Iran. J. Chem. Chem. Eng. Research Article Vol. 39, No. 1, 2020 Archive of SID
Thermodynamic Assessment and Optimization of Performance of Irreversible Atkinson Cycle
Ahmadi, Mohammad Hossein*+ Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, I.R. IRAN
Pourkiaei, Seyed Mohsen; Ghazvini, Mahyar, Pourfayaz, Fathollah*+ Department of Renewable Energies and Environmental, Faculty of New Sciences and Technologies, University of Tehran, Tehran, I.R. IRAN
ABSTRACT: Although various investigations of Atkinson cycle have been carried out, distinct output power and thermal efficiencies of the engine have been achieved. In this regard, thermal efficiency, Ecological Coefficient of Performance (ECOP), and Ecological function (ECF) are optimized with the help of NSGA-II method and thermodynamic study. The Pareto optimal frontier which provides an ultimate optimum solution is chosen utilizing various decision- making approaches, containing fuzzy Bellman-Zadeh, LINMAP, and TOPSIS. With the help of the results, interpreting the performances of Atkinson cycles and their optimization is enhanced. Error analysis has also been performed for verification of optimization and determining the deviation in the study.
KEYWORDS: Atkinson cycle; Thermodynamic analysis; Power; Ecological Coefficient of Performance; Thermal efficiency; Entropy generation.
INTRODUCTION The main purpose of designing the cycle is to present of combining with an increment stroke and/or decreased the performance of the system by the mean of input volume of the combustion chamber, causes the ratio of power. In the Atkinson cycle, the intake, compression, expansion to outstrip the ratio of compression, throughout power, and exhaust strokes of the four-step take place the time of maintaining a regular compression pressure. by one piston sweep. In the Atkinson cycle, the compression So, it is favorable for enhanced cost efficiency, since ratio is less than the expansion ratio, due to the linkage, in a spark-ignition engine, the octane rating ratio limits which results in higher efficiency related to the engines the compression. So, a longer power stroke is the result utilizing the alternative Otto cycle. Also, four-stroke of a high expansion ratio, provides higher expansion ratios, engines are referred to as the Atkinson cycle. In these followed by the reduction of heat associated with arrangements, the intake step takes longer time to fill the waste in the exhaust [1]. the intake manifold with fresh air. Its result is the reduction A growing number of thermodynamic investigations of efficient compression ratio, and on the condition has focused on determining the limits of performance * To whom correspondence should be addressed. + E-mail: [email protected] , [email protected], [email protected] 1021 -9986/2020/1/267-280 14/$/6.04
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in thermal systems as well as optimizing thermodynamic interchanges in the objective function area. With respect cycles and processes containing finite-rate, finite-time, to this, multi-objective optimization of various processes and finite-size constraints [2-7]. Also, The performance has been examined in plenty of researches [43-83]. analysis for internal combustion engine cycles by using In the present research, an irreversible Atkinson cycle finite time thermodynamics were also performed by other is optimized in line with performance improvement papers [8-12]. Recently, abundant performance of the system. In this scenario, for maximizing the thermal evaluations of the Atkinson heat engine have been carried efficiency, ECOP and ECF parameters, a multi-objective out founded on irreversible, endo-reversible, and optimization solution is applied. For the sake of reversible cycle arrangements with utilizing various evaluating eventual answers’ precision in different objective functions including power output, the generated decision-making approaches, error analysis is carried out. power and performance, etc [13-25]. The engine sizes’ effect associated with the investment cost is not taken Cycle model and analysis into account in the analyses of performance contributed Fig. 1 shows an air standard Atkinson cycle diagram. to the mentioned optimization criteria. In this way, The