Tortuosity in anode-supported proton 
conductive solid oxide fuel cell found from 
current flow rates and dusty-gas model
Authors: C.-L. Tsai and V. Hugo Schmidt
NOTICE: this is the author’s version of a work that was accepted for publication in Journal of 
Power Sources. Changes resulting from the publishing process, such as peer review, editing, 
corrections, structural formatting, and other quality control mechanisms may not be reflected in 
this document. Changes may have been made to this work since it was submitted for publication. 
A definitive version was subsequently published in Journal of Power Sources, VOL# 196, ISSUE# 
2, (2011), DOI# 10.1016/j.jpowsour.2010.08.005.
C.-L. Tsai and V.H. Schmidt, “Tortuosity in anode-supported proton conductive solid oxide fuel 
cell found from current flow rates and dusty-gas model,” Journal of Power Sources 196, 692-699 
(2011), doi: 10.1016/j.jpowsour.2010.08.005
Made available through Montana State University’s ScholarWorks 
scholarworks.montana.edu 
Tortuo
curren
Chih-Lon
Department of
1. Introdu
Strong e
are in prog
600–800 ◦C
thebenefit o
steel, and a
sion, atomi
is made as
ions throug
to 20m ra
anode, mus
cell.
With a
electrolyte
strongly aff
current den
parameter i
The definiti
tortuosity = t
∗ Correspon
E-mail add
0378-7753/$ –
doi:10.1016/j.sity in anode-supported proton conductive solid oxide fuel cell found from
t flow rates and dusty-gas model
g Tsai ∗, V. Hugo Schmidt
Physics, Montana State University, Bozeman, MT 59717, USA
a b s t r a c t
Ba(Ce0.8Y0.2)O3−ı anode-supported proton conductive solid oxide fuel cells were fabricated and tested.
By changing the H2 partial pressure at the anode side, the effect of anodic concentration polarization on
open-circuit voltage of the cell was observed. Saturation current densities under concentration polar-
ization were obtained from different anode thickness cells and were used for tortuosity calculation. The
calculation is based on the dusty-gasmodelwhich includes Knudsen diffusion and Stefan–Maxwell equa-
tion terms. The tortuosity value for our supporting anode is 1.55±0.1 which is in a physically reasonable
range for modern porous anode materials. The tortuosity that we found is independent of the cell testing
temperature and anode thickness, which is consistentwith the fact that tortuosity is a geometric factor of
the anode structure. The derived equation also can be used for predicting the effect of varying the anode
thickness, porosity and pore size. Also, the concentration of the gases as a function of position across the
anode is determined.
meti
sity f
reme
ode p
ode
nvok
igh t
ty is
imp
tion
ding
ty to
dvan
usi
M),
e evi
–4.0ction
fforts of modern solid oxide fuel cell (SOFCs) designs
ress to extend the operating temperature down to the
range. The low operating temperature would increase
f using lowcost interconnectmaterial, such as stainless
lso reduce problems associated with thermal expan-
c migration and corrosion. Accordingly, the electrolyte
thin as possible to reduce ohmic loss from transferring
h electrolyte. The electrolyte thickness is usually down
nge and requires that an electrode layer, usually the
t be made thick enough to mechanically support the
thicker anode, gas concentrations at the anode–
interface need to be calculated carefully because they
ect the terminal voltage V as a function of electrolyte
sity i, especially in the high current density range. A key
n determining the pressure gradient is the tortuosity .
on of ‘tortuosity’ is
and so
as
tortuo
Measu
the an
Manym
 and i
Such h
porosi
The
centra
mislea
integri
ity is a
anodes
(FIB-SE
provid
is 1.33ypical diffusion path between two points through the pores
straight distance betwee the same two points
ding author. Tel.: +1 406 994 6150; fax: +1 406 994 4452.
ress: tsai@physics.montana.edu (C.-L. Tsai).
finding tort
solid oxide
sity under
dusty-gas m
cate that h
also provid
modeling.
see front matter © 2010 Elsevier B.V. All rights reserved.
jpowsour.2010.08.005me is confused with ‘tortuosity factor’ which is defined
actor = (tortuosity)2
nt of its value is needed for determining the quality of
ore configuration and necessary for any V(i) modeling.
rn SOFCmodels donot adequately calculate the valueof
e anode tortuosities in the range as high as 10–17 [1,2].
ortuosities do not seem physically reasonable when the
usually higher than 30% of the supporting anode.
