# Water electrolysis at near-neutral pH: A semi-empirical modelling strategy to assess the benefit of pressure on cell voltage

Junzhe Sun, Luc Etcheverry, Alain Bergel

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Junzhe Sun, Luc Etcheverry, Alain Bergel. Water electrolysis at near-neutral pH: A semi-empirical modelling strategy to assess the benefit of pressure on cell voltage. Electrochimica Acta, 2025, 530, pp.146264. 10.1016/j.electacta.2025.146264. hal-05064586

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# Water electrolysis at near-neutral pH: A semi-empirical modelling strategy to assess the benefit of pressure on cell voltage

Junzhe Sun,*, Luc Etcheverry, Alain Bergel

Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France

## ARTICLE INFO

### Keywords:

Gas evolving electrode

Hydrogen

Bubble

Ohmic drop

Diphasic electrolyte

Power savings

## ABSTRACT

Water electrolysis for hydrogen production is being deployed using three main technologies: alkaline electrolysis, proton exchange membrane electrolysis, and solid oxide electrolysis. However, electrolysis at near-neutral pH offers an alternative that is safer, more eco-friendly, and requires less capital expenditure. This paper presents the first experimental and theoretical analysis of near-neutral pH water electrolysis under pressure. Pressure can affect the electrolysis cell voltage in three ways: i) it increases the equilibrium voltage, ii) it reduces the negative impact of bubbles that shield part of the electrode surface, and iii) it decreases the negative impact of bubbles on the electrolyte ionic conductivity. Each of these contributions was characterized for gas-evolving electrodes implemented in phosphate pH 7.2 or carbonate pH 10.0, with freely rising bubble layers. In this context, the cell voltage decreased with pressure. Semi-empirical correlations were combined with a simple numerical approach. The resulting model perfectly correlated with the experimental data for pressures ranging from 1 to 4 bar and current densities from 500 to 2000 A m⁻². The model demonstrated that the beneficial effect of pressure on the freely rising bubble layers was the predominant contribution, and that the corresponding gain increases with the applied current density. At 2000 A m⁻², it was possible to achieve up to a 400 mV reduction in cell voltage by operating under 4 bar instead of atmospheric pressure. The power savings can reach 0.7 kWh per Nm³ of hydrogen produced, representing a considerable gain in comparison to the energy generally consumed by water electrolysis.

## 1. Introduction

Hydrogen will become a major energy carrier only if it can be produced at low cost from non-fossil resources, with low CO₂ emission [1–3]. For this purpose, water electrolysis is sometimes seen as the philosopher's stone of hydrogen production, provided that the necessary electrical energy is produced sustainably at sufficiently low cost [4]. Water electrolysis facilities have already been developed at large scales using three major technologies: alkaline, proton exchange membrane (PEM), and solid oxide electrolysis cells [5–7]. Each of these technologies involves extreme operating conditions - in alkalinity, acidity or temperature, respectively - imposing high equipment costs, particularly to withstand exacerbated corrosion, and requiring dedicated facilities run by qualified staff.

Water electrolysis at near-neutral pH is an alternative attracting growing interest because it uses mild operating conditions that reduce the equipment expenditure and make it much more eco-friendly than the major technologies developed so far [8–12]. In addition, water

electrolysis at near-neutral pH may open the way to using seawater instead of freshwater [13,14]. It may also pave the way to cutting-edge technologies such as photo-electrochemical water splitting [15] and microbial electrolysis [16–18].

Water electrolysis at near-neutral pH is unlikely to replace mature water electrolysis technologies for large-scale production of hydrogen. Several rate-limiting steps have yet to be overcome. The requirement for buffered media, to mitigate the negative effect of pH gradients at the surface of the electrodes, is a major issue [19]. Another is the low ionic conductivity of the electrolytes. For example, alkaline water electrolysis can use KOH solutions 20–40 wt% with conductivities greater than 50 S m⁻¹ at 25 °C [20]. In contrast, potassium phosphate and potassium carbonate at 1.5 M, the most used electrolytes at near-neutral pH, have conductivities <15 S m⁻¹. Nevertheless, using electrolytes at near-neutral pH offers remarkable advantages if the goal is to design small, easy-to-use electrolysers that can be used in everyday settings by unskilled operators. It is easier for an inexperienced user to drain an electrolyser when the electrolyte is potassium phosphate at around pH 7

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: alain.bergel@toulouse-inp.fr (A. Bergel).</footnote>
rather than concentrated potassium hydroxide at pH 14.

Among the many advantages of water electrolysis is that it can supply pressurized hydrogen directly from the electrolyser outlet. Operating electrolyzers under pressure eliminates the need for the first mechanical compression stages downstream, typically ranging from 1 to a few bar, which are the most energy-demanding [6]. The potential benefit of reaching very high pressures, up to several hundred bar, directly at the electrolyser outlet is a subject of discussion [21–23]. Nevertheless, for pressures of a few bar, there is unanimous agreement that the total power consumption of a pressurized electrolyser is lower than that of an atmospheric electrolyser coupled with a downstream gas compressor [23–26]. For example, it has been claimed that total power consumption could be reduced by >70 % by running an electrolyser at 5 bar, despite using an atmospheric electrolyser followed by mechanical compression [27]. Moreover, with the objective to design small, easy-to-use electrolyzers, it is advantageous to produce hydrogen under moderate pressure, directly from the electrolyser outlet, without the need for a compressor. Unfortunately, despite the considerable advantage in terms of power consumption, the behaviour of water electrolysis cells under pressure remains difficult to grasp.

Basically, the Nernst law applied to water electrolysis:

$$H_2O \rightleftharpoons H_2 + \frac{1}{2} O_2 \qquad (1)$$

gives:

$$U_{eq} = U_{eq}^{0} + \frac{RT}{2F} \ln \left( \frac{f_{H2} f_{O2}^{1/2}}{a_{H2O}} \right) \qquad (2)$$

where $U_{eq}$ is the equilibrium cell voltage, $U_{eq}^0$ is the equilibrium cell voltage at standard conditions (1.23 V), R is the gas constant (8.314 mol⁻¹ K⁻¹), T is the absolute temperature, F is the Faraday constant (96 485 C mol⁻¹), fH₂ and fO₂ are the hydrogen and oxygen fugacities, and aH₂O is the water activity. This equation shows that the equilibrium cell voltage increases with the pressure at which the gases are produced. This increase in cell voltage (U) with pressure has sometimes been observed experimentally [28,29] and confirmed numerically [30].

