## 2 Materials and Methods
### The Numerical Simulation Methods  
To speed up the evaluation process for an arrangement of checkers in a fixed-bed heat regenerator, our numerical simulations are based on simplified 1-dimensional partial differential equations as in Refs. [1; 2; 3]. This 1-dimensional model has been discussed in Refs. [11; 15; 57]. (See Ref. [58] for the fundamental theory of Continuum Mechanics.) These 1-dimensional partial differential equations were further developed in Refs. [4; 5]. Let us consider the two-phase cycle of a fixed-bed heat regenerator, which is placed on the _right_ end of a regenerative combustion system, as shown in Figure 7. In Phase 1, the high-temperature exhaust gas, from the central furnace, flows into the _right_ heat regenerator and _heats up_ the _checkers_ arranged there. Then Phase 2 starts at the _Phase Switch_ point of time. In Phase 2, the above process is _reversed_. The cool fresh air flows into the _right_ heat regenerator and is _preheated_ by the _checkers_ arranged there, which already absorbed heat from the high-temperature exhaust gas expelled from the central furnace during Phase 1. Then the _preheated_ fresh air flows into the central furnace for combustion with natural gas. Let \(t_{sw}\) denote the _switch time_ of the operation cycle of the fixed-bed heat regenerator shown above. Thus  
\[2t_{sw}=t_{sw}+t_{sw}\]  
is the _period_ of the operation cycle. When \(0\leq t\leq t_{sw}\), Phase 1 operates. When \(t_{sw}\leq t\leq 2t_{sw}\), Phase 2 operates. To evaluate the efficiency of an arrangement of checkers on a fixed-bed heat regenerator, this operation cycle must be performed periodically until we reach an approximately _steady state_. We assume that the fixed-bed heat regenerator is located on the interval \([0,l]\) of the \(x\)-axis. See Figure 7 for details. We use the subscript \(ar\) to indicate functions related to the _fresh air_ flow. We use the subscript \(ex\) to indicate functions related to the _exhaust gas_ flow. We use the subscript \(s\) to indicate functions related to the _checkers_ (solid porous medium). Let \(n\) be a nonnegative integer. When  
\[n\cdot 2t_{sw}+0\leq t\leq n\cdot 2t_{sw}+t_{sw},\]  
we have the following 1-dimensional partial differential equations:  
\[\epsilon\frac{\partial(\rho_{ex}C_{ex}T_{ex})}{\partial t}+u\frac{\partial( \rho_{ex}C_{ex}T_{ex})}{\partial x}=\epsilon\frac{\partial}{\partial x}\bigg{(} k_{ex}\frac{\partial T_{ex}}{\partial x}\bigg{)}-ha_{s}(T_{ex}-T_{s}) \tag{21}\]  
and  
\[(1-\epsilon)\frac{\partial(\rho_{s}C_{s}T_{s})}{\partial t}=(1-\epsilon) \frac{\partial}{\partial x}\bigg{[}k_{s}\frac{\partial T_{s}}{\partial x} \bigg{]}+ha_{s}(T_{ex}-T_{s})-q_{d} \tag{22}\]  
Figure 7: Two-phase cycle of a fixed-bed heat regenerator. satisfying the following boundary conditions  
\[\left[\frac{\partial T_{s}}{\partial x}\right]_{x=0}=0=\left[\frac{\partial T_{s}} {\partial x}\right]_{x=l},\ \left[\frac{\partial T_{ex}}{\partial x}\right]_{x=l}=0\ \text{and}\ T_{ex}(0,t)\; given. \tag{23}\]  
**Remark 1**. : _Here, in (21) and (22), we omit the Darcy dissipation. The Darcy dissipation may be estimated through experiments or through the Ergun equation for the pressure drop. See page 86 of Refs. [11; 15; 57] for a discussion of the Ergun equation. See, for examples, Refs. [59; 60] for certain experimental investigations on the pressure drop. For the fluid flows (fresh air or exhaust gas) considered in our simulations, we have_  
\[0.5\;(\mathrm{m}/\mathrm{s})\leq\text{superficialflowvelocities}\leq 16.0\;( \mathrm{m}/\mathrm{s}).\]  
_In the CFD simulations on Ansys Fluent by [PERSON], Ref. [10], it is found that, for the following arrangement of checkers_  
\[(\text{Cordierite,Cordierite,Cordierite,Cordierite,Mullite,Mullite})\]  
_at 6 positions in a fixed-bed heat regenerator, the pressure drop is_  
\[110\text{Pa}\;(\text{less than}\;0.0011\;\text{Atmospheric Pressure}).\]  
_The material parameters of these checkers, considered above, are listed in Table 1 below.