act of such high tortuosity (10–17) is to produce con-
polarization easily when running the cell. With this
analysis, one might think sacrificing the structural
decreasing the anode thickness or increasing its poros-
tageous. However, recent studies of SOFC supporting
ng focused ion beam scanning electron microscopy
X-ray computed tomography and gas counter-diffusion
dence that the tortuosity for typical supporting anodes
[3–5]. The present work provides another way for
uosity of the supporting anode for proton conductive
fuel cells (H-SOFCs) by finding saturation current den-
concentration polarization of H-SOFC and using the
odel. Tortuosity calculations from our tested cells indi-
igh tortuosity values may not be correct. The results
e more accurate information for SOFC design and V(i)
Nomenclature
a anode forward attempt current density (A cm−2)
a0 open-circuit value of a (A cm−2)
a01 open-circuit value of a when pH2 = 1atm (Acm−2)
a3
ele
/3 unit volume of oxygen ion in unit lattice of elec-
trolyte material (cm3)
DKi Knudsen diffusion coefficient for component i
(cm2 s−1)
Dij binary diffusion coefficient (cm2 s−1)
Dij1 binary diffusion coefficient when total pressure
P1 =1 atm (cm2 s−1)
fa fraction of oxygen sites at the anode–electrolyte
interface
i SOFC current density (A cm−2)
ias saturation current density under anodic concentra-
tion polarization (A cm−2)
Ji molecular flow densities of gas i (molec s−1)
ji,m metered inflow rates of gas i (molec s−1)
ji,net net inflows into the plenum of gas i (molec s−1)
k Boltzmann’s constant, 1.38×10−23 (J K−1)
Mi molar mass of the diffusing gas i (gmol−1)
mi molecule mass of component i (gmolec−1)
ni concentration of component i (molec cm−3)
n1 total gas concentration in the plenum when
P=1atm (molec cm−3)
ni,p concentration of component i in the plenum
(molec cm−3)
nH2,as H2 concentration at the anode–electrolyte interface
under saturation current density (molec cm−3)
ni,p,as concentration of gas i in the plenum at the outer
anode surface under saturation current density
(molec cm−3)
nH,ele proton concentration in the electrolyte material
(cm−3)
P total pressure
q charge transfer per reaction, 3.2×10−19 (C)
r¯ mean pore radius (m)
T absolute temperature (K)
Ua enthalpy released when the reaction at
anode–electrolyte interface occurs (J kgel−1)
V SOFC terminal voltage (V)
Va potential difference across the anode–electrolyte
interface (V)
vele probability of oxygen ion vacancy in electrolyte
material
w anode thickness (cm)
Greek letters
ij average collision diameter of molecules i and j (Å)
 tortuosity

˝
2. Experim
The Ba(C
reaction. St
99.8%), cer
99.99%) we
hours. Then
the perovsk
with Cu K
Ba(Ce0.8Y0.2
repeated un
X-ray diffra
Toprepa
was weighe
tion to cera
was added
former. The
formgreen
interlayer, w
the electrol
the inks pre
ethylcellulo
drying of el
die press an
formawell
La0.8Sr0.2M
paint brush
between ca
A home
was used fo
were used
The total in
MKS mass fl
water vapo
fuel gas thr
ments were
cells were
N2 mixture
performanc
tures, i.e. at
the total flo
The curren
area.
Porosity
Archimede
broken into
the cathode
bath in eth
for more th
weight, Ww
sured using
ethanol, the
for 10min
sured in air
‘Porosity’ w
Porosity (%
oth
Field
ile H
crost
ined
.
ty-gporosity of supporting anode
dimensionless collision integral
ental procedures
e0.8Y0.2)O3−ı powder was prepared from solid state
The
ished.
Versat
the mi
determ
images
3. Dus
oichiometric amounts of barium carbonate (BaCO3,
ium oxide (CeO2, 99.97%), and yttrium oxide (Y2O3,
re mixed by an agate auto-grinding machine for two
, the powder was calcined at 1100 ◦C for 15h to form
ite phase. An X-ray diffraction (Scintag, XGEN-400)
pores
The mu
erally desc
Stefan–Max(=1.5418 A˚) was used for checking the formation of
)O3−ı. The auto-grinding and calcining processes were
til a single phase of the material was confirmed by the
ction.
re the supporting anodes, theBa(Ce0.8Y0.2)O3−ı powder
d and mixed with NiO at volume ratio 1 to 2. In addi-
mic powders, 8wt% cornstarch of the total solid load
to the powder and auto-grinded for 2h to serve as pore
powders were uniaxial die-pressed with a 3/4′′ die to
pellets to serveas the supportinganode. Then, theanode
ith a 1:1 volume ratio of Ba(Ce0.8Y0.2)O3−ı to NiO, and
yte were paint brushed on the supporting anode using
pared by mixing the solid powders with alpha terpinol,
se, oleic acid and xylene using a three rollmill. After the
ectrolyte ink, the pelletswere re-pressed by the uniaxial
d sintered in air at a temperature of 1400 ◦C for 5h to
-bonded electrolyte–anode structure. Cathodematerial,
nO3 (LSM), was then applied on the sintered pellets by
and fired to 1000 ◦C in air for 2h to form a good bond
thode and electrolyte.
-made seal-less testing system made from Inconel 600
r the measurements, Fig. 1. Silver mesh and nickel foam
as cathode and anode current collectors, respectively.
put gas flow rates on each side were controlled by
ow controllers at 200mlmin−1 in all experiments. The
r concentration in fuel gas was about 3% by flowing the
ough a water bubbler at room temperature. Measure-
carried out at 700 and 800 ◦C in ambient pressure. All
reduced in situ at high temperature in a 60% H2 +40%
for more than an hour prior to the measurements. All
e of the cells was measured using various fuel gas mix-
various partial pressures of H2/N2 ratio in fuel gaswhile
w rate of the gas was kept constant at 200mlmin−1.
t densities were calculated based on the cathode
of the supporting anode was measured using
s’ method. Each of the tested cells, after reducing, was
two pieces. One of the pieces was polished to erase
and electrolyte layers and cleaned with an ultrasonic
anol. The sample was then kept in a dry oven at 95 ◦C
an 2h to evaporate the ethanol. Dry weight, Wdry, wet
et, and weight saturated with ethanol, Wsat, were mea-
a high accuracy balance. Prior to measuring Wwet in
sample was immersed in ethanol and kept in vacuum
to remove possible air in the pores. The Wsat was mea-
soon after the surface of the sample was shaken dry.
