the cathode causes a decrease in cell voltage. Methanol reaching the cathode also results in decreased fuel efficiency, thus lowering the energy density of the system. Methanol crossover in DMFC has been extensively studied both experimentally and theoretically. Narayanan et al. [42] and Ren et al. [43] measured the methanol crossover flux with different membrane thickness and showed that methanol crossover rate is inversely proportional to membrane thickness at a given cell current density, thus indicating that diffusion is dominant. In addition, Ren et al. [44] compared diffusion with electro-osmotic drag processes and demonstrated the importance of electro-osmotic drag in methanol transport through the membrane. In their analysis, methanol electro-osmotic drag is considered a convection effect and the diluted methanol moves with electro-osmotically dragged water molecules. Valdez and Narayanan [45] studied the temperature effects on methanol crossover and showed that the methanol crossover rate increases with cell temperature. Ravikumar and Shukla [30] operated a liquid-feed DMFC at an oxygen pressure of 4 bars and found that cell performance is greatly affected by methanol crossover at methanol feed concentrations greater than 2 M, and that this effect increases with increased operating temperature. Wang et al. [46] analyzed chemical composition of the cathode effluent of a DMFC using a mass spectrometer. They found that methanol crossing over the membrane is completely oxidized to CO2 at the cathode in the presence of a Pt catalyst. Additionally, the cathode potential is influenced by the mixed potential phenomenon due to simultaneous methanol oxidation and oxygen reduction as well as poisoning of Pt catalysts by methanol oxidation intermediates. Kauranen and Skou [47] presented a semi-empirical model to describe the methanol oxidation and oxygen reduction reactions on the cathode and concluded that the oxygen reduction current is reduced in the presence of methanol oxidation due to surface poisoning. Kuver and Vielstich [48] studied the dependence of crossover on reaction conditions, such as temperature and methanol concentration. Additionally, a cyclic voltammetry technique was presented which allows the evaluation of methanol

334 Transport Phenomena in Fuel Cells crossover in a fuel cell under operating conditions. Scott et al. [49] investigated the effect of cell temperature, air cathode pressure, fuel flow rate and methanol concentration on power performance. Gurau and Smotkin [50] used gas chromatography to measure crossover variation with temperature, fuel flow rate and concentration. Heinzel and Barragan [51] gave an extensive review of the state-of-the-art of methanol crossover in DMFC. Influence of methanol concentration, pressure, temperature, membrane thickness and catalyst morphology have been discussed. Development of novel membranes with low methanol crossover would surely increase cell performance and fuel efficiency [14, 16, 52–54]. Alternatively, Wang and co-workers [9, 10, 55] proposed to modify the anode backing structure to mitigate methanol crossover. It was demonstrated that a compact microporous layer can be added in the anode backing to create an additional barrier to methanol transport, thereby reducing the rate of methanol crossing through the polymer membrane. Both practices to control methanol crossover by increasing mass transport resistance, either in the anode backing or in the membrane, can be mathematically formulated by a simple relation existing between the crossover current, Ic , and anode mass-transport limiting current, IA,lim . That is:  I , (23) Ic = Ic,oc 1 − IA,lim where Ic,oc is the crossover current at open circuit, and I the operating current. In the conventional approach using thick membranes with low methanol crossover, Ic,oc is low and IA,lim is set to be high. In contrast, setting up a barrier in the anode backing is essentially reducing the anode limiting current, IA,lim , and making Ic,oc a significant fraction of IA,lim , about 50–80%. Two immediate advantages result from this latter cell design principle. One is that more concentrated fuel can be used thus leading to much higher energy density of the DMFC system. Lu et al. [10] successfully demonstrated the use of 8 M methanol solution as the anode feed, and Pan [55] most recently reported a DMFC operated with 10 M (or 30% by volume) methanol fuel solution. Secondly, this type of DMFC permits use of thin membranes such as Nafion 112, which greatly facilitates water back flow from the cathode to anode [10, 56], thus addressing another major challenge of portable DMFC to be discussed in the next subsection. As an example, Fig. 13 shows the polarization curve of a DMFC design based on the above new concept and using a very thin membrane, Nafion 112. The cell was operated at an anode stoichiometry of 2 and a cathode stoichiometry of 4 at a current density of 150 mA/cm2 . It is evident from Fig. 13 that the mass transport limiting current density, IA,lim , in this cell is equal to 205 mA/cm2 . Figure 14 displays the polarization behavior of the cell when using humidified nitrogen in the cathode to obtain the crossover rate at the open circuit, which gives Ic,oc = 157 mA/cm2 . According to eqn (23), the crossover current at the operating current density of 150 mA/cm2 is Ic = 42 mA/cm2 . Therefore, the fuel efficiency defined in eqn (8) reaches 150/(150 + 42) = 78%. This experiment provides direct evidence that it is possible to obtain very high fuel efficiency even when using a very thin membrane, Nafion 112, provided that the cell is well designed. Finally, this cell yields a power

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Figure 13: Polarization and power density curves for a DMFC designed for portable application.

