A non-isothermal PEM fuel cell model including
two water transport mechanisms in the
membrane
K. Steinkamp
J.O. Schumacher
[email protected]
[email protected]
Fraunhofer Inst. for Solar Energy Systems Institute for Computational Physics
Heidenhofstrasse 2, 79110 Freiburg
Zuercher Hochschule Winterthur
Germany
PO Box 805, CH-8401 Winterthur
Switzerland
F. Goldsmith
M. Ohlberger
[email protected]
[email protected]
Institute for Applied Mathematics
Massachusetts Institute of Technology
77 Massachusetts Ave., Cambridge
Albert-Ludwigs-Universitaet Freiburg
USA
Herrmann Herderstr 10, 79110 Freiburg
Germany
C. Ziegler
[email protected]
Fraunhofer Institute for Solar Energy Systems
Heidenhofstrasse 2, 79110 Freiburg
Germany
A dynamic two-phase flow model for proton exchange membrane (PEM) fuel cells is presented. The two-dimensional
model includes the two-phase flow of water (gaseous and
liquid) in the gas diffusion layers (GDLs) and in the catalyst layers (CLs), as well as the transport of the species in
the gas phase. The membrane model describes water transport in a perfluorinated sulfonated acid ionomer (PFSA)
based membrane. Two transport modes of water in the membrane are considered, and appropriate coupling conditions to
the porous catalyst layers are formulated. Water transport
through the membrane in the vapor equilibrated transport
mode is described by a Grotthus-Mechanism, which is included as a macroscopic diffusion process. The driving force
for water transport in the liquid equilibrated mode is due to
a gradient in the hydraulic water pressure. Moreover, electroosmotic drag of water is accounted for. The discretisation
of the resulting flow equations is done by a mixed finite element approach. Based on this method, the transport equations for the species in each phase are discretised by a finite
volume scheme. The coupled mixed finite element/finite volume approach gives the spatially resolved water and gas saturation and the species concentrations. In order to describe
the charge transport in the fuel cell the Poisson equations
for the electrons and protons are solved by using Galerkin
finite element schemes. The electrochemical reactions in the
catalyst layer are modeled with a simple Tafel approach via
source/sink terms in the Poisson equations and in the mass
balance equations. Heat transport is modelled in the GDLs,
the CLs, and in the membrane. Heat transport through the
solid, liquid, and gas phases is included in the GDLs and
the CLs. Heat transport in the membrane is described in the
solid and liquid phases. Both heat conduction and heat convection are included in the model.
1
Introduction
Energy conversion to electrical energy by polymer electrolyte membrane (PEM) fuel cells offer several
benefits, such as the pollution-free operation, the high
power density of fuel cell systems, and a high energy
conversion efficiency. Nevertheless, several key issues
need to be resolved before fuel cells can become competitive to other energy conversion techniques. One of
these key issues is water management.
Water management is critical to achieve the optimum
performance of a PEM fuel cell. The protonic conductivity of perfluorinated sulfonic acid (PFSA) based
membranes strongly depends on its water content.
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Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Ohmic losses occur within membrane regions with a
low water content. On the other hand, the presence
of excess water in the gas diffusion layer (GDL) blocks
the transport of hydrogen and oxygen, and it can lead
to the coverage of the active catalytic surface, thereby
hindering the electrochemical reaction. Thus, the prediction of the liquid water saturation in the porous media of a PEM fuel cell is essential for design of the fuel
cell components, the analysis of fuel cells, and for the
development of operation strategies.
Several models have been published that describe
water and heat management of PEM fuel cells without
accounting for the transport of liquid water. Springer
et al. parameterised the electrical conductivity, the electroosmotic drag coefficient, and the water diffusion coefficient of Nafion membranes by fitting of polynomial
expressions to measurement data [1]. A drag/diffusion
model for the water transport through the membrane
was introduced, thereby developing a simple but extremely useful water-management model of PEM fuel
cells. Water transport through polymer electrolyte
membranes was also adressed in [2] and [3], which
served as the basis for a large number of refined models that treat liquid water effects in a semi-empirical
way without solving a transport equation for liquid
water. Nguyen and White published an along-thechannel model for water and heat management of PEM
fuel cells, which predicts the temperature and current
distribution along the one-dimensional flow channels
on the anode and cathode side [4]. A stationary alongthe-channel model was presented by Gurau et al. [5].
Their one-phase model accounts for mass transport of
gaseous species in all layers of the PEM fuel cell. The
Navier-Stokes equations were solved in the gas channels. Moreover, the model includes the effect of heat
transport in the gas channels and in the porous media.
An isothermal, steady-state model for multicomponent
gas transport in the porous electrode of a PEM fuel cell
with an interdigitated gas distributor structure was investigated by Yi and Nguyen [6].
Recently, a number of models were developed that
account for liquid water transport in the porous layers of a PEM fuel cell. The hydrodynamics of capillary,
two-phase flow in hydrophobic single- and two-layer
porous media was investigated by Nam [7]. A threedimensional, non-isothermal, two-phase flow model
was published by Berning and Djilali [8]. In this approach the two-phase flow inside the porous media
is described by unsaturated flow theory (UFT). The
multi-phase mixture (M 2 ) formalism was applied by
Pasaogullari [9] to include the effects of counter gasflow as a contribution to oxygen transport. Particularly, the influence of the properties of a micro-porous
layer (e.g. thickness and wettability) on two-phase
transport was adressed. A two-dimensional model of
a PEM that includes liquid water transport was presented by Siegel et al. [10]. The model accounts for
gas species transport, charge- and heat transport, and
water dissolved in the ion conducting polymer. Liquid
water is assumed to be transported by capillary pressure within the GDLs and the catalyst layers and by
advection within the gas channels. The catalyst layers are described by an agglomerate model. The removal of water droplets at the gas diffusion layer /
gas flow channel interface was investigated by Chen
et al. [11] both by modelling and experiment. Dropletinstability diagrams were computed, and the influence
of the droplet shape was quantified. A non-isothermal,
one-dimensional model of the cathode side of a PEM
that includes two-phase transport was published by
Shah et al. [12]. The catalyst layer and the gas diffusion layer are characterised by several measurable microstructural parameters in this model.
The focus of this paper is to formulate a timedependent model that accounts for two-phase flow of
water in the gas diffusion layers, the catalyst layers,
and in the membrane. The model equations are described in detail in Section 2. The GDL and the catalyst layer description is given in Section 2.1. The
equations that describe water and heat transport in the
membrane are described in Section 2.2. Proton transport in the membrane and in the catalyst layers is outlined in Section 2.3. An essential part of the model is
the coupling conditions between the subdomains and
the boundary conditions, which are described in Section 2.4. A model discussion is given in Section 3.
Details of the numerical discretisation methods can be
found in Section 4. Finally, it is demonstrated in Section 5 that the system of coupled partial differential
equations (PDEs) can be solved with these discretisation methods.
2
PEM fuel cell model
The PEM fuel cell model consists of several submodels that describe the transport processes for:
• liquid water phase and gas phase,
• gas species (hydrogen, oxygen, water vapour and
nitrogen),
• heat,
• charge carriers (electrons and protons).
As shown in Fig. 1, the modeled PEM fuel cell is made
up of five layers: the gas diffusion layers on the anodic
and cathodic side, the catalyst layers on both sides, and
the membrane in the center of the cell. Three modeling
regions are distinguished:
• Region I consists of the gas diffusion layers (GDL)
and the catalyst layers.
• Region II consists of the membrane.
• Region III includes the membrane and the catalyst
layers.
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Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
wetting angle θ of the liquid on the surface of the capillary
pc =
Figure 1.
