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A time transient technique for performance characterization and degradation diagnostics in solid oxide… Hoff, Brian David 2006

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A TIME TRANSIENT TECHNIQUE FOR PERFORMANCE CHARACTERIZATION A N D DEGRADATION DIAGNOSTICS IN SOLID OXIDE FUEL CELLS By BRIAN DAVID HOFF B . E n g . , T h e U n i v e r s i t y o f V i c t o r i a , 2 0 0 4 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR THE D E G R E E OF MASTER OF APPLIED SCIENCE In THE F A C U L T Y OF G R A D U A T E STUDIES (Mechanical Engineering) UNIVERSITY OF BRITISH COLUMBIA August 2006 © B r i a n D a v i d H o f f , 2 0 0 6 Abstract Solid oxide fuel cells (SOFCs) have demonstrated higher fuel-to-electricity conversion efficiency than any competing device. However, their commercialization is hindered by poor reliability, poor durability, and high manufacturing and materials costs. Degradation diagnostics in operating SOFCs by non-invasive methods remains a challenge. Diagnostic technique developments might cultivate improvements in S O F C reliability and durability. Ohmic-, activation- and concentration-related power losses in SOFCs are determined using transient techniques. Electrochemical impedance spectroscopy (EIS) is the leading non-destructive technique for this purpose. Time domain transient techniques including current step and galvanostatic current interruption (GCI) have been applied to some extent in S O F C research. Currently, no technique exists to elucidate S O F C power losses from time transient data with resolution comparable to EIS. If such a technique is made available, time domain techniques might transcend EIS with reduced testing durations and equipment costs. In this aspect, time transient techniques might be more suitable candidates for onboard degradation monitoring of commercial S O F C stacks; simplification of auxiliary electronics and reduction of diagnostic testing times might yield significant cost savings and operational benefits. This work consisted of two parts: 1. the assessment of perturbation via load resistance switching as a means of electrochemical characterization, and 2. the evaluation of spectroscopic interpretation of fuel cell power losses via Laplace inversion of time domain data. The electrochemical characteristics of an S O F C were evaluated via D C polarization and EIS at temperatures ranging from 600-900°C. A circuit was constructed to rapidly switch the fuel cell 's load. Current and voltage responses to load switching were acquired at 800°C. Rather than inducing a voltage response to a current step, load switching yielded mutual relaxations of both current and voltage. Due to this complication, neither overpotential decays nor the complex impedance response were derived using this technique. Nevertheless, responses to small load steps yielded ohmic data comparable to that of EIS. Analysis of the Laplace inversion of simulated transient data suggested that even with extremely small noise levels, a lack of confidence exists in the resolution of electrochemical constituents in time transient responses. i i Table of Contents Abstract..; ii List of Tables vii List of Figures viii Acknowledgements xi Chapter 1: Introduction 1 1.1 Overview of Fuel Cells 1 1.2 Synopsis of SOFC Technology 3 1.3 SOFC Degradation Mechanisms 9 1.4 Scope and Background: SOFC Degradation Diagnostics 10 Chapter 2: Electrochemical Fundamentals 13 2.1 Introduction...: 13 2.2 Fuel Cell Polarization 13 2.3 Ohmic Polarization 15 2.4 Current Overpotential 16 2.4.1 General 16 2.4.2 Activation Polarization 18 2.4.3 Concentration Polarization 20 2.5 Leakage Polarization 22 2.6 Overall Cell Potential at Closed Circuit 22 2.7 Electrical Double Layer 22 Chapter 3: Literature Review of Electrochemical Transient Techniques 24 3.1 Introduction 24 3.2 Cell Polarization Measurement 24 3.2.1 Two-Electrode Configuration 24 3.2.2 Three-Electrode Configuration 25 3.2.3 iR\j Compensation 27 3.2.4 Potentiostatic and Galvanostatic Control 28 3.3 Galvanostatic Current Interruption and Current Step 29 3.3.1 Theoretical Development 29 3.3.2 Experimental Development 32 iii 3.4 Potential Step 35 3.4.1 Theoretical Development 35 3.4.2 Experimental Development 36 3.5 Electrochemical Impedance Spectroscopy 37 3.5.1 Theoretical Development 37 3.5.2 Experimental Development 42 3.5.3 Data Interpretation 43 3.6 Combination of Transient Techniques 46 Chapter 4: Applications of Transient Techniques to SOFCs and Other Systems 48 4.1 Introduction 48 4.2 Early Contributions 48 4.3 Anode Investigations 49 4.4 Cathode Investigations 51 4.5 Summary of Electrochemical Relaxation Times in SOFCs 53 4.6 Examples of Applications to Other Electrochemical Systems 56 4.6.1 Introduction 56 4.6.2 Molten Carbonate Fuel Cells (MCFCs) 56 4.6.3 Polymer Electrolyte Fuel Cells (PEFCs) 58 4.6.4 Lead-Acid Batteries 59 Chapter 5: Research Objectives 62 5.1 Motivation 62 5.2 Summary of Key Points from the Literature Review 62 5.3 Requirements 63 5.4 Scope of Contribution 63 Chapter 6: Numerical Techniques for Exponential Analysis of Transients 65 6.1 Introduction .65 6.2 Application of Spectroscopic Exponential Analysis (SEA) to SOFCs 66 6.3 Principle of SEA 66 6.4 Tikhonov Regularization 67 6.5 CONTIN 68 Chapter 7: Experimental Procedure 70 7.1 Introduction 70 iv 7.2 Test Cell 70 7.3 Cell Test Station 72 7.4 Cell Test Conditions 73 7.5 EIS and DC Polarization Experiments 74 7.6 Measurement System for Transient Recording 74 7.7 Load Switching Circuit 75 7.8 Characterization of Signal Noise and Interference 78 7.9 Algorithm for Transient Recording 79 7.10 Procedure, Terminology, and Conventions in the Use of CONTlN 80 Chapter 8: Results and Discussion: SOFC Performance, Evaluation of Load Switching. 82 8.1 Introduction 82 8.2 Polarization and Power Density 82 8.3 Electrochemical Impedance Spectroscopy 83 8.4 Dynamic Characterization of the Load Switching Circuit 87 8.5 Prediction of an SOFCs Transient Response to Load Switching 89 8.6 Transient Signals Acquired in Load Switching Experiments 93 8.7 Estimation of Ohmic Resistance from Transient Signals 95 8.8 Calculation of Ohmic Resistance from Published Data 98 8.9 Analysis of Signal to Noise Ratio (SNR) in Voltage Transients 100 8.10 Analysis of Noise in Transient Signals 101 8.11 Repeatability of Transient Experiments 103 Chapter 9: Results and Discussion: Spectroscopic Analysis of Simulated Transients ...106 9.1 Introduction 106 9.2 Simulation I: Strategy Development for Isolation of Transient Features 106 9.3 Simulation II: Sensitivity of Time Constant Distribution Features 113 9.4 Simulation III: Effects of Noise on Resolution Capabilities 119 9.5 Implications of Simulation Predictions on Required SNRs 125 9.6 Suggestions for Pre-processing Transient Data for CONTIN 126 Chapter 10: Conclusions 128 10.1 SOFC Performance and Perturbation by Load Switching 128 10.2 Spectroscopic Interpretation of Simulated Transient Data 130 10.3 Final Remarks 131 v Chapter 11: Recommendations for Future Work 132 Bibliography 134 Appendix: Sample CONTLN Input Block 141 vi List of Tables Table 1: GCI Hardware Specifications Presented by Nardella et al. [47] 34 Table 2: Reported Electrochemical Relaxation Time Constants 54 Table 3: Measured Thicknesses of Functional Test Cell Layers 72 Table 4: Cell Performance Properties Obtained from DC Polarization Analysis 83 Table 5: Properties of Impedance Spectra at 800°C 85 Table 6: Parameters Used in the Equivalent Circuit Model of the Test Cell and Load Switching Circuit 91 Table 7: Relative Error of Peak Centers and Moments (Solution domain: 0 to 15 s) 122 Table 8: Relative Error of Peak Centers and Moments (Solution domain: 0 to 1.5 s) 124 vii List of Figures Figure 1: SOFC Operating Principle 4 Figure 2: Two-Electrode Test Cell Schematic 25 Figure 3: Three-Electrode Test Cell Schematic 25 Figure 4: Ring-Shaped Reference Electrode 27 Figure 5: Equivalent Circuit of an Electrode-Electrolyte Interface Measured in a 3-Electrode Configuration 38 Figure 6: Impedance Plane Plot - Z w Negligible 41 Figure 7: Impedance Plane Plot - Zw Non-Negligible 41 Figure 8: Full Cross Section of Test Cell (100 x Magnification) 71 Figure 9: Cross Section of Test Cell (1000 x Magnification) 71 Figure 10: Cutaway View of Cell Support Fixture (not drawn to scale) 73 Figure 11: Schematic Diagram of Measurement System for Transient Recording 75 Figure 12: Schematic Diagram of Load Switching Circuit 76 Figure 13: Load Switching Circuit Construction 78 Figure 14: Polarization and Power Density of Test Cell at Various Temperatures 82 Figure 15: Complex Impedance Response of Test Cell - Open Circuit Voltage 84 Figure 16: Complex Impedance Response of Test Cell - Operating Load 84 Figure 17: Complex Impedance Response of Test Cell - O C V and Operating Load at 800°C 85 Figure 18: Dynamic Response of the Load Switching Circuit - Magnitude of Impedance 88 Figure 19: Dynamic Response of the Load Switching Circuit - Phase Angle of Impedance.. 88 Figure 20: Equivalent Circuit Model of Test Cell and Load Switching Circuit 90 Figure 21: Transient Response of the Equivalent Circuit Model 92 Figure 22: Voltage Transients of Various Step Sizes 93 Figure 23: Current Transients of Various Step Sizes 94 Figure 24: Rapid Transient Responses and Extrapolation of Ohmic Data (dR = 5 ohms) 95 Figure 25: Estimations of Ohmic Resistance at Various Perturbation Magnitudes 97 Figure 26: Effect of Coaxial Cable Length on the Rapid Transient Response 98 Figure 27: SNR of Filtered and Unfiltered Voltage Transients .100 viii Figure 28: Steady State Noise (Time Domain) 101 Figure 29: Frequency Response Amplitudes of Signal Noise in the Range of 50 to 500 kHz 102 Figure 30: Repetition of Five Load Switching Experiments (dR = 0.33 ohms) 104 Figure 31: Effect of High Resolution Sampling on Time Constant Distributions of Simulated Transients 107 Figure 32: Effect of Intermediate Resolution Sampling on Time Constant Distributions of Simulated Transients (Solution domain: 0 to 10 s, 10 samples per time domain decade) 108 Figure 33: Effect of Low Resolution Sampling on Time Constant Distributions of Simulated Transients 109 Figure 34: Time Constant Distributions of Simulated Transients at High Sample Resolution 110 Figure 35: Truncation of Transient Data with Inclusion and Exclusion of a DC Offset 111 Figure 36: Time Constant Distributions of Truncated Simulated Transients 112 Figure 37: Sensitivity of Time Constant Distribution Features from Simulated Transients.. 114 Figure 38: Sensitivity of Time Constant Distribution Features from Simulated Transients ..115 Figure 39: Sensitivity of Time Constant Distribution Features from Simulated Transients ..116 Figure 40: Time Constant Distribution Features from Simulated Transients using Various a Values 118 Figure 41: Time Constant Distribution Features from Noisy Simulated Transients, SNR = 10000, 5000, 2000, & 1000 (Solution domain: 0 to 15 s, 100 samples per time domain decade) 120 Figure 42: Time Constant Distribution Features from Noisy Simulated Transients, SNR = 500, 400, 300, & 200 (Solution domain: 0 to 15 s, 100 samples per time domain decade) 121 Figure 43: Time Constant Distribution Features from Noisy Simulated Transients, SNR = 10000, 5000, 2000, & 1000 (Solution domain: 0 to 1.5 s, 100 samples per time domain decade) : 123 Figure 44: Time Constant Distribution Features from Noisy Simulated Transients SNR = 500, 400, 300 & 200 (Solution domain: 0 to 1.5 s, 100 samples per time domain decade) ........ 124 ix Figure 45: Suggested Preprocessing Steps - (a): original data, (b): discarding of data before perturbation, (c): referencing about the settling value, (d): inversion about the horizontal axis 126 x Acknowledgements I would like to acknowledge my supervisor, Dr. Olivera Kesler for her ongoing support in this project, as well as her enthusiasm and commitment to clean energy and sustainability. I would also like to acknowledge Dr. Cyrille Deces-Petit and Xinge Zhang of the National Research Council of Canada - Institute for Fuel Cell Innovation (NRC - ICFI) for their expert advice and guidance throughout this entire project, particularly in times of experimental crises. Furthermore, I would like to thank Javier Gazzarri for his expert advice on issues ranging from Laplace inversion to Argentinean slang. Finally, I would like to thank my family for their ongoing support and encouragement. My accomplishments would not have been possible without the support of these people. I would like to thank the NRC - ICFI for their generosity in training me and allowing me to use their laboratory, as well as the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial assistance. xi Chapter 1: Introduction 1.1 Overview of Fuel Cells Fuel cells are electrochemical conversion devices that generate power with minimal emission of pollutants. A spectrum of advantages and challenges arise from the implementation of fuel cells. Fuel cell technology is currently in the development stage, and is presently being researched and progressed by both academic and commercial organizations throughout the world. The implementation of fuel cells has been under investigation in recent decades for transportation, stationary power generation and portable electronics. This direction offers a major advantage to human health and the environment through the reduced emission of greenhouse gas, particulates, and other harmful substances that are found in combustion-related emissions. Hydrogen fuel offers an alternative to some of the adverse facets of petroleum-based fuel use. Hydrogen can be made by techniques such as water electrolysis and steam reforming of fossil fuels. High temperature fuel cells have also been demonstrated to run on methane, methanol, other organic compounds, and carbon monoxide. Widespread power generation by stationary fuel cell systems can lead to reduced transmission losses, avoidance of grid failure hindrances, and enhanced power system security regarding threats such as terrorism. Further development of small scale fuel cells might lead to their commercialization for use in portable electronics. Even with the most advanced secondary battery technologies, greater energy density is desirable. Fuel cell power systems may eventually be able to provide superior energy storage capacity. Fuel cell technology must compete with combustion technology in order for its widespread implementation in the areas of transportation and stationary power generation. In past centuries, society has adopted coal and petroleum-based fuels as sources of energy to service needs such as vehicle transportation and heating. These fuels have also had a major role in the progression of electric utilities. As a result of past developments, much of society's energy infrastructure is dominated by petroleum-based fuels. The consumer economy favours petroleum-based fuels over costly alternative fuels such as hydrogen. Current fuel cell systems lack the reliability and durability of their combustion-driven competitors. However, a major 1 advantage of fuel cells over heat engines is the fact that fuel cells are not subjected to the Carnot limitation [1]. This is a restriction on the thermal efficiency of combustion engines and turbines that imposes a significant constraint on performance at lower operating temperatures. In comparison, fuel cells demonstrate a theoretical advantage over heat engines at relatively low temperatures. Aside from the theoretical performance, fuel cells offer advantages over heat engines from a practical perspective because they have no intrinsic moving parts. In stationary applications, high temperature fuel cells offer electrical power and serviceable heat. This heat can be used to drive turbines and generate additional power. Such integration is referred to as a bottoming cycle. Solid oxide fuel cell (SOFC) systems have demonstrated the highest fuel-to-power efficiency of any known device, and higher efficiencies are expected to accompany future developments [2]. This point suggests that fuel cell power is advantageous over combustion-based power from a fuel savings perspective even when fuel is produced from petroleum-based products. There are several different types of fuel cells, each with its inherent advantages and disadvantages. The alkaline fuel cell (AFC) and proton exchange fuel cell (PEFC) are two types that are most commonly fuelled by hydrogen and operate at temperatures below 100°C. The AFC was the first type of fuel cell implemented in modern applications. It was used in the Apollo and Shuttle Orbiter spacecraft voyages. A problem with the A F C is that it is sensitive to CO2, and thus, requires sufficient CO2 removal strategies when air is used as an oxidant. The PEFC is an intrinsically simple type of fuel cell. This type is being implemented as an electrical power source for automotive propulsion since it is considered the most suitable type of fuel cell for the characteristic load changes during vehicle operation. Disadvantages of the PEFC include its sensitivity to CO [3], its requirement for scrupulous water balance in the electrolyte, and high platinum/ruthenium catalyst costs. Much effort is currently being devoted to mitigate these problems. The phosphoric acid fuel cell (PAFC) is a type of fuel cell that operates at moderate temperatures (~200°C). It is a relatively reliable and maintenance-free fuel cell [1]. This was the first type to be produced in commercial quantities. Disadvantages of the PAFC include relatively low efficiency, limited lifetime, and expensive catalysts. Fuel cells that operate at temperatures greater than 600°C include the molten carbonate fuel cell (MCFC) and the solid oxide fuel cell (SOFC). The M C F C has been utilized for large-scale power generation; however, its poor durability, long start-up times, and electrolyte 2 instability have prevented its widespread commercialization. Nevertheless, M C F C power modules are commercially available, and improvements are currently underway. The S O F C generates high quality waste heat and is thereby an excellent candidate to be used in combined heat and power (CHP) systems. Also , the S O F C shows remarkable efficiency when used in co-generation of electrical power with turbomachinery (bottoming cycle). The S O F C demonstrates superior simplicity in its solid state design. It has been utilized in large-scale power modules, generating electric power on the order of hundreds of kilowatts. Disadvantages of the S O F C include long start-up times, poor reliability, poor durability, and high manufacturing costs. The most common overall reaction that occurs in a fuel cell is the oxidation of hydrogen and the reduction of oxygen to produce water, as follows: H 2 (g ) + | 0 2 ( g ) - > H 2 0 ( g / l ) (1.1) In this reaction, two electrons are transferred through an external circuit. This transport of electrons is manifested as electrical power. The state of the product water depends on the temperature. For example, water vapour is produced in SOFCs as a result of their high operating temperatures, and liquid water is produced in P E F C s and A F C s at operating temperatures below 100°C. 1.2 Synopsis of SOFC Technology The S O F C is a solid state device that consists of two porous electrodes (anode and cathode) that are separated from each other by an ionically conductive solid ceramic region (electrolyte). A single cell is shown in Figure 1. The cell generates electric power via the concurrent oxidation of a fuel gas and exchange of oxide-ions through the electrolyte. The electrolyte is ionically conductive at temperatures typically between 600°C and 1000°C. At the cathode, the oxidant, which is either air or pure O2, is fed through the porous medium. The gaseous O2 molecule catalytically reduces to form two O 2 " ions which can pass through the electrolyte. The fuel is passed through the porous anode, where it oxidizes and the product H2O (and/or CO2, depending on the fuel) is formed. The anode and cathode reactions are 3 electronically balanced by a transfer of two electrons per 0 " ion through an external electrical circuit, thus providing electrical power. Cathode Electrolyte Anode H 2 0 , Combustible C O ; Fuel Figure I: SOFC Operating Principle The most common S O F C fuel is hydrogen (H2); however, any fuel that can be oxidized by accepting an oxide ion can, in theory, be used as a fuel. Methane (CH4) is a promising option, as it can readily be supplied through the natural gas infrastructure existing in developed countries. Recent work by Gorte et al. [3] has shown that methane can be directly oxidized in an SOFC under certain conditions without adverse effects on performance. Furthermore, a range of fuels including carbon monoxide (CO) and gasoline have been considered. Cel l component materials have been conscientiously selected in order to reach the current state of cell performance. Singhal [4] has emphasized the importance of material selection based on suitable electrical conductivity, adequate structural and chemical stability at SOFC operating and fabrication temperatures, minimal reactivity and interdiffusion among various cell layers, and thermal expansion match between adjacent cell components. Other important considerations include porosity of electrode materials, electrical conductivity of the electrolyte, as well as density of the electrolyte and interconnect to prevent gas leakage. The following paragraphs describe the individual cell components. Electrolyte. Electrolyte materials include fluorite-structured yttria-stabilized zirconia (YSZ) , rare-earth-doped ceria and rare-earth-doped bismuth oxide [4]. Y S Z has been the most successful electrolyte material. Zirconia does not conduct ions sufficiently in its natural monoclinic crystal form. Its fluorite-structured phase can be stabilized by doping with divalent or trivalent oxides, which gives rise to high oxide mobility at elevated temperatures. The 4 addition of each mole of yttria creates a mole of oxygen vacancies according to the following reaction (which is expressed in the Kroger-Vink notation): Y 2 0 3 ^ 2 Y ; + 3 0 X 0 + V ( o (1.2) As the cell temperature is raised above approximately 500°C, oxygen vacancies in the YSZ bulk become mobile to a sufficient degree to support cell operation. In general, a solid electrolyte must have relatively high ionic conductivity and ionic transport number in the operating temperature and pressure range [5]. A notable challenge in SOFC development is to reduce the resistance of the electrolyte while maintaining structural integrity and avoiding gas permeability. Ceria-based electrolytes have shown higher O2" ion conductivity than YSZ, but their lower ionic transference numbers makes them mixed ionic and electronic conductors (MIECs), implying an electronic short circuit. Thus, implementation of these materials allows for lower SOFC operating temperatures at the expense of significant loss of power due to electronic leakage. New dopants are being investigated to reduce the electronic conductivity of ceria-based electrolyte candidates [6]. A n o d e : The porous anode material consists of a cermet (metal and ceramic mixture) which contains a catalytic metal and a ceramic electrolyte material [6,7]. The most commonly used anode material is a mixture of nickel and YSZ. The nickel phase is electronically conductive as well as electrocatalytically active for the oxidation of gaseous fuels. Presence of YSZ results in ionic conductivity throughout the anode, as well as enhanced structural support and thermal expansion compatibility with the adjacent cell material. The anode's porosity gives access to the gas phase in which fuel can enter and diffuse to the triple phase boundary (TPB) zones at which the gas phase (pore space), electrolyte phase (ionic conductor), and metal phase (electronic conductor) come into contact. Here, the oxide ions can react with the fuel to form the product. Fuel reacts as follows for the cases of hydrogen (Equation 1.3), carbon monoxide (Equation 1.4), and methane (via direct oxidation) (Equation 1.5) as fuel: H2(g) + 0* - s z ^H 2 0(g ) + 2e-CO(g) + 0 ^ s z ^ C 0 2 ( g ) + 2e-(1.3) (1.4) 5 C H 4 (g) + 40 2 Y- S Z 2 H 2 0 ( g ) + C 0 2 (g) + 8e" (1.5) In the case of direct hydrocarbon oxidation (Equation 1.5), nickel tends to catalyze hydrocarbon cracking (formation of solid carbon) when inadequate reactant proportions are present [3,8]. Park et al. [8] reported cells with n ickel -YSZ and nickel-ceria-YSZ anodes to become completely deactivated within 30 minutes of operation using dry methane as fuel. They observed build-up of carbon fibres resulting from hydrocarbon cracking. They investigated the replacement of nickel with copper, and found that copper-YSZ was basically inert to methane, facilitating neither direct oxidation nor internal reforming. However, they also reported that the addition of ceria to the copper-YSZ resulted in stable direct oxidation of methane with no observable deposition of carbon. Copper-ceria-YSZ may have a major role as a future S O F C anode material. A n S O F C is able to facilitate hydrogen production within the anode by direct internal steam reforming of hydrocarbons [9,10]. This is a virtue of SOFCs , as external hydrogen production consumes power and reduces the overall fuel-to-power efficiency of the system. Steam reforming of hydrocarbons occurs as follows: This reaction is endothermic and thereby requires a heat input. In an S O F C , the generated heat from electrochemical reaction is available to activate the reforming reaction. Methane has been shown to be a suitable candidate for internal reforming. Steam is mixed with the fuel in order to drive the reforming reaction. In practical applications, carbon has been seen to inadvertently deposit on the electrochemically active surfaces of the anode. Inadequate proportions of steam lead to hydrocarbon cracking. Cathode. The cathode consists of a porous mixture of electronically conductive phase and electrolyte, or a ceramic M I E C . A s a result of the oxidizing conditions at the cathode, the electronically conductive phase can only consist of noble metals or electronically conducting oxides [6,7]. The most commonly used cathode material is strontium-doped lanthanum manganite ( L S M ) mixed with Y S Z . L S M is a preferred material because of its high electronic conductivity in oxidizing environments. Lanthanum manganite is a perovskite oxide that C „ H m (g) + « H 2 0 ( g ) -> «CO(g) + [n + \m\L2 (g) (1.6) V * J 6 exhibits p-type electronic conductivity. When doped with a divalent cation such as strontium, the electronic conductivity of lanthanum manganite is enhanced. The conductivity of LSM is dependent upon the oxygen partial pressure (P02) [11]; as P02 is decreased to values substantially lower than the normal cathodic operating range of an SOFC, conductivity decreases due to decomposition of the lanthanum-manganite phase. Addition of YSZ to LSM enhances the cathode's ionic conductivity. It has also been shown to improve the thermal expansion match between the electrode and electrolyte layers. L S M not only acts as an electronic conductor, but it also serves as an electrocatalyst for the oxygen reduction reaction (ORR), which takes place as follows: ^0 2(g) + 2 e - ^ 0 2 - (1.7) A problem with the use of L S M arises due to the thermal expansion mismatch. This problem can be mitigated by substituting the lanthanum with a smaller cation, such as calcium. At temperatures above 1400°C diffusion of manganese takes place, causing adverse reactions with the electrolyte. This problem restricts LSM-YSZ cathode manufacturing temperatures to below 1400°C. The use of solid solutions containing chromium has been suggested to solve this problem [6]; however, chromium deposition on the cathode could be a problem if it volatilizes from the oxide and deposits at the reactive sites. Strontium-doped lanthanum cobaltite (LSC) is more conductive than L S M ; however, LSC is less stable. It reacts with YSZ to form poorly conducting intermediate phases. Nevertheless, LSC is compatible with ceria-based electrolytes, which makes it a candidate for use in lower temperature SOFCs. Interconnect: The SOFC requires an interconnect material to join adjacent cells to form a fuel cell stack. The interconnect material must have high electronic conductivity, good stability in the SOFC environment, compatibility with other SOFC materials, and adequate density to prevent gas leakage. The most common interconnect material is doped lanthanum chromite, which is a perovskite oxide with p-type electronic conductivity. A variety of cations can be used to dope lanthanum chromite. The most common dopants are calcium, magnesium, and strontium. The dopant selection determines properties such as the phase stability, conductivity, and thermal expansion coefficient. A major problem with lanthanum chromite is its difficulty to sinter. This presents a hindrance with regard to the interconnect density 7 requirement. Traditionally, lanthanum chromite has been sintered at temperatures greater than 1600°C under low oxygen partial pressures in order to achieve adequate density. Processing techniques are being investigated to reduce sintering temperatures for doped lanthanum chromite manufacturing [6]. Several different cell design configurations have been employed [6,12]. The segmented-cell-in-series design .involves cells of non-planar shape, which are placed in series. The bell-and-spigot configuration is an example of a segmented-cell-in-series design, as described by Minh [6]. Another design is the tubular configuration, which has been demonstrated in stationary power applications. In this configuration, tubular cathode-supported cells are bundled together to make a stack. Both the segmented-cell-in-series and tubular configurations present relatively tortuous paths for electrons to travel, and as a result, they exhibit high resistance losses. Another variation is the monolithic configuration, in which the cell components are manufactured in a corrugated shape. This provides a simple flow-field design to allow reactant gasses to reach the reaction sites. The planar configuration is a very common design that consists of flat plates that are sandwiched together and connected in series. Planar cells provide the benefit of simplicity of manufacturing and high power densities. An important feature of an SOFC design is the selection of the structural support for each cell. The supporting layer is one of the functional layers (anode, cathode, electrolyte, interconnect) which is made substantially thicker than the others to facilitate structural integrity. By increasing the layer's thickness, the layer's resistance is also increased. The anode is commonly allocated as the supporting layer due to its mechanical strength and superior electrochemical characteristics. As a result of the fast reaction kinetics and high rate of hydrogen diffusion, the anode resistance is lower than that of the cathode. This makes the anode the best candidate for a structural support. The electrolyte is a poor candidate; it must be made very thin in order to minimize resistance. The reader is encouraged to visit the work of Singhal and Kendal [13] for thorough coverage of the current state of SOFC technology, particularly for topics related to SOFC manufacturing which are beyond the scope of this review. 