Energy Conversion and Management 207 (2020) 112502
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Energy Conversion and Management
journal homepage: www.elsevier.com/locate/enconman
Investigation of performance heterogeneity of PEMFC stack based on 1+1D
and flow distribution models
T
Zirong Yang, Kui Jiao, Zhi Liu, Yan Yin, Qing Du
⁎
State Key Laboratory of Engines, Tianjin University, 135 Yaguan Road, Tianjin 300350, China
ARTICLE INFO
ABSTRACT
Keywords:
PEMFC stack
Flow distribution
Performance heterogeneity
Voltage overestimation
Manifold geometric parameter
Flow distributions among proton exchange membrane fuel cell (PEMFC) stacks remain an important issue owing
to the great influences on stack performances. To simultaneously consider flow distributions as well as reactions,
phase changes, and transport processes inside every fuel cell, a comprehensive stack model is developed based
on the integration of a 1+1 dimensional multiphase stack sub-model and a flow distribution sub-model.
Frictional and local pressure drops due to diverging, converging, and bending configurations are calculated.
After rigorous model validation, differences between the uniform flow assumption and the authentically nonuniform distribution are quantitatively investigated, including the distribution of voltage, mass flow rate,
temperature, reactant concentration, and other parameters. Results show that the uniform assumption not only
overestimates the stack output performance but also underestimates the cell voltage variations. Besides, the
uniform assumption may lead to higher predictions of the overall stack temperature and lower predictions of
temperature variations among different fuel cells. Even though the total amount of air seems abundant, it is still
possible for some fuel cells to suffer from the local reactant starvation. Therefore, a higher cathode stoichiometry
is preferable since it increases the inlet mass flow rate for middle cells. Increasing the inlet pressure contributes
negligibly to the uniformity of reactant distribution, but it improves the stack performance. A larger manifold
cross-sectional area leads to more uniform reactant distributions among the stack and less local current density
variations inside single cells.
1. Introduction
During the past decades, there is growing concerns about energy
and environment around the world. In addition to the limited storage,
fossil fuels may cause environmental problems in actual applications
such as automobile vehicles. Therefore, researchers are paying more
and more attention to the environmentally friendly vehicles nowadays
(e.g. hybrid electric vehicles, battery electric vehicles, and fuel cell
vehicles) [1–4]. Among these vehicles, proton exchange membrane fuel
cell (PEMFC) has the advantages of long cruising distance, low operating temperature, and zero emissions [5–7]. Several motor corporations have already issued their fuel cell vehicles, such as Toyota Mirai
[8], Honda Clarity [9], and Hyundai Nexo [10]. To enhance the output
power, numerous PEMFCs are typically stacked together. For Mirai, 370
single fuel cells are stacked in single-line, which achieves the maximum
output at 114 kW [8]. During the scaling up process, the uneven flow
distribution remains an important issue because it not only affects the
overall stack output power, but also significantly influences the stack
reliability and durability [11–13]. Therefore, it is necessary to
⁎
comprehensively investigate the mechanisms and influences of flow
distributions within the whole stack.
After being supplied into the stack inlet manifold, reactants are
subsequently distributed into each individual fuel cell where the hydrogen–oxygen reaction occurs [14]. Due to the pressure losses along
with reactants movement, the mass flow rates into each fuel cell are
typically different, leading to the performance heterogeneity such as
output voltage, temperature, and other parameters. To investigate the
uneven flow distribution phenomenon and its influences, many experimental and numerical studies have been conducted. Kim and Hong
[15] evaluated the effects of operating conditions on the performance
of a PEMFC stack with 10 cm2 active area and 10 single cells. The
output voltage of each individual fuel cell was measured. It was found
that the voltage variations among different cells could reach 5% under
the tested conditions. Yi et al. [16] used the same amorphous carboncoated 304 stainless steel as bipolar plates in a single cell and a 100 Wclass short stack. The results showed that the peak power density was
1150.6 mW cm−2 for the single cell and 1040.1 mW cm−2 for the stack,
indicating that the maximum power density deteriorated after scaling
Corresponding author.
E-mail address: duqing@tju.edu.cn (Q. Du).
https://doi.org/10.1016/j.enconman.2020.112502
Received 4 October 2019; Received in revised form 11 December 2019; Accepted 12 January 2020
Available online 08 February 2020
0196-8904/ © 2020 Published by Elsevier Ltd.
Energy Conversion and Management 207 (2020) 112502
Z. Yang, et al.
Nomenclature
a
A
ASR
c
Cp
D
Dh
EW
F
f
G
h
I
l
k
K
m
M
p
p
R
Re
s
S
ST
t
t
T
u
V
X
Y
κ
water activity
effective geometric area (m2)
area specific resistance (Ω m2)
mole concentration of gas species (mol m−3)
specific heat capacity (J kg−1 K−1)
diffusivity (m2 s−1)
hydraulic diameter (m)
equivalent weight of membrane (kg kmol−1)
Faraday’s constant (C mol−1)
function
Gibbs free energy (J mol−1)
heat transfer coefficient (W m−2 K−1)
current density (A m−2)
length (m)
thermal conductivity (W m−1 K−1)
permeability (m2)
mass flow rate (kg s−1)
molecular weight (g mol−1)
pressure (Pa)
pressure drop (Pa)
universal gas constant (J mol−1 K−1)
Reynolds number
volume fraction
source terms (kmol m−3 s−1, kg m−3 s−1, W m−3), entropy (J mol−1 K−1)
stoichiometry
time (s)
time step size (s)
temperature (K)
velocity (m s−1)
voltage (V)
mole fraction of gas species
mass fraction of gas species
Subscripts and superscripts
a
act
atm
bend
bot
BP
c
CH
cir
CL
con
conc
cool
div
eff
EOD
eq
FC
fr
g
GDL
H2
l
lq
MEM
mw
MPL
N2
O2
ohmic
out
pc
per
react
rect
ref
sat
surr
vp
m-l
m-v
v-l
Greek letters
ε
ζ
λ
ξ
μ
ρ
ω
δ
θ
conductivity (S m−1)
friction factor
porosity
water transfer rate (s−1), pressure drop coefficient
membrane water content
stoichiometry ratio
dynamic viscosity (kg m−1 s−1)
density (kg m−3)
volume fraction of ionomer
thickness (m)
contact angle (°)
up. Pei et al. [17] measured the temperature distribution characteristic
in a 200 cm2 stack with 46 cells by embedding micro-thermocouples
into the cathode plate of four marked cells (cell 1, 11, 23, and 46). Nine
thermocouples were placed at different positions for the aforementioned cells. It was found that the highest temperature difference among
the marked cells could reach approximately 11 °C. Besides, the temperature variation inside the single cell could reach around 8 °C. To
simultaneously measure the voltage and temperature distributions,
Devrim et al. [18] conducted a research with an air-cooled 500 W stack
consisting of 24 fuel cells. The highest temperature difference among
the individual cells was around 9 °C when the stack was operated at
65 °C.
