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.

1.
Swift
,
G. W.
, 2002,
Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators
,
ASA
,
Melville, NY
.
2.
Rott
,
N.
, 1980, “
Thermoacoustics
,”
Adv. Appl. Mech.
0065-2156,
20
, pp.
135
175
.
3.
Swift
,
G. W.
, 1988, “
Thermoacoustic Engines
,”
J. Acoust. Soc. Am.
0001-4966,
84
, pp.
1145
1180
.
4.
Arnott
,
W. P.
,
Bass
,
H. E.
, and
Raspet
,
R.
, 1991, “
General Formulation of Thermoacoustics for Stacks Having Arbitrarily Shaped Pore Cross Sections
,”
J. Acoust. Soc. Am.
0001-4966,
90
, pp.
3228
3232
.
5.
Xiao
,
J. H.
, 1995, “
Thermoacoustic Heat Transport and Energy Transformations Part 1: Formulation of the Problem
,”
Cryogenics
0011-2275,
35
, pp.
15
19
.
6.
Watanabe
,
M.
,
Prosperetti
,
A.
, and
Yuan
,
H. A.
, 1997, “
Simplified Model for Linear and Nonlinear Processes in Thermoacoustic Prime Movers. Part I. Model and Linear Theory
,”
J. Acoust. Soc. Am.
0001-4966,
102
, pp.
3484
3496
.
7.
Guoqiang
,
L.
, and
Ping
,
C.
, 2000, “
Friction Factor and Nusselt Number for Thermoacoustic Transport Phenomena in a Tube
,”
J. Thermophys. Heat Transfer
0887-8722,
14
, pp.
566
573
.
8.
Adeff
,
J. A.
,
Hofler
,
T. J.
,
Atchley
,
A. A.
, and
Moss
,
W. C.
, 1998, “
Measurements With Reticulated Vitreous Carbon Stacks in Thermoacoustic Prime Movers and Refrigerators
,”
J. Acoust. Soc. Am.
0001-4966,
104
, pp.
32
38
.
9.
Roh
,
H.
,
Raspet
,
R.
, and
Bass
,
H. E.
, 2007, “
Parallel Capillary-Tube-Based Extension of Thermoacoustic Theory for Random Porous Media
,”
J. Acoust. Soc. Am.
0001-4966,
121
(
3
), pp.
1413
1422
.
10.
Mahmud
,
S.
, and
Fraser
,
R. A.
, 2009, “
Therporaoustic Convection: Modeling and Analysis of Flow, Thermal, and Energy Fields
,”
ASME J. Heat Transfer
0022-1481,
131
, p.
101011
.
11.
Vafai
,
K.
, and
Tien
,
C. L.
, 1981, “
Boundary and Inertia Effects on Flow and Heat Transfer in Porous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
24
, pp.
195
203
.
12.
Bejan
,
A.
, 1984,
Convection Heat Transfer
,
Wiley
,
New York
.
13.
Kinsler
,
L. W.
,
Frey
,
A. R.
,
Coppens
,
A. B.
, and
Sanders
,
J. V.
, 2000,
Fundamental of Acoustic
,
Wiley
,
New York
.
14.
Tasnim
,
S. H.
, and
Fraser
,
R. A.
, 2009, “
Modeling and Analysis of Flow, Thermal, and Energy Fields Within Stacks of Thermoacoustic Engines Filled With Porous Media
,”
Journal of Thermal Science
1003-2169, submitted.
15.
Mahmud
,
S.
, and
Fraser
,
R. A.
, 2005, “
An Analytical Solution and Computer Simulation for a Multi-Plate Thermoacoustic System
,”
Int. J. Exergy
1742-8297,
2
, pp.
207
230
.
16.
Liu
,
J.
, and
Garrett
,
S. L.
, 2006, “
Relationship Between Nusselt Number and the Thermoviscous (Rott) Functions
,”
J. Acoust. Soc. Am.
0001-4966,
119
(
3
), pp.
1457
1462
.
17.
Landau
,
L. D.
, and
Lifshitz
,
E. M.
, 1982,
Fluid Mechanics
,
Pergamon
,
New York
.
18.
Cao
,
N.
,
Olson
,
J. R.
,
Swift
,
G. W.
, and
Chen
,
S.
, 1996, “
Energy Flux Density in a Thermoacoustic Couple
,”
J. Acoust. Soc. Am.
0001-4966,
99
, pp.
3456
3464
.
19.
Burmeister
,
L. C.
, 1992,
Convective Heat Transfer
,
Wiley
,
New York
.
20.
2008, ERG Materials and Aerospace Corporation, 900 Stanford Avenue, Oakland, CA.
21.
Richardson
,
E. G.
, and
Richardson
,
E. T.
, 1929, “
The Transverse Velocity Gradient Near the Mouths of Pipes in Which an Alternating or Continuous Flow of Air is Established
,”
Proc. R. Soc. London, Ser. A
0950-1207,
42
, pp.
1
15
.
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