Multiphysics Modeling and Performance Optimization of CO₂/H₂O Co-Electrolysis in Solid Oxide Electrolysis Cells: Temperature, Voltage, and Flow Configuration Effects

Abstract

This study developed a two-dimensional multiphysics-coupled model for co-electrolysis of CO₂ and H₂O in solid oxide electrolysis cells (SOECs) using COMSOL Multiphysics, systematically investigating the influence mechanisms of key operating parameters including temperature, voltage, feed ratio, and flow configuration on co-electrolysis performance. The results demonstrate that increasing temperature significantly enhances CO₂ electrolysis, with the current density increasing over 12-fold when temperature rises from 923 K to 1423 K. However, the H₂O electrolysis reaction slows beyond 1173 K due to kinetic limitations, leading to reduced H₂ selectivity. Higher voltages simultaneously accelerate all electrochemical reactions, with CO and H₂ production at 1.5 V increasing by 15-fold and 13-fold, respectively, compared to 0.8 V, while the water–gas shift reaction rate rises to 6.59 mol/m³·s. Feed ratio experiments show that increasing CO₂ concentration boosts CO yield by 5.7 times but suppresses H₂ generation. Notably, counter-current operation optimizes reactant concentration distribution, increasing H₂ and CO production by 2.49% and 2.3%, respectively, compared to co-current mode, providing critical guidance for reactor design. This multiscale simulation reveals the complex coupling mechanisms in SOEC co-electrolysis, offering theoretical foundations for developing efficient carbon-neutral technologies.

1. Introduction

The urgency of global climate change and carbon neutrality goals has driven the exploration of efficient carbon dioxide (CO₂) resource utilization technologies [1]. Traditional carbon capture and storage (CCS) strategies face challenges related to long-term storage risks and economic viability [2,3], while carbon capture and utilization (CCU) technologies—which convert CO₂ into high-value fuels and chemicals (such as syngas, methane, and methanol)—demonstrate significant potential [4]. Among various CCU approaches, electrochemical catalytic CO₂ reduction has become an international research focus due to its advantages of mild reaction conditions, controllable products, and direct coupling with renewable electricity. It is worth noting that, in recent years, metal–organic frameworks (MOFs) have attracted widespread attention as porous crystalline materials. Owing to their exceptionally high surface areas, tunable pore sizes and active sites, and outstanding CO₂-adsorption/enrichment capability, MOFs offer a unique platform for electro-, thermo-, and photocatalytic CO₂ reduction. The latest review by Ahmed et al. [5] systematically summarizes the breakthrough advances achieved with MOFs and their derivatives in electro-, thermo-, and photocatalytic CO₂ conversion—as well as in photoelectrocatalytic and photothermal/electrothermal CO₂-reduction technologies—and clarifies how structure–activity relationships guide catalyst design. This study further corroborates the critical value of micro-structurally tunable catalytic materials for enhancing CO₂ activation and selectivity in high-temperature electrolysis systems. Solid Oxide Electrolysis Cells (SOECs) [6,7,8,9], with their unique high-temperature operation (700–1000 °C), show revolutionary potential for efficient CO₂ electrolysis and CO₂/H₂O co-electrolysis. Compared to low-temperature electrolysis technologies [10,11] (e.g., alkaline and proton exchange membrane electrolyzers), SOECs’ high-temperature environment significantly: (1) Reduces reaction activation energy; (2) Enhances ionic conductivity; (3) Improves electrode kinetics; (4) Provides an ideal platform for thermochemical synergy reactions (such as the water–gas shift reaction), thereby substantially increasing energy conversion efficiency and product selectivity. Through electrolysis of H₂O to produce H₂ and CO₂ to produce CO, SOECs can convert surplus solar and wind energy into storable chemical energy. The resulting H₂ and CO can be catalytically converted into methanol and gasoline for direct use in transportation networks, establishing SOECs as key technology for seasonal energy storage [12]. When functioning as reversible solid oxide cells (R-SOCs), they generate electricity in fuel cell mode (SOFC) and perform electrolysis in SOEC mode. Therefore, advancing SOEC technology is critical for achieving carbon peak and neutrality goals, significantly contributing to sustainable societal development.

SOEC technology originates from its reverse process—the Solid Oxide Fuel Cell (SOFC) [13,14]. While SOFC directly converts the chemical energy of fuels (such as H₂ and CO) into electricity [15,16], SOEC uses electrical energy to drive high-temperature electrochemical reactions that decompose H₂O or CO₂ into H₂, CO, or syngas (H₂ + CO). This technological commonality manifests in highly consistent core components: (1) The electrolyte, typically yttria-stabilized zirconia (8YSZ; e.g., (ZrO₂)₀.₉₂(Y₂O₃)₀.₀₈ in this model), exhibits excellent oxygen ion (O²⁻) conductivity at high temperatures; (2) Porous electrodes, where the cathode (fuel electrode) commonly employs Ni-YSZ cermet, and the anode (oxygen electrode) typically utilizes perovskite materials (such as LSCF); (3) The triple-phase boundary (TPB): This electrode-electrolyte-gas interface serves as the central site for electrochemical reactions.

