modeling the effective thermal conductivity of an
TRANSCRIPT
1 Copyright © 2011 by ASME
Proceedings of ASME 2011 5th International Conference on Energy Sustainability & 9th Fuel Cell Science, Engineering and
Technology Conference
ESFuelCell2011
August 7-10, 2011, Washington, DC, USA
ESFuelCell2011-54364
MODELING THE EFFECTIVE THERMAL CONDUCTIVITY OF AN ANISOTROPIC
GAS DIFFUSION LAYER IN A POLYMER ELECTROLYTE MEMBRANE FUEL CELL
J. Yablecki University of Toronto
Toronto, ON, Canada
A. Bazylak University of Toronto
Toronto, ON, Canada
ABSTRACT
The anisotropic and heterogeneous effective thermal
conductivity of the gas diffusion layer (GDL) of the polymer
electrolyte membrane fuel cell was determined in the through-
plane direction using an analytical thermal resistance model.
The geometry of the GDL was reconstructed using porosity
profiles obtained through microscale computed tomography
imaging of four commercially available GDL materials. The
effective thermal conductivity increases almost linearly with
increasing bipolar plate compaction pressure. The effective
thermal conductivity was also seen to increase with increasing
GDL thickness as bulk porosity remained almost constant. The
effect of the heterogeneous through-plane porosity distribution
on the effective thermal conductivity is discussed. The
outcomes of this work will provide insight into the effect of
heterogeneity and anisotropy of the GDL on the thermal
management required for improved PEMFC performance.
NOMENCLATURE
A Area (m2)
a,b major and minor semi axes of elliptical contact area
(m)
C Contact point
d Fiber diameter (m)
E Young’s modulus (Pa)
E’ effective elastic modulus (Pa)
F Contact force (N)
F1 Integral function of ρ
H Height of modeling domain (m)
k Thermal conductivity (W/m K)
K(η) Elliptical integral
L Length of modeling domain (m)
l Length of fiber (m)
N Number of fibers
P Pressure (Pa)
q Heat flux (W/m2)
R Thermal Resistance (K / W)
T Temperature (K)
t Thickness of modeling domain layer (m)
V Volume (m3)
W Width of modeling domain (m)
Greek Symbols
ε Porosity
ρ’, ρ’’ Major and minor relative radii of curvature (m)
ρe Equivalent radius of curvature (m)
θ Angle between two fibers (radians)
υ Poisson’s ratio
ξ Strain (m / m)
Subscripts
bp Bipolar plate
C Contact point
co Constriction resistance
cond Conduction
eff effective
GDL Gas Diffusion Layer
sp spreading resistance
s solid fiber
t thickness
2 Copyright © 2011 by ASME
INTRODUCTION
Achieving proper heat management within a polymer
electrolyte membrane (PEM) fuel cell is critical for improving
its performance and lifetime. Heat produced from the
electrochemical reaction and water phase change results in
temperature gradients across a single cell and fuel cell stack (1-
5). The temperature within the fuel cell affects the relative
humidity, membrane water content, saturation pressure, and
reaction kinetics. Careful control of the temperature throughout
the cell is also important to avoid membrane dehydration and
degradation at elevated temperatures and detrimental humidity
levels (4). Modeling the heat transfer rates and temperature
distributions in a fuel cell requires the knowledge of the
thermal transport properties, in particular, the effective thermal
conductivity.
The main path for heat removal from the PEM fuel cell
membrane to the current collectors is through the gas diffusion
layer (GDL), and the rate of heat removal is therefore largely
dependent upon the thermal transport properties of the GDL.
Although a number of analytical correlations exist for the
effective thermal conductivity of a composite material with
different geometric considerations (6), the GDL thermal
conductivity can be estimated but not accurately represented by
a single analytical correlation due to its anisotropic (7) and
heterogeneous nature (8). Through a compact analytical model
of the GDL, Sadeghi et al. (9) calculated the through-plane
effective thermal conductivity of the GDL using an idealized
repeating unit cell to represent the GDL structure. The
analytical model accounted for: the conduction through the
solid fibrous material, gas phase, the thermal constriction
resistance at the regions of fiber-to-fiber contact, and gas
rarefaction in the microgaps to determine the effective thermal
conductivity. The results from their model (9) and parametric
study showed that the thermal constriction resistance dominates
the total thermal resistance in the determination of the effective
thermal conductivity.
