High Current Density Operation of a Proton Exchange Membrane Fuel Cell with Varying Inlet Relative Humidity—A Modeling Study

Abstract

This paper focuses on proton exchange membrane fuel cell (PEMFC) operation at current densities in the order of 6 A/cm². Such high current densities are conceivable when the traditional carbon fiber papers are replaced with perforated metal plates as the gas diffusion layer to enhance waste heat removal, and at the same time the relative humidity inside the fuel cell is kept below 100% by applying appropriate operating conditions as was found in previous one-dimensional modeling work. In the current paper, we applied a three-dimensional, multi-phase computational fluid dynamics model based on Ansys-CFX to obtain additional insight into the underlying physics. The calculated pressure drops are in very good agreement with previous one-dimensional modeling work, and the current densities in all case studies are in the order of 5–6 A/cm², but different from the previous one-dimensional study, the results suggest that the relative humidity is very close to 100% throughout the entire channel length when the inlet relative humidity is 100%, ensuring best hydration cell conditions and hence best performance. Importantly, the model results suggest that fuel cell performance at a high current density in conjunction with relatively low stoichiometric flow ratios around 1.5–2 is possible.

1. Introduction

In low-temperature proton exchange membrane (LTPEM) fuel cells, hydrogen (H₂) reacts with oxygen (O₂) in air to produce water (H₂O) and electricity with by-product heat. The basic structure and operation principle of the LTPEM fuel cells are illustrated in Figure 1.

The electrochemical reaction mechanisms are as follows:

Cathode:  O₂ + 4H⁺ + 4e⁻ => 2H₂O

Anode:  2H₂        => 4H⁺ + 4e⁻

Combined:  O₂ + 2H₂     => 2H₂O

By comparison to combustion engines, the advantage is the environmental friendliness. Meanwhile, the efficiency is usually higher than 60% [1,2].

With the rapid development of PEM fuel cell commercialization, there is a need for high power and high current operation to further reduce the capital expenses [3] and increase the gravimetric and volumetric power density. The Toyota Mirai (First Generation 2014) uses a PEM fuel cell drive module, which has a maximum power of 114 kW with a limiting current density of 2.5 A/cm² [4]. In the second generation (2020), the limiting current density was increased to 3.5 A/cm² [5], which may be the highest among commercialized fuel cell vehicles. The New Energy and Industrial Technology Development Organization (NEDO) has set a goal of increasing the volumetric power density to 6.0 kW/L by 2030 [6]. That means that the fuel cell needs to be operated at a very high current density to achieve such a high power density [7]. The objectives for the limiting current density by 2030 and 2040 stand at 3.8 A/cm² at 0.66 V and 4.4 A/cm² at 0.85 V, respectively [8].

As the fuel cell current density is increased significantly, the waste heat coming from electrochemical reactions becomes a problem. Moreover, there has always been a need for proper water management [9,10] inside the PEM fuel cell. On the one hand, the proton exchange membrane needs an appropriate amount of water to ensure the transfer of protons. On the other hand, insufficient water management fuel cells may cause flooding, which has a detrimental effect on mass transfer. To this end, dew point diagrams were previously devised to identify suitable fuel cell operating conditions [11].

For traditional PEM fuel cells, carbon fiber papers (CFP) have been used to distribute the fuels (hydrogen, oxygen, and water vapor) to the reaction sites, conduct electrons, and allow for capillary water transport. Therefore, its physical properties of mass transfer are compatible with fuel cells. However, the thermal conductivity of CFP is in the range of 0.5–2 W/mK [12,13,14], which may not meet the requirements of high current density. More studies [15,16] have focused on thermal management because increasing current density and power density is imperative. Moreover, fuel cell manufacturing technology is also developing rapidly. Hussain et al. [17] replaced the CFP by thin, perforated metal sheets with a high thermal conductivity where round holes in the order of 80–120 microns are manufactured by a chemical etching technique. When such a metal gas diffusion layer (GDL) is used, the micro-porous layer (MPL) takes on the role of distributing the reactants to the under-land areas because a metal GDL only allows for transport in the through-plane direction.

In a modeling study employing computational fluid dynamics (CFD), the holes in the metal GDL were predicted to remain dry while liquid water was found in the MPL [18]. Therefore, all product water would leave the fuel cell in the vapor phase, which is also beneficial for the thermal management.

In the current study, our group combines our previous work that included a one-dimensional Engineering Equation Solver (EES) model [3], which investigated the possibility of operating a proton exchange membrane fuel cell at a high current density, and a three-dimensional CFD model based on Ansys CFX [19,20]. The 1D EES model provided the boundary conditions for the cathode side (stoichiometric flow ratio, temperature, current density, pressure, and relative humidity). For the anode side, a low stoichiometry operation is chosen, as was previously proposed by our group [19]. It is the main goal of the current study to verify with a CFD model whether the high limiting current density as suggested by the one-dimensional EES model is indeed feasible and to study the detailed reactant and water distribution inside the fuel cell.

