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Journal of Power Sources 145 (2005) 30–39
A high dynamic PEM fuel cell model with temperature effects
ଝ
Yuyao Shan
∗
, Song-Yul Choe
Department of Mechanical Engineering, Auburn University, Auburn, 36830 AL, USA
Received 26 November 2004; accepted 23 December 2004
Available online 20 March 2005
Abstract
Safe and reliable operation of a fuel cell requires proper management of the water and heat that are produced as by-products. Most of
the current models for the cell used for an analysis of the

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Journal of Power Sources 145 (2005) 30–39
A high dynamic PEM fuel cell model with temperature effects
Yuyao Shan
∗
, Song-Yul Choe
Department of Mechanical Engineering, Auburn University, Auburn, 36830 AL, USA
Received 26 November 2004; accepted 23 December 2004Available online 20 March 2005
Abstract
Safe and reliable operation of a fuel cell requires proper management of the water and heat that are produced as by-products. Most of the current models for the cell used for an analysis of the fuel cell system are based on the empirical polarization curve and neglect thedynamic effects of water concentration, temperature and reactant distribution on the characteristics. The new model proposed in this paperis constructed upon the layers of a cell, taking into account the following factors: (1) dynamics in temperature gradient across the fuel cell;(2) dynamics in water concentration redistribution in the membrane; (3) dynamics in proton concentration in the cathode catalyst layer; (4)dynamics in reactant concentration redistribution in the cathode GDL. Simulations have been performed to analyze the effects of load currentson the behaviors of the fuel cell. In the future, the fuel cell model will be extended to a stack model and integrated with system models. Allof the models will be implemented on a real time system that optimizes the computation time by a parallelization of solvers, which providesan environment to analyze the performance and optimize design parameters of the PEM fuel cell system and components.© 2005 Published by Elsevier B.V.
Keywords:
PEMFC; Dynamic; Temperature; Water; Efﬁciency; Startup
1. Introduction
The PEM fuel cell is a strong candidate for use as analternative power source in future vehicle and power condi-tioning applications. The effects of electric loads on tem-perature, water in the stack and reactants are crucial is-sues that must be considered for the optimum design of fuel cell powered systems. Currently, fuel cell stack mod-els are being employed to analyze these effects. However,the simulation results do not incorporate either the dynamicor transient aspects of the fuel cell system in operatingenvironments.As a matter of fact, the dynamic power output and ef-ﬁciency proﬁle of a PEMFC is strongly inﬂuenced by thevariation of the temperature, reactant and product transfer inthe fuel cell caused by a current load.
This paper was presented at the 2004 Fuel Cell Seminar, San Antonio,TX, USA.
∗
Corresponding author. Tel.: +1 334 220 6533.
E-mail addresses:
shanyuy@auburn.edu (Y. Shan),choe@eng.auburn.edu (S.-Y. Choe).
Firstly, the temperature signiﬁcantly affects the perfor-mance of a fuel cell by inﬂuencing the water removal andreactants activity, etc. A current proposed model assumes aconstant working temperature[1],which does not incorpo-
rate the reality that this working temperature dynamicallyvaries at different load currents, as well as during startup andshut-down of the fuel cell system. Some authors proposedimproved models, with Amphlett et al.[2]using the ﬁrst em-pirical thermal model, and Gurski et al.[3]considering thereactant ﬂows and coolant control based upon the previousmodel. Others proposed models calculating the temperaturevariationofthestack,cell[4–10]ortwoelectrodesandMEA[11,12]. B. Wetton et al.[13]proposed an explicit thermal
model to analyze the temperature gradient of different layersinthefuelcellstackconsideringthestackasymmetriceffects,which does not include dynamics. Recently, M. Sundaresanpublished the most detailed 1D thermal dynamic model[14].However, the ﬂow of species at the inlet must be the sameas that at the outlet. Thus, no ﬂuid dynamics is considered inthe model.Secondly, the proton transport in the membrane and itsassociated ohmic losses mainly determine the characteristics
