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High-fidelity simulation of a standing-wave thermoacoustic–piezoelectric engine

Published online by Cambridge University Press:  26 October 2016

Jeffrey Lin ,
Carlo Scalo  and
Lambertus Hesselink
Jeffrey Lin*
Affiliation:
Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA
Carlo Scalo
Affiliation:
School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
Lambertus Hesselink
Affiliation:
Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: linjef@stanford.edu
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Abstract

We have carried out wall-resolved unstructured fully compressible Navier–Stokes simulations of a complete standing-wave thermoacoustic–piezoelectric engine model inspired by the experimental work of Smoker et al. (J. Appl. Phys., vol. 111 (10), 2012, 104901). The model is axisymmetric and comprises a 51 cm long resonator divided into two sections: a small-diameter section enclosing a thermoacoustic stack and a larger-diameter section capped by a piezoelectric diaphragm tuned to the thermoacoustically amplified mode (388 Hz). The diaphragm is modelled with multi-oscillator broadband time-domain impedance boundary conditions (TDIBCs), providing higher fidelity over single-oscillator approximations. Simulations are first carried out to the limit cycle without energy extraction. The observed growth rates are shown to be grid convergent and are verified against a numerical dynamical model based on Rott’s theory. The latter is based on a staggered grid approach and allows jump conditions in the derivatives of pressure and velocity in sections of abrupt area change and the inclusion of linearized minor losses. The stack geometry maximizing the growth rate is also found. At the limit cycle, thermoacoustic heat leakage and frequency shifts are observed, consistent with experiments. Upon activation of the piezoelectric diaphragm, steady acoustic energy extraction and a reduced pressure amplitude limit cycle are obtained. A heuristic closure of the limit cycle acoustic energy budget is presented, supported by the linear dynamical model and the nonlinear simulations. The developed high-fidelity simulation framework provides accurate predictions of thermal-to-acoustic and acoustic-to-mechanical energy conversion (via TDIBCs), enabling a new paradigm for the design and optimization of advanced thermoacoustic engines.

JFM classification

Acoustics: Acoustics Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 808 , 10 December 2016 , pp. 19 - 60
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Copyright
© 2016 Cambridge University Press 