It should be noted that the performance heterogeneity mainly results from the non-uniform reactant distributions within the stack.
However, the mass flow rate of each fuel cell is hard to experimentally
measure since the mass flow sensor could not be placed in the limited
anode
activation
atmosphere
bending
bottom
bipolar plate
cathode, capillary
flow channel
circular
catalyst layer
converging
concentration
coolant
diverging
effective
electro-osmotic drag
equilibrium
fuel cell
frictional
gas phase
gas diffusion layer
hydrogen
liquid phase
liquid water
membrane
membrane water
micro-porous layer
nitrogen
oxygen
ohmic
output
phase change
permeation
reaction
rectangular
reference state
saturation
surroundings
water vapor
membrane water to liquid water
membrane water to water vapor
water vapor to liquid water
space of stack manifolds or fuel cell channel inlet regions. Therefore,
numerical models have been developed to quantitatively depict the
reactant and coolant distributions. Baschuk and Li [19] investigated the
distributions of pressure and reactants based on a hydraulic network
approach. The frictional pressure drops in stack manifolds and fuel cell
channels were calculated. Mass conservation analysis for hydrogen, air,
and water vapor was conducted for the zero-dimensional steady-state
stack model. It was found that the power density of a stack is lower than
that of a single cell, which was resulted from unequal reactant distributions. However, local pressure drops owing to the bending, converging, and diverging configurations were not considered. To supplement the model, the local pressure drops and heat transport processes
were added in the work of Park and Li [20]. The number of heat
transfer unit (NTU) method was adopted to calculate the temperature of
every fuel cell. To investigate the thermal and electrical behaviors of a
water-cooled PEMFC stack, Cozzolino et al. [21] developed a one2
Energy Conversion and Management 207 (2020) 112502
Z. Yang, et al.
dimensional electrochemical model, which was integrated with thermal
analysis based on mass and energy balance. The experimental tests
were also conducted to validate and modify the stack model. However,
the pressure drops and non-uniform reactant distribution were not
considered in the study. Similar researches were improved by Salva
et al. [22], which calculated the frictional pressure drops through the
mass and pressure balance. Amirfazli et al. [23,24] established an
analytical model to study the temperature distributions and coolant
channel designs with the U-type and Z-type stacks consisting of 65 fuel
cells. The concept of index of uniform temperature (IUT) was used to
evaluate the degree of temperature uniformity. However, the mass
transport processes and electrochemical reactions inside fuel cells were
not involved. There were several three-dimensional (3D) stack models
developed by commercial computational fluid dynamics (CFD) software
[25–28]. Macedo-Valencia et al. [26] established a single-phase 3D
stack model with five 31.2 cm2 fuel cells. The reactants concentration,
local current density, and temperature distributions among the stack
were presented. Luo et al. [27] investigated the cold start performance
of a 3-cell stack with the transient multiphase model. However, the
inlet mass flow rates of all fuel cells were assumed identical, and the
stack manifolds were not taken into consideration. To improve the
modeling studies, Lim et al. [28] developed a steady-state single-phase
stack model, and the manifold structures were further included in the
computational domains. The performances of single-cell, dual-cell,
quad-cell, and hexa-cell were comparatively studied. Despite aforementioned studies, the researches which simultaneously consider nonuniform flow distributions, electrochemical reactions, and phase
changes are rarely presented. Note that the phase changes among water
vapor, membrane water, and liquid water have great influences on the
components of mixture gases, which affect the pressure drops inside
single cells and correspondingly influence the reactant distributions
within the whole stack. Besides, comparison between the uniform flow
assumption and the non-uniform flow distribution has not been quantitatively investigated, which remains an important issue for stack
modeling studies.
In the study, a flow distribution sub-model is integrated with a
1+1D multiphase stack sub-model to develop the comprehensive stack
model, which is capable of simultaneously demonstrating non-uniform
reactant and coolant distributions among the stack as well as electrochemical reactions, phase changes, and transport processes inside every
fuel cell. Detailed correlations for frictional and local pressure drop
coefficients are presented. The established stack model is rigorously
validated against experimental data. Differences between the uniform
flow assumption and the authentically non-uniform distribution are
quantitatively investigated, including the distribution of cell voltage,
mass flow rate, temperature, reactant concentration, and other parameters, which are rarely presented in literature but of great significance
for stack researches. To alleviate non-uniform flow distributions and
thereby enhance stack performances, the effects of stack operating
conditions (e.g. stoichiometry, pressure) and manifold geometric
parameters are further studied, which gives valuable suggestions for the
stack performance improvement.
2. Model development
The schematic diagram of a proton exchange membrane fuel cell
(PEMFC) stack with the Z-type configuration is shown in Fig. 1. The
stack configuration is a U-type if the flow direction in the outlet
manifold is designed contrary to the inlet manifold. To investigate the
uneven flow distributions among different fuel cells, a flow distribution
sub-model is developed based on the hydraulic network approach. As
regards heat and mass transfer processes inside every fuel cell, a 1+1
dimensional multiphase stack sub-model is developed. The two submodels are subsequently integrated to develop the comprehensive stack
model. Detailed methodologies about the two sub-models are explained
in Subsection 2.1 and 2.2. The calculation procedure, initial and
boundary conditions are explained in Subsection 2.3.
