@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix dc: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Mechanical Engineering, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Gazzarri, Javier Ignacio"@en ; dcterms:issued "2007-12-10T18:13:25Z"@en, "2007"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description "A numerical model of the steady state and alternating current behaviour of a solid-oxide fuel cell is presented to explore the possibilities to diagnose and identify degradation mechanisms in a minimally invasive way using impedance spectroscopy. This is the first report of an SOFC impedance model to incorporate degradation, as well as the first one to include the ribbed interconnect geometry, using a 2-D approximation. Simulated degradation modes include: electrode/electrolyte delamination, interconnect oxidation, interconnect/electrode interface detachment, and anode sulfur poisoning. Detailed electrode-level simulation replaces the traditional equivalent circuit approach, allowing the simulation of degradation mechanisms that alter the shape of the current path. The SOFC impedance results from calculating the cell response to a small oscillatory perturbation in potential. Starting from the general equations for mass and charge transport, and assuming isothermal and isobaric conditions, the system variables are decomposed into a steady-state component and a small perturbation around the operating point. On account of the small size of the imposed perturbation, the time dependence is eliminated, and the original equations are converted to a new linear, time independent, complex-valued system, which is very convenient from a numerical viewpoint. Geometrical and physical modifications of the model simulate the aforementioned degradation modes, causing variations in the impedance. The possibility to detect unique impedance signatures is discussed, along with a study of the impact of input parameter inaccuracies and parameter interaction on the presented results. Finally, a study of pairs of concurrent degradation modes reveals the method’s strengths and limitations in terms of its diagnosis capabilities."@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/218?expand=metadata"@en ; dcterms:extent "1285025 bytes"@en ; dc:format "application/pdf"@en ; skos:note "i IMPEDANCE MODEL OF A SOLID OXIDE FUEL CELL FOR DEGRADATION DIAGNOSIS by JAVIER IGNACIO GAZZARRI Ingeniero Mecánico, Universidad de Buenos Aires, 1998 M.A.Sc., The University of British Columbia, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA December 2007 © Javier Ignacio Gazzarri, 2007 ii Abstract A numerical model of the steady state and alternating current behaviour of a solid-oxide fuel cell is presented to explore the possibilities to diagnose and identify degradation mechanisms in a minimally invasive way using impedance spectroscopy. This is the first report of an SOFC impedance model to incorporate degradation, as well as the first one to include the ribbed interconnect geometry, using a 2-D approximation. Simulated degradation modes include: electrode/electrolyte delamination, interconnect oxidation, interconnect/electrode interface detachment, and anode sulfur poisoning. Detailed electrode-level simulation replaces the traditional equivalent circuit approach, allowing the simulation of degradation mechanisms that alter the shape of the current path. The SOFC impedance results from calculating the cell response to a small oscillatory perturbation in potential. Starting from the general equations for mass and charge transport, and assuming isothermal and isobaric conditions, the system variables are decomposed into a steady-state component and a small perturbation around the operating point. On account of the small size of the imposed perturbation, the time dependence is eliminated, and the original equations are converted to a new linear, time independent, complex-valued system, which is very convenient from a numerical viewpoint. Geometrical and physical modifications of the model simulate the aforementioned degradation modes, causing variations in the impedance. The possibility to detect unique impedance signatures is discussed, along with a study of the impact of input parameter inaccuracies and parameter interaction on the presented results. Finally, a study of pairs of concurrent degradation modes reveals the method’s strengths and limitations in terms of its diagnosis capabilities. iii Table of Contents Abstract……………………………………………………………………………………ii Table of Contents…………………………………………………………………………iii List of Tables……………………………………………………………………………... v List of Figures……………………………………………………………………………. vi Nomenclature……………………………………………………………………..…….. xii Acknowledgements…………………………………………………………………….. xiv Dedication………...…………………………………………………………………….. xv Co-Authorship Statement………………………………………………………………. xvi 1. Chapter One. Introduction and Background. ............................................................... 1 1.1 Introduction......................................................................................................... 1 1.2 The Solid Oxide Fuel Cell (SOFC)..................................................................... 1 1.2.1 Principle of Operation..................................................................................... 2 1.3 Degradation Phenomena in Solid Oxide Fuel Cells ........................................... 5 1.4 SOFC Modeling .................................................................................................. 8 1.5 Scope of this thesis............................................................................................ 13 1.6 References......................................................................................................... 15 2. Chapter Two. Mathematical Model ............................................................................ 18 2.1 Introduction....................................................................................................... 18 2.2 General Description and Assumptions.............................................................. 18 2.3 Mathematical Formulation................................................................................ 23 2.3.1 Steady State Equations.................................................................................. 24 2.3.1.1 Steady-State Charge Balance................................................................ 24 2.3.1.2 Steady-State Mass Balance ................................................................... 28 2.3.2 AC Equations ................................................................................................ 30 2.3.2.1 AC Charge Balance............................................................................... 30 2.3.2.2 AC Mass Balance.................................................................................. 33 2.4 Calculation of the Impedance ........................................................................... 36 2.5 Model Parameters ............................................................................................. 37 2.5.1 Electrical conductivity .................................................................................. 39 2.5.2 Gas species diffusion .................................................................................... 40 2.5.3 Electrochemistry ........................................................................................... 41 2.6 On the Validity of the Model Equations and Results ....................................... 44 2.7 Conclusions....................................................................................................... 45 2.8 References......................................................................................................... 47 3. Chapter Three. Electrode Delamination .................................................................... 51 3.1 Introduction [ ] ............................................................................................ 51 3.2 Results and Discussion ..................................................................................... 53 3.2.1 Electrolyte-supported button cell .................................................................. 53 3.2.1.1 Intact cell............................................................................................... 53 3.2.1.2 Cathode Delamination .......................................................................... 56 3.2.1.3 The capacitance of the delamination gap.............................................. 59 3.2.2 Electrolyte supported rectangular cell .......................................................... 60 3.2.2.1 Intact cell............................................................................................... 60 3.2.2.2 Cathode Delamination .......................................................................... 63 iv 3.3 The Normalized Series and Polarization Resistances....................................... 66 3.4 The Influence of the Supporting Configuration................................................ 69 3.5 Anode Delamination ......................................................................................... 71 3.6 On the Aspect Ratio .......................................................................................... 73 3.7 Comparison with Previous Experimental Observations ................................... 74 3.8 Conclusions....................................................................................................... 75 3.9 References......................................................................................................... 77 4. Chapter Four. Degradation of the Interconnect - Electrode Interface ....................... 80 4.1 Introduction ............................................................................................... 80 4.2 Results and Discussion ..................................................................................... 81 4.2.1 Oxide Layer .................................................................................................. 81 4.2.1.1 The effect of interconnect geometry ..................................................... 84 4.2.2 Rib Detachment between the Interconnect and the Cathode ........................ 86 4.2.2.1 The effect of interconnect geometry ..................................................... 90 4.3 Conclusions....................................................................................................... 91 4.4 References......................................................................................................... 93 5. Chapter Five. Sulfur Poisoning................................................................................... 94 5.1 Introduction....................................................................................................... 94 5.2 Results and Discussion ................................................................................... 100 5.3 Conclusions..................................................................................................... 106 5.4 References....................................................................................................... 107 6. Chapter Six. Sensitivity Analysis and Combined Degradation Mode Scenario...... 108 6.1 Introduction..................................................................................................... 108 6.2 Sensitivity of the Method to Inaccuracies in the Input Data and Interaction among Parameters....................................................................................................... 109 6.2.1 Delamination............................................................................................... 110 6.2.2 Oxide Layer Growth ................................................................................... 114 6.2.3 Sulfur poisoning.......................................................................................... 115 6.2.4 Interconnect detachment ............................................................................. 120 6.2.5 Summary..................................................................................................... 123 6.3 On the Simultaneous Occurrence of Multiple Degradation Modes................ 124 6.3.1 Tracking an individual electrode process ................................................... 126 6.3.2 Tracking the Degradation Path ................................................................... 128 6.4 Conclusions..................................................................................................... 134 6.5 References....................................................................................................... 136 7. Chapter Seven. Conclusions and Outlook. .............................................................. 137 7.1 Conclusions..................................................................................................... 137 7.2 Outlook ........................................................................................................... 143 7.3 Concluding remarks ........................................................................................ 144 7.4 References....................................................................................................... 145 I. Appendix One. Geometry and Mesh Generation..................................................... 146 II. Appendix Two. On the linearization of nonlinear PDEs around the steady state point ................................................................................................................................. 162 III. Appendix Three. Fitting to an experimental polarization curve. ............................. 164 IV. Appendix Four. Uniform design parameter grids .................................................... 166 v List of Tables Table 1-1. Traditional materials used in planar SOFCs ...................................................4 Table 1-2. Study of SOFC degradation ............................................................................8 Table 1-3. Recent research work on impedance modeling of SOFC-relevant electrochemical systems................................................................................11 Table 2-1. Modeled variables.........................................................................................20 Table 2-2. Standard operating conditions and geometry for typical simulations...........38 Table 2-3. Electrical conductivity and related properties...............................................39 Table 2-4. Literature review of correction factors used for the calculation of the effective conductivity of phase k ..................................................................40 Table 2-5. Gas diffusion properties................................................................................41 Table 2-6. Anode and cathode experimental data used to calculate the exchange current and the double layer capacitance by fitting ......................................42 Table 2-7. Electrochemical data adopted from the literature and results of the fitting.............................................................................................................43 Table 2-8. Reported values for the polarization resistance of LSM-YSZ cathodes reported by researchers at Risø, Denmark .....................................44 Table 3-1. Series, polarization, and normalized resistances calculated based on reported degradation attributed to electrode delamination. ..........................75 Table 5-1. Resistance change (ohm cm2) reported by Xia and Birss (Table 1 in [2]). Conditions: 800ºC, OCV, 18 hs exposure to 10 ppm H2S, 3% H2O, balance H2. Three-electrode measurement. ........................................98 Table 5-2 Series and anodic polarization resistance (Ohm), and normalized resistances, before and after exposure to different concentrations of hydrogen sulfide, at two different temperatures, at OCV, Pt counter- electrode [3]. Balance stream composition: 79% H2, 21% H2O..................98 Table 5-3. Kinetics of sulfur poisoning according to [3] ...............................................99 Table 5-4. Series and polarization resistance (ohm cm2) according to [1] before and after exposure to 1ppm H2S in H2. .........................................................99 Table 5-5. Estimated anodic Rp increase as a function of H2S concentration [1]. ......100 Table 6-1. Extent of degradation used in the example of combined degradation. .......126 Table 6-2. Range of variability of input parameters for the sensitivity analysis of the results ................................................................................................112 Table 6-3. Average and percent deviation in normalized series and polarization resistance for the 30 Uniform Design runs, for each degradation mode, varying 21 factors according to Table 6-2. Cases 1 and 2 indicate before and after locking the parameter identified as primary source of deviation. Gray cells indicate relevant dispersion values affected by this change................................................................................124 Table 7-1. Summary of the qualitative changes expected in the series and polarization resistance, and in the peak frequency, for the four degradation modes under study. .................................................................141 vi List of Figures Figure 1-1. Principle of operation of an SOFC unit cell.................................................3 Figure 1-2. Cross-flow configuration for a planar-stack unit cell ..................................4 Figure 2-1. Modeled geometry for the button-cell configuration. Variables anot, elyt, and catt are the anode, electrolyte, and cathode thicknesses, and they vary according to the supporting configuration. Values for typical simulations are L = 10mm, anot (ESC) = 40 microns, anot (ASC) = 1mm, catt = 40 microns, elyt (ESC) = 150 microns, elyt (ASC) = 10 microns, where ASC and ESC stand for anode-supported cells and electrolyte-supported cells, respectively.....................................................21 Figure 2-2. The rectangular cell geometry includes the ribbed interconnect plates. This two-dimensional approximation is valid for co- and counter-flow configurations.......................................................................................22 Figure 3-1. Solid line: Polarization curve for the electrolyte-supported cell configuration. Model parameters detailed in Tables 2-2 to 2-7. Dashed line: Ohmic component of the polarization curve. Kinetic polarization accounts for approximately 75% of the total polarization losses, as also indicated in Figure 3-2. ..............................................................................................54 Figure 3-2. Nyquist (bottom) and imaginary impedance vs. logarithmic frequency plots of the impedance corresponding to the ESC configuration. The curved arrow indicates the direction of increasing frequency. VCELL = 0.7V...........................................................................................55 Figure 3-3. A gap with the dielectric characteristics of air simulates cathode delamination...............................................................................................................57 Figure 3-4. Effect of cathode delamination on the impedance spectrum. Both series and polarization resistances increase, without change in relaxation frequencies. VCELL = 0.7V. ......................................................................57 Figure 3-5. Ionic current density (A/cm2) within the electrodes and the electrolyte, after a concentric, circular cathode delamination of radius a in a button cell of radius L. The shadowing effect caused by the delamination makes the current density within the affected zone close to zero. VCELL = 0.7V................................................................................................58 Figure 3-6. Electronic current density profile from the centre to the edge along the cathode current collecting line. ..................................................................59 Figure 3-7. Reactant partial pressure contours inside the porous cathode (left) and anode (right), in atm. The limited access of reactant to regions underneath the interconnect rib causes local depletion. The lines of equal reactant concentration appear to be straight because of the large aspect ratio of the cell. Note the difference in colour scale range. The picture shows half the rib and half the channel for symmetry reasons. Axes indicate length in metres. ..................................................61 Figure 3-8. Polarization curve corresponding to the ribbed interconnect geometry (thick line), calculated for an equivalent set of parameters as vii Figure 3-1 (thin line). The maximum current density is not high enough for the polarization curve to show mass transport limitation........................62 Figure 3-9. A new time constant, whose frequency depends on rib and channel size, appears at 10Hz as an anodic diffusional effect (A) in the intact case, for a ribbed interconnect geometry. Processes B and C correspond to anodic and cathodic electrochemical reactions, respectively. Labels indicate rib and channel half-width, both equal in the three cases shown. Numbers below the real axis indicate the high and low frequency intercept impedance values. VCELL = 0.7V.................................63 Figure 3-10. Electronic current density magnitude (colour scale, A cm-2) and streamlines (red lines) for a cathode-electrolyte delamination the length of 2 rib widths and 2 channel widths (extending 0.008 m from the left). Axes indicate spatial dimensions in meters. ch indicate channels, r indicate ribs. The zone affected by delamination is completely deactivated. Mid-way across the intact ribs (r4 and r5) the current density value is small (~0.25 A/cm2) because of the reactant depletion underneath the rib, which makes the current density bend towards the channels, where reactant concentration is higher. At the corners of ribs r3, r4, r5, and r6 with their neighbour channels, the current density is higher than 0.5 A/cm2 (top of scale), thus the region appears white. This scaling strategy was needed for better visualization of the overall figure....................................................................................................64 Figure 3-11. Vertical component of the ionic current density at the mid- plane in the electrolyte for the delamination case shown in the previous figure. The first eight millimetres show almost no ionic species flow. ....................65 Figure 3-12. Impedance change caused by a cathode delamination the length of two rib widths and two channel widths. VCELL = 0.7V. ............................66 Figure 3-13. Normalized series and polarization resistances as a function of normalized degradation for a circular concentric delamination of radius a, on a button cell of total radius L. Both resistances scale with delamination area. ......................................................................................................68 Figure 3-14. Normalized resistances corresponding to the case of delamination of the cathode-electrolyte interface for ESC rectangular cells including the interconnect. The series resistance is affected uniformly with increasing loss of contact. The polarization resistance shows an edge effect, by which the shadowing is lesser for delamination of the cathode between the first ribs, where the electrochemical activity is lower in the intact case due to limited access of reactant. Labels refer to rib and channel numbers (see Figure 3-10) under which the electrode delamination from the electrolyte occurs. .......................69 Figure 3-15. The influence of cathode delamination on the impedance spectrum for an anode-supported cell. VCELL = 0.7V. .............................................70 Figure 3-16. Normalized resistances corresponding to the anode-supported configuration. The series resistance shows an increase that is larger than proportional to the delaminated area. The origin of this “overshadowing” is not clear at present.....................................................................71 viii Figure 3-17. A delamination of the electrolyte from the anode causes essentially the same effect as a delamination of the cathode from the electrolyte in this simulation for an anode-supported button cell. VCELL = 0.7V. .......................................................................................................................72 Figure 3-18. Vertical component of the electronic current density for cathode (top) and anode (bottom) delamination in an ASC button cell. Note that the position of the electrodes is reversed for computational convenience, hence the inversion in the direction of the current. Axes indicate distance in metres. Scale bar shows electronic current density in A/cm2. ....................................................................................................................73 Figure 3-19. Normalized series and polarization resistance as a function of the ratio of button cell radius to electrolyte thickness, for 30% cathode delamination. Aspect ratios ≥ 20 approach R0/R = 0.7. 0.7 is the normalized resistance expected for 30% cathode delamination for the case of complete shadowing. Labels indicate radii in mm of button cells. ...........................................................................................................................74 Figure 4-1. Change in current magnitude and streamline distribution produced by a 20 micron thick oxide layer between the cathode and the interconnect................................................................................................................82 Figure 4-2. Impedance spectrum change produced by the growth of a 20 micron thick layer of chromia on the interconnect surface in contact with the cathode. VCELL = 0.7V. ...............................................................................83 Figure 4-3. Normalized resistance behaviour for oxide layer growth between the interconnect and the cathode. The total degradation state definition is arbitrary, and corresponds to a thickness of 20 microns, the expected growth for 40,000 hrs at 850°C in an oxidant atmosphere. ........................84 Figure 4-4. The effect of interconnect rib and channel width on the normalized resistances for oxide layer growth, for the case of equal widths. Performance deterioration is almost insensitive to variations in rib and channel width.................................................................................................85 Figure 4-5. The effect of interconnect rib and channel width on the normalized resistances for oxide layer growth, for the case of total width equal to 4 mm. Fractional numbers in the label indicate rib/channel width in mm. The series resistance increases to a larger extent, the narrower the rib. .......................................................................................86 Figure 4-6. Redistribution of the electronic current density upon the detachment of the first two cathode interconnect ribs from the left. Red lines: current density path. Colour scale indicates A/cm2. The current density increases at the first intact rib as a result of this rib taking up the current produced at the electrodes within zones above and below the first two ribs. ........................................................................................................87 Figure 4-7. Impact on the impedance of the rib detachment shown in Figure 4-6. The reduction of the conductive path increases the series resistance. The high electronic conductivity of the electrodes redistributes the current density, thereby limiting the shadowing effect. ix Almost no change in characteristic frequency is observed. VCELL = 0.7V............................................................................................................................88 Figure 4-8. Normalized series and polarization resistances as a function of the number of interconnect ribs detached from the cathode. .....................................89 Figure 4-9. Ionic current density at the mid-plane in the electrolyte. Compared with the effect caused by electrode delamination (Figure 3-11, Chapter Three), the detached zone deterioration is less severe. .......................90 Figure 4-10. Influence of rib and channel width in the relative performance loss. Left: Ribs and channels of equal width, 1 mm, 2 mm, and 5 mm. Relative deterioration increases with increasing width. Right: Constant rib+channel width equal to 4 mm. Relative deterioration increases with narrower ribs and wider channels. .............................................................................91 Figure 5-1. The influence of anode surface area loss on the impedance spectrum of an electrolyte supported SOFC. The anodic process arc increases in diameter with decrease in anode active surface area. The anode time constant changes due to an overall change in polarization. This change would have not been observed if the calculation had been done at OCV. Calculated using a 1-D ESC geometry, i.e., with no interconnect. VCELL = 0.7V. .....................................................................................101 Figure 5-2. Normalized series and polarization resistances as a function of loss of active area in the anode, for the ESC configuration. The deterioration is less severe than for cathode delamination, for equivalent fractional degradation.............................................................................102 Figure 5-3. Thick electrodes are relatively less fully utilized than thin electrodes (calculation performed using a 1-D version of the model, ESC configuration). Top: normalized local Faradaic current density as a function of the normalized distance from the electrolyte. Bottom: local Faradaic current density as a function of the distance from the electrolyte. Legends show electrode thickness in microns. ....................................104 Figure 5-4. Surface area loss has a stronger impact on thin electrodes than on thick electrodes, because thin electrodes are more fully utilized in the intact state, whereas thick electrodes experience a redistribution of the electrochemical activity away from the electrolyte, thereby tolerating larger uniform surface area loss. ESC configuration..............................105 Figure 6-1. Normalized resistance results for 30% cathode area delamination (button cell) and for a delamination of 36% of the cathode length (two ribs and two channel widths in a rectangular cell) are almost insensitive to input parameter variation, and no unexpected interaction is apparent. Complete list of values corresponding to each run # are in Appendix IV, Table IV-1, page 162. ....................................................113 Figure 6-2. Some variability is observed in the normalized resistance results for 5 micron interconnect oxide layer growth. The variability in polarization resistance is associated with the non-symmetric polarization behaviour of the cathode. Series resistance variability occurs because electrolyte thickness is among the varied input parameters. If electrolyte thickness is fixed at the default value (thin x line), the variability decreases. Complete list of values corresponding to each run # is in Appendix IV, Table IV-2, page 163...............................................115 Figure 6-3. While the normalized series resistance shows, as expected, no variability, a high level of interaction among parameters yields a large variability in the results for polarization resistance. The kinetic parameters, inlet partial pressures, and the operating point are responsible for much of this large variability. This variability decreases when these parameters are fixed. The simulation shows 90% loss in the anode active area. Complete list of values corresponding to each run # is Appendix IV, Table IV-2, page 162. ..................................................116 Figure 6-4. Uniform design experiment corresponding to 90% surface area loss due to sulphur poisoning, for the anode-supported configuration. After fixing the parameters responsible for the scatter, the magnitude of this scatter is smaller than it was in the ESC case. Complete list of values corresponding to each run # is in Appendix IV, Table IV-3, page 163............................................................................................................................117 Figure 6-5. Cathode polarization resistance dependence on overall (external) cathodic overpotential vs. 1.12V. Bold line: standard case of non-equal cathodic charge transfer coefficients (1.5 and 0.5). Thin line: Equal cathodic charge transfer coefficients (0.5 and 0.5)................................119 Figure 6-6. Both series and polarization resistance show a variability that is comparable to that of oxide layer growth. Cathode porosity and thickness, the factors that affect cathode conductance the most, are responsible for the series resistance variability (solid circles). When these parameters are locked at the default values, the series resistance variability decreases (thin line). Complete list