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References

Anton, S. R. & Sodano, H. A. 2007 A review of power harvesting using piezoelectric materials (2003–2006). Smart Mater. Struct. 16 (3), R1.CrossRefGoogle Scholar
Bermejo-Moreno, I., Bodart, J., Larsson, J., Barney, B. M., Nichols, J. W. & Jones, S. 2013 Solving the compressible Navier–Stokes equations on up to 1.97 million cores and 4.1 trillion grid points. In Proc. Intl Conf. on High Performance Computing, Networking, Storage and Analysis, pp. 62:1–62:10. ACM.Google Scholar
Ceperley, P. H. 1979 A pistonless stirling engine – the traveling wave heat engine. J. Acoust. Soc. Am. 66 (5), 1508–1513.CrossRefGoogle Scholar
Chen, X., Xu, S., Yao, N. & Shi, Y. 2010 1.6 V nanogenerator for mechanical energy harvesting using PZT nanofibers. Nano Lett. 10 (6), 2133–2137.CrossRefGoogle ScholarPubMed
De-Yi, S. & Bu-Xuan, W. 1990 Effect of variable thermophysical properties on laminar free convection of gas. Intl J. Heat Mass Transfer 33 (7), 1387–1395.CrossRefGoogle Scholar
Dowling, A. P. & Williams, J. E. F. 1983 Sound and Sources of Sound. Ellis Horwood.Google Scholar
Feldman, K. T. Jr. 1968 Review of the literature on Sondhauss thermoacoustic phenomena. J. Sound Vib. 7 (1), 71–82.CrossRefGoogle Scholar
Fung, K.-Y. & Ju, H. 2001 Broadband time-domain impedance models. AIAA J. 39 (8), 1449–1454.CrossRefGoogle Scholar
Fung, K.-Y. & Ju, H. 2004 Time-domain impedance boundary conditions for computational acoustics and aeroacoustics. Intl J. Comput. Fluid Dyn. 18 (6), 503–511.CrossRefGoogle Scholar
Gardner, D. & Swift, G. W. 2003 A cascade thermoacoustic engine. J. Acoust. Soc. Am. 114 (4), 1905–1919.CrossRefGoogle ScholarPubMed
Gedeon, D. 2014 Sage User’s Guide: Stirling, Pulse-Tube and Low-T Cooler Model Classes. Gedeon Associates.Google Scholar
Ham, F., Mattsson, K., Iaccarino, G. & Moin, P. 2007 Towards Time-Stable and Accurate LES on Unstructured Grids, Lecture Notes in Computational Science and Engineering, vol. 56, pp. 235–249. Springer.Google Scholar
Hartley, R. V. L.1951 Electric power source. U.S. Classification 290/1.00R, 333/141, 60/39.77, 116/137.00A, 322/3, 116/DIG.220; International Classification F03G7/00, H02N11/00; Cooperative Classification F03G7/002, H02N11/002, Y10S116/22; European Classification H02N11/00B, F03G7/00B.Google Scholar
Idelchik, I. E. 2003 Handbook of Hydraulic Resistance, 3rd edn. CRC Press.Google Scholar
Jensen, B. L., Sumer, B. M. & Fredsøe, J. 1989 Turbulent oscillatory boundary layers at high Reynolds numbers. J. Fluid Mech. 206, 265–297.CrossRefGoogle Scholar
Kirchhoff, G. 1868 Über den Einfluss der Wärmeleitung in einem Gase auf die Schallbewegung. Poggendorfs Annal. 134, 177–193.Google Scholar
Kramers, H. A. 1949 Vibrations of a gas column. Physica 15 (971), 971–984.CrossRefGoogle Scholar
Marrison, W. A.1958 Heat-controlled acoustic wave system. U.S. Classification 60/516, 116/137.00R, 340/384.7, 116/DIG.220, 310/27, 290/1.00R, 116/137.00A, 60/531, 310/306; International Classification G08B17/04, F25B9/14, F03G7/00, H02N11/00; Cooperative Classification F03G7/002, F25B2309/1407, F02G2243/52, H02N11/002, G08B17/04, F25B2309/1403, F25B9/145, Y10S116/22; European Classification F25B9/14B, G08B17/04, H02N11/00B, F03G7/00B.Google Scholar
Matveev, K. I., Wekin, A., Richards, C. D. & Shafrei-Tehrany, N. 2007 On the coupling between standing-wave thermoacoustic engine and piezoelectric transducer. In ASME 2007 Intl Mechanical Engineering Congress and Exposition, pp. 765–769. ASME.Google Scholar
Müller, U. A. & Rott, N. 1983 Thermally driven acoustic oscillations, part VI: excitation and power. Z. Angew. Math. Phys. 34, 609–626.CrossRefGoogle Scholar
Nouh, M., Aldraihem, O. & Baz, A. 2014 Transient characteristics and stability analysis of standing wave thermoacoustic-piezoelectric harvesters. J. Acoust. Soc. Am. 135 (2), 669–678.CrossRefGoogle ScholarPubMed
Priya, S. 2007 Advances in energy harvesting using low profile piezoelectric transducers. J. Electroceram. 19 (1), 167–184.CrossRefGoogle Scholar
Rayleigh, J. W. S. 1878 The explanation of certain acoustical phenomena. Nature 18, 319–321.CrossRefGoogle Scholar