working fluid of the most cycle models Sahin et al. [26,27] presented a novel optimization is considered to be as an ideal gas with the characteristic criteria named the maximum power density examination of fixed specific heats. However, this presumption could for the sake of including the engine size’s effects. be authentic only on the condition of small temperature Optimum operational states of reversible [26] and difference. So, in practical cycle, in which there is large irreversible [27] non-regenerative Joule-Brayton cycle temperature difference, the mentioned presumption have been studied by utilizing the maximum power cannot be implemented. As ref. [84] indicated, under density (MPD) criterion. In the study, the power density the condition of temperature range between 200-1000 K, has been maximized and model elements at MPD states the specific heat capacity with fixed pressure is as follows: have been found. These conditions lead to more efficient 4 and smaller setups. Many investigations have been performed C(..TP 3 56839 6 788729 10 (1) the MPD method to various models of heat engines 1.T.T 5537 106 2 3 29937 10 12 3 [28-38]. 15 4 466.T)R 395 10 g Added to this, Chen et al. [39] and Wang and Hou [40]
performed the technique of MPD to the Atkinson For temperatures between 1000 and 6000 K, the Cp cycle. They presented that the MPD efficiency is higher is calculated as: than the MP efficiency. Also, Wang and Hou [40] 4 investigated an Atkinson cycle linked to a variable C(..TP 3 08793 12 4597 10 (2) temperature heat source at MP and MPD. The analysis 0.T.T 42372 106 2 67 4775 10 12 3 revealed that an engine which is designed for MPD 15 4 3.T)R 97077 10 g conditions, has smaller size than a MP design based engine. For temperatures between 200 and 600k, Eqs. (1) For the sake of unraveling enigma of this general and (2) can be used which the rage is too wide category, during the whole of the mid-eighties, for the temperature range (300-3500 K) of pragmatic engine. Evolutionary Algorithms (EA) were basically utilized Thus, for describing the specific heat model a single [41]. Determining a cluster of answers, each of which equation has been utilized. The presumption associated implements the objectives on the condition of a gratifying with this is air must be an ideal gas. degree without being overshadowed by any other answers 11 2 7 1. 5 C(.T.TP 2 506 10 1 454 10 (3) is a pragmatic answer to a multi-objective puzzle [42]. 7 5 0. 5 Issues contributed to multi-objective optimization 0.T.T. 4246 10 3 162 10 1 3303 regularly perform as an achievably innumerable group 1.T.T.T) 512 104 1. 5 3 063 10 5 2 2 212 10 7 3 of answers which is called Pareto frontier, where investigated vectors indicate primary feasible CCRV P g (4)
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T TT 43 (9) e TT43s 3 Cv c And e can present the internal irreversibility 2S 2 of the procedures. Due to the dependency of CP and CV 4 4S on temperature, adiabatic element k C C varies Cp PV 1 by the changes of temperature. Thus, Eq. (10) is not valid for adiabatic stages. Nonetheless, regarding to refs. [90-101], a proper estimation for reversible adiabatic procedure
S with k can be carried out, i.e. this procedure can be split to many infinitesimally small stages that each k is Fig. 1: Schematic of an Atkinson cycle diagram. considered fixed. For instance, between states i and j, every reversible adiabatic process could be considered as containing plentiful infinitesimally small processes with According to Eq (4) the C at fixed volume is v fixed k. On the condition of volume dV of the working determined as follows: fluid takes place, and an infinitesimally small change 11 2 7 1. 5 in temperature dT, for any of these processes, the equation C(.T.TV 2 506 10 1 454 10 (5) for reversible adiabatic process with variable k 0.T.T 4246 107 3 162 10 5 0. 5 can be presented as follows. 1..T 0433 1 512 104 1. 5 TVkk11 (T dT)(V dV) (10) 3.T.T) 063 105 2 2 212 10 7 3 The heat added in constant-volume process The received heat by the working fluid in through is calculated as follows: 2 → 3 is: Qin C(T V j T) i TS i j TClnTT V j i . So T3 12 3 Qin M C V dT M(8 . 353 10 T 5 (6) T one has T (T T ) ln(j ) , where T is the equivalent T2 ji Ti 8 2.. 