Numerical Study of SOECs for Hydrogen Production Using Different Continuity Equations

Numerical investigation of solid oxide electrolysis cells for hydrogen
production applied with different continuity expressions
Ji-Hao Zhang
a,b,c
, Li-Bin Lei
d
, Di Liu
e
, Fu-Yun Zhao
a,b,c,
⇑
, Fanglin Chen
d
, Han-Qing Wang
f
a
Key Laboratory of Hydraulic Machinery Transients (Wuhan University), Ministry of Education, Wuhan, Hubei Province, China
b
Hubei Key Laboratory of Waterjet Theory and New Technology (Wuhan University), Wuhan, Hubei Province, China
c
School of Power and Mechanical Engineering, Wuhan University, Wuhan, Hubei Province, China
d
Department of Mechanical Engineering, University of South Carolina, Columbia, USA
e
College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao, Shandong Province, China
f
School of Civil Engineering, University of South China, Hengyang, Hunan Province, China
article info
Article history:
Received 27 April 2017
Received in revised form 4 July 2017
Accepted 5 July 2017
Available online 7 August 2017
Keywords:
SOECs modeling
Source imbalance
Current densities
Computational fluid dynamics (CFD)
abstract
A dynamic SOEC (Solid Oxide Electrolysis Cell) model is proposed to investigate the transient response
and steady performance of a planar SOEC. Three representative types of continuity equation expressions
are systematically compared for the simulation of source terms introduced by electrochemical reactions.
For the conservative form of continuity equation (Type A), reasonable predictions at both sides of cathode
and anode cannot be achieved, as the diffusion effect is neglected. The non-conservative form of continu-
ity equation (Type B) can obtain reasonable prediction for the cathode side but poor prediction for the
anode side. The Type C of continuity equation, newly proposed by the authors for modeling the SOECs,
is based on the law of volume conservation. It could achieve the volume increment (oxygen produced)
at the anode compartment and good agreements with the analytical ones. It is also found that continuity
equations significantly influence the fluid flow and mass transport, whereas their effects on the electrical
characteristics are negligible when the global current density is not high.
Ó2017 Elsevier Ltd. All rights reserved.
1. Introduction
Hydrogen is considered as a promising alternative fuel for the
future. It can be directly used in fuel cells to efficiently generate
electrical power for stationary or mobile applications. Among
many hydrogen production methods, water electrolysis is a well-
established technique, which is emission-free if it is driven by
renewable or nuclear energy. The electrolysis reaction can be con-
ducted below 100 °C with liquid water (Alkaline Electrolyser, AEL,
and Proton Exchange Membrane Electrolyser, PEMEL) and higher
than 500 °C with steam (Solid Oxide Electrolysis Cells, SOECs) [1].
SOECs have great potentials in energy saving in comparison with
AEL and PEMEL due to reduced electrical energy consumption
and fast electrochemical reaction kinetics at a high operating tem-
perature. Moreover, high operating temperature enables SOECs to
utilize the waste heat from the nuclear power plant or from the
various industries.
Most research works concerning SOECs for hydrogen produc-
tion were experimental in nature, with an emphasis on the devel-
opment of new materials to cope with the high operating
temperature and strong reductive/oxidized environments or the
promotions on the performance, efficiency and durability of the
cell [2–4]. Comparing with experimental approach, mathematical
modeling is an economical and powerful tool to understand the
fundamental processes during electrolysis. After a mathematical
model is validated, the performance of the SOECs could be fully
predicted by adjusting the corresponding governing parameters.
Moreover, the mathematical model could provide very detailed
information, which is generally difficult to be obtained from exper-
imental investigations.
Recent years, mathematical models for SOECs have been devel-
oped by several academic research groups [5–9]. Theoretical model
on the SOECs generally could be divided into three relatively inde-
pendent sections: an electrochemical model, a fluid dynamical
model and a chemical model. The electrochemical model, contain-
ing equilibrium voltage, concentration overpotential, activation
overpotential and ohmic overpotential, is mainly utilized to predict
the current density-voltage characteristics of the SOECs. For the
fluid dynamical model, the CFD (Computational Fluid Dynamics)
http://dx.doi.org/10.1016/j.enconman.2017.07.013
0196-8904/Ó2017 Elsevier Ltd. All rights reserved.