In the late 1960s, Spacil [17] began using ZrO₂ as an electrolyte for water electrolysis to produce hydrogen, conducting preliminary studies on SOEC thermodynamics and kinetics. Subsequently, the consecutive oil crises of 1973 and 1979 [18,19] intensified global focus on sustainable energy technologies, spurring renewed research in the early 1980s. When SOECs perform co-electrolysis by simultaneously introducing CO₂ and H₂O, their advantages become particularly prominent: (1) Enhanced energy efficiency: Under high-temperature conditions (>800 °C), electrical energy demand is significantly lower than in low-temperature electrolysis, with thermal energy substituting partial electrical requirements (ΔG < ΔH); (2) Flexible product control: By adjusting feed ratios (CO₂:H₂O), temperature, and voltage, the syngas H₂/CO ratio can be precisely tuned for Fischer-Tropsch or methanol synthesis; (3) Reaction synergy: Three key cathode reactions occur: H₂O electrolysis, CO₂ electrolysis, and the water–gas shift reaction (WGSR). The WGSR rapidly reaches equilibrium at high temperatures, dynamically optimizing product distribution. Virkar [20] developed an oxygen-electrode kinetics model for SOECs based on local thermodynamic equilibrium concepts, analyzing oxygen chemical potential and partial pressure near electrodes in both SOFC/SOEC modes using equivalent circuit modeling. Zhang [21] established a steady-state model for oxygen electrochemical potential distribution in 8YSZ-electrolyte SOECs, analyzing one-dimensional distributions of oxygen, electron, and oxygen ion electrochemical potentials plus oxygen partial pressure within the electrolyte. Zhang [22] created a Weibull-theory-based numerical model simulating LSM oxygen electrode degradation kinetics, finding increased current density and operating temperature both reduce electrode lifespan. Yashima [23] investigated phase diagrams of doped ZrO₂ (including YSZ and ScSZ) and conductivity-temperature relationships. ZrO₂ conductivity is strongly influenced by dopant elements and concentrations, with peak ionic conductivity occurring at specific doping levels. Common fuel electrode materials include Ni-YSZ cermet, which exhibits excellent catalytic activity for hydrogen oxidation (SOFC mode) and hydrogen evolution (SOEC mode). Hubert [24] conducted 1000–9000 h tests at 750 °C and 850 °C in both SOFC/SOEC modes. Results showed significant Ni particle coarsening within 2000 h at 850 °C, reducing TPB length and decreasing the Ni-to-gas specific surface area ratio by ~40%, while the Ni-YSZ contact area remained stable. The YSZ network inhibits Ni agglomeration—a finding corroborated by Chen-Wiegart [25]. Chen [26] observed nanoscale ZrO₂ particles forming on Ni grains in Ni-YSZ fuel electrodes during 800–850 °C water/co-electrolysis. Udagawa [27] simulated SOEC stack temperature responses to current density step changes using a 1D model, proposing temperature control strategies via air excess ratio and gas inlet temperature adjustments. Wang [28] analyzed this strategy’s applicability in SOEC co-electrolysis through 3D continuum modeling. Xing [29], when optimizing SOEC hydrogen system steady-state operations, incorporated thermal properties (stack temperature/gradients) as efficiency optimization constraints, identifying maximum hydrogen production points at different load powers. Recent progress has expanded the SOEC research landscape in several complementary directions: (1) Flow-configuration and thermo-mechanical analysis. Liu et al. [30] used finite-element modeling to explore how co-current, counter-current, and cross-flow arrangements influence temperature fields and thermal stress in planar SOECs, providing guidance for robust stack design. (2) Lifetime-oriented system studies. Beyrami et al. [31] developed a comprehensive framework that predicts performance and degradation of SOEC stacks under galvanostatic and potentio-galvanostatic modes, showing that lifetimes beyond 50,000 h are attainable when operating windows are optimized. (3) Catalyst-integrated co-electrolysis. Błaszczak et al. [32] demonstrated high-temperature CO₂/H₂O co-electrolysis combined with direct methanation over Co-impregnated SOECs, revealing a beneficial bimetallic synergy between Co and Ni for in situ fuel upgrading. (4) High-resolution diagnostics. Zaghloul et al. [33] employed embedded fiber-optic sensors to obtain centimeter-scale temperature profiles across operating solid-oxide cells, furnishing valuable experimental data for validating coupled thermal and electrochemical models.

Despite SOEC co-electrolysis’s promising prospects, its industrialization faces complex challenges due to strong multi-physics coupling: (1) Electrochemical-mass transfer coupling: Gas diffusion within porous electrodes (governed by porosity ε = 0.4 and tortuosity ξ = 3) must satisfy reactant supply and product removal requirements; (2) Thermal management issues: Temperature gradients induced by Joule heating, electrochemical reversible heat, and activation loss heat affect reaction equilibrium and material stability (e.g., thermal stress cracking); (3) Competitive reaction mechanisms: At the cathode, CO₂ and H₂O compete for adsorption sites, with their electrolysis current density regulated by local potential η_m and gas partial pressures; (4) Sensitivity to operating conditions: Minor variations in temperature (T = 1073 K), voltage (V = 1.5 V), and feed composition (xCO₂ = 0.5, xH₂O = 0.498) significantly impact product distribution and current density profiles.

Traditional experimental methods struggle to analyze these multi-scale, multi-physics coupling processes in real time. Numerical simulations have thus become essential for optimizing SOEC design—particularly the COMSOL Multiphysics platform based on the finite element method. This integrated environment solves: (1) Charge conservation equations; (2) Mass/momentum transport equations; (3) Energy equations; (4) Electrochemical kinetics; (5) Homogeneous reactions. Luo [34] developed a 2D model analyzing H₂O/CO₂ co-electrolysis performance in tubular SOECs, revealing that the reverse water–gas shift reaction significantly influences co-electrolysis outcomes. Chen [9] demonstrated that increasing pressure enhances CH₄ conversion rates up to an optimal 3 bar; beyond this threshold, only temperature reduction further improves conversion. Chi [35] compared water electrolysis technologies including Alkaline water electrolysis, Proton exchange membrane water electrolysis, Solid oxide water electrolysis, and Alkaline anion exchange membrane water electrolysis. Zhao [36] employed 3D modeling to study metal foam applications in SOECs, showing gas conversion rates increase from 72.21% to 76.18% at 10,000 A/m² when using metal foam flow fields. Zhang [37] designed a novel flow channel and validated it through COMSOL simulations against conventional designs. Results confirmed the new channel significantly enhances electrolysis performance while maintaining structural simplicity, accelerating its commercialization potential.