Figure 14: Measurement of methanol crossover current density at open circuit using humidified nitrogen in the cathode. The limiting current density signifies the crossover rate. density of 56 mW/cm2 at 60 ◦ C under operating conditions eminently suited for portable systems. 4.2 Water management in portable DMFC systems Water management emerges as a new significant challenge for portable DMFC systems. Constrained by the methanol crossover problem, the anode fuel solution

336 Transport Phenomena in Fuel Cells has been very dilute, meaning that a large amount of water needs to be carried in the system and therefore reduces the energy content of the fuel mixture. In addition, for each mole of methanol, one mole of water is needed for methanol oxidation at the anode and 2.5 × 6 moles of water are dragged through a thick membrane such as Nafion 117 towards the cathode, assuming that the electro-osmotic drag coefficient of water is equal to 2.5 water molecules per proton. This then causes 16 water molecules to be lost from the anode for every mole of methanol. Water in the anode must therefore be replenished. On the other hand, inside the cathode, there are 15 water molecules transported from the anode due to electro-osmosis and 3 additional water molecules produced by consuming six protons generated from oxidation of one methanol molecule. Presence of a large amount of water floods the cathode and reduces its performance. The difficult task of removing water from the cathode to avoid severe flooding and supplying water to the anode to make up water loss due to electro-osmotic drag through the membrane is referred to as innovative water management in a portable DMFC. Traditionally, a high cathode gas flow rate (high stoichiometry) is employed to prevent flooding. This strategy not only increases parasitic power consumption but also removes excessive water from the fuel cell, making external water recovery more difficult; see the discussion in Section 3.2.1. How to minimize water removal from the cathode and subsequent recovery externally to replenish the anode without causing severe cathode flooding becomes an important engineering issue. A greater understanding and ability to tailor water flow in the cell is of fundamental interest for portable DMFC systems. This is an area where DMFC modeling plays an important role. In the open literature, Blum et al. [57] proposed a concept of a water-neutral condition under which the anode does not need water supply and the cell maintains perfect water balance by losing exactly 2 moles of water per mole of methanol consumed. Apparently, this condition corresponds to α = −1/6. Most recently, Peled et al. [56] reported experimental data with α being small and even negative at low current densities by using a nonporous proton-conducting membrane and oxygen at 3 bars as the oxidant. It was postulated that the convection effect induced by a hydraulic pressure differential across the membrane can offset the electro-osmotic drag, leading to α being much smaller than the electro-osmotic drag coefficient of approximately 3 at 60 ◦ C. Low-α DMFCs are highly desirable from the water management standpoint as the anode does not require an excessive amount of water and, in conjunction with the methanol transport barrier concept suggested in Section 4.1, it becomes possible to use highly concentrated methanol fuel at the anode. In addition, there is less or no need to recover water from the cathode exhaust, thus eliminating or reducing the condenser in a portable system. Based on our theory of liquid water transport in PEFCs [19, 58], we have designed a unique MEA structure which utilizes the microporous layer, as shown in Fig. 4, to build up the hydraulic pressure on the cathode side and which then uses a thin membrane (i.e. Nafion 112) to promote the water back flow under this hydraulic pressure difference. Such MEAs exhibit extraordinarily low α and hence are generally termed low-α MEA technology.

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Figure 15: Evolutions in cell voltage during constant current loading to measure the net water transport coefficient α. Figure 15 shows the evolution of cell voltage at constant current load in a series of experiments to measure the net water transport coefficient, α, at various temperatures using a moisture trap [59]. The steady-state power density reaches 16 mW/cm2 at 23 ◦ C, 33.3 mW/cm2 at 40 ◦ C and 56 mW/cm2 at 60 ◦ C, respectively. At low current densities, e.g. 50 mA/cm2 at 23 ◦ C and 100 mA/cm2 at 40 ◦ C, the cell voltages remain stable for an extended period of operation. While at a high current density such as 150 mA/cm2 at 60 ◦ C and 70 ◦ C, the cell voltage occasionally experiences a sharp fluctuation once a slug of CO2 gas produced by the large current density blocks the anode channels temporarily, causing a “short-lived” mass transport limitation on the anode side. Paying attention to gas management in the design of anode flowfield will likely remove this voltage oscillation. Figure 16 displays the net water transport coefficient α measured at different temperatures. It is seen that α is only 0.05 at room temperature and α is equal to 0.64 at 60 ◦ C for the air stoichiometry of 4 and methanol stoichiometry of 2. The significance of this set of experiments, shown in Figs 15 and 16, is the fact that commercially available Nafion membranes and MEA materials were used and the cell operated with ambient air without pressurization.

5 Heat transport Thermal management in DMFCs is intimately tied to water and methanol transport processes. First, heat generation in DMFC is much higher than H2 /air PEFC due to a much lower energy efficiency (only 20–25% when the cell is operated between 0.3–0.4 V). That is, for a 20 W DMFC system, 60–80 W of waste heat is produced. The waste heat is typically removed from DMFC by liquid fuel on the anode side

338 Transport Phenomena in Fuel Cells

Figure 16: The net water transport coefficient α at different temperatures with constant flow rates of methanol and ambient air. and by water evaporation on the cathode side. The latter also determines the amount of water loss from a DMFC and the load of water recovery by an external condenser. Therefore, while a higher cell temperature promotes the methanol oxidation reaction, it may not be practically feasible from the standpoint of water evaporation loss. Moreover, the higher cell temperature increases the methanol crossover rate, thereby reducing the fuel efficiency and the system energy density. Argyropoulos et al. [60] present a one-dimensional thermal model for direct methanol fuel cell stacks. In this model, the variation of the temperature of the various components in the stack as well as the heat flow inside the system has been assessed. Dohle et al. [61] described the heat and power management of a direct methanol fuel cell by taking the power for auxiliary equipment into consideration. However, a heat flow analysis carried out for portable DMFC systems is absent in the open literature. The total waste heat produced from DMFC reactions has been derived in Section 2.2, i.e. eqn (11). In addition, a heat sink term exists due to liquid water evaporating into the gas phase in the cathode. This can be expressed as: Q = −
Hvapor = −je
hlg ,

