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Home , Engine efficiency, Stirling engine, Thermal efficiency

Sa¯dhana¯ Vol. 41, No. 11, November 2016, pp. 1321–1331 Ó Indian Academy of Sciences DOI 10.1007/s12046-016-0553-0

Optimal design of Stirling using an advanced optimization algorithm

R V RAO1, K C MORE1,*, J TALER2 and P OCłON´ 2

1 Department of Mechanical Engineering, S.V. National Institute of Technology, Surat 395007, India 2 Instytut Maszyn i Urza˛dzen´ Energetycznych, Politechnika Krakowska, Al. Jana Pawła II 37, 31-864 Krako´w, Poland e-mail: [email protected]

MS received 9 March 2015; revised 20 April 2016; accepted 23 May 2016

Abstract. The presents an excellent theoretical output equivalent to the output of Carnot engine and it is less pollutant and requires little maintenance. In this paper, Stirling is considered for optimization with multiple criteria. A recently developed advanced optimization algorithm namely ‘‘teaching– learning-based optimization (TLBO) algorithm’’ is used for maximization of output , minimization of losses and maximization of the thermal efficiency of the entire solar Stirling system. The comparisons of the proposed algorithm are made with those obtained by using the decision-making methods like linear programming technique for multi-dimensional analysis of preference (LINMAP), technique for order of pref- erence by similarity to ideal solution (TOPSIS) and fuzzy Bellman–Zadeh method that have used the Pareto frontier gained through non-dominated sorting genetic algorithm-II (NSGA-II). The comparisons were also made with those obtained by the experimental results. It is found that the TLBO algorithm has produced comparatively better results than those given by the decision-making methods and the experimental results presented by the previous researchers.

Keywords. Stirling heat engine; multi-objective optimization; thermal efficiency; pressure loss; teaching– learning-based optimization algorithm.

1. Introduction determine the parameters of a 200 W Stirling engine by measuring the thermal-flux and mechanical-power losses to The Stirling engine has huge prospective to be useful for maximize the engine efficiency. Orunov et al [4] presented exchanging heat into the mechanical effort with a high a method to calculate the optimum parameters of a single- thermal efficiency. Its thermal efficiency might be as high cylinder Stirling engine for maximizing the efficiency. as the Carnot efficiency. Stirling engine can be powered by Abdalla and Yacoub [5] used saline feed raw water as the different heat sources and heat. It can utilize com- cooling water and had improved the thermal efficiency of pressible fluid as a working fluid. The schematic diagram of the Stirling engine. Hirata et al [6] evaluated the perfor- Stirling engine is shown in figure 1 [1]. The compression mance of a small 100 W displacer-type Stirling engine and expansion processes of the cycle take place in power Ecoboy-SCM81. An analysis model using an isothermal cylinder with a power . A displacer piston shuttles the method considering the pressure loss in the regenerator, the working fluid back and forth the heater, regenerator and buffer space loss, and the mechanical loss of the prototype cooler at constant . For providing a closed engine engine, was developed to improve the engine performance. chamber, the circuit of displacer piston is closed using Costea and Feidt [7] studied the effect of variation of the the tubes. The displacer and the power piston have same overall coefficient on the optimum distribution dimensions. In mono- operation, the power piston of the heat transfer surface conductance or areas of the contacts the engine chamber pressure on one side and Stirling engine . Wu et al [8] analyzed the ambient pressure on the other side. optimal performance of a Stirling engine. The influences of The Stirling engine has three dissimilar configurations heat transfer, regeneration time and imperfect regeneration namely alpha, beta and gamma configurations and is shown on the optimal performance of the irreversible Stirling in figure 2 [2]. Markman et al [3] conducted an experiment engine cycle were discussed. In another , Wu et al [9] using the beta-configuration of the Stirling engine to showed that the quantum was different from the classical thermodynamic one because of different *For correspondence characters of the working fluids. The authors had studied

