Mench M.M. Fuel Cell Engines
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250 Transport in Fuel Cell Systems
0
0.5
1.0
100
S
W0
S
NW0
S
NW
(%)
S
W
(%)
0
0
100
k
r,w
, k
r,nw
k
r,w
k
r,nw
k
r,w
+ k
r,nw
Figure 5.31 Typical relative permeability as function of saturation. (Adapted from [41].)
typical relative permeability curve as a function of saturation. The relative permeability
curve for the liquid (nonwetting) phase shows a modest increase at low saturations, because
at low saturations only a small portion of the pores is occupied by the liquid water, whereas
a large portion of the pores is still available for the gas-phase transport. Hence, there is
no significant impact of saturation on the reactant flow. However, at high saturations, the
relative permeability exhibits a considerable increase with small increases in saturation,
because as the available pores are nearly all filled by the liquid water, gas-phase transport
is highly restricted.
In PEFC modeling studies, a typical relative permeability expression for noncon-
solidated sands is adapted to describe the phase transport in DM due to its simplicity
[k
r
=(s
nw
)
3
]. However, recent work has indicated a slightly different relationship is more
appropriate [42]:
k
r−nw
= s
2.16
nw
(5.96)
Since this relationship is a function of morphology, a value of ∼2.5 is suggested for the
value of n unless other data are available.
Four well-established empirical correlations along with the empirical fit by [42] are
presented in Table 5.13.
Table 5.13 Summary of Relative Permeability Models
Relative Permeability
Models Nonwetting Phase Wetting Phase
Wyllie model k
r,nw
= (s
e,nw
)
3
k
r,w
= (1 − s
e,nw
)
3
Corey model k
r,nw
= [1 − (s
e,w
)
2
][1 − s
e,w
]
2
k
r,w
= (s
e,w
)
4
Brooks–Corey model k
r,nw
= (1 − s
e,w
)
2
[1 − (s
e,w
)
(2+λ)/λ
] k
r,w
= (s
e,w
)
(2+3λ)/λ
Van Genuchten model k
r,nw
= (1 − s
e,w
)
1/3
[1 − (s
e,w
)
1/m
]
2m
k
r,w
= (s
e,w
)
0.5
[1 − (1 − (s
e,w
)
1/m
)
m
]
2
Kumbur et al. [42] k
r,nw
= (s
nw
)
2.16
k
r,nw
= (1 − s
nw
)
2.16
a
s
e
represents the effective saturation defined as s
e
= (s − s
o
)/(1 −s
o
).
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5.5 Multiphase Mass Transport in Channels and Porous Media 251
Water
Solid
Solid
r’, r’’ are radii
of curvature
of water–air
interface
r’
r’’
Figure 5.32 Equilibrium at curved interface. (Reproduced from [41].)
5.5.4 Capillary Transport in Fuel Cell Diffusion Media
Capillary Pressure Whenever two or more fluids coexist in a system, there exists a
pressure difference at the interface between the phases due to the interfacial tensions caused
by the imbalance of the molecular forces at the line of contact. The difference between
the pressures of any two phases at the interface is referred as the capillary pressure. The
capillary pressure between wetting and nonwetting phases is defined as
P
c
= P
nw
− P
w
= γ
1
r
+
1
r
=
2γ
r
∗
(5.97)
Here, r
and r
are the two principal radii of the interface between the solid pore surface
and the liquid surface curvature, and r is the mean radius of curvature, and on the same
order as the pore size as shown in Figure 5.32.
Equation (5.97) is also called the Laplace equation for capillary force. The surface
tension of water can be correlated as a function of temperature:
γ
(
N/m
)
=−1.78 × 10
−4
(T ) + 0.1247 (5.98)
where temperature is in Kelvin. Most modeling studies of fuel cells adopt a capillary tube
model, in which the porous media is composed of capillary tubes of different radii when the
mean radius of the tubes is r
∗
. Using this formulation, the variable contact angle produced
by a hydrophobic additive can be included in the model. The capillary force for the tube
model can be written as
P
c
= P
nw
− P
w
=
2γ cos θ
r
∗
(5.99)
where the r
*
is taken as a negative value to assure the signs match between formulations.