2.1. Flow distribution sub-model
During the motion of fluid, two types of pressure losses should be
considered, including the frictional pressure loss and the local pressure
loss. The frictional losses result from the friction between fluids and
walls. The local pressure losses are usually caused by the converging,
diverging and bending of the flow. It should be noted that the two types
of pressure losses are arbitrarily divided for the sake of calculation
while the total pressure losses are inseparable [29].
ptotal = pfr + plocal
1
pfr = 2 fr u2
1
p local = 2 local u2
(1)
where ptotal (Pa) is the total pressure loss, pfr (Pa) is the frictional
pressure loss, plocal (Pa) is the local pressure loss, fr , local are the
pressure loss coefficients, (kg m−3) is the density, and u (m s−1) is the
velocity.
To determine the total pressure losses, it is necessary to obtain the
pressure loss coefficients which are related to flow regimes, mass flow
rates, and structural designs. The frictional loss coefficient is defined as
[29]:
fr =
l
Dh
(2)
where is the friction factor which depends on the Reynolds number
and the wall roughness, l (m) is the length, and D h (m) is the hydraulic
diameter.
For circular tube with smooth walls under stabilized flow conditions, the friction factor is calculated as [29]:
Fig. 1. Schematic diagram of a Z-type PEMFC stack.
3
Energy Conversion and Management 207 (2020) 112502
Z. Yang, et al.
uD h
µ
Re =
f1,div2 = p00 + p10 x + p01y + p20 x 2 + p11xy + p02 y 2 + p30 x 3
+ p21x 2y + p12 xy 2 + p03 y 3 + p40 x 4
64
Re
(Re < 2000)
+ p31x 3y + p22 x 2y 2 + p13 xy 3 + p04 y 4 + p50 x 5 + p41x 4y + p32 x 3y 2
(2000
Re < 4000)
0.3164
Re 0.25
(4000
Re < 105)
(1.8 lg Re
1.64) 2
(Re > 4000)
f1 (Re )
cir =
x = A2 /A1 , y = m2 /m1
(3)
+ p23 x 2y 3 + p14 xy 4 + p05 y 5
f1,div3 = 0.969(1
g =
RT
i
Yi
Mi
1
=
RT
MiX i
µg =
i
j
ij =
1.0
= 0.9(1
0.55
(6)
y)2
)
y2
(11)
(x < 0.35)
y ) (x > 0.35, y < 0.4)
(x > 0.35, y > 0.4)
(12)
x = A2 A1 , z = a b
f1bend = p00 + p10 z + p01x + p20 z 2 + p11zx + p02 x 2 + p30 z 3 + p21z 2x + p12 zx 2 + p03 x 3
(2000
Re < 4000)
(13)
(7)
Table 1
Schematic diagram and calculation of local pressure losses [29].
The sum of squares due to error (SSE) is 7.94 × 10−-8, and the
coefficient of determination (R-square) is 0.9994, indicating good fitting accuracy.
If the cross-section of stack manifold differs from circular tubes (e.g.
rectangular, elliptical, trapezoid), the structural calibration is necessary
for the corresponding friction factor. For rectangular conduit, the actual
friction factor is calculated as [29]:
Type
Correlations
p2div =
1 div
2 1,2
u12 ,
1 div
2 1,3
u12
A2
A1
m2
m1
div
div
1,3 = f 1,3
m3
m1
1 con
2 1,2
u12 ,
p3con =
A2
A1
m2
m1
1 bend
2 1
u2
a
b
A2
A1
div
div
1,2 = f1,2
p3div =
( , ),
( )
(8)
cir
rect
is the calibration coefficient which typically depends on the
where f cali
Reynolds number and cross-sectional designs. The calibration coefficient is also obtained by curve fitting methods based on the presented
figures and tables in the reference [29].
( ) + 1.96( )
0.0781( ) + 0.203( )
b
rect
f cali
=
2(1
1.49 × 10 12Re 3 + 1.036 × 10 8Re 2
1.561 × 10 5Re + 0.0331
rect
rect = f cali
y 2
x
For the first fuel cell near the inlet manifold or the last fuel cell near
the outlet manifold, the local pressure drop owing to the bending
configuration should also be taken into consideration. For rectangular
section elbows with 90° sharp corners, the friction factor f1bend is calculated as
The function f1 in Eq. (4) is obtained by polynomial curve fitting
methods based on the original reference [29].
f1 (Re ) =
(1 + ( )
f1,con
3 = 1.55y
[1 + (µi µ j )0.5 (Mj Mi )0.25] 2
[8(1 + Mi Mj)] 0.5
−1
x = A2 A1 , y = m2 m1
where pg (Pa) is the gas pressure, R (J mol−1 K−1) is the universal gas
constant, T (K) is the temperature, Yi is the mass fraction of gas species
i, Xi is the mole fraction, and Mi (g mol−1) is the molecular weight. The
mixture dynamic viscosity is calculated as [37]:
Xi µi
,
Xj ij
y ) + 0.728
0.282(1
where A (m ) is the cross-sectional area, and m (kg s ) is the mass flow
rate. The multinomial coefficients of curving fitting results are listed in
Table 2. The R-square of the two functions are 0.9988 and 0.9683.
For the converging configuration, the corresponding functions are
calculated as [29]:
(5)
i
y) 2
2
f1,con
2 =
pg
0.709(1
(10)
(4)
where µ (kg m−1 s−1) is the dynamic viscosity.
It should be noted that different types of gas are involved in stack
manifolds and fuel cell channels, including hydrogen, oxygen, nitrogen,
and water vapor. Therefore, the density and dynamic viscosity refer to
the corresponding mixed gases. Based on the ideal gas law, the mixture
density is calculated as
pg
y) 3
0.707 a0
3
b0
a0
3
b0
a0
0
b0
a0
2
p2con =
con
con
1,2 = f1,2
( )
0.224( ) + 1.10 (Re > 2000)
( , ),
1 con
2 1,3
u12
con
con
1,3 = f1,3
m3
m1
( )
b
1.87 a0 + 1.50 (Re < 2000)
0
2
b0
a0
(9)
where b0 (m) is the cross-sectional width, and a0 (m) is the cross-sectional length. The R-square of the two functions are 0.9998 and 0.9983,
respectively.
The local pressure loss coefficients due to diverging, converging,
and bending configurations are related to the cross-sectional area and
the mass flow rate. The schematic diagram and calculation of local
pressure losses are summarized in Table 1.
