MODELLING A 500W POLYMER ELECTROLYTE MEMBRANE FUEL CELL

Transcription

1 University of Technology, Sydney Faculty of Engineering MODELLING A 500W POLYMER ELECTROLYTE MEMBRANE FUEL CELL by Andrew Richard Balkin Student Number: Major: Electrical Engineering Supervisor: Dr Joe Zhu Co-supervisor: Mr Brett Holland A 1 Credit Point Project submitted in partial fulfilment of the requirement for the Degree of Bachelor of Engineering A June 00

2 SYNOPSIS The primary task of this project was to model a 500W Polymer Electrolyte Membrane (PEM) fuel cell. A model based on an actual 500W hydrogen fuelled PEM fuel cell at UTS is given. The modelling of the PEM fuel cell requires a multi disciplinary approach that includes the study of electrochemistry, heat transfer, fluid dynamics, electroplating and electronics. This thesis assists the UTS Masters student, Mr Brett Holland, who is currently implementing the 500W PEM fuel cell project. His project includes building a fuel cell system, modelling the system and validating the model against measured data. This thesis has focused on the second stage of the Masters project, specifically the modelling of the PEM fuel cell itself and not the entire fuel cell system which includes auxiliary components. At the time of writing this report the fuel cell power system was not ready for validation of this fuel cell model. This task is to be completed when the fuel cell system is operational. This thesis includes a comprehensive literature review of PEM fuel cell modelling and states the fuel cell model that is to be implemented for the fuel cell system simulation. Parameter identification for the model has been addressed. The methods required to evaluate the constants essential to the model are described. This thesis is a reference for PEM fuel cell modelling in general and also specifies a complete model of the 500W fuel cell at UTS. Capstone Project A0-080 ii

3 DECLARATION OF ORIGINALITY The work contained in this project, other than that specifically attributed to another source, is that of the author. I have not used fragments of text from other sources without proper acknowledgment. Theories, results and designs of others have been appropriately referenced and all sources of assistance have been acknowledged. Andrew Richard Balkin Capstone Project A0-080 iii

4 ABSTRACT The primary task of this project was to model a 500W Polymer Electrolyte Membrane (PEM) fuel cell. A model was required for the design and control of the 500W hydrogen fuelled PEM fuel cell at UTS. The modelling of the PEM fuel cell requires a multi disciplinary approach including the study of electrochemistry, heat transfer, fluid dynamics, electroplating and electronics. This project was conducted in collaboration with Mr. Brett Holland, a UTS Masters student, who is currently implementing a 500W PEM fuel cell testing rig. A survey was conducted on fuel cell modelling from both an analytical and empirical perspective. The development and comparative analysis of PEM fuel cell modelling techniques are presented and a model was proposed that includes both analytical and empirical features. Initially a general PEM fuel cell model was given. This model was then applied specifically to the 500W fuel cell at UTS. The analytical data and experimental procedures required for determining the parameter values used in the proposed model are specified. In addition, three modelling packages CFD-ACE, Trnsys and Engineering Equation Solver were investigated for incorporation of the model into the fuel cell system design. Capstone Project A0-080 iv

5 ACKNOWLEDGEMENTS I firstly acknowledge the universal being that is the creator of this world. Although I do not fully understand the process I acknowledge the awesome presence of the creator. I would like to thank Mr Brett Holland who has spent time with me beyond his duty and without his help I would have been wasting my time. Dr Joe Zhu has been a guide for the general direction and organisation of this project and I am grateful for his ongoing help. Lastly I would like to thank my friends and family, in particular my parents who have guided me through university. Capstone Project A0-080 v

6 TABLE OF CONTENTS SYNOPSIS... II DECLARATION OF ORIGINALITY...III ABSTRACT...IV ACKNOWLEDGEMENTS... V TABLE OF CONTENTS...VI LIST OF FIGURES...IX LIST OF TABLES...XI NOMENCLATURE...XII 1. INTRODUCTION...1. POLYMER ELECTROYTE MEMBRANE FUEL CELLS Fundamental Operation...4. Structure Polymer Electrolyte Membrane Electrodes Gas Diffusion Backing Bipolar Plate Fuel Cell Voltage Reversible Cell Voltage Activation loss Ohmic Loss Concentration Loss Internal Current Loss Efficiency Water Management Self Humidification Modelling Terminology Stoichiometry Partial Pressure Temperature Relative Humidity Charge Double Layer UTS Fuel Cell LITERATURE REVIEW ON PEM FUEL CELL MODELLING Analytical Simple Analytical Models Complex Analytical Models Empirical Historical Background Empirically Modelling the Entire Polarisation Curve...31 Capstone Project A0-080 vi

7 3..3 Empirically Modelling each Loss Electrical Equivalent Circuit Experimentation General Results Current Interrupt Method Electrochemical Impedance Spectroscopy Fast Auxiliary Pulse PROPOSED MODEL... ERROR! BOOKMARK NOT DEFINED. 4.1 Reversible Voltage Gibbs Free energy for V reversible Pressure Variation Temperature Variation Ohmic Loss Extreme Operating Conditions Activation Loss Cathode Activation Loss Anode Activation Loss Exchange Current Concentration Loss Internal Current Stoichiometric Ratio Air and Oxygen Hydrogen Water Transport Relative Humidity of Hydrogen Relative Humidity of Air General Model Summary W Fuel Cell Model Simplest Model Charge Double Layer Without Concentration Loss With Concentration Loss PARAMETER DETERMINATION Current Interrupt Method Manual Interrupt Valid Results DC Offset Molten Carbonate Fuel Cell Electrochemical Impedance Spectroscopy Simple Method Commercial Products Result Analysis Reversible Voltage Resistance Resistance with an Internal Current...93 Capstone Project A0-080 vii

8 5.5 Activation Loss and Internal Current Ideal Activation Loss Non-Ideal Activation Loss Simplified Non-ideal Activation Loss Concentration Empirical Techniques MODELLING PLATFORMS CFD-ACE Trnsys Engineering Equation Solver CONCLUSION AND RECOMMENDATIONS REFERENCES APPENDICES Appendix 1 Equivalent Terms Appendix Schematic of UTS Fuel Cell System Appendix 3 Membrane types by Dupoint Appendix 4 Mathematica Code, Parameter Determination of Activation Loss...18 Capstone Project A0-080 viii

9 LIST OF FIGURES Figure 1: Basic PEM Fuel Cell Operation...5 Figure : Simplified Fuel Cell []...6 Figure 3: Typical Fuel Cell Stack...7 Figure 4: Typical Polarisation Curve...10 Figure 5: Typical Losses in a PEM Fuel Cell...10 Figure 6: Normal PEM With External Water - Saturator...15 Figure 7: Self-Humidifying PEM...16 Figure 8: Simple Equivalent Circuit...19 Figure 9: Photo of the UTS 500W PEMFC + the Control Unit...0 Figure 3 1: Empirical Fit of Polarisation Curve [39]...3 Figure 3 : Fit of Voltage With Activation and Ohmic Losses [39]...33 Figure 3 3: Fit of Concentration Loss [39]...33 Figure 4 1: Concentration Loss...64 Figure 4 : Effect of Relative Humidity of Hydrogen on the Polarisation Curve.68 Figure 4 3: Effect of Relative Humidity of Air on the Polarisation Curve...69 Figure 4 4: Polarisation Curve...7 Figure 4 5: Equivalent Circuit Including Concentration Loss...75 Figure 5 1: Current Interruption on a PEM Fuel Cell...78 Figure 5 : Current Interrupt Test Set-Up...79 Figure 5 3: Interruption Techniques...80 Figure 5 4: Testing Circuit...81 Figure 5 5: DSO Screen...8 Figure 5 6: DC Offset Set-Up...8 Figure 5 7: New DSO Screen...83 Figure 5 8: Ohmic Loss [46]...84 Figure 5 9: Activation Loss [46]...84 Figure 5 10: Concentration Loss [46]...85 Figure 5 11: Log Scale Overview [46]...85 Figure 5 1: Basic EIS Circuit...86 Figure 5 13: EIS With Commercial Products...88 Figure 5 14: EIS at Cathode and Anode...89 Figure 5 15: EIS of Total Fuel Cell at Low Currents...89 Figure 5 16: EIS of Total Fuel Cell at High Currents...90 Figure 5 17: Polarisation Curve for a Solid Oxide Fuel Cell...90 Figure 5 18: Equivalent Circuit for a Solid Oxide Fuel Cell...91 Figure 5 19: Results of an Equivalent Circuit...91 Figure 5 0: Examples of Nyquist and Bode Plot Plane Plots...9 Figure 5 1: Ideal Activation Loss With and Without Internal Current...96 Figure 5 : Ideal Activation Loss With and Without Internal Current (Zoom)...97 Figure 5 3: Activation Loss vs Ln(Current) Without Internal Current...97 Figure 5 4: Activation Loss vs Ln(Current+Internal Current)...98 Figure 5 5: UTS Fuel Cell Activation Loss With Modelled Data Figure 5 6: UTS Fuel Cell Polarisation Curve With Modelled Data Capstone Project A0-080 ix

10 Figure 5 7: SS for Various Internal Current Values Figure 5 8: SS for Various Internal Current Values (Zoom) Figure 5 9: Polarisation Curve With Upper and Lower Worst Fit Figure 5 30: Replot of Figure Figure 5 31: Activation Loss Example Figure 5 3: Activation With Internal Current Figure 5 33: Activation Loss With No Internal Current Figure 5 34: Expected Polarisation Curve Using the Simpler Method Figure 5 35: Concentration Loss Effect Figure 6 1: Effect of Temperature on the Stack Figure 6 : Trnsys Interface Figure 6 3: Predicted Polarisation Curves Figure 9 1: Schematic of UTS Fuel Cell System...17 Figure 9 : Legend for Schematic of UTS Fuel Cell System...17 Capstone Project A0-080 x

11 LIST OF TABLES Table 1: Definition of Reversible Voltage and Losses...9 Table 3 1: Kim et al. Empirical Equation Terms...3 Table 4 1: Terms Relevant to Free Energy...40 Table 4 : Affect of Operating Conditions Table 9 1: Terms Equivalent to PEM Fuel Cell Table 9 : Terms Equivalent to the Losses in a Fuel Cell...16 Table 9 3: Nafion Membrane Thicknesses Table 9 4: Output Data from the Mathematica Program...19 Capstone Project A0-080 xi

12 NOMENCLATURE α x x E loss F e F ec Charge transfer coefficient of the anode or cathode Change in variable x Equivalent to the concentration loss Standard free energy of activation for the cathode reaction Standard free energy of activation for chemical absorption 0 g f Change in Gibbs free energy of formation per mole at STP g K Change in Gibbs free energy per mole at STP Change in Gibbs free energy per mole at temp T and standard P g T h f Molar enthalpy of formation h K Change in the molar enthalpy of formation at STP h T h at temperature T and the standard pressure f s f s K s K Change in the molar entropy at STP s T Change in the molar entropy at T and the standard pressure δ x Empirical coefficients for V actloss ε x φ x λ x Empirical coefficients for V ohmloss Relative humidity of hydrogen or air Stoichiometry for hydrogen or air Capstone Project A0-080 xii

13 ζ x σ σ b σ m Empirical coefficients for V CELLactLOSS Calculated uncertainty for each y value Uncertainty in the intercept b Uncertainty in the slope m A Area of each cell in the stack (cm - ) b B a x A stack A T b b K Cd C proton C x EES EIS y-intercept in least squares method Empirical coefficient for V conloss Activity of the substance Tafel slope of the stack Tafel slope of one cell Predicted y-intercept by the least squares method Tafel slope (mv / decade), Kim et al. Double layer capacitance Total concentration of protons in the membrane Concentration of H, O or H O Engineering Equation Solver Electrochemical Impedance Spectroscopy F Faraday s constant ( C mol -1 ) G G f products G f reactants G f Gibbs free energy Gibbs free energy of formation. Gibbs free energy of all products Gibbs free energy of all reactants h f Molar enthalpy of formation i Current of the fuel cell or stack (A) Capstone Project A0-080 xiii

14 i L i n i 0.x i n Limiting current for the concentration loss of the stack (A) Internal current of the stack (A) Exchange current of the fuel cell at the anode or cathode (A) Internal current of the stack (A) j Current density of the fuel cell or stack (A/cm ) j 0 Exchange current density of the fuel cell or stack (ma.cm - ) k x 0 m Intrinsic Rate constant for the anodic or cathodic reaction Predicted linear slope by the least squares method m * x Mass flow of air or hydrogen, in kg.s -1 MEA m K n Membrane Electrode Assembly Concentration loss coefficient (mv) from Kim et al. Number of data points N A Avagadro s Number, 6.0*10 3 n x Number of electrons transferred per mole of the electrolysed component of the anodic or cathodic reaction N cell n K Number of cells in the stack, 3 for UTS stack Concentration loss coefficient (cm / ma) from Kim et al. * p x P 0 PEM PEMFC Partial pressure of species x Standard pressure in the pressure units used Polymer Electrolyte Membrane Polymer Electrolyte Membrane Fuel Cell R Universal gas constant, J K -1 mol -1 R CELLelectronic Resistance of the non-ideal electrodes of each cell (Ω) Capstone Project A0-080 xiv

15 R CELLmembrane Resistance of the membrane of each cell (Ω) R cell,k Ohmic resistance (Ω.cm ) from Kim et al. Ract Rct Re Activation equivalent resistance charge transfer resistance (Ω) electrolyte and contact resistance (Ω) R electronic Resistance of the non-ideal electrodes of the stack (Ω) R membrane Resistance of the membrane of the stack (Ω) Rohm Resistance of the membrane and electrodes R ohmic Total resistance of the fuel cell stack (Ω) s Molar entropy SS i n Differential of SS with respect to i n SS STP T Sum of the Squares of the residuals Standard temperature and pressure, 98.15K and 101.3kPa Temperature of the fuel cell (K) T ref Reference temperature, 98.15K V 0 Reversible fuel cell voltage at STP, 1.9V V 0,K Open circuit potential of the cell (mv) from Kim et al. V actloss Activation loss of the stack V cell Cell voltage (V) V CELLactLOSS Activation loss of a fuel cell (V) V CELLconLOSS Concentration loss of the a fuel cell (V) V CELLohmLOSS Ohmic loss of a fuel cell (V) V CELLReversible Reversible voltage of each cell (V) Capstone Project A0-080 xv

16 V conloss V fuelloss V ohmloss V Nernst V Reversible difference V T V stack y y c Zd Concentration loss of the fuel cell stack (V) Loss due to wasted fuel in the cell (V) Ohmic loss of the fuel cell stack (V) Reversible cell voltage at the standard temperature (V) The maximum theoretical (reversible) voltage of a stack (V) Change in V CELLReversible due to the temperature of the stack (V) Experimental voltage of the fuel cell stack (V) Activation loss of the stack Calculated y value by least squares method Concentration impedance Capstone Project A0-080 xvi

17 1. INTRODUCTION Fuel cells have been five years behind a mainstream application since the 1960 s. It has always been said that the fuel cell technology will take off in five years. However while fuel cells have advanced other technologies have as well and fuel cells have remained a few years behind. Fortunately fuel cells are now coming into the market. One of the promising areas for fuel cells is the automotive industry which already have buses and cars running on fuel cells. Polymer Electrolyte Membrane (PEM) fuel cells are now an efficient and commonly used power source. The goal now is to fully penetrate well-established markets. PEM fuel cells are non-emission energy conversion devices, and their main byproduct is water. Hydrogen and oxygen (air) are used as the fuel and oxidant respectively. Hydrogen does not occur naturally so it must be converted from another source. The desire is to use existing sustainable energy techniques such as wind power to produce the hydrogen using electrolysis. The advantages of fuel cells are many. They have, the potential for zero emissions, high efficiency, quiet operation, high energy density, plus high reliability and long life due to few moving parts. For this report a comprehensive literature survey has been conducted. The findings of the literature survey include fuel cell models that fit into two broad categories, that is, analytical and empirical. Analytical models are theoretically based, whereas empirical models are experimentally based. In this report a detailed model specific to the 500W fuel cell at UTS is given. The model outlines characteristics of a PEM fuel cell of 500W but it does not relate directly to the dimensions of the UTS fuel cell. It was found to be difficult to obtain information specific to the dimensions and the make up of the UTS fuel cell as this was considered proprietary information by the manufacturer. In reply to our queries the manufacturer explained, We cannot offer details on the makeup. These details are sometimes the same as any other fuel cell stack. Capstone Project A

18 In respect to the 500W PEM fuel cell model the experiments necessary for parameter determination are given. This ensures that the model is complete and practical. Furthermore this thesis summarises three packages used for detailed fuel cell modelling, CFD-ACE, Trnsys and Engineering Equation Solver. Conclusions and recommendations are expressed that summarise the project and propose future work to be carried out. This document has nine main sections, each section flows on from the last. This thesis gives a detailed review on past PEM fuel cell modelling, details of the proposed model with parameter determination, example platforms of low level modelling with conclusions and future directions of the project. The nine main sections are: 1. Introduction: An introduction to this capstone project and fuel cells are given here, the basic flow of the document is also discussed.. Polymer Electrolyte Membrane Fuel Cells: The main concepts of PEM fuel cells and fuel cell modelling are introduced here. This section is required to give a background to the terms and concepts so that the literature review and proposed model can be properly understood. 3. Literature Review on PEM Fuel Cell Modelling: A comprehensive literature review of PEM models is given. Analytical and empirical models are examined along with their validity and application. 4. Proposed Model: A general PEM fuel cell model and a model specific to the 500W UTS fuel cell are given. Detailed background, justification and assumptions are specified. This model contains analytical modelling techniques, and where these are not practical, the model uses empirical methods. 5. Parameter Determination: To reinforce that the proposed model is not only valid but also practical, the experiments and procedures to determine the parameters in the proposed model are given. Capstone Project A0-080