2γ cos θ
.
r
(4)
Because the effective radius r changes with liquid saturation sw , the capillary pressure is not constant, but
depends on sw . The relationship pc (sw ) describes the
capillary pressure required to obtain a given liquid water phase saturation in a hydrophobic GDL. GDLs have
a distribution of pore throat sizes, so as more pressure is applied to the liquid water phase, increasingly
smaller pore openings are invaded. The capillary pressure curve is important for understanding saturation
distribution in GDL and affects imbibition and twophase flow through the GDL.
In this paper, the so-called global pressure formulation of the two-phase equations is used. It can be obtained by applying elementary transformations to Eq.
(1), cf. [13,14]. In this formulation there is one equation
for the global velocity
Layer assembly of a PEM fuel cell. Five layers of a PEM
fuel cell are modeled: cathodic gas diffusion layer (GDL), anodic
GDL, cathodic catalyst layer, anodic catalyst layer, and membrane.
The two gas channels are taken into account as boundary conditions
into the model.
(5)
~uglob = ~uw +~ug
2.1 GDL and catalyst model
2.1.1 Two-phase flow in the GDLs and the catalyst
layers
Two-phase flow in region I, which is made up of
porous material, is modeled with Darcy’s law. In the
following formulation, the index i = w is the liquid water phase and i = g the gas phase. The gas phase itself
consists of several gas species.
Darcy’s equations for the two phases can be written as
∂t (φκ ρi si ) + ∇ · (ρi~ui ) = qi ,
as well as for the global pressure pglob . Neglecting the
influence of gravity, one obtains for constant densities
ρw , ρg
∇ ·~uglob = 0 ,
(6)
~uglob = −Kκ λtot (sw , T ) ∇pglob .
(7)
Another equation results for the liquid water saturation sw ,
(1)
~ui = −Kκ λi (sw , T )(∇pi − ρi~g)
φκ ∂t (ρw sw ) + ∇ · ( fw (sw , T )~uglob )
− ∇ · (Kκ d(sw , T )∇sw ) = qw .
φκ is the porosity of the layer κ relating to Fig. 1, e.g.
κ = Ωagdl for the anodic gas diffusion layer. The absolute permeability Kκ also depends on the layer. ρi is the
mass density, ~ui the velocity, pi the pressure, si the volume saturation and λi the mobility of phase i. qi is the
source term of phase i and will be discussed later.
The following supporting equations [13] are used
and
sw + sg = 1
pg − pw = pc (sw ) .
(8)
The global pressure is defined by
pglob (pg , sw , T ) =
Z sw
(9)
λg − λw 0
1
= pg −
pc (sw ) −
pc (sw ) d sw .
∗
2
λtot
sw
(2)
Thus, the value of the global pressure strongly depends
on the capillary pressure and its derivative with respect
to the water saturation. A Brooks-Corey parametrisation for the capillary pressure curve [15] is used
(3)
pc (sw ) is the capillary pressure curve. In fluid dynamics, capillary pressure is the difference in pressure
across the interface between two immiscible fluids, cf.
Eq. 3. The pressure difference is proportional to the
surface tension γ, and inversely proportional to the effective radius r of the interface, it also depends on the
1
pc (sw ) = −pd (1 − sw )− bc ,
3
(10)
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
The fractional flow rate fi is defined by
λw (sw , T )
and
λtot (sw , T )
fg (sw , T ) = 1 − fw (sw , T ) .
fw (sw , T ) =
(15)
(16)
Eq. (8) contains the capillary diffusion coefficient d
d(sw , T ) = − fw (sw , T )λg (sw , T )p0c (sw ) .
Again, the derivative of the capillary pressure with respect to the liquid water saturation has a strong influence on the transport equations.
The source term qw in Eq. (8) for liquid water is given
by the evaporation rate and the condensation rate, respectively [19]
Figure 2. Plot of the capillary pressure as a function of the liquid
water saturation pc (sw ) according to the Brooks-Corey model. The
gas diffusion layers are hydrophobic, and therefore the capillary pressure is negative. The intersection point with the y-axis denotes the
threshold pressure pd .
qw = kc φκ sg
kr,i (sw )
ηi (T )
for i = w, g ,
with the Heavyside function H(x). Assuming water
vapour as an ideal gas, its partial pressure is given by
pH2 O =
cH2 Ov
pg ,
MH2 O
(19)
where MH2 O is the molar mass and cH2 Ov is the mass
fraction of water vapour. The saturation vapour pressure can be parametrised [20] as
(11)
−6094.464
+ 21.124995 − 2.72455 · 10−2 T
T
+ 1.68534 · 10−5 T + 2.45755 · ln(T ) .
(20)
ln (psat (T )) =
where ηi is the viscosity of phase i and kr,i (sw ) is the relative permeability of phase i as a function of the liquid
water saturation. In the global pressure formulation of
two-phase flow the total mobility λtot is used, which is
the sum of the two phase mobilities,
2.1.2
λtot (sw , T ) = λw (sw , T ) + λg (sw , T ) .
ρg
(pH2 O (cH2 Ov ) − psat (T ))H(pH2 O − psat )
pg
+ kv φκ sw ρw (pH2 O (cH2 Ov ) − psat (T ))H(psat − pH2 O ) ,
(18)
where bc is the Brooks-Corey exponent and pd the
threshold pressure. These parameters were determined by fitting Eq. (10) to numerical data which was
obtained with the pore-morphology method [16]. This
method is based on the 3D reconstruction of the microstructure of the GDL by synchrotron-tomography.
The method predicts the 3D liquid water saturation
distribution and the corresponding capillary pressure.
The mobility [17] of phase i is
λi (sw , T ) =
(17)
(12)
Gas species flow
The gas phase is assumed to consist of four gas
species:
The relative permeabilities of the two phases can be
parametrised [18] as
m 2
1
√
m
kr,w (sw ) = sw 1 − 1 − sw
kr,g (sw ) =
p
and
h
i
1 m 2
1 − sw 1 − 1 − (1 − sw ) m
.
• hydrogen, supplied at the anodic gas channel,
• oxygen and nitrogen (air), supplied at the cathodic
gas channel,
• water vapour, produced either by the electrochemical reaction or by evaporation.
(13)
(14)
Hence, there are four transport equations needed to describe gas species flow. Assuming Fickean diffusion,
Again, the value of the parameter m was obtained by
fitting these expressions to numerical data from [16].
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Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
one obtains [6]
The mass fraction of oxygen decreases during the
reduction-reaction in the cathodic catalyst layer
φκ ∂t (ρg sg ck ) + ∇ · (ρg ck~ug )
− ∇ · φκ ρg sg Dκ (~uglob , sw , T )∇ck = qkg (21)
1
O2 + 2H + + 2e− → H2 O
2
v
with k = H2 , O2 , H2 O ,
(22)
1. in Ωacat :
The temperature dependence of the density of the gas
phase follows the ideal gas law
ρg (T ) =
Mg pg
.
RT
cH2 Mg pg
(1 − α)F
ia = sg A j0,a
exp
ηa (φe , φ p ) ,
[cH2 Mg pg ]re f
RT
(29)
2. in Ωccat :
(23)
ic = sg A j0,c
The molar mass Mg of the gas phase is calculated by
averaging the molar masses of the four gas species
harmonically and weighting them with the respective
mass fraction
1
=
Mg k=H
∑
2 ,O2 ,H2 O,N2
ck
.
Mk
(24)
3
= D0 (φκ (1 − sw )) 2 (1.41 · 10−7 · T − 2.07 · 10−5 ) .
(31)
2
qO
g
(32)
v
(25)
MH2 ia
,
2F
MO ic
=− 2 ,
4F
MH2 O ic
=
− qw .
