Detailed Dynamic Solid Oxide Fuel Cell Modeling for Electrochemical

Journal of Power Sources 195 (2010) 53205339

Contents lists available at ScienceDirect

Journal of Power Sources

journa l homepage: www.e lsev ier .com

Detailed dynamic Solid Oxide Fuel Cell modelingimpedance spectra simulation

Ph. Hofmanna, K.D. Panopoulosb,

a Laboratory of ring SeHeroon Polytecb Institute for S km. PP.O. Box 95, GR

a r t i c l

Article history:Received 17 JuReceived in reAccepted 13 February 2010Available online 24 February 2010

Keywords:Solid oxide fuel cell (SOFC)ImpedanceEISgPROMSTM

Simulation

le my-stabeha

impedance spectroscopy (EIS). Themodel isbasedonphysico-chemical governingequations coupledwitha detailedmulti-component gas diffusionmechanism (Dusty-GasModel (DGM)) and amulti-step hetero-geneous reaction mechanism implicitly accounting for the water-gas-shift (WGS), methane reformingand Boudouard reactions. Spatial discretization can be applied for 1D (button-cell approximation) upto quasi-3D (full size anode supported cell in cross-ow conguration) geometries and is resolved withthe nite difference method (FDM). The model is built and implemented on the commercially available

1. Introdu

1.1. The sol

An operpower by cthe rest isglobal hydr

Abbreviatioary conditionscentered nitFDM, nite difVolume Methcatalytic reactstirred reactor

CorresponE-mail add

0378-7753/$ doi:10.1016/j.modeling and simulations platform gPROMSTM. Different fuels based on hydrogen, methane and syngaswith inert diluents are run. The model is applied to demonstrate a detailed analysis of the SOFC inherentlosses and their attribution to the EIS. This is achieved by means of a step-by-step analysis of the involvedtransient processes such as gas conversion in the main gas chambers/channels, gas diffusion through theporous electrodes together with the heterogeneous reactions on the nickel catalyst, and the double-layercurrent within the electrochemical reaction zone. The model is an important tool for analyzing SOFC per-formance fundamentals as well as for design and optimization of materials and operational parameters.

2010 Elsevier B.V. All rights reserved.

ction

id oxide fuel cell

ating solid oxide fuel cell (Fig. 1) produces electricalonverting part of the chemical energy of a fuel whilerejected as heat due to the oxidation reactions. Theogen oxidation reaction, which is assumedly the fastest

ns: AC, alternating current; ASC, anode supported cell; B.C., bound-; BC, base case; BFDM, backward nite difference method; CFDM,e difference method; DC, direct current; DGM, Dusty-Gas Model;ference method; FFDM, forward nite difference method; FVM, Finiteod; EIS, electrochemical impedance spectrum; HCR, heterogeneousion; I.C., initial conditions; OCV, open circuit voltage; PSTR, perfectly; SOFC, solid oxide fuel cell; TPB, triple phase boundary.ding author. Tel.: +30 210 6501771; fax: +30 210 6501598.ress: [email protected] (K.D. Panopoulos).

electrochemical reaction within an SOFC is:

H2 +12O2 H2O (1)

The electrical potential reaches its theoretical maximum Erev(=reversible potential) at electrochemical equilibrium, i.e. zero cur-rent operation (unpolarized cell or open circuit voltage OCV) andchemical equilibrium of reactants and products. This is related tothe Gibbs free energy of the electrochemical reaction through thefollowing equation:

Erev = GnF = G

nF RT

nFln Q (2)

The rst part of the right hand side equation is the temperature-dependent standard potential E and the second part describes theinuence of reactants activities (here partial pressures) expressedthrough the reaction quotient Q of the electrochemical reaction.Substituting the reactionquotientQwithpartial pressure termsandthe rst part of the right hand side equation with the temperature-dependent standard potential E, Eq. (2) results in the well-known

see front matter 2010 Elsevier B.V. All rights reserved.jpowsour.2010.02.046Steam Boilers and Thermal Plants, School of Mechanical Engineering, Thermal Engineehniou 9, 15780 Athens, Greeceolid Fuels Technology and Applications, Centre for Research and Technology Hellas, 4th502, 50200 Ptolemais, Greece

e i n f o

ly 2009vised form 21 January 2010

a b s t r a c t

This paper presents a detailed exibwhich allows the simulation of stead(Vj) curves, and dynamic operation/ locate / jpowsour

for electrochemical

ction, National Technical University of Athens,

tolemais-Mpodosakeio Hospital, Region of Kouri,

athematical model for planar solid oxide fuel cells (SOFCs),te performance characteristics, i.e. voltagecurrent densityvior, with a special capability of simulating electrochemical

Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195 (2010) 53205339 5321

Nomenclature

aO2 activityofbulkoxygen ions (KroegerVinknotation)aNi specic surface area of Nickel catalyst (cm2 cm3)aV

Oactivity of electrolyte bulk vacancies (KroegerVinknotation)

A area (m2)Acell solid oxide fuel cell active area (m2)Ach gas channel cross-section area (m2)Ai pre-exponential factor of reaction i (units vary)

(mol, cm, s)cel,k gas phase species concentration (mol cm3) or sur-

face species concentration (mol cm2)C electrical double-layer capacitance (F cm2)dan anode thickness (m)dca cathode thickness (m)delectrolyte electrolyte thickness (m)Dk,j binary diffusion coefcient (cm2 s1)DKN,k Knudsen diffusion coefcient (cm2 s1)E voltage (V)Ea,i activation energy of reaction i (Jmol1 K1)Ecell electrical potential of the SOFC (V)E standard potential (temperature-dependent) (V)Ei activation energy for electrolyte conductivity

(Jmol1 K1)Ep amplitude of alternating cell voltage output (V)f frequency (Hz)F Faraday constant =6.0231023 1.6021019

(Cbmol1)G molar Gibbs free energy change of reaction (1)

(Jmol1)G standardmolarGibbs free energy change of reaction

(1) (Jmol1)h height (m)I current (A)Iall number of irreversible elementary reactionsIad number of adsorption reactionsIgain gain current amplitude for EIS (A)j current density (A cm2)j0,el exchange current density (A cm2)Jbias bias current density for EIS (A cm2)jF,el Faradaic current density (A cm2)Kan number of chemical species at the anode sideKg,an number of gaseous chemical species at the anode

sideKs number of surface chemical species at the anode

sidel cell length (m)mch mass ow (kg s1)MWk molecular weight of species kn number of electrons transferred in reaction (1)nk molar ux of species k (mol s1 cm2)N volume ow (L s1)pi partial pressure of component i (atm)Pel,tot total pressure of electrode channel (bar)Pop the SOFC operating pressure (bar)Q reaction quotientri adsorption reaction rates (mol cm2 s1)rpore pore diameter (m)rTPB electrochemical reaction rate (mol cm2 s1)R area-specic resistance (Ohmcm2)Rg ideal gas constant (8.314 Jmol1 K1)Rohm ohmic electric area-specic resistance (ohmcm2)

sk Species net molar production rate (mol cm2 s1)t time (s)T temperature (K)uch gas velocity in channels (ms1)w width (m)Wch width of channel plus part under interconnect rib

(m)Uf fuel utilization factor ()Uo oxygen utilization factor ()V voltage (V)Vch gas channel/chamber volume (m3)V0m standard molar volume (mol L

1)x dimension xX mole fraction ()y dimension xY mass fraction ()z dimension zZ impedance (Ohmcm2)

Greek lettersan,el anodic symmetry factor for ButlerVolmer equation

()ca,el cathodic symmetry factor for ButlerVolmer equa-

tion ()i temperature exponent () i sticking coefcient () available surface site density (mol cm2) porosity ()i surface site fraction-dependent activation energy

()ohm,act,conc overpotentials due to ohmic, activation, concen-

tration losses (V) surface site fraction () period (s1)ki the difference between stoichiometric coefcients

of products and reactants of the kth species in theith reaction.

ch gas density in channel (kgm3)

i electrolyte conductivity (S cm1)

0 parameter for electrolyte conductivity (SK1 cm1) tortuosity () phase angle ()e,el electrode (anodeor cathode)electronicpotential (V)i,el electrode (anode or cathode) ionic potential (V)el potential step in electrode (anode or cathode) (V)

Subscriptsan anodebias biased variableca cathodecell total cellch channel (i.e. anode or cathode side)e electronicel electrode: el = an for anode and el = ca for cathodeeq equilibriumdl double layerF Faradaici ionic phase (when used in )i reaction counterin inputk species counterohm ohmic

5322 Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195 (2010) 53205339

p peak (when used with f)rev reversibletot

Superscriin

Nernst equa

Erev = E +

When thacurrent iscells potenwhich depmechanism

Ohmic resphases (ephases (Now.

