1. Introduction
Fuel cells are power-generation devices that directly convert chemical energy into electrical energy at high efficiency through electrochemical oxidation reactions. Solid oxide fuel cells (SOFCs) are potential candidates for large-scale stationary and co-power generation due to low noise, better fuel impurity tolerance, and flexibility [
1,
2]. SOFCs are categorized into two major types, which are planar and tubular configurations. The planar SOFCs have the specific advantages of manufacturing considerations and higher power density, but they require high matching of the thermomechanical properties and sealing between the component materials [
3]. Several variations of the planar SOFC designs are possible, such as electrode-supported SOFCs, electrolyte-supported SOFCs, and symmetrical SOFCs [
4,
5]. Symmetrical SOFCs can improve compatibility between cell components and simplify cell configuration. However, the further development of electrode materials and novel structures is still the key to the practical use of symmetrical SOFCs [
4]. The anode-supported SOFCs can effectively reduce ohm loss, thereby reducing the operating temperature of the SOFC, and the thickened anode can realize the function of internal reforming [
6].
However, the reaction mechanisms and transport processes that take place in the micro/nanoporous electrode pores and TPBs (TPBs are thin boundaries between the electrode and electrolyte micrograins surrounded by gaseous phases that fill the pores) inside SOFCs are extremely complex [
7,
8]. In addition, the thermal stress caused by the mismatch of coefficients of thermal expansion between materials and the uneven temperature distribution can easily lead to cracks in the electrode layer or seal failure [
9,
10,
11]. It is difficult to make real-time measurements of complex transport phenomena and related physical quantities inside SOFCs. With the development of computers and commercial software, the reliability and accuracy of numerical simulations have been improved and verified. Numerical simulations have gradually become an important approach to studying SOFCs [
12,
13,
14,
15]. Lee et al. [
8] established a three-dimensional single-channel numerical model with hydrogen as the fuel to study the thermal-fluid reacting environment and local thermodynamic state in SOFCs. The results showed that high fuel utilization induces a narrow and nonuniform reaction zone near the fuel inlet and substantial species concentration gradients along the cell. Ong et al. [
16] established a one-dimensional MEA (membrane electrode assembly) model and demonstrated that part of the CO also participates in the electrochemical reaction due to the equilibrium limitations of water–gas shift reaction (WGSR) under high-temperature conditions. Xie et al. [
17] investigated the methane-reforming mechanism and multitransport processes by the finite element method. Numerical results showed that the reaction processes in the anode of direct methane-fueled SOFC are more sluggish than the oxygen reduction process in the cathode, which is the opposite case of the hydrogen-fueled SOFC. It is noted that the reaction rate mismatch between the endothermic methane steam-reforming reaction (MSR) and the exothermic electrochemical reaction tends to cause a temperature gradient [
7]. As researched by Tseronis et al. [
18], the cooling effect of the methane steam-reforming reaction leads to a significant local temperature drop near the anode inlet and a large temperature difference inside the SOFC. The uneven temperature will cause thermal stress, even the proper materials are chosen at the operating temperature, as reported by Chiang et al. [
19]. Many researchers have used a linear intrinsic structure model to describe the relationship between stress and strain inside the cell, which assumes that the components of the cell undergo elastic deformation and that the mechanical theory is following Hooke’s law [
20,
21,
22]. Xu et al. [
23] numerically examined the thermal stress of anode-supported planar SOFCs fueled by methane syngas. The results show that due to the endothermic MSR in the porous nickel-based anode, there is a significant temperature gradient between the anode support layer and the anode active layer, resulting in a sudden stress change between the electrode layers. Additional stress from the final arrangement and fixation of the cells in the SOFC stack is probably one of the causes for crack of the cell.
From the literature research above, it is clear that the complex reforming reaction system inside the porous nickel-based anode of hydrocarbon-fueled SOFCs leads to a different internal transport mechanism, as well as thermal stress distribution from that of hydrogen-fueled SOFCs. Furthermore, many scholars have studied only a single-channel unit (periodic unit consisting of PEN, channels, and interconnect) as the object due to the limitations of previous methods. Based on the previous research, a comprehensive thermo–electro–chemo–mechanical coupled three-dimensional model (including a frame, sealant, interconnect, and manifold) has been applied to elucidate the thermofluid reacting environment and thermal stress inside the SOFCs when fueled with hydrogen and methane syngas.