as calculated using the equation
) = Wsat − Wdry
Wsat − Wwet × 100% (1)
er part of the cell was hardened in an epoxy and pol-
Emission Scanning Electron Microscopy (SUPRATM 55
igh Performance FE-SEM, Zeiss) was used to examine
ructure of the cell. The mean pore radius of the cell was
by quantitative measurements of the pore size on SEM
as model for gas flow in anode-supported H-SOFClti-gas diffusion process in pore structure is gen-
ribed by the dusty-gas model, which includes the
well equation and Knudsen terms. The equation in
t syst
molecular u
Ji
DKi
+ (Jinj −
D
where Ji an
j (molec s−1
for compon
and the nkT
toDij, the bi
any position
of compone
and T is the
in Eq. (2) is
across the a
the product
The Knu
between ga
diffusion co
DKi =
2
3
(
8

where r¯ is
diffusing ga
N2 and H2O
Table 1
Calculated (a)
and 800 ◦C wit
(a)
DKi (cm2/s)
800 ◦C
700 ◦C
(b)
Dij1 (cm2/s)
700 ◦C
800 ◦C
bin
an–E
.86 ×
˝
d–Jo
les i
and
.2an
culat
ed in
exp
o the
gas m
it is
he fl
(A c
(3.2Fig. 1. Schematic of fuel cell tes
nits is [6]
Jjni)kT
ij1P1
= −∂ni
∂x
(2)
d Jj are molecular flow densities of components i and
), respectively, DKi is the Knudsen diffusion coefficient
ent i,Dij1 is the binary diffusion coefficient at P1 =1atm,
-type terms in the numerator divided by P1 convert Dij1
nary diffusion coefficient, at the actual total pressure at
x along the anode pore, ni and nj are the concentration
nts i and j, k is Boltzmann’s constant, 1.38×10−23 J K−1,
absolute temperature. It is important to clarify that x
the coordinate along a typical gas diffusion path, not
node. Therefore, the total diffusion path length is w,
of the tortuosity and the anode thickness.
dsen diffusion coefficient considers the collisions
s molecules and the wall. The equation for the Knudsen
efficient from the kinetic theory of gases is
kT
Mi
)1/2
r¯ (3)
The
Chapm
Dij =
1
where
Lennar
molecu
gases i
Table2
the cal
are list
The
tures t
H2/N2
so that
used. T
density
chargethe mean pore radius and Mi is the molar mass of the
s. The calculated Knudsen diffusion coefficients for H2,
at 700 and 800 ◦C are listed in Table 1(a).
Knudsen and (b) binary diffusion coefficients for various gases at 700
h r¯ = 1.41m.
H2 N2 H2O
31.68 8.47 10.56
30.17 8.06 10.06
H2–N2 H2–H2O N2–H2O
5.35 6.65 1.87
6.10 7.70 2.21
in the react
The 2/
density JH2
completely
a simple m
section whi
anode–elec
divided by a
perpendicu
the pore, so
plane paral
the total are
factor. The
but the flow
which is sm
 to the enh
final factorem.
ary diffusion coefficient, Dij, is calculated using the
nskog equation from Cussler [7]
10−3T2/3
(
1
Mi
+ 1Mj
)1/2
P˝2
ij
(4)
is a dimensionless collision integral, based on the
nes potential, ij is the average collision diameter of
and j (in Å), Mi and Mj are molar masses of diffusion
j, respectively, and P the total pressure (in atm). Using
d2.3 inCussler for˝andij and total pressureP=1atm,
ed Dij1 for various gases at temperature 700 and 800 ◦C
Table 1(b).
eriments were performed with flow of H2/N2 gas mix-
anodeofBa(Ce0.8Y0.2)O3−ı anode-supportedSOFCs. The
ixture picked up H2O (steam) through a water bubbler
a ternary system, or a binary system when 100% H2 was
ow density of H2 is JH2 = i2/q, where i is the current
m−2 =C cm−2 s−1) in the solid electrolyte and q is the
×10−19 Cmolec−1) carried per H2 molecule annihilated
ion at the anode/electrolyte interface.
factor is an enhancement factor by which the flow
is enhanced compared to its value if the anode were
porous (i.e.  =  =1). To derive this result, consider
odel for which the anode has pores of circular cross-
ch are tilted at an angle  away from the normal to the
trolyte interface. The tortuosity  (actual path length
node thickness) is 1/cos. The flowvelocity component
lar to the interface is a fraction cos of the velocity in
this contributes a factor  to the enhancement factor. A
lel to that interface has pore area fraction compared to
a, thereby contributing a factor 1/ to the enhancement
pore area intercepted by this plane has elliptical shape,
is perpendicular to the pore’s circular cross-section
aller by a factor cos, so this contributes another factor
ancement factor. In earlier work [6] we overlooked this
and incorrectly used / as the enhancement factor.