The work of Sadeghi et al. (9) was extended by the same
authors to include an experimental and analytical study on the
effective thermal conductivity and thermal contact resistance of
the GDL under a compressive load (10). Increases in the
bipolar plate pressure were found to increase the thermal
conductivity. Further to this approach, Sadeghi et al. (11)
recently published an experimental and analytical study on the
in-plane effective thermal conductivity and thermal contact
resistance of the GDL. The compact analytical model
accounted for heat conduction through randomly oriented
fibers, the contact area between fibers, and
polytetrafluoroethylene (PTFE) covered regions. Their work
(11) was the first combined analytical and experimental
approach to determining the in-plane thermal conductivity of
the GDL.
Recently, Pfrang et al. (12), Zamel et al. (13), and Veyret
et al. (14) used a commercial simulation software package
(GeoDict) to determine the anisotropic thermal conductivity of
the GDL. Pfrang et al. (12) calculated the effective thermal
conductivity of three commercially available GDLs with
GeoDict based on three-dimensional (3-D) reconstructions
from x-ray computed tomography (CT) visualizations. Zamel
et al. (13) used GeoDict to construct a 3-D pore morphology of
dry carbon paper GDL with no PTFE by providing the GDL
geometric parameters as the initial conditions. The thermal
conductivity was determined using two different commercial
solvers: ThermoDict and Fluent. Both programs yielded the
same results for the effective thermal conductivity, but
ThermoDict was found to have significantly less computational
requirements than Fluent.
The effective thermal conductivity of the GDL can be
determined experimentally in-situ from the temperature of an
operating fuel cell or from ex-situ experiments. Ex-situ
experiments for the GDL are more common due to the
complexity of coupled processes within an operating fuel cell,
which must be considered for in-situ measurements (15). The
first experimentally determined GDL thermal conductivity was
reported by Vie and Kjelstrup in 2003 (16). Burheim et al. (15)
reported experimental measurements for the anisotropic
through-plane effective thermal conductivity of dry and wet
Nafion membranes and compressed GDLs from ex-situ
experiments. Compression was shown to cause a nearly linear
increase on the effective thermal conductivity with applied
compression load. The thermal conductivity of the GDLs with a
residual water content of 25% was found to increase by almost
70% when compared with a dry sample. Burheim et al. (17)
recently reported an extensive study on the effective through-
plane thermal conductivity of a number of GDL materials. The
authors (17) studied the effect of residual water, PTFE content,
and compression on the through-plane thermal conductivity and
the thermal contact resistance between the GDL and aluminum
bi-polar plate.
Sadeghi et al. (9) provided insight into the through-plane
thermal conductivity of the GDL with a compact analytical
model but only considered the case of a periodic ordered GDL
structure. In practice, the GDL is anisotropic and carbon paper
GDL materials are composed of randomly oriented fibers.
Recent work by Fishman et al. (8) indicates that the porosity of
the GDL is heterogeneous. Burheim et al. (17) have attributed
experimentally found trends in the through-plane thermal
conductivity with the GDL thickness of carbon paper GDL
materials to this heterogeneity. It is important to investigate the
effect of heterogeneous porosity distributions on the effective
through-plane thermal conductivity of the GDL in more detail.
In this work, an analytical model is used to determine the
effective through-plane thermal conductivity of four
commercially available GDL materials based on the
heterogeneous porosity profiles presented in (8). The effects of
the heterogeneous porosity distribution and GDL compression
on the effective thermal conductivity are investigated. The
results of the model are compared with experimental data
presented in (17).