In contrast, the Fateev research group, working with PEM electrolysis cells, has warned of behaviours that are more complex. For example, it was observed that U increased when the pressure was increased from 1 to 25 or 50 bar at low current densities (less than around 0.5 to 0.75 A cm⁻²), while it decreased at higher current densities [21,31]. At constant current density, the same group observed U values that decreased for pressures increasing from 1 to 30 bar and then increased for pressures up to 100 [32]. Another group has reported that U increased with pressure up to 100 bar at low current densities (< 0.8 A cm⁻²) and no longer increased at higher current densities [33].

The deviations of experimental observations from the thermodynamic predictions have been explained by multiple possible phenomena, which can result in U decreasing with pressure. Some possible explanations are linked to the specific PEM electrolyser technology. For instance, studies have found that high pressures improve water mass transport within the PEM cell membrane and the porous structure of PEM anodes [33,34]. Pressure may also improve the electric contact between the catalytic layer and the current collector [34].

A group of more general explanations refers to the effect of pressure on the bubbles. The bubbles evolving at the surface of the electrodes mask a part of their surface area [35]. The actual active electrode surface area available for electron exchange is thus diminished by the presence of bubbles. In addition, in the bulk of the electrolyte, the presence of bubbles decreases the electrolyte apparent ionic conductivity, because it reduces the cross-section available for ion transport. The presence of bubbles in the electrolytes therefore increases the ohmic drop [36,37]. Under pressure, the volume occupied by bubbles is reduced, which consequently mitigates both effects, i.e. electrode surface shielding and ohmic drop increase [6,27]. The virtuous effect of pressure on the

bubbles may offset the negative impact on thermodynamics.

Pressure may also have an impact on the kinetics parameters. It has been claimed that pressure does not impact the Tafel slope but may increase the exchange current density [33]. However, it is difficult to distinguish whether the effect on the exchange current density is due to a direct action on the electron exchange reaction or is an indirect effect resulting from the impact on the electrode structure, or includes surface shielding by the bubbles. Additional complexity may lie in factors such as the hydrodynamics created or disturbed by the formation of bubbles [38–41] and the formation of gas plumes [42].

In short, sound general rules on the effect of pressure on the performance of water electrolysis cells are yet to be established, notably because the steps to be investigated are numerous, intricate, and interconnected, while studies remain scarce, address different electrolysis types, and consider various pressure and temperature ranges.

The situation is even more obscure in the context of water electrolysis at near-neutral pH as, to the best of our knowledge, the possible effect of pressure had never before been considered, except in terms of thermodynamics. The study presented here aims to offer first insight in this area. A number of studies have focused on the basics of bubble formation, growth, and release on gas-evolving electrodes, using specific experimental set-ups such as narrow platinum strips [43], small-diameter platinum [44], tungsten and stainless steel [45] disks, horizontal electrodes with bubbles generated downwards [46] or upwards [47], pit-ring systems [48], and graphite paper electrodes [49]. All of these studies have provided valuable basic information, but which is difficult to apply directly to the analysis of actual electrolysis cells because the electrode materials, set-ups, and operating conditions were often a long way from industrial conditions.

In order to help bridge the gap between basic studies and the analysis of an electrolysis cell, we used an experimental system that was close to a parallel plate electrolysis cell. The electrodes were arranged vertically, face to face, with a separator in between. No forced convection was established inside the cell, because this will be the most likely situation in the context of near-neutral pH electrolyzers. In order to design small, easy-to-use electrolyzers that can be used in a wide range of settings, it is attractive to avoid the need for a pump. For instance, the absence of a pump will make the electrolyser more autonomous as it can be powered directly by photovoltaic panels, in remote areas far from the grid.

The electrodes were made of solid, non-porous materials, to avoid possible effects of pressure acting on the electrode structure. The cathode was made of stainless steel and the anode of a mixed metal oxide deposited on a titanium basis, so-called dimensionally stable anode (DSA). These electrode materials are likely not the best possible catalysts for water electrolysis, but they are industrial materials that could easily be used for large-scale implementation. The main difference between our analytical cell and a commercial design was the inter-electrode distance, which was chosen to be large enough to enable the bubbles to rise freely without mechanical stress imposed by the separator. Nevertheless, the inter-electrode distance did not affect the experimental data and modelling since the whole approach was based on cell voltage differences between values under pressure minus the value at atmospheric condition. The contributions of the separator and the part of the electrolyte not containing bubbles were thus eliminated. Based on the finding that most of the gain in total power consumption occurs when electrolyzers operate at pressures of a few bar [27], we investigated the range of pressures from 1 to 4 bar.

Recent review articles, which analyzed the many catalysts designed for the hydrogen evolution reaction (HER) [10] or the oxygen evolution reaction (OER) [50] at near neutral pH, has shown that the most commonly used buffers include phosphate buffer (pH around 7), borate (pH around 8.5), and (bi)carbonate (pH 9.2 to 10.6). Several studies by the Takanabe group dedicated to electrolyte engineering have shown that phosphate and carbonate buffers were also appropriate to promote the oxygen evolution reaction (OER) at the anode [9,51,52]. They also demonstrated that high concentrations, in the molar range, were
necessary to obtain high current densities under reasonable cell voltages [8,53–55]. Accordingly, phosphate and carbonate buffers were used here. Borate was not selected because of its toxicity in high concentration, which could be detrimental to large-scale implementation. The concentration of 1.5 M was chosen, because it offers a safety margin in relation to saturation to avoid the risks of local crystallization.

Consistent with previous findings reported in the literature on PEM electrolysis cells, it was observed here that, in contradiction with the thermodynamic prediction, cell voltage *U* did not increase with pressure. It remained almost constant or even significantly decreased with pressure for the highest values of the current density. These first results put in light the unexpected advantage of running water electrolysis at near-neutral pH under pressure. A step-by-step modelling strategy was consequently designed, based on semi-empirical correlations found in the literature combined with a specific numerical approach for each of the three components by which pressure can affect cell voltage.