5 7 2 5 1 5 .T.T.T816 10 2 123 10 2 108 10 temperature of the procedure. When CV is the subordinate 4 0. 5 1.T.T 0433 3 024 10 of temperature, the CV (T) could be considered as mean 3.T.T) 063 105 1 1 106 2 T3 specific heat of fixed volume. T2 From eq. (10), one gets The rejected heat is calculated as follows: T V C ln(j ) R ln(i ) (11) T4 Vg 12 3 TVij Qout M C P dT M(8 . 353 10 T (7) T1 The temperature in CV calculation is logarithmic 8 2. 5 7 2 5.T.T 816 10 2 123 10 Tj 5 1.. 5 4 0 5 T (Tji T ) ln( ) . Also, the compression ratio 2.T.T.T 108 10 1 3303 3 024 10 Ti 3.T.T) 063 105 1 1 106 2 T4 is determined as: T1 V 1 (12) The adiabatic compression and expansion V2 performance of the cycle in 1 → 2 and 3→4 procedure Thus, equations contributed to reversible adiabatic are determined as follows [85-91]: processes 1 → 2S and 3 → 4S are: TT 21s (8) T c C ln(2s ) R ln (13) TT21 Vg T1
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T4s T1 P CV ln( ) R g ln( ) R g ln (14) th (22) TT34s QQin leak
Heat transfer losses are not considered for an ideal T2S can be achieved by Eq. (13) while Rg, T1, T3, c,
Atkinson design. Nonetheless, heat transfer irreversibility and e are provided. Next, substituting T2S into Eq. (8)
between cylinder wall and working fluid cannot be leads to measure T2, T4S with the help of Eq. (14).
neglected in a real Atkinson cycle. It is presumed The last step is to determine T4 by substituting T4S into Eq. (9). that mean temperature of the working fluid and the cylinder The power and efficiency can be obtained by Substituting
wall mutually and the heat transfer via the cylinder wall T2 and T4 into Eqs. (21) and (22). (i. e. the heat leakage loss) are relative and the wall Entropy generation, ECOP and ECF (kW) of
temperature is equal to T0 (K). If the generated heat the Atkinson cycle can be determined as following as:
by combustion procedure per second be A1 (kW) and Qout Qin the heat leakage factor of the cylinder wall be B1 [kJ/kg.K], Sgen (23) TT the applied heat flow rate is calculated as follows [102-105]: LH P (T2 T 32 T 0 ) ECOP (24) QABin 11 (15) 2 TS0 gen
Eq. (15), shows that Qin contains two terms, the first ECF P T0 Sgen (25) part is A , and the other one is the heat loss per second, 1 which can be defined as: Multicriteria optimization
Qleak B(T2 T 3 2 T 0 ) (16) For optimization purposes, Genetic Algorithms (GA) were proposed by Holland (1960) [44]. The evolution where B = B1/2. generally begins a population which each individuals In order to consider the friction loss [92] we have: generated accidentally. The fitness rate associated with dx each individual of the population is investigated. f (17) dt Many random individuals are developed to make a new population. At the next step, the new population Where [Ns/m] is a friction parameter and x is is applied to the following iteration of the GA. Regularly,
the piston displacement. P, which represents the lost on the condition of achieving an favorable fitness level of power is defined as: the population or creating the highest value of generations, the GA stops. Privious works have been presented more dW 2 details of GA [42,46]. P (18) dt Furthermore, during the past years with persistent The total displacement of the piston in each cycle investigations on multipart mathematical puzzles and of four-step engines is calculated as: on pragmatic engineering problems, MOEAs were extracted. Also, throughout the examinations, it was concluded that
44L (x12 x ) (19) the complexity of conventional approaches can be excluded [42,46]. Fig. 2 depicts the construction The mean velocity of the piston is as follows contributed to the MOEA applied in this paper [44,46]. (N cycles): It is worth stating that the actual amounts of decision elements were employed instead of their binary coded. 4LN (20) Three objective functions are utilized in this Then, the power output and the cycle performance optimization: thermal efficiency, ECOP and ECF, efficiency are determined as follows: described by Eqs. (22), (24) and (25), respectively. Also, five decision variables are considered: PQQP (21) in out temperatures of state points 1 and 3, the Expansion
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Fig. 2: Algorithm steps applied in the study [44,46].
