Many-objective thermodynamic optimization of Stirling heat engine

Highlights

• Many-objective (i.e. four objective) optimization of Stirling engine is investigated.

• MOHTS algorithm is introduced and applied to obtain a set of Pareto points.

• Comparative results of many-objective and multi-objectives are presented.

• Relationship of design variables in many-objective optimization are obtained.

• Optimum solution is selected by using decision making approaches.

Abstract

This paper presents a rigorous investigation of many-objective (four-objective) thermodynamic optimization of a Stirling heat engine. Many-objective optimization problem is formed by considering maximization of thermal efficiency, power output, ecological function and exergy efficiency. Multi-objective heat transfer search (MOHTS) algorithm is proposed and applied to obtain a set of Pareto-optimal points. Many objective optimization results form a solution in a four dimensional hyper objective space and for visualization it is represented on a two dimension objective space. Thus, results of four-objective optimization are represented by six Pareto fronts in two dimension objective space. These six Pareto fronts are compared with their corresponding two-objective Pareto fronts. Quantitative assessment of the obtained Pareto solutions is reported in terms of spread and the spacing measures. Different decision making approaches such as LINMAP, TOPSIS and fuzzy are used to select a final optimal solution from Pareto optimal set of many-objective optimization. Finally, to reveal the level of conflict between these objectives, distribution of each decision variable in their allowable range is also shown in two dimensional objective spaces.

Introduction

Thermodynamic optimization of any system is important to improve its performance and reduce pollution caused by that system [1]. In recent years, Stirling heat engines draw a lot of attentions due to its high theoretical efficiency and energy conservation [2]. A Stirling heat engine is an external combustion, closed cycle engine, which uses external heat source for power generation [3]. Performance parameters of a Stirling heat engine like power output, thermal efficiency, exergy efficiency, pressure drop etc. are sensitive to various geometric and operating parameters such as piston diameter, regenerator diameter, regenerator length, engine's rotation speed, mean effective pressure etc. Further, the effect of each operating parameter on various performance parameters may not be similar [4]. Thus, optimization of a Stirling heat engine is many-objective in nature and it is necessary to carry out simultaneous optimization of all objectives, so Pareto solutions provide more insights into the competing objectives.

Earlier, researchers carried out an extensive work related to the development of thermal model of Stirling heat engines. Babaelahi and Sayyaadi [5] developed thermal model of Stirling engine for thermal simulation of its prototype. Hosseinzade et al. [6] also proposed thermal model of a Stirling engine and obtained its simulation and optimization results. Araoz et al. [7] presented a thermodynamic model for the performance analysis of a Stirling engine.

Recently, researchers reported the work related to optimization of Stirling heat engine using metaheuristic algorithms. Ahmadi et al. [8] employed finite-speed thermodynamic analysis to obtain an optimum power output and pressure loss of a Stirling heat engine. Simultaneous optimization of thermal efficiency, power output and entropy generation rate of a solar dish-Stirling engine was performs by adapting finite-time thermodynamics and NSGA-II [9]. Thermo-economic optimization of a solar dish-Stirling based on dimensionless thermo-economic objective function was demonstrated by using NSGA-II [10]. Optimization of output power, thermal efficiency and total pressure losses are reported using NSGA-II and finite speed thermodynamic [11]. Optimize value of absorber and working fluid temperature was obtained for the maximization of output power and thermal efficiency of solar power Stirling engine using NSGA-II [12]. Maximization of dimensionless output power, thermal efficiency and entransy rate of the solar-dish Stirling system was obtained using thermodynamic analysis and NSGA-II algorithm [13].

Li et al. [14] analyzed and optimized mechanical power, thermal efficiency and entropy generation rate of Stirling engine using finite physical dimensions thermodynamics and genetic algorithm. Ferreira et al. [15] performed a thermo-economic optimization of Stirling engine used for micro-cogeneration purpose. The authors optimized geometric and operational parameters of engine for thermo-economic consideration. Multi-objective optimization of Stirling engine based on non-ideal adiabatic analysis [16] and third order thermodynamic analysis [17] was reported using NSGA-II. Patel and Savsani [18] investigated a variant of teaching-learning based optimization algorithm for maximizing thermal efficiency, power output and minimizing total pressure drop of the Stirling engine simultaneously. Duan et al. [19] performed multi-objective optimization between power output and efficiency of Stirling engine with cycle irreversibility consideration. Zare and Tavakolpour-Saleh [20] adopted genetic algorithm and presented an optimized design of frequency based free piston Stirling engine. Campos et al. [21] performed optimization of Stirling engine under different operating and design conditions and reported the behaviour of cycle efficiency under these conditions.

Punnathanam and Kotecha [22] optimized thermal efficiency, output power and entropy generation rate of Stirling engine by adapting NSGA-II. Arora et al. [23] investigated NSGA-II for thermo-economic optimization of solar parabolic dish Stirling heat engine. The author considered power output, efficiency and economic function of the engine for optimization. Hooshang et al. [24] obtained optimized value of output power and efficiency of Stirling engines based on neural network concepts. Luo et al. [25] performed a multi-objective optimization of a GPU-3 Stirling engine and reported an output power of more than 3 kW with 5% rise in thermal efficiency. Many other works related to multi-objective optimizations are reported by the researchers. For example, Pareto optimal design of stand-alone wind/PV/FC generation micro grid System was obtained using particle swarm optimization algorithm [26]. Multi-objective solutions for optimal allocation of multi-type flexible alternating transmission system were reported in Ref. [27].

Thus, it can be observed from literature survey that, works related to thermodynamic optimization of Stirling engine are reported either for single objective or multi-objective (two or three objective) consideration. However, many-objective thermodynamic optimization of Stirling engine is yet to be reported in the literature. In order to fulfill the gap, present work reports many-objective (i.e. four-objective) thermodynamic optimization of Stirling heat engine. Further, as an optimization tool, heat transfer search (HTS) algorithm [28] is implemented in the present work. Heat transfer search is a recently developed meta-heuristic algorithm based on the natural law of thermodynamics and heat transfer [28]. Researchers reported the application of heat transfer search algorithm for truss topology optimization [29], optimization of semi-active vehicle suspension system [30], sizing optimization of truss structure [31] etc. In this work, a multi-objective variant of heat transfer search (MOHTS) algorithm is introduced to address many-objective optimization problem of a Stirling heat engine.

Main objectives and contributions of the present work are: (i) To develop many-objective thermodynamic optimization problem of Stirling heat engine to maximize thermal efficiency, power output, exergy efficiency and ecological function. (ii) To propose multi-objective variant of the heat transfer search (MOHTS) algorithm and using it to solve many-objective optimization problem of Stirling heat engine (iii) To compare and analyze the results of many-objective (i.e. four-objective) optimization with multi-objective (i.e. two-objective) optimization. (iv) To demonstrate the underlying relationship of the decision variables during many-objective (i.e. four-objective) optimization (v) To select the final optimal solution from the Pareto optimal set of the many-objective optimization with the help of LINMAP, TOPSIS and fuzzy decision making approaches and (vi) To perform the quantitative assessment of obtained Pareto solutions.

The remainder of this paper is organized as follows. Section 2 presents the thermal-modeling and the objective functions formulation of Stirling heat engine. Section 3 describes the heat transfer search algorithm. Section 4 explains proposed multi-objective heat transfer search algorithm. Section 5 presents the application example of Stirling heat engine. Section 6 describes the results-discussion. Finally, conclusions are presented in Section 7.