Numerical Determination of Two-Phase Material Parameters of a
Gas Diffusion Layer Using Tomography Images
Jurgen Becker,∗ Volker Schulz, and Andreas Wiegmann
Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany
(Dated: March 22, 2007)
Abstract
In this paper we give a complete description of the process of determining two-phase material
parameters for a gas diffusion layer: Starting from a 3D tomography image of the gas diffusion
layer the distribution of gas and water phase is determined using the pore morphology method.
Using these 3D phase distributions, we are able to determine permeability, diffusivity and heat
conductivity as a function of the saturation of the porous medium with comparatively low numerical
costs. Using a reduced model for the compression of the gas diffusion layer, the influence of the
compression on the parameter values is studied.
Keywords: Fuel Cell, Gas diffusion layer, Microstructure, Two-phase parameters
∗Electronic address: [email protected]
1
I. INTRODUCTION
As a tool for converting hydrogen directly, and thus pollution free, into electrical energy,
polymer electrolyte fuel cells (PEFC) have gained widespread interest in research and indus-
try. A PEFC is a layered structure with the proton exchange membrane (PEM) at its centre.
On both sides of the PEM a catalyst layer is attached. The electrochemical reaction occur-
ring at the cathode catalyst layer combines protons, which result from hydrogen oxidation at
the anode catalyst layer and have passed the PEM, with oxygen to produce water. Located
between the catalyst layer and the flow field is the gas diffusion layer (GDL). According to
Mathias et al.[10], the gas diffusion layer has to comply with the following functionality:
reactant permeability, product permeability, electronic conductivity, heat conductivity, and
mechanical strength.
These material functions are described by physical material parameters: permeability,
diffusivity, heat conductivity, and electric conductivity, which are in general not only depen-
dent on the fibre structure of the GDL but also on the water saturation level. Simulation
schemes modeling the whole PEFC usually treat the GDL as a homogeneous porous medium
and take the abovementioned material parameters as input variables (see e.g. [2, 8]). To
determine these averaged parameters from a model of the GDL’s fibre structure is a well-
known numerical task if there is only one fluid (air or water) present. This method has been
applied, for example, by Pharoah [12] to gas diffusion layers or Ngo and Tamma [11] to more
general porous media. However, also the relative, i.e. saturation dependent parameters are
needed. As two-phase flow calculations are numerically costly, to our knowledge no numer-
ically calculated relative values have been published for a GDL layer yet. Rather, they are
determined with the help of physical experiments, as can be found in Acosta et al. [1]. Here,
we overcome these problems by combining the pore morphology method and single phase
simulations to calculate these relative parameters.
II. TOMOGRAPHY IMAGE
The calculations presented in this paper are based on a three dimensional model of the
GDL. In general, there are two different ways to obtain such a model. One way is to create it
virtually using methods of stochastic geometry [14], which has been applied to model a Toray
2
paper in [13]. The second way is to base the model upon a three dimensional tomography
image of a gas diffusion layer. Such an image can be obtained in an appropriate resolution
via Synchrotron-Radiation Imaging [3, 4]. (Note the difference between the 3D image and
the 3D model which is obtained from the image by image processing.)
Here, a sample of a carbon paper GDL was imaged by ANKA GmbH at the European
Synchrotron Radiation Facility (ESRF) in Grenoble. The image resolution was 0.7µm per
voxel (grid point). The obtained tomography image is a three dimensional grey-valued
image. To turn the grey-valued image into a 3D model, it has to be binarised into solid
parts and void parts by choosing a certain grey-value threshold or by using other binarisation
methods. The choice of a threshold value is not automated, but has to be done manually.
Criteria for the choice of the threshold value used here were the porosity of the resulting
model and visual inspection of the resulting model.
