Analytical Modelling of Fuel Cells

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Preface A theory if you hold it hard enough And long enough gets rated as a creed Robert Frost

The idea of intelligent and efficient two-step fuel combustion with the generation of charged particles as intermediates was conceived in 1838 by a Swiss professor of chemistry Christian Friedrich Schoenbein and realized by Sir William Grove (Bossel, 2000). In spite of this long history, huge worldwide interest and multimillion investments, fuel cells (FCs) are still far from conquering the market for power sources. The reason is a lack of knowledge of many aspects of FC operation. Generally speaking, there are two levels of FC research: (i) study of the fundamental physical and electrochemical processes (reaction kinetics, ionic transport, capillary phenomena etc.) and (ii) engineering design of cells, stacks and FC systems. The coupling between these two levels is provided by the physical modelling of fuel cells. This relatively new discipline aims at understanding the basic transport and kinetic phenomena in a real cell and stack environment. Physical modelling uses first-level data to pave the way for better cell, stack and system designs. This discipline is the subject of this book. Equations describing the physics and chemistry of fuel cells are extremely complicated and, in general, only a numerical solution of these equations is possible. The exponential growth of computer power and the appearance of commercial software for the solution of hydrodynamic problems facilitate the development of numerical FC models. The literature on numerical FC simulations is growing at an alarming rate. However, information resulting from numerical calculations is in many respects limited. A numerical result represents just one point in the multidimensional space of parameters determining FC operation. What happens in the system if we change one parameter? To answer this question another calculation has to be done. Since the number of parameters is large and each variant is a v

vi

PREFACE

time-consuming procedure, a parametric study with the numerical models is exhausting. Analytical solutions resulting from a good model simply show the parametric dependencies. These solutions may significantly advance our understanding of FC operation; sometimes they predict novel effects. In physics, analytical modelling has long been considered a key component of research, together with numerical modelling and experiment. For historical reasons, however, in FC science analytical methods are much less developed, they still remain “exotic goods”. Several years ago the author received a negative response to a paper he submitted to a wellrespected electrochemical journal. One of the reviewer’s remarks was: “The work contains 52 equations!”. This book is written by a physicist and it contains many equations. The goal of analytical modelling is to derive simple relations describing different aspects of FC operation in special cases. The book considers these special cases. Fuel cell science is a rapidly growing field: it includes overlapping domains of chemistry, physics and fluid mechanics. It is, therefore, hardly possible to discuss all the models and approaches used in FC studies under one heading. In this book we demonstrate the basic analytical solutions describing coupled kinetic, transport and electric phenomena in catalyst layers, cells and stacks. However, the aim of this book is not a simple collection of results, but rather a step-by-step demonstration of how they are derived. The author’s greatest reward would be if young readers were motivated to apply the analytical instruments discussed below in their daily work. The author is grateful to Alexei Kornyshev, Anthony Ku´cernak, Michael Eikerling and Jiri Divisek for numerous discussions. Some of the models discussed in this book were validated in unique experiments designed by Klaus Wippermann. Invaluable experimental material was provided by Ayhan Egmen and Harald Scharmann. Many thanks to colleagues and friends from the Institute of Energy Research—Fuel Cells of Forschungszentrum J¨ ulich who, over the years, helped the author to understand obscure points of fuel cell science. Many thanks to Alexander Tikhonravov and other friends and colleagues from the Research Computing Centre of Moscow State University for the unforgettable atmosphere for making science there. Thanks to Janet CarterSigglow for revising the English text. This book would have never been written without indispensable support from my wife Maria.

Introduction The fuel cell effect is simple: two electrochemical reactions are separated in space by an ionic conductor (electrolyte). One of these reactions produces ions and electrons, and the other consumes them. Ions move through the electrolyte, while electrons perform useful work in the external load. A fuel cell is easy to assemble at home. All you need is a membraneelectrode assembly (a one millimetre-thick piece of high-tech material), two metallic or graphite plates with channels for feed gas supply and a cartridge with metal hydride which liberates hydrogen upon heating. Clamp a 10 × 10 cm piece of MEA between current collector plates, connect a hydrogen source to the channels in the anode plate and blow air through the channels in the cathode plate. The fuel cell is now ready to provide you with several tens of watts of DC current. However, this apparent simplicity hides a lot of complex and poorly understood processes inside the cell. Feed molecule ionization and ion recombination run on a surface of precious catalyst particles (Pt, as a rule), which is in contact with the electrolyte. A giant electric field arises at the Pt/electrolyte interface. The electrochemical reactions in this environment have been intensively investigated for more than fifty years, but their mechanisms still remain controversial. Current produced by the cell increases with the active surface of catalyst. To increase this surface a huge number of tiny catalyst particles are mixed with ionic and electron conductors in the catalyst layers. A void pore in the close vicinity of the catalyst particle is needed to provide fast delivery of feed molecules. Over the past decade, an enormous work has been done to engineer stable and efficient catalyst layers. However, even in the best layers a large amount of precious particles are disconnected from the electrolyte or located far from the void pore and are, therefore, “dead” for the reaction. Ionic current between the anode and the cathode is transported through an ionic conductor. In low-temperature fuel cells, internal charge carriers are protons and the proton conductor is a solid polymer electrolyte membrane r (typically Nafion ). It took more than 10 years to clarify the mechanism of proton transport in this membrane. However, the membrane structure and its dependence on water content are still unclear. xiii

xiv

INTRODUCTION

In high-temperature cells, the internal ionic current is transported by doubly charged oxygen ions in a special ceramic material. The mechanism of ion transport in this material is not fully understood. Fuel cells consume fuel and oxygen and generate current, heat and reaction products. Feed molecules have to be distributed over the FC surface as uniformly as possible and reactants should be removed without significant losses. At this level, a FC appears as the chemical reactor which needs to be optimized with respect to geometry, operating conditions, etc. To increase power density FCs are assembled in stacks. In a stack new problems arise: the most significant of these are removal of superfluous heat and reaction products, and uniform distribution of feed molecules between the individual cells. The harsh market of power sources requires FC stacks to operate reliably many thousands of hours. In the past few years, attention has been drawn to problems of FC durability. The power of FC-based energy sources ranges from a few milliwatts in implantable devices to hundreds of kilowatts in stationary applications. The diversity of FC types and a wide range of applications has led to a large number of different FC designs. Fortunately, all fuel cells are based on the same fundamental principles and hence they have many similarities. The structure of this book aims to emphasize the similarity of fuel cells of different types. Nowadays, when the number of publications on FC science and technology is growing at an alarming rate, the fuel cell community tends to separate into sub-domains interested in only one type of cell. In an attempt to counteract this unfortunate trend the book is structured according to the hierarchical principle (from catalyst layers to stacks), rather than according to the cell type. SOFC, PEFC and DMFC communities can learn a lot from each other. Chapter 1 contains an overview of the basic principles of FC operation and of the processes in a FC. Models of catalyst layers for all types of cells are collected in Chapter 2. This chapter is followed by Chapter 3 on the through-plane modelling of PEFC, DMFC and SOFC. Then we proceed to quasi-2D cell modelling, which takes into account feed molecule exhaustion along the channel (Chapter 4). The final Chapter 5 is devoted to modelling of DMFC and SOFC stacks (the ideas in this Chapter can be directly applied to model PEFC stacks as well).

Notations Throughout the book a dimensionless form of equation is extensively used. Dimensionless variables greatly simplify the analysis of equations. In many cases, just a simple scaling analysis of dimensionless parameters suggests fruitful simplifications.

INTRODUCTION

xv

In each case, the dimensionless variables are chosen to provide the simplest form of the resulting equations. Thus, in the various sections these variables generally differ. To avoid misunderstandings, the lists of dimensionless variables are shown in red-violet. To find the list currently in effect the reader should find the nearest red-violet fragment moving backwards along the text (See also the entry “dimensionless variables” in the book index). Optional material (Section 3.4.6) is typeset in blue font. Generally, the mathematical notation is uniform throughout the book. However, fuel cell modelling deals with numerous substances and the duplication of symbols is unavoidable. The nomenclature section is placed at the end of the book and in case of doubt, this section should be consulted.

Maple codes Some problems considered in this book have been solved using a r powerful software for analytical calculations Maple . The Maple codes can be downloaded from doi:10.1016/B978-0-444-53560-3.00003-4. The codes contain comments to help readers use them.

Chapter 1

Fuel cell basics In this chapter, we introduce the basic concepts for fuel cell modelling. There are indispensable books where some of these concepts are discussed in detail (Bard and Faulkner, 2001; Newman, 1991). The aim of this chapter is to outline the physics which stand behind the equations used in the subsequent chapters.

1.1 Fuel cell thermodynamics 1.1.1

The physics of the fuel cell effect

Consider two Pt plates separated by an electrolyte (Figure 1.1). Suppose that we put two hydrogen molecules onto the Pt/electrolyte interface on the left (anode) side and one oxygen molecule onto the electrolyte/Pt interface on the right (cathode) side (Figure 1.1). What happens in the system? On the anode side, each hydrogen molecule splits into two protons and two electrons: 2H2 → 4H+ + 4e− .

(1.1)

Four protons move to the plasma of the electrolyte, while four electrons stay in the metallic Pt (Figure 1.1). As a result, on the anode side we get a charged capacitor (Figure 1.1). On the cathode side, the oxygen molecule captures four protons from the electrolyte and four electrons from the metal (Pt). When all these species come together on the Pt surface, two water molecules result: O2 + 4H+ + 4e− → 2H2 O. 1

(1.2)

2

CHAPTER 1. FUEL CELL BASICS

Figure 1.1: Schematic of the fuel cell effect. The hydrogen molecule at the anode/electrolyte interface splits into two protons and two electrons (only one of the two molecules is shown). Protons shift to the electrolyte while electrons stay in the metal. Due to the separation of charges the anode part of the cell appears to be a charged capacitor. On the cathode side, the oxygen molecule captures four protons from the electrolyte (leaving this part of the electrolyte negatively charged) and four electrons from the metal (thereby charging the metal positively). This results in two water molecules and another charged capacitor. The anode and the cathode capacitors determine the cell open-circuit voltage. Depletion of protons in the electrolyte results in a negative charge on the electrolyte side, whereas removal of four electrons charges Pt positively. Thus, on the cathode side we also get a charged capacitor (Figure 1.1). The three-layer sandwich in Figure 1.1 transforms to the two charged capacitors connected in series, which results in a voltage drop between the Pt plates. This system is a hydrogen fuel cell under open-circuit conditions; the voltage drop between the anode and the cathode Pt plates is called the cell open-circuit voltage (OCV). Can we use this voltage to generate electric power? Thermodynamics says, “Yes, we can”1 . For any reaction running under constant pressure and temperature the following relation holds ∆H = T ∆S + ∆G

(1.3)

where ∆H is the enthalpy change in the reaction, T is the absolute temperature, ∆S is the entropy change and ∆G is the change in the Gibbs free energy. Thermodynamics states that ∆G can be used as useful work. 1A

concise and clear introduction to fuel cell thermodynamics is given in Li (2007).

1.1 FUEL CELL THERMODYNAMICS

3

Figure 1.2: Schematic of a hydrogen fuel cell in operation. To support the currents, hydrogen has to be continuously supplied to the anode and oxygen to the cathode. Protons move through the electrolyte separator, while electrons perform useful work in the external load. Consider direct hydrogen-oxygen combustion: 2H2 + O2 → 2H2 O.

(1.4)

In this reaction temperature and/or pressure increases and ∆H ultimately transforms into heat: ∆H = ∆Q. This means that in the direct reaction (1.4), the chance of an in situ conversion of ∆G into the useful work is lost. We need to build a heat machine (turbine) to produce useful work from the heat energy ∆Q. In hydrogen cells the reaction (1.4) is split into two half-reactions (1.1) and (1.2) running at constant temperature and pressure. The spatial separation of these reactions enables a direct conversion of ∆G into electric energy: proton current flows between the anode and the cathode through a separator (electrolyte membrane), while electron current generates useful work in the external circuit (Figure 1.2). To support these currents we have to continuously supply hydrogen to the anode and oxygen to the cathode. Thus, in a fuel cell the purely chemical combustion of neutral molecules (1.4) is split up into two electrochemical reactions (1.1) and (1.2), which run with the participation of charged particles. Basically, any combustion reaction can be split up into a pair of electrochemical half-reactions and hence any fuel can be utilized in a fuel cell for direct conversion of ∆G into electric energy. For the reasons discussed below, in a system with only a few molecules shown in Figure 1.1 the OCV is very small. If the number of hydrogen and oxygen molecules is large enough, this voltage is about 1.2 V. The opencircuit cell potential can be calculated from thermodynamic relations, as discussed in the following sections.

4

1.1.2

CHAPTER 1. FUEL CELL BASICS

Open-circuit voltage

The thermodynamic potentials of a system in the initial state A and final state B do not depend on the way the system is transferred from A to B, provided that the transfer proceeds through a series of equilibrium states (reversibility). In particular, the quantities appearing in Eq. (1.3) remain the same if we produce water from H2 and O2 either in hypothetical reversible direct combustion (1.4), or in the half-cell reactions (1.1) and (1.2)2 . Therefore, ∆G available in fuel cells can be found in standard thermodynamic tables of fuels. In fuel cells, ∆G is related to the cell opencircuit voltage Voc by the equation Voc = −

∆G ne F

(1.5)

where ne is the number of electrons transferred per mole of fuel (in PEFCs, ne = 2, and in DMFCs, ne = 6) and F is the Faraday constant3 . Rewriting Eq. (1.5) in a form |∆G| = ne F Voc we see that this is simply an expression for electrostatic energy ∆G stored in a capacitor with the charge ne F at potential Voc . This shows that the physics of fuel cells indeed involve charging/discharging of capacitors, as discussed in Section 1.1.1. Note that energy and charge in Eq. (1.5) are calculated per mole of fuel.

1.1.3

Nernst equation

∆G does not depend on the trajectory between the initial and final states, but it does depend on species concentrations in these states. The reference point for Voc is defined under standard conditions (standard atmospheric pressure and temperature of 298 K). In this state, the variables in Eq. (1.5) are denoted by the superscript 0: 0 Voc =−

∆G0 . ne F

(1.6)

What happens to Voc when the concentrations of reactants and products change? To answer this question we need to rationalize the dependence of 2 Here “reversible” means that the entropy change is due to the creation of new (water) molecules only (see also Section 1.5). 3 In all combustion processes, ∆G < 0, which means that this energy is available for useful work.

1.1 FUEL CELL THERMODYNAMICS

5

Gibbs free energy on concentration. Thermodynamics states that dG = −SdT + V dP

(1.7)

where V and P are the volume and pressure of the reagent of interest, respectively. Under isothermal conditions, dT = 0 and Eq. (1.7) reduces to dG = V dP.

(1.8)

Using the ideal gas law P V = mRT we can exclude V from (1.8), which yields dG = mRT

dP = RT d(ln P m ) P

where m is the number of moles of the reagent. Integrating this relation from the standard state we find G = G0 + RT ln



P P0

m 

' G0 + RT 0 ln

 c m  c0

(1.9)

where c is the reagent molar concentration and c0 is this concentration in the standard state. In the last equation we used the ideal gas law in a form P = cRT and took into account that the process is isothermal, T = T 0 . We see that the Gibbs free energy logarithmically increases with species concentration. The number of moles m (stoichiometry factor in the reaction) appears as an exponent under the logarithm sign. For the fuel cell reaction (1.4), Eq. (1.9) takes the form (for details see Li (2007)) ∆G = ∆G0 − RT 0 ln



√  c˜H2 c˜ox c˜w

(1.10)

where c˜ = c/c0 is the normalized molar concentration of the respective component. Dividing Eq. (1.10) by 2F yields the Nernst equation Voc =

0 Voc

RT 0 + ln 2F



√  c˜H2 c˜ox . c˜w

(1.11)

Generally, reactants and/or products may appear in a liquid form. In that case, molar concentrations in the Nernst equation should be replaced by the activity coefficients (for details see Bard and Faulkner (2001)).

6

1.1.4

CHAPTER 1. FUEL CELL BASICS

Temperature dependence of OCV

Cell open-circuit voltage depends on temperature. To rationalize this dependence let us now consider the reaction under constant pressure. Setting dP = 0 in (1.7) we arrive at dG = −SdT. Integrating this relation from the standard state and assuming that the temperature dependence of reaction entropy S is weak, we find G = G0 − S(T − T 0 )

(1.12)

where T 0 = 298 K is the standard temperature. Writing (1.12) for the final and initial states of the reaction Gi = G0i − Si (T − T 0 ) Gf = G0f − Sf (T − T 0 ) and subtracting these relations we get ∆G = ∆G0 − ∆S(T − T 0 ). Dividing this by ne F and using (1.5) we finally obtain 0 Voc = Voc +


∆S T − T0 . ne F

(1.13)

Thus, to a good approximation the dependence of cell OCV on temperature is linear. Note that in a hydrogen fuel cell, ∆S < 0 (Li, 2007) and with the growth of temperature, Voc decreases. However, due to the presence of a large factor ne F in the denominator of Eq. (1.13) this decrease is small (about 10−3 V/K). Above we have assumed that the entropy change in the reaction does not depend on T . This approximation is quite accurate, provided that the difference T − T 0 is not large.

1.2 Potentials in a fuel cell Most of the fuel cell reactions run on the surface of precious catalysts. However, so far none of the catalyst properties has appeared in the equations. The relations above describe the maximum possible cell opencircuit voltage, which is determined solely by the thermodynamics of species involved in the reaction4 . 4 This maximal OCV is obtained with the ideal pair “catalyst + electrolyte”. A poor catalyst or electrolyte may significantly reduce the OCV of a real cell.

1.2 POTENTIALS IN A FUEL CELL

7

However, if we draw current from the fuel cell, the reactions shift from the equilibrium state and the properties of the reaction environment immediately come into play. This is the case of fuel cell kinetics. Before proceeding to the discussion of kinetic relations it is advisable to consider the potentials in a fuel cell. At zero current, fuel cell electrodes provide the thermodynamic OCV: Vcell = Voc . Connection of a load induces current I in the cell and reduces Vcell by the value of voltage loss Vloss (I). Thus, current drawn from the fuel cell “costs” some potential; the thermodynamic OCV Voc is essentially the “capital” at our disposal. It is convenient to eliminate the cell active area A introducing the mean current density J = I/A. The quantities Vcell and Vloss are then functions of J: Vcell (J) = Voc − Vloss (J). A central question in fuel cell modelling is: Which processes contribute to Vloss (J) and how large is each contribution? Typically, the largest voltage loss in a cell occurs in the catalyst layers. Electrochemical reactions run at the catalyst particle/electrolyte interface. Consider for definiteness, Pt particles. The total number of molecules produced or consumed in the reaction is proportional to the surface area of the Pt/electrolyte interface. The plain interface shown in Figure 1.2 would give a very small current suitable for electrochemical studies, but would be of no practical interest for energy conversion devices. In practical fuel cells, the Pt/electrolyte interface should be as large as possible. This is achieved by mixing tiny carbon-supported catalyst particles with the polymer electrolyte and filling voids of carbon cloth or paper with this mixture. The matrix of carbon threads provides mechanical stability and electronic contact between catalyst particles. Note that the Pt/electrolyte mixture does not fully fill the voids: some residual porosity is needed to provide transport of reactants to the catalyst sites. A detailed discussion of the catalyst layer structure and useful references are given in Eikerling et al. (2007) and Promislow and Wetton (2009) (see also Section 2.9). The driving forces for charged particles in this environment are modelled by two continuous potentials: the electrolyte phase potential ϕm and the carbon phase potential ϕ. The gradient of ϕm drives protons in the electrolyte phase while the gradient of ϕ induces electron current in the Pt-carbon phase. Physically, in the catalyst layer both carbon and electrolyte phases are porous structures. However, in modelling, porous carbon and electrolyte clusters are usually replaced by the two interpenetrating continuum media (phases). The idea of representing a “porous” distribution of potentials by continuous functions of space is usually referred to as a macrohomogeneous approach. Strictly speaking, this approach has no rigorous justification. However, theoretical results obtained within the scope of this approach can be compared with available experimental data. So far this comparison has given us no good reasons to doubt this idea.

8

CHAPTER 1. FUEL CELL BASICS

Figure 1.3: Schematic for the calculation of voltage loss in a fuel cell (for discussion see text). ACL and CCL are the abbreviations for the anode and cathode catalyst layers, respectively. Yellow shaded areas indicate the local polarization voltage η. For simplicity, the proton conductivity of catalyst layers is taken to be equal to the proton conductivity of the bulk membrane (otherwise the curve ϕ0m loses smoothness at the membrane interfaces). Note that the half-cell voltage loss is given by the value of the overpotential at the catalyst layer/membrane interface. The polymer electrolyte can be thought of as a plasma in which positive charges (protons) can move under the action of the electric field, whereas the negative charges are fixed at the polymer backbone. In the presence of feed molecules, positive and negative charges at the Pt/electrolyte interface separate and an electric double layer forms (Figure 1.1). The difference ϕm − ϕ models the jump of potential on the double layer. The potential of catalyst particles does not differ from the potential of carbon threads; thus the double layer may be thought of as being extended over the whole volume of the catalyst layer. In the following, ϕ will be furnished with the superscripts a and c to distinguish between the anode and the cathode side, respectively. In fuel cell modelling, it is convenient to use the picture of potential distribution across the cell sketched in Figure 1.3 (Kulikovsky et al., 2000). In this picture, the physical potentials of the anode carbon phase and of the membrane phase are shifted as a whole along the V -axis. This shift, however, does not change the anode and cathode half-cell overpotentials, which are of primary interest in modelling. Let the cathode side be grounded, i.e. the carbon phase potential ϕc is zero. Thanks to the high electronic conductivity of this phase, the variation

1.3 RATE OF ELECTROCHEMICAL REACTIONS

9

of ϕc along x is small (Figure 1.3). Suppose for a moment that the cell open-circuit voltage is zero and consider an imaginary fuel cell driven by the external battery with the voltage Vloss . In this cell, the carbon phase a potential on the anode side is ϕ0 = Vloss and the membrane phase potential 0 is ϕm (Figure 1.3). Figure 1.3 shows that Vloss is the sum of three components: Vloss = η c + Rm j + η a

(1.14)

a

where η a = ϕ0 − ϕ0m and η c = ϕ0m − ϕc are the anode and cathode halfcell polarization voltages (overpotentials). Note that the overpotentials appearing in (1.14) are calculated at the respective catalyst layer/membrane interface (Figure 1.3). The values of overpotential in the “bulk” of the respective catalyst layer (CL) determine the local rate of the electrochemical reaction5 . In Chapters 2–4 we will show that η c at the membrane/CCL interface comprises all the types of voltage loss on the cathode side (the ORR activation, the oxygen transport loss in the GDL, CCL and in the flow field, and the voltage loss due to poor proton transport in the CCL). The same is true for η a at the ACL/membrane interface. In other words, any transport loss in a fuel cell translates into a higher η c or η a . Returning to the real fuel cell, we see that the real cell voltage is simply Vcell = Voc − Vloss and the real carbon phase potential on the anode side is a a ϕa ≡ Vcell = Voc − ϕ0 (Figure 1.3). The introduction of ϕ0 is convenient a in modelling: to calculate the losses we may ignore the OCV and use ϕ0 instead of the real carbon phase potential on the anode side. The cell voltage a is then obtained from the relation Vcell = Voc − ϕ0 . Suppose that the current j in the cell decreases. What happens to the a potentials in Figure 1.3? ϕ0 drops and the variation of ϕ0m becomes smaller, in agreement with the relation j = −σt ∂ϕ0m /∂x (σt is the ionic conductivity a of the electrolyte phase). When j = 0, we have ϕ0 = ϕ0m = ϕc = 0, i.e. the total voltage loss is zero and according to Figure 1.3, Vcell = Voc .

1.3 Rate of electrochemical reactions 1.3.1

Butler-Volmer equation

The kinetics of the hydrogen oxidation reaction (HOR) (1.1) are very fast and the respective polarization voltage can be ignored. However, 5 In electrochemical studies, voltage loss due to a transport of reactants to the catalyst sites is usually marginal and η is called overpotential, or activation overpotential. “Polarization voltage” is a more general term, which includes transport loss (see below). In the following we will use the two terms as synonyms.

10

CHAPTER 1. FUEL CELL BASICS

activation of the oxygen reduction reaction (ORR) requires quite significant overpotential. Like many other electrochemical reactions, ORR is a multistep process. In spite of several decades of intense research, the exact kinetic scheme of this reaction is poorly known. The situation is complicated by experimental evidence that the reaction pathways depend on the structure of the catalyst surface and they are different for different values of the overpotential (Markovi´c and P. N. Ross, 2002). At low overpotentials (small currents) the reaction scheme is thought to be (Kuhn et al., 2007): 1. Oxygen adsorption onto the surface of catalyst particles k

1 O2 −→ Oad + Oad .

2. Electron attachment to adsorbed oxygen atoms k

2 Oad + e− −→ O− ad .

3. Recombination of oxygen ion and proton k

3 + O− ad + H −→ HOad .

4. Associative recombination of electron and proton k

4 HOad + H+ + e− −→ H2 O.

(1.15)

Inspection of the rate constants ki of steps 1-4 shows that the following relation holds6 : k4 ' 10−1 k3 ' 10−8 k2 ' 10−12 k1 . Thus, the rate constants of individual steps differ by more than 10 orders of magnitude. Clearly, in the chain of events 1-4 the rate-determining step (RDS) is (1.15). The other steps can be considered as infinitely fast. The dramatic difference in rate constants allows us to ignore the fast steps and consider the ORR (1.2) as an equivalent single-step single-electron transfer whose rate constant is determined by the RDS (1.15). Note that the other electrons are transferred in fast steps “for free”, i.e. their transfer does not require any significant overpotential. Nonetheless, these electrons must be taken into account in the balance of charges by the respective stoichiometry coefficient. 6 For a correct comparison the rate constants in (Kuhn et al., 2007), which have a dimension mol cm s−1 , have to be multiplied by the proton concentration in Nafion, cH+ ' 1.2 · 10−3 mol cm−3 , since the respective reactions involve H + .

1.3 RATE OF ELECTROCHEMICAL REACTIONS

11

The rate of this equivalent single-step reaction Qf (the number of charges consumed in unit volume per second) is Qf = kf (η)

cox cref

(1.16)

where kf (η) is the rate constant of step (1.15) which depends on the cathode half-cell overpotential η (in the present section the superscript c is omitted). This dependence reflects the fact that the RDS involves charged particles. For the reasons discussed below, the dependence kf (η) is exponential   αF η kf = i∗ exp (1.17) RT where i∗ is the volumetric exchange current density (A m−3 ), the reaction rate at equilibrium (at η = 0). Physically, the exponential dependence (1.17) follows from the general Arrhenius form of the reaction rate constant:   Eact (1.18) kf = iref exp − RT where Eact is the activation energy and iref is the reference exchange current density7 . To participate in the reaction, one of the charged species must overcome the activation barrier Eact . Let at equilibrium (open-circuit conditions) the eq activation barrier be Eact (this value will be calculated in the next section). To draw current from the reaction this barrier must be lowered by the value αF η: eq Eact = Eact − αF η.

(1.19)

The transfer coefficient α ' 0.5 (sometimes called the symmetry factor ) takes into account the details of the shape of the activation barrier (Bard and Faulkner, 2001). Using (1.19) in (1.18) we get       E eq αF η αF η kf = iref exp − act exp = i∗ exp (1.20) RT RT RT which coincides with (1.17). Equation (1.20) shows that   E eq i∗ = iref exp − act . RT

(1.21)


= i0∗ exp Eact /(RT 0 ) , where the superscript 0 marks the values under standard conditions. 7i

ref

12

CHAPTER 1. FUEL CELL BASICS

The exponential dependence (1.21) is one of the most important temperature dependencies in fuel cells. It enables current production to be dramatically increased by increasing the cell temperature. Close to equilibrium, the ORR (1.2) is balanced by the reverse reaction of water electrolysis: 2H2 O → O2 + 4H+ + 4e− .

(1.22)

The RDS of this reaction is also a single-electron transfer and hence the reaction rate can be expressed as  Qr = i∗

2

cw cwref



βF η exp − RT

 (1.23)

where cw and cwref are the available and reference water molar concentrations, respectively. Note the change in sign of the exponent power: positive overpotential facilitates the direct reaction (1.2) and retards the reverse reaction (1.22). Subtracting (1.23) from (1.16) we get the total rate of ORR " Q = i∗

cox cref



 exp

αF η RT



 −

cw cwref

2

#  βF η . exp − RT

(1.24)

This is the famous Butler-Volmer equation, which describes the overall rate of the half-cell electrochemical reaction (ORR). If the RDSs of direct and reverse reactions are the same, the transfer coefficients of these reactions add up to yield 1: α + β = 1 (Bard and Faulkner, 2001). In this case, in the second exponent we can set β = 1 − α. In fuel cell studies, Eq. (1.24) can be simplified further if we take into account that the second exponent in (1.24) makes a noticeable contribution to the total rate Q only close to equilibrium, when the rates of direct and reverse reactions are nearly the same. However, near equilibrium the oxygen and water concentrations are close to their reference values and hence the respective concentration factors in Eq. (1.24) are close to 1. This allows us to write Eq. (1.24) in the following approximate form  Q = i∗

cox cref

     αF η (1 − α)F η exp − exp − . RT RT

(1.25)

Indeed, close to equilibrium we have cox ' cref , cw ' cwref , and Eqs (1.25) and (1.24) coincide. Far from equilibrium, the second exponent in Eq. (1.25)

1.3 RATE OF ELECTROCHEMICAL REACTIONS

13

is negligible and hence the concentration factor at this exponent is of no interest8 . It is worth noting that Eq. (1.25) is valid if the whole cell surface is run in a current-generating (fuel cell) mode. If part of the cell experiences strong oxygen depletion, the respective domain may turn into the electrolysis mode (Kulikovsky et al., 2006). In that case, the exact Butler-Volmer equation in the form (1.24) has to be used. Equation (1.25) can be written in an equivalent form  Q = i∗

cox cref

   η η − exp − exp b br

(1.26)

where b=

RT αF

and br =

RT (1 − α)F

(1.27)

are Tafel slopes of the direct and reverse reactions, respectively. For the transfer coefficient, kinetic theory gives a value α = 0.5 (Bard and Faulkner, 2001). We then have α = 1 − α = 0.5 and b = br . With this Eq. (1.26) can be written in a more compact form  Q = 2i∗

cox cref

 sinh

η b

.

(1.28)

For many practical catalysts, b is known from experiments. The characteristic values of b for the low- (PEFC and DMFC) and hightemperature (SOFC) fuel cells are listed in Table 1.1. Note that in the electrochemical literature the dependence of Q on η is often represented as a power of 10, rather than the exponent; the respective value b10 ' 2.3b.

1.3.2

Butler-Volmer and Nernst equations

At equilibrium the Butler-Volmer equation should transform into the eq Nernst equation. This condition gives the activation energy Eact in Eq. (1.21). 8 In the fuel cell literature, another form of the approximate Butler-Volmer equation is sometimes used

 Q = i∗

cox cref



 exp

αF η RT



  (1 − α)F η − exp − . RT

Using the same arguments, it is easy to show that this equation also approximates the exact equation (1.24).

14

CHAPTER 1. FUEL CELL BASICS

Table 1.1: Characteristic values of the Tafel slope b for low- and hightemperature fuel cells. In electrochemical studies, a value b10 ' 2.3b (Tafel slope per decade) is often used. PEFC, DMFC 50 115

b (mV) b10 (mV/dec)

SOFC 150 345

Substituting (1.20) into (1.16) and setting η = 0 we get the forward reaction rate near equilibrium Qeq f = iref



cox cref



  E eq exp − act RT

(1.29)

where the concentration factor cox /cref & 1. Equation (1.29) describes the conditions when the reaction is slightly shifted forward, toward reduction.
At equilibrium Qeq = iref exp −|∆G0 |/(RT ) (Bard and Faulkner, f 2001). Using this in Eq. (1.29) we come to 

cox cref



E eq exp − act RT 



  |∆G0 | = exp − . RT

(1.30)

Setting here eq Eact = F Voc

(1.31)

after simple algebraic transformations we arrive at Voc

|∆G0 | RT = + ln F F



cox cref

 .

0 Taking into account that for the half-cell reaction |∆G0 | = F Voc , we finally find

0 Voc = Voc +

RT ln F



cox cref

 (1.32)

which is the Nernst equation for the forward single-electron transfer reaction. Thus, the reaction activation barrier at open circuit is given by Eq. (1.31) or, equivalently, by eq Eact = |∆G|.

1.3 RATE OF ELECTROCHEMICAL REACTIONS

1.3.3

15

Tafel equation

The values in Table 1.1 suggest that in typical operating conditions of low-temperature fuel cells, the second exponent in Eq. (1.24) can be neglected. Indeed, the typical working overpotential on the cathode side is η ' 300 mV. Clearly, in PEFC and DMFC, η/b  1 and hence exp(η/b)  exp(−η/b). Note that the last inequality already holds at η ' b ' 50 mV. Thus, in modelling low-T cell regimes of practical interest we can safely neglect the reverse reaction and Eq. (1.28) reduces to the Tafel equation:  Q = i∗

cox cref

 exp

η b

.

(1.33)

In many electrochemical reactions, the rate-determining step is a singleelectron transfer, analogous to (1.15). This explains why in a large range of polarization voltages, the half-cell reactions exhibit Tafel behaviour. Thus, Butler-Volmer or Tafel equations are very good starting points for any fuel cell model. The great advantage of these equations is their simplicity: they characterize reaction kinetics by just two parameters, b and i∗ . To illustrate the use of the Tafel equation, consider the following simple problem. Let the proton current density j0 enter the cathode catalyst layer from the membrane. In the CCL, this current is converted into the electron current and at the CCL/GDL interface the proton current j1 = 0. The proton current conservation equation reads ∂jp = −Q ∂x

(1.34)

where jp is the local proton current density and Q is given by (1.33). Suppose that j0 is small, so that we can neglect the variation of η along x: η ' η0 = const. We will assume that oxygen transport across the CCL is ideal and hence cox /cref is also independent of x. For these ideal conditions the solution to (1.34) is a linear function of x:  jp = (lt − x) i∗

cox cref

 exp

η  0

b

(1.35)

where lt is the CCL thickness. Setting in (1.35) x = 0 we get a relation of cell current density j0 and overpotential η0 , which is the polarization curve of the CCL:  j0 = lt i∗

cox cref

 exp

η  0

b

.

(1.36)

16

CHAPTER 1. FUEL CELL BASICS

We see that the current (1.36) converted in the CCL exponentially depends on overpotential. To obtain this current we have to multiply the constant reaction rate Q (1.33) by the CCL thickness lt . Solving (1.36) for η0 we find   j 0  (1.37) η0 = b ln   ox j∗ ccref where j ∗ = lt i ∗

(1.38)

is the superficial exchange current density (A cm−2 ). The volumetric exchange current density i∗ (A cm−3 ) should not be confused with the superficial exchange current density j∗ (A cm−2 ). The first value is a local characteristic of a CCL, proportional to the local concentration of catalyst particles (to be precise, to their active surface)9 . In contrast, parameter j∗ is an integral characteristic of the CL. This parameter includes CL thickness; under variable i∗ (x) the generalization of (1.38) is obvious Z lt j∗ = i∗ dx. (1.39) 0

Importantly, both i∗ and j∗ are scalar values. In cell and stack modelling, the catalyst layer is often treated as an electrical resistance with nonlinear voltage-current characteristic of the form η  0 (1.40) j0 = j∗∗ exp b (here j∗∗ absorbs the concentration factor cox /cref ). As discussed above, this implies that the overpotential η is nearly constant across the layer, which is justified if the cell current is sufficiently small and the layer is not very thick. In Chapter 2 we will see that at large current the CL polarization curve strongly differs from (1.40).

1.4 Mass transport in fuel cells 1.4.1

Overview of mass transport processes

A fuel cell is a two-scale system: the characteristic in-plane size of a typical cell is about 10 cm, while the thickness of the membrane-electrode assembly 9 In this section, i is constant; however, this is not necessarily the case. In Section 2.6 ∗ we will see that the gradient of catalyst loading is beneficial for CL performance.

1.4 MASS TRANSPORT IN FUEL CELLS

17

Figure 1.4: Left: sketch of the membrane-electrode assembly and channels for the distribution of reactants over the cell surface (flow field). Right: small fragment of a hydrogen fuel cell. Note that both panels are not to scale: the total MEA thickness is typically in the order of the channel height above the plane. (MEA) is in the order of 0.1 cm (Figure 1.4). Delivery of reactants to the catalyst sites thus involves (i) the distribution of reactants over the cell active surface and (ii) the transport of reactants through the gas-diffusion layer (GDL) to the catalyst layer10 . The distribution of reactants over the cell surface is usually performed through a system of channels called the flow field (Figure 1.4). The channels with a typical cross-section of 1 mm × 1 mm are machined in the metallic or graphite current collectors (Figure 1.4). In stacks, the role of the current collector is taken over by the bipolar plate, which plays the role of the cathode and the anode of two adjacent cells (Chapter 5). In cells of various types, the flow in the channels can be purely gaseous (SOFC and anode of PEFC), two-phase (PEFC and DMFC cathodes, and DMFC anode) or purely liquid (DMFC anode at low current). A typical inlet flow velocity of liquid methanol-water solution in the anode channel of the DMFC varies between 0.1 and 1 cm s−1 . The velocity of gaseous flow in fuel cells is between 10 and 103 cm s−1 . With the typical channel diameter in the order of 0.1 cm, the Reynolds number varies in the range of 100-1000 and hence the flows are laminar. The speed of sound in atmospheric pressure air at T = 350 K is about 3 · 104 cm s−1 and it increases with temperature. Therefore, gaseous flow in the channel is always deeply subsonic and hence incompressible (Section 4.1). This means that the pressure in the channel varies mainly due to the viscous friction of the flow over the channel walls. This friction leads to the formation of a velocity boundary layer. In this layer, the axial (longitudinal) flow velocity drops from the maximal value at 10 In the DMFC, the anode GDL is usually called the backing layer since it transports both gaseous CO2 and liquid methanol.

18

CHAPTER 1. FUEL CELL BASICS

the channel axis to zero at the channel walls. In short channels the pressure drop due to viscous forces is not large and to a good approximation the pressure (and hence total molar concentration of species) can be assumed to be constant. In PEFCs, the liquid water produced in the cathodic reaction enters the cathode channel and mixes with air to form a two-phase flow. In this flow, liquid droplets may form and be destroyed; computer modelling shows complicated two-phase flow patterns (Le and Zhou, 2009). Detailed study of two-phase flows in fuel cell channels is still in its infancy; some knowledge has been gained from computational fluid dynamics (CFD) simulations (Wang, 2004; Gurau et al., 2008b; Le and Zhou, 2009). Being distributed over the cell surface, reactants are transported through the GDL/backing layers to the catalyst layers, where the electrochemical reactions occur (Figure 1.4). The main mode of reactant transport through the GDL is usually diffusion due to the concentration gradient. In low-temperature fuel cells, an important process is water transport through the polymer membrane. Water in the membrane is transported by diffusion due to the concentration gradient and by electro-osmosis: the proton current in the membrane “drags” water molecules from the anode to the cathode. Electro-osmosis pumps water through the membrane, drying the anode side of the membrane and flooding the cathode.

1.4.2

Stoichiometry and utilization

The molar flux N of any neutral species (mol cm−2 s−1 ) can be expressed in electric units by multiplying N by the Faraday constant: jN = nF N (n is the number of electrons transferred in the electrochemical reaction per mole of a reactant). Note that although jN has the dimension of the current density (A cm−2 ), it is not a real current, since N does not transport charge. By definition, the stoichiometry factor λ of the feed molecules is the ratio of incoming total molar flux of these molecules expressed in electric units to the cell current (A): λ=

nF N hw . I

(1.41)

Here h and w are the height and width of the feed channel. For example, oxygen stoichiometry λ = 2 means that in a fuel cell, 50% of oxygen flux is converted to current. If flow velocity in the channel is constant, λ = 2 also means that 50% of oxygen molecules are consumed. The SOFC community usually uses hydrogen utilization u instead of hydrogen stoichiometry. In this book we accept the following definition: u = 1 − c1 /c0 , where c0 and c1 are the inlet and outlet molar concentrations of feed molecules. It is easy to show that at constant flow velocity u = 1/λ (Section 4.6.5).

1.4 MASS TRANSPORT IN FUEL CELLS

19

Figure 1.5: The idea of quasi-2D approximation.

1.4.3

Quasi-2D approximation

Assuming for the estimate that the oxygen concentration drops linearly along the channel and taking the oxygen stoichiometry λ = 2, for the channel length of 10 cm, we get a concentration gradient, along the channel, of 0.5/10 = 5 · 10−2 cm−1 . The GDL thickness is typically in the order of 10−3 cm. Assuming again that only 50% of the oxygen is consumed in the ORR, for the throughplane gradient we have 0.5/10−3 = 5 · 102 cm−1 . We see that the gradient of oxygen concentration across the cell is 4 orders of magnitude greater than this gradient along the channel. Consider now a small box in the porous GDL and in the catalyst layer (Figure 1.5). In this box, the through-plane gradient of concentration (along the x-axis) is much greater than the in-plane gradient (along the z-axis). In other words, the x-velocity of species transport is much greater than the zvelocity (Figure 1.5). Therefore, to a good approximation we can neglect the transport along the z-axis in this box and consider the transport problem in the MEA as purely 1D. In the channel we have the opposite situation: the dominant direction of species transport is along the z-axis, whereas the diffusion velocity along the x-direction is much smaller. Thus, in the channel we may consider the flow as 1D along the z-axis (Figure 1.5). An accurate analysis of leading terms in the full system of hydrodynamic equations (Birgersson et al., 2003) confirms this simple reasoning. These arguments are of fundamental importance in fuel cell modelling. They allow us to split the fully 2D problem in the x-z plane in Figure 1.5 into two 1D problems: one for the channel (channel problem) and the other for the MEA (MEA problem). The two problems are coupled by the mass transport through the channel/GDL interface (red line in Figure 1.5)11 . The solution of the channel problem returns the distribution of species concentration along the channel. These concentrations are used to solve the MEA problem, which gives the shape of the local current density along z. 11 Strictly speaking, the mass transport is accompanied by the momentum transfer through this interface. In Section 4.1, we will show that in purely gaseous cells this momentum transfer is negligible.

20

CHAPTER 1. FUEL CELL BASICS

In numerical modelling, the two problems must be solved iteratively to obtain the self-consistent distributions of species concentration and local current. In analytical modelling, splitting is realized in the following way. We consider first the MEA problem, assuming that the local current j and oxygen concentration ch in the channel are fixed. The solution gives the distribution of local values along the x-axis in the MEA (Figure 1.5). Then we consider j and ch as functions of z and we solve the mass conservation equation in the channel (Figure 1.5). This equation relates ch to the inlet concentration c0h . The local current j is related to the mean current density in the cell J by the condition of electrode equipotentiality. The resulting shapes j(x) and ch (x) are then used to derive the polarization curve of a cell. This approach will be used in Chapters 3 and 4.

1.4.4

Mass conservation equation in the channel

Correct splitting of a 2D problem in Figure 1.5 requires an accurate account of the mass transfer through the channel/GDL interface. In this section, we derive the general mass conservation equation for the cathode channel of a PEFC or DMFC. No volume sources of mass exist in the fuel cell channel and hence the continuity equation in the channel reads ∇ · (ρv) = 0.

(1.42)

Here ρ is the mass density of the flow and v is the flow velocity. However, flow in the channel exchanges molecules with the MEA. To calculate the respective fluxes consider an elementary box of thickness dz along the channel (Figure 1.6). Integrating (1.42) over the box volume V and applying Gauss’ theorem we obtain Z Z ∇ · (ρv) = ρvdS = 0, (1.43) V

S

where S = wh is the channel cross-section (Figure 1.6). The side and top surfaces of the box are impermeable to gas, whereas the bottom surface is in contact with the GDL. Electrochemical reactions under the GDL induce the flux ρvx , directed perpendicular to this surface. Equation (1.43) then gives [(ρv)z+dz − (ρv)z ] S − (ρv)x w dz = 0 or ∂(ρvz ) 1 = (ρvx ) ∂z h

(1.44)

1.4 MASS TRANSPORT IN FUEL CELLS

21

Figure 1.6: An elementary volume of the channel. The red area is an interface with the GDL. where it is assumed that the flux ρvx is directed inward (into the channel volume). The flux ρvx is related to the local current density j(z). At the cathode the electrochemical reaction is (1.2) and hence the following relations hold: jMox , 2F jMw jMw ρ w vw = + αw 2F F

2ρox vox = −

(1.45) (1.46)

where vox and vw are mass velocities of oxygen and water fluxes, respectively, through the channel/GDL interface. The first term on the right side of Eq. (1.46) is the stoichiometric flux of water due to the ORR. The second term is the electro-osmotic flux of water through the membrane, where αw is the effective transfer coefficient of water molecules through the MEA12 . The right side of (1.44) is, therefore, j 1 (ρvx ) = ρox vox + ρw vw = h h



2(1 + 2αw )Mw − Mox 4F

 .

(1.47)

With this (1.44) takes the form j ∂(ρvz ) = ∂z h



2(1 + 2αw )Mw − Mox 4F

 .

(1.48)

12 The following relation holds: 0 ≤ α w ≤ nd , where nd is the drag coefficient. The parameter αw = nd , if the back diffusion of water from the cathode to the anode is negligible; αw = 0 when the electro-osmotic flux is fully compensated for by the back diffusion. In the latter case, there is no net flux of water through the membrane and all water produced in the cathode catalyst layer enters the cathode channel. Intermediate values of αw describe the situation when the electro-osmotic flux of water is partially compensated for by the back diffusion. Water transport in the membrane will be discussed in more detail in Section 1.4.8.

22

CHAPTER 1. FUEL CELL BASICS

Typically, of great interest are the partial mass conservation equations for feed molecules (oxygen) and water in the channel. These equations are ∂(cox vz ) j =− ∂z 4F h 2j(1 + 2αw ) ∂(cw vz ) = . ∂z 4F h

(1.49) (1.50)

Here we take into account that ρox = cox /Mox and ρw = cw /Mw . Summing up (1.49) and (1.50) and taking into account the relation of mass density and molar concentration, we come to (1.48). Each of the equations (1.48)–(1.50) contains two unknowns: species density or concentration and the flow velocity vz . Generally, vz is determined by the Navier-Stokes equation; however, in the fuel cell channel, simplifications are possible, as discussed in the next section. The structure of Eqs (1.48)–(1.50) is typical of a 1D formulation of the channel problem: the variation of flux along the channel (left-hand side) is balanced by the fluxes coming in and out of the channel in the perpendicular direction. These fluxes appear as the source terms in the 1D mass conservation. The same approach is used in the problem of heat transport along the channel (Section 5.1).

1.4.5

Flow velocity in the channel

Laminar flow in the channel is a Poiseuille flow with the constant velocity proportional to the pressure gradient (Landau and Lifshits, 1987) vz = −

kp ∇p µ

(1.51)

where kp is the channel hydraulic permeability, µ is the flow viscosity, ∇p = (p1 − p0 )/L is the pressure gradient, L is the channel length and the superscripts “0” and “1” indicate the values at the channel inlet and outlet, respectively. The constancy of vz is provided by the balance of forces acting on an elementary fluid volume shown in Figure 1.6. The force −∇p due to the pressure gradient is exactly compensated for by the viscous force µvz /kp of friction with the channel walls. Equation (1.51) is valid for the flow in the channel with impermeable walls. In the fuel cell channel there is mass and momentum transfer through the channel/GDL interface due to species consumption and production in the electrochemical reactions. The effect of mass transfer was considered in Section 1.4.4; here we discuss momentum balance in the channel. Consider the cathode channel of a PEFC and suppose that the water transfer coefficient through the membrane αw is zero. Then, in the channel,

1.4 MASS TRANSPORT IN FUEL CELLS

23

Figure 1.7: In the channel, each oxygen molecule consumed is replaced by two water molecules. To keep pressure in the elementary volume constant, the front face velocity of this volume must exceed the rear face velocity. each oxygen molecule consumed is replaced by two water molecules. Gaseous pressure is proportional to species number density (the number of molecules per unit volume). As discussed above, pressure variation along the channel is usually small and hence the number density must remain constant. Consider an elementary fluid volume moving at the flow velocity along the channel. Due to incoming water molecules the number of species in this volume increases; thus, to keep pressure constant this volume must expand in the flow direction. This means that the flow velocity must increase along z (the front face of the elementary volume moves faster than the rear face, Figure 1.7). However, in typical conditions flow acceleration is small (Section 4.1). This forms the other cornerstone of fuel cell analytical modelling, the approximation of plug flow . The flow in the channel is considered to be ideally mixed, with constant velocity. Constancy of vz enables the mass conservation equation to be decoupled from momentum balance, which greatly simplifies the channel problem. The amount of liquid water in PEFC channels is usually not large and in many cases the effect of liquid droplets on flow velocity can be ignored. In the DMFC anode channel the effect of gaseous bubbles on the flow velocity is typically large and two-phase nature of the flow cannot be ignored13 . In spite of all the difficulties caused by the two-phase effects, channel flows in fuel cells are understood better than the flows in porous layers. Channel flows are subject to fluid dynamics equations with known transport coefficients. Modern commercially available CFD packages 13 Accurate modelling of the dynamics of two-phase flow in PEFC and DMFC channels requires numerical calculations. Dynamics of a two-phase mixture can be described by equations for average density and velocity (M2 model, (Wang, 2004)). However, this approach does not give a detailed picture of liquid bubble formation and transport. In addition, the validity of M2 model is still debated (Gurau et al., 2008a; Wang, 2009; Gurau, 2009). Direct simulation of bubble formation and tracing the bubbles is a difficult task (Le and Zhou, 2009). A simple M2-like model of slug flow in the anode channel of DMFC predicts the variation of flow velocity due to gaseous bubbles and the respective lowering of the time-average methanol concentration (Kulikovsky, 2005a, 2006a).

24

CHAPTER 1. FUEL CELL BASICS

contain numerical codes for two-phase problems in various formulations (for more details see (Wang, 2004; Gurau et al., 2008b)). In contrast, transport properties of porous layers are poorly known, which makes modelling of two-phase flows in these layers more difficult and less reliable.

1.4.6

Mass transport in gas-diffusion/backing layers

Fick’s diffusion Fuel cells usually operate at equal pressures on the anode and cathode sides. In these conditions, the main mechanism of feed molecule transport through the GDL to the catalyst layer is diffusion due to the concentration gradient. Modern gas-diffusion medium in low-temperature fuel cells is typically a highly porous carbon paper with porosity in the range of εGDL = 0.6-0.8 and with the mean pore radius in the order of 10 µm (10−3 cm). By the order of magnitude, the mean free path of molecules in atmospheric pressure air is lf = 1/(NL σkin ), where NL = 2.686 · 1019 cm−3 is the Loschmidt number (number of molecules in a cubic centimetre of atmospheric pressure gas at standard temperature) and σkin ' 10−14 cm2 is the molecular cross-section for kinetic collisions. With this data we get lf ' 3 · 10−6 cm, or 3 · 10−2 µm. Obviously, mean pore radius in the GDL is nearly 3 orders of magnitude greater than lf and the physical mechanism of molecule transport is binary molecular diffusion. Oxygen constitutes 21% of air and to a good approximation it may be treated as a small component. The diffusion flux of a small component in a mixture of gases is described by Fick’s law: Nox = −Dox ∇cox .

(1.52)

Here Nox is the oxygen molar flux (mol cm−2 s−1 ), Dox is the oxygen diffusion coefficient (cm2 s−1 ) in the nitrogen-oxygen mixture (air) and cox is the oxygen molar concentration. In liquid-fed DMFCs, methanol typically also constitutes a small fraction of water-methanol solution and Eq. (1.52) can be written for methanol flux NM = −DM ∇cM

(1.53)

where the subscript M denotes the values related to methanol. Fick’s equation is also valid for the self-diffusion of gases, e.g. it can be written for the oxygen flux in the GDL when the cell is fed with pure oxygen. In that case, Dox in Eq. (1.52) should be replaced by the coefficient of oxygen self-diffusion. The relations above are valid for the species flux in a free space. However, porous medium lowers the diffusion coefficient in Eqs (1.52) and (1.53)

1.4 MASS TRANSPORT IN FUEL CELLS

25

ef f to the value Dox . This lowering is usually described by the Bruggeman correction (Bruggeman, 1935) 3

ef f 2 Dox = εGDL Dox .

(1.54)

ef f Here Dox is the effective oxygen diffusion coefficient in a dry GDL of porosity εGDL . At large current densities, the cathode GDL in PEFCs and DMFCs can be significantly flooded. In that case Dox should be corrected for liquid saturation, s, which is a fraction of the GDL volume filled with liquid ef f water. The exact parametric dependence of Dox on s is poorly known; the correction usually used has the form 3

ef f 2 Dox = sm εGDL Dox

(1.55)

where m ranges from 1 to 3. Typically, oxygen flux through the membrane is negligibly small and the diffusion flux of oxygen in the GDL is related to the cell current. Stoichiometry prescribes ef f Dox ∇cox =

j . 4F

(1.56)

This equation determines the linear profile of the oxygen concentration across the GDL. In DMFCs, the methanol flux through the membrane is large and it cannot be ignored. The flux of methanol in the anode backing layer of DMFC obeys an equation ef f DM ∇cM =

j 3 + Ncross . 6F 2

(1.57)

ef f Here DM is the effective diffusion coefficient of methanol in the backing layer, and Ncross is the molar flux of methanol in the membrane. The expression for Ncross will be derived in Section 3.3. Equations (1.56) and (1.57) follow from the general mass conservation equation, which states that in the absence of mass sources the divergence of diffusion flux is zero:

∇ · (D∇c) = 0.

(1.58)

Integrating (1.58) once and taking into account conditions at the GDL/CL interface one arrives at the balance of fluxes, Eq. (1.56) or (1.57). Usually it is convenient to deal with the balance of fluxes directly, rather than to start from the general equation (1.58).

26

CHAPTER 1. FUEL CELL BASICS

Stefan-Maxwell diffusion The diffusion of species i in a mixture of gases is a fundamental process which homogenizes the ith species concentration and increases system entropy. However, diffusion also redistributes mass over the system volume. Since mass density must be constant14 , the diffusion flux of species i induces counter-diffusion of the other mixture components. If a fraction of component i is not small, this counter-diffusion has to be taken into account. The general relation for the coupled fluxes of diffusing species is given by the Stefan-Maxwell equations X ξi Nk − ξk Ni i

Dik

= −ctot ∇ξk .

(1.59)

Here ξk is the molar fraction of the kth component, Nk is the molar flux, Dik is the binary diffusion coefficient of species i in the mixture with species k, and ctot is the total molar concentration of the mixture. Equations (1.59) written for all mixture components form a linear system of equations with respect to the molar fluxes. However, the determinant of this system is zero, since these equations are not linearly independent. Summing up Eqs (1.59) over k we get a trivial relation ! −ctot ∇

X

ξk

=0

k

P which holds since k ξk = 1. Physically, in Eqs (1.59), the reference frame for species motion is not fixed i.e. these equations describe the relative fluxes only. Eqs (1.59) require an independent closing relation, which establishes this reference frame (Frank-Kamenetskii, 1969). In fuel cells, nitrogen does not participate in the reactions and hence the molar flux of nitrogen is zero: NN 2 = 0. This relation is usually used to close the system (1.59) in modelling of the cathode side fed with air. Generally, the increased complexity of the model based on StefanMaxwell relations does not justify improvement in model accuracy, which in most cases is redundant. Complex equations can be tolerated in numerical modelling; however, in analytical modelling, clarity and simplicity of the resulting expressions are of the highest priority. Below we will use a simple Fick’s law of diffusion to describe species transport in GDLs and in catalyst layers. 14 The variation of gas mass density causes a respective variation in pressure, which induces flow homogenizing pressure and density.

1.4 MASS TRANSPORT IN FUEL CELLS

1.4.7

27

Mass transport in catalyst layers

Mean pore radius in the CL is rcl ' 3·10−2 µm. Thus, the mean free path of molecules, lf (Section 1.4.6), is in the order of rcl . In this situation Knudsen diffusion contributes to the total diffusion flux of species. Formally, the expression for the ith species flux due to the Knudsen diffusion has the form of Fick’s law: NiK = −DiK ∇ci .

(1.60)

The fluxes due to Fick’s molecular and Knudsen diffusion add up yielding the total flux   Ni = − Dief f + DiK ∇ci . (1.61) When the Knudsen diffusion is accompanied by the Stefan-Maxwell diffusion, the following combination of (1.59) and (1.60) is used: X ξ i Nk − ξ k Ni Nk + = −ctot ∇ξk . Dik DkK i

(1.62)

Note that the Knudsen flow is determined with respect to the laboratory system of coordinates and thus the system (1.62) does not require additional closing relations. However, if Knudsen terms in Eqs (1.62) are small, the system becomes ill-posed and a numerical solution of this system requires special methods (Kulikovsky et al., 1999). The mass conservation equation in the catalyst layer reads Q ∂Ni = ∂x nF

(1.63)

where Q is the rate of electrochemical conversion given by, for example (1.25) or (1.33). If Ni has the form of Fick’s law (1.61), Eq. (1.63) reduces to Ditot

∂ 2 ci Q = 2 ∂x nF

(1.64)

where Ditot = Dief f + DiK is the total diffusivity of the ith species. Equation (1.64) can be simplified further. Consider for definiteness the cathode catalyst layer. Proton current density jp obeys a conservation equation ∂jp /∂x = −Q. Using this relation in (1.64) we get tot ∂ Dox

2

cox =− ∂x2



1 nF



∂jp . ∂x

28

CHAPTER 1. FUEL CELL BASICS

Integrating once we arrive at the balance of fluxes tot Dox

∂cox j − jp = . ∂x nF

(1.65)

At x = 0 (at the membrane/CCL interface) jp = j and the diffusion flux of oxygen is zero. At the CCL/GDL interface we have jp = 0 and the diffusion flux of oxygen is determined by the cell current j. Equation (1.65) will be considered in more detail in Chapter 2. Catalyst layers are filled with an ionic (proton) conductor; thus, porosity of CL, εcl , is typically much smaller than that of GDL. In Nafion-based cells, εcl depends on Nafion content; for typical contents in the range of 30-40%, εcl varies in the range of 0.3-0.1 (Wang et al., 2004). Therefore, the Bruggeman correction (1.54) lowers the molecular oxygen diffusion coefficient in the CL by a factor in the range 0.1-0.03. Moreover, flooding ef f ' 10−3 Dox . can further decrease this coefficient making it as low as Dox The Knudsen diffusion coefficient is proportional to the product of the mean pore radius rcl and the average thermal velocity vT of molecules: N K ∼ ζrcl vT , where ζ ' 1 is the correction factor. With the well-known formula for vT we get r 8RT K Di = ζrcl . (1.66) πMi Here Mi is the molecular weight of the ith species.

1.4.8

Proton and water transport in membranes

An extensive literature is devoted to the physics of water channels formation and to the mechanism of proton and water transport in membranes. In this section, we give a brief overview relevant to the analytical modelling of fuel cells. A detailed review of phenomenological membrane transport models is given in (Weber and Newman, 2007). Atomistic modelling and experiments on proton transport in membranes are reviewed in (Kreuer et al., 2004). Recent advances in mesoscopic membrane modelling are discussed in (Promislow and Wetton, 2009). The reader is referred to these works for a detailed discussion of the transport processes in polymer membranes. Polymer electrolyte membranes contain a long polymer backbone with the HSO3 groups attached to the side chains. Upon water uptake, the backbone rolls up and forms water-filled channels. In these channels, the proton leaves the HSO3 group to move in water, while the SO− 3 ion stays attached to the channel wall. Typical membrane thickness is in the order of 100 µm (10−2 cm). Current in the external circuit creates in the membrane an electric field in the

1.4 MASS TRANSPORT IN FUEL CELLS

29

order of 101 -102 V cm−1 . In this field, protons move along the water-filled channels. The membrane is, therefore, a special case of plasma, in which positive species can move while the negative species are immobile15 . Let the anode side of a hydrogen cell inject a single proton into the plasma of the membrane. The excess proton forms a Zundel ion H5 O+ 2 with two H2 O molecules, or an Eigen ion H9 O+ 4 with three H2 O molecules. The big ion then dissociates into neutral H2 O molecules, while the other proton shifts along the direction of the electric field and forms a new big ion (Kreuer, 2000; Kreuer et al., 2004). This mechanism is accompanied by a diffusion of the H3 O+ ion (Walbran and Kornyshev, 2001). Some contribution to the current gives direct proton hopping between the two SO− 3 groups located at the pore wall. In analytical modelling the membrane is usually treated as a continuum with the prescribed transport parameters. (Alternative approaches are discussed in Weber and Newman (2004, 2007)). The proton current in membrane j obeys Ohm’s law j = σm Em

(1.67)

where Em is the electric field strength in the membrane. The membrane proton conductivity σm depends on membrane water content. The latter is usually characterized by parameter λw , which is the number of water molecules per SO− 3 group in the membrane. In Nafion membranes, λw varies between zero in dry state and 22 in a fully hydrated state. Measurements (van Bussel et al., 1998) give a linear dependence of σm on λw : σm = 0.005738λw − 0.007192,

Ω cm−1 .

(1.68)

From this equation it follows that σm = 0 at the critical λcrit ' w 719/574 ' 1.25, which corresponds to a percolation threshold in the waterfilled cluster. Note that σm exponentially depends on temperature; the experimental dependence is given in (Springer et al., 1991). Protons in the membrane drag water molecules. The total flux of water in the membrane can be written as Nw = −Dw (λw )∇cw +

nd (λw )j F

(1.69)

where nd is the drag coefficient, the number of water molecules transported by a single proton. Here the first term describes water transport due to diffusion and the second term accounts for the electro-osmotic drag. 15 This statement is correct on a time scale in the order of proton transport through the membrane. On larger time scales, membrane morphological changes lead to a redistribution of negative SO− 3 groups.

30

CHAPTER 1. FUEL CELL BASICS (a)

(b)

Figure 1.8: Nafion conductivity σm , water drag coefficient nd and diffusion coefficient of liquid water Dw . The data are from van Bussel et al. (1998) and the fitting equations from Kulikovsky (2003a). Generally, both transport coefficients in Eq. (1.69) depend on membrane water content. For Dw and nd , the following parametrizations based on the measurements of van Bussel et al. (1998) can be used (Kulikovsky, 2003a):     λw − 2.5 λw −6 Dw = 4.1 · 10 1 + tanh , (cm2 s−1 ) (1.70) 25 1.4  1, λw < 9 nd = (1.71) 0.117 · λw − 0.0544, λw ≥ 9. The dependencies of σm , Dw and nd on membrane water content are shown in Figure 1.8. As can be seen, the variation of the drag coefficient with water content is not large and to a first approximation it can be neglected. However, the variation of proton conductivity and of the diffusion coefficient with λw is significant and, in general, it should not be ignored. Importantly, at zero λw the water diffusion coefficient in the membrane vanishes. To illustrate the effect of vanishing Dw we will approximate Dw with a less accurate but simpler relation    λw Dw = D0 1 − exp − , λ∗ where D0 is the “bulk” diffusion coefficient of water in pure water and λ∗ is the characteristic value of water content. This relation retains the two main features of the experimental curve: it tends to zero at λw = 0 and it tends to the bulk value D0 at λw  1 (cf. Figures 1.8b and 1.9a). Suppose that the net flux of water in the membrane is zero, i.e. the electro-osmotic drug is fully compensated for by the back diffusion. Equating Nw , Eq. (1.69) to zero we get    λw nd j D0 1 − exp − ∇cw = . (1.72) λ∗ F

1.4 MASS TRANSPORT IN FUEL CELLS (a)

31

(b)

Figure 1.9: (a) The model diffusion coefficient and (b) the respective shape of the membrane water content (Eq. (1.75)). The anode is at x ˜ = 0 and the cathode is at x ˜ = 1. Indicated are the values of parameter β in Eq. (1.75); parameter λ∗ = 2. The fully hydrated membrane corresponds to λ = 22. Thus, cw is related to λw as cw = cl λw /22, where cl is the molar concentration of pure liquid water. With this Eq. (1.72) transforms to    λw 22nd j 1 − exp − ∇λw = . λ∗ D0 F cl

(1.73)

Let the axis x with the origin at the anode side of the membrane be directed toward the cathode. Introducing the dimensionless coordinate x ˜ = x/lm , where lm is the membrane thickness, we can rewrite Eq. (1.73) in the form    λw ∂λw 1 − exp − =β (1.74) λ∗ ∂x ˜ where β=

22nd jlm D0 F cl

is independent of x ˜. An interesting situation arises when the anode side of the membrane is dry, λw (0) = 0. The solution to (1.74) with this boundary condition is     βx ˜ λw (˜ x) = λ∗ 1 + W − exp − −1 + βx ˜ λ∗

(1.75)

where W is the Lambert function. This solution is depicted in Figure 1.9. As can be seen, λw (˜ x) exhibits an infinite gradient ∂λw /∂ x ˜ at x ˜ = 0. This is not surprising: the membrane transports proton current and, according to

32

CHAPTER 1. FUEL CELL BASICS

the assumption, the respective electro-osmotic flux of water must be fully compensated for by the back diffusion. However, at x ˜ = 0 the diffusion coefficient vanishes and hence finite diffusion flux at this point can be achieved with an infinite gradient of λw only. Note that the non-zero total flux of water in the membrane does not qualitatively change this result16 . An important practical conclusion is that if the anode side of the membrane gets dry, there is no way to wet it with the water produced on the cathode side. To wet the membrane the anodic feed stream has to be humidified.

