Finite time exergy analysis and multi-objective ecological optimization of a regenerative Brayton cycle considering the impact of flow rate variations

Energy Conversion and Management 103 (2015) 790–800

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Finite time exergy analysis and multi-objective ecological optimization of a regenerative Brayton cycle considering the impact of flow rate variations Mohammad Mahdi Naserian, Said Farahat ⇑, Faramarz Sarhaddi Department of Mechanical Engineering, University of Sistan and Baluchestan, Zahedan, Iran

a r t i c l e

i n f o

Article history: Received 24 April 2015 Accepted 7 July 2015 Available online 18 July 2015 Keywords: Finite time Finite size Power Ecological function Regenerative Brayton cycle Exergy Optimization

a b s t r a c t In this study, the optimal performance of a regenerative Brayton cycle is sought through power and then ecological function maximization using finite-time thermodynamic concept and finite-size components. Multi-objective optimization is used for maximizing the ecological function. Optimizations are performed using genetic algorithm. In order to take into account the finite-time and finite-size concepts in current problem, a dimensionless mass-flow parameter is introduced deploying time variations. The variations of output power, total exergy destruction of the system, and decision variables for the optimum state (maximum ecological function state) are compared to the maximum power state using the dimensionless parameter. The modified ecological function in optimum state is obtained and plotted relating to the dimensionless mass-flow parameter. One can see that the modified ecological function study results in a better performance than that obtained with the maximum power state. Finally, the appropriate performance zone of the heat engine will be obtained. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Thermodynamic processes take place in finite-size components during finite time; therefore, Carnot heat efficiency is not of great importance as thermodynamic equilibrium is unreachable and irreversibility appears. Literature is rife with attempts made to improve analysis techniques to account for both internal and external irreversibility in heat engines. External irreversibility is due to temperature difference between fluid flows of the heat exchangers. The internal irreversibility is present inside the system’s boundaries and can be carried out by the working fluid. Curzon and Ahlborn [1] investigated the effect of external irreversibility on Carnot cycle’s output power and thermal efficiency. Thermal efficiency was expressed as in Eq. (1) at the maximum output power (T L and T H are temperatures at cold and hot heat exchangers, respectively). Systems with internal reversibility are known as endoreversible cycles. Bejan [2] applied endoreversible concept to Brayton cycle and concluded that the efficiency of the cycle is independent of thermal conductance distribution. However, the effect of internal irreversibility cannot be neglected on thermodynamic performance of heat engine. In a separate study, comparison was established between endoreversible ⇑ Corresponding author. Tel.: +98 9151412276. E-mail address: [email protected] (S. Farahat). http://dx.doi.org/10.1016/j.enconman.2015.07.020 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.

Carnot cycle and the same system with both internal and external irreversibility [3]. It was shown that internal irreversibility reduces power and efficiency. For finite-rate heat transfer and finite-capacity thermal reservoir conditions, efficiency at the maximum power output depends on thermal reservoir’s temperature and other system variables such as reservoir capacity and working fluid heat capacity [4].

gCA

sffiffiffiffiffiffi TL ¼1 TH

ð1Þ

Heat engines operate in finite time; therefore, the realistic study of their optimal performance (the highest efficiency at the maximum power) is feasible through the concept of finite-time thermodynamics [5]. This method was applied to the optimization of regenerative endoreversible Brayton cycle for finite thermal capacitance rates of the heat reservoirs [6]. In the undertaken research, application of regenerators led to decrease in the maximum power and thermal efficiency. Further analyses were performed on regenerative and irreversible models of Brayton heat engines [7,8]. All irreversibility sources associated with finite time heat transfer process were taken into account in irreversible model, along with variable temperature at heat reservoir. Optimization of real systems is confined to thermal performance and physical constraints. Bejan [9] established two

M.M. Naserian et al. / Energy Conversion and Management 103 (2015) 790–800

791

Nomenclature T _ W S_ E_ e cP P E_ _ m Q_ A U F R PR r NTU c

temperature (K) power (kW) entropy rate (kW/K) ecological function (kW) specific heat (kJ/kg K) pressure (bar) exergy rate (kW) mass flow rate (kg/s) heat transfer rate (kW) area (m2) overall heat transfer rate (kW/m2 K) dimensionless mass flow rate specific gas constant (J/kg K) pressure ratio thermal conductance ratio number of transfer unit capacity ratio

