Proton exchange membrane (PEM) fuel cells are devices that directly convert the chemical energy of fuel into electricity using an electrochemical process . PEMFC fuel cells use dihydrogen to generate electricity. Ahmad et al. (2024) examined the potential of green hydrogen as a solution to the climate crisis. They showed that green hydrogen technology will dominate the sector, not only because of its zero-carbon potential, but above all because of government commitments to achieve carbon neutrality by 2050. Alaswad et al. 2020 showed that PEM fuel cells are a promising alternative energy source for portable, automotive and stationary applications due to their relatively low operating temperature (60-80°C), high efficiency (40-60%) and high-power density (0.5-1.0) W/cm², comparable to that of existing internal combustion technology . PEM fuel cells have attracted the attention of the automotive industry as candidates for power generation applications with a lower environmental impact . Several parameters, such as entropy, contribute to improving the chemical stability of materials that conduct protons for fuel cell operation. This is the case with Ullah et al. 2023 ., who synthesized for the first time a new type of multicomponent proton-conducting oxide with a hexagonal perovskite structure capable of integrating up to eight different rare earth elements. They demonstrated that increasing the configurational entropy significantly improves the chemical stability of the material against CO₂, without degrading its proton conductivity. This work broadens the field of crystal structures compatible with the “high entropy” approach and paves the way for the systematic study of the link between entropy, conductivity, and chemical stability. On the other handThe, GDL consisting of a porous microfibrous substrate combined with a micro-nanoporous layer, improves gas distribution . GDLs are made from porous materials that have good electrical conductivity and an anisotropic porous microstructure . Its main function is to efficiently remove water from fuel cells during operation . It also facilitates the diffusion of reactants and the removal of reaction products while ensuring uniform current distribution . Due to its fibrous structure, GDL material has high anisotropy, which differentiates the transport of gas species, electrons and heat, as well as water . Nabovat et. (2014) have shown that tortuosity affects the path of protons in the catalytic layer and electrons in the carbon fibres of the GDL. Due to the many difficulties associated with observing and measuring the movement and distribution of species within a PEM fuel cell, numerical modelling and simulations have become indispensable tools . Software such as COMSOL Multiphysics are powerful tools for studying the effects of tortuosity on PEM fuel cell performance in numerical modelling . These models enable accurate simulation of transport and reaction processes in GDLs, providing valuable information for optimising their structure. Experimental studies have demonstrated the significant effect of GDL structural parameters and material characteristics on species transport . GDL tortuosity describes the complexity of the path that gas molecules must take through the porous structure and has a direct impact on the transport of reactive species . Yu et al. 2012 have shown that high tortuosity can lead to increased transport resistances, limiting the diffusion of reactants to active sites and the removal of generated water, while too low tortuosity can compromise water management in the cell. Wang and al. (2017); Zhao and all. (2018); Jing and all. (2024) have shown that tortuosity affects the diffusion of reactants such as oxygen and hydrogen, which has an impact on cell efficiency. In the case of anisotropic tortuosity, i.e. tortuosity that varies depending on the direction in the GDL, this can influence the directional distribution and transport efficiency of species (oxygen, hydrogen, water) . This means that the performance of PEM fuel cells can be better or worse depending on the orientation of the GDL relative to the gas flow. Anisotropy induced by tortuosity can lead to differences in gas diffusion, affecting the distribution of reactants on the catalytic surface . This leads to areas that are underfed with oxygen, creating ‘hot spots’ where the electrochemical reaction is less efficient. Although tortuosity is a key factor in the transport of reactive gases and reaction products, previous work has often neglected it or modelled it in a simplified manner . Fick's diffusion models, the Stefan-Maxwell model, convective diffusion, Boltzmann and numerical models (CFD: Computational Fluid Dynamics) are used to study the transport of gaseous species in GDLs . This study aims to model the behaviour of PEM fuel cells by analysing the impact of the GDL microstructure on performance, including mass transport losses (via Maxwell-Stefan equations for gaseous reactants: H₂, O₂, H₂O, N₂); pressure losses (via Darcy's generalised law governing gas flows) and the essential role of tortuosity (τ) in these phenomena. We are therefore primarily studying the impact of tortuosity on species transport within the gas diffusion layers (GDL) of a proton exchange membrane fuel cell. The study explores the influence of GDL tortuosity on PEM fuel cell performance.