8 1.3 SOFC Degradation Mechanisms The degradation of SOFC performance is a major issue that hinders commercialization. Degradation is often measured as a reduction of voltage over time or an increase in polarization over time while operating at a steady current density. Some of the major modes of SOFC degradation include carbon deposition, sulphur poisoning, electrode sintering, and cracking. As described previously, deposition of solid carbon results from catalyzed hydrocarbon cracking in the anode and it impedes fuel cell performance by blocking critical reaction sites. Extensive carbon deposition is irreversible, as no practical method of removing large amounts of solid carbon has been devised. Sulphur naturally occurs in hydrocarbon mixtures such as natural gas extracted from beneath Earth's surface. Furthermore, sulphur is added to natural gas as an odorant in the form of H2S. Sulphur has a tendency to adsorb to the catalytic surfaces in SOFC anodes. These surfaces are contaminated by even small amounts of sulphur in the fuel. As operating temperatures decrease, sulphur poisoning becomes more of a problem. This leads to an increase in electrode resistance. Matsuzaki and Yasuda [14] quantified the increase in nickel-YSZ electrode resistance resulting from introduction of various concentrations of H2S and S2 gasses into the fuel stream. They found that a particular threshold of sulphur poisoning could be identified, below which the poisoning effects could be reversed by passing sulphur-free fuel through the anode for sufficient time. Electrode sintering is a spontaneous degradation mechanism that occurs most commonly in the cermet anode. Sintering causes the metallic phase to coarsen and agglomerate. This results in a reduced TPB length, leading to a diminution in electrochemical performance. In the context of SOFC anodes, percolation refers to the pathways by which electronic charge can pass through the electrode to the current collector as well as the pathways by which gas and ionic species can travel to the TPB sites. When nickel particles sinter, performance is reduced due to reduced electronic percolation. Sintering of nickel either opens new pores or enlarges existing pores. Ioselevich et al. [15] considered two types of metal sintering separately: (i) weak sintering caused by one particle being physically disconnected, becoming electrochemically inactive, and (ii) strong sintering, when a particle is substituted with a pore space. They identified two important parameters that characterize sintering: the number of 9 bonds per active particle, and the probability that a sintering act opens a pore. It is poorly understood how these parameters can be controlled during cell manufacturing in order to reduce degradation from sintering. Thermal stress is a major source of SOFC degradation which leads to layer delamination and cracking. Excessive thermal stresses are often a result of thermal expansion mismatch between adjacent layers. This is an inherent problem related to the intensive thermal cycling to and from relatively high manufacturing and operating temperatures. Much effort has been devoted to matching the thermal expansion coefficients of the various SOFC layers by modifying material structures. A crack is essentially an insulator to ionic and electronic current. Partial electrode delamination causes large polarizations as a result of the reduced contact area. Degradation is not only limited to the four described mechanisms; other modes are known. For example, chromium poisoning at the cathode is a common degradation mechanism in SOFCs that contain chromium-rich materials. More advanced fuel cell diagnostic techniques can help researchers isolate and observe the severity of these degradation mechanisms in operating cells. 1.4 Scope and Background: SOFC Degradation Diagnostics Sophisticated diagnostic techniques are required to improve the current state of SOFC development in two ways: 1. to promote longer service life by distinguishing reversible from irreversible degradation mechanisms during operation, and 2. to endorse further advancements in SOFC technologies through the characterization of new materials and designs. It is essential to have convenient and practical methods for in-situ evaluation of performance and degradation tendencies of SOFCs over their entire life cycles. Such an approach will provide researchers with a thorough understanding of the ramifications of new developments and operating strategies as well as opportunities to extend SOFC service life. Innovations in diagnostic technologies might inspire advancements in core fuel cell technologies and might help accelerate the commercialization of SOFCs. Several techniques have been developed for the nondestructive examination of electrochemical processes. DC polarization analysis is the leading method for quantifying electrochemical characteristics of SOFCs; however, this method lacks the resolution 10 capabilities of transient techniques. Electrochemical impedance spectroscopy (EIS) is a frequency-domain transient technique that is commonly used to elucidate electrochemical processes. EIS requires relatively long testing times and costly equipment. Furthermore, interpretation of impedance data has been shown to be problematic, as overlap of various processes is often observed, and the common data interpretation technique of equivalent circuit fitting has inherent ambiguity. For these reasons, conventional EIS might be a poor candidate for implementation onboard of SOFC stacks for in-situ evaluation. A technique that can be performed rapidly with lower-cost equipment might be more suitable. Time domain transient techniques can be implemented relatively quickly with simple hardware. Their applications to SOFCs are limited, although galvanostatic current interruption (GCI) has been used frequently to quantify the ohmic losses in SOFCs. The conventional strategy of GCI has been relatively invasive to cell steady-state operation; by interrupting the current, the fuel cell is rapidly driven to the open circuit state. The investigation of the time transient response is an endeavour that has seldom been explored. It is unclear what state the response characteristics, such as the decay of electrochemical activation-related voltage relaxations, are attributed to: the power generating state, the open circuit state, or some intermediate state. If one wishes to investigate sources of fuel cell losses in operation, it is imperative to evaluate degradation at the power generating state. Rather than interrupting the current completely, an alternative is to perturb the fuel cell by an amount that is small enough so that the change of state is negligible, but large enough to create a transient signal that can be recorded and interpreted. Transient techniques can be used for the diagnosis of fuel cell degradation. The philosophy is as follows: If a transient technique can resolve independent governing electrochemical processes within a fuel cell, then changes in the presence, scale, features, or dominance of these processes can serve as indicators of degradation. Before a transient technique can diagnose degradation, the un-degraded (baseline) electrochemical characteristics must be well understood. This philosophy has been demonstrated through the diagnosis of degradation via interpretation of changing shapes in impedance spectra. The first part of this work is a literature review of transient techniques including EIS, GCI, and the current and potential step techniques. Following the review, research objectives are stated. Next, the development of a transient technique ultimately intended for the diagnosis of degradation in SOFCs is presented. The technique is based on fuel cell perturbation via 11 rapid load resistance switching, voltage and current signal transient recording, and data interpretation via spectroscopic exponential analysis (SEA). Some aspects of the proposed technique have been applied in experiments involving an SOFC. Simulated data has also been employed to some extent in this early state of development. The feasibility of extracting electrochemical performance information from time domain data acquired in a relatively non-invasive manner has been investigated. The author anticipates that this development could ultimately be integrated into a diagnostic instrumentation toolkit that can be used to monitor the in-situ performance of commercial SOFCs throughout their life cycles. 12 Chapter 2: Electrochemical Fundamentals 2.1 Introduction An understanding of solid state electrochemical phenomena is a prerequisite to the topic of SOFC diagnostics. Centuries of experimentation in the field of aqueous electrochemistry have provided a platform of knowledge from which solid state electrochemical principles have been derived. The reader is encouraged to visit the foundations of aqueous electrochemistry, reviewed by Bard and Faulkner [16] and Prentice [17], as well as the principles of solid state chemistry by West [18]. In this section, a brief overview of fuel cell electrochemical phenomena is provided, with particular emphasis on SOFCs. The following is a review of the literature found in the aforementioned sources. 2.2 Fuel Cell Polarization The reversible equilibrium potential Eth is the theoretical electrical potential across the cell when the electrodes are electronically isolated from each other (open circuit conditions). The electrolyte is ideally a barrier to the transport of both gas molecules and electrons. In reality, the electrolyte experiences some electronic current and fuel crossover. Based on the assumption of isothermal operation, Eth is derived from the Gibb's free energy of reaction, AG, which yields the following relationship: in 7-T -nF Here, n is the number of electrons transferred per molecule of fuel and F is Faraday's constant (defined as the charge of one mole of electrons). In the case of hydrogen oxidation, the H2 molecule donates 2 electrons (n = 2). The definition of the theoretical cell potential is further expanded by accounting for electrochemical potential; when adjacent phases are in equilibrium with each other, their electrochemical potential is balanced. Based on electrochemical potential equilibrium of the 13 anode, electrolyte, and cathode, can be expressed in terms of reactant and product chemical activities. This is known as the Nernst equation, and Eth is referred to as the Nernstpotential. The generalized form of the Nernst equation is: r £<h =£zero- — l n nb Tlx, products UxJ reactants v J J (2.2) where x, and Si are the activity and stoichiometric coefficient of species /, respectively. Ez^0 is the Nernst reference potential, measured at the standard state. Equation 2.2 thereby predicts a theoretical potential that decreases with increasing temperature and product activity, and increases with reactant activity. Actual cell potential at equilibrium, referred to as open circuit potential (OCV), is always less than the theoretical maximum. For a hydrogen/oxygen fuel cell, the Nernst equation is: t^h ~ z^ero RT. + ln! IF { y V l H , 0 (2.3) For gaseous species i, adequate accuracy is often achieved by substituting x; with the partial pressure of /, which is interchangeably referred to in this work as the concentration. When the fuel cell is connected to an external circuit, electrons are driven from one electrode to the other through the circuit, causing an electrochemical potential imbalance. This imbalance is counteracted by the electrochemical phenomenon of catalytic oxidation in which the fuel species at the anode donates its valence electrons to the restoration of equilibrium. Hence, the fuel species is the source of electrons in a fuel cell. In order for the donation of electrons to occur, an oxidant must be supplied to the cathode to accept electrons. The balance of charge is established via the transport of an ionic species through the electrolyte. In an SOFC, the ionic species is the oxide ion (O "), and it reacts with protons (H ) at the anode to form product water (as described by Equation 1.1). Faraday's equation expresses the rate of reaction of the oxidized species in relation to the current density i as: 14 (2.4) Current density is defined as the electronic current per unit electrode area. A reaction whose rate is determined by Equation 2.4 is known as a Faradaic reaction. Accompanying the flow of current (and the generation of power), four intrinsic loss mechanisms are responsible for the decrease of cell potential below the Nernst potential. These include: ohmic, activation, concentration, and leakage overpotential. The term current overpotential encompasses both the activation and concentration polarizations. Note that the terms overpotential and polarization are interchangeable, and both signify an adverse reduction of cell potential. 2.3 Ohmic Polarization Ohmic polarization is the linear portion of voltage loss that increases with current density. It is attributed to ionic and/or electronic resistivity innate in conductive materials (excluding ideal superconductors). Ohmic polarization accounts for substantial losses in the SOFC electrolyte layer where the ionic resistivity is much higher than the resistivity of the other layers. Ohmic polarization E0hm is governed by Ohm's law, which relates E0hm to current density i and conductivity a as follows: The integral is evaluated along the path that the charged species travels. This is primarily across the thickness of the electrolyte in SOFCs. Conductivity is the reciprocal of resistivity and for any material and charged species m, conductivity is defined as: CT (2.5) m (2.6) m 15 where nm is the concentration of charge carriers in species m, em is their charge, and fim is their mobility. 2.4 Current Overpotential 2.4.1 General For fuel cells, electric and/or ionic current is expressed as a normalized quantity (current density) in units of amperes (or milliamperes) per unit area and denoted as i. Consider the following homogeneous reversible electrode reaction in which n electrons are exchanged for each O molecule involved: V O + ne^R (2-7) The rate of the forward (/) and backward (b) processes are determined by: R a t e / [mol m"2 s"1 ] = kf-C0 (0,0 = YnF (2-8) Rate, [molm-2 s"1] = kb • CR(0,t) = *a/np (2-9) Here, ^and kb are the rate constants of the forward and backward directions of Equation 2.7, respectively, and Cm(x,t) denotes the concentration of species m at distance x away from the electrochemically active surface at time t. In the context of SOFCs, Cm{x,t) is the concentration in either the gas phase or the ionically conductive phase, depending on which reaction is under consideration. Cathodic and anodic current densities are denoted by ic and ia, respectively. Cathodic current density is the flow of electrons per unit area from the electronic conducting phase to the reaction sites where they are available to reduce species O to form R. As a convention of electrochemistry, cathodic current density is considered to be negative in sign. Anodic current density refers to the opposite of cathodic current density, namely, the flow of 16 electrons per unit area that are donated by species R and enter the electronic conducting phase. In an SOFC, this would be the nickel phase in a Ni-YSZ cermet anode. The total current density is: / = ia + ic = nF[kbCR (0,0 + kfCo(0,t)) (2.10) Now, assuming the electrode is initially at a fixed potential, E\, consider a hypothetical change in electrode potential from E\ to E2. The resultant change in associated energy per mole of electrons that reside on the electrode is: • nF(E2 -Ex) = -nFAE (2.11) If AE is negative (E2 < E\), the activation barrier for reduction of species O (cathodic reaction) decreases by a fraction of the total change in electron energy. This fraction is called the symmetry factor, denoted by /?. In addition, the activation barrier for the reverse reaction (oxidation of species R) increases by l-B. Rate constants change exponentially with electrode potential, and they can be expressed in terms of the symmetry factor as: kf - -kf° exp| K = K e x P | BnFAE RT (\-B)nFAE RT . (2.12) (2.13) where kf and kf are the standard rate constants for the forward and backward reactions, respectively. Current density can be expressed in terms of these quantities as follows: f i = nF\ kb°CR(0,t)exp (l-S)nFAE' RT -kfCo(0,t)exp JnFAE RT (2.14) Now, consider the change from a reference potential to the equilibrium potential Eeq. AE is the equilibrium potential, and is denoted by Ee. At equilibrium, the net current density is zero, and electrode surface concentrations of species O and R are equal to the bulk concentrations away 17 from the electrochemically active surface, denoted by Co* and CR*. From Equation 2.10 and 2.14: L =-ic=nFkb°CR * e x P Here ia is a quantity known as the exchange current density. i0 is a reflection of the electrode's dynamic equilibrium, which is largely influenced by the electrode catalyst performance as well as the reactant activity. RT nFkf°C0 * exp! PnFEe RT = i (2.15) 2.4.2 Activation Polarization Electrode reactions are probabilistic in nature, and their rates strongly depend on electrode potential as seen in Equation 2.12 and 2.13. A species requires sufficient activation energy to overcome the energy barrier associated with a particular reaction. The loss associated with reactant species overcoming this barrier is manifested as activation (or surface) polarization na. Consider an electrode reaction in which the involved species mix very rapidly, so the surface and bulk gas concentrations are equal at all times. For this scenario, the only contribution to current overpotential is due to the activation energy barrier. na is the electrode potential measured with respect to the equilibrium potential. According to common convention, rja is negative for net cathodic current density and positive for net anodic current density. Equation 2.14 can be expressed in terms of iQ and na as follows: i = i exp RT exp RT (2.16) This is known as the Butler-Volmer equation. For relatively large values of \na\, Equation 2.16 can be approximated by: i = i0 exp oFfh RT (2.17) 18 where a is the charge transfer coefficient. For na > +50 mV, the electrode polarization can be considered to be predominantly anodic with acceptable accuracy, a is expressed as: a = aa=n(l-j3) (2.18) For na < -50 mV, the electrode polarization can be considered to be predominantly cathodic and a becomes: a = ac=nB (2.19) Equation 2.17 can be rearranged to give the generalized Tafel approximation of activation overpotential as follows: U U ^ l n l / l - ^ l n / ^ ^ i n M (2.20) aF aF aF i This form is representative with desirable accuracy when ia is less than 1% of ic for a net cathodic current density, or ic is less than 1% of ia for a net anodic current density. For values of \na\ less than 25 mV, the Butler-Volmer equation can be expanded using the Taylor-MacLaurin series and simplified by a linear form as follows: i = i o ^ (2.21) 0 RT In SOFCs, the ORR exhibits a relatively large activation energy barrier in comparison to the facile kinetics of the hydrogen oxidation reaction (HOR). Thus, the Tafel approximation is commonly used to model the activation polarization at the cathode. The linear approximation is often used to model the activation polarization at the anode when hydrogen is used as a fuel. 19 2.4.3 Concentration Polarization Electrode reactions become limited when mass transport rates are insufficient to adequately supply reactants or remove products. Mass transport of species m is governed by the Nernst-Planck equation, which expresses total molar flux Jm(x) [mol s"1 cm"2] in a simple one-dimensional form as follows: Diffusion Migration f f)C f F)F(r\\ Convection JM = \ - D m ^ ^ y [ - z m ^ m F C m ^ \ (Cmv(*)) (2.22) dx J J The diffusion term shows that species move to locations of lower concentration. Dm is the diffusion coefficient. The migration term governs transport of positively charged species towards locations of more negative polarity as well as transport of negatively charged species towards more positive polarity. zm is the valence and /um is the mobility of mobile species m. The convection term incorporates the velocity v(x) of a volume unit of fluid. This term is null in the case of transport through solid phases in SOFCs. Diffusion can be approximated by assuming a linear concentration gradient within a zone close to the reaction surface. This region is Nernst diffusion layer, which is assumed to extend a distance dm away from the surface to a region at which the bulk concentration is encountered. In SOFC studies, the migration and convection terms in Equation 2.22 are typically neglected. Considering only the diffusion term in Equation 2.22 and Faraday's law, the current density can be expressed as follows: i_nFDm[Cm*-Cm(x = 0)} Here, sm is the stoichiometric coefficient for species m in the electrochemical reaction. By convention, sm is positive if m is an oxidant and negative if m is a reductant. If m diffuses to a surface and reacts, a decreasing concentration profile is established towards the electrochemically active surface. A limiting case occurs when the current is so large that 20 Cm(x=0) approaches zero. A larger current density cannot flow because m cannot diffuse at a higher rate. The limiting current density ii_m for species m is defined as: nFDmCm* s.S. (2.24) Combining Equation 2.23 and 2.24, Cm(x = 0) 1 i c • l~T~ (2-25) S» ll,m If m is a product, then the evolution of current depends on the diffusion of m away from the reaction surface and a limiting current is reached when m cannot diffuse fast enough to allow a higher current density. This limiting current density is also described by Equation 2.25. The concentration (or diffusion) overpotential nc is the polarization that results from a concentration gradient across the Nernst diffusion layer. Considering a momentary equilibrium that exists between the electrochemical surface and bulk for species m, nc,m can be described by the Nernst equation as follows: 17 = ^ l n lc,m j~, nt ' C m ( * = 0)A V C * nF In 1 — (2.26) This equation models concentration polarization within accepted accuracy for / < z'/,m. Based on the sign conventions for current density and stoichiometric coefficients, the concentration polarization is positive for an anodic reaction and negative for a cathodic reaction. In SOFCs, the oxide ion diffusion rate is very high in the solid electrolyte, so concentration polarization is very low for oxide ion transport in comparison to oxygen, fuel, and water transport. 21 2.5 Leakage Polarization In actuality, an electrolyte layer is neither an ideal gas barrier nor an ideal electrical insulator. SOFC electrolytes are very thin in order to mitigate ohmic polarization, and this leads to susceptibility of fuel crossover from the anode to the cathode. It is common practice to measure the gas leakage rate across the electrolyte prior to performing polarization experiments. Although electrolyte materials are generally classified as electronic insulators, some materials display non-negligible MIEC behaviour. Electronic current can be measured experimentally with ion-blocking electrodes. The contributions of fuel crossover and internal current are often lumped together and referred to as the leakage polarization l^eakage- This quantity reduces the cell potential below the Nernst potential even at null overall current. 2.6 Overall Cell Potential at Closed Circuit The four polarization mechanisms contribute to a reduced operational cell potential as follows: c^ell = ~~ o^hm ~ 7 l n ~ r l z ~ l^eakage ( 2 . 2 7 ) With the use of experimental electrochemical methods, the ohmic, activation, and concentration polarizations can be separated and their contributions to loss of performance in fuel cells can be studied. 2.7 Electrical Double Layer The electrode/electrolyte interfaces in SOFCs demonstrate capacitor-like behaviour. Consider equilibrium (or steady state) conditions in an SOFC: the excess electrical charge qe resides on the outer surfaces of the electronically conductive phases. Similarly, excess charge qi from positively charged oxide vacancies resides at the outer surfaces of the electrolyte phase. 2 2 Based on conservation of charge when different phases are in contact, the following balance exists: q e = - a i = q ' , (2.28) This charged layer is called the electrical double layer. It is characterized by its double layer capacitance d\, defined as follows: (2.29) In this case, E is the hypothetical potential difference between the bulk electronic and ionic conducting phases at equilibrium. Under load current, charge is exchanged across the interface via electrochemical reaction and multiple mechanisms contribute to a surface capacitance that differs from the equilibrium value. For example, concentration gradients near the electrochemically active surface exhibit relaxation behaviour, and thus, contribute to an interfacial capacitance. This capacitor-like behaviour is responsible for the observed time relaxations following an induced perturbation. 23 Chapter 3: Literature Review of Electrochemical Transient Techniques 3.1 Introduction Transient techniques are based on disruption of a system's steady state operation. The time response is a characteristic of charge transfer, mass transport, heat transfer, and other aspects of the system. If the transient behaviour is well understood, such as in the case of lead-acid batteries, governing equations can be fit to transient data to yield quantitative indications of performance and degradation. This approach has led to rapid improvements in the development of secondary batteries (Section 4.6.4). Time transient behaviour of SOFCs is not as well understood. The electrochemical impedance spectroscopy (EIS) technique has offered widespread benefits to SOFC development (Chapter 4); however, data interpretation has been shown to be problematic, and different research groups have presented conflicting viewpoints regarding EIS findings [19]. Galvanostatic current interruption (GCI) has prevailed as a technique for the separation of the instantaneous response from various relaxation processes in SOFCs. Other electrochemical transient techniques have been developed for applications other than SOFCs, as described by Bard [16] and MacDonald [20]. This section reviews basic concepts in SOFC testing, and it provides the foundations, capabilities, and challenges associated with GCI, current and potential step techniques, and EIS. This offers a background for further development of time transient techniques for diagnosis of degradation in SOFCs. 3.2 Cell Polarization Measurement 3.2.1 Two-Electrode Configuration For the purpose of polarization measurements in fuel cells and batteries, current and potential across the cell electrodes can be easily accessed with a low impedance ammeter (I) and a high impedance voltmeter (V), as depicted in Figure 2. 24 LOAD Figure 2: Two-Electrode Test Cell Schematic 3.2.2 Three-Electrode Configuration If one wishes to decouple the characteristics of each electrode, the three-electrode test cell is used, as shown schematically in Figure 3. The working electrode (WE) and counter electrode (CE) represent the two electrochemically active electrodes. The third electrode is the reference electrode (RE). With this configuration, electrochemical characteristics of the working electrode can be isolated. LOAD; Figure 3: Three-Electrode Test Cell Schematic In aqueous electrochemical experiments, a remote reference electrode is often coupled to the cell via a Luggin capillary; however, for cells with solid state electrolytes, a permanent 25 reference electrode is deposited (or placed in contact) strategically on the electrolyte. The measured potential between the working andreference electrode is: - ^ W R ~ ^ W E ~ _ " ^ R E (3-1) The term iR\j is the ohmic potential difference between the reference electrode and the working electrode. The subscript U stands for "uncompensated". This nomenclature originated from early experiments in which this term was not compensated for and false interpretations resulted. iR\j arises primarily from the potential gradient established from current passage between the working and counter electrodes. As the cell current changes, iR\j not only scales with current, but it can also vary nonlinearly due to current distribution effects. It is common practice to measure and subtract the iR\j component from £ W R . Nagata et al. [21] investigated the placement of reference electrodes on SOFC electrolytes. They used both disk-shaped and rectangular YSZ electrolytes and they experimented with Pt reference electrodes deposited in various configurations. The disk-shaped electrolytes included two types of reference electrode configurations: a) deposited laterally around the circumference of the electrolyte, and b) deposited on the cathode side. On the rectangular electrolyte, reference electrodes were deposited on the cathode side at various distances away from the cathode. They concluded that accurate electrode measurements are not possible when the anode and cathode are significantly misaligned with respect to each other. From the rectangular electrolyte studies, it was found that as one places the reference electrode farther from the working electrode, the apparent overpotential of the working electrode becomes larger whereas the apparent overpotential of the counter electrode becomes smaller. They proposed the ring-shaped reference electrode shown in Figure 4 as a configuration that yields more accurate overpotential measurements than their test cases. 26 I _ _ I Flee riolyre Electrode Figure 4: Ring-Shaped Reference Electrode Winkler et al. [22] proposed two practical guidelines for reference electrode placement on solid electrolytes: 1. Error in electrode polarization resistance resulting from non-uniform current distribution should be no more than 5%, and 2. The potential difference across the electrolyte region in contact with the reference electrode should be no more than 10"3 times the total potential drop across the cell. They speculated that if criterion 2 is satisfied, electrochemically driven reactions will not interfere significantly with three-electrode polarization measurements. Winkler et al. developed a numerical simulation of current distributions in an SOFC in order to realize the implications of their guidelines. They found that for working and counter electrodes with identical area specific polarization resistance, Guideline 1 is satisfied when these electrodes are misaligned by no more than 0.1 electrolyte thicknesses. They believed that this level of alignment accuracy was not achievable with manufacturing processes available at the time (1998). This postulation implies a lack of reliability in some reported SOFC findings (see Chapter 4). A pellet-shaped reference electrode configuration was developed at the Riso National Laboratory in Denmark [23]. This is a reliable alternative to planar three-electrode cells which exhibits a relatively uniform current distribution between working and counter electrodes. This configuration conforms to the criteria presented by Winkler et al. [22] under a broader range of conditions than predicted for planar geometry. 3.2.3 iRv Compensation There are several strategies for the compensation of iRu, and thus, a more correct interpretation of polarization losses across an electrode-electrolyte interface. Methods mentioned by MacDonald [20] include direct measurement with an A C bridge (not applicable 27 to solid electrolytes), electronic simulation and correction, direct feedback compensation, and GCI. The technique of electronic simulation and correction was described by Hayes and Reilley [24]. This technique requires a predetermined value of the R0hm- Overcompensation can be a problem. Direct feedback compensation also requires a predetermined R0hm [20]. This value is tuned with a variable resistance in the feedback circuit. Overcompensation and feedback instability can cause errors, as the circuit tends to oscillate around the compensation point. Of these different methods, GCI has prevailed as an iRv compensation technique for SOFC experiments. GCI is a relatively simple technique that does not require predetermination of Rohm- GCI is further explained in Section 3.3. EIS is also a common technique for iRu compensation. Application of a high-frequency sinusoid between the reference and working electrodes offers a means to isolate R\j; however, inductance in the experimental apparatus can complicate this measurement. 