# Measurement of Heat Flow Transmitted through a Stacked-Screen Regenerator of Thermoacoustic Engine
## 4 Results and Discussion
### Stacked-Screen Regenerator  
Figure 3 presents the relation between the heat flow rate \(|Q|\) and the velocity amplitude \(|\langle V\rangle|\) for the mesh screens in Table 1, for 200 Hz frequency of the oscillatory flow. For comparison with the empirical equations, we show the measured \(|Q|\) values in the four graphs Figure 3a-d. Solid symbols, each of which corresponds to meshes listed in Table 1, show that \(|Q|\) increases quadratically with \(|\langle V\rangle|\), as in the case of the regular flow channel regenerator. Figure 3a compares the measured \(|Q|\) with that obtained by inserting [PERSON]'s effective radius \(r_{0}\) into Equations (3) and (4), which shows good agreement with #50, #60, and #80 meshes, but slight deviation is visible for #20 and #30 meshes. Although the effective radius \(r_{0}\) proposed by [PERSON] et al. was obtained from measurements at uniform temperature, the present experimentally obtained results  
Figure 2: Comparison between the experimental heat flow rate \(|Q|\) and theoretical calculations is given by Equation (3) for the ceramic honeycomb catalyst, where experimentally obtained results are shown as symbols with error bars. The theoretical calculations of \(|Q|\) and \(|Q_{\text{D}}|\) are shown, respectively, as solid and dashed curves. confirm the usefulness of evaluating the heat flow in the stacked-screen regenerator when Ar is the working gas. Comparison with results of the simple harmonic analysis derived by [PERSON] and [PERSON] is presented in Figure 3b, where curves are shown using Equations (5)-(8). For Equations (6), \(\theta_{T}\) was estimated from Equation (4.68) of Reference [17] using \(d_{\mathrm{h}}/2\). Figure 3b shows that good agreement is obtained, except for the #80 mesh. Because the frequency of 200 Hz gives the thermal penetration depth \(\delta_{\alpha}=\sqrt{2\alpha/\omega}=0.15\) mm, these results indicate that the empirical expression of [PERSON] and [PERSON] is suitable for derivation of \(|Q|\), when \(d_{\mathrm{h}}/(2\delta_{\alpha})\) is in the range of \(d_{\mathrm{h}}/(2\delta_{\alpha})>1\), even though they assumed good thermal contact between the oscillation gas and the solid wall of the flow channel [24]. Heat flows estimated using the empirical equations of [PERSON] et al. and Gedeon and Wood are shown, respectively, in Figure 3c,d, where we used Equations (9) and (10) to derive the temperature \(\langle T\rangle_{Nu}\) in Equation (11). Then, we used Equations (7) and (8). All curves in Figure 3c show agreement with the measured \(|Q|\) to a similar degree to those portrayed in Figure 3a,b. Actually, Figure 3d shows poor agreement with meshes of #30, #50, and #60, but the estimation is not bad overall. These results indicate that these empirical equations are applicable to thermoacoustic Stirling engines that operate at higher oscillation frequencies than the mechanical Stirling engines [3; 4]. To assess the frequency dependence of the heat flow \(Q\), the regenerator stacked with #30 mesh screens was tested with frequencies of 140 Hz and 180 Hz in the manner described above. Results are presented in Figure 4a-d, where the experimental \(|Q|\) values increased slightly because of the decrease of the frequency. Figure 4a shows that the agreement with the effective radius of [PERSON] et al. was satisfactory. The Swift-Ward formulation in Figure 4b also works well, probably because of the value \(d_{\mathrm{h}}/(2\delta_{\alpha})\) beyond 2 with those frequencies. The empirical equation of [PERSON] et al. in Figure 4c agrees with the measured \(|Q|\), although only the tendency of \(|Q|\) for the frequency change is reproduced by the empirical equation of [PERSON] and [PERSON], as shown in Figure 4d. Figure 3: Experimental heat flow rate \(|Q|\) and predictions given by respective empirical equations cited in this study for stacked-screen regenerators tested in pressurized helium gas with 0.45 MPa and with oscillation frequency of 200 Hz. Each panel presents identical experimentally obtained results by symbols with error bars.