Greek symbols g efficiency e effectiveness Subscripts L low H high CA Curzon and Ahlborn g generation, gas 0 dead state me maximum ecological mp maximum power C Carnot W working fluid T total D destruction F fuel P product

optimization approaches based on these two elements: (i) improving thermal performance subject to physical size constraints (e.g. the minimum entropy generation) and (ii) physical size minimization subject to specified thermodynamic performance. He concluded that both approaches lead to the same physical configuration. Herrera et al. [10] used heat exchanger size and admissible pressure drop as design constraints for irreversible regenerative Brayton cycle. In the model, finite-time thermodynamics and optimization was used to determine the maximum power and minimum entropy generation, along with the global maximum net power. In various studies, thermal parameters used as criterion for thermodynamic optimization of heat engines are different. Power density and exergy density are two objective functions used in irreversible Brayton cycle with regeneration/cogeneration [11,12]. Dimensionless power density and thermal efficiency were optimized for Braysson cycles using NSGA algorithm and finite time thermodynamic analysis [13]. Angulo-Brown [14] proposed an ecological criterion as Eq. (2) for finite-time Carnot heat-engines, _ is the where TL is the temperature of the cold heat-reservoir, W power output and S_ g is the entropy-generation rate. Yan [15] discussed the results of Angulo’s study and suggested that the definition of ecological function as Eq. (3) is more reasonable. Thus it can be claimed that ecological function indicates the contrast between output power and exergy destruction resulting from entropy generation within the system and its surroundings, as well as the contrast between output power and thermal efficiency. The thermal efficiency at maximum ecological function was found as it is given in Eq. (4).

_  T L S_ g E_ e ¼ W

ð2Þ

_  T 0 S_ g E_ e ¼ W

ð3Þ

1 2

ð4Þ

gme  ðgC þ gmp Þ

Cheng and Chen applied the ecological optimization criterion given in Eq. (2) to optimize the power output of an endoreversible [16] and irreversible [17] closed JB cycle. The decision variables of the optimization problem were thermal conductance ratio and the adiabatic temperature ratio. They claimed that with the introduction of the ecological function, the second-law efficiency and

thermal efficiency are improved noticeably. Five different thermodynamic optimization criteria used for assessing heat engines, were compared by Açıkkalp [18]. According to the results, the ecological function criterion was defined as the most convenient optimization method. Huang et al. [19] carried out an ecological exergy optimization for an irreversible open cycle JB engine with a finite capacity heat source. They concluded that even though the work output was reduced, the ecological exergy study results in a better performance than that obtained with the maximum power output conditions. The effect of the regenerator effectiveness on the optimal performance for the maximum ecological function conditions was discussed by Ust et al. [20]. It was shown that the entropy-generation rate and thermal-efficiency advantages of the heat engine increase as the regenerator effectiveness increases. Chen [21] investigated the optimal ecological exergy performance of a generalized linear phenomenological heat-transfer law irreversible Carnot-engine by taking the exergy-based ecological optimization criterion as the objective function. Their study showed the optimization decreased greatly the entropy generation of the cycle with the cost of a little decrease in power output. Ust et al. [22] introduced a new objective function criterion called ecological coefficient of performance and applied to gas turbines based on irreversible Brayton cycle [22] and irreversible regenerative-Brayton [23] models. The objective function was defined as the ratio of power output to the loss rate of availability. Durmusoglu and Ust [24] optimized an irreversible regenerative closed Brayton cycle using a thermoeconomic objective criterion which was defined as the ratio of net power output to the total cost rate. Ecological performance analysis of an endoreversible modified Brayton cycle and ecological optimization of an irreversible Brayton cycle with regeneration, inter-cooling and reheating were conducted by Wang et al. [25] and del Rio Oliveira et al. [26], respectively. Sadatsakkak et al. [27] performed a multi objective optimization on irreversible regenerative closed Brayton cycle using three objective functions which included the power output, the ecological coefficient of performance and thermoeconomic criterion. Açıkkalp [28] investigated irreversible refrigeration cycle by using exergetic sustainability index. In that study, exergetic sustainability indicator was presented for the refrigeration cycle and its relationships with other thermodynamics parameters including ecological function and work input were assessed. It was recommended that exergetic sustainability index an important parameter for the environmental effects of any system. Ahmadi et al.

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[29] presented a developed ecological function for absorption refrigerators with four-temperature level. Moreover, the refrigerator was optimized via multi objective evolutionary approaches on the basis of NSGA-II method. The objective of the current study, is to investigate an irreversible closed-cycle of a regenerative Brayton cycle by power and then ecological function optimization. The optimal performance of the model is sought through power maximization and maximum ecological function using finite-time thermodynamic concept and finite-size components. In order to take into account the finite-time and finite-size concepts in current problem, a dimensionless mass-flow parameter is introduced. Finally, The modified maximum ecological function is obtained and plotted relating to the dimensionless mass-flow parameter and the appropriate performance zone of the heat engine is obtained, which provide a reliable, adoptable, and optimum system even if with daily variable power demand.