(24)

where
hlg is the latent heat of evaporation. The evaporation water flux, je , can be calculated from:  ξ psat (T ) I − 0.5 , (25) je = 0.84 ptotal F where the first term accounts roughly for the amount of water vapor present in the cathode exhaust and the second term describes the water vapor produced by ORR

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at the cathode. Therefore, the net heat generation from a DMFC is given by:  I + Ix over ξ psat (T ) I Qtotal = −
h · − 0.5 (26) − IVcell −
hlg . 6F 0.84 ptotal F This amount of heat must be removed either by circulating aqueous fuel solution (fuel cooling) or by air cooling from the stack peripherals. Much work remains to be done to analyze heat management in a DMFC and to understand how heat flow affects water management. It is expected that heat management will become a major technological challenge in development of portable DMFC power systems.

6 Mathematical modeling and experimental diagnostics While attempts continue to elucidate the fundamental electrochemical reaction mechanisms, to explore new compositions and structures of catalysts, and to develop new membranes and methods to prevent methanol crossover, important system issues relevant to DMFC are emerging, such as water management, gas management, flow field design and optimization, and cell up-scaling for different applications. A number of physicochemical phenomena take place in liquid-feed DMFC, including species, charge, and momentum transfer, multiple electrochemical reactions, and gas-liquid two-phase flow in both anode and cathode. Carbon dioxide evolution in the liquid-feed anode results in strongly two-phase flow, making the processes of reactant supply and product removal more complicated. All these processes are intimately coupled, resulting in a need to search for optimal cell design and operating conditions. A good understanding of these complex, interacting phenomena is thus essential and can most likely be achieved through a combined mathematical modeling and detailed experimental approach. It is apparent that three of the four technical challenges for portable DMFC systems discussed in Section 1 require a basic understanding of methanol, water and heat transport processes occurring in DMFC. This provides a great opportunity to exercise fundamental modeling. Another good topic for modeling is the micro DMFC system. Both anode carbon dioxide blockage and cathode flooding are especially acute in microsystems due to the small channel length scale involved, low operating temperature, dominance of surface tension forces, and requirement for low parasitic power losses in these systems [62–66]. More discussion on the general characteristics of micro DMFCs is presented in Section 7. In addition, DMFC technology is a system requiring a high degree of optimization. A multitude of operating parameters affect the performance of a DMFC. These variables include cell temperature, molarity of aqueous methanol solution, cathode pressure, anode and cathode stoichiometry, and flow-field design. Higher cell temperatures improve catalytic activity, but increase water loss from the cathode. Efficient removal of carbon dioxide gas bubbles and liquid water produced on the anode and cathode, respectively, must be maintained to allow reactants to reach catalyst sites. Removal of carbon dioxide “slugs” and prevention of cathode

340 Transport Phenomena in Fuel Cells “flooding” can be attained by increasing flow rates. However, increasing flow rates requires more pumping power. Too high a flow rate on the cathode will dry out the polymer membrane, decreasing proton conductivity and hence cell performance. An understanding of the interdependence of these parameters plays a key role in optimizing the performance of a DMFC. DMFC modeling thus aims to provide a useful tool for the basic understanding of transport and electrochemical phenomena in a DMFC and for the optimization of cell design and operating conditions. This modeling is challenging in that it entails the two-phase treatment for both anode and cathode, and in that both the exact role of the surface treatment in backing layers and the physical processes which control liquid phase transport are unknown. 6.1 Mathematical modeling In the literature, Scott et al. [67–69] developed several simplified single-phase models to study transport and electrochemical processes in DMFC. Baxter et al. [70] developed a one-dimensional mathematical model for a liquid-feed DMFC anode. A major assumption of this study was that the carbon dioxide is only dissolved in the liquid and hence their anode model is a single-phase model. Using a macrohomogeneous model to describe the reaction and transport in the catalyst layer of a vapor-feed anode, Wang and Savinell [71] simulated the effects of the catalyst layer structure on cell performance. Kulikovsky et al. [72] simulated a vapor-feed DMFC with a two-dimensional model and compared the detailed current density distributions in the backing, catalyst layer, and membrane of a conventional to a novel current collector. In another paper, Kulikovsky [73] numerically studied a liquid-feed DMFC considering methanol transport through the liquid phase and in hydrophilic pores of the anode backing. In both publications of Kulikovsky, the important phenomenon of methanol crossover was ignored. Dohle et al. [74] presented a one-dimensional model for the vapor-feed DMFC and included the crossover phenomenon. The effects of methanol concentration on cell performance were studied. In a three-part paper [75–77], Meyers and Newman developed a theoretical framework that describes the equilibrium of multicomponent species in the membrane. The transport of species in the membrane based on concentrated-solution theory and membrane swelling were taken into consideration in their model. The transport phenomena in the porous electrodes were also included in their mathematical model. However, the effect of pressure-driven flow was not considered. In addition, the transport of carbon dioxide out of the anode was neglected by assuming that the carbon dioxide was dilute enough to remain fully dissolved in liquid. Nordlund and Lindbergh [78] studied the influence of the porous structure on the anode with mathematic modeling and experimental verification. In their model, they also assumed that carbon dioxide does not evolve as gas within the electrode. Recently, Wang and Wang [79] presented a two-phase, multicomponent model. Capillary effects in both anode and cathode backings were accounted for. In addition to the anode and cathode electrochemical reactions, the model considered

