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Research Papers

Control-Oriented Modeling of the Dynamics of Stirling Engine Regenerators

[+] Author and Article Information
Mitchel Craun

Department of Mechanical Engineering,
University of California—Santa Barbara,
Santa Barbara, CA 93106
e-mail: craunm@gmail.com

Bassam Bamieh

Department of Mechanical Engineering,
University of California—Santa Barbara,
Santa Barbara, CA 93106
e-mail: bamieh@engr.ucsb.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 28, 2015; final manuscript received August 24, 2017; published online November 10, 2017. Assoc. Editor: Beshah Ayalew.

J. Dyn. Sys., Meas., Control 140(4), 041001 (Nov 10, 2017) (13 pages) Paper No: DS-15-1652; doi: 10.1115/1.4037838 History: Received December 28, 2015; Revised August 24, 2017

We develop a first-principles model of the regenerator component of a generic Stirling engine. The model is based on the Euler equations of one-dimensional gas dynamics coupled with its convective/conductive heat transfer with the embedded mesh material. We investigate various methods for deriving simpler and low-order control-oriented models from this first principles model, the basic criterion being high fidelity representation of the dynamics of the regenerator when coupled to other dynamic components of the engine. We identify several nondimensional parameters that potentially categorize different modes of operation, and investigate the corresponding time-scale separation. A hierarchy of singularly perturbed models is derived in which acoustic dynamics are eliminated, periodic mesh dynamics are averaged, and the shape of the distributed regenerator gas state is approximated. In addition, since the reduced model is to be operated cyclically when connected to other parts of the engine, we develop such a feedback-aware model reduction algorithm based on a proper orthogonal decomposition (POD) with a chirped signal input (chirp-POD). This algorithm yields reduced models that are accurate over a range of engine operating frequencies.

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Figures

Grahic Jump Location
Fig. 1

A conceptual diagram of a generic Stirling engine. The displacer piston shuttles gas (through the regenerator) between the hot and cold sections. This causes the average temperature (and consequently the pressure) of the working gas to oscillate. These pressure oscillations drive the power piston, which performs mechanical work on a load. Depending on the type of engine, kinematic or dynamic linkages use a small amount of that work to in turn drive the displacer piston, thus setting up a limit cycle. The geometry shown for linkages is conceptual.

Grahic Jump Location
Fig. 2

A block diagram of the various components of our Stirling engine model. The displacer actuation input is only relevant to the case of the actively controlled engine. The piston dynamics block has the positions and velocities of the pistons as states (together with any kinematic constraints) while the gas section blocks have the pressures and densities of each section as states. Lines with the port symbol (•−) indicate interactions whose directions switch depending on the sign of velocity at the boundaries. The only block with a distributed state is that of the regenerator, to which we apply the model reduction techniques described in this paper.

Grahic Jump Location
Fig. 3

A diagram of the first-principles model of the gas sections and regenerator interactions. Each gas section is considered as a well-mixed compartment with a lumped state. The regenerator is modeled using the Euler equations of one-dimensional compressible gas dynamics interacting through convective heat transfer with a spatially distributed metal mesh. The spatial coordinate axis is used for the distributed regenerator state only and not the lumped gas sections.

Grahic Jump Location
Fig. 4

A diagram depicting the coupling of one-dimensional gas dynamics (Eq. (6)) with the distributed mesh temperature dynamics (Eq. (7)) through spatially distributed convective and conductive heat exchange. Although the gas temperature T is not explicitly a state of the gas dynamics, it can be considered as an output using the ideal gas law T=R ρ/p.

Grahic Jump Location
Fig. 5

Conceptual diagram of a beta-type Stirling engine with flywheel kinematic connections. P0 and P1 are the pressures in the hot and cold sections, respectively, while Pex is the pressure on the external side of the power piston.

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Fig. 6

A qualitative comparison of time histories of the sections' states for both the full (blue) and QSS models (red). The top diagram is for the case of a moderate value of ε1εf for which the QSS model is a crude approximation, while the bottom diagram is for a smaller value of ε1εf, for which the QSS model is a relatively more accurate approximation.

Grahic Jump Location
Fig. 7

(Left) Time histories of the hot section states for full (with Acoustics, in blue) and chirp-POD reduced model (in dashed red) with three regenerator states. The two sets of trajectories are indistinguishable. (Right) The trajectories produced when the same chirp-POD reduced model is coupled to gas sections with wall conduction/convection coefficients of twice the magnitude of the sections used to generate model reduction POD trajectories. Note that despite the increase in engine frequency, the reduced models still match closely, indicating the efficacy of the chirp-POD technique over a wide range of engine parameters. The cold sections' states are not shown, but their behavior is similar to the above.

Grahic Jump Location
Fig. 8

The first four POD modes for the density (left) and pressure (right) profiles of the driven engine model. These modes were obtained with a chirp input driving signal. The horizontal axis is the element number in the spatial discretization of the no-acoustics model of Sec. 4.1. Note the approximate odd reflection symmetry (between pressure and density) of modes 1 and 2, and the approximate even reflection symmetry of modes 3 and 4.

Grahic Jump Location
Fig. 9

Frequency response comparison of the full (blue) and a four-state chirp-POD model (red). The horizontal axes are the frequency of the displacer input, which is a pure sinusoid. The vertical axes are the amplitudes of the first four harmonics of the respective outputs. The bottom figure is the response of the systems to an input with amplitude 50% larger than the top figure. The two figures would not necessarily be similar since this is the frequency response of a nonlinear system.

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