Multiphysics-Driven Structural Optimization of Flat-Tube Solid Oxide Electrolysis Cells to Enhance Hydrogen Production Efficiency and Thermal Stress Resistance

Abstract

The solid oxide electrolysis cell (SOEC) has potential application value in water electrolysis for hydrogen production. Here, we propose an integrated multi-scale optimization framework for the SOEC, addressing critical challenges in microstructure–property correlation and thermo-mechanical reliability. By establishing quantitative relationships between fuel support layer thickness, air electrode rib coverage, and Ni-YSZ volume ratio, we reveal their nonlinear coupling effects on the hydrogen production rate and thermal stress. The results show that when the fuel support layer thickness increases, the maximum principal stress of the fuel electrode decreases, and the hydrogen production rate and diffusion flux first increase and then decrease. The performance is optimal when the fuel support layer thickness is 5.4 mm. As the rib area decreases, the hydrogen production rate and thermal stress gradually decrease, but the oxygen concentration distribution becomes more uniform when the rib area portion is 42%. When the Ni volume fraction increases, the hydrogen production rate and the maximum principal stress gradually increase, but the uniformity of H₂O flow decreases. When the Ni volume fraction is lower than 50%, the uniformity of H₂O flow drops to 20%. As the volume fraction of nickel (Ni) increases, the fuel utilization gradually increases. When the volume fraction of Ni is between 50% and 60%, the fuel utilization reaches the range of 60–80%. This study indicates that the fuel support layer thickness, rib area, and Ni-YSZ ratio have different effects on the overall performance of the SOEC, providing guidance for the optimization of the flat-tube SOEC structure.

1. Introduction

Against the backdrop of energy transition, hydrogen energy, with its potential for being renewable and sustainable, has become an important direction for energy technological transformation [1,2,3]. Among various hydrogen production technologies, electrolytic hydrogen production has the advantages of being clean and having high purity [4,5]. The SOEC can produce green hydrogen using renewable energy [6]. As the SOEC operates at a high temperature and pressure, it enhances the thermodynamic efficiency of water splitting [7]. Moreover, the SOEC eliminates concerns over electrolyte evaporation, corrosion, and pollutant accumulation, achieving a higher current density and outstanding efficiency performance [8]. It is one of the most promising technologies for large-scale hydrogen production.

Typically, the SOEC consists of electrolytes sandwiched between porous fuel electrodes, air electrodes, and bipolar plates. These electrodes are important to promote the kinetics of electrochemical reactions and provide mechanical support for cell components. Therefore, optimizing the structural design parameters of the electrodes is essential for achieving superior performance [9]. Previous SOEC structural studies have mostly focused on the active layer [10,11] and the shape of the channel [12,13]. However, the fuel support layer also has an impact on SOEC performance. For example, Pan et al. observed that an increased water vapor content enhances mass transfer resistance in the fuel support layer, which negatively impacts mass transfer efficiency [14]. The current density of the fuel cell is normalized in relation to length and width, and the fuel enters the flow channel, diffusing mainly in the thickness direction to reach the three-phase boundary (TPB) for the reaction. An excessive fuel support layer thickness will increase mass transfer resistance, emphasizing the importance of understanding its impact on SOEC performance. While current research has concentrated on changes in rib shape due to alterations in channel geometry, the rib area also significantly affects various physical parameters of the cell [15]. Additionally, microstructural conditions can influence SOEC performance. For example, Michael Keane et al. highlighted that an increased Ni content can lead to rapid migration and agglomeration of Ni, which, in turn, affects the electrochemical performance of the SOEC [16]. However, previous studies have largely neglected the impact of the Ni content on the thermal stress of the SOEC, which can affect their durability when thermal stress levels become excessive. Additionally, a deeper understanding of the chemical reactions, mass transfer, and heat transfer processes within the cell is crucial, as these factors play a significant role in enhancing SOEC performance. While experimental methods provide valuable insights into SOEC performance, the complexity and high cost of experimental studies make it challenging to directly observe the chemical reactions, mass transfer, and heat transfer processes occurring inside the cell.

Numerical simulation is a powerful tool that enables a thorough and detailed study of the physical and chemical processes within the SOEC. As a result, it has been extensively applied in research on chemical reactions [17,18], charge transfer mechanisms [19,20], and the structure of the SOEC [21,22,23]. To study the internal mechanisms during the operation of the SOEC, several numerical models have been developed. Li et al. established a 1D model to examine the performance of the SOEC, highlighting the influence of electrode thickness and porosity on the current density, oxygen production, and concentration polarization [10]. Wang et al. developed an integrated model to study cell degradation and the limitations to cell durability caused by microstructural changes during SOEC operation. Their results indicated that significant depletion of Ni led to an increase in the overall overpotential of the cell [23].

Moreover, compared with the conventional flat-plate SOEC, the flat-tube SOEC fuel channel is embedded in the fuel support layer, which is more conducive to the heat transfer inside the SOEC, and at the same time, is more conducive to the discharge of electrolysis gas, which can promote electrochemical reactions. However, current research on the flat-tube SOEC predominantly focuses on investigating the effects of operating conditions. For instance, Luo et al. [24] studied the influence of the ratio of seawater steam to hydrogen, temperature, and current density on the performance of flat-tube SOECs. Liu et al. [25] explored the impact of constant-current and wide-load-fluctuation electrolysis on the stability and energy consumption of flat-tube SOECs. However, the structure of the SOEC also has a significant impact on its performance. Xu et al. established a three-dimensional SOEC model of a flat plate. They studied the influence of rectangular, triangular, trapezoidal, and semi-circular flow channels on the performance of the cell. The study found that the SOEC with trapezoidal flow channels had the highest hydrogen production rate [26]. Tu et al. simulated the spatial distributions of reactant/product concentrations, temperatures, current densities, and thermal stresses in square, trapezoidal, and rectangular channels. They found that the uniformity of thermal stress distribution in the rectangular channel was inferior to that in the other two configurations [27]. Kim et al. established SOECs with different stack structures and flow configurations and investigated the influence of the structures and flow configurations on stress. The research shows that the co-flow configuration and metal foam structure are beneficial for reducing thermal stress [28].