19 6. Modelling Platforms: Three low level modelling programs CFD-ACE, Trnsys and Engineering Equation Solver are discussed including their background and application to fuel cell modelling. 7. Conclusions and Recommendations: Final remarks on the capstone project and recommendations of the future direction of fuel cell development at UTS are discussed here. 8. References: This section gives a complete list of references used in this project and report. 9. Appendices: Relevant information complementing the body of the report such as tables and programming code is given here. Capstone Project A

20 . POLYMER ELECTROYTE MEMBRANE FUEL CELLS A Polymer Electrolyte Membrane Fuel Cell (PEMFC) or an equivalent term (shown in appendix 1) is a fuel cell that uses a thin ion conducting solid electrolyte. A solid electrolyte, the polymer electrolyte membrane has its advantages over liquid electrolytes in that it has a high power density and reduced corrosion. The low temperature of these cells ensures a quick start up time and its wide application range includes the transport industry. One of the drawbacks to PEMFCs in the past was the fact that they required expensive platinum metal catalysts. Developments in recent years mean that only minute amounts of platinum are now used and the cost of platinum is now a small part of the total price of a PEM fuel cell [1]. The large number of applications for PEMFCs is mainly due to the enormous power range and versatility of the fuel cell. The power range is from a few watts to several hundred kilowatts, with temperature only varying from 5 0 C to C. PEMFCs are particularly suited to the transport industry because they can start quickly due to a low operating temperature. Also there are no corrosive fluid hazards and the cell can work in any orientation. These factors set the PEMFCs apart from other fuel cells. This and the potential for PEMFCs to produce zero emissions creates a great prospect for clean energy in the transport industry..1 Fundamental Operation PEM fuel cells use hydrogen and oxygen to produce electricity, heat and water. The physics of the PEM fuel cell can be considered as the opposite of electrolysis. In electrolysis an electric current is passed through water to produce hydrogen and oxygen. In the fuel cell, hydrogen and oxygen gases are passed either side of a polymer electrolyte membrane which produces an electric current, heat and water. The electro-chemical reactions for the PEM fuel cell are shown below. Anode reaction H H e + + ( 1) Cathode reaction + O + H + e H O 1 ( ) Overall reaction 1 O + H H O ( 3) Capstone Project A

21 A PEMFC primarily consists of three components: a negatively charged electrode (cathode), a positively charged electrode (anode), and a polymer electrolyte membrane. The simple chemistry of the fuel cell involves hydrogen being oxidized on the anode and oxygen reduced on the cathode. Protons (H + ) are transported from the anode to the cathode through the polymer electrolyte membrane and electrons are carried to the cathode through an external circuit. On the cathode, oxygen reacts with protons and electrons forming water and producing heat. Both the anode and the cathode contain a catalyst, usually platinum, to speed up the electrochemical processes. The basic PEM fuel cell operation is seen from figure -1 below, modified from []. Figure 1: Basic PEM Fuel Cell Operation In a PEM fuel cell the main inputs to the system are the fuel, hydrogen and the oxidant, oxygen found in air. Outputs include DC electricity, heat and water. This is shown overleaf in figure -. Capstone Project A

22 Figure : Simplified Fuel Cell [] A fuel cell system includes a fuel cell stack and auxiliary components: air blower, control system, inverter, reformer, heat exchanger and sometimes a compressor. It is this entire system that will be built and modelled by Mr Brett Holland. A schematic for this system can be seen in appendix.. Structure A normal PEM fuel cell will produce about one volt. To create a larger system the area of each cell (A) can be increased and/or several cell units can be connected in series to create a stack. The output voltage depends on the voltage of each cell and the number of cells in the stack. A typical PEMFC stack structure is shown [3] in figure -3 overleaf. The fuel cell stack is shown on the right of the figure and an exploded view of one cell on the left. A MEA is a Membrane Electrode Assembly which consists of the polymer electrolyte membrane and two electrodes. Capstone Project A

23 Figure 3: Typical Fuel Cell Stack..1 Polymer Electrolyte Membrane As discussed the polymer electrolyte membrane (sometimes called just membrane) allows proton conduction to complete the electric circuit in the fuel cell. It also has several other key functions. The membrane hinders electron conduction so the electricity flows through the external circuit that is through the load. Ideally, the membrane prevents any hydrogen or oxygen from being transported through it. Nafion by Dupoint is used as the de facto standard for polymer electrolyte membranes. It is made with various thicknesses which are shown in appendix 3. Polymer electrolyte membranes must be hydrated to be conductive. This limits the operating temperature of PEM fuel cells to the boiling point of water and makes water management a key issue in PEM fuel cell development [3]. Some membranes are self-humidifying membranes that make use and actually encourage a small fuel and oxidant crossover. Non-ideal properties in the membrane are the cause for losses in the fuel cell. Although each key function of the membrane is essential to the operation of the fuel cell, every variation from the ideal membrane performance is the root of an energy loss. The most significant imperfection is a finite ion conduction rate. This gives the membrane an effective resistance that changes with temperature, hydration and current. Another non-ideal property is fuel and oxidant conduction (fuel crossover). This results in a reaction of hydrogen and oxygen without resulting in an external Capstone Project A

24 current. The last and least important imperfection in the membrane is electron conduction. This only occurs at minute levels... Electrodes The bulk of the electrochemical reactions occur on the surface of the two electrodes, the anode and cathode of the fuel cell. The electrodes are made highly porous. This has the affect of increasing the surface area by hundreds and even thousands of times of the straightforward length by width of the surface [1]. This microstructure design is also improved with the use of catalysts within the electrodes. These aid the speed of the electrochemical reactions that take place on each electrode. Virtually always platinum or an alloy of platinum and other noble metals are used as the catalyst [3]. These greatly reduce the activation losses that occur at the electrode surface. The reaction at the cathode is particularly slow. The catalyst platinum assists this reaction, although a significant activation loss is still seen...3 Gas Diffusion Backing As can be seen in figure -3 the membrane electrode assembly has a gas diffusion backing either side of the membrane. This provides three main functions: firstly, it makes an electrical contact between the bipolar plates; secondly, the diffusion backing also distributes the hydrogen and air (oxygen) to the electrodes; and lastly, allows the product water to flow between the electrode and the flow channels...4 Bipolar Plate In a fuel cell stack, bipolar plates separate the reactant gases of adjacent cells, connect the cells electrically, and act as a support structure [3]. The bipolar plate has two sides as its name suggests. On one side it allows hydrogen to flow to the diffuse backing and electrode and provides the electrical connection for each separate electrode. On the other side it allows air to flow to the diffuse backing and provides the electrical connection for the cathode. Capstone Project A

25 .3 Fuel Cell Voltage The cell voltage is the most important aspect of fuel cell modelling. This is because the cell potential changes with different loads and operating conditions so it is essential that the cell voltage is known. The individual cell voltage and then the stack voltage are found by modelling the maximum cell voltage and the major voltage losses. The losses include the activation, ohmic, concentration and internal current losses. The internal current losses are modelled by an internal current integrated into the other loss mechanisms. These terms are described in table -1. Table 1: Definition of Reversible Voltage and Losses. V Reversible V actloss V ohmloss V conloss V fuelloss Reversible cell voltage. The maximum theoretical voltage of a stack. The activation loss, which is the loss related to driving the chemical reaction that occurs on the electrodes. Ohmic losses are due to resistance to proton flow across the membrane, also included in this loss is the resistance originating from the nonideal electrodes and the electrode interconnections. Concentration or mass transport losses, this loss relates to the reduction of the fuel s concentration in the gas channels. The fuel concentration at the inlet valve is greater than at various points in the gas channel. This loss relates to the wasted fuel that passes through the membrane that does not produce any useful work and to a lesser extent electron conduction in the membrane. This loss is equivalent to an internal current. A polarisation curve is a characteristic graph of voltage versus current for a set of operating conditions. This curve is the most common output of models and is seen as the most important performance criteria. Modelling this graph is the primary goal of the model proposed. A typical polarisation curve from Wood et al. [4] is shown in figure -4. Capstone Project A

26 Figure 4: Typical Polarisation Curve The polarisation curve in figure -5 below gives an indication of the typical contribution of the losses [5]. Figure 5: Typical Losses in a PEM Fuel Cell Capstone Project A

27 A typical error in PEM fuel cell modelling is shown in figure -5. The open circuit potential of the cell is a little below one volt and yet the open circuit potential is quoted as the same potential as the reversible voltage, 1. volts. The activation loss is evident at a zero external current because of an internal current in the fuel cell. This fact is often skimmed over because of the difficulty in determining the internal current. Also missing from this polarisation curve is the concentration loss. This omission is through no fault of the author. Some fuel cells are not operated at high enough currents to ever see the effects of this loss..3.1 Reversible Cell Voltage The reversible cell voltage V CELLreversible, is the maximum voltage that each cell in the stack can produce at a given temperature with the partial pressure of the reactants and products known. The reversible stack voltage is equal to the number of cells in the stack multiplied by the reversible voltage for each cell. The maximum cell voltage is derived from the maximum amount of energy that is available to do useful work, that is the change in Gibbs free energy per mole. This derivation as well as a full discussion of the losses is in section 4 of this report..3. Activation loss The activation loss of the fuel cell relates to the slowness of the reactions that take place on the surface of the electrodes. A proportion of the voltage generated is lost in driving the chemical reaction at the electrodes. It is called an activation loss because it relates to the activation energy required at both the anode and cathode of the fuel cell. In a PEM fuel cell the activation loss at the hydrogen-anode is much smaller than the oxygen-cathode so it is often neglected. The chemical processes contributing to the activation loss are complex. They involve absorption of reactant species, transfer of electrons across the fuel cell s double layer, desorption of product species, and the nature of the electrode surface []. A scientist, Tafel, gave a simplified semi-empirical equation that describes this loss, as can be seen in figure -5 (this loss levels out somewhat at higher currents). Capstone Project A

28 .3.3 Ohmic Loss The ohmic loss is due to the resistance of ions through the polymer electrolyte membrane and the resistance of imperfect electrodes. For a complete electrical circuit the membrane is required to conduct protons. An ideal membrane would freely conduct H + ions. Some PEM fuel cells have only a very small electrode resistance and it can be omitted from the model. The slope of the cell voltage in the middle section of the polarisation curve is due to the ohmic loss. The loss in the fuel cell is approximately linear after the activation loss levels out and before the concentration loss becomes significant. The ohmic loss of the PEMFC is known to be slightly non-linear and variable due to the characteristics of ionic conduction at different conditions. Some models solve this problem empirically and others omit this fact..3.4 Concentration Loss A concentration loss relates to the reduction of the reactant s concentration in the gas channels. The fuel and oxidant are used at the surface of the electrodes. The incoming gas must then take the place of the used reactant. The concentration of the fuel and oxidant is reduced at the various points in the fuel cell gas channels and is less than the concentration at the inlet value of the stack. This loss becomes significant at higher currents when the fuel and oxidant are used at higher rates and the concentration in the gas channel is at a minimum. This is sometimes called the gas-diffusion loss or diffusion loss because the concentration of the inlet gases are linked to how quickly they travel through the Gas Diffusion Backing. The term diffusion is explained further in section Internal Current Loss The internal current is due to the wasted fuel and oxidant that passes through the membrane and does not produce any useful work. It is also to a lesser extent, due to electron conduction in the membrane. This loss affects the fuel cell s performance most at open circuit and is insignificant at higher currents. It is mathematically represented in each of the other losses and often absent from PEM fuel cell models. An internal current loss means that excess chemical activity in the fuel cell is always Capstone Project A

29 occurring. Even at open circuit voltage, losses are evident when ideally there are none. Since the activation loss is the most pronounced loss at low currents, the internal current affects the activation loss the most significantly..3.6 Efficiency Electrical and heat energy is produced by the overall reaction, equation (-3). Theoretically, the Gibbs free energy of the reaction is available as electrical energy and the rest of the reaction enthalpy is released as heat. In practice, a part of the Gibbs energy is also converted into heat via the loss mechanisms [3]. The thermal efficiency of an energy conversion device is defined as the amount of useful energy produced relative to the change in stored chemical energy (commonly referred to as thermal energy) that is released when a fuel is reacted with an oxidant []. The amount of useful electrical energy in the system is the Gibbs free energy of the reaction. Some energy in the system is not available to do useful work and is always output as heat. The maximum efficiency of a fuel cell reaction is [1], Maximum Efficiency = Maximum _ Electrical _ Energy Thermal _ Energy = g h f f ( 4) By convention the value of the change in Gibbs free energy for the fuel cell reaction g and the value of thermal energy, commonly called the calorific value or change f in enthalpy of formation h, are always negative. Because of this the numerator f and denominator in equation (-4) are both positive. There is a discrepancy to the amount of thermal energy in the fuel cell reaction. It relates to the product of the combustion reaction for hydrogen and oxygen. Capstone Project A

30 When the product is steam the value of h is referred to as the lower heating f value. H + O H O( steam) h = kj/mole f When liquid water is the product, h is called the higher heating value because a f higher amount of heat can be output from the reaction. H + O H O( liquid) h = kj/mole f It is more appropriate to define the maximum efficiency in terms of the higher heating value (HHV). If all the energy from the enthalpy of formation (using the HHV) were transferred into electrical energy the potential of the cell would be 1.48 volts, so the cell efficiency can now be defined as, Cell Efficiency = V cell 1.48 ( 5).4 Water Management As discussed in section., the membrane must be hydrated to ensure it can conduct protons. This is a key issue in the operation of the fuel cell. Water management issues are the most important design problems that occur in the PEMFC. Various problems arise if the membrane is under hydrated or if the fuel cell is flooded. Initial development of PEMFCs by NASA and General Electric was discontinued because of water management issues and also at that time a lot of platinum was required as a catalyst. Ballard Power Systems has been at the forefront of PEMFC technology and has overcome both the water management issues and high catalyst s costs. A fully saturated membrane without containing too much excess water in the fuel cell is the main goal of water management. This ensures that the membrane is most conductive and that the fuel cell operates efficiently. In practice the membrane is not always fully saturated so it loses conductivity, typically at higher currents. Capstone Project A

31 Contributing factors to water transport in the membrane are the water drag through the cell, back diffusion from the cathode, and the diffusion of any water in the fuel stream through the anode []. Another issue is that the water content of the membrane is constantly being removed by the flowing gases in contact with the membranes [6]. The water transport is a function of cell current, the flow and humidity of the inlet gases, and the characteristics of the individual membrane and electrodes. Water drag is the amount of water that is pulled along with the conducting proton in the membrane. It is estimated that between 1 and.5 molecules are dragged with each proton. The ion exchanged can be envisioned as a hydrated proton, H + (n H O) []. Water drag (from the anode to the cathode) is proportional to proton flow so this phenomenon increases at higher currents. Back diffusion reverses this effect somewhat. It is usually a back flow of the water produced at the cathode. In rare cases, for example, occasionally when the hydrogen fuel is hydrated, back diffusion will occur from the anode to the cathode..4.1 Self Humidification To combat water management problems some membranes are self-humidified. The operation of a normal PEMFC running with saturated fuel and oxidant is shown below in figure -6 [6]. Figure 6: Normal PEM With External Water - Saturator Capstone Project A

32 From this figure the water flow can be seen. The hydrated proton H + (n H O), back diffusion from the cathode and hydrated gases all flow in and around the membrane. Also shown in figure -6 is the gas crossover that is the cause of the internal current loss. Usually this loss is suppressed in the design stage. There are some problems with this method. There is a large amount of latent heat absorbed when the water is vaporized before the inlet. This lowers the cell s efficiency and creates difficulty in start up or rapid changes [6]. An obvious downfall as well is the need to humidify one or both of the inlet gases. A self-humidified stack aims to overcome these problems. Its basic operation is shown in figure -7. Figure 7: Self-Humidifying PEM An internal current in a self-humidified stack is not suppressed but is slightly encouraged with the use of thin membranes. The fuel reacts in the polymer electrolyte membrane with the help of platinum which is impregnated in it as shown in figure -7. Oxides such as TiO or SiO are added to the membranes to retain the membrane water content. This simple method is used so that the inlet fuel no longer needs to be humidified. The disadvantage to this method is that there is an extra loss in the fuel crossover. However, the simplification in the experimental set-up and the proposed start up advantages are meant to counteract this problem. Although the UTS fuel cell is self-humidified, the option of humidifying the hydrogen fuel is included in the design for flexibility. Capstone Project A

33 .5 Modelling Terminology There are a few terms that need to be known to understand the current literature on PEMFC modelling. These terms define the most important of the operating conditions of the fuel cell..5.1 Stoichiometry The stoichiometry ratio λ is the proportion of each inlet of the gases input into the fuel cell stack compared to the amount used. There is a stoichiometric ratio for hydrogen and air. The ratio is optimised for different operating conditions depending on the cooling rate, humidity level and required mass flow..5. Partial Pressure The partial pressure of the reactants alters the fuel cell s performance. In the case of hydrogen the partial pressure is equal to the pressure. For oxygen this is not the case because oxygen is a constituent of air. Increased partial pressure increases the reversible voltage and it also generally decreases the effects of the activation and concentration losses. It may seem that the higher the partial pressure of the fuel the better the efficiency of the stack. The operating voltage of the stack generally increases at higher pressures although the power required running the compression equipment quite often is more than the energy gained. The air compressor generally absorbs more than 40% of the power generated from the stack [7]. So the overall efficiency can be improved by operating the fuel cell at low pressure using a blower (as used on the UTS fuel cell) instead of a compressor. If a compressor is used a variable flow compressor is more efficient because it can be used at variable power levels as required..5.3 Temperature Increased temperature of the stack reflects higher cell efficiency up to a point. It decreases the reversible voltage but overall the efficiency is improved by decreasing the activation loss and by increasing the exchange current. PEM fuel cells have a limited operating temperature. This is partly due to the requirement of liquid water in the membrane. Capstone Project A