## 2. Experimental

The electrolysis cell was a parallelepiped Plexiglass reactor 8 cm wide, 12 cm long, and 10 cm high (inner size), containing around 800 mL of electrolyte, i.e. with a gas headspace around 1.5 cm high (Scheme 1). The cell was divided into two compartments by a polyphenylene sulphide separator (PPS, 6TMIC France) held in place by a Plexiglas frame with an open area of 80 cm². The volume of the cathode compartment thus defined was twice that of the anode compartment. A Plexiglas cover was fixed to the top of the cell, with a flat rubber gasket placed in between, by 10 threaded rods with the necessary nuts. The cathode and anode, each with a surface area of 3 × 3 cm², were set face to face on either side of the separator, with the cathode and the anode 6 and 2.5 cm away from the separator, respectively. The cathode was made of 316 L stainless steel, and the anode was a dimensionally stable anode made of titanium with a deposited catalyst layer of iridium oxides (Di Noer Technology Co., Ltd., China). Each electrode was embedded in insulating resin (EpoFix Resin, Struers), leaving only its surface exposed to the electrolyte to avoid any edge effect. The electrodes were electrically connected by 2 mm-diameter screwed titanium rods, with the part exposed to the electrolyte insulated by heat-shrink tubing.

The phosphate buffer solution (PBS) at pH 7.2 was made of KH₂PO₄ and K₂HPO₄ 1.5 M. The carbonate buffer solution (CBS) was made of KHCO₃ 1.5 M, and the pH was adjusted to 10 with KOH (pH-meter HQ

Series Hach). The value of the ionic conductivity of the electrolyte σ₀ used in the calculations was 11.5 S m⁻¹, which corresponded to the value measured with PBS (HQ Series Hach conductimeter).

Current intensities of 0.45, 0.9, 1.35, and 1.8 A were applied between the two electrodes with a potentiostat (Biologic SA, France) in order to obtain current densities of 500, 1000, 1500, and 2000 A m⁻². The voltage was recorded every 5 s. Each compartment had a gas outlet equipped with a valve. To operate under pressure, the two valves were closed and the pressure was measured as a function of time. The rate of hydrogen production at the cathode is twice that of oxygen production at the anode (Reaction 1). As the volume of the cathode compartment was twice that of the anode compartment, the pressure remained equal in both compartments. Otherwise, a pressure imbalance would have been created on either side of the separator, which would disturb the electrolyte levels.

The diameter of the bubbles was determined as a function of the pressure by taking photos with a camera (Canon EOS 70D) equipped with a macro lens (Canon Macro 100 mm) along an axis parallel to the surface of the electrode, from one of its edges, with an exposure time of 1/2500 s in order to freeze the motion of the bubbles. The large inter-electrode distance chosen for hydrodynamic reasons was also necessary to position the camera at an angle that allowed the bubbles to be isolated on the images. The photos were processed using ImageJ, taking the anode plate thickness of 1 mm as the reference to convert the number of pixels to millimetres. The average diameters and standard deviations resulted from 20 measurements each.

In addition, a second electrochemical cell was used, which was left in the open air and was not equipped with a separator. This open-to-air cell was used to measure the thickness of the layer of bubbles on the surface of the electrodes at atmospheric pressure. This cell had thinner walls than the electrolysis cell used under pressure, in order to obtain the sharpest possible photos. The same electrodes, facing each other, 1.8 cm apart, and the same experimental protocols were used.

The “current density – voltage (J-U)” curves were also recorded at atmospheric pressure in the open-to-air cell. The cell voltage was measured at different values of applied current density. For each value, electrolysis was performed for 30 min and the measurement used was the average voltage value over the five last minutes. The electrolyte resistance between the two electrodes was measured by electrochemical impedance spectroscopy (EIS) performed in potentiostatic mode. The value of the resistance (1.22 Ω) was the average of 3 measurements.

<figure><img src="image_3.png" /><figcaption>**Scheme 1.** Experimental set-up.</figcaption></figure>
# 3. Results and discussion

## 3.1. Experimental results

Electrolysis was performed at atmospheric pressure in an open-to-air cell, in galvanostatic conditions at applied current densities ranging from 100 to 1000 A m⁻². The bubbles formed a layer that rose along the surface of each electrode at an apparently uniform velocity. This layer widened with height, so that it had a triangular cross-section (Schemes 2.A and 2.B). The bubbles were released at the free surface, and the electrolyte flowed back down in the centre of the system, without apparently interfering with the curtain of bubbles along the electrodes.

The bubble layer was a few millimetres thick at the top of the electrodes, so that the pattern of the bubble layer was not affected by the presence of the separator, which was at least 2.5 cm away from the electrode. The thickness of the bubble layer was measured at the top of the electrode, i.e. at a height of 3 cm. The measures performed with phosphate and carbonate gave identical results. For the cathode, the thickness increased linearly with the applied current density starting at a current density of 100 A m⁻² (Fig. 1.A). For the anode, the linearity range of the bubble layer thickness started at 200 A m⁻² (Fig. 1.B).

Electrolysis was performed in the closed electrolysis cell with current densities ranging from 500 to 2000 A m⁻². The value of the cell voltage (U) was recorded as a function of time as illustrated in Fig. 2.A. The cell was hermetically sealed and had two outlets controlled by valves.

When the valves remained open during electrolysis, the reactor operated at atmospheric pressure. After a few minutes of stabilization with open valves, U was approximately stable (Fig. 2.B). When the valves were closed, the pressure slowly increased with time. Just after closing the valve, U increased by around 20 mV for a few minutes and then continuously decreased. All raw “U - time” curves are reported in the Supplementary Information file. The raw “U - time” curves were used to extract the variation of U as a function of pressure (Fig. 2.C). Both phosphate and carbonate electrolytes showed similar behaviour and similar U values. The two buffers have equivalent capacities to promote water electrolysis with the electrode materials used here, i.e. stainless steel at the cathode and DSA at the anode.

The values of U were high because the distance between the electrodes was large (8.5 cm), with a separator in between. This choice was made in order to allow the gas bubbles to move freely without being disturbed by the presence of the separator, but it resulted in high ohmic drops. The value of the voltage at the atmospheric pressure (U₁) was taken just when the valves were closed, after the initial stabilization period. These values increased linearly with current density (Fig. 2.D), as expected for a process limited by the ohmic drop. The value of the slope, of 0.0023 Ω m², gave a rather high electrolyte resistance of 2.56 Ω (by factoring in a surface area of 9 10⁻⁴ m²).