1.5 Sources of heat in a fuel cell The rates of virtually all transport and kinetic processes in fuel cells exponentially depend on temperature. Local overheat by several degrees Kelvin changes the rate of electrochemical reactions, enhances the diffusion transport of reactants and shifts liquid-vapour equilibrium in low-T cells. In SOFCs, improper heat management leads to thermal degradation of cell components (heat problems in SOFC will be considered in Section 5.1). Heat management is especially important in fuel cell stacks where dense packing of cells retards heat removal. Generally, cell warming occurs due to heat generated in the electrochemical reactions and due to Joule electric power dissipated by the currents. The fuel cell sandwich supports two types of currents: (i) electron current in the carbon threads of the GDL and CLs, and (ii) proton (ionic) current in the bulk membrane (electrolyte) and in the electrolyte phase dispersed in the CL. For the sake of brevity, we will consider the proton current; however, the discussion below is applicable to the ionic current in SOFCs as well. In the CL, the electron conductivity of the carbon phase is much greater than the proton conductivity of the electrolyte phase. Since maximal electron and proton current densities in the CL are equal, the major proportion of Joule heat there is released in the electrolyte phase. Further, the typical proton conductivity of the CL is several times lower than the conductivity of the bulk membrane (Xie et al., 2005; Havranek and Wippermann, 2004) and thus the dissipation of electric power in the CL exceeds this dissipation in the bulk membrane. Taking into account that the reaction heat is also released in CLs we conclude that active layers are the major source of heat in the fuel cell. Consider for definiteness the cathode catalyst layer of a low-temperature fuel cell. The thermodynamic heat (J mol−1 ) released in the ORR is T ∆S, where ∆S (J mol−1 K−1 ) is the entropy change in the ORR. 16 Large gradient of water content at the anode side of the membrane has been detected in experiment (B¨ uchi and Scherer, 2001).

1.5 SOURCES OF HEAT IN A FUEL CELL

33

Generally, ∆S depends on pressure and temperature. We will assume that the pressure variation due to the reaction is negligible. At constant pressure the temperature dependence of ∆S is given by T

(cP w − cP ox ) dT T T0   T ' ∆S0 + (cP w − cP ox ) ln T0 Z

∆S(T ) = ∆S0 +

(1.76)

where ∆S0 ≡ ∆S(T0 ) is the entropy change at the standard temperature T0 and cP w and cP ox are the specific heats of water and oxygen, respectively. Note that the contribution of protons and electrons to the second term in (1.76) is negligible due to their small mass. An estimate shows that in the temperature range typical of low-T fuel cell operation (300-360 K), the second term on the right side of (1.76) is much smaller than the first one. Physically, the entropy change ∆S0 due to the creation of new molecules (water) is much greater than the entropy change due to species heating. This is also true for the half-cell reactions in SOFCs. In the following we thus will take ∆S = ∆S0 = const. The local rate of thermodynamic heat production RS (W m−3 ) in the catalyst layer is17 RS =

T ∆S Q 4F

(1.77)

where Q is the local rate of the electrochemical reaction (the proton charge consumed in the reaction per second per unit volume, A m−3 ). Generally, Q is a function of the distance across the CL x (Section 2.3). The decay of proton current along x is described by (1.34). Integrating this equation from x = 0 to x = lt and taking into account that jp (lt ) = 0, we get Z

lt

Q dx = j

(1.78)

0

where j ≡ jp (0) is the cell current density. Note that Eq. (1.78) expresses the conservation of charge and thus it is valid regardless of the actual relation for Q. In a well-designed CCL, Q is almost constant along x (Chapter 2) and Eq. (1.78) simplifies to Q'

j . lt

(1.79)

17 Here and below we consider ∆S of the ORR as a positive value. The respective term in the heat transport equation represents the rate of cell heating.

34

CHAPTER 1. FUEL CELL BASICS

ORR occurs at the catalyst surface, which has the potential of the electron-conducting phase ϕ in the CL. Protons, however, move in the polymer electrolyte phase, which has the potential ϕm . The schematic in Figure 1.3 shows that, on the cathode side, to reach the catalyst surface the proton must overcome the potential jump η = ϕ0m − ϕ (overpotential)18 . The rate of heat production due to this so-called irreversible heating is Rη = η c Q.

(1.80)

Another source of heat in the CL is the Joule electric power dissipated due to currents. Since CL supports electron and proton currents, the total rate of Joule heating is RJ = −je

∂ϕ0 ∂ϕ − jp m , ∂x ∂x

(1.81)

where je and jp are the electron and proton current densities, respectively. Taking into account Ohm’s law je = −σe ∂ϕ/∂x, jp = −σt ∂ϕ0m /∂x we may rewrite (1.81) as RJ =

jp2 je2 + . σe σt

(1.82)

Here σe and σt are the electron conductivity of the electron-conducting (carbon) phase and the proton conductivity of the electrolyte phase, respectively. The local rate of heat generation in the CCL is, therefore,  RS + Rη + RJ =

T ∆S + ηc nF



jp2 j j2 + e + . lt σe σt

(1.83)

A more accurate relation will be derived in Section 2.8.

1.6 Types of cells considered in this book 1.6.1

Polymer electrolyte fuel cells (PEFCs)

Probably the most widely studied are polymer electrolyte fuel cells, in Europe abbreviated as PEFC, in USA and Canada as PEM (polymer 18 Physically, the proton should overcome the voltage barrier V c − η, which separates oc c is the cathode half-cell the potentials of the membrane and carbon phases. Here Voc open-circuit voltage. However, the irreversible dissipation of energy occurs only when c . Further transport the proton acquires the voltage η required to reach the potential Voc c to the Pt particle is a thermodynamically reversible of the proton with the potential Voc process.

35

1

1

1.6 TYPES OF CELLS CONSIDERED IN THIS BOOK

41

000

000

41

Figure 1.10: Typical design of a PEFC.

electrolyte membrane) fuel cells or PEMFC. These cells utilize hydrogen as a fuel and oxygen (air) as the cathode oxidant. The electrochemical reactions in these cells are given by (1.1) and (1.2); the operating temperature varies in the range of 30-90 ◦ C. The design of a typical laboratory-scale PEFC is shown in Figure 1.10 (Kulikovsky et al., 2005c). The membrane-electrode assembly is clamped between two graphite plates with serpentine channels for the hydrogen and air supply. The channels 1 mm deep and 1 mm wide are separated by ribs of 1 mm width; anode and cathode flow fields are identical. In our experiments, the cell “sandwich” was clamped between two gold-plated stainless steel endplates (not shown) and tightened by the set of rods along the cell periphery (Kulikovsky et al., 2005c). The peak performance of this cell with air as oxidant is about 800 mW cm−2 . The cell active area is about 16 cm2 , which gives peak power in the order of 13 W. Thus, three such cells can feed a laptop computer (40 W). The greatest problems of PEFC technology are the need for developing more efficient and stable catalysts for ORR and understanding the laws of liquid water transport in these cells. Water is produced in the ORR and wets the membrane, which conducts protons only in a wet state. On the other hand, the excessive water blocks the pores in the GDL and retards oxygen transport to the catalyst sites. Accurate water management requires a compromise between the two issues. Note that the requirements of automotive applications are more stringent; in particular, PEFCs for vehicles should withstand multiple start-stop cycles and low temperatures.

36

CHAPTER 1. FUEL CELL BASICS

Figure 1.11: Schematic of DMFC and SOFC.

1.6.2

Direct methanol fuel cells (DMFCs)

The other member of the family of low-temperature fuel cells utilizing liquid methanol as a fuel and oxygen/air as oxidant is the direct methanol fuel cell (DMFC). The half-cell electrochemical reactions in DMFC are CH3 OH + H2 O → CO2 + 6H+ + 6e− , anode 3 O2 + 6H+ + 6e− → 3H2 O, cathode 2

(1.84) (1.85)

(Figure 1.11). As can be seen, the cathodic reaction is the same as in PEFCs (1.2). On the anode side, the ionization of the methanol-water solution yields six protons and six electrons. In contrast to PEFC, the anodic reaction (1.84) is sluggish and requires quite a significant portion of the cell open-circuit voltage to run. Consequently, DMFC efficiency is lower than that of PEFC. The DMFC design is quite similar to that of a PEFC (Figure 1.10). As in a PEFC, in a DMFC the anode and cathode are separated by a polymer r electrolyte membrane, typically Nafion . These cells also operate in the ◦ temperature range of 30-90 C. Unlike the PEFC, the DMFC is fed with a liquid methanol-water solution, which makes the DMFC-based power sources much more compact. However, liquid methanol easily penetrates the membrane into the cathode side, where it is directly oxidized. This parasitic process dramatically reduces the DMFC open-circuit voltage. Another problem specific to DMFCs is the large amount of gaseous CO2 on the cathode side. At temperatures above 60 ◦ C, the solubility of CO2 in water is small and CO2 bubbles enter the anode channel, where they strongly disturb the flow. Worldwide interest in DMFCs is increasing due to their high volumetric power density which makes them ideal candidates to replace Li-ion batteries in mobile devices. Since 1991, the number of publications on DMFC science

1.6 TYPES OF CELLS CONSIDERED IN THIS BOOK

37

Figure 1.12: Number of publications on DMFC science and technology r (data from Scopus ; the search string is “DMFC”). and technology has doubled every two years (Figure 1.12). This incarnation of Moore’s law is characteristic of an emerging technology.

1.6.3

Solid oxide fuel cells (SOFCs)

Working temperature of solid oxide fuel cells (SOFC) varies in the range of 600-900 ◦ C. A unique feature of these cells is their ability to utilize methane or other hydrocarbons as fuels. In combination with high power density (about 1 W cm−2 ), this makes SOFCs a very attractive power source for residential applications. Half-cell reactions in SOFC are H2 + 2O2− → H2 O + 4e− , −

O2 + 4e → 2O

2−

,

anode

(1.86)

cathode

(1.87)

(Figure 1.11). The internal charge carrier between the anode and the cathode is a doubly charged O2− ion (Figure 1.11). The transport of O2− ions occurs in a ceramic electrolyte, typically yttria-stabilized zirconia (YSZ), which has sufficient ionic conductivity at temperatures above 600 ◦ C. At temperatures of 600-900 ◦ C the conversion of hydrocarbons (methane) to hydrogen can be organized in situ so that no special reforming equipment is necessary. This conversion is a purely chemical process, which does not involve charged particles. In principle, conversion can be organized directly in the electrochemical section of the cell shown in Figure 1.11. However, it is more convenient to arrange a special conversion section so that the electrochemical cell is fed with converted hydrogen. The SOFC design differs from that of low-temperature fuel cells. Figure 1.13 shows the so-called planar anode-supported SOFC developed

38

CHAPTER 1. FUEL CELL BASICS

Figure 1.13: SOFC design of the Forschungszentrum J¨ ulich (Blum et al., 2007). c 2007, John Wiley & Sons, Inc. Publishers. Reprinted with permission.

at Forschungszentrum J¨ ulich. In this design, the mechanical stability of the MEA sandwich is provided by a thick ('1 mm) anode (anode substrate and anode layer, Figure 1.13). Next to the anode there is a 10-µm layer of ionic conductor (electrolyte) and the cathode is located on the other side of the electrolyte (Figure 1.13). The cell sandwich is clamped between metallic interconnects and tightened by a glass sealant (Figure 1.13). During operation, the cell may experience local overheating and to avoid cracking, individual layers should have close coefficients of thermal expansion. A useful introduction to the electrochemistry and engineering aspects of these cells is given in (Bagotsky, 2006).

Chapter 2

Catalyst layer performance Cathode and anode catalyst layers (CLs) are key elements of any fuel cell. Participants in electrochemical reactions are charged particles (positive and negative) and neutral molecules in the gaseous or liquid form. Any catalyst layer is thus a composition of three ingredients: electronic and ionic conductors and voids for gas/liquid supply. The reaction itself occurs on a surface of precious metal particles (typically Pt), which must be connected to the avenues for charged and neutral species transport. Functional models of such a complex structure are based on the idea of macrohomogeneous approach. This means that the microscopic details of a CL structure are ignored and CL is considered as a continuum with prescribed transport properties for ions, electrons and neutrals. These transport properties are usually taken from experiments or from structural models. The latter are beyond the scope of this book; the reader is referred to (Eikerling et al., 2007) for a review of structural models. Functional models discussed below predict the distribution of local currents, reaction rate and species concentration in the CL. The distributions of local values are of great theoretical interest; however, these distributions cannot be directly measured, at least with the present level of experimental techniques. Measurable quantities are the total current converted in the CL and the respective voltage loss (CL polarization curve). The primary goal of functional models is to derive the CL polarization curve. In some cases, this curve can be found without solving a problem for parameter distribution (Section 2.3). Comparison of theoretical and experimental polarization curves helps us to understand the regime of CL operation. 39

40

CHAPTER 2. CATALYST LAYER PERFORMANCE

r The first polymer ionic conductor Nafion was invented in the 1960s. Until that time electrochemistry had worked with liquid electrolytes. The polarization behaviour of catalyst layers with aqueous electrolyte had been studied in numerous works (see Striebel et al. (1995) and Perry et al. (1998) and the literature cited therein). r The invention of Nafion radically changed the situation in lowtemperature fuel cell technology. Solid (polymer) proton conductors made it possible to design porous catalyst layers with much better access for feed gases (oxygen). Since the 1960s, catalyst layers have been thin porous structures and thanks to better gaseous transport they exhibit superior performance. Nonetheless, novel active layers required the development of new models. The cathode side of all fuel cells is fed with oxygen. In PEFCs and DMFCs, the oxygen reduction reaction (ORR) completely neutralizes charged particles, while in the SOFC cathode ORR converts electrons into negative oxygen ions. The thickness of the cathode catalyst layer (CCL) in low-temperature fuel cells varies from ten micrometres in modern PEFCs to hundred of micrometres in direct methanol fuel cells (DMFCs). The CCL in DMFCs operates at a high level of flooding, which reduces ORR efficiency. The considerable thickness of the active layer facilitates electrochemical conversion under these conditions. In the anode-supported SOFC, the anode thickness is about 1 mm (1000 µm) to provide mechanical stability of the cell sandwich. Large thickness leads to a number of interesting effects specific to this type of cell; these effects are considered in Section 2.3.6. In PEFCs the cathode side makes the largest contribution to voltage loss. This explains the great interest in CCL performance in these cells. We begin the analysis of CL performance with the cathode catalyst layer of a low-temperature hydrogen fuel cell. However, it should be emphasized that the performance of other catalyst layers of the cells considered in this book can be described by similar equations. The features of particular layers are taken into account by the expression for the rate of the electrochemical reaction. Regardless of fuel cell type, in a certain range of polarization voltages this rate is well approximated by the Butler-Volmer or Tafel equations. A better (non-Tafel) approximation of the reaction rate for DMFC anode is considered in Section 2.7. The pre-exponential concentration dependence in the kinetic equations may differ for different half-cell reactions. However, if the transport loss in the CL is negligible, the concentration factor is constant. Thus, the relations for the case of negligible transport loss derived below are valid for all types of cells.

2.1 BASIC EQUATIONS

41

Figure 2.1: Schematic of the cathode catalyst layer and the dimensionless shapes of the proton current density ˜j and oxygen concentration c˜. Note that in Chapter 1 j is denoted as jp .

2.1 Basic equations 2.1.1

The general case

ORR participants arrive at the CCL from different sides: protons come from the polymer electrolyte membrane, while electrons and oxygen come from the gas-diffusion layer (GDL, Figure 2.1). Let the proton current density entering the CCL be j0 (note that this value coincides with the cell current density) and the axis x be directed from the membrane to the CCL/GDL interface (Figure 2.1). Due to the ORR, proton current j decreases along x and at the CCL/GDL interface, j = 0 (Figure 2.1). Oxygen coming from the GDL is consumed in the reaction; thus the oxygen concentration decreases toward the membrane (Figure 2.1). Following (Perry et al., 1998), CCL performance is governed by equations:   ∂j(x) c = −2i∗ φ(η) ∂x cref ∂η(x) j(x) = −σt ∂x j0 − j ∂c = . D ∂x 4F

(2.1) (2.2) (2.3)

Here j(x) is the local proton current density, i∗ is the volumetric exchange current density (the number of charges produced in unit volume per second, A cm−3 ), c is the molar concentration of oxygen, cref is the reference oxygen concentration, φ is the conversion function, η is the local polarization voltage, σt is the proton conductivity of the CCL, D is the effective oxygen diffusion coefficient and j0 is the cell current. Equation (2.1) expresses the decay of proton current along x at a rate given by the right side of this equation. Equation (2.2) is Ohm’s law relating

42

CHAPTER 2. CATALYST LAYER PERFORMANCE

proton current to the overpotential gradient1 . Equation (2.3) says that the local diffusion flux of oxygen is equal to the local electron current density j0 − j to be converted in the ORR. The stoichiometry factor 4F in Eq. (2.3) transforms current density to molar flux. The conversion function φ in Eq. (2.1) expresses the dependence of the ORR rate on the local polarization voltage η. This dependence is usually well approximated by Butler-Volmer, or (when far from equilibrium) by the Tafel equation      αF η (1 − α)F η 1 exp − exp − (2.4) φBV = 2 RT RT   αF η 1 . (2.5) φT = exp 2 RT As discussed in Section 1.3.1, the first exponent in Eq. (2.4) describes the direct (oxygen reduction) reaction, while the second exponent represents the reverse reaction of water electrolysis. At η & RT /(αF ) the second exponent in Eq. (2.4) can be neglected and φBV reduces to φT (Section 1.3.3). Physically, far from equilibrium the rate of reverse reaction is negligibly small. Following (Bard and Faulkner, 2001), the transfer coefficient for the single-electron transfer through the symmetrical activation barrier is α = 0.5. To simplify the analysis, we will adopt this value in the Butler-Volmer conversion function (2.4): α = 1 − α = 0.5, whereas in the Tafel function (2.5) α may have an arbitrary value. Note that the restriction of α = 0.5 is important only in the region of small overpotentials; at large η the results for Butler-Volmer and Tafel functions coincide. We introduce dimensionless variables x η c jlt x ˜ = , η˜ = , c˜ = , ˜j = (2.6) lt b ct σt b where lt is the CCL thickness, ct is the oxygen concentration at the CCL/GDL interface (at x/lt = 1, Figure 2.1) and b=

RT αF

(2.7)

is the Tafel slope. With these variables, Eqs (2.4) and (2.5) transform to φBV (˜ η ) = sinh η˜ 1 φT (˜ η ) = exp η˜ 2

(2.8) (2.9)

1 Note that η = ϕ0 − ϕc , where ϕ0 and ϕc are potentials of ion- and electronm m conducting phases in the CCL (Figure 1.3). Ohm’s law for ionic (proton) current is ∂ϕ0

∂η j(x) = −σt ∂xm = −σt ∂x , since the electronic conductivity of CCL is high and the c variation of ϕ with x is small.

2.1 BASIC EQUATIONS

43

and Eqs (2.1)–(2.3) take the form ε2

∂ ˜j =− ∂x ˜



c˜

 φ(˜ η)

c˜ref ˜j = − ∂ η˜ ∂x ˜ ∂˜ c ˜ = ˜j0 − ˜j. D ∂x ˜

(2.10) (2.11) (2.12)

Here ε=

l∗ , lt

(2.13)

is the ratio of the reaction penetration depth l∗ (Newman, 1991) r σt b l∗ = 2i∗

(2.14)

to the CCL thickness lt and ˜ = 4F Dct D σt b

(2.15)

is the dimensionless diffusion coefficient. Boundary conditions for the system (2.10)–(2.12) are ˜j(0) = ˜j0 ,

˜j(1) = 0 c˜(1) = 1.

(2.16) (2.17)

Without loss of generality, throughout Section 2.2 we will put c˜ref = 1. If necessary, this factor can be restored replacing ε2 → ε2 c˜ref

(2.18)

or  i∗ → i∗

ct cref

 (2.19)

in the results of this section. This substitution will be done in all dimension equations below to explicitly indicate the concentration dependence. Equations (2.10)–(2.12) describe the CCL performance in the general case of finite transport losses. We see that the problem is controlled by ˜ and ˜j0 . three parameters: ε, D In the general case, this system can hardly be solved. However, in the limiting cases of ideal oxygen or proton transport the reduced problems are solvable. Before proceeding to these cases we will establish a general conservation equation, which follows from Eqs (2.11) and (2.12).

44

CHAPTER 2. CATALYST LAYER PERFORMANCE

2.1.2

First integral

Equations (2.11) and (2.12) can be integrated. Using (2.11) in (2.12) we get  ∂ ˜ D˜ c − η˜ = ˜j0 . ∂x ˜

(2.20)

  ˜ c − η˜ = ˜j0 x ˜ c0 − η˜0 . D˜ ˜ + D˜

(2.21)

Integrating we obtain

  ˜ − η˜1 = ˜j0 + D˜ ˜ c0 − η˜0 . Expressing D˜ ˜ c0 − η˜0 Setting here x ˜ = 1 we get D from this equation and using the result in (2.21) we find ˜ − c˜) + (˜ D(1 η − η˜1 ) = ˜j0 (1 − x ˜).

(2.22)

Setting x ˜ = 0 here we get a useful relation for the voltage drop over the CL ˜ − c˜0 ). ∆˜ η ≡ η˜0 − η˜1 = ˜j0 − D(1

(2.23)

Care should be taken when using this relation in the limiting case of ideal ˜ → ∞, while c˜0 → 1 so that the product oxygen transport. In this case D ˜ D(1 − c˜0 ) is finite.

2.2 Ideal oxygen and proton transport This is the case of an ideal catalyst layer with no transport losses. In Eq. (2.10) we set c˜ = c˜ref = 1 and φ(˜ η ) ' φ(˜ η0 ) ≡ φ0 , and omit Eqs (2.11) and (2.12). The system discussed here reduces to a single equation ε2

∂ ˜j = −φ0 . ∂x ˜

(2.24)

The solution to this equation is a straight line ˜j = φ0 (1 − x ˜). ε2

(2.25)

Setting x ˜ = 0 here we get the polarization curve of the catalyst layer ε2 ˜j0 = φ0 .

(2.26)

With φ0 given by (2.8) we finally obtain
η˜0 = arcsinh ε2 ˜j0 .

(2.27)

2.3 IDEAL OXYGEN TRANSPORT

45

This is the general polarization curve of an ideal catalyst layer. If ε2 ˜j0 ≥ 2 the function arcsinh can be replaced by the logarithm of the double argument and we get
(2.28) η˜0 = ln 2ε2 ˜j0 , ε2 ˜j0 ≥ 2. In the dimensional form this relation reads   j0 η0 = b ln     i∗ ccreft lt

(2.29)

where Eq. (2.19) was used. Note that in Eq. (2.29) parameters D and σt do not appear, since this is the limit of infinite oxygen diffusivity and proton conductivity of the CL. Equation (2.29) is a well-known Tafel form of the activation polarization voltage (or simply activation polarization), which is often used in the fuel cell literature. In Section 2.3.4, the limits of validity of this approximation will be established.

2.3 Ideal oxygen transport 2.3.1

Basic equations

In this case we only set c˜ = c˜ref = 1 and omit Eq. (2.12). The system (2.10)–(2.12) simplifies to ε2

∂ ˜j = −φ(˜ η) ∂x ˜ ∂ η˜ = −˜j ∂x ˜

(2.30) (2.31)

where φ(˜ η ) is given by one of the relations (2.8) and (2.9).

2.3.2

Integral of motion

Multiplying together Eqs (2.30) and (2.31) we obtain ε2 ˜j

∂ ˜j ∂ η˜ = φ(˜ η) ∂x ˜ ∂x ˜

or ε2

∂(˜j 2 ) ∂Φ(˜ η) =2 ∂x ˜ ∂x ˜

(2.32)

46

CHAPTER 2. CATALYST LAYER PERFORMANCE

where Z Φ(t) =

φ(t)d t.

(2.33)

Integrating Eq. (2.32) we find 2Φ − ε2 ˜j 2 = 2Φ0 − ε2 ˜j02 = 2Φ1 .

(2.34)

Thus, the sum 2Φ − ε2 ˜j 2 is constant along x ˜. Note also that the equation ε2 ˜j02 = 2Φ0 − 2Φ1

(2.35)

which follows from (2.34) is the general form of the CCL polarization curve at high currents (see below). Equation (2.34) with the Butler-Volmer and Tafel conversion functions takes the forms 2 cosh(˜ η ) − ε2 ˜j 2 = 2 cosh(˜ η0 ) − ε2 ˜j02 = 2 cosh(˜ η1 ), Butler-Volmer (2.36) 2 ˜2 exp(˜ η ) − ε j = exp(˜ η0 ) − ε2 ˜j02 = exp(˜ η1 ), Tafel. (2.37)

2.3.3

Equation for proton current

For further calculations we will use the Butler-Volmer conversion function: φ = sinh(˜ η ). With this, Eq. (2.30) takes the form ε2

∂ ˜j = − sinh(˜ η ). ∂x ˜

(2.38)

To analyse the system of Eqs (2.31) and (2.38) it is convenient to transform the system into a single equation for proton current. Differentiating (2.38) over x ˜ we get ε2

∂ 2 ˜j ∂ η˜ = − cosh(˜ η) . ∂x ˜2 ∂x ˜

(2.39)

η ) − sinh2 (˜ η ) = 1 we come to Using Eq. (2.31) and the identity cosh2 (˜ ε2

q ∂ 2 ˜j ˜j 1 + sinh2 (˜ η ). = ∂x ˜2

Excluding sinh2 (˜ η ) with Eq. (2.38) we obtain s  2 2˜ ∂ j ∂ ˜j 2 4 ˜ ε 1 + ε = j . ∂x ˜2 ∂x ˜

(2.40)

2.3 IDEAL OXYGEN TRANSPORT

47

Figure 2.2: The shapes of the dimensionless proton current density ˜j and overpotential η˜ across the cathode catalyst layer for the two indicated values of parameter ε and ˜j0 = 0.1. The general analytical solution to this equation hardly exists. However, in the limiting cases of small and large currents it can be solved.

2.3.4

Low cell current

If the cell current is not large, in (2.40) we may neglect the term with ε4 under the root sign (a rigorous condition for this approximation will be obtained below). The resulting equation ε2

∂ 2 ˜j = ˜j, ∂x ˜2

˜j(0) = ˜j0 ,

˜j(1) = 0

˜j0 sinh sinh




(2.41)

can easily be integrated to yield ˜j(˜ x) =

1−x
ε 1 ε

.

(2.42)

The shape of η˜ now follows from Eq. (2.38): η˜ = arcsinh

ε˜j0 cosh sinh

1−x
ε 1 ε


! .

(2.43)

The functions (2.42) and (2.43) are depicted in Figure 2.2. Setting x ˜ = 0 in (2.43) we get the polarization curve of the CCL    1 ˜ η˜0 = arcsinh εj0 coth . (2.44) ε When ε  1 we have coth(1/ε) ' 1, and Eq. (2.44) reduces to
η˜0 = arcsinh ε˜j0 .

(2.45)

48

CHAPTER 2. CATALYST LAYER PERFORMANCE

If, in addition, ˜j0 ≤ 1 (that is ε˜j0  1), we obtain simply η˜0 = ε˜j0 .

(2.46)

Thus, when ε is small and ˜j0 ≤ 1 the polarization curve of the CCL is linear. Equation (2.46) in dimension variables takes the form η0 = Ract j0

(2.47)

where Ract

v u u =t

2σt i∗

b 

ct cref



(2.48)

is the activation resistivity of the CL. Here Eq. (2.19) was taken into account. In the opposite limit of ε  1 we have coth(1/ε) ' ε, and Eq. (2.44) takes the form
η˜0 = arcsinh ε2 ˜j0 . (2.49) This relation was obtained in Section 2.2 for the case of an ideal catalyst layer. Thus, ideal proton transport means that ε  1. If ε2 ˜j0 ≥ 2 the function arcsinh can be replaced by the logarithm of double argument and we get
η˜0 = ln 2ε2 ˜j0 , ε  1, ε2 ˜j0 ≥ 2. (2.50) In dimension variables this equation has the form (2.29), obtained above for an ideal catalyst layer. As seen, Eq. (2.29) holds when oxygen transport is ideal, ε  1 and ε2 ˜j0 ≥ 2. The notion of “ideal oxygen transport” will be rationalized in Section 2.5.

2.3.5

High cell current

In the opposite limit of large cell current, we may neglect 1 under the square root in Eq. (2.40) (a rigorous condition for this approximation will be derived below). This equation reduces to ∂ 2 ˜j ∂ ˜j = −˜j 2 ∂x ˜ ∂x ˜ or
∂ ˜j 2 ∂ 2 ˜j 2 2+ = 0. ∂x ˜ ∂x ˜

(2.51)

2.3 IDEAL OXYGEN TRANSPORT

49

Figure 2.3: The shapes of the dimensionless proton current density ˜j, ˜ and overpotential η˜ across the cathode catalyst layer for reaction rate Q large cell currents. Parameter ε = 1. Note that β → π as ˜j0 → ∞ (Eq. (2.61)). Note that ∂ ˜j/∂ x ˜ < 0 and hence we took a negative value for the square root in Eq. (2.40). Integrating (2.51) once we find 2

∂ ˜j ˜2 ˜ 0 − ˜j02 ) = −2Q ˜1 + j = −(2Q ∂x ˜

(2.52)

where we introduced the dimensionless rate of electrochemical conversion ˜ Q: ˜ η) ˜ ≡ − ∂ j = φ(˜ Q . ∂x ˜ ε2

(2.53)

The solution to (2.52) is ˜j = β tan



 β (1 − x ˜) , 2

(2.54)

where β=

q

˜ 0 − ˜j 2 . 2Q 0

Using (2.54) in Eq. (2.38) we get η˜:  2 2   β ε β η˜ = arcsinh 1 + tan2 (1 − x ˜) . 2 2

(2.55)

(2.56)

˜ x) are shown in Figure 2.3 for two values of The shapes ˜j(˜ x), η˜(˜ x) and Q(˜ parameter β and ε = 1. This figure shows the physical origin of the Tafel slope doubling in the high-current regime (see the next page). Poor proton transport induces a

50

CHAPTER 2. CATALYST LAYER PERFORMANCE

peak of the reaction rate at the membrane interface, where the expenditure for proton transport is lower. The nonuniform conversion of proton current is costly in terms of η˜. The high-current polarization curve of the CCL follows from Eq. (2.54). Setting x ˜ = 0 in this equation we get   ˜j0 = β tan β . (2.57) 2 Since ˜j0 is large, parameter β ' π. Taking into account the expression for β (2.54) we find ˜ 2 − ˜j 2 = π 2 . 2Q 0 0 Neglecting π 2 as compared to ˜j02 we come to ˜0. ˜j02 = 2Q Using (2.53) we obtain ε2 ˜j02 = 2φ(˜ η0 ). At large currents the Butler-Volmer conversion function reduces to Tafel one (2.9) and we finally find
η˜0 = 2 ln ε˜j0 .

(2.58)

In dimension variables this equation takes the form   j0 η0 = 2b ln t jσ

(2.59)

where s jσt

=

 2i∗ σt b

ct cref

 (2.60)

is the characteristic current density2 . Note that here substitution (2.19) was performed. The factor on the right side of Eq. (2.59) is 2b instead of b. Equation (2.59) exhibits the effect of apparent Tafel slope doubling discussed above. Equation (2.57) enables β to be expressed through ˜j0 . For β . π the tan function is well approximated by tan(β/2) ' 2/(π − β). Using this 2 Below

j0 > 4 coth we will show that Eq. (2.58) is valid provided that ε˜

1 ε




.

2.3 IDEAL OXYGEN TRANSPORT

51

Figure 2.4: The general polarization curve of the catalyst layer with ideal transport of the reactant (oxygen). Solid lines: analytical solutions, points— the exact numerical curve. Parameter ε = 0.1 (left panel) and ε = 10 (right panel). Linear domain is described by η˜0 = ε˜j0 , Tafel region is given by
˜ η˜0 = arcsinh εj0 coth (1/ε) and double-Tafel law is η˜0 = 2 ln(ε˜j0 ). expansion in (2.57) and solving for β we get β'

2.3.6

π˜j0 . 2 + ˜j0

(2.61)

The general polarization curve

The results above allow us to plot the general polarization curve of the catalyst layer with ideal oxygen transport (Figure 2.4). At ε  1 the four distinct regions are clearly seen (Figure 2.4, right panel). At small currents η˜ is linear (Eq. (2.46)). This linear growth is followed by the Tafel dependence (Eq. (2.44)) with the apparent Tafel slope being equal to the kinetic value (Figure 2.4, right). The region ε . ε˜j0 . 4ε is the transition region, and at ε˜j0 > 4ε we have a high-current region with doubling of the Tafel slope; see Eq. (2.58) (Figure 2.4, right). These types of polarization curves exhibit the cathode sides of PEFC and SOFC, and both sides of DMFC. At small ε the Tafel region disappears and the transition region directly links the linear and high-current domains (Figure 2.4, left). This type of curve is characteristic of the anode sides of PEFC and SOFC. In PEFC anode, the smallness of ε is provided by large exchange current densities. (Note that the PEFC anode always operates in the linear mode.) In the anode of the anode-supported SOFC, the condition ε  1 holds due to the large anode thickness. This feature of SOFC anode operation is discussed in Section 4.6. Equations (2.44) and (2.58) allow us to calculate the length of the transition region. At y > 2 we have arcsinh(y) ' arccosh(y) ' ln(2y).

52

CHAPTER 2. CATALYST LAYER PERFORMANCE

Using this in (2.44) and equating the result to (2.58) we get 

ln 2ε˜j0 coth

 
1 = ln ε2 ˜j02 ε

or ε˜j0 = 2 coth

  1 . ε

(2.62)

This relation determines the position of the intersection of the low- and high-current polarization curves (Figure 2.4). Note that for ε  1 we have coth(1/ε) ' 1 and the intersection is located at ε˜j0 = 2 (Figure 2.4, left plot)3 . Equation (2.62) and Figure 2.4 give us the following accurate limits of validity of low- and high-current approximations:   1 , low-current limit ε   1 ε˜j0 > 4 coth , high-current limit. ε ε˜j0 < coth

(2.63) (2.64)

Importantly, the regime of catalyst layer operation depends on parameter ε. The transition from low- to high-current mode occurs in the transition region     1 1 ˜ ≤ εj0 ≤ 4 coth . coth ε ε For small and large ε this inequality reduces to 1 ≤ ε˜j0 ≤ 4, 1 ≤ ˜j0 ≤ 4,

ε1 ε  1.

Note that for ε  1 parameter ε does not affect the transition region position.

3 With

Eq. (2.62), Eq. (2.44) takes the form    1 η˜0 = arcsinh 2 coth2 . ε

The minimal value of the argument of this function is achieved when ε  1. For these ε we have η˜0 = arcsinh(2) ' ln(4). This justifies the use of relation arcsinh(x) ' ln(2x) in the derivation of Eq. (2.62).

2.4 IDEAL PROTON TRANSPORT

2.3.7

53

Condition of negligible oxygen transport loss

A sufficient condition for the approximation of ideal oxygen transport is infinity of oxygen diffusivity D → ∞. A numerical study of the full system (2.10)–(2.12) (Section 2.5) enables us to formulate a more accurate criterion. The contribution of oxygen transport to the total voltage loss can be neglected if ˜ >D ˜ ∗ = ˜j0 . D

(2.65)

This condition is discussed in Section 2.5.

2.4 Ideal proton transport 2.4.1

Basic equations

Ideal proton transport means that the overpotential gradient is small. Setting in the system (2.10)–(2.12) η˜ ' η˜0 and omitting Eq. (2.11) we arrive at ∂ ˜j = −˜ cφ(˜ η0 ) ∂x ˜ c ˜ ∂˜ D = ˜j0 − ˜j. ∂x ˜

ε2

(2.66) (2.67)

Here φ(˜ η ) is given by one of the relations (2.8) and (2.9).

2.4.2

The x-shapes and polarization curve

Differentiating (2.66) with respect to x ˜ and excluding ∂˜ c/∂ x ˜ with Eq. (2.67) we get ! ˜ ∂ 2 ˜j ε2 D = −(˜j0 − ˜j), ˜j(0) = ˜j0 , ˜j(1) = 0 (2.68) φ0 ∂x ˜2 where φ0 ≡ φ(˜ η0 ). The solution to this equation is   ˜) ˜j = ˜j0 1 − sinh (ζ x sinh (ζ)

(2.69)

where r ζ=

φ0 . ˜ ε2 D

(2.70)

54

CHAPTER 2. CATALYST LAYER PERFORMANCE

Figure 2.5: The shapes of the local current density and oxygen molar concentration across the catalyst layer for the diffusion-limited case. ˜ = 0.1, ε = 1, and ˜j0 = 1. Parameters: φ0 = 10, D Solving (2.66) for c˜ we get  c˜ = −

ε2 φ0



∂ ˜j =− ∂x ˜



1 ˜ 2 Dζ



∂ ˜j . ∂x ˜

Using here (2.69) we find c˜ =

˜j0 cosh(ζ x ˜) . ˜ Dζ sinh ζ

(2.71)

The shapes of the local current and oxygen concentration across the CCL are shown in Figure 2.5 for the case of small oxygen diffusivity. Equation (2.71) is valid for ˜j0 and η˜0 related by the polarization curve. At x ˜ = 1 we must have c˜ = 1; using this in (2.71) we get ˜ tanh ζ. ˜j0 = Dζ

(2.72)

With (2.70) this equation is the general polarization curve for the CCL with ideal proton transport. Of particular interest are the two limiting cases.

2.4.3

Large cell current (ζ  1)

˜ Using (2.70) In this case, tanh ζ ' 1 and Eq. (2.72) reduces to ˜j0 = Dζ. and taking into account that φ0 = sinh η˜0 , we get  η˜0 = arcsinh

ε2 ˜j02 ˜ D

 .

(2.73)

2.5 OPTIMAL OXYGEN DIFFUSION COEFFICIENT

55

Note that the polarization curve (2.73) exhibits doubling of the Tafel slope. This is immediately seen if we replace arcsinh(x) by ln(2x) in (2.73): r ! 2 ˜ η˜0 ' 2 ln εj0 . (2.74) ˜ D ˜ is small and ˜j0 (Unless ε is very small, this replacement is justified since D is large.) In the dimensional form Eq. (2.74) is   j η0 = 2b ln D j∗ where the characteristic current density j∗D is given by p j∗D = 4F Dct i∗ .

(2.75)

Remarkably, doubling of Tafel slope arises when at least one of the transport processes in the CCL (proton or oxygen transport) is poor. This makes it difficult to distinguish the physical origin of doubling.

2.4.4

Small cell current (ζ  1)

˜ 2 . With (2.70) In this case, tanh ζ ' ζ and Eq. (2.72) reduces to ˜j0 = Dζ we get ε2 ˜j0 = φ0 . Taking into account (2.8) we obtain
η˜0 = arcsinh ε2 ˜j0 . (2.76) Equation (2.76) coincides with Eq. (2.49) derived in the case of ideal oxygen transport at small current. Small ζ is equivalent to small η˜0 or a large diffusion coefficient; in both cases oxygen transport does not contribute to the voltage loss.

2.5 Optimal oxygen diffusion coefficient In this section, we return to the full system of equations for CL performance, (2.10)–(2.12). We study numerically the effect of the oxygen diffusion coefficient D on the CCL polarization voltage η0 for the general case of the Butler-Volmer conversion function (Kulikovsky, 2009a).

2.5.1

Reduction of the full system

For numerical calculations, it is convenient to exclude η˜ from the system (2.10)–(2.12). Differentiating (2.10) with φ(˜ η ) = sinh(˜ η ) with respect to x ˜

56

CHAPTER 2. CATALYST LAYER PERFORMANCE (a)

(b)

Figure 2.6: The x-shapes of oxygen concentration c˜, proton current density ˜ Parameters are: ˜j0 = 1, ε = 1, ˜j, overpotential η˜ and reaction rate Q. ˜ = 1, (b) D ˜ = 0.2. The membrane is at x c˜ref = 1, and (a) D ˜ = 0. we get ε2

sinh(˜ η ) ∂˜ c˜ cosh(˜ η ) ∂ η˜ c ∂ 2 ˜j =− − . 2 ∂x ˜ c˜ref ∂ x ˜ c˜ref ∂x ˜

Using the identity cosh(˜ η) =

∂ 2 ˜j − ε2 ε ∂x ˜2 2



˜j0 − ˜j ˜c D˜



∂ ˜j − ∂x ˜

q 1 + sinh2 (˜ η ), (2.11) and (2.12), we arrive at



c˜ c˜ref



s ˜j 1 + ε4



c˜ref c˜

2 

∂ ˜j ∂x ˜

2 = 0. (2.77)

Equations (2.77) and (2.12) with the boundary conditions (2.16) and (2.17) fully describe the problem. The presence of η˜ in Eqs (2.10) and (2.11) makes the numerical solution of the system (2.10)–(2.12) much more complex. The problem is that Eq. (2.11) requires a boundary value of η˜ at x ˜ = 0 or x ˜ = 1. Both these values are not known a priori ; they arise as a result of problem solution. In contrast, the boundary conditions (2.16) and (2.17) for the system (2.77) and (2.12) are fully determined. When c˜ and ˜j are found, the shape η˜(˜ x) follows from Eq. (2.10):     c˜ref ∂ ˜j . η˜(˜ x) = −arcsinh ε2 c˜ ∂x ˜

2.5.2

(2.78)

Optimal oxygen diffusivity

The shapes of oxygen concentration c˜, local current density ˜j, polarization ˜ are depicted in voltage η˜ and the rate of electrochemical reaction Q ˜ ˜ Figure 2.6 for the values of D = 1 and D = 0.2 (the other parameters are listed in the caption to Figure 2.6).

2.5 OPTIMAL OXYGEN DIFFUSION COEFFICIENT (a)

57

(b)

Figure 2.7: Voltage loss vs oxygen diffusion coefficient in the catalyst layer for the indicated values of parameter ε. Total current density in a cell is ˜ ∗ = ˜j0 are indicated ˜j0 = 0.5 (left) and ˜j0 = 2 (right). Optimal values of D by red dashed lines.

Figure 2.8: Voltage loss vs oxygen diffusion coefficient in the catalyst layer for indicated values of dimensionless cell current ˜j0 and ε = 1. For each ˜ ∗ = ˜j0 is indicated by a red dashed line. curve the optimal value D ˜ = 1 the total voltage loss η˜0 ' 1.5 (Figure 2.6(a)); it rises to η˜0 ' 3 At D ˜ at D = 0.2 (Figure 2.6(b)). This is the result of poor oxygen transport in ˜ = 1 oxygen concentration drops across the CCL the catalyst layer: with D ˜ = 0.2 this drop exceeds two orders of by a factor of two, whereas with D magnitude (Figure 2.6(a) and (b)). Low oxygen concentration at the membrane interface shifts the whole curve η˜(˜ x) to a higher value and leads to formation of a distinct peak of ORR rate at the CCL/GDL interface, where more oxygen is available (Figure 2.6(b)). Qualitatively, at x ˜ = 1 oxygen concentration is high and large η˜(1) maximizes the reaction rate at this point. However, high η˜(1) shifts the whole curve η˜(˜ x) upward, since the derivative ∂ η˜/∂ x ˜ = −j must be negative. ˜ for various ˜j0 and The dependencies of η˜0 on the oxygen diffusivity D ε are depicted in Figures 2.7 and 2.8. These figures display an important feature of CCL performance.

58

CHAPTER 2. CATALYST LAYER PERFORMANCE

˜ which is an CCL polarization voltage decreases with the growth of D, obvious manifestation of lower transport loss at higher oxygen diffusivity ˜ which has two (Figures 2.7 and 2.8). Less obvious is the shape of η˜0 (D), ˜ turns into slopes (Figure 2.7). For all calculated variants, rapid decay η˜0 (D) a slow fall at ˜ ∗ = ˜j0 D

(2.79)

(Figures 2.7 and 2.8). In dimension variables Eq. (2.79) takes the form D∗ =

j 0 lt . 4F ct

(2.80)

At D < D∗ the polarization voltage decreases rapidly with the growth of ˜ whereas for D > D∗ lowering of η˜0 due to the increase in D is marginal D, (Figures 2.7 and 2.8). Higher diffusivity means higher CCL porosity, which is usually achieved at the cost of lower Nafion content and thus of lower proton conductivity. Equation (2.80) thus gives an optimal oxygen diffusion coefficient in the CCL, and D∗ can be used as a reference point for optimal CCL design in terms of porosity and related Nafion content. Equation (2.80) shows that D∗ increases with the growth of cell current and/or CCL thickness. The growth of current requires a larger oxygen flux across the CCL and hence larger D∗ . Evidently, larger CCL thickness also retards oxygen transport toward the CCL/membrane interface thereby increasing the optimal D∗ .

2.6 Gradient of catalyst loading 2.6.1

Model

Figure 2.3 (page 49) suggests that nonuniform catalyst loading can be beneficial for CL performance. Indeed, under uniform loading the peak of current production is located at the membrane interface. It seems reasonable to increase catalyst loading close to the membrane at the cost of lower loading at the GDL interface, where the rate of electrochemical conversion is anyway low. Catalyst loading appears in the system (2.1)–(2.3) as a factor i∗ in Eq. (2.1). In the previous sections this factor was constant. In this section we will see what happens if i∗ varies across the CCL (Kulikovsky, 2009e). To understand the effect we replace i∗ by i∗ g(x), where g is the “profile” function for the catalyst concentration in the CL.

2.6 GRADIENT OF CATALYST LOADING

59

To simplify calculations we will assume that • oxygen transport is ideal, and • the cell operates in the high-current regime. The second assumption means that we can employ Tafel conversion function (2.9). It is convenient to formally treat g(x) as a function of overpotential: g(x) = g(η(x)). The system of governing equations for CL performance then follows from (2.30) and (2.31): 2ε2

∂ ˜j = −g(˜ η ) exp η˜ ∂x ˜ ˜j = − ∂ η˜ . ∂x ˜

(2.81) (2.82)

Multiplying together these equations we find 2ε2 ˜j

∂ η˜ ∂ ˜j = g(˜ η ) exp η˜ ∂x ˜ ∂x ˜

and thus ε2

∂(˜j 2 ) ∂ (exp η˜) = g(˜ η) . ∂x ˜ ∂x ˜

(2.83)

Integrating both sides we get 2

Z

˜ j0

ε

0

d(˜j 2 ) =

Z

η˜0

g(˜ η )d (exp η˜) .

(2.84)

η˜1

The left side equals ε2 ˜j02 . The right side can be integrated by parts: Z

Z

∂g exp η˜ d˜ η ∂ η˜ Z ∂g = g exp η˜ − d(exp(˜ η )) ∂ η˜ Z 2 ∂g ∂ g = g exp η˜ − exp η˜ + exp(˜ η ) d˜ η ∂ η˜ ∂ η˜2   ∂g ∂2g = g− + 2 − · · · exp η˜. ∂ η˜ ∂ η˜

g(˜ η ) d(exp η˜) = g exp η˜ −

We therefore obtain Z g d(exp η˜) = Fg (˜ η ) exp η˜

(2.85)

60

CHAPTER 2. CATALYST LAYER PERFORMANCE

where Fg = g −

∂g ∂2g + 2 − ···. ∂ η˜ ∂ η˜

(2.86)

Using this in (2.84) we get η˜0

η0 ) exp η˜0 − Fg (˜ η1 ) exp η˜1 . ε2 ˜j02 = [Fg exp η˜]η˜1 = Fg (˜

(2.87)

When loading is uniform we have Fg = g = 1 and Eq. (2.87) reduces to ε2 ˜j02 = exp η˜0 − exp η˜1 .

(2.88)

Neglecting the second exponent on the right side we come to η˜0 = 2 ln(ε˜j0 )

(2.89)

which is a high-current polarization curve of a CCL (cf. Eq. (2.58)). We will require that for all η˜ Fg (˜ η) = χ

(2.90)

where χ is constant. The rationale for this condition is that at uniform loading we have Fg ≡ g = χ = 1 for all η˜ and thus Eq. (2.90) provides a correct transition to the limiting case of uniform loading. Note that the choice (2.90) is not unique; e.g. we could set Fg (˜ η ) = χ exp η˜.

(2.91)

With (2.91) the polarization curve (2.87) takes the form ε2 ˜j02 = χ exp(2˜ η0 ) − χ exp(2˜ η1 ). Neglecting the second exponent we obtain  η˜0 = ln

ε˜j0 χ



which is the Tafel law. Therefore, nonuniform loading which obeys (2.91) switches cell operation from the high-current regime (with the Tafel slope being doubled) to the “normal” Tafel regime (this is discussed in more detail below). Options (2.90) and (2.91) lead to very close results and for simplicity we take (2.90) for further calculations. For g(˜ η ) we therefore have an equation g−

∂g ∂2g + 2 − ··· = χ ∂ η˜ ∂ η˜

(2.92)

2.6 GRADIENT OF CATALYST LOADING

61

where the constant χ is determined by the normalization condition Z

1

g(˜ x) d˜ x = 1.

(2.93)

0

Equation (2.93) requires that for any shape g(˜ x) the total amount of catalyst remains the same. We will assume that the contributions of third and higher derivatives in Eq. (2.92) are small (this assumption will be validated below). With this we get an approximate equation ∂g ∂2g ∂g g(˜ η) − + 2 = χ, g(˜ η1 ) = g1 , =0 (2.94) ∂ η˜ ∂ η˜ ∂ η˜ η˜1 where g1 ≡ g|x˜=1 = g|η˜=˜η1  1 is the small catalyst loading at the CCL/GDL interface (g1 6= 0). Zero derivative of g at x ˜ = 1 (at η˜ = η˜1 ) is required in accordance with zero derivative of η˜ at x ˜ = 1. The solution of Eq. (2.94) is "



g1 g(˜ η) = χ 1 + 1 − χ √ − cos



 exp

η˜ − η˜1 2 !!#

3 (˜ η − η˜1 ) 2



1 √ sin 3

√

! 3 (˜ η − η˜1 ) 2

.

(2.95)

However, transformation g(˜ η ) to g(˜ x) requires η˜(˜ x). The equation for η˜(˜ x) follows from (2.81) with ˜j from Eq. (2.82): 2ε2

∂ 2 η˜ = g(˜ η ) exp η˜, ∂x ˜2

η˜(1) = η˜1 ,

∂ η˜ = 0. ∂x ˜ x˜=1

(2.96)

Equations (2.96), (2.95) and (2.93) represent a problem for coupled distributions of overpotential and catalyst loading. Note that iterations are required to obtain a solution satisfying (2.93). The resulting shape of catalyst loading and the distributions of local current and overpotential are shown in Figure 2.9(b). For comparison, Figure 2.9(a) shows these shapes for the case of uniform loading (g = 1). The total voltage loss η˜0 is the same in both cases. Note that the total current is almost twice as high in the case of nonuniform loading (Figure 2.9). Note also the much flatter distribution of overpotential in the case of nonuniform loading (Figure 2.9). Figure 2.9(b) also shows the shape of catalyst loading, which follows from Eq. (2.91) (long-dashed line). This shape leads to distributions of ˜j and η˜ which are indistinguishable from those shown in Figure 2.9(b). Thus, the two options (2.90) and (2.91) lead to the very close results. Figure 2.9(b)

62

CHAPTER 2. CATALYST LAYER PERFORMANCE (a)

(b)

Figure 2.9: (a) Current ˜j and overpotential η˜ distribution for uniform loading (g(˜ x) = 1). (b) ˜j and η˜ for optimal shape of catalyst loading g(˜ x) across the catalyst layer (short-dashed line). The total voltage loss η˜(0) in (a) and (b) is the same. Long-dashed line: the shape of catalyst loading following from (2.91). The respective shapes of ˜j(˜ x) and η˜(˜ x) are indistinguishable from those shown in the figure. Parameters ε = 1 and g1 = 0.1; the resulting parameter χ = 44.45. Note the almost twice higher total current ˜j(0) in the case of nonuniform loading. explains why is this so. The variation of the overpotential along x ˜ in this figure is small and thus exp(˜ η ) in Eq. (2.91) is almost a constant factor, which simply rescales the parameter χ.

2.6.2

Polarization curve

With (2.90), Eq. (2.87) takes the form ε2 ˜j02 = χ (exp η˜0 − exp η˜1 ), or χ exp η˜0 = χ exp η˜1 + ε2 ˜j02 .

(2.97)

At uniform loading the polarization curve is given by Eq. (2.35), which for the Tafel conversion function reduces to exp η˜0u = exp η˜1u + ε2 ˜j02 .

(2.98)

Here the superscript u marks the values at uniform loading. Subtracting Eqs (2.97) and (2.98) we get χ (exp η˜0 − exp η˜1 ) = exp η˜0u − exp η˜1u .

(2.99)

Since χ  1, the variation of overpotential across the CL with nonuniform loading is smaller than in the uniform case. The polarization curve of the CL with the optimal shape of catalyst loading is compared in Figure 2.10 to those curves for the CL with uniform

2.6 GRADIENT OF CATALYST LOADING

63

Figure 2.10: Upper solid: analytical high-current polarization curve for uniform loading. Crosses: the exact numerical polarization curve for uniform loading. Lower solid: polarization curve of the active layer with optimal shape of catalyst loading. Short-dashed curve: nonuniform loading; third and fourth order derivatives in Eq. (2.92) are taken into account. loading. Remarkably, the benefit from optimal loading increases with the cell current and with b = 50 mV it reaches 150 mV at the right side of this plot. Note that taking into account the third and fourth derivatives of g in Eq. (2.92) has marginal effect on the results (Figure 2.10). To understand the performance of the optimally loaded CL we note first that the cell current ˜j0 appears in the polarization curve (2.97) in the combination ε˜j0 = j0 /jσ , where jσ =

p

2i∗ σt b.

(2.100)

Above we have seen that the regime of uniformly loaded CL operation is determined by the relation between j0 and jσ . When j0  jσ , the activation polarization has the form η0 ∼ b ln(j0 ) (the Tafel law, the low-current regime), while at j0  jσ the polarization curve transforms to η0 ∼ 2b ln(j0 ) (doubling of the Tafel slope, the high-current regime). Typically, the cell works in the high-current or intermediate (j0 & jσ ) regime; the curves in Figure 2.10 are displayed for this range of current densities. Doubling of the Tafel slope dramatically increases voltage loss in the uniformly loaded CL. Thanks to the high catalyst concentration at the membrane interface, the optimal loading effectively switches CL function to the low-current mode with the normal Tafel kinetics. Indeed, with g(x) shown in Figure√2.9(b) the parameter jσ at the membrane interface increases by a factor of 13 ' 3.6. Almost four times higher jσ switches the regime of the high-loaded domain operation to the low-current mode (j0 /jσ < 1). Further, since the largest portion of proton current is converted in the high-loaded domain, the current entering the low-loaded domain appears to be much smaller than ˜j0 (Figure 2.9(b)). Therefore, in the low-loaded

64

CHAPTER 2. CATALYST LAYER PERFORMANCE

domain the ratio of incoming current density to jσ decreases and this domain also turns into a low-current (Tafel) mode. The small variation of overpotential in Figure 2.9(b) confirms this concept. The optimal shape in Figure 2.9(b) may falsely suggest removal of the low-loaded domain, since this domain converts only a minor part of the total proton current. This removal, however, will increase the polarization voltage. CL design in Figure 2.9(b) works well due to the fact that the boundary condition ˜j(1) = 0 is located far enough from the high-loaded domain. Attempts to remove the low-loaded domain will transfer this boundary condition closer to the membrane and the flat shape of overpotential η˜(˜ x) shown in Figure 2.9(b) will become much more nonuniform. Further numerical tests show that the cell polarization curve is rather insensitive to the exact shape of g(x): a small variation of g(x) leads to a small variation in η˜(˜j0 ). This means that in practical applications there is no need to reproduce the theoretical shape of g(x) with high accuracy.

2.7 DMFC anode One of the key problems in DMFC technology is sluggish kinetics of methanol oxidation. In spite of several decades of studies (Bagotzky and Vasilyev, 1967; Gasteiger et al., 1993; Kauranen et al., 1996), the reaction mechanism of methanol oxidation is still not fully understood. This mechanism includes the relatively slow potential-independent step of methanol adsorption on the catalyst surface (Gasteiger et al., 1993). In DMFC modelling, kinetics of methanol oxidation is often described by the Tafel of Butler-Volmer relations (see e.g. (Baxter et al., 1999; Kulikovsky, 2002b)). The presence of a potential-independent step in the chain of reaction events violates the validity of this approach at large polarizations. In some references (Kauranen et al., 1996; Nordlund and Lindbergh, 2004; Birgersson et al., 2004) performance of the anode catalyst layer (ACL) was studied numerically, taking into account non-Tafel effects. In this section we will construct a simple analytical model of DMFC anode performance taking into account the methanol adsorption step (Kulikovsky, 2005d).

2.7.1

The rate of methanol oxidation

The simplest way to take the adsorption step into account in the kinetics of methanol oxidation is as follows. Consider the two-stage reaction scheme: k

a CH3 OH + Site −→ CH3 OHads

CH3 OHads + H2 O

k∗ exp(η/b)

−→

CO2 + 6H+ + 6e− .

(2.101) (2.102)

The first step (2.101) describes the methanol adsorption on the catalyst surface; the second step (2.102) represents the electrochemical conversion.

2.7 DMFC ANODE

65

Parameters ka and k∗ exp(η/b) are the rate constants (cm3 mol−1 s−1 ) of adsorption and conversion, respectively. The fraction of occupied sites Θ on the catalyst surface are subject to the equation A

η dΘ = ka ct A(1 − Θ) − k∗ exp cw AΘ dt b

(2.103)

where A is the molar concentration of active catalyst particles, and ct and cw are the molar concentrations of methanol and water, respectively. Equation (2.103) implies that η is large enough so that the electrochemical conversion is shifted toward oxidation and the reverse reaction can be neglected. The first term on the right side of Eq. (2.103) is the rate of free sites population due to the adsorption step (2.101). The second term represents the rate of occupied sites depopulation due to the electrochemical conversion (2.102). In the steady state dΘ/dt = 0 and (2.103) yields Θ=

ka ct . ka ct + k∗ cw exp(η/b)

(2.104)

The rate of electrochemical conversion Q (A cm−3 ) is given by the second term on the right side of (2.103). Substituting (2.104) into this term and multiplying the result by 6F we find Q=

6F ka ct k∗ cw A exp(η/b) . ka ct + k∗ cw exp(η/b)

(2.105)

A similar equation was derived by Meyers and Newman (2002) from a much more detailed reaction scheme.

2.7.2

Basic equations and the conservation law

We will assume that the methanol concentration does not vary significantly across the ACL. Figure 2.11 illustrates the system of coordinates and the typical profile of the overpotential η(x) in the ACL. To emphasize that we are on the anode side, the axis x ˜ is now directed toward the membrane so that the polarization curve of the ACL is η1 (j1 ) (Figure 2.11). The system of equations for the ACL performance is ∂j = Q, ∂x ∂η j = σt , ∂x where Q is given by Eq. (2.105).

(2.106) (2.107)

66

CHAPTER 2. CATALYST LAYER PERFORMANCE

Figure 2.11: The system of coordinates and the sketch of overpotential in the anode catalyst layer.

In this section we will use slightly different scaling for the proton current: ˆj = j , j∗

ˆ = lt Q Q j∗

(2.108)

2σt b lt

(2.109)

where j∗ =

˜ = x/lt . is twice as large as in (2.6). As before, η˜ = η/b and x With (2.108), Eqs (2.106) and (2.107) take the forms ∂ ˆj ˆ =Q ∂x ˜ ∂ η˜ 2ˆj = ∂x ˜

(2.110) (2.111)

where ˆ= Q

exp(˜ η) . ω∗ + ωa exp(˜ η)

(2.112)

The system is thus controlled by two dimensionless parameters: ω∗ =

j∗ 6F lt k∗ cw A

(2.113)

is the inverse characteristic current of electrochemical conversion and ωa =

j∗ 6F lt ka ct A

(2.114)

2.7 DMFC ANODE

67

is the inverse characteristic current of methanol adsorption. In the following we will also use parameter ψ, the ratio of conversion to adsorption currents: ψ≡

ωa k∗ cw = . ω∗ k a ct

(2.115)

Multiplying together Eqs (2.110) and (2.111) we get ∂ ˆj ˆ ∂ η˜ =Q 2ˆj ∂x ˜ ∂x ˜ or ∂(ˆj 2 ) 1 ∂H(˜ η) = ∂x ˜ ωa ∂ x ˜

(2.116)

H(˜ η ) = ln (ω∗ + ωa exp(˜ η )) .

(2.117)

where

Integrating Eq. (2.116) we find a conservation law (Kulikovsky, 2003b) H(˜ η ) − ωa ˆj 2 = H(˜ η1 ) − ωa ˆj12 = H(˜ η0 ).

(2.118)

The subscripts “0” and “1” mark the values at the backing layer/ACL interface and at the membrane surface, respectively (Figure 2.11).

2.7.3

The general form of the polarization curve

Equation (2.118) allows us to obtain the polarization curve η˜1 (ˆj1 ). From (2.117) and (2.118) we get s ˆj1 =

1 ln ωa



 1 + ψ exp(˜ η1 ) . 1 + ψ exp(˜ η0 )

(2.119)

To exclude η˜0 we need an equation for η˜. Taking the square of Eq. (2.111) and multiplying both sides by ωa we get 4ωa ˆj 2 = ωa



∂ η˜ ∂x ˜

2 .

(2.120)

68

CHAPTER 2. CATALYST LAYER PERFORMANCE

From Eq. (2.118) we find ωa ˆj 2 = ln



1 + ψ exp(˜ η) 1 + ψ exp(˜ η0 )

 .

(2.121)

Using this in Eq. (2.120) we get the equation for η˜  ωa

∂ η˜ ∂x ˜

2

 = 4 ln

1 + ψ exp(˜ η) 1 + ψ exp(˜ η0 )

 .

(2.122)

Since ∂ η˜/∂ x ˜ > 0 we may take a square root of this equation, which yields √

s   1 + ψ exp(˜ η) ∂ η˜ ωa = 2 ln . ∂x ˜ 1 + ψ exp(˜ η0 )

(2.123)

Separating variables and integrating we find √

ωa 2

1−x ˜=

Z

η˜1

η˜

−1/2   1 + ψ exp(τ ) dτ. ln 1 + ψ exp(˜ η0 )

(2.124)

Setting x ˜ = 0 here we find the implicit relation of η˜0 and η˜1 √

ωa 2

Z

η˜1

η˜0

−1/2   1 + ψ exp(τ ) dτ = 1. ln 1 + ψ exp(˜ η0 )

(2.125)

This equation allows us to exclude η˜0 from (2.119), which then gives the desired relation ˆj1 (˜ η1 ).