The aforementioned research studies have only focused on hydrogen production performance or thermal stress, respectively. However, hydrogen generation performance is an important indicator for the SOEC, which can be used to evaluate the energy conversion efficiency of the SOEC [29]. Meanwhile, thermal stress can affect the reliability of the SOEC during long-term operation [30]. These two aspects cannot be ignored for the development of the SOEC. Therefore, to improve SOEC performance, it is essential to consider both the hydrogen production rate and thermal stress simultaneously. Despite their importance, there are few studies that address this dual consideration. In this study, a three-dimensional model of a flat-tube SOEC is developed to investigate the effects of the macrostructure and microstructure on the overall performance of the SOEC, with a focus on both the hydrogen production rate and thermal stress. The model incorporates the effects of electrochemical reactions, mass transfer, heat transfer, and thermal stress. Specifically, this study examines the effects of the fuel support layer thickness, air electrode rib area, and Ni-YSZ ratio on SOEC performance, with a particular emphasis on the impact of the Ni-YSZ ratio on the thermal stress within the cell. This analysis is crucial for optimizing both the performance and structural integrity of the SOEC.

2. Model Description

In this paper, a 3D finite electrochemical–thermal-stress coupling model is developed, which fully takes into account the electrochemical reactions, mass transfer, momentum transfer, and energy transfer during the reaction processes [31]. In this section, the geometrical structure of the model, the governing equations, and the boundary conditions will be described in detail.

2.1. Geometry

Figure 1 shows the geometrical schematic of a flat-tube SOEC. As shown in Figure 1a, 17 fuel gas channels are embedded inside the fuel support layer. The fuel active layer, electrolyte, and air active layer form the electrode structure, with connectors at both ends. Figure 1b shows the structure of the air channels, with arrows indicating the direction of gas flow. Figure 1c shows a localized schematic of the channels.

Table 1. Geometric model parameters [32].

Parameter	Length × Width × Height (mm)
Fuel support layer	150 × 64 × 5
Fuel active layer	120 × 50 × 0.02
Electrolyte	120 × 50 × 0.015
Air active layer	120 × 50 × 0.02
Interconnector	125 × 50 × 2.5

shows the geometric parameters of the model.

2.2. Governing Equations

In order to clearly illustrate these processes, it is necessary to combine the models of electrochemical reactions, gas flow, species diffusion, heat transfer, and mechanics [33]. The governing equations of the models are given below.

2.2.1. Electrochemical Reaction Models

In this model, hydrogen and oxygen are produced by the following two reactions. At the fuel electrolyte,

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

(1)

At the air electrolyte,

$${\mathrm{O}}^{2-}\to {\displaystyle \frac{1}{2}}{\mathrm{O}}_2+2{\mathrm{e}}^{-}$$

(2)

Overall reaction:

$${\mathrm{H}}_2\mathrm{O}\to {\mathrm{H}}_2+{\displaystyle \frac{1}{2}}{\mathrm{O}}_2$$

(3)

For the SOEC, the operating voltage E [34] can be determined using the equilibrium potential and various overpotential losses:

$$\mathrm{E}={\mathrm{E}}_{\mathrm{e}\mathrm{q}}{+\mathsf{\eta }}_{\mathrm{a}\mathrm{c}\mathrm{t}}+{\mathsf{\eta }}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}+{\mathsf{\eta }}_{\mathrm{o}\mathrm{h}\mathrm{m}}$$

(4)

where ${\mathrm{E}}_{\mathrm{e}\mathrm{q}}$ is the thermodynamic equilibrium potential, and ${\mathsf{\eta }}_{\mathrm{a}\mathrm{c}\mathrm{t}}$, ${\mathsf{\eta }}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}$, and ${\mathsf{\eta }}_{\mathrm{o}\mathrm{h}\mathrm{m}}$ represent the activation overpotential, concentration overpotential, and ohmic overpotential, respectively.

$${\mathrm{E}}_{\mathrm{e}\mathrm{q}}={\mathrm{E}}_0+{\displaystyle \frac{\mathrm{R}\mathrm{T}}{2\mathrm{F}}}\mathrm{l}\mathrm{n}\left[{\displaystyle \frac{{\mathrm{P}}_{{\mathrm{H}}_2}·{\left({\mathrm{P}}_{{\mathrm{O}}_2}\right)}^{\frac{1}{2}}}{{\mathrm{P}}_{{\mathrm{H}}_2\mathrm{O}}}}\right]$$

(5)

where ${\mathrm{E}}_0$ is the standard potential, which can be calculated by Equation (6) [35]:

$${\mathrm{E}}_0=1.253-0.00024516\mathrm{T}$$

(6)

The Butler–Volmer equation can be employed to represent the relationship between activation overpotential and current density [36]:

$$\mathrm{i}={\mathrm{i}}_0\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{{\mathsf{\alpha }}_{\mathrm{a}}\mathrm{n}\mathrm{F}{\mathsf{\eta }}_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{a}}}{\mathrm{R}\mathrm{T}}}\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{-{\mathsf{\alpha }}_{\mathrm{c}}\mathrm{n}\mathrm{F}{\mathsf{\eta }}_{\mathrm{c}\mathrm{a}\mathrm{t},\mathrm{c}}}{\mathrm{R}\mathrm{T}}}\right)\right]$$

(7)

where ${\mathrm{i}}_0$ is the exchange current density, n is the number of electrons generated from a single electrochemical reaction, $\mathsf{\alpha }$ is the charge transfer coefficient from the single electrochemical reaction, and F is the Faraday constant from a single electrochemical reaction.

Expression for the charge conservation equation:

$$-\nabla ·\left({\mathsf{\sigma }}_{\mathrm{e}\mathrm{l}}\nabla {\mathsf{\Phi }}_{\mathrm{e}\mathrm{l}}\right)=\nabla {\mathrm{i}}_{\mathrm{e}\mathrm{l}}$$

(8)

$$-\nabla ·\left({\mathsf{\sigma }}_{\mathrm{i}\mathrm{o}}\nabla {\mathsf{\Phi }}_{\mathrm{i}\mathrm{o}}\right)=\nabla {\mathrm{i}}_{\mathrm{i}\mathrm{o}}$$

(9)

where $\mathrm{i}$ is the current density, and $\mathsf{\sigma }$ and $\mathsf{\Phi }$ represent the effective conductivity and potential, respectively. The subscripts $\mathrm{e}\mathrm{l}$ and $\mathrm{i}\mathrm{o}$ denote electrons and ions, respectively, where the conductivity of the material is shown in

Table 2. Electronic/ionic conductivity [32].

