A critical review of modeling hydrogen production using water electrolyzer

1.1 Background and significance

Since the Industrial Revolution of the 18th century, use of fossil fuels has severely polluted the atmosphere with greenhouse gases (GHGs). Fossil fuels have environmental damage that is already cause of ozone depletion, worldwide climate change, and the ecosystem due to nitrogen oxides (NO_x), carbon monoxide (CO), and carbon dioxide (CO₂). According to the Intergovernmental Panel on Climate Change (IPCC), CO₂ concentrations (e.g., in the Paris Agreement 2015) are to limit global warming and keep the average air temperature below 1.5 °C. Also, global average temperatures are expected to rise by 4.8 °C, and sea levels are expected to increase by 6.4 m by 2,100 unless emissions are controlled, and countries that follow net greenhouse gas emissions should be zero by 2050 (Church et al. 2013; Collins et al. 2013; International Energy Agency (IEA) 2019; Jang et al. 2021).

The use of hydrogen fuel as a sustainable alternative has reduced some of these concerns. Hydrogen is also considered an energy source of the future due to its minimal carbon emissions and the possibility of making it clean (Zakaria and Kamarudin 2021; Zakaria et al. 2021). Hydrogen system can be integrated with other renewable energy and system. In one such hybrid system designed for a geothermal-powered desalination plant, hydrogen storage showed its capability for large-scale energy supply, up to 14,920 kWh/yr. This formed part of a robust backup with batteries to minimize system shutdowns. The main desalination process itself had a high recovery rate of 55 % at a specific energy consumption of 4.78 kW/m³, proving that such integrated systems can be efficient (Hoseinzadeh et al. 2025). Nevertheless, its current usefulness is circumscribed by the lack of infrastructure and its significant cost. With the rapid growth in the use and development of production and storage technologies, costs are expected to decrease. Hydrogen production from electrical energy using renewable sources is needed for the sake of sustainability. Now with this switch, hydrogen will be a cheaper, greener, and cleaner source of energy. Recent techno-economic analyses confirm that underground hydrogen storage can become a profitable part of the energy grid, and optimization studies reveal substantial revenues can be made, for example, as high as $5.2 million per year from natural gas blending, which in turn provides the necessary flexibility and economic rationale to support such extensive demand (Y. Chen et al. 2024). This economic viability is further supported by real-world request. Applications of hydrogen in transportation and the energy sector alone are anticipated to expand, so it can be regarded as an energy source that may replace petroleum in vehicles. These trends demonstrate the considerable potential of the hydrogen economy (Tarhan and Çil 2021). The estimated growth in the hydrogen economy determines its transformative role in a variety of industries by 2050, with a total demand of ≅793 Mt. Power accounts for 30 % of the demand, transport follows at 23 %, heavy industry at 21 %, chemicals at 15 %, buildings at 7 %, and refining at 4 % (Future hydrogen uses and demand 2022). New technologies like hydrogen fuel cells, green ammonia manufacturing, and zero-emission vehicles back all these sectors. All these efforts are being spearheaded by corporations like CEMEX, H₂ Green Steel, and ZeroAvia, pointing towards the decarbonization shift of the world (Future hydrogen uses and demand 2022).

The most important challenge in using hydrogen as a fuel in the coming years is the cost of hydrogen production. With this in mind, water electrolysis/water splitting (Molaei 2024), photolysis (Bosu and Rajamohan 2024), fermentation (Lou et al. 2024), biomass gasification (Rubinsin et al. 2024), and reformers (Di Nardo et al. 2024) are the main strategies for hydrogen production. Even though steam methane reforming is the most commonly used technology for hydrogen production, this technique presents environmental problems, such as dependence on fossil fuels [e.g., methane and natural gas (Cho et al. 2022)], high carbon emissions due to the utilization of natural gas or other hydrocarbons as fuel, and catalyst deactivation.

A recent study (N. S. Hassan et al. 2024) showed that among the other methods of hydrogen production using renewable sources, water electrolysis is the leading method because of its clean and effective use of energy sources. This method also produces hydrogen with higher purity compared to the other methods mentioned. Despite these inherent advantages, commercial application of water electrolysis remains challenging in several respects: efficiency/performance, cost, durability, and scalability. In addition to these challenges, substantial R&D efforts are under way in two main complementary tracks: experimental research to advance the state of material science and component design, and sophisticated modeling techniques aimed at system optimization.

Experimental techniques, such as radiography and optical imaging can capture two-phase flow images (Selamet et al. 2013). But both radiography and optical imaging techniques often provide two-dimensional data, which can be insufficient for fully understanding the complex three-dimensional nature of two-phase flow. Modeling techniques such as numerical can provide there-dimensional information for two-phase flow. Performance of water electrolysis is affected by several factors such as the gas-liquid two-phase flow temperature, pressure, gas-liquid two-phase flow type in the electrolyzer, etc. In this purpose, realistic modeling of such systems should be developed for a powerful and scalable design of its performance and optimization (Jiang et al. 2023; Zhang et al. 2025).

With the existent technological pathways being so varied and the challenges in water electrolysis so persistent, there have been many review articles that have systematically and comprehensively discussed important aspects such as cost-reduction strategies, performance optimization, and comparative analyses among different electrolyzer technologies including but not limited to proton exchange membrane electrolyzer, alkaline Water electrolysis, and solid oxide electrolyzer cells. Following section will present a critical review of the existing reviews to synthesize their findings and identify research gaps clearly that this present comprehensive review aims at filling, especially in advancing integrated modeling approaches across several electrolyzer technologies.

