Higher order modeling of a free-piston Stirling engine: analysis and experiment

International Journal of Energy and Environmental Engineering https://doi.org/10.1007/s40095-018-0267-7 ORIGINAL RESEARCH Higher order modeling of a free‑piston Stirling engine: analysis and experiment S. H. Zare1 · A. R. Shourangiz‑Haghighi1 · A. R. Tavakolpour‑Saleh1 Received: 13 November 2017 / Accepted: 21 February 2018 © The Author(s) 2018 Abstract This paper focuses on higher order modeling and design of the free-piston Stirling engine (FPSE) based on Ant Colony Optimization (ACO). First, the governing thermodynamics and dynamical equations of the engine have been derived. Then, the design parameters of the engine are selected taking into account the finite heat transfer coefficient (resulting in a fifth-order model) and pressure drop (resulting in a sixth-order model) in the dynamical system and the corresponding differential equations are derived in detail. In the following, the mentioned methods and their performance in modeling the FPSE dynamics are investigated. The simulated results show that the effect of the pressure drop on the places of the closed-loop poles of the system is not significant, while the heat transfer coefficient has a considerable effect on the engine dynamics. Accordingly, a fifth-order model along with ACO algorithm is proposed to justify the FPSE behavior. To validate the presented modeling scheme, the prototype engine SUTECH-SR-1 was experimented. It is found that the values of parameters obtained from the proposed design method are close to those of the experiment. Besides, the presented higher order model predicts the engine behavior with an acceptable accuracy through which the validity of the design technique is affirmed. Keywords Free-piston Stirling engine (FPSE) · Ant Colony Optimization · Higher order model List of symbols A (Cross-sectional area of the piston and displacer ) 2 m ( 2) Ar Cross-sectional area ( 2of ) the displacer rod m Ac Area of heat sink m( ) Ah Area of heat source m2 ( ) Awall Area on the chamber between the hot and cold m2 b Damping coefficient(between )displacer rod and power piston potion N s m−1 bd (Damping)coefficient of the displacer piston N s m−1 ( ) bp Damping coefficient of the power piston N s m−1 H Enthalpy (J) ( ) hc Heat transfer coefficient of cold sink watt K−1 m−2 of heat source hh (Heat transfer coefficient ) watt K−1 m−2 hwall Heat transfer coefficient of the)chamber between ( the hot and cold watt K−1 m(−2 ) Kd Spring stiffness of displacer N m−1 * A. R. Tavakolpour‑Saleh 1 ( ) Kp Spring stiffness of power piston N m−1 M Total mass of the gas in the engine (kg) Md Mass of displacer (kg) Mp Mass of the power piston (kg) P Linear pressure (Pa) P0 Initial pressure of working gas (Pa) Pw Power generation (J/s) ̂ Nonlinear pressure (Pa) P ( ) R Ideal gas constant J kg−1 K−1 T Temperature (K) Th Gas temperature in compression space (K) Tc Gas temperature in compression space (K) W Work (J) ( ) Vh Volume of expansion space m3 ( ) 3 Vh0 Initial volume of expansion space ( 3 )m Vc Volume of compression space m ( ) 3 Vc0 Initial volume of compression ( 3 )space m Vr Volume of the regenerator m x Displacer position ((m) ) ẋ Displacer velocity m s−1 y Power piston position ((m) ) ẏ Power piston velocity m s−1 Department of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz, Iran 13 Vol.:(0123456789) International Journal of Energy and Environmental Engineering Greek symbols 𝛾 Heat capacity ratio 𝜔 Engine frequency (rad∕s) Introduction Free-piston Stirling engines (FPSEs) are one of the novel converters for converting solar energy into other types of energy [1, 2]. It was first developed by Robert Stirling in the early 1810s in Scotland [3]. Up to now, many researchers have conducted extensive works to design and analyze these types of engines. High efficiency, self-starting, and long operating life can be noted as one of the major advantages [4, 5]; however, despite these advantages, designing and setting up these types of engines has been a major challenge for scientists. The dynamic structure of the FPSEs is such that the motion of the pistons of the system must reach the unstable or marginally stable (in linear analysis) and in nonlinear analysis achieve a stable limit cycle [6, 7], but unfortunately this condition involves a lot of complexity in the design procedures of Stirling engines. On the other hand, not only parameters such as power, frequency are significant, but also startup condition must be remarked. So far, various practical methods have been documented in literature to analyze and design the FPSEs. The first method is related to the linear analysis so that the behavior of all engine parameters is assumed linearly. The advantage of such systems is their simplicity and reliability simultaneously [8, 9]. Linear methods can be discussed and investigated in two directions. The first approach is employed for linear methods in order to analyze and set up FPSE, which is based on the principle that the system is unstable or marginally stable [10]. Truthfully, in these methods, the dynamic analysis of the engine is based on the places of dominant poles of the closed-loop system, and working condition for starting the engine is that the location of closed-loop poles are either on the imaginary axis of the s-plane or on the right side of imaginary axis. The first analysis was carried out to set up and analyze the FPSE employing the wellknown linear method known as the fourth-order model [11, 12], which tries to examine the effects of unknown parameters on the engine behavior. The basis of the fourth-order model is devised by Schmidt et al. (also called Schmidt’s theory [12], in which the pressure variations in the entire space of the engine compartment are supposed to be, and the heat transfer coefficient in the chamber are also assumed to be infinite. However, the other linear theory presented is known as the fifth-order model, in which the finite heat transfer coefficient is assumed to make it feasible to analyze the behavior of FPSE more precisely [13]. In the case of the second approach, it is used for linear methods to design these engines. Providing a valid design methodology as a 13 means for predicting the critical parameters of the design procedure is crucial although the fact that the FPSEs are passive. Thus far, the most popular method still used is the first method, and there are still few reports regarding the second method. However, in a recent study, Zare and Tavakolpour-Saleh [14] presented an innovative design based on the fourth-order model to design, which has been able to predict optimally the design parameters of the Stirling engine and led to design and construct these engines at a lower cost. But another concern that should be addressed in the design procedure of the desired FPSE is to provide not only an advanced design methodology, but also a more trustworthy method than the fourth-o (...truncated)