Catalytic Difunctionalization of Unactivated Alkenes with Unreactive Hexamethyldisilane through Regeneration of Silylium Ions

Article abstract of DOI:10.1002/anie.201911282

A metal-free, intermolecular syn-addition of hexamethyldisilane across simple alkenes is reported. The catalytic cycle is initiated and propagated by the transfer of a methyl group from the disilane to a silylium-ion-like intermediate, corresponding to the (re)generation of the silylium-ion catalyst. The key feature of the reaction sequence is the cleavage of the Si?Si bond in a 1,3-silyl shift from silicon to carbon. A central intermediate of the catalysis was structurally characterized by X-ray diffraction, and the computed reaction mechanism is fully consistent with the experimental findings.

Full text of DOI:10.1002/anie.201911282

Received: 13 June 2019
DOI: 10.1002/er.4869
Revised: 5 August 2019
Accepted: 21 August 2019
R E S E A R C H A R T I C L E
Global sensitivity analysis of uncertain parameters based on
D modeling of solid oxide fuel cell
2
1
2
1
1
Chengru Wu | Meng Ni | Qing Du
| Kui Jiao
1
State Key Laboratory of Engines, Tianjin
Summary
University, 135 Yaguan Road, Tianjin,
China, 300350
In this study, we propose an enhanced quasi–two‐dimensional and
nonisothermal model for solid oxide fuel cell (SOFC) parametric simulation
and optimization. The dependence of effective properties on microstructural
parameters is fully considered in this model. Besides, an elementary effect
2
Building Energy Research Group
Department of Building and Real Estate
The Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong, China
(EE) approach based on Monte Carlo experiments is adopted to comprehen-
Correspondence
sively evaluate the sensitivity of totally 24 parameters. A two sample
Kolmogorov‐Smirnov (K‐S) test is carried out to evaluate the ability of EE
method for robust and accurate sensitivity analysis. The investigation focuses
on the important microstructural parameters of the composite anode/cathode
function layers (AFL/CFL). With this research, relative volume fractions of
conducting materials in the AFL/CFL are the most sensitive factors among
all input parameters while particle radius is found to be the least sensitive
microstructural parameters. The particle size ratio of electronic particles to
ionic particles is found to be much more sensitive than particle size due to
its significant effect on effective conductivity. The cathodic electrochemical
parameters reflect cell performance more significantly than the anodic ones.
To further elucidate the role of input factors, this study provides a principle
for parametric sensitivity classification as well. Besides, impacts of current den-
sity variation on parametric sensitivity are comprehensively considered. Nega-
tive or positive effects of parameters on cell performance and their influencing
mechanisms are also discussed. Furthermore, global sensitivity analysis for sin-
gle parameters at different positions is performed to assess the probability of
structural optimization along the channel length. Then, a feasible nonuniform
distribution method in allusion to function layers is proposed for further
K. Jiao and Q. Du, State Key Laboratory of
Engines, Tianjin University, 135 Yaguan
Road, Tianjin, China, 300350.
Email: kjiao@tju.edu.cn; duqing@tju.edu.
cn
Funding information
Funding support from research Grant
Council, University Grants Committee,
Hong Kong SAR, Grant/Award Number:
PolyU 152214/17E; The National Key
Research and Development Program of
China, Grant/Award Number:
2017YFB0601904
2
−3
2
−1
−1
Nomenclature: A, Cell geometric area (m ); C, Concentration (mol m ); D, Diffusion coefficient (m s ); Eact, Activation energy (J mol ); E, Open
−
1
−2 −1
−2
circuit voltage (V); F , Faraday's constant (C mol ); h, Surrounding heat transfer coefficient (W m
K
); i, Exchange current density (A m ); I,
2
−3
−1
Current density (A m ); J, Electrochemical reaction rates (A m ); k, Pre‐exponential factor; M, Molecular weight (kg mol ); p, Pressure (Pa); R,
−
1
−1
Universal gas constant (J mol
g
K ); r , Hydraulic pore radius (μm); r, Radius (μm); δ, Thickness (μm); T, Temperature (K); q, Particle‐radii ratio;
S, Source term; ST, Stoichiometry ratio; V, Voltage (V)
−
1
−3
−1
Greek: λ, Three phase boundary length (m ); ρ, Density (kg m ); τ, Tortuosity; ε, Porosity; σ, Conductivity (S m ); β, Transfer coefficient; θ, Contact
angle (°); φ, Relative volume fraction; η, Overpotential (V)
Subscripts and superscripts: a, Anode; AFL, Anode function layer; ADL, Anode diffusion layer; BP, Bipolar plate; c, Cathode; CFL, Cathode
function layer; CH, Channel; eff, Effective; ele, Electronic; ed, Electronic‐conducting particle; el, Ionic‐conducting particle; env, Environment; FL,
Function layer; ion, Ionic; rev, reversible; TPB, Three phase boundary length; 0, Standard state
Int J Energy Res. 2019;1–19.
wileyonlinelibrary.com/journal/er
© 2019 John Wiley & Sons, Ltd.
1

