Proceedings of the Sixth International ASME Conference on Nanochannels, Microchannels and Minichannels
ICNMM2008
June 23-25, 2008, Darmstadt, Germany
ICNMM2008-62201
A CRITICAL REVIEW OF WATER TRANSPORT MODELS IN GAS DIFFUSION MEDIA OF PEM
FUEL CELL
Jacob LaManna
[email protected]
Mechanical Engineering Department
Kate Gleason College of Engineering
Rochester Institute of Technology, Rochester, NY USA
ABSTRACT
Proton Exchange Membrane (PEM) fuel cells are
gaining popularity as a replacement to the internal combustion
engine in automobiles. This application will demand high
levels of performance from the fuel cell making it critical that
proper water management is maintained. One of the areas of
interest in water management is the transport of water through
the Gas Diffusion Medium (GDM) on the cathode side of the
cell. Research is currently being conducted to understand how
water moves through the porous structure of the GDM. Due to
the small scale of the GDM, most work done is analytical
modeling. This paper will focus on reviewing current models
for water transport within the GDM of a PEM fuel cell to
address state of the art and provide recommendations for future
work to extend current models.
1 INTRODUCTION
Hydrogen fuel cells are gaining viability as replacements
for current fossil fuel powered internal combustion engines.
This viability is in part due to the efficiency of fuel cells and
also their power densities. Proton Exchange Membrane (PEM)
fuel cells offer the best chance for internal combustion engine
replacement in the automotive sector.
Because of the high demand that the automotive sector
would place on PEM fuel cells it is imperative to understand
the internal operation of the cell completely. Performance of
PEM fuel cells is strongly related to the quantity of water
within the cell making water management one of the most
important areas of current research.
There are currently several methods that are being pursued
to model the water transport behavior in the Gas Diffusion
Medium (GDM), an area vital to proper water management.
Extensive research within the past few years has resulted in
many models and modeling methods for analyzing the modes
of transport of water through the GDM and its effect on fuel
cell performance.
Satish G. Kandlikar
Mechanical Engineering Department
Kate Gleason College of Engineering
Rochester Institute of Technology, Rochester, NY USA
One modeling method developed to understand flow in the
GDM is the pore network method. Pore network modeling was
adapted from soil mechanics and percolation theory [6, 8, 59,
60] where it would track liquid flow in porous soil. One of the
first adaptations to fuel cells of the pore network model was
performed by Nam and Kaviany [28]. Here the structure for
the network was changed to a square grid to represent the
structure in the GDM formed by the carbon fibers. Since this
adaptation several other models have been developed using this
method [10, 11, 24, 39, 45, 46].
Another popular modeling method is the multiphase
mixture model (M2) original developed by Wang and Cheng
[52]. This model is capable of tracking two-phase flow within
a capillary porous media, specifically the GDM of a PEM fuel
cell. The M2 model has been implemented in several models
since this initial model [7, 12, 16, 20, 32, 34, 48, 61].
Since most models either describe the GDM from the
macroscopic level in many two-phase flow models with
parameters such as effective porosity and permeability or
develop generalized pore structures as in pore network models,
work has been performed to use actual GDM structure data.
Niu et al. [29] developed a lattice Boltzmann model that
tracked the two-phase flow through an X-ray tomography
reconstructed model of actual GDM. Schultz et al. [40] used a
stochastic approach for reconstructing the form for the GDM.
Other modeling approaches have included continuity and
energy based models, electrochemical coupled models, and
mixed domain models. Continuity and energy based models
such as by Matamoros and Brüggemann [25] and Rawool et al.
[38] base transport through the GDM on the continuity equation
and conservation equations. Electrochemical coupled models
such as by Hwang [14], Ju et al. [18], and Sinha et al. [47]
account for the electrochemical reactions occurring at the
catalyst sites to model the entire fuel cell. This method allows
for the construction of an i-V curve to show the effects of twophase flow on performance within the fuel cell. Mixed domain
models by Meng [27] and Vynnycky [50] split the fuel cell into
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Copyright © 2008 by ASME
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Copyright © 2008 by ASME
110°
Not given
Air
Air,
water
vapor
Air,
water
vapor
2-p flow using full
morphology model
2-p flow using
multiphase mixture
model
2007
Pasaogullari et
2007
al.
Koido et al.
Wang et al.
[54]
[20]
Ju et al.
