Towards a model concept for coupling porous gas diffusion layer and gas distributor in PEM fuel cells

Transcription

1 Universität Stuttgart - Institut für Wasserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung Prof. Dr.-Ing. Rainer Helmig Diplomarbeit Towards a model concept for coupling porous gas diffusion layer and gas distributor in PEM fuel cells Submitted by Lena Walter Matrikelnummer Stuttgart, December 10th, 2008 Examiners: Prof. Dr.-Ing. Rainer Helmig, Dr.-Ing. Holger Class Supervisor: Dr.-Ing. Holger Class, Dr.-Ing. Steffen Ochs

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3 Abstract The water management in a polymer electrolyte membrane fuel cell (PEMFC) has an outstanding importance for the performance of the fuel cell. In this study, the influence of the conditions in the gas channel on the water distribution in the gas diffusion layer and on the performance of the fuel cell is investigated. Additionally, the impact of the chosen capillary pressure-saturation equation is tested. For an adequate consideration of the transport processes in the porous gas diffusion layer, a 3D, two-phase, three component model with a conventional gas distributor is developed. Different coupling approaches for the porous gas diffusion layer with the flow channel are discussed and the implementation of Darcy s law in both domains is chosen for the simulation. The equations for the electrochemical reactions are added to this model and the limiting influence of the water saturation on the current density is additionally considered. Variations of the parameters as the pressure gradient, the permeability or the water vapour content in the gas channel are performed and show a great influence on the water distribution in the gas diffusion layer and on the performance of the fuel cell. Simulations with different capillary pressure-saturation relations for the hydrophobic gas diffusion layer illustrate a high influence of the choice of capillary pressure on the water distribution. An extreme coupling between capillary pressure, saturation and current density is demonstrated. Summing up, the presented model can provide good information how the conditions of the gas channel have to be chosen to improve the performance in addition to this it demonstrates that reliable data for the capillary pressure of the hydrophobic gas diffusion layer is essential to model the transport processes in an exact way. The model also presents a basis for further coupling approaches.

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5 Acknowledgement We thank the German Research Foundation (DFG) for the funding within the International Research Training Group Non-Linearities and Upscaling in Porous Media (NUPUS).

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7 Contents 1 Motivation Aims of this thesis Fundamentals of a PEMFC The operating mode of the PEMFC The assembly of the PEMFC The gas distributor The gas diffusion layer The reaction (catalyst) layer The polymer electrolyte membrane Thermodynamic and electrochemical fundamentals of the fuel cell The fuel cell under open loop conditions The fuel cell under load Coupling theories for flow in combined free and porous domains Beavers and Joseph slip condition Brinkman equation Origin of the Brinkman equation Modification of Brinkman equation by Rosenzweig Coupling Brinkman equation with Stokes equation Darcy-equation in both domains Modelling of fuel cells State of the art of common PEMFC models Existing models for coupling gas diffusion layer and gas distributor The simulation model Fundamentals of multiphase flow New parameters on the macroscale Capillarity Conceptual model for the porous layers in the PEMFC Mathematical model Domain description and mesh generation Model equation for the multiphase flow Model equations for the electrochemical reaction Primary variables and boundary conditions V

8 VI Contents 5.4 Numerical model The numerical simulator MUFTE-UG Discretisation method D-Simulation with the Darcy approach Parameter definitions and assumptions General assumptions Parameter for the porous gas diffusion layer and reaction layer Assumptions and parameters for the gas channel The transport processes Variation of different parameters Effect of different pressure gradients Effect of different permeability in the gas channel Effect of different water vapour mole fraction in the gas channel Effect of different capillary pressure Discussion Validation of the Darcy approach Robustness of the simulation Liquid water in the channel Drying-out of the membrane Influence of the water saturation on the current density Summary and Outlook 78

9 List of Figures 2.1 Basic assembly and principle of operation of a PEMFC [You and Liu(2005)] Different gas distributors [Friedrich(2007)] The three phase zone in the reaction layer Theoretical efficiency of Carnot engine and Fuel Cell Polarisation curve Velocity distribution in a horizontal channel coupled with porous media Dominating terms (red) of the Brinkman equation in each region REV at the interface between free flow and porous domain Laminar flow through a small channel k r S w relation after Brooks and Corey [Helmig(1997)] Contact angle for a wetting and a nonwetting fluid p c S w relation after Brooks and Corey [Helmig(1997)] The two phase, three component, non-isothermal model concept Water as nonweeting fluid (left side) and water as wetting fluid (right side) Front and back side of the model domain Modelling domain: Two half channels and the bar with a length in z-direction of 5mm Generated mesh in 3D Boundary conditions, shown from the front (or back) side of the domain Primary FEM and secondary FVM mesh Illustration of the implementation of sink and source terms Distribution of the oxygen mole fraction x o g [-] in the 3D domain Distribution of the gas pressure and the oxygen mole fraction in the reference scenario at y= m and assumed steady state Distribution of different parameters in the reference scenario at z=0.0025m and assumed steady state Gradients and transport mechanism in the reaction layer (RL) and the GDL Comparison of the water mass in the system for different p Comparison of the saturation distribution of simulations with different pressure gradient, slice at z=0.025m (assumed steady state) Comparison of the current density of simulations with different pressure gradient, slice at z=0.025m (assumed steady state) VII

10 VIII List of Figures 6.8 Comparison of the oxygen mole fraction distribution of simulations with different pressure gradient, slice at z=0.025m (assumed steady state) Comparison of the water mass in the system Comparison of the saturation distribution of simulations with different permeability in the channel, slice at z=0.025m (assumed steady state) Comparison of the current density of simulations with different permeability in the channel, slice at z=0.025m (assumed steady state) Comparison of the water mass in the system for different mole fraction in the channel Comparison of the saturation of simulations with different water vapour mole fraction in the channel, slice at z=0.025m (assumed steady state) Comparison of the current density of simulations with different water vapour mole fraction in the channel, slice at z=0.025m (assumed steady state) Comparison of the water mass in the system for different capillary pressure Comparison of the saturation distribution of simulations with different capillary pressure, slice at z=0.0025m (assumed steady state) Comparison of the saturation distribution of simulations with higher capillary pressure, slice at z=0.0025m (assumed steady state) Comparison of the current density of simulations with different capillary pressure, slice at z=0.0025m (assumed steady state) Dependency of the different parameters Comparison of the capillary pressure distribution of simulations with different capillary pressure, slice at z=0.0025m (assumed steady state) Capillary pressure saturation relations of the different scenarios Additional scenario with p c = p Acosta c /1000 and x w,inlet g = Saturation distribution of two scenarios with p c = p Acosta c /

11 List of Tables 2.1 Thermodynamic data for the H 2 /O 2 fuel cell for p=1bar and T = 25 C [Wöhr(1998)] Dimensions of the different layers in (x)-, (y)- and (z)-direction Parameters of the reaction layer and gas diffusion layer Parameters for the electrochemical reaction Parameters of the gas channel for the reference scenario Operational parameter for the reference scenario Varied pressure gradients in the different scenarios Saturation and current density in the middle of the reaction layer at (0/0.001/0.0025) Varied permeability in the different scenarios Saturation and current density in the middle of the reaction layer at (0,0.001,0.0025) Varied mole fractions at the inlet of the gas channel in the different scenarios Saturation and current density in the middle of the reaction layer at (0,0.001,0.0025) Varied capillary pressure in the different scenarios Dynamic and kinematic viscosity of air at 2 bar and 70 C Velocities and Reynold s numbers for the different permeabilities and pressure gradients IX

12 Nomenclature symbol meaning dimension F Faraday s constant = [C/mol] G change of Gibb s free energy [J/mol] H change of enthalpy [J/mol] H rev height of REV (Rosenzweig and Shavit) [m] Q change of heat [J/mol] S change of entropy [J/mol] U change of internal energy [J/mol] G # 0 free activation enthalpy [J/mol] Deff c effective diffusion coefficient [m 2 /s] D c binary diffusion coefficient of component c [m 2 /s] JD c diffusive flux [mol/sm 2 ] K intrinsic permeability [m 2 ] K f hydraulic conductivity [m/s] Q seepage velocity [m/s] R specific gas constant = [J/molK] R spec specific resistance [Ωcm 2 ] S α saturation of phase α [ ] S wr residual saturation of water [ ] T temperature [K] T 0 reference temperature [K] T ref reference temperature [K] U 0 open circuit voltage [V ] U rev reversible voltage [V ] U th thermal voltage [V ] U cell cell voltage [V ] W el electrical work [J/mol] b channel width [m] c ox concentration of the oxidised species [mol/m 3 ] c s specific heat capacity of the soil grain [J/kgK] d diameter of the capillary [m] f v catalyst surface increasing factor [ ] g gravity [m/s 2 ] X

13 Nomenclature XI h height of the water surface [m] h α specitfic enthalpy of phase α [kj/kg] i current density [A/cm 2 ] i 0 exchange current density [A/cm 2 ] i ref 0 reference exchange current density [A/cm 2 ] i 0 current density at a special temperature [A/cm 2 ] k rα relative permeability of phase α [m 2 ] p pressure [P a] p c capillary pressure [P a] p np pressure of the nonwetting fluid [P a] p wp pressure of the wetting fluid [P a] q c source/sink term [mol/sm 2 ] q h energy source/sink term [J/sm 2 ] p O2 current oxygen pressure [P a] p inlet O 2 reference inlet pressure of oxygen [P a] p ref O 2 oxygen reference pressure [P a] t time [s] t H2 O water transport number [ ] u velocity (Beaver & Joseph) [m/s] u B slip velocity (Beaver & Joseph) [m/s] u f averaged intrinsic velocity (Rosenzweig and Shavit) [m/s] v seepage velocity/darcy velocity [m/s] x c α mole fraction of component c in phase α [ ] z amount of transferred electrons [ ] α load transfer number [ ] α BJ dimensionless value of Beavers & Joseph [ ] β coefficient of Beavers & Joseph [1/m] γ contact angle [ ] η th thermodynamic maximum efficiency [V ] η act activation overvoltage [V ] η C concentration overvoltage [V ] η equ equilibrium overvoltage [V ] η ohmic ohmic overvoltage [V ] λ pm equivalent heat conductivity of the soil [W/Km] µ dynamic viscosity [kg/ms] µ effective viscosity (Brinkman) [kg/ms] ν kinematic viscosity [m 2 /s] νi s stoichiometric coefficient [-] Φ BJ fractional increase in mass flow (Beaver & Joseph) [ ] φ i porosity [ ] ϱ mass density [kg/m 3 ] ϱ s the soil grain density [kg/m 3 ] ϱ mol molar density [mol/m 3 ]

14 XII Nomenclature σ surface tension [N] σ BJ coefficient of Beavers & Joseph [ ] τ tortuosity [ ] subscript meaning 0 reference condition α phase g gas phase w water phase ch channel GDL gas diffusion layer Hg mercury M membrane R reaction zone superscript meaning c n o w inlet ref component component nitrogen component oxygen component water inlet condition reference conditon

15 1 Motivation Nowadays, the development of new alternative, more effective power sources becomes more and more important, since a further increase of the world wide energy demand is still expected in the next years. The efforts of the industrial countries to reduce the energy demand won t be sufficient and the emerging and developing countries are growing rapidly and their energy demand will rise extremely. Today s energy supply is mainly based on fossil fuels, which are only available in a limited amount. The combustion of fossil fuels releases much CO 2 emissions to the atmosphere, which were originally kept in the fossil fuels under the earth. CO 2 is one of the most important contributors to global warming. By accumulation in the atmosphere, CO 2 enforces the greenhouse effect. Today s research reports promise an increase of the world wide mean temperature in the next years with an extreme impact for the environment. If the CO 2 emissions aren t reduced in the next years the temperature will rise more and more. Consequently, the decrease of the CO 2 emissions is a very important task nowadays. One way to comply with the increasing energy demand and the reduction of the CO 2 emissions is to develop more efficient power generators, like fuel cells. Fuel cells can be used in many different applications e.g. in cars or mobile phones or even as combined heat and power systems. The principle of the fuel cell was already discovered in 1839 by Sir William Grove. In the following years some sporadic investigations have been made, but significant applications weren t developed. In 1937, Francis T. Bacon began to develop practical applications. But the first real application was developed for a space program not until the late 1950s. The first PEMFC was invented by General Electric in 1955 for NASA s Gemini space program. Several space programs followed but investigations of fuel cells for other applications were forgotten until the 1990s. In 1989, the first PEMFC powered submarine was presented by Perry Energy Systems and the first passenger car powered by PEMFC was developed in Nowadays, almost all car manufactures have already presented a prototype of a fuel cell car and there are great investigations in advancing this technology. But a great potential for improvement of the fuel cell technology is still given. For a more detailed description of the history see [Barbir(2005)]. Fuel cells are good alternatives to combustion engines due to the high efficiency combined with low emissions. The efficiency of a fuel cell is much higher than the one of a combustion engine because it isn t limited by the Carnot-efficiency. In a fuel cell, the chemical energy is directly transformed in electrical energy. The step of transferring the chemical energy first in thermal energy and then in electrical energy, like it is done in a combustion engine, is omitted. The earlier mentioned advantage of lower emissions is an important fact nowadays. The polymer electrolyte membrane fuel cell (PEMFC), which is considered in this thesis, can be used with pure hydrogen. Then the only emerging product is water. So, this fuel cell has 1

16 2 1 Motivation zero emissions. The only emissions, which have to be taken into account, are the emissions (e.g. CO 2 ) produced while generating the hydrogen. To reduce these CO 2 emissions, there have been thoughts about using solar energy for the hydrogen production. The fact that the PEMFC emits only water would be a good alternative in cars to avoid the emissions in the cities in combination with a higher efficiency. Additionally, the PEMFC also fulfills other requirements for automobiles, as a low operating temperature, a low weight and a good cold start (see e.g. [Barbir(2005)]). Another advantage is the simplicity of fuel cells. The essentials for the construction are relatively simply due to only few moving parts. Therefore, the system can last a long time but the life time of the used material as the membrane and the gas diffusion layer have to be considered. Additionally to mention is that the fuel cell works very quiet in comparison to a combustion engine. The biggest disadvantage of a fuel cell is the high costs. The used material in a PEMFC is very expensive, especially the platinum catalyst. One way to lower the costs is to improve the performance of the fuel cell. The performance is highly dependent on the water management in the fuel cell. At the cathode side, water emerges in the reaction zone during the reaction. It has to be transported through the gas diffusion layer to the gas channel. If it isn t removed properly it can cover the reaction sites or it can block the pores in the porous gas diffusion layer and the reaction layer. This can cause an inhibition of the reactant supply through the gas diffusion layer and the reaction can be limited. Otherwise, water is needed for the transportation of the protons through the membrane. The risk of membrane drying out, mainly on the anode side but also on the cathode side can occur. On the cathode side the risk isn t so high, since water is produced but it has to be assured that not too much water is removed. Therefore, the parts as the membrane, the gas diffusion layer and the reaction layer have to be constructed carefully and the conditions in the gas channel have to be well specified. The investigation of the material used in fuel cells by experimental studies is difficult, due to the small geometry and the expensive material. Therefore, numerical models can help to improve the material and the performance of the fuel cell. With the help of numerical models, relations between the different properties can be found out or the conditions in the gas channel to improve the performance can easily be tested. 1.1 Aims of this thesis The aim of this thesis is to develop a numerical model to investigate the water management in the gas diffusion layer of the cathode and how it influences the performance of the PEMFC. One main focus will be the impact of the flow in the gas channel on the water management. In this thesis an existing two-dimensional model of the gas diffusion layer of the cathode will be expanded to a three-dimensional model including the gas distributor. An adequate method to couple the gas diffusion layer with the gas channel will be identified and discussed. A multiphase, multicomponent approach will be used in the model to correctly describe the transport processes in the porous gas diffusion layer. It is important to consider a liquid-gas two phase flow in the porous gas diffusion layer, since it affects the amount of water in the reaction layer and the membrane and the reactant supply from the gas diffusion layer to the reaction layer. The equations of the electrochemical reaction will be added to this approach. Here, the limiting effect of the emerging water on the current density by the

17 1.1 Aims of this thesis 3 covering effect of the reaction sites will be implemented. Then, the model will be used to find out how different parameters like pressure gradient, permeability and the mole fraction of water vapour in the gas channel influence the water distribution in the gas diffusion layer and the reaction layer and how the named parameters affect the performance of the fuel cell. It will also be investigated how the chosen capillary pressure-saturation relation for the hydrophobic gas diffusion layer influences the water transport and the performance. The issues of flooding of the channel and membrane drying out will also be discussed.