es 196
We kno
anode, i.e. J
∂nH2
∂x
= −
(
∂nN2
∂x
= i
2
q
∂nH2O
∂x
= i

In review
paper apply
Nevertheles
culations w
as
Deff
i
= 

Di
where Deff
i
of species i
FIB-SEM an
diffusion-ba
confusion b
it appears th
be “tortuos
2/ agrees
Adding E
n and its so
∂n
∂x
= −
(
i

where ntotal
is used in th
across the a
total pressu
interface to
We can
nN2 = nN2,p
nH2O = nH2O
nH2 = nH2,p
−nH2
where nN2,p
H2 in the p
in the plen
n1 =7.559×
T=1073K. W
the pore fro
If the ce
ization at t
“constants”
position x/
at the anod
and are de
condition.
find
ur a
ensit
con
t de
ual
he e
f an
2− ↔
, we
of
H+ +
our
H2O
ctrol
s sim
vide
ree e
ode–
e–el
hode
clos
ard
ies. T
e bet
is de
ion s
ur m
whi
ward
cribe
e˛/2
is
n su
elec
actio
×10−
elec
d fo
H2
(
mH2
lecu
direc
− a3
el
n att
n ca
y in
unit
tion
lectr
the a
sion
ettin
n be
− cH
− nHC.-L. Tsai, V.H. Schmidt / Journal of Power Sourc
w that there is no N2 and H2O flow in the supporting
N2 = 0 and JH2O = 0. Eq. (2) can then be written as
i2
q
)(
1
DK,H2
+ nN2kT
DH2,N2,1P1
+ nH2kT
DH2,H2O,1P1
)
(5a)
nN2kT
DN2,H2,1P1
and (5b)
2
q
nH2OkT
DH2,H2O,1P1
(5c)
ing mass transport modeling literature, none of the
this enhancement factor in their calculation [8–11].
s, an effective diffusion coefficientwas used for the cal-
ithout further explanation, which is in general defined
(6)
and Di are effective and general diffusion coefficient
, respectively. However, after carefully reviewing the
d XCT articles [3,5], and considering discussion of
sed tortuosity analysis by Moldrup et al. [12] and of
etween tortuosity and tortuosity factor by Epstein [13],
at the “tortuosity” in these articles and in Eq. (6) should
ity factor” which is 2. Then our enhancement factor
with the ones used in Refs. [3,5] and [8–11].
qs. (5a)–(5c) yields the equation for total concentration
lution
2
q
1
DK,H2
)
and n = ntotal,p −
i2x
qDK,H2
≡ ntotal,p − cH23
(8)
,p is the total gas concentration in the plenum and x= y
e equation, where y is position on a straight line going
node. From the ideal gas law, P=nkT, we see that the
re P decreases linearly with x from the anode–plenum
the solid electrolyte.
solve Eq. (5a), (5b) and (5c) for nH2 , nN2 and nH2O
exp
(
i2x
n1qDH2,N2,1
)
≡ nN2,p exp(cN23) (8a)
,p exp
(
i2x
n1qDH2,H2O,1
)
≡ nH2,p exp(cH2O3) and
(8b)
− cH23 − nN2,p(exp(cN23) − 1)
O,p(exp(cH2O
3) − 1) (8c)
, nH2O,p and nH2,p are concentrations for N2, H2O and
lenum, respectively, n1 is the total gas concentration
um when P=1atm=1.015×105 Nm−2, n1 =P1/kT and
1018 cm−3 for T=973K and n1 =6.855×1018 cm−3 for
e see that nN2 and nH2O increase exponentially along
m plenum to electrolyte.
ll current density is saturated by concentration polar-
he anode, designated as ias, then i become ias in the
To
from o
rent d
on the
curren
the eq
from t
face o
2H+ +O
model
instead
tion, 4
with f
form 2
the ele
proton
to pro
are th
the an
cathod
the cat
For
of forw
densit
ferenc
which
a react
In o
events
The for
be des
iaf = a
Here a
reactio
anode–
the re
is 3.2
anode–
require
a ≡ 1
2
n
where
H2 mo
pore)
	ele)(1
a proto
reactio
vacanc
ion in
centra
from e
tion at
discus
By s
(8c) ca
nH2,p,cH2 , cN2 and cH2O, which are really linear functions of
across the anode. For x/ = w (the anode thickness)
e–electrolyte interface, they really become constants
signated as cH2,as, cN2,as and cH2O,as for the saturated
All cons
cally for .
The con
boundary c
trations in t (2011) 692–699 695
nH2 at the anode–electrolyte interface, we use results
nalysis of H-SOFC voltage V as a function of cur-
y i. The analysis of the I–V curve for H-SOFC is based
cept of exchange current density, i0. The exchange
nsity is defined for open-circuit operation in which
and opposite current densities in the system result
qual forward and reverse reaction rates at the inter-
ode–electrolyte, H2 ↔2H+ +2e−, cathode–electrolyte,
H2O, and cathode-pore, 1/2O2 +2e− ↔O2−. In our
propose a two-step cathode reaction mechanism
a one-step mechanism. For a one-step cathode reac-
O2 +4e− ↔2H2O, an O2 molecule is required to react
protons and four electrons at a cathode TPB site to
molecules. However, the concentration of protons in
yte is low (<20%), which makes the probability of four
ultaneously being within reaction distance is too low
the dominant reaction mechanism. Accordingly, there
xchange current densities, one from the reactions at
electrolyte interface, another from the reactions at the
ectrolyte interface and the third from the reactions from
–pore interface.
ed-circuit operation, we continue to use the concept
and reverse reactions and their corresponding current
henet currentdensity across the cell comes fromthedif-
ween the forward and reverse current densities, each of
scribed as the product of an attempt current density and
uccess probability.