3.2.4 Potentiostatic and Galvanostatic Control Common three-electrode polarization measurements require either control of the cell current for potential measurements (galvanostatic control), or control of the cell potential for current measurements (potentiostatic control). For these purposes, a voltage follower circuit known as apotentiostat was developed [25]. A simple potentiostat can be constructed with an operational amplifier. The task of the potentiostat is to supply whatever current is necessary in order to maintain a controlled potential between its two terminals. Connected to an electrochemical cell, the potentiostat drives the potential difference between two electrodes to a reference value, which is often provided by a function generator. The galvanostat is an offshoot of the described potentiostat; instead of controlling the potential between the working and reference electrode, it controls the potential across a reference resistor in series with the cell, thereby controlling the current through the cell. Modern potentiostats have evolved in complexity far beyond the single operational amplifier concept; however, their functionality generally remains the same. When used for dynamic operations, potentiostats have finite settling times which limit their effectiveness in step response applications. 28 3.3 Galvanostatic Current Interruption and Current Step 3.3.1 Theoretical Development Chronopotentiometry is the measurement of voltage response to a controlled current with time. When the cell current is interrupted by a fast switch, the ohmic contribution to electrode potential disappears instantaneously and the ohmic resistance R0hm can be determined as follows: Rh (3.2) ohm v ' where AE{=o is the instantaneous change in potential following a current step AI. Other attributes, such as concentration gradients, decay with time. Separation of ohmic polarization from all other polarizations is the premise upon which chronopotentiometric techniques have been applied to fuel cells and batteries. However, the voltage-time response to current changes has implications beyond ohmic compensation. Consider an ideal (discrete) step in current, controlled between a working and counter electrode with a galvanostat. After the step, the constant cell current / is composed of two time variant components: the faradaic current If and the double layer charging current Ic (assuming the electrode-electrolyte interface behaves as a parallel plate capacitor). Thus, dF I = I,{t) + Ie(t) = I,{t) + C ^ (3.3) Although the sum of current components is controlled, the addends change with time as the double layer is charged to a new steady state. Some researchers have devised methods to step only //between discrete values, thereby investigating only the faradaic reaction and neglecting the capacitive behaviour [20]. Under these circumstances, chronopotentiometry techniques have been considered problematic, but elimination of Ic has been achieved in aqueous systems (via compensation with a dummy cell that is inert to faradaic reaction). This strategy has not been applied to SOFCs; therefore, interactions involving Ic and I/are poorly understood from a theoretical standpoint in SOFC current step experiments. 29 Newman [26] derived time constants that characterize individual relaxation processes at a generalized electrochemical interface. He recognized that immediately after interruption, the double layer remains charged, and it may then discharge via faradaic reaction. An approximate time constant for this decay is defined as: (3.4) Fi An important consideration is that the double layer is not necessarily uniformly charged when ohmic overpotential is large compared to current overpotential. This scenario of non-uniformity is referred to as a primary current distribution, and a second time constant arises from the transport of charge after interruption to equilibrate the charge gradients on the electrode surface charge. This time constant is: r 2 = - ^ L (3.5) cr Here, a is the electronic conductivity of the electrolyte and rQ is the characteristic length of the electrode. Newman gave little description of the physical meaning of this length. In other literature which describes current distribution characteristics [17], the characteristic length has been presented as the radius of a disk-shaped electrode. Newman inserted values into Equations 3.4 and 3.5 and showed that time constants TI and T2 are on the same order of magnitude; thus, they would be difficult to resolve in transient data. Aside from charge distribution characteristics, another time constant can be attributed to the relaxation of concentration gradients, as follows: r 3 = ^ (3.6) Here, Sm is the diffusion path length for diffusing species m. Again, Dm is the diffusion coefficient. This time constant was presented by Wruck et al. [27]. They performed periodic current interruption on batteries and cells involving rotating disk electrodes. They observed 30 some features in the transient voltage response, but they did not correlate them to any of the three time constants x\, T2, and T3. However, in the case of the rotating disk electrode, they used the changing slope of the voltage transient at times greater than 0.05 seconds after interruption to deduce a mass-transfer relaxation time. These time constant relationships have been used in recent years to model the transient response of fuel cells and batteries (Section 4.6). Governing equations for diffusion-controlled potential relaxation have been developed to model various simple electrochemical systems. Theoretical formulations have been reviewed by Paunovic [28] and MacDonald [20] in which the cell potential is derived in terms of current density and time after perturbation. In the case that diffusion is the only transport mechanism of reactants and products to and from reaction sites, the concentration time dependence is given by Fick's second law of diffusion. By solving Fick's second law with suitable boundary conditions and substituting the time-dependent concentrations into the Nernst equation, one can arrive at the time-dependent concentration overpotential. This formulation was elaborated to consider parallel and series faradaic reactions [20,28]. Lorenz incorporated electrode surface adsorption into the governing diffusion equations of an aqueous electrode-electrolyte interface during a current step experiment [29-34]. This theoretical approach has helped support hypotheses regarding the nature of electrochemical reactions involving adsorbed species. Kinetic theory has also yielded analytical models of voltage-time relationships. Harrington and Conway [35] presented an analytical model of potential decay transients subsequent to GCI. Their investigation dealt with the hydrogen evolution reaction on a platinum electrode submerged in acid. Their formulation neglected mass transfer effects and assumed Langmuir monolayer adsorption (only one type of adsorption site exists) and Tafel behaviour. From this formulation they arrived at a time-dependent activation overpotential equation. This relationship was fitted to experimental data in order to acquire unknown electrochemical characteristics. Larminie [36] described a method of analytical modeling of a fuel cell's transient response using an equivalent circuit model. He claimed that this is a more suitable approach than EIS when relatively slow responses are of interest. Overlap of anode and cathode transient responses yielded ambiguous interpretations. Several models have been developed to simulate transient voltage responses of an SOFC stack to load change [37-40]. These models have all assumed that an ideal current step results from a change in the power demand (i.e. a step in the load resistance) of an operating 31 SOFC. By Faraday's equation, a change in fuel cell current causes different gas flow conditions. On this premise, the mass and energy balances pertaining to the specific SOFC under investigation have been solved, yielding transient temperature and concentration relationships. Equilibrium potential and ohmic losses have been included in the models of cell potential. Sedghisigarchi and Feliachi [40] elaborated this approach to include Tafel activation overpotential (Equation 2.20), and concentration overpotential (Equation 2.26) in their model. Inclusion of an unsteady temperature in these relationships yielded transient overpotential responses. All of these models examined a relatively large timescale on the order of 102 s or more. Hence, slowly responding temperature effects were of principal interest and rapid electrochemical responses were not taken into account. This section has illustrated examples in which analytical expressions have been formulated to model potential responses to current steps or interruptions in electrochemical systems. Although voltage transients have been modeled in SOFCs, the models have not been elaborated far enough to consider rapidly decaying transient features. 3.3.2 Experimental Development GCI has become a popular method used in battery and fuel cell electrode polarization experiments. Literature has shown that it has been used primarily for the purpose of separating the ohmic response from electrochemical relaxations. The following section reviews developments in CS and GCI hardware for electrochemical applications. In 1960, Kordesch and Marko [41] described a mechanical switching circuit for testing battery polarization and internal resistance. A charging capacitor was used to retain the OCV. Disadvantages of this approach included capacitance round-off, mechanical contact bouncing, and excessive ringing. These problems led Kordesch and Marko to investigate injection of a 60 Hz sinusoidal half-wave current across the cell (in parallel with the capacitor). In this way, the voltage response could be used to determine ohmic resistance. This circuit was used to test oxygen sensors, Ni-Cd batteries, AFCs, and MCFCs. Kordesch and Marko claimed that the concept could be applied to measure ohmic resistance within good agreement with square wave measurements. Although this concept negated the requirement of a mechanical switch, it did not offer significant contributions to interpretation of electrochemical transients. 32 Warner and Schuldiner [42] presented a technique to acquire rapid potential responses to current steps between periods of potentiostatic control. They used a pulse generator to intermittently reverse-bias a diode, causing a current step response. They reported "on" and "off switching times of 0.2 to 1.0 microseconds and 6 to 20 microseconds, respectively. They anticipated excessive potential ringing in their circuit. This technique depended highly upon the specifications of the potentiostat used, and it was expected to not work for all potentiostats. Mclntyre and Peck [43] presented an operational amplifier-based circuit to interrupt the potentiostatic control and induce constant current conditions (hence a current step). They speculated that pulses lasting 5 to 10 microseconds, repeated at frequencies of 0.1 to 1 kHz caused relatively non-invasive perturbations. They encountered a degree of instability in potentiostatic control. This disadvantage limited the technique's applications at higher cell currents. Gsellmann and Kordesch [44] presented an interrupter circuit for battery testing based on solid state circuit elements. Their design consisted of four main modules: the potentiostat, the test cell, the interrupter, and the recording device (digital oscilloscope). They discussed the application of instrumentation amplifiers for voltage and current acquisition, sample and hold integrated circuits, a digital logic circuit, and an analog output for potentiostatic or galvanostatic control. Current stepping was achieved via generation of a pulse train reference signal using semiconductor switch networks. This signal was applied to the potentiostat reference input. Gsellmann and Kordesch discussed the shape of the current step and they noted that a current rise time should be in the range of microseconds for practical electrochemical applications. They believed that this rise time was faster than activation and concentration overpotential relaxations which can take milliseconds to minutes to reach equilibrium. Wruck et al. [27] used similar circuitry to perform periodic current interruption with rise times in the order of microseconds and current-on duty cycles from 40% to 99.9%. They achieved fast digital switching with solid state switches with clocked gate pulses. Wruck et al. tested the accuracy of their hardware on RC circuits resembling electrochemical cells. The digital storage oscilloscope (DSO) has become the most widely used device for high-speed transient recording of electrochemical responses. A DSO offers desirable features such as display auto-ranging, pre- and post-triggering, and signal processing. Advanced DSOs allow for the user to program logic triggering so that recording can be triggered by digital patterns such as a pulse of predefined width [45]. 33 Soares et al. [46] dealt with the challenge of steady-state disruption in applications of GCI to large-area electrolytic cells drawing large currents. They introduced the concept of probe electrodes, which are isolated electrode segments devoted to diagnostic testing via GCI. This allowed for GCI to be performed on a large cell without departing significantly from steady state operation. Application of this concept to SOFC cells and stacks could offer insight into the operating characteristics in a non-invasive manner. Nardella et al. [47] presented GCI hardware which was suitable for ohmic resistance measurements in SOFCs. Highlights of the hardware are summarized in Table 1. Tab le 1: G C I H a r d w a r e Specif icat ions Presented by Na rde l l a et al. [47] A n a l o g Input M a x . Da ta Points Resolut ion T i m e Resolut ion 64000 8 bit 0.1 us 32000 16 bit Vol tage Range 10 uV to 200 V C u r r e n t Range 10 uA to 1 A Swi tch ing T i m e <0.1 us Similar to Wruck et al, Nardella et al. validated their hardware via GCI tests on RC circuits representing solid electrolyte cells. It was shown that circuit resistances could be measured to within 2% of their actual values by GCI. Biichi et al. [48] investigated the use of current steps in PEFC electrolyte resistance measurements. They predicted that switching and recording within the nanosecond range is essential for accurate separation of ohmic resistance from electrochemical processes. They superimposed short current pulses from an auxiliary current source onto the fuel cell's operating current using two DMOS transistor analog switches (switching time of 5 ns). Ringing was observed within 200 nanoseconds after the current pulse onset. For this reason, the ohmic polarization could not be read directly from the potential data and it had to be back-extrapolated. They assumed that no faster electrochemical processes were hidden in the region of oscillations. A relatively poor signal-to-noise ratio was experienced at small current pulse amplitudes. Biichi et al. hypothesized that inductance in the fuel cell load loop was a limiting factor in the reduction of current switching time. They related the minimum pulse decay time At to the inductive loop parameters of the fuel cell and load loop as follows: 34 &t>L A i Jloop (3.7) Here, L is the loop inductance and isi0op is the potential in the current loop. Equation 3.7 indicates that the experiment should be designed in order to minimize L when fast electrochemical processes are present. This can be done by either minimizing wire lengths or maximizing the potential drop in the loop. In the aforementioned studies, several different active electronic devices have been used to impose a perturbation onto the fuel cell load. The disadvantage of this method is the possible overlap of transient responses of the power source and the test cell. It is known that no power source is ideal, and all exhibit a rise time; thus, only power sources with very small rise times should be used. 3.4 Potential Step 3.4.1 Theoretical Development Chronoamperometry involves the application of a controlled voltage signal to an electrochemical cell and measurement of the current-time response. The potential step technique has been used extensively in aqueous electrochemistry to study potential-dependent reactions. In theory, a discrete potential step yields an instantaneous current response which can be used to extract the cell's ohmic resistance, similar to applications of GCI. However, the potential step technique has been less popular than GCI for this purpose for reasons that have not been thoroughly explained in literature. One reason might be the simplicity of interrupter circuits used in GCI. Similar to the case of a current step, Fick's second law of diffusion can be solved to yield a current-time transient relationship [16]. Several variations of the analytical diffusion scenario at reaction interfaces have been reviewed by MacDonald [20] for the case of a potential step. Johnson and Newman [49] used an analytical potential step model in the investigation of electrochemical desalination techniques. From this model, they estimated double layer 35 capacitance in porous electrodes. Tiedemann and Newman [50] elaborated this concept for the case of porous electrodes and they suggested some graphical techniques to deduce an electrochemical system's double layer capacitance from its response to a potential step. 3.4.2 Experimental Development In potential step experiments, a potentiostat is used to control the potential between the selected electrodes. A signal generator is commonly used as a potential input to supply the step function, and similar to current step hardware, a recording oscilloscope or other data acquisition device is used to record the current-time response. Vielstich and Delahay [51] studied the current response to a potential step at the cadmium amalgam electrode to estimate electrochemical parameters such as exchange current density via graphical techniques. They applied a small voltage step (AE - 2 to 5 mV) to maintain near-equilibrium conditions. They claimed that the principal advantage of their method over others was the experimental simplicity; a Luggin probe was not required, and thus, difficulties in ohmic compensation were avoided. However, a disadvantage of this technique was its requirement for long measurement durations to allow for double layer charging currents to dissipate. Underkofler and Shain [52] presented two different circuit designs for potential step experiments which involved mechanical switches and operational amplifiers. They claimed that their circuit was not useful for rapid recording of experimental times of 100 milliseconds after perturbation due to the rise time of the mechanical switches. Krischer and Osteryoung [53] applied short voltage pulses to germanium electrodes to investigate double layer capacitance. They derived an analytical current relaxation relationship assuming capacitor charging behaviour. In an attempt to measure the electrode's dynamic characteristics, they superimposed periodic voltage pulses with 3 us durations onto a linear voltage sweep signal. They claimed that accuracy of double layer capacitance measurements were within ±20%. Error was believed to result from poor resolution of the oscilloscope display as well as the potentiostat's non-ideal switching capabilities. Tiedemann and Newman [50] applied potential steps to Pb and Pb02 electrode plates manufactured for automotive lead-acid cells. They estimated double layer capacitance by fitting the analytical model presented by Vielstich 36 and Delahay to experimental data. Several sources of error were supposed, including systematic error from the model assumption of negligible Faradaic current. For the purpose of SOFC diagnostics, the potential step technique might provide insight into capacitive behaviour at the electrode-electrolyte interface as it has for the other applications described above. However, this has not yet been demonstrated. 3.5 Electrochemical Impedance Spectroscopy 3.5.1 Theoretical Development The measurement of impedance across two electrodes using a sinusoidal perturbation is an effective method for the elucidation of ohmic resistance, activation polarization processes, concentration polarization processes, and double layer capacitance associated with electrode-electrolyte interfaces. This technique is known as electrochemical impedance spectroscopy (EIS) when impedance is analyzed as a function of frequency (Bode plot) or its resistance and reactance components are plotted in the complex plane throughout a frequency range (Nyquist plot). Examination of impedance at different frequencies indicates the timescales at which different relaxation phenomena occur. A thorough review of EIS principles has been provided in the literature [16 (ch.l0),20,54]. Cell impedance Z is a complex value with real (resistive) and imaginary (reactive) components. Z can be determined at a frequency co by applying a sinusoidal voltage V (potentiostatic mode) across the electrodes of an electrochemical system. The sinusoidal current response is measured and impedance is determined by Ohm's law as: I(ca) The current response I is characterized by a magnitude and phase angle, both of which are governed by the system impedance. Alternatively, sinusoidal current can be imposed and impedance can be determined from voltage response (galvanostatic mode). A standard convention in electrochemistry is to invert the imaginary axis in the impedance plane (Nyquist 37 plot) so that the positive axis represents negative imaginary impedance. This has been done in all impedance plots in this work. As a prerequisite to the theory of EIS, an introduction to electrical circuit representation of electrochemical systems is given here. Equivalent circuit representations typically consist of resistors and capacitors, but they have been elaborated to include other elementary building blocks specific to electrochemistry. Figure 5 shows a general equivalent circuit representation of one electrolyte-electrode interface in an electrochemical cell in operation (Vis measured between a working and reference electrode). This is known as the Randies cell. The term Ru has been discussed in Section 3.2.2. Cai is the double layer capacitance of the electrode-electrolyte interface and Z/is the impedance ascribed to the faradaic reaction. Figure 5: Equivalent Circuit of an Electrode-Electrolyte Interface Measured in a 3-Electrode Configuration In Figure 5, the cell current consists of a transient double layer charging current ic and a faradaic current //. The faradaic impedance Z/is a major source of complexity, as its value changes with frequency. Also the components Ru and di can show some degree of variance with frequency; the sources of these dependencies are not thoroughly understood. It is desirable to separate Z/from Ru and Cdi in order to better understand the faradaic process. Zf can be depicted as a series combination of the charge transfer resistance Rct and the Warburg impedance Zw, as follows: w (3.9) 38 Rct is attributed to charge transfer kinetics and Zw represents diffusion behaviour. With a sinusoidal voltage input at a known frequency, elementary circuit theory yields the following expression [16]: Zw =Rlv+j/(a}Cw)=c™-l/2 -j(cro)-i/2) (3.10) Here j = V-T, Rw is a purely resistive term attributed to the diffusion process, a is a diffusion-related function, and Cw is known as the pseudocapacitance. With the conjecture that the electrochemical interface can be modeled accurately with the described circuit model, the magnitude of the real component of impedance is: (e>«-,/2 + \) + co2Cd2 (Rct + aco"2 J If the reversible electrode reaction of Equation 2.7 is under investigation, the function a is defined as: <7 = dE nFAyfl dCo(0,t) i,CJ0,t) dE dCR{0,t) i,Co[0,t) D{/2 (3.12) JJ Here, A is the electrode area. This function changes when different reactions are under consideration. The magnitude of the imaginary component of impedance is: When EIS is performed with the mean electrode potential at equilibrium (steady state at a particular DC offset) and the perturbation amplitude is relatively small, the prediction of linear kinetics is applicable. Based on this prediction, the following relationship can be deduced: 39 (3.14) A value of Rct can be extracted from impedance plots. Hence, the exchange current density can be extracted via the EIS technique. For the limiting case of co —• 0, Equation 3.11 and 3.13 can be simplified and the impedance can be expressed as: Z^=ZRe-Ru-Rcl + 2a2Cai (3.15) The frequency dependence of this relationship arises from the Warburg impedance term. This term is a representation of a diffusion process, so the linear impedance regime indicates that the electrode process is diffusion-controlled. For the limiting case of co —> oo, Equation 3.11 and 3.13 can be simplified and the following relationship can be derived: zRe-Ru—f~ +zl (3.16) Under these circumstances, the Warburg impedance is negligible and the faradaic impedance reduces to the resistive element Rct. This represents a kinetic-controlled regime. The Nyquist plot yields a circular arc with a center at Ru + RJ2. Figure 6 illustrates this impedance behaviour. Here the arc has been extrapolated to co = 0 (diffusion behaviour has been neglected). The arc diameter yields Rct, and the frequency at the arc apex yields Cd\. 40 •Zjjii. Figure 6: Impedance Plane Plot - Zw Negligible On the other hand, if diffusion behaviour is non-negligible, then a combination of the two limiting cases occurs and the impedance plane appears as seen in Figure 7 . Zlm Kinetic control Increasing® Figure 7: Impedance Plane Plot - Zw Non-Negligible Two distinct regimes are observed, pertaining to kinetic and mass transfer control. In actual impedance plots, data can overlap and electrochemical parameters can be indistinct. Also, the center of an impedance arc often lies below the real axis for reasons described in detail by MacDonald [54]. Nevertheless, if an arc is visible in the impedance plane, it represents a particular polarization process with admissible accuracy, and by extrapolation of the semicircular arc as shown by the dashed lines in Figure 7 , characteristics of the electrode process can be extracted. Arcs in impedance data are of paramount importance in the analysis of SOFCs. Often, multiple arcs are seen, as discussed in Chapter 4, and each arc represents a unique relaxation process within a known frequency regime. Furthermore, it has been shown that the height of a 41 Mass ; transfer > control fuel cell's high frequency arc is inversely proportional to the square root of i0 (Yuh and Selman [55]). A theoretical premise of this claim is beyond the scope of this review; nevertheless, observations such as this are invaluable in the characterization of fuel cell performance. 3.5.2 Experimental Development EIS was developed for the characterization of electrical circuits and it has been later adapted to electrochemistry. There are various ways in which impedance can be measured. Technique development is briefly reviewed here. It has been described thoroughly by MacDonald [54]. There are several different ways by which the frequency response can be measured. Application of white noise is one option, although generation of true white noise is problematic. An unequivocal approach is to measure impedance of a chosen frequency directly, using one of a variety of instruments, such as an impedance bridge, transformer arm ratio circuit, potentiostat, or phase-sensitive device. In most cases, discrete Fourier transform (DFT) of transient data is required in order to process the response. A simple potentiostatic method of direct impedance measurement consists of a potentiostat with an A C signal superimposed on a DC voltage. This potential is applied across a working and reference (or counter) electrode. The current response can be displayed on a two-beam oscilloscope, along with the sinusoidal voltage input, and the impedance amplitude and phase angle can be graphically determined. This is a tedious procedure, as the process needs to be repeated for every frequency under investigation. Phase sensitive devices (PSDs) are common in EIS instruments. These devices are based on sequential operation of multiplexing and time-averaging circuits. They produce an output that reflects the phase angle between the sinusoidal input signal and a reference signal. A PSD can be designed to output a signal that represents cell impedance at a certain frequency. Frequency response analyzers (FRAs) are digitally-demodulated impedance meters that are based on PSD technology. By automating PSD operation at stepped frequencies, FRAs can generate a high precision sinusoidal wave and impedance can be measured over a broad frequency range. Often the range of interest lies between the mHz and MHz range. An FRA imposes certain requirements on the electrochemical system under investigation. One 42 important requirement is time invariance. It is often necessary to let the system dwell at steady state prior to EIS measurements. Data acquisition can take significant time, especially at low frequencies. These considerations can lead to lengthy experiments. FRAs are relatively expensive devices and interpretation of results might require some expertise. New innovations are being presented in acquisition of frequency domain data. A technique known as instantaneous impedance spectra measurement has been developed as an alternative to the stringent requirements of an FRA. The technique's theoretical foundation and applications have been described in the literature [56-60]. Instantaneous impedance spectra measurement uses a recently developed algorithm known as the short time Fourier transform (STFT), in which data is sampled in consecutive time windows, and for each time window, a discrete Fourier transform is carried out. This strategy allows for fast characterization of processes such as electrode degradation or electrochemical deposition, which can be the source of system time-variance. In addition, Adler [61] presented a technique involving the measurement of second and higher order voltage harmonics resulting from sinusoidal current perturbations. His objective was to improve identification of physical processes via their nonlinearity and extent of coupling. He pointed out that in SOFCs, electrochemical impedance is complicated by nonlinear coupling between kinetics and mass transfer. Adler proposed to elucidate nonlinearities via the application of a sinusoidal perturbation at frequencies too high to disrupt electrochemical interface concentrations, then by probing the nonlinear response through perturbation amplitude modulation. Harmonics provide an indication of coupled electrochemical processes. Adler claimed that this technique requires further development before it can compete with conventional EIS in terms of accurate resolution of processes. Despite the mentioned disadvantages, EIS is the most common and accurate electrochemical diagnostic technique used to expound the various processes in SOFCs. 