The local pressure loss coefficients are determined based on the
original tables and figures [29]. It should be noted that the cross-sectional area of the straight passage is identical, namely, A1 is equal to A3
as shown in Table 1. For the diverging configuration, the functions f1,div2 ,
are calculated as
p1bend =
bend
= f1bend
1
( , ) (a, b refers to the
length and width of the inlet cross-section
A1 )
4
Energy Conversion and Management 207 (2020) 112502
Z. Yang, et al.
Table 2
Multinomial coefficients in the diverging configuration.
Parameter
Value
Parameter
Value
Parameter
Value
Parameter
Value
p00
p10
p01
p20
p11
p02
8.747
−139.7
27.15
825.5
−172.1
5.251
p30
p21
p12
p03
−2132
388.8
77.77
−41.55
p40
p31
p22
p13
p04
2485
−475.6
17.53
−119.7
72.27
p50
p41
p32
p23
p14
p05
−1046
225.1
−58.86
71.95
8.671
−25
The multinomial coefficients in the bending configuration are listed
in Table 3. The SSE is 0.02034 and the R-square is 0.993. It should be
noted that the fitting results apply to the conditions with smooth walls
and the Reynolds number larger than 2 × 105. However, appropriate
coefficients are not available if the Reynolds number is out of the above
range.
In stack manifolds, the gas velocity can be obtained by the corresponding mass flow rate, density, and cross-sectional area.
liquid pressure because the distinct hydrophobicity may cause liquid
water jump phenomenon at the interface of neighboring layers [36].
The water saturation in porous layers are subsequently calculated by
Leverett-J function once the capillary pressure is obtained. Detailed
transport properties can be found in our previous studies [35].
m
u=
A
where s lq is the liquid water saturation, pl (Pa) is the liquid pressure,
and Slq (kg m−3 s−1) is the liquid water source term. The hydraulic
permeation caused by pressure differences between the anode CL and
the cathode CL is also considered [37].
(
(14)
EW
t
=
MEM
EW
·(
1.5D
mw
mw ) + Smw
Sreact Sm v Sm l + SEOD (in cathode CL)
Sm v Sm l SEOD
(in anode CL)
pl + Slq
µlq
pc = pg
(17)
Sper (in cathode CL)
Sv l
(in anode CL)
(18)
(in MPL, GDL)
pl
(19)
( ) [1.42(1 s ) 2.12(1 s ) + 1.26(1 s ) ]( < 90 )
cos( ) ( ) [1.42s
2.12s + 1.26s ]
( > 90 )
0.5
cos( )
pc =
lq
K0
lq
0.5
lq
K0
lq
2
lq
2
lq
3
3
0
0
(20)
For gas species, the ideal gas law is adopted. The governing equations of gas species are solved in porous layers and flow channels.
t
t
( (1
s lq ) g Yi ) =
( g Yi ) +
·( g Dieff Yi ) + Si
·( g ug Yi ) =
·( g Dieff
(in CL, MPL, GDL)
Yi ) + Si
(in CH)
(21)
where Yi is the mass fraction of gas species i (H2, O2, N2, vp), g (kg
m−3) is the mixture gas density, and Si (kg m−3 s−1) is the source term.
The energy equation is calculated in the whole fuel cell.
t
(( cp )eff
fl,sl T ) =
ST =
eff
·(k fl,sl
T ) + ST
(22)
I 2ASRCL
+ IVact,a + Spc
3 CL
(in CLa)
I 2ASRCL
3 CL
(in CLc)
T S
I + IVact,c + Spc
2F
I 2ASR
I 2ASR
(in MEM, BP)
+ Spc
(in MPL, GDL, CH)
(23)
−3
where T (K) is the temperature, and ST (W m ) is the heat source term,
including activational heat, reversible heat, ohmic heat, and latent
heat.
(15)
where mw is the membrane water content, Dmw (m2 s−1) is the
membrane water diffusivity, and Smw (kmol m−3 s−1) is the source
term, which consists of electrochemical reactions, electro-osmotic drag
effect, and phase changes (listed in Table 5).
Smw =
lq
Sm l MH2O + Sv l
A typical PEMFC stack consists of many single cells to increase the
output voltage and power. A single PEMFC is usually composed of 11
layers, including proton exchange membrane, catalyst layer (CL),
micro-porous layer (MPL), gas diffusion layer (GDL), flow channel
(CH), and bipolar plate (BP). The coolant is designed to flow through
cooling channels inside every bipolar plate to improve the temperature
distribution uniformity [31]. To depict the parameter variations (e.g.
reactant concentration, temperature, local current density, and velocity) from the channel inlet region towards the outlet region, the fuel
cell is divided into several nodes in the flow direction. The schematic
diagram and detailed explanation about the stack sub-model can be
found in our previous studies [32,33]. Table 4 lists the structural
properties and operating conditions.
The governing equations include membrane water, liquid pressure,
gas species, and temperature. The generated water is assumed to be
membrane water since it is stated that the liquid water and vapor
production assumptions should be cautiously adopted [34].
mw )
Klq
·
Slq = Sm l MH2O + S v l + Sper
2.2. 1+1 dimensional multiphase stack sub-model
(
=
t
The pressure drops in fuel cell channels are calculated in the similar
way as shown in Eq. (1). It should be noted that the pressure drops in
fuel cell channels are also influenced by the possible presence of liquid
water [30]. However, specific correlations between pressure drops and
the gas–water two-phase flow seem unavailable in literature. Therefore,
only single-phase pressure drops are calculated in the study. Note that
the gas velocity varies in the flow direction owing to reactants consumption and water vapor generation. Besides, the mixture gas density
and viscosity are also changing. Therefore, the density, viscosity, and
velocity for the pressure drop calculation are treated as the average
value of the inlet status and the outlet status.
MEM
lq s lq )
Table 3
Multinomial coefficients in the bending configuration.
(16)
In porous layers, the transport process is assumed to be dominated
by diffusion, and the previous studies also use the assumption [27,35].
The liquid water volume fraction is solved based on the continuity of
5
Parameter
Value
Parameter
Value
Parameter
Value
p00
p10
p01
0.8257
−0.1395
−0.1586
p20
p11
p02
0.07811
−0.01717
0.1637
p30
p21
p12
p03
−0.04289
−0.004508
0.01079
−0.06378
Energy Conversion and Management 207 (2020) 112502
Z. Yang, et al.
Table 4
Stack structural properties and operating conditions.