0378-7753/$ – see front matter © 2005 Published by Elsevier B.V.doi:10.1016/j.jpowsour.2004.12.033
Y. Shan, S.-Y. Choe / Journal of Power Sources 145 (2005) 30–39
31
Nomenclature
Alphabetsa
species activity
A
area (m
2
)
C
mass concentration (kgm
−
3
)
D
diffusion coefﬁcient (m
2
s
−
1
)
F
faraday number
G
gibbs free energy (Jmol
−
1
)
H
enthalpy (Jmol
−
1
)
i
current density (Am
−
2
)
j
exchange current density (Am
−
2
)
l
thickness (m)
m
mass (kg)
M
mole mass (kgmol
−
1
)
n
d
electro-osmotic drag coefﬁcient
N
mole ﬂux (mols
−
1
m
−
2
)
P
pressure (partial pressure) (Pa)
R
universal gas constant
R
mem
proton transfer resistance (
)
R
ab
electrical resistance (
)
S
entropy (Jmol
−
1
K
−
1
)
T
temperature (K)
W
mass ﬂux (kgs
−
1
m
−
2
)
Greek symbols
ε
porosity
λ
water uptake coefﬁcient
ρ
density (kgm
−
3
)
τ
tortuosity
Superscripts and subscripts
an anodeca cathodecv control volumed gas diffusion layerg gas
i
indexl liquidmem membrane layerref reference valuesat saturationsou sourceof ohmic polarization. The proton conductivity has been re-gardedasconstant,temperaturedependent[1]ortemperature
and water concentration dependent variables[15].Recently,
Pukrushpan et al.[16]proposed the most comprehensivemodel that considers the dependence of the proton conduc-tivity on the water concentration and temperature. However,thewaterconcentrationofthemembraneisobtainedfromthemembrane relative humidity (RH) on an average of the an-odeandcathodeRH.Infact,theRHintheanodeandcathodevariesrapidly,whiletheRHinthemembranedoesslowlybe-cause the amount of water residing in both sides is relativelyless than in the membrane[15].Thirdly,theoxygenconcentrationintheGDLonthecath-ode side is continuously changing in operating environmentsand signiﬁcantly affects the performance of the cell. There-fore,plentyofmodelsconsideringmulti-phasemulti-specieshave been employed to investigate the transport phenom-ena in the GDL. However, those models do not consider thedynamics. Recently, Pukrushpan et al. proposed a dynamicmodel with lumped parameters to predict the gas dynamicsin a cathode electrode, which does not consider the effects inthe GDL[16]. In this paper, we use a 1D single-phase modelto represent the dynamics present in the GDL.
2. Model setup and assumptions
The model has been developed on the basis of layers in acell that consist of a MEA, two gas diffusion layers and twogas channels sandwiched by two coolant channels, as showninFig. 1.The input variables for the model are current load,
mass ﬂow rate, the gas components fraction, temperature,pressure and relative humidity of reactants as well as thetemperature and velocity of coolants at the inlets.The main assumptions made for the new model are asfollows:1. Reactants are ideal gases.2. Thereisnopressuregradientbetweentheanodeandcath-odeside;itmeansnoconvectionbutonlydiffusionforgastransport is considered.3. There is no gas pressure drop from the inlet to the outletof the gas channel.4. The temperature gradient is linear across the layers in afuel cell.5. The thermal conductivity for the materials in a fuel cell isconstant.6. There is no contact resistance.7. Anodic over-potential is negligible.
Fig. 1. Schematic simulation domain.
32
Y. Shan, S.-Y. Choe / Journal of Power Sources 145 (2005) 30–39
8. There is no current density gradient across the cathodecatalyst layer; it means that the reactants completely re-acted as soon as it reaches to the cathode catalyst layersurface.9. Based on these assumptions, ﬁve sub-models have beendeveloped and are described in the following sections.
3. Model description
3.1. Electrochemical model
Generally, the overall chemical reaction of the PEM fuelcell can be described by using the following expressions,illustrating that a chemical reaction of hydrogen and oxy-gen molecules produces electricity, water and heat as a by-product:H
2
+
12
O
2
→
H
2
O
+
Q
res
+
V
cell
The output cell voltage
V
cell
is the difference between theopen circuit voltage (OCV)
E
0
and over-potentials
η
and
V
ohm
.
