A Computational Fluid Dynamics Analysis of Multiphase Flow in the Anode Side of a Proton Exchange Membrane Electrolyzer

Abstract

This work describes an innovative three-dimensional model of a proton exchange membrane electrolyzer. For the first time, a multi-phase model has captured segregated channel flow together with multiphase flow in a porous medium, as well as heat transfer and phase change employing an Eulerian multiphase model. The novel electrolyzer design investigated employs a symmetrical, interdigitated flow field to facilitate even water distribution. In the current case, a hot spot is predicted with a temperature increase of 7 °C at a current density of 1.0 A/cm². The flow field plates are horizontally oriented, and it is shown that gravity plays an important role in the electrolyzer design and orientation. A parametric study shows, for the first time, the effect of operating a PEM electrolyzer at sub-ambient anode pressure to favorably adjust the concentration ratio between water vapor and oxygen in the anode compartment. This ratio is increased by a factor of 5.6 when the pressure is decreased from one bar to 500 mbar.

1. Introduction

In recent years, there has been growing interest in generating hydrogen via water electrolysis in order to store intermittent energy sources like solar and wind in chemical form. Among the different types of water electrolyzers, proton exchange membrane (PEM) water electrolyzers have the advantage of quick start-up times, very pure hydrogen generation, and an operating temperature of around 80 °C, which makes them suitable to couple to a district heating grid. Importantly, it is possible to generate hydrogen at a pressure of several hundred bars, which eliminates the need for a costly and service-intensive compressor. The main disadvantage of this technology is the need for precious metal catalysts such as iridium and platinum, which leads to high capital costs [1]. Therefore, reducing the catalyst loading and replacing the precious metals is nowadays one of the prime research objectives. The efficiency of PEM electrolyzers is in the range of above 70%, ideally above 80%, so that operating expenses are moderate [2]. A comparison of the different electrolyzer types was given by Schmidt et al. [1]. On balance, if PEM electrolyzers can operate at the storage pressure, and the amount of catalyst can be reduced, then this technology could become the preferred choice for small- and intermediate-scale operations.

The half-cell and full-cell reactions in a PEM electrolyzer are as follows:

$$\begin{array}{}\hfill \mathrm{(1)}& \hfill & \mathrm{Anode}:\hfill & \hfill {\mathrm{H}}_2\mathrm{O}& \Rightarrow 2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}+{\displaystyle \frac{1}{2}}{\mathrm{O}}_2\hfill \hfill \left(2\right)& \hfill & \mathrm{Cathode}:\hfill & \hfill 2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}& \Rightarrow {\mathrm{H}}_2\hfill \hfill \left(3\right)& \hfill & \mathrm{Combined}:\hfill & \hfill {\mathrm{H}}_2\mathrm{O}& \Rightarrow {\mathrm{H}}_2+{\displaystyle \frac{1}{2}}{\mathrm{O}}_2\hfill \end{array}$$

Water is fed to the anode side, where it is split into electrons, protons, and oxygen. The protons migrate through the membrane, most likely in the form of hydronium ions, and recombine at the cathode side to form hydrogen. A recent review of the status of PEM electrolyzer technology was given by Wang et al. [3]. A schematic of a PEM electrolyzer is shown in Figure 1 [4].

The heat and mass transfer processes inside the electrolyzer, especially inside the anode compartment where the expensive iridium-based catalyst is used, are complex. Clearly, a fundamental understanding of the multiphase flow phenomena can lead to an improved performance and a reduced catalyst loading.

In the past, computational fluid dynamics (CFD) studies of PEM electrolyzers were conducted in order to shed light on the distribution of water, oxygen, and the temperature in an operating electrolyzer. Grigoriev et al. were among the first to show the water distribution in their GenHy^® 1000 PEM electrolyzer stack [5], calculated with COMSOL Multiphysics (www.comsol.com). However, the details of their model were not disclosed.

In 2011, Nie and Chen applied Fluent v6.3 to study multiphase flow in the anode side of a PEM electrolyzer [6]. These authors employed a mixture model, meaning that one set of Navier–Stokes equations was solved for both phases, and applied an algebraic slip velocity that depended on the local Reynolds number. The water flow rate was fixed, while the oxygen flow rate was varied. The flow field geometry investigated was a square bipolar plate with straight channels.

In 2013, Ruiz et al. published a CFD analysis of different flow field patterns such as parallel, serpentine and multi-pass serpentine in a high temperature (HT) PEM electrolyzer [7]. They employed a non-isothermal, single-phase and single-domain model implemented in Ansys Fluent, and they concluded that the multi-pass flow field was superior. However, the findings of the HT PEM electrolyzer cannot be transferred to a low-temperature PEM electrolyzer, where multi-phase flow is expected.

Olesen et al. modeled the water and oxygen distribution in the anode flow field plate of the Primolyzer^® design by the Danish manufacturer IRD A/S [8]. They used the Eulerian approach, where a set of transport equations is solved for each phase. Their non-isothermal results indicated that flow maldistribution can lead to temperature discrepancies of up to 8 °C, which will limit the nominal operating temperature to around 80 °C due to material restrictions of the Nafion membrane. The modeling tool was the commercial software Ansys CFX Version 12.0 (Ansys Inc., Canonsburg, PA, USA). Later on, Lafmejani et al. employed the Volume of Fluid approach implemented in Ansys Fluent (version 16.0) to study the multiphase flow in an expanded cathode flow field [9]. In 2019, Olesen et al. published a detailed study of the multiphase flow in the round, interdigitated flow fields used in the Primolyzer^® design [10].

In 2018, Toghyani et al. published numerical studies of different flow fields [11,12]. They compared a parallel flow field with a single-pass and multi-pass serpentine flow field, and they concluded that the latter was superior. However, similar to the study by Ruiz et al. [7], this model was single-phase and ignored gravity, and therefore, there is uncertainty about these results.

In 2019, Zinser et al. published a study on mass transport in an anodic porous transport layer (A-PTL) in a PEM water electrolyzer, identifying regions of dry-out, where the electrolyzer cannot function [13]. Their model was self-developed and coded in MATLAB (www.mathworks.com). In addition, they experimentally determined material properties of a Ti sintered fiber PTL.

