1. Introduction
In all types of polymer electrolyte fuel cells (PEFC), also called proton exchange membrane (PEM) fuel cells, the gas diffusion layers (GDLs) are components with high relevance for efficient operation of fuel cells. Two major requirements can be identified. One is the electric contact to be provided between the bipolar plates and the catalyst layer (CL). This determines the choice of the material. The second requirement is to facilitate efficient mass transport. For this purpose, an appropriate microstructure is required. To meet the requirements, carbon fibers are typically used for the fabrication of different types of GDLs, e.g., paper, woven, or non-woven textiles. The lattice Boltzmann method (LBM) is widely used for the simulation of mass transport in GDLs. Weber et al. [
1] reviewed the many aspects of the modeling activities of PEM fuel cells.
One topic that was identified as a critical issue to be investigated was the area between domains of different spatial scales; the interface between the GDL and channels is one of them. Andersson et al. [
2] investigated multiphase modeling of PEFCs on the cell level. In their review, they highlighted the relevance of the GDL/channel interface. Various methods are available to obtain the microstructure of the GDL, e.g., X-ray synchrotron, focused ion beam (FIB)/SEM and stochastic reconstruction [
3]. The gas transport in a GDL was simulated by Froning et al. [
4,
5,
6] and van Doormaal and Pharoah [
7] by means of LBM. Uncompressed GDLs, as well as compressed GDLs were studied. In more detail, Froning et al. [
4,
5] simulated gas transport in paper-type GDL with its microstructure based on the stochastic geometry model of Thiedmann et al. [
8]. The gas flow at the GDL exit was analyzed from a statistical viewpoint [
5]. Furthermore, the macroscopic properties of permeability and tortuosity were analyzed statistically from the viewpoint of compression and the variation of the microstructure [
4]. Froning et al. [
4] used the Kozeny–Carman equation to verify the calculated effective permeability and tortuosity. The Kozeny–Carman equation is widely used by researches investigating gas flow in porous structures, e.g., by Mangal et al. [
9] in their experimental studies. Salomov et al. [
10] used the LBM for their transport simulations in woven GDLs of high-temperature PEFCs (HT-PEFCs). The catalyst layer (CL) was reconstructed mimicking the clusterization of carbon particles statistically. Nabovati et al. [
11] reconstructed Toray GDL and investigated the addition of PTFE to the microstructure with varying total amount and spatial distribution. They studied the influence of the PTFE on surface area and volume and calculated the permeability from single-phase flow simulations using the LBM. Based on microstructures reconstructed from X-ray tomographic microscopy (XTM) scans, Ros
n et al. [
12] in turn simulated liquid transport, and they obtained effective properties of the GDL. Eller et al. [
13] completed the knowledge of the microstructure of a GDL with the distribution of binders. Simaafrookhteh et al. [
14] obtained the characteristics of the porous structure of paper-type GDL from transport simulations in geometries, which were reconstructed according to the distribution of the orientation of the fibers.
Experimental investigations by several research groups complement the simulations mentioned above. The gas permeability of a GDL was measured by Tamayol et al. [
15]. They considered different levels of compression. Other properties of GDLs—diffusion, thermal, and electrical conductivity—were measured by Zamel and Li [
16]. The in-plane diffusivity of several types of GDL was measured by Rashapov and Gostick [
17]. Chen et al. [
18], meanwhile, investigated the impact of compression on commercial GDLs. Taira and Liu [
19] used two adjacent channels of a flow field as an application-oriented experimental setup. In order to obtain the in-plane permeability of the GDL, they analyzed the cross-flow under the ribs. The permeability of the GDL was measured by Reshetenko et al. [
20], and its impact on the fuel cell efficiency was analyzed.
The focus in the studies mentioned above was on GDLs, in particular their material properties and the mass flow in the microstructure. The simulation of entire fuel cells and stacks spans more than such small components. The consideration of all components covers multiple scales. Continuum-based approaches are often used in cell and stack modeling [
21,
22,
23]. Such methods use effective volumetric properties of porous materials, e.g., the permeability. Surface characteristics can be found in experimental work; for instance, the roughness of the GDL surface was analyzed by Yuan et al. [
24]. Interfaces between the regions of different properties have been investigated by several researchers. Breitwieser et al. [
25] presented a review of the membrane/catalyst layer interface. The interface between the electrode and GDL was also addressed by Froning et al. [
26].