18 2 Fundamentals of a PEMFC In this chapter, the basic fundamentals of a polymer electrolyte membrane fuel cell (PEMFC) are introduced. First, it is explained how a PEM fuel cell works, then the basic components and their functions are described. In the third part, the thermodynamic and electrochemical fundamentals are pointed out. 2.1 The operating mode of the PEMFC Fuel cells convert chemical energy directly in electrical energy. The step of converting the chemical energy in thermal energy as it is done in heat engines is omitted. Therefore power generation in fuel cells isn t limited by the Carnot efficiency. In polymer electrolyte membrane fuel cells (PEMFC) the H 2 /O 2 reaction is separated in two part-reactions, one at the anode and one at the cathode, by a polymer-electrolyte membrane. This membrane is impermeable for gases, only protons and H 2 O can pass the membrane. The following reactions take place in the PEM fuel cell: anode: H 2 2H + + 2e 1 cathode: O H + + 2e H 2 O total reaction: H 2 + 1O 2 2 H 2 O (2.1) H 2 gas is supplied at the anode side and is transported through the gas diffusion layer to the catalyst layer. Here, H 2 reacts to 2H + and 2e at the catalyst. Emerging protons (H + ) move through the membrane and electrons are supplied across the extern electric circuit to the cathode side. Pure oxygen or air is fed on the cathode side and the oxygen, which is transmitted through the gas diffusion layer to the catalyst layer, reacts with the protons and the electrons to water. Figure 2.1 clarifies the principle of the operation. The electrical current flow occurs due to driving forces because of different Galvani-potentials of both partreactions. Thus, a part of the reaction enthalpy ( H) is directly transformed into electrical energy and can be used in the external electric circuit. The other part is converted and lost in heat. 2.2 The assembly of the PEMFC The technical assembly of a PEMFC consists of a polymer membrane situated between two electrodes, the anode and the cathode. This unit is called membrane-electrode assembly (MEA). Both electrodes consist of a gas diffusion layer (GDL) and a reaction layer, which is also called catalyst layer. Again, the MEA is situated between two plates, the gas distributors. Each component of the cell has to achieve different requirements in order to assure 4

19 2.2 The assembly of the PEMFC 5 Figure 2.1: Basic assembly and principle of operation of a PEMFC [You and Liu(2005)] an effective operation of the fuel cell. The necessary properties and interactions of each component to provide a reliable and effective operation are explained in this section The gas distributor Beside the mechanical load transmission, the gas distributor has to assure a consistent distribution of reaction gases over the area of the gas diffusion layer and a good electrical current contact to the reaction layer. Thus, the used material has to provide a good electrical conductivity. Additionally, the contact resistance between gas distributor and gas diffusion layer should be as small as possible. If corrosion occurs at the distributor the resistance will strongly increase. Due to this the material used has to be corrosion-resistant. To guarantee a consistent distribution, the inside is provided with a special structure, which has to combine the consistent distribution with a low pressure drop. Figure 2.2 shows different types of these structures. The straight distributor has a low pressure drop but the distribution of gases is limited by condensed water, which may block several channels. This means that whole areas of the gas diffusion layer can be separated from the gas supply. A better distribution provides the meander by avoiding a blocking, because the stream is forced to follow the meanders. Therefore, a high pressure drop, which results in a high energy demand, isn t preventable because of the necessary high velocities due to the large channel length. Isotropic distributors, which don t channel gases into a specific direction, can have a low pressure drop, depending on the ratio of open volume to the solid [Wieser(2001)]. Additionally, the bars between the channels of the gas distributor have to fulfill the task of cooling the fuel cell. Consequently, the bars have to be as wide as possible for good cooling effects. A fuel cell stack consists of several cells, which are connected by bipolar plates. These plates have to separate reliably the reaction zones of two cells from each other. They have a flow structure at both sides to provide good gas distribution and have the same requirements as gas distributor plates.

20 6 2 Fundamentals of a PEMFC Figure 2.2: Different gas distributors [Friedrich(2007)] The gas diffusion layer The porous gas diffusion layer is situated between gas distributor and reaction layer. The tasks of this layer are to provide an efficient distribution and transport of reaction gases and emerging water between reaction zone and gas distributor, to assure a high electrical conductivity between reaction zone and gas distributor and to provide mechanical stability. To fulfill the first task of a good gas distribution, the material needs a high porosity. Thus, the gas can also be distributed beneath the bars of the gas distributor. But a high porosity compromises a good electrical conductivity. Due to the opposite requirements of these two tasks, a material has to be found, which has such a porosity and such properties that both tasks can be performed satisfactorily. Today, gas diffusion layers of carbon fibre cloth or carbon paper are a good compromise and they are established in most applications due to the fact that these materials provide a high electrical conductivity having a high porosity. Both materials also possess a good mechanical stability. Carbon paper is used when the cell has to be as small as possible in design. In contrast, carbon fibre cloth is thicker, absorbs more water and simplifies the design of the cell [Larminie and Dicks(2006)]. A problem poses the removal of emerging water at the cathode side. By avoiding the blocking of the pores through flooding by water, the diffusion layer is made water-repellent with polytetrafluorethylene (PTFE). Blocking of the pores can cause an inhibition of the transport of reaction gases through the layer to the reaction zone. The hydrophobic property has to be adjusted in such way that on the one hand a flooding is avoided but on the other hand the membrane doesn t run dry and the electrical conductivity is still met. After [Thoben(2006)], a PTFE-content of 10% to 30% shows the best performance. All properties of the gas diffusion layer are also strongly dependent on the interface conditions between gas distributor and gas diffusion layer and reaction layer and gas diffusion layer.

21 2.2 The assembly of the PEMFC 7 Therefore, the construction of the components has to be optimised as a whole. E.g., if the fibre structure of the cloth is exposed to surface pressure, it can swell and block the channels of the gas distributor [Wieser(2001)]. [Lee et al.(1999)] compare three different materials for gas diffusion layers and determine the behaviour by varying the compression. It is shown that this parameter highly affects the performance of fuel cells The reaction (catalyst) layer The reaction layers border directly on both sides of the membrane and consist of a threedimensional, porous structure. Here, the real reaction takes place. In the reaction layer of a PEMFC, platinum is typically used as a catalyst, which is supported on larger carbon particles. Platinum is the most suitable catalyst for both reactions, the oxidation of hydrogen and the reduction of oxygen [Wieser(2001)]. In this layer, the transport of the reaction gases to the active centres has to be assured. So, the emerging water has to be removed, in order to avoid a blocking of the pores. Therefore, PTFE is added in the reaction layer as it is done in the diffusion layer to provide a hydrophobic character. Since reaction gases, electrons, protons and catalyst are required, reactions only take place in the three phase zone, where gas transport can occur and the contact to reaction layer and membrane is assured. Thus, the three phase zone has to contact void space (the pores) of the porous media to assure that the reaction gases can reach the zone. Electrons travel through electrically conductive solid in the reaction layer, which also contains the needed catalyst. So, contact to the solid phase of the reaction layer is necessary. The contact to the membrane is required because the protons reach the reaction zone through that layer (see e.g. [Barbir(2005)]). The three phase zone with all needed components is shown in figure 2.3. ionomer H + platinum catalyst O 2 H O 2 gasphase carbon e Figure 2.3: The three phase zone in the reaction layer The three phase zone is often thin and so only the part of the catalyst, which borders directly on the membrane takes part in the reaction. Some methods are developed to enlarge the depth of the reaction zone in the reaction layer. Two methods are described in [Wieser(2001)]: Ionomer is added in the production step of the reaction layer or the ionomer is additionally impregnated.

22 8 2 Fundamentals of a PEMFC The polymer electrolyte membrane An electrolyte has to fulfill three requirements: first separate the two reaction zones, second provide a good ionic conduction and third act as an electronic isolator between anode and cathode. Today, Nafion, produced by the company DuPont is mainly used as an electrolyte. The construction of the electrolyte is done in three main steps [Larminie and Dicks(2006)]: As basic material polyethylene is used. To make it more durable and resistant to chemical attacks it is modified by substituting the H-atoms with F-atoms (perfluoration). This material is called PTFE (polytetrafluoroethylene) or Teflon. Teflon is a hydrophobic material. Completely hydrophobic behaviour isn t desirable in the electrolyte since water is needed to transport the H + ions through the membrane. So, another modification is needed. This is done by adding a side chain with a sulphonic acid ending on the PTFE polymer. The bonding between the SO3 and the H + ion has ionic character and sulphonic acid is highly hydrophilic and it generates cluster within the structure. Thus, hydrophilic zones exist, where water is attracted and the H + ions are not so strongly bonded and can move through the material. This is an important property since hydrogen ions, which are produced on the anode side, have to be transported through the membrane to the cathode. Due to the reason that the ions are transported in hydrate covers by electro-osmotic drag, the amount of ions transported is dependent on the water content in the membrane. If the membrane dries out, the transport of ions will be inhibited. But too high water content can lead to a flooding of the reaction layer and gas diffusion layer of the cathode, which results in an inhibition of the supply of reactants. So, the water content of the membrane has a strong impact on the performance of the cell. On the one hand the water has to be removed at the cathode side, where it is produced and supplied through the membrane to avoid a flooding. On the other hand the anode has to be humidified in sufficient manner to avoid the drying out of the membrane. But also a complete dry out of the cathode has to be avoided. Therefore, the incoming reaction gases often are humidified and the membrane has to be as thin as possible, so that water can diffuse back to the anode. 2.3 Thermodynamic and electrochemical fundamentals of the fuel cell To provide a basic understanding of PEMFCs, some thermodynamic and electrochemical fundamentals are necessary. Therefore, the main fundamentals are described in this section The fuel cell under open loop conditions The maximum cell voltage, which arises under open loop conditions, is called reversible voltage U rev and can be calculated from thermodynamic data as followed. Considering the second law of thermodynamics, the difference in enthalpy H in the H 2 /O 2 reaction can t be transferred completely in electrical work, because the entropy S is lost as heat. Only the free energy G can be converted in work. For Gibbs free energy at constant temperature yields: dg = dh T ds (2.2)

23 2.3 Thermodynamic and electrochemical fundamentals of the fuel cell 9 With the first law of thermodynamic at constant pressure du = dq dw = T ds pdv dw el (2.3) and the enthalpy at constant pressure dh = du + pdv (2.4) following equation for the electrical work can be defined: dw el = T ds dh (2.5) From this equation, it follows that the change of free energy equals the negative change of electrical work dw el : dg = dw el (2.6) The electrical work W el can easily be calculated by [Mench(2008)]: W el = zf U (2.7) where z is the amount of transferred electrons and F the Faraday s constant =96485 C/mol. Thus, the reversible cell voltage can be defined by combining equation 2.6 with equation 2.7: U rev = G zf (2.8) In the H 2 /O 2 reaction, two electrons are transferred and the resulting reversible cell voltage is shown in table 2.1. H 2 /O 2 reaction z 2 H [kj/mol] S [J/mol K] G [kj/mol] U th [V ] U rev [V ] η th [%] Table 2.1: Thermodynamic data for the H 2 /O 2 fuel cell for p=1bar and T = 25 C [Wöhr(1998)] If the total amount of enthalpy H is transformed in electrical energy without loses, the so called thermal voltage would arise: U th = H zf As earlier mentioned, this isn t possible due to the second law of thermodynamic. (2.9)

24 10 2 Fundamentals of a PEMFC To judge an energy converter, the efficiency is the most important value. As already mentioned in section 2.1, fuel cells aren t limited by the Carnot efficiency. But the thermodynamic maximum efficiency of a fuel cell is the ratio of maximum theoretic reachable electrical work to reaction enthalpy [Mench(2008)]: η th = G H = U rev U th (2.10) Table 2.1 also shows the maximum efficiency (η th ) of the H 2 /O 2 fuel cell. The reversible cell voltage is dependent on the temperature and the pressure, since the free energy G is a function of temperature and pressure. By differentiating equation 2.8 with respect to the temperature (T) at constant pressure and adopting a reference temperature T 0, following equation for the temperature dependence can be found [Wieser(2001)]: U rev = U rev (T 0 ) + S zf (T T 0) (2.11) Considering that, the entropy change ( S) is always negative, the reversible cell voltage and the efficiency decrease with increasing temperature. This means that the efficiency of PEMFCs shows an opposite behaviour than the efficiency of heat engines, which increases with increasing temperature. Therefore, PEM fuel cells have high advantages at low temperatures. The different characteristics of the efficiencies are shown in figure 2.4. It can be seen that the fuel cell has advantages until a temperature of circa 1000 C is reached. 100 efficiency (%) H 2 /O 2 Fuel cell Carnot ( T low =100 C) temperature ( C) Figure 2.4: Theoretical efficiency of Carnot engine and Fuel Cell The pressure dependence can also be expressed by differentiating equation 2.8 with respect to the pressure p at constant temperature and using the ideal gas law [Wieser(2001)]: U rev = U rev (p 0 ) RT ( ) pi ν ln (2.12) zf i p 0,i with p i and p 0,i the pressure of species i and the reference pressure of species i in [Pa], R the universal gas constant and ν stoichiometric coefficient [-]. For the H 2 /O 2 reaction i ν is negative, thus the pressure should be as high as possible to reach a high efficiency.