In 2020, Upadhyay et al. employed Ansys Fluent (version 19.2) to simulate the flow through a round anode flow field plate [14]. The model was transient and single-phase, and the flow was entering from one side of the bipolar plate and leaving on the other side, as, e.g., in the electrolyzer design by Millet et al. [15] or the Primolyzer^® design modeled by Olesen et al. [8]. Later on, Upadhyay et al. employed the Ansys Fluent (version 2020R2) module to study different channel configurations in a straight channel flow field [16]. The model was single-phase and ignored the effect of gravity, but the comparison with experimental performance data was very good.

Ma et al. employed the multiphase mixture model by Ansys Fluent (version 19.2) to study a three-pass serpentine flow field under different operating conditions [17]. The total cell area was 25 cm². Each flow channel was apparently discretized by a 5 × 5 mesh, and the total cell count was 500,000. While the cell appeared to be horizontally oriented, gravity terms were ignored. The gas phase was assumed to be in droplet form, so that the drag coefficients were implemented using the Schiller–Naumann equation.

In 2022, Lin et al. published a modeling and experimental study of three different flow fields: parallel, serpentine, and a point flow field [18]. They concluded that contact resistances, neglected in most modeling studies, are important to account for, and that the parallel flow field exhibited the best performance in the experiments. The model that was used was also single-phase.

Corda et al. [19] reproduced the Ansys Fluent model by Ma et al. [17] with STAR-CCM+ (version 2022.2, SIEMENS DISW). They also studied a three-pass serpentine flow field and obtained identical results. The flow channels were discretized by a 4 × 4 mesh, and the total cell count was 1.3 mio. Similar to Ma et al. [17], this group ignored the effect of gravity, and the mixture model was used to simulate multiphase flow.

In 2024, Su et al. published a CFD study of different rectangular flow field geometries [20]. Very recently, Mu et al. compared a parallel flow field, a serpentine flow field and an annular flow field based in Murray’s law, suggesting that the latter had superior performance [21].

Overall, the number of studies of PEM electrolyzers that entailed CFD are few compared to PEM fuel cells, and the fewest employed a multiphase solver, counting the commercial Ansys Fluent module as single-phase. Moreover, buoyancy effects in modeling have often been ignored, although they are particularly important in electrolyzer design, which is why alkaline electrolyzers are usually horizontally oriented cylinders with vertical flow field plates.

In this work, a multi-fluid approach is applied in order to model the liquid water, water vapor and oxygen distribution under different operating conditions. The investigated flow field was designed at Aalborg University, Denmark, and it is shown in Figure 2.

A detailed description of the electrolyzer design is given in reference [22]. In this design, the plates are oriented horizontally, which is different from most other PEM electrolyzers. The preferred orientation of the plates is currently the subject of investigation.

The anode side flow channels are radial, where the water enters from the center hole and the multiphase flow leaves at the outside. This figure suggests that the anode channels face upward and are placed below the porous medium, forcing the water through it. The cathode channels are spiral, and the hydrogen escape hole is shown on the right in Figure 2. The electrolyzer stack is placed inside a vessel so that the mixture of oxygen and water is diverted to the bottom, where a heat exchanger/recuperator is located. The phases are separated, and the water is recirculated while saturated oxygen leaves from a separate hole.

The goal of the current work is to gain a fundamental understanding of the multiphase flow at the anode side of this electrolyzer, especially in terms of the effect of gravity and the length of the anode inlet and outlet channels. In addition, an improved basic understanding of the thermal management of PEM electrolyzers is needed to attain higher current densities. It was shown by Olesen et al. that the amount of waste heat increases nonlinearly with current density, and, therefore, proper thermal management is imperative to attain high current densities [8]. In order to obtain an improved understanding, a multi-fluid model was developed in the software CFX-4.4, a formerly commercial CFD package that employs an IJK-structured grid and has strong multiphase flow modeling capacities.

2. Model Description

2.1. Assumptions

The following assumptions and simplifications were made in this model:

• The problem is steady-state.
• The flow is laminar.
• The gas phase is an ideal gas.
• The liquid phase contains only water.
• The porous medium is isotropic.
• In the channel, the flow is segregated and in the Stokes regime.
• The catalyst layer is a thin interface and its thickness is ignored.

The fact that the model is steady-state essentially rules out inherently transient flow phenomena such as slug flow, where larger gas bubbles are periodically driven out of the channel when the pressure gradient is sufficiently high. It will be shown that the operating conditions investigated lead to wavy-stratified flow and even slug flow, where the first may still be captured with a steady-state model, but the latter may not. The implications of this will be discussed below.

The last assumption listed was also made by Olesen et al. [8]. The channel length investigated here is in the order of 50 mm, while the CL thickness is in the order of 10 microns. It was found in prior work, e.g., by Marshall et al. and Millet et al., that the electrolyzer performance depends strongly on the type of catalyst used [23,24], but in the current study the operating current and voltage are taken as input parameters, and the output is the multiphase behavior and the thermal management under the conditions investigated. From this, insights can be gained as to how to improve the bipolar plate design and the cell performance.

2.2. Modeling Domain and Computational Grid

The model geometry and the computational grid are shown in Figure 3. The anode side flow field consists of straight channels where the water is fed from a central hole and is flowing radially outward. The flow field is interdigitated, which means that the water is forced through the porous transport layer. In the case presented here, the plates are actually turned around compared to Figure 2, meaning that the anode flow channels are placed below the porous medium and the water is forced to flow upward. Both plate orientations were studied, and the upward facing channels seem to provide better flow characteristics that will be discussed in detail below.

The mesh consists of a total of 20 Blocks, where Blocks 1–4 constitute the feed channel with the inlet, where a velocity condition for the liquid water is applied, and blocks 5–8 constitute the escape channel with the outlet. The channels are 0.5 mm deep and 1 mm wide, but only half the width is modeled due to symmetry. In the first hardware design, shown in Figure 2, the escape channels started close to the center of the plate [22]. The effect of the channel length is the subject of future investigation. Blocks 9–20 constitute the porous transport layer, which is around 200 microns thick. The total number of cells is 63,000, and they are all hexahedral. The cross-section of the flow channels is discretized by a 10 × 10 grid, as compared to a 5 × 5 grid used by Ma et al., and a 4 × 4 grid used by Corda et al. [17,19]. Because the cell section investigated is just 7.5 degrees, a Cartesian grid was used rather than a cylindrical grid. The cell layer in the bottom represents the catalyst layer, where the water vapor is consumed and oxygen is produced. This means that the CL is assumed to be a thin interface.

