Exergoeconomic multi objective optimization and sensitivity analysis of a regenerative Brayton cycle

Energy Conversion and Management 117 (2016) 95–105

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Exergoeconomic multi objective optimization and sensitivity analysis of a regenerative Brayton cycle 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 6 January 2016 Accepted 5 March 2016

Keywords: Finite-time Exergoeconomic Regenerative Brayton cycle Optimization Sensitivity analyses

a b s t r a c t In this study, the optimal performance of a regenerative Brayton cycle is sought through power maximization and then exergoeconomic optimization using finite-time thermodynamic concept and finitesize components. 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 used deploying time variations. The decision variables for the optimum state (of multi objective exergoeconomic optimization) are compared to the maximum power state. One can see that the multi objective exergoeconomic optimization results in a better performance than that obtained with the maximum power state. The results demonstrate that system performance at optimum point of multi objective optimization yields 71% of the maximum power, but only with exergy destruction as 24% of the amount that is produced at the maximum power state and 67% lower total cost rate than that of the maximum power state. In order to assess the impact of the variation of the decision variables on the objective functions, sensitivity analysis is conducted. Finally, the cycle performance curve drawing according to exergoeconomic multi objective optimization results and its utilization, are suggested. Ó 2016 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. The 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 Ahlbom [1] investigated the effect of external irreversibility on Carnot cycle 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,

⇑ Corresponding author. Tel.: +98 9151412276. E-mail address: [email protected] (S. Farahat). http://dx.doi.org/10.1016/j.enconman.2016.03.014 0196-8904/Ó 2016 Elsevier Ltd. All rights reserved.

comparison was established between endoreversible 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 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 a regenerative endoreversible Brayton cycle with two finite thermal capacity heat reservoirs [6]. In that 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 processes were taken into account in irreversible model, along with variable temperature at heat reservoirs.

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Nomenclature T _ W S_ C_ E_ e cP P E_ _ m Q_ A U F R PR r C c N Z AF LHV

temperature (K) power (kW) entropy rate (kW/K) cost rate ($/s) ecological function (kW) specific heat (kJ/kg.K) pressure (bar) exergy rate (kW) mass flow rate (kg/s) heat transfer rate (kW) area (m2) overall heat transfer rate (kW/m2 K) dimensionless mass flow rate specific gas constant (J/kg K) pressure ratio thermal conductance ratio cost ($) capacity ratio number of hours purchase cost ($) air fuel ratio lower heating value

/

maintenance factor

Subscripts L low H high CA Curzon and Ahlbom 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 gt gas turbine CC combustion chamber AC air compressor R heat regenerator HTHE high temperature heat exchanger LTHE low temperature heat exchanger op optimum

Greek symbols g efficiency e effectiveness

Optimization of real systems is confined to thermal performance and physical constraints. Bejan [9] established two 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 that model, finite-time thermodynamics and optimization was used to determine the maximum power and minimum entropy generation, along with the global maximum net power. The performance of an air-standard rectangular cycle with heat transfer loss and variable specific heats of working fluid was analyzed using finite time thermodynamics by Wang et al. [11]. They found that the effects of heat transfer loss and variable specific heats of working fluid on the cycle performance were obvious. Agnew et al. [12] presented a finite time analysis of a trigeneration cycle that was based upon coupled power and refrigeration Carnot cycles. In various studies, thermal parameters considered as objectives 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 [13,14]. Dimensionless power density and thermal efficiency were optimized for Braysson cycles using Nondominated Sorting Genetic Algorithm (NSGA) and finite time thermodynamic analysis [15]. Açıkkalp [16] investigated irreversible refrigeration cycle by using exergetic sustainability index. In that study, it was recommended that exergetic sustainability index an important parameter for the environmental effects of any system. Angulo-Brown et al. [17] proposed an ecological criterion as Eq. (2) for finite-time Carnot heat-engines, where TL is the temperature of the cold heat_ is the power output and S_ g is the entropyreservoir, W generation rate. Yan [18] discussed the results of Angulo’s study

and suggested that the definition of ecological function as Eq. (3) is more reasonable

