## 2 Methods
### COP Estimation for Thermoacoustic Devices
(2018), rewriting \(Q_{cn}\) and \(W_{n}\) in dimensionless form by using dimensionless parameters as  
\[Q_{cn}=-\bigg{[}\frac{\delta_{xn}D^{2}\sin(2x_{n})}{8\gamma(1+ \sigma)\Lambda}\bigg{]}\\ \times\bigg{[}\frac{\Delta T_{mn}\tan(x_{n})}{(\gamma-1)BL_{sn}} \frac{1+\sqrt{\sigma}+\sigma}{1+\sqrt{\sigma}}-\big{(}1+\sqrt{\sigma}-\sqrt{ \sigma}\delta_{xn}\big{)}\bigg{]} \tag{3}\]  
\[W_{n}=\bigg{[}\frac{\delta_{xn}D^{2}L_{sn}(\gamma-1)B\cos^{2}(x_ {n})}{4\gamma}\bigg{]}\\ \times\bigg{[}\frac{\Delta T_{mn}\tan(x_{n})}{BL_{sn}(\gamma-1) \big{(}1+\sqrt{\sigma}\big{)}\Lambda}-1\bigg{]}\\ -\bigg{[}\frac{\delta_{xn}L_{sn}D^{2}}{4\gamma}\times\frac{\sqrt {\sigma}\sin^{2}(x_{n})}{B\Lambda}\bigg{]} \tag{4}\]where \(\Lambda=\left(1-\sqrt{\sigma}\delta_{xn}+\frac{1}{2}\sigma\delta_{xn}^{2}\right)\), \(\delta_{xn}=\delta_{x}/y_{0}\) is the dimensionless thermal penetration depth, \(B=y_{0}/(y_{0}+l)\) is the regenerator blockage ratio, \(y_{0}\) is half the distance between the stack plates, \(l\) is half the thickness of the regenerator planes (see Figure 2), \(\sigma\) is the Prandtl number, \(\Delta T_{mn}=\Delta T_{m}/T_{m}\) is the dimensionless temperature difference, \(\gamma\) is the ratio of isobaric to isochoric specific heats, \(L_{sn}=kL_{s}\) is the dimensionless stack length, \(x_{n}=kX_{s}\) is the dimensionless stack position, \(k\) is the wave number, and \(D=p_{0}/P_{m}\) is the drive ratio (\(p_{0}\) depends on \(D\) and \(P_{m}\)). If the gas and operating frequency are fixed, the parameters \(\delta_{xn}\), \(\delta_{yn}\), \(\gamma\), and \(\sigma\) in Table 1 can be determined, so the \(COP\) depends on (A, B, C, D, E), defined in Table 2. \(\Lambda\) plays a crucial role in describing the behavior of acoustic waves and their interaction with thermal systems, particularly in the stack and heat exchanger components of thermoacoustic engines and refrigerators. [PERSON] [1] explains that thermal penetration depth plays a role in optimizing heat transfer between the oscillating gas and the stack plates, and that the stack plate spacing should ideally be proportional to the thermal penetration depth to maximize heat transfer efficiency. When the thermal penetration depth matches the spacing of the stack plates, heat transfer is most efficient. Its role in heat transfer efficiency dictates the interaction between acoustic oscillations and thermal energy. Specifically, optimal stack spacing is typically in the range of 2 to 4 times the thermal penetration and enhances the \(COP\) and overall efficiency of the thermoacoustic refrigerator. Moreover, \(\Lambda\) decreases as the frequency of the acoustic wave increases (frequency dependence). This means that for higher-frequency waves, the thermal oscillations occur over a shorter distance, affecting the design considerations of the stack. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Definitions** \\ \hline Operational parameters & Drive ratio \(D=p_{0}/P_{m}\) \\  & Dimensionless cooling power \(Q_{cn}=Q_{c}/P_{m}aA\) \\  & Dimensionless acoustic power \(W_{n}=W/P_{m}aA\) \\  & Dimensionless temperature difference \(\Delta T_{mn}=\Delta T_{m}/T_{m}\) \\ \hline Gas parameters & Prandtl number \(\sigma=(\delta_{v}/\delta_{x})^{2}\) \\  & Dimensionless thermal penetration depth \(\delta_{xn}=\delta_{x}/y_{0}\) \\  & Dimensionless viscous penetration depth \(\delta_{m}=\delta_{v}/y_{0}\) \\  & Ratio of isobaric to isochoric specific heats \(\gamma\) \\ \hline Geometrical parameters & Dimensionless stack length \(L_{sn}=kL_{s}\) \\  & Dimensionless stack position \(x_{n}=kX_{s}\) \\  & Blockage ratio \(B=y_{0}/(y_{0}+l)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for optimizing thermoacoustic refrigerator [13].