Materials	Electric/Ionic Conductivity (S/m)
Ni	9.5 × 107/T × exp (−1150/T)
LSCF	4.2 × 107/T × exp (−1200/T)
YSZ	3.34 × 104 × exp (−10,300/T)

.

2.2.2. Gas Flow Model

Gas flow takes place in channels and porous regions. In channels, the continuity and compressible Navier–Stokes equations can be expressed as follows:

$$\nabla ·\left(\mathsf{\rho }\mathrm{v}\right)=0$$

(10)

$$\mathsf{\rho }\mathrm{v}·\nabla \mathrm{v}=-\nabla \mathrm{p}+\nabla ·\left[\mathsf{\mu }\left(\nabla \mathrm{v}+\mathrm{v}\nabla \right)-{\displaystyle \frac{2}{3}}\mathsf{\mu }\nabla \mathrm{v}\mathrm{I}\right]$$

(11)

The electrode structure differs from the flow channel, as the electrode has a large number of micropores. Therefore, the Navier–Stokes equation modified by the Darcy term is commonly employed to illustrate the conservation of momentum in porous electrodes [37]

$$\nabla ·\left(\mathsf{\rho }\mathrm{v}\right)={\mathrm{S}}_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}$$

(12)

$$\mathsf{\rho }{\displaystyle \frac{1}{\mathsf{\epsilon }}}\mathrm{v}·\left(\nabla \mathrm{v}\right){\displaystyle \frac{1}{\mathsf{\epsilon }}}=-\nabla \mathrm{p}+\nabla ·\left[\mathsf{\mu }{\displaystyle \frac{1}{\mathsf{\epsilon }}}\left(\nabla \mathrm{v}+\mathrm{v}\nabla \right)-{\displaystyle \frac{2}{3}}\mathsf{\mu }\left(\nabla ·\mathrm{v}\right)\mathrm{I}\right]-{\displaystyle \frac{\mathsf{\mu }\mathrm{v}}{\mathrm{k}}}$$

(13)

where $\mathrm{v}$ is the velocity vector, ${\mathrm{S}}_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}$ is the mass source term, $\mathsf{\epsilon }$ is the porosity, and $\mathrm{k}$ is the permeability. In the channel, the gas flow is independent of the porosity. Therefore, the Darcy term can be neglected.

In addition, $\mathsf{\rho }$ and $\mathsf{\mu }$ are dependent on the total gas density and the gas kinetic viscosity of the gas components. The values of $\mathsf{\rho }$ and $\mathsf{\mu }$ are calculated using the following equations [38]:

$$\mathsf{\rho }={\displaystyle \frac{\mathrm{p}\sum {\mathrm{x}}_{\mathrm{i}}{\mathrm{M}}_{\mathrm{i}}}{\mathrm{R}\mathrm{T}}}$$

(14)

$$\mathsf{\mu }=\sum {\mathrm{x}}_{\mathrm{i}}{\mathsf{\mu }}_{\mathrm{i}}$$

(15)

$${\mathsf{\mu }}_{\mathrm{i}}=(\mathrm{A}+\mathrm{B}\times \mathrm{T}-\mathrm{M}\times {10}^{-5}\times {\mathrm{T}}^2)\times {10}^{-7}$$

(16)

where ${\mathrm{x}}_{\mathrm{i}}$, ${\mathrm{M}}_{\mathrm{i}}$, and ${\mathsf{\mu }}_{\mathrm{i}}$ are the mole fraction, molecular weight, and kinetic viscosity of component i. The kinetic viscosity of each gas can be calculated by Equation (16), where the values of A, B, and C are shown in

Table 3. Parameters for Equation (16).

μi	A	B	C
H2O	−36.826	0.429	1.62
H2	27.758	0.212	3.28
N2	42.606	0.475	9.88
O2	44.224	0.562	1.13

[39].

2.2.3. Mass Transfer Model

Electrochemical reactions occur near the interface between the electrolyte and the electrode [40]. To reach the reaction site, the gas must be transported through the porous electrodes. To better describe the gas diffusion phenomenon in porous electrodes, Fick’s law combined with Knudsen diffusion is used [41]:

$$\nabla ·{\mathrm{J}}_{\mathrm{i}}+\mathsf{\rho }\left(\mathsf{\epsilon }\mathrm{v}·\nabla \right){\mathsf{\omega }}_{\mathrm{i}=}{\mathrm{S}}_{\mathrm{i}}$$

(17)

$${\mathrm{J}}_{\mathrm{i}}=-\mathsf{\rho }{\mathrm{D}}_{\mathrm{i}}^{\mathrm{m}\mathrm{k}}\nabla {\mathsf{\omega }}_{\mathrm{i}}-\mathsf{\rho }{\mathsf{\omega }}_{\mathrm{i}}{\mathrm{D}}_{\mathrm{i}}^{\mathrm{m}\mathrm{k}}{\displaystyle \frac{\nabla \mathrm{M}}{\mathrm{M}}}+\mathsf{\rho }{\mathsf{\omega }}_{\mathrm{i}}\sum _{\mathrm{k}}{\displaystyle \frac{{\mathrm{M}}_{\mathrm{i}}}{\mathrm{M}}}{\mathrm{D}}_{\mathrm{k}}^{\mathrm{m}\mathrm{k}}\nabla {\mathrm{x}}_{\mathrm{i}}$$

(18)

$${\mathrm{D}}_{\mathrm{i}}^{\mathrm{m}\mathrm{k}}={{\displaystyle \frac{\mathsf{\epsilon }}{\mathsf{\tau }}}\left({\displaystyle \frac{1}{{\mathrm{D}}_{\mathrm{i}}^{\mathrm{m}}}}+{\displaystyle \frac{1}{{\mathrm{D}}_{\mathrm{i}}^{\mathrm{k}}}}\right)}^{-1}$$

(19)

$$\mathrm{M}={\left(\sum _{\mathrm{i}}{\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{i}}}{{\mathrm{M}}_{\mathrm{i}}}}\right)}^{-1}$$