Despite considerable research on SOEC models, significant limitations persist. Most models either neglect water–gas shift reaction (WGSR) kinetics or assume equilibrium conditions, thus underestimating its impact on product selectivity. One-dimensional models fail to capture concentration gradients, temperature distributions, and current density variations along flow channels. The interactive effects of temperature, voltage, feed ratios, and flow configurations (co-current vs. counter-current) remain insufficiently studied, particularly lacking quantitative analysis of enhancement mechanisms in counter-current operations. This work develops a high-fidelity 2D multiphysics SOEC co-electrolysis model to reveal coupling mechanisms within the reaction network. The model integrates kinetics of H₂O electrolysis, CO₂ electrolysis, and WGSR to quantify each pathway’s contribution. It analyzes parameter influence patterns including: (1) Temperature’s nonlinear regulation (923–1423 K) of CO/H₂ selectivity and current density threshold effects; (2) Voltage’s synergistic effects (0.8–1.5 V) on reaction rates and WGSR activity; (3) Feed ratio steering (CO₂:H₂O = 2:8 to 8:2) of product distribution. Innovatively, this study verifies counter-current advantages by establishing through full-scale modeling—for the first time—that counter-current configuration increases H₂ yield by 2.49% and CO yield by 2.3%, providing critical theoretical support for reactor design. Furthermore, it guides material/structural optimization by visualizing local overpotential, temperature, and concentration fields to identify performance-limiting regions such as cathode concentration polarization zones.

2. Solid Oxide Electrolysis Cell Co-Electrolysis Model

The structure of the 2D model of Solid Oxide Electrolysis Cell Co-Electrolysis described in this paper is illustrated in Figure 1. During the SOEC co-electrolysis process, a mixture consisting of H₂O, CO₂, H₂, CO is supplied to the inlet of the SOEC cathode, while air is simultaneously supplied to the anode channel. The reversible water gas shift reaction (WGSR, Equation (1)) is considered to potentially occur in the porous cathode layer. In the reaction, H₂O and CO₂ molecules diffuse through the porous cathode to the triple phase boundary (TPB) at the cathode-electrolyte interface, resulting in the production of H₂ and CO, as well as O²⁻ (Equations (2) and (3)). The generated O^2- are transported through the solid electrolyte to the triple phase boundary (TPB) at the anode-electrolyte interface, where they lose electrons to form O² (Equation (4)).

$$CO+H_{2}O\rightleftarrows {CO}_2+H_2$$

(1)

$$H_{2}O+2e^{-}\to H_2+O^{2-}$$

(2)

$${CO}_2+2e^{-}\to CO+O^{2-}$$

(3)

$$2O^{2-}\to O_2+4e^{-}$$

(4)

In this study, a 2-D Solid Oxide Electrolysis Cell Co-Electrolysis model has been developed based on the working principles to investigate the heat and mass transfer, as well as electrochemical reactions during the co-electrolysis of H₂O and CO₂. The model includes the water electrolyzer model, the heat transfer in solids and fluids model, and the electrochemical heating multi-physics model coupling electrochemical and thermal processes. It allows for calculating current density distribution at given operating potentials and simulating the reversible water gas shift reaction process. The model was built using the COMSOL Multiphysics software to simulate the electrochemical, heat transfer, and mass transfer processes of Solid Oxide Electrolysis Cell (SOEC) co-electrolysis of CO₂ and H₂O for performance analysis. The main parameters of the model are shown in

Table 1. List of main parameters for the Solid Oxide Electrolysis Cell Co-Electrolysis Model.

Parameter	Value
Cathode thickness (Cathode-supported) dc (um)	500
Electrolyte thickness, L (mm)	100
Anode thickness, da (mm)	100
Height of gas flow channel (mm)	1
Length of the planar SOEC (mm)	20
Thickness of interconnect (mm)	0.5
Electrode porosity, ε	0.4
Electrode tortuosity, $\xi$	3
Gas permeability (m2)	1 × 10−10
Operating pressure, P (bar)	1
Operating temperature, T (K)	1073
Exchange current density for H2O electrolysis (A/m2)	1
Exchange current density for oxygen reaction (A/m2)	1
Exchange current density for CO2 electrolysis (A/m2)	1
Inlet mass flow rate (kg/s)	0.0005
Molar fraction of inlet H2O	0.498
Molar fraction of inlet CO2	0.5
Molar fraction of inlet H2	0.001
Molar fraction of inlet CO	0.001

. The electrolyte of the Solid Oxide Electrolysis Cell (SOEC) is composed of 8YSZ, a solid solution comprising zirconia (ZrO₂) and yttria (Y₂O₃), with a chemical formula expressed as (ZrO₂)0.92-(Y₂O₃)0.08. Arrhenius plots of the electrical conductivity of 8YSZ are shown in Figure 2.

2.1. Electron Electrolyte Phase

In solid oxide electrolysis cells (SOEC), electron conduction primarily takes place within the anode and cathode materials. This relationship between current density, electric potential, and effective electric conductivity can be represented by the theory of electron conducting phases, expressed as:

$$i_s=-{\sigma }_{s,\mathrm{e}\mathrm{f}\mathrm{f}}\nabla {\varphi }_s$$

(5)

Here, i_s denotes the current density in the phase (A/m²), ${\sigma }_{s,\mathrm{e}\mathrm{f}\mathrm{f}}$ represents the effective electric conductivity (S/m), and ${\varphi }_s$ is the dependent variable for electric potential (V).

The charge balance equation in the phase is

$$\nabla \cdot i_s=-i_{v,\mathrm{t}\mathrm{o}\mathrm{t}}$$

(6)

where $i_{v,\mathrm{t}\mathrm{o}\mathrm{t}}$ (A/m³) is the sum of a volumetric current density contributions of the electrode reactions occurring in the gas diffusion electrode domains.

For current collectors and gas diffusion layer domains,

$$\nabla \cdot i_s=0$$

(7)

For solid oxide electrolysis cells (SOEC), the electrolyte is typically a solid oxide material, such as yttria-stabilized zirconia (YSZ). In SOECs, the electrolyte layer is located between the anode and cathode, responsible for conducting oxygen ions (O²⁻) during the electrolysis process.

