# A Multiscale Approach to the Numerical Simulation of the Solid Oxide Fuel Cell
## 4 Mathematical Model
### Transport Model
#### 4.2.3 Charge Conservation  
The charge conservation is based on the Ohm's law and may be formulated as:  
\[0=\mathbf{\
abla}\cdot\left(\sigma_{i}^{\text{eff}}\mathbf{\
abla}\,\phi_{i}\right)+j_{ \text{PB}}, \tag{22}\]  
where \(\phi_{i}\) (V) is the electric potential of the appropriate phase \(i\), \(\sigma_{i}^{\text{eff}}\) (S m\({}^{-1}\)) stands for the effective electronic or ionic conductivity of phase \(i\) and \(j_{\text{PB}}\) (A m\({}^{-3}\)) is the source term.  
The effective conductivities are microstructure-relative and they are computed with the following formula:  
\[\sigma_{i}^{\text{eff}}=\frac{\psi_{i}}{\tau_{i}}\sigma_{i}. \tag{23}\]  
The bulk conductivities \(\sigma_{i}\) (S m\({}^{-1}\)) are calculated from the curve fit to the data published in the open literature [28; 68; 69; 70]:  
\[\sigma_{\text{el,ano}} =1.25867\cdot 10^{6}\cdot\exp\left(\frac{544.744}{T}\right), \tag{24a}\] \[\sigma_{\text{ion,ano}} =32605\cdot\exp\left(-\frac{10028}{T}\right),\] (24b) \[\sigma_{\text{el,cath}} =0.3716T^{2}-782.7924T+4.7534\cdot 10^{5},\] (24c) \[\sigma_{\text{ion,cath}} =-\frac{8F^{2}\vartheta_{\text{cath}}}{RT}C. \tag{24d}\]  
In Equation (24d), \(\vartheta_{\text{cath}}\) stands for the specific molar volume of the LSCF, \(\vartheta_{\text{cath}}\) = 28,170 mol m\({}^{-3}\). The constant \(C\) is calculated using the following formula [28]:  
\[C=\left(2.4939\cdot 10^{-14}\cdot T^{2}-4.5604\cdot 10^{-11} \cdot T+2.0978\cdot 10^{-8}\right)\\ \left(-3.3626\cdot 10^{-5}\cdot T+2.59403\cdot 10^{-2}\right). \tag{25}\]  
It is assumed that the potential of the phase is changed by the electrochemical reactions, therefore the change is proportional to the current calculated by the Butler-Volmer equation (Equations (3)). The summary of the sources of the electric potential is shown in Table 12.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Element** & **Anode** & **Electrolyte** & **Cathode** \\ \hline
**Joule heat - electrons flow**\(\vartheta_{V}^{\text{Joule,el}}\) (W m\({}^{-3}\)) & \(\frac{i_{\text{el}}^{2}}{\sigma_{\text{el}}^{\text{eff}}}\) & — & \(\frac{\ell_{\text{el}}^{2}}{\sigma_{\text{el}}^{\text{eff}}}\) \\  & & & \(\frac{\ell_{\text{el}}^{2}}{\sigma_{\text{el}}^{\text{eff}}}\) \\ \hline
**Joule heat - ions flow**\(\vartheta_{V}^{\text{Joule,ion}}\) (W m\({}^{-3}\)) & \(\frac{i_{\text{ion}}^{2}}{\sigma_{\text{ion}}^{\text{eff}}}\) & \(\frac{i_{\text{ion}}^{2}}{\sigma_{\text{ion}}^{\text{eff}}}\) & \(\frac{i_{\text{ion}}^{2}}{\sigma_{\text{ion}}^{\text{eff}}}\) \\ \hline
**activation heat**\(\vartheta_{V}^{\text{act}}\) (W m\({}^{-3}\)) & \(\eta_{\text{act}}^{\text{anoano}}/\text{TPB}\) & — & \(\eta_{\text{act}}^{\text{cath}}/\text{TPB}^{\text{cath}}\) \\ \hline
**heat of water formation**\(\vartheta_{V}^{\text{H,O}}\) (W m\({}^{-3}\)) & \(-\frac{\Lambda_{\text{H,2O}}h}{2F}/\text{TPB}\) & — & — \\ \hline \hline \end{tabular}
\end{table}
Table 11: Heat sources for Equation (19).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Charge** & **Electronic-Anode** & **Electronic-Cathode** & **Ionic-Anode** & **Ionic-Electrolyte** & **Ionic-Cathode** \\ \hline
**source** & \(-j_{\text{PB}}\) (A m\({}^{-3}\)) & \(-j_{\text{ITB}}^{\text{ano}}\) & \(j_{\text{DPB}}^{\text{cath}}\) & \(j_{\text{ITB}}^{\text{ano}}\) & 0 & \(-j_{\text{DPB}}^{\text{cath}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 12: Source terms for Equation (22).  
The equation for the electronic phase potential is required to be solved within the anode and cathode and the equation for the ionic phase potential should be solved in all of the elements of the PEN.