⇑
Corresponding author at: School of Power and Mechanical Engineering, Wuhan
University, Luo-Jia-Shan, 430072 Wuhan, Hubei Province, China.
E-mail addresses: [email protected] (D. Liu), [email protected] (F.-Y. Zhao).
Energy Conversion and Management 149 (2017) 646–659
Contents lists available at ScienceDirect
Energy Conversion and Management
journal homepage: www.elsevier.com/locate/enconman

technique is used for simulating the fluid flow, heat and mass
transports inside the SOECs. For the third section, the chemical
model is used to calculate the chemical reaction rates and reaction
heats, which are used as source terms in the fluid dynamic model.
Primarily, a theoretical model of the SOECs is established in
Fig. 1(a), which consists of a sandwiched three-layer solid structure
(porous cathode, electrolyte and porous anode). The length of this
planar SOEC is assumed infinite in the horizontal direction. Simul-
taneously, the steam concentration at the top free opening and the
oxygen concentration at the bottom free opening are maintained
constant. At the Triple Phase Boundaries close to the cathode-
electrolyte interface, the steam is electrolyzed into hydrogen and
Nomenclature
Cvolume fraction of species, m
3
/m
3
d
a
thickness of anode, l m
d
c
thickness of cathode, l m
d
e
thickness of the electrolyte, l m
Ddiffusion coefficient of species, m
2
/s
Eequilibrium potential, V
Llength of the planar SOEC, mm
Jcurrent density, A m
2
Kpermeability of the electrode
Mmolecular weight of species, kg/mol
Poperating pressure, bar
Qvolume flux, m
3
/s
Runiversal gas constant, 8.3145 J/(mol K)
ssource term
Ttemperature, K
uvelocity, m/s
V volume, m
3
x,yCartesian coordinates, mm
Greek symbols
kbinary parameter
e electrode porosity
c the pre-exponential factors
g
con,a
concentration overpotential at anode, V
g
con,c
concentration overpotential at cathode, V
g
act,a
activation overpotential at anode, V
g
act,c
activation overpotential at cathode, V
g
act,c
ohmic overpotential of the electrolyte, V
l viscosity of the fluid, kg/(m s)
q density, kg/m
3
s time, s
Subscripts
mgas mixture
ispecies i
in (out) into (out of) the channel
a (c) at the anode (cathode) side
Superscripts
’ updated value
Ielectrode-electrolyte interface
0initial condition
H2
Cathode
H
2
O
O2
Pure Fluid
Porous
Solid
Anode
Electrolyte
(b) The geometry of a two-dimensional planar SOEC for applications
H
2O; H2
H
2O; H2
Air (O
2; N2
)
Air (O
2; N2
)
x, u
x
y, u
y
Infinite
H2
Cathode e
n
w
H
2
O
O2
C
H2O = Constant
C
O2 = Constant
Pure Fluid
Porous
Solid
Anode
Electrolyte
(a) A theoretical model for a SOEC in the situation of fully diffusion
s
x, u
x
y, u
y
Fig. 1. A theoretical model for a SOEC in the situation of fully diffusion (a), and the geometry of a two-dimensional planar SOEC for applications (b).
J.-H. Zhang et al. / Energy Conversion and Management 149 (2017) 646–659 647

oxygen ions under the effect of external electrical field. Then the
oxygen ions subsequently pass through the electrolyte and gener-
ate oxygen and electrons at the Triple Phase Boundaries for the
anode side.
It is the oxygen ions instead of the fluid transporting through
the electrolyte. This causes the CFD modeling of fuel cells/elec-
trolyzers to be difficult, as the fluid flow model could not describe
the transport process in the electrolyte. As an alternative solution,
some researchers [10–12] tried to simulate the H
2
-H
2
O and O
2
-N
2
fluid flow field separately to intentionally skip the region of elec-
trolyte. In addition, some investigators [7,13–15] modeled the
electrolyte as an extremely dense porous medium (solid medium),
where fluid subsides and mass transfer does not occur accordingly.