3.5.3 Data Interpretation Problems inherent to impedance measurements include overlap of arcs in the impedance plane, as well as presence of inductance artefacts. Also, fuel cell current loops exhibit an extent of inductive behaviour, as described by Biichi et al. [48], which can complicate interpretation of impedance arcs. For these reasons, strategies have been devised to 43 interpret electrochemical process characteristics from impedance data. The equivalent circuit fitting technique has been adopted to overcome challenges associated with process overlap; however, ambiguity lies in this method. Equivalent circuit fitting and other methods for impedance data interpretation are described here. Equivalent circuit fitting is commonly applied to extract parameters from impedance data, such as polarization resistance and double layer capacitance. The basis of the equivalent circuit fitting method relies on the analogy between electrochemistry and electric circuit theory. Circuit analogies offer a way to represent and compare experimental data. To obtain adequate representation of impedance spectra, the equivalent circuit must possess the same frequency response as the actual electrochemical cell. A problem lies in the fact that an equivalent circuit solution is not unique. In fact, a multitude of parameter sets and configurations can be used to represent the system behaviour under the specific test conditions. Furthermore, when an experimental parameter such as temperature is varied, the rate-limiting phenomenon can change altogether, and a different circuit configuration might be required to accurately represent the same electrochemical system. Thus, an equivalent circuit is too ambiguous to truly represent the physical principles of electrochemical phenomena in fuel cells; however, their use is advantageous in some circumstances from an engineering perspective. Actual passive circuit elements used in electrochemical representations include the resistor, capacitor, and inductor. The Warburg impedance element is a fictitious circuit element that represents a diffusion-controlled process. Furthermore, the constant phase element (CPE) is a generalized circuit element whose impedance is represented by the following relation: Z C P E = / * ( » - (3-17) Here, A is the proportionality constant, n is the phase-displacement-related exponent, and co is the angular frequency of the perturbation signal. If n equals 1, 0, or -1, then the CPE represents a capacitor, resistor or inductor, respectively. If n has a rational value, then the CPE becomes a generalized voltage-current transfer function with a fractional derivative. This type of element has been used to generate accurate fits to the frequency responses of SOFC electrodes under certain conditions. A problem with this approach is that it represents neither the physics of the fuel cell nor the physics of an electrical circuit. Van Heuveln [62] represented the cathode-electrolyte interface in an SOFC with a resistor/CPE parallel pair. Although he claimed that 44 this representation has no satisfactory chemical or physical explanation, he hypothesized that the good correlation could result from distributed isolated reaction zones throughout the cathode. This might result in charging behaviour that is neither continuous nor homogeneous throughout the electrode, and thus, not well represented by discrete RC pairs. Van Heuveln used the CPE circuit model to derive an analytical solution to the voltage-time response following current interruption. Boukamp [63] described a nonlinear least squares fit (NLLSF) procedure to fit equivalent circuit parameters to impedance (or admittance) data. By this technique, a simulation function is generated and modified via iterative convergence to match the experimental data as closely as possible. Boukamp presented software called "EQIVCT", which performs NLLSF on electrochemical impedance data based on circuit description code. The user can attempt to resolve different impedance arcs in SOFC data by constructing a circuit with building blocks including the described circuit elements, plus additional elements that represent different conditions of mass-transfer control. The circuit is constructed by the user so that its frequency response vaguely resembles that of the SOFC. The program then adjusts the element parameters to fit to the impedance data accurately. This approach is problematic; it requires a priori assumptions of the circuit layout. Equivalent circuit fitting has allowed for convenient presentation of SOFC transient data, but it has offered no insight into SOFC degradation from a first principles perspective. A technique known as deconvolution has been used to resolve impedance data in SOFC experiments [64]. Deconvolution refers to algebraic division of two mathematical functions in Fourier transform space, followed by an inverse Fourier transform. The power of the deconvolution method for electrochemical analysis lies in the fact that no a priori assumptions of an equivalent circuit are required. Instead, this technique assumes that the electrochemical process is represented by an infinite number of parallel RC pairs, connected with each other in series, thus forming a continuous time constant distribution y(f). The following relationship implicitly defines y(r): CO (3.18) 0 45 The objective of the deconvolution method is to solve for y(r) and analyze time constant distribution peaks. Each peak can be thought to represent a distinct physicochemical process whose area can be correlated to a charge transfer resistance. A description of this procedure can be found in the literature [64-65]. Analysis of time constant distribution peaks offers an alternative to equivalent circuit fitting. The former is based directly on acquired experimental data, whereas the latter requires a priori assumptions. A challenge associated with the deconvolution technique is the scrupulous selection of a digital filter. In this selection, compromise must be made between data resolution and error suppression. Smirnova et al. [64] demonstrated that deconvolution offers much higher resolution than equivalent circuit fitting. Schichlein et al. [65] also demonstrated the high resolution of the deconvolution method. They generated artificial impedance data from a circuit model. Deconvolution of the frequency response yielded three distribution peaks within one decade of the time constant distribution, all of which correlated well with the calculated time constants of the circuit model. This level of resolution had not been achieved using equivalent circuit fitting of impedance data. However, time constant distribution peaks have not yet been ascribed to known electrochemical processes with confidence. Deconvolution of EIS data has not yet been used to correlate time domain characteristics to those found in time transient experiments. This endeavour could offer the benefit of experimental validation of SOFC time transient experiments. 3.6 Combination of Transient Techniques In the diagnosis of SOFC degradation, maximum information can be extracted through the combination of different transient techniques. For example, GCI and current step techniques can be used to evaluate ohmic overpotential and acquire electrochemical responses. The GCI technique is thought to be relatively invasive to cell operating conditions as it disturbs the steady state operation of the cell. This causes ambiguity in measured electrochemical properties; however, this technique has been widely adopted to isolate ohmic polarization. Perturbation via small amplitude current steps offers an alternative to steady state disruption. EIS offers additional information regarding electrochemical phenomena. Application of these techniques in combination has seldom been reported in the literature. Complementary 46 information from multiple techniques might offer a new insight in both the interpretation of characteristics pertaining to operating SOFCs, a well as the development of new methods for diagnosing degradation in SOFCs. 47 Chapter 4: Applications of Transient Techniques to SOFCs and Other Systems 4.1 Introduction A limited amount of literature has been found that exclusively describes the application of time transient techniques to SOFC diagnostic research. A larger foundation of literature has been reviewed in which GCI and EIS have been applied along with DC polarization, material characterization and other experimental techniques in order to study the highly questioned reaction mechanisms governing SOFC performance. EIS and GCI have become common practice for the extraction of ohmic-free overpotentials. 4.2 Early Contributions Olmer [66] pioneered the use of small step perturbations to investigate SOFC behaviour. He used a 1000 Hz square wave to separate ohmic losses from electrochemical relaxations. He claimed that the perturbation signal should be small enough so the cell remains in the linear polarization region. Badwal and Ciacchi [67] used EIS to study oxygen sensors with Pt electrodes and YSZ electrolytes. This study focused primarily on Pt electrode microscopy. As previously mentioned, Nardella et al. [47] applied GCI to SOFCs. They discussed how ohmic heating and electrode decomposition can cause distortion in measured electrode resistance. They performed electrode polarization measurements and GCI experiments using a three-electrode cell configuration. They also considered capacitive effects within the electrolyte. These had not been considered in previous works. Badwal and Nardella [68] used GCI to investigate the application of a high magnitude square wave voltage to the cell to manipulate the microstructures of noble metal electrodes. GCI was used to extract ohmic-free electrode overpotentials. This study exemplified the convenience of the GCI technique for separating ohmic polarization from electrochemical processes in SOFC studies. In these early investigations as well as many of the studies that followed, efforts were restricted to the separation of ohmic losses from other losses, and no progress was made to isolate multiple overpotential relaxations from the transient responses. 48 4.3 Anode Investigations In comprehensive investigations, transient techniques have been used to better understand performance limitations in SOFC anodes [69-82]. Mizusaki et al. [69] used both GCI and EIS to investigate reaction kinetics on Ni-patterned anodes. They observed only one arc in impedance data. From GCI, they generated ohmic-free polarization curves and investigated reaction orders. Eguchi et al. [70] used the GCI technique to isolate the anodic overpotential in the concentration-limited polarization region. They claimed that under their particular test conditions (1000°C, pure O2 at the cathode) neither GCI nor EIS could effectively separate concentration from activation overpotential. Jiang and Badwal [71] used both the GCI and EIS techniques simultaneously to reinforce hypotheses regarding the nature of the HOR on Ni and Pt anodes. This study set precedence for quantitative comparisons between the EIS and the GCI techniques. Jiang and Ramprakash [72] performed GCI tests at different temperatures, fuel humidities, and H2 concentrations in order to determine whether the general polarization behaviour of the Ni-YSZ anode is limited by concentration or activation. Due to lack of confidence in curve fitting with analytical polarization relationships, they concluded that the HOR is not a simple charge transfer process, and at least two steps might limit the reaction. This study illustrated that analysis of ohmic-free polarization characteristics obtained via the GCI method cannot yield the resolution required to identify multiple rate-limiting reaction steps. Jiang, Ramprakash, and Love [74] investigated the apparent reaction orders and activation energies of the electrode reactions in SOFCs using both the EIS and the GCI techniques. Poor correlation between EIS and GCI techniques was observed for the anode behaviour; both the reaction orders and activation energies obtained from EIS were approximately twofold higher than those obtained from GCI. Good correlation was seen between the two techniques for the cathode properties. A difference was thought to exist in pmo at anode TPBs for the two different techniques. Jiang et al. observed an improvement in the anode behaviour (as measured by both techniques) at a higher pmo- The GCI technique requires a DC load prior to perturbation, which results in a higher pmo (from the reaction product) at the anode reaction sites than in the case of EIS measured at OCV. Therefore, the differences in experimental parameters between the two different techniques in the anode studies were likely a result of the differences in pmo at TPBs. Jiang and Ramprakash [75] further compared EIS and GCI measurements of anode reaction parameters. They 49 investigated systematic error in the GCI technique and they predicted that the error was not large enough to explain the observed deviations between techniques. Although these comparisons provoked much speculation into the nature of the techniques, better correlation between the EIS and GCI techniques may have been observed if experiments were performed at the same DC load (initial steady state load in the case of GCI). Jiang and Badwal [76] performed GCI and EIS to evaluate anode performance using different proportions of Ni and YSZ. Open circuit impedance spectra revealed two distinct arcs which were hypothesized to be two series steps in the HOR. An equivalent circuit model was realized based on the two relaxation processes. In this study, GCI and EIS experiments elucidated a decreasing trend in overpotential with increased YSZ content in the Ni/YSZ cermet. Primdahl and Mogensen [77] performed EIS studies to clarify the nature of the HOR at the Ni/YSZ anode. They observed three distinct impedance arcs. Geyer et al. [78] performed EIS on anode symmetric cells (common electrode material and gas atmosphere at both sides of the electrolyte). Open circuit impedance yielded two relaxation processes. Cells were contaminated with a small stream of hydrogen sulfide (H2S) and EIS was used to interpret the degradation of performance. Primdahl and Mogensen [79] used EIS to isolate gas conversion impedance (impedance due to concentration overpotential in a two-atmosphere system). They predicted this phenomenon to appear as a low frequency arc in EIS experiments. Predictions were validated via comparison of impedance spectra of a full cell test with those of an anode symmetric cell test (no expected gas conversion impedance). They claimed that gas conversion impedance is not related to the anode performance itself, but relates to the gas flow channel geometry and gas flow rates. Furthermore, Primdahl and Mogensen [80] used EIS to investigate mass transport limitations from flow restrictions in the gas delivery pathway to the anode. They compared impedance spectra for different current collector configurations (each presenting a variation in gas flow restriction). Two relaxation processes were consistently observed in all EIS data. Primdahl and Mogensen [81] used EIS to investigate SOFC degradation over time. They performed EIS at different intervals on cells operating at steady state, and for cells undergoing thermal cycles (100 to 1000°C). They quantified degradation as a change in electrode polarization resistance per unit time under steady state operation or per thermal cycle. Bieberle and Gauckler [82] stressed the inconsistencies and disagreements between different research groups regarding the number of impedance arcs, primary electrode process 50 activation energies, and the rate-limiting steps for the HOR at the Ni/YSZ anode. They claimed that the different studies attempted to compare the fundamentals of the HOR, even though different manufacturing techniques were used in each study. They felt that microstructure had a large influence on the observable characteristics of the HOR, and consequently, they conducted a study of anodes manufactured by several different techniques, including sputtering, binding of woven wire gauze, and screen printing. From EIS results at different temperatures and overpotentials, anodes under consideration were all dominated by two thermally-activated relaxation processes. 4.4 C a t h o d e I n v e s t i g a t i o n s Parallel to studies of SOFC anodes, transient techniques have also played a major role in explaining the performance of SOFC cathodes [65,83-95]. Takeda et al. [83] studied the ORR on several different perovskite oxide cathodes. GCI was used to generate ohmic-free polarization curves. Hammouche et al. [84] studied the ORR at LSM cathodes. They tested pin-shaped (low surface area) electrodes using current and potential step techniques. Transient responses from both techniques showed a distinct transition between relaxation regions. The time scale of interest was in the order of 101 s to 102 s. After this transition, the relaxation reversed in direction (i.e. the slope changed in sign) and settled asymptotically. Inoue et al. [85] used GCI and EIS to compare ohmic-free electrode polarizations of L S M cathodes on YSZ and SDC electrolytes. Under the experimental conditions, the polarizations appeared exceptionally similar for the different electrolyte materials. Ostergard and Mogensen [86] used EIS to study L S M cathodes. They observed several impedance arcs which varied in size with po2- Several hypotheses were drawn regarding the nature of the arcs; however, Ostergard and Mogensen conceded that more work was necessary to prove these hypotheses. They used equivalent circuit fitting to model observed cathode processes. In their circuit model, they included a series inductance to represent artefacts originating in their experimental apparatus; however, they did not state the magnitude of this inductance. Siebert et al. [87] investigated L S M cathodes using EIS over a wide range of polarizations. A transition in cathodic polarization was observed which they attributed to the generation of oxygen vacancies in the LSM. At cathodic overpotentials higher 51 than the transition, slow polarization relaxations were observed which took several days to settle. Siebert et al. believed that these were due to a redox reaction involving Mn. They stated that L S M behaves similar to a storage battery when the cathodic polarization range is traversed. The impedance spectrum observed at high cathodic polarizations was composed of two distinct arcs. In both high and low polarization regions, a high frequency relaxation was consistently observed in EIS data. Van Herle et al. [88] used EIS to compare electrode performance of porous versus dense L S M cathodes. For the porous cathodes, they observed a single impedance arc over a range of conditions. The dense cathodes showed a Warburg diffusion line. 0stergard et al. [89] used EIS to investigate the electrode performance of LSM-YSZ composite bilayer cathodes (a catalytic layer and a current collector layer). EIS was used here to measure total polarization resistance. They estimated the inductance of the electrochemical apparatus to be between 0.5 and 1 uH. Juhl et al. [90] used EIS to study the effects of layer thickness on cathode performance in composite cathodes. At lower temperatures (700°C and 850°C), they could not resolve overlapping arcs. At 1000°C, two distinct arcs were visible. Hart et al. [91] studied electrode performance of functionally graded YSZ/LSM cathodes with LSC current collector layers. In their cathode symmetric cell tests, they observed three distinct impedance arcs; however, they stated that more work was required in order to correlate each arc to an electrochemical process. They measured the inductance of their apparatus to be 0.7 x 10"7 H. Godickemeier et al. [92] used GCI to compare the electrochemical performance of LSM and LSC porous cathodes on SDC and gadolina-doped ceria (GDC) electrolytes. They claimed that electrode overpotential is not directly accessible with the GCI technique for the case of a MIEC electrolyte (via a reference electrode). This is because the reference electrode is polarized even at open circuit conditions as a result of the steady state ionic/electronic currents. Jiang et al. [93] used GCI and EIS to study the effect of applied cathodic current treatment on cathode performance. They isolated the LSM/YSZ interface via a reference electrode and observed three distinct cathode impedance arcs. After current treatment with 250 mA/cm for 3 hours, GCI revealed a 50% reduction of cathode polarization. After several hours, the low frequency impedance arc disappeared, and the other two arcs became smaller. They attributed these observations to a passive layer on L S M grain surfaces that is rich in La or Sr relative to Mn. They thought that the passage of cathodic current removes the deleterious 52 layer. Jiang [94] further explored the effects of varying polarization and current density on cathode resistance. He used EIS to study L S M cathodes made by co-precipitation and solid state reaction methods. Jiang presumed that the microstructure and interface properties of the LSM cathode are constantly changing and evolving under SOFC operating conditions. Martin and Petric [95] further investigated cathodic current treatment on L S M cathodes. With the use of GCI, they observed a similar decrease in cathodic overpotential as reported by Jiang et al. They claimed that the cathode alteration caused by current treatment is reversible when the applied current is switched on and off at a period on the order of hundreds of hours (105 s). Smirnova et al. [64] interpreted EIS data from tubular SOFC electrodes using the previously described deconvolution technique. Time constant distribution peaks elucidated four distinct polarization processes. EIS was performed at several different temperatures. Polarization resistance and double layer capacitance were extracted from distribution peaks. Smirnova et al. did not assign known electrochemical processes to the observed peaks. Schichlein et al. [65] applied deconvolution to simulated EIS data representing the SOFC cathode-electrode interface. They simulated variation in poi- By deconvolution of their artificial frequency response, they were able to resolve a resultant change in one of the distribution peaks, thereby (artificially) resolving a P02 dependent process. However, they could not extract rate constants for the simulated oxygen reduction reaction. 4.5 Summary of Electrochemical Relaxation Times in SOFCs Table 2 summarizes the orders of magnitude of characteristic response times of electrochemical relaxation processes in SOFCs reported in the literature. Most of these figures were not given as process time constants; rather, they were stated as the peak frequencies of impedance arcs. Time constants have been deduced as the reciprocals of the peak frequencies. This approach provides a general order of magnitude of the response times that one might expect to observe in time transient data. 53 Table 2: Reported Electrochemical Relaxation Time Constants Anode Investigations Reference Relaxation Time (s) [Order of Magnitude] Hypothesized Process Details of Experiment Jiang, Badwal [76] IO"5 Anode charge transfer EIS: 1000°C, 2 % H 2 0 in H 2 , O C V IO"2 H 2 dissociation/adsorption Primdahl, Mogensen [77] 10"5 Anode charge transfer and ionic transport resistance, charged double layer behaviour EIS: 850°C to 1000°C, 3% H 2 0 in H 2 , O C V to 50 mV anode overpotential IO"2 N / A 10° Anode bulk gas concentration polarization Geyer et al. [78] IO"3 Anode charge transfer and ionic transport resistance EIS: Anode symmetric cell, 700°C to 1000°C, 2% to 90% H 2 0 in H 2 , O C V IO"1 Anode gas diffusion in flow field channels Primdahl, Mogensen [79] 10° Gas conversion impedance EIS: 1000°C, 3 % H 2 0 i n H 2 , O C V Primdahl, Mogensen [80] IO"3 N / A EIS: 1000°C, 3 % H 2 0 in H 2 , O C V 10"2 to 10"' Anode gas diffusion in 1mm layer outside of porous anode Bieberle, Gauckler [82] 10"2to 10"4 Anode charge transfer EIS: 400 to 900°C, 0.3% H 2 0 in H 2 / N 2 , O C V to 400 mV anode overpotential N / A (relatively slow) Desorption of H 2 0 at anode TPBs 54 Table 2, continued Cathode Investigations Reference Relaxation Time (s) [Order of Magnitude] Hypothesized Process Details of Experiment Hammouche et al. [84] 101 Cathode gas diffusion Current & potential steps: 960°C, 0 2 in Ar, Step amplitudes: AI = 400 uA, A V = 350 mV 102 Cathode microstructure change Ostergard and Mogensen [85] N / A (relatively fast) O2" transfer from surface to lattice EIS: 1000°C, cathode polarization: -300 mV (cathodic) to +50 mV (anodic), N / A (intermediate) O2 dissociation N / A (relatively slow) Surface diffusion Siebert et al. [87] io-4 N / A EIS: 960°C, P02 = 10"3atm, O C V N / A (several days) Mn reduction / oxide ion . vacancy formation DC test: application of high cathodic polarization Van Herle et al. [88] 10u 0 2 dissociation EIS: 700-900°C, O C V to 400 mV cathode overpotential, P02 = l-10"4atm Jiang et al. [93] 10"J N / A EIS: 800°C, O C V m2 N / A IO"1 Surface adsorption / dissociation on L S M passive layer Smirnova et al. [64] IO"4 N / A EIS: 950°C, OCV, interpreted via deconvolution 10"Jto IO - 1 N / A 10 uto 10"' N / A As can be seen in Table 2, observations of electrochemical relaxation transients vary substantially between different SOFC researchers. The reviewed transient experiments have been performed under different conditions. SOFC materials, manufacturing procedures, and experimental variables, such as temperature, Pm, Pmo, P02, and cell polarization are believed to be intricately related to cell performance. Lack of uniformity in the sample manufacturing procedures and testing conditions throughout the literature is likely the reason for the lack of consensus regarding the nature of the observed electrochemical responses. Also, some of the 55 described experiments discussed the use of reference electrode placement without details of the stringent placement requirements for accurate polarization resistance measurement (as described by Winkler et al. [22]). This might be a significant source of inconsistency amongst different research groups. Studies conducted at the Riso National Laboratory [23,77,79-81,86,89,90] have consistently reported on strategic placement of reference electrodes for accurate EIS measurements. 4.6 Examples of Applications to Other Electrochemical Systems 4.6.1 Introduction Transient techniques have been used to diagnose performance losses in other types of electrochemical systems. The various types of fuel cells range in their stages of development. Some types of batteries, such as the lead-acid cell, are highly developed and commercialized electrochemical systems for which extensive progression of transient diagnostic technology has been described in the literature. In this section, applications of transient techniques to MCFCs, PEFCs, and lead-acid batteries are reviewed. These three systems have been selected as examples in order to offer more insight into transient electrochemical behaviour than what is available in SOFC-related literature. Applications of transient techniques extend well beyond these fuel cells and batteries, but review of further applications has been excluded for conciseness. 4.6.2 Molten Carbonate Fuel Cells (MCFCs) Transient techniques have had a significant presence in M C F C research [36,55,96-101]. A thorough introduction to MCFCs has been presented by Larminie and Dicks [1]. As previously described, Kordesch and Marko [36] used half sine waves to measure the transient response of MCFCs. Nishina et al. [96] used EIS along with the potential step technique to study various M C F C anode materials. They examined a 100 microsecond window after perturbation. They mentioned that transient measurements might have been complicated 56 by overlap of the double layer discharge current along with the slow response of the potentiostat within this time window. Yuh and Selman [55,97] used GCI and EIS to study the electrochemical processes in MCFCs. They described the electrode reactions in MCFCs as being under mixed control. For this reason, they anticipated difficulties in the interpretation of time transients. In the first part of their work [97], they measured cell polarization under various temperatures and gas compositions. They performed GCI with a custom electronic switch with switching time of less than 2 microseconds. Qualitative observations were made regarding the transient responses. In the second part of their work [55], they presented time transient relaxations along with impedance spectra. They stated that the transient response to GCI offers a useful qualitative assessment of rate-limiting processes; however, they did not attempt to extract quantitative data from the relaxations. They claimed that a descriptive model needed to be developed before this could be achieved. Lagergren et al. [98] presented an analytical model of GCI in MCFCs. They modeled the electrodes as porous structures made of equally sized agglomerates (a detailed model description is given in the literature). By applying chosen boundary conditions anticipated in a GCI experiment and solving the mass and charge balances, potential-time transients were simulated. Lagergren et al. correlated the apparent double layer capacitance to the time rate of change immediately after interruption. They claimed that their modeling approach was not restricted to M C F C cathodes, but it was applicable to any porous electrode system. They varied several parameters in order to understand sensitivities of relaxation rates. EIS has had a significant role in the investigation of M C F C materials. For example, Zhu et al. [99] used EIS along with polarization studies to investigate corrosion rates of bipolar plate alloys in molten carbonate melts under anodic and cathodic conditions. Ganesan et al. [100] used EIS to evaluate the performance of an alternative M C F C cathode material under different gas compositions. Huang et al. [101] applied EIS and equivalent circuit fitting to compare NiO cathode behaviour at different durations after being submerged in a molten salt. Many more studies can be found in which EIS has been applied in the development of MCFC technologies. 