Rienstra, S. W. 2006 Impedance models in time domain, including the extended Helmholtz resonator model. In 12th AIAA/CEAS Aeroacoustics Conference. AIAA.Google Scholar
Rijke, P. L. 1859 LXXI. Notice of a new method of causing a vibration of the air contained in a tube open at both ends. Phil. Mag. 4 17 (116), 419–422.CrossRefGoogle Scholar
Rott, N. 1969 Damped and thermally driven acoustic oscillations in wide and narrow tubes. Z. Angew. Math. Phys. 20, 230–243.CrossRefGoogle Scholar
Rott, N. 1973 Thermally driven acoustic oscillations, part II: stability limit for helium. Z. Angew. Math. Phys. 24, 54–72.CrossRefGoogle Scholar
Rott, N. 1974 The influence of heat conduction on acoustic streaming. Z. Angew. Math. Phys. 25, 417–421.CrossRefGoogle Scholar
Rott, N. 1975 Thermally driven acoustic oscillations, part III: second-order heat flux. Z. Angew. Math. Phys. 26, 43–49.CrossRefGoogle Scholar
Rott, N. 1976 Ein ‘Rudimentarer’ Stirlingmotor. Neue Zurecher Ztg. 197 (210).Google Scholar
Rott, N. 1980 Thermoacoustics. Adv. Appl. Mech. 20, 135–175.CrossRefGoogle Scholar
Rott, N. 1984 Thermoacoustic heating at the closed end of an oscillating gas column. J. Fluid Mech. 145, 1–9.CrossRefGoogle Scholar
Rott, N. & Zouzoulas, G. 1976 Thermally driven acoustic oscillations, part IV: tubes with variable cross-section. Z. Angew. Math. Phys. 27, 197–224.CrossRefGoogle Scholar
Scalo, C., Bodart, J. & Lele, S. K. 2015a Compressible turbulent channel flow with impedance boundary conditions. Phys. Fluids 27, 035107.CrossRefGoogle Scholar
Scalo, C., Lele, S. K. & Hesselink, L. 2015b Linear and nonlinear modeling of a theoretical traveling-wave thermoacoustic heat engine. J. Fluid Mech. 766, 368–404.CrossRefGoogle Scholar
Smoker, J., Nouh, M., Aldraihem, O. & Baz, A. 2012 Energy harvesting from a standing wave thermoacoustic-piezoelectric resonator. J. Appl. Phys. 111 (10), 104901.CrossRefGoogle Scholar
Sondhauss, C. 1850 Ueber die Schallschwingungen der Luft in erhitzten Glasröhren und in gedeckten Pfeifen von ungleicher Weite. Ann. Phys. 155 (1), 1–34.CrossRefGoogle Scholar
Swift, G. W. 1988 Thermoacoustic engines. J. Acoust. Soc. Am. 84 (4), 1145–1180.CrossRefGoogle Scholar
Swift, G. W. 1992 Analysis and performance of a large thermoacoustic engine. J. Acoust. Soc. Am. 92 (3), 1551–1563.CrossRefGoogle Scholar
Swift, G. W. 2002 Thermoacoustics: A Unifying Perspective for some Engines and Refrigerators. Acoustical Society of America through the American Institute of Physics.Google Scholar
Tam, C. K. W. & Auriault, L. 1996 Time-domain impedance boundary conditions for computational aeroacoustics. AIAA J. 34 (5), 917–923.CrossRefGoogle Scholar
Tijani, M. E. H. & Spoelstra, S. 2011 A high performance thermoacoustic engine. J. Appl. Phys. 110, 093519.CrossRefGoogle Scholar
Ward, B., Clark, J. & Swift, G. 2012 Design Environment for Low-amplitude Thermoacoustic Energy Conversion: Users Guide. Los Alamos National Laboratory.Google Scholar
Yazaki, T., Iwata, A., Maekawa, T. & Tominaga, A. 1998 Traveling wave thermoacoustic engine in a looped tube. Phys. Rev. Lett. 81 (15), 3128–3131.CrossRefGoogle Scholar
Yu, Z., Jaworski, A. J. & Backhaus, S. 2012 Travelling-wave thermoacoustic electricity generator using an ultra-compliant alternator for utilization of low-grade thermal energy. Appl. Energy 99, 135–145.CrossRefGoogle Scholar
Zouzoulas, G. & Rott, N. 1976 Thermally driven acoustic oscillations, part V: gas-liquid oscillations. Z. Angew. Math. Phys. 27, 325–334.CrossRefGoogle Scholar

Lin et al. Movie 1

Instantaneous visualizations of fluid temperature (see colorbar) surrounding the stack, showing streaming of hot fluid out of the stack, and of high vorticity magnitude (white), showing entrance effects. Data taken under limit cycle conditions for temperature setting 5, grid-resolution/stack-type C/I.

Download Lin et al. Movie 1(Video)
Video 8.4 MB

Lin et al. Movie 2

Instantaneous visualizations of fluid temperature (see colorbar) surrounding the resonator area change, showing streaming of hot fluid, and of high vorticity magnitude (white), showing entrance effects. Data taken under limit cycle conditions for temperature setting 5, grid-resolution/stack-type C/I.

Download Lin et al. Movie 2(Video)
Video 1.4 MB