## 2 Materials and Methods
### The Numerical Simulation Methods
It should be pointed out that the CFD simulations on Ansys Fluent by [PERSON], [10], are found to be compatible with the experimental results provided by the Metal Industries Research & Development Centre (MIRDC) of Taiwan. Thus, for arrangements of 6 checkers, whose material parameters are not too far from those shown above, it is reasonable to omit the Darcy dissipation in our numerical simulations. In fact, experimental data of MIRDC supports this omission in our optimization search. See Section 3 for a comparison of our 1-dimensional numerical simulations with the experimental results provided by MIRDC._  
When \(n\cdot 2t_{sw}+t_{sw}\leq t\leq n\cdot 2t_{sw}+2t_{sw}\), we have the following 1-dimensional partial differential equations:  
\[2t_{sw}=t_{sw}+t_{sw}\]0  
and  
\[2t_{sw}=t_{sw}+t_{sw}\]1 \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Cordierite** & **Mullite** \\ \hline Size (mm) & 150 \(\times\) 150 \(\times\) 100 & 150 \(\times\) 150 \(\times\) 100 \\ Pore size (mm) & 4.9 & 3.0 \\ Wall thickness (mm) & 1.04 & 0.70 \\ Porosity & 0.62 & 0.64 \\ Specific surface area (\(1/\mathrm{m}\)) & 501 & 853 \\ Density (kg/m\({}^{3}\)) & 2200 & 2500 \\ Specific heat capacity (J/kgK) & 1000 & 1200 \\ Thermal conductivity (W/mK) & 2 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters of checkers. satisfying the following boundary conditions  
\[\left[\frac{\partial T_{s}}{\partial x}\right]_{x=0}=0=\left[\frac{\partial T_{s}} {\partial x}\right]_{x=l}\text{, }^{\prime}\left[\frac{\partial T_{ar}}{\partial x}\right]_{x=0}=0\text{and }T_{ar}(l,t)\text{\emph{given}}. \tag{26}\]  
Here \(u(\text{m}/\text{s})\) is the superficial flow velocity of the gas/air. \(\rho(\text{kg}/\text{m}^{3})\) and \(\rho_{s}(\text{kg}/\text{m}^{3})\) are, respectively, the densities of the gas/air and the solid porous medium (checkers). \(C(\text{J}/\text{kg}\text{K})\) and \(C_{s}(\text{J}/\text{kg}\text{K})\) are, respectively, the specific heat capacities of the gas/air and the solid porous medium (checkers). \(k(\text{W}/\text{m}\text{K})\) and \(k_{s}(\text{W}/\text{m}\text{K})\) are, respectively, the thermal conductivities of the gas/air and the solid porous medium (checkers). \(T(\text{K})\) and \(T_{s}(\text{K})\) are, respectively, the temperatures of the gas/air and the solid porous medium (checkers). \(\varepsilon\) is the void fraction (_porosity_) of the solid porous medium (checkers). \(a_{s}(\text{m}^{2})\) is the specific surface area of the porous ceramic medium (checkers) exposed to gas/air per unit volume. \(h(\text{W}/\text{m}^{2}K)\) is the _heat transfer coefficient_ of the solid porous medium (checkers). \(q_{d}\) is the _heat dissipation_ through the _external surface_ of a fixed-bed heat regenerator. We assume that \(q_{d}=0\). We have the following relation for the Nusselt number of the fluid (gas/air) flow:  
\[\text{Nu}=\frac{h\cdot d}{k} \tag{27}\]  
in which \(d\) is the _characteristic length._ Let Re and Pr denote, respectively, the Reynolds number and the Prandtl number of the fluid (gas/air) flow. We will use the following classic relation  
\[\text{Nu}=1.86\cdot\left(\frac{d}{L}\cdot\text{Re}\cdot\text{Pr}\right)^{ \
icefrac{{1}}{{3}}}\cdot\left(\frac{\mu_{b}}{\mu_{w}}\right)^{0.14} \tag{28}\]  
of Sieder and Tate to calculate the Nusselt number. Here \(\mu_{b}\) and \(\mu_{w}\) are, respectively, the viscosity of fluid at the mean bulk temperature and the viscosity of fluid at the wall. See page 583 of Ref. [6]. Note that, for a positive real number \(M>0\), we have  
\[M^{0.14}=e^{(0.14)\cdot\ln M}\approx e^{0}=1\]  
when \(|\ln M|\) is _not_ large. Thus, when performing numerical simulations, we will use the simplified formula  
\[\text{Nu}=1.86\cdot\left(\frac{d}{L}\cdot\text{Re}\cdot\text{Pr}\right)^{ \
icefrac{{1}}{{3}}} \tag{29}\]  
Let \(M_{ex}(\text{kg}/\text{s})\) and \(M_{ar}(\text{kg}/\text{s})\) be, respectively, the mass flow rates of the _exhaust gas_ flow and of the _fresh air_ flow.