1.2 Related surveys and present contribution

The recent review papers published in 2020–2024 are focused on the hydrogen production progress and evaluation of water electrolyzers. Niblett et al. (2024) reviewed and evaluated the progress made and persisting issues faced with respect to the creation of hydrogen using offshore wind energy via the process of electrolysis of water. Also, it can easily be scaled up and on the comprehension of the offshore conditions that affect production efficiency. Arsad et al. (2024) conducted a review to assess advancements in water electrolysis for hydrogen production, with a particular emphasis on bibliometric analysis, publication trends, and other significant technological developments in the field. El-Shafie (2023) evaluated several approaches toward producing hydrogen using electrolysis of water with special emphasis on hydrogen as a clean energy carrier devoid of carbon. It explains how proton exchange membrane electrolyzer (PEMWE) and alkaline Water electrolysis (AWE) technologies work by outlining their advantages and disadvantages. The anion exchange Membrane technology appears to be more feasible in terms of cost, while solid oxide electrolyzer cells (SOEC) are more efficient, although they work at elevated temperatures. Martinez Lopez et al. (2023) and Li et al. (2022) explained the dynamics, fundamentals, and progress of low-temperature water electrolyzers and their integration with photovoltaic systems, focusing on alkaline electrolyzers, low-temperature PEMWE, and anion exchange membrane water electrolyzer (AEMWE) technologies. N. S. Hassan et al. (2024) reviewed the advancements in green hydrogen production via water electrolysis, focusing on technological innovations, economic evaluations, environmental impacts, and the role of nanomaterials in enhancing efficiency and reducing costs. The review by Tarhan and Çil (2021) is a critical evaluation or assessment of a subject, mostly summarizing the existing literature, analyzing findings, and providing insights or recommendations from the material reviewed. Ham et al. (2024) provided a review of water electrolysis technology, the main goal being to reach hydrogen at a production cost of 1$/kgH₂. The authors covered system types and limitations, recent material developments, and future research toward efficient hydrogen production. Grigoriev et al. 2020) reviewed the status, research trends, and challenges of the present water electrolysis technologies, proton exchange membrane, and solid oxide electrolyzer, naming their performance, cost components, and future applications. A literature survey conducted from 2020 to 2024 focused exclusively on reviewing prior research on specific types of water electrolysis cells, as categorized in Table 1.

1.3 Motivation, innovation and contribution of this review

Although the literature reviewed offers the different way for developing water electrolyzer technologies, there is a significant gap in the approaches modeling. Previous reviews have focused largely on single-phase flow models, even though two-phase flow has been established as essential for accurately representing the complex phenomena within water electrolysis cells (Falcão and Pinto 2020). Also, some review papers were not covered in fundamentals and mentioned modeling, such as analytical, numerical, Artificial intelligence (AI), and empirical modeling. For instance, Shash et al. (2025) did not cover lattice Boltzmann modeling, empirical modeling, or governing equations in computational fluid dynamics (CFD) in a water electrolyzer. In another survey that was done by (Vedrtnam et al. 2025), although they pointed out some governing equations and modeling techniques, they did not cover empirical and lattice Boltzmann, two-phase modeling approaches. Also, AI approaches and design of experiments for empirical modeling were not pointed out by Busam et al. (2025). As a view, these gaps lead to an urgent need for a review that will bring modeling of different types of water electrolyzers together such as two-phase flow modeling with approaches such as CFD, lattice-Boltzmann method and AI, and then take the next step to provide practical guidance both on empirical correlations and analytical modeling. Also, this should be compared all of these methods with each other.

The purpose of the current study is to circumvent these limitations by reviewing and synthesizing a variety of modeling strategies in the field of water electrolysis research. Specifically, the following classes of models are examined:

Classical models: These are empirical, semi-empirical, and mechanistic models that rely on the fundamental, underlying electrochemical principles and fluid dynamics. Although these models are important, they typically converge to a level of detail that is too general to adequately model the details of two-phase flow and electrochemical coupling.

Artificial intelligence (AI)-based models: The use of machine learning (ML) and deep learning (DL) algorithms is continuing to expand for forecasting and optimization of water electrolyzer performance. This data-driven approach also has the potential to circumvent some of the limitations of the traditional approach, especially for nonlinear and high-dimensional systems.

Statistical models: Design of experiments (DoE) and meta-heuristic optimization techniques are employed in order to optimize operating conditions and the optimization of the system.

The purpose of this paper is to conduct a critical review of the robustness, limitations, and general adequacy of these modeling approaches to problems of water electrolysis system systems using a comprehensive and critical review of such modeling methods. In addition, it highlights the need to incorporate two-phase flow models into the simulation architecture in order to model the water electrolyzer behavior with as realistic detail as possible according to the field applications. In particular, these studies could be a basis for an improved reliability, performance and scalability of such models towards the development of water electrolysis technology for green hydrogen production.

1.4 Objectives and roadmap of the study

This paper presents a critical review of the different modeling schemes that have been used for water electrolyzers, with a specific focus on applications involving complex nonlinear systems. The main objectives of this review are including: 1) to comprehensive review and classify the modeling techniques applied to various water electrolyzers; 2) to critically investigate the challenges and advances in modeling the critical phenomenon of two-phase flow; 3) to analyze the emerging role of AI in both modeling and optimization; and 4) to review established optimization strategies, including the DoE for integrated systems and empirical equation based on correlations. To achieve these goals, the paper is organized in a manner that transitions the reader from basic concepts to state-of-the-art, coupled methodologies. Section 2 lays the groundwork by describing water electrolyzer types and configurations since the physical system dictates the dominant physical phenomena and hence the appropriate modeling approach. Section 3 explores canonical modeling approaches (empirical, semi-empirical, mechanistic).

The review follows this with more complex numerical techniques. Section 4 deals with the critical issue of the modeling of two-phase flow, discussing how computational fluid dynamics and the lattice Boltzmann method (LBM) offer high-fidelity insights at different scales but at a considerable computational cost. This sets the scene for Section 5, which describes how AI and machine learning are revolutionizing this field by developing data-driven surrogate models that can circumvent this computational bottleneck and give rapid optimization and control, albeit having their own demands in terms of extensive training data.

Finally, Section 6 synthesizes these modeling approaches by reviewing the optimization strategies leveraging them, from statistical techniques to meta-heuristic methods. Section 7 concludes by presenting a synthesized perspective on future research directions. Logical flow within the paper is depicted in Figure 1, showing this progression from fundamental principles through sophisticated, integrated analysis to a future outlook.

Figure 1:

Schematic representation of the review’s structure and methodology: (a) thematic roadmap and (b) sequential research process.

2.1 Different types of water electrolyzers

Water electrolyzers can be systematically categorized in several ways, as shown in Figure 2. The main classifications are electrolyte type, temperature, configuration, applications, and electrochemical process (electrochemical water electrolyzers and bioelectrochemical water electrolyzers). Other classifications of researchers can be determined as a Subcategory of water electrolyzers, which are included in the main classification, for example, based on the membrane (membranes (membrane-free) water electrolyzer and a common water electrolyzer with a membrane, also known as a membrane-based electrolyzer).

Figure 2:

The main classification of water electrolyzer based on electrolyte type, temperature, configuration, and application.

It should be noted that another classification is based on operative parameters such as temperature and pressure. Classification based on temperature includes low and high-temperature, for example, low-temperature PEMWE, and AEMWE; higsh-temperature, for instance, high-temperature PEMWE and SOEC (Chandrasekar et al. 2021). Classification based on pressure includes low-pressure electrolyzers and high-pressure electrolyzers (Dang et al. 2023) can produce hydrogen with a pressure of up to 12 MPa (Dang et al. 2022).

