The planar SOFC with a reverse channel was used. The basic unit is composed of a positive electrolyte negative (PEN), seal layer, and collector plate. PEN is a single cell with an anode, electrolyte, and cathode. The PEN is connected to the collector plate by bonding with glass–ceramic. The geometric model of the basic repetitive unit is shown in Figure 1. The length and width of SOFCs are 10 and 10 cm, respectively. The materials of the cathode, electrolyte, anode, seal, and collector plate are LSCF, YSZ, NiO-YSZ, GC-9, and 430 stainless steel, respectively. The thickness of the cathode, electrolyte, anode, and seal is 40, 10, 600, and 100, respectively. Both the height and width of the air/fuel channel in the collector plate are 500 μm. The total height of the collector plate is 2000 μm. The cooling channel with a height of 200 μm is located in the center of the collector plate.

The SOFC stack is heated to 650°C at a heating rate of 1°C/min. Then, the stack is operated at 650°C for 50,000 h. The assembly pressure is 0.1 MPa. During the operation, the creep effect at high temperatures is considered. In addition, the SOFC stack is assumed to be stress-free at the bonding temperature due to the stress relaxation at high temperatures. The residual stresses generated from bonding temperature to room temperature are also considered. The variations in thermal stresses, creep damage, and failure probability with time of the SOFC are calculated during the operation. Temperature-dependent thermophysical and mechanical properties are used, which are obtained from Wang et al. (2019).

Because the SOFC stack is a periodic repeating structure, the finite element model is simplified as a two-dimensional (2D) plain-strain model to save computing time. The finite element meshing for the conventional and new SOFC stacks is shown in Figure 2. The element width is 100 μm, and the element height is changed from 5 μm to 100 μm for different component thicknesses. The element type is a four-node plane reduction integrating element (CPS4R). During the stress analysis, in order to limit the model displacement and have freedom, the displacements of points A and C were restrained along the X-direction, and the displacements of points B and D were restrained along the Y-direction, as shown in Figure 2.

The thermal stress is calculated by thermoelastoplastic analysis. For the operation stage of the SOFC stack, the total strain rate can be decomposed into four components as follows: ε ˙ = ε ˙ e + ε ˙ p + ε ˙ t s + ε ˙ c , (1) where ε ˙ e , ε ˙ p , ε ˙ t s , and ε ˙ c stand for the rate of elastic strain, plastic strain, thermal strain, and creep strain, respectively. For the start-up stage, the creep effect is ignored due to the short time, and the total strain in the SOFC is induced by the elastic, plastic, and thermal strains. Elastic strain is calculated using the isotropic Hooke’s law with temperature-dependent Young’s modulus and Poisson’s ratio. Thermal strain is calculated using the temperature-dependent coefficient of thermal expansion (CTE). For plastic strain, a rate-independent plastic model with von Mises yield criteria and a linear isotropic hardening model is employed. The isotropic hardening model shows that the yield surface expands with the accumulated plastic strain, while its center remains in the same place of the stress field, as shown in Figure 3.

Creep strain should be calculated accurately to predict the creep damage under a high-temperature state. The creep strain during the operation stage is calculated based on the Wen–Tu creep damage constitutive model (Wen et al., 2013): ε ˙ i j = 3 2 B σ e q n − 1 S i j 1 + β 0 σ 1 σ e q 2 n + 1 2 , (2) β 0 = 2 ρ n + 1 + 2 n + 3 ρ 2 n n + 1 2 + n + 3 ρ 3 9 n n + 1 3 + n + 3 ρ 4 108 n n + 1 4 , (3) ρ = 2 n + 1 π 1 + 3 / n ω 3 / 2 , (4) where ε ˙ i j c is the creep strain rate tensor, B is the creep strength coefficient related to the steady state, σ e q is the von Mises stress, S i j is the deviatoric stress tensor, ρ is the damage parameter of microcracks, n is the stress exponent related to the steady state, β 0 is the parameter related to ρ and n, and ω is the damage state variable ranging from 0 to 1.