The relevance of the interface between the GDL and channels is reflected by experimental and modeling work. In the measurements of Kaneko et al. [
27], effective properties of the GDLs were investigated, as well as the mass flow at the GDL/channel interface. Many investigations of this interface focus on the liquid water transport in low-temperature PEFCs. Yoon et al. [
28] observed in their experiments the behavior of droplets on a GDL surface. They studied the removal of liquid water droplets in a channel from the surface of different types of GDL. Wang et al. [
29], meanwhile, identified the GDL surface near the channel as a topic of high interest for the simulation of gas flow on the channels of the flow field of a fuel cell. Niu et al. [
30] simulated two-phase flow in fuel cell channels, assuming a static contact angle at the GDL surface. Kim et al. [
31], in turn, simulated the hydrodynamics of water droplets in the gas channels. They investigated droplets leaving the GDL surface at two distinct positions. Koz and Kandlikar [
32] simulated the inhibition of oxygen transport in flow channels in the presence of liquid water. They worked with regular patterns at the GDL interface where the water was entering the channel. The position of liquid water transported from the GDL into the gas channel can be inherently transferred by coupling the simulation domains of both spatial scales, as was done by Chen et al. [
33]. They found that such tight coupled simulations can require enormous computational resources.
Two adjacent regions of a fuel cell are sometimes simulated on different spatial scales. From a macroscopic view, the interface is a 2D element. The role of the GDL/channel interface in this scenario is illustrated in
Figure 1. The GDL/channel interface was investigated by Yu et al. [
34], who analyzed irregular contact angles of water droplets at the GDL surface and their pattern when they passed the GDL/channel interface at several positions. The relevance of the interface for multi-scale simulations was shown by Qin et al. [
35,
36] and Aghihi et al. [
37], who used pore network modeling (PNM) to bridge the scales (still on water transport in low temperature PEFCs). Niu et al. [
38] coupled the LBM in the porous GDL structure with OpenFOAM simulations in the air channel.
The relevance of the characterization of the interface for multi-scale simulations still holds for other types of fuel cells, e.g., HT-PEFCs. Yang et al. [
39] analyzed the gas flow in a channel over a regular porous structure for different Reynolds numbers. The interface was included in the comprehensive analytical studies of Kulikovsky [
40]. The studies of Chevalier et al. [
41] showed that the Peclet number at the GDL/channel interface is relevant for the current density profile along the channel. They focused their studies on oxygen transport in the GDL and channel, as well as on charge transport in the membrane.
In this manuscript, through-plane transport in GDLs is simulated with the LBM. The stochastic geometry model of Thiedmann et al. [
8] is used to create 25 representations of the microstructure; Froning et al. [
4,
5] sed the same geometries. The statistical variation of the microstructure was completed with various compression levels. For this manuscript, virtual microstructures were selected from the studies mentioned above, both uncompressed and compressed. The focus of the new investigations is the analysis of the two-dimensional region between the GDL and gas channels; this is called the GDL/channel interface. Areas are classified according to the total amount of gas leaving the GDL with the highest velocity. The location of areas at the GDL surface where the most gas is flowing is the resulting 2D information on the GDL surface. The knowledge can possibly assist the development of new methods in the field of channel/cell-level simulations.
3. Results
Through-plane transport was simulated in 25 realizations created by the geometry model, oriented in reverse order to distinguish the GDL/channel surface from the GDL/electrode surface presented by Froning et al. [
26]. The operating conditions, according to
Section 2.3, are summarized in
Table 1. The conditions led to a Reynolds number of
. This in turn led to velocity vectors at the GDL exit that were almost parallel to the through-plane direction. The free space downstream of the GDL mentioned in
Section 2.3 was required to arrange the velocity vectors properly behind the porous structure.
The variation of the volumetric characteristics—permeability
and tortuosity
—is shown in
Figure 3a. A comprehensive study of these was presented by Froning et al. [
4,
5]. The quantile levels
z were implicitly defined by
, as introduced by Equation (
4). They define the total quantile areas:
that can be summarized to the values illustrated in
Figure 3b. Like the volumetric characteristics, they showed statistical variation, because the simulation results were based on statistical microstructures. In addition to the total quantile area defined by Equation (
6), the size of the largest of these regions is shown in
Figure 3c. For this purpose, the function
from Equation (
5) needs to be applied on the largest area identified by the contour levels
from Equation (
4), which was done via the visualization tool paraview and an external R [
50] script.