3 Copyright © 2011 by ASME
MODEL DEVLEOPMENT
GDL Reconstruction To model the effective through-plane thermal conductivity
of the GDL, a modeling domain, shown in Figure 1, with width,
W, and length, L, was constructed of a number of randomly
oriented fibers, Nt, in the xy plane. The fibers are immersed in
stagnant air, each with a length and diameter of l and d,
respectively. Layers of the fibers are stacked vertically through
the entire thickness or height, H, of the GDL modeling domain.
The through-plane porosity distributions reported in (8) of
four GDL materials are employed in this work to establish the
number of fibers, in each layer, t, of the modeling domain. The
porosity of each layer can be found from:
�� � 1 � ��������� �1�
where Vsolid is the total volume of the solid carbon fibers, and
Vtotal is the total volume of modeling domain. Based on the
volumes of the modeling domain used, the porosity is
determined from:
�� � 1 � �� ���4 ���� �2�
where d and l are the diameter and length of the carbon fiber,
respectively.
In previous analytical modeling work, Sadeghi et al. (9)
found that the primary path for through-plane heat conduction
in the GDL is through the contact area formed by two fibers
overlapping with an orientation angle, θ.
Based on the assumption that both fibers are smooth, a
smooth, non-conforming contact area between the two fibers is
formed, which is a function of the force applied and the angle
between the two fibers. When cylindrical fibers contact each
other eccentrically, the contact region is close to elliptical (18).
The Hertzian theory of contact is used to predict the shape of
contact between solids and how it grows under an increasing
load (18). Following the analytical modeling approach for the
GDL presented by Sadeghi et al. (9), the application of the
Hertzian contact theory for non-conforming smooth cylinders is
used to define the contact area as (18):
� � ��"�′3 �!4"′ #
$/& $�3�
' � � (�′�")�/&
�4�
where a and b are the major and minor semi-axies of the
elliptical contact region, respectively, and F is the contact load.
Figure 1: Schematic illustrating a section of the geometry in the
computational domain, which consists of solid, continuous fibres
oriented in a planar direction a) x-y plane orientation, b) x-z plane
orientation, and c) isometric view.
The major and minor relative radii of curvature at the contact
points, ρ' and ρ'’ are given as (18):
�** � �+2�1�,-.2/ 0 2�5�
�* � 124�3 � 2 1�**3
�6�
E’ is a function of the Young’s modulus, E and Poisson’s ratio,
υ (18):
"* � (1 � 5$�"$ 0 1 � 5��"� )�7�
where the subscripts, 1 and 2, are used to denote the two
materials in contact. For this model the contact area is formed
between two fibers of the same material. ρe is the equivalent
radius of curvature expressed as (18):
�! � +�*�**�8�
4 Copyright © 2011 by ASME
Sadeghi et al. (9) proposed the following relationship, accurate
within 0.8% for the complex integral function of ρ' and ρ'’, F1:
$ � 19.1√;1 0 16.76√; 0 1.34;�9�
where:
; � �*�** �10�
In our model, the number of contact points, Ct, for a single
fiber in layer, t, and the adjacent layer, t+1, is a function of the
porosity of both layers and the orientation angle of the fiber.
The fibers are randomly oriented with a uniform distribution in
the modeling domain with an angle between 0 and 90 degrees.
The maximum number of contact points for a given fiber in
each layer occurs when the fibre orientation angle is 90°, as
shown in the periodically ordered fiber arrangement (Figure 2).
The minimum number of contact points is seen when θ is 0°
between the two layers. A non-linear relationship was
determined for the number of contact points for a single fiber in
a layer with porosity εt based on its orientation angle. The
average number of contact points for a single fiber, Ct in a layer
with porosity εt was determined from this relationship and is
used as an input into the model. The average number of contact
points was assumed to vary linearly with porosity and is
interpolated for each porosity value, εt. The contact load, F, for each contact point can be
expressed in terms of the pressure within the GDL, PGDL, the
cross-sectional area of the unit cell defined by L and W, and the
number of contact points for a given layer, Ct (9):
=�>,� � @ABC��D� �11�
The pressure within the GDL will affect the uncompressed
values for the GDL thickness measured in (8). The strain, ξ,
from the GDL compression is assumed to be elastic and is
found from Hooke’s law:
E � "ABC@ABC �12�
where EGDL is the Young’s modulus of the GDL material.
