# A Multiscale Approach to the Numerical Simulation of the Solid Oxide Fuel Cell
## 4 Mathematical Model
### Boundary Conditions  
To close the mathematical mode, the boundary conditions are required. In this model, every one of the subdomains is treated separately and the transmission conditions at the interfaces between the subdomains are used. At the cathode channel inlet and the anode channel inlet:  
* for Equation (13): the given inlet composition--\(p_{\text{O}_{2}}\) and \(p_{\text{N}_{2}}\) (cathode), and \(p_{\text{H}_{2}}\), \(p_{\text{H}_{2}\text{O}}\) and \(p_{\text{N}_{2}}\) (anode);
* for Equation (19): the given inlet temperature--\(T_{\text{in}}\). At the cathode channel outlet and the anode channel outlet:  
* for Equation (13): the outflow condition--\(\boldsymbol{\
abla}\;p_{i}=0\);
* for Equation (19): the outflow condition--\(\boldsymbol{\
abla}\;T=0\). At the walls:  
* for Equation (13): walls impermeable for mass transport--\(\dfrac{D_{i}^{\text{eff}}}{RT}\;\boldsymbol{\
abla}\;p_{i}=0\);
* for Equation (19): the constant temperature--\(T_{\text{wall}}\). At the cathode surface:  
* for Equation (13): the transmission condition--\(p_{i,\text{cath channel}}=p_{i,\text{cathro}}\left(\dfrac{D_{i}^{\text{eff}}}{RT}\; \boldsymbol{\
abla}\;p_{i}\right)_{\text{cath channel}}=\left(\dfrac{D_{i}^{ \text{eff}}}{RT}\;\boldsymbol{\
abla}\;p_{i}\right)_{\text{cath}}\);
* for Equation (19): the transmission condition--\(T_{\text{cath channel}}=T_{\text{cath}}\), \(\left(k^{\text{eff}}\;\boldsymbol{\
abla}\;T\right)_{\text{cath channel}}=\left(k^{\text{eff}}\;\boldsymbol{\
abla}\;T\right)_{\text{cath}}\);
* for Equation (22) for the electrons: the potential equals the terminal voltage--\(\phi_{\text{el}}=E_{\text{OP}}\);
* for Equation (22) for the ions: no current flow--\(\phi_{\text{ion}}^{\text{eff}}\;\boldsymbol{\
abla}\;\phi_{\text{ion}}=0\). At the anode surface:  
* for Equation (13): the transmission condition--\(p_{i,\text{ano channel}}=p_{i,\text{ano}}\left(\dfrac{D_{i}^{\text{eff}}}{RT}\; \boldsymbol{\
abla}\;p_{i}\right)_{\text{ano channel}}=\left(\dfrac{D_{i}^{ \text{eff}}}{RT}\;\boldsymbol{\
abla}\;p_{i}\right)_{\text{ano}}\);
* for Equation (19): the transmission condition--\(T_{\text{ano channel}}=T_{\text{ano}}\), \(\left(k^{\text{eff}}\;\boldsymbol{\
abla}\;T\right)_{\text{ano channel}}=\left(k^{\text{eff}}\;\boldsymbol{\
abla}\;T\right)_{\text{ano}}\);
* for Equation (22) for the electrons: the potential equals the open circuit voltage \(\phi_{\text{el}}=E_{\text{OC}}\);
* for Equation (22) for the ions: no current flow--\(\phi_{\text{ion}}^{\text{eff}}\;\boldsymbol{\
abla}\;\phi_{\text{ion}}=0\). Between the cathode and the electrolyte:  
* for Equation (13): electrolyte impermeable for mass transport--\(\dfrac{D_{i}^{\text{eff}}}{RT}\;\boldsymbol{\
abla}\;p_{i}=0\);
* for Equation (19): the transmission condition--\(T_{\text{cath}}=T_{\text{elec}}\left(k^{\text{eff}}\;\boldsymbol{\
abla}\;T \right)_{\text{cath}}=\left(k^{\text{eff}}\;\boldsymbol{\
abla}\;T\right)_{ \text{elec}}\);
* for Equation (22) for the electrons: electrolyte impermeable--\(\phi_{\text{el}}^{\text{eff}}\;\boldsymbol{\
abla}\;\phi_{\text{el}}=0\);* for Equation (22) for the ions: the transmission condition--\(\phi_{\text{ion,cath}}=\phi_{\text{ion,elec}}\), \(\left(\sigma_{\text{ion}}^{\text{eff}}\,\boldsymbol{\
abla}\,\phi_{\text{ion}} \right)_{\text{ion,cath}}=\left(\sigma_{\text{ion}}^{\text{eff}}\,\boldsymbol{ \
abla}\,\phi_{\text{ion}}\right)_{\text{ion,elec}}\).