odel, we treat all the reactions at interfaces as collision
ch can be described by classical (Boltzmann) statistics.
current density at the anode–electrolyte interface can
d as
cosh˛, and ˛ ≡ (Ua − qVa)/2kT
forward attempt current density and e˛/2cosh˛ is
ccess probability for the reaction, H2 →2H+ +2e−, at
trolyte interface. Ua is the enthalpy released when
n occurs, q is the charge transfer per reaction which
19 C, and Va is the potential difference across the
trolyte interface. Thus, (Ua −qVa) is the net energy
r the reaction to occur. The expression for a is
kT
mH2
)1/2
faq(1 − vele)(1 − a3elenH,ele/3)
2
, (9)
is the H2 molecule mass, and (kT/mH2)
1/2 is the
le average velocity component (through the anode
ted toward the triple phase boundary (TPB). (1 −
e
nH,ele/3)
2
is the probability that an O2− ion without
ached to it at the anode TPB so that the H2 dissociation
n occur in which vele is the probability of oxygen ion
electrolyte material, a3
ele
/3 is the unit volume of oxygen
lattice of electrolyte material and nH,ele is proton con-
in the electrolyte, and fa is the fraction of oxygen sites
olyte that sit on the TPB and are available for the reac-
node–electrolyte interface. The detailed derivation and
of our V(i) model will be published elsewhere.
gnH2 = nH2,as under the anode saturation condition, Eq.
rewritten as
2,as
3 − nN2,p,as(exp(cN2,as3) − 1)
2O,p,as(exp(cH2O,as
3) − 1) − nH2,as = 0 (10)tants in Eq. (10) are positive and can be solved numeri-
stants in Eq. (10) have been defined except for the
onditions nH2,p,as, nN2,p,as and nH2O,p,as for gas concen-
he plenum at the outer anode surface and nH2,as for the
H2 concentr
we conside
all the expe
the water b
of 200mlm
atmosphere
is, from the
PBozeman
dVi
dt
Themet
inflow rate
ance of the
the inflow
the inflow
room temp
has metere
jN2,m = pN2
2.107 × 101
To find
consider th
of H2 into
q=3.2×10−
inflow rate
analysis for
jH2,net = pH
jH2O,net = p
Our test
the plenum
where. Acc
anode satur
ntotal,p =
PB
nN2,p,as =
(
nH2O,p,as =
The rem
centration
saturation c
changes, be
of i. For ope
nH2,0 = pH2
into Eq. (9) p
value of a,
510.60 and
anode satur
to a, we hav
nH2,as =
nH
Inserting
yields the fo(
jH2,net,a
jtotal,net,
−nH2O,p
ults
the
ickne
1m
mi
supp
knes
ckne
port
was
whic
expe
of th
ls fo
elec
ted H
sts w
b), un
ant
anc
ation
pres
ivati
to th
nnin
fect f
creas
ses, F
ir ga
ut 0
ox r
osp
lyte
.8Y0.2
expl
tical
edwh
elec
en examining the I–V curves from Ba(Ce0.8Y0.2)O3−ı anode-
ted cells, we notice the current density increases rapidly at
ltage, e.g. at voltage lower than0.05V for 10%H2 partial pres-
Figs. 3(b) and Fig. 5(a). The same feature is reproducible and
so seen on the anode thickness of 1.34 and 1.84mm but notation at the anode–electrolyte interface. To find nH2,p,as
r the total flow in and out of the anode plenum. For
riments, the total gas inflow rate before going through
ubbler is 200mlmin−1. The total metered inflow rate
in−1 at temperature 21 ◦C and at Bozeman, Montana
(4900 ft. above sea level, PBozeman =8.534×104 Nm−2)
ideal gas law,
n = kTin
dNin
dt
, so
dNin
dt
≡ j1 = 7.012 × 1019 molec s−1
(11)
ered inflowH2 is foundbymultiplying the totalmetered
by the partial pressure fraction, and N2 is the bal-
total inflow. Because H2O is obtained by flowing
gas through a water bubbler, the flow rate of H2O is
rate multiplied by saturated water partial pressure at
erature (3%). For example, a 20% H2 input gas flow
d flow rates: jH2,m = pH2 j1 = 1.404 × 1019 molec s−1,
j1 = 5.618 × 1019 molec s−1 and jH2O,m = pH2O,sat j1 =
8 molec s−1.