As mentioned, the active area of the investigated domain is a section of 7.5°, where the active area stretches in a channel length from 10 mm to 50 mm. With this, the active area investigated is:

$$A_{act}={\displaystyle \frac{7.5}{360}}\pi \left(5^2-1^2\right){\mathrm{cm}}^2=1.57{\mathrm{cm}}^2$$

(4)

2.3. Model Equations

The model employs the Euler–Euler approach, where two full sets of conservation equations are solved to model the transport in the gas phase and liquid phase.

The continuity equation is:

$$\nabla ·\left(\epsilon r_{\alpha }{\rho }_{\alpha }{\mathbf{U}}_{\alpha }\right)=\epsilon S_{c\alpha }+\left({\dot{m}}_{\alpha \beta }-{\dot{m}}_{\beta \alpha }\right)$$

(5)

where $\epsilon$ is the porosity of a porous medium, the subscript $\alpha$ represents different phases, and r stands for the volume fraction. In the region of the channels, the porosity is set to one. Furthermore, $\rho$ is the density, $\mathbf{U}$ is the three-dimensional velocity vector $(U,V,W)$, S denotes source terms, for example, due to the electrochemical reaction at the catalyst layer. The last bracket denotes mass exchange between phases due to phase change.

Likewise, the momentum equations are:

$$\begin{array}{cc}\hfill & \nabla ·\left(\epsilon r_{\alpha }{\rho }_{\alpha }{\mathbf{U}}_{\alpha }\otimes {\mathbf{U}}_{\alpha }-\nabla {\tau }_{\alpha }\right)=\epsilon r_{\alpha }{\rho }_{\alpha }\mathbf{g}+\hfill \\ \hfill & \epsilon r_{\alpha }\left({\mathbf{B}}_{\alpha }-\nabla P\right)+c_{\alpha \beta }^d\left({\mathbf{U}}_{\beta }-{\mathbf{U}}_{\alpha }\right)+\left({\dot{m}}_{\alpha \beta }{\mathbf{U}}_{\beta }-{\dot{m}}_{\beta \alpha }{\mathbf{U}}_{\alpha }\right)\hfill \end{array}$$

(6)

where P is the pressure, which is assumed to be the same for both phases, and ${\mathbf{B}}_{\alpha }$ is the so-called body force term. The third term on the right-hand side accounts for interfacial drag, and the last term for momentum exchange due to phase change of species. These terms will be explained in detail below. The first term on the right-hand side accounts for gravity, and in these simulations it was found to be significant.

The stress tensor ${\tau }_{\alpha }$ for a Newtonian fluid is:

$${\tau }_{\alpha }={\mu }_{\alpha }\left[\nabla {\mathbf{U}}_{\alpha }+{(\nabla {\mathbf{U}}_{\alpha })}^T\right]$$

(7)

where ${\mu }_{\alpha }$ is the dynamic viscosity of phase $\alpha$.

In addition to the Navier–Stokes equations that calculate the pressure distribution and the three-dimensional, two-phase velocity fields, one species transport equation is solved for. It is convenient to track the species that is undergoing phase change, in this case the water. Thus, a transport equation for water vapor is accounted for, while the background fluid in the gas phase is oxygen.

$$\nabla ·\left({\rho }_g{\mathbf{U}}_g{Y}_{H_{2}O}\right)-\nabla ·({\rho }_g{D}_{H_{2}O-O_2}\nabla Y_{H_{2}O})=S_i$$

(8)

where $Y_{H_{2}O}$ indicates the mass fraction of water vapor, and $D_{H_{2}O-O_2}$ is the binary diffusion coefficient of water vapor in air, specified below.

The system of equations is closed by the ideal gas law:

$$\rho ={\displaystyle \frac{pW}{RT}}$$

(9)

where W is the molecular weight of air (28.84 g/mol) and R is the universal gas constant. For the liquid phase, the density is taken to be 972 kg/m³.

Finally, the energy equation is solved for to calculate the temperature distribution, written as:

$$\nabla ·\left({\rho }_{\alpha }{\mathbf{U}}_{\alpha }{H}_{\alpha }\right)-\nabla ·({\lambda }_{\alpha }\nabla T_{\alpha })=S_{h_{\alpha }}$$

(10)

where H is the total enthalpy, given in terms of static (thermodynamic) enthalpy h by:

$$H_{\alpha }=h_{\alpha }+{\displaystyle \frac{1}{2}}{\mathbf{U}}_{\alpha }^2$$

(11)

2.4. Source Term Implementation

The above conservation equations include various source terms that are important to define in user subroutines provided by CFX-4.4. The detailed implementation will be defined next.

2.4.1. Calculation of the Local Current Density

At the lowest cell in the $z$-direction, where the interface between the porous GDL and the catalyst layer is located, oxygen is produced in the gas phase. This is accounted for by a source term that is applied to the gas phase continuity equation (Equation (5)), according to:

$$S_{cg}={\displaystyle \frac{i_{local}}{4F}}{M}_{O_2}$$

(12)

In this model, oxygen is considered the background species in the gas phase, and the transport equation of water vapor is solved. Likewise, there is a sink term for water vapor, which is accounted for in the species transport equation, the continuity equation and the volume fraction equation:

$$S_{cg}=-{\displaystyle \frac{i_{local}}{2F}}{M}_{H_{2}O}$$

(13)

In order to determine the local current density, the average concentration of water vapor in the lowest computational cell of the domain (low-z), where the GDL/CL interface is located, is used, according to:

$$i_{local}=i_{ave}{\displaystyle \frac{c_{H_{2}O,local}}{c_{H_{2}O,ave}}}$$

(14)

where $i_{ave}$ is the average, nominal, current density, here 1.0 A/cm².