Figure 1 shows SEM-like pictures of the complete 3D model. As can be seen in the
picture, the diffusion layer does not fill the circular imaged sector completely. Therefore the
calculations were performed on a usable rectangular cut-out based at the centre of the imaged
layer. Figure 1 shows a cut-out of size 1024 × 1024 × 300 grid points, which corresponds
to 716.8µm × 716.8µm × 210µm. This cut-out has a porosity of 74.66%, which is close to
the average porosity of 73% stated by the manufacturer. As can be seen in figure 1, the
fibre structure is highly anisotropic. One can clearly distinguish between in-plane (x and y)
directions and through-plane (z) direction.
Due to the manufacturer, the paper consists (in weight) of 80% fibre material and 20%
PTFE/soot coating. In the synchrotron imaging based model, it is impossible to distinguish
between fibres and coating. Therefore, in the numerical calculations we always assume that
the solid material is homogeneous. In particular, we assume that the material surface has
the same wetting properties everywhere.
III. CAPILLARY PRESSURE AND WATER SATURATION
Once the three dimensional model of the gas diffusion layer is set up, the stationary
distribution of wetting and non-wetting phase for an arbitrary capillary pressure pc can be
determined using the pore morphology method [5]. This method makes use of the Young-
3
Laplace equation
pc =2γ cos θ
r, (1)
which relates capillary pressure pc to pore radius r, surface tension γ and contact angle θ.
In particular, equation (1) states that a pore is only accessible to the non-wetting fluid if its
radius is at least of size r.
To give the notion of ’pore radius’ a meaning also in the case of non-cylindrical pores
(see e.g. figure 1), the pore morphology method uses the morphological opening
Or(X) =⋃
Br,x⊆X,x∈X
Br,x, (2)
where X represents the pore space and Br,x is a sphere with radius r and centre point x.
In other words, Or is that part of the pore space, where the structuring element Br fits in.
Thus, Or is defined to be the space of pores with radius r′ ≥ r. Combining (1) and (2) we
see that a relation between non-wetting phase saturation snw and capillary pressure pc is
given by
snw(pc) =|O2γ cos θ/pc(X)|
|X|. (3)
Physically, this formula corresponds to the assumption, that interfaces between wetting and
non-wetting phase are of spherical shape or at least can be approximated by a superposition
of spheres. Furthermore, wetting and non-wetting phase are distributed freely in this model,
which can be seen as an equilibrium state reached by repeated drainage and imbibition of
the wetting phase.
When simulating a drainage process of an initially wetting phase saturated GDL, the
accessibility of a pore for the non-wetting phase is not only determined by the pore radius,
but also by it’s connectivity to a non-wetting phase reservoir. Therefore, the quasi-static
primary drainage curve pc(sw) has to be determined differently from (3). Recalling that the
Opening Or(X) equals a dilation of an erosion of the pore space with the same radius r,
Or(X) = Dr(Er(X)), (4)
where
Dr(X) =⋃x∈X
Br,x, (5)
and
Er(X) = x : Br,x ⊆ X , (6)
4
we proceed as follows: We define a subset R ⊆ ∂X to be connected to the non-wetting
phase reservoir (usually R is one of the six sides of a rectangular sample) and by
CRX = x ∈ X : ∃x0 ∈ R : ∃ path in X connecting x and x0. (7)
Then the space of all pores with a radius larger than r connected to the non-wetting phase
reservoir is given by Dr(CR(Er(X))), thus a capillary pressure - saturation relation for the
primary drainage is given by
snw(pc) =|D2γ cos θ/pc(CRE2γ cos θ/pc(X))|
|X|. (8)
Capillary pressure curves related to the equations (3) and (8) and thus modelling primary
drainage and repeated drainage/imbibition are shown in figure 2. As parameters we used
γ = 0.07272 N/m and θ = 40.
Via the primary drainage simulation the bubble point of the GDL can be determined
by finding the pressure needed for the non-wetting phase entering from top of the GDL to
reach the bottom of the GDL. Here, the bubble point was 8.3 kPa, which corresponds to a
pore diameter of 26.8 µm.
Figure 3 shows the simulated water distribution at the bubble point; the overall water
saturation in the GDL reaches 17%. Experimentally obtained pictures of water distribution
at the bubble point looking similar to the pictures on the right hand side of figure 3 can be
found in [9].