2F
2
qH
g =−
2O
qH
g
(33)
2.1.3 Thermal model
In addition to the liquid (index w) and the gas (index g) phases, the solid phase (index s) in the gas diffusion layers and catalyst layers, i.e. the framework of
carbon fibers, must be considered in order to describe
heat flow. The thermal energy density ε contains these
three phases
Ddi f f ,κ is the diffusion coefficient in layer κ. Its temperature [21] and saturation [13] dependencies are parameterised with
Ddi f f ,κ (sw , T ) =
cO2 Mg pg
αF
exp
−2
η
(φ
,
φ
)
.
c e p
[cO2 Mg pg ]re f
RT
(30)
The source densities of the three gas species are
In Eq. (21) Dκ is a diffusion-dispersion matrix, which
can be written as
Dκ (~uglob , sw , T ) = Ddi f f ,κ (sw , T ) + Ddisp,κ (~uglob ) .
(28)
Furthermore, this reaction specifies a source for water
vapour. Evaporation and condensation act as source
and sink of water vapour, respectively. Using the Tafel
approximation [22, 23] the reaction rates of the two reactions can be written as
where ck is the mass fraction of species k. After solving
these three differential equations, the mass fraction of
nitrogen can be obtained by
cN2 = 1 − (cH2 + cO2 + cH2 Ov ) .
in Ωccat .
(26)
ε = (1 − φκ )ρs Hs + φκ
∑
ρi si Hi (T ) ,
(34)
i=w,g
Ddisp,κ is the Scheidegger dispersion tensor (for details
see [13]).
Sinks for hydrogen are located only in the anodic catalyst layer. There the oxidation of hydrogen leads to a
decrease of the hydrogen mass fraction
H2 → 2H + + 2e−
in Ωacat .
where Hs , Hw and Hg are the specific enthalpies of the
individual phases and ρs , ρw and ρg are the respective
mass densities. A variation over time of the energy
density occurs due to heat conduction and convective
heat flux
(27)
!
∂t
(1 − φ)ρs Hs (T ) + φ
∑
ρi si Hi (T ) +
i=w,g
!
+∇·
∑
i=w,g
5
ρi~ui Hi (T ) − ∇ · (κT (sw )∇T ) = qT .
(35)
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
~uw and ~ug are the phase velocities, and κT is the heat
conductivity. The latter depends on the water saturation sw and can be parameterised in various ways [24].
For simplicity in this model a linear approximation has
been chosen
κT (sw ) =
0.49sw + 0.21 in Ωa,c
gdl
0.26sw + 0.26 in Ωa,c
cat
W
.
Km
(36)
There are several heat sources and sinks which are discussed briefly below.
Figure 3.
• The condensation heat q phase is the product of the
phase transition rate qw (cf. Eq. (18)) and the latent
heat ∆Hlat
a network of inverted micelles (drawn as circles) arises around the
sulfonic acid groups of nafion. Water molecules can be transported
through this network by building H3 O+ ions together with protons.
This transport of hydrated protons through the membrane is discribed
(37)
q phase = qw ∆Hlat .
by the Grotthus-Mechanism and can be modeled macroscopically
like a diffusion (cf. [26]). 3) Liquid equilibrated transport mode: for
• The reaction heat qa,c
rxn is generated during the electrochemical reactions in the catalyst layers, which
depends on the anodic and cathodic reaction entropies ∆Sa,c , the respective reaction rates ia,c , the
overpotential ηa,c and the temperature T
qa,c
rxn
= ia,c
T
∆Sa,c
ηa,c −
2F
14 6 λ 6 22 more and more connections between the micelles are
expanded to channels, which are filled with liquid water. A coherent
liquid phase with well-defined hydraulic pressure is formed. The fraction of already expanded channels in a considered volume is labeled
with S.
.
(38)
rates (cf. Eqs. (29), (30) )

a,c
 0 in Ωgdl
qe = −ic in Ωccat

ia in Ωacat
e,p
• The ohmic heat qohm is produced from the electronic and protonic current ∇φe,p and can be expressed as
e,p
qohm = σe,p (∇φe,p ) .
 e
+ q phase
in Ωa,c
qohm
gdl
p
qT = qeohm + qohm + q phase + qarxn in Ωacat .
 e
p
qohm + qohm + q phase + qcrxn in Ωccat
(40)
Electron transport
The electronic potential φe can be described using
a Poisson equation
∇ · (−σeκ ∇φe ) = qe
in Ωa,c
κ ,
κ = gdl, cat .
(42)
2.2 Membrane model
2.2.1 Water transport
The model of water transport in the membrane is
based on articles from Weber and Newman [25], [26].
They developed a detailed steady-state model of a perfluorinated sulfonic acid ionomer. In addition to water
transport, structural effects of the membrane are integrated in the model, e.g. membrane swelling by water
uptake. That model has been extended in order to obtain a dynamic model of water transport in the membrane.
Depending on the membrane water content λ, water
transport is described by two different transport mechanisms (Fig. 3):
(39)
The total source term in Eq. (35) is modeled layerdependent as
2.1.4
The two transport modes of water in the membrane are
illustrated ([25]). 1) For λ 6 2, the membrane is nearly impermeable
for water. 2) Vapor equilibrated transport mode: For 0 6 λ 6 14,
• For 2 6 λ 6 14, water transport is driven by the
gradient ∇µH2 O of the chemical potential of water vapour. This is called the vapour equilibrated
transport mode.
• For 14 6 λ 6 22, water transport is driven by the
gradient of the hydraulic pressure of liquid water
∇pl . This is called the liquid equilibrated transport
mode.
(41)
The electronic conductivity σeκ is assumed to be a layerdependent constant. In the anodic and cathodic catalyst layers, electrons are produced and consumed in
the electrochemical reactions respectively. The source
term qe is linked directly to the corresponding reaction
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Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
In a macroscopic representative volume these two
transport mechanisms occur in parallel. They are superposed linearly with the fraction of expanded channels, S , as weighting factor. In this manner one obtains
the following expression for the total water flux in the
membrane
ξl,v are the electroosmotic coefficients. A more detailed
description of these transport parameters can be found
in [26].
In order to obtain the partial differential equations that describe the dynamic water transport, a timedependent continuity equation is considered
~NH O =
2
= S[−Dl ∇pl − Kl ∇φ p ] + (1 − S)[−Dv ∇µH2 O − Kv ∇φ p ] ,
{z
} |
{z
}
|
~N v
H2 O
~N l
H2 O
(43)
∂t cH2 O + ∇ · ~NH2 O = 0 .
where NHl,v2 O are the two contributions to the water flux,
corresponding to the liquid equilibrated and vapour
equilibrated transport mode, respectively. The fraction
of expanded channels S(r) can be calculated as an integral over the normalised differential volume of channels V (ρ)
(50)
The molar water concentration cH2 O in the membrane
is related to the membrane water content λ by
cH2 O =
λ
,
Vmλ
(51)
with Vmλ = Vm + λ VH2 O .
Z ∞
S(r) =
(44)
V (ρ) dρ ,
VH2 O and Vm are the molar volume of water and the molar volume of the dry membrane, respectively. According to the two transport mechanisms, the membrane
water content λ is composed of two parts
r
where r is a specific channel radius, and φ p is the protonic potential (Section 2.2). The transport parameters
Dl,v and K l,v are given by
Dl (pl ) = αl +
Dv (cv ) = αv +
K l (pl ) =
!
p
σm,l ξ2l
F2
p
σm,v ξ2v
F2
p
σm,l ξl
F
λ(pl , cv ) = λv (cv ) + S(pl ) · (λmax
− λmax
v ).
l
VH2 O ,
(45)
,
(46)
and
(47)
λv (cv ) =
(48)

C·Θ c
n
n+1
 λ [ 1−Θaa cvv ][1−(n+1)(Θa cv ) +n(Θa cv ) ]
m
1+(C−1)Θa cv −C·(Θa cv )n+1
λm C·n(n+1)
1 λv (cv ) = 2
1+C·n
Θa
λmax = lim
v
cv →
.