Figure 1 shows a 50-watt fuel cell device as hardware. It consists of three main compartments. Block 1 is the electrolysis compartment, block 2 is the electronics compartment, and block 3 is the fuel cell compartment.

As shown in this figure, the demineralized water contained in the tank is sent to the proton exchange membrane electrolyzer using a pump. This produces a reaction that breaks down the water into oxygen and protons at the electrodes (anode and cathode). Then, thanks to the current flowing through the bipolar titanium plates, the protons pass through the solid membrane made of Nafion 115 and converge towards the cathode. The oxygen obtained and the excess water are returned to the water tank. The second reaction occurs at the cathode. Two electrons flow through the bipolar plate and recombine with two H+ protons that exit the membrane. Finally, hydrogen and water are obtained in a mixture. The mixture then converges towards the gas separator, where the hydrogen is separated from the water. The electrons generate current flowing through the bipolar plates. These plates distribute the gases evenly across the electrodes and separate the gases between the anode and cathode. The protons then pass through the solid membrane. The final reaction inside the fuel cell occurs at the iridium cathode, where oxygen from the air recombines with electrons and protons to produce water vapor through an exothermic reaction according to the following equations:

The modelled section of the fuel cell is composed of three main parts: (a) an anode (Ωa), (b) a proton exchange membrane (Ωm) and (c) a cathode (Ωc) as shown in Figure 2. In that figure, the electrodes are in contact with a gas distributor, which has an inlet channel (∂Ωa, inlet), a current collector (∂Ωa, cc) and an outlet channel (∂Ωa, outlet).

The feed gases (humidified hydrogen and humidified air) are treated as ideal gases and are transported by diffusion and convection. The electrodes are treated as homogeneous porous media with uniform morphological properties such as porosity and permeability. The gas inside each electrode exists as a continuous phase, so Darcy's law applies. An agglomerate model describes the electrode reactions in the active catalyst layers. The agglomerates are composed of catalyst particles and carbon embedded in a polymer electrolyte. The equations for the agglomerate model are derived from the analytical solution of a diffusion-reaction problem in a spherical porous particle. The different models used in the simulation are presented below:

The modelling of gas flows is based on Darcy's law in the interface of the PEM fuel cell, where the gas velocity is given by the continuity equation :

$∆.\left(\mathrm{\rho u}\right)=0$ in ${\mathrm{\Omega }}_{\mathrm{a}}$ and ${\mathrm{\Omega }}_{\mathrm{c}}$ with $\mathrm{u}=-\frac{{\mathrm{k}}_{\mathrm{p}}}{\mathrm{ղ}}{\nabla }_{\mathrm{p}}$and

${\mathrm{k}}_{\mathrm{p}}$ is the electrode permeability (m2), ղ is the gas viscosity (Pa.s), p is the pressure (Pa), R is the gas constant (J/mol.K), T is the temperature (K), M is the molar mass (kg/mol), and $x$ is the molar fraction. At the inputs, we have:

At the thin electrode gas diffusion boundary for the anode and cathode, the gas velocity is automatically calculated by the hydrogen fuel cell interface from the total mass flow given by the electrochemical reaction rate and the stoichiometric coefficients using Faraday's law.

The hydrogen fuel cell interface is used to model the potential distributions in the three domains, which local current density expressions, for the anode and cathode are given by the following equation :

Where the subscript e represents « a » (anode) or «c» (cathode), ${\mathrm{R}}_{\mathrm{agg}}$is the radius of the agglomerate in m and ${\mathrm{j}}_{\mathrm{agg},\mathrm{a}}$ and ${\mathrm{j}}_{\mathrm{agg},\mathrm{c}}$ (in A/m³) are the current densities given by the agglomerate model, ${\mathrm{\epsilon }}_{\mathrm{mac}}$ is porosity (the macroscopic porosity) The specific surface area of the fine gas diffusion electrode (in 1/m) can be calculated by the following formula:

In electrodes and conductive GDL, electrons move according to Ohm's law:

where:

${\mathrm{i}}_{\mathrm{S}}$ is electron current density [A/m²]

${\sigma }_s$ is electrical conductivity of the solid [S/m]

${\varnothing }_s$ is electrical potential in the solid phase [V]