2
WU ET AL.
improvement of cell performance according to global SA results of single
parameters along the channel direction.
KEYWORDS
elementary effect, global sensitivity analysis, microstructure, non‐uniform, SOFC
1
2
1
| INTRODUCTION
sensitivity analysis techniques have been developed.
Wagner firstly proposed the idea of global sensitivity, and
Solid oxide fuel cell (SOFC) is a promising energy conver-
sion device with a lot of advantages such as flexible fuel
choice, wide power range, low emission, and high effi-
meanwhile, a multicomponent‐based global sensitivity
13
analysis method was presented. Sobol proposed a precise
quantitative approach to evaluate the importance of input
parameters with the assumption of independent input fac-
tors, while large computational resources would be needed
1,2
ciency. Despite those excellent features, extensive efforts
are needed before successful commercialization of SOFC,
for example, further SOFC performance improvement by
14-16
to get the reliable results.
Hence, it is not an applicable
3,4
design optimization of electrode structure. Compared
with experimental investigation on cell structure optimi-
zation, numerical simulation and optimization can serve
as a powerful, economic, and reliable tool.
approach for computational models containing numerous
parameters. Herschel et al presented two types of high‐
dimensional model representations (ANOVA/Cut HDMR)
aiming at achieving a dramatic reduction of computa-
tional complexity for capturing input‐output relationships,
and it was found that Cut HDMR shows significantly
The operation of SOFC always involves complex
physical/chemical phenomena such as multicomponent
diffusion, spontaneous reforming reaction, and electro-
chemical reaction process. All these processes are highly
coupled and usually behave nonlinearly. In addition,
their geometric scale ranges from microscale to macro-
scale. Empirical parameters are generally employed to
ensure consistence of SOFC macroscopic behaviors such
as the I‐V curves between the numerical data and the
experimental data, which is commonly used for model
validation. In fact, empirical parameters that are hard to
measure experimentally would inevitably increase model
freedom degree thus damage the results reliability at the
17
higher efficiency than ANOVA HDMR does. Neverthe-
less, due to the negligence of input order correlating
effects, the HDMR method is unable to provide a descrip-
tion of model structure. Borgonove introduced a moment
independent and variance‐based global uncertainty indi-
cator to evaluate parametric importance of independent
parameters. With all input factors free to vary in their
uncertain ranges, the computational stability is hard to
guarantee, especially those multiparameters coupling
18
models. Morris presented an effective screening global
sensitivity measure and an efficient factorial sampling
strategy to identify the importance of specific factors in
multiple variance models. This method is dependent on
the calculation of increasing ratio for each uncertain fac-
tors, namely elementary effect (EE). Average and variance
value are obtained to evaluate overall sensitivity informa-
5
same time. Thus, it is highly necessary to identify groups
of factors that are most responsible for the results reliabil-
ity and voltage fluctuation during the numerical predica-
6
tion process. Modelers and practitioners have reached a
consensus that sensitivity analysis (SA) method should
be applied as a minimum, necessary component to
answer the question of how uncertain of those factors
19
tion of inputs. To further enhance computational effi-
ciency, Campolongo et al further improved the EE
method by raising new sampling strategy and refining
7-9
are and where those uncertainties come from.
20
Sensitivity analysis is usually implemented in fuel cells
to evaluate how uncertainties of inputs affect the predicted
cell performance. Various methods have been proposed in
recent studies. Most of published literatures on sensitivity
analysis of SOFC are dependent on “one‐factor‐at‐a‐time”
the measure of absolute mean value of EE.
Even through sensitivity study has been extensively
used in various scientific researches, systematically global
SA studies for SOFC are rarely found in the literatures.
Chan et al performed a sensitivity analysis focused on var-
iation of cell layer thickness. Anode support fuel cells are
found to be more efficient than cathode support ones
(OAT) method with the assumption of model linearity and
additivity which is obviously irrational for most of multi-
10
21
ple variables coupled computational models. The biggest
defect of the OAT method is its incapability to reflect
global impacts of entire parametric space on cell perfor-
under high pressure conditions. Campanari and Iora
established a finite volume SOFC model to conduct sensi-
tivity analysis to elucidate the effects of different polariza-
tion losses with merely four empirical electrochemical
parameters varied in a relatively wide range, thus
11
mance. To overcome the inherent shortcomings of
OTA method, a progressively increasing number of global

WU ET AL.
3
2
2
demonstrated the importance of appropriate parameters.
proposed to evaluate the parametric effects of SOFC. To
the best of authors' knowledge, previous sensitivity studies
are based on either isothermal assumption or noniso-
thermal model with simplified electrochemical pro-
Nagel et al conducted a sensitivity analysis to gain insights
of internal charge and mass process as well as performance
23
fluctuation under different fuel compositions. Based on
an electrochemical model in which exchange current den-
sity and gas diffusion coefficients were completely depen-
dent on cell microstructures, Ni et al systematically
analyzed the effects of structural/operational parameters
on the individual overpotentials and operation efficiency
of SOFC. It should be noted that all the researchers of
above literatures prefer “OAT” sensitivity analysis (OAT‐
SA) method, in which all factors are assumed to be inde-
pendent from one another while the interaction among
factors are irrationally neglected. Moreover, recent studies
of Vijay et al proposed a two‐stage validation strategy for
SOFC model in which global sensitivity analysis was
implemented to evaluate parametric uncertainties con-
nected with the second kind of error. Macroscopic opera-
tion parameters such as effective conductivity, heat
transfer coefficient, and diffusion coefficient were directly
assigned to illustrate their uncertainties. However, above
microscopic properties were determined by microstruc-
tural parameters to a great extent and actually not the orig-
26,27
cess.
Different from the previous studies, the mass
transfer process along the channel direction is taken into
account in this study which is always neglected in sensitiv-
ity analysis in the literatures to simplify computation. A
film percolation theory is implemented to evaluate effec-
tive mass and charge transport properties inside electrode.
Elementary effects (EE) method which has been proved to
be very promising for global sensitivity analysis with mul-
24
28
tiple input factors is applied to evaluate the global sensi-
tivity of totally 24 parameters. Totally 24 pivotal structural
and empirical parameters are selected to examine how and
how much their variation affects cell performance. Then,
the coupling effects of different locations on cell perfor-
mance are discussed. Based on the conclusions above, a
feasible nonuniform distribution method in allusion to
function layers is proposed for further improvement of cell
performance.
25
25
inal inputs for most of computational models. In general,
establishment of high computational efficiency model and
adoption of appropriate sensitivity analysis techniques are
the key factors to conduct the sensitivity analysis of SOFC.
In this study, a quasi‐two dimensional and noniso-
thermal model with high computational efficiency is
2 | MODEL DEVELOPMENT
2.1 | Physical problem
The basic structure and computational domain of SOFC
are illustrated in Figure 1, including the bipolar plate,
FIGURE 1 Illustration of anode‐
supported solid oxide fuel cell structure
[
Colour figure can be viewed at
wileyonlinelibrary.com]