As the reader will already have noticed, these approaches not only describe capillary
pressure - saturation relations but also a pore size distribution is calculated as a by-product.
Using the morphological opening (2), a distribution of pore volume fractions is determined
and shown in figure 4.
The drainage approach can be used to model mercury intrusion porosimetry (MIP) mea-
surements (details on the experimental method can be found in [7]), where the pore size
distribution is determined experimentally by pressing mercury into the pores. Mercury is
non-wetting to most materials, so this corresponds to a drainage of the wetting phase, where
the intruding mercury is connected to a reservoir on top and (or) bottom of the sample. The
results of the simulated MIP measurement are also presented in figure 4. As can be seen
in the figure, MIP tends to underestimate the amount of larger pores, which indicates that
these are hidden behind narrower bottlenecks in the interior of the GDL.
5
IV. MATERIAL PARAMETERS FOR THE UNSATURATED MEDIUM
The material parameters were computed on a cut-out of the GDL of size 600 × 600 ×
300 voxels (420 µm× 420 µm× 210 µm) as domain Ω:
Ω = (0, d1)× (0, d2)× (0, d3), d1 = d2 = 420 µm, d3 = 210 µm. (9)
Thus, the computational domain consists of 108 million grid cells.
A. Thermal Conductivity
Heat transfer in the GDL takes place due to thermal diffusion and is governed on the
microscopic level by the Poisson equation
div (β∇u) = 0 in Ω, (10)
where we have neglected possible source terms on the right hand side which might be intro-
duced to describe ohmic heating or phase change effects. Here, β(x) is the local (isotropic)
heat conductivity at position x ∈ Ω. In the computation, this means that for any given grid
cell, a direction-independent conductivity value is used. Due to the geometric anisotropy
of the gas diffusion layer, the effective thermal conductivity β∗ of the layer is given by the
3× 3 coefficient tensor which on the macroscopic level satisfies Fourier’s law of conduction
j = −β∗∇T (11)
for a heat flux j and a temperature gradient ∇T . Following homogenisation theory [6], this
tensor β∗ can be determined by solving three auxiliary problems in Ω corresponding to the
three space directions ~e1, ~e2, ~e3:
div (β(x)(∇ul + ~el)) = 0 in Ω, l ∈ 1, 2, 3 (12)
with periodic boundary values
ul(x + id1 ~e1 + jd2 ~e2 + kd3 ~e3) = ul(x) ∀i, j, k ∈ Z. (13)
With the three solutions u1, u2, u3 the components of the coefficient tensor β∗ can be found
by integration
β∗ij =1
d1d2d3
∫Ω
〈~ei, β(x)(∇uj + ~ej)〉dx, i, j ∈ 1, 2, 3, (14)
6
which is symmetric by construction [6]. To solve (12)-(13), we used the EJ-Heat solver of
Wiegmann and Zemitis [17].
Using the parameters β = 17 W/mK as conductivity value of the carbon fibres and
β = 0.0262 W/mK as conductivity of the air, the diagonal entries of the heat conductivity
tensor of the GDL are
β∗ =
1.75
2.05
0.296
W
m K. (15)
As the fibres in the GDL are mainly oriented in the xy-plane and isotropic in this plane, the
off-diagonal entries are negligible.
B. Diffusivity
On the microscopic level, diffusion is governed by Laplace’s equation
−∆u = 0 in X, (16)
where u denotes the concentration and X denotes the pore space of Ω. We consider Neu-
mann boundary conditions on the fibre surfaces, given concentrations on the boundaries in
the diffusion gradient direction and periodic boundary conditions in the two perpendicular
directions. Similar to the conductivity tensor β∗, a 3× 3 diffusivity tensor D∗ exists, which
fulfills on the macroscopic level Fick’s first law
j = −D∗∇c, (17)
where ∇c denotes the concentration gradient and j the concentration flux. As before, we
follow [6] and solve three auxiliary boundary value problems to obtain all D∗ij. Again, we use
an explicit jump method for solving eq. (16), for details we refer to [17] and the forthcoming
paper [18].