Thermal Model
The effective thermal conductivity is a property of porous
media that accounts for the contributions of the thermal
conductivity of each phase (19). For determination of the
effective thermal conductivity of the GDL, the two phases that
are generally considered are carbon fibers and air with values
of thermal conductivity of 120 W/m K and 0.03 W/m K,
respectively (7). Results from analytical modeling work by
Sadeghi et al. (9) revealed that the dominant path for heat flow
Figure 2: Schematic illustrating a periodically ordered fiber
arrangement with θ = 90°
in the GDL is through fiber-to-fiber contact, and heat flow
through the air can be neglected. This was validated in
experimental work (10), where the calculated through-plane
thermal conductivity of the GDL in vacuum and ambient
conditions resulted in thermal conductivity values that differed
by less than 3%.
For the analytical model, it is assumed that the only heat
transfer is steady-state one-dimensional (1-D) conduction in the
through-plane direction of the GDL. With the calculation of the
Grashof and Peclet numbers, Ramousse et al. (7) showed that
natural convection and convective heat transfer are negligible
compared with conduction in the GDL. Radiative heat transfer
can be neglected for temperatures below 1000 K (20), which is
well above the operating range of a PEM fuel cell. Heat is
transferred through the modeling domain from fiber-to-fiber
contact only. The thermal resistance in the conduction along a
fiber is ignored as it is assumed to be negligible compared with
the thermal constriction and spreading resistances (10).
A thermal resistance network or circuit can be constructed
for 1-D heat transfer with no internal energy generation and
constant properties (21). The thermal resistance, Rt, for
conduction in a plane wall is defined by:
F�G�H� � ∆JK � �
LM�13�
where ∆T is the difference in temperature across the wall, q is
the heat flux, L is the length of the plane wall, k is the thermal
conductivity of the wall, and A is the cross sectional area of the
wall. An equivalent thermal circuit with thermal resistances in
parallel and series is analogous to an electrical circuit governed
by Ohm’s law.
The dominant thermal resistance for heat conduction
through the GDL is the thermal constriction and spreading
resistance (9). The thermal spreading resistance, Rsp, is
equivalent to the thermal constriction resistance, Rco, and
accounts for the thermal energy that is transferred between the
two fibers at the contact interface (9). When the dimensions of
the contact area are very small compared with the dimensions
of the contacting bodies, the heat transfer through the contact
5 Copyright © 2011 by ASME
area is constrained, and the solution can be modeled as a heat
source on a half-space (22). In the case of the GDL, the contact
area between the two fibers is the heat source, and the much
larger carbon fibers are the half space.
The thermal constriction resistance is then a function of
the elliptical contact area calculated with the Hertizan contact
theory and described by the major and minor semi-axis of the
elliptical contact area, a and b, respectively. Using the work of
Yovanovich and Marotta (23) that resulted in half-space
solutions for different contact interfaces, Sadeghi et al. (9)
expressed the thermal constriction resistance analytically to be:
FG� � 12�L�' N�O��14�
where K(η) is the complete elliptical integral and a function of
the contact area dimensions, a and b.