57 4.6.3 Polymer Electrolyte Fuel Cells (PEFCs) Major work has been done in recent years using transient techniques to investigate performance of PEFCs [48]. A thorough introduction to PEFCs has been presented by Larminie and Dicks [1]. As previously discussed, Biichi et al. [48] used a high speed semiconductor switch circuit with an auxiliary power source to superimpose current steps onto the DC load of a PEFC. Btichi and Scherer [102] applied the CS technique to investigate the relationship between current density and membrane resistance in PEFCs. In this study, they superimposed a current pulse train with a 5% duty cycle onto the DC load. After each current pulse, ohmic resistance was measured. Mennola et al. [103] used GCI to measure ohmic losses in PEFCs as a diagnostic technique to identify parasitic surface reactions, poor assembly, and membrane poisoning. They interrupted current across individual cells as well as across the entire stack. Noise and voltage overshoot were major drawbacks in data interpretation. Mennola et al. investigated the reliability of their testing procedure by comparing the sum of ohmic losses of individual cells to the ohmic loss measured across the entire stack. These two figures were in relatively good agreement. They speculated the possibility that electrochemical relaxation processes may have been hidden in the overshoot period, resulting in systematic errors. Development of analytical models has led to contributions in PEFC transient response interpretation. Jaouen and Lindbergh [104] applied a spherical agglomerate model to simulate GCI and EIS experiments at the PEFC cathode, similar to the approach of Lagergren et al. [98]. For the GCI model, Jaouen and Lindbergh solved activation overpotential and O2 concentration relationships numerically using different sets of boundary conditions. Simulated potential transients were characterized by their slopes at different time periods after interruption. Jaouen and Lindbergh demonstrated that this type of characterization is effective for separating regimes within the time domain controlled by mixed diffusion and migration, diffusion only, and reaction kinetics only. Parameters were varied, including steady state current density, cathode thickness and P02. Influences of these parameters in the time domain were reported. Similarly, an EIS model was derived, in which relations for frequency dependent O2 concentration, current density, and potential were expressed and impedance was solved using numerical methods. Jauoen, Lindbergh, and Wiezell [105] performed GCI and EIS ' 58 experiments on PEFC cathodes in order to validate the described analytical models. Both the GCI and EIS models were fitted to experimental results using several presumed parameter values. Jaouen et al. remarked that the drawback of GCI is its lack of accurate data within the microsecond range after interruption. They found the EIS numerical model less cumbersome to solve than the GCI model. Schindele et al. [106] applied both EIS and GCI to study the influence of parasitic current ripple from power conditioning devices on PEFC efficiency and life. From EIS data, they realized an equivalent circuit in an attempt to understand whether the parasitic load fluctuations perturbed the cell within the linear (ohmic) control region. They applied periodic current interruption to individual cells in a 5 cell stack using 50% duty cycle pulses with a 2 s period. Qualitative observations revealed that after 28 days of continuous operation, distinctly different transient shapes were seen among different cells. Schindele et al. [107] later described two strategies for the measurement of in situ membrane resistance: 1. to use the intrusive 10 kHz current ripple from the power converter as a perturbation to measure voltage response (and thereby calculate membrane resistance via the instantaneous voltage response), and 2. to periodically switch the load and extract membrane resistance from steady state (relaxed) current and voltage values after switching. For the latter strategy, they did not propose a tactic for separation of non-ohmic resistance from the measured potential difference. 4.6.4 L e a d - A c i d B a t t e r i e s Transient techniques have provided useful insight in research and development of secondary batteries [50,108-114]. The lead-acid battery represents one of the most highly developed, commercialized, and well understood practical electrochemical power systems. As previously described, Tiedemann and Newman [50] applied potential steps to lead-acid batteries to measure double layer capacitance. Riietschi and Cahan [108] presented a novel galvanostatic control circuit for both GCI and DC loading in their study of battery support grid corrosion. With their circuit, they isolated ohmic-free electrode polarizations and they observed electrode potentials at which O 2 and H 2 gas formation occurred. Subsequently, Riietschi, Angstadt, and Cahan [109] used this circuit to investigate potential relaxations at the P b 0 2 electrode. They were able to identify different relaxation signatures for different PbC>2 59 morphologies. This led them to speculate on the free energy of the different structures. In the investigation of the O 2 evolution reaction, they identified four distinct relaxations from the transient data. They associated each with a specific electrochemical process. These relaxations 3 3 ranged in time scale from 10" to 10 s after perturbation. Ho et al. [110] further investigated the O2 evolution reaction at the P b 0 2 electrode using GCI and EIS techniques. Potential relaxation slopes were evaluated at various times after interruption. A unique mathematical approach was presented to calculate the apparent electrode capacitance based on these slopes. Ho et al. used EIS to validate findings from time transient experiments. In addition to studies of electrode kinetics and microstructure, time transient modeling and experimental validation have been very important in the evolution of lead-acid battery designs for hybrid and electric vehicles. Tiedemann and Newman [50] performed potential step experiments with lead-acid battery electrodes and applied their time transient mathematical relationships to estimate double layer capacitance. This approach served as a premise for further development in battery simulations. Gu et al. [Ill] presented a comprehensive numerical model encompassing electrochemical kinetics, fluid flow, ion transfer by convection, varying electrode porosity, state-of-charge cycling, property variability, and multidimensional configuration. This robust model utilized a computational fluid dynamics technique to solve a coupled set of nonlinear differential equations, producing a transient simulation of battery behaviour. The goal of this work was to predict charge and discharge performance as well as to identify performance constraints. The model yielded a range of predictions such as acid concentration profiles at various charging times and fluid velocity profiles. Experimental measurements showed close correlation to model results for several validation tests. This model was later expanded to account for liquid electrolyte displacement and capillary flow, oxygen gas transport, and non-isothermal conditions [112]. Kim and Hong [113] used a similar numerical model to simulate the transient response to rest periods and depolarization pulses during the charging of lead-acid batteries. Simulation results suggested that these strategies can help abate adverse heating and concentration gradients and can lead to faster charging times. Srinivasan et al. [114] used another mathematical model to elaborate on the processes occurring after GCI in a lead-acid battery. A numerical model of a battery consisting of a series and parallel combination of several lead-acid cells was constructed. GCI simulation yielded concentration profiles and potential-time responses of individual cells at different times after interruption. It was estimated that the sluggish reaction kinetics of the positive electrode were 60 the major limitation of the overall battery performance. Model results were experimentally validated by G C I ; they showed good correlation. 61 Chapter 5: Research Objectives 5.1 Motivation Poor reliability and durability remain crucial barriers that limit the widespread commercialization of SOFCs. In efforts to mitigate these barriers, the origins of degradation must first be explored using diagnostic techniques; only then can methods be devised to overcome these issues. As demonstrated in the literature review of existing diagnostic techniques, possibilities for higher-level characterization of SOFC performance exist. With a better insight into SOFC degradation, reliability and durability issues might be alleviated and SOFCs might play a more significant role in the enhancement of energy efficiency and the reduction of pollution and greenhouse gasses in the future. The general objective of this work has been to investigate the feasibility of time transient techniques for the diagnosis of SOFC degradation. 5.2 Summary of Key Points from the Literature Review The following key points identified in the literature review have inspired the development and examination of a specific time transient testing protocol: • EIS has been used extensively to investigate sources of polarization losses in SOFC research; this technique can assist in the development of an alternative testing method. • Interpretation of EIS data via equivalent circuit fitting is semi-empirical and also presents a source of ambiguity. • Interpretation of EIS data via deconvolution yields a time constant distribution, which might be a very valuable dataset for comparison purposes in the development of a time transient technique. This method is still in the development stage. • GCI has had a major role in SOFC research, but only for the quantification and removal of ohmic losses. Signal noise has been thought to be a major hindrance in the interpretation of non-ohmic relaxations from GCI data. • Capabilities of the potential and current step transient techniques for the characterization of transient responses in SOFCs other than the ohmic response have not yet been investigated. 62 • SOFC design materials, manufacturing techniques, and experimental conditions have not been consistent amongst different research groups; likely as a result, there is a general lack of agreement upon the specific nature and characteristics of electrochemical processes expected in a "well-working" (un-degraded) SOFC. A new testing technique should take this into account and it should be compatible with the different timescales of all reported SOFC transient behaviours. • Hardware developed for SOFC current step experiments has utilized active circuit components to inject or draw current. No hardware scheme has been reported that utilizes rapid switching of passive load resistance to perturb an SOFC for characterization purposes. • Several simulations of SOFC responses to load switching have assumed that an ideal current step is a good representation of a load switching operation. Thus, these models have neglected double layer charging/discharging currents accompanying load switching. 5.3 Requirements In order to play a major role in improving SOFC reliability and durability, a newly developed diagnostic technique should use lower-cost hardware, test durations should be shorter, and resolution should be equivalent to or higher than EIS. Furthermore, the technique should not significantly disrupt the steady state operation of the fuel cell (as GCI does) because significant digression from steady state invalidates any attempts to characterize the true in-situ performance of the fuel cell. 5.4 Scope of Contribution In an effort to meet these criteria, the concept of fuel cell perturbation via load resistance switching has been devised. This is a different approach than the reviewed techniques of current step and voltage step perturbation. At this point, interpretation of current and voltage transient responses to rapid load switching has not been used as a means to interpret an SOFCs performance characteristics. Although work in the literature has discussed the presence of discharge currents from capacitor-like behaviour in electrochemical systems 63 being a nuisance in time domain methods, other literature work has suggested that load switching can be well represented by an ideal current step. This presents some confusion that can be clarified by experimentation. If electrochemical performance data could be adequately interpreted from responses to load switching in a non-invasive manner, then this method could become a very cost effective and rapid strategy for diagnosing degradation in operating SOFCs. The first major endeavour of this work has been the development and experimentation of an algorithm for current and voltage transient recording. As another stage of this work, the strategy of spectroscopic interpretation of transient data has been explored as a means to interpret electrochemical data from time transient responses. The feasibility of this approach has been investigated via analysis of simulated transient data. The overall contribution of this work has been the presentation of concepts and preliminary feasibility investigations relating to a testing method that, if proved successful, might one day be used to diagnose degradation in operating SOFC stacks. In-depth model developments, SOFC characterization via transient analysis, and diagnosis of degradation have been left as future endeavours. 64 Chapter 6: Numerical Techniques for Exponential Analysis of Transients 6.1 Introduction Exponential decays are observed in a variety of disciplines, including physics, biology, and engineering. If the system under investigation is linear, the principle of superposition applies; a transient response to a change of state can be characterized as a sum of individual responses. Several different numerical techniques have been developed for the identification and analysis of multiple exponential response components. The most common techniques have been reviewed thoroughly by Istratov and Vyvenko [115]. Exponential analysis is generally divided into three categories: mono-exponential, multi-exponential, and spectroscopic exponential analysis (SEA) . Mono-exponential analysis assumes that a transient response/(0 is characterized by a single exponential decay with amplitude A and time constant z as follows: Multi-exponential analysis assumes that the transient is a sum of n exponential decays as follows: S E A , which is the most general case, assumes that the transient is composed of a continuous distribution of exponential decays whose weights are defined by the spectral distribution function g(r). The overall transient response is then described as follows: /(f) = ^-exp(--) (6.1) r / ( 0 = Z 4 - e x p ( - - ) (6.2) (6.3) The goal of S E A is to solve for g(r). The characteristics of the spectral function (peaks, spikes, etc.) can provide useful insight into the nature of the system being studied. 65 6.2 Application of Spectroscopic Exponential Analysis (SEA) to SOFCs Exponential analysis has never been used to the author's knowledge to investigate electrochemical transient behaviour in SOFCs. As discussed in Section 4.5, an SOFC transient response is known to include in most cases at least one relatively fast electrochemical kinetic response, as well as a relatively slow mass transport response. Furthermore, many authors have discussed the suitability of using constant phase elements (CPEs) to model impedance in SOFCs. Van Heuveln [62] has suggested that in many types of fuel cell and battery experiments, a voltage transient from a GCI experiment is spectroscopic in nature rather than consisting of discrete exponential responses. Based on this suggestion, mono- or multi-exponential analysis techniques would not be suitable methods to interpret an SOFCs voltage transient response; SEA would be more appropriate. In addition to transients related to electrochemical and mass transport phenomena, it is known that an experimental transient response can include non-negligible artefacts of the test system (from ohmic resistance, inductance, and capacitance in the electronic devices and connections). A transient analysis technique should be capable of isolating electrochemical from parasitic transient responses. 6.3 Principle of SEA In principle, g(r) is solved by taking the inverse Laplace transform of the transient data J[t) [116]. This involves integration in the complex plane, as follows: Here, c is a real constant and j = . However, since exponentials are not orthogonal functions along the real axis, the contribution of an exponential component in the complex plane cannot be known only from knowledge of data along the real axis as it can with (6.4) 66 sinusoidal base functions. In practice, Equation 6.4 cannot be solved. Alternatively, g(r) must be solved implicitly using Equation 6.3. This problem belongs to a general class known as Fredholm integral equations of the first kind. Such equations are ill-posed, meaning that the solution g(f) may not be unique, may not exist, or may not be continuously dependent on the experimental data. The error in a solution to an ill-posed problem is generally unbound. This nature of the problem has significant ramifications when noise is present in the experimental data; the addition of very small noise levels toJ[t) (the real axis image) can cause a significant perturbation in g(z). For this reason, the presence of noise in transient data determines the resolution capabilities and therefore the computational feasibility of SEA techniques. Noise is typically expressed as the signal to noise ratio (SNR), which is the maximum signal value divided by the standard deviation of the noise [117]. 6.4 Tikhonov Regularization Istratov and Vyvenko [115] reviewed and compared six common strategies for solving g(r) from the Fredholm integral equation of the first kind (Equation 6.3). The strategies that they covered include: the sampling method, correlation method, approximation by orthogonal functions, Fourier (Gardner) transform, Tikhonov regularization, and the method of maximum entropy. Of these six strategies, they concluded that Tikhonov regularization enables one to solve g(r) with the highest resolution for a given SNR. This is particularly important with regard to the goals of this work; as discussed in Chapter 5, it is desirable to perturb an SOFC by a relatively small amount in a transient experiment. A small perturbation implies a relatively low SNR. For this reason, the Tikhonov regularization method has been selected in this work. It is reviewed briefly here. A general regularization method for solving a Fredholm integral equation of the first kind was presented by Tikhonov [118]. In this strategy, an approximate solution for g(r) is found via the iterative minimization of the following functional Ma via an optimization algorithm: \K{t,T)-g(T)-dT-f(t) + a (6.5) 67 Here, Q is the regularizing (i.e. smoothing) functional and K is the exponential kernel. The double brackets indicate the norm. K takes the form: K = exp (6.6) V <• J The most common forms of the regularizing functional are: dt (6.7) and (6.8) In Equation 6.8, k andp are positive smoothing functions, a and b are constants. Many forms for Q. are possible for implementation of Equation 6.5. Some Tikhonov regularization algorithms try all possible forms of Q that are relevant to the problem. Many forms can be eliminated based on the specific constraints. The regularization parameter a determines the balance between accuracy and stability. The null value a = 0 gives a minimum Ma, but the solution might not be stable with respect to noise. A large value of a might be stable, but physical relevance of the solution might be lost. Several methods for the selection of a have been discussed by Davies [119]. 6.5 C O N T I N Several software applications have been developed to perform Tikhonov regularization. An open-source software package called CONTIN was developed by Provencher [120,121]. 68 This software package has been used in this work for the analysis of simulated and experimental SOFC transients. A very brief description of CONTIN is given here. CONTIN was designed to solve an abundance of problems including Fredholm integral equations of the first kind. It chooses the form of Q using a combination of three strategies: 1. absolute prior knowledge of constraints (for example, g(z) > 0), 2. statistical prior knowledge (for example, noise level), and 3. the principle of parsimony, which states that of all possible decisions, the one should be chosen that is simplest and reveals the least amount of detail that was not already expected. CONTIN regularizes the problem via a strategy that is similar to ordinary constrained weighted least-squares fit. Its computational algorithm is complex and further details are beyond the scope of this study. Such details can be found in the literature [120,121]. CONTIN outputs a series of solutions which vary between a small a (artefacts are present) and a large a (excessively smoothed/biased solution), as well as a chosen solution which is a compromise between these two extremes. Provencher has advised users to not blindly accept the chosen solution. Alternatively, all possible solutions should be examined and a solution should be accepted based on the type of problem being solved and the user's experience gained through simulations. In addition to the solution g(r), CONTIN also performs a numerical integration of the solution via a quadrature approximation. It identifies individual peaks in the solution and computes each peak's moment (area under the peak). This integral value is analogous to the discrete amplitude of a single exponential decay component. As a consequence of CONTIN's ability to handle noisy data via regularization, a true Dirac delta function is smoothed into a distributed peak in the solution. This limits CONTIN's resolving capabilities. However, this is not necessarily a hindrance in transient analysis; resolution limits can be understood with the use of noise-free simulated data. Irresolvable regions of the solution space can be identified. Furthermore, the limiting separation between two Dirac delta functions can be identified at which two peaks can no longer be resolved. 69 Chapter 7: Experimental Procedure 7.1 Introduction A series of experiments were conducted to explore the presented concepts in this work. These include the following main components: 1. characterization of a test SOFC via microscopy, DC polarization analysis, and EIS, 2. design, construction, and characterization of a custom electrical circuit for load switching and transient recording, and 3. processing and interpretation of simulated transient signals via Laplace inversion using the software CONTIN. This chapter describes the procedures, specifications, and rationale regarding time transient experiments and data interpretation strategies. 7.2 Test Cell A circular "button" SOFC was supplied by X. Zhang of the National Research Council of Canada - Institute for Fuel Cell Innovation (NRC-IFCI) for testing purposes. Figure 8 and Figure 9 are SEM micrographs showing a cross sectional fracture surface of the test cell taken after all other tests were performed. 70 Figure 8: Full Cross Section of Test Cell (100 x Magnification) Figure 9: Cross Section of Test Cell (1000 x Magnification) The cell was composed of the four functional layers labelled in Figure 9. Thicknesses of these layers are listed in Table 3. These thicknesses were measured with a ruler from the micrograph. 71 Table 3: Measured Thicknesses of Functional Test Cell Layers Anode Support Layer Anode Functional Layer Electrolyte Cathode 690 ± 10 nm 11 ± 2 urn 21 ± 1 um 8 ± 2 um The Ni/YSZ cermet anode support layer was manufactured via tape casting. All other layers were screen-printed on top of the support layer. The anode functional layer was also made of Ni/YSZ cermet, but with smaller particles than the support layer, allowing for a larger volumetric density of reaction sites. The electrolyte was made of dense YSZ. All YSZ was of the form (Y2O3).08(ZrC»2).92 (denoted as 8YSZ). The anode bilayer and electrolyte were co-sintered at 1400°C for 3 hours. The cathode was composed of a 4:1 weight ratio mixture of Lao.6Sro.4Coo.8Feo.2O3 (LSCF) to Smo.2Ceo.80i.9 (SDC). This layer was screen printed on the electrolyte and sintered at 1100°C for 2 hours. The overall cell diameter (anode support layer, anode functional layer, and electrolyte) was approximately 17 mm. The cathode covered a circular surface area of 0.47 ± .05 cm2. This area was centered on the electrolyte surface. It has been used as a normalizing factor in the computation of current density, power density, and area-specific resistances (the cathode was assumed to be completely electrochemically active). 7.3 Cell Test Station An Amel Inc. fuel cell test station was used to support the cell, supply reactant gasses, and control cell temperature during all fuel cell tests. This station consisted of a spring-loaded cell support fixture shown in Figure 10, enclosed in a furnace, with a mass flow control module for delivery and monitoring of gasses. Flow rates and furnace temperature were controlled via a PC using Amel RealCommand™ software. The cell support fixture applied a contact force between Pt mesh current collectors and cell electrodes. Ceramabond™ ceramic adhesive was applied around the perimeter of the test cell to prevent gas leakage between the anode and cathode environments. A small amount of cathode slurry containing a 9:1 weight ratio mixture of LSCF to SDC was applied between the cathode and current collector to reduce electrical contact resistance. Voltage and current signals were measured with a four-probe configuration (four electrical connections shown in Figure 10) consisting of two Pt signal leads per electrode; one for voltage measurement and the other for current measurement. The two wires were 72 welded together at the Pt mesh on each side. This configuration allowed for simultaneous high-accuracy voltage and current measurements. Ceranmbond ™ Pt Mesh • Outer Tube • Inner Tube z Electrolyte Anode Electrical Connections F i g u r e 10: C u t a w a y V i e w o f C e l l S u p p o r t F i x t u r e ( n o t d r a w n to s ca le ) 7.4 C e l l T e s t C o n d i t i o n s D C polarization and EIS experiments were performed at temperatures between 600°C and 900°C. Cel l temperature was monitored via a thermocouple in close proximity to the cell. Temperature fluctuations within approximately ± 1°C of the mean value were observed during tests. EIS and time transient tests were performed at power generating conditions. EIS was also performed at O C V for comparison purposes. For experimental consistency, it was desirable to adopt a selection criterion for choosing the operating load (current density and voltage). This would serve as a homogeneous relative operating point for experiments at different temperatures at which significantly different peak power conditions would be expected. The operating load was chosen to be the load at which the current density equals 80% of its value at maximum power density. This ensured that cell performance would be characterized at a load that is close to the maximum power generating capabilities of the cell, but without excessive hindrance from mass transport losses. Prior to transient tests, the test cell was maintained at the test temperature and operating load for at least 20 minutes prior to testing to allow for conditions to stabilize. 73 Supply gasses were delivered at a mass flow rate of 40 seem to each side of the cell. Air was used as a reactant at the cathode. A mixture of 50%moi H2 in N2 was supplied at the anode. 7.5 EIS and D C Polarization Experiments EIS and DC polarization experiments were performed using a Zahner IM6 electrochemical workstation. EIS was performed in potentiostatic mode with a perturbation signal amplitude of 20 mV and a frequency range of 0.1 Hz to 5 MHz. At each temperature, EIS was performed first at OCV. Next, DC polarization was performed using a voltage sweep at a rate of 10 mV/s. Operating load was determined calculated from this data. EIS was then performed at operating load. 7.6 Measurement System for Transient Recording Transient signals were recorded with a National Instruments Inc. PCI-6115 DAQ multifunction input/output device (denoted PCI-6115). This device was chosen for its ability to record multiple analog signals at sample rates of up to 10 mega-samples per second (MS/s) and to concurrently deliver a digital output signal that can be used for simultaneous gating of an analog switch and self-triggering of data acquisition (DAQ). The rise time of this signal was inspected via transient recording and it was seen to be on an order less than hundreds of nanoseconds (i.e. faster than the maximum resolution of the transient recording device). The PCI-6115 was equipped with 50 kHz and 500 kHz 3-pole Bessel low pass filters which were used in investigations of noise. Figure 11 shows a schematic representation of the transient measurement system. Here the SOFC is shown mounted inside the furnace of the Amel test station and connected to the load switching circuit via coaxial cables. Each coaxial cable running from the Amel test station to the load switching circuit was one foot in length. Short cables helped to minimize the inductance of the fuel cell current loop. The voltage and current signals were connected to the PCI-6115 where they were converted to digital signals via a 12-bit analog-to-digital converter (ADC) within the PCI-6115. For all transient recordings, the analog current and voltage signal input ranges were set to -500 mA to +500 mA and -1 V to +1 V, respectively. These ranges 74 provided the highest signal resolution for the given signal levels when both were referenced to a ground at the S O F C s anode. Based on the ranges and A D C precision, the signal precision was 244 uA and 488 u V for the current and voltage signals, respectively. A l l signals were saved on a P C for later analysis. The analog switch also required an input voltage, highest and lowest voltage references, and a ground reference, which are not shown in Figure 11. PC SOFC Figure 11: Schematic Diagram of Measurement System for Transient Recording 7.7 Load Switching Circuit The circuit shown in Figure 12 was designed, constructed, and used in all time transient experiments in this work. 75 Vo l tage S i gna l —*>• SOFC Anode (-) Cathode (+) A n a l o g S w i t c h . P a t h A P a r h B S w i t c h Gate, ; S i gna l Figure 12: Schematic Diagram of Load Switching Circuit The purpose of this device was to rapidly perturb the fuel cell via quickly switching the electrical resistance connected between the fuel cell 's terminals from Path A (R + R s ) to Path B , a higher value (R + R s + dR). The two signals of interest were: 1. the voltage signal across the terminals of the fuel cell, and 2 . the current signal through the fuel cell. A s mentioned, the voltage signal was measured via independent leads terminating at the S O F C s electrodes (see Figure 10). The current signal was calculated via Ohm's law based on the voltage across the current sense resistor R s . Other types of current sensors were considered, but none were found with suitable power handling capabilities or enhanced dynamic ranges. A n important requirement of the circuit design was the ability to tune the current and voltage to operating load. This was achieved with the use of a potentiometer. It can be understood from fuel cell polarization curves that an increase in load resistance causes the signals to shift closer to their open circuit values by an amount that depends on the magnitude of dR. Driving the load in this direction was desired; the alternative would induce excessive mass transport losses as well as undesirable degradation from extreme current cycling. This was thought to cause more harm to the cell, and thus, could not be applied as a non-invasive cell perturbation mode. A n important feature of the circuit was that current interruption could be performed simply by disconnecting dR from the current path. In this manner, the circuit was used for G C I 76 experiments at various temperatures. This allowed for the comparison of transient signals acquired from load switching with various dR values and from GCI. A prototype of the load switching circuit was constructed using common electronic components. For R, a 5 ohm wirewound potentiometer was used. Various dR values were used to obtain different extents of load perturbation. These values included 0.05, 0.10, 0.25, 0.50, 0.75, 1.0, 2.0, 3.0, 5.0, and 10 ohms. An R s value of 1 ohm was used for simplicity. All resistors were 0.25 W carbon film type. All selected components had sufficient power ratings for the anticipated fuel cell load. Two Analog Devices ADG819 multiplexer/single pole double throw CMOS switches were combined in parallel to switch the load resistance. Various other analog switch models were considered. The chosen model was selected for its low on-resistance of 0.5 ohms and its high maximum continuous current of 200 raA. The use of two analog switches in parallel allowed for twice the rated current of a single switch, which offered protection from current spikes that might accompany load perturbations. The penalty for the parallel configuration was that the equivalent switch on-capacitance doubled to make a total of 600 pF. The rated on and off times of the ADG819 were 45 and 16 nanoseconds, respectively. The gating signal from the PCI-6115 was branched to the gates of the parallel analog switches to control them in a synchronous fashion. Figure 13 shows the physical construction of the circuit. The circuit was built on a solderless prototyping breadboard which was fastened to the inside of a metal box. Since the ADG819 switches were not available in pin-mount packaging, they were surface mounted (with multipurpose glue) to a rectangular piece of PC board. Lead wires were soldered to all essential switch connections. These wires were routed to various locations on the breadboard. BNC jacks were used to couple the coaxial cables from the fuel cell current loop to the load switching circuit. Also, a BNC jack was used to couple the current signal leads to a coaxial cable running to the PCI-6115. All of these details, along with the other features of the circuit construction, can be seen in Figure 13. 