For the GDL examined here, we obtained the following diagonal entries for the dimen-
sionless diffusivity tensor:
D∗ =
0.602
0.619
0.518
. (18)
7
The diffusivity tensor is normalized. In empty space, that is without the presence of fibres,
the diffusivity is the three-dimensional identity matrix, D∗ = 1. That means D∗ has to
be multiplied by the diffusion constant of the considered gas species to obtain the physical
values.
C. Permeability
The permeability matrix K is defined by Darcy’s law
u = − 1
µK∇p, (19)
where u denotes the average flow velocity, µ the viscosity and p the pressure. If the pressure
gradient is parallel to the ith axis, Darcy’s law reduces to the scalar equations
uj = − kji
µLδp, j = 1, 2, 3, (20)
where δp is the pressure drop along the ith axis and L the length of the sample in i-direction.
To determine the coefficients kji it is thus sufficient to determine the average flow velocity
for a given pressure drop. To determine the whole matrix, this has to be done for all three
space directions.
The average flow velocity is found by solving the governing flow equation in the pore
space, namely the stationary Stokes equation
−µ∆u +∇p = 0, div u = 0, (21)
where the pressure drop along the ith axis is a boundary condition. Periodic boundary
conditions are used in directions perpendicular to the pressure drop. To obtain the velocity
u, we used the FFF-Stokes solver of Wiegmann [16].
As the average flow in directions perpendicular to the pressure gradient is negligible here,
so are the off-diagonal entries kij, i 6= j. The obtained diagonal entries are:
K =
13.51
13.66
9.24
· 10−12m2. (22)
All computations of thermal conductivities, relative diffusivities and permeabilities were
done with the software tool GeoDict [15] with its integrated fast partial differential equation
8
solvers [16–18]. Due to the effectivity of these solvers, all computations could be performed
on a 64-bit AMD Opteron desktop system needing less than 13 GB of memory for the per-
meability calculations and less than 8 GB of memory for conductivity and relative diffusivity
computations.
V. TWO-PHASE FLOW PARAMETERS
The basic idea is to combine pore morphology method and single phase simulations to
obtain relative, i.e. saturation dependent, material parameters. First, the pore morphology
method is used to determine the distribution of air and water at a given capillary pressure
pc. Then, this distribution is assumed to be stationary and flow and diffusivity simulations
are carried out in the corresponding pore space only. Figure 5 illustrates this approach.
As described in chapter III, the pore morphology can be used to model either drainage
(8) or repeated drainage and imbibition (3). The two phase flow parameters calculated
here were determined using the repeated drainage / imbibition approach, which – from a
physical point of view – corresponds to phase distributions due to arbitrary mobile wetting
and non-wetting fluids.
A. Relative Permeability
In case of a partly saturated porous medium, the permeability K of Darcy’s law (19) is no
longer constant. Rather, it is depending on the saturation of the medium. These saturation
dependent or relative values kij(s) of the Darcy-Buckingham law
u = − 1
µK(s)∇p, (23)
can be calculated using the approach described above. So for a chosen capillary pressure
pc the water distribution is determined with the pore morphology method as described in
section III, here using the equations (2) and (3). Then the Stokes equation (21) is solved
only in the space occupied by the non-wetting (gas) phase, where water and solid parts are
treated as immobile. Note, that this implies the assumption of no-slip boundary conditions
between water and gas. Thus, we neglect that the phases may rearrange by the flow of air.
Figure 6 shows the relative permeability values obtained by this method. The presence of
9
water strongly reduces the permeability as it fills the large pores first and thus reduces the
size of the pores available for gas flow. Theoretically, we expect K ∼ r2, where r denotes
the effective pore radius of the gas flow, but the relation of r to the saturation s is in general
dependent on the material.
Remark, that each point shown on the curves required the solution of a Stokes problem.