H1 and L1 flows are considered 1200 K and 300 K, respectively. The dead-state condition occurs at P 0 ¼ 1:01 bar and T 0 ¼ 293:15 K. Heat capacity of gas at constant pressure cP;gas varies with temperature as indicated in Eq. (5) [30].

cP;gas ðTÞ ¼ 0:93750 þ

0:01215 102

Tþ

0:01670 105

T2 

0:07164 109

T3

ð5Þ

Energy analysis is performed on each component of cycle separately. Energy balance equations are derived for compressor (Eq. (6)), turbine (Eq. (7)), high temperature heat exchanger (Eq. (8)) and low temperature heat exchanger (Eq. (9)) as follows.

8 2 39  cg 1 < = 1 4 P 2 cg T2 ¼ T1 1 þ  15 : ; gc P1 8 2 39  1c cg = < g P 4 5 T 5 ¼ T 4 1 þ ggt 41  : ; P5

ð6Þ

ð7Þ

2. Heat engine model The schematic of the heat engine is presented in Fig. 1. The model represents an irreversible regenerative closed Brayton cycle. Heat reservoirs and heat exchangers have finite capacitance rates and finite total conductance respectively. The system comprises of an irreversible adiabatic compressor, a heat regenerator, a pair of heat exchangers and an irreversible adiabatic turbine. Heat exchangers are used to transfer heat from high temperature reservoir to the system and from the system to the low temperature reservoir. The entire analysis of system can be broken down into two sub-analyses: (i) Thermodynamic model and energy analysis (ii) exergy analysis.

_  cP;H  ðT H1  T 8 Þ ¼ m _  cP;W1  ðT 4  T 3 Þ Q_ H ¼ m _  cP;HTHE min  ðT H1  T 3 Þ ¼ eHTHE  m _  cP;L  ðT 7  T L1 Þ ¼ m _  cP;W2  ðT 6  T 1 Þ Q_ L ¼ m _  cP;LTHE min  ðT 6  T L1 Þ ¼ eLTHE  m

A couple of fundamental assumptions are made to develop thermodynamic model. All processes take place at steady state. Heat exchangers, compressor and turbine are adiabatic; hence, no heat transfers between these components and the ambience. Moreover, it is assumed that heat sources have limited heat capacity. The gas (is considered as a product of combustion process) follows the principle of ideal-gas mixture. The temperature of working flow is variable and heat exchangers are configured in counter-flow direction. The chemical exergetic term does not vary in turbine, compressor and heat exchangers models. The kinetic and potential exergetic terms are neglected. The temperature of

ð9Þ

where eHTHE and eLTHE are the effectiveness of high temperature and low temperature heat sources simultaneously. The embedded parameters in above relations are described in Eqs. (10)–(15).

  cP;HTHE min ¼ min cP;H ; cP;W1

ð10Þ

  ¼ min cP;L ; cP;W2

ð11Þ

cP;CTHE min

2.1. Thermodynamic model and energy analysis

ð8Þ

cP;H ¼ cP;g

cP;L ¼ cP;g



T H1 þ T 8 2



  T L1 þ T 7 2

ð12Þ

ð13Þ

cP;W1 ¼ cP;g

  T3 þ T4 2

ð14Þ

cP;W2 ¼ cP;g

  T6 þ T1 2

ð15Þ

Fig. 1. The schematic of the system.

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Energy analysis of heat regenerator is detailed in Eqs. (16)–(19).

_ P;HR1 ðT 3  T 2 Þ ¼ mc _ P;HR2 ðT 5  T 6 Þ mc

eHR

ð16Þ

T3  T2 ¼ T5  T2

cP;HR1

cP;HR2

ð18Þ

  T6 þ T1 ¼ cP;g 2

ð19Þ

ð20Þ

_ net ¼ eHTHE  m _  cP;HTHE min  ðT H1  T 3 Þ W _  cP;LTHE min  ðT 6  T L1 Þ  eLTHE  m

ð21Þ

The novelty of current research relies on the utilization of physical exergy and exergy destruction of fluid streams in finite-time thermodynamics concept. Given the constant total thermal conductance of system heat exchangers, a dimensionless gas mass flow rate (F) is defined as Eq. (22). The parameter embeds time variable and physical constraint of the problem. Parameter F is preferable to time as it is restricted between finite values, while time varies from zero to infinity. In fact, F includes the finite-time and finite size concepts at the same time. ðUAÞT cP;min

ð26Þ



cP;com ¼ cP;g

_ net ¼ Q_ H  Q_ L W

i;

_  cP;gt  ðT 4  T 5 Þ E_ 10 ¼ m cP;gt ¼ cP;g

  T3 þ T2 ¼ cP;g 2

_ m

ð25Þ

ð17Þ

Finally, the produced net power in the system is presented in Eqs. (20) and (21).