The agglomerate model describes the current density in an active layer made up of agglomerates of ionic conductive material and electric conductive particles partially coated with catalyst. The resulting equations for the current density in the anode and cathode are given by the following formula :

In these equations, ${\mathrm{D}}_{\mathrm{agg}}$ is the gas diffusivity of the sinter (m²/s), ${\mathrm{n}}_{\mathrm{e}}$ is the number of «charge transfers» (1 for the anode and -2 for the cathode), S is the specific surface area of the catalyst inside the sinter (1/m), and F is the Faraday constant (C/mol), the ${\mathrm{c}}_{\mathrm{i},\mathrm{ref}}$ are the reference concentrations of the species (mol/m³), ${\mathrm{c}}_{\mathrm{i},\mathrm{agg}}$ are the corresponding concentrations in the surface of the cluster (mol/m³), ${\mathrm{i}}_{0\mathrm{a}}\mathrm{ and }{\mathrm{i}}_{0\mathrm{c}}\mathrm{ }$are the exchange current densities (A/m²), R is the gas constant, T is the temperature (K), and the overvoltage’s at the anode and cathode are given by the following formula where ${\mathrm{E}}_{\mathrm{eq}}$ represents the equilibrium voltage:

The concentrations of hydrogen and oxygen dissolved at the surface of the agglomerates are related to the molar fractions of the respective species in the gas phase by Henry's law, where K is Henry's constant (Pa.m³/mol) :

The boundary conditions for the charge equilibrium potential are:

The model takes into account two species at the anode H₂ and H₂O and three at the cathode O₂, H₂O and N₂. The interface of the PEM fuel cell uses Maxwell-Stefan multi-component diffusion governed by the equations :

To solve (13) for mass fractions$\mathrm{ }{\mathrm{w}}_{\mathrm{i}}$, this PEM fuel cell model assumes that temperature-induced diffusion is insignificant and sets the source term R to zero. The binary Maxwell-Stefan diffusion coefficients, ${\mathrm{D}}_{\mathrm{ij}}$ (m²/s), are calculated automatically by the interface.

In porous GDLs, the effective binary diffusion coefficients,${\mathrm{D}}_{\mathrm{ij},\mathrm{eff}}$ must be taken into account using a diffusivity correction for porous media based on porosity ${\mathrm{\tau }}_{\mathrm{g}}$and tortuosity ${\mathrm{\epsilon }}_{\mathrm{g}}$which defines the binary diffusion coefficient according to the relationship:

Here, tortuosity is defined as a tensor in the form of a 2 x 2 diagonal matrix according to the formula:

In steady state, all temporal variations disappear:

This simplified form is used to model gas transport in porous GDLs. The mole fractions of the feed gas are reported at the inputs. At the boundaries of the fine gas diffusion electrodes, the mass fluxes of the species are automatically determined from the electrochemical reaction rate and the stoichiometric coefficients, using Faraday's law. Contributions to the Stefan velocity are also automatically calculated by the interface. The membrane transport functionalities of the hydrogen fuel cell interface are used to model the transport of water in the ionomer phase in the membrane domain.

Water transport in the membrane of a PEMFC cell is governed by two main mechanisms: diffusive transport due to water concentration gradients in the membrane, and electro-osmotic transport induced by hydrated protons (H₃O₊) crossing the membrane from the anode to the cathode. The average number of water molecules transported per proton, denoted nd, is called the electro-osmotic entrainment coefficient. The molar flux density of water transported in the membrane is then given by the following equation:

where:

${\varnothing }_m$ is the electrical potential in the membrane [V]

${\sigma }_m$ is the ionic conductivity of the membrane [$S{.m}^{-1}$]

F is Faraday's constant, [$C{.\mathit{mol}}^{-1}$]

$D_{H_{2}O}^m$ is the diffusion coefficient of water in the membrane [${.m}^{-2}{.S}^{-1}$]

$C_{H_{2}O}$ is the molar concentration of water in the membrane [$\mathit{mol}{.m}^{-3}$]

The flow of ionic charges in the membrane follows the local Ohm's law:

The reaction layers are the areas where electrochemical reactions take place. Due to the sintering process, a fraction of the catalyst can partially penetrate the membrane. Since these layers are much thinner than the other components of the cell (such as the channels, gas diffusion layers, and membrane), they are generally treated as interfaces or boundary conditions in mathematical models. To determine the local distribution of current density on the catalytic surface, the Butler-Volmer kinetic equation is applied in both static and dynamic approaches. At very low current densities, this equation correctly reproduces the cell voltage.

