application of fluid flows through porous media in fuel

16

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS L. MA*, D. B. INGHAMt and M. C. FGURKASHANIAN^ * Centre for Computational Ruid Dynamics, University of Leeds, Leeds, LSI 9JT, UK email: [email protected]

^Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK email: aint6dbi©maths. leeds .ac.uk * Energy Resources Research Institute, University of Leeds, Leeds, LS2 9JT, UK email: f ue6mtz@leeds .ac.uk

Abstract A fuel cell is a multicomponent power generating device which relies on chemistry, rather than combustion, to convert chemical energy into electricity. The key components of the fuel cell are made of porous materials through which fuel and oxidant are delivered to the active site of the cell where electrochemical reactions take place to generate power, heat and water Fuel cell technology presents a huge economical and environmental potential in the future power markets, from small portable cells to large residential power plants. However, at present, there are numerous technical barriers that prevent fuel cells from becoming commercially competitive and the fluid flow and reactant transport in the porous electrodes are major issues in the fuel cell design. This chapter aims at providing a general introduction to the fluid flows through the porous media in fuel cells with emphasis being placed on the numerical modelling of the convective and diffusive processes of the fluid flow, species transport, heat/mass transfer and the electrical potential. The challenges and the areas that need further investigations in the modelling of fuel cells are discussed where appropriate. Keywords: fuel cell, CFD model, SOFC, PEMFC, porous media, electrochemical reactions, catalysts, power generation

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L.MAETAL.

16.1

419

INTRODUCTION

Fuel cells offer a unique combination of high efficiency and ultra-low emissions for converting hydrocarbon fuels into electricity. At present, a conversion efficiency of about 26% may be achieved by an internal combustion engine and about 30% in a CHP system using traditional power generation techniques. However, fuel cells are expected to achieve an energy efficiency of over 40% and a remarkably higher efficiency of about 80% in a fuel cell turbine hybrid CHP system, see, for example, George (2000) and Veyo et ah (2002). In addition, fuel cells have virtually no emissions if pure hydrogen is used as the fuel. This is because the by-products of the fuel cells consist primarily of water and steam, see, for example, Hoogers (2003). Unfortunately, today's fuel cells are still too expensive for general commercial use and have many technique barriers to overcome before they could compete with the well developed traditional power system. In recent years, significant research and development in the fuel cell technology have been carried out in terms of reducing production cost and increasing power density of the fuel cell products in order to make them viable for commercial applications and thus benefit from their advantages. It is anticipated that with the further development and improvement in fuel cell materials and cell designs, fuel cells will be able to compete with combustion engines in transportation applications and for stationary power supply at locations off the electrical grid, even when they can also compete head-on with grid power with an advanced CHP hybrid system.

16.2 OPERATION PRINCIPLES OF FUEL CELLS The concept of the fuel cell was invented in 1839, see, for example, Singhal and Kendall (2003). The distinct difference from a combustion engine is that the operation of the fuel cell relies on the chemistry, instead of combustion, to combine fuel and oxygen to create electricity and therefore it is more efficient than the combustion engine. At present, there exist several different types of fuel cells and all have similar geometrical structures and share the same basic operating principles. A fuel cell typically consists of two electrodes, namely the anode and the cathode, and an electrolyte which is sandwiched in between the two electrodes, see the schematic diagram in Figure 16.1. The fuel and oxidant are fed at the side of the electrodes. The electrodes are made of porous materials so that the fuel and oxidant can penetrate to the location near to the surface of the electrolyte where the electrochemical reactions take place in the presence of the catalysts. The fuel, usually hydrogen or hydrogen rich hydrocarbons, transports through the porous gas diffusion layer to the catalyst layer on the anode side of the electrolyte where electrons are stripped from the fuel and form ions and electrons. In the case of a PEMFC, the electrons make their way from the anode to the cathode through an external circuit to drive a load and the remaining ions travel through the ionic conducting electrolyte to the cathode. At the same time, the oxygen is fed from the cathode gas channel, penetrates the cathode gas diffusion layer and reaches the catalyst layer in the cathode side next to the electrolyte. The oxygen combines with the emerging hydrogen

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APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

Gas channel

Anode

Electrolyte

Cathode

Gas channel

Figure 16.1 A schematic diagram of a hydrogen-oxygen fuel cell.

ions and the free electrons coming form the anode to from the by-products, typically heat and water. Figure 16.2 shows an electron micrograph of the cross-section of an SOFC developed by Siemens Westinghouse, see Ormerod (2003), where the distinguished structures of the electrodes and the electrolyte can be clearly observed. The SOFC uses an oxygen ion conducting electrolyte which works at a relatively high temperature. Therefore, in the case of the SOFC, it is the oxygen ions in the cathode side that is conducted to the anode side and the main reduction reaction occurs at the anode side to form water. Although detailed chemistry may vary in different types of fuel cells, see, for example, Larminie and Andrew (2000), the fundamental overall reaction is the same, i.e. the

Fuel electrode ^ f

J^ . .