of values corresponding to each run # are in Appendix III, Table III-2, page 162. ........................................121 Figure 6-7. Both Rp magnitude and Rp scatter increase when the number of detached ribs increase from 2 to 3. Equivalently to the 2-rib case, the scatter in Rs decreases upon locking the parameters relevant to cathode electronic conductance: porosity and thickness Complete list of values corresponding to each run # are in Appendix III, Table III-2, page 162. ................122 Figure 6-8. Interconnect detachment of two ribs on the anode side presents a lower scatter than it does on the cathode side, because of the very high electronic conductance of the anode. Complete list of values for each run # in Appendix IV, Table IV-3, page 163...........................................................122 Figure 6-9. Impedance spectra corresponding to the first three stages of combined sulfur poisoning + interconnect oxidation degradation shown in Table 6-3. Above: Complete spectrum. Below: Detail of the anodic contribution. VCELL = 0.7V. ....................................................................................127 Figure 6-10. Approximate overall degradation path followed by the mechanisms under study, when acting individually. ...............................................128 Figure 6-11. Degradation path plot showing the normalized resistance behaviour of:  Interconnect oxidation plus sulfur poisoning, and  cathode delamination. Although the final degradation state is approximately the same for both cases, the difference in degradation xi history presents a possible way of distinguishing between the two cases. The error bars correspond to the scatter calculated in section 6.2. ESC. .........................................................................................................................130 Figure 6-12. Degradation path plots showing two different degradation scenarios: combination of delamination with interconnect oxidation (filled circles), and sulfur poisoning (filled triangles). The delamination, oxide layer growth, and sulfur poisoning characteristic paths are shown for comparison. Electrolyte-supported configuration. The error bars correspond to the scatter calculated in section 6.2. ..........................131 Figure 6-13. Degradation plot for the same combinations shown in Figure 6-12, for the anode-supported configuration. The error bars were recalculated by re-running the 30 uniform design runs for the anode- supported configuration. ..........................................................................................132 Figure 6-14. Normalized resistance of the two combined degradation scenarios of the example above, as a function of the result of their direct addition, according to equation (3), with the assumption of no interaction. The error bars correspond to the scatter calculated in section 6.2. ...............................................................................................................134 Figure I-1. Geometry details. Upon a change in physical properties, a rectangle between the rib and the cathode becomes an oxide layer or a detachment, and a rectangle between the cathode and the electrolyte becomes a delamination...........................................................................................147 Figure I-2. Global meshing strategy and detail showing densification around the rib/cathode/channel corner.....................................................................148 Figure III-1. Polarization curve fitting of the present model (line) to experimental data (dots) presented by Kim et al. Adapted from Fig 3a, J. Electrochem Soc. 146 (1) 69-78 (1999). ..............................................................165 xii Nomenclature Note: The mathematical model uses SI units throughout. However, it is customary in the SOFC community to normalize currents using centimeters, because the relevant electrode areas of both test cells and commercial prototypes are of the order of a few square centimeters. Therefore, cm2 is the preferred unit to express output values for current densities and area-specific resistances. Likewise, millimeters occasionally describe geometrical quantities such as lengths and widths. Symbol units anot anode thickness m ASC anode supported cell APU auxiliary power unit elyt electrolyte thickness m catt anode thickness m dlC electrochemical double layer capacitance F m -2 cy concentration of species y mol m-3 c0y reference concentration of species y mol m-3 pc steady state concentration of species p mol m -3 pc ~ oscillatory concentration of species p mol m-3 Deffp-q binary effective diffusivity m2 s-1 ESC electrolyte supported cell f frequency Hz F Faraday’s constant Coul mol-1 pqf RT Fpqα V-1 i current density A m-2 or A cm-2 iF Faradaic current density A m-2 iION ionic current density A m-2 or A cm-2 iELE electronic current density A m-2 or A cm-2 i0,q exchange current density electrode q A m-2 j imaginary unit - k generic conductivity S m-1 kp,q effective conductivity of species p, electrode q S m-1 L cell length or radius m M number of concurrent degradation modes - n number of electrons transferred in a reaction - n normal unit vector - Np flux of species p [species] m-2 s-1 NTOT total (bulk) species flux [species] m-2 s-1 xiii OCV open circuit voltage V PDE partial differential equation p system gas pressure Pa or atm pq partial pressure of gas q Pa or atm Q generic species source term [species] m-3 s-1 r radial coordinate Rg universal gas constant J mol-1 K-1 R* universal gas constant cal mol-1 K-1 Rs series resistance ohm cm2 Rp polarization resistance ohm cm2 sR normalized series resistance - pR normalized polarization resistance - S electrochemically active surface area m-1 tp thickness of layer p microns T absolute temperature K CELLV operating cell potential V xy molar fraction of species y - xp,q volume fraction of p phase in electrode q - z axial coordinate Z area specific impedance ohm cm2 αpq charge transfer coefficient, electrode p, direction q - pε porosity fraction of electrode p - pη local overpotential of electrode p V η REFIONELE Φ−Φ−Φ V η~ IONELE Φ−Φ ~~ V pτ gas phase tortuosity of electrode p - Φ generic electric potential V ELEΦ electronic potential V IONΦ ionic potential V REFΦ reference potential V ELEΦ steady-state electronic potential V ELEΦ ~ AC electronic potential V IONΦ steady-state ionic potential V ON ~ IΦ AC ionic potential V 0 ~Φ∆ perturbation amplitude V ω angular frequency s-1 xiv Acknowledgements I am most grateful to my research supervisor, Dr. Olivera Kesler, for her trust and support throughout the duration of my PhD project. Thank you, Olivera, for so much time, patience and intellectual generosity dedicated to my research work. The knowledge, experience, and patience of Professor Colin Oloman were essential in shaping my research work and thesis. Thank you very much to all my other Committee members: Drs. Walter Mérida, Brian Wetton, and Gary Schajer for their input throughout the years. This work would have never been possible without the unconditional love and understanding of my wife Verónica and my son Matias. xv a Verónica y Matias xvi Co-Authorship Statement The research work presented in this thesis (and the publications stemming from it) corresponds to the work of the author, under the supervision of Dr. Olivera Kesler, co- author of the publications. The author of the thesis was responsible for the development of the equations, and code programming, and Dr. Kesler provided continuous academic guidance, critical review of the results, and supervision. 1 1. Chapter One. Introduction and Background. 1.1 Introduction Modern lifestyles have led to a relentless increase in energy consumption. Traditional ways to generate power include combustion of fossil fuels and coal, and hydroelectric and nuclear energy conversion. An undesirable consequence of fossil fuel and coal combustion, the most widespread power generation methods, is environmental contamination, as a result of the products of combustion released to the atmosphere. In addition, the uneven distribution of the finite fossil fuel sources worldwide causes geopolitical unrest. Therefore, there is a need for better ways to satisfy the energy demands of society. Fuel cells appear as an attractive alternative to traditional power generation methods. A fuel cell is a reactor that generates electricity by combining a fuel and an oxidant electrochemically, rather than thermochemically, as is the case with fossil fuel combustion. Electrochemical power generation has many advantages over fossil fuel combustion, including higher efficiency, low pollution, low equipment maintenance, and modularity. The fuel cell receives the reactants that take place in the energy conversion process in a continuous manner, unlike batteries, which use chemical energy that is stored within the electrodes. A fuel cell based stationary generation plant could also provide independence of a power distribution grid, an attractive choice for remote settlements and mining operations. Since the first demonstration of the fuel cell principle described by Sir William Grove in 1839, many types of fuel cells have been developed. Among them, Solid Oxide Fuel Cells (SOFC) appear especially suitable for stationary electricity generation, both for power plant scales, and for small domiciliary scale. 1.2 The Solid Oxide Fuel Cell (SOFC) The SOFC is a completely solid-state type of fuel cell that presents many advantages as an energy conversion device, especially for stationary applications. The main components of a fuel cell are the electrolyte and the electrodes. The electrolyte is 2 an ionically conducting ceramic membrane that prevents direct combination of fuel and oxidant. This membrane is sandwiched between two porous, electronically (or mixed) conducting electrodes: the anode, or fuel electrode, and the cathode, or air electrode. The electrolyte is a dense, gas-tight ceramic layer, traditionally made of yttria-stabilized zirconia (YSZ), whose high mobile vacancy content makes it a good conductor of oxide ions, O2-. An electronically conductive, gas tight interconnect plate provides the series connection of unit cells, constituting an SOFC stack. Because ionic conductivity in ceramics is a thermally activated process, high operating temperatures are required to obtain sufficient overall conductivity in the fuel cell. YSZ exhibits acceptable conductivity in the 700º - 1100ºC temperature range. The oxide ion as the main charge carrier allows the use of a variety of fuels other than hydrogen, such as carbon monoxide or even hydrocarbons, either by direct oxidation or via steam reforming. Another important feature of the high operating temperature is the possibility to use the exhaust heat in a power generating bottoming cycle, using either a gas or steam turbine. These features give the SOFC great versatility compared to other fuel cell types, as well as creating the possibility of generating electrical power at very high efficiencies. On the other hand, the elevated temperature of operation requires long start-up periods, which are inconvenient for mobile applications, and results in large thermal stresses on the delicate ceramic components during thermal cycling to and from the operating temperature, which jeopardizes the device performance. In addition, the uneven distribution of electrochemical reaction sites also generates large thermal gradients that are a source of stress. Finally, fabrication of the cell components also requires high temperatures that can cause detrimental residual stresses within the cell components. 1.2.1 Principle of Operation Oxygen molecules in the cathode pores dissociate and combine with free electrons coming from the external circuit, and the resulting oxygen ions migrate through the electrolyte toward the anode. In the anode, the oxygen ions combine with hydrogen, carbon monoxide, or hydrocarbon molecules to form water (the only exhaust product if 3 H2 is used) and CO2. The electrons released in the oxidation reaction have to escape through an external circuit, creating usable electricity and closing the loop. Figure 1-1 depicts the case of hydrogen being used as a fuel. H2 H2O O2 O2- e- cathode electrolyte anode 1/2 O2 + 2e-  O2- H2+ 1/2 O2-  H2O + 2e- e- Figure 1-1. Principle of operation of an SOFC unit cell A unit cell generates approximately 1V at open circuit conditions, with exact voltage depending on temperature, oxygen partial pressure, and reactant and product concentration. This value is commonly known as open-circuit voltage, or OCV. Thus, most practical applications require a series connection between adjacent cells to increment this voltage. The series connection is provided by the interconnect layer, which also acts as a gas separator in the planar configuration. Under operating conditions, the delivered voltage is smaller than the OCV due to irreversible losses or cell polarization. In ceramic fuel cells, the most important contribution to these losses is typically the ohmic polarization, i.e. the voltage drop due to internal resistance. Table 1-1 shows the most commonly used materials for each cell unit of planar SOFC stacks. Each stack component is the subject of a great deal of research, aimed at optimizing its chemical, mechanical and micro-structural characteristics and compatibility with other cell components, as well as improving the fabrication methods and long-term stability of the cells. Ivers-Tiffée et al. published a very good summary of SOFC material properties in [1]. 4 Table 1-1. Traditional materials used in planar SOFCs Component Material Denomination Acronym Anode Ni - Yx Zr1-x O2-x/2 nickel-yttria stabilised zirconia Ni-YSZ Electrolyte Yx Zr1-x O2-x/2 yttria stabilised zirconia YSZ Cathode SrxLa1-xMnO3-δ + Yx Zr1-x O2-x/2 doped lanthanum manganite LSM-YSZ Interconnect stainless steel - A great variety of techniques has been developed to fabricate each component. Traditionally, each layer was deposited on top of the substrate (typically the electrolyte or the anode) by a wet method, which consists in painting the surface with a well-mixed slurry and subsequently drying it in a furnace and consolidating it by sintering, or creating pores by adding a pore-former to the slurry [2] in the case of the electrodes. cell interconnect air fuel Figure 1-2. Cross-flow configuration for a planar-stack unit cell Internal or external manifolds can be used to transport reactants to the reaction sites. The reactants spread over the plane of the cell by means of channels of varying thickness and pattern. These channels are usually grooved on the interconnect surface. In planar SOFC stacks, the reactants can be manifolded in different ways: co-flow, counter-flow, cross-flow (Figure 1-2), or radial flow. 5 1.3 Degradation Phenomena in Solid Oxide Fuel Cells The widespread commercialization of SOFCs is subject to the solution of problems related to their durability, reliability, and cost. Associated with durability and reliability is the important problem of cell degradation. The severe environmental conditions within an SOFC limit the choice of suitable materials and poses important challenges to the stability of the different cell components. The mismatch in thermal expansion characteristics of the cell’s different layers is a major cause of stresses during thermal cycling, during both fabrication and operation. In addition, the uneven thermodynamics of chemical reactions throughout the cell also generates large thermal gradients that are also a source of stress. A consequence of the aforementioned thermo-mechanical phenomena is the degradation of the contact between adjacent cell layers, namely the electrolyte-electrode interface, or the interconnector- electrode interface. This degradation can lead to detachment of two adjacent layers, with the consequent loss of conductive area. Another example of a degradation mechanism affecting SOFCs is the growth of an electrically less conductive oxide layer between the interconnect plate and the electrodes, especially the cathode. A consequence of this phenomenon is also an increase in the cell’s internal resistance, due to the increased Ohmic resistance along the conductive path. The elevated temperature of operation favours the kinetics of chromium oxide growth on the surface of the stainless steel interconnect plate. Extended periods of operation at elevated temperature and exposure to contaminants contained in the reactants are responsible for another important phenomenon: microstructural degradation. The electrochemical combination of reactants in a fuel cell occurs inside the porous electrodes. In state-of-the-art SOFCs, these reactions are not restricted to the interface with the electrolyte, but take place within a few microns to tens of microns into the volume of the electrode. The (3D) electrode microstructure is of great importance in providing adequate active surface area for electrochemical reaction and sufficient porosity for gas species transport. Consequently, degradation mechanisms that affect electrode microstructure can compromise the cell performance, stability, and durability. Examples of microstructure degradation mechanisms include: 6 1. Sintering of the electrode microstructure, typically in the nickel-YSZ cermet anode [3,4]. Solid mobility phenomena are thermally favoured. The fine-grained nickel in the porous anode shows a tendency to agglomerate after long exposure to high temperatures. This evolution results in loss of active surface area and changes in the porosity distribution, thereby lowering overall performance. 