2.7.4

Small variation of overpotential in the active layer

Let the variation of overpotential across the active layer be small: δ˜ ≡ η˜1 − η˜0  1 (conditions where this inequality is fulfilled will be derived below). This case is of particular interest since it enables us to obtain a closed-form solution to the problem. Substituting τ = η˜0 + δ˜ into (2.125), expanding logarithm over δ˜ and performing integration over 0 ≤ δ˜ ≤ η˜1 − η˜0 we get η˜1 = η˜0 +

exp(˜ η0 ) . ω∗ + ωa exp(˜ η0 )

(2.126)

2.7 DMFC ANODE

69

We see that δ˜ =

exp(˜ η0 ) . ω∗ + ωa exp(˜ η0 )

(2.127)

For η˜0 → ∞ we find 1 . δ˜lim = ωa

(2.128)

Substituting (2.126) into (2.119) we obtain v    u r u   ω∗ + ωa exp η˜0 + δ˜ u 1 1 ˜ ˆj1 = t   ' ln ln 1 + ωa δ˜2 ' δ, ωa ω∗ + ωa exp(˜ η0 ) ωa (2.129) where we used ωa δ˜2  1 (see below). We therefore get ˆj1 =

exp(˜ η0 ) . ω∗ + ωa exp(˜ η0 )

Since η˜1 − η˜0  1, we may write η˜1 ' η˜0 and hence with a good accuracy (2.130) is the polarization curve of the ACL: ˆj1 =

exp(˜ η1 ) . ω∗ + ωa exp(˜ η1 )

(2.130)

Equation (2.130) is evident: when η˜ is almost constant, total proton current j1 is simply a product of constant reaction rate Q and the ACL thickness lt . In the dimensionless form this results in Eq. (2.130). Solving (2.130) for η˜1 we find     η˜1 = ln ω∗ ˆj1 − ln 1 − ωa ˆj1 .

(2.131)

In terms of ψ (2.115), Eq. (2.131) is     η˜1 = ln ωa ˆj1 − ln ψ − ln 1 − ωa ˆj1 .

(2.132)

70

CHAPTER 2. CATALYST LAYER PERFORMANCE

The last logarithm in Eq. (2.132) shows that the limiting current density is4 ˆj1lim = 1 . ωa

(2.134)

Physically, the rate of methanol adsorption limits the cell current. Remarkably, the limiting term in Eq. (2.132) has the same logarithmic form as the diffusion-limiting term in Eq. (3.5) (Section 3.1). Polarization curves (2.132) for ωa = 10 are depicted in Figure 2.12a along with the exact numerical curves calculated from Eqs (2.119) and (2.125) (see the next section). When ψ . 10−2 , the curves (2.132) coincide with the numerical ones and exhibit regular behaviour: close to ˆj1 = 0 they have a Tafel-like shape, η˜1 ∼ ln(ˆj1 ) and close to the limiting current they show limiting behaviour, η˜1 ∼ ln(1 − ωa ˆj1 ). Equation (2.132) shows that variation of ψ shifts the polarization curve as a whole along the voltage axis (Figure 2.12). Physically, the rate constant k∗ is proportional to the exchange current density: k∗ ∼ i∗ ; therefore, at a fixed ωa , larger ψ corresponds to larger i∗ . The larger the i∗ , the less the overpotential required for current production; this effect is clearly seen in Figure 2.12. However, as ψ increases, the numerical polarization curves exhibit very rapid variation of ˆj1 with η˜1 in a region of small currents (dashed line in Figure 2.12(a)). This behaviour is not described by Eq. (2.132). With the growth of ψ this irregular region extends to larger currents (Figure 2.12(a)). For ψ & 1 the numerical polarization curves “jump” to a high current at very small overpotentials η˜1 (Figure 2.12(a)). Formally, at small η˜1 we have η˜0 = 0, so that Eq. (2.132) no longer approximates the polarization curve and hence this equation does not describe the jump. As ψ increases, the numerical curves in Figure 2.12(a) manifest a new mechanism of active layer operation. Our simplified model of methanol oxidation (Section 2.7.1) is valid when η˜ & 1. In the case of ωa = 10 shown in Figure 2.12(a), the new mechanism is seen at small η˜1 and thus it is beyond the scope of this model. However, at small ωa this mechanism appears at η˜1 & 1 and hence it can be analysed. 4 At η ˜ η0 ) is monotonic, for arbitrary ˜0 = 0, Eq. (2.127) gives δ˜ = 1/(ω∗ + ωa ). Since δ(˜ η˜0 we have

1 1 ≤ δ˜ ≤ . ω∗ + ωa ωa

(2.133)

The inequality δ˜  1 is thus guaranteed for any η0 when ωa  1. With (2.133) we see that for ωa  1 we have ωa δ˜2 ≤ 1/ωa  1, which justifies the expansion in terms of ωa δ˜2 in (2.129).

2.7 DMFC ANODE (a)

71 (b)

Figure 2.12: (a) Polarization curves of the anode catalyst layer for ωa = 10 and indicated values of ψ = ωa /ω∗ . Solid lines—Eq. (2.132), and dashed lines—numerical calculations for the general case. At η˜1 & 0.1, analytical and numerical curves are indistinguishable. Numerical curve for ψ = 10 is emphasized in red. (b) Numerical polarization curves of the catalyst layer for ωa = 0.1 and indicated values of ψ. Dashed line—analytical solution (2.141) for ψ  1. Local overpotential and proton current density in the points indicated by filled circles are shown in Figures 2.13 and 2.14.

2.7.5

Active layer of variable thickness

In the general case of arbitrary δ˜ (arbitrary ωa ), the procedure of calculation of polarization curve is as follows. For given η˜1 , Eq. (2.125) is solved for η˜0 . Then the pair (˜ η1 , η˜0 ) is substituted into (2.119) to calculate ˆj1 . The results of numerical calculations for ωa = 0.1 are shown in Figure 2.12(b). At very small ψ = 10−3 we have qualitatively the same result as in the previous section: Tafel-like behaviour, η˜1 ∼ ln(ˆj1 ), at small currents and a limiting shape, η˜1 ∼ ln(1 − ωa ˆj1 ), close to the limiting current, ˆj1 = 1/ωa (Figure 2.12(b)). However, as ψ increases, polarization curves become non-logarithmic at small currents (Figure 2.12(b)). At ψ = 10 the whole curve changes its shape (Figure 2.12(b)). The new mechanism provides much faster growth of current with the polarization voltage. To understand this effect it is advisable to plot the shapes of local parameters η˜(˜ x) and ˆj(˜ x) as we move along the polarization curve. Consider first these shapes in the several points on the “regular” curve for ψ = 10−3 (Figure 2.12(b)). In the regular case, local current is generated over the entire thickness of the catalyst layer (Figure 2.13(b)). At small currents, local overpotential is almost constant along x ˜; with the growth of current the curve η˜(˜ x) shifts upward and the difference η˜1 − η˜0 increases (Figure 2.13(a)). The shapes of η˜(˜ x) and ˆj(˜ x) on the limiting curve (Figure 2.12(b), the curve for ψ = 10) are quite different. As long as η˜1 is below a certain limiting value (˜ η1 < η˜1∗ ' 9.6) the electrochemical reaction covers only part

72

CHAPTER 2. CATALYST LAYER PERFORMANCE (a)

(b)

Figure 2.13: The distribution of local (a) overpotential and (b) current density across the catalyst layer at different points on the polarization curve for ψ = 10−3 (Figure 2.12(b)). Indicated are the values of the total voltage loss η˜1 . The membrane is at x ˜ = 0. (a)

(b)

Figure 2.14: The same as in Figure 2.13 for ψ = 10 and ωa = 0.1 (see also Figure 2.12(b)). Note the much larger current density ˆj1 for the same overpotentials η˜1 as in Figure 2.13. of the ACL (Figure 2.14(a)). There is a point x ˜0 , where local current and overpotential vanish: η˜(˜ x0 ) = ˆj(˜ x0 ) = 0 (Figure 2.14(b)). The growth of total current in this regime results from the increase in the thickness of the current-generating domain (by the shift of x ˜0 to the left, Figure 2.14) at a cost of minimal growth of overpotential η˜1 . In other words, as ˆj1 increases the electrochemical reaction occupies a larger portion of the active layer thickness, moving toward x ˜ = 0 (Figure 2.14). This is the regime with “variable thickness” (VT regime) of the ACL. When x ˜0 reaches zero, η˜1 = η˜1∗ and the ACL enters the regular regime. The whole thickness of the ACL is now covered by the reaction (Figure 2.14). Further growth of the total current is supported by the simultaneous increase in η˜0 and η˜1 , which is much more expensive in terms of voltage loss. For the case of ψ  1, the functions ˆj(˜ x) and η˜(˜ x) can be obtained analytically. From the conservation law (2.118) we get the relation of local

2.7 DMFC ANODE

73

ˆj and η˜: s ˆj =

1 ln ωa



 1 + ψ exp(˜ η) . 1 + ψ exp(˜ η0 )

(2.135)

Neglecting both units under the logarithm sign in (2.135) and putting η˜0 = 0 we find η˜ = ωa ˆj 2

(2.136)

or ˆj =

r

η˜ . ωa

(2.137)

Using (2.137) in (2.111) and solving the resulting equation with the boundary condition η˜(˜ x0 ) = 0 we get η˜ =

(˜ x−x ˜0 )2 . ωa

(2.138)

Equation (2.111) then yields ˜−x ˜0 ˆj = x . ωa

(2.139)

Therefore, in the VT-mode, local current density increases linearly with x ˜, and η˜ has a parabolic shape along x ˜. Numerical solutions in Figure 2.14 confirm this result. Substituting x ˜ = 1 into Eq. (2.139) we get the thickness δ˜j = 1 − x ˜0 of the current-generating domain as a function of total current: δ˜j = ωa ˆj1 .

(2.140)

Thus, δ˜j increases linearly with the cell current density. From Eq. (2.136) we immediately get the polarization curve in the VT regime η˜1 = ωa ˆj12 .

(2.141)

Therefore, instead of a characteristic for regular regime logarithmic dependence (the first term on the right side of Eq. (2.132)), we have a parabolic law (2.141). This is seen in Figure 2.12(b): below η˜1∗ ' 9.6 the polarization curve for ψ = 10 is close to a parabola. Furthermore, for ψ  1 the polarization curve does not depend on ψ. The curve for ψ = 10 is close to the limiting curve (the dashed line in

74

CHAPTER 2. CATALYST LAYER PERFORMANCE

Figure 2.12(b)). The latter curve is a parabola (Eq. (2.141)) in the range 0 ≤ η˜ < 1/ωa and it becomes a vertical line ˆj1 = 1/ωa at η˜ = 1/ωa (cf. Eq. (2.134)). With a growing demand for current, the reaction domain expands toward x ˜ = 0. This expansion is accompanied by a slow rise in voltage loss η˜1 = ωa ˆj12 . As soon as the reaction covers the whole thickness of the active layer, the parabolic dependence (2.141) disappears and further growth of the overpotential with ˆj1 becomes much more rapid. The VT solutions arise when the exchange current density i∗ ∼ k∗ is large and/or the rate of methanol adsorption ka ct is small. In this case, the first term in the denominator of Eq. (2.105) is small and the rate of methanol oxidation Q ' 6F ka ct A

(2.142)

does not depend on the overpotential. Physically, Eq. (2.142) is a rate of methanol adsorption on the catalyst surface. This is the limit of fast electrochemical conversion: any methanol molecule which manages to sit on the catalyst surface is immediately oxidized. In this regime, methanol oxidation behaves like a “chemical” rather than an electrochemical reaction. However, since the overall reaction generates protons, it is activated close to the membrane, where the transport of protons is “cheaper”. It is advisable to write Eq. (2.140) in the dimensional form: δj =

j1 , 6F ka ct A

(2.143)

where δj ≡ δ˜j lt . In the opposite limit of Tafel kinetics, when the rate of methanol adsorption 6F ka ct A is large, δj tends to zero and the VT regime disappears. Using (2.142), Eq. (2.143) can be written as j1 ' δj Q. Physically, reaction rate Q in the current-generating domain is constant and hence the total current produced is simply the product of Q and the domain thickness δj . Voltage loss in the VT regime is much smaller than in the regular regime (Figure 2.12). The reason is that in the VT regime protons should be transported only through a narrow current-generating domain, whereas in the regular regime the reaction covers the whole thickness of the catalyst layer and expenditure for proton transport is much larger.

2.8 Heat balance in the catalyst layer In this section, we will study heat production and transport in the catalyst layer. The mechanisms of heat transport in fuel cells were briefly discussed

2.8 HEAT BALANCE IN THE CATALYST LAYER

75

in the Introduction. Here we derive the expressions for temperature shape across the CL and for the heat flux emitted by the CL (Kulikovsky, 2006d).

2.8.1

Heat transport equation in the CL

The velocity of gas flow in the CL is very small; this allows us to neglect the convective term in the heat transport equation. The heat balance in the CL is, therefore, given by −

∂ ∂x

 λeff

∂T ∂x

 = RS + Rη + RJ .

Here λeff is the CL thermal conductivity (W m−1 K−1 ) and the source terms RS , Rη and RJ describe the rate of heating due to entropy change in the electrochemical reaction, irreversible heating due to proton transport through the double layer at the metal/electrolyte interface and Joule heating, respectively. Taking into account the expressions for these rates (1.77), (1.80) and (1.82) we get ∂ − ∂x

 λeff

∂T ∂x



 =

 T ∆S j2 j2 +η Q+ e + . 4F σe σt

(2.144)

This is the general equation for heat transport in the cathode catalyst layer of a hydrogen fuel cell. Equation (2.144) does not take into account the heat due to phase transformation of water in the CL. This process can, in principle, be accounted for as described in Nguyen and White (1993) and Birgersson et al. (2005). Note, however, that evaporation/condensation processes include mass transfer between the liquid and vapour phases (Natarajan and Nguyen, 2001), which complicates the problem. The derivation of Eq. (2.144) employs only the general laws of heat and charge conservation. The only detail in (2.144) specific to the ORR is the stoichiometry coefficient 4F . Clearly, Eq. (2.144) can be generalized to describe the heat balance in any CL as follows: ∂ − ∂x

 λeff

∂T ∂x



 =

 T ∆S j2 j2 +η Q+ e + . nF σe σt

(2.145)

Here Q is the local rate of the electrochemical reaction, ∆S is the respective entropy change and n is the number of electrons transferred in the reaction (e.g. n = 4 for ORR and n = 6 for methanol oxidation).

76

2.8.2

CHAPTER 2. CATALYST LAYER PERFORMANCE

Reduction to the boundary condition

In the next section, we will see that the temperature variation across the CCL is small. For that reason, numerical models of cells and stacks usually ignore the details of heat transport in CLs and assume that these layers are infinitely thin interfaces generating heat (Berning et al., 2002; Senn and Poulikakos, 2005; Liu et al., 2005; Freunberger et al., 2006). This approach requires the expression for the heat flux from the CL. In this section, we show how the heat balance equation in the CL can be reduced to the boundary condition for the “external” problem of heat transport in the other fuel cell components. This procedure gives the exact expressions and makes it possible to establish their limits of validity. The exact boundary condition for external problems can be obtained from Eq. (2.145) using the following assumptions. Suppose that the feed molecule concentration does not vary significantly across the CL, the cell operates in the low-current regime and the reaction penetration depth is large (these assumptions are discussed in detail in Sections 2.1–2.4). In that case, the electron and proton current densities vary linearly with the distance across the CL:   x j = j0 1 − (2.146) lt x je = j0 . (2.147) lt Furthermore, in that case, η does not vary significantly across the CL and we may put in (2.145) η ' const. Temperature variation across the CL is small (see the next section); thus on the right side of Eq. (2.145) we put T ' const. Integrating this equation over the CL thickness we get     ∂T ∂T T ∆S j02 lt 1 1 −λeff + η j0 + + (2.148) + λeff = ∂x lt ∂x 0 nF 3 σe σt where (1.78), (2.146) and (2.147) are taken into account. The electron conductivity of CL, σe , is several orders of magnitude larger than the ionic (proton) conductivity σt . Thus 1/σe on the right side of (2.148) can be neglected. Physically, Joule heat in the active layer is generated mainly in the electrolyte phase. The left-hand side of (2.148) is the sum of the two one-sided heat fluxes leaving the CL in opposite directions: the first one is directed to the GDL and the other to the membrane (Figure 2.1). Clearly, the total flux is a sum of one-sided fluxes:   ∂T T ∆S j 2 lt −λeff ≡ qtot = + η j0 + 0 . (2.149) ∂x tot nF 3σt

2.8 HEAT BALANCE IN THE CATALYST LAYER

77

Equation (2.149) is the desired boundary condition for external problems. The sign “minus” on the left-hand side means that the temperature drops along x and the heat flux is directed toward the GDL (Figure 2.1); this sign should be chosen in accordance with the external problem formulation. The first term on the right side of (2.149) represents the heat flux due to the electrochemical reaction. The second term is the heat flux due to Joule heating in the proton-conducting phase. Generally, the terms in (2.149) are of the same order of magnitude and none can be neglected. The coefficient 1/3 in the second term on the right side of (2.149) arises due to the linear dependence j(x) (2.146). If the cell operates in a highcurrent regime or ε  1 (for ε see Eq. (2.13)), the function j(x) strongly deviates from linear law. The respective coefficient in Eq. (2.149) can then be calculated using the relations of Section 2.3. Note also that if j(x) is nonlinear, the overpotential is no longer constant across the CL and the integration of Eq. (2.145) with η(x) from Section 2.3 leads to a more complicated expression. Physically, in a CL with, for example, poor proton conductivity a peak of the reaction rate at the membrane interface may lead to larger overheat, especially when the CL thermal conductivity is not large. The Joule term in (2.149) is proportional to the square of cell current density. The distribution of local current over the cell surface is usually very nonuniform and we may expect the effects due to temperature variation along the cell surface. Furthermore, in stacks with poor heat management these thermal nonuniformities may be further enhanced. Though temperature variation across the CL in the stack is still small, the absolute temperatures at different points of the cell surface may differ quite strongly. The respective temperature fields and effects can be studied with the models of a higher dimensionality (Chapter 5). These models usually utilize the boundary condition (2.149), where T , η and j0 are considered as local values.

2.8.3

Solution to the heat transport equation

In this section we return to the general equation (2.145). Above we have shown that heat in the CL is generated mainly in the electrolyte phase. Therefore, the term je2 /σe in (2.145) can be neglected. The linear shape of proton current density in the CL (2.146) is equivalent to the constancy of the ORR rate Q across the layer (Section 2.2). Thus Eq. (1.78) reduces to j0 = Qlt . Taking into account this relation and Eq. (2.146) we may rewrite Eq. (2.145) as −

∂ ∂x

     2 ∂T T ∆S j0 j2 x λeff = +η + 0 1− . ∂x nF lt σt lt

(2.150)

78

CHAPTER 2. CATALYST LAYER PERFORMANCE Assuming that λeff is constant and introducing dimensionless variables x ˜=

x , lt

ˆj0 = j0 j∗

(2.151)

where j∗ is given by Eq. (2.109), Eq. (2.150) transforms to ∂2T − 2 = ∂x ˜



nF η T+ ∆S



  22  j∗ lt ∆S ˆ j∗ lt 2 ˆj 2 (1 − x j0 + ˜) . (2.152) nF λeff λeff σt 0

The left-hand side of (2.152) has the dimension of temperature (K); therefore, the expression in square brackets on the right side is the dimensionless parameter: α=

j∗ lt ∆S 2σt b∆S = nF λeff nF λeff

(2.153)

where Eq. (2.109) is used. Introducing the dimensionless temperature and overvoltage T T˜ = , T∗

η˜ =

η b

(2.154)

where T∗ =

nF b , ∆S

Eq. (2.152) transforms to −

  ∂ 2 T˜ 2 ˆj0 T˜ + η˜ + 2αˆj02 (1 − x = α ˜) . ∂x ˜2

(2.155)

Parameter α is small, which means that heating due to the sources on the right side of this equation is small. In other words, heat conductance effectively homogenizes T (x), and temperature variation across the CL is small. Therefore, on the right side of Eq. (2.155) we can safely replace T˜ by the temperature at x ˜ = 1, T˜1 : −

  ∂ 2 T˜ 2 ˆj0 T˜1 + η˜ + 2αˆj02 (1 − x = α ˜) . ∂x ˜2

(2.156)

To illustrate the solutions to Eq. (2.156), we will assume that heat flux to the membrane is negligible, i.e. ∂ T˜/∂ x ˜|x˜=0 = 0; heat flux at x ˜ = 1 (GDL interface) is then the total heat flux from the CL. At x ˜ = 1 we fix the temperature: T˜(1) = T˜1 . Note that the actual value of T˜1 should be determined from the solution of an external problem.

2.9 REMARKS ON CHAPTER 2

79

Figure 2.15: Temperature profile across the catalyst layer of a low-T fuel cell. Solid line: the solution to Eq. (2.156), and dashed line: the exact shape (solution to Eq. (2.155)). Both curves are practically indistinguishable. The solution to (2.156) subject to these boundary conditions is 
αˆj0  ˜ αˆj 2 T˜(˜ x) = T˜1 + T1 + η˜ (1 − x ˜2 ) + 0 (1 − x) 4 − (1 − x)3 . (2.157) 2 6 The plot of (2.157) with the parameters from Table 2.1 is shown in Figure 2.155 . For comparison the exact solution to Eq. (2.155) is shown (dashed line; both curves are practically indistinguishable). As seen, temperature variation across the CL is less than 0.1 K and it can safely be ignored. However, the temperature gradient at x ˜ = 1 is not small. This gradient (given by Eq. (2.149)) determines the heat flux from the CL to the GDL.

2.9 Remarks on Chapter 2 In this chapter the scope of our discussion was restricted by the macrohomogeneous model of CL performance and its derivatives. The first numerical macrohomogeneous models of CCL for a PEM fuel cell were developed by Springer and Gottesfeld (1991) and by Bernardi and Verbrugge (1991). These models included the diffusion equation for oxygen transport, the Tafel law for the rate of ORR and Ohm’s law for the proton transport in the electrolyte phase. A similar approach was then used by Perry, Newman and Cairns (Perry et al., 1998) and by Eikerling and Kornyshev (1998) for combined numerical and analytical studies. 5 In low-T cells, CCL generates water. We may expect that in typical situations the thermal conductivity of the CCL does not differ much from that of water and for the estimate we put λeff ' λw . In a dry catalyst layer, due to the lower thermal conductivity the temperature variation can be larger.

80

CHAPTER 2. CATALYST LAYER PERFORMANCE

Table 2.1: Parameters for the problem of temperature distribution in the cathode catalyst layer of a PEM fuel cell. Proton conductivity of the electrolyte phase, σt (S m−1 ) Tafel slope b (V) (Xie et al., 2005) Entropy change in the ORR, ∆S (J mol−1 K−1 ) (Lampinen and Fomino, 1993) Thermal conductivity of water, λw (W m−1 K−1 ) (Lide, 1998) Current density j0 (A m−2 ) Overpotential η (V) Catalyst layer thickness lt (m) Temperature at x = lt (K) T∗ (K) j∗ (A m−2 ) α ˆj0 η˜

1 0.05 326.36 0.58 104 0.5 10−5 350 59.14 104 1.458 · 10−4 1 10

Perry, Newman and Cairns (Perry et al., 1998) obtained a numerical solution to the problem of CCL performance and provided the asymptotic analytical solutions for large and small cell current densities j0 . Eikerling and Kornyshev (Eikerling and Kornyshev, 1998) derived the explicit analytical solution to the system for the case of small overpotentials; in the general case they reported numerical results. In low-temperature fuel cells, CL is usually filled with 10-20 nm carbon black particles (carbon support) decorated with 3-5 nm particles of catalyst (Pt or Pt/Ru). Certain techniques of preparation lead to the formation of large-scale (100-1000 nm) clusters of carbon particles. These clusters (agglomerates) may be fully covered by Nafion film. In this situation an “elementary conversion unit” of the CL is a cluster, which itself may exhibit significant resistance to feed molecule transport. To describe the function of these CLs in PEM fuel cells, in many works a flooded agglomerate model (FAM) is utilized (Baschuk and Li, 2000; Jaouen et al., 2002; Schwarz and Djilali, 2007). In recent years, FAM has been used as a sub-model in the CFD modelling of fuel cells (Schwarz and Djilali, 2007; Kamarajugadda and Mazumder, 2008). In the FA model, an elementary conversion unit is a spherical cluster of carbon support immersed in Nafion. The cluster may or may not be covered by a thin electrolyte film. In any case, to reach the catalyst surface an oxygen molecule must dissolve and diffuse in Nafion. Since oxygen

2.9 REMARKS ON CHAPTER 2

81

Figure 2.16: Scanning electron microscope picture of a catalyst layer for HT-PEFC (courtesy of Dr. C. Wannek, Forschungszentrum J¨ ulich). The typical agglomerate diameter is about 100 nm (note the scale bar). diffusivity in Nafion is low, a large cluster has quite significant transport resistance. Within the scope of this approach, the CL problem transforms to (i) an auxiliary problem of single agglomerate performance and (ii) the problem of CCL performance with the conversion function derived from (i) (Karan, 2007; Harvey et al., 2008). Clearly, in the limit of small transport losses in the agglomerate, its conversion function reduces to the Tafel law (Pisani et al., 2003). Should FAM always be preferred to a macrohomogeneous model? One of the key factors in answering this question is agglomerate size. Agglomerate radius is sometimes assumed to be between 0.5 µm (Jaouen et al., 2002) and 10 µm (Baschuk and Li, 2000). However, this radius depends on the preparation technique. A SEM picture of a catalyst layer produced at Forschungszentrum J¨ ulich shows that agglomerate radius is about 100 nm (Figure 2.16). In this situation, transport loss inside the agglomerate is marginal and the agglomerate conversion function reduces to that function of Pt species (Tafel or Butler-Volmer law). Another factor seemingly underestimated in FAM is gaseous oxygen diffusivity. In large (1-10 µm) agglomerates, small void pores with a radius in the order of 0.1-1 µm may exist. Gaseous diffusion in these pores may dramatically improve the transport properties of the agglomerate, thereby reducing its conversion function to the Tafel law. Both macrohomogeneous and flooded agglomerate models predict doubling of the Tafel slope in the proton- and oxygen-limiting regimes. Thus, polarization curves cannot be used to distinguish between these two models. The decisive experiment would be to measure the distribution of

82

CHAPTER 2. CATALYST LAYER PERFORMANCE

the reaction rate across the catalyst layer. In the regimes with doubling of the Tafel slope, FAM predicts a uniform reaction rate across the CCL, while MHM predicts a peak of this rate at the interface where the limiting species are in excess. Recently, a new class of stochastic CL models has been developed (Mukherjee and Wang, 2006). These models simulate species transport in a small 3D domain of the catalyst layer. The domain is subdivided into elementary computational cells representing either a void space or an electrolyte/carbon phase. The structure of this domain is obtained by the stochastic reconstruction of micro-images of real catalyst layers. 3D macroscopic transport equations are then solved in this domain with the transport coefficients, which take into account the nature of every computational cell (direct numerical simulation (DNS) model). Physically, this approach enables to calculate transport parameters of a porous media, for example, porosity or the Bruggemann exponent (Mukherjee and Wang, 2006; Wang et al., 2006a). An extension of the macrohomogeneous model, which takes into account the dependence of transport parameters on CCL composition, has recently been developed by Eikerling (2006). The model includes the dependence of species diffusivity on pore size distribution and incorporates a model of water management. So far, however, some challenging questions about the CL structurefunction relations remain unanswered. The macrohomogeneous approach utilizes effective transport parameters of CL obtained experimentally. Generally, this limitation needs to be relaxed. Ideally, a CL model should generate all the necessary transport data self-consistently. The DNS model is the step toward this goal. This ambitious task raises complicated nano-scale problems of CL structure-composition-function relations (Promislow and Wetton, 2009). Nonetheless, thanks to their relative simplicity, in the foreseeable future, variants of macrohomogeneous models will play an important role in cell and stack modelling.

Chapter 3

One-dimensional model of a fuel cell A typical membrane-electrode assembly consists of two catalyst layers, two gas-diffusion layers and a membrane. So far we have considered voltage loss in the catalyst layers only. In this chapter, we will include the losses in the GDL and membrane1 . Transport of reagents through the GDL to the catalyst layer and transport of protons through the membrane “cost” some potential. The solution of a full through-plane problem yields the distribution of reactant concentrations and the expressions for the respective voltage losses. These expressions enable us to construct a one-dimensional (1D) polarization curve of a fuel cell. A 1D curve ignores losses due to reactant transport in the feed channels. If these losses are negligible, the equations derived in this chapter form a model for cell performance. A necessary (though not sufficient) condition for small voltage loss in the flow field is the large stoichiometry of feed gases. In the present context, we are no longer interested in the details of proton current and overpotential distribution across the catalyst layer; all required information about the layer is contained in the dependence η˜0 (˜j0 ). In other words, with the polarization curve of a catalyst layer in hand we can consider this layer as a thin interface with the prescribed voltage-current 1 On the cathode side of low-T cells, there is usually a microporous layer (MPL) between the CL and GDL. The role of this layer is to prevent the leakage of water produced in the ORR in order to keep the membrane well humidified. In this book we will not consider MEAs with MPLs; however, transport loss in the MPL can be calculated using the relations from this chapter.

83

84

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

Figure 3.1: Schematic of the oxygen concentration profile across the cathode catalyst and gas-diffusion layers. In this chapter, the oxygen concentration in the catalyst layer is assumed to be constant: c ≡ ct . characteristics. Thus, there is no need to retain the subscript 0 in the symbols ˜j0 and η˜0 , and this subscript will be omitted in the following. We begin with the calculation of voltage loss due to oxygen transport through the GDL. This loss is of particular interest for low-temperature fuel cells, in which it usually limits the cell current. In addition, the model below provides a method for calculating transport losses in other situations, e.g. on the anode side of DMFC or SOFC.

3.1 Voltage loss due to oxygen transport in the GDL Consider the cathode side of a hydrogen cell (Figure 3.1). Let oxygen concentration in the channel ch be fixed. In this section, we neglect oxygen transport losses in the CCL, i.e. the oxygen concentration in the CCL ct is assumed to be constant along x (Figure 3.1). This approximation holds for modern highly porous catalyst layers in PEFC. Evidently, the slope of the oxygen concentration profile across the GDL depends on the current density: the larger the j the larger the slope (Figure 3.1). At a certain limiting current density jD , ct appears to be zero and the further growth of current is not possible. We will describe oxygen flux through the GDL by Fick’s equation (1.52) with the effective diffusion coefficient Db . As discussed in the Introduction, in a dry GDL, Db corresponds to the oxygen binary diffusion coefficient corrected for GDL porosity. For a partially flooded GDL, Db should be corrected for liquid saturation, as discussed below. A more rigorous approach is based on Stefan-Maxwell relations, which take into account the effect of nitrogen and water vapour fluxes on oxygen transport (Section 1.4.6). However, oxygen constitutes a small fraction of air (21%) and hence Fick’s formula provides quite reasonable accuracy.

3.1 TRANSPORT LOSS IN THE GDL

85

Importantly, in contrast to the cumbersome Stefan-Maxwell relations, Fick’s simple equation leads to a transparent physical picture of the process, which is our primary goal. Mass conservation prescribes that the diffusion flux of oxygen in the GDL must be proportional to the local cell current density j. We therefore have Db

j ∂c = , ∂x 4F

(3.1)

where c is the local oxygen molar concentration in the GDL. Solving this equation with the boundary condition c|x=lt +lb = ch (Figure 3.1) and substituting x = lt into the solution, we get ct :   lb j j ct = ch − = ch 1 − . (3.2) 4F Db jD Here lb is the GDL thickness and jD =

4F Db ch lb

(3.3)

is the limiting current density: when j = jD we get ct = 0 and the electrochemical conversion in the CCL stops. The value of ct appears as c in the expression for the rate of the ORR (2.1). The polarization curve of the cathode side can now be obtained simply by replacing c ≡ ct in the equations of Chapter 2 with the expression (3.2). Consider first the low-current polarization curve for Tafel kinetics (2.29). In this equation we should replace      ct ch j → 1− . (3.4) cref cref jD Making this substitution, we obtain the low-current polarization curve of the cathode side, which takes into account oxygen transport in the GDL:     j j η = b ln h − b ln 1 − (3.5) j∗ jD where j∗h

 = i∗ lt

ch cref

 .

(3.6)

Substituting (3.4) in a high-current polarization curve, Eq. (2.59), we get  η = 2b ln

j jσh



  j − b ln 1 − jD

(3.7)

86

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

Figure 3.2: The high-current polarization curve of the cathode side of a PEM fuel cell, Eq. (3.7). Note the limiting current density at j = jD . Table 3.1: Parameters for the curve in Figure 3.2. b (V) i∗ (A cm−3 ) σt (Ω−1 cm−1 ) lt (cm) jD (A cm−2 ) jσh (A cm−2 ) 0.05 10−3 10−2 10−3 1 10−3 where s jσh

=

 2i∗ σt b

ch cref

 .

(3.8)

Note that in Eqs (3.5) and (3.7) the terms describing transport loss in the GDL appear to be the same. This is not necessarily the case for other regimes: e.g. substitution of (3.4) in the general low-current polarization law (2.44) leads to a more complicated relation for transport loss. The function (3.7) is shown in Figure 3.2. The parameters for this plot are listed in Table 3.1. As can be seen, the characteristic current density jσh is small (the last column in Table 3.1) and hence this set of parameters prescribes the high-current regime of cell operation.

3.2 1D polarization curve of a cell The expressions in the previous section allow us to construct the polarization curve of a fuel cell with negligible losses on the anode side (PEFC, HT-PEFC). Generally, this curve is given by Vcell = Voc − η − Rj.

(3.9)

In this equation, Vcell is the cell voltage, Voc is the open-circuit voltage, η is the voltage loss on the cathode side and R is the cell resistivity, which comprises membrane and contact resistivities.

3.3 1D MODEL OF DMFC

87

Ideally, the cell should operate in the Tafel regime, when η is given by (3.5). With this relation Eq. (3.9) takes the form  Vcell = Voc − b ln

j j∗h



  j + b ln 1 − − Rj. jD

(3.10)

If the cell operates in a high-current mode, η is given by (3.7) and we obtain  Vcell = Voc − 2b ln

j jσh



  j + b ln 1 − − Rj. jD

(3.11)

Thus, the total voltage loss is the sum of the activation, transport and resistive components (the second, third and fourth terms on the right side of Eqs (3.10) and (3.11)). Equation (3.10) or (3.11) can be used for fitting the experimental polarization curves, provided that the effects due to oxygen exhaustion in the flow field are small2 . Note that the transport voltage loss is sometimes called concentration overpotential.

3.3 1D model of DMFC The binary diffusion coefficient of hydrogen is roughly four times higher than this coefficient for oxygen. Thanks to high H2 diffusivity, the transport losses on the anode side of hydrogen cells typically are negligibly small. The DMFC is fed with the liquid methanol-water solution. The diffusivity of methanol in water is low and the respective voltage loss cannot be neglected. Another source of voltage loss specific to DMFC is methanol crossover through the membrane. Polymer electrolyte membranes offer a high 2 It is sometimes argued that since in a working fuel cell, oxygen concentration in the catalyst layer is reduced, the cell open-circuit voltage should be corrected for this lowering. The correction is calculated by substituting oxygen concentration (3.2) in the Nernst equation (1.11). This leads to an apparent lowering of OCV by a value

ηconc =

  RT 1 ln 1 − . nF jD

(3.12)

This reasoning is incorrect. The Nernst equation describes the dependence of OCV on the oxygen concentration only at equilibrium. This equation is not applicable in the situation when the cell generates current. Thus, Eq. (3.12) does not represent any real voltage loss in a cell. A true voltage loss in non-equilibrium conditions can be calculated from the kineticequations;  as discussed above, this leads to the transport loss in the form j ηconc = RT ln 1 − . Note that in Eq. (3.12) the factor is RT /(nF ), while in the αF j D

transport term in Eqs (3.10) and (3.11) the factor is b = RT /(αF ). Depending upon the half-cell reaction, n = 2-6, whereas α ' 0.5-1; thus, Eq. (3.12) strongly underestimates ηconc .

88

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

resistance to crossover of gases. However, the mechanism of proton transport in these membranes is inherently related to water molecules and PEM membranes are highly permeable to water. Methanol and water molecules are similar and hence methanol easily permeates these membranes. On the cathode side, permeated methanol reacts directly with oxygen; the presence of catalyst particles facilitates the direct combustion. This parasitic reaction consumes methanol and lowers the amount of oxygen available for useful electrochemical conversion (Gottesfeld, 2007). Thus, any realistic model of DMFC should include crossover. Assuming that the rate-determining step of the reaction (1.84) is a singleelectron transfer and neglecting the effect of the finite rate of methanol adsorption onto the catalyst surface (Section 2.7), we may apply the Tafel formula for the reaction rate. The system of equations for the DMFC anode then differs from Eqs (2.1)–(2.3) only by the stoichiometry factor in Eq. (2.3), where 4F should be replaced by 6F . With this in mind, in the case of zero crossover we can use the relations for voltage loss η obtained in Chapter 2 for the anodic voltage loss. The value ct in these relations should be treated as the methanol concentration in the ACL. However, oxygen and methanol fluxes in the DMFC are affected by crossover. Thus, to construct a full 1D model of this cell we should write the balance of methanol and oxygen fluxes taking crossover into account. From this balance we will deduce the methanol (cat ) and oxygen (cct ) concentrations in the respective catalyst layer. Making a substitution similar to (3.4), we will obtain the half-cell polarization voltages and then construct the polarization curve of the whole cell.

3.3.1

Feed molecule concentration in the active layers

Figure 3.3 sketches the profiles of the methanol and oxygen concentrations across the cell. To simplify the analysis, cat and cct are assumed to be constant across the respective active layer (Figure 3.3). However, cat and cct depend on current density and on the rate of methanol permeation through the membrane. Note that the superscripts a and c denote the anodic and cathodic values respectively.

Methanol The diffusion flux of methanol in the anode backing layer should cover the expenditure for current production and crossover: Dba

∂ca j =− − Ncross ∂x 6F

(3.13)

3.3 1D MODEL OF DMFC

89

Figure 3.3: Sketch of methanol and oxygen concentrations in DMFC. BL stands for backing layer, and CL denotes catalyst layer. where Dba is the methanol diffusion coefficient in the anode backing layer, ca is the methanol molar concentration, and Ncross is the molar flux of methanol through the membrane (crossover flux). To calculate Ncross we assume that (i) methanol is transported through the membrane due to diffusion and electro-osmosis and (ii) the diffusion coefficient of methanol in the membrane Dm is constant (Ren et al., 2000a). The latter assumption is justified since in DMFC the membrane is fully hydrated. Under these assumptions Ncross = −Dm

∂ca + nd ∂x



cat caw



j F

(3.14)

where nd is the drag coefficient (the number of molecules of methanol-water mixture transported by one proton), and caw is the molar concentration of water. Here we also assume that the proton does not distinguish between methanol and water molecules; the drag term in (3.14) is then proportional to the methanol molar fraction in the ACL3 . Neglecting the methanol concentration in the CCL we obtain Ncross '

Dm cat j + nd lm F



cat caw

 (3.15)

where lm is the membrane thickness. Substituting (3.15) into (3.13), solving the resulting equation with the boundary condition ca |x=0 = cah and substituting x = lba (Figure 3.3) into the solution we get the molar 3 Note that the drag term in Eq. (3.14) is overestimated. A more accurate expression m is the local methanol concentration in the for this term is nd (cm /ca w )j/F , where c membrane. However, below we will show that this term is negligible.

90

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

concentration of methanol in the ACL as a function of current density:   −1 j j 1 + β + nd cat = cah 1 − a , jD jw

(3.16)

where cah is the methanol concentration in the anode channel, Dm lba Dba lm

(3.17)

jw =

F Dba caw lb

(3.18)

a jD =

6F Dba cah lba

(3.19)

β= is the crossover parameter,

and

a , the methanol is the methanol-limiting current density: if j = jD concentration in the ACL is zero. With (3.16) the crossover flux (3.15) can be written as

Ncross

ja = D 6F

β + nd jjw 1 + β + nd jjw

!

j 1− a jD

 .

(3.20)

The drag terms nd j/jw in (3.20) are small as compared to β. To show this we note that typically lm ' lba and Dm ' Dba ; this gives β ' 1. Further, with Dba = 2×10−5 cm2 s−1 , caw ' 2×10−2 mol cm−3 and lba = 1×10−2 cm we get jw ' 5 A cm−2 . Since nd ' 2 and the current density in DMFC usually does not exceed 0.2 A cm−2 , we find that nd j/jw  β and hence the terms with nd in (3.20) may be omitted. Physically, this means that unless the cell current density is large, the dominant mechanism of methanol transport through the membrane is diffusion. Equation (3.20) thus simplifies to Ncross

a β∗ j D = 6F

  j 1− a , jD

(3.21)

where β∗ =

β . 1+β

(3.22)

3.3 1D MODEL OF DMFC

91

It is convenient to introduce the equivalent crossover current density jcross   j a jcross ≡ 6F Ncross = β∗ jD 1 − a . (3.23) jD a Clearly, at j = jD crossover is zero, since there is no methanol in the ACL4 . Equation (3.23) shows that jcross linearly decreases with the current a at j = 0 to zero at density from the maximal value jcross (0) = β∗ jD a j = jD . This behaviour correlates well with the experimental data (Ren et al., 2000b; Jiang and Chu, 2004; Gogel et al., 2004).

Oxygen Following Vielstich et al. (2001), we assume that methanol permeating through the membrane is consumed on the cathode side in a direct catalytic combustion: 2CH3 OH + 3O2 = 2CO2 + 4H2 O. The diffusion flux of oxygen in the cathode backing layer is thus the sum of the fluxes required to convert the proton current j and to burn permeated methanol: Dbc

j 3 ∂cc = + Ncross . ∂x 4F 2

(3.24)

Substituting (3.21) into (3.24), solving the resulting equation with the boundary condition c|x=x2 = cch , substituting x = x1 into the solution and taking into account that x2 − x1 = lbc (Figure 3.3), we get the oxygen concentration in the CCL in the presence of methanol crossover:   j + jcross cct = cch 1 − , (3.25) c jD where cch is the oxygen concentration in the cathode channel and c jD =

4F Dbc cch lbc

(3.26)

is the oxygen-limiting current density. Comparing (3.25) to Eq. (3.2) we see that in (3.25) instead of j we have a sum j + jcross . This means that crossover increases the oxygen transport loss in the cell. Physically, in Eq. (3.25) useful and crossover current densities add up since both these currents consume oxygen. In particular, the oxygen concentration in the c CCL approaches zero when j + jcross = jD . 4 To avoid confusion it should be noted that j cross is simply the methanol flux expressed in units of electric current; physically this is not a current since no charge is transported with this flux.

92

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

3.3.2

1D polarization curve of DMFC

Suppose that both sides of the DMFC operate in a low-current regime. The half-cell polarization voltages are then determined by Eq. (2.29), where on the anode side ct = cat is given by (3.16) and on the cathode side we should set ct = cct (Eq. (3.25)). Making these substitutions in Eq. (2.29), we get the half-cell polarization voltages     ηa j j = ln a − ln 1 − a + ln (1 + β) (3.27) ba j∗ jD     j ηc j + jcross = ln c − ln 1 − (3.28) c bc j∗ jD where j∗a j∗c

=

cah caref

!

ia∗ lta

=

cch ccref

!

ic∗ ltc

(3.29)

.

(3.30)

If one of the sides operates in the high-current mode, the first (activation) logarithm in the respective equation (3.27) or (3.28) should be replaced by 2 ln(j/jσh ), where jσh is given by (3.8). On the anode side, the crossover simply shifts η a by a constant value ln(1 + β) (Eq. (3.27)). Physically, on the anode side, crossover diminishes the amount of methanol in the ACL at zero current and this term describes the anodic contribution to lowering of the cell OCV. On the cathode side, the effect of crossover is more subtle. The crossover current jcross increases the sum j + jcross in the transport logarithm in Eq. (3.28). Therefore, due to jcross the cathode operates closer to the limiting current. Below we will consider several consequences of this effect. Equating the argument of the last logarithm in (3.28) to zero and taking into account (3.23), after simple manipulations we find that the limiting c a c current density of the cathode side is jD − β (jD − jD ). The overall limiting current density of the cell is thus a c a c jD = min {jD , jD − β (jD − jD )} .

(3.31)

a c From (3.31) it follows that the maximal jD is attained when jD = jD . Note a c that when jD = jD the value of jD does not depend on β, i.e. the crossover does not affect cell operation. In Section 4.7.4, we will see that this “crosslinked” regime can be organized under more general conditions, when local current density depends on the distance along the feed channel.

3.4 HEAT TRANSPORT IN THE MEA OF A PEFC

93

The cell voltage in the DMFC is Vcell = Voc − η a − η c − Rj,

(3.32)

where the last term on the right side comprises all ohmic losses (contact and membrane resistance). Relation (3.32) with η a (3.27) and η c (3.28) forms a 1D model of a DMFC. This model can be used for the analysis of cell polarization curves if special precautions are taken to provide a uniform distribution of reactants over the cell surface. However, even at moderate current densities these nonuniformities cannot be neglected. The reason is the gaseous CO2 bubbles, which disturb the flow in the anode channel. Nevertheless, Eqs (3.32) with (3.27) and (3.28) can be used to analyse the DMFC polarization curves at low current densities. Moreover, these equations provide the basis for the construction of a more detailed quasi2D model of DMFC (Section 4.7).

3.4 Heat transport in the MEA of a PEFC The Arrhenius law prescribes an exponential temperature growth of the rates of all transport and kinetic processes in a fuel cell. The laws of temperature distribution are thus of great interest for cell and stack design. The temperature profile across the MEA of a PEFC has been studied numerically by several authors (Rowe and Li, 2001; Berning et al., 2002; Ramousse et al., 2005; Birgersson et al., 2005). Numerical calculations indicate that local overheat may reach several degrees Kelvin (Rowe and Li, 2001; Ramousse et al., 2005). This value, however, strongly depends on the thermal conductivity of the catalyst layers λ, which is poorly known. The literature data on λ in the CCL of a PEFC differ by two orders of magnitude (Rowe and Li, 2001; Vie and Kjelstrup, 2004). In this section, we derive exact analytical solutions to the heat transport equations in the catalyst layers and membrane of a PEFC (Kulikovsky, 2007b). The results are illustrated by two cases: when the temperature on the outer sides of the MEA is kept fixed and when the cathode side of the MEA is thermally insulated. The solutions yield simple relations for the temperature on both sides of the membrane. These relations suggest a method for measuring λ and λm in the working fuel cell environment. Finally, we derive the exact expressions for the one-sided heat fluxes from the MEA.

3.4.1

General assumptions

The model is based on the following assumptions. 1. Heat in the MEA is transported due to conduction only. Energy transport through the membrane due to the flux of water is neglected.

94

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL 2. Joule heating by electron current is neglected. 3. Proton current density in the catalyst layers linearly depends on the distance across the layer.

The first assumption is justified since the electro-osmotic flux of water from the anode to the cathode is usually almost fully compensated for by the back diffusion. Thus, the net transport of water through the membrane is typically not large and the respective energy flux can be neglected. The smallness of the total water flux through the membrane is realized in a quite wide range of cell operating conditions (Janssen and Overvelde, 2001; Sui et al., 2008b). An accurate estimate of the thermal effect of water flux through the membrane is given in Section 3.5.1. The second assumption is discussed in Section 2.8.2. The third and last assumption means that the rates of the electrochemical reactions on both sides of the cell are constant across the respective catalyst layer. This assumption is fulfilled if transport of feed molecules and protons in both catalyst layers is close to ideal (Chapter 2). Note that in the case of poor reactants or proton transport in the CL the reaction rate and the respective source of heat are distributed nonuniformly (Chapter 2). In this chapter we ignore these nonuniformities. The solutions for the case of ideal transport derived below provide a reference point for more complicated studies.

3.4.2

Equations

Under the assumptions of Section 3.4.1, the heat balance in the catalyst layers and membrane is given by   2 Ta ∆Sa j j 2 xa + ηa + , in the ACL (3.33) 2F la σ t la   ∂ ∂Tm j2 − λm = , in the membrane (3.34) ∂xm ∂xm σm      2 ∂ ∂Tc Tc ∆Sc j j2 xc − λ = + ηc + 1− , in the CCL. (3.35) ∂xc ∂xc 4F lc σt lc −

∂ ∂xa



λ

∂Ta ∂xa





=

Here ∆S is the entropy change in the half-cell reaction, η is the half-cell polarization voltage, j is the mean current density in the cell, l is the thickness of the respective catalyst layer, σt is the proton conductivity of the catalyst layer, σm is the proton conductivity of the bulk membrane, and λ and λm are the thermal conductivities of the catalyst layers and membrane, respectively. Note that the thermal conductivities of the ACL and CCL are assumed to be the same.

3.4 HEAT TRANSPORT IN THE MEA OF A PEFC

95

Figure 3.4: The system of coordinates in the membrane-electrode assembly. Each functional layer is equipped with its own x-axis.

Up to the end of this chapter the values related to the anode side, the cathode side and the membrane are indicated by the subscripts a, c and m, respectively. To simplify the equations, each catalyst layer is equipped with its own system of coordinates: in the ACL the origin of the coordinates is at the catalyst/GDL interface, in the membrane, xm is counted from the ACL/membrane interface and in the CCL, xc is counted from the membrane (Figure 3.4). The factor j/l in the first term on the right sides of (3.33) and (3.35) is the rate of the respective electrochemical reaction. The constancy of this rate is equivalent to the linear shapes of the local proton current in the catalyst layers: ja = jxa /la and jc = j(1 − xc /lc ) (Section 2.2). The second term on the right sides of (3.33) and (3.35) is the electric power dissipated in the membrane phase of the respective catalyst layer. In the bulk membrane, the proton current is constant and the Joule term is j 2 /σm (Eq. (3.34)). For a detailed discussion of the terms in Eqs (3.33)–(3.35) see Section 2.8. It is convenient to introduce dimensionless variables x ˜=

x , lc

T T˜ = , T∗

˜j = j , j∗

η˜ =

η . bc

(3.36)

Here lc is the thickness of the CCL, bc is the Tafel slope on the cathode side, and 4F bc ∆Sc 2σt bc j∗ = lc

T∗ =

are the characteristic temperature and current density, respectively.

(3.37) (3.38)

96

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL With these variables, Eqs (3.33)–(3.35) transform to −kl2

  ∂ 2 T˜a ˜0 + η˜a ˜j + 2α˜ = αk k T x2a ˜j 2 , l S a ∂x ˜2a T˜a (0) =

T˜a0 ,

∂ T˜a ∂x ˜

= fa (3.39) x ˜a =kl

∂ 2 T˜m 2α˜j 2 − = ∂x ˜2m kσ kλ −

(3.40)

  ∂ 2 T˜c ˜1 + η˜c ˜j + 2α(1 − x = α T ˜c )2 ˜j 2 , c ∂x ˜2c ∂ T˜c ∂x ˜c

= fc ,

T˜c (1) = T˜c1 . (3.41)

x ˜c =0

Here α=

σt bc ∆Sc 2F λ

(3.42)

and kl =

la , lc

kS =

2∆Sa , ∆Sc

kλ =

λm , λ

kσ =

σm . σt

(3.43)

Note that the temperatures on the anode and the cathode side of the MEA are fixed at T˜a0 and T˜c1 , respectively. The case of thermally insulated CCL is considered in Section 3.4.4. Parameter α is small (Table 3.3, page 99). This means that the rate of CL heating is much smaller than the rate of temperature homogenization due to heat conduction. In other words, temperature variation in the catalyst layers is small and we may safely replace Ta by Ta0 on the right side of Eq. (3.33) and Tc by Tc1 on the right side of Eq. (3.35). This has been taken into account in Eqs (3.39) and (3.41). Parameters fa and fc in (3.39) and (3.41) are the slopes of the curve T˜(˜ x) (heat fluxes) at the ACL/membrane and membrane/CCL interfaces, respectively (Figure 3.4). These slopes are related. Indeed, the general solution to Eq. (3.40) is a parabola T˜m = −



α˜j 2 kσ kλ



x ˜2m + C1 x ˜m + C0

where C1 and C0 are the constants to be determined. The continuity of the heat flux prescribes   kλ ∂ T˜m /∂ x ˜m = fa x ˜m =0

(3.44)

3.4 HEAT TRANSPORT IN THE MEA OF A PEFC 

kλ ∂ T˜m /∂ x ˜m

 x ˜m =km

97

= fc

where km =

lm lc

(3.45)

(Figure 3.4). Differentiating (3.44) with respect to x ˜m and multiplying the result by kλ we obtain     α˜j 2 −kλ 2 x ˜m + kλ C1 = fa kσ kλ x ˜m =0     α˜j 2 x ˜m + kλ C1 = fc . −kλ 2 kσ kλ x ˜m =km From the first equation we get kλ C1 = fa ; the second one then gives fc = fa −

2αkm ˜j 2 . kσ

(3.46)

Physically, the second term on the right side of (3.46) is the heat flux produced in the membrane due to Joule heating.

3.4.3

Exact solutions

The general solutions to Eqs (3.39)–(3.41) are as follows. The temperature in the anode catalyst layer: 
α˜j x α˜j 2 x ˜a ˜a  ˜ 0 3 3 T˜a = T˜a0 + 4k − x ˜ + T k + η ˜ ˜ a ) + fa x ˜a . a (2kl − x l a a S 6kl2 2kl (3.47) The temperature in the cathode catalyst layer:  2  α˜j 12km 1 3 ˜ ˜ Tc = Tc + 4+ − (1 − x ˜c ) 6 kσ   α˜j  ˜1 + Tc + η˜c (1 + x ˜c ) − fa (1 − x ˜c ). 2

(3.48)

The temperature in the membrane is given by (3.44) with C1 = fa /kλ : T˜m = −



α˜j 2 kσ kλ



x ˜2m +

fa x ˜m + C0 . kλ

(3.49)

98

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL (a)

(b)

Figure 3.5: Temperature (solid lines) and heat flux (dashed lines) profiles across the MEA for (a) base case conditions and (b) for ten times lower thermal conductivity of the catalyst layers (λ = 0.16 W m−1 K−1 ). Temperature on both sides of the MEA is fixed at 350 K. Solutions (3.47)–(3.49) contain two as yet indefinite constants fa and C0 . These constants are obtained from the continuity of temperature at the ACL/membrane and membrane/CCL interfaces (Figure 3.4): T˜a = T˜m (3.50) x ˜a =kl x ˜ =0 m = T˜c . (3.51) T˜m x ˜m =km

x ˜c =0

From (3.50) we get C0 = T˜a0 +

 αk 2 ˜j 2 αkl ˜j  ˜0 l Ta kS + η˜a + + fa kl . 2 2

Using this C0 in (3.51) we find     
αkλ ˜j ˜j kl2 − 1 + kl kS T˜a0 + η˜a − T˜c1 + η˜c fa = − 2 (km + kλ (kl + 1))   ˜a0 − T˜c1 k T 2 ˜ λ αj km (km + 2kλ ) + − . kσ (km + kλ (kl + 1)) km + kλ (kl + 1)

3.4.4

(3.52)

(3.53)

Temperature profiles

Figure 3.5(a) displays the temperature shapes across the MEA for the base case conditions. Figure 3.5(b) shows what happens when the thermal conductivity of the catalyst layers, λ, is ten times lower. The base case physical parameters are listed in Table 3.2. The respective dimensionless parameters, which appear in (3.47) and (3.48) are collected in Table 3.3. The fixed temperature on both sides of the MEA corresponds to a single stand-alone cell with an external heating system.

3.4 HEAT TRANSPORT IN THE MEA OF A PEFC

99

Table 3.2: Parameters for calculations of heat transport in the MEA. Entropy change for liquid water formation at 100 ◦ C is ∆Sw = 156.21 J mol−1 K−1 (Barin, 1995). Therefore, ∆SHOR = ∆Sw − ∆SORR /2 ' −6.97 J mol−1 K−1 , which is negligibly small compared to ∆SORR . Proton conductivity of the electrolyte phase in the catalyst layer, σt (S m−1 ), (Xie et al., 2005) Proton conductivity of the bulk membrane, σm (S m−1 ), (Xie et al., 2005) Tafel slope on the cathode side, bc (V), (Xie et al., 2005) Entropy change in the ORR, ∆Sc (J mol−1 K−1 ), (Lampinen and Fomino, 1993) Entropy change in the HOR, ∆Sa (J mol−1 K−1 ) Thermal conductivity of the catalyst layer, λ (W m−1 K−1 ) (Ramousse et al., 2005) Thermal conductivity of the membrane, λm (W m−1 K−1 ) (Ramousse et al., 2005) Current density in the cell, j (A m−2 ) Overpotentials (ηa , ηc ) (V) Catalyst layer thickness (m) Membrane thickness (m) Cell temperature (K) j∗ (A m−2 ) T∗ (K)

2.5 10 0.05 326.36 0 1.6 0.34 104 (0, 0.6) 10−5 5 · 10−5 350 2.5 · 104 59.2

Table 3.3: Dimensionless parameters. ˜j 0.4

α 1.32 · 10−4

kl 1

kS 0

km 5

kλ 0.34/1.6

kσ 4

(˜ ηa , η˜c ) (0, 12)

The set of parameters in Table 3.2 corresponds to the situation when the cell operates close to the limiting current and heat production is thus close to maximum. Even in these “stressed” conditions the temperature variation across the MEA is small ('0.03-0.3 K). Local through-plane overheat in the cell running with proper heat management is, therefore, negligibly small. In the base case, the heat flux in the membrane is close to zero (Figure 3.5(a)). Due to its relatively low thermal conductivity, the membrane serves as a thermal insulator, which separates temperature fields on the anode and the cathode side. The heat flux from the CCL to the

100

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

(a)

(b)

Figure 3.6: Same as Figure 3.5 for the case when the cathode side of the MEA is thermally insulated. cathode GDL is, however, not small: this flux is roughly ten times greater than the flux from the ACL to the anode GDL (Figure 3.5(a)). With λ = 0.16 W m−1 K−1 (ten times lower), the situation with the heat fluxes is similar to that in the base case (Figure 3.5(b)). The heat flux in the membrane is not large. The fluxes in the membrane and on the anode side are much smaller than the flux from the CCL to the GDL (Figure 3.5(b)). However, in a stack environment the temperature distribution may change dramatically. To model the worst-case scenario in a stack we will assume that the cathode side of the cell is thermally insulated. This is the limiting case of no special cooling system and poor heat transport to the adjacent cell. With the thermally insulated cathode, local overheat in the MEA increases by a factor of 40 (Figure 3.6(a)) and reaches ' 2 K. Physically, membrane thermal conductivity is not high enough to efficiently remove the heat produced in the CCL and the temperature on the cathode side rises dramatically. In this situation, the value of thermal conductivity of the catalyst layer plays a minor role: a reduction of λ by a factor of 10 changes the local overheat by a factor of less than two (Figure 3.6(b)). With the thermally insulated cathode, the total heat flux is directed towards the anode GDL (Figure 3.6(a), (b)). The temperature shapes in the MEA components are close to linear. Clearly, if heat removal on the anode side is also hindered, local overheat of the cathode side would increase further.

3.4.5

How to measure thermal conductivities of MEA layers

In this section, we will assume that the thicknesses of the ACL and CCL are the same, i.e. kl = 1. This particular case is realized in a majority of MEAs for low-temperature fuel cells.

3.4 HEAT TRANSPORT IN THE MEA OF A PEFC

101

The method for measuring λ and λm is based on measurements of the temperature of the anodic T˜a1 ≡ T˜a |x˜a =kl =1 and cathodic T˜c0 ≡ T˜c |x˜c =0 sides of the membrane as a function of cell current, provided that the external sides of the MEA are kept at the same temperature T0 . Simple expressions for T˜a1 and T˜c0 follow from (3.47) and (3.48). In a PEFC, the thermodynamic heat on the anode side is small and the anodic overpotential is also small; we may thus set kS = 0 and η˜a = 0. Making these substitutions in (3.47), (3.48), and setting T˜a0 = T˜c1 = T˜0 , x ˜a = kl in (3.47) and x ˜c = 0 in (3.48), we get: αkλ (˜ ηc + T˜0 )˜j α(2km + kσ )˜j 2 T˜a1 = T˜0 + + 2(km + 2kλ ) 2kσ ˜ ˜ α(2km + kσ )˜j 2 α(k + k )(˜ η + T ) j m λ c 0 + . T˜c0 = T˜0 + 2(km + 2kλ ) 2kσ

(3.54) (3.55)

Remarkably, in (3.54) and (3.55) the quadratic in ˜j terms are the same; thus the temperature drop across the membrane ∆T˜m ≡ T˜c0 − T˜a1 does not contain ˜j 2 . Subtracting (3.54) from (3.55) we find

∆T˜m =

  αkm η˜c + T˜0 ˜j 2(km + 2kλ )

=

−1  α˜j  2kλ η˜c + T˜0 1 + . 2 km

(3.56)

Note that Eq. (3.56) does not contain the parameter kσ , which is difficult to measure. Physically, the terms with ˜j 2 in (3.54) and (3.55) describe the contribution of Joule heating in the catalyst layers. When the thicknesses of the CLs are the same, the respective contributions to ∆T˜m cancel out. In other words, Joule heating in the CLs shifts the curve T˜m (˜ xm ) along the temperature axis as a whole, which does not affect the temperature difference ∆T˜m . The latter is determined only by thermodynamic and irreversible heating in the CCL. In dimension variables Eq. (3.56) takes the form ∆Tm

lc j = 2λ



T0 ∆Sc ηc + 4F

 −1 2lc λm 1+ . lm λ

(3.57)

Equation (3.57) is further simplified for a thick membrane, when 2lc λm /(lm λ)  1. Physically, in that case the transport of heat through the membrane is negligible. The term in the square brackets in (3.57) is then close to 1 and this equation reduces to ∆Tm (lm

lc j → ∞) ' 2λ



T0 ∆Sc ηc + 4F

 .

(3.58)

102

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

This equation does not contain λm , since the membrane serves as an ideal thermal insulator. In the opposite limit of a thin membrane, we may neglect 1 in the square brackets in (3.57) and this equation transforms to ∆Tm (lm

lm j → 0) ' 4λm



T0 ∆Sc ηc + 4F

 .

(3.59)

This equation does not contain λ. In the MEA with a thin membrane, the heat transport properties of the catalyst layers do not affect the temperature gradient across the membrane. Equations (3.58) and (3.59) provide a means of measuring the thermal conductivities of the catalyst layers and membrane in a working fuel cell environment. The temperature difference on both sides of the thick and thin membranes and simultaneously the half-cell polarization voltage ηc have to be measured as a function of cell current. In the region of currents where ηc is approximately constant, the slope of the straight line ∆Tm versus j would give λ and λm in experiments with thick and thin membranes, respectively. If the variation of ηc with j is not small, it can be taken into account; this would only slightly complicate the procedure. It is important to note that both sides of the MEA should be kept at the same temperature T0 . In practice, the temperature of the flow fields may be kept constant, since GDL thermal conductivity is usually high.

3.4.6

One-sided fluxes from the MEA

The above results allow us to write the exact expressions for one-sided heat fluxes on both sides of the MEA. The dimensionless heat flux is q˜ = −

∂ T˜ ∂x ˜

(3.60)

where q˜ is nondimensionalized according to q˜ =

qlc λT∗

(3.61)

and T∗ is given by (3.37). Consider first the flux from the ACL to the anode GDL. Differentiating (3.47) with respect to x ˜a and setting ˜a = 0 in the resulting expression we  x 0 ˜ obtain heat flux q˜ = − ∂ Ta /∂ x ˜a from the anode side of the MEA: a

q˜a0

x ˜a =0

  2kl (2km (km + 2kλ ) + kσ kλ (1 − kl2 )) 2 ˜ = αj − − 3 2kσ (km + kl kλ + kλ )

3.4 HEAT TRANSPORT IN THE MEA OF A PEFC  

+ α˜j − kS T˜a0 + η˜a

+

  kλ T˜a0 − T˜c1 km + kl kλ + kλ



103

      kλ kl kS T˜a0 + η˜a − T˜c1 + η˜c  + 2(km + kl kλ + kλ )

.