V
cell
=
E
0
−
η
−
V
ohm
(1)By neglecting the dependence of the OCV on the reactantpressure,therelationshipbetweentheOCVandthetempera-turecanbesimpliﬁedwiththeempiricalparameterd
E
0
/d
T
.If thereactantisideal,itsactivitycanbedescribedbyusingEq.(2),where index
i
indicates H
2
and O
2
, while
P
i
is the partialpressure of gas components, and
P
0
is the overall pressure of both the anode and cathode side. Then,
E
0
can be derived bymodifying the Nernst Eq.(3).
a
i
=
P
i
P
0
(2)
E
0
=
E
ref
+
d
E
0
d
T
(
T
−
T
ref
)
+
RT
2
F
ln(
a
1H
2
a
0
.
5O
2
) (3)The anodic over-potential is negligible; while the
η
repre-sents the over-potential of the cathode catalyst layer. Underthe further assumptions that the asymmetric parameter of thereaction is(1)and(8), the Butler–Volmer equation leads to
Eq.(4)that describes the over-potential on the cathode side.
i
=
j
0
A
cata,eff
A
cell
p
O
2
p
O
2
,
ref
[H
+
][H
+
]
ref
exp
FηRT
−
1
(4)The ohmic over-potential
V
ohm
is determined by the productof the current density and the proton resistance
R
mem
accord-ing to Ohm’s law(5).
V
ohm
=
iR
mem
(5)
3.2. Thermal model for a cell
If a cell is assembled with cubic layers whose thermo-physical properties are isotropic and constant, and then ac-cording to the energy conservation equation, the total energychanges in a controlled volume equals the sum of the energyexchangeatboundariesandinternalenergyresources.Infact,the energy exchanges at boundaries occur by three factors:(a) the mass ﬂow into each volume; (b) the conduction heattransferacrossthecell;(c)theconvectionheattransferoccur-ringbetweenbipolarplateswiththecoolantandthereactants.Thus, the thermal dynamic behavior can be described withthe following energy conservation equation:
i
Cp
i
C
i,
mass
A
cell
l
cv
d
T
cv
d
t
=
˙
m
in
A
cell
Cp
j
(
T
in
−
T
cv
)
mass ﬂow in
+
˙
Q
Conv
A
cell
convection heat transfer
+
˙
Q
Cond
A
cell
conduction heat transfer
+
˙
Q
sou
sources
(6)On the other hand, the internal energy source is composed of theentropylossandthechemicalenergyrequiredforprotonsto overcome the barrier of the over-potentials in both catalystlayers (Eq.(7)). In addition, other heat sources are ohmiclosses caused by a transport of electrons and protons in thecell:˙
Q
sou
=
iA
cell
−
TS
4
F
+
η
+
iA
cell
R
ele
(7)In fact, the change of entropy due to the electrochemical re-action (Eqs.(8)and(9))in both of the catalysts sides pre-
dominantly inﬂuences the energy sources term according tothe calculation shown below.H
2
2H
+
(aq)
+
2e
(pt)
−
(8)O
2
+
2H
+
+
2e
−
H
2
O
(l)
(9)In order to obtain the entropy change of these reactions, thezero point of semi-absolute entropy is taken as a referenceaccording to[17]:
s
[H
+
(aq)
]
≡
0 (10)Theentropyofanelectronobtainedfromthestandardhydro-gen electrode results in the following equations[17]:
S
SHE
=
H
SHE
−
G
SHE
T
=
0 (11)
s
[e
(pt)
−
]
=
12
s
[H
2
]
=
65
.
29Jmol
−
1
K
−
1
(12)Therefore,theentropychangeofthecathodereactionisequalto the sum of that in water, oxygen and electron:
S
ca
=
s
[H
2
O
(l)
]
−
12
s
[O
2
]
−
2
s
[e
(pt)
−
]
=
69
.
91
−
12
×
205
.
03
−
2
×
65
.
29
=−
163Jmol
−
1
K
−
1
(13)
Y. Shan, S.-Y. Choe / Journal of Power Sources 145 (2005) 30–39
33
If the anode is assumed as a standard electrode, the anodicentropy change becomes 0.