When an external voltage is applied in SOECs, oxygen ions within the electrolyte layer are released at the anode and transported through the electrolyte layer to the cathode. At the cathode, oxygen ions combine with electrons, undergoing oxygen reduction reactions to produce oxygen gas. The primary function of the electrolyte layer is to ensure the efficient transport of oxygen ions between the anode and cathode while preventing gas mixing. Therefore, the electrolyte plays a crucial role in SOECs by facilitating the transport of oxygen ions and enabling electrochemical reactions to occur.

To avoid the common assumption of constant conductivity, the present model employs a temperature-dependent mixed ionic–electronic conductivity,

$$\sigma \left(T\right)={\sigma }_{0}exp\left(-E_a/RT\right)$$

(8)

The prefactor ${\sigma }_0$ and activation energy $E_a$ are fitted to four-point-probe measurements. This relation is implemented in COMSOL via the General PDE interface.

The current density ${\mathbf{i}}_l$ (A/m²) in the electrolyte is expressed as:

$${\mathbf{i}}_l=-{\sigma }_{l,\mathrm{e}\mathrm{f}\mathrm{f}}\nabla {\varphi }_l$$

(9)

where ${\sigma }_{l,\mathrm{e}\mathrm{f}\mathrm{f}}$ is the effective electrolyte conductivity (S/m).

The charge balance equation in the electrolyte phase is defined as

$$\nabla \cdot {\mathbf{i}}_l=i_{v,\mathrm{t}\mathrm{o}\mathrm{t}}$$

(10)

For domains such as membrane, separator, and gas-electrolyte compartment:

$$\nabla \cdot {\mathbf{i}}_l=0$$

(11)

2.2. Electrochemical Reaction Model

The electrochemical reaction rate in SOEC can be described by relating it to the activation overpotential, denoted as ${\eta }_{\mathrm{m}}$, for an electrode reaction indexed as m.

$${\eta }_m={\varphi }_s-{\varphi }_l-E_{\mathrm{eq}, m}$$

(12)

where $E_{\mathrm{eq}, m}$ denotes the equilibrium potential for reaction m.

For Equation (1), we need to consider the relationship between CO and CO₂. This relationship can be represented by Equation (3) and its reverse equation. We can then use the Nernst equation to calculate the equilibrium potential.

$$E_{\mathrm{CO}+{\mathrm{H}}_2\mathrm{O}}=E_{\mathrm{CO}+{\mathrm{H}}_2\mathrm{O}}^0-\frac{RT}{2F}\ln \frac{p_{{\mathrm{CO}}_2}{p}_{{\mathrm{H}}_2}}{p_{\mathrm{CO}}{p}_{{\mathrm{H}}_2\mathrm{O}}}$$

(13)

For Equations (2)–(4), we can directly apply the Nernst equation.

$$E_{{\mathrm{H}}_2\mathrm{O}}=E_{{\mathrm{H}}_2\mathrm{O}}^0-\frac{RT}{2F}\ln \frac{p_{{\mathrm{H}}_2}}{p_{{\mathrm{H}}_2\mathrm{O}}}$$

(14)

$$E_{{\mathrm{CO}}_2}=E_{{\mathrm{CO}}_2}^0-\frac{RT}{2F}\ln \frac{p_{\mathrm{CO}}}{p_{{\mathrm{CO}}_2}}$$

(15)

$$E_{{\mathrm{O}}_2}=E_{{\mathrm{O}}_2}^0-\frac{RT}{4F}\ln p_{{\mathrm{O}}_2}$$

(16)

where $E_{{\mathrm{H}}_2\mathrm{O}}^0$, $E_{\mathrm{CO}+{\mathrm{H}}_2\mathrm{O}}^0$, $E_{{\mathrm{O}}_2}^0$ and $E_{{\mathrm{CO}}_2}^0$ are the standard electrode potentials, R is the gas constant (8.314 J/(mol·K)), T is the temperature (K), F is the Faraday constant (96,485 C/mol), p_H2, p_H2O, p_CO, p_CO2 respectively represent the partial pressure of H₂, H₂O, CO, and CO₂.

Within the Secondary Current Distribution interface, electrochemical reactions are represented as a function of the overpotential. The physics interface employs various relationships between charge transfer current density and overpotential, including Butler-Volmer and Tafel expressions. The most comprehensive expression follows the Butler-Volmer formulation.

$$i_{\mathrm{l}\mathrm{o}\mathrm{c},\mathrm{m}}=i_0\left(\exp \left({\displaystyle \frac{{\alpha }_{a}F\eta }{RT}}\right)-\exp \left({\displaystyle \frac{-{\alpha }_{c}F\eta }{RT}}\right)\right)$$

(17)

where $i_{\mathrm{l}\mathrm{o}\mathrm{c},\mathrm{m}}$ is the local charge transfer current density for reaction m, $i_0$ is the exchange current density, ${\alpha }_a$ is the anodic transfer coefficient, ${\alpha }_c$ the cathodic charge transfer coefficient.

2.3. Conservation of Momentum, Mass, and Energy

In a Solid Oxide Electrolysis Cell (SOEC), fluid flow and mass transfer take place within the cathode/anode channels and porous electrodes. Additionally, heat transfer occurs throughout the entire computational domain. The governing equations for mass conservation, momentum conservation, and energy conservation are summarized below.

$$\rho {\displaystyle \frac{\partial }{\partial t}}\left({\omega }_i\right)+\rho \left(\mathbf{u}\cdot \nabla \right){\omega }_i+\nabla \cdot {\mathbf{j}}_i=R_i+S_{\mathrm{o}\mathrm{p}\mathrm{f},i}$$

(18)

$$\rho {\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+\rho \mathbf{u}\cdot \nabla \left(\mathbf{u}\right)=-\nabla p+\nabla \cdot \left(\mu \left(\nabla \mathbf{u}+\nabla {\mathbf{u}}^{\mathrm{T}}\right)\right)+\mathbf{F}$$

(19)

$$\kappa \left(T,t\right)={\kappa }_0\left({\displaystyle \frac{1-\varphi }{{\varphi }^3}}\right)\exp \left[-k_{shr}\mathit{texp}\left(-{\displaystyle \frac{Q_{shr}}{RT}}\right)\right]$$