34 In one-dimensional models, as is the proposed model, the temperature of the entire stack is usually assumed to be constant. In two and three dimensional simulation models the temperature is not assumed to be constant and detailed techniques are used to calculate the temperature..5.4 Relative Humidity The relative humidity of the fuel, and partly the oxidant are of importance to the cell s performance. Stacks that are not self-humidified and are externally humidified have a particular dependence on the relative humidity of the inlet gases. In general hydrated inlet gases improve stack performance although stack flooding must be avoided..5.5 Charge Double Layer A complex electrode phenomenon exists in fuel cells. A double layer occurs in many reactions and can be studied at great length. Three references [1,8,9] are ideal for further study on this process. When the reactions shown in equations (-1, -) occur there are charges and charge densities existing in the MEA. The probability of these reactions will depend on the charge densities of the electrons and protons on the electrode and polymer electrolyte membrane surfaces - the greater the charge, the greater the reactions. However, any collection of charge, such as electrons and H + ions at the electrode/membrane interface will generate an electrical voltage [1]. Without a concentration loss the voltage at the electrode/membrane interface is the activation voltage loss and this explains why the loss is sometimes called the charge transfer resistance loss. This shows that the platinum catalyst is required on the electrodes so that it will increase the probability of the reaction so that a higher current will flow without a large build up of charge. It can be shown that when a large current is flowing, there will be a larger build up of charge. There will also be an slight increase in the activation loss as shown in the cathode activation loss in figure.5. The collection of charge at the electrode/membrane interface acts as a capacitor. If the concentration loss is negligible and the internal current loss is built into the activation loss, we have the equivalent circuit shown in figure -8. Here Ract is the activation equivalent resistance, Rohm the resistance of the membrane and electrodes, Cd the double layer capacitance and Vreversible is the reversible Capstone Project A

35 voltage of the cell. This is the equivalent circuit for the UTS fuel cell. Ract, Rohm and Vreversible values change with the operating conditions as depicted in the proposed model. Figure 8: Simple Equivalent Circuit The affect of the double charge capacitor is that it gives the fuel cell a good dynamic performance. When the load of the fuel cell changes, the voltage of the fuel cell has an initial IR loss from Rohm then slowly moves to the new potential..6 UTS Fuel Cell The fuel cell at UTS is a 500W PEMFC. It is a self-humidified 3 cell stack that can be run with varied inlet gases. At UTS we have chosen to operate the fuel cell with industrial grade hydrogen and mostly using air as the other inlet gas. There is going to be an option to humidify the hydrogen gas and to use oxygen instead of air. To cool the stack air or water is to be used depending on the power output of the stack. Although this fuel cell has not been operated at UTS there is one polarisation curve given in the operator s manual [10] that will be used in later analysis of the proposed model. Capstone Project A

36 The 500W PEM fuel cell at UTS is shown below [10]. Figure 9: Photo of the UTS 500W PEMFC + the Control Unit The stack has an electrode area (A) of 64 cm, a maximum operating temperature of 65 0 C and an operating pressure of atm for air and atm for hydrogen. The membrane is an altered form of Nafion 11 that is changed to be a selfhumidifying membrane. Capstone Project A

37 3. LITERATURE REVIEW ON PEM FUEL CELL MODELLING In 1838 Christian Friedrich Schoenbein discovered the fuel cell effect [11] that marked the beginning of fuel cell research. Later on William Robert Grove invented the first fuel cell [11] which started the development of the clean energy converter that we see today. Researchers have since attempted to model the performance of the fuel cell for an insight into the improved design and application of the fuel cell. The main body of fuel cell modelling work has been conducted over the past ten years and it is this period that the fuel cell modelling review is focused. It is this period that has seen an increase in investigation of fuel cell performance forecasting. Much of the modelling centres on the fuel cell s polarisation curve, that is, the voltage output over a range of load currents and a specific set of operating conditions. A model that accurately predicts the polarisation curve for a range of operating conditions is desired. Another aspect of the modelling is the dynamic performance of the fuel cell. PEMFC models can be divided into two broad categories, analytical and empirical. Analytical models are based on theory while empirical models are based on experimentation. A simple, diverse, accurate analytical model is the objective of fuel cell modelling. This would ensure that the performance of the fuel cell could be predicted over a large range of operating conditions after determining the model s parameters. Unfortunately, fuel cells are complex systems and are difficult to completely model analytically. Many analytical models primarily have a theoretical basis while still holding some empirical features. Empirical models, on the other hand, have some theoretical basis while many of the components of the model are simply curve-fitting tools. Because these curve-fitting tools have little correspondence to the theory or physical understanding they are generally accurate over a small range of operating conditions. If an analytical or empirical model omits an operating condition such as membrane hydration then the model is only valid when the omitted condition is held constant. Voltage performance depends on many dependent variables including: current, operating temperature, pressure, hydration content of the membrane, fuel cell dimensions, constituent material properties and stoichiometry of the inlet gases. The Capstone Project A

38 interdependence of these variables makes it difficult to model and is the reason why empirical models were introduced. Thirumalai and White [1] completed a sensitivity analysis on Nguyen and White s earlier model [13] to find the important variables in the fuel cell. They found that the following factors were particularly important: Gas mass flows (pressure and stoichiometry of the inlet gases) Operating temperature of the fuel cell. Relative humidity. In regards to the relative humidity they found that the relative humidity of the anode and not the cathode was important. These and the cell current are the dominant terms that determine the voltage performance of the fuel cell. They are frequently used in fuel cell modelling. A model that describes the reversible voltage and the losses should contain these terms. In regards to relative humidity, if this was kept constant for all practical applications of the fuel cell then it could be incorporated into the constants of the model instead of being referenced as a variable. 3.1 Analytical This section investigates the background of the fundamental physics related to the performance of PEM fuel cells. The advantages and disadvantages of the analytical models are presented with the conditions for their application. A useful analytical model that includes every performance variable has not been found. The models that have attempted this have not included every important variable and have been so complex and specific that they are of little use. Other models have given simple analytical equations and included empirical aspects when this could not be done. It is these types of models that are the most useful to the UTS project and hence these are the models that have been focused on. Capstone Project A0-080

39 3.1.1 Simple Analytical Models Today simple modelling techniques are readily adopted. Empirical parameters must be used to reduce the complexity of the model when the theoretical background to the model has not been fully comprehended or the parameters required in the model cannot be found Historical Background This section describes the historical development of the relatively simple analytical fuel cell modelling technique. These models attempt to classify the operation of the fuel cell into short equations that reflect the reversible voltage and any losses. The general trends and advances of analytical modelling will be examined. The analytical models are based around the reversible voltage and major losses as described in section.3. Some earlier work by groups in the 1960 s (eg. Berger) set the groundwork for the major losses and the theoretical background. Various groups aiming to describe the losses in an easy to use model while still being accurate developed these further. The more important groups are described below Berger Berger edited a book in 1968 [14] that summarised the understanding of fuel cell theory and experimentation up to the date of publication. A number of terms and analysis were given that are still relevant today. The basic structure of fuel cell modelling was outlined. The reversible voltage and the activation, ohmic and concentration losses were described. The reversible voltage was given in the same form that is used today for low temperature PEM fuel cell models. The Tafel equation was also used to model the activation loss. Berger was only one of the early editors - another is Williams [15] Military College of Canada The team in the Department of Chemistry and Chemical Engineering at the Royal Military College of Canada have completed a substantial body of work with specific regard to the Ballard fuel cell. This team has had contributions from Mann, Amphlett, Baurmert, Hooper, Jensen, Peppley, Roberge, Rodrigues plus Harris from Capstone Project A

40 Queen s University in Canada. This group has concentrated on studying the Ballard Mark IV fuel cell from the late 1980 s [16,17], the Mark V fuel cell from the 1994 [18-0] and more recently attempted to generate a general PEM fuel cell model [1]. They have used a semi-empirical approach to the activation loss, and have given theoretical background on the losses but stated that the parameters were inherently difficult to measure. An empirical approach was used to find some of the constants in their model. They proposed an individual solution to the mass transport effects [16] but this was seen to be inaccurate for other fuel cell manufacture types []. In an earlier model [18] the Royal Military College group gave a thorough theoretical description of the voltage output heat losses and stack temperature in unsteady state conditions of the Ballard Mark V fuel cell stack. This model is not directly related to this project although it could be useful if the start-up and shut down performance of the fuel cell is required. One point to note is that this model is complicated and the methodology used may be outside the scope of the UTS fuel cell model Larminie and Dicks The book Fuel Cells Explained [1] outlines the operation of fuel cells with respect to the reversible voltage and four losses, namely the activation, ohmic, concentration and fuel crossover loss. The fuel crossover loss is described here as a loss relating to the wasted fuel that passes through the electrolyte without producing an external current. It is also used to model electron transfer through the electrolyte. A lot of time is devoted to giving a theoretical background to the reversible voltage and the losses. Some experimental guidelines for finding the model s constants are also given Fuel Cell Handbook There are many recent textbooks on fuel cells, this is one of them and is another very good reference. It outlines the operation and the design of the major fuel cell types and gives some useful information in relation to water management for PEMFCs Others Other simple analytical research groups include the Corrêa and Maggio parties. Corrêa et al. [3] summarised the Amphlett et al. [1] and Larminie and Dicks [1] models but do not propose any new ideas. Maggio et al. [4] give a different Capstone Project A

41 approach to the concentration loss. They call it a diffusional loss and add a convection loss that is usually assumed to be zero. They find the convection loss to be around 1% of the entire losses in the PEM fuel cell Reversible Voltage The reversible voltage of the fuel cell was described by Berger [14]. His book relates the theoretical maximum voltage to the temperature of the fuel cell and partial pressure of the reactants. The reversible voltage also required a list of well defined constants consisting of the universal gas constant, Faraday s constant, the change in Gibbs free energy at STP and the molar entropy of the fuel cell reaction equation (-3). His equation describes the reversible voltage of each cell and is still used in existing fuel cell modelling. It is shown below. V = * * 5 ( 4.308*10 ) * ln 4 T p p ( 8.453*10 )( T 98.15) Re versible H O Activation Loss Activation loss is primarily described by the Tafel equation reported in Later more background was given to this equation that was initially empirical. The Tafel equation is now regarded as a simplification of Butler-Volmer model. One of the forms for the Tafel loss is, V CELLactLOS S = A T i ln i 0 A T was initially described as an empirical constant, i is the cell current and i 0 is the exchange current - the current that flows back and forth for the reversible reactions at the each electrode. Theoretical background to this equation can be found in many electrochemical textbooks (eg. [5]). This loss was described analytically by Amphlett et al. [16] to determine the activation loss in terms of temperature, concentration of O and current. Later [1], they added a term that included the concentration of hydrogen. The concentrations (partial pressure with a certain stoichiometry) used were average values in the stack not the concentrations at the inlet. Parameters for the equation Capstone Project A

42 were obtained empirically and the values are described in their papers. Papers of various groups in later years tend not to publish parameter values of their models as this is regarded a proprietary information. Most other models do not propose how the constants in the Tafel equation relate to the operating conditions so this is an important paper for activation loss background. Maggo et al. [4] propose an equation different to [16] for i 0 although it is derived from the exchange current of platinum and does not give its relationship to the operating conditions of the fuel cell Ohmic Loss Generally, research papers quote the ohmic loss which is due to membrane and electrode resistance as a simple relationship of IR, where R is constant. It is known that membrane resistance is not strictly constant because it depends on the water concentration in the membrane, temperature and cell current. There is difficulty in creating an accurate model of membrane resistance, this is summarised by Büchi and Scherer [6], One of the most disputed questions, which is of great importance for applications, is the dependence of the resistance on the current density. This dependence is very difficult to model, as many parameters either in the membrane itself (such as electroosmotic water drag or back diffusion of water) or outside the membrane (such as humidity, pressure and flow of gases, hydrophobicity and porosity of the electrodes or flow field design) may be involved. Ganssen and Overvelde [7] propose a method of calculating the water flow associated with the membrane for a non self-humidified stack. This could be of assistance except for the need of equipment integrated into the stack. Subsequently their method is predominately useful for fuel cell designers although [7] provides helpful background reading. Despite these problems there has been some effort in creating an analytical relationship for the membrane resistance of some Nafion products for different currents and water contents in the membrane. Some empirical techniques were used in deriving this relationship. Capstone Project A

43 Resistance of some Nafion membranes. Springer et al [8,9] from the Los Alamos National Laboratory in New Mexico completed preliminary work to find that the water content was a major factor in the resistance of the proton flow of the membrane. This was used by [16] and later by [3]. A semi-empirical equation was used to find the resistance of the proton flow that related to the operating current and temperature. For variation of the water content in the membrane a constant was inserted. This constant is given an estimated range of 14 to 3 depending on the water content of the membrane. The exact dependence on this variable is not fully known and thus it must be calculated for varying operating conditions. It was initially stated as the effective water content of the membrane in a ratio of H O/SO 3. For example, a value of 14 corresponds to 14 H O molecules per charge site in the membrane Mass Transfer Loss I have specified the title to this section as mass transfer loss because concentration loss is really a sub set of the mass transfer loss. Mass transfer loss can occur through three independent mechanisms: migration, diffusion and convection. Migration is the movement of a charged body under the influence of an electric potential. Losses from migration only occur at particular situations and are not significant to fuel cells. Diffusion is the movement of a species under the influence of a concentration gradient [5]. This is linked to the concentration loss that occurs for the fuel and oxidant of the fuel cell. Another mechanism is convection. This is the amount of stirring in the fluid and the type of flow. Convection and diffusion (concentration) losses were both used instead of just a diffusion loss by Maggio [4]. It was found that the convection loss only accounted for about 1% of the total losses in the system when the fuel cell had a high and low load. I will not deal with this loss for this reason. Most modellers disregard this loss as well. Mass transport loss mostly comprises of a concentration loss so these terms are often interchangeable in fuel cell modelling. Capstone Project A

44 Concentration loss is usually derived from Nernst diffusion layer thickness called Fick s first law. V conloss = i B ln 1 i L B is an empirical constant, i is the cell current and i L a limiting current - it is a point when the gas is used at a rate equal to the maximum supply speed. This loss is used in many models although the strict dependence on the important operating conditions (stoichiometry, pressure, humidity and temperature) is unclear. The theoretical value of B is RT/F for the hydrogen gas and RT/4F for oxygen. Historically this does not match the experimental values in the fuel cell. This is mainly because the activation loss is neglected when the theoretical values were found. In reality the activation loss is always present when there is a concentration loss for fuel cells Fuel Crossover and Internal Current Loss Many researches find that the open circuit voltage is not the same as that expected by the activation, ohmic and concentration losses. They state that their models are valid for currents above a certain value and forget about this difference. Larminie and Dicks [1], Chu et al. [30], and a few others acknowledge that this loss that occurs mainly at open circuit is due to a small amount of hydrogen and oxygen that travels through the membrane. The strict dependence of the important operating parameters on internal current on is unknown at this stage Complex Analytical Models Complex analytical models have been found but usually they can only be used by the authors or groups that have years of fuel cell modelling experience. At UTS it may be more efficient to learn and use simulation programs such as CFD-ACE or Trnsys rather than developing complex analytical models. Because of the complex phenomenon of water management much time has been spent modelling this, other complex analytical models look at the overall operation of Capstone Project A

45 PEM fuel cells. On the whole, water management models do not deal with selfhumidified stacks which is vital for the UTS model Water Management Models Rowe and Li [31] propose a one-dimensional non-isothermal model that includes membrane hydration, the reacting gases, phase change of water in the electrodes and an energy equation that describes the temperature of the cell. They solved these detailed equations with an algorithm developed by another author. Janssen [45] developed a steady state two-dimensional model that used solution theory to describe water transport in the membrane. Further studies by Eikerling et al. [33], Yi and Nguyen [34] and Nguyen and White [13] provide insight into the electro-osmotic effect of water, and examine the overall water management issues in the membrane. The electro-osmotic effect is the phenomenon of protons that carry water molecules when they are conducted in the membrane. They describe the amount of H O molecules that are conducted for each proton for various conditions Overall Models Murgia et al. [35] describe a modified version of the Bernardi and Verbrugge model [5,36,37] that is quicker to simulate. The Bernardi and Verbrugge model gives detailed analytical equations of the anode catalyst layer, cathode catalyst layer, membrane and cathode gas diffusion layer. It gives these in equations that often require parameters that are hard or almost impossible to acquire. Baschuk and Li [38] describe the PEM fuel cell voltage in terms of the three aspects: 1) reversible voltage; ) activation and mass transport loss of the cathode, and 3) ohmic losses of the electrode, membrane and flow channel plate. This may seem to be in the category of a simple analytical model until Baschuk and Li go into detailed modelling in respect to specific dimensions and attributes of the fuel cell. For example, the fixed-charge concentration of the membrane, width of the flow channels and the amount of platinum on the carbon support in the cathode catalyst layer are all required for the model. This shows some of the application problems for complex Capstone Project A

46 analytical models at UTS. We have struggled to receive basic structural information on the fuel cell stack so details such as these are near impossible. 3. Empirical Because of the complexity and interdependence of variables on the performance of the fuel cell an empirical equation can be used to predict the polarisation curve (voltage versus current) of the fuel cell stack for a given set of operating conditions. The advantage of this approach is that it accurately predicts the polarisation curve for a given set of operating conditions. Its disadvantage is that the polarisation curve must be recalculated for a change in an operating condition (eg. humidity) that the empirical model has omitted. It is this reliance on experimental data that is the major disadvantage for empirical models. For example, an empirical model has omitted temperature from the modelled activation loss. If the temperature of the fuel cell was changed from 45 0 C to 55 0 C then an entire set of experimental measurements would have to be made so that a new empirical model could be found for the new polarisation curve. A comprehensive empirical model will need to identify the correct dependence of the most important performance factors that is the temperature, pressure, stoichiometry, relative humidity and current of the fuel cell. So far this has not been achieved although the empirical models presented include a number of these variables Historical Background There are primarily three different approaches to empirical modelling. On one hand, the Royal Military College of Canada (Amphlett et al.) [17] modelled various parameters empirically, the structure for the model was found analytically. Secondly, Kim et al. [39] viewed the polarisation curve as a whole and used the important parameters to reflect that trend. The resemblance of the empirical terms to the losses was secondary. Both approaches empirically model the PEM fuel cell accurately for a given set of steady state operating conditions and are both used extensively. The third approach involves an electrical circuit equivalent using electrochemical impedance spectroscopy (EIS) or current interruption. EIS involves measuring the Capstone Project A