1321 1322 R V Rao et al

irreversibility were described. Petrescu et al [17] presented a method to calculate the efficiency and power of a Stirling engine. The method was based on the first law of thermo- dynamics for processes with finite speed and the direct method for closed systems. Martaj et al [18] studied the thermodynamic analysis of a low differential Stirling engine at steady state operation, and , and balances. Timoumi et al [19] developed a numerical model based on Figure 1. Schematic diagram of the Stirling engine [1]. lumped analysis approach. Formosa and Despesse [20] implemented an isothermal model for presenting engine power output and efficiency due to dead volume. Li et al [21] optimized the absorber temperature and corresponding thermal efficiency and developed a mathematical model for the overall thermal efficiency of solar-powered high tem- perature differential dish Stirling engine. Tlili [22] studied the effects of regenerating effectiveness and heat capaci- tance rate of external fluids in heat source/sink at maximum power and efficiency. Ahmadi et al [23] used NSGA-II algorithm for opti- mization of Stirling heat engine. Maximization of output power and overall thermal efficiency and minimization of Figure 2. Basic mechanical configurations for Stirling engine. the pressure loss were considered as objective functions. (a) Alpha-configuration. (b) Beta-configuration. (c) Gamma-con- Ahmadi et al [24] investigated the optimal power of an figuration [2]. endoreversible Stirling cycle with perfect regeneration and genetic algorithm (GA) was used for the optimization of the optimal performance of forward and reverse quantum this endoreversible Stirling engine. Ahmadi et al [25] used Stirling cycles. Costea et al [10] studied the finite speed of NSGA-II algorithm for dimensionless thermo-economic the piston; and throttling in the regenerator to optimization of solar dish-Stirling engine. In another work, validate the design of a solar Stirling . Ahmadi et al [26] used NSGA-II algorithm for optimization Wu et al [11] presented the finite-time exergoeconomic of solar dish-Stirling engine. Maximization of output power optimal performance of a quantum Stirling engine. The and overall thermal efficiency and minimization of the rate maximum exergoeconomic profit, the optimal thermal of entropy generation were considered as an objective efficiency and power output corresponding to performance functions. bound of an endoreversible quantum Stirling engine were From the literature review, it is observed that only very presented. Organ [12] presented the optimization of Stirling few advanced optimization algorithms like GA and NSAG- engine regenerator using the effects of various parameters II had been used by the researchers for the optimization of such as diameter, length and materials on regenerator per- Stirling heat engine. However, these algorithms require formance, irreversibility, and temperature gradient. tuning of their specific parameters. For example, GA Kaushik and Kumar [13] investigated the finite time ther- requires crossover probability, mutation probability and a modynamic analysis of an endoreversible Stirling heat selection operator; NSGA-II requires crossover probability, engine and used finite time to optimize the mutation probability, real-parameter SBX parameter and power output and the corresponding thermal efficiency of real-parameter mutation parameter. The proper tuning of the endoreversible Stirling heat engine. In another study, the algorithm-specific parameters is very essential factor Kaushik and Kumar [14] applied finite time thermody- which affects the performance of the optimization algo- namics to optimize the power output and the corresponding rithms. The improper tuning of algorithm-specific param- thermal efficiency of an irreversible Ericsson and Stirling eters either increases the computational effort or yields the heat engines. local optimal solution [27]. In addition to the algorithm- Gu et al [15] attempted to design a high efficiency specific parameters of an algorithm, common control Stirling engine using a composite working fluid, e.g. two- parameters such as population size and the number of component fluid: gaseous carrier and phase-change com- iterations are to be considered. The common control ponent and single multi-phase fluid, together with super- parameters are common to any population-based algorithm critical heat recovery process. Hsu et al [16] studied the and these also need proper tuning. The designer’s burden integrated system of a free-piston Stirling engine and an will be reduced if there is no need to tune at least some of incinerator. The efficiency and the optimal power output, the parameters required by the algorithm. Hence, to over- including the effect induced by internal and external come the problem of tuning of algorithm-specific Optimal design of Stirling heat engine 1323 parameters, a recently developed algorithm-specific constant temperature (TC)toheatsinkwhichhasa parameter-less algorithm known as teaching–learning- constant temperature (TL). Then the working fluid crosses based optimization (TLBO) algorithm [28, 29] is used in over the regenerator and is warmed up to Th in an iso- the present work for the multi-objective optimization of choric process 2-3. In process 3-4, the working fluid Stirling engine. expands at a constant temperature Th, and obtains the Multi-objective optimization has been defined as finding heat from the heat source at a constant temperature (TH). a vector of decision variables, while optimizing (i.e. mini- The last process 4-1 is an isochoric cooling process mizing or maximizing) several objectives simultaneously, where the regenerator absorbs heat from the working with a given set of constraints. In the present investigation, fluid. In an actual cycle it is impractical to have an ideal multi-objective optimization is carried out to achieve the heat transfer in the regenerator in which the entire optimization of three objective functions: maximization of amount of absorbed heat (in process 4-1) is transferred to output power and the thermal efficiency of the entire solar the working fluid into the isochoric heating process Stirling system and minimization of pressure losses. The (process 2-3). multi-objective optimization is conducted with eleven The ideal Stirling engine cycle is shown on p–V and T–S decision variables including the of heat source diagrams in figure 3. The T–S diagram is modified to and and their difference with working fluids, include the effects of heat transfer through a temperature rotation speed, mean effective pressure, , heat source difference at the source and sink and for incomplete temperature, heat sink temperature, number of gauzes of regeneration. Additional heat from the external source is the matrix, regenerator’s length, piston diameter, and shown to be needed in the process, Qx-3, due to incomplete regenerator diameter are considered. regeneration. Similarly, the unregenerated heat is shown In the present work, an attempt is made to see if there is being rejected in the process, Qy-1. Fluid friction of the gas any improvement possible in the design of Stirling engine causes most of the pressure loss as it passes through the by using an advanced optimization algorithm known as regenerator. teaching–learning-based optimization (TLBO) algorithm. The mathematical model presented by Ahmadi et al [23] The reason for choosing the TLBO algorithm is that it is is considered in the present work and the details are given simple, robust, algorithm-specific parameter-less and gives below. optimal solutions comparatively with less number of The pressure losses are presented as [7, 10, 17, 31], function evaluations and with less computational effort X ¼ þ þ ; ð Þ [28]. The TLBO algorithm is applied in the present work DPi DPthroat Dpf Dpw 1 for simultaneously optimizing the three objectives consid- ered by Ahmadi et al [23] for the design of Stirling engine. where DPthroat is the pressure drop due to the internal The next section presents the details of the analysis of the friction produced in the regenerator. Stirling engine cycle with irreversibilities and the consid- 15 p D2 ered design problem. DP ¼ m N c : ð2Þ throat ðÞþ ðÞþ 2 c 2R s 1 TL DTL NRDR