Concentrathe electrary TPBby slow dthrough t

Activationbecause ein the descathode a

The opertion of the d

Ecell = Erev= Erev

Accordincontributiothese voltagsteps. In refuel cell sys

potential e,ca, the ionic phase (electrolyte) potential i and theanode electronic phase (electrode) potential e,an. The potentialdifference between cathode and anode constitutes the operatingcell potential:

e,ca

sults

he c

ca =at t

an =

ce thnd eonic)a certentn zotantal re

electsituad exing

phasunpriumuctions Eqhode

2e

ca = E

de:total

ptinput

Fig. 1. Schematic representation of an SOFC operating.

tion:

RT

2Fln

(pH2p

1/2O2

pH2O

)(3)

e fuel cell is connected to a load through a closed circuit,produced through theelectrochemical reactionsand thetial is reduced by internal non-reversible voltage lossesend on the current and derive from the following threes:

Ecell =It re

(1) at t

(2) and

Sintrode a(electrwithinThis poreactioall reacchemicin theTPB isface andependzone),

Theequilibthe redreactio

Cat

12O2 +

eq,

Ano

istance losses ohm: which occur in the solid electrolyte.g. YSZ or GDC) due to ions ow and in the electrodei, LSM, etc.) and metallic interconnects due to electrons

tion overpotentials conc: reduced Nernst potential atochemically active reaction zone (triple phase bound-) due to depletion of charge carrying reactants causediffusion from the bulk of the gas chambers/channelshe porous electrodes.overpotentials act: reduced electrochemical potentialnergy is needed to drive the electrochemical reactionsired forward direction, i.e. reduction of oxygen at thend oxidation of hydrogen at the anode.

ating cell potential thus can be expressed as a subtrac-ifferent losses from the reversible potential [14]:

(j) ohm conc,an conc,ca act,an |act,ca| (4)

g toBessler [5], Eq. (4) gives indeedagoodpictureof then from the different kinds of loss mechanisms, howevere losses do not represent physicalmeaningful potentialality, three different potential levels exist within thetem. These are the cathode electronic phase (electrode)

H2 + O2

eq,an =

Their difthe Nernst

The dropelectrolyteand cathodgen ions mpart in theinto the eleoxygen ionhave typicamembranetrolyte/elecelectrode thhigh electroapproximat

The cobetween thchamber (oan,TPB atin the differsion induce e,an (5)from two potential steps occurring:

athode/electrolyte interface:

e,ca i,ca (6)he anode/electrolyte interface:

e,an i,an (7)

e state-of-the art SOFC electrodes contain both elec-lectrolyte phases in form of distributed particles (e.g. Niand YSZ (ionic) in the anode), the potential steps varytain depth of the porous anode and cathode electrodes.ial distribution is conned to the electrochemical activene, the so-called triple phase boundary (TPB), wheres and products can meet and proceed with the electro-actions: Ionic O2 (in electrolyte phase), gas reactantsrode pores and electrons (in the electrode phases). Theted near the electrolyte membrane and electrode inter-tents typically a few ten microns [6] into the electrodeon parameters such as the TPB length (active reactione conductivities and gas phase activities.olarized cell is in electrochemical equilibrium, and thepotential steps given by Eqs. (9) and (11) arise fromn potentials of the respective half-cell electrochemicals. (8) and (10) [7]:

:

O2 (8)

O2/O

2 RT

2Fln

(aO2

p0.5O2 aV

O

)(9)

H2O + 2e (10)

EH2O/H2

RT2F

ln

(pH2 aO2pH2O aVO

)(11)

ference equals the reversible cell potential Erev given byEq. (3) for the global reaction Eq. (1).in ionic phase potential i occurs mainly in the dense

membrane but also to a certain extentwithin the anodicic electrochemical reaction zones (TPB) where the oxy-igrate from and into the respective electrodes to takedistributed charge-transfer. The more the TPB extendsctrode, the higher are the ionic ohmic losses because theneeds to pass through the electrolyte particles whichlly much smaller conductivity than the bulk electrolytedue to the porous and distributed nature of the elec-trode cermet. The drop in electrode potentials along theickness due to electrons transfer is negligible due to thenic conductivity, thuse,ca ande,an can be consideredely constant.ncentration overpotentials represent a differencee larger potential step an,b at the electrode/gasr channel) interface and the smaller potential stepthe TPB. Their cause canbededucted fromFig. 2 and liesent half-cell reduction potentials eq,el due to diffu-d reactants partial pressure gradients. Fig. 2 additionally


Fig. 2. PartialH2/H2O and ca

shows thattial pressuroxygen depstant duringthe half-celindependen

Activatiothe potenti(TPB) due tchemical reare in equithe higher tavailable TPelectrodes a

act,ca = act,an =

The abofromthe thrimplicitly thhalf-cell red

1.2. Electro

Electrocfor solid ox[3,810]. Thwith equivaall performphysical souoverlapping

In theorhave its owDue to thecertain timedition. In pra sinusoidavaried for acapacitancereactor volufor heat traport betwee

In SOFC impedancemeasurements, the dependent output signal(in this work the voltage) has the same frequency as the perturbinginput signal (here the current) but due to the capacitances is shiftedby a negative phase angle and thus manifests itself as arcs on the

ve imis exort procesal kitiveley exsporchars.ractisurees orferend. W0) isto thhavioary ae reltakent mis aeudose at]. Thinputpartrve, Eausein thExterremeancelowl err

them

devarteda qupressure distribution within the porous electrodes for the anodicthodic O2/N2 systems (1D).

the electrode/gas chamber (or channel) interface par-es differ from the inlet partial pressures due to fuel andletion (utilization) when the inlet ows are kept con-polarization. This results in an additional reduction of

l reduction potentials and is a material and geometryt purely thermodynamic loss.n overpotentials given by Eqs. (12) and (13) decrease

al difference within the electrochemical reaction zoneo the additional energy required to drive the electro-actions into the desired forward reaction. These losseslibrium with the ionic ohmic losses within the TPB:he electrochemical reaction rates and/or the larger theB, the less do the oxygen ions need to travel into thend vice versa.

ca eq,ca whereca < eq,ca (12)

an eq,an where an < eq,an (13)ve described origin of the cell potential Ecell resultingeedifferent potential levelswithin the fuel cell includese different losses given in Eq. (4) which reduce the twouction potentials.

chemical impedance spectroscopy

negatireal axtranspport pchemicrespecand thlar tranof theproces

Inpor meaent sizthe difguishe(/t=adaptary beimaginthus ththis isdiffere

EIS(or psresponnal [11of thelinearVj-cumon ca driftstate.measuimpedand/orimenta

2. Ma

Thewas stwherehemical impedance spectroscopy is a widely used toolide fuel cell (SOFC) performance and materials analysise common approach of tting the impedance spectralent electrical circuit models is good enough for over-ance comparison, but lacks accuracy in explaining therce of the different losses, especially due to the usuallyarcs of the spectrum.

y, each transport process occurring in the SOFC shouldn arc in the electrochemical impedance spectrum (EIS).ir capacitive nature, the transport processes need ato relax when perturbed by a changing boundary con-

actical SOFC impedance measurements, this is typicallyl AC current or voltage on top of a DC bias which isrange of frequencies in order to generate the EIS. Thes for themain transport processes aremass (function ofme and mass density) for mass transport, heat capacitynsport and double-layer capacitance for charge trans-n ionic and electronic conductive phases.

sented by tmembraneume MethoSOFC modewas built inSection 1, adetailed anand EIS sim2D and quwhich onlyis a good apups. For thecathodic syresenting pthe boundacounter-oincludes eldiscretizatition in the yaginary impedance axis. The width of these arcs on thepress the relaxation time distributions of the respectiverocesses and are related to their resistances: the trans-s are inhibited by convective and diffusive velocities,netics, heat conductivity and charge-transfer kinetics,y. These arcs have the shape of a semi-circle (or similar)press the range of frequencies for which the particu-t process is sensitive. The peak frequency is the inverseacteristic relaxation time of the underlying transport

ce, theelectrochemical impedancespectrum(simulatedd) manifests itself as a superimposition of arcs of differ-iginating from the underlying transport processes. Thust overpotential contributions cannot clearly be distin-hen the transient term of a transport process equationset to zero, the equations output values immediately

e varying input signal. This simulated periodic station-r with no capacitive inertia results in no signal on thexis of the EIS. The real axis however is not affected andevant process resistances are still effective. In thiswork,n advantage of in order to break down the EIS into theain contributing loss mechanisms.rather sensitive measurement method. Only a linear-linear) system results in a sinusoidal phase-shiftedthe same frequency as the sinusoidal perturbation sig-e cells response is pseudo-linear when the amplitudesignal is small and measurements are done in pseudo-of the Vj-curve. In the highly non-linear part of theIS spectra can loose their linear behavior. Another com-of problems in EIS measurements and their analysis ise system being measured due to non-stationary initialnal factors such as wiring of the current and voltagent leads can cause additional capacitive or inductivefeatures in EIS measurements often observed as highfrequency artifacts [10,12]. A detailed analysis of exper-ors in EISmeasurement is givenbyCimenti et al. [13,14].

atical model description

elopment of a distributed model of single planar SOFCson the EESTM simultaneous equation solver platformasi-2D steady-state model was implemented as pre-he authors in [15], in which spatial distribution in theplane (x- and y-direction)were solvedvia the FiniteVol-d (FVM). In the current work, a more complex dynamicl capable of simulating 1D, 2D and quasi-3D geometriesgPROMSTM. The potential step approach presented indetailed porous electrode gas diffusion mechanism,

ode and cathode activation overpotential descriptionulation routines were included. Fig. 3 shows that theasi-3D models are spatial extensions of the 1D caseconsiders the distributed electrodes (z-direction) andproximation of so-called button-cell experimental set-2D models, the equations of both the anodic and the

stemsare additionallydistributed in the x-direction rep-arallel fuel and oxidizer (air) channels. Depending onry conditions and discretization methods, both co- andw congurations can be simulated. The quasi-3D modelectrodes discretization in the z-direction, fuel channelon in the x-direction and oxidizer channel discretiza--direction resulting in a cross-owcongurationwhere


Fig. 3. A quasnode point in

the channelquasi-3Dmables can bnite differ

The follobasis of the

H2 is con(due to itand activEq. (1).

