Index
AAccad, Y., 45Adsorption time scales, 153–154Advani, S.G., 360Advective timescale
large Peclet number, 150–151small Peclet number, 149–150
Amphlett, J.C., 370Anderson, B.D.O, 469Antoniou, A, 426Araki, H., 56Arato, E., 362, 387Astrom, K.J., 428Atomic units
Bohr radius, 37correction factor, 38
BBaker, J.D., 49Balakrishnan, A., 60Baldwin, K.G.H., 60Belevitch, V, 426Bellanger, M.G., 426Bergeson, S.D., 60Berg, P., 358Bethe, H.A., 44Bhatia, A.K., 53Bingel, W.A., 43
Bockris, J.O.M., 312Bohr–Sommerfeld quantum theory, 44Bradean, R., 379, 380Brinkman equation, 300Bruggeman expression, 301, 337Brug, G.J., 95, 125, 129Buchi, F.N., 358Burgers, A., 49Butler–Volmer equation, 86, 285
CCarlin, H.J., 426Carman–Kozeny equation, 322, 326Carnes, B., 291, 294Catalyst-layer modeling, PEFC
active phase volume fraction, 309agglomerate-type structure, 307–308impedance models
electrochemical impedancespectroscopy (EIS), 317, 319
equivalent-circuit of porouselectrode, 317
modeling equationsCL flooding approaches, 314–315electrocatalyst and electrolyte
interface, 311embedded agglomerate model,
311–313
525
526 Index
ionomer, 310kinetic expressions, 310–311surface concentration, 313–314
optimization analysisCL and GDL capillary properties,
316–317macrohomogeneous approach,
315–316two-phase and three-phase interface,
308Catalyst (platinum) utilization
cathode catalyst layer, 234effectiveness factor, 231, 232, 234Faradaic current density, 232, 233oxygen reduction reaction, 232parameterization, 232
Cathode catalyst layeragglomerates, 225capillary equilibrium, 231composition, 225, 226current-voltage curve, 230pore size distributions, 229rate of vaporization, 224sensitive dependencies, 229stability diagram, 231three state model, 228water balance modeling, 225Young–Laplace equation, 227
Cathode humidification temperature,276–277
Cell-design strategiesalternate cooling approaches,
364–365gas-flow direction
CFD models, 358counterflow and coflow, 357–358flow orientation manipulation, 357
interdigitated flow fields (IDFF)CFD models, 361gas-diffusion layer (GDL), 360–361mass-transport limitation reduction,
360pressure-drop calculation, 361water-transport plates, 363–364
optimal cell hydration, 357reactant flow and cross flow
orientation, 358–359
Chen, F.L., 367, 370Cheng, K.T., 55Chen, K.S., 354, 355Chang, S.M., 373Choe, S.Y., 371Choi, P., 294Chu, H.S., 349, 373Client–server model, 29Cold-start process
automotive process, 376–377frozen state startup process
bootstrap start, 384–386cell-level models, 388–392procedural strategies, 386–387stack-level models, 387–388
shutdown and freezingcell-level models, 379–384stack-level models, 378–379
Constant phase element (CPE) modelCPE behavior, 126–127disk electrodes, 127–130distribution function, 124–125double layer capacitance, 125–126faradaic reactions, 124fractal electrode model, 119–120Hull cell simulations, 127kinetic dispersion, 131–133porous electrode model, 94–95
Continuous porous modeldiffusion pores equation, 114–115impedance evaluation, 115polymer fuel cell, 117–118principle, 113solution theory, 115–117
Corey, A.T., 326Corrosion
carbon, 261–262catalyst support, 260PEMFC, 260–262
Costa, P., 362Cyclic voltammetry
electrochemical system, 435interaction potential, 440, 441Nernst approximation, 440, 441open-circuit voltage, 440potential sweep rate, 435–436
Index 527
Cylindrical pore electrode modelde Levie impedance equation, 69–70equivalent electrical circuit, 73–74phasors ratio, 70–71principle, 68–69transmission line impedance
equation, 72–73
DDarcy’s law, 300, 302, 362Darling, R.M., 363Datta, R., 294de Brujin, F.A., 394De Francesco, M., 387de Levie impedance equation, 69–70Desorption timescale, 153–154Devan, S., 115DeVidts, P., 312Diffusive timescale, 164–165Direct methanol fuel cell (DMFC)
technology, 170Disk electrodes, 127–130Djilali, N., 291, 294, 316, 322, 345, 364Douglas, M., 58Drachman, R.J., 53Drake, G.W.F., 49, 59, 61
EEESS. See Electrochemical energy-
storage systemEides, M.I., 55Eikema, K.S.E., 60Eikerling, M., 178, 294, 315Electrochemical cell interface (ECI),
504Electrochemical energy-storage system