For this porous assumption, one-domain approach could be
adopted to simulate the entire SOECs. As expected, latter one-
domain approach is more flexible such that it could include the
flow and temperature fields. According to the literature reviews
[13,15], following continuity equation of the CFD model was ever
since adopted for the one-domain approach and it could be written
as,
e
@ q
m
@ s
þ r ð q
m
u
!m
Þ¼S
m
ð1Þ
It is referred to Type A, which was widely adopted in the mod-
eling of the solid oxide fuel cells (SOFC) [16–19] and the proton
exchange membrane fuel cell (PEMFC) [20–23]. The source term
s
m
accounts for the mass balance due to the electrochemical reac-
tions at the electrode–electrolyte interface and
e is the porosity of
the material. However, this conservative form of the continuity
equation does not conform to the reality in the region of the
cathode-electrolyte interface, as shown by the control volume in
Fig. 1(a). Particularly, transient term in Eq. (1) would become zero
in a steady state, when the fraction of species is constant,
e
@ q
m
@ s
¼0ð2Þ
The second term on the left-hand side of Eq. (1) could become
zero when the flow velocity subsides in the situation of fully diffu-
sion (Appendix A),
r ð q
m
u
!m
Þ¼0ð3Þ
For the source term, it only depends on the local current den-
sity, as the oxygen ions pass through the electrolyte to the anode.
s
m
¼ J
x
2F D yM
O
2
–0ð4Þ
where J
x
represents the local current density along the planar elec-
trolyte and M
O
2
is the molecular mass of the oxygen ions. When
substituting Eqs. (2)–(4) into Eq. (1), two sides of the equation turn
to be unequal.
Several researchers implemented another non-conservative
form of continuity equation for modeling SOFC [24–27] and PEMFC
[28–31], where the influence of electrochemical reaction on conti-
nuity equation was totally neglected. According to their researches,
the second form of continuity equation, Type B in the present
paper, can be written as
e
@ q
m
@ s
þ r ð q
m
u
!m
Þ¼0ð5Þ
According to Ref. [28], the non-conservative form implied that
no source term appeared on the right-hand side of Eq. (5) provided
more stability to a system of equations and ameliorated overall
convergence behavior. However, this simplification would not be
physically conservative and lead to poor performance especially
for the anode side.
To the authors’ best knowledge, there are no publications
regarding of the choice as well as the influence upon the SOECs
modeling of these two types of continuity equations. In this paper,
an effective continuity equation is proposed, which facilitates the
accurate numerical simulation of the source term caused by oxy-
gen ion transport.
Aforementioned three continuity equations will be compared
through the SOECs modeling results. Thorough and rigorous
descriptions on these continuity equations will be theoretically
demonstrated, accompanying with detailed numerical simulations.
Spatial distributions of axial velocity, species volume fractions,
local currents and local overpotentials will be compared for differ-
ent continuity equations. Simultaneously, global current density
will be set as the operating condition. Following that, the analytical
volumetric fluxes and species volume fractions at the outlet will be
presented to validate the numerical methodology with different
continuity equations. Furthermore, a full 2D (two-dimensional)
dynamic model will be developed to predict the transient response
of SOECs for H
2
O electrolysis. The Brinkman-Forchheimer extended
Darcy model will be incorporated into the dynamic fluid model,
where the Beavers and Joseph problem will be considered as the
benchmark one to validate present CFD model.
2. Mathematical descriptions for the SOECs
A 2D view of a planar SOEC and the computation domain [7] are
illustrated in Fig. 1(b), which includes two flow streams and a
sandwiched three-layer solid structure. Steam, hydrogen gas mix-
ture and air are introduced to the cathode flow channel and the
anode flow channel, respectively. The detailed dimensions and
governing parameters are summarized in Table 1.
2.1. The chemical model for the SOECs
During operation, an electric potential sufficient for water-
splitting is applied to the SOECs. At the cathode side, steam mole-
cules are transported through the porous cathode layer to the Tri-
ple Phase Boundaries of the cathode, where they receive electrons
and decompose to hydrogen gas and oxygen ions. This chemical
reaction can be expressed as follows [4,10],
H
2
O!
2e
H
2
þO
2
ð6Þ
As expected, the produced hydrogen is transported out of the
porous cathode to the cathode fluid channel. Simultaneously, the
oxygen ions pass through the dense electrolyte to the anode where
they produce oxygen and electrons. This chemical reaction can be
written as,
Table 1
Parameters used in the 2D CFD modeling analyses of a planar SOEC [7,34].