_  T L S_ g E_ e ¼ W

ð2Þ

_  T 0 S_ g E_ e ¼ W

ð3Þ

Huang et al. [19] and Chen et al. [20] carried out ecological exergy optimization for heat engines. 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 condition. The effect of the regenerator effectiveness on the optimal performance for the maximum ecological function conditions was discussed by Ust et al. [21] and Kumara et al. [22]. Ecological performance analysis of an endoreversible modified Brayton cycle was conducted by Long and Liu [23] and Wang et al. [24]. Ecological optimization of an irreversible Brayton cycle with regeneration, inter-cooling and reheating was performed by del Rio Oliveira et al. [25]. Naserian et al. [26] sought the optimal performance of a regenerative Brayton cycle through power and then multi-objective ecological function maximization using finite-time thermodynamic concept and finite-size components. A dimensionless parameter, which embeds time variable was defined by them. Durmusoglu et al. [27] 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. Sadatsakkak et al. [28] 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. Thermoeconomic analysis of irreversible refrigeration and heat pump systems was demonstrated for specified cooling and heating loads, by Qureshi and Zubair [29]. As a conclusion, they found that endoreversible models provide a reasonable basis for initially understanding real systems. Ahmadi et al. [30] carried out a

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thermo-economic optimization of irreversible Stirling heat pump cycles that include both internal and external irreversibilities. Thermoeconomic optimization of irreversible power systems with finite thermal capacitances for design situation was performed by Qureshi [31]. The investigation was made with respect to the case of specified power output. Sahraie et al. [32] performed a thermodynamic modeling and thermo-economic optimization of an irreversible absorption heat pump. The specific heating load, coefficient of performance and the thermo-economic benchmark were considered as objective functions. They also conducted sensitivity analysis and error analysis in order to improve understanding of the system performance. There are other studies that used, Finite time thermodynamic (FTT), power optimization, thermoenvironmental, and ecological methods in evaluation and comparison of thermal cycles [33–37]. In current study, an irreversible closed-cycle Brayton engine with regeneration is investigated. The main originality of the current research is as follows:  A dimensionless parameter includes the finite-time and size concepts is used in exergoeconomic analysis and multi objective optimization of the system.  Physical exergy and cost concepts of fluid streams are inserted into finite-time thermodynamics.  The system performance based on the optimum point of exergoeconomic optimization is compared with maximum power and maximum ecological function states.  Sensitivity analysis is performed to determine the impact of decision variables on the objective functions.  The cycle performance curve drawing and utilization were suggested.

entrance of heat exchanger. For simplicity, all the three flows are assumed to flow at the same mass flow rate and have same material properties. The entire analysis of system can be broken down into two sub-analyses: (i) Thermodynamic model and energy analysis (ii) exergy analysis. 2.1. Thermodynamic model and energy analysis The other objective of the current study is to prepare finite time thermodynamics for studying more practical systems. However, some fundamental assumptions are made yet to develop thermodynamic model which restrict the study of real systems. 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 exergy does not vary in turbine, compressor and heat exchangers. The kinetic and potential exergetic terms are neglected. The temperature of H1 (The state of the hot flow at the entrance of high temperature heat exchanger) and L1 (The state of the cold flow at the entrance of low temperature heat exchanger) flows are considered 1200 K and 300 K, respectively. The fuel used in the cycle is considered methane, and the lower heating value (LHV) of methane is 50,000 kj/kg [40]. Heat loss from the combustion chamber ðQ_ l;cc Þ is considered to be 2% of LHV. The dead-state condition occurs at P0 ¼ 1:01 bar and T 0 ¼ 293:15 K. Heat capacity of gas at constant pressure C P;gas varies with temperature as indicated in Eq. (4) [26].

cP;gas ðTÞ ¼ 0:93750 þ 2. Heat engine model The schematic of the heat engine is presented in Fig. 1. The model represents an irreversible regenerative Brayton cycle. The heat engine is an external combustion gas cooled gas turbine powerplant. The system comprises of an irreversible adiabatic compressor, a heat regenerator, a pair of heat exchangers and an irreversible adiabatic turbine. High temperature heat exchanger is used to transfer heat ðQ_ H Þ from hot flow coming from the combustion chamber, at H1 state, to the system working fluid while low temperature heat exchanger is used to transfer heat ðQ_ L Þ from system working fluid to cold flow which is, at L1 state, at the

0:01215 102

Tþ

0:01670 105

T2 

0:07164 109

T3

ð4Þ

Energy analysis is performed on each cycle component separately. Energy balance equations are derived for compressor (Eq. (5)), turbine (Eq. (6)), high temperature heat exchanger (Eq. (7)) and low temperature heat exchanger (Eq. (8)) as follows:

8 2 39  cg 1 < = 1 4 P2 cg T2 ¼ T1 1 þ  15 : ; gAC P1 8 2 39  1c cg = < g P 4 5 T 5 ¼ T 4 1 þ ggt 41  ; : P5

Fig. 1. The schematic of the system.