Dpf is the pressure drop due to mechanical resistance of 2. Analysis of the Stirling engine cycle parts of engine and is experimentally found by using the with irreversibilities following equation: 5 Stirling cycle consists of four major processes as shown ð0:94 þ 0:0015snrÞ10 1 Dp ¼ 1 : ð3Þ in figure 3 [30]. Process 1-2 is an in f 3l0 k which the compressing working fluid rejects the heat at 1 l0 ¼ 1 : ð4Þ 3k The pressure drop due to the piston speed is given by rffiffiffi snr 4pm klnk c 1 Dpw ¼ : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 60 ð1 þ kÞð1 þ sÞ k 1 R T þ DT "#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L L T DT 1 þ H H : TL þ DTL ð5Þ

DQR is heat loss through two regenerative processes in the Figure 3. diagrams of ideal Stirling cycle cycle and is calculated as [30]. 1324 R V Rao et al

DQR ¼ mgCvgXðTH DTH TL DTLÞ: ð6Þ where y is the second adjusting coefficient and X1, X2 are the regenerative losses coefficient values representing The heat released between heat source and working fluid ‘‘optimistic’’ and ‘‘pessimistic’’ evaluation respectively. (Qh) is given by The value of y = 0.72 allowed for better match between the analytical and experimental results [7, 10, 17, 31]: In Dpw:ðk þ 1Þðs þ 1Þ bDpthroat f Dpf Qh ¼ mg 1 Eq. (19), 4pm 2pm pm ð Þ : þ þ B R TH DTH ln k ¼ 1 2M e : ð Þ X1 ð þ Þ 20 ð7Þ 2 1 M The net heat released from heat source (Q ) is given by M þ eB H X ¼ : ð21Þ 2 1 þ M QH ¼ Qh þ DQR: ð8Þ mgCvg The pure flux transferred to working fluid is M ¼ : ð22Þ mRCR : n ¼ð þ Þ r : ð Þ 2 2 Q Qh DQR 9 p DRLdqst H 60 m ¼ : ð23Þ R 16ðb þ dÞ Substituting Eqs. (6–8) into Eq. (9): hA 30 B ¼ð1 þ MÞ R : ð24Þ Dp ðÞk þ 1 ðÞs þ 1 bDp f Dp w throat f mgCvg nr QH ¼ 1 4pm 2pm pm The convection coefficient h is computed as ðÞ þ ðÞ nr : RTH DTH ln k XmgCvg TH DTH TL DTL 60 ÀÁ: 4pm s:nr 0 424 0:576 0:395 : Cpv ð Þ RTL 30 10 h ¼ : ð25Þ ð þ Þ ðpÞ 0:576 0:667 The output power of engine is given by [17, 31]: 1 s 1 ðÞ½ðÞb þ DR Pr 4 d 1 þ ¼ _ ¼ TL DTL _ : ð Þ On the other hand, power gQH 1 gII;irrQH 11 TH DTH 2 2 p DRL ¼ : ð Þ AR ¼ : ð26Þ DTL Tc TL 12 4ðb þ dÞ