2.4.2. Implementation of Porous Media Resistance

The body force is written as:

$${\mathbf{B}}_0+{\mathbf{B}}_P·\mathbf{U}=\nabla P$$

(15)

where ${\mathbf{B}}_0$ and ${\mathbf{B}}_P$ are the parameters to be specified, and they can be used to approximate the phenomenological Darcy equation [25,26]. Written in terms of the advection velocity U, Darcy’s law for flow in the $x$-direction reads:

$$U=-{\displaystyle \frac{1}{\epsilon }}{\displaystyle \frac{K}{\mu }}\Delta P_x$$

(16)

where K is the hydraulic permeability in ($m^2$). Thus, reformulation yields:

$${\mathbf{U}}_{\alpha }={\displaystyle \frac{1}{{\mathbf{B}}_{P,\alpha }}}·\nabla P-{\displaystyle \frac{{\mathbf{B}}_{0,\alpha }}{{\mathbf{B}}_{P,\alpha }}}$$

(17)

so that

$${\mathbf{B}}_{0,\alpha }=0\wedge {\mathbf{B}}_{P,\alpha }=-\epsilon {\displaystyle \frac{{\mu }_{\alpha }}{K_{\alpha }}}$$

(18)

which are the specified body force terms in the porous media. For the specification of the relative permeabilities, a cubical form is chosen:

$$\begin{array}{cc}\hfill K_l& =s^{3}K\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill K_g& ={(1-s)}^{3}K\hfill \end{array}$$

(20)

where s is the liquid saturation, which is equal to the liquid phase volume fraction $r_l$.

2.4.3. Implementation of Drag Terms

The second term on the right-and side in Equation (6) is used to specify the drag term in the flow channels. In order to determine the correct drag coefficients, the flow regime has to be identified first. To this end, the well-known modified Baker diagram has been consulted [27,28], shown in Figure 4. Here, the G values represent mass fluxes with $\lambda =1$ for air and $\psi =1$ for water.

In the current cases, the liquid flow rate is constant to represent a stoichiometry of $\xi$ = 350 at a current density of i = 1.0 A/cm². Thus, the liquid flux is:

$$\begin{array}{cc}\hfill G_l& =350\times {\displaystyle \frac{1.0\frac{\mathrm{A}}{{\mathrm{cm}}^2}\times 1.57{\mathrm{cm}}^2\times 0.018\frac{\mathrm{kg}}{\mathrm{mol}}}{2\times 96485\frac{\mathrm{As}}{\mathrm{mol}}\times 0.25\times {10}^{-6}{\mathrm{m}}^2}}=205{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^2\mathrm{s}}}\hfill \end{array}$$

(21)

The liquid flow rate has been kept constant in these simulations, and the amount of water that evaporates is small. For a current density of i = 1.0 A/cm², the oxygen flux at the outlet is:

$$G_g={\displaystyle \frac{1.0\times \frac{\mathrm{A}}{{\mathrm{cm}}^2}\times 1.57{\mathrm{cm}}^2\times 0.032\frac{\mathrm{kg}}{\mathrm{mol}}}{4\times 96485\frac{\mathrm{As}}{\mathrm{mol}}\times 0.25\times {10}^{-6}{\mathrm{m}}^2}}=0.52{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^2\mathrm{s}}}$$

(22)

This places all likely operating points into the stratified flow regime, but close to the plug/slug flow regime in the Baker diagram. A more detailed calculation using the complex chart by Taitel and Dukler places the flow regimes investigated in-between the wavy stratified and the slug flow regimes [29]. By comparison, the flow field plates investigated by Olesen et al. were of similar size as the ones investigated here, but because they were oriented vertically, the flow regime was slug/plug flow [8]. The quick analysis above only included the oxygen flow in the gas phase, and the addition of water vapor moves the flow regime even closer to the unsteady slug flow.

In fact, during the calculations it was observed that the water flux and the pressure in the outlet channel were slightly pulsating, i.e., the solution failed to reach a strict steady state, as the outlet flow continued to oscillate slightly with further iterations. But it was also observed that the key results investigated here, such as the oxygen and water vapor concentration, did not vary for the converged solution, i.e., the flow in the outlet channel was decoupled from the PTL. Such behavior can be expected when the pressure drop over the PTL is significantly higher than the pressure drop along the outlet channel. This observation gives confidence in the CFD results, as it predicts slug/plug flow when it should, while the key results can still be interpreted, because they are independent of the flow pattern in the outlet channel.

Having thus established that the flow is expected to be in the stratified flow or slug flow regime, the drag coefficients can be determined. These are the coefficients $c_{\alpha \beta }$ in the momentum equation as well as the interphase transfer coefficient in the energy equation and the species equation for water vapor. According to the CFX-4.4 subroutine USRIPT, for laminar, stratified flow the drag coefficient can be specified as:

$$c_{\alpha \beta }^d=0.5\times C_f\times {\rho }_m\times A_i\times u_{slip}$$

(23)

where

$$C_f=C/\mathrm{Re}=24/\mathrm{Re}$$

(24)

in the Stokes flow regime. Re is the Reynolds number, defined as:

$$\mathrm{Re}={\rho }_m{u}_{slip}H/{\mu }_m$$

(25)

where $u_{slip}$ is the slip velocity between both phases. The mixture density is calculated out of the phasic volumes:

$${\rho }_m=r_l{\rho }_l+r_g{\rho }_g$$

(26)

where $r_i$ is again the volume fraction of phase i in each computational volume. Combining the above yields for the drag coefficient yields:

$$c_{\alpha \beta }^d={\mu }_m{C}_d{A}_i$$

(27)

where, according to the Isbin equation, the mixture viscosity ${\mu }_m$ is calculated in a similar way as the mixture density. Furthermore,

$$C_d=0.5{\displaystyle \frac{C}{H_T}}$$

(28)

where $C=24$ and $H_T$ is the length scale, here taken to be 1 mm. Finally, the interfacial area $A_{lg}$ is determined using the following equation:

$$A_{lg}=r_l{r}_g{\displaystyle \frac{V_{CV}}{D_P}}$$

(29)

Here, $V_{CV}$ is the physical volume of each control element, and $D_P$ is the interface length scale, here also taken to be 1 mm. With this, the drag coefficient in the momentum equation is defined, and it will be seen in the results that the channel flow is segregated, as intended.