2. Poisoning of the cathode microstructure with chromium products outgassed from the interconnect plates [5,6]. This electrochemically driven degradation mode has been extensively reported, and constitutes a major disadvantage of stainless steel interconnectors. The chromium rich compound deposits preferentially on the most electrochemically active sites, resulting in a loss of active area. 3. Carbon deposition in the anode [7]. An undesirable reaction favoured by nickel, which catalizes coking in addition to hydrogen oxidation. Solid carbon deposits on nickel, affecting electrochemistry, diffusion, and structural integrity. 4. Sulfur poisoning of the anode, a consequence of the traces of sulphur present in hydrocarbon fuels or hydrogen produced by reforming [8,9]. Sulfur present in small quantities as H2S or thiols adsorbs on nickel as a molecular monolayer, blocking electroactive sites. Nonetheless, unlike the two mechanisms previously mentioned, sulfur desorbs upon re-establishment of a sulfur-free fuel stream, making it reversible at low concentrations. Furthermore, it does not affect the anode gas diffusion properties, at least at early stages of poisoning. Despite their difference in nature, the effect of these degradation modes on cell performance measured in long-term degradation testing is common to all of them: a loss in available potential at constant current load, or a drop in delivered current density at constant voltage. In other words, it is not possible to identify a specific degradation mechanism or combination of mechanisms by observing only changes to the DC behaviour of the cell. Thus, there is a need to develop a diagnostic technique that allows the identification of specific degradation mechanisms of SOFCs in a minimally invasive way. Such a technique would be a useful tool for diagnosis of both a cell in service or under research, where identifying the nature of a degradation mechanism could save the disassembly time required for a direct observation. In the case of reversible degradation, the method would indicate the possibility to correct the failure while the fuel cell is in 7 operation. In the case of irreversible degradation, the method would aid in identifying the specific cause of failure of a component that needs replacement, so that operating conditions or cell or stack materials or designs could potentially be adjusted in subsequent tests to minimize further degradation. At a research stage, a diagnostic tool of this kind would be a useful complement to post-mortem microstructural observation. A minimally invasive tool that provides a wealth of information about electrochemical systems, and that has become a standard in fuel cell research, is electrochemical impedance spectroscopy [10]. Impedance spectroscopy is a well-known technique used to study electrochemical characteristics of systems such as batteries, capacitors, and fuel cells, and which is widely used in disciplines such as corrosion and materials science. It consists of measuring the impedance of a system at different frequencies by superimposing a small voltage or current perturbation onto the voltage or current operating point. The underlying idea is that individual processes will appear in the impedance spectrum at different frequencies, according to their inherent rate. For example, a charge transfer process is likely to appear at higher frequencies than a slower diffusional process. This technique is convenient because it is minimally invasive, and it can be used on systems in operation. Its resolution in frequency allows the separate study of simultaneous cell processes, potentially enabling their individual optimization. As mentioned above, a common observation during SOFC single-cell testing at R&D stages is a change in performance over time. Regardless of its origin, this change is usually manifested as a decrease in output power, and, if this measurement is available, as an evolution of the impedance spectrum, which varies in shape and size over time. Although the state-of-the-art knowledge provides general ideas about the nature of the degradation mechanism that is taking place within the cell or stack, there is still no systematic procedure to link a certain degradation mode to its effect on the impedance spectrum. A technique that is capable of providing information about the nature of the degradation modes affecting the cell or stack would be most useful to a researcher who is testing new materials, geometries, or fabrication procedures. In summary, SOFC degradation is an important research area, and successful commercialization of SOFCs depends to a large extent on the understanding of the 8 degradation phenomena. Table 1-2 gives a summary of recent research done in the area of SOFC degradation. This thesis focuses on degradation modes associated with contact impedance, namely electrode delamination, interconnect oxidation, and interconnect detachment, together with the simplest case of microstructural degradation, sulfur poisoning. Details of relevant previous work done in this area will appear in the appropriate chapters. Table 1-2. Study of SOFC degradation Research Group Main Researchers Field of Study Ref Illinois I. of T. Y. Hsiao, R. Selman Degradation of SOFC electrodes [11] Risø Nat. Lab. A. Hagen, R. Barfod, P. Hendriksen, Y. Liu, S. Ramousse, S. Primdahl, M. Mogensen Delamination Chromium poisoning Anode stability [12,13] FZ Jülich D. Simwonis, F. Tietz, D. Stöver Nickel coarsening [3] Fuji Electric T. Iwata Nickel coarsening [4] Univ. Calgary S. Paulson, V. Birss Chromium poisoning [5] FZ Jülich E. Konysheva et al. Chromium poisoning [6] Univ. Calgary S. Xia, V. Birss Sulfur poisoning [8] Georgia Inst. Tech. S. Zha, Z. Cheng, M. Liu Sulfur poisoning [9] EPFL D. Larrain, J. van Herle, D. Favrat Interconnect oxidation [14] Tokyo Gas K. Fujita et al. Interconnect oxidation [15] Georgia Inst. Tech. G. Nelson, C. Haynes Delamination [16] Univ. Karlsruhe M. Heneka, N. Kikillus, E. Ivers-Tiffeé Delamination [17] PNNL S. Simner et al. Delamination [18] Tohoku Univ. K. Sato, J. Mizusaki, et al. Delamination and cracking detection using acoustic emmission [19] FZ Jülich J. Malzbender, T. Wakui, R. Steinbrech, L. Singheiser Interconnect detachment [20] Univ. Florida J. Smith, E. Wachsman Interfacial reactions [21] 1.4 SOFC Modeling Increasing computational power at affordable costs has permitted the widespread use of modeling as an invaluable tool for design, optimization and analysis of fuel cells at many levels, from detailed electrochemistry electrode level to system, cell and balance- of-plant level. Models can be used to predict performance variations upon the change of 9 a certain design parameter(s), as well as to identify the relative importance of operational variables. Advantages of modeling analysis include cost reduction, both from time and materials saving, and flexibility of parameter variations. Disadvantages include difficulty in the interpretation of results and in conducting unambiguous experimental validation, and the challenges involved in the choice of reliable input parameters. A fuel cell is a complex nonlinear system where several interacting processes occur simultaneously in a highly convoluted manner, and care must be taken in the choice of input parameters since the same overall results may be obtained using different combinations of these parameters. A core contribution of this thesis is the combination of an SOFC impedance model with degradation. Modeling the impact of degradation on performance and on the impedance of a working SOFC requires, in the first place, modeling the intact cell. Several numerical simulation studies currently published assume lateral invariance of the modeled quantities (e.g. [24,27,29,31,33]), leading to the important simplification of the governing equations to one dimension, i.e. the main direction of current flow. Although this assumption is useful to study the cell performance under certain conditions, it cannot account for effects that are inherently two- or three-dimensional, such as the uneven reactant concentration distribution within the porous electrode resulting from the ribbed interconnect geometry. Furthermore, many degradation modes do not have a uniform impact on the cell components such that they can be represented using a one-dimensional model. A clear example of this statement is electrode delamination, which leads to an in- plane redistribution of the current within the cell, destroying any initial in-plane invariance of the system. A requirement for modeling cell degradation is to have detailed information about the cell processes at the electrode level. The need to know the current redistribution after degradation is an example that supports this need. Modeling at the electrode level provides information about space distribution of operating variables such as currents, temperature, and concentrations. The charge, energy, and mass conservation equations describe the behaviour of these quantities within the different cell components. In solving these equations, the macrohomogeneous modeling approach is especially suitable if no a priori assumption is made on the relative importance of the transport processes. 10 Newman and Tobias first described this approach in 1962 [22]. In that work, the authors treat the porous electrodes as continuum materials characterized by effective properties, corresponding to their averaged porous composite properties. Every phase, ionic conductor, electronic conductor, and pore, coexists at every point in the modeled electrode in proportion to its presence in the real electrode. A balance equation is solved for each species throughout the electrode, with appropriate boundary conditions. The microscopic details of each of the reactions that take place within the SOFC electrodes remain largely unknown despite decades of research in the field. Although there is general consensus in that the electrochemical combination of fuel and oxidant takes place at or near the triple-phase-boundary in a porous solid electrode, there is disagreement on the details of the elementary processes that constitute the overall reactions. The characteristics of these processes seem to depend on the type of reactants, the electrode and electrolyte materials and microstructure, the operating temperature, and the power demand. From a modeling perspective, these uncertainties pose the need for a number of assumptions about the nature and characteristics of the modeled processes. Furthermore, the difficulty (or impossibility) in measuring certain quantities results, unavoidably, in the need to treat some input variables as free parameters, whose values would be determined by fitting to experimental data. Several research groups have developed or are currently developing SOFC impedance models. Table 1-3 summarizes this information, indicating the main topic of their modeling work. In [24], Adler et al. postulate mechanisms other than charge transfer as responsible for the Gerischer-type impedance observed in solid mixed conducting electrodes. Their simulation is exemplified with a model of an LSFC cathode on SDC. In [25], Fleig and Maier study the influence on the impedance of imperfect contact between porous electrodes on solid electrolytes, using finite elements. The anode reaction is studied in [26], where Bieberle and Gauckler propose a series of steps as responsible for the overall electrochemical oxidation of hydrogen. Also interested in the anode reaction, Bessler [27] modeled the impedance of patterned and porous anodes, in an attempt to assess the validity of the mechanism postulated in [26]. In [28], Bessler applies a simpler approach to reproduce the conversion impedance arc reported by Primdahl and Mogensen [23] in their anode experiments. Using a two-dimensional 11 geometry that describes the fuel impingement on the anode, the author investigates the influence of gas flow rates and concentration variations on the impedance. Table 1-3. Recent research work on impedance modeling of SOFC-relevant electrochemical systems Research Group Main Researchers Field of Study Ref Ceramatec, Imperial College S. Adler, J. Lane, B. Steele Electrode kinetics of porous mixed ionic- electronically conductive electrodes [24] Max-Planck Int. Festkörperforsch. J. Fleig, J. Maier Influence of current constriction on impedance [25] ETH Zürich A. Bieberle, L. Gauckler Anode processes, state-space modeling [26] Univ. Heidelberg W. Bessler Anode processes, concentration effects [27, 28] SINTEF S. Sunde Monte Carlo simulations, porous electrodes [29] Univ. Karlsruhe H. Schichlein, E. Ivers-Tiffeé Deconvolution of impedance spectrum [30] LEPMI J. Deseure, Y. Bultel, L. Dessemond Cathode processes, influence of electrode parameters [31] Univ. Twente B. Boukamp Equivalent circuit approach to study impedance behaviour of alternative anodes [32] Colo. Sch. Mines H. Zhu, R. Kee 1-D Time-domain analysis of button cells using H2 or CH4 [33] Sunde [29] utilized an alternative to the equivalent circuit approach, using Monte Carlo simulations to calculate the impedance of porous anodes. His original approach, albeit computationally intensive, allowed him to account for non-percolating clusters in the porous microstructure. Schichlein et al. [30] developed a promising technique to increase the resolution of impedance deconvolution, proposing an infinite string of RC parallel circuits as an alternative to the traditional equivalent circuit approach that lumped each electrode’s impedance properties into a more or less complicated set of circuit elements. The group at LEPMI simulated the impedance behaviour of one-phase mixed conducting cathodes [31]. In their 1-D model, the authors solved the O2- transport equations assuming Langmuir adsorption of oxygen and solid-state diffusion as the governing processes, with charge transfer in equilibrium. The authors predict a non- monotonic decrease-increase in polarization resistance with increasing cathodic bias (Figs 2b, 3b in [31]), and a monotonic increase in polarization resistance with increasing oxygen partial pressure (Fig 5a, [31]). This prediction is in contradiction with respect to 12 the commonly encountered behaviour of Butler-Volmer kinetic electrodes (e.g. Fig 3 in [34], and Fig 4 in [35]), probably because of the fact that the assumed limiting process in [31] is solid-state diffusion. The milestone work described in [32] describes the analysis of electrochemical systems by linking a circuit element to electrochemical processes such as charge transfer, double-layer charging, and diffusion. Values for the circuit elements are assigned using complex non-linear least-squares fitting. The work of Zhu and Kee [33] describes a 1-D model of the impedance of a button cell where the authors consider methane reforming among the modeled reactions. They use finite volumes to solve the problem in the time domain, letting the response stabilize after a certain number of periods. The system is, however, assumed to respond linearly to the small amplitude of the perturbation. The exchange current is a fitting parameter to reproduce typical performance, and the charge transfer coefficients are taken as 1.5 and 0.5 for the anodic and cathodic directions, respectively, for both electrodes. A common feature of all these modeling approaches, except for the recent anode model published by Bessler [28], is the assumption of in-plane uniformity, leading to the simplification of the system to one dimension: the main direction of current. This assumption is not applicable in the present case, since degradation modes such as electrode delamination and interconnect detachment dramatically modify the direction of the current path, making the problem inherently two- or three-dimensional. Moreover, the presence of the ribbed interconnect in contact with the electrode also produces local reactant depletion underneath the ribs, thereby invalidating the assumption of in-plane invariance, as will be shown in later chapters of this work. Modeling in two or three dimensions is clearly essential in view of the objective sought by this thesis. A second important aspect is that the electrode-level impedance simulations have mostly focused on one electrode at a time. This is true for all the cited previous work except for [33]. Using the method proposed in this thesis in practical-size, operating SOFCs requires modeling the full cell, and also not counting on a reference electrode to separate the individual contributions to the impedance of each electrode, since reference electrodes are not feasible in practical SOFC systems, especially those with thin electrolytes. A third aspect to take into account is computational cost. In the first place, 13 gradients close to the electrochemical interfaces in SOFCs require local densification of the numerical grid. Secondly, two-dimensional modeling demands slender elements due to the large aspect ratio of the planar SOFC. In the third place, having flexibility in the creation of the modeled geometry was an attractive option to consider, so as to test the model results on diverse supporting configurations, as well as to perform important parametric studies. These requirements do not pose major difficulties to one-dimensional models, since the number of total degrees of freedom in such models is limited. In the present case, the equations, the geometry, and the meshing were developed taking into account the computational requirements that were specific to the problem being solved. Isothermal and isobaric conditions are an assumption common to all the aforementioned research work, as well as to the work presented in this thesis. 