Parameter
Value
Number of fuel cells
Gas manifold cross-sectional length, width
Coolant manifold cross-sectional length,
width
Effective area of single cells
Cooling area of single cells
Channel length, width, depth, rib width
Thickness of MEM, CL, MPL, GDL, BP
Density of MEM, CL, MPL, GDL, BP
5
10, 6 mm
10, 5 mm
Specific heat capacity of MEM, CL, MPL,
GDL, BP
Thermal conductivity of MEM, CL, MPL,
GDL, BP
Electric conductivity of CL, MPL, GDL, BP
Ionomer volume fraction in CL
Contact angle of CL, MPL, GDL
Porosity of CL, MPL, GDL
Expected stack operating temperature
Stack manifold inlet pressure
Stoichiometry ratio
Inlet relative humidity
Inlet gas temperature and surrounding
temperature
Inlet mass flow rate and temperature of
coolant
Heat transfer coefficient between endplate
and environment
Heat transfer coefficient between fuel cell
and coolant
120 cm2
60 cm2
100, 1.0, 1.0, 1.0 mm
0.025, 0.01, 0.03, 0.2, 2 mm
1980, 1000, 1000, 1000,
1000 kg m−3
833, 3300, 2000, 568,
1580 J kg−1 K−1
0.95, 1.0, 1.0, 1.0, 20 W m−1 K−1
5000, 5000, 5000, 20,000 S m−1
0.4
100°, 120°, 120°
0.3, 0.4, 0.6
60 °C
pa = 1.5 atm, pc = 1.5 atm
ST a = 1.5, ST c = 2.0
RHa = 1, RHc = 1
60 °C, 25 °C
0.2 kg s−1, 25 °C
100 W m−2 K−1
200 W m−2 K−1
The output voltage of a single fuel cell is calculated based on Tafel
equations for simplification [37,38]. The stack output voltage can be
obtained by adding all single cell voltages.
Vout = VNernst
Vact
Vohmic
G
S
VNernst =
+
(T
2F
2F
Fig. 2. Calculation procedure of the integrated steady-state PEMFC stack
model.
(24)
Vconc
The inlet mass flow rate of individual fuel cells is initially guessed
which satisfies the mass conservation at each diverging, converging,
and bending point. The pressure drops in stack manifolds and fuel cell
channels are calculated, including frictional pressure drops and local
pressure drops as shown in Fig. 1. It should be noted that the total
pressure drop in each closed loop should be equal to zero if the total
mass flow rate is correctly distributed into single fuel cells. Hardy cross
method is adopted to update the mass flow rate of gases and coolant
until the convergence standard is reached [20].
c H2,CLa
c O2,CLc
RT
1
Tref ) +
ln
+ ln
2F
c H2,ref
2
c O2,ref
(25)
where Vact , Vohmic , Vconc (V) are the activation loss, ohmic loss, and
concentration loss. Detailed calculation equations can be found in our
previous studies [35].
2.3. Calculation procedures
Max { ploop,1 , ..., ploop,N 1 } < 10 3 Pa
As stated previously, the two sub-models are strongly related. The
calculation procedure of the integrated stack model is shown in Fig. 2.
The total mass flow rate of reactant is obtained based on the required
stoichiometry and operating current density.
mHtotal
=
2
Nstack IAcell STa MH2
N
IA ST M
total
, mair
= stack cell c air
2F × 1000
4F × 0.21 × 1000
(27)
After the flow distribution sub-model is solved, the inlet conditions
for each fuel cell are set (e.g. mass flow rate and pressure). The 1+1
dimensional stack sub-model is subsequently calculated. For single
cells, the explicit formulation calculation method is adopted. More
detailed information is presented in our previous study [14]. In the
channel direction inside single cells, the voltage is assumed to be
(26)
Table 5
Phase change functions.
Phase change
Correlation
Membrane water and water vapor
Sm v =
Membrane water and liquid water
Sm l =
Water vapor and liquid water
Sv l =
Unit
MEM
(1
EW
MEM
(1
m l EW
v l
(1
l v s lq
Phase change rates [5]
m v,
s lq )( m
eq )
kmol m−3 s−1
s lq )( m
eq )
kmol m−3 s−1
m v
s lq )
(pg Xvp
(pg Xvp
psat )
RT
psat )
RT
m l = 1.0 , l v ,
6
v l = 1000
(pg Xvp > psat )
kg m−3 s−1
(pg Xvp < psat )
s−1
Energy Conversion and Management 207 (2020) 112502
Z. Yang, et al.
identical for all nodes since bipolar plate has good electrical conductivity [14,33]. Therefore, the local current density distribution
along the channel direction is obtained through the following nonlinear
equations.
data, including the stack polarization curve and the cell voltage distribution [16]. In the experiment, the 100 W-class short stack consists of
three fuel cells with 120 cm2 active area each. 30% humidified hydrogen and air are supplied at 80 °C. The anode stoichiometry and
cathode stoichiometry are 1.2, 2.0, respectively. The stack temperature
is maintained at 60 °C through the water cooling. The overall polarization curve is shown in Fig. 3(c), and it shows good agreement.
Meanwhile, the cell voltage variations at 2.1 V, 1.8 V, and 1.5 V are also
correspondingly compared as shown in Fig. 3(d). The voltage distribution matches reasonably, and the sum of squares due to error is
0.007. Besides, rigorous validation of the 1+1 dimensional stack submodel has been presented in our previous studies [32,35].
n
(28)
where n is the number of nodes in the channel direction.
For different fuel cells among the stack, the total current is identical
since the fuel cells are stacked in a single line [8]. The heat loss to the
surroundings happens in the end bipolar plates.
Qsurr = hsurr Acell (T
(29)
Tsurr )
The heat exchange between fuel cells and coolant is calculated by
the heat convection.