47 impedance of the fuel cell at difference frequencies then equating an electrical circuit equivalent using capacitors and resistors. Unconventional approaches include those used by Lee and Lalk [40]. Using an iterative finite element approach Lee and Lalk developed a transient fuel cell model. They proposed an empirical relation of the cell voltage that depended on the operating current and the temperature. The model calculates the temperature of the stack at various points in the stack along the gas channels. It uses these temperatures, the relation of voltage to current at each point and the previously modelled value of cell voltage to estimate the next value of the cell voltage. With the limited analysis and background that can be given in one paper of this complex model an approach such as this one could only be used by the authors. The conventional empirical methods are discussed below. 3.. Empirically Modelling the Entire Polarisation Curve Kim et al. A landmark empirical equation was presented in 1995 by Kim et al. [39] from background work by various groups (eg. Springer et al. [41]). Without complete electrochemical consideration an empirical equation was presented that gave an accurate prediction of the polarisation curve. The empirical equation given by Kim et al. [39] is given below. V cell = V R j b ln( j) m exp( n ) (3 1) 0, K cell, K K K K j Although initially the empirical equation mentioned did not have a detailed consideration of the physical relevance of the models terms, the terms in the equation can be seen to represent the losses in the fuel cell. Capstone Project A

48 The set of terms in the model are shown in table 3-1 []. Table 3 1: Kim et al. Empirical Equation Terms. V cell - Cell voltage (mv) j - Current density (A/cm ), R cell,k - Ohmic resistance (Ω.cm ) b K - Tafel slope (mv / decade) V 0,K - Open circuit potential of the cell (mv) R cell,k I - Ohmic loss term (mv) b K ln(j)- Activation loss term (mv) m K exp(n K j) - Concentration loss term (mv) m K - concentration loss coefficient (mv) n K - concentration loss coefficient (cm / ma) This gives a good fit of the polarisation curve (see figure 3-1) but must be recalculated for different operating conditions (eg. hydrogen stoichiometry). Figure 3 1: Empirical Fit of Polarisation Curve [39] Capstone Project A

49 The equation given gives a good fit of the activation, ohmic and concentrations losses as shown in figures 3- and 3-3. Figure 3 : Fit of Voltage With Activation and Ohmic Losses [39] Figure 3 3: Fit of Concentration Loss [39] This equation has been used in the work of Øystein Ulleberg and team. I have reviewed the simulation of his models in Trnsys that uses the empirical equation (3-1). His group proposed a paper [], which gives an example of how equation (3-1) can be applied to various fuel cell systems. This, along with justification from other groups, ensures that this empirical model is useful to a range of PEM fuel cells. Capstone Project A

50 3... Lee and Lalk Lee, Lalk and Appleby propose a model [4] similar to equation (3-1) but adding a log p p O term. They state that the empirical constants depend on stoichiometry, pressure, temperature, current and they use the humidity of the air and not hydrogen. The constants are stated as functions of these operating parameters but no further details are given Pisani et al. Another group that have proposed an empirical equation of the entire polarisation curve is Pisani et al [43]. They propose an equation slightly different again and provide some mathematical insight into the origin of the empirical parameters Empirically Modelling each Loss The team at the Royal Military College of Canada [17] completed a body of work that relates to the empirical features of their Ballard model. When the analytical constants were not fully understood they went about finding an empirical equation for each parameter and for the entire IR loss. This technique is used as a supplement to analytical models and will be used in the proposed model Electrical Equivalent Circuit Amphlett et al. [44] give details of an electrical circuit model for a PEMFC. The model appears to be operating in a region where there is no concentration loss. That is at currents lower than when the concentration loss becomes significant. This is evident because the section of the circuit that would normally describe a concentration loss is missing. This is the same circuit described earlier by Uosaki et al. [45], later by Larminie, and Dicks [1] and is shown in the previous section.5.5 as figure -8. It was proposed by Larminie and Dicks [1] that the concentration loss would be added to in series with the activation resistance. As depicted in the results of Lee et al [46] this simplification is not always valid. The concentration loss would be an impedance rather than purely resistive. Another equivalent circuit used mostly in Capstone Project A

51 modelling batteries (which has many parallels to fuel cells) will be discussed later in section 4. Discussed in section 5 will be the results of Wagner et al. [47]. Two other equivalent circuits are of interest but little use for the UTS project. The first is from [48], it proposes an equivalent circuit for an entire generation system. They model the fuel reformer, stack, DC-DC converter and load. This gives a good background to the equivalent circuit technique but the over simplified stack circuit is of little use. Another oversimplified circuit is described by Standaert et al. [49]. 3.3 Experimentation The results of some experimental techniques were discussed in the section above but some insight into the procedure is required, this has been discussed here. The main techniques for experimental determination of the PEM fuel cell are current interruption, Electrochemical Impedance Spectroscopy and fast current pulses. In addition to the experimental results in the modelling papers already discussed there are some research papers specifically tailored to describing the PEMFC in various conditions. These papers give trends to the polarisation curve when some operating parameters (eg. temperature, humidity etc.) are changed. These papers are discussed in the General Results section that follows General Results Jiang and Chu [30,50,51] do not specify techniques to find individual losses but provide a range of polarisation curves for different operating conditions. General trends in the polarisation curve can be seen for different temperatures, inlet gas flow rates, Nafion membranes and relative humidities. Basically a thinner membrane, higher temperature, higher gas flow rate and higher humidity are optimum operating conditions. A system of measuring individual cells in a stack and not experimental results were proposed by Webb and Møller-Holst [5]. They built a measuring system that connected to their stack that used multiplexers to read individual cell voltages in the stack. This is done because cell performance can vary depending on where it is placed in the stack. Local humidity levels, catalyst activity and temperature change Capstone Project A

52 in the stack and give rise to an individual voltage change. At high currents especially, the voltage of individual stacks may fall considerably from the average. This paper gives insight into the operation of each cell in the stack and gives hints to avoid possible cell damage. The experimental technique is aimed more at manufacturers than organisations like UTS, who use an already built system. Büchi and Srinivasan [53] give experimental results of the effects of water management of humidified and non-humidified inlet gases with different stoichiometric ratios and temperatures. Surprisingly, the systems that use nonhumidified gases also do not use self-humidification in the membrane. Despite this Büchi and Srinivasan discuss the advantages of stack design simplification and performance without humidified inlet gases Current Interrupt Method Current interruption is a widely used technique for finding out parameters to models although not many papers discuss the finer aspects of this technique. The experimental method involves a sudden change current with analysis of the dynamic voltage output of the fuel cell. Lee et al. [46] and Larminie and Dicks [1] give a good background to this method. This, as well as electrochemical impedance spectroscopy, will be discussed in full later Electrochemical Impedance Spectroscopy Electrochemical Impedance Spectroscopy (EIS) is an experimental method that involves inputting an AC current of varying frequencies into the fuel cell and measuring its impedance. It is common nowadays to use frequency analysers (which are also generators) like Solartron. A simple technique for carrying out EIS can be used which is slower but also cheaper than using the commercial products. There have been few papers that give results and analysis of EIS and even less that give some indication of the experimental set-up. By and large EIS has been conducted on individual cells. The technical notes of Solarton [54,55] are the most concise of the references. Springer et al. [56,57] give some useful example results of EIS for a PEMFC but do not provide much insight into the experimental set-up. Wagner et al. [47] give results for a PEMFC and solid oxide fuel cell at different cell Capstone Project A

53 potentials. Wagner et al. perform EIS to give the losses for the cathode and then the anode (as separate losses) using half cells. At UTS we do not have access to the points required for these detailed tests although we can perform EIS on the entire fuel cell stack. An equivalent circuit is given in [47] although for a solid oxide fuel cell. This gives some indication of the equivalent circuit required for a PEMFC. Mueller et al. [58] also give an example of EIS for a different kind of fuel cell, a direct methanol fuel cell, and provide a block diagram of the experimental set-up Fast Auxiliary Pulse An innovative technique [6,59] uses fast 5A current pulses. This gives accurate resistance measurements but is expensive to set up and time consuming. This technique is mostly used on single cells although Mikkola [3] suggest this technique is useful for stacks that operate above 10A. Because this technique is moderately expensive and time consuming it is not suggested for use at UTS. Capstone Project A

54 4. PROPOSED MODEL Analytical models focus on the theoretical voltage potential of the fuel cell and the major losses. The major losses are the activation, ohmic and concentration losses (all with an internal current). PEMFC models are much simpler without water management issues. This sets the PEMFC modelling apart from other cells. I will now propose a model that has an analytical background where possible, and uses empirical techniques when this is not possible. Modelling a stack that we could not yet operate at UTS posed some problems. The model that I propose to be used at UTS is in reality one of a few models depending on how simple the model needs to be. The amount of simplicity (while still being accurate) of the model will only be known when the stack is operated and the performance data is analysed. If the simplest model is accurate then the modelling will stop there. If not, other techniques that are stated will be used. Even the best simple analytical model [16] does not deal with water content in the membrane. They use the assumption, that the gas flow rate and the design of the gas flow fields are sufficient to guarantee removal of excess liquid water. The water content of the stack is not specifically referred to, and the membrane is assumed to be well hydrated without being flooded. This should be the case for the stack at UTS because the water content issues of the membrane are meant to be solved by the selfhumidifying membrane. BSC Technology, the manufacture of the UTS PEMFC stated that no humidification of the reactant gases were required because it is a selfhumidified stack. The option for humidification of the hydrogen gas is a precautionary measure. These facts combined with the practical values of the hydrogen relative humidity should ensure that the stack has a well hydrated membrane as required. Capstone Project A

55 If the membrane is too dry the operation of the fuel cell should be stoped as this can damage the membrane. One checking technique for determining this problem is to perform a current interrupt test when the load current is set to a new level. This procedure was suggested by Andrew Dicks [60]. If the membrane resistance becomes higher than expected the membrane is most likely dry and damage will occur if the stack continues to operate in the same way. An initial estimate of the membrane resistance is its value at open circuit. Proposed here are empirical techniques to give a relationship of the water content of the membrane to the stack voltage for the case that this is required. The water content in the membrane primarily depends on the relative humidity and mass flows of the air and hydrogen. The relative humidity of the air is the least significant of these. If the temperature of the stack were operated beyond the recommended limit of 65 0 C the stack may have water management problems because the water in the membrane would be a gas not a liquid. 4.1 Reversible Voltage Reversible cell voltage (V CELLreversible ) is the maximum theoretical potential of the cell. This is independent of the load on the fuel cell. V Nernst is the theoretical potential when the fuel cell is operating at the standard temperature, 98K. The reactant pressure variations are included in V Nernst. The temperature variation away from V Nernst will be described later. V CELLreversible = V Nernst +V T difference (4 1) Capstone Project A

56 4.1.1 Gibbs Free energy for V reversible There are many relevant terms in this section. Table 4-1 shown below shows a summary of these terms. Table 4 1: Terms Relevant to Free Energy V CELLReversible Reversible cell voltage. The maximum theoretical voltage of an individual fuel cell. V Reversible V Nernst G G f G G f The reversible voltage of the stack. Reversible cell voltage at the standard temperature. Gibbs free energy. The free energy available in a reaction to do useful work. Gibbs free energy of formation. Gibbs free energy with a specific zero point which is that of an element s normal state at STP. The change in Gibbs free energy in a reaction, the Gibbs free energy of the products minus the reactants. The change in Gibbs free energy of formation in a reaction. g f The change in Gibbs free energy of formation in a reaction in per mole form. To understand these terms the logical beginning is to start at a definition of G the Gibbs free energy. The Gibbs free energy is the energy available to do external work, neglecting any work done by changes in pressure and/or volume [1]. This definition states that the Gibbs free energy neglects any work done in a change in pressure. At specific pressures, the Gibbs free energy is quoted at numerous values. It is the work done as the pressure changes from one value to the next that is neglected. The value of Gibbs free energy different pressures changes. The free energy is higher for a reaction when the reactant partial pressures are a higher value. The zero potential energy point for the Gibbs free energy can be defined at any arbitrarily defined position. When it is defined for pure elements at the elements Capstone Project A

57 normal state at STP (98.15K and 1 atm) then G is defined as the Gibbs free energy of formation G f. We cannot determine the absolute amount of free energy a substance has [61]. However it is the change in free energy that is important. We wish to know how much free energy has been used when a reaction has occurred. G f is the change in the Gibbs free energy of formation, the difference in the energy of the products minus the reactants. As the only difference between G and G f is the zero reference point then G is equal to G f. G f = G products reactants f G f Warn and Peters [6] tell us that the change in Gibbs free energy G is one of the most important concepts in thermodynamics. G is the criterion for deciding whether or not a change of any kind will tend occur in the reaction. Chemical reactions that have a negative value of G under the accommodating conditions are spontaneous, a positive value tells us that the reverse reaction tends to occur, whereas a zero value implies equilibrium. The fuel cell reaction, equation (-3), is likely to occur in the presence of a catalyst, and G always has a negative value. To simplify later analysis the value of G f is usually given in the per mole form. This is shown by a ( ) over the lower case g that is g f. In the hydrogen fuel cell reaction the change in the Gibbs free energy of formation per mole becomes: g f = g f g 1 g f H O H f O (4 ) g changes for different molecular states of the materials in the fuel cell and at f different fuel cell temperatures. If the material changes state then g f must change accordingly. Fortunately the states for the reactants and products in the fuel cell do not change. These are shown: H O gas gas H O liquid Capstone Project A

58 Note that the water product is expected to be a liquid and gas mixture but the gas content is small enough to be neglected. Nearly all the models in literature use V CELLreversible, although most do not describe its origin in detail. I will first detail the form of the equation for V CELLreversible. Next I will describe how V CELLreversible varies with different reactant and product partial pressures. Lastly the temperature variation of V CELLreversible away from the standard temperature will be given. Larminie and Dicks [1] give a detailed description of the reversible voltage. This is also explained to a minor extent in many papers. The reversible voltage for a reaction that produces two electrons per molecule of fuel as with the PEM reaction is: V CELLreversible g f = (4 3) F where: V CELLreversible = Reversible voltage of a hydrogen fuel cell at the standard temperature, g f = Change in Gibbs free energy of formation per mole, F = Faraday s constant ( C mol -1 ). The reversible voltage is more complex than it seems at first inspection. As stated the change in Gibbs free energy varies when the pressure of the reactants (H and O ) vary and when the fuel cell is operating at different temperatures. Research papers describe these changes although they perform this with different approximations and diverse methods. When looking at the reversible voltage equation (4-3), the complexity of it resides in Gibbs free energy of formation per mole. The reason for the two on the denominator can be proven as in [1]. It denotes that two electrons are transferred for each molecule of the fuel. In the operation of the hydrogen fuel cell two moles, N A electrons are conducted in the external circuit for each mole of H and half mole of O used - N A is Avagadro s number, the number of fundamental particles in a mole. Capstone Project A

59 Each mole of H and half mole of O used produce a mole of H O and two moles of electrons. As stated, two moles of electrons are equivalent to N A electrons. Therefore the charge that will flow in the external circuit in this case is shown: Total Charge = Number of electrons * charge of electron = N A * -e = N A e Coulombs Total Charge = F Coulombs (4 4) Remembering that the Faraday s constant is the charge on one mole of electrons. If the reaction is reversible then the electrical work done in moving the charge F will be equal to the change in usable energy of the reactant and product chemicals ( g f ). The reactants, H and O have a certain amount of usable energy and the reactants combine to form one product H O. If the H O product has less usable energy then we can say that some of potential energy of H and O has been used. The most efficient reaction would change all of the potential energy available to do useful work to electrical energy. The electrical energy is the electrical work done in moving the charge F. Electrical work done = charge x voltage Electrical work done = F * V CELL Re versible (4 5) Also, Electrical work done = g f Rearranged this is: V CELL Re versilbe g f = as in Equation (4-3). F Capstone Project A

60 4.1. Pressure Variation The pressures of the products and reactants play a part in effecting the reversible voltage of the fuel cell. Basically increasing the pressure of the reactants increases the amount of reactants on the surface of the electroplates and so increases the frequency of chemical activity. The change in Gibbs free energy varies with different pressures and temperatures of the fuel cell. This in turn changes the reversible voltage of each cell. I will describe how g varies from its standard value which f is stated at STP. The effect of pressure on g is displayed in literature with slight variations. f Sometimes they approximate one bar to equal one atm, or vice versa. The difference is that 1 atm is equal to bar. Although this is not much, it is worthwhile minimising errors when possible. In some papers such as [7] researchers do not make any assumptions but state their standard value of 0 g f is stated at 1 bar, not 1 atm. The deviation of pressure from 0 g f assumes that V CELLreversible is at the standard temperature. In this condition V CELLreversible is equal to V Nernst. The effect of pressure on V Nernst as shown in literature is [7], 1 * * 1 p H p O 0 RT 1 V = + Nernst V ln * * (4 6) F p P0 H 0 where: V 0 = Reversible fuel cell voltage at STP (V), 1.9V, * x p = The partial pressure of the matter using a chosen pressure unit, P 0 = Standard pressure in the pressure units used, STP = Standard temperature and pressure, 98.15K and 101.3kPa (1 atm), R = Gas constant J K -1 mol -1. Capstone Project A

61 As a simplification the partial pressures of H, O and H O can be stated in atmospheres which reduces the term 1 P0 1 to 1. Larminie and Dicks [1] gave an excellent derivation of the pressure effect but made the mistake that the standard pressure is at 1 bar not 1 atm. Most other research papers (eg [1,3]) used the pressure unit of atm for H, O and H O but had little explanation of the form of their equation. A short derivation of equation (4-6) can be found in [1], as is summarised below. In the ideal gas case the thermodynamic term, activity is defined as: p a = (4 7) P 0 where: p P 0 = Pressure or partial pressure of the gas, = Standard pressure, 101.3kPa (or 1 atm). As described in [1], the change in Gibbs free energy of formation per mole can be described in terms of activity. g f = g ( a ) 1 a H O RT * ln (4 8) ah O 0 * f where: 0 g f = Change in Gibbs free energy of formation per mole at STP which has a value of 37. *10 3 J mol -1, R = Gas constant J K -1 mol -1, T a x = Temperature of the fuel cell (K), = Activity of the substance. Capstone Project A

62 Capstone Project A substituting a, equation (4-7) into equation (4-8) we get, = 1 0 * 1 * * 0 1 * * *ln P p p p RT g g O H O H f f (4 9) Using F g V f 0 0 = and substituting the equation above (4-9), into equation (4-3) we get: + = 1 0 * 0 1 * * 0 1 * ln P p p p F RT V V H O H Nernst (4 10) This gives the reversible voltage at the standard temperature and varying pressure. Most references (eg. [1,1,3]) use the pressure of water in the fuel cell, * H O p as one atmosphere. Naso et al. [3] retain * O H p in their reversible voltage equation although it is ambiguous whether or not this assumption was made in their analysis. Because liquid water is the main product H O can be assumed to be 1 atmosphere. If this assumption is made, standard values are used and all partial pressure values are in atmospheres then the reversible voltage at the standard temperature reduces to: + = 1 * * ln.9 1 O H Nernst p p F RT V (4 11) The activity (a x ) assumes that the gases are ideal so it must be remembered variation in V CELLReversible due to the pressure difference also uses this assumption. A paper that does not assume this has not been found. This assumption has little consequence on V CELLReversible.