(20)

where ${\mathsf{\omega }}_{\mathrm{i}}$ is the mass fraction of $\mathrm{i}$, ${\mathrm{J}}_{\mathrm{i}}$ is the mass flux of $\mathrm{i}$, $\mathsf{\epsilon }$ is the porosity, $\mathsf{\tau }$ is the tortuosity, $\mathrm{M}$ is the molar mass, $\mathrm{d}$ is the average pore size of the porous electrode, and ${\mathrm{D}}_{\mathrm{i}}^{\mathrm{m}\mathrm{k}}$ is the total diffusion coefficient, which can be calculated from the molecular diffusion coefficient (${\mathrm{D}}_{\mathrm{i}}^{\mathrm{m}}$) and the Knudsen diffusion coefficient (${\mathrm{D}}_{\mathrm{i}}^{\mathrm{k}}$) [42,43]. Furthermore, for the species diffusion model, the mass conservation equation is [41]

$${\mathrm{S}}_{{\mathrm{H}}_2}={\mathrm{M}}_{{\mathrm{H}}_2}{\displaystyle \frac{\mathrm{i}}{2\mathrm{F}}}$$

(21)

$${\mathrm{S}}_{{\mathrm{H}}_{2\mathrm{O}}}=-{\mathrm{M}}_{{\mathrm{H}}_2\mathrm{O}}{\displaystyle \frac{\mathrm{i}}{2\mathrm{F}}}$$

(22)

$${\mathrm{S}}_{{\mathrm{O}}_2}=-{\mathrm{M}}_{{\mathrm{O}}_2}{\displaystyle \frac{\mathrm{i}}{4\mathrm{F}}}$$

(23)

where ${\mathrm{S}}_{\mathrm{i}}$ is the mass source term for component $\mathrm{i}$, and diffusion, convection, and reaction-induced changes are considered in the equation.

2.2.4. Heat Transfer Model

The classical heat transfer control equation is

$$\mathsf{\rho }{\mathrm{C}}_{\mathrm{p}}\mathrm{v}·\nabla \mathrm{T}+\nabla ·\left(-\nabla {\mathsf{\lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}\mathrm{T}\right)=\mathrm{Q}$$

(24)

$${{\mathsf{\lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\left(1-\mathsf{\epsilon }\right)}_{{\mathsf{\lambda }}_{\mathrm{s}}}+\mathsf{\epsilon }{\mathsf{\lambda }}_{\mathrm{g}}$$

(25)

where ${\mathrm{C}}_{\mathrm{p}}$ is the specific heat at a constant pressure, ${\mathsf{\lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is the thermal conductivity, and Q is the heat source term.

2.2.5. Electrode Mesoscopic Structure and Mechanical Model

The tuning of the Ni-YSZ ratio was achieved by varying the volume fraction of Ni to calculate the effective parameters of the porous composite electrode material based on the model proposed by Nemat-All [44]:

$${\mathrm{E}}_0={\mathrm{E}}_1\left[{\displaystyle \frac{{\mathrm{E}}_1+\left({\mathrm{E}}_2-{\mathrm{E}}_1\right){\mathrm{V}}_2^{2/3}}{{\mathrm{E}}_1+\left({\mathrm{E}}_2-{\mathrm{E}}_1\right)\left({\mathrm{V}}_2^{2/3}-{\mathrm{V}}_2\right)}}\right]$$

(26)

$${\mathsf{\alpha }}_0={\mathsf{\alpha }}_1{\mathrm{V}}_1+{\mathsf{\alpha }}_2{\mathrm{V}}_2+{\displaystyle \frac{{\mathrm{V}}_1{\mathrm{V}}_2\left({\mathsf{\alpha }}_2-{\mathsf{\alpha }}_1\right)\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)}{{\mathrm{K}}_1{\mathrm{V}}_1+{\mathrm{K}}_2{\mathrm{V}}_2+\left(3{\mathrm{K}}_2{\mathrm{K}}_1/4{\mathrm{G}}_1\right)}}$$

(27)

$${\mathrm{v}}_0={\mathrm{v}}_1{\mathrm{V}}_1+{\mathrm{v}}_2{\mathrm{V}}_2$$

(28)

$${\mathrm{K}}_1={\displaystyle \frac{{\mathrm{E}}_1}{3\left(1-2{\mathrm{v}}_1\right)}},{\mathrm{G}}_2={\displaystyle \frac{{\mathrm{E}}_1}{2\left(1+{\mathrm{v}}_1\right)}},{\mathrm{G}}_2={\displaystyle \frac{{\mathrm{E}}_1}{2\left(1+{\mathrm{v}}_1\right)}}$$

(29)

$${\mathrm{E}}_{\mathrm{p}}={\displaystyle \frac{{\mathrm{E}}_0\left(1-\mathrm{p}\right)}{1+\mathrm{p}\left(5+8{\mathrm{v}}_0\right)/\left[8\left(1+{\mathrm{v}}_0\right)\left(23+8{\mathrm{v}}_0\right)\right]}}$$

(30)

$${\mathsf{\alpha }}_{\mathrm{p}}={\mathsf{\alpha }}_0$$

(31)

$${\mathrm{v}}_{\mathrm{p}}={\mathrm{v}}_0$$

(32)

where ${\mathrm{E}}_{\mathrm{p}}$, ${\mathsf{\alpha }}_{\mathrm{p}}$, and ${\mathrm{v}}_{\mathrm{p}}$ are the effective modulus of elasticity, coefficient of thermal expansion, and Poisson’s ratio of the porous composites, respectively. $\mathrm{p}$ is the porosity, and V₁ and V₂ represent the volume fractions of Ni and YSZ, respectively.