To this end, four different cases are investigated: relative humidity (RH) at the cathode inlet is 100%, 90%, 80%, and 70%, respectively, with correspondingly high current density. The focus is predominantly put on the predicted oxygen concentration, pressure drop, velocity, liquid water distribution, and relative humidity in both in-plane and through-plane directions. The model is implemented as a standalone Ansys CFX document, which could provide a starting point for more complex model development.

2. Model Description

In this section, a three-dimensional, steady-state multifluid model is presented with development based on the previous single-phase model [21]. The flow in the channels is assumed to be laminar. Diffusion is dominant in porous mediums, and capillary force is the driving force for the liquid phase.

2.1. Geometry

Figure 2 shows the geometry of the PEM fuel cell for this simulation. Taking advantage of the symmetrical structure of the fuel cell, only half of it is calculated to reduce computation time. Symmetry boundary conditions are applied on the left/right side (YZ plane) of the geometry (red markers in Figure 2). The entrances for both sides are opposite; that is, the inlet of the anode and the outlet of the cathode are on the same side. The thickness of 100 microns for the perforated metal GDL is higher than in the work by Hussain et al. [17], who used 30 microns. Therefore, the results for the limiting current density in the current work are conservative estimations. Moreover, in the work by Hussain et al., the combined thickness of the metal GDL and the MPL was only 50 microns. It is noted that the limiting current density in the work by Hussein et al. was in the order of 2.5 A/cm² for the operating conditions and channel geometry investigated, but the operating conditions were different from this work.

The geometrical parameters are given in

Table 1. Geometrical parameters [3]. All units are in mm.

Parameters	Symbol	Value
Channel width	${Geo}_{\mathrm{w}\mathrm{i}\mathrm{d}}$	0.3
Channel height	${Geo}_{hei}$	0.3
Channel length	${Geo}_{\mathrm{l}\mathrm{e}\mathrm{n}}$	50
Land	${Geo}_{\mathrm{l}\mathrm{a}\mathrm{n}}$	0.3
Metal GDL thickness	${\delta }_{\mathrm{G}\mathrm{D}\mathrm{L}}$	0.1
MPL thickness	${\delta }_{\mathrm{M}\mathrm{P}\mathrm{L}}$	0.075
CL thickness	${\delta }_{\mathrm{C}\mathrm{L}}$	0.03
Membrane thickness	${\delta }_{\mathrm{M}\mathrm{e}\mathrm{m}}$	0.01

, and the structure/size for both anode and cathode are the same. These values were taken from the former one-dimensional modeling work using the Engineering Equation Solver V11.653 (2023).

2.2. Computational Mesh

Figure 3 shows the computational mesh. The meshes are all coupled through appropriate boundary exchange terms and an iterative solution procedure. It was ensured that the number of layers in each domain is greater than five as in previous work, with a refined mesh at the domain interface, i.e., the interface between the channel and the GDL.

Two increasing grid numbers of 54,100 and 108,200 were used to check mesh independence. The results showed that the values of the pressure at the inlet corresponding to these grid numbers are 15,943 Pa and 15,972 Pa, respectively. The difference is less than 0.19%, which is a very small error, but the simulation time increased from 14 h to 25 h. In order to save the calculation time of the numerical simulation, the number of meshes of 54,100 was selected to divide the geometric model. The cases presented here were calculated on a workstation with a 3.10 GHz processor and 20 cores. The number of iterations required to obtain converged solutions ranged from 70,000 to 100,000; the latter required around 24 h of CPU time for each operating condition.

2.3. Model Assumptions

The PEM fuel cell model is usually a complex system of equations including both convection and diffusion transport processes with many sets of conservation equations. A previous CFD analysis suggested that the fuel cell with metal GDL operates almost isothermally, and there is almost no temperature gradient between the membrane and the flow channels [18].

To create a numerically tractable 3D model, it is necessary to make some simplifying assumptions:

• The fuel cell operates under steady-state conditions.
• The gases are ideal gases.
• The current density and therefore the mass sink and source terms are uniform.
• The flow in the channels is considered laminar.
• Neglecting the influence of pressure on the porous media, and therefore the permeability is isotropic.

The fact that the current density is assumed to be uniform made these simulations more challenging than a realistic case might be. The inlet air enters at an elevated pressure, and therefore the oxygen concentration is higher. In practice, this will lead to an uneven current distribution with higher values at the inlet and lower values at the outlet. However, in this work, we wanted to compare the three-dimensional results with the one-dimensional model based on the Engineering Equation Solver [3]. It should be kept in mind that the realistic case is likely to yield better results in terms of the oxygen concentration at the channel outlet, where it is known that the under-the-land region can suffer from oxygen depletion.