(EESS)adaptive filter, 426–427composite power system, 420control system architectures, 424conventional lithium ion technology,
428electrochemical modeling
constant power operation, 442–456cyclic voltammetry, 435–442equivalent circuit, 431–434
equilibrium voltage, 428–429
state estimatorsalgorithm verification and
validation, 502–511generalized weighted recursive
least squares, 466–475method of least squares, 456–466regression analysis, 475–477variable forgetting factors, 493–502
state of charge (SOC), 418, 419state of health (SOH) and state of
power (SOP), 419weighted recursive least squares
(WRLS), 425Electrochemical impedance spec-
troscopy (EIS), 67, 317, 319Electrochemical modeling
constant power operationacetonitrile-based capacitor, 445Arrhenius relationship, 449, 453constant-power discharge and
charge, 447–450Coulombic capacity, 452current and voltage histories, 443,
444Lambert W function, 442, 443lithium ion cell, 446NiMH module, 455
cyclic voltammetryelectrochemical system, 435Nernst approximation, 440, 441open-circuit voltage, 440potential sweep rate, 435–436
equivalent circuit, 431–434Electrochemical reactions
constant phase element (CPE) modelCPE behavior, 126–127disk electrodes, 127–130distribution function, 124–125double layer capacitance, 125–126faradaic reactions, 124fractal electrode model, 119–120Hull cell simulations, 127kinetic dispersion, 131–133porous electrode model, 94–95
continuous porous modeldiffusion pores equation, 114–115impedancies evaluation, 115
528 Index
polymer fuel cell, 117–118principle, 113solution theory, 115–117
cylindrical pore electrode modelde Levie impedance equation,
69–70equivalent electrical circuit, 73–74phasors ratio, 70–71principle, 68–69transmission line impedance
equation, 72–73fractal electrode model
CPE behavior, 119–120distribution function, 123faradaic reaction, 121–122quasi-random surfaces, 120–121von Koch line segments, 118–119
red-ox porous electrodeabsence of dc current, 82–84concentration and potential
gradient, 105–110gradient concentration, 95–105pores distribution, 110–113presence of dc current, 85–95
V-grooved pore electrodesac signal penetration length, 79–81electronic resistivity, 79model kinetics, 78–79pore geometry, 74–75pore shape and size, 75–78
Electrochemical systemcomputer engineering aspects, 27–30
constructing modeling systems, 27data communication, 28software introduction, 29
mathematical modelingdefinition, 2geometric and physical properties
specification, 4–6postprocessing and analysis, 14–15solution method specification, 6–13solution process, 13–14
software designing, 26–27Electronic resistivity, 79Elliott, J.A, 178, 187Equivalent circuit
battery, 433, 434, 455, 457
regressed capacitance and resistance,510
symmetric supercapacitor, 432, 455Equivalent electrical circuit model,
73–74Ergun equation, 326Euler, J., 312
FFaradaic impedances
flat electrodes, 102–103polarization resistance, 101semicircle formation, 104–105Thiele modulus, 103–104
Faradaic reactions, 124Faraday’s law, 289, 304Farhat, Z.N., 309Ferreira, P.J., 256, 259Fickian diffusion model, 147, 152–153Fick’s equation, 96–97Fick’s law, 309, 314Fimrite, J., 291Finite-difference method
approximtion values, 17electrochemical setup, 2linear boundary value problem, 16
Finite element methodsblending functions, 25–26electrochemical setup, 2electrostatic problem, 18five-point sampling Laplace equation,
21–23Hermite polynomials and B splines,
24–25interpolating polynomials, 23–24weighted-residual formulation, 19–21
Fractal electrode modelCPE behavior, 119–120distribution function, 123faradaic reaction, 121–122quasi-random surfaces, 120–121von Koch line segments, 118–119
Freund, D.E., 49Friede, W., 367Frost heave, 382–383Frozen state startup process
bootstrap start, 384–386
Index 529
cell-level models1-D model, 389–390nonisothermal models, 390–392semi-empirical approach, 388–389
procedural strategies, 386–387stack-level models, 387–388
Fuel starvationelectrode potentials, 262, 263localized, 263–264
Fuller, T.F., 371Fundamental governing equations
conservation equationscharge conservation, 289energy conservation, 289–290mass conservation, 288–289principal equation types, 288
