A Lumped-parameter Dynamic Model of a Thermal Regenerator for Free-piston Stirling Engines

This paper uses lumped parameter dynamic equations to model the mass flow, piston dynamics, and control volume behavior inside a free-piston Stirling engine. A new model for a Stirling engine thermal regenerator that incorporates a dynamically changing temperature gradient is presented. The use of graphite as a regenerator matrix material is justified despite its limited background by comparing the functional requirements of regenerators to heat exchangers where graphite use is commonplace. Experimental results are used to characterize a graphite regenerator and validate the dynamic model. INTRODUCTION Stirling engines provide clean, reliable, mechanical power when provided only with a temperature gradient. Unlike combustion engines, Stirling engines do not require a distilate fuel like gasoline and can therefore run on heat from any source such as geothermal, solar, biomass or nuclear energy. The Stirling cycle operates by shuttling a compressible fluid (gas) between two chambers at two distinct temperatures. A lightweight piston called the displacer piston is responsible for the transport of this gas. The temperature variation of the gas causes a pressure fluctuation which in turn performs work on a second piston called the power piston. A “free piston” Stirling engine is a Stirling engine in which the two pistons are connected through dynamics as opposed to a kinematic linkages seen in more traditional kinematic Striling engines. Free piston Stirling engines offer low noise, low maintenance operation at scales and power outputs that can rival more traditional internal combustion engines. [1] Stirling engines are theoretically capable of Carnot efficiency, but are not used in mainstream power production today because they have typically been heavy and/or inefficient compared to their alternatives. To understand why this has been the case in the past, and why a new approach holds the potential to improve their performance, a brief description of the Stirling cycle is in order. FIGURE 1: EXAMPLE SCHEMATIC DIAGRAM OF A FREEPISTON STIRLING CYCLE ENGINE WITH A LINEAR ALTERNATOR FOR ENERGY EXTRACTION AND A DAMPING MECHANISM OF INTERACTION BETWEEN THE DISPLACER AND POWER PISTON. As seen in Fig. 1, the displacer piston in a Stirling engine serves to transport heat carried in a compressible fluid from a heat source on one side of the engine to a sink on the other. When the displacer piston is nearer to the bottom, more of the gas is on the hot side of the engine, and the pressure inside of the engine increases. When the displacer is nearer to the top, the majority of the gas is cooled and the pressure decreases. Adding to these effects is the movement of the power piston which interacts directly with the displacer via some amount of damping or spring force, or indirectly by via pressure. By delicately balancing the area and masses of the pistons, the dynamic relationship between the pistons, the mass flow restriction from one side to the other, the heat transfer, and the load dynamics, a self sustaining cycle can be obtained to transform heat into useful work that is extracted from the power piston. Most Stirling engines use a regenerative heat exchanger or simply “regenerator” to increase efficiency. The regenerator works as a thermal capacitor where heat is absorbed and released from the gas as it passes from one chamber to the other. This heat transfer occurs cyclically at the operating frequency of the engine. Traditionally, Stirling engines were designed using a purely kinematic and thermodynamic approach, but these approaches are insufficient and inappropriate for free-piston Stirling engines because the relationship between analytical models and true physical performance currently remains inaccurate to the point of reducing most design efforts to a process of trial and error. Often times in the past, engines with designs that appeared feasible on paper would not run when built [2]. Using system dynamic modeling and analysis, as opposed to purely thermodynamic/kinematic analysis, has been shown to accurately reproduce the behavior of existing engines. In order to design and construct a Stirling engine from a system dynamics point of view that can additionally utilize control design tools, it is necessary to have an accurate model of the Stirling engine that can be reproduced physically. PRIOR WORK IN DYNAMIC MODELING OF FREEPISTON STIRLING ENGINES Recent efforts in modeling free-piston Stirling engines as dynamic systems by Riofrio [2] have suggested that there is a correlation between the locations of the closed loop poles in the right half plane and the operating characteristics of an engine. This interpretation reveals that a free-piston Stirling engine produces power and maintains self-sustained oscillation if the system has unstable 2 closed-loop poles as shown in Fig. . This linearized interpretation is valid about a small operating region of the equilibrium values, whereas the real nonlinear dynamics will limit the amplitude of oscillation as the states depart far enough from the equilibrium. Real Axis Im ag in ar y A xi s O pe ra tin g Fr eq ue nc y