N�O� = P ��+Q1 − O�.RS��T
U/�
V�15�
where:
O = W1 − 2� 'X 3��16�
The total thermal resistance for each layer, Rt, within the
modeling domain can be found from the summation of each
individual thermal resistance of each contact point in parallel:
1F� =Y 1
FG�,Z + F�[,Z\
V�17�
The total resistance in the modeling domain, Rtotal, can be found
as a series summation of the thermal resistances of each layer:
F���� =YF� �18��]^
V
The effective thermal conductivity can be found from the total
thermal resistance:
L!__ = �����F����M =` − �
F�������19�
RESULTS
The analytical model was implemented with
heterogeneous through-plane porosity distributions for Toray
carbon paper TGP-H-030, 060, 090, and 120 obtained through
x-ray microscale computed tomography (µCT) experiments
conducted in (8). The GDL materials considered were
uncompressed and did not have a microporous layer (MPL) or
PTFE treatment. The spatial resolution for the µCT data
Table 1: Unit Cell Geometry Properties
Fiber
diameter, d
Fiber
length, l
Unit cell
width, W
Unit cell
length, L
7.32 µm 325 µm 1500 µm 1625 µm
Table 2: Carbon Fiber Properties
Thermal
Conductivity, k
Poisson’s
ratio, υ
Young’s
Modulus, E
120 W/m K (3) 0.3 (13) 210 GPa (13)
gathered is 2.44µm (8); however, for the purpose of this model,
measurements for porosity are interpolated from the
experimentally determined data set at every 7.32 µm through
the thickness of the GDL. The details of the microscale
computed tomography visualization are presented in reference
(8). The geometric parameters of the modeling domain and the
material properties used in the analytical model are shown in
Tables 1 and 2. A value of 17.9 MPa for the Young’s modulus
of the Toray carbon paper GDL is used (24).
The effective through-plane thermal conductivity obtained
from the analytical model is shown in Table 3 for three
different compression pressures, Pbp. The results from the
analytical model are also plotted in Figure 3 and compared with
experimental data for Toray carbon paper GDL materials
presented by Burheim et al. (17). The results from the
analytical model are in agreement within 11.1%, 8.9%, and
3.1% of the experimentally obtained results from (17) for Toray
TGP-H-060, 090, 120, respectively. Similar trends can be seen
between both sets of data.
The effective thermal conductivity is observed to increase
with increasing compression pressure. As noted by Burheim et
al. (15), and shown in Figure 3, the effective thermal
conductivity increases almost linearly with increasing
compression pressure. The results from the analytical model
also display another trend noted in the experimental work by
Burheim et al. (17) for Toray carbon GDL material with
varying thickness. The results in Figure 3 show an increase in
the effective thermal conductivity with increasing GDL
thickness, even as the average bulk porosity remains
approximately consistent. This is the trend for all three GDL
materials presented by Burheim et al. (17). The results
presented in Figure 3 for the analytical model follow this trend
except for Toray TGP-H-060 and 090. The effective thermal
conductivity of Toray TGP-H-060 is an average of 3% higher
than 090. To investigate this trend further, the effective thermal
conductivity for each GDL material is plotted throughout the
thickness of the GDL in Figure 4.
6 Copyright © 2011 by ASME
Table 3: Effective thermal conductivity of Toray paper GDLs
Toray TGP-H-030
Compression
Pressure
(kPa)
Thickness
(µm)
Bulk
Average
Porosity
(%)
Effective thermal
conductivity
(W/m K)
460 114.11
0.2601
930 111.03 82.93 0.3327
1390 108.02
0.3811
Toray TGP-H-060
460 213.96
0.4822
930 208.18 82.09 0.6038
1390 202.54
0.6722
Toray TGP-H-090
460 290.03
0.4678
930 282.2 82.60 0.5758
1390 274.55
0.6664
Toray TGP-H-120
460 349.46
0.6450
930 340.03 78.69 0.8180
1390 330.81
0.9292
As shown in Figure 4, the thermal conductivity through
the thickness of the GDL is strongly dependent upon the local
value of the porosity, εt. For all four GDL materials presented,
the Toray carbon paper materials display local porosity minima
and maxima throughout the thickness (8). At the location of a
porosity minimum, a maximum local value of thermal
conductivity is observed. The four materials investigated have a
local maximum value of thermal conductivity between 2.5 and
3 W/m K.
DISCUSSION
The effective thermal conductivity is observed to increase
with increasing compression pressure. This result has been
previously shown in experimental (9, 15, 17, 25) and analytical
work (9, 10) and can be explained with the analytical model
presented. As the compression pressure increases, the elliptical
contact area between two fibers increases. Increases in the
contact area cause a decrease in the thermal constriction and
spreading resistance, and subsequently, an increase in the
overall thermal conductivity. Compression will also cause a
decrease in the overall thickness of the GDL material that has
been accounted for in this analytical model. As the thickness of
the GDL decreases, the overall effective thermal conductivity
will also decrease. The effect of the compression on the
Figure 3: Comparison of measured average effective thermal
conductivity with compression pressure and with values from
Burheim et al. (17).