In order to get a direct view of the impact of pressure on U, the difference was calculated between the cell voltage under pressure (Uₚ) and the cell voltage at atmospheric pressure (U₁):

<figure><img src="image_4.png" /><figcaption>**Scheme 2.** A) Photography and B) scheme of the bubble layer pattern. C) Integration scheme. D) Main hypothesis of the modelling strategy: under pressure, the gas fraction ε of the bubble layer remains constant and its thickness decreases.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 1. Experimental results. A) Bubble layer thickness at the top of the electrode (H = 3 cm) at atmospheric pressure with phosphate 1.5 M (circles) or carbonate 1.5 M (triangles) for the cathode. B) As A) for the anode. C) Bubble diameter as a function of the pressure measured at the cathode in carbonate 1.5 M for J = 1000 A m⁻² and theoretical ideal gas model (dashed line); each value resulted from 20 measurements.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. Experimental results. A) Example of experimental record of raw U as a function of time obtained with PBS at the applied current density J = 1000 A m⁻². B) Zoom on initial period. C) Raw U as a function of the logarithm of the pressure p obtained with CBS (circles) or PBS (triangles) at various applied current densities. D) Initial values of U (U₁) as a function of the applied current density for PBS (circles) and CBS (triangles). E) ΔU as a function of the logarithm of the pressure p at various applied current densities. Same colour chart for C and E: blue 500 A m⁻²; pink 1000 A m⁻²; grey 1500 A m⁻²; orange 2000 A m⁻².</figcaption></figure>

$$ \Delta U_p = U_p - U_1 \quad (3) $$

Fig. 2. E gives an example of ΔUₚ plotted as a function of the logarithm of the pressure. These results showed that it was possible to considerably decrease U when operating under a few bar of pressure. The decay was almost linear as a function of the pressure logarithm, and the rate of decay increased with increasing current density. Thus, our finding is that the effect of pressure on the bubbles, which tends to reduce cell voltage, was greater than the thermodynamic effect, which

tends to increase cell voltage, and this difference increased with current density.

## 3.2. Modelling

The value of the cell voltage is the sum of four terms:

$$ U = U_{eq} + |\eta_C| + \eta_A + R_T I \quad (4) $$
where $η_C$ and $η_A$ are the cathode and anode overpotentials, $R_T$ is the total ionic resistance of the electrolyte(s), including that of the separator if applicable, and $I$ is the current intensity. The term $R_T I$ represents the total ohmic drop.

As discussed in the introduction, pressure can have some impact on the overpotentials by acting on the electrode structure. However, here, compact electrode materials were used so that this type of effect did not apply. Three groups of effects remained valid here: thermodynamics, electrode surface shielding by bubbles, and effect of bubbles on ohmic drop. Each is analysed below.

### 3.2.1. Thermodynamic effect of pressure

The thermodynamic effect of pressure on cell voltage can be expressed as:

$$ \Delta U_{thermo,p} = U_{eq,p} - U_{eq,1} \qquad (5) $$

where $U_{eq,p}$ and $U_{eq,1}$ are the equilibrium cell voltages at the pressure $p$ and at 1 bar, respectively. Using the Nernst Eq. (2), assuming that the fugacities can be taken equal to the pressures and that hydrogen and oxygen had the same pressure $p$, leads to:

$$ \Delta U_{thermo,p} = \frac{RT}{2F} \frac{3}{2} \ln \left( \frac{p}{p_0} \right) \qquad (6) $$

where $p_0$ is the reference pressure equal to 1 bar.

### 3.2.2. Effect of pressure on ohmic drop

A diphasic electrolyte, made up of solution and dispersed bubbles, has a higher resistivity than a solution without bubbles because the bubbles reduce the cross-section available for ion transport. The pressure reduces the volume occupied by the bubbles and consequently reduces the resistivity of the electrolyte. The effect of the pressure on the volume occupied by the bubbles was confirmed experimentally by measuring the diameter of the hydrogen bubbles under pressures of 1 to 3 bar (Fig. 1.C).

The ideal gas law applied to a gas bubble:

$$ p V_b = n_b R T \qquad (7) $$

where $n_b$ is the number of gas moles contained in the bubble and $V_b$ is the volume of the bubble of diameter $d$:

$$ V_b = \frac{\pi d^3}{6} \qquad (8) $$

indicates that the bubble diameter should vary as $p^{-1/3}$:

$$ d_p = d_1 \left(\frac{p}{p_0}\right)^{-1/3} \qquad (9) $$

where $d_p$ and $d_1$ are the bubble diameters at pressure $p$ and 1 bar, respectively. The dashed line drawn in Fig. 1.C with Eq. (9) fully correlated with the experimental data. The ideal gas law and Eq. (9) that shows that the bubble diameter varies as $p^{-1/3}$ were consequently adopted in the following.

It was observed that the gas bubbles evolved and rose along the electrode, forming a layer whose thickness increased linearly with height (Scheme 2). This triangular layer of bubbles was well established for current densities above a threshold of around 100–200 A m⁻² and up to the maximum value explored here of 2000 A m⁻². This pattern of bubble layers over the surface of the cathode and anode was used to calculate the ionic resistance of the electrolyte $R_T$. According to this scheme, the part of the ionic resistance $dR_T$ of a slice of thickness $dh$ at height $h$ (Scheme 2.C) is:

$$ dR_T = \frac{\rho_0 L - (\rho_0 - \rho_e)h(\tan(\alpha_C) + \tan(\alpha_A))}{W dh} \qquad (10) $$

where $ρ_0$ is the ionic resistivity of the electrolyte, $ρ_e$ is the ionic resistivity of the bubble layer, and $L$ is the distance between the electrodes. The differential slices of thickness $dh$ impose parallel resistances, so the total resistance of the electrolyte is calculated by integrating the inverse of the differential resistances from the bottom ($h=0$) to the top ($h=H$) of the electrode:

$$ \frac{1}{R_T} = \int_{0}^{H} \frac{W dh}{\rho_0 L - (\rho_0 - \rho_e)h(\tan(\alpha_C) + \tan(\alpha_A))} \qquad (11) $$

which gives:

$$ R_T = \frac{-(\rho_0 - \rho_e)(\tan(\alpha_C) + \tan(\alpha_A))}{W Ln \left( 1 - \left( 1 - \frac{\nu_e}{\rho_0} \right) \left( \tan(\alpha_C) + \tan(\alpha_A) \right)^{\frac{H}{L}} \right)} \qquad (12) $$