ð5Þ

ð6Þ

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M.M. Naserian et al. / Energy Conversion and Management 117 (2016) 95–105

The efficiency of the compressor (gAC ), is assumed to be 0.9, and the efficiency of turbine (ggt ), is assumed to be equal to 0.95 [10].

_  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

ð7Þ

_  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

ð8Þ

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. (9)–(14).

  cP;HTHE min ¼ min cP;H ; cP;W1

ð9Þ

  cP;CTHE min ¼ min cP;L ; cP;W2

ð10Þ

cP;H ¼ cP;g

cP;L

  T H1 þ T 8 2

ð11Þ

  T L1 þ T 7 ¼ cP;g 2

ð12Þ

cP;W1 ¼ cP;g

  T3 þ T4 2

ð13Þ

cP;W2 ¼ cP;g

  T6 þ T1 2

ð14Þ

Energy analysis of heat regenerator is detailed in Eqs. (15)–(18).

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

eHR

T3  T2 ¼ T5  T2

cP;HR1

ð15Þ ð16Þ

  T3 þ T2 ¼ cP;g 2

ð17Þ

  T6 þ T1 2

ð18Þ

cP;HR2 ¼ cP;g

Combustion chamber is energetically analyzed in Eqs. (19)– (21).

_ F; _ ¼m _ aþm m

_a m AF ¼ _F m

ð19Þ

_ F LHV ¼ mh _ H1 þ Q_ l;cc _ a cP;air þ m m

ð20Þ

_ F LHVð1  gcc Þ Q_ l;cc ¼ m

ð21Þ

In this analysis, taking into account the assumption of constant temperature of combustion chamber outlet (equals to 1200 K), and constant temperature of air at the inlet of combustion chamber (equals to 293 K), the air–fuel ratio is constant and calculated as follows:

_  AF _ m m _ P;g T H1 cP;air T 1 þ g LHV ¼ mc AF þ 1 AF þ 1 CC AF ¼

gCC LHV  cP;g T H1 cP;g T H1  cP;air T 1

¼ 41:8

ð22Þ

ð23Þ

Finally, the produced net power in the system is presented in Eqs. (24) and (25).

_ net ¼ Q_ H  Q_ L W

ð24Þ

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

ð25Þ

In order to enter the finite size constraint to the problem, total thermal conductance of system heat exchangers (ðUAÞT ) is considered constant [2,10,26]. Moreover, assuming that the mass flow rate is finite and non-zero, time will be constrained, too. Therefore, in order to take into account the time and the size constraints in current problem, using the dimensionless mass-flow parameter (F) is the perfect choice. The parameter is defined by Naserian et al. [26], as Eq. (26).

F¼h

_ m

i;

ðUAÞT cP;min

0
ð26Þ

where cP;min is the least heat capacity of the gas in the system. 2.2. Exergy analysis of system 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:

Z _ E_ ¼ m

Z

T

T

cP ðTÞ  dT  T 0

T0

cP ðTÞ  T0

dT P  R ln T P0

 ð27Þ

Physical exergy of gas flows (based on Eq. (4)) is expressed in Eqs. (28)–(32) [26]. For i ¼ H1; L1; 1; 2; 3; 4; 5; 6; 7; 8

 0:006075 2 0:00567 3 0:01791 4 0:93750T þ T þ T  T 105 102 109  T i 0:01215 0:00835 2 0:02388 3 T þ T  T T 0 0:93750lnðTÞ þ 105 102 109 T0  Pi ð28Þ þT 0 R ln P0