[16]
2008
2007
2007
Air,
water
vapor
Air
3-D, multiphase
model
2-p flow using
multiphase mixture
model
Air
2-p flow using
multiphase mixture
model
[36] Pharoah et al. 2006 2-D Cathode model
[32]
Schulz et al.
[40]
Air
2006
2-p flow using mmrt
lattice Boltzmann
model
Niu et al.
[29]
Capillary
diffusion
Capillary
diffusion
Darcy Law flow
Potential driven
transport
Capillary
transport
Capillary forces
Diffuse interface
theory
Comment
Does not take actual GDL
morphology into account
Results
Relative permeability and capillary
pressure on heterogeneity of
porous material
Model was able to track 2phase interface in 3D GDL
structure
Methods for calculating capillary The authors used a good mix
3D reconstruction of pressure and relative permeability
of experimental and
actual diffusion
matched experimental curves well.
numerical techniques to
media
Shows applicability of these
verify the models purposed in
this paper
models
Provides reliable capillary pressure
Further evaluation of
data that can feed into full cell
compression model needed
models
Anisotropic properties of the GDL Many other models assume
Effective Porosity greatly affect the temperature and isotropic properties which can
affect the results collected
liquid water distributions
Anisotropic properties of the GDl
can greatly affect effective mass
Should be extended into to
Effective Porosity
diffusivity, thermal conductivity,
3D to show anisotrpoic
electron conductivity, and liquid
differences in more depth
permeability
Agrees with NR but shows
Model compares well with neutron
the weakness of NR in it
Effective Porosity
radiography data. Shows actual
inability to separate anode
distribution of water in cell.
and cathode water
distributions
Compared the performance
Data shows the importance
characteristics of carbon paper
of knowing operational
Effective Diffusivity
and carbon cloth, found that
conditions when designing a
carbon cloth performs better under cell and selecting between
high humidity will carbon paper is
cloth and paper.
Stochastic
reconstruction of
two media
3D reconstruction of Authors show that the mmrt lattice
actual diffusion
Boltzmann model is effective for
media
use in GDL 2-phase flow modeling
Further research is needed to
Regular cubic
Most models most likely over
validate the results of
network, calibrated
estimate gas phase flow
possible overestimation
to two samples
Shows flows governed by capillary Further research needed in
Regular cubic
the study of mixed wet
network, calibrated fingering, thereforce Darcy's Law
cannot be used
properties
to two samples
Crossover from fractal fingering to Further Research into mixed
Regular cubic
stable front morphology occurs
wet effects and new
network, calibrated
with increasing hydrophillic fraction manufacting processes to
to two samples
supporting the use of Darcy's Law control hydrophillic fraction
Regular square
network
GDL characterizing
Table 1: Summary of Selected Articles
162°
110°
92°
120°
60° and 120°
Capillary and
viscous forces
Distribution
range of 60°120°
Air
Capillary and
viscous forces
diffusion
Liquid flow
mechanism
invasion
percolation
algorithm w/
110°
[46] Sinha & Wang 2008 Pore network model
100° for SGL
and 98° for
Toray
Not given
Air
Air
GDL Contact
angle
[45] Sinha & Wang 2007 Pore network model
Gostick et al.
[11]
Capillary network
model
Gas
phase
Air
2007
Markicevic et
al.
[24]
Model approach
2007 Pore network model
Year
Author
Ref
#
two separate domains. This allows for the removal of
complicated boundary conditions between the catalyst layers
and the membrane therefore simplifying the model.
This article will focus on discussing several pore network
and two-phase flow models. A table summarizing selected
articles from this review is given in Table 1.
NOMENCLATURE
GDM – gas diffusion medium
CCL – cathode catalyst layer
PEM – proton exchange membrane
MEA – membrane electrode assembly
MPL – microporous layer
2 DISCUSSION
2.1 GOSTICK ET AL. [11] PORE NETWORK MODEL
Model Description:
Gostick et al. [11] developed a pore network model to
investigate the relative permeability of water and gas, and the
effective gas diffusivity. The authors modeled the fibrous
GDM as regular cubic pores.
These cubic pores are
interconnected by narrowing throats that are treated as square
ducts. The basic framework of the model can be seen below in
Figure 1. A Weibull cumulative distribution was used to assign
the pore body size distribution as the network was constructed.