At atmospheric pressure, the values of $\tan(\alpha_1)$ were calculated on the basis of the experimental observations. For the cathode, the thickness of the layer at the top, i.e. at a height of 3 cm, was 1.2 and 2.8 mm for 100 and 1000 A m⁻², respectively (Fig. 1.A). Accordingly, $\tan(\alpha_C)$ was calculated for the different current density values as:

$$ \tan(\alpha_C) = 0.040 + (J - 100) 5.93 \times 10^{-5} \qquad (13) $$

where $J$ is expressed in A m⁻². Similarly, for the anode, the bubble layer thickness was 1 mm at 200 A m⁻² and 1.9 mm at 1000 A m⁻² (Fig. 1.B), leading to:

$$ \tan(\alpha_A) = 0.033 + (J - 200) 3.75 \times 10^{-5} \qquad (14) $$

At this stage, the unique unknown parameter that remained in the calculation of $R_T$ was the ionic resistivity of the bubble layer $ρ_e$. The ionic resistivity of diphasic electrolytes can be calculated using the Brugge-man equation, which correlates electrolyte conductivity with the gas fraction ($ε$):

$$ \sigma_e = \sigma_0 (1 - \varepsilon)^{3/2} \qquad (15) $$

where $\sigma_e$ is the conductivity of the bubble layer and $\sigma_0$ is the conductivity of the electrolyte in the absence of bubbles. The reliability of this equation to assess the conductivity of diphasic flow patterns against gas evolving electrodes has long been established [56] and is widely used [27,57,58]. In terms of resistivity, this equation gives:

$$ \rho_e = \rho_0 (1 - \varepsilon)^{-3/2} \qquad (16) $$

The resistance of the electrolyte $R_T$ (Eq. (12)) can thus be estimated at atmospheric pressure with the gas fraction ($ε$) as the single unknown parameter.

It is then necessary to explore how $R_T$ behaves under pressure. To this aim, the bubble layer was modelled by making a differential balance on the gas volume (Scheme 2.C). At the cathode, where hydrogen was produced:

$$ 2 H_2O + 2 e^- \rightarrow H_2 + 2 OH^- \qquad (17) $$

the differential balance on the gas volume in a slice of thickness $dh$ is:

$$ \dot{V}_g|_{h+dh} = \dot{V}_g|_h + \frac{J W dh}{2F} \frac{RT}{p} \qquad (18) $$

where $\dot{V}_g$ is the gas volume flow rate, $J$ is the current density, $W$ is the electrode width, and $h$ is the height from the bottom of the electrode. The gas volume flow rate at height $h$ is:

$$ \dot{V}_g|_h = u \varepsilon W h \tan(\alpha_C) \qquad (19) $$

where $u$ is the bubble rinsing velocity, $ε$ is the gas fraction in the bubble layer, and $\alpha_C$ is the angle between the boundary of the bubble layer and the surface of the cathode. Assuming that the bubble rising velocity and
the gas fraction do not vary with height gives:

$$ \tan(\alpha_C) = \frac{J}{2F} \frac{RT}{p \varepsilon u} \qquad (20) $$

Similarly, at the anode where oxygen evolved:

$$ 2 H_2O \rightarrow O_2 + 4 H^+ + 4 e^- \qquad (21) $$

assuming that the bubble rising velocity and the gas fraction were identical to those of the cathode gives:

$$ \tan(\alpha_A) = \frac{J}{4F} \frac{RT}{p \varepsilon u} \qquad (22) $$

It is worth noting the relevance of the Eqs. (20) and (22), which show that the tangents are proportional to the current density J, in line with what was observed experimentally (Fig. 1.A and B). The expressions of tan(α) indicate that the tangents were affected by pressure in two ways: first, the pressure appears explicitly in the tan(α) expressions; second, the rising velocity u depends on p. For a rough assessment, we can assume that u should vary with p in the same way as the terminal rising velocity u_t calculated from Stokes' law:

$$ u_t = \frac{\Delta \rho_m g d^2}{18 \mu} \qquad (23) $$

where Δρ_m is the difference of the mass densities between gas and fluid, g is the gravitational field strength, and μ is the dynamic viscosity. Obviously, the terminal rising velocity u_t gives the maximal rising velocity of a single bubble in a stagnant solution, which was not the case here. This is why we did not use this velocity expression here, but only assumed that the actual velocity u should vary as a function of pressure in the same way as the Stokes' limit velocity. Recalling that d varies with the pressure as p⁻¹/³ (Eq. (9)), it was considered that u_t varied with pressure as p²/³ (Eq. (23)), so that it can finally be concluded that tan(α) varies with pressure as p⁻¹/³ (Eq. (20) or 22):

$$ \tan(\alpha)_p = \tan(\alpha)_1 p^{-1/3} \qquad (24) $$

It is should be highlighted that the gas fraction of the bubble layers ε was assumed to remain constant when the pressure increased. This is a major hypothesis of the modelling strategy proposed here (Scheme 2.D). It was assumed that, while the pressure reduced the diameter of the bubbles (Fig. 1.C), the physical-chemical processes that control the dispersion of bubbles within the electrolyte remained unchanged with pressure. Accordingly, when the diameter of the bubbles was reduced, the fraction of gas within the bubble layer remained constant and the thickness of the film was reduced.

Finally, the effect of pressure on the ohmic drop (ΔUohmic,p) was assessed by calculating tan(α_C)p and tan(α_A)p with Eq. (24) and then R_T,p with Eq. (12), so that:

$$ \Delta U_{ohmic,p} = (R_{T,p} - R_{T,1})I \qquad (25) $$

In this component, the gas fraction of the bubble layer ε was the unique parameter that we were not able to measure or easily estimate, which remained to be adjusted numerically.