For 1D: Theled as pelayer. For[1618,4]

For 2D anchannelsand axial

Hydro-dychambers

The DArcGas Modeit does noTransportto the sm

Equi-potethe neglig

Activationelled witdistributeat the Trelectrodesidered bfor anodecomplexelectroch

Mean elparametetrodes is c

Isothermafor singleof SOFC smodel ne

The following subsections give a detailed presentation ofall necessary equations of the model which can be bundledinto several main groups: mass transport equations in gaschambers/channels and in the porous electrodes, a detailed multi-

nentrogery gach, Btion.ainpar

nt eqthe

her o. The thens ofdardtratimolated ftweee theVj-to zelledent tthe Vg st

Mass

. Tramod. Butby adelepos

2D aanne) to oeeqhe aulatee inlell.all mi-3D SOFC in cross-ow conguration where each control volume orthe xy plane produces 1D results as in Fig. 2.

s are perpendicular to each other. With both the 2D andodels, full size SOFCswith their spatial variation of vari-e simulated. Spatial discretization is resolved with theence method (FDM).wing main assumptions and simplications are themodel:

sidered as the only electrochemical active compounds fast reaction kinetics) and thus the Nernst potentialation losses only depend on the H2 oxidation reaction

e gas chambers above the porous electrodes are mod-rfectly stirred reactors (PSTR) with no gas stagnationfurther readings on gas stagnation layer effects see

.d quasi-3D: The gas ow in the anode and cathode gasis modeled as plug ow neglecting boundary layer owdiffusion. For axial diffusion effects see [19,20].namics were neglected, thus no pressure loss in gas/channels is considered.y viscous ux term (pressure driven ux) of the Dusty-l (DGM) for the porous electrodes was neglected sincet have any signicant effect on performance results.limitations are in the diffusion-controlled regime due

compoa hetementaapproasimula

Domtainingtransiewell asare eitof bothto closdomai

Stanconcention Y,presene.g. beto mak

Forare setis modto prestivelyawaitin

2.1.1.

2.1.1.1The

Table 1imatedaremogas com

Forgas chEq. (16ow, thSince tto calccathodsized c

For

all pore sizes [21,22].ntial current collection is a common assumption due toible electronic ohmic losses within the electrodes.overpotentials due to charge-transfer kinetics is mod-

h a modied ButlerVolmer type approach and nod charge-transfer is considered (H2 oxidation occursiple Phase Boundary (TPB) which is reduced to the/electrolyte membrane interface). This approach is con-y Zhu and Kee [23] to be accurate enough, especiallysupported cells (ASC) in comparison with the more

distributed charge-transfer and additional elementaryemical kinetics approach.d approximation: No distribution of microstructuralrs such as pore size, particle size and tortuosity in elec-onsidered.l operation is modelled which is a good approximationbutton and full size cell experiments. For modellingtacks, temperature distributions occur and a thermaleds to be appended as in [15].

tion in z-di(17) togethmolar uxethrough anwithin theon the bouninterface. Tchemistry tare the speEq. (25) and

Fuel and(20).

2.1.1.2. PorThe por

vides the linand the gasthe SOFC istation of thporous media diffusion mechanism for the electrodes,neous catalytic reforming mechanisms (HCR) of ele-s-surface and surface reactions, detailed potential steputlerVolmer type activationoverpotentials and theEIS

s, boundary conditions (for the transport equations con-tial spatial derivatives) and initial conditions (for theuations) are given in the respective equations tables asspecies (k) and reaction (i) counter variables. Domainspen, denoted by brackets (), closed [], or a combinatione boundary conditions (B.C.) are additional equationsdomains. Initial conditions (I.C.) are valid within the

the respective equations.equations converting between the different forms of

on and partial pressures (molar fraction X, mass frac-r concentration c, density , partial pressure pi) are notor the sake of brevity. Also the unit conversion factors,n kmol and mol and min and s, etc. are left out in orderequations more readable.

curve simulation, the transient parts of the equationsero to obtain steady-state equations. Everything elsewith the same equations so that it is not necessary

he steady-state performance model explicitly. Alterna-j-curves can be simulated with the dynamic model

eady-state for each current set-point.

transport

nsport equationsels governing mass transport equations are given inton-cell experimental set-ups can be very well approx-1Dmodelwhere the gas chambers above the electrodesdasperfectly stirred reactors (PSTR)with auniformbulkition, given by Eq. (14).nd quasi-3D models, the species conservation in thels is evaluated as plug ow by Eq. (15) together withbtain total mass conservation. For 2D co- and counter-uationsandvariablesarebothdistributed inx-direction.node and cathode channels are parallel, it is sufcientone channel with correspondingly reduced anode and

et ows in order to obtain the same results as for a full

odels, the porous media transport equations (distribu-rection) are considered purely diffusive as given by Eq.er with Eq. (18) for the total mass balance. The speciess nk are evaluated by the Dusty-Gas Model (DGM)implicit relationship with the concentration gradientsporous electrodes described in Eq. (21), and dependdary conditions at the electrode/electrolyte membranehese connect the mass transport model to the electro-hrough Faradays law Eq. (35). The mass sources/sinkscies net molar production rates from the HCR given byare only applicable for methane/syngas fuels.oxygen utilization can be calculated with Eqs. (19) and

ous media diffusion: Dusty-Gas Modelous media diffusion mechanism given in Table 2 pro-k between the electrochemistry taking place at the TPBchambers/channels system above the electrodes wherefed with fuel and oxidizer gases. A schematic represen-e partial pressure distribution of the H2 and H2O fuel


Table 1Governing equations for the main mass transport processes in the gas chambers (for 1D) and gas channels (for 2D and quasi-3D) (both denoted by subscript ch) and withinthe porous electrodes (el) for anode (an) and cathode (ca) side respectively. The domains for equations and variables as well as initial conditions (I.C.) and boundary conditions(B.C.).

Species cons

Ych,kt

=m

ch(14)

I.C.: Ych,k = Yfor k=1 to K

Species cons

(15

k

Ych,k =

I.C.: Ych,k = YDomains:

2D co-ow

2D counte

Quasi-3D

B.C.:2D co-ow

2D counte

Quasi-3D

Porous med

cel,kt

=

I.C.: cel,k = cchk

Xel,k = 1

sk = 0 forfor k=1 to

Domains: ze

B.C.: for anoXel,k(0) =XPel,tot(0) =Pnk(del) =

reaction

Uf =2F Nina

Uo =Itot

4F Ninc

gas withincathode is p

The Dusauskas [21]uxes nk foDGM is nowa comparisofusion mod

The binaaccording toaccurate mthe Knudse[29] and decan get signcients areervation (s1) anode/cathode gas chambers (for 1D) [24]:

inch

Vch(Y inch,k Ych,k) +

AcellchVch

(Ych,k

Kgi=1

ni(0) MWi nk(0) MWk

)

inch,k

g, where Kg =number of gas phase species anode/cathode.

ervation (kg m3 s1) anode/cathode gas channels (for 2D and quasi-3D) [25]:

1

in , for k = 1 toKg 1ch,k

r-ow

cross-ow

r-ow

cross-ow

ia transport anode/cathode (mol cm3 s1) [25]:nkz

+ aNi sk (17),k

(18)

H2/H2O/N2 anode atmospheres and for cathodeKg

l = (0: del), x= [0: lx] (for 2D), y= [0: ly] (for quasi-3D)

de/cathodech,k (for k=1 to Kg 1)op

k rTPB (for k=1 to Kg) where k is the stoichiometric coefcient for the electrochemical, i.e. +1 for H2, 1 for H2O and +0.5 for O2 and 0 for others

Itot V0mn(X

inH2

+ X inCO + 4X inCH4 ), fuel utilization () (19)

V0m

a X inO2, oxygenutilization () (20)

the anode and the O2 and N2 oxidizer gas within theresented in Fig. 2.ty-Gas Model (DGM), developed by Mason and Malin-, is given by Eq. (21) which evaluates the species molarr the porous media transport equations (Eq. (17)). Theadays employed in most detailed SOFC models [2] andn by Suwanwarangkul et al. [22] between different dif-els found the DGM most applicable for SOFC modeling.ry diffusion coefcients given by Eq. (22) are evaluatedFuller et al. [26,27]whichwas found out to be themost

ethod for SOFC conditions by Todd and Young [28]. Forn diffusion coefcients, Eq. (24) was taken from Millspends on the pore diameter and Knudsen diffusion thaticant at pore diameters below1m.All diffusion coef-corrected in Eq. (17) by the porosity and tortuosity to

account forman and Ythe effectivand applied

2.1.2. Hetersyngas (HCR

A multiof 42 irrev[33,4] wasthe methatrodes. Thiapplicationitly accoun)

(16)x= (0: lx] BFDM discretization (anode and cathode)

x= (0: lx] (BFDM, anode) and x= [0: lx) (FFDM, cathode)

x= (0: lx] and y= [0: ly] (BFDM, anode)x= [0: lx] and y= (0: ly] (BFDM, cathode)

For k=1 to Kg 1Anode/cathode: Ych,k(0) = Y inch,k uch(0) = uinchAnode: Ych,k(0) = Y inch,k uch(0) = uinchCathode: Ych,k(lx) = Y inch,k uch(lx) = uinchAnode: Ych,k(0, y) = Y inch,k uch(0, y) = uinchCathode: Ych,k(x,0) = Y inch,k uch(x,0) = uinch

the free gas pathways in the pores. According to Haber-oung [30], the tortuosity has a quadratic inuence one diffusion coefcients which was later on conrmedby DeCaluwe et al. [31].

ogeneous reaction mechanism for methane and)

-step heterogeneous reaction mechanism consistingersible elementary reactions (Iall =42) as reported inemployed to evaluate the source and sink terms of

ne/syngas mass transport through the porous elec-s mechanism, validated for Ni-YSZ cermets in SOFCs for temperatures between 220 and 1700 C, implic-ts for the water-gas-shift (WGS), methane reforming


Table 2Equations and parameters for the Dusty-Gas diffusion model (DGM) and binary and Knudsen diffusion coefcients. All equations and variables are distributed within theclosed domains zan = [0: dan], x= [0: lx] (for 2D), y= [0: ly] (for quasi-3D). They account for anode and cathode and valid for k,j=1 to Kg gas phase species.