(54)
The two cases correspond to S = 0 and S > 0 , respectively. Substituting Eqs. (51) and (53) into Eq. (50) leads
to
pl is the hydraulic pressure of liquid water, and cv is
the vapour saturation of water vapour, which is linked
directly to the relative humidity RH of water vapour
via RH = Θa · cv . Here Θa is the activity coefficient of
water vapour. The hydraulic pressure pl is related to a
critical radius rc by
2 σH2 O cos θm
.
rc
(53)
Following Thampan [27], λv (cv ) can be parameterised
using a Brunauer-Emmett-Teller (BET) equation. C and
n are parameters of the BET model, see [27] for details.
p
σ ξ
K v (cv ) = m,v v .
F
pl = −
(52)
Vm
∂t [λv + (λmax
− λmax
v ) · S(pl )] +
l
(VH2 O · λ +Vm )2
+ ∇ · (~NHl O + ~NHv O ) = 0 .
2
(55)
2
(49)
On a microscopic scale, the two transport modes do not
exist in parallel. A connection between two micelles
(cf. Fig. 3) is either collapsed or expanded, nothing
in between. In the first case water is transported in
the vapour equilibrated mode; in the second case it is
transported in the liquid equilibrated mode. Water that
is transported in different modes cannot be influenced
by each other. Hence Eq. (55) can be separated into two
equations, one for each transport mode. With respect to
σH2 O is the surface tension of water. For a specific
value of pl all channels in the hydrophobic membrane with radius r > rc are filled with liquid water.
Consequently the fraction of expanded channels is
a function of the hydraulic pressure, S = S(pl ). The
intrinsic coefficients αl,v describe the remaining water
p
transport if the protonic conductivity σm,l,v vanishes.
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Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
max
(λmax
l −λv )Vm
2
· S0 (pl )
2
· λ0v (cv )
swelling
Bl (pl , cv ) =
coefficients
Bv (pl , cv ) =
diffusion
Dl (pl ) = S(pl ) · Dl (pl )
coefficients
Dv (pl , cv ) = (1 − S(pl )) RTcvΘa Dv (cv )
convection
Kl (pl ) = S(pl ) · K l (pl )
(VH2 O ·λ(pl ,cv )+Vm )
Vm
(VH2 O ·λ(pl ,cv )+Vm )
NHl 2 O and NHv 2 O are the water flux in the liquid and the
vapour equilibrated transport mode, respectively. Both
contributions to the water flux are given by Eq. (43).
κmem
is the heat conductivity in the membrane. Since
T
there is no gas phase in the membrane, there is no associated latent heat for the condensation and evaporation of water in the membrane. Furthermore, no chemical reactions take place in the membrane. Thus, the
only heat source term in the membrane originates from
p
Ohmic heating qohm (Section 2.1.3)
p
coefficients
Table 1.
in Ωmem .
qT = qohm
Kv (pl , cv ) = (1 − S(pl )) · K v (cv )
(62)
Coefficients for the membrane model (56), (57)
2.3
the parameterisations of λv (cv ) and S(pl ) one obtains
Bl (pl , cv )∂t pl − ∇ · [Dl (pl )∇pl + Kl (pl )∇φ p ]
In analogy to Eq. (41) for the electronic potential,
the distribution of the protonic potential φ p in the catalyst layers is described by a Poisson equation
= 0,
(56)
Bv (pl , cv )∂t cv − ∇ · [Dv (pl , cv )∇cv + Kv (pl , cv )∇φ p ] = 0 .
(57)
p
∇ · (−σcat ∇φ p ) = q p
Vr,s (λ) = 1 −Vr,w .
(63)
p
2.2.2 Heat transport in the membrane
In contrast to the gas diffusion and catalyst layers
there is no gas phase in the membrane. The thermal
energy density is proportional to the two condensed
phases: the solid phase (index s), i.e. the nafion framework, and the liquid phase (index w) within the membrane
and
a,c
in Ωcat
.
The protonic conductivity σcat is assumed to be constant throughout the catalyst layers. The source term is
given by the reaction rates (Eqs. (29), (30) )
The transport parameters in these equations are listed
in the Table 1.
ε = Vr,s (λ)ρs Hs +Vr,w (λ)ρw Hw (T ) ,
MH2 O
with Vr,w (λ) =
ρw (Vm + λVH2 O )
Proton transport in the membrane and in the catalyst layers
qp =
ic in Ωccat
.
−ia in Ωacat
(64)
In the membrane the transport of protons is linked to
the water transport [25]. Similar to Eq. (43) for the
water flux, the expression for the flux of protons ~N+
includes both a liquid and a vapour equilibrated term
[26]. Both contributions are linearly superposed, with
the fraction of expanded channels S as a weighting factor. The protonic current density~i p is linked to the protonic flux via ~i p = F ~N+ .
(58)
(59)
(60)
p
F ~N+ =~i p = S[−K +
l ∇pl − σm,l ∇φ p ]+
p
+ (1 − S)[−K +
v ∇µH2 O − σm,v ∇φ p ] .
The two contributions to the thermal energy density ε
are weighted by the relative volume Vr,i of phase i compared to the total volume of the membrane Vmλ , Eq. (52).
Taking into account heat conduction and convective
heat flux (Eq. (35)) one obtains the heat balance equation for the temperature T
(65)
The two parameters K +
l,v reflect the influence of the
electroosmotic drag on the transport of protons in the
two transport modes
p
K +l (pl ) =
∂t (Vr,s ρs Hs (T ) +Vr,w ρw Hw (T )) +
+ ∇ · ~NHl 2 O Hw (T ) + ~NHv 2 O Hg (T ) − ∇ · (κmem
T ∇T ) = qT .
σm,l (pl )ξl
VH2 O ,
F
p
σ (c )ξ
K +v (cv ) = m,v v v .
F
(61)
8
(66)
(67)
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
p
The protonic conductivities σm,(l,v) are parameterised
using an Arrhenius approach

10−9h


i

1
1
15000
1.5
p
−
σm,(l,v) = 50( fl,v − 0.06) exp
.
R
Tre f i
T
h


15000
1
1
1.5
 50(0.39) exp
R
Tre f − T
(68)
The three cases correspond to

fl,v < 0.06

0.06 6 fl,v 6 0.45 .

fl,v > 0.45
Starting from a solenoidal Poisson equation ∇ ·~i p = 0,
Eq. (65) is applied. Rearranging yields
p
∇φ p + Kl+ ∇pl + Kv+ ∇cv = 0
∇ · − σmem
in Ωmem ,
(69)
where the transport parameters are given by
Kl+ (pl ) = S(pl )K +
l (pl )
Kv+ (cv ) =
Coupling diagram of the PEM fuel cell model. The trans-
port mechanisms and the solution variables (state variables) of the
and
(1 − S(pl ))RT Θa K +
v (cv )
cv
Figure 4.
(70)
.
corresponding PDEs are written in the boxes. Each arrow indicates
a coupling between two PDEs, the coupling state variables that are
(71)
contained in the PDEs are noted at the arrows.