1.5 Scope of this thesis This thesis explores, from a modeling viewpoint, the possibility of using impedance spectroscopy as a minimally invasive tool to diagnose degradation in a working SOFC. The work focuses on contact resistance degradation modes, namely electrode delamination, interconnect detachment, and oxide layer growth between the interconnect and the electrode, and one microstructural degradation mode, sulfur poisoning. Chapter Two presents a novel SOFC impedance model based on the finite element method. In subsequent chapters, geometrical, physical, and electrochemical parameter modifications simulate different degradation conditions. This is the first time that an SOFC model combines impedance behaviour and degradation. Chapter Three focuses on electrode delamination, describing its effects, electrochemical implications, and impedance signatures. A comparison with published experimental observations supports the modeling results. Chapter Four describes two further contact degradation problems, this time related to the interconnect-electrode interface: interconnect oxidation and interconnect detachment. An analysis of their impact on performance and on the impedance spectrum reveals interesting differences with delamination, as well as possible ways to diagnose them. Reducing the anode electro-active surface area simulates sulfur poisoning in Chapter Five, and an analogous analysis is presented in terms of diagnosis. Chapter Six studies a scenario of two simultaneous degradation modes and assesses the 14 method’s capability to diagnose them. Secondly, it presents a sensitivity analysis of the presented results to variations in the model’s input parameters. Chapter Seven presents the overall conclusions, and recommendations for future work stemming from this thesis. Original research contributions of this thesis include: 1. The development of the first two-dimensional impedance model of a working full-cell SOFC, and the first one to include the ribbed interconnect geometry. 2. The incorporation of degradation features into an impedance model of an SOFC. 3. The linearization of the time-dependent AC equations around the operating point to yield a linear, time-independent, complex-valued system of equations for the charge and material balance for an SOFC. 4. The definition of novel and convenient non-dimensional variables that represent extent of degradation, and that are indicative of the degradation mechanism present in the SOFC. 5. A study of the impact of inaccurate estimation of, and of the interaction among input parameters on the model predictions. 6. A study of possible interactions among different degradation modes and input parameters. 7. The coding of an automated generic geometry and mesh generator for use in any finite element two-dimensional analysis of a fuel cell comprising a ribbed interconnect and a cell, using high aspect ratio rectangular elements. Not restricted to just SOFCs, this code is expected to be useful in performing parametric studies involving geometrical and physical variables on a variety of fuel cell types. 15 1.6 References [1] E. Ivers-Tiffée, Q. Weber, D. Herbstritt, “Materials and technologies for SOFC- components”, J. Eur. Ceram. Soc., 21, 1805-1811 (2001). [2] M. Boaro, J. Vohs, R. Gorte, “Synthesis of highly porous YSZ by tape-casting methods”, J. Am. Ceram. Soc, 86 [3], 395-400 (2003). [3] D. Simwonis, F. Tietz, and D. Stöver, “Nickel coarsening in annealed Ni/8YSZ anode substrates for solid oxide fuel cells”, Solid State Ionics 132, 3-4, 2 (2000) 241-251. [4] T. Iwata. “Characterization of Ni-YSZ Anode Degradation for Substrate-Type Solid Oxide Fuel Cells”, J. Electrochem. Soc. 143 (1996), 1521. [5] S. Paulson and V. Birss, “Chromium Poisoning of LSM-YSZ SOFC Cathodes I. Detailed Study of the Distribution of Chromium Species at a Porous, Single-Phase Cathode”, J. Electrochem. Soc., 151, 11, (2004) A1961-A1968 [6] E. Konysheva et al, “Chromium Poisoning of Perovskite Cathodes by the ODS Alloy Cr5Fe1Y2O3 and the High Chromium Ferritic Steel Crofer22APU”, Journal of The Electrochemical Society, 153 (4) A765-A773 (2006) [7] H. He and J. Hill, “Carbon deposition on Ni/YSZ composites exposed to humidified methane”, Applied Catalysis A: General, 317, 2, (2007), 284-292. [8] S. Xia and V. Birss, “Deactivation and recovery of Ni-YSZ anode in H2 fuel containing H2S”, Proc. SOFC IX, Vol. 2, 2005, 1275-1283. [9] S. Zha, Z. Cheng, and M. Liu, “Sulfur Poisoning and Regeneration of Ni-Based Anodes in Solid Oxide Fuel Cells”, Journal of The Electrochemical Society, 154 (2) B201-B206 (2007) [10] E. Barsoukov and J. MacDonald (editors), “Impedance Spectroscopy”, 2005. [11] Y. Hsiao and R. Selman, “The degradation of SOFC electrodes”, Solid State Ionics 98, (1998) 33-38 [12] R. Barfod, M. Mogensen, T. Klemensoe, A. Hagen, Y. Liu, “Detailed characterisation of anode supported SOFCs by impedance spectroscopy”, Proc. SOFC IX, Vol. 1, 2005, 524-533. 16 [13] A. Hagen, R. Barfod, P. Hendriksen, and Y. Liu, “Effect of operational conditions on long term stability of SOFCs”, Proc. SOFC IX, Vol. 1, 2005, 503-513. [14] D. Larrain, J. van Herle, D. Favrat, “Simulation of SOFC stack and repeat elements including interconnect degradation and anode reoxidation risk”, J. Power Sources 161, (2006) 392-403 [15] K. Fujita et. al., “Relationship between electrochemical properties of SOFC cathode and composition of oxide layer formed on metallic interconnects”, J. Power Sources 131, (2004) 270-277 [16] G. Nelson, “Solid oxide cell constriction resistance effects”, MSc Thesis, GIT, 2006. [17] M. Heneka, N. Kikillus, and E. Ivers-Tiffée, “Design of experiments for lifetime modelling of SOFC” , Proc. SOFC IX, Vol. 1, 2005, 908-919. [18] S. Simner, M. Anderson, M. Engelhard, and J. Stevenson, “Degradation Mechanisms of La–Sr–Co–Fe–O3 SOFC Cathodes”, Electrochemical and Solid- State Letters, 9 (10) (2006) A478-A481 [19] K. Sato et al. “Mechanical damage evaluation of SOFC under simulated operating conditions”, J. Ceram. Soc. Japan, 113 (8) 2005 562-564 [20] Malzbender, J., Wakui, T. and Steinbrech, R. W., “Deflection of planar solid oxide fuel cells during sealing and cooling of stacks”. Proceedings of the sixth Euro. SOFC Forum, vol 1 (2004) 329–338. [21] J. Smith and E. Wachsman, “Effect of harsh anneals on the LSM/YSZ interfacial impedance profile”, Electrochimica Acta 51 (2006) 1585–1591 [22] J. Newman and C. Tobias, “Theoretical analysis of current distribution in porous electrodes”, J. Electrochemical Soc., Vol. 109, No. 12, 1962, 1183-1191. [23] S. Primdahl and M. Mogensen, “Gas Conversion Impedance: A Test Geometry Effect in Characterization of Solid Oxide Fuel Cell Anodes”, J. Electrochem. Soc., Volume 145, Issue 7, pp. 2431-2438 (1998) [24] S. Adler, J. Lane, and B. Steele, \"Electrode Kinetics of Porous Mixed-Conducting Oxygen Electrodes\", J. Electrochem. Soc., 144, (5), (1997) 1884-1890 17 [25] J. Fleig and J. Maier, “The influence of current constriction on the impedance of polarizable electrodes”, J. Electrochem. Soc., 144, (11), (1997) L302-L305 [26] A. Bieberle and L. Gauckler, “State-space modeling of the anodic SOFC system Ni, H2–H2OYSZ”, Solid State Ionics,146, 1-2, (2002), 23-41 [27] W. Bessler, “A new computational approach for SOFC impedance based on detailed electrochemical reaction-diffusion models,” Solid State Ionics 176, (2005) 997-1011. [28] W. Bessler, \"Gas concentration impedance of solid oxide fuel cell anodes. I. Stagnation point flow geometry,\" J. Electrochem. Soc. 153, (2006) A1492-A1504. [29] S. Sunde, \"Calculations of Impedance of Composite Anodes for Solid Oxide Fuel Cells\", Electrochimica Acta, 42, (1997) 2637 [30] H. Schichlein, A. Müller, M. Voigts, A. Krügel and E. Ivers-Tiffée, “Deconvolution of electrochemical impedance spectra for the identification of reaction mechanisms in solid oxide fuel cells”, Journal of Applied Electrochemistry 32 (2002) 875–882 [31] J. Desseure, Y. Bultel, L. Dessemond, and E. Siebert, “Modelling of dc and ac responses of a planar mixed conducting oxygen electrode”, Solid State Ionics, 176, 3-4 (2005) 235-244. [32] B. Boukamp, “A package for impedance/admittance analysis”, Solid State Ionics 18-19 (1986) 136-140. [33] H. Zhu and R. Kee, “Modeling Electrochemical Impedance Spectra in SOFC Button Cells with Internal Methane Reforming”, J. Electrochem. Soc., 153 (9) (2006) A1765-A1772 [34] C. Xia, W. Rauch, F. Chen and M. Liu, “Sm0.5Sr0.5CoO3 cathodes for low- temperature SOFCs” Solid State Ionics, 149, 1-2, (2002), 11-19 [35] A. Co, S. Xia, and V. Birss, “A kinetic study of the oxygen reeduction reaction at LaSrMnO3-YSZ composite electrodes”, J. Electrochemical Soc., Vol. 152, No. 3, A570-A576 (2005) 18 2. Chapter Two. Mathematical Model1 2.1 Introduction Several physical phenomena take place simultaneously within a working SOFC. The reactants flow along the distribution channels and diffuse into the porous electrodes toward the electrolyte, driven by the concentration gradient established due to electrochemical consumption. A result of this reaction is the production or consumption of ionic (O2-) or electronic (e-) species, which move through the ionically- or electronically-conductive phases. Ions travel across the electrolyte, and electrons flow through an external circuit, producing usable electricity. In addition, there is production or consumption of heat, depending on the type of chemical or electrochemical reaction taking place within the porous electrode, as well as heat transfer among the different cell components. These many concurrent physical processes pose important difficulties in modeling fuel cell systems, since the fundamental partial differential equations that describe the involved variables form a coupled, nonlinear system, with many parameters not independently measurable. A further complication appears when a perturbation is imposed that causes the system to evolve dynamically. Impedance studies are an example of this situation, where the system oscillates around the operating point in response to a small sinusoidal perturbation. 2.2 General Description and Assumptions The simulation of localized degradation mechanisms such as partial electrode delamination or interconnect detachment requires modeling at the electrode level, to provide information about changes in the current flow patterns. This requirement rules 1 This chapter constitutes the mathematical formulation used in the models published as: a. J. I. Gazzarri and O. Kesler, “Non-destructive delamination detection in solid oxide fuel cells”, Journal of Power Sources, 167 (2) 430-441 (2007). b. J. I. Gazzarri and O. Kesler, “Electrochemical AC impedance model of a solid oxide fuel cell and its application to diagnosis of multiple degradation modes”, Journal of Power Sources, 167 (1) 100-110 (2007). c. J. I. Gazzarri and O. Kesler, “Short stack modeling of degradation in solid oxide fuel cells - Part I: Contact degradation”, Journal of Power Sources, doi: 10.1016 / j.jpowsour.2007.10.047. d. J. I. Gazzarri and O. Kesler, “Short Stack Modeling of Degradation in Solid Oxide Fuel Cells - Part II: Sensitivity and Interaction Analysis”, Journal of Power Sources, doi : 10.1016 / j . jpowsour . 2007.10.046. 19 out one-dimensional approaches, such as equivalent circuit (e.g. [1]) or 1-D transmission- line [2] modeling, which inherently assume no variation in the plane perpendicular to the current flow. The present model describes the cell behaviour by solving the material and charge balance within the porous electrodes. Hydrogen and water are the only species considered at the anode, while oxygen diffuses through nitrogen at constant partial pressure at the cathode. The electrochemical reactions modeled at each electrode are, then: ( ) ( ) ( ) −− −− →+ +→+ 2 2 2 2 2 2 2 1 2 OecathodeO eanodeOHOanodeH ( 1 ) Hydrogen flowing through anode pores combines with an oxide ion coming from the ionically conductive phase, forming water and releasing two electrons. Oxygen gas at the cathode is reduced to oxide ions when the molecule accepts the electrons coming from the external circuit, and the oxide ion fills a vacancy within the ionic conductor, and then travels towards the anode through the electrolyte. Within the electrodes, two electrical processes take place in parallel, and with no mutual interaction (this statement implies that the double-layer capacitance is assumed independent of the local overpotential. Chapter Six of this thesis explores the impact of this and other assumptions on the reported results): 1. A charge transfer current is established between the ionic and electronic phases. This current is known as Faradaic current, and its magnitude depends on the local overpotential. Ions and electrons migrate according to the generalized Ohm’s Law, i.e. at a rate proportional to the gradient in ionic and electronic potential, respectively. 2. A non-Faradaic current flows between the two phases as a result of the charge and discharge of the interfacial double layer. The magnitude of this current depends on frequency. The electrochemical and diffusional processes are linked via the concentration dependence of the reaction rates. The nature of the rate-limiting step(s) in electrode reactions has been the subject of numerous research studies, and consensus in this regard has yet to be achieved. In the context of this model, it is sufficient to describe this 20 mechanism as a polarization- and concentration-dependent process whose result is the electrochemical (as opposed to direct) combination of hydrogen and oxygen. At least one step of the overall process at each electrode involves a charge transfer between phases. Later chapters in the thesis will present a discussion of the implications of these assumptions. Table 2-1 summarizes the fundamental and derived variables solved throughout the SOFC, the corresponding domain, and the coupling mechanisms among them. Ionic and electronic potentials are the variables solved for in the charge balance equations. These potentials determine the local overpotential within each electrode, which determines the local current density via the Butler-Volmer relationship. Table 2-1. Fundamental and derived variables solved over the model domains Fundamental Variable link Derived Variable Domain ionic potential Ohm's law ionic current density anode, cathode, electrolyte electronic potential Ohm's law electronic current density anode, cathode, interconnect hydrogen concentration mass conservation water concentration anode oxygen concentration cathode charge conservation concentration dependence of electrochemical reaction rates Coupling Internal reforming is not considered, as well as the presence of any non-reacting species other than nitrogen at the cathode. This assumption justifies the use of binary diffusion as the mass transfer process that describes the concentration profiles within the electrodes. The model presented in this thesis is restricted to two dimensions, with two different modeled configurations: a. A small circular button cell, representative of typical laboratory-scale cell geometry (Figure 2-1), resulting in an axisymmetric model. b. A small rectangular cell, including a ribbed-interconnect / current collector, which constitutes a representative unit cell in a small planar SOFC stack in co- or counter-flow configurations (Figure 2-2). The two dimensions represent the through-thickness direction (the main current path), and a transversal direction. The latter represents the radial dimension in the axisymmetric case, or an in-plane direction perpendicular to the interconnect reactant flow channels in the rectangular case. 21 L cathode electrolyte anode an o t ca tt el yt r z (axisymmetry) Figure 2-1. Modeled geometry for the button-cell configuration. Variables anot, elyt, and catt are the anode, electrolyte, and cathode thicknesses, and they vary according to the supporting configuration. Values for typical simulations are L = 10mm, anot (ESC) = 40 microns, anot (ASC) = 1mm, catt = 40 microns, elyt (ESC) = 150 microns, elyt (ASC) = 10 microns, where ASC and ESC stand for anode-supported cells and electrolyte- supported cells, respectively. The intact cell behaviour is adequately described using a single half-rib / half- channel repeating unit (inset in Figure 2-2), at substantially less computational cost compared to the geometry in Figure 2-2. Using symmetry boundary conditions along the lateral edges, this geometry is representative of a cell consisting of the side-by-side concatenation of the repeating unit. However, this configuration cannot be used to describe localized, non-uniform degradation, as will become clear in the next chapter. 