25 2.3 Thermodynamic and electrochemical fundamentals of the fuel cell The fuel cell under load In practice the reversible cell voltage isn t even reached under open circuit condition. Measured values of the open circuit voltage are often lower. Different reasons are discussed in the literature. E.g. two reasons are mentioned in [Mench(2008)]: electrical short circuits in the cell and crossover of reactions through the electrolyte and subsequent mixed-potential reaction at the opposite electrode. The first reason mentioned, which can occur if the cell is poorly constructed, doesn t actually influence low-temperature fuel cells as PEMFCs. But the second reason limits the open circuit voltage U 0 to approximately 1 V instead of 1.2V for the H 2 /O 2 fuel cell. If the fuel cell is under load, the cell voltage will further decrease. A typical polarisation curve for a fuel cell is shown in figure 2.5. Urev Fuel cell voltage (V) Uo I II Current density (A/ cm 2 ) III Figure 2.5: Polarisation curve The curve, which shows voltage versus current density, can be divided into three parts, where different losses, so called overvoltages predominate. In region I, the activation losses dominate and the main overvoltage is called activation overvoltage. This region is kinetically determined and the losses are affected by the velocity of the reactions in the electrodes. The size of the losses depends on the reactants and the kind of catalyst. Region II is dominated by ohmic losses. The internal electron resistance and the resistance of the ion conduction in all layers are the main reasons for losses. A linear run is typically for this region. In region III, mass transport limitations and diffusion processes dominate. The so called concentration overvoltage and equilibrium overvoltage are determined [Wieser(2001)]. At a highly electrical current, the transport of the reactants to the reaction zone will be limited; this results in a depletion of reactants.

26 12 2 Fundamentals of a PEMFC Several approaches for the overvoltages exist in literature. E.g. [Acosta et al.(2006)], [O Hayre et al.(2006)], [Wieser(2001)] and [Wöhr(1998)] show similar approaches, which differ in some points. But in all approaches, the cell voltage is calculated by subtracting the sum of the overvoltages η i from open loop voltage (U 0 ). U cell = U 0 i η i (2.13) [Wieser(2001)] divides the overvoltages in four different kinds: activation overvoltage, concentration overvoltage, equilibrium overvoltage and ohmic overvoltage. At this point, some approaches differ, e.g. [Wöhr(1998)] doesn t consider the equilibrium overvoltage. In this thesis the definitions and equations for the overvoltages derived by [Wieser(2001)] are used and they are shortly described in the following: Activation overvoltage Activation overvoltage is the voltage loss, which results from the activation energy, which has to be raised to start an electrochemical reaction. These losses are based on the inhibition of the charge transfer at the boundary between electrode and electrolyte. The inhibition occurs since a so called double layer is formed at the interface. To explain the formation of the double layer, the anode side is considered. Here, protons move into the electrolyte and electrons are left in the electrode under open circuit conditions. The separation of the charge causes an electrical field. Under load, the protons have to overcome the resistance of this electrical field before the reaction can occur. The voltage, which is necessary for overcoming this resistance is called the activation overpotential or overvoltage (see [Mench(2008)] and [Tüber(2004)]). An expression to describe activation overvoltage against current density is given by the Butler-Volmer equation. For the current density i at the cathode yields: ( ) ( )] (1 α)zf αzf i = i 0 [exp η act exp RT RT η act (2.14) with α load transfer number [-]. i 0 is called the exchange current density, which occurs under open circuit conditions (i=0). Under these conditions the two reverse current density flows are equal. For clarification of i 0, a simple redox reaction is considered at the anode and cathode Ox + ze = Red (2.15) where the species Ox is reduced to Red. In equilibrium, when open circuit conditions prevail, the reaction streams in both directions are equal. This means that the current density has the same, but opposite value i 0 in both directions, which implies that they neutralize each other. When the reaction in the fuel cell takes place, the reduction reaction will get faster and the oxidation reaction will get slower at the cathode and vice versa at the anode, so a current density can occur. The exchange current density is dependent on the temperature T and the concentration of reactants. This can be combined by i 0: i 0 = i ref 0 f v exp [ G # 0 R ( 1 T 1 T ref ) ] (2.16)

27 2.3 Thermodynamic and electrochemical fundamentals of the fuel cell 13 where i ref 0 is the reference exchange current density at fixed concentration c ref i and a reference temperature T ref, f v the catalyst surface increasing factor, and G # 0 the free activation energy for the charge transfer. Thus, the Butler-Volmer equation can be expressed with i ref 0, depending on the current concentration c ox. Additionally, the current density at the anode can be neglected for high current densities at the cathode. Then the Butler-Volmer equation for the cathode ends in: i = i 0 c ox c ref ox ( ) αzf exp RT η act (2.17) The current density of the anode will also be neglected in this thesis and will be discussed again in chapter 5. Using the ideal gas law c = p/rt, following equation yields for the activation overvoltage: η act = RT αzf ln i i 0 + RT αzf ln p O 2 p ref O 2 (2.18) where p O2 is the current oxygen pressure and p ref o 2 conditions at T ref and c ref i. the oxygen pressure at the defined reference Ohmic overvoltage The ohmic overvoltage describes losses due to charge transport and contact resistance. Charge transport losses occur while electrons are transported through the electrodes and ions move through the membrane. Contact resistance can be found between membrane and electrodes or between gas distributor and gas diffusion layer. To determine these potential losses, ohmic law is used: η ohmic = ir spec (2.19) R spec is the sum of electronic, ionic and contact resistance. After [Barbir(2005)], a typical value for the resistance is 0.1 to 0.2 Ω/cm 2. Concentration overvoltage The influence of mass transport is considered in the term of concentration overvoltage. In case of high current densities, the concentration of reactants in the reaction zone will decrease, due to inhibition of mass transport. Both, mass transport to the reaction zone and removal of emerging products are limited by the diffusion velocity. For high reduction streams, the concentration overvoltage at the cathode can be defined as: η C = RT αzf ln p O 2 p inlet O 2 (2.20) A new reference condition () inlet at the inlet is defined because the thermodynamic condition at the inlet can differ from the reference condition () ref.

28 14 2 Fundamentals of a PEMFC Equilibrium overvoltage While the concentration overvoltage describes the influence on the mass conversion, the equilibrium overvoltage specifies the influence of the concentration on the potential. This is explained by considering that the integral polarisation curve is a superposition of all local characteristic curves. Thus, the open circuit voltage, which is described by the concentration at the inlet isn t the reference voltage for the local curves. The difference between the voltage at the reaction zone and the open circuit voltage is described by the equilibrium overvoltage. η equ = RT zf ν i ln p i p inlet i i = RT zf 1 2 ln p O 2 p inlet O 2 (2.21) where ν i is the stoichiometry coefficient and i ν i is the sum of all stoichiometry coefficients of reaction i. For the H 2 /O 2 reaction H 2 + 1/2O 2 H 2 O, i ν i is 1 1 1/2 = 1/2.

29 3 Coupling theories for flow in combined free and porous domains Free flow coupled with flow in porous media can be found in a wide range of processes in the natural environment as well as in industrial applications. Some examples for natural processes are flows in fractured rocks, coupling between surface water flow and groundwater flow and flow over the surface during rainfall. Coupled flows in industrial applications can be found during filtration, cooling and drying processes and solidification of metals in casting industries (see [Das et al.(2001)] and [Rosenzweig and Shavit(2007)]). Furthermore, an important application of coupling free flow with flow in porous media is found in the assembly of PEM fuel cells. Here, the gas channel with free flow has to be coupled with the porous gas diffusion layer. Flows through porous media have been investigated by many researches and there are also studies about coupling between free flow and flow in porous media. To model the flow at the interface between free flow in e.g. a channel and flow in a porous domain is a difficult task since the equations, normally used to describe the flow in both domains are extremely different. Darcy s law, which is only first order for velocity, is typically used to model the flow in porous media. For the simple Darcy law yields [Helmig(1997)]: v = K µ p (3.1) where v is the velocity, K the permeability of the porous medium, µ the viscosity of the fluid and p the pressure. In the open flow channel, Navier-Stokes equation is normally used to describe the flow [Helmig and Class(2004)]: ϱg p + µ 2 v = ϱ v t + ϱ(v )v (3.2) where ϱ is the density. The Navier-Stokes equation is second order derivative of velocity (see third term of equation 3.2). Coupling of equations of different orders in different regions generates mathematical problems. All components of velocity, the normal and tangential velocity, are needed for the Navier-Stokes equation. Thus, two boundary conditions are required for this second order derivative equation. But only one boundary condition for Darcy s law has to be defined, so the velocity components at the interface can t be specified. Thus, problems for the conditions at the interface between the free flow and porous media occur due to the difference of both equations. Either the porous region is over-determined while using two boundary conditions or the open flow region is under-determined while having only one boundary condition. Thus, discontinuities occur at the interface [Das et al.(2006)], 15

30 16 3 Coupling theories for flow in combined free and porous domains [Rosenzweig and Shavit(2007)],[Salinger et al.(1994)]. Two main approaches have been developed to address this problem at free/porous flow interfaces. The first approach was derived by [Brinkman(1947)], who modified Darcy s law in order to use it in both domains, the free flow and the porous domain. The second approach was found by [Beavers and Joseph(1967)]. They combined Darcy flow in the porous medium with Poiseuille motion in the free domain (derived with Navier-Stokes equation) over a slip condition at the interface. Both approaches will be discussed in more detail in section 3.1 and 3.2. Research in the following years was mainly based on one of these approaches by implementing several modifications. Beavers and Joseph slip condition was investigated in many studies. [Layton et al.(2003)] investigate the mathematical model of transport of substances between the surface and the ground water. Here, Stokes equation was set in the free fluid region coupled with Darcy s law in the porous domain. For coupling across the interface Beavers and Joseph slip condition was used. The slip condition theory was also studied by [Murdoch and Soliman(1999)] and [Das et al.(2001)]. The Brinkman equation was investigated and modified by [Rosenzweig and Shavit(2007)]. The modification is also explained in more detail in section [Das et al.(2001)] discussed the advantages and disadvantages of using the Brinkman equation in comparison to using Darcy s equation. They stated that the Brinkman equation is valid for higher porosity materials (> 0.6) and higher Reynolds numbers (Re > 10), when inertial forces in the pores may become more significant. [Salinger et al.(1994)] compared two approaches, the typical Beavers & Joseph approach (NS in the free flow domain and Darcy s law in the porous medium coupled through the slip condition) and the Brinkman approach for the porous domain combined with Navier Stokes equation in the free flow region. This approach is explained in section 3.3. They found out that the Beavers & Joseph slip formulation is not as easy to implement as the Brinkman formulation but it shows better solution accuracy, higher algorithmic robustness and lower computational costs. In the following sections, four different approaches to couple free flow and flow in porous media are described: The Beavers and Joseph slip condition, the original Brinkman equation and the modified Brinkman equation by [Rosenzweig and Shavit(2007)], coupling Brinkman and Stokes equation and an approximation to use Darcy s law in both domains. 3.1 Beavers and Joseph slip condition [Beavers and Joseph(1967)] showed in their experimental studies that the efflux of a Poiseuille flow over a permeable block instead of an impermeable wall significantly increases. This increase has been explained by the presence of a boundary layer at the interface of free flow region and the porous medium. Within this boundary layer, a change of the velocity occurs from Darcy value in the porous medium to a slip value near the interface. Figure 3.1 illustrates the situation. The basic idea of this theory is to replace the boundary layer effect with a slip condition, which is proportional to the velocity gradient in the free flow.

31 3.1 Beavers and Joseph slip condition 17 y flow channel Poiseuille b u B boundary layer x Q porous media Darcy Figure 3.1: Velocity distribution in a horizontal channel coupled with porous media In the experiment, a two-dimensional Poiseuille flow (laminar flow of a homogeneous, viscous fluid) over a saturated, permeable block is used to analyse the characteristic of the tangential flow in the boundary layer region of the permeable interface. The flow channel is formed by the permeable porous medium on the lower side (y=0) and an impermeable wall on the upper side (y=h). Porous Medium In the proposed flow model, Darcy s law is used for flow in homogeneous and isotropic porous media, which is highly valid in the interior of the porous domain, but not near the interfacial area. Thus, the seepage velocity Q in the porous medium is defined as: Q = K µ dp dx (3.3) where K is the permeability of the porous medium and µ the dynamic viscosity of the fluid. Free flow A simplification of Stokes equation, Poiseuille equation, describes the free flow domain. Poiseuille equation can be derived from Stokes equation and the continuous equation for an incompressible, parallel stationary flow in a channel. External and gravity forces are neglected. d 2 u dy = 1 dp 2 µ dx where u is the velocity in x direction. (3.4)

32 18 3 Coupling theories for flow in combined free and porous domains Boundary layer At the interface between free flow and porous medium, the tangential velocity has to be defined. [Beavers and Joseph(1967)] assumed that the slip velocity at the interface is not equal to the mean filter velocity in the porous domain and that the shear effects are transferred into the interior of the porous medium through a boundary layer. In their theory, a sudden change of velocity occurs from filter velocity of Darcy s law to a slip value u B at the interface. Another assumption made is that the slip velocity for free flow is proportional to the shear rate at the permeable boundary. [Beavers and Joseph(1967)] developed the following boundary condition for the slip velocity related to the free flow: du dy (y = 0 +) = β(u B Q) (3.5) By posting the assumption that the flow in the channel is parallel and the boundary layer is laminar, [Beavers and Joseph(1967)] stated that β is independent of x. Due to this independence on x, β can only be a function of fluid properties or properties of the porous medium. To obtain an expression for β a dimensional analysis is made, which shows that β can t be dependent on the viscosity but only on the properties of the porous medium. The following expression was shown by [Beavers and Joseph(1967)]: β = α BJ K (3.6) where α BJ is a dimensionless value, which depends on properties of the porous medium in the boundary layer. The final [Beavers and Joseph(1967)] slip condition is expressed by: du dy (y = 0 +) = α BJ K (u B Q) (3.7) With the boundary conditions u=0 at y=b and equation 3.7 at y=0, the solution of equation 3.4 is given by: u(y) = u B (1 + α BJ K y) + 1 2µ (y2 + 2α BJ y K) dp dx (3.8) By integrating equation 3.8 in [0,b] and multiplying with mass density ϱ, the mass flux can be derived by ( ) M = ϱb3 dp 12µ dx ϱb3 σbj + 2α BJ dp (3.9) 4µσ BJ 1 + α BJ σ BJ dx with σ BJ = b/ K. Following equation can be derived for the fractional increase in mass flow rate through the channel with a permeable lower wall: Φ BJ = 3(σ BJ 2α BJ ) σ BJ (1 + α BJ σ BJ ) (3.10) [Beavers and Joseph(1967)] have tested this theory by experimental studies and they found out that the theoretical mass flux is consistent with the measured mass flux.