Dusty-Gas Model equation (mol cm4) [21] without the viscous ux (pressure driven) term:

cel,kz

= Kg

j /= k

cel,j nk cel,k njcel,tot (/2) Dk,j

nk(/2) DKN,k

(21)

Fuller et al. expression [27] for the binary diffusion coefcient (cm2 s1)

Dk,j = 0.00143 T1.75

Pel,tot MW0.5k,j (V1/3k

+ V1/3j

)(22)

Fuller et al. diffusion volumes [28]:H2: 6.12, H2O: 13.1, CH4: 25.14, CO2: 26.7, CO: 18.0, O2: 16.3, N2: 18.5

Binary molecular weight for binary diffusion coefcient evaluation according to Fuller et al:

MWk,j = 2(

1MWk

+ 1MWj

)1(23)

Knudsen diffusion coefcients determined from kinetic theory [29]:

DKN,k =2 rpore

3(

8RgT MWk

)0.5(24)

and Boudouard reactions. The mechanism describes the adsorp-tion (Iad = 6) and desorption reactions of the 6 gas phase species(Kg,an = 6) H2, CO, CH4, CO2, H2O, O2, and surface reactions of 13surface species (Ks = 13) including the free Nickel catalyst sites, i.e.Hs, Os, OHs, HCOs, Cs, CHs, CH2,s, CH3,s, CH4,s, COs, CO2,s, H2Os andNis. It is assumed that surface adsorption is limited to a monoatomic layer. In total, the system includes 19 chemical species(Kan = 19=Kg,an +Ks) which take part in the 42 reactions. The reac-tion mechanism complies with the mass balances according to thelaw of mass-action kinetics with the formalism described in detailin [32] and given in brevity in Table 3. The 42 elementary reactionswith the cotor), i (temsite fraction[33,4].

2.1.3. Electr

2.1.3.1. PotThe pote

sented in d

distributed charge-transferwas applied, so that the charge-transferand potential steps only occur lumped at the interfaces of elec-trodes (Ni-YSZ anode or LSM cathode) and electrolyte membraneas the assumed TPB.

The electrochemical model equations are given in Table 4. Thetotal cell current Eq. (34) is the applied alternating current dur-ing EIS simulation from Eq. (45) and equals the sum of the locallydistributed currents for the 2D and quasi-3D approach. The totalcurrent (or current density j given by Eq. (35)) originates from twodifferent sources during transient operation. The Faradaic currentIF is directly proportional to the electrochemical reaction rate given

daypendtran).

. But-tranBut

ial ed Bu

Table 3Basic elements and vlx] (for 2D), y=

Species net m iometrith reaction:

sk =Ialli=1

ki

Adsorption r 1), whreaction, wh th reac(mol cm2):

ri =100 i mi

Arrhenius tycoverage-de

ri = AiTi ex

Transient sukt

= sk

,

I.C.: k =1EConservation

k = 1,Surface speccan,k = k,rresponding model parameters Ai (pre-exponential fac-perature exponent), Eai (activation energy), i (surface-dependent activation energy) and i can be found in

ochemical model

entials and currentntial step approach for cell potential evaluation is pre-etail in Section 1.1. For the models in this work, no

byFararent deduringEq. (37

2.1.3.2charge

Thepotentmodi

of HCR reforming kinetics from [32] according to law of mass-action. All equations[0: ly] (for quasi-3D).

olar production rate in (mol cm2 s1), where ki is the difference between stoich

ri, for k = 1, . . . , Kan

eaction rates (mol cm2 s1) evaluated with sticking coefcient i (between 0 andere mi is the sum of stoichiometric coefcients of surface species reactants in the i

RgT

2MWi

Kank=1

cki

an,k, for i = 1, . . . , Iad

pe reaction rate (mol cm2 s1) dependent on surface site fraction CO (pre-exponential fpendent activation energy ea,i parameters from Maier et al. [33]), where ki is the stoichi

p

(Ea,iRgT

)exp

(ea,iCORgT

)

Kank=1

cki

an,k, for i = Iad + 1, . . . , Iall

rface site fraction (s1) distribution:

for k = 1, . . . , Ks 17of surface site fractions:

for k = 1, . . . , Ksies concentration (mol cm2):for k = 1, . . . , Kss law inEq. (36). Theelectricaldouble-layer inducedcur-s on the double-layer capacitances Cdl and only exists

sient change of the half-cell potential steps as given by

lerVolmer type activation overpotentials forsfer reactionslerVolmer equation (40) relates the activation over-act to the Faradaic current density jF. In this work, atlerVolmer type approach developed by Zhu et al. [34]

ariables are distributed within the closed domains zan = [0: dan], x= [0:

ic coefcients of products and reactants of the kth species in the

(25)

ere ki is the stoichiometric coefcient of reactant k in the ithtion, and where is the available surface site density(26)

actor A, temperature coefcient , activation energy Ea andometric coefcient of reactant k in the ith reaction:

(27)

(28)

(29)

(30)


Table 4Equations for the electrochemical model [5]. All equations and variables are distributed within the closed domains x= [0: lx] (for 2D) and y= [0: ly] (for quasi-3D). They accountfor anode (el = an) and cathode (el = ca).

Cell voltage (in V) is not distributed because of equi-potential assumption:Ecell = e,ca e,an (5)B.C.: e,ca = 0

Potential step between electron (e) and ion (i) conducting phases (in V):

el = e,el i,el (6,7)B.C.: i,an =i,ca + Rohmj

Cathodic half-cell reduction potential (in V) for the half-cell reaction in Eq. (8):

eq,ca = EO2/O2 RT

2Fln

(aO2

p0.5O2 aV

O

)(9)

Anodic half-cell reduction potential (in V) for the half-cell reaction in Eq. (10):

eq,an = EH2O/H2 RT

2Fln

(pH2 aO2pH2O aVO

)(11)

Standard electromotive force (in V) at standard pressure depending only on temperature:

Eel =

Gel

2F(31)

withGel = Hel TSel (32)

Nernst potential (in V) of the global electrochemical hydrogen oxidation reaction in Eq. (1):ENernst = eq,ca eq,an (33)Relationship between activation overpotential (in V) and potential steps:act,el = el eq,el (12, 13)

Total cell current (in A) where I is the local current in distributed models (2D/quasi-3D):Itot = I for 1D (34)Itot =

I for 2D and quasi-3D

Current density (in A cm2):

j = IA

= jF,el + jdl,el (35)for local current density 2D: A=Acell/(# of discretization intervals in x+1)for local current density quasi-3D: A=Acell/(# of discretization intervals (x+1)(y+1))

Faradays law is the relation between electrochemical reaction rate and Faradaic current density (in mol cm2 s1):

rTPB = jF2F (36)Double-layer current density (in A cm2) [7]:

jdl,el = Cdl,el (el)

t, ( for cathode) (37)

I.C.: ((el)/t) = 0Ohmic resistance (in Ohmcm2) through the dense electrolyte membrane:

Rohm =delectrolyte

i(38)

Electrolyte conductivity (in S cm1):

i =

0T

exp

( Ei

RgT

)(39)

Table 5Equations and parameters for the modied ButlerVolmer-type activation overpotential due to charge-transfer kinetics [34]. All equations and variables are distributedwithin the closed domains x= [0: lx] (for 2D) and y= [0: ly] (for quasi-3D). They account for anode (el = an) and cathode (el = ca).

ButlerVolmer equation (A cm2):

jF,el = j0,el[exp

(an,elF act,el

RgT

) exp

(ca,elF act,el

RgT

)](40)

with an,el = 1.5 and ca,el = 0.5

Anodic exchange current density (A cm2):

j0,an = kH2 exp(

EH2RgT

) (pH2 (dan)/pH2 )0.25(pH2O(dan))0.751 + (pH2 (dan)/pH2 )

0.5p in atm (41)

withpH2 =AH2

2

2 Rg T MWH210 0 exp

(

EdesH2RgT

)in atm (42)

with AH2 = 5.59E + 19 cm2 mol1s1, =2.6E+9mol cm2, EdesH2 = 88,120 Jmol

1, 0 = 0.01, kH2 = 207,000Acm2, EH2 = 87,800 Jmol1

Cathodic exchange current density (A cm2):

j0,ca = kO2 exp(

EO2RgT

) (pO2 (dca)/pO2 )0.251 + (pO2 (dca)/pO2 )

0.5p in atm (43)

withpO2 = AO2 exp(

EdesO2RgT)

)in atm (44)

with AO2 = 4.9E + 8 atm, EdesO2 = 200,000 Jmol1, kO2 = 51,900Acm2, EO2 = 88,600 Jmol

1


Table 6Model equations for the simulation of the electrochemical impedance spectra (EIS).