3.3. Proton conducting model for membrane
The membrane resistance is a function of the temperatureand water content in a membrane layer, which is describedas follows[16]:
R
mem
=
l
mem
(
b
11
λ
mem
−
b
12
)exp
b
2
1303
−
1
T
mem
(14)where the temperature
T
mem
can be derived from the previ-ous Eq.(6),while the membrane water content
λ
mem
can bedescribedbyusingthewatermassconcentration[15]andthe
mass conservation equation[16]:
λ
mem
=
C
H
2
O
,
mass
/M
H
2
O
ρ
dry
,
mem
/M
mem
−
bC
H
2
O
,
mass
/M
H
2
O
(15)d
m
water
d
t
=
d(
C
H
2
O
,
mass
A
cell
l
mem
)d
t
=
W
ele
,
mem
,
an
−
W
ele
,
mem
,
ca
+
W
diff
,
mem
,
ca
+
W
diff
,
mem
,
an
(16)The electro-osmotic driving force created by the differentelectrochemical potential at the anode and cathode deter-mines the water mass ﬂows of
W
ele,mem,an
and
W
ele,mem,ca
at the boundaries of the membrane layer. In addition, thediffusion caused by the water concentration gradient at thetwo boundaries makes up the mass ﬂows of
W
diff,mem,an
and
W
diff,mem,ca
. Those relationships are described by Eqs.(17)–(19), proposed by Spriger[24].
n
d
=
0
.
0029
λ
2mem
+
0
.
05
λ
mem
−
3
.
4
×
10
−
19
(17)
W
ele
,
mem
,i
=
M
water
A
cell
n
d
,i
iF
(18)
W
diff
,
mem
,i
=
M
water
A
cell
D
water
C
i
−
C
mid
l
mem
(19)Hence, the diffusion coefﬁcient
D
water
and the water concen-tration
C
i
arecalculatedfromtheempiricalEq.[24]expressed
as a function of membrane water content
λ
mem
:
D
water
=
D
(
λ
mem
)exp
2416
1303
−
1
T
mem
(20)
D
(
λ
mem
)
=
10
−
6
2
> λ
mem
10
−
6
(1
+
2(
λ
mem
−
3)) 3
≥
λ
mem
≥
210
−
6
(3
−
1
.
67(
λ
mem
−
3)) 4
.
5
> λ
mem
>
31
.
25
×
10
−
6
λ
mem
≥
4
.
5(21)The boundary water content
λ
i
is a function of water activity
a
i
, which is calculated from the water vapor partial pressure:
λ
i
=
0
.
043
+
17
.
81
a
i
−
39
.
85
a
2
i
+
36
a
3
i
1
≥
a
i
>
014
+
1
.
4(
a
i
−
1) 3
≥
a
i
>
116
.
8 3
≤
a
i
(22)
a
i
=
P
v,i
P
sat
,i
(23)
3.4. Proton dynamic model in the cathode catalyst layer
Thedynamicbehaviorofafuelcellataloadisinvestigatedby experiments. When the output current changes abruptly,the output voltage of the fuel cell reacts with an overshoot[18].Thesedynamicsresultfromdifferentphysicalphenom-ena of reactants and their chemical reaction in the cell, suchas dynamics ﬁlling in the gas ﬂow channel, diffusing reac-tants through the GDL and reacting process in the doublelayer at the interface of electrodes. Ceraolo et al. explainedthe dynamic effects with a relationship between the numberof mobile protons and water content[1].As a matter of fact,
whenthecurrentdensityincreases,thehydrationofthepoly-meric electrolyte near the cathode catalyst tends to rise aswell; consequently, the proton concentration near the cath-ode catalyst increases rapidly. On the other hand, the protonconcentration will decrease slowly at a decrease of current.Thus, the dynamics can be described by the following differ-ential equation using the proton concentration as a variable[1]:
δ
−
∂
˙
C
H
+
∂t
∂
˙
C
H
+
∂t
+
˙
C
H
+
τ
H
+
=
1
+
α
H
+
i
3
τ
H
+
(24)˙
C
H
+
=
[H
+
]
/
[H
+
]
ref
is the dimensionless proton concentra-tion,
δ
() the Heaviside function, and
τ
H
+
and
α
H
+
are empir-ical parameters.Fig. 2shows the calculated response. The voltage de-creasesquicklywhenthecurrentdensityincreases.However,
Fig. 2. Voltage response by a consideration of proton concentration.

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