(20)

$$\rho C_{\mathrm{p}}{\displaystyle \frac{\partial T}{\partial t}}+\rho C_{\mathrm{p}}\mathbf{u}\cdot \nabla T=\nabla \cdot \left(k\nabla T\right)+Q$$

(21)

where ρ is the density, kg/m³; ${\omega }_i$ represents the mass or molar concentration of species i per unit volume, mol/m³; $\mathbf{u}$ denotes the velocity field, m/s; ${\mathbf{j}}_i$ is the flux of species i, mol/(m³·s); $R_i$ stands for the reaction term, mol/(m³·s); $S_{\mathrm{o}\mathrm{p}\mathrm{f},i}$ represents the source term associated with other physical phenomena for species i. mol/(m³·s). p is pressure, Pa. $C_{\mathrm{p}}$ is the specific heat capacity at constant pressure, J/(kg·K). $k$ is the viscous stress tensor, Pa. T is the absolute temperature, K. Q contains the heat sources, W/m³. F is the volume force vector, N/m³, φ is the local porosity. This ordinary differential equation is solved through the Distributed ODE feature, thereby coupling microstructural shrinkage to gas-flow resistance.

In a multicomponent mixture, the mass flux relative to the mass average velocity, ${\mathbf{j}}_i$ can be defined by the generalized Fick equations (Ref. [1]):

$${\mathbf{j}}_i=-\rho {\omega }_i\sum _{\begin{array}{c}k = 1\end{array}}{\tilde{D}}_{ik}{\mathbf{d}}_k-D_i^T\nabla \mathrm{l}\mathrm{n}T$$

(22)

where ${\tilde{D}}_{ik}$ are the multicomponent Fick diffusivities, m²/s; $D_i^T$ are the thermal diffusion coefficients, kg/m·s; ${\mathbf{d}}_k$ is the diffusional driving force acting on species k, 1/m.

$${\mathbf{d}}_k=\nabla x_k+{\displaystyle \frac{1}{p}}\left[\left(x_k-{\omega }_k\right)\nabla p-\rho {\omega }_k{\mathbf{g}}_k+{\omega }_k\sum _{l=1}^Q\rho {\omega }_l{\mathbf{g}}_l\right]$$

(23)

where ${\mathbf{g}}_l$ is an external force (per unit mass) acting on species k. In the case of an ionic species the external force arises due to the electric field, m/s²; p is the total pressure, Pa; mole fraction $x_k$s given by

$$x_k={\displaystyle \frac{{\omega }_k}{M_k}}M$$

(24)

the mean molar mass M, kg/mol.

2.4. Water Gas Shift Reaction Model

The water gas shift reaction (WGSR) is a key reaction in the SOEC co-electrolysis of CO₂ and H₂O, occurring at the anode. This study adopts the reversible water gas shift reaction rate model provided by COMSOL Multiphysics. Unlike previous studies that treat the water–gas-shift reaction (WGSR) as being at equilibrium, the reverse WGSR is modeled here as a finite-rate homogeneous reaction. The Arrhenius form used in Equations (24) and (25) is calibrated to the CO₂/H₂ data reported in Ref. [20] and is directly coupled to the Butler–Volmer source term, so that electrochemical and homogeneous pathways compete for the same reactants.

$$r_{\mathrm{W}\mathrm{G}\mathrm{S}\mathrm{R}}=k_{\mathrm{s}\mathrm{f}}({\displaystyle \frac{p_A}{p_{\mathrm{r}\mathrm{e}\mathrm{f}}}}{)}^2\left(x_{\mathrm{C}\mathrm{O}}{x}_{\mathrm{H}20}-\exp \left({\displaystyle \frac{{\mathsf{\Delta }}_{\mathrm{r}}G\left(\tau \right)}{RT}}\right)x_{\mathrm{C}\mathrm{O}2}{x}_{\mathrm{H}2}\right)$$

(25)

where ${\mathsf{\Delta }}_{\mathrm{r}}G\left(\tau \right)$ is corresponding change in Gibbs free energy of a reaction, $x_{\mathrm{C}\mathrm{O}}$ and $x_{\mathrm{H}20}$ respectively represent the molar fractions of CO and H₂O in the reaction. The reaction coefficients are taken from the calculation model in reference [20].

$$k_{\mathrm{s}\mathrm{f}}=0.0171\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{-103191}{RT}}\right){\mathrm{m}\mathrm{o}\mathrm{l}·\mathrm{m}}^{-3}·{\mathrm{P}\mathrm{a}}^{-2}·{\mathrm{s}}^{-1}$$

(26)

2.5. Electrochemical Heating

The heat source in the electrochemical heating multiphysics coupling includes the sum of irreversible losses (Joule heating and activation losses) and reversible heat from the electrochemistry interface. For the porous electrodes of SOEC, the total heat source in the domain is obtained by summing up the Joule heating and electrochemical sources.

$$Q_{tot}=\sum _m{a}_{v,m}{Q}_m+{ Q}_{JH}$$

(27)

The formula for calculating the Joule heating source term generated by charge transport in the electrode material and electrolyte of SOEC is as follows:

$$Q_{JH} = -\left({\mathit{i}}_s\cdot \nabla {\varphi }_s + {\mathit{i}}_l\cdot \nabla {\varphi }_l\right)$$

(28)

The formula for calculating the heat generated by electrochemical reactions is

$$Q_m=\left({\eta }_{m,tot}+{\displaystyle \frac{T\mathsf{\Delta }{S}_m}{n_{m}F}}\right)i_m$$

(29)

where the first term represents the irreversible activation losses, and the second term is the reversible heat change due to the net change in entropy in the conversion process. ${\eta }_{m,\mathrm{t}\mathrm{o}\mathrm{t}}$ defined as

$${\eta }_{m,tot}={\varphi }_s-{\varphi }_l-E_{eq,m}$$

(30)

2.6. Numerical Methodology

In COMSOL Multiphysics, a 2D geometric model was constructed to simulate an SOEC for the co-electrolysis of CO₂ and H₂O. Following this, the components were equipped with the physics fields of water electrolyzer (we) and Heat Transfer in Solids and Fluids (ht), which were coupled via electrochemical heating (ech). Subsequently, boundary conditions were specified, and the model underwent mesh generation.