C , ^ /

^.^N*^*-^'J Electrolyte

Air electrode Figure 16.2 An electron micrograph of the cross-section of an SOFC developed by Siemens Westinghouse, see Ormerod (2003).

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L.MAETAL.

exothermic hydrogen-oxygen reaction, and it may be expressed as follows: (16.1)

2H2 + 0 2 - ^ 2 H 2 0 + A E ,

where AE" denotes the heat produced in the reaction. The maximum electrochemical work that a fuel cell can do may be determined using the Nemst equation as follows:

'^=^-5'° n^r

(16.2)

where E is the electric potential that is produced by the electrochemical reaction, E^ is the potential under a standard reference pressure, usually p^ = 1 atm, and may be calculated from the changes in the free Gibbs energy of the reaction, R is the gas constant, T is the temperature, pk is the partial pressure of the reactant k in the mixture, m is a constant related to the stoichiometric coefficient of the species, Ue is the number of electrical charges (electrons or protons) transferred in the reaction and F is the charge carried by a mole of electrons (or protons), known as Faraday's constant. The Nemst equation (16.2) represents the ideal maximum voltage potential that can be achieved in a fuel cell. However, in reality, there are certain irreversible losses that reduce the magnitude of the maximum potential that a fuel cell can produce. Typically, there are three major losses in the system, namely the activation losses due to the kinetic resistance, the ohmic losses due to the material ohmic resistance and the concentration losses due to the limitations of the species diffusion through the porous electrodes, see Figure 16.3. Thus the voltage produced by a fuel cell is always lower than the ideal voltage predicted

1.25 • Open-circuit losses

Irreversible ideal voltage

Porosity increase

Concentration losses

0.25

0.25

Figure 16.3

0.5 0.75 Current density [A/cm^j

1.25

Schematic diagram of the potential polarisation in a fuel cell.

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by the Nemst equation and it is typically expressed as follows: Kell — E - r/act - ôhmic " ^con ,

(16.3)

where Keii is the cell voltage, E is the reversible cell potential given by the Nemst equation, ôhmic is the ohmic overpotential, r/act is the kinetic overpotential at the anode and the cathode, and r/con denotes the concentrate overpotential. The kinetic or the activation losses are the major losses when the fuel cell operates at a low current density whilst the other two losses are small. For a well-designed fuel cell working at the designed operational condition, additional losses in cell potential are mainly due to the ohmic resistance of electrical conducting material and it is proportional to the current density in the fuel cell. However, at high current density, when the speed of the consumption of the reactants is faster than the speed of the delivery of the fuel and/or oxidant by the system of species diffusion, then the limiting effect of the concentration of the reactants on the performance of the fuel cell will appear. It is clear that the performance of the fuel cell ultimately relies on the effective delivery of the fuel and oxidant through the porous electrodes to the active catalytic reacting site where the electrochemical reactions take place to produce electricity, heat and the by-product, water. If a fuel cell is poorly designed, in particular if when the reactants could not be efficiently delivered to the reacting site then the speed of the diffusion process, at which the fuel and oxygen species are transported through the diffusion layer to the reaction surface of the catalyst layer, will control the electrochemical processes. Figure 16.3 schematically shows the effect of the species transport in the electrodes on the performance of the fuel cell. Usually, the limitations of the reactant transport to the cell power output are not noticeable until a large operating current density occurs. However, if the permeability of the porous media of the electrodes is too small, or the pores of the electrodes are blocked by water, then an insufficient supply of oxidant to the catalyst site will appear. This results in an earlier deterioration of the performance of the fuel cell. In the worst situation, if no reactant is able to reach the active site, such as when the pores of the electrodes are flooded by the presence of an excessive amount of water, then the fuel cell will lose its function. Therefore, the investigations on the fluid flows and heat and mass transfer in the porous electrodes are of vital importance in the design and the optimization of fuel cells. There are a number of distinctive differences in the nature of the fluid flow in the porous electrodes of the fuel cells over the traditional fluid flow through porous materials, such as the fluid flows through packed bed materials or in underground oil/water flows. Firstly, the length scales of the porous media in the fuel cells are extremely small and they vary considerably within different components of the fuel cell. Secondly, due to the nature of the manufacturing process and the material that is used to make up the fuel cell, the porous media used in some fuel cells, such as the PEMFC, shows significant anisotropy and/or spatial inhomogeneity. This significantly influences the behaviour of thefluidflow in the porous electrodes. Thirdly, species transport in fuel cells typically involves more than two species and therefore there exists a multicomponent species transport where the concentration of one species may significantly influence the transport of the others and this only adds to the complexity of the species transport process.