DTH ¼ TH Th: ð13Þ The effect of mechanical friction, piston speed and pressure drop in regenerator are calculated as þ ¼ TL DTL : ð Þ gc 1 14 P 1 Dpi TH DTH 3l p1 g ; ð Þ ¼ 1 : ð27Þ II irr Dp By substituting Eqs. (12–14) in Eq. (11), we have g0 TH DTH ln k TLþDTL ¼ : ð Þ g gcgII;irr 15 0 ¼ ðÞ g gII;irrðXÞ gC 28 : ¼ Q; ð Þ power gcgII;irr 16 4pm H p ¼ : ð29Þ 1 ð1 þ kÞð1 þ sÞ where gc is Carnot efficiency and gII;irr is the second law efficiency and is given by The above analysis shows that the pressure losses and their effect on the efficiency and output power of the engine ¼ : ð Þ gII;irr gII;irrðXÞ gII;irrðDpÞ 17 depend on the piston speed and are thereby to the engine rotational speed. The deficient regenerating is calculated as Three objective functions considered for the present study are the same as those considered byP Ahmadi et al [23] ¼ 1 ð Þ ðÞ gII;irrðXÞ 18 and these are: system pressure losses Dpi , the output þ ðXÞ 1 ððc1Þ ln kÞ gc power (Power) and the Stirling engine thermal efficiency ðgÞ. The number of decision variables is also same as those with considered by Ahmadi et al [23] and these are given as: engine’s rotation speed (nr); mean effective pressure (Pm); X ¼ yX1 þð1 yÞX2; ð19Þ stroke (s); temperature of heat source (TH); temperature of Optimal design of Stirling heat engine 1325 heat sink (TL); temperature difference between heat source The output in TLBO algorithm is considered in terms of and working fluid (DTL); temperature difference between results or grades of the learners which depend on the quality heat sink and working fluid (DTH); number of gauzes of the of teacher. So, teacher is usually considered as a highly matrix (NR); regenerator’s length (L); piston diameter (DC); learned person who trains learners so that they can have and regenerator diameter (DR). better results in terms of their marks or grades. Moreover, The constraints are nothing but the ranges of variables learners also learn from the interaction among themselves and are given by which also helps in improving their results. TLBO is a population-based method and a group of learners is con- : ð Þ 1200 nr 3000 rpm 30 sidered as population and different design variables are considered as different subjects offered to the learners and 0:69 P 6:89 MPa: ð31Þ m learners’ result is analogous to the ‘‘fitness’’ value of the 0:06 S 0:1m: ð32Þ optimization problem. In the entire population, the best solution is considered as the teacher. The flowchart of 800 TH 1300 K: ð33Þ TLBO algorithm is shown in figure 4 [32]. The working of TLBO is divided into two parts, ‘‘teacher phase’’ and : ð Þ 288 TL 360 K 34 ‘‘learner phase’’. For more details on TLBO algorithm and its code, one may refer to https://sites.google.com/site/ 64:2 DT 237:6K: ð35Þ H tlborao and Rao [28]. 5 DT 25 K: ð36Þ Like many population-based optimization algorithms, the TLBO algorithm also requires certain population size and 250 NR 400: ð37Þ number of iterations (but it does not have any algorithm- specific parameters to tune and hence, the burden on the 6 10 3 L 73 10 3: ð38Þ designer to tune the algorithm-specific parameters is relieved). Improper selection of population size and number : : : ð Þ 0 05 Dc 0 14 m 39 of iterations may also lead to sub-optimal solutions. The population size and number of iterations affect the perfor- 0:02 D 0:06 m: ð40Þ R mance of the algorithm and hence to get optimal values of The values of the constraints are in a way somewhat the design variables and thereby the objective functions, arbitrary, in the sense that other aspects which need to be one has to execute the optimization algorithm with different considered like those from fabrication viewpoint are not population sizes and number of iterations. In this paper, given due importance. Especially, the upper value of tem- after conducting various trials, it has been observed that a perature (TH = 1300 K) puts a severe constraint on the population size of 50 and 40 number of iterations have selection of material of construction for the hot end of the given the optimal results of the design variables (and engine (heat absorbing part/heat receiver). The designer has thereby the optimum values of the objective functions). The to be careful while deciding the values of the constraints. number of function of evaluations required by TLBO However, the constraints represented by Eqs. (30–40) were algorithm to achieve optimum solution is equal to 4000 (i.e. considered by Ahmadi et al [23] and hence, the same 2 9 40 9 50 as the evaluations are done in both the tea- constraints are considered in the present work for making cher’s and learner’s phases). The same optimum results are fair comparison. obtained for increased number of function evaluations also. The next section presents the details of the TLBO However, less number of function evaluations reduces the algorithm. computational time and effort. In this paper, the TLBO algorithm is applied for the design optimization of Stirling heat engine. The three 3. Teaching–learning-based optimization objectives are combined into scalar objective via a weight algorithm vector. For this problem, the combined objective function (Z) is defined as follows: Teaching–learning-based optimization (TLBO) algorithm is a teaching–learning process inspired algorithm proposed Maximize ¼ = þ = = ; by Rao et al [29] based on the effect of influence of a Z w1 Z1 Z1max w2 Z2 Z2max w3 Z3 Z3min teacher on the output of learners in a class. The algorithm ð41Þ mimics the teaching–learning ability of teachers and learners in a classroom. Teacher and learners are the two where Z1, Z2 and Z3 are the objective functions of power, vital components of the algorithm and describe two basic efficiency and pressure loss respectively and w1,w2 and w3 modes of the learning, through teacher (known as teacher’s are the weights given to the objective functions 1, 2 and 3 phase) and interacting with the other learners (known as respectively. The designer can give any weight to the learner’s phase). objective functions. But the condition is that summation of 1326 R V Rao et al

