# Measurement of Heat Flow Transmitted through a Stacked-Screen Regenerator of Thermoacoustic Engine
## 2 Axial Heat Flow Estimated by Empirical Equations  
In thermoacoustic theory [17; 18; 19], heat transport by gas oscillations along the \(x\)-axis with angular frequency \(\omega\) is given by a result of hydrodynamic transport of entropy as  
\[Q=\frac{1}{2}\rho_{\mathrm{m}}T_{\mathrm{m}}\int\mathrm{Re}[s\widetilde{V}]dA, \tag{1}\]  
where \(s\) and \(V\), respectively, represent the complex amplitude of entropy oscillations and axial velocity oscillations of the gas, and \(\rho_{\mathrm{m}}\) and \(T_{\mathrm{m}}\), respectively, denote the temporal mean density and temperature. In addition, \(\mathrm{Re}\left[\cdots\right]\) and \(\widetilde{\cdots}\), respectively, show the real part and conjugate of a complex number. The surface integration is done over the cross-sectional area of the flow channel. The complex amplitude \(s\) in Equation (1) is expressed using complex amplitudes \(P\) and \(T\) of the gas pressure and temperature as  
\[s=-\frac{\beta}{\rho_{\mathrm{m}}}P+\frac{c_{\mathrm{p}}}{T_{\mathrm{m}}}T, \tag{2}\]  
where \(\beta\) signifies the coefficient of thermal expansion and \(c_{\mathrm{p}}\) stands for the isobaric heat capacity per unit mass. For flow channels with a regular shape, \(T\) is obtained analytically. It can be expressed in a simpler form when the gas is assumed as an ideal gas. In such a case, the heat flow is given explicitly as  
\[Q=-\frac{1}{2}A\mathrm{Re}\Big{[}gP\widetilde{\langle V\rangle} \Big{]}+\frac{1}{2}A\frac{\rho_{\mathrm{m}}c_{\mathrm{p}}}{\mathrm{Im}[g_{ \mathrm{D}}]}\frac{dT_{\mathrm{m}}}{dx}\left|\langle V\rangle\right|^{2} \tag{3}\] \[\mathrm{with}\quad g=\frac{\chi_{\alpha}-\widetilde{\chi_{v}}}{ \left(1+\sigma\right)\left(1-\widetilde{\chi_{v}}\right)}\quad\mathrm{and} \quad g_{\mathrm{D}}=\frac{\chi_{\alpha}+\sigma\widetilde{\chi_{v}}}{\left(1- \sigma^{2}\right)\left|1-\chi_{v}\right|^{2}},\]  
where \(\sigma\), \(\omega\), and \(A\), respectively, denote the Prandtl number of the gas, angular frequency of the oscillation, and the cross-sectional area of the flow channel. In the equation, \(\langle\cdots\rangle\) denote the cross-sectional average. Furthermore, \(\left|\cdots\right|\) and \(\mathrm{Im}\left[\cdots\right]\) signify taking the absolute value and imaginary part of a complex number; \(\chi_{j}\) (\(j=\
u,\alpha\)) is the thermoacoustic function associated with the thermal diffusivity \(\alpha\) and kinematic viscosity \(\
u\) of the working gas. For a cylinder with radius \(r_{0}\), \(\chi_{j}\) is given as  
\[\chi_{j}=\frac{2J_{1}\left[\left(i-1\right)\frac{r_{0}}{\beta_{j}}\right]}{ \left(i-1\right)\frac{r_{0}}{\beta_{j}}J_{0}\left[\left(i-1\right)\frac{r_{0}} {\beta_{j}}\right]}, \tag{4}\]  
where \(J_{1}\) and \(J_{0}\), respectively, represent the first-order and zeroth-order Bessel functions of the first kind, \(i\) signifies the imaginary unit, and \(\delta_{j}\) (\(=\sqrt{2j/\omega}\)) denotes the thermal or viscous penetration depth. We test the applicability of effective radius \(r_{0}=\sqrt{d_{\mathrm{h}}d_{w}}/2\) by inserting it into the equation above. Moreover, we compare \(Q\) obtained by Equation (3) with the experimental heat flow that exists in the stacked-screen regenerator. [PERSON] and [PERSON] [22] adopted the heat transfer coefficient of the steady flow and assumed the relation of \(\left\langle T\right\rangle=\left\langle VT\right\rangle/\left\langle V\right\rangle\) to obtain the cross-sectional average \(\left\langle T\right\rangle\) in the stacked-screen regenerator as  
\[\left\langle T\right\rangle_{\mathrm{S}}=\frac{\beta T_{\mathrm{m}}}{\rho_{ \mathrm{m}}c_{\mathrm{p}}}\gamma P-\frac{1}{i\omega}\frac{dT_{\mathrm{m}}}{dx}G \left\langle V\right\rangle, \tag{5}\]  
with  
\[