The scale of use is significant in choosing a suitable water electrolyzer for proper application, so large-scale [e.g., AEW, SOEC, and PEM (Mohammadi and Mehrpooya 2018)] and small-scale [i.e., integrated into renewable resources (RES (Tebibel 2021))]. Water electrolyzers can be classified. in addition to their configuration, they can either be single-cell electrolyzers (J. Wang et al. 2023) or stack electrolyzers. Both single-cell and stack electrolyzers have their advantages when considering the application requirements. Single-cell electrolyzers are most appropriate for smaller or experimental installations, while stack electrolyzers are preferable for large-scale industrial hydrogen production since these systems usually have greater efficiency and production capacity (Nguyen et al. 2019; J. Wang et al. 2023).

In electrochemistry, various electrolytes exist for industrial and experimental applications, including polymer electrolyte membrane (PEM), alkaline electrolyte, anion exchange membrane (AEM), and solid oxide (Figure 3). The water electrolyzer can be classified based on the type of electrolyte employed.

Figure 3:

Schematic of different types of water electrolyzers: (a) AWE, (b) SOEC, (c) PEMWE, (d) AEM. Reprinted from (Shiva Kumar and Lim 2022), under the CC-BY license.

2.2 Water electrolyzer configuration

2.2.2 Flow pattern

In electrochemical devices, e.g., SOEC and fuel cells, the geometry of gas flow passages plays an important role in the determination of the system performance. To get the optimum balance in terms of reactant distribution, heat removal, and mass transfer efficiency, various channel configurations, including serpentine, parallel, and interdigitated configurations, are employed. A two-phase mixture flow regime is the physical configuration of its two phases in the pipe or channel (Coskun Avci and Toklu 2022). The geometry of the system, the absolute and relative magnitudes of the two phases’ flow rates, and the interaction of the forces acting on the phases, in particular, all influence the flow regime that will occur under specific circumstances (Coskun Avci and Toklu 2022). Also, different flow field designs, such as parallel or serpentine, affect mass transfer and current distribution, with parallel flow fields that are important to design for the best performance (Zheng et al. 2023). The different path, which enhances the contact area between the gas and the electrode, and thus the reaction efficiency, at the cost of increased pressure drop. By contrast, parallel channels offer continuous flow gas and a very small pressure drop, and therefore are more suitable for situations where energy efficiency counts. Interdigitated patterns that steer gas access to the active electrode material significantly improve the ability to access reactant species and mass transport, but complicate the fabrication process. Hence, the choice of channel configuration is a critical dimension of water electrolyzer design, which dictates to what degree the efficiency, lifetime, and operational stability of the system are maximized. Studies have demonstrated that the maximum design of such architectures has the potential to yield significant performance improvements with regard to both the electrochemical performance and underscores the importance of channel geometry to the development of energy transduction fields (Chen et al. 2022; Su et al. 2024; Teuku et al. 2021; T. Zhou et al. 2024). The different flow patterns of the water electrolyzer, visualized in Figure 4a, and the gas flow direction in SOEC are illustrated in Figure 4b.

Figure 4:

Schematic representations of key electrolyzer design features. (a) Types of channel configuration (Chen et al. 2022; Su et al. 2024; Teuku et al. 2021; T. Zhou et al. 2024), (b) the gas-flow configurations for SOEC with counter-flow and cross-flow (Xu et al. 2017) (with Elsevier’s permission).

2.3 Fundamental challenges in water electrolyzer modeling

Researchers and engineers can guide design and optimization efforts by using accurate modeling to predict electrolyzer performance under a variety of operating conditions. While the choice of modeling approach depends on the specific electrolyzer type (Section 2.1) and exhibits sensitivity to system configuration (Section 2.2) that was discussed before. Several fundamental challenges also remain. Particularly, PEMWE can exhibit complex transient phenomena. These include voltage overshoots and fluctuations associated with dynamic operations. The intermittency and randomness of renewable energy supplies, which mainly come from wind or solar sources, further exacerbate these issues by undermining the operational efficiency and shortening the life expectancy of electrolyzers (He et al. 2024; Rauls et al. 2024; J. Wang et al. 2024).

Modeling of the basic electrochemical processes, particularly the sluggish kinetics of OER, is highly challenging and thus restricts efficiency in hydrogen production. It requires both thorough mechanistic insight and a well-developed approach to advanced electrocatalysts (Grimaud 2019; Li et al. 2022).

Conventional modeling approaches tend to oversimplify the critical components by assuming homogeneous porous transport and catalyst layers, thus ignoring the inherent spatial variability in real electrode structures. Also, variations in material properties, such as porosity distributions, significantly influence the internal pressure dynamics, species transport, and gas evacuation efficiency under high-load conditions (Bayat et al. 2025).

Most of the system-level transient models do not take into consideration the tightly coupled thermal and electrical dynamics, which has resulted in overestimation of response capability and can lead to equipment safety issues. Temperature and pressure setpoints are hard to maintain precisely, wherein deviations will have direct influence on the system efficiency and stability of operation during dynamic conditions (Rauls et al. 2024; Yuan et al. 2026).

Multi-scale integration requires to solve simultaneously from nanoscale bubble dynamics to macroscale system performance. This also includes interactions among coupled electrochemical reactions, multiphase fluid flow, heat transfer, and mass transport at disparate spatial and temporal scales (Wu et al. 2025; B. Xu et al. 2024; Zhang et al. 2025).

Operational conditions such as current density, water inlet temperature, and flow rates have profound effects on overpotential and efficiency, which are not easy to optimize (Wang et al. 2024; Zhang et al. 2025). Environmentally, added complications stem from impurities in feed water and the need for impurity-tolerant membranes, further complicating modeling accuracy and operational stability (Lindquist et al. 2020).

These challenges are cause of comprehensive classification framework enables a critical assessment of how different modeling paradigms overcome specific electrolyzer challenges, which it is presented in Section 2.4.

2.4 Modeling framework and classification

Water electrolyzer modeling represents an important activity toward advancing the technology of hydrogen production, central to resolving energy sustainability and environmental issues. Among the array of modeling approaches available, numerical methods stand out because they capture the detailed electrochemical, fluid, and thermal phenomena that occur during electrolysis. Unlike analytical models, which offer simpler solutions for well-defined problems, numerical methods can handle complex geometries, nonlinear behavior, and Multiphysics interactions inherent in water electrolyzer systems. In Figure 5, three provide a comprehensive overview of the different water electrolyzer modeling approaches categorized into numerical, analytical, semi-empirical, and artificial intelligence (AI). Table 2 provides examples in which each methodology contributes to emphasizing the merits/demerits and limitations of each methodology.