efficiency (e), the compression efficiency (c) and the Power output (P) reduced by increasing the T1 the compression ratio . at different amounts of the T3. Although the decision variables might be different According to Fig. 4a, the ECF of the system reduced in the optimizing plan, but they typically need to be fited significantly with augmenting the T1 at different rates of in a sensible range. Thus, the objective functions the Expansion efficiency (e). According to Fig. 4b, are determined by reletive succeeding limits: ECOP decreased considerably with T1 at different rates of the Expansion efficiency (e). As depicted in Fig. 4c, 325 T 400 K (26) 1 the thermal efficiency (th) reduced considerably with rising of T at different rates of the Expansion efficiency ( ). 1300 T 19 00K (27) 1 e 3 As it is seen in Fig. 4d, the Power output (P) decreased
significantly with rising of T1 at distinct rates of 0.85 e 0 . 97 (28) the Expansion efficiency (e).
0. 85 c 0 . 97 (29) As depicted in Fig. 5a, the ECF reduced by increasing the T1 at different rates of the compression efficiency
6 12 (30) (c). As it is illustrated in Fig. 5b, the ECOP reduced by enhancing T1 at different rates of the compression efficiency (c). As shown in Fig. 5c, the thermal
RESULTS AND DISCUSSION efficiency (th) reduced with increasing the T1 at different
In this section, the sensitivity of the objective rates of the compression efficiency (c). functions for decision parameters is investigated. As shown in Fig. 5d, the Power output (P) reduced
Following Ref. [106], the following parameters are used with increasing the T1 at different rates of the −2 −2 here: x1 = 8×10 m, x2 = 1×10 m, = 12.9 (Ns/m), compression efficiency (c). TH = 2200 K, TL = 300 K , T0 = 300 K, N = 30 , γ = 8.5 , In this study, the thermal efficiency, ECOP and ECF −3 M = 4.553 × 10 kg/s, B=0.2 (kj/kgK), e = 0.97, of the Atkinson cycle are maximized concurrently
and c= 0.97. utilizing multi-objective optimization based on the According to Fig. 3a, the ECF reduced with NSGA-II approach. The objective functions are
augmenting the T1 at different values of T3. As illustrated illustrated by Eqs. (22), (24) and (25) and the limitations
in Fig. 3b, the ECOP decreased considerably with T1 by Eqs. (26) -(30).
at different values of the T3. According to Fig. 3c, The decision parameters of optimization are
the thermal efficiency (th) reduced by augmenting the T1 as follow: T1, T3, the Expansion efficiency (e),
at different values of the T3). According to Fig. 3d, the compression efficiency (c) and the compression ratio .
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1.2 1.5
1.1
1 1
0.9
0.5 0.8
0.7 0
0.6
ECF ECF (kW) 0.5 ECF (kW) -0.5
0.4 -1 0.3
0.2 -1.5 320 330 340 350 360 370 380 390 400 320 330 340 350 360 370 380 390 400
T1 (K) T1 (K)
2.3 2.4
2.2 2.2
2.1 2
2 1.8
1.9 1.6
1.8 1.4
ECOP 1.7 ECOP 1.2
1.6 1
1.5 0.8
1.4 0.6
1.3 0.4 320 330 340 350 360 370 380 390 400 320 330 340 350 360 370 380 390 400
T1 (K) T1 (K)
0.34 0.35
0.32 0.3 0.3
0.28 0.25
th th 0.26
0.24 0.2
0.22 0.15 0.2
0.18 0.1 320 330 340 350 360 370 380 390 400 320 330 340 350 360 370 380 390 400 T1 (K) T1 (K)
2.2 2.2
2 2
1.8 1.8
1.6 1.6
1.4 1.4
P (kW) P (kW) 1.2 1.2 1 1 0.8 0.8 320 330 340 350 360 370 380 390 400 320 330 340 350 360 370 380 390 400 T1 (K) T1 (K)
Fig. 4: Effects of the T1 and the expansion efficiency (e)
Fig. 3: Effects of T1T3 on the (a) ECF (b) ECOP, (c) thermal on the (a) ECF (b) the ECOP, (c) thermal efficiency,
efficiency , (d) Power output in e = 0.97, c = 0.97. (d) Power output in T3 = 1900 K, c = 0.97.