kineticsButler–Volmer expression, 285–286electrochemical reaction, 285electrode overpotential, 286–287hydrogen-oxidation reaction
(HOR), 287oxygen-reduction reaction (ORR),
287–288platinum metal electrode, 286
thermodynamics, 284–285efficiency definition, 285thermoneutral potential, 284
GGas channels, liquid water
droplet models and GDL/gas-channelinterface
boundary condition, 352contact-angle hysteresis, 355drag force, 354droplet-specific studies, 353droplet-stability diagrams, 355–356Reynolds number, 354–356surface-tension force, 353–354
gas-channel analysiscathode inlet relative humidity
effects, 349droplets effect on cell performance,
350dynamics, 352multiphase models, 348–349
water movement, flow channel,350–351
liquidwater transport mechanisms,347–348
reactant starvation, 347Gas-diffusion layers (GDL), water
movement, 320–321macroscopic analysis
anisotropic properties, 334–338compression, 338–343microporous layers, 328–331temperature-gradient (heat-pipe)
effect, 331–334two-phase-flow parameter
determination, 325–327microscopic treatments
capillary-pressure-saturationcurves, 323
capillary-tree and channelingmechanism, 318, 321
dominant water pathways, 321–322microscopic models, 322mixed-wettability system, 324relative permeability, 322–323
Gauss, C.F., 427Ge, J.B., 340Geometric specification, 4–5Ghausi, M.S., 426Gibbs free energy, 284, 385Gibbs–Thomson equation, 381Giordano, A.B., 426Goldman, S.P., 49, 61Gostick, J.T., 335Grotch, H., 55Guilin, H., 374Guilminot, E, 257Guo, Q.Z., 319Guvelioglu, G.H., 344
HHaddad, A., 367Hagstrom, S.A., 49, 50Hammer, B., 195Hardware-in-the-loop (HWIL) system,
503, 504Hartree–Fock method, 39Haykin, S., 426, 427
530 Index
Heat-pipe effect. See Temperature-gradient effect
Heat-transfer coefficient, 387–388He, G.L., 357Helium atom
basic setsadvantages, 49doubling, advantages, 46exponential scale factors variation,
47ground state, convergence study, 49principles, 47screened hydrogenic energy, 48
calculation methodsBohr–Sommerfeld quantum theory,
44configuration interaction (CI)
calculation, 44–45ground state energy, 44–45
computational methods, 49–50coordinate system, 39correlated variational basis sets
basic set members, 41Hylleraas–Undheim–McDonald
theorem, 42–43Pekeris shell, 41Rayleigh-Schrodinger variational
theorem, 42trial wave function, 44
Hartree–Fock method, 39Schrodinger equation, 38, 40
Henry, K.S., 383He, S., 383HEV. See Hybrid electric vehicleHickner, M.A., 279Higher temperature operation
advantages and disadvantages, 394cathode layers (CLs), 396novel membrane synthesis, 394–396polybenzimidazole (PBI) system,
396–397procedures, 392–394
High precision atomic theoryatomic units
Bohr radius, 37correction factor, 38of energy, 37
correction methodsmass polarization, 52–53quantum electrodynamic, 55–59relativistic, 53–55
helium energy levelsP-state ionization energy, 61QED breakdown, 62S-state ionization energy, 60
Kepler’s laws of planetary motion, 34mass polarization
center-of-mass (CM) frame, 52normal and specific isotope shift,
52perturbation approach, 52–53
nonrelativistic helium atombasis sets principles, 47calculations, 44–46computational methods, 49–50coordinate system, 39correlated variational basis sets,
40–44doubling the basis set, 46exponential scale factors variation,
47ground state, convergence study, 49Hartree–Fock method, 39Schrodinger equation, 38screened hydrogenic energy, 48variational basis sets, 50
nonrelativistic hydrogen atomclassical mechanics, 35gravitational interaction energy, 34Hamiltonian appraoch, 35ideas and concepts, 34radial function Rnl (ρ), 37Rydberg formula, 36Schrodinger’s equation, 36
quantum electrodynamic correctionsBethe logarithm, 56–57electron-electron QED, 57electron self energy, Feynman
diagram, 55relativistic corrections
Briet interactions, 53–54finite nuclear mass and recoil terms,
55Hill, R.N., 49
Index 531
Himanen, O., 342Hishinuma, Y., 390, 391Hitz, C., 76Horgorvorst, H., 60Hottinen, T., 342HPSP. See Hybrid powertrain simulation
programHuang, V.M.W., 127Huang, W., 371Hu, J.W., 397Hull cell simulations, 127Hwang, J.J., 290, 364HWIL. See Hardware-in-the-loopHybrid electric vehicles (HEVs)
electric-traction system, 421electrochemical cells, 428NiMH state estimator, 472propulsion system architecture,