Where j is local current density; $j_{\mathit{ref}}$ is reference exchange current density; α is charge transfer coefficient; C is concentration of reactive species; $n_{\mathit{act}}$ is activation overpotential; R is ideal gas constant; F is Faraday constant.

Boundary conditions are defined on all external boundaries of the calculation domain in order to describe the behaviour of the fluid and chemical species.

At the channel inlet, a Dirichlet condition is applied: it directly sets the values of the main variables, such as velocity, pressure and species concentrations. This means that the characteristics of the gas mixture are known at the inlet.

At the outlet of the channels, a Neumann condition is used. This assumes that the velocity, pressure and concentration gradients in the direction of flow are zero. In other words, the variables no longer vary when they leave the domain, which corresponds to a free outlet of the fluid.

Finally, on the channel walls and at the ends of the MEA, the conditions applied to pressure, velocities and concentrations are also Neumann-type. This reflects the absence of material flow and pressure variation across these surfaces, which are considered impermeable.

The table below provides information on the boundary conditions used in this study.

Figure 3 illustrates the impact of tortuosity configurations on current density in the porous medium of the PEM fuel cell. Four cases are examined according to the x and y directions, namely low isotropic tortuosity (tortx=1.1, torty=1.1) and high tortuosity (tortx=2.8, torty=2.8) and anisotropic cases (tortx=1.1, torty=2.8) and (tortx=2.8, torty=1.1). After analysis, it was found that low tortuosity produces a uniform current density, while high tortuosity (2.8, 2.8) results in a non-uniform distribution with localised reductions due to impeded transport of reactants. In anisotropic cases, the case (tortx=1.1, torty=2.8) facilitates transport along x but hinders it along y, resulting in a higher current density along x. The inverse asymmetry occurs for (tortx=2.8, torty=1.1), with easier transport along y.

Figure 4 presents the impact of tortuosity on the mass fractions of reactants (H₂ at the anode and O₂ at the cathode) in the PEM fuel cell. The previous four different configurations are compared. In the case of low tortuosity (tortx=1.1, torty=1.1), the mass fractions of the reactants are high and uniformly distributed, indicating efficient gas transport. In contrast, when tortuosity is higher (tortx=2.8, torty=2.8), the overall mass fractions decrease, leading to increased resistance to gas transport in both directions.

For anisotropic tortuosity, an asymmetry in the distribution is observed. This results in a significant reduction in the mass fractions along the x- and y-directions in the cases of (tortx=1.1, torty=2.8) and (tortx=2.8, torty=1.1), respectively. Thus, reactants are less accessible under anisotropic tortuosity, leading to increased concentration gradients. This phenomenon can cause a local decrease in the mass fraction of reactants, particularly in regions where tortuosity is highest.

Figure 5 illustrates the influence of tortuosity on the hydrogen mass fraction.

This figure shows that in the cases of isotropic tortuosity (tortx=1.1, torty=1.1) and (tortx=2.8, torty=2.8), the tortuosity is low and high respectively in the x and y directions. In the case (tortx=1.1, torty=1.1), hydrogen diffuses easily through the GDL while in the case (tortx=2.8, torty=2.8), hydrogen encounters strong transport resistance in both directions, which significantly reduces the mass fraction near the membrane. In the case of anisotropic tortuosity, i.e. (tortx=1.1, torty=2.8) and (tortx=2.8, torty=1.1), there is an asymmetric distribution of the mass fraction with under-utilised zones in the direction of high tortuosity.

Figure 6 shows the influence of tortuosity on the oxygen mass fraction.

As in the case of the study of the influence of tortuosity on the hydrogen mass fraction, similar results are obtained for the influence of tortuosity on the oxygen mass fraction. Indeed, low tortuosity, i.e. (tortx=1.1, torty=1.1), after analysis is ideal for a uniform and high distribution of the oxygen mass fraction, which maximises the performance of the cell. Anisotropic tortuosity (tortx=1.1, torty=2.8) or (tortx=2.8, torty=1.1) also results in an asymmetric mass fraction distribution, with under-utilised areas in the direction of high tortuosity. Analysis of the figure also shows that high tortuosity (tortx=2.8, torty=2.8) is the least favourable because it reduces the overall oxygen mass fraction, thus increasing losses by concentration.

