## 4 Conclusions  
In this paper, a multi-physical and electrochemical coupling model was well established for the typical PCFC with H\({}^{+}\)/e\({}^{-}\)/O\({}^{2-}\) mixed conducting cathode. Taking a typical Ni-BZCY/BZCY/BZCY-LSCF PCFC as an example, the validation of the model was verified by good agreements with the experimental results, and the following conclusions were achieved.  
1. Compared with the O\({}^{2-}\)-SOFC, the ionic conductivity of PCFC greatly decreased, and the electronic conductivity only slightly changed, while the working temperature decreased from 800 \({}^{\circ}\)C to 550 \({}^{\circ}\)C. Thus, the cathodic electrochemical active zones for both the oxygen reducing and vapor producing half reactions would be further compressed to a smaller distance (i.e., 8 \(\upmu\)m in the current situation) away from the electrolyte than that of O\({}^{2-}\)SOFC to achieve the minimal total energy loss (Figures 3, 4, and 9). The electrochemical contribution of the extending thickness would be negligible.
2. Different from the O\({}^{2-}\)-SOFC, O\({}^{2-}\) within the PCFC cathode is only an intermediate transform substance between the electrons and protons. Thus, there is a peak oxygen ion current distribution within the composite cathode of PCFC (Figure 4). The oxygen reduction half reaction on the cathode side of PCFC is found to be a key factor to dominate the total PCFC voltage loss at intermediate temperature zone (Figure 6).
3. Compared with the O\({}^{2-}\)-SOFC, the concentration polarization of anode-supported PCFC is small. This result is attributed to the reason that the vapors are generated in the cathode side of PCFC instead of the anode side (Figure 5). This means that high mole fraction of hydrogen can be maintained over the anode-electrolyte interface of PCFC, even if many hydrogens are consumed.  
Figure 9: The dependences of the distributions of the local oxygen ions and protons generation rates on different ionic conductivities. (**a**) Different oxygen ionic conductivity; (**b**) different protonic conductivity.  
It should be noted that the model is not limited to Ni-BZCY/BZCY/BZCY-LSCF, but also can be further used for other PCFCs with similar H\({}^{+}\)/e\({}^{-}\)/O\({}^{2-}\) mixed conducting characteristics.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app12083889/s1](https://www.mdpi.com/article/10.3390/app12083889/s1)]([https://www.mdpi.com/article/10.3390/app12083889/s1](https://www.mdpi.com/article/10.3390/app12083889/s1)), Supplementary: Multi-physical and Electrochemical Coupling Model for the Proteomic Ceramic Fuel Cells with H\({}^{+}\)/e\({}^{-}\)/O\({}^{2-}\) mixed conducting cathodes.  
Data curation, D.Y., W.W., S.J. and R.L.; Investigation, L.L. and D.C.; Writing--Original Draft Preparation, D.Y. and R.L.; Writing--Review and Editing, A.L. and D.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China (22179054), Ministry of Science and Technology of China (CU03-10), and Postgraduate Research and Practice Innovation Program of Jiangsu Province (KYCX21_3463).  
Not applicable.  
The authors declare no conflict of interest.