The effective protonic conductivity is again modelled
by superposing the two membrane transport mechanisms
gas channels is assumed to be removed by the gas flow
immediately. Consequently, the boundary value of the
liquid water saturation sw at the boundaries RI,II is set
to zero. The boundary value of the electronic potential at RI is the electric fuel cell potential, φe = Ucell . At
the boundary RII the electronic potential is set to zero,
φe = 0 . It is assumed that the temperature of the bipolar plates and the gas temperature in the gas channels
is constant. Therefore, the boundary value of the temperature at RI,II is constant. Similarly, T is set to room
temperature at boundary RIII . For the flow ~Nw of liquid water and the flow ~Ngk of the gas species k (with
k = H2 , O2 , H2 Ov , N2 ), Neumann no-flow conditions are
defined at RIII
p
p
p
σmem
(pl , cv ) = S(pl )σm,l (pl ) + (1 − S(pl ))σm,v
(cv ) . (72)
2.4
Coupling and boundary conditions
Table 2.4 gives an overview of the transport
mechanisms that are described in Sections 2.1 to 2.3. In
addition to the solution variables and the subdomains
of the model, the numerical discretisation methods are
listed.
In Fig. 4 the coupling between the various submodels
is illustrated.
The subdomain boundaries R1 to R4 are plotted as
dashed lines in Fig. 5. Outer boundaries of the model
domain are indicated by RI to RIII .
~Nw ·~nIII = 0
~Ngk ·~nIII = 0
Outer boundaries The interfaces between the gas
channels and gas diffusion layers are labelled RI and
RII (Fig. 5). The gas channels are not spatially resolved
in the fuel cell model. A linear drop of the gas pressure
pg is taken into account between the boundaries RI,II .
Assuming counterflow conditions, the gradient in gas
pressure at RI is orientated in opposite direction to that
at the domain boundary RII . Liquid water reaching the
with ~Nw = φ ρw sw ~uw ,
with ~Ngk = φ ρg sg ck ~ug .
(73)
(74)
~nIII is the outer normal unit vector on RIII . These equations lead to boundary conditions for the water and
gas velocities and thereby for the global velocity (Section 2.1). Furthermore, no electrons and protons can
leave the fuel cell across RIII . Hence, two more Neuman no-flow conditions are defined for the electronic
9
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
transport mechanism
solution
subdomain
variables
two-phase flow
discretisation
No.
sw
a,c
Ωgdl
~uglob , pglob
cH2 , cO2
a,c
∪ Ωcat
I
method
finite volume
mixed
finite elements
a,c
a,c
Ωgdl
∪ Ωcat
I
finite volume
p l , cv
Ωmem
II
finite volume
proton transport
φp
Ωmem ∪ Ωa,c
cat
III
finite elements
electron transport
φe
a,c
a,c
Ωgdl
∪ Ωcat
I
finite elements
T
a,c
a,c
Ωgdl
∪ Ωmem ∪ Ωcat
I+II
finite volume
gas species flow
cH2 Ov , cN2
memb. H2 O transp.
heat transport
(energy balance)
Table 2. The transport mechanisms and corresponding solution variables are shown in the table. Different discretisation mechanisms are
applied depending on the transport mechanism.
Assuming that water from the membrane cannot leave
the fuel cell across RIII , two more Neuman no-flow
conditions for each of the transport modes within the
membrane can be formulated
~NHl O ·~nIII = 0 ,
2
~NHv O ·~nIII
2
and
(77)
(78)
= 0.
Using Eq. (76) and the definitions of the water fluxes
from Eq. (43), one obtains
∇pl ·~nIII = 0 ,
and
∇cv ·~nIII = 0
(79)
(80)
as boundary conditions at RIII for the hydraulic pressure pl and the vapour saturation cv .
Figure 5. The model domain is shown.
R1 to R4 indicate bound-
aries between subdomains. The solid lines correspond to the outer
boundaries RI to RIII .
Inner boundaries R1 and R4 represent the boundaries
between the gas diffusion layers and the catalyst layers
on the cathodic and anodic side, respectively. The gas
diffusion layer does not conduct protons, and therefore
the protonic current across R1,4 vanishes
and the protonic current, respectively
~ie ·~nIII = 0
~i p ·~nIII = 0
with ~ie = −σeκ ∇φe ,
p
with ~i p = −σκ ∇φ p .
~i p ·~n1,4 = 0
(75)
p
with ~i p = −σcat ∇φ p .
(81)
Additionally, the protonic current i p · n is set to zero
at these boundaries. The two-phase flow, gas species
(76)
10
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
flow, electron transport and thermal submodels are defined within both the gas diffusion and catalyst layers (modeling region I). Hence, there are no boundary
conditions necessary for the solution variables of these
submodels at R1,4 . In the same manner no boundary
conditions are needed for the thermal and the proton
transport submodels at the membrane borders R2 and
R3 . At R2,3 the two membrane variables pl and cv are
coupled to the variables of the two-phase flow and gas
species flow submodels.
The continuity of the capillary pressure pc was chosen
as a boundary condition at R2,3 [28]
Rearranging this expression and using Eq. (83), a
boundary condition for the vapour saturation is obtained
2VH2 O
Θcat
a
.
exp
p
g mem
Θa
RT
cat
cv |mem = cH2 Ov |cat ·
In addition to Eqs. (83) and (87), the liquid water flux
across R2,3 must be continuous
~NHl O ·~n
2
mem
pc |mem = pc |cat .
pl |mem = −pc (sw )|cat
~NHv O ·~n
2
(83)
to the water saturation sw in the catalyst layer.
Assuming that the water on the catalyst side is in thermodynamical equilibrium with the water on the membrane side, the chemical potential must be continuous
across the boundaries R2,3
µH2 O |mem = µH2 O |cat .
mem
(88)
.
v
= −~NgH2 O ·~n
cat
.
.
and
(89)
(90)
3 Model discussion
3.1 Two-phase flow in the porous regions and in the
membrane
Two-phase flow of water is described in our model
with the so-called global pressure formulation of the
two-phase equations. With this ”full” two-phase approach it is possible to quantitatively model the water
transport, even if the water saturation is high in certain
parts of the cell. On a spatial scale that is relevant for
technical fuel cell applications, parts of the cell can be
dry, and other parts can be flooded with liquid water. If
the liquid water saturation is high the resulting water
transport mode is mainly driven by the liquid equilibrated transport mechanism, that is, the water transport is driven by a gradient in the hydraulic pressure
and a gradient in the protonic potential. Within a dry
part of the cell the dominating water transport contribution is driven by a gradient in the membrane water
concentration. Such a situation is well-described by the
model.
The parameters for the two-phase flow in the gas diffusion layers were determined by fitting Eqs. (10), (13)
and (14) to numerical data which was obtained from
micro scale simulations [16]. This method is based
on the 3D reconstruction of the microstructure of the
GDL by synchrotron-tomography. The method predicts the 3D liquid water saturation distribution for a
given channel pore radius. As a result a parameteri-
(84)
g
= (µwH2 O + µH2 O )
cat
~n|lay denotes the normal unit vector on the boundaries
R2,3 pointing outwards layer lay .
In the catalyst layers the chemical potential can be
seperated into the contributions of the liquid phase and
gas phase. The chemical potential in the membrane is
composed of the contributions of the liquid mode and
the vapour mode
(85)
Using the thermodynamic relations [21]
•
•
•
•
cat
The water vapour flux across R2,3 must also be continuus. However, as there exists no gas phase in the membrane, the orthogonal component of the gas phase velocity ~ug is zero. That is, water vapour flux across R2,3
is caused only by diffusion
~ug ·~n|cat = 0
mem
= −~Nw ·~n
(82)
As there exists no coherent gas phase in the membrane,
the gas pressure is set to zero. Therefore, the capillary
pressure in the membrane, pc = pg − pl = −pl equals the
negative hydraulic pressure, which then is linked via
(µlH2 O + µvH2 O )
(87)
µHl 2 O = VH2 O pl ,
µvH2 O = RT ln(Θmem
a cv ) ,
µwH2 O = VH2 O pw ,
and
g
cat
µH2 O = RT ln(Θa cH2 Ov ) +VH2 O pg ,
the activity coefficient of the membrane is calculated
ln(Θmem
a cv )|mem =
VH2 O VH2 O
cat
(pw + pg ) −
pl
.
