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Weakly nonlinear thermoacoustics for stacks with slowly varying pore cross-sections

  • P.H.M.W. IN 'T PANHUIS (a1), S. W. RIENSTRA (a1), J. MOLENAAR (a2) and J. J. M. SLOT (a1)

A general theory of thermoacoustics is derived for arbitrary stack pores. Previous theoretical treatments of porous media are extended by considering arbitrarily shaped pores with the only restriction that the pore cross-sections vary slowly in the longitudinal direction. No boundary-layer approximation is necessary. Furthermore, the model allows temperature variations in the pore wall. By means of a systematic approach based on dimensional analysis and small parameter asymptotics, we derive a set of ordinary differential equations for the mean temperature and the acoustic pressure and velocity, where the equation for the mean temperature follows as a consistency condition of the assumed asymptotic expansion. The problem of determining the transverse variation is reduced to finding a Green's function for a modified Helmholtz equation and solving two additional integral equations. Similarly the derivation of streaming is reduced to finding a single Green's function for the Poisson equation on the desired geometry.

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Arnott, W. P., Bass, H. E. & Raspet, R. 1991 General formulation of thermoacoustics for stacks having arbitrarily shaped pore cross sections. J. Acoust. Soc. Am. 90, 32283237.
Atchley, A. A., Hofler, T., Muzzerall, M. L., Kite, M. D. & Ao, C. 1990 Acoustically generated temperature gradients in short plates. J. Acoust. Soc. Am. 88, 251.
Auriault, J. L. 1983 Heterogeneous medium. Is an equivalent macroscopic description possible? Intl J. Engng Sci. 29, 785795.
Auriault, J. L. 2002 Upscaling heterogeneous media by asymptotic expansions. J. Engng Mech. 128, 817822.
Backhauss, S. & Swift, G. W. 2000 A thermoacoustic Stirling heat engine: detailed study. J. Acoust. Soc. Am. 107, 31483166.
Bailliet, H., Gusev, V., Raspet, R. & Hiller, R. A. 2001 Acoustic streaming in closed thermoacoustic devices. J. Acoust. Soc. Am. 110, 18081821.
Buckingham, E. 1914 On physically similar systems: illustrations of the use of dimensional equations. Phys. Rev. pp. 345–376.
Chapman, C. J. 2000 High Speed Flow. Cambridge University Press.
Chapman, S. & Cowling, T. G. 1939 The Mathematical Theory of Non-uniform Gases; an Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases. Cambridge University Press.
Duffy, D. G. 2001 Green's Functions with Applications. Chapman & Hall.
Garrett, S. L. 2004 Thermoacoustic engines and refrigerators. Am. J. Phys. 72, 1117.
Gifford, W. E. & Longsworth, R. C. 1966 Surface heat pumping. Adv. Cryog. Engng 1, 302.
Gusev, V., Bailliet, H., Lotton, P. & Bruneau, M. 2000 Asymptotic theory of nonlinear acoustic waves in a thermoacoustic prime-mover. Acustica 86, 2538.
Hornung, U. (ed.) 1997 Homogenization and Porous Media. Springer.
Kamiński, M. M. 2002 On probabilistic viscous incompressible flow of some composite fluids. Comput. Mech. 28, 505517.
Kirchhoff, G. 1868 Ueber den Einfluss der Wärmteleitung in einem Gas auf die Schallbewegung. Annln Phys. 134, 177.
Kramers, H. A. 1949 Vibrations of a gas column. Physica 15, 971.
Kröner, E. 1986 Modeling Small Deformations of Polycrystals, chap. Statistical modeling. Elsevier.
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.
Licht, W. Jr., & Stechert, D. G. 1944 The variation of the viscosity of gases and vapors with temperature. J. Phys. Chem. 48, 2347.
Mattheij, R. M. M., Rienstra, S. W. & ten Thije Boonkkamp, J. H. M. 2005 Partial Differential Equations: Modeling, Analysis, Computation. SIAM, Philadelphia.
Merkli, P. & Thomann, H. 1975 Thermoacoustic effects in a resonant tube. J. Fluid Mech. 70, 161.
Nyborg, W. L. M. 1965 Physical Acoustics, vol. IIB, chap. Acoustic streaming, p. 265. Academic.
Olson, J. R. & Swift, G. W. 1994 Similitude in thermoacoustics. J. Acoust. Soc. Am. 95, 14051412.
Olson, J. R. & Swift, G. W. 1997 Acoustic streaming in pulse-tube refrigerators: tapered pulse tubes. Cryogenics pp. 769–776.
Poesse, M. E. & Garrett, S. L. 2000 Performance measurements on a thermoacoustic refrigerator driven at high amplitudes. J. Acoust. Soc. Am. 107, 24802486.
Quintard, M. & Whitaker, S. 1993 Transport in ordered and disordered porous media: volume-averaged equations, closure problems, and comparison with experiments. Chem. Engng Sci. 48, 25372564.
Rayleigh, Lord 1945 Theory of Sound, Vol. 2 II, Dover.
Rienstra, S. 2003 Sound propagation in slowly varying lined flow ducts of arbitrary cross-section. J. Fluid Mech. 495, 157173.
Rijke, P. L. 1859 Notiz über eine neue Art, die in einer an beiden Enden offenen Rohre enthaltene Lift in Schwingungen zu versetzen. Annln Phys. 107, 339.
Roh, H., Raspet, R. & Bass, H. E. 2007 Parallel capillary-tube-based extension of thermoacoustic theory for random porous media. J. Acoust. Soc. Am. 121, 14131422.
Rott, N. 1969 Damped and thermally driven acoustic oscillations in wide and narrow tubes. z. Angew. Math. Phy. 20, 230243.
Rott, N. 1973 Thermally driven acoustic oscillations. Part II: Stability limit for helium. z. Angew. Math. Phy. 24, 5472.
Rott, N. 1974 The influence of heat conduction on acoustic streaming. z. Angew. Math. Phy. 25, 417421.
Rott, N. 1975 Thermally driven acoustic oscillations. Part III: Second-order heat flux. z. Angew. Math. Phy. 26, 4349.
Rott, N. 1980 Thermoacoustics. Adv. Appl. Mech. 20, 135175.
Rott, N. & Zouzoulas, G. 1976 Thermally driven acoustic oscillations. Part IV: Tubes with variable cross-section. z. Angew. Math. Phy. 27, 197224.
Sondhauss, C. 1850 Ueber die Schallschwingungen der Luft in erhitzten Glasröhren und in gedeckten Pfeifen von ungleicher Weite. Annln Phys. 79, 1.
Swift, G. W. 1988 Thermoacoustic engines. J. Acoust. Soc. of Am. 84, 11461180.
Swift, G. W. 1992 Analysis and performance of a large thermoacoustic engine. J. Acoust. Soc. Am. 92, 15511563.
Swift, G. W. 2002 A Unifying Perspective for Some Engines and Refrigerators. Acoustical Society of America.
Taconis, K. W. 1949 Vapor–liquid equilibrium of solutions of 3He in 4He. Physica 15, 738.
Van Dyke, M. 1987 Slow variations in continuum mechanics. Adv. Appl. Mech. 25, 145.
Waxler, R. 2001 Stationary velocity and pressure gradients in a thermoacoustic stack. J. Acoust. Soc. Am. 109, 2739.
Wheatley, J. C., Swift, G. W. & Migliori, A. 1986 The natural heat engine. Los Alamos Science pp. 2–33.
Zaoui, A. 1987 Homogenization Techniques for Composite Media, chap. Approximate statistical modelling and applications. Berlin.
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