Parameter Value
Operating temperature, T(K) 1073
Operating pressure, P(bar) 1.0
Electrode porosity,
e 0.4
Electrode tortuosity, n3.0
Average pore radius, r
p
( l m) 0.5
Cathode-supported SOEC
Anode thickness, d
a
( l m) 100
Electrolyte thickness d
e
( l m) 100
Cathode thickness d
c
( l m) 500
Height of gas flow channel, H(mm) 1.0
Length of the planar SOEC, L(mm) 20
Inlet velocity 0.2
Anode inlet gas molar ratio, C
in
(O
2
/N
2
) 0.21/0.79
Cathode inlet gas molar ratio, C
in
(H
2
O/H
2
) 0.8/0.2
648 J.-H. Zhang et al. / Energy Conversion and Management 149 (2017) 646–659

O
2
!0:5O
2
þ2e ð7Þ
The generated oxygen is transported out of the porous anode
layer to the anode flow channel and the produced electrons are
transported to the cathode side via the external circuit to complete
the cycle. Finally, the net reaction of SOECs can be expressed as,
H
2
O!H
2
þ0:5O
2
ð8Þ
Concerning the CFD modeling, the source terms in the continu-
ity equations can be determined by the local current density.
Mathematically, the mass source terms for hydrogen, steam and
oxygen are respectively written as [17,19],
s
H
2
¼J
x
2F D yM
H
2
;s
H
2
O
¼ J
x
2F D yM
H
2
O
;s
O
2
¼J
x
4F D yM
O
2
ð9Þ
where F represents the Faraday constant (9.6485 10
4
C mol
1
)
and
D yis the width of the control volume in ydirection at the
electrode-electrolyte interface.
2.2. The electrochemical model for the SOECs
The global operation voltage applied to the SOECs could be
written as follows [8,32],
E
global
¼Eþ g
conc;c
þ g
conc;a
þ g
act;c
þ g
act;a
þ g
ohmic
ð10Þ
The equilibrium voltage Eof the SOECs can be expressed by
Nernst equation [33],
E¼1:253 2:4516 10
4
TþRT
2Fln P
0
H2
ðP
0
H2
Þ
1=2
P
0
H2O
"# ð11Þ
where Ris the universal gas constant (8.3145 J mol
1
K
1
); Tis the
absolute temperature (K). The concentration overpotential is caused
by the resistance to the transport of reactant species approaching
the reaction site and the transport of product species leaving the
reaction site. For the SOECs, the concentration over potentials can
be expressed in terms of the gas concentration difference between
the electrode surface and the electrode–electrolyte interface as fol-
lows [33,34],
g
conc;c
¼RT
2Fln P
I
H2
P
0
H2O
P
0
H2
P
I
H2O
!
;
g
conc;a
¼RT
2Fln ðP
I
O2
=P
0
O2
Þ
1=2
hi
ð12Þ
where P
0
is the equilibrium pressure at open circuit and P
I
is the
partial pressures during operation, respectively. Depending on the
incompressible gas assumption, the ratio of the partial pressure
equals to that of the volume fraction.
The activation overpotentials are related to the electrode kinet-
ics at the reaction sites. They represent the overpotential incurred
due to the activation necessary for charge transport. For water
electrolysis, the activation overpotential of an electrode can be
expressed explicitly as [32,35],
g
act;i
¼RT
Fsinh
1
J
x
2J
0;i
!
¼RT
Fln J
x
2J
0;i
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
J
x
2J
0;i
!
2
þ1
v
u
u
t
2
6
43
7
5;i¼a;c:
ð13Þ
In electrochemistry, J
0,i
represents the readiness of an electrode
to proceed with electrochemical reaction. The exchange current
density can be expressed as,
J
0;a
¼ c
a
expðE
act;a
=RTÞ;J
0;c
¼ c
c
expðE
act;c
=RTÞð14Þ
where c
a
and c
c
represent the pre-exponential factors of anode and
cathode, respectively; E
act,a
and E
act,c
are the activation energy
levels at the anode and cathode, respectively. The values of
E
act,a
and E
act,c
are found to be 1.210
5
and 1.010
5
J mol
1
,
respectively. Simultaneously,
c
a
and c
c
are found to be
2.051 10
9
and 1.344 10
10
Am
2
, respectively [32,33].
