## 5 Result and Discussion
### Effect of the Relative Position of the Cooler Stack  
[PERSON] et al. reported that the position of the cooler stack influences the performance of the system [1]. Therefore, the relative position \(L/L_{loop}\) was changed. The radius of the narrow channels of the engine and cooler stacks was set as 0.27 mm, which corresponds to the same value as that in [PERSON]'s experimental setup [1]. Given this value, \(r/\delta\) in the stacks approximately corresponds to 1.5 at \(T_{A}=301\) K.  
Figure 3: \(|P|\), \(|U|\), and \(\phi\) as a function of \(x/L_{loop}\). The symbols show the experimental results obtained from the article [1]. The calculated results with one-wavelength mode (\(\omega/(2\pi)\sim 360\) Hz) are shown in Figure 4. The results indicated that the efficiency of the engine \(\eta_{e}\) and the coefficient of performance of the cooler COP\({}_{c}\) depended on the relative position \(L/L_{loop}\), and that the maximum value of \(\eta_{e}\) and that of COP\({}_{c}\) corresponded to 0.31 and 2.45 at \(L/L_{loop}=0.54\), respectively. These values are 1.9 and 105 times larger than the values of \(\eta_{e}\) and COP\({}_{hp}\) at \(L/L_{loop}=0.50\), respectively. Given that \(L_{loop}\) corresponds to 2.8 m, the difference between 0.54 and 0.50 times of \(L_{loop}\) corresponds to 0.11 m. Hence, it is considered that the optimization of \(L/L_{loop}\) is significant in increasing the performance of the cooler. Conversely, \(\eta_{tube}\) as shown in Figure 4c monotonically decreased from 0.46 to 0.21, when \(L/L_{loop}\) was increased from 0.50 to 0.56. This means that there is a tradeoff between the dependence of \(\eta_{tube}\) on \(L/L_{loop}\) and those of \(\eta_{e}\) and COP\({}_{c}\) when \(0.50<L/L_{loop}<0.54\). As the dependence of \(\eta_{tube}\) was weak when compared to that of the COP\({}_{c}\), the COP\({}_{total}\) became maximum ( 0.21) at \(L/L_{loop}=0.54\), as shown in Figure 4d. This result is consistent with that in the study by [PERSON] et al. [1]. Additionally, \(P\) and \(U\) were calculated along the looped tube and the results revealed that the distributions of \(P\) and \(U\) were not fixed but were influenced by \(L/L_{loop}\). Figure 4e,\(f\) show \(|Z/(\rho a)|\) and \(\phi\) at the ends of the engine and cooler stacks as a function of \(L/L_{loop}\), where \(a\) is the adiabatic sound speed and \(\phi\) is the argument of \(|Z/(\rho a)|\). As indicated by the open triangles in Figure 4e, \(|Z_{e}/(\rho_{e}a_{e})|\) in the engine stack corresponded to a maximum at \(L/L_{loop}=0.54\). This contributes to the maximum \(\eta_{e}\) obtained at \(L/L_{loop}=0.54\) because high \(|Z_{e}/(\rho_{e}a_{e})|\) decreases the viscous loss. It is also found that \(\phi_{e}\)s at both of the ends of the engine stack took negative values above \(L/L_{loop}=0.54\), as indicated by open-circles and squares in Figure 4e. This means that \(\phi\) was negative along the engine stack. This result also contributes to the increase of \(\eta_{e}\)[19]. This is because the energy conversion caused by the standing wave works positively only when \(\phi<0\), whereas the energy conversion caused by the traveling wave works positively when \(-\pi/2<\phi<\pi/2\) (as shown in Equations (12) and (13) in the previous study [3]). In the cooler stack, \(|Z_{c}/(\rho_{c}a_{c})|\) weakly depended on \(L/L_{loop}\), whereas \(\phi_{c}\) depended on \(L/L_{loop}\) as shown in Figure 4f. At \(L/L_{loop}=0.54\), at which COP\({}_{c}\) became maximum, \(\phi_{c}\) was not near zero but close to \(60^{\circ}\). We consider the discrepancy between the optimum \(\phi_{c}\) for the cooler stack and that for the engine stack can be attributed to the difference of the temperature gradient along them: the temperature gradient along the cooler stack was much lower than that along the engine stack.