# Measurement of Heat Flow Transmitted through a Stacked-Screen Regenerator of Thermoacoustic Engine
## 2 Axial Heat Flow Estimated by Empirical Equations
\begin{split}& Y=\frac{\epsilon_{\mathrm{s}}+(g_{\mathrm{c}}+e^{2i \theta_{\mathrm{p}}}g_{\mathrm{r}})\epsilon_{\mathrm{h}}}{1+\epsilon_{\mathrm{ s}}+(g_{\mathrm{c}}+e^{2i\theta_{\mathrm{r}}}g_{\mathrm{r}})\epsilon_{ \mathrm{h}}},G=\frac{\epsilon_{\mathrm{s}}+(g_{\mathrm{c}}-g_{\mathrm{r}}) \epsilon_{\mathrm{h}}}{1+\epsilon_{\mathrm{s}}+(g_{\mathrm{c}}+e^{2i\theta_{ \mathrm{r}}}g_{\mathrm{r}})\epsilon_{\mathrm{h}}},\\ &\epsilon_{\mathrm{s}}=\frac{\phi\rho_{\mathrm{m}}c_{\mathrm{p}}} {(1-\phi)\rho_{\mathrm{s}}c_{\mathrm{s}}},\epsilon_{\mathrm{h}}=\frac{i}{2b \omega^{1/3}}\left(\frac{d_{\mathrm{h}}}{\delta_{\mathrm{a}}}\right)^{2},\\ & b=3.81-11.29\phi+9.47\phi^{2},\\ &\theta_{\mathrm{P}}=\mathrm{phase}\left[\left\langle V\right\rangle \right]-\mathrm{phase}\left[P\right],\theta_{\mathrm{T}}=\mathrm{phase}\left[ \left\langle V\right\rangle\right]-\mathrm{phase}\left[\left\langle T\right\rangle \right],\\ & g_{\mathrm{c}}=\frac{2}{\pi}\int_{0}^{\pi/2}\frac{dz}{1+Re_{ \mathrm{h}}^{3/5}\cos^{3/5}(z)},g_{\mathrm{r}}=-\frac{2}{\pi}\int_{0}^{\pi/2} \frac{\cos 2 zdz}{1+Re_{\mathrm{h}}^{3/5}\cos^{3/5}(z)},\end{split}  \tag{6}\]  
where \(\phi\) represents the volume porosity, and where both the solid density \(\rho_{\mathrm{s}}\) and heat capacity \(c_{\mathrm{s}}\) are safely assumed as much larger than those of the gas; \(Re_{\mathrm{h}}\) denotes the Reynolds number given as \(Re_{\mathrm{h}}=\left|\left\langle V\right\rangle\right|d_{\mathrm{h}}/\
u\) (\(\left|\left\langle V\right\rangle\right|\) represents the magnitude of the complex velocity \(\left\langle V\right\rangle\)). The heat flow in Equation (1) is then given for an ideal gas as  
\[Q\cong\frac{1}{2}A\rho_{\mathrm{m}}T_{\mathrm{m}}\mathrm{Re}\left[\left\langle \mathrm{s}\right\rangle\widehat{\left\langle V\right\rangle}\right], \tag{7}\]  
with  
\[\left\langle\mathrm{s}\right\rangle=-\frac{1}{\rho_{\mathrm{m}}T_{\mathrm{m}}} P+\frac{c_{\mathrm{P}}}{T_{\mathrm{m}}}\left\langle T\right\rangle_{\mathrm{S}}. \tag{8}\]  
We also compare \(Q\) given by Equation (7) with the value obtained from experimentation. In contrast to the Swift-Ward formulation, [PERSON] et al. and [PERSON] experimentally obtained the Nusselt number \(Nu\) for oscillatory flows in the frequency range of the mechanical Stirling engine. [PERSON] et al. obtained the Nusselt number from measurements conducted with a maximum oscillation frequency of 10 Hz [3], which is expressed as  
\[Nu_{\mathrm{T}}=0.33\mathrm{Re_{h}}^{0.67}. \tag{9}\]  
[PERSON] and [PERSON] also proposed their \(Nu\)-\(Re_{\mathrm{h}}\) correlation [4] with oscillation frequency up to 120 Hz. They incorporated it into a Stirling machine design software code--Sage [25], which is given as  
\[Nu_{\mathrm{G}}=\left(1+0.99\left(Re_{\mathrm{h}}\sigma\right)^{0.66}\right) \phi^{1.79}. \tag{10}\]  
Applicability of these empirical equations should be tested, but incorporating them into the thermoacoustic theory presents a difficult problem for several reasons [22]. Therefore, as a tentative method, we determine \(\left\langle T\right\rangle\) by inserting these empirical Nusselt numbers into the following equation:  
\[\left\langle T\right\rangle_{Nu}=\left(\rho_{\mathrm{m}}c_{\mathrm{p}}+\frac{ 16}{i\omega}\frac{kNu}{d_{\mathrm{h}}^{2}}\right)^{-1}P-\left(i\omega+\frac{16} {\rho_{\mathrm{m}}c_{\mathrm{p}}}\frac{kNu}{d_{\mathrm{h}}^{2}}\right)^{-1} \frac{dT_{\mathrm{m}}}{dx}\left\langle V\right\rangle. \tag{11}\]  
Equation (11) is obtained from Equation (27) of Reference [22] by introducing complex notation for \(\left\langle T\right\rangle\) and by replacing the heat transfer coefficient \(h\) with \(h=4 kNu/d_{h}\) while using thermal conductivity \(k\) of the gas. The heat flow can be estimated by inserting \(\left\langle T\right\rangle_{Nu}\) into Equation (8) and by then usingEquation (7). Although this derivation would not qualify as a legitimate procedure, as we show later, it yields heat flows adequately to a similar degree to those of other methods tested in this study.