Itot(t) = Ibias Igain sin( t) sinusoidal alternating current (45)Time

t= 1

= 2 f = 1

fper

Ep cos =

Z1t

= Ecell(Ep cos =

Ep sin =

Z2t

= Ecell(Ep sin = 2

Re(Z) = Acel

Im(Z) = A

|Z| =

Z2rea

was employdensities j0,kinetics for

2.1.3.3. EISThe mod

In Fig. 4, this presentedthe frequengiven by EqThe gPROMequations (given. No dmodeling srequire expTime, needsgiven by Eqwith the int

The simuand (50.a)the sinusoiimaginary (equations athe generaloutput for acalculationFirst, the ti(49.b) and (end of the p(50.c) to evathe systembe enough)for a given f

2.1.4. Comp

Fig. 4 giprocedurevalues formaterials ptions from

gPROMSTM solvers DASOLV (differential algebraic solver for thetransient problem with absolute and relative tolerance 1E12 and1E10) and SPARSE (non-linear algebraic solver for the spatially

ized problem with convergence tolerance of 1E7). Foric diifferpolyFDM:

entrFDMckwde c

renceckwa

etaile a ce alt

ulat

j-cu

analing Selecans oith tharedeoperiedted wilizatfor tel anambeus thechadue tableto ceI.C. : Time = 0 (46)angular frequency in radians (47)

iodof the sinusoidal signal (48)

2f

0

Ecell(t) sin( t)dt in degrees (49.a)

t) sin( t) integratedbygPROMSTM for t = 0 to (49.b)2f Z1() evaluatedbygPROMSTM at t = (49.c)

2f

0

Ecell(t) cos( t)dt in degrees (50.a)

t) cos( t) integratedbygPROMSTM for t = 0, . . . , (50.b)f Z2() evaluatedbygPROMSTM at t = (50.c)

l 10 Ep cos Igain

(Ohmcm2) (51)

cell 10 Ep sin Igain

(Ohmcm2) (52)

l+ Z2

im(Ohmcm2) (53)

ed, who derived expressions for the exchange currentel (Eqs. (41) and (43)) from elementary electrochemicalthe assumed rate limiting reaction steps (Table 5).

modelel equations for the EIS simulation are given in Table 6.e simulation schedule of the computational procedure: the steady-state current Ibias needs to be switched tocy- and time-dependent alternating sinusoidal current. (45). It serves as the input signal for the EIS calculation.STM dynamic solver implicitly integrates all transientdenoted by $ in gPROMSTM) when a time schedule isirect access to the time variable itself is allowed in theection [35]. Since the equations for the EIS evaluationlicitly the time variable, a dummy time variable, e.g.to be introduced whose time derivative equals 1 as

. (46). This dummy time variable proceeds isochronicernal gPROMSTM time variable.ltaneous evaluation of the two time integral Eqs. (49.a)determines the phase angle and the amplitude of

discretnumernite dwith alowingchosen

zan: c zca: C x: ba

cathodiffe

y: ba

A dproducf for th

3. Sim

3.1. V

Theoperatin theby merun wwhichcase is(ASC)humidevaluafuel utoptioninlet fugas chand thsion mlosses(preferrelateddal cell voltage Ep with which the real (Eq. (51)) andEq. (52)) parts of the impedance Z are calculated. Thesere obtained through trigonometric transformation ofexpression for the sinusoidal phase-shifted cell voltagegiven sinusoidal current input [24]. In gPROMSTM, thisis achieved by splitting up the problem into two steps.me derivatives of Eqs. (49.a) and (50.a), given in Eqs.50.b), are integrated over a full period and then at theeriod (Time=) the results are used in Eqs. (49.c) andluate the impedanceparameters. It has tobenoted, thatneeds to be run for several periods (10 periods proved toto reach a periodic steady-state, before the impedancerequency can be evaluated.

utational procedure

ves a schematic representation of the computationalfor the EIS simulation. The model is fed with inputthe desired operational parameters, SOFC geometry,roperties, etc. All governing and constitutive equa-Tables 16 are solved simultaneously with the inbuilt

sible sinceow rates sfrequenciesand biomawith the 2Section 3.6.

An overvwhere itsVactivation oevaluated wtial curves (zones (tripchambers apotential fotionwhile tonly affectethecurrentcentration ogradients wcan furtherevaluatingscretization of the spatially distributed equations, theence method (FDM) was chosen in its different formsnomial degree of 2. For the different domains, the fol-methods and number of discretization intervals were

ed nite difference method (CFDM), 10 intervals, 4 intervals

ard nite difference method (BFDM), 10 intervals forhannel in counter-ow conguration the forward nitemethod (FFDM) was chosen, 10 intervalsrd nite difference method (BFDM), 10 intervals

ed simulation schedule has to be specied in order toomplete EIS from the solutions of a range of frequenciesernating current Eq. (45) as the input signal.

ion results and discussion

rve and EIS of a base case simulation

ysis of the inherent voltage losses (overpotentials) of anOFC and the breaking down of these losses appearingtrochemical impedance spectrum (EIS) is carried outf a base case dened in Table 7. The simulations aree 1D model approximating button-cell experiments

commonly used for new materials testing. The basened for a typical laboratory scale anode supported cellating at a common SOFC temperature of 800 C withhydrogen. The simulations results presented here wereith a xed fuel and air inlet ow and thus varying

ion Uf and oxidizer (oxygen/air) utilization Uo. Anotherhe analysis is the xation of Uf and Uo by adjusting thed air ow rates to the current density. Thiswouldx ther bulk partial pressures for all points on the Vj-curvee inlet boundary values for the porous electrode diffu-nism. Although the reversible Nernst potential relatedo gas depletion in the gas chambers would be inhibitedsince these are purely thermodynamic losses and notllmaterial), this technique is not experimentally acces-the required mass ow controllers cannot adjust theo quickly to the sinusoidal current perturbation (withup to 1MHz). Section 3.5 shows also the EIS ofmethane

ss derived syngas fuelled SOFCs. EIS results obtainedD and quasi-3D models are comparatively shown in

iew of the base case performance is presented in Fig. 5j-curve togetherwith the curves for anode and cathodeverpotentials from Eq. (40) and ohmic overpotentialith Eq. (38) are given. In addition, the Nernst poten-Eq. (33)) evaluated for partial pressures at the reactionle phase boundary (TPB) at dan and dca) and bulk gasre shown. The former is the actual electrochemicalr the global electrochemical hydrogen oxidation reac-he latter represents the theoretically availablepotential,d by bulk gas depletion (due to changingUf andUo withdensity). Theirdifferenceaccounts for the so-calledcon-verpotential due to diffusion induced partial pressureithin the porous anode and cathode electrodes. Thisbe split up into anodic and cathodic contributions bythe respective half-cell potentials (Eqs. (9) and (11))


for TPB andconcentratib shows thcurrent denoverpotentrent densiti

Fig. 5. Base coverpotentialsFig. 4. Computational procedure.

bulk gas partial pressures. Thus anode and cathodeon overpotentials are plotted likewise in Fig. 5a ande corresponding course of area-specic resistances vs.sity which can be evaluated by dividing the respectiveials by the current density: Ri =i/j. For selected cur-es, simulated electrochemical impedance spectra (EIS)

obtainedwplot in Fig.

The shapNernst poteThis typicadensities an

ase (BC) steady-state performance characteristics. (a) Vj-curve and the different Nern, their sum the total resistance Rtot and the differential resistance Ecell/j (slope of the Vith the full lossmodel are presented in formof a Nyquist6.e of the Vj-curve more or less follows the shape of thential curve evaluated for partial pressures at the TPB.

l shape of high non-linearity at low and high currentd a rather linear part in between originates from the

st- and overpotentials; (b) area-specic resistances of the differentj-curve).


Table 7Model input values for a base case (BC) and the parametric analysis.

Parameter Unit Base case Parametric analysis Comment

Operational parametersPop bar 1.013 T C 800 Jbias A cm2 2.75 02.85Igain A 0.1 Nan,in ltmin1 0.5 0.52XH2,in 0.97 0.485XH2O,in 0.03 0.15XN2,in 0 0.5Nca,in ltmin1 2 XO2,in 0.2 XN2,ca,in 0.8

Cell geometrylcell cm 3.16 lx cm 3.16 ly cm 3.16 delectrolyte m 10 dan m 500 0500dca m 30 han, hca mm 3 wan,wca mm 2 Acell cm2 10 Van E7Vca

Materials prrpore

0EiCdlj0,an

change of thThe most dcathodic action overpothe resistanrent densityand most ddecline atdensity duepressure pHactivation ocurrent den

The ohmelectrolyte)Rohm indepconc,ca is ra

Fig. 6. Electrosen from the b

ratiomallomprentn fo

ct, thm3 1.517E06 1.5E5, 1.5m3 1.517E06

operties 0.35 0.4, 0.45 3.5 2, 3.1m 0.5 0.75, 1SK1 cm1 3.6E+5 Jmol1 80,000 F cm2 1E4 1E1, 1E7Acm2 Eq. 3.41 0.1, 10

e partial pressure ratio pH2/pH2O with fuel utilization.ominant overpotentials for all current densities are thetivation overpotential act,ca and the anodic concentra-tentialconc,an which can also be seen from the course ofces. While act,ca increases rather linearly with the cur-, is highly non-linear (Nernst potential shape)

excess(only s

By cent curbe drawmon faconc,an

ominant at low and high current densities. The sharphigh current density constitutes the limiting currentto diffusion induced depletion of the hydrogen partial

2 at the TPB. Additionally, close to OCV also the anodicverpotential is high but drops quickly with increasingsity j.ic overpotential has a medium effect (due to the thinand increases linearly due to its constant resistance

endent of j. The cathode concentration overpotentialther negligible in this base case due to the high oxygen

chemical impedance spectra (EIS) for different current densities cho-ase case Vj-curve of Fig. 5.

