1. Introduction
Proton Exchange Membrane Fuel Cell (PEMFC) has garnered significant attention as a promising green energy device, attracting extensive research efforts [
1,
2]. The commercial viability of PEMFCs depends on their electrical performance, stability, and durability. Improving the power density of fuel cells has an important driving effect on their application [
3,
4].
Figure 1a gives a schematic illustrating the components of PEMFC. Among these components, the bipolar plate (BP) is a focal point for many researchers because the structural design of BP has a significant impact on the electrical performance of PEMFC [
1]. The BP is composed of the gas channel (GC) and the rib. The coupling effect between the BP and the gas diffusion layer (GDL) has a significant impact on the heat and mass transfer behavior of fuel cells. During the assembly process of PEMFC, a compressive force is applied to ensure the sealing of the structure. As known, the GDL is the thickest porous layer in the membrane electrode assembly (MEA) and possesses the lowest elastic modulus within the MEA [
5]. Consequently, the GDL is susceptible to deformation and damage under high compressive forces. This deformation can cause the GDL under the GC to intrude into the GC, thereby altering the heat and mass transfer characteristics within the PEMFC [
6]. As one of the essential components, the GDL plays the role of draining water out of the catalyst layer (CL) and providing enough gas pathways. The large-scale commercialization of PEMFCs requires higher power and current densities; however, at high operating current densities, the massive accumulation of liquid water in the GDL will lead to flooding and impede gas diffusion, resulting in rapid degradation of cell performance [
7]. Accordingly, improving GDL’s water management ability is imperative for pursuing better cell output. The output performance of PEMFC is significantly affected by the heat and mass transfer behaviors of GDL. Thus, investigating the relationship between the GC and the output performance of PEMFCs under GDL deformation holds great importance.
The channel-to-rib width ratio (CRWR, the channel width divided by rib width) is usually used to describe the relation between the GC and rib [
8]. Researchers have focused on improving the electrical performance of PEMFCs by optimizing CRWR. Acar et al. [
9] investigated the temperature distribution in a fuel cell under five CRWRs (0.33, 0.6, 1, 1.66, and 3). They found that the highest temperature distribution uniformity was achieved at CRWR = 3, leading to a high heat stability of the fuel cell. Pan et al. [
10] found that the optimal electrical performance and oxygen utilization rate were achieved at CRWR = 2.4 among five CRWRs (0.8, 1.2, 1.6, 2, and 2.4). Qiu et al. [
11] found that the best electrical performance was achieved at CRWR = 3 among seven CRWRs (0.09, 0.14, 0.33, 0.6, 1, 1.67, and 3). Zeng et al. [
12] optimized the CRWR to maximize the power density. They found that the power density of the fuel cell in the optimized case (CRWR = 1.45) was 8.09% higher than that in the conventional case of CRWR = 1. These researchers investigated the CRWR from the perspective of fuel cell electrical performance, heat, and mass transfer characteristics. Large CRWR implies better heat and mass transfer characteristics and higher electrical performance. However, large CRWR signifies large GDL stress, and excessive CRWR may destroy the structure of GDL. This brings the necessity of investigating the stress behavior exerted on the GDL.
The material properties of GDL play a crucial role in the assembly of BP and GDL. In the deformation of GDL, the elastic modulus, strain, and stress of GDL are the key parameters. The elastic modulus of GDL is related to the type of GDL. The strain of GDL is represented by compression ratio (CR). The CR serves as an indicator of sealing performance since different component materials and types of GDL have different stress-strain properties [
13,
14]. A large CR of GDL indicates a better sealing performance of PEMFC. The stress of GDL is proportional to the elastic modulus and strain of GDL. Moreover, the stress of GDL is inversely proportional to compressive area. The compressive area of GDL is dependent on CRWR. When the CRWR increases, the compressive area of GDL decreases. When the stress of GDL is larger than the compressive strength of GDL, the GDL breaks. Simon et al. [
14] and Mason et al. [
15] found that most of Toray’s GDLs experienced structural collapse under compressive stresses of 2.5 MPa by experiment. Excessive compressive stress causes GDL fibers to break, leading to structural damage to the GDL. These broken fibers may puncture PEM, allowing hydrogen to cross and form a short circuit. Excessive compressive stress causes irreversible structural damage to GDL, which affects its porosity, thickness, electrical resistance, and gas permeability [
16]. Niblett et al. [
17] found that a broken GDL attenuates the mass transfer properties within the PEMFC, leading to an increase in concentration polarization. The electrical performance of PEMFC is reduced when the GDL is broken. Yu et al. [
18] measured the power density of PEMFC and provided scanning electron microscope images of GDL when the GDL is broken/unbroken. They found that the peak power density of broken GDL was 33.33% lower than that of unbroken GDL. Moreover, Kang et al. [
19] found that the peak power density of broken GDL was 30.43% lower than that of unbroken GDL in their experiment study. Therefore, it is necessary to capture the broken conditions of the GDL exactly for cell performance analysis, which requires an accurate mechanical relationship between GDL and BP.
