## 3 Results  
Before starting our optimization process using the tree search, we test the validity of our 1-dimensional partial differential equations. This 1-dimensional model has been discussed in Refs. [11; 15; 57]. It has also been utilized in Refs. [1; 2; 3; 4; 5]. We test our 1-dimensional modeling for two types of checkers, Cordierite and Mullite, with the material parameters shown in Table 1 of Section 2. We consider the following arrangement of checkers  
\[(\text{Cordierite,Cordierite,Cordierite,Cordierite,Mullite,Mullite})\]  
at 6 positions in a fixed-bed heat regenerator. These checkers are provided by the Metal Industries Research & Development Centre (MIRDC) of Taiwan. It is known that the CFD simulations on Ansys Fluent by [PERSON], [10], are compatible with the experimental results of MIRDC. Thus, it is natural to compare our 1D simulations with the 3D CFD simulations on Ansys Fluent for the arrangement of checkers described above. We assume that the temperature of exhaust gas, expelled from the central furnace, is 900 \({}^{\circ}\)C (1173.15 K). We assume that the temperature of fresh air, at the Inlet, is 40 \({}^{\circ}\)C (313.15 K). We assume that the time of the _Phase Switch_ is 30 s. We assume that the superficial flow velocity of the exhaust gas is 2.52 m/s. We assume that the superficial flow velocity of the fresh air is 0.56 m/s. The comparison of our 1D simulations with the 3D CFD simulations on Ansys Fluent for the arrangement of checkers  
\[(\text{Cordierite,Cordierite,Cordierite,Cordierite,Mullite,Mullite})\]  
at 6 positions, in a fixed-bed heat regenerator, is shown in Figure 8.  
In Figure 8, the Inlet temperature of exhaust gas (expelled from the central furnace) and the Outlet temperature of fresh air (into the central furnace), based on our 1D simulations using C++, is shown in black. In Figure 8, the Inlet temperature of exhaust gas (expelled from the central furnace) and the Outlet temperature of fresh air (into the central furnace), based on 3D CFD simulations on Ansys Fluent, is shown in red. In Figure 8, the Outlet temperature of exhaust gas (expelled from the right heat regenerator) and the Inlet temperature of fresh air (into the right heat regenerator), based on our 1D simulations using C++, is shown in blue. In Figure 8, the Outlet temperature of exhaust gas (expelled from the right heat regenerator) and the Inlet temperature of fresh air (into the right heat regenerator), based on 3D CFD simulations on Ansys Fluent, is shown in green. It can be observed that, in the _long term_, our numerical simulation results, based on the 1-dimensional partial differential equations, are consistent with the simulation results based on CFD (Computational Fluid Dynamics) on _Ansys Fluent_. See Figure 8.  
We also test the validity of our 1-dimensional model by comparing our simplified 1-dimensional numerical simulation results with the experimental results provided by the Metal Industries Research & Development Centre (MIRDC) of Taiwan. There are 3 types of checkers, \(A\), \(B\) and \(C\), used in an arrangement  
\[(A,A,C,C,C,C,A,C,B,B,B,B,B)\]  
of checkers at 13 positions. The material parameters of these 3 types of checkers, \(A\), \(B\) and \(C\), are shown in Table 2.  
There are two experiments provided by MIRDC. The operating parameters for Experiments 1 and 2 are shown in Table 3. The _Phase Switch_ time is 30 s for both experiments. The superficial flow velocity of the exhaust gas is 15.9 m/s. The superficial flow velocity of the fresh air is 3.5 m/s.