22 cathode electrolyte rib cathode current collector anode current collector channel anode rib channel distance (mm) di st an ce (m m ) 0 5 10 15 20 2 2 2 2 r1 r2 r3 r4 r5 r6ch1 ch2 ch3 ch4 ch5 rj : rib j chj : channel j (interconnect) (interconnect) half-rib / half-channel geometry Figure 2-2. The rectangular cell geometry includes the ribbed interconnect plates. This two-dimensional approximation is valid for co- and counter-flow configurations. The system is assumed to be isothermal. Previous modeling work has revealed that the main temperature variation in a planar SOFC occurs in the in-plane direction [3,4]. Temperature gradients are the result of the progressive variation in reaction rate, and consequently heat generation, caused by reactant consumption. The much smaller through-thickness dimension is expected to present far less temperature variation, especially if internal reforming is not considered. Ackmann et al calculated a maximum difference of 1.5ºC for a ribbed geometry, considering methane reforming [4]. The main temperature variation is expected to occur along the channel direction. The impact of this three-dimensional effect on the results presented in this work will be addressed in Chapter Six. For a two-dimensional model, the addition of energy balance is not 23 expected to improve the predictive capabilities due to the relatively uniform in-plane temperature. Energy balance will be needed if the present model is expanded to three dimensions. The simulation of the impedance of an SOFC requires calculating the response of the system to a small sinusoidal perturbation. If this perturbation consists of a small sinusoidal voltage, then the response will be a small sinusoidal current of the same frequency, generally shifted in phase due to the capacitive behaviour of the electrochemical interfaces within the cell. The small amplitude of the perturbation is a requirement for the linearity in the response, i.e. the absence of induced harmonics. All the variables detailed in Table 2-1 will have a steady state and an oscillatory component, as will be described later in this chapter. As far as the author knows, this is the first report of a two-dimensional model of the impedance behaviour of a working SOFC, and also the first one to include the interconnect plates in the geometry. Up to the present, SOFC impedance modeling work has aimed at elucidating the influence on the impedance spectrum of the many factors that determine the cell behaviour, or at finding optimal parameters characterizing the system, but no attempt has been made to develop a numerical model that uses the impedance behaviour as a diagnostic tool. 2.3 Mathematical Formulation Each cell component is treated as a continuum with quantities that obey material and charge transport laws. The general equation for the transport of species x is [5] xx QNt x +⋅∇−= ∂ ∂ ( 2 ) This fundamental equation states that the time variation in the amount of species x equals the negative divergence of its flux, Nx, plus the rate of production or consumption of x, Qx, within an elemental volume. This general equation takes the form of mass and charge conservation equations to solve for electric potential and concentrations throughout the cell components. For example, if x represents ionic or electronic charge per unit volume (Coul/m3), Nx will represent a current density (A/m2), and Qx will take the form of a Faradaic current 24 source or sink (A/m3). In this case, the fundamental variable to be solved is the electric potential, and the accumulation term on the left hand side will describe the charge of an interfacial double layer, given by a time change in potential. The current density is proportional to the gradient in electric potential using Ohm’s law, using an effective conductivity as the proportionality constant. If x represents concentration of gaseous species within the electrode pores (mol/m3), Nx will become a diffusive mass flux (convection within small pores can be neglected), and Qx will take the form of a Faradaic production or consumption of gas species. Species concentration is the fundamental variable to solve; the accumulation term indicates the time variation of concentration within an elemental volume. The diffusive mass flux is, in this case, proportional to the concentration gradient with the effective diffusivity as the proportionality constant. A core feature of the present model is the partial decoupling of the steady state and oscillatory component of all the variables solved for. This mathematical approach results in a convenient and efficient computational formulation, useful for systems of the size presented here, on the order of 30,000 degrees of freedom. As will be demonstrated in Section 2.3.2, the oscillatory variables can be calculated separately from the steady state variables, once the steady state variables are known. It is a requirement to start by solving for the steady state equations of the system, and solve the oscillatory equations secondly. For simplicity, the next section first presents the steady state formulation, and the oscillatory equations immediately after. 2.3.1 Steady State Equations 2.3.1.1 Steady-State Charge Balance The general form of the ionic or electronic charge balance for an elemental volume within a porous electrode is [6, 7] )( ηη Fdl iSitCS +⋅−∇=∂ ∂ − ( 3 ) which states that the time variation of charge density equals the negative divergence of the current density, and the production or consumption of current density via electrochemical reactions. The time change in charge density results from the charging 25 and discharging of the interfacial double layer. This phenomenon can be visualized considering the porous electrode as a two-path transmission line, in which the paths are connected via parallel RC elements. These elements are commonly known as leaky capacitors. The capacitive element is the only source of time dependence, and is responsible for the phase shift between an input signal and the system’s response. In this expression, i (A m-2) is the ionic or electronic current density, S (m-1) is the electroactive surface area per unit volume of the porous electrode, iF (A m-2) is the Faradaic current density, established between electronically- and ionically-conductive phases, Cdl (F m-2) is the double layer capacitance of the (distributed) interface between electronic and ionic conductors, and REFIONELE Φ−Φ−Φ=η is the local overpotential (V), with respect to a reference electrode of the same kind as the one under study. This formulation for porous electrodes was first described by Newman and Tobias [8], and states that the local overpotential equals the potential difference between the electronic and ionic phases. In a full-cell model, one of the reference potentials can be arbitrarily set to zero (the anode reference potential, in this work), and the other reference potential must be equal to the open-circuit potential of the system. The term on the l.h.s. of equation ( 3 ) is null for DC operation because the time derivative vanishes at steady state. Combined with Ohm’s law, which relates current density and potential gradient as Φ∇−= ki , we obtain an expression as a function of the potential only: )( 2 ηFiSk =Φ∇− ( 4 ) where it is assumed that the effective conductivity k (S m-1) is independent of position. Here Φ (V) denotes ionic or electronic potential, the fundamental unknowns for which the equations are solved. Applying Ohm’s law to the solid phases in the electrodes assumes that the transport of ionic and electronic species occurs mainly by migration. This is a good assumption for electronic transport and solid-state ionic transport [9,10]. Another fundamental principle involved in the calculation is the conservation of charge, which states that the divergence of the electronic ( ELEi ) and ionic ( IONi ) current densities must add to zero, since there is a continuous conversion of current from electronic to ionic, from the current collector toward the electrolyte. Equation ( 5 ) 26 indicates that the source terms for the ionic and electronic species are equal in magnitude and opposite in sign. Equivalently, a source of ions is a sink of electrons. 0=⋅∇+⋅∇ ELEION ii ( 5 ) The Faradaic current density constitutes the current source term of equation ( 3 ). A function of the local overpotential, its magnitude reflects the electrochemical activity at every point in the electrode. In this work, we assume a Butler-Volmer relationship between overpotential and Faradaic current density, modified to include mass transport effects. The source term is positive or negative, indicating production or consumption of the corresponding current. For example, the cathode is a source of oxide ions, and consequently a sink of ionic current (according to the convention of positive current being the flow of positively charged species). The source term for ionic current will be negative for cathodic polarization. Therefore, the Faradaic ionic current generated in the DC cathodic reaction is given by [11, 12]                 −−         = CAT g CATCAT O O CAT g ANOCAT CATCATIONF TR F c c TR F ii ηαηαη ,0 , ,0,, expexp)( 2 2 ( 6 ) where i0,CAT is the cathodic exchange current density (A m-2), αij is the charge transfer coefficient for the reaction with the first sub-index indicating the electrode and the second sub-index indicating the anodic or cathodic direction, F is the Faraday's constant (A s mol-1), Rg is the universal gas constant (J mol-1 K-1), T is the absolute temperature (K), cO2 (mol m-3) is the local oxygen concentration, and c0O2 (mol m-3) is the reference oxygen concentration, at which i0,CAT is measured, and it is equal to the channel or bulk concentration in this model. The second term on the r.h.s. of ( 6 ) is larger than the first term for cathodic polarization, resulting in a negative source term. The local reactant concentrations are unknown, and they must be solved for simultaneously using the mass transport equations explained in the next section. The temperature dependence of the exchange current density is not stated explicitly because the model is isothermal; i.e. i0,CAT is a constant for the single operating temperature considered in this model. Expression ( 6 ) incorporates both activation and concentration overpotentials [13]. The cathodic (second on the r.h.s.) term becomes very small at sites of reactant depletion. The concentration ratio of products (oxide ions) is approximated as one, since the oxide 27 ion concentration is largely independent of the atmosphere for YSZ, because its ionic conductivity is an extrinsic property. Combining expression ( 6 ) with expression ( 4 ), we obtain the DC ionic potential equation for the cathode.                 −−         =Φ∇− CAT g CATCAT O O CAT g ANOCAT CATCATIONCATION TR F c c TR F iSk ηαηα ,0 , ,0 2 , expexp 2 2 ( 7 ) The product CATCAT iS ,0 constitutes a volumetric exchange current. The two parameters CATS and CATi ,0 are, in the context of this model, inseparable, and their product reflects the microstructure dependence of the electrode kinetics. This point is revisited in the Model Parameters section of this chapter. An analogous expression applies to the electronic current, using the appropriate electronic conductivity parameters, and equation ( 5 ) to calculate the source term:                 −−         −=Φ∇− CAT g CATCAT O O CAT g ANOCAT CATCATELECATELE TR F c c TR F iSk ηαηα ,0 , ,0 2 , expexp 2 2 ( 8 ) The anode is a sink of O2- ions, thus, a source of ionic current. The DC ionic expression results:                 −−         =Φ∇− ANO g CATANO OH OH ANO g ANOANO H H ANOANOIONANOION TR F c c TR F c c iSk ηαηα ,0 , 0,0 2 , expexp 2 2 2 2 ( 9 ) with a positive source term for anodic polarization. Finally, the electronic current equation is:                 −−         −=Φ∇− ANO g CATANO OH OH ANO g ANOANO H H ANOANOELEANOELE TR F c c TR F c c iSk ηαηα ,0 , 0,0 2 , expexp 2 2 2 2 ( 10 ) In the purely ionically conductive electrolyte, no reaction takes place, and therefore the source term is absent: 02 =Φ∇ ION ( 11 ) Equivalently, the expression for the interconnect domain becomes 02 =Φ∇ ELE ( 12 ) Prescribed potential boundary conditions for the steady state equations are only required for the electronic potential, along the current collecting lines at the top of the cathode interconnect and at the bottom of the anode interconnect, as seen in Figure 2-1: 28 0 collectorcurrent anode collectorcurrent cathode =Φ =Φ ELE CELLELE V ( 13 ), where VCELL is the operating voltage of the cell. 2.3.1.2 Steady-State Mass Balance The driving force for reactant transport within a porous electrode is the concentration gradient between the electrode/channel and the electrode/electrolyte boundaries. The change in reactant or product concentration at a point within an electrode is proportional to the divergence of the negative flux of that species, and to its electrochemical production or consumption. nF iSN t c F k k )( η+⋅−∇= ∂ ∂ ( 14 ) Here, ck is the concentration of species k (mol m-3), and Nk is the species’ molar flux (mol s-1m-2), n is the number of electrons participating in the relevant electrochemical production or consumption reaction per molecule of reactant, and the other terms are as defined above. Equation ( 14 ) indicates that accumulation of species within a volume element can be the result of species flux and of electrochemical production or consumption. For binary gas mixtures, Fick’s law of diffusion gives an adequate representation of the diffusive flux, under the assumption that Knudsen diffusion effects are small for pore sizes of about 1 micron: TOTkkkk NxcDN +∇−= ( 15 ) where Dk is the effective diffusivity of the gas in the binary mixture (m2 s-1), xk is the molar fraction of species k, and NTOT is the total flux for all species. At the anode, hydrogen diffuses toward the electrolyte, while gaseous water diffuses away from it. This combination obeys the laws of binary equimolar diffusion, since the sum of molar hydrogen and water fluxes is null at every point because one hydrogen molecule is transformed into one water molecule: 0 22 =+= OHHTOT NNN ( 16 ) 2222 HOHHH cDN ∇−= − ( 17 ) 29 The following steady state mass balance equation for the hydrogen at the anode results, assuming that the effective diffusion coefficient is spatially invariant, i.e., that the porosity and microstructure are invariant throughout the electrode, and independent of gas composition within the range of interest (Table II in [14]):                 −−         −=∇− − ANO g CATANO OH OH ANO g ANOANO H HANOANO HOHH TR F c c TR F c c F iS cD ηαηα ,0 , 0 ,02 expexp 2 2 2 2 2 222 ( 18 ) The water concentration is calculated using the relation: 1 22 =+ OHH xx ( 19 ) with xk indicating molar fraction of species. At the cathode, with dry air as the oxidant, the situation is slightly different, since only oxygen takes part in the reaction, while the nitrogen remains unreacted. The net total flux at a reaction site is equal to the oxygen flux, since the net flux of nitrogen at any specific reaction site is null: 222222 OOONOO NxcDN +∇−= − ( 20 ) Simplifying and substituting for the mole fraction xO2 from the ideal gas law, 2 222 2 222 2 11 O g ONO O ONO O c p TR cD x cD N − ∇− = − ∇− = −− ( 21 ) where p is the total pressure of the system, often very close to atmospheric pressure. The resulting mass transport equation, again assuming that the diffusivity is spatially invariant, is:                 −−         =             − ∇ ⋅∇− − CAT g CATCAT O O CAT g ANOCATCATCAT O g O NO TR F c c TR F F iS c p TR c D η α η α , 0 ,,0 expexp 4 1 2 2 2 2 22 ( 22 ) The boundary conditions for hydrogen and oxygen concentrations are prescribed concentrations at the boundaries between the electrodes and channels, and no-flux everywhere else: 22 0 HH cc = at the channel / anode boundary 0 2 =⋅∇ ncH everywhere else on the anode side 30 22 0 OO cc = at the channel / cathode boundary 0 2 =⋅∇ ncO everywhere else on the cathode side Here n is the unit vector normal to each boundary. The DC mass transport equations are solved within the porous electrodes, simultaneously with the DC charge transport equations. The equations are nonlinear and coupled. The nonlinearity arises from the exponential dependence of current density on overpotential. The coupling is a result of the concentration dependence of the Butler-Volmer source terms. Comsol Multiphysics (Comsol, Inc.) general purpose PDE solver was used to solve the equations using finite elements. The choice of initial guess is of importance to achieve convergence. The default value for the initial guess is zero for the Comsol package, but a better value is the Dirichlet boundary condition value. The Analytical Jacobian method used by Comsol makes it possible to input the term inside the divergence in equation ( 22 ) directly. 