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16.3 GOVERNING EQUATIONS FOR THE FLUID FLOWS IN POROUS ELECTRODES From a mathematical modelling point of view, the fluid flow and the heat and mass transfer in the fuel cell can be described by the typical conservation equations for the thermal reacting fluid flows, such as the Navier-Stokes, energy, mass conservation and the associated chemical reaction equations. For the electric current flow that is generated in the fuel cell, the conservation equation of charge may be employed and it governs the current density distributions and the overall potentials in the fuel cell. All the transport processes occurring in fuel cells are ultimately determined by the operating conditions of the fuel cell, i.e. the electric current that goes through the fuel cell and this is usually defined as a boundary condition to the governing set of the transport equation for the fuel cells. 16.3.1 Equations for the fluid flow and mass transfer in fuel ceUs Mathematically, the fluid flow in the porous electrodes may be expressed by the well known Navier-Stokes and the mass conservation equations. In writing these equations, as well as doing mathematical calculations, two representative fluid velocities are in use, namely the superficial fluid velocity, or the Darcy averaged velocity, and the physical fluid velocity, or the real velocity. The superficial fluid velocity is defined as if there is no solid matrix present. Thus the relation between the superficial fluid velocity and the physical fluid velocity may be given as follows: ^superficial — T^physical j

(lo.4)

where 7 is the ratio of the void area to the whole cross-sectional area. In a homogenous matrix, 7 is equal to the porosity of the matrix. Employing different representative velocities produces slight differences in the form of the equations. In this chapter, the superficial fluid velocity is employed. Written in tensor notation in the Cartesian coordinate system, the Navier-Stokes and the continuity equations for the fluid flow through an isotropic porous medium may be expressed as follows: 1 djpu') e dt

1 djpu'u^) _ e^ dx^

dp dx^

d dx^

7 \dx^^'^^)

+ Su,

(16.5)

where x^ is the component of the Cartesian coordinate system, t is the time, u\p, p and p are the fluid velocity components, pressure, density and molecular viscosity of the fluid flow, respectively, and e is the porosity of matrix. These equations are also valid for the fluid flows through gas channels of the fuel cells where we have e == 1.

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The first term on the left-hand side of equation (16.5) represents the rate of change of momentum per unit volume with time and the second term represents the change of momentum resulting from the convective motion. On the right-hand side of equation (16.5), the first term is the force resulting from the pressure differences in the fluid flow and the second term is the viscous shear forces resulting from the motion of the fluid. Further, Su denotes the extra resistance force on the fluid flow due to the presence of the porous matrix. This momentum sink contributes to the pressure gradient in the porous media, creating a pressure drop that is usually proportional to the fluid velocity in the application of the fuel cells. The source Sm on the right-hand side of equation (16.6) is the mass added to the continuous phase from, for example, the source of the chemical species, or the change in mass due to the condensation/vaporization of a second phase such as water. Depending on the strategies employed in a mathematical model to connect different regions of thefluidflowsin a fuel cell, Sm may be used to reflect the effect of the electrochemical reactions and/or mass transfer across the boundaries of each component. Since the porosity and the permeability of the porous part of the electrodes, such as the gas diffusion and the catalyst layers, are usually very small, this produces a very large resistance to the fluid flow and results in an extremely low fluid velocity. Thus the convective acceleration and the diffusion terms appearing in the Navier-Stokes equation are relatively small and thus can often be ignored. Then the momentum equation (16.5) for the fluid flow through the porous matrix of the electrodes reduces to the Darcy law as follows:

g = -5„^-|uS

(16.7)

where K is the permeability of the matrix. Whilst the Navier-Stokes equation and Darcy's equation are primarily responsible for the convective transport of the reactants, the diffusion processes, which are significant in the transport of chemicals in a gradient species concentration field in fuel cells, are modelled by the species transport equation. The transport of reacting and non-reacting species throughout the fuel cell follows the conservation law of chemical species which can be written as follows:

djpYk) Idjpu'Yk) d f dYk\ -dr^-e—d^ = dî[f^''-d^)+^'^

fc

= l,2,...,Ar-l,

(16.8)

where Yk and Sk denote the mass fraction and the production/consumption of the species k, respectively, Dk is the diffusivity of the species k, and A^ denotes the total number of chemical species present in the system and is dependent on the types of the fuel and oxidant that feed the fuel cell. For a simple hydrogen-oxygen fuel cell, the species in the system typically include as least hydrogen, oxygen and water, and in this situation N = S. The production/consumption of the fuel and oxidant due to electrochemical reactions are represented in the species and mass conservation equations by the source/sink terms. These terms are proportional to the electronic current that is produced by the electrochemical reaction. For hydrogen, oxygen and water species the source/sink terms may be expressed.