4. Results and Discussion
The temperature distributions inside SOFCs are shown in
Figure 3. As expected, the temperature profiles of the different fuels are rather different. It is found in
Figure 3a (hydrogen fuel) that the high-temperature zone (1117 K) of the PEN is mainly concentrated in the central region of units 5, 6, and 7, and then gradually decreases from this region towards the surrounding walls. The first reason is that the initial molar concentration of H
2 and O
2 is larger, and the electrochemical reaction gradually increases along the x-axis, releasing a large amount of heat in the central region, and then the concentration of the gas species decreases and the electrochemical reaction weakens. The second reason is that there is a significant temperature difference between the inside of the cell and the outside environment, resulting in heat being radiated from the outside wall through electromagnetic waves, while the outside ambient air also takes away some of the heat through thermal convection. As for the methane syngas fuel (
Figure 3b), the temperature in the latter part of units 5, 6, and 7 is the lowest (1032 K), and then gradually increases from this region toward the surrounding walls. The low temperature occurs because the initial molar concentration of H
2 is small, the initial concentration of CH
4 and H
2O is large, and the heat absorbed by the endothermic MSR reaction is greater than that released by electrochemical reactions. The gradual increase in temperature toward the surrounding walls is due to the absorption of energy radiated from outside by the outside walls and the partial heat brought by the thermal convection of outside air. From
Figure 3a, it also found that the temperature distribution trend of each layer along the x-axis is almost the same, and the same trend is also found from
Figure 3b. The temperature difference is mainly due to the different thermal conductivities as well as the thickness of each component. In addition, since no chemical reactions occur in the manifold and only the diffusion of gas species is involved, the manifold temperature changes are small.
The velocity distributions inside SOFCs are shown in
Figure 4. It is found that the velocity field distributions are the same for different fuels. The velocity field distribution is influenced by various factors, such as the temperature difference between the channel wall and the gas inside the channel, the concentration variation of the gas species (mass diffusion of gas species and the participation of gas species in electrochemical and chemical reactions), the gas mass flow rate, etc. It can be seen that in the y-z plane x = 0.024 m; the velocity in the center of the channel is greater than that near the wall. This is due to the viscosity factor of the fluid itself, which causes the flow of gas species to form a boundary layer at the wall of the channel. It can be seen also that the velocity in the cathode channel is larger than that in the anode channel. This is because enhancing the thermal convection effect of the manifold and channels by increasing the airflow is usually used as the main way to cool the SOFC while providing enough air to ensure the reduction reaction rate so that the reduction reaction occurs fully, so the air flow rate is generally much higher than the fuel gas flow rate and the air velocity on the cathode side is higher than the fuel velocity on the anode side. In addition, the transfer of gas species in porous media follows Darcy’s law, which relies mainly on the pressure drop caused by additional resistance to percolation and is influenced by the pore size and particle scale of porous electrodes, while the transfer of gas components in SOFC manifolds and gas channels is mainly by convection, so the velocity gradient in gas channels is significantly larger than that in porous media. It is found that the average velocity of channels 4 and 5 is significantly higher than that of channels 1, 2, 6, and 7, and the average velocity of anode channel 4 (1.35 m·s
−1) is the highest, and channel 1 is the slowest. As can be seen in
Figure 2b, the inlet/outlet of channels 4 and 5 are closer to the inlet/outlet of the anode manifold, and the fuel in channels 4 and 5 can come up and flow directly from the inlet manifold to the outlet manifold, while the flow to channels 1 and 2 requires a longer diffusion path and slows down the velocity.
The molar fractions inside the hydrogen-fueled SOFC are shown in
Figure 5. The general trend of the molar fraction of H
2 decreases gradually along the fuel flow direction (x-direction), while the molar fraction of H
2O increases gradually. This is due to the H
2-H
2O electrochemical reaction that takes place in the anode active layer, in which there is a depletion of H
2 and a production of H
2O. It can be seen that in the channels (x-y plane, z = 0.0015 m), the molar fraction of H
2 in channels 1 and 2 is significantly smaller than that in the central region of the cell. This is consistent with the velocity field analysis above, where H
2 comes up from the anode manifold and diffuses to the edge channels, requiring a longer transport path. Also from the middle section of the anode active layer (x-y plane, z = 0.00238 m) in
Figure 5b, it can be found that the higher molar fraction region of H
2O is located at the latter part of units 6 and 7. This is because the temperature in this region is higher and the charge exchange process is stronger, which drives the acceleration of the electrochemical reaction and consumes more hydrogen to generate water, while the region is also a marginal region where the generated water is difficult to transfer out. The molar fractions inside the methane syngas-fueled SOFC are shown in
Figure 6. As can be seen from
Figure 6, the molar fraction distributions of H
2 and H
2O in the methane syngas-fueled SOFC change significantly from the anode support layer to the active layer compared to the hydrogen-fueled SOFC. This is due to the simultaneous electrochemical reactions of methane reforming and H
2-H
2O in the anode activation layer. CH
4 is the main reactant of the reforming reaction on the anode side, and the continuous reforming reaction causes the molar fraction of CH
4 to gradually decrease along the direction of gas flow (positive direction of the x-axis). CO
2 is the dual product of the water vapor conversion reaction and CO-CO
2 electrochemical reaction, and will gradually accumulate along the direction of the main flow channel as the above reaction proceeds. When the gas flow rate is low, most of the reactants (H
2, CO) are consumed in the TPB of anode active layer, resulting in low molar fractions of H
2 and CO in the anode active layer. If the gas flow rate is too large, only some of the methane is reformed, resulting in a higher methane molar fraction in the anode support layer. Combining
Figure 5 and
Figure 6 for analysis, it found that the uneven distribution of gas components is mainly concentrated in the electrode area corresponding to the channels and the ribs.