(3.62)

˜c and setting x ˜c = 1 in the Differentiating (3.48) with respect to x 1 ˜ ˜c from the resulting expression we obtain heat flux q˜c = − ∂ Tc /∂ x x ˜c =1

cathode side of the MEA: q˜c1


! 2km (km + 2kλ ) + kσ kλ (1 − kl2 ) 2 2km = αj + − 3 kσ 2kσ (km + kl kλ + kλ )        kλ kl kS T˜a0 + η˜a − T˜c1 + η˜c  + α˜j T˜c1 + η˜c + 2(km + kl kλ + kλ )   kλ T˜a0 − T˜c1 . (3.63) + km + kl kλ + kλ ˜2

The heat fluxes q˜a0 and q˜c1 have opposite directions and hence their difference is the total heat flux leaving MEA: q˜tot = q˜c1 − q˜a0 . Subtracting (3.62) from (3.63) we get q˜tot

= α˜j 2



2(kl + 1) 2km + 3 kσ



  + α˜j T˜c1 + η˜c + kS T˜a0 + η˜a . (3.64)

The first term on the right side describes the heat flux due to Joule heating in the catalyst layers and membrane. The second term is a flux due to thermodynamic and irreversible heating in the active layers. This helps us to understand the meaning of the first two terms in Eqs (3.62) and (3.63). These terms arise due to asymmetry in the heat transport coefficients, conductivities and thicknesses of the anode and the cathode catalyst layers. In other words, these terms take into account the “crossover” of Joule heat (the terms with α˜j 2 ) and of reaction heat (the terms with α˜j) through the membrane. As it should be, in the expression for total heat flux (3.64) the crossover terms cancel out. The last term in Eqs (3.62) and (3.63) describes the heat flux due to the temperature difference on both sides of the MEA. Note that this flux does not contain the small parameter α. In the dimension variables, Eqs (3.62) and (3.63) take the form qa0

lc j 2 = 2σt



2kl (2km (km + 2kλ ) + kσ kλ (1 − kl2 )) − − 3 2kσ (km + kl kλ + kλ )



104

CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL 



+ j −

+

Ta0 ∆Sa + ηa 2F

λ Ta0 − Tc1 lc


!



  0   1   T ∆S T ∆S kλ kl a2F a + ηa − c4F c + ηc  + 2(km + kl kλ + kλ )

kλ km + kl kλ + kλ

(3.65)

and qc1


! 2km (km + 2kλ ) + kσ kλ (1 − kl2 ) 2 2km + − 3 kσ 2kσ (km + kl kλ + kλ )

lc j 2 = 2σt

  0   1   T ∆S T ∆S kλ kl a2F a + ηa − c4F c + ηc 1 ∆S T c  + ηc + +j  c 4F 2(km + kl kλ + kλ ) 

+

3.4.7

λ Ta0 − Tc1 lc


!

kλ . km + kl kλ + kλ

(3.66)

Heat crossover through the membrane

It is advisable to evaluate the discussed above heat “crossover” through the membrane. In low-T cells, membrane thermal conductivity is two to five times smaller than the thermal conductivity of the catalyst layers; as an estimate we take kλ = 5. With the 50-µm membrane and 10-µm catalyst layers we have km = 5 (Table 3.3). With the parameters from Table 3.3 we get    T˜c1 + η˜c 2 ˜0 ˜1 ˜ 23α j 53˜ η a  α˜j + Ta − Tc q˜a0 = − − + 2 54 54 27     ˜c1 + η˜c 53 T 2 ˜ 23α j η ˜ T˜0 − T˜c1 a q˜c1 = + +  α˜j + a . 2 54 54 27 Thus, the membrane perfectly insulates the heat produced on the anode and the cathode side: 53/54 of the reaction heat flux produced on the cathode side is emitted from the CCL to the cathode GDL and only 1/54 of this flux penetrates through the membrane and leaves the MEA as a part of the anode flux. Note that Joule heat flux is symmetrical due to the symmetry of the MEA geometry and properties.

3.5 HEAT TRANSPORT IN THE MEA OF A DMFC

105

Consider now the case of a five times thinner membrane, i.e. km = 1. With the other parameters from Table 3.3 we find    ˜c1 + η˜c T 2 ˜0 ˜1 ˜ 11αj 13˜ ηa  α˜j + Ta − Tc q˜a0 = − − + 12 14 14 7     ˜1 11α˜j 2  13 Tc + η˜c η˜a T˜0 − T˜c1 1 q˜c = + +  α˜j + a . 12 14 14 7 The situation practically does not change: the largest fraction (13/14) of the reaction heat flux produced on the cathode side is still emitted to the cathode GDL and only 1/14 of this flux contributes to the anode flux. As a final example, consider the 10-µm membrane (km = 1) of the same thermal conductivity as that one of the catalyst layers i.e. kλ = 1. With these parameters we get    ˜c1 + η˜c T 2 ˜0 ˜1 ˜ 11αj 5˜ ηa  α˜j + Ta − Tc q˜a0 = − − + 12 6 6 3     5 T˜c1 + η˜c 2 ˜ T˜0 − T˜c1 11α j η ˜ a q˜c1 = + +  α˜j + a . 12 6 6 3 Even in that case, 5/6 of the reaction heat flux is emitted by its “own” CL and only 1/6 permeates through the membrane. We conclude that in all cases the membrane almost perfectly separates the heat produced on the cathode and the anode side.

3.5 Heat transport in the MEA of a DMFC Unlike in a PEFC, in a DMFC both the anodic and cathodic sides produce heat. The effect which is specific to DMFCs is permeation of methanol through the polymer electrolyte membrane (methanol crossover). Permeated methanol is catalytically oxidized on the cathode side in a direct reaction with oxygen (K¨ uver and Vielstich, 1998; Paganin et al., 2005) 3 CH3 OH + O2 = CO2 + 2H2 O, 2

∆H298 = 639 kJ mol−1 .

(3.67)

As can be seen, the amount of heat released in this reaction is high (Lindstr¨ om and Pettersson, 2003). Thus, (3.67) is quite a significant source of heat in a DMFC.

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The effect of crossover on cell performance is twofold. Though mass fluxes induced by crossover always lead to voltage loss, heating due to (3.67) improves the kinetics of the oxygen reduction reaction. The flux of methanol through the membrane decreases linearly with the increase in cell current (Eq. (3.23)). At small currents, cell depolarization due to crossover is maximal, but the positive effect of cathode heating due to (3.67) is also maximal. In other words, at low currents, heating due to reaction (3.67) accelerates ORR and helps to heat up the cell to the working temperature. Close to the methanol-limiting current density, both cell depolarization and heating due to (3.67) tend to zero. Therefore, the overall effect of crossover on cell performance is quite different in different regions of the polarization curve. In DMFC modelling, the effect of cathode heating due to reaction (3.67) is not always accurately accounted for. Most of the existing DMFC models are isothermal (Baxter et al., 1999; Birgersson et al., 2004; Garcia et al., 2004; Senn and Poulikakos, 2006; Li and Wang, 2007; Xu et al., 2008; Miao et al., 2008). The two-dimensional CFD model (Divisek et al., 2003) seemingly includes heat release due to crossover; however, the results do not clarify the role of this process in the cell heating. It is worth mentioning that the vast majority of DMFC models are numerical. In this section, we construct a model of heat transport in the DMFC MEA, which takes into account the thermal effect of crossover. We derive the exact analytical solution to model equations. The solution is greatly simplified under open-circuit conditions. As in a PEFC, the respective relations suggest a method for in situ measurements of the thermal conductivities of the catalyst layers and membrane.

3.5.1

Assumptions

1. Heat in the MEA is transported due to conduction only. Energy transport through the membrane due to water flux is neglected. 2. Joule heating by electron current is neglected. 3. Proton current density in the catalyst layers linearly depends on the distance across the layer. 4. Heat flux due to water evaporation-condensation is neglected. The first assumption follows from the following estimate. The energy flux (J cm−2 s−1 ) due to water flux through the membrane is Uw = Mw Nw cP w T , where Mw is the molecular weight of water, Nw is the total molar flux of water through the membrane, cP w is the water heat capacity at constant pressure and T is the temperature. The respective rate of heat production in the membrane (J cm−3 s−1 ) is ∂Uw /∂x = Mw Nw cP w ∂T /∂x (x is directed across the membrane). It is convenient to represent the overall flux of water in the membrane as Nw = αw j/F , where αw is the coefficient of water transport through the membrane.

3.5 HEAT TRANSPORT IN THE MEA OF A DMFC

107

The rate of Joule heating in the membrane is j 2 /σm , where σm is the proton conductivity of Nafion. Equating ∂Uw /∂x and j 2 /σm we get ∂T jF = . ∂x σm Mw αw cP w

(3.68)

This relation gives the temperature gradient across the membrane, at which the heat source due to water transport equals Joule heating. Evidently, if the temperature gradient is much smaller than (3.68), the heating due to water transport is negligible. Substituting the values from Table 3.4 into (3.68) we find ∂T /∂x ' 1.3 × 103 K cm−1 . In the membrane with a thickness of 10−2 cm this value corresponds to a temperature drop of about 13 K. A typical temperature drop across the membrane does not exceed 2 K (see below); thus the term due to water transport in the energy balance equation can be neglected5 . The second assumption is fulfilled since the electron conductivity of the carbon phase in a cell is much greater than the proton conductivity of the electrolyte phase. The third assumption is equivalent to the constancy of the rate of electrochemical reaction in the catalyst layers (for a more detailed discussion, see Section 2.3). The last assumption is justified if (i) water produced in the ORR evaporates in the GDL (Weber and Newman, 2006), or (ii) cell temperature is not high so that the rate of water evaporation is small (Birgersson et al., 2005).

3.5.2

Equations

The sketch of the MEA and the system of coordinates is shown in Figure 3.4. Under the assumptions of Section 3.5.1, the heat balance in the catalyst layers and membrane is given by   ∂ ∂Ta − λ = ∂xa ∂xa   ∂ ∂Tm λm = − ∂xm ∂xm   ∂ ∂Tc − λ = ∂xc ∂xc



Ta ∆Sa + ηa 6F



j j2 + la σt



xa la

2

j2 σm    2 Tc ∆Sc j j2 xc + ηc + 1− 4F lc σt lc   Tc ∆S∗ jcross + . 6F lc

(3.69) (3.70)

(3.71)

Equations (3.69)–(3.71) are written for the ACL, membrane and the CCL, respectively. The symbols used are those of Section 3.4.2. 5 Note, however, that the temperature drop across the membrane depends on the thermal conductivity of the catalyst layers λ, which is poorly known (see below). Lower values of λ lead to a higher value of the temperature gradient across the MEA and to a higher thermal effect due to the water flux.

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CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

Table 3.4: Parameters for calculations of heat transport in the DMFC. Proton conductivity of the electrolyte phase in the catalyst layer, σt (S m−1 ) (Havranek and Wippermann, 2004) Proton conductivity of the bulk membrane, σm (S m−1 ) (Xie et al., 2005) Tafel slope on the cathode side, bc (V) (Xie et al., 2005) Entropy change in the ORR, ∆Sc (J mol−1 K−1 ) (Lampinen and Fomino, 1993) Entropy change in the reaction (3.67), ∆S∗ (J mol−1 K−1 ) (Lindstr¨ om and Pettersson, 2003) Thermal conductivity of the catalyst layer, λ (W m−1 K−1 ) (Ramousse et al., 2005) Thermal conductivity of the membrane, λm (W m−1 K−1 ) (Ramousse et al., 2005) Heat capacity of water, cP w (J g−1 K−1 ) (Lide, 1998) Maximal water transfer coefficient in the membrane, αw (Lu et al., 2005) Cell current density, j (A m−2 ) Diffusion coefficient of methanol in the anode backing layer (m2 s−1 ) (Kulikovsky, 2002b) Catalyst layer thickness (m) Membrane thickness (m) Cell temperature (K) j∗ (A m−2 ) T∗ (K) α β β∗ γ φ

0.1 10 0.05 3.26 · 102 2.144 · 103 1.6 0.34 4.18 1 103 10−9 10−4 10−4 350 103 59.2 5.28 · 10−6 1 0.5 1.34 · 10−3 2.59 · 10−2

The last term on the right side of (3.71) describes the effect of the reaction (3.67). Here ∆S∗ is the entropy change in (3.67) and jcross is the equivalent current density of the methanol crossover given by Eq. (3.23). Discussion of the other terms is given in Section 3.4.2. With the dimensionless variables (3.36), Eqs (3.69)–(3.71) transform to −kl2

  ∂ 2 T˜a ˜a + η˜a ˜j + 2α˜ = αk k T x2a ˜j 2 l S ∂x ˜2a

(3.72)

3.5 HEAT TRANSPORT IN THE MEA OF A DMFC

109

2α˜j 2 ∂ 2 T˜m = (3.73) ∂x ˜2m kσ kλ     ˜j ∂ 2 T˜c 2 − 2 = α T˜c + η˜c ˜j + 2α (1 − x ˜c ) ˜j 2 + γβ∗ 1 − T˜c . (3.74) ˜ ∂x ˜c jD

−

Here σt bc ∆Sc 2F λ ∆S∗ Da c0M γ= λ

α=

(3.75) (3.76)

and kl =

la , lc

kS =

2∆Sa , 3∆Sc

kλ =

λm , λ

kσ =

σm . σt

(3.77)

The solution to (3.72)–(3.74) subject to the conditions which express the continuity of the heat flux and temperature at the ACL/membrane and membrane/CCL interfaces (Figure 3.4): ∂ T˜a = fa , ∂x ˜a x ˜ =k a l ˜ ∂ Tm = fa , kλ ∂x ˜m x˜m =0 ∂ T˜c = fc , ∂x ˜c

T˜a |x˜a =kl = T˜m |x˜m =0

(3.78)

∂ T˜m kλ ∂x ˜m

= fc

(3.79)

T˜c |x˜c =0 = T˜m |x˜m =km

(3.80)

lm . lc

(3.81)

x ˜m =km

x ˜c =0

where km =

The procedure quite analogous to that of Section 3.4.2 leads to the following relation between fc and fa : fc = fa −

2αkm ˜j 2 . kσ

(3.82)

Physically, the second term on the right side of (3.82) is the gain in the heat flux produced in the membrane due to Joule heating. The solutions to (3.72)–(3.74) are given in Section 3.5.6. Below we will consider the case of open-circuit conditions, when current in the cell is zero and the terms with ˜j in (3.72)–(3.74) can be omitted.

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CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

3.5.3

Heat transport under open-circuit conditions

Equations With an open circuit we may set ˜j = 0 in (3.72)–(3.74) and these equations reduce to ∂ 2 T˜a =0 ∂x ˜2a ∂ 2 T˜m =0 − ∂x ˜2m ∂ 2 T˜c − 2 = γβ∗ T˜c . ∂x ˜c −

(3.83) (3.84) (3.85)

Solutions to these equations are subject to conditions (3.78)–(3.80). At ˜j = 0, Joule heating in the membrane is zero; in view of Eq. (3.82) we have fc = fa . The only source of heat in a DMFC with an open circuit is direct oxidation of permeated methanol on the cathode side (the term on the right side of Eq. (3.85)). At j = 0 this term is maximal (cf. Eq. (3.74)), since no methanol is consumed in the useful electrochemical reaction and the flux of methanol through the membrane is maximal. The system of equations (3.83)–(3.85) and (3.78)–(3.80) is not complete; it should be supplemented by the boundary conditions at the outer sides of the MEA. We will consider the cases when one side is thermally insulated while the other side is kept at a fixed temperature. These conditions suggest a simple method for in situ measurements of thermal conductivities of the catalyst layers and membrane in a DMFC.

The anode side is thermally insulated In that case, we have



∂ T˜a ∂x ˜a

x ˜a =0

= 0 for Eq. (3.83) and T˜c (1) = T˜0 for

(3.85). Solving (3.83)–(3.85) with these conditions and taking into account (3.78)–(3.80), we get T˜0 T˜a = T˜m = cos φ ˜ T0 cos(φ˜ xc ) T˜c = cos φ

(3.86) (3.87)

where φ≡

p γβ∗ .

(3.88)

3.5 HEAT TRANSPORT IN THE MEA OF A DMFC

111

The temperature of the ACL and membrane is constant: T˜0 / cos φ. The temperature shape in the CCL is part of a cosine (Eq. (3.87)).

The cathode side is thermally insulated In that case for Eq. (3.83) we have T˜a (0) = T˜0 and for (3.85) we have ∂ T˜c = 0. The solutions are ∂x ˜c x ˜c =1

T˜a = T˜0 + T˜m = T˜0 +

T˜0 kλ φ sin (φ) x ˜a kλ cos φ − (km + kl kλ ) φ sin φ T˜0 φ sin (φ) (˜ xm + kl kλ )

kλ cos φ − (km + kl kλ ) φ sin φ ˜ T0 kλ cos (φ(1 − x ˜c )) T˜c = . kλ cos φ − (km + kl kλ ) φ sin φ

(3.89) (3.90) (3.91)

Temperatures in the ACL and the membrane are linear functions of distance. The temperature in the CCL is proportional to cos(φ(1 − x ˜c )). The temperature of the outer side of the CCL is T˜c1 ≡ T˜c (1) =

3.5.4

T˜0 kλ . kλ cos φ − (km + kl kλ ) φ sin φ

(3.92)

How to measure thermal conductivities of MEA layers

Parameter φ is small (Table 3.4, page 108); this allows us to expand Eqs (3.86) and (3.92) in terms of φ. The temperature of the thermally insulated side of the MEA is then   β∗ γ 0 Ta = T0 1 + (3.93) 2     2km β∗ γ Tc1 = T0 1 + 1 + 2kl + . (3.94) kλ 2 Note that Eqs (3.93) and (3.94) imply that the other side of the MEA is kept at a temperature T0 (Tc1 = T0 in Eq. (3.93) and Ta0 = T0 for Eq. (3.94)). Equations (3.93) and (3.94) suggest a means of measuring thermal conductivities of the catalyst layers and membrane. Analogously to Section 3.4.5, the method is based on measurements of the temperature of the thermally insulated side of the MEA when the other side is kept at a fixed temperature. With measured values Ta0 , Tc1 and T0 , Eqs (3.93) and

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CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

(3.94) give the parameter kλ and the product β∗ γ, which contains λ. The simplest way to calculate λ and λm is as follows. It is useful to rewrite Eqs (3.93) and (3.94) in terms of overheats θa and θc . By the definition Ta0 −1 T0 T1 θc = c − 1 T0

θa =

(3.95) (3.96)

θa is the overheat of the thermally insulated anode side, when the cathode side is kept at the temperature T0 . θc is the overheat of the thermally insulated cathode side when the anode side is kept at T0 . From (3.93) and (3.94) we get θa = θc =

β∗ γ 2 1 + 2kl +

(3.97) 2km kλ



β∗ γ . 2

(3.98)

We see that the ratio θc /θa does not contain β∗ γ: θc 2km = 1 + 2kl + . θa kλ

(3.99)

Equation (3.99) means that the ratio of overheats in the two experiments does not depend on the rate of heat production in the CCL. This ratio is determined solely by the heat transport properties of the catalyst layers and membrane6 . Equation (3.99) provides a simple means of measuring kλ . Measuring Ta0 in the experiment with the thermally insulated anode, from Eq. (3.95) we obtain the parameter θa . Measuring Tc1 in the experiment with the thermally insulated cathode we can calculate θc with (3.96); Eq. (3.99) yields kλ (the parameters kl and km are given by (3.77) and (3.81)). Equations (3.97) and (3.76) lead to the following relation for λ: λ=

β∗ ∆S∗ Da c0M . 2θa

(3.100)

Calculation with (3.100) requires literature data on Da and Dm 7 . Finally, with λ and kλ in hand we get λm = kλ λ. 6 This

result is a consequence of the linearity of Eqs (3.83)–(3.85). the estimate, Da = Dm = DM w may be taken, where DM w is the diffusion coefficient of methanol in water. 7 For

3.5 HEAT TRANSPORT IN THE MEA OF A DMFC

113

Figure 3.7: Temperature (solid line) and heat flux (dashed line) profiles across the MEA. The temperature on the cathode side of the MEA is fixed at 350 K; the anode side is thermally insulated.

3.5.5

Temperature profiles and discussion

The parameters used for the calculations are listed in Table 3.4. The entropy change in the reaction (3.67) is ∆S∗ =

∆H298 = 2.144 × 103 J mol−1 K−1 . T298

Taking this value and the data from Table 3.4 we get γ = 1.34 × 10−3 . The temperature shapes for the case of the thermally insulated anode, (3.86)–(3.87), and the respective profile of the heat flux are shown in Figure 3.7. The growth of temperature in the MEA is small, about 0.15 K (Figure 3.7). Heat produced in the CCL is efficiently removed through the CCL/backing layer interface (Figure 3.7). In the case of the thermally insulated cathode, the overall heating of the MEA is ten times greater (about 1.5 K, Figure 3.8). In that case the heat flux generated in the CCL is transported through the membrane and the ACL; the insulating effect of these two layers leads to a growth of CCL temperature. Overheat θ depends on the thermal conductivity of the catalyst layers, λ. According to (3.97) and (3.76), in the case of the thermally insulated anode, θa ∼ 1/λ. From (3.98) it follows that in the case of the thermally insulated cathode, θc = p + q/λ, where p and q are constants. Therefore, in both cases the decrease in λ increases the overheat. In the experiment suggested for measuring λ and λm , care should be taken to prevent the formation of the electrolytic domain in a cell. This can be done by keeping the oxygen (air) flow rate at a sufficiently high level. For a more detailed discussion of this issue, see Section 4.7. The literature data on λ are contradictory. Measurements (Vie and Kjelstrup, 2004) performed in the hydrogen cell environment gave a λ value,

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CHAPTER 3. ONE-DIMENSIONAL MODEL OF A FUEL CELL

Figure 3.8: Same as Figure 3.7, except that the cathode side is thermally insulated. The temperature on the anode side is fixed at 350 K.

two orders of magnitude lower than the value reported in (Ramousse et al., 2005) and used here. If we take λ ten times smaller, the temperature of the thermally insulated cathode would increase by 10 K. This may dramatically increase the rate of ORR on the cathode side. If the anode side is thermally insulated, the heat flux through the membrane is zero (Figure 3.7). In other words, heat generated in the CCL is removed through the CCL/backing layer interface. In that case, membrane thermal conductivity is of no significance and it does not appear in Eq. (3.93). In the case of the thermally insulated cathode, the heat flux is directed in the opposite way, through the membrane and the ACL; for that reason the relation (3.94) involves kλ ≡ λm /λ. Suppose that λ is fixed; according to (3.94) the decrease in λm then dramatically increases Tc1 . In a DMFC with a thermally insulated cathode, the “spots” of low membrane thermal conductivity are very dangerous: heat of the reaction (3.67) appears to be “trapped” and the adjacent domain of the CCL may be strongly overheated.

3.5.6

Exact solutions for finite current

Parameters α and γ are small. Similar to the situation in a PEFC (Section 3.4.2), this means that the rate of temperature homogenization due to heat conductance is much greater than the rate of heating. In other words, temperature variation across the MEA is small and we may safely replace Ta by Ta0 ≡ Ta (0) on the right side of Eq. (3.69) and Tc with Tc1 ≡ Tc (ltc ) on the right side of Eq. (3.71). These equations transform to −kl2

  ∂ 2 T˜a ˜a0 + η˜a ˜j + 2α˜ = αk k T x2a ˜j 2 l S ∂x ˜2a

(3.101)

3.5 HEAT TRANSPORT IN THE MEA OF A DMFC

115

2α˜j 2 ∂ 2 T˜m = (3.102) ∂x ˜2m kσ kλ     ˜j ∂ 2 T˜c 2 − 2 = α T˜c1 + η˜c ˜j + 2α (1 − x T˜c1 . (3.103) ˜c ) ˜j 2 + γβ∗ 1 − ˜ ∂x ˜c jD

−

For fixed temperatures on both MEA sides T˜a |x˜a =0 = T˜a0 ,

T˜c |x˜c =1 = T˜c1 .

(3.104)

the solutions to (3.101)–(3.103) are as follows. The temperature of the anode catalyst layer:    α˜j x ˜2 ˜j x ˜2 + 3kl kS T˜a0 + η˜a T˜a = T˜a0 − 6kl2     2α˜j 2 kl 0 ˜ ˜ + αj kS Ta + η˜a + fa . + (3.105) 3 The temperature of the cathode catalyst layer: T˜c = T˜c1   2˜j 2 (3k + k ) ˜j 2 (1 − x ˜j(1 + x ˜)3 ˜)  ˜ 1 m σ Tc + η˜c + − 2 3kσ 6   1 ˜ ˜ j γβ∗ Tc + 1− (1 − x ˜2 ) − (1 − x ˜)fa . (3.106) ˜jD 2 

+ α(1 − x ˜)

The temperature of the membrane: α˜j 2 x ˜2m fa x ˜m T˜m = − + + C0 kσ kλ kλ

(3.107)

 αkl2 ˜j 2 αkl ˜j  ˜0 + C0 = T˜a0 + kS Ta + η˜a + kl fa . 2 2

(3.108)

where

Parameter fa in Eqs (3.105)–(3.108) is given by
α˜j 2 kσ kλ (1 − kl2 ) + 2km (2kλ + km ) fa = 2kσ (kl kλ + kλ + km )      α˜jkλ kl kS T˜a0 + η˜a − T˜c1 + η˜c − 2(kl kλ + kλ + km )     ˜ 1 ˜ γβ∗ Tc kλ 1 − ˜jj − 2kλ T˜a0 − T˜c1 D + . 2(kl kλ + kλ + km )

(3.109)

Chapter 4

Quasi-2D model of a fuel cell In this chapter, we construct quasi-2D models of fuel cells. These models take into account the variation of local parameters (reactant and product concentration, current density, and overpotentials) along the feed channel. A typical membrane-electrode assembly is a two-scale structure: its inplane size ('10 cm) exceeds its thickness ('0.1 cm) by three orders of magnitude. The reactants are usually distributed over the cell surface through a system of feed channels (flow field). Reactants are consumed in electrochemical reactions and hence part of the cell located close to the channel inlet receives more feed than the “remote” part at the outlet. Nonuniformity of the feed concentration along the channel leads to a specific voltage loss, which is not captured by 1D models. In PEFCs, another in-plane source of voltage loss is the nonuniform distribution of water. The proton conductivity of the membrane increases linearly with water content and in PEFCs the feed gases are usually humidified to avoid membrane drying. However, water is produced in the ORR and the “remote” part of the cell may suffer from an excess of liquid water (flooding). In this chapter, we construct quasi-2D models of the PEFC, DMFC and SOFC, which take into account various nonuniformities along the feed channel. For simplicity, we will restrict our consideration to a cell equipped on both sides with single straight channels (Figure 1.5, page 19; see also Figure 4.25 on page 174). Clearly, the cell with meander channels can always be cut and transformed into such an equivalent cell1 . This 1 In

the case of n parallel meanders, this procedure gives n identical cells.

117

118

CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

transformation implies that we neglect reactant “crossover” through the gas-diffusion (backing) layer between the two adjacent turns of the meander. This is justified unless a very narrow channel and/or highly permeable GDLs are used. Since channel length greatly exceeds MEA thickness, the characteristic length of variation of any parameter along the channel is much larger than the MEA thickness. This allows us to neglect the z-component of all fluxes in the MEA (Figure 1.5). The dominant transport in MEA thus occurs along the x-axis, whereas the flow in the channel transports reactants/products along the z-axis (Figure 1.5). Therefore, the 2D problem shown in Figure 1.5 can be split into two 1D problems: the channel problem along the z-axis and the internal problem in the MEA along the x-axis. The channel problem gives the “boundary conditions” (reagent concentrations) for the internal problem. The latter returns the local current density required to calculate the concentrations of reactants along the channel. Following the accepted terminology, this is a quasi-2D (or 1D + 1D) approach. Physically, quasi-2D effects due to nonuniformity of the oxygen concentration in the channel arise when the oxygen stoichiometry λ is not large. Low λ is typical of many real systems, since large stoichiometry requires more energy for pumping. In the channel, we will assume plug flow conditions, i.e. well-mixed flow with constant velocity directed along the z-axis. The rationale for this assumption is discussed in the next section.

4.1 Gas dynamics of channel flow In a fuel cell, feed gas consumption can affect the hydrodynamics of channel flow. For example, in PEFCs hydrogen atoms are consumed in the anode channel and appear in the cathode channel as a component of the water molecules. The electro-osmotic effect causes the transfer of water from the anode channel to the cathode channel. Due to these fluxes the mass of the flow changes, which may affect flow density and velocity. A rigorous approach to modelling the channel flows is based on 2D or 3D classical equations of viscous fluid dynamics (Dutta et al., 2000; Um et al., 2000; Wang et al., 2001; Wang, 2004). Numerical calculations, however, do not give parametric dependencies in the problem. The aim of this section is to understand the features of single-phase flow in the cathode channel of a PEFC or DMFC. The model below takes into account mass and momentum transfer through the channel/GDL interface. The model gives exact solutions and helps in clarifying how the electrochemical reactions and electro-osmotic effect affect the flow in the fuel cell channels (Kulikovsky, 2001).

4.1 GAS DYNAMICS OF CHANNEL FLOW

4.1.1

119

Momentum balance in the cathode flow

In practical devices, the channel usually has a rectangular cross-section (Figure 1.6, page 21). The steady flow in a long channel with impermeable walls is a Poiseuille flow at the constant velocity determined by the pressure gradient. However, flow velocity may vary due to mass and momentum transfer through the channel/GDL interface. To understand these effects we will neglect viscous forces, assuming, however, that the axial component of flow velocity at the channel/GDL interface is zero. The solutions below give the conditions when the effects of mass and momentum transfer through the channel/GDL interface are small and the plug flow assumption is justified. Note that we ignore the possible presence of liquid droplets in the channel and assume that the cathode flow is purely gaseous. The mass conservation equation in the cathode channel is (1.48). The momentum transfer (Euler) equation has the form ρ(v∇)v = −∇p.

(4.1)

On the right side of Eq. (4.1), we should add a term describing the momentum flux through the channel/GDL interface. Each oxygen molecule consumed removes momentum from the cathode flow and each water molecule injected into the flow must be accelerated to the average flow velocity v. Therefore, the flux of ith molecules through the channel/GDL interface induces the momentum flux ρi vx

∂vz vz ' ρ i vx . ∂x h

(4.2)

Since at the channel wall, vz ' 0, the derivative ∂vz /∂x is estimated here as vz /h (the system of coordinates is shown in Figure 1.6, page 21). Summing up the contributions of oxygen and water with the proper signs and taking into account (1.45) and (1.46) we obtain 1 jvz (ρox vox − ρw vw ) vz = − h h



2(1 + 2αw )Mw + Mox 4F

 .

This term should be added to the right side of (4.1) and the momentum balance equation takes the form   ∂vz ∂p jvz 2(1 + 2αw )Mw + Mox ρvz =− − . (4.3) ∂z ∂z h 4F 2 The gas law p p = ρRT /M = cs ρ allows us to exclude pressure from (4.3). Here cs = RT /M by the order of magnitude is the speed of sound and

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

M is the mean molecular weight of the mixture defined as ρ ρox ρw ρN 2 = + + . M Mox Mw MN 2

(4.4)

Equation (4.3) then takes the form jvz ∂ρ ∂vz + c2s =− ρvz ∂z ∂z h



2(1 + 2αw )Mw + Mox 4F

 .

(4.5)

In the following the subscript z will be omitted, that is v ≡ vz . Expanding the derivative on the left side of (1.48) and excluding ∂ρ/∂z from (1.48) and (4.5) we arrive at c2s

 
∂v jc2s 2(1 + 2αw )Mw − Mox −v = ∂z ρh 4F   2 jv 2(1 + 2αw )Mw + Mox + . ρh 4F 2

(4.6)

The first term on the right side of Eq. (4.6) describes the velocity variation due to the change in the flow mass. The second term describes this variation due to the momentum loss through the channel/GDL interface. Excluding in a similar manner ∂v/∂z from (1.48) and (4.5) we obtain c2s − v 2


∂ρ jv (1 + 2αw )Mw =− . ∂z h F

(4.7)

Equations (4.6) and (4.7) form a system of two nonlinear equations for two unknowns: v and ρ. Flow velocity is typically much less than the speed of sound. In the next section, this inequality is used to simplify and solve this system.

4.1.2

The limit of low flow velocity

When s v  cs

2(1 + 2αw )Mw − Mox . cs , 2(1 + 2αw )Mw + Mox

(4.8)

(which also means that v 2  c2s ), the second term on the right side of (4.6) can be neglected. Physically, due to the smallness of the flow velocity, momentum loss associated with the mass transfer through the channel/GDL interface is small.

4.1 GAS DYNAMICS OF CHANNEL FLOW

121

The term v 2 on the left side of (4.6) and (4.7) can also be neglected and these equations reduce to   ∂v j 2(1 + 2αw )Mw − Mox = ∂z ρh 4F   ∂ρ jv (1 + 2αw )Mw =− 2 . ∂z hcs F

(4.9) (4.10)

Thus, the variation of flow velocity and density is due to the mass transfer through the channel/GDL interface. The contribution of momentum transfer to these variations is negligible. At a temperature of about 100 ◦ C, the speed of sound cs is about 300 m s−1 . In practice, the inlet flow velocity usually does not exceed 10 m s−1 and the condition 3v  cs ,

(4.11)

which is necessary for further calculations is guaranteed. It is convenient to transform (4.9) and (4.10) to a dimensionless form. Introducing z˜ =

z , h

v˜ =

v , v0

ρ˜ =

ρ , ρ0

˜j =

j MH , ρ0 v 0 F

c˜ =

c v0

(4.12)

where v 0 and ρ0 are the inlet velocity and density, and MH is the molecular weight of hydrogen atom, we obtain ˜j ∂˜ v =ξ ∂ z˜ ρ˜ ˜j˜ ∂ ρ˜ v = −2(ζ + 8) 2 ∂ z˜ c˜s

(4.13) (4.14)

where ζ = 1 + 18αw .

(4.15)

Dividing (4.13) by (4.14), integrating the result and taking into account that at the inlet v˜ = 1 and ln ρ˜ = 0, we find v˜2 +



ζ ζ +8



c˜2s ln ρ˜ = 1.

(4.16)

Equation (4.16) is the first integral of the system (4.13) and (4.14). This equation is the analogue of the Bernoulli equation in classical hydrodynamics. Note that (4.16) does not contain the local current density.

122

CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Physically, v˜ and ρ˜ variations with z˜ are proportional to ˜j; hence their relative variation is independent of ˜j 2 . In the case of αw = 0 (zero net transport of water through the membrane), we have ζ = 1 and Eq. (4.16) takes the form  2 c˜s v˜2 + ln ρ˜ = 1. (4.17) 3 Equations (4.13) and (4.14) show that v increases and ρ decreases along z. The decrease in density is small due to the presence of a large factor c˜2s in the denominator of (4.14). However, due to the same factor c˜2s in (4.16) and (4.17), a small variation in ρ induces a large variation in flow velocity. This effect can be explained as follows. Even at αw = 0, one oxygen molecule in the cathode flow is replaced by two water molecules, that is, the total mass and the number of molecules in the flow increase. If αw > 0, the water flux from the anode side further increases the flow mass. Flow density, however, remains almost constant along z. This occurs due to the growth of velocity with z: an elementary volume of gas expands along z while it moves. To show this explicitly we express ρ˜ from (4.16) and substitute it into (4.13). This gives   2  ∂˜ v ˜ ζ +8 v˜ − 1 = j(˜ z )ζ exp , v˜(0) = 1. (4.18) ∂ z˜ ζ c˜2s This equation can be integrated to yield s ! v˜ ζ + 8 Erf c˜s ζ s !   Z z˜ 1 ζ +8 2 p ζ +8 ˜j d˜ = Erf + √ ζ(ζ + 8) exp − 2 z (4.19) c˜s ζ ζ˜ cs c˜s π 0 where Erf(u) is the error function. p In view of (4.15), the largest value of parameter (ζ + 8)/ζ is 3 (this value is achieved at αw = 0). From (4.11) it follows that the inequality 3/˜ cs  1 is fulfilled. Therefore, the argument of the Erf-function is small and√Eq. (4.19) can be further simplified. For small u, we have Erf(u) ' 2u/ π. Further, the exponential factor in the last term on the right side of 2 Solving

Eq. (4.16) for ζ we get  −1 c2 ln ρ˜ ζ = 8 1 − s2 . v˜ − 1

Thus, by measuring v˜ and ρ˜ we can determine the local water transfer coefficient αw along the cathode channel.

4.2 A MODEL OF PEFC

123

Eq. (4.19) can be replaced by 1. This finally gives Z z˜ Z ˜j d˜ v˜ = 1 + ζ z = 1 + (1 + 18αw ) 0

z˜

˜j d˜ z.

(4.20)

0

Comparing this to Eq. (4.13), we see that Eq. (4.20) is simply the solution of (4.13) with ρ˜ = 1. Thus, cathodic flow can be treated as incompressible, i.e. in the continuity equation (1.48) we may safely set ρ = const. Note that in Eq. (4.16), ρ is not constant: a small variation in ρ˜ is multiplied by the large factor c˜2s and gives a finite variation in v˜. In Section 4.2.1 we will show that at constant stoichiometry λ the local current density in the cathode channel of a PEFC varies as3   z/L 1 1 j = −Jλ ln 1 − 1− . λ λ Using this equation in (4.20) we get "  z/L # ˜ 1 LλJζ 1− 1− v˜ = 1 + h λ "  z/L # 4MH ξ 0 ζ 1 = 1+ 1− 1− M λ

(4.21)

where ξ is the oxygen molar fraction in the flow (in air, ξ = 0.21). Here we took into account that λJ˜ = 4hMH ξ 0 /(LM ), which follows from (4.12) and the definition of λ = 4F hv 0 c0h /(LJ). In the regime with constant stoichiometry, the variation of flow velocity along the channel is independent of the mean current in the cell. The function (4.21) is illustrated in Figure 4.1 for several values of parameter αw . We see that under realistic operating conditions (when αw ≤ 0.5) the velocity variation does not exceed 15% (Figure 4.1). Thus, to a good approximation we may simply set v = v 0 in model equations; this relation will be extensively used below. The effect of large water crossover on the cell performance will be considered in Section 4.2.3.

4.2 A model of PEFC Throughout this section we will assume that the cell is run in the lowcurrent regime, so that the polarization voltage of the cathode side is given by (3.5). 3 Strictly speaking, this solution is obtained under the condition of constant flow velocity. However, it can still be used if the velocity variation along the channel is not large.

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.1: The shape of the flow velocity along the air channel for the indicated values of the water transfer coefficient αw .

4.2.1

Oxygen concentration and local current along the channel

Consider the 1D expression for voltage loss (3.5). Following the general procedure of transition from the 1D to the quasi-2D model, we now treat oxygen concentration in the channel ch and current density j in this expression as local values, which depend on z. These values obey the following equations. ch is governed by the oxygen mass balance equation v0

j ∂ch =− , ∂z 4F h

(4.22)

which describes oxygen consumption at a stoichiometric rate proportional to the local current density j(z). The equation for j(z) follows from the equipotentiality of the cell electrodes. In the case of an ideally humidified membrane, equipotentiality means that η = E0 ,

(4.23)

where η is given by (3.5) and E0 = Voc − Vcell − RJ is the total voltage loss, constant along z. Equation (4.23) stems from the following arguments. Generally, the cell voltage is given by (3.9). Approximating the resistive term in this equation as RJ, we get Vcell = Voc − η − RJ. Voc and RJ in (4.24) do not depend on z, which gives us (4.23).

(4.24)

4.2 A MODEL OF PEFC

125

It is convenient to introduce dimensionless variables z˜ =

z , L

˜j = j 0 jD

ch , c0h

c˜h =

(4.25)

where L is the channel length, 0 jD =

4F Dbc c0h lbc

(4.26)

and the superscript “0” marks the values at the channel inlet. With these variables, Eq. (4.22) takes the form λJ˜

∂˜ ch = −˜j, ∂ z˜

c˜h (0) = 1.

(4.27)

Here 4F hv 0 c0h LJ

λ=

(4.28)

is the oxygen stoichiometry. Equation (3.5) in dimensionless variables (4.25) takes the form  η˜ = ln

˜j ˜j∗h c˜h



˜j − ln 1 − c˜h 

 (4.29)

where ˜j∗h =

i∗ ltc lbc . 4F Dbc cref

(4.30)

The solution to the system (4.27) and (4.29) for η independent of z˜ is (Kulikovsky, 2004a)  z˜ 1 c˜h (˜ z) = 1 − λ  z˜ 1 ˜j(˜ z ) = fλ J˜ 1 − , λ

(4.31) (4.32)

where4   1 fλ = −λ ln 1 − . λ

(4.33)

4 Inspection of Eq. (4.29) shows that for constant η ˜ the ratio ˜ j/˜ ch must also be constant. Using this in Eq. (4.27) we arrive at Eqs (4.31) and (4.32).

126 (a)

CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL (b)

Figure 4.2: The profiles of (a) oxygen concentration and (b) local current density along the oxygen channel for the indicated values of oxygen stoichiometry λ. (a)

(b)

Figure 4.3: Experimental (points) and theoretical (curves in (b)) profiles of local current density for the indicated values of oxygen stoichiometry. Experimental current distribution curves were obtained along a single flow channel (1 mm × 1 mm cross-section) at a cell potential of 0.492 V and cell temperature of 30 ◦ C. For further details of the experiment see Kulikovsky et al. (2005a). Equations (4.31) and (4.32) show that c˜ and ˜j/J˜ are universal functions of z˜, controlled by the single parameter λ. These functions are depicted in Figure 4.2. Physically, at lower λ, oxygen is consumed along z˜ faster and the local current also decays faster with z˜. Note that the ratio ˜j/˜ ch = fλ J˜ does not depend on z˜. The activation and transport terms in (4.29) are therefore constant along z˜. Figure 4.3(a) shows the experimental results from Ku`cernak’s group (Brett et al., 2001; Kulikovsky et al., 2005a). The distribution of local current density along the channel is given for different inlet flow rates, which correspond to different oxygen stoichiometries. Figure 4.3(b) shows the same results replotted in a dimensionless form. The agreement between theory and experiment is very good. Note that in this experiment, special measures have been taken to keep the membrane well humidified.

4.2 A MODEL OF PEFC (a)

127 (b)

Figure 4.4: (a) Function fλ and (b) the effect of indicated λ on cell polarization curve. Infinite λ corresponds to uniform distribution of oxygen concentration c˜(˜ z ) along the channel (Figure 4.2). As λ decreases, the nonuniformity of c˜(˜ z ) increases (Figure 4.2) and the apparent limiting current density decreases by a factor fλ .

4.2.2

Cell polarization curve

Substituting (4.31) and (4.32) into (4.29) yields a simple formula for the polarization voltage of the cathode side at finite oxygen stoichiometry (Kulikovsky, 2004a): !   fλ J˜ ˜ . η˜ = ln − ln 1 − f J (4.34) λ ˜j∗h Comparing (4.34) and (4.29), we see that the effect of finite λ reduces to re-scaling of the mean current density J˜ by a factor fλ . The function fλ is depicted in Figure 4.4(a). It tends to infinity as λ → 1 and it tends to 1 as λ → ∞. At large λ, fλ → 1 and Eq. (4.34) reduces to (4.29) (˜j ≡ J˜ in that case). However, as λ → 1, fλ grows rapidly and the apparent 0 limiting current density jD /fλ decreases (Figure 4.4(b)). The reason for this decrease is the nonuniformity of oxygen concentration along the channel (Figure 4.2(a))5 . With (4.34) the cell polarization curve (4.24) takes the form     fλ J fλ Jcref + b ln 1 − − RJ. (4.35) Vcell = Voc − b ln 0 i∗ lt c0h jD This equation perfectly fits the experimental polarization curves measured for different oxygen concentrations (Figure 4.5). For further details see Kulikovsky et al. (2005c). 5 Physically, the total limiting current density is the average of the local limiting currents (3.3) over z˜. This can be shown by direct averaging of (3.3) over z with ch = c0h (1 − 1/λ)z/L .

128

CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.5: Experimental polarization curves of a PEFC (points) and the fitting equation (4.35) (curves) for the indicated values of the oxygen molar fraction (%) in the cathode flow. For details see Kulikovsky et al. (2005c).

4.2.3

Water crossover and the polarization curve

Equations (4.34) and (4.35) are obtained assuming that the flow velocity in the channel is constant. However, large water crossover may induce variation of the flow velocity. The results of Section 4.1 allow us to rationalize the effect of water crossover on cell performance. Setting in Eq. (4.9) ρ = ρ0 and using the dimensionless variables (4.25) we get λJ˜ fα

!

∂˜ v ˜ =j ∂ z˜

(4.36)

where fα =

2(1 + 2αw )Mw − Mox M

(4.37)

and M is given by (4.4). At constant stoichiometry and variable flow velocity, the oxygen mass conservation equation reads λJ˜

∂(˜ v c˜h ) = −˜j. ∂ z˜

(4.38)

Equations (4.36) and (4.38) determine a relation between v˜ and c˜h . Dividing (4.38) by (4.36) we get fα

∂(˜ v c˜h ) = −1. ∂˜ v

(4.39)

4.2 A MODEL OF PEFC

129

Integrating we find fα (˜ v c˜h − 1) = 1 − v˜

(4.40)

since c˜h = 1 at v˜ = 1. Solving Eq. (4.40) for v˜ we obtain v˜ =

1 + fα . 1 + fα c˜h

(4.41)

Expanding the derivative on the left side of (4.38) and excluding v˜ with the help of (4.41), we get the equation for c˜h :   1 + fα ∂˜ ch = −˜j. (4.42) λJ˜ 2 (1 + fα c˜h ) ∂ z˜ Equation (4.42) reduces to (4.27) if the expression in the square brackets equals 1. This expression tends to 1 when fα → 0. With (4.37) we obtain the respective αw0 : αw0 =

Mox 1 1 − =− . 4Mw 2 18

(4.43)

Physically, for every proton transported through the membrane, an equivalent amount of water must be returned to the anode by back diffusion. In the opposite limit of large fα , Eq. (4.42) transforms to ! λJ˜ ∂˜ ch = −˜j. (4.44) fα c˜2h ∂ z˜ This is the case of significant water crossover6 . As discussed above, the equipotentiality of cell electrodes means that η˜ is constant. From Eq. (4.29) it follows that η˜ is constant if the local current density is proportional to the oxygen concentration: ˜j = k˜ ch , 6f

α

(4.47)

 1 means that 2(1 + 2αw )Mw − Mox  M , which is equivalent to αw 

M + Mox 1 − . 4Mw 2

(4.45)

For dry air, from (4.4) we get 1 ξox ξN 2 = + . M Mox MN 2

(4.46)

In dry air M ' 28.75 and (4.45) gives αw  0.34. Thus, this analysis is valid for αw & 2.

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

where k is constant. Using this in Eq. (4.44), we get ! λJ˜ ∂˜ ch = −k˜ ch , c˜h (0) = 1. 2 fα c˜h ∂ z˜

(4.48)

Solving this equation we find 1 . c˜h = q 1 + 2fλαJ˜k˜z

(4.49)

The constant k can be found if we substitute (4.49) into (4.47), integrate both R 1 sides of the resulting equation over z˜ and take into account that ˜ This gives ˜j d˜ z = J. 0 k = J˜ (1 + ψ)

(4.50)

where ψ=

fα . 2λ

(4.51)

With this k we finally find 1 c˜h = p 1 + 4ψ (1 + ψ) z˜ ˜ ˜j = p J (1 + ψ) . 1 + 4ψ (1 + ψ) z˜

(4.52) (4.53)

We see that both c˜h and ˜j are one-parametric functions of the distance z˜. These functions are depicted in Figure 4.6 for several values of ψ. Larger αw means larger fα and ψ; thus, according to Eqs (4.52) and (4.53) large αw leads to a dramatic decay of the oxygen concentration and local current density in a small domain close to the oxygen channel inlet (Figure 4.6). Significant water crossover thus increases the nonuniformity of local current. Physically, water accelerates flow in the channel, which lowers the molar concentration of oxygen. Note that to compensate for this effect a higher oxygen stoichiometry is needed (since ψ ∼ fα /λ). ˜ + ψ) into Eq. (4.29) we get the cathode Substituting ˜j/˜ ch = J(1 polarization voltage !   ˜ + ψ) J(1 ˜ + ψ) . η˜ = ln − ln 1 − J(1 (4.54) ˜j∗h We see that the limiting current density is lowered by a factor 1 + ψ. This is illustrated in Figure 4.7.

4.2 A MODEL OF PEFC

131

(a)

(b)

Figure 4.6: (a) Local oxygen concentration (4.52) and (b) current density (4.53) along the cathode channel for the indicated values of parameter ψ.

Figure 4.7: Cathodic voltage loss (4.54) for the indicated values of parameter ψ. Parameter ˜j∗h = 10−3 .

4.2.4

Local polarization curves

In many situations the cell is run at a constant inlet flow rate of oxygen, rather than at a constant stoichiometry. In that case, it is convenient to rewrite Eq. (4.27) as γ

∂˜ ch = −˜j ∂ z˜

(4.55)

hv 0 lbc . LDbc

(4.56)

where γ=

Under constant v 0 , parameter γ is also constant.

132

CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

To solve the system (4.29) and (4.55), we rearrange Eq. (4.29) as follows c˜h 1 = h + 1. ˜j ˜j∗ exp η˜ The right side of this equation is independent of z˜. Introducing the notation  a=

1 +1 ˜j∗h exp η˜

−1 (4.57)

we get ˜j = a˜ ch .

(4.58)

Substituting this into (4.55) and solving the resulting equation we get 

 a˜ z c˜h = exp − . γ

(4.59)

Using this in (4.58) we finally obtain   z ˜j = a exp − a˜ γ

(4.60)

where a(˜ η ) is given by (4.57). Relation (4.60) is the local polarization curve at a distance z˜ from the channel inlet. The set of curves for different distances along the channel is shown in Figure 4.8 together with the experimental curves obtained by Ku`cernak’s group at Imperial College (Kulikovsky et al., 2005a). We see that this simple model correctly reproduces the experimental picture. Close to the channel inlet the curves are monotonous, whereas close to the outlet they exhibit a distinct maximum (Figure 4.8). These maxima result from the effect of “oxygen starvation”. Physically, the points near the inlet always operate at an excess of oxygen and their local polarization curves are monotonous. However, the points located far from the inlet obtain the required amount of oxygen only at low overvoltages η˜ . 10-12 (Figure 4.8). At larger η˜, oxygen consumption upstream from the given point dramatically reduces the amount of oxygen in the downstream (“remote”) points and their polarization curves start “folding back” (Figure 4.8). In other words, remote points cannot produce the required current, since a large amount of oxygen has been consumed upstream. This S-shape effect only appears in the case of constant inlet velocity. It is easy to show that in the case of constant λ, local polarization curves are monotonous.

4.3 A MODEL OF PEFC WITH WATER MANAGEMENT

133

(a) (c)

(b)

Figure 4.8: (a), (b) Experimental voltage-current curves at different points along a single flow channel (1 mm×1 mm cross-section) on the cathode side of a fuel cell, for an air flow rate of 20 sccm (a), and 50 sccm (b). Hydrogen flow rate on anode of 25 sccm. Cell temperature of 30 ◦ C. Dimensionless distances along the channel are listed below each curve. Dotted line, average cell performance. (c) Theoretical local polarization curves (4.60) for the indicated dimensionless distances from the channel inlet. Parameters for these curves are listed in Tables 3.1 and 4.1. Table 4.1: Parameters for the curves in Figure 4.8. Oxygen molar concentration corresponds to air at T = 30 ◦ C. The other parameters required are listed in Table 3.1. h (cm) 0.1

v 0 (cm s−1 ) 2

L (cm) 10

c0h (mol cm−3 ) 8.31 · 10−6

c˜ref 1

4.3 A model of PEFC with water management So far the models in this chapter assume that the membrane is well humidified. However, such an ideal water management is difficult to achieve, especially in fuel cell stacks. A PEFC needs water to maintain the membrane in a wet state and it may suffer from excess of water. Due to insufficient humidification of the air flow, close to the channel inlet the membrane can dry up, thus limiting local current density. Close to the outlet, the excess of liquid water produced in the cell may retard oxygen transport to the catalyst sites (flooding). In recent years, tremendous efforts have been made to model two-phase water transport in PEFCs (see Meng and Wang (2005); Ye and Nguyen (2007); Sui et al. (2008a,b); Gurau et al. (2008b); earlier work is reviewed in Weber and Newman (2004); Wang (2004), while mathematical aspects of modelling are discussed in Promislow and Wetton (2009)). However, so far CFD models have only been able to predict the distribution of local

134

CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

parameters in the cell with an accuracy in the order of 100% (Sui et al., 2008b). In this section, we construct a model of a PEFC which couples transport of water and oxygen across the cell with oxygen depletion and water accumulation along the air channel (Kulikovsky, 2004c). The model helps us to understand the effect of water on cell performance without timeconsuming CFD calculations.

4.3.1

Model and governing equations

Proton current induces the electro-osmotic flux of water, which dries out the anode side of the membrane thereby increasing the membrane resistance. Below we will see that voltage loss in the membrane reduces the local polarization voltage of the cathode side and hence the local rate of oxygen consumption. Thus, local oxygen and water fluxes across the cell are strongly coupled. Furthermore, since oxygen is distributed nonuniformly along the MEA surface, the magnitude of these fluxes depends on the position on the cell surface. To take into account this coupling of fluxes, the model below employs the following assumptions. 1. Liquid water in the cathode GDL affects the transport of oxygen through lowering of the effective oxygen diffusion coefficient. 2. The oxygen and water vapour diffusion coefficients in the cathode GDL coincide. 3. The diffusion coefficient of liquid water in the membrane is constant. 4. Liquid water in the membrane is equilibrated with the water vapour in the cathode CCL according to the water sorption isotherm. 5. The total flux of water in the membrane is zero. The first assumption means that the effect of liquid water in the GDL is reduced to lowering of the effective oxygen diffusion coefficient. This assumption is supported by CFD calculations (Sui et al., 2008b and Ye and Nguyen, 2007; the latter work shows that in a wide range of mean current densities, liquid saturation in the GDL is constant). The last assumption means that the local electro-osmotic flux of water in the membrane is exactly counterbalanced by the back diffusion. Recent studies have shown that in a wide range of operating conditions the total transfer coefficient of water from the anode to the cathode does not exceed 0.2 (Janssen and Overvelde, 2001; Berg et al., 2004; Sui et al., 2008b). On the other hand, the electro-osmotic drag coefficient in Nafion is '1.5 (Fuller and Newman, 1992). Thus, the average over the cell surface electro-osmotic flux of water in the membrane is almost fully compensated for by the back diffusion. Note that the local value of total water flux in the membrane may significantly deviate from the surface-averaged value, e.g. close to the

4.3 A MODEL OF PEFC WITH WATER MANAGEMENT

135

outlet of the oxygen channel (Berg et al., 2004). Nevertheless, assumption 5 seems to be a reasonable approximation. Under these assumptions, the local voltage loss in the membrane Vm is given by (Kulikovsky, 2004b) 

 j

 , Vm = −bm ln 1 −  ψ j 0 h D r 2ξ +j 0

(4.61)

h

where ψh is the local water molar fraction in the channel, ξh0 is the oxygen molar fraction at the channel inlet, bm =

F Dw cH+ , σm1 nd

(4.62)

is the characteristic voltage loss, Dw is the diffusion coefficient of liquid water in the membrane, σm1 is the membrane conductivity at unit water content (here we adopt a simple linear dependence of membrane conductivity on water content λw : σm = σm1 λw ), nd is the electro-osmotic drug coefficient and cH+ is the molar concentration of protons/sulphonic groups in the membrane. The oxygen-limiting current density at the inlet 0 is given by (4.26). The dimensionless parameter r in (4.61) is jD r=

Dw lb Kw cH + , 4Db nd lm csat w

(4.63)

where Kw is the average slope of the membrane-water sorption isotherm (Kulikovsky, 2004b), lm is the membrane thickness and csat w is the molar concentration of saturated water vapour. Physically, r is proportional to the ratio of the mass transfer coefficient of liquid water in the membrane to the mass transfer coefficient of water vapour in the GDL. The parameter r thus describes the competition of two water fluxes of opposite direction: back diffusion, which wets the anode side of the membrane and water leakage through the cathode GDL to air channel, which facilitates membrane drying. Below we will see that r controls the local water-limiting current density. Water produced in the ORR enters the cathode channel, where it is transported toward the outlet. Therefore, the model should account for the effects due to variable composition of the mixture in the cathode channel. For that reason, in all relations the dependencies on ξh and ψh are shown explicitly. For the local polarization voltage of the cathode side we will employ the low-current relation (3.5), which can be transformed to η = ln bc



j ξh jT



  jξh0 − ln 1 − . 0 ξh jD

(4.64)

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Here  jT = lt i∗

c

 (4.65)

cref

and c is the total molar concentration in the channel. Comparing Eqs (4.65) and (3.6), we see that j∗h = ξh jT . The problem is hence governed by the following system of equations: first the equation of oxygen mass balance in the channel (4.22), and second the condition of equipotentiality of the cell electrodes. Now, however, voltage loss in the membrane varies along the channel and this condition reads η(z) + Vm (z) = E0 ,

(4.66)

where E0 is the total voltage loss independent of z (cf. Eq. (4.23)). Constant pressure in the channel means constant c; the sum of the molar fractions is thus also invariant along z: ψh (z) + ξh (z) + ζh = 1,

(4.67)

where ζh is the molar fraction of bulk gas (nitrogen). The system of equations (4.22), (4.61) and (4.64)–(4.67) forms the model for cell performance.

4.3.2

Solution at constant flow velocity

Introducing dimensionless variables ξh ξ˜h = 0 , ξh

ψh ψ˜h = 0 , ξh

V V˜ = c , b

˜j = j 0 jD

(4.68)

and substituting (4.61) and (4.64) into (4.66), Eqs (4.22) and (4.66) take the forms ∂ ξ˜h = −˜j ∂ z˜    ˜j ln − ln 1 − ξ˜h q γ

(4.69) ˜j ξ˜h







2˜j ˜0 . (4.70)  = E − p ln 1 −  r ψ˜h + 2˜j

˜0 = E0 /bc . Note that here we assume a As before, z˜ = z/L, η˜ = η/bc , E constant inlet flow rate; the case of constant oxygen stoichiometry will be considered in Section 4.3.7.

4.3 A MODEL OF PEFC WITH WATER MANAGEMENT

137

The behaviour of the system is thus governed by four parameters: γ, q, p and r. Parameter γ is given by (4.56), lt lb i∗ 4F Db cref bm p= c b q=

(4.71)

˜0 determines the working point and r is given by Eq. (4.63). Parameter E on the cell polarization curve. Equation (4.70) can be rewritten as −p  ˜j 1 1 1 2 1 −   . = + ˜j ˜0 ξ˜h q ξ˜h exp E r ψ˜h + 2˜j

(4.72)

˜0 ; the shape of the solution is thus determined As seen, q simply re-scales E by three parameters: γ, r and p. The system of equations (4.69), (4.72) and (4.67) determines the profiles of oxygen, water fractions and local current density along the channel. In the general case, this system has to be solved ˜0 this system has an analytical solution. numerically. However, for large E

4.3.3

˜0 → ∞) Close to the limiting current density (E

˜0 means that the cell operates close to the limiting Large voltage loss E current density. This case is of particular interest, since in this regime the effects of water management are best seen. Furthermore, solution at the limiting current helps us to understand the character of the solution in the general case. Equation (4.70) contains two terms exhibiting limiting behaviour: the second and the third logarithms on the left side. Equating the expressions under these logarithms to zero we find two asymptotic solutions to Eq. ˜0 → ∞: (4.70) at E ox ˜jlim = ξ˜h lim w ˜jlim = fr ψ˜h lim ,

(4.73) (4.74)

where fr =

r 2(1 − r)

(4.75)

˜0 → ∞. and the subscript “lim” denotes the values at E The first solution corresponds to the oxygen-limiting regime, when the local limiting current is determined by oxygen transport through the GDL.

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ox In that case both ˜jlim and ξ˜h lim exponentially decrease with z˜; this follows immediately if we substitute ˜j = ξ˜h into Eq. (4.69). The solution to this equation with the initial condition ξ˜h (0) = ξ˜hin is

  z˜ ox in ˜ ˜ ˜ ξh lim = jlim = ξh exp − . γ

(4.76)

Note that Eq. (4.76) can be obtained directly from Eqs (4.59) and (4.60) in the limit of η˜ → ∞ (aη → 1). Note also that, in general, ξ˜hin does not coincide with ξ˜h0 = 1 (see below). The second solution corresponds to the water-limiting regime, when local current is limited by membrane drying. This solution arises if fr > 0 or, equivalently, r < 1. When r ≥ 1 this solution disappears and for all z˜ we have an oxygen-limiting regime. Physically, at r ≥ 1 insufficient membrane humidification only reduces the cell potential, but does not affect the limiting current density. Thus, the case of r < 1 gives a new type of solution, which is of particular interest. Substituting   ˜j = fr 1 − ζ˜h − ξ˜h (4.77) ξh0 into Eq. (4.69) and solving the resulting equation with the initial condition ξ˜h (0) = 1 we obtain     fr z˜ 0 ˜ ˜ −1 . ξh lim (˜ z ) = 1 − ψh exp γ

(4.78)

Plugging this into Eq. (4.77) we find w ˜jlim (z) = fr ψ˜h0 exp



 fr z˜ . γ

(4.79)

For r < 1 parameter fr is positive; thus, in the water-limiting regime the local current density exponentially increases with z˜. Physically, in this case the local current is limited by membrane drying. The growth of ˜j with z˜ is due to the accumulation of water in the feed channel, which results in increasing membrane conductivity with z˜. Clearly, in both the oxygen- and water-limiting cases the oxygen concentration monotonically decreases with z˜, while the water concentration monotonically increases. Local current density, however, increases with z˜ in the water-limiting regime and decreases in the oxygen-limiting regime. We therefore have three cases. 1. Oxygen-limiting regime everywhere along z˜ (˜jlim decreases with z˜). 2. Water-limiting regime everywhere along z˜ (˜jlim grows with z˜).

4.3 A MODEL OF PEFC WITH WATER MANAGEMENT

139

3. Mixed case: water-limiting close to the inlet and oxygen-limiting in the rest of the cell (close to the inlet, ˜jlim increases with z˜ and then decreases). Local current density profiles in the first and the second case are described by (4.76) with ξ˜hin = 1 and by (4.79), respectively. In the mixed case, close to the inlet, ˜jlim is given by (4.79); along the rest of the channel it is given by (4.76). The point z˜max , which in the mixed case separates the water- and oxygen-limiting domains, is obtained from the continuity of local currents and oxygen concentrations at this point. Equating (4.78) to (4.76) and (4.79) to (4.76), we get       fr z˜max z˜max 1 − ψ˜h0 exp − 1 = ξ˜hin exp − γ γ     fr z˜max z˜max fr ψ˜h0 exp = ξ˜hin exp − . γ γ

(4.80)

Equating the left sides of these equations we find z˜max =

    γ ξ0 ln 1 + h0 − ln (1 + fr ) . fr ψh

(4.81)

On the right side, the sign “tilde” is omitted, since ξ˜h /ψ˜h ≡ ξh /ψh . Clearly, a mixed case is realized if 0 < z˜max < 1. This leads to the following condition fr <

ξh0 < (1 + fr ) exp ψh0



fr γ

 − 1.

(4.82)

Note that the onset of the mixed regime and the position of the point z˜max depend on the ratio ξh0 /ψh0 , rather than on ξh0 and ψh0 separately. Figure 4.9 shows the profiles of ˜jlim and ξ˜h lim in the mixed regime (the respective curves are marked by the symbol “∞”). max At z˜max , local current density reaches the maximum, ˜jlim . This value is obtained if we substitute Eq. (4.81) into Eq. (4.79): max ˜jlim =

fr 1 + fr

 1+

ψh0 ξh0

 .