3. Results and Discussion

3.1. Model Validation

Mogensen’s team [13,14] reported detailed experimental data on the J-V characteristics of CO₂ electrolysis using SOEC to produce O₂. Therefore, their experimental data were utilized for model evaluation in this study. In their experiments, J-V data of a cathode-supported SOEC were measured.

The parameters from

Table 2. Comparison of simulation results with experimental data [13,14] for model validation.

Potential (V)	Experimental Current Density (50/50) (A/m2)	This Work’s Current Density (50/50) (A/m2)	Err. %	Experimental Current Density (50/50) (A/m2)	This Work’s Current Density (50/50) (A/m2)	Err. %
1.0	872.87	996.29	14.1	1892.23	2396.4	26.7
1.1	3314.3	2764.7	–16.6	4556.09	4392.3	–3.6
1.2	5246	4558.7	–13.1	6820.61	6554.5	–3.9
1.3	6446.57	6593.2	2.3	8663.84	9139.6	5.5

* Average absolute percentage error (MAPE): 50% CO₂/50% CO = 11.5%; 70% CO₂/30% CO = 9.9%. * Root-mean-square error (RMSE): 0.45 kA m⁻² and 0.38 kA m⁻², respectively.

were used in the computational model of this study, with an inlet mass flow rate of 0.0005 kg/s. A comparison between the calculated values and the experimental data from the literature is presented in Table 2, showing consistent trends between the two datasets.

3.2. Simulation Results

We utilize the established model for simulation to analyze the characteristics of CO₂ and H₂O co-electrolysis. The meshing is shown in Figure 3. The boundary conditions are set as follows: at the cathode inlet, the molar fractions of H₂O, CO₂, CO, and H₂ are 0.498, 0.5, 0.001, and 0.001, respectively. At the anode inlet, the molar fractions of N₂ and O₂ are 0.79 and 0.21. The SOEC operating potential is set to 1.5 V. Mass flow rates at the cathode and anode inlet boundaries are both set to 0.005 kg/s. The computed components, temperature, pressure, and water–gas shift reaction rate are illustrated in Figure 4, Figure 5 and Figure 6. Figure 4c,d show an increase in both the mole fraction of CO and the mole fraction of H₂ along the length of the electrolyzer. The mole fraction of H₂ increases from 0.001 at the inlet to 0.064787 at the outlet, while the mole fraction of CO increases from 0.001 to 0.44059. The concentration difference between CO₂ and H₂ is considerably higher between the hydrogen gas diffusion electrode and the hydrogen gas flow channel. This difference is attributed to the slower diffusion of carbon dioxide compared to hydrogen, resulting in a greater concentration difference between the two regions. Additionally, Figure 4a illustrates a decrease in the mole fraction of H₂O from 0.498 at the inlet to 0.43405 at the outlet, and Figure 4b depicts a decrease in the mole fraction of CO₂ from 0.5 to 0.44069 along the length of the electrolyzer. Furthermore, Figure 4e shows an increase in the mole fraction of O₂ from 0.21 at the inlet to 0.25296 at the outlet, while Figure 4f indicates a decrease in the mole fraction of N₂ from 0.79 to 0.74704 along the length of the electrolyzer. These values represent the average values across the inlet and outlet cross-section.

Figure 5 illustrates the calculated pressure and temperature distribution. In Figure 5a, it can be observed that the average pressure decreases from 36.117 Pa at the inlet to 0 Pa at the outlet. Figure 5b indicates that the average temperature rise increases from 5.9 K to 35.37 K. Figure 6 represents the water–gas shift reaction rate. The average water–gas shift reaction rate at the anode inlet decreases from 7.5 mol/m³·s to 6.7294 mol/m³·s.

The total cathodic (negative) current density, which is integrated along the y-direction for each grid point along the x-direction using the general projection operator, comprises two components. The total current density for electrolyzing CO₂ and H₂O can be calculated using Equation (16). Figure 7 illustrates the distribution of H₂O and CO₂ current density in the y-direction along the length of the electrolyzer during the cathode electrolysis process. It can be observed that the absolute value of CO₂ electrolysis current density gradually decreases along the length direction, descending from 12,094 A/cm² to 11,016 A/cm², and then slowly increases near the outlet to 11,173 A/cm² (at a length of 18 mm). Meanwhile, the absolute value of H₂O electrolysis current density gradually increases along the length direction, rising from 6582.5 A/cm² at the inlet to 8380 A/cm² (at a length of 17 mm) before slowly decreasing to 8177.1 A/cm² at the outlet.

3.3. The Effect of SOEC Operating Temperature on the Co-Electrolysis of CO₂ and H₂O

Figure 8 illustrates the curve of the calculated molar fraction of substances at the outlet of the electrolyzer as a function of SOEC operating temperature. It can be observed from the graph that as the SOEC operating temperature increases, the molar fraction of hydrogen reaches its peak value of 0.087 at 1173 K, then decreases with further temperature increase, reaching 0.074 at 1423 K. The molar fraction of CO at the outlet increases monotonically with increasing temperature. Before 1173 K, the increase rate is approximately 0.03 for every 50 K increase in temperature, and then the increase slows down, with a rate of approximately 0.01 for every 50 K increase in temperature. The molar fractions of water and CO₂ gradually decrease with increasing temperature. The molar fraction of CO₂ decreases from 0.49 at 923 K to 0.35 at 1423 K, while the molar fraction of H₂O decreases from 0.47 at 923 K to 0.43 at 1423 K.