The electrical connecting plates and electrodes generally have
much higher electrical conductivity than the electrolyte. Therefore,
their effects on the overall ohmic overpotential could be neglected
[36]. According to Ohm’s law, the ohmic overpotential of the SOECs
can be expressed as [32,33],
g
ohmic
¼2:99 10
5
J
x
d
e
exp 10;300
T
 ð15Þ
where d
e
represents the thickness of the electrolyte in micron (mm).
2.3. The fluid dynamical model for the SOECs
According to the definition of mixture density (Appendix A), the
continuity equation can be obtained from the species equation. For
the hydrogen and steam in mixture gas, their species equations in
porous medium can be respectively written as [37],
e
@ q
H2
C
H2
@ s
þ r ð q
H2
u
!
C
H2
Þ¼ r D
eff
H2
r q
H2
C
H2
ðÞ þs
H2
ð16aÞ
e
@ q
H2O
C
H2O
@ s
þ r ð q
H2O
u
!
C
H2O
Þ¼ r D
eff
H2O
r q
H2O
C
H2O
ðÞ þs
H2O
ð16bÞ
Combining the above equations, a precise continuity equation
can be derived as,
e@qm
@sþ
r ð q
mu
!Þ¼
r ðDeff
H2
r ð q
H2CH2ÞÞþ
r Deff
H2O
r ð q
H2OCH2O Þ
þsm
ð17Þ
The source term in the former equation could be calculated by
the following formulation,
s
m
¼s
H2
þs
H2O
ð18Þ
Comparing with Eq. (1), the diffusion term is reserved in the
precise continuity equation, Eq. (17). With the approach of one
domain, the electrolyte is treated as extremely dense porous med-
ium without fluid and mass penetrations. This means there would
be no mass transfer at the interface sof the control volume as
shown in Fig. 1(a). Therefore, the diffusion effect at the interface
ncan not be balanced out with the diffusion effect at the interface
s. In other words, such diffusion effect should not be omitted in the
expression of the former continuity equation stated in Eq. (1).
Furthermore, a more compact form of continuity equation could
be proposed if the incompressible gas assumption is valid. As the
component densities and effective diffusion coefficients are con-
stants, the species equations (Eqs. (16a)–(16b)) for the hydrogen
and steam can be respectively written as,
e
@C
H2
@ s
þ r ðu
!
C
H2
Þ¼D
eff
H2
r ð r C
H2
ÞþV
H2
ð19Þ
e
@C
H2O
@ s
þ r ðu
!
C
H2O
Þ¼D
eff
H2O
r ð r C
H2O
ÞþV
H2O
ð20Þ
where the mass source turns to be the volumetric one,
V
H2
¼s
H2
= q
H2
;V
H2O
¼s
H2O
= q
H2O
ð21Þ
In the binary fluid mixture, the volume fractions of hydrogen
and steam volume will satisfy the following relations,
C
H2
þC
H2O
¼1; r C
H2
¼ r ð1C
H2O
Þ¼ r C
H2O
ð22Þ
Hence, following equations could be obtained,
J.-H. Zhang et al. / Energy Conversion and Management 149 (2017) 646–659 649

e
@C
H2
@ s
þ@C
H2O
@ s
¼0ð23aÞ
r ðu
!
C
H2
Þþ r ðu
!
C
H2O
Þ¼ r u
!
ð23bÞ
D
eff
H2
r ð r C
H2
ÞþD
eff
H2O
r ð r C
H2O
Þ
¼ðD
eff
H2O
D
eff
H2
Þ r ð r C
H2O
Þð23cÞ
The volume source for the mixture gas is defined as,
V
m
¼V
H2
þV
H2O
ð24Þ
Further summarizing above two equations, the form of the con-
tinuity equation can be written as,
r u
!
¼D
eff
H2O
D
eff
H2
r r C
H2O
ðÞ
þV
m
ð25Þ
According to the assumption of incompressible gases, the bin-
ary mixture diffusion coefficients are identical for different phases
(details could be found in Appendix B). Therefore, the diffusion
effect can be subtly avoided in this new form of continuity equa-
tion. Consequently, the continuity equation, hereby defined as
Type C, could be written as
r u
!
¼V
m
ð26Þ
It is indicated that the volume still satisfies the conservation
law although the oxygen ions are not included in the CFD model.