the net flow of each gas into the plenum, we must
e fuel gas, H2, outflow into the anode. The inflow rate
anode is jH2 = −iS/q, where S is the active area and
19 C. When the cell is running at saturation current, the
into the anode is designated as jH2,as. To summarize the
net inflows into the plenum,
2 j1 −
iS
q
, jN2,net = pNj1,
H2O,sat j1 and jtotal,net = jH2,net + jN2,net + jH2O,net
(12)
system is open to space andwe assume that the gases in
are well mixed and have the same mole fraction every-
ordingly, in the plenum the gas concentrations under
ation conditions are
ozeman
kT
, nH2,p,as =
(
jH2,net,as
jtotal,net,as
)
ntotal,p
jN2,net
jtotal,net,as
)
ntotal,p and
(
jH2O,net
jtotal,net,as
)
ntotal,p
(13)
aining parameter to determine is nH2,as, the H2 con-
at the anode–electrolyte interface under the anode
ondition. FromEq. (9),nH2 remainsproportional to a as i
causeweassume theother parameters are independent
n-circuit (i=0) conditions, nH2 at the interface equals
ntotal,p. This known value for open-circuit nH2 inserted
rovides a knownvalue a0 = pH2a01 for the open-circuit
where a01 is the open-circuit value for pH2 = 1 and is
486.22Acm−2 for 700 and 800 ◦C, respectively. For the
ation condition, a= ias, and because nH2 is proportional
e
2,0aas
a0
= pH2ntotal,pias
pH2a01
= ntotal,pias
a01
(14)
the nH2,as value and the nH2,p,as value into Eq. (10)
llowing equation for 3 in terms of known parameters
s
)
3 3
4. Res
For
ent th
1.85 to
section
anode-
in thic
The thi
the sup
graphs
∼34%,
in the
pores
channe
The
suppor
The te
Fig. 3(
as oxid
perform
domin
partial
the act
pared
was ru
tion ef
the de
decrea
of H2/a
by abo
the red
ing atm
electro
Ba(Ce0
which
theore
report
BCY as
Wh
suppor
lowvo
sure in
was alas
ntotal,p − cH2,as − nN2,p,as(exp(cN2,as ) − 1)
,as(exp(cH2O,as
3) − 1) − ntotal,pias
a01
= 0 (15)
Fig. 2. Micros
image.and discussion
purpose of tortuosity investigation, a series of differ-
ss anode-supported Ba(Ce0.8Y0.2)O3−ı H-SOFCs, from
m, were made and tested. Fig. 2 shows the cross-
crostructure of one of the tested Ba(Ce0.8Y0.2)O3−ı
orted cells from SEM. The dense electrolyte is ∼20m
s and adheres to a ∼35m anode interlayer very well.
ss of porous cathode is∼45m.Themeanpore radiusof
ing anodes estimated quantitatively from SEM micro-
∼1.41m. The porosity of the supporting anode was
h was measured using Archimedes’ method described
rimental procedures. This porosity only refers to open
e supporting anode since closed pores do not provide
r gas diffusion.
trochemical performances of aBa(Ce0.8Y0.2)O3−ı anode-
-SOFC with a 1.5mm-thick anode are shown in Fig. 3.
ere done at temperature 700 ◦C, Fig. 3(a), and 800 ◦C,
der different H2 partial pressures while air was used
at the cathode. The near linear I–V curves of the cell
e at high H2 partial pressures, >40% H2, indicate the
of ohmic polarization of the cell outputs. For low H2
sures, less than 30%, the convex I–V curvatures indicates
on and ohmic polarization loss of the cell are small com-
e concentration polarization, especially when the cell
g at high current density. The concentration polariza-
rom changing H2 partial pressures also shows up with
ing of open-circuit voltage when H2 partial pressure
ig. 4. When comparing the open-circuit voltage (OCV)
s inputs, our OCV is lower than the theoretical value,
.1V at 700 ◦C and 0.122V at 800 ◦C. This is because of
eaction of Ce3+/Ce4+ in Ba(Ce0.8Y0.2)O3−ı under reduc-
here, which increases the electronic conductivity of the
and results in lower OCV. The electronic conductivity of
)O3−ı increaseswith increasingoperating temperature,
ains the bigger voltage difference of OCV between the
EMF and measured OCV at 800 ◦C. Similar results were
enusing lanthanum-doped ceria and zirconium-doped
trolyte for SOFCs [14–19].tructure of Ba(Ce0.8Y0.2)O3−ı anode supported H-SOFC from SEM
Fig. 3. Hydro
Ba(Ce0.8Y0.2)O
700 ◦C and (b)
on 1mm ce
ing anode g
specifically
than 13h a
the cell can
the electrol
i.e. the elec
cell an extr
down I–V cu
Fig. 4. Open-
1.5mm anode
up all of the
a saturation
faster than
Our specula
along Ni su
Ni surface
and provide
i.e. 2H+ +O2
SOFC, 8mol
meet anO2−
the anode s
surface diff
distance an
[4,8]. There
Due to th
e of H
esent
cur
rrent
rve m
ssur
sity c
.8Y0.2
84m
Table 2
List of saturati
pH2
(a)
10%
15%
20%
30%
(b)
10%
15%
20%
30%gen partial pressure dependent electrochemical performance of
3−ı anode supported H-SOFC with 1.5mm anode thickness at (a)
800 ◦C.
lls. It seems the “tail” gets bigger when the support-
ets thicker. The cell with 1.5mm anode thickness was
run with 20% H2 partial pressure under 0.1V for more
t 800 ◦C before the test system was shut down. Since
ing rat
nH2 pr
uration
the cu
I–V cu
tial pre
tortuo
Ba(Ce0
ness 1.be run for a long period, it excludes the possibility that
yte was giving up its oxygen ion for the SOFC reaction,
trolyte was not decomposing and cannot provide the
a oxygen source. Conventionally, we expect a straight
rve when concentration polarization dominates, using
H2 partial p
but are limi
below 20%.
encounter s
above 50%.
on current densities for different anode thicknesses which were tested under different H
ias (A cm−2) Anode 1.84mm ias (A cm−2) Anode 1.50mm ia
0.506 0.522 0.
0.782 0.816 0.
1.055 1.091 1.
1.489 1.537 1.
0.543 0.517 0.
0.721 0.782 0.