F¼h

_  cP;c  ðT 2  T 1 Þ E_ 9 ¼ m

0
T4 þ T5 2

 ð27Þ

  T1 þ T2 2

ð28Þ

2.2.1. Exergy of cycle components The fuel and product exergy for system components are tabulated in Table 1. Identification of exergy destruction in different components leads to the solutions for the improvement of system and its components performance. In order to maximize the overall efficiency of system, exergy destruction should be minimized. Exergy destructions of system components are formulated in Eqs. (29)–(34) [31].

E_ D;k ¼ E_ F;k  E_ P;k

ð29Þ

E_ D;1 ¼ E_ F;1  E_ P;1 ¼ E_ 9  ðE_ 2  E_ 1 Þ

ð30Þ

E_ D;2 ¼ E_ F;2  E_ P;2 ¼ ðE_ 5  E_ 6 Þ  ðE_ 3  E_ 2 Þ

ð31Þ

E_ D;3 ¼ E_ F;3  E_ P;3 ¼ ðE_ H1  E_ 8 Þ  ðE_ 4  E_ 3 Þ

ð32Þ

E_ D;4 ¼ E_ F;4  E_ P;4 ¼ ðE_ 4  E_ 5 Þ  E_ 10

ð33Þ

E_ D;5 ¼ E_ F;5  E_ P;5 ¼ ðE_ 6  E_ 1 Þ  ðE_ 7  E_ L1 Þ

ð34Þ

ð22Þ

where cP;min is the least heat capacity of the gas in the system. Based on some decision variables, an optimization will be conducted, and for certain values of F, output power will be maximized. Variations of maximum power, first and second law efficiencies, and maximum ecological function of cycle corresponding to F, are going to be plotted. Finally, regarding the results, the appropriate performance zone of the heat engine will be obtained.

The total exergy destruction is calculated as follows:

E_ D;T ¼

5 X

E_ D;k

ð35Þ

k¼1

3. Optimization study

2.2. Exergy analysis of system

3.1. Mono-objective function optimization

Exergy is determined by the first and second laws of thermodynamics. It depends on thermodynamic quantities (i.e. enthalpy and entropy) although it is not a thermodynamic quantity itself. Exergy analysis assists with developing efficient thermodynamic processes and the results of such analysis lead to operational and technology improvements. In order to carry out an exergy analysis of an energy system, an exergy model must be provided. Physical exergy for temperature dependent specific heat is defined as follows:

For certain values of F, in order to obtain the variation of maximum power with parameter F, the system net output power is maximized. A simple genetic algorithm code is used in this investigation to determine the optimal values. The optimization code

Z _ E_ ¼ m

T

T0

cP ðTÞ  dT  T 0

Z

T

T0

cP ðTÞ 

dT P  R ln T P0

 ð23Þ

Physical exergy of gas flows (based on Eq. (5)) is expressed in Eqs. (24)–(28).

Table 1 Fuel and product exergy of the components. No.

Component

Fuel exergy

Product exergy

1

Compressor

2

Heat regenerator

3

High temperature heat exchanger

E_ 9 E_ 5  E_ 6 E_ H1  E_ 8

E_ 2  E_ 1 E_ 3  E_ 2 E_ 4  E_ 3

4

Turbine

5

Low temperature heat exchanger

E_ 4  E_ 5 E_ 6  E_ 1

E_ 10 E_ 7  E_ L1

For i ¼ H1; L1; 1; 2; 3; 4; 5; 6; 7; 8 h    iT i 2 Pi 0:00567 3 0:01791 4 0:01215 0:00835 2 0:02388 3 _  0:93750T þ 0:006075 E_ i ¼ m T þ T  T  T 0:93750 lnðTÞ þ T þ T  T þ T R ln 0 0 P0 105 105 102 109 102 109 T0

ð24Þ

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Table 2 Tuning parameters.

Table 4 The bounds used for decision variables.

Parameter

Value

Decision variable

Range

Population size Selection process Tournament size Selection strategy Crossover fraction Mutation fraction

400 Tournament 2 Elitist 0.8 0.01

Pressure ratio of compressor r HTHE rLTHE

[1,20] [0,1] [0,1]

was written by MATLAB. The objective function for this problem is given by.