2.4. Model Equations

The general set of equations solved by Ansys CFX are the unsteady Navier–Stokes equations in their conservation form. The multi-fluid model [22] with the Euler–Euler method solves a complete set of conservation equations for each phase (continuity equation, momentum equation, and species equation), which is the main difference between the homogeneous model and the more complex Eulerian model. The generic advection–diffusion equation is:

$$\nabla ·\epsilon s_{\mathsf{\alpha }}\left({\mathsf{\rho }}_{\mathsf{\alpha }}{U}_{\mathsf{\alpha }}{\varphi }_{\mathsf{\alpha }}-{\Gamma }_{\mathsf{\alpha }}\nabla {\varphi }_{\mathsf{\alpha }}\right)=\epsilon s_{\mathsf{\alpha }}{S}_{\mathsf{\alpha }}$$

(1)

where $\epsilon$ is the porosity of porous medium (in the channel $\epsilon =1$), α stands for different phases, $s$ is the saturation, $\rho$ is the density, $U$ is the velocity vector $U=(u, v, w)$, $\Gamma$ is the product of the diffusion coefficient and the mass density, and $S$ is the source term, which may be mass sources or momentum sources depending on the physical process. $\varphi$ is the transported quantity, for example, a velocity component.

The steady-state equations of mass and momentum can be written as follows in a stationary frame.

Continuity equation:

$$\nabla ·\left(\epsilon s_{\mathsf{\alpha }}{\rho }_{\mathsf{\alpha }}{U}_{\mathsf{\alpha }}\right)=\epsilon s_{\mathsf{\alpha }}{S}_C$$

(2)

Momentum equation:

$$\nabla ·\left({\rho }_{\alpha }{U}_{\mathsf{\alpha }}⨂U_{\mathsf{\alpha }}\right)=-\nabla p+\nabla ·\tau +S_M$$

(3)

where $\tau$ is the stress tensor, related to the strain rate by:

$$\tau =\mu (\nabla U_{\mathsf{\alpha }}+{\left(\nabla U_{\mathsf{\alpha }}\right)}^T-{\displaystyle \frac{2}{3}}\delta \nabla {·U}_{\mathsf{\alpha }})$$

(4)

Species equation:

$$\nabla ·\epsilon s_{\mathsf{\alpha }}{\mathsf{\rho }}_{\mathsf{\alpha }}\left(U_{\mathsf{\alpha }}{Y}_{\mathsf{\alpha }}-{\mathrm{D}}_{\mathsf{\alpha }}\nabla Y_{\mathsf{\alpha }}\right)=\epsilon s_{\mathsf{\alpha }}{S}_S$$

(5)

$S_M$ is the body force that may arise by the porous (Darcian) resistance, $p$ is the pressure, $\mathsf{\mu }$ is the viscosity, and $S$ is the source term. Finally, $D$ and $Y$ are diffusivity and mass fraction, respectively. The parameters used for the simulations are shown in

Table 2. Operating conditions.

Property	Symbol	Value
Working temperature	$T$	80 °C
Anode/cathode working pressure	$P$	1 atm
Anode inlet relative humidity	${RH}_A$	1.0
Cathode inlet relative humidity	${RH}_C$	1.0, 0.9, 0.8, 0.7
Anode stoichiometric flow ratio	${\xi }_A$	1.01
Cathode stoichiometric flow ratio	${\xi }_C$	1.451, 1.857, 2.576, 3.755

.

Moreover, we have the algebraic constraint that the volume/mass fractions sum to unity:

$$\sum _{\alpha =1}^N{r}_a=1$$

(6)

$$\sum _{i=1}^N{Y}_i=1$$

(7)

where $r$ is the volume fraction and $i$ stands for different species in one phase.

For the gas phase, the density is calculated via the ideal gas law:

$${\mathsf{\rho }}_{\mathrm{g}}={\displaystyle \frac{P_{\mathrm{g}}M}{RT}}$$

(8)

where $R$ is the ideal gas constant, $8.314 \mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{K}$, and $M$ is the molecular weight ($M_{O_2}=32 \mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l},M_{H_2}=2 \mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l}$, $M_{H_{2}O}=18 \mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l},{\mathrm{a}\mathrm{n}\mathrm{d} M}_{N_2}=28 \mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l}$).

For a multifluid flow, one more equation is needed to close the system, and this is usually given by an algebraic constraint on the pressure, that is, all phases share the same pressure field. Note that the different pressures for each phase are determined by the momentum source term in the momentum equations for the porous medium.

2.5. Boundary Conditions

At the anode inlet, dry hydrogen is entering, that is $Y_{H_2}=1,Y_{H_{2}O}=0$, without liquid water.

The mass flow rate is:

$${\dot{m}}_{inlet,a}={\xi }_a\times I\times {\displaystyle \frac{M_{H_2}}{2F Y_{H_2}}}$$

(9)

where $F$ is the Faraday constant (96,487 C/mol) and $I$ is the current (A). $\xi$ is the stoichiometry, which is the ratio of gas provided to gas consumed; $a$ stands for anode. In the cases investigated, the stoichiometric flow ratio at the anode side is 1.01 [21].