33 3.2 Brinkman equation Brinkman equation Origin of the Brinkman equation [Brinkman(1947)] showed the origin of his well known equation in his paper A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Here, the viscous force, which is exerted by a flowing fluid on a spherical particle, embedded in a dense swarm of particles, is calculated. The dense swarm of particles is described by a porous mass. The Brinkman equation, which is a modification of the Darcy equation was developed to specify the flow through this porous mass. He emphasized that Darcy s equation does not include a term for accounting the viscous interaction between the fluid particles, which occur in the boundary layer near the interface between the free flow (Brinkman was calling this region holes between the particles) and porous media domain (see [Salinger et al.(1994)]). Brinkman argued that if the first order Darcy law describes the flow in porous media and second order Navier-Stokes equation is valid in the free channel, difficulties will arise to formulate boundary condition. He suggests the following approach to avoid this problem. By stating equilibrium on a volume element of fluid between the three forces: pressure gradient, the divergence of viscous stress tensor and the damping force caused by the porous mass, Brinkman proposed following equation for the whole region: p = µ K v + µ v (3.11) with v the Darcy velocity, µ the dynamic fluid viscosity and µ an effective parameter for the viscosity, which is different from the fluid viscosity. As a first approximation, Brinkman has chosen µ = µ. There are also further options, e.g. µ > µ or even µ < µ. This is discussed in [Rosenzweig and Shavit(2007)]. Equation 3.11 results for high permeability K, which is valid in the free flow channel, in the Stokes equation. Since the first term on the right hand side becomes very small. p = µ v (3.12) For low permeability as in the porous medium, the second term on the right hand side of equation 3.11 becomes very small in comparison to the first term. Therefore, the first term (the Darcy term) of equation 3.11 dominates in the porous domain. The dominating terms of the Brinkman equation in each region are shown in figure 3.2. Brinkman proposed the following condition at the interface between the two domains: continuity of the normal components of the stress tensor and continuity of the normal velocity and the tangential velocity. For assuring continuity of the tangential velocity, the boundary between the porous mass and the free flow is replaced by a transition region. In this region the permeability varies from K in the porous medium to in the free flow region. By assuming that the transition region is infinitesimally small, the tangential velocity is continuous at the boundary.

34 20 3 Coupling theories for flow in combined free and porous domains Figure 3.2: Dominating terms (red) of the Brinkman equation in each region Modification of Brinkman equation by Rosenzweig In some studies, the Brinkman equation is criticised since no real derivation of the equation exists. But, [Rosenzweig and Shavit(2007)] showed that a similar approach can be derived by starting on the pore scale and they developed the so called modified Brinkman equation (MBE). Based on the microscopic Navier-Stokes equation, they followed the generalized volume averaging method presented by [Quintard and Whitaker(1994)] and derived the spatially averaged MBE. The MBE is valid with the assumption of a steady state, unidirectional, laminar flow, a Newtonian fluid, a stagnant solid phase and constant fluid properties. For an exact derivation it is referred to [Rosenzweig and Shavit(2007)]. Here, the resulting MBE ends in: ( ) p + µ 2 u f = 0 z Hrev x z 2 2 ((( ) ) ) p + µ 1 φ x H rev z + 1+φ 2 u f + 2(1 φ) u f φ 2 z 2 H rev z k uf H = 0 rev z Hrev (3.13) 2 2 ( ) p + µ φ 2 u f φ x z 2 k uf = 0 z Hrev 2 where H rev is the height of the representative elementary volume (REV), z the vertical coordinate with z=0 at the interface, u f the averaged intrinsic free flow velocity in x-direction and φ the porosity inside the porous medium. For clarification, figure 3.3 shows the REV at the interface. The first one of the three linear differential equations is similar to the Stokes equation in the free flow channel outside the porous domain (for z H rev /2), the last equation equals the Brinkman equation in the porous domain (for z H rev /2). In the boundary layer between the porous and the free flow domain, the second equation, a transition of both flows is valid (for H rev /2 z H rev /2). 3.3 Coupling Brinkman equation with Stokes equation Brinkman equation is often combined with Stokes equation. Here, the interface flow is modelled by using Brinkman equation in the porous domain and Stokes equation in the free flow domain, (see e.g. in [Das et al.(2001)] and [Salinger et al.(1994)]). This coupling strategy is also called two-domain approach in comparison with using Brinkman equation for

35 3.4 Darcy-equation in both domains 21 REV free flow channel Hrev 2 z=0 interface porous medium Hrev 2 Figure 3.3: REV at the interface between free flow and porous domain the whole domain (one-domain approach) (see [Rosenzweig and Shavit(2007)]). For porous media, the following variation of Brinkman equation can be written. p = ϱ dv dt + µ K v + µ 2 v (3.14) For the free flow channel, Stokes equation can be defined as followed [Das et al.(2001)]: p = ϱ dv dt + µ 2 v (3.15) The gravity is neglected in this equation and the derivative of the velocity with respect to time, which accounts for the storage within the domain isn t considered in the most approaches (steady state assumption). Coupling the two equations poses no special difficulties. All variables, pressure, normal and tangential velocity can be defined as continuous functions, since both equations have secondorder velocity terms. A problem, which can occur is that the boundary layer requires complicated mesh resolutions [Das et al.(2001)]. Two boundary conditions at the free flow/porous interface are needed to define the equations. Thus, shear stress and velocity continuity are set as boundary conditions. 3.4 Darcy-equation in both domains Another possibility to couple porous and free flow domains is, to use Darcy s law for both, the porous and the free flow domain. In the porous medium, the general Darcy equation obtain. For an incompressible fluid yields: v = ϱgk µ h = K µ p (3.16) where v [m/s] is the Darcy velocity, K [m 2 ] the permeability of the porous medium and h the height of the water surface. For the free flow channel, following approach can be used [Silberhorn-Hemminger(2002)]: Hagen-Poiseuille-law describes the flow between two parallel plates. In this case, the width

36 22 3 Coupling theories for flow in combined free and porous domains z=+h z=0 z= H v(z) b=2h Figure 3.4: Laminar flow through a small channel of the channel is much smaller than the length. Figure 3.4 illustrates this situation. For the velocity in the channel, this equation can be derived from the Navier-Stokes equation for a laminar one-phase-flow of an incompressible, Newton fluid. v(z) = ϱg 2µ [ d dx ( p ϱg + z) ] (H 2 z 2 ) (3.17) The maximum velocity in the middle of the channel (z=0) can be calculated by: v max = v(z = 0) = ϱg [ 2µ H2 d dx ( p ] ϱg + z) Thus, for the mean velocity of a parabolic velocity field yields: v = 2 3 v max == ϱg [ H 2 d µ 3 dx ( p ] ϱg + z) }{{} h (3.18) (3.19) By using the channel width b = 2H, the mean velocity in the channel can be written as: v = b2 ϱg 12 µ h = ϱg µ K h = K µ p (3.20) This equation looks similar to Darcy s law, whereas the permeability K in the channel can be expressed by the channel width: K = b2 12 [m2 ] (3.21) Thus, both regions can be described by Darcy s law. By using this method, problems of different orders of the equations can be avoided and only one boundary condition is necessary. But, the method is limited by the Reynold s number: Re = vd ν (3.22) with ν the kinematic viscosity. For small Reynold s numbers (Re < 10), the flow is laminar and creeping and Darcy s law is valid [Helmig and Class(2004)]. For 10 < Re < 60 70, the flow is still laminar, but no longer creeping. If higher Reynold s numbers are reached, the flow is turbulent. Thus, Darcy s law is not valid in these two cases.

37 4 Modelling of fuel cells Recently, modelling of fuel cells has been strongly investigated. In this chapter, a short review of the existing modelling aspects in the commom literature is given. In the first section, the development from one dimensional to multidimensional models and from singlephase to multiphase approaches is shown. Additionally, the consideration of the hydrophobic character of the GDL is discussed. In the second section, some models, which include the gas channel, are regarded more closely. Mainly discussed is how the mathematic coupling of the channel and the GDL is implemented in the model. 4.1 State of the art of common PEMFC models As already mentioned, the flooding of the gas diffusion layer of the cathode and the dryout of the membrane are critical issues in PEMFCs. The improvement of the mass-transport problems in the GDL, is an important task and has great potential for development. Due to the small geometry, the actual concentration of reactants and products in the reaction and gas diffusion layer are difficult to measure. Therefore, mathematical models have been developed for understanding the transport processes in fuel cells. Since the cathode has a greater limiting factor on the performance of PEMFCs, researchers mostly focus only on the cathode. But even some models of the whole PEMFC have been published. Many previous studies only consider a one phase system. They assume a simple gas phase, where the presence of liquid water is neglected. Some of these models are one dimensional, with the dimension normal to the plane of the reactive catalyst surface, see e.g. [Bernardi and Verbrugge(1992)] and [Springer et al.(1993)]. [Springer et al.(1993)] present a 1D model for the cathode side of the PEMFC to show how different parameters affect the polarisation curve. Since these models neglect the water phase, they can t make a statement about the water management in the PEMFC and the influence of water on the gas transport. Moreover, these 1D models can t compare the different behaviour of the gas distribution due to different structures of the gas distributors and can t take into account mass consumption or production. But the one dimensional models provide a good basic for further PEMFC modelling. [Gurau et al.(1998)] was one of the first, who analysed two dimensional effects in the entire fuel cell sandwich. However, gas-liquid phases are only considered in the membrane and the catalyst layer. In the residual domain, single gas phase transport is assumed. [Yi and Nguyen(1998)] also included the flow in the gas channel in their 2D model and analyzed multicomponent transport on the cathode side connected with an interdigitated 23

38 24 4 Modelling of fuel cells flow channel. [Dutta et al.(2000)] presented a three-dimensional, steady state, isothermal model, which includes the anode and the cathode side and the flow channels on both sides. However liquid water isn t considered in the model. Therefore, all these models are only valid in the absence of water, since they mainly consider only the gas phase in the porous electrode and do not account for phase change of water within the electrode. In later studies, accounting for liquid water became more important. [Shimpalee and Dutta(2000)] accounted for liquid water in their three-dimensional model of the whole fuel cell, but water was only treated as a component of the gas mixture and not as a separate phase. Therefore, the influence of liquid water distribution in the gas diffusion layer on the electrochemical reaction and on the transport of the reactants to the reaction zone can t be investigated with this model. The reason for considering liquid water was the idea to include the energy exchange of phase changes. Thus, this model is one of a few models, which include an energy balance. There have also been published several studies on two-phase, multidimensional transport in fuel cells. [He et al.(2000)] developed a two dimensional, two phase model, which accounts for capillary transport of liquid water. The governing equation of liquid water and the gas phase are coupled by an interfacial mass-transfer rate. The model included an interdigitated flow field and a hydrophilic gas diffusion layer is assumed. Based on this model, [Natarajan and Nguyen(2001)] modelled two phase flow in the cathode with a conventional gas channel in their two dimensional isothermal model. Other important models are based on Wang s multiphase mixture theory (M 2 ). In this theory, an individual mass conservation equation for each phase is derived, but only one single momentum equation is solved. E.g. [Wang et al.(2000)] used this approach for a two dimensional, two phase model for the air cathode of PEMFC with a hydrophilic membrane. They investigated two-phase hydrodynamics and transport of air and liquid water to assess the influence on the performance of PEMFCs. The problem of the M 2 model is the limitation to flows without phase change and that it can t account for mass production or consumption, which occur in PEMFCs. Thus, this model can t describe two phase flow in the porous reaction layer, where mass is produced or consumed (see [Gurau et al.(2008)]). The gas diffusion layers in the PEMFC have the property to enhance the area, which can be reached from the reactants. Therefore, a distribution of the current density on the membrane in the direction of the bulk flow and the direction orthogonal to the flow, parallel to the membrane can occur. This can t be shown by two dimensional models, which are discussed aboove (see [Al-Baghdadi et al.(2007)]). Additionally, the flow in the gas channel can t be suffiently modelled, if only two dimensions are considered. The study of [Berning and Djilali(2003)] has been one of the first, which included all three dimensions in a two-phase flow model. A momentum equation for gas and liquid phase was accounted and exchange terms between both phases were also included. The model includes GDL and gas channel on anode and cathode side, but it excluded the membrane. Thus, water transport through the membrane was ignored.

39 4.2 Existing models for coupling gas diffusion layer and gas distributor 25 All models mentioned above have the disadvantage that they don t consider that the gas diffusion layer material has hydrophobic character. The effects of the hydrophobic property of the GDL on the water transport and gas supply to the reaction zone is neglected. [Nam and Kaviany(2003)] and [Pasaogullari and Wang(2005)] were one of the first who considered the hydrophobic properties in their two-phase transport models. [Nam and Kaviany(2003)] investigated the influence of fiber diameter, porosity and capillary pressure on the liquid water distribution in a one dimensional model. They assumed a constant gas-phase pressure, thus the influence of gas flow on liquid water flow is neglected. [Pasaogullari and Wang(2004)] developed a systematic theory of liquid water transport through a hydrophobic GDL and presented a one dimensional solution of liquid water and oxygen transport in the GDL. One year later, [Pasaogullari and Wang(2005)] presented a multidimensional, two-phase model and investigated the influence of inlet humidification and flow rates on the performance of a PEMFC. There are only few measured data of the hydrophobic GDL available in the published literature. [Ihonen et al.(2004)] measured data like contact angle, permeability, porosity and conductivity of GDLs. Since the dimensions of GDL are so small, conventional methods to measure capillary pressure can t be adapted. Therefore, data of capillary pressure can only be found in a few publications. [Acosta et al.(2006)] presented a method to determine the capillary pressure saturation relation of an usual GDL and used this date in a two-dimensional, non-isothermal, two-phase model of the cathode GDL. High values for the capillary pressure were measured. Different GDL materials were investigated by [Gostick et al.(2005)]. The measured curves of the capillary pressure-saturation relation of the different materials show only small differences. But smaller values than in the study of [Acosta et al.(2006)] were presented. [Nguyen et al.(2008)] also developed a method to measure the relation of a typical GDL material. They predicted much smaller values for the capillary pressure than in the both studies mentioned before. [Koido et al.(2007)] used the porous diaphragm method to measure capillary pressure and predicted it by the pore network model. But they only measured values for a small range of the saturation between 0 and 0.1. These values again strongly differ from the values of the other studies. All values given for the capillary pressure by the different studies vary strongly. So, there is still the problem of reliable data for the capillary pressure of hydrophobic GDLs. 4.2 Existing models for coupling gas diffusion layer and gas distributor One main interest in this thesis is to couple the porous GDL with a gas channel. Many studies consider only the GDL and neglect the influence of the flow in the channel on the transport in the GDL and the performance of the cell, e.g. [Wang et al.(2000)] and [Acosta et al.(2006)]. In this section some studies, which include the gas channel in their model will be presented and the coupling of the two regions will be discussed. Some studies mentioned above also included the gas channel. One of the first models, which accounted for fully transport in the gas channel, was pub-