To accurately describe the mechanical relationship between GDL and BP structures, many researchers have developed mathematical models to investigate the structure and electrical performance of PEMFC. The traditional approach is to use the linear-elastic theory to describe the stress-strain relationship between the GDL and the BP [
5] and then combine it with the electrochemical, heat, and mass transfer models of the PEMFC for simulation and analysis. The porosity, conductivity, and gas diffusion coefficient of GDL are determined by the CR of GDL, which significantly impacts the heat and mass transfer behavior of PEMFC [
20]. Zhou et al. [
21] used a linear-elastic model to simulate the performance of the PEMFC of GDL at different stresses and found that the polarization curves for the case of a stress of 1 MPa were closest to the experimental data. The average current density deviation of their model from an experiment was 8%. In 2021, Zhang et al. [
5] proposed a linear-elastic model to analyze the relationship between the CR of the GDL and the electrical performance of the PEMFC, and the polarization curves of their simulations deviated from the experimental data by 7.8%. Chen et al. [
20] used a linear-elastic model to investigate the effect of GDL on heat and mass transfer behavior under different CRs and stress conditions. The polarization curves of their simulations deviated from the experimental data by 6%, but they ignored the concentration loss in the polarization curve, resulting in limited applicability. Other researchers have described the deformation of GDL based on linear-elastic theory, and the average current density deviations of their simulation from experimental data were 9.3% [
22] and 7.3% [
23], respectively. Therefore, using the linear-elastic theory to describe the deformation of GDL leads to a significant current density deviation of simulation from the experiment.
However, Zhang et al. [
24], Meng et al. [
25], and Yan et al. [
26] revealed through mechanical experiments that GDL is a nonlinear-elastic material. The nonlinear deformation of GDL is a complex phenomenon, including carbon fiber slip, fracture, and elastic-plastic deformation [
27]. The nonlinear-elastic theory is more accurate for describing the stress and strain of GDL than the traditional linear-elastic theory. Xiao et al. [
28] proposed an improved nonlinear-elastic model applicable to carbon paper GDL and verified it experimentally. They found that the average stress deviation of mechanical simulation from the experiment at a GDL porosity of 0.75 was reduced from >35% to 15%, using nonlinear-elastic theory. In 2023, Afrasiab et al. [
29] proposed a correction factor for describing the nonlinear deformation of GDL based on the Timoshenko beam theory in the macroscopic case and showed that the average stress deviation of the simulation from the experiment was 10%. However, they focused on the mechanical properties of the ex-situ GDL (without PEMFC assembly), which is different from the in-situ GDL (with PEMFC assembly). Besides, some researchers coupled the nonlinear-elastic theory with the PEMFC model to investigate the effect of the stress-strain of GDL on the performance of PEMFC. Li et al. [
30] applied a nonlinear-elastic theory in the in-plane direction and through-plane direction of GDL and found that the average current density deviation of the simulation from the experiment was 6.5%. Yan et al. [
26] investigated the deformation of GDL using a nonlinear-elastic theory and found that the average current density deviation of the simulation from the experiment was extremely small. However, they did not consider the impact of stresses on the GDL. The maximum compressive stresses of the GDL in their experiments were 5.0 MPa [
30] and 3.5 MPa [
26], which may damage the GDL structure [
31].
Despite the impressive research on the elastic theory of GDL deformation, a comprehensive investigation of the CRWR of BP, elastic modulus, strain, and stress of GDL is necessary for improving the electrical performance of PEMFC. In this study, a three-dimensional PEMFC model incorporating nonlinear-elastic theory is proposed and validated. It can be used to calculate the GDL mechanical parameters and the corresponding fuel cell electrical performance for different CRWR cases. The mechanical, electrochemical, heat, and mass transfer behaviors of the PEMFC during the assembly process are solved using COMSOL Multiphysics. Stress-strain analysis is used to reveal the correlation among CRWR of BP, stress of GDL, strain of GDL, and elastic modulus of GDL. Based on the correlation, specific GC shapes are established to analyze the quantitative relationship among electrical performance, heat, and mass transfer behaviors inside PEMFC.