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Research Papers

Hopf Instabilities in Free Piston Stirling Engines

[+] Author and Article Information
Farhan Choudhary

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: farhanch@mit.edu

Balakumar Balachandran

Fellow ASME
Department of Mechanical Engineering,
University of Maryland,
College Park, MD 20742
e-mail: balab@umd.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 3, 2012; final manuscript received July 10, 2013; published online September 12, 2013. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 9(2), 021003 (Sep 12, 2013) (11 pages) Paper No: CND-12-1138; doi: 10.1115/1.4025123 History: Received September 03, 2012; Revised July 10, 2013

Free piston Stirling engines (FPSEs) are examples of closed cycle regenerative engines, which can be used to convert thermal energy into mechanical energy. These engines are multidegree-of-freedom dynamical systems, which are designed to operate in a periodic manner. Traditionally, for design purposes, linear models are used and the associated periodic orbits are meta-stable, making the system operations sensitive to disturbances. A preferred operating state would be a stable limit cycle, which can make the system dynamics robust to disturbances. To this end, in this article, it is investigated as to how to engineer Hopf bifurcations of an equilibrium solution in the β and double acting α FPSE configurations that could lead to attracting periodic solutions. Weakly nonlinear analyses are conducted and analytical relations governing the periodic motions are obtained and studied in the vicinity of Hopf bifurcation points. The analytical predictions are confirmed through numerical simulations that are based upon reported engine parameters. The overall analytical-numerical approach pursued here could serve as a tool for using nonlinearity in the design of FPSEs, thereby enhancing the robustness of device operations.

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References

Figures

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Fig. 1

Schematics of two different FPSE configurations are shown with the key parameters listed in Table 1. Heat addition takes place across the heater, heat removal occurs across the cooler, and the regenerator is in between these two elements. Additionally, the work extraction is represented by a damping element in the bounce space. (a) The FPSE in a β configuration. (b) The FPSE in a double acting α configuration; it was found that a minimum of three cylinders is required for a successful engine [4]. The expansion process and the compression process for each of the individual thermodynamic cycles is located in adjacent cylinders, which dynamically couples the associated pistons.

Grahic Jump Location
Fig. 2

Parametric plot of the nondimensional steady-state piston and displacer displacements. The parameter values are based on the Sunpower RE-1000, as reported in Ref. [2] and adapted to satisfy the condition of proportional damping. As predicted, in the limit, the system migrates to a limit cycle. The analytical approximation (dashed line) is shown for comparison with the numerical solution of Eq. (9) (solid line); the difference between the two is attributable to the neglected higher order terms.

Grahic Jump Location
Fig. 3

(a) Variations of the nondimensional piston amplitude (solid line) and amplitude ratio (dashed line). (b) Variation of the phase angle. (c) Variations of the corrected frequency (solid line) and uncorrected frequency (dashed line). (d) Variation of the nondimensional power. The variation in the parameter γ corresponds to a hot side temperature range of 585 ≤ Th < ∞ (K).

Grahic Jump Location
Fig. 4

(a) Variations of the nondimensional amplitude (solid line) and amplitude ratio (dashed line). (b) Variation of the phase angle. (c) Variations of the corrected frequency (solid line) and uncorrected frequency (dashed line). (d) Variation of the power output. The parameter αp is varied over a range corresponding to a mean pressure range of 57 ≤ Pm < 142 (bar).

Grahic Jump Location
Fig. 5

(a) Plot of the steady-state displacements of the three pistons with respect to time; all three plots are identical, except for a phase shift of 120 deg. The cylinder 1 response is depicted by a solid line, the cylinder 2 response is depicted by a bold dashed line, and the cylinder 3 response is depicted by a light dashed line. (b) The three-dimensional parametric plot of the steady-state displacements with the numerical solution is shown by using a solid line and the analytical solution is shown by using a dashed line.

Grahic Jump Location
Fig. 6

(a) Variation of the amplitude. (b) Variation of the frequency. (c) Variation of the power output. The variation of the parameter γ corresponds to a heater temperature range of 407 ≤ Th < ∞ (K).

Grahic Jump Location
Fig. 7

(a) Variation of the amplitude. (b) Variation of the frequency. (c) Variation of the power output. The variation in the parameter α corresponds to a charge pressure range of 0 ≤ Pm < 3.2 (bar).

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