0.846 1.015 0.
1.005 1.338 1.circuit voltage of Ba(Ce0.8Y0.2)O3−ı anode supported H-SOFC with
thickness.
fuel gas at the anode–electrolyte interface, and giving
current density. This behavior indicates a mechanism
gas diffusion dominated supply of fuel gas to the TPB.
tion on this mechanism is surface diffusion of protons
rface to the TPB. For H-SOFC, protons diffusing along
allows them to incorporate into electrolyte at the TPB
s enough protons for the reaction on the cathode side,
− ↔H2O. The same mechanism was not seen on our O-
% yttria-stabilized ZrO2 cells, because twoprotonsmust
at the TPB at the same time for the reaction to occur on
ide. However,Williford et al. and Lee et al. point out that
usion of protons along Ni surface occurs only over small
d lower speed when compared to bulk gas diffusion
fore, the reason for the “tail” behavior remains unclear.
e fact that awasobtained fromthemodelingof imping-
2 onto the triple phase boundary (TPB), the amount of
at the anode–electrolyte interface determines the sat-
rent density when V goes to zero. Therefore, we took
density as ias when the voltage drops to zero in the
easurements. Values of ias for various hydrogen par-
es appear in Table 2. Fig. 5 shows the results from our
alculations and the electrochemical performances of a
)O3−ı anode-supported H-SOFC with an anode thick-
m. The shown data is for tests at 800 ◦C under different
ressures. The tortuosity results range from 2.94 to 1.50,
ted to a value around 1.50 when H2 concentrations are
As shown in Fig. 5(a), the cell performances did not
erious concentration polarization at H2 partial pressure
For H2 partial pressure above 50%, the main loss was
2 partial pressures at (a) 800 ◦C and (b) 700 ◦C.
s(A cm−2) Anode 1.34mm ias (A cm−2) Anode 1.00mm
709 0.854
955 1.174
193 1.410
493 1.686
619 0.757
802 1.058
950 1.294
117 1.590
Fig. 5. Ba(Ce0 Y )O anode-supported H-SOFC (a) the electrochemical perfor-
mance at 800 ◦
for its anode.
from ohmic
at voltage d
concentrati
the same va
concentrati
shows the c
when the “t
solid lines
then increa
Ba(Ce0.8
thickness, b
the tortuos
are shown
centration a
independen
Compared t
at higher H2
because of
the cells en
pressure.
We com
vious result
2.3±0.6 [6,
shouldhave
Fig. 6. Calcula
and (b) 700 ◦C
tuos
erage
dius
, even
rosit
sys
ith
cm−
ch l
lenum
sity i
ce to
ize r
hereb
low
milar
put
ts of
pres
7. Th
comp.8 0.2 3−ı
C under different H2 partial pressures and (b) calculated tortuosities
polarization so that the calculated tortuosity using ias
rop to zero should not be considered. Note that when
onpolarization is severe, the tortuosity values approach
lue. Therefore, the tortuosity calculated from lowest H2
ons should be a more accurate number. Fig. 5(b) also
alculated tortuosity for H2 partial pressure under 40%
ail” effect is ignored, using the intersect value between
and current density axis in Fig. 5(a). The tortuosity is
sing from ∼1.50 to ∼1.60 at low H2 concentration.
Y0.2)O3−ı anode-supported cells with different anode
ut the same porosity and pore radius, were tested and
ities of their anodes were also calculated. The results
in Fig. 6. The tortuosities obtained from 10% H2 con-
ll fall into the same range, 1.55±0.1, and are thickness
t, as they should be, for the tests both at 800 and 700 ◦C.
o 700 ◦C, the tortuosity values approach their final value
concentration when cells were tested at 800 ◦C. This is
the higher fuel utilization of cells at 800 ◦C that makes
counter concentration polarization at higher H2 partial
The tor
ous av
pore ra
radius
54% po
ternary
Fig. 4 w
1.055A
our mu
fromp
tortuo
interfa
pore s
wall, t
ing in
also si
ray com
[3–5].
Plo
partial
in Fig.
of thepare the result presented here, 1.55±0.1 with our pre-
s from analyzing Jiang–Virkar’s experiment which was
14]. Ifwe take into account that the enhancement factor
been 2/, then thenewnumber inRef. [6] is 1.74±0.3.
nential effe
appear mor
porting ano
a supportinted tortuosities for different thickness anode cells tested at (a) 800 ◦C
.
ity valuepresentedhere is a little smaller than theprevi-
value but within the error. This results from our bigger
, 1.41m,whencompared to Jiang–Virkar’s 0.5mpore
though our porosity is lower, 34%, compared to their
y. When looking at Jiang–Virkar’s 20%H2–80%N2-H2O
tem, its saturation current ias is 0.89Acm−2. The test in
20% H2 partial pressure delivered a saturation current
2 or 0.877Acm−2 when the “tail” is not considered. For
ower porosity but delivering similar amount of H2 fuel
to anode–electrolyte interface, itmakes sense thatour
s lower, i.e. the pores are straighter fromplenum–anode
anode–electrolyte interface. Furthermore, the bigger
educes the collision rate of gas molecules with the
y reducing the effect from Knudsen diffusion, result-
er tortuosity of our supporting anode. This value is
to the results which were obtained from FIB-SEM, X-
ed tomography and gas counter diffusion,  =1.33–4.0
gas concentrations across the anode for six different H2
sures under anode limiting currents at 700 ◦C are shown
e total concentrations only decrease slightly because
ensation from N2 concentration increasing. The expo-
cts on the gas concentrations appear insignificant. They
e like linear changes, due to the relatively “thin” sup-
de. To see the exponential effects, it is necessary to have
g anode thicker than 1 cm and high H2 partial pressure.