It should be noted that upon initially saturating a porous medium, for example, by conden-
sation into a PEFC DM, liquid water remains unconnected and unable to transmit pressure
and drive liquid flow. As saturation level increases, the isolated droplets become connected
between pores, and capillary flow can be initiated.
Capillary pressure increases as pore radius decreases, so that the very small pores
in a catalyst layer can have very high capillary pressure. It is important to emphasize
that, even for very homogeneous porous media, these relationships cannot be applied
on a microscopic level to the entire media due to the complex and highly varied pore
structure in PEFCs, and attempts at modeling the effects are considered global and meant
to capture the qualitative trends using bulk parameters derived that are representative of
the bulk media. Therefore, these parameters must be derived experimentally and are highly
dependent on material properties, and in fuel cells, interfacial effects are not treated in the
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252 Transport in Fuel Cell Systems
Wetting
fluid
Upward
surface
tension force
θ θ
Nonwetting
fluid
Downward
surface
tension force
θθ
Figure 5.33 Illustration of capillary action in a column for (a) hydrophilic walls and (b) hydropho-
bic walls.
bulk approach. Nevertheless, we can understand the basic principles involved in capillary
flow by considering a single capillary column, as shown in Figure 5.33. If the walls of the
capillary tube are hydrophilic to the fluid, a net capillary suction force will draw fluid up
the walls of the tube until the gravity force balances the surface tension force. If the walls
of the tube are hydrophobic to the fluid, the meniscus will be concave down, and the air
will be the wetting fluid.
In hydrophobic DM, the liquid pressure (nonwetting phase) is larger than the gas
(wetting) phase pressure, whereas in hydrophilic DM, gas-phase pressure is higher than
liquid pressure:
p
c
= p
nw
− p
w
p
c
> 0 ⇔ p
nw
> p
w
for θ>90
◦
(i.e., hydrophobic)
p
c
< 0 ⇔ p
nw
< p
w
for θ<90
◦
(i.e., hydrophilic)
(5.100)
The resulting pressure imbalance can cause water to flow from locations of high p
c
to low
p
c
. In fuel cell studies, it is reasonable to assume immiscible flow, meaning we treat the
gas phase and liquid phase as nonmixing (even though some amount of gas phase will
dissolve into the liquid phase). This allows us to separate the phases into nonwetting and
wetting components. In fuel cell DM, once the gas phase is fully saturated with water
vapor, assuming evaporation and condensation does not take place in the DM, liquid water
flow becomes the only mode of water transport across the DM [5]. Water generation in
the catalyst layer, water transport across the membrane, and condensation or evaporation
within the DM pores cause nonuniformity in saturation distribution. As hydrophobic DM
pores are filled by the liquid water, the liquid-phase pressure increases, eventually driving
the liquid water from higher to lower liquid pressure regions. As a result, the liquid water
in the DM is driven via capillary action produced by the liquid saturation gradients in DM.
In an operating PEFC, the higher saturation generally occurs in the catalyst layer, and it
decreases in the vicinity of the channel, implying that capillary transport takes place from
high- to low-saturation regions in DM [43, 44], as depicted in Figure 5.34.
Example 5.13 Comparison of Capillary and Gravitational Effects Compare the expected
capillary pressure force on a 0.1-mm water droplet (ρ = 971 kg/m
3
) in a hydrophobic (θ =
105
◦
) diffusion media pore with the same diameter to the gravitational force on the same
droplet at 80
◦
C.
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5.5 Multiphase Mass Transport in Channels and Porous Media 253
Reactant flow channel
Diffusion media
Catalyst
layer
Liquid saturation, s
l
01
Water
Generation
01
Gas saturation, s
g
Liquid flow
Reactant flow
Water
generation
Figure 5.34 Capillary-induced liquid transport in a hydrophobic fuel cell DM.
SOLUTION The gravitational body force is simply calculated as
F
g
= ρVg = (971 kg/m
3
)
4
3
π
0.1
1000
mm
m
3
9.8m/s
2
= 4 × 10
−8
N
The capillary pressure is estimated as
p
c
= p
nw
− p
w
= γ
1
r
+
1
r
≈
2γ
r
= 2 ×
0.062 N/m
0.1/1000 m
= 124 N/m
2
To solve for the resultant force on the droplet, we multiply by the cross-sectional area of
the droplet:
F
cap
= (124 N/m
2
)π(0.1/1000m)
2
= 3.9 × 10
−6
N
So the capillary force on the droplet is about two orders of magnitude larger than the
gravitational force, meaning we can neglect the gravity force in this circumstance.