63 4.1.3 Temperature Variation The last consideration for V CELLreversible is the fuel cell temperature variation from the reference temperature 98.15K. Most background sources calculate the difference in V CELLreversible due to a change in temperature (V difference T ) using an assumption that is applicable for low temperature PEM fuel cells (5 0 C to 80 0 C). I have included here for completeness a more accurate method of calculating V difference T and explain how this method is simplified using the low temperature assumption readily used. The accurate method can also be used for other fuel cells that operate at higher temperatures where the product is steam and the assumption used previously cannot be made. Eventually both methods find the variation of V CELLreversible for a PEMFC due to a temperature change as: V difference T s f = F ( T T ) ref (4 1) where: f s = Molar entropy change of equation in equation (-3) at STP, F = Faraday s constant ( C mol -1 ), T T ref = Temperature of the Fuel Cell (K), = Reference temperature, 98.15K. Most research papers do not state their assumptions or calculations so the derivation behind the assumption in the temperature variation of V CELLreversible was found from another source. The reason behind the temperature variation including the assumption used can be found in [63]. Capstone Project A

64 The Gibbs free energy can be defined as a function of enthalpy (h), entropy (s) and temperature (T): g f = h f T s (4 13) where: f g = Gibbs free energy per mole, h f = Molar enthalpy of formation, s T The change in = Molar entropy, = Temperature of the fuel cell (K). g is: f g f = h f T * s s* T (4 14) If the chemical change is at a constant temperature (isothermal) the change in Gibbs free energy can be simplified. In a fuel cell, the chemical reaction can be considered to be at a constant temperature [1], when the temperature is changing the effect of T on g f can be neglected. Remember the value of f is still different for g different temperatures as shown by the T * s term. It is the value of f at a g temperature, T, and not a reaction with the temperature varying that is investigated here. The change in Gibbs free energy of formation in a reaction in per mole form is then: g f = h f T * s (4 15) At the standard temperature 98.15K this equation is relatively easy. g 98.15K = h 98.15K T * s K (4 16) where: g K = Change in Gibbs free energy per mole at STP, h K = Change in the molar enthalpy of formation at STP Capstone Project A

65 s K = J mol -1, = Change in the molar entropy at STP = J mol -1 K -1, T = Temperature of the fuel cell (K), in this case it is 98.15K. Substituting the standard values for h K and s K the above equation gives: g = * ( ) K = g f = 37.* 10 g J mol -1 as given before. If temperature is varied the change in Gibbs free energy can be calculated as below: g T = h T T s T where: g = Change in Gibbs free energy per mole at temp T and standard P, T h T = Change in the molar enthalpy of formation at temperature T and the standard pressure, s T = Change in the molar entropy at temperature T and standard pressure, Although T h T = Temperature of the fuel cell (K). s and T are required to calculate T. Brady and Holum [63] g describe how this process can be simplified: The magnitudes of the h T s and T for a reaction are relatively insensitive to the temperature. This is because the enthalpies and entropies of both the reactants and products increase about equally with increasing temperature, so their difference, h T s h 15 and T, remain nearly the same. As a result, we can use 98. K and s K as reasonable approximations of T and T. h s Capstone Project A

66 This means g can be written as: T g T h 98.15K T s K (4 17) since 0 g f = h 98.15K 98.15* s K the effect temperature has on 0 g f can be shown: 0 g f 0 (temp effect of) = g f g T = h 98.15K 98.15* s 98.15K h 98.15K T s K T s 15 (4 18) = ( ) 98. K Using V g = the voltage difference of V CELLreversible at different temperatures is: F V difference T s = F 98.15K ( T 98.15) This is the same form as equation (4-1). Substituting this and equation (4-11) into the reversible voltage equation (4-1), then V CELLreversible is seen with temperature and pressure variations, p RT + ln F s F 1 * * 0 1 H p O g 1 f * * CELLrevers ible = F p P0 H K ( T 98.15) (4 19) V The equation above can be simplified using, STP values for 0 g f s 15, R, F, 98. K Using the pressure units of atm for all partial pressures Assuming that * H 0 p for water is 1 atm. Capstone Project A

67 1 * * ( 4.308*10 ) * ln V = + T p p ( 8.453*10 )( T 98.15) (4 0) CELLrevers ible H O This is the same equation as shown in recent literature. I have explained their assumptions and given background calculations where none was specified. Assumptions: The gases are ideal for the pressure variation. The temperature of the fuel cell is constant at all points in the fuel cell stack. g T h 98.15K T s K (4.17), i.e. the fuel cell variations are isothermal plus the change in the molar enthalpy and the change in molar entropy at varying temperature are equal to their value at the standard temperature. The product consists of pure liquid water and no steam content. Using the pressure units of atm for all partial pressures. * p for water is 1 atm. H O The reversible voltage changes marginally with a change in reactant pressure and stack temperature. In the operating ranges of the UTS stack the combined pressure changes of the reactant gases will alter the reversible voltage of the stack by around 0. volts. Change in temperature from 55 0 C to 65 0 C will change the reversible voltage of the stack by 0.6 volts. Capstone Project A

68 Strict Temperature Variation If the change in enthalpy h T and the change in entropy T were to be strictly s calculated the following method could be used [1]. The values of h T s and T shown below would then be substituted into the equation (4-15) to get the change of Gibbs free energy at various temperatures. g T = h T T s T h T = h T h 1 h T H O H T O (4 1) s T = s T s 1 s T H O H T O (4 ) For each molar enthalpy of formation following empirical equation can be used. h T x, namely for x = H O, H and O the h T x T ( c p ) = h dt (4 3) x x Similarly, for each molar entropy s the equation below can be used. s T x = s x + T T ( c ) p x dt (4 4) The constant used in these equations, c p, is not in fact a constant. It changes with the temperature. Larminie and Dicks [1] state that this has been well documented and give the values for c p for the PEMFC reaction. They also state that when liquid H O is the product of the fuel cell reaction the value of c p for all products and reactants can be assumed to be the constant, the value c p is at 98.15K. This is because the fuel cell stack has a small temperature variation away from the standard temperature in low temperature PEM fuel cells. This in effect is the same as the assumption g T h 98.15K T s K. Capstone Project A

69 4. Ohmic Loss The ohmic loss of the fuel cell relates to the current, temperature and water content in the membrane. The water content in the membrane relates to the humidity, pressure and stoichiometry of the inlet gases. The relative humidity of hydrogen is of particular importance to water content. In this section it is assumed that the water content has been kept at a reasonable level. This means that the water content in the membrane is saturated without flooding the reactant gas flow. The water content in the membrane should be at this level, one of the operating goals of the fuel cell is for this to occur. The resistance of the membrane is expected to increase with current. This is because the water flow in the membrane increases as the current increases. The current is proportional to the electro-osmotic drag. This phenomenon dehydrates the anode side of the membrane and decreases its performance. The specific consequence that the current and temperature have on the membrane at UTS is unknown at this stage and will have to be found experimentally. If the self-humidifying stack performs as expected there will be little change in the resistance due to the current and humidity of the inlet gases. This is assuming practical values of humidity levels of hydrogen and air. The example polarisation graph in the fuel cell manual given in section 4.9 shows that the current does not affect the resistance of the stack enormously. As a first approximation the stack resistance can be said to be constant. The temperature and current will slightly affect the stack resistance as shown later in this section and will need to be modelled empirically. Extreme non-linear ohmic losses will occur with unreasonable humidity levels of hydrogen or oxygen that flood or dry out the stack. Stack dehydration is the main concern. The stack is operated under a comparatively small temperature range so the temperature effect on the resistance will be small but noticeable. The resistance of the Nafion 117 membrane changes slightly with a change in temperature and current [16]. The Nafion membrane resistance equation as discussed in the literature review cannot be used as it seems only to apply to externally humidified membranes. Amphlett et al. [1] question the accuracy of their Capstone Project A

70 easy to use membrane resistance equation, even for externally humidified membranes. The proposed ohmic loss of the stack is shown, ohmloss CELLohmLOSS ( R R ) V = i + (4 5) cell CELLohmLOSS CELLelectronic CELLmembrane ( Relectronic + Rmembrane ) irohmic V = N V = i = (4 6) R electronic is the resistance of the non-ideal electrodes. Sometimes it is neglected, at other times it cannot be because its value is 5% of the total resistance as in [4]. Because we are using empirical methods to find the stack resistance the specific value of R electronic is not important. R electronic is lumped with R membrane to give R ohmic so its value in relation to R membrane does not matter. As found in [16,17] the cell resistance only depends on the linear fit of the current (i) and temperature (T) of the stack. There was a fit to the terms i, T and it as well but these terms did not contribute to the cell resistance. For an initial fit it is expected that only linear relationships of current and temperature are required for the resistance of the stack (R ohmic ). If this is found to be invalid the terms i, T and it should also be used. Following this the empirical form of V ohmloss is, ( ε + ε i + T ) V ohmloss = i 1 ε 3 (4 7) As a rough estimate of results produced in [17], the R ohmic will change by 10% with a change in temperature from 55 0 C to 65 0 C and 30% with a change in current over the specified operating range. An increase in current is expected to increase the stack resistance and an increase in temperature is expected to decrease it as found in [17,1,8,9]. Despite the best efforts for optimised stack design and optimised operating conditions the membrane resistance will increase slightly as the current is increased. This is because of the electro-osmotic water drag slightly dries the membrane at higher currents at a level that is not damaging. Capstone Project A

71 4..1 Extreme Operating Conditions The concentration of the reactants will affect the current of the fuel cell and in turn this will affect R membrane (by about 30%). As long as the water content is at an adequate level in the membrane the concentration of the inlet gases will not directly affect R membrane, but it will affect the current (which is measurable) which in turn affects the resistance of the membrane. There is usually no requirement for the concentration of the reactants to be in the R membrane equation. This is seen in all the simple analytical models. This however is not strictly true. In extreme conditions water content in the membrane will be dependant on the flow (concentration) of the reactant gases to some extent. Very high reactant pressure both of the oxygen and particularly of the hydrogen fuel can cause membrane dehydration, even if the anode gas stream is fully humidified [31]. High hydrogen pressures results in a reduction in the hydration effect of the hydrogen gas. Similarly, if air was used with a large mass flow then this could dry out the stack and affect the membrane resistance. At UTS limits are stated in the fuel cell manual on the hydrogen and air pressure so the drying effect of the reactant gases should not be a problem. In summary the resistance of the stack will be a function of the stack temperature and current. As a first approximation the resistance of the stack is assumed constant. A better approximation is that it depends on the stack temperature and current. In extreme operating conditions, the pressure, stoichiometry and humidity of oxygen and hydrogen will affect the stack resistance. Capstone Project A

72 4.3 Activation Loss The activation loss of the fuel cell relates to the slowness of the reactions taking place on the surface of the electrodes [1]. Energy is lost breaking the oxygen and hydrogen bonds at these electrodes. In 1905 Tafel observed that this voltage loss is proportional to the logarithmic of current. Later on, the theoretical justifications of his constants were given. The activation voltage loss of one cell is described now: j i V = CELLactLOS S = AT ln AT ln (4 8) j0 i0 where: A T i i 0 = Tafel slope, = Current of the fuel cell (A), = Exchange current of the fuel cell (A), j = Current density of the fuel cell (ma.cm - ) = i / A, A = Area of each cell in the stack (cm - ), j 0 = Exchange current density of the fuel cell (ma.cm - ). The units for current density and the exchange current density are often quoted in ma.cm -. In the Tafel equation these terms combine to form a ratio so as long as the same units are used for both this is not important. Because of this fact it does not matter if current or current density is used in equation (4-8). Note that this equation only holds for i>i 0. If i was smaller than i 0 then the activation loss term would be negative, indicating a gain in voltage instead of a loss. Electrochemical reactions occur at each electrode, so there will be an activation loss at each electrode, namely the anode (hydrogen) and cathode (oxygen). The reactions at each electrode have been given previously. Anode reaction H H e + + ( 1) Cathode reaction + O + H + e H O 1 ( ) Capstone Project A

73 Let use take the reaction at the cathode, shown in equation (-). At zero current there is really an equilibrium between the products and the reactants, so a reaction is taking place but also the reverse reaction is taking place, so at zero net current the cathode reaction is shown below. Cathode reaction + O + H + e H O 1 The current that occurs in this equilibrium reaction is called the exchange current. It is the current that continually transpires between the products and reactants. When the cathode reaction in equation (-) moves in one particular direction (i.e. to the right) then the reaction is shifting not starting. If the reaction is quite active there will be a higher value of i 0 and the losses involved in changing an existing reaction will be less than starting a slow reaction. It can be seen from this that a high exchange current is desired. The exchange current changes in orders of magnitude and is the dominant variable in the activation loss, the value of A T plays a secondary role. In the case of the hydrogen anode the exchange current is much larger and thus produces a small loss compared to the oxygen cathode. For a typical low pressure PEMFC operating at ambient pressure j 0 for the hydrogen anode would be around 00 ma.cm - when the oxygen cathode is in the order of 0.1 ma.cm - [1]. As stated the hydrogen anode is more active and has smaller losses compared to the cathode. This is why the activation loss of the anode is often neglected. As discussed the constant A T has been given theoretical justification and it can be given by [1]. A T RT = (4 9) αf where: A T = Tafel slope, R = Gas constant J K -1 mol -1, F = Faraday s constant ( C mol -1 ), α = charge transfer coefficient. The charge transfer coefficient (α) is the proportion of the electrical energy applied that is harnessed in changing the rate of an electrochemical reaction [1], its value is Capstone Project A

74 between 0 and 1 depending on the reaction and electrode material. When the current is low for example, the electrochemical reaction at the electrode is at a set rate, when the load current increases, the rate at which the electrochemical reaction takes place must also increase. The charge transfer coefficient is the proportion of the electrical energy input to the fuel cell reaction that is used to change the rate of the electrochemical reaction. The charge transfer coefficient is not strictly set. Among other variables it is slightly dependent on fuel partial pressure. Increasing the temperature of the fuel cell will increase the constant A T as shown in equation (4-9). The minimising effect of increasing the temperature of the stack on i 0 will dominate. Therefore, the effect of raising the temperature will decrease the activation loss as is seen with PEM fuel cells operated at a higher temperature. Ideally the exchange current i 0 can be experimentally found from a range of operating conditions. In other words the exchange current is ideally known depending on the fuel partial pressure, stoichiometry and fuel cell temperature, and it is then accurately estimated for any operating condition. Most papers experimentally find the value of i 0 for a every new set of operating conditions, without describing the dependence on the important parameters Cathode Activation Loss The activation loss for the oxygen cathode of PEM fuel cells dominates. The activation loss is a simple relation of the exchange current density, gas constant, Faradays constant, the temperature of the fuel cell, and the current density as shown below. V CELLactLOSS at cathode = RT α F c ln i i 0, c (4 30) where: RT α c F = Tafel slope of the cathode, i i 0,c α c = Current density of the fuel cell (A), = Exchange current of the fuel cell cathode (A), = charge transfer coefficient of the cathode. Capstone Project A

75 4.3. Anode Activation Loss The anode activation loss will be dealt with here. I will include the anode activation loss in the proposed model despite its smaller value. When the activation loss is included in other models the charge transfer coefficient of hydrogen is often assumed to be 0.5, as in [14]. For this model the activation loss parameters will be experimentally found so this assumption is not applicable or important. The anode activation loss is similar to that at the cathode. V CELLactLOSS at anode = RT α F a ln i i 0, a (4 31) where: RT α a F = Tafel slope of the cathode, i i 0,a α a = Current density of the fuel cell (A), = Exchange current of the fuel cell anode (A), = charge transfer coefficient of the anode. Because the anode activation loss is lower than the cathode activation loss the concentration of the hydrogen inlet gas is not as important to the activation loss of the cell Exchange Current Berger [14] proposed the analytical relationship of the exchange current of the anode (i 0,a ) and cathode (i 0,c ) He gave the exchange currents in terms of temperature and the concentration of hydrogen and oxygen. Amphlett et al. [16,1] used this relationship along with equations (4-30) and (4-31) to give the activation loss in terms of the concentration of hydrogen and oxygen, temperature of the stack, the current and several constants that can be experimentally found. Capstone Project A

76 Oxygen Cathode Berger stated that the exchange current of the cathode could be described as below [14,16], i 1α F c αc 1 αc e ( C ) ( C ) ( C ) exp 0 0, c = ncfakc proton HO O RT (4 3) where: T = Temperature of the Fuel Cell (K), C O = Concentration of oxygen, F = Faraday s constant ( C mol -1 ), R A α c n c = Universal gas constant, = Area of each cell in the stack, = Charge transfer coefficient of the cathode, = number of electrons transferred per mole of the electrolysed component of the cathode reaction [16], k c 0 = Intrinsic Rate constant, C = Concentration of water, H O C proton = Total concentration of protons in the membrane, F e = Standard free energy of activation for the cathode reaction in J/mol. The constants F and R are known and n c, C and F e are constants that are initially H O unknown. The parameters α c, k c 0 and C proton are approximately constant for the reaction and are assumed to be constant as in [16]. The exchange current for the 0 cathode is a function of T, C O and the quoted constants F, R, n c, C H O, F e, α c, k c and C proton. Capstone Project A