_ net ¼ E_ 10  E_ 9 W

ð36Þ

Maximize Eq: ð36Þ subject to constraints (Eqs. (40) and (41)). The decision variables for the optimizations of the system are the compressor pressure ratio (PRcomp), thermal conductance ratio of high temperature heat exchanger to the entire system (Eq. (37)), and thermal conductance ratio of low temperature heat exchanger to the entire system (Eq. (38)). The bounds used for decision variables is provided in Table 4. The stopping conditions used for solving the optimization problems are maximum number of generation and cumulative function tolerance which shown in Table 3.

r HTHE ¼

ðU  AÞHTHE ðU  AÞT

ð37Þ

r LTHE ¼

ðU  AÞLTHE ðU  AÞT

ð38Þ

in this study, is a controlled elitist genetic algorithm (a variant of NSGA-II (Elitist Non-dominated Sorting Genetic Algorithm)). The controlled NSGA-II has a better convergence property than the original NSGA-II. Since diverse non-dominated fronts are also maintained in the population, controlled NSGA-II also produces a better distribution of solutions than the original NSGA-II [32]. The structure of the multi objective genetic algorithm used in this study is illustrated by Deb [33]. The optimization code is written by MATLAB. In order to obtain the best performance of the optimization method, tuning of parameters has been performed according to Grefenstette [34]. The tuning parameters values are given in Table 2. 3.3. Constraints Actual designs have physical constraints that must be considered for optimization. Two significant constraints are taken into account in this study: (i) heat exchangers’ size and (ii) admissible pressure ratio (PR). The total thermal conductance of heat exchangers is constant [2,10,17]. This assumption results in the following relations (Eqs. (40) and (41)), where rR is described as Eq. (42). The bounds used for decision variables are listed in Table 4.

ðU  AÞTOT ¼ Constant

ð40Þ

3.2. Multi-objective functions optimization

rHTHE þ rLTHE þ r R ¼ 1 0 < r < 1

ð41Þ

In most of previous related papers [14–17], the optimization of the ecological function were conducted via a mono objective optimization of Eq. (3), which yields only one trade off solution _ net and T 0 S_ g . between W

rR ¼

In this paper, for certain values of F, multi objective ecological function optimizations are carried out to obtain a range of trade off solutions along a Pareto front for probable modification of ecological function depends on problem. For certain values of F, the net output power of the system should be maximized (it is equivalent to minimize its negative), and the total exergy destruction (Eq. (35)) should be minimized via multi-objective optimization simultaneously. In this study, the negative of the first objective function, and the second objective function are minimized simultaneously, as follows.

_ net  E_ D;T E_ e ¼ W

ð39Þ

Maximize Eq: ð39Þ ) Minimize

Eq: ð36Þ Eq: ð35Þ

subject to constraints (Eqs. (40) and (41)). The decision variables for the optimizations of the system are same as the ones in the mono function optimization. The bounds used for decision variables are provided in Table 4. The multi objective genetic algorithm is used

Table 3 Stop criteria for the optimization algorithms. Stop criterion

Mono objective

Multi objective

Number of generation Function tolerance

1000 1e7

5000 1e5

ðU  AÞR ðU  AÞT

ð42Þ

The power optimization and multi-objective optimization of ecological function are conducted. The results will be analyzed in the following section. 4. Results and discussion 4.1. Comparison of maximum power state with optimum state of ecological function considering the impact of F variations Variation of dimensionless net power with parameter F is depicted in Fig. 2 at the maximum power state. The slope of power increases gradually with F up to the maximum of maximum power value at F = 0.3. Afterward, the dimensionless net power slopes down sharply. This behavior is identical to power variation with time. As time approaches infinity (F converges to zero), the produced power contiguously decreases to zero. The produced power also equals zero at time zero. Therefore, the maximum of maximum net power is located between the two limits. Parameter F is preferable to time as it is restricted between finite values, while time varies from zero to infinity. In order to verify the accuracy of optimization method for maximum power state, a four-variable optimization was implemented in which F is added as the fourth decision variable to the decision variables. Results of optimization show that the maximum point of the maximum power state is obtained again as the maximum point of the new optimization. The effects of F on dimensionless power in the maximum power state, and dimensionless power in the optimum state (maximum

M.M. Naserian et al. / Energy Conversion and Management 103 (2015) 790–800

Fig. 2. Effect of F on the value of dimensionless maximum net power.