Figure 5:

Classification of water electrolyzer modeling.

Specific attention will be given to numerical methods, including the finite element method (FEM) (Fu et al. 2023), finite volume method (FVM), LBM (J. Liu et al. 2023; Zhang et al. 2024), and CFD (Church et al. 2013; Collins et al. which collectively enhance our ability to simulate the electrolysis process accurately. Two types of packages were used to simulate the CFD of water electrolyzer: open-source packages [e.g., OpenFOAM (Rho et al. 2020)] and commercial packages such as ANSYS Fluent, ANSYS CFX, COMSOL, etc., as well as the formulation in the past in the form of coding in programming languages such as Fortran (Han et al. 2015; Ni 2009; Ulleberg 2003), which is now less commonly used and is almost obsolete. This section aims to elucidate these diverse modeling strategies and highlight their significance in advancing the field of water electrolysis and optimizing the efficiency of renewable hydrogen production. A semi-empirical model, which represents a hybrid approach that combines theoretical principles with empirical observations, can be conceptually formulated using either differential or algebraic equations, thereby enabling the characterization of experimental outcomes in the assessment of the performance metrics associated with the electrolyzer (Ma et al. 2021). Currently, the development of water electrolyzer technologies necessitates a combination of both techniques at different stages of development, despite the notable distinctions between 1D analytical and CFD modeling techniques. The design of controllers, thermal management algorithms, comprehensive performance evaluation across the entire operational range, and design optimization depend heavily on computationally efficient 1D analytical models. In contrast, high-fidelity CFD models offer greater accuracy for validating configurations optimized based on 1D analytical models and 3D spatial analysis for geometric component optimization (Berasategi et al. 2024). The spatial discretization distinguishes the 0D, 1D, 2D, and 3D simulations. Complexity decreases and usually requires less computing power as the number of degrees of freedom increases. However, a 2D CFD model offers more information than a 1D model for intricate ejector flow-field analyses with geometric influences. In terms of non-rotationally symmetric influences, such as inlet and outlet piping a 3D CFD model offers even more information, but the 2D CFD model requires less computing power. For instance, the 3D CFD model has a 360-fold larger mesh than the 2D CFD model, which considerably lengthens the simulation time if the 3D CFD simulation considers only the rotationally symmetric portion and has one cell per degree angle (Singer et al. 2023). To shorten simulation time, complementary modeling approaches leads to the analytical methods that will be discussed in the following section.

3.1 Electrochemical model

The nature of water electrolysis is an electrochemical process. Using electricity, water molecules are electrochemically split into hydrogen and oxygen via a process known as water electrolysis. The two half-cell reactions comprise the cathodic hydrogen evolution reaction (HER) and the anodic oxygen evolution reaction (OER). Ionic conduction is permitted, but the separator prevents direct electrical contact between the two electrodes (Puranen et al. 2024).

In electrolysis, a power source generates a potential difference between the electrodes to start half-cell processes. Total cell voltage is divided into a set of overpotentials, including:

1. Ohmic loss or ohmic overpotential

2. Activation overpotential or activation loss

3. Concentration overpotentials, mass transfer loss, diffusion voltage (Sezer et al. 2025)

4. Reversible or open-circuit potential (V_con or η_mass)

The water electrolyzer’s static voltage response was modeled using this technique.

Cell voltage can be calculated as (Daoudi and Bounahmidi 2024):

where E_cell and E_rev are the total cell voltage and the reversible voltage of the electrolyzer, respectively. Also, ∑η represents the sum of overpotentials, including activation, ohmic, and concentration overpotentials.

3.2 Thermodynamic solution

3.3 Empirical correlations

The design of the experiment (DOE) is a statistical way to gain a formula from the regression process. The most common DOE methods are response surface methods [RSM (Duan, Xiang, et al. 2024)] and the Taguchi method (Ozdemir et al. 2023). The central composite design (CCD) is the most common method of DOE analysis. The experimental design utilizes the CCD approach, while the response surface model employs the genetic aggregation method. A multi-objective genetic algorithm can be attached to the CCD for optimization to achieve the optimal global solution. In this framework, the Pareto solution sets enhance the likelihood of superior individuals being passed on to subsequent generations (Duan et al. 2024). DOE entails using statistical techniques to model and optimize reaction variables to increase efficiency and lower expenses. Traditional DOE considers only one factor as a variable while keeping the others constant, leading to unexplored interactions between factors and incomplete representation of their effects on the trend. Moreover, a substantial amount of testing is needed, resulting in a time-consuming and costly procedure (Assareh et al. 2023). To tackle this obstacle, RSM was implemented for optimization. It involves using mathematics to create empirical models, utilizing a collection of statistical techniques. These initiatives focus on maximizing the reaction, which relies on several separate variables. An experiment involves a series of trials referred to as implementation. The input parameters were modified in every trial to identify the reasons for variations in response. RSM is trending in the studies of integrating systems involving water electrolyzers (Assareh et al. 2023; Rahimi-Esbo et al. 2024; Sun et al. 2022).

DOE utilizes a regression model analysis of variance (ANOVA) that is developed and implemented. The relationship between variables used in decisions and the functions used to achieve objectives is referred to as follows (Sun et al. 2022):

where m, c, x,y, ε, and m are objective functions, the coefficients, the decision variable, the statistical error, and the number of factors, respectively.

The empirical correlations gained by the researcher are presented in Table 3.

3.4 Critical discussion: analytical and empirical correlation in water electrolyzer modeling

Researchers created analytical solutions from complex mathematical models that incorporate different physical and chemical parameters, so they provide precision and comprehensiveness on how the system will behave. However, the main disadvantage to it is complexity and time in development. Requirement for validation against experimental data to be accurate. In contrast, it is empirical correlations such as DOE and RSM that provide a very easy and efficient way to optimize processes through statistical analysis of multiple factors and their interaction. It can also predict accurately with the experimental range, but is less accurate when extrapolating from this range. Also, it will depend on application requirements in choosing one of these methodologies, in terms of detail modeling versus efficiency in rapid optimization (Assareh et al. 2023; Ozdemir et al. 2023 2024; Rahimi-Esbo et al. 2024; Sánchez et al. 2018; Sun et al. 2022a). Finally, these inherent limitations, especially towards handling complex geometries and coupled multiphysics phenomena, bring out the need for advanced numerical modeling techniques. Indeed, methodologies such as computational fluid dynamics presently resolve these issues through discretization of the domain to obtain high-fidelity spatially resolved solutions. A detailed dissection of these numerical methods is presented in Section 4.