Figure 7 illustrates the influence of tortuosity on relative humidity.

The analysis of this figure shows that the water produced during the electrochemical reaction, along with the water from the humidification of the reactant gases, diffuses easily through the GDL when the tortuosity is isotropic and low (i.e. tortx=1.1, torty=1.1). This promotes good proton conductivity within the membrane. Conversely, when the isotropic tortuosity is high (tortx=2.8, torty=2.8), water transport is significantly hindered in both directions, resulting to dryness near the membrane. A reduction in proton conductivity could therefore be observed, leading also to a reduction in cell performance. As in the case of other species, anisotropic tortuosity (tortx=1.1, torty=2.8) or (tortx=2.8, torty=1.1) leads to an asymmetric distribution of relative humidity, with under-utilised areas in the direction of high values of tortuosity. Similarly, increased tortuosity leads to less pronounced concentration gradients, which can limit the efficiency of charge transfer.

Figure 8 illustrates the variation of current density within the anode active layer as a function of cell height (i.e. Vertical position). Here, the cell height represents the spatial coordinate used to examine the distribution of different tortuosity configurations on this distribution.

In this figure, an uneven distribution of current density is observed across the layer for both configurations. Despite the variations, the curves for each configuration follow a similar overall trend, with the current density showing noticeable variation along the x and y directions due to tortuosity effects. A comparison between the two tortuosity reveals distinct differences. The highest current density is achieved when the tortuosity is low and isotropic (tortx=1.1, torty=1.1). Conversely, the lowest current density occurs in the configuration with high isotropic tortuosity (tortx=2.8, torty=2.8). Thus, low tortuosity, particularly when it is isotropic, favours optimum anodic performance by facilitating mass transport and ensuring a more homogeneous distribution of current. These results indicate that reducing the GDL tortuosity enhances reactant transport.

Figure 9 illustrates the effect of gas diffusion layer (GDL) tortuosity on the polarization curve of the proton exchange membrane fuel cell (PEMFC). The different curves, corresponding to various tortuosity configurations, exhibit similar general shapes and can be divides into three characteristic performance regions: activation losses, ohmic losses, and mass transport (concentration) losses.

Activation zone (0-0.2 A/m²): a sharp drop in voltage is observed at low current density. This phenomenon is linked to the activation potential, which reflects the kinetics of electrochemical reactions at the anodic and cathodic interfaces.

Ohmic zone (0.2-0.8 A/cm²): Linear decrease in voltage with increasing current density, reflecting ohmic losses due to resistance to proton transport across the membrane and electron conduction through the electrodes and interconnections.

Mass transport zone (0.8-1.4 A/cm²): The cell voltage drops rapidly with increasing current density. This results from mass transport limitations, where depletion of reactants (mainly oxygen) at the catalytic sites hinders the electrochemical reaction, leading to concentration. These different results imply that the best performance is achieved for low isotropic tortuosity (tortx = 1.1, torty = 1.1), where reactant transport is most efficient and voltage losses are minimized.

In order to assess the consistency of the model developed, a qualitative comparison was made between the numerical results and the experimental data from the PEMFC stack device available in the laboratory shown in Figure 10. This comparison focuses mainly on the shape of the polarization curve, representing the variation in cell voltage as a function of current density. The objective is not to quantitatively validate the model, but to evaluate its ability to reproduce the general trends observed experimentally.

Observation of the experimental and simulated polarization curves highlights the three distinct zones characteristic of the operation of a PEMFC fuel cell:

Activation region: For low current densities, the voltage drops rapidly. This decrease is mainly due to activation losses associated with electrochemical reactions at the electrodes. The model correctly reproduces this trend, although the initial slope may differ slightly due to the approximation of the kinetic parameters used.

Ohmic region: In the intermediate zone, the voltage decreases almost linearly with current density. This part is dominated by ohmic losses due to the ionic resistance of the membrane and electrical contacts. The numerical results show a slope similar to that observed experimentally, reflecting a good representation of the overall resistance of the system.

Mass transport region: At high current densities, the voltage drops sharply due to diffusion limitations of the reactants in the diffusion layers and the catalytic layer. This drop is also reproduced by the model, although the transition to this zone may appear more or less pronounced depending on the operating conditions considered.

