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: becker@itwm.fhg.de

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

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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