## 2 Structure, Theory, and Numerical Model
### Governing Equations  
The 3D one-cell stack model in this paper couples mass conservation, momentum conservation, energy conservation, species conservation, and quasi-electrochemical control equations. Coupling calculation is a complex process, and how to calculate efficiently is a difficulty. The specific description of the control equations is as follows. The continuity equation for the mixture gas is:  
\[\
abla\cdot(\epsilon\rho\overrightarrow{u})=S_{m}. \tag{1}\]  
All fluid flow processes follow this basic rule. In the equation, \(\epsilon\) is the porosity of the porous electrode (porosity refers to the proportion of the void space in the porous to the total volume of the  
Figure 3: Meshes of the one-cell SOFC stack model and the local enlarged picture illustrates the mesh quality. Figure 2: (**a**) Interdigitated design for the fuel flow rib channels; (**b**) The schematic diagram of the interdigitated design in the y-direction view; (**c**) Species transports among the fuel inlet channels, exhaust collecting channels, and the porous anode. electrode), \(\overrightarrow{u}\) is the fluid velocity, and \(\rho\) is the density of the fluid. The higher the porosity, the higher is the molecular pass rate. This equation is effective for both incompressible and compressible flows:  
\[\
abla\cdot(\frac{1}{\varepsilon^{2}}\rho\overrightarrow{u}\overrightarrow{u}) =-\varepsilon\
abla p+\
abla\cdot\left(\varepsilon\mu_{eff}\
abla \overrightarrow{u}\right)+S_{u}. \tag{2}\]  
Equation (2) is the momentum conservation equation for the fluid part of the stack: \(p\) is the pressure of the fluid, \(\mu_{eff}\) is the effective viscosity coefficient of the fluid, and \(S_{u}\) is the source item, which does not exist in the momentum conservation equations in the manifold and flow channel zones. \(S_{u}\) only exists at the interface between the electrode and dense electrolyte. The gas consumption and production occur at this interface, and this process will change the fluid concentration [26]. \[\
abla\cdot\left(\overrightarrow{u}\overrightarrow{C}_{k}\right)=\
abla \cdot\left(D_{k,eff}\
abla C_{k}\right)+S_{k}. \tag{3}\]  
Equation (3) is the conservation of species where \(C_{k}\) is used to represent the mass fraction of species \(k\); \(D_{k,eff}\) is the effective diffusion coefficient of species \(k\); and \(S_{k}\) is the source item at the interface between the electrode and dense electrolyte. \(S_{k}\) for species \(k\) in the cathode and anode sides can be respectively estimated as:  
\[ \begin{array}{c}S_{\text{O}_{2}}=-i_{\text{out}}\frac{M_{\text{O}_{2}}}{4 \text{F}},\quad S_{\text{N}_{2}}=0\\ S_{\text{H}_{2}}=-i_{\text{out}}\frac{M_{\text{H}_{2}}}{2\text{F}},\quad S_{ \text{H}_{2}\text{O}}=-i_{\text{out}}\frac{M_{\text{H}_{2}\text{O}}}{2\text{F }}\end{array}  \tag{4}\]  
\(S_{k}\) is 0 in the manifold and flow channel zones. \[\frac{\partial}{\partial t}(pE)+\
abla\cdot\left(\overrightarrow{v}(pE+p) \right)=\
abla\cdot\left(k_{eff}\
abla T-\sum_{j}h_{j}\overrightarrow{j}+ \left(\overrightarrow{\pi}_{eff}\cdot\overrightarrow{v}\right)\right)+S_{h}. \tag{5}\]  
Equation (5) is the energy conservation equation where \(K_{eff}\) is the effective thermal conductivity (\(k+k_{t}\), where \(k_{t}\) is the turbulent thermal conductivity, as defined by the turbulence mode) and \(\overrightarrow{J}_{j}\) is the diffusion flux of species \(j\). The first three items on the right side of the equation represent the energy transfer due to the thermal conduction of the gas, the diffusion of species, and the viscous dissipation. Generally, it is quite difficult to consider the electrochemical reaction details around the three phase boundary sites within the composite electrode. Thus, the heat sources corresponding to the reaction and ohmic heats are assumed to be uniformly produced within the whole porous anode and cathode. In this case, \(S_{h}\) includes the heat of the chemical reaction and any other volumetric heat source.