4
WU ET AL.
channel, a thick Ni/yttria‐stabilized zirconia (Ni/YSZ)
anode diffusion layer (ADL), a thin Ni/scandium stabi-
lized zirconia (Ni/ScSZ) anode function layer (AFL), a
thin LSM/ScSZ cathode function layer (CFL), and a pure
ScSZ electrolyte layer (ELE) in between. The directions
both perpendicular to the channel and along the channel
are treated as the major directions of mass and heat trans-
fer. Electrochemical reaction merely occurs at the inter-
face of three phase boundary (TPB) area. Owing to
particular aspect ratio of SOFC, namely thin electrode
thickness but large channel length, a quasi‐two dimen-
sional model is proposed in this study which couples
the 1D heat and mass transfer process through electrode
and the 1D transport process along the channel direc-
∂
Ci;channel
I
uy
¼ κi
(1)
∂y
niFδchannel
−
3
where Ci, channel (mol m ) represents the gas concentra-
tion of species i along the channel, κ is the stoichiometric
coefficient (1 for H and O , −1 for H O), n is the charge
transfer number of species i, and I (A m ) refers to the
current density distribution along the channel. u (m s )
y
represents the flow velocity which depends on stoichio-
metric ratio and operating current density:
i
2
2
2
i
−
2
−
1
I0Aact
uy ¼ ST
(2)
2
FCinlet;iAinlet
−
2
29
where ST is the stoichiometric ratio and I
(A m ) is the
0
tion. This is a popular method since it provides the dis-
tributions of current density, species, and temperature
distributions in two directions with manageable compu-
tational resources compared with traditional 3D model.
It is essential to remark that the quasi‐two dimensional
model here is not like the traditional segment models that
simply integrate specific number of 1D model and con-
nect them by the channel flow. The mass and heat trans-
fer phenomena as well as charge transport process inside
electrode and channel are integrated together to capture
the real physical process along the channel direction to
a great extent. Detailed geometry and operating condi-
operating current density. F is the Faraday's constant
−
1
(
96 458 C mol ).
Sherwood number (Sh) is introduced to describe the
gas transport process from channel to porous interface
in analogy with heat transfer process. The interfacial con-
centration at electrode interface can be expressed as
IAact
Ci;interface ¼ Ci;channel −
(3)
2ShDeff;iFd A
h
inlet
2
−1
where Deff, i (m s ) is the effective diffusion coefficient
in channel. d (m) is hydraulic diameter of channel.
h
30-33
tions of a basic case are shown in Table 1.
Model assumption
2.2.2 | Mass transfer in electrode
(
1) The gas velocity along the flow channel is regarded
Diffusion processes in both directions are considered
while the convection effects are small in the porous elec-
trodes and can be reasonably neglected. Then, the mass
diffusion process in electrode is formulated as
as constant.
(
(
2) Ideal gas law is considered.
3) The gas flow in the channel is dominated by convec-
tion and the diffusion process is neglected.
4) The pressure loss along the channel direction is
small and can be neglected.
2Ci
∂x2
2
i
∂
∂ C
(
(
(
(
x
eff;i
y
eff;i
D
þ D
¼ Si
(4)
2
∂y
5) Contact resistance between two adjacent layers is
ignored.
−
3
−1
where S (mol m s ) represents the source terms of
i
species i which are comprehensively listed in Table 2.
6) The radiation heat transfer inside the fuel cell is not
considered.
x
eff;i
2
−1
y
2 −1
D
(m s ) and D (m s ) are the effective diffusion
eff;i
7) Flow is in steady state and laminar inside channel.
coefficients in two directions, respectively, which is calcu-
34
lated by Bosanquet equation, and the detailed computa-
34
tional formula are summarized in Table 3.
2
2
.2 | Conservation equations
2.2.3 | Energy balance
.2.1 | Mass transport in channel
The energy balance equation is solved over the entire fuel
cell domain. As the convection in the porous electrode is
neglected, the governing equation of temperature distri-
bution is expressed as
In this model, the pressure drop is safely neglected due to
relatively short channel and small Reynold's number. The
assumption above simplifies the model a lot by avoiding
complicated momentum equation solving process with
great numerical accuracy. Then, the mass conservation
equation can be described as
2
∂
2T
∂x2
∂ T
x
eff
y
eff
k
þ k
¼ ST
(5)
2
∂y

WU ET AL.
5
3
0-33
TABLE 1 Cell structures and basic operation conditions
Parameters
Unit
Value
Reference
3
3
3
3
3
3
3
3
3
0
0
1
1
1
1
2
2
2
Channel length, width, depth
Thickness of bipolar plate
mm
mm
μm
100; 1; 1
2.5
Thickness of ADL, AFL, electrolyte
Thickness of CFL
680; 20; 20
μm
20
Porosity of ADL, AFL, CFL
Tortuosity of ADL, AFL, CFL
Particle‐radii ratio
0.335; 0.335; 0.335
2.5; 2.5; 2.5
1
Electronic‐conducting particle radius of ADL, AFL, CFL
Operating pressure
μm
atm
K
1; 0.2; 0.2
1
Operating temperature
1073
1073
‐
Inlet temperature of fuel and air
Air composition
K
‐
3
1
1
2 2
21% O , 79% N
3
Fuel composition
2
98.3% H2, 3.07% H O,
−
2
Cell current density
A m
W m
°
3000
‐
32
−
2
−1
−1
Heat transfer coefficient between cell and surroundings
Contact angles of particles in AFL, CFL
Anode and cathode stoichiometry ratio
K
50
3
1
15, 15
1.5; 1.5
‐
32
−
1
Thermal conductivity of anode, cathode, electrolyte,
interconnect
W m
K
6.23; 9.6; 2.7; 30
−
1
33
Active energy of anode, cathode
kJ mol
120; 160
0.
3
3
1
0
2 2 2
Reaction order of H , H O, O
Transfer coefficient of anode, cathode
0.5; 0.5
TABLE 2 Source terms
Source Term
Unit
−
kg m
3
−1
Ji
s
Si ¼ κi
ðIn anode=cathodeÞ
niF
8
−3
I2
I2
ηact
δAFL
I
TΔSc hside⋅ðTa − TenvÞ
−
W m
>
>
þ
þ
−
þ
þ I
þ I
ðIn anodeÞ
>
>
eff
eff
>
σ
σ
2F
δa
>
ele
ion
>
>
>
>
<
I2
I2
ηactI
δCFL
TΔSc hside⋅ðTc − TenvÞ
ST ¼
þ
−
ðIn cathodeÞ
eff
ele
eff
ion
>
σ
σ
2F
δc
>
>
>
>
>
ꢀ
ꢁ
>
2
>
hside TBP=CH=ELE − Tenv
δBP
>
I
>
:
ðInBP=Channel=electrolyteÞ
eff
ion
σ
x
eff
−2
y
eff
−2
−2
wherek (W m K) andk (W m K) represent the effec-
where h (W m K) represents the convective thermal
tive thermal conductivity in two directions. T (K) is the
coefficient.
−
3
temperature and S (W m ) is the source term of heat
T
transport that is listed in Table 2. The heat exchange
between fuel cell and surrounding environment can be
expressed as
2
.3 | Electrochemical model
32
The output voltage is described as
Q ¼ hAwallðTwall − TenvÞ
(6)
Vout ¼ Erev − ηact − ηohm
(7)