2.4.4. Implementation of Phase Change

The amount of mass undergoing phase change, ${\dot{m}}_{\alpha \beta }$, is either water vapor condensing from the gas phase into the liquid phase, or evaporation from the liquid phase into the gas phase. In the current model, the source term for oxygen production is in the gas phase conservation equation, while pure liquid water enters the electrolyzer. The created gas phase volume leads to evaporation of water, calculated as [25,30]:

$${\dot{m}}_{\alpha \beta }=k_{xm}{A}_{lg}{M}_{H_{2}O}{\displaystyle \frac{p_{H_{2}O}-p_{sat}\left(T\right)}{RT}}$$

(30)

where $k_{xm}$ is the convective mass transfer coefficient, and $A_{lg}$ is the interfacial area, both described below. $M_{H_{2}O}$ is the molecular weight of water, and $P_{H_{2}O}$ is the partial pressure of water vapor inside a given control volume. The saturation pressure $P_{sat}\left(T\right)$ corresponds to the temperature calculated in the respective control volume, according to Antoine’s equation:

$$P_{sat}\left(T\right)=\exp \left(A-{\displaystyle \frac{B}{T+C}}\right)$$

(31)

where A = 23.196, B = 3816.44, C = −46.13, and T is the temperature in [K]. The resulting pressure is then in [Pa].

Similar to Olesen et al., in case of condensation, the convective mass transfer coefficient in the porous region is calculated out of the mean molecular speed ${\overline{u}}_m$ [31,32]:

$$k_{xm}={\mathsf{\Gamma }}_u{\overline{u}}_m={\mathsf{\Gamma }}_u\sqrt{8RT/\pi M_w}$$

(32)

where ${\mathsf{\Gamma }}_u$ is a dimensionless uptake coefficient which was estimated to be 0.006 [32], ${\overline{u}}_m$ is the mean molecular gas speed, and $M_w$ is the mixture molecular weight. With this, the unit for $k_{xm}$ is [m/s].

In case of evaporation, the convective mass transfer coefficient $k_{xm}$ is calculated via [30]:

$$k_{\mathrm{xm}}=c_m{\displaystyle \frac{D_{H_{2}O-O_2}}{D_c}}{\mathrm{Sh}}_m=c_m{\displaystyle \frac{D_{H_{2}O-O_2}}{2r_c}}{\mathrm{Sh}}_m$$

(33)

Here, $c_m$ is the mixture concentration in [mol/m³]:

$$c_m=P_g/\left(R_{u}T\right)$$

(34)

where $R_u$ is the universal gas constant (8.314J/mol-K). For the binary diffusion coefficient $D_{H_{2}O-O_2}$, the relation for water vapor in air was used [33]:

$$D_{H_{2}O-O_2}=1.87\times {10}^{-10}\times {\displaystyle \frac{T^{2.072}}{P_{atm}}}$$

(35)

where $P_{atm}$ is the total pressure in [atm] and T is the temperature in [K]. Finally, the Sherwood number ${\mathrm{Sh}}_m$ is given by:

$${\mathrm{Sh}}_m=2+0.6{\mathrm{Re}}^{1/2}{\mathbf{Sc}}^{1/3}$$

(36)

where the Reynolds number is defined as:

$$\mathrm{Re}=u_{slip}{\rho }_g{D}_c/{\mu }_g$$

(37)

The slip velocity $u_{slip}$ is calculated out of the relative velocity components:

$$u_{i,rel}=(u_{i,g}-u_{i,l})$$

(38)

where i denotes the three different directions x, y, and z. The slip velocity is then:

$$u_{i,slip}=\sqrt{\sum _i{u}_{i,rel}^2}$$

(39)

The characteristic pore diameter is described below, and the Schmidt number Sc is defined as:

$$\mathrm{Sc}={\displaystyle \frac{{\nu }_g}{D_{H_{2}O-O_2}}}={\displaystyle \frac{{\mu }_g}{{\rho }_g{D}_{H_{2}O-O_2}}}$$

(40)

The specific interfacial area is calculated out of the liquid volume fraction, assuming the liquid body is split up into droplets that fill the pores, with an average radius corresponding to the characteristic radius $r_c$:

$$r_c={\left({\displaystyle \frac{K}{\epsilon }}\right)}^{0.5}$$

(41)

For a dry permeability of K = ${10}^{-11}$ m² and a porosity of $\epsilon =0.78$, this results in a characteristic pore diameter of around 10 microns. Grigoriev et al. determined such properties for various different PTL types, and the characteristic pore size was in this order of magnitude [34]. The pore size was then used to calculate the interfacial area $A_{lg}$. In the porous media, the volume fraction of the liquid phase in each control volume was divided into assumed droplets that occupied pores with the characteristic pore size [25]. The number of droplets in a given control volume is thus:

$$N_{drop}={\displaystyle \frac{r_l{V}_{CV}}{\frac{4}{3}\pi r_c^3}}$$

(42)

and the interfacial area is then calculated according to:

$$A_{lg}=N_{drop}{\displaystyle \frac{\pi r_c^2}{4}}={\displaystyle \frac{3}{16}}{\displaystyle \frac{r_l{V}_{CV}}{r_c}}$$

(43)

where care must be taken that $r_l$ is by naming convention the dimensionless volume fraction of the liquid phase, whereas $r_c$ is the characteristic radius [m].

It can, however, happen that the liquid volume fraction becomes so large that the interfacial area decreases again. Therefore, the same procedure was carried out for the gas phase occupying the pores, and the interfacial area was set to be the minimum of the two. Combining all of the above yields the mass transfer rate per computational cell ${\dot{m}}_{\alpha \beta }$ in [kg/s]. The results show that the gas phase was at an average relative humidity between 99% and 100% wherever liquid phase is present.

Finally, in order to determine the mass transfer in the flow channels, the Reynolds analogy is employed, yielding:

$$\mathrm{Sh}=h_{mass}{\displaystyle \frac{L}{D_{H_{2}O-O_2}}}={\displaystyle \frac{C_D}{2}}\mathrm{Re}={\displaystyle \frac{24}{2·\mathrm{Re}}}\mathrm{Re}=12$$

(44)

where L is a characteristic length scale. The thus calculated convective mass transfer coefficient:

$$h_{mass}={\displaystyle \frac{12}{L}}{D}_{H_{2}O-O_2}$$

(45)

has the units [m/s] and is therefore equivalent to $k_{xm}$.