77 V CELLactLOS S The value of i 0,c can then be placed into the cathode activation loss equation (4-30). This rearranged gives the activation loss at the cathode, = α c R F F e Rln c Tlni + α cf α cf 0 ( n FAk ) 1α α R( 1α ) c C proton c c c C H ln O T T α cf The activation loss of the cell at the cathode can be described as, V CELLactLOSS ( C ) = ζ T ln i + ζ + ζ T + ζ 4T ln (4 34) 1 3 O The relationship of the cell loss to the stack loss can then be used, V = V * N (4 35) actloss CELLactLOSS cell ( C ) O (4 33) Hydrogen Anode Berger also stated that the exchange current of the anode. This is [14,16], 0 Fec i0, a = na FAka CH exp (4 36) RT where: T = Temperature of the Fuel Cell (K), C H = Concentration of hydrogen, α a n a = Charge transfer coefficient of the anode, = number of electrons transferred per mole of the electrolysed, component of the anode reaction, k a 0 = Intrinsic Rate constant in the anodic reaction, F ec = Standard free energy of activation for chemical absorption for the cathode reaction in J/mol [16]. The constants F and R are known and n a and F ec are constants that are initially unknown. The parameters α a, k 0 a are approximately constant for the reaction and assumed to be constant. Capstone Project A

78 The value of i 0,a can then be placed into the anode activation loss equation (4-31). This rearranged gives the activation loss at the anode as, V CELLactLOSS = α a R F Fec T ln i + α a F R ln α a F 0 ( na FAka ) T ln( C ) H (4 37) Therefore the activation loss of the cell at the anode can be described as, V CELLactLOSS ( C ) = ζ T ln i + ζ + ζ 7T ln (4 38) 5 6 H The relationship of the cell loss to the stack loss can then be used, V = V * N (4 39) actloss CELLactLOSS cell Combined The activation loss for the cathode and the anode can now be combined in the form of a semi-empirical equation. V actloss () i + δ + δ T + δ T ln( C ) δ T ( C ) = δ 1T ln 3 4 O + ln 5 H (4 40) The physical meaning of δ 1, δ, δ 3, δ 4, δ 5 are, δ 1 = R R + (4 41) α F F c α a δ = Fc Fec + α F α F c a (4 4) δ 3 = R ln 0 ( n FAk ) c 1α c αc C proton CH O α c F (4 43) δ 4 = ( 1α ) R c (4 44) α F c 0 ( n FAk ) R ln a δ 5 = α a F (4 45) Amphlett et al. made accurate predictions of the activation loss with this form of the activation loss as shown in [1]. One difference with their approach is that the Capstone Project A

79 concentrations of the reactant gases were the averages of the stack concentration in the gas flow channels. In the UTS model the concentration at the input to the stack are used instead. A well hydrated stack was assumed in this model so if the hydration levels of the stack are greatly altered then V actloss will also have to be found with dependence on the humidity levels of the reactant gases. Under normal operating conditions equation (4-40) will hold. The activation loss of the stack can be used in the form as below. V actloss i = Astack ln = Astack ln() i Astack ln( i0 ) i (4 46) 0 A stack is the Tafel slope of the stack. Equation (4-40) can also be put into this form, V actloss () i ( δ + δ T + δ T ln( C ) δ T ( C ) = δ 1T ln 3 4 O + ln 5 H (4 47) The parameters T, I, C O and C H will be known when the stack is being used. A stack, and i 0 will be found by examining the Tafel plots by looking at each polarisation curve. δ 1, δ, δ 3, δ 4, δ 5, will be found by recalculating A stack, and i 0 with different temperatures, and concentrations of oxygen and hydrogen. concentration of oxygen will be the operating condition that most affects V actloss. The 4.4 Concentration Loss As described in the literature review the concentration loss is, V conloss = i B ln 1 i (4 48) L The dependence of V conloss for all the operating conditions must be found experimentally. It is fortunate that the fuel cell at UTS does not exhibit the concentration loss. This is shown in the parameter determination section of this report. The most important parameter for this loss is the concentration of the oxygen gas, followed by, the temperature, concentration of hydrogen and the humidity of Capstone Project A

80 hydrogen. Maggio et al. [4] state the dependence of i L on the temperature of the stack, gas mole fraction of oxygen to air, and a term called the gas porosity in the gas-diffusion layer. It is unclear whether all of the relevant terms have been captured with this approach. An example of the shape of the cell voltage with purely a concentration loss is illustrated in figure 4-1. It shows the cell voltage in volts versus the current density of the cell (ma/cm ). The limiting current i L is 1000 ma/cm with B = and 0. volts. Figure 4 1: Concentration Loss 4.5 Internal Current The internal current acts on all the losses described. The current for the loss mechanisms are increased because of the internal current. Effectively this means the current used in the voltage loss equations is equal to the external current i plus the internal current i n. The losses are then, V actloss V ( i + i )( ε + ε i + T ) = (4 49) ohmloss n 1 ε 3 ( i + i ) ( δ + δ T + δ T ln( C ) δ T ( C ) = δ 1T ln n 3 4 O + ln 5 H (4 50) V conloss = i + in B ln 1 (4 51) il Capstone Project A

81 In practice i 0 is always greater than i n so equation (4-50) holds for all i. The term ε i was not equal to ε (i+i n ) in equation (4-49) because the empirical constant is calculated with respect to the observable external current. Unfortunately the omission of the internal current from many models means that it has not been studied in depth. The dependence on the important operating conditions will have to be found empirically. The internal current will primarily depend on the concentration of the reactant gases. The internal current affects the activation loss the most. This is the loss that occurs at low currents. At higher currents i n has only a minor effect in proportion to the losses. In the case of the ohmic loss equation (4-49), the voltage difference i n will impose on this loss will be i n *R ohmic. Both i n and R ohmic are be small so this difference is not significant. Therefore the following approximation is valid, V ( i + i )( ε + ε i + ε T ) i( ε + ε i + T ) = (4 5) ohmloss n ε 3 For the concentration loss the current at which this loss becomes significant is very high so that i+i n will be approximately equal to i. The concentration loss is then, V conloss = i + in B ln 1 i i L B ln 1 i (4 53) L If the internal current is too difficult to predict the internal current needs to be omitted from the model. This would mean the model is not accurate near the open circuit condition. 4.6 Stoichiometric Ratio As stated in the section.5.1, the stoichiometric ratio λ x is the ratio of the input gases (hydrogen or air) compared to the amount of the gases used. Calculations of this ratio for air and hydrogen are now shown. Capstone Project A

82 4.6.1 Air and Oxygen As seen by equation (4-4) the charge produced by half a mole of oxygen is shown, Total Charge = -F Coulombs For each mole of O, Total Charge = -4F Coulombs (4 54) Dividing by time, rearranging and using conventional current (which is opposite to the actual charge flow) for each cell we get, O used per cell = i 4F moles.s -1 (4 55) For the entire stack this is, Total O used = in cell 4F moles.s -1 (4 56) where: i = Current of the fuel cell (A), N cell = Number of cells in the stack. The units for this flow rate is not in a conventional form so it is often expressed in kg.s -1 or another unit. Also the expression must be in terms of air not oxygen since air is the inlet gas. Using the molar proportion of oxygen to air of 0.1 and the molar mass of air as 8.97*10-3 kg.mole -1 [1], the amount of air required is; Air usage = *10 * 0.1* 4F in cell = 3.57*10-7 *i*n cell kg.s -1 (4 57) The stoichiometric ratio for air λ air is then, λ air = * mair (4 58) *10 * in cell where: m * air is the mass flow of air, in kg.s -1. Capstone Project A

83 4.6. Hydrogen The stoichiometric ratio can be found similarly. The total charge per mole of H is shown in equation (4-4). For the convenience of discussion, it is rewritten below. Note that the total charge is for a mole of hydrogen gas where it is half a mole for O. Total Charge = -F Coulombs For the entire stack this is, Total H used = in cell F moles.s -1 (4 59) using the molar mass of hydrogen as.0*10-3 kg.mole -1, this is, H usage = 3.0 *10 * F in cell = 1.05*10-8 *i*n cell kg.s -1 (4 60) The stoichiometric ratio for H, λ H is then, * mh λ H = (4 61) *10 * in cell * where: m H is the mass flow of air, in kg.s Water Transport Water Transport issues have been around as long as the PEMFC. This is still the key issue for PEMFC manufactures as can be seen in the amount of research papers aimed at modelling water transport. Larmine and Dicks [1] give calculations relating to the relative humidity of air, which is useful for a qualitative view of the drying effects of air. The main concern however is the water content in the hydrogen input gas. This is mainly due to the electro-osmotic water drag that dries the membrane at the anode and increases its resistance. Capstone Project A

84 4.7.1 Relative Humidity of Hydrogen The effect of relative humidity of hydrogen can be seen by the experiments completed by Chu and Jiang [30]. They used a 6 cell air breathing stack at a temperature of around 35 0 C. Here they use an externally humidified stack at varying hydrogen relative humidities. The hydrogen used was wet and dry. The effect of this on the stack is shown in figure 4-. This is an externally humidified stack so the effect of the relative humidity of hydrogen will be greater than that seen on the UTS self-humidified stack. Figure 4 : Effect of Relative Humidity of Hydrogen on the Polarisation Curve The performance does not change greatly when the stack is not self-humidified. The stack at UTS is self-humidified and this is meant to maintain a well hydrated stack. The assumption that the water content in the UTS membrane would not change considerably seems valid Relative Humidity of Air The effect of relative humidity of air was also viewed by Chu and Jiang [30]. They operated their stack under vastly changing relative humidities of air. In these extreme conditions the water content in the membrane and the gas flow channel are not at their optimum levels. The affect of the performance of the stack at varying air humidity levels can be seen in figure 4-3. Capstone Project A

85 Figure 4 3: Effect of Relative Humidity of Air on the Polarisation Curve As shown in figure 4-3 the concentration loss greatly affects the performance of this stack when the relative humidity of the air is 10%. Remember this is an externally humidified stack where the water management issue is more of a problem. A concentration loss is not evident at air relative humidities of 30% to 85%. At these varying levels the resistance of the stack increases. The relative humidity is varied considerably, and this is for an externally humidified stack. This graph gives a qualitative view of the problems of an externally humidified stack operating at ultra low humidity levels of air. In practice the relative humidity would not reach this level. Typical relative humidities vary from about 30% in the ultra-dry conditions of the Sahara desert to about 70% in New York on an average day [1]. 4.8 General Model Summary Logically the operating voltage for the stack is the operating voltage for each cell multiplied by the number of stacks. Most models describe the operating voltage for each cell. One of the papers that confirms this simple relationship is Chu and Jiang [30]. V stack = N cell * V cell (4 6) Capstone Project A

86 V Calculating the stack voltage can be done as shown, V stack = N cell (V CELLreversible -V CELLohmLOSS -V CELLactLOSS -V CELLconLOSS ) (4 63) V stack = V reversible -V ohmloss -V actloss -V conloss (4 64) These terms have the internal current loss incorporated into them. These are, = N ( ) ( )( ) * * *10 T *ln ph p 8.453* O T Re versible cell (4-0) V actloss ( ε + ε i + T ) V ohmloss = i 1 ε 3 (4-5) ( i + i ) ( δ + δ T + δ T ln( C ) δ T ( C ) = δ 5T ln n 1 3 O + ln 4 H (4-50) V conloss = i B ln 1 i (4-53) L where: i T = Fuel cell current in amps, = Temperature of the fuel cell stack in Kelvin, * H p = The partial pressure of H in atm, * p = The partial pressure of O in atm, O C H = Concentration of hydrogen, C O = Concentration of oxygen. Major assumptions with this model are the stack has a reasonable amount of water and the stack is isothermal. Table 4- overleaf shows how each operating parameter affects the reversible voltage and losses. Also stated are the assumptions when an operating parameter is neglected from the loss. Remember when viewing this table that the concentration of the reacting gases are dependant on both the pressure and stoichiometry of the gas. Capstone Project A

87 Table 4 : Affect of Operating Conditions. Operating Parameter V reversible V actloss V ohmloss V conloss I N/A Yes Yes Yes T Yes Yes Yes B, i L p * O Yes Yes (4) B, i L p * H Yes Yes, minor (5) B, i L λ N/A Yes (6) B, i O L λ N/A Yes, minor (7) B, i H L φ O N/A (1) (8) (11) φ H N/A () (9) (1) i n N/A Yes, (3) (10) (13) LEGEND Yes: Indicates that the operating parameter can been incorporated into the model as can be seen in the pervious equations. φ x : Relative humidity of hydrogen or air. N/A: The parameter does not apply in any circumstances. B, i L : The empirical constants will be affected by the operating condition in a fashion that is unknown. COMMENTS (1), (), (8), (9), (11), (1): It is assumed that the stack is operated so these parameters do not affect the model. The stack should be operated so that the membrane is well hydrated without stack flooding. This assumption may not hold in (1) and (), in this case the resistance would be empirically modelled in regard to significant operating parameters. Capstone Project A

88 (3): The activation loss is dependant on the internal current. In the case where i n is too difficult to model i n should be omitted from the model. If this were done the model would only be valid for currents above ~0.3A. (4-7): These parameters will only affect the performance of V ohmloss when the stack is run in extreme cases. At this stage the limits of these extreme cases are unknown. If one of these parameters greatly affects the stack resistance then this parameter should not be operated to that level when the resistance starts to change. (10,13): i n does not change these losses significantly as described previously in section W Fuel Cell Model The model specific to the 500W PEM fuel cell model is now be presented. The single polarisation curve shown in the 500W PEM fuel cell manual is shown below in figure 4-4 [10]. Figure 4 4: Polarisation Curve Capstone Project A

89 The operating condition for the polarisation curve in figure 4-4 is as follows: Temperature Pressure of air Pressure of hydrogen 60 0 C C 0 10 psi, approx atm 0 psi, approx atm V Air stoichiometry Hydrogen stoichiometry 1. The first aspect of the stack is that the open circuit voltage per cell (~0.97V) is not equal to the reversible voltage (~1.V). The internal current is therefore required to make the model accurate for open circuit and low currents. If the modelling of the internal current proves to be hard then the internal current would have to be omitted from the model. The model would be valid for currents above about 0.3 amps. The second aspect is that the concentration loss does not come into play for these operating conditions and currents. The maximum current specified for the load is around 8A. Using the electrode area of 64 cm, this correlates to current density of 434 ma/cm. Therefore the model for the fuel cell at UTS should not contain a concentration loss term. Practical models (eg. [30]) also omit this term when the individual stack does not exhibit the concentration loss. A model that is accurate and as simple as possible is required. The equations for the model are shown, = V stack = V reversible -V ohmloss -V actloss (4 65) 1 ( ) ( )( ) * * *10 T * ln ph p 8.453* O T Re versible V actloss ( ε + ε i + T ) V ohmloss = i 1 ε 3 (4 67) ( i + i ) ( δ + δ T + δ T ln( C ) δ T ( C ) (4 66) = δ 5T ln n 1 3 O + ln 4 H (4 68) Capstone Project A

90 where: i T = Fuel cell current in amps, = Temperature of the fuel cell stack in Kelvin, * H p = The partial pressure of H in atm, * p = The partial pressure of O in atm, O C H = Concentration of hydrogen in chosen units, C O = Concentration of oxygen in chosen units. V Simplest Model The simplest model that should be used in the initial stages of modelling the UTS stack is now shown. Expected deviations from this would result in more detailed empirical modelling. The resistance loss is the first factor if any that would be altered to empirically fit the data. In the following section a constant resistance value was fit to the data so the effect of current did not play a significant role. The resistance would normally be found from the current interrupt method later described and not estimated from the polarisation curve. The simplest equations for the modelling of the PEM fuel cell at UTS are shown below. The resistance R ohmic is assumed to be constant for all temperatures and currents. = V stack = V reversible -V ohmloss -V actloss (4-65) 1 ( ) ( )( ) * * *10 T * ln ph p 8.453* O T Re versible (4 69) V actloss V ohmloss = ir ohmic (4 70) ( ζ T ln i + ζ + ζ T + ζ T ln( C ) = 3 4 (4 71) 1 3 O The activation loss is purely calculated from the loss at the cathode. An internal current is omitted so the open circuit voltage of the stack would not be predicted if these equations were used. Capstone Project A

91 4.10 Charge Double Layer The charge double layer capacitance is a function of the electrode and individual stack properties. This value does not change for varying operating conditions [60] Without Concentration Loss An electrical model that encompassed the activation loss is shown in figure -8. This will be the equivalent circuit for the UTS fuel cell. The values Vreversible, Ract and Rohm will change according to the equations (4-66) to (4-68). In the case of Ract, we know the voltage of the loss at a certain current so the effective resistance can be used with Ohm s law. This predicts the dynamic performance of the fuel cell. This circuit is simple because the circuit only has ohmic, activation losses and an internal current With Concentration Loss For completeness I will illustrate the equivalent circuit that includes the concentration loss. This circuit is mostly used in the field of batteries. The parallels in the performance of fuel cells and batteries mean there are a lot of similarities in the equivalent circuits. The equivalent circuit for this model is shown in figure 4-5 below, where Re is the electrolyte and contact resistance (=Rohm), Rct is the charge transfer resistance (it is this the same as activation loss), Cd is the double charge layer capacitance, Zd is the diffusion impedance also called the concentration loss or Warburg Impedance. The form of the equivalent circuit is figure 4-5 is shown on page of [47]. Figure 4 5: Equivalent Circuit Including Concentration Loss Capstone Project A

