# Measurement of Heat Flow Transmitted through a Stacked-Screen Regenerator of Thermoacoustic Engine
## 4 Results and Discussion
### Discussion  
We have presented, through comparisons with experimentally obtained values, that the empirical equations proposed up to this point are useful as a first approximation to the heat flow in the stacked-screen regenerators, in spite of the considerable difference between their derivations. Among them, [PERSON]'s formulation is the simplest to handle because of the absence of the Nusselt number \(Nu\), which changes with the Reynolds number, \(Re_{\text{h}}\), through the velocity amplitude \(\left|V\right|\). Hereinafter, to strengthen the applicability of \(r_{0}\) further, we investigate the function \(\text{Im}\left[g_{\text{D}}\right]\) from the experimental \(\left|Q\right|\). As described in Section 3, the heat flow component \(Q_{\text{A}}\) is negligibly small compared to \(Q_{\text{D}}\) for all experimental conditions. Therefore, we approximate \(Q\) with \(Q_{\text{D}}\), to express \(\text{Im}\left[g_{\text{D}}\right]\) as  
\[\text{Im}\left[g_{\text{D}}\right]=\frac{2}{A}\frac{\omega}{\rho_{\text{m}}c _{\text{p}}}\frac{Q}{\left(dT_{\text{m}}/dx\right)\left|\left\langle V\right \rangle\right|^{2}}. \tag{14}\]  
All factors in Equation (14) were evaluated using the measured values and the thermal properties of the gas at the hot end of the regenerator (\(T_{\text{H}}\)=543 K). The resulting \(\text{Im}\left[g_{\text{D}}\right]\) is shown in Figure 5, after averaging over different values of \(\left|\left\langle V\right\rangle\right|\). Additionally, the error bars attached on the symbols denote the standard deviations, which show the degree of the influence of \(\left|\left\langle V\right\rangle\right|\). The curve represents \(\text{Im}\left[g_{\text{D}}\right]\) given using \(\sigma=0.66\). It is apparent that \(\text{Im}\left[g_{\text{D}}\right]\) obtained by Equation (14) roughly approximates the theoretical value for the cylindrical flow channel, which means that non-dimensional quantity \(r_{0}/\delta_{\alpha}\) captures the mechanism of the oscillation-induced heat flow in the stacked-screen regenerator. This finding is consistent with the results presented by [PERSON] et al. [21], who used air at atmospheric pressure as the working gas. Therefore, we consider that the local heat transfer between the screen meshes and the working fluid would be governed mostly by the non-dimensional parameter \(r_{0}/\delta_{\alpha}\). Therefore, the axial heat flow through the stacked screen regenerator is fundamentally described by Equation (3). Figure 4: Comparisons between the experimental heat flow \(\left|Q\right|\) and predictions given by the empirical equations. The mesh number of the stacked screens is #30. The working gas is Ar gas with 0.45 MPa. The test frequencies are 140, 180, and 200 Hz. Every panel presents the same experimentally obtained results by symbols with error bars, and empirical predictions are given respectively as the following: (**a**) for [PERSON] et al. ; (**b**) for [PERSON] and [PERSON]; (**c**) for [PERSON] et al. ; and (**d**) for [PERSON] and [PERSON]. Thermoacoustic theory describes that the flow resistance of the regular flow channel is a constant [17; 18; 19], independent of the velocity amplitude, whereas that in the porous media with tortuous flow channels is shown to increase with the velocity [28; 29; 30] or Reynolds number [2; 3; 4; 5; 6; 7]. From measurements of the acoustic field, we have proposed modification of the effective radius \(r_{0}\) to have velocity dependence [16] as  
\[r_{\mathrm{eff}}=2\delta_{v}\left(\frac{0.8v}{\omega d_{\mathrm{h}}^{2}}Re_{ \mathrm{h}}+\frac{16\delta_{v}^{2}}{d_{\mathrm{h}}d_{\mathrm{w}}}\right)^{- \frac{1}{2}}. \tag{15}\]  
The effective radius \(r_{\mathrm{eff}}\) approaches \(r_{0}\) when the velocity is negligibly small. It decreases concomitantly with increasing velocity.