COMMENTS: Comparison of the gravitational to surface tension forces is commonly
done through the Bond number, which is a dimensionless comparison of the gravitational
and surface tension forces:
Bo =
ρgl
2
γ
(5.101)
Here, l is the representative length scale of the fluid. Gravitational and surface tension
forces are comparable when Bo = 1, which will occur when l =
√
γ/ρg. For an air–water
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254 Transport in Fuel Cell Systems
surface at 80
◦
C, this length is approximately
l =
γ
ρg
=
0.0626
971 × 9.81
= 2.6mm
Therefore PEFC diffusion media and catalyst layers will be dominated by surface tension
forces, with gravitational forces playing a real role only in the in-plane direction in flow
channels and in manifolds where length scales can exceed a few millimeters. However,
surface tension effects in the channels are also important, especially in the direction normal
to the catalyst layer surface.
Leverett Function The most important relationship that must be established for an ac-
curate prediction of the liquid–gas transport in the DM is the relationship between the
capillary pressure and the liquid saturation. Capillary pressure depends on a variety of
parameters, including temperature through surface tension and the morphology, tortuosity,
and pore size of the material but most importantly liquid saturation.
Several empirical and semiempirical expressions are available which attempt to
describe the behavior of capillary pressure in terms of a porous media and fluid properties.
A generic Leverett function from soil science has been commonly employed to describe
the capillary transport behavior of the porous media in multiphase models. Since many
porous media share similar characteristic behavior, for PEFC gas diffusion layer material,
a Leverett-type function has been used to represent this behavior as a first step toward
achieving an accurate two-phase transport model. Udell [45] used Leverett’s approach [46]
to develop a semiempirical relation correlating capillary pressure and saturation data for
clean unconsolidated sands of various permeability and porosity by means of defining a
capillary pressure function:
P
c
= γ cos θ
φ
k
1/2
J (s) (5.102)
where k, φ, and θ are the permeability and porosity of the porous media and a representative
contact angle, respectively, and J(s) represents the Leverett function for scaling drainage
capillary pressure curves:
J (s
l
) =
1.417(1 − s
l
) − 2.120(1 − s
l
)
2
+ 1.263(1 − s
l
)
3
if θ<90
◦
⇒ Hydrophilic
1.417s
l
− 2.120s
2
l
+ 1.263s
3
l
if θ>90
◦
⇒ Hydrophobic
(5.103)
The Leverett approach incorporates the effects of interfacial tension but uses a simple
relation for the average pore radius (k/φ)
0.5
, basically ignoring the tortuous nature of porous
media [37]. Although this approach serves as a useful starting point, the applicability of a
generic Leverett function to the highly anisotropic thin-film DM is unclear and unverified.
Specific concerns center around the differences in fuel cell DM and conditions under
which the Leverett function was derived.
In particular, the definitions of wetting and nonwetting phases used to determine cap-
illary pressure and surface tension angle are taken as a statistical average over the entire
medium, obscuring local effects which differ from the whole. Because the droplet/bubble
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5.5 Multiphase Mass Transport in Channels and Porous Media 255
sizes are on the same order of magnitude as the DM thickness, a volume-averaged ap-
proach may not be appropriate. In addition, most porous PEFC material is impregnated
with an anisotropic coating of hydrophobic material (PTFE or other), thus yielding a com-
plex bimodal (hydrophobic and hydrophilic) pore size distribution. With heterogeneous
wettability, certain regions of the pore space will have a greater affinity for liquid, while
other regions will tend to repel water, so that within the bulk media liquid water will be
both the wetting (on untreated DM fiber) and nonwetting (on PTFE) phases. This situation
obviously cannot be dealt with using a single function. The chemical nature of the surface
and the spatial distribution of the wettability significantly affect the capillary transport
characteristics of these thin-film media. Instead of a single function to describe capillary
pressure as a function of liquid saturation, it may be more appropriate to construct a model
based on two or more functions acting in parallel in the DM that emulate the transport of
both phases through the discrete flow paths.
