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Research Papers

# Modeling and Analysis of Flow, Thermal, and Energy Fields Within Stacks of Thermoacoustic Engines Filled With Porous Media: A Conjugate Problem

[+] Author and Article Information
Syeda Humaira Tasnim1

Department of Mechanical and Mechatronics Engineering, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canadashtasnim@engmail.uwaterloo.ca

Roydon Andrew Fraser

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

1

Corresponding author.

J. Thermal Sci. Eng. Appl 1(4), 041006 (Jun 24, 2010) (12 pages) doi:10.1115/1.4001747 History: Received August 28, 2009; Revised May 06, 2010; Published June 24, 2010; Online June 24, 2010

## Abstract

In this paper, an analytical study has been conducted on the flow and energy transfer of an unsteady compressible oscillating flow through channels filled with porous media representing stacks in thermoacoustic engines and refrigerators. The flow in the porous material is described by the Darcy momentum equation. The thickness of the channel wall is considered to be nonzero, and the entire problem is treated as a conjugate heat transfer problem, i.e., by considering conduction heat transfer inside the channel walls. Analytical expressions for the oscillating temperature, complex Nusselt number, and energy flux density are obtained after linearizing and solving the governing differential equations with long wave, short stack, and small amplitude oscillation approximations. To verify the present study, the energy flux density expression derived in this paper is compared with the expression available in the existing thermoacoustic literature. The two expressions match quantitatively for the limiting case of infinitely large pores. For infinitely large pore limits, the Nusselt number (nondimensional heat transfer between the porous media and the channel wall) obtained in the present study also agrees quantitatively with the nonporous medium expression reported in the literature. The present study indicates that refrigeration performance comparable to that of a traditional plastic parallel plate stack is achievable using reticulated vitreous carbon foam ($ϕ=0.95$, $Lck=2.11$) as a porous medium, which is also supported by other researchers. The system of equations developed in the present study is a helpful tool for thermal engineers and physicists to design porous stacks for thermoacoustic devices.

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## Figures

Figure 1

(a) Schematic diagram of a typical thermoacoustic refrigerator with different parts; (b) coordinate system and dimensions of the analytical domain in the porous medium

Figure 2

The real and imaginary parts (a) of tanh((1+i)σy0/δk) for the porous medium and (b) of tanh((1+i)l/δs) for the solid region

Figure 3

Normalized porous medium temperature as a function of nondimensional transverse distance at (a) ϕ=1 and Lck=0.5−2.076, (b) at ϕ=1 and Lck=5−10, and (c) at ϕ=0.83 and Lck=2.11

Figure 4

Magnified view of a gas parcel as it completes an acoustic cycle: cases (a) 1, (b) 2, and (c) 3

Figure 5

Heat flux density as a function of y/y0 (a) at σ=11.47 and (b) at σ=1

Figure 6

(a) Normalized solid temperature inside the stack as a function of the transverse distance; (b) real part of complex Nusselt number versus Lck at different values of σ and εs

Figure 7

Imaginary and real parts of fk as a function of Lck (a) at σ=1 and (b) at σ=11.47 in the porous medium

Figure 8

Distribution of normalized E2 as a function of Lck

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