The creep damage is calculated by the ductility exhaustion approach (Wen and Tu, 2014). In this approach, when the local creep strain accumulation reaches the creep ductility (creep fracture strain) value, the damage reaches the critical value. The ductility exhaustion approach is expressed in Eq. 5. In addition, the multiaxial creep fracture strain is calculated using the modified microscopic cavity growth model, as shown in Eq. 6 ω ˙ = ∫ 0 t ε ˙ c ε f * , (5) ε f * = ε f ⁡ exp 2 3 n − 0.5 n + 0.5 / exp 2 n − 0.5 n + 0.5 σ m σ e q , (6) where σ m is the hydrostatic stress, ε ˙ c is the equivalent creep strain rate, ε f is the uniaxial creep failure strain, representing the creep resistance at high temperature, and ε f * is the multiaxial creep failure strain. When the damage reaches 0.99, the material is considered to be damaged completely, and crack initiation occurs. The creep and Weibull parameters of every parts of SOFC at 600 °C are shown in Table 1 (Wang Y. et al., 2022).

Currently, the failure calculation of SOFC is mainly based on the Weibull statistical method. Weibull (1939) proposed a more flexible mathematical distribution model to fit the spread of brittle material failure strength. The traditional generalized form of the Weibull distribution model is usually expressed as (Anandakumar et al., 2010): P = 1 − exp − ∫ V j σ σ 0 m d V j V 0 , (7) where P is the survival probability, σ ₀ is the characteristic strength that represents a scale parameter for the distribution, and m is the Weibull modulus. V ₀ is a reference volume linked to the characteristic strength. V _j is the volume of the material. The Weibull method shows that the failure probability is controlled by volume and applied stress. However, since the applied stresses in a SOFC are relaxed gradually during creep, it can also cause the growth of the pre-existing flaws and nucleation of new cavities and cracks by the creep damage. In this study, the failure probability is calculated based on the time-dependent failure probability model proposed by Zhang et al. (2018): P = 1 − exp − ∫ V j ε e η m V j V 0 , (8) where η is the scale parameter that expresses the characteristic strain. The corresponding failure probability is obtained at each increment by importing the equivalent creep strain. Here, we assume that the Weibull parameters are the same before and after creep. The Weibull parameters of every part of the SOFC at 600°C are shown in Table 1 (Wang Y. et al., 2022).

The thermal stresses are generated because of temperature variation from room temperature to operation temperature. Figure 4 shows the contour distribution of CEEQ, creep damage, and failure probability after 50,000 h of operation. The maximum CEEQ and creep damage are all located at the sealing layer. The failure probability is located at the sealing layer between the anode and bipolar plate. The CEEQ, creep damage, and failure probability at other positions are very small. The distribution of creep damage is similar to CEEQ. The maximum creep damage and failure probability are 0.68 and 6.81E-4, respectively. The marginal area of the SOFC stack, especially in the sealing layer, is the key focus area.

In order to analyze the stress and damage in the marginal area clearly, Figure 5 shows the CEEQ, creep damage, and failure probability along path P1 after 50,000 h of operation. Here, path P1 is the distance from a collector plate→sealing layer→PEN→another collector plate. The maximum CEEQ and creep damage are all located in the sealing layer, while the maximum von Mises stress is located in the cell. The variation law of creep damage is the same as that of the CEEQ. The von Mises stress in the sealing layer is very small, so it indicates that the creep damage and failure probability are determined by the CEEQ rather than the von Mises stress.