The throat size is determined by using the diameter of the
smallest adjacent pore. Saturation levels were varied from 0 to
1 for liquid water to find the affects on gas transport.
Anisotropic properties were analyzed in this article. To
account for this in the model, throats were additional
constricted in specific directions. This constriction was
calibrated to match the porosity of Toray 090 and SGL 10BA.
The authors selected a method to describe how the nonwetting phase (water) would move through the diffusion media.
This simulation began with low capillary pressure and the
network was analyzed to find any pores that could be
penetrated. Then the throats that are connected to these pores
are counted and all are marked as open (available for nonwetting phase). The algorithm then increases the capillary
pressure and repeats the procedure.
Two limiting cases were developed to analyze the twophase transport within the porous material. The first case
stipulates that once a pore is penetrated by liquid water, the
pore is no long capable of accepting the gas phase. The second
case allows the wetting phase to maintain connections with
neighboring pores as the current pore is penetrated. Case 1 was
designed to give the worst case for gas transport.
Model Results:
The model was used to calculate the relative permeabilities
of each phase. This relative permeability accounted for the
interaction between phases.
Relative permeability was
calculated in the x, y, and z directions to determine what the
anisotropic effects were. It was found that there is a
preferential spreading of water in the in-plane direction at the
surface of the GDM. This spreading of liquid water can greatly
reduce the available area for through-plane diffusion of the
reactant gases.
The authors found that current models most likely
overestimate the amount of gas phase transport that occurs and
suggested new models that agree with their results.
Overestimation of the gas phase transport within the cell results
in underestimation of the quantity of water saturation within the
GDM. This can result in much lower current densities being
calculated due to the fact that a dry GDM can support much
higher current densities than a saturated GDM.
Need for Future Work:
The authors, Gostick et al. [11] call for an urgent need for
experimental research into the effects of water saturation on
water permeability and gas diffusion. This need is to verify the
results of this experiment since it claims that most models
overestimate the gas phase transport. Also this model could be
extended to the actual topology of the GDM, which could lead
to results that better represent the actual physical structure
within a fuel cell.
2.2 SINHA AND WANG [45], [46] PORE NETWORK
MODEL
Figure 1: Pore network used by Gostick et al. [5]
Capillary pressure was modeled using the Young-Laplace
equation due to the choice of modeling the pores as square
cross-sections. A uniform contact angle of 115° was chosen for
the entire network.
Model Description:
Sinha and Wang [45] developed a pore network model
based on the frame work as determined by Nam and Kaviany
[28]. This method models the GDM as randomly stacked
regular fiber screens that form cubic pore shapes and square
cross-sectional throats; this can be viewed below in Figures 2
and 3. A cut-off log normal distribution was selected to
represent the pore size distribution. The authors assume that
only one fluid can inhabit a throat at a time.
The algorithm selected begins with the pore network
saturated with air with a liquid water reservoir at the inlet face.
The authors also account for land regions and only allow water
to leave the GDM in areas that reside above channels. The
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Copyright © 2008 by ASME
Young-Laplace equation is used to govern liquid water
movement. Liquid water injection rate is equivalent to the
water production at 2.0 A cm-2.
within the GDM and that this must be taken into account.
Darcy’s law is applicable in cases where there is compact
invasion at high capillary numbers and therefore cannot
describe the fractal capillary fingering at low capillary numbers
described in this model. They suggest that modeling methods
need to be developed that are capable of applying this
knowledge to the typical macroscopic descriptions used for fuel
cells.
Figure 2: Pore Network Development by Nam and Kaviany
[28].
Figure 4: Example of Results from Sinha and Wang [45].
Figure 3: Methods of Diffusion [28]
Model Results:
The authors, Sinha and Wang [45] found that with the
fractal capillary fingering transport of water through the GDM
that there is no surface coverage by the water on the inlet side.
Water moves in several clusters due to the fractal nature of
transport. These two effects can be seen below in Fig. 4.
Because capillary pressure determines the direction of the flow,
areas of dead ends are formed. These dead ends occur when
water contacts a throat where the capillary pressure to invade is
too high for the water to continue forward motion. To
determine the active pore for water transport, the authors
graphed only the pores with non-zero volumetric flow rates
(Fig. 5). This shows that the vast majority of the water within
the GDM is non-transporting and only blocks gas flow.
