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Research Papers

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

[+] Author and Article Information
Shohel Mahmud, Roydon Andrew Fraser

Department of Mechanical and Mechatronics Engineering, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

J. Heat Transfer 131(10), 101011 (Jul 31, 2009) (12 pages) doi:10.1115/1.3180705 History: Received September 29, 2008; Revised June 03, 2009; Published July 31, 2009

## Abstract

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.

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## Figures

Figure 1

(a) A multiplate porous thermoacoustic system and (b) coordinate system and dimensions of analytical domain in the porous medium

Figure 2

Dimensionless velocity as functions of Swift numbers in the clear fluid limit

Figure 3

Dimensionless velocity as functions of Darcy numbers at Sw=2.0

Figure 5

Dimensionless temperature as functions of Swift numbers and Darcy numbers

Figure 6

Darcy limit of the complex Nusselt number at different Swift numbers

Figure 7

Profiles of complex Nusselt numbers (Eq. 43) at different Swift and Darcy numbers

Figure 8

Distribution of normalized E2∗ as functions of Da and S¯w for Γ0∗=10 (conduction energy flux is neglected in this figure)

Figure 9

Mesh-contour plot showing that the Swift number distribution corresponds to the maximum energy transfer as a function of Darcy number (Da) and temperature gradient ratio (Γ0) for σ=10 (note: log10(0.5)≈−0.3 and log10(5)≈0.7)

Figure 10

Performance plot corresponds to Fig. 9 showing the magnitude of the Swift number at maximum energy transfer as a function of Darcy number and Γ0

Figure 4

Dimensionless temperature as functions of Swift numbers in the clear fluid limit

Figure 11

Mesh-contour plot showing that the Swift number distribution corresponds to the maximum energy transfer as a function of Darcy number (Da) and temperature gradient ratio (Γ0) for ERG AL 40 in helium (σ=279)

Figure 12

Performance plot corresponds to Fig. 1 showing the magnitude of the Swift number at maximum energy transfer as a function of Darcy number and Γ0 for ERG AL 40 (41) in helium

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