## 2 Methods
### COP Estimation for Thermoacoustic Devices
\begin{table}
\begin{tabular}{c l c c} \hline \hline
**Factor** & **Name** & \
\begin{tabular}{c} **Level** \\ **Low–High** \\ \end{tabular} &
\begin{tabular}{c} **Units** \\ \end{tabular} \\ \hline A & Drive ratio (\(D\)) & \(-\) & \(+\) & N/A \\ B & Dimensionless stack position (\(x_{n}\)) & \(-\) & \(+\) & N/A \\ C & Dimensionless temperature difference (\(\Delta T_{mn}\)) & \(-\) & \(+\) & N/A \\ D & Blockage ratio (\(B\)) & \(-\) & \(+\) & N/A \\ E & Dimensionless regenerator length (\(L_{sn}\)) & \(-\) & \(+\) & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Low and high levels by factor used in the DOE. Figure 2: Structure diagram of a parallel stack. Thus, the performance of the regenerator depends explicitly on dimensionless variables, as shown in Equations (3) and (4). It is important to note that the Prandtl number \(\sigma\) is proportional to the ratio of the thermal \(\delta_{\kappa}\) and viscous \(\delta_{v}\) penetration depths; see Equations (1) and (2), respectively. The introduction of \(\sigma\) as the Prandtl number in thermoacoustic theory would make sense if the relationship between this term and the classical Prandtl number was clarified. The Prandtl number (\(Pr\)) is a dimensionless number used in fluid dynamics and heat transfer:  
\[Pr=\frac{\
u}{\alpha}=\frac{\mu_{v}c_{p}}{k} \tag{5}\]  
where the following apply:  
* \(\
u\) is the kinematic viscosity (moment diffusivity);
* \(\alpha\) is the thermal diffusivity;
* \(\mu_{v}\) is the dynamic viscosity;
* \(c_{p}\) is the specific heat at constant pressure;
* \(k\) is the thermal conductivity. In thermoacoustic theory, the behavior of gas within a stack (or heat exchangers) is governed by both the thermal and viscous effects defined above. The \(\delta_{\kappa}\) and \(\delta_{v}\) are critical in defining how heat and momentum are transferred near solid boundaries. The Prandtl number \(\sigma\) helps to relate these two penetration depths. If \(\sigma\) is large, the viscous effect dominates, meaning the moment boundary layer is thicker than the thermal boundary layer. If \(\sigma\) is small, thermal effects dominate, meaning heat diffuses more effectively. Therefore, the design of the stack in thermoacoustic devices often takes into account the thermal penetration depth, which is related to the Prandtl number, when choosing the appropriate blockage ratio. The Prandtl number is an essential parameter for understanding and analyzing the convective heat transfer characteristics of fluids. The Prandtl number and the blockage ratio are not directly related in a mathematical sense, but both affect the overall performance of thermoacoustic systems. The Prandtl number influences the boundary layer thickness, while the blockage ratio is a geometric parameter controlling the interaction between the gas and the solid boundaries. For the analysis in this study, five of these dimensionless variables were selected from Table 1: the drive ratio \(D\), the blockage ratio \(B\), the dimensionless temperature difference \(\Delta T_{mn}\), the dimensionless stack position \(x_{n}\), and the dimensionless stack length \(L_{sn}\). The dimensionless viscous penetration depth \(\delta_{vn}\) was implicit in the \(COP\) calculation, even though it is not present in Equations (3) and (4). The analysis focused on those variables defined above in Equations (3) and (4) through the performance of the stack in terms of the \(COP\) as  
\[COP=\frac{Q_{cn}}{W_{n}} \tag{6}\]  
Our analysis considered a quarter-wavelength resonator using air as a working gas [31].