(4.83)

max Note that ˜jlim does not depend on γ. The increase in γ simply shifts max z˜max toward the outlet leaving ˜jlim intact. The decrease in r also shifts max ˜ z˜max toward the outlet, but jlim then decreases. Physically, lower r means a higher rate of water removal through the GDL to the channel, which max facilitates membrane drying and reduces ˜jlim .

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

(a)

(b)

Figure 4.9: (a) Local current density and (b) oxygen concentration for the ˜0 . The other parameters are r = 0.5, indicated values of total voltage loss E β = 0.5, q = 10−3 , ζ = 0.6, ξh0 = 0.2, ψh0 = 0.2, p = 1. With Eq. (4.81) we can calculate ξ˜hin from Eq. (4.80),

ξ˜hin =

ψ0 fr 0h ξh

0 ξh 0 ψh



1+



1 + fr

r  1+f fr



.

(4.84)

w ox In the pure oxygen-limiting case, ˜jlim (0) = ˜jlim (0); thus fr ψh0 /ξh0 = 1 and hence z˜max = 0, which in turn implies that 1 + ξh0 /ψh0 = 1 + fr . Equation (4.84) then gives ξ˜hin = 1. The meaning of ξ˜hin is clear from Figure 4.9(b): this is the inlet oxygen concentration, which is obtained if the exponential limiting shape of the oxygen fraction at z˜ > z˜max had been extended down to z˜ = 0.

4.3.4

˜0 ) The general case (finite E

˜0 , the shapes of oxygen and local current density along the At finite E channel are given by the system of equations (4.69), (4.70) and (4.67). ˜0 are Numerical solutions to this system for several values of parameter E 7 depicted in Figure 4.9 . We see that the numerical solutions qualitatively reproduce the behaviour of the limiting curve: at the inlet, the current increases with z˜ (water-limiting domain) and then decreases (oxygen˜0 the numerical solution limiting domain). As expected, with the growth of E tends to the limiting analytical curve. 7 To

simplify numerical calculations it is useful to differentiate Eq. (4.72) with respect to z˜ and to transform the result into a differential equation for ˜ j using Eq. (4.69). The initial condition ˜ j(0) is obtained from (4.70) or (4.72). A numerical solution of the system of two first-order ODEs can easily be obtained with any mathematical software (see the respective Maple code).

4.3 A MODEL OF PEFC WITH WATER MANAGEMENT

141

Figure 4.10: Dimensionless overpotential η˜ (solid curves) and voltage loss in membrane V˜m (dashed curves) for the indicated values of total voltage ˜0 . The other parameters are the same as in Figure 4.9. loss E ˜0 , Figure 4.9(a) shows that the numerical plots ˜j(˜ z ) are smooth at finite E but they tend to a limiting curve, which is non-differentiable at z˜max . The disappearance of the derivative ∂ ˜jlim /∂ z˜ at z˜max suggests singularity at this point. The plots of η˜ and membrane voltage loss V˜m as a function of z˜ are shown in Figure 4.10. Although the sum η˜ + V˜m remains constant along z˜, near z˜max both η˜ and V˜m exhibit a distinct gradient, which increases with ˜0 (Figure 4.10). At large E ˜0 , near z˜max a large z˜-component the growth of E of the proton current density arises. Thus, operation in the mixed regime close to the limiting current induces this additional overhead. Physically, the z˜-gradient of η˜ is induced by the growth of membranewater content with z˜. This growth means the decrease in membrane voltage loss. Since η˜+ V˜m is constant, the decrease in V˜m means the increase in η˜. As ˜0 → ∞, the gradient of polarization voltage at z˜max also tends to infinity E (Figure 4.10). This means an infinite z-component of proton current density in the CCL and membrane. Large ∂ η˜/∂ z˜ means large oxygen consumption to support this parasitic proton current. Evidently, the quasi-2D model fails to describe the details of this effect. In the vicinity of z˜max , a true 2D distribution of the local proton current arises. An accurate description of this situation requires numerical calculations.

4.3.5

Model validation

Berg et al. (Berg et al., 2004) developed a numerical quasi-2D model of a PEFC and validated it against measured along-the-channel profiles of local current density. To verify our model we fitted the solution of equations (4.69), (4.70) and (4.67) to the experimental data (Berg et al., 2004). At a given feed composition, the system (4.69), (4.70) and (4.67) contains ˜0 , j 0 , p and r; for simplicity we take p = 1. The 6 fitting parameters: γ, q, E D

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.11: The shapes of local current density for the two dew points of the humidifier (two compositions of the cathode feed). Points: experimental data of (Berg et al., 2004). Squares correspond to the dew point of 0 ◦ C, and crosses—dew point of 43 ◦ C. Solid lines—model. Parameters for the ˜0 = 12, β = 0.8; r = 0.8 for dew point 0 ◦ C curves are p = 1, q = 10−3 , E 0 and r = 0.93 for dew point 43 ◦ C. In both cases jD = 2 A cm−2 . curves were fitted to the data by the trial-and-error method; the results are shown in Figure 4.11 (fitting parameters are listed in the caption to this figure). As can be seen, the model fits the experiment reasonably well. In both cases shown in Figure 4.11, the local current has a maximum, i.e. the cell operates in a mixed regime. The rise in humidifier temperature from 0 ◦ C to 43 ◦ C means a 10-fold increase in the inlet fraction of water. At lower water content, the current in the water-limiting domain rises faster with distance (squares in Figure 4.11). For further details see Kulikovsky (2004c).

4.3.6

Limiting current and optimal feed composition

Integration of ˜jlim (4.76) and (4.79) over z˜ yields the respective overall limiting current density of the cell J˜lim . For the water-limiting, oxygenlimiting and mixed cases, respectively, we write: w J˜lim =

Z

1 w ˜jlim d˜ z

(4.85)

ox ˜jlim d˜ z

(4.86)

0 ox J˜lim =

Z

1

0 mix J˜lim =

Z 0

z˜max

w ˜jlim d˜ z+

Z

1 ox ˜jlim d˜ z.

(4.87)

z˜max

Polarization curves are usually measured at a constant oxygen stoichiometry ratio λ rather than at a constant γ. Integrals (4.85)–(4.87)

4.3 A MODEL OF PEFC WITH WATER MANAGEMENT

143

can be transformed to the case of constant λ using the identity γ = J˜lim λ. Finally this gives   −1 S w J˜lim = fr λ ln 1 + (4.88) λ −1   1 ox ˜ (4.89) Jlim = − λ ln 1 − λ      −1 λfr 1+S mix J˜lim = fr λ fr ln + (1 + fr ) ln . (4.90) S (λ − 1) 1 + fr Here we denote S = ξh0 /ψh0 , the ratio of oxygen to water molar fractions at the inlet. Note that Eq. (4.90) is valid only if the condition of the mixed regime (4.82) is fulfilled; otherwise one of the equations (4.88) and (4.89) should be used. In particular, one has to be careful in varying the parameters in Eq. (4.90), since variation of λ, fr and S shifts the position of z˜max . ox As expected, J˜lim does not depend on water content. In the water-limiting and mixed regimes, J˜lim depends on the ratio ξh0 /ψh0 rather than on ξh0 and ψh0 separately. It follows that if we vary ξh0 ∼ ψh0 , the value of the dimensionless limiting current density J˜lim in these regimes does not change. Note that Jlim in the dimensional form is proportional to ξh0 , since the 0 normalization factor jD ∼ ξh0 . In the mixed case there is an optimal feed composition, which provides mix mix maximal J˜lim . This composition is a root of equation ∂ J˜lim /∂S = 0. The solution to this equation is S = fr or  0 ξh r = . (4.91) 0 ψh opt 2(1 − r) From (4.81) it is seen that for ξh0 /ψh0 = fr we have z˜max = 0, or equivalently, ˜j w (0) = ˜j ox (0). The optimal feed composition (4.91), therefore, provides lim lim equality of water- and oxygen-limiting currents at the channel inlet. This means that everywhere along z˜ the current is not limited by membrane drying. In other words, ψh0 = ξh0 /fr is the optimal water fraction preventing membrane dehydration. mix on ξh0 /ψh0 for different r is illustrated in The dependence of J˜lim Figure 4.12. We see that for r . 0.5 the curve exhibits a sharp maximum. To the left of the maximum the cell suffers from oxygen “starvation”, and to the right it experiences “thirst”.

4.3.7

Constant oxygen stoichiometry

In this section, we re-formulate the model equations for the case of constant oxygen stoichiometry and obtain the criterion of ideal membrane humidification.

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.12: Limiting current density in the mixed regime as a function of the ratio of inlet oxygen to water fractions for the indicated values of r. Oxygen stoichiometry λ = 2.

Modification of model equations If the cell is run under constant oxygen stoichiometry λ, Eq. (4.69) should be modified. Using the identity γ = λJ˜ we get λJ˜

∂ ξ˜h = −˜j, ∂ z˜

ξ˜h (0) = 1.

(4.92)

Taking into account (4.67), Eq. (4.92) yields ˜j ∂ ψ˜h ∂ ξ˜h =− = . ∂ z˜ ∂ z˜ λJ˜

(4.93)

For further analysis we will convert (4.70) into a differential equation for ˜j. Differentiating (4.70) with respect to z˜ and using (4.93) we come to h i 2 2 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ j −r ψ + 2(1 − 2r) j ψ + 2p j ξ + 2(2 − 2r − p) j h h h ∂ ˜j h i, = ∂ z˜ λJ˜ rξ˜h ψ˜h2 + 2(2r − 1 + p)˜j ξ˜h ψ˜h − 2p˜j 2 ψ˜h − 4(1 − r)˜j 2 ξ˜h ˜j(0) = ˜j 0 . (4.94) An equation for the local current density at the inlet ˜j 0 is obtained if we substitute inlet values ξ˜h = 1, ψ˜h = ψ˜h0 and ˜j = ˜j 0 into (4.70):  ln

˜j 0 q



  0 ˜
2 j ˜0 ,  = E − ln 1 − ˜j 0 − p ln 1 −  0 0 ˜ ˜ r ψh + 2j

where ψ˜h0 = 1/ξ 0 − 1 − ζ˜h .

(4.95)

4.3 A MODEL OF PEFC WITH WATER MANAGEMENT

145

The system (4.92), (4.94), (4.95) determines the oxygen, water vapour and local current density profiles along the channel in the case of constant λ. Note that ˜j must satisfy the following constraint 1

Z

˜ ˜j d˜ z = J.

(4.96)

0

Condition of ideal membrane humidification For small ˜j the system (4.92), (4.94), (4.95) has an explicit solution. Expanding the right side of (4.94) over ˜j and retaining the first two nonvanishing terms we get " # ˜j 2 2p(ξ˜h + ψ˜h )˜j ∂ ˜j =− 1− . λ ∂ z˜ J˜ξ˜h rψ˜h2

(4.97)

The second term in the square brackets describes the effect of water management (see below). This term is negligible if ˜j 

rψ˜h2 . 2p(ξ˜h + ψ˜h )

(4.98)

Since ξ˜h + ψ˜h = ξ˜h0 + ψ˜h0 ≡ 1 + ψ˜h0 and ψ˜h ≥ ψ˜h0 , in (4.98) we can replace ξ˜h by 1 and ψ˜h by ψ˜h0 , which yields ˜j 

r(ψ˜0 )2  h . 2p 1 + ψ˜h0

(4.99)

This condition holds for sufficiently large water concentrations at the inlet ψ˜h0 and/or large parameter r. Physically, large r means a high rate of liquid water back diffusion in the membrane and/or a low rate of water vapour “leakage” to the cathode channel. In both these cases, the upper limit of the current density which satisfies (4.99) increases. Taking into account (4.71) and (4.26), Eq. (4.99) in the dimensional form reads jc (ψh0 )2 ψh0 + ξh0

(4.100)

bc σm1 Kw c lm csat w

(4.101)

j where jc =

146

CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

is the characteristic current density. Note that csat w rapidly increases with the cell temperature, so that at high T much more water is needed to keep the membrane well humidified. If (4.99) is fulfilled we can neglect the second term in the square brackets in Eq. (4.97). This equation then takes the form λ

˜j 2 ∂ ˜j =− , ∂ z˜ J˜ξ˜h

˜j(0) = ˜j 0 .

(4.102)

The oxygen fraction ξ˜h is still governed by (4.92). The solution to (4.92) and (4.102) is   z˜ ˜ ξh = exp − (4.103) µ   ˜j = ˜j 0 exp − z˜ , (4.104) µ ˜ ˜j 0 . The local current density at the inlet ˜j 0 is determined where µ = λJ/ R1 ˜ Using (4.104) and calculating the integral we by the condition 0 ˜j d˜ z = J. find ˜ ˜j 0 = fλ J. where fλ is given by (4.33). The characteristic scale µ is then −1   1 . µ = − ln 1 − λ

(4.105)

(4.106)

The solutions (4.103)–(4.106) do not contain the water management parameters r and p. Furthermore, the exponential shapes of ξ˜h and ˜j/J˜ are governed by a single parameter, the oxygen stoichiometry λ. Eqs (4.103) and (4.104) with µ (4.106) and ˜j 0 (4.105) coincide with the solutions (4.31), (4.32) in Section 4.2.1, where ideal membrane humidification is assumed. This means that the current which obeys (4.99) does not produce any significant nonuniformity of membrane resistance along z˜. Equation (4.99) or (4.100) is thus the condition of ideal membrane humidification. Physically, if the inlet water concentration is large enough and ˜j obeys (4.100), the water produced in the ORR has practically no effect on the profiles of oxygen concentration and local current along the channel. Membrane resistance and voltage loss in the membrane Vm are then almost constant along z˜. Since Vm is constant, it can be included in the “contact” resistances and the model above reduces to that of Section 4.2.1. It is worth mentioning that the model discussed ignores local 2D effects due to nonuniformity of oxygen and water distribution under the channel/rib (Kornyshev and Kulikovsky, 2001). This nonuniformity also reduces the effective diffusion coefficient of oxygen in 1D or quasi-2D

4.4 DEGRADATION WAVE

147

models. There is evidence that these effects can be approximately accounted for in quasi-2D models by a simple correction of the oxygen diffusion coefficient (Kulikovsky, 2005c).

4.4 Degradation wave Like living organisms, fuel cells also experience ageing. The proton current, flux of water and thermal effects inevitably change the structure and composition of cell components, thereby causing a degradation of cell performance. The main factors determining cell performance are: (i) kinetics of electrochemical reactions, (ii) conductivity and permeability of membrane, and (iii) transport properties of the catalyst and backing layers. Ageing, therefore, may be thought of as an irreversible change of one or several kinetic or transport parameters. In this section we show that the quasi2D model makes it possible to predict general scenarios of the degradation process, not specifying its microscopic origin. To simulate conditions in a stack, in life test experiments the cell is usually run under fixed total current (galvanostatic regime). In this regime, degradation manifests itself as a decrease in cell voltage with time. This decrease is not linear: a gradual change is usually followed by a very rapid drop to zero (Ahn et al., 2002; Kulikovsky et al., 2004). The model in Section 4.2 helps in explaining this behaviour. Qualitatively, we may expect that the rate of degradation is higher in the regions where the local current density j is high. We will assume that this rate is a stepwise function of j. In other words, there is a critical current density jcrit , at which the rate of degradation jumps from zero to a certain finite value. Let the characteristic time of local degradation be τd , i.e. when τd expires the region where j > jcrit no longer produces current. These assumptions and a model in Section 4.2 lead to an interesting phenomenon. Suppose that at time t0 the local current density exceeds jcrit (Figure 4.13). The coloured domain at t0 is then subject to local degradation (Figure 4.13). In time τd this domain no longer produces current. At t1 = t0 +τd , the peak of the local current density shifts to a new position and a new domain is “exposed” to degradation (Figure 4.13). Since total current is fixed, the length of this domain increases with time (Figure 4.13). Clearly, this mechanism causes propagation of the degradation wave (DW). In this section, we obtain the equation for the DW propagation and investigate the features of these waves (Kulikovsky et al., 2004).

4.4.1

Model

Let zw (t) be the instant position of the wavefront (vertical lines in Figure 4.13). The instant length of the current-generating domain is L−zw ;

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.13: Sketch of the degradation wave. The domain where local current density exceeds the critical value (dashed areas at t0 , t1 , . . . , tn ) is subject to degradation. The degradation wave propagates toward z/L = 1. a simple generalization of Eq. (4.32) for that case is    z−zw     L 1 L−zw f J 1 − , z ≥ zw λ j(z) = L − zw λ  0, z < z . w

(4.107)

Consider the case of fixed total current Itot and λ (galvanostatic regime under constant oxygen stoichiometry), which is of practical importance. The profile (4.107) provides the invariance of Itot with time. Indeed, L

Z Itot (t) = dh

L

Z j(z) dz = dh

0

j(z) dz = dh LJ = Itot (0), (4.108) zw

where dh is the in-plane width of the channel. The length of the domain exposed to degradation lw (Figure 4.13) is determined by condition j(zw + lw ) = jcrit , or J fλ 1 − z˜w



1 1− λ

l˜w  1−˜ z

w

= jcrit

(4.109)

where the dimensionless length and distance are ˜lw = lw , L

z˜w =

zw . L

Solving Eq. (4.109) for ˜lw , we get ˜lw =

(1 − z˜w ) ln



ln 1

jcrit (1−˜ zw ) fλ J
− λ1

 .

(4.110)

4.4 DEGRADATION WAVE

149

Since the logarithm in the denominator is negative (λ > 1), the logarithm in the numerator must also be negative. At t = 0, we have z˜w = 0; the condition of DW generation is then a≡

fλ J > 1, jcrit

or fλ J > jcrit .

(4.111)

Setting in (4.107), z = zw = 0, we get fλ J = j(0). Relation (4.111), therefore, is equivalent to j(0)|t=0 > jcrit .

(4.112)

In other words, the wave is initiated when the maximum (inlet) local current density exceeds jcrit .

4.4.2

Wave propagation

The velocity of the wave is vw = lw /τd . The wavefront propagates according to the equation ∂zw /∂t = vw . With (4.110) we get   1 − z˜w ∂ z˜w = − (1 − z˜w ) ln , z˜w |t˜=0 = 0, (4.113) a ∂ t˜ where the dimensionless time is t t˜ = , τw

  1 . τw = −τd ln 1 − λ

(4.114)

The solution to (4.113) is8
z˜w = 1 − a exp − ln(a) exp t˜ .

(4.115)

We see that in the dimensionless variables, wave propagation is controlled by a single “start-up” parameter a (4.111). The dependence z˜w (t˜) is depicted in Figure 4.14 for several values of a. This figure shows that when ln(a) is not small, the wave moves very fast (as a double exponent of time; see Eq. (4.115)). If, however, ln a → 0 (a → 1) we have the two distinct phases of wave propagation. Near z˜ = 0 the wave moves very slowly due to the small coefficient ln(a) in Eq. (4.115) (slow phase). As time progresses, exp t˜ in Eq. (4.115) increases and the slow phase is followed by the fast propagation phase, when the wave velocity reaches the maximum (Figure 4.14). 8 In fact, the wave propagation is discrete: the wavefront waits until the domain 0 → z + l . Transition j > jcrit is degraded and then jumps to a new position zw w w to continuous velocity (4.113) means that we interpolate discrete front positions with a continuous function of time.

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Figure 4.14: Dynamics of the degradation wave for indicated values of parameter a. Solid lines—position of the wavefront z˜w , and dashed lines— wave velocity v˜w . The duration of the slow propagation phase can be estimated as follows. Differentiating (4.115) we get the dimensionless wave velocity v˜w = ∂ z˜w /∂ t˜:   v˜w = a ln(a) exp t˜ − ln(a) exp t˜ .

(4.116)

t˜v max = − ln(ln a)

(4.117)

At

the wave velocity (4.116) reaches a maximum v˜max = a exp(−1) ' 0.368 a.

(4.118)

The distance z˜v max travelled before v˜w reaches the maximum is obtained by substituting t˜ = t˜v max into (4.115). This gives z˜v max = 1 −

a . exp(1)

(4.119)

Note that z˜v max > 0 if 1 < a ≤ exp(1); however, if a > exp(1) we have z˜v max < 0, t˜v max < 0, and the maximum of velocity is achieved at t˜ = 0. In that case, v˜w decreases with distance, which corresponds to the onset of the fast propagation phase at t˜ = 0. The time (4.117) it takes for the velocity to reach the maximum by the order of magnitude is the time of slow propagation. To show this we substitute a = 1 + ε into (4.116) and expand the result over ε. Retaining the quadratic term we get     1 v˜w ' exp(t˜) 1 + − exp(t˜) ε ε + O(ε3 ). 2

(4.120)

4.4 DEGRADATION WAVE Equating the expression in square brackets to zero we find   1 1 t˜slow = ln + ' − ln ε 2 ε

151

(4.121)

since ε  1. At t˜ = t˜slow , the linear and quadratic in ε terms in (4.120) cancel out and the velocity is very small, v˜w ∼ O(ε3 ). For t˜ < t˜slow , the function v˜w (t) (4.116) increases monotonically; therefore for all t˜ < t˜slow , the wave velocity is even smaller. With a = 1 + ε, Eq. (4.117) gives t˜v max = − ln(ln a) ' − ln(ε) and we see that t˜v max ' t˜slow . Qualitatively, for small ε, the time of fast propagation is much smaller than the time of slow propagation, and t˜v max and t˜slow are close to each other. Physically, for ε  1, the peak j(0) only slightly exceeds jcrit (the coloured area at t0 in Figure 4.13 is small). In this regime, the initial wave velocity is small since the length of the domain exposed to degradation is small. Indeed, putting in (4.110), a = 1 + ε and z˜w = 0, and expanding the logarithm we get lw = −ε/ ln (1 − (1/λ)), i.e. lw ∼ ε. The wave must slowly travel a certain distance until lw is sufficiently large to provide further fast propagation.

4.4.3

Cell potential

From (4.107) and (4.115), we find the instant mean current density Jw ahead of the wavefront: Z L 1 J JL Jw = = j dz = L − zw zw L − zw 1 − z˜w      J t = exp ln(a) exp . (4.122) a τw This shows that Jw increases rapidly with time. When a → 1, rapid growth is preceded by the phase of a slow rise. The growth of current ahead of the wave is accompanied by the rise of the half-cell polarization voltage η. Evidently, with the growth of current the transport losses dominate and to rationalize the dependence η˜(t˜) we have to consider the transport term in Eq. (4.34):   η fλ Jw ' − ln 1 − . (4.123) b jD Substituting Jw (4.122) into (4.123) and taking into account (4.111) we get the dependence of η on time:   η jcrit = − ln 1 − ∗ (4.124) b jD (t)

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.15: Cathode polarization voltage as a function of dimensionless time for the indicated values of parameter a. where the limiting current density    t ∗ jD (t) = jD exp − ln(a) exp τw

(4.125)

decreases with time. Physically, the cell potential is determined by the voltage loss due to oxygen transport in the working domain of a cell. This loss rapidly increases with time, since more and more oxygen must be transported through the GDL into the unspoilt domain to support the growing current density. Equating the expression under the logarithm in (4.124) to zero and taking into account (4.125) we get the cell lifetime    jD t˜lif e = − ln(ln a) + ln ln = t˜slow + t˜D jcrit

(4.126)

   jD t˜D = ln ln jcrit

(4.127)

where

is discussed below. Physically, as t˜ → t˜lif e , η → ∞, that is, the cell potential drops to zero. The function (4.124) is shown in Figure 4.15; the catastrophe at t˜ = t˜lif e is clearly seen. Suppose that tslow is positive, that is, 1 < a < exp(1); we then have two cases. If 1 < jD /jcrit < exp(1), the term t˜D is negative. This means that the cell potential “dies” before the end of the slow propagation phase. Physically, the large transport loss in the GDL leads to a faster drop of the cell voltage. If jD /jcrit > exp(1) (small transport loss), t˜D in Eq. (4.126) is positive and the time of cell operation somewhat exceeds t˜slow .

4.4 DEGRADATION WAVE (a)

153 (b)

Figure 4.16: The two scenarios of fuel cell degradation. (a) λ − 1  1: local current density exceeds the critical level at cell start-up. The degradation wave starts immediately. (b) λ − 1 & 1: the critical current density initially exceeds j0 (0). Due to the slow process of ageing, jcrit decreases with time and at some moment in time the condition ˜j0 (0) = jcrit is fulfilled. This initiates slow and then fast propagation of the degradation wave.

4.4.4

Two scenarios of cell performance degradation

No assumptions were made on the microscopic mechanism of local degradation under the overcurrent conditions. Rather, we took the phenomenological ansatz that it takes a time τd to “spoil” the domain in which the local current exceeds jcrit . Parameters jcrit and τd are determined by the physical mechanism of cell degradation. The scaling of the t˜-axis and the shapes of the curves in Figure 4.15 contain information on τd . Thus, using Eq. (4.124) for fitting the experimental voltage-time curves may give information on τd and hence this may help us to guess the microscopic mechanism of degradation. If degradation is caused by thermal effects, τd would be inversely proportional to the local current density. Since local current in front of the wave increases with time, τd would decrease with time. Qualitatively, this would lead to even faster wave propagation. Suppose that a fresh MEA is characterized by jcrit associated with some physical mechanism of degradation. Depending upon the oxygen stoichiometry λ, we may have two regimes of cell operation: • λ − 1  1 (λ is close to 1 ). In that case, j(0) may exceed jcrit at the cell start-up. DW then starts immediately (Figure 4.16(a)) and the cell voltage immediately begins to decrease. • λ − 1 & 1 (λ is about or exceeds 2 ). In that case, jcrit most probably exceeds j(0) (Figure 4.16(b)). However, jcrit may itself decrease with time due to some slow process of ageing. At some moment in time, j(0) would slightly exceed jcrit (or, equivalently, at this moment a = 1+ε). This initiates slow and then fast propagation of DW, which finally results in a catastrophic drop of cell potential.

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

In the second scenario, we have three characteristic times: the time of ageing (slow decrease of critical current density) tageing , the time of slow wave propagation tslow and the time of fast propagation tf ast . In view of (4.114), tslow and tf ast are proportional to the local time of degradation τd . The role of λ is two-fold. For fixed mean current density, an increase in λ flattens the profile j(z) and lowers j(0) (Figure 4.2), thus giving more time for ageing. On the other hand, larger λ shortens the period of slow propagation (see (4.121)). At large λ, once the local current exceeds jcrit , the cell potential will drop very fast, on a time scale of the order of τd . Nonuniformity of any parameter along the channel (e.g. partial cell flooding or membrane drying) increases nonuniformity of local current density and facilitates DW generation. If some region of the cell is flooded, local current in the non-flooded region increases and local j can exceed jcrit . The flooding itself may follow one of the degradation scenarios. If a local increase in current density leads to local flooding, we have the conditions for generation of the wave of flooding.

4.4.5

Accelerated testing of ageing phenomena

There is a rapidly growing literature on various aspects of fuel cell ageing (see the review (Borup et al., 2007)). In this section, we discuss a general approach to the problem of accelerated ageing and illustrate this approach by a simple example. The lifetime of a well-designed PEFC stack can reach 20,000 hours, which is equivalent to more than 2 years of continuous operation. The cost of such a long experiment is prohibitively high and various engineering approaches to accelerate stack or cell ageing using “stressed” conditions have been developed (Wu et al., 2008). However, care has to be taken when mapping the time in stressed experimental conditions to the real time under standard operating conditions. The relation between “accelerated” and real times should be established on a modelling basis. Importantly, “stressed” experimental conditions must obey modelling assumptions. To illustrate the problem, consider the following example. Suppose that the catalyst layer ageing occurs due to a certain electrochemical process, characterized by the Tafel rate Q(T, η). Typically, the electrochemical ageing process can be considered as secondary, i.e. this process is controlled by the distribution of overpotential η, which is determined by the useful half-cell reaction. Suppose that the ageing occurs uniformly over the CL volume and the reaction products are immediately removed. Assuming for simplicity that the ageing reaction is first-order in the molar concentration c of the

4.4 DEGRADATION WAVE

155

component subject to ageing, the continuity equation for c reads ∂c Q(T, η) =− ∂t nF





c

.

cref

(4.128)

The solution to this equation is  c = c0 exp −

Qt nF cref

 (4.129)

where c0 = c(0) is the initial concentration of the component. The temperature dependence of the reaction rate usually takes the Arrhenius form (1.21); therefore, the reaction rate is Q(T, η) =

i0ref



T∗ exp − T

 exp

η b

where i0ref is the reference rate (A cm−3 ) of the reaction under consideration and b is the Tafel slope of the useful half-cell reaction. With this Q, Eq. (4.129) takes the form    η i0∗ T∗ c = c0 exp − exp exp − t . nF cref b T 

(4.130)

With (4.130) in hand, we can force time to run “faster”. Consider first the case when the stressing variable is T . Writing (4.130) for the standard operating temperature Tstd and time tstd , then for the enhanced experimental temperature Tenh and time tenh , and equating the resulting expressions we get  tstd = tenh exp

T∗ T∗ − Tstd Tenh

 .

(4.131)

At Tenh > Tstd , the power of the exponent is positive and hence tenh < tstd . Thus, increasing the temperature to Tenh we can simulate faster ageing. Equation (4.131) provides the mapping of “accelerated” and real times. Another option would be to use the overpotential as a stressing variable; in other words we could accelerate the time by increasing η. A simple calculation gives  tstd = tenh exp With ηenh > ηstd , we get tstd > tenh .

ηenh − ηstd b

 .

(4.132)

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

It should be emphasized that these results are valid, provided: • the process runs uniformly over the testing volume; • the reaction rate constant increases with the stressing variable; • the transport of all species involved in the reaction is fast. The example above suggests that the simple mapping of experimental and real times can be obtained only in the simplest case of a uniform in space reaction. A concerted action of species diffusion and reaction makes the mapping problem much more difficult. The situation of simultaneous diffusion and reaction, however, is typical of ageing processes. One thing is obvious: accelerated ageing experiments should always be preceded by a theoretical analysis.

4.5 Gradient of catalyst loading along the oxygen channel Unless special measures are taken, the distribution of local current over the surface of a fuel cell is nonuniform. Inhomogeneity in local current is detrimental for stable long-term operation of a cell. A large local current is accompanied by an increase in local polarization voltage and heating. These processes accelerate the ageing of cell components thereby reducing cell lifetime. Inhomogeneity in j is typically caused by uneven distribution of reactants. In Section 4.2.1, it was shown that in a non-flooded cell, both oxygen concentration cox and j decrease exponentially along z. However, the ORR rate is also proportional to the local exchange current density. This value includes an active surface of catalyst particles, which in turn is proportional to the catalyst concentration (loading). Is it possible to compensate for the decrease of cox along the channel by the increasing catalyst loading along z? Santis et al. managed to significantly flatten the distribution of local current using the nonuniform Pt loading (Santis et al., 2006b). However, local current homogenization was achieved at the cost of decreasing cell performance. Prasanna et al. showed that the nonuniform distribution of Pt enables total Pt loading to be reduced without sacrificing the cell performance (Prasanna et al., 2007). In both these works, various shapes of Pt loading growing toward the outlet of the oxygen channel were tested. It should be noted that in Santis et al. (2006b) and Prasanna et al. (2007) these shapes were chosen rather arbitrarily; they did not follow from theory. In this section, we derive a simple analytical expression for the optimal shape of catalyst loading along the oxygen channel. This shape appears to be a one-parametric function of the distance z, with the parameter being the oxygen stoichiometry. The model shows that the optimal shape of Pt loading homogenizes the local cell current without affecting the

4.5 CATALYST GRADIENT ALONG THE CHANNEL

157

cell polarization curve. Importantly, in the regime with the fixed oxygen stoichiometry, this optimal shape does not depend on the total cell current.

4.5.1

Low cell current

Consider first the low-current regime of CCL operation. The low-current polarization curve of a CCL is given by (2.44). To simplify calculations we will assume that parameter ε (2.13) is large, so that coth(1/ε) ' ε (this situation is typical of PEFCs). Equation (2.44) then reduces to ε2 ˜j = sinh η˜

(4.133)

(the subscript 0 is omitted). The model is completed by oxygen mass balance in the channel (Eq. (4.27)). Note that this equation is valid for any nondimensional currents. Here we use the dimensionless variables (2.6). The catalyst concentration appears in Eq. (4.133) through the factor i∗ in the expression for ε (Eqs (2.13) and (2.14)). To calculate the optimal shape of catalyst loading along z˜, we will replace i∗ by i∗ gz (˜ z ), where gz is a profile function. Note that this function obeys a normalization condition Z 1 gz (˜ z ) d˜ z=1 (4.134) 0

which means that for any g(˜ z ) the total amount of catalyst is constant. To take into account the concentration dependence, according to the convention accepted in Chapter 2, in Eq. (4.133) we should replace ε→ r gz

ε0 

(4.135) c˜t c˜ref



where s ε0 =

σt b 2¯i∗ lt2

(4.136)

and ¯i∗ is the average exchange current density. Making this substitution we obtain   c˜t ε20 ˜j = gz sinh η˜. (4.137) c˜ref Suppose for a moment that the oxygen transport across the GDL is ideal. Then c˜h = c˜t = c˜ and Eq. (4.27) with the polarization curve (4.137) determine the optimal shape of catalyst loading along the channel.

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

We will require that the optimal gz provides constant local current along the channel. Thus, gz is obtained from Eqs (4.137) and (4.27) if we set ˜ With this, Eq. (4.27) takes the form ˜j = J. λ

∂˜ c = −1, ∂ z˜

c˜(0) = 1.

(4.138)

z˜ . λ

(4.139)

The solution to this equation is c˜ = 1 −

Using this relation and setting ˜j = J˜ in (4.137) we get ε20 J˜ =

gz c˜ref

 1−

z˜ λ

 sinh η˜.

(4.140)

Solving this for gz we obtain gz =

ε20 c˜ref J˜
. sinh(˜ η ) 1 − λz˜

(4.141)

The function (4.141) must satisfy the normalization condition (4.134). Calculating the integral (4.134) we get sinh(˜ η) J˜ = 2 ε0 c˜ref fλ

(4.142)

where fλ is given by (4.33). Equation (4.142) is the polarization curve of a cell with variable catalyst loading. Using (4.142) in (4.141) we finally find gz =

1 1−

z˜ λ




fλ

.

(4.143)

Remarkably, the shape of catalyst loading does not depend on cell current; gz is a one-parametric function of z˜ with the parameter λ.

4.5.2

High cell current

The high-current polarization curve is given by (2.58). Making a substitution (4.135) in Eq. (2.58), we get s ε0 ˜j =

 gz

c˜ c˜ref

 exp η˜.

(4.144)

4.5 CATALYST GRADIENT ALONG THE CHANNEL

159

Figure 4.17: Optimal shapes of catalyst particle loading (4.148) along the oxygen channel for the indicated values of oxygen stoichiometry. Requiring again that the optimal loading provides constant ˜j = J˜ we see that Eq. (4.139) remains unchanged and it should be substituted ˜ With this we obtain into (4.144) together with ˜j = J. ε0 J˜ =

s

gz c˜ref

  z˜ exp η˜. 1− λ

(4.145)

Solving this for gz we find gz =

ε20 J˜2 c˜ref exp(−˜ η) . z˜ 1− λ

(4.146)

This result should obey the normalization condition (4.134); calculating the integral we get the high-current polarization curve of a cell with variable catalyst loading: J˜2 =

exp η˜ . ε20 c˜ref fλ

(4.147)

Using this in (4.145) we finally obtain gz =

1 1−

z˜ λ




fλ

.

(4.148)

This result coincides with the low-current shape (4.143). The shapes (4.143) or (4.148) are depicted in Figure 4.17 for several values of λ. Qualitatively, the lower the λ, the faster the oxygen consumption along z˜ and the faster the catalyst loading should grow with z˜ to compensate for the oxygen depletion and to homogenize the local current. At high stoichiometries the function (4.148) tends to unity, i.e. constant loading is optimal.

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

The polarization curves of cells with uniform and nonuniform catalyst loadings are the same. This is seen from Eqs (4.142) and (4.147), which do not contain gz . Thus, the shape of catalyst loading (4.148) does not change the cell performance. Rather, this shape homogenizes the distribution of local current density. To illustrate the effect of shaping we can compare the local current at the inlet of the uniformly loaded cell with the mean current density. Setting in (4.32), z˜ = 0, we get ˜j(0) = fλ . J˜

(4.149)

For λ close to 1 the ratio ˜j(0)/J˜ is large (Figure 4.4(a)). Thus, the homogenization of local current by nonuniform catalyst loading protects the cell from large overcurrent at the oxygen channel inlet.

4.5.3

The effect of transport loss in the GDL

Above we have neglected oxygen transport loss in the GDL. It is easy to show that accounting for this loss does not change the result (4.148). Consider the low-current polarization curve of the catalyst layer (4.140). In the presence of transport loss, the oxygen concentration in the catalyst layer c˜t is related to this concentration in the channel c˜h by Eq. (3.2). In dimensionless variables this equation reads  c˜t = c˜h

˜j 1− ˜jD

 .

(4.150)

Substituting (4.150) into (4.137), setting ˜j = J˜ and taking into account that c˜h = 1 − z˜/λ, we get gz =

ε2 c˜ J˜  0 ref  .
˜ 1 − λz˜ 1 − ˜jJ sinh(˜ η)

(4.151)

D

˜ ˜jD in the This result differs from (4.141) by a constant factor 1 − J/ denominator. However, after normalization (4.134) this factor vanishes and we again arrive at (4.143). A similar procedure for the high-current regime leads to (4.148). Thus, accounting for oxygen transport loss in the GDL does not affect the optimal shape of catalyst loading along the channel.

4.6 A model of SOFC anode The polarization curve of a solid oxide fuel cell (SOFC) is close to linear (Mogensen and Hendriksen, 2003). This fact justifies the introduction of a

4.6 A MODEL OF SOFC ANODE

161

cell area-specific resistivity R, which is the slope of the line Vcell = Voc − RJ.

(4.152)

In spite of its simplicity (or rather because of it), Eq. (4.152) raises several questions. To a good approximation, the kinetics of electrochemical reactions on both sides of the cell follow the Butler-Volmer law, which establishes exponential dependence of cell current on the respective halfcell overpotential. Why then is the resulting polarization curve linear? Does this mean that the resistive losses in SOFC dominate and the contribution of activation polarizations to the overall voltage loss is small? In recent years, many CFD models for SOFC performance have been developed. Some of these models rely on the empirical notion of area-specific resistivity (ASR), not detailing the kinetics of electrochemical reactions (Yakabe et al., 2001; Xue et al., 2005). The others utilize the Butler-Volmer equation for the calculation of activation losses (Iwata et al., 2000; Larrain et al., 2003; Aguiar et al., 2004; Yuan and Liu, 2007; Wang et al., 2007; Ho et al., 2008; Zhu and Kee, 2008). However, all these models are numerical and they do not give an irrefutable answer to the questions above. The analytical model (Costamagna et al., 1998) describes the performance of the SOFC electrode in the limit of the small overpotential η. The smallness of η allows us to linearize the Butler-Volmer equation, which greatly simplifies the problem. This approximation, however, does not give us a picture of electrode function at large η, neither does it describe the polarization curve in the transition region from small to large anodic overpotentials. Besides, the model (Costamagna et al., 1998) is 1D and it ignores the effect on the anode performance of hydrogen depletion along the channel. A 1D analytical model of the SOFC sandwich (Pisani and Murgia, 2007) leads to implicit expressions for the electrode voltage-current relation in the limits of low and high current densities (overpotential). This model is designed to assist in the multidimensional CFD simulations of SOFC; in the CFD environment the solution of implicit equations poses no problems. However, the relations (Pisani and Murgia, 2007) are not suitable for an analysis of the effect of hydrogen utilization on cell performance. A model of the SOFC anode developed by the Imperial College group (Aguiar et al., 2004) includes Butler-Volmer kinetics and transport losses on both sides of the cell. The results (Aguiar et al., 2004) show that for a cell temperature 800 + 273 K, the diffusion of reactants to the catalyst sites has a minor effect on cell performance. Activation polarizations calculated in Aguiar et al. (2004) are quite significant, and they are comparable to the resistive voltage loss in a cell. Thus, understanding the nature of activation loss is of great interest for SOFC technology. In this section, we construct a simple model of an SOFC anode aiming to answer the questions raised above (Kulikovsky, 2009b).

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.18: Schematic of SOFC anode and the shape of ionic current.

4.6.1

Basic equations and the local polarization curve

Measurements show that at large anodic overpotentials, hydrogen oxidation follows the Tafel law (Holtappels et al., 1999). The hydrogen diffusion coefficient in the anode material is large and we will ignore hydrogen transport losses. A detailed model (Hussain et al., 2005) shows that the hydrogen concentration variation across the anode thickness is marginal. However, the ionic conductivity of the cermet is small and voltage loss due to ion transport should be taken into account. Local anode performance is thus governed by two equations:      cH2 αF η (1 − α)F η exp − exp − (4.153) cref RT RT ∂η(x) . (4.154) j(x) = σt ∂x

∂j(x) = i∗ ∂x



Here the axis x is directed from the interconnect to the electrolyte (Figure 4.18), j(x) is the local ionic current density, i∗ is the volumetric exchange current density, cH2 is the molar concentration of hydrogen in the anode, cref is the reference hydrogen concentration, α is the transfer coefficient, η is the polarization voltage of the anode side and σt is the ionic conductivity of the anode. Equation (4.153) expresses the decay of the ionic current from the anode/electrolyte interface to the anode bulk (Figure 4.18). Equation (4.154) is Ohm’s law relating the ionic current to the gradient of overpotential. Based on the reaction stoichiometry, the rate of hydrogen oxidation Q is taken to be proportional to the hydrogen concentration cH2 . Impedance measurements show that Q may also depend on water concentration cw (see Costamagna and Honegger (1998) and the literature cited therein). A modelling study (Costamagna and Honegger, 1998) indicates that Q ∼ −0.5 cw ; however, this dependence is weak and to a first approximation it can be ignored.

4.6 A MODEL OF SOFC ANODE

163

Following Bard and Faulkner (2001), for the reaction with the RDS involving single-electron transfer, parameter α is 0.5. Measurements (Holtappels et al., 1999) gave α ' 0.7; however, to simplify calculations we will adopt the value α = 1 − α = 0.5. Note that this choice of α is important only in the region of small currents, when both exponents in the Butler-Volmer equation contribute to the activation polarization. At larger currents the second exponent in Eq. (4.153) is small and the results below are valid for arbitrary α. With the dimensionless variables (2.6), the system (4.153) and (4.154) takes the form ε2

∂ ˜j = sinh(˜ η) ∂x ˜ ˜j = ∂ η˜ ∂x ˜

(4.155) (4.156)

where ε=

l∗ lt

(cf. Eq. (2.13)), lt is the anode thickness and s   σt b cref l∗ = 2i∗ cH2

(4.157)

(4.158)

is the characteristic reaction penetration depth into the anode (the thickness of the reaction zone near the electrolyte). The system (4.155) and (4.156) coincides with the system (2.30) and (2.31) if we take into account that the x-coordinate in Figure 4.18 is related ˜→1−x ˜. to the x-coordinate of Section 2.1 as x A feature of the anode-supported SOFC design is its large anode thickness (lt ' 1 mm). Experiments show that the reaction penetration depth into the anode is in the order of 10 µm (Mogensen and Hendriksen, 2003). Thus, for the anode-supported SOFC, ε is a small parameter: ε ' 0.01. The polarization curve of the SOFC anode is, hence, equivalent to the curve of the PEFC cathode with ε  1 (Section 2.3.6). The latter curve is represented by Eqs (2.46) and (2.58). Omitting the subscript, these relations are η˜ = ε˜j, ε˜j < 1
η˜ = 2 ln ε˜j , ε˜j > 4.

(4.159) (4.160)

In the transition region 1 . ε˜j . 4 the curve has no exact analytical representation. The polarization curve (4.159), (4.160) is depicted in Figure 2.4(a) (page 51). Direct proportionality between cell current and overpotential

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.19: Anode channel and schematic of hydrogen concentration profile. in the low-current region justifies the introduction of anode resistivity Ra . The expression for Ra is given by (2.48) with ct = cH2 : s   b cref Ra = . (4.161) 2σt i∗ cH2 In dimension variables, Eqs (4.159) and (4.160) take the form q η = Ra j, j < jσ cH2 /cref   q jRa , j > 4jσ cH2 /cref η = 2b ln b

(4.162) (4.163)

where jσ =

4.6.2

p

2i∗ σt b.

(4.164)

Hydrogen concentration in the channel

The relations above include the hydrogen concentration cH2 . This concentration is constant along the anode channel if the cell is run at low hydrogen utilization. However, real cells are usually run at high fuel utilization and cH2 decreases toward the anode channel outlet. To rationalize the effect of H2 utilization on anode voltage loss, consider an SOFC with the linear anode channel (Figure 4.19). The assumptions we will use are as follows. 1. Cell temperature is uniform. This assumption is justified when air flow rate is large, so that the cooling effect of air is also high. 2. Flow velocity in the anode channel is constant (plug flow).

4.6 A MODEL OF SOFC ANODE

165

Under these assumptions the mass balance equation for H2 concentration in the channel is 0 vH2

∂cH2 j =− ∂z 2F hc

(4.165)

0 is the anode where z is the coordinate along the channel (Figure 4.19), vH2 flow velocity and hc is the channel height. It is convenient to introduce the dimensionless variables:

z˜ =

z , L

ˆj = j , jσ

c˜ =

˜ = Rjσ R b

cH2 , c0H2

(4.166)

where L is the channel length, c0H2 is the inlet hydrogen molar concentration and jσ is given by (4.164)9 . With these variables, Eq. (4.165) takes the form λa Jˆ

∂˜ c = −ˆj ∂ z˜

(4.168)

where the hydrogen (anode) stoichiometry is λa =

4.6.3

0 2F hc vH2 c0H2 . LJ

Cell voltage

Cell voltage V˜cell is OCV V˜oc minus losses: ˜ ˆj. V˜cell = V˜oc − η˜ − R

(4.169)

˜ comprises all the resistivities, except the anodic η˜. Here R ˜ includes the cathodic resistivity, which is assumed to be Note that R independent of local cell current and thus independent of z˜. Typically, due to small cathode thickness the respective concentration polarization for currents below 1 A cm−2 is negligible (Zhao and Virkar, 2005). The situation with the activation polarization on the cathode side is more subtle. The assumption of constant activation resistivity is justified for cathodes with a high exchange current density iORR . For sufficiently ∗ large iORR , the critical current density (4.164) greatly exceeds the working ∗ 9 Note

that the following relation holds: s ˆ j = ε0 ˜ j,

where

ε0 =

σt b . 2i∗ lt2

(4.167)

166

CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

current density and the cathode operates in the linear mode, with currentindependent activation resistivity. It is worth mentioning that the literature data on iORR vary by several orders of magnitude (see e.g. Pisani and ∗ Murgia (2007) and Kakac et al. (2007)), which reflects the large sensitivity of this parameter to cathode composition and the preparation technique. The conductivity of the interconnect is high, i.e. the cell electrodes are equipotential. Thus, the sum ˜ ˆj = E ˜ η˜ + R

(4.170)

is constant along the channel. Here ˜ = V˜oc − V˜cell E is the total voltage loss.

4.6.4

Low current: z-shapes

In the low-current mode the anode polarization curve is given by (4.162). With (4.166), Eq. (4.162) takes the form r c˜ref η˜ = ˆj . (4.171) c˜ The system of equations (4.168), (4.171) and (4.170) determines the shape of the hydrogen molar fraction and local current along the channel. Using (4.171) in (4.170) and solving the resulting equation for ˆj we find ˆj =

˜ E p . ˜ + c˜ref /˜ R c

(4.172)

Substituting this into (4.168) we get the equation for c˜ λa Jˆ

˜ ∂˜ c E p =− , ˜ + c˜ref /˜ ∂ z˜ R c

c˜(0) = 1.

(4.173)

The solution to this equation is v  !2 u u ˜ z˜ p 1  p E ˜ R ˜ + 2 c˜ref −  . − c˜ref + tc˜ref + R c˜(˜ z) = ˜2 R λa Jˆ

(4.174)

The shape of the local current along z˜ is given by (4.172) with c˜ (4.174). ˜ and Jˆ are related by the polarization curve. In the lowNote that E ˜ ∼ Jˆ (see the next section) and hence the functions c˜(˜ current limit, E z)

4.6 A MODEL OF SOFC ANODE (a)

167 (b)

Figure 4.20: The shapes of (a) hydrogen concentration and (b) local current density along the anode channel for the indicated values of the hydrogen stoichiometry λa . ˜ = k Jˆ and ˆj(˜ z )/Jˆ do not depend on Jˆ (this is seen if we substitute E ˆ into Eqs (4.174) and (4.172)). Thus, c˜(˜ z ) and ˆj(˜ z )/J are the universal oneparametric functions of z˜ (Figure 4.20).

4.6.5

Low current: Polarization curve

Generally, theR cell polarization curve results from the equation for mean cell 1 current Jˆ = 0 ˆj d˜ z . However, above we derived the expression (4.174) for c˜(˜ z ), which is simpler than the expression (4.172) for ˆj. From Eqs (4.168) and (4.174), another useful integral relation follows, which leads to the polarization curve. Integrating (4.168) over z˜ we find Z 1 Z 1 ∂˜ c a ˆ ˆj d˜ d˜ z=− z. λ J ˜ 0 ∂z 0 R1 ˆ we have z = J, Since 0 ˆj d˜
λa 1 − c˜1 = 1 (4.175) where c˜1 ≡ c˜(1) is the hydrogen concentration at the channel outlet. Note that Eq. (4.175) determines the relation between stoichiometry and utilization. By definition, hydrogen utilization is u = 1 − c˜1 . Using this in (4.175) we get u=

1 . λa

(4.176)

Solving (4.175) for c˜1 we find c˜1 = 1 −

1 . λa

(4.177)

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Setting now z˜ = 1 in (4.174) and equating the resulting expression to 1 − (1/λa ) we get v !2 u u ˜ p p 1  ˜ R ˜ + 2 c˜ref − E  = 1 − 1 . − c˜ref + tc˜ref + R 2 ˜ λa R λa Jˆ 

˜ we finally find the anode polarization curve Solving this for E   ˜= R ˜+R ˜ a Jˆ E

(4.178)

where ˜ a = φλ R

p c˜ref

(4.179)

is the anode activation resistivity at finite λa and ! r 1 φλ = 2λa 1 − 1 − a . λ

(4.180)

˜ is still proportional to the mean We see that the total voltage loss E ˆ ˜ is the sum of the anode current density J. The total cell resistivity R ˜ activation resistivity Ra and the total resistivity of the other functional ˜ layers R: ˜=R ˜ a + R. ˜ R In the dimension variables, Eq. (4.179) takes the form Ra = φλ Ra0

(4.181)

where s Ra0

=

b 2i∗ σt



cref c0H2

 (4.182)

is the anode resistivity at zero utilization (cf. Eq. (4.161)). In terms of hydrogen utilization u = 1/λa , Eq. (4.181) reads Ra =

√
2 1 − 1 − u Ra0 . u

(4.183)

In the limiting cases of zero and 100% utilization, Eq. (4.183) yields Ra (0) = Ra0 , u = 0 Ra (1) = 2Ra0 , u = 1.

4.6 A MODEL OF SOFC ANODE

169

Figure 4.21: The dependence of
anode resistivity on hydrogen utilization √ u (the function 2 1 − 1 − u /u). When utilization is above 80%, the resistivity grows dramatically. We see that 100% hydrogen utilization increases anode resistivity by a factor of 2.
√ The function fu = u2 1 − 1 − u is depicted in Figure 4.21. Below u ' 0.8, the growth of anode resistivity with u is almost linear; however, above u ' 0.8, this growth is dramatic. Running the cell at H2 utilization above 80% is, thus, detrimental for anode performance.

4.6.6

High current: z-shapes and polarization curve

In coordinates (4.166), Eq. (4.163) for the high-current polarization voltage takes the form ! r c ˜ ref . (4.184) η˜ = 2 ln ˆj c˜ Hydrogen consumption is still governed by Eq. (4.168), and Eq. (4.170) completes the system. In this section, to simplify the analysis we will omit the resistive term in Eq. (4.170). This helps us to understand the effect of hydrogen stoichiometry/utilization on the cell polarization curve without cumbersome calculations. ˜ = 0, Eq. (4.170) reduces to η˜ = E. ˜ Taking into account (4.184) With R we get ! r c˜ref ˜ ˆ 2 ln j = E. (4.185) c˜ Equations (4.168) and (4.185) determine the shapes of the hydrogen concentration and local current density along the hydrogen channel.

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Table 4.2: Parameters for calculations. Note that here we take cH2 = cref . T K

l∗ cm

i∗ A cm−3

σt Ω cm−1

b V

jσ A cm−2

Ra Ω cm2

973 1073

8.47 · 10−3 4.52 · 10−3

9.86 102.1

8.44 · 10−3 2.26 · 10−2

0.168 0.185

0.167 0.924

1.0 0.2

To obtain these shapes we solve (4.185) for ˆj: s ˆj =

˜ c˜ exp E . c˜ref

(4.186)

Using this in (4.168) and solving the resulting equation with the condition c˜(0) = 1 we get q 2 ˜ z˜ exp E  . c˜ = 1 − p 2λa Jˆ c˜ref 

(4.187)

The cell polarization curve can now be obtained using Eq. (4.177). Setting in Eq. (4.187), z˜ = 1, and equating the result to 1 − (1/λa ) we obtain 

q

˜ exp E

2

1 − p  =1− 1 . a ˆ λa 2 c˜ref λ J ˜ we finally find Solving this for E   p ˜ = 2 ln Jˆ c˜ref + 2 ln (φλ ) . E

(4.188)

In terms of hydrogen utilization u = 1/λa , Eq. (4.188) takes the form    p  √
2 ˜ ˆ E = 2 ln J c˜ref + 2 ln 1− 1−u . (4.189) u We see that the effect of hydrogen utilization reduces to a constant shift of the polarization curve as a whole along the voltage axis. The value of this shift is given by the second term in Eq. (4.189). As in the low-current case, this shift dramatically increases above u ' 80% and reaches the value 2b ln 2 ' 235 mV at u = 1 (Figure 4.22 and Table 4.2). The effect of hydrogen utilization on the anode polarization curve is summarized in Figure 4.23. This figure shows the range of anode polarization voltage variation when hydrogen utilization varies from 0 to 100%.

4.6 A MODEL OF SOFC ANODE

171

Figure 4.22: Voltage loss due

hydrogen utilization u in the high√ to finite current regime 2 ln 2 1 − 1 − u /u . When utilization is above 80%, the voltage loss grows dramatically.

Figure 4.23: The effect of hydrogen utilization on the anode voltage loss η˜. Jˆ is the dimensionless mean current density.

4.6.7

Remarks

The key parameter which determines the regime of SOFC anode operation is jσ (4.164). If the working current density J < jσ , the anode operates in a linear regime with the anodic polarization voltage proportional to J. If J ≥ 4jσ , the anode works in a nonlinear regime with the logarithmic dependence of η on J and with the apparent Tafel slope, which is twice as large. In the intermediate region J & jσ , the analytical solution to the problem discussed hardly exists. Parameter jσ can be expressed as a function of temperature (Figure 4.24) (Kulikovsky, 2009b). The typical working current density of SOFC is about 0.5 A cm−2 ; Figure 4.24 shows that jσ ' 0.5 A cm−2 at T ' 750 + 273 K. Therefore, running the cell above 800 + 273 K secures the inequality J < jσ

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.24: Characteristic current density as a function of temperature (Kulikovsky, 2009b).

and anode operation in the linear mode. At temperatures below 700+273 K the anode works in the intermediate or in the high-current regime. Note that Figure 4.24 was plotted for the standard YSZ-Ni anodes. The goal of materials science is to design an anode material with higher jσ . The most difficult situation arises when hydrogen utilization is large. In that case, c˜H2 varies from 1 at the channel inlet to almost p zero at the outlet. This means that the local value of the product jσ cH2 /cH2 decreases toward the channel outlet. This may lead to a situation where the domain at the channel inlet works in the linear mode, while the domain close to the outlet operates in a nonlinear mode. The analysis of this mixed regime requires numerical calculations.

4.7 A model of DMFC In low-T hydrogen cells, there is quite a large range of parameters where the effects of liquid water are small, or they can be taken into account by simple reduction of the oxygen diffusion coefficient in the GDL. Thus, many features of PEFCs can be understood using a single-phase approximation. Direct methanol fuel cells are more difficult to model, since in the most interesting regimes the DMFC is essentially a two-phase object. Carbon dioxide bubbles produced in the methanol oxidation reaction enter the anode channel and disturb the flow of the methanol-water solution. The direct effect of gaseous bubbles is a dramatic acceleration of the flow: experiments show that in typical situations the outlet flow velocity exceeds the inlet velocity by an order of magnitude (Yang et al., 2005). This is easy to understand: gas density is much lower than the density of liquid and simple mass conservation prescribes that due to an increasing concentration of gaseous bubbles, the two-phase flow must accelerate.

4.7 A MODEL OF DMFC

173

The effect of gaseous bubbles can, however, be neglected in two cases: (i) when the cell operating temperature is low (below ' 30 ◦ C) and (ii) when the cell current is small. At low temperature the solubility of CO2 in water is high and the product CO2 remains dissolved in water. At small currents, the rate of bubble formation is small. In this section, we will construct a quasi-2D model of DMFC neglecting two-phase effects (Kulikovsky, 2005b). This model reveals many interesting features of DMFC operation in a low-current regime. Furthermore, this model can be used to take into account the effect of bubbles (Kulikovsky, 2006a). With these remarks in mind, in the next section we will assume plug flow conditions in the channels on both sides of the cell; that is, pressure and flow velocity in the channel are constant and the reactant concentration is uniform across the channel.

4.7.1

Continuity equations in the feed channels

Following the idea of a quasi-2D approach, the solutions to the 1D problem given in Section 3.3 are now considered as local. These solutions contain the current density j and the methanol and oxygen concentration in the channel cah and cch , respectively. In this section, these variables are considered as functions of the distance along the channel z˜. Since the total molar concentration of the anode and cathode flows is constant, it is convenient to introduce the molar fractions ψ and ξ of methanol and oxygen, respectively. In terms of molar fractions, the equations of methanol and oxygen mass balance in the respective channel read j ∂ψ =− − Ncross ∂z 6F ∂ξ j 3 hc v c cc =− − Ncross ∂z 4F 2

ha v a ca

(4.190) (4.191)

where the axis z is directed along the channels (co-flow conditions; see Figure 4.25), h is the channel height, v is the flow velocity, c is the total molar concentration of the mixture in the channel, j(z) is the local current density and Ncross is given by Eq. (3.21). In this section, the superscripts a and c mark the anode and cathode variables, respectively. It is convenient to introduce dimensionless variables z˜ =

z , L

ψ ψ˜ = 0 , ψ

ξ ξ˜ = 0 , ξ

˜j = j , a0 jD

η˜ =

η ba

(4.192)

where a0 jD =

6F Dba ca ψ 0 lba

(4.193)

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.25: Schematic of the cell. ACL and CCL abbreviate anode catalyst layer and cathode catalyst layer, respectively. is the methanol-limiting current density at the inlet (co-flow conditions are assumed) and the superscript 0 marks the inlet values. Consider the practically important case when DMFC is run at constant methanol and oxygen stoichiometries λa and λc , respectively. By definition λa =

6F ha v a ca ψ 0 , LJ

λc =

4F hc v c cc ξ 0 . LJ

With these relations, (4.190) and (4.191) transform to
∂ ψ˜ = − ˜j + ˜jcross ∂ z˜
∂ ξ˜ = − ˜j + ˜jcross λc J˜ ∂ z˜

λa J˜

(4.194) (4.195)

where   ˜jcross = 6F Ncross = β∗ ψ˜ − ˜j a0 jD

(4.196)

is the dimensionless equivalent crossover current density (cf. Eq. (3.23)). With (4.196), Eqs (4.194) and (4.195) are   ∂ ψ˜ = − ˜j + β∗ (ψ˜ − ˜j) , ∂ z˜   ∂ ξ˜ λc J˜ = − ˜j + β∗ (ψ˜ − ˜j) , ∂ z˜

λa J˜

˜ ψ(0) =1

(4.197)

˜ = 1. ξ(0)

(4.198)

˜ ξ˜ and ˜j. The The system (4.197), (4.198) contains three unknowns: ψ, ˜ equation for j follows from the equipotentiality of the cell electrodes. Assuming that (i) DMFC electrodes are ideally conductive (equipotential)

4.7 A MODEL OF DMFC

175

˜ of voltage and (ii) the resistive term can be approximated as RJ, the sum E losses on both sides of the cell must be constant along z˜: ˜ = η˜a (˜ E z ) + η˜c (˜ z ).

(4.199)

˜ immediately follows from the expression for cell Constancy of E voltage (3.32) with Rj = RJ 10 : ˜ J. ˜ V˜cell = V˜oc − η˜a − η˜c − R

(4.200)

Low-current polarization voltages of the anode and cathode sides are given by (3.27) and (3.28), respectively. In dimensionless variables (4.192), Eqs (3.27) and (3.28) take the form    ˜j ˜j + ln(1 + β) − ln 1 − ψ˜ q ψ˜      ˜j ˜j + ˜jcross c η˜ = p ln − ln 1 − αq ξ˜ γ ξ˜

η˜a = ln



(4.201) (4.202)

where the parameters are given by lta ia∗ lba 6F Dba caref   ltc ic∗ caref cc ξ 0 α= aa c lt i∗ cref ca ψ 0   2Dbc lba cc ξ 0 γ= 3Dba lbc ca ψ 0 bc p = a. b q=

(4.203) (4.204) (4.205) (4.206)

With Eqs (4.201) and (4.202), the equipotentiality condition (4.199) takes the form     ˜j ˜j − ln 1 − + ln(1 + β) ln q ψ˜ ψ˜      ˜j ˜j + ˜jcross ˜ + p ln − ln 1 − = E. (4.207) αq ξ˜ γ ξ˜ Equations (4.197), (4.198) and (4.207) form a quasi-2D model of DMFC: the relation between the local current density and voltage loss (4.207)

10 In this book, the effects due to variable V ˜oc in DMFCs are not considered (for a discussion of these effects, see Kulikovsky (2008c)).

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

couples the mass transport equations (4.197), (4.198) for methanol and oxygen in the respective channel. An analysis of these equations for an arbitrary set of parameters will be performed in Section 4.8. In the next section, we start with the analysis of the case of equal oxygen and methanol stoichiometries, when the system discussed has an exact solution (Kulikovsky, 2005b).

4.7.2

Solution for the case of λa = λc

If λa = λc , the solution to the system (4.197), (4.198) and (4.207) is ψ˜ = ξ˜ ˜ ˜j = ˜j 0 ψ.