_  E_ i ¼ m

_  cP;c  ðT 2  T 1 Þ E_ 9 ¼ m

ð29Þ

_  cP;gt  ðT 4  T 5 Þ E_ 10 ¼ m

ð30Þ

  T4 þ T5 ¼ cP;g 2

ð31Þ

cP;gt

cP;com ¼ cP;g

  T1 þ T2 2

ð32Þ

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

Component

Product exergy

Fuel exergy

1

Compressor

2

Heat regenerator

3

High temperature heat exchanger

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

E_ 9 E_ 5  E_ 6 E_ H1  E_ 8

4

Turbine

5

Low temperature heat exchanger

E_ 10 E_ 7  E_ L1

E_ 4  E_ 5 E_ 6  E_ 1

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2.2.1. Exergy of cycle components The fuel and product exergy for system components are tabulated in Table 1. Although combustion chamber is one of the main sources of exergy destruction [38], as the inlet and outlet thermodynamic properties of combustion chamber are considered fixed, the exergy analysis of combustion chamber is neglected in optimization process. 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 destruction rates of system components are formulated in Eqs. (33)–(38) [39].

E_ D;k ¼ E_ F;k  E_ P;k

ð33Þ

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

ð34Þ

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

ð35Þ

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

ð36Þ

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

ð37Þ

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

ð38Þ

The total exergy destruction is calculated as follows:

E_ D;T ¼

5 X E_ D;k

ð39Þ

k¼1

2.3. Cost analysis Cost analysis is conducted on each system component separately. Cost analysis involves cost balances equations. A cost balance applied to the kth component shows that the sum of cost rates associated with exiting exergy streams (product cost rate) equals the sum of cost rates of all entering exergy streams (fuel cost rate) plus the cost rate associated with capital investment and the maintenance cost which is denoted by Z_ k . The cost associ-

ated with the fuel (C_ F ) and product (C_ P ) of a component are _ given in Table 1 with cost obtained by replacing exergy rates (E) _ rates (C). Accordingly, for the kth component:

C_ P;k ¼ C_ F;k þ Z_ k

C_ 2  C_ 1 ¼ C_ 9 þ Z_ AC

ð41Þ

C_ 3  C_ 2 ¼ C_ 5  C_ 6 þ Z_ AP

ð42Þ

C_ 5 C_ 6 ¼ E_ 5 E_ 6

C_ 4  C_ 3 ¼ C_ H1  C_ 8 þ Z_ HTHE cH1 ¼ c8 )

C_ H1 C_ 8 ¼ E_ H1 E_ 8

_f ¼ m

_ m AF þ 1

ð46Þ

where cf is fuel cost which is considered to be 0.004 $/Mj in our analysis [40].

C_ 10 ¼ C_ 4  C_ 5 þ Z_ gt c9 ¼ c10 )

c4 ¼ c5 )

C_ 9 C_ 10 ¼ E_ 9 E_ 10

C_ 4 C_ 5 ¼ E_ 4 E_ 5

ð47Þ ð48Þ

ð49Þ

C_ 7  C_ L1 ¼ C_ 6  C_ 1 þ Z_ LTHE

ð50Þ

C_ L1 ¼ 0

ð51Þ

c6 ¼ c1 )

C_ 6 C_ 1 ¼ E_ 6 E_ 1

ð52Þ

For more details see Appendix section. Cost balance equation of the components form a system of linear equations as follows: 3 32 _ 3 2 2 1 1 0 0 0 0 0 0 1 0 Z_ AC C1 7 7 6 6 6 0 1 1 0 1 1 0 0 0 _ 7 6 7 0 7 Z_ AP 76 6 7 6 C2 7 6 7 6 7 6 _3 7 6 0 0 0 0 E_ 6 E_ 5 0 0 0 0 76 0 C 7 7 6 6 76 6 7 7 6 7 6 0 0 1 1 _ _ _ 7 7 6 6 0 0 0 1 0 0 7 C4 C þ Z 6 H1 HTHE 7 6 7 6 7 6 7 6 _ _ _ 6 0 0 0 0 0 0 0 E_ H1 0 0 76 C 5 7 6 E8 C H1 7 7 76 6 ¼ 7 7 6 6 6 0 0 0 1 1 7 6 0 0 0 0 1 7 Z_ gt C_ 6 7 76 6 7 7 6 6 7 6 7 6 7 _ 76 6 0 0 0 E_ 5 E_ 4 0 0 0 0 0 0 C7 7 6 7 76 6 7 6 7 76 6 0 0 0 0 _ _ _ 7 7 6 6 0 0 0 0 E  E 0 10 9 76 C 8 7 6 7 6 76 6 7 7 6 4 1 0 0 0 0 1 1 0 0 0 54 C_ 9 5 4 Z_ LTHE 5 E_ 6 0 0 0 0 E_ 1 0 0 0 0 0 C_ 10 The system of equations is solved and the cost of the streams is obtained. Finally, the cost destruction of components are calculated from Eq. (53).