Active specific surface area formula [45]:

$${\mathrm{S}}_{\mathrm{TPB}}=\mathsf{\pi }{\sin }^2\mathsf{\theta }{\overline{\mathrm{r}}}^2{\mathrm{n}}_{\mathrm{t}}{\mathrm{n}}_{\mathrm{e}\mathrm{l}}{\mathrm{Z}}_{\mathrm{e}\mathrm{l}-\mathrm{i}\mathrm{o}}{\mathrm{P}}_{\mathrm{e}\mathrm{l}}{\mathrm{P}}_{\mathrm{i}\mathrm{o}}$$

(33)

The changes in the volume fractions of Ni and YSZ will lead to variations in the coefficient of thermal expansion and elastic modulus, thereby directly causing changes in thermal stress. When the components of the SOEC are subjected to thermal expansion, thermal stresses are generated. We assume the materials of the components are elastic, and the thermal strain can be calculated using

$${\mathsf{\delta }}_{\mathrm{t}\mathrm{h}}=\mathsf{\alpha }\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)$$

(34)

where $\mathsf{\alpha }$ is the coefficient of thermal expansion for different components, and ${\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is the reference temperature. Generally, the thermal stress is at its minimum value before the cell is heated. However, during the reduction process after heating, the cell will generate and accelerate creep, so the thermal stress in the cell will be relaxed and actually become zero [46]. The operating temperature of this study is 1073 K. After the heating and reduction process, the thermal stress became zero. Therefore, 1073 K is chosen as the stress-free temperature.

The correlation between thermal stress and thermal strain can be formulated as

$$\mathsf{\sigma }=\mathrm{D}{\mathsf{\delta }}_{\mathrm{e}\mathrm{l}}+{\mathsf{\sigma }}_0$$

(35)

where $\mathsf{\sigma }$, $\mathrm{D}$, ${\mathsf{\delta }}_{\mathrm{e}\mathrm{l}}$, and ${\mathsf{\sigma }}_0$ represent thermal stress, the elastic matrix, elastic strain, and initial thermal stress, respectively. Since the stress-free temperature is close to the operating temperature, the influence of initial stress is neglected in this paper.

2.3. Boundary

To describe the operating state of the SOEC, it is essential to choose appropriate boundary conditions. The parameters and boundary conditions of the model are shown in Table 4 and Table 5. Meanwhile, the following assumptions are used in the model of this paper:

(1)

Assuming that the ion/electron phase is homogeneous and isotropic [47];

(2)

Assuming that the gas is ideal [47];

(3)

Assuming that thermal radiation is ignored [48];

(4)

The fluid in the SOEC flows via laminar flow [49].

Table 4. Boundary [50].

Distribution	Current Transport	Mass Transfer	Momentum Transport	Heat Transfer	Thermal Stress
Fuel inlet	/	Mole fraction	Mass flow rate	Temperature	Free
Fuel outlet	/	Convection	Pressure	Heat convection	Free
Air inlet	/	Mole fraction	Mass flow rate	Temperature	Free
Air outlet	/	Convection	Pressure	Heat convection	Free
Fuel interconnector face	0 V	/	/	Heat isolation	Roller
Air interconnector face	Operating voltage	/	/	Heat isolation	Roller

Table 4. Boundary [50].

Distribution	Current Transport	Mass Transfer	Momentum Transport	Heat Transfer	Thermal Stress
Fuel inlet	/	Mole fraction	Mass flow rate	Temperature	Free
Fuel outlet	/	Convection	Pressure	Heat convection	Free
Air inlet	/	Mole fraction	Mass flow rate	Temperature	Free
Air outlet	/	Convection	Pressure	Heat convection	Free
Fuel interconnector face	0 V	/	/	Heat isolation	Roller
Air interconnector face	Operating voltage	/	/	Heat isolation	Roller

Table 5. Material parameters [51].

Parameters	Porosity	Permeability (m2)	Thermal Conductivity $(\mathbf{W}·{\mathbf{m}}^{-1}·{\mathbf{K}}^{-1}$)	Thermal Capacity $(\mathbf{J}·{\mathbf{K}\mathbf{g}}^{-1}·{\mathbf{K}}^{-1}$)
Fuel active layer	0.23	1 × 10−12	6	450
Fuel support layer	0.46	1 × 10−12	6	450
Electrolyte	/	/	2.7	550
Air active layer	0.3	1 × 10−12	11	430
Interconnector	/	/	20	550

Table 5. Material parameters [51].

Parameters	Porosity	Permeability (m2)	Thermal Conductivity $(\mathbf{W}·{\mathbf{m}}^{-1}·{\mathbf{K}}^{-1}$)	Thermal Capacity $(\mathbf{J}·{\mathbf{K}\mathbf{g}}^{-1}·{\mathbf{K}}^{-1}$)
Fuel active layer	0.23	1 × 10−12	6	450
Fuel support layer	0.46	1 × 10−12	6	450
Electrolyte	/	/	2.7	550
Air active layer	0.3	1 × 10−12	11	430
Interconnector	/	/	20	550

2.4. Model Setup

This paper mainly focuses on the optimization of the fuel electrode support layer thickness, air electrode rib area, and Ni-YSZ ratio.

(1)

The thickness of the fuel support layer is increased from 5.0 mm to 5.5 mm at an interval of 0.1 mm.

(2)

The air rib area portion is increased from 17% to 50%.

(3)

The ratio of Ni is increased from 20% to 60%.

3. Results and Discussion

3.1. Model Validation

The simulation results of the 3D SOEC model are then compared with experimental data, and

Table 6. Model validation parameters [52].

Parameters	Value	Unit
Operating Voltage	1.0–1.4	V
Operating Temperature	1073–1123	K
Operating Pressure	1	atm
Fuel Composition	H2:H2O = 50%:50%	/
Air Composition	O2:N2 = 21%:79%	/
Fuel Outlet	/	Convection

shows the relevant parameters during model validation. As shown in Figure 2, the I-V curves from the model closely match the experimentally measured I-V curves [52], confirming the accuracy of the SOEC model and the reliability of the solution method.

3.2. Macrostructural Thickness of Fuel Support Layer

First, we investigate the effect of fuel support layer thickness on the performance of the SOEC at a fuel inlet temperature of 1073 K and an operating voltage of 1.4 V, conditions commonly observed in practical applications. The hydrogen production rate is a key indicator for evaluating the performance of SOEC technology [50], as it is directly related to both the hydrogen production capacity and efficiency. Initially, we examine the effect of the fuel support layer thickness on the hydrogen production rate considering different fuel channel pore sizes.

$$\mathrm{Hydrogen}\mathrm{production}\mathrm{rate}={\int \int }_{\mathrm{A}}\left|\overrightarrow{\mathrm{u}}\right|\dot{\mathrm{y}}{\mathrm{H}}_{2,\mathrm{o}\mathrm{u}\mathrm{t}}\mathrm{d}\mathrm{A}$$

(36)

where A is the outlet area, and $\dot{\mathrm{y}}{\mathrm{H}}_{2,\mathrm{o}\mathrm{u}\mathrm{t}}$ is the ${\mathrm{H}}_2$ mole fraction at the outlet.