77 A software application was developed using Lab V I E W for the synchronization of transient recording and analog switch gating. This program combined features of two examples found in the National Instruments example library; one for analog recording and the other for counter pulse generation. 7.8 Characterization of Signal Noise and Interference Noise was thought to be a major encumbrance in the analysis of fuel cell transient data. For this reason, S O F C signal noise and other sources of dynamic interference were examined 78 at operating load at 800°C. Noise in steady state signals was examined via transient recording with appropriate analog filters enabled. This strategy ensured adequate attenuation of any possible periodic noise constituents at frequencies larger than the Nyquist frequency (i.e. half of the sample frequency). Sampled noise was then transformed to the frequency domain via Matlab's fft( ) function and inspected for periodic characteristics. The possibility of dynamic interference of transient signals from induced currents originating from the furnace heating coils was studied. G C I experiments were performed at 800°C first with the furnace heating elements periodically on, and then with the elements completely off. Also , G C I was performed using different cable lengths running from the Amel test station to the load switching circuit. This allowed for the signal ringing associated with current loop inductance to be examined on a qualitative basis. 7 . 9 Algorithm for Transient Recording In all transient recordings, the fuel cell 's current and voltage signals were acquired at a common sample rate and 1000 pre-trigger data points were captured prior to perturbation. This allowed for investigation of steady state signal characteristics immediately prior to perturbation. Both current and voltage transient signals were analyzed qualitatively and their instantaneous responses were interpreted to provide an estimation of ohmic resistance. Because decays of electrochemical overpotentials were sought, voltage signals were analyzed in more detail than current signals; thus, the algorithm developed for transient recording was more steered towards the acquisition of voltage signals. The following algorithm was conceived to ensure that all relevant transient data were captured. First, an excessively large initial window (in time) was guessed during which transient data would be recorded. In almost all cases, 20 s was sufficient. Transient data were captured for this duration at a sample rate that was sufficient to interpret the shape of the transient. In all cases, 1000 S/s was used. The voltage transient was then inspected on the P C screen. If the transient appeared to settle within the initial window, a smaller critical window was chosen by inspection that encompassed the entire transient plus a small duration of the settled signal. This additional portion ensured that the entire transient was encompassed within the critical window. The signal was assumed have settled when no further changes in the mean 79 signal value could be resolved by inspection on the PC screen (with default scaling of 0 to 1 V). If a settling time could not be seen in the initial window, the process was repeated with a larger initial window size. Once the critical window had been established, sampling was performed at the critical sample rate, which was a value slightly higher than the frequency at the high frequency intercept in the corresponding impedance plot at operating load. Recording at an excessive sample rate was thought to present the problem of excessively large data files and slow PC operations. These proposed strategies were thought of as a way to include maximum time domain electrochemical data in a practical manner; however, their validity requires further examination. After recording of the critical window, another transient was recorded at 10 MS/s (the upper limit of the PCI-6115) for 10 milliseconds for the purpose of characterizing the ohmic response of the test cell. 7.10 Procedure, Terminology, and Conventions in the Use of C O N T I N In this work, CONTIN was used for Laplace inversion of various simulated transient responses. Several essential input parameters were used in all runs of CONTIN to define the Laplace inversion problem stated in Equation 6.3. These can be seen in the Appendix. Some other input parameters have been changed from one run to the next, such as the number of data points, the solution domain limits, or in some cases, the specific regularization parameter. The procedure for using CONTIN to solve Laplace inversions can be found in the User's Manual [122]. A sample input list is shown in the Appendix. In all inquiries, the term sample window has been used to describe the portion of the simulated transient response that has been strategically sampled and used in CONTIN. A sample window specification of the form a to b indicates that data have been examined from time a to b at a fixed sample rate per decade, starting with the most highly resolved time decade. In most inquiries, only one decade has been examined, in which case the sample window has contained data at equally spaced time intervals. The solution domain refers to the bounds of the discretized time constant distribution, solved by CONTIN. The term moment has been adopted here to describe the area under a peak in the solution domain (calculated by CONTIN using a quadrature algorithm). As input, CONTIN requires a definition of the solution domain bounds and the number of solution grid 80 points. In this work, the solution domain has been specified in the form 0 to c. The 0 starting point has been stated for simplicity, but the grid actually begins at one grid division (dx) in order to avoid a divide-by-zero error in CONTIN's computation. Istratov and Vyvenko [115] mentioned that if the solution domain is not known a priori, one can use a zooming technique in which the domain is first assumed to be excessively large, and then it is reduced to an adequate size via zooming in on the segment of the domain where a solution can be seen. The discretized solution is linearly spaced. Thus, in order to "see" the slower features in the solution domain, the fast features need to be condensed into the boundary of the domain. Furthermore, in order to "see" the faster features, the slower features must be excluded from the domain. Provencher [122] has warned that boundary conditions can affect a few grid points at each end of the solution domain and cause erroneous interpretation. To understand capabilities and limitations of CONTIN, a series of experiments have been conducted using simulated transient data. These have included comparison of sample window resolutions, selectivity of transient features via strategic inclusion or exclusion of transient data, and sensitivity of time constant distribution features to solution domain and sample window bounds. Provencher has recommended several settings for using CONTIN to solve Laplace inversion problems. He has suggested that up to 200 grid points be used in the solution domain, and he has warned that using more than 500 grid points causes numerical instability. Based on these recommendations, 200 grid points have been adopted for the solution in many inquiries. In some of the sensitivity studies, this value has been slightly adjusted in order to maintain a constant solution domain resolution. Nevertheless, it has never exceeded 300. As previously discussed, Provencher [122] warned that the chosen solution should not be blindly accepted. For inverse Laplace problems, he has advised selection of the solution with the lowest a value that has the same number of peaks as the chosen solution. This criterion has been adopted in this work. All further details regarding procedures of Laplace inversion using CONTIN are presented in Chapter 9. 81 Chapter 8: Results and Discussion: SOFC Performance, Evaluation of Load Switching 8.1 Introduction This chapter presents the experimental characterization of the test cell via D C polarization and EIS, the findings regarding load switching circuit limitations, the predictions of transient behaviour using a simplified model, and finally, the analysis of actual experimental S O F C transient responses to load switching. Characterization of the test cell and the method of perturbation have provided important implications regarding the feasibility of this technique. A discussion of the strengths and shortcomings of this transient technique has been provided here. 8.2 Polarization and Power Density Figure 14 shows the polarization behaviour of the test cell in the temperature range from 600°C to 900°C. 1.000 0.900 0.800 0.700 0.600 1° 0.500 0.400 0.300 0.200 0.100 0.000 0 100 200 300 400 -2 YV"" """""•••v... \ \ \ \ \ C--.. 800 \ • • • - : ; ) • "V" \ _ J ' S X \ 1 y \ / 700°C \ \ 1 '% ^ ~<! ^ t C. /T f 600°C j '\ Voltage er-Density Pow 300 250 200 5 0 0 500 600 150 £• V 100 S o C H Current Density / mA'cm" Figure 14: Polarization and Power Density of Test Cell at Various Temperatures The polarization curves (current density versus voltage) show an activation region near O C V , inferred from the concave slope. This region was more pronounced at 600°C and 700°C. A t 82 higher temperatures, this region was harder to distinguish, but close inspection showed concavity. A region of mass transport control was seen at high current densities, as indicated by the apparent convergence of polarization curves to a limiting current density. This region was less pronounced at 600°C. None of the polarization curves showed a distinguishable region exclusively controlled by ohmic behaviour. For this reason, ohmic resistance could not be interpreted graphically. Table 4 lists relevant properties obtained from D C polarization behaviour. Table 4: Cell Performance Properties Obtained from DC Polarization Analysis Current Density at Voltage at Temperature OCV Max. Power Density Operating Load Operating Load (°C) (V) (m\V/cm2) (mA/cm2) (V) 600 .993 50 96 .502 700 .986 127 200 .581 800 .968 253 335 .708 900 .932 275 342 .736 Based on the maximum power densities, it can be seen that the cell performed relatively well at temperatures of 800°C and higher. The current and voltage at operating load listed in Table 4 were used in EIS and load switching experiments. 8.3 Electrochemical Impedance Spectroscopy Figure 15 and Figure 16 are complex impedance plots showing the area-specific impedance response of the test cell at O C V and at operating load, respectively, for temperatures ranging from 600 - 900°C. In accordance with the convention typically used in electrochemistry, the negative of the imaginary part of the impedance has been plotted on the positive vertical axis in each complex impedance plot. 83 6 0 1 2 3 4 5 6 7 8 Real Impedance / ohm»cm 2 Figure 15: Complex Impedance Response of Test Cell - Open Circuit Voltage 3.0 Real Impedance / ohm*cm Figure 16: Complex Impedance Response of Test Cell - Operating Load A relatively high frequency arc can be seen in both of these plots at all temperatures. This arc showed significant thermal activation, as indicated by its significant reduction in size with increasing temperature. A t O C V , another thermally activated arc was seen at lower frequencies 8 4 at 600°C and 700°C. This arc was not well distinguishable at the two higher temperatures; only a main arc was seen which appeared to be an amalgamation of the two arcs. At operating load, a low frequency feature was distinguishable at all temperatures. Impedance spectra measured at 800°C have been examined in greater detail. These spectra are shown in Figure 17. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R e a l I m p e d a n c e / o h m > c m 2 Figure 17: Complex Impedance Response of Test Cell - OCV and Operating Load at 800°C Some important properties of these spectra are listed in Table 5. R0hm, FHFU and Fp denote the area-specific ohmic resistance, the frequency at the high frequency axis intercept, and the frequency at the peak of the main arc, respectively. Rp is the area-specific polarization resistance of the main arc, inferred via extrapolation of the arc to the real axis near its low frequency end using an approximate fit with a semicircle. Therefore, the Rp values given here are representative of the amalgamation of possibly several individual polarization processes. Table 5: Properties of Impedance Spectra at 800°C Rohm (ohnvcm2) FHFI (kHz) FP (Hz) RP (ohm'cm2) OCV 0.208 23.3 67.7 0.641 Operating Load 0.213 19.8 169 0.262 85 In some of the reviewed literature (summarized in Table 2), it has been suggested in both anode and cathode studies that rate-limiting electrochemical processes occur at peak frequencies in the vicinity of those seen here. However, a precise correlation to processes suggested in the literature has not been attempted due to experimental inconsistencies discussed in Section 4.5. Nevertheless, it is expected that at both operating states the observed main arc consisted of overlapping electrochemical activation processes from both the anode and cathode. Considering that the S O F C cathode reaction has higher activation energy than the anode reaction with H2 as the fuel, the main arc was likely most influenced by cathode electrochemical kinetics. Based on suggestions of mass transport relaxation behaviour discussed in the literature, it is likely that the low frequency feature seen at operating load was a result of a mass transport limitation. Its absence at O C V was in agreement with the findings from D C polarization studies. In comparing the impedance spectra at the two operating conditions, an improvement in the kinetics was seen at operating load (Figure 17). The main arc at O C V showed a larger, more drawn-out plateau shape in its peak region, likely resulting from suggested amalgamation of two thermally-activated processes. In contrast, the main arc at operating load was smaller and it had a more circular appearance at its peak region, suggesting that the lower-frequency rate-limiting process at O C V was not detectable at operating load. The origins of the impedance arc features might have been clarified to a greater degree through the estimation of activation energies via equivalent circuit fitting. However, only a qualitative interpretation of these impedance features was considered necessary for this introductory investigation. These differences in the electrochemical kinetic responses have implications in the interpretation of G C I data. If similar electrochemical responses were found at the two states, one might expect the decay of the kinetic overpotential accompanying current interruption to reflect dominant electrochemical processes at operating load or any other load between the two states. In that case, a spectroscopic analysis of the activation overpotential decays resulting from current interruption might be an effective way to characterize the dominant processes at the operating load. However, because of the observed differences, one would expect the actual voltage decay to be an artefact of changing dominant electrochemical processes. If one is interested in the dominant electrochemical process at operating load, current interruption would not result in the same behaviour as a minimally invasive perturbation, as the cell voltage 86 settles in a regime in which the rate-limiting electrochemical process shows different behaviour and results in greater power losses than at operating load. This argument also applies to other types of relatively invasive perturbations such as large current steps or sinusoidal perturbations with excessively large amplitudes. 8.4 Dynamic Characterization of the Load Switching Circuit Imperfections in the load switching circuit including inductance and capacitance of electronic components limit the dynamic range in which interpretation of electrochemical transient responses should be attempted. This dynamic range was investigated via frequency response analysis of the impedance across the circuit's current and voltage terminals with the fuel cell disconnected. The frequency response was measured with the switch in position of Path A (Figure 12). It was speculated that non-ideal characteristics of the potentiometer might have had a significant influence on the dynamic range of the load switching circuit. Accordingly, R was tuned to two different values for the circuit's dynamic characterization, namely 2.5 and 5 ohms (half and all of its maximum resistance, respectively). In load switching experiments performed after this circuit characterization, it was found that R needed to be tuned to approximately 3.5 ohms in order to maintain the fuel cell at its operating load. Thus, the range of R values used in circuit characterization encompasses the value used for load switching tests. Figure 18 and Figure 19 show the magnitude and phase angle of the circuit's complex impedance response, respectively. 87 6 0 5 0 ° 4 0 o C 3 0 £ _ 2 0 1 0 0 — R = 5 ohms — R = 2 . 5 ohms 1 . E + 0 0 l . E + 0 1 l . E + 0 2 l . E + 0 3 l . E + 0 4 l . E + 0 5 l . E + 0 6 l . E + 0 7 F r e q u e n c y / H z Figure 18: Dynamic Response of the Load Switching Circuit - Magnitude of Impedance 1 0 0 8 0 Z 6 0 ^ 4 0 • R = 5 o l u n s R = 2 . 5 o h m s F r e q u e n c y 7 H z fc I . E f O O l . E + 0 1 l . E + 0 2 l . E + 0 3 l . E f 0 4 l . E + 8 5 / l . E J t - 0 6 l . E t f - 0 7 - 2 0 - 4 0 - 6 0 Figure 19: Dynamic Response of the Load Switching Circuit - Phase Angle of Impedance In Figure 18 it can be seen that the impedance was consistent with the DC resistance measured with a multimeter up to a frequency of approximately 105 Hz. An interesting observation is that for both R values, the magnitude of impedance descended to a common value of approximately 1.8 ohms at 250 kHz. Figure 19 shows that the voltage and current were approximately in phase up to a frequency of about 104 Hz. A transition can be seen at 33 kHz at which the phase angle diverged from zero by a few degrees. This was more pronounced with R = 2.5 ohms. At 88 frequencies greater than IO 5 the phase angle diverged significantly. Similar to the magnitude diagram, a transition point can be seen at 250 k H z in the phase angle diagram. These findings are likely a result of capacitance across the resistors as well as inductance in conductive pathways. The observation at 250 k H z appears to be a transition between capacitance effects (short circuiting across resistors) and inductance effects (open circuiting across conductive paths). These observations suggest that the fast electrochemical processes with characteristic response frequencies equal to or greater than approximately 33 k H z would be difficult to interpret in the time domain with the use of this load switching device, due to interference from parasitics. F H F I values were less than this upper limit; however, the close proximity of the upper limits of electrochemical and parasitic responses suggests that differentiation of these relaxations via a spectroscopic technique might be very cumbersome or impossible, especially in the presence of signal noise. Critical transient windows at 800°C were sampled at a rate of 50 kS/s, which is likely a fast enough rate to capture a large portion of the circuit's parasitic transient response. Before hypotheses can be drawn regarding the nature of electrochemical responses to load switching, the circuit response should be characterized and identified in the rapidly decaying portion of the transient data. This can be done by measuring the response of the load switching circuit to a rapid square wave. 8.5 Prediction of an S O F C s Transient Response to Load Switching A basic simulation was performed to predict the current and voltage transient responses to load switching on a qualitative basis. A s discussed in Section 3.3.1, models of the transient response of an S O F C to load switching were presented in the literature, assuming that an ideal current step was the result of load switching. The objective of this simulation was to predict whether load switching would yield an ideal current step, or whether the discharge of capacitor-like interfaces in the test cell might yield a non-negligible current transient response to accompany the voltage response. A simple equivalent circuit model was generated using PowerSIM™ simulation software to crudely represent the test cell at 800°C at operating load. The load switching circuit 89 was also modeled and some of the system imperfections were included. The equivalent circuit model is shown in Figure 20 along with its parameter values. V o l t a g e : C I C 2 1 . 0 6 e - 2 F 0 . 5 F R p i : II R p 2 ^AAAH OCV ; ; ; : R o h m i -(J2h— ^AAAr-9 6 8 V . 4 5 3 o h m s ! . 5 6 o h m s 5 o h m s L l e - 6 H • : : : : : ( S O F C M O D E L ) i • : • • C u r r e n t : : . ; : . ; ; : : : : ; : : : . ; : . : : : : ; : : : : ; . : : . : ; : : : : . : : • : . ( S W I T C H : l ) ( S W I T C H 2) 1 k R s : : 1 o h m ; R •3 . 5 o h m s mar :f m R o n = 0 . 5 o h m s C o n = 6 0 0 p F : P a t h A d R 0 . 5 o h m s Path B Figure 20: Equivalent Circuit Model of Test Cell and Load Switching Circuit The rudimentary S O F C power source model was conceived based on a model used in the literature. M i n h and Takahashi [123] used an equivalent circuit schematic representation of an operating S O F C to help explain conductive pathways in M I E C materials. This representation consisted of a parallel combination of: 1. an ideal voltage source in series with an ionic resistance, and 2. an electronic resistance. This model was adapted here with two distinct modifications. First, the electronic resistance in the electrolyte was assumed infinite, reducing the configuration to an ideal voltage source in series with an ionic resistance (R0hm)-This was done because Y S Z is known to have an electronic transference number close to zero. B y representing the cell voltage as an ideal voltage source, an unsteady Nernst potential was neglected. Second, two R C pairs were placed in series with the power source to abstractly represent the activation and concentration limitations seen in the impedance spectrum. Little information was obtained about the presumed concentration limitation due to lack of frequency response data below 0.1 H z . For this reason, the R C pair representing a concentration limitation was purely hypothetical, representing an arbitrary slow transient response. 90 The load switching circuit was also incorporated into the model using some of the features of PowerSIM™. Each analog switch was modeled as a parallel combination of two M O S F E T s , each with opposite gating logic. Gating was specified so that at a certain instance, a normally-on M O S F E T connected to Path A turned off and a normally-off M O S F E T connected to Path B turned on. Each M O S F E T was given an on-resistance and capacitance equal to the manufacturer's specifications for the A D G 8 1 9 . Also included in the switching circuit model were the resistances R, R s , and dR. Furthermore, an inductance L was added to the overall circuit. Inductance was not measured in the actual fuel cell current loop, so a value reported in the literature for an S O F C test system was assumed [89]. Table 6 summarizes all model properties and the sources or justifications by which their values were selected. Table 6: Parameters Used in the Equivalent Circuit Model of the Test Cell and Load Switching Circuit Model Parameter Source of Model Parameter Value O C V Measured O C V of the test cell Rohm Measured ohmic resistance from EIS (with cell area = 0.47 cm 2 ) R p l Polarizat ion resistance estimated v ia extrapolation o f main impedance arc to the real axis with a semicircular fit CI Approximat ion assuming that the main impedance arc in the impedance spectrum was an ideal semicircle; C I = 1/ ( R p l • Fp) where Fp is the peak frequency o f the main arc R p 2 Gross approximation v ia semicircular fit o f the low frequency arc segment in the impedance spectrum C 2 Arbitrary value; unknown due to lack o f EIS data below 0.1 H z . Chosen so that Rp2 • C 2 = 2.5 s Rs Consistent with Rs used in the load switching circuit (which was arbitrarily chosen) dR Arbitrary value R Consistent with R used in the load switching circuit S W I T C H 1 / S W I T C H 2 On-resistance and on-capacitance consistent with reported value for the A D G 8 1 9 analog switch used in the actual load switching circuit (Ron = 0.5 ohms, C o n = 600 pF total for both switches) L Value reported by Ostergard et al. [89] It can be seen in Table 6 that some of these model parameter values were vaguely approximated or were chosen in a completely arbitrary fashion. Furthermore, the discussed 91 features that were thought to have limited the circuit's dynamic range were only partially represented in the model. For example, a capacitor was not included in parallel to the potentiometer. Despite the mentioned shortcomings of this equivalent circuit representation, the model was thought to have represented the fundamental system characteristics. In light of the lack of confidence in some of the quantitative parameters, only qualitative observations of the simulated transient response were considered in the scope of this work. A more accurate model development has been left as a future endeavour. Figure 21 shows the simulated transient response of the equivalent circuit model to load switching using the parameters listed in Table 6. 0.46 0.44 > 0.42 > 0.40 0.38 0.36 Voltage Transient Current Transient 4 6 Time / s 0.16 0.14 0.12 < 0.10 ^ 0.08 0.06 10 Figure 21: Transient Response of the Equivalent Circuit Model Rather than inducing a step response, both of the voltage and current waveforms were seen to relax to settling values after the initial ohmic step. The transient simulation was repeated first with the inductor removed from the model, and then with the switch capacitances removed. These modifications had no apparent effect on waveforms. These simulated transients have helped to clarify important aspects of interpretation of transient responses to load switching: First, response of the idealized SOFC model to load switching is neither a current step nor a voltage step. Second, the relaxation of current to a higher value after the ohmic step is neither 92 caused by inductance in the current loop nor by parasitic capacitance of the switch. Rather, it is a result of the discharge of the capacitors in the Randies cells. 8.6 Transient Signals Acquired in Load Switching Experiments Voltage and current transient responses to load switching were acquired at 800°C within a critical window of 10 s for all the tested dR values as well as for the case of current interruption. Responses are shown in Figure 22 and Figure 23. Time / s Figure 22: Voltage Transients of Various Step Sizes 93 160 140 120 100 e 80 3 60 40 20 dR = 0.05 ohms "W^ToIu ; dR=5 ohms;. i Current Interruption 4 6 Time / s 10 Figure 23: Current Transients of Various Step Sizes For all dR values, mutual relaxation of both the current and voltage signals was seen. The fact that this behaviour was predicted suggests that there is merit in including interfacial charging behaviour in an analytical SOFC transient model. Relaxation behaviour differed significantly in the case of current interruption. Here the current was driven to the null value with no observable transient behaviour aside from some ringing. The voltage response to current interruption exhibited a different shape altogether from the load switching transient responses. With regard to the discussed differences in rate-limiting loss mechanisms between operating load and OCV, one might expect the voltage response to GCI to be slightly more delayed than the response to a small ideal current step, because a lower frequency rate-limiting process was seen at O C V that was absent at operating load. However, the response to GCI settled much more quickly than load switching responses, suggesting that in the case of load switching responses, the voltage response is delayed by the unsettled current behaviour. Although load switching has provided a simple, cost-effective means of fuel cell perturbation, ambiguity of the bilateral relaxations presents a major shortcoming of this technique. Unlike the response to a pulse, step, or frequency sweep, the dual responses to load switching present a more indirect task of interpretation; since the current signal is not a controlled input of a known form, the transfer function (complex impedance) cannot be 94 extracted in a straightforward manner by treating the voltage data as the only output. In the case of a voltage response to a current step, theory suggests that the complex impedance can be derived mathematically as the Z-transform of the first derivative of the step response. In the case of load switching, it would be more realistic to assume both the current and voltage signals are output responses to a controlled input: the load resistance. A method of extracting the complex impedance response from the mutually decaying transients might be conceivable; however, it is beyond the scope of this work. 8.7 Estimation of Ohmic Resistance from Transient Signals The area-specific ohmic resistance of the test cell was estimated based on the apparent instantaneous steps seen in transient responses to load switching recorded at the maximum sample rate of the PCI-6115(10 MS/s). Figure 24 shows the transient responses to load switching within 100 us after perturbation. The response acquired with dR = 5 ohms has been shown here for illustrative purposes. Responses to other load steps showed similar characteristics. 7751 i 1 1 1 ! 