C_ D;k ¼ cF;k E_ D;k

ð53Þ

where

cF;k ¼ C_ F;k =E_ F;k

ð54Þ

ð40Þ

Cost balance equations are derived for compressor (Eq. (41)), heat regenerator (Eqs. (42) and (43)), high temperature heat exchanger (Eqs. (44)–(46)) turbine (Eqs. (47)–(49)), and low temperature heat exchanger (Eqs. (50)–(52)) as follows [39]. As a rule, n  1 auxiliary equations are required for components with n exiting exergy streams [39]. Eqs. (43), (45), (48), (49), and (52) represent the auxiliary equations for the components. More descriptions can be found in [39].

c5 ¼ c6 )

_ f cf LHV _ m C_ H1 ¼ þ Z CC ; 1000

ð43Þ ð44Þ ð45Þ

3. Optimization study In some of previous related studies [15,26,30,32,41–45], the optimization of the thermodynamic problems were conducted via Genetic algorithm (GA). A simple GA works as follows [46]:     

Encoding the objectives functions; Defining a fitness function or selection criterion; Initializing a population of individuals; Assessing the fitness of all the individuals in the population; Creating a new population by performing crossover, and mutation, fitness proportionate reproduction etc.;  Evolving the population until certain stopping criteria are met;  Decoding the results to obtain the solution. Some of the advantages of GA which make it one of the most suitable choice for thermodynamic optimization, are as follows [35]:  Optimizes with continuous or discrete variables;  Doesn’t require derivative information;

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 Simultaneously searches from a wide sampling of the cost surface;  Deals with a large number of variables;  Is well suited for parallel computers;  Optimizes variables with extremely complex cost surfaces (they can jump out of a local minimum). According to foregoing discussion, Genetic algorithm is chosen and used for optimization purpose. 3.1. Mono-objective function optimization In order to obtain the maximum power, the system net output power is maximized by a single objective optimization algorithm. A simple genetic algorithm code is used in this investigation to determine the optimal value of power. The optimization code was written by MATLAB. The objective function for this problem is given by.

_ net ¼ E_ 10  E_ 9 W

ð55Þ

Maximize Eq. (55) subject to constraints (Eqs. (58) and (59)). The decision variables for the optimization of the system are the compressor pressure ratio (PRcomp), thermal conductance ratio of high temperature heat exchanger to the entire system (Eq. (56)), thermal conductance ratio of low temperature heat exchanger to the entire system (Eq. (57)), and dimensionless mass flow rate (Eq. (26)). 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

ð56Þ

r LTHE ¼

ðU  AÞLTHE ðU  AÞT

ð57Þ

Table 3 Stop criteria for the optimization algorithms. Stop criterion

Mono objective

Multi objective

Number of generation Function tolerance

1000 1e7

5000 1e5

Minimize

Eq:ð55Þ Eq:ð58Þ

subject to constraints (Eqs. (59) and (60)). The decision variables for the optimization 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 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 [47]. The structure of the multi objective genetic algorithm used in this study is illustrated by Deb [48]. The optimization code is written by MATLAB. The tuning parameters values for multi objective optimization 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 [2,10,20]. This assumption results in the following relations (Eqs. (59) and (60)), where rR is described as Eq. (61). The bounds used for decision variables are listed in Table 4.