ness
5. Conclus
A series
supported H
pressures,
observed. F
ties under c
tortuosity c
supporting
independen
of the anod
The accu
of the total
the average
the Jiang–V
ity and big
with a low
of H2 gas fr
result is als
nologies fo
Therefore, w
range. The d
drops acros
SOFC electr
tools for m
voltage–cur
and for ana
and electro
Acknowled
The auth
facilities. Th
rgy,
Nort
1830
nces
. Kim,
99) 69
ostam
Iwai,
ashi,
oshid
. Willi
03) A1
Izzo J
ctroch
. Schm
. Cussl
ss, Cam
. Lee,
. DeCa
8.
Lehne
Izzo
.
oldru
5 (200
pstei
iang, A
. Xio
okaw
. Park, H.I. Yoo, Phys. Chem. Chem. Phys. 11 (2009) 391.
Mogensen, N.M. Sammes, G.A. Tompsett, Solid State Ionics 129 (2000)
ing, X. Xue, X. Liu, G. Meng, J. Power Sources 195 (2010) 775.Fig. 7. Plots of gas concentration vs. position in anode for 1.84mm anode thick
ions
of different anode thickness Ba(Ce0.8Y0.2)O3−ı anode-
-SOFC were made and tested. By changing H2 partial
the anodic concentration polarization effect can be
or low H2 partial pressures, saturation current densi-
oncentration polarization were obtained and used for
alculation. The result of tortuosity calculation for the
anode is 1.55±0.1, which is temperature and thickness
t, as it should be since tortuosity is a geometric factor
e structure.
rate tests bring down the error between cells to±0.1
value. The value of 1.55±0.1 is a little smaller than
number, which we found previously from analyzing
irkar results, 1.74±0.3. However, the lower poros-
ger pore size of our supporting anode are consistent
er tortuosity to deliver the measured similar amount
om the plenum to the anode–electrolyte interface. Our
o consistent with the values found by different tech-
r modern anode material with typical porosity >30%.
e can say that our value is in the physically reasonable
eveloped equations also give information for pressure
s the electrodes and the gas concentrations at the H-
ode–electrolyte interface which provides the required
odeling the voltage–current density expression. Such
rent modeling gives information for new SOFC designs
lyzing performance of existing designs in both fuel cell
lysis modes.
of Ene
Pacific
76RL0
Refere
[1] J.W
(19
[2] P. C
[3] H.
Hay
H. Y
[4] R.E
(20
[5] J.R.
Ele
[6] V.H
[7] E.L
Pre
[8] W.Y
[9] S.C
B53
[10] W.
[11] J.R.
200
[12] P. M
J. 6
[13] N. E
[14] Y. J
[15] Y.-P
Yok
[16] S.-H
[17] M.
63.
[18] H. Dgements
ors thank Dr. Stephen W. Sofie for the use of all his
isworkwas supportedby theUnitedStatesDepartment
[19] L. Yang, Z
471.under anode-limiting current conditions at 700 ◦C.
as a subcontract from Battelle Memorial Institute and
hwest National Laboratory under Award No. DE-AC06-
.
A.V. Virkar, K.Z. Fung, K. Mehta, S.C. Singhal, J. Electrochem. Soc. 146
.
agna, K. Honegger, J. Electrochem. Soc. 145 (1998) 3995.
N. Shikazono, T. Matsui, H. Teshima, M. Kishimoto, R. Kishida, D.
K. Matsuzaki, D. Kanno, M. Saito, H. Muroyama, K. Eguchi, N. Kasagi,
a, J. Power Sources 195 (2010) 955.
ford, L.A. Chick, G.D. Maupin, S.P. Simner, J. Electrochem. Soc. 150
067.
r., A.S. Joshi, K.N. Grew, W.K.S. Chiu, A. Tkachuk, S.H. Wang, W. Yun, J.
em. Soc. 155 (2008) B504.
idt, C.-L. Tsai, J. Power Sources 180 (2008) 253.
er, Diffusion-Mass Transfer in Fluid Systems, Cambridge University
bridge, MA, 1984.
D. Wee, A.F. Ghoniem, J. Power Sources 186 (2009) 417.
luwe, H. Zhu, R.J. Kee, G.S. Jackson, J. Electrochem. Soc. 155 (2008)
rt, J. Meusinger, F. Thom, J. Power Sources 87 (2000) 57.
Jr., A.A. Peracchio, W.S.K. Chiu, J. Power Sources 176 (2008)
p, T. Olesen, T. Komatsu, P. Schjönning, D.E. Rolston, Soil Sci. Soc. Am.
1) 613.
n, Chem. Eng. Sci. 44 (1989) 777.
.V. Virkar, J. Electrochem. Soc. 150 (2003) A942.
ng, H. Kishimoto, K. Yamaji, M. Yoshinaga, T. Horita, M.E. Brito, H.
a, Electrochem. Solid-State Lett. 13 (2010) B21.. Liu, S. Wang, Y. Choi, C. Zuo, M. Liu, J. Power Sources 195 (2010)