### 3.2.3. Effect of pressure on bubble coverage of the electrode surface

The pressure also acts on the cell voltage by reducing the diameter of the bubbles that cover the surface of the electrodes. The bubble coating shields a part of the electrode surface area. The active surface area of the electrode is reduced and the electrode must consequently operate at an "actual average current density" [59] (J<sub>actual</sub>) higher than the nominal current density J. By reducing the size of the bubbles, the pressure reduces the impact of the bubble coverage. In the literature, it is difficult to find theoretical tools to calculate the surface coverage on gas-evolving electrodes [60]. Nevertheless, a useful empirical equation has been designed by Vogt et Balzer on a solid basis of experimental data [61]:

$$ \theta_1 = \left( \frac{J}{J_M} \right)^{0.3} \qquad (26) $$

where θ_1 is the bubble coverage ratio under 1 bar, and J_M is an adjusted value of the current intensity that would correspond to the entire coverage of the electrode surface by adhered bubbles. Vogt and Balzer have shown that the J_M value of 3 10⁵ A m⁻² gave a correct match of Eq. (26) with a large basis of experimental data.

The shielding effect of the bubble coverage is proportional to the projected surface area of the bubbles adhered on the electrode surface, i. e. to πd²/4. As the bubble diameter d varies with the pressure as p⁻¹/³, it can be concluded that θ varies with pressure as p⁻²/³:

$$ \theta_p = \theta_1 \left(\frac{p}{p_0}\right)^{-2/3} \qquad (27) $$

where θ_p is the bubble coverage ratio under the pressure p.

In the presence of bubbles adhered on the electrode surface, the part of the electrode surface area that remains active is A(1-θ), and the actual current density relating to this active part is:

$$ J_{actual}(1 - \theta) = J \qquad (28) $$

To calculate the actual current density under 1 bar, Eqs. (26) and (28) were combined:

$$ J_{actual,1} = \frac{J}{1 - \left(\frac{J}{J_M}\right)^{0.3}} \qquad (29) $$

Then, Eq. (27) was plugged into Eq. (29) to calculate the actual current density under the pressure p:

$$ J_{actual,p} = \frac{J}{1 - \left(\frac{J}{J_M}\right)^{0.3} \left(\frac{p}{p_0}\right)^{-2/3}} \qquad (30) $$

J<sub>actual,p</sub> is smaller than J<sub>actual,1</sub>, which corresponds to a gain in overvoltage.

This gain (ΔU<sub>surf,p</sub>) was measured using an experimental curve that gave J<sub>actual</sub> as a function of the cell voltage U. This involved using the raw experimental J-U curve, by first subtracting the ohmic drop (Fig. 3). However, this simple operation was tricky because when operating at high current density, any small experimental error in the value of the resistance has a considerable impact on the correction of the curve. The value of the ionic resistance determined by electrochemical impedance spectroscopy (EIS) (1.22 Ω) was used here to correct the J-U curve, which became almost vertical. The error bars on the diagram represent ±10 % on the resistance value. The verticality of the J-U curve corresponds to the fact that there was no limiting step, due to the transport of the reagent, since the reagent was water [19]. Furthermore, as the electrolyte was strongly buffered (1.5 M), the pH gradients at the surface

<figure><img src="image_7.png" /><figcaption>Fig. 3. Experimental results under atmospheric pressure. J as a function of U. Raw measures in PBS (blue points), corrected from ohmic drop (red points) with error bars corresponding to ±10 % on the ionic resistance, and J<sub>actual</sub> (orange squares); the dashed line is the fitted Eq. (31).</figcaption></figure>
of the electrodes were not yet too steep at 2000 A/m² to limit the current. Then, the curve corrected from the ohmic drop was transformed into “J<sub>actual</sub> -U” curve by using Eq. (29).

The value of ΔU<sub>surf,p</sub> was the difference on cell voltage measured on the J<sub>actual</sub>-U curve when the actual current density decreased from J<sub>actual,1</sub> to J<sub>actual,p</sub>. To make this calculation easier, the J<sub>actual</sub>-U curve was transformed into U-J<sub>actual</sub> curve, which was smoothed using logarithmic regression as:

$$U = 0.269 \ln(J_{\text{actual}}) + 1.03 \qquad (31)$$

leading to:

$$\Delta U_{\text{surf,p}} = 0.269 \ln \left( \frac{J_{\text{actual,p}}}{J_{\text{actual,1}}} \right) \qquad (32)$$

Finally, the total effect of the pressure p on the cell voltage was the sum of the three contributions:

$$\Delta U_p = \Delta U_{\text{thermo,p}} + \Delta U_{\text{ohmic,p}} + \Delta U_{\text{surf,p}} \qquad (33)$$

In the calculation process that results in the final Eq. (33), only one parameter remained unknown and had to be adjusted numerically: the gas fraction in the bubble layer ε. This parameter was firstly adjusted numerically to match with a set of 4 experimental curves plotted as ΔU<sub>p</sub> as a function of the pressure p (Fig. 4). The value of ε of 0.485 gave a satisfactory representation of the overall shape of the set of curves and correct orders of magnitude. The model gave a linear variation of ΔU<sub>p</sub> with the pressure logarithm. This linear trend was not observed experimentally for the lowest values of the pressure and the lowest values of the applied current density. This was the major deviation between model and experimental data, but is most likely due to the difficulty of accurately stabilizing and measuring the lowest pressures.

To reach reliable conclusions encompassing a whole range of experimental results, average values of ΔU<sub>p</sub> were calculated from different experiments and plotted as a function of the applied current density (Figs. 5.A-C). The set of average values was thus based on 43 different measurements for the pressures of 2, 3, and 4 bar (Table 1). The experimental data showed almost linear variations of ΔU<sub>p</sub> with the applied current density, with slopes that depended on the pressure. The model perfectly correlated with this behaviour when using the single ε value of 0.456 for the entire set of data.

The sensitivity of the model to the adjustable parameter ε was tested by plotting the curves ΔU<sub>p</sub> as a function of the applied current density with ε values of 0.456 ± 10% (Figure S.1 in Supplementary Data). These tests showed that adjusting the gas fraction ε significantly affected the gain in cell voltage. When ε was decreased by 10%, ΔU<sub>p</sub> at 2000A<sup>-2</sup> was decreased (in absolute value) by 18% and, inversely, when ε was increased by 10%, ΔU<sub>p</sub> was increased by 21%. Logically, when the gas fraction increases, the ohmic drop due to the diphasic character of the electrolyte has a greater impact on the cell voltage; consequently, the

<figure><img src="image_8.png" /><figcaption>Fig. 4. Calculated values of ΔU for the case plotted in Fig. 2.E (same colour chart), obtained with ε = 0.48.</figcaption></figure>

gain that can be achieved under pressure is greater. The gas fraction ε, used as the unique adjustable parameter of the model, was consequently a relevant parameter to control the behaviour of the cell under pressure.