Decide number of students (population size), subjects (decision variables), termination criterion

Calculate the mean of each design variable

Identify the best solution (teacher)

Modify solution based on best solution Teacher X'j,k,i = Xj,k,i + ri(Xj,kbest,i – Mj,i) phase

No Is new solution better Yes Reject Accept than existing?

Keep previous solution Select two solutions randomly X'total-P,i and X'total-Q,i

No Is X'total-P,i better than Yes X'total-Q,i

X''j,P,i = X'j,P,i+ri(X'j,Q,i –X'j,P,i) X''j,P,i = X'j,P,i + ri(X'j,P,i –X'j,Q,i) Student Phase

NoIs new solution better Yes Reject Accept than existing?

Keep previous solution

Is termination No criterion satisfied?

Yes Final value of solution

Figure 4. Flowchart of TLBO algorithm [32].

all weight values should be equal to 1. Equal weights are piston diameter is from 50 to 140 mm. The specifications of considered in this problem for demonstration, i.e. the Stirling engine are considered as given below [23]: w1 ¼ w2 ¼ w3 ¼ 1=3. The values of Z1max, Z2max and Z3min ÀÁ N ¼ 8; t ¼ 3:249 105 m2=s ; c ¼ 1:667; C ¼ 3115:6 are the maximum power, maximum efficiency and mini- ÀÁ v mum pressure loss respectively, which are obtained by Jkg1 K1 ; P ¼ 0:71; m ¼ 0:001135ðÞ kg ; ÀÁr g ÀÁ individually considering the respective single objective q ¼ 8030 kg m 3 ; C ¼ 5193 J kg 1 K 1 ; functions (ignoring the other objective functions) and st ÀÁpg 1 1 5 solving the considered individual objective function under CR ¼ 502:48 J kg K ; k ¼ 1:2; b ¼ 6:88 10 ; the same constraints. The values of Z1max, Z2max and Z3min f ¼ 0:556; d ¼ 4 10 5 ðÞm : are shown in table 1 and these are obtained by applying the TLBO algorithm considering the individual objective Ahmadi et al [23] used NSAGA-II for achieving Pareto functions under the same constraints. optimal frontier of three objective functions, output power, thermal efficiency of Stirling engines and system pressure loss. Each point on the Pareto frontier represents the opti- 4. Results mal solution of the problem. The decision-making methods were used to find out the final optimal solution from the The case study considered in this paper is adopted from Pareto frontier which was gained through NSGA-II algo- Ahmadi et al [23]. The objective functions, constraints and rithm. Well-known decision-making procedures including the ranges of variables considered in the present work are the fuzzy Bellman–Zadeh, LINMAP and TOPSIS methods same as those considered by Ahmadi et al [23]. The range were used by Ahmadi et al [23] to select the best solution of speed is 1200 rpm to 3000 rpm and the range of tem- from among many Pareto optimal solutions given by the perature of heat source is 800–1300 K and the range of NSGA-II algorithm. The Bellman–Zadeh procedure Optimal design of Stirling heat engine 1327