Fig. 3. Variation of dimensionless net power in the maximum power and optimum states with F.

ecological function state) are shown in Fig. 3. As can be seen in the figure, the trend of power curve in the optimum state is the same as that in the maximum power state. In addition, in the optimum state power is maximized at F = 0.3. The only difference between these two curves is that for all values of F, power values at the optimum state is less than that in the maximum power state. The changes in dimensionless exergy destruction for the optimum (multi-objective optimization) and the maximum power states with F, are shown in Fig. 4. It is revealed that exergy destruction is increased with increment of F, and for all values of F, exergy destruction value in the optimum state is less than or equal to the maximum power state. Figs. 5 and 6 show the variations in the first and the second law efficiencies, according to F. The first and the second law efficiencies decrease as F gets higher and finally, they tend to zero. For low values of F, the first and the second law efficiencies of maximum power state are less than that in optimum state. However, for high values of F, the first and the second law efficiencies of the optimum state are a little less than the maximum power state. The reason for these observations is discussed in Fig. 7. The percentage reduction of output power, received heat, and received exergy are shown in Fig. 7. As can be seen for low values of F, in the optimum state, the proportion of net output power of the system decrease is much lower than the received heat and exergy to the system, and therefore the first and the second law

795

Fig. 4. Variations of dimensionless exergy destruction in the maximum power and optimum states with F.

Fig. 5. Effects of F on the first law efficiency for the maximum power and optimum states.

Fig. 6. Effects of F on the second law efficiency for the maximum power and optimum states.

efficiencies of the system are higher compared to the maximum power state. Nonetheless, for high values of F, the proportion of system net output power decrease is much higher than the

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M.M. Naserian et al. / Energy Conversion and Management 103 (2015) 790–800

Fig. 7. The percentage reduction of output power, received heat, and received exergy in the optimum state compare to the maximum power state.

Fig. 9. The variations of the system pressure ratio for the optimum and the maximum power states.

received heat and exergy to the system, and therefore the first and the second law efficiencies of the system are lower. Moreover, in order to study the reason for significant increase of the efficiencies and the decrease of the exergy destruction for low values of F, changes of thermal conductivity of heat regenerator and the system pressure ratio for the optimum and the maximum power states are considered in Figs. 8 and 9. In the maximum power state the amount of thermal conductivity of regenerator is less than 0.1 for all considered values of F, while for the optimum state, the amount of thermal conductivity increases with the decrease of F and tends to 0.9. Therefore, according to the definition of thermal effectiveness and number of transfer units (Eqs. (43) and (44)) the value of regenerator effectiveness increases with the decrease of F. The results of Figs. 3, 6 and 8 are corresponded well to Ust et al. [20] and del Rio Oliveira et al. [26]. In that researches, when the regenerator effectiveness equals to 0.5, the system works as the current system for F = 0.07. For low values (less than 0.5) of the regenerator effectiveness and high values (more than 0.07) of F, the second law efficiency for maximum power state and maximum ecological function almost have the same values. However, for high values (more than 0.5) of the regenerator and low values (less than 0.07) of F, the second law efficiency and exergy destruction rate for maximum ecological

function are, respectively, higher and lower than those of maximum power state. One can see from Fig. 9 that the compressor pressure ratio in the maximum power state is at least two times greater than that of the optimum state. Therefore, considering Figs. 8 and 9, it can be concluded that using heat regenerator, compensates decrease of the net output power in the optimum state which is dependent on the significant decrease of system pressure ratio. So, it plays an pivotal role in increasing the efficiency and decreasing the total exergy destruction.

Fig. 8. The variations of thermal conductivity of heat regenerator for the optimum and the maximum power states.

e¼

1  exp½NTU  ð1  cÞ ; 1  c  exp½NTU  ð1  cÞ

NTU ¼

c¼

C min C max

UA C min

ð43Þ

ð44Þ

4.2. Maximum ecological function curve In order to calculate the maximum ecological function, the optimal results of multi-objective optimization are used. In each F, the value of total exergy destruction should be subtracted from the value of net output power. The obtained curve is the maximum ecological function curve. However, before drawing the maximum ecological function curve dimensionless power and exergy destruction curves in the optimum state should be modified, based on the following issues. The multi-objective optimization of ecological function is carried out for different values of F. It is observed in Fig. 10 that for intermediate values of F, Pareto front is convex, so that the points near the two ends of the curve have almost the least distance from the ideal point. This means that in optimum point decreasing exergy destruction and decreasing net output power do not act the same. For example, in F = 0.1, decreasing only 5% of exergy destruction decreases net output power by 20%. This shows that decreasing net output power for the sake of decreasing exergy destruction is not efficient, and the points at the two ends of the Pareto are appear to be better choices than the optimum points. Of course, the minimum point of exergy destruction is not practically suitable due to the fact that net output power equals zero for this point. Therefore, for intermediate values of F, the maximum power state acts better than the optimum state of multi-objective optimization of ecological function. For small values of F (except for the intermediate point), Pareto front is concave (Fig. 11). It can be concluded that multi-objective

M.M. Naserian et al. / Energy Conversion and Management 103 (2015) 790–800

Fig. 10. The Pareto front for F = 0.1.

797

Fig. 12. The modified dimensionless power and exergy destruction curves.