2.4.5. Energy Equation

The source term in the energy equation, $S_H$, accounts for the waste heat generation due to the electrochemical reaction, as well as the cooling or heating due to the phase change of water. As the water enters the electrolyzer in the liquid phase and the oxygen is produced in the gas phase, there is water evaporating into the gas. This happens predominantly inside the porous media, and it leads to a cooling effect.

Inside the CL, the waste heat is generated, the magnitude of which depends on the local current density that was calculated out of the local water vapor concentration. To this end, an electrolyzer voltage of 1.85 V at the specified current density was assumed, taken from Olesen et al. [8]. It is well known that the performance depends strongly on the type and the loading of the catalyst, and therefore, it was not attempted to calculate the performance in the current version of the model.

The heating term at the lower PTL interface, where the CL is located, is [8]:

$${\dot{q}}^{″}=\left(E_{Cell}-{\displaystyle \frac{\Delta H}{2F}}\right)·i_{local}$$

(46)

Employing a cell voltage of $1.85$ V and using $\Delta H_r$ = 241,820 J/mole in case the water is consumed in the form of vapor [35], this yields a total heat source of 0.94 W for the active area of 1.57 cm² investigated here. The corresponding electrical efficiency is 1.482 V/1.85 V = 80%, and the total power in this section is 2.9 W.

2.5. Boundary Conditions

At the inlet, the velocity and temperature of the incoming water are specified. The velocity is calculated according to Equation (21) out of the stoichiometric flow ratio of $\xi =350$ and the current density of 1.0 A/cm², where the flux $G_l$ has to be divided by the water density. The inlet temperature is equal to the operating temperature of 80 °C. A zero-gradient condition was specified for the inlet pressure.

At the outlet, the absolute pressure of 1 atm was applied for the base case, and zero-gradient conditions were used for all other transport equations.

The inlet and outlet are located at the low-x and the high-x interfaces. At the low-y and the high-y interfaces (left and right in Figure 3), symmetry conditions are applied so that only half a channel width needs to be modeled. At all other outer boundaries, wall boundary conditions are applied which prevent any species or heat fluxes.

2.6. Material Properties

Main input parameters for the model were the material properties of the PTL, listed in

Table 1. Key material properties.

Property	Symbol	Unit	Value
Porosity	$\epsilon$	[-]	0.75
Tortuosity	$\tau$	[-]	5
Hydraulic Permeability	K	[m2]	2 × ${10}^{-11}$
Thermal Conductivity	$\lambda$	[W/m-K]	5.0

. These are standard values, although the tortuosity was chosen somewhat higher than the usual value of around $\tau =3$.

For the PTL, various materials can be used, from carbon fiber paper, which is said to lead to carbon corrosion, to sintered titanium powder, which has a higher thermal conductivity. The results discussed below are only mildly affected by the values listed above. With the permeability and the porosity given, the characteristic pore size can be calculated, according to Equation (41).

2.7. Solution Procedure

The calculations presented here were challenging, and a solution procedure had to be developed to by and by switch on the different terms.

(1)

Run the case with liquid water only, where water is running through the geometry at a prescribed combination of current density and stoichiometry. This is similar to experiments, where first water is run through the stack without drawing current, and then the current density is step-wise increased while keeping the flow rate constant.

(2)

Switch on the oxygen source term, assuming the current density is uniform. The thus obtained results already yield useful insights in the gas–liquid distribution.

(3)

Switch on phase change of water. The results then indicate to which degree the gas phase is saturated with vapor, and they can be used to determine the concentration ratio between oxygen and water vapor. However, only mass transfer is considered, not yet the energy terms.

(4)

Adjust the local current density to account for the water vapor distribution, and switch on the sink term for water vapor.

(5)

Switch on the sink and source terms in the energy equation to obtain the temperature distribution.

2.8. Numerical Procedure

It was found that the algebraic multigrid (AMG) solver, while taking more time per iteration, results in the most robust convergence behavior. Moreover, the AMG solver is known for producing a nearly grid-independent solution, especially when there are no items in the geometry that could only be accounted for by a finer mesh. For the results shown below, it was ensured that they were all grid-independent.

For the pressure velocity coupling, the PISO algorithm was employed. For all transport equations, a combination of under-relaxation factors and false time steps were used. Note that for all these numerical parameters, default values are automatically specified in CFX-4.4.

3. Results

The Results section is divided into two parts: a detailed presentation of the base case results, and a parametric study section, where the operating pressure of the electrolyzer was stepwise reduced from 1 bar to 0.5 bar. The plate orientation was such that the channels are facing upward and the water is pressed through the porous medium from below.

3.1. Base Case Results

The base case was calculated for ambient pressure at the anode outlet. The resulting relative pressure distribution at the low-z interface, where the electrochemical reaction occurs, is shown in Figure 5a. The relative pressure is equal to zero at the outlet, and the calculated relative pressure at the inlet is around 1418 Pa. Across the porous medium, there is a smooth drop-off in pressure, and the value is nearly constant underneath the channels. Such a smooth and intuitive distribution gives confidence in the overall results.

The calculated temperature distribution in the same layer is shown in Figure 5b. The nominal cell temperature is 80 °C, hence 353.15 K, and there is a predicted peak temperature increase of 7 °C, which can be found before the outlet channel section begins, and at the outer edge near the outlet channel. This means that both hot spots are predicted in regions, where either the outlet channel has not yet started, or the inlet section has ended, and the region between the two channels experiences only a mild temperature increase of 3 °C. This observation could be important for the placement of the catalyst, and it deserves further investigation. The outlet channel begins some distance from the inner radius, and this was done to avoid short-circuiting the water, but if this leads to a hot spot, the outlet channel should begin closer to the center. The reason for the existence of the hot spot is that there is a predicted gas bubble here with no liquid phase present. By contrast, the region between two channels is predicted in the multiphase regime, and the evaporative cooling prevents hot spots, as will be seen below.

The distribution of the gas phase volume fraction and the local evaporation rate is shown in Figure 6. By definition, positive values indicate evaporation and negative values condensation. As was mentioned above, there is a region with only gas phase at the inner part of the cell before the outlet channel starts. This coincides with the region of the inner hot spot. Moreover, all-dry regions are predicted underneath the channels and near the outer edge, where the other hot region is located. These are also the regions in the cell where the evaporation rate is zero, while evaporation is highest between the two channels and nearer to the inlet channel, where the temperature was predicted to stay constant. It appears desirable to operate an electrolyzer isothermally, and it is obvious that this local evaporation supports that. From these figures, it is concluded that the region between the channel is favored and should be as large as possible, i.e., the radius of the plates can be larger and both inlet and outlet channel should be as long as possible.

