Carnot’s theorem as noether’s theorem for thermoacoustic engines

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 58, Issue 3, 1998, Pages 2818-2832

  Smith, Eric ^a,b  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ACOUSTIC DEVICES; APPROXIMATION THEORY; EIGENVALUES AND EIGENFUNCTIONS; EQUATIONS OF MOTION; HEAT ENGINES; LINEAR SYSTEMS; OSCILLATIONS; PERTURBATION TECHNIQUES; PHONONS; THERMAL CONDUCTIVITY; THERMODYNAMIC STABILITY; VARIATIONAL TECHNIQUES;

CARNOT THEOREM; NOETHER THEOREM; THERMOACOUSTIC ENGINES; THERMOACOUSTIC ONSET;

STATISTICAL MECHANICS;

EID: 0032161080     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.58.2818     Document Type: Article

References (35)

1. N. Rott, Z. Angew. Math. Phys. 26, 43 (1975).ZAMPA8
2. N. Rott, Z. Angew. Math. Phys. 24, 54 (1973); ZAMPA8
3. N. RottAdv. Appl. Mech. 20, 135 (1980).
4. J. Wheatley, T. Hofler, G. W. Swift, and A. Migliori, J. Acoust. Soc. Am. 71, 153 (1983).JASMAN
5. G. W. Swift, J. Acoust. Soc. Am. 84, 1145 (1988).JASMAN
6. A. A. Atchley, H. E. Bass, T. J. Hofler, and H.-T. Lin, J. Acoust. Soc. Am. 91, 734 (1992).JASMAN
7. A. A. Atchley, J. Acoust. Soc. Am. 95, 1661 (1994).JASMAN
8. P. H. Ceperley, J. Acoust. Soc. Am. 66, 1508 (1979).JASMAN
9. P. H. Ceperley, J. Acoust. Soc. Am. 72, 1688 (1982)
10. J. Acoust. Soc. Am.P. H. Ceperley77, 1239 (1985)
11. J. Acoust. Soc. Am.P. H. Ceperley85, S48(A) (1989)
12. U. S. Patent No. 4355517 (26 October 1982).
13. R. Raspet, H. E. Bass, and J. Kordomenos, J. Acoust. Soc. Am. 94, 2232 (1993)
14. J. Acoust. Soc. Am.see also R. Raspet, 100, 673(E) (1996) for the appropriate scope of this work. The importance of longitudinal conduction along the stack in implementing the Stirling regenerator is discussed in Appendix A.
15. H.-T. Lin and A. Atchley, J. Acoust. Soc. Am. 100, 2646(A) (1996).JASMAN
16. See Ref. 4, Fig. 4, p. 1148.
17. K. G. Wilson and J. Kogut, Phys. Rep., Phys. Lett. 12C, 75–200 (1974), Introduction, Sec. 2.1, pp. 87–89.PRPLCM
18. See Ref. 12, Introduction, pp. 78–86.
19. This has been attempted, unsuccessfully, in the course of other work, but not pursued; A. Atchley (private communication).
20. S. Coleman, Aspects of Symmetry (Cambridge University Press, New York, 1985), Chap. 5, pp. 113–184
21. S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1996), Vol. II, Chap. 19.
22. This association may not be intrinsic though. For a counterexample, in the case of flow away from an unstable critical point, see the treatment of critical localization in A. J. McKane and M. Stone, Ann. Phys. (N.Y.) 131, 36 (1981).APNYA6
23. S. Coleman, Aspects of Symmetry (Ref. 15), p. 118, and references therein.
24. S.-K. Ma, Modern Theory of Critical Phenomena (Benjamin/Cummings, London, 1976), p. 67.
25. L. E. Reichl, A Modern Course in Statistical Physics (University of Texas Press, Austin, 1987), Chap. 17, pp. 623–654.
26. Enrico Fermi, Thermodynamics (Dover, New York, 1956).
27. Gerald D. Mahan, Many-Particle Physics (Plenum, New York, 1990), Chap. 3, pp. 133–238.
28. For definitions and properties see A. J. Organ, Thermodynamics and Gas Dynamics of the Stirling Cycle Machine (Cambridge University Press, New York, 1992), Chap. 5, pp. 76–92.
29. A traveling-wave engine capable of onset has not yet been built because the comparable width of viscous and thermal boundary layers in conventional stack engines creates flow losses greater than the stack gain. Due to the generically O(1) value of the Prandtl number for gases, this is a serious problem, but is arguably a technical limitation of solid/gas plate stacks, which other regeneration methods may overcome.
30. This combination of assumptions is necessary to isolate the symmetries and conservation laws of a reversible limit. Though unrealistic for conventional engines, it is not inconsistent with the definition of the reversible cycle as a limiting behavior.
31. K. Huang, Statistical Mechanics (Wiley, New York, 1987), Sec. 6.4, pp. 136–138.
32. If the separation of scales between long-wavelength and short-wavelength modes is not wide, the notion of local corrections to the dynamical equations loses its meaning. In that case, more complex corrections to the correlation functions can be obtained by replacing the Matsubara diagram expansion with that of Keldysh: see E. M. Lifshitz and L. P. Pitaevsky, Physical Kinetics (Pergamon, New York, 1980), Chap. 10, pp. 391–412. The actions (2) and (5), however, are appropriate for this discussion.
33. See the review of Swift 4 for a more comprehensive list.
34. J. R. Olson and G. W. Swift, J. Acoust. Soc. Am. 95, 1405 (1994).JASMAN
35. The factor of γ relating the boundary layer thickness in this calculation to δκ is an artifact of the choice to expand in ρ and v instead of the p and u of Ref. 4. The difference term of order (γ-1)δκ is comparable to bulk conduction terms that are ignored.