Saturation levels for liquid water were determined based on
capillary number and varied from 0-0.95.
Sinha and Wang found that water flow through a
homogeneously hydrophobic GDM is governed by fractal
capillary fingering due to the extremely low capillary numbers
encountered in fuel cells. The authors conclude that two-phase
Darcy’s law cannot properly describe the conditions occurring
Figure 5: Steady State Flow Clusters [45]
Model Description:
Later, Sinha and Wang [46] extended the model described
above to investigate the effects of mixed-wet properties within
a GDM. This mixed-wet model combines areas that are
hydrophobic with areas that are hydrophilic. The distribution
of hydrophilic regions is assumed in this analysis since no
method of calculating the actual distribution currently exists.
The authors assumed a random distribution for the contact
angle assigning larger diameter pores larger contact angles.
Assigning smaller contact angles to the smaller pores
approximates what could occur during actual teflonation
process since PTFE may not actually coat small diameter pores
which leaves them hydrophilic.
Model Results:
The authors Sinha and Wang [46] accounted for the effect
of hydrophilic/hydrophobic interfaces within a pore. They
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Copyright © 2008 by ASME
found that water takes preference towards the hydrophilic areas
within the GDM. As the concentration of hydrophilic regions
increase the movement of water shifts from a finger-like
structure to more cylindrical-like structures. This results in
more localized flow which goes against the current model of
finger-like distribution since the flow transitions to a stablefront morphology (Fig. 6). Saturation levels for this model
varied from 0-0.5 based on contact angle. The saturation levels
seen here are lower than the previous model because water
takes the preferential routes through the hydrophilic regions of
the GDM.
wettability of the GDM during manufacturing should occur
since there is no current method to precisely control the
hydrophobic distribution in the GDM and if this method could
be produced then custom mixed-wet GDMs could be
engineered.
Figure 7: Hydrophilic distributions in Mixed-Wet GDM
Analysis [46]
Figure 6: Stable-Front Water Transport in mixed-wet
GDM [45]
Sinha and Wang [46] found that two-phase Darcy’s law is
still applicable for use in mixed-wet GDMs which is in
opposition to results found by the authors in their previous pore
network model stated above. They find that the mixed-wet
conditions better describe the actual conditions of GDM
materials since the contact angle for carbon varies and the
PTFE distributions are not constant.
Sinha and Wang [46] also analyze for a non-uniform
distribution of hydrophilic areas. It is found that high
distributions of hydrophilic properties towards the center of the
GDM thickness (Fig. 7) results in higher liquid water saturation
in the center of the GDM. This results in higher levels of mass
transport losses due to the fact that the gas diffusion paths for
the reactants are much more tortuous. The authors describe the
need to optimize the wettability distribution to limit the effects
of mass transport losses.
Need for Future Work:
The current models by Sinha and Wang are assumed
isothermal and therefore does not account for phase change of
water. This needs to be addressed due to the temperature
gradients across the GDM in non-isothermal analyses.
Temperature gradients and phase change could significantly
affect the liquid water distributions found in this analysis. The
actual morphology of the GDM should be taken into account to
better represent the actual structure of the GDM. This could
affect liquid water distributions due to the pore size
distributions within the GDM. As suggested by the authors,
there is a need to experimentally analyze the actual properties
of GDM pore walls since this will give true insight into the
distribution of hydrophilic properties. Research into controlled
2.3 PASAOGULLARI ET AL. [32] TWO PHASE MODEL
Model Description:
Pasaogullari et al. [32] developed a two-phase flow model
to analyze the effects of anisotropic properties of the GDM on
water transport. The authors constructed a 2-D, two-phase
flow, non-isothermal model developed for the cross-section of
the GDM. The main focus of the model was to investigate how
modeling the GDM with anisotropic properties due to its
inherent structure would affect current density, temperature
distribution, reactant fluid velocities, and liquid water content.
Due to the stacking of carbon fibers, GDM has higher thermal
conductivity in the in-plane direction along the fibers whereas
the thermal conductivity is lower through-plane because of
crossing through fibers.
The model developed by the authors is based on the
multiphase mixture model (M2). This computational method is
capable of explicitly tracking the interface between single- and
two-phase regions. The Brinkman extension to Darcy’s Law is
used to govern conservation of momentum. Due to the low
solubility of oxygen and nitrogen in water, water is assumed to
be the only liquid in the mixture.