Qcool = hcool A cool (T
ave
Tcool
)
3.2. Comparison between uniform assumption and non-uniform distribution
As stated previously, reactant and coolant mass flow rate of individual fuel cells are assumed to be equal in numerous stack modeling
studies [26,27], which may lead to inaccurate predictions of stack
performances. Therefore, differences between the uniform flow assumption and the authentically non-uniform distribution are quantitatively investigated, which are rarely presented in literature. Fig. 4(a)
shows the stack voltage under various operating current densities. It is
seen that the uniform flow assumption leads to higher performance
predictions. The stack voltage difference becomes more significant with
the stack current increasing. For current lower than 108 A, the relative
difference is calculated to be less than 0.5%. The relative difference
reaches 1.2% if the current increases to 156 A, which refers to the
current density of 1.3 A cm−2. In other words, assuming the reactants
and coolant to be uniformly distributed into individual fuel cells
overestimates the stack output performance. For modeling studies, if
the correlation of stack voltage difference and current density is obtained, the performance predictions of an authentic stack model can be
calculated based on the modification of a simplified stack model with
the uniform flow assumption, which largely enhances the calculation
efficiency and improves the prediction accuracy. Although the correlation is likely to change with different setting of structural designs and
operating conditions, the same method is theoretically applicable. The
(30)
ave
where Tcool
(K) is the average of coolant inlet temperature and outlet
temperature.
To guarantee the convergence of the explicit calculation method,
the time step size for the stack sub-model is set as 10−6 s. To modify the
corresponding pressure drops in the flow distribution sub-model, the
mixture gas and coolant properties are updated, such as temperature,
density, viscosity, and velocity. The flow distribution sub-model is recalculated, and the new inlet mass flow rates are obtained. Iterations
continues until the convergence of the integrated stack model is
reached.
V , T,
, s lq, Ygas
in
in
mHin2 , mair
, mcool
< 2 × 10 6
(31)
where is the relative error. For the flow distribution sub-model, is
calculated as the differences between two iterations divided by the
present value. For the stack sub-model, a transient mode rather than a
steady-state one is developed. Note that the time step size is set as 10−6
s, and the differences between a time step could be extremely small.
Therefore, the relative error is calculated based on the interval of 10−3
s, and it is selected as the maximum value of all nodes in the stack submodel. The influences of convergence criteria are studied, and the results are presented in the model validation subsection. All sub-models
are developed with self-written codes in Matlab 2018a.
3.5
8
Relative error (%)
3.0
3. Results and discussion
3.1. Model validation
2.5
2.0
1.5
1.0
0.5
For the integrated proton exchange membrane fuel cell (PEMFC)
stack model, the influences of node number in the fuel cell flow direction and convergence criteria are firstly investigated. As shown in
Fig. 3(a), the output voltage, inlet mass flow rate, and outlet gas velocity are compared when the node number increases from 1 to 10.
Note that the relative error is calculated as the maximum value of all
nodes in the stack, which consists of 5 fuel cells. Compared with numerical results in the case of 10 nodes, the relative error falls below
0.5% if the node number reaches 4, and it drops below 0.25% if the
node number is increased to 5. Further increasing the node number
from 5 to 10 leads to negligible changes. Therefore, the node number in
fuel cell flow direction is set as 5 in the study. The influences of convergence criteria are shown in Fig. 3(b). The output voltage, inlet mass
flow rate, oxygen concentration, local current density, temperature,
and water volume fraction are compared. It is found that the relative
error drops below 0.5% if the criteria is set at 2 × 10−6. Decreasing the
criteria to 1 × 10−6 requires around 200-hour calculation time, which
is almost twice the time than that of the criteria 2 × 10−6. Therefore,
the convergence criteria are set as 2 × 10−6. To validate the integrated
stack model, simulation results are compared with the experimental
Output voltage
Anode inlet flow rate
Oxygen concentration
Current density
Temperature
Water volume fraction
7
Output voltage
Anode inlet mass flow rate
Cathode inlet mass flow rate
Anode outlet gas velocity
Cathode outlet gas velocity
Relative error (%)
Ii = n·I
i
6
5
4
3
2
1
0.0
0
1
10
2 3 4 5 6 7 8 9 10
Node number in fuel cell flow direction
9
8
0.9
Stack voltage (V)
6
5
4
3
2
1
(b)
3.0
0.8
Experimental data
Simulation results
2.5
7
Convergency criteria ( x10-6 )
(a)
Cell voltage (V)
f (Ii ) = Vout,
2.0
1.5
Exp. 2.1 V
Exp. 1.8 V
Exp. 1.5 V
Sim. 2.1 V
Sim. 1.8 V
Sim. 1.5 V
The sum of squares
due to error: 0.007
2
3
0.7
0.6
0.5
0.4
0.3
1.0
0
20
40
60
80
100
0.2
1
Current (A)
Cell number
(c)
(d)
Fig. 3. Comparison between simulation results and experimental data. (a)
Effects of node number in the fuel cell channel direction. (b) Effects of convergence criteria. (c) Stack polarization curve [16]. (d) Cell voltage variations
at different stack voltages [16].
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Energy Conversion and Management 207 (2020) 112502
Z. Yang, et al.
4.3
0.84
4.2
Fuel cell voltage (V)
Stack voltage (V)
0.82
Uniform flow assumption
Non-uniform distribution
4.1
4.0
3.9
3.8
3.7
3.6
0.78
0.6 A cm-2
1.0 A cm-2
1.3 A cm-2
Uniform
Uniform
Uniform
Distributed
Distributed
Distributed
0.76
0.74
0.72
0.70
3.5
3.4
0.80
48
60
72
84
0.68
96 108 120 132 144 156 168
1
2
3
Stack current (A)
Fuel cell number
(a)
(b)
4
5
Fig. 4. Comparison between the uniform flow assumption and the non-uniform distribution. (a) Stack voltage. (b) Fuel cell voltage distribution at 0.6, 1.0, and 1.3 A
cm−2.
cell voltage distributions at 0.6, 1.0, and 1.3 A cm−2 are further presented in Fig. 4(b). With the uniform flow assumption, the cell voltages
are almost identical under the same current density. For the distributed
stack model, the output voltages of middle cells are found to be lower
than end fuel cells, which are similar to parabola curves and consistent
with previous studies [19,20]. The highest voltage locates in the fuel
cell 1, and the lowest voltage happens in the cell 2. Therefore, the
largest voltage difference is calculated to be 1.1% at 0.6 A cm−2, and it
rises to 4.5% at 1.3 A cm−2. Higher current densities mean higher inlet
mass flow rates of the stack, which results in larger pressure drops
during the motion of reactants, leading to more uneven flow distribution among individual fuel cells. In general, neglecting the non-uniform
flow distribution not only overestimates the overall stack performance
but also underestimates the single cell voltage variations.