In general, the shape of the simulated curve follows that of the experimental curve, which shows that the model adequately captures the main physical phenomena influencing the electrochemical behavior of the cell. Nevertheless, discrepancies may remain, attributable to simplifications in the model.

Figure 11 illustrates the polarisation curves for four electrical conductivity values, namely 20 S/m, 100 S/m, 1000 S/m and 2000 S/m. It can be seen that the increase in conductivity significantly improves the cell voltage for the same current density, reflecting a notable reduction in ohmic losses in the solid phase. This behavior is particularly pronounced in the ohmic region of the curve, where the electronic resistance of the GDL plays a predominant role. Thus, increasing conductivity from 20 S/m to 100 S/m reduces ohmic losses by approximately 30% (calculated from voltage differences at 0.6 A/m²), as mass transport resistance is negligible. Higher conductivity therefore allows for more efficient transport of electrons to the reaction sites, thereby reducing the potential drop associated with the passage of current. However, it should also be noted that the benefits of improved conductivity become progressively less significant above 1000 S/m. The curves associated with conductivities of 1000 S/m and 2000 S/m overlap, indicating that performance improvement saturates beyond a certain threshold. It appears that moderate conductivity, in the order of 1000 S/m, seems to be a satisfactory compromise between performance and material feasibility, while that values below 100 S/m cause significant losses and should be avoided in realistic battery configurations. The analysis shows that a conductivity between 100 and 1000 S/m is sufficient to ensure optimal performance.

Figure 12 illustrates the impact of porosity on the polarization curve of a proton exchange membrane fuel cell. The simulation were conducted at a constant temperature of 80°C and pressure of and 1 atm.

As illustrated by the polarization curves in the figure for porosities of 0.2, 0.4, 0.6, and 0.8, the influence of GDL porosity on PEMFC performance is reflected in a characteristic change in voltage as a function of current density. For each porosity value, the voltage decreases gradually as the current density increases, which corresponds to typical polarization behavior. However, there is a noticeable improvement in performance as porosity increases: at equal current density, the voltage is higher. This gain is particularly significant between porosities of 0.2 and 0.6. In this range, the increase in porosity facilitates the diffusion of reactive gases (particularly oxygen) to the active sites of the cathode. Increased permeability thus reduces the limitations due to mass transport, which become predominant at high current densities. Beyond a porosity of 0.6, the improvement in performance diminishes; at 0.8, it becomes negligible. This is because higher porosity promotes better gas distribution in the GDL and the catalytic layer, which increases oxygen availability and improves the removal of water produced by the reaction. However, when porosity exceeds a certain threshold (approximately 0.8), the solid fraction of the porous layer is reduced excessively. This reduction can then compromise the cell's power output, despite the advantages it brings to gas transfer and water removal.

Figure 13 explores the influence of gas pressure on the polarization behaviour of the PEMFC under the same operating temperature of 80°C. This figure aims to highlight how variations in operating pressure affect the cell’s voltage-current characteristics.

This figure shows that at a constant current density of 1 A/m², the cell voltage increases by approximately 6% when the pressure rises from 1 atm to 4 atm. This increase in pressure is necessary to maintain a constant current density as the pressure increases. The increase in pressure causes an increase in the partial pressures of the reactive gases, which increases the Nernst potential. Above 4 atm, the improvement in cell performance becomes marginal, revealing the existence of a critical pressure threshold. Finally, it is important to note that high-pressure operation can induce technical constraints, risk of membrane drying, premature aging of components (mechanical stress), and additional energy consumption related to gas compression.

Figure 14 highlights the influence of relative humidity on PEM stack performance.

This figure shows the polarisation curves for different concentrations of oxygen and hydrogen, with relative humidity (RH) levels between 40% and 100%. At an operating voltage of approximately 0.7 V, there is a significant increase in current density, from 0.22 A/m² to 0.3 A/m² when RH increases from 40% to 80%. Optimal performance is achieved when RH is maintained between 60% and 80%. At low RH values (e.g. 40%), dehydration of the membrane occurs, leading to a reduction in proton conductivity, and high RH levels can cause membrane clogging.