ðU  AÞT ¼ Constant ðIn this study ¼ 1Þ

ð59Þ

rHTHE þ rLTHE þ rR ¼ 1 0 < r < 1

ð60Þ

3.2. Multi-objective functions optimization In this paper, two functions including power, and the total cost rate of product and exergy destruction impact are considered as objective functions for multi objective Exergoeconomic optimization purpose. The net output power of the system should be maximized (it is equivalent to minimize its negative), and the total cost rate (Eq. (58)) should be minimized. The multi objective optimization is the modified version of multi objective ecological function optimization performed by Naserian et al. [26]. In the current research the influence of cost is added to the second objective function, which makes the results more acceptable for industrial use. In this study, the negative of the first objective function, and the second objective function are minimized simultaneously, as follows:

C_ T ¼ C_ H1 þ

n n X X Z_ k þ C_ D;k k¼1

ð58Þ

k¼1

rR ¼

ðU  AÞR ðU  AÞT

ð61Þ

The results of the optimizations will be analyzed in the following section. 4. Results and discussion The power maximization and multi-objective optimization are conducted. Fig. 2 shows the Pareto frontier solution with the objective functions in the multi objective optimization section. In this figure, while the maximum power of the cycle increases to about 26 kW, the total cost rate increases moderately. Increasing the maximum power from 26 kW to a higher value leads to a drastic increase of the total cost rate. According to the figure, it can be concluded that multi-objective optimization function means decreasing the total cost rate and yet having suitable net output power. Therefore, decreasing net output power for the sake of decreasing

Table 2 Tuning parameters. Parameter

Value

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

400 Tournament 2 Elitist 0.8 0.01

Table 4 The bounds used for decision variables. Decision variable

Range

Pressure ratio of compressor r HTHE rLTHE F

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

M.M. Naserian et al. / Energy Conversion and Management 117 (2016) 95–105 E_ D;op E_ D;me

¼ 6:18 ¼ 1:02 6:06

C_ T;op C_ T;me

¼ 0:000605 ¼ 0:97738 0:000619

_ op W _ me W

¼ 19:37 ¼ 0:98 19:82

101

The both methods lead to almost the same result. Despite a small increment in total exergy destruction rate, using exergoeconomic optimization causes a small reduction in total cost rate, and net output power. As can be seen in Table 7 for both points, the main origin of exergy destruction rates of the system are the high temperature and the low temperature heat exchangers and also the low temperature heat exchanger has the most exergy destruction rate. 5. Sensitivity analysis Fig. 2. The Pareto front for multi-objective optimization function.

the total cost rate is efficient which means that the multi-objective optimization function is reasonable. The closest point of Pareto frontier to the ideal point (The ideal point is defined as a hypothetical point which has the most optimum value of each objective function individually at the same time), which leads to 0.000605 _ respectively, is consid$/s and 19.33 kW values for the C_ T and W ered as the desirable optimum solution. However, selection of the optimum solution from Pareto frontier depends on the preferences of the decision maker. The optimum values of decision variables for the mono objective and multi objective optimizations are shown in Table 5. According to Table 5 and Eq. (60) the value of rR is almost equal to zero for maximum power state. However, heat regenerator usage has significant impact on the system performance in multi objective optimization optimum point. Although for both points, r LTHE is more than the r HTHE , their values are near 0.5. For better comparison of maximum point of maximum power with optimal point of multi objective optimization, detailed data of system flows are brought in Tables 6 and 7. The ratio of total exergy destruction, total cost rate, and net power of optimum point of multi objective exergoeconomic optimization to maximum power state are as follows: E_ D;op E_ D;mp

6:18 ¼ 25:37 ¼ 0:24

C_ T;op C_ T;mp

0:000605 ¼ 0:001805 ¼ 0:33

_ op W _ mp W

¼ 19:37 ¼ 0:71 27:39

System performance based on the optimum point of multi objective optimization yielded 71% of the maximum power, but only with exergy destruction as 24% of the amount that is produced at the maximum power state and 67% lower total cost rate than that of the maximum power state. The ratio of total exergy destruction, total cost rate, and net power of optimum point of multi objective exergoeconomic optimization to maximum point of maximum ecological function (reported by Naserian et al. [26]) are as follows:

Table 5 The decision variables in optimal state of multi objective optimization and maximum power. Maximum power