6
WU ET AL.
3
4
TABLE 3 Diffusion coefficients
corresponding to each FL segment, average activation
loss at middle FL can be calculated by an inverse function
Parameter
Correlation/Value
of B‐V equation under a constant current density
ꢄ
ꢅ
−
1
Molecular weight of
species i
32,35
1
1
Mi; j ¼ 2
þ
mode
:
Mi
Mj
ꢂ
ꢂ
ꢂ
ꢃ
ꢂ
ꢃꢃ
Mean characteristic
length of species i
and j
σi þ σj
2β Fη
2ð1 − β ÞFη
eff;a
a
act;a
a
act;a
σi; j ¼
Ja ¼ i0;aλtpb exp
− exp −
2
RT
RT
(
10)
Diffusion collision
integral
1:06036
0:19300
ΩD ¼
þ
þ
Γ0:15610 expð0:47635ΓÞ
ꢂ
ꢃ
ꢂ
ꢃꢃ
4
βcFη
4ð1 − β ÞFη
1
:03587
1:76474
eff;c
act;c
c
act;c
þ
Jc ¼ i0;cλtpb exp −
− exp
expð1:52996ΓÞ expð3:89411ΓÞ
RT
RT
kBT
(11)
Γ ¼ ꢀ
Di;K ¼
Di; j ¼
ꢁ
0
:5
υiυj
−
3
−3
ꢄ
ꢅ
where J (A m ) and J (A m ) represent the electro-
a c
0
:5
Knudsen diffusion
coefficient
2
3
8
πMi
RT
rg
:5
chemical reaction rates while the corresponding current
density can be obtained by multiplying thickness of func-
0:00266T1
PM0 σ Ω
Binary diffusion
coefficient
eff;a
eff;c
2
−3
:5
2
tion layer. λtpb and λtpb (m m ) refers to effective three
i; j i; j
D
ꢃ
phase boundary length of anode and cathode. i0 and i0,c
(A m ) are the reference exchange current density in
,a
Effective diffusion
coefficient
εꢂ 1
Di; j
1
D ei ff ¼ ξ
þ
−2
Di;K
anode and cathode sides which are calculated by the fol-
Density of gas mixture
1
33
ρ ¼
N
lowing formulas :
∑
Yi=ρi
i¼1
ꢂ
ꢂ
ꢃ
ꢂ
ꢃ
Density of species i
298
γ
H2
γ
H2O
ρi ¼ ρi;298K T
CH ;bRT
2
CH O;bRT
2
i0:a ¼ k0;a
(12)
(13)
p0
p0
ꢂ
ꢃꢃ
where Vout (V) represents the output voltage. ηact (V) is
the total activation loss, and ηohm (V) is the total ohmic
loss. Erev (V) is the reversible voltage following thermo-
dynamic principles and can be evaluated using Nernst
equation [32]:
Eact;a
R
1
T
1
Tref
exp −
−
ꢂ
ꢃγ
ꢂ
ꢂ
ꢃꢃ
O2
CO2;bRT
Eact;c
R
1
T
1
i0:c ¼ k0;c
exp −
−
p0
Tref
!
RT
pH2O
:5
−
4
where k0,a and k0,c are the pre‐exponential factors in the
cause of fitting experiment data, and E and Eact,c (J
mol ) represent the active energy for anodic and cathodic
Enernst ¼ 1:253 − 2:4516 × 10 T −
ln
(8)
0
2
F
p p
H2 O2
act,a
−
1
−
1
where R is the universal gas constant (8.314 J mol
K ). p , p , and p (Pa) represent partial pressures
electrochemical reaction. γ , γH2O, and γO2 represent the
H2
−
1
H2O
H2
O2
reaction order of H , H O, and O , respectively.
2
2
2
of water vapor, hydrogen, and oxygen, respectively.
The concentration overpotential is included in the
reversible voltage as the particle pressure is obtained
by multiplying the operating pressure and species mole
fraction at the reaction site.
For simplification, the control volume along the chan-
nel length is regarded as parallel; hence, the potential of
bipolar plate interface remains the same. Different gas
concentration and temperature distribution along the
channel direction result in a variation of current density
distribution. Under the constant current density mode,
the value of output voltage and local current density dis-
tribution can be solved based on the following system of
The ohmic loss caused by charge transfer resistance is
calculated by ohm's law:
ꢀ
ꢁ
ηohm ¼ R
þ RADL;a þ RFL;a=c þ RCH;a=c þ RELE I (9)
36
BP;a=c
nonlinear equations :
8
where ηohm represents ohmic activation. RBP,a/c,RADL,a
,
½0ꢀ
½0ꢀ ½0ꢀ
>
0 ¼ E − Vout − I R
>
rev
>
RFL,a/c,RCH,a/c and RELE refer to the conductivity resis-
tance for different parts of SOFC. I is the local current
density distribution.
The Butler‐Volmer (B‐V) equation illustrates the rela-
tionship between electrochemical reaction rates and acti-
vation loss. As there is only an average gas concentration
>
>
½nꢀ
½nꢀ ½nꢀ
>
0 ¼ E − Vout − I R :::
>
rev
>
<
½
N−1ꢀ
½N−1ꢀ ½N−1ꢀ
− Vout − I R :::
0
¼ E
rev
(14)
>
>
n¼N − 1
>
>
½nꢀ
>
∑ I
>
>
>
:
n¼0
N
¼
I

WU ET AL.
7
½
nꢀ
[n]
[n]
ꢀ
ꢁ
ψl=rl
where E , I , and R represent the reversible voltage,
rev
2
k
2
l
Zk;l ¼ 0:5 1 þ r =r Z
(20)
2
current density, and total resistance of n th segment.
∑
ψ =rk
k
k¼1
2.4 | Film percolation microstructure
model
2.4.2 | Effective conductivity
In this section, an improved film percolation model is
applied to investigate the effective characteristics of
Effective conductivity of electronic and ionic conducting
materials can be estimated as :
37
37,38
microstructures.
Porous structures are assumed to
consist of random mixture of spherical electronic and
ionic conducting particles in this theory. By changing pri-
mary microstructural parameters, including average par-
ticle radius, particle radii ratio, relative electronic/ionic
volume fraction as well as porosity, the basic characteris-
tics of microstructures can be significantly affected.
Several important effective properties estimated by perco-
lation theory are listed below:
ꢂ
ꢃ
ψ −ψt
k
1 þ ε=ð1−εÞ−ψk
2
eff
0
k
σ
¼ σ
(21)
k
k
t
t
k
where ψ represents the percolation threshold volume
fraction, which can be calculated by equations set :
3
7
t
ψ
ed
Z
Z
ꢀ
ꢁ
¼ 1:764
¼ 1:764
(22)
(23)
t
ed
t
ed
ψ
þ 1 − ψ
q
2.4.1 | Three phase boundary length
t
ψ
el
ꢀ
ꢁ
t
el
t
el
ψ þ 1 − ψ =q
Active three‐phase boundary length is actually formed
with percolated clusters and pores. There are two types
of percolated clusters thought to be conductive in the
Z equals to 6 here, and q represents the particle radii
ratio. The pure conductivity can be calculated as fol-
37,38
composite electrode, namely cluster I and cluster II.
32
lows :
The active three phase boundary (TPB) length is depen-
dent on the probability of particle that belongs to cluster
I and II, namely percolated probability which can be cal-
ꢂ
ꢃ
−
9681
0
4
σ
¼ 6:92 × 10 exp
(24)
(25)
ScSZ
T
37,38
culated as
:
ꢂ
ꢃ
4
:2 × 107
−1150
"
#
0
ꢂ3
:764−Zk;k
ꢃ
0:4
2
:5
σ
¼
exp
LSM
T
T
PI;k ¼ 1−
(15)
(16)
2
ꢂ
ꢃ
−
10300
0
4
σ
¼ 3:34 × 10 exp
(26)
(27)
YSZ
T
ꢆ
ꢇαðN−1Þ
ꢀ
ꢁ
Zk;k
Z
ꢀ
ꢁ
γ
PII;k ¼ 1 − PI;k
β
1−PI;k
0
Ni
6
σ
¼ 3:27 × 10 − 1065:3T
where Zk,k is the average number of contacts. Then the
37,38
total probability is
:
2.4.3 | Hydraulic pore radius
Pk ¼ PI;k þ PII;k
:
(17)
(18)
The hydraulic pore radius in electrode is determined by
microstructure of conductive materials :
37
32
The total TPB length can be estimated as :
2
rg ¼
:
(28)
3
ðψ =red þ ψ =relÞð1 − εÞ
eff
tpb
v
ed
v
el
ed
el
λ
¼ led;eln Zed;elPedPel ¼ lel;edn Zel;edPedPel
where lk,l is the contact perimeter between two types of
contacting particles, and Zk,l is the coordination number.
2.5 | Elementary effects method
37
In this section, the EE approach proposed by Morris is
introduced to perform global sensitivity analysis. For
1
9
lk;l ¼ 2πminðrk; rlÞsinθ (19)