decrease in the thermal resistance is greater than the effect on
the decrease in the GDL thickness, therefore, causing an overall
increase in the effective thermal conductivity with increasing
compression.
The analytical model assumes that compression is applied
uniformly within the GDL. This is not the case during fuel cell
operation as there will be higher compression in areas under the
lands of the bipolar plate than in areas under the channels (15).
The Toray carbon paper materials display local porosity
minima and maxima throughout the thickness that affect the
local thermal conductivity value and the overall effective
thermal conductivity. Fishman et al. (8) noted that the
heterogeneous porosity distributions for all four GDL materials
are distinct but each display three different segments: two
transitional surface regions and a core region. The transitional
surface region extends linearly between the outer surfaces and
the local porosity minima and the core area is between the two
transitional surface regions (8). For the thinnest GDL
investigated, Toray TGP-H-030, this transitional surface region
accounts for approximately 66% of its total thickness (8).
Fishman et al. (8) observed that the transitional surface region
accounts for 45%, 33%, and 28% of the material thickness for
060, 090, and 120, respectively. The variation of the overall
effective thermal conductivity with GDL thickness observed in
Table 3 can be attributed to the amount of transitional surface
region. The thermal conductivity appears to be strongly
affected by the higher porosity values in the transitional surface
regions, regardless of the overall bulk porosity value or the
local maximum thermal conductivity value.
The results from the analytical model for the effective
thermal conductivity observe an increase in effective thermal
conductivity with increasing GDL thickness, same results
7 Copyright © 2011 by ASME
presented by Pharaoh et al. discrepancy in analytical model for
Toray 060 and 090. The discrepancy between our analytical
results and the experimental results from literature stems from
the small variability in our experimentally obtained porosity
profiles of commercially available materials. Batch-specific
variations associated with the manufacturing process have been
observed in these commercially available materials (26). In this
work, only one sample for each different material was
employed as an input in our analytical model. Here, we see that
the results from the analytical model are quite sensitive to the
porosity values in the transitional regions, or locations of
porosity maximums. Future work should include a larger
number of porosity distributions in order to fully characterize
the trends on the effective thermal conductivity.
CONCLUSION
An analytical model to determine the effective thermal
conductivity of the GDL is presented. The model is used to
calculate the effective thermal conductivity of four
commercially available Toray carbon paper GDL materials
based on heterogeneous porosity profiles obtained from
microscale computed tomography imaging (8). The effect of
the heterogeneous porosity distribution and GDL compression
on the effective thermal conductivity is investigated. The
effective thermal conductivity increases almost linearly with
increasing bipolar plate compaction pressure. The effective
thermal conductivity was also seen to increase with increasing
GDL thickness as bulk porosity remained almost constant. The
effective thermal conductivity is affected by the heterogeneous
porosity distribution. The outcomes of this work will provide
insight into the effect of heterogeneity and anisotropy of the
GDL on the thermal management required for improved
PEMFC performance.
ACKNOWLEDGMENTS
The Natural Sciences and Engineering Research Council
of Canada (NSERC), Canada Foundation for Innovation (CFI),
Bullitt Foundation, and University of Toronto are gratefully
acknowledged for their financial support. Alexander Graham
Bell Graduate Scholarship from Natural Sciences and
Engineering Research Council of Canada (NSERC) is also
gratefully acknowledged.
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Figure 4. Through-plane effective thermal conductivity and
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a)
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8 Copyright © 2011 by ASME
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