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diffusion and convection of both gas and liquid phases in backing layers and flow channels. The model fully accounted for the mixed potential effect of methanol oxidation at the cathode as a result of methanol crossover caused by diffusion, convection and electro-osmosis. The model of Wang and Wang was solved using a computational fluid dynamics technique and validated against experimental polarization curves. Results indicated the vital importance of gas phase transport in the DMFC anode. Divisek et al. [80] presented a similar two-phase, two-dimensional model of DMFC. Two-phase flow and capillary effects in backing layers were considered. In addition, detailed, multi-step reaction models for both ORR and MOR were developed. Murgia et al. [7] described a one-dimensional, two-phase, multicomponent steady-state model based on phenomenological transport equations for the catalyst layer, diffusion layer, and polymer membrane for a liquid-feed DMFC. Despite the fact that much effort has been made to model the DMFC system, considerable work remains, particularly in support of the emerging portable designs and systems. Few studies have treated the dominating effects of two-phase flow. No model to date has sufficient detail to provide a microfluidic theory for portable systems including effects of channel geometry and wettability characteristics of the GDL on fluid flow in the anode or cathode. Modeling studies are needed to fully elucidate the intricate couplings of methanol, water and heat transport processes. This understanding is key to successful design and operation of portable DMFC systems. Finally, although the important role of a micro-porous layer in DMFC and its tailoring to control the flow of methanol and water have begun to be recognized, much remains to be done. 6.2 Experimental diagnostics Similarly, experimental diagnostics are an important component of DMFC development. Diagnostic techniques for DMFC have included: • • • • •

• •

cyclic voltammetry (CV) to determine the electrochemically active area of the cathode, CO stripping to determine the electrochemically active area of the anode, electrochemical impedance spectroscopy (EIS), anode polarization characterization via a CH3 OH/H2 cell as proposed by Ren et al. [28], methanol crossover rate measurement by CO2 sensing in the cathode (via FITR, GC or infrared CO2 sensors), or by measuring the limiting current in a CH3 OH/N2 cell (Ren et al. [28]), current distribution measurements via a segmented cell in conjunction with a multi-channel potentiostat (Mench and Wang [40]), and material balance analysis of CH3 OH and H2 O (Narayanan et al. [17] and Muller et al. [18]).

In addition, two-phase visualization of bubble dynamics [5, 31, 32] on anode and liquid droplet dynamics on cathode [32] as described in Sections 3.1.1 and 3.2.2, respectively, is a useful tool for cell design and optimization.

342 Transport Phenomena in Fuel Cells Mench and Wang [40] described an experimental technique to measure current distribution in a 50 cm2 instrumented DMFC based on a segmented cell and multichannel potentiostat. In this method, separate current collector ribs are embedded into an insulating substrate (e.g. Lexan plate) to form a segmented flowfield plate. The resulting flowfield plates for both anode and cathode are then assembled with a regular MEA to form a fuel cell with independently controllable subcells. All subcells are connected to a multi-channel potentiostat to undergo potentiostatic experiments simultaneously. The subcell currents measured thus provide information on the current density distribution for the full-scale fuel cell. The spatial and temporal resolution of this method depends on the number of channels available and capabilities of the potentiostat. Current density distribution measurements were made for a wide range of cathode flow rates in order to elucidate the nature of cathode flooding in the DMFC. Figure 17 displays the current density distributions for a high and a low cathode air flow rates, respectively. In the case of high cathode stoichiometry (Fig. 17(a)), it can be seen that the current is rather uniform for all three cell voltages. As expected, the current density increases as the cell voltage decreases. In the case of low cathode stoichiometry (still excessive for the oxygen reduction reaction), Fig. 17(b) clearly shows that a portion of the cathode towards the exit is fully flooded, leading to almost zero current. The information provided in Fig. 17 can be used to identify innovative cathode flowfield designs and enables the development of MEA structures with improved water management capabilities. Material balance analysis proves to be a critical diagnostic tool for the development of portable DMFC systems. In this analysis, methanol balance on the anode side along with the methanol crossover rate typically measured by an infrared CO2 sensor is conducted. In addition, water balance on both anode and cathode sides is performed, where cathode water is carefully collected by a moisture trap and measured [17, 18, 59]. From such analyses, Müller et al. [18] revealed that the

Figure 17: Current density distributions in a 50 cm2 DMFC for: (a) high cathode air flowrate (stoichiometry of 85 @ 0.1 A/cm2 ) and (b) low cathode air flowrate (stoichiometry of 5 @ 0.1 A/cm2 ).