Figure 9 shows the variations in electrolysis current density of CO₂ and H₂O along the electrolysis cell (y-direction) under different operating temperatures of the SOEC. At an operating temperature of 923 K in the SOEC, the CO₂ electrolysis current density at a position of 10 mm is 1652.9 A/m². At temperatures of 1023 K, 1123 K, 1223 K, 1323 K, and 1423 K, the corresponding values are 5426 A/m², 11,631 A/m², 16,683 A/m², 19,324 A/m², and 21,695 A/m², respectively (Figure 9a). At an operating temperature of 923 K in the SOEC, the electrolysis current density of H₂O at a position 10 mm away is 4189.7 A/m². Meanwhile, at temperatures of 1023 K, 1123 K, 1223 K, 1323 K, and 1423 K, the respective values are 8647.53 A/m², 13,818.4 A/m², 15,399.4 A/m², 14,273.2 A/m², and 13,426.4 A/m² (Figure 9b). In the process of co-electrolyzing CO₂ and H₂O in SOEC, the current density is used to measure the rate of product generation. Higher current density typically implies a higher reaction rate, which may result in increased production of CO and H₂, while lower current density may lead to more CO₂ and H₂O being reduced to other products. As the operating temperature increases, the current density of co-electrolyzed CO₂ generally increases. However, the current density of H₂O may decrease after reaching a certain temperature.

3.4. The Effect of Operating Voltage on the Co-Electrolysis of CO₂ and H₂O

Figure 10 illustrates the variation trend of the molar fractions of outlet substances from the SOEC electrolyzer cell under different operating voltage conditions. It can be observed that with the increase in operating voltage of the SOEC, the average molar fractions of CO₂ and H₂O at the outlet decrease. For instance, at an operating voltage of 0.8 V, the molar fractions of CO₂ and H₂O are 0.49739 and 0.4917, respectively, while the molar fractions of H₂ and CO are 0.0046653 and 0.0037694, respectively. This indicates that the electrochemical reaction is very weak at this voltage. Additionally, when the operating voltage increases to 1.1 V and 1.5 V, respectively, the molar fractions of CO₂ and H₂O further decrease, while those of H₂ and CO increase. For example, at an operating voltage of 1.1 V, the molar fractions of CO₂ and H₂O are 0.48527 and 0.47693, respectively, while the molar fractions of H₂ and CO are 0.02191 and 0.015888, respectively. At an operating voltage of 1.5 V, the molar fractions of CO₂ and H₂O decrease to 0.44059 and 0.43405, respectively, while the molar fractions of H₂ and CO increase to 0.064787 and 0.060582, respectively.

Figure 11 shows the computed distribution of current density along the length of the electrolyzer under different operating voltages. From Figure 11a, it can be observed that at an operating voltage of 0.8 V, the current density of CO₂ at a position 10 mm along the length of the electrolyzer is 374 A/m². This current density increases to 1377.8 A/m² at an operating voltage of 1.0 V, and further increases to 8176 A/m² at 1.5 V. From Figure 11b, it can be seen that at an operating voltage of 0.8 V, the current density of H2O at the same position is 610 A/m², while at 1.0 V, it is 2349 A/m², and at 1.5 V, it is 11,163 A/m².

Figure 12 shows the computational results of the water–gas shift reaction rate at operating voltages of 0.8 V and 1.5 V, respectively. At an operating voltage of 0.8 V, the average rate of the cathodic water–gas shift reaction is 0.1213 (mol/m³·s), while at an operating voltage of 1.5 V, it increases to 6.5884 (mol/m³·s). Higher operating voltages typically lead to higher current densities, which may accelerate the rate of the water–gas shift reaction.

Analyzing Figure 10, Figure 11 and Figure 12, it can be concluded that the operating voltage can influence the rate of electrolysis reactions and product selectivity. Higher operating voltages typically lead to higher current densities, thereby accelerating the reaction rate. This may increase the production rate of products such as CO and H₂.

3.5. The Effect of Changes in the Molar Fraction of Inlet Substances

Table 3. Variations in the output of CO and H2 with changes in the ratio of inlet CO₂ to H₂O.

	Molar Fraction of H2O	Molar Fraction of CO2	Molar Fraction of CO	Molar Fraction of H2
CO2/H2O: 2/8	0.71792	0.18163	0.019477	0.080968
CO2/H2O: 4/6	0.52663	0.35552	0.045637	0.07221
CO2/H2O: 6/4	0.3433	0.52462	0.076542	0.055538
CO2/H2O: 8/2	0.16721	0.68988	0.11123	0.031683

shows the variations in the output of CO and H₂ with changes in the ratio of inlet CO₂ to H₂O. When SOEC co-electrolyzes CO₂ and H₂O, as the inlet CO₂ concentration increases, the molar fraction of CO in the outlet also gradually increases. From Table 3, it can be observed that when CO₂/H₂O: 2:8, the molar fraction of CO at the outlet is 0.18163, whereas when CO₂/H₂O: 8/2, the molar fraction of CO increases to 0.68988. Meanwhile, with the increase in CO₂ concentration, the available oxygen atoms in the reaction also increase. Consequently, a portion of the water splitting reaction is inhibited, leading to a decrease in the production of H₂. The molar fraction of H₂ at the outlet decreases from 0.080968 to 0.031683.

Figure 13 shows the distribution of current density along the y-axis for the electrolysis of CO₂ and H₂O at the inlet of a Solid Oxide Electrolysis Cell (SOEC) under different ratios of CO₂ and H₂O.

It can be observed that with the increase in the molar fraction of inlet CO₂, the electrolysis current density of CO₂ also increases. For instance, at 0.01 m, when the CO₂/H₂O ratio is 2/8, the current density is 0.33703 A/cm², which increases to 1.288 A/cm² when the ratio becomes 8/2. Meanwhile, the current density of electrolyzing H₂O decreases with the decrease in the molar fraction of H₂O. It decreases from 1.7311 A/cm² to 0.47116 A/cm².