8
WU ET AL.
simplification, cell performance is regarded as the func-
tion of total k input factors:
V ¼ f ðX ; X ; :::; X ; :::; X Þ
(29)
1
2
n
k
where X , X , … , X represent the standardized input var-
1
2
k
iables that ranges from 0 to 1. Each standardized variable
is discretized into L levels. The conversion relationship
between the actual value and no dimensional value can
19
be estimated as :
si ¼ Xiðsmax − sminÞ þ smin
(30)
where s represents the real value of input factors. Then, a
i
sample space of k dimensional and L level is constructed.
19
Elementary effects for i th input factors are defined as :
(A)
f ðX ; X ; :::X ± Δ; :::X Þ − f ðX ; X ; :::X ; :::X Þ
1
2
i
k
1
2
i
k
EE ¼
i
±
Δ
(
31)
where EE represents the EEs involving with two groups
i
of parameters which is obtained from sample space ran-
domly. Each parameter range is evenly divided into L
portions while △ takes the value of L/2(L − 1) to ensure
equal probability during the sampling process. In order to
ensure that the selected sampling points could cover the
whole sampling space, Morris proposed a randomized
19
sampling strategy to get the sampling matrix :
ꢀ
ꢁ
*
*
*
B ¼ J
x þ ΔB P
(32)
(33)
kþ1;1
(
B)
ꢈꢀ
ꢁ
ꢉ
*
ΔB ¼ ðΔ=2Þ 2B − Jkþ1;k D þ Jkþ1;k
FIGURE 2 Comparison between modeling prediction and
experiments data. A, Current density‐voltage curves; B,
overpotential [Colour figure can be viewed at wileyonlinelibrary.
com]
*
39
where B is a (k + 1) × k sampling matrix in which each
row represents a k‐dimensional sampling point corre-
sponding to one EE factor. Jk + 1,1 refers to a (k + 1) ×
1
matrix of 1's, while J
refers to a (k + 1) × k matrix
k + 1,k
*
of 1's. P represents a random permutation matrix that
selected to obtain r EEs for each factor. By averaging these
each row and column only has one entry of 1 and 0's else-
where. D is a diagonal matrix of −1's and 1's with equal
EEs of each factor, the sensitivity of factor i can be evalu-
*
19,20
ated using formulas set
:
probabilities which determines the increase or decrease of
factors along the trajectory. B is a (k + 1) × k matrix of 0's
and 1's with the limitation that two rows of B only differ
r
1
j
i
μi ¼ r ∑ EE
(35)
(36)
(37)
j¼1
19
in their i th value :
2
6
3
7
r
1
r
0
; 0; 0; :::0
*
j
i
μ ¼ ∑ ∣EE ∣
i
j¼1
6
1; 0; 0; :::0 7
6
6
7
7
B ¼ 1; 1; 0; ::: 0 :
(34)
6
6
4
7
7
5
r ꢄ
ꢅ
2
1
:::
2
j
i
σ ¼
∑ EE −μ
i
r − 1
j¼1
1
; 1; 1; ::: 1
*
i
2
The sampling strategy above merely corresponds to one
where μ
i
, μ , and σ refer to the mean value, absolute mean
randomly generated trajectory. There are r trajectories
value, and sampling variance of EEs.

WU ET AL.
9
3
3
| RESULT AND DISCUSSION
.1 | Solution procedures and experiment
validation
The computational process is implemented by in‐house
codes based on Matlab platform. There are totally 8 ×
1
0 nodes defined in the entire computational domain,
while finite difference method and fourth‐order Runge‐
Kutta approach are utilized to discretize and solve the
−
8
governing equations with a numerical precision of 10 .
Figure 2 shows the comparison between modeling pre-
39
diction and experiment data. Humidified hydrogen
96%) is supplied as fuel to the anode. Air is used in cath-
(
ode. Effects of temperature distribution (1023/1073/1123
K) are taken into account as well. Both the global polari-
zation curves and various voltage losses show reasonable
agreement with experiment data although there still
exists slight inconformity of polarization curve at 1023
K. This small difference between modeling results and
the experimental data might result from inaccurate exper-
imental measurements or the use of several assumptions
in the model. It is worth to point out that this study
focuses on the global sensitivity of input parameters
under the normal working conditions, while some
extreme working condition such as extreme current den-
sity is not taken into account. Duo to the fact that fuel or
oxygen starvation cause divergence, the output voltage of
(A)
40
this model cannot go down to 0 V.
Moreover, careful evaluation is necessary to ensure
that global sensitivity approach is reliable and applicable.
A
two sample Kolmogorov‐Smirnov (K‐S) test is
employed to validate the computational stability of SA
method by examining whether two independently calcu-
lated results have notable difference. The K‐S statistics
for two given cumulative distribution function G(x) is
(
B)
FIGURE 3 Feasibility assessment of elementary effects by two
sample Kolmogorov‐Smirnov (K‐S) test. A, Comparison between
standard result 1 and selected result 2; B, comparison between
standard result 1 and selected result 3 [Colour figure can be viewed
at wileyonlinelibrary.com]
D ¼ sup∣G ðResult jÞ − G ðResult kÞ∣
(38)
i
i
where G (Result j) and G (Result k)represent the absolute
i
i
distribution of EEs for two independently calculated
results. D represents the standard deviation. Two ran-
domly selected cases are compared with standard case 1
under exactly same conditions. There is no space to go
into detail for all factors. An example of CFL particle
radius is plotted in Figure 3A,B. The P value is treated
as a criteria to judge whether the result is credible. Larger
P value than significant level 0.05 is a desirable result as it
indicates the stability and reliability of the calculation
results. As can be seen from Figure 3A,B, the P value of
each plot is much higher than 0.05; hence, the sensitivity
analysis scheme is accurate.
3.2 | Global sensitivity analysis
3.2.1 | The necessity to do sensitivity
analysis
There are three cases summarized in Table 4 to demon-
strate the necessity to do global sensitivity analysis. Several
parameters are defined to be different among these cases.
Case 2 has absolutely different structural parameters com-
pared with cased 1, while case 3 has different empirical
parameters and relative volume fraction compared with
case 1 and case 2. As can be seen from Figure 4, there are

