Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T05:39:05.573Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  10 January 2009

P.H.M.W. IN 'T PANHUIS ,
S. W. RIENSTRA ,
J. MOLENAAR  and
J. J. M. SLOT
P.H.M.W. IN 'T PANHUIS*
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands
S. W. RIENSTRA
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands
J. MOLENAAR
Affiliation:
Biometris, Wageningen University, PO Box 100, 6700 AC, Wageningen, The Netherlands
J. J. M. SLOT
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 618 , 10 January 2009 , pp. 41 - 70
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

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, 3228–3237.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Auriault, J. L. 1983 Heterogeneous medium. Is an equivalent macroscopic description possible? Intl J. Engng Sci. 29, 785–795.CrossRefGoogle Scholar
Auriault, J. L. 2002 Upscaling heterogeneous media by asymptotic expansions. J. Engng Mech. 128, 817–822.Google Scholar
Backhauss, S. & Swift, G. W. 2000 A thermoacoustic Stirling heat engine: detailed study. J. Acoust. Soc. Am. 107, 3148–3166.CrossRefGoogle Scholar
Bailliet, H., Gusev, V., Raspet, R. & Hiller, R. A. 2001 Acoustic streaming in closed thermoacoustic devices. J. Acoust. Soc. Am. 110, 1808–1821.CrossRefGoogle ScholarPubMed
Buckingham, E. 1914 On physically similar systems: illustrations of the use of dimensional equations. Phys. Rev. pp. 345–376.CrossRefGoogle Scholar
Chapman, C. J. 2000 High Speed Flow. Cambridge University Press.Google Scholar
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.Google Scholar
Duffy, D. G. 2001 Green's Functions with Applications. Chapman & Hall.CrossRefGoogle Scholar
Garrett, S. L. 2004 Thermoacoustic engines and refrigerators. Am. J. Phys. 72, 11–17.CrossRefGoogle Scholar
Gifford, W. E. & Longsworth, R. C. 1966 Surface heat pumping. Adv. Cryog. Engng 1, 302.Google Scholar
Gusev, V., Bailliet, H., Lotton, P. & Bruneau, M. 2000 Asymptotic theory of nonlinear acoustic waves in a thermoacoustic prime-mover. Acustica 86, 25–38.Google Scholar
Hornung, U. (ed.) 1997 Homogenization and Porous Media. Springer.CrossRefGoogle Scholar
Kamiński, M. M. 2002 On probabilistic viscous incompressible flow of some composite fluids. Comput. Mech. 28, 505–517.Google Scholar
Kirchhoff, G. 1868 Ueber den Einfluss der Wärmteleitung in einem Gas auf die Schallbewegung. Annln Phys. 134, 177.CrossRefGoogle Scholar
Kramers, H. A. 1949 Vibrations of a gas column. Physica 15, 971.CrossRefGoogle Scholar
Kröner, E. 1986 Modeling Small Deformations of Polycrystals, chap. Statistical modeling. Elsevier.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Licht, W. Jr., & Stechert, D. G. 1944 The variation of the viscosity of gases and vapors with temperature. J. Phys. Chem. 48, 23–47.CrossRefGoogle Scholar
Mattheij, R. M. M., Rienstra, S. W. & ten Thije Boonkkamp, J. H. M. 2005 Partial Differential Equations: Modeling, Analysis, Computation. SIAM, Philadelphia.CrossRefGoogle Scholar
Merkli, P. & Thomann, H. 1975 Thermoacoustic effects in a resonant tube. J. Fluid Mech. 70, 161.CrossRefGoogle Scholar
Nyborg, W. L. M. 1965 Physical Acoustics, vol. IIB, chap. Acoustic streaming, p. 265. Academic.Google Scholar
Olson, J. R. & Swift, G. W. 1994 Similitude in thermoacoustics. J. Acoust. Soc. Am. 95, 1405–1412.CrossRefGoogle Scholar
Olson, J. R. & Swift, G. W. 1997 Acoustic streaming in pulse-tube refrigerators: tapered pulse tubes. Cryogenics pp. 769–776.CrossRefGoogle Scholar
Poesse, M. E. & Garrett, S. L. 2000 Performance measurements on a thermoacoustic refrigerator driven at high amplitudes. J. Acoust. Soc. Am. 107, 2480–2486.CrossRefGoogle Scholar
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, 2537–2564.CrossRefGoogle Scholar
Rayleigh, Lord 1945 Theory of Sound, Vol. 2 II, Dover.Google Scholar
Rienstra, S. 2003 Sound propagation in slowly varying lined flow ducts of arbitrary cross-section. J. Fluid Mech. 495, 157–173.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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, 1413–1422.CrossRefGoogle ScholarPubMed
Rott, N. 1969 Damped and thermally driven acoustic oscillations in wide and narrow tubes. z. Angew. Math. Phy. 20, 230–243.CrossRefGoogle Scholar
Rott, N. 1973 Thermally driven acoustic oscillations. Part II: Stability limit for helium. z. Angew. Math. Phy. 24, 54–72.CrossRefGoogle Scholar
Rott, N. 1974 The influence of heat conduction on acoustic streaming. z. Angew. Math. Phy. 25, 417–421.CrossRefGoogle Scholar
Rott, N. 1975 Thermally driven acoustic oscillations. Part III: Second-order heat flux. z. Angew. Math. Phy. 26, 43–49.CrossRefGoogle Scholar
Rott, N. 1980 Thermoacoustics. Adv. Appl. Mech. 20, 135–175.CrossRefGoogle Scholar
Rott, N. & Zouzoulas, G. 1976 Thermally driven acoustic oscillations. Part IV: Tubes with variable cross-section. z. Angew. Math. Phy. 27, 197–224.CrossRefGoogle Scholar
Sondhauss, C. 1850 Ueber die Schallschwingungen der Luft in erhitzten Glasröhren und in gedeckten Pfeifen von ungleicher Weite. Annln Phys. 79, 1.CrossRefGoogle Scholar
Swift, G. W. 1988 Thermoacoustic engines. J. Acoust. Soc. of Am. 84, 1146–1180.CrossRefGoogle Scholar
Swift, G. W. 1992 Analysis and performance of a large thermoacoustic engine. J. Acoust. Soc. Am. 92, 1551–1563.CrossRefGoogle Scholar
Swift, G. W. 2002 A Unifying Perspective for Some Engines and Refrigerators. Acoustical Society of America.Google Scholar
Taconis, K. W. 1949 Vapor–liquid equilibrium of solutions of 3He in 4He. Physica 15, 738.Google Scholar
Van Dyke, M. 1987 Slow variations in continuum mechanics. Adv. Appl. Mech. 25, 1–45.CrossRefGoogle Scholar
Waxler, R. 2001 Stationary velocity and pressure gradients in a thermoacoustic stack. J. Acoust. Soc. Am. 109, 2739.CrossRefGoogle Scholar
Wheatley, J. C., Swift, G. W. & Migliori, A. 1986 The natural heat engine. Los Alamos Science pp. 2–33.Google Scholar
Zaoui, A. 1987 Homogenization Techniques for Composite Media, chap. Approximate statistical modelling and applications. Berlin.Google Scholar