92 The concentration loss is an impedance rather than a straight resistance. The dynamic response of this can be seen in [54]. The impedance Zcon is initially unknown. The real part of the impedance can be found by using the concentration loss equation (4-53). Because this is a simple circuit for a complex reaction it is not perfect for every fuel cell. The predictions of the equivalent circuit are made extremely accurate by adding a host of other electrical components that vary for each fuel cell. This is as seen in the Electrochemical Impedance Spectroscopy literature. Capstone Project A

93 5. PARAMETER DETERMINATION The parameter determination for the general proposed PEMFC model is described in this section. Included in this description is the process for the parameter determination for the 500W PEM fuel cell. I have also shown how the parameter determination is affected when simplifications mentioned in section 4 are applied. Outlined below are the experimental techniques and the specific details of the data analysis required to determine the parameters in the proposed model. The experimental techniques used are the current interrupt method and Electrochemical Impedance Spectroscopy (EIS). The current interrupt method will be frequently used in the testing of the fuel cell at UTS, and the EIS technique may be used to provide a more detailed analysis of the fuel cell. By using these techniques the parameters for the resistance, activation and concentration losses plus the internal current can be found. 5.1 Current Interrupt Method The current interrupt method is the most widely used and simplest experimental procedure for fuel cell parameter determination. When the concentration loss is insignificant the ohmic and activation loss can easily be found by this straightforward technique. The double layer capacitance is also found from this method. The resistance of fuel cell stacks can be easily measured using the current interrupt method. The resistance loss is determined from the sudden change in voltage which results from the interrupt test as shown in figure 5-1. The V=IR relationship of the resistance voltage loss and current gives the resistance of the stack. The external current used in this simple calculation is the value it was just before the current of the stack was changed to zero. Capstone Project A

94 Figure 5-1 below [1] shows an example of experimental results from a current interrupt test. The output of the current interrupt test shown is for a low temperature, ambient pressure, PEM fuel cell. The x axis on the graph below represents time at 0. sec/div, the y axis is voltage with the scale unspecified. The current density at the time of the test was 100 ma.cm -. Figure 5 1: Current Interruption on a PEM Fuel Cell From this figure the ohmic loss called Vr is observed. After the step rise the stack voltage slowly rises to its open circuit value. The double layer capacitance can be found from this graph by looking at the first order response of the activation loss and the equivalent circuit of the fuel cell; this is illustrated in figure -8. The current interrupt method can be applied by using a Digital Storage Oscilloscope (DSO) and an electronic load for rapid switching, both of which are available at UTS. The experimental set-up for this test is shown overleaf in figure 5-. Capstone Project A

95 Figure 5 : Current Interrupt Test Set-Up As well as being the current interrupt test set-up figure 5- is also the simplified circuit diagram for the operation of the fuel cell. The electronic load shown can be one of the Lybotech loads on level 18 of Building 1. The Lybotech load is a controllable Inverter rated at 00V and 10A coupled with three resistive loads. A 10A load for the fuel cell is the maximum current to be switched with this set-up, as it is limited to the Lybotech inverter rating. Andrew Dicks stated that the electronic load was the only hindrance at higher currents. The 500W fuel cell can switch all currents provided that the electronic load is rated sufficiently. Therefore modification is required for performing the tests when the fuel cell load is greater than 10A Manual Interrupt When the current is required to be above 10A an alternate load method needs to be used. There are a number of methods to create a manual load for the interrupt test. These methods are preferred over the other option of buying a new expensive electronic load that has a higher current rating. Capstone Project A

96 Example tests and techniques are shown in figure 5-3 [54], where a) Single Interruption technique, b) periodic interruption technique, c) use of a relay for a battery, and d) use of a diode on a battery. Figure 5 3: Interruption Techniques In addition to these techniques a MOSFET can be used for the switching. This is the method that the frequency analyser Solartron 186 uses. Capstone Project A

97 5.1. Valid Results There are two ways to check whether results are valid for the current interrupt method. The first way is to operate the fuel cell at open circuit and measure the resistance of the fuel cell with an accurate four-wire resistance meter. Resistance can be estimated as approximately constant over the full range of operating conditions for the purpose of this check. Next the resistance of the stack is measured at a current (eg. 5A) using the current interrupt method. If the resistance was, for example, 10 times smaller or larger than the open circuit resistance, then there must be serious flaws in the procedure or experimental set-up. A second checking system required a dummy circuit to be constructed. Background to this method can be found in [64] although this method is quite simple. A dummy circuit can be used as illustrated in figure 5-4, with known resistor and capacitor values input to the test circuit. Figure 5 4: Testing Circuit The test values of the resistors and capacitor will be experimentally found by the current interrupt technique. If they correspond with the actual values in the test circuit then the current interrupt technique has passed the test. Capstone Project A

98 5.1.3 DC Offset At low currents when the voltage drop from the interrupt test is small there may be difficulty with accurately recording the small voltage loss. This issue was brought to my attention by Andrew Dicks [60]. The problem arises because a DSO has a limit to the DC offset it can nullify. This issue is illustrated in figure 5-5. Figure 5 5: DSO Screen To counteract this problem a battery can be placed in the DSO circuit as shown in figure 5-6. Figure 5 6: DC Offset Set-Up Capstone Project A

99 The oscilloscope can now be set to a smaller range to display clearly the small voltage variation as shown in figure 5-7. Figure 5 7: New DSO Screen Molten Carbonate Fuel Cell There is not a lot of published data on the current interrupt method. It is seen as an old technique without need for further research papers. Because of this fact there are not many recent research papers available on PEMFC interrupt tests. One paper that does go into detail on this technique is Lee et al. [46] on the Molten Carbonate Fuel Cell. The results of the tests from [46] should be taken with the realisation that different fuel cells have different dynamic responses. The Molten Carbonate Fuel Cell has a much quicker dynamic response compared to the PEMFC because of the double layer capacitor value of these cells. This fact is shown by the time the activation loss takes to settle. For the PEMFC in figure 5-1 the activation loss takes approximately 0.8 seconds to settle whereas the activation loss of the Molten Carbonate Fuel Cell shown in figure 5-9 takes 00 ms. Capstone Project A

100 The figures 5-8 and 5-9 below are from Lee et al. [46]. The test conditions for these figures were for an interrupt test where the current switched was 10A. Figure 5-8 depicts the resistance voltage loss occurring in 0µs. The activation loss is dominant from 100µs to 00ms. This is shown in figure 5-9. Figure 5 8: Ohmic Loss [46] Figure 5 9: Activation Loss [46] A voltage peak occurs at 10µs in figure 5-8, this may occur with the testing of the UTS fuel cell. If it occurs this peak should be disregarded from the ohmic voltage loss. This individual fuel cell exhibits a concentration loss. After 500ms the concentration loss (shown as E loss) takes over. This is shown overleaf in figure Capstone Project A

101 Figure 5 10: Concentration Loss [46] It can be seen in figure 5-10 that the concentration loss shows a slower response compared to the activation loss. This indicates that the concentration loss is an impedance in parallel with the double layer capacitor as shown previously in figure 4-5. A view of the entire test as a log-linear graph is shown in figure Figure 5 11: Log Scale Overview [46] 5. Electrochemical Impedance Spectroscopy Electrochemical Impedance Spectroscopy (EIS) is an AC impedance experimental technique. During EIS experiments, small amplitude (5-0 ma) AC signals of varying frequencies are applied to the fuel cell being studied. This small signal is applied with a DC bias that is equal to the DC voltage of the fuel cell. The impedance of the fuel cell is then recorded. This can be done when the fuel cell is in open circuit condition or in the presence of a DC load. Capstone Project A

102 The advantages and disadvantages of EIS with particular emphasis on resistance measurements are summarised by Mikkola [3] as below, Current interruption method is more straightforward than the AC impedance method, since the magnitude of the ohmic loss can be obtained directly from the data. The AC impedance method reveals more information about the cell, but the analysis of the data is more complicated, because resistance is not measured directly, i.e. the resistance is not a function of terminal voltage. Instead, the resistance is evaluated from simultaneous amplitude and phase measurements by fitting the data into an equivalent circuit model of the studied cell. In addition, most error components are difficult to eliminate, because they have the same frequency as the useful information Simple Method UTS does not have the equipment that is readily used for industry standard EIS measurements. This equipment is very expensive and is usually used but not strictly limited to single and half-cells. However simple EIS measurements can be made. These measurements are less accurate and more time consuming compared to the measurements with proprietary instruments but are more affordable. The basic concept and the simple EIS measurement is shown below in figure 5-1. Figure 5 1: Basic EIS Circuit Capstone Project A

103 An electronic load is controlled to produce a fixed external current from the fuel cell. An AC voltage source (Vac) is used to produce an AC current of about 5-0mA that is set by varying the resistor Rac. A DC bias voltage (Vdc) is included in the source voltage that is equal to the DC voltage of the fuel cell. The phase and amplitude of the AC voltage of the fuel cell would then be recorded on the DSO. This is done for a range of AC input frequencies. The impedance of the fuel cell is then calculated by looking at the voltage magnitude and phase of the fuel cell with respect to the voltage signal across the resistor Rac. The concept of this technique has been presented although the finer details of the circuit are not yet known. This method compared with the current interrupt technique is more complicated and susceptible to noise. 5.. Commercial Products One way to conduct the experiments is to have an Electrochemical Interface that inputs the signal to the fuel cell and an analyser that records the impedance. This is usually done with expensive commercial products such as Solarton. Impedance measurements and the analysis are much more automated with these products. The main disadvantage with this method is the high costs of the commercial products. The Electrochemical Interface is a high-bandwidth potentiostat which provides the DC cell bias voltage and a small sinusoidal AC signal (typically a few millivolts). The Frequency Response Analyser records the results for a range of input frequencies. An experimental set-up with Solarton products is shown in figure 5-13 [65]. The set-up here is not for a fuel cell but for a connection with a battery in a,3 or 4 terminal configuration. The fuel cell would replace the battery in this set-up. Capstone Project A

104 Figure 5 13: EIS With Commercial Products 5..3 Result Analysis Analysing the results of the EIS spectrums can be complex and this is why commercial products are used to measure and analyse the EIS data. The following section provides an introduction to the EIS analysis. As stated, EIS is often done with cells and half cells. In these cases the anode and the cathode are accessible. The impedance measurements for the anode and the cathode can be taken separately. Data for an EIS experiment both on the anode and the cathode for a PEMFC are shown overleaf in figure 5-14 [47]. The arrow at the top of the screen in figure 5-14 indicates that the curve is the cathodic impedance. The arrow at the bottom of the screen specifies that the curve is the anode impedance. Curves that are not referenced are the phases of the impedance. The top of these is the cathodic phase. Capstone Project A

105 Figure 5 14: EIS at Cathode and Anode As shown in figure 5-14 the impedance and hence the voltage losses are larger at the cathode as expected. The entire cell can also be examined at different external currents. Figures 5-15 and 5-16 [47] show the impedance spectrum for currents ranging from 0 to ma. Figure 5 15: EIS of Total Fuel Cell at Low Currents Capstone Project A

106 Figure 5 16: EIS of Total Fuel Cell at High Currents Note that the impedance of the cell rises when the current increases. This is expected because the voltage loss of the cell increases with an increased current. Figure 5-17 below shows the accuracy of a predicted polarisation curve. This graph is for a solid oxide fuel cell from [47]. Figure 5 17: Polarisation Curve for a Solid Oxide Fuel Cell EIS is often used because it produces a complex equivalent circuit that is a very good predicting tool. This accuracy is not displayed in figure 5-17 although this is the exception to the rule. The equivalent circuit that was produced from this graph is displayed in figure 5-18 [47]. Capstone Project A

107 Figure 5 18: Equivalent Circuit for a Solid Oxide Fuel Cell A common graph that is analysed from the output of the EIS is a graph of the real part of the impedance versus the imaginary part. Information about the equivalent circuit can be extracted from this graph. An example of this graph depicting the response of the proposed equivalent circuit (figure 4-5) with a concentration loss is shown in figure Figure 5 19: Results of an Equivalent Circuit Capstone Project A

108 Different shapes of the type of graph in figure 5-19 determine the equivalent circuit. Example shapes are displayed in figure 5-0 [55]. Figure 5 0: Examples of Nyquist and Bode Plot Plane Plots In summary, the current interrupt test can provide simple resistance results and relatively simple equivalent circuits. EIS on the other hand is time consuming and expensive to set-up but provides detailed equivalent circuits that are very accurate. 5.3 Reversible Voltage The calculation of the reversible voltage is relatively simple. To calculate the reversible voltage, experimentally measure the pressure of the inlet gases, temperature of the cell and calculate the reversible voltage from equation (4-0). Capstone Project A

109 5.4 Resistance As discussed in the current interrupt section, section 5.1, the resistance of the stack can be found when the stack is running. Practical tips for the resistance measurements are dealt with thoroughly in [3]. It is proposed here that the current interrupt technique be used for the resistance measurement, because it is simple and accurate. The resistance will be put into a form that was described in section 4. ( ε + ε i + T ) V ohmloss = i 1 ε 3 At open circuit the four-wire resistance method can be used. The resistance measurement would be done with the accurate resistance meters provided by UTS. This will be measured with the fuel cell operating in open circuit condition and at varying temperatures. The four-wire method does not work when there is an external current as the current changes the resistance measurement. Once the current interrupt technique is proven to be an accurate resistance measurement method then the empirical fit of the stack resistance can be determined. The stack will be operated at varying currents and constant temperature. Next the temperature will be varied and resistance measurements taken for a set of external currents. An empirical fit of the data should then be carried out with the aid of a curve fitting package. Microsoft Excel is sufficient for this Resistance with an Internal Current As discussed in section 4 of this report, the ir loss is approximately equal to (i+i n )R. It will be strictly different by an amount of i n R. For the values calculated from the polarisation curve in the BSC fuel cell operator s manual this value is *0.19 = 0.01 volts for the stack volts is 0.07% of the smallest operating voltage (17.85V). Therefore ir (i+i n )R. Capstone Project A

110 5.5 Activation Loss and Internal Current The methodology for calculating the activation loss is as follows. Firstly the stack voltage equation must be examined, V stack = V reversible -V ohmloss -V actloss (4-65) When the stack is running the values of stack temperature along with the partial pressures of hydrogen and oxygen will give the reversible voltage of the stack. In the previous section the ohmic loss was calculated so the only unknown in the equation above is the activation loss. The activation loss for the polarisation curve is now identified so the parameters for this loss can be determined. The parameters for the activation loss are easily calculated when no internal current is considered. This is one of the reasons why many of the models in literature do not include the internal current. At low fuel cell operating currents (approximately four times the internal current), the activation loss is similar with and without the internal current included in the model. The model described here has included the internal current and is therefore accurate for the open circuit condition where other simple analytical models are not. The parameters for the activation loss are described by Amphlett et al [16,17] and the parameters for the internal current are described by Larminie and Dicks [1]. However in both cases they do not depict a practical way to identify the parameters for the activation loss with an internal current. The method for calculating the Tafel slope with the exchange current once the exchange current is determined (or is approximated as zero) is well described in literature [1]. The method described for calculating the internal current is original material by this author. Larminie and Dicks [1] suggest that the internal current can be found by calculating the small fuel usage when the fuel cell is in open circuit. They also admit that this method cannot be found with mass flow meters because the fuel usage is so small, in the order of cm 3.sec -1 for hydrogen. Therefore, another method must be devised to calculate the internal current of the fuel cell. An innovative technique is described in this section that calculates the internal current (i n ), Tafel slope of the stack (A stack ) exchange current (i 0 ) and their Capstone Project A

111 uncertainties. It involves a mathematical program using Mathematica and the method of least squares. A person could guess these parameters by a trial and error method that can result in a reasonable fit of the data points. This method is tedious, lengthy and unscientific. The innovative method describes the most accurate way to identify the parameters. The problems in determining the internal current, Tafel slope and exchange current occur because there are three independent variables on a single graph. Usually in x-y graphs there are only two independent variables. The Tafel slope and exchange current are commonly calculated from the slope and y-intercept of the graph of activation loss versus ln(i 0 ). The problem here is that the slope and y-intercept are dependant on the internal current i n. Which i n should we use? The answer to that is shown in the section that follows. Later I propose a simpler method that is checked against the most accurate fit. Both fits give approximately the same results so the simpler method should be used. The advantage of proposing the complex method is that there are grounds for justification of the simpler method. Results from the simpler method can now be checked and proven to be approximately valid. Once the values of i n, A stack and i 0 are known for a set of conditions the concentration of oxygen and hydrogen and the temperature of the stack are varied so that the empirical constants shown in equation (4-68) can be found Ideal Activation Loss The activation loss of the fuel cell at currents when the concentration loss is negligible is the reversible voltage minus the cell voltage minus the ohmic loss. Figure 5-1 shows a graph of ideal values for the activation loss with typical values of internal current, Tafel slope and exchange current that would be seen for the fuel cell at UTS. The typical values are in the same range as those calculated from the example polarisation curve given in the BSC fuel cell operator s manual. The real losses will have similar values but the activation loss graph will not be as smooth as the ideal graph. The typical values for the graphs are A stack = 1.V, i n = 0.006A, i 0 = A. Capstone Project A

112 Without an internal current the activation loss is described by the equation below. V actloss = A stack i ln i 0 When there is an internal current this is, V actloss = A stack i + i ln i0 n 16 Ideal Activation Loss With and Without Internal Current 14 1 Activation Loss (V) No internal current With internal current Current (A) Figure 5 1: Ideal Activation Loss With and Without Internal Current What the most noticeable aspect of figure 5-1 is that the activation loss when the internal current is omitted is very close to the loss with the internal current. The curves appear as if they completely overlap. The main variance in the different graphs is the loss around open circuit, and this is shown in figure 5-. Capstone Project A