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1.3
1.2 1.23
1.1 1.22
1 1.21
1.2 0.9 1.19 0.8 1.18 0.7 ECF ECF (kW) 1.17
ECF ECF (kW) 0.6 1.16
0.5 2.8 2.7 0.326 0.4 2.6 0.324 2.5 0.32 0.322 2.4 0.318 2.3 0.316 320 330 340 350 360 370 380 390 400 2.20.314 ECOP T1 (K) th 2.3 Fig. 6: Pareto optimum frontier in objective space. 2.2
2.1
2 The Pareto optimal frontier of objective functions
1.9 (the thermal efficiency, ECOP and ECF) is depicted in Fig. 6.
1.8 Selected points with different decision-making
1.7
ECOP approaches are depicted, as well. 1.6 Table 1 outlines and compares the optimal results 1.5
1.4 associated with decision elements and objective functions
1.3 utilizing LINMAP, TOPSIS, and Bellman-Zadeh 320 330 340 350 360 370 380 390 400 decision-making approaches. Analyses results are presented T1 (K) in Table 2, as well. The achieved results 0.34 are favorable and it is predicted that the present investigation improves interpreting the optimum model of the Atkinson 0.32 cycle. Based on the Mean Absolute Percent Error (MAPE) 0.3 approach, an error analysis was performed to define 0.28
th the average error for thermal efficiency of solutions gathered 0.26 by decision making approaches. Along with results, these
0.24 errors are 0.071%, 0.079% and 0.138% for TOPSIS,
0.22 LINMAP and FUZZY, respectively. This study
0.2 demonstrated that the average error for the ECOP 320 330 340 350 360 370 380 390 400 are 0.163%, 0.122% and 0.237% for TOPSIS, LINMAP T1 (K) and FUZZY, respectively. This study demonstrated that 2.1 the average error for the ECF of solutions are 0.237%,
2 0.234% and 0.169% for TOPSIS, LINMAP and FUZZY, 1.9 respectively. 1.8
1.7
1.6 CONCLUSIONS
P (kW) 1.5 A thermodynamic optimization procedure
1.4 has been employed for determining the thermal efficiency,
1.3 ECOP and ECF of the Atkinson cycle. The Expansion
320 330 340 350 360 370 380 390 400 efficiency (e), the compression efficiency (c), T1,T3, and the
T1 (K) compression ratio are studied utilizing the NSGA-II
method. For the sake of designing and evaluating Fig. 5: Effects of the T1 and the compression efficiency (c) the performance and robustness of Atkinson cycle, the results on the (a) ECF (b) ECOP, (c) thermal efficiency , (d) Power can be implemented. By utilizing decision making output in T3 = 1900 K, e = 0.97.
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Table 1: Decision making results of this study.
Decision variables Objectives Decision Making Method T1 (K) T3 (K) e c th ECOP ECF (kW)
TOPSIS 325/003 1900 0/970 0/970 11/888 0/314 2/696 1/217
LINMAP 325/003 1900 0/970 0/970 11/813 0/315 2/688 1/217
Fuzzy 325/004 1900 0/970 0/970 10/348 0/320 2/520 1/213
Table 2: Error analysis based on MAPE for this study.
Decision Making Method TOPSIS LINMAP Fuzzy
Objectives th ECOP ECF th ECOP ECF th ECOP ECF
Max Error % 0.141 0.310 0.464 0.156 0.251 0.466 0.234 0.240 0.380
Average Error % 0.071 0.163 0.237 0.079 0.122 0.234 0.138 0.237 0.169
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