1 1 1 r 1 1 T i m e / u s Figure 24: Rapid Transient Responses and Extrapolation of Ohmic Data (dR = 5 ohms) 95 A region of ringing was observed immediately after perturbation, which is a common characteristic of transient behaviour in a second order system. In this particular system, the second order behaviour originated from the presence of both inductance and capacitance. The source of inductance was likely the signal lead wires and electronic components rather than the fuel cell itself. Ringing is thought to have interfered with interpretation of electrochemical data to a certain extent. Btichi and Scherer [102] made similar observations of ringing in their study of in-situ PEFC membrane resistance, except the ringing that they observed persisted for approximately 300 ns, whereas the ringing seen here persisted for approximately 5 to 15 us. This implies a significant difference in system dynamics. Btichi and Scherer used linear extrapolation of transient data back to the instance of perturbation to expound ohmic characteristics. They fit the data from 300 to 900 ns after perturbation (twice the ringing duration). A similar approach has been adopted here, as shown by the fit lines in Figure 24. The ringing duration / r j n g was estimated via graphical inspection. Data ranging from trmg to 3/Ving was fit with a line. The line was then back-extrapolated to the instance of perturbation. For each of the two signals, the ohmic step was interpreted as the difference between the back-extrapolated value at the instance of perturbation (Figure 24) and the signal value immediately before perturbation. To accommodate for noise, the signal value before perturbation was approximated as the mean of 100 data points prior to perturbation. For each load step, i ? 0 hm was calculated from these figures via Ohm's law. Figure 25 shows the estimated R0hm values for all load steps as well as for current interruption. These values were plotted against the final steady state current density after perturbation to maintain uniformity with DC polarization plots: Error bars indicate ± standard deviation of the dataset. 96 o ii 1.0 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 0 . 4 0 .3 0 .2 0 , 1 4 0 . 0 0 Average R olun_ Initial Steady State Current Density (before pel tui:bation)_ 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 Final Steady State Current Density (after perturbation) / mA cm" Figure 25: Estimations of Ohmic Resistance at Various Perturbation Magnitudes An average i? 0 h m value of 0.255 ohnvcm was estimated for the various load steps including current interruption. This value was significantly higher than the observed 0.213 ohm»cm 2 determined by EIS. The inconsistency alludes to systematic errors in the method of interpretation of the apparent instantaneous changes in the transient signals. An interesting observation is that R0um differed the most from that of EIS for the case of current interruption. Furthermore, R0hm estimated by load switching converged to very close to the value estimated by EIS as the load step was reduced to a very small size. Further investigation of fast electrochemical transient responses using a system with reduced inductance is required before fast electrochemical responses can be quantified. The influence of cable inductance on ringing was explored by comparing current interruption transient responses using different coaxial cable lengths connecting the fuel cell to the load switching circuit. Total coaxial cable length was varied between 2 and 15 ft. Figure 26 shows a pronounced difference in the ringing characteristics resulting from the use of these different cable lengths. 97 a) 2 ft Coaxial Cable 1.2r 1 it 0.8-« •*•» [© 0.6->> 0.4 -2 1.2 1 b O S -? 0.6 0.4 _ J ± _ J L 6 8 10 12 14 16 18 20 Time / us b) 15 ft Coaxial Cable -2 0 2 4 6 8 10 12 14 16 18 20 Time / us Figure 26: Effect of Coaxial Cable Length on the Rapid Transient Response Two distinguishable regimes of ringing were visible. For all cable lengths, a region of relatively slow oscillations was observed. This behaviour appeared to be independent of cable length; thus, it was likely an artefact of the load switching circuit. A cable length dependent, rapidly oscillating region was seen immediately after perturbation, as shown in Figure 26. This region was most prominent when 15 ft of cable were used. Oscillations in this region would likely make electrochemicaf transient responses very difficult to identify, suggesting that it is very important to minimize cable length when interpreting rapid electrochemical effects. 8.8 Calculation of Ohmic Resistance from Published Data To clarify inconsistencies in ohmic data, R0hm was estimated using published electrolyte conductivity data and compared to the experimental values. Van Heuveln [124] measured the A C conductivity of 8YSZ at various temperatures using three different measurement configurations. From data interpolation he determined the pre-exponential factor o0 and activation energy Ea in the conductivity equation of the form: 98 a I E \ a = ^ e x p (8.1) T \ RT) To predict i?0hm, Equation 8.1 was solved at 800°C with Van Heuveln's fitting parameters, yielding a conductivity of 41.9 ± 8 S/m. Based on this conductivity and the dimensional measurements of the test cell, an area-specific electrolyte resistance of 0.050 ± .011 ohm»cm was estimated. Conductivity of YSZ was also published by Minh and Takahashi [123]. They presented a graph of YSZ conductivity versus the mole fraction of Y2O3. A conductivity of 12.6 ± 3 S/m was interpreted from this graph for 8YSZ. Using this conductivity value, an area-specific electrolyte resistance of 0.168 ± .03 ohm»cm 2 was calculated. It is clear that the calculated electrolyte resistance depends strongly on the assumed conductivity. The manufacturing process, density, impurity content, and other characteristics determine the true electrolyte conductivity. Since high purity 8YSZ was used to make the test cell's electrolyte and near-fully dense YSZ was observed in micrographs, it is likely that the true conductivity was comparable to published values mentioned above; however, its exact value was not measured directly. In addition to electrolyte resistance, the electrode resistances and contact resistances also contributed to R0hm- Electrical conductivity of Ni and L S M are known to be on the order of 107 S/m and 104 S/m, respectively. Thus, electrode resistances were likely very small and probably did not have much contribution to i?0hm- However, it is expected that contact resistances did make a significant contribution. Jiang [125] was able to isolate contact resistances between Pt mesh current collectors and electrodes of an SOFC. He measured area-specific contact resistances to be 0.16 ohm»cm and 0.65 ohm*cm at the Ni/YSZ anode and LSM/YSZ cathode, respectively. Considering that in this work, additional cathode slurry was applied between the cathode and Pt mesh of the test cell, the contact resistance was likely less than that observed by Jiang. If the test cell's electrolyte conductivity is assumed equal to the highest published value discussed (yielding the lowest i?0hm) and an area-specific anode contact resistance comparable to Jiang's observations is assumed to be the only additional contribution, the resultant i?0hm is within 3% of the values determined both by EIS and transient responses to the two smallest 99 load steps. This suggests that the actual R0hm was comparable to the value predicted by EIS and transient responses to small load steps. 8.9 Analysis of Signal to Noise Ratio (SNR) in Voltage Transients The S N R was calculated as the maximum absolute value of the signal level divided by the standard deviation of noise, assuming a signal datum at the transient's final settling value. Noise characteristics were extracted from the last 100 samples settled in the transient tail. Only the voltage S N R values were analyzed. Figure 27 shows the S N R values obtained for filtered and unfiltered data. Figure 27 shows a linear fit of the S N R data. The filtered response to current interruption (polarization step: 263 mV) was excluded from the fit because it significantly offset the slope of the linear fit. It is unclear why this point did not behave linearly. Figure 27 exemplifies the drastic improvement of the S N R when a filter is used. However, one should be aware that filtration smoothes rapidly decaying signals as well as 100 noise. Such manipulation could be detrimental if one wishes to characterize rapid features. This point applies to digital as well as analog filtering. 8.10 Analysis of Noise in Transient Signals Prior to the examination of signal noise, it was speculated that the source of the noise might be from adjacent equipment or communication signals, in which case an opportunity might exist to isolate the test system and improve the SNR. Figure 28 illustrates typical noise that was present in the current and voltage signals at steady state. These data were sampled within a time window of 10 ms at a rate of 10 MS/s to capture rapid fluctuations. T T 1 p 1 j "1 1 1 •iii»www»i^mnwii»iiiLiiiiilMiiiil mn II m m i mmm niimrn i mi \l !i j i j i ii i j "'*''() 0,001 0.002 0,003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Time As Q'iO&t j ; i j j 1 j ! 1 r 1 1 0.66 4/ 0.648 i Y ^ - ^ T ' ' f ' - ^ - r - ' — - •' -0.644 h ' 1 ^ ' 4 — 1 ' 0 0;001, 0.002 0:003 01004 0;005D;006 0.007 0.008 0;009 0.01 Tiine / s Figure 28: Steady State Noise (Time Domain) The current signal noise within this time window had a range and standard deviation of 2.20 mA and 0.241 mA, respectively. The voltage signal had a range and standard deviation of 4.90 and 0.535 mA, respectively. Effects of digitization can be seen in Figure 28. The frequency responses were calculated for the range of 50 to 500 kHz from steady state signals sampled at 1 MS/s for 0.1 s. These responses are shown in Figure 29. For these 101 0.166 -g 0M62 t 0.158 0.154 recordings, the 500 kHz analog filter was enabled in order to attenuate periodic signals at frequencies greater than 500 kHz (and thus attenuate aliased features). The response showed no significant periodic constituents in this frequency range except for a few low intensity peaks (less than 0.1 mV / 0.06 mA amplitude) within the radio frequency range, as shown in Figure 29. The process was repeated for the range of 0 to 50 kHz using signals sampled at 100 kS/s for 1 s with the 50 kHz analog filter enabled. No periodic features were observed. .10 08 c s I i 0 6 i ' . i -04 II -.02 .0Q.1 a) Current Signal Noise T r 250 300 Frequency / kHz b) Voltage Signal Noise 500 50 100 150 200 250 300 350 400 450 500 Frequency / k H z Figure 29: Frequency Response Amplitudes of Signal Noise in the Range of 50 to 500 kHz A noise recording was performed with null signal connections and a noise level roughly half of that for the voltage signal shown in Figure 28 was observed for both analog input channels, indicating that the recording device alone contributed to the signal noise. The observed fluctuations in the signal noise with the fuel cell connected might have been due to periodic noise signals at frequencies higher than 500 kHz; however, higher frequencies were not examined. These fluctuations might have also been due to resistor noise (thermal, contact, and shot noise). These noise sources are omnipresent in all electronic devices to a certain degree. If resistor noise was the dominant source of these fluctuations, noise would not be 102 reduced by signal shielding or isolation of test equipment. Rather, noise reduction could be achieved by using low noise resistors (metal film, large size, large wattage, etc.). In a comparison made between transient responses to current interruption with the furnace heating elements on and off, no difference in waveform shape or noise level could be seen. Thus, it is probable that the furnace heating elements had a negligible effect on the quality of the transient signals, possibly because the current signals were relatively small and significantly separated from the furnace heating elements. In the case of larger fuel cell, larger furnace element currents, or closer spacing, signal distortion from mutual inductance might be a problem. 8.11 Repeatability of Transient Experiments A strategy to improve the SNR of a signal is to repeat the experiment several times and compute the average signal value. Istratov and Vyvenko [115] mentioned that the SNR is improved by a factor of A; if k repeated experimental transients are averaged. However, averaging experimental signals with poor repeatability could cause detrimental distortion. Instabilities in experimental parameters could yield an average signal that poorly represents fuel cell properties. To investigate the repeatability of the load switching technique, five perturbations were performed consecutively with a delay time of approximately two minutes between each run. As mentioned, the cell was maintained at operating load for 20 minutes prior to each set of transient experiments. During repeated tests, the switch was in the position corresponding to Path B (Figure 12) for approximately 17% of the time. Figure 30 shows five consecutive voltage transient responses. Here a dR value of 0.33 ohms was chosen arbitrarily for illustrative purposes; similar trends were seen for all load step sizes. 103 0,725 0.720 > 0.715 a 0.710 0.705 0.700 ! 1 1 j ! fiiist sample I : \ ] i _ last sample f i . r i i Average i i i i Time / s Figure 30: Repetition of Five Load Switching Experiments (dR = 0.33 ohms) As seen in Figure 30, the DC offset dropped over the course of the five repeated experiments. This drift was observed for all load step sizes as well as for current interruption. A drift of up to 10 mV was observed. No correlation was seen between the range of drift and the load step size. Steady state signal values were examined over the course of several hours and small oscillations were observed within a range of 5 mV. These oscillations might have been caused by fluctuating furnace temperature or gas flow rates. Conversely, the drifts seen in repeated experiments differed from these; they fluctuated much more rapidly and both voltage and current levels decreased over the course of several repeated tests. This observation suggests that drifts were caused by neither an unaccounted slow polarization response, nor a shift in load resistance from self-heating of the circuit components. Otherwise, the downward drift in voltage would likely be accompanied by a constant or upward-drifting current. Drifts might have been the result of degradation of electrochemical performance over the course of repeated tests. Degradation might have been caused by large current ringing oscillations accompanying perturbation that converged too rapidly to be seen in transient recordings. It is unclear how such current spikes could cause the observed degradation. After repeated 104 perturbations, the cell settled back to operating load from the final drift values over the course of less than an hour, suggesting that the observed drifts were not caused by irreversible damage. Although the average of five repeated transient experiments produced a signal with a much reduced noise level (Figure 30), the signal'inherited the invasive characteristics of the drift. 105 Chapter 9: Results and Discussion: Spectroscopic Analysis of Simulated Transients 9.1 Introduction As a prerequisite to the spectroscopic interpretation of SOFC transient data, a study of simulated transients was carried out to clarify some of the specific capabilities and limitations of transient signal processing with CONTIN. Simulated data were generated in Matlab from a discretized multi-exponential decay function with chosen time constants and amplitudes. In various inquiries, the exponential decay function was evaluated or "sampled" at time values of interest to give a set of simulated transient decay signal values. Sample resolution, sample window limits, and solution domain limits were varied and their influences on interpretation of transient signal constituents were examined. Noise was introduced to simulate a real transient experiment and its ramifications were studied. It should be understood that in all cases the simulated data has consisted of exponential decays with discrete time constants and CONTIN has solved for these discrete values with tolerance to noise by assuming they are distributed peaks in the solution domain. In the following analysis, the moment (area under a peak) has frequently been compared to its "true value" (magnitude of the discrete time constant). The wording "well resolved" has been used in the following to indicate that the center and moment of a feature have been resolved within less than 1% error from the known time constant and amplitude, respectively. 9.2 Simulation I: Strategy Development for Isolation of Transient Features As a first inquiry, simulated noise-free transient data was generated from a multi-2 2 0 0 exponential decay function with time constants [TI,T2,T3,T4] = [1x10' , 5x10" , 1x10 , 5x10 ]. All four amplitudes [ai,a2,a3,a4] were set to 0.25 to produce a convenient sum of unity. In this inquiry, resolution capabilities of both x's that are separated from each other by two orders of magnitude in time, and x's that are within the same decade were investigated. First, a broad solution domain of 0 to 10 s was investigated to see if all of the transient features could be interpreted within one window. It was unclear what influence the resolution of data sampling would have at times immediately after perturbation. To help clarify this point, 106 two different sample window limits were investigated: First, data was sampled starting at 10"4 s, and then at 10" s after perturbation. 10 samples per decade were included to provide strategic sampling over a relatively long time period. Hence, for data sampled starting at 10"4 s, data was sampled at 10 kS/s between 10"4 and 10"3 s, providing slightly more information within the rapidly decaying portion of the simulated transient response. Figure 31 shows the time constant distributions obtained for these two cases. 35 30 Vl ii - a 25 © vi S •8 c £ 10 S a m p l e W i n d o w .0001 t o 100 s .001 t o 100 s A i l 3 4 5 6 7 Time Constant Distribution / s 8 10 Figure 31: Effect of High Resolution Sampling on Time Constant Distributions of Simulated Transients (Solution domain: 0 to 10 s, 10 samples per time domain decade) Peaks at 13 and 14 were not distinguishable from each other in Figure 31; the peak seen between 3 and 5 s was presumably an amalgamation of these two peaks. However, the centers and moments of the observed peak reflected neither independent characteristics of any of the true peaks, nor a linear combination of the two. The peak shifted to a lower center value when less rapidly decaying transient data was included. The sharp spike close to 0 s in the solution domain appears to be an artefact caused by the presence of n and T2 in the sample window. These transient features decayed too rapidly to be resolved within this solution domain. 107 Based on these findings, it appears that the presence of rapidly decaying transient components interferes with interpretation of slowly decaying components. The amount of high resolution data appears to influence the apparent characteristics of the slowly decaying features. An attempt was made to interpret the slowly decaying transient components by further reducing the amount of rapid transient data to a level in which the effects of the rapidly decaying components would become negligible. Results for three sample windows are shown Figure 32. 10 VI VI •ii - a .2' = ii & 4 -V3 a 0 S a m p l e W i n d o w ; .01 t o 100 s 0.1 t o 100 s 1 t o 100 s 3 4 5 6 7 Time Constant Distribution / s 10 Figure 32: Effect of Intermediate Resolution Sampling on Time Constant Distributions of Simulated Transients (Solution domain: 0 to 10 s, 10 samples per time domain decade) From the .01 to 100 s sample window, a peak similar to the one seen at higher maximum resolutions was observed, but also, another peak emerged close to 10 s. From the 0.1 to 100 s sample window, two distinct peaks were seen in the proximity of the true time constant locations. Finally, from the 1 to 100 s window, the peak centers and moments were well resolved. Next, the time constant distribution was studied from a 10 to 100 s sample window in order to see if this transient tail contained enough information to resolve the peaks at T3 and x4. The solution is shown in Figure 33. 108 VI Vl ii Vl c a B VI 2.0 1.5 = 1.0 JS 0.5 0.0 0 small peaks Sample Window 10 to 100 s — i i 3 4 5 6 7 Time Constant Distribution / s i 9 10 Figure 33: Effect of Low Resolution Sampling on Time Constant Distributions of Simulated Transients (Solution domain: 0 to 10 s, 10 samples per time domain decade) As seen in Figure 3 3 , the peak at T4 was well resolved. On the other hand, there was no indication of the peak at 13. Also, a cluster of small evenly spaced erroneous peaks emerged between 0 and 1 s. The foregoing inquiry was repeated using 100 samples per decade within the same sample window limits (i.e. data was sampled for longer periods at each resolution). No significant changes were observed in the resolution capabilities. The use of more data points at a fixed resolution (and hence, a greater density of data over several decades) penalizes computational time, although the penalty was not a relevant loss in these inquiries. Nevertheless, the simpler approach of 10 samples per decade was adopted until this comparison was later revisited in the presence of noise. These inquiries have suggested that by excluding the rapidly decaying transient features in the sample window, one can resolve the slower transient features. In application, the best resolution was achieved when the transient was sampled at a resolution on the same order of magnitude as the time constants of the slow transient features. 109 Next, a solution domain of 0 to 0.1 s was examined in attempt to resolve TI and T2 from the wide-ranging sample window of .001 to 100 s. The solution is shown in Figure 34. S a m p l e i W i n d o w : .0001 t o 100 s 4500 4000 J 3500 13 .2 3000 I 2500 ^ 2000 £ 1500 I 1000 500 0 ~r" i i i i 1 1 i 1 r 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Time Constant Distribution / s Figure 3 4 : Time Constant Distributions of Simulated Transients at High Sample Resolution (Solution domain: 0 to 0.1 s, 10 samples per time domain decade) Figure 3 4 shows two erroneous peaks close to the limits of the solution domain. Hence, the rapidly decaying features were irresolvable from this sample window. It was speculated that the rapidly decaying features might be expounded by excluding slow features. Accordingly, the upper limit of the sample window was reduced and the effects of truncating the window were examined in the solution domain of 0 to 0.1 s. It was found that no peaks could be resolved until the problem was redefined to allow CONTIN to interpret an unknown DC offset in the time domain. This approach can be understood by examining the arbitrarily chosen sample window shown in Figure 35 . The transient, which would otherwise decay to zero, has been truncated at some premature time. The problem has been defined such that the software fits the data with a transient response decaying to an asymptote close to the signal value at truncation. The result is a satisfactory fit of the data close to the time of perturbation (roughly the first 1 0 % of the sample window), and a less accurate fit for the remainder of the sample window. Although this approach is clearly accompanied by a 110 significant loss of accuracy, it has yielded much greater resolution capabilities than inspection of rapid solution domain features from the all-encompassing sample window (Figure 34). 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 S a m p l e W i n d o w / s Figure 35: Truncation of Transient Data with Inclusion and Exclusion of a DC Offset Truncation effects were examined in a solution domain encompassing the rapidly decaying transient features. At this point it was unclear what upper solution domain limit should be specified. It was speculated that this selection might not be trivial, and the upper limit might influence the resolution capabilities. For this reason, solution domain upper limits ranging from 0.10 to 0.15 s were examined. Figure 36 shows the results of this inquiry. I l l VI VI-ii ii VI a-c 120 100 80 60 40 20 0 Solution Domain (Is 0s to i—»- to .10 s ! .15 s S a m p l e W i n d o w : . 0 1 t o 0 .1 | 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 Time Constant Distribution / s Figure 36: Time Constant Distributions of Truncated Simulated Transients (Solution domain: 0 to .10 s through 0 to .15 s, 10 samples per time domain decade) From Figure 36 it can be seen that i i has been well resolved and was insensitive to small adjustments in the solution domain. This is likely a result of CONTIN's accurate data fit close to the time of perturbation when the transient response is truncated and a DC offset is interpreted, as shown in Figure 35. The peak centered close to .04 s showed nonlinear sensitivity to the solution domain. Both its center and moment appeared to converge as the solution domain upper limit was increased. The peak in the upper half of the solution domain showed linear dependence on the solution domain upper bounds. This peak was thought to be an artefact of the slowly decaying constituents. Its erroneous appearance in a rapid solution domain was likely associated with the reduced accuracy accompanying the interpretation of an unknown DC offset at increasing time after perturbation, as seen in Figure 35. The aforementioned analysis of simulated noise-free transient data has demonstrated that the simulated transient features are irresolvable in a single solution domain when a large portion of the transient response is examined. However, slowly decaying transient features can be resolved by excluding the rapidly decaying portion of the transient data. Furthermore, 112 rapidly decaying features can be resolved by excluding slowly decaying features, although the method proposed to exclude slow transient features appears to inherently limit the degree of resolvability. It is clear that specifications of both the sample window and solution domain bounds are not trivial. These issues require further examination. 9.3 Simulation II: Sensitivity of Time Constant Distribution Features In a more intricate simulation, spectroscopic resolution capabilities were examined to a greater depth. Noise-free transient data were generated with the following specifications: [TI,T2,T3,T4] = [5xl0~2, 2x10"', 8x10"', 5x10°]. Again, all amplitudes were assigned the value of 0.25. These specifications yielded a combination of slowly, intermediately, and rapidly decaying transient features. The strategies for isolating transient features suggested in Section 9.2 were applied here. Sensitivities of time constant distribution features to sample window and solution domain limits were examined to a greater degree than in the foregoing simulation. Also, sensitivities to the sample window resolution and the regularization parameter were examined. Since isolation of transient features could not be achieved by examining several orders of magnitude of time domain data, only one order of magnitude was examined at a time here. For this reason, the mentioned strategic sampling scheme was abandoned and a constant sample rate was used throughout the sample window. First, several solution domain upper limits were tried, ranging from 0.7 x 10kto 1.5 x 10k for k = -1, 0, and 1. Initially, a constant sample rate of 10 points per sample window was used. Also, a consistent regularization parameter of a = 1 x 10" was used. This value was chosen because of a satisfactory compromise between stability and smoothness in the fitting of noise-free data. Figure 37 shows the solution domains, varied from [0 to 7 s] to [0 to 15 s] from a sample window of 1.0 to 10 s. 113 2.5 VI Vl 0) - a o Vl S ij 2.0 A 1.5 1.0 Vl C 4> 0.5 0.0 S a m p l e W i n d o w : a = 1 . 0 K 1.0 t o 10 s| 7 — i r 1 1 i 1 1 T 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time Constant Distr ibution / s Figure 37: Sensitivity of Time Constant Distribution Features from Simulated Transients (Solution domain: 0 to 7 s through 0 to 15 s, 10 samples per time domain decade) Three peaks were seen. The two rightmost peaks were well resolved interpretations of the peaks at T3 and T4. The center of the leftmost peak was resolved within 1% of 12; however, its moment varied from 19% to 47% error of a2 as the solution domain upper limit was increased from 7 to 15 s. This poor resolution was likely a result of solution domain boundary effects, as Provencher had warned about [122]. Also, it might have related to interference caused by the erroneous interpretation of the peak at xi. Figure 38 shows the time constant distribution from a 0.1 to 1.0 s sample window within a solution domain varied from [0 to 0.7 s] to [0 to 1.5 s]. 114 ii I 200 180 160 140 120 100 80 60 40 20 0 S a m p l e W i n d o w : 6.1 t o 7 a = 1 . 0 E 1 . 0 s S o l u t i o n D o m a i n i 0 to 0.7 s ! Oto r 1.5 s 1 U & 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Time Constant Distribution / s Figure 38: Sensitivity of Time Constant Distribution Features from Simulated Transients (Solution domain: 0 to 0.7 s through 0 to 1.5 s, 10 samples per time domain decade) In the 0 to 1.5 s solution domain, the center of the leftmost peak was well resolved (in accordance with ii). However, its moment was in 10% error of a i . The adjacent peak's center and moment were in 5.5% and 6.4% error of T2 and a2, respectively. The feature seen in the upper half of the solution domain shifted linearly with the solution domain's upper limit until approximately 1.2 s, where it appeared to converge. Figure 39 shows the time constant distribution from a .01 to .10 s sample window for a solution domain varied from [0 to .07 s] to [0 to .15 s]. 115 2000 1800 1600 1400 1200 H 1000 800 600 400 200 -f 0 I S a m p l e W i n d o w : . O l -i o - 1 . 0 E - 7 t o 0 . 