4.1 Computational fluid dynamics

Computational fluid dynamics as a numerical method can simulate water electrolyzers in a very almost accurate way. Simulations can also be solved taking into account electrochemical equations. So, these simulations required enough previous knowledge of the electrochemistry of the water electrolyzer, as well as the equations governing fluids and heat and mass transfer, including continuity, momentum, mass transfer, and energy survival. It’s also in research or solving fluids that researchers don’t solve electrochemical equations. However, they cannot find the function of the electrolyzer. Analytical solving is also merely solving electrochemical equations alongside thermodynamic laws, including the first and second laws of thermodynamics and the Gibbs law. Analytically, the problem is that they cannot see the current, or they cannot see the temperature, pressure, and speed distribution. So, the computational fluid dynamics method is the way to break these trends. So, to get acquainted with CFD in the electrolyzer, you need to know the simulation steps in CFD.

4.1.1 Steps of CFD modeling

Based on Figure 6, the analysis typically involves three essential stages in CFD simulations of water electrolyzers that are discussed in the following sub-sections.

Figure 6:

Steps of CFD modeling.

4.1.1.1 Pre-processing

To start with, there is a phase called pre-processing, which consists of setting up the simulation space and outlining the essential parameters to be used in the exercise. In this stage, several important steps include importing geometry, and mesh/grid generation [e.g., using ICEM and ANSYS meshing (Ma et al. 2021)], physical models selection, boundary conditions, initial conditions, and choosing material properties. The physical geometry of the water electrolyzer is made using computer-aided design (CAD) software [e.g., CATIA (Tijani et al. 2015), SolidWorks (Sánchez-Molina et al. 2021), AutoDesk Inventor (O’Neil et al. 2016)] or special modeling packages. This includes electrodes, electrolyte flow fields, and other system components if necessary. After importing/creating geometry, the modeling of the water electrolyzer is limited to the selection of electrochemical properties (i.e. conductivity, initial current density, etc.) and thermophysical properties of the materials (viscosity, thermal conductivity, convective heat transfer coefficient) used in its implementation, for this purpose parameters such as conductivity, viscosity, volume fraction, input current value, input mass flow rate/velocity, porosity value, etc. They should be defined according to the circumstances of the problem. The description of boundary conditions, namely inlet velocity, temperature, and concentration conditions, and initial conditions, referring to the status of the system at the beginning of modeling, is needed (Figure 7). This is the most important step regarding model behavior setting and response against any kind of input. In the step of physical model selection, in the process of populating the electrolyzer and its interiors with appropriate features, suitable physical models are utilized.

Figure 7:

A simple example for boundary conditions (Zhuang et al. 2024) (with Elsevier’s permission).

4.1.1.2 Processing (solving)

The physics in question, which is separated by meshing methods, is implemented and solved at this stage. This may include as:

1. Fluid dynamics (Navier–Stokes equations, Darcy equation, Darcy–Forchheimer equation)

2. Electrochemistry (Bulter–Volmer, charge conversion)

3. Two-phase model application (gas–liquid, mixture, volume of fluid)

4. Heat transfer (energy balance)

Depending on the dimension of the problem-solving domain, the governing equations for electrolyzer are often expressed as multidimensional models. To express the problem, such as all CFD simulations, requires assumptions that specify the scope of the problem. These assumptions can be categorized in Table 4.

Table 4:

Principles and manners in CFD simulations.

Principles	Manners	Application	References
Time dependency	Steady-state condition/Stationary	Neglecting time changing in time-invariant cases	Rodríguez and Amores (2020), Özdemir and Taymaz (2022)
	Unsteady/Transient	Solution till steady state time
Properties of materials	Homogeneity – Homogeneous – Heterogeneous	Porous layers are homogeneous and isotropic. Exist gases are incompressible and ideal gases	Zhang and Xing (2020)
	Direction-dependent properties – Isotropy – Anisotropy
	Fluid compressibility – Compressible – Incompressible
	Fluid gas – Ideal gas – Non-ideal gas
Simulation	Dynamic simulation	Simulation in startup time, and operation dynamics; analyzes cell voltage and energy consumption	Sakas et al. (2022), H. Liu et al. (2023), J. Yang et al. (2024)
	Static simulation	Simulation of both analytical and numerical for a regular behavior without dynamics	–
Domain	Single-dimensional domain – 1D	Simulates startup and operation dynamics; analyzes cell voltage and energy consumption in most cases in steady-state conditions	García-Salaberri (2022), Laoun and Kannan (2024)
	Multi-dimensional domain – 2D – 3D	Analyzing fluid dynamics and electrochemical phenomena in more detail	Lopata et al. (2022), Singer et al. (2023)
Flow regime	Laminar – Single/Double serpentine flow – Cascade flow – Bubbly flow – Slug flow – Annular flow – Microhannels flow in laminar flow	Evaluation performance impacts, improvement of mixing, enhancement of efficient gas evolution, and improved reactant distribution and use of various patterns in laminar flow	Li et al. (2018), Rho et al. (2020), Wong et al. (2021), Dang and Zhou (2024), Kim and Jung (2024)
	Transition – Menter’s γ – BC transition	Analysis between laminar and turbulence regime	–
	Turbulent – RANS equations with k-ε model – k-ω model – k-ε Euler-Euler model	Chaotic and mixed flow, pressure swings and gas bubble removal with turbulence	Bakker and Vermaas (2019), Zarghami et al. (2020), Gao et al. (2023)
Coupling	Multi-scale model	Integrating various modeling scales into a comprehensive electrolyzer model remains a challenge	Xu et al. (2024)
	Pseudo-coupling methods	including a pseudo-coupling method to account for the prediction of the effects of two-phase flow on transport phenomena and cell performance at the interface between the flow channel and the liquid/gas diffusion layer (L/GDL) for a novel double-layer flow	Xu et al. (2021)
	Electrochemical coupling	Understanding gas bubble dynamics and multiphase flow needs this coupling	Dreoni et al. (2022), Xue et al. (2024)
	Multiphysics coupling	Fluid dynamics, heat and mass transfer, electrochemical reactions coupling for comprehensive simulation of complex interactions, for instance, bubble dynamics, and flow characteristics	Dreoni et al. (2022)
	AI/ML and optimization algorithm coupling – Predictive model – Multiple inputs and Single output (MISO) Model – Optimization of operational parameters	Predict complex relationships between input parameters and output parameters with high precision and enable the fine-tuning of these parameters to achieve optimal performance and efficiency	Sirat et al. (2024), Y. Wang et al. (2024)
Phase flow type	Singe-phase flow – Single-phase modeling – Mixture modeling	Understanding internal fluid dynamics and optimization and investigation of pressure drops, velocity profiles, and concentration distributions in most cases of steady-state condition	Castañeda et al. (2017), Lopata et al. (2020), Jia et al. (2021), Özdemir and Taymaz (2022)
	Multi-phase flow – Eulerian-Eulerian two-fluid model – VOF model – Multi-phase CFD with AI/ML integration – Six-field two-fluid model – Euler-Lagrange approach	Accurate simulation of two-phase flow, improved performance through novel designs, enhanced understanding of bubble dynamics, optimized flow channel design, detailed phase distribution, improved physical interpretation	Wu et al. (2021), Lee et al. (2022), Zhou et al. (2023), Duan, Zhou et al. (2024), Mohamed Mohsin et al. (2024), Xue et al. (2024)
Electrochemical	On-electrochemical model	In multiphysics approaches and detailed phenomena, for instance, non-isothermal with 3D simulation and necessity of transport equation solving for cases such as gas, liquid phase, and electric current conversion	Dreoni et al. (2022), Lee et al. (2024a,b), Mohamed Mohsin et al. (2024), Saidi and Chekir (2024), X. Yang et al. (2024)
	Off-electrochemical model	For simple cases, simplification is necessary. For instance, specific component simulation
Heat transfer	Isothermal solution – Single-phase isothermal model – Two-phase isothermal model – Isothermal continuum model	Suitable for steady-state analyses and simpler systems and necessary for less computational time cost	Lopata et al. (2020, 2022), Özdemir and Taymaz (2022), Hassan et al. (2023), Y. Wang et al. (2024)
	Non-isothermal solution	Indispensable for detailed studies of performance, efficiency, and safety in dynamic systems
Crossover	With crossover – Hydrogen crossover – Oxygen crossover – Water crossover	Considering crossover phenomenon can reduce efficiency and pose safety risks due to potential flammability hazards	J. S. Lopata et al. (2021), Wrubel et al. (2022), Franz et al. (2023), Kim et al. (2024), Mohamed Mohsin et al. (2024)
	Without crossover	For the cases that need to be simplified the simulation
Solution	Implicit	Solution of the governing equation at the next time step is calculated directly from the known information at the current time step	Ni (2009), Lafmejani et al. (2017), Berasategi et al. (2024)
	Explicit	Solving governing equations for the unknowns at the next step often requires iterating to find a solution
	Semi-implicit	Combination of both explicit and implicit