Fig. 13. The diagrams of dimensionless power changes in terms of the first and the second law efficiencies after modifications. Fig. 11. The Pareto front for F = 0.02.

optimization function means decreasing the exergy destruction and yet having suitable net output power. Therefore, decreasing net output power for the sake of decreasing exergy destruction is efficient. For high values of F, although the percent of exergy destruction decrease is low compared with net output power decrease, the amount of this decrease is significant compared to other values of F (for example, 30% decrease in exergy destruction for F = 0.8 equals 80% decrease in exergy destruction for intermediate values of F). Therefore, similar to small and initial values of F, for high values of F, multi-objective optimization function is reasonable. Based on the mentioned issues, for intermediate values of F, the curves of power and exergy destruction of the maximum power state should be respectively replaced by power and exergy destruction curves in the optimum state, so that the modified curves are obtained as Fig. 12. The diagrams of dimensionless power changes in terms of the first and the second law efficiencies after modifications of intermediate values of F, are as Fig. 13. Comparing the curves, it is observed that for certain values of net output power, the second law efficiency is more than the first law efficiency. In addition, changes in intervals of second law efficiency is more than that of the first law efficiency. Apparently, the maximum values of both first and second law efficiencies occur at the zero value of net output power.

Fig. 14. Effects of F on dimensionless maximum ecological function.

Fig. 14 shows variations of dimensionless maximum ecological function versus F. As can be seen, the maximum value of maximum ecological function for all values of F occurs at F = 0.1. It was predictable that this point would be located at small and initial values

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M.M. Naserian et al. / Energy Conversion and Management 103 (2015) 790–800

Fig. 15. The flowchart of obtaining the proposed maximum ecological function curve.

of F because in this area, the first and the second law efficiencies are high. The effects of power and exergy destruction are considered simultaneously in the definition of maximum ecological function curve. Therefore, performance evaluation of a system based on it, is closer to reality in comparison with maximum power curve. System performance based on the maximum point of maximum ecological function (F = 0.1) yields 72% of the maximum of maximum power (F = 0.3), but only with exergy destruction as 24% of the amount that is produced at the maximum of maximum power. The results are consistent with the results of Angulo-Brown [14] and Wang et al. [25]. According to the Fig. 14, the area between F = 0 and F = 0.3 because of positive ecological function is suggested as the appropriate performance zone of the system. Therefore, in order to have optimal conditions at any time, depending on different power demands, the appropriate F, and decision variables could be proposed. Fig. 15 depicts the flowchart of obtaining the proposed maximum ecological function curve. Finally, for better comparison of maximum point of maximum power with maximum point of maximum ecological function, detailed data of system flows and components exergy destructions are brought in Tables 5–7.

E_ D;me 6:06 ¼ ¼ 0:24 E_ D;mp 25:37 _ me 19:83 W ¼ ¼ 0:72 _ W mp 27:39

Table 5 The decision variables in maximum of maximum ecological function and maximum power. Maximum of maximum ecological function

Maximum of maximum power

PR

r HTHE

r LTHE

PR

r HTHE

r LTHE

9.1159

0.4709

0.5291

4.7904

0.4471

0.5529

Table 6 Values of the temperatures and exergies for fluid flows. Flow

1 2 3 4 5 6 7 8

T

E_

Maximum of maximum ecological function

Maximum of maximum power

Maximum of maximum ecological function

Maximum of maximum power

362.1147 655.9902 655.9902 1095.132 655.9912 655.9912 597.9414 771.1289

442.4857 676.0575 676.0575 974.0981 676.0576 676.0576 541.123 916.3439

0.626768 32.42796 32.42796 66.22964 13.31939 13.31939 9.935943 20.91866

8.412867 84.32656 84.32656 150.513 43.68998 43.68998 20.90596 95.68374

799

M.M. Naserian et al. / Energy Conversion and Management 103 (2015) 790–800 Table 7 Fuel exergy, product exergy, and exergy destruction for the system components. Component

Compressor Regenerator HTHE Turbine LTHE

Maximum of maximum ecological function

Maximum of maximum power

Fuel exergy

Product exergy

Exergy destruction

Fuel exergy

Product exergy

Exergy destruction

31.8327 0 35.8566 52.9102 12.6926

31.8012 0 33.8017 51.6535 9.9350

0.2291 0 2.0549 1.0591 2.7567

76.7340 0 74.6421 106.8230 35.2771

75.9137 0 66.1864 104.12489 20.9049

1.2407 0 8.4557 2.2778 14.3712

Fig. 16. The Pareto front for multi-objective optimization function based on four decision variables.