40 26 4 Modelling of fuel cells lished by [Gurau et al.(1998)]. Previous models mainly used averaged values and assumed a well-mixed concentration in the gas channel, e.g. [Springer et al.(1993)]. [Gurau et al.(1998)] stated that the concentration along the gas distributor-gdl interface depends on the diffusion-advection transport and the electro kinetics. The gas distributor and the GDL were considered as one single domain. An equation similar to the Brinkman equation (see section 3) was used to avoid special boundary conditions at the interface between gas distributor and GDL, which would be necessary if Navier-Stokes and Darcy s law was used. This model really shows an approach to couple gas distributor and GDL. But since only two dimensions are considered, distribution of the different parameters can t be shown because the width of the gas channel is ignored. Another point is that although the model considers gas and liquid phase it doesn t consider the interactions between the phases. As earlier mentioned, [Dutta et al.(2000)] published a three dimensional approach, which includes the gas channel to provide a real distribution of the current density. They solved the three dimensional Navier-Stokes equation in the whole domain, the gas channel and the GDL. Based on Darcy s law, a source term for the porous domain is added to the momentum equation. In the porous domain, the momentum equation is reduced to this source term since all the other terms become negligible. Thus, this model also presents a kind of coupling of the two different domains, but as already mentioned the second phase, liquid water is neglected. [Wang et al.(2000)] also modified the momentum balance equation in their M 2 model, which is reduced to Darcy s law in the GDL and to Navier-Stokes equation in the gas channel. All these approaches are in some way similar to the Brinkman equation, but the difference of the viscosty as it is considered with the help of effective viscostiy in the Brinkman equation, isn t taken into account. Newer studies consider mostly both: the gas diffusion layer and the gas channel. [Al-Baghdadi et al.(2007)] mainly focused on the liquid water impacts. But it seems that values for a hydrophilic material are used. They solve the steady-state Navier-Stokes equations for each phase in the gas channel. Here, phase change can t occur and sinks and sources don t exist. These mechanisms are added in the GDL. Therefore, mass balance and momentum equations change in the GDL. For the momentum equation of gas phase in the GDL, Darcy s law is used. Here, the problem of coupling at the interface between the gas channel and the GDL occurs again, since Darcy s law is only first order of derivation and the Navier-Stokes equation is second order of derivation for the velocity. In this study, discontinuities at the interface, which can arise due to this problem, are not mentioned. An other point is that both phases exist in the channel, which isn t realistic since normally only gas is provided to the channel. [Gurau et al.(2008)] modelled two-phase flow in PEMFC cathodes. A main momentum equation is considered for the whole domain, where different parts of the equation become negligible small in either the channel or the GDL. In the reaction layer and the GDL, liquid water is transported by the sum of gas phase pressure and capillary pressure gradient. The forces, which drive liquid water in the gas channel, are gravity and two-phase drag. But the diffenrence of the viscosity in the parts of the momentum equation, as Brinkman mentioned,

41 4.2 Existing models for coupling gas diffusion layer and gas distributor 27 is also not considered in this study. The approach to use Darcy s law in both domains hasn t ever been tried to implement. But, [Wang et al.(2008)] simulated two-phase flow only in the gas channel by using Darcy s law. They figured out a flow analogy between fuel cell channels and porous media, since they stated that fuel cell channel flow is laminar and a linear relationship between pressure drop and velocity can be found. Thus, they argued that Darcy s law is valid for this case. The expanded version for two phase flow of Darcy s law is used and the absolute permeability of the gas channel is also calculated by Hagen-Poisseuille equation. But Darcy s law is only valied until a Reynold s number Re=10. This isn t considered in this study. The coupling the gas channel and GDL with the Beaver and Joseph slip condition can t be found in the published literature. This is an issue, which needs to be further investigated. In this thesis, different aspects of the studies mentioned above will be put together to focus on the influence of the gas channel on the transport processes in the GDL and the performance of the fuel cell. Additionally, the influence of the chosen capillary pressure-saturation relation will be investigated. A three-dimensional, nonisothermal, two-phase, three components model of the cathode side including the gas channel is being developed in this thesis. For the GDL hydrophobic material is assumed, based on the measurements of [Acosta et al.(2006)]. The gas channel and the GDL are coupled by using Darcy s law in both domains. This can be done by considering some assumptions for the gas channel, which are explained in chapter 6. Addionally, the gas channel is modelled in such a way that only gas phase exit, which is realistic since only gas is provided to the channel.

42 5 The simulation model In this chapter, the model for the simulation of the transport phenomena in the porous reaction and gas diffusion layer and the added gas distributor and the electrochemical reactions are described. First, the fundamentals of multiphase flow are shortly explained. Then, the conceptual, mathematical and the numerical model are presented. 5.1 Fundamentals of multiphase flow In this section multiphase flow is shortly explained in general and some definitions and explanations are discussed. For a detailed description it is referred to [Helmig(1997)]. Multiphase flow will occur, if several substances of same or different aggregate state (gas, liquid, solid) exist in a system and affect each other. Each one of these substances is called a phase. Here, multiphase flow in a porous media, which consists of a matrix of solid material and voids in between, which are also called pores, is considered. The term phase implicates separated, homogeneous areas, wherein no jumps of physical properties arise. Two phases differ in their physical properties, are not miscible and are separated by a sharp interface. Components are parts of phases, which are independent substances. So, a phase can consist of several components. Between the phases mass and energy is transferred. Some of these processes are evaporation, condensation or dissolution. Thus, each phase can contain small parts of all components of the system New parameters on the macroscale The multiphase problem can be considered from different scales. At micro scale a point within a phase or at the interface is considered and discontinuities can clearly be identified. To enable the calculation of multiphase problems, the switch from the micro scale to the macro scale is necessary since too much time and computing power would be needed for complex multiphase problems calculations on the micro scale. This is done by averaging the properties of the micro scale within a representative elementary volume (REV). By the averaging process new equations (e.g. Darcy s law) and parameters (porosity, permeability and saturation) are created. The first mentioned parameter, the porosity is defined as the ratio of pore space within the REV and the total volume of the REV: φ = volume of pore space within the REV total volume of the REV (5.1) If flow through a porous media occurs, the fluid interacts with the solid phase. On the macroscale, these interaction can be expressed by the hydraulic conductivity K f. If more 28

43 5.1 Fundamentals of multiphase flow 29 than one fluid phase exists, it is defined as: K f = K k rα ϱ α g µ α (5.2) where K is the intrinsic permeability, which depends only on the porous media and k rα is the relative permeability, which is dependent on the saturation of the particular phase, representing the influence of one phase to another. g is the gravity, ϱ α the density and µ α the dynamic viscosity of phase α. The typical dependence of the relative permeability on the saturation for a wetting and nonwetting phase is shown in figure 5.1. It means that the space for movements of one fluid is dependent on how much of the other fluid is in the system. The meaning of wetting and nonwetting phase will be described in the next section. Figure 5.1: k r S w relation after Brooks and Corey [Helmig(1997)] As mentioned before saturation is the third new parameter, which is important while considering flow through porous media on the macroscale. Following definition for saturation is made: volume of phase α within the REV S α = (5.3) pore volume Capillarity At the sharp interface between two immiscible phases, a surface tension occurs, which arises by reason of cohesive forces in the phases and adhesion forces at the interface between the phases. To keep equilibrium condition, a discontinuity of the pressure occurs. The pressure difference between the two fluids is called capillary pressure. p c = p np p wp (5.4)

44 30 5 The simulation model where p c is the capillary pressure, p nw is the pressure of the non-wetting fluid and p wp the pressure of the wetting fluid. At microscopic consideration, the capillary pressure can be calculated by the Laplace equation: p c = 4σ cos γ d (5.5) where σ is the surface tension, γ is the contact angle and d the diameter of the capillary. For explanation of the contact angle, following situation will be considered: If two fluids contact a solid surface, a contact angle occurs between the fluids and the surface (see figure 5.2). The contact angle is always measured in the denser fluid and the convention is done that for 0 < γ < 90 the fluid is called the wetting fluid since it wets the solid. For 90 < γ < 180, the fluid is called the non-wetting fluid. nonwetting fluid γ>90 γ<90 wetting fluid Figure 5.2: Contact angle for a wetting and a nonwetting fluid The macroscopic consideration of the capillarity leads to the so called capillary-pressure saturation relation. p c = p c (S w ) (5.6) Several parameterisations of the p c -S w -relation are developed, e.g. Brooks and Corey or Van Genuchten derived such an approach, which can be seen in figure 5.3. A more detailed description can be found in [Helmig(1997)]. 5.2 Conceptual model for the porous layers in the PEMFC A conceptual model is the basis to derive a numerical model. Therefore, the conceptual model should represent all relevant processes, which influence the system, but at the same time, the complexity should be limited to a minimum. The used multiphase, multicomponent model is based on the model derived by [Class et al.(2002)] and [Class and Helmig(2002)], which was developed for modelling non-isothermal processes during the subsurface remediation of contaminations via steam injection. [Acosta et al.(2006)] adapted this model and implemented the processes and electro-chemical reactions in a PEMFC in a 2D model of

45 5.2 Conceptual model for the porous layers in the PEMFC 31 Figure 5.3: p c S w relation after Brooks and Corey [Helmig(1997)] the gas diffusion layer. This is adequate since the GDL is a highly porous medium, where multiphase flow occurs. In this thesis, the model is expanded and some changes were made. As already mentioned in chapter 2, water management in a PEMFC is a difficult task since on the one hand the membrane has to be humidified for better transport of protons, on the other hand accumulation of water in the GDL has to be avoided in order to ensure free oxygen transport. Humidifying the incoming gases and emergence of water results in a gasliquid two-phase flow in the porous gas diffusion and reaction layers. For the simulation of the problem a three dimensional, non-isothermal two-phase three-component model is used, with gas and water as the two phases and H 2 O, O 2 and N 2 as the three components. The mass and energy transfer between the phases occurs through condensation, evaporation and dissolution of gas in the liquid phase. Figure 5.4 clarifies the conception of the two phase system and the transfer processes between the phases. As explained in section the used material for the gas diffusion layer is hydrophobic. This means that water with a contact angle γ > 90 is the nonwetting phase and the gas is the wetting phase in this system. The different behaviour of water in the presence of hydrophilic and hydrophobic solid material is shown in figure 5.5. Additionally, the model has to be adjusted by an electrochemical model to account for the processes occurring by the reactions. Here, the approaches of [Wieser(2001)] are used. The effect that water can cover the reaction sites is additionally considered, when the current density is calculated.

46 32 5 The simulation model liquid phase (n) (w) (o) dissolution volatilization condensation gaseous phase (n) (w) (o) thermal energy components: water (w) nitrogen (n) oxygen (o) Figure 5.4: The two phase, three component, non-isothermal model concept γ>90 γ<90 water water hydrophobic material hydrophilic material Figure 5.5: Water as nonweeting fluid (left side) and water as wetting fluid (right side) 5.3 Mathematical model Domain description and mesh generation The 3D-model in this work includes the reaction layer, the gas diffusion layer and the gas distributor of the cathode of PEMFCs. There is a special focus on the cathode due to its higher limiting effect to the performance of PEMFCs. This can be explained by looking at the kinetics of the reactions on both sides. The kinetics of the hydrogen oxidation reaction (HOR) at the anode is much faster than the oxygen reduction reaction (ORR) at the cathode. Thus the overpotential on the anode side is neglected in this model, since the influence on the performance isn t high. The existing 2D-model of [Acosta et al.(2006)] consists only of the reaction layer and the gas diffusion layer of the cathode. In this thesis, the existing domain of the model, is expanded to three dimensions and the gas distributor is added. As written before, the gas distributor has a special structure for providing a good distribution of gases. Here, a conventional gas distributor is used for modelling. The model domain includes one collector and two halfchannels. The width and the height of channels and collectors are 1 mm. Figure 5.6 shows the model domain in 2D. As mentioned before, this domain represents only one part of the

47 5.3 Mathematical model 33 Part of the domain length (x) nodes (x) length (y) nodes (y) length (z) nodes (z) reaction layer m m m 11 gas diffusion layer m m m 11 half gas channel m m m 11 Table 5.1: Dimensions of the different layers in (x)-, (y)- and (z)-direction fuel cell, the cathode. On the left side of the GDL, the membrane is situated and then the reaction, gas diffusion layer and gas distributor of the anode would follow, if the anode was also taken into account. The transport of water through the membrane is considered as a source term in the reaction layer. reaction layer y=0.002 gas diffusion layer 1/2 gas channel y= current collector y= /2 gas channel y=0 x=0 x= x=00015 Figure 5.6: Front and back side of the model domain For 3D mesh generation, Ansys ICEM-CFD is used. Herewith, meshes for complex domains can be generated. The mesh for the combined GDL and gas distributor contains hexahedrons. In table 5.1, the dimensions of the chosen domain are listed. The length of the channel was first chosen to 5 cm in z-direction, including 20 elements. This length would be a realistic value for the whole length of the gas channel, but due to the high ratio between the length and the width of elements (1:25), numerical problems occur during the simulation. Therefore, the length of the used mesh is only 5mm and contains of 10 elements in z-direction. This means that only a section in the middle of the whole cathode in z- direction is considered. This is illustrated in figure 5.7. The main physical processes can even be shown with this domain. In figure 5.8, the generated mesh in 3D is shown.