The solution of pressure and velocity should be coupled by a solver such as SIMPLE, PISO, PIMPLE, etc. In the FVM method with Ansys Fluent, the PISO scheme for pressure-velocity coupling is used for SOEC flow solution by Liang et al. (Liang et al. 2023). This is initialized at the first step through a transient simulation with the steady-state solution at a given voltage. The ramp time is defined as the time during which the voltage linearly rises from the initial to a prescribed final value and remains constant until the end of the simulation time. The transient simulation continues until a new steady state has been reached or until the maximum simulation time is reached. Convergence criteria include: energy residuals less than the user-defined threshold, residuals for Y_H2 below a different user-defined limit, and the electrical potential residuals below another user-defined threshold value.

(19) Δ t n + 1 = Δ t ramp n ramp 0 ≤ t < t ramp Δ t n . MIN δ T ave T ave n − 1 − T ave n , δ Y H 2 , ave Y H 2 , ave n − 1 − Y H 2 , ave n t ramp ≤ t < t newsteady Variation of temperature per step : δ T ave Variation of hydrogen molde fraction per step : δ Y H 2 , ave Time of new steady : t newsteady Time ivision : n ramp

The ramp times investigated vary widely. It would be computationally prohibitive to cover such a wide range using a constant time-stepping strategy as this would involve an enormously large number of time steps for the simulation. Accordingly, an adaptive time-stepping approach is adopted to achieve an optimum balance between temporal resolution and computational efficiency. Firstly, at least ten-time steps are taken during the voltage change in order to properly capture the ramp.

The size of the next time step is then determined using the average temperature and hydrogen mass fraction at the current and previous time steps. This adaptive approach strives for a controlled change of temperature and hydrogen mass fraction per step. In this way, it is possible to carry out simulations with such a large time interval to completion with far fewer steps than would be required without such a time-step size algorithm (Liang et al. 2023).

4.1.1.3 Post-processing

At this point, it displays and analyzes the results obtained from the solution, including common CFD posts, TecPlot software, and other ways that researchers used. The steps, such as validation, grid independence study, illustrating contours, and parametric studies of water electrolyzer are for this section (e.g., Figures 8 and 9). One of the post-processing techniques that helps the reader to detect and follow the investigation is a parametric study. Most parametric studies are about polarization curves in water electrolyzers. The I–V curve can help the investigators recognize optimal regions in the range of input parameters. For example, in the stack management of water electrolyzer, boosting the stack temperature in PEMWE results in enhanced performance as current density improves with cell voltage.

Figure 8:

Numerical model verification procedure. (a) Example of validation (G. Li et al. 2024), (b) mesh independence based on the computing domain, (c) current density computed @1.5 V using different meshes; (d) water content computed @1.5 V using different meshes (Jiang et al. 2023) (with Elsevier’s permission).

Figure 9:

Different flow patterns for (a) current density, (b) volume fraction, (c) velocity, (d) transverse velocity (T. Zhou et al. 2024) (with permission from Elsevier).

4.2 Lattice Boltzmann method

The lattice Boltzmann method has become a helpful tool for simulating how different fluids flow through materials with porous media such as the porous transport layer of a water electrolyzer. LBM can use real images of the empty spaces found through special scanning techniques. This allows it to work well with very uneven pore shapes (Bhaskaran et al. 2022). LBM differs from traditional CFD by solving the Boltzmann equation on discrete grids instead of the Navier–Stokes equations. LBM functions at a level between microscopic and macroscopic by employing distribution functions to analyze macroscopic behaviors. Two primary frameworks, single relaxation time (SRT) and multiple relaxation time (MRT), are employed for representing collisional terms. Even though SRT-LBM may lack stability with low viscosity, MRT-LBM provides better stability and has a well-defined physical foundation (Dreoni et al. 2022).