Because the maximum creep damage and failure probability are all located in the sealing layer, variations occur in CEEQ, creep damage, von Mises stress, and failure probability during the operation with time of a node in the sealing layer, as shown in Figure 6. It obviously shows that the variation curves are divided into three stages. At stage I, the CEEQ, creep damage, and failure probability are greatly increased, while the von Mises stress is greatly decreased. At stage II, the CEEQ, creep damage, and failure probability are increased gradually, while the von Mises stress is decreased gradually. At stage III, the CEEQ, creep damage, and failure probability continue to increase at a smaller rate, while the von Mises stress decreases slightly. More attention should be paid to the thermal stress and creep damage of the SOFC stack in the early service period, especially within 10,000 h.

The cooling channel was employed in the SOFC stack. Figure 7 shows the contour distribution of CEEQ, creep damage, and failure probability after 50,000 h of operation for the optimized SOFC stack with a cooling channel at 600°C. Compared with Figure 4, the CEEQ, creep damage, and failure probability were all decreased. The CEEQ and creep damage were decreased by 65% and 72%, respectively. The failure probability was decreased by 10 times. It is proved that the cooling channel can reduce creep damage and failure probability, thus increasing the service life at high temperatures.

The marginal area is the potential area for failure to occur, and the CEEQ, creep damage, and von Mises stress along path P1 in the marginal area after 50,000 h of operation for the optimized SOFC stack are shown in Figure 8. Similar to the conventional SOFC stack, the maximum CEEQ and creep damage are located in the sealing layer, where the von Mises stress is the minimum. Furthermore, the CEEQ and creep damage are the same for the two types of sealing layers (anode/bipolar plate and cathode/bipolar plate). The maximum CEEQ and creep damage are 0.021 and 0.135, respectively. The maximum von Mises stress is 69 MPa, which is located in the bipolar plate adjacent to the cathode side. The maximum von Mises stress is transferred from the cathode side to the bipolar plate side.

Figure 9 shows the variation in CEEQ, creep damage, von Mises stress, and failure probability of a node in the sealing layer during operation with time for the optimized SOFC stack. The variation curves are also divided into three stages, which are similar to the curves of the conventional SOFC stack. The difference is that the CEEQ, creep damage, and failure probability have almost no changes at stages II and III. After 50,000 h of service, the creep damage and failure probability for the optimized SOFC stack are still smaller than those for the conventional SOFC stack. It indicates that the creep damage and failure probability mainly accumulate at the first stage within 6,000 h for the optimized SOFC stack. The attenuation of the SOFC stack due to creep damage is mainly concentrated in the early stage, and it is negligible in the late stage.

Based on the aforementioned analysis, the cooling channel can reduce creep damage and failure probability of the SOFC stack. The temperature of the cooling channel is set at 600°C. In order to analyze the cooling channel temperature on the controlling effect, the CEEQ, creep damage, and failure probability of the optimized SOFC stack with a low cooling temperature (450°C) were also discussed. Figure 10 shows the CEEQ, creep damage, and failure probability after 50,000 h of operation for the optimized SOFC stack at a cooling temperature of 450°C. The distributions of CEEQ, creep damage, and von Mises stress are similar to those for the higher temperature (600°C). Compared to 600°C, the CEEQ and creep damage at 450°C decreased from 0.23 to 0.17 and from 0.14 to 0.11, respectively. The maximum von Mises stress is transferred from the bipolar plate side to the cathode side, and its value in the bipolar plate is increased.

Figure 11 shows the variation in CEEQ, creep damage, von Mises stress, and failure probability of a node in the sealing layer during the operation with time for the optimized SOFC stack at a low cooling temperature. It can be seen that the variation curves are similar to those of the higher cooling temperature. The creep damage and failure probability have no changes in stage III. The creep damage and failure probability are also decreased with the increase in the cooling temperature.

Table 2 lists the comparisons of maximum CEEQ, creep damage, and failure probability for the conventional and optimized SOFC stack. The maximum CEEQ, creep damage, and failure probability are all decreased by setting the cooling channel. Decreasing the cooling channel temperature is beneficial to further reducing the creep damage and failure probability. The cooling channel temperature can be realized by internal reforming or passing the cooling medium.