In contrast to the display of the key properties in
Figure 3, the detail of the two realizations, Nos. 5 and 15, is depicted in
Figure 4. With the decrease of the quantile level
q, the areas resulting from the contour lines became smaller, while the number of areas decreased, as well. The contour levels addressed in
Section 2.4 look completely different in detail, which depicts the large variation of the areas summarized in
Figure 3c.
3.1. Analysis of the GDL Surface
The GDL/channel interface is at the location where the through-plane flow leaves the GDL. This interface was analyzed by evaluating the mass-related quantiles defined by Equation (
4). For two realizations, the 70%, 50%, and 20% quantiles are shown in
Figure 4. With lower quantile levels, the sizes of the areas illustrated by the contour lines in the post-processing step decreased, and some characteristic situations could be identified. Areas that are concentric for the three quantile levels with similar shapes are marked with labels, e.g., “a” in
Figure 4. In a similar manner, the areas can still be concentric, but with a modified shape of the inner area (for
q = 20%), labeled with “b”. Another situation is marked with “c”. The inner area was split into two, identifying two small pores in the GDL that were hidden, when higher quantile levels were shown. Positions where only one or two areas were nested—labeled as “d”—showed pores, where less gas left the GDL. Furthermore, positions where “b” and “c” are combined are labeled accordingly.
The gas within the mass-related quantiles was flowing through a certain fraction of the total quantile area of the GDL, which is shown in
Figure 3b. The sizes of the largest area show large variation in
Figure 3c.
Through-plane transport was simulated in microstructures as specified in
Section 2.1 and
Section 2.3. They were applied to 25 realizations of the geometry model.
Table 2 summarizes the detailed results of the analysis of the mass-related quantiles obtained from the velocity field. The total quantile area
is presented for
70%, 50%, and 20% and, also, the number of areas (number of regions, #reg.) belonging to the quantiles and the average
and maximum size
of the areas. The maximum size is also presented in
Figure 3c. The largest area can be of interest for fuel cell and stack modeling when the location of the pores emphasized in the GDL surface is required. The large spread of the values in
Figure 3c is in line with the variation coefficient of
in
Table 2.
3.2. Statistical Evaluation
The transport simulations provided velocity fields that enabled the calculation of volume-based properties—
and
—and surface properties, as discussed in
Section 3.1. All of these showed statistical variations and were different in their ranges of values. For the volume-based properties of porous material, many approaches have been developed that describe the relationships between several properties of the material. The Kozeny–Carman relation is only one of them [
47]. It is desired to have also a simple approach to calculate the surface properties from volume-based characteristics. As a first step, the statistical correlation between the results can be calculated. Hedderich and Sachs [
51] provided not only the fundamentals of various statistical tests, but also criteria for the proper choice of the right test for the desired evaluation. Kendall’s correlation test [
51] allows for correlation coefficients to be calculated under the conditions mentioned above.
Table 3 shows the coefficients
,
that were obtained using the R software [
50,
52].
The total quantile areas
for
were chosen as candidates for the correlation test with the volumetric properties: permeability
and tortuosity
. Correlation coefficients can indicate a positive
or negative
relationship. Every value other than an extreme one
can only be used for comparisons. For this reason, the absolute value
was chosen as an indicator of whether a correlation coefficient indicates a possible physical relation. It was already shown by Froning et al. [
4] that
and
are non-linear with respect to the Kozeny–Carman relation:
with only the geometric properties of the microstructure on the right side: the porosity
, total volume
, and inner surface
. In
Table 3, the correlation between
and
is
. The absolute value of every correlation coefficient between the surface-related properties
and
or
was much lower than this value. This indicates that there was no statistical correlation between surface-based and volume-based properties, although all of them were calculated from the same velocity field. Therefore, it is still necessary to perform transport simulations on the 3D structure to obtain the 2D properties presented in
Section 3.1. The same conclusion was found in the analysis of the other interface (GDL/electrode) by Froning et al. [
26].
3.3. Impact of the Compression
The through-plane transport was simulated under 30% compression.
Figure 5 shows the velocity extracted at the GDL/channel interface with contour levels according to the mass-related quantiles
z.
The velocity is colored in the same range as is used in
Figure 4. Although the transport properties—permeability and tortuosity—changed under compression according to the Kozeny–Carman equation, the distribution of the relative total quantile areas at the interface did not change. The permeability and tortuosity are shown in
Figure 6a. Compared to the uncompressed properties in
Figure 3a, the tortuosity increased and the permeability decreased under compression, which is consistent with earlier studies [
4]. Related to the quantiles of the total mass flow,
Figure 6b,c shows the relative total quantile areas of the gas flow and the largest of these.