4.3 Critical discussion CFD and LBM in water electrolyzer modeling

The comparison of both the LBM and classical CFD concerning their capabilities and differences in approach can show the application of these two fields, especially in their application to the research water electrolyzer. LBM uses a mesoscopic treatment, relying on kinetic theory, thereby enabling one to model complex boundary conditions and multiphase flows occupying microscale phenomena in the simulations at high resolutions. Hydrogen production processes can be modeled easily with particular methods along with micromorphological optimization of the catalytic layers for efficient electrochemical reactions. CFD, on the contrary, is a commercial method that follows the principle of macroscopic by solving Navier–Stokes equations but boasts of relatively well-established techniques and broad applicability in industry. CFD excels in general flow simulations in which some of those aspects are measured by heat transfer. However, because it relies solely on continuum mechanics, hash values often pose a challenge to studies that may need detail. LBM thus makes it favorable for particular optimization tasks to be carried out and require a higher level of detail, while CFD provides good strength for the wide-scale clear analysis and design of electrolyzer systems, giving an alone call toward an integrated understanding of fluid dynamics for water electrolyzer by highlighting the necessity of integrating both. LBM emerges excellently from the oxygen removal phenomenon from the anode PTL in the PEM electrolyzer owing to its mesoscopic nature which efficiently covers complex boundary conditions and microstructural features. It helps in modeling transport phenomena within highly intricate porous structures by providing high-fidelity results even at small length scales where conventional models falter. It would be critical for multiphase flow modeling behavior of oxygen bubbles during the different gas removal stages as well as for seamless integration with other models such as the immersed boundary method and the phase field model on bubble morphology and gas saturation. With this method, it is possible to analyze how wettability and porosity vary in the PTL concerning oxygen transport and improve the processes by which energy conversion efficiency is realized in the performance of electrolysis. The method is proven to be better than conventional methods because it simulates the variations in pressure and mass conversion due to reactions, thus making it very effective in optimizing oxygen removal methods in PEM electrolyzer (Q. Li, He, Zhang, Pan, et al. 2024; Q. Li, He, Zhang, Sun, et al. 2024; Sourya et al. 2024; Zhang, Guan and Yang 2024). However, CFD-based optimization requires the implementation of algorithms. This will be discussed in the next section.

5.1 Machine learning

Machine learning has a lot of algorithms and methodologies for predicting single-target or multi-target outcomes in water electrolyzers. The type of machine learning is for classification tasks or regression tasks. The water electrolyzer, because they do not need any labels, almost all regression models are more common. The process of machine learning to predict the output of the water electrolyzer included (a) feature extraction (b) data splitting (c) data scaling (d) performance evaluation criteria (e) ML architecture optimization (f) ML training evaluation (T. Wang et al. 2024).

In the feature extraction section, the data is extracted from a water electrolyzer modeling module or an experimental test of the water electrolyzer. The extracted data from the water electrolyzer is divided into a test set and a training set. Both training and test sets should be scaled, such as min/max data scaling or scaled between two numbers (e.g. between [-1,1] or [0,1]), which is optional in the setup of machine learning for water electrolyzer. This scaling will help the network to train more easily. The performance will be evaluated by the criteria (Alibeigi et al. 2024; Hayatzadeh et al. 2024; Shangguan et al. 2024).

1. Sum of squared errors:

1. Total sum of squares:

1. Mean-squared error:

1. Root-mean-squared error or root mean squared deviation:

1. Mean absolute error:

1. Mean absolute percentage error:

1. Determination factor, determination coefficient:

1. Adjusted R-squared:

In which, N, K, y _i , y ˆ i , and y ̅ are the number of samples, the number of predictor/independent parameters in the model, actual value/observed value, predicted value, and mean (yi), respectively.

The activation function is essential since it contains nonlinearity, and general nonlinearity enables machine learning to approximate complex functions. The following is listed among the activation functions of sigmoid, hyperbolic tangent (Tanh), and rectified linear unit (ReLU) (Wang et al. 2024). Among these activation functions, RELU is exploited more (J. Chen et al. 2024). Specifically, Table 8 concretely illustrates the application of the ML algorithm in the modeling of a water electrolyzer.

Most applications of statistical and artificial intelligence can be classified as:

1. Machine learning-based prediction

2. Machine learning-based optimization.

Many mathematical-statistical studies are conducted on water electrolyzers. Among these, optimization (López-Fernández et al. 2022), analysis of variance (ANOVA), uncertainty analysis (Karyofylli et al. 2024), profitability analysis (Lee et al. 2017), and sensitivity analysis (Muhsen et al. 2024; Sakas et al. 2024) are only a few of them. Yet increasingly important research on optimization has been and will be methodological. Optimization of water electrolyzers is being carried out in many applications, both as experiments and as modeling, both in numerical experiments and as analytic calculations. Because of the multitude of functions that have to be maximized and minimized, two optimization paradigms exist, i.e., single-object (single-target) and multi-object (multi-target) functions in electrolyzer design. Multi-objective optimization design encompasses experiment design, response surface modeling, and optimization algorithms (Duan, Xiang, et al. 2024).

5.2 Optimization with AI-aided

Optimization can be approached by AI. In this regard, to clarify the usage of AI-aided optimization in water electrolyzer optimization, a summary of the research methodology for using artificial intelligence to help optimize is given in Figure 10. First, there is a target/target to optimize. This requires communication between the input and output/output. To this end, first, the inputs (e.g., the effects of inlet water temperature on electrolyzer safety, reaction rate, and efficiency of hydrogen evolution, etc.) An analysis is performed using a computing platform (e.g., COMSOL software). In order to train the neural network and optimize the genetic algorithm, MATLAB/Python software was used with the data set obtained by the CFD program. To improve the three dependent variables, a set of predictive data sets was obtained, which provides a useful guide to improving the performance parameters of the water electrolyzer (Y. Wang et al. 2024). This optimization procedure is observed not in water electrolyzers, but in other devices, such as fuel cells, in the literature (Alibeigi et al. 2024).

Figure 10:

A flowchart example for employing artificial intelligence alongside of optimization (Y. Wang et al. 2024) (with Elsevier’s permission).

The evaluation of the regression capabilities of several submodels of ANN has been done by Chen et al. (J. Chen et al. 2024). The predictions made by these submodels are for the training, validation, and testing. The correlation value (R) reflects the relationship between the predicted values as well as the target values, where values close to 1 indicate a high correlation. The fitting accuracy of their model was over 0.95. Also, they were obtained from the training datasets while the general fitting accuracy was above 0.98.