For this reason, the use of a fast and efficient solver [16] is essential.
B. Relative Diffusivity
The saturation dependent diffusivities D∗(s) are determined similar to the diffusivity of
the unsaturated media as described in the previous chapter. The main difference is that the
pore space available for the gas diffusion is reduced by the presence of water.
So for an arbitrary capillary pressure pc, the water saturation and distribution is deter-
mined by the pore morphology approach using the equations (2) and (3). Then the Laplace
equation (16) is solved in the space occupied by the wetting (=gas) phase, where water and
fibres are treated alike. So the water is treated as immobile and impassable for the gas
particles and Neumann boundary conditions are applied on both gas-water and gas-solid in-
terfaces. From the obtained solution, the tensor D∗ is calculated as described in the previous
chapter.
Repeating these steps for different capillary pressures pc, and thus for different saturation
levels, the results presented in Figure 7 are obtained. The resulting profiles look almost linear
for low water saturations, which is explained by the fact that one expects the diffusivity to
be related by D ∼ εt
to porosity ε and tortuosity t. As the water – for low saturations – fills
only the centres of the large open spaces between the fibres and thus can easily be bypassed
by the diffusing gas, the tortuosity stays almost constant. So, approximately, D is linearly
dependent on ε and thus also on the saturation s.
C. Relative Heat Conductivity
In order to obtain the absolute heat conductivity coefficients β∗ in (15) , the equations
(12)-(13) were solved in the whole domain Ω with a discontinuous local conductivity coeffi-
cient β(x), which varied between solid and gas phase. With the appearance of water in the
10
GDL a third value is needed for the local coefficient, so we simply set
β(x) =
0.0262 W/mK if x ∈ gas phase
0.606 W/mK if x ∈ water phase
17 W/mK if x ∈ solid
(24)
and determine β∗ with the same algorithm. Here, the stationary water and gas phase distri-
butions are obtained by the pore morphology approach (3). Figure 8 shows the corresponding
heat conductivity results. The resulting in-plane values show only a slight dependence on
the saturation, as due to the choice of β in (24) the conduction mainly takes place inside
the in-plane oriented fibres. This is not true for the through-plane direction as here the heat
conduction is perpendicular to the fibre orientation, so conduction through air and water
phase plays a more important role here.
VI. COMPRESSION
When used inside the PEFC, the gas diffusion layer is compressed due to the clamping
pressure. The mathematical modelling of this is still a challenging task, but as we are mainly
interested in the determination of diffusivity and permeability parameters, we can apply a
reduced model for the compression of the layer.
Here, we assume that a GDL with initial height h is compressed with a factor 0 < c < 1
to a height of (1 − c)h. This is done numerically by assigning to each solid voxel a new
height, i.e. a new z-coordinate, by setting z′ = [(1− c)z], where the square brackets indicate
rounding to the closest integer value. Of course, solid voxels are not allowed to penetrate
into each other. To prevent this, a block of solid voxels is shifted as a whole by taking a
new z-coordinate only for it’s centre.
With this reduced model it is not possible to find a relation between compression ratio
and external load or to address problems of elasticity or mechanical strength of the GDL.
Nevertheless, for smaller compression ratios (approx. c < 0.4) this approach leads to a
realistic 3D morphology of the compressed GDL. It sustains the basic structural features
but reduces the pore space in between the fibres.
In figure 9 the relative diffusivity and permeability values for the uncompressed sample
and a 20% compressed sample are compared. Both diffusivity and permeability values are
of course lower for the compressed than for the uncompressed sample. The reduction of
11
the permeability is greater than the reduction of the diffusivity, as the permeability value is
dependent on the square of the effective pore radius, which is lowered by the compression,
whereas the diffusivity is dependent on the tortuosity, which stays almost constant during
compression.
This effect can also be seen in figure 10, where the absolute permeability and absolute
diffusivity values are shown for various compression levels. Here, it becomes visible, that due
to the compression direction the in-plane values are reduced more than the through-plane
values.