Optimal state of multi objective optimization

F

r LTHE

r HTHE

PR

F

r LTHE

r HTHE

PR

0.3000

0.5529

0.4471

4.7904

0.0982

0.5144

0.4713

7.8287

A comprehensive sensitivity analysis is performed to determine the impact of the variation of the decision variables on the objective functions, and to improve understanding of the system performance. The effects of F variation on the objective functions are shown in Fig. 3. It is revealed that for all values of F, total cost rate is increased with increment of F, however, this increase leads to increase in the power at first, followed by a decrease in the power. So variations in this parameter in the range of 0 to 0.3 cause a conflict between the objective functions. The area between F = 0 and F = 0.3 was already suggested as the appropriate performance zone of the heat engine by Naserian et al. [26]. Fig. 4 shows the variation of both objective functions when the compressor pressure ratio varies in the defined interval. It is shown that the slope of power increases sharply with PR up to the maximum power value at PR = 9. Afterwards, the dimensionless net power slopes down gradually. As can be seen in the Figure, for low values of PR, increase in this design parameter leads to decrease in the total cost rate up to the minimum cost value at PR = 6, then more increasing of PR results in increase of total cost rate. Therefore, total cost rate exhibits a reverse trend compared with that for dimensionless power. The variations of the objective functions with r LTHE and rHTHE are depicted in Figs. 5 and 6. As it is shown in these figures, the increase in thermal conductance ratio of heat exchangers lead to improvement of both objective functions. Therefore, variations of these parameters do not cause a conflict between two objective functions. In addition it can be concluded that for better performance of the system, the values of r LTHE and rHTHE must be almost identical (near 0.5). The sensitivity of objective functions to decision variables are analyzed more precisely in Fig. 7. The effects of same changes in decision variables relating to their optimum values on objective functions are investigated. According to the figure the objective functions are most and least sensitive to parameter F and rHTHE , respectively. Except for F, variations of the decision variables have more influences on the output power than the total cost rate. In addition, negative values of changes in variables, cause more variation in the objective functions than that of the positive ones. Comparing Fig. 7c with Fig. 7d, it can be concluded that the objective functions are more sensitive to r LTHE . 6. The cycle performance curve Having the optimal conditions at any time, depending on different power demands, the appropriate decision variables could be proposed. The heat engine design is desirable, if it is based on two conditions: (i) the maximum output power, and (ii) the minimum total cost rate conditions. Then, in order to draw the cycle performance curve, the results of exergoeconomic optimization

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

E_ (kW)

T (K)

1 2 3 4 5 6 7 8

C_ ($/s) ⁄ 1.0e+03

Optimal state

Maximum power

Optimal state

Maximum power

Optimal state

Maximum power

366.01 636.95 641.72 1094.32 678.63 673.86 612.40 758.15

442.49 676.06 676.06 974.10 676.06 676.06 541.12 916.34

0.6879 29.2459 29.5244 63.4924 14.4603 14.1659 10.5537 19.6485

8.41287 84.3266 84.3266 150.5130 43.6900 43.68900 20.9060 95.6837

0.0073 0.3390 0.3589 0.6731 0.1533 0.1502 0.1606 0.1619

0.0915 0.9880 1.0038 1.6340 0.4756 0.4756 0.4000 0.7882

Table 7 Purchase cost rate, Cost rate of exergy destruction, and exergy destruction rate for the system components. Component

Compressor Regenerator HTHE Turbine LTHE

Optimal state

Maximum power

Purchase cost rate ($/s) ⁄ 1.0e+05

Cost rate of exergy destruction ($/s) ⁄ 1.0e+06

Exergy destruction rate (kW)

Purchase cost rate ($/s) ⁄ 1.0e+05

Cost rate of exergy destruction ($/s) ⁄ 1.0e+06

Exergy destruction rate (kW)

0.9200 1.6763 1.6857 1.9806 1.7706

2.3172 0.1840 22.654 9.5106 31.002

0.2067 0.0159 2.1368 0.8971 2.9242

1.3094 1.7358 1.6857 4.6066 1.7506

14.4960 0 93.0700 25.0690 158.0100

1.2407 0 8.4557 2.2778 14.3712

Fig. 3. The effects of F variation on the objective functions.

Fig. 5. The variations of the objective functions with thermal conductance ratio of high temperature heat exchanger to the entire system.

Fig. 4. The effects of pressure ratio variation on the objective functions.

Fig. 6. The variations of the objective functions with thermal conductance ratio of low temperature heat exchanger to the entire system.

are used. The multi-objective exorgoeconomic optimizations are carried out for different values of F. The decision variables for the optimizations are the compressor pressure ratio, thermal conduc-

tance ratio of the high temperature heat exchanger to the entire system (Eq. (56)), thermal conductance ratio of the low temperature heat exchanger to the entire system (Eq. (57)). The area

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Fig. 7. The sensitivity of objective functions to decision variables.

between maximum power (F = 0.3) and F = 0 was suggested as the appropriate performance zone by Naserian et al. [26]. The performance curve of the system studied in the research is plotted in Fig. 8.