It is well known that cell voltage variations among the stack mainly
result from the uneven reactant and temperature distribution. Fig. 5
shows the anode and cathode inlet mass flow rate. It is found that
variations of mass flow rate increase with the current density increasing. Besides, the degrees of cathode variations are more significant
than that of the anode. The inlet mass flow rates of the end cells (cell 1,
5) are larger than the middle cells (cell 2, 3, 4), which partly leads to
higher output voltages, and it is consistent with previous studies
[20,23]. At 1.3 A cm−2, the real cathodic stoichiometry for cell 1 is
calculated to be around 3.31, and it is approximately 1.25 for cell 2.
The non-uniformity is supposed to aggravate if the current density
further increases. Therefore, it is inferred that the cathodic reactant
distributed to middle fuel cells might be insufficient under extremely
high current densities even though the overall cathode stoichiometry is
2 in the study. In other words, supplying abundant air is an important
issue for avoiding the possible oxygen starvation caused by the uneven
reactant distribution [39,40]. The distribution of coolant follows similar trends as the reactants, and negligible change is seen under different current densities since the total inlet mass flow rate is set as
constant.
Apart from reactant distributions, stack performances are also influenced by the temperature distribution. Meanwhile, the temperature
uniformity is of vital importance for the stack durability because temperature gradients may lead to thermal stresses and consequently result
in the deformation of components. Fig. 6 shows the temperature differences between the uniform flow assumption and the non-uniform
distribution, which is plotted as the temperature of the latter subtracted
by the former. Note that the layer thickness in the figure is not to scale
as the real thickness. At 0.6 A cm−2, the temperature of distributed
stack model is observed to be around 0.7 °C higher than that of the
uniform stack model. Besides, temperature differences do not change
noticeably among the five cells. At 1.0 A cm−2, the highest temperature
difference reaches around 1.7 °C which happens in the cell 3 and cell 4.
The temperature difference in the anode channel of cell 1 and the
cathode channel of cell 5 are found to be −0.4 °C. At the current
density of 1.3 A cm−2, temperature differences in the cell 1 are approximately −1.0 °C while temperature differences in the cell 3 and cell
4 reach about 3.6 °C. It is obvious that the distributed stack model leads
to more significant temperature variations at relatively high current
densities. The reason is that reactants and coolant mass flow rate of end
cells are much larger than middle cells, which takes away more heat
after exiting the fuel cells. Besides, the output voltage of middle fuel
Fig. 5. Inlet mass flow rate of reactants among different fuel cells. (a) Anode. (b) Cathode.
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Energy Conversion and Management 207 (2020) 112502
Z. Yang, et al.
BPc
0.6 A cm-2
CL MPL GDL CH
2.17
1.0 A cm-2
1.46
0.74
0.03
1.3 A cm-2
-0.69
Distributed, STc = 1.5
9.35
Flow
2.89
Temperature difference (K)
Flow
3.60
8.23
7.10
Distributed, STc = 1.8
5.98
4.85
3.73
Distributed, STc = 2.0
2.60
-1.41
1.48
-2.12
Cell 1
Cell 2
Cell 3
Cell 4
Oxygen concentration (mol m-3)
BPa
0.35
Cell 5
Cell 1
Fig. 6. Temperature differences between the uniform flow assumption and the
non-uniform distribution.
Cell 2
Cell 3
Cell 4
Cell 5
Fig. 8. Effects of cathode stoichiometry on the oxygen distribution, 1.2 A cm−2.
cells is lower than end cells, which implies that more heat is generated
owing to larger polarization voltage losses. Hence, neglecting the nonuniform flow distribution may lead to higher predictions of the overall
stack temperature and lower predictions of the temperature variations
among individual fuel cells.
changes. The corresponding cathode inlet mass flow rate is shown in
Fig. 7(b). With the stoichiometry increasing, the inlet mass flow rate for
all fuel cells rises although the non-uniformity becomes more obvious.
At the stoichiometry of 1.5, the real stoichiometry for cell 1 is around
2.23 while it is approximately 1.07 for cell 2. At the stoichiometry of
1.8, the real stoichiometry for cell 1 is about 2.82, and it is about 1.20
for cell 2. The reason that a higher cathode stoichiometry is preferable
because it increases the amount of reactant distributed into middle fuel
cells, which consequently results in the major performance improvement. Generally speaking, the key to stack performance enhancement is
to guarantee sufficient reactants for every fuel cell.
The oxygen concentration distribution inside each single fuel cell is
further presented in Fig. 8. Note that the layer thickness in the figure is
not to scale as the real thickness of porous layers and gas channel. The
oxygen concentration keeps decreasing towards the channel outlet
owing to electrochemical consumptions. Besides, the oxygen concentration in the cell 1 and cell 5 are much higher than other cells,
which results from larger inlet mass flow rates as shown in Fig. 7(b). At
the stoichiometry of 1.5, the oxygen concentration of cell 2 and cell 3 in
the outlet regions are approximately 0.5 mol m−3, indicating that
oxygen is almost completely consumed. Suppose that the stack stoichiometry is less than 1.5, the cell 2 and cell 3 is possible to suffer from
the local reactant starvation. The outlet oxygen concentration of cell 2
rises to 1.5 mol m−3 at the stoichiometry of 1.8, and it increases to
2.6 mol m−3 at the stoichiometry of 2.0. Therefore, a larger cathode
stoichiometry also helps reduce the possibility of air starvation, which
3.3. Effects of stack operating conditions
It is widely acknowledged that the operating conditions have great
influences on stack performances. Based on Fig. 5, it is found that the
cathode inlet mass flow rate varies significantly among different fuel
cells while variations in the anode are much less noticeable. Therefore,
effects of cathode stoichiometry on the flow distribution and stack
performances are investigated while the current density is set as 1.2 A
cm−2. Fig. 7(a) shows the fuel cell voltage distribution at the cathode
stoichiometry of 1.5, 1.8, and 2.0 under the uniform flow assumption
and the non-uniform distribution. It is seen that increasing the stoichiometry is beneficial to the stack output voltage. The performance
improvement is more noteworthy when the stoichiometry increases
from 1.5 to 1.8 in comparison with the cases when the stoichiometry
rises from 1.8 to 2.0. It implies that further increasing the cathode
stoichiometry may no longer lead to performance enhancement once it
has reached the specific value. For the distributed stack model, the
voltage of middle fuel cells rises remarkably with the increment of
stoichiometry while the voltage of end cells only increases slightly,
indicating that middle cells are more sensitive to stoichiometry
Fig. 7. Effects of cathode stoichiometry on stack performances, 1.2 A cm−2. (a) Fuel cell voltage. (b) Cathode inlet mass flow rate.