VII. CONCLUSION
Starting from a tomography image of the GDL, we were able to determine numerous
effective material parameters numerically. Table I gives an overview of the parameters
found for the dry set-up and compares the uncompressed with the 20% compressed case.
The figures 7, 6 and 8 show the saturation dependence of gas permeability, gas diffusivity and
thermal conductivity and the effect of compression on these curves is studied in figure 9. As
e.g. the effective heat conductivity β∗ is strongly dependent on the chosen input parameters
(24), a comparison with experimental values is desirable and will be addressed in the future.
The same methods can also be applied to determine the relative water permeability or the
electric conductivity of the GDL, but for the sake of brevity, these topics are not included
in this paper.
The obtained quantities are necessary as input parameters in macroscopic PEM simula-
tions. Being able to calculate them numerically not only allows to avoid more complicated
physical experiments but also helps when designing new media as the physical properties of
any virtually created model can now be predicted.
Acknowledgments
We would like to thank L. Helfen from ANKA GmbH, Karlsruhe for fruitful discussions
on volume image processing.
The authors acknowledge financial support for this work from the BMBF-project PEMDe-
sign (03SF0310A). Andreas Wiegmann is grateful for partial support by the Kaiserslautern
12
Excellence Cluster Dependable Adaptive Systems and Mathematical Modeling.
[1] M. Acosta, C. Merten, G. Eigenberger, H. Class, R. Helmig, B. Thoben, and H. Muller-
Steinhagen. Modeling non-isothermal two-phase multicomponent flow in the cathode of PEM
fuel cells. J. Power Sources, 159:1123–1141, 2006.
[2] J.J. Baschuk and X. Li. A general formulation for a mathematical PEM fuel cell model. J.
Power Sources, 142:134–153, 2004.
[3] P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker. Quanti-
tative phase contrast tomography using coherent synchrotron radiation. In U. Bonse, editor,
Proceedings SPIE: Developments in X-Ray Tomography III, volume 4503, pages 82–91, 2002.
[4] L. Helfen, T. Baumbach, K. Schladitz, and J. Ohser. Determination of structural properties
of light materials by three-dimensional synchrotron-radiation imaging and image analysis.
Imaging & Microscopy, 5:55–57, 2003.
[5] M. Hilpert and C. Miller. Pore-morphology-based simulation of drainage in totally wetting
porous media. Adv. Water Resour., 24:243–255, 2001.
[6] U. Hornung. Homogenization and Porous Media. Springer, 1997.
[7] A. Jena and K. Gupta. Characterisation of pore structure of filtration media. Fluid Particle
Separation Journal, 4(3):227–241, 2002.
[8] K. Kuhn, M. Ohlberger, J. O. Schumacher, C. Ziegler, and R. Klofkorn. A dynamic two-phase
flow model of proton exchange membrane fuel cells. Proceedings of the 2nd European PEFC
Forum, 1:283–296, 2003.
[9] S. Litster, S. Sinton, and N. Djilali. Ex situ visualization of liquid water transport in PEM
fuel cell gas diffusion layers. J. Power Sources, 154:95–105, 2005.
[10] M. Mathias, J. Roth, J. Fleming, and W. Lehnert. Diffusion media materials and charac-
terisation. In W. Vielstich, H. Gasteiger, and A. Lamm, editors, Handbook of Fuel Cells -
Fundamentals, Technology and Applications, volume 3. Wiley, 2003.
[11] N. D. Ngo and K. K. Tamma. Complex three-dimensional microstructural permeability pre-
diction of porous fibrous media with and without compaction. Int. J. Numer. Meth. Engng,
60:1741–1757, 2004.
[12] J.G. Pharoah. On the permeability of gas diffusion media used in PEM fuel cells. Journal of
13
Power Sources, 144:77–82, 2005.
[13] V. P. Schulz, P. P. Mukherjee, J. Becker, A. Wiegmann, and C. Y. Wang. Modelling of
two-phase behaviour in the gas diffusion medium of PEFCs via full morphology approach. J.