Finally, the cycle performance curve drawing according to exergoeconomic optimization results and its utilization, were suggested to have the optimal conditions at any time, depending on different power demands. Appendix A

7. Conclusion The optimal performance of a regenerative Brayton cycle was sought through power maximization and then exergoeconomic optimization using finite-time thermodynamic concept and finite-size components. Optimizations were performed using genetic algorithm. The novelty of the research was relied on using a dimensionless parameter (F), which embeds time variable, in exergoeconomic study and optimization. Moreover, the modified ecological function concept (defined by Naserian et al. [26]), was entered in the exergoeconomic analysis of the cycle. The exergoeconomic objective function had the advantages of both modified ecological function and minimum total cost rate of the cycle. System performance based on the optimum point of multi objective optimization yielded 71% of the maximum power, but only with exergy destruction as 24% of the amount that was produced at the maximum power state and 67% lower total cost rate than that of the maximum power state. According to the sensitivity analysis, the objective functions were most and least sensitive to parameter F and r HTHE . Except for F, variations of the decision variables had more influences on the output power than the total cost rate.

For a cost analysis of a system, it is needed to consider the annual cost of fuel and the annual cost associated with owning and operating each plant component. The expressions for obtaining the purchase costs of the components (Z) are as follows [40]:

Z CC ¼



 _a C 21 m ½1 þ expðC 23 T H1  C 24 Þ C 22  ð1  DPCC Þ

Z gt ¼

_ C 31 m C 32  ggt 

Z AC ¼

!

_a C 11 m C 12  gAC 

Z R ¼ C 41

 ln

 P4 ½1 þ expðC 33 T 4  C 34 Þ P5

    P2 P2 ln P1 P1

_ 5  h6 Þ mðh ðUAÞR ðDTLMÞR 

Z HTHE ¼ C 41

ðA1Þ

ðA2Þ

ðA3Þ

0:6

_ H1  h8 Þ mðh ðUAÞHTHE ðDTLMÞHTHE

ðA4Þ 0:6 ðA5Þ

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M.M. Naserian et al. / Energy Conversion and Management 117 (2016) 95–105 Table A1 Constants used in purchase cost equations of components. Component

Constant

Compressor Combustion chamber

C 12 ¼ 0:95 C 22 ¼ 0:995 DP CC ¼ 0:05 C 24 ¼ 26:4 C 32 ¼ 0:97 C 34 ¼ 54:4

Turbine Heat regenerator High temperature heat exchanger Low temperature heat exchanger

C 11 ¼ 39:5 ð$=ðkg=sÞÞ C 21 ¼ 25:6 ð$=ðkg=sÞÞ C 23 ¼ 0:018 ðK1 Þ C 31 ¼ 266:3 ð$=ðkg=sÞÞ C 33 ¼ 0:036 ðK1 Þ

C 41 ¼ 2290 ð$Þ

References

Fig. 8. The performance curve of the system.

 Z LTHE ¼ C 41

_ 6  h1 Þ mðh ðUAÞLTHE ðDTLMÞLTHE

0:6 ðA6Þ

where

ðDTLMÞR ¼

ðT 5  T 3 Þ  ðT 6  T 2 Þ

 3 Ln TT 56 T T 2

ðA7Þ

ðDTLMÞHTHE ¼

ðT 8  T 4 Þ  ðT H1  T 3 Þ

 4 Ln TTH18 T T 3

ðA8Þ

ðDTLMÞLTHE ¼

ðT 5  T 7 Þ  ðT 6  T L1 Þ

 T 7 Ln TT65T L1

ðA9Þ

Based on the costs, the general equation for the cost rate associated with capital investment and the maintenance cost for the kth component is:

Z_ k ¼ Z k  CRF  /=ðN  3600Þ

ðA10Þ

Here, CRF is the annual capital recovery factor (CRF = 18.2%), N represents the number of the hours of plant operation per year (N = 8000 h), and / is the maintenance factor ð/ ¼ 1:06Þ. The constants for calculating the purchase costs of the components are tabulated in Table A1.

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