Table 1. Optimization results obtained by TLBO algorithm for Stirling heat engine.

Optimum value of the objective function Optimum value of the objective function when the combined Objective functions when attempted individually objective function is attempted Maximization of power 15.62 6.89 (kW) Maximization of thermal 22.01 15.07 efficiency (%) Minimization of total 11.65 18.70 power loss (kPa)

implements the fuzzy non-dimensionalization, while the [23] using TOPSIS and LINMAP methods are exactly other methods (i.e. LINMAP and TOPSIS) used Euclidian same. The results of optimization using TLBO algorithm non-dimensionalization. have given the optimum values of engine’s rotation speed The proposed TLBO algorithm is now applied to the multi- (nr) as 1800 rpm, which is lower than the speeds sug- objective optimization (containing three objectives) of power gested by other methods. The optimum value of mean Stirling heat engine problem. The number of function eval- effective pressure (Pm) is 2585.83 kPa which is higher uations required by TLBO algorithm is equal to 4000. The than that given by other methods except the experimental output power and thermal efficiency of the Stirling engine are results. The optimum value of stroke (s) is 60 mm which maximized along with the minimization of the pressure los- is almost same as that given by other methods except the ses of the Stirling engine simultaneously. Table 1 shows the experimental results. The optimum value of temperature optimum values of the three objective functions when they of heat source (TH) is 1200 K which is higher than those were attempted individually (i.e. considering only one given by other methods. As compared to the results given objective at a time ignoring the other objectives but consid- by other methods and experimental results, the optimum ering the same set of constraints) and combinedly (i.e. con- values obtained by TLBO algorithm for the temperature of sidering the combined objective function and solving the heat sink (TL) and temperature difference between heat same for the same set of constraints). It may be observed that sink and working fluid (DTL), number of gauzes of the optimum values of the individual objective functions are matrix (NR) and piston diameter (DC) are comparatively better than those when attempted simultaneously. It is due to higher. the reason that an increase or decrease in the value of one The optimum value of the temperature difference objective may lead to decrease or increase in the value of between heat source and working fluid (DTH) is lower than another objective. This happens in multi-objective opti- those given by the other methods. The optimum value of mization problems and the values of the three objective regenerator’s length (L) is 71 mm which is almost similar to functions obtained by solving the combined objective func- those given by the other methods, but much higher than that tion represent the compromise solutions, i.e. these are the given by the experimental results of Petrescu et al [17]. The solutions obtained by satisfying all the constraints consid- optimum value of regenerator diameter (DR) is 60 mm and ering all the objectives simultaneously. this is almost similar to those given by the other methods, Table 2 shows the comparison of optimal results of but lower than that given by Petrescu et al [17]. The opti- power Stirling heat engine obtained by using TLBO algo- mum values suggested by the application of TLBO algo- rithm, TOPSIS method, LINMAP method, fuzzy Bellman– rithm are well within the given ranges of the variables Zadeh method and experimental results of Petrescu et al considered by Ahmadi et al [23]. Thus, the values of power, [17]. Petrescu et al [17] compared the analytical results efficiency and pressure losses obtained by the TLBO with experimental results. Two objectives namely; output algorithm are meaningful as the obtained values of the power and thermal efficiency of Stirling engines were design variables satisfy the constraints and these constraints considered. However, Petrescu et al [17] had not consid- are nothing but the ranges of variables. ered the system pressure loss in their work. They showed It is observed from table 2 that the optimum values of the the accurate analysis of the performance in terms of power three objective functions suggested by the TLBO algorithm and efficiency for Stirling engines over a range of condi- by solving the multi-objective optimization problem (by tions. In table 2 the values of DTL and DTH are missing converting it into a combined objective function) are better corresponding to experimental results because these values than those suggested by the other methods and the experi- are related to system pressure loss and were not considered mental approach. For the maximum power, the results by Petrescu et al [17]. obtained by using the TLBO algorithm are 74.05, 18.02 and It may be observed from table 2 that the optimum 42.54% better than those given by the TOPSIS and LIN- values of the design variables suggested by Ahmadi et al MAP methods, fuzzy Bellman–Zadeh method and 1328 R V Rao et al