Electrochem. Soc., 154(4):419–426, 2007.
[14] S. Torquato. Random Heterogeneous Materials. Springer, New York, 2002.
[15] A. Wiegmann. GeoDict virtual micro structure simulator and material property predictor
http://www.geodict.com.
[16] A. Wiegmann. FFF-Stokes: A fast fictitious force 3d Stokes solver. In preparation, 2006.
[17] A. Wiegmann and A. Zemitis. EJ-HEAT: A Fast Explicit Jump harmonic averaging solver
for the effective heat conductivity of composite materials. Technical Report 94, Fraunhofer
ITWM Kaiserslautern, 2006.
[18] A. Wiegmann and A. Zemitis. EJ-DIFFUSION: A Fast Explicit Jump harmonic averaging
solver for the effective diffusivity of porous materials. In preparation, 2007.
14
FIG. 1: SEM-like images of the GDL model obtained from the tomography data. The pictures
on the left show cuts through the whole dataset, the pictures on the right show the 716.8µm ×
716.8µm× 210µm large cut-out used in the calculations.
15
FIG. 2: Capillary pressure curves showing drainage from top (Z+) and bottom (Z-) of the GDL
and repeated drainage/imbibition. The calculations were performed on the cut-out shown on the
right hand side of figure 1.
16
FIG. 3: Water distribution at the bubble point. The pictures show a SEM-like view of the GDL
seen from the dry side with fibres (on the left) and without fibres (on the right).
17
FIG. 4: Pore size distribution of the 716.8 µm× 716.8 µm× 210 µm cut-out of the medium. The
grey curve shows the geometrical pore size distribution, the black one shows the result of the
simulated MIP.
18
FIG. 5: Illustration of the method. The first picture shows a tiny 3D model, here a virtually
generated fibre structure. The second picture demonstrates compression with c = 0.2. In the third
step the water distribution is determined for a given capillary pressure pc, here the water is shown
in blue. The last picture shown illustrates the determination of air permeability (through-plane
direction): the streamlines shown are the solution of Stokes’ equation.
19
FIG. 6: Relative permeabilities. The chart shows the diagonal entries of K. K11 and K22 are
in-plane directions. The permeability values are given in 10−12 m2.
20
FIG. 7: Saturation dependent gas diffusivity values. The graphs show the diagonal entries of
the diffusivity tensor D∗. In-plane (D11 and D22) and through-plane (D33) values are clearly
distinguishable.
21
FIG. 8: Relative heat conductivity. The chart shows the diagonal entries of β∗ using the parameters
from (24).
22
FIG. 9: Effect of compression on relative diffusivity and permeability values. The charts shows
in-plane and through-plane values for both the uncompressed (c = 0) and the 20% compressed
(c = 0.2) layer. Through-plane values are averages over x and y direction. Permeability values are
given in 10−12 m2.
23
FIG. 10: Effect of compression on absolute diffusivity and permeability. The charts show the
in-plane and through-plane values for the unsaturated medium in dependence on the compression
factor c. Through-plane values are averages over x and y direction. Permeability values are given
in 10−12 m2.
24
c = 0 c = 0.2
Porosity 74.66% 68.32%
Bubble Point 8.3 kPa 9.5 kPa
Max. through pore diam. 26.8 µm 23.3 µm
Gas diffusivity (through-plane) 0.518 0.438
Gas diffusivity (in-plane) 0.611 0.522
Gas permeability (through-plane) 9.24 · 10−12 m2 5.49 · 10−12 m2
Gas permeability (in-plane) 13.59 · 10−12 m2 7.88 · 10−12 m2
Thermal conductivity (through-plane) 0.296 W/mK 0.388 W/mK
Thermal conductivity (in-plane) 1.90 W/mK 2.55 W/mK
TABLE I: Overview of the determined data for the dry GDL both for the uncompressed (c = 0)
and the 20% compressed (c = 0.2) case.
25