420–421zero-emission-vehicle (ZEV) range,
430Hybrid powertrain simulation program
(HPSP), 505Hydrogen atom
center-of-mass plus relativecoordinates, 35
classical mechanics, 35gravitational interaction energy, 34ideas and concepts, 34radial function Rnl (ρ), 37Rydberg formula, 36Schrodinger’s equation, 36
Hylleraas, E.A., 40, 44Hylleraas–Undheim–McDonald
theorem, 42–43
IIhonen, J., 340Inoue, G., 362In situ visualization, PEFC
gas-diffusion layers, 280–281imaging techniques, 278–279
JJianren, F., 374Jiao, K., 350Jorcin, J.B., 127Ju, H., 345Ju, H.C., 345
KKabir, P.K., 56, 57Karan, K., 331Kaviany, M., 299Kazim, A., 361Keiser, H., 75Kelvin equation, 305Kernel identification, 30Khandelwal, M., 335Klahn, B., 43Knudsen diffusion coefficient, 154Knudsen number, 153–154Kornyshev, A.A., 185Korobov, V., 58Korobov, V.I., 49Kreuer, K.D., 291Krishna, R., 148Kroll, N.M., 58Kulhavy, R., 469
LLai, M.C., 350Laker, K.R., 426Lambert W function, 442, 443Lasia, A., 76Lattice-Boltzmann model, 322–323Least square methods
algorithm, 458, 464cell hysteresis voltage, 459NiMH battery, 457–459open-circuit voltage, 459, 460regression voltage, 461robustness, 465
Lee, C.I., 349Lee, S.J., 397Lennard–Jones (LJ) potential, 207, 212Leverett J -fuction, 322Levy, R., 426Liquid-phase transport
frost heave, 382–383Gibbs–Thomson equation, 381–382pure-diffusion model, 383–384
Lithium-ion cellalgorithm convergence test, 485, 490discharge power test, 486, 487electrochemical parameters, 489, 491open-circuit potential, 480, 481, 488power capability projections, 492
532 Index
recursive skewness analysis, 483skewness, determinant, and voltage
error, 48912-V Panasonic HV1255 VRLA
module, 480weight factor, 484
Lithium, variational basic sets, 50Litster, S., 280, 364Liu, H.T., 361Li, X.G., 360Ljnug, L., 428, 469Loch, J.P.G., 383Lucatorto, T.B., 60
MMacroscale (bulk) transport
general formulationsflux vector components, 146mass conservation, 145–146physical model assumptions,
147–148scale analysis
diffusional model, 148diffusion timescale, 151large Peclet number, 150–151small Peclet number, 149–150
Macroscopic analysis, GDLanisotropic properties
collective anisotropies, 338, 339electronic and thermal conductivity,
335in-plane and through-plane
permeability, 335–338relative Knudsen diffusivity, 334
GDL compressionconsequence of, 339contact resistances, 340current-density distribution, 342and gas channel expansion,
340–341modeling complexity, 342–343in situ state vs. ex situ state, 338
microporous layer (MPL)advantages, 328half-cell and full-cell models, 331hydrophilic pore fraction, 330–331oxygen transfer limitations, 329
PEFC performance, 328–329water pressure and saturation
profiles, 329–330temperature-gradient effect
liquid-saturation contours, 333mass-transport limitation, 333–334nonisothermal modeling, 331–332water and thermal management
coupling, 332two-phase-flow parameter
determinationabsolute permeability, 325–326Carman–Kozeny equation, 326effective permeability, 325Leverett J -function, 327relative permeability, 326–327
Manke, I., 280Mao, L., 391Marangos, J.P., 60Markicevic, B., 322Mass transport
description and representation,142–144
PS gas sensor timescales, 163–164sensor properties, 144–145
Mathematical modelingcomputer modeling, 3geometric and physical properties
specification, 4–6plate elecrode geometrics, 4postprocessing and analysis, 14–15solution method specification, 6–13
analytic solutions, 6finite-difference methods, 8–10finite-element methods, 10–11Galerkin method, 12–13Laplace equation, 6–7sampling theory, 7–8
solution process, 13–14Mathias, M., 261McIlrath, T.J., 60Membrane modeling, PEFC
concentrated solution theorybinary friction model, 291–292Schlogl’s equation, 293transport equations, 292water content calculation, 292–293
Index 533
membrane microstructure, 291membrane pretreatment, 290other transport through membrane,
297–298water content and properties
constraint treatment, 295–296membrane coefficient, 296–297molecular-dynamic-type models,
295water-uptake isotherm models,