Table 2. Optimization results for Stirling heat engine.

Objective functions Decision variables Z1 Z2 PZ3 nr Pm TH DTH DTL Dc DR Power Dpi (rpm) (kPa) S (mm) (K) TL (K) (K) (K) NR L (mm) (mm) (mm) (kW) g (%) (kPa) TOPSIS [23] 2120 2550.3 60.5 989.6 298.4 74.4 11.8 339 70 101.6 59.5 6.076 14.56 19.69 LINMAP [23] 2120 2550.3 60.5 989.6 298.4 74.4 11.8 339 70 101.6 59.5 6.076 14.56 19.69 Fuzzy Bellman– 2056 2437 60.5 989.3 299.5 76.4 12.1 338 71.8 106.1 58.9 5.84 14.51 18.82 Zadeh [23] Experimental 2500 4130 100 977 288 - - 308 22.6 22.6 69.9 3.96 12.7 - results [17] TLBO (present 1800 2585.83 60 1200 300.87 64.2 13.5 350 71 122.2 60 6.89 15.07 18.70 work)

The numbers in bold indicate better values. experimental results respectively. In maximization of Figure 7 shows the effects of the designP variables on thermal efficiency ðgÞ, the results obtained by using the the three objectives, i.e. P, g and DPi of Stirling TLBO algorithm are 3.5, 3.85 and 18.66% better than the engine. It may be observed that increasing mean effec- TOPSIS and LINMAP methods, fuzzy Bellman–Zadeh tiveP pressure increases P and g and slightly decreases method and experimental result respectively.P Similarly, in DPi (figureP7(a)). Increasing engine speed decreases g minimization of total power loss ð DpiÞ, the results and increases DPi and P first increases then decreases obtained by using the TLBO algorithm are 5.02 and 0.679% and again increasesP with increasing engine speed (fig- better than the TOPSIS and LINMAP methods and fuzzy ure 7(b)). The DPi increases with the increase in Bellman–Zadeh method respectively. stroke,however,theg increases only slightly and the P Figure 5 shows the convergence of the TLBO algorithm remains almost constant withP increase in the stroke for single objective optimization of Stirling heat engine (figure 7(c)). The P, g and the DPi increase with the problem. It shows the convergence rate of the TLBO temperature of heat source (figure 7(d)) and decrease algorithmP for the power (P), efficiency (g) and pressure with the temperatureP of heat sink (figure 7(e)). The val- losses ð DPiÞ when considering the individual objectives ues of g and DPi almost remain constant and P and the convergence takes place after 15th, 19th and 12th decreases with increasing temperature difference of heat iteration of the TLBO algorithm, respectively. Figure 6 source and working fluid (figure 7(f)). The valuesP of g shows the convergence rate of the TLBO algorithm for the and P remain constant and the value of the DPi combined objective function and the convergence takes slightly decreases with increasing temperature difference place after 7th iteration of TLBO algorithm. It may be of heat sink and working fluid (figureP 7(g)). Figures 7(h– observed that the TLBO algorithm takes less number of k) show that the values of P and DPi remain almost iterations for convergence. constant and g increases with increasing number of gauzes of the matrix, regenerator’s length, piston diam- eter and regenerator diameter. It is observed that in the case of power P, the most 30 important design variables are mean effective pressure (Pm), engine speed (nr), temperature of heat source (TH) 25 and temperature difference of heat source and working fluid (DTH). Similarly, in the case of efficiency g, the 20 important design variables are mean effective pressure p