(4.208) (4.209)

Indeed, for λa = λc = λ, Eqs (4.197) and (4.198) coincide. The boundary ˜ ˜ = 1 also coincide; thus ψ˜ = ξ. ˜ Substituting ˜j = ˜j 0 ψ˜ conditions ψ(0) = ξ(0) into (4.197) we get λJ˜



∂ ψ˜ = − ˜j 0 + β∗ 1 − ˜j 0 ψ˜ ∂ z˜

(4.210)

where ˜j 0 is the local current density at the inlet. Solving (4.210) we find   z˜ , ξ˜ = ψ˜ = exp − µ

(4.211)

˜ Jλ . ˜j 0 + β∗ (1 − ˜j 0 )

(4.212)

where µ=

Using (4.211) in (4.209) we find   ˜j = ˜j 0 exp − z˜ . µ

(4.213)

Substituting (4.211) and ˜j = ˜j 0 ψ into (4.196) we get   ˜jcross = β∗ (1 − ˜j 0 ) exp − z˜ . µ

(4.214)

All the variables thus decrease exponentially along z˜ with the same characteristic scale µ (4.212). Note that the exponential-like decay of the

4.7 A MODEL OF DMFC (a)

177 (b)

Figure 4.26: Voltage-current curves of DMFC with equal stoichiometries of methanol and oxygen. (a) λ = 2; indicated are the values of the crossover parameter β. (b) β = 0.333; indicated are the values of λ. local current density along the channel was obtained in 2D numerical calculations (Birgersson et al., 2003, 2004). Substituting ˜j = ˜j 0 ψ˜ into (4.201) and ˜j = ˜j 0 ξ˜ = ˜j 0 ψ˜ into (4.202) we find  ˜j 0
− ln 1 − ˜j 0 + ln(1 + β) q  0   ˜j ˜j 0 + β∗ (1 − ˜j 0 ) ηc = ln − ln 1 − . bc αq γ

ηa = ln ba



(4.215) (4.216)

We see that the polarization voltages on both sides of the cell depend only on ˜j 0 and thus they are constant along z˜. ˜0 R 1The local current at the channel inlet j is obtained from the condition ˜ Using here (4.213) and calculating the integral we find ˜j d˜ z = J. 0 ˜ With (4.212), after simple manipulations we get ˜j 0 µ[1 − exp(−1/µ)] = J. 0 ˜ the equation for j : ˜j 0 + β∗ (1 − ˜j 0 ) ˜   = −J. ˜0 ˜ j0) λ ln 1 − j +β˜j∗0(1− λ

(4.217)

Note that ˜j 0 depends on stoichiometry λ. Equations (4.200) and (4.215)–(4.217) determine the cell polarization ˜ β and λ, a solution to (4.217) gives ˜j 0 . Eqs (4.215) curve. For given J, and (4.216) then give η a and η c ; finally, the cell voltage is calculated from (4.200). Polarization curves for different β and λ are shown in Figure 4.26 (the other parameters are listed in Table 4.3). An increase in the rate of crossover β reduces the cell open-circuit voltage Voc (Figure 4.26(a)). Qualitatively, this is what we could expect since methanol crossover reduces the amount of oxygen in the CCL.

178

CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Table 4.3: Parameters for the curves in Figure 4.26. Anode side Temperature (C) Pressure (atm) Db (cm2 s−1 ) i∗ (A cm−3 ) cref (mol cm−3 ) Oxygen molar fraction, ξ 0 Methanol molar fraction, ψ 0 Tafel slope, b (V) Catalyst layer thickness, lt (cm) Backing layer thickness, lb (cm) Membrane thickness, lm (cm) Dm (cm2 s−1 ) for β = 0.1, 0.333, 1

Cathode side

70 70 2 2 −5 2 · 10 3 · 10−3 −2 10 1 10−3 7 · 10−5 – 0.21 0.032 (1M) – 0.05 0.05 10−3 10−3 2 · 10−2 2 · 10−2 10−2 −6 1 · 10 , 3.33 · 10−6 , 10−5

Unexpectedly, however, Voc depends on stoichiometry: with the growth of λ, Voc increases (Figure 4.26(b)). At λ = ∞, the cell open-circuit voltage T − E0∞ , where E0∞ . 100 mV is the voltage loss reaches the value Voc under open-circuit conditions (see below). The effect of Voc lowering due to methanol crossover (mixed potential) is well known in DMFC studies. The model above allows us to explain the dependence of OCV on feed molecule stoichiometry λ.

4.7.3

Cell depolarization at zero current: Mixed potential

Of greatest interest is the function ˜j(˜ z ) (4.213), since ˜j 0 appears in polarization voltages (4.215), (4.216). Parameter ˜j 0 is a solution to Eq. (4.217); µ is given by (4.212). The behaviour of ˜j 0 and µ as J˜ → 0 determines the cell voltage at zero current. To understand this behaviour, consider first the limiting cases. In the case of zero crossover we put β∗ = 0 and Eqs (4.217) and (4.212) reduce to   ˜ ln 1 − 1 ˜j 0 = −Jλ (4.218) λ   −1 1 µ = − ln 1 − . (4.219) λ Thus, as J˜ tends to zero, local current at the inlet ˜j 0 also tends to zero, whereas µ does not change. If, therefore, an ideal DMFC (without crossover)

4.7 A MODEL OF DMFC

179

is run at a constant λa = λc = λ, the z˜-shape of the local current density is the same for all points on the polarization curve. A variation of J˜ simply re-scales the whole curve ˜j(˜ z ). Crossover, however, adds an additional degree of freedom. If β∗ 6= 0, the ˜ This can be shown explicitly characteristic scale of exponent µ varies with J. for the case of large λ: the logarithm in (4.217) can then be expanded and ˜ With this ˜j 0 , from (4.212) we find we get ˜j 0 = J. µ=

˜ Jλ . ˜ ˜ J + β∗ (1 − J)

(4.220)

Now µ → 0 as J˜ → 0. Therefore, as the mean current density tends to ˜ local zero, both ˜j 0 and µ in (4.213) tend to zero. In other words, at small J, ˜ current decreases with z˜ faster than at large J. Most interesting is the case of finite λ. The numerical solution to (4.217) for various λ is shown in Figure 4.27. As J˜ → 0, we again have µ → 0, whereas ˜j 0 now tends to the non-zero value ˜j00 ≡ ˜j 0 |J=0 (Figure 4.27). The ˜ value of ˜j00 is a solution to (4.217) with J˜ = 0. Clearly, as J˜ → 0, the expression under the logarithm in (4.217) should tend to zero. For ˜j00 , we thus have an equation 1−

˜j00 + β∗ (1 − ˜j00 ) = 0. ˜j00 λ

The solution is ˜j00 =

β∗ β = . λ − 1 + β∗ λ(1 + β) − 1

(4.221)

With ˜j 0 = ˜j00 , Eq. (4.212) gives µ=

λ (λ(1 + β) − 1) J˜ , β(1 + λ)

(4.222)

and thus µ → 0 as J˜ → 0. Therefore, as J˜ → 0 near the inlet a narrow “jumper” of local current with the finite amplitude ˜j00 is formed. The width of the jumper µ vanishes ˜ whereas its amplitude ˜j 0 does not. with J, 0 This is illustrated in Figure 4.28. In the case of zero crossover, ˜j 0 (4.218) decreases with J˜ and µ (4.219) remains constant (Figure 4.28(a)). In the logarithmic scale, the curves in Figure 4.28(a) are parallel straight lines. ˜ while ˜j 0 In the case of non-zero crossover, µ (4.222) tends to zero with J, 0 ˜ (4.221) tends to a constant value j0 (Figure 4.28(b)). Furthermore, mean

180

CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL (a)

(b)

Figure 4.27: Characteristic scale of the exponent µ and local current density at the inlet ˜j 0 as a function of mean current density J˜ for the indicated values of λ. Crossover parameter β = 0.333. (a)

(b)

Figure 4.28: The shape of the local current density along the channel for the indicated values of mean current density J˜ in a cell. (a) the case of zero crossover (β = 0), and (b) non-zero crossover (β = 0.333). In both cases, λ = 1.5. Arrows indicate the evolution of the curve as J˜ → 0. current density in the jumper J˜jumper remains finite as J˜ → 0. Indeed, 1 J˜jumper ' µ

1

Z 0

     1 0 ˜j00 exp − z˜ d˜ ˜ z = j0 1 − exp − . µ µ

Since µ → 0 as J˜ → 0, we have J˜jumper → ˜j00 =

β∗ . λ − 1 + β∗

(4.223)

The half-cell polarization voltages (4.215) and (4.216) depend on the local current density at the inlet ˜j 0 rather than on the total current in the system. Thus, the formation of the jumper induces finite η a and η c in the cell. Furthermore, since η a and η c are constant along z, the cell potential immediately “feels” the jumper.

4.7 A MODEL OF DMFC

181

Table 4.4: Cell voltage drop (mV) at zero current for the indicated values of β and λ. β 0.1 0.333 1 E0 (λ = 8), mV 520 637 738 E0 (λ = ∞), mV 13 43 135

Total voltage loss at open circuit E0 can be calculated from (4.207) using inlet values ˜j 0 = ˜j00 and ψ˜ = ξ˜ = 1:    0 ˜j
E0 = ba ln 0 − ln 1 − ˜j00 + ln(1 + β) q     0 ˜j 0 + β∗ (1 − ˜j00 ) ˜j0 − ln 1 − 0 , + bc ln αq γ

(4.224)

where ˜j00 is given by (4.221). With the parameters in Table 4.3 and λ = 8 we find the values of E0 , shown in the second row of Table 4.4. The jumper reduces the DMFC opencircuit voltage to 0.5-0.7 V, which agrees with the experiments (Ravikumar and Shukla, 1996; Qi and Kaufman, 2002; Jiang and Chu, 2004). It should be emphasized that the OCV decreases as soon as the cell is connected to any load resistance. The jumper already forms at very small (strictly speaking, at infinitesimal) total current in the load. Care thus should be taken to prevent leakage current in a wet environment: this current allows the jumper to form. If the cell is polarized, in a wet environment fuel and oxygen will be consumed by the jumper, even in the absence of a useful load. Equation (4.221) does not contain the kinetic parameters of electrochemical reactions. This means that the jumper arises regardless of these kinetic details; the nature of the jumper is determined by electrostatic and transport phenomena. However, the value of voltage loss E0 (4.224) due to the jumper includes the parameters of the reactions. Relation (4.221) shows that ˜j00 tends to zero as λ → ∞. Therefore, the voltage loss E0 can be reduced by increasing λ at small currents (Figure 4.26(b)). Physically, the increase in λ increases the characteristic size of the jumper µ (4.222) (Figure 4.27(a)), thereby “smearing” the jumper over a larger surface.

4.7.4

Cross-linked feeding

Consider cell operation at the oxygen-limiting current density. Formally, ˜ operation at the limiting current is equivalent to infinite voltage loss E. The expression under the last logarithm on the left side of (4.207) should

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˜ or ˜j + β∗ (ψ˜ − ˜j) = γ ξ. ˜ Using here tend to zero and we get ˜j + ˜jcross = γ ξ, 0 ˜ ˜ ˜ ˜j = ˜j ψ (4.209) and ψ = ξ we find ˜j 0 + β∗ (1 − ˜j 0 ) = γ. If γ = 1 we get ˜j 0 = 1 and thus ˜jcross = β∗ (1 − ˜j 0 )ψ = 0 regardless of β∗ . This case is of particular interest. c0 a0 , or 6F Dba ca ψ 0 /lba = = jD Condition γ = 1 is equivalent to jD 4F Dbc cc ξ 0 /lbc . In other words, at the inlet the methanol flux in the anode backing layer equals the oxygen flux in the cathode backing layer. Both fluxes are maximal since they provide limiting current density. The concentrations of oxygen and methanol in the respective catalyst layer thus tend to zero. The fact that ˜jcross = 0 regardless of β means that under λa = λc , equality of oxygen and methanol transverse fluxes at z˜ = 0 provides their equality at any z˜. The condition a0 c0 λa = λc and J = jD = jD

(4.225)

thus describes the regime of cross-linked feeding: at any z˜, methanol and oxygen are fully consumed, so that regardless of β, crossover is exactly zero. With ˜j 0 = 1, the characteristic scale µ (4.212) is ˜ µ = λJ.

(4.226)

˜ over z˜ and equating the result to J, ˜ after Integrating ˜j = exp(−˜ z /(λJ)) simple transformations we find   −1 1 ˜ J = − λ ln 1 − . λ Limiting current density in a cell is thus a function of λ only. Substituting this J˜ into (4.226) we find   −1 1 µ = − ln 1 − . λ

(4.227)

This value coincides with those obtained for β∗ = 0 (cf. (4.219)). Physically, the regime of cross-linked feeding is equivalent to the case of an ideal membrane with zero crossover. Using (4.227) in (4.211) and (4.213), we get ψ˜ = ξ˜ = ˜j =



1 1− λ

z˜ .

All variables in this regime are universal functions of λ only.

(4.228)

4.7 A MODEL OF DMFC

4.7.5

183

Oxygen and methanol utilization, and mean crossover current density

Mass balance equations (4.194) and (4.195) enable us to formulate integral relations, which are valid regardless of the distribution of local current density. Equations (4.194) and (4.195) show that the rates of oxygen and methanol consumption differ only by a constant factor. Equating the left sides of (4.194) and (4.195) and integrating the result over [0, z˜] we get λa (ψ˜ − 1) = λc (ξ˜ − 1).

(4.229)

Introducing local oxygen and methanol utilization according to uc (˜ z) = ˜ z ) and ua (˜ ˜ z ), we immediately find that 1 − ξ(˜ z ) = 1 − ψ(˜ ua (˜ z) λc = . uc (˜ z) λa

(4.230)

Though ua and uc depend on z˜, their ratio does not. In the case of λc = λa , oxygen and methanol utilizations are equal: ua = uc . Putting z˜ = 1 in (4.230), we get the ratio of total methanol and oxygen utilizations in a cell: ua (1)/uc (1) = λc /λa . R1 Integrating (4.195) over z˜ ∈ [0, 1] and taking into account that 0 ˜j d˜ z= ˜ after simple calculations we find J, J˜cross = λc uc (1) − 1 = λa ua (1) − 1 J˜

(4.231)

R1 z and the last equality was obtained with (4.230). where J˜cross = 0 ˜jcross d˜ The flux of methanol through the membrane is usually calculated by measuring the amount of CO2 in the cathodic exhaust (Dohle et al., 2002; Jiang and Chu, 2004). This method, however, is not reliable since CO2 permeates through the membrane from the anode to the cathode side (Dohle et al., 2002). Equation (4.231) provides a simple means for evaluating of the overall crossover current in DMFC: by measuring oxygen and/or methanol concentration at the outlet, one can calculate Jcross with (4.231). When crossover is zero, from (4.231) we find uc (1) =

1 , λc

ua (1) =

1 λa

(4.232)

(cf. Eq. (4.176)). These relations do not depend on the type of feed molecule and hence are valid for any fuel cell.

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

4.7.6

Remarks

Voltage loss at open circuit E0 cannot be explained within the scope of a 1D model: the formation of the jumper is a 2D effect. This can be seen from the expression for ˜j00 (4.221), which contains the parameter of the flow λ. Note that the experiments show strong dependence of the cell open-circuit voltage on the air flow rate (Qi and Kaufman, 2002). However, low OCV does not necessarily mean poor DMFC performance in the whole range of currents. If conditions (4.225) are satisfied, the cell with large E0 exhibits excellent performance near the limiting current density. The model in this section, ignores the formation of CO2 bubbles on the anode side. At high current density these bubbles may seriously affect the transport of liquid methanol in the channel. Conditions (4.225) should, therefore, be considered as a qualitative indication of how to minimize crossover. The regime close to cross-linked feeding was presumably realized in the experiments of Ren et al. (2000b) and Thomas et al. (2002). Reported in Thomas et al. (2002) the voltage loss due to crossover was less than 20 mV at 100 mA cm−2 . This small value indicates low methanol concentration in the anode catalyst layer.

4.8 DMFC: The general case of arbitrary λa and λc 4.8.1

Equation for local current

In the general case of arbitrary stoichiometries the problem has to be solved numerically (Kulikovsky, 2004d). To simplify calculations it is convenient to convert (4.207) into the differential equation for ˜j. Differentiating (4.207) with respect to z˜ and using (4.197) and (4.198) to ˜ z˜ and ∂ ξ/∂ ˜ z˜, after algebraic manipulations we arrive at exclude ∂ ψ/∂ ∂ ˜j AJ = , ∂ z˜ BJ

˜j(0) = ˜j 0

(4.233)

where h ˜ γ(1 + β)λc ξ˜ + (pγ(1 + β)λa − β(1 + p)λc ) ψ˜ AJ = ˜j(˜j + β ψ) i + ((pβ − 1)λc − pγ(1 + β)λa ) ˜j (4.234) h  ˜ + β) pγ(1 + β)ξ˜ + (1 − pβ)ψ˜ ˜j BJ = λa λc J(1   i + (1 + p) β ψ˜ − γ(1 + β)ξ˜ ψ˜ . (4.235)

4.8 DMFC: THE GENERAL CASE OF ARBITRARY λA AND λC 185 The equation for ˜j 0 is obtained if we substitute inlet values ˜j = ˜j 0 and ˜ ψ = ξ˜ = 1 into (4.207):  ˜j 0
− ln 1 − ˜j 0 + ln(1 + β) ln q     0 ˜j ˜j 0 + β∗ (1 − ˜j 0 ) ˜ − ln 1 − = E. + p ln αq γ 

(4.236)

Note that (4.233) does not contain α and q. These parameters appear in the solution only through the value of ˜j 0 (4.236). In other words, α and q simply re-scale the curve ˜j(˜ z ) as a whole, but do not affect its shape. In the general case of arbitrary λa and λc , DMFC performance is governed ˜ determine by the system (4.197), (4.198) and (4.233). Parameters J˜ and E the point on the polarization curve. The mean current density in a cell is ˜ the solution to (4.233) must, therefore, obey the relation J; Z

z˜min

˜j d˜ z = J˜

(4.237)

0

where z˜min = min{˜ z0 , 1} (see below). The strategy for solving the system (4.197), (4.198) and (4.233) is as follows. ˜ and solve (4.236) for ˜j 0 . • Fix E • Solve the system (4.197), (4.198) and (4.233). This is an initial-value problem, which can be solved using the Runge-Kutta method or any mathematical software. ˜ E) ˜ gives a point on the cell • Calculate J˜ with (4.237). The pair (J, polarization curve.

4.8.2

Numerical solution

The numerical solution to the system (4.197), (4.198) and (4.233) is obtained by the standard Runge-Kutta method. Figure 4.29(a) shows the ˜ as a function of J˜ for λa = 4, λc = 2 and three values total voltage loss E of parameter β. At small currents, all curves form the plateau (Figure 4.29(a)) which indicates the formation of the jumper. The length of the plateau increases with β (Figure 4.29(a)). The shapes of the local current density in the several points on the polarization curve for β = 1 (Figure 4.29(a)) are depicted in Figure 4.29(b). In contrast to the exponential shape of local current at λa = λc (Figure 4.28(b)), in the case of λa > λc the jumper has finite spread over the cell surface (Figure 4.29(b)). At z˜ = z˜0 , the local current vanishes (Figure 4.29(b)).

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL (a)

(b)

˜ vs mean current density J˜ for the Figure 4.29: (a) Total voltage loss E indicated values of crossover parameter β. Stoichiometries are λa = 4 and λc = 2. The other parameters are α = 20.04, γ = 2.803, and q = 1.727 × 10−5 . Diamonds: the points where the jumper forms (˜ z0 = 1; see the text). (b) The profiles of local current density in the black points in (a). As J˜ decreases, the point z˜0 moves to zero, whereas ˜j 0 remains constant (Figure 4.29(b)). This jumper “shrinking” is analogous to that discussed in the previous section. To understand the effect of a localized current on cell performance, in the next section we derive a “low-current” analytical solution to the system (4.197), (4.198) and (4.233) in the limiting case of large methanol stoichiometry.

4.9 DMFC: Large methanol stoichiometry and small current 4.9.1

The shape of the jumper

To further investigate the effect of jumper formation it is advisable to consider the case of large methanol stoichiometry and small cell current. In the limit of small current we expand the right side of Eq. (4.233) over ˜j and retain only the linear term. This yields h i c a c ˜ ˜ ˜j β λ γ(1 + β) ξ + (pλ γ(1 + β) − λ β(1 + p)) ψ ∂ ˜j h i = . ∂ z˜ ˜ + β)(1 + p) β ψ˜ − γ(1 + β)ξ˜ λa λc J(1

(4.238)

Since γ & 1, p & 1, ξ˜ ≤ 1 and λa  λc , we can omit terms with λc in the numerator of (4.238). Further, large methanol stoichiometry means that

4.9 LARGE METHANOL STOICHIOMETRY, SMALL CURRENT 187 ψ˜ ' 1. Making these transformations we arrive at λc J˜

∂ ˜j βpγ ˜j h i. = ∂ z˜ (1 + p) β − γ(1 + β)ξ˜

(4.239)

The equation for ξ˜ in this limit is obtained from (4.198) if we put ψ˜ = 1 and neglect ˜j on the right side: λc J˜

∂ ξ˜ = −β∗ , ∂ z˜

˜ = 1. ξ(0)

(4.240)

This equation is valid provided that ˜j  β∗ . Physically, the useful current ˜j should be much less than the crossover current density; oxygen consumption is then mainly determined by crossover. The solution to (4.240) gives a linear decrease of oxygen concentration with the distance: z˜ ξ˜ = 1 − . z˜ox

(4.241)

Here z˜ox =

λc J˜ β∗

(4.242)

is a point where ξ˜ vanishes. Substituting (4.241) into (4.239) and solving the resulting equation we find p   1+p ˜j = ˜j 0 1 − z˜ z˜0

(4.243)

where z˜0 = λ J˜ c



1 1 − β∗ γ

 .

(4.244)

Local current density (4.243) for several values of J˜ is shown in Figure 4.30(a) together with the numerical solutions of the full system of equations (4.197), (4.198) and (4.233). Note that the current is localized in the domain 0 ≤ z˜ ≤ z˜0 (Figure 4.30(a)). This localization explains the nature of a jumper. According to (4.196), the crossover current ˜jcross increases with the decrease in ˜j. Suppose that we are moving along the cathode channel from the inlet toward the outlet. Local current ˜j decreases with z˜ due to oxygen consumption. There is an

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

(a)

(b)

Figure 4.30: (a) Evolution of local current density profile upon variation of the total current J˜ (indicated) in the cell. Solid lines—analytical solution (4.243) of reduced system, and dashed lines—numerical solution of the full system of equations (4.197), (4.198) and (4.233). Parameters are β = 1, λa = 100, λc = 8, p = 1, and γ = 2. The other parameters required for the numerical solution are α = 20.04 and q = 1.727 × 10−5 . (b) The shapes of the useful and crossover current densities along the channel. excess of methanol (ψ˜ = 1); thus, according to (4.196) the local crossover current increases with z˜ (Figure 4.30(b)). As we are moving along z˜, more oxygen is expended to burn the permeated methanol and thus less oxygen is left for useful current production. At z˜0 (4.244) the local current is zero, whereas ˜jcross is maximal (Figure 4.30(b)). Starting from that point, all available oxygen is consumed in the reaction with permeated methanol. This scenario takes place for λa  λc , when, everywhere along the channel, ψ˜ = 1. In the case of λa = λc , the local fraction of methanol ψ˜ exponentially decreases along z˜ and thus ˜jcross also decreases with z˜. A certain amount of oxygen is then available for current production everywhere; this leads to a non-zero local current along z˜. According to (4.241), to the right of z˜ox there is no oxygen at all. Note that the length of the domain where all oxygen is consumed in the reaction ˜ with permeated methanol is ∆˜ z = z˜ox − z˜0 = λc J/γ. This length increases ˜ with the mean current density J. R z˜ Current density at the inlet ˜j 0 is obtained from the condition 0 0 ˜j d˜ z= ˜ J. Integrating (4.243) we find ˜j 0 =

γβ∗ (1 + 2p) . λc (γ − β∗ )(1 + p)

(4.245)

˜ whereas z˜0 Thus, the local current at the inlet ˜j 0 does not depend on J, ˜ Figure 4.30(a) confirms this result. (4.244) is proportional to J. R z˜ As before, the mean current density in the jumper J˜jumper = z˜10 0 0 ˜j d˜ z ˜ remains constant as J → 0. Using here (4.243) and calculating the integral

4.9 LARGE METHANOL STOICHIOMETRY, SMALL CURRENT 189 we find   −1 (1 + p)˜j 0 1 1 c ˜ Jjumper = = λ − . 1 + 2p β∗ γ

(4.246)

˜ We see that J˜jumper does not depend on J. Parameter γ is proportional to the mass transfer coefficient of oxygen through the cathode backing layer. Equations (4.244) and (4.246) show that when γ = β∗ the jumper has zero thickness and infinite local current density. Therefore, at γ < β∗ the cell does not work at all due to insufficient flux of oxygen through the GDL. Since β∗ ≤ 1, we conclude that with γ > 1 the cell generates current at any rate of crossover. What does small current physically mean? In this section, all currents are normalized to the methanol-limiting current density at the inlet, a0 jD = 6F Dba ca ψ 0 /lba . If the methanol concentration is large enough, current in the cell is limited by the oxygen side. Small J˜ then may be in the order of the oxygen-limiting current density in the system. The solution shown in Figure 4.30 then gives a picture of DMFC operation in quite a large range of operating current densities. Note, however, that the effect of gaseous bubbles in the anode channel has been neglected; thus, the solutions discussed are valid if the rate of bubble production is small.

4.9.2

Plateau

Cell polarization curves for λa  λc are qualitatively similar to those for λa > λc shown in Figure 4.29(a). Suppose that we are moving along ˜ As soon as z˜0 any of the curves in Figure 4.29(a) from large to small J. equals 1, the mean current density in the current-carrying domain [0, z˜0 ] ceases to change (Eq. (4.246)). From that moment on, this domain (jumper) maintains constant the mean current density at the constant ˜j 0 . Equation (4.244) allows us to calculate the mean current density J˜∗ when the jumper forms. Equating z˜0 to 1 we get   −1 1 1 c ˜ J∗ = λ − . β∗ γ

(4.247)

As should be, J˜∗ coincides with the mean current density in the jumper (4.246). For all J˜ ≤ J˜∗ , the local current density at the channel inlet does not vary with J˜ and thus the cell potential does not change either. J˜∗ is hence the point where the cell polarization curve enters the plateau (Figure 4.29(a)). The respective loss in the cell open-circuit voltage is given by (4.236) with ˜j 0 (4.245).

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CHAPTER 4. QUASI-2D MODEL OF A FUEL CELL

Figure 4.31: The shape of the jumper for the indicated values of oxygen stoichiometry λc . Parameters β = 1, J˜ = 0.1 and λa = 100; parameters α, γ and q are the same as in Figure 4.29. ˜ on λc and Equations (4.236) and (4.245) give also the dependencies of E c β. The shapes of the jumper for different λ are depicted in Figure 4.31. The increase in λc reduces ˜j 0 and “smears out” the jumper over the larger domain. In view of (4.236), lower ˜j 0 means higher cell open-circuit voltage.

4.9.3

Critical air flow rate

Equation (4.247) determines the critical current density J˜∗ at the constant oxygen stoichiometry λc . In the general case of variable stoichiometry, one should speak about the critical air flow rate. Indeed, the product λc J˜ is proportional to the air flow rate fair . By definition, λc is λc =

4F fair c0h , LwJ

(4.248)

where Lw is the cell active area and fair = hwv 0

(4.249)

is the air flow rate (cm3 s−1 ). Using (3.19), (4.205) and (4.248), it is easy to verify that λc J˜ = γ f˜air ,

(4.250)

where the dimensionless air flow rate f˜air = fair /f∗ and f∗ =

LwDbc . lbc

(4.251)

4.9 LARGE METHANOL STOICHIOMETRY, SMALL CURRENT 191 (a)

(b)

Figure 4.32: (a) Model polarization curves of DMFC for the indicated values of oxygen stoichiometry λc and λa = 100. Shown is the dimensionless ˜ as a function of the mean current density in a cell total voltage loss E ˜ J. Crossover parameter β = 1, and the other parameters are the same as in Figure 4.29. (b) Experimental polarization curves of DMFC for the indicated values of oxygen stoichiometry λc and λa  λc . Diamonds—the points where the jumper forms. Multiplying (4.247) by λc , we get the critical air flow rate at which the jumper forms: crit f˜air =



γ −1 β∗

−1 .

(4.252)

DMFC is usually run at γ & 3; typical β∗ . 0.5. This allows us to neglect 1 in Eq. (4.252) and this relation simplifies to β∗ crit f˜air ' . γ

(4.253)

In dimension variables this relation takes the form crit fair =

4.9.4

a0 β∗ jD Lw 3β∗ Dba c0M Lw = . a 0 2lb ch 4F c0h

(4.254)

Experimental verification

The analysis above shows that the jumper is best seen when methanol stoichiometry greatly exceeds oxygen stoichiometry. Cell polarization curves calculated for the three values of λc with an excess of methanol (λc  λa ) are shown in Figure 4.32(a). As discussed above, the jumper manifests itself as a plateau at a small current density. Furthermore, Eq. (4.247) shows that the product λc J˜∗ must remain constant. Numerical curves in Figure 4.32(a) confirm this result.

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To verify these predictions the measurements of DMFC polarization curves in conditions λc  λa , J˜  1 were performed (Kulikovsky et al., 2005b). The results are shown in Figure 4.32(b). Comparing Figures 4.32(a) and 4.32(b), we see that the experimental polarization curves reproduce, remarkably well, the two main features of the model curves: the drastic change in the slope at a current density J∗ , where the jumper forms (large diamonds in Figure 4.32(b)) and the constancy of the product λc J∗ : J∗ ’s for λc = 8, 4 and 2 are related nearly as 1:2:4. For J ≤ J∗ , the model predicts constant cell voltage (Figure 4.32(a)); the experimental curves, in contrast, exhibit a decrease in cell voltage with the decrease in J (Figure 4.32(b)). The effect is caused by the formation of the electrolytic domain in a cell. For a detailed discussion of this effect, see Kulikovsky and Wippermann (2009).

Chapter 5

Modelling of fuel cell stacks A typical laboratory fuel cell generates power in the range of 10-100 W. To increase the power of the FC system and to achieve high volumetric power density, fuel cells are assembled in stacks. A stack is a layered structure consisting of membrane-electrode assemblies (fuel cells) clamped between metal or graphite bipolar plates (BPs) with channels for feed gas supply (Figure 5.1). Individual cells in a stack are connected in series, i.e. the total current R I = j dS crossing each cell is the same (j is the local current density and S is the cell active area). Due to high conductivity, the potential of the bipolar plate does not vary much over the BP surface. However, below we will show that small variations of BP voltage indicate quite significant inhomogeneities in transversal (through-plane) current in the fuel cell and thus these variations are of great interest. Fuel cell stacks are devices of tremendous complexity. The overall thermal effect of electrochemical reactions in cells is heat production. The functioning of an individual fuel cell requires a continuous supply of feed gases or liquids and the removal of excess water and heat. Water and heat management in a stack is much more difficult than in a single stand-alone cell. Basically, there are three approaches to stack modelling. The first is fully 3D modelling, taking into account the hydrodynamics of flows in channels, heat transport and potential distribution over the stack volume. This approach leads to extremely time-consuming CFD codes; to our knowledge there is only one model of that type (Liu et al., 2006). Another option is to model a single cell in a stack taking into account the stack environment by using periodic or adiabatic boundary conditions 193

194

CHAPTER 5. MODELLING OF FUEL CELL STACKS

Figure 5.1: Sketch of part of the stack. The membrane-electrode assembly is clamped between two bipolar plates with the channels machined for feed gas supply.

Figure 5.2: Sketch of the stack repeating element: the single linear air channel in the BP. From the top and bottom the element is heated by membrane-electrode assemblies (not shown). for the temperature of cell surfaces. This approach has been widely used in SOFC modelling (see e.g. Achenbach, 1995; Recknagle et al., 2003; Haberman and Young, 2005). Models of that type take into account details of individual cell geometry and result in temperature and current density fields over the cell surface. However, a lot of useful knowledge can be gained from much simpler models, which consider just a single repeating element of a stack: a piece of bipolar plate with the channel. As an example, Figure 5.2 shows an element of an SOFC stack with a single straight air channel. The stack element is surrounded by the other elements and hence in the simplest case a zero heat flux to the neighbouring elements can be assumed. The thermal model for the element then reduces to a 1D model, which can be solved using the asymptotic technique. This approach is utilized in Section 5.1. A model for a single element cannot describe large-scale effects in a stack. For example, large temperature nonuniformity spanning several elements is beyond the scope of this model. To rationalize the large-scale 3D effects, a promising approach is a hybrid of analytical and CFD calculations. Below we will show that fully 3D Laplace equations for voltage and temperature distribution over the stack volume can be split into a number

5.1 TEMPERATURE FIELD IN PLANAR SOFC STACKS

195

of 2D Poisson equations for the voltages and temperatures of individual bipolar plates. The cells (MEAs) can be represented as thin interfaces with prescribed equations for voltage-current characteristics and for heat fluxes. Furthermore, a physical 2D model can be formulated for a single element as in Figure 5.2. This model can then be replicated to the number of processors equal to the number of elements in a stack. This leads to very efficient parallel models for stack simulation (Sections 5.5 and 5.6). We begin this chapter with the analytical modelling of heat transport in a single element of an SOFC stack. This problem illustrates the advantages and limitations of 1D models.

5.1 Temperature field in planar SOFC stacks The high operating temperatures impose strict requirements on materials used for SOFC stacks. Especially dangerous are temperature nonuniformities: they induce inhomogeneities in local current density j and the domains with higher T and j experience larger thermal stress and a faster rate of ageing. Thus, one of the most important issues in SOFC design is minimizing temperature gradients over the stack volume. To our knowledge, no local measurements of temperature distribution along the surface of individual cell in SOFC stack have been performed so far. These measurements are difficult: they require the incorporation into the stack of segmented cells equipped with temperature sensors, which raises numerous problems with sealing, electrical connections etc. In this situation, a simulation gives invaluable information on the behaviour of the temperature field in a stack. The modelling of temperature distribution in an SOFC stack is usually based on numerical calculations (Recknagle et al., 2003; Iora et al., 2005; Ji et al., 2006; Zhu et al., 2005; Haberman and Young, 2006). It is worth noting that due to the lack of experimental data, none of the papers on modelling contains a comparison of calculated temperature profiles with the measurements. Below we develop a simple model of heat transport in a stack element shown in Figure 5.2 (Kulikovsky, 2009d). An analysis of dimensionless equations reveals the dominant term: the rate of heat exchange between the BP and air in the channel. This allows us to construct an asymptotic solution to a problem.

5.1.1

General assumptions

The model of heat transport in the element (Figure 5.2) is based on the following assumptions. 1. Heat fluxes along the y- and z-axes are negligible; the dominant role in heat removal is played by air flow in the channel (Figure 5.2).

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

2. Flow in the hydrogen channel does not affect the heat balance of the element. Heat management in the SOFC stack is usually performed by varying the air inlet temperature and flow rate; fuel is supplied in a pre-heated form. 3. All transport coefficients (except the Nusselt number) are constant and independent of temperature. 4. The radiative heat transfer is neglected. 5. The stack is run under high hydrogen stoichiometry. In this section, the effects due to hydrogen depletion along the channel are ignored. 6. A stack current is not large and the contribution of Joule heating is small. The last assumption is justified by the following estimate. Ionic conductivity of the active layers is usually much smaller than the conductivity of the bulk electrolyte. Thus, maximal Joule heating is produced by ionic current in the active layers. Consider for definiteness the cathode active layer. The heat flux qJ due to Joule heating in this layer is given by the last term in Eq. (2.149) qJ =

j 2 ltc . 3σi

Here ltc is the thickness of the cathode catalyst layer1 and σi is the ionic conductivity of the layer. For j = 1 A cm−2 and σi = 0.01 Ω cm−1 , we get qJ = 0.03 W cm−2 . The heat flux due to irreversible heating in the electrochemical reactions is qη = ηj. With η = 0.3 V, we find qη = 0.3 W cm−2 . Therefore, for typical current densities in the SOFC below 1 A cm−2 , we have qη  qJ and the Joule term can be safely neglected. Note, however, that below 600+273 ◦ C, the conductivity of the YSZ electrolyte dramatically decreases and the contribution of Joule heating grows (Kulikovsky, 2009f).

5.1.2

The general equation for bipolar plate temperature

Consider a small box with the sides δx and δy in the BP of thickness hp separating the cells α and β in a stack (Figure 5.3). Let the heat flux incoming to the box from the cell β be qβ and the flux leaving the box to the cell α be qα (Figure 5.3). The fluxes qN , qS , qE and qW are “in-plane” heat fluxes through the north, south, east and west end faces of the box (Figure 5.3). 1 On the anode side of a planar anode-supported cell, instead of lc the thickness of t the current-generating domain l∗ should be used.

5.1 TEMPERATURE FIELD IN PLANAR SOFC STACKS

197

Figure 5.3: Sketch of the heat fluxes in a small box in the BP.

The balance of fluxes in the box gives (qE − qW )hp δy + (qN − qS )hp δx + (qα − qβ )δxδy = 0.

(5.1)

Note that the flux directed outward is accounted for as positive (Figure 5.3). Dividing (5.1) by δxδy and noting that qE − qW = δqx , qN − qS = δqy , where qx and qy are the x- and y-components of in-plane heat flux, we get δqx δqy 1 + = (qβ − qα ). δx δy hp

(5.2)

According to Fourier’s law, qx = −λpx ∂T /∂x and qy = −λpy ∂T /∂y, where λpx and λpy are the BP thermal conductivities in the x- and y-directions, respectively. For plate temperature T we thus find ∂ ∂x

    ∂T ∂ ∂T 1 λpx + λpy = (qα − qβ ). ∂x ∂y ∂y hp

(5.3)

If BP thermal conductivity is isotropic and constant (λpx = λpy = λp ), Eq. (5.3) reduces to ∂2T ∂2T 1 + = (qα − qβ ) . 2 ∂x ∂y 2 hp λp

(5.4)

Equation (5.4) can be transformed into the standard Laplace equation for plate temperature. Indeed, since the plate is thin, we can write qα − qβ ∂qz . ' hp ∂z

(5.5)

Taking into account that qz = −λp ∂T /∂z we come to a well-known result ∂2T ∂2T ∂2T + + = 0. ∂x2 ∂y 2 ∂z 2

(5.6)

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

This helps in clarifying the meaning of Eq. (5.4). This equation describes heat transport in a thin plate, for which the z-derivative of heat flux in the Laplace equation (5.6) can be approximated using the finite-difference formula (5.5). A great advantage of Eq. (5.4) is lower dimensionality.

5.1.3

Heat balance in the air channel

Now we return to the stack element in Figure 5.2. The heat flux (W m−2 ) transported by the air flow along the channel is qx = ρair vcP a Tair

(5.7)

where ρair is the air density, v is the flow velocity, cP a is the air heat capacity and Tair is the flow temperature. In Eq. (5.7) the heat flux due to the kinetic energy of the flow ρair vv 2 /2 is omitted which is justified for low subsonic flows. The divergence of qx equals the rate of heat exchange with the BP. The equation of heat balance in the channel, therefore, is T − Tair ∂qx = Hair (x) ∂x hc

(5.8)

where hc is the characteristic length for the heat exchange (channel height) and Hair (W m−2 K−1 ) is the coefficient of heat transfer between air in the channel and the BP. Hair is a rapidly varying function of x, which can be approximated as Hair

  x = + − exp − lu   0    Hair x 1 = Hair 1 + 1 − 1 exp − l Hair u 1 Hair

0 Hair

1 Hair




(5.9)

where the superscripts “0” and “1” mark the values at the channel inlet and outlet, respectively. Physically, the variation of Hair with x is due to the development of the thermal boundary layer on the channel walls. At 0 the inlet, where no boundary layer exists yet, Hair is much larger than 1 1 Hair ; the value of Hair is achieved at the characteristic length of boundary layer development x ' lu (Haberman and Young, 2005). Note that typically lu  L, where L is the channel length. A subsonic flow of gas is incompressible (Section 4.1). Any density disturbance in such a flow induces a respective disturbance in pressure; the latter generates an acoustic wave which neutralizes the initial density disturbance. We therefore may set ρair ' ρ0air .

5.1 TEMPERATURE FIELD IN PLANAR SOFC STACKS

199

At low oxygen stoichiometry, flow velocity v decreases with x. Indeed, the concentration of molecules in the flow determines the pressure. Oxygen is consumed by an electrochemical reaction; thus, with a decreasing number of molecules flow velocity must decrease with x (the decrease in v provides compression of the elementary gas volume thereby keeping pressure in this volume constant). However, if oxygen stoichiometry is large (which is typical of SOFC stacks), this effect is negligible and we may put v ' v 0 = const. Therefore, ρair v ' ρ0air v 0 and Eq. (5.8) reduces to2 ρ0air v 0 cP a

5.1.4

T − Tair ∂Tair = Hair (x) . ∂x hc

(5.10)

Heat balance in the BP

For the element in Figure 5.2, Eq. (5.4) takes the form 1 ∂2T = (qα − qβ ) . ∂x2 hp λp

(5.11)

The total heat flux qT emitted by the MEA is determined by thermodynamic and irreversible heat released in the electrochemical reactions (Section 2.8):  qT =

T ∆S 2F

 j + ηj.

(5.12)

Here ∆S is the entropy change in the overall reaction, j is the local current density in the cell, and η is the sum of half-cell polarization voltages: η = ηa + ηc . The total flux qT is split into the cathode- and anode-directed fluxes (Figure 5.4(a)). Let the cathode- and anode-directed fractions be κqT and (1 − κ)qT , respectively (Figure 5.4(a)). Taking into account the signs of the fluxes we write qα − qβ = −(1 − κ)qT − κqT = −qT .

(5.13)

In other words, Figure 5.4(a) is equivalent to Figure 5.4(b), where only one side of the BP is heated by the total flux qT while the other side is thermally insulated3 . 2 Detailed calculations (Iora et al., 2005) show that the variation of the product ρ air v in the SOFC air channel does not exceed 2%, though ρair and v exhibit 10% variations. 3 In this section, we assume that the total heat flux along the stack axis z is zero. This assumption will be relaxed in Section 5.6.

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CHAPTER 5. MODELLING OF FUEL CELL STACKS (a)

(b)

Figure 5.4: Schematic of the heat fluxes onto the cathode and anode sides of the BP. When stack temperature and current do not vary along the z-axis the situation in (a) is equivalent to the situation in (b), where the total flux falls onto one side of the plate while the other side is thermally insulated. Using (5.13) in (5.11) and taking into account (5.12) we obtain −λp

∂2T T j∆S ηj = + . ∂x2 2F hp hp

(5.14)

Equation (5.14) is, however, not complete: the rate of heat transfer between the air channel and BP must be added to the right side. Finally we find −λp

T j∆S ∂2T ηj T − Tair = + − Hair (x) ∂x2 2F hp hp hc

(5.15)

where Hair was discussed above. For further calculations, it is convenient to introduce the dimensionless variables: x ˜=

x , L

T T˜ = , T298

˜j = j , jref

η˜ =

η bc

(5.16)

where T298 is the standard temperature (298 K), jref the reference current density and bc the Tafel slope on the cathode side. With these variables, Eqs (5.15) and (5.10) take the form    ∂ 2 T˜  2 ˜ + ω T + φ2 η˜ ˜j = ξ 2 fu (˜ x) T˜ − T˜air 2 ∂x ˜   ∂ T˜air ψ2 = ξ 2 fu (˜ x) T˜ − T˜air . ∂x ˜

(5.17) (5.18)

Here s ω= s φ=

L2 jref ∆S 2F hp λp

(5.19)

L2 bc jref T298 hp λp

(5.20)

5.1 TEMPERATURE FIELD IN PLANAR SOFC STACKS s ξ= s ψ=

201

1 L2 Hair hc λp

(5.21)

Lρ0air v 0 cP a λp

(5.22)

  x ˜ fu = 1 + (ru − 1) exp − ˜lu 0 0 Nu H = ru = air 1 Hair Nu 1

(5.23) (5.24)

and Nu =

Hair hc λair

(5.25)

is the Nusselt number; λair is the thermal conductivity of air. Comparing (5.21) and (5.25) we see that 2



ξ =

λair λp



Nu 1 2

(5.26)

where =

5.1.5

hc  1. L

(5.27)

Cell polarization curve

Equation (5.17) for the BP temperature contains voltage loss η˜ and local current ˜j. These values are related by the cell polarization curve. In an SOFC, this curve is well approximated by a linear function ˜ ˜j V˜cell = V˜oc − η˜ = V˜oc − R

(5.28)

where V˜cell = Vcell /bc is the cell voltage, V˜oc = Voc /bc is the open-circuit voltage, ˜ ˜j η˜ = R

(5.29)

is the total voltage loss, R is the cell area-specific resistance (Ω cm2 ) and ˜ = Rjref . R bc

(5.30)

˜ is a strong (exponential) function of local temperature; The resistance R ˜ T˜) is specified below. the actual dependence R(

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

The electric conductivity of BP is large and in this section the BP surface is assumed to be equipotential (the effects due to non-equipotentiality of BPs are considered in Sections 5.5–5.7). In other words, cell voltage V˜cell is constant along the channel. Neglecting the weak dependence of V˜oc on temperature, the constancy of V˜cell means that η˜ is constant along x ˜. We therefore have η˜ = η˜0 ; taking into account (5.29) we get ˜j =

η˜0 . ˜ T˜) R(

(5.31)

Substituting (5.31) into (5.17) we finally find the equation for BP temperature    η˜0 ∂ 2 T˜  2 ˜ 2 2 0 ˜ ˜ = ξ f (˜ x ) T − T + ω T + φ η ˜ u air . ˜ T˜) ∂x ˜2 R(

(5.32)

Parameter η˜0 fixes a point on the polarization curve: for given η˜0 , the temperature distribution T˜(˜ x) determines the local current ˜j(˜ x) (5.31) and the average stack current density J˜ =

Z

1

˜j d˜ x.

(5.33)

0

5.1.6

Boundary conditions

We will assume that the thermal insulation of a stack is ideal, i.e. heat flux to the ambient space is zero. However, at the inlet and outlet, heat flux from the BP to air in the channel is not zero. The temperature of the BP is, therefore, subject to the boundary conditions ∂T 0 −λp = −Hair (T − Tair ) ∂x x=0 ∂T 1 −λp = Hair (T − Tair ). (5.34) ∂x x=L In dimensionless variables this pair of equations takes the form   ∂ T˜ 0 − = −kNu 0 T˜ − T˜air ∂x ˜ x˜=0   ∂ T˜ 1 − = kNu 1 T˜ − T˜air ∂x ˜ x ˜=1

(5.35)

(5.36)

5.1 TEMPERATURE FIELD IN PLANAR SOFC STACKS

203

where k=

Lλair . hc λ p

(5.37)

Physically, Eqs (5.35)–(5.36) express the heat transfer between the BP and air flow at the channel inlet and outlet, respectively. The boundary condition for the air temperature is merely 0 T˜air |x˜=0 = T˜air .

(5.38)

The problem (5.32) and (5.18) with the boundary conditions (5.35), (5.36)–(5.38) contains seven parameters (ω, φ, ξ, ψ, lu , Nu 0 and Nu). This complicates the numerical analysis of the problem. Fortunately, one of these parameters is large; this allows us to construct the approximate analytical solution using the method of asymptotic expansion.

5.1.7

Method of asymptotic expansion

To illustrate this method, consider the following problem. Suppose that we are seeking an approximate positive solution to the equation γx2 + x − 1 = 0

(5.39)

where γ  1 is a small parameter. Setting in (5.39), γ = 0, we find the leading term x = 1. To find the next approximation we write x = 1 + γf + · · · where f is a constant to be determined. Substituting this relation in (5.39) and collecting the terms with the same powers of γ we get γ(1 + f ) + 2γ 2 f + γ 3 f 2 = 0.

(5.40)

The key trick in the methods of asymptotic expansion is that now we separately equate the terms with various powers of γ to zero (this procedure is called equating like powers of γ)4 . Taking the first-order term in Eq. (5.40), 4 In

other words, the solution to the equation g0 (x) + γg1 (x) + γ 2 g2 (x) + · · · = 0

where γ is a small parameter, is sought in the form g0 = 0 g1 = 0 g2 = 0 ...

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

we write γ(1 + f ) = 0, or f = −1. Thus, the first-order solution to (5.39) is x ' 1 − γ.

(5.41)

This result can, of course, be obtained if we directly solve Eq. (5.39) and then expand the result in series over γ. Note, however, that the procedure above leads to (5.41) without the knowledge of the exact solution of the equation to be solved. For nonlinear algebraic and differential equations, this advantage of the method is crucial5 .

5.1.8

Asymptotic solution

Neglecting the effect of the variable Nusselt number, we set fu = 1 in Eqs (5.32), (5.18) and these equations take the form6  η˜0   ∂ 2 T˜  2 ˜ 2 0 2 ˜ ˜air + ω T + φ η ˜ T − T = ξ ˜ T˜) ∂x ˜2 R(   ∂ T˜air = ξ 2 T˜ − T˜air . ψ2 ∂x ˜

(5.42) (5.43)

Parameter ξ is large (in typical conditions ξ ' 16; see Table 5.2). Therefore, the parameter γ≡

1 ξ2

(5.44)

is small. Dividing (5.42) and (5.43) by ξ 2 we get   η˜0 ω 2 T˜ + φ2 η˜0 2˜ ∂ T  = T˜ − T˜air γ 2 + ˜ T˜) ∂x ˜ R( 

γψ 2

∂ T˜air = T˜ − T˜air . ∂x ˜

(5.45)

(5.46)

5 The methods of asymptotic expansion are described in an indispensable book by Van Dyke (Dyke, 1964). 6 The characteristic length l of the Nusselt number variation in Eq. (5.23) is usually u small compared to a typical channel length (lu ' 0.3 cm, while L & 10 cm). In numerical calculations this allows us to put fu = 1 in Eqs (5.32), (5.18) keeping, however, a high value of the inlet Nusselt number Nu 0 in the boundary condition (5.35).

5.1 TEMPERATURE FIELD IN PLANAR SOFC STACKS

205

At γ = 0, the left sides of Eqs (5.45), (5.46) vanish and we get T˜ = T˜air . Thus, at the leading order, the stack and air temperatures coincide. To find the first-order correction to this relation, we employ the following regular asymptotic expansion T˜ = T˜air + γf1 (x) + · · ·

(5.47)

where f1 (x) is a function to be determined. Substituting (5.47) into (5.43) and using (5.44) we get ψ2

∂f1 ∂ T˜ − γψ 2 = f1 . ∂x ˜ ∂x ˜

Equating like powers of γ, we find f1 = ψ 2

∂ T˜ ∂x ˜

∂f1 = 0. ∂x ˜

(5.48)

Differentiating the first of Eqs (5.48) we find ∂ 2 T˜ 1 ∂f1 = 2 . ∂x ˜2 ψ ∂x ˜ Using this relation and (5.47) in (5.42) we get   η˜0 ω 2 T˜air + γω 2 f1 + φ2 η˜0 1 ∂f1   + = f1 . ψ2 ∂ x ˜ ˜ T˜air + γf1 R Setting here γ = 0 (i.e. collecting the terms with γ 0 ≡ 1) we come to   η˜0 ω 2 T˜air + φ2 η˜0 1 ∂f1 − 2 + f1 = . ˜ T˜air ) ψ ∂x ˜ R(

(5.49)

Parameter ψ 2  1 (Table 5.2). Thus, in (5.49) the term with the derivative ∂f1 /∂ x ˜ can be safely omitted and we get   η˜0 ω 2 T˜air + φ2 η˜0 f1 =

˜ T˜air ) R(

=

  0 η˜0 ω 2 T˜air + φ2 η˜0 ˜ T˜0 ) R( air

.

The last relation holds since in view of (5.48) we have f1 ' const.

(5.50)

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

From (5.47) we now find f1 T˜ = T˜air + 2 ξ

(5.51)

where the constant f1 is given by the last relation (5.50). The stack and air temperatures thus differ by a constant value. To find the dependencies of T˜ and T˜air on x ˜ we substitute (5.51) into (5.43), which yields ψ2

∂ T˜air = f1 . ∂x ˜

(5.52)

With constant f1 the solution is f1 0 T˜air = T˜air + 2x ˜. ψ

(5.53)

With this, Eq. (5.51) finally gives f1 T˜ = T˜0 + 2 x ˜ ψ

(5.54)

f1 0 T˜0 = T˜air + 2. ξ

(5.55)

where

We see that the air and stack temperatures are parallel straight lines.

5.1.9

Local current

The distribution of local current follows from Eq. (5.31). Using (5.54) in (5.31) we find ˜j =

η˜0 

˜ T˜0 + R

f1 x ˜ ψ2

.

(5.56)

An estimate with the data from Table 5.2 shows that f1 /ψ 2  T˜0 . Expanding (5.56) over f1 and retaining the linear term we get ! ! 0 ˜ η ˜ ∂ ln R f x ˜ 1 ˜j = 1− . (5.57) ˜ T˜0 ) ψ2 R( ∂ T˜ ˜0 T

˜ this is a straight line with the slope proportional to Since f1 ∼ 1/R, −3 ˜ 0 ˜ ˜ T˜0 ) ' R( ˜ T˜0 )). Thus, any variation of the inlet R (Tair ) (here we put R( air

5.1 TEMPERATURE FIELD IN PLANAR SOFC STACKS

207

˜ air temperature has a strong effect on the distribution of local current. R ˜ usually exponentially decreases with T (see the next section); the increase 0 in T˜air thus dramatically increases the nonuniformity of local current along the channel.

5.1.10

Example: Oxide-dominated stack resistivity

˜ T˜). To illustrate these results we have to specify the dependence R( ˜ Typically, R is determined by one (or a combination) of the following factors: (i) the resistivity of the electrolyte, (ii) the resistivity of the oxide layer covering the BP surface, and (iii) the resistivity corresponding to the activation of the anode electrochemical reaction (Section 4.6). ˜ follows the inverse In all cases, the temperature dependence of R Arrhenius law with the linear in T˜ pre-exponential factor (Zhu et al., 2005; Huang et al., 2000) ! ! ˜ ˜∗ T T ˜ T˜) = R ˜∗ R( exp (5.58) T˜∗ T˜ where T∗ is the characteristic temperature (activation energy). For the YSZ electrolyte, T∗ is in the order of 104 K (Zhu et al., 2005); for the oxide layers it falls into the range of 3000-5000 K (Huang et al., 2000). In (Yakabe et al., 2001), measurements of cell resistivity were fitted by a similar type of dependence with T∗ = 7185 K. Below we will consider the oxide-dominated type of stack resistivity (Table 5.1). ˜ For ∂(ln R)/∂ T˜ we have   ˜∗ − T˜ T ˜ ∂(ln R) =− . (5.59) ∂ T˜ T˜2 The SOFC working temperature is far below 3000 K; hence this derivative is always negative. Therefore, cell resistivity decreases with increasing temperature. ˜ (5.58) takes the form Equation (5.54) with R ! ! ! 0 η˜0 T˜∗ T˜∗ ω 2 T˜air + φ2 η˜0 0 ˜ ˜ T =T + exp − 0 x ˜. (5.60) ˜ ∗ T˜0 ψ2 R T˜air air In dimension variables, (5.60) reads T = T0 +



η0 0 hp ρair v 0 cP a R∗



T∗ 0 Tair



  0  T∗ Tair ∆S exp − 0 + η 0 x. Tair 2F (5.61)

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Table 5.1: The physical parameters. Air temperature at the inlet, Tair (K) Bipolar plate thickness, hp (m) Characteristic length for heat exchange, hc (m) Channel length, L (m) Bipolar plate thermal conductivity, λp (W m−1 K−1 ) Air thermal conductivity, λair (W m−1 K−1 ) Entropy change in hydrogen-oxygen reaction, ∆S (J mol−1 K−1 ) Specific heat of air at 700 ◦ C, cP a (J kg−1 K−1 ) Air density at 700 ◦ C, ρair (kg m−3 ) Flow velocity in the channel, v 0 (m s−1 ) Characteristic temperature, T∗ (K) (Huang et al., 2000) Characteristic area-specific resistivity, R∗ (Ω m2 ) Voltage loss, η (V) Tafel slope on the cathode side, bc (V) Reference current density, jref (A m−2 ) Nusselt number in a developed flow, Nu ≡ Nu 1 Nusselt number at the channel inlet, Nu 0 Characteristic length of Nusselt number decay, lu (m)

600 + 273 10−3 10−3 10−1 12 0.073 44.3 1160 0.32 6 3280 0.116 × 10−4 0.3 0.15 104 4 40 5 × 10−3

Comparison of the approximate relation (5.61) with the numerical solution of the full system of equations (5.32), (5.18), (5.35), (5.36), (5.38) and (5.58) is given in Figure 5.5. Parameters for the calculations are displayed in Table 5.1; the respective dimensionless parameters are listed in Table 5.2. We see that the linear shape (5.61) describes the numerical temperature profile quite well. Local currents calculated with the relation (5.31) using the numerical and analytical temperature shapes agree even better (Figure 5.6).

5.1.11 Remarks In view of Eq. (5.26), ξ ∼ 1/; thanks to the smallness of , parameter ξ is large (ξ & 10, Table 5.2) and ξ 2 ' 102 . Thus, the leading term in the heat transport equations is the rate of heat exchange between air in the channel and the stack. Due to this high rate, T and Tair are close to each other and their x-dependencies are well described by a simple linear law. A numerical model (Iora et al., 2005) supports this conclusion. For large ξ, the boundary conditions (5.35) and (5.36) have minor effect on the shape of T˜(˜ x). These conditions describe the “fine structure” of the

5.1 TEMPERATURE FIELD IN PLANAR SOFC STACKS

209

Table 5.2: The dimensionless parameters for calculations and their meanings. The physical parameters are listed in Table 5.1. kNu 0 kNu = kNu 1 ˜lu ˜ R T˜0 air

ω φ ξ ψ

(a)

146 14.6 0.03 2.32 2.93 1.383 2.048 15.60 4.308

Factorized Nusselt number at the inlet Factorized Nusselt number in developed flow Length of Nusselt number relaxation Cell resistivity Inlet air temperature Reversible heating in reactions Irreversible heating Heat transfer between stack and air flow Heat transport with the air flow

(b)

Figure 5.5: (a) Solid lines—numerical solution to the full system of equations (5.32), (5.18), (5.35), (5.36), (5.38). Dashed line—approximate analytical solution (5.61). Parameters for calculations are listed in Table 5.1. (b) Numerical (solid curves) and approximate (dashed lines) stack temperatures for the indicated values of voltage loss (V). curve T˜(˜ x) in the vicinity of the boundary points (Figure 5.5(a)); in the “bulk” region, T˜(˜ x) is well described by the linear law. Equation (5.61) shows that the voltage loss η 0 (or mean current, which is proportional to η 0 ) increases the slope of T (x). This effect has been reported in numerical calculations (Inui et al., 2006). One of the consequences of this effect is that the thermal stress due to dynamic variation of load is maximal at the air channel outlet. Flow velocity appears in the denominator in Eq. (5.61). Therefore, an increase in v 0 is always beneficial, since it flattens the temperature slope. Moreover, if J varies with time, a synchronous variation of v 0 such that v 0 ∼ J can completely cancel temporal variations of T near the channel outlet. This idea was suggested in Inui et al. (2006) based on numerical calculations; we see that it immediately follows from Eq. (5.61).

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Figure 5.6: Local current density along the air channel (5.31) calculated with the numerical shape of T (x) (solid curves) and with the analytical relation (5.60) (dashed curves). Indicated are the values of total voltage loss (V). In real stacks, air stoichiometry is large since air plays the role of a cooling medium. The linear temperature shape can be distorted by fuel exhaustion along the channel. If hydrogen utilization is large, the domain close to the outlet of the respective channel suffers from fuel starvation; the local current there is lowered and we may expect a flattening of the temperature shape in the middle of the channel (here we discuss co-flow feeding).

5.2 Temperature gradient in SOFC stack Equation (5.61) was derived under the assumption of low hydrogen utilization. This assumption can be relaxed; a simple equation for the temperature gradient along the air channel valid for an arbitrary value of hydrogen utilization is obtained in this section (Kulikovsky, 2009f). The mass balance equation for the hydrogen concentration along the channel (4.168) has to be taken into account. This equation leads to the general integral relation of hydrogen stoichiometry and outlet concentration (4.175) which will also be used below.

5.2.1

Stack and air temperatures

In Eq. (5.42), we return to the local cell current:    ∂ 2 T˜  2 ˜ 2 0 ˜ 2 ˜ ˜air = ω T + φ η ˜ j + ξ T − T ∂x ˜2

(5.62)

where ˜j is the local current density. The equation for the air flow temperature is still (5.43).

5.2 TEMPERATURE GRADIENT IN SOFC STACK

211

As discussed in Section 5.1.8, due to the large value of parameter ξ, at the leading order the stack and air flow temperature coincide. Using again the expansion (5.47) and repeating the procedure of Section 5.1.8, we find that the equation for f1 , Eq. (5.50), takes the form   f1 = ω 2 T˜ + φ2 η˜0 ˜j.

(5.63)

With (5.63), Eq. (5.43) reduces to ψ2

  ∂ T˜air = ω 2 T˜ + φ2 η˜0 ˜j. ∂x ˜

(5.64)

Dividing Eq. (5.64) by Eq. (4.168) we obtain   ψ 2 ∂ T˜air = − ω 2 T˜ + φ2 η˜0 c λa J˜ ∂˜ where c˜ is the hydrogen concentration. Note that x ˜ in this section is equivalent to z˜ in Eq. (4.168). At the leading order, T˜ ' T˜air ; thus, to a good approximation, on the right side we can replace T˜ by T˜air . This gives   ψ 2 ∂ T˜air = − ω 2 T˜air + φ2 η˜0 , c λa J˜ ∂˜

T˜air

c˜=1

0 = T˜air .

(5.65)

Note that Eq. (5.65) is valid for an arbitrary value of hydrogen utilization. Separating in (5.65) variables and integrating we write ψ2 λa J˜

Z

T˜air

0 T˜air

dT˜air 

Z

ω 2 T˜air + φ2 η˜0

 =−

c˜

d˜ c 1

which yields ψ2 ln λa J˜

ω 2 T˜air + φ2 η˜0 0 + φ2 η ω 2 T˜air ˜0

! = 1 − c˜.

Solving this for T˜air , we finally find T˜air

  φ2 η˜0 0 exp = T˜air + 2 ω

˜ 2 (1 − c˜(˜ λa Jω x)) ψ2

! −

φ2 η˜0 . ω2

(5.66)

This equation relates the x ˜-shapes of hydrogen concentration c˜(˜ x) and air flow temperature T˜air (˜ x).

212

5.2.2

CHAPTER 5. MODELLING OF FUEL CELL STACKS

Temperature gradient

Setting in Eq. (5.66), x ˜ = 1, and taking into account Eq. (4.175), we obtain 1 the outlet air flow temperature T˜air : 1 T˜air =



0 T˜air

φ2 η˜0 + 2 ω

˜ 2 Jω ψ2

 exp

! −

φ2 η˜0 . ω2

(5.67)

For the change in air flow temperature over the whole channel length we find ! !   ˜ 2 φ2 η˜0 Jω 1 0 0 ˜ ˜ ˜ ˜ ∆Tair ≡ Tair − Tair = Tair + 2 exp − 1 . (5.68) ω ψ2 At the leading order T˜ = T˜air ; thus, this equation also approximates the variation of stack temperature ∆T˜: ∆T˜ ' ∆T˜air .

(5.69)

In dimension variables the latter equation reads  ∆T '

0 Tair +

2F η 0 ∆S

  exp

JL∆S 2F hp ρ0air v 0 cP a



 −1 .

(5.70)

If the power of an exponent is small (which is typically the case), the exponent can be expanded and we obtain  ∆T '

0 Tair ∆S + η0 2F



JL

. hp ρ0air v 0 cP a

(5.71)

The polarization curve of SOFC is linear; this means that η 0 and J are related through η 0 = RJ, where R is the cell resistivity. Thus, Eq. (5.71) transforms to  0  ∆T Tair ∆S J ' + RJ (5.72) L 2F hp ρ0air v 0 cP a which is the main result of this section. Typical temperature gradients resulting from Eq. (5.72) are depicted in Figure 5.7 as a function of stack current density (solid lines) for the three values of cell resistivity R (Ω cm2 ). For simplicity, here we ignore the temperature dependence of cell resistivity; in practical estimates with Eq. (5.72), this dependence can be taken into account using the literature data. If R is small, the growth of ∆T with J is almost linear and slow (Figure 5.7). However, when cell resistivity is large, the quadratic in J

5.2 TEMPERATURE GRADIENT IN SOFC STACK

213

Figure 5.7: The average gradient of the stack temperature along the air channel (K cm−1 ; see Eq. (5.72)) as a function of mean current density for the indicated values of cell resistivity (Ω cm2 ). Solid lines: Eq. (5.72), and points: the exact numerical solution to Eqs (5.62) and (5.43). Parameters for this plot are listed in Table 5.1, except flow velocity, which is 10 m s−1 here. term in Eq. (5.72) increases and the temperature gradient grows with J much faster. For comparison, the exact numerical solution to Eqs (5.62) and (5.43) is shown by points in Figure 5.77 . As can be seen, the analytical solution is in excellent agreement with the numerical result. From the analysis above it follows that Eq. (5.72) is valid for any distribution of hydrogen concentration along the channel. Equation (5.72) only requires that the integral relation (4.175) be satisfied; the latter relation holds regardless of the details of the hydrogen concentration shape. Though this shape is a function of the hydrogen utilization u, Eq. (5.72) is valid for arbitrary values of u. The validity of Eq. (5.72) essentially rests on large values of parameters ξ 2 and ψ 2 . Large ξ 2 provides equality of stack and air flow temperatures at the leading order, while large ψ 2 simplifies the expression for the first-order correction f1 . Equation (5.72) does not include the interconnect thermal conductivity. Physically, this parameter plays a minor role in the heat balance due to the dominating rate of heat exchange between the interconnect and air flow. Note also that the temperature gradient over the channel length increases 0 . with the growth of inlet air flow temperature Tair The model ignores radiative heat transfer in the stack. In the temperature range from 700 to 800 ◦ C, this process merely shifts the temperature curve 7 In Eq. (5.62), we replaced η ˜ J˜ and ˜ ˜ The last replacement is ˜0 with R j with J. justified by the following arguments. Along the channel, hydrogen concentration drops and temperature rises; both effects partially compensate for each other and the variation of local current along x ˜ is not large (Figure 5.6).