113 Zoom of Ideal Activation Loss With and Without Internal Current in Activation Loss (V) i i0 No internal current With internal current Current (A) Figure 5 : Ideal Activation Loss With and Without Internal Current (Zoom) There is a 3.6V difference in the modelled voltage loss at open circuit in the example shown in figure 5-. This is the difference between the curve without internal current No internal current and the curve with an internal current With internal current at open circuit. When the activation loss without an internal current is graphed against the natural logarithm (ln) of the current when there is no internal current in the fuel cell the graph is linear. This is shown in figure 5-3. Activation Loss vs the Natural log of Current - Without Internal Current Activation Loss (V) ln current (ln[a]) Figure 5 3: Activation Loss vs Ln(Current) Without Internal Current Capstone Project A

114 The slope of the graph in figure 5-3 is the Tafel slope - in this case it is 1.V. When the internal current in the fuel cell is known then we can get the same graph shown above. We get this by plotting V actloss versus the natural logarithm of the internal current plus the internal current versus the activation loss (i.e. ln( i + i ) ). n When the activation loss is i A stack ln we get a linear graph by plotting V actloss i 0 versus ln(i). When the activation loss is plotting V actloss versus ln( i + i ). n i + in A ln stack we get a linear graph by i0 Athough the internal current is initially unknown the correct internal current will produce a linear graph of V actloss vs ln( i + i ) when the activation loss has an internal current present. The graphs of V actloss vs ln( i + i ) for the internal current estimated as: 1) too small (or zero), ) too large, or 3) the correct value would look similar to graph in figure 5-4. n n Activation Loss with Internal Current vs the Natural log of Current Activation Loss (V) Correct in Low in High in ln Current+internal_i (ln[a]) Figure 5 4: Activation Loss vs Ln(Current+Internal Current) Capstone Project A

115 For ideal values of the activation loss equation (4-46), the graph of V actloss vs ln( i + ) will produce a straight line when the chosen internal current is correct. If a in linear fit is applied to the data for the ideal case, then the line estimates each point on the V actloss vs ln( i + i ) graph without any error. If the internal current is not chosen n correctly a linear fit does not describe the data points. For the case where the values perfectly fit equation (4-46) the correct internal current can be found by calculating values for when there is no deviation in the linear fit and the activation loss data points Least Squares Method A recognised method for determining a linear fit is the method of least squares. This method finds the best line through the set of points by minimising the sum of the squares of the residuals (SS). A residual is the experimental f(x) value minus the calculated f(x) value for a given x value. The linear equation needs to be in the form y = mx + b and hence the activation loss of the fuel cell is rearranged in this form. The activation loss for each cell is shown by equation (4-8) which is rewritten below. V CELLactLOS S = A T ln j j 0 = A T ln i i 0 The activation loss for the stack including the internal current is shown below. V = A i + i ln i0 n actloss stack (5 1) where: V actloss i i n i 0 = Activation Loss for the stack, = Current (A), = Internal current (A), = Exchange current (A). Capstone Project A

116 Rearranged equation (5-1) is shown in the form y = mx + b, V actloss ( i + i ) A *ln( ) = A * ln i (5 ) stack n stack 0 where: x = ln(i + i n ), (5 3) y = V actloss, (5 4) m = slope of the graph = A stack = N cell *A T, (5 5) b = y intercept of the graph = * ln( i ) A stack. (5 6) The data for the fuel cell, for activation loss versus current is in the form: ( i y )(, i, y )(, i, y ),...,(, ) 1, i q y q 0 where: i y = Current in amps, = Activation loss of the stack. In the method of least squares we take the data points to be plotted as ( { + i }, y ), ( ln{ i + i }, y ), ln{ i + i }, ( y ),...,( ln{ i i } y ) ln i 3, 1 n 1 n 3 n q + where i n is still unknown. These can be put into the matrices (bold character represents a matrix): x = [x 1, x, x 3,,x q ], where x q = ln(i q +i n ). y = [y 1, y, y 3,,y q ], where y q = V actloss. Using the method of least squares the following formulas can be used [66]: n q Note that all summations are equivalent to q 1. for the slope, for the y intercept, xq n xq ( xq ) n yq yq m = (5 7) x q xq ( xq ) n xq yq xq yq b = (5 8) n x q where, n = number of data points, x... + q = x1 + x + x3 + x q, Capstone Project A

117 y... + q = y1 + y + y3 + y q, x q yq = x1 y1 + x y + x3 y xq yq, x q = x1 + x + x xq. The slope and y-intercept are usually a function of x and y, in this case they are a function of x, y and i n. The slope and y-intercept, and hence the Tafel slope and exchange current are a function of the experimental data points and the internal current. The internal current must be calculated. The calculated y values are y c = mx + b using the m and b values as described above. The sum of the squares of the residuals (SS) is a measure of how well the data points match the linear fit. ( y c y ) SS = (5 9) q As i n changes, SS also changes. The best value of i n is when SS is a minimum. That SS is when = 0 and it is minimum, not a maximum point. In the ideal case SS will i n be zero at the minimum point. In the real case the minimum of SS will be a positive number because a perfect fit does not occur in practice. q SS and SS i n will be a complex function of the data points for current of the stack, the activation loss and internal current. The expression for SS i n is 33 pages long with just eight experimental data points of current and activation loss. Experimental data from UTS experiments will have more than eight data points. A maths program should be used to calculate the minimum to the curve SS. The program Mathematica (version 3.0) was used here, which applies numerical methods to find when SS equal to zero. The code and further explanation of the code for this solution is described in Appendix 4. The program is set-up so that a set of data points from the UTS fuel cell tests can be easily input and then calculated. The user would enter these values, follow the simple procedure described and the program will calculate i n is Capstone Project A

118 the internal current (i n ), Tafel slope of the stack (A stack ) exchange current (i 0 ) and the uncertainties. The equations for the uncertainties for the slope and the y intercept are shown below [66]. 1 1 σ = * SS n (5 10) 1 σ ( n) σ = (5 11) m 1 [ n x ( ) ] q xq 1 σ ( x q ) 1 [ n x ( ) ] q xq σ = (5 1) b The slope with uncertainties is now m±σ m, and the intercept is b±σ b. The closer the data points fit the linear expression y = mx + b, the smaller the uncertainties in the slope and the intercept Calculating i 0 The calculation of i 0 is quite straightforward. Once the intercept b is known i 0 can be calculated using equation (5-6) which is rewritten below. b = * ln( i ) A stack 0 Rearranged this is, b A stack i 0 = e (5 13) where b, A stack are now known. The uncertainty range of i 0 can be calculated from the uncertainty of the y-intercept. The minimum value of i 0 is, The maximum value of i 0 is, i ( bσ ) b A stack = e 0 (5 14) i ( b σ ) + b A stack = e 0 (5 15) Capstone Project A

119 5.5. Non-Ideal Activation Loss From the polarisation data of the UTS PEM fuel cell shown in figure 4-4 a fit of the internal current, Tafel slope and exchange current parameters has been calculated. An estimated value of stack resistance was used in this section. Ordinarily the resistance of the stack would be found by applying the current interrupt or EIS method. If these methods are not used a simpler technique can be used. The method involves attributing the slope in the middle portion of the polarisation curve to the IR losses only and approximating R as linear. The activation loss levels out after low currents and the concentration loss are insignificant in this middle region. The linear region in the polarisation curve for the UTS fuel cell is from 15A to 8A as shown in figure 4-4. This was used for the approximate resistance value of the stack, which was calculated as 0.19 Ω. In practice SS is never zero for the V actloss vs ln( i + i ) graph. The correct internal current occurs when SS is minimum, that is, when the differential of SS is zero and n the point is a minimum not a maximum. point is a minimum not a maximum point. SS i n = 0 occurs at one i n value and this The experimental activation loss and modelled activation loss for the UTS fuel cell is described below in figure Fuel Cell Activation Loss Activation Loss (V) Experimental Estimated Current (A) Figure 5 5: UTS Fuel Cell Activation Loss With Modelled Data Capstone Project A

120 The stack polarisation curve and the modelled polarisation curve including the activation loss are shown in figure Fuel Cell Polarisation Curve 30 5 Voltage (V) 0 15 Experimental V Estimated V Current (A) Figure 5 6: UTS Fuel Cell Polarisation Curve With Modelled Data The SS values for a range of internal current values given in the UTS fuel cell operator manual are shown in figure 5-7. The plot is a log-linear plot of the sum of the squares of the residuals versus the estimated internal currents. This plot was carried out using Microsoft Excel. SS for estimated Internal currents Sum of the Squares of the Residuals Internal current (A) Figure 5 7: SS for Various Internal Current Values Capstone Project A

121 We can see from figure 5-7 that SS does indeed come to a minimum; it occurs around 0.06A. A closer look at this is shown below in figure Zoom of SS for estimated Internal currents Sum of the Squares of the Residuals Internal current (A) Figure 5 8: SS for Various Internal Current Values (Zoom) The minimum is around 0.064A. The plots of SS versus ln( i + i ) in Microsoft Excel take a lot of time to produce in comparison with the mathematical program developed in Mathematica. The program takes minutes to run and set-up whereas using Microsoft Excel will take hours to set-up. For the fuel cell operator s manual data, the Mathematica program calculates i n to be amps, Tafel slope of stack A stack = volts, i 0 = amps, σ m = volts, value of i 0 at min uncertainty = amps and value of i 0 at max uncertainty = amps. It may seem that the uncertainty of i 0 is considerable but in actual fact, the uncertainty of A stack plays a slightly more significant role. Note that if i n changes slightly then the open circuit voltage will change considerably. This change is also described by Larminie and Dicks [1]. n Figure 5-9 shows the upper and lower limit of the worst fit of A stack and i 0 in the uncertainty ranges. The lower curve of the graph below is with the Tafel slope of the stack equal to A stack + σ m and the exchange current as A, the upper curve is with A stack - σ m and A. Capstone Project A

122 35 Fuel Cell Polarisation Curve 30 5 Voltage (V) 0 15 Experimental V low fit high fit Current (A) Figure 5 9: Polarisation Curve With Upper and Lower Worst Fit The lower and upper fits of the polarisation curves are close to the experimental values. It is expected that when the number of data points are increased the uncertainty of A stack and i 0 will decrease Simplified Non-ideal Activation Loss Once the most appropriate technique is formulated any simplifications made can be checked and evaluated for validity. An assumption is made here that is found to be valid. Therefore the easier technique described below can be used. The advantage in having both methods is that the method shown above can be easily used, as the Mathematica program is already set-up. The major assumption here is that the Tafel slope at higher currents without an internal current is approximately equal to the Tafel slope when the internal current is added. Papers on previously modelled fuel cells probably used the same assumption although this was not specified. The technique described below is used in the modelling of batteries. The technique is used because the Tafel equation is invalid for low values of the activation loss not because of an internal current. The activation loss is never close to zero in the operation of PEMFC because of the internal current. Capstone Project A

123 In figure 5-4 it can be seen that when the estimate of i n is zero or very low the data points in the graph are similar to the correct curve using the proper value of i n. At higher currents the graph using ln( i + i ) is approximately equal the one using ln(i). n Basically if only the middle to higher current range is considered the slope and y-intercept of V actloss versus ln( i + i ) is approximately equal to V actloss verses ln(i). This is shown in figure n Activation Loss with Internal Current vs the Natural log of Current Activation Loss (V) SIMILAR SLOPE and INTERCEPT Correct internal current Low internal current ln Current+internal_i (ln[a]) Figure 5 30: Replot of Figure 5-4 This concept is in agreement with the activation loss graph in figure 5-31 which is taken from []. Figure 5 31: Activation Loss Example Capstone Project A

124 Figure 5-31 is a plot of current versus activation loss and does not include an internal current. Despite any differences at lower currents the graph of current versus activation loss has a linear slope at higher currents. The graph also depicts why the term polarisation is not a very good one. By definition a polarisation is negative but because it is associated with a loss it is sometimes incorrectly used as a positive value. The graph of activation loss verus ln(current+internal current) is shown in figure 5-3. Activation Loss versus ln(current+internal current) y = 1.357x Activation Loss (V) WITH internal current Linear (WITH internal current) ln(current) [ln(a)] Figure 5 3: Activation With Internal Current The graph of activation loss versus ln(current+internal current) and with internal current equal to zero is shown in figure Only data points higher than A are shown. This corresponds to 0.69 on the ln(current) x axis. The zero current data point can never be shown on the log graph when the internal current is zero because the log of zero is negative infinity. The zero data point is not required regardless because it is low current value. Capstone Project A

125 Activation Loss versus ln(current) Activation Loss (V) y = 1.175x NO Internal current Linear (NO Internal current) ln(current) [ln(a)] Figure 5 33: Activation Loss With No Internal Current The second method gives an A stack value of 1.175, and i 0 of exp( /1.175) = Now we can do one of two things with the new values of A stack and i 0. Firstly, if we wanted to make the model simpler the internal current would be neglected and the model would not be valid for small currents. And secondly, we could estimate the internal current with the new values of A stack and i 0. If the simple method was used i n can still be calculated. The predicted activation loss should match the actual activation loss at open circuit condition. The value of i n is the value of current in the following equation, V actloss at open circuit = in A ln stack (5 16) i = 1.175ln i n i n = amps Capstone Project A

126 From the simpler method the parameters for the activation loss are: A stack = 1.175, i 0 = and i n = This is similar to the values calculated using the complex least squares method where: A stack = , i 0 = and i n = A graph of the polarisation curve with the values of A stack, i 0 and i n calculated from the simpler method is almost identical to the previous one. This is shown in figure Fuel Cell Polarisation Curve 30 5 Voltage (V) 0 15 Experimental V Estimated V Current (A) Figure 5 34: Expected Polarisation Curve Using the Simpler Method From this graph it is shown that the simpler method can be accurately used. The simpler method gives a slightly worst fit to the data points but this is not significant. 5.6 Concentration The method for determining the concentration loss is briefly described here although it is not required for the UTS fuel cell. The concentration loss can be shown by equation (4-48) V conloss = i B ln 1 i L Capstone Project A

127 If the stack at UTS did exhibit the concentration loss then the voltage at higher currents would sharply fall off. For the cases where this occurs there could be two reasons. Firstly, it could be because the stack is dehydrated. And secondly, it could be because there is a reduction in the fuel and oxidant at the anode and cathode. If the UTS fuel cell had a concentration loss the polarisation curve would look similar to figure Polarisation Curve with Concentration Loss 30 5 Voltage (V) 0 15 Con Loss Experimental V With Loss Current (A) Figure 5 35: Concentration Loss Effect The normal experimental data without the concentration loss is shown as Experimental V. With Loss is the curve that would describe the polarisation curve with the addition of the concentration loss. Con Loss is the value of the concentration loss as described by equation (4-48). The values of B=0.9 and i L =9 have been used in figure The experimental value of V conloss can be calculated by, V conloss = V reversible - V ohmloss - V actloss - V stack (5 17) V reversible, V ohmloss, and V actloss are calculated as in the sections above. V stack is the operating voltage of the stack. Once this is done a match of the parameters B and i L can be fitted to the data. Capstone Project A

128 5.7 Empirical Techniques Outlined here are the basic techniques for empirical modelling. The expected form for the losses is set primarily using analytical methods. It may be determined that purely empirical methods are required for the data fits for the UTS fuel cell. This needs to be done when the dependence of an operating condition to a parameter is not known. This is the case for the internal current in regard to the important parameters of the PEM fuel cell. The operating conditions that affect the parameter in question are found by keeping all other conditions constant and then changing one operating condition (eg. stoichiometry of air). When the other conditions cannot be held constant more complicated techniques are required. The operating conditions that affect a chosen parameter are shown below. Parameter = (a 0 +a 1 A+a A..+a n A n )*(b 0 +b 1 B+b B +b n B n )*(c 0 +c 1 C+c C +c n C n )... The lower case letter represents the coefficient and the upper case letter represents the operating condition. Each term will be found by holding all but one operating condition constant and then performing a fit to the data. This will be done for all operating conditions for the parameter in question. Capstone Project A

129 6. MODELLING PLATFORMS Although Microsoft Excel, Mathematica and Matlab can be used for the data analysis of the simple models, more detailed modelling programs are used to model fuel cell systems or fuel cell stacks at a low level. I have reviewed three of these modelling programs. In the initial stages of this project more time was allocated for this review and simulation of the CFD-ACE package but the viability of this task was reduced due to time constraints. 6.1 CFD-ACE CFD-ACE is a low level three-dimensional modelling program. It requires detailed information about the stack s structure and characteristics. Once this information is input into the program CFD-ACE is a very powerful simulation program. Apart from detailed structure information typical inputs into the stack include: Inlet flow speeds and humidity ratio Catalyst loading Membrane and Electrode Porosity and Permeability Reference current density and open circuit voltage Coolant temperature Tafel constants for the electrode CFD-ACE is aimed at manufacturers of fuel cells. Manufacturers have the input data necessary for the model and benefit most from the powerful capabilities of CFD-ACE. Below is an excerpt from the CFD-ACE user manual [67], CFD analysis can provide performance characteristics of a cell or reformer under various operating conditions, catalysts, membranes, etc. This reduces the cost of development significantly by eliminating costly experiments and/or by narrowing down the range of parameters over which the final set of experiments need to be performed. Capstone Project A

130 The predictive power of CFD-ACE is immense. The polarisation curves can be re-evaluated for many types of changes in the stack structure and changes to material properties of the stack. Figure 6-1 shows how the polarisation curve changes with a change in temperature in Kelvin. Figure 6 1: Effect of Temperature on the Stack CFD-ACE is not able to model the fuel cell system as it cannot model auxiliary components. Therefore this modelling program will not be used extensively for the fuel cell project at UTS. Current interrupt tests and EIS also cannot be performed using CFD-ACE. Capstone Project A

131 6. Trnsys A modelling program that has the ability to model the entire fuel cell system is Trnsys. Currently the Trnsys package is being evaluated and it is likely that the package will be procured for modelling the fuel cell system. Trnsys is a modular program that uses Fortran. Once the user becomes familiar with the modelling environment Trnsys is a very powerful modelling tool for systems. The modular structure of Trnsys maintains flexibility and provides the user with access to a large data base of predefined modules. Other modules (like Ulleberg s modules [68]) can be used provided that they are not regarded as proprietary information. An example of the TRNSYS user interface is shown below in figure 6-. Figure 6 : Trnsys Interface Capstone Project A