Modified Leverett Function Appropriate for Fuel Cell Media The standard Leverett
function has been found to be appropriate for qualitative matching of the flow character-
istics through the media; however, actual measurements of PEFC diffusion media show
different quantitative behavior. Kumbur et al. [47] presented a modified Leverett func-
tion appropriate for thin-film fuel cell DM to estimate the capillary pressure as a func-
tion of liquid saturation and hydrophobic additive content. This empirical fit was derived
from the direct measurement of capillary pressure–saturation for different types of DMs
(cloth and paper) with PTFE content ranging from 0 to 20% of weight and a microp-
orous layer. Figure 5.35 depicts the measured capillary pressure (P
c
) versus nonwetting
liquid saturation for carbon paper DM tailored with 20% PTFE content. The nature of
the capillary pressure–saturation curves exhibits a continuous “S” shape, rather than “J”
shape, and yields four inflection points. For saturation leads under 0.5, the capillary pres-
sure in the DM was fit to a modified Leverett function, appropriate for the hydrophobic
pores:
P
c
= γ
φ
k
1/2
M(s
nw
) (5.104)
where
M(s
nw
) =
%wt(0.0469 − 0.00152 × %wt − 0.0406s
2
nw
+ 0.143s
3
nw
) + 0.0561 ln s
nw
for 0 < s
nw
< 0.50
where %wt and s
nw
represent the PTFE weight percentage of the DM and nonwetting liquid
saturation, respectively. For hydrophilic pores, liquid water must be removed by evaporation
or convective forces since inhibition, not drainage, will be spontaneous. A key feature of
this empirical function is that the contact angle parameter is implicitly embedded in the
adjustable PTFE parameter, which enables successful iteration of the capillary pressure as
a function of the hydrophobic content of the DM. Compression and temperature also play
an important role in the capillary pressure saturation relationship. For more information,
the reader should consult ref. [47].
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256 Transport in Fuel Cell Systems
Figure 5.35 Measured capillary pressure (P
c
) and predicted P
c
by Leverett function versus satura-
tion for carbon paper DM tailored with 20% PTFE content [48].
Example 5.14 Determination of Capillary Pressure Using Leverett Approach Capillary
pressure can be represented by Leverett approach for hydrophobic DM pores. Determine
the capillary pressure as a function of the given DM properties at a liquid saturation of 0.4
and briefly explain:
(a) What is the expected change in capillary pressure when contact angle is changed
from 110
◦
to 150
◦
? Explain why the given change occurs.
(b) Determine the change in capillary pressure for an increase in porosity to 0.9.
(c) Determine the qualitative trend expected if the temperature is increased.
P
C
= γ cos θ
φ
k
1/2
J (s
l
)
J (s
l
) = 1.417(s
l
) − 2.120(s
l
)
2
+ 1.263(s
l
)
3
Properties
Contact Angle (deg) 110
Porosity, φ 0.7
Permeability, k (m
2
)10
−12
Surface tension (N/m) 0.062
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5.5 Multiphase Mass Transport in Channels and Porous Media 257
SOLUTION
P
C
= (0.062 N/m)
|
cos 110
|
0.7
10
−12
m
2
1/2
J (0.4)
J (0.4) = 1.417(0.4) − 2.120(0.4)
2
+ 1.263(0.4)
3
= 0.308
P
C
= 5.4kPa
(a) If the contact angle is changed from 110
◦
to 150
◦
, then the new capillary pressure
becomes
P
C
|
θ=110
◦
P
C
|
θ=150
◦
=
cos 110
cos 150
P
C
|
θ=150
◦
= 2.53 P
C
|
θ=110
◦
= 13.8kPa
As the DM becomes more hydrophobic, the representative contact angle increases. The
increase in contact angle leads to an increase in capillary pressure due to the increase in
hydrophobicity. Therefore, rendering the DM more hydrophobic enhances the liquid water
drainage from the DM by increasing the driving force for water motion.
