# Performance and Parameter Sensitivity Analysis of the PEMFC Flow Channel with Porous Baffles
## 3 Establishment of Numerical Model
### Governing Equations  
In order to establish the PEMFC model, the mass conservation equation, the Brinkman equations, the transport equations of concentrated species, the multi-phase flow multi-matter equation, the water conservation equation in membrane, and the secondary current density equation were established. \begin{table} \begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline Channel length (\(L_{ch}\)) & 20 & mm \\ Channel width (\(W_{ch}\)) & 1 & mm \\ Channel height (\(H_{ch}\)) & 1 & mm \\ Rib width (\(W_{rid}\)) & 1 & mm \\ Operating pressure (\(p_{a}^{in}\), \(p_{c}^{in}\)) & 1 & atm \\ Operating temperature (\(T\)) & 343.15 & K \\ Relative humidity of inlet flows & 100\% & - \\ The anode excess coefficient & 3 & - \\ The cathode excess coefficient & 3 & - \\ \multicolumn{3}{c}{} \\ GDL thickness (\(H_{gdl}\)) & 0.3 & mm \\ CL thickness (\(H_{cl}\)) & \(1.29\times 10^{-2}\) & mm \\ Membrane thickness (\(H_{mem}\)) & 0.108 & mm \\ GDL porosity (\(H_{ch}\)) & 0.4 & 1 \\ GDL permeability (\(K_{gdl}\)) & \(1.78\times 10^{-11}\) & m\({}^{2}\) \\ GDL electrical conductivity (\(\sigma_{gll}\)) & 2000 & S/m \\ CL porosity (\(\varepsilon_{gdl}\)) & 0.4 & 1 \\ CL permeability (\(K_{cl}\)) & 3.56 \(\times 10^{-12}\) & m\({}^{2}\) \\ CL electrical conductivity (\(\sigma_{cl}\)) & 5000 & S/m \\ Membrane electrical conductivity (\(\sigma_{mem}\)) & 18.75 & S/m \\ Relative humidity of inlet flows & 100\% & - \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \multicolumn{3}{c}{} \\ \end{tabular} 
\begin{tabular}{c} \(\quad\)Mass conservation equation:  
\[\frac{\partial(\varepsilon_{d}\rho)}{\partial t}+\
abla\cdot\left(\varepsilon_{d} \rho\overrightarrow{u}\right)=S_{m} \tag{1}\]  
The Brinkman equation in porous media:  
\[\frac{1}{\varepsilon_{d}}\rho\frac{\overrightarrow{u}}{\partial t}+\frac{1}{ \varepsilon_{d}}\rho\left(\overrightarrow{u}\cdot\
abla\right)\overrightarrow {u}\frac{1}{\varepsilon_{d}}=\
abla\cdot\left[p_{i}\overrightarrow{I}+ \overrightarrow{K}\right]-\left(\mu\kappa^{-1}\right)\overrightarrow{u} \tag{2}\]  
The transport equation of concentrated species:  
\[\rho\frac{\partial w_{i}}{\partial t}+\
abla\cdot\overrightarrow{j_{i}}+\rho \left(\overrightarrow{u}\cdot\
abla\right)w_{i}=R_{i} \tag{3}\]  
where \(\overrightarrow{j_{i}}\) is expressed by Maxwell-Stefan diffusion:  
\[\overrightarrow{j_{i}}=-\left(\rho w_{i}\sum D_{ik}^{M}\overrightarrow{d}_ {k}\right) \tag{4}\]  
where \(D_{ik}^{M}\) is the Maxwell-Stefan diffusion coefficient. Under assumption 7, \(D_{ik}^{M}\) is expressed as follows [17]:  
\[D_{ik}^{M}=D_{ik} \tag{5}\]  
where \(D_{ik}\) is the Fick's diffusion coefficient, The correlation diffusion coefficient was obtained from reference [15]. In Equation (4), \(\overrightarrow{d}_{k}\) is the diffusional driving force and was expressed as follows:  
\[\overrightarrow{d}_{k}=\
abla x_{k}+\frac{1}{p}[(x_{k}-w_{k})\
abla p] \tag{6}\]  
\[x_{k}=\frac{w_{k}}{M_{k}}M_{n}M_{n}=\left(\sum\frac{w_{i}}{M_{i}}\right)^{-1} \tag{7}\]  
The water conservation equation in the membrane:  
\[\frac{\partial c_{v}}{\partial t}+\
abla\cdot\overrightarrow{j_{v}}=R_{v} \tag{8}\]  
where \(\overrightarrow{j_{v}}\) is the ionized water flux and is expressed as follows:  
\[\overrightarrow{j_{v}}=-D_{ionomer}\cdot\
abla c_{v}+z_{i}u_{i}Fc_{v}\
abla V \tag{9}\]  
\[\lambda=\frac{c_{ion}EW}{\rho_{mem}} \tag{10}\]  
\[\lambda_{eq}=\left\{ \begin{array}{c}0.043+17.8a-39.85a^{2}+36.0a^{3}\;0<a \leq 1\\ 14+1.4(a-1)\;1<a\leq 3\end{array} \right.