### 3.2.4. Discussion

The ability of the model to match the set of experimental data with a single value of the unique adjustable parameter ε supports its physical relevance. The semi-empirical strategy that was developed here can therefore be used to progress in understanding the effect of pressure on electrolyser behaviour. The experimental data were obtained with an analytical set-up that involved a large inter-electrode distance. This choice did not reduce the applicability of the strategy, which was based on the difference (ΔU<sub>p</sub>) between cell voltages measured at given values of pressure minus the cell voltage at atmospheric pressure. The ohmic drop due to 1) the separator and 2) the part of the electrolyte not containing bubbles was therefore not involved in the data analysis and calculations. This strategy can therefore be applied to any electrolyser operating with freely rising bubble layers.

The condition of “freely rising bubble layer” is the most important limiting factor for the direct application of the results of this work. Actually, forced convection is generally imposed in industrial reactors, which should modify the bubble patterns and their response to pressure. Here, in contrast, the bubble behaviour was investigated without forced convection because this will be the most likely situation in the context of near-neutral pH electrolysers. As evoked in the introduction, the objective of this technology is to design small, easy-to-use electrolysers for niche applications. In this context, it is important to think in terms of the simplest possible set-ups and therefore to avoid the use of pumps in order to produce equipment that is as autonomous as possible, even capable of operating in remote areas, far from the grid.

The model gives the contribution of the various factors that affect the cell voltage when pressure increases. Fig. 5.D, plotted for a pressure of 3 bar, shows that the variation with pressure of the impact of bubble coverage was small, while most of the effect of pressure was on the ohmic component. If the objective is to optimize the energy savings that can be achieved by increasing the operating pressure, the model shows that the primary effort should be to manage the resistivity ρ<sub>ε</sub> of the diphasic part of the electrolyte. This conclusion was drawn here by using flat electrodes for the experimental work and, accordingly, using the correlation reported in the literature for flat electrodes [61]. However, this observation can be extended to the zero-gap electrode set-ups, which are often used in water electrolysis cells [62]. In this case, the membrane electrode assembly (MEA) is designed to minimize the contact time between the bubbles and the electrode surface. The bubble coverage of the electrode surface of an MEA should therefore be even less than on flat electrodes. As concluded here for flat electrodes, with an MEA set-up, the resistivity of the diphasic part of the electrolyte should remain the main mechanism by which bubbles affect the cell voltage.

The model also provides clues to understanding the slight rise in potential observed just after the valves were closed (Fig. 2.A and 2.B). The total effect of the pressure p on the cell voltage was the sum of the three contributions (Eq. (33)): ΔU<sub>thermo,p</sub>, which tends to increase the cell voltage, and ΔU<sub>ohmic,p</sub> and ΔU<sub>surf,p</sub>, which tend to decrease the cell voltage. Deriving the thermodynamic contribution (ΔU<sub>thermo,p</sub> Eq. (6)):

$$d(\Delta U_{\text{thermo,p}}) = \frac{3RT}{4F} \frac{1}{p} dp \qquad (34)$$

shows that the most rapid evolution of ΔU<sub>thermo,p</sub> with pressure is just at the beginning of the pressure increase at p = 1 bar. For instance, using a first order approximation shows that increasing the pressure from 1 to 1.1 bar should increase U by 3RT/4F 0.1 bar, while increasing the pressure from 2 to 2.1 bar should have half the effect: 3RT/4F 1/2 0.1. Accordingly, the thermodynamic effect was visible at the start of the pressure rise, just after the valves were closed (small increase to the cell voltage), but was then outweighed by the effect of the pressure on the bubbles.
<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Theoretical/experimental data. A) Average values of ΔU as a function of the applied current density at *p* = 2 bar; green squares: experimental data; dashed line: linear regression of the experimental data; red points: calculated values with ε = 0.46. B) Similar as A for *p* = 3 bar. C) Similar as A for *p* = 4 bar. D) Calculated contributions of the different parts of ΔU for *p* = 3 bar.</figcaption></figure>

<table><caption>**Table 1**

Experimental values of ΔU<sub>p</sub> with standard deviations and the number of independent measures in parenthesis.</caption><thead><tr><th colspan="5">ΔU<sub>p</sub> (mV)</th></tr><tr><th>J (A m<sup>-2</sup>)</th><th>2 bar</th><th>3 bar</th><th>4 bar</th><th></th></tr></thead><tbody><tr><td>500</td><td>-13 ± 5 (3)</td><td>-18 ± 9 (3)</td><td>-8 (1)</td><td></td></tr><tr><td>1000</td><td>-61 ± 29 (5)</td><td>-113 ± 23 (5)</td><td>-122 ± 12 (2)</td><td></td></tr><tr><td>1500</td><td>-93 ± 34 (5)</td><td>-186 ± 46 (5)</td><td>-256 ± 89 (4)</td><td></td></tr><tr><td>2000</td><td>-184 ± 87 (4)</td><td>-311 ± 82 (4)</td><td>-395 ± 92 (4)</td><td></td></tr></tbody></table>

The experimental data and the model confirmed the considerable effect that pressure can have in decreasing the cell voltage required to sustain a given current density. For instance, when operating at 2000 A m⁻², a gain of around 400 mV in cell voltage can be achieved by increasing the pressure to 4 bar, as compared to working at atmospheric conditions (Fig. 5.C). This effect increases with the applied current density, so that the gain is even if we analyse it in terms of consumed power. The power density that is saved by increasing the pressure from atmospheric condition to *p*:

$$ \text{Saved power density} = \Delta U_p J \quad (35) $$

is plotted in Fig. 6. The experimental data and the model are in agreement in showing the considerable energy savings that can be made by working at the highest possible current density. For example, at 3 bar, the power saving was multiplied by a factor of 2 by going from 1500 to 2000A m⁻².