(b) If the porosity is increased from 0.7 to 0.9:
P
C
|
φ=0.7
P
C
|
φ=0.9
=
0.7
0.9
0.5
P
C
|
φ=0.9
= 4.76 kPa
The capillary pressure is reduced through increased porosity. In the extreme of 100%
porosity, the capillary pressure is the lowest, which is one factor that drives flow to the
channel from the hydrophobic pores of the DM.
(c) Surface tension is a function of temperature and is given as
γ (N/m) =−1.78 × 10
−4
(T ) + 0.1247
Therefore, if the temperature is increased, surface tension will decrease, thereby yielding
lower capillary pressure at given conditions.
COMMENTS: The values for capillary pressure solved for here are on the order of
0.1 atm. This is the driving force for liquid motion. The areas with high capillary pressure
drive the liquid to areas of low capillary pressure, which are areas of higher pore size, or
low hydrophobicity such as the flow channels or hydrophilic pores.
Example 5.15 Effect of Microporous Layer Often, to enhance water transport, a special
highly hydrophobic coating between the DM and catalyst layer will be used, called the
microporous layer (MPL). This layer has an average pore size somewhere between that of
the catalyst layer and the macro-DM. Consider a bilayered DM coated with an MPL. In
equilibrium, there will be a liquid saturation jump at the interface between the MPL and
the DM, because of the discontinuity in pore sizes. That is, in order to have a liquid phase
pressure balance at the MPL–DM interface, as required for equilibrium, the saturation in the
MPL will be lower, since the pore size is lower, increasing the capillary pressure. Assume
c05 JWPR067-Mench January 23, 2008 18:58 Char Count=
258 Transport in Fuel Cell Systems
that at the interface, the DM has a local saturation value of s
int
. What will be the boundary
condition to calculate the saturation jump at the interface and write down the necessary
equation? Use the Leverett approach in this example:
P
C
= γ cos θ
φ
k
1/2
J (s
l
)
SOLUTION Liquid pressure across the interface of the DM and MPL should be contin-
uous in equilibrium. However, the different material properties of these two layers cause a
discontinuity in the liquid saturation across the interface, even though the capillary pressure
at the interface is equal.
Imposing the pressure balance at the interface yields
P
DM
C
|
DM−MPL Interface
= P
MPL
C
|
DM–MPL Interface
where
P
C
= γ cos θ
φ
k
1/2
J (s
l
)
cos θ
MPL
φ
MPL
k
MPL
1/2
J
s
MPL
int
= cos θ
DM
φ
DM
k
DM
1/2
J
s
DM
int
COMMENTS: One can solve the saturation jump using this equation at given conditions.
The magnitude of the discontinuity or the jump in the saturation strongly depends on the
c05 JWPR067-Mench January 23, 2008 18:58 Char Count=
5.5 Multiphase Mass Transport in Channels and Porous Media 259
material properties of these two layers and the amount of liquid water flux at the interface.
In general, the medium that is more hydrophobic and has smaller pore sizes has a reduced
saturation, and flow is driven to areas of high pore size or increased wettability. In this case,
with an MPL, the DM will have a much higher saturation than the MPL since the pore size
is smaller and the structure is more hydrophobic in the MPL.
Example 5.16 Local Saturation in Diffusion Media of PEFC Consider net hydrophobic
fuel cell DM. The fuel cell is operating at 80
◦
C, supplying 1 A/cm
2
, and water generated
(N
w
= iA/2F) is in the liquid phase. Assume that the only water flux coming into the DM
is the generated water from the catalyst layer. According to the given DM properties and
necessary correlations below, find the local position (distance from the catalyst layer) in
the DM where the liquid saturation values are 0.2, 0.3, and 0.4:
u
l
=
−k
µ
∇P
C
(Darcy law)
P
C
= γ cos θ
φ
k
1/2
J (s
l
)
k
r
= s
3
l
(relative permeability)
Note that the liquid water flowing inside the DM is assumed to be Newtonian and incom-
pressible, and the flow in the pores of the DM is presumed to be one dimensional, steady,
and laminar. Inside the DM, the gas-phase pressure is assumed to be constant (P
g
≈ const).
Properties
Thickness (µm) 300
Contact angle (degs) 150
Porosity, φ 0.7
Permeability, k(m
2
)10
−12
Surface tension at 80
◦
C (N/m) 0.062