1 s Oto .07 s S o l u t i o n D o m a i n Oto .15 s 0.000 0.025 0.050 0.075 0.100 Time Constant Distribution / s 0.125 0.150 Figure 39: Sensitivity of Time Constant Distribution Features from Simulated Transients (Solution domain: 0 to .07 s through 0 to .15 s, 10 samples per time domain decade) A very small peak emerged and its center shifted as the solution domain was enlarged. This feature appeared to be converging to a location which could not be deciphered with certainty in the range of solution domain limits studied here. Also, a linearly sensitive feature was seen in the upper half of the solution domain. Again, this was believed to be an artefact caused by one or more of the more slowly decaying constituents. Due to the ambiguity of the converging peak seen in Figure 39, a more sophisticated scheme was devised to estimate the convergent properties. A nonlinear regression fit of the centers and moments was tried using the software GraphPad Prism™ with a mono-exponential fitting function and an unknown DC offset. This was an arbitrarily chosen fitting function; no correlation to the exponential nature of the simulated data was implied. By this manner of extrapolation, the projected convergences of the center and moment were in 1.0% and 17% error of their true values, respectively. Hence, this estimation method could not resolve the peak at x\ with the accuracy achieved by inspecting both the sample window and solution domain on a higher order of magnitude. Due to the lack of confidence in extrapolated convergence values, this technique was not applied in the remainder of this work. 116 Up to this point, the sample window had been varied by one order of magnitude from one inquiry to the next. It was speculated that resolution capabilities might be dependent upon small changes in the sample window limits. Accordingly, both the upper and lower limits of the sample window were independently incremented by several steps. Only the 0.1 to 1.0 s sample window was investigated within the 0 to 1.5 s solution domain, because these conditions yielded the best achievable resolution of the peak at i 2 . No significant improvement trends could be seen. A more comprehensive investigation of solution sensitivity to the sample window limits might suggest further opportunities for improvement in resolution of relaxation features. A tenfold increase in transient sample rate (to 100 samples per window) was tried to see if better resolution of xi and X2 could be achieved. Again, a 0 to 1.5 s solution domain and a 0.1 to 1.0 s sample window were examined. Once again, the center of the leftmost peak was well resolved (in accordance with x i ) . The moment of this peak was resolved to within 4.4% of its true value (an improvement from 10%). However, no significant improvement was seen in the resolution of the adjacent peak. Again, an artefact in the upper half of the solution domain was seen. High resolution sampling introduced small erroneous peaks scattered throughout the solution domain, suggesting a penalty in solution stability resulting from higher resolution sampling. In the case of noisy transient data, the use of a very large sample rate might yield an unstable solution; a greater extent of time domain noise characterization might complicate the Laplace inversion. Nevertheless, the increase in sample resolution enhanced the resolvability of the rapidly decaying features. For this reason, the higher sample resolution was used in all further inquiries. It was speculated that the limited resolution of X2 might have been related to the choice of regularization parameter. Accordingly, a was varied within several orders of magnitude. Again, only the 0.1 to 1.0 s sample window was inspected within a 0 to 1.5 s solution domain. Sensitivity of the time constant distribution to a is shown in Figure 40. 117 Vi Vi - a e • Vi = 12 10 8 6 4 2.-0 S a m p l e W i n d o w : 0 a :=, 1 E - 7 ol= IE 1 t o , 0 s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Time Constant Distribution / s Figure 40: Time Constant Distribution Features from Simulated Transients using Various a Values (Solution domain: 0 to 1.5 s, 100 samples per time domain decade) Figure 40 reveals a very important characteristic: The error in resolution of the center of i 2 depended largely upon the regularization parameter. As a was reduced, the center of T2 was seen to converge. The solution could not be interpreted for a values of 1x10" or less due to solution instability. Solution sensitivity to a clarifies the importance of selecting an adequate level of smoothing. In the presence of noise, a relatively large a is required to yield a stable solution, but it can be seen in Figure 40 that large a values falsify the apparent characteristics of transient features. This is an intrinsic drawback of the ill-posed problem. This investigation has proved that slow, intermediate, and fast transient features can be resolved to a certain degree through a process of zooming concurrently in the sample window and in the solution domain. All centers and moments were resolved within 7% of their true values. However, resolution was sensitive to sample window limits and resolution, solution domain limits, and the regularization parameter. No features appeared to be resolvable in the upper half of the solution domain. Only distorted artefacts were seen here which shifted dramatically when the solution domain limits were modified. Interpretation of simulated noisy 118 transient data remains a very important prerequisite to the analysis SOFC transient data using this methodology. 9.4 Simulation III: Effects of Noise on Resolution Capabilities Spectroscopic analysis of the previous multi-exponential decay ([xi, T 2 , T 3 , T 4 ] = [5x10" , 2x10"', 8x10"', 5x10°]) was repeated with the inclusion of simulated signal noise. Noise was generated using Matlab's randn( ) function which produces an array of random numbers with a normal distribution centered about zero with a standard deviation of unity. The noise was scaled accordingly and added to the multi-exponential decay. A variety of SNR values were examined, ranging from values exceedingly high to values comparable to those encountered in the load switching experiments. Only the simulation conditions that yielded the best resolution in the noise-free analysis were examined here. First, the time constant distribution of the 1 to 10 s sample window was inspected within the 0 to 15 s solution domain. Figure 41 and Figure 42 show the time constant distributions for the various SNR values. The noise-free solution has been included in Figure 41 for comparison. Also, the regularization parameter values have been shown to indicate the extent of regularization required to interpret the noisy data. 119 1.4 1.2 m • a 1 0 o r. | 0.8 ^0 .6 5 0.4 c 0.2 0.0 0 1 — N o N o i s e ( a = 3 . 3 6 E - 7 ) S N R = 1 0 0 0 0 ( a = 4 . 8 3 E - 6 ) S N R = 5 0 0 0 ( a = 1 . 2 7 E - 6 ) - - S N R = 2 0 0 0 ( a = 3 . 5 6 E - 5 ) S N R = 1 0 0 0 ( a = 6 . 9 4 E - 5 ) 4 5 6 7 8 9 10 11 12 13 14 15 Time Constant Distribution / s Figure 41: Time Constant Distribution Features from Noisy Simulated Transients, SNR = 10000, 5000, 2000, & 1000 (Solution domain: 0 to 15 s, 100 samples per time domain decade) 120 0 1 *1—•—I ~ I 4 5 6 7 8 9 10 11 12 13 14 15 Time Constant Distribution / s Figure 42: Time Constant Distribution Features from Noisy Simulated Transients, SNR = 500,400,300, & 200 (Solution domain: 0 to 15 s, 100 samples per time domain decade) Even very small amounts of noise have caused ambiguity in the interpretation of transient features, particularly those situated close together in the solution domain. For S N R = 2000 and less, the two leftmost peaks in the solution domain merged into one, as seen in the leftmost region of Figure 41. Table 7 lists the relative error of the peak centers and moments. Amalgamated peaks were not included. 121 Table 7: Relative Error of Peak Centers and Moments (Solution domain: 0 to 15 s) Relative Error: Relative Error: Relative Error: First Peak [from left] (%) Second Peak (%) Third Peak (%) SNR Center Moment Center Moment Center Moment 10 k < 1 7.4 6.3 3.2 3 1.2 5k < 1 61 < 1 < 1 1.0 < 1 2 k 3 1.6 1 k 22 6.7 500 6 1.5 400 20 5.8 300 19 200 11 As seen in Table 7, the peaks were resolved relatively well for SNR = 5 k and 10 k, although the moment of the second peak was in significant error for SNR = 5 k. The moment of the third peak was relatively well resolved (less than 7% error) for all SNR values greater than 300. Next, the time constant distribution of the 0.1 to 1.0 s sample window was inspected within the 0 to 1.5 s solution domain. Figure 43 and Figure 44 show the time constant distributions for various SNR values within this solution domain. 122 Sample Window: 0.1 to 1.0 s — No Noise (a = 3.36E-7) SNR = 10000 (a = 6.94E-5) SNR = 5000 (a = 6.94E-5) SNR = 2000 (a = 6.94E-5) SNR = 1000 (a = 6.94E-5) ~ i — • — i — 1 1 1 1 1 1 r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Time Constant Distribution / s Figure 43: Time Constant Distribution Features from Noisy Simulated Transients, SNR = 10000, 5000, 2000, & 1000 (Solution domain: 0 to 1.5 s, 100 samples per time domain decade) 123 12.0 10.0 2 8.0 A 3 6.0 is = 0 OB = = -3 = S 4.0 2.0 0.0 Sample Window: 0.1 to 1.0 s — SNR = 500(a - 3.56E-5) — SNR = 400 (a = 5.12E-4) SNR = 300 (a = 7.36E-3) — SNR = 200 (a = 7.36E-3) 1 , 1 .—| p 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Time Constant Distribution / s 1.5 Figure 44: Time Constant Distribution Features from Noisy Simulated Transients SNR = 500,400,300 & 200 (Solution domain: 0 to 1.5 s, 100 samples per time domain decade) Again, distortion was seen even for very large S N R values. In this case, the leftmost peaks remained separable from each other for S N R values above 300. For S N R = 200 and 300, these peaks merged into one. Table 8 lists the relative error of the properties of these peaks. Only the two leftmost peaks were examined here. The rightmost peak appeared to be an artefact that was largely distorted and did not reflect true properties of any feature in the simulated data. Table 8: Relative Error of Peak Centers and Moments (Solution domain: 0 to 1.5 s) Relative Error: Relative Error: First Peak [f rom left] (%) Second Peak (%) SNR Center Moment Center Moment 10 k < 1 29 25 8 5k < 1 28 25 7.5 2 k < 1 27 25 6.0 1 k 10 27 28 3.0 500 40 58 7.5 21 400 40 17 63 4 300 200 124 The presence of noise has resulted in significantly higher error in the interpretation of these rapidly decaying transient features. An interesting observation is that the second peak's center could not be resolved accurately even for an excessively large SNR. In the noise-free analysis, the poorest resolution was encountered for this peak. Analysis of noisy simulated transients has revealed a major limitation of the spectroscopic analysis of transient responses; significantly large SNR values would be required to resolve some of the simulated transient features, and even with an SNR as high as 10 k, some features could not be resolved with adequate confidence. 9.5 Implications of Simulation Predictions on Required SNRs To exemplify the tolerable noise level for a relatively non-invasive fuel cell perturbation experiment, the following scenario has been reflected on: Consider a transient voltage signal with a magnitude of 20 mV. This might be acceptable as a relatively non-invasive perturbation; its magnitude is much less than that for a current interruption experiment from a practical SOFC operating load. Assuming the same data acquisition device used in the experimental portion of this work is employed (12 bit ADC, range: -1 to +1 V), the uncertainty due to digitization is ± 244 uV. Even if noise is completely removed from the analog signal, the digitization error inevitably persists. Thus a SNR ratio of no more than roughly 80 can be achieved from the 20 mV perturbation. Based on the findings for noisy simulated transients, SNR = 80 is too low to achieve adequate resolution of any of the transient signal features considered in the simulation of noisy transients. This upper limit can be greatly improved by using a 16 bit A D C and with a more confined analog input range. If the range can be reduced to 0 to 100 mV, the uncertainty due to digitization with a 16 bit A D C is reduced to ± 0.763 uV. This yields an upper SNR limit of roughly 26 k. Thus, if the noise can be well attenuated with an analog filter, it might be possible to acquire a 20 mV perturbation with a SNR on the order of 104. With the instrument used in load switching experiments, this SNR could not have been achieved due to digitization error alone. 125 9.6 Suggestions for Pre-processing Transient Data for C O N T I N Some pre-processing steps might greatly reduce processing time and improve resolution capabilities when using CONTIN to interpret SOFC transient signals. The ohmic step might have an adverse effect on the interpretation of spectroscopic data; finite sample resolution might lead to a false interpretation of the instantaneous ohmic voltage step as a rapidly decaying transient feature. To avoid this problem, all data before and including the instance of perturbation should be discarded prior to Laplace inversion. Provencher [122] warned about the increased instability of solutions that are not constrained to only positive values. Processing of transients that settle to higher signal values yield negative ordinate values in the time constant distribution. It might be an advantageous precautionary measure to invert voltage transients about the time axis prior to processing with CONTIN. The step-by-step inversion sequence is illustrated in Figure 45. It is not anticipated that this sort of manipulation would distort the solution because the procedure does not change . the shape of the transient signal and CONTIN supports the interpretation of an arbitrary DC offset. However, this strategy might introduce some round-off/truncation error in the signal, and the tradeoffs between noise reduction and stability pertaining to this strategy should be examined. v(t) v(t) 4 v(t) v(t) -*/t (a) (b) (c) Figure 45: Suggested Preprocessing Steps - (a): original data, (b): discarding of data before perturbation, (c): referencing about the settling value, (d): inversion about the horizontal axis The critical transient data sets acquired in load switching experiments contained up to 500,000 data points, logged at a consistent sample rate. This was far too much data for CONTIN to handle. The penalty of capturing high resolution data immediately after perturbation in this manner was an abundance of unnecessary data characterizing the transient 126 tail. Since CONTIN requires a lesser concentration of data points at increasing times after perturbation to generate an accurate fit, it is recommended that data is either logged at equal intervals in logarithmic time (as suggested by Provencher [122]), or a strategy be employed to reduce the sample rate at consecutive time intervals. The latter recommendation can be easily implemented by reconstructing a strategically re-sampled dataset from the original using software such as Matlab. This might greatly reduce processing time. 127 Chapter 1 0 : Conclusions 10.1 SOFC Performance and Perturbation by Load Switching In the experimental portion of this work, the performance of an SOFC test cell was characterized via DC polarization and EIS. Polarization data exposed activation and concentration control regimes. No distinct ohmic polarization region was seen, indicating a region of mixed control. EIS experiments at 800°C at both O C V and operating load showed pronounced activation responses. Data at operating load showed a more pronounced mass transport response. Improved electrochemical kinetics was observed at operating load. It appeared that a thermally activated loss mechanism was rate limiting at O C V and was absent at operating load. This observation suggested a consequence in the interpretation of activation overpotential relaxations in the response to current interruption or other relatively invasive perturbations: analysis of the decay of activation overpotentials might project an obscured perception of a dominant process at operating load because the transient response would likely be an artefact of changing dominant processes. Such elusiveness cannot be gauged at this point without further experimentation. Nevertheless, in view of these considerations, it is clear that small magnitude perturbations would bypass this concern and likely provide a more definite means of interpreting the rate-limiting electrochemical processes pertaining to operating conditions. From the dynamic characterization of the load switching circuit along with EIS studies, a possible overlap of fast electrochemical transients from the test cell and parasitic transients from the load switching circuit has been inferred. Thus, the particular circuit construction was thought to produce too sluggish of a transient response for this particular transient recording application. Rapid load switching was considered a candidate for rapid fuel cell perturbation with low-invasiveness involving only passive circuit elements. SOFC transient load switching models were presented in the literature which assumed that an ideal current step is an adequate approximation of the dynamic effects of load switching. This assumption might have been suitable to predict the relatively slow transient behaviour under investigation. However, in the examination of experimental transient responses to load switching on the order of seconds or less, an ideal current step was not seen. Rather, mutual transient decays of both current and 128 voltage were observed. A basic model formulation suggested that this behaviour is independent of the inductance in the current loop and the non-ideal characteristics of the analog switches. Literature has suggested that this type of behaviour originates from the discharge of capacitor-like interfaces in electrochemical systems. Ambiguities lie in the observed mutual transient behaviour accompanying load switching; the unsteady voltage behaviour is convoluted by both decaying overpotentials and unsteady current behaviour. For this reason, interpretation of the fuel cell's complex impedance response from the mutually decaying transients would not likely be a straightforward task. It seems that perturbation by small current steps is a much more practical approach. In this sense, decays of overpotentials could be interpreted directly and the complex impedance might be derivable from the step response. The (area-specific) ohmic resistance estimated from load switching by small load steps was in good agreement with that measured via EIS. However, when larger load steps were used, the estimated ohmic resistance exceeded that measured by EIS. This inconsistency was likely related to interference of fast transient responses from signal ringing, which likely became more pronounced when larger load steps were used. This ringing appeared to be bimodal; a rapidly ringing region was seen as a result of the coaxial cable inductance, as indicated by the region's sensitivity to cable length. Another region of slower oscillations was likely a result of imperfections in the load switching circuit. In transient experiments, the use of relatively short coaxial cables has minimized ringing to a certain degree; however, a reconstructed circuit with less inductance might permit more accurate isolation of ohmic behaviour from fast electrochemical or parasitic responses. Periodic noise constituents in current and voltage signals were examined in the range of 0 to 500 kHz; peaks were seen in the frequency response with amplitudes of less than approximately 20% of the standard deviation of the noise. The noise fluctuations might have also originated from both resistor noise (thermal, contact, or shot noise) as well as interference from periodic signals above 500 kHz. The recording device alone (null connections) was seen to exhibit signal noise. The fuel cell's current and voltage noise levels were well reduced with the use of an analog filter with cut-off frequency of 50 kHz. After filtering, the remaining noise consisted predominantly of digitization error. However, this strategy raised a particular concern: A poorly specified analog filter jeopardizes the resolution of rapidly decaying electrochemical transients. The possibility of transient signal interference from the furnace heating coils was 129 also examined; results suggested that such interference did not play a significant role in distorting the transient signals. By examining the transients from several successively repeated load switching tests, a lack of repeatability was observed: downward drifts of both the current and voltage signals were seen. These drifts could not be explained at this point and they present a concern regarding the stability of fuel cell perturbation via rapid load switching. Longer settling times between repeated experiments might have reduced or eliminated the signal drifts. However, prolonged test durations would cause this technique to fall short of one of its major development criteria: to provide rapid testing. If the repetitions were more consistent, averaging of repeated transients could serve as a further means to enhance the SNR. 10.2 Spectroscopic Interpretation of Simulated Transient Data This work has presented the concept of spectroscopic interpretation of SOFC transient data via solving the Laplace inversion as a Fredholm integral equation of the first kind using the software CONTIN. This concept was explored using simulated transient data consisting of a multi-exponential decay with time constants that spanned several decades, as well as time constants in close proximity within one decade. By modifying the lower limit of the sample window (i.e. beginning to sample at various times after perturbation), it was discovered that high resolution data can be excluded and the slower transient constituents can be resolved in the solution domain. The introductory noise-free investigation suggested that best resolution can be achieved when the transient is sampled starting at a time after perturbation that is on the same order of magnitude as the time constant of the particular slow transient feature under investigation. In this noise-free investigation, all centers and moments were resolved within 7% of their true values. The largest error was encountered in the interpretation of the more rapid of the two intermediate transient features specified in close proximity in the time constant distribution. Resolvability of peaks depended on sample window limits and resolution, solution domain limits, and regularization parameter selection. Simulated noisy transient data were interpreted employing the strategies suggested in the noise free analysis. Resolution capabilities were greatly reduced in the presence of noise. 130 Even for an SNR of 10 k, relative error of up to 25% and 29% were encountered for resolution of the center and moment of transient features, respectively. SOFC transient responses to load switching were not analyzed via Laplace inversion in this work due to lack of confidence in the resolution capabilities of this technique. Further efforts to reduce signal noise and improve resolution capabilities of the Laplace inversion method might lead to an accurate method to interpret spectroscopic characteristics of SOFC transient data. 10.3 Final Remarks The current and voltage responses to rapid load switching of an operating SOFC have demonstrated a mutual transient nature: both signals showed unsteady regions of relaxation. For this reason, the basic load switching demonstrated in this work is not an advisable method to perturb an operating fuel cell if one intends to stimulate a response from which the electrochemical characteristics of the fuel cell can be inferred. Furthermore, the load switching circuit construction used in this work has shown parasitic behaviour that would likely interfere with the interpretation of fast electrochemical responses nested in time transient data. The spectroscopic technique that has been examined in the simulation portion of this work has not been found to be a suitable method for interpreting SOFC transient responses with noise levels comparable to those encountered in the experimental part of this work; the technique's stringent requirements for low signal noise levels would invalidate any interpretations of transient electrochemical responses. At this point in time, no conclusions can be drawn regarding the advantages of time transient fuel cell diagnostic techniques over other fuel cell diagnostic techniques. EIS remains the most precise method of elucidating electrochemical performance and diagnosing degradation in SOFCs. Further development of non-invasive time domain perturbation techniques is required before time domain techniques can be considered an alternative to the more expensive and time consuming technique of EIS. Despite the pitfalls encountered, this work is the first of its kind to suggest a method for spectroscopic interpretation of time transient SOFC data. This preliminary investigation has been intended to inspire further developments in fuel cell diagnostic technique development. 131 Chapter 1 1 : Recommendations for Future Work The findings of this work have suggested that many possibilities for improvement of this overall technique exist. A first avenue of research might be the vivid characterization of an SOFC by EIS and the deconvolution of impedance spectra. Aside from providing a high-resolution method of interpreting SOFC loss mechanisms without dependence on a priori assumptions, time constant distributions obtained from deconvolution would be very useful for direct comparison with those obtained by Laplace inversion of time transient data. One could test both half cells and full cells and expound time constant distribution characteristics originating at each electrode-electrolyte interface. One could examine the temperature and gas concentration dependence of all observed time constant distribution features. This level of cell characterization would provide a strong premise to which correlations might be made when interpreting time domain data. It would also provide valid information that could yield a more versatile analytical SOFC transient model than those presented in the literature. Following a thorough fuel cell performance characterization, a next endeavour might be to develop hardware to perform rapid current step perturbations. One could use an aftermarket electrochemical test stand with rapidly responding galvanostatic control capabilities; however, long-term advantages would be implied if one could develop a relatively low-cost custom perturbation, transient recording, and signal processing instrument. It is recommended that operating load be made tuneable with a discrete network of low-capacitance resistors rather than a wirewound potentiometer. This would reduce the parasitic behaviour from that of the circuit used in this work. The dynamic range of the instrument should be well beyond the fastest reported electrochemical response, and its parasitic transient response should be well characterized by step response methods. Possible sources of noise should be examined and minimized in the instrument design. Noise filtering schemes should be employed. A DAQ device with a 16-bit A D C should be used. The DAQ device should be capable of recording transient data at equally spaced intervals in logarithmic time, with a minimum time resolution on the order of nanoseconds. With accessibility to an instrument with these specifications, the ambiguities in signal interpretation encountered in this work could be greatly improved. 132 Another important avenue of research would be to gauge the invasiveness of various levels of perturbation. This might be achievable by assessing the long-term fuel cell performance losses resulting from periodic current step perturbations of various magnitudes. It would be very beneficial to establish a threshold between low-invasiveness and adequate signal magnitude for transient signal processing. It would also be beneficial to repeat EIS experiments at a multitude of different loads in order to elaborate on speculations regarding the ambiguity of interpreting transient responses to large-magnitude perturbations. Furthermore, any observations of signal drift resulting from perturbation should be examined in detail and efforts should be made to mitigate such effects. It is recommended that algorithms for Laplace inversion be examined in greater detail. The parameter space should be explored and optimized to obtain maximum resolution capabilities for fuel cell applications. This might involve the creation of a new customized software tool. If the recommended endeavours yield an accurate method comparable to EIS, a next step would be to use this method to characterize degradation in SOFCs . Different modes of degradation should be induced in controlled experiments, and the resultant time constant distributions should be analyzed. 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Provencher, CONTIN (Version 2) User's Manual, E M B L Technical Report DA07, European Molecular Biology Laboratory (1984) ch. 3.1-4.3. 123]. N. Q. Minh, T. Takahashi, Science and Technology of Ceramic Fuel Cells, Elsevier Science B.V., Amsterdam, The Netherlands (1995) p. 30, 81. 124]. F. Van Heuveln, Characterization of Porous Cathodes for Application in Solid Oxide Fuel Cells, Ph.D. Thesis, Universiteit Twente, The Netherlands (1997) section 2.3.2. 125]. S. P. Jiang, "Resistance Measurement in Solid Oxide Fuel Cells" Journal of the Electrochemical Society 148 (2001) p. A887-A897. 140 Appendix: Sample C O N T I N Input Block The following is a sample of an input data set used in the spectroscopic analysis of a simulated transient. Its format is explained in CONTESTS User Manual [122]. Following the parameter definition, the first array of numerical data (91 points) defines the time values, and the subsequent array defines the corresponding signal values. TRANSENT TEST DATA SIMULATION LAST -1. NG 200.E+0 GMNMX 1 . 5E + 0 GMNMX 2 1. OE + 2 NINTT 0 . NLINF 1. IFORMY (5E14.6) IFORMT (5E14.6) DOUSNQ -1. DOMOM • 1. IQUAD 3 . IGRID 1. IUSER 10 4 . RUSER 21 1. RUSER 22 -1. RUSER 23 0. END NY 91 1.000000E+001 1 .100000E + 001 1. 200000E+001 1.500000E+001 1.600000E + 001 1. 700000E+001 2.000000E+001 2.100000E+001 2 . 200000E+001 2.500000E+001 2 .600000E + 001 2 . 700000E+001 3.000000E+001 3.100000E+001 3 . 200000E+001 3.500000E+001 3.600000E+001 3.700000E+001 4.000000E+001 4.100000E+001 4 . 200000E+001 4.500000E+001 4 .600000E + 001 4 . 700000E+001 5.000000E+001 5.100000E+001 5 . 200000E+001 5.500000E+001 5 .600000E + 001 5 . 700000E+001 6.000000E+001 6.100000E+001 6.200000E+001 6.500000E+001 6.600000E+001 6.700000E+001 7.000000E+001 7.100000E+001 7 . 200000E+001 7.500000E+001 7.600000E+001 7.700000E+001 8 .000000E+001 8.100000E+001 8.200000E+001 8.500000E+001 8.600000E+001 8 . 700000E+001 9.000000E+001 9.100000E+001 9.200000E+001 9.500000E+001 9.600000E+001 9.700000E+001 1.000000E+002 3 .296962E-002 2 .436989E-002 2 .293023E-002 1 .482860E-002 1 .256888E-002 8 .268051E-003 4 .205493E-003 5 .200475E-003 1. 892702E-003 1 .912349E-003 3 .512678E-003 1. 247708E-003 141 1 . 3 0 0 0 0 0 E + 0 0 1 1 . 8 0 0 0 0 0 E + 0 0 1 2 . 3 0 0 0 0 0 E + 0 0 1 2 . 8 0 0 0 0 0 E + 0 0 1 3 . 3 0 0 0 0 0 E + 0 0 1 3 . 8 0 0 0 0 0 E + 0 0 1 4 . 3 0 0 0 0 0 E + 0 0 1 4 . 8 0 0 0 0 0 E + 0 0 1 5 . 3 0 0 0 0 0 E + 0 0 1 5 . 8 0 0 0 0 0 E + 0 0 1 6 . 3 0 0 0 0 0 E + 0 0 1 6 . 8 0 0 0 0 0 E + 0 0 1 7 . 3 0 0 0 0 0 E + 0 0 1 7 . 8 0 0 0 0 0 E + 0 0 1 8 . 3 0 0 0 0 0 E + 0 0 1 8 . 8 0 0 0 0 0 E + 0 0 1 9 . 3 0 0 0 0 0 E + 0 0 1 9 . 8 0 0 0 0 0 E + 0 0 1 1 . 9 1 4 3 7 7 E - 0 0 2 7 . 4 8 5 5 1 5 E - 0 0 3 6 . 8 7 9 3 3 1 E - 0 0 3 7 . 3 3 1 6 9 1 E - 0 0 4 1 . 4 0 0 0 0 0 E+001 1 .900000E+001 2 . 4 0 0 0 0 0 E+001 2.900000E+001 3 . 4 0 0 0 0 0 E+001 3 .900000E+001 4 . 4 0 0 0 0 0 E+001 4 .900000E+001 5 . 4 0 0 0 0 0 E+001 5.900000E+001 6 . 4 0 0 0 0 0 E+001 6.900000E+001 7 . 4 0 0 0 0 0 E+001 7 .900000E+001 8 . 4 0 0 0 0 0 E+001 8.900000E+001 9 . 4 0 0 0 0 0 E+001 9.900000E+001 1 . 2 9 0 9 5 8 E - 0 0 2 5 . 9 4 1 9 7 1 E - 0 0 3 1 . 7 8 4 6 4 5 E - 0 0 3 9 . 0 7 8 1 0 2 E - 0 0 4 1 208510E- 003 -2 165006E- 003 1 844038E- 003 3 5 8 7 2 1 6 E - 0 0 3 - 1 105108E- 003 1 943964E- 003 2 694649E- 003 -3 034646E- 003 -2 7 5 6 8 1 6 E - 003 1 244729E- 003 7 159055E- 004 1 4 4 8 6 5 8 E - 003 1 687461E- 003 1 469843E- 003 2 618183E- 003 1 368053E- 003 2 4 0 6 9 3 6 E - 003 -2 384233E- 003 -2 2 6 4 6 9 3 E - 0 0 5 - 2 995717E- 004 3 196821E- 003 5 2 3 9 0 1 0 E - 004 -2 105338E- 003 2 836512E- 0 0 3 - 1 605081E- 003 1 061661E- 003 4 420599E- 004 -1 841004E- 003 -4 339057E- 0 0 3 - 1 164995E- 004 2 019731E- 003 1 2 3 0 1 8 4 E - 003 1 016511E- 003 3 385703E- 003 1 183255E- 003 1 286625E- 003 7 611372E- 004 -2 017852E- 003 -3 871122E- 0 0 5 - 9 618767E- 005 2 942659E- 007 -6 355487E- 004 2 190147E- 003 -3 747866E- 003 8 564600E- 004 1 791353E- 003 1 461977E- 003 1 155766E- 003 8 067003E- 005 1 354213E- 003 1 137829E- 003 -5 112678E- 004 -7 549191E- 004 -5 917588E- 004- 2 950256E- 003 4 679977E- 004 2 368981E- 004 6 296250E- 004 2 887022E- 0 0 3 - 7 019448E- 004 1 246472E- 003 1 598100E- 003 1 881782E- 003 -1 984181E- 003 4 240720E- 004 4 757655E- 004 -2 015526E- 003 -1 484089E- 003 2 164591E- 003- 2 629988E- 004 7 797615E- 004 142 


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