293–295Mench, M.M., 335, 383Meng, H., 344, 353Meyers, J.P., 115, 253, 254, 257, 364Microscale transport, 161–163Miller, R.D., 382Modeling process, 2Model reference adaptive system
(MRAS), 424–425Moore, J.B., 469Moore, R.M., 388Morgan III, J.D., 49MRAS. See Model reference adaptive
systemMueller, F., 366, 372Multiscale formulation
diffusive timescale, 164–165mass transport timescales, 163–164simple adsorption model, 165–166
Munroe, N.D.H., 397
NNam, J.H., 299Nanoscale transport
anlytical solutionorthogonal series expansion
solution, 159–161steady-state problem, 158–159
nanoporesdiffusion model, 154–155simplified model, 155–156transient properties, 157–158
nanopores continuum assumptionadsorption and desorption time
scales, 153–154molecular length and time scale,
152–153
Navier–Stokes equations, 300, 361Nazarov, I., 295Nernst equation, 284, 286, 440, 441Newman, J., 277, 294, 295, 297, 312,
332, 345, 358, 371, 431Neyerlin, K.C., 287Nguyen, T.V., 361Nonnenmacher, W., 312Nørskov, J.K., 195Nyikos, L., 119, 120
OO’Brian, T.R., 60Ohm’s law, 292, 306, 309Okada, T., 294Ong, I.J., 431Oszcipok, M., 388Ota, K.I., 256
PPachucki, K., 51, 58, 61, 63, 64Paddison, S.J., 178, 187Pajkossy, T., 119, 120Parallel hybrid propulsion system, 421Park, J., 360Pasaogullari, U., 328, 331, 337Peclet number, 149–151Pekeris, C.L., 45Pekeris shell, 41Peltier coefficient, 290PEMFC. See Proton-exchange
membrane fuel cellsPenetrability coefficient, 111Peng, J., 397Perry, M.L., 263, 265Pesaran, A.A., 376, 378, 387Petersen, M.K., 178Phaes field model, 26–27Pharaoh, J.G., 335, 336Phasors ratio, 70–71Pillar, S., 426Plackett, R.L., 427Plate electrode geometry, 4Platinum nanoparticle catalyst
carbon-supportedcyclic voltammograms, 251electrochemical oxidation, 250
534 Index
chemical statePourbaix diagrams, 251solubility, 252
dissolutionequilibrium concentration vs.
electrode potential, 254potential cycling, 256, 257
particle growth, 257–260Poiseulle’s equation, 326Polymer electrolyte fuel cells (PEFCs)
basic methodologygeometric dimensionality, 281–282macroscopic and microscopic
models, 281pseudo-dimensional models, 282
catalyst-layer modelingactive phase volume fraction, 309agglomerate-type structure,
307–308impedance models, 317–319modeling equations, 309–315optimization analyses, 315–317two-phase and three-phase
interface, 308cell-design strategies
alternate cooling approaches,364–365
gas-flow direction, 357–359interdigitated flow fields, 360–363optimal cell hydration, 357water-transport plates, 363–364
cold-start processautomotive process, 376–377frozen state startup process,
384–392shutdown and freezing, 377–384
continuous porous model, 117–118electron transport, 306–307fundamental governing equations
conservation equations, 288–290kinetics, 285–288thermodynamics, 284–285
gas channels, liquid water, 347–348droplet models and GDL/gas-
channel interface, 352–357gas-channel analyses, 348–352
liquidwater transport mechanisms,347–348
reactant starvation, 347higher temperature operation
advantages and disadvantages, 394cathode layers (CLs), 396novel membrane synthesis,
394–396polybenzimidazole (PBI) system,
396–397procedures, 392–394
hydrogen, 170, 171low-relative-humidity operation
3-D velocity profiles, 345membrane-cathode interface,
345–347reactant stream humidification,
343–344macroscopic modeling, 277materials modeling
complex process, 176length scales, 175, 176multi-scale phenomena, 176
membrane modelingconcentrated solution theory,
291–293membrane microstructure, 291membrane pretreatment, 290other transport through membrane,
297–298water content and properties,
293–297model implementation and boundary
conditions, 319–320multilayered design, 171, 172nonuniformities, 343PEM and catalyst layer
agglomerates, 205Carbon–Nafion–Water–Solvent
(CNWS), 208coarse-grained molecular dynamics
(CG-MD), 205, 208, 211complex interactions, 204computational approach, 206Coulombic interaction, 207Derjaguin–Landau–Verwey–
Overbeek (DLVO), 213
Index 535
hydrated Nafion membrane, 211,212
interaction parameters, 210Lennard–Jones (LJ) potential, 207,
212microstructure and pore size
distribution, 204site–site radial distribution
function, 209solvent dielectric constant, 210structural complexity, 206structural formation process, 205structure–performance relationship,