As shown in Figure 3, under the same flow channel pore size, when the thickness of the fuel support layer increases from 5.0 mm to 5.4 mm, the hydrogen production rate gradually increases. However, as the thickness of the fuel support layer continues to increase, the hydrogen production rate shows a declining trend. Additionally, when the thickness of the fuel support layer is the same, the hydrogen production rate gradually increases as the flow channel pore size decreases. As can be seen from Equation (36), the hydrogen production rate is closely related to the velocity of the fuel, along with the H₂ mole fraction at the outlet. The thickness of the fuel support layer can affect the fuel diffusion process [53], thereby influencing the amount of fuel participating in the reaction, which in turn affects the molar fraction of H₂ at the outlet. The diffusion of the fuel along the thickness direction can be represented by the diffusive flux [54].

$$\mathrm{J}\equiv {\displaystyle \frac{\mathrm{d}\mathrm{m}}{\mathrm{A}\mathrm{d}\mathrm{t}}}=\mathrm{D}{\displaystyle \frac{\partial \mathrm{C}}{\partial \mathrm{Z}}}$$

(37)

where D is the diffusion coefficient, ∂C is the concentration gradient, and ∂Z is the distance gradient. To adapt this equation to this model, it can be modified to Equation (37) as below [54]:

$$\mathrm{J}=\mathrm{D}{\displaystyle \frac{{\mathrm{C}}_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}}-{\mathrm{C}}_{\mathrm{f}\mathrm{u}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}}}{{\mathsf{\delta }}_{\mathrm{f}\mathrm{u}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}}}}$$

(38)

where ${\mathrm{C}}_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}}$ is the concentration in the channel, ${\mathrm{C}}_{\mathrm{f}\mathrm{u}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}}$ is the concentration in the fuel active layer, and ${\mathsf{\delta }}_{\mathrm{f}\mathrm{u}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}}$ is the thickness of the fuel support layer. Diffusion flux directly indicates the mass transfer effectiveness. Diffusion flux will be directly affected by the thickness ${\mathsf{\delta }}_{\mathrm{f}\mathrm{u}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}}$.

Figure 4 shows the variation in diffusion flux along the flow direction for different fuel support layer thicknesses. First, along the flow channel direction, the diffusion flux of H₂O under different fuel support layer thicknesses initially increases and then decreases. First, along the flow channel direction, the diffusion flux of H₂O under different fuel support layer thicknesses initially increases rapidly, but as the reaction progresses, the rate of increase slows down, and finally, the diffusion flux decreases. At the beginning of the diffusion process, the concentration difference between the channel and fuel active layer is significant. Therefore, at the beginning of the diffusion process, the diffusion flux shows an increasing trend. As the diffusion process progresses, the H₂O concentration difference between the flow channel and the fuel active layer gradually decreases along the direction of the flow channel thickness, thus slowing the rate of increase in diffusion flux. The decrease in diffusion flux near the outlet is due to the fact that as the diffusion process progresses, the concentration of the substance gradually approaches the equilibrium state at the outlet. As the concentration gradient becomes smaller, the diffusion flux decreases because it is proportional to the concentration gradient [55,56].

As the fuel support layer thickness increases from 5.0 mm to 5.5 mm, the diffusion flux first increases, reaching a maximum of 5.4 mm, and then decreases. However, further increasing the thickness of the fuel support layer results in a decrease in the diffusion flux. While a thicker fuel support layer enhances the mass flux of reactants [57], thereby increasing the diffusion flux, it also increases the transport resistance of water vapor within the support layer [58].

Moreover, the porosity and permeability of the fuel support layer determine the gas transport efficiency within the fuel electrode, affecting the electrochemical reaction rate. To examine the impact of porosity and permeability on SOEC performance with varying fuel electrode support layers, we investigated the change in H₂O partial pressure along the thickness of the fuel active layer under different porosity and permeability conditions of the fuel support layer. Figure S1a depicts a calculation position of Z = 0 mm. As shown in Figure S1b,c, along the flow direction, the partial pressure of H₂O first increases and then decreases. This is because, on the side of the active layer near the channel, H₂O diffuses from the channel through the porous electrode into the interior. Since the reaction has not yet occurred at this point, the diffusion rate of H₂O is higher than the consumption rate of the reaction, resulting in an increase in partial pressure. However, as one moves deeper into the active layer, the consumption rate of H₂O exceeds the diffusion replenishment rate, causing the partial pressure to decrease. As the thickness of the fuel support layer increases, the H₂O partial pressure along the thickness direction rises under higher porosity and permeability conditions of the fuel electrode support layer. This is because an increase in porosity and permeability facilitates gas flow within the porous electrode, thereby enhancing mass transfer efficiency.

During SOEC operation, the fuel active layer is subjected to high temperatures, high current densities, and stresses generated by electrochemical reactions. Excessive maximum first principal stresses can result in cracking, breakage, or deformation of the fuel active layer material, thereby compromising its structural stability. D-R represents the distance between the fuel pole channel and the fuel support layer. Since the fuel channel is embedded within the fuel pole support layer, a smaller channel aperture and a thicker support layer thickness will increase the distance between the channel and the support layer; that is, the value of D-R will increase. As shown in Figure 5, with the decrease in the thickness of the fuel electrode support layer and the increase in the flow channel aperture, that is, D-R gradually decreases, the temperature gradient gradually increases. This increase reduces the temperature gradient, leading to a decrease in the maximum first principal stress. Figure 6 shows the variation in the maximum first principal stress of the fuel active layer as a function of the fuel support layer thickness. As shown in Figure 6, the maximum first principal stress decreases with an increase in the thickness of the fuel support layer and a reduction in the fuel channel aperture.