At the cathode inlet we specify the relative humidity (RH), i.e., 1.0, 0.9, 0.8, or 0.7. The molar fraction of water vapor can be calculated out of the given RH:

$$\mathrm{R}\mathrm{H}={\displaystyle \frac{P_{H_{2}O}}{P_{sat}\left(T\right)}}$$

(10)

$$P_{H_{2O}}=P_{in}{x}_{H_{2O}}$$

(11)

where $P_{H_{2}O}$ is the partial pressure of water vapor and $P_{sat}$ is the saturation pressure that depends on temperature alone. A first guess for the absolute pressure at the inlet was taken from the one-dimensional EES model.

The saturation pressure of water vapor from the Antoine equation [23] has been approximated by:

$$P_{sat}=133.233\times 10^(8.07131-{\displaystyle \frac{1730.63}{T-39.724}})$$

(12)

where $T$ is in K. In this work, all cases were calculated for a cell temperature of 80 °C.

The cathode inlet includes oxygen, water vapor, and nitrogen. The mass flow rate is:

$${\dot{m}}_{inlet,c}={\xi }_c\times I\times {\displaystyle \frac{M_{O_2}}{4F Y_{O_2}}}$$

(13)

The mass fraction $Y_i$ can be calculated out of the corresponding molar fraction:

$$Y_i={\displaystyle \frac{x_i{M}_i}{\sum x_i{M}_i}}$$

(14)

The molar fractions sum to unity:

$$\sum _{i=1}^N{x}_i=1$$

(15)

According to the electrochemical reaction, the molar flow rates of oxygen, nitrogen, and water vapor at the inlet are [24]:

$${\dot{N}}_{O_2}={\xi }_c\times {\displaystyle \frac{I}{4F }}$$

(16)

$${\dot{N}}_{N_2}={\xi }_c\times {\displaystyle \frac{I}{4F }}\times {\displaystyle \frac{79}{21}}$$

(17)

$${\dot{N}}_{H_{2}O}={\xi }_c\times {\displaystyle \frac{I}{4F }}\times (1+{\displaystyle \frac{79}{21}})\times {\displaystyle \frac{x_{H_{2}O}}{1-x_{H_{2}O}}}$$

(18)

The outlet boundary condition for both sides is static pressure, and it depends on the reference pressure in this model. Here, the static pressure is atmosphere (1 atm).

In addition to the aforementioned inlet, outlet, and symmetry boundary conditions, all others are wall boundary conditions, which are solid and impermeable boundaries to fluid flow.

2.6. Sink and Source Terms

At the anode side, hydrogen is consumed in the CL, and the hydrogen sink term is defined as follows:

$${Sink}_{H_2}=-I{\displaystyle \frac{M_{H_2}}{2F }}$$

(19)

Oxygen is consumed at the cathode CL, and its sink term is defined as follows:

$${Sink}_{O_2}=-I{\displaystyle \frac{M_{O_2}}{4F }}$$

(20)

Meanwhile, liquid water is produced here, and the source term is:

$$Sourc{e}_{H_{2}O}=I{\displaystyle \frac{M_{H_{2}O}}{2F }}$$

(21)

Here, the current density I is the volumetric average of the prescribed current density for each case.

In order to determine the effective drag coefficient $r_d$, it is assumed that the hydrogen is saturated with water vapor at the outlet. This yields low, negative values for $r_d$, and this was found to be a good approximation [24].

Darcy’s Law is commonly used for flows in the porous medium (CL and MPL here). The permeability, K (m²), of the porous medium used is $18\times {10}^{-12} {\mathrm{m}}^2$ with the material of SGL 10BA plain paper [25].

For the gas phase:

$$Sourc{e}_{gas}=-{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{g}}}{K_{rel,g} }}{U}_{\mathrm{g}}$$

(22)

where $K_{rel,g}$ is the gas relative permeability and $Sourc{e}_{gas}$ stands for the momentum source of the gas phase.

The relative permeability depends on the liquid phase saturation, according to:

$$K_{rel,g}=K(1-s{)}^q$$

(23)

where $s$ is the liquid water saturation (volume fraction) and $q$ is an exponent, and it is taken as 3.0 here [26].