The model can thus be a useful tool for determining an optimal operating pressure. Calculating the energy savings for high pressures (Fig. 6) showed that there is no longer any significant benefit in increasing the pressure beyond a certain value, which depends on the applied current density. For example, there is no interest to go beyond 5 bar with a current density of 500 A.m⁻². Furthermore, at this low current

<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Power density saved by increasing the pressure.</figcaption></figure>

density, the power savings remained limited. In contrast, at 2000 A m⁻², the saved power becomes considerable as the pressure increases and it seems possible to save even more power by increasing the pressure beyond 50 bar.

Assuming a Faradic yield of 100 %, the volumetric hydrogen production rate per unit electrode surface area (V̇), expressed in Nm³ h⁻¹ m⁻², is:

$$ \dot{V} = \frac{J 22.4 10^{-3}}{2F} 3600 \quad (36) $$

We can deduce the corresponding energy cost expressed in Wh per Nm³ hydrogen:

$$ \text{Energy cost} = U \frac{2F}{22.4 \times 3.6} \quad (37) $$

An electrolyser running at 2000 A m⁻² would produce 0.84 Nm³ h⁻¹ m⁻² hydrogen. Here, using the open-to-air analytical cell, applying 2000 A m⁻² required a cell voltage of 5.234 V (Fig. 3). The energy cost was
therefore around 12.52 kWh $Nm^3$. This value is considerably higher than the energy costs reported for the current commercial facilities of the order of 4 to 5 kWh per $Nm^3$ of hydrogen produced [63,64]. As evoked in the introduction, the low ionic conductivity of the electrolyte used at near-neutral pH is a major cause of the high power cost. In addition, here, the choice was made to use industrial electrode materials without specific catalyst. The large number of studies dealing with specific catalysts for HER [10] or OER [50] at near-neutral pH can provide great help in lowering the power cost.

Fig. 6 shows that, at 2000 A m⁻², a gain of the order of 600 W m⁻² was achieved by operating at a pressure of 3 bar instead of the atmospheric pressure, which corresponds to a gain of the order of 0.7 kWh per $Nm^3$. Furthermore, the model suggests that gains of up to 1.4 to 1.8 kW m⁻² may be reached by operating at 10 or 50 bar, respectively, which would correspond to gains of 1.7 and 2.1 kWh per $Nm^3$ of hydrogen produced. In conclusion, there are several ways of significantly reducing the cost of water electrolysis at near-neutral pH, and operating under pressure is one of them.

Finally, it must be recalled that the experiments and the numerical approach were carried out here with a system that allowed the bubbles to rise freely along the surface of the electrode. This design, including cathode and anode compartments large enough not to disturb the natural rise of the bubbles, was chosen to allow an easier theoretical approach in the context of the pioneering work presented here. A commercial electrolyser would have narrower compartments in order to decrease the ohmic drop. Even with a reduced inter-electrode distance, the general strategy designed here will remain valid, and notably the ways of calculating the thermodynamic ($\Delta U_{thermo,p}$) and surface ($\Delta U_{surf,p}$) contributions can be used without modification. However, in narrower compartments, the hydrodynamics driven by the bubbles may be different from that established here, because of the mechanical constraint due to the closeness of the separator.

The concept developed here, of a free rising bubble layer that expands with height, will have to be adapted to the hydrodynamic conditions. In many commercial reactors, an external pump forces the electrolyte circulation. In this context, an expanding layer of bubbles similar to ours has been observed or modelled [42,65,66]. The “free rising bubble layer” concept can be adapted easily to these cases by factoring in the appropriate average rising velocity as imposed by the electrolyte motion, in place of the Stokes’ limit velocity we used here.

Finally, it must be kept in mind that the concept of a free rising bubble layer is based on the hypothesis that the pressure affects only the thickness of the bubble layer (Scheme 2.D). By reducing the volume of the bubbles, the pressure reduced the thickness of the bubble layer, while the bubble layer kept the same gas fraction $\epsilon$ and, consequently, the same ionic conductivity $\rho_\epsilon$. The reliability of this assumption for the experimental cell used here was supported by the strong correlation between the calculations and the experimental data. Nevertheless, this part of the approach should be modified for narrower compartments, which would constrain free expansion of the bubble layer. We aim to continue our research in this direction, still combining the experiment approach with semi-empirical modelling.

## 4. Conclusion

This study establishes that operating water electrolysis at near-neutral pH under pressure has an intrinsic beneficial effect by mitigating the negative effect of the presence of bubbles in the electrolyte. A semi-empirical approach was designed, which perfectly correlated with the experimental data. It appears that considerable power savings are achievable by operating under a few bar of pressure, especially when the current density is of the order of a few kA m⁻².

The literature includes numerous basic studies dealing with isolated bubbles, horizontal electrodes, and analytical and microfluidic systems, but there is still a large gap between these accurate laboratory studies and the practical needs of an engineer who wants to optimize the

operating pressure of a water electrolysis cell, especially when operating at near-neutral pH. Here, we have noted four major contributions of pressure on cell voltage: thermodynamic effect, impact on the electrode structure (not dealt with here), decrease of the masking effect of the bubble coverage, and decrease of the resistance of the diphasic part of the electrolyte. We used the empirical correlation established by Vogt and Balzer and the concept of “actual average current density” to assess the contribution of the bubble coverage. The concept of a “free rising bubble layer” was developed to assess the contribution of the bubbles in the electrolyte.

The search for correlations or simple models, which was necessary for each step of the strategy proposed here, highlighted the lack of any established procedure that could be applied to electrolysis conditions approaching an industrial context. We hope that this work provides advances in this direction and will prompt the further research needed to optimize water electrolysis at near-neutral pH under pressure.

## CRediT authorship contribution statement

**Junzhe Sun:** Writing – original draft, Validation, Methodology, Data curation, Conceptualization, Investigation. **Luc Etcheverry:** Supervision, Methodology, Investigation, Conceptualization. **Alain Bergel:** Writing – review & editing, Writing – original draft, Validation, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This research was conducted within the framework of the “HyTEAM” joint laboratory co-operated by the Laboratoire de Génie Chimique (LGC, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France) and the company 6TMIC (www.6t-mic.com, Castanet-Tolosan, France), with financial support from the French Région Occitanie in the framework of the “Dispositif GRAINE”.

## Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.electacta.2025.146264.

## Data availability

Data will be made available on request.

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