= ln(Θa cH2 Ov ) +
RT
RT mem
cat
(86)
11
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
sation of the capillary pressure pc (sw ) as a function of
the liquid water saturation is obtained from the poremorphology method.
Phase transitions are accounted for in the model. However, constant evaporation and condensation rate constants are included in Eq. 18. The description of the
evaporation and condensation processes is oversimplified in the model.
Water transport in the membrane is described by the
superposition of two transport modes [26], the liquid
equilibrated and the vapour equilibrated mode. Furthermore, the swelling behaviour of the membrane due
to water uptake is accounted for. The transport processes for water and protons in the membrane are well
described for PFSA-based membrane materials. The
following coupling conditions between the membrane
and the catalyst layers were assumed: continuous flux
of liquid water at the interface, continuous diffusive
flux of water vapour, continuity of the capillary pressure, and the continuity of the electrochemical potential.
pressure curves, relative permeabilities, diffusion constants), (iii) the inclusion of the multi-step reaction for
the oxygen reduction reaction (eg. as described in [29]),
(iv) account for varying coverage fractions of the catalyst sites with intermediate reaction products.
3.4
Heat transport
The heat transport is modelled in all three phases
in the gas diffusion layers and in the catalyst layers;
heat transport in the membrane is modelled in the solid
and liquid phases only. Two heat transport mechanisms are accounted for, heat conduction and heat convection. It is assumed that heat conduction in the solid
phase is dominant in comparison to heat conduction
in the two other phases. The temperature of the three
phases are assumed to be equal. The source terms
for the temperature equation includes three contributions: latent heat due to the phase transition of water,
reaction heat due to the electrochemical reaction, and
ohmic heat production due to the electronic and protonic current. The heat conductivity of the solid phase
is assumed to be linearly dependent on the water saturation.
3.2
Gas species flow
Mass transport limitation due to blockage of the
gas pores is described accurately. Multicomponent diffusion is not considered in the model at this stage,
although mixture-dependent diffusion coefficients are
important to account for.
4
Discretisation methods
The discretisation of the full fuel cell model, described in Section 2 is based on mass conservative
mixed finite element and finite volume schemes for
mass, momentum, and energy balance equations. Standard Galerkin finite element methods for the discretisation of the electron and proton potential equations
were used.
Concerning the discretisation of the two phase flow
equations in a porous media we refer to [30–33].
The reactive transport equations for the species in the
gas phase are discretised by using a self adaptive finite
volume scheme that is based on rigorous a posteriori
error estimates for so called weakly coupled systems.
For details we refer to [34–37].
Finally, these transport equations are coupled to the
potential equations for the proton and electron flow
through the reactive source terms. The potential equations are solved by a Galerkin finite element method
using standard piecewise linear basis functions. In order to cope with the strongly non-linear source term, a
time relaxation of the stationary potential equations is
used. This time stepping approach also incorporates a
non-linear fixed point iteration for the solution of the
resulting non-linear coupled system.
3.3
Charge transport
The electric potential is modeled by two seperate
potential fields for the electrons and the protons. This
allows for the calculation of locally distributed effects
in the membrane electrode assembly.
The source terms for the potential equations are modeled by the Tafel approach. Thereby, the best convergence behaviour of the model is obtained although the
description of the transport processes and the electrochemical reactions in the catalyst layers is oversimplified. In order to describe the electrochemical reaction rates more accurately it is straight-forward to replace these expressions by the Butler-Volmer equations
for the reaction rates on the cathode and anode side.
Thereby, it is assumed that the electrode structure is homogeneous.
However, real porous gas diffusion electrodes exhibit
an agglomerate structure on the microscale, that is, an
agglomeration of the carbon support, the ionomer, and
the platinum particles occurs. A further step for the
model refinement is the implemementation of an analytical agglomerate model of the catalyst layers into our
numerical model. Further effects that are of importance
for the model refinement with respect to the catalyst
layer description are: (i) account for the diffusion of
oxygen to the reaction sites in the dissolved form (Henrys law), (ii) the consideration of seperate two-phase
flow parameterisations in the catalyst layers (capillary
5
Numerical experiments
In this section results of numerical experiments
with the numerical PEM fuel cell model that is described in Section 2 are presented. The simulations
were performed on the 2D grid shown in Fig. 6 .
12
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Figure 6. The grid geometry used for simulations. The figure shows
the five layers of the fuel cell.
Figure 8. The mass fraction cH2 of hydrogen after 0.03 s and
0.10 s. Due to the electrochemical reaction, a depletion of hydrogen in the anodic catalyst layer can be observed. (a) and (b) show
Distribution of the liquid water saturation sw at different
points in time. (a) The evaporation process is already visible after
0.001 s. Figures (b) and (c) show the further progression of the evap-
Figure 7.
the spatial dissolved distributions of cH2 . In (c) the hydrogen mass
fraction is displayed along a cross section through the anodic gas
diffusion and catalyst layers.
oration process. Due to the boundary conditions, a residual saturation remains at the left and right outer boundaries. In the membrane,
the liquid water saturation is not defined, and thus its value there is
influence of water production by the electrochemical
reaction by far within the first 0.1 s .
Fig. 8 shows the distribution of the hydrogen mass
fraction cH2 on the anode side. A depletion of cH2
towards the catalyst layer can be seen. This is caused
by the dissipation of hydrogen during the hydrogen
oxydation reaction in the anodic catalyst layer.
The corresponding distributions of the oxygen mass
fraction cO2 are depicted in Fig. 9 . Analogous to cH2 at
the anode side, a decrease in cO2 towards the cathodic
catalyst layer is obtained, which is due to the oxygen
reduction reaction taking place within that layer.
Alongside the dissipation of oxygen, water vapour is
produced in the cathodic catalyst layer. This can be
seen by an increase of the water vapour mass fraction
in Fig. 10 .
Figs. 10a) to c) show the distribution of the water
vapour mass fraction cH2 Ov . Water vapour is produced
in the cathodic catalyst layer and transported through
the gas diffusion layer. The catalyst layer is also
spatially resolved, and therefore, it is possible to see
that water vapour is not produced homogeneously
within the catalyst layer, but particularly in the left half
of it. There the electrochemical reaction rate exceeds
the values that are reached nearby the membrane. Furthermore, water vapour is transported from the anode
side across the membrane to the cathode side. This
happens due to the higher initial value for the water
vapour mass fraction on the anode side. Note that the
transport of water vapour across the membrane is not
arbitrary and with no relevance to the simulation.
5.1
Liquid water saturation
Fig. 7 shows the spatial distribution of the liquid
water saturation sw at three different times. One obtains a decrease of sw due to the evaporation of liquid
water. Additionally, a small amount of liquid water
enters the membrane from the anodic side. This can
be seen by the slight decrease of sw in the anodic catalyst layer in Fig. 7c) . The initial value of the water
vapour mass fraction corresponds to a dry fuel cell.
Consequently, the evaporation process dominates the
behaviour of sw on the examined time scale of about
0.1 s . The achieved simulation duration of 0.16 s is not
long enough to investigate condensation processes of
the water vapour produced by the chemical reactions
(Section 5.2).