For the liquid phase:

$$Sourc{e}_{liquid}=-{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{l}}}{K_{rel,l} }}{U}_{\mathrm{l}}-\nabla P_{\mathit{cap}}$$

(24)

where $K_{rel,l}$ is the liquid relative permeability; $Sourc{e}_{liquid}$ stands for the momentum source of the liquid phase; and $\nabla P_{\mathit{cap}}$ is the gradient of capillary pressure.

$$K_{rel,l}=K{s}^q$$

(25)

For the metal GDL, the Hagen–Poiseuille equation is used for the through-plane resistance term.

$$Sourc{e}_{GDL}=-{\displaystyle \frac{1}{n_{pores}^{″}}}{\displaystyle \frac{8}{\pi }}{\left(1/r_c\right)}^{2}U$$

(26)

where $n_{pores}^{″}$ is the number of pores in the cell area in the Y-direction, $1.96\times {10}^7 {\mathrm{m}}^{-2}$, and $r_c$ is the hole radius, 0.06 mm. In the in-plane directions of the metal GDL, the resistance was set to a high value to prevent any in-plane transport.

The porosity for GDL is calculated as follows:

$${\epsilon }_{GDL}=D_{Hole}^2{\displaystyle \frac{\pi }{4{\psi }^2}}$$

(27)

where $D_{Hole}$ is the diameter of the holes, and pitch $\psi$ is the distance between the center of two holes, which was 0.17 mm.

For most multiphase simulations, the capillary pressure depends strongly on the liquid water volume fraction. To describe capillary pressure in a CFD model, the Leverett equation [27] is frequently used here, similar to the work by Wang et al. [26,28,29]:

$$P_{\mathit{cap}}=-\sigma \mathit{cos}\theta {\left({\displaystyle \frac{\epsilon }{K}}\right)}^{0.5}J\left(S_{eff}\right)$$

(28)

where $\sigma$ is the surface tension (0.0625 Pa/m), $\theta$ is the contact angle (108°), $\epsilon$ is the porosity, $S_{eff}$ is the effective saturation, and $J\left(S_{eff}\right)$ is the Leverett function described by:

$$J\left(S_{eff}\right)=\left\{\begin{array}{c}1.417\left(1-S_{eff}\right)-{2.12\left(1-S_{eff}\right)}^2+{1.263\left(1-S_{eff}\right)}^3, \theta <90°\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} 1.417S_{eff}-{2.12S_{eff}}^2+{1.263S_{eff})}^3, \theta \ge 90°\end{array}\right.$$

(29)

The effective saturation ($S_{\mathit{eff}}$) is a function of the irreducible saturation ($S_{\mathit{irr}}$) and liquid water saturation in the porous medium [26]:

$$S_{\mathit{eff}}={\displaystyle \frac{S-S_{\mathit{irr}}}{1-S_{\mathit{irr}}}}$$

(30)

Water vapor condenses on either a pre-existing liquid spot or a hydrophilic surface. On the contrary, liquid water evaporates only from a pre-existing liquid surface area.

In order to model the phase change rate, which is similar to mass sources, the specific liquid–gas interfacial area of mass transfer $A_{lg}$ needs to be determined:

$$A_{lg}={\Gamma }_s{A}_{\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{e}}$$

(31)

where ${\Gamma }_s$ is the surface accommodation coefficient, 0.01. Inside the porous medium, it is reasonable to assume that the specific interfacial area depends on the pore surface area $A_{\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{e}}$, $1\times {10}^5 {\mathrm{m}}^{-1}$.

In this model, the equations from Nam and Kaviany are used [30], which describe the phase change rate:

$${\dot{m}}_{\alpha \beta }=f\left(x\right)=\left\{\begin{array}{c}{k}_{xm}{A}_{lg}{M}_{H_{2}O}{\displaystyle \frac{p_{H_{2}O}-p_{sat}}{RT}}, p_{H_{2}O}>p_{sat}\\ k_{xm}{A}_{lg}{M}_{H_{2}O}{\displaystyle \frac{p_{sat}-p_{H_{2}O}}{RT}}, p_{sat}>p_{H_{2}O}\end{array}\right.$$

(32)

where $k_{xm}$ is the convective mass transfer coefficient $(\mathrm{m} {\mathrm{s}}^{-1})$; $\alpha \mathrm{a}\mathrm{n}\mathrm{d} \beta$ represent different phases.

$$k_{xm}={\Gamma }_u{\overline{u}}_m$$

(33)

${\Gamma }_u$ is the overall uptake coefficient, 0.006, and ${\overline{u}}_m$ is the is the mean molecular speed, according to:

$${\overline{u}}_m=\sqrt{{\displaystyle \frac{8RT}{\pi M_{H_{2}O}}}}$$

(34)

For the species transport equation in the gas phase, the diffusivity of the species needs to be corrected by temperature, pressure, porosity, and tortuosity. The experimentally obtained binary diffusivities ($D_{a-b}^{bin}(T_0,p_0)$) are taken from Cussler [31].

$$D_{a-b}^{bin}(T, p)=D_{a-b}^{bin}(T_0,p_0)\left({\displaystyle \frac{P_0}{P}}\right){\left({\displaystyle \frac{T}{T_0}}\right)}^{{\displaystyle \frac{2}{3}}}$$

(35)

For the porous medium in a PEM fuel cell, the expression is modified into the effective diffusivity ($D_{a-b}^{eff}$), according to:

$$D_{a-b}^{eff}={{\displaystyle \frac{\epsilon }{\tau }}D}_{a-b}^{bin}(T, p)$$

(36)

where $\tau$ is tortuosity, ${\tau }_{GDL}$ is 1, and ${\tau }_{CL}$ is 3. An overview of all material properties is given in

Table 3. Material properties.