204platinum nanoparticle electrocatalysis
active site model, 200, 202adsorption energies, 194catalyst poison, 197chronoamperometric current
transients, 200, 202complex surface reaction
mechanism, 196heterogeneous surface model, 198hydrogen reduction kinetics, 195kinetic modeling, 199kinetic Monte Carlo (kMC)
simulations, 201methanol electrooxidation, 195orbital Free DFTcalculation, 196,
197specific exchange current density,
193spillover effect, 194Tafel-plots, 201, 203transient current, 199
polarization curve, 274–275polymer electrolyte membrane
(PEM), 172proton transport, 182–193typical 7-layer structure, 171–172
random heterogeneous mediacomposite porous catalyst layers,
213fractal internal surface, 214, 215membrane electrode assemblies
(MEAs), 213
scales reconcilingcatalyst utilization, 231–235cathode catalyst layer, 223–231water management, 219–223
shutdown and freezing, 377cell-level models, 379–384stack-level models, 378–379
in situ visualization of water,278–281
startup from frozen state, 384–387cell-level models, 388–392stack-level models, 387–388
structure and waterblock-copolymer systems, 217electro-osmotic drag effect, 209Joule heating, 217mass transport phenomena, 217membrane dehydration, 216pore size distributions, 218two phase models, 218
transient operation and load changessingle-phase-flow models, 367–372time-constant analysis, 366two-phase-flow models, 372–376water-management strategies, 365
two-phase flowgas-diffusion layer, 298–299gas-phase transport, 300–303liquid and gas phase coupling,
303–306liquid-phase transport, 300
Polymer electrolyte membrane (PEM)proton transport
activation energy, 183, 184Car-Parinello molecular dynamics,
191charge transfer theory, 185conductivity, 183Coulomb barrier, 186empirical valence bond (EVB)
approach, 184–185formation energy, 189, 190microscopic mechanism, 184molecular modelling, 187objectives, 192Poisson–Boltzmann theory, 185sulfonate ions, 185
536 Index
water binding and molecularmechanisms, 186
Zundel-ion, 187, 188structural evolution, 182, 183typical 7-layer structure, 171–172water management
diffusion models, 221, 222electro-osmotic coupling, 219Gibbs free energy, 223hydraulic permeation model, 221,
222molar flux, 220pressure gradient, 220proton conductivity, 219proton current density, 220
Porous electrodescontinuous porous model
diffusion pores equation, 114–115impedancies evaluation, 115polymer fuel cell, 117–118principle, 113solution theory, 115–117
cylindrical pore electrode model,67–74
definition, 67red-ox and double layer capacitance
absence of dc current, 82–84concentration and potential
gradient, 105–110pores distribution, 110–113presence of concentration gradient,
95–105presence of dc current, 85–95
V-grooved pore electrodes, 74–81Porous silicon (PS) gas sensors
analytical solutions, 142chemical sensors, 141–142dissolution process, 139–140macroscale (bulk) transport
general formulations, 145–148scale analysis, 148–151
mass transportdescription and representation,
142–144sensor response, 144–145
microscale transport, 161–163multiscale formulation, 163–166
nanoscale transportanalytical solution, 158–161nanopores continuum assumption,
152–154nanopores diffusion model,
154–155nanopores simplified model,
155–156nanopore transient response,
157–158visible photoluminescence (PL),
140–141Postprocessing and analysis modeling,
14–15Press, W.H., 467Promislow, K., 295, 299Proportional-integral-differential (PID)
schemes, 424–425Proton-exchange membrane fuel cells
(PEMFCs)alloy effects
crystallinity and exchange currentdensities, 267
degradation rate, 268phosphoric acid system, 267stability, 268
corrosioncarbon, 261–262catalyst support, 260
fuel starvationelectrode potentials, 262, 263localized, 263–264
platinum nanoparticle catalystcarbon-supported, 250–251chemical state, 251–253dissolution, 253–257particle growth, 257–260
start/stop cycling, 264–265temperature and relative humidity,
266Proton transport (PT)
activation energy, 183, 184Car-Parinello molecular dynamics,
191charge transfer theory, 185conductivity, 183Coulomb barrier, 186
Index 537
empirical valence bond (EVB)approach, 184–185