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Energy Conversion and Management 207 (2020) 112502
Z. Yang, et al.
Fig. 9. Effects of inlet pressure on stack performances, 1.2 A cm−2. (a) Fuel cell voltage. (b) Cathode inlet mass flow rate.
Fig. 10. Effects of manifold geometric parameter on stack performances, 1.2 A cm−2. (a) Fuel cell voltage. (b) Cathode inlet mass flow rate.
is important for the stack durability.
Since the reactant distribution is associated with pressure drops, the
effects of stack inlet pressure are also studied. The anode and cathode
inlet pressure are simultaneously changed from 1.4 atm to 1.6 atm. The
cell voltage distribution is presented in Fig. 9(a), and the cathode inlet
mass flow rate is shown in Fig. 9(b). It is seen that a higher inlet
pressure contributes to the stack performance improvement, which
results from the increased reactant concentration. However, the increase of inlet pressure does not lead to better uniformity of cell voltage
distribution. For different inlet pressures, the cathode inlet mass flow
rates are almost identical. The reason is that the reactant distribution is
strongly coupled with pressure drops rather than the absolute pressure.
3.4. Effects of stack manifold geometric parameters
Since the reactant distribution is strongly dependent on the stack
manifold structural designs, the effects of manifold geometric parameters are investigated to alleviate the non-uniform distribution phenomena. The operating current density is selected as 1.2 A cm−2. The
cross-sectional sizes of reactant manifold are changed, and the corresponding results are presented in Fig. 10. In the case of 10 × 6 mm2,
the largest voltage difference among the five fuel cells is calculated to
be 3.5% while it decreases to about 0.8% in the case of 10 × 12 mm2.
The cell voltage variations significantly decrease with larger reactant
manifold areas. This is because increasing the sectional dimensions
reduces the gas velocities, which consequently decreases the frictional
pressure drops in stack manifolds. Besides, the local pressure drops
caused by the diverging and converging configurations are also influenced. Note that the maximum voltage difference between the uniform
assumption and the case of 10 × 12 mm2 is reduced to 0.4%. The air
mass flow rates of individual fuel cells are shown in Fig. 10(b). With the
increment of manifold sizes, the air distribution is found to be more
uniform, and it is the same with the anode reactant distribution. The
variations of reactant flow rate among the stack decreases, remarkably
enhancing the output voltage of middle fuel cells.
Fig. 11. Local current density distribution along the channel direction of the
fuel cell 2, 1.2 A cm−2.
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Energy Conversion and Management 207 (2020) 112502
Z. Yang, et al.
The local current density distribution of the fuel cell 2 in the
channel direction is shown in Fig. 11. Note that the node number 1
represents the channel inlet region while the number 5 represents the
outlet region. The local current density keeps decreasing from the upstream channel towards the downstream channel, and it is caused by
the reactants consumption. It is obvious that the distributed stack
model leads to more severe non-uniformity of the local current density
distribution compared with the uniform assumption. However, the
current density variations are decreased with the cross-sectional area
increasing, which actually results from the increased inlet reactant mass
flow rate. In other words, supplying enough reactants for every fuel cell
contributes to the uniformity of local current density inside single cells.
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4. Conclusion
In the study, the flow distribution and performance heterogeneity are
investigated based on a 1+1 dimensional multiphase proton exchange
membrane fuel cell (PEMFC) stack sub-model integrated with a flow distribution sub-model. Frictional pressure drops as well as local pressure
drops due to the diverging, converging, and bending configurations are
calculated. For every fuel cell, transport processes are considered in the
through-plane direction and the gas channel direction. Phase changes
among the membrane water, liquid water, and water vapor are taken into
account. The integrated stack model has been validated against experimental data, including the polarization curve and the cell voltage distributions. Differences between the uniform flow assumption and the nonuniform distribution are quantitatively compared, including the distribution of cell voltage, mass flow rate, temperature, reactant concentration,
and local current density. Results show that the uniform flow assumption
not only overestimates the stack output performance but also underestimates the fuel cell voltage variations. Besides, neglecting the nonuniform flow distribution may lead to higher predictions of the overall
stack temperature and lower predictions of the temperature variations
among different fuel cells. The inlet reactant mass flow rate of middle fuel
cells is found to be lower than end fuel cells, and the cell voltage distribution follows the similar trend. Even though the total amount of air
supplied to the stack is abundant, it is still possible for some fuel cells to
suffer from the local reactant starvation. Therefore, a higher cathode
stoichiometry is preferable because it increases the inlet mass flow rate for
middle fuel cells, which significantly contributes to the stack performance
improvement. Increasing the inlet pressure contributes negligibly to the
uniformity of reactant distribution, but it is favorable for the stack performance owing to enhanced electrochemical reactions. A larger stack
manifold cross-sectional area is suggested because it not only contributes
to the more uniform reactant distribution among the stack but also helps
decreasing the local current density variations inside individual fuel cells.
CRediT authorship contribution statement
Zirong Yang: Methodology, Software, Validation, Writing - original
draft. Kui Jiao: Conceptualization, Validation, Resources, Writing review & editing. Zhi Liu: Data curation, Writing - review & editing.
Yan Yin: Writing - review & editing, Funding acquisition. Qing Du:
Writing - review & editing, Supervision, Project administration.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This work is supported by the National Key Research and
Development Program of China (2018YFB0105601) and the National
Natural Science Foundation of China (grant No. 51976138).
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