Property	Symbol	Value
Permeability in CL/MPL	K	$18\times {10}^{-12}{\mathrm{m}}^2$
Exponent in relative permeability	q	3.0
Number of pores per unit area in the metal GDL	$n_{\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{s}}^{″}$	$1.96\times {10}^7 {\mathrm{m}}^{-2}$
Hole radius	$r_c$	0.06 mm
Pitch	$\psi$	0.17 mm
Surface tension	$\sigma$	0.0625 Pa/m
Contact angle	$\theta$	108°
Channel porosity	${\epsilon }_{CH}$	1
CL porosity	${\epsilon }_{CL}$	0.75
MPL porosity	${\epsilon }_{MPL}$	0.75
CL/MPL tortuosity	${\tau }_{CL}, {\tau }_{MPL}$	3
GDL tortuosity	${\tau }_{GDL}$	1
Irreducible saturation	$S_{\mathrm{irr}}$	0

.

3. Results and Discussion

The goal of this study is to elucidate the feasibility of operating the PEM fuel cell at a high current density. The particularly important issue is the question of whether the membrane would be sufficiently hydrated and whether the oxygen would diffuse to the reaction interface under the given operating conditions from the previous EES model, especially when the stoichiometric flow ratio is quite low.

In this part, it was decided to reduce the operating inlet RH from 1.0 to 0.7 with intervals of 0.1. The outlet relative humidity is set to 1.0 for all cases. In the prior work by Bielefeld et al. [3], it was found that when both inlet and outlet relative humidity were fixed at an RH of 1, the gas phase became supersaturated in the middle of the channel. This would lead to undesired condensation and possible channel blocking. However, the model used in that work was a simplified one-dimensional model assuming a perfect mixture of the gas in the channel. It is one goal of the current work to shed additional light into such cases and to verify whether a CFD model will predict similar behavior.

Bielefeld et al. also investigated the behavior with a reduced inlet RH while keeping the outlet RH at one. This was performed to avoid channel condensation. It is understood that an outlet RH of one means that all the product water leaves the fuel cell in the vapor phase, which reduces the load on the coolant. Reducing the inlet RH has several advantages. Firstly, the overall mass flow rate and thus the compressor work are reduced, and it is known that the compressor is the biggest parasitic power loss in a fuel cell system [1,2]. Using the model by Bielefeld et al., it was also found that the stoichiometric flow ratio is reduced when reducing the inlet RH, and thus the overall mass flow rate and compressor work are reduced by two separate effects.

Finally, it is shown in

Table 4. Matrix of the four cases. The operating condition is 80 °C.

Inlet RH	Limiting Current Density iL (A/cm2)	Working Current Density (A/cm2)	Stoichiometry at iL	Working Stoichiometry	Pressure Drop (Pa)
1.0	6.10	4.90	3.462	3.755	10,450
0.9	5.88	4.70	2.528	2.576	6100
0.8	5.58	4.50	1.833	1.857	4100
0.7	5.26	4.20	1.437	1.451	2900

below that the pressure drop is also reduced in the case of lower inlet RH, again reducing the compressor work. In general, operating a PEM fuel cell at a reduced inlet RH can lead to membrane dehydration. This effect could not be investigated with the simplified model by Bielefeld et al., and therefore a focus in the current CFD analysis is put on calculating the resulting RH distribution inside the cathode CL and thus drawing insights into the expected membrane hydration level.

The working temperature in these cases is 80 °C, as the high temperatures can result in lower stoichiometry [3]. Because it is impossible to operate a fuel cell at the limiting current density, we are using a value of 80% limiting current density. Only the results of the cathode side are investigated because the boundary conditions for the anode side are the same with previous work [17] and the results are similar.

The matrix of cases investigated is shown in Table 4. All these data points have been identified with the help of the EES model. By comparison to the work by Bielefeld et al. [3], these values for the limiting current densities are in fact fairly low. The reason for this is that the operating temperature chosen was quite high, 80 °C, and the thickness of the metal GDL was 100 microns in this work, while the values found by Bielefeld et al. assumed a thinner GDL, similar to the one made by Hussein et al. [17].

Finally, Bielefeld et al. assumed a very fine land and channel area of 150 microns, as that resulted in the highest calculated limiting current densities, demonstrating that finer channels and land areas result in an increased limiting current density. However, it is currently questionable whether the stamping process employed to make the metallic bipolar plates allows for such a fine channel and pitch. Therefore, in the current work, larger and more realistic values are taken. It is noted that limiting current density of 5–6 A/cm² is still above the target of 4.4 A/cm² for the year 2040, as was defined in reference [8].