To account for the anisotropic structure of the GDM,
effective diffusivity is calculated for both the in-plane and
through-plane directions. The calculated diffusivities show that
the in-plane diffusion is favored since it is much higher than the
through-plane direction. Permeability and capillary pressure
are also calculated for both the x and y directions to account for
anisotropy.
Assumptions made include analyzing water content in the
membrane and anode based on a function of water content and
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temperature within the GDM given that the anode side is fully
saturated, negligible anode overpotential, and the catalyst layer
is a thin layer between the GDM and membrane. The first
assumption was made to simplify the model and to reduce the
model to just the cathode GDM as it was assumed that there
would be a net zero water transport coefficient.
Model Results:
To highlight the effects of anisotropic properties, the
authors ran two identical simulations where one simulation had
anisotropic properties while the second had isotropic properties.
These tests were calibrated to Toray TGPH carbon paper.
It was found that temperature effects can vary significantly
between anisotropic and isotropic properties as seen in Fig. 8.
Because of the higher thermal conductivity in the in-plane
direction the temperature profile is significantly more linear
when moving from the catalyst layer to the bi-polar plate. This
change in temperature distribution greatly affects the water
transport and saturation levels as seen in Figure 9. Liquid
water concentration increases in the areas above the lands due
to the lower temperatures seen there in the anisotropic trials
when compared to the isotropic trials.
Overall lower temperatures are predicted for the
anisotropic conditions which result in higher liquid water
content in the GDM. It was also found that the capillary
transport of liquid water and diffusion of water vapor aid each
other in the through-plane direction but oppose each other in
the in-plane direction since water is condensing near the
channel land areas which cause a shift between phases. The
authors call for the strong need for coupled, anisotropic twophase heat and water transport modeling to further explore this
phenomenon and its effect on fuel cell performance.
Figure 9: Liquid Water Saturation Distribution Comparison at
0.6V [32]
Need for Future Work:
The model should be applied to a cell wide model so that
the actual water content within the membrane and anode can be
calculated. This will eliminate the assumption of water content
and link the effects of anisotropic properties to the overall cell
performance. Extension into compression effects should be
considered since compression could make the properties more
isotropic since the resistance in the through-plane direction
would be reduced due to the contact pressure. Also, a move to
3D should be performed for this model to test for the existence
of 3D anisotropy and its affect on cell performance.
2.4 PHAROAH ET AL. [36] TWO PHASE MODEL
Figure 8: Temperature Distributions in GDM at 0.6V
Comparing Anisotropic and Isotropic Properties [32]
Model Description:
Pharoah et al. [36] performed an extensive review into the
methods of accounting for anisotropic properties and then
developed a 2D cathode model to analyze the effects of
anisotropic properties.
The review that was performed
concluded that most current models assume that volumetric
averages are appropriate and use isotropic properties. The
authors ran their model with both isotropic and anisotropic
properties to compare the results between the two.
The model developed is a 2D electrochemical model that
tracks the multi-component diffusion of species within the
GDM, electron transport across the GDM, and the
electrochemical reactions taking place at the catalyst layer.
Diffusion through the GDM is governed by the Maxwell-Stefan
equation. The electrochemical reaction is governed by the
Butler-Volmer equation modified to account for catalyst
microstructure and the charge transfer is governed by Ohm’s
Law.
Model Results:
The authors compare data for isotropic effective diffusivity
found using the Bruggeman relation, isotropic effective
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diffusivity found using percolation theory, and anisotropic
effective diffusivity found using percolation theory.
It was found that anisotropic properties have considerable
effect on current density distribution when compared to
isotropic properties as can be seen in Fig. 10. The difference
between anisotropic and isotropic properties increases as
porosity decreases. The reason for this is the oxygen transport
is higher with greater porosity which increases the overall
reaction. It is also shown that the Bruggeman relation typically
overestimates the current density by 5-6%.
As shown in Fig. 11, there is minimal influence on the
calculated polarization curve for different material models.
What is important to note is that the current density distribution
is affected.
More work should be performed to understand the
differences that arise when modeling with isotropic or
anisotropic properties. This is important since it is often
assumed that properties are isotropic in many current models
while the actual structure of the GDM lends itself to anisotropic
properties. Also, the model should be adapted to two-phase
flow so that saturation levels could be obtained with reference
to the diffusivities through the GDM.