10
WU ET AL.
TABLE 4 Three selected groups of parameters
Parameters
r
ed,AFL
q
AFL
φ
AFL
δ
ELE
k
0,a
k
0,c
−
−
−
3
3
4
−6
Case 1
Case 2
Case 3
0.2 μm
0.15 μm
0.2 μm
1
0.5
20 μm
25 μm
20 μm
2.17 × 10
2.17 × 10
1.17 × 10
3.115 × 10
3.115 × 10
1.115 × 10
−6
−5
1.5
1
0.55
0.45
their variation affects cell performance, while the cou-
pling effects of those factor are also considered in this sec-
tion. In previous sensitivity analysis literatures, input
parameters are always irregularly enlarged to adapt more
41
working conditions. Such procedures have hidden dan-
gers that wide range of parameters itself can also greatly
impact the cell performance while effects of interactions
among all parameters might be completely or partly cov-
ered. As different selected ranges of parameters might sig-
nificantly influence sensitive degree, input parameters
are defined to fluctuate around the standard value with
a unified wave range of ±15%, which are listed in
−
2
−2
Table 5. Current density of 1500 A m and 7500 A m
are selected as the basic operating condition in this study.
For a more comprehensive study of parametric sensitiv-
ity, particle radii ratio (q) is employed to describe the rela-
FIGURE 4 Simulation results for three selected cases [Colour
figure can be viewed at wileyonlinelibrary.com]
tive size of electronic particle radius (r ) and ionic particle
ed
radius (r ). r is defined as the input variable to reflect the
el
ed
overall effects of particle size while rel varies as particle
radii ratio changed. Theoretically, particle radii ratio does
not merely refer to effects of ionic particle radius but also
represents impacts of relative size of electronic and ionic
particle radius, which can be expressed as
two main evidences that emphasize the importance to con-
duct sensitivity analysis. On the one hand, we can see that
different combinations of parameters may result in identi-
cal polarization curves, which arises the problem that
polarization curves sometimes are insufficient for valida-
tion. On the other hand, parametric variation amplitudes
among three cases are quite different. For instance, when
we compare case 3 with case 1, as relative volume fractions
decrease by 10%, the pre‐exponential factors change more
than 10‐fold to fit the I‐V curve, which signifies that the
importance of different parameters on cell performance
are diverse, and obviously, positive or negative effects of
parameters are different as well.
In this study, a global sensitivity analysis of totally 24
input parameters is performed based on an EE method.
Sensitive properties of input factors and their immanent
causes are comprehensively investigated. To further
explore the potential of microstructure optimization, sensi-
tivity examinations for single parameters at different posi-
tion along the channel length are also conducted in the last
segment.
red
q ¼
:
(39)
rel
We demonstrate nominal value of particle radii ratio
as 1 aimed at two main tasks: (a) To obtain a reasonable
range of particle radii ratio including both less than 1 part
and larger than 1 part. (b) To get a relatively wide value
space of relative volume fraction owing to the facts that
too large particle radii ratio would greatly narrow effec-
30
tive range of relative volume fraction.
Researchers generally reach a consensus that a multi-
layer anodic structure should be proposed to enable an
anode function layer (AFL) to restrain active loss by max-
42,43
imizing effective TPB length.
Microstructure design of
function layers is supposed as one of the fundamental
34
challenges for electrode optimization. In this regard,
sensitivity analysis of microstructural parameters inside
FL shows guiding significance both on the structural opti-
mization and model improvement. It is natural that para-
metric sensitivity analysis for function layers is taken into
consideration in the first section.
3
.2.2 | Sensitivity analysis of diverse input
parameters
In this section, totally 24 widely used parameters in SOFC
models are selected to examine how and to which extent

WU ET AL.
11
TABLE 5 Parametric ranges for global sensitivity analysis.
Model Input
Parameters
Nominal
Value
Range of
Variation
Number
Unit
1
2
3
4
5
6
7
8
9
Particle radii ratios of AFL/CFL (q)
Relative volume fraction AFL/CFL (φ)
Particle radius of electronic conductors of AFL/CFL (red
Thickness of AFL/CFL (δ)
Porosity
1
0.5
±15%
±15%
±15%
±15%
±15%
±15%
±15%
±15%
±15%
±15%
±15%
±15%
±15%
±15%
±15%
±15%
‐
)
0.2
μm
20
μm
0.3
‐
Thickness of ADL
680
500
3
μm
Thickness of BPa/BPc
μm
Sherwood number
‐
Tortuosity
3
‐
10
11
12
13
14
15
16
Channel length
100
150
5e − 4
5e − 6
0.5
mm
−
1
Active energy of anode/cathode
Pre‐exponential factor of anode
Pre‐ exponential factor of anode
J mol
‐
‐
‐
‐
‐
Reaction order of H
2
2
Reaction order of H
O
0.25
0.25
Reaction order of O
2
Sensitivity characteristics for microstructural factors at
AFL/CFL are illustrated in Figure 5A‐5D with the distri-
is interesting to find that thickness of both sides is not
that important as expected. In view of analyses above,
the microstructural factors should be given higher prior-
ity than FL thickness in electrode optimization.
bution properties of three specific sensitivity measure-
*
2
ment factors, μ and μ and σ . A basic condition of 1500
−
2
A m is discussed as an example while the sensitive
characteristics are compared with an operating condition
Even through there is an impressive improvement of
−
2
−2
sensitivity at 7500 A m compared with 1500 A m , rel-
ative sensitivity sequence among different factors seems
to not change. To further characterize the effects of cur-
rent density variation, Figure 6A‐6D exhibit a comparison
of different geometric parameters at both sides. Consider-
ing that sensitivity variation tendencies of several param-
eters slightly fluctuate at high current density, a fitting
curve method is applied here to clarify the sensitivity var-
iation tendency. Sensitivities of thickness, relative volume
fraction, and particle radii ratio show almost linear
growth with the increase of current density. Sensitivity
disparities between several AFL and CFL factors, includ-
ing relative volume fraction and particle radii ratio, grad-
ually widened at high operating current density, while
increase of current density, to some extent, promotes
the sensitivity consistency between AFL and CFL thick-
ness. Differently, with increasing current density, impact
of particle radius at CFL remains almost unchanged
while sensitivity of particle radius at AFL still remains
far less than that of CFL.
−
2
of 7500 A m . It is important to point out that relative
volume fractions at AFL/CFL sides are the most sensitive
*
2
factors in light of the largest value of both μ and σ .
Ohmic overpotential is the essential factor governing
the sensitivity of relative volume fraction. More signifi-
cant parametric variation effects of particle radii ratios
on cell performance than electronic particle radius should
be noted. Moreover, particle radius at AFL side exhibits
the lowest importance on cell performance considering
both mean value and sample variance of EE. Activation
loss plays a major role for insensitivity of electronic parti-
cle radius. Considering that electronic and ionic particle
radius contribute same proportion to active overpotential
by influencing effective TPB length, it is concluded that
sensitivity of particle radii ratio largely involves with
ohm loss. In other words, small ionic particle radius
shows more significant influence than small electronic
particle radius does by way of influencing effective
conductivity.
Electronic particle radius at CFL shows larger impacts
on cell performance than that of AFL side. A vital mech-
anism contributing to the sensitivity of electronic particle
radius at CFL is relatively small oxygen reaction rates. It
Figure 7A‐7D presents sensitivity properties for the
rest of vital geometric parameters. Although electrolyte
thickness ranks the third most sensitive factor among this
*
2
group of parameters, the values of μ and σ are merely a