Some advantages could be listed for above compact form of the
continuity equation,
The unsteady term can be avoided in the continuity equation
even for the instantaneous electrolysis process.
The mass source term has been transformed into volume source
term. It is useful and convenient to present the volumetric
source terms explicitly.
For instance, at the cathode-electrolyte interface, H
2
O molecule
decomposes to the same volume of hydrogen gas, and the continu-
ity equation can be further written as [38],
r u
!
¼0ð27Þ
At the anode-electrolyte interface, only oxygen is produced and
the continuity equation could then be written as,
r u
!
¼V
O
2
ð28Þ
Through introducing the binary parameter k, the conservation
equations for the fluid and porous layers can be unified into a set
of equation from the continuity of the physical quantities. Accord-
ing to the Brinkman-Forchheimer extended Darcy model [26,39–
40], the two dimensional governing systems of transport equations
are as follows,
Continuity equation,
r u
!
¼V
m
ð29Þ
Momentum balance equation,
@ q
m
u
!
@tþk e
þð1kÞ r ð q
m
u
!
u
!
Þ
¼
e kþð1kÞ½ r pþ r l
m
r u
!
½ks
u
ð30Þ
Species equation,
e kþð1kÞ½
@C
i
@tþ
r ðu
!
C
i
Þ¼ r ðD
eff
i
r C
i
Þþs
c;i
ð31Þ
The binary parameter could be stated as,
k¼0in the fluid layer
1in the porous layer
Source term s
u
could be obtained by the following relation,
s
u
¼
el
m
Ku
!
þ1:75
q
m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
150
e K
pju
!
ju
!
ð32Þ
3. Finite volume methodology and numerical procedure
In terms of CFD applied in the present investigation, finite vol-
ume methodology will be employed to integrate the corresponding
governing equations on the staggered grid system.
3.1. Boundary conditions for numerical modeling
At the bottom and the top boundaries of the computation
domain, non-slip velocity boundary conditions and concentration
adiabatic conditions are imposed. At x= 0, a constant flow velocity
u=u
in
and total species flux C=C
in
are maintained, where convec-
tion and diffusion effects are specified at the inlet. Mathematically,
the transient species volume fraction C
i
can be obtained by the fol-
lowing relation,
u
in
C
in
¼u
in
C
i
D
eff
i
@C
i
@xð33Þ
In some circumstances, u
in
at the inlet for the porous electrodes
could turn to zero if the low Darcy permeability are assumed. The
detailed model governing parameters can be found in Table 1.At
the outlet of the computation zone, free vent boundary conditions
are imposed as follows [41],
p¼p
atm
þ0:5u
2
x
;@u
y
=@x¼0;@u
x
=@xþ@u
y
=@y¼0;
ð@C=@xÞ
out
¼0ð34Þ
3.2. Numerical methodology and solution procedure
The transient solution procedure for the SOECs is demonstrated
in Fig. 2, where the CFD model, the electrochemical model and the
chemical model are taken into account. In practice, instantaneous
SOECs hydrogen production is sensitive to the initial conditions.
In this work, the inlet gases are supplied in advance and the fluid
flow has been fully developed before a constant global current den-
sity (J
global
) is applied for electrolysis.
J
global
¼Z
L
J
x
dx =Lð35Þ
At first, CFD model is called to obtain the species volume frac-
tions at the electrode-electrolyte interface. In the process of CFD
model, the governing equations described in the aforementioned
sections are discretized into algebraic equations by the utilization
of Finite Volume Method (FVM). In the course of discretization, the
third-order deferred correction QUICK scheme and a second order
central difference scheme are respectively implemented for the
convection and diffusion terms. And the fully implicit scheme is
applied to the unsteady term. The pressure-velocity coupling is
treated by the SIMPLE algorithm with the incompressible form of
pressure-correction equation. After the velocity field is obtained,
the discretized species equations are solved. It should be men-
tioned that the source terms in the governing equations will
depend on the local current density (J
x
), which has been demon-
strated in Section 2.1.
Afterwards, the electrochemical model developed in the Sec-
tion 2.2 is utilized to determine the local current density for a given
global current density. Definitely, the concentration overpotentials
650 J.-H. Zhang et al. / Energy Conversion and Management 149 (2017) 646–659