## 2 Simulation Model  
The numerical transient model of the thermal regenerator is implemented using the software Matlab with the Simulink tool. As in the real system, the numerical model is composed of two RCs sub-models.  
Each RC model is modeled with a solid domain representing the refractory and with a fluid domain that models the cold or the hot fluid phase. In order to reproduce the temperature profile, the solid and the fluid domains are discretized with N finite cells, as shown in Figure 2. For each cell, a transient energy balance equation is added to describe the heat and mass transfer between the adjacent cells using the following hypothesis:  
* No heat transfer through the casing with the external environment,
* No seepage or flux leakage from the casing,
* Fluid thermal properties function of temperature only,
* Incompressible flow.  
Figure 3 shows a sketch of the energy interaction between adjacent cells. The energy balance of the generic \(i^{th}\) fluid cell can be evaluated by Equation (1): the internal energy time variation (left side  
of the equation) is equal to the sum of the contributions of the heat exchanged with the adjacent solid cell and the net heat transported by the mass flow from the inlet to the outlet of the fluid cell.  
The energy balance of the generic \(i^{th}\) solid cell can be evaluated by Equation (2): the internal energy time variation (left side of the equation) is equal to the sum of the contributions of the heat exchanged with the adjacent fluid cell and upper and lower solid cells.  
\[\begin{split}\text{For }i=1,\cdots,N\\ m_{f,i}cv_{f,i}\frac{\dot{dT}_{f,i}}{dt}=\dot{m}_{f}\Big{(}cp_{f,i- 1}T_{f,i-1}-cp_{f,i}T_{f,i}\Big{)}+\Big{(}U_{sf,i}A_{sf,i}\Big{)}\Big{(}T_{s,i }-T_{f,i}\Big{)}\end{split} \tag{1}\]  
Figure 3: Heat flow for the ith layer.  
Figure 2: Sketch of a single chamber domain discretization.  
\[m_{s,i}cv_{s,i}\frac{\dot{m}_{s,i}}{dt}=(U_{ss,i-1}A_{ss,i-1})(T_{s,i-1}-T_{s,i})+(U _{ss,i+1}A_{ss,i+1})(T_{s,i+1}-T_{s,i}) \tag{2}\]  
where the subscript \(f\) and \(s\) refer to the fluid and solid domain, respectively, \(\dot{m}_{s}\) is the cell mass, \(\dot{m}_{f}\) is the mass flow, \(cv\) and \(cp\) are respectively the specific heat flow at constant volume and pressure. The parameters \(A_{sf,i}\) and \(A_{ss,i}\) are respectively the heat transfer surface at the solid/fluid interface and at the solid/solid interface of the \(i^{th}\) layer. \(U_{sf,i}\) and \(U_{ss,i}\) are respectively the thermal transmittances between the solid and the fluid cells and between the two solid cells, defined in the following Equations (3) and (4).  
\[\begin{array}{c}\text{For }i=1,\cdots,N\\ U_{sf,i}=\frac{1}{\frac{1}{h_{tot,i}}+\frac{(h}{h_{s}}}\ \ \left[\frac{\text{W}}{ \text{m}^{2}\cdot\text{K}}\right]\\ U_{ss,i}=\frac{k_{s}}{H/N}\ \ \ \left[\frac{\text{W}}{\text{m}^{2}\cdot\text{K}}\right] \end{array} \tag{3}\]  
where \(h_{tot}\) is the total heat transfer coefficient; it includes the convective and radiant effects between the gas flow and the checkers. \(th\) is half of the refractories thickness, \(k_{s}\) is its thermal conduction coefficient, and \(H\) is the total checkers height.  
The equation system described by Equations (1) and (2) can be re-arranged as in Equations (5) and (6).  
\[\frac{d}{dt}\Big{[}T_{f}\Big{]}=\Big{[}a_{f}\Big{]}\Big{[}T_{f}\Big{]}+\Big{[} b_{f}\Big{]}[T_{s}]+\Big{[}c_{f}\Big{]}T_{f,ini} \tag{5}\]  
\[\frac{d}{dt}[T_{s}]=[a_{s}][T_{s}]+[b_  
\[b_{s_{s}(i,i)}=\frac{U_{sf,i}A_{sf,i}}{m_{s,i}cv_{s,i}} \tag{14}\]  
\[\text{For }i=2,\cdots,N \tag{15}\]  
\[a_{s,(i,i-1)}=\frac{U_{sf,i-1}A_{sf,i-1}}{m_{s,i}cv_{s,i}} \tag{16}\]  
\[\text{For }i=1,\cdots,N-1 \tag{17}\]  
\[a_{s,(i,i+1)}=\frac{U_{sf,i+1}A_{sf,i+1}}{m_{s,i}cv_{s,i}} \tag{18}\]