12
WU ET AL.
(
A)
(C)
(
B)
(D)
FIGURE 5 Sensitivity characteristics of structural parameters of anode/cathode function layers. A, and B, Geometric parameters at FL,
−
2
−2
1
500 A m ; C, and D, geometric parameters at FL, 7500 A m [Colour figure can be viewed at wileyonlinelibrary.com]
(
A)
(C)
(
B)
(D)
FIGURE 6 Influence of current density variation on parametric sensitivity. A, Thickness; B, electronic conducting particle radius; C,
relative volume fraction of electronic conducting materials; D, particle radii ratio [Colour figure can be viewed at wileyonlinelibrary.com]

WU ET AL.
13
(
A)
(B)
(D)
(F)
(
C)
(
E)
(
G)
(H)
FIGURE 7 Sensitivity characteristics for rest of geometric parameters and electrochemical parameters; A, and B, Geometric parameters at
−
2
−2
−2
all zones, 1500 A m ; C, and D, geometric parameters at all zones, 7500 A m ; E, and F, electrochemical parameters, 1500 A m ; G, and H,
−2
electrochemical parameters, 7500 A m [Colour figure can be viewed at wileyonlinelibrary.com]

14
WU ET AL.
bit bigger than the least sensitive factors in the function
layers. Sherwood number is always employed to measure
the proportion of convection and diffusion process from
channel to electrode interface, and it is found that its
impact on performance is not as significant as expected.
Sensitivity of tortuosity, which largely involves with
diffusion resistance, is much lower than porosity on
account of low diffusion loss in the porous electrode.
Porosity is found to be the most sensitive factor in this
group owing to the facts that porosity is not only related
to diffusion resistance but also tightly affects the effective
TPB length and conductivity. Collectively, these facts and
speculations reveal that the nature of porosity optimiza-
tion for SOFC is to acquire appropriate effective proper-
ties of TPB length and conductivity while diffusion
coefficient can be regarded as the secondary factor. Con-
sidering that the entire effective reaction area and most
of charge resistance exist in function layers, above inves-
tigation unmasks a dominating effect of function layers
on overall sensitivity of porosity.
Sensitive characteristics of electrochemical parameters
are plotted in Figure 7E‐7H. Reaction order and pre‐
exponential factor at cathode side are the most sensitive
empirical factors, while active energy of both side shows
minimal effects considering average and variance value
of EE. Not surprisingly, empirical factors in cathode affect
cell performance more significantly on account of much
(
A)
lower electrochemical reaction rate of O . Notable
2
−
2
increase of sensitivity is observed at 7500 A m com-
−
2
pared with that of 1500 A m .
Sensitivity tendencies for several high sensitivity fac-
tors toward increase of current density are illustrated in
Figure 8A,B. Sensitivities of geometric and pre‐
exponential factor at anode side show approximately lin-
ear growth as operating current density rising. Duo to the
limitation of oxygen diffusion rates, sensitivities of O2
reaction order as well as cathodic pre‐exponential factors
gradually tend to be flat at high current density.
Figure 9 exhibits the histogram distribution of μ and
*
μ to better understand the sensitivity properties of input
(
B)
parameters. Negative value of μ means negative effects
*
on cell performance. Histogram lengths of μ and μ are
FIGURE 8 Influence of current density variation on parametric
utilized as the standard to judge the monotonicity of
parameters. Exactly same histogram heights of μ and μ
sensitivity. A, Thickness of electrolyte and porosity; B, reaction
*
order of O
2
and pre‐exponential factor [Colour figure can be viewed
at wileyonlinelibrary.com]
mean monotonous effects while different heights
2
FIGURE 9 Histograms of sensitivity distribution for all input parameters, 1500 A m [Colour figure can be viewed at wileyonlinelibrary.
com]

WU ET AL.
15
correspond to nonmonotonous effects on cell perfor-
mance. Except that increase of particle radii ratios
improves the cell performance, all the rest of micro-
scopic parameters in function layers exhibit monotonous
negative impacts on cell voltage. Due to the presence of
more significant mass transfer and charge transport
resistance, increase of layer thickness shows a negative
effect on cell performance in most cases. By affecting
active polarization, pre‐exponential factors show monot-
onous positive effects while reaction orders of gas species
work in the opposite direction.
TABLE 6 Sensitive degree of 24 parameters
Sensitive Degree
Factors
Very sensitive factors
Rather sensitive factors
Insensitivity factors
r
ed,CFL,qAFL,qCFL,ε
a c
,ε ,φAFL,φCFL
k0;c; γO ; δAFL; δCFL; δELE
2
red;AFL; δADL; δBPa; δBPc; Sh; τ; Lcell;
Eact;a; Eact;c; k0;a; γH2O; γH2
*
According to the sensitivity measurement factor μ ,
uncertain parameters are clarified into very sensitive fac-
tors, rather sensitive factors and insensitive factors in
terms of principles below:
*
μ > 0.017 very sensitive factors
*
0
.005 < μ < 0.017 rather sensitive factors
*
μ < 0.005 insensitive factors
Detailed classifications are listed in Table 6. The
Monte‐Carlo approach is applied to validate the accuracy
and reliability of numerical model and sample strategy.
As can be seen in Figure 10, all factors, sensitive factors
and insensitive factors are individually tested 1200 times.
Sensitive factors result in an identical voltage distribution
area with all factors while voltage range produced by
insensitive factors is very narrow compared with that of
all factors. It is clear that sensitive factors are the primary
causes for fluctuation of cell performance; hence, great
efforts are needed on optimizing those truly sensitive fac-
tors rather than insensitive ones. It should be demon-
strated that the model is efficient enough to conduct
global sensitivity analysis. The computational time of
one global sensitivity analysis process is about 5 minutes.
The operation of Monte‐Carlo approach takes longer time
since all factors, sensitive factors and insensitive factors
are individually tested 1200 times, and this process takes
about 1 hour.
(
A)
(B)
FIGURE 10 Scatterplots for comparison of output voltage owing
to sensitive factors, insensitive factors, and all factors. A,
Comparison between insensitive factors and all factors; B,
comparison between sensitive factors and all factors [Colour figure
can be viewed at wileyonlinelibrary.com]
parameters at different positions are regarded as indepen-
dent factors to investigate the coupling effects of different
locations on cell performance in this approach. The cell
structure is divided into 10 segments along the channel
to obtain a 11 × 10 sampling matrix, each row of which
represents a nonuniform distribution of the specific struc-
tural parameters while the other parameters remain
unchanged. In a nutshell, a combination of EE and OAT
method is applied to evaluate the significance of specific
parameters at different locations.
3
.2.3 | Sensitive analysis for single param-
eters along the channel length
In the previous section, totally 24 parameters have been
classified into sensitive and insensitive factors based on
the EE method. Microstructural parameters in the func-
tion layers have been proven to be most sensitive for cell
performance. In the following work, global sensitivities
of single parameters at FL along the channel direction
are tested using the EE method to assess the probability
of structural optimization along the channel length. In
contrast to the previous sampling strategy, single
Due to the facts that sensitivity distribution regularities
of AFL are quite similar with that of CFL, sensitivities of

