## 3 Applications to a Conventional Glass Thermal Regenerator  
The regenerator model is applied to analyze the system performance with different size and operative conditions. As reported in [21], the performance of a periodic thermal storage can be evaluated using three non-dimensional parameters:  
(1) The thermal efficiency \(\eta_{T}\), i.e., the ratio between the air outlet and the exhaust gas inlet temperatures (Equation (26)):  
\[\eta_{T}=\
icefrac{{T_{char,out}}}{{T_{char,in}}} \tag{26}\]  
(2) The storage effectiveness \(R\), i.e., the ratio between the energy absorbed by the air in a cycle and the energy available from the exhaust gas flow (Equation (27)):\[R=\frac{\dot{m}_{air}c_{pair}\tau_{air}\left(\widetilde{T}_{air,out}-T_{air,in}\right) }{\dot{m}_{ex}c_{pex}\tau_{ex}\left(\widetilde{T}_{ex,out}-T_{ex,in}\right)} \tag{27}\]  
(3) The capacitance utilization \(CU\), i.e., the ratio between the energy absorbed by the air in a cycle and the maximum energy that can be stored in the solid matrix (Equation (28)):  
\[CU=\frac{\dot{m}_{air}c_{pair}\tau_{air}\left(\widetilde{T}_{air,out}-T_{air,in }\right)}{m_{s}c_{s}(T_{ex,in}-T_{air,in})} \tag{28}\]  
In glass furnace regenerative systems, \(\eta_{T}\) can commonly approach values close to one. This is possible due to the mass flow and specific heat imbalance between the air and the waste gas phases. From Equation (27), it is clear that if \(\eta_{T}\) is equal to one, \(R\) reaches its maximum value (Equation (29)). With this assumption, a specific effectiveness \(\varepsilon\) can be defined (Equation (30)) as the ratio between the actual storage effectiveness \(R\) and \(R_{max}\). \[R_{max}=\dot{m}_{air}c_{pair}/\dot{m}_{ex,in}c_{p,sh} \tag{29}\]  
\[\varepsilon=\
icefrac{{R}}{{R_{max}}} \tag{30}\]  
The analyses are performed with the following reference case: a thermal regenerator where each RC has a total volume of 100 m\({}^{3}\), \(F_{f}\) equal to 0.7, and a total heat transfer area equal to 2900 m\({}^{2}\). The regenerator is set up to work with a reverse time \(\tau\) equal to 20 min. The regenerator is fed with an exhaust mass flow rate equal to 4.88 kg/s at a temperature of 1135 \({}^{\circ}\)C and with an air mass flow rate equal to 3.77 kg/s at a temperature of 145 \({}^{\circ}\)C. The chemical composition of the air and exhaust gases are set as reported in Table 1. A constant linear profile of the heat exchange coefficients is set for each gas phase, as reported in [17]. Figure 4 shows the time-averaged temperature profile of the two flows and of the refractories. Figure 5 shows the same variables on the top and the bottom layers of the RC over a regeneration cycle. From the above data, the \(\tilde{\eta}_{T}\) is 0.883 with an oscillation of about 0.02 over the cycle, and the \(R\) value is 0.482, which means \(\varepsilon\) equal to 0.834. Figure 4: Average temperature profile across the regenerative chamber (RC) height (reference case). The unsteady regenerator model is then applied to compare three regenerative chambers of varying size, as detailed in Table 2. The heat transfer surface and the volume are gradually reduced. The inversion time equal to 20 min is kept. Figure 6 shows the value of the regenerative performance coefficients for each case. The \(\widetilde{\eta}_{T}\) increases with the RC size. The increase rate is more pronounced for changes in small RC. The \(R\) shows a trend similar to \(\widetilde{\eta}_{T}\), and the \(CU\) value decreases by increasing the RC volume. In fact, referring to Equation (28), the increase of the thermal capacity of the storage \(m_{s}c\
u_{s}\) is more effective than the increase of the regenerator energy performance. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Case A** & **Case B** & **Case C** & \\ \hline \(A_{sf,tot}\) & 2900 & 1930 & 970 & m\({}^{2}\) \\ \(V_{tot}\) & 100 & 66.6 & 33.3 & m\({}^{3}\) \\ \(F_{f}\) & 0.7 & 0.7 & 0.7 & \\ \hline \end{tabular}
\end{table} 
Table 2: Geometrical parameters of simulated cases.