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

T (x) by 20-30 K as a whole, but does not affect the temperature gradient (VanderSteen and Pharoah, 2006).

5.3 Thermal waves in SOFC stack Temperature variation over the surface of small “button” cells with an active area ' 1 cm2 is not large (Jiang and Virkar, 2003) and these cells are well described by 1D through-plane models (Pisani and Murgia, 2007). However, as shown above, for application-relevant cells with the active area in the order of 100-1000 cm2 the in-plane temperature gradient cannot be neglected. The analytical model (Section 5.1) and CFD calculations help one to understand the shape of the temperature and current density fields in large stacks (Achenbach, 1994, 1995; Recknagle et al., 2003; Iora et al., 2005; Ji et al., 2006; Zhu et al., 2005; Haberman and Young, 2005). Stationary models (Recknagle et al., 2003; Iora et al., 2005; Ji et al., 2006; Zhu et al., 2005; Haberman and Young, 2005), however, do not answer the fundamental questions: are steady-state solutions stable? What is the fate of small temperature disturbances in the stack? The response of the SOFC stack to the step change in total current at constant feed utilization has only been studied numerically in one work (Achenbach, 1995) (see the discussion below). In this section, we study the evolution of a small perturbation of stack temperature in a stack element shown in Figure 5.2. The analysis shows that the temperature distribution along the channel is unstable with respect to small long-wave perturbations (Kulikovsky, 2008b). Instability manifests itself as a slow thermal wave moving along the direction of air flow with an exponentially growing amplitude. Some consequences of this effect for stack operation and for numerical modelling of stacks are discussed.

5.3.1

Basic equations

Consider again the stack element in Figure 5.2. Our goal is understanding the basic physics of transient thermal effects in a stack. Thus, many details important for engineering calculations (e.g. hydrocarbon fuel reforming, co- and cross-flow feeding etc.) will be omitted; accounting for these details is a task for CFD calculations. In this section, we return to the simplest though realistic situation of a stack fed with hydrogen at small utilization. This regime is usually used as a reference point in the development of real SOFC stacks. The set of assumptions for the model below is listed in Section 5.1.1. As before, we will assume that the velocity and thermal boundary layers are fully developed, so that the Nusselt number of the flow is constant along the channel. Model assumptions lead to Eqs (5.32) and (5.18), which describe the BP and air flow temperature, respectively. To rationalize the response of the

5.3 THERMAL WAVES IN SOFC STACK

215

element temperature to small perturbations in T or j, we have to add the terms with time derivatives to these equations. The non-stationary dimension version of Eq. (5.32) for bipolar plate temperature T is ρp cP p

∂T ∂2T − λp 2 = ∂t ∂x



T ∆S + η0 2F



η0 T − Tair − Hair . hp R hc

(5.73)

Here ρp is the BP density and cP p is the BP heat capacity. The heat balance in the air channel is described by ρ0air cP a

∂Tair ∂Tair T − Tair + ρ0air v 0 cP a = Hair . ∂t ∂x hc

(5.74)

Introducing the dimensionless time tv 0 t˜ = L

(5.75)

and using the dimensionless variables (5.16), Eqs (5.73) and (5.74) take the form     η˜0 ∂ T˜ ∂ 2 T˜ 2 ˜ 2˜ 2 0 ˜air (5.76) − ξ T − T − = ω T + φ η ˜ ˜ T˜) ∂x ˜2 ∂ t˜ R(   ∂ T˜air ∂ T˜air ψ2 + ψ2 = ξ 2 T˜ − T˜air (5.77) ∂x ˜ ∂ t˜ µ2

where s µ=

ρp v 0 cP p L λp

(5.78)

and the other dimensionless parameters are given by (5.19)–(5.22). In typical situations, ψ  µ (Tables 5.2 and 5.4). This means that air flow temperature varies much faster than the stack temperature. Physically, air enthalpy is 4 orders of magnitude smaller than the enthalpy of BP and the largest time scale in the system is determined by BP heating (Achenbach, 1995). Therefore, flow temperature can be considered as a quasi-stationary variable i.e. the time derivative in Eq. (5.77) can be omitted. For T˜air we thus get ψ2

  ∂ T˜air = ξ 2 T˜ − T˜air . ∂x ˜

(5.79)

For numerical calculations the system (5.76), (5.79) should be supplemented by the initial and boundary conditions. However, for a stability

216

CHAPTER 5. MODELLING OF FUEL CELL STACKS

analysis the explicit expressions for boundary conditions are not needed. We assume that the zero-order stationary solutions T˜0 and T˜air 0 (see below) obey these conditions, while the early stage of evolution of small perturbation is not affected by the boundary effects.

5.3.2

Stability analysis

To analyse stability of the system (5.76) and (5.79), it should be linearized. Let T˜0 and T˜air 0 be the stationary solutions and T˜1 and T˜air 1 the small perturbations: T˜ = T˜0 + T˜1 , T˜1  T˜0 T˜air = T˜air 0 + T˜air 1 , T˜air 1  T˜air 0 .

(5.80) (5.81)

Substituting (5.80) and (5.81) into (5.76), (5.79), and omitting the small term with T˜2 we get equations for the disturbances T˜1 and T˜air 1 : 1

˜ ∂ 2 T˜1 η˜0 − = µ ˜0 ∂x ˜2 ∂ t˜ R 2 ∂ T1

 ∂ ln R ˜ ω − ω T˜0 + φ2 η˜0 ˜ ∂T 2



2

!

  T˜1 − ξ 2 T˜1 − T˜air 1 (5.82)

  ∂ T˜air 1 = ξ 2 T˜1 − T˜air 1 . ψ2 ∂x ˜

(5.83)

Here ˜ 0 ≡ R( ˜ T˜0 ) R and we used the expansion 1 1 ' ˜ ˜ ˜ T˜0 + T˜1 ) ˜ R( R0 + ∂∂R T T˜ 1 1

1  ' = ˜ ˜ ˜ ∂ ln R ˜ R 0 R0 1 + ∂ T˜ T1

˜ ∂ ln R 1− T˜1 ∂ T˜

! .

The system (5.82), (5.83) is linear and its formal solution can be written in terms of Fourier series. To analyse the stability of this system we substitute the Fourier component of the solution
T˜1 = T˜1 exp i(k˜ x − ν t˜)
T˜air 1 = T˜air 1 exp i(k˜ x − ν t˜) (5.84) where T˜1 and T˜air 1 on the right side are complex amplitudes of the disturbances, k is the wave vector and ν is the complex frequency. The

5.3 THERMAL WAVES IN SOFC STACK

217

same notations for the disturbances and their amplitudes do not introduce ambiguity, since in all the equations below, complex exponents cancel out. The Fourier series expansion with the components (5.84) means that the solution of the system discussed is sought in terms of the sum of harmonics with the frequencies ν and wave vectors k 8 . In the following we will refer to the wave with small k as “low-k wave” or “long wave” (the wave with large λ). With (5.84), time- and x ˜-derivatives turn into products, e.g. ∂ T˜1 = −iν T˜1 , ∂ t˜

∂ T˜1 = ik T˜1 , ∂x ˜

∂ 2 T˜1 = −k 2 T˜1 . ∂x ˜2

Using (5.84) in (5.82) and (5.83) we get a system of linear equations for the amplitudes: a11 T˜1 + a12 T˜air 1 = 0 a21 T˜1 + a22 T˜air 1 = 0

(5.85)

where the matrix elements are a11

η˜0 = k 2 + ξ 2 − iµ2 ν + ˜0 R



 ∂ ln R ˜ ω 2 T˜0 + φ2 η˜0 − ω2 ∂ T˜

!

a12 = −ξ 2 a21 = −ξ 2 a22 = iψ 2 k + ξ 2 .

(5.86)

Nontrivial solutions to the system (5.85) exist if the determinant of the matrix on the left side is zero a11 a22 − a12 a21 = 0.

(5.87)

The solution to this equation gives the dispersion relation ν(k) of the problem. Physically, ν(k) describes the time response of a system to the disturbance with the wave vector k. Note that solutions with the positive imaginary part =(ν) > 0 mean an unstable regime of stack operation: if =(ν) ≡ κ > 0, we have exp(−iν t˜) ∼ exp(−i(iκ)t˜) = exp(κt˜), which grows with time.

5.3.3

Flow temperature is constant

To understand the general dispersion relations it is advisable to consider the case of constant flow temperature first. In that case T˜air 1 = 0, the 8 The

wave vector is inversely proportional to the wavelength λ: k = 2π/λ.

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

second equation in the system (5.85) vanishes and this system reduces to a11 = 0.

(5.88)

˜ 0 = ˜j0 , we Using the expression for a11 , Eq. (5.86) and the relation η˜0 /R get " !#   ˜ i 2 2 2 2˜ 2 0 ∂ ln R ˜ ν = − 2 k + ξ − j0 ω − ω T0 + φ η˜ (5.89) µ ∂ T˜ where ˜j0 is the undisturbed current density. We see that ν has only an imaginary part, i.e. R(ν) = 0. Physically this means that the disturbance does not propagate along the channel (see below). ˜ T˜ is negative (Section 5.1.10); thus the term with The derivative ∂ ln R/∂ ˜j0 in Eq. (5.89) is negative !   ∂ ln R ˜ 2 2 2 0 < 0. −˜j0 ω − ω T˜0 + φ η˜ ∂ T˜ However, parameter ξ 2 is large. With the parameters from Table 5.4, it appears that for all k, the parameter ξ 2 provides a positiveness of expression in the square brackets in (5.89): !   ∂ ln R ˜ 2 2 2 2 2 0 k + ξ − ˜j0 ω − ω T˜0 + φ η˜ > 0. (5.90) ∂ T˜ We conclude that due to large ξ 2 , =(ν) is always negative, i.e. stack temperature T˜ is stable with respect to small-amplitude perturbations of any wavelength. Physically, since the rate of heat transfer between air and BP is high, cooling media (air flow) with constant temperature effectively damp disturbances in T˜.

5.3.4

Solution: The general case

In the general case, Eq. (5.87) leads to quite a cumbersome expression for ν. However, thanks to large ξ this expression can be simplified to R(ν) =

ψ2 k µ2

1 =(ν) = − 2 µ

(5.91) "

ψ4 1+ 2 ξ



k 2 − ˜j0

  ∂ ln R ˜ ω 2 − ω 2 T˜0 + φ2 η˜0 ∂ T˜

!# . (5.92)

These relations are valid provided that k 2 ψ 4  ξ 4 , which is fulfilled for parameters from Table 5.4 and k . 10.

5.3 THERMAL WAVES IN SOFC STACK

219

Figure 5.8: Imaginary part of complex frequency ν versus wave vector k. Shown are the exact curve (dashed line) and the approximate analytical relation (5.92) (solid line); the two curves are practically indistinguishable. The imaginary part of ν is positive when the expression in the square brackets in (5.92) is negative: 

ψ4 1+ 2 ξ



k − ˜j0 2

 ∂ ln R ˜ ω − ω T˜0 + φ2 η˜0 ˜ ∂T 2



!

2

< 0.

(5.93)

Comparing the left sides of (5.93) and (5.90) we see that in (5.93) the large term ξ 2 has disappeared. This means that the air flow no longer provides absolute stabilization of disturbances in stack temperature. Indeed, for sufficiently small k, Eq. (5.93) is fulfilled:

k < kcrit

v u u = t˜j0

 ∂ ln R ˜ ω 2 − ω 2 T˜0 + φ2 η˜0 ∂ T˜ 

! 1+

ψ4 ξ2

−1 . (5.94)

˜ T˜ < 0, the expression under the root sign in Eq. (5.94) is Since ∂ ln R/∂ always positive. This means that kcrit is real, i.e. the stack temperature is unstable with respect to disturbances with k < kcrit . Figure 5.8 shows a comparison of approximate equation (5.92) with the exact solution to Eq. (5.87) for the set of parameters listed in Table 5.4. As can be seen, both curves are practically indistinguishable (Figure 5.8). We see that for the parameters in Table 5.4, kcrit ' 1. Since k = 2π/λ, for kcrit = 1 we get λcrit = 2π. Roughly speaking, this means that in order to generate instability the disturbance must occupy more than half of the channel length. Equations (5.91) and (5.92) help one to understand the physics of instability. The real part of ν is proportional to k (Eq. (5.91)); this is the signature of convective heat transport in the air channel. Air flow induces propagation of disturbance along the channel with the same group and

220

CHAPTER 5. MODELLING OF FUEL CELL STACKS

Table 5.3: The physical parameters. The other parameters are listed in Table 5.1. Specific heat of steel bipolar plate, cP p (J kg−1 K−1 ) Steel density, ρp (kg m−3 )

500 7900

Table 5.4: The dimensionless parameters. For ω, φ, ξ and ψ, see Table 5.2. The physical parameters are listed in Tables 5.1 and 5.3. µ ˜ ∂ ln R ∂ T˜

ε

428.9 −0.9411 0.01

Inverse time scale for stack temperature variation

phase velocities: v˜wave =

∂R(ν) R(ν) ψ2 = = 2. ∂k k µ

In dimension variables for vwave we find  0  ρ a cP a vwave = v0 . ρp cP p

(5.95)

(5.96)

Since ρp cP p  ρ0a cP a , the wave velocity vwave is much smaller than the flow velocity v 0 . With the data from Table 5.3 we find vwave ' 10−4 v 0 . For v 0 in the range of 1-10 m s−1 we get vwave = 10−4 -10−3 m s−1 or 0.1-1 mm s−1 . In other words, the disturbances in T˜ with the wave vectors satisfying (5.94) generate slow travelling waves with an exponentially growing amplitude. Physically, this thermal wave is generated due to the following mechanism. A local increase in stack temperature reduces local ASR, thereby increasing the local current j = η 0 /R. This leads to further growth of temperature, since the rate of heat production in Eq. (5.76) is proportional to the product T˜˜j. The disturbance in T moves toward the channel outlet due to heat transport with the air flow; however, the velocity of propagation is small, since the wave needs considerable time to heat up the BP domains located downstream. A schematic of wave propagation is shown in Figure 5.9. Note that this figure is for illustrative purposes only: the actual shape of the wave cannot be derived from the equations above—this is a task for numerical modelling. The analysis above only gives a criterion for wave generation and the wave velocity. To generate instability the initial disturbance must occupy a sufficiently large domain of a stack, i.e. k has to be small enough. Short-wave

5.3 THERMAL WAVES IN SOFC STACK

221

Figure 5.9: Sketch of the thermal wave at the successive moments in time; “0” is the initial undisturbed state. Note that in general, T in this state is not necessarily constant.

Figure 5.10: Schematic of small- and long-wave disturbances. In the first case (top) the temperature gradient along the x-axis is large and these disturbances rapidly die out due to in-plane heat transport in the bipolar plate. For long-wave disturbances (bottom) this gradient is small and the disturbances grow and propagate. disturbances (e.g. small hot spots) are effectively damped by the inplane heat transport in the bipolar plate (Figure 5.10). In contrast, the temperature gradient induced by a long-wave disturbance is small (Figure 5.10) and these disturbances grow and propagate. The maximal instability increment =(ν) is achieved when k = 0 (Eq. (5.92) and Figure 5.8). For the characteristic time of instability growth τ˜i = 1/=(ν) from (5.92) we find "

˜j0 τ˜i = µ2

 ∂ ln R ˜ ω 2 − ω 2 T˜0 + φ2 η˜0 ∂ T˜ 

!#−1 .

(5.97)

˜ 0 = 0.227 we obtain With the parameters from Table 5.4 and ˜j0 = η˜0 /R 4 τ˜i ' 5 × 10 . Taking into account Eq. (5.16), for typical channel length

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

L = 0.1 m and inlet velocities in the range of 1-10 m s−1 we get τi = 5005000 s. Thus, instability manifests itself on a time scale in the order of 10 min-2 hours.

5.3.5

Role of boundary conditions

The analysis above was performed ignoring the thermal boundary conditions at the air channel inlet and outlet. However, the inlet boundary condition may have quite a strong effect on the fate of temperature disturbance in a stack. Indeed, air temperature at the inlet is fixed and thanks to a high Nusselt number at x = 0, the inlet rate of heat transfer between air flow and bipolar plate is very high. For this reason, at the inlet the stack temperature is strongly coupled to the temperature of air flow. In other words, the inlet boundary condition for stack temperature (the first of Eq. (5.34)) can be replaced by the condition 0 T˜0 ' T˜air

(5.98)

where T˜0 ≡ T˜|x˜=0 . Equation (5.98) has a stabilizing effect on T˜: due to this condition any disturbance arising close to the channel inlet will be rapidly damped. Furthermore, if the stack temperature increases far from the inlet, Eq. (5.98) sooner or later will damp development of instability due to the wave of cooling propagating from the inlet. Above we have seen that thermal waves are transported along the stack at the velocity given by (5.95). Suppose that the stack temperature has increased by a constant value δ T˜ (which is equivalent to the disturbance with zero k). The boundary condition (5.98) will initiate a wave of cooling propagating from the inlet to the outlet. This wave will reach the channel outlet at the time τ˜w =

1 v˜wave

.

(5.99)

Thus, the stack temperature at the outlet T˜1 will grow if the characteristic time of instability growth is smaller than τw . In other words, T˜1 will grow if τ˜i < 1. τ˜w With Eqs (5.97), (5.99) and (5.95) this equation transforms to τ˜i ψ2    <1 =  τ˜w ˜j0 ω 2 − ω 2 T˜0 + φ2 η˜0 ∂ ln R˜ ∂ T˜

(5.100)

5.3 THERMAL WAVES IN SOFC STACK

223

or τ˜i ψ2   = τ˜w ˜j0 ω 2 1 − T˜0 +

φ2 η˜0 ω2



˜ ∂ ln R ∂ T˜

 < 1.

(5.101)

The ratio ψ2 va0 ∼ . ˜j0 ω 2 j0 L Therefore, stacks with longer channels at lower air flow rates and/or higher stack currents are more prone to instability. In contrast, stacks with short air channels and high air flow rates are expected to be more stable. Note also that the larger characteristic temperature T∗ lowers the time τ˜i and hence it facilitates the growth of instability at the channel outlet. Figure 5.11 shows the numerically calculated evolution of a small disturbance in stack temperature with time. As can be seen, the initial growth of T˜1 with time is then damped due to the wave of cooling propagating from the channel inlet.

5.3.6

Remarks

The analysis above is performed for a stack operated in the potentiostatic regime (constant η˜0 ). In this regime, propagation of the thermal wave causes growth of the stack current; the respective variation of the mean current density J˜ can be obtained from ˜ t˜) = η˜0 J(

Z 0

1

d˜ x  . ˜ T˜(˜ R x, t˜)

(5.102)

The fate of small temperature perturbations in a stack operating at constant total current (galvanostatic regime) is not obvious. Fixing the total current may damp instability or diminish the domain of parameters where the system is unstable. Numerical calculations show instability of galvanostatically operated stacks with a channel length of 40 cm. This instability is damped by the wave of cooling from the channel inlet, similar to that shown in Figure 5.11. At a nonlinear stage, the evolution of waves can be damped by heat transport to the chamber walls, which has been ignored in the analysis above. We assumed that at the initial stage the effect of heat transport to the chamber walls is small; this assumption led to instability. However, as the stack temperature increases, heat flux to the chamber walls also increases and this may damp the further instability growth. This scenario will cause oscillations of cell and stack voltage. The oscillations could be identified by their period, which is in the order from ten minutes to several hours.

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Figure 5.11: Evolution of disturbance in stack temperature. At time t = 0 s, stack temperature is disturbed by a constant value of 20 K. Note the very fast (in the order of 10 s) decay of disturbance at x ˜ = 0 (channel inlet, frame 33 s). Due to the instability, the stack temperature at the outlet (˜ x = 1) grows by 30 K (frames 33-494 s). Further temperature growth is damped by a slow wave of cooling propagating from the inlet (frames 164-987 s).

Equation (5.92) shows that =(ν) ∼ 1/µ2 ∼ 1/v 0 . Thus, by increasing the air flow velocity one can decrease the instability increment at the cost of accelerated wave propagation (cf. Eq. (5.96)). Note that v 0 appears in (5.92) as a factor (1/µ2 ∼ 1/v 0 ) rather than as a separate term. This means that by increasing flow velocity one can only reduce the instability increment, but not prevent instability development. Physically, this is due to the fact that the heat flux between the stack and air flow is proportional to the difference T − Tair . A small increase in T rapidly increases Tair and the difference T − Tair appears to be restored.

5.3 THERMAL WAVES IN SOFC STACK

225

This is a disadvantage of a cooling system with flow-through channels, which is essentially based on a surface mechanism of heat transfer through the interface between heated and cooling media. In contrast, in lowtemperature fuel cells water evaporation is a volume mechanism of energy loss, which effectively damps temperature disturbances (Kulikovsky, 2008a). The absence of this mechanism in SOFC stacks makes them, in general, thermally unstable. Long-wave disturbances in T can easily be produced during the startstop cycles or in the regimes with variable load. Varying current in the external load induces the variation of local current over the whole surface area. Formally this is equivalent to the disturbance in local current with small wave vectors k. The disturbance in j translates into the disturbance in T , which may trigger propagation of a thermal wave. Available experimental data on thermal (R.Steinberger-Wilkens et al., 2006) and current (Bujalski et al., 2007) cycling do not reveal the oscillations of stack voltage discussed above. However, the measurements cited were performed with short two-cell stacks; in such stacks both cells are “boundary” cells and the cooling heat flux from the cell to the chamber walls could damp the disturbances. The model above is formulated for a stack element located far enough from the boundary cells. In other words, thermal waves are expected to develop in large stacks with several tens of cells, in which core elements in the middle of the stack are thermally well insulated from the chamber walls. Achenbach (Achenbach, 1995) studied the response of a single cell in a stack to step load change using a 3D transient model. His calculations reveal the relaxation of stack temperature to a steady state on a time scale of the order of 300 s. These calculations were performed under constant H2 utilization, i.e. the hydrogen flow rate was increased prior to the step change in stack current. It is not clear whether the air flow rate was also increased. If this was the case, increased air flow rate could stabilize the thermal instability induced by the current step. In stacks fed with hydrocarbons, an important factor in the heat balance is cooling due to hydrocarbon-to-hydrogen reforming. In our model this process is neglected. However, numerical calculations cited at the beginning of Section 5.3 show that the reforming length is usually small compared to the total length of the air channel. Moreover, in some cases the reforming section is separated from the electrochemical section. In any case, the largest portion of the channel operates as a hydrogen-fed cell and for this portion the model above is valid. Thermal instability may cause convergence problems in numerical steady-state SOFC models, which include thermal effects. Roundoff errors in numerical calculations may occasionally take the form of long-wave perturbation; this initially negligible disturbance would then grow exponentially during the iteration process and it may destroy the convergence of the numerical scheme.

226

CHAPTER 5. MODELLING OF FUEL CELL STACKS

5.4 Heat effects in DMFC stack The efficiency and lifetime of the DMFC stack depend on a large number of design and operational parameters; stack temperature T is among the most important. The rates of kinetic and transport processes in DMFC rise exponentially with T . Improper thermal management increases current nonuniformity over the stack volume, which may dramatically lower stack performance and lifetime. Modelling of DMFCs is complicated by several factors. The transport properties of porous layers in DMFCs are poorly known. In spite of several decades of studies, the kinetics of electrochemical reactions on both sides of the cell are still controversial. Last but not least, due to a large amount of gaseous CO2 produced in the methanol oxidation reaction the flow in the anode channel is usually two-phase in nature. A review of approaches and challenges in DMFC modelling is given in Wang (2004) and Kulikovsky (2007a). In recent years, a number of CFD models of DMFCs have been developed (Divisek et al., 2003; Wang and Wang, 2003; Birgersson et al., 2003, 2004; Yang and Zhao, 2007). In some papers (Wang and Wang, 2003; Birgersson et al., 2003, 2004; Yang and Zhao, 2007; Li and Wang, 2007) thermal effects have been ignored and the cell was assumed to be isothermal. This assumption is usually fulfilled in experiments with a stand-alone cell under external thermal control (Krewer and Sundmacher, 2005). However, heat management in stacks is much less efficient, since cooling is provided by “internal” mechanisms of heat transfer to the flow in the anode channel and of liquid water evaporation. The energy equation was included in Divisek et al. (2003); this model, however, describes a cell fragment in a plane perpendicular to the anode channel, and the temperature distribution along the channel is beyond the scope of the model. In this section, we construct a thermal model of an element of a DMFC stack with a straight anode channel (Kulikovsky, 2008a). We derive an approximate asymptotic solution to model equations and compare it to numerical results. In low-T cells, cooling due to evaporation of liquid water gives significant contribution to the heat balance. Below the role of this process is discussed in detail.

5.4.1

General assumptions

The model of heat transport in the DMFC element (Figure 5.12) is based on the following assumptions. • Heat exchange with the air flow in the cathode channel is negligible. The heat capacity of a liquid methanol-water mixture is much larger than the heat capacity of air and thus at moderate oxygen stoichiometries the contribution of air flow to the overall heat balance is small.

5.4 HEAT EFFECTS IN DMFC STACK

227

Figure 5.12: Fragment of DMFC stack with the straight anode channel. • The element in Figure 5.12 is located in the middle of the stack, so that heat transport along the y- and z-axes is negligible. Heat exchange with the ambient space is also assumed to be small. The main factors which affect stack temperature distribution along the channel are local heating due to the reactions in the MEA and cooling due to water evaporation and due to heat exchange with the anode flow. • For typical DMFC currents (j . 0.2 A cm−2 ) Joule heating is small. • Methanol and oxygen stoichiometries are large, so that the respective concentrations do not vary significantly along x. • The current is distributed uniformly along the channel, j = J. At moderate current densities and large stoichiometries this approximation is not far from reality. • Evaporation occurs mainly in the porous layers of the membraneelectrode assembly (MEA). This assumption is well justified taking into account the much larger surface area of liquid water in the porous layers as compared to this area in the channel. The regime with high methanol and oxygen stoichiometries is often used in a process of stack development and testing. This regime is a useful reference point, since it gives confidence that the effects due to methanol and oxygen depletion are excluded.

5.4.2

Equations for stack and flow temperature

Equation for the bipolar plate temperature T is     ∂2T T ∆S j T ∆Scross jcross a c −λp 2 = +η +η + ∂x 6F hp 6F hp   0 εBL sρs T − Ts − ∆Hevap Kevap (psat w − pw ) − Hs Mw hc

(5.103)

where λp is the BP thermal conductivity, hp is the BP thickness, ∆S is the entropy change in the overall fuel cell reaction, η a and η c are the anodic and

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Table 5.5: Coefficients in Eqs (5.105) and (5.115). a0 −2.1794 · ln(10)

a1 0.02953 · ln(10)

a2 −9.1837e-5·ln(10)

a3 1.4454e-7·ln(10)

cathodic half-cell polarization voltages, respectively, ∆Scross is the entropy change in a direct methanol-oxygen combustion on the cathode side, jcross is the equivalent current density of crossover, ∆Hevap is the enthalpy of water evaporation, Kevap is the evaporation rate constant, εBL is the backing layer porosity, s is the liquid saturation, ρ0s is the inlet density of methanolwater solution, Mw is the molecular weight of water, psat w is the pressure of saturated water vapour, pw is the partial pressure of water vapour in the stack, Hs is the heat transfer coefficient between the bipolar plate and the flow in the anode channel, Ts is the temperature of the methanol-water solution in the anode channel, and hc is the channel height. As before, the superscript 0 indicates the values at the channel inlet. The first term on the right side of Eq. (5.103) describes the reversible BP heating due to the entropy change in the useful electrochemical reactions and due to the irreversible heat of methanol oxidation and oxygen reduction. The second term takes into account the heat generated on the cathode side of the MEA due to a direct catalytic combustion of permeated methanol; the respective rate of stack heating is proportional to jcross (3.23). The third term describes BP cooling due to evaporation of liquid water in the porous layers of the MEA (Natarajan and Nguyen, 2001). The last term in (5.103) accounts for the heat exchange with the flow in the anode channel. The pressure of saturated water vapour is given by psat ˜sat (T ) w = p0 p

(5.104)

where p0 = 1 atm and
p˜sat (T ) = exp a0 + a1 (T − 273) + a2 (T − 273)2 + a3 (T − 273)3 (5.105) describes the temperature dependence (the coefficients a0 , . . . , a3 are given in Table 5.5). We will assume that the partial pressure of water vapour in a stack is negligible: pw = 0. Under this assumption the rate of stack cooling due to evaporation is maximal and is controlled by a single parameter s. The effect of finite pw on the results is discussed in Section 5.4.4. We introduce dimensionless variables x ˜=

x , L

˜j = j , a0 jD

T T˜ = , T298

η˜ =

η ba

(5.106)

a0 where L is the channel length, jD is given by (4.193) and ba is the Tafel slope on the anode side.

5.4 HEAT EFFECTS IN DMFC STACK

229

With these variables, Eq. (5.103) takes the form    ∂ 2 T˜  2 ˜ 2 2 ˜˜jcross = χ2 p˜sat (T˜) + ξ 2 T˜ − T˜s . (5.107) ˜j + ωcross + ω T + φ η ˜ T ∂x ˜2 Here η˜ = η˜a + η˜c is the sum of the polarization voltages, ˜jcross is given by (4.196) and s ω= s ωcross =

a0 ∆S L2 jD 6F hp λp

(5.108)

a0 ∆S L2 jD cross 6F hp λp

(5.109)

s

a0 L2 ba jD T298 hp λp s  λ0s Nu ξ= λ p ε2

φ=

Hs hc λ0 ss L2 ∆Hevap Kevap εBL sρ0s p0 χ= λp Mw T298

Nu =

(5.110)

(5.111) (5.112) (5.113)

are parameters, Nu is the Nusselt number, λ0s is the thermal conductivity of the liquid solution at the inlet, ε=

hc L

(5.114)

and   2 3 p˜sat (T˜) = exp a0 + a1 T298 (T˜ − u) + a2 T298 (T˜ − u)2 + a3 T298 (T˜ − u)3 (5.115) where u=

273 ' 0.9161. 298

(5.116)

Heat transport in the anode channel is governed by the equation ∂ (ρs vs cP Ts ) T − Ts = Hs ∂x hc

(5.117)

where ρs is the density, vs is the velocity of the two-phase flow, cP is the

230

CHAPTER 5. MODELLING OF FUEL CELL STACKS

liquid heat capacity and Ts is the flow temperature. The product ρs vs cP Ts is the heat flux transported along the anode channel with the flow. Variation (divergence) of this flux equals the rate of heat exchange with the solid BP (the right side of Eq. (5.117)). Mass flux ρs vs is constant, hence we can replace it by ρ0s vs0 and factor it out from the derivative sign9 . The heat capacity of the two-phase flow is equal to the heat capacity of a liquid phase, since the mass of the gaseous phase is much smaller than the mass of the liquid phase. In other words, the energy of a two-phase flow is concentrated in the liquid phase. With the dimensionless variables (5.106), Eq. (5.117) takes the form ψ2

  ∂ T˜s = ξ 2 T˜ − T˜s ∂x ˜

(5.118)

where s ψ=

Lρ0s vs0 cP . λp

The boundary conditions for T˜ and T˜s are   ∂ T˜ = κNu T˜ − T˜s0 ∂x ˜ x˜=0   ∂ T˜ = κNu T˜ − T˜s (1) − ∂x ˜ x˜=1 ˜ ∂ T ∂ T˜ = = 0. ∂ y˜ ∂ y˜ y˜=0

(5.119)

(5.120)

(5.121)

(5.122)

y˜=1

Here κ=

Lλ0s . hc λp

(5.123)

Physically, Eqs (5.120) and (5.121) describe heat transfer between the bipolar plate and the anode flow at the channel inlet and outlet. The boundary condition for the methanol solution temperature is T˜s |x˜=0 = T˜s0 .

(5.124)

The system (5.107), (5.118) with the boundary conditions (5.120)–(5.122) and (5.124) describes the heat balance in the DMFC stack element. Note 9 Gaseous CO bubbles in the flow do not violate mass conservation in the form 2 ρs vs = ρ0s vs0 . The bubbles reduce the average flow density and increase flow velocity, so that the product ρs vs remains constant.

5.4 HEAT EFFECTS IN DMFC STACK

231

that this system is valid for an arbitrary distribution of local current density along the channel. In the general case, this system should be completed with the equation for ˜j(˜ x) (Section 4.7). However, in the limit of constant ˜j this system can be solved using the asymptotic technique, as described below.

5.4.3

Asymptotic solution: The general case

The method of solution of the system (5.107) and (5.118) is analogous to the method utilized in Section 5.1.8. It is based on the fact that in typical situations, parameter ξ is large (Table 5.6) and thus parameter γ≡

1 ξ2

(5.125)

is small. Physically, the rate of heat exchange between the BP and the anode flow dominates in the heat balance. Dividing Eq. (5.107) by ξ 2 and equating γ to zero we see that all terms except the last one vanish and a zero-order solution to (5.107) is T˜ = T˜s . Thus, due to a large rate of heat exchange between the BP and the anode flow, at the leading order, the stack and flow temperatures coincide. In the next approximation we write T˜ = T˜s + γf (x)

(5.126)

where f is a function to be determined. Substituting T˜s = T˜ − γf into (5.118) we get ψ2

∂ T˜ ∂f − γψ 2 = f. ∂x ˜ ∂x

Equating here like powers of γ we obtain f = ψ2

∂ T˜ ∂x ˜

∂f = 0. ∂x Differentiating (5.127) with respect to x ˜ we come to ∂ 2 T˜ 1 ∂f = 2 . ∂x ˜2 ψ ∂x ˜ Using this result and (5.126) in (5.107) we get   1 ∂f 2 ˜ 2 ˜j + ω ( T + γf ) + φ η ˜ s ψ2 ∂ x ˜ 2 + ωcross (T˜s + γf )˜jcross − χ2 p˜sat (T˜s + γf ) = f.

(5.127) (5.128)

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Table 5.6: The dimensionless parameters for the calculation. The physical parameters are listed in Table 5.7. ψ

φ

ξ

ω

ωcross

χ

15.47

0.979

38.51

0.894

4.91

2.79

Table 5.7: The physical parameters. Methanol solution temperature at the inlet, Ts0 (K)

70 + 273

Bipolar plate thermal conductivity, λp (Poco graphite) (W m−1 K−1 )

70

Water thermal conductivity, λw (W m−1 K−1 ) ◦

Specific heat of liquid water at 70 C, cP (J kg Liquid water density,

ρ0s

(kg m

−3

0.58 −1

K

−1

)

4190 103

)

Molecular weight of water, Mw (kg mol

−1

)

0.018

Entropy change in electrochemical methanol-oxygen reaction, ∆S (J mol−1 K−1 ) (Appleby and Foulkes, 1989)

80.9

Entropy change in a direct methanol-oxygen combustion, ∆Scross (J mol−1 K−1 ) (Appleby and Foulkes, 1989)

2438

Enthalpy of water evaporation, ∆Hevap (J mol−1 ) Evaporation rate constant, Kevap (atm (Yang and Zhao, 2007)

−1

−1

s

)

41.7 × 103 5 × 10−3

Backing layer porosity, εBL (Wang and Wang, 2003)

0.7

Average liquid saturation in the stack, s

0.5

Characteristic pressure, p0 (atm) Working current density, j (A m

−2

1 )

Total voltage loss, η (V) Inlet flow velocity,

vs0

(m s

2 × 103 0.7

−1

)

Nusselt number in the anode channel, Nu (Senn and Poulikakos, 2006)

0.02 2.7

Bipolar plate thickness, hp (m)

10−3

Channel height, hc (m)

10−3

Channel length, L (m)

0.2

5.4 HEAT EFFECTS IN DMFC STACK

233

Dropping here the terms with γ, at the leading order we find −

  1 ∂f 2˜ 2 2 ˜j + ωcross + f = ω T + φ η ˜ T˜s ˜jcross − χ2 p˜sat (T˜s ). s ψ2 ∂ x ˜

˜ Parameter ψ 2  1 (Table 5.6). Thus, the term with the derivative ∂f /∂ x can be neglected and we get   2 f = ω 2 T˜s + φ2 η˜ ˜j + ωcross T˜s ˜jcross − χ2 p˜sat (T˜s )   2 ' ω 2 T˜s0 + φ2 η˜ ˜j + ωcross T˜s0 ˜jcross − χ2 p˜sat (T˜s0 ). (5.129) The last relation holds, since in view of Eq. (5.128) f is approximately constant. Equation (5.118) is now ψ2

∂ T˜s = f, ∂x ˜

T˜s (0) = T˜s0

(5.130)

where f is given by (5.129). The solution to (5.130) is fx ˜ T˜s = T˜s0 + 2 . ψ

(5.131)

Using this in (5.126) we get the stack temperature fx ˜ T˜ = T˜0 + 2 ψ

(5.132)

f T˜0 = T˜s0 + 2 . ξ

(5.133)

where

We see that both T˜ and T˜s are straight lines with the slope f /ψ 2 . Physically, when the local current density is constant along the channel, heat flux from the MEA to the bipolar plate is also nearly constant along x ˜, which leads to a linear growth of temperature toward the outlet. The approximate temperature shapes (5.132) and (5.131) are shown in Figure 5.13 together with the numerical solution to a full system of equations (5.107), (5.118), (5.120)–(5.122), (5.124). As can be seen, the quality of the analytic solution is quite good: the deviation of the analytic curve from the exact numerical curve does not exceed 0.5 K (Figure 5.13). In dimension variables Eq. (5.132) is T = T0 +



 0 1 Ts j∆S ηj T 0 (j a0 − j)β∗ ∆Scross + + s D 0 0 0 ρ s v s cP 6F hp hp 6F hp    0  0 ∆Hevap Kevap εBL sρs p0 Ts sat − p˜ x. (5.134) Mw T0

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Figure 5.13: Temperature of the bipolar plate T and of the anode flow Ts . Solid lines—numerical solution to a full system of equations and dashed lines—asymptotic relations. The terms in square brackets represent stack heating by reversible heat of the useful electrochemical reactions (the term with ∆S), by irreversible heat (the term with η), by direct methanol-oxygen combustion (the term with ∆Scross ), and stack cooling due to evaporation (the term with ∆Hevap ). The product ρ0s vs0 c0P in the denominator of (5.134) describes stack cooling by the anode flow. Importantly, the contributions of two cooling processes to (5.134) are quite different. The parameters of the anode flow appear in the denominator in (5.134); thus, heat transfer to the flow can only reduce the slope of the line T (x), but it cannot fully homogenize the temperature distribution. In contrast, cooling due to evaporation appears as a separate term on the right side of (5.134) and it can fully compensate for the heating, as discussed below.

5.4.4

Optimal stack temperature

Equations (5.131) and (5.132) show that both T˜ and T˜s are uniform along x ˜ if f = 0. Equating (5.129) to zero we find 

 2 ω 2 T˜s0 + φ2 η˜ ˜j + ωcross T˜s0 ˜jcross = χ2 p˜sat (T˜s0 ).

(5.135)

This equation determines the optimal inlet temperature of the methanol solution T˜s0 , which provides uniform stack and solution temperatures. Physically, the temperature dependencies of the local rate of heating and the local rate of cooling are different, so that equality of these rates at the inlet means that the stack and flow temperatures do not vary along x ˜. The exponential term in the expression for p˜sat (5.115) does not allow us to write an analytical solution to (5.135). A numerical solution to this

5.4 HEAT EFFECTS IN DMFC STACK

235

Figure 5.14: Optimal stack temperature versus average liquid saturation for the indicated values of stack current density (A cm−2 ). equation can easily be obtained by making 3-4 iterations according to Eq. (5.136), which results from the application of Newton’s method to Eq. (5.135): T˜s0 [n + 1] = T˜s0 [n] +


2 ˜jcross T˜s0 [n] − φ2 η˜˜j χ2 p˜sat (T˜s0 [n]) − ω 2 ˜j + ωcross . (5.136) 2 ˜jcross − χ2 ∂ p˜sat ω 2 ˜j + ωcross ∂ T˜ ˜ ˜ 0 T =Ts [n]

Iterations start with T˜s0 [0] = (273 + 100)/298 ' 1.2517, which corresponds to 100 ◦ C. Here the symbol n in square brackets enumerates the iterations. The solution to Eq. (5.135) as a function of liquid saturation in a stack is depicted in Figure 5.14 for the three values of mean current density. Consider first the parametric dependence of optimal T on j. The higher the stack current, the higher the optimal working temperature. In view of Eq. (5.115) the rate of stack cooling due to evaporation rapidly increases with T . Clearly, the faster the stack heating due to the reactions, the larger must be the rate of cooling in order to keep T uniform. With the growth of liquid saturation, optimal T , however, decreases (Figure 5.14). The rate of cooling due to evaporation is proportional to the amount of liquid water in the stack. Thus, the higher the saturation, the faster the stack cooling and the lower the optimal temperature. At optimal conditions we have f = 0 and hence stack and methanol solution temperatures coincide and equal the inlet temperature of the flow: T = Ts = Ts0 10 . This condition provides a simple means for checking the optimal regime: it is achieved if Ts = T . Note that both temperatures should be measured far from the channel inlet and outlet to avoid the influence of boundary effects. 10 The

heat flux from the stack to the anode flow is zero in that case.

236

CHAPTER 5. MODELLING OF FUEL CELL STACKS

Figure 5.15: Schematic of current flow in a two-cell stack with the resistive spot in cell A. The model neglects the effect of partial pressure of water vapour in a stack on the rate of evaporation. An accurate account of a finite pw requires a numerical approach. Qualitatively, finite pw would lead to a lower value of parameter χ in Eq. (5.135). Lower χ means a lower rate of cooling due to evaporation; this would only slightly shift the optimal temperature of stack operation to a higher value. Indeed, water vapour saturation pressure rapidly increases with T and hence even a small increase in T would be sufficient to compensate for the effect of lower χ (see Eq. (5.135)).

5.5 Mirroring of current-free spots in a stack The role of the bipolar plate in a stack is twofold: it distributes reactants along the surface of adjacent cells and transports current from one cell to another. A serious problem in stacks is the formation of highly resistive (current-free) spots. These spots arise due to local corrosion of BP material, due to numerous processes which increase the local resistivity of the membrane-electrode assembly (e.g. delamination of the catalyst layer from the membrane (Liu et al., 2004)), or simply due to uneven distribution of clamping pressure. In this section, we will see that bipolar plates serve as electrostatic mirrors: they replicate the disturbance induced by the resistive spot to a number of adjacent cells (Kulikovsky, 2007c). An analysis of the governing equations shows that the number of cells affected by this mirroring is inversely proportional to the square root of the BP electric conductivity. The CFD models of SOFC stacks cited in Sections 5.1 and 5.3 do not consider resistive spots. The majority of low-T stack models deal with stacks of PEFCs (Thirumalai and White, 1997; Lee et al., 1998; Lee and Lalk, 1998; Baschuk and Li, 2004; Karimi et al., 2005). All these models assume that reactants and local current are uniformly distributed over the cell surface. To illustrate the effect of a resistive spot consider a stack of two cells A and B separated by a bipolar plate (Figure 5.15). Suppose that cell A has a defect, a spot of high resistivity (Figure 5.15). Current produced in cell B

5.5 MIRRORING OF CURRENT-FREE SPOTS IN A STACK

237

Figure 5.16: Sketch of a stack fragment with the square cross-section. Latin letters denote bipolar plates and Greek letters indicate cells. Each bipolar plate is equipped with its own system of coordinates (x, y). under the spot is not allowed to cross cell A in the through-plane direction; this current will be transported along the bipolar plate to bypass the spot (Figure 5.15). The spot thus induces in-plane current in the BP. In-plane current is supported by the variation of the potential along the BP surface. This variation in turn causes the variation of local potentials of the adjacent cells. Therefore, the spot in the defective cell induces a redistribution of currents and potentials at least in the nearest neighbours. Below we show that a spot is mirrored by the bipolar plates in the several adjacent cells and we calculate the length of mirroring.

5.5.1

Equation for bipolar plate potential

Consider the fragment of a stack: the three square cells separated by the bipolar plates (Figure 5.16). We will ignore the details of BP structure (geometry of channels and ribs) and assume that each BP is a thin rectangular plate. Generally, the voltage Va of BP a is described by a 3D Laplace equation ∂ 2 Va ∂ 2 Va ∂ 2 Va + + = 0. 2 2 ∂x ∂y ∂z 2

(5.137)

The thinness of the plate along the z-axis (Figure 5.16) allows us to transform ∂ 2 Va /∂z 2 as follows: !   ∂ 2 Va ∂ ∂Va 1 ∂Va ∂Va jβ − jα = ' − = 2 ∂z ∂z ∂z hp ∂z z=α ∂z z=β σp hp where jα and jβ are local current densities leaving and entering, respectively, the BP a (Figure 5.16), hp and σp are the BP thickness and electric

238

CHAPTER 5. MODELLING OF FUEL CELL STACKS

conductivity, respectively, and we use Ohm’s law j = −σp

∂Va . ∂z

(5.138)

Here we assume that the cell currents jα and jβ have only the z-component, which is typical of fuel cells. Equation (5.137), therefore, reduces to ∂ 2 Va ∂ 2 Va + = rp (jα − jβ ) ∂x2 ∂y 2

(5.139)

which is formally a 2D Poisson equation. Here rp =

1 σp hp

is the BP resistivity (Ω). It is convenient to equip each BP with its own system of coordinates (x, y) (Figure 5.16). The potential of each BP is calculated with respect to its value at (0, 0). For voltage Vβ of the cell β we have 0 0 Vβ = Vab + Va − Vb = Vab + δVβ

(5.140)

0 where Vab is the voltage drop between plates a and b at (0, 0), Va and Vb are potentials of plates a and b, respectively, and δVβ ≡ Va − Vb (Figure 5.16). Note that Latin and Greek subscripts indicate bipolar plates and cells, respectively. For further calculations we introduce dimensionless variables

x ˜=

x , L

y˜ =

y , L

V V˜ = c , b

˜j = j . 0 jD

(5.141)

Here L is the length of the side of the BP, bc is the Tafel slope on the 0 cathode side and jD is given by (4.26). With these variables, Eq. (5.139) takes the form ˜p
∂ 2 V˜b ∂ 2 V˜b R + = 2 ˜jβ − ˜jγ . 2 2 ∂x ˜ ∂ y˜ 

(5.142)

Here 0 ˜ p = hp jD R σp bc

is the dimensionless BP resistivity and =

hp . L

(5.143)

5.5 MIRRORING OF CURRENT-FREE SPOTS IN A STACK The boundary conditions for Eq. (5.142) are ∂ V˜b ∂ V˜b = 0, = 0, V˜b (0, 0) = 1. ∂x ˜ ∂ y˜ x ˜=0,1

239

(5.144)

y˜=0,1

The first two conditions express the absence of normal current through the BP end faces. The last condition establishes an arbitrary reference point for BP potential; for the stability of the numerical calculations a non-zero reference voltage should be taken.

5.5.2

Cell polarization curve

To rationalize the origin of large-scale electrostatic effects in the stack we employ the simplest linear polarization curve of individual cells: Vβ = Voc − Rβ jβ .

(5.145)

Here Voc is the open-circuit voltage (the same for all cells), and Rβ (x, y) is the local internal resistivity (Ω cm2 ) of the cell β. Equation (5.145) gives the basic physics due to the spots in stacks, which is our primary goal in this book. In the dimensionless variables, Eq. (5.145) takes the form ˜ β ˜jβ V˜β = V˜oc − R

(5.146)

0 ˜ β = Rβ jD . R bc

(5.147)

where

Equation (5.140) in dimensionless form is 0 V˜β = V˜ab + δ V˜β .

(5.148)

Equating the right sides of (5.148) and (5.146) we get ˜β0 − R ˜ β ˜jβ . δ V˜β = E

(5.149)

0 ˜β0 = V˜oc − V˜ab E

(5.150)

Here

is independent of coordinates. Equation (5.149) is a local polarization curve of the cell; it includes a ˜ 0 . To calculate this parameter we equate two yet undefined parameter E β

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expressions for the useful electric power generated by the individual cell: Z  Z    0 ˜ ˜ ˜ ˜ β ˜jβ ˜jβ dS. ˜ ˜ Vab + δ Vβ jβ dS = V˜oc − R (5.151) ˜ S

˜ S

Here S˜ = (L/L)2 = 1 is the dimensionless cell active area. Equation (5.151) is obtained if we multiply Eqs (5.148) and (5.146) by ˜jβ , integrate the results over S˜ and equate the resulting expressions. Equation (5.151) can be rearranged to yield Z  Z    0 ˜ ˜ β ˜jβ + δ V˜β ˜jβ dS. ˜ V˜oc − V˜ab jβ dS˜ = R (5.152) ˜ S

˜ S

Taking into account (5.150), from (5.152) we find Z   1 0 ˜ β ˜jβ + δ V˜β ˜jβ dS˜ ˜ R Eβ = J˜ S˜

(5.153)

where J˜ =

Z ˜ S

˜jβ dS˜

(5.154)

is the mean current density in the stack (the same for all cells). Equations (5.149) and (5.153) allow us to calculate the local current in the cell β provided that δ V˜β is known (see below). From (5.149) we get ˜jβ (˜ x, y˜) =

˜ 0 − δ V˜β (˜ E x, y˜) β . ˜ Rβ (˜ x, y˜)

(5.155)

˜ 0 = J, ˜ we can calculate ˜jβ from (5.155). Taking the initial approximation E β 0 ˜ ˜ 0 we can The result can be used to update Eβ with (5.153); with the new E β ˜0. calculate the new ˜jβ etc. Usually 2-3 iterations give the exact value of E β

5.5.3

Spot shape and numerical details

Consider the eight-cell stack (the fragment of the stack is shown in ˜ 0 , while at Figure 5.16). Cells 1-3 and 5-8 have a constant resistivity R the centre of cell 4 we have a spot of high resistivity with the Gaussian shape:   x − 0.5)2 (˜ y − 0.5)2 ˜4 = R ˜0 + R ˜ s exp − (˜ R − . (5.156) s2 s2 ˜ s is the peak resistivity at the centre of the spot (˜ Here R x = 0.5 and y˜ = 0.5) and s is the characteristic spot radius.

5.5 MIRRORING OF CURRENT-FREE SPOTS IN A STACK

241

Table 5.8: Parameters for calculations. Small value of R0 is taken to simulate the Ohmic cell resistivity. Bipolar plate thickness, hp (cm) MEA thickness, hmea (cm) Bipolar plate conductivity, σp (Ω−1 cm−1 ) BP dimension, L × L (cm) Cell resistivity, R0 (Ω cm2 ) Peak resistivity of the spot, Rs (Ω cm2 ) Tafel slope of the cathode side, bc (V) 0 Limiting current density, jD (A cm−2 ) ˜0 R ˜s R ˜p R s J˜

0.1 0.1 200 10 × 10 0.05 100 0.05 1 1 100 10−2 0.05 0.5

The system of equations (5.142) and (5.149) written for all cells and BPs determines cell currents and BP potentials in the stack. Iterations are required to find the self-consistent distribution of currents and potentials. Starting from uniform potentials we calculate cell currents using Eqs (5.155) and (5.153), as described in the previous section. These currents are then used to calculate new BP potentials with Eq. (5.142). This procedure is repeated until convergence is achieved. The most computationally intense procedure is a solution of the 2D Poisson equation (5.142). The algorithm of stack simulation can thus be effectively parallelized: each stack module “cell + BP” can be solved on a separate processor. Upon completion of an iteration step, adjacent modules exchange with the information (the cell currents and BP potentials) required for the next iteration.

5.5.4

Numerical results

The physical and the respective dimensionless parameters for the calculations are listed in Table 5.8. The results are depicted in Figure 5.17. The spot in the fourth cell generates a noticeable disturbance of local current in cells 2, 3 and 5, 6 (Figure 5.17). These disturbances are accompanied by disturbances in the BP potentials (Figure 5.17). Physically, the spot in cell 4 blocks the direct through-plane propagation of current. The disturbance of potential of BP 4 redistributes the current produced under the spot to the unaffected parts of the fourth cell. Nonuniformity of the potential of BP 4 leads to the formation of a spot image in the third cell (Figure 5.17). The spot image in cell 3 is “seen” by cell 2 etc. A similar mechanism provides spot mirroring in cells 5 and 6 (Figure 5.17).

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Figure 5.17: The excess of cell local current density ˜j over the mean value J˜ = 0.5 (left column) and potential of the bipolar plate minus 1 (right column) in cells/plates (Nos 2-6). Cell No. 4 has a spot of high local resistivity at the centre, which results in a spot of low current (left column, cell 4). Mirroring of this spot in the adjacent cells is clearly seen (left column). All variables are dimensionless. The disturbance of BP voltage translates into the disturbance of cell potentials (Figure 5.18). In the defective cell (No.4), the cell voltage in the spot decreases (Figure 5.18). The decrease in V4 is caused by the formation of a ring of enhanced current around the spot (Figure 5.17; see also (Kulikovsky, 2006c)). Due to this ring, the product R4 j4 in the region of the spot appears to be higher than outside the spot, which means lower cell voltage inside the spot. However, in the adjacent cells the cell voltage in the spot increases (Figure 5.18). The resistivity of these cells is not disturbed and hence lower local current in the spot images corresponds to higher local cell voltage. This effect can provoke stack ageing (Section 5.5.6). Figures 5.17 and 5.18 show that the amplitude of the disturbance decays along the stack axis. What is the characteristic length of this damping and how does it depend on stack parameters?

5.5 MIRRORING OF CURRENT-FREE SPOTS IN A STACK

243

Figure 5.18: Dimensionless cell voltage in cells 2-6. In the defective cell (No. 4) cell voltage in the spot decreases. In the adjacent cells, local voltage in the spot images increases.

5.5.5

Analysis of equations: The length of mirroring

Equation for stack potential To analyse the length of disturbance penetration along the stack axis z we need a governing equation for cell voltage. Equation (5.139) for plate b (Figure 5.16) reads ∂ 2 Vb ∂ 2 Vb + = rp (jβ − jγ ) . 2 ∂x ∂y 2

(5.157)

Subtracting (5.157) from (5.139) we obtain ∂ 2 Vβ ∂ 2 Vβ + = rp (jα − 2jβ + jγ ) ∂x2 ∂y 2

(5.158)

0 where we used Eq. (5.140) (Vab is constant)11 . The expression in brackets on the right side of Eq. (5.158) is a well-known finite-difference approximation of the second derivative of stack current

11 A

1D variant of Eq. (5.158) was obtained in (Kim et al., 2005).

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

along the z-coordinate: jα − 2jβ + jγ = (hp + hmea )2

∂ 2 jβ (jα − 2jβ + jγ ) ' (hp + hmea )2 2 (5.159) 2 (hp + hmea ) ∂z

where hmea is the thickness of the cell (MEA). Physically, in (5.158) current is a stepwise function of z, which may jump between the cells due to the resistive spots. Equation (5.159) means that we smooth this jump over the thickness hp + hmea of the elementary module “MEA + BP”. Using (5.159) in (5.158) we find ∂ 2 Vβ ∂ 2 Vβ ∂ 2 jβ + = rp (hp + hmea )2 2 . 2 2 ∂x ∂y ∂z

(5.160)

This is the general relation valid for an arbitrary form of the cell polarization curve. If this curve is linear, Eq. (5.160) can be further simplified. Differentiating (5.145) twice with respect to z we obtain ∂ 2 Vβ ∂ 2 jβ = −Rβ . 2 ∂z ∂z 2

(5.161)

Using this in (5.160) and taking into account the relation for BP resistivity Rp = rp h2p = hp /σp (Ω cm2 ), we get ∂ 2 Vβ ∂ 2 Vβ + + 2 ∂x ∂y 2



Ch2 Rp Rβ



∂ 2 Vβ =0 ∂z 2

(5.162)

where Ch = 1 +

hmea . hp

(5.163)

is a geometric factor. In general, cell resistivity Rβ is a function of coordinates x and y. If Rβ = R0 is constant for all cells, we may introduce a stretched coordinate s z∗ = z

R0 . Ch2 Rp

(5.164)

With (5.164), Eq. (5.162) transforms to ∂ 2 Vβ ∂ 2 Vβ ∂ 2 Vβ + + = 0. ∂x2 ∂y 2 ∂z∗ 2

(5.165)

Equation (5.165) is the 3D Laplace equation, in which the z-coordinate is stretched according to (5.164).

5.5 MIRRORING OF CURRENT-FREE SPOTS IN A STACK

245

Figure 5.19: Schematic of the resistivities along the stack. The module is the bipolar plate and the MEA. Rp is the resistivity of the bipolar plate, and R1 , R2 , . . . are the cell resistivities. Physically, the transition from Eq. (5.158) to Eq. (5.162) or Eq. (5.165) means that the distribution of individual cell potentials stepwise along z is replaced by the smooth stack potential Vβ . Note that in Eq. (5.162) the resistivities Rβ are not smoothed: they form a layered structure (Figure 5.19). Each cell is characterized by its own Rβ (x, y) and the adjacent cells are separated by BPs with the constant resistivity Rp (Figure 5.19). In the general case, (5.162) is the equation with the stepwise coefficient Ch2 Rp /Rβ : along the first module “BP + MEA” this coefficient is Ch2 Rp /R1 , and along the second module it is Ch2 Rp /R2 etc.

The damping length of small-amplitude disturbance In order to calculate the damping length, consider the stack of circular cells and plates of radius L. For the rest of this section, the subscript β in the symbol Vβ is omitted; V is thus the local stack potential. In cylindrical coordinates, Eq. (5.162) takes the form 1 ∂ r ∂r

   2  2 ∂V Ch Rp ∂ V r + = 0. ∂r Rβ ∂z 2

(5.166)

Suppose that all cells in the stack (except one) have the same constant resistivity R0 , while the cell at z = 0 has a small disturbance of resistivity R|z=0 = R0 + R1 (r),

R1  R0 .

(5.167)

R1 induces the disturbance in stack potential: V = V 0 (z) + V 1 (r, z)

(5.168)

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

where V 1 is also small. Note that the undisturbed potential V 0 is a function of z only12 . Substituting (5.167) and (5.168) into (5.166), taking into account that ∂ 2 V 0 /∂z 2 = 0 and neglecting small terms, for V 1 we get an equation with constant coefficients    2  2 1 1 ∂ ∂V 1 Ch Rp ∂ V r + = 0. (5.169) r ∂r ∂r R0 ∂z 2 With the dimensionless coordinates r r˜ = , L

z z˜∗ = L

s

R0 Ch2 Rp

(5.170)

Eq. (5.169) transforms to the Laplace equation 1 ∂ r˜ ∂ r˜

∂ V˜ 1 r˜ ∂ r˜

! +

∂ 2 V˜ 1 = 0. ∂ z˜∗2

(5.171)

The solution to Eq. (5.171) is subject to the following boundary conditions ∂ V˜ 1 ∂ r˜

∂ V˜ 1 ∂ r˜

= 0, r˜=0

=0

(5.172)

r˜=1

and V˜ 1 |z˜∗ =0 = V˜01 (˜ r),

V˜ 1 |z˜∗ =∞ = 0.

(5.173)

Equations (5.172) mean symmetry at r˜ = 0 and the absence of normal current through the side surface of the stack, respectively. Equations (5.173) describe the disturbance of potential at z˜∗ = 0 and the decay of this disturbance at large distance. We assume that the damping length is much smaller than the stack length. We seek a partial solution of the problem (5.171)–(5.173) in the form V˜ 1 = ζ(˜ z∗ )ρ(˜ r).

12 It

(5.174)

is easy to show that V 0 is constant along r and increases linearly with z: V0 =

where H is the stack length.

zVstack H

5.5 MIRRORING OF CURRENT-FREE SPOTS IN A STACK

247

Substituting (5.174) into (5.171) and separating variables we find   1 ∂ ∂ρ 1 ∂2ζ = −ν 2 r˜ =− ρ˜ r ∂ r˜ ∂ r˜ ζ ∂ z˜∗2 where ν is constant. We, thus, have the following system   ∂ρ ∂ρ ∂ρ 1 ∂ 2 = 0, =0 r˜ + ν ρ = 0, r˜ ∂ r˜ ∂ r˜ ∂ r˜ r˜=0 ∂ r˜ r˜=1 ∂2ζ − ν 2 ζ = 0, ∂ z˜∗2

ζ|z˜∗ =0 = ζ0 ,

ζ|z˜∗ =∞ = 0

(5.175)

(5.176) (5.177)

where ζ0 is constant (see below). The solution to (5.177) is an exponent: ζ(˜ z∗ ) = ζ0 exp(−ν z˜∗ ).

(5.178)

The eigenvalues ν are found from (5.176). The solution to Eq. (5.176) is the Bessel function of the first kind: ρ(˜ r) = ρ0 J0 (ν r˜),

(5.179)

where ρ0 is constant. This solution must obey the boundary conditions (5.176). The first condition is satisfied automatically. The second condition leads to (∂J0 (ν r˜)/∂ r˜)r˜=1 = 0; the solution to this equation is νn = an (n = 1, . . . , ∞), where an are zeros of the derivative of the Bessel function. The first two zeros are located at a1 = 3.8317 . . . and a2 = 7.0155 . . ., respectively. The general solution to (5.171)–(5.173) is a sum of partial solutions: V˜ 1 (˜ r, z˜∗ ) =

X

cn J0 (an r˜) exp(−an z˜∗ )

(5.180)

n

where the constants cn can be determined from the series expansion of V˜01 (˜ r) in terms of the Bessel functions. At fixed z˜∗ , the terms in (5.180) represent the modes of the r˜-shape of the potential. The Bessel function oscillates and thus an ’s are analogous to frequencies in the Fourier series expansion. Equation (5.180) shows that each mode exponentially decays with the distance z˜∗ ; the characteristic damping length is ˜ln = 1/an , or, in the dimensional form LCh ln = an

r

Rp . R0

(5.181)

The values an rapidly increase with n; thus the “high-frequency” modes (the modes with n > 1) die out along z˜ faster. The maximal damping length

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

exhibits the mode with the lowest “frequency” (n = 1); for this length we have r LCh Rp l1 = . (5.182) 3.8317 R0 We see that l1 is proportional to the stack radius L; the larger the L, the more the cells affected by the disturbance. Thus, to minimize mirroring, the cross-sectional size of the stack should be made as small as possible. Note that l1 does not depend on the radial size of the spot s. Thus, the lengths of mirroring of small and large spots are the same; parameter s determines only the size of spot images in the radial direction. Equation (5.182) also shows that the damping length increases with the bipolar plate resistivity Rp and decreases with the growth of cell resistivity R0 . The first result is quite obvious while the second is not. Physically, Rp is the in-plane resistivity, while R0 is the through-plane resistivity. The plate with small Rp effectively redistributes the defect in cell local current and hence it “screens” the disturbance. On the contrary, in a cell with small R0 a small variation of potential induces a large variation of local current; thus, the amplitude of the spot image in such a cell is larger. This means that good stacks with small losses in the cells are more prone to mirroring. Finally, we note that the disturbance spreads in both directions along z and the total number of cells affected by mirroring is N1 = 2l1 /(hmea + hp ), or r r 2LCh 2L Rp Rp N1 ' = . (5.183) 3.83 (hmea + hp ) R0 3.83 hp R0 Taking the values from Table 5.8 and L = 5.64 cm (the effective “radius” of a square stack with a side length of 10 cm) we get N1 = 2.94 ' 3 cells. The numerical result in Figure 5.17 shows that 4 cells are affected by mirroring. The agreement is good, taking into account that N1 is the characteristic number of affected cells.