Next, the distribution of the molar oxygen and water vapor concentration is shown in Figure 7. Not surprisingly, there are local maxima in the dry regions of the cell, while minima occur in the wet region between the two channels. It is expected that the oxygen concentration is decreased when the operating pressure at the anode side is decreased [22], and that the ratio between the water vapor concentration and the oxygen molar concentration is accordingly increased. Detailed results of the water vapor-to-oxygen ratio will be presented in the parametric study section. However, it is not yet known whether or how this concentration ratio affects the electrochemistry, and therefore, these results are only of qualitative nature.

Figure 8a shows the calculated speed of the gas phase near the CL. The magnitude is in the order of a few cm/s, and a local speed maximum is predicted before the beginning of the outlet channel. In the region between the channels, the flow is slightly accelerating, which is a consequence of the gradual increase in gas volume fraction. The distribution of the gas volume fraction, shown in Figure 8b, is very interesting. The inlet channel is predicted to be almost exclusively in the liquid phase, as was intended in our design. With the current orientation, where the channels are facing up and the water is pushed into the porous medium from below, the inlet channel is almost entirely filled with water, while there is segregated flow in the outlet channel, as expected. This is caused by the implementation of the drag forces here, Equations (23)–(29), and it is encouraging to see that it leads to the desired results. The water stays in the bottom of the channel, but, as described, the flow is wavy and slightly pulsating. By analyzing these results, it becomes obvious that the flow might be steadied by inclining the outlet flow channel, i.e., making it deeper at the outside than at the beginning. This can be accomplished when the channels are facing upwards, but not downwards.

There is some gas phase predicted in the inlet channel, before the outlet channel starts, in the region where the gas bubble was predicted. The coexistence of segregated channel flow and multiphase flow in porous media together with heat transfer and phase change exemplified the numerical challenge of these cases, and they were difficult to converge. However, seeing the complexity of the model, it is quite robust, and once a first converged solution was obtained, it was easy to conduct a parametric study. A result such as the pressure distribution agrees with expectation and gives confidence, and interesting findings such as the appearance of the gas bubble before the outlet channel begins have to be verified, but they suggest that there is room for improvement of the bipolar plate design.

3.2. Parametric Study Results

In the base case, the anode side was operated at 1.013 bar, prescribed at the outlet. It was previously suggested that a reduction in anode side pressure with help of a vacuum pump is expected to favorably adjust the ratio between the water vapor concentration and the oxygen concentration [22]. Therefore, calculations were run with a different outlet pressure of 0.8 bar, 0.6 bar and 0.5 bar, respectively. At the operating temperature of 80 °C, the saturation pressure of water vapor is around 48 kPa (0.48 bar) and, if thermodynamic equilibrium prevails, the water vapor will be at this saturation pressure, corresponding to a relative humidity of 100%.

Generally, the distributions as shown and discussed for the base case remained very similar. An observation was that for a decreased operating pressure, there is a larger gas fraction in the entire compartment. This is not unexpected because a lower-pressure gas can accommodate a higher amount of water vapor, and therefore, the evaporation terms have increased. This higher gas fraction leads to an increase in the gas phase mass flow rate, and thus to an increased pressure drop. While the observed pressure drop was around 1418 Pa in the base case, this value increased to over 3300 Pa for a back pressure of 0.5 bar, i.e., it more than doubled, which is caused by the higher gas phase velocity.

The key outcome of the parametric study is the calculated ratio between the oxygen and the water vapor concentration in the gas phase, because this is likely to have an effect on the electrolyzer performance. According to the Butler–Volmer equation and general reaction kinetics, a concentration ratio between reactant and product leads to more favorable reaction kinetics.

Table 2. Water vapor and oxygen concentrations at the GDL/CL interface.

Back Pressure [kPa]	${\mathit{C}}_{{\mathit{H}}_2\mathit{O},\mathit{g}}$ [mol/m3]	${\mathit{C}}_{{\mathit{O}}_2}$ [mol/m3]	${\mathit{C}}_{{\mathit{H}}_2\mathit{O},\mathit{g}}$:${\mathit{C}}_{{\mathit{O}}_2}$ [-]
101.3	13.07	17.29	0.76
80.0	14.36	11.45	1.25
60.0	15.22	5.07	3.00
50.0	14.71	3.44	4.27

lists the average water vapor and oxygen concentrations at the PTL/CL interface, as well as their ratio, which increased by a factor of 5.6 between the lowest and the highest outlet pressure. These values are likely to change when different material parameters of the porous medium are chosen.

Finally, it is important to analyze the energy balance. In the electrolyzer, the only heat source is the waste heat produced. As mentioned above, it was assumed that the operating voltage is 1.85 V, which leads to a heat source of 0.94 W for the investigated cell area of 1.57 cm². In contrast to this heating term, there is the evaporative cooling. The amount of evaporation was calculated as part of the simulations, and a good first approximation is to assume that the oxygen gas leaving the cells is fully saturated, i.e., the relative humidity $\phi$ is equal to one:

$$\phi ={\displaystyle \frac{p_{H_{2}O}}{p_{sat}\left(T\right)}}=1\Rightarrow p_{H_{2}O}=p_{sat}\left(T\right)$$

(47)

According to Dalton’s law, it holds that:

$${\displaystyle \frac{p_{H_{2}O}}{p_{tot}}}={\displaystyle \frac{{\dot{n}}_{H_{2}O,out}}{{\dot{n}}_{O_2,out}+{\dot{n}}_{H_{2}O,out}}}$$

(48)

out of which follows:

$$({\dot{n}}_{O_2,out}+{\dot{n}}_{H_{2}O,out})\times p_{sat}\left(T\right)={\dot{n}}_{H_{2}O,out}\times p_{tot}$$

(49)

This can be reformulated into the desired expression for the molar flow rate of water vapor at the outlet:

$${\dot{n}}_{H_{2}O,out}={\dot{n}}_{O_2,out}{\displaystyle \frac{p_{sat}\left(T\right)}{p_{tot}-p_{sat}\left(T\right)}}={\displaystyle \frac{I}{4F}}{\displaystyle \frac{p_{sat}\left(T\right)}{p_{tot}-p_{sat}\left(T\right)}}$$

(50)

where I is again the total current (here 1.57 A) and F is Faraday’s constant. This yields a molar oxygen flow of $4.07\times {10}^{-6}$ mole/s at the outlet, and the molar water vapor flow rate varies accordingly. To approximate the evaporative cooling, this molar flow rate has to be multiplied with the molar heat of phase change at 80 °C, r = 41.58 kJ/mole. Using a saturation pressure of water of 47.415 kPa at that temperature, the amount of evaporative cooling can be calculated as shown in

Table 3. Molar flow rates and cooling power for thermodynamic equilibrium (gray cells) and CFD results (last two columns).