The study conducted in this work explores the influence of the microstructure of porous layers, in particular tortuosity, as well as the effect of certain operating parameters on the overall performance of a proton exchange membrane fuel cell (PEMFC). The analysis focused on four tortuosity configurations (low isotropic: tortx = 1.1, torty = 1.1; high isotropic: tortx = 2.8, torty = 2.8; anisotropic: tortx = 1.1, torty = 2.8 and tortx = 2.8, torty = 1.1) and, on the other hand, on the effects of porosity, electronic conductivity, feed pressure, and reagent humidity. The results show that tortuosity plays a decisive role in the transport of reactive species and water. Low isotropic tortuosity allows for homogeneous distribution of the reactants. and humidity, promoting efficient mass transport and uniform current density distribution. Conversely, high or anisotropic tortuosity induces marked concentration gradients, water accumulation in certain areas, and degradation of the electrochemical performance of the cell. These observations are consistent with the conclusions of Ceballos and al. (2025) , who demonstrated that increased tortuosity in gas diffusion layers significantly limits transport efficiency and increases concentration losses. At the same time, the study highlighted the influence of certain operating parameters. The electronic conductivity of the GDL, when increased from 20 S/m to 1000 S/m, reduces ohmic losses and improves cell voltage, in line with the pioneering work of Springer and al. (1991) . Porosity, when increased in moderate proportions around 0.6, promotes gas distribution and evacuation. water, which improves fuel cell performance, as shown by Jing et al. (2024) . Reagent humidity is also a critical parameter, influencing both the proton conductivity of the membrane and water management. Insufficient humidity causes the membrane to dry out, while excess humidity can cause flooding of the porous layers, both of which are detrimental to cell operation. These observations are consistent with the experimental results reported by Zhao et al. (2018) , which emphasizes the importance of water balance in PEMFCs. This result also relates to the findings of Anyanwu et al. (2021) , who experimentally analysed and compared the behavior of corrugated flow field designs with those of serpentine, interdigitated, and straight parallel flow fields. An increase in humidity levels is accompanied by improved charge transfer, which influences performance. The results also indicate that the corrugated flow field configuration uses less parasitic charge, with better humidity tolerance, improved mass transport, and increased performance compared to other conventional flow field designs. Finally, the increase in Gas feed pressure (from 1 to 3 atm) improves cell voltage by increasing the partial pressures of H2 and O2 and the associated Nernst voltage. However, it must be carefully controlled to avoid excessive mechanical stress on the membrane, in accordance with the findings of Wang et al. (2004) . This study shows that PEMFC performance is highly dependent on the interaction between microstructural parameters and operating conditions. Low tortuosity and moderate porosity promote good mass transport, while high conductivity and optimized moisture management reduce ohmic losses. This is also the case with the findings of Ullah et al. 2026 , which showed that integrating high-precision processes with more economical methods, optimized by artificial intelligence, is essential to making SOFC fuel cells commercially viable.

These results are consistent with those of Jourdani et al. (2017) , who evaluated the combined effect of operating parameters on cell performance, although their study did not take tortuosity into account. Recent work by Ceballos and al. (2022) recommends a coupled multi-parameter approach to model the actual behavior of fuel cells more accurately, given the interdependence between these variables. Thus, for effective PEMFC optimization, it is essential to adopt a strategy that takes into account the microstructure of materials, thermophysical properties, and operating conditions.

This study, based on modeling species transport in PEM fuel cells using COMSOL Multiphysics, highlights the essential role of the porous structure of gas diffusion layers (GDL), particularly tortuosity, on the performance of a proton exchange membrane fuel cell (PEMFC), as well as evaluating the effect of certain operating parameters on the transport of reactive species. The results show that tortuosity, as a geometric parameter of the GDL, strongly influences the distribution of the mass fraction of reactants, relative humidity, and current density. Low isotropic tortuosity allows for better reagent supply, more homogeneous humidity, higher current density, and therefore improved cell performance. Conversely, high or anisotropic tortuosity induces marked concentration gradients, poor water distribution, and diffusion losses. The impact of other parameters such as porosity, pressure, relative humidity, and proton conductivity has also been highlighted. These parameters act synergistically with tortuosity and influence both ohmic losses and diffusion limitations. This knowledge opens up promising prospects for the design of advanced GDLs, contributing to greater durability and improved fuel cell performance in practical applications.

The authors declare no conflicts of interest.