, Δ 15 Δp (Pm), engine speed (nr), temperature of heat source (TH),

P, η P temperature of heat sink (TL), temperature difference of 10 η heat source and working fluid (DTH), regenerator’s length (L) and regenerator diameter (DR). In the case of pressure 5 P losses DPi, the important design variables are engine 0 speed (nr), stroke (s), temperature of heat sink (TH) and 0 5 10 15 20 temperature of heat sink (TL). It is observed that the No. of iterations TLBO algorithm gives better results than the TOPSIS, Figure 5. Convergence graphs of TLBO algorithm for power, LINMAP, fuzzy Bellman–Zadeh method and the experi- thermal efficiency and pressure losses. mental results. Optimal design of Stirling heat engine 1329

1.4 (a) 40 (b) 30 35 25 1.2 30 20 p 25 p

1 , Δ 20 P 15 P , Δ η 15 η 10 η P,

10 P, η 0.8 Δp 5 Δp 5 0 0 TLBO 0.6 05 1100 2100 3100 Mean effective pressure (P ) (MPa) Engine's rotation speed (n ) 0.4 m r

Combined objective function 0.2 (c) 30 (d) 25 0 25 20 0 5 10 15 20 20 p

p 15

No. of iterations , Δ P

, Δ 15 P 10 η 10 η P, η Figure 6. Convergence graph of the TLBO algorithm for the P, η 5 Δp 5 Δp combined objective function. 0 0 700 900 1100 1300 0.055 0.075 0.095 5. Conclusions Stroke (s) Temperature of heat source (TH) (e) (f) 20 Multi-objective optimization is a very important research 25 area in engineering studies, because of real-world design 20 15 p

15 p 10 P problems that require the optimization of a group of , Δ P , Δ η 10 objectives. Multiple, often conflicting, objectives arise η η P, 5

5 P, η naturally in most real-world optimization scenarios. Adding Δp Δp 0 0 more than one objective to an optimization problem adds 280 330 380 50 150 250 complexity. In this paper, three different objective func- Temperature of heat sink (TL) ΔTH tions are considered for optimization of the design of the Stirling heat engine. The three different objective functions (g) (h) considered are maximum power, maximum thermal effi- 20 20 15 15

ciency and minimum pressure loss. The experimental p p Δ , results and the mathematical models for the design of , Δ 10 P 10 P η η η P, Stirling engine which were earlier attempted by other P, η 5 5 researchers are considered in the present work. For opti- Δp Δp 0 0 mization of design of Stirling heat engine, a recently 0102030 225 275 325 375 425 Number of gauzes of the matrix (N ) developed advanced optimization algorithm namely, ΔTL R teaching–learning-based optimization (TLBO) algorithm, is considered. As the TLBO is an algorithm-specific param- (i) (j) 25 eter-less algorithm, the performance of this technique is not 20 20 affected by the algorithm parameters. It makes this algo- 15 p p 15

, Δ P rithm applicable to real life optimization problems, easily , Δ 10 P 10 η η P, η and effectively. P, η 5 5 Δp Δp The TLBO algorithm has shown its ability in solving the 0 0 0.045 0.095 0.145 individual objective functions as well as multi-objective 0 255075 optimization problem by using a combined objective Regenerator's length (l) Piston diameter (DC) function. The convergence behaviour of the TLBO algo- rithm to a near global solution is observed in less number of (k) 20 iterations and the algorithm more effective than the deci- 15 sion making methods of TOPSIS, LINMAP and fuzzy p

Bellman–Zadeh that have used the results of NSGA-II , Δ 10 P η algorithm to select the best optimum values of the design P, η 5 Δp variables and the objective functions. The optimum results 0 suggested by the TLBO algorithm are also better than those 0.015 0.035 0.055 0.075 given by the experimental results. The TLBO algorithm can Regenerator diameter (DR) be quite promising and a reliable choice for the optimiza- tion of design of the Stirling heat engine. The output of this Figure 7. (a–k) Effects of design variables on power, efficiency and pressure losses of Stirling engine. work may help the designer to make effective use TLBO 1330 R V Rao et al algorithm for the design optimization of other heat engines. t of the working gas (m2 s-1) The application of the TLBO algorithm may be extended to solve multi-objective optimization problems of different thermal engineering systems and devices. Subscripts

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