formation energy, 189, 190microscopic mechanism, 184molecular modelling, 187objectives, 192Poisson–Boltzmann theory, 185sulfonate ions, 185water binding and molecular
mechanisms, 186Zundel-ion, 187, 188
QQuan, P., 350
RRand, D.A., 256Rao, R.M., 368Rayleigh-Schrodinger variational
theorem, 42Red-ox porous electrode
absence of dc currentintermediate length pores, 82–83semi-infinite pores, 83–84shallow pores, 83transfer resistance, 82
concentration and potential gradientdiffusion coefficients, 105–106electroreduction process, 107–108impedance complex, 109–110limitations, 106–107
gradient concentrationcurrent density–potential relation,
95–96faradaic and double layer
impedances, 101–105Fick’s equation, 96–97limitations, 98–99linearized current, 99–100Thiele modulus, 97
pores distributiondistribution functions, 111–112Fredholm integral equation,
112–113transmission line ladder network,
110–111presence of dc current
Butler–Volmer equation, 86current density, 85–86
electrode impedances, 88–90semi-infinite length pores, 87simulated impedances, 93–95skewed impedances, 90–92Tafel curves, 87–88
Reiser, C.A., 263, 265Rempel, A.W., 383Rengaswamy, R., 368Reverse current mechanism, 265Reynolds number, 354–356Roen, L.M., 261Rolston, S.L., 60Rost, J.-M., 49Roudgar, A., 178
SSalpeter, E.E., 44, 56, 57Sansonetti, C.J., 60Santhanagopalan, S., 426Sapirstein, J., 55Scanlan, J.O., 426Schiff, B., 45Schlogl’s equation, 293Schroder’s paradox, 296Schrodinger’s equation, 36, 38, 40Schulz, V.P., 323Schwartz, C., 49Semi-infinite pores plot, 83–84Shah, A.A., 312, 316, 374Shallow pores plot, 83Shan, Y.Y., 371, 372Shelyuto, V.A., 55Shutdown and freezing process
cell-level modelsliquid-phase transport, 381–384vapor-phase transport, 379–381
stack-level models, 378–379Sims, J.S., 49, 50Single-phase-flow models
isothermal transient model, 367–370lumped model
0-D, 1-D, 2-D models, 3683-D isothermal model, 368–370membrane hydration effects,
367–368nonisothermal transient model,
370–372steady-state performance, 370–372
538 Index
Sinha, P.K., 323Smith, K.A., 426SOC. See State of chargeSoderstrom, T., 428, 469SOH. See State of healthSolidification process, 27Solid oxide fuel cells (SOFCs), 170Solution method specification
analytic solutions, 6finite-difference methods, 8–10finite-element methods, 10–11Galerkin method, 12–13Laplace equation, 6–7sampling theory, 7–8
Song, D.T., 316, 374Song, H.K., 111SOP. See State of powerSpohr, E., 178, 185Springer, T.E., 117, 319, 389Srinivasan, S., 312State estimators
algorithm verification and validationcapacitor voltage, 506hardware-in-the-loop (HWIL)
system, 503, 504maximum discharge power, 508test protocol, 509velocity vs time relationship, 509,
510generalized weighted recursive least
squaresalgorithm, 466, 470instantaneous error, 466matrix system of equations,
469–470parameter models, 473–475weight factor, 468
least square methodalgorithm, 458, 464cell hysteresis voltage, 459NiMH battery, 457–459open-circuit voltage, 459, 460regression voltage, 461robustness, 465
lithium-ion cellalgorithm convergence test, 485,
490
discharge power test, 486, 487electrochemical parameters, 489,
491open-circuit potential, 480, 481,
488power capability projections, 492recursive skewness analysis, 483skewness, determinant, and voltage
error, 48912-V Panasonic HV1255 VRLA
module, 480weight factor, 484
regression analysisdeterminant value, 475–476skewness, 476–477
state of power (SOP)constant-voltage, 478–480maximum discharge power, 477
variable forgetting factorshigh-power-density lithium ion
battery, 493optimized values, 499power projections, 501
State of charge (SOC)experiment-theory comparison, 438,
439regressed combined and voltage-
based, 465schematic representation, 418
State of health (SOH)definition, 501electrochemical parameters, 491schematic representation, 419
State of power (SOP)composite power system, 420constant-voltage, 478–480maximum discharge power, 477
Stearns, S.D, 426Stefan–Maxwell equations, 291,
300–302Stenger, H.G., 344St-Pierre, J., 294, 358Sucher, J., 56Sundaresan, M., 388Surface diffusion coefficient, 155
Index 539