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water balance on the DMFC anode is highly negative, thus calling for membrane development with low water crossover in addition to low methanol crossover. 6.3 Model validation Experimental validation of the two-phase DMFC model of Wang and Wang [79] has been carried out for a 5 cm2 graphite cell. A brief description of the cell geometry, MEA compositions and operating conditions is given in Fig. 18. Figure 18(a) illustrates the capability of the model to predict the polarization curves at two cell temperatures. Excellent agreement is achieved not only in the kinetic- and ohmiccontrolled regions of the polarization curves but also in the mass transport controlled region, where the methanol oxidation kinetics is modeled as a zero-order reaction for molar concentrations above 0.1 M, but a first-order reaction for a molarity below 0.1 M. This shift in the reaction order and the molarity of transition is consistent with direct kinetics measurements. A lower mass transport limiting current density at 50 ◦ C, seen from Fig. 18(a), is caused by the lower diffusion coefficients in both liquid and gas phases and the lower saturation methanol vapor concentration in the gas phase at lower temperatures. Using the same model and physical property data, Fig. 18(b) shows equally satisfactory agreement in the polarization curves between numerical and experimental results for different methanol feed concentrations. In accordance with these experiments, the model prediction for the 2 M case shows a slightly lower performance (due primarily to higher methanol crossover) and an extended limiting current density. Similar success in validating global I-V curves was also reported by Murgia et al. [7], among others. While the model validation against cell overall performance data has been satisfactory and encouraging, as evident from Fig. 18, the ultimate test of these highly sophisticated two-phase models is comparison with detailed distribution

Figure 18: Comparisons of 2-D model predictions with experimental data for a DMFC with: (a) temperature effect and (b) concentration effect.

344 Transport Phenomena in Fuel Cells 0.7

(a) Exp. Data

x/L = 0.16 x/L = 0.37 x/L = 0.58 x/L = 0.68 x/L = 1.0 Average

0.5 0.4 0.3 0.2 0.1 0.00

0.6 1 0.4

0.2 0.05

0.10

0.15

0.20

Current Density (A/cm2)

(b) Numerical Simulation

0.8

Voltage (V)

Voltage (V)

0.6

0.25

0

0.7

0.6

0.4 0.05

average 0.1

x/L= 0.15 0.15

0.2

0.25

Current Density (A/cm2)

Figure 19: Comparison of localized polarization curves between experiments (a) and model predictions (b) for a 50 cm2 DMFC with the anode flow stoichiometry of 27 and cathode air stoichiometry of 5 @ 0.1 A/cm2 . measurements. Figure 19 presents such an attempt toward developing high-fidelity, first-principles models for DMFC. Figure 19(a) shows a set of localized polarization curves measured using the current distribution measurement technique of Mench and Wang [40], and Fig. 19(b) displays the same set of polarization curves predicted from the DMFC two-phase model of Wang and Wang [79]. A low air stochiometry of 5 (although not low for the electrochemical reaction requirement) was deliberately chosen so that cathode GDL flooding may occur and a non-uniform current density distribution results. The two graphs in Fig. 19 share a qualitative similarity. For example, both experiment and model results indicate that the local polarization curves near the dry air inlet exhibit a monotonic function between the voltage and current. Also, the shape of the polarization curves near the exit, from both experiment and simulation, is clearly evidence of flooding in the cathode GDL. Another interesting observation is that the average cell polarization curves, measured and predicted, do not exhibit any sign of cathode flooding, indicating that detailed distribution measurements are absolutely required in order to discern complex physicochemical phenomena occurring inside the cell. Finally, it can be seen from Fig. 19 that a satisfactory quantitative comparison between experiment and model is lacking on the detailed level. Difficulties in obtaining good quantitative agreement between predicted and measured distribution results are indicative that model refinements as well as an improved property data base will be needed before accurate quantitative predictions of not only overall polarization curve but also detailed distributions within a DMFC may be obtained.

7 Application: micro DMFC Micro-power sources are a key technology in future integrated micro-systems that enable sensing, computing, actuation, control, and communication on a single chip.

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Due to such advantages as easy storage of liquid fuel, ambient temperature operation, and simple construction, the direct methanol fuel cell has received much attention as a leading candidate for micro-power sources of the future [17]. Thanks to the integrated-circuit (IC) fabrication technology, micro-channel patterns of DMFC bipolar plates into which reactants are fed can be featured on the silicon wafer with high resolution and good repeatability. Kelley et al. [81] reported a 0.25 cm2 micro DMFC using silicon (Si) wafer as the substrate. The anode catalyst in their micro DMFC was prepared by coelectrodepositing a Pt-Ru alloy onto a carbon coated Si wafer. Using 0.5 M methanol solution, the micro DMFC was tested and yielded an output current density very close to that of large-scale DMFC. In a subsequent paper [62], they reported that a prototype cell (12 mm3 in volume) micro-fabricated on Si substrates and featuring electrodeposition of Pt-Ru as the anode catalyst successfully demonstrated a lowering of catalyst loading to 0.25 mg/cm2 without loss of performance. Pavio et al. instead explored low-temperature co-fired ceramic (LTCC) material as an alternative for the bipolar plates of micro fuel cell system, and a DMFC prototype, packaged using LTCC, was reported in their paper [64]. In addition, micro PEM fuel cells based on a Si wafer using microelectromechanical system (MEMS) technology have been under extensive development [66, 82–85]. Lee et al. [66] reported a micro fuel cell design in which a planar array of cells are connected in series in a “flip-flop” configuration. Maynard and Meyers [82] proposed a conceptual design for a miniaturized DMFC for powering 0.5–20 W portable telecommunication and computing devices. Heinzel et al. [83] demonstrated a prototype miniature fuel cell stack to power a laptop computer using hydrogen and air as reactants running at ambient pressure and temperature. Most recently, Yu et al. [84, 85] fabricated a miniature twin-fuel-cell connected in series by sandwiching two membrane-electrode-assemblies between two Si micromachined plates. In the recent work of Lu et al. [10], the fabrication process of the silicon wafer is illustrated in Fig. 20. Figure 21 shows a picture of the silicon wafer with fabricated flow channel pattern. The fluid channels in the Si wafer have a depth of 400 µm. Both the flow channel and the rib separating two neighboring channels were 750 µm wide, with the channel length of 12.75 mm. There were a total of nine channels with serpentine flowfield, forming a cell effective area of approximately 1.625 cm2 . In order to collect current and to minimize contact resistance between the MEA and the Si wafer, Ti/Cu/Au (with thickness of 0.01/3/0.5 µm) was deposited on the front-side of each wafer by electron beam evaporation. Figure 22 shows a series of cell polarization curves operated at different temperatures using 1 M methanol solution under ambient pressure in the Si-based micro DMFC [10]. The flow rate of non-preheated air was 88 mL/min and the methanol feed rate was 2.83 mL/min. The maximum power density of the cell reached 14.27 mW/cm2 at a voltage of 0.196 V at room temperature (i.e. 23 ◦ C). The maximum power density was 24.75 mW/cm2 at a voltage of 0.214 V when the temperature increased to 40 ◦ C. This is because the kinetics of electrodes, particularly methanol oxidation at the anode, is enhanced at elevated temperatures. For the