5.2
Gas mass fractions
The following figures show the simulated distributions of the mass fractions of the various gas species
at different points in time. Here, the phase transitions
of liquid water and water vapour were not accounted
for. By doing so it was possible to investigate the
effect of the electrochemical processes more clearly:
if the phase transitions are taken into account, then
the evaporation of liquid water and the resulting
increase of the water vapour mass fraction exceeds the
13
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Figure 9. The mass fraction cO2 of oxygen after 0.03 s and 0.10 s.
Due to the electrochemical reaction, a depletion of oxygen in the cathodic catalyst layer can be observed. (a) and (b) show the spatial
dissolved distributions of cO2 . In (c) the oxygen mass fraction is dis-
Figure 10.
The mass fraction
cH2 Ov of water vapour neglecting
phase transitions. In (a) to (c) the spatial distributions of cH2 Ov are
shown at three different times, 0.005 s, 0.030 s and 0.100 s. Water
played along a cross section through the cathodic gas diffusion and
catalyst layers.
vapour is produced in the cathodic catalyst layer as a result of the
oxygen reduction reaction. Additionally, water vapour is transported
a gas diffusion process, but corresponds to the vapour
equilibrated transport mechanism in the membrane, as
described in section 2.2.1.
through the membrane as can be seen by the water vapour depletion
on the anode side. (d) shows the water vapour mass fraction along a
cross section through the cell.
6
Conclusions and outlook
Our model accounts for all important transport
processes in a PEM fuel cell. Due to the high model
complexity of the system of coupled nonlinear partial
differential equations, the computational costs for the
simulation are rather high. Therefore, the simulation
domain of this model is restricted to small geometrical dimensions. It is beyond the scope of this paper to
demonstrate numerical solutions that are relevant for
technical applications. Even if performed on a parallelised computer cluster, simulations of fuel cells with
realistic spatial dimensions on time scales that are relevant for technical applications are expensive to accomplish with our model implementation. At present, the
model can be applied to analyse and explore the importance of the various transport processes occuring
in a PEM fuel cell. Solving the numerical model on a
larger discretisation mesh allows for comparison of the
simulation results with time-dependent measurement
data obtained with small test fuel cells. From this comparison it is possible to study what transport processes
are important under certain operating conditions. A
further future step is the development of reduced twophase models with fewer degrees of freedom that still
5.3
Temperature
Fig. 11 shows the temperature distribution at two
different times. It can be seen that the main source
of heat is localised inside the cathodic catalyst layer.
This is due to the reaction heat of the oxygen reduction
reaction. Phase transitions of liquid water and water
vapour were neglected in this simulation. Therefore,
no temperature changes due to latent heat can be observed. The produced heat is transported through the
cathodic gas diffusion layer and across the membrane
to the anode side. Due to the higher heat conductivity of the gas diffusion layer, the heat transport process
there is faster than in the membrane. Since the magnitude of the temperature profile across the cell is so
small, the results of this work are consistent with previous assumptions that a single steady-state PEM fuel
cell operates under isothermal conditions. However,
the apparent lack of temperature dependence may be
due to the short simulation time. If longer time scales
were computationally feasible, then one would expect
a more significant temperature profile. This local temperature variation could play a more significant role in
PEM stacks, for example, where time-dependent nonisothermal behaviour is observed.
14
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
[5]
[6]
[7]
[8]
Figure 11. The temperature distribution after
0.024 s and 0.090 s
[9]
are shown. As can be seen in Figs. (a) and (b), the main heat source
is located in the cathodic catalyst layer. This is due to the exothermal
reaction occuring there. (c) shows the respective cross sections of
the temperature distribution through the cell.
[10]
capture the dominating two-phase transport processes
in all sub-domains. These reduced two-phase models
require less computational costs, and can be used to
perform parameter studies for fuel cells that are relevant for technical applications.
[11]
Acknowledgements
The authors would like to acknowledge the important contributions by Karsten Kuehn who is now with
the BMW group.
[12]
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Paper No. FC-07-1193
A
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Parameters and constants
Symbol
Explanation
Value
Unit
Ref.
1.0 · 107
m
m
est.
A
specific active surface
bc
fitted Brooks-Corey exponent
1.7
−
[16]
C
fitting parameter of the BET parametrisation
150
−
[27]
4182.0
J
kg K
[21]
710
[21]
[26]
Cw
Cs
D0
Dm,v
H2 O
e
specific heat capacity of water
specific heat capacity of graphite
(used for the specific heat capacity of
the GDL, the catalyst layers and
the Nafion membrane)
gas diffusion coefficient
diffusion coefficient of water
1.39 · 10−3
J
kg K
m
s
in the membrane (vapour equilibrated mode)
1.8 · 10−5
m
s
1.60219 · 10−19
As
0.909
kg
mol
As
mol
elementary charge
[13]
EW
equivalent weight of the dry membrane
F
|~g|
Faraday’s constant
absolute value of the
acceleration of gravity
96485.3
j0,a
anodic exchange current density
1.0 · 104
j0,c
cathodic exchange current density
7.8 · 10−3
kc
Kcat
condensation rate of water vapour
absolute permeability of the
catalyst layers
100
m
s
A
m
A
m
1
s
2.5 · 10−12
m
est.
absolute permeability of the GDL
7.5 · 10−12
m
est.
Kgdl
9.81
17
[26]
est.
est.
Paper No. FC-07-1193
Symbol
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Value
Unit
Ref.
in the membrane (liquid equilibrated mode)
1.8 · 10−18
m
[26]
kv
evaporation rate of water
98.7 · 10−5
1
Pa s
[19]
m
Van Genuchten exponent
0.95
−
[16]
[21]
[21]
Ksat
Explanation
convection coefficient of water
MH2
molar mass of hydrogen
0.002
MO2
molar mass of oxygen
0.032
MH2 O
molar mass of water and water vapour
0.018
MN2
molar mass of nitrogen
0.028
kg
mol
kg
mol
kg
mol
kg
mol
the pore walls (BET model)
13.5
−
[27]
pd
threshold pressure (Brooks-Corey)
7500
Pa
[16]
R
ideal gas constant
8.314
J
K mol
n
[21]
[21]
maximum number of water layers on
∆Sa
reaction entropy in anodic catalyst layer
−130.7
J
mol K
∆Sc
reaction entropy in cathodic catalyst layer
−65.0
J
mol K
Tre f
reference temperature (25C)
298.15
K
[26]
molar volume of water
1.81 · 10−5
m3
mol
[21]
molar volume of the dry membrane
4.6 · 10−4
m3
mol
[26]
VH2 O
Vm
18
Paper No. FC-07-1193
Symbol
α
ηw
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Explanation
Value
charge transfer coefficient
viscosity of liquid water
κmem
T
heat conduction coefficient of the membrane
λmax
l
maximum water content of the
liquid equilibrated membrane
λm
Ref.
0.11
−
1.0 · 10−3
Ns
m
[21]
0.19
W
Km
est.
