## 3 Results and Discussion
### DC Analysis  
In the second set of experiments, the DC analysis was performed. A comparison between calculated and measured electric performance of SOFC was done to address more realistic degradation due to the coarsening of Ni particles within the anode. A fraction of backscattered electron (BSE) microscope pictures were taken from Figure 3a in Ref. [10] to define an approximation to the real structure of porous anode. The experimental degradation test was performed for 1000 h at the temperature of 900 \({}^{\circ}\)C and at current density of 0.45 A cm\({}^{-2}\), whereas 3 vol% humidified H\({}_{2}\) was supplied to the anode and air was supplied on the cathode side. The flow rates of hydrogen and air were 350 cm\({}^{3}\) min\({}^{-1}\) and 1500 cm\({}^{3}\) min\({}^{-1}\), respectively [10]. Due to this, the temperature and current density of the modeled SOFC was set to \(T_{\rm cell}\) = 900 \({}^{\circ}\)C and \(J_{cell}\) = 0.45 A cm\({}^{-2}\), respectively, during the simulation from 0 to 1000 h. Time increment \(\Delta t\) = 100 h was used since the experimental data was defined at 100 h intervals in Ref. [10]. The anode exponential coefficient (\(\kappa_{\rm a}\) = 3) and the anode growth coefficient (\(k_{\rm a}\) = 2.1 \(\times\) 10\({}^{-21}\) m\({}^{3}\) h\({}^{-1}\)) were manually fitted to match the relative TPB density of the measured SOFC in Ref. [10]. Figure 7a shows relative TPB density in the anode (TPB\({}_{\rm a}\)) and in the cathode (TPB\({}_{\rm c}\)) as a function of time. The calculated relative TPB\({}_{\rm a}\) density had a similar profile as that in Figure 8d in Ref. [10]. The initial absolute value of TPB\({}_{\rm a}\) (\(t\) = 0 h) was approximately four times lower (0.7 \(\upmu\)m\({}^{-2}\) vs. 2.9 \(\upmu\)m\({}^{-2}\)), but another study revealed [30] that the density of active TPBs may be lower (0.74 \(\upmu\)m\({}^{-2}\)). Since the TPBs that lie in the third dimension (i.e., in the plane of 2-D grey-scale image) cannot be resolved with our algorithm, it was obvious why the obtained density of TPBs from 2-D grey-scale image was lower. Although the TPB density was lower in our case than it should have been (or than it was in the real SOFC), it could be compensated for with higher activity of TPBs (i.e., higher exchange current density, \(i_{0,\rm a/c}\)) to achieve a similar electric performance of modeled SOFC. It is important to note the presented model approximated the spatial distribution of TPBs in real SOFC. Due to this, it could be expected that the simulation results may be more reliable compared to those obtained with 0-D or 1-D models. Figure 7b shows the calculated voltage (\(V_{\rm da}\)) of SOFC, with degraded anode, which is almost the same as the reference voltage (\(V_{\rm Ref. }\)[10] in Figure 8a in Ref. [10]) from 0 to 100 h, whereas a higher voltage difference (0.03-0.06 V) were noticed from 200 h to 1000 h. This difference may be partly attributed to a noticeable drop of open circuit voltage (\(\Delta\)OCV = 0.02 V) that occurs in real SOFC at \(t\) = 200 h, as seen in Figure 8a in Ref. [10], and partly to some other degradation processes that possibly were going on concurrently. The \(\Delta\)OCV was attributed to different agents. Since OCV = \(V_{0,\rm c}-V_{0,\rm a}\) depends on mass fraction of hydrogen (\(y_{0,\rm H_{2}}\)), referring to Equation (17), it can be speculated that \(y_{0,\rm H_{2}}\) was slightly decreased due to increased humidity in the fuel or hydrogen gas leak. To achieve this, the input molar fraction of hydrogen (\(x_{0,\rm H_{2}}\)) was adjusted to 0.955 in order to simulate \(\Delta\)OCV at \(t\) = 200 h.  
The voltage (\(V_{\text{da+\Delta OCV}}\)) of SOFC, with degraded anode and \(\Delta\)OCV, followed the reference voltage (\(V_{\text{Ref.