346 Transport Phenomena in Fuel Cells

Figure 20: Fabrication process flow of the µDMFC.

Figure 21: A silicon wafer with flow channels. same reason, maximum power density was 47.18 mW/cm2 at a voltage of 0.258 V and temperature of 60 ◦ C. A problem emerging in Si-based micro DMFCs is that the silicon substrate is too fragile and it becomes difficult to compress the cell tightly for sealing and to reduce the contact resistance between the MEA and flow plates. In addition, a thick gold layer has to be coated on Si substrate to improve the conductivity as a current collector. Alternatively, one can use photochemical etching of stainless steel to fabricate the flow plate/current corrector instead of silicon wafer [86]. Stainless steel provides much higher conductivity than silicon, thus avoiding thicker metal coating on the surface. Also, photochemical etching is a simple, high-quality, fast-turnaround, low-cost process for micro-fabrication of flow channels on stainless steel.

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Figure 22: Polarization and power density curves at different temperatures using 1 M methanol solution with the air flow rate of 88 mL/min and methanol flow rate of 2.83 mL/min at atmospheric pressure.

Figure 23: Polarization curve using 2 M methanol solution at 22 ◦ C, air flow rate of 161 mL/min, methanol flow rate of 2.2 mL/min, and ambient pressure. Figure 23 shows the polarization and power density curves using 2 M methanol solution at 22 ◦ C and atmospheric pressure. The flow plates in this DMFC were made of stainless steel using the photochemical etching method. The flowfield and cell size were identical to the Si-based micro DMFC reported before [10]. The methanol flow rate was 2.2 mL/min, and the air flow rate was 161 mL/min. At room temperature, the current density reach 90 mA/cm2 at 0.3 V and the maximum power density was 34 mW/cm2 at 0.23 V. Figure 24 displays the polarization and power density curves of the same stainless steel cell using 2 M methanol solution

348 Transport Phenomena in Fuel Cells

Figure 24: Polarization curve using 2 M methanol solution at 40 ◦ C, air flow rate 161 mL/min, methanol flow rate 2.2 mL/min, and ambient pressure.

Figure 25: Polarization curve using 2 M methanol solution at 60 ◦ C at different air flow rates, methanol flow rate 2.2 mL/min, and ambient pressure. at 40 ◦ C and atmospheric pressure. The cell performance reach about 200 mA/cm2 at 0.3 V and the maximum power density was 62.5 mW/cm2 at 0.26 V. Figure 25 shows the polarization and power density curves using different air flow rates. The methanol flow rate was fixed at 2.2 mL/min. At low current densities, the cell voltages are almost identical under different air flow rates. The overall cell performance improves with the air flow rate increasing. At the air flow rate of 375 mL/min, the cell performance reach 330 mA/cm2 at 0.3 V and the maximum power density was 100 mW/cm2 at 0.28 V. Figure 26 displays the anode polarization behaviors at different temperatures when hydrogen was used in the cathode. As expected, the anode overpotential

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Figure 26: Anode overpotential with hydrogen cathode, 2 M methanol with flow rate of 2.2 mL/min, hydrogen flow rate 161 mL/min, and ambient pressure.

Figure 27: Methanol crossover rate at open circuit voltage, 2 M Methanol with flow rate of 2.2 mL/min, nitrogen flow rate 161 mL/min, and ambient pressure. decreased as the temperature increased. Using this H2 -evolving counter electrode, the anode overpotential may deviate somewhat from the real kinetic results using dynamic hydrogen electrode [28]. However, using the H2 -evolving cathode is still a simple and useful method to evaluate and screen the anode characteristics in an assembled DMFC. Figure 27 shows the polarization behavior when air was replaced by humidified nitrogen in the cathode. Such cells, consisting of MeOH anode and N2 cathode,

350 Transport Phenomena in Fuel Cells allow measurement of the methanol crossover rate. As methanol diffuses from the anode side to the cathode side, the electrochemical reaction at the N2 cathode becomes: CH3 OH + H2 O → CO2 + 6H+ + 6e− . Protons produced from the above reaction then migrate back to the anode side of the membrane where they are combined to evolve H2 . That is: 6H+ + 6e− → 3H2 . Thus, the methanol crossover rate at the open circuit voltage can be determined from the limiting current density observed from the polarization behavior of such a cell [28]. It is seen from Fig. 27 that the methanol crossover rate is relatively large with the small MEA area due to edge leakage [86]. To summarize, micro DMFCs fabricated by photochemical etching of stainless steel have attained impressive performance (i.e. 100 mW/cm2 at 60 ◦ C) for highpower applications. For future development, it is necessary to further lower the air and fuel feeding rates so as to reduce the pumping power. In this regard, a selfactivated micro DMFC holds much promise where the cathode is air breathing and the anode features a pumpless delivery of liquid fuel. Further development in microDMFCs is to decrease the system volume and increase the methanol concentration in the fuel tank and hence the energy density.