3.6. The Effect of Co-Directional Flow and Counterflow

The Solid Oxide Electrolysis Cell (SOEC) conducts simultaneous H₂O and CO₂. The mixed gases of H₂O and CO₂ flow into the cathode, while nitrogen (N₂) and oxygen (O₂) pass through the anode. We further investigated the impact of counterflow and co-directional flow at the cathode and anode on the co-electrolysis performance.

Figure 14 and Figure 15 illustrate the variations in the molar fractions of CO and H₂ at the outlet of a solid oxide electrolysis cell (SOEC) under co-directional and counterflow conditions.

Table 4. Variations in the output of CO and H₂ with changes under conditions of co-directional and counterflow.

	Molar Fraction of H2O	Molar Fraction of CO2	Molar Fraction of CO	Molar Fraction of H2
Counterflow	0.43243	0.43828	0.062886	0.066402
Co-directional flow	0.43405	0.44059	0.060582	0.064787

displays the average molar fractions of each substance component at the outlet, as shown in Figure 14 and Figure 15.

In Table 4, it can be observed that under counterflow conditions, the molar fractions of H₂ and CO produced by electrolyzing H₂O are higher compared to co-directional flow conditions. Specifically, the molar fraction of H₂ at the outlet is 0.064787 under co-directional flow conditions, whereas it is 0.066402 under counterflow conditions, resulting in an increase of 2.49% in the production of H₂ during counterflow. Conversely, under co-directional flow conditions, the molar fraction of CO at the outlet is 0.060582, while it is 0.062886 under counterflow conditions, leading to an increase of 2.3% in the production of CO during counterflow.

Figure 16 shows the distribution of current density along the y-direction under co-directional and counterflow conditions. It can be observed that during the co-electrolysis of CO₂ and H₂O, the current density components of CO₂ and H₂O are larger under counterflow conditions, especially near the inlet. However, towards the outlet, they become more consistent.

3.7. Parametric Sensitivity: Voltage vs. Temperature vs. Flow Mode

To clarify which operating knob delivers the greatest performance return, we benchmarked each variable against the same reference case (1073 K, 1.5 V, counter-current). The outlet CO mole fraction was chosen as the performance metric because it tracks overall syngas quality and correlates well with total current density. The normalized sensitivity S

$$S={\displaystyle \frac{∆Y/Y_0}{∆X/X_0}}$$

where Y is the outlet CO mole fraction and X is the tested variable. Baseline: 1073 K, 1.5 V, counter-current flow. The analysis of normalized sensitivity is presented in

Table 5. Normalized sensitivity of CO outlet yield to the three key operating variables.

Variable	Scan Window	Gain in CO Yield ($∆\mathit{Y}/{\mathit{Y}}_0$)	Relative Change in Variable ( $∆\mathit{X}/{\mathit{X}}_0$)	Normalized S	Practical Impact
Voltage (V)	0.8 → 1.5 V	×15.0	0.88	≈1.0	One extra 0.1 V raises CO~70%; strongest lever.
Temperature (T)	923 → 1423 K	×4.3	0.46	≈0.8	High T cuts activation losses; second most effective.
Flow mode (F)	Co-flow → Counter-flow	+2.3%	discrete	≈0.03	Helps outlet utilization; minor but additive.

.

Voltage is the most powerful lever: because the Butler–Volmer relationship couples over-potential to reaction rate exponentially, even a small voltage rise can boost current density by an order of magnitude, although excessive voltages increase ohmic heating and demand more aggressive stack cooling. Temperature is the second-strongest variable. Raising the cell from 1073 K to roughly 1223 K nearly triples the O²⁻ conductivity of 8YSZ and speeds up the water–gas-shift reaction, yet practical constraints such as seal creep and Ni particle coarsening generally limit operation to below 1273 K. Switching from co-flow to counter-flow provides a smaller but still valuable gain—typically 2–3%—by maintaining higher reactant partial pressures near the outlet without extra energy input. These variables are interactive: combining a moderate voltage increase with higher temperature can overshoot kinetic targets and create local hot spots. Accordingly, designers should first maintain tight power-supply control (±20 mV) to exploit voltage sensitivity, then operate near the material “sweet-spot” temperature of 1100–1150 K, and finally adopt counter-current routing as a low-risk way to capture the remaining performance margin once voltage and temperature are optimized.

4. Conclusions

This paper establishes a co-electrolysis model for solid oxide electrolysis cells (SOECs) for CO₂ and H₂O using COMSOL Multiphysics and performs performance simulations to reveal its behavior under various operating conditions. The simulation results show that both operating temperature and voltage have a significant influence on the electrolysis reaction rate and product selectivity in the SOEC co-electrolysis process. Specifically, as the temperature increases from 920 K to 1273 K, the CO production rate increases approximately tenfold under a 50% CO₂/50% H₂O feed, and about fourfold under a 70% CO₂/30% H₂O feed, highlighting the strong temperature dependence of the reaction kinetics and selectivity. Under different CO₂ and H₂O feedstock ratios, the simulation results show that the CO production increases with increasing CO₂ concentration, while the H₂ production decreases accordingly. When the CO₂/H₂O ratio increases from 2/8 to 8/2, the CO production mole fraction increases from 0.0195 to 0.111, while the H₂ production decreases from 0.081 mol to 0.302 mol. During the co-electrolysis process, the CO and H₂ productions under counter-current conditions are higher than under co-current conditions, further demonstrating the influence of operating conditions on co-electrolysis performance. Under counter-current conditions, the CO production is 2.3% higher than under co-current conditions, while the H₂ production increases by 2.49%. These insights—obtained through a fully coupled heat-, charge- and mass-transfer calculation implemented with a finite-element platform such as COMSOL Multiphysics—provide quantitative guidance that is difficult to obtain experimentally, and they clarify how temperature, voltage, feed ratio and flow direction interact to govern the behavior of high-temperature co-electrolysis systems. Future work will extend the model to include microstructural evolution, multiscale mass-transfer effects, and a Monte-Carlo–based uncertainty analysis, so that the influence of porosity, detailed gas diffusion, and three-dimensional heat transport can be quantified alongside the temperature, voltage, and flow-configuration effects reported here.