48 34 5 The simulation model Air flow in Modelling domain in the y z plane Air flow out Figure 5.7: Modelling domain: Two half channels and the bar with a length in z-direction of 5mm Figure 5.8: Generated mesh in 3D

49 5.3 Mathematical model Model equation for the multiphase flow The equations, which are presented in this section are based on the equation used by [Acosta et al.(2006)] and they are valid for the porous reaction and gas diffusion layer. Since the Darcy coupling approach is chosen for the simulation in this thesis, the equations are also used to calculate the processes in the gas channel. The assumptions, which are made for the channel flow will be described in chapter 6. For calculating the two-phase three-component problem, a system of equations has to be defined. Local thermal equilibrium is assumed. According to the Gibbsian phase rule after e.g. [Atkins(1996)], the system has four degrees of freedom, which implicates that four equations are needed to determine the problem. Thus, three mass balance equations, one for each component and one energy equation are defined. The mass balance equation for each component of the two-phase three-component model can be written as: ) ( α ϱ mol,α x c α S α φ }{{ t } storage [ϱ mol,α x cα v α ] + div α }{{} advective ( ) div D c eff ϱ mol,g grad (x c g) }{{} diffusive q }{{} c = 0, sink/source (5.7) where Φ is the porosity, S α the saturation of phase α, ϱ mol,α the molar phase density, x c α the mol fraction of component c in phase α, v α the velocity, which is defined by Darcy s law, D c eff the effective diffusion coefficient of component c and qc are source or sink terms. In the following, the four different terms of the mass balance equation will be shortly explained. The first term accounts for the mass, which can be stored in the system. The second term describes the advective transport of species, which is caused by differences in pressure or even density. Darcy s law, extended for multiphase flow is used to calculate the velocity v α (see [Helmig(1997)]). v α = k rα µ α K(gradp α ϱ α g) (5.8)

50 36 5 The simulation model where p α represents the pressure of phase α, K is intrinsic permeability, k rα relative permeability and g is the gravity vector. The diffusive transport, due to differences in the concentration is described by the third term As it can be seen in equation 5.7, following equation yields for the diffusive flux: J c = D c effϱ α grad x c α (5.9) To consider the effect, that the mean path length in a porous media is reduced, the diffusion coefficient is multiplied by the quotient of porosity and torturosity. Additionally, D c eff is a function of the saturation of the gas phase S g, since the mean path length depends on the available free path length. D c eff = Φ τ S gd c (5.10) where τ is the tortuosity and D c the binary diffusion coefficient, which can be calculated by: D i = 1 xi g x w g Diw + xn g Din i [o, w] (5.11) In this model the assumption is made, that the diffusive flux is considered through the binary diffusion model, which is actually only valid for diffusion of a trace component (x i g << 1) in gas. For an exact consideration a ternary diffusion model should be added. The last term of equation 5.7 accounts for the sinks and sources. In the considered system, sinks and sources occur due to the electrochemical reaction and transport of water through the membrane. This will be described in the next section. Only a single energy equation is needed due to the assumption of local thermal equilibrium, which implicates that all phases have the same temperature. Thus, heat exchange terms between the phases are not needed. The following balance equation for the energy is used [Acosta et al.(2006)]:

51 5.3 Mathematical model 37 φ α (ϱ mass, α u α S α ) t } {{ } storage fluids ( ) div λ pm grad(t ) } {{ } conduction + (1 φ) (ϱ s c s T ) t }{{} storage solid [ k r α div ϱ mass,α h α K ( grad(p α ) ϱ mass,α g ) ] µ α α }{{} convection q }{{} h = 0. sink/source (5.12) where u α is the specific internal energy of phase α, ϱ mass, α the phase density, ϱ s the soil grain density, c s the specific heat capacity of the soil grains, T the temperature, λ pm the equivalent heat conductivity of the soil and h α the specific enthalpy of phase α. The different terms account for storage, heat conduction, heat convection and sink and sources similar as in the mass balance equation. The storage term is divided into two parts: the storage in the fluids and in the solid. Considering the conventional flow field, the main transport mechanisms for oxygen and nitrogen is diffusion and liquid water is transported by capillary forces from regions with higher water saturation to regions with lower saturation. For the capillary pressure, the general equations and relations, which are described in section 5.1 are valid. The properties, as contact angle, torturosity, porosity, relative permeability and capillary pressure-saturation relationship for a commercial GDL are taken from [Acosta et al.(2006)], who determined the different properties and relations. Here the determination of the capillary pressure-saturation relation and the relative permeability relation will be briefly described. For the other parameters it is referred to [Acosta et al.(2006)]. As mentioned in chapter 4, measuring of the capillary pressure relationship of the small hydrophobic GDL is a difficult task and only few data is available. [Acosta et al.(2006)] determined the p c (S) relationship with the help of the mercury injection porosimetry. The relation for a Hg-air-system can be converted to the H 2 O-air-system with following equation: p c (Hg air) p c (H 2 O air) = σ w cos θ w σ Hg cos θ Hg (5.13) with σ i is the surface tension and θ the contact angle. The following parametrization for the capillary pressure-saturation curve was made: p Acosta c = A exp(b S w + C) + D(1 S w ) (5.14)

52 38 5 The simulation model with A = B = 8.5 C = 0.2 D = 700 (5.15) The measured data for the capillary pressure relation may not be realistic for the given material. This issue will be addressed in the next chapter and some variations of p c will be made. In general, the equation developed by [Acosta et al.(2006)] diveded by is used for calculating the capillary pressure. The division by is done since it seems that the [Acosta et al.(2006)] determined the parametrisation for p c for the whole cathode, including the reaction layer and not only for the GDL. Therefore, a too high capillary pressure was assumed. But the division value will be discussed and varied in section For the determination of the relative permeability the Burdine approach as it is described in [Acosta et al.(2006)] is used. In this model, a capillary pore model of parallel tubes with constant cross section in flow direction is assumed and for the Burdine approach follows: [ ] 2 Sw Sw S wr 0 k rw = 1 S wr 1 0 ds w p 2 c ds w p 2 c (5.16) where k rw is the relative permeability of the water phase. The relative permeability of the gas phase results in: k rw = [ 1 S w S wr 1 S wr ] 2 Sw ds w p 2 c ds w p 2 c (5.17) Model equations for the electrochemical reaction In this section the equation for calculating cell voltage and current density are shown. For the determination of the cell voltage, all overpotential, concerning the cathode side are subtracted from the open loop voltage. As mentioned before, the equations for the overvoltage derived by [Wieser(2001)] are used in this model. Thus, the equations for cell voltage and current density vary from the ones used by [Acosta et al.(2006)]. For the cell voltage yields: U cell = U 0 i η i = U 0 η act η equ η C η ohmic (5.18) Here, the reversible cell voltage U rev is used for the open cell voltage. Inserting all overvoltages, which are described in section 2.3.2, following equation for the cell voltage results: U cell = U rev ir spec RT αzf with the partial pressure of oxygen p o g = x o g p g ( ln i i 0 ln po g p o,ref g ln p o g α p o,inlet g 2 ln p o ) g p o,inlet g (5.19) (5.20)

53 5.3 Mathematical model 39 the inlet pressure of oxygen p o,inlet g = x o,inlet g and the reference pressure of oxygen p o,ref g = x o,ref g p inlet g. (5.21) p ref g. (5.22) For a given constant cell voltage, an implicit relation for the current density can be formulated: i = (1 S [ w) U rev U cell RT ( ln i ( R spez αzf i 0 ln po g α ) p o,ref g p o )] g ln (5.23) p o,inlet g The saturation S g = 1 S w of the gas phase is implemented in the equation to consider the influence of the emerging water in the reaction zone. The emerging water can cover several catalyst sites and can block the reactant gases to reach this sites. Here, several modifications were necessary in comparison with [Acosta et al.(2006)], since the saturation S w seems to have a higher influence than it was considered by [Acosta et al.(2006)]. Therefore, S w was considered as a factor in front of the bracket. The sinks and sources in this model are oxygen consumption, water production and heat production, occurring in the reaction zone. Additionally treated as a source term is the water flux through the membrane due to electro-osmotic drag. Oxygen consumption rate and water production rate are calculated by Faraday s law: q c = i z c F (5.24) where z c is the amount of transferred electrons and F is the Faraday constant. For oxygen reduction rate follows: q o R = i 4F and for water production rate yields: q w R = i 2F The water flux through the membrane is determined by a water transport number t H2 O: qm w i = t H2 O F (5.25) (5.26) (5.27) The index R or M labels the region of the sink or source. R stands for the reaction zone and M for the membrane. In the reaction zone, heat is also produced. For the heat production rate following equation yields: q h R = (U th U cell )i (5.28) The H + protons are assumed to be always present in a sufficient amount for the reaction in the reaction zone.

54 40 5 The simulation model Primary variables and boundary conditions This non-isothermal two-phase three-component system with four governing equations requires four independent primary variables. For the choice of the primary variables two cases are considered: (i) both phases, gas and liquid phase exist and (ii) only gas phase exists. In both cases gas pressure p g and temperature T are the first two primary variables. For case (i), oxygen mole fraction in the gas phase x O g and liquid water saturation S w are the other chosen primary variables. In case (ii), when no liquid water is available, nitrogen mole fraction in the gas phase x n g and water fraction in the gas phase x w g are the third and fourth primary variables. As initial conditions, only gas phase exists in the gas channel and both phases are present in the GDL. This implicates the primary variables for channel and GDL. For the initial conditions in the GDL yields: IC for GDL: x O g = x O,inlet g, S w = Sw inlet, p g = p inlet g, T = T inlet (5.29) In the gas channel, following initial conditions are used: IC for GC: x w g = x w,inlet g, x n g = x n,inlet g, p g = p inlet g, T = T inlet (5.30) Two kinds of boundary conditions are available in the numerical simulator MUFTE-UG: Dirichlet boundary condition and Neumann boundary condition. At all boundaries except for z=0m and z=0.005m, Neumann no-flow boundary conditions are chosen. At z=0m and z=0.005m, the outlet and inlet of the gas channel is situated. Here, Dirichlet boundary conditions are chosen. For the GDL at z=0m and z=0.005m, Dirichlet boundary conditions are also chosen, since the considered domain is an section of the whole length of the cathode. x O g = x O,inlet g, S w = Sw inlet, p g = p inlet g, T = T inlet (5.31) x O g = x O,outlet g, S w = Sw inlet, p g = p outlet g, T = T inlet (5.32) For the Diriclet boundary conditions at the inlet and outlet of the gas distributor yields: x w g = x w,inlet g, x n g = x n,inlet g, p g = p inlet g, T = T inlet (5.33) x w g = x w,outlet g, x n g = x n,inlet g, p g = p outlet g, T = T inlet (5.34) The boundary conditions are shown in a 2D illustration in figure 5.9. They are similar for the front and back side of the domain. 5.4 Numerical model For solving complex fluid dynamic problems the method of computational fluid dynamics (CFD) has prevailed. CFD has the aim to solve problems with approximative, numerical methods. The most common methods are finite volume method (FVM), finite difference method (FDM) and finite element method (FEM). To combine the advantages of FVM and FEM the Box-method has been developed and used in this simulation [Helmig(1997)]. The Box-method will be described below. For solving the system of the highly coupled, nonlinear partial differential equations resulting of the considered problem, the equations are implemented in the numerical simulator MUFTE-UG, which was developed for multiphase problems in porous media [Helmig(1997)].

55 5.4 Numerical model 41 Neumann=0 Dirichlet Neumann=0 Neumann=0 Neumann=0 Dirichlet Neumann=0 Neumann=0 Dirichlet Neumann=0 Neumann=0 Figure 5.9: Boundary conditions, shown from the front (or back) side of the domain The numerical simulator MUFTE-UG MUFTE-UG is a joint-venture development of the Institute of Hydraulic Engineering (IWS), University of Stuttgart (part MUFTE) and the Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg (part UG). UG, which stands for Unstructured Grid provides powerful numerical solution techniques to solve partial differential equations on unstructured grids. Several solvers are available to find the best solution for each problem. MUFTE stands for Multiphase Flow Transport and Energy model and provides several methods and applications to solve isothermal and non-isothermal multiphase multicomponent flow and transport processes in porous structured media. MUFTE is organised in different modules, which can be combined to find a problem-specific solution. The simulation of the PEMFC is a non-isothermal, two-phase, three component problem Discretisation method As mentioned before, MUFTE provides several methods for the spatial discretisation e.g. Fully Upwind Box-Method or Control-Volume Finite-Element Method. The Box-method has been used in several non-isothermal, multiphase, multicomponent models and is also an adequate method to solve the considered system. The Box-method combines FEM and FVM by constructing a secondary mesh. This is done by assigning a box to each node of the primary FEM-mesh. For the construction of the boxes, the centers of gravity of the elements are connected with the midpoints of the elements edges. So, each element is divided in several sub control volumes and each box can be assigned to a specific node [Helmig(1997)]. The primary FEM-mesh and the secondary FVM-mesh for quadratic boxes are shown in figure

56 42 5 The simulation model The method can also be used with irregular meshes. FVM mesh FEM mesh FEM nodes sub control volume Figure 5.10: Primary FEM and secondary FVM mesh The time discretisation is done by a fully implicit Euler scheme, which is unconditionally stable. For linearization of the highly non-linear system of equations, the Newton-Raphson method is used. One special feature has to be taken into account due to the implementation of the electrochemical model. For the implementation of the sink and source terms in section 5.3.3, two conversions of the calculated current density have to be performed. The unit of the determined current density of equation 5.23 is A/cm 2. For the correct calculation of the sink/source terms the current density has to be performed to A/m 2. This is done by the first factor in equation 5.35 i new c = i c (5.35) The second conversion is due to the fact that the sink and source terms are multiplied by the volume of the box. In the calculation of the current density only the relevant area at x=0m, where the reaction occurs is considered and it is given per m 2. To give the current density per volume, it has to be divided by the half length of the box in x-direction before the sink and source terms can be calculated. Otherwise the values for the sink and source terms would be too low. For clarification, figure 5.11 shows the section of the domain direct at the reaction layer. The the relevant area for current density calculation and the box volume are shown.