gC ¼ 63:8%; gmp ¼ 26:3%; gme ¼ 38:3% According to Table 5. and Eq. (41) the value of r R is almost equal to zero. However, for low values of F in optimum state, usage of heat regenerator has significant impact on the system performance. Although for both maximum of maximum power and maximum of maximum ecological function points, r LTHE is more than the r HTHE , their values are near 0.5. As can be seen in Table 7 for both points, the main origin of exergy destructions of the systems are the high temperature and the low temperature heat exchangers and also the low temperature heat exchanger has the most exergy destruction rate. 4.2.1. Verification of new maximum ecological function To verify the accuracy of the obtained results, multi-objective optimization function is implemented with equal objective functions, but based on four decision variables in which F is added as the fourth decision variable to the decision variables. F can vary between 0 and 1. Based on the results of the new optimization, the maximum point of maximum ecological function of the previous method is re-obtained directly (Fig. 16). The interesting point about this optimization is that the value of objective function in the optimum state (F = 0.1) are exactly equal to the results after the modifications on the maximum ecological function. This confirms the validity of ecological function modification. 5. Conclusion The optimal performance of a regenerative Brayton cycle was sought through power and then a multi-objective ecological function maximization using finite-time thermodynamic concept in finite-size components. Optimizations were performed using genetic algorithm.

The novelty of current research was relied on defining a dimensionless parameter (F), which embeds time variable. Parameter F is preferable to time as it is restricted between finite values, while time varies from zero to infinity. Moreover, utilization of physical exergy and exergy destruction of fluid streams in finite-time thermodynamics concept, and also multi-objective optimization of ecological function were the other significant novelty of the study. The obtained ecological function had the advantages of both maximum power (for intermediate values of F) and maximum ecological function states (for other values of F). For low values of F usage of heat regenerator had significant impact on the system performance in optimum state. Although for both maximum of maximum power and maximum of maximum ecological function points, r LTHE was more than the r HTHE , their values were near 0.5. For both points, the main origin of exergy destructions of the systems were the low temperature and the high temperature heat exchangers, respectively. System performance based on the maximum point of ecological function (F = 0.1) yielded 72% of the maximum of maximum power (F = 0.3), but only with exergy destruction as 24% of the amount that was produced at the maximum of maximum power. According to the ecological function curve, the area between F = 0 and F = 0.3 was suggested as the appropriate performance zone of the heat engine on the curve. One could see that the curve of maximum ecological function provided not only one optimum point but also a range of optimum results, which provide a reliable, adoptable, and optimum system even if with daily variable power demand. References [1] Curzon FL, Ahlborn B. Efficiency of a Carnot engine at maximum power output. Am J Phys 1975;43(1):22–4. [2] Bejan A. Theory of heat-transfer irreversible power-plants. Int J Heat Mass Transfer 1988;31(6):1211–9. [3] Wu C. Power optimization of a finite time Carnot heat engine. Energy 1988;13(9):681–7. [4] Gordon JM. Observations on efficiency of heat engines operating at maximum power. Am J Phys 1990;58(4):370–5. [5] Wu C, Kiang RL. Finite-time thermodynamic analysis of a Carnot engine with internal irreversibility. Energy 1992;1(12):1173–8. [6] Cheng CY, Chen CK. Power optimization of an endoreversible regenerative Brayton cycle. Energy 1996;2(4):241–7. [7] Cheng CY, Chen CK. Power optimization of an irreversible Brayton heat engine. Energy Sources 1997;1(5):461–74. [8] Chen LG, Sun FR, Wu C, Kiang RL. Theoretical analysis of the performance of a regenerative closed Brayton cycle with internal irreversibilities. Energy Convers Manage 1997;3(9):871–7. [9] Bejan A. Thermodynamic optimization alternatives: minimization of physical size subject to fixed power. Int J Energy Res 1999;23:1111–21. [10] Herrera Carlos A, Sandoval Jairo A, Rosillo Miguel E. Power and entropy generation of an extended irreversible Brayton cycle: optimal parameters and performance. J Phys D: Appl Phys 2006;39:3414–24. [11] Chen LG, Zheng JL, Sun FR, Wu C. Power density analysis and optimization of a regenerated closed variable-temperature heat reservoir Brayton cycle. J Phys D Appl Phys 2001;3(11):1727–39. [12] Ust Y, Sahin B, Yilmaz T. Optimization of a regenerative gas-turbine cogeneration system based on a new exergetic performance criterion: exergetic performance coefficient. Proc IMechE Part A: J Power Energy 2007;221:447–58. [13] Sadatsakkak SA, Ahmadi MH, Bayat R, Pourkiaei SM, Feidt M. Optimization density power and thermal efficiency of an endoreversible Braysson cycle by

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