## 3 Results
\begin{table}
\begin{tabular}{l c c c} \hline  & **Mullite \(A\)** & **Mullite \(B\)** & **Mullite \(C\)** \\ \hline Size (mm) & 100 \(\times\) 100 \(\times\) 100 & 150 \(\times\) 150 \(\times\) 100 & 150 \(\times\) 150 \(\times\) 100 \\ Pore size (mm) & 17 & 4 & 6 \\ Wall thickness (mm) & 0.8 & 1.5 & 2 \\ Porosity & 0.182 & 0.524 & 0.524 \\ Specific surface area (1/m) & 42.7 & 523.8 & 349.2 \\ Density (kg/m\({}^{3}\)) & 2200 & 2200 & 2200 \\ Specific heat capacity (J/kgK) & 836 & 836 & 836 \\ Thermal conductivity (W/mK) & 1.8 & 1.8 & 1.8 \\ \hline \end{tabular}
\end{table} 
Table 2: Material parameters of checkers. Figure 8: Comparison of 1-D simulations with 3-D CFD simulations on Ansys Fluent. It can be observed through Table 4 that, in the long term, our numerical simulation results, based on the 1-dimensional model, are compatible with the experimental results of MIRDC. Now we apply our simple tree search method with the Mutation Expansion operator to solve the optimization problem for 3 types of checkers to be arranged at 6 positions in a fixed-bed regenerator. The material parameters of these 3 types of checkers, \(A\), \(B\) and \(C\), are shown in Table 2. These checkers are provided by MIRDC. The operating parameters for a regenerative combustion system with two fixed-bed heat regenerators are shown in Table 3.  
Thus, we have \(S=\{1,2,3,4,5,6\}\). We choose the _strictly increasing_ sequence  
\[\alpha_{0}=\{1,2,3\}\prec\alpha_{1}=\{1,2,3,4,5,6\} \tag{33}\]  
of _Partitions_ of \(S=\{1,2,3,4,5,6\}\). Thus, we have the following decomposition  
\[S=\{1,2\}\cup\{3,4\}\cup\{5,6\}\]  
of \(S\) into a union of _Interval Components_  
\[I_{1}=\{1,2\},\ I_{2}=\{3,4\},\ I_{3}=\{5,6\}\]  
with respect to the Partition \(\alpha_{0}=\{1,2,3\}\). We select the top _15_ arrangements in \(\Omega_{\alpha_{0}}\) as the _\"mother (root) nodes\"_ in \(\Omega_{\alpha_{1}}\) to create _\"child nodes\"_ (_qualified_ arrangements with respect to the _Partition_\(\alpha_{1}\)) in \(\Omega_{\alpha_{1}}\). For a _complete_ evaluation process for the total  
\[729=3^{6}\]  
possible arrangements of checkers, based on 1D simulations, it takes 63 h to finish this evaluation process. The top 3 arrangements are shown in Table 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Experiments by** & **Inlet** & **Computer of** & **Computer of** & **Experimental** & \\
**MIRDC** & **Temperature of** & **Exhaust Gas (K)** & **Temperature of** & **Data of Outlet** & **Error (\({}^{\circ}\)C)** \\
**of Taiwan** & **Exhaust Gas (K)** & **(\({}^{\circ}\)C) by Our** & **Temperature of** & **Exhaust Gas** & \\  & & **1D Simulations** & & \\ \hline
1 & 1050 & 83.82 & 82.80 & 1.02 \\
2 & 1150 & 137.4 & 124.60 & 12.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of 1D simulation results, in the long term, with the experimental results. \begin{table}
\begin{tabular}{l l} \hline \hline Total horizontal length of stacking (mm) & 200 \\ \hline Total vertical length of stacking (mm) & 300 \\ Natural gas flow (\(\mathrm{Nm^{3}/hr}\)) & 17.65 \\ Air-fuel ratio (mm) & 15.9 \\ Inlet temperature of exhaust gas (K) & 1050 (Experiment 1) \\ Inlet temperature of fresh air (K) & 1150 (Experiment 2) \\ Time of Phase Switch (sec) & 313 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Operating parameters for Experiments 1 and 2. For the optimization process by our simple _tree search_ method, it takes only 6 h to evaluate  
\[42=3^{3}+15\]  
possible arrangements of checkers, based on 1D simulations. The top three arrangements, found by this simple _tree search_ method, are shown in Table 6.  
Thus, empirical evidence shows that our _tree search_ method is helpful and _efficient_. An algorithm based on our simple tree search method, with the Mutation Expansion operator, is shown in Figure 10.