Fuel utilization is also an important indicator for evaluating the performance of the SOEC. The volume fraction of Ni is 30–40% in this section. The relationship between fuel utilization and the thickness of the fuel support layer is shown in Figure 7. As can be seen from the figure, with the increase in the thickness of the fuel anode support layer, the fuel utilization rate first increases and then decreases, reaching the maximum at 5.4 mm, which is consistent with the excellent mass transfer effect when the thickness of the fuel support layer is 5.4 mm.

3.3. Macrostructure Air Electrode Rib Area

To explore the influence of diverse rib areas on the overall performance of the SOEC, this step of the study was conducted under operating conditions of 1.4 V and a working temperature of 1073 K. The rib design is critical in ensuring the uniform distribution of physical quantities, as demonstrated in a previous study [59]. Therefore, optimizing the rib design is a key strategy for enhancing both the electrochemical performance and the long-term stability of the SOEC. In this study, we quantify the rib area using the rib area portion, which is defined and described by Equation (39) [60], for the purpose of a comparative analysis. For the convenience of calculation, a flow channel is selected here to define the rib area portion.

$$\mathrm{R}\mathrm{i}\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}={\displaystyle \frac{\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}}{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}+\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}}}={\displaystyle \frac{\mathrm{r}}{\mathrm{d}+\mathrm{r}}}\times 100\mathrm{\%}$$

(39)

To eliminate the influence of other factors, this section controls the value of d + r so that it remains constant while changing the value of d to adjust r, thereby altering the rib area portion when studying the influence of rib area ratio on SOEC performance.

Table 7. Rib area.

Channel Width (mm)	Rib Width (mm)	Rib Area Portion (%)
1.5	1.5	50
1.75	1.25	42
2.0	1	33
2.25	0.75	25
2.5	0.5	17

shows the rib area portion. We first investigated the hydrogen production rate of the SOEC under varying rib area portions, which correspond to different rib areas. As shown in Figure 8, the hydrogen production rate gradually decreases as the rib area increases. This phenomenon is linked to the reduction in the molar fraction of hydrogen at the outlet as the channel width expands. In Figure 8, it is evident that the current density decreases with a decreasing rib area portion. The reduction in current density leads to a decrease in the intensity of the electrochemical reaction, which subsequently causes a drop in the hydrogen production rate.

Since the concentration of reactants is related to the current density, to explore the influence of rib area portion on the current density, we analyzed the relationship between the diffusion coefficient and temperature, as shown in Equations (40) and (41).

$${\mathrm{D}}_{\mathrm{i}}^{\mathrm{m}}={\displaystyle \frac{1-{\mathrm{x}}_{\mathrm{i}}}{\sum _{\mathrm{i}\ne \mathrm{j}}\left(\frac{{\mathrm{x}}_{\mathrm{i}}}{{\mathrm{D}}_{\mathrm{i}\mathrm{j}}}\right)}}$$

(40)

$${\mathrm{D}}_{\mathrm{i}\mathrm{j}}=0.0143{\mathrm{T}}^{1.75}{\displaystyle \frac{{[\frac{1}{{\mathrm{M}}_{\mathrm{i}}}+\frac{1}{{\mathrm{M}}_{\mathrm{j}}}]}^{\frac{1}{2}}}{\mathrm{P}{\left[{\mathrm{v}}_{\mathrm{i}}^{\frac{1}{3}}+{\mathrm{v}}_{\mathrm{j}}^{\frac{1}{3}}\right]}^2}}$$

(41)

where $\mathrm{x}$ is the mole fraction, ${\mathrm{D}}_{\mathrm{i}\mathrm{j}}$ is the binary diffusion coefficient, and $\mathrm{v}$ is the diffusion volume.

After the reduction in the air electrode rib area portion, the channel width will increase, which in turn will carry away more heat and lead to a decrease in temperature [61]. As can be seen from Equation (41), a change in temperature will result in a change in the diffusion coefficient. With the decrease in temperature, the diffusion coefficient will also decrease. Under the same conditions, it becomes more difficult for H₂O to reach the TPB, leading to a reduction in the concentration of reactants, which in turn causes a decrease in current density.

Uneven oxygen concentration distribution can lead to local overheating. The high-temperature areas may cause uneven thermal expansion of the material, increasing thermal stress and even leading to structural damage or the formation of cracks. In addition, changes in the rib area portion will directly affect the stress in the air electrode active layer. Uneven oxygen concentration distribution can lead to local overheating. The high-temperature areas may cause uneven thermal expansion of the material, increasing thermal stress and even leading to structural damage or the formation of cracks. In addition, changes in the rib area portion will directly affect the stress in the air electrode active layer. Then, we examine the distribution of O₂ concentration in the rib with varying area portions to assess the uniformity of oxygen distribution. Figure 9 shows the distribution of oxygen molar concentration for different rib area portions. From the figure, it can be seen that the trend of O₂ concentration change along the flow direction is consistent. As the rib area ratio increases, the oxygen production at the outlet gradually increases, which is consistent with the trend of current density. However, as the rib area decreases, a more uniform oxygen concentration distribution in the air electrode under the ribs and in the channel is observed in the model with rib area portions of 42%, 33%, 25%, and 17% compared to the model with a rib area portion of 50%, which can be attributed to the diffusion of oxygen ions—produced by the electrolysis of water—through the YSZ electrolyte membrane to the TPB near the active layer of the air electrode. At the TPB, oxygen ions lose electrons and are converted into oxygen. As a result, the air active layer exhibits the highest oxygen concentration [27]. The absence of channels beneath the rib restricts gas diffusion, leading to localized high-concentration areas. When the rib area is larger, it results in more extensive high-oxygen-concentration zones, contributing to a more uneven concentration distribution. The most significant change in oxygen molar concentration was observed at the rib area portions of 50% and 42%, due to the higher current density and more intense reactions in the larger rib areas, leading to increased oxygen production. In comparison to the 50% rib area portion, the oxygen concentration under the 42% rib area portion exhibited less variation with rib area, and the oxygen concentration was more uniformly distributed, stabilizing around 42%.