TTafel curves, 87–89Tate, E.D., 425Temkin, A., 53Temperature-gradient effect, 331–334Thermoneutral potential, 284Thiele modulus, 97, 102–103Thomas-Alyea, K.E., 312Three-phase electric-traction system,
421Tiedemann, W., 312Tobias, C.W., 312Transfer resistance, 82Transient operation, load changes
single-phase-flow models, 367–372time-constant analysis, 366water-management strategies, 365
Transmission line impedance equation,72–73
Tretter, S.A., 427Two-phase-flow models
cell uniform temperature, 372–3741-D and 3-D CFD model, 374–375gas-diffusion layer, 298–299gas-phase transport
dusty-gas model, 302–303gas-phase volume fraction, 301Knudsen diffusion, 302Stefan–Maxwell equations,
300–301liquid and gas phase coupling
capillary pressure, 303cathode GDL/CL interface, 304Kelvin equation, 305multiphase mixture model, 306
liquid-phase transport, 300oxygen mole fraction, 375–376
UUbachs, W., 60Udell, K.S., 323, 327
VVahidi, A., 494Van Valkenburg, M.E., 426Van Zee, J.W., 345, 368, 374Vapor-phase transport, 379–381Vassen, W., 60
V-grooved pore electrodesac signal penetration length, 79–81electronic resistivity, 79model kinetics, 78–79pore geometry, 74–75pore shape and size, 75–78
Visible photoluminescence, 140–141Vogel, H.J., 323Von Koch line segments, fractal model,
118–119Vorobev, A., 367Voth, G.A., 178, 185
WWalbran, S., 185Wang, C.Y., 277, 323, 328, 332, 344,
345, 353, 368, 391, 396Wang, G.Q., 308Wang, L., 361Wang, Q.P., 316Wang, Y., 332, 345, 368Warshel, A., 184Weber, A.Z., 294, 295, 297, 299, 315,
332, 345, 358, 363, 373Weighted recursive least squares
(WRLS)algorithm, 466, 470application, 425–426instantaneous error, 466lead acid and lithium ion cell
characteristics, 472matrix system equations, 469–470parameter models, 473–475parameter regression, 425statistics, 483step-by-step comparison, 428weight factor, 468, 484
Weighted-residual formulation, 19–21Wen, J., 60Wesselingh, J.A., 148Westbrook, N., 60White, R.E., 312, 319, 426Widrow, B., 426Wiegman, H.L.N., 425Wiezell, K., 319Wilkinson, D.P., 358Williams, M.V., 335
540 Index
Wintgen, D., 49Wittenmark, B., 428WRLS. See Weighted recursive least
squaresWyllie equation, 322
YYamada, H., 361Yan, W.M., 361, 368Yan, Z.-C., 59Yasuda, K., 257
Yelkhovsky, A., 58Yi, J.S., 349Young–Laplace equation, 227Yu, H.M., 367Yu, P.T., 263, 265
ZZhang, F.Y., 354, 357Zhan, Z.G., 350Ziegler, C., 367, 373Zou, J., 361
Color Plates
Length scale (m)
10−10 10−9 10−8 10−6 10−5 10−4 10−2
Sec.III
Sec. IV
Sec. V & VI
Sec. VII
Figure 2. Multi-scale phenomena in PEFC, from fundamental proton transport inPEM (Sect. III), to kinetic mechanisms at nanoparticle electrocatalysts (Sect. IV),to structure formation (Sect. V) and effective properties (Sect. VI) of complexcomposite materials, to transport, reaction and performance at the macroscopicdevice level (Sect. VII).
(a)
21 3
Figure 7. (a) Molecular mechanism of interfacial proton transfer at the minimallyhydrated array.
Active Pt siteInactive Pt site
Support
reactants (e.g. O2)wateradsorbents
(e.g. OH) nanoparticle
substrate
Figure 8. Mapping of surface structure of supported nanoparticle onto a 2D regularhexagonal array, distinguishing active and inactive sites.
Figure 10. Chronoamperometric current transients and surface processes on cata-lyst model surface during COad electrooxidation.
Solvent
C/Pt agglomerated C/Pt
micelle
ionomer adsorption cross-linking
Ionomer
polar
apolar
Figure 14. Schematic representation of structural formation processes during thefabrication of conventional catalyst layers in PEFC.
Figure 16. Equilibrium structure of a catalyst blend composed of Carbon (black),Nafion (red), Water (green) and implicit solvent. Hydrophilic domains are notshown in (b) for better visualization.
Figure 17. Site–site radial distribution functionsfor the CNWS system (C carbon; P polymer back-bones; W water; H cluster containing hydronium).
Figure 19. (a) Snapshots of the final microstructure in hydrated Nafion membrane atdifferent water contents. Hydrophilic domain (water, hydronium and side chains) isshown in green, while hydrophobic domain is in red. (b) Site–site RDF showing theseparation of hydrophilic and hydrophobic domains in Nafion membrane. W water;S side chain; H hydronium; B ionomer backbone.