2.5 NIU ET AL. [29] TWO PHASE MODEL
Model Description:
The authors constructed a 3D, 2-phase flow model using
the multiphase, multiple-relaxation-time lattice Boltzmann
model. The model is based on the mean-field diffuse interface
theory. This model is capable of tracking the Liquid-Gas Phase
Interface through the GDM and varies liquid saturation levels
from 0-0.6.
GDM structure was represented by a 3D
reconstruction using the Dissipative Particle Dynamics (DPD)
method as seen below in Fig. 12.
Figure 10: Comparison of Effective Diffusivity on Current
Density [36]
Figure 12: Dissipative Particle Dynamics Reconstruction of
GDM. [29]
The MRT lattice Boltzmann method solves the NavierStokes and Cahn-Hilliard equations for N discrete velocities.
The benefit of the lattice Boltzmann method over other solution
methods is that it can solve the Navier-Stokes equations over
complex geometries like that of a GDM. The forces in the
interface of the liquid and gas are also analyzed in this method.
Figure 11: Comparison of Polarization Curves [36]
Model Results:
The MRT lattice Boltzmann method is found to give better
simulation results when compared to the Bhatnagar-GrossKrook (BGK) method which is typically used for this type of
analysis. The calculated liquid interface can be seen in Figure
13 where the GDM image has been removed for better
visibility.
The authors used the MRT lattice Boltzmann method to
calculate absolute and relative permeabilities. The calculated
Need for Future Work:
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results compare well with current numerical methods including
other two-phase flow methods and pore network methods.
Need for Future Work:
Integration of water vapor in gas phase and the effects of
temperature gradients could turn this modeling method into a
powerful tool. Currently, the model only analyzes air as the gas
phase when water vapor could become important under certain
operational ranges. Further work should be performed to verify
this modeling method since it is new to fuel cell analysis.
typically the method of capillary flow of water through the
GDM in most current models and was shown to transition to
stable-front flow by Sinha and Wang [46]. This transition to
stable-front flow allows for the justification of the use of Darcy
Law transport modeling. Anisotropic property effects were
analyzed by Pasaogullari et al. [32] and Pharoah et al. [36].
These effects were shown to significantly affect liquid water
and current density distributions. This is significant because
many current models volume-average properties and assume
isotropic properties. Niu et al. [29] presented a new modeling
method for water transport models with the use of the
multiphase MRT lattice Boltzmann method. The use of this
new method allows for the tracking of the actual liquid-gas
interface in a complex 3D reconstructed model of the GDM.
4 CONCLUSION
This paper has given a detailed review of several modeling
methods currently employed to investigate water management
within a PEM fuel cell. It has been shown through each of the
models how temperature gradients within the GDM can affect
the water phase and saturation level. Therefore, it is critical to
couple heat transfer with water transport to properly model
phase change and saturation levels of water in the GDM. As
models account for phase change of water with greater detail, a
better understanding of the actual operational conditions in the
fuel cell will be obtained.
Several recommendations for improvement to the current
models addressed within this paper have been made. All
models discussed in this paper lack a combination of two-phase
flow, phase change of water, and an actual representation of the
GDM. The three of these combined could result in an
improved model of the GDM. The phase change of water
needs to be implemented in models since it can effect the actual
liquid water saturation in the GDM possibly resulting in underor over-estimation of saturation levels when not accounted for.
Actual GDM structure needs to be implemented with phase
change since the actual structure of the GDM could
significantly influence condensation and evaporation with the
random pore sizes that exist. These recommendations could
improve the current knowledge of PEM operation and result in
an overall improvement in fuel cell performance.
The current models combined with futures models based
off of the recommendations made in this paper will lead to
better water management techniques that will improve overall
performance of the fuel cell. This increased performance will
ensure that the PEM fuel cell will be capable of performing in
its role of internal combustion engine replacement.
ACKNOWLEDGMENTS
This work was supported by the US Department of Energy
under contract No. DE-FG36-07G017018.
Figure 13: Graphical Representation of Liquid-Gas Interface.
[29]
2.6 MODEL SUMMARY
The models reviewed within this paper all present
innovative and/or critical items for water transport modeling.
Gostick et al. [11] discovered the possible overestimation of
gas phase transport in most current models and discussed the
need to address this in future modeling. Fractal fingering is
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9
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11
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