Arabian Journal for Science and Engineering https://doi.org/10.1007/s13369-018-3677-1
RESEARCH ARTICLE - MECHANICAL ENGINEERING
Applying Particle Swarm Optimization to Study the Effect of Dominant Poles Places on Performance of a Free Piston Stirling Engine Sh. Zare1 · A. R. Tavakolpour-Saleh1 Received: 7 December 2017 / Accepted: 3 December 2018 © King Fahd University of Petroleum & Minerals 2018
Abstract In this paper, the effects of dominant closed-loop poles places of a free piston Stirling engine (FPSE) on the engine performance are investigated. First, linear and nonlinear formulations of an FPSE are presented. Then, based on the presented formulations, dynamic behavior of a prototype free piston Stirling engine, namely SUTECH-SR-1, is studied. Afterward, a part of the simulation results is compared with those of an experimental work to validate the mathematical models. Consequently, the obtained reasonable agreement between the experimental and simulation results affirms the validity of the proposed nonlinear theorem. Next, according to linear and nonlinear formulations, the necessary condition for startup state of the engine is investigated. In addition, it is demonstrated that existence of a stable limit cycle is the necessary condition for engine startup based on theory of nonlinear systems. Subsequently, the effects of two important factors including real and imaginary parts of the dominant closed-loop poles of the FPSE on the engine performance are investigated using particle swarm optimization. According to the obtained simulation outcomes, the free piston Stirling converter is more sensitive to variation of the real component of the dominant closed-loop poles than that of the imaginary part. Keywords Free piston Stirling engine · Particle swarm optimization · Dominant poles
List of Symbols A Crosscut area of the piston and displacer (m2 ) Crosscut area of the displacer rod (m2 ) Ar b Damping coefficient (N s m−1 ) F Position of particle (m) K Spring stiffness (N m−1 ) M Piston mass (kg) m Gas mass (kg) P Pressure (Pa) Q Velocity of particle (m s−1 ) R Gas constant for dry air (J kg−1 K−1 ) s Random vector T Temperature (K) t Time (s) u Random vector V Volume (m3 ) x Position of displacer piston (m) y Position of power piston (m)
B 1
Subscript and superscript c Cold gas d Displacer piston h Hot gas i Particle L Linearity k Iteration r Regenerator 0 Initial value N Nonlinearity Greek symbols α Constant β Constant θ Phase difference (◦ ) ω Frequency (rad s−1 )
1 Introduction A. R. Tavakolpour-Saleh
[email protected] Department of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz, Iran
Limitation of fossil fuel resources has prompted scientist to find an appropriate, reliable, and interminable energy supply to meet the energy requirements of future world. Solar
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energy is one of the most applicable solutions to this problem. Numerous methods have been utilized to generate electrical energy from solar radiation, one of which is Stirling engines. The Stirling engines are categorized into several different groups according to their working conditions and their different applications. Generally, these engines have two types, namely dynamic and kinematic engines. In the kinematic Stirling engines, power and displacer pistons are connected to each other via linkages, while in the dynamic type, there is no geometrical constraint between two pistons and they are coupled through the pressure dynamics [1–5]. Dynamic type of Stirling engines can be passive and active. In passive dynamic engines, it is assumed that no external electrical energy is applied in order to run the engine. One of the main complexities of passive FPSE is due to its dynamic behavior. Accordingly, several different theories have been proposed for simulation of this kind of engine, either linear or nonlinear. Originally, primitive linear theories were based on Schmidt principle [6–8]. However, this theory was more useful for kinematic Stirling engines [9–11]. Indeed, Schmidt theory cannot describe the system behavior in the startup moment, while this subject is a key issue in the dynamic type of Stirling engines. As a result, researchers shift their attentions to other linear methods to investigate the startup state of such engines. In this regard, studying the location of closed-loop roots of the system (root locus method) is a common procedure to study the performance of dynamic Stirling engines. Riofrio et al. [12] proposed a dynamical model for a linear FPSE and examined the engine instability through root locus method. Hofaker et al. [13] presented a dynamic model in order to investigate FPSE. In the developed model, finite heat transfer is assumed to take place between heat exchangers. Given this assumption, the problem of piston movement upon marginal stability and instability was studied. On the other hand, linear theories can only predict dynamic behavior of FPSE in the startup state. In fact, based on the linear analysis, whenever the dynamic system experiences instability, the pistons stroke increases monotonically as time elapses. In other words, it is not possible to study the performance of the realistic engine and evaluate accurately the important parameters such as phase difference, output work, and output power in the steady-state conditions. Therefore, it is essential to propose a method to resolve this drawback. Nonlinear analysis of FPSEs can be an effective way to fulfill these latest requirements. Karabulut et al. [14] proposed a nonlinear method to model and evaluate FPSE. In this model, it is shown that pressure variation inside engine cylinder of FPSE is extremely sensitive to damping coefficient. Jian et al. [15] studied performance of the FPSE in terms of the phase difference. In their conducted research, the analysis was nonlinear due to the nonlinearity in the pressure model. Tavakolpour-Saleh et al. [16] proposed another nonlinear analysis based on perturbation technique
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to explore FPSE behavior. In the proposed model, the springs are assumed nonlinear. Zare et al. [17] investigated nonlinear dynamics of FPSE in which the pressure difference is assumed different between hot and cold chambers unlike previous researches. Design and optimization of FPSEs are other important issues beside their analysis. Up to now, the optimization of the kinematic Stirling engines has received considerable attention from researchers. Accordingly, design parameters can be optimized so as to meet a certain goal [18–20]. Similarly, it can be done to optimize dynamic FPSE as well. PSO is a suitable method for finding an optimal performance of these engines. This method starts with a group of random positions and velocities of particles. Afterward, the optimization process continues via updating particles’ position. In addition, the initial position of particles is chosen in a random manner. Then, based on position and velocity of particles, next situation of particles is calculated [21–24]. Generally, PSO has numerous advantages such as simple concept, easy implementation, robustness to control parameters, computational efficiency, insensitive to scaling of design variables, simple implementation, easily parallelized for concurrent processing, derivative free, very few algorithm parameters, and very efficient global search algorithm [25–31]. In this research, PSO is employed in order to enhance the performance of a nonlinear FPSE. Accordingly, the influences of the imaginary and real parts of the dominant closed-loop poles of the engine on the system performance are investigated. Indeed, by studying the effect of variation of real and imaginary parts of the dominant poles, design parameters such as the mass of pistons, the springs stiffness, and the rod area are obtained using PSO. Next, the engine performance is studied based on the proposed nonlinear method. Finally, a sensitivity analysis is carried out to highlight the effect of real and imaginary parts of the closed-loop poles on the engine performance. Eventually, simulation results are compared with the experimental data of a prototype engine, i.e., SUTECH-SR-1.
2 Fundamental of Free Piston Stirling Engines Free piston Stirling engines consist of several components such as power piston, displacer piston, and engine cylinder (Fig. 1). In addition, these engines have two chambers, namely expansion and compression spaces. The power generation is carried out by the power piston, while the gas pressure varies due to the motion of the displacer piston. Besides, FPSE operates based on Stirling cycle [7,8]. Stirling cycle is a closed thermodynamic cycle composed of two constant-temperature and two constant-volume regenerative processes. The thermodynamic efficiency of the Stirling