8 Summary and outlook The fundamental transport processes of methanol, water and heat occurring in DMFCs for micro and portable applications have been reviewed, along with a summary of recent DMFC models and diagnostic techniques. Significant challenges still exist before a DMFC can compete with the latest Li-ion battery technology. We have stressed in this chapter that a better understanding of the basic transport phenomena achieved through combined flow visualization studies and transport simulations is essential to overcome these challenges and to inspire new design concepts. We demonstrated that, contrary to conventional wisdom, a DMFC based on thin Nafion 112 membrane can reach a fuel efficiency of ∼80% and a water crossover coefficient lower than unity while still maintaining a power density of 56 mW/cm2 at 60 ◦ C and ambient air. Two-phase modeling capabilities for DMFC have emerged, which unravel the importance of gas phase transport of methanol as compared to the liquid phase transport. In addition, much effort is being directed towards developing a coupled model for methanol, water and heat transport processes simultaneously in a DMFC. Such models are extremely useful for the discovery of unique design and operation regimes of the DMFC system for portable application, where the high energy density entails using highly concentrated methanol (preferably pure methanol), maintaining low water and methanol crossover, and improving high-voltage performance. The latter two factors will result in high efficiency DMFCs. It is expected that the DMFC model development will be directed less towards refining model accuracy

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or improving computational speed, but more towards applying the models to invent new cell designs and pinpoint areas of improvement. Other important research issues such as an accurate materials data base were discussed in Wang [87].

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Transport Phenomena in Fuel Cells

Nomenclature ba bc D db dp
E F g
g
h
hlg I IA,lim Ic,oc Io,a Io,c Ix over jair je jH2 O jO2 Km MH2 O n nd pH2 O psat (T ) ptotal Q RH
s T ub Vcell Wact Wmax

Tafel slope of anode methanol oxidation reaction Tafel slope of cathode oxygen reduction reaction diffusion coefficient bubble diameter backing layer pore size thermodynamic equilibrium potential Faraday constant gravitational acceleration Gibbs free energy change per mole of fuel enthalpy change per mole of fuel latent heat of evaporation operating current density anode mass-transport limiting current density crossover current at open circuit anode exchange current density cathode exchange current density methanol crossover current density air molar flow rate at the inlet cathode water evaporation flux water flux consumption rate of oxygen membrane hydraulic permeability water molecular weight number of electrons transferred for each molecule of fuel electro-osmotic drag coefficient of water water partial pressure saturation pressure operating pressure heat generation rate relative humidity entropy change per mole of fuel temperature bubble drift velocity through the liquid cell voltage electric work maximum possible work

Greek symbols α δm η ηa

net water transport coefficient through the membrane membrane thickness total energy efficiency anode activation overpotential

357

358 Transport Phenomena in Fuel Cells ηfuel ηrev ηvoltaic µ µl θ ρl ρg σ ξ ξcri

fuel efficiency thermodynamic efficiency voltaic efficiency dynamic viscosity liquid water viscosity surface contact angle liquid density gas density liquid/gas interfacial tension the stoichiometry defined at the current density of I critical air stoichiometry

Exergy Method Technical and Ecological Applications J. SZARGUT, Silesian University of Technology, Poland The exergy method makes it possible to detect and quantify the possibilities of improving thermal and chemical processes and systems. The introduction of the concept “thermo-ecological cost” (cumulative consumption of non-renewable natural exergy resources) generated large application possibilities of exergy in ecology. This book contains a short presentation on the basic principles of exergy analysis and discusses new achievements in the field over the last 15 years. One of the most important issues considered by the distinguished author is the economy of non-renewable natural exergy. Previously discussed only in scientific journals, other important new problems highlighted include: calculation of the chemical exergy of all the stable chemical elements, global natural and anthropogenic exergy losses, practical guidelines for improvement of the thermodynamic imperfection of thermal processes and systems, development of the determination methods of partial exergy losses in thermal systems, evaluation of the natural mineral capital of the Earth, and the application of exergy for the determination of a proecological tax. A basic knowledge of thermodynamics is assumed, and the book is therefore most appropriate for graduate students and engineers working in the field of energy and ecological management. Series: Developments in Heat Transfer, Vol 18 ISBN: 1-85312-753-1 2005 192pp £77.00/US$123.00/€115.50