Back Pressure [kPa]	${\dot{\mathit{n}}}_{{\mathit{O}}_2,\mathbf{out}}$ [mole/s]	${\dot{\mathit{n}}}_{{\mathit{H}}_2\mathit{O},\mathbf{out}}$ [mole/s]	${\mathit{P}}_{\mathbf{evap}}$ [W]	${\dot{\mathit{n}}}_{{\mathit{H}}_2\mathit{O},\mathbf{evap}}$ [mole/s]	${\mathit{P}}_{\mathbf{evap}}$ [W]
101.3	$4.07\times {10}^{-6}$	$3.58\times {10}^{-6}$	−0.15	$1.18\times {10}^{-5}$	−0.49
80	$4.07\times {10}^{-6}$	$5.92\times {10}^{-6}$	−0.25	$1.51\times {10}^{-5}$	−0.63
60	$4.07\times {10}^{-6}$	$1.53\times {10}^{-5}$	−0.64	$2.40\times {10}^{-5}$	−0.99
50	$4.07\times {10}^{-6}$	$7.47\times {10}^{-5}$	−3.10	$2.83\times {10}^{-5}$	−1.18

, where the gray shaded cells correspond to these thermodynamic equilibrium calculations.

Note that these values correspond to an outlet temperature of 80 °C, and are therefore only an approximation. The calculated cooling rates in Table 3 have to be compared with the heating term of 0.94 W due to the waste heat production. However, it has to be borne in mind that these values were calculated assuming thermodynamic equilibrium at the outlet at a temperature of 80 °C. If the temperature increases, the amount of evaporation will be higher, as seen from the CFD results. Overall, the amount of phase change cooling at a high electrolyzer operation temperature is in the same order as the waste heat produced, and it can even exceed it so that the cell is cooled down. The last two columns in Table 3 summarize the CFD results. Because of temperature variations and mass transport limitations, these values deviate from the values calculated assuming thermodynamic equilibrium.

It is also noted from Equation (50) that the amount of evaporation water shows a high sensitivity to the operating temperature; the saturation pressure depends exponentially on the temperature according to Antoine’s equation, and an increase in temperature leads to both an increase in the numerator of Equation (50) and a decrease in the denominator. As the outlet pressure approaches the saturation pressure, this fraction becomes very large so that the amount of water vapor leaving the cell greatly exceeds the amount of oxygen, which only depends on the current density.

It can be concluded that reducing the outlet pressure enhances the evaporative cooling, and in the current case, at a back pressure of 60 kPa the calculated heat of evaporation almost exactly balances the waste heat. At the lowest pressure investigated, the cell is in fact cooled down, which makes this mode of operation unsustainable, because the waste heat from the anode outlet is meant to preheat the incoming water in the integrated heat exchanger. This means that the operating conditions have to be carefully chosen in order to obtain sustainable electrolyzer operation, and future work is directed into identifying suitable operating conditions that end up in a temperature increase of a few °C while avoiding hot spots. Finally, it can be seen from Equation (50) that the amount of water that can evaporate is highly temperature-dependent, and a small decrease in operating temperature will significantly reduce the amount of water that evaporates.

Finally, Figure 9 shows the calculated temperature distribution, where the temperature distribution for 600 mbar and the base case with 1 atm back pressure are directly compared. Reducing the pressure leads to an overall cooler cell, but the minimum temperature remains at the nominal operating temperature, demonstrating the efficiency of evaporative cooling. Compared to the base case, the isothermal region has greatly expanded. The hot spot at the inlet section is predicted for both cases, but it is less severe for the lower back pressure. This also holds true for the hot region at the outer edge of the plate where the peak temperature is around 5 °C lower for the reduced back pressure. For the hot region near the inlet, it can be expected that when the beginning of the outlet channel is located closer to the center, i.e., the outlet channel is longer, the hot spot will be less severe, and the section of the cell that experiences an almost uniform temperature will be expanded. The hot spot near the outlet appears to be caused by the higher outlet temperature, and it can be reduced by adjusting the back pressure.

4. Conclusions

In this work, a three-dimensional multiphase model for the anode side of a proton exchange membrane electrolyzer has been developed. The model is based on the Eulerian approach and thus solves a complete set of transport equations for each phase, in this case gas and liquid. The liquid phase consists of pure water, while the gas phase contains oxygen and water vapor. Importantly, the model is non-isothermal and thus allows for the prediction of hot spots.

The presentation of results included the distribution of the pressure and the temperature, local rates of phase change and the distribution of gas phase and liquid phase. A hot spot was predicted near the inlet, and another hot area near the outlet, while the cell section between the flow channels showed a uniform temperature distribution and is thus preferred. The flow was shown to be segregated in the outlet channel with the water accumulating in the bottom of the channel, highlighting the effect of gravity that was ignored in several prior studies. Based on this, it was suggested to manufacture the outlet channel in an angle to let gravity assist in the otherwise pulsating outlet flow. This effect will be investigated in the future.

The parametric study focused on the effect of the back pressure in the cathode compartment. It was found that the concentration ratio between water vapor and oxygen can be favorably adjusted towards the water concentration. Moreover, the cell was observed to experience a larger cooling effect due to the higher evaporation rates. However, if the outlet pressure is too low, the amount of cooling can exceed the waste heat production, and the outlet is at a lower temperature than the inlet, which is unsustainable.

In the future, attention will be paid to identify suitable and sustainable operating conditions, and to better understand the impact of the plate orientation and channel geometry. Moreover, experimental verification is required.