2.3.2 AC Equations 2.3.2.1 AC Charge Balance One premise in impedance spectroscopy analysis is that the system responds linearly to a small perturbation. If the imposed perturbation is small, all system variables can be decomposed into a steady state, or DC, component, and an oscillatory component. tj OHOHOH tj OOO tj HHH tj IIONION tj ELEELEELE eccc eccc eccc e e ω ω ω ω ω 222 222 222 ~ ~ ~ ~ ~ ON += += += Φ+Φ=Φ Φ+Φ=Φ ( 23 ) Here e is the base of the natural logarithm, ω is the angular frequency of the applied perturbation, and j is the imaginary unit. Appendix II provides a demonstration of this statement for a general, nonlinear PDE. In ( 23 ), the bar indicates the steady state part, and the tilde indicates the oscillatory part. The oscillatory component is a complex variable of small magnitude that represents the phase shift and the excursion from the 31 equilibrium point of the corresponding variable. In practical terms, “small” means “lower than the system thermal voltage”, a temperature dependent quantity given by: KmV F TR V gTH 1123@97≈= ( 24 ) In practice, 20mV is an adequate compromise between linearity and acceptable signal-to-noise ratio. The AC charge balance at the anode results from placing the anode- relevant expressions from ( 23 ) in ( 2 ). The equation for ionic charge is: ( ) ( ) ( )( ) ( )( )         +− + −+ + =Φ+Φ∇−+ ∂ ∂ − tj ac OH tj OHOHtj aa H tj HH ANOANO tj IIONANOION tj ANOdlANO ef c ecc ef c ecc iS eke t CS ω ω ω ω ωω ηηηη ηη ~exp ~ ~exp ~ ~ ~ 2 22 2 22 00,0 ON 2 ,, ( 25 ) Here, for notation simplicity, the following replacements were made: TR Ff RT Ff g CATANO ac ANOANO aa IONELE REFIONELE , , ~~ ~ α α η η = = Φ−Φ= Φ−Φ−Φ= ( 26 ) Equation ( 25 ) contains terms that can be expressed in simpler form, on account of the small amplitude of the perturbation. Linear order Taylor expansion of the exponential functions yields: ( )( ) ( )( ) ( )( ) ( )( )tjacactjac tj aaaa tj aa effef effef ωω ωω ηηηη ηηηη ~1exp~exp ~1exp~exp −−≈+− +≈+ ( 27 ) Substituting simplified expressions ( 27 ) in ( 25 ) and expanding the time derivative, 32 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                           −+ +−− +−+ +−− =Φ∇−Φ∇−− ηηηη ηη ηηηη ηη ηω ωωωω ωω ωω ωω ~exp ~ ~exp ~ exp ~ exp ~ ~exp~exp expexp ~ ~ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 00 00 00 00 ,0 ON 2 , 2 ,, ac tj ac tj OH OH aa tj aa tj H H tj ac OH OHtj aa H H tj acac OH OHtj aaaa H H ac OH OH aa H H ANOANO I tj ANOIONIONANOION tj ANOdlANO fefe c cfefe c c ef c c ef c c eff c c eff c c f c cf c c iS ekkejCS ( 28 ) All terms in equation ( 28 ) that do not have the pulsation tje ω as a factor add up to zero because they are identical to those of the steady state equation, whose solution we already have (equation ( 9 )). Furthermore, the last two terms in brackets are of second order, since they contain the product of two oscillatory terms, η~~ 2H c and η~~ 2OHc , each of small magnitude, and can therefore be neglected with respect to the other terms. These two simplifications allow the removal of the tje ω term from all the remaining terms, yielding a linear, time independent PDE with respect to the oscillatory potential and concentration amplitudes [15, 16]: ( ) ( ) ( ) ( )            −− +−+ =Φ∇−− ηη ηηηη ηω ac OH OH aa H H acac OH OH aaaa H H ANOANO IANOIONANOdlANO f c cf c c ff c cff c c iS kjCS exp ~ exp ~ ~exp~exp ~ ~ 2 2 2 2 2 2 2 2 00 00 ,0 ON 2 ,, ( 29 ) The first two terms in the brackets correspond to the Faradaic local current that appears as a consequence of the imposed AC overpotential η~ . The last two terms in the brackets indicate the AC concentration contribution to the source term. The first term on the l.h.s. constitutes the non-Faradaic contribution to the local current density, assumed to be additive with the Faradaic component, and without mutual interaction [16]. This formulation is numerically very convenient, since the DC component of the solution is decoupled from the AC component of the solution. On account of the small size of the AC perturbation, it does not have any influence on the DC solution. This fact allows the solution of the AC equations based on the previously and independently 33 obtained DC solution. The steady state equations are then solved using a nonlinear iterative routine yielding values for η , and the steady state solution is then used as a base on which the AC solution is superimposed. No iterative method is required for the latter, since the equations are linear in the oscillatory unknowns. The electronic potential equation is obtained similarly, as explained in the previous section: ( ) ( ) ( ) ( )            −− +−+ − =Φ∇−− ηη ηηηη ηω ac OH OH aa H H acac OH OH aaaa H H ANOANO ELEANOELEANOdlANO f c cf c c ff c cff c c iS kjCS exp ~ exp ~ ~exp~exp ~ ~ 2 2 2 2 2 2 2 2 00 00 ,0 2 ,, ( 30 ) The derivation of the cathode ionic charge balance equation is omitted, but it is obtained in the same way, resulting in: ( ) ( ) ( )      −−−+ =Φ∇−− ηηηηη ηω cc O O cccc O O cacaCATCAT ICATIONCATdlCAT f c cff c cffiS kjCS exp ~ ~exp~exp ~ ~ 2 2 2 2 00,0 ON 2 ,, ( 31 ) Finally, a similar expression corresponds to the electronic charge balance at the cathode: ( ) ( ) ( )      −−−+− =Φ∇−− ηηηηη ηω cc O O cccc O O cacaCATCAT ELECATELECATdlCAT f c cff c cffiS kjCS exp ~ ~exp~exp ~ ~ 2 2 2 2 00,0 2 ,, ( 32 ) Prescribed potential boundary conditions for the AC equations are only required for the electronic potential: 0collectorcurrent anode 0collectorcurrent cathode ~~ ~~ Φ∆+=Φ Φ∆−=Φ ELE ELE ( 33 ) where 0 ~Φ∆ is the perturbation amplitude. Neumann boundary conditions apply everywhere else for the electronic potential, and everywhere for the ionic potentials. 2.3.2.2 AC Mass Balance In order to solve for the oscillatory concentrations we need another set of equations to be solved simultaneously with equations ( 30 ) to ( 33 ). The decomposition 34 shown in eq. ( 23 ) can be applied to the mass balance equations, in a similar way as shown for the charge balance. The equation describing the distribution of hydrogen concentration at the anode can be calculated starting with the general form of the mass balance within the pores at the anode: ( ) ( ) ( ) ( )( ) ( ) ( )( )      −− −∇= ∂ ∂ − txf c txc txf c txc F iS txcD t txc ac OH OH aa H HANOANO HOHH H ,exp , ,exp , 2 , , 2 2 2 2 222 2 00 ,0 2 ηη ( 34 ) Here, the concentration and the overpotential depend on time t and on the spatial coordinates ( )yxx ,= . Replacing them using the expressions ( 23 ): ( ) ( ) ( )( ) ( )( )         +− + −+ + −+∇= ∂ +∂ − tj ac OH tj OHOHtj aa H tj HHANOANO tj HHOHH tj HH ef c ecc ef c ecc F iS eccD t ecc ω ω ω ω ω ω ηηηη ~exp ~ ~exp ~ 2 ~ ~ 2 22 2 22 2222 22 00 ,0 2 ( 35 ) After a simplification equivalent to that shown for the charge balance, equation ( 35 ) results: ( ) ( ) ( ) ( )            −− +−+ −∇= − ηη ηηηη ω ac OH OH aa H H acac OH OH aaaa H H ANOANO HOHHH f c cf c c ff c cff c c F iS cDcj exp ~ exp ~ ~exp~exp 2 ~~ 2 2 2 2 2 2 2 2 2222 00 00 ,0 2 ( 36 ) The derivation of the AC equation for oxygen concentration on the cathode side is more complicated, because of the presence of the extra term in the expression for the oxygen flux (eq. ( 20 ), second term on the r.h.s.). Revisiting eq. ( 21 ), 2 222 2 1 O g ONO O c p TR cD N − ∇− = − ( 21 ) it is apparent that we cannot directly incorporate the AC terms as before, because of the presence of the concentration in the denominator. However, it is possible to expand the 35 2 1 1 O g c p TR − term around some convenient value for 2O g c p TR . If the expansion is performed around zero, the following expression holds if we neglect second order terms and higher:       +∇−≈ − ∇− − − 2222 2 222 1 1 O g ONO O g ONO c p TR cD c p TR cD ( 37 ) as long as 1 2 200E-6 mano = 1-fliplr([0:b1:b2 b2+0.01:0.01:b2+b3 b2+b3+0.2:0.2:1]); end s1 = 0.05; % fraction of the landing that determines the vertical mesh refinement at the landing edges s2 = 0.1; % fraction of the landing over which the mesh is refined s5 = 0.2; % equivalent to s1 for the first and last landings s6 = 0.9; % equivalent to s2 for the first and last landings mlan = [0:s1:s2 0.2:0.2:0.8 (1-s2):s1:1]; % Locally defined element distribution for a landing mlanf = [0:s5:s6 (1-s2):s1:1]; % Locally defined element distribution for first landing mlanl = [0:s1:s2 s5:s5:1]; % Locally defined element distribution for last landing % Subdomain concatenation requires each block to end one element before unity to avoid repetition mlan1 = [0:s1:s2 s5:s5:s6 (1-s2):s1:1-s1]; % Same as above for concatenation mlan1f = [0:s5:s6 (1-s2):s1:1-s1]; % first landing mlan1l = [0:s1:s2 s5:s5:1-s5]; % last landing s3 = 0.05; % fraction of the gap that determines the vertical mesh refinement at the gap edges s4 = 0.1; % fraction of the gap over which the mesh is refined mlan2 = [0:s3:s4 0.2:0.2:0.8 (1-s4):s3:1]; % Locally defined element distribution for a gap mlan3 = [0:s3:s4 0.2:0.2:0.8 (1-s4):s3:1-s3]; % Same as above for concatenation 158 % ms is the meshing definition structure. The # elements is 2 x # edges. The odd numbered positions are occupied by the numbers from 1 to #edges. The even numbered % positions give the normalized vectors that define the element distribution on the corresponding % edge nedge = 8+2*nlan*7+2*(nlan-1)+6+2*(nlan-1)+2*nlan+2*(2*nlan-1)-1; % total number of edges nsub = 3+4*nlan+2+2*nlan-1; % total number of subdomains aux1 = []; aux2 = []; if nlan>2 aux1 = [48:24:48+24*(nlan-3)]; aux2 = [53:24:53+24*(nlan-3)]; end % vertical edges belonging to the interlayer edgeil = nonzeros([5 24 35:24:35+24*(nlan-2) aux1 nedge-7 15 29 41:24:41+24*(nlan-2) aux2 nedge-2]); % vertical edges belonging to the interconnect aux1 = []; aux2 = []; if nlan>2 aux1 = [46:24:46+24*(nlan-3)]; end edgeic=nonzeros([3 22 33:24:33+24*(nlan-2) aux1 nedge-8 17 31:24:31+24*(nlan-2) 43:24:43+24*(nlan-2) nedge-1]); aux1 = []; aux2 = []; if nlan>2 aux1 = [50:24:50+24*(nlan-3)]; end edgedel = nonzeros([11 26 38:24:38+24*(nlan-2) aux1 nedge-4]); % initialization of mesh vector ms = cell(1,2*nedge); for i=1:nedge ms(2*i-1) = {i}; ms(2*i ) = {1}; end for i=1:nedge if i==7 | i==nedge-6 ms(2*i)={mano}; % anode elseif i==9 | i==nedge-5 ms(2*i)={4}; % electrolyte elseif i==13 | i==nedge-3 ms(2*i)={mcat}; % cathode elseif i==1 | i==nedge-9 | i==19 | i==nedge ms(2*i)={5}; % current collectors elseif i==10 | i==2 | i==21 % electrolyte, top and bottom current collecting lines aux1=[]; for kk=1:nlan-1 if kk==1 aux1=[aux1 mlan1f*p+(kk-1)*(p+q) kk*p+mlan3*q+(kk-1)*q]; else aux1=[aux1 mlan1*p+(kk-1)*(p+q) kk*p+mlan3*q+(kk-1)*q]; end end aux1=[aux1 mlan1l*p+(nlan-1)*(p+q)]; ms(2*i)={[aux1 L]/L}; end for kk=1:4*nlan if i==edgeil(kk) ms(2*i) = {2}; elseif i==edgeic(kk) ms(2*i) = {5}; 159 end end for kk=1:2*nlan if i==edgedel(kk) ms(2*i) = {2}; end end end ms(2*([4 6 8 12 14 16 18 20]))={mlanf}; for i=1:nlan-1 ms(2*([23 25 27 28 30 32] +24*ones(1,6)*(i-1))) = {mlan2}; ms(2*([34 36 37 39 40 42 44 45]+24*ones(1,8)*(i-1))) = {mlan}; end ms(2*([34 36 37 39 40 42 44 45] +24*ones(1,8)*(nlan-2))) = {mlanl}; %%% Incorporation of the mesh into the fem structure %%% fem.mesh=meshmap(fem,'edgelem',ms); %%% Plotting of the mesh %%% figure(20) axes('FontName','Helvetica','FontSize',14); axis off meshplot(fem); %ylim([-0.2*(tT+r),1.2*(tT+r)]); xlim([-p/10,L+p/10]); Each equation needs to be solved separately, specifying the relevant domains and boundary conditions. As an example, consider the DC electronic potential equations. The information is stored in the fem.appl substructure. clear appl % Clears appl from previous definitions appl.mode.class = 'FlPDEG'; % General PDE formulation (recommended) appl.dim = {'Ve','Ve_t'}; appl.name = 'DCele'; % Name appl.assignsuffix = '_DCele'; clear bnd % Boundary conditions bnd.r = {'-Ve','-Ve','Vcell-Ve'}; bnd.type = {'dir','neu','dir'}; % Type of boundary condition bnd.ind = 2*ones(1,nedge); bnd.ind(edgea) = 1; % edgea contains the anode subdomains bnd.ind(edgec) = 3; % edgea contains the cathode subdomains appl.bnd = bnd; clear equ %%% Source terms: cathodic and anodic DC Faradaic current equ.f = {'-ictc','0','-icta','0','0' ,'0','0'}; %%% Flux expression equ.ga = {{{'-kec*Vex' ;'-kec*Vey'}},... {{'-kit*Vex' ;'-kit*Vey'}},... {{'-kea*Vex' ;'-kea*Vey'}},... {{'-kila*Vex' ;'-kila*Vey'}},... {{'-kaira*Vex';'-kaira*Vey'}},... {{'-kilc*Vex' ;'-kilc *Vey'}},... {{'-kairc*Vex';'-kairc*Vey'}}}; %%% equ.ind(where to apply the equation) = what equation to apply equ.ind(subano) = 3; % anode equ.ind(suba) = 2; % anode interconnect equ.ind(subail(1:nilr)) = 5; % anode interlayer delaminated equ.ind(subail(nilr+1:nlan)) = 4; % anode interlayer intact equ.ind(subcat) = 1; % cathode equ.ind(subc) = 2; % cathode interconnect 160 equ.ind(subcil(1:nilr)) = 7; % cathode interlayer delaminated equ.ind(subcil(nilr+1:nlan)) = 6; % cathode interlayer intact %%% Initial value to begin the Newton iteration equ.init = {{'Vcell'},0,0,0,0,{'Vcell'},{'Vcell'}}; appl.equ = equ; fem.appl{3} = appl; COMSOL labels the geometry objects and their edges following a bottom-to-top and left- to-right criterion. Keeping the subdomain and edge labels in a vector (e.g. subail and edgeil) is very convenient and recommended in order to refer to these labels from anywhere in the code. The key point is to define the logic of label assignment for a general geometry. For this task, it is advisable to start with a simple geometry, e.g. three ribs / two channels, let COMSOL label the geometry, and find the general labeling pattern. The source term definition uses the indexing described above, in essentially the same way: fem.expr = {'faa' ,'alphaaa*F/R/T',... 'fac' ,'alphaac*F/R/T',... 'fca' ,'alphaca*F/R/T',... 'fcc' ,'alphacc*F/R/T',... 'ictcAC','Sc*i0c*(fca*overAC*exp(fca*overpot) +cO/cOref*fcc*overAC*exp(-fcc*overpot)-cOAC/cOref*exp(fcc*overpot))',... 'ictaAC','Sa*i0a*(cH/cHref*faa*overAC*exp(faa*overpot) +cH2O/cWref*fac*overAC*exp(-fac*overpot)+cHAC/cHref*exp(+faa*overpot)- cWAC/cWref*exp(-fac*overpot))',... 'ictc' ,'Sc*i0c*(exp(fca*overpot)-cO/cOref*exp(-fcc*overpot))',... 'icta' ,'Sa*i0a*(cH/cHref*exp(faa*overpot)-cH2O/cWref*exp(- fac*overpot))',... 'tdcsa' ,'Sa*(Cdla*j*2*pi*f) *(VeAC-ViAC)',... % time dependent current source anode 'tdcsc' ,'Sc*(Cdlc*j*2*pi*f) *(VeAC-ViAC)'}; Each expression is defined globally, and the equ.ind structure assigns the source term to the appropriate domain. The solution mode depends on the equation to be solved. As explained in Chapter Two, the DC equations are nonlinear because the source term depends on the unknown potential in an exponential form. The AC equations are linear, because they result from the linearization of the time-dependent conservation equations. The procedure is as follows: 1- Solution of the DC potentials and DC concentrations 2- Parametric solution of the AC equations, using the DC solution as datum, and using 161 the frequency as the parameter. Usually, five or six frequencies per decade is a reasonable compromise between resolution and computing time. 3- Integration of the AC current along the current collectors, for each frequency 4- Impedance calculation COMSOL functions for nonlinear and linear solving are femnlin and femlin, respectively. fem.sol=femnlin(fem, ... 