independenaxis as the h

The rscant loss wit must beThe secondlarge at OCthe anodiccentration oanodic conctwo remaintrend of Rcosities and opartially su

Theprelin Fig. 7 bythe differenThis is carriprocesses (3current (jdland gas chaand cathodthem intoare still evameans thatthe periodi

The resurst high-frCell pressure chamber/channelsCell TemperatureBias current density for EISGain current amplitude for EISInlet anode volume owInlet anode molar fraction hydrogenInlet anode molar fraction steamInlet anode molar fraction nitrogenInlet cathode volume owInlet cathode molar fraction oxygenInlet cathode molar fraction nitrogen

Length/width of cellCell length in x-direction (for 2D and quasi-3D)Cell length in y-direction (for quasi-3D)Electrolyte thicknessAnode thicknessCathode thicknessHeight anode/cathode channelsWidth anode/cathode under channelActive cell areaVolume anode chamberVolume cathode chamber

Electrode porosityElectrode tortuosityElectrode average pore radiusParameter for electrolyte conductivityParameter for electrolyte conductivityAnode double-layer capacitanceExchange current density

(low Uo) and thin cathode porous electrode thicknessdiffusion induced pO2 gradient) employed.aring the resistances fromFig. 5bwith the EIS for differ-densities given in Fig. 6, rough conclusions can alreadyr the assignation of the different visible arcs. It is a com-at the ohmic resistance R has constant impedanceohmt of the frequency [11]. It manifests itself on the Re(Z)igh-frequency intercept of the EIS.

t visible high-frequency arc (left) describes a signi-hose resistance is increasing with current density. Thusrelated to the cathodic activation overpotential act,ca.high-frequency arc (to the right of the rst arc) isV, then decreases with j; thus it can be related to

activation overpotential act,an. Since the cathodic con-verpotential conc,ca is negligible small, the remainingentration overpotential conc,an must be related to theing middle and low frequency arcs. These follow thenc,an which is very high at low and high current den-f similar magnitude in between. The origin of the twoperimposed arcs will be investigated further below.iminaryndingsof aboveare investigated inmoredetailmeans of a step-by-step reduction of the inuence oft transport processes on the imaginary part of the EIS.ed out by setting the transient parts of the six transportfor cathode (ca) and 3 for anode (an)), i.e. double-layer

in Eq. (37)), porous electrode diffusion (c/t in Eq. (17))mber species conservation (Y/t in Eq. (14)) for anodee, respectively, successively to zero and thus turningstationary equations. The losses of these mechanismsluated, but their capacitive nature is taken away, whichthe partial pressures are instantaneously adapting to

cally varying current density.lts conrm the above made preliminary analysis. Theequency arc on the left (peak frequency fp 25MHz) is


Fig. 7. Base ca ting btransient parts

related to tpears whenAs will be sspecic parrelated arc.the anodic aconcentratianode concmiddle-freqAdditional(fp 4Hz) icathodic gavation exprSince it is refuel and oxalso called g

The relaing from th(presentedtra is not stconstitutestheVj-curvno clear relaand the widcesses canbresistanceslatively plothe impeda

Theprevmain lossesther investia parametrifor the anodequivalentpled systemorder to supiedparamein Table 7.

Fig. 8 shing the dyn

les ato pith tin Tshiftidalcursed b

ancehighy jdl,ansitase scellxcep

to thion ofreqse EIS at j=2.75Acm2 (in the non-linear part of the Vj-curve close to current limiof the transport equations.

he cathodic activation overpotential because it disap-setting the cathodic double-layer layer current to zero.hown later on, the variation of activation overpotentialameters inuence the magnitude of the double-layerLikewise, the second arc (fp 100kHz) can be related toctivation overpotential. As it was assumed, the cathodeon overpotential turns out to be negligible while theentration overpotential dominates and is related to theuency arc (fp 40Hz), also called diffusion impedance.information on the nature of the low frequency arcs revealed here. It remains as an effect of the anodic ands chamber (perfectly stirred reactor) species conser-essions where the cathodic contribution is negligible.lated to the changing Nernst potential due to changing

ygen utilization, it can be called Nernst impedance. It isas conversion impedance in literature [19,16,36].

tionship between steady-state cell performance result-e Nernst potential reduced by different resistances

variabfor twized w(givenphase-sinusoshift ocdescribimped

Atdensitrent deare phshiftedslight esitiveactivatathighasVj-curves) and theelectrochemical impedance spec-raightforward. The Re(Z)-axis low frequency interceptthe total impedance Ztot which is equal to the slope ofeEcell/j at the investigated current [19,10]. However,tionship between calculated resistances given in Fig. 5bth of the impedance arcs of the different transport pro-e established. Fig. 7 additionally provides the calculatedRi and differential resistances (i/j) which are cumu-tted at the Re(Z)-axis. Both of them do not collide withnce arcs intercepts.ious analysis revealedwhich anode and cathode relatedcan be seen in the EIS. For the sake of brevity, the fur-gation of the three main loss mechanisms by means ofc analysis in the following sections will only be carriedic system. The nature of the cathode related losses are

to the anode and both can be regarded as two decou-s. Thus all cathodic transient terms are set to zero, inpress their imaginary impedance components. The var-ters for the analysis of the threemain losses are included

ows how the impedance arcs come about by present-amic periodic variation of important SOFC operation

related to tAt medi

gas chambcourse. ThejF,an is in linfor the phaTPB pH2,TPBattributed tlosses throu

At low fHere, the g

Table 8Variable value

Variable

jEcellpH2,bpH2,TPBjF,anjdl,anehavior) with the full loss model and the subsequent reduction of the

four different frequencies f. The variables are plottederiods during the EIS simulation and are normal-he respective value at the beginning of each periodable 8). The origin of the impedance arcs are theed cell potential Ecell (output signal) when applying aalternative current density j (input signal). The phasedue to capacitive behavior of the different subsystems,y the transport equations, and produces imaginary

.frequency (100kHz), a double-layer induced currentn occurs which adds up together with the Faradaic cur-y jF,an to the total current density j. Both jdl,an and jF,anhifted with respect to j and thus account for a phase-potential Ecell. All other important variables (with ation of pH2,TPB) are constant because they are not sen-e high frequency. Since jF,an is directly related to theverpotential, the emerging double-layer impedance arcuenciesdue toelectrical double-layer capacitance is also

he activation overpotentials.um frequency (100Hz), all variables except the anodicer hydrogen partial pressure pH2,ch show a periodicdouble-layer induced current is almost zero and thuse with j. Here, the phase-shifted variable responsible

se-shifted Ecell is the hydrogen partial pressure at the. The intermediate frequency impedance arc thus can beo concentration overpotentials due to diffusion inducedgh the porous anode electrode.requency (5Hz), all variables show a periodic course.as chamber hydrogen partial pressure pH2,ch is phase

s at the beginning of each period during EIS simulation.

Unit 100kHz 100Hz 5Hz 0.1Hz

Acm2 2.75 2.75 2.75 2.75mV 560.08 559.76 559.60 560.13bar 0.59481 0.59482 0.59414 0.59476bar 0.0834 0.0828 0.0826 0.0834Acm2 2.755 2.750 2.750 2.750Acm2 4.96E03 0 0 0


Fig. 8. Dynamthe respective

shifted in aquency Ner

At very lically in line

3.2. Variati

This secber relatedNernst imp

Fig. 9 shume Van. Dvalue shiftsing with thaffected duchamber voalmost comarcs wherefrequency.Nernst impto zero. ThimpedancerelationshipMogensen [

The Nerntion is keptic variation of some important variables for two periods during EIS simulation shown forvalue at the beginning of the period.

ddition to pH2,TPB and is responsible for the low fre-nst impedance arc.ow frequency (0.1Hz), all variables are varying period-with j and thus no phase shift exists.

on of anode gas chamber related parameters

tion describes how the variation of anode gas cham-parameters affects the EIS, especially the low frequencyedance arc.ows the inuence of changing anode gas chamber vol-ecreasing the volume to 10% of the base case (BC)the Nernst impedance arc to higher frequencies merg-e diffusion impedance arc. The Nernst impedance ise to the changing fuel gas ow velocity when the gaslume is varied. A ten times larger Van results in anplete separation of diffusion and Nernst impedancethe latter is shifted towards a ten times lower peakTwo cases are presented additionally where only theedance arc occurs by setting all other transient termse variation of Van however does not affect the totalZtot and thus also not SOFC performance. A numericalfor the variation of this arc is given by Primdahl and

16].st impedance arc does not exist when the fuel utiliza-constant (by setting the gas chamber partial pressures

to the outlepart of the gthe gas chaand diffusioapproached

Fig. 10 sNan,in. An inimpedancediffusion imvelocities).smaller fueThis resultsimpedance

Fig. 11 sthe base cdiffusion inhydrogen incompoundof hydrogention) as formore than t

3.3. Variati

The gasThe semi-cifour different frequencies f. The variable values are normalized with

t values of the steady-state results). When the transientas chamber transport equation (Eq. (14)) is set to zero,

mber mass fractions follow instantaneously the currentn and Nernst impedance arcs merge. This extreme iswhen Van is set to very small values.

hows the inuence of changing the anode gas inlet owcreaseof the inlet fuel owresults in a shift of theNernstarc towards higher frequencies until it merges with thepedance arc at very high ows (due to the high gas owHigher fuel ows at constant current densities implyl utilization and thus higher hydrogen partial pressures.in decreased diffusion losses and thus also smaller totalZtot.hows the inuence of changing fuel composition. Whenase gas composition is diluted with N2 by 50%, theduced losses increase signicantly due to the smallerlet partial pressure and the additional large molecule

N2. When the diluted ow is doubled, the same amountis entering the SOFC (allowing the same fuel utiliza-

the base case, however diffusion induced losses are stillwice as large.

on of diffusion mechanism related parameters

diffusion induced impedance arc has a typical shape.rcle exhibits an almost linear slope (45 angle) at the


Fig. 9. EIS of base case and for the variation of anode gas chamber volume Van. Additional cases with only the Nernst impedance arc and without the Nernst impedance arcare shown.