5.5.6

Remarks

In our model the cell polarization curve Vcell (j) is linear. This approximation is quite reasonable if the cell operates in the linear “resistive” region of the real curve. In that case, all previous relations are valid provided that R0 is replaced by the differential cell resistivity −∂Vcell /∂j. The polarization curve of low-temperature cells is, however, strongly nonlinear in the region of small and large currents (Section 3.2). At low currents, cell voltage is determined by the reaction activation loss, which exponentially depends on potential.
Close to the limiting current the cell 0 voltage decreases as − ln 1 − j/jD , which is also a rapid function of j when

5.6 HYBRID 3D MODEL OF SOFC STACK

249

0 j → jD . Therefore, at both ends of the polarization curve the analysis above should be modified taking into account the actual dependence Vcell (j). Qualitatively, we may expect that in the regions of small and large current the effect of mirroring is less pronounced due to the larger differential resistivity of the cell. In our model, the geometric details of the bipolar plate (channels for feed gas/liquid supply and ribs for current collection) are ignored. Qualitatively, these details would lead to distortion of the shape of spot images due to the following effect. Suppose that bipolar plates have parallel straight channels and ribs directed along the x-axis (the coordinates are shown in Figure 5.16). The effective in-plane resistivity of such a plate in the direction parallel to the ribs is lower than the in-plane resistivity in the perpendicular direction. This means that the damping length for the disturbance along the x-axis would be lower than this length for the disturbance along the y-axis. The images of a circular spot would, therefore, take the form of ellipses with the large axis directed along the y-axis. Detailed calculations of this effect require a numerical approach. Mirroring was observed in the experiment with the three-cell stack (Santis et al., 2006a). It has been found that the disturbance of the local current in cell 1 spreads to the two other cells. Furthermore, the disturbance induced in the adjacent cell 2 was larger than the disturbance in cell 3. Mirroring can accelerate stack ageing due to the following mechanism. Local cell voltage in the spot images increases (Figure 5.18). The rate of Pt dissolution in the PEFC and the DMFC cathode exponentially depends on Vcell (Wang et al., 2006b; Darling and Meyers, 2003); the spot of enhanced Vcell will, therefore, “wash out” the catalyst on the cathode side faster. Furthermore, mirroring provides the mechanism for propagation of this defect along the stack axis: as soon as the spot image turns into a real resistive spot, it generates spot image of enhanced Vcell in the next neighbouring cell and so on.

5.6 Hybrid 3D model of SOFC stack So far we have considered electric phenomena in an isothermal stack or thermal phenomena in a stack element with ideally conductive bipolar plates. In reality the electric and thermal phenomena in a stack are coupled. This coupling is due to the strong (exponential) temperature dependence of the electric conductivity of various stack layers and of the half-cell exchange current densities. In this section, we will construct a 3D coupled thermoelectric hybrid model of an SOFC stack (Kulikovsky, 2009g). The model is valid for stacks with parallel straight air and hydrogen channels. Following the approach discussed in previous sections, we formulate equations for the repeated stack element (Figure 5.2, page 194). The stack model is then “assembled” by replicating the single-element model to parallelize calculations.

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Figure 5.20: To the calculation of heat fluxes irradiating the nth BP (interconnect). As an example we will calculate the thermal effect of resistive spot in the SOFC stack. In Section 5.5 we have seen that the disturbance in local current induced by a resistive spot is replicated in 2-3 adjacent cells. The thermal effect of a spot penetrates the stack much more deeply.

5.6.1

Thermal model

As before, heat transport in the BP is described using a two-fluid approximation. The heat problem is given by the two coupled equations for stack temperature T and air flow temperature Tair    ∂2T qtot T − Tair ∂2T + = − H (x) air ∂x2 ∂y 2 hp hc   T − Tair ∂Tair ρ0air v 0 cP a = Hair (x) ∂x hc 

−λp

(5.184) (5.185)

where qtot is the sum of the heat fluxes from the adjacent MEAs (see below), x, y are the coordinates on the bipolar plate (Figure 5.2, page 194), and hc is the channel height (the characteristic length for heat transfer); for other notations see Section 5.1. In this section, we are interested in the 3D heat transport and, therefore, qtot should take into account the heat flux due to the temperature gradient across the MEA. The total heat flux falling onto the nth BP is a sum of one-sided heat fluxes from the nth and (n − 1)th MEAs (Figure 5.20):  qtot =

  n  T n ∆Sa T ∆Sc n−1 n−1 n + ηa j + + ηc j n 2F 4F

χa χe + ke χa + ka χe + χe χa

λc T n+1 − 2T n + T n−1 lc


! .

(5.186)

Here ∆S is the entropy change in the half-cell electrochemical reaction, η is the half-cell polarization voltage, the subscripts a, c and e indicate the anode, the cathode and the electrolyte, respectively, and the superscript n enumerates the MEAs (Figure 5.20). Parameters χa , χe , ke and ka are

5.6 HYBRID 3D MODEL OF SOFC STACK

251

given by λa λe , χe = λc λc la le ka = , ke = lc lc

χa =

(5.187) (5.188)

where λ is the thermal conductivity and l is the thickness of the respective layer. The first term on the right side of Eq. (5.186) is the heat flux from the anode of the (n−1)th cell (Figure 5.20). The second term is the cathodic flux from the nth cell (Figure 5.20). The last term describes the heat flux due to the temperature gradient across the adjacent nth and (n−1)th MEAs. This expression follows from the exact solution of the heat transport problem in MEA (Section 3.4.6). Note that the terms which describe the crossover of reaction heat through the electrolyte (Section 3.4.7) and the Joule terms are small and they are neglected. Note also that in Eq. (5.186) we assume that the variation of temperature across the BP (interconnect) is small, i.e. the reaction terms on the anode and cathode sides of the interconnect are calculated using the same temperature T n . At this point it is convenient to introduce dimensionless variables x y T Tair , y˜ = , T˜ = , T˜air = , L L T298 T298 ˜ = Rjref . ˜j = j , η˜ = η , R jref bc bc

x ˜=

(5.189)

With these variables, the system (5.184) and (5.185) takes the form −

    ∂ 2 T˜n ∂ 2 T˜n − = ω 2 kS T˜n + φ2 η˜an−1 ˜j n−1 + ω 2 T˜n + φ2 η˜cn ˜j n 2 2 ∂x ˜ ∂ y˜     + θ2 kT T˜n+1 − 2T˜n + T˜n−1 − ξ 2 fu (˜ x) T˜n − T˜n air

(5.190)   ∂ T˜n n x) T˜n − T˜air . ψ 2 air = ξ 2 fu (˜ ∂x ˜

(5.191)

Here fu (˜ x) is given by (5.23) and s

L2 λc lc λp hp χa χe kT = ke χa + ka χe + χa χe θ=

(5.192) (5.193)

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

are the dimensionless parameters (the expressions for the other parameters are given in Section 5.1.4). The boundary conditions for the heat problem (5.190) and (5.191) are (5.35), (5.36) and (5.38). The stack is assumed to be thermally insulated, that is, in each cell ∂ T˜ ∂ T˜ = 0, =0 ∂ y˜ ∂ y˜ y˜=0

y˜=1

at the leftmost and rightmost elements, respectively.

5.6.2

Electric problem

Equation (5.190) contains local current densities and half-cell overpotentials in the MEAs n and n + 1. These values are calculated using the equations from Section 5.5. n The dimensionless potential V˜ n of the BP n and cell potential V˜cell are governed by Eqs (5.142) and (5.146), respectively. In the notations of this section these equations read ˜ pn
R ∂ 2 V˜ n ∂ 2 V˜ n ˜j n − ˜j n−1 + = 2 2 2 ∂x ˜ ∂ y˜ ε n ˜ ˜ ˜ n ˜j n . Vcell = Voc − R

(5.194) (5.195)

The boundary conditions for Eq. (5.194) are given by (5.144). Cell resistivity is calculated according to (5.58). Equations (5.194), (5.195) and a closing relation n V˜cell = V˜ n+1 − V˜ n

(5.196)

form a self-consistent problem for the distribution of stack voltage and local current densities in the individual MEAs. If the stack mean current density J˜ is fixed, the problem is solved iteratively to obtain local current densities of individual cells satisfying the relation Z 0

1

Z

1

˜ ˜j n (˜ x, y˜) d˜ xd˜ y = J,

n = 1, . . . , Ncell

(5.197)

0

where Ncell is the number of cells in the stack. Once ˜j n is obtained, the total local overpotential in the nth cell η˜n is calculated according to ˜ n ˜j n . η˜n = R

(5.198)

5.6 HYBRID 3D MODEL OF SOFC STACK

253

Figure 5.21: Schematic of the simulated 15-cell stack. MEA (cell) 8 has the resistive spot at the centre.

5.6.3

Numerical details

The model is used to study the effect of a resistive spot on temperature and current distribution in a 15-cell fragment of a planar SOFC stack (Figure 5.21). Two variants have been calculated. In the base case, the stack contained no spots; this variant provides the reference undisturbed distribution of stack temperature Tbase . In the second variant, a spot with the Gaussian distribution of resistivity ˜ {8} of the 8th cell is placed at the centre of the 8th cell. The resistivity R has the form    −(˜ x−x ˜s )2 − (˜ y − y˜s )2 {8} ˜ ˜ R = R0 1 + 240 exp . (5.199) s2 Here ˜ ˜0 = R ˜ ∗ T exp R T˜∗

T˜∗ T˜

!

is the undisturbed cell resistivity and factor 240 describes the peak dimensionless resistivity at the spot centre with the coordinates (˜ xs , y˜s ). The physical and operational parameters are listed in Table 5.9. The respective dimensionless parameters are collected in Table 5.10. In a 10×10 cm2 stack the characteristic spot radius was taken to be s = 0.5 cm. A variant of the periodic boundary conditions for stack temperature and voltage along the z˜-axis was imposed: the first cell in Figure 5.21 gets the (n − 1)th data (temperature and voltage) from the 15th cell and the 15th cell gets the (n + 1)th data from the first cell. Each cell in the stack was split into 8 elements (equivalent to 8 air channels per cell) and the problem was solved using 8×15 = 120 processors (Kulikovsky, 2009c). The computational mesh in every element had 400×50 nodes. A large number of nodes are needed to resolve the sharp boundaries of the resistive spot (see Eq. (5.199)). The wall clock time of simulation on the J¨ ulich IBM cluster JUMP (year 2009) was about one hour.

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Table 5.9: The physical parameters. 0 Air temperature at the inlet, Tair (K) Flow velocity in the air channel, va0 (m s−1 )

600 + 273 6

Bipolar plate thermal conductivity, λp (W m−1 K−1 ) Bipolar plate electric conductivity, σp (Ω m2 ) Bipolar plate mass density, (kg m−3 ) Specific heat of bipolar plate, cP p (J kg−1 K−1 )

12 3000 7900 500

Air thermal conductivity, λair (W m−1 K−1 ) Air density at 700 ◦ C, ρair (kg m−3 ) Specific heat of air at 700 ◦ C, cP a (J kg−1 K−1 )

0.073 0.32 1160

Entropy change in the cathodic half-cell reaction, ∆Sc (Takehara et al., 1989) (J mol−1 K−1 ) Entropy change in the anodic half-cell reaction, ∆Sa (Takehara et al., 1989) (J mol−1 K−1 )

108

Nusselt number in a developed flow, Nu ≡ Nu 1 Nusselt number at the channel inlet, Nu 0 Characteristic length of Nusselt number decay, lu (m) Tafel slope on the cathode side, bc (V) Reference current density (A m−2 ) Characteristic area-specific resistivity, R∗ (Ω m2 ) Characteristic temperature, T∗ (Huang et al., 2000) (K) Anode thermal conductivity, λa (W m−1 K−1 ) Electrolyte thermal conductivity, λe (W m−1 K−1 ) Cathode thermal conductivity, λc (W m−1 K−1 ) Channel length, L (m) Cell size (m2 ) Characteristic length for heat exchange, hc (m) Anode thickness, la (m) Cathode thickness, lc (m) Electrolyte thickness, le (m) Bipolar plate thickness, hp (m) Spot radius, s (m)

−27

4 40 5 · 10−3 0.15 104 1.39 · 10−5 3280 11 2.7 6 10−1 0.1 × 0.1 10−3 10−3 10−5 10−5 10−3 0.005

5.6 HYBRID 3D MODEL OF SOFC STACK

255

Table 5.10: The dimensionless parameters. ˜ R

ψ

φ

ξ

ω

θ

kT

kS

J˜

2.32

4.31

2.05

15.6

2.16

707.1

0.0173

−0.5

0.4

5.6.4

Numerical results

Figure 5.22 shows the local current density and the temperature disturbance induced by the spot ∆T = Tspot − Tbase over the stack volume (Tspot is the temperature obtained with the spot). The physical resistive spot in cell 8 is clearly seen on the map of local current: in the spot region, j drops to zero (Figure 5.22, left column). The physical spot in the 8th cell induces two current spot images (the regions of reduced local current) in the adjacent cells 7 and 9 (Figure 5.22, left column). This is a manifestation of the electrostatic “mirroring” of resistive spots in a stack discussed in Section 5.5. Note that the disturbance in local current rapidly decays with the distance along the stack axis: in cells 6 and 10 this disturbance vanishes (Figure 5.22, left column). In this variant, the length of mirroring Eq. (5.182) is smaller than that in Figure 5.17 (page 242) due to larger bipolar plate conductivity. In contrast, the temperature disturbance spreads along the stack axis much more deeply (Figure 5.22, right column). The amplitude of the temperature cavity ∆Tpeak in the first cell is only 40% lower than that in the defective cell 8 (Figure 5.23). To understand the physics of high stack thermal transparency the governing equations need to be analysed.

5.6.5

Analysis of governing equations

3D equation for stack temperature Equation (5.190) for stack temperature takes into account the layered structure of the stack. To analyse this equation it is advisable to transform it into a more familiar form with the second derivative of temperature along z. The term with θ2 on the right side of this equation describes heat transport through the MEA due to the temperature gradient. This term has the form of a finite-difference approximation for the second derivative of T˜ along z: ∂2T T n+1 − 2T n + T n−1 ' . ∂z 2 h2z

(5.200)

In our reduced model the size of the “mesh” hz along the z-axis is the thickness of the module hmea + hp . Multiplying and dividing the term

256

CHAPTER 5. MODELLING OF FUEL CELL STACKS

Figure 5.22: Local current density (left column) and temperature disturbance due to the spot ∆T = Tspot −Tbase (right column). The physical resistive spot is located at the centre of cell 8.

5.6 HYBRID 3D MODEL OF SOFC STACK

257

Figure 5.23: Points: Peak temperature disturbance (K) as a function of cell number. Both anode- and cathode-directed branches are well described by the exponent ∆T = 1.6 exp (−|8 − n|/12.5) (dashed line), which follows from the theory. discussed by 2 ˜ 2 = (hmea + hp ) , h z L2

taking into account (5.200) and setting fu = 1 (Section 5.1.8), we arrive at −

2 ˜n ∂ 2 T˜n ∂ 2 T˜n ˜ 2 θ2 ∂ T − − k h T z ∂x ˜2 ∂ y˜2 ∂ z˜2       n = ω 2 kS T˜n + φ2 η˜an−1 ˜j n−1 + ω 2 T˜n + φ2 η˜cn ˜j n − ξ 2 T˜n − T˜air .

(5.201) The left side of Eq. (5.201) transforms to the 3D Laplacian of T˜ if we introduce the stretched z-coordinate according to zˆ =

z˜ √ . ˜ hz θ kT

(5.202)

 √ −1 ˜ z θ kT Physically, the factor h is the anisotropy coefficient for stack thermal conductivity along the zˆ-axis. With Eq. (5.202) we get −

∂ 2 T˜n ∂ 2 T˜n ∂ 2 T˜n − − 2 2 ∂x ˜ ∂ y˜ ∂ zˆ2       n = ω 2 kS T˜n + φ2 η˜an−1 ˜j n−1 + ω 2 T˜n + φ2 η˜cn ˜j n − ξ 2 T˜n − T˜air . (5.203)

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

The damping length To simplify the further analysis consider a circular stack of radius L. Radial symmetry implies that the air channels are directed along r, which differs from the geometry used in the calculations above. However, we may expect that the fundamental heat transport properties of the circular and square stacks are the same. Numerical results show that the amplitude of the cold spot is small. We may therefore write T˜n = T˜0 + T˜1 , T˜1  T˜0 n T˜air = T˜air 0 + T˜air 1 , T˜air 1  T˜air 0

(5.204)

where T˜0 and T˜air 0 are the undisturbed solutions and T˜1 and T˜air 1 are the small-amplitude perturbations caused by the spot. The disturbance in local current decays much faster than the temperature disturbance (Figure 5.22). We may therefore set ˜j n ' ˜j n+1 ' J˜ and η˜n ' η˜n−1 . Substituting Eqs (5.204) into (5.203) we get the equation for T˜1 : !   ∂ T˜1 ∂ 2 T˜1 1 ∂ 2 ˜T˜1 − ξ 2 T˜1 − T˜air 1 (5.205) − r˜ − = ω (1 + k ) J S r˜ ∂ r˜ ∂ r˜ ∂ zˆ2 where r˜ =

r L

(5.206)

is the dimensionless radial coordinate. ∗ Parameter ξ is large (Table 5.10). Thus, at the leading order, T˜1∗ = T˜air 1 (here the superscript ∗ denotes the leading-order solution)13 . The equation for T˜1∗ is obtained from Eq. (5.205) setting T˜air 1 = T˜1 ' T˜1∗ : ! 1 ∂ ∂ T˜1∗ ∂ 2 T˜1∗ r˜ + = −ω 2 (1 + kS )J˜T˜1∗ . (5.207) r˜ ∂ r˜ ∂ r˜ ∂ zˆ2 We seek a solution to Eq. (5.207) in the form T˜1∗ = ρ(˜ r)ζ(˜ z ).

(5.208)

Substituting this into (5.207) and separating variables we get   1 ∂ ∂ρ 1 ∂2ζ r˜ =− − ω 2 (1 + kS )J˜ = −ν 2 ρ˜ r ∂ r˜ ∂ r˜ ζ ∂ zˆ2 13 This

result is immediately seen if we divide Eq. (5.205) by ξ 2 . Since ξ 2 ' 100, all ∗ . terms in this equation except the last one vanish and we get T˜1∗ = T˜air 1

5.6 HYBRID 3D MODEL OF SOFC STACK

259

which is equivalent to the following system of equations   1 ∂ ∂ρ ∂ρ ∂ρ = 0, =0 (5.209) r˜ + ν 2 ρ = 0, r˜ ∂ r˜ ∂ r˜ ∂ r˜ r˜=0 ∂ r˜ r˜=1  ∂2ζ  2 2 ˜ ζ = 0, ζ|zˆ=0 = ζ 0 , ζ|zˆ→∞ = 0. (5.210) − ν − ω (1 + k ) J S ∂ zˆ2 The boundary conditions to Eq. (5.209) express solution symmetry at r˜ = 0 and the adiabatic stack surface at r˜ = 1. The condition at infinity to Eq. (5.210) expresses the decay of thermal disturbance. Below we will see that this condition does not always hold. The solution to (5.209) is the Bessel function of the first kind: ρ(˜ r) = ρ0 J0 (an r˜)

(5.211)

where an are zeros of the first derivative of J0 (page 247). The solution to Eq. (5.210) is the exponent:  q  ζ = ζ 0 exp −ˆ z ν 2 − ω 2 (1 + kS )J˜

(5.212)

where ζ 0 is the amplitude of disturbance at zˆ = 0, where the physical resistive spot is assumed to be located. The general solution to Eq. (5.205) is (Section 5.5.5) T˜1∗ =

X

 q  cn J0 (an r˜) exp −ˆ z a2n − ω 2 (1 + kS )J˜

(5.213)

n

where cn are constants. We see that the modes with large an rapidly decay with the distance zˆ. Therefore, the damping length is determined by the “low-frequency” mode n = 1. Taking into account (5.202), for this length we find √ ˜ z θ kT h ˜l1 = q . a21 − ω 2 (1 + kS )J˜

(5.214)

In SOFC, kS = −0.5 (Table (5.189)) and this equation takes the form s ˜l1 = h ˜zθ

a21

kT . − 0.5ω 2 J˜

(5.215)

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Taking into account (5.192) and (5.143) we get (5.215) in the dimensional form s kT l1 = (hmea + hp ) θ . (5.216) 2 a1 − 0.5ω 2 J˜ With the data from Table 5.9 we obtain l1 ' 2.86 cm. The thickness of the stack element MEA + BP is 0.2 cm; thus, 2.8 cm is equivalent to 14 cells. The fitting exponent for the numerical shape of ∆T along the stack axis has the characteristic length of 12.5 cells (Figure 5.23); therefore, the agreement between theory and numerical experiment is very good14 . In Eq. (5.216), θ ∼ L and ω ∼ L. Therefore, if the term with ω is small, we have l1 ∼ L, i.e. the damping length is proportional to the transversal size of the stack. In this situation the stacks with large cell area are more transparent for temperature disturbances.

5.6.6

Temperature stratification

The exponential decay of temperature disturbance along z takes place unless the expression in the denominator in Eq. (5.216) is zero or negative. Equating this expression to zero and taking into account (5.19) we get L2 J∆S = a21 . 4F hp λp

(5.217)

This relation determines the limiting mean current density in a stack Jlim =

4F hp λp a21 . L2 ∆S

(5.218)

When J → Jlim the damping length tends to infinity, that is to say the temperature disturbance is constant along z. Furthermore, for J > Jlim the power of the exponent of the first (long-wave) mode in Eq. (5.212) becomes complex, which means that this mode oscillates along z. This effect is a manifestation of the thermal instability expressed by Eq. (5.207). The origin of this instability is similar to that discussed in Section 5.3: the right side of Eq. (5.207) is proportional to T˜1∗ , i.e. the rate of heating increases in domains where the temperature disturbance T˜1∗ is positive and it decreases in domains, where T˜1∗ is negative. At high currents this leads to a stratification of the temperature disturbance: along the stack axis, temperature valleys alternate with temperature peaks. 14 The numerical result is obtained for the square cells in the stack, while Eq. (5.216) is derived for circular cells. To correct Eq. (5.216) for the square √ stack we equate the surface areas of a square and circular cells. This gives L → L/ π. Thus, the value of L √ from Table 5.9 was divided by π.

5.6 HYBRID 3D MODEL OF SOFC STACK

261

Note that for a given current density J, Eq. (5.217) determines the limiting stack radius (transversal size): r 4F hp λp Llim = a1 . (5.219) J∆S At L > Llim , stratification takes place, while for L < Llim , the disturbance exponentially decays with z. Thus, stacks of a large transversal size are fully transparent to temperature disturbances15 .

5.6.7

The mechanism of anomalous heat transport

Physically, the anomalous transport of a cold spot is provided by the following mechanism. The rate of thermodynamic heating in electrochemical reactions is proportional to local stack temperature T times local current j (Eq. (5.190)). The resistive spot in a defective cell lowers local current and hence the local rate of heating. As a result the local stack temperature decreases. Lower temperature means lower heat flux to the adjacent cells, which lowers the local temperature in these cells. In this way, the temperature disturbance propagates along the stack axis. This propagation is further facilitated by thermal insulation of the stack side surface and by high thermal conductivity of MEA and BP. The adiabatic side surface of the stack reduces the effect of cold spot smoothing by the conductance mechanism of heat transport. Literally speaking, the adiabatic side surface “traps” the cold spot in a stack volume. At the same time, the large thermal conductivity of the stack components along z facilitates spot penetration. Parameter ω determines the “sensitivity” of the local rate of heating to the temperature variation in adjacent cells. If this sensitivity is large enough, the damping length tends to infinity, i.e. the cold spot amplitude does not decay. A further increase in ω makes cold spot propagation unstable: instead of a long cold “needle”, a garland of alternating cold and hot spots forms along the stack axis. Analogously, hot domains in this temperature wave are supported by the higher rate of heating there. Heat removal in SOFCs is provided by the air flow only. The air flow tends to stretch the temperature valley in the flow direction, leaving intact the spot size in the perpendicular direction (Figure 5.22). Moreover, Eq. (5.207) for the temperature disturbance does not contain the air flow temperature. This means that the effect of air on the temperature disturbance is of the second order, i.e. the air flow practically does not affect the disturbance amplitude. 15 Note that if a2 −0.5ω 2 J˜ is negative and the absolute value of this expression is small, 1 the disturbance decreases with z as a part of cosine.

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Figure 5.24: The origin of voltage loss in the bipolar plate (Kulikovsky, 2006b). Local current in cell A linearly decreases with x; in cell B local current is constant. Current jA (x) generated at x in cell A cannot pass through cell B directly at x; the fraction jA (x)−jB (x) has to be transported along the bipolar plate to x0 , where the condition jA (x) − jB (x) + jA (x0 ) = jB (x0 ) is fulfilled.

5.7 Power generated and lost in a stack 5.7.1

The nature of voltage loss in bipolar plates

As discussed above, transport of current through the BPs results in power loss. BPs account for about 80% of the stack weight and up to 45% of the cost (Hermann et al., 2005); thus an accurate calculation of power loss in the BPs is of vital importance for stack design. To minimize cost and weight the BPs in PEFC and DMFC stacks are usually made of expanded graphite. Graphite plates have good chemical stability and mechanical strength (Besmann et al., 2000; Cunningham et al., 2005). However, the conductivity of graphite is much lower than that of a metal and power loss in a graphite BP may be quite significant. If in all cells in a stack, the distribution of current over the cell surface is the same, voltage loss in the BPs is negligible. Indeed, current crosses the BPs only in a z-direction in that case and since the BPs are thin, the respective voltage loss is small. Significant power loss in the BP may arise due to the different distributions of local current over the surface of adjacent cells. To illustrate this effect, consider a stack of two 1D cells A and B (Figure 5.24). Suppose that local current in cell A linearly decreases with the distance x, whereas current in cell B is constant (Figure 5.24). Since jA 6= jB , current generated at x in cell A cannot pass through cell B at the same point (Figure 5.24). Part of jA (x) (shaded bar) must be transported along the bipolar plate to x0 , where it sums with jA (x0 ) to

5.7 POWER GENERATED AND LOST IN A STACK

263

yield jB (Figure 5.24). Figure 5.24 suggests that the current transported along the bipolar plate is jA − jB . For numerous reasons the local balance of currents in adjacent cells is usually not fulfilled and a significant amount of stack current is transported along the bipolar plate (in-plane current). In-plane current induces a variation of potential over the BP surface. Suppose that the potential distribution over the BP surface and currents in the adjacent cells are measured. Are these data sufficient to calculate the useful power generated by the cells and the power lost in the bipolar plate? If yes, how should we calculate these values? The answer to these questions is the subject of this section (Kulikovsky, 2007d).

5.7.2

Power dissipated in a bipolar plate

The general relation for electric power W dissipated in a conductive medium is Z W = j E d3 r (5.220) P

where j is the local current density, E is the electric field strength and integration is performed over the conductor volume P . It is convenient to eliminate j and E from (5.220) using Ohm’s law j = −σp ∇3 V and the relation E = −∇3 V . Here σp is the conductivity of BP material, V is the BP potential and the subscript 3 indicates that the respective operator acts in a 3D space. These substitutions yield Z 2 W = σp (∇3 V ) d3 r. (5.221) P

Thus, in the general case to calculate W the distribution of potential over the plate volume is needed. BP potential obeys the Laplace equation ∆3 V = 0.

(5.222)

The boundary conditions for this equation provide local current densities in the adjacent fuel cells (Figure 5.25) ∂V ∂V −σp = jα , −σp = jβ (5.223) ∂z S ∂z S 0 and the absence of normal current through the BP end faces ∂V =0 ∂n Γ

(5.224)

where ∂V /∂n is calculated along the normal to the end face Γ (Figure 5.25).

264

CHAPTER 5. MODELLING OF FUEL CELL STACKS

Figure 5.25: To the boundary conditions for the 3D Laplace equation for bipolar plate potential.

Figure 5.26: Sketch of the stack fragment. Left: individual cells (α, β) are separated by bipolar plates (a, b, c). Right: plate b and volume P bounded by the side surface Γ and by the surfaces S on the top and bottom.

We see that the straightforward approach leads to a 3D problem (5.222)–(5.224). However, since BPs are thin the calculation of W can be simplified. Consider the two cells α and β and the three bipolar plates a, b and c (Figure 5.26). To describe the potential of BP we introduce a system of coordinates (x, y) (Figure 5.26). As before, we will use the plate potential at (0, 0) as a reference point, i.e. Va (0, 0) is the reference point for potential Va , Vb (0, 0) is the reference point for Vb and so on. The smallness of BP potential variation in the through-plane direction allows us to reduce the 3D Laplace equation for Vb to the 2D Poisson equation (5.139) (Section 5.5.1): ∆2 Vb = rp (jα − jβ )

(5.225)

where the subscript 2 indicates that the operator ∆2 acts in a 2D space: ∆2 ≡

∂2 ∂2 + . ∂x2 ∂y 2

5.7 POWER GENERATED AND LOST IN A STACK

265

Once Vb is determined from Eq. (5.225), we can treat the BP as a plate with zero gradient of potential along the z-axis16 . Thus, the relation for power loss in the BP (5.221) is simplified to Z ∂ ∂ 2 W = σp hp (∇2 V ) dS, ∇2 ≡ i +j (5.226) ∂x ∂y A where integration is performed over the cell active area A. Thus, the smallness of BP thickness allows us to reduce the dimensionality of the problem. In numerical simulations, V and ∇V are usually calculated with high accuracy, so that power loss in the BP can be obtained directly from Eq. (5.226). In experiments, however, measured values of V usually contain “noise”. Numerical differentiation of “noisy” data leads to large errors. Moreover, Eq. (5.226) contains the square of ∇V , which further increases these errors. Below we derive the general relation for power dissipated in the BP, which does not contain derivatives. This relation is then used to calculate the power available for use in the external circuit (the useful power) generated by the individual cell. Calculations with these relations are illustrated by a specially constructed 1D case.

5.7.3

Power dissipated in a thin bipolar plate

Hereinafter, the subscript “2” is omitted and the symbols ∆ and ∇ denote 2D operators. Consider a bipolar plate b (Figure 5.26), and the volume P = hp S bounded by the side surface Γ and by the surfaces S on the top and bottom (Figure 5.26, right). According to (5.226), power dissipated in the volume P is 2  2 # Z Z " ∂V ∂V 2 WP = hp σp (∇V ) dS ≡ hp σp + dS. (5.227) ∂x ∂y S S The following theorem forms the basis for further calculations. Theorem. If the normal current through the side surface Γ is zero, i.e. if jn |Γ = −σp (∇V )n |Γ = 0, then the following relations hold Z Z 2 WP = hp σp (∇V ) dS = − V (jα − jβ ) dS, S

(5.228)

(5.229)

S

where jα and jβ are local transversal current densities on both sides of plate b (Figure 5.26). 16 Indeed, Eq. (5.225) does not contain BP conductivity in the through-plane direction σ⊥ : this conductivity is “hidden” in local currents jα and jβ . Therefore, we may formally put σ⊥ = ∞ and neglect ∂V /∂z.

266

CHAPTER 5. MODELLING OF FUEL CELL STACKS

Proof. We introduce the dimensionless variables x ˜=

x , hp

y˜ =

y , hp

˜j = j , J

S S˜ = 2 , hp

V σp V˜ = , Jhp

(5.230)

where J is the mean current density in a stack. With these variables, Eq. (5.229) transforms to17 Z Z  2 ˜ ˜ ˜ (5.231) dS = − V˜ (˜jα − ˜jβ ) dS. ∇V ˜ S

˜ S

To prove Eq. (5.231) we note that with the dimensionless variables (5.230), Eq. (5.225) takes the form ∆V˜ = ˜jα − ˜jβ .

(5.232)

Substituting ˜jα − ˜jβ = ∆V˜ into (5.231) we get Z  Z 2 ˜ ˜ ˜ ∇V dS = − V˜ ∆V˜ dS. ˜ S

(5.233)

˜ S

The right side of this equation can be transformed and integrated by parts: Z Z   − V˜ ∆V˜ dS˜ = − V˜ ∇ · ∇V˜ dS˜ ˜ ˜ S ZS  Z   2 ˜ ˜ ˜ dS. dΓ + ∇V˜ = − V ∇V n

Γ

˜ S

Using this in (5.233) we find Z  Z Z  2   2 ˜ ∇V˜ dS˜ = − V˜ ∇V˜ dΓ + ∇V˜ dS. ˜ S

n

Γ

(5.234)

˜ S

If Eq. (5.228) is fulfilled, the first term on the right side vanishes and the resulting identity proves the theorem.  Equation (5.229) states that if the in-plane current does not cross the side surface of the volume P , power dissipated in P can be calculated in two ways: either directly, as the power dissipated by electric field in P , or as the power dissipated by the current jα − jβ in potential V . Note that jα and jβ are the transversal currents, whereas jα − jβ is the in-plane current in the volume P . Remark 1. Since the in-plane current through the end faces of the whole plate is zero, Eq. (5.229) always holds for the whole bipolar plate. 17 Up

to the end of Section 5.7.3, ∇ and ∆ are dimensionless: ∇≡i

∂ ∂ +j , ∂x ˜ ∂ y˜

∆≡

∂2 ∂2 + . 2 ∂x ˜ ∂ y˜2

5.7 POWER GENERATED AND LOST IN A STACK

267

Remark 2. The addition of an arbitrary constant to the potential in Eq. (5.231) does not change the value of the integrals. Indeed, the substitution V˜ → V˜ + C does not change the left side of Eq. (5.231) since the integral involves ∇V˜ . Making the substitution V˜ → V˜ + C on the right side of this equation we get Z  Z Z  ˜ (5.235) V˜ + C (˜jα − ˜jβ ) dS˜ = V˜ (˜jα − ˜jβ ) dS˜ + C (˜jα − ˜jβ ) dS. ˜ S

˜ S

˜ S

Since no current crosses the side surface of PR, the total Rcurrent through the upper and the lower surfaces S is the same: S ˜jα dS = S ˜jβ dS. Therefore, the last term on the right side of Eq. (5.235) is zero. This means that relations (5.229) are insensitive to the choice of a reference point for the BP potential. Therefore, to calculate W with any of the relations (5.229) it is sufficient to solve Eq. (5.225) with the boundary condition (∇V )n |end faces = 0 and e.g. V (0, 0) = 1. However, with a special choice of the reference point for V , the last expression in Eq. (5.229) can be further simplified.

5.7.4

Useful power generated by the individual cell

Consider now cell α (Figure 5.26). The local voltage of this cell is 0 Vα = Vab + Va (x, y) − Vb (x, y)

(5.236)

0 where Vab is the voltage drop between plates a and b at the origin of the coordinates. Multiplying (5.236) by jα and integrating over the cell active area A, we get the total power Wα generated by the cell α: Z Z Z
0 0 Wα = Vab + Va − Vb jα dS = Vab jα dS + (Va − Vb ) jα dS A A A Z 0 = AJVab + (Va − Vb ) jα dS. (5.237) A

The first term on the right side looks like the power available for use 0 in the external circuit. This term, however, contains Vab , the voltage drop between bipolar plates a and b at the arbitrarily chosen point (0,0). Clearly, useful power must not depend on the choice of the origin of coordinates. To understand the meaning of the terms in (5.237) consider the identity (Va − Vb ) jα = −Vb (jα − jβ ) + Va jα − Vb jβ . Integrating both sides over A we get Z Z Z (Va − Vb ) jα dS = − Vb (jα − jβ ) dS + (Va jα − Vb jβ ) dS. A

A

A

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

According to (5.229), the first term on the right side is the power loss Wbloss in plate b. We thus have Z Z (Va − Vb ) jα dS = Wbloss + (Va jα − Vb jβ ) dS. (5.238) A

A

Now for simplicity we will assume that the endplates in a stack are ideally conductive, i.e. VN = V0 = 0. If we sum up (5.238) over all the cells in the stack the second term on the right side of the resulting equation disappears, since N X

(Vi ji − Vi−1 ji−1 ) = VN jN − V0 j0 = 0.

i=1

Summing up (5.238) we, therefore, get N Z X i=1

(Vi − Vi−1 ) ji dS =

A

N X

Wiloss .

(5.239)

i=1

Thus, the sum on the left side of (5.239) is the total power loss in all the bipolar plates in the stack. This suggests that with a proper choice of the reference point for potentials, the left side of Eq. (5.238) would represent the power loss in the single plate b. Clearly, the required transformation of potentials should make the integral on the right side of Eq. (5.238) vanish. This can be achieved as follows. Since the potential is defined within an arbitrary constant, we can “shift” Vi by a constant Ci Vi0 = Vi + Ci

(5.240)

to fulfil the condition Z

Vi0 ji dS = 0.

(5.241)

A

Multiplying (5.240) by ji , integrating the result and using (5.241), for Ci we find Z 1 Ci = − Vi ji dS. (5.242) J A Now suppose that the potentials of all the plates are transformed according to Eqs (5.240) and (5.242). The last integral in Eq. (5.238) then vanishes and this equation simplifies to Z loss Wb = (Va0 − Vb0 ) jα dS. (5.243) A

5.7 POWER GENERATED AND LOST IN A STACK

269

0 0 0 = (Vab ) − C 0 into (5.237) we get Substituting Vi = Vi0 − Ci and Vab

Z

−C + ((Va0 − Ca ) − (Vb0 − Cb )) jα dS A Z 0 0 = AJ(Vab ) + (Va0 − Vb0 ) jα dS − AJC 0 + A(Cb − Ca )J. (5.244)

Wα = AJ

0 0 (Vab )

0




A

Taking C 0 = Cb − Ca

(5.245)

we find 0 0 Wα = AJ(Vab ) +

Z

(Va0 − Vb0 ) jα dS = Wαload + Wbloss .

(5.246)

A

The second term on the right side is the power lost in bipolar plate b and the first term there is the power available for use in the external load: 0 0 Wαload = AJ(Vab ).

(5.247)

These results allow us to formulate the following procedure for evaluating Wαload in practical applications: • Select two points (origin of coordinates) on BPs a and b and measure 0 the voltage drop Vab between these points. The points must be on the perpendicular connecting the plate surfaces. • Measure the distributions of local current density and potential over the surfaces of plates a and b. The potential of each plate should be measured with respect to the selected point. • Using (5.242), calculate Ca , Cb and then C 0 = Cb − Ca . 0 0 0 • Take (Vab ) = Vab − C 0 and calculate the useful power with (5.247).

With (5.243) and (5.241) we get the following equivalent relations for power dissipated in the BP: Wbloss

Z

2

= hp σp (∇Vb ) dS Z A =− Vb (jα − jβ ) dS A Z =− Vb0 jα dS. A

Here Vb0 = Vb + Cb and Cb is calculated from (5.242).

(5.248) (5.249) (5.250)

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CHAPTER 5. MODELLING OF FUEL CELL STACKS

Power dissipated in the bipolar plate determines the voltage loss induced by this plate. The voltage loss is V loss =

W loss JA

(5.251)

where W loss is given by any of the equations (5.248)–(5.250). Physically, V loss is the voltage loss on the equivalent resistor, which dissipates the same power and supports the total current JA in the cell.

5.7.5

Illustration: A 1D case

To illustrate these results, consider a narrow rectangular plate of width w  L, where L is the BP length along the x-axis. For simplicity we will assume that the current is distributed uniformly along the plate width, so that the plate potential varies only along the x-axis. Suppose that there are two points x1 and x2 where the in-plane current is zero: ∂ V˜b ∂ V˜b (5.252) = = 0. ∂x ˜ ∂x ˜ x ˜1

x ˜2

According to Eqs (5.248)–(5.250), the power dissipated between these points is ˜ Ploss = w W ˜

Z

x ˜2

x ˜1

Z

x ˜2

= −w ˜ x ˜1 Z x˜2

= −w ˜

∂ V˜b ∂x ˜

!2 d˜ x

(5.253)

V˜b (˜jα − ˜jβ ) d˜ x

(5.254)

V˜b0 ˜jα d˜ x

(5.255)

x ˜1

˜ = W σp /(J 2 h3p ) and Eq. (5.255) is valid provided that where w ˜ = w/hp , W R x˜2 0 ˜1 = 0 V˜ ˜j d˜ x = 0. Note that the conditions (5.252) always hold at x x ˜1 b β and x ˜2 = 1 (the BP end faces). To illustrate Eqs (5.253)–(5.255), we take ˜jα = 1 + sin 4π˜ x ˜jβ = 1

(5.256) (5.257)

R1 R1 (see Figure 5.27). Note that 0 ˜jα d˜ x = 0 ˜jβ d˜ x = 1, i.e. as it should be, the total current in cells α and β is the same.

5.7 POWER GENERATED AND LOST IN A STACK

271

Figure 5.27: Model current densities in the adjacent cells α and β. Using these ˜jα and ˜jβ in (5.232), we get the following equation for V˜b : ∂ 2 V˜b = sin (4π˜ x) . ∂x ˜2

(5.258)

Solving Eq. (5.258) with the boundary conditions ∂ V˜b ˜ Vb (0) = =0 ∂x ˜

(5.259)

0

we get 1 V˜b = 4π



sin 4π˜ x x ˜− 4π

 .

(5.260)

˜ loss with (5.253) or (5.254). To This potential can be used to calculate W use Eq. (5.255), we have to “shift” the potential to satisfy the condition Z 1  V˜b + C˜b ˜jβ d˜ x = 0. (5.261) 0

Using here (5.260) and (5.257), we find 1 C˜b = − . 8π With this C˜b , we finally find 1 V˜b0 = V˜b + C˜b = 4π



sin 4π˜ x 1 x ˜− − 4π 2

 .

(5.262)

The plot of V˜b0 is shown in Figure 5.28. We see that ∂ V˜b0 /∂ x ˜ = 0 at the three points: x ˜ = 0, 1/2 and 1. Therefore, Eqs (5.253)–(5.255) are valid at    the three intervals [˜ x1 , x ˜2 ]: 0, 12 , 12 , 1 and [0, 1].

272

CHAPTER 5. MODELLING OF FUEL CELL STACKS

Figure 5.28: The distribution of dimensionless voltage V˜b0 along the bipolar plate and the product V˜b0 ˜jα . The function −V˜b0 ˜jα is shown in Figure 5.28. Using (5.256), (5.257) and (5.262) in (5.253)–(5.255) and calculating the integrals we find Z 0

1/2 

∂ V˜b0 /∂ x ˜

2

Z d˜ x=−

1/2


V˜b0 ˜jα − ˜jβ d˜ x=−

0

Z 0

1/2

V˜b0 ˜jα d˜ x=

3 . 64π 2

The  1 same result is obtained for the respective integrals over the interval 2 , 1 . Clearly, all three integrals over [0, 1] (the total power loss) are twice as large: ˜ loss = W

3 . 32π 2

(5.263)

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Abbreviations ACL BL BP CL CCL CFD DMFC FAM FC GDL HOR MEA OCV ORR PEMFC PEFC RDS SOFC

Anode catalyst layer Backing layer Bipolar plate Catalyst layer Cathode catalyst layer Computational fluid dynamics Direct methanol fuel cell Flooded agglomerate model Fuel cell Gas-diffusion layer Hydrogen oxidation reaction Membrane-electrode assembly Open circuit voltage Oxygen reduction reaction Polymer electrolyte membrane fuel cell (=PEMFC) Polymer electrolyte fuel cell Rate-determining step Solid oxide fuel cell

283

Nomenclature ˜ ˆ A a a1 b bm Ch C0 c ch c0h cH+ cH2 cl cref cP cP a cP p cP ox cP w ct cs cw csat w D Db Dm Dw dh

Indicates dimensionless variables Indicates dimensionless variables Cell active area (m2 ); molar concentration of active catalyst particles (Section 2.7.1), (mol m−3 ) Dimensionless parameter (4.57) and (4.111) 3.8317, first root of the derivative of the Bessel function Tafel slope (V) Characteristic voltage loss in the membrane (4.62) (V) Geometrical constant (5.163) Dimensionless constant (3.52) and (3.108) Molar concentration (mol m−3 ) Molar concentration in the channel (mol m−3 ) Molar concentration at the channel inlet (mol m−3 ) Proton molar concentration in the membrane (mol m−3 ) Hydrogen molar concentration in the anode channel (mol m−3 ) Molar concentration of liquid water (1/18 mol m−3 ) Reference molar concentration (mol m−3 ) Heat capacity (J kg−1 K−1 ) Heat capacity of the air flow (J kg−1 K−1 ) Heat capacity of the bipolar plate (J kg−1 K−1 ) Heat capacity of oxygen (J kg−1 K−1 ) Heat capacity of liquid water (J kg−1 K−1 ) Molar concentration in the catalyst layer (mol m−3 ) Speed of sound (m s−1 ) Molar concentration of liquid water in the membrane (mol m−3 ) Molar concentration of the saturated water vapour (mol m−3 ) Diffusion coefficient (m2 s−1 ) Diffusion coefficient in the GDL (m2 s−1 ) Methanol diffusion coefficient in the membrane (m2 s−1 ) Water diffusion coefficient in the membrane (m2 s−1 ) Channel width (m) (4.108)

285

286 E E0 Eact F Fg f fa , fc fr fα fλ fu f1 G g gz H Hair , Hs h, hc hp hz i∗ I, Itot J J0 Jcross Jlim j jcrit jcross jD 0 jD je jp jσ jσh jσt j∗

NOMENCLATURE Total voltage loss in the cell (V); electric field strength (V m−1 ) Total voltage loss (V); voltage loss at zero current, Section 4.7.3 Activation energy (J) Faraday constant (9.6495 × 104 Coulomb mol−1 ) Dimensionless function (2.86) Dimensionless constant; air flow, Section 4.9.3 Dimensionless heat flux at the ACL/membrane and the membrane/CCL interface, respectively Dimensionless parameter (4.75) Dimensionless parameter (4.37) Dimensionless parameter (4.33) Dimensionless function in Section 4.6.5; (5.23) First-order correction (5.47) Gibbs free energy (J mol−1 ) Fraction of catalyst loading across the catalyst layer (0 ≤ g ≤ 1) Fraction of catalyst loading along the channel (0 ≤ g ≤ 1) Enthalpy (J mol−1 ); dimensionless integral of motion (2.117); stack length (m) Heat transfer coefficient (W m−2 K−1 ) Channel height above the GDL (m) Bipolar plate thickness (m) Geometrical constant in Section 5.6.5 Volumetric exchange current density (A m−3 ) Total current in the external load (A) Mean current density in the cell (A m−2 ) Bessel function of the first kind Average crossover current density (A m−2 ) Limiting current density (A m−2 ) Local proton current density in the catalyst layer, Chapter 2; local current density in the cell (A m−2 ) Critical current density, Section 4.4 (A m−2 ) Equivalent crossover current density (A m−2 ) Diffusion-limiting current density (A m−2 ) Diffusion-limiting current density at the channel inlet (A m−2 ) Electron current density (A m−2 ) Local proton current density in the catalyst layer, Chapter 1 (A m−2 ) Characteristic current density (4.164) (A m−2 ) Characteristic current density (3.8) (A m−2 ) Characteristic current density (2.60) (A m−2 ) Superficial exchange current density (A m−2 ); characteristic current density (A m−2 )

NOMENCLATURE j∗D j∗h j0 j0 Kevap Kw k ka ke kl km kp kS kλ kσ k∗ L l lb le lm lt lu lw l1 l∗ M N N1 Nu n, ne nd P p psat w Q Qr q qT R R∗

287

Characteristic current density (2.75) (A m−2 ) Characteristic current density (3.6), (A m−2 ) Proton current density in the membrane (A m−2 ), Chapter 2 Local current density at the channel inlet, (A m−2 ) Evaporation rate constant (atm−1 s−1 ) Average slope of the membrane water sorption isotherm (Kulikovsky, 2004b) Rate constant (mol cm s−1 ) (Section 1.3.1); dimensionless parameter; dimensionless wave vector Rate constant of methanol adsorption (2.101), (cm3 mol−1 s−1 ); dimensionless parameter (5.188) Dimensionless parameter (5.188) la /lc , (3.43) lm /lc , (3.45) Hydraulic permeability (m2 ) 2∆Sa /∆Sc , (3.43) λm /λ, (3.43) σm /σt , (3.43) Characteristic rate constant of methanol oxidation (2.102), (cm3 mol−1 s−1 ) Channel length (m); radius of a circular stack (m) Catalyst layer thickness (Sections 3.4 and 3.5) Thickness of the GDL or backing layer (m) Electrolyte thickness (m), Section 5.6 Membrane thickness (m) Thickness of the catalyst layer (m) Length of the Nusselt number relaxation in the channel (m) Length of the domain exposed to degradation, Section 4.4 (m) Damping length of the disturbance in the stack Reaction penetration depth (2.14) and (4.158) (m) Molecular weight (kg mol−1 ) Molar flux (mol m−2 s−1 ) Number of cells affected by the disturbance in the stack Nusselt number Number of electrons transferred in the reaction Electroosmotic drag coefficient Pressure (Pa); bipolar plate volume (m3 ) Ratio of Tafel slopes bc /ba or bm /bc ; pressure (Pa) Pressure of saturated vater vapour (Pa) Rate of electrochemical reaction (A m−3 s−1 ) Rate of the reverse reaction (1.23), (A m−3 s−1 ) Heat flux (W m−2 ); dimensionless constant (4.71) and (4.203) Total heat flux due to the electrochemical reaction (W m−2 ) Gas constant (8.314 J mol−1 K−1 ); cell resistivity (Ω m2 ) Characteristic cell resistivity (Ω m2 )

288 Ra RJ RS Rη Rm Rp r rcl rp S s T T∗ Tair Ta , Tc Ts u V Vab Vcell Vloss Voc Vstack v vH2 vw , vwave vmax W w x

y z zmax zw zox

NOMENCLATURE Anode resistivity (Ω m2 ) Rate of Joule heating (W m−3 ) Rate of heating due to the reversible thermodynamic heat released in the electrochemical reaction (W m−3 ) Rate of heating due to the irreversible loss in the electrochemical reaction (W m−3 ) Membrane resistivity (Ω m2 ) Bipolar plate resistivity (Ω m2 ) Dimensionless parameter (4.63); radial coordinate (m) Mean pore radius in the catalyst layer (m) Bipolar plate resistivity ((1/(σp hp )), Ω) Entropy (J mol−1 K−1 ); cell active area (m2 ); ξh0 /ψh0 , Section (4.3.6) Liquid saturation; characteristic radius of the resistive spot (m) Cell or stack temperature (K) Characteristic temperature (K); activation temperature (K) Air flow temperature in the cathode channel (K) Temperature of the anode and cathode side of the MEA, respectively (K) Temperature of the anode flow (methanol-water solution) in DMFC (K) Feed molecule utilization (u = 1/λ); ratio 273/298 Voltage (V); volume (m3 ) Voltage drop between bipolar plates a and b (V) Cell voltage (V) Total voltage loss (V) Cell open-circuit voltage (V) Stack voltage (V) Flow velocity (m s−1 ) Hydrogen flow velocity in the anode channel (m s−1 ) Wave velocity (m s−1 ) Maximal velocity of the degradation wave (4.118) (m s−1 ) Electric power (W) Channel width (m) Coordinate across the cell (m); coordinate along the channel, Chapter 5; in-plane coordinate on the bipolar plate In-plane coordinate on the bipolar plate Coordinate along the channel (m); coordinate along the stack axis, Chapter 5 Point separating the water- and oxygen-limiting regimes (4.81) (m) Position of the wavefront (m) Point where the oxygen concentration vanishes (4.242), (m)

NOMENCLATURE z0 z∗

289

Point where the local current vanishes (4.244), (m) Stretched z-coordinate (5.164), (m)

Greek symbols α αw β

β∗ γ ∆H ∆S ∆Scross δ ε εBL εGDL ε0  ζ

η ηconc Θ θ κ λ λm λa λc λe λp λw λ∗ µ

Kinetic transfer coefficient; Dimensionless parameter (4.204), (2.90), (3.42) and (2.153) Transfer coefficient of water in membrane Kinetic transfer coefficient of reverse reaction; dimensionless parameter (Section 1.4.8); dimensionless parameter (2.55); dimensionless parameter of methanol crossover (3.17) β/(1 + β), dimensionless parameter of methanol crossover Dimensionless parameter (3.76), (4.56) and (4.205) Enthalpy change in the reaction (J mol−1 ) Entropy change in the reaction (J mol−1 K−1 ) Entropy change in the direct methanol oxidation (J mol−1 K−1 ) Variation of overpotential across the catalyst layer (V) (2.127) Dimensionless parameter (2.13) and (4.157) Backing layer porosity GDL porosity Dimensionless parameter (4.136) and (4.167) Small dimensionless parameter (5.27) Correction factor (1.66); dimensionless parameter (2.70) and (4.15); nitrogen molar fraction; axial dependence of stack voltage or temperature Half-cell or total polarization voltage (overpotential, V) Concentration overpotential (V) Fraction of occupied sites on the catalyst surface (Section 2.7.1) Overheat (3.95) and (3.96); dimensionless parameter (5.192) Dimensionless parameter (5.123) Flow stoichiometry; thermal conductivity (W m−1 K−1 ) Membrane thermal conductivity (W m−1 K−1 ) Stoichiometry of the anode flow Stoichiometry of the cathode (air) flow Electrolyte thermal conductivity (W m−1 K−1 ), Section 5.6 Bipolar plate thermal conductivity (W m−1 K−1 ) Membrane water content (a number of water molecules per sulphonic group) Characteristic water content of the membrane (Section 1.4.8) Fluid viscosity (kg m−1 s−1 ); dimensionless characteristic length (4.212); dimensionless parameter (4.106) and (5.78)

290 ν ξ ρ σ, σe σp σm σt τ τd τi τw Φ φ ϕ ϕm φλ χ χa , χe ψ ω ω∗ ωa

NOMENCLATURE Dimensionless frequency (5.84) Molar fraction; oxygen molar fraction; dimensionless parameter (5.21) and (5.111) Mass density (kg m−3 ); radial function for stack voltage or temperature Electron conductivity of the carbon phase (Ω m−1 ) In-plane electron conductivity of the bipolar plate (Ω m−1 ) Proton conductivity of the bulk membrane (Ω m−1 ) Proton conductivity of an electrolyte phase in the catalyst layer (Ω m−1 ) Dummy variable; characteristic time (s) Local time of degradation (s) Characteristic time of instability growth (s) RCharacteristic time (4.114) (s) φ, where φ is the conversion function (2.4) or (2.5) Dimensionless parameter (3.88), (5.20) and (5.110); conversion function (2.4) or (2.5) Carbon phase potential (V) Membrane phase potential (V) Function of hydrogen stoichiometry (4.180) Dimensionless constant in (2.90); dimensionless parameter (5.113) Dimensionless parameters (5.187) Water molar fraction; methanol molar fraction; dimensionless parameter (2.115), (5.22) and (5.119) Dimensionless parameter (5.19) and (5.108) Dimensionless parameter (2.113) Dimensionless parameter (2.114)

Superscripts 0 1 a c eff h K load sat tot max ox w

channel inlet; standard state channel outlet anode cathode effective channel Knudsen in the load saturation total maximal oxygen water

NOMENCLATURE Subscripts * 0 1 a a, b, . . . air BV BL b c crit cross D E, W, N, S e ef f enh evap f ast H2 H+ h l lif e lim M max N2 oc ox opt p s slow std T t tot w α, β, . . .

characteristic membrane/CCL interface; anode GDL/ACL interface CCL/GDL interface; ACL/membrane interface anode; adsorption Enumerate bipolar plates in the stack air Butler-Volmer backing layer backing layer cathode critical crossover diffusion-limiting East, West, North and South electron-conducting phase in the catalyst layer; electron; electrolyte effective enhanced (under stressed conditions) evaporation fast hydrogen proton channel liquid water life (time) limiting methanol maximal nitrogen open-circuit oxygen optimal proton; bipolar plate (interconnect) methanol-water solution slow phase of the degradation wave propagation standard conditions Tafel; thermal catalyst layer total wave; water enumerate cells in the stack

291

Index cell lifetime, 152 cell resistivity, differential, 248 cell voltage, 165 CFD, 18 channel problem, 19, 118 characteristic current density, 50, 55 CL performance, 40 CL polarization curve, 39 CL porosity, 28 CL, heat transport, 75 CL, proton conductivity, 41 CL, temperature shape, 79 CL, transport properties, 39 computational fluid dynamics, 18 concentration factor, 12, 16 concentration overpotential, 87 continuity equation, 20 conversion function, 42 cross-linked feeding, 92, 182 crossover current density, 91 crossover flux, 89, 90 crossover parameter, 90 current collectors, 17

accelerated testing, 155 ACL, current-generating domain, 72 ACL, polarization curve, 67, 70, 73 ACL, variable thickness, 73 activation barrier, 11 activation energy, 11 activation polarization, 45 activation resistivity, 48 activity coefficient, 5 anode-supported SOFC, 37 aqueous electrolyte, 40 area-specific resistivity, 161, 201 Arrhenius equation, 155 Arrhenius law, 207 Arrhenius rate constant, 11 back diffusion, 32, 94 backing layer, 17 Bernoulli equation, 121 Bessel function, 247, 259 binary molecular diffusion, 24 bipolar plate, 17 Bruggeman correction, 25 Butler-Volmer equation, 9, 12, 40, 42, 55

damping length, 245, 248, 260 degradation wave, 147 dimensionless variables, 42, 43, 66, 78, 95, 121, 125, 136, 137, 148, 149, 157, 163, 165, 173, 200, 215, 228, 238, 246, 251, 266 direct methanol fuel cell, 36 dispersion relation, 217 DMFC, 36 DMFC, anode polarization voltage, 177 DMFC, cathode polarization voltage, 177 DMFC, cell voltage, 175 DMFC, critical air flow rate, 190 DMFC, design, 36

carbon phase, 7 carbon phase potential, 7 catalyst layer, 39 cathode catalyst layer, 40 CCL performance, 41, 43 CCL polarization curve, 15 CCL, heat balance, 94 CCL, high-current regime, 51 CCL, polarization curve, 46, 47, 51, 53, 55, 60, 62 CCL, proton conductivity, 94 CCL, thermal conductivity, 94 cell active area, 7

293

294 DMFC, half-cell reactions, 36 DMFC, Joule heating, 227 DMFC, jumper, 179, 185, 186, 188 DMFC, mixed potential, 178 double layer, 8 drag coefficient, 29, 89 Eigen ion, 29 electro-osmosis, 18, 89 electro-osmotic drag, 29 electro-osmotic flux of water, 21, 94 electrolyte phase, 7 electrolyte phase potential, 7 endplate, 35 enthalpy, 2 entropy, 2 entropy change, 6, 32, 33, 75, 94, 113, 250 equilibrium, 12 Euler equation, 119 exchange current density, 11, 16, 41, 70 Fick’s diffusion, 24, 84 flooded agglomerate model, 80 flooding, 25, 117 flow field, 17, 35 flow velocity, 17, 22 fuel cell kinetics, 7 galvanostatic regime, 147, 148, 223 gas-diffusion layer, 17, 24, 83 Gauss’ theorem, 20 Gaussian shape, 240, 253 GDL, 17, 24 GDL porosity, 24 GDL thickness, 19 GDL, transport loss, 85, 160 Gibbs free energy, 2, 5 gradient of catalyst loading, 16, 58 half-cell polarization voltage, 92 heat flux, 199 high-current polarization curve, 50, 86, 158 high-current regime, 92 hydraulic permeability, 22 hydrogen fuel cell, 2, 3 hydrogen mass balance, 165 hydrogen oxidation reaction, 9 hydrogen utilization, 18 ideal membrane humidification, 145 ideal proton transport, 53 incompressible flow, 17, 123

INDEX irreversible heating, 34, 75, 199, 228, 234 Joule heating, 34, 75, 94, 106, 196, 227 kinetics, 7 Knudsen diffusion, 27 Knudsen diffusion coefficient, 28 Laplace equation, 237, 246, 263 limiting current density, 70, 84, 85, 90, 127, 142, 152 liquid saturation, 25, 235 load, 7 local current density, 21 local polarization curve, 132 Loschmidt number, 24 low-current polarization curve, 45, 85, 157, 175 macrohomogeneous approach, 7 macrohomogeneous model, 39, 82 mass conservation equation, 20, 27 mass conservation equations in the channel, 22 mass transfer, 20, 22 mass transport, 16, 24, 27 mass velocity, 21 MEA, 17, 19, 38, 117 MEA problem, 19, 118 mean current density, 7 mean free path of molecules, 24 membrane, thermal conductivity, 94, 100 membrane-electrode assembly, 17, 83 methanol adsorption, 64, 70 methanol crossover, 87, 105, 182, 187 methanol crossover, heating due to, 228 methanol flux, 24 methanol mass balance, 25 methanol oxidation, 64 methanol stoichiometry, 174 methanol utilization, 183 microporous layer, 83 mirroring, 241, 249 mixed potential, 178 momentum transfer, 22 Nafion, 40 Navier-Stokes equation, 22 Nernst equation, 5 number of transferred electrons, 18, 75 Nusselt number, 201, 204, 229 occupied sites, 65 OCV, 2, 6, 7

INDEX Ohm’s law, 29, 34, 42, 238, 263 open-circuit conditions, 2 open-circuit voltage, 2, 4, 6, 7, 9 ORR, 10, 34, 40, 41 ORR kinetics, 10 ORR rate, 12, 156 overheat, 112 overpotential, 9, 15 oxygen flux, 18 oxygen mass balance, 25 oxygen mass conservation, 128 oxygen reduction reaction, 10, 40 oxygen stoichiometry, 18 oxygen transport in the CL, 42 oxygen utilization, 183 particle/electrolyte interface, 7 PEFC, 34 PEFC, design, 35 PEM fuel cell, 35 PEMFC, 35 plug flow, 23 Poiseuille flow, 22, 119 Poisson equation, 238, 241, 264 polarization curve, 86, 92, 127, 163 polarization voltage, 9, 41, 94, 127 polymer electrolyte fuel cell, 34 potential distribution in a fuel cell, 8 potentials in a fuel cell, 7 potentiostatic regime, 223 pressure gradient, 22 pressure of saturated water vapour, 228 proton concentration in Nafion, 10 proton current conservation equation, 41 proton current density, 41 Pt/electrolyte interface, 7 rate constant, 10 rate of electrochemical conversion, 27, 33, 49 rate of the ORR, 11 rate-determining step, 10 RDS, 10, 11 reaction penetration depth, 43, 163 reaction steps, 10 reference molar concentration, 12 reference oxygen concentration, 41 reverse reaction, 12 reversible (thermodynamic) heating, 75, 199 reversible heating, 228, 234 Reynolds number, 17

295 single-electron transfer, 10 SOFC, 37 SOFC, anode resistivity, 169 SOFC, design, 38 SOFC, half-cell reactions, 37 SOFC, instability increment, 221 SOFC, thermal instability, 219 SOFC, thermal wave, 220 solid oxide fuel cells, 37 speed of sound, 17, 121 stack cooling, 234 stack potential, equation, 243 standard conditions, 4 standard temperature, 6 Stefan-Maxwell diffusion, 27 Stefan-Maxwell equations, 26, 84 stoichiometry factor, 5, 18 stressing variable, 155 subsonic flow, 17 superficial exchange current density, 16 symmetry factor, 11 Tafel equation, 15, 40, 42, 88 Tafel slope, 13, 42, 95, 155 Tafel slope doubling, 49–51, 60, 63, 82, 171 thermodynamic reversibility, 4 total molar flux, 18 transfer coefficient, 11, 13, 42 transition region, 51, 52, 163 transport in membrane, 28 two-phase flow, 18, 23 utilization, 18, 183 velocity boundary layer, 17 viscous friction, 17 voltage loss, 7, 9 voltage-current characteristic, 16 volumetric exchange current density, 11, 16 water water water water

crossover, 128 electrolysis, 12 management, 35 transfer coefficient, 21, 23

YSZ, 37 yttria-stabilized zirconia, 37 Zundel ion, 29