Therporaoustic Convection: Modeling and Analysis of Flow, Thermal, and Energy Fields

The problem of therporacoustic (thermal-porous-acoustic) convection near a porous medium, representative of a stack in a thermoacoustic engine/refrigerator, is modeled and analyzed in this paper. Assumptions (e.g., long wave, short stack, and small amplitude oscillation) are made to enable simplification of the governing unsteady-compressible-viscous forms of the continuity, momentum, and energy equations to achieve analytical solutions for the fluctuating velocity and temperature and the complex Nusselt number. Boundary walls are assumed to be very thin in thickness and the conduction heat transfer inside the boundary walls are neglected in this paper. The derived analytical results are expressed mainly in terms of the Darcy number (Da), critical temperature gradient ratio $(Γ0)$⁠, Swift number $(Sw)$⁠, Prandtl number (Pr), and modified Rott’s and Swift’s parameters (⁠$fν$ and $fk$⁠). The real part of the fluctuating flow complex Nusselt number approaches to the steady result, as reported in the literature, at the zero frequency limit. While in the high frequency limit, the real part of the complex Nusselt number matches well with the limit obtained by other oscillating flow researchers with slight differences explained by additional terms included in this work. A wave equation for the pressure fluctuation is modeled by combining the continuity, momentum, and energy equations and subsequent integrations which, in the inviscid no-stack limit, approaches the Helmholtz wave equation. Based on the derived energy flux density equation performance plots are proposed, which give the Swift number at the maximum energy transfer $(Sw0)$ for a given $Γ0$ and Da.