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.