The reaction rate of methane syngas-fueled SOFC in the anode layer is shown in
Figure 7. In
Figure 7a (MSR), it is clear that the general trend of the MSR reaction rate decreases gradually along the fuel flow direction (x-direction), and the maximum value of the MSR reaction rate appears in the first half of unit 1 and the minimum value appears in the latter part of unit 7. The reaction rate is mainly influenced by the supply of reactants involved in the reaction and the amount of heat required for the reaction. From the temperature field distribution above, it is clear that the temperature of unit 1 near the wall is higher, coupled with the larger initial molar fraction of H
2O and CH
4 at the entrance of unit 1, so the reaction rate in this region is higher. This is because the MSR near the gas channel inlet is strong and provides a large amount of CO for the WGSR reaction, so the WGSR reaction rate in this region is large and then decreases gradually along the gas flow direction. It can also be seen that the reaction rate of WGSR becomes negative from the anode support layer (x-y plane, z = 0.00218m) to the anode active layer (x-y plane, z = 0.00238 m). The first reason is that the permeability of the anode active layer is smaller than that of the anode support layer (the relevant physical parameters of the solid structure are in
Table 3). The second reason is that the CO-CO
2 electrochemical reaction in the anode active layer also consumes a large amount of CO continuously, prompting the WGSR at this place to carry out the inverse reaction to reach the equilibrium, and the reaction rate of the WGSR becomes negative.
The first principal stress is normally the maximum stress normal to a plane, which may reduce thermal cracks in the ceramic cell. Therefore, this study focuses on comparing the first principal stresses, with positive and negative values representing tensile and compressive stresses, respectively [
23].
The first principal stress in the middle section of each electrode layer of the PEN is shown in
Figure 8. The maximum values of the first principal stresses (comparing the magnitude of the stress extremes, positive and negative only represent the direction) for the anode support layer, electrolyte layer, diffusion barrier layer, and cathode current collection layer fueled by hydrogen are 13.60 MPa, 57.80 MPa, 15.50 MPa, and 2.89 MPa, respectively. The maximum values of the first principal stresses fueled by methane are 7.7 MPa, 20.9 MPa, 6.8 MPa, and 4.76 MPa, respectively. It can be found that the maximum stress variation gradient between electrode layers is larger for hydrogen SOFCs (44.2 MPa) than for methanol syngas SOFCs (14.1 MPa). It can also be seen that the maximum value of the first principal stress in the electrolyte is much higher than that in the anode and cathode. This is because the electrolyte layer has the lowest coefficient of thermal expansion and the highest Young’s modulus of all components (the thermomechanical parameters of each component in
Table 5) and is susceptible to mechanical constraints from adjacent components. It is found in
Figure 8a (hydrogen fuel) that the stress distribution trends of the anode support layer, electrolyte layer, and barrier layer are the same, with larger stress values in the center region of unit 1 and smaller stress values in the inlet and outlet side of unit 1. As for the methane syngas fuel (
Figure 2b), it is revealed that each electrode layer is subjected to both tensile and compressive stresses. Combining
Figure 8a,b for analysis, it found that there is a significant stress gradient in the y-axis direction in the anode support layer and the cathode current density collection layer. This is because the positive corresponding electrode part of the ribs is constrained by the fixed stress of the ribs as well as the upper and lower connectors, while the gas channel cannot provide support. The first principal stress of assembly components is shown in
Figure 9. As expected, the thermal stress distribution of different fuels is rather different. It is noted that the remaining components of the methane syngas-fueled SOFC have a greater range of stress variation than the hydrogen-fueled SOFC, and the stress extremes occur in the edge regions where the adjacent assembly components are in contact. It is found, in
Figure 9 (interconnect), that the higher stress zone located at the lower interconnect near the anode manifold inlet is subject to tensile stress, while the methane syngas-fueled SOFC is mainly subject to compressive stress in this region. It is also found that the sealant was subjected to the lowest stress value of all components in
Figure 9 (sealant). In addition, the interconnect and the metal frame are subjected to both tensile and compressive stresses, which may lead to delamination or severe deformation, resulting in gas leakage and other conditions. In this case, it can be improved by increasing the external load of the stack or by finding a more compatible sealing material.