Fig. 10. EIS of base case and for the variation of anode gas inlet ow Nan,in.

Fig. 11. EIS of base case and for different anode gas mixture with additional nitrogen.


Fig. 12. EIS of base case and for the variation of anode thickness dan.

higher frequency part of the arc (left side). This phenomenon iscalled Warburg impedance [10].

Fig. 12 shows the inuence of changing the porous anode elec-trode thickness dan. A decrease of dan results in a shift of thediffusion immass capacfusion resisthe middle-double-laye

Figs. 13properties wtrodes. Highsmaller torpass througimpedance

3.4. Variatioverpotentia

Concernlayer capac

order of magnitude given in literature [24] and Cdl,ca was arbi-trarily set 1000 times smaller in order to see separated arcs inthe impedance spectra. Fig. 16 shows the variation of the twodouble-layer capacitances and their effect on the EIS. Setting

aramancempee topedaeakithann thby thtancetent-layesincis theer obous additipedancearc towardshigher frequencies (due to smalleritance) and a reduction in the arc size due to smaller dif-tance. An innitely thin anode (0m)does not exhibitfrequency diffusion impedance arc anymore. Only ther and Nernst impedance arcs remain.15 show the effect of varying characteristic materialhich affect the gas diffusion through the porous elec-er porosity , larger average pore diameter rpore and

tuosity (non-linear pathway which the gas needs toh) all result in a decrease of the size of the diffusionarc due to decreased diffusion losses.

on of double-layer capacitance and activationl related parameters

ing the choice of base case values for the two double-itances (anodic and cathodic), Cdl,an was chosen in an

both pimpedlayer iincreasthe imcally sphigherbetweeposedcapacioverpodoublementswhichshoulderroneof an aFig. 13. EIS of base case and for the variation of aneters to equals values results in only one double-layerarc due to superimposition of the two arcs. The double-dance arcs shift proportionally with the capacitanceswards lower frequencies and start overlapping alsonce arcs of the other transport processes. Theoreti-ng, if realistic double-layer capacitance valueswould be, e.g. 1E3F cm2, it would get difcult to differentiatee different losses since all arcs would be superim-e double-layer impedance arc. For small double-layers, it is possible to relate the arcs to the activationials. The high-frequency arcs due to cathodic or anodicr capacitances might not appear in real EIS measure-e they can occur at frequencies higher than 100kHztypical highest frequency measured. A high-frequencyserved in experiments is sometimes interpreted as anrtifact, but in fact could possibly be the beginning partonal double-layer impedance arc.ode porosity .


Fig. 14. EIS of base case and for the variation of anode average pore radius rpore.

Fig. 15. EIS of base case and for the variation of anode tortuosity an.

Fig. 16. EIS of base case and for the variation of anodic and cathodic double-layer capacitances Cdl,an and Cdl,ca. The high-frequency arcs peaks are given next to the legendwhere the rst value is due to the anodic and the second due to the cathodic double-layer contribution.


Fig. 17. EIS of base case and for the variation of anodic exchange current density j0,an.

Fig. 17 shows the inuence of changing anodic exchange cur-rent density j0,an which describes the kinetics of the anodiccharge-transfer reaction. Decreasing j0,an to 10% of the base case(BC) value shifts the double-layer impedance arc to lower fre-quencies alarger j0,antowards higthat doubleoverpotenti

3.5. Syngas

This subderived procase (BC) hfor each fueof the baseand CO shifa steam-to-avoid carbo

water-gas-shift reactions are evaluated with the detailed hetero-geneous catalytic reaction mechanism (HCR) given in Table 3. Animportant parameter determining the catalytic reaction rates is thespecic nickel catalyst surface aNi which allows higher production

henthes varelec

rm othickon imgh leer ans thanodance

mee ret of celecnd increases its resistance. Accordingly a ten timesresults in a shift of the double-layer impedance archer frequencies with decreased resistance. This shows,-layer impedance arcs are related to the activationals.

1D EIS

section presents the inuence of methane and biomassducer gas fuels on the EIS in comparison with the baseumidied hydrogen fuel. The anode inlet ow Nan,inl was adjusted to match the hydrogen inlet mass owcase when considering complete methane reformingt. The steam diluted methane is fed to the SOFC withcarbon ratio of 2.5 which is a typical composition tondepositionproblems [37]. Themethane reformingand

rates wFor

dan waporoussink teanodediffusiAlthoua thinnexplainof theimpeda largemust bamounfor theFig. 18. EIS of a methane fuelled SOFC where the anodethe value is high.simulations presented in Fig. 18, the anode thicknessied in order to study the relation of the HCR with thetrode diffusion to which it is linked via the source andf the porous media transport Eq. (17). A decrease of theness by a factor of 10 decreases the middle-frequencypedance related arc and also the total impedance Ztot.ss catalytic active sites for reforming are available inode, the decreased diffusion resistance dominates ande increase in SOFC performance. A further decreasee thickness by a factor of 10 results in a large totalZtot. Although the diffusion impedance is close to zero,dium to low frequency impedance arc occurs whichlated to the slow reforming rates due to the smallatalytic active sites resulting in less available hydrogentrochemical reaction.thickness dan is varied.


Fig. 19. EIS of a methane fuelled SOFC with a steam-to-carbon ratio (H2O:CH4) =2.5 where the specic nickel surface aNi is varied. Also the hydrogen fuelled base case isgiven for comparison.

Fig. 19 shows the inuence of varying the specic nickel cat-alyst surface aNi allowing for slower or faster catalytic reactionsproducing hydrogen. The higher the amount of active catalyticsites, the faster is the production of hydrogen and the totalimpedanceSmaller vato less avaarc relatedsignicantlrole.

Finally ahydrocarboin Fig. 20. T(2) a biomabed downdcontent andidized bed

to steam used as gasication agent. The inlet gas compositions forthose cases are given in Table 9.

It can be seen that the producer gas fuel from the Viking gasiershows by far the highest total impedance due to a large diffu-

pedance arc. The cause of this is the high content of theolecules nitrogen, carbon monoxide and carbon dioxide

increase the diffusion resistance within the porous anode.

syngas fuel compositions.

] Viking gasier [38] Gssing gasier [39]

23.1 25.812.93 15.01.57 6.0

Fig. 20. EIS offrom the GssZtot of the hydrogen fuel base case EIS is approached.lues of aNi result in increased total impedance dueilable hydrogen. In this case, the middle-frequencyto the porous media transport (diffusion) increases

y where the catalytic reactions must play a major

comparison of the EIS obtained for three differentn containing fuels at same operating conditions is givenhese are (1) the steam diluted methane from above,ss producer gas from the air-blown two-stage xedraft biomass gasier Viking [38] with high nitrogen(c) a biomass producer gas from the circulating u-

gasier at Gssing [39] with high steam content due

sion imlarge mwhich

Table 9Biomass

[mol%

XH2XCOXCH4XCO2XH2OXN2three cases with different kinds of fuels. H2O:CH4 =2.5, biomass derived producer gas fing gasier [403.42]. Also the hydrogen fuelled base case is given for comparison.14.08 12.013.0 40.035.3 1.2rom the Viking gasier [393.41] and biomass derived producer gas


d cou

The steamshows decrimpedanceis higher fohumidiedsmaller hyd

3.6. EIS of 1

Fig. 21 sgurationswith a homco-ow andchannel pachannels in(quasi-3D)x-directioncompared.much small

4. Conclus

This papplanar SOFCsimulationsits capabilit(EIS), whichysis and diaboundary amodel, are

The modexible tomethane toplanar SOFCexperimenta 1D modelthe porous eow (quasisized cells.

The modSOFC inherlute the immain transp

er, gss thodeloveNern(due-freqing rtranly exion oe obss.variift oion ttanceove tsultse trapedatal imntia

speciFig. 21. EIS of the base case produced with the 1D, 2D (co- an

diluted producer gas fuel from the Gssing gasiereased diffusion impedance and approaches the totalof the steam diluted methane fuel. The total impedancer all hydrocarbon fuels compared to the operation onhydrogen due to increased diffusion impedance androgen partial pressures reducing the Nernst potential.