It should be kept in mind that, in general, the stoichiometric flow ratios should be kept as low as possible in order to minimize the compressor work. In addition, it is generally desired to operate the fuel cell at a low inlet RH to minimize the size of the humidifier. Also, compressed air requires fewer moles of water than uncompressed air to reach the same RH. While some of the values for the stoichiometric flow ratio shown here appear to be small considering the high current densities that the values have been calculated in the EES model with the well-known equation [1,2]:

$$i_L={\displaystyle \frac{4F{C}_B}{R_{tot}}}$$

(37)

$$R_{tot}=R_{GDL}+R_{MPL}$$

(38)

$$R_{GDL}={{\delta }_{GDL}/D}_{GDL}^{eff}$$

(39)

where $i_L$ is the limiting current density (A/cm²), $C_B$ is the bulk concentration of oxygen (mol/m³) in the channels, $R_{tot}$ is the total diffusion resistance (s/m) included resistance in GDL and CL, and $\delta$ is the diffusion length (m). The thus calculated values for the limiting current densities in the work by Bielefeld et al. [3] could even exceed 10 A/cm².

On the other hand, prior experimental work has suggested that it is the oxygen transport inside and through the electrolyte phase in the CL that is limiting the current density [32,33]. However, in contrast to that work, the oxygen mass transport inside the ionomer of the CL itself is neglected in the current work, which is in accordance with textbook definitions of the limiting current density [1]. Therefore, all the usual mass transport resistances are accounted for.

For the convenience of displaying the results below, figures have been reduced by ten times in the Z-direction (channel length). Moreover, Figure 4 is for demonstration purposes, indicating that this is a complete fuel cell result with both the anode and the cathode sides. As mentioned before, we have studied the anode side in a publication [21], so the figures in this work only focus on the cathode side for ease of display.

Figure 5 shows the relative pressure at YZ plane, X = 0.3 mm, and the outlet pressure is 1 atm as a boundary condition. As mentioned above, the general finding is that decreasing the inlet RH yields a lower pressure drop from the inlet to the outlet. This is mainly because decreasing inlet RH will cause a smaller inlet mass flow (see Equation (11)). On the other hand, the pressure exhibits an almost one-dimensional distribution in the XY plane; that is, the value changes slightly from the channel to the porous medium.

Figure 6 shows the volume fraction of liquid water. For traditional PEM fuel cells with CFPs, liquid water was predicted in all porous media [23]. However, in this study with a metal GDL, the liquid water is predicted to stay out of holes. This is in very good agreement with prior, more detailed simulations where the holes themselves were discretized [18]. With the decrease of inlet RH, the liquid water decreases at the beginning of the inlet. Moreover, the maximum volume fraction also decreases slightly from 0.1093 to 0.1011. The volume fraction of liquid water corresponds closely to the irreducible saturation that was specified, and it can be seen that there is almost no gradient in the liquid water volume fraction, which means that capillary transport is low. As expected, the amount of liquid water at the inlet region is reduced as the inlet RH is decreased, but it appears that the liquid phase occupies wide regions of the MPL and CL.

Figure 7 shows the oxygen molar concentration distribution, which decreases along the channel because of oxygen consumption by the electrochemical reaction. Moreover, it also decreases in the Y-direction from the channel to the CL due to oxygen sink term here. With the decrease of inlet RH, oxygen molar concentration at the inlet increases synchronously; that is, the component of inlet gases will be affected by the inlet RH. Note that, when the inlet RH is decreasing, there is a risk of oxygen molar concentration reducing to zero (near the outlet), which is highly dependent on the diffusivity of oxygen. When considering these figures, one must not disregard the decrease in the stoichiometric flow ratio from 3.755 in the first case to just 1.451 in the last case.

The RH in through-plane is shown in Figure 8. Basically, the RH increases along the channel from the inlet to the outlet due to the produced water, and the RH is lower than 100% at the outlet slightly because there is a small part of liquid water leaving the cell. This is different from the model by Bielefeld et al. [3], who assumed a perfectly mixed gas in their 1-D model. In this more refined CFD study, there is a boundary layer of water vapor predicted that leads to liquid water leaving the cell, similar to the work by Berning [16]. Although the decrease in pressure can lead to a decrease in RH, a significant increase in water molar fraction plays a dominant role here, which can be derived from the expression of RH (Equations (9) and (10)).

Obviously, due to the generation of water in the CL and its diffusion into the channel, the value of RH increases from the CL to the channel. Moreover, when the inlet RH is 100%, the distribution of RH is almost uniform (close to 100%), which matches the results of EES very well.

Figure 9 shows the RH distribution at the interface of the CL and the membrane (XZ plane, Y = 0.205 mm). Similar to the through-plane results, the RH increases along the channel from the inlet to the outlet. All cases show that the most area of the membrane is hydrated, that is, it is kept wet. However, as the inlet RH decreases, the RH also decreases, especially at the beginning of the inlet (although the RH is still very high, RH = 86.84%).