22
−
[26]
1.8
−
[26]
14
−
[26]
est.
reference water content of the membrane
(BET model)
λmax
v
Unit
maximum water content of the
vapour equilibrated membrane
φcat
porosity of the catalyst layers
0.40
−
φgdl
porosity of the gas diffusion layers
0.78
−
Θcat
a
activity coefficient in the
catalyst layers
1
−
[21]
activity coefficient in the membrane
1
−
[21]
Θmem
a
θm
contact angle of water in the membrane
ρs
mass density of graphite
[26]
90.02
(used for the GDL, the catalyst
ρw
σH2 O
[21]
998.2
kg
m
kg
m
0.07
N
m
[21]
layers and the Nafion membrane)
2000
mass density of liquid water
surface tension of water
19
[21]
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Symbol
Explanation
Unit
aH2 O
water activity
−
kg
m3
kg
m3
kg
bar m
kg
m
Bl
membrane swelling factor (liquid equilibrated case)
Bv
membrane swelling factor (vapour equilibrated case)
Bl
membrane swelling coefficient (liquid equilibrated case)
Bv
membrane swelling coefficient (vapour equilibrated case)
cH2
mass fraction of hydrogen
−
cO2
mass fraction of oxygen
−
mass fraction of water vapour
−
cN2
mass fraction of nitrogen
−
cv
vapour saturation in the membrane
−
cvH2 O
molar water concentration (vapour equilibrated case)
cH2 O
total molar water concentration
mol
m3
mol
m3
cH2 Ov
Cs
specific heat capacity of solid
carbon matrix
J
kg K
Cg
spec. heat capacity of gas mixture
J
kg K
Cw
spec. heat capacity of liquid water
J
kg K
d
d j,l
capillary diffusion coefficient
diffusive numerical flux across an edge S j,l
1
s
-
Ddisp
Scheidegger dispersion tensor
m
s
Ddi f f
diffusion coefficient
m
s
D
diffusion-dispersion matrix
Dl
diffusive transport coefficient (liquid equilibrated case)
Dv
diffusive transport coefficient (vapour equilibrated case)
Dl
pressure diffusion coefficient (liquid equilibrated case)
Dv
diffusion coefficient (vapour equilibrated case)
m
s
kg
bar m s
kg
J ms
kg
bar m s
kg
ms
f
right hand side
fl
volume fraction of water (liquid equilibrated case)
−
fv
volume fraction of water (vapour equilibrated case)
−
fw
fractional flow rate of liquid water
−
fg
fractional flow rate of gas mixture
−
-
20
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Symbol
Explanation
Unit
~g
gravitational acceleration
Hs
specific enthalpy of solid
m
s
carbon matrix
J
kg
Hg
specific enthalpy of gas mixture
J
kg
Hw
specific enthalpy of liquid water
J
kg
Latent heat
J
ms
ia
reaction rate in anodic catalyst layer
A
m
ic
reaction rate in cathodic catalyst layer
A
m
~ie
electronic current density
A
~i p
protonic current density
A
ja
anodic partial current density
A
m
jc
cathodic partial current density
A
m
kr,w
relative permeability of liquid water
−
kr,g
relative permeability of gas
−
K
absolute permeability
m
Kl
convective transport coefficient (liq. equil. case)
kg
V ms
Kl+
Flux parameter (liq. equil. case)
Kv
convective transport coefficient (vapour equil. case)
m2
Vs
kg
V ms
Kv+
Flux parameter (vapour equil. case)
∆Hlat
Kl
convection coefficient (liq. equil. case)
Kl+
potential parameter (liq. equil. case)
Kv
convection coefficient (vapour equil. case)
Kv+
potential parameter (vapour equil. case)
Mg
Molar mass of gas mixture
21
mol
V ms
kg
V ms
m
Vs
kg
V ms
Jm
V mol s
kg
mol
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Symbol
Explanation
Unit
~Nw
liquid water flux in membrane
~N+
molar protonic flux
kg
ms
kg
m2 s
kg
m2 s
kg
m2 s
kg
ms
mol
s
pc
capillary pressure
Pa
pg
gas pressure
Pa
pglob
global pressure
Pa
pH2 O
partial pressure of water vapour
Pa
pl
hydraulic flux in the membrane
bar
psat
saturation pressure of water
Pa
pw
liquid water pressure
Pa
qe
source term of electronic current
qg
source density of gases due to evaporation
source density of water vapour
A
m
kg
ms
kg
ms
kg
ms
kg
ms
ohmic heat
J
ms
source term of protonic current
A
m
heat source term, phase transition
J
ms
qa,c
rxn
heat of reaction in anod./cath. catalyst layer
J
ms
qT
source density of temperature
qw
source density of liquid water due to condensation
J
ms
kg
ms
r
radius of liquid water channel
m
rc
critical radius of water channel
m
RH
relative humidity
−
Rohm
ohmic resistance
Ω
r phase
phase transition rate
−
~Ngk
flux of k-th gas component
~N l
H2 O
water flux (liq. equil. case)
~N v
H2 O
water flux (vapour equil. case)
~NH O
2
total water flux in membrane
2
qH
g
source density of hydrogen
2
qO
g
source density of oxygen
2
qH
g
Ov
e,p
qohm
qp
q phase
22
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Symbol
S
Eplanation
Unit
fraction of expanded channels
−
∆Smol
molar reaction entropy
J
mol K
sw
liquid water saturation
−
s∗w
residual saturation
−
sg
gas saturation
−
T
temperature
K
~uw
velocity of liquid phase
m
s
~ug
velocity of gas phase
m
s
~uglob
global velocity
m
s
∆Vmol
change of molar volume
V (r)
differential volume of water channels in membrane
1
m
Vr,s
relative volume of solid phase (membrane)
−
Vr,w
relative volume of liquid phase (membrane)
−
v
xH
2O
molar fraction of water (vapour equil. case)
−
charges transferred per molecule
−
z
23
J
mol Pa
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Symbol
αl,v
Explanation
Unit
intrinsic transport coefficient
(liquid-/vapour equil. case)
mol 2
J ms
ε
thermal energy density
J
m
ηa
overvoltage, anodic catalyst layer
V
ηc
overvoltage, cathodic catalyst layer
V
ηg
viscosity of gas mixture
Ns
m
κT
heat transfer coefficient of porous layers
W
Km
λ
total membrane water content of membrane
−
λg
mobility of gas mixture
m
Ns
λtot
total mobility
m
Ns
λv
membrane water content (vapour equil. case)
−
λw
mobility of liquid water
m
Ns
chemical potential of water
J
mol
µH2 O
chemical potential of water vapour
µlH2 O
chemical potential of water (liquid transport mode)
µvH2 O
chemical potential of water (vapour transport mode)
µwH2 O
chemical potential of liquid water
J
mol
J
mol
J
mol
J
mol
φ
porosity
−
φe
electronic potential
V
φp
protonic potential
V
equilibrium potential difference
V
density of solid carbon matrix (graphite)
kg
m
kg
m
µH2 O
g
∆φeq
ρs
ρglob
global pressure
24
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Symbol
σeκ
p
σκ
p
σm,l
Explanation
Unit
electronic conductivity of layer κ
S
m
protonic conductivity of layer κ
S
m
protonic conductivity of membrane
S
m
(liquid equil. case)
p
σm,v
protonic conductivity of membrane
(vapour equil. case)
S
m
p
effective protonic conductivity of membrane
S
m
ξl
electro-osmotic drag coefficient (liquid equil. case)
−
ξv
electro-osmotic drag coefficient (vapour equil. case)
−
Ω
simulation domain
-
Ωacat
sub-domain: anodic catalyst layer
m
Ωccat
sub-domain: cathodic catalyst layer
m
Ωagdl
sub-domain: anodic gas diffusion layer
m
Ωcgdl
sub-domain: cathodic gas diffusion layer
m
Ωmem
sub-domain: membrane
m
σmem
25
Paper No. FC-07-1193
Steinkamp, Schumacher, Goldsmith, Ohlberger, Ziegler
Index
Explanation
a
anodic
c
cathodic
cat
catalyst layer
e
electronic
g
gas mixture
gdl
gas diffusion layer
glob
global (in a mathematical sense)
H2
hydrogen
H2 O
water
H2 Ov
water vapour
l
liquid equilibrated transport mode
m
membrane
mem
membrane
mol
molar
N2
nitrogen
O2
oxigen
ohm
ohmic
p
phase
re f
protonic
phase transition (liquid water/vapour)
reference
s
solid phase
sat
saturation
tot
total
v
vapor equilibrated transport mode
w
liquid water
+
protonic
26