57 5.4 Numerical model 43 relevant area for ic calculation FEM mesh FVM Box x 2 Figure 5.11: Illustration of the implementation of sink and source terms

58 6 3D-Simulation with the Darcy approach In this chapter, simulations with the Darcy coupling approach are described. Different parameters are varied and the influence of the conditions in the channel on the water distribution and the performance of the PEMFC is tested. Additionally, the influence of the capillary pressure is investigated by varying the capillary pressure-saturation relation. In this approach, the GDL and the gas channel are coupled by implementing Darcy s law in both domains as it is described in section 3.4. The Darcy coupling approach is chosen since no discontinuities occur at the interface of the two domains and no mathematic problems arise. This approach will be only valid, if several assumptions are considered. This issue will be discussed later on. For multiphase systems Darcy s law is extended to account for the interaction of two different phases. Thus, following modified Darcy law, as already illustrated in chapter 5 is used [Helmig(1997)]: v α = k rα ν α Kgradp α (6.1) where k rα is the relative permeability of phase α, which depends only on the phase not on the porous medium and accounts for the interaction of one phase on the other, K is the absolute permeability, which describes the porous medium. The gravity term is neglected in these simulations. 6.1 Parameter definitions and assumptions In this section, assumptions for the simulation are described and the used parameters for the different parts of the PEMFC and for the reaction are presented General assumptions The implementation of Darcy s law in both domains is limited to small Reynold s numbers. In the porous media of the GDL and reaction layer, small velocities prevail and the Reynold s numbers are always smaller than 10. Consequently, no limitation exists for using Darcy s law in the GDL. Assuming a Reynold s number smaller than 10, Darcy s law can also be used in the gas channel. This assumption requires also low velocities in the channel. Here, experimental investigations would be necessary, since no measured data for velocities in gas channel of PEMFCs are available, but this wasn t possible in the scope of this study. In chapter 7 will be discussed, if the Reynold s number in the different scenarios lies in the 44

59 6.1 Parameter definitions and assumptions 45 Parameters of the gas diffusion layer Absolute permeability K m Porosity φ 0.78 Tortuosity τ 3.02 Residual saturation 0.12 Capillary pressure p c = p Acosta c /10000 Parameters of the reaction layer Absolute permeability K m Porosity φ 0.07 Tortuosity τ 5 Residual saturation 0.12 Capillary pressure p c = p Acosta c /10000 Table 6.1: Parameters of the reaction layer and gas diffusion layer range, where Darcy s law is valid. During the simulation process, numerical problems occur as soon as a primary variable change tries to arise. As initial conditions, gas phase exits in the gas channel and both phases in the GDL. A change of the primary variables can happen either, if nodes in the GDL dry out and the state switches from both phases to gas phase, or if water attains the gas channel and a state change from gas phase to both phases occurs. As soon as a variable change occurs the solver doesn t converge anymore. At the beginning of the most simulations the nodes of the GDL near the gas channel change from both phases to gas phases. Therefore, the simulation stopped at an early time. Due to this, an assumption is made, which forbids the change from both phases to gas phase. This is achieved by setting a small value for the saturation vapour pressure, if the saturation decreases under Consequently, the saturation can only decrease to a value briefly under 0.05 until equilibrium is reached. The convergence problems due to the switch from gas phase to both phases are avoided by increasing the switch criterion. Therefore, the variable switch would first occur, if the criterion was clearly reached. The convergence problems couldn t be solved completely and several simulations didn t run into steady state. The robustness of the simulation will be further discussed in section Parameter for the porous gas diffusion layer and reaction layer The used parameter in the GDL and the reaction layer are taken from [Acosta et al.(2006)]. The relation for relative permeability and capillary pressure are also based on the measurements of [Acosta et al.(2006)]. Since the relation for the capillary pressure of [Acosta et al.(2006)] is much higher than other values in the common literature, e.g. [Gostick et al.(2005)] or [Nguyen et al.(2008)], the values are divided by 10,000 in the reference scenario. This value will be varied and tested in section In table 6.1, all parameters for the GDL and the reaction layer can be found. The parameters, which are needed for the electro-chemical reaction are shown in table 6.2.

60 46 6 3D-Simulation with the Darcy approach Reversible voltage at K, U rev 1.191V Thermal voltage at K, U th V Cell voltage, U cell 0.5V Reference exchange current density A/cm 2 Reference temperature, T ref K Free activation enthalpy, G # J/mol Reference partial pressure, p ref g P a Reference mole fraction for oxygen, x o,ref g 1 Load transfer number, α 0.5 Surface increasing factor, f v 60 Specific resistance, R spec 0.25Ωcm 2 Universal gas constant, R 8.314J/mol K Faraday constant F C/mol Number of electrons transfer, z 2.0 Electro-osmotic drag coefficient, t H2 O Table 6.2: Parameters for the electrochemical reaction Assumptions and parameters for the gas channel Using Darcy s law in the gas channel requires several assumptions and parameter determinations. For the gas channel a higher permeability has to be chosen, since it is much more permeable than the GDL. Due to the fact that the robustness of the model is strongly affected by the difference of the permeability between the two domains a value, which is only 10 times higher than the value in the GDL is chosen for the reference simulation. However, the influence of the permeability is investigated by variations in section The porosity of the gas channel is also higher than in the GDL and is set to the value one, since no solid media is present in the channel. For the residual saturation the same value as in the GDL is used. Experimental studies could help to find out, which value would be realistic. But the value was chosen, since some water can be blocked by the edges at the contact to the GDL. Therefore, a certain amount of water can also be immobile in the channel. Diffusion in the gas channel is higher due to the higher porosity in the channel. Since an adequate value for the tortuosity isn t available for the gas channel the diffusion is assumed to be 10 times higher than the diffusion in the GDL. For the relative permeability and the capillary pressure, the same relations as in the GDL are used. Here, improvements are possible by using separate relations for the gas channel. But the fault that occurs when using the same relations is not really high, since only one phase exists in the channel during the simulations. Therefore, the capillary pressure and the relative permeability aren t relevant in the gas channel. All parameters, which differ from the values in the GDL are shown in 6.3. Absolute permeability K ch m Diffusion of all species i 10 DGDL i Porosity φ 1 Table 6.3: Parameters of the gas channel for the reference scenario

61 6.2 The transport processes The transport processes For a better understanding of the processes in the considered system, the main transport processes are explained in this section. A reference simulation scenario is considered for the illustration of the transport processes. All parameter for the GDL, the reaction layer and the gas channel, described above are valid for this scenario. The operational parameters for the considered scenario are shown in table 6.4. Pressure at inlet, p inlet g P a Pressure at outlet, p outlet g P a Temperature at the inlet, T inlet K O 2 mole fraction at the inlet, x o,inlet g Water saturation at the inlet,sw inlet 0.12 Water vapour mole fraction at inlet, x w,inlet g 0.11 N 2 mole fraction at inlet, x n,inlet g Capillary pressure p c = p Acosta c /10000 Table 6.4: Operational parameter for the reference scenario Figure 6.1: Distribution of the oxygen mole fraction x o g [-] in the 3D domain Figure 6.1 shows the distribution of the oxygen mole fraction in the 3D domain. The figure demonstrates that oxygen is transported in z-direction and to the reaction zone, where it is consumed. First the transport mechanism in z-direction is discussed. The predominant transport mechanism in negative z-direction for all components is advection due to the applied pressure gradient. This is demonstrated in figure 6.2, where the pressure gradient and the oxygen mole fraction are shown at a cut through the channel at y= m. The high

62 48 6 3D-Simulation with the Darcy approach value of oxygen at the lower boundary occurs since the boundary conditions are the same for inlet and outlet. All other species are also transported in z-direction mainly due to advection. (a) Gas pressure (p g ) (b) Oxygen mole fraction in the gas phase (x o g) Figure 6.2: Distribution of the gas pressure and the oxygen mole fraction in the reference scenario at y= m and assumed steady state For the explanation of the transport mechanism in x-direction a section in the middle of the domain is cut at z=0.0025m. This cut is chosen, since the distribution of the different parameter in the GDL can be well compared and the boundary conditions have the lowest influence at this cut. The distribution of the saturation, the oxygen mole fraction in the gas phase, the temperature and the water vapour mole fraction in the gas phase at this cut are shown in figure 6.3. The saturation in figure 6.3(a) shows a typical distribution in the GDL. The highest values occur directly in the reaction layer, where water is produced during the reaction. In the middle of the reaction layer, where the gas channels are farthest away, the saturation reaches the highest value. Near to the channel the lowest values are reached. The distribution of x o g in figure 6.3(b) is also typical, since oxygen is consumed during the reaction. Therefore, the lowest values occur directly in the reaction layer. Figure 6.3(c) shows the temperature distribution in the system. Heat is produced during the reaction, thus the highest temperature is reached in the reaction layer. Only a small temperature gradient occurs. It needs to be mentioned that cooling effects at the bars of the distributor aren t considered in this model. The distribution of the water vapour mole fraction in the gas phase depends on the temperature distribution. Due to the reason that the differences in the GDL are very small, figure 6.3(d) only shows the distribution in the GDL since the legend is adjusted to the small range in the GDL. This distribution continues in the channel with lower values up to Now the transport processes of these most important species through the GDL in x-direction are briefly described. Figure 6.4 demonstrates the gradients of the different parameter and the predominated transport mechanism. Since water is produced in the reaction layer, a

63 6.2 The transport processes 49 (a) Saturation (S w ) (b) Oxygen mole fraction in the gas phase (x o g) (c) Temperature (T ) (d) Water vapour mole fraction in the gas phase (x w g ) Figure 6.3: Distribution of different parameters in the reference scenario at z=0.0025m and assumed steady state saturation difference arises and water is transported to the gas channel predominately due to capillary forces. The incoming air moves through the membrane mainly due to diffusion. Thus, oxygen is transported to the reaction layer by diffusion since oxygen is consumed in the reaction layer and a concentration gradient occurs. The heat production during the reaction results in a small temperature gradient from the reaction layer to the gas channel. The gradient of water vapour in the gas phase is dependent on the temperature gradient, since more water can be taken up by warmer gas. Thus, also a small water vapour gradient occurs and water vapour in the gas phase diffuses to the gas channel.

64 50 6 3D-Simulation with the Darcy approach water flow diffusion Saturation RL GDL x oxygen in the gas phase RL GDL x heat transport diffusion temperature water in the gas phase RL GDL x RL GDL x Figure 6.4: Gradients and transport mechanism in the reaction layer (RL) and the GDL

65 6.3 Variation of different parameters Variation of different parameters In this section, four of the most important parameters concerning the water management are varied. These parameters are the pressure gradient between inlet and outlet, the permeability in the gas channel, the mole fraction of water in the gas channel and the capillary pressure. The scenario of section 6.2 is considered as a reference scenario in each variation. As mentioned before, all parameters described above are valid for this scenario. The changed parameters of the variations are described at the beginning of each section. In all scenarios, the saturation distribution and the current density will be compared, since the saturation can give important information about the water management in the GDL and the reaction layer and the current density can show, how the performance of the fuel cell is influenced. The saturation and the current density are strongly connected, since more water is produced, if a higher current density prevails. Otherwise, if too much water accumulates in the reaction layer or the GDL, the current density will be lower, since the water can block the pores in both layers and it can cover the reaction sites in the reaction layer. Consequently, not enough reaction gas can reach the reaction zone. This is implemented in the equation for the current density, which was presented in chapter 5. For comparison, the same cut as in section 6.2 is illustrated in each scenario. For each scenario the simulation time when steady state or assumed steady state was reached is given in the figures ( t). If real steady state wasn t reached the time when the simulation stopped is given. In these cases steady state is assumed but it is taken into account in the discussion that the real steady state wasn t reached Effect of different pressure gradients The pressure gradient between inlet and outlet of the gas channel is varied in this section. Four different scenarios are regarded, which are listed in table 6.5. The respective pressure gradient is calculated by dividing the pressure difference by the length of the channel. p = p m (6.2) Scenario inlet pressure outlet pressure (a) p = 0P a P a P a (b) p = 1P a P a P a (c) p = 5P a P a P a (d) p = 10P a P a P a Table 6.5: Varied pressure gradients in the different scenarios Figure 6.5 shows how the liquid water mass, which exists in the system, changes during the simulation time. In the end of the simulation a constant value should arise, which implicates that steady state is reached. The figure demonstrates that simulation (a) and (b) stopped before steady state was reached. This happened due to the fact that the simulations weren t

66 52 6 3D-Simulation with the Darcy approach robust and numerical problems arose. This issue has to be taken into account, while discussing the results and will be further discussed in chapter 7. But the figure clearly shows that more water exists in the system, if a higher pressure gradient prevails. Figure 6.5: Comparison of the water mass in the system for different p Looking at the saturation distribution of the different scenarios in figure 6.6, it can be seen that the pressure gradient between inlet and outlet has a great influence on the amount and distribution of the saturation. The highest values of the saturation, occurring in the middle of the GDL directly at the reaction layer will be strongly decreased, if the pressure gradient increases. Additionally, more and more nodes near the gas channel dry out. As mentioned before, the time ( t), which is given in the figures is the simulation time when steady state or assumed steady state was reached. In figure 6.7, the current density occurring in the reaction zone is illustrated. For a better visualisation, only the section in the middle of the GDL between y=0.005 and y= is shown. If the pressure gradient in the channel increases the current density in the reaction zone will also increase. Scenario saturation current density (a) p = 0P a A/cm 2 (b) p = 1P a A/cm 2 (c) p = 5P a A/cm 2 (d) p = 10P a A/cm 2 Table 6.6: Saturation and current density in the middle of the reaction layer at (0/0.001/0.0025) In table 6.6, the values for the saturation and the current density in the middle of the reaction layer are shown. As mentioned before, the current density is directly influenced by the saturation. The physical explanation is that if a lower water saturation prevails in the reaction layer and the GDL the transport of the reaction gases through the layers is facilitated. More space is available for the gas phase to move through the porous layers, since less

67 6.3 Variation of different parameters 53 (a) Saturation, p = 0P a, t = 197.4s (b) Saturation, p = 1P a, t = 180.6s (c) Saturation, p = 5P a, t = 147.1s (d) Saturation, p = 10P a, t = 102.7s Figure 6.6: Comparison of the saturation distribution of simulations with different pressure gradient, slice at z=0.025m (assumed steady state) pores are blocked by water. Therefore more gas phase can reach the reaction layer. Beside the transport effect, the covering effect of the catalyst sites by liquid water is important. If a smaller amount of water accumulates in the reaction zone, more gas phase is available and less catalyst sites are covered by liquid water Therefore oxygen, which is needed for the reaction can reach more catalyst sites. In this part the different scenarios are looked at more closely. If no pressure gradient is present as in scenario (a), over 80% water exists in the middle of the GDL and the current density is very low. The high amount of saturation allows only a little amount of the gas phase to reach the reaction zone. Water covers the reaction sites and blocks the gas phase to reach the catalysts. Even, if the mole fraction of oxygen in the gas phase is still high enough, the reaction is limited, since not enough gas phase can reach the catalyst sites.