Figure 10 shows a typical velocity distribution in fuel cells. The velocity in the porous medium is zero, and in the channel is laminar flow with an obvious boundary layer, consistent with the model’s previous assumption. When the inlet RH is decreasing, the average velocity at the outlet decreases. Note that the CFD results are smaller than the calculated EES results. For example, when the inlet RH is 100%, the average velocity at the outlet is 36.28 m/s in Ansys-CFX and 37.73 m/s in EES, respectively. Because in the EES model all fluids are assumed to be single-phase instead of multi-phase.

Finally,

Table 5. Matrix of key results.

Inlet RH	Average Oxygen Concentration in the CL (mol/m3)	Average RH in the CL	Mole of Water Vapor Inlet per Mole of Product Water
1.0	1.310	100.00	6.543
0.9	1.157	99.95	4.017
0.8	0.944	99.91	2.470
0.7	0.909	98.74	1.605

summarizes the average oxygen concentration in the CL, the average RH in the CL, and the mole of water vapor inlet per mole of product water. The oxygen concentration in the CL has the same trend with decreasing inlet RH, with the minimum value being 0.909 mol/m³. The average RH values in the CL as listed in Table 5 are generally very high, exceeding 98% in all cases, even when the inlet RH is only 70%. This suggests that the membrane would be sufficiently hydrated under all conditions investigated here, and it leaves room for a further reduction in the inlet RH, thus affording a smaller humidifier.

The last column in Table 5 lists the calculated moles of water entering the cell to reach the desired inlet RH, compared to the amount of product water. This information is important for the dimensioning of the humidifier. The values listed here are taken directly from the CFD results, and it is striking that these are probably too high for successful fuel cell operation. Historically, this work is a follow-up of the experiments conducted by Hussein et al. [17] under laboratory conditions, and they used such high inlet RH values in their work. Clearly, future work should focus on identifying operating conditions to reduce both the inlet RH and the stoichiometry such that the water required to enter the cell is brought to realistic levels.

4. Conclusions

A modeling study on PEM fuel cells has been presented to elucidate appropriate operating conditions to run a fuel cell at a high current density in order to improve the power density. Based on the previous work, we focused on the feasibility of high current density and low stoichiometric flow ratio in this paper. The following conclusions can be drawn:

• In all cases investigated, using metal GDL, the limiting current density can vary in the order of 5–6 A/cm². The maximum limiting current density is 6.1 A/cm², which is a very high value, and the CFD results suggest that it can be run under appropriate conditions (i.e., geometry, stoichiometry, inlet gas component, temperature, and pressure).
• The minimum stoichiometry is 1.451 when the inlet RH is 0.7, which reduces the power of the air compressor. Given a current density of 5.26 A/cm² for this case, this result is encouraging in that it suggests that the combination of a high current density and a low stoichiometry might be feasible.
• In this work, the inlet conditions had to be found by trial-and-error because the inlet pressure was not known a priori. With knowledge of the inlet pressure, the mass fraction of water vapor was adjusted, and finally the amount of water leaving the fuel cell was such that the outlet RH was also close to 100% on average. However, different from the prior one-dimensional model, which assumed a perfect mixture in the channel, we have observed a gradient in the water distribution at the outlet owing to a boundary layer between the channel and the metal GDL. At the interface between the CL and membrane, the RH distribution suggests that the membrane is hydrated. However, as the inlet RH decreases, the RH at the membrane also decreases slightly at the beginning of the inlet.
• The pressure decreases along the channel from inlet to outlet, and the pressure drop is consistent with the previous 1D EES model. Additionally, the modeling results indicate that the liquid volume fractions for all four cases show that liquid water stays in the MPL and CL and does not penetrate the porous metal GDL.
• It appears problematic that the amount of water that needs to enter the fuel cell under any conditions investigated here exceeds the amount of product water. Future work will thus focus on identifying operating conditions to further reduce this value.

In the outlook and future work, we can add a temperature gradient from the inlet to the outlet. For example, if the cathode inlet is the cold end, the RH is higher for the same inlet water vapor condition. Moreover, we can use the co-flow channel instead of the counter-flow design and have the dry hydrogen also entering at the cold end.

This work assumes only one value for the permeability of porous media. In this regard, different values can be evaluated depending on the chosen materials. Additionally, we can also investigate a metal GDL with an adjusted pore-size distribution from cathode inlet to outlet. At the dry inlet, where the oxygen concentration is high, there can be fewer pores to keep the water at the membrane and also humidify the incoming dry hydrogen. Towards the outlet, the porosity can be increased because the membrane is wet here anyway, but there is a lower oxygen concentration. With the porous metal GDL, it should be possible to engineer such desired porosity distribution.