-
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
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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
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Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339 5323
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
-
5324 Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339
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
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Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339 5325
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
-
5326 Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339
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)
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Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339 5327
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
-
5328 Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339
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))
-
Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339 5329
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).
-
5330 Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339
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
-
Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339 5331
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
-
5332 Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339
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
-
Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339 5333
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.
-
5334 Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339
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
.
-
Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339 5335
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.
-
5336 Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339
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.
-
Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339 5337
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
-
5338 Ph. Hofmann, K.D. Panopoulos / Journal of Power Sources 195
(2010) 53205339
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
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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