## 3 Results
### Source Terms  
The definitions of the source term for one species in different domains are different due to the varied sites of the reactions. In ACL, the hydrogen/oxygen consumption rate is related to the current density based on [PERSON]'s law [37]. The source terms for hydrogen, oxygen consumption, and water production can be determined by the quantitative relationship in the hydrogen and oxygen redox formula. The source term of hydrogen and oxygen can be defined as follows:  
\[S_{H_{2}}=\left\{ \begin{array}{ll}-\frac{i}{2F}&\text{in ACL}\\ 0&\text{in AGDL}\end{array} \right. \tag{13}\]  
Figure 3: (**a**) The relationship between electrolyte conductivity and water absorption; (**b**) The relationship between water absorption and water activity in ionomers. \[S_{O_{2}}=\left\{ \begin{array}{ll}-\frac{i}{4\pi}&\text{in CCL}\\ 0&\text{in CGDL}\end{array} \right. \tag{14}\]  
The source term of electrons and protons in ACL and CCL are represented by \(S_{electron}\) and \(S_{proton}\) as  
\[S_{electron}=\left\{ \begin{array}{ll}-i&\text{in ACL}\\ i&\text{in CCL}\\ 0&\text{in AGDL and CGDL}\end{array} \right. \tag{15}\]  
\[S_{proton}=\left\{ \begin{array}{ll}i&\text{in ACL}\\ -i&\text{in CCL}\\ 0&\text{in PEM}\end{array} \right. \tag{16}\]  
The interfacial water transport mainly involves the process of absorption and desorption. The process of water absorption and desorption on the ionic polymer is realized on an hourly time scale, and water is generated from the electro-chemical reaction of the cathode side not in vapor phase due to the fuel cell working temperature lower than boiling point [38]. The water produced at the surface of the solid catalyst is in the form of dissolved water from the electro-chemical reaction. The way water molecules leaving the polymer phase depends on whether the gas phase is saturated, and sufficiently high-energy to enable water to evaporate. Alternatively, when the gas phase is saturated, the water leaves the interface as a liquid, and which is forced to be transported away by the capillary diffusion or other forced mechanisms. In this study, the source term of water can be described as follows:  
\[S_{H2O}=\left\{ \begin{array}{ll}-\frac{k_{i}}{LV_{m}}\big{(}\lambda_{eq}- \lambda\big{)}&\text{in ACL}\\ -\frac{k_{m}}{LV_{m}}\big{(}\lambda_{eq}-\lambda\big{)}-\gamma_{i}\big{(} \chi_{H2O}-\chi_{sat}\big{)}&\text{in CCL}\\ -\gamma_{i}\big{(}\chi_{H2O}-\chi_{sat}\big{)}&\text{in CGDL}\\ 0&\text{in AGDL}\end{array} \right. \tag{17}\]  
Liquid water exists in the polymer's pores, and protons can migrate through liquid water in principle, enabling electrochemical reactions to occur in the liquid phase, which may produce water directly rather than as dissolved water. However, the diffusion of protons occurring in liquid water is considerably smaller than the ionomer. Hence, the mechanism of dissolved water should be considered. The dissolved water is expressed by \(J_{\lambda}\), which is contributed mainly by back diffusion and electro-osmotic drag. Back diffusion is caused by the moisture content gradient of \(\bigtriangledown\lambda\) as shown in Table 1. Electro-osmotic drag occurs in the action of an electric field, the solution moving in a certain direction. The flux of the dragged water is related to Faraday constant and current as shown in Equation (18). The produced dissolved water at the platinum-ionomer phase is affected by the consumption rate [28], which is given by the following:  
\[S_{\lambda}=\left\{ \begin{array}{ll}\frac{k_{i}}{LV_{m}}\big{(}\lambda_{eq} -\lambda\big{)}&\text{in ACL}\\ \frac{k_{i}}{LV_{m}}\big{(}\lambda_{eq}-\lambda\big{)}+\frac{i}{2\pi}&\text{ in CCL}\end{array} \right. \tag{18}\]  
Absorption and desorption in the membrane are accompanied by phenomena with exothermic and endothermic processes. Water vapor absorption and vapor condensation are considered as the processes of heat-releasing [39]. By contrast, the water molecules desorption from the membrane and the evaporation of water should absorb heat to drive them.