*3.5. Maximum Output Performance at a Given pO*<sup>2</sup>

In the application of fuel cell vehicle, not only the quantity but also the quality of energy should be considered. Figure 7 shows the relation curve between power density and thermodynamic efficiency under operating temperature *T* (453 K), hydrogen and oxygen intake pressure *pO*<sup>2</sup> , *pO*<sup>2</sup> (3 atm), membrane thickness *lm* (20 μm) and membrane acid doping level *X* (10). In order to improve the calculation accuracy, *P*/*Pmax* is chosen to transform the engineering problem into a mathematical problem. The curve is the willow leaf curve going back to the origin. As shown in Figure 7, when *P* = *PB*, *η* = *ηmax* and when *η* = *ηA*, *P* = *Pmax*. Thus, the optimal region of HT-PEMFC can be obtained.

$$P\_B \le P \le P\_{\max}, \; \eta\_A \le \eta \le \eta\_{\max}$$

**Figure 7.** Optimization relationship between *P* and *η*.

Curve OABO is an optimization curve derived from the power and efficiency model. When the operating point of the cell is located on the curve AB, its performance reaches the best. When the running point is located in region <sup>1</sup> , it has better performance; when the running point is located in the region <sup>2</sup> , it has the worst performance. Regions <sup>3</sup> <sup>4</sup> <sup>5</sup> are unstable region, because these three regions are outside OABO curve; there exists no operation points in the region <sup>6</sup> because *P* > *Pmax*, *η* > *ηmax*.

#### **4. Discussion**

In this paper, a finite time thermodynamic model of HT-PEMFC is established, which takes the irreversibility caused by polarization and leakage current into account. The influences of operating temperature, proton membrane thickness, proton membrane phosphoric acid doping level, hydrogen intake pressure and oxygen intake pressure on *Pmax* and *ηmax* at a given temperature are studied. The results show that *Pmax* and *ηmax* both increase with the increase of temperature. When the operating temperature is 433 K, with the decrease of proton membrane thickness, *Pmax* improves greatly, but the decrease of membrane thickness has little effect on *ηmax*. As the doping level of proton membrane phosphoric acid increases, *Pmax* increases by 84% at the temperature of 373 K, and by 52% at the temperature of 473 K. However, the increase of phosphate doping level has little influence on *ηmax*. The increase of hydrogen intake pressure and oxygen intake pressure will increase *Pmax* and

*ηmax*. The optimal relationship between power density and thermodynamic efficiency of HT-PEMFC is also studied. The optimal interval of power density and thermodynamic efficiency is *PB* ≤ *P* ≤ *Pmax*, *η<sup>A</sup>* ≤ *η* ≤ *ηmax*.

**Author Contributions:** Conceptualization, Z.M.; formal analysis, B.X.; investigation, B.X. and D.L.; project administration, Z.M.; resources, M.Z.; software, D.L.; supervision, Z.M.; validation, M.Z.; visualization, Y.L.; writing—original draft, B.X.; writing—review & editing, Z.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** We gratefully acknowledge the financial support of the National Natural Science Foundation of China (No. 51176069) and Scientific Research Foundation of Nanjing Forestry University (No. GXL2018004).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