'nullfun','flspnull', ... 'solcomp',{'Vi','Ve','cH','cO'},... % What variables to solve 'outcomp',{'Vi','Ve','cH','cO'},... % What variables to output 'maxiter',35); % Maximum number of Newton iteration fem0 = fem; fem.sol=femlin(fem, ... 'init',fem0.sol, ... 'nullfun','flspnull', ... 'solcomp',{'ViAC','VeAC','cHAC','cOAC'},... 'outcomp',{'Vi','Ve','ViAC','VeAC','cH','cO','cHAC','cOAC'},... 'pname','f', ... % Parameter 'plist',[logspace(-3,6,Nf) 10^10]) % Parameter range 162 II. Appendix Two. On the linearization of nonlinear PDEs around the steady state point The small size of the AC perturbation imposed on the system in an impedance study allows the linearization of the transport equations, resulting in linear, time- independent, complex-valued equations, largely advantageous from a numerical standpoint. This simplification is possible for a general nonlinear PDE when: I. The steady state solution exists II. The perturbation from it is small Consider the nonlinear PDE in F: )(2 FfF t F +−∇= ∂ ∂ ( 46 ) where f is any differentiable function, and the time dependence is only the result of a small sinusoidal perturbation of frequency ω. If F is the steady state function, known to exist, that satisfies )(0 2 FfF +−∇= ( 47 ) Then the general solution can be approximated as the sum of the steady state solution plus a small sinusoidal variation, of the same frequency as the perturbation tj eFFF ω~+≈ ( 48 ) Replacing ( 48 ) in ( 46), )~(~~ 22 tjtjtj eFFfeFFFej ωωωω ++∇−−∇= ( 49 ) The first order expansion of the non-homogeneous term around F is ( ) ( ) ( )tjtj eFFfFfeFFf ωω ~ !1 )~( ′+≈+ ( 50 ) which, replaced in equation ( 49 ) results in ( ) ( )( )tjtjtj eFFfFfeFFFej ωωωω ~~~ 22 ′++∇−−∇= ( 51 ) Given that the steady state solution exists, the terms without the oscillatory factor vanish, leaving only ( )( )tjtjtj eFFfeFFej ωωωω ~~~ 2 ′+−∇= ( 52 ) 163 The common time dependent term cancels out as well, ( )FfFFFj ′+−∇= ~~~ 2ω ( 53 ) F is already known, so ( )Ff ′ is a constant; then, this equation is linear in F~ . 164 III. Appendix Three. Fitting to an experimental polarization curve. For the purpose of comparison, the DC behaviour of the present model is fitted to the experimentally measured polarization curve presented by Kim et al in J. Electrochem Soc. 146 (1) 69-78 (1999). The authors of this excellent work measured polarization curves of anode-supported button cells within the 650ºC-800ºC range, using the same materials as those simulated in the present model. The cell anode consisted of a 750 micron thick Ni-8YSZ cermet, with 38% porosity. The electrolyte was a thin 10 micron thick layer of 8YSZ. The cathode was an LSM-8YSZ composite with approximately 40% porosity, 50 microns in thickness. The button cells were tested in humidified hydrogen / air conditions, and the polarization curve at 800ºC is reproduced in the following graph, along with the result of the fitting of the present model, using the cathode volumetric exchange current density (Scat i0,c = 10E9 A/m3) and the hydrogen effective diffusivity (DH = 0.199E-4 m2/s, calculated by the authors) as the fitting parameters, and considering the anode as reversible (Sano i0,a = 1E14 A/m3 >> Scat i0,c), i.e., the cathode being the main contributor to the cell polarization. The authors suggest that the anode works in the linear regime, although it is not possible to validate this assumption without individual measurement of each electrode’s polarization. There is, therefore, an ambiguity in assigning the activation polarization contribution to each electrode when fitting to a full-cell polarization curve. The open-circuit voltage in the experiment is approximately 0.1V lower than the theoretical OCV, and it was used directly as the OCV. Finally, the charge transfer coefficients were assumed to be the same as those used at 850ºC in this model. The cathode polarization curves at different temperatures presented by Co, Xia, and Birss in J. Electrochem. Soc., 152 (3) A570- A576 (2005) (Fig 8, pA574) suggest this assumption to be reasonable. 165 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 1 2 3 4 5 6 i (A/cm2) V C EL L (V ) Kim et al this work Figure III-1. Polarization curve fitting of the present model (line) to experimental data (dots) presented by Kim et al. Adapted from Fig 3a, J. Electrochem Soc. 146 (1) 69-78 (1999). 166 IV. Appendix Four. Uniform design parameter grids The following tables detail the parameter combinations used in the Uniform Design numerical experiments of Chapter Six. Table 6-1 lists the ranges of the 20 parameters (21 for the case of delamination) varied in the experiments. In the next two matrices, the 30 rows correspond to each numerical run, and the 20 (or 21) columns correspond to the varied parameters. IV. 1. Delamination: anot catt elyt p0a p0c V a1 a2 a3 a4 Sa Sc pora porc tora torc Cdla Cdlc i0A i0C t # m m m atm atm - - - - - 1/m 1/m - - - - F/m2 F/m2 A/m2A/m2 m 1 3.17E-05 9.38E-05 1.51E-04 0.19 0.10 0.39 0.9 0.3 1.3 0.8 4.72E+05 8.76E+05 0.33 0.16 4.4 3.9 24.0 0.4 16 42 7.76E-07 2 5.03E-05 3.79E-05 1.09E-04 0.73 0.10 0.88 1.8 0.7 0.5 0.7 4.41E+05 5.03E+05 0.66 0.56 6.5 6.7 4.5 2.8 36 12 1.28E-06 3 3.48E-05 7.21E-05 1.28E-04 0.46 0.14 0.55 1.7 0.7 1.4 0.6 1.00E+05 3.79E+05 0.37 0.70 2.2 3.0 0.1 0.1 92 1000 2.69E-07 4 2.86E-05 1.62E-05 1.54E-04 0.91 0.11 0.66 1.1 0.5 1.5 0.5 6.90E+05 1.62E+05 0.14 0.45 6.1 5.4 62.1 0.2 108 728 3.82E-06 5 5.34E-05 5.03E-05 1.96E-04 0.25 0.15 0.45 0.7 0.4 0.6 0.7 8.14E+05 4.41E+05 0.64 0.35 6.7 2.3 0.2 0.5 386 281 3.31E-06 6 6.59E-05 5.97E-05 1.20E-04 0.40 0.20 0.20 0.6 0.6 0.9 0.2 4.10E+05 2.24E+05 0.43 0.37 3.2 6.3 7.3 0.1 30 30 4.16E-06 7 1.62E-05 8.14E-05 1.24E-04 0.88 0.18 0.77 1.4 0.9 0.8 0.9 5.97E+05 9.07E+05 0.45 0.20 5.1 6.1 0.2 0.2 204 240 2.30E-06 8 9.07E-05 2.55E-05 1.36E-04 0.61 0.12 0.74 1.0 0.3 0.8 0.8 1.93E+05 4.10E+05 0.22 0.14 3.4 4.9 0.1 24.0 67 36 3.48E-06 9 7.83E-05 2.86E-05 1.01E-04 0.82 0.13 0.28 1.4 0.4 1.1 0.3 3.79E+05 2.55E+05 0.49 0.18 5.3 3.2 11.7 5.7 853 204 6.07E-07 10 4.10E-05 5.34E-05 9.76E-05 0.13 0.17 0.80 1.5 0.3 1.1 1.0 9.38E+05 1.00E+05 0.16 0.49 2.5 3.4 2.2 1.7 12 127 2.13E-06 11 4.72E-05 4.10E-05 1.89E-04 0.70 0.17 0.34 0.8 0.8 1.8 0.8 2.24E+05 5.34E+05 0.18 0.27 2.9 6.8 5.7 0.9 530 386 1.45E-06 12 8.76E-05 8.45E-05 1.58E-04 0.94 0.20 0.42 1.8 0.4 1.6 0.7 5.03E+05 2.86E+05 0.12 0.47 4.2 2.7 0.3 3.6 329 10 1.96E-06 13 5.66E-05 4.72E-05 9.00E-05 0.49 0.18 0.61 0.5 0.5 1.9 0.9 1.31E+05 9.38E+05 0.41 0.58 7.0 3.7 78.8 78.8 281 92 2.47E-06 14 8.45E-05 4.41E-05 1.92E-04 0.79 0.16 0.72 1.3 0.9 1.0 0.4 2.86E+05 8.14E+05 0.29 0.64 4.9 2.5 18.9 0.3 14 16 3.14E-06 15 7.52E-05 6.90E-05 2.00E-04 0.52 0.16 0.99 2.0 0.4 1.9 0.6 7.83E+05 3.48E+05 0.39 0.22 4.6 6.5 14.9 100.0 19 853 2.80E-06 16 6.90E-05 0.0001 9.38E-05 0.64 0.13 0.31 1.6 0.9 2.0 0.3 9.69E+05 5.66E+05 0.24 0.29 6.8 4.4 1.1 0.7 49 67 2.97E-06 17 1.93E-05 9.07E-05 1.70E-04 0.67 0.18 0.36 1.6 0.2 0.7 0.4 3.17E+05 6.28E+05 0.60 0.53 3.6 4.6 48.9 14.9 57 329 3.65E-06 18 8.14E-05 1.31E-05 1.32E-04 0.34 0.19 0.91 1.2 0.2 1.8 0.3 7.21E+05 6.59E+05 0.62 0.62 4.1 5.1 0.3 0.3 174 108 1.11E-06 19 3.79E-05 7.52E-05 1.05E-04 0.76 0.15 0.53 0.9 0.3 1.3 0.4 8.45E+05 7.83E+05 0.27 0.66 4.8 7.0 0.9 30.4 621 19 4.32E-06 20 2.24E-05 2.24E-05 1.77E-04 0.55 0.19 0.23 1.1 0.9 1.0 0.8 8.76E+05 3.17E+05 0.35 0.60 5.4 4.8 0.7 48.9 26 79 4.38E-07 21 5.97E-05 1.00E-05 1.47E-04 0.16 0.14 0.50 1.9 1.0 0.6 0.3 5.34E+05 8.45E+05 0.10 0.43 2.0 5.8 3.6 62.1 240 174 2.63E-06 22 6.28E-05 3.48E-05 1.39E-04 0.97 0.21 0.58 1.0 0.7 1.4 0.6 9.07E+05 1.00E+06 0.70 0.12 3.0 3.6 2.8 4.5 10 452 5.00E-06 23 2.55E-05 1.93E-05 1.13E-04 0.37 0.15 0.47 1.7 0.8 1.7 0.9 7.52E+05 4.72E+05 0.56 0.24 3.9 2.0 38.6 9.2 149 14 3.99E-06 24 1.31E-05 5.66E-05 1.85E-04 0.85 0.11 0.83 0.6 0.6 1.7 0.2 6.28E+05 6.90E+05 0.47 0.39 2.3 2.2 0.5 18.9 42 57 1.62E-06 25 9.69E-05 6.28E-05 1.43E-04 0.22 0.12 0.25 1.3 0.6 1.6 1.0 3.48E+05 7.52E+05 0.51 0.51 5.6 6.0 0.4 7.3 22 530 4.49E-06 26 9.38E-05 7.83E-05 1.17E-04 0.31 0.19 0.85 0.8 0.8 0.7 0.5 6.59E+05 5.97E+05 0.20 0.33 5.8 2.9 30.4 11.7 79 621 9.45E-07 27 1.00E-05 3.17E-05 1.66E-04 0.28 0.21 0.96 1.9 0.6 1.2 0.5 1.62E+05 7.21E+05 0.31 0.31 6.0 4.2 1.4 1.4 1000 49 4.83E-06 28 7.21E-05 9.69E-05 1.62E-04 0.58 0.12 0.94 0.7 1.0 1.2 0.9 5.66E+05 1.93E+05 0.58 0.68 3.7 4.1 9.2 2.2 728 149 4.66E-06 29 4.41E-05 8.76E-05 1.73E-04 0.10 0.16 0.69 1.2 0.8 1.5 0.4 2.55E+05 1.31E+05 0.68 0.10 6.3 5.3 1.7 38.6 127 22 1.79E-06 30 0.0001 6.59E-05 1.81E-04 0.43 0.13 0.64 1.5 0.5 0.9 0.6 1.00E+06 9.69E+05 0.53 0.41 2.7 5.6 100.0 1.1 452 26 1.00E-07 167 IV. 2. Interconnect oxidation, detachment, and sulfur poisoning, ESC: anot catt elyt p0a p0c V a1 a2 a3 a4 Sa Sc pora porc tora torc Cdla Cdlc i0A i0C # m m m atm atm - - - - - 1/m 1/m - - - - F/m2 F/m2 A/m2A/m2 1 3.48E-05 2.55E-05 1.66E-04 0.13 0.43 0.45 1.0 0.3 1.0 0.9 6.90E+05 8.45E+05 0.68 0.43 3.6 6.3 18.9 0.4 452 36 2 5.34E-05 6.90E-05 1.54E-04 0.20 0.28 0.31 0.7 0.9 1.8 0.2 4.41E+05 7.83E+05 0.29 0.24 3.7 4.8 14.9 1.1 22 12 3 9.07E-05 7.21E-05 1.73E-04 0.13 0.25 0.34 1.6 0.4 1.7 0.7 5.03E+05 2.55E+05 0.41 0.47 6.7 4.9 48.9 0.7 621 853 4 9.38E-05 5.97E-05 1.47E-04 0.17 0.13 0.96 1.4 0.6 1.5 1.0 1.00E+06 6.59E+05 0.49 0.60 2.0 3.0 1.7 5.7 329 10 5 6.59E-05 9.69E-05 1.24E-04 0.16 0.40 0.85 1.4 0.6 0.7 0.3 1.31E+05 8.14E+05 0.53 0.12 6.5 5.6 38.6 11.7 386 67 6 6.28E-05 5.34E-05 1.39E-04 0.19 0.94 0.36 1.1 0.4 0.7 0.6 8.14E+05 7.52E+05 0.33 0.16 2.5 4.6 0.4 38.6 1000 1000 7 5.03E-05 8.45E-05 1.28E-04 0.19 0.37 0.23 1.9 0.3 0.9 0.4 5.66E+05 3.48E+05 0.64 0.58 3.2 2.3 7.3 24.0 49 204 8 5.97E-05 3.48E-05 1.89E-04 0.10 0.22 0.80 1.3 0.4 1.2 0.5 8.45E+05 3.17E+05 0.12 0.18 3.9 4.1 78.8 62.1 26 42 9 3.79E-05 1.31E-05 1.01E-04 0.21 0.88 0.88 1.8 0.3 1.6 0.5 3.79E+05 4.10E+05 0.27 0.41 4.4 3.7 24.0 3.6 728 26 10 1.93E-05 3.79E-05 1.51E-04 0.16 0.91 0.99 1.1 1.0 0.8 1.0 5.34E+05 1.62E+05 0.37 0.14 6.8 2.5 9.2 1.4 67 174 11 5.66E-05 0.0001 1.58E-04 0.15 0.67 0.55 1.2 0.2 1.6 0.2 9.07E+05 1.00E+05 0.20 0.37 5.4 2.0 0.1 0.2 281 92 12 4.41E-05 4.10E-05 1.92E-04 0.19 0.10 0.53 2.0 0.8 1.4 0.9 2.55E+05 6.28E+05 0.22 0.33 5.8 7.0 3.6 30.4 240 386 13 8.14E-05 2.24E-05 1.32E-04 0.13 0.49 0.69 0.6 0.8 1.7 0.4 5.97E+05 1.00E+06 0.62 0.27 4.9 2.2 2.8 100.0 530 281 14 2.55E-05 1.62E-05 1.05E-04 0.15 0.16 0.50 1.5 0.9 0.8 0.3 9.69E+05 5.34E+05 0.51 0.39 4.2 5.3 0.1 2.8 14 728 15 8.76E-05 1.93E-05 1.81E-04 0.17 0.79 0.25 0.8 0.6 0.9 0.4 6.59E+05 3.79E+05 0.16 0.53 6.3 5.8 1.4 7.3 174 19 16 0.0001 5.66E-05 1.43E-04 0.10 0.97 0.47 1.8 1.0 1.2 0.3 2.86E+05 6.90E+05 0.70 0.70 5.6 4.4 2.2 0.5 57 57 17 4.72E-05 8.76E-05 9.38E-05 0.11 0.85 0.28 0.8 0.6 1.4 0.9 9.38E+05 5.97E+05 0.47 0.35 6.1 3.4 30.4 9.2 10 127 18 4.10E-05 6.28E-05 2.00E-04 0.21 0.58 0.83 0.5 0.7 0.5 0.6 8.76E+05 5.03E+05 0.58 0.66 5.1 4.2 100.0 0.3 108 452 19 1.31E-05 5.03E-05 1.09E-04 0.18 0.19 0.42 0.7 0.5 1.3 0.7 1.00E+05 2.86E+05 0.60 0.20 5.3 3.9 0.7 0.1 127 22 20 1.00E-05 8.14E-05 1.13E-04 0.15 0.64 0.39 1.5 0.9 1.9 0.8 7.52E+05 9.07E+05 0.10 0.64 4.8 5.1 4.5 78.8 149 49 21 8.45E-05 2.86E-05 1.20E-04 0.18 0.70 0.58 1.3 0.7 1.1 0.8 1.62E+05 8.76E+05 0.18 0.49 3.0 2.7 62.1 0.2 16 329 22 2.86E-05 4.41E-05 1.36E-04 0.12 0.76 0.61 0.9 0.5 1.8 0.3 2.24E+05 1.31E+05 0.43 0.62 2.2 6.5 11.7 18.9 92 530 23 7.21E-05 3.17E-05 9.76E-05 0.18 0.34 0.77 1.0 0.4 1.9 0.6 7.21E+05 7.21E+05 0.39 0.68 7.0 6.8 0.3 0.9 30 108 24 6.90E-05 1.00E-05 1.96E-04 0.14 0.61 0.20 1.6 0.7 2.0 0.8 3.17E+05 4.72E+05 0.56 0.10 2.3 3.6 0.3 2.2 79 79 25 1.62E-05 9.07E-05 1.77E-04 0.11 0.31 0.74 0.9 0.9 1.1 0.5 1.93E+05 5.66E+05 0.24 0.51 2.9 3.2 0.5 1.7 853 149 26 3.17E-05 4.72E-05 1.70E-04 0.12 0.55 0.66 1.7 0.3 0.6 0.7 4.10E+05 9.38E+05 0.35 0.56 6.0 2.9 0.2 14.9 19 14 27 7.52E-05 6.59E-05 9.00E-05 0.12 0.52 0.64 1.9 0.8 0.6 0.6 7.83E+05 1.93E+05 0.31 0.29 2.7 6.7 5.7 0.3 204 16 28 9.69E-05 7.52E-05 1.17E-04 0.14 0.46 0.94 0.6 0.2 1.0 0.9 3.48E+05 4.41E+05 0.14 0.31 4.6 5.4 0.9 4.5 42 621 29 2.24E-05 7.83E-05 1.85E-04 0.16 0.82 0.91 1.7 0.5 1.5 0.4 6.28E+05 9.69E+05 0.45 0.22 3.4 6.0 1.1 0.1 36 240 30 7.83E-05 9.38E-05 1.62E-04 0.20 0.73 0.72 1.2 0.8 1.3 0.8 4.72E+05 2.24E+05 0.66 0.45 4.1 6.1 0.2 48.9 12 30 168 IV. 3. Interconnect detachment and sulfur poisoning, ASC: anot catt elyt p0a p0c V a1 a2 a3 a4 Sa Sc pora porc tora torc Cdla Cdlc i0A i0C # m m m atm atm - - - - - 1/m 1/m - - - - F/m2 F/m2 A/m2A/m2 1 6.38E-04 2.55E-05 1.53E-05 0.13 0.43 0.45 1.0 0.3 1.0 0.9 6.90E+05 8.45E+05 0.68 0.43 3.6 6.3 18.9 0.4 452 36 2 7.41E-04 6.90E-05 1.38E-05 0.20 0.28 0.31 0.7 0.9 1.8 0.2 4.41E+05 7.83E+05 0.29 0.24 3.7 4.8 14.9 1.1 22 12 3 9.48E-04 7.21E-05 1.64E-05 0.13 0.25 0.34 1.6 0.4 1.7 0.7 5.03E+05 2.55E+05 0.41 0.47 6.7 4.9 48.9 0.7 621 853 4 9.66E-04 5.97E-05 1.28E-05 0.17 0.13 0.96 1.4 0.6 1.5 1.0 1.00E+06 6.59E+05 0.49 0.60 2.0 3.0 1.7 5.7 329 10 5 8.10E-04 9.69E-05 9.66E-06 0.16 0.40 0.85 1.4 0.6 0.7 0.3 1.31E+05 8.14E+05 0.53 0.12 6.5 5.6 38.6 11.7 386 67 6 7.93E-04 5.34E-05 1.17E-05 0.19 0.94 0.36 1.1 0.4 0.7 0.6 8.14E+05 7.52E+05 0.33 0.16 2.5 4.6 0.4 38.6 1000 1000 7 7.24E-04 8.45E-05 1.02E-05 0.19 0.37 0.23 1.9 0.3 0.9 0.4 5.66E+05 3.48E+05 0.64 0.58 3.2 2.3 7.3 24.0 49 204 8 7.76E-04 3.48E-05 1.84E-05 0.10 0.22 0.80 1.3 0.4 1.2 0.5 8.45E+05 3.17E+05 0.12 0.18 3.9 4.1 78.8 62.1 26 42 9 6.55E-04 1.31E-05 6.55E-06 0.21 0.88 0.88 1.8 0.3 1.6 0.5 3.79E+05 4.10E+05 0.27 0.41 4.4 3.7 24.0 3.6 728 26 10 5.52E-04 3.79E-05 1.33E-05 0.16 0.91 0.99 1.1 1.0 0.8 1.0 5.34E+05 1.62E+05 0.37 0.14 6.8 2.5 9.2 1.4 67 174 11 7.59E-04 0.0001 1.43E-05 0.15 0.67 0.55 1.2 0.2 1.6 0.2 9.07E+05 1.00E+05 0.20 0.37 5.4 2.0 0.1 0.2 281 92 12 6.90E-04 4.10E-05 1.90E-05 0.19 0.10 0.53 2.0 0.8 1.4 0.9 2.55E+05 6.28E+05 0.22 0.33 5.8 7.0 3.6 30.4 240 386 13 8.97E-04 2.24E-05 1.07E-05 0.13 0.49 0.69 0.6 0.8 1.7 0.4 5.97E+05 1.00E+06 0.62 0.27 4.9 2.2 2.8 100.0 530 281 14 5.86E-04 1.62E-05 7.07E-06 0.15 0.16 0.50 1.5 0.9 0.8 0.3 9.69E+05 5.34E+05 0.51 0.39 4.2 5.3 0.1 2.8 14 728 15 9.31E-04 1.93E-05 1.74E-05 0.17 0.79 0.25 0.8 0.6 0.9 0.4 6.59E+05 3.79E+05 0.16 0.53 6.3 5.8 1.4 7.3 174 19 16 1.00E-03 5.66E-05 1.22E-05 0.10 0.97 0.47 1.8 1.0 1.2 0.3 2.86E+05 6.90E+05 0.70 0.70 5.6 4.4 2.2 0.5 57 57 17 7.07E-04 8.76E-05 5.52E-06 0.11 0.85 0.28 0.8 0.6 1.4 0.9 9.38E+05 5.97E+05 0.47 0.35 6.1 3.4 30.4 9.2 10 127 18 6.72E-04 6.28E-05 2.00E-05 0.21 0.58 0.83 0.5 0.7 0.5 0.6 8.76E+05 5.03E+05 0.58 0.66 5.1 4.2 100.0 0.3 108 452 19 5.17E-04 5.03E-05 7.59E-06 0.18 0.19 0.42 0.7 0.5 1.3 0.7 1.00E+05 2.86E+05 0.60 0.20 5.3 3.9 0.7 0.1 127 22 20 5.00E-04 8.14E-05 8.10E-06 0.15 0.64 0.39 1.5 0.9 1.9 0.8 7.52E+05 9.07E+05 0.10 0.64 4.8 5.1 4.5 78.8 149 49 21 9.14E-04 2.86E-05 9.14E-06 0.18 0.70 0.58 1.3 0.7 1.1 0.8 1.62E+05 8.76E+05 0.18 0.49 3.0 2.7 62.1 0.2 16 329 22 6.03E-04 4.41E-05 1.12E-05 0.12 0.76 0.61 0.9 0.5 1.8 0.3 2.24E+05 1.31E+05 0.43 0.62 2.2 6.5 11.7 18.9 92 530 23 8.45E-04 3.17E-05 6.03E-06 0.18 0.34 0.77 1.0 0.4 1.9 0.6 7.21E+05 7.21E+05 0.39 0.68 7.0 6.8 0.3 0.9 30 108 24 8.28E-04 1.00E-05 1.95E-05 0.14 0.61 0.20 1.6 0.7 2.0 0.8 3.17E+05 4.72E+05 0.56 0.10 2.3 3.6 0.3 2.2 79 79 25 5.34E-04 9.07E-05 1.69E-05 0.11 0.31 0.74 0.9 0.9 1.1 0.5 1.93E+05 5.66E+05 0.24 0.51 2.9 3.2 0.5 1.7 853 149 26 6.21E-04 4.72E-05 1.59E-05 0.12 0.55 0.66 1.7 0.3 0.6 0.7 4.10E+05 9.38E+05 0.35 0.56 6.0 2.9 0.2 14.9 19 14 27 8.62E-04 6.59E-05 5.00E-06 0.12 0.52 0.64 1.9 0.8 0.6 0.6 7.83E+05 1.93E+05 0.31 0.29 2.7 6.7 5.7 0.3 204 16 28 9.83E-04 7.52E-05 8.62E-06 0.14 0.46 0.94 0.6 0.2 1.0 0.9 3.48E+05 4.41E+05 0.14 0.31 4.6 5.4 0.9 4.5 42 621 29 5.69E-04 7.83E-05 1.79E-05 0.16 0.82 0.91 1.7 0.5 1.5 0.4 6.28E+05 9.69E+05 0.45 0.22 3.4 6.0 1.1 0.1 36 240 30 8.79E-04 9.38E-05 1.48E-05 0.20 0.73 0.72 1.2 0.8 1.3 0.8 4.72E+05 2.24E+05 0.66 0.45 4.1 6.1 0.2 48.9 12 30 "@en ; edm:hasType "Thesis/Dissertation"@en ; vivo:dateIssued "2008-05"@en ; edm:isShownAt "10.14288/1.0066175"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Mechanical Engineering"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "Attribution-NonCommercial-NoDerivatives 4.0 International"@en ; ns0:rightsURI "http://creativecommons.org/licenses/by-nc-nd/4.0/"@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Impedance model of a solid oxide fuel cell for degradation diagnosis"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/218"@en .