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in a volumetric basis, respectively, as follows:

2

Q

J

*-^H2 — ~7^2F 5

C

'^02 — ~T^

•'

C

'

*^H20 —

2F

(16.9)

where j is known as the electronic charge transfer current density representing the charges generated by the electrochemical reactions. The electrochemical charge transfer process in an electrode is an activation-controlled process and the charge transfer current density it generates is determined by the effectiveness of the catalysts and the concentrations of the reactants that are available at the reaction sites. The kinetics of the catalytic electrochemical reaction in the fuel cell is usually described by the Butler-Volmer equation. Taking into consideration the effect of the concentrate polarisations, the Butler-Volmer equation for the anode and the cathode of a hydrogen-oxygen fuel cell may be expressed, respectively, as follows, see, for example. Bard and Faulkner (1980) and Beming et ai (2002): . _ .ref / £Hi

Ja — Jo,a I êf

. _.reff£0^'

Jc - Jo,c I ref

1/2

exp exp

agF âct,a RT

agF RT âct,c

exp

- exp

OcF - RT ^ a c t , i

(16.10) (16.11)

where c denotes the concentration of the species at the reaction site, a is known as the anodic transfer coefficient (0 < a < 1) and it usually takes the value of 0.5,77act is the active overpotential representing the voltage loss due to electrochemical reaction, and the subscripts a and c denote the anode and cathode of the fuel cell, respectively. The value of the reference current density JQ^^ is catalyst material and operational-condition dependent and it usually has to be obtained experimentally. In order to calculate the transfer current density produced in an electrochemical reaction, the magnitude of the active overpotential of the reaction must be known which varies with the catalyst that is used and the operational condition of the fuel cell. At the moment, information on the overpotential for a particular catalyst has to be obtained experimentally. However, it can be estimated from the data of the open circuit voltage of the fuel cell as follows, for the anode and the cathode, respectively, see, for example, Meng and Wang (2004): (16.12) âct,a = ^ e - ^ i âct,c = ^ e - îo K, where $e and $ion are the electrical potential and the ionic potential, respectively, at the interface between the solid catalyst layer and the membrane, Foe is the cell open circuit potential that has to be obtained experimentally. 16.3.2 Heat generation and transfer in fuel cells Heat generation, transfer and balance are important issues in both low and high temperature fuel cells. A low temperature PEMFC could not withstand a high temperature as the

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polymer electrolyte relies on liquid water to keep it active. Heat balancing in high temperature SOFC becomes even crucial as severe nonuniform or sudden changes in the temperature may cause crack or even a structure failure to the whole fuel cell system. The heat transfer in the fuel cell is a combination of the heat transfer through the fluid flow and the heat transfer through the solid matrix and it is governed by the energy conservation equation which may take the following form:

(16.13) where p, Cp and k are the density, specific heat and the heat conductivity of the fluid, respectively, ps, Cp^s and kg are the density, specific heat and the heat conductivity of the solid matrix, respectively, and SE is the energy source that is generated within the fuel cell. In the gas channel, the convective heat transfer will be the dominate heat transfer process. However, in the solid electrical conducting elements, such as the solid current collector, the conduction is the sole heat transfer process. In the porous electrodes and/or membrane, both conduction and convection will be important. Heat generations in a fuel cell may come from a number of mechanisms. Most of the heat comes from the exothermal chemical reactions, which combine hydrogen and oxygen to form water, and the electrochemical activation (kinetic) losses that split species into electrons and ions in the catalyst layer. These heat sources are also major sources of the non-uniformity of the temperature distribution across the cell. In addition, the ohmic losses in the electronic conducting elements, due to ohmic resistance of the solid materials, also produce a significant amount of heat, particularly when the cell is operating at a high current density. Clearly, all these heat generation mechanisms are linked to the current density of the fuel cell in operation. The heat generation due to the chemical reactions may be estimated by the change in the entropy, A 5 , of the system and the current density as follows, see, for example, Lampinen and Fomino (1993) and Beming et al. (2002):

The heat source due to the activation losses is given by Ssict = m c t ,

(16.15)

where a^^ is the effective electronic conductivity of the cell, and the ohmic losses are directly proportional to the current density and are given by Soh-^,

(16.16)

where i is the local current density. In the situation where a phase change occurs, such as those that exist in low temperature fuel cells where water may experience condensation and/or evaporation, then additional

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heat source/sink terms may be required to represent the heat release and absorption during the process of phase changing. 16.3.3 The electric field in fuel cells The ions and electrons generated during the electrochemical reactions in the fuel cell produce an ionic potential field within the ionic conducting elements and an electrical potential field within the electronic conducting elements. The transport of the ions and electrons within the fuel cell are governed by the conservative law of ions and charges. Thus the ionic potential, $ion, and the electric potential, $e»should follow the conservation equations which may be written in the form, respectively, as follows: _d_

dx^

' dx^

+ S