Figure 10 shows the distribution of the first principal stress in the air active layer, where positive values represent tensile stress and negative values represent compressive stress. From the figure, it can be seen that the active region in contact with the ribs is mainly subjected to compressive stress. As the rib area portion increases, the value of the maximum compressive stress in this region gradually decreases. Moreover, as the rib area portion decreases, the compressive stress gradually decreases, and the maximum first principal stress also gradually decreases. This is because, as the rib area portion decreases, the amount of oxygen generated gradually decreases, and the oxygen concentration distribution becomes more uniform, thereby reducing thermal stress.

Figure 11 shows the relationship between fuel utilization and the rib area portion. The volume fraction of Ni is 30–40% in this step. As the rib area portion increases, the fuel utilization gradually rises. This can be attributed to the increase in current density, which leads to an increase in reaction rate and thus greater fuel utilization.

The above analysis shows that varying the air rib area portion can cause a change in SOEC performance, and that the performance is better when the rib area portion is about 42%.

3.4. Microstructure-Ni-YSZ Ratio

Based on Equations (26)–(33), the specific surface area, coefficient of thermal expansion (CTE), and elastic modulus used in the model setup of this study can be calculated.

Table 8. STPB, CTE, and elastic modulus.

Volume of Ni (%)	STPB (m2/m3)	CTE (10−6 K−1)	Elastic Modulus (GPa)
20	0.51 × 105	9.062	130
30	1.43 × 105	9.473	131
40	3.11 × 105	9.884	132
50	6.00 × 105	10.295	133
60	11.2 × 105	10.706	135

shows the specific surface area, CTE, and elastic modulus of activity corresponding to different Ni volume fractions.

Next, we investigate the effect of varying Ni volume fractions on the overall performance of the SOEC under an operating voltage of 1.4 V and a working temperature of 1073 K. Figure 12 illustrates the relationship between the hydrogen production rate and Ni volume fraction. As shown, the hydrogen production rate increases progressively with an increasing Ni volume fraction. This trend mirrors the current density, which also increases with a higher Ni content. The increase in hydrogen production rate can be attributed to the corresponding rise in current density. As indicated in Table 8, the active specific surface area increases with the Ni volume fraction, providing a larger reaction surface. This enhancement promotes electrochemical reactions, leading to a higher current density and, consequently, an increased hydrogen production rate.

The uniformity of gas flow distribution will affect the temperature gradients [62], which in turn influence thermal stress. Figure S2 illustrates the flow uniformity of H₂O along the flow direction under different Ni volume fractions. As shown in the figure, the homogeneity of gas distribution generally exhibits a first increase followed by a decreasing trend along the flow direction. Additionally, as the Ni volume fraction increases, the flow uniformity gradually decreases, which in turn leads to an increasing temperature gradient.

The SOEC has a sandwich structure. The electrolyte layer is sandwiched in the middle, with the lowest thickness. Moreover, the electrolyte layer has a lower thermal expansion coefficient, making it more susceptible to failure due to high thermal stress [62]. Therefore, it is particularly important to study the thermal stress of the electrolyte. Figure 13 shows the temperature distribution of the electrolyte layer at different nickel (Ni) volume fractions. It can be seen from the figure that as the Ni volume fraction increases, the temperature of the electrolyte gradually rises. The voltage selected in this section is 1.4 V, which is higher than the thermionic neutral voltage. Therefore, the SOEC is in a state of exothermic reaction. The increase in Ni volume fraction leads to an increase in the TPB, making the reaction more intense and causing the temperature to rise. The increase in temperature will lead to changes in thermal stress. Figure 14 shows the relationship between fuel utilization and the maximum first principal stress of the electrolyte layer with varying Ni volume fractions. It can be seen from the figure that as the Ni volume fraction increases, the maximum first principal stress of the electrolyte also increases gradually. This is because the increase in Ni volume fraction leads to a rise in temperature, which in turn increases the thermal stress. Additionally, the increase in Ni volume fraction also results in a higher thermal expansion coefficient, exacerbating the thermal stress caused by the mismatch in thermal expansion coefficients between the fuel active layer and the electrolyte. Therefore, as the Ni volume fraction increases, the maximum first principal stress of the electrolyte gradually increases.

Furthermore, it can be observed from Figure 14 that as the Ni volume fraction increases, fuel utilization also increases. This is related to the increase in TPB due to the higher Ni volume fraction. The larger TPB leads to a more intense reaction, as more H₂O is consumed, resulting in a corresponding increase in fuel utilization. In solid oxide cells (SOCs), the fuel utilization typically ranges from 60% to 80% [63]. It can be seen in the figure that when the Ni volume fraction is between 50% and 60%, the fuel utilization meets the standard.

Through the above analysis, it was found that the increase in Ni volume fraction leads to an increase in the hydrogen production rate. In addition, as the Ni volume fraction increases, the uniformity of H₂O flow along the flow direction gradually weakens, and this decline becomes particularly evident when the Ni volume fraction reaches 60%. Furthermore, as the Ni volume fraction increases, the first principal stress gradually increases. Considering the hydrogen production rate, thermal stress, and flow uniformity, it can be concluded that when the Ni volume fraction is 50–60%, the SOEC exhibits a relatively excellent performance.

4. Conclusions

To investigate the effects of macrostructure and microstructure on the performance of the SOEC, a coupled multiphysics field model for SOEC electrolysis of water for hydrogen production was established. The following conclusions are drawn from the analysis of factors such as the thickness of the fuel support layer, the rib area portion, and the Ni-YSZ ratio:

(1)

As the thickness of the fuel support layer increases, the hydrogen production rate and diffusion flux first increase and then decrease, reaching the optimum when the thickness of the fuel electrode support layer is 5.4 mm. Additionally, as the thickness of the fuel support layer increases, the maximum first principal stress gradually decreases.

(2)

Although the oxygen generation and current density increase with the increase in the rib area portion, the distribution of oxygen concentration is more uniform, and the thermal stress is lower when the rib area portion is 42%. Therefore, the performance of the SOEC is optimal when the rib area is 42%.

(3)

When the Ni volume fraction increases from 20% to 60%, both the hydrogen production rate and thermal stress increase, and the hydrogen production rate increases by 86%. The flow uniformity of H₂O gradually decreases. Notably, when the Ni volume fraction is below 50%, the decline in flow uniformity is more significant. After a comprehensive analysis, it can be concluded that the SOEC exhibits a better performance when the Ni volume fraction is 50%.