D, 2D and quasi-3D models

hows the inuence of different planar SOFC ow con-on simulated EIS. A button-cell approximation (1D)ogeneous gas composition above the porous electrodes,counter-ow channel congurations (2D) where gas

rtial pressures are distributed along the length of thex-direction and a cross-ow channel conguration

where anode partial pressures are distributed along theand cathode partial pressures along the y-direction areThe 2D and quasi-3D cases are comparable and shower total impedance than obtained for the 1D case.

ions

er presents a successful implementation of a dynamicmodel on the commercially available modeling and

Howevto asseThe mport abcalledtrodesmiddlereformchargeand onactivatarc. Thsystem

Thein a shrelaxatcapacinels abthus recies. Ththe imThe to(differe

Its

platformgPROMSTM. The special featureof themodel isy to simulate electrochemical impedance spectroscopyis a common experimental SOFC performance anal-

gnostic tool. All the necessary equations, parameters,nd initial conditions, that alloweasy reproduction of thepresented.el based on physico-chemical governing equations issimulate different fuels ranging from hydrogen oversyngas, e.g. biomass derived producer gas. Differentgeometries can be investigated: button cells which areally used to evaluate new materials (approximated byonly discretized in the gas diffusion direction throughlectrodes), co- and counter-ow (2Dmodel) and cross--3D) gas channel congurations which describe real

el was applied in a detailed parametric analysis of theent losses (overpotentials) in an attempt to deconvo-pedance spectrum of an SOFC. Each of the consideredort processes can be attributed to an impedance arc.

spectroscopSOFC fundaparametersa reduction

Acknowled

The auththe Nationathe develop

References

[1] R. Bove,and Techence+Bu

[2] S. Kakac,(2007) 76

[3] M.A. Khahal, K. KeDesign annter-ow) and quasi-3D models.

eneral overlapping of several arcs makes it difculte underlying loss mechanisms in EIS measurements.facilitates the interpretation of EIS. The mass trans-

the electrodes produces a low frequency impedance arcst impedance. Mass transport through the porous elec-to diffusion) causes the concentration impedance arc inuency range of the EIS. In case of hydrocarbon fuels, theeactions additionally affect this arc. The double-layersport induces a current at the triple phase boundaryists during dynamic SOFC operation. It is related to theverpotentials and is responsible for the high-frequencyervations account for the anodic as well as the cathodic

ation of the transport processes capacitances resultsf the process specic peak frequencies (characteristicime) but does not affect the resistances. Increasing the(e.g. mass or volume in the gas chambers and chan-

he electrodes)makes the process relaxation slower andin a shift of the impedance arc towards lower frequen-nsport processes resistances are related to the width ofnce arcs but no direct relationship could be obtained.pedance coincides with the slope of the Vj-curves

l resistance) at a specic current.al capability of simulating electrochemical impedance

y makes the model an important tool for analyzingmentals as well as for design, materials and operationaloptimization and SOFC failure diagnosis. This allows forin the amount of costly experiments.

gements

ors would like to thank Prof. Emmanuel Kakaras froml Technical University of Athens for his support duringment of the work.

S. Ubertini, Modeling Solid Oxide Fuel Cells: Methods, Proceduresniques (Fuel Cells and Hydrogen Energy), rst ed., Springer Sci-siness Media B.V., 2008.A. Pramuanjaroenkij, XiangYang Zhou, Int. J. Hydrogen Energy 321786.leel, J. Robert Selman, Cell, stack and system modeling, in: S.C. Sing-ndall (Eds.), High Temperature Solid Oxide Fuel Cells: Fundamentals,d Applications, Elsevier, Oxford, 2003, pp. 291331.


[4] V.M. Janardhanan, O. Deutschmann, J. Power Sources 162 (2006) 11921202.[5] W.G. Bessler, Electrochemistry and Transport in Solid Oxide Fuel Cells, Univer-

sitt Heidelberg, Habilitationsschrift, 2007.[6] R.J. Kee, H. Zhu, A.M. Sukeshini, G.S. Jackson, Sci. Technol. 180 (2008)

12071244.[7] W.G. Bessler, S. Gewies, M. Vogler, Electrochim. Acta 53 (2007) 17821800.[8] B.A. Boukamp, Solid State Ionics 169 (2004) 6573.[9] Q. Huang, R. Hui, B. Wang, J. Zhan, Electrochim. Acta 52 (2007) 81448164.

[10] J. Ross Macdonald, Impedance Spectroscopy: Theory, Experiment and Applica-tions, second ed., Wiley, 2005.

[11] Electrochemical Impedance Spectroscopy Primer, Gamry Instruments, 2005,Available at http://www.gamry.com.

[12] Daria Vladikova, The technique of the differential impedance analysis. Part1. Basics of the impedance spectroscopy, in: Proceedings of the InternationalWorkshop Advanced Techniques for Energy Sources Investigation and Test-ing, Soa, Bulgaria, September 49, 2004.

[13] M. Cimenti, A.C. Co, V.I. Birss, J.M. Hill, Fuel Cells 7 (2007) 364376.[14] M. Cimenti, V.I. Birss, J.M. Hill, Fuel Cells 7 (2007) 377391.[15] P. Hofmann, K.D. Panopoulos, L.E. Fryda, E. Kakaras, Energy,

doi:10.1016/j.energy.2008.09.015.[16] S. Primdahl, M. Mogensen, J. Electrochem. Soc. 145 (1998) 24312438.[17] S. Primdahl, M. Mogensen, J. Electrochem. Soc. 146 (1999) 28272833.[18] W.G. Bessler, J. Electrochem. Soc. 153 (2006) A1492A1504.[19] K. Takano, S. Nagata, K. Nozaki, A.Monma, T. Kato, Y. Kaga, A. Negishi, K. Kato, T.

Inagaki, H. Yoshida, K. Hosoi, K. Hoshino, T. Akbay, J. Akikusa, J. Power Sources132 (2004) 4251.

[20] W.G. Bessler, S. Gewies, J. Electrochem. Soc. 154 (2007) B548B559.[21] E.A. Mason, A.P. Malinauskas, Gas Transport in Porous Media: the Dusty-Gas

Model, American Elsevier, New York, 1983.[22] R. Suwanwarangkul, E. Croiset, M.W. Fowler, P.L. Douglas, E. Entchev, M.A.

Douglas, J. Power Sources 122 (2003) 918.[23] H. Zhu, R.J. Kee, J. Electrochem. Soc. 155 (2008) B715B729.[24] H. Zhu, R.J. Kee, J. Electrochem. Soc. 153 (2006) A1765A1772.

[25] R. Bove, S. Ubertini, J. Power Sources 159 (2006) 543559.[26] E.N. Fuller, P.D. Schettler, J.C. Giddings, Ind. Eng. Chem. 58 (1966) 1927.[27] B.E. Poling, J.M. Prausnitz, J.P. OConnell, The Properties of Liquids & Gases, fth

ed., McGraw-Hill, New York, 2000.[28] B. Todd, J.B. Young, J. Power Sources 110 (2002) 186200.[29] A.F. Mills, Heat and Mass Transfer, Irwin, Chicago, 1995.[30] B.A. Haberman, J.B. Young, J. Fuel Cell Sci. Technol. 3 (2006) 312321.[31] S.C. DeCaluwe, H. Zhu, R.J. Kee, G.S. Jackson, J. Electrochem. Soc. 155 (2008)

B538B546.[32] R. Kee, R. Coltrin, P. Glarborg, Chemically Reacting Flow: Theory and Practice,

Wiley and Sons, New York, 2003.[33] L. Maier, V.M. Janardhanan, B. Schaedel, O. Deutschmann, Surface Reactions:

Methane Reforming Kinetics Within A NiYSZ SOFC Anode Support, Ver-sion 1.2, ITCP, University of Karlsruhe, Germany, 2006, March. Available at:http://www.detchem.com/mechanisms.

[34] H. Zhu, R.J. Kee, V.M. Janardhanan, O. Deutschmann, D.G. Goodwin, J. Elec-trochem. Soc. 152 (2005) A2427A2440.

[35] gPROMSTM Documentation, Release 3.0.4, Process Systems Enterprise Limited,Bridge Studios, London, 2007, November 2007.

[36] A. Momma, Y. Kaga, K. Takano, K. Nozaki, A. Negishi, K. Kato, T. Kato, T. Inagaki,H. Yoshida, K. Hosoi, K. Hoshino, T. Akbay, J. Akikusa, M. Yamada, N. Chitos,Solid State Ionics 174 (2004) 8795.

[37] R.M. Ormerod, in: S.C. Singhal, K. Kendall (Eds.), High-temperature Solid OxideFuel Cells: Fundamentals, Design and Applications, Elsevier Science, Oxford,2003, pp. 333361.

[38] P. Hofmann, A. Schweiger, L. Fryda, K.D. Panopoulos, U. Hohenwarter, J.D.Bentzen, J.P. Ouweltjes, J. Ahrenfeldt, U. Henriksen, E. Kakaras, J. Power Sources173 (2007) 357366.

[39] R. Rauch, H. Hofbauer, Zweibett-Wirbelschichtvergasung in Gssing (A) mit2MWel/4,5MWth; Konzept, Betriebserfahrungen und Wirtschaftlichkeit, in:Proceedings of 7 Holzenergiesymposion, ETH Zrich, Switzerland, October18, 2002, 2002, See also: http://members.aon.at/biomasse/zuerich.pdf (in Ger-man).

Detailed dynamic Solid Oxide Fuel Cell modeling for electrochemical impedance spectra simulationIntroductionThe solid oxide fuel cellElectrochemical impedance spectroscopy

Mathematical model descriptionMass transportTransport equationsPorous media diffusion: Dusty-Gas Model

Heterogeneous reaction mechanism for methane and syngas (HCR)Electrochemical modelPotentials and currentButlerVolmer type activation overpotentials for charge-transfer reactionsEIS model

Computational procedure

Simulation results and discussionVj-curve and EIS of a base case simulationVariation of anode gas chamber related parametersVariation of diffusion mechanism related parametersVariation of double-layer capacitance and activation overpotential related parametersSyngas 1D EISEIS of 1D, 2D and quasi-3D models

ConclusionsAcknowledgementsReferences

Date post:	13-Sep-2015
Category:	Documents
Upload:	saifaltai9493
View:	238 times
Download:	3 times

Detailed Dynamic Solid Oxide Fuel Cell Modeling for Electrochemical

Documents