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Condensation Heat Transfer Enhancement V.G. RIFERT, Thermodistillation, Kiev, Ukraine and H.F. SMIRNOV, State Academy of Refrigeration, Odessa, Ukraine Professors V.G. Rifert and H.F. Smirnov have carried out research on heat transfer enhancement by condensation and boiling for over 30 years and have published more than 200 papers on the topic. In this book they provide research results from the former USSR to which there has previously been little access and also describe different theoretical models of the condensation process. Partial Contents: Theoretical Principles of Heat Transfer at Film Condensation; Condensation on Horizontal Low-Finned Tubes – Theoretical Models; Experimental Study of Condensation Heat Transfer on Finned Tubes; Condensation on Vertical Profiled Surfaces and Tubes; Heat Transfer Enhancement at Film Condensation Inside Tubes; Condensation in the Electric Field; Hydrodynamics and Heat Transfer at Film Condensation of Rotating Surfaces. Series: Developments in Heat Transfer, Vol 10 ISBN: 1-85312-538-5 2004 392pp £144.00/US$230.00/€216.00

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Heat and Fluid Flow in Thermal Analysis of Microscale and Nanoscale Welds N.T. NGUYEN, ETRS Pty Ltd., HRL Structures Services, Mulgrave, Australia Editors: M. FAGHRI, University of Rhode Island, USA and B. SUNDÉN, Lund Institute of Technology, Sweden Presenting state-of-the-art knowledge in heat transfer and fluid flow in micro- and nanoscale structures, this book provides invaluable information for both graduate researchers and R&D engineers in industry and consultancy. All of the chapters are invited contributions from some of the most prominent scientists in this active area of interdisciplinary research and follow a unified outline and presentation to aid accessibility. Contents: Miniature and Microscale Energy Systems; Nanostructures for Thermoelectric Energy; Heat Transport in Superlattices and Nanowires; Thermomechanical Formation and Thermal Detection of Polymer Nanostructures; Two-Phase Flow Microstructures in Thin Geometries – MultiField Modeling; Radiative Energy Transport at the Spatial and Temporal Micro/ Nanoscales; Direct Simulation Monte Carlo of Gaseous Flow and Heat Transfer in a Microchannel; DSMC Modeling of NearInterface Transport in Liquid-Vapor PhaseChange Processes with Multiple Microscale Effects; Molecular Dynamics Simulation of Nanoscale Heat and Fluid Flow. Series: Developments in Heat Transfer, Vol 13 ISBN: 1-85312-893-7 2004 392pp £117.00/US$187.00/€175.50

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This book is written for postgraduate students, and welding, mechanical design and research engineers who deal with problems such as residual stresses and distortions of welds, design of welded structures, micro-structure modelling of welds and optimisation of welding sequences and procedures. The subject is approached in a simplified way, focusing on heat conduction and stepby-step derivation of the analytical solutions using fundamental calculus. Solutions are given in closed form and are ready for use by means of simple integrals, while applications are demonstrated through easily accessible case studies. Special features include new analytical solutions developed by the author for vaious 3D heat sources and a CD-ROM containing Visual Basic programs combined into the WHEATSIM (weld heat simulation) package. Contents: Introduction to Welding Processes and Heat Sources; Methods of Analysis; Analytical Solutions for Basic Heat Sources; Analytical Solutions for 2D Gaussian-Distributed Heat Sources; Analytical Solutions for Spherical Heat Sources; Analytical Solutions for SingleEllipsoidal Density Heat Source; Analytical Solutions for Double-Ellipsoidal Density Heat Source; Application in Weld-Pool Shape Simulation; Thermal Stresses and Distortions; Modelling of Residual Stresses in Welded Joints; Microstructure Modelling of Fusion Welds; Appendices. Series: Developments in Heat Transfer, Vol 14 ISBN: 1-85312-951-8 2004 352pp+CD-ROM £124.00/US$198.00/€186.00

Heat Transfer in Gas Turbines

Thermal Conversion of Solid Fuels

Editors: B. SUNDÉN, Lund Institute of Technology, Sweden and M. FAGHRI, University of Rhode Island, USA

B. PETERS, Research Centre Karlsruhe, Karlsruhe, Germany

Containing invited contributions from some of the most prominent specialists working in this field, this unique title reflects recent active research and covers a broad spectrum of heat transfer phenomena in gas turbines. All of the chapters follow a unified outline and presentation to aid accessibility and the book provides invaluable information for both graduate researchers and R&D engineers in industry and consultancy. Partial Contents: Heat Transfer Issues in Gas Turbine Systems; Combustion Chamber Wall Cooling - The Example of Multihole Devices; Conjugate Heat Transfer - An Advanced Computational Method for the Cooling Design of Modern Gas Turbine Blades and Vanes; Enhanced Internal Cooling of Gas Turbine Airfoils; Computations of Internal and Film Cooling; Heat Transfer Predictions of Stator/Rotor Blades; Recuperators and Regenerators in Gas Turbine Systems. Series: Developments in Heat Transfer, Vol 8 ISBN: 1-85312-666-7 2002 536pp £159.00/US$247.00/€238.50

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“This is a very good text that is worthy of purchase.” ENERGY SOURCES “...an invaluable text for students examining solid fuel combustion, the detailed exploration of different aspects of packed bed combustion also makes [it] an ideal reference for experienced engineers and postgraduates.” PETROLEUM SCIENCE AND TECHNOLOGY This book deals with the complex process, and various aspects, of the thermal conversion of solid fuel beds. All three major sub-processes, computational fluid dynamics, motion of granular media and combustion, are introduced thoroughly and their contribution to the process described. The text differs from other approaches in that the packed bed is considered as an ensemble of fuel particles. The entire conversion process is composed of the sum of the particle processes and this represents its structure more realistically than traditional approaches. Series: Developments in Heat Transfer, Vol 15 ISBN: 1-85312-953-4 2003 220pp £79.00/US$122.00/€118.50

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