X. Yang et al. (2024) 800 random porosity distributions, with 700 sets designated for training and 100 for system testing. The expected values are closely aligned with the computed ones, showing RMSE values of 0.18 %, 1.31 %, and 0.05 %. DeepONet accurately forecasts the electrolytic cell’s current density, temperature, and oxygen mole fraction. The temperature homogeneity factor improves as porosity increases. However, the minimum value of current density non-uniformity is reached at the average porosity equal to 0.5. Porosity has an effect on oxygen diffusion as well as conduction in the diffusion layer, thus affecting the internal electrochemical reactions and the transfer of electrons. The optimization of the temperature uniformity index is presented in Figure 11, section optimal, which shows lower voltage and energy consumption and current density non-uniformity index compared to the uniform porosity case. However, increasing temperature uniformity reduces the uniformity of current density. This is because of the high porosity at the entrance and low porosity at the upper left corner, which results in a non-uniform temperature distribution. A decrease in porosity is beneficial for the current transport, but not for the oxygen transportation, which leads to a smaller amount and not more uniformity in the distribution. It can be said that the temperature uniformity preferred leads to less porosity of the anode porous transport layer when compared to the case of current density uniformity preferred.

Figure 11:

Predicted current density, oxygen mole fraction, and temperature distributions, and optimal porosity for different average APTL porosities for (a) minimizing current density uniformity index, (b) minimizing temperature uniformity (X. Yang et al. 2024) (with Elsevier’s permission).

5.3 Optimization without machine learning

Before the machine learning method became common, optimization was by connecting the optimization algorithm to electrolysis cell simulation, whether simulated numerically or analytically. This optimization was done without using trained data and directly using the governing equations solved in the software. For example, Duhn et al. (2016) coupled with Monte–Carlo optimization with CFD. This type of optimization is still observed in research. Zhuang et al. (2024) did not use a predictive model for optimization. They used a genetic algorithm by coupling MATLAB with COMSOL. One of the aims of optimization is to give an optimal performance. One way is to add a blockage to the flow channel without a meta-heuristic algorithm (Yuhao Xu et al. 2024b). Another way is to simulate another type of flow pattern or channel to reach optimal performance which was discussed before.

5.4 Critical discussion: AI and optimization in water electrolyzer modeling

Techniques such as machine learning and artificial neural networks can be used to optimize electrolyzer performance and efficiency and increase the accuracy of the prediction, decrease time and experimental cost, the management of high-dimensional data, and, accordingly, experimental research. For instance, to support the XGBoost-based application and the Sub-model additivity with SHapley Additive exPlanations (SHAP) method, the implementation produced 67.9 % lower time costs, and the output matched the case of full-variable optimization (Y. Zhang et al. 2024). Conversely, traditional optimization algorithms are always a hands-on, cumbersome, iterative, trial-and-error-based, inaccurate, and non-predictive way that involves a vast amount of experimental data and is thus costly. The review demonstrates that although AI appears as the promise of the future of optimization of water electrolyzers, traditional methods are surprisingly not capable of this purpose due to their iterative nature, but also because they are not predictive. Nonetheless, further studies are needed in order to better accommodate AI for this new realm and evaluate its potential broad applicability in many of the technologies considered in electrolyzer fabrications towards the ultimate goal of sustainability in hydrogen generation (Hayatzadeh et al. 2024; Ozdemir and Pektezel 2024; Yuhao Xu et al. 2024a; Y. Zhang et al. 2024).

6.1 Challenges in modelling

Due to the complex interactions between physical and chemical phenomena, the modeling of water electrolyzers faces severe challenges. Such challenges vary depending on the electrolyzer technology and directly impact the ability to optimize performance, scale systems, and interface with renewable energy sources. For PEMWE, one of the main drawbacks is the utilization of oversimplified thermodynamic models that fail to comprehensively quantify the effect of temperature and pressure fluctuations on efficiency factors. Such deviations are critical under real industrial conditions, and without being able to capture them accurately, models are unable to predict system behavior or optimize performance under varying conditions. Furthermore, electrochemical models that are presently available do not contain real-gas behavior equations needed to describe high-pressure operating conditions. This absence limits the accuracy of performance predictions and impedes the design and control of PEMWEs used in high-pressure hydrogen generation. AWE possesses a separate array of model challenges. To begin with, their inherent design to operate on low current densities results in generating bulk, lower-density designs, and therefore it’s problematic to up-scale and increase connected infrastructure costs. It’s still difficult to model the interface between electrolyte characteristics, electrode geometry, and operational parameters for increased current density enhancement. Secondly, alkaline systems exhibit sluggish dynamic response due to high ohmic losses in the liquid electrolyte and diaphragm, and also gas crossover effects between electrodes. These phenomena reduce system efficiency and hinder rapid adaptation to varying power inputs, especially from variable renewable energy sources. Currently, comprehensive dynamic models capable of including all the concerned physicochemical processes, including gas crossover and two-phase flow, are not available. New technologies also create important modeling problems. SOECs are high-temperature systems that require coupled models that include electrochemical reactions, heat transfer, and thermal stress. Thermal gradients within SOECs can cause material degradation, such as electrode delamination, that cannot be well-represented with existing models. SOECs also prefer reversibility in their operation as fuel cells and as electrolyzers, which demand dynamic transient models in order to represent their reversible operation modes. Capturing degradation mechanisms like nickel oxidation and ceramic cracking remains a persistent modeling shortcoming. AEMWEs suffer from model uncertainty in ion transport, particularly hydroxide ion mobility at fluctuating humidity and pH. Catalyst stability prediction in operation is challenging, particularly under high current density where dissolution or oxidation may occur. Furthermore, membrane durability towards mechanical stress and gas crossover, particularly under low hydration, is not well parameterized in existing models.

6.2 Research gaps

In the modeling of water electrolyzers, several gaps should be considered for future direction. To solve these issues, researchers could develop novel ideas that could not be modeled before. For instance, multiphysics and multiscale models need to be developed that integrate multiphysics and multiscale paradigms to reproduce complex interactions in electrolyzers from the microscopic level of bubble dynamics to macroscopic system response. For example, CFD modeling of two-phase flow in an alkaline water electrolyzer targets the issue of simulating bubble curtain spreading, which indicates a lack of adequate CFD models for two-phase flow. This is important in generating realistic performance forecasts under varied conditions (R. Ding et al. 2022; Martinez Lopez et al. 2023). Dynamic operation models are another gap in recent studies. with transient renewable energy inputs are required by the transience of resources like solar and wind. It has been indicated a requirement for models that can handle the swift dynamics, highlighting the lack of such models in the literature (Rocha et al. 2024). Models should be coupled with power-to-X technologies (Power-to-gas, power-to-heat, power-to-mobility) to optimize system performance in different applications. It had been pointed out that the lack of modeling frameworks for these integrations was a critical gap (Cozzolino and Bella 2024).

6.3 Future research direction

The future development of water electrolyzer modeling will be driven by two primary forces: the urgent need for green hydrogen production and rapid advancements in computational methods. Trends indicate a paradigm shift in how researchers simulate and design electrolyzers, with three areas of innovation that are essential.