The overall picture of the diagrams in
Figure 4 looks very similar to that in
Figure 6, but with reduced absolute values. It is noticeable that also the permeability and tortuosity in
Figure 6 changed their absolute values compared to the numbers in
Figure 3, which was already discussed by Froning et al. [
4,
5]. Detailed surface characteristics of the compressed GDL are shown in
Table 4. The GDL was compressed here by 30%.
The size of the total area
was smaller under compression, i.e., the average value was reduced from 167,000
(
Table 2) to 150,000
for the 70% quantile level. The percentage of
related to the total surface of 590,000
is shown in
Table 5. The upper limit of the total quantile area was roughly the porosity of the GDL, which was 78% in the representations of the geometric model [
4,
8]. In particular,
was limited by the local porosity of the upper fiber layer, which may vary slightly from the average. A similar trend can be observed on the other average values in the tables. The reduced porosity of a compressed porous structure leads (on average) to a higher absolute velocity under the condition of a fixed total amount of gas to be transported through the GDL. The higher velocity is caused by smaller pore sizes. The correlation between the surface characteristics of the compressed and uncompressed material is presented in
Table 6. Because the range of the values changed under compression, Kendall’s test was again chosen as the test method. The entries in the table are the correlation coefficients of the given property, i.e.,
for the 70% quantile in the first line, first column, obtained from simulation results on uncompressed and compressed microstructures. As before, the value of 0.593—the correlation between
and
from
Table 3—was taken as a lower limit for judging two characteristics as being related to each other or not. On the basis of this value, the total quantile area
was evaluated as being correlated under compression for all quantile levels, as well as the average sizes
for the quantile levels of 70% and 50%. The average size
was uncorrelated for the 20% quantile level, and the largest area
was not correlated in any case.
The transfer of the detailed information from transport simulations in the GDL to larger scales of cell and stack simulations requires additional investigations of the channel and flow field modeling. Simulation domains of GDLs were typically much smaller than any flow field of real fuel cells, which is illustrated in
Figure 1. Furthermore, the GDL/channel interface connected a region of deterministic geometry with an irregular microstructure. As a consequence, many simulations on stochastic equivalent representations of the microstructure are required to consider the results from GDL transport simulations in the operating or boundary conditions of the cell-level models. Transport simulations on the microstructure can be evaluated statistically to characterize material or surface properties.
4. Discussion
The use of the volumetric effective properties of porous materials is the state of the art [
53]. This is reflected by simulations that use continuum-based approaches for porous regions. These methods are used, e.g., in fuel cell simulations based on commercial CFD software [
21,
22,
54]. Other investigations, not only fuel cell related, used open source CFD software [
55,
56,
57].
Domain sizes for the pore-scale simulations of GDLs are often in the mm range [
4,
5,
6], while cell and stack simulations require domain sizes of many cm, the size of real fuel cell stacks [
21]. The tight coupling of both scales is therefore impossible because of computational resources. The GDL/channel interface was already addressed by PEFC modeling reviews focusing on microfluidics [
58] and cell-scale modeling [
2]. Some kinds of simulation tasks use boundary conditions at the GDL/channel interface. Weber et al. [
1] identified the GDL/channel interface as being of high significance when they discussed two-phase phenomena in PEFCs. The path of further evaluation of the GDL/channel interface depends on its use for upscaling the results to cell-/stack-level simulations. In PNM approaches, the GDL can be represented by regularly-located pores and their characterization by randomly-distributed radii and flow behavior at the interface [
35,
37]. This increases the relevance of the knowledge about the sizes and positions of such pores. Cai et al. [
59] studied meander-type flow field channels by placing inlet regions on the GDL/channel interface of their PEFC model. Further investigations in the analysis of the exit surface from transport simulations in GDL can improve such cell-level simulations. The combination of this kind of transport simulations leads to multi-scale approaches in fuel cell modeling.
The approach of analyzing surfaces can potentially also be applied to different simulation techniques. Modeling approaches based on PNM have the potential to cover multiple scales [
36,
37], and the interfaces between domains of different spatial scales are of high interest for such investigations. In the field of multi-scale simulations, the presented methods can be a vehicle for combining the simulation domains, especially when domains of different spatial scales are connected.