## 5 Result and Discussion
### Optimization of Radii in Stacks  
At \(L/L_{loop}=0.54\) where COP\({}_{total}\) corresponded to a maximum, \(T_{H}\) corresponded to 674 K as shown in Figure 4e. Given that \(T_{A}\) and \(T_{C}\) were set to 301 K and 251 K, respectively, the thermodynamic upper limit of \(\eta_{e}\), namely [PERSON]'s efficiency is as follows:  
\[\eta_{Carnot}=1-\frac{T_{A}}{T_{H}}. \tag{13}\]  
Additionally, [PERSON]'s COP is as follows:  
\[\text{COP}_{Carnot}=\frac{T_{C}}{T_{A}-T_{C}}. \tag{14}\]  
Hence, \(\eta_{e}\) and COP\({}_{c}\) corresponds to 0.55 and 5.0, respectively. This implies that the calculated maximum \(\eta_{e}\) and COP\({}_{c}\) corresponds to 57% and 49% of the [PERSON]'s values, respectively. It was considered that there was some scope for improvement with respect to these values. It is widely known that the radius of a stack influences the efficiency of the energy conversion caused by an acoustic wave [8]. Hence, the calculation described in Section 4.2 was performed by varying \(r_{e}\) and \(r_{c}\) and keeping \(L/L_{loop}=0.54\) and \(T_{C}=251\) K. In order to compare the obtained \(\eta_{e}\), COP\({}_{c}\), and COP\({}_{total}\) with their thermodynamic upper limit values, the efficiencies are defined in the following equations:  
\[\eta_{2,e}=\frac{\eta_{e}}{\eta_{Carnot}}, \tag{15}\]  
\[\eta_{2,c}=\frac{\text{COP}_{c}}{\text{COP}_{Carnot}}, \tag{16}\]  
\[\eta_{2,total}=\frac{\text{COP}_{total}}{\eta_{Carnot}\text{COP}_{Carnot}}. \tag{17}\]  
As shown in the contour plot of Figure 5a, \(\eta_{2,e}\) improved when \(r_{e}/\delta\) and \(r_{c}/\delta\) were decreased and increased, respectively. The plot also shows that \(\eta_{2,e}\) exceeded 70% when \(r_{e}/\delta\) and \(r_{c}/\delta\) were determined as the values in the hatched area of Figure 5a. The value of \(\eta_{2,e}\) was comparable to the efficiency of the most efficient thermoacoustic and Stirling engines constructed to date [15; 21; 22]. Conversely, \(\eta_{2,c}\) increased with increasing \(r_{e}/\delta\) in the calculated ranges as shown in Figure 5b. This is opposite to the dependence of \(\eta_{2,e}\) on \(r_{e}/\delta\). Hence, it is difficult to simultaneously achieve high \(\eta_{2,c}\) and \(\eta_{2,e}\) by controlling for \(r_{e}/\delta\) and \(r_{c}/\delta\). Furthermore, \(r_{e}/\delta\) and \(r_{c}/\delta\) should be optimized simultaneously. In order to consider the reason as to why the decrease of \(r_{e}/\delta\) caused an improvement in \(\eta_{2,e}\) and the deterioration in \(\eta_{2,c}\), the acoustic impedance \(Z\) was re-calculated at the engine and cooler stacks. The results indicated that a decrease in \(r_{e}/\delta\) increased \(|Z|\) at the engine stack and decreased \(|Z|\) at the cooler stack. This is caused by the fact that when \(r_{e}/\delta\) was decreased, the sweet spot area was narrowed, in which \(|Z|\) is high and the argument of \(Z\) is near zero [23], and, then, the position of the cooler stack departed from the sweet spot area. This implied that the optimum relative position shifted toward \(L/L_{loop}=0.50\). Hence, after considering the optimum values of \(r_{e}/\delta\) and \(r_{c}/\delta\), the relation between \(\eta_{2,total}\) and \(L/L_{loop}\) was re-calculated. In Figure 5c, the contour plot of the calculated \(\eta_{2,total}\) is shown as a function of \(r_{e}/\delta\) and \(r_{c}/\delta\). It is possible to determine the optimum point to increase \(\eta_{2,total}\). With respect to the optimum point (\(r_{e}/\delta=0.95\), \(r_{c}/\delta=1.06\)), \(\eta_{2,total}\) was approximately 10%. As shown in Figure 5d, the temperature \(T_{H}\) increased when \(r_{e}/\delta\) and \(r_{c}/\delta\) decreased. This means that, in the case where the value of \(T_{H}\) is determined as a given condition, such as in the case of the utilization of waste heat, the optimum \(r_{e}/\delta\) and \(r_{c}/\delta\) depend on the given \(T_{H}\) value. As described above, the results indicated that the optimum relative position \(L/L_{loop}\) changed when \(r_{e}/\delta\) decreased.