68 54 6 3D-Simulation with the Darcy approach (a) Current density, p = 0P a, t = 197.4s (b) Current density, p = 1P a, t = 180.6s (c) Current density, p = 5P a, t = 147.1s (d) Current density, p = 10P a, t = 102.7s Figure 6.7: Comparison of the current density of simulations with different pressure gradient, slice at z=0.025m (assumed steady state) Figure 6.8 demonstrates that in scenario (a) with no pressure gradient and in scenario (d) with the highest pressure gradient, the oxygen mole fraction is still high at the reaction layer although the reaction is limited. If a pressure gradient exists in the system, the saturation will decrease, since more water can be removed from the GDL and taken up by the gas in channel. This means that more water can be transported away in the channel by advection, because a higher flow prevails. At the same time a higher current density occurs since less water blocks the pores and more reaction gas can reach the reaction zone. The simulation with 1P a/5mm pressure gradient already shows a lower saturation and a higher current density. The step to 5P a/5mm shows the highest change. The further increase of the pressure gradient, e.g. to 10P a/5mm doesn t show a high difference to 5P a/5mm. It seems that much higher pressure gradients don t improve the performance significantly. The given explanations fit with the decrease of the

69 6.3 Variation of different parameters 55 (a) Oxygen mole fraction, p = 0P a (b) Oxygen mole fraction, p = 10P a Figure 6.8: Comparison of the oxygen mole fraction distribution of simulations with different pressure gradient, slice at z=0.025m (assumed steady state) water mass in the system while increasing the pressure gradient. It has to be mentioned that the amount of water mass of scenario (a) and (b) could be even higher, since the simulations aren t yet in steady state. This means an even higher decrease of the current density could occur. Another interesting point is that the GDL dries out increasingly, if the pressure gradient is higher. In the regions near the gas channel, the saturation decreases under 0.05 in scenario (c) and (d). Due to the earlier expressed assumption that no variable switch from both phase to gas phase can occur, the saturation can t decrease any further. But it is highly probable that at these nodes normally only gas phase would exist. This could cause a drying-out of the membrane, if the pressure gradient was further increased or the inlet gases weren t humidified. As earlier mentioned this would also cause an inhibition of the reaction, since water is needed in the membrane to transport the protons. This isn t considered in the model, since the protons are assumed to be always present in a sufficient amount for the reaction. The main focus of this thesis is on the influence of the water distribution in the GDL and blocking of the catalyst sites on the performance, but the issue will be theoretical discussed in chapter 7. Finally, the optimum pressure gradient for the given system should lie between the values of the discussed scenarios. Since the performance wouldn t further increase, if the pressure gradient was higher. Furthermore the risk of a drying-out of the membrane would be prevented, if the chosen pressure gradient wasn t too high. Additionally it needs to be mentioned that the saturation distribution in figure 6.6 shows that water isn t transported sufficiently from the membrane through the GDL to the gas channel. Even, if the pressure gradient is high, much water remains in the reaction zone directly at the membrane and parts of the GDL run dry. In conventional gas distributors,

70 56 6 3D-Simulation with the Darcy approach water transport is dominated by capillary forces. It is guessed that the assumed capillary pressure could be too low. This issue will be considered by variations of the capillary pressure in section Effect of different permeability in the gas channel In the previous section, the real influence of the flow in the gas channel could not exactly be shown since the pressure gradient also changed in the GDL. Therefore, a change of the permeability can help to find out how the water saturation and the current density are directly influenced by the gas channel. So far, permeability 10 times higher than in the GDL is assumed for the gas channel. This isn t a realistic value, since the permeability of the channel has to be much higher. But this lower value has to be assumed, due to the robustness of the simulation. Although this isn t a realistic value, the main processes and physics can be shown, if other parameters are changed. A realistic value for the permeability can be determined with Hagen-Poisseuille s law as it is described in chapter 3.4. Thus it appears that the permeability of a small channel can be calculated by following equation: K ch = b2 ch 12 (6.3) The width of the used gas channel is b = 0.001m, thus a permeability of K ch = m 2, which is ca times higher than in the GDL would be realistic. The scenario with this realistic K ch and a third one with a permeability in the channel 100 times higher than in the GDL are compared in this section. The changed parameters for each scenario are shown in table 6.7. Since the initial condition for the saturation in the GDL was assumed to be equal with the residual saturation, problems occur in the scenarios with a higher K ch because the water is immediately mobile at the interface between the GDL and the gas channel. This water can attain the gas channel with the high permeability at the beginning of the simulation and an immediate variable switch would occur. As already mentioned, convergence problems arise and the simulation ends if a variable switch occurs. Therefore, a smaller value was chosen for the initial condition Sw inlet to avoid the switch at the beginning of the simulation. This switch isn t relevant for the later steady state condtion since the nodes try out again. Scenario permeability in the channel Sw inlet as IC and BC (a) K ch = 10 K GDL m (b) K ch = 100 K GDL m (c) K ch = b 2 ch / m Table 6.7: Varied permeability in the different scenarios The comparison of the water mass in figure 6.9, shows that less water will exist in the system, if a higher permeability in the channel prevails. Again, it has to be considered that scenario (c) isn t run into steady state. As earlier mentioned the simulation with a high difference in the permeability of the GDL and the gas channel wasn t robust and stopped before steady

71 6.3 Variation of different parameters 57 Figure 6.9: Comparison of the water mass in the system Scenario saturation current density (a) K ch = 10 K GDL A/cm 2 (b) K ch = 100 K GDL A/cm 2 (c) K ch = b 2 / A/cm 2 Table 6.8: Saturation and current density in the middle of the reaction layer at (0,0.001,0.0025) state was reached due to numerical problems. But the tendency that less water exists in the domain can even be illustrated. It is also shown that the difference of the water mass between scenario (a) and (b) is much higher than the difference between scenario (b) and (c), although the permeability change is higher. It follows that the water mass doesn t linearly decrease with decreasing permeability in the channel and a still higher permeability in the channel wouldn t influence the performance significantly. In figure 6.10 the saturation distribution demonstrates that an increase of the permeability in the flow channel, implicates a decrease of the saturation, especially in the middle of the reaction layer. Additionally, the GDL dries out increasingly near the channel contact. Nodes with saturation below 0.05 are again assumed to have actually changed to gas phase only. The influence on the current density isn t very high. To demonstrate the small change a smaller slice of the GDL is taken and shown in figure It proves that the current density will be higher, if a higher permeability prevails in the channel. The difference between scenario (b) and (c) is not as high as between (a) and (b). This fits with the theory earlier mentioned, which explains that the change of saturation with decreasing permeability has no linear character. For a better illustration of the small changes the values of the saturation and the current density in the middle of the reaction layer are presented in table 6.8. The variation of the permeability in the channel shows the same results as the variation of the pressure gradient. A higher permeability in the channel results in a better removal of water from the GDL and the reaction layer since a higher flow prevails in the channel. As a result the current density increases and an improvement of the performance of the PEMFC

72 58 6 3D-Simulation with the Darcy approach (a) Saturation, K ch = 10 K GDL, p = 5P a, t = s (b) Saturation, K ch = 100 K GDL, p = 5P a, t = 196s (c) Saturation, K ch = b 2 /12, p = 5P a, t = 117.1s Figure 6.10: Comparison of the saturation distribution of simulations with different permeability in the channel, slice at z=0.025m (assumed steady state)

73 6.3 Variation of different parameters 59 (a) Current density, K ch = 10 K GDL, p = 5P a, t = s (b) Current density, K ch = 100 K GDL, p = 5P a, t = 196.0s (c) Current density, K ch = b 2 /12, p = 5P a, t = s Figure 6.11: Comparison of the current density of simulations with different permeability in the channel, slice at z=0.025m (assumed steady state)

74 60 6 3D-Simulation with the Darcy approach is reached. An important issue is that these variations prove that the flow in the gas channel directly influences the water distribution in the GDL and even the performance of the fuel cell. This couldn t be shown by the variation of the pressure gradient, since the gradient prevails also in the GDL Effect of different water vapour mole fraction in the gas channel To find out how the composition of the incoming gas mixture influences the water management and the performance of the fuel cell, variations of the water vapour mole fraction in the channel are compared in this section. So far, a water vapour fraction at the inlet and outlet of the gas channel of 11% is assumed in the simulations. The remaining fraction is divided in oxygen and nitrogen in the ratio of 1/4. This ratio is retained in all variations. The four scenarios are shown in table 6.9. It has to be mentioned that values are valid at the inlet and the outlet of the gas channel. Scenario inlet mole fraction of water inlet mole fraction of nitrogen (a) x w g = 0.09 x w,inlet g = 0.09 x n,inlet g = (b) x w g = 0.11(ref) x w,inlet g = 0.11 x n,inlet g = (c) x w g = 0.13 x w,inlet g = 0.13 x n,inlet g = (d) x w g = 0.14 x w,inlet g = 0.14 x n,inlet g = Table 6.9: Varied mole fractions at the inlet of the gas channel in the different scenarios Figure 6.12: Comparison of the water mass in the system for different mole fraction in the channel In figure 6.12, the water mass of the different scenarios is compared again. The overall mass of liquid water increases with increasing water vapour mole fraction in the gas channel. This can be also seen, when the water distribution of the different scenarios in figure 6.13 are compared. Scenario (a) has a lower water mole fraction in the gas channel, thus more water can be taken up by the gas steam and transported away. The comparison of this scenario with the reference scenario (b), shows that the saturation directly in the reaction layer and

75 6.3 Variation of different parameters 61 in the overall GDL is lower. Thus, a direct influence of the water vapour content in the gas channel on saturation distribution is assessed. More water can be taken up by the gas in the channel, so more water can be transported through the GDL by capillary pressure and less water accumulates in the reaction layer and the GDL. Scenario (c) and (d), with higher water vapour mole fraction in the gas channel, shows the opposite behaviour. More water accumulates in the middle of the reaction layer, since less water can be taken up by the gas in the channel. Consequently, a smaller amount of water can be transported away. Therefore, the transport of the water through the GDL is limited. A flooding of the gas channel can t be observed with the chosen values for the water vapour content in the inlet gas. A higher value for the x g,inlet g is not reasonable because the water vapour in the incoming gas would immediately condensate, since saturation vapour pressure would be reached. The issue of emerging water in the gas channel will be discussed again in chapter 7. The influence on the current density is shown in figure Less water vapour in the incoming gas leads to an increase of the current density, since less water accumulates in the reaction layer. The other way around, if the water vapour content in the gas channel is increased the current density will be decreased. These phenomena are again due to the already discussed relation between saturation and current density (see 6.3.1). For a more detailed comparison, table 6.10 shows again the values of the saturation and the current density in the middle of the reaction layer. Summing up, the model shows that a lower water vapour mole fraction in the channel results in a better performance. Finally, this could infer that the best solution is to inject dry air to the cathode. But decreasing the water content of the inlet gas can result again in drying-out of the membrane, since too much water is removed from the GDL. In scenario (a) several nodes near the gas channel already run dry. This has to be taken into account, while choosing an adequate value for x g,inlet w. For a more detailed discussion see chapter 7. Scenario saturation current density (a) x w g = A/cm 2 (b) x w g = 0.11(ref) A/cm 2 (c) x w g = A/cm 2 (d) x w g = A/cm 2 Table 6.10: Saturation and current density in the middle of the reaction layer at (0,0.001,0.0025)

76 62 6 3D-Simulation with the Darcy approach (a) Saturation, x g,inlet w = 0.09, p = 5P a, t = 165.8s (b) Saturation, x g,inlet w = 0.11, p = 5P a, t = s (c) Saturation, x g,inlet w = 0.13, p = 5P a, t = 121.8s (d) Saturation, x g,inlet w = 0.14, p = 5P a, t = 116.4s Figure 6.13: Comparison of the saturation of simulations with different water vapour mole fraction in the channel, slice at z=0.025m (assumed steady state)

77 6.3 Variation of different parameters 63 (a) Current density, x g,inlet w = 0.09, p = 5P a, t = 165.8s (b) Current density, x g,inlet w = 0.11, p = 5P a, t = s (c) Current density, x g,inlet w = 0.13, p = 5P a, t = 121.8s (d) Current density, x g,inlet w = 0.14, p = 5P a, t = 116.4s Figure 6.14: Comparison of the current density of simulations with different water vapour mole fraction in the channel, slice at z=0.025m (assumed steady state)

78 64 6 3D-Simulation with the Darcy approach Effect of different capillary pressure For the capillary pressure of GDL, very little data is available, since no adequate measuring technique for the very small dimension of the GDL exists and there are only measuring techniques for hydrophilic materials available, which may not provide adequate data for the hydrophobic GDL. As mentioned before, the determined values for the p c (S w ) relationship of [Acosta et al.(2006)] are very high in comparison of data reported by other authors for similar material (see [Gostick et al.(2005)] or [Nguyen et al.(2008)]). Therefore, p c (S w ) is divided by 10,000 in the reference scenario. Additionally, this factor is varied between 1,000 and 10,000 in this section to investigate the influence of the capillary pressure saturation relation on the water distribution in the GDL and the performance of the overall fuel cell. Scenario (a) p c /10000 (ref) (b) p c /8000 (c) p c /5000 (d) p c /2000 (e) p c /1000 capillary pressure p c = p Acosta c /10000 p c = p Acosta c /8000 p c = p Acosta c /5000 p c = p Acosta c /2000 p c = p Acosta c /1000 Table 6.11: Varied capillary pressure in the different scenarios Figure 6.15: Comparison of the water mass in the system for different capillary pressure Figure 6.15 shows the water mass over the time in the system of each scenario. The variation of the capillary pressure (p c ) doesn t show a typical change of the water content in the system as may be first assumed. To describe the behaviour, the best way is to start from scenario (e) with the highest capillary pressure. Decreasing of p c results first in an increase of the liquid water mass in the system (see scenario (d) and (c)). But if p c is further decreased, the overall water mass in the system will decrease again. This issue will be now discussed in detail but for a better explanation, the saturation distribution and current density of the different scenarios will be shortly described first.

79 6.3 Variation of different parameters 65 (a) Saturation, pc = pc Acosta /10000, p = 5P a, t = s (b) Saturation, pc = pc Acosta /8000, p = 5P a, t = 180.3s (c) Saturation, pc = pc Acosta /5000, p = 5P a, t = 145.0s (d) Saturation, pc = pc Acosta /2000, p = 5P a, t = 137.1s (e) Saturation, pc = pc Acosta /1000, p = 5P a, t = 126.6s Figure 6.16: Comparison of the saturation distribution of simulations with different capillary pressure, slice at z=0.0025m (assumed steady state)

80 66 6 3D-Simulation with the Darcy approach In figure 6.16, it can be seen that the highest saturation occurs in the reference scenario (a), where the lowest p c prevails. The difference between the highest and lowest value in the GDL is about 0.3. Near the gas channel, the GDL dries out and normally gas phase would occur. Scenario (b) shows a similar distribution. The parts near the gas channel run dry and the ratio between highest and lowest value is about The distribution changes in scenario (c) to (e). It can be clearly seen that the parts near the channel still contain water and wouldn t even change to gas phase, if the prohibition of variable switch wasn t made. Another point is that the differences between highest and lowest value are much smaller in theses scenarios. In scenario (c) the difference is only 0.12 and in (d) and (e) the value is only about For a better visualisation of scenario (c), (d) and (e), figure 6.17 shows the saturation distribution in a smaller range. (a) Saturation, pc = pc Acosta /5000, p = 5P a, t = 145.0s (b) Saturation, pc = pc Acosta /2000, p = 5P a, t = 137.1s (c) Saturation, pc = pc Acosta /1000, p = 5P a, t = 126.6s Figure 6.17: Comparison of the saturation distribution of simulations with higher capillary pressure, slice at z=0.0025m (assumed steady state)

81 6.3 Variation of different parameters 67 (a) Current density, pc = pc Acosta /10000, p = 5P a, t = s (b) Current density, pc = pc Acosta /8000, p = 5P a, t = 180.3s (c) Current density, pc = pc Acosta /5000, p = 5P a, t = 145.0s (d) Current density, pc = pc Acosta /2000, p = 5P a, t = 137.1s (e) Current density, pc = pc Acosta /1000, p = 5P a, t = 126.6s Figure 6.18: Comparison of the current density of simulations with different capillary pressure, slice at z=0.0025m (assumed steady state)