16
WU ET AL.
anodic parameters are plotted as an example to exhibit the
impacts of position on cell performance in Figure 11A‐11D.
Apparently different sensitivities of single parameters at
different position lie in an interaction of temperature and
species distribution along the channel length. It can be
seen from Figure 11 that the highest sensitivity is obtained
at the location near the center. The counter‐flow operation
has the advantage in power output, and the temperature
3.3 | Nonuniform microstructure
distribution along the channel length
According to the approximately parabolic tendency of
sensitivity distribution along the channel length, we
design an uneven distribution strategy for microscopic
parameters in the function layers where microstructure
parameters are distributed in a parabolic pattern as
shown in Figure 12A. The nonuniform distribution strat-
egy for microscopic parameters can be formulated as
44,45
near the center achieves peak,
which significantly
influences effective conductivity, reversible voltage, and
exchange current density. Theoretically, these properties
are essential factors governing cell performance that is also
greatly affected by microstructural parameters. The com-
bined effects of temperature and microstructural parame-
ters acting together amplify the sensitivity of different
location on voltage fluctuation.
4
SN χ þ ω
2
P ¼ −
x−Lx þ SN 1 þ χ
(39)
(40)
2
L
cell
4
SN ðχ þ ωÞ
2
N ¼
ðx−LxÞ þ SN ð1 − ωÞ
2
L
cell
In conclusion, investigation carried out above has fur-
ther revealed a feasible microstructural optimization
method along the channel direction, namely nonuniform
distribution of microstructure factors. The core mecha-
nism of this approach is placed on the construction of
nonuniform distribution along the channel length in
terms of SA results.
where SN represents nominal parameters as mentioned
before. χ and ω are introduced to specify upper and lower
limits of parameters. P and N express the distribution
strategy for positive and negative effect factors. L repre-
x
sents the extreme parabola coordinates. Three selected
cases with different χ and ω are listed in Table 7. Case 4
(
A)
(C)
(
B)
(D)
FIGURE 11 Sensitivity distribution of single parameters along the channel direction. A, Particle radius; B, porosity; C, relative volume
fraction; D, particle radii ratios [Colour figure can be viewed at wileyonlinelibrary.com]

WU ET AL.
17
Lcell/2 here. Examples of optimized nonuniform distribu-
tion for microscopic factors are illustrated in Figure 12B,
the voltage output of which are compared with that of
uniform distribution. Attention should be paid to the
findings of this investigation that sensitivity analysis is
employed as only basis for nonuniform structural optimi-
zation. We can see that there is an obvious improvement
of cell performance when nonuniform distribution is uti-
lized along the channel length.
The positive or negative effects of χ and ω are depen-
dent on their corresponding parameters. For negative
effect factors, larger value of ω and smaller value of χ
are beneficial to performance improvement. For negative
effect factors, smaller value of ω and larger value of χ are
suitable. However, overall ranges of χ and ω are limited
owing to coupling effects among parameters for forming
an effectively conductive connection. Choosing appropri-
ate value of χ and ω is essential for structural optimiza-
tion along the channel length.
(
A)
Evidently, structural optimization focusing on high
sensitivity position is more efficient than equivalent opti-
mization of all locations. It is confirmed that nonuniform
distribution approach based on sensitivity analysis
rewards more attention in future research.
4
| CONCLUSION
In this study, a computationally efficient quasi‐two
dimensional model is developed, considering detailed
heat and mass transfer as well as electrochemical process
in both directions, which are not fully considered in pre-
vious sensitivity analysis of SOFC. An EE approach pro-
posed by Morris and improved by Campolongo is
employed in this paper to conduct global sensitivity
analysis of totally 24 input parameters. By creatively com-
bining EE and OAT approach together, sensitivity distri-
bution of single parameters along the channel direction
is also evaluated. This study is designed to process an
EE method to conduct global sensitivity analysis and
explain how uncertainties of input factors impact cell per-
formance and the conclusions are listed as follows:
(
B)
FIGURE 12 Nonuniform distribution of microstructural factors
along the channel direction. A, Uneven distribution strategy of
microscopic parameters along the channel length; B, effects of
uniform and nonuniform microstructure distribution on cell
performance [Colour figure can be viewed at wileyonlinelibrary.com]
TABLE 7 Nonuniform distribution of microscopic structure
along the channel length
Particle Radii Relative
Ratios of
AFL/CFL
Volume
Fraction Radius
Particle
Porosity
Case 4
Case 5
Case 6
χ
ω
0
0
0
0
0
0
0
0
1
. Microstructural parameters of anode and cathode
function layers play a decisive role in fuel cell perfor-
mance with seven very sensitive factors. Relative vol-
ume fraction of conductive materials is found to be
most sensitive factors, while particle radius at AFL
exhibits lowest importance on cell performance
among microscopic factors. Ionic particle radius is
much more sensitive than electronic particle radius
by means of influencing effective conductivity rather
than TPB length. Almost all cathode side factors
χ
ω
15%
15%
15%
15%
15%
15%
15%
15%
χ
ω
15%
0
0
15%
0
15%
0
15%
represents uniform distribution. Case 5 refers to a non-
uniform distribution with same value of χ and ω, while
case 6 chooses different values of χ and ω. L equals
x

18
WU ET AL.
contribute more to voltage fluctuation than anode
side factors. Function layer thickness is not that
important as expected; hence, it is suggested that fur-
ther study should concentrate on microstructure
optimization.
. Impacts of current density variation on sensitivity are
also observed in this study. Except that most factors
increase linearly as current density growing, varia-
tion tendencies of particle radius and other empirical
parameters at cathode side gradually become flat
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. Seven microstructural parameters are classified into
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k ; γ ; δAFL; δCFL; δ
are set as rather sensitive
0
;c
O
ELE
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2
sections, while rest of empirical and geometric param-
eters are regarded as insensitive ones. According to the
classifications above, the Monte Carlo approach is
introduced to assess the reliability of simulation.
. Sensitivity analysis of single parameters at different
locations is presented as the basis for structural opti-
mization along the channel length. For counter flow,
highest sensitivity is obtained at the position nearby
the channel center due to the combination effects of
temperature and microstructural parameters. Then,
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This work is supported by the National Key Research and
Development Program of China (2017YFB0601904). M.
Ni thanks the funding support (Project Number: PolyU
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How